Reflection and transmission of light at a curved interface: coherent state approach. N.I. Petrov

Transcription

1 Reflecto ad trasmsso of lght at a curved terface: coheret state approach N.I. Petrov Lea Street, 9-39, Istra, Moscow Rego 435, Russa Phase-space procedure based o coheret state represetato s proposed for vestgato of reflecto ad trasmsso of lght beams at a curved delectrc boudary. Numercal smulatos of reflecto ad trasmsso of lght at varous boudares separatg two dfferet delectrcs are carred out. Sgfcat fluece of wave-frot curvature ad polarzato of cdet beam o the reflectace ad trasmttace s show. Itroducto: Mcro-optcal elemets are wdely used moder optcal systems, such as lght homogezers, mcro-les arrays, etc. For derato of large aperture mcro-optcal systems, covetoal methods, such as physcal optcs, become overly tme-umg. Beam-mode represetatos do ot provde rght alteratve, sce the beam modes caot be tracked smply through the arbtrarly curved surfaces. Dscrete phase-space methods [, ] have bee proposed as a effcet alteratve. These methods represet felds as dscrete ad fte superpostos of elemetary Gaussa beams that ca be traced easly a complcated evromet. However, the method usually proposed to dscrete felds, the so-called Gabor represetato, has bee show umercally ustable [3, 4].

2 I ths paper, the approach based o coheret state represetato s proposed for aalyss of reflecto ad trasmsso of lght beam at a curved delectrc surface profle. Coheret states are used may dfferet areas of physcs [5-7]. I [8] the coheret state method was used for derato of oparaxal propagato ad focusg of wave beams a graded-dex medum. The term coheret states was troduced by Glauber [5, 6] for a oedmesoal steady-state quatum oscllator coecto wth problems quatum optcs. Such states were tructed ad vestgated already by Schrodger 96 [9] order to establsh a relatoshp betwee the classcal ad quatum approaches. Formulato of the problem: Cosder the curved boudary betwee two dfferet delectrc meda Fg.. For smplcty, the two-dmesoal perodcally corrugated terface y depedet wth the perod d >> λ s dered, but the exteto to the arbtrary profle of 3D case s straghtforward. The calculato procedure of the reflected ad trasmtted powers sts of the followg steps. At frst, the cdet beam feld s expaded to coheret states, represetg elemetary Gaussa beams wth axs dsplacemet ad tlt. Note, that a sese such Gaussa beams are smlar to the complex rays used for smulato of reflecto ad trasmsso of beams at a curved terface []. Coheret states CS form a full set of fuctos π d, so the arbtrary cdet feld Ex, ca be expaded to a set of CS: x, π E d x f, where x s gve by the expresso 3, d d Re d Im s the elemetary phasespace volume, f dx x E x, are the ampltudes of the expaso.

3 Coheret state ca be determed as the egefucto of the ahlato operator a : a, ˆ x kw where a + pˆ x, px. w k x A explct form of CS s gve by a Gaussa beam fucto πw 4 x exp w x + w, 3 x kw where the complex egevalues + px exp ϑ determe the tal coordates w x of the ceter of the elemetary beam ad the agle of ts clato p x s to the z- axs, s the refractve dex of medum, w s the elemetary beam wdth, k π/λ s the waveumber. Note, cotrast to Fourer-expaso, there s o requremet for orthogoalty of fuctos, owg to CS form overfull fucto system. The square of the modules f E determe the cdet beam power dstrbuto betwee the elemetary beams CS. For the 4 cdet Gaussa beam E x, / a x a π exp the ampltudes of expaso have the form w / a w / a f exp + ϑ. 4 + w / + / a w a The Gaussa elemetary beams CS pass through the terface as determed by the correspodg Fresel coeffcets that vary accordg to the agle of cdece. Usually the Fresel formulae are kow from the plae wave lmt. However, these formulae ca be used also for localzed wave beams wth the beam wasts w > λ [, ]. 3

4 4 Fally, total reflected ad trasmtted powers are determed by a summato of powers of all elemetary beams. The reflectace ad trasmttace are defed as the ratos of reflected ad trasmtted powers to the cdet power, accordgly: f d R f d P P r r, f d T f d P P t t, 5 where R ad T are the reflecto ad trasmsso coeffcets, s the cdet agle. The reflecto ad trasmsso coeffcets for TE ad TM learly polarzed cdet beams, accordgly, are gve by the expressos s s E R +, s s 4 E T + ; s s + H R, s s 4 + H T, 6 where ad are the refractve dexes of meda ad, accordgly. Results: For the surface-relef profle zsx, the ray CS wth tal coordates x, strkes the terface at the pot x, z, where the correspodg cdece agle s uquely determed. For example, the cdet agle ca be expressed as ϕ, where [ ] arcta x s ϕ, dx x ds x s s the dervatve of the surface profle fucto wth respect to the x coordate.

5 Results of smulato are preseted for dfferet surface profles, wavefrot curvature raduses ad polarzatos of cdet beam. The parameters for cdet beam, surface-relef, ad elemetary beam are the rato a > d >> w > λ, where d s the dameter of the sgle elemet corrugated surface Fg.. Fg. ad Fg. 3 show the reflectace ad trasmttace as fucto of sag h of the surface relef s x h πx / d wth the perod d5µm for TE ad TM polarzed beams wth dfferet wavefrot curvature raduses. Aalogcal depedeces are obtaed for the parabolc surface profle s x x x m s, where Rsc RL s R, R L d/ sc ad R sc are the radus ad curvature radus of the sgle elemet, x m s the ceter coordate of the sgle elemet ot show. Lesser sestvty of the reflecto ad trasmsso to the sag h ad wavefrot curvature radus R f chages s observed for parabolc surface profle. Reflectace creases ad trasmttace decreases wth the crease of sag h. The reflectace ad trasmttace are sestve to the wavefrot curvature radus ad polarzato of the cdet beam. For lower-hgher dex terface the reflectace s lower ad trasmttace s hgher for TM polarzed beam f h < d. For hgher-lower dex terface there s o evdet dfferece betwee TE ad TM polarzatos. It follows from the smulatos that the decrease of the reflectace ad crease of the trasmttace take place wth the crease of the sag h for TM polarzato owg to Brewster agle effect. The proposed method s a effcet alteratve to model aperture fuctos ad flat-topped laser beams. Ulke the superposto of off-axs Gaussa fucto compoets used [3], CS decomposto represets the superposto of learly shfted ad spatally rotated beams, formg the full set of fuctos. Trasfer-matrx method [4] ca be used to calculate the 5

6 propagato of off-axs Gaussa beams optcal systems wth tlted, dsplaced ad curved optcal elemets. Cocluso: The CS approach ca be used for smulato of testy dstrbuto of lght dffracted by the mcro-les array [5, 6]. As llustrated by smulatos, the optcal effcecy trasmttace of such systems strogly depeds o the aspect rato h/d. For h/d < the trasmttace of mcro-les arrays exceeds 9%, whch s good agreemet wth the exstg expermetal data. The proposed method s a effcet alteratve to model aperture fuctos ad flat-topped laser beams. Thus, the CS based cotuous decomposto has bee preseted for smulato of lght beam reflecto ad trasmsso at arbtrarly curved surfaces. Reflectace ad trasmttace of lght beam at dfferet surface-relefs wth depths h >> λ are vestgated. Ths method provdes effectve algorthm to model complcated structures delectrc ateas, leses, multple terfaces, etc. wth the help of combato of wave propagato ad ray-tracg procedures. Refereces. P.D. Ezger, S. Raz, ad M. Shapra, J. Opt. Soc. Am. A 3, B.Z. Steberg, E. Heyma, ad L.B. Felse, J. Opt. Soc. Am. A 8, I. Daubeches, IEEE Tras. IT-36, D. Lugara, ad C. Letrau, Electrocs Letters 34, R.J. Glauber, Phys. Rev. 3, R.J. Glauber, Phys. Rev. Lett., J.R. Klauder ad B.S. Skagerstam, Coheret states, Sgapore: World Scetfc,

7 8. N.I. Petrov, Optcs Express 9, E.S. Schrodger, Naturwss. Bd. 4, S Y.Z. Rua, ad L.B. Felse, J. Opt. Soc. Am. A 3, N.I. Petrov, Optcs Letters 9, N.I. Petrov, J. Mod. Opt. 5, A.A. Tovar, J. Opt. Soc. Am. A 8, A.A. Tovar, ad L.W. Casperso, J. Opt. Soc. Am. A, N. Ldle, J. Opt. A: Pure Appl. Opt. 4, S. 6. H. Urey, ad K.D. Powell, Appl. Opt. 44,

8 Fgure captos Fg. Geometrcal cofgurato ad coordate system for a boudary. Fg. Reflectace r sold curves ad trasmttace t dashed curves versus depth h at lower/hgher dex terface for dfferet values of cdet wavefrot curvature raduses: left - TE polarzed beam, rght TM polarzed beam; curves, R f 5 µm; curves, plae wavefrot. Fg. 3 Reflectace r sold curves ad trasmttace t dashed curves versus depth h at hgher/lower dex terface for dfferet values of cdet wavefrot curvature raduses: left - TE polarzed beam, rght TM polarzed beam; curves, R f 5 µm; curves, plae wavefrot. 8

9 Fgure z x, zsx h d x 9

10 Fgure r, t,,8,6,4,..5 TE, h, µm r, t..5 TM,,8,6,4,, h, µm

11 Fgure 3,.5. TE,.5. TM,8,8,6,6 r, t r, t,4,4,, h, µm,, h, µm