Two-dimensional Guided Mode Resonant Structures For Spectral Filtering Applications

Transcription

1 Unversty of Central Florda Electronc Theses and Dssertatons Doctoral Dssertaton (Open Access) Two-dmensonal Guded Mode Resonant Structures For Spectral Flterng Applcatons 7 Sakoolkan Boonruang Unversty of Central Florda Fnd smlar works at: Unversty of Central Florda Lbrares Part of the Electromagnetcs and Photoncs Commons, and the Optcs Commons STARS Ctaton Boonruang, Sakoolkan, "Two-dmensonal Guded Mode Resonant Structures For Spectral Flterng Applcatons" (7). Electronc Theses and Dssertatons Ths Doctoral Dssertaton (Open Access) s brought to you for free and open access by STARS. It has been accepted for luson n Electronc Theses and Dssertatons by an authorzed admnstrator of STARS. For more nformaton, please contact lee.dotson@ucf.edu.

2 TWO-DIMENSIONAL GUIDED MODE RESONANT STRUCTURES FOR SPECTRAL FILTERING APPLICATIONS by SAKOOLKAN BOONRUANG B.S. Kng Mongkut s Insttute of Technology Ladkrabang, M.S. Unversty of Central Florda, 4 A dssertaton submtted n partal fulfllment of the requrements for the degree of Doctoral of Optcs n the College of Optcs and Photoncs/CREOL at the Unversty of Central Florda Orlando, Florda Fall Term 7 Major Professor: M.G.Moharam

3 7 Sakoolkan Boonruang

4 ABSTRACT Guded mode resonant (GMR) structures are optcal devces that consst of a planar wavegude wth a perodc structure ether mbedded n or on the surface of the structure. The resonance anomaly n GMR structures has many applcatons as delectrc mrrors, tunable devces, sensors, and narrow spectral band reflecton flters. A desrable response from a resonant gratng flter normally ludes a nearly % narrowband resonant spectral reflecton (transmsson), and a broad angular acceptance at ether normal dence or an oblque angle of dence. Ths dssertaton s a detaled study of the unque nature of the resonance anomaly n GMR structures wth two-dmensonal (-D) perodc perturbaton. Clear understandng of the resonance phenomenon s developed and novel -D GMR structures are proposed to sgnfcantly mprove the performance of narrow spectral flters. In -D gratng dffracton, each dffracted order nherently propagates n ts dstt dffracton plane. Ths allows for coupled polarzaton dependent resonant leaky modes wth one n each dffracton plane. The nature of the nteracton between these non-collnear gudes and ts mpact on spectral and angular response of GMR devces s nvestgated and quantfed for -D structures wth rectangular and heagonal grds. Based on the developed understandng of the underlyng phenomenon, novel GMR devces are proposed and analyzed. A novel controllable mult-lne guded mode resonant (GMR) flter s proposed. The separaton of spectral wavelength resonances supported by a two-dmensonal GMR structure can be coarse or fne dependng on the physcal dmensons of the structure and not the materal propertes. Multple resonances are produced by weakly guded modes ndvdually propagatng along multple planes of dffracton. Controllable two-lne and three-lne narrow-band reflecton

5 flter desgns wth spectral separaton from a few up to hundreds of nanometers are ehbted usng rectangular-lattce and heagonal-lattce gratng GMR structures, respectvely. Broadenng of the angular response of narrow band two-dmenson guded mode resonant spectral flters, whle mantanng a narrow spectral response, s nvestgated. The angular response s broadened by couplng the dffracted orders nto multple fundamental guded resonant modes. These guded modes have the same propagaton constant but propagatng n dfferent planes nherent n multple planes of dffracton of the -D gratngs. The propagaton constants of the guded resonant modes are determned from the physcal dmensons of the gratng (perodcty and duty cycle) and the dent drecton. A fve-fold mprovement n the angular tolerance s acheved usng a gratng wth strong second Bragg dffracton n order to produce a crossed dffracton. A novel dual gratng structure wth a second gratng located on the substrate sde s proposed to further broaden the angular tolerance of the spectral flter wthout degradng ts spectral response. Ths strong second Bragg backward dffracton from the substrate gratng causes two successve resonant bands to merge producng a resonance wth symmetrc broad angular response. v

6 To my parents and famly v

7 ACKNOWLEDGMENTS I would lke frst to thank my advsor, Dr. M. G. Moharam, for hs contnuous support and gudance durng my perod of study n CREOL. He had provded me wth a great research envronment and a contnuous help and valuable advces that facltate my progress towards fnshng my thess work. I also would lke to acknowledge my colleagues Dr. Don Jacob, Dr. Andy Greenwell, and George Sganaks for the constructve dscussons. I want to thank my parents and famly back n Thaland for ther never-endng emotonal support that eased the harshness of beng away from them, especally at the begnnng. Also, I want to thank my new famly, Dr. Waleed Mohammed, who always beleved n me and gave me lots of help n research and lfe, and my frends who made me feel home. Fnally, I would lke to thank NECTEC who supported my fellowshp and Dr. Sarun Sumrddetchkajorn for recommendng me CREOL. v

8 TABLE OF CONTENTS LIST OF FIGURES... LIST OF TABLES... LIST OF ACRONYMS/ABBREVIATIONS... CHAPTER : INTRODUCTION... CHAPTER : BACKGROUND.... The dffracton theory of the gratngs..... One-dmensonal (-D) gratng Two-dmensonal (-D) gratng Three-dmensonal rgorous coupled wave analyss (3-D RCWA) a Three-dmensonal coupled wave equatons (3-D CWE) b Boundary condton: transmsson matr approach (TMA) c Permttvty/permeablty epansons d RCWA formulaton for a fnte beam...6. Delectrc slab wavegude Multlayer slab wavegude mode solvng method Feld dstrbuton Delectrc gratng-wavegude structures Scatterng matr approach Bloch mode approach...4 CHAPTER 3 : RESONANT ANOMALY IN GUIDED MODE RESONANT (GMR) STRUCTURES...45 v

9 3. Prple of resonance anomaly n GMR Resonances n -D GMR Couplng nto a sngle-dffracted order resonant mode n -D GMR GMR wth Lorentzan resonant response De-phasng manners at off-resonant wavelength and off-resonant angle Couplng nto a two-dffracted order resonant mode n -D GMR at n-plane mountng At normal dence At oblque dence Couplng nto a two-dffracted order resonant mode n -D GMR at concal mountng Summary...75 CHAPTER 4 : ANALYSIS OF TWO DIMENSIONAL-GUIDED MODE RESONANT (-D GMR) STRUCTURES Resonances n -D GMR D GMR usng rectangular-lattce gratng Couplng of resonant modes at normal dence Couplng of resonant modes at oblque dence D GMR usng arbtrary crossed gratng D GMR usng heagonal-lattce gratng Couplng of resonant modes at normal dence Couplng of resonant modes at oblque dence Calculatons of the leaky mode dsperson Summary...6 v

10 CHAPTER 5 : MULTI-LINE GMR FILTERS Overvew Mult-lne -D GMR Flters Desgn of controllable mult-lne -D GMR flters Mult-lne -D GMR Flters Desgn of controllable mult-lne -D GMR flters Results and dscusson a Controllable two-lne -D GMR flters usng a rectangular-lattce gratng b Controllable three-lne -D GMR flters usng a heagonal-lattce gratng Fabrcaton tolerances of Mult-lne -D GMR Flters Summary...45 CHAPTER 6 : BROADENING THE ANGULAR TOLERANCES IN -D GMR AT OBLIQUE INCIDENCE Overvew Couplng characterstcs to enhance the angular tolerances of -D GMR Multple resonant modes wth the same β n -D GMR at oblque dence Three resonant modes wth the same β n a rectangular-lattce gratng Four resonant modes wth the same β n a heagonal-lattce gratng Results and dscussons Broadenng the angular tolerances n a sngle- gratng -D GMR Broadenng the angular tolerances n a dual- gratng -D GMR Summary...75

11 CHAPTER 7 : CONCLUSION...77 LIST OF REFERENCES...8

12 LIST OF FIGURES Fgure.: Confguraton of the gratng and the dffracton drectons... Fgure.: Dffracton n -D gratng at (a) n- plane (b) fully concal (c) concal mountng...4 Fgure.3: A general two-dmensonal crossed dffracton gratng and the dffracton pattern...5 Fgure.4: Confguratons of (a) a Gaussan beam at arbtrary dent (θ and φ ) and ts polarzaton (b) Gaussan beam s parameters...8 Fgure.5: (a) confguratons of three-layer slab wavegude (b)-(d) plots of the feld ampltudes of the frst three vertcally guded modes, m=,, and...3 Fgure.6: Confguraton of multlayer slab wavegude...3 Fgure.7: Feld dstrbuton of fundamental mode n a multlayer slab wavegude wth (a) sngle (b) two-layer of hgh nde...37 Fgure.8: Confguratons of (a) the corrugated slab wavegude (b) the gratng wavegude...38 Fgure.9: (a) an artfcal perodcty used n Bloch mode approach (b) a unt-cell computaton of the gratng-wavegude structure....4 Fgure 3.: Confguratons of GMR devces (a) refractve nde modulated wavegude (b) surface-relef gratng coupler (c) coated delectrc gratng...46 Fgure 3.: The confguratons of phase matchng condtons n the GMR structure at resonance47 Fgure 3.3: The confguratons of (a) -D GMR, (b) -D GMR, (c) GMR structure consdered as multlayer wavegude when applyng the HWA...5 Fgure 3.4: Dffracton/gudance planes and phase matchng dagrams of the resonant modes n -D GMR at n-plane mountng (a) sngle-propagaton (b)/(c) counter-propagaton and (d) two resonant modes at concal mountng....53

13 Fgure 3.5: (a) Plot of resonances (calculated by the HWA) versus the flm thckness (t f ), and the spectral responses (calculated by the RCWA) of resonances n-d GMR (b) t f =.5 μm (c)/(d) t f =.594 μm...55 Fgure 3.6: Scatterng of the sngle-dffracted order resonant mode n -D GMR at oblque dence...57 Fgure 3.7: Plots of resonance response appromately calculated by Eq. (3.)...58 Fgure 3.8: Contour plots of R and η d versus gratng depth and fllng factor...59 Fgure 3.9: (a) plots of R and η d versus gratng depth when f=.85 (b) Spectral responses of -D GMR wth the gratng A (t g =.5 µm, t f =.53 µm), B (t g =.3 µm, t f =.58 µm) and C (t g =.9 µm, t f =.57µm)....6 Fgure 3.: (a) Confguratons of the multlevel AR gratng (b) plots of R and η d versus the N- level (c) Spectral responses of -D GMR wth the -, 4-, 7-, and -level AR gratng6 Fgure 3.: Scatterng felds of a sngle-dffracted order resonant mode n -D GMR (a) at-off resonant wavelength (b) at off-resonant angle (c)/(d) spectral and angular responses6 Fgure 3.: Scatterng of the two-dffracted order resonant mode n -D GMR at normal dence...65 Fgure 3.3: Scatterng felds of a two-dffracted order resonant mode n -D GMR (a) at-off resonant wavelength (b) at off-resonant angle (c)/(d) spectral and angular responses66 Fgure 3.4: Contour plots of R, η d, and η B versus gratng depth and fllng factor...67 Fgure 3.5: (a) plots of R, η d, and η B versus gratng depth for.4-fllng factor (b) plots of Δθ FWHM angular/spectral lnewdth rato,, versus the gratng depth...68 ΔλFWHM / λ Fgure 3.6: Scatterng of a two-dffracted order resonant mode n D-GMR at oblque dence69 Fgure 3.7: (a) Scatterng felds of two-dffracted order resonant modes wth dfferent β n -D GMR at off-resonant angle (b) comparson of the angular responses of resonances.7

14 Fgure 3.8: (a) Confguratons of the gudng planes and related resonant modes n -D GMR at concal mountng (b) HWA calculatons of resonances versus the shft of dent plane ( θ = o, ϕ ) n -D GMR (Fgure 3.5) wth t f =.5 µm...7 o o Fgure 3.9: (a) Spectral responses of -D GMR at concal mountng θ =, ϕ = 45 (b) Devaton of reflectvtes at resonances plotted versus dent polarzaton...7 Fgure 3.: Phase matchng dagram and decomposton of TE and TM dent beam correspondng to TE and TM resonant mode for a fully concal mountng...73 o o Fgure 3.:(a) Spectral responses of -D GMR at concal mountngθ, ϕ = 9 (b) = plots of the feld magntude ( E ) of the dffracted waves versus the θ for the TE and TM polarzed beam...74 o o Fgure 3.: Spectral responses of -D GMR at concal mountng θ 5, ϕ = 9 versus = the devaton of fllng factor (FF) ( Λ =.6 μm.)...75 Fgure 4.: -D GMR structure confguraton...79 Fgure 4.: Frst-order dffracton/gudance planes n -D GMR wth (a) rectangular- (b) arbtrary- (c) heagonal-lattce gratng...8 Fgure 4.3: Scatterng of feld nsde the structure wth a rectangular-lattce gratng at resonance (a) forward dffracton (b) backward and crossed dffracton....8 Fgure 4.4: (a) Phase matchng dagram of TE resonant mode ( β ) for an unpolarzed dent beam (b) plots of normalzed powers dsspated to the normal ( η ) and n-plane a ( η d,// ) feld components by a square-lattce gratng versus ψ...84 d, Fgure 4.5: RCWA calculatons (a) spectral response for ψ = o, 45 o, 9 o ; (b) and (c) polarzaton rotaton ( Δ ψ R = ψ R ψ ) and phase ( Δ φ R ) of the reflected beam at TM resonance Fgure 4.6: (a) Phase matchng dagram of TE resonant mode (β ) for an unpolarzed dent beam, when Λ a > Λ a (b) plots of resonances versus shft of perod Λ a, calculated by HWA...85

15 Fgure 4.7: Comparson between reflectvty (R) at TE,β resonance calculated by RCWA and cos( ψ ) versus ψ...86 Fgure 4.8: Reflectvty spectrum of GMR wth rectangular-lattce gratng ( Λ > Λ )...87 Fgure 4.9: Phase matchng dagram of the resonant modes n -D GMR wth a square-grd o o gratng at oblque dence (a) ϕ = ; (b) ϕ = Fgure 4.: Plots of resonances n of GMR wth square-lattce gratng ( Λ ϕ o = 45 versusθ a b a = Λ ) at, calculated by homogeneous wavegude approach...89 Fgure 4.: Reflectvty spectrum of GMR wth square-lattce gratng ( Λ ( o, 45 o ) a b = Λ ) at ( θ, ) =...9 ϕ Fgure 4.: Closed-up plots of reflectvty spectrum n Fgure 4. (a) TM,β (b) TE,β (c) TM,β (d) TE,β resonance...9 Fgure 4.3: Comparson of devaton of gratng couplng strengths ( Δ ηd ) for TE and TM dent beam when couplng to the same mode (sold-lne: TM mode, dash-lne: TE o o mode) (a) ϕ = ; (b) ϕ = b Fgure 4.4: (a) Frst-order dffracton/gudance planes by an arbtrary crossed gratng; phase o o o matchng dagram (b) ζ < 3 (c) ζ = 3 and (d) ζ > 3, when Λ a = Λ b...93 Fgure 4.5: Phase matchng dagram of TE resonant mode n -D GMR wth an arbtrary o o o crossed gratng ( Λ = Λ, ζ 3 ) at normal dence (a) ψ = (b) ψ = 9 94 a b Fgure 4.6: Plots of (a) phase msmatches of wave (-,) (b) the devaton of the couplng strength, when applyng the TE and TM polarzed beam, versus the ar-hole radus o (r)of the crossed gratng wth ζ = n comparson to the square-lattce gratng o ( ζ = )...95 Fgure 4.7: (a) Plot of the flm thckness (t f ) for achevng the TE resonance at λ. 475μm o versus the change of the ar-hole radus (r) of the crossed gratng wth ζ = (b) spectral response when r=.95 µm and t f = ~.8 µm...96 = v

16 Fgure 4.8: Scatterng of feld nsde the structure wth a heagonal-lattce gratng at resonance (a) forward dffracton (b) backward and crossed dffracton Fgure 4.9: Phase matchng dagram of TE resonant modes (β ) n -D GMR usng a regular heagonal-lattce gratng for the unpolarzed beam...99 Fgure 4.: Plots of (a) polarzaton drecton and (b) normalzed powers dsspated to frst-order waves by the regular heagonal-lattce gratng at normal dence versus ψ... Fgure 4.: Spectral responses versus the polarzaton drecton ψ = o to 9 o (a) TE resonance: ~.7μm (b) TM resonance: ~.55μm.... t f t f Fgure 4.: Plot of (a) polarzaton rotaton ( Δ ψ R = ψ R ψ ) and (b) phase φ R = φe r φe ) of the reflected beam at λ versusψ... ( Δ,// r, Fgure 4.3: (a) Phase matchng dagram of TE resonant mode (β ) for an unpolarzed dent beam, when Λ a < Λ a (b) plots of resonances versus shft of perod Λ a, calculated by HWA...3 Fgure 4.4: Spectral responses of the structure n Fgure 4.9b havng Λ a < Λ a, when dent polarzaton s ψ =º and 9º...4 Fgure 4.5: (a) Symmetrc planes of dence; phase matchng dagram of the resonant modes n D-GMR wth a regular heagonal-lattce gratng at oblque dence o o (b) ϕ = ; (c) ϕ = Fgure 4.6: Plots of resonances n of GMR wth heagonal-lattce gratng ( Λ o ϕ = (b) ϕ o = 9 versusθ a = Λ ) at (a), calculated by HWA...6 Fgure 4.7: Reflectvty spectrum of GMR wth heagonal-lattce gratng ( Λ ( o, 9 o ) a b = Λ ) at ( θ, ) =...7 ϕ Fgure 4.8: Phase matchng dagram of four resonant modes n -D GMR usng an rregular o lattce gratng at oblque dence ϕ = 9 and couplng characterstcs for (a) TE and (b) TM polarzed beam...8 b v

17 Fgure 4.9: Plots of TE resonant wavelengths as a contrbuton of waves (,)/(-,) and (,)/(-,) versus the dent angle θ n -D GMR usng an rregular heagonallattce gratng ( Λ =.5μm and Λ =.7μm )...9 a a o o Fgure 4.3: Reflectvty spectrum of GMR n Fgure 4.9 at (a) θ = (b) θ.45 and (c) o θ =... Fgure 4.3: Unt cells of D-GMR wth a rectangular-lattce gratng (rectangular-shape hole) Fgure 4.3: Dsperson of leaky modes (a) normalzed R { n eff } Re { n eff } (b) normalzed I { n eff } of TM guded modes (c) normalzed I { n eff } of TE guded modes...3 Fgure 4.33: Spectral response of the resonances at (a) normal dence/(c) at o o ( ϕ, ) = (45,9.4 ) and (b) band dagram of structure n Fgure θ Fgure 4.34: (a) Spectral response and (b) matchng plots of normalzed modal nde (TE /TM ) and the normalzed ± waves (calculated from Eq.(4.3)) at ( θ, ) = ( o, o ϕ ), and TE-polarzed beam...5 Fgure 5.: The spectral response of the two-lne -D GMR flter at normal dence: gratng (n g =.5, Λ a =3 nm FF=.6, t g =5 nm), hgh nde flm (n f =.7, t f =353 nm), an AR coatng ( n AR = n f ns, t AR = λ / 4n AR ), a substrate (n s =.47), and an ar superstrate (n c =)... Fgure 5.: Resonance separaton ( Δ λ λ ) calculatons of TE and TE resonances n two-lne D-GMR flters (Fgure 5.) versus perod of the gratng ( Λ a ) and flm thckness (t f ) (a) flm nde (n f )=.7 (b) flm nde (n f )= Fgure 5.3: Resonance separaton ( Δλ λ for both eact and appromated Eq. (5.5) for two-lne D-GMR flters (n Fgure 5.) versus flm refractve nde (n f ) and flm thckness (t f )...5 Fgure 5.4: Resonances separaton ( Δ λ λ ) calculatons of TE and TM resonances n -D GMR flters usng a square-grd gratng versus gratng perod and flm thckness (t f ) (a) flm nde (n f )=.7 (b) flm nde (n f )= v

18 Fgure 5.5: Spectral response of the two-lne -D GMR flters wth a rectangular-lattce structure. (a) Λ b /Λ a =.89; (b) Λ b /Λ a =.96; (c) Λ b /Λ a =.3; (d) Λ b /Λ a = Fgure 5.6: (a) Shft n the locaton of the second resonance (b) resonance locatons n two-lne - D GMR flters wth a rectangular-lattce structure versus the rato of gratng perods ( Λ a = 8 nm )...34 Fgure 5.7: Resonance locatons n two-lne -D GMR flters wth a rectangular-lattce structure versus the rato of gratng perods ( Λ a = nm )...36 Fgure 5.8: (a) The change of the gratng perods Λ b versus Λ a to f two resonances ( λ and λ ), whle shft the other ( λ 3 ). (b) The locaton of resonances n three-lne -D GMR flters usng a heagonal-lattce gratng versus the rato of the two gratng perods ( Λ Λ )...37 b a Fgure 5.9: Spectral response of the three-lne -D GMR flters usng a heagonal-lattce structure (a.) Λ b /Λ a =.96; (b.) Λ b /Λ a =.7; (c.) Λ b /Λ a =.4; (d.) Λ b /Λ a =...38 Fgure 5.: Devaton of the resonant wavelengths whle changng one of each structure dmenson from the desgn (a) TM resonance (b) TE resonance...4 Fgure 5.: Devaton of the spectral lnewdth whle changng one of each structure dmenson from the desgn (a) TM resonance (b) TE resonance...4 Fgure 5.: Contour plot of the shft (nm) of TM resonance (a) devaton of gratng depth (Δt g ) and ar-hole radus (Δr) (b) devaton of hgh nde flm thckness (Δt f ) and ar-hole radus (Δr)...43 Fgure 5.3: (a) A confguraton of the gratng wth slanted sde wall (b) the contour plot of the shft (nm) of TM resonance wth the change of ar-hole radus (Δr) and slanted sde wall angle (Δθ)...44 Fgure 6.: Confguratons of dffracton/gudance planes at oblque dence and phase matchng dagram of resonant modes wth the same β n the -D GMR wth (a) a rectangular-(b) a heagonal-lattce gratng...5 Fgure 6.: Confguratons of dffracton/gudance plane and phase matchng dagram of three resonant modes at off-resonant angle (a) on- (b) off-dent plane v

19 Fgure 6.3: Confguratons of dffracton/gudance plane and phase matchng dagram of four resonant modes at off-resonant angle (a) on- (b) off-dent plane Fgure 6.4: (a) Phase matchng condtons (b) spectral (c) and (d) angular resonant response along and perpendcular to plane of dence of -D GMRs n Table Fgure 6.5: Comparson of de-phasng compensatons n GMR# and GMR# (Fgure 6.4) calculated from Eqs. (6.3)-(6.6) (a) on- dent plane (b) at off-dent plane...59 Fgure 6.6: (a) Scatterng of four resonant modes at resonance (A: forward dffracton, B: a crossed dffracton) and de-phasng manners of a crossed dffracton at off-resonance (b) θ +Δθ (c) φ +Δφ...6 Fgure 6.7: η B and η d of the gratng structure of GMR# n Table 6. versus the gratng depth, where gratng has r=.3 μm, λ =.98μm θ ~.4º...63 Fgure 6.8: (a)/(e) Angular responses of four TM resonant modes located at λ =.98μm, θ =.4º and dsperson plots calculated by RCWA (a)-(b) t g =.5 μm, t f =.53 μm and (e)-(d) when t g =.5 μm, t f =.44μm (c) dsperson plot calculated by HWA for t g =.5 μm...64 Fgure 6.9: Resonant band dagram of the structure n Table 6. calculated by RCWA and angular responses at λ, (a)-(d) t g =.5 μm, (b)-(e) t g =.35 μm, and (c)-(f) t g =.55 μm...65 Fgure 6.: The angular -wavelength lnewdth rato vs. gratng depth...66 Fgure 6.: Dual-gratng -D GMR confguraton...67 Fgure 6.: (a) Scatterng of four resonant modes at resonance (A: forward dffracton, B: a crossed dffracton by the substrate gratng) and de-phasng manners of a crossed dffracton (B) at off-resonance (b) θ +Δθ (c) φ +Δφ...69 Fgure 6.3: The angular-wavelength lnewdth rato vs. substrate gratng depth, t g, (resonances n Table 6.3)...7 Fgure 6.4: Resonant band dagram of the structure n Table 6.3 calculated by RCWA (a) t g =.5 μm (b) t g =.5 μm (c) t g =.34 μm (d) plot of angular responses at λ of the structures wth t g =.5-.4 μm...7 v

20 Fgure 6.5: The angular-wavelength lnewdth rato vs. substrate gratng depth, t g, (resonances n Table 6.4)...73 Fgure 6.6: Resonant band dagram of the structure n Table 6.4 calculated by RCWA (a) t g =.5 μm (b) t g =.5 μm (c) t g =.4 μm (d) plot of angular responses at λ of the structures wth t g = μm...74

21 LIST OF TABLES Table 6.: Spectral and angular lnewdth of resonances (λ =.98 μm) by a counter-propagaton resonant mode n D-GMR at oblque dence...57 Table 6.: Spectral and angular lnewdth of resonances (λ =.98 μm) n a heagonal-lattce gratng resonance structures (Λ a =.55 μm, Λ b =.6 μm, and r =.3 μm) at oblque dence (φ =9º, ψ =º)...65 Table 6.3: Spectral and angular lnewdth of resonances (λ =.98 μm) n a dual heagonal-lattce gratng resonance structures (Λ a =.55 μm, Λ b =.6 μm, superstrate gratng: nh/nl=/, t g =.55 μm, r =.3 μm, substrate gratng: nh/nl=/.47, r =. μm ) at oblque dence (φ =9º, ψ =º, θ =4 º)...7 Table 6.4: Spectral and angular lnewdth of resonances (λ =.98 μm) n a dual heagonal-lattce gratng resonance structures (Λ a =.55 μm, Λ b =.6 μm, superstrate gratng: nh/nl=/, t g =.35 μm, r =.3 μm, substrate gratng: nh/nl=/.47, r =. μm ) at oblque dence (φ =9º, ψ =º)...7

22 LIST OF ACRONYMS/ABBREVIATIONS -D One-Dmensonal -D Two-Dmensonal 3-D Three-Dmensonal CWE GMR HWA RCWA S-matr TE TM TMA Coupled Wave Equatons Guded Mode Resonant Homogeneous Wavegude Approach Rgorous Coupled Wave Analyss Scatterng matr Transverse Electrc Transverse Magnetc Transmsson Matr Approach

23 CHAPTER : INTRODUCTION Hstory of resonance anomaly n a gratng-wavegude structure developed from the dscovery of a dscrete spectrum n the hgher-order dffracton reflected waves by R.W. Wood n 9 [.]. Although, the theory of gratng dffracton was well establshed, the unepected phenomenon could not be eplaned at that tme. Se then, several researchers [-9] have attempted to eplan the effect, the so-called Wood s anomales. Lord Raylegh n 97 showed that the passng off or the emergence of a hgher dffracted order results n the redstrbuton of energy manfested tself as a sngularty n one or more of the propagatng dffracton orders. The wavelengths correspondng to ths transton of the dffracted orders from evanescent-to propagatng are now known as the Raylegh wavelengths [.] However, Wood n 9 [3] observed a dfferent type of anomales not assocated wth Raylegh wavelengths. Wood attrbuted the anomales, whch was observed lght polarzed perpendcular to the gratng groove (S polarzaton), to a resonant effect wthn the gratng grooves. In 938, Fano descrbed the lneshape of the anomales and hs approach was based on the estence of surface waves n a metallc gratng [4]. Ths was the frst step n better physcal understandng of the resonant phenomena. In 95, Palmer found the anomales when the dent beam s polarzed n the plane parallel to the gratng grooves (P polarzaton) for the frst tme [5]. He showed that P anomales dsappear when the gratng groove depth s much less than the wavelength. Ths eplaned why the theoretcal descrptons up to that tme cannot predct the P-polarzaton anomales. Lppmann later verfed ths dscusson by analyzng the effect of Raylegh s appromaton for the deep gratng [6.] Through the 95 s and early 96,

24 the theoretcal works focused on clarfyng the anomales usng multple scatterng approaches [7-8] after the well-developed treatments of electromagnetc scatterng from varous obstacles. In 965, a new understandng the Wood s anomales based on the nfluence of the guded wave was presented by Hessel and Olner [9.] They dentfed for the frst tme two mechansms for Wood s anomales. The frst s the well-known Raylegth s wavelength type. The second s assocated wth the couplng of the dffracted order to ecte of weakly guded modes supported by the structure. The felds leaked from the structure remans n phase and constructvely nterferes thereby producng the anomaly. Ths s clearly a resonant effect (as Wood suspected). The Hessel and Olner s work was further etended by Nevere n 97. He nvestgated the resonances assocated wth delectrc coated metallc gratngs []. The resonances of P polarzaton were observed n a shallow gratng wth a delectrc layer. The resonances were related to the modes supported n the structure when a delectrc flm s ntroduced. Later, he proposed a rgorous formalsm for gratng coupler wavegude system and assocated the comple poles and zeros wth the mama and mnma of the appromated Lorentzan resonance spectra [-.] In 986, Popov and Mastev were the frst to theoretcally report a % effcent resonance n the reflected zero order as a result of the guded mode ectaton n coated delectrc gratngs [3]. Later, Avrutsky and Sychugov related the radatve loss n the structure to the resonance spectral and angular lnewdths [4]. In 99, Magnusson and Wang s nvestgaton of a planar delectrc gratng wavegude usng rgorous coupled wave analyss (RCWA) also demonstrated % effcent narrowband spectral peaks n the reflected zero order [5]. The spectral lnewdths were shown to be drectly proportonal to the gratng modulated ampltude.

25 The devce was referred to as guded mode resonance (GMR) flter. In 99, they suggested a potental use of these structures as polarzaton elements, electro-optc swtches and narrowband flters [6]. After these nvestgatons, a number of researchers have analyzed the resonance characterstcs [7-4], related resonances to the structure parameters [-, 5], attempted to fabrcate devces [6-8] and mproved the responses [9-34]. In the last few years, delectrc GMR devces were etensvely nvestgated. Norton et al. [7] formulated an appromate closed-form epresson for the radatve couplng loss of the planar delectrc gratng wavegude usng the coupled mode theory. The radatve couplng losses along wth the appromated Lorentzan functon resonance response establshed by Nevere [] were shown to accurately predct the spectral/angular lnewdth epresson verfed by the RCWA. Its nverse epresson as well defned mnmum sze of the structure requred a % reflectvty. A better physcal understandng of the resonance characterstcs were later ntroduced by Sharon usng the nterference approach [8-9.] The approach was used to nvestgate the scatterng/absorpton loss on the resonance lneshape and shft of the resonances n lossy delectrc structure. Rosenblatt [] demonstrated broaden and flat angular responses of resonances at normal dence by usng coupled wave approach together wth the ray pcture model to study the mpact of the couplng strength of the two counter-propagatng felds on the angular response. Combnng the studes of Sharon [8-9] and Rosenblatt [], Jacob et al. proposed two new separate models, based on the nterference-wavegude approach, for a structure at oblque [] and at normal dence []. Resonance responses were connected to the gratng couplng strengths assocatng wth the gratng physcal dmensons. Other approaches to nvestgate the resonance effect lude: eamnng the feld and power flu nsde 3

26 the gratng usng rgorous modal soluton of the scatterng problem by Tamr and Zhang [3], and addressng the leaky stop bands assocated wth resonant structures [4.] Several GMR multlayer confguratons were nvestgated and proposed for spectral flterng applcatons whch requre a symmetrc narrow spectral lneshape and low baselne reflecton [9-38]. Due to the hgh dependence of the flter response on the structure propertes and the drecton of dent angle, the GMR applcatons etended lude sensors, rotatonal tunable flters, electro-optc tunable flters and also optcal swtches [39-43]. The numercal and epermental results were ntensvely reported. Whle GMR flters have been observed to possess a very narrow spectral response, there angular response may not broad enough to operaton wth fnte beams. The dvergence nature of fnte beams results n reduced peak reflecton at resonance due to the narrow acceptance angle of the devce. Ths s partcularly more pronounced at oblque dence. Increasng the resonance angular acceptance requres engneerng the couplng characterstcs of the leaky mode. Lemarchand et al.[44] was the frst to propose a structure wth double-perod gratngs n 998. Usng ths confguraton, the couplng between the counter-propagatng modes was strengthened and therefore broadenng the angular lnewdth at normal dence. To reduce the fabrcaton process complety, the confguraton was later re-modfed to b-atom or asymmetrc gratng [45]. They recently appled ths structure at oblque dence [46]. Another study by Jacob et al. [] reported the rement of the angular tolerance at normal dence by optmzng the gratng dmensons to have a strong second Bragg dffracton. He also proposed a new approach by couplng the dent beam to a hgher-order mode n order to mnmze the dependency of the spectral and the angular lnewdths []. 4

27 Whle the resonance anomaly n the gratng-wavegude structure was thoroughly nvestgated n the past twenty years, most of the studes focused on the feld scatterng and the modal nde n one-dmensonal (-D) gratng structure. Only few studes referred to the nfluences of the double dffracton usng two-dmensonal (-D) gratng structure on the resonances. The frst study was epermentally demonstrated by Peng et al. n 995 [49]. He later theoretcally nvestgated the scatterng of the feld nsde the structure at resonance usng the three-dmensonal (3-D) rgorous coupled wave analyss [5]. Two potental applcatons were proposed to beneft from the double perodcty at normal dence: polarzaton-ndependent narrow-band flter and dual narrow band flter. Mzutan et al. [5] proposed a rhombc-lattce gratng structure as a non-polarzng GMR flter at oblque dence wthout eplanng the mechansm of the flter. Fehrembach and Sentenac later presented a phenomenologcal theory of dffracton gratngs made by perturbng a planar wavegude [5-53]. They consdered the Hermtan reflecton matr of the second order lnkng the orthogonal polarzed magntudes of the reflected order and the dent wave. Two uncoupled Egen modes were presented but do not est at the same wavelength for the orthogonal polarzed dent waves. The unpolarzed flter requres smultaneous ectaton of both Egen modes usng a certan value of the Fourer coeffcent of the permttvty, or a gratng geometry, for a partcular dent drecton [53.] Ths eplaned the mechansm of the unpolarzed flter proposed by Mzutan [5.] Fehrembach and Sentenac combned ther studes of polarzaton ndependence [53] and the broadenng of the angular lnewdth [46] of the resonances at oblque dence and proposed an unpolarzed structure wth hgh angular tolerances. The structure usng the square-lattce gratng wth nonunform gratng-groove sze was desgned to have two pars of frst-order dffracted waves 5

28 smultaneously coupled to two guded modes havng dfferent propagaton constant [54]. Recently, Clausntzer et al. [55] fabrcated an unpolarzed flter based on the desgn presented by Mzutan [5.] To rease the desgn tolerance, he used a double-sded crossed gratng nstead. The epermental results were n agreement to the numercal models. Two-dmensonal crossed gratngs dffract the dent lght nto multple dffracted orders n concal confguraton wth each dffracted order occupyng a dstt dffracton plane. Therefore, the dent beam could be coupled to the multple polarzaton-dependent leaky modes each n ts own plane (that assocated wth relevant dffracted). The capablty n producng of the varous symmetrc/asymmetrc couplng confguratons for a certan dent drecton allows engneerng both resonance locaton and response drectly through the gratng parameters (duty cycle and perodcty.) Ths offers a huge advantage of the two dmensonal structures where resonances are less dependent on the structure materal propertes. However, a clear through - understandng of the resonant couplng characterstcs n the two-dmensonal structure has not been provded. The objectve of ths dssertaton s to provde a detaled study of the unque nature of the resonance anomaly n GMR structures wth two-dmensonal (-D) perodc perturbaton. Clear understandng of the resonance phenomenon s developed and novel -D GMR structures are proposed to sgnfcantly mprove the performance of narrow spectral flters. The focus wll be on nvestgatng and quantfyng nature of couplng between the polarzaton dependent resonant leaky modes, each n ts plane. The mpact of ths couplng on spectral and angular response of GMR devces wth rectangular and heagonal grd -D gratng structures wll be nvestgated. 6

29 Based on the developed understandng of the underlyng phenomenon, novel GMR devces are proposed and analyzed. In the net chapter, the bascs elements of resonance anomaly n GMR structures: dffracton and wave gudng wll be dscussed to provde the background for the analyss of GMR devces. The theory of the gratng dffracton wth dfferent gratng geometres, the wave gudng characterstcs of multlayer homogenous slab wavegudes, and the wave gudng/scatterng n the gratng-wavegude structures are presented. The numercal tool used to determne the feld magntude of the dffracton waves namely the eact three-dmensonal rgorous coupled wave analyss (3-D RCWA) [56-59] wll be descrbed. The numercally stable transmsson matr approach (TMA) [6] used for gratng stacked wll be presented. Etenson of the formulaton to fnte beams s luded. Formulaton for the analyss multlayer slab wave gudng structures wll be developed. The nature of leaky modes n e gratng-wavegude structure s nvestgated usng the scatterng matr method [-] and the Bloch mode approach [6-6.] Formulatons for the two technques wll be presented. In Chapters 3, the general concept of the resonant anomaly based on the scatterng and couplng characterstcs of the resonant modes n GMR devces havng the one-dmensonal gratng structure wll be revewed. An approach where the homogenous wavegude approach (HWA) [], whch does not account for the leaky mode, s used to develop a very good estmate for the structure dmenson and the resonance locaton, and the rgorous coupled analyss (RCWA) to determne the eact structure parameters wll be developed. The mpact of the structure parameters and gratng couplng strengths wll be studed to develop a conceptual understandng that can be used how to control the spectral and angular lneshapes. The analyss 7

30 wll help evaluatng the capablty and the lmtaton of the flter due to the restrcton of the modes n a common plane at n-plane dence. The polarzaton dependency of resonances s nvestgated n the concal mount and GMR flters for unpolarzed lght wll be demonstrated. In Chapter 4, the resonance anomaly n the GMR structure usng a -D gratng wll be studed. The effect of the nherent multple dffracton/gudance planes on the resonance s nvestgated n detals. Three -D GMR grd structures, rectangular lattce, arbtrary cross lattce, and heagonal lattce wll be consdered. The feld scatterng and the phase matchng condton at resonance are studed n order to gan a better understandng of the couplng characterstcs ludng the mpact of lght polarzaton and the effect of the gratng couplng strengths on the resonance responses. The Bloch mode approach s used to calculate the actual comple modal nde related to the formaton of the resonances n the -D GMR. Investgatng the resonances of unpolarzed lght can be smplfed by the decomposton of the dent beam nto two components each assocated wth a polarzaton-dependent leaky mode n ts dffracton planes. The developed understandng of the couplng mechansms through the dent drecton and the gratng vectors ntroduces the concept n order to acheve % reflecton for the polarzed and unpolarzed beam. As wll be shown n the chapter, achevng the total reflecton and polarzaton ndependent resonances requres at least two resonance modes to whch both orthogonal polarzed beams can be coupled and the structure havng the approprate gratng dmensons (duty cycle.) Heagonal-lattce gratng are shown to provde better characterstcs for unpolarzed flters. A novel structure s proposed for the frst tme where the dent beam s coupled nto four resonant modes havng the same propagaton constant. A polarzaton ndependence of the resonance at oblque dence s acheved wthout the requrement of 8

31 changng the gratng duty cycle. At the end of the chapter, the resonance phenomenon n the GMR structure wll be nvestgated utlzng the comple modal ndces determned by the Bloch mode approach. The dsperson of the leaky resonant modes counter-propagatng along the dffracton planes wll be produced and ths ndcates the nteracton of resonant modes and hence the structure set up requred for the formaton of specfc resonances. In Chapters 5, a novel -D GMR structures wll be proposed as a controllable mult-lne passve optcal flter [63]. Multple spectral resonances are obtaned by phase matchng the frst dffracted orders at specfed spectral locatons to weakly guded modes propagatng along dfferent planes of dffracton each wth ts dstt propagaton constant. The locaton and the separaton between the resonances are drectly controlled va the physcal dmensons of the structures: the gratng perods, and the geometrc shape of the unt cell. Controllable two-lne and three-lne narrow-band reflecton flter desgns wth spectral separaton from few up to hundreds nanometers are demonstrated usng rectangular-lattce and heagonal-lattce gratng GMR structures, respectvely. The strong dependency of resonance locatons on the gratng perod requres a good control of the fabrcaton processes. The fabrcaton tolerance and the mpact of each structure dmenson on the resonance locatons as well as the resonance lnewdths are characterzed at the end of the chapter. In Chapter 6, broadenng of the angular tolerance n two dmensonal GMR spectral flters at oblque dence, whle mantanng the narrow spectral lnewdth, s nvestgated. Symmetrc and broad angular responses are demonstrated by phase matchng the frst order dffracton to four fundamental resonant modes havng the same propagaton constant n a heagonal gratng structure. The angular lnewdth can be reased by a factor of two 9

32 comparng to when couplng to hgher-order modes wth unequal magntude of propagaton constant. A fve-fold mprovement n the angular tolerance s acheved usng a gratng wth strong second Bragg dffracton n order to produce a crossed dffracton. A novel dual gratng structure wth a second gratng located on the substrate sde s proposed to further broaden the angular tolerance of the spectral flter wthout degradng ts spectral response. Ths strong second Bragg backward dffracton from the substrate gratng causes two successve resonant bands to merge producng a resonance wth symmetrc broad angular response.

33 CHAPTER : BACKGROUND In ths chapter, two phenomena partcpatng n guded mode resonant devces, dffracton and leaky mode n wave gudng structure are dscussed. The dffracton part focuses on the effect of the gratng geometry/perodcty to the dffracted waves where the dffracted feld magntudes are analyzed usng the eact three-dmensonal rgorous coupled wave analyss (3-D RCWA) [56-59] together wth the transmsson matr approach [6] n the lnear regme. The formulatons of the approaches are summarzed for both a plane wave and a fnte beam. They are also manly used to nvestgate the resonances n GMR structure n ths dssertaton. For the wavegude secton, the content ludes the mode solvng approaches n both multlayer slab (homogeneous) wavegudes and the gratng-wavegude structure. Two approaches, the scatterng matr approach [-] and the Bloch mode approach [6-6], are brefly revewed n order to solve the comple propagaton constant n the GMR structure.. The dffracton theory of the gratngs Gratngs are perodc structures wth modulated permttvty (a volume/phase gratng), absorpton (an ampltude gratng), or surface (a surface relef gratng). Gratngs dffracts dent lght nto multple drectons dependng on the gratng vectors. Each dffracton order corresponds to the drecton where the scattered waves constructvely nterfere regardless of the gratng type [64.]

34 In general, when an dent lght wth a tangental wave vector, k r perodc structure wth a tangental gratng vector, K r dffracted order ( k r p, tan ) s wrtten as tan, tan, s modulated by a, the tangental wave vectors of the p th r k r r = (.) p, tan k,tan pktan The dffracton drectons defned n Eq. (.) are demonstrated n Fgure., where the dffracted waves have tangental wave vectors k p, tan < kn, ( k s the free space wave vector and n, s the refractve ndces of the meda and outsde the gratng regon.) The number of dffracted waves s lmted by both the dent drecton and the tangental gratng vector ( K r.) tan Fgure.: Confguraton of the gratng and the dffracton drectons The gratng dffracts lght n the desred drectons wth a proper perodcty as wll be shown for the one- and two-dmensonal gratng cases n Secton.. and.., respectvely. The dffracton effcences are determned usng the rgorous coupled wave analyss (RCWA). The

35 calculaton technque n Secton..3 and the dscusson follows support the surface relef gratng case whch s manly utlzed n ths dssertaton... One-dmensonal (-D) gratng For the one-dmensonal gratng n Fgure., the structure s perodc along the -as wth a perod Λ. The gratng vector ( K r ) along the k -as, then, has a magntude π Λ. For ths a settng, the dffracton equatons (.) are generally represented as follows. a k = k n π snθ cosϕ p, p, Λa k y, p = kn sn θ sn ϕ. (.) where k, p and y p k. are the wave vector components (tangental to the boundary) of the p-th dffracted order, respectvely. θ, ϕ ) ( are the radal dent angles. n s the refractve nde of the dent medum. The drectons of the dffracted waves can be wrtten as + k p k, y, p θ p = tan, kz, p k y, p ϕ p = tan, k, p z, p knout k, p ky, p and ( ) k =. (.3) 3

36 where k, s the wave vector of the p-th dffracted order normal to the boundary (-y). As z p shown, the drectons of the dffracted waves depend on the output regon (refractve nde n out.) The dffracted waves can be wthn the common plane at the n-plane mountng (Fgure.a), or ndvdual planes at concal mountng (Fgures..b and c show the rotaton of the dffracted planes wth respect to the dent plane). The drawngs n Fgure. show the dffracton planes when assumng that the dffracton occurs only n the transmtted medum. (a) (b) (c) Fgure.: Dffracton n -D gratng at (a) n- plane (b) fully concal (c) concal mountng.. Two-dmensonal (-D) gratng For a crossed gratng, the structure s perodc along two drectons wth perods Λ a and Λ b as llustrated n Fgure.3a. The dffracton pattern s along recprocal planes [65] desgnated n Fgure.3b wth unt lengths C a and C b. In ths structure, the gratng vectors are wrtten as r r r K = C a + C b (.4) a b 4

37 The dffracton orders, p and q, are defned as harmoncs along gratng vectors C r a and C r b, respectvely. Fgure.3: A general two-dmensonal crossed dffracton gratng and the dffracton pattern For the generalzed case, the gratng vectors are rotated along k r and k r y drecton as follows v C a, p p π r k v C q π r π r tan( ζ ) k q, k. (.5) =, and b q = + y Λb cos( ζ ) Λa Λa The gratng vectors n the new coordnates are r K pq = K, pq = p Λ r k b + K y, pq r k y π π r π r + q tan( ζ ) k + q k cos( ζ ) Λa Λa 5 y (.6)

38 The wave vectors of the dffracted waves are determned from the two-dmensonal gratng equatons. π π k = ( ζ ) + ( ζ ), pq k, p sec q tan (.7) Λb Λa and, π k y, pq = k y, q. (.8) Λ a where k y, pq, s the wave vector component (tangental to the boundary) of a dffracted order desgnated by p n a-drecton and q n b-drecton. Unlke the -D gratng at concal mountng, the crossed dffracton gratng (Fgure..3b) dffracts the dent beam more symmetrcally. In two-dmensonal gratng structures, the dffracted orders are n multple planes of dffracton and multple waves can propagate n the common plane. Eq s (.7) and (.8) show that each plane contans all the dffracted orders desgnated by < p < for a gven value of q...3 Three-dmensonal rgorous coupled wave analyss (3-D RCWA) Feld magntudes of each dffracted wave are accurately calculated usng the rgorous coupled wave analyss (RCWA) [56-59]. Ths technque s based on plane wave epanson and t s an eact soluton of Mawell s equatons n the frequency doman. Then, the accuracy reles on the convergence of the solutons and hence the number of harmoncs under calculatons. The space harmoncs of the tangental electrc and magnetc feld Egen modes are calculated wthn the gratng regon from the coupled wave equatons, and then proper boundary condtons are 6

39 appled through the transmsson matr approach [6] to calculate the feld outsde the structure. The calculatons here are carred for the orthogonal Cartesan coordnate systems, however to serve the arbtrary (non-orthogonal) gratng perodctes, the gratng vectors are modfed correspondng to the calculatng coordnate as n Eq. (.6). The approach determnes an dent plane wave. The response of a fnte beam s nvestgated by the plane wave decomposton [65] as wll be presented by the end of ths secton. The normalzed dent electrc feld s wrtten as r E r r = u ep y, z, k r [ j( k + k y + k z) ], u, = (.9) where u v s normalzed dent electrc feld vector. The electromagnetc felds nsde the perodc meda are represented by summatons of the space-harmonc components (p,q). r E g = p q r S pq ( z) e j ( k + k y), pq y, pq (.) and, r g r j( k, pq+ k y pq y) = jη U ( z) e. (.) H, p q pq r where, S pq (z) r and U pq (z) are the z-spatal components of the normalzed electrc and magnetc felds respectvely, and η = μ ε, μ = 4π H / m s permeablty of free space, ε = F / m s permttvty of free space. The reflected ( E r ) and transmtted ( E r ) felds are also wrtten as summatons of the R T 7 space harmoncs, 7

40 r r E R = Er, pq ep, pq y, pq, z, p, p q [ j( k + k y k z) ] q (.) r r E T =, Et, pq ep[ j( k, pq + k y, pq y + k, z pqz) ] p q (.3) where, k, z, pq and k, z, pq are the wave vector normal components of the dffracted orders n the dent medum and the substrate respectvely. r E r, pq and r E t, pq are the electrc feld of the reflected and transmtted waves. The calculatons of these felds wll be presented n the followng sectons...3.a Three-dmensonal coupled wave equatons (3-D CWE) The frst step of calculatng the transmtted and reflected felds usng ths approach s determnng the feld Egen modes nsde the gratng through the coupled wave equatons. In general 3-D CWE deals wth all three vector components of both electrc and magnetc felds. Formulatng the couple wave equatons s mportant to the convergence rates of the solutons. The equatons are mplemented to preserve the contnuty of the EM felds along the dscontnuty of the gratng s permttvty/permeablty. Therefore, they are reformulated for the non-magnetc materal as a form [59.] Sy ( kz) K y E K = S I + K E K ( k z) I K K y E E K y U K y U y, 8

41 and U y ( kz) K = U K ( k z) y K E y E K K K y y S S y. (.4) Matrces E y = E + ( α ) A α and E = α A + ( α ) E are wrtten as the average of the harmonc coeffcent matrces of the relatve permttvty Ε and nverse relatve permttvty A addressng the unformty of the tangental electrc feld at the gratng-groove surfaces. Factors α and α are the two postve numbers n the nterval [,.] They ndcate the relatve strength between matr E and A. Wth the proper values of α and α, the convergence rate s mamzed and that reduces the mnmum number of space harmoncs, M, under the calculatons for the accurate results. tangental wave vectors I M M s an dentty matr. K and k, pq and y pq k, (Eqs. (.7) and (.8)), respectvely. K y are dagonal matrces of the Here, to smplfy the formulaton, the tangental components of the felds are analyzed along new coordnates parallel (//) and perpendcular ( ) to the planes of dffracton. Usng these new coordnates, the coupled wave equatons, n terms of the electrc and magnetc feld harmoncs (tangental components) can be wrtten as follows. S ( k z) A = S// A ( k z), //. A A,// //,// U U //, A, =, A,// = I, A = K n E K n I, and A //,// =. (.5a) //. 9

42 = // //,// //.,//, // S S B B B B ) ( ) ( U z k U z k, ' y ' ' y ' y, K E K K E K B =, ' ' ' y y ' y //, K E K K E K B + =, ' y ' y ' y ' n //. K E K K E K K B =, and ' ' y ' y y ' // //, K E K K E K B + =. (.5b) In (.5a) and (.5b), ' K, ' K y, and ' K n are the dagonal matrces : M M pq y pq pq k k k + =,,, ' K, M M pq y pq pq y k k k + =,,, ' y K, and M M pq y pq k k k + =,, n K Combnng the two coupled wave equatons n (.5a) and (.5b), the second-order dfferental matr equatons are [ ][ ] = = // // //,// //.,//, //,// //.,//, // S S B A S S B B B B A A A A ) ( S ) ( S z k z k (.6) That can be re-wrtten n a form of an Egen value problem [ ][ ] = // // S S B A S S μ (.7)

43 Based on the solutons of the Egen-value problem n (.7), the space harmoncs of electrc felds ( S r (z) ) can be wrtten as S S // We = We kμz k μz W e k μ(zd) k μ(zd) W e C C + (.8) μ s the Egen value matr and [ W] = s the Egen vector matr. The where [ ] M M space harmoncs of magnetc felds U r (z) are W W M M U U // Ve = Ve kμz k μz V e k μ(zd) k μ(zd) V e C C +, V V and [V] = = [A] [ μ][ W] M M (.9) In Eqs. (.8) and (.9), C + and C - represent the magntude coeffcents of the electrc felds when propagatng n the forward and backward drectons nsde the perodc medum wth thckness d. μ pq = k z, pq, gratng k ndcates the propagatng waves nsde the gratng regon (when μ pq s purely magnary). The effectve ndces of the subwavelength gratng for the dent beam polarzed normal to ( n ) and n ( n // ) the plane of dence are drectly estmated by μ [59, 67.] The two unknown parameters, C + and C -, wll be subtracted when applyng the boundary condton to estmate the dffracted feld outsde the gratng.

44 ..3.b Boundary condton: transmsson matr approach (TMA) For a sngle-level gratng, the tangental components of electromagnetc felds at nterfaces z= and z=d can be wrtten as E, δ Z,ηH, δ jηh, δ Y,ηE, δ I + jy,pq jz,pq I E jηh r,pq, r,pq, W W = V V WX W X C VX C VX +, at z=, and W X WX V X VX W W C V C V + = I jy,pq jz I,pq E jηh t,pq, t,pq,, at z=d (.) where X, Z l, pq, and, pq Z l, pq = [ k l,z,pq k n l ], and Y = [ ] M, pq k l,z,pq k M M transmtted regons.) [ ] M M k Y l are the dagonal matrces: X [ μd e ] M M M =, l (l = and represent the reflected and δ s Kronecker delta functon matr. As shown, calculatng the dffracted felds ( E r pq and H r pq ) confronts the operaton of nverse matrces. That may cause numercal nstablty, especally for a deep gratng havng very small eponental term X. Ths can be avoded usng the transmsson matr approach 39 whch elmnates the use of X from the calculatons. For an N-level gratng structure, the calculatons ntally separate the X from the man nverse matr [ A] N N (Eq.(.)) Therefore, the fnal matr equaton of the tangental electromagnetc felds s modfed as follow.

45 T g f t,pq, t,pq,,pq,pq [A] r,pq, r,pq,.,,,,pq,,, H jη E jy I jz I V V V V W W W W II X V X V V X V W X W W X W V V V V W W W W ΙΙ X V X V V X V W X W W X W H jη E δ H jη δ E jy jy I I jz jz I I + + = N N N N N (.) where II s an dentty matr wth dmensons M M. The rght hand sde of Eq (.) s then reduced to [ ] T g f, where sub-matrces [ ] f and [ ] g are calculated from known [ ] f + N and[ ] g + N as defned n Eq. (.). That can be wrtten as follows, assumng that N N N N T X a T + = n order to elmnate X N. N N N N N N N N T X a b I X V V V X V W X W W X W T b a II X V X V V X V W X W W X W T g f N N N = = +,

46 and a b N W W = V V W W V V N f g N +. (.) f f,.,. Substtutng [ f ] =, and [] =.3 f f.4 g g g n Eq. (.), the matr T s wrtten as g.3 g.4 E t,, g, jy,pqf, g,4 jy,pqf,3 jy,p,qe, δ, T = = (.3) jηh t,, f, jz,pqg, f,4 jz,pqg,3 Z,p,qηH, δ, And hence, the reflected and transmtted felds are T E t,pq, N + = = a N X Na NX Na NX N...a XT jηh, t,pq, and E jηh r,pq, r,pq, f,e = g,e t,, t,, + f + g,3,3 jη H jη H t,, t,, E H', δ,, δ,. (.4) For a normalzed power, E, = cosψ and j ηh, = jn snψ, where ψ s the polarzaton of the dent feld wth respect to an as normal to the plane of dent. The dffracton effcences of the reflected and transmtted waves are k, z, pq k, z, pq DER = Er, pq, Re + jη H r, pq, Re (.5) k, z, n k, z, k, z, pq k, z, pq DET = Et, pq, Re + jη H t, pq, Re (.6) k, z, n k, z, 4

47 ..3.c Permttvty/permeablty epansons As these calculatons assume that the perodc structure s nfnte along - and y as, the space harmonc coeffcents of both permttvty, (, y) ε, and nverse permttvty, (, y) ε, of a bnary -D gratng wth any unt cell can be epanded n two-dmensonal Fourer seres of the form [65] j ( K, j + K y, j y ) f (, y) = d e, j j and d j = A A f (, y) e j ( K, j + K y, j y ) d dy. (.7) In Eq. (.7), A s the area of the unt cell. th th d, s the ( j ) j, Fourer component of the functon ( y) f, whch dentfes the gratng s groove/permttvty profle wthn the unt cell. In the coupled wave equatons (Secton..3a), the harmoncs (, j) n Eq. (.7) are replaced by ( p p, q q ) correspondng to each feld harmonc ( p, q) where p = [ p( )... p( M )] [ q( )... q( M )] and q =. For a skew perodc functon, the spatal frequences ( K, j, K y, j ) are defned as n Eq. (.6). For the surface relef gratng, the same approach as a bnary gratng can be appled after slcng the gratng nto multple levels [6,] where each level has a constant fll factor. The permttvty/nverse permttvty epanson, hence, s proportonal to the gratng profle. The coupled wave calculatons are solved for each level ndvdually. The numbers of the levels reases the accuracy of the calculatons. 5

48 ..3.d RCWA formulaton for a fnte beam In practce, the dent beam s fnte wth certan dvergence angle. The above formulatons are not suffcent unless the beam s perfectly collmated. Theoretcally, the fnte beam s a composton of plane waves wth symmetrc angular components around the drecton of propagaton of the beam. Usng Fourer analyss, the magntudes of each plane wave are determned [66.] The reflected/ transmtted felds of each plane wave wth scaled magntude are ndvdually calculated usng RCWA. The total feld s calculated by recombnng the reflected/transmtted felds through the nverse Fourer transform. Here, the plane wave decomposton s carred by determnng the Fourer transform of the dent beam. Assumng that the dent beam E (, y) s ntally sampled n spatal doman by N ponts along and y wth Δ and Δ y samplng spacng, the sampled dent beam s then E, y ), where = n Δ, and y = n Δy. In the angular doman, the feld s ( n ny n sampled by M ponts along k and k y wth Δk and Δk y samplng spacng. The angular components of the dent beam are calculated usng the dscrete Fourer transform. n y y E N y N j( mδk n + yδ y n ) ( Δ, Δ ) = (, ) m k y k m k E y e y m Δ Δy (.8) y y n = N n = N y y n n y L ( u u It s worth mentonng that the samplng theory [65] s satsfed n our calculatons π Δk and π Δu, where L u and K u are the calculaton range of the spatal and K u angular doman, and u represents and y). The reflecton and transmsson of each scaled plane wave s then calculated usng RCWA. The total reflected/transmtted beams at dstance z away 6

49 7 from the structure surface (superstrate or substrate) are fnally reconstructed by ntally combnng the multple reflected/transmtted waves, pq t r E, /, for each angular component and then composng them to the spatal doman through the numercal nverse Fourer transform. ( ) ( ) ( ) ( ) = = Δ + Δ Δ Δ Δ Δ = / /, E,, y y y n y y y n y M M m M M m y y k m k m j y y T R n n T R k k e k m k m z y E π, and ( ) ( ) ± + = Δ Δ p q z k y k k j pq t r y y T R pq z ny pq y n pq e k m k m,,,, / / E, E. (.9) The composton can be ndvdually performed at z= nterface for each dffracted wave (p,q) when calculatng the dffracton effcences. In ths chapter, we consder the dent beam wth Gaussan profle. Eq. (.3) shows a general epresson for an oblquely dent normalzed Gaussan beam (Fgure..4) at the structure surface (z=) [ ] { } u y k y k j w y w z y E y y v v + + = = ep ep ),, ( Where, the beam rad are ' ' ' ' cos sn cos sn cos cos ϕ ϕ θ ϕ ϕ θ y y y w w w w w w + = = (.3) The sub-plane waves have angular components defned as, cos sn ϕ θ n k k m k m + Δ =,

50 and k y, m = m yδk y + k n sn θ sn ϕ y (.3) (a) (b) Fgure.4: Confguratons of (a) a Gaussan beam at arbtrary dent (θ and φ ) and ts polarzaton (b) Gaussan beam s parameters The polarzaton of the sub-plane waves remans the same as that of the beam. Rotatng the dent polarzaton along the dent plane of each sub-plane wave, the polarzaton can be wrtten as // u m =, my ψ m, m tan, y um, my u // y z m, m u cosϕm, m cosθm, m + u snϕm, m cosθm, m u sn y = θ, y y y y m, m y and y u m, m = u y snϕ m, m + u y cosϕm, my. (.3) In Eq. (.3), u, y u, and These vectors can be wrtten s follows. z u are the unt vectors of the dent beam ( v y z u u ˆ u yˆ = + + u zˆ ). 8

51 u ϕ, = snψ cosθ cosϕ cosψ sn u ϕ, y = snψ cosθ snϕ + cosψ cos u θ. (.33) z = snψ sn Here, ψ s the polarzaton of the dent beam. In ths secton, we have eplaned brefly the dffracton analyss of the general -D surface relef gratng part of the GMR structure. As mentoned earler, the resonance effect n GMR structures rely on both dffracton by the gratng and gudance n the slab wavegude layer beneath the gratng. In the followng secton, we wll hghlght the gudng calculaton n general delectrc slab wavegude structures.. Delectrc slab wavegude Wave gudng n planar slab wavegude s acheved when lght propagates n a regon surrounded by lower -refractve nde meda, and t nteracts to each nterface wth an angle larger than the crtcal angle ( θ = sn ( n n ) c, where n and n are refractve ndces of the surroundng and gudng meda respectvely). Once the multple total reflected waves are constructvely nterfered, lght s perfectly bounded wthn the gudng regon. The smple wavegude structure, called three-layer slab wavegude, s demonstrated Fgure.5a. 9

52 (a) m= m= m= z (μm).5 z (μm).5 z (μm) E norm E norm E norm (b) (c) (d) Fgure.5: (a) confguratons of three-layer slab wavegude (b)-(d) plots of the feld ampltudes of the frst three vertcally guded modes, m=,, and In the structure, lght s vertcally confned n the flm regon (only along z-drecton) when assumng that the structure s nfnte n - and y-drecton. The guded waves propagate from the upper flm nterface to the lower flm nterface and reflects back agan wth a total phase shft of mπ (m s an nteger number). These dscrete waves form the so-called guded modes (m). The frst three guded modes feld profles are plotted n Fgures.5b-d. Based on these condtons, the characterstc equaton for the slab wavegude s epressed as [68] κ d φ φ = mπ m =,,,3,..., (.34) m s c 3

53 In Eq. (.34), m = k n eff, m m n f m β s the modal propagaton constant, κ ( k ) β = s the transverse propagaton constant wthn the flm. φc and φs are the phase shfts due to the total reflecton at the flm-cover and flm-substrate nterfaces, respectvely. The formulaton s more nvolved when the structure comprses of composte cover and composte substrate, referred to as multlayer slab wavegude... Multlayer slab wavegude mode solvng method For an N-layer slab wavegude (as llustrated n Fgure.6,) the structure may comprse of sngle or multple hgh nde layers. Fgure.6: Confguraton of multlayer slab wavegude 3

54 3 The wavegude modes can be determned usng the same approach, where the gudng regon s the hghest nde layer, -p. The phases ganed by the total reflecton at upper/lower nterfaces (p- /p and p/p+) are calculated from the ampltude reflecton coeffcents, whch are determned by applyng the boundary condton to match the tangental felds from the outmost nterfaces, / and N/N+, towards the nner nterfaces, p-/p and p/p+, respectvely. As depcted n Fgure.6, the felds n each layer bounce upward and downward wth magntude coeffcents desgnated as + C and C. For the TE-case, the tangental felds at each layer are defned as ( ) ( ) < < + + < = = = + + = = = = N jβ z γ N p p jβ z d z jβ ) d (z γ z γ jβ z γ y, d, for e e C d z d, d z for, e e X s e C or e e X c e C z for, e e C E N N ) ( γ γ (.35a) and,

55 33 ( ) ( ) < < + < = = = = = = = N N j z N N j z d z z j d z z j z, d, for z e e j C d z d, d z for, e e X s e j C or e e X c e j C z for, e e j C H N N ) (, ) (,, β γ β γ γ β γ γ β γ μ ωμ γ ωμ γ ωμ γ ωμ γ (.35b) The felds are wrtten as a functon of the magntude reflecton coeffcents, X C C c + at the upper nterface, and X C C s + at the lower nterface, where d X e γ =. d and n are the thckness and the refractve nde of each layer. μ s the relatve permeablty of each nterface. γ ( n k = β γ ) s the transverse propagaton constant wthn each layer. At the th -layer, matchng the tangental felds ( Φ ) at the lower nterface, () ( ) d Φ = Φ, and at the upper nterface, ) ( ) ( = Φ + Φ d, the ampltude reflecton coeffcent at the left and rght nterfaces ( s and c ) are ( ) ( ),,,,, + = + = s s p s X s X s s γ γ μ γ μ γ μ γ μ γ (.36) and,

56 34 ( ) ( ),,,,, = + = c c c X c X c c γ γ μ γ μ γ μ γ μ γ (.37) Se there s no upward wave n the substrate ( = + C and = s ), p s s successvely calculated from s to p s. Meanwhle, there s no downward wave n the cover ( = N + C and N + = c ), p c s consecutvely calculated from N + c to p+ c. At the calculated gudng layer,-p, p p p j n k j κ β γ = =. Therefore, the ampltude reflecton coeffcents at the gudng nterfaces are TE s j p s p p p p s p p p p e j j s,,,,, φ μ γ μ κ μ γ μ κ = + =, and TE c j p c p p p p c p p p p e j j c,,,,, φ μ γ μ κ μ γ μ κ = + = (.38) where, the phase shft due to the total nternal reflecton at the flm-composte substrate and flm-composte cover nterfaces are =,,, tan p s p p p TE s μ γ μ κ φ,and = + +,,, tan p c p p p TE c μ γ μ κ φ (.39)

57 35 For the TM-guded modes, the formulatons are smlar to the TE case. After applyng the boundary condtons, the ampltude reflecton coeffcent s and c are ( ) ( ),,,,, + = + = s s s X s X s s γ γ ε γ ε γ ε γ ε γ (.4) and, ( ) ( ),,,,, = + = c c c X c X c c γ γ ε γ ε γ ε γ ε γ (.4) Therefore, the ampltude reflecton coeffcents at the gudng nterfaces are TM s j p s p p p p s p p p p e j j s,,,,, φ ε γ ε κ ε γ ε κ = + =, and TM c j p c p p p p c p p p p e j j c,,,,, φ ε γ ε κ ε γ ε κ = + = (.4) The phase shfts, due to the total nternal reflecton at the lower and upper nterfaces, are

58 κ p ε, p φ = s, TM tan,and s γ p ε, p κ p ε, p φ = c, TM tan (.43) c γ p+ ε, p+ where, ε s the relatve permttvty of each nterface... Feld dstrbuton To determne the feld dstrbuton of the guded mode (m), t s necessary to defne the ntal magntude coeffcents, etherc or + p C p, of the feld nsde the calculated gudng layer,-p. Based on these values, the feld magntude coeffcents + / C of the other layers are consecutvely estmated by applyng the proper boundary condtons as mentoned earler. Therefore, the feld coeffcents of the layers above the gudng layer, = p +,..., N, are wrtten as ( + c ) X C = C (.44) ( + c X ) + + And those of the layers beneath the gudng layer, =,..., p, are wrtten as ( + s ) X C = C (.45) ( + s X ) 36

59 The plots n Fgure.7 show the feld dstrbuton of the fundamental mode n multlayer slab wavegude wth sngle (Fgure.7a) and multple (Fgure.7b) hgh nde layers usng ths approach..5.5 z (μm).5 player z (μm).5 player E norm E norm Fgure.7: Feld dstrbuton of fundamental mode n a multlayer slab wavegude wth (a) sngle (b) two-layer of hgh nde In general, delectrc slab wavegudes wth unform delectrc layers confne the guded mode n the hgher nde layer(s). Introducng non-unformty n the gudng regon or n the surroundng layers would cause lght to leak outsde the structure and hence reduce ts confnement. Ths effect wll be brefly llustrated n the followng secton..3 Delectrc gratng-wavegude structures Slab wavegudes comprsed of corrugated surface, as llustrated n Fgure.8a, or wth nhomogeneous wavegude n Fgure..8b, are referred to as gratng-wavegude structures. Due the presence of the modulated surface or permttvty the guded mode n these structures s scattered outsde by the gratng. Hence the mode s not perfectly bound as t was for the slab wavegude n Secton.. Ths type of mode s called a leaky mode. The power confned n 37

60 the gudng regon s slowly reduced along the drecton of propagaton. The leaky mode, as a result, has a comple propagaton constant, β = β j β, where the magnary term( β ) ndcates the leakage, and hence / β defned as the dstance whch the leaky mode propagates n the structure before beng completely scattered outsde. Hence t ndcates the mnmum sze of the GMR structure, wll be dscussed n Chapter 3, requred to acheve total reflecton at resonances. (a) (b) Fgure.8: Confguratons of (a) the corrugated slab wavegude (b) the gratng wavegude Solvng the leaky modes n ths type of structure requres ludng the scatterng by the gratng n the calculatons. Formng a characterstc equaton as n Secton. s not suffcent. Hence, two calculaton approaches are presented here: scatterng matr approach [-, 69] and the Bloch mode approach [6-6.] 38

61 .3. Scatterng matr approach The formulatons are based on applyng the boundary condtons at each layer nterface as t s the case n Secton.. In spte of formng the characterstc equatons, the formulatons group the matchng of the tangental felds from the frst (/) to the last (N/N+) nterfaces n a form of a scatterng matr[ S ][69.] For the N-layer structure depcted n Fgure.6, the feld equaton s wrtten as C + CN + = C C C C N+ C CN+ [ S( β) ] [ S( β) ] = + + N+ + (.46) C N+ Wthout an eternal ectaton from the substrate nor the cover ( = ), non-zero solutons, + C whch ndcate the presence of modes, can est f ( β ) = S (.47) In equaton (.46), C + and C - are the ampltude coeffcents of the forward and backward propagatons along each layer, where subscrpt - and -N+ represent the substrate and cover, respectvely. S denotes the determnant of the nverse of the S-matr. For the gratng-wavegude structures, the felds wthn the structure are scattered due to the perturbaton by the gratng. To form the S-matr, the scatterng felds along each layer are ntally determned by solvng the coupled-wave equatons (.4) to (.9) n Secton..3. For the homogenous layer, the Egen vector matr[ W ] s the dentty matr and each element of the 39

62 Egen value matr [ ] layer. q s pq n h k tan, pq k, where n h s the refractve nde of the homogeneous Then, proper boundary condtons are appled at each nterface. For the nterface of layers a and -b, the S-matr [69] s wrtten as C C A B C D ab ab ab a + b ab = = = = = b [ S ] [ S ] ab C C + a [( V b Va + W b W a ) ( X b )] [( V b Va + W b W a ) ( V b Va X a W b W a X a [( Va V b + W a W b ) ( Va V b X b W a W b X b )] ( V V + W W ) ( X ) [ a b a b a ] ab A = C ab ab B D ab ab (.48) Where, the matrces[ W ], [ V ], and [ X ] have been defned n Secton..3. The S-matr of the entre structure (coverng from frst to fnal nterfaces) s the (*) product of all sub-s matrces. [] [ S ]* [ S ]* * [ ] S,, S N, N + = K (.49) Where the (*) s referred as the S-matr multplcaton defned as [] S = [ S ][ * S ] A = B = C = D = ab {( A ab A bc ) ( A ab B bc )( C ab B bc I) ( C ab A bc )} {( B ab ) ( A ab B bc )( C ab B bc I) ( D ab )} {( C bc ) ( D bc C ab )( B bc C ab I) ( A bc )} ( D D ) ( D C )( B C I) ( B D ) { } bc ab bc A = C bc ab B D bc ab bc ab (.5) 4

63 Here I s a unty matr wth dmensons M M (M s the total number of harmoncs). Se the structure s perodc, the solutons are appled va the tangental wave vectors as defned by gratng equatons. For a -D gratng case, the comple solutons can be wrtten as 7 π β β β p = p j Λ (.5) β and β are the real and magnary terms of the propagaton constant of the leaky mode. Λ s the gratng perod. Achevng the comple solutons n ths approach requres smart root fndng computaton, otherwse t s a tme consumng procedure. Besdes, the numercal nstablty ests due to the nverse S-matr operaton. These drawbacks can be avoded when usng the Bloch mode approach..3. Bloch mode approach In ths approach, the actual propagaton constants of the guded or leaky mode are calculated va the Bloch mode (perodc mode) formulatons [6-6.] An artfcal perodcty s formed n the drecton of the mode confnement as depcted n Fgure.9a. The nteracton between the modes from these adjacent wavegudes s elmnated by nsertng a perfect matched layer (PML) [7] between them. Usng the plane wave epanson as t s the case for RCWA, the Egen modes (Bloch modes) are determned. Wthn the propagaton length Λ, only the modes wth null magnary part of propagaton constant are guded. 4

64 Fgure.9: (a) an artfcal perodcty used n Bloch mode approach (b) a unt-cell computaton of the gratng-wavegude structure. The guded mode formulatons start wth formng a scatterng matr along the propagaton length, nterface and n Fgure.9a, where modes at each nterface eperences the same medum. For the gratng-wavegude structure, the computaton takes place along a symmetrc N unt cell as llustrated n Fgure.9b. That reduces the S matr multplcaton by a factor, where N s the total number of nterfaces n the unt cell. The S-matr s wrtten as 4

65 C C + = S S 3 S S 4 C C + (.5) Se each mode propagates for a dstance Λ, each one of them wll accumulate a phase of β Λ. Here, β s the propagaton constant of each mode. Therefore, the scatterng matr n Eq.(.5) s rearranged as follows I S S 4 C C + S = μ S 3 C I C + (.53) where μ = e jβ Λ. The propagaton constant of each Bloch mode s then calculated from the Egen values ( μ ). In general, the propagaton constant s consdered to be comple ( β β + β = j ). Hence, { β} Im s drectly determned by the magntude of the Egen value and Re { β} s calculated from the phase of the Egen values (φ ). ln μ β =, and Λ φ + π, forward propagaton modes Λ ' β = (.54) φ π, backward propagaton modes Λ Fnally, the guded modes are the ones whch have the real part of the propagaton constants, ( β ), wthn the cut off range ( ma[ k nsuperstrate, knsubstrate ] < β < knhgh nde flm ). A mode wth null magnary part ( β ) s a bound mode. Otherwse t s a leaky mode. Usng ths approach, the eact propagaton constants of both guded and leaky modes can be accurately determned wth no numercal nstablty. Ths approach s utlzed n ths 43

66 dssertaton to determne the comple propagaton constant of the leaky modes n the gratngwavegude structures whch wll be analyzed n ths dssertaton. 44

67 CHAPTER 3 : RESONANT ANOMALY IN GUIDED MODE RESONANT (GMR) STRUCTURES As mentoned earler n Chapter, guded mode resonant (GMR) structures produce a very narrowband resonance as a consequence of phase matchng between dffracted waves and slowly leaky modes. The structures sut for a narrowband spectral flterng applcatons. In ths chapter, the structure geometry and the resonance mechansm are presented and nvestgated manly n a one-dmensonal GMR structure. That constructs the man concept of achevng and controllng the desred resonance responses. An appromated computaton, homogeneous wavegude approach (HWA,) s developed and used to relate resonances to the wavegude modes. That leads to a better understandng of the couplng confguratons. The desgn technques to mprove the resonance performances (symmetrc spectral response wth low base-lne and narrow lnewdth, large angular tolerance, and the polarzaton ndependence) are presented and dscussed for the specfc use. 3. Prple of resonance anomaly n GMR Guded mode resonant (GMR) structures are optcal devces that consst of a planar wavegude wth a perodc structure. In general, GMR structures are categorzed nto three confguratons as depcted n Fgure 3. for the gratng mbedded n, on the surface and on both surface and substrate sde of the structure. Lke the structure mentoned n Secton.3, the wavegude modes estng n the GMR structure are not perfectly bounded due to the scatterng by the gratng. The 45

68 amount of energy leaked out from the structure s proportonal to the gratng couplng strength. Hence, the mode has comple propagaton constant. β = β j β (3.) The leaky modes are manly guded n the nhomogeneous layer for the frst geometry n Fgure 3.a. For the other two geometres (Fgure 3.and 3.b), the leaky modes are guded n the homogeneous hgh nde layer. Fgure 3.: Confguratons of GMR devces (a) refractve nde modulated wavegude (b) surface-relef gratng coupler (c) coated delectrc gratng 46

69 Fgure 3.: The confguratons of phase matchng condtons n the GMR structure at resonance To acheve a % reflecton at resonance, the gratng has to be n the subwavelength scale and dffracts only frst-order waves nsde the gudng regon, not outsde the structure. Therefore, at resonance, the dffracted waves are phase matched to the wavegude mode. The leakng felds n the dent regon hence constructvely nterfere and the dent beam s totally reflected. The phase matchng condton at resonance s depcted by the ray model n Fgure 3. when - dffracted wave s phase matched to the wavegude mode. In general, the phase matchng condton between the frst-order dffracted waves and the wavegude modes at resonance can be appromately wrtten as β (3.) kdff, st where kdff st s the tangental wave vector of the frst-order dffracted waves whch are defned, by the gratng vectors and the drecton of the dent beam as dscussed n Secton.. The resonance locaton can be drectly predcted f the modal propagaton constant ( β ) s known from the gratng equatons (.) for a -D gratng structure and Eqs. (.7)-(.8) for a - 47

70 D gratng structure. The magnary part of the modal propagaton constant ( β ) represents the leakage of the wavegude mode. Ths does not only determne the propagaton length of the modes but the resonance lnewdths as well. The connecton of β and the resonance lnewdths ( Δ λfwhm and Δ θ FWHM ) can be determned by usng an appromate Lorentzan epresson of the reflectvty [, 7], whle mantanng the resonance as a comple pole. The reflectvty s then wrtten as [ ( π Λ β )] + [ β ] k,tan c R =. (3.3) a where c s an arbtrary constant. The angular lnewdth can be drectly formulated from Eq. (3.3) as β Δ θ FWHM. (3.4) kn cosθ θ s the drecton of the dent beam at resonance. To determne the spectral lnewdth, the reflectvty n equaton (3.3) s wrtten as a functon of the wavelength [7.] R = c ( Λaλ π ) [ λ Λ ( snθ n )] + [ Λ n ] a a. (3.5) 48

71 where λ s the resonant wavelength. n and n are the real part and magnary part of the comple modal nde at the resonant wavelength. Then, the spectral lnewdth s defned as the followng. λ Λa ΔλFWHM β, (3.6) π The resonance lnewdths n Eq. (3.4) and (3.6) may not be accurate when the resonance lneshape s not Lorentzan, especally when copng wth the multple resonant modes. For ths case, the resonance responses depend on the couplng characterstcs. That wll be nvestgated n the net sectons for the -D GMR and n Chapter 4 for the -D GMR. The resonance responses however are eactly determned usng the RCWA calculatons as dscussed n Secton. Avodng the nvolved computatons of the comple propagaton constant, the wavegude mode (TE m, or TM m ) propagaton constant can be determned usng the homogeneous wavegude approach (HWA) [.] As ths approach does not lude the scatterng by the gratng, only the real part of the propagaton constant s determned whch gve a very good appromaton for the locaton of resonances. Consderng the GMR structure as the homogeneous slab wavegude n Fgure 3.3 where the gratng, both -D and -D cases, s replaced by a homogeneous flm wth an effectve refractve nde (n g ). The accuracy of the approach then reases by determnng the gratng effectve nde usng the RCWA. The error s sgnfcant when dealng wth the gratng wth the strong couplng strength. 49

72 Fgure 3.3: The confguratons of (a) -D GMR, (b) -D GMR, (c) GMR structure consdered as multlayer wavegude when applyng the HWA The mode solvng method apples the formulatons n Secton.. The phase,, accumulated by the total reflecton at flm-composte substrate nterface n Secton. can be smplfed as follows φs κ φ s = tan A, s γ AR s + s e γ s AR γ AR = γ AR γ ARe AR AR d d AR AR, and s AR A γ s AR s =. (3.7) A γ s AR γ + γ s The total reflecton phase φc at the flm-composte cover nterface s wrtten as follow: κ φ c = tan B, c γ g c γ g + c γ g d g e c γ g = γ g d g e g g, and c g Bcγ g γ c =. (3.8) B γ + γ c g c 5

73 The coeffcents n Eq. (3.7) and (3.8) are defned for TE modes as A A =, and B B =. = s = c For the TM modes, A = ε AR ε f, A s = εs ε AR, B = ε g ε f, and B c = εc ε g. For the subscrpts - AR, -s, -f, and -g represent the ant-reflecton (AR) flm, the substrate, the gudng layer, and the gratng, respectvely. The rest of the parameters have been desgnated n Secton.. Due to the smple and fast computatons, the HWA s used n ths dssertaton to estmate the locaton of resonances and obtan a very good estmaton of the structure parameters. The RCWA s then used to refne the structure parameters and to determne the spectral and angular responses of the resonance. In net sectons, the dscusson focuses on the resonances n -D GMR structure. Based on the couplng characterstcs, the resonance modes are classfed and nvestgated usng the HWA and the RCWA. Later, the phase confguratons of each resonance modes type at on- and off- resonance are dscussed and related to the resonance responses. 3. Resonances n -D GMR For -D GMR structure, the frst-order dffracton s lmted along two waves (±) propagatng wthn a common plane at n-plane mountng (Fgure.a) or separated planes at concal mountng (Fgure.b/c). Satsfyng the resonant condton, the gratng perod must meet three requrements. Frst, there s no dffracted order outsde the structure; second, an dent beam dffracts only frst order waves n the gudng regon whch refractve nde 5 n f and fnally, the frst order waves s above the cut-off range of the wavegude mode. Therefore, the mnmum and mamum lmts of the gratng perod ( Λ ) are wrtten as follow: a

74 λ Λ a,mn =, n + s [ n sn θ sn ϕ] + n sn θ cosϕ and λ Λ a,ma =. (3.9) n + f [ n sn θ sn ϕ ] + n snθ cosϕ Where wavelength. n f and n s are the refractve ndces of the flm and the substrate, and λ s the resonance Hence, the resonant modes n -D GMR structure are classfed nto three categores when resonance s a contrbuton of a sngle-dffracted order resonant mode (Fgure 3.4a), two- dffracted order resonant mode n a common plane (Fgure 3.4b at normal dence and Fgure 3.4c at oblque dence), and two-dffracted order resonant mode n the separated planes (Fgure 3.4d). 5

75 Fgure 3.4: Dffracton/gudance planes and phase matchng dagrams of the resonant modes n -D GMR at n-plane mountng (a) sngle-propagaton (b)/(c) counter-propagaton and (d) two resonant modes at concal mountng. 53

76 Usng the HWA approach, structure dmensons needed to acheve each type of these resonances are estmated. The eact resonance responses are then determned by the RCWA. As an llustraton, resonances due to the phase matchng condton of + and - dffracted waves to the wavegude modes are calculated and plotted versus the flm thckness (t f ) n Fgure 3.5a. The calculatons are consdered at the n-plane oblque dence (, ) = ( o, o ) θ for a resonance ϕ wavelength λ. 98μm. The structure has a flm whch refractve nde deposted on a substrate whch refractve nde.47. The -D bnary gratng wth a fll factor of.5 and.-μm depth s drectly fabrcated on the flm. The gratng perod calculated from Eq. (3.9) s lmted wthn the range of.45 Λ.59μm. Here, the gratng perod s selected to be.55μm and a hence the resonant spectral range of nterest s.95 λ. 5μm. As shown, the resonances of a sngle-propagaton mode by + dffracted wave are located at two dfferent wavelengths for the phase matchng of TM and TE wavegude modes. They are ndvdually ected when the dent beam s TM and TE polarzed, respectvely. That can be seen n the resonance responses n Fgure 3.5b, where the structure (t f =.5 μm) supports only fundamental modes. The resonances shft towards longer wavelengths when reasng the flm thckness. For the multmode structure, the +-dffracted wave s also phase matched to the hgher order modes and t ntroduces the resonances located at the shorter wavelengths. The resonance responses plotted n Fgure 3.5c show the resonances relatng to the wavegude modes for the cases of TM (sold lnes) and TE (dash lnes) dent beams. 54

77 t (μm) f.5.5 B A.75.5 (d).5 TM,+ TE,+ TM,+ TE,+ TM,+ TE,+ (c) (b) λ (μm) (a) TE,+ TE,+ TE,+ TE, TE, TM,+ TM,+ TM,+ TM, TM,.8 t f =.5 μm TE TM.8 t f =.594 μm TE TM.8 t f =.594 μm TE TM Reflectvty.6.4. Reflectvty.6.4. Reflectvty λ (μm) λ (μm) λ (μm) (b) (c) (d) Fgure 3.5: (a) Plot of resonances (calculated by the HWA) versus the flm thckness (t f ), and the spectral responses (calculated by the RCWA) of resonances n-d GMR (b) t f =.5 μm (c)/(d) t f =.594 μm Resonant modes by - dffracted wave wth larger propagaton constants can be formed when the flm s adequately thck. In beneft of the resonance responses (wll be dscussed n net secton), resonances by - dffracted wave can be located at the same wavelength as the ones by + dffracted wave when desgnng the structure properly. That can be seen n Fgure 3.5a, where resonances n desgn A (TE mode) and B (TM mode) are for the two-dffracted order resonant modes n a common plane. For the sngle-dffracted order resonant mode n desgn (d), resonance becomes polarzaton ndependent. The resonant mode s coupled along + dffracted wave by 55

78 the TM polarzed beam. For the TE polarzed beam, the resonant mode s ected by - dffracted wave and t s located at the same wavelength ( λ ~. 957μm ). In spt the fact that the resonances of each polarzaton are at the same wavelength, the resonance responses plotted n Fgure 3.5d do not completely overlap due to the dfference n the phase matchng condtons. It has shown here that the resonance locaton and type can be effcently estmated usng the HWA approach. However, the couplng confguratons and the feld scatterng have an effect on the resonance responses (resonance lnewdths, resonance lneshape, the spectral sde lobes and the polarzaton senstvty.) The effect of the scatterng feld for each type of resonance modes and the desgn methodologes to mprove the resonance responses n -D GMR wll be dscussed n the net sectons. 3.3 Couplng nto a sngle-dffracted order resonant mode n -D GMR The scatterng of the feld wthn the -D GMR structure, when + dffracted wave satsfes the resonance condton at oblque dence, s shown n Fgure 3.6. In the fgure, the propagatng dffracton orders are presented nsde the flm. Only the sold arrows, representng the frst-order dffracted wave (+,) are totally reflected at the flm/substrate nterface and accumulate the constructve phases of the wavegude modes. As t nteracts wth the gratng, the mode s partally leaked out from the structure and also dffracted back to the flm. The estence of other scatterng felds, the non-dffracted wave () and the - dffracted wave, contrbutes to the three sources of Fresnel reflectons (R at the cover/gratng nterface, R f and R f at hgh nde flm/substrate nterface). These reflectons deform the resonance response and cause asymmetrc lneshape and hgh base lne []. 56

79 Fgure 3.6: Scatterng of the sngle-dffracted order resonant mode n -D GMR at oblque dence Wthout these unwanted scatterngs, the leakng felds of the + dffracted wave are nterfered and hence the reflectvty, when the -D GMR has an nfnte sze, can be wrtten as [] η d R = (3.) φ η d + 4( η d ) sn The constant η d, referred to here as the gratng strength, s defned as the normalzed power that s coupled to the resonant mode and t s the leakng power at each ndvdual round trp. φ s the phase ganed each round-trp propagaton. As seen, the reflectvty s unty at resonance or φ = mπ. The resonance response, on the other hand, has a phase lnewdth Δφ FWHM = 4sn ( η η ) d d whch s drectly proportonal to the gratng strength ( η d ). Fgure 3.7 presents the resonance response of the structure for varous values of η d. The graph shows the Lorentzan resonance response havng a hgh phase selectvty and low base lne when the gratng strength s small. 57

80 η d =% η d =4% η d =7% R φ/π Fgure 3.7: Plots of resonance response appromately calculated by Eq. (3.) Here, the gratng strength, η d, s the man parameter that nfluences both the spectrum base lne and lnewdth as t s the case n the Fabry Perot cavty. Therefore, the GMR structure s consdered as an nverted Fabry Perot cavty [75.] 3.3. GMR wth Lorentzan resonant response It s desrable for GMR structure to have a symmetrc spectral response, narrow resonant lnewdth and low baselne reflecton. Ths can be acheved by usng the gratng that produces a small η d and a small Fresnel reflecton R at the cover/gratng nterface (Fgure 3.6.). An antreflecton flm s nserted at the hgh nde flm/substrate nterface to elmnate the Fresnel reflectons R f and R f. Before applyng the HWA to estmate the flm thckness, the gratng s desgned to produce the mnmzed couplng strength and Fresnel reflecton at the resonance wavelength. The dffracton effcences of dffracted and non-dffracted waves by the gratng are determned 58

81 whle the flm s consdered as a half-space. As defned by the drawng n Fgure 3.8, the Fresnel reflecton R and the gratng strength η d are the dffracton effcences of reflected th wave and transmtted + st wave, respectvely. For the bnary gratng of the GMR consdered earler n Secton 3., the R and η d are determned usng the RCWA and plotted versus the change of the gratng fll factor ( f ) and the gratng depth ( t g ) n Fgure 3.8. Gratng depth : t g Fresnel Reflecton : R Gratng strength : η d Fllng factor : f Fllng factor : f Fgure 3.8: Contour plots of R and η d versus gratng depth and fllng factor The contour plots show that for a fllng factor between.8 and.9 (that may not be feasble to fabrcate) and depth between.5 and.5 μm, the gratng effcences are mnmzed (R <. and η d <.8.) Here, the objectve s to show the mpact of the gratng effcences on the 59

82 resonance responses. A -D GMR structures wth a.85-fllng factor gratng s used n the nvestgaton. As shown n the graph (Fgure 3.9a,) the Fresnel reflecton R s gradually reduced when reasng the gratng depth. In contrary, the gratng strength becomes stronger. As a result, the resonances n the structure wth shallower gratng are epected to have narrower resonant lnewdth, but they have the hgh baselne reflecton. Applyng three gratng desgns wth depths A, B, and C (reported n Fgure 3.9a), the structure dmensons can be estmated for each desgn n order to acheve the resonance due to the same TE resonant mode. The spectral responses plotted n Fgure 3.8b show that the spectral responses are proportonal to the gratng effcences as dscussed earler...8 R A B C.4 η d Gratng depth : t g Reflectvty A B C λ(μm) (a) (b) Fgure 3.9: (a) plots of R and η d versus gratng depth when f=.85 (b) Spectral responses of -D GMR wth the gratng A (t g =.5 µm, t f =.53 µm), B (t g =.3 µm, t f =.58 µm) and C (t g =.9 µm, t f =.57µm). A second approach to reduce the Fresnel reflecton R s to have the gratng behavng as a multlevel AR subwavelength structure [67] as depcted n Fgure 3.9a. Usng the formulatons based on Chebyshev polynomal [7], the effectve refractve nde of each gratng level s determned, 6

83 and that defnes the fll factor, where each gratng level s a quarter-wave thck. Increasng the number of levels broadens the ant-reflecton band. That allows achevng resonances wth large spectral separatons and neglgble base-lne reflecton between resonances. Smlar to the bnary gratng, removng the Fresnel reflecton confronts the fact that the gratng strength ( η d ) reases as well as the resonances lnewdth. (a).4 R η.8 d number of levels : N Reflectvty Reflectvty. TE TE. E-3 E-4 E E λ (μm) N= N=4 N=7 N= (b) (c) Fgure 3.: (a) Confguratons of the multlevel AR gratng (b) plots of R and η d versus the N- level (c) Spectral responses of -D GMR wth the -, 4-, 7-, and -level AR gratng For the GMR usng a multlevel AR gratng, t appears that the effectve gratng strength s reduced when the number of the gratng levels s reased. The Fresnel reflecton however does not sgnfcantly rease. Comparng the spectral lnewdths of TE resonance n GMRs 6

84 wth -, 4-, 7-, and -level AR gratng, Fgure 3.c shows narrower and more symmetrc lneshape when reasng the number of levels. In addton, the baselne reflecton between resonances s also mnmzed due to the broader ant-reflecton band as demonstrated n Fgure 3.c De-phasng manners at off-resonant wavelength and off-resonant angle For couplng of one dffracted order nto the wavegude mode, the de-phasng characterstcs have observed smlar behavor for the devaton of ether wavelength or the dent angle. (a) (b).8.8 Reflectvty.6.4 Δλ FWHM Reflectvty.6.4 Δθ FWHM λ (μm) θ ( o ) (c) (d) Fgure 3.: Scatterng felds of a sngle-dffracted order resonant mode n -D GMR (a) at-off resonant wavelength (b) at off-resonant angle (c)/(d) spectral and angular responses Fgures 3.a and 3.b show that the + dffracted wave accumulates a de-phasng of Δφ + at both off-resonant wavelength ( λ + Δλ ) and off-resonant angle ( θ + Δθ ) but n opposte sgns. 6

85 Hence, both spectral and angular responses have a Lorentzan lneshape, as plotted n Fgure 3.c and 3.d, respectvely. The de-phasng rates ( Δ φ Δλ and Δ φ Δθ ) ndcate the reducton of the reflectvty at off-resonance. Therefore, n addton to the mode leakage ( β ) or the gratng strength ( η d ), the spectral/angular lnewdths of ths resonance type are drectly proportonal to the de-phasng rates as follows [.] dφ dθ dβ dφ dφ k θ dθ dβ cos dβ, dφ k dλ Λ a ( k n ) dφ + d f f β dβ, where dφ β d f + +. (3.) dβ κ γ c γ s Assumng that β k + and the structure s a smple three-layer slab wavegude ecludng the gratng and the AR flm from the calculatons, the resonant mode has a phase φ defned by Eqs. (3.7) and (3.8). The parameters n Eq. (3.) are desgnated as n Eqs. (3.7) and (3.8). Eq. (3.) clarfes the dependence of the de-phasng rates and hence the spectral and angular lnewdths n a sngle-dffracted order resonant mode. Both lnewdths are drectly proportonal to the drecton of the dent beam and also the modal propagaton constant. Ectng a hgher-order mode requres thcker flm and hence the resonance has narrower 63

86 lnewdths. The dependency of both lnewdths ( Δ λfwhm, Δ θ FWHM ) become stronger when usng shorter gratng perod. That can be wrtten as [] ΔλFWHM Δ θfwhm (3.) Λ cosθ a Where θ s the dent angle at resonance. Therefore, achevng very narrow-band resonances by a sngle propagaton mode confronts a lmted angular tolerance. 3.4 Couplng nto a two-dffracted order resonant mode n -D GMR at n-plane mountng A dfferent resonance behavor s observed when two frst-order dffracted waves (±) are phase matched to two resonant guded modes propagatng wthn a common plane. At normal dence, the dffracton s symmetrc and the two counter-propagatng resonant modes have the same propagaton constant. For ths type of resonance behavor at oblque dence, each one of the frst-order waves s coupled to a dfferent wavegude mode At normal dence The scatterng of the resonant mode at normal dence s demonstrated n Fgure 3.. The two sources of Fresnel reflectons are the non-dffracted waves, R and R f. As shown, the small leakages wth constructve phases are a contrbuton of two modes wth the same propagaton constants along + and - waves. The backward scatterng by the gratng (long-dash red arrow) causes these two modes to nteract nfluencng the resonance responses as wll be dscussed later 64

87 n the secton. The strength of the nteracton reases when the gratng has stronger second Bragg dffracton ( η B ) [.] Fgure 3.: Scatterng of the two-dffracted order resonant mode n -D GMR at normal dence To acheve a symmetrc and low sde-lobe spectral lneshape, the same approach used n the case of the sngle-dffracted order resonant mode s utlzed. However, the gratng strength ( η d ) s a combnaton of the dffracton effcency of both + and -. The calculaton n Eq. (3.) s vald only f the nteracton between modes s neglgble. At off-resonant wavelength (Fgure 3.3a), both modes are de-phased n the same drecton. The spectral lneshape n Fgure 3.3c then remans the same as n the case of a sngledffracted order resonant mode (Fgure 3.c). At off-resonant angle, on the other hand, the modes are de-phased wth opposte sgns, as shown n Fgure 3.3b (sold-blue arrows). Ths dephasng compensaton of these two modes at off-resonant angle produces a gradual reducton of the reflectvty at off-resonant angle. 65

88 .8 (a) (b) Reflectvty.6.4. Δλ FWHM λ (μm) (c) (d) Fgure 3.3: Scatterng felds of a two-dffracted order resonant mode n -D GMR (a) at-off resonant wavelength (b) at off-resonant angle (c)/(d) spectral and angular responses Therefore, the angular lneshape has a flat-top and broader lnewdth (Fgure 3.3d). That makes the resonance by ths type more angularly tolerant. In addton, the de-phasng compensaton can be further mproved when the gratng has a strong second Bragg dffracton []. As a result, the angular lnewdth s epected to be even broader and more ndependent on the spectral lnewdth. Strengthenng the second Bragg dffracton or the backward gratng strength, η B, mght mpact the gratng strength, η d, and the Fresnel reflecton, R, as demonstrated n Fgure 3.4. The contour plots of these gratng dffracton effcences are calculated based on the devaton of the gratng depth and fllng factor when applyng the structure n Secton 3.3. In order to mantan R { β} of the resonant mode, the gratng at normal dence has a perod of.6-µm. 66

89 Gratng depth : t g (μm) Fresnel reflecton : R Gratng strength : η d Backward couplng strength : η B Fllng factor : FF Fgure 3.4: Contour plots of R, η d, and η B versus gratng depth and fllng factor In comparson, the leakage of the counter-propagatng modes reases roughly twce of that of the sngle propagaton mode. Strengthenng η B n ths structure results n a hgh R causng asymmetrc and hgh sdelobe resonant lneshape. That occurs when the gratng has <.5-fllng factor and >.-μm thckness. To demonstrate the mprovement of the resonant response due to the nteracton of two counter-propagatng modes, the TE resonance response of GMR structures wth gratngs of.4- fllng factor are calculated when reasng the depth from.5 to.5 μm. 67

90 R η d η B optmum range Gratng depth : t_g (μm) Δθ FWHM /(Δλ FWHM /λ ) : rad 4 8 Δ ~ tmes Gratng depth : t g (μm) (a) (b) Fgure 3.5: (a) plots of R, η d, and η B versus gratng depth for.4-fllng factor (b) plots of Δθ FWHM angular/spectral lnewdth rato,, versus the gratng depth Δλ / λ FWHM The gratng dffracton effcences are plotted n Fgure 3.5a. The graphs n Fgure 3.5b depct the angular/spectral lnewdth rato versus the gratng depth. It s shown that when the gratng has suffcently strong second Bragg dffracton (t g ~ μm), the dependency of spectral/ angular lnewdth s sgnfcantly reduced. The optmum desgn shows an mprovement of the angular/spectral rato by a factor of two At oblque dence The scatterng of the resonant modes at oblque dence depcted n Fgure 3.6 represents the phase matchng of two frst-order (±) waves to two dfferent wavegude modes. The - wave s coupled to the lower-order mode wth a larger propagaton constant. The leakages from both modes contrbute to the resonance. Therefore, the resonant lnewdths are drectly proportonal to 68

91 the summaton of the gratng strengths η d,+ and η d,. The two gratng strengths may not be dentcal. Fgure 3.6: Scatterng of a two-dffracted order resonant mode n D-GMR at oblque dence At off-resonance, each resonant mode accumulates de-phasng n the same manner as the one at normal dence. However, the de-phasng of these counter-propagaton modes performs asymmetrcally at off resonant angle as depcted n Fgure 3.7a. Ths s due to the devaton of the propagaton constant ( Δ β ). Ths msmatch ( Δφ Δθ Δφ Δθ ) degrades the de-phasng + compensaton and hence the angular lnewdth. Based on Eq. (3.), the de-phasng compensaton can be appromately wrtten as Δφ φ κ β β κ β γ c γ s + Δ Δ Δ Δ Δ t f k cosθ θ θ κ γ c γ c κ γ c γ (3.3) Δ Δ s Eq. (3.3) shows that de-phasng msmatch reases wth the propagaton constant dfference ( Δ β ) and the flm thckness (t f ). However, the angular lneshape remans the same but wth narrower lnewdth compared to the case at normal dence. Lkewse, the angular lnewdth s also broadened due to the strong second Bragg dffracton. 69

92 Reflectvty at θ= o :TE at θ= o :TE,- &TE, Δλ : μm Δθ : degree (a) (b) Fgure 3.7: (a) Scatterng felds of two-dffracted order resonant modes wth dfferent β n -D GMR at off-resonant angle (b) comparson of the angular responses of resonances Achevng ths type of resonance requres the proper gratng perod and the flm thckness for the desred resonance. The structure dmenson can be estmated usng the HWA approach demonstrated n Fgure 3.5a. As seen, the resonance n desgn A s a contrbuton of the phase matchng of - and + waves to TE and TE mode, respectvely. Usng the RCWA, the spectral and angular lnewdths of ths resonance are plotted n Fgure 3.7b compared to the one wth smlar spectral response at normal dence. It s notceable that the angular lnewdth s reduced by a factor of two. The angular response appears to be asymmetrc se the resonance s located near the cut off range of the - wave. Ths can be avoded usng a longer gratng perod. 3.5 Couplng nto a two-dffracted order resonant mode n -D GMR at concal mountng At concal mountng, the frst-order dffracton waves do not share the same plane as demonstrated n Fgure 3.8a. Due to the fact that the resonant modes propagate along two tlted planes, both TE and TM modes could be ected smultaneously. However, t s not guaranteed 7

93 that the dent beam s totally reflected at resonances. Ths depends on the rato of the feld components coupled to each mode and hence the dent beam polarzaton. The resonant modes propagatng n two dfferent planes could be smultaneously ected at the same wavelength once the structure s set at fully concal mountng ( ϕ = 9 ). Otherwse, resonances relatng to modes wth dfferent propagaton constants ( β + and β ) cause a shft n wavelengths. Ths s shown n Fgure 3.8b usng the HWA. (a) (b) Fgure 3.8: (a) Confguratons of the gudng planes and related resonant modes n -D GMR at concal mountng (b) HWA calculatons of resonances versus the shft of dent plane ( θ = o, ϕ ) n -D GMR (Fgure 3.5) wth t f =.5 µm. Resonances due to modes TE m,p and TM m,p are plotted versus the shft of the dent plane, ϕ, away from the gratng vector. The subscrpt -p represents the dffracton wave whch phase matches to the mode m. For the case of o ϕ 9, resonances are ndvdually formed by sngle-propagaton modes. At resonance, the dent beam s totally reflected only f t s polarzed perpendcular or parallel to the related dffracton plane. The resonance lnewdths are proportonal to the power 7

94 dsspated normal to or wthn the plane of dffracton for TE and TM mode, respectvely. In Fgure 3.9a, the resonant responses are calculated usng the RCWA when o ϕ = 45. Resonances wth partal reflectons occur at the locatons wth correspondence to Fgure 3.7b, o o when the dent beam s TE ( ψ = ) and TM ( ψ = 9 ) polarzed. The reflectvty s proportonal to the dent beam polarzaton. Fgure 3.9b demonstrates the change of reflectvty at each resonance when rotatng the dent beam polarzaton. Reflectvty Ψ = o Ψ =9 o TE, TE, TM,+ TE, λ (μm) (a) TM, TE, Reflectvty at λ Ψ ( o ) (b) TE, TM,+ TE,+ o o Fgure 3.9: (a) Spectral responses of -D GMR at concal mountng θ =, ϕ = 45 (b) Devaton of reflectvtes at resonances plotted versus dent polarzaton For the case of ϕ o = 9, the dent beam dffracts along two waves symmetrcally tlted away from the dent plane. Therefore, resonances are formed due to the phase matchng of two resonant modes wth the same propagaton constant. At ths settng, the polarzed dent beam s totally reflected at resonances ndependent of the polarzaton [35.] However, the phase matchng of the resonant modes for the TE and TM beam performs dfferently as the demonstraton n Fgure 3., whch s based on the feld decomposton. Ths wll produce the 7

95 shft of resonances. Resonances are formed by ant-symmetrc modes ( E = E, ) when, + couplng the dent beam to a mode wth cross polarzaton (phase matchng of TM dent beam and TE wavegude mode, for eample.) Otherwse, resonance satsfes the phase matchng of symmetrcal modes ( E, = E, + ). That may ntroduce a phase msmatch and hence shfts the o resonances as plotted n Fgure 3.a for the structure calculated n Fgure 3.8 atϕ = 9. Fgure 3.: Phase matchng dagram and decomposton of TE and TM dent beam correspondng to TE and TM resonant mode for a fully concal mountng As seen, the spectral responses do not only shft both resonances (TM and TE ) under TE (red dash-lne) and TM (black sold-lne) dent beams, but also dffers n the spectral lnewdths as well. That s eplaned n the calculatons of the feld magntudes of the dffracted waves n Fgure 3.b. The results ndcate stronger gratng strength when couplng the dent beam to the ant-symmetrc mode. The dfferences are however less sgnfcant when operatng the 73

96 structure at larger angle. Achevng almost dentcal leakage for both symmetrc and ant- o symmetrc modes requres phase matchng of TM mode for the structure at θ > 5. TM.8 TE Ψ = o Ψ =9 o E //,+..5 Ψ= o Ψ=9 o Reflectvty.6.4. E, λ (μm) θ ( o ) (a) (b) o o Fgure 3.:(a) Spectral responses of -D GMR at concal mountngθ =, ϕ = 9 (b) plots of the feld magntude ( E ) of the dffracted waves versus the θ for the TE and TM polarzed beam. Although the resonances at fully concal mountng are senstve to the dent polarzaton due to the phase msmatch of the symmetrc and ant-symmetrc modes, the slght shft of the resonance can be compensated by optmzng the gratng fllng factor, whch drectly compensates the phase msmatch. When properly desgnng the gratng, the resonances are overlapped at the same wavelength as seen n the calculaton results n Fgure 3.. Here, the structure s set at o θ = 5 and the -D gratng perod s adjusted to be.6 µm n order to o acheve the smlar resonances as n the prevous calculatons atθ = keepng the other parameters constant. 74

97 Reflectvty.5.5 FF=.54 TE FF=.65 FF=.75 Ψ= o Ψ=9 o.5 TM λ (μm) o o Fgure 3.: Spectral responses of -D GMR at concal mountng θ = 5, ϕ = 9 versus the devaton of fllng factor (FF) ( Λ =.6 μm.) a As shown, optmzng the structure to remove the polarzaton dependency can be ndvdually done for each resonance (FF=.54 for TE resonance and FF=.75 for TM resonance) but not both n the same structure. Besdes, the resonance performance s better when matchng the TM mode, where the spectral responses are almost completely overlapped. 3.6 Summary In ths chapter, the concepts of resonant anomaly wthn the GMR structure were presented. GMR structures are nvestgated for the desred resonances usng the RCWA together wth the HWA. The resonant response s drectly related to the scatterng of the resonant modes and the de-phasng characterstcs at off resonance. In general, the resonance response s symmetrc wth low sde lobes when the leakng felds are purely a contrbuton of the resonant mode. The lnewdth s on the other hand dependent on the gratng strength ( η d ). The gratng was properly desgned to have neglgble 75

98 Fresnel reflecton and meanwhle low gratng strength n order to acheve the narrow band resonances. To do so, GMR wth the mult-level AR gratng was shown to serve the requrements. For -D GMR at n-plane mountng, the frst-order dffracton s lmted along a sngle plane. Therefore, resonance s senstve to the dent polarzaton. At oblque dence, resonances formed by the sngle-dffracted order resonant mode appear to have strong dependency on the angular and spectral lnewdths. Resonances due to the two-dffracted order resonant modes are more desrable. Both modes gan de-phasng n opposte drectons at offresonant angle. The de-phasng compensatons reduce the degradaton of reflectvty. De-phasng can be further reduced when the gratng has a strong second Bragg dffracton and the resonant modes are more laterally confned. That ntroduces broader angular lnewdth and less dependency to the spectral lnewdth. However, couplng nto the two-dffracted order resonant modes at oblque dence confronts a de-phasng msmatch due to matchng of dfferent wavegude modes. The angular lnewdth s then not as broad as the one at normal dence. Polarzaton ndependent resonance s feasble. Resonances are formed by couplng each one of the counter-propagaton waves to dfferent polarzed modes at the same wavelength when the structure s set at oblque dence. For -D GMR at concal mountng, the resonant mode propagates along two planes. In general, resonance s a contrbuton of a sngle dffracted wave, unless t s fully concal mountng ( ϕ o = 9.) In advantage, t s possble to ecte the same resonant mode when usng TE and TM beam. However, there ests a slght shft of resonance due to the phase msmatch of 76

99 symmetrc and ant-symmetrc modes. For the proper gratng desgn, the shft s removed and that makes resonances at ths settng polarzaton ndependent. 77

100 CHAPTER 4 : ANALYSIS OF TWO DIMENSIONAL-GUIDED MODE RESONANT (-D GMR) STRUCTURES In ths chapter, the resonance anomaly n the GMR structure usng a -D gratng s studed. The effect of the nherent multple dffracton/gudance planes on the resonance s nvestgated n detals. The feld scatterng and the phase matchng condton at resonance are studed n order to gan better understandng of the couplng characterstcs ludng the mpact of lght polarzaton and the effect of the gratng couplng strengths on the resonance responses. The Bloch mode approach s used to calculate the actual comple modal nde related to the formaton of the resonances n the -D GMR. 4. Resonances n -D GMR For -D GMR (Fgure 4.), the dffracted orders, and hence the gudng planes, ntrnscally propagate n multple planes due to the double perodcty of the -D gratng as dscussed n Secton... Phase matchng of a dffracted wave to a guded resonant mode can be acheved n varous confguratons (planes) dependng on the perodcty of the structure and the drecton of the dent wave. 78

101 Fgure 4.: -D GMR structure confguraton In general, the GMR structure comprses of a subwavelength gratng. For a -D gratng, the range of the gratng perods that meets the resonance condton s determned from the gratng equatons (.7) and (.8). For the perod gratng as n Eq. (3.8). The mamum and mnmum values of Λ a, the range s smplfed n the form of -D Λ b can be wrtten as follows Λ b,mn = n s + λ sec( ζ ) [ n snθ snϕ ] + n snθ cosϕ and f λ sec( ζ ) [ n snθ snϕ ] + n snθ cosϕ Λ b,ma =. (4.) n + Where n s and n f are the refractve ndces of the substrate and the flm respectvely. The other parameters n Eq. (4.) are desgnated n Fgure 4. Se the couplng characterstcs rely on the frst-order dffracton, for the -D gratng n Fgure 4., the perodcty can be classfed nto three categores (rectangular-, arbtrary-, and 79

102 heagonal-lattce crossed gratng.) These classfcatons are based on the gratng skew angle (ζ.) The fundamental frst-order dffracton of each pattern s depcted n Fgure 4.. Fgure 4.: Frst-order dffracton/gudance planes n -D GMR wth (a) rectangular- (b) arbtrary- (c) heagonal-lattce gratng Resonance locatons relatng to each frst-order wave are ndvdually estmated usng the HWA approach. These resonance locatons are shfted from the actual ones obtaned by the RCWA as the homogeneous wavegude approach does not lude the scatterng by the -D gratng. Ths scatterng and the multplcty of the dffracton and gudance n -D structure stress three mportant parameters mpactng the resonances characterstcs; the related frst-order dffracted waves, polarzaton of the dent beam and the polarzaton of the leaky mode. In net sectons, 8

103 the resonance characterstcs n -D structures wll be analyzed based on the scatterng of the feld relatng to resonances at both normal and oblque dence when applyng dfferent gratng patterns. The resonant responses wll be dscussed and the advantages of havng the hgher-order fundamental multple planes wll be shown to allow for new GMR devses to mprove the resonant performances. 4. -D GMR usng rectangular-lattce gratng o The estence of two orthogonal gratng vectors ( ζ = ) n the rectangular-lattce gratng (Fgure 4.a) allows the frst-order dffracton to be along four waves (p,q) = (,), (,-), (,), and (-,). The tangental wave vectors of these wave, reduced from Eqs. (.7) and (.8), can be wrtten as π k, pq = k, p, and Λ b π k y, pq = k y, q. (4.) Λ a The dent drectons θ, ) and the gratng perods Λ, Λ ) defne the wave vectors, and ( ϕ ( a b Re β k + k ). hence the propagaton constant of the weak leaky resonant mode ( { }, pq, pq 4.. Couplng of resonant modes at normal dence At normal dence, the dent beam s symmetrcally dffracted along two orthogonal fundamental planes as depcted n Fgure 4.3a. Each plane sustans two counter-propagatng frstorder waves wth the same propagaton constant that phase matches the leaky wavegude mode at resonance. However, t s not guaranteed that resonances are always a contrbuton of four 8

104 resonant modes at the same wavelength. Ths s defned by the dent beam polarzaton as wll be dscussed later. In all cases, at each round trp a backward dffracton wth the same ampltudes of the wave vectors as well as a crossed dffracton s produced due to the symmetrc dffracton as demonstrated n Fgure 4.3b. The nfluence of these dffractons wll be consdered when nvestgatng a structure wth broad angular tolerance n Chapter 6. Fgure 4.3: Scatterng of feld nsde the structure wth a rectangular-lattce gratng at resonance (a) forward dffracton (b) backward and crossed dffracton. The orthogonal planes of dffracton/gudance make resonances a contrbuton of both TE and TM resonant modes. These two modes are decoupled when the dent beam s lnearly polarzed along one of these gratng vectors. Then, at least two fundamental resonances from TE and TM modes are epected. Couplng to both polarzed modes propagatng along two orthogonal planes does not prevent the total reflecton to be formed at resonances as n the case of -D GMR. The wave vector of each mode defnes the resonance wavelength. The spectral 8

105 lnewdths are proportonal to the power dsspated to the feld normal ( η ) and n-plane ( η // ) components relatng to TE and TM resonances, respectvely. In advantage, the double dffracton reduces the power dsspated to each resonant mode. Hence, resonances n -D GMR may have narrower lnewdths than the ones n -D GMR. d, d, Usng a square-lattce gratng ( Λ a = Λ ) wth symmetrc fllng factors, four resonant b modes have the same propagaton constant. Therefore, phase matchng of both polarzed modes can be accomplshed n both planes. That produces resonances whch are ndependent of the dent beam polarzaton. The dffracton/couplng characterstcs relatng to the TE resonant mode s demonstrated n Fgure 4.4a based on the decomposton of the dent beam along each plane, v E v v = E + Ey y. E v s coupled to the resonant modes counter-propagatng along the dffracted waves (, ) and (,-), whle E v y s coupled to the resonant modes va the dffracted waves (, ) and (-, ) and vce versa for the TM resonant mode. The power couplng to each dffracted wave devates correspondng to the dent beam polarzaton ( ψ ) as shown n Fgure 4.4b. The total power couplng to each polarzed mode however remans constant. That results n the conservaton of both resonant locaton and resonant lnewdth. The analyss s verfed usng the RCWA as shown n Fgure 4.5. The structure has an arhole gratng pattern fabrcated on a delectrc flm wth a refractve nde of. The gratng has perods Λ a =Λ b =5 nm, depth t g = 5 nm, and hole-radus nm. A hgh nde flm has a refractve nde of n f =.5 and a thckness t f =~85 nm. An ant-reflecton flm (n AR = n f n s, t AR = 475/4n AR nm.) s nserted at the hgh nde flm/substrate (n s =.47) nterface. The graphs show an nvarant resonance response, whle the dent beam s polarzed from to 9 degree wth 83

106 respect to the dent plane for TM resonant mode. In addton, the polarzaton of the reflected beam mantans the same drecton and state as that of the dent beam. η d,, η d,, Ση d, η d,,// η d,,// Ση d,// Ψ ( o ) Ψ ( o ) (a) Fgure 4.4: (a) Phase matchng dagram of TE resonant mode ( β ) for an unpolarzed dent beam (b) plots of normalzed powers dsspated to the normal ( η ) and n-plane ( η d,// ) feld components by a square-lattce gratng versus ψ (b) d, Reflectvty Ψ = o.5 Ψ =45 o.5 Ψ =9 o λ (μm) (a) Δ Ψ R o at λ = 475 nm o Ψ (b) Δφ R o at λ = 475 nm o Ψ (c) Fgure 4.5: RCWA calculatons (a) spectral response for ψ = o, 45 o, 9 o ; (b) and (c) polarzaton rotaton ( Δ ψ R = ψ R ψ ) and phase ( Δ φ R ) of the reflected beam at TM resonance. 84

107 In contrary, partal reflecton s epected at resonance for a rectangular-lattce gratng GMR, Λ a Λ, unless the dent beam s lnearly polarzed along one of the dffracton b planes. The phase matchng dagram n Fgure 4.6a shows the possble modes wth two dfferent propagaton constants ( β and β )..3 TE,β.8 TM,β TE,β Λ a (μm).6 Λ a >Λ b Λ a =Λ b TM,β λ (μm) (a) (b) Fgure 4.6: (a) Phase matchng dagram of TE resonant mode (β ) for an unpolarzed dent beam, when Λ a > Λ a (b) plots of resonances versus shft of perod Λ a, calculated by HWA. Therefore, four resonances correspondng to these fundamental modes may be formed. Resonances calculated by HWA are plotted versus the shft of perod Λ a n Fgure 4.6b. That manly mpacts the propagaton constant ( β ) of the resonant modes propagatng along waves (, ±.) The calculaton results demonstrate the relocaton of the resonances to the longer wavelength due to the decrement of the modal propagaton constant (larger perod), whle the other resonances ( β ) reman almost at the same wavelengths. The dent beam polarzaton defnes the couplng characterstcs as dscussed. Here, resonances of TE resonant mode wth 85

108 propagaton constant β s a contrbuton of only E v coupled along dffracted waves ( ±, ) as y n Fgure 4.6a. Hence, the dent beam s partally reflected at resonance where the reflectvty s proportonal to the polarzaton angle ψ as an appromated form ( ) cos ψ. In comparson to the actual reflectvty, the dfference appears more sgnfcant when polarzaton angle s larger as n the graph Fgure 4.7. R cos(ψ ) Ψ ( o ) Fgure 4.7: Comparson between reflectvty (R) at TE,β resonance calculated by RCWA and cos( ψ ) versus ψ The resonance response of a structure wth Λ a = 6 nm s plotted n Fgure 4.8, when the dent beam polarzaton s set at, 45 and 9 degrees. As seen, resonances are polarzaton dependent. The separaton between the resonances can be changed over a wde range by changng the nput polarzaton. The resonance locatons are adjustable by changng the perod as demonstrated n the graph n Fgure 4.6b. 86

109 Reflectvty TM,β TM TE,β,β TE,β Ψ = o Ψ =45 o Ψ =9 o λ(μm) Fgure 4.8: Reflectvty spectrum of GMR wth rectangular-lattce gratng ( Λ > Λ ) a b 4.. Couplng of resonant modes at oblque dence At oblque dence, the symmetry of the frst-order dffracton s broken. The leaky mode can be coupled by any of the four frst-order dffracted waves that match ts propagaton constant ( Re{ β} ~ of the leaky mode.) The tangental wave vectors of the dffracted orders can be completely dfferent. As dscussed n Secton 3.5 for -D GMR at concal mountng, total reflecton s acheved when the dent beam s perfectly polarzed normal to or along a dffracton plane. Total reflecton happens also when all the components of an unpolarzed dent beam are coupled to the resonant modes wthout phase msmatch. At least two resonant modes havng the same propagaton constant propagate symmetrcally around the plane of o o dence. That can be accomplshed when the dent plane s set at ϕ = and ϕ = 45 as demonstrated n Fgure 4.9 for structure wth square-lattce gratng. 87

110 (a) (b) Fgure 4.9: Phase matchng dagram of the resonant modes n -D GMR wth a square-grd o o gratng at oblque dence (a) ϕ = ; (b) ϕ = 45 Although total reflecton forms at resonance for TE and TM polarzed beam, resonances occur at dfferent wavelengths due to the phase msmatches between the symmetrc and ant-symmetrc modes, thus degradng ths reflectvty for an unpolarzed beam. Ths dfference n the resonance locaton can be elmnated by changng the gratng dmensons (fllng factor and perod.) For -D GMR wth square-lattce gratng ( Λ a = Λ ), havng two orthogonal fundamental b planes of dffracton/gudance makes t possble to acheve four fundamental resonances wth total reflecton at oblque dence. For eample, when the dent plane s set at o = 45 dsplayed n Fgure 4.9b, two pars of frst-order waves, (,-)/(-,) and (,)/(,), dffract symmetrcally around the dent plane and phase match to resonant modes wth shfted propagatng constants β and β, respectvely. Usng HWA, resonances correspondng to the resonant modes are appromately determned. ϕ, as 88

111 3 θ ( o ) TE,β TM,β TE,β TM,β λ (μm) Fgure 4.: Plots of resonances n of GMR wth square-lattce gratng ( Λ a = Λ b ) at o = 45 versusθ, calculated by homogeneous wavegude approach. ϕ The calculaton results n Fgure 4. demonstrate the fundamental resonances locatng at four dfferent wavelengths. Tltng the structure wth large angle ( θ ) results n the rement of β and hence related resonances movng toward the shorter wavelength. In contrary, resonances of modes wth the smaller β are located at the longer wavelengths. The structure consdered here has the same materal as n Secton 4... The gratng has a square-lattce pattern wth perod 3 nm, hole-radus 9 nm, and depth t g = 5 nm. A hgh nde flm has a thckness t f =7 nm. Unlke the structure wth rectangular-lattce gratng at normal dence, all of the four fundamental resonances are formed and have a total reflecton when the dent beam s TE or TM polarzed, as demonstrated n Fgure 4.. The structure s calculated atθ o =. Lke the -D GMR at oblque dence, the shft of resonances by orthogonal polarzed beams s epected unless the gratng s well optmzed to remove the phase msmatch of the symmetrc and ant-symmetrc modes. As the optmum gratng dmenson s sutable for ndvdual 89

112 resonances, achevng all resonances ndependent to the dent polarzaton s not feasble at oblque dence. Reflectvty Reflectvty.5 TM TE TM,β,β,β TE,β Ψ = o Ψ =9 o λ (μm) Fgure 4.: Reflectvty spectrum of GMR wth square-lattce gratng ( Λ ( θ, ) = ϕ ( o, 45 o ) a = Λ ) at b The enlarged spectra n Fgure 4. ndcate that only TM resonance due to the β resonant modes ((,-)/(-)) s polarzaton ndependent, whle the other three resonances are not. The appearance of the slght shft (~-3 nm) between the orthogonal polarzaton beam components results n the partal reflecton and deformed spectral lneshapes (Fgure 4.b to d: red-sold plots) when usng an unpolarzed beam. 9

113 Reflectvty TM,β Ψ = o Ψ =9 o Ψ =45 Reflectvty TE,β Δ~3 nm Ψ = o Ψ =9 o Ψ = λ (μm) (a) λ (μm) (b) Reflectvty TM,β Δ~ nm Ψ = o Ψ =9 o Ψ =45 Reflectvty TE,β Δ~ nm Ψ = o Ψ =9 o Ψ = λ (μm) (c) λ (μm) (d) Fgure 4.: Closed-up plots of reflectvty spectrum n Fgure 4. (a) TM,β (b) TE,β (c) TM,β (d) TE,β resonance For unpolarzed GMR flter, t s mportant to equalze the gratng couplng strength for the symmetrc and the ant-symmetrc modes so that resonant lnewdths are smlar and the lneshape remans unchanged. Even though the spectra may not be completely overlapped due to the Fresnel reflectons, the spectral lneshape for an unpolarzed beam s not deformed. The calculaton results n Fgure 4.3 ndcate that the structure settng at o ϕ = 45 s more 9

114 desrable. The devaton of power leakages ( Δη = η to roughly 3 tmes of the one at ϕ o = d d ψ o o ψ =9 = η d ) s sgnfcantly reduced. Achevng resonances wth dentcal lnewdths at o ϕ = s stll possble when couplng to TM resonant modes but at the large dent angle nearby the cut-off lmt TM mode TE mode TM mode TE mode 4 4 Δη d 3 Δη d 3 3 θ ( o ) 3 θ ( o ) (a) (b) Fgure 4.3: Comparson of devaton of gratng couplng strengths ( Δ η d ) for TE and TM dent beam when couplng to the same mode (sold-lne: TM mode, dash-lne: TE o o mode) (a) ϕ = ; (b) ϕ = D GMR usng arbtrary crossed gratng Havng two non-orthogonal gratng vectors as n the arbtrary crossed-gratng (Fgure 4.b) produces four frst-order dffracted waves wthn two tlted planes. The hgher-order dffracted waves (,-) and (-,), n Fgure 4.4a, wll have smaller tangental wave vectors (defned n Eq. (.7) and (.8)) when the gratng skew angle reases (Fgures 4.4b-d.). Therefore, the hgher order modes mght phase-match to the leaky mode and hence contrbute the resonances. 9

115 Havng resonant modes propagatng along three tlted planes, couplng characterstcs are more complcated than the prevous case as wll be dscussed n the net secton. Fgure 4.4: (a) Frst-order dffracton/gudance planes by an arbtrary crossed gratng; phase o o o matchng dagram (b) ζ < 3 (c) ζ = 3 and (d) ζ > 3, when Λ a = Λ b o For a -D GMR usng a crossed gratng wth skew angle ζ < 3, the hgher-order waves (,-) and (-,) are cut off. In ths case, the presence of two tlted fundamental dffracton planes makes the couplng characterstcs at normal dence smlar to a structure usng an orthogonallattce gratng at oblque dence. Therefore, ths confguraton does not mprove the resonant performances and resonance wth total reflecton s stll lmted by the beam polarzaton. Decreasng the polarzaton dependence requres symmetrc dffracton around the plane of dence when couplng to resonant modes havng the same propagaton constant. Otherwse, the plane of dent has to be algned along one of the gratng vectors. Then, the actve resonant modes wll propagate along a drecton overlappng wth the dent plane as n the case of a rectangular-lattce gratng. 93

116 The couplng characterstcs of the TE resonant mode at normal dence are demonstrated n Fgure 4.5 for the cases of TE and TM dent beams when the crossed gratng has dentcal perods ( Λ a = Λ ). As seen, the dent beams n both cases are b symmetrcally coupled to each wave vector. However, only normal components of the dffracted waves contrbute to the TE resonant mode. (a) (b) Fgure 4.5: Phase matchng dagram of TE resonant mode n -D GMR wth an arbtrary o o o crossed gratng ( Λ = Λ, ζ 3 ) at normal dence (a) ψ = (b) ψ = 9 a b The decomposton of the TE and TM polarzed beam wth respect to the dffracton planes performs smlarly to the square-lattce gratng case. Due to couplng to the tlted planes, beams wth orthogonal polarzatons are ndvdually coupled to the ant-symmetrc or symmetrc modes va dfferent pars of frst-order waves whle the dffracton angles between them ( ϕ / for TE beam, and ϕ / for TM beam, as n Fgure 4.5 when couplng to ant-symmetrc modes) are devated proportonally to the polarzaton drecton. As a result, the frst-order dffracted waves accumulate phase msmatches and hence the resonance s located at dfferent wavelengths for the 94

117 TE and TM beam. As dscussed earler, the phase msmatches can be removed when the gratng parameters are optmzed. The graph n Fgure 4.6a shows the phase msmatch of the frst order wave (-,) caused o by a crossed gratng wth skew angle ζ = n comparson to the one caused by a square-lattce gratng when changng the gratng fllng factor. At the optmum gratng parameters (r~. and.88 µm), there ests no phase msmatch. Resonances n the structure then are epected to be polarzaton ndependent. Nevertheless, the devaton of the couplng strengths n Fgure 4.6b s consderably large, and hence t prevents the spectral responses to be perfectly overlapped as wll be seen n Fgure 4.7b for the structure usng the gratng wth.95-µm ar-hole radus..3. ζ= o ζ= o ζ= o ζ= o Δ (φ, ).. Δ (Ση d, ). polarzaton ndependence r (μm) r (μm), (a) (b) Fgure 4.6: Plots of (a) phase msmatches of wave (-,) (b) the devaton of the couplng strength, when applyng the TE and TM polarzed beam, versus the ar-hole radus o (r)of the crossed gratng wth ζ = n comparson to the square-lattce gratng o ( ζ = ) 4 Fgure 4.7a shows the fll factor (hole sze) and flm thckness that wll produce resonances at a specfc wavelength for two orthogonal polarzatons. It s shown that for a gratng wth the proper parameters smultaneous resonance s located at the same wavelength for 95

118 both polarzatons and therefore unpolarzed lght. The shft of resonances agrees wth the phase msmatches when adjustng the gratng fll factor n Fgure 4.6a. t f (μm)..8.4 polarzaton ndependence Ψ = o Ψ =9 o Reflectvty Ψ = o Ψ =9 o Ψ =45 o r (μm) λ (μm) (a) (b) Fgure 4.7: (a) Plot of the flm thckness (t f ) for achevng the TE resonance at λ =. 475μm o versus the change of the ar-hole radus (r) of the crossed gratng wth ζ = (b) spectral response when r=.95 µm and t f = ~.8 µm 4.4 -D GMR usng heagonal-lattce gratng Heagonal-lattce gratng s formed when the perodcty has a skew angle = sn ( ) ζ Λ a Λ b, as shown n Fgure 4.c. The gratng generates fundamental dffracton along three crossed planes; two of them are equally slanted around the other. Ths confguraton produces s frstorder dffracted waves (p,q) =(,), (,-), (,), (-,), (-,), and (,-). The tangental frstorder wave vectors are wrtten as k, pq = k, p 4Λ 4π b Λ a + q 4Λ π b Λ a, and π k y, pq = k y, q. (4.) Λ a 96

119 The dffracted waves have the same length once both perods are dentcal. In the case of equal perods the gratng s referred to as regular-heagonal lattce. Otherwse, the tangental wave vectors of four waves ((,), (,-), (-,), and (,-)) along the symmetrcally tlted planes are equal but dfferent than the other two ((,) and (-,)). Ths gratng confguraton s referred to as rregular-heagonal lattce Couplng of resonant modes at normal dence The symmetrc dffracton by the heagonal-lattce gratng at normal dence results n phase matchng the resonant modes wth the same propagaton constants counter-propagatng along the common plane. Due to the hgher-order fundamental dffracton, couplng to ths type of resonant modes could occur n the three crossed planes, as demonstrated by the feld scatterng at resonance n Fgure 4.8. Lkewse, the backward and the addtonal crossed dffractons mantan the same paths as the forward dffractons. That reases the nteracton between the modes and hence mprovng the resonance angular tolerance. As the dffracton/gudance planes are not orthogonal, resonances n ths structure are a contrbuton of resonant modes propagatng wthn at least two crossed planes. Nonetheless, couplng to the counter-propagaton resonant modes wthn only one plane can be obtaned when usng an rregular heagonal-lattce gratng. 97

120 Fgure 4.8: Scatterng of feld nsde the structure wth a heagonal-lattce gratng at resonance (a) forward dffracton (b) backward and crossed dffracton. o For a regular heagonal-lattce gratng ( Λ = Λ and ζ = 3 ), the dent beam s partally coupled to all of the s frst-order waves havng the same tangental wave vectors. These waves contrbute to the resonances unless some of the dffracted waves are polarzed orthogonal to the resonant-mode polarzaton. In general, the couplng characterstcs of an unpolarzed dent beam can be consdered as the decomposton of the feld correspondng to v TE and TM modes along the dffracton/gudance planes, E = Ea aˆ + Ea aˆ + Ea3 aˆ 3. The drawng n Fgure 4.9 s for the case of couplng to the TE resonant mode, where the unt vectors, a, a and a 3, are perpendcular to each plane. Conceptually, the unpolarzed beam s totally reflected at resonances as long as all of the feld components are drectly coupled to the resonant modes wthout rotatng the polarzaton. a b 98

121 Fgure 4.9: Phase matchng dagram of TE resonant modes (β ) n -D GMR usng a regular heagonal-lattce gratng for the unpolarzed beam Due to the non-orthogonal perodcty n the heagonal-lattce gratng, the dent beam s dffracted nto each wave wth a slght rotaton of the polarzaton and change n the couplng strengths ( η,(, ) for TE modes and η //,(, ) for TM modes) proportonal to the dent beam d, p q d, p q polarzaton drecton, as plotted n Fgure 4.. The structure calculated here apples the same desgn as n Secton 4. but usng the regular heagonal-lattce gratng wth the same dmensons (perods, hole-radus, and depth.) The calculaton results report small rotatons of the dffractedwave polarzaton when the dent beam s polarzed at ψ o o o =, 3, and 9, wth respect to one of the dffracton planes, by Δ ψ o o ~.,.7, and.7 o, respectvely. As a result, these addtonal polarzaton components of the dffracted waves are phase matched to the resonant modes and contrbute to the constructve nterference at the dent regon. Ths reflected wave 99

122 becomes ellptcally polarzed wth slghtly reduced magntude. The reducton of reflectvty s drectly proportonal to the degree of rotaton. Ψ ( o ) (p,q) 8 (,) (,) ΔΨ~.7 o ΔΨ~.7 o 6 (,) 4 ΔΨ~. o ΔΨ= o Ψ ( o ) (a) η d, Ση d, (,) (,) (,) Ση d,// η d,// Ψ ( o ) Ψ ( o ) (b) Fgure 4.: Plots of (a) polarzaton drecton and (b) normalzed powers dsspated to frst-order waves by the regular heagonal-lattce gratng at normal dence versus ψ Whle rotatng the dent beam polarzaton, the total couplng strengths ( η d, and η d,// ) reman almost constant. The devaton becomes slghtly larger when couplng to the TM resonant mode. Therefore, resonances have lnewdths almost ndependent of the dent polarzaton.

123 The resonance locatons however mght be slghtly shfted due to the phase msmatch when couplng to three non-orthogonal planes. Obtanng both TE and TM resonances located at the same wavelength can be acheved n the -D GMR by shftng the flm thckness to satsfy the phase matchng of the partcular resonant mode whle keepng the other dmensons constant. The resonance responses are calculated and plotted versus the dent beam polarzaton n Fgure Ψ = o Ψ = o.8 Ψ = o Ψ = o Reflectvty.6.4 Reflectvty λ (μm) λ (μm) Reflectvty TE resonances λ (μm) Ψ o o o 3 o 4 o 5 o 6 o 7 o 8 o 9 o Reflectvty TM resonances λ (μm) Ψ o o o 3 o 4 o 5 o 6 o 7 o 8 o 9 o (a) (b) Fgure 4.: Spectral responses versus the polarzaton drecton ψ = o to 9 o (a) TE resonance: ~.7μm (b) TM resonance: ~.55μm. t f t f The RCWA calculatons show an almost nvarant spectral response to the dent polarzaton. The shft of resonance response s more notceable when couplng to the TM resonance. The zoom-n plots of both cases agree n the fact that total reflecton s accomplshed only when

124 ψ o = 6. That ndcates no phase msmatch and no rotaton of the polarzaton (Fgure 4.a), as the dent beam s totally coupled nto resonant modes along two planes. The reducton of reflecton appears more sgnfcant when the polarzaton devated away from 6 degrees, especally n the TM resonance case. That corresponds to the fact that the polarzaton s rotated by the dffracton of the gratng. The partal reflected beam has a polarzaton slghtly devated from the dent beam (Fgure 4.a) n addton to the change of the polarzaton state (Fgure 4.b.) ΔΨ R at λ 4 TE resonance TM resonance Δφ R at λ TE resonance TM resonance Ψ ( o ) Ψ ( o ) (c) (d) Fgure 4.: Plot of (a) polarzaton rotaton ( Δ ψ = ψ ψ ) and (b) phase φ ) of the reflected beam at λ versusψ. ( Δ R = φe r φ,// Er, R R The crossed couplng between these resonant modes n three crossed planes nstead of two reduces the phase msmatches. Therefore, ths polarzaton dependence s not as sgnfcant as the case when couplng to modes propagatng along sngle plane n -D GMR or propagatng along two-tlted planes n -D GMR wth rectangular-lattce gratng at oblque dence or wth an arbtrary crossed gratng, as dscussed earler.

125 Usng an rregular heagonal-lattce gratng ( Λ a Λ ), the resonance s senstve to the b dent polarzaton due to the shft of the propagaton constants of the resonant modes ( β ) along a plane (±,) away from that of the resonant modes ( β ) along two tlted planes (,±) and (,-)/(-,). The phase matchng dagram of TE resonant mode ( β ) s depcted n Fgure 4.3a. Hence, the unpolarzed beam s partally reflected at resonant wavelength, as n the case of counter-propagaton resonant modes n -D GMR usng a rectangular lattce gratng. In addton, the fact that the dent beam s also coupled to modes propagatng along two tlted planes, polarzaton ssues can be removed when desgnng the structure properly as dscussed..3 TE,β.8 TM,β TE,β Λ a (μm).6 Λ a =Λ b TM,β.4 Λ a <Λ b λ (μm) (a) (b) Fgure 4.3: (a) Phase matchng dagram of TE resonant mode (β ) for an unpolarzed dent beam, when Λ a < Λ a (b) plots of resonances versus shft of perod Λ a, calculated by HWA Usng the HWA calculatons, resonances relatng to each resonant mode are calculated n Fgure 4.3b when shftng one perod Λ a. That produces four fundamental resonances. Changng one of the perodctes results n the shft of the tangental wave vectors of the all frst-order waves smultaneously. Therefore, all resonances are shfted n manners defned by the change of the 3

126 modal propagaton constants. As seen, the tangental wave vectors k y,± (Eq. (4.3)) rease when desgnng the gratng wth larger perod Λ. Consequently, the related resonances move toward the shorter wavelengths. a The man advantage of ths structure s that at least three fundamental resonances, nstead of two, wth total reflectons are formed when the dent beam s polarzed normally or n plane of the dffracted waves (-,) and (,). One resonance s a contrbuton of TE or TM mode wth a propagaton constant β va the dffracted waves (-,) and (,). The others are due to both fundamental modes wth propagaton constants β va the dffracted waves (, ), (,-), (-, ), and (,-)..5 TM TE,β,β Ψ = o TE,β Reflectvty TM,β TM,β Ψ =9 o TE,β.5 ~ nm ~ nm λ (μm) Fgure 4.4: Spectral responses of the structure n Fgure 4.9b havng polarzaton s ψ =º and 9º Λ a < Λ a, when dent Usng the RCWA calculatons, the resonance responses of both dent polarzatons are plotted n Fgure 4.4. The graphs show a slght shft of the resonances for modes havng propagaton constant β, when rotatng the dent polarzaton by 9º. 4

127 4.4. Couplng of resonant modes at oblque dence In the heagonal-lattce gratng desgn, there ests s possble planes of dence along whch total reflecton at oblque dence s guaranteed when the dent beam s polarzed along one of the dffracton planes (sold lne n Fgure4.5a) or one of the planes between them (dash lne n Fgure 4.5a.) In any case, the tangental wave vectors are devated whle the symmetry of dffracton s mantaned around the plane of dence. The phase matchng dagram forϕ o and ϕ = 9 are ehbted n Fgures 4.5b and 4.5c respectvely. o = (a) (b) (c) Fgure 4.5: (a) Symmetrc planes of dence; phase matchng dagram of the resonant modes n D-GMR wth a regular heagonal-lattce gratng at oblque dence o o (b) ϕ = ; (c) ϕ = 9 o Resonant modes wth four possble propagaton constants created when ϕ = can contrbute to eght fundamental resonances. However, not all of them are phase matched due to the presences of two sngle-propagaton modes along the waves (,) and (-,). Thereby, the resonances wth total reflecton at ths settng are located at s wavelengths nstead, and so are the ones when ϕ o = 9. S fundamental resonances are formed by the phase matchng of 5

128 resonant modes propagatng along three pars of frst order waves when ϕ o = 9. In both cases, the separaton of resonances s further broadened and notceable for large dent angle as shown n Fgure 4.6. Two fundamental resonances (TE and TM ) relatng to each propagatng constants are calculated versus the change of the dent angle ( θ ) usng the homogeneous wavegude approach. Resonances move toward the longer wavelengths when decreasng the modal propagaton constants. As a result, the hgher-order resonances mght appear near resonances at a shorter wavelength. That can be seen n the calculaton results n Fgure 4.6b (dash plots). TE,β TE,β 5 TM,β TE,β 5 TM,β TE,β θ ( o ) TM,β TE,β3 θ ( o ) TM,β TE,β3 TM,β3 TM,β3 5 TE,β4 5 TE,β TM,β4 TM,β λ (μm) λ (μm) (a) (b) Fgure 4.6: Plots of resonances n of GMR wth heagonal-lattce gratng ( Λ o o ϕ = (b) ϕ = 9 versusθ, calculated by HWA. a = Λ ) at (a) b That eplans the presence of the etra resonance when calculatng the spectral responses usng RCWA n Fgure 4.7. In ths calculaton, the plane of dent s set at o ϕ = 9 and the dent angle s θ o =. Lke the other structures at oblque dence, resonances are 6

129 nherently polarzaton dependent. Remove ths dependency requres optmzng the gratng desgn to elmnate the phase msmatches. Reflectvty Reflectvty TE,β.5 TM,β3 TE,β3 Ψ = o TM TM,β TE TE,β,β,β Ψ =9 o λ (μm) Fgure 4.7: Reflectvty spectrum of GMR wth heagonal-lattce gratng ( Λ ( θ, ) = ϕ ( o, 9 o ) a = Λ ) at b In advantage, the frst-order dffracton n ths structure at oblque dence can be manpulated n a number of symmetrc manners. Couplng to four resonant modes wth the same propagaton constant s possble. Ths s accomplshed by properly adjustng the perodctes and the dent angle. The couplng characterstcs then perform smlarly to the polarzaton ndependent GMR flter usng a square-lattce gratng at normal dence, as dsplayed n Fgure 4.8. The resonant modes do not propagate n perpendcular drectons to each other, hence the phase msmatches when applyng TE and TM polarzed beams stll est. 7

130 (a) (b) Fgure 4.8: Phase matchng dagram of four resonant modes n -D GMR usng an rregular o lattce gratng at oblque dence ϕ = 9 and couplng characterstcs for (a) TE and (b) TM polarzed beam The nteracton of the forward waves (,)/(-,) and the backward waves (,)/(-,) wth respect to the plane of dence prevents the phase matchng of all of the four resonances from formng at the same wavelength. The graph n Fgure 4.9 shows the openng of the resonant band gap (calculated by RCWA: sold-lne plots) around the pre-determned resonant angle o θ ~ (usng HWA: dash-lnes plots.) As a result, resonances are located at the upper and lower resonant band. The shft of the band for TE and TM polarzatons dent beam s notceable. Achevng polarzaton-ndependent resonances s then possble and can be even more practcal compared to the prevous desgns. As seen n the plot, at the proper dent angle, the resonances at the upper resonant band of the TE beam and the lower resonant band of 8

131 the TM beam are shfted and properly ntersect at the same wavelength. The phase msmatches are then removed wthout the need of a precse gratng fllng factor optmzaton..8.6 HWA : TE, HWA : TE, RCWA : Ψ = o θ ( o ).4. Band gap RCWA : Ψ = o RCWA : Ψ =9 o RCWA : Ψ =9 o λ (μm) Fgure 4.9: Plots of TE resonant wavelengths as a contrbuton of waves (,)/(-,) and (,)/(-,) versus the dent angle θ n -D GMR usng an rregular heagonallattce gratng ( Λ =.5μm and Λ =.7μm ) a a The resonance responses of the structure at three dent angles n Fgure 4.3 ndcate the shft of the TE resonances for TE and TM dent beams. The resonances overlap at the proper angle. The spectral lnewdths on the other hand can be adjusted va the gratng dmensons ndependently wthout ntroducng phase msmatches as was the stuaton for the other cases mentoned earler. 9

132 Reflectvty θ = Δ~ nm Ψ = o Ψ =9 o Ψ = λ (μm) (a) Reflectvty θ =.45.5 Ψ = o Ψ =9 o Ψ = λ (μm) (b) Reflectvty θ = Δ~ nm Ψ = o Ψ =9 o Ψ = λ (μm) o o Fgure 4.3: Reflectvty spectrum of GMR n Fgure 4.9 at (a) θ = (b) θ.45 θ o = (c) and (c) 4.5 Calculatons of the leaky mode dsperson Leaky-mode means havng a comple propagaton constant and that can be drectly related to the resonance lnewdth as dscussed earler. For the -D structure, there are a number of couplng characterstcs relant on the dent beam (polarzaton and dent drecton) and the gratng

133 pattern. Therefore, the resonance-lnewdth calculatons through the comple propagaton constant (Eq. (3.)) only may be not accurate as was the case n -D GMR. The lnewdth depends on the gratng couplng strengths regardng to the dent beam polarzaton as well. Nevertheless, the calculaton of the modal dsperson s of nterest. Ths demonstrates the actual nteracton between the modes as well as the actve resonant modes for a partcular dent angle. The Bloch mode approach (Secton.3.) s used to calculate the comple propagaton constants of the leaky modes n -D GMR usng a rectangular-lattce gratng. Usng ths approach requres engneerng the calculaton unt cell along the drecton of propagaton. Hence, the calculaton s effectve only for modes propagatng n an ndvdual dffracton/gudance plane when the gratng has dfferent perods ( Λ a Λ b ). The nteractons of the modes n the orthogonal plane are also taken nto consderaton whle formng the scatterng matr wthn the unt cell. To smplfy the calculatons, the surface gratng used here has a rectangular groove profle. Besdes, the unt cell s selected n a symmetrc manner as shown n Fgure 4.3. There est three ndvdual sldes wthn the unt cell. Two of them are dentcal. The scatterng matr along the calculaton regon s then formed by the symmetrc S-matr multplcaton as mentoned n Secton.3.. For the -D case, the sub-s-matr s ndvdually calculated by consderng each one of these sldes to be perodc along y- and z-as. The propagatons of the Bloch modes are determned by solvng the Egen value problem n Eq. (.53) and (.54). Only comple propagaton constants wth small magnary parts actually propagate and may cause resonances.

134 Fgure 4.3: Unt cells of D-GMR wth a rectangular-lattce gratng (rectangular-shape hole) Calculatng the comple propagaton constants of the leaky modes supported by a -D GMR wth square-lattce gratng ( Λ a = Λ b =.5μm ) n Secton 4.., the crcular-hole gratng ( r =.μm or FF=.5) s transformed to be a square-hole gratng wth dmensons of nm mantanng the gratng effectve nde. The dsperson graphs n Fgure 4.3 represent the modes supported n ths structure. The calculaton results are plotted n term of the real and the magnary parts of the normalzed modal ndces (n eff ) to the rato Λ λ when each one of them propagates n opposte drectons, called the forward and the backward modes. It s shown that the structure supports two fundamental modes, TE and TM. The dsperson plots of each mode demonstrate the formaton of the band gap due to the nteracton of the forward and the backward modes n the calculaton plane. The addtonal band gap s due to the modes counterpropagatng along the orthogonal plane. Ths band-gap s shfted from the actual ones under the calculaton. The dsperson plots also ehbt the overlap of the forward-tm mode and the a

135 backward-te mode ndces at the same wavelength where the resonant band gap s generated when applyng the proper settng. Forward modes : TM Forward modes : TE Backward modes : TM Backward modes : TE TM /TM TM band:.5 calculated.53 TM band:.5 orthogonal plane.5 TM /TE.5.5 Λ a /λ TE /TE TE band: orthogonal plane.48 TE band:.47 calculated R (n ) *Λ /λ : Forward modes eff a R (n eff ) *Λ a /λ + : Backward modes I (n eff ) *Λ a /λ : TM I (n 3 eff ) *Λ a /λ : TE (a) (b) (c) Fgure 4.3: Dsperson of leaky modes (a) normalzed R { n eff } Re { n eff } (b) normalzed { n eff } TM guded modes (c) normalzed I { n eff } of TE guded modes I of Achevng the resonances correlatng to the normalzed modal ndces, the dent drecton can be smply estmated usng the phase matchng condton of the frst-order waves and the leaky modes as R Λ a Λ a { n eff } n snθ cosϕ m λ ± λ, b b and R{ n eff } n sn sn Λ λ ± θ ϕ Λ m λ (4.3) 3

136 At normal dence, equaton (4.3) nsures that two resonances are located around the TE and TM band gaps, where the normalzed modal ndces of the leaky modes equal to. Both frstorder counter-propagatng waves ( ±,) or (, ± ) are coupled to the leaky mode. Achevng resonance by couplng to the forward and backward modes wth orthogonal polarzatons, the normalzed modal ndces happen to be ~.57 and ~.943. In ths case, the plane of dence o needs to be set at ϕ = 45 n order that the polarzed dence beam can coupled to both TE and TM mode. Thus, the frst order waves (-/-) and (/) are phase matched to the forward TE modes and backward TM mode, respectvely. The dent angle calculated from Eq.(4.3) sθ o = 9.4. The RCWA calculatons n Fgure 4.33 demonstrate the agreement of the resonant responses to the modal dsperson and the phase matchng condton n Eq. (4.3) for both the cases at normal (Fgure 4.33a) and oblque (Fgure 4.33c) dences. The couplng characterstcs at oblque dence however are polarzaton selectve especally when the dent plane s along one of the fundamental dffracton/gudance planes, as dscussed earler n ths chapter. Fgure 4.34 llustrates resonances at oblque dence (, ) = ( o, o ) θ relatng to the modal dsperson plot, when the dent beam s TE ϕ polarzed. As shown, only TE resonances are formed by (-, ) and (,) waves. They are phase matched to TE forward and backward modes and located at dfferent wavelengths. The frst order waves (, ±) on the other hand are phase matched to both TE and TM modes located nearby ther ndvdual band gaps. That reports the estence of the nteracton between the forward and the backward modes at these resonances. 4

137 TM.55 TM,(/) TM /TM TM Λ a /λ TM /TE.5.5 TE,(/) /TM,(/) TE Reflectvty TE /TE TE TE,(/).8..5 R (n eff ) *Λ a /λ : Forward modes Reflectvty R (n eff ) *Λ a /λ + : Backward modes (a) (b) (c) Fgure 4.33: Spectral response of the resonances at (a) normal dence/(c) at o o ( ϕ, ) = (45,9.4 ) and (b) band dagram of structure n Fgure 4.3 θ TM TM,(,+/) TE (,+/) (,) (,) TE,(,).5 Λ a /λ TE,(,+/) TE,(,) Reflectvty Re(n ) *Λ /λ eff a (a) (b) Fgure 4.34: (a) Spectral response and (b) matchng plots of normalzed modal nde (TE /TM ) and the normalzed ± waves (calculated from Eq.(4.3)) at ( θ ) ( o o, ϕ =, ), and TE-polarzed beam. 5

138 4.6 Summary In ths chapter, the couplng characterstcs of the dent beam to the leaky modes contrbutng to the resonances of the -D GMR structure usng three gratng patterns: rectangular lattce, arbtrary cross lattce and heagonal lattce were demonstrated. In all cases, the dscussons are smplfed based on the decomposton of the dent beam wth respect to the frst-order dffracton pattern and the resonant mode. Therefore, resonance wth total reflecton occurs once all components of the dent beam are phase matched to the leaky modes propagatng n a common plane or dfferent planes at the same wavelength. The polarzaton of the reflected beam remans unchanged. Usng symmetrc gratng patterns, a square- or a regular heagonal-lattce gratng, at normal dence, the total reflecton s obtaned at resonances, and resonances are polarzaton ndependent as well. Due to the slght rotaton of the dffracton-wave polarzaton by the regular heagonal-lattce gratng, the dent beam s smultaneously coupled to the leaky modes propagatng along the three tlted planes nstead of two planes based on the dent beam decomposton. Ths ntroduces a rotaton of the reflected beam polarzaton, however the reducton of the reflectvty s not that sgnfcant (<%) as well as the spectral responses reman almost unchanged whle rotatng the dent beam polarzaton. The dent beam polarzaton becomes the man concern once the symmetry of dffracton s broken by ether perodcty or the dent drecton. It s necessary to select the plane of dent n such a way that the polarzed dent beam s symmetrcally dffracted around t. The phase msmatches when couplng the TE and TM beams to the same resonance mode can be removed by controllng the gratng fll factor. Another approach presented was the 6

139 use of the rregular heagonal-lattce gratng to couple the dent beam along four dffracted waves wth equal tangental wave vectors. The nteracton of these modes forms the resonant band gaps but shfted when utlzng orthogonal polarzed dent beam. At the proper dent angle, the ntersecton of the upper and the lower bands of the TE and TM beams, respectvely or vce versa, ntroduces resonance ndependent of the dent polarzaton. The requrements of achevng the precse gratng fll factor are then mnmzed. In addton to the polarzaton ssue, the advantage of havng multple dffracton/gudance planes n -D GMR was shown to have more degree of freedom n achevng multple fundamental resonances wth total reflectons at both normal and oblque dence. A number of fundamental resonances rease proportonal to the dffracton/gudance planes. Resonance locatons are adjustable drectly through the gratng perods or the dent drecton when applyng the proper polarzed beam. The controllablty of resonance responses s practcal. Therefore, n net chapter, -D GMR wll be proposed as controllable mult-lne narrow-band spectral flters. The desgn methodology ludng the lmtaton wll be demonstrated. 7

140 CHAPTER 5 : MULTI-LINE GMR FILTERS In ths chapter, multple wavelength narrow-band reflecton flters are developed based on the resonance anomaly n the guded-mode resonant (GMR) structure. Multple resonances are produced once the structure supports multple leaky modes that could lude the hgher-order wavegude modes or only the fundamental wavegude modes when usng a -D gratng for nstance. The capablty of controllng resonances s necessary and very useful n many applcatons. The -D GMR s proposed and demonstrated for the controllable mult-lne spectral flters wth spectral separaton n a wde range whch s not possble wth -D GMR as the lmtaton of the structure materal. 5. Overvew Mult-lne resonant flters have been generated usng -D GMR structure [36-38], where the multple resonances are acheved by couplng to multmode leaky wavegude structures. These have been done by reasng the thckness of the gudng regon or addng ether thn-flm layers or gratngs nto the orgnal GMR structure. Mult-lne resonances are acheved by nsurng that each dffracted order s coupled to a guded mode n ths structure. However, as wll be shown, t s not possble to control the locaton of all the resonances (and ther separaton n a broad spectral range) wthout requrng specfc change n the materal refractve nde whch may not be feasble. 8

141 Mult-lne reflecton GMR flters are developed and the resonance locatons are controlled by adjustng the physcal dmensons of the structures (thckness, and perods, etc) and not ts materal propertes (refractve nde). Ths s accomplshed by utlzng the dffracton propertes of -D GMR flters, where the presence of multple planes of dffracton generated by the -D gratng makes multple fundamental resonances. The locatons of these resonances can be controlled by adjustng the perodctes of the two-dmensonal gratng structure and the geometrc shape of the unt cell. Frst, the lmtatons of -D GMR structures n specfyng the locaton of the mult-lne resonances owng to the requrement to specfy the refractve nde of the materal for a gven resonance locaton s demonstrated. Meanwhle, the materal dependency of a controllable multlne -D GMR flter s also brefly dscussed. The utlzaton of mult-plane dffracton characterstcs n -D GMR structures wll be nvestgated and contrasted wth -D GMR structures. A desgn approach and methodology s presented for mult-lne flters n -D GMR structures. Two-lne reflecton flter desgns n the vsble spectrum wth spectral lne wdths less than nm and a controllable spectral separaton up to 3 % of the short resonant wavelength are presented usng rectangular-lattce gratng GMR structures. Three-lne flters are desgned n a heagonal-lattce gratng GMR structure. Fnally, the resonance characterstcs and the advantages of each approach are dscussed. In addton, the last secton nvestgates the fabrcaton tolerance and the mpact of each structure dmenson on the resonance lnewdth as well as the resonance locaton. 9

142 5. Mult-lne -D GMR Flters As dscussed n Chapter 3, the resonances depend on the guded modes supported n the structure as well as the dffracton characterstcs of the gratng. One dmensonal gratng at an n-plane o dence ( ϕ = ) dffracts the dent beam along a common plane regardless of the drecton of each dffracted order as shown n Fgure 3.4. Ths restrcts all of the leaky modes n the GMR structure to propagate along the same plane. Hence, the mplementaton of multple-resonance structure s accomplshed by couplng each frst-order dffracted wave to a dfferent guded (hgher) order mode. At oblque dence, two fundamental resonances va + and - waves are ndvdually formed as a contrbuton of a sngle-propagaton mode. Resonances however are not tolerant to the dent drecton. Desgnng the structure at normal dence s more desrable and manly mplemented n ths work. The symmetry of the dffracton pattern results n phase matchng of counter-propagaton resonant modes wth the same propagaton constant. Ths can be ntally defned by the gratng perod. Lke the tradtonal three-layer slab wavegude, the number of guded modes depends on the geometry and materal propertes of the structure. More modes propagate as the wavegude normalzed frequency, or V number [7], reases. V = k t n n (5.) f f s Ths can be done by reasng the thckness of the gudng regon (t f ), or enhancng the nde contrast between the flm (n f ) and the substrate (n s ). The desgn approach [] s to numercally optmze varous structure parameters, such as the gratng perod, the refractve nde contrast of

143 the structure and the thckness of the gudng regon, to obtan the resonances at the desred locatons. Fgure 5. shows the spectral response of a -D GMR that supports the frst two leaky modes (TE and TE ) wthn the vsble spectrum. In ths graph, a TE polarzed normal dent beam s consdered. Then, the frst resonance ( λ = 475 nm ) s due to the phase matchng between the frst order dffracted waves and the TE mode. The second resonance ( λ = 5 nm ) corresponds to the TE mode. Fgure 5.: The spectral response of the two-lne -D GMR flter at normal dence: gratng (n g =.5, Λ a =3 nm FF=.6, t g =5 nm), hgh nde flm (n f =.7, t f =353 nm), an AR coatng ( n AR = n f ns, t AR = λ / 4nAR ), a substrate (n s =.47), and an ar superstrate (n c =) The separaton between these two resonances depends on the modal dsperson of the structure. When usng a shallow gratng (weak gratng strength), the GMR structure can be consdered as a three-layer slab wavegude, where the propagaton constants of the modes are defned by the gratng equaton (.). The phase matchng condton can be re-wrtten as

144 Re{ β } snθ and { β } + π λ n c π Λ a π π Re n c snθ + (5.) λ Λ a At normal dence, both propagaton constants code. Modes propagate n opposte drectons wth the same propagaton constants. To determne the separaton between the resonance wavelengths due to two successve modes, one needs to solve the dervatve of the TE mode characterstc equaton (.34) to (.39)) when dm =. t dκ κ + γ κ + γ ( κ dγ γ dκ ) ( κ dγ γ dκ ) = π f c c c s s s (5.3) Consderng two successve modes wth the same propagaton constant, d β =, k n f dk dκ =, κ d k n dk =, and c γ c γ c d k n dk = (5.4) s γ s γ s The wavelength separaton ( d λ ) s then estmated by substtutng d κ, dγ s, and dγ c from Eq. (5.4) n Eq. (5.3). dλ dk = λ k = t k f n f π κ + β γ c + γ s (5.5) It s shown that the wavelength separaton s not only nversely proportonal to the hgh nde flm parameters (both refractve nde and thckness) but also to the propagaton constant as well. Ths mples that the more modes supported nsde the structure, the narrower the

145 separaton of resonances can be. Changng each of these parameters wll change the resonance wavelengths; however the separaton of resonances s lmted by the nde contrast of the structure that wll be seen n net secton. 5.. Desgn of controllable mult-lne -D GMR flters Whle t s possble to obtan multple resonances n -D GMR structures, t s crtcal to be able to control the locaton of (and the separaton between) each resonant wavelength. As ponted out, the resonance locatons depend on the modal dsperson of the structure. Both the thckness of the gudng layer and ts materal propertes nfluence the resonance. In -D GMR structures, controllng the locatons of the resonances n a broad spectrum requres changng the refractve ndces of the flm or the substrate accordng to the thckness of the hgh nde flm. A typcal approach n desgnng a mult-lne -D GMR flter s to ntally estmate the structure parameters under the cut-off condtons for the frst resonance wavelength. For eample, a two-lne reflecton flter can be desgned by allowng only the frst two guded modes (TE and TE ). To acheve the frst resonance at 475 nm, and to guarantee that the second resonance corresponds to the lower guded mode at the desred wavelength, one needs to adjust the gratng perod and the flm thckness whle fng the other parameters. The second resonance appears at a longer wavelength n ths case. To control the second resonance whle fng the frst resonance, the thckness of the hgh nde flm s reduced whle reasng the gratng perod. The wavelength separaton reases when reducng the thckness of the hgh nde flm as seen n Eq. (5.5). The lmtaton of the nde contrast between the flm and the substrate restrcts the locaton of the second resonance. Ths can be seen n Fgure 5.. Varyng 3

146 the gratng perod wthn the resonant range ( λ < Λ < n, Eq. (3.8)) from 9 to 3 n f a λ nm reases the second resonance separaton from 3% to % of the frst resonance wavelength when the flm refractve nde s.7. For a flm wth hgher refractve nde of.5, the resonant range of the gratng perod s broadened and hence the separaton changes from 4% up to 4%. The appromated results, calculated usng Eq.(5.5), show smlar trend when changng these two parameters (t f and Λ a ). A mamum error of ~8% s eperenced at larger perods. s 8 t f Δλ/λ : Eact Δλ/λ : Appromated t f Δλ/λ : Eact Δλ/λ : Appromated t f (nm) 6 Δλ/λ (%) t f (nm) Δλ/λ (%) Λ a (nm) : n f = Λ (nm) : n =.5 a f (a) (b) Fgure 5.: Resonance separaton ( Δ λ λ ) calculatons of TE and TE resonances n two-lne D-GMR flters (Fgure 5.) versus perod of the gratng ( Λ a ) and flm thckness (t f ) (a) flm nde (n f )=.7 (b) flm nde (n f )=.5 To be able to control the second resonance over a broader range from few nanometers up to hundreds nanometers, one needs to optmze the flm thckness, flm nde and the gratng perod. The structure wth hgher flm nde gves a tunable range of the second resonance n a broader separaton. However, the lmtaton of the cut-off resonant range prevents the resonances to have very narrow separaton (< nm.) Fgure 5.3 shows the effect of changng both the flm thckness and the flm nde when desgnng a two-lne flter wth the frst resonance fed at 475 4

147 nm. The gratng has perod at the mnmum resonant range (9 nm) for the flm nde.7. The graphs show resonance separatons n a broader spectrum from 3 % to 4 % of the frst resonance. Lkewse, the appromated calculatons show smlar trend (mamum error of 5 %). Δλ/λ : Eact 5 8 t f Δλ/λ : Appromated 4 t f (nm) Δλ/λ (%) n : Λ =9 nm f a Fgure 5.3: Resonance separaton ( Δλ λ for both eact and appromated Eq. (5.5) for two-lne D-GMR flters (n Fgure 5.) versus flm refractve nde (n f ) and flm thckness (t f ) Realzaton of these structures, however, requres materals wth refractve ndces between.7 and.6. Ths, n addton to changng the physcal dmensons of the structure, make mult lne - D GMR flters not practcal. The resonances tend to be more ndependent f each leaky mode s completely guded along a separate plane. GMR structure wth a crossed gratng adds more degrees of freedom to manpulate resonances as wll be presented n the net secton. 5.3 Mult-lne -D GMR Flters In -D GMR flters, multple resonances appear ntrnscally due to the multple planes of dffracton. As dscussed n Chapter 4, the number of the fundamental resonances reases 5

148 dependng on the degree of symmetry n the gratng grd geometry. As a result, modes correspondng to multple resonances do not necessarly share the same planes of gudance. Ths eases the controllablty of the resonances. Each resonance can be manpulated by modfyng the value of the propagaton constant ndependently through the perodctes of the gratng at any dent drecton. These multple planes of dffracton rease the degree of freedom n couplng the dent wave to both TE and TM guded modes of the structure. The normal components of dffracton waves potentally have phase matched to the TE guded mode, and vce versa for the TM guded mode case. Lke the analyss of -D GMR, the wavelength separaton between successve modes can be calculated by dfferentatng the characterstc equaton, when the structure s consdered as a three-layer slab wavegude. The analyss of -D GMR here copes wth crossed-polarzaton modes nstead. Then, the characterstc equaton s generally wrtten as κ t n f γ n c f p tan + tan p nc κ ns γ s + κ f = mπ (5.6) Where p ndcates the polarzaton of the guded modes (p = for TE m modes, and p = for TM m modes). The constant m s the mode number. Consderng the separaton of resonance wavelength between TE m and TM m modes wth the same mode number, the dervatve of Eq. (5.6) s presented n Eq. (5.7), where dm = and dp =. 6

149 7 { } {} { } = + + II d II I d I t d f κ (5.7) where κ γ c c f n n p I + =, and κ γ s s f n n p II + = (5.8) At normal dence, propagaton constants of both resonant TE m and TM m modes are fed by the gratng perods. Havng the same propagaton constant, = β d, κ d, c dγ, and s dγ are ntally defned n Eq. (5.4). The wavelength separaton calculated wth respect to the TM mode resonance ( = p ) s shown n Eq. (5.9). λ s the resonance wavelength relatng to the TM mode and λ π = k = Δ S n C n t S n C n n k s s c c f s s c c f wg γ γ β γ γ κ λ λ, ( ) c f c f n n k n n C + = β, and ( ) s f s f n n k n n S + = β. (5.9) The wavelength separaton between TE and TM modes of the same wavegude s sgnfcantly dependent on the structure parameters. Wth the same mode number, the wavelength separaton s narrower compared to two successve modes wth the same polarzaton. The wavelength separaton s nversely proportonal to the hgh nde flm parameters (both nde and thckness).

150 The larger the propagaton constant s, the narrower the separaton can be. Fng the frst resonance at λ requres reducng the hgh nde flm thckness and reasng the gratng perod. 4 Δλ wg /λ : Eact 8 4 Δλ wg /λ : Eact 4 3 t f Δλ wg /λ : Appromated 6 t f Δλ wg /λ : Appromated t f (nm) 4 Δλ wg /λ (%) t f (nm) Δλ wg /λ (%) Λ a = Λ b (nm) : n f = Λ a = Λ b (nm) : n f =.5 (a) (b) Fgure 5.4: Resonances separaton ( Δ λ λ ) calculatons of TE and TM resonances n -D GMR flters usng a square-grd gratng versus gratng perod and flm thckness (t f ) (a) flm nde (n f )=.7 (b) flm nde (n f )=.5 Fgure 5.4 shows a comparson between the appromated spectral separaton between TE and TM resonances and the eact RCWA calculatons when reasng the perods of the gratng over the resonance range. The structure consdered here s etended from the -D GMR n Fgure 5., where the -D gratng s replaced by a square-lattce -D gratng havng -nm depth and ~. fllng factor (rato of ar-hole and unt cell area). Ths appromaton ndcates more sgnfcant errors especally when the gratng perods approach the upper lmt of the resonance range. In ths case, the resonance separaton tends to be narrower when reasng the gratng perod. Ths s not trval for the wavegude concept due to the complety of couplng along two planes n the GMR structure. The RCWA results for two -D GMR structures wth flm ndces n f =.7 and.5 agree n the fact that the resonance separaton s adjustable va the 8

151 gratng perod. However, the separaton s lmted when applyng the low nde contrast structure. Controllng resonances n -D GMR confronts the same constrants unless each one of these fundamental resonant modes reles on a separate frst-order wave. Thereby, t s possble to ndependently adjust the modal propagaton constant wthn the desred spectra by changng the physcal dmensons of the structure (gratng perod, geometry, and depth; and flm thckness f t s necessary) wthout the need to change the refractve nde of the structures as n the case of mult-lne -D GMR flters Desgn of controllable mult-lne -D GMR flters Avodng the dependency of the mode propagaton constants that contrbute to the resonances, the mult-lne -D GMR flter s set at normal dence. Thus, the dent beam can be coupled to the ndvdual leaky modes along dfferent fundamental dffracton planes for the proper polarzatons. For a rectangular-lattce crossed gratng [49-5] wth equal perods, the structure conssts of two dentcal planes of dffracton orthogonal to each other. Ths mples that modes propagatng along one of these planes have the same propagaton constant at normal dence. Both (TE and TM) guded-mode resonances present at any dent wave nput polarzaton. It s possble to separate these two modes to be guded along dfferent orthogonal planes f the dent beam s polarzed along ether one of these perodctes. Then, the resonance characterstc s equvalent to two supermposed -D GMR flters and the dent wave could be guded nto both TE and TM wavegudes at each resonance wavelength at orthogonal drectons. 9

152 However, se one wavegude supports a TE mode and the other supports TM modes, equal propagaton constants requre dfferent resonance wavelength. The spectral shft depends on the dsperson of the wave gudng structure. Changng the gratng perod n one drecton, the dffracton n that plane results n a dfferent mode propagaton constant and hence shfts the resonance wavelength. Increasng the perod, whch corresponds to the TE mode (second resonance), has an effect on the broader resonance separaton and vce versa. The shft of the flm thckness n order to mantan the frst resonance may not be necessary here as long as the cross couplng between modes s neglgble. The mnor shft of the frst resonance (TM mode) s always compensated by the gratng perod related to that resonance. For a regular heagonal-lattce crossed gratng, the structure has three planes of dffracton wth heagonal symmetry that corresponds to the fundamental dffracted orders. The resonance characterstc s equvalent to the square-lattce crossed gratng, but the dent beam could be guded nto each of the three planes of dffracton. An etra resonance wll occur f the skew angle (ζ ) of the heagonal grd symmetrcally slants from 3 degrees. The symmetry of the skew angle depends on the perodctes of the heagonal grd, where = sn ( ) ζ Λ a Λ b. Thus, changng the skew angle wll change the perodctes. Unlke the rectangular-grd crossed gratng, the gudance along two symmetrcally tlted planes (k y ' and k y '' drectons n Fgure 4.c) are not completely ndependent. The thrd resonance s created f the perod along y-drecton Λ, n Fgure 4.) s dstorted from the perods along ' and '' drectons ( Λ n Fgure 4.). ( a b Consequently, two resonances due to TE and TM guded modes along k y '-k z and k y ''-k z dffracton planes are generated when the dent electrc feld s algned on or normal to k -k z dffracton plane. The dffracted waves along k -as cause the thrd resonance, where ths 3

153 resonance s controllable by adjustng one of the gratng perods correspondng to the skew angle (ζ ). In ether case, the resonances are roughly determned by applyng the homogeneous wavegude approach [] for each plane of dffracton. The eact locatons of the resonances are determned by adjustng the perodcty along each dffracton plane Results and dscusson Usng two approaches dscussed n the prevous secton results n multple resonances at the desred wavelengths. Here, the flters are desgned at normal dence consderng an nfnte dent plane wave and nfnte GMR structure. The structure s comprsed of a gratng, a flm and an AR coatng deposted on the substrate wth a refractve nde of.47 as depcted n Fgure 4.. The gratng (wth refractve nde of.5) s also desgned to behave as an AR coatng 5 to reduce the Fresnel refecton at the ar-gratng nterface. In addton, the gratng s desgned such that the power transferred to the dffracted waves s small enough to narrow the spectral lnewdth at each resonance [.] The thckness of the AR coatng between the flm and the substrate s desgned as a quarter-wave flm. A two-lne reflecton flter at any arbtrary o wavelengths s desgned wth rectangular-grd gratng ( ζ = ) GMR structure. Controllable three resonances are acheved usng the heagonal-grd gratng ( ζ = sn ( ) Λ a Λ b ) GMR flter. The gratng regon of the mult-lne -D GMR structure s desgned to elmnate the Fresnel reflecton at the ar-gratng nterface. Ths requres a gratng depth of nm and a holeradus of 5 nm for the.5 gratng nde. Two resonances wll occur n the structure due to the frst two guded modes (TE and TM ). The -D GMR flter conssts of a wavegude wth both 3

154 TE and TM guded modes. TM mode refers to the case where the guded wave s polarzed n the plane of dffracton where the mode s guded. TE mode s when the guded wave s polarzed normal to the plane of dffracton. Each wavegude s optmzed to correspond to one resonance. Se both sngle-mode wavegudes have the same flm refractve nde and flm thckness, the locaton of the two resonances are controlled by adjustng the gratng perods. The ablty to adjust the crossed gratng perodcty n an asymmetrc manner makes t feasble to desgn arbtrary resonances. It s mportant to note that the two resonances are caused by couplng one nput polarzaton (TE or TM) nto the TE and TM guded modes. It s necessary that the dent feld s ether TE or TM polarzed to acheve % reflectvty. Otherwse, the reflecton peaks wll be formed at four locatons wth reflectvty less than % when the perods are dfferent. In the followng calculatons, the dent beam s polarzed along the -as as depcted n Fgure a Controllable two-lne -D GMR flters usng a rectangular-lattce gratng In the rectangular-lattce GMR flter, the two planes of dffracton are orthogonal. The structure s ntally desgned to acheve the frst resonance ( λ ) at 475 nm as a result of the TM mode (guded wave polarzed n the plane of dffracton). Here, the flm wth a refractve nde of.5 has a fed thckness of 5 nm. Therefore, the gratng perodcty parallel to the dent electrc feld relatng to the TM resonance s defned at 8 nm. The second resonance due to the mode propagatng along the orthogonal plane (TE ) occurs at a longer wavelength (guded wave polarzed normal to the plane of dffracton). Increasng the perod of the gratng n the 3

155 orthogonal drecton ( Λ ) up to b λ ns ~3 nm, the second resonance s mamally shfted from the frst resonance. The spectral response of two-lne GMR flter s shown n Fgure 5.5a to 5.5d. Fgure 5.5: Spectral response of the two-lne -D GMR flters wth a rectangular-lattce structure. (a) Λ b /Λ a =.89; (b) Λ b /Λ a =.96; (c) Λ b /Λ a =.3; (d) Λ b /Λ a =. The response s symmetrc wth low sde lobe (< %) wth very narrow lne wdth (less than nm). Spectral separatons of 3 nm up to nm are shown. That s about <% to 3% of the short resonance wavelength. Ths s acheved by changng the gratng perod n one orthogonal drecton relatve to the other. 33

156 Fgure 5.6a shows the shft of the second resonance as a functon of the rato of the two gratng perods n the orthogonal drecton ( Λ = 8 nm and λ = 475 nm). a.3.5 Δλ/λ.. λ (μm) ~ 3 nm Λ b /Λ a Λ b /Λ a (a) (b) Fgure 5.6: (a) Shft n the locaton of the second resonance (b) resonance locatons n two-lne - D GMR flters wth a rectangular-lattce structure versus the rato of gratng perods ( Λ a = 8 nm ) The spectral separaton may be appromated by: Δλ Λ = C λ Λ b a Δλ + λ wg, C = λ d d( Δλ) ( Λ Λ ) b a (5.) Δλ wg s the spectral separaton when the rato of the two gratng perods are equal. Δλwg and C depend on the modal dsperson of the wavegude between the TE and the TM modes wth the same propagaton constant. For the structure presented here, C s.9 and Δλ wg λ s.. Near zero spectral shft occurs when Λ = Λ ( Δλ C ) b a wg λ. Due to the nteracton of resonant modes whch are phase matched at the same wavelength, resonances do not code as shown n Fgure 5.6b. The estence of a band structure results n a mnmum spectral separaton for a 34

157 structure wth Λ b = 47 nm of ~ 3 nm or <% of the short resonance wavelength. The mamum spectral shft s determned by the mamum useable gratng perod whch s Λ b = λ n s. Ths s to nsure that the dffracton orders n the nput regon and the substrate are suppressed. For the structure consdered here, the mamum spectral separaton s 3% of the short resonance wavelength. Ths s acheved by changng the gratng perod and wthout the need to change nether the refractve nde nor the thckness of the wave gudng regon. Wder spectral separaton s possble wth further optmzed the wavegude structure. Desgnng a structure wth gratng perods that cover the resonance range, n f Λ λ n s λ < <, can broaden the spectral separaton up to 3-4 % of the short wavelength. The graph n Fgure 5.7 represents the resonance locatons n a structure desgned wth the gratng perod Λ a set at the mnmum lmt ( nm.) The second resonance at a longer wavelength, ~ -nm larger than the frst resonance, s acheved when shftng the gratng perod Λ b to the mamum lmt (3 nm.) However, the presence of the hgher-order resonance (TE ) at a shorter wavelength lmts the mamum perod mamum separaton s ~4 nm or ~ 3% of the TE resonance wavelength. Λ b to ~8 nm only, and hence the 35

158 λ (μm) λ TE λ TM λ TE ~4 nm ~8 nm Λ b /Λ a Fgure 5.7: Resonance locatons n two-lne -D GMR flters wth a rectangular-lattce structure versus the rato of gratng perods ( Λ a = nm ) 5.3..b Controllable three-lne -D GMR flters usng a heagonal-lattce gratng In the heagonal-lattce gratng GMR structure, there are two dentcal dffracton/ wavegude planes symmetrcally slantng wth arbtrary angle n addton to the fundamental dffracton plane (k -k z plane). Therefore, three resonances wll occur n the GMR structure f the dent feld s polarzed perpendcular to or n the fundamental dffracton plane -z. The frst and the second resonances, due to the slanted wavegudes (k y '-k z plane and k y ''-k z plane), can be eplaned by the decomposton of the dent polarzaton nto two components. One component s parallel to each of the dffracton planes producng a TM guded mode and the other component s normal to the two dffracton planes producng a TE guded mode. The frst resonance, (desgned to be at 475 nm), s then due to the guded TM mode. The resonance due to the TE mode of these gratngs appears at a longer wavelength of ~5 nm. The propagaton constants of these two modes are dentcal. The spectral shft depends on the modal dsperson of the wavegudes. Meanwhle, the thrd resonance s due to the fundamental dffracton plane (k - 36

159 k z plane). Ths resonance s adjustable by changng one of the perodctes ( Λ ), whch sgnfcantly shfts the propagaton constant of the mode along ths plane. However, a slght shft of the propagaton constants of the frst two modes s always found. Ths can be eplaned by the gratng equaton (4.). In the recprocal plane, the wave vectors of the dffracted waves are not orthogonal. Fng the two resonances due to the modes propagatng along the tlted planes b requres optmzng the other gratng perod Λ a whle keepng the other structure dmensons (flm thckness) constant. The flm thckness s fed at ~95 nm. A structure wth equal gratng perods of 8 nm produces two resonances. Controllng the thrd resonance by changng Λ b requres a slght shft of the perod Λ a nversely proportonal to Λ b as shown n Fgure 5.8a Λ a λ 9 6 λ Λ a (nm) 8 λ (nm) 55 λ Λ b (nm) Λ b /Λ a (a) Fgure 5.8: (a) The change of the gratng perods Λ b versus Λ a to f two resonances ( λ and λ ), whle shft the other ( λ 3 ). (b) The locaton of resonances n three-lne -D GMR flters usng a heagonal-lattce gratng versus the rato of the two gratng perods ( Λ Λ ). b a (b) The calculaton results n Fgure 5.8b show the shft of the thrd resonance when the dent fled s polarzed normal to the k -k z dffracton plane. Ths makes the thrd resonance to be 37

160 caused by the TE mode of the thrd dffracton plane. The thrd resonance can have a wavelength at a short wavelength, between the frst two resonances, or even at a longer wavelength. Fgure 5.9: Spectral response of the three-lne -D GMR flters usng a heagonal-lattce structure (a.) Λ b /Λ a =.96; (b.) Λ b /Λ a =.7; (c.) Λ b /Λ a =.4; (d.) Λ b /Λ a =. It s shown n Fgure 5.9a through Fgure 5.9d that the resonance response of each desgn wth dfferent perod along '-as ( Λ ) has symmetrc and narrow (less than nm) spectrum b around each resonance wavelength. Equaton (5.) may be used to estmate the spectral separaton between the resonances caused by the non-slanted wavegude and any of the other two 38

161 from the slanted wavegudes. slanted wavegudes. Δλwg s the separaton between TE and TM guded modes of the The advantages of the double perodcty of the two-dmensonal gatng GMR structures are that the gudance propertes of the leaky-wavegude structure (GMR flter) can be completely controlled by adjustng the perods of the gratngs n order to acheve the desred resonances. The dependency of two resonances due to the non-orthogonal wavegudes lmts the desgn of the three-lne reflecton flters. However the locatons of the three resonances are much better controlled usng the heagonal -D gratng GMR structure compared to the -D GMR flter. In all these structures, a shft of resonance always ests due to the fabrcaton error. Trade-off between the structure dmensons helps to sustan the desred resonances, as wll be dscussed n net sectons. 5.4 Fabrcaton tolerances of Mult-lne -D GMR Flters The error n fabrcatng these structures mght cause a devaton n the locatons of the resonances ludng ther spectral responses as well. Hence, t s necessary to study the fabrcaton tolerance of each ndvdual structure parameter and ts effect on the resonance locatons and spectral response. As presented prevously, the locatons of the resonances are senstve to the gratng perod and the hgh nde flm thckness. Good control of the gratng fabrcaton ludng flm deposton s requred. In practce, precse control of the flm thckness mght requre costly growth technques (EBM). Usng cheaper technques such as E-beam or thermal evaporaton are usually accompaned wth certan errors. These errors can be corrected for by the changng the gratng parameters such as gratng perod and ar-hole radus as wll be 39

162 shown later n ths secton. Besdes, the dffculty of fabrcatng a hgh aspect raton gratng wth vertcal sde walls can be always compensated by alterng the gratng-hole radus. Here, -D GMR structure wth a square-grd gratng (ar hole gratng) s nvestgated. The gratng s etched n a materal wth refractve nde of. The hgh nde flm refractve nde s n f =.5. The AR flm ( n = n n ) wth quarter wave thckness s nserted between the hgh AR f s nde flm and the substrate (refractve nde n s =.47) to neglect the Fresnel reflecton at the flm/substrate nterface. The structure s desgned to support two fundamental modes (TE and TM ) wthn the vsble spectrum. The frst resonance s located at λ = 475 nm and the second one s at λ = 5.6 nm. The hgh nde flm s.8 nm thck. The gratng has a perod of 3 nm wth an ar-hole radus of 65 nm, and a gratng depth of 8 nm. The beam s normally dent from free space to the GMR structure. To determne the fabrcaton tolerances of each structure dmenson, one calculates the resonance response whle shftng each one of these structure dmensons from the orgnal desgn. That s shown n Fgure 5. and Δλ (nm) Δt g Δr Δt f ΔΛ Δλ (nm) Δt g Δr Δt f ΔΛ 4 Δ (nm) (a) (b) Fgure 5.: Devaton of the resonant wavelengths whle changng one of each structure dmenson from the desgn (a) TM resonance (b) TE resonance 4 4 Δ (nm)

163 The calculatons n Fgure 5. show the shfts of resonance locatons when changng the structure parameters (a gratng depth: t g, an ar-hole radus: r, thckness of a hgh nde flm: t f, and a gratng perod: Λ) from the orgnal desgn. Fgures 5.a and 5.b are the shfts of TM ( λ ) and TE ( λ ) resonances, respectvely. Both graphs show lnear shft of resonances wth the most sgnfcant shft by changng the gratng perod ( Δ λ ΔΛ s ~.7 and λ ΔΛ s ~.9). Besdes, resonances are also senstve to the thckness of the hgh nde flm ( Δ λ Δt s ~.8 Δ f and λ Δt s ~.6). Changng the gratng depth and ar-hole radus shows the least effect. Δ f The results pont out some shft of the TM resonance ( λ Δr s ~.3 and λ Δr s ~.9) when changng the ar-hole radus. The change n ar-hole radus affects the effectve nde of the gratng. The phases of the TM guded modes are more senstve to the structure refractve ndces than the TE ones. Then, changng the gratng fll factor or ar-hole sze n ths case has more sgnfcant mpact on the locaton of the TM resonance than the TE resonance. The plots n Fgure 5. llustrate the devaton of the spectral lnewdth ( Δ Δ Δ λfwhm ). The data are calculated at the relocated resonances when changng each ndvdual parameter. It s seen that the spectral lnewdth reases dramatcally wth the ar-hole radus. Ths s, however, obvously formed n the TE resonance. The depth and perod of the gratng show no major nfluence n alterng the spectral lnewdth. 4

164 .6.6 Δ [ Δλ FWHM of λ ] (nm).3.3 Δt g Δr Δt f ΔΛ Δ [ Δλ FWHM of λ ] (nm).3.3 Δt g Δr Δt f ΔΛ.6 Δ (nm).6 Δ (nm) (a) (b) Fgure 5.: Devaton of the spectral lnewdth whle changng one of each structure dmenson from the desgn (a) TM resonance (b) TE resonance The calculaton results correspond to the prevous dscusson n Chapter 3. The spectral lnewdths of the resonances rely on the leakage of the modes ( Im { β} ) or the gratng strength ( η d ). Ths s manly adjusted by the gratng parameters (depth and fll factor). Therefore, the spectral lnewdth tends to be senstve to the gratng depth and fll factor. The calculaton results n Fgure 5. show that the spectral lnewdth s more tolerant to the gratng depth compared to the fll factor. In addton to the gratng strength, the thckness of the hgh nde flm also nfluences the spectral lnewdth of the resonances. It s seen that the thckness of the hgh nde flm s crtcal to both resonance locaton and ts spectral lnewdth. Ths, however, can be compensated by the gratng parameters (perod and fll factor). For eample, f the thckness of the hgh nde flm s off by + nm, the frst resonance s relocated at a wavelength ~7.4-nm longer than the desgn. Achevng the resonance at the desred wavelength can be done by reducng the gratng perod by ~4 nm. The data s analyzed by the curve-fttng equaton: Δλ ~.73ΔΛ. 95. Meanwhle, the spectral lnewdth 4

165 4 6 s also slghtly narrowed by ~.6 nm. The ar-hole radus needs to be ~5-nm bgger than the orgnal desgn ( Δ( Δλ ) ~.Δr +.34Δr. 9 FWHM.)Ths fes both the resonance locaton and the spectral lnewdth as well. Fabrcaton process would always cause some error n the dmensons of the structure, especally the thckness of each layer. Fgure 5. show that ths error can always be corrected by the gratng fll factor (ar-hole radus n ths desgn). The contour plot n Fgure 5.a and 5.b shows the shft of the TM resonance whle changng the gratng depth versus the ar-hole radus and changng the thckness of hgh nde flm versus the ar-hole radus, respectvely. The plots show that the resonance locaton can be fed at the desred wavelength by adjustng the hole-radus for a specfc thckness. As mentoned n the prevous paragraph, the resonance locaton s more tolerant to the gratng depth. Then, t requres a slght shft of the ar-hole radus from the orgnal desgn. Δr (nm) Δt g (nm) Δr(nm) Δt (nm) f (a) (b) Fgure 5.: Contour plot of the shft (nm) of TM resonance (a) devaton of gratng depth (Δt g ) and ar-hole radus (Δr) (b) devaton of hgh nde flm thckness (Δt f ) and ar-hole radus (Δr). 43

166 The other concern of the gratng fabrcaton s the dffculty to acheve vertcal sde walls. A cone shape gratng (Fgure 5.3a) s usually formed when the gratng s desgned to have hgh aspect rato (depth: ar-hole dameter). (a) Δθ(deg) Δr(nm) (b) Fgure 5.3: (a) A confguraton of the gratng wth slanted sde wall (b) the contour plot of the shft (nm) of TM resonance wth the change of ar-hole radus (Δr) and slanted sde wall angle (Δθ). In ths case, the resonance s potentally shfted from the desgn. The effectve nde of the subwavelength gratng depends on the percentage of the materal left wthn the unt cell. The cone shape gratng has more gratng materal, and hence, a hgher effectve nde. Therefore, reasng the gratng-hole sze can compensate for the cone shape. The resonance appears at the same wavelength as long that the absolute gratng materal s conserved wthn the unt cell. The 44

167 prevous dscusson s confrmed by the contour plot of the shft of the TM resonance n Fgure 5.3b. Wth the cone shape gratng, the resonance remans at the same wavelength only f the arhole has larger radus from the desgn. 5.5 Summary A novel mult-lne flter usng two-dmensonal guded mode resonant (GMR) flter was proposed. The flter concept utlzes the multple planes of dffracton produced by the twodmensonal gratng. Multple resonances are obtaned by matchng the guded modes n the dfferent planes of dffracton to dfferent wavelengths. It was shown that the locaton and the separaton between the resonances can be specfcally controlled by modfyng manly the perodcty of the gratng and other physcal dmensons of the structure f t s necessary. It was shown that one-dmensonal GMR flters where multple resonance are obtaned by couplng to hgher order modes, where the locaton of the resonances s materal refractve nde dependent, are mpractcal. Two types of two-dmensonal gratngs GMR desgns, rectangular grd and heagonal grd, were presented to obtan controllable two-lne and three-lne reflecton flters, respectvely. For the rectangular grd gratng, a two-lne GMR flter was constructed where the resonance separaton was controlled over the regon from <% to 3% of the short resonance wavelength by changng one of the gratng perods n the resonant range λ < Λ < n whle n f λ keepng the other structure parameters constant. Fnally, a three-lne GMR flter usng the heagonal gratng structure was presented. Smlarly, the lne separatons were controlled through alterng the perodcty of the gratng correspondng to the skew angle (ζ ). s 45

168 The locatons of the resonances are sgnfcantly dependent on the perod of gratng and the thckness of the hgh nde flm. The gratng depth and hole-sze have more nfluence on the spectral lnewdth of each resonance. In practce, precse structure dmensons are not always guaranteed. Ths defntely results n the dstorton of the resonances from the desgn. However, slght shft of the dmensons can always be compensated by the fll factor of the gratng. As shown, the errors from both thcker delectrc flm and the cone shape gratng were corrected by reasng the gratng-hole sze. 46

169 CHAPTER 6 : BROADENING THE ANGULAR TOLERANCES IN -D GMR AT OBLIQUE INCIDENCE Important consderaton n the GMR devces s to have large angular tolerance n order to nsure hgh reflecton for fnte beams. Whle these devces have been observed to possess an angular bandwdth broad enough to accommodate dent fnte beams at normal dence, the angular bandwdth of the resonant response at oblque dence s consderably narrower. In ths chapter, broadenng of the angular response of -D GMR spectral flters at oblque dence s nvestgated. Two dfferent approaches are consdered. The frst s by couplng nto multple fundamental guded resonant modes havng the same propagaton constant and enhancng the second Bragg dffracton of the -D gratng structure. A second approach utlzng a dual -D gratng confguraton wth a second gratng on the substrate sde s proposed and demonstrated. 6. Overvew In the lterature, there est some attempts [, 44-46] to broaden the angular lnewdth whle mantanng a narrow spectral lnewdth. At normal dence, the tangental components of both the + and - dffracted orders are phase-matched to leaky modes wth equal magntude propagaton constants travelng n opposng drectons. In ths stuaton, the spectral bandwdth of the resonant gratng remans very narrow, n the order of a few angstroms, whle the angular bandwdth can often become qute broad, many angular degrees. Havng a counter propagaton resonant mode results n formng the resonant band gap and a dspersonless resonant mode s localzed at the upper or lower frequency of the resonant band gap. Ths condton of resonant 47

170 mode always occurs at normal dence due to the symmetry of the dffracton. The angular tolerances are further mproved by enhancng a strong second Bragg dffracton lke the case of the doubly perodc structure proposed by Lemarchand et al [44] or by optmzng the gratng dmensons to have a strong backward couplng as presented by Jacob et al [.] Broader angular response can be formed by mergng the sde bands n multmode structures where resonance s a contrbuton of a hgher-order wavegude mode. At oblque dence, the dffracton orders are not symmetrc and the resonant band gap s formed by two counter-propagatng comple modes of dfferng magntudes. Recently, Sentenac et al [46] proposed a desgn of -D GMR multmode structure to match two frst-order waves ndvdually to hgher-order TE and TE modes. Ths concept was etended to the -D structure n order to acheve polarzaton ndependent resonances and hgh angular tolerance [54.] Resonance s acheved by couplng the dent wave ( ϕ 48 o = 45 ) to two pars of hgher resonant modes propagatng n dfferent planes. Havng dfferent propagaton constant resonant modes, resonance was not as tolerant as at normal dence. In -D GMR devces, the two-dmensonal gratng s used to control the resonant mode propagaton constants, and hence the resonance locatons, by adjustng the perodcty and the dent drecton [63] as demonstrated n Chapter 5. In ths chapter, broadenng of the angular response of -D GMR narrow spectral flters at oblque dence s nvestgated and demonstrated. Couplng to multple fundamental resonant modes havng the same propagaton constant but propagatng n dfferent planes s shown to produce sgnfcant broadenng of the angular tolerance at oblque dence whle mantanng the narrow spectral lnewdth. The phase msmatch between these frst-order dffracton waves at off- resonant angles s mnmzed,

171 and that substantally mproves the angular tolerances. Broadenng of the angular tolerances s symmetrc when couplng to four resonant modes n a structure wth an rregular heagonal gratng pattern. Further broadenng s obtaned by enhancng the second Bragg dffracton of the -D gratng structure. That s done by adjustng the gratng s physcal dmensons. A second approach utlzng a dual -D gratng confguraton wth a second gratng on the substrate sde s nvestgated. The mprovement n the lateral confnement results n mergng of two successve resonant bands and further mprovement of the angular/spectral lnewdth rato. 6. Couplng characterstcs to enhance the angular tolerances of -D GMR In -D GMR structure, the fundamental dffracton/gudance planes are along two orthogonal planes for the rectangular-lattce gratng and along three crossed planes for the heagonal-lattce gratng. Resonance could be caused by multple waves propagatng n dfferent planes. The resonant modes are controlled by the gratng perodctes along the dent drecton. Achevng resonant response wth broad angular lnewdth, the dent beam s coupled to resonance modes havng de-phases compensated at off resonant angle. Hence, the dent beam s smultaneously coupled to the forward and backward propagatng modes correspondng to the dent plane. They do not essentally code n the same plane. Ths ntrnscally occurs at normal dence, where de-phases are absolutely compensated at off resonance. Oblque dence, on the other hand, requres that the dent plane be along a symmetrc plane (parallel to one of the gratng vectors or n the mddle between two of them.) Couplng of the forward and the backward propagatng modes can be n varous manners. Multple resonant modes, where 49

172 each mode has the same propagaton constant, at oblque dence can be acheved only n two confguratons. Fgure 6.: Confguratons of dffracton/gudance planes at oblque dence and phase matchng dagram of resonant modes wth the same β n the -D GMR wth (a) a rectangular-(b) a heagonal-lattce gratng In the frst confguraton, the resonant modes propagate along three frst-order dffracton waves as n Fgure 6.a for the rectangular-lattce gratng. Second, the resonant modes propagate along four frst-order dffracton waves as n Fgure 6.b for the heagonal-lattce gratng. For the rectangular-lattce gratng, the dent wave has to be along one of the two fundamental planes. One of the resonant modes s then n the same plane va the dffracted wave (,). The other resonant mode propagates along the planes of the two dentcal dffracted waves (,) and (,-) as shown n Fgure 6.a. All of these three waves can have the same propagaton constant and contrbute to a resonance at the same wavelength once the perodctes and the dent drecton meet the followng condton 5

173 λ Λ b snθ = (6.) Λ b Λ a For the heagonal-lattce gratng, the dent plane has to be algned n the mddle of two of the three crossed fundamental planes (dash planes n Fgure 6.b.). The resonant modes then propagate along two pars of symmetrc dffracted waves (,/-, and,/-,). In order to couple them smultaneously to the wavegude mode wth the same propagaton constant, the gratng perods and the drecton of dence need to satsfy the followng equaton: λ 3 snθ = (6.) Λ a Λ b 4 Λ a In both cases, the resonant modes are located n dfferent planes. The de-phasng at offresonant angle s n an asymmetrc manner due to the dffracton and not the wavegude modes ( Δ β ). Although the de-phasng compensaton s less than that at normal dence, the angular tolerance at oblque dence s epected to be mproved compared to the resonant modes wth dfferent propagaton constants as wll be llustrated n Secton Multple resonant modes wth the same β n -D GMR at oblque dence In -D structures, the de-phasng msmatch ests due to the asymmetry of the dffracton, but t s less than that of the case of -D GMR. The degree of symmetry s proportonal to the dffracton pattern. That wll be chroncally dscussed n Secton 6.3. for the rectangular-lattce 5

174 gratng and n Secton 6.3. for the heagonal-lattce gratng. The resonant responses of both cases wll be presented n Secton Three resonant modes wth the same β n a rectangular-lattce gratng In ths case, these resonant modes propagate wth the same propagaton constant along three dffracton planes as n Fgure 6.a. The de-phasng at off-resonant angle s demonstrated n Fgure 6.a for on-dent plane and Fgure 6.b for off-dent plane where the dashdffracton planes and thn dot- wave vectors represent the case at resonance. For the on-dent plane case, the de-phasng compensaton behaves smlarly to the -D structure, but along two pars of waves, (,/,) and (,/,-). The de-phasng msmatch s formed due to the dffracton and not due to couplng to dfferent modes. The forward waves (, ) and (,-) have dentcal de-phasng rate whch s dfferent from the backward waves (, ). That can be appromately wrtten as dφ+ k snθ dφ k cosθ, dθ k dβ y, + and dφ dφ k cosθ (6.3) dθ dβ where the subscrpt +/- represent the forward (,) and (,-) /backward (,) waves. k ( k snθ ) + ( Λ ) y, + = π a s the forward wave vectors wth respect to the dent plane. As shown, the msmatch reduces when the gratng has larger perod or the resonant mode has smaller propagaton constant. 5

175 Fgure 6.: Confguratons of dffracton/gudance plane and phase matchng dagram of three resonant modes at off-resonant angle (a) on- (b) off-dent plane. For the off-dent plane case, the de-phasng rates of both forward waves and the backward wave are not eactly the same. However, the symmetrc dffracton along the waves /- results n slghtly devatng de-phasng rates accumulatng n opposte sgns. They are notceably larger than the (,)-wave de-phasng rate as appromately wrtten n Eq. (6.4). dφ dϕ k snθ cosϕ π dφ m, k Λ dβ ± y,± a and dφ dϕ k snθ snϕ k y, π dφ. (6.4) Λ dβ b The de-phasng compensatons are along two pars of waves, / and /-. Due to less dephasng accumulaton along the wave (,) compared to the others, the de-phasng compensaton 53

176 at off-dent plane are degraded from that at on-dent plane causng asymmetrc angular responses Four resonant modes wth the same β n a heagonal-lattce gratng In the heagonal gratng GMR structure, the resonant modes propagate along four dffracton planes as shown n Fgure 6.b. The de-phasng at off-resonant angle s shown to be more symmetrc as depcted n the drawngs n Fgure 6.3a for on-dent plane and Fgure 6.3b for off-dent plane. The de-phasng at off-angle along the plane of dence s compensated va four dfferent pars of waves, (,/), (,/-,), (-,/-,), and (-,/,). The forward waves (, ) and (-, ) have the same de-phasng rate, but dfferent from the backward waves (, ) and (-, ). These rates can be appromately wrtten as dφ+ k snθ dφ k cosθ, dθ k dβ y, + and dφ dθ π k snθ Λ a dφ k cosθ. (6.5) k dβ y, y, = 4π 4Λ b Λ a + k snθ where the forward wave vectors are ( ) ( ) k + and the backward y, b a a wave vectors are k = ( π 4Λ Λ ) + ( k snθ π Λ ). The de-phasng rate of the backward waves s somewhat smaller than that of Secton 6.3. under the smlar crcumstances 54

177 (dent angleθ and gratng perods). Hence the de-phasng between the forward and backward waves s better compensated. Resonances are then epected to be more angular tolerant along the dent plane than the rectangular structure. Fgure 6.3: Confguratons of dffracton/gudance plane and phase matchng dagram of four resonant modes at off-resonant angle (a) on- (b) off-dent plane. The degree of de-phasng compensatons at off-dent plane tends to mprove as well when havng four par of waves, (,/-,), (,/-,), (,/-,) and (,/-,) n spte the fact that the de-phasng rates are slghtly dfferent. The de-phasng compensaton of dffracton waves are more effectve along the symmetrc dffracton waves (,/-,) and (,/-,) where the dephasng rates are appromately defned by Eq. (6.6). 55

178 dφ dϕ ± k θ π sn 4 dφ m snϕ, k y, ± dβ 4Λ b Λ a and dφ / dϕ k θ sn ± k y,/ 4Λ π b Λ a snϕ π dφ ϕ a cos. (6.6) Λ dβ Consequently, the asymmetry of the angular responses s mnmzed n ths confguraton whch s more desrable n broadenng of the angular tolerances of GMR structure at oblque dence Results and dscussons In ths secton, the resonant responses of the desgn approaches mentoned n Secton 6.3. are calculated and compared to the one n Secton To show the mprovement of the angular tolerance; the structure n Secton 6.3. s also modfed to have forward and backward waves wth dfferent β. For all cases, the structures are desgned to have resonances located at a wavelength λ =.98 μm wth almost dentcal spectral lnewdth (Δλ FWHM ). The structure s comprsed of a refractve nde flm of deposted on a substrate wth a refractve nde of.47. The gratng s drectly fabrcated on the flm as depcted n Fgure 4.. The gratng has ar crcular-holes wth radus r and depth t g. The actual thckness of the man gudng layer s t f. In these calculatons, the Fresnel reflecton (that causes an asymmetrc and hgh sdelobe resonant lneshape) at the flm/substrate nterface s mnmzed by nsertng an ant-reflecton (AR) flm wth a refractve nde of n AR = n n f s and thckness t AR=λ /4n AR. The calculated resonance lnewdths of three dfferent desgns are presented n Table

179 Table 6.: Spectral and angular lnewdth of resonances (λ =.98 μm) by a counter-propagaton resonant mode n D-GMR at oblque dence GMR Incdent beam Structure parameters (μm) Δλ FWHM Δθ FWHM Δφ FWHM # Θ φ Ψ Λ a Λ b ζ r t g t f (nm) at φ at θ 9º º 9º º º.º.9º 9º º º º.45º 3.5º 9º º º º.8º In Table 6., GMR # and # match the frst-order dffracted waves to the TM mode by the gratng structure and the settngs n Secton 6.3. and 6.3., respectvely. The gratng perods of these GMRs are selected n order to acheve an dentcal modal nde. The thrd desgn on the other hand matches the (,/-,) and (,/-,) waves to the TM and TM modes at the same wavelength, respectvely. The gratng has the same pattern as GMR # but dfferent gratng perods. The spectral and angular responses of each case are shown n Fgure 6.4. To estmate the mnmum beam sze allowed n the structures, the angular responses are calculated here as a functon of the devaton Δk along ( // k ) and perpendcular to ( k ) the dent drecton. 57

180 Fgure 6.4: (a) Phase matchng condtons (b) spectral (c) and (d) angular resonant response along and perpendcular to plane of dence of -D GMRs n Table 6. The calculaton results n Table 6. show an mprovement of the angular tolerances when the resonant modes have the same β as n GMR # and GMR #. The angular lnewdths ( Δ θ FWHM and Δ ϕ FWHM ) are about twce of that n GMR #3 when the resonant modes have dfferent β. That s n agreement wth Eqs. (6.3)/ (6.4) and (6.5)/ (6.6) where the de-phasng msmatches at off-resonance are mnmzed. Based on the equatons, the de-phasng compensaton n GMR# and GMR# at off-resonant angle s plotted n Fgure 6.5a for ondent plane and n Fgure 6.5b for off-dent plane. 58

181 (a) (b) Fgure 6.5: Comparson of de-phasng compensatons n GMR# and GMR# (Fgure 6.4) calculated from Eqs. (6.3)-(6.6) (a) on- dent plane (b) at off-dent plane The calculatons n Fgure 6.5a ndcate better de-phasng compensaton n GMR# wth the heagonal structure although ts angular lnewdth Δ θ FWHM n Table 6. s narrower than that of GMR# wth the rectangular structure. Ths represents more nteracton between modes n GMR# when the gratng has a strong second Bragg dffracton as wll be dscussed n Secton 6.4. In spte of the broader angular lnewdth of GMR # (3 modes wth the same β), the asymmetrc angular response shown n Fgures 6.4c and 6.4d lmts the mnmum sze of the dent beam to ln( ) / ΔkFWHM ~ 34-μm n dameter. Ths s also observed n GMR #3 (dfferent β). A more symmetrc dffracton pattern n the heagonal structure results n less de-phasng msmatches between /- and /- n Fgure 6.5b for off-dent planes. The angular response (Fgure 6.4) then appears more symmetrc for GMR # (4 modes wth the same β). Ths broad angular tolerance allows an dent beam as small as 3-μm dameter, whch s about half of the other two desgns. 59

182 Lke the resonances at normal dence, the angular tolerances at oblque dence could be even broader by strengthenng the second Bragg dffracton of the gratng. Therefore, n Secton 6.4 and 6.5, resonances by four modes wth the same β n GMR # wll be further mplemented. 6.4 Broadenng the angular tolerances n a sngle- gratng -D GMR The approach developed for the normal dence [] case s used to rease the angular tolerances of resonances at oblque dence. However, the asymmetrc dffracton at oblque dence must be taken nto consderaton. As n normal dence, the resonance has a broader angular lnewdth when the resonant mode s more laterally confned [.] That requres a gratng wth a strong second Bragg dffracton. The resonant modes at oblque dence are n four planes. They are cross-dffracted and coupled to each other once the gratng has a strong second Bragg dffracton as depcted n Fgure 6.6a. The strength of ths dffracton can be appromately measured n term of the backward dffracton effcency ( η B ), where the flm s assumed as a half-space. 6

183 Fgure 6.6: (a) Scatterng of four resonant modes at resonance (A: forward dffracton, B: a crossed dffracton) and de-phasng manners of a crossed dffracton at off-resonance (b) θ +Δθ (c) φ +Δφ At off-resonant angle along the dent plane (θ +Δθ) (as shown n Fgure 6.6a,) the crossed dffractons of the dffracted wave (-, ) are llustrated n Fgure 6.6b. The dffracted wave has accumulated a de-phasng of Δφ as presented n Fgure 6.3a. As a result, the crossed dffractons eperence less de-phases ( Δφ = Δφ + Δφ+ ) along two backward dffracted waves, (b) and (c). At an off-resonant angle whch s devated away from the dent plane (φ +Δφ n Fgure 6.3b), the dffracted wave (-, ) de-phases by Δφ. The crossed dffractons add less de-phasng along the backward dffracted waves (a) and (b) as shown n Fgure 6.6c. Broadenng the angular tolerances n ths structure behaves n a symmetrc manner f the three η Bs are 6

184 smultaneously mamzed. η B, ( a) manly mpacts the angular lnewdth Δ ϕfwhm and η broadens the angular lnewdth Δ θ FWHM. Broadenng the angular lnewdth can be done by B, ( c) optmzng the gratng dmensons. Changng the gratng dmensons modfes the gratng strength ( η d ), and hence the spectral lnewdth [.] Thus the gratng s desgned to have strong backward dffracton effcences and weak gratng strengths n order to further mprove the angular tolerance whle keepng a narrow spectral lnewdth. The nfluence of the crossed dffracton on the resonant response s nvestgated by calculatng the resonance response whle gradually reasng the gratng depth. The structure consdered n ths secton has smlar desgn as GMR # (Table 6.,) but wthout the AR flm s nserted between the flm and the substrate. Therefore, resonances wth hgh baselne and asymmetrc lneshape are epected. The dent drecton whch corresponds to the gratng perodctes can be estmated from Eq. (6.). In order to acheve a resonant mode by the same wavegude mode (TM ) at λ =.98μm, the dent drecton, θ, has to be ~.4º. At ths settng, the backward dffracton effcences ( η B, ( a)/( b)/( c) ) and the gratng strengths ( η d +/ ), where the subscrpt + represents the dffracted waves (-,) and (,) and the subscrpt represents the dffracted waves (-,) and (,), are calculated and plotted versus the gratng depth n Fgure 6.7. Based on the calculatons, the desred range of the gratng depth that broadens the angular lnewdth not the spectral lnewdth s from.4 to.6 μm ( t g λ ). 6

185 Fgure 6.7: η B and η d of the gratng structure of GMR# n Table 6. versus the gratng depth, where gratng has r=.3 μm, λ =.98μm θ ~.4º. In the calculatons, the resonant mode as well as the resonant wavelength s fed whch requres changng the flm thckness when reasng the gratng depth. The flm thcknesses and ther resonant responses are eactly determned usng the RCWA calculaton. For the structure wth a shallow gratng, t g =.5 μm, the angular response ehbts resonance separaton as shown n Fgure 6.8a. The result ndcates a phase msmatch between the two pars of the resonant modes, + and -, due to the scatterng by the gratng. That s nvestgated by plottng the dsperson of the resonances n Fgure 6.8b versus the wavelength ( λ ) and the dent angle ( θ ). In comparson to the homogeneous wavegude approach [] (Fgure 6.8c.), the ntersectons of the resonances as a contrbuton of + and resonant modes are opened due to the scatterng by the gratng creatng a band gap whch s referred to n ths thess as the resonance band gap. Ths band gap shows that the homogeneous wavegude approach, Eq. (6.), s no longer an accurate tool. The results ndcate a resonance at λ ~. 98μm located nearby the shorter wavelength sde of the desred band gap (TM + /TM - ). The scatterng by the gratng shfts 63

186 the resonant band gap towards a longer wavelength. Therefore, the desgn resonance s not eactly at the top of the lower resonance band. The shft reases wth the gratng strength as seen n Fgures 6.8d and 6.8e for a structure wth t g =.5 µm. Fgure 6.8: (a)/(e) Angular responses of four TM resonant modes located at λ =.98μm, θ =.4º and dsperson plots calculated by RCWA (a)-(b) t g =.5 μm, t f =.53 μm and (e)-(d) when t g =.5 μm, t f =.44μm (c) dsperson plot calculated by HWA for t g =.5 μm The gratng strength on the other hand opens the resonant band gap wder and the band s flattened. Ths nsures broader angular tolerances for deeper-gratng structures when movng the resonance to the top of the lower band by changng the dent angle. Ths s shown n Table 6. where slghtly tltng the dent drecton ( θ ) and smultaneously changng the gratng depth and flm thckness results n sgnfcant mprovement n the rato between the angular tolerance to the spectral lnewdth. 64

187 Table 6.: Spectral and angular lnewdth of resonances (λ =.98 μm) n a heagonal-lattce gratng resonance structures (Λ a =.55 μm, Λ b =.6 μm, and r =.3 μm) at oblque dence (φ =9º, ψ =º) GMR # θ (º) Structure parameters (μm) t g t f Δλ FWHM (nm) Δθ FWHM at φ (º) Δφ FWHM at θ (º) Δθ/(Δλ/ λ ) (rad) Δφ/(Δλ/ λ ) (rad) Fgure 6.9: Resonant band dagram of the structure n Table 6. calculated by RCWA and angular responses at λ, (a)-(d) t g =.5 μm, (b)-(e) t g =.35 μm, and (c)-(f) t g =.55 μm 65

188 The flattenng of the resonant band s shown n Fgures 6.9a- 6.9c for gratng depths.5,.35, and.55 mm respectvely. The graphs llustrate a gradual openng of the band gap belongng to TM + /TE -. The upper band of ths band gap slowly moves and algns almost at the same wavelength as the lower band of the TM + /TM - band gap. The angular responses, n Fgure 6.9d-6.9f, show a flat-top lneshape. Fgure 6. shows the calculated angular/spectral lnewdth ratos ( Δ θ ( Δλ ), Δ ϕ ( Δλ ) λ λ ). The plots ndcate that structures wth gratng depth between.45 and.55 μm have good resonance performances n agreement wth the ntal estmaton of the gratng strengths n Fgure 6.7. Fgure 6.: The angular -wavelength lnewdth rato vs. gratng depth (resonances n Table 6.) In addton to a wde angular tolerance ( Δ θ,) t s mportant to have a symmetrc angular response. For an angle of dence around 4 o, symmetrc response s acheved for an angular lnewdth rato Δϕ : Δθ = cosθ / snθ : 4 :. For a.55-μm deep gratng, the optmum desgn has resonance wth a very narrow spectral lnewdth ~. nm and angular lnewdth as large as Δ θ ~.5º and Δ ϕ ~4.3º. These angular lnewdths correspond to a beam wast as small 66

189 as ~6-μm dameter. The angular/spectral lnewdth ratos ( Δ θ ( Δλ ), Δ ϕ ( Δλ ) λ λ ) are (~75%, ~3%). Ths rato for a deep gratng s much larger than that for a very shallowgratng structure where the angular wdth s proportonal to the spectral lne wdth. 6.5 Broadenng the angular tolerances n a dual- gratng -D GMR Dual-gratng -D GMR structures consdered n ths secton are comprsed of gratngs wth the same perodctes located above and beneath the hgh nde flm as depcted n Fgure 6.. The resonant mode s scattered by both gratngs. Ths modfes the spectral and angular lnewdths. Better resonance performances can be obtaned as wll be dscussed n ths secton. Fgure 6.: Dual-gratng -D GMR confguraton Dual-gratng -D GMR structures were proposed to control the spectral lnewdth by Jacob et al. [74] Due to the dffculty of achevng the desred gratng dmensons, the spectral lnewdth s controlled va the flm thckness. Havng two gratngs along the slab wavegude, the mode ectaton at resonances s smultaneously eecuted by both gratngs where the leakng felds have phase dfferences accumulated by the non-dffracted wave nsde the flm. Therefore, 67

190 the effectve leakage, and hence the spectral lnewdth, can be adjusted proportonally to ths phase. Ths s controlled by the flm thckness. Increasng the second Bragg dffracton may rease the gratng strength and degrades the resonance performance as mentoned n Secton 6.4. Ths constrant was removed once the gratng was properly desgned to have t. In dual gratng structures, the spectral g λ lnewdth could be narrowed wthout the lmt of the gratng thckness. Besdes, the double scatterngs by two gratngs wth the same perodcty enhance the lateral confnement of the resonant mode f the substrate gratng s suffcently strong. Ths mproves the resonance performances compared to the sngle gratng GMR structure. The lateral confnement by the substrate gratng s accomplshed dfferently from the superstrate gratng due to the etraordnary backward scatterng paths as depcted n drawng B n Fgure 6.a at resonance and n Fgure 6.b and 6.c at off-resonant angles Δ θ and Δ ϕ. At resonance, the mode s smultaneously ected by the superstrate and the substrate gratngs. The ordnary scatterng by the superstrate gratng s llustrated n the dffracton/gudance planes A (Fgure 6.a.) As the substrate gratng has a strong second Bragg dffracton, the frst dffracted waves by the superstrate gratng (dash arrows n Fgure 6.a) are cross-dffracted to the others wth a total phase of ~mπ, as demonstrated n the dffracton/gudance plane B (Fgure 6.a.) At off-resonant angle (both Δ θ and Δ ϕ ), each one of these waves de-phases by a half cycle ( Δ φ / ). The phase msmatch n the ected mode s nearly compensated by the crossed dffractons whch are along waves (b)/(c) n Fgure 6.b and along waves (a)/ (b) n Fgure 6.c. The de-phasng s reduced to roughly half of that of the sngle gratng. Ths effectvely flattens the resonant band as wll be seen n followng calculatons. 68

191 Fgure 6.: (a) Scatterng of four resonant modes at resonance (A: forward dffracton, B: a crossed dffracton by the substrate gratng) and de-phasng manners of a crossed dffracton (B) at off-resonance (b) θ +Δθ (c) φ +Δφ The structures consdered here are smlar to those analyzed n the prevous secton. The substrate gratng s drectly fabrcated on the substrate before depostng the flm. The superstrate gratng has the optmum dmensons n Secton 6.4 (t g =.55 μm). The crossed dffracton of the substrate gratng s mproved by reasng the gratng depth. Smlar to the calculatons n Secton 6.4, the resonance band dagram and the angular response at resonant wavelength are calculated. The structure dmensons ludng the spectral/angular lnewdths are shown n Table 6.3 for dfferent substrate gratng thcknesses (-.35 μm). Table 6.3 shows that both angular lnewdths along the dent plane ( Δ θ ) and ts orthogonal plane ( Δ ϕ ) smultaneously rease. Ths mantans the symmetry of the angular response. The spectral lnewdth of ths structure gratng s suffcently narrow regardless of the 69