Selective Mode Excitation In Specialty Waveguides Using Micro Optical

Transcription

1 University of Central Florida Electronic Theses and Dissertations Doctoral Dissertation Open Access Selective Mode Excitation In Specialty Waveguides Using Micro Optical 004 Waleed Mohaed University of Central Florida Find siilar works at: University of Central Florida Libraries Part of the Electroagnetics and Photonics Coons, and the Optics Coons STARS Citation Mohaed, Waleed, "Selective Mode Excitation In Specialty Waveguides Using Micro Optical" 004. Electronic Theses and Dissertations This Doctoral Dissertation Open Access is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized adinistrator of STARS. For ore inforation, please contact

2 SELECTIVE MODE EXCITATION IN SPECIALTY WAVEGUIDES USING MICRO OPTICAL ELEMENTS by WALEED SOLIMAN MOHAMMED B.S. Cairo University, 996 M.S. Cairo University, 999 M.S. University of Central Florida, 00 A dissertation subitted in partial fulfillent of the requireents for the degree of Doctoral of Optics in the College of Optics and Photonics/CREOL at the University of Central Florida Orlando, Florida Fall Ter 004

3 004 Waleed Mohaed ii

4 ABSTRACT Although optical fibers and specialty waveguides are the base of ajority of today s teleco and light delivery applications, fabrication deforation, nonlinearity and attenuation liit the bandwidth of the data being transitted or the aount of power carried by these systes. One-way to overcoe these liitations without changing the fibers design or fabrication is to engineer the input light in order to excite a certain ode or a group of odes with unique optical properties. Diffractive and icro optics are highly effective for selectively coupling light to specific odes. Using icro optics, ode selective coupling can be achieved through several atching schees: phase only, phase and aplitude, or phase, aplitude and polarization. The ain scope of this work is the design and fabrication of novel optical eleents that overcoe the liitations of these light delivery systes, as well as the characterization and analysis of their perforance both experientally and using nuerical siulation. iii

5 To y parents iv

6 ACKNOWLEDGMENTS First, I would like to acknowledge y advisor, Dr. Eric Johnson, for all the help and support during the years of y Ph.D. work. I was lucky to start with hi fro scratch. I have learned a lot during the establishent of our labs. But the eories of the epty lab with a table in there still have their own beauty. I would like also to thank fro inside y heart the people who becae close to e and ade these years uch easier specially: Sakoolkan Boonruang, Robert Iwanow, Fuiyo Yoshino, Martina Atanassova, Ference Szcepoc, Mahesh Pitchuani, Te-Yuan Chung, Mohaed Abdel Rahan, Aar Aouri, Cailo Lopez, George Siganakis, Adela Apostol, Erde Ultrain, and Bojan Resan. v

7 TABLE OF CONTENTS LIST OF FIGURES... ix LIST OF TABLES... xvi LIST OF ACRONYMS/ABBREVIATIONS... xvii CHAPTER ONE: INTRODUCTION... CHAPTER TWO: VISION OF RESEARCH... 7 CHAPTER THREE: MODELING OF SPECIALTY FIBERS Multiode Fibers Nuerical Exaple: Few Modes Step Index Fiber Multiode Interference Scattering Matrix ethod for odal calculations of Multi-core fiber Scattering Matrix Method Nuerical Exaple: Super Modes in Multicore fiber Hollow waveguide Modal calculations in cylindrical hollow-glass waveguide HGW Loss echanis in hollow waveguide Nuerical Exaple: calculations of the bending loss for the TE 0 and TM 0 odes in a 300 µ core HGW at 980 n CHAPTER FOUR: DESIGN OF OPTICAL COUPLING METHODS Single ode fiber to ultiode fiber direct coupling vi

8 4... Multi ode interference SMF to MMF Coupling Re-iaging Conditions Fiber based wavelength tunable condensing lens Single Phase Eleents Dual eleents Least Square Error ethod Method of Projection MOP Sub-wavelength periodical structure for coupling to hollow waveguide Polarization converter eleent for coupling to TE 0 ode CHAPTER FIVE: FABRICATION TECHNIQUES Photolithography Exaple of Photolithography: Fabrication of Single phase eleents Electron Bea Lithography Introduction to E-Bea Lithography Fabrication of Polarization Converter Eleent Focused ion bea syste for prototyping icro-optical eleents Focused Ion Bea as an iaging syste Spot size liitation Interaction of ions with the substrate FIB as a tool for icro-fabrication... 5 vii

9 Subtractive Milling Process Integrated Mirror Exaple CHAPTER SIX: EXPERIMENTAL VERIFICATION OF SELECTIVE COUPLING SCHEMES MMF as a Condensing Lens Excitation of LP and LP in Large Core Fiber Using Single Phase Eleent Nuerical analysis of the selective excitation of the highest order superode in seven core fiber Selective Coupling to TE 0 ode in HGW Measureent of the Transission and Bending Loss in HGW Characterization of the Fabricated Polarization Conversion Eleent Characterization of the Transission Loss on the HGW When Using the Polarization Converter Eleent Characterization of the icro lens integrated on a fiber v-groove using the FIB syste CHAPTER SEVEN: CONCLUSION LIST OF REFERENCES viii

10 LIST OF FIGURES Figure. : Cross sections of three specialty fibers hollow waveguide, photonic crystal fiber and ultiode fiber.... Figure 3. : Intensity distributions of the first three higher order odes a LP b LP c LP 0 and phase distributions of the first three higher order odes d LP e LP and f LP Figure 3. : Intensity distribution representation of the light propagating along the MMF due to ultiode interference Figure 3. 3: The construction of a standard seven cores ulticore fiber... 6 Figure 3. 4: Geoetry representation of a general ulticore fiber structure... 8 Figure 3. 5: Fields representation around the cores of the ulticore fiber... 9 Figure 3. 6: The evaluation of M β over the range of the effective refractive indices MCF between ε and II ε I Figure 3. 7: The fundaental superodes for seven core fiber of fused silica cladding and doped fused silica cores at.33 µ operating wavelength. The spacing between the cores is 0.5 µ and core radii of 3.5 µ Figure 3. 8: Dielectric coated etallic waveguide structure and aterials paraeters at ix

11 .55 µ wavelength Figure 3. 9: Siplified representation of the HGW by a etallic cylindrical waveguide.3 Figure 3. 0: The a E x and b E y field distribution for the TE0 ode. c The electric field polarization of TM0 and d TE0 respectively Figure 3. : The optical path of a guided ode in the fiber Figure 3. : Fresnel reflection fro the ultilayer fiber cladding Figure 3. 3: Transission of a 300 µ bore diaeter hollow waveguide Figure 3. 4: The two odes of ray and angles corresponding to: a reflection fro both walls of the fiber and b colliding with one side only. c The geoetry used to calculate the angles for both odes... 4 Figure 3. 5: Transittance of the TE 0 and TM 0 odes versus the curvature of the fiber /R for a bore Figure 4. : Geoetry of the device, SMF fusion spliced to a MMF. b Aplitude distribution of the light inside the MMF for the fibers paraeters entioned in table Figure 4. : Coupling efficiency as function of ode nuber for MMF core radii of 5.5µ and 9.5µ respectively Figure 4. 3: Change of L MMF versus the MMF radius a. L MMF is considered to be 0.95 of the re-iaging distance inside the MMF Figure 4. 4: The coupling schee showing the phase eleent placed in the path of the x

12 input bea... 7 Figure 4. 5: The calculated a Aplitude and b phase profiles of the field at the input facet of the larger core fiber Figure 4. 6: The coupling efficiency to the LP ode as a function of the nuber of expansions Figure 4. 7: Coupling Efficiency vs. the effective focal length of the few odes fiber objective using exact nuerical integration in equation 4.4 and the analytical solution in equation Figure 4. 8: Dependence on values of p and p on effective focal length showing increasing relationship for both paraeters Figure 4. 9: Dual eleent structure Figure 4. 0: Dual eleents coupling schee Figure 4. : Scheatic diagra of the diffractive eleent design ethod Figure 4. : a The first phase eleent, b the second phase eleent c phase c and intensity d distributions of the output field at the input facet of the fiber Figure 4. 3: Scheatic diagra of MOP Figure 4. 4: a aplitude and b phase profiles of the field at the input facet of the seven core fiber when using the dual phase eleents c and d Figure 4. 5: Sub-wavelength grating structure that fors an artificial birefringent crystal Figure 4. 6: The proposed design of polarization converter eleent to couple the light at xi

13 the input of the hollow waveguide to the TE 0 ode Figure 4. 7: Calculated a Field and b polarization distributions of the output of the polarization converter eleent Figure 5. : The two ajor photolithography techniques: a Contact aligner and b stepper syste... 0 Figure 5. : a and b Zygo iages of the first and second phase eleents for coupling to LP and LP odes respectively, fabricated in PR805 photoresist Figure 5. 3: Scheatic diagra of e-bea syste Figure 5. 4: ZEP50 resist profile Figure 5. 5: a the GDS pattern of gratings of varying periods b the pattern written in the e-bea resist Figure 5. 6: Map of the quality of the written patterns in ters of dose values and feature size Figure 5. 7: SEM iages of the varying period gratings using 90 µc/c after developing and coating with 00 A o layer of AuPd.... Figure 5. 8: The polarization converter pattern transferred to the ZEP50 e-bea resist.... Figure 5. 9: Cross section SEM iages of gratings of different periods etched in GaAs using 500 n of ZEP50 as a ask and BCl 3 /Ar/N plasa for inutes. The grating periods are a 600 n, b 800 n, c 000 n, and d 400 n xii

14 Figure 5. 0: SEM iage of the pattern transfer etched in GaAs using ZEP 50 as a ask Figure 5. : SEM cross sectional iage of the polarization converter pattern transfer etched in the GaAs substrate using a process for 4 inutes, b process 3 for 5 inutes, c process 5 for 6 inutes, and d process 4 for 7 inutes... 7 Figure 5. : a Representation of the FIB as an iaging syste. b Scheatic picture of a typical FIB syste Figure 5. 3: FIB syste as a tool for icro fabrication... 7 Figure 5. 4: a Quantization of the spherical irror b subtractive illing technique; N concentric circles are illed sequentially starting by the biggest one results in a ulti-level spherical irror... 9 Figure 5. 5: a The geoetry of the fiber inside the v-groove. b The location of the fiber relative to the icro lens integrated on the tilter surface of the v-groove... 3 Figure 5. 6: SEM iage of the icro spherical irror fabricated on the wall of the v- groove Figure 6. : Power, noralized to the input power, across an area equals to the SMF core along the MMF Figure 6. : Calculation of intensity inside ulti-ode fiber λ.55 µ using equations 5 and 5 for the field inside and outside the MMF. The MMF is polished slightly less than 0 corresponds to 3L p. [SMF of 4.5 µ core radius xiii

15 and MMF core radius of 5.5 µ.] Figure 6. 3: Device connectorized in FC connector and 0x icroscope iage of end facet of MMF quality Figure 6. 4: the syste setup showing the SMF fused splice to MMF Figure 6. 5: The experientally easured back-coupled power versus the wavelength Figure 6. 6: FD-BPM, experiental easureents, and first order results for focal point location as a function of wavelength... 4 Figure 6. 7: Siulated field distribution out of the MMF facet for four different wavelengths deonstrating wavelength dependence of focal position Figure 6. 8: Spot size HWHM at focal location as a function of wavelength for BPM, first order solutions, and experiental results Figure 6. 9: Depiction of exiting bea at focus location overlaid on actual size of MMF end facet Figure 6. 0: Observed far field intensity distributions of the light at the out of the fiber using the a first phase eleent and b the second phase eleent Figure 6. : Coupling efficiency to the last superode in seven core fiber against the tilt and shift in the input bea Figure 6. : SEM iages of the HGW showing the dielectric AgI and the etal Ag layers... 5 Figure 6. 3: Experiental setup for easuring transission and bending losses in HGW. xiv

16 ... 5 Figure 6. 4: The easured bending loss in HGW circles represent three different easureent and the dashed line is the average and the calculated bending loss using equation Figure 6. 5: Transittance through the GaAs wafer as a function of wavelength Figure 6. 6: Experiental setup for testing the perforance of the polarization converter eleent Figure 6. 7: The far field intensity distribution of the light out of the polarization converter eleent Figure 6. 8: Far field intensity distributions for different polarization angles for the light out of the polarization converter eleent Figure 6. 9: Measured polarization distribution of the output field Figure 6. 0: The polarization profiles of the output field for two input polarization states: vertical top and horizontal botto... 6 Figure 6. : The Calculated solid and easured dashed transittance as a function of the input polarization angle easured relative to the horizontal axis... 6 Figure 6. : The easured and siulated x and y bea waists at a distance of.5 above the v-groove using a single ode fiber at 63 n xv

17 LIST OF TABLES Table 3. Optical Fiber Characteristics Table 5. GaAs dry etching process when using ZEP 50 as a ask... 6 Table 5. Typical values for the spherical and chroatic abberrations diaeters for a LMIS focusing colus with the paraeters entioned in the above paragraph... Table 5. 3 Typical values of the sputtering rates and total illing rates for different substrates with the ion bea paraeters entioned above... 5 xvi

18 LIST OF ACRONYMS/ABBREVIATIONS AVA BAA CCD EBL EBPG FD-BPM FIB HGW HWHM ICP IPA IR LMIS LP MCF MIBK MMF MOP PCF Autoatic Variable Aperture Bea Acceptance Aperture Charge-Coupled Device Electron Bea Lithography Electron Bea Pattern Generator Finite Difference Bea Propagation Method Focused Ion Bea Hollow-Glass Waveguide Half Width Half Maxiu Inductively Coupled Plasa Isopropanole Infra-Red Liquid Metal Ion Source Linearly Polarized Multicore Fiber Methyl Isobutyl Ketone Multiode Fiber Method Of Projection Photonic Crystal Fiber xvii

19 PMMA PR RIE SEM SMF SMM STEM TE TEM TM UV Polyethyl Methacrylate Photo-Resist Reactive Ion Etching Scanning Electron Microscope Single Mode Fiber Scattering Matrix Method Scanning Transission Electron Microscope Transverse Electric Transission Electron Microscope Transverse Magnetic Ultra Violet xviii

20 CHAPTER ONE: INTRODUCTION In 930 Heinrich La was the first to report the transission of light through a bundle of transparent rods. His work was followed by several experients by Holger Moeller in 95 and Abraha van Heel in 954. All early fibers were bare. Van Heel was the first to introduce the cladding region. By 960, glass-clad fiber had an attenuation of decibel per eter. Since the deonstration of the first low-loss optical fiber in 97 by Corning Glass Works, there has been a continuous strea of technological iproveents designed to reduce ipairents due to propagation loss, nonlinearity and dispersion. This strea of fiber technology led the industry fro ultiode fibers to single ode fibers, and on to specialty fibers. Specialty fibers are defined as optical fibers designed to have special functions that are different fro standard telecounication fibers. Unique aterials and fiber structures along with new processing techniques allowed the refractive index to vary across the fiber, to have several cores cobined within the sae cladding, and air core fibers to be coated with thin layers of etal/dielectric. Exaples of specialty fibers are: graded index fiber, polarization aintaining fiber, ulti core fiber, hollow waveguides and photonic crystal fibers. When guiding light in these specialty fibers, the light propagates in different odes. Each ode is characterized by its aplitude, phase and polarization profile. Soe of these odes have unique properties that can be useful for certain applications such as

21 telecounication and light delivery systes. Therefore, it is iportant to develop odeling techniques to investigate the odal properties of these fibers. Accordingly, several studies have been carried out in odeling the light propagating inside different specialty fibers as well as their odal properties. Figure. : Cross sections of three specialty fibers hollow waveguide, photonic crystal fiber and ultiode fiber. Photonic Crystal Fibers PCF s are exaples of specialty fibers that attracted uch interest in the recent years due to their unusual wave-guiding properties []. PCF is fored of periodical pattern of air rods in a dielectric ediu. The light is guided or localized in a deforation produced intentionally in the pattern. Quite a few analytical techniques have been published in this field. Figotin et al. [] used plane wave expansion to study the localization of light in a three diensional photonic crystal structure when a defect is introduced inside this structure. E Centeno et al. [3] used a full vectorial scattering atrix ethod to study the localization of light inside a two diensional photonic crystal. Knight et al. [4] reported the waveguiding properties of the PCF s using

22 the effective index ethod. Ferrando et al. [5] presented a full-vectorial analysis of the guided higher-order odes in PCF s. In coparison to PCF s, ulticore fiber MCF consists of a few cores placed close to each other to increase the coupling between the. This coupling between the cores odifies the odal property in each core correspondingly. Chang et al. [6] deonstrated a full vectorial odal solution of MCF structures using circular haronic expansions. Wrage et al. [7] presented nuerical and experiental deterination of the coupling constant in a MCF when the cores are placed in a ring geoetry for ode locked high power fiber laser applications. However, nonlinearity of the cores aterial restricts the axiu transitted power. Hollow waveguides present an attractive alternative to other solid-core infra-red IR fibers such as step index fiber and MCF. Key features of hollow guides are: their ability to transit wavelengths well beyond 0 µ; their inherent advantage of having an air core for high-power laser delivery; and their relatively siple structure and potential low cost [8]. Initially these waveguides were developed for edical and industrial applications involving the delivery of CO laser radiation, but ore recently they have been used to transit incoherent light for broadband spectroscopic and radioetric applications. In general, hollow waveguides enjoy the advantages of high laser power thresholds, low insertion loss, no end reflection, ruggedness, and sall bea divergence. Potential disadvantages, however, include an additional loss on bending and a sall NA. Nevertheless, they are today one of the best alternatives for both cheical and 3

23 teperature sensing as well as for power delivery in IR laser surgery or in industrial laser delivery systes Researchers have done uch work in the area of light and ode coupling in these specialty fibers [8]. In 984 M. Miyagi et al. [9] presented a design of a dielectric coated etallic cylindrical hollow waveguide with the ephasis of low-loss transission of the HE ode. Modal structures and transission properties were clarified for this hollow waveguide by using a noralized surface ipedance and adittance. M. Miyagi et al. [0] and S. Abe et al. [] calculated the bending losses in hollow glass waveguides by relating the power attenuation coefficient to the bending curvature and the noralized ipedance of the waveguide. For the specialty fibers presented above, selective excitation of certain odes with proper optical properties can iprove the perforance of this syste or add an extra functionality to it. This requires designing proper coupling techniques. Kyung Shick et al. [] presented analytically that the polarization ode coupling between any two linearly polarized LP core odes is possible by appropriately adjusting the grating paraeters. The grating was created by pressing a two-ode fiber with a groove plate. In this schee, the ode coupling is achieved by odifying the propagation constant of the light to atch that of the desired ode. Derived fro this concept, Laurent Vaissie et al [3] presented another selective ode excitation technique through a non-axial evanescent coupling between two highly asyetric side polished fibers: single ode fiber SMF and graded index fiber. Side polishing the fibers allows one to anipulate 4

24 the evanescent field near the core of the fiber. Selective excitation of high order aziuthal odes is achieved by appropriately tilting the SMF. Another application of ode excitation in side polished SMF was presented by Russel et al. [4]. The key eleent in their design is a photoresist grating placed on the surface of a side polished SMF. This grating fiber-coupler was used as a high resolution spectroeter. These coupling schees require introducing changes in the existing fiber syste such as side polishing and writing gratings in the core or the side of the fiber. Engineering the input light technique, in contrast, can be applied to the existing fiber systes without changing their structure. It only requires adding extra optical coponents in front of the input facet of the fiber. The purpose of these eleents is to alter the phase distribution of the input light at one or ore planes along the path of the input light. This phase odulation of the input light anipulates the properties of the field at the input facet of the fiber such as: phase, aplitude and polarization profiles. If one or ore of these properties atches that of a certain ode or a set of odes, selective ode excitation is achieved. For this to occur, proper design of the optical coponents and selection of there locations are required. Micro eleents play a great role in realizing these optical coponents required to achieve ode selective excitation. For instance, Ghafoori et al. [5] and Fu Yong [6] deonstrated the analysis and fabrication of icro lenses to couple light to single ode fiber. The ai of these icro lenses is to axiize the power coupled to the fundaental ode in a SMF. Conversely, Eric Johnson et al [7] presented a novel diffractive vortex 5

25 lens to couple the light to higher order aziuthal odes to avoid the central deforation of the refractive index profile in graded index fibers. 6

26 CHAPTER TWO: VISION OF RESEARCH In the previous chapter we introduced the ode selective excitation by engineering the light at the input of the fiber syste. This excitation is achieved through placing proper optical coponents in the path of the input light. The design of these optical coponents depends on the structure of the fiber in use as well as the desired ode or set of odes. Choosing this ode or set of odes is a key feature in iproving the perforance of any optical fiber syste, that requires odal analysis and odeling of the light guided in this syste. Accordingly, the process of achieving selective ode excitation can be divided into four ain parts: odal analysis and odeling of the guided light in these fibers, designing the proper optical coponents, fabrication of these coponents, and characterization of their perforance in the syste. This work focuses on the selective excitation of certain odes in ultiode fiber, few odes fibers, ulticore fibers, and hollow cylindrical waveguides for different applications. For exaple, a ultiode fiber, when attached to a single ode fiber, can be utilized as a condensing lens if only radially syetric odes are excited and the fiber is cut to a proper length. In addition, it can be used as a displaceent sensor by easuring the light coupled back to the single ode fiber. This coupling schee does not require additional optical coponents. In coparison, coupling the light to a certain higher order linearly polarized ode in few odes fibers requires at least one phase eleent. Aong the first higher order odes, LP and LP preserve a inial intensity 7

27 in the central region of the fiber. If selectively excited, these odes can be used to iprove the gain properties of the optical fiber aplifiers. Another area of interest where specialty fibers play a great role is light delivery systes. However, these fibers suffer ajor drawbacks that, in any cases, reduce their perforance. The hollow cylindrical waveguide, for instance, suffers of high bending and transission loss. One way to reduce these losses is to selectively couple the input light to the TE 0 ode which sustains a TE polarization around the inner wall of the core. This requires a coupling schee that alters aplitude, phase and polarization due to the degeneracy of the TE 0 with the TM 0 ode. Alternatively, ulticore fibers suffer of core aterial nonlinearity that sets a liit on the axiu aount of power being transitted. This effect can be reduced if the power is spread over a larger area across the fiber. The spreading of the power is ade possible by selective coupling to a certain high order ode with field profile which satisfies this power criterion. Finally, integration of icro-optical eleents into actual devices presents nuerous challenges to the optical engineer. Therefore, prototyping integrated icro-optics is soewhat prohibitive due to cost and coplexity. A natural solution to this is the ability to write the optical eleent into the device without resorting to the prototyping of an optical eleent followed by an active alignent. One such ethod of fabrication is through the use of a Focused Ion Bea FIB. The ain strategy of this research is to first realize the proper odes in each of these fiber systes. The second step is to design optical coponents that axiize the excitation of these odes. Developing a fabrication procedure for the designed coupling 8

28 coponents coes next. The final phase is to characterize the fabricated coponents experientally and confir the results with nuerical siulations. 9

29 CHAPTER THREE: MODELING OF SPECIALTY FIBERS In this chapter we develop several calculation ethods with the ai of deterining the guided odes in the different fiber systes entioned in the previous chapters. For the ultiode fiber MMF, one can obtain an analytical odal solution using the linearly polarized odes approxiation. On the other hand, ulticore fiber and hollow waveguide require a full vectorial solution. In this section we briefly illustrate three different ethods to calculate the odal properties of the ultiode fiber, ulticore fiber and hollow cylindrical waveguide. We will also explain the three diensional finite difference bea propagation ethod. 3.. Multiode Fibers The ultiode fiber is a standard optical fiber structure that supports a large nuber of odes due to its large core size. Each ode is defined by its propagation constant, β, and field profile. These profiles and propagation constants can be calculated by solving a boundary condition proble at the core cladding interface taking into consideration the continuity of the tangential field coponents. Applying the separation of variables and solving for linearly polarized odes, the field distribution of the guided odes, ψ r, θ,, can be written in ters of the core radius, r a, and propagation v, z constant, β, as follows [8] 0

30 > a r x e r r w K w K u J A a r x e r r u A J z r z i a v v v z i a v v v v ˆ cos ˆ cos,,,,,,,,, β β θ θ θ ψ. 3. In equation 3., u v, and w v, are the noralized transverse propagation constants inside the core, and in the cladding respectively. They are related to the fiber paraeters and the propagation constant, β, through the following relation,,,, clad o a core o a n k r w and n k r u ν ν ν ν β β, 3. where n core and n clad are the core and cladding refractive indexes respectively. In addition, the suffixes ν and are the indices for the guided radial and aziuthal coponents respectively. N and M+ are the total nuber of radial and aziuthal odes respectively. The values of these noralized transverse propagation constants for each guided ode can be obtained by solving the following characteristic equation [8] w wk w K u uj u J. 3.3

31 3... Nuerical Exaple: Few Modes Step Index Fiber In these calculations, we considered a single ode Corning SMF 8 which is a single ode at 550 n, and supports four odes at a wavelength of 63.8 n. Using equations 3. and 3.3, the aplitude and phase distributions of the first three higher order linearly polarized odes are depicted in Figure 3.. Figure 3. : Intensity distributions of the first three higher order odes in few odes fiber at λ633 n. a LP b LP c LP 0 and phase distributions of the first three higher order odes d LP e LP and f LP 0.

32 For a larger core fiber, the nuber of guided odes increases. For a standard ultiode fiber the nuber of odes is approxiated by V /, where V is the, so called, v nuber of the fiber [8]. Modes are not excited equally. The strength of each ode is strongly dependant on the incident bea profile. This dependence is represented by the coupling efficiency, which is defined as the aount of power coupled to each ode relative to the input one. The square root of this quantity is the field excitation constant. Thus, for a specific input bea profile the field distribution at a specific plane across the fiber is a superposition of the guided odes profiles weighted with the excitation constants taking into consideration the phase gained due to propagation as depicted in Figure 3.. This superposition is referred to as ultiode interference. In Figure 3. the Intensity distribution at distance z differs fro the one at L although the excitation coefficients of the guided odes do not change while propagating. This difference in the intensity profiles is due to the fact that each ode acquires a different phase while propagating a distance L-z as each travels with different propagation constant β as presented in equation 3.. 3

33 Figure 3. : Intensity distribution representation of the light propagating along the MMF due to ultiode interference Multiode Interference At any distance, z, along the MMF, the total field distribution is represented as a linear suation of the guided odes in this fashion > a r x e r r w K w K u J c a r x e r r u J c z r E N v M M z i a v v v v N v M M z i a v v v v,,,,,, ˆ cos ˆ cos,,,, β β θ θ θ. 3.4 In equation 3.4, c ν, is the field excitation constant. The value of this constant is the 4

34 square root of the power coupling coefficient, η v,, which is defined as follows π r, θ ψ s v, 0 0 η, π π v. 3.5 E ψ r, θ rdrdθ 0 0 s E r, θ rdrdθ 0 0 r, θ rdrdθ v, In equation 3.5, E s r,θ is the field distribution at the input facet of the MMF. At the entrance of the MMF, z0, the source field is apped into the different guided odes, neglecting the cladding odes. Each ode is weighted with the excitation coefficient described above. If E s is radially syetric then the value of the overlap integral in 3.5 goes to zero for the odes with a nonzero value of. This is due to the cosine ter in equation 3.4. Oitting these odes, only radial odes will be excited. This is the key feature of utilizing the MMF as a condensing lens as will be described later. 3.. Scattering Matrix ethod for odal calculations of Multi-core fiber Multicore fiber is a collection of step index cores sharing the sae cladding region as shown in Figure 3.3. The separation between these cores is very sall. Due to the strong coupling between the cores, the individual odes of each fiber split into several sub-odes called superodes. The nuber of these super odes depends on the nuber of cores, syetry of the structure and the ode order. 5

35 Figure 3. 3: The construction of a standard seven cores ulticore fiber. To utilize this specialty fiber structure a full vectorial wave propagation equation has to be solved in an inhoogeneous ediu 0,,,,, z r r z r o φ φ ε µ ω φ E E. 3.6 The solution of equation 3.6 can be carried by solving two hoogeneous wave equations inside and outside the individual cores as follows 0,,,, 0,,,, II I z r z r z r z r o o φ ε µ ω φ φ ε µ ω φ E E E E. 3.7 In equation 3.7 ε I and ε II are the perittivity of the core and cladding regions. These two equations are coupled through the boundary conditions around each core-cladding 6

36 interface. One way to solve these equations is to expand the field using a set of orthogonal functions. The selection of these functions depends on the geoetry of the structure. Besides, the nuber of these expansions depends on the required precision. For exaple, a Fourier expansion represents the field as a su of plane waves with different direction of propagations and coplex aplitudes. Typically, one requires a large nuber of expansion ters in order to achieve reasonable precision. However, selecting another set of expansion functions ight reduce this nuber draatically, thus the calculations tie and coplexity. For instance, the ulticore fiber structure can be represented by a set of cylinders placed at certain locations. To accurately calculate the field distribution across this structure using a Fourier expansion, one requires a large nuber of plane waves. On the other hand, the scattering atrix ethod SMM uses cylindrically syetric expansion functions. The selection of cylindrically syetric functions reduces the nuber of expansions and the coputation effort. Moreover, SMM siplifies the effect of the structure geoetry to one atrix the so called scattering atrix. Thus, the field at any point of space can be calculated using this atrix. That eliinates sapling and resolution constraints while nuerically coputing the field distribution across the fiber Scattering Matrix Method Scattering Matrix Method SMM is used to solve the set of coupled equations 7

37 3.7 for the guided superodes [9] to []. Using SMM, the structure is represented by cylinders of infinite length. Each cylinder is defined by its center coordinates, radius and refractive index as depicted in Figure 3.4. Figure 3. 4: Geoetry representation of a general ulticore fiber structure. In this structure each core, of index j, is defined by its location vector r j easured fro a reference point O, a refractive index n j and a radius R j. Using this representation the total field around each core is the su of two ain coponents: the local incident field, E loc, and cladding field, E s. The local incident field is the su of the evanescent field fro the surrounding cores that couples back to that core. The cladding field is the evanescent field due to the guidance inside that core. Figure 3.5 depicts the field representation around two cores. 8

38 Figure 3. 5: Fields representation around the cores of the ulticore fiber. Due to the geoetry of the structure the fields are represented by cylindrical syetric expansion functions as follows [],,, z t i e r F z r F β ϖ φ φ. 3.8 In equation 3.8, F represents the z coponents of both the Electric and Magnetic Fields z z H E F. The propagation constant, β, corresponds to the z coponent of the wave vector 9

39 according to the coordinates shown in Figure 3.5. The figure shows three field coponents around rod j, the cladding light, the local incident light, and the transitted, or guided, light. The fields are represented by their respective Fourier-Hankel expansions [] in equations 3.9 through 3. rods j N j h e N N i j s b b b e r H b r F,, φ χ φ, 3.9 j N N h e i j loc j a a a e r J a r F,, φ χ φ, 3.0 h e N N i j t j c c c e r J c r F j,, φ χ φ. 3. The superscripts e and h represent the electric and agnetic fields respectively. The angle φ j is easured fro the reference point O to the center of rod j. χ and χ are the tangential coponents of the wave vector outside and inside the core respectively. They are related to β through the following equation,, i n k i o i β χ. 3. However, we ust apply the boundary conditions to each core. In order to calculate the 0

40 field coponents in equations 3.9 through 3., we need first to apply the boundary conditions at the core-cladding interface at each core, rr, as shown in Figure 3.4. Applying the continuity of the z coponents of the fields we get the following set of equations c b R J R H R J R H a R J R J R J R J χ χ χ χ χ χ χ χ. 3. Fro the continuity of the φ coponents of the electric and agnetic fields, expressed as a function of the z coponents of the fields, we get the following equations c R J R J R i R J R i R J b R H R H R i R H R i R H a R J R J R i R J R i R J + ' ' ' ' ' ' χ χ ωµ χ χ β χ χ β χ χ ωε χ χ ωµ χ χ β χ χ β χ χ ωε χ χ ωµ χ χ β χ χ β χ χ ωε. 3.3

41 We can cobine equations 3. and 3.3 in a atrix for taking into account all the expansion coefficients to get the following vector equations c b B a A +, 3.4 c C b B a A +, 3.5 where h M h e M e a a a a a : :, h M h e M e b b b b b : :, h M h e M e c c c c c : :,,,, B A B A and C are square atrixes and M is the nuber of expansion coefficients. These atrixes are defined as R J R H R J R H B R J R J R J R J A χ χ χ χ χ χ χ χ, 3.6

42 ' ' ' ' ' ' R J R J R i R J R i R J C R H R H R i R H R i R H B R J R J R i R J R i R J A χ χ ωµ χ χ β χ χ β χ χ ωε χ χ ωµ χ χ β χ χ β χ χ ωε χ χ ωµ χ χ β χ χ β χ χ ωε, :. : :. : 0... M M M M H H H and J J J. Solving equations 3.4 and 3.5, a linear relation between the cladding light expansion coefficients vector, b, and the local incident one, a, can be obtained as follows a S b, 3.8 where A A C B B C S

43 In equations 3.8 and 3.9, S is a square atrix of diensions N + N N + N which represents the scattering atrix of the rods rods ulticore fiber. This atrix relates the guided field to the local incident one on each core. The geoetry of the structure is iplicitly included in the atrix S as the fields in 3.9 through 3. are expanded around the centers of each core. The field inside the cores can be related to the local incident field through the following relation c S a, 3.0 where S A B C B B A B. 3. Although the inforation regarding the geoetry of the structure is hidden in S, we still need to calculate the cross talk between the cores. Recalling that the locally incident field on core j is defined as the suation of the cladding light around all the cores except j and the incident field, one can the following forula for the local field E M loc j a e J k j E s k M iqφ j e rj e bqhq χrk k j q χ e iqφ j. 3. 4

44 In equation 3. the input field is set to zero in order to solve for the allowed propagating odes. Applying the expansion of H q χ r e k iqφ j around rj, [] H q M iφ rr iφ r i q a r r e J ar e H q ar e φ r 3.3 to equation 3. we get the following relation between the local incident and the cladding field coefficients a k j M q b T jk i q φ j q j k T q j k e kl, H q r,,,,,, χ k. 3.4 This can be written in the following vector for a T b. 3.5 In equation 3.5, T represents the cross talk between the cores. Recalling the linear relation between a and b in equation 3.8, we can write equation 3.5 as follows M β. b s 0,, 3.6 M β I S T s 5

45 where I is the identity atrix. Equation 3.6 represents an Eigen-value proble for which the Eigen-value is zero. Using the linear relation between the local incident field and the guided field in equation 3.0, equation 3.6 can be rewritten in ters of the guided field coefficient as follows M MCF β. c 0, 3.7 where M MCF I S T S S. 3.8 The solution of the Eigen-value proble in 3.7 can be accoplished by first finding the zeros of the deterinant of M MCF β. This requires applying an iterative technique to search for the coplex roots of the deterinant. To accelerate the convergence process, we first evaluate the deterinant, M β, over the range MCF π λ ε π ε I < β <. 3.9 λ II For the values of β near the coplex zero, the aplitude of the deterinant drops draatically as depicted in Figure

46 Figure 3. 6: The evaluation of M β over the range of the effective refractive indices MCF between ε and ε. II I The graph in Figure 3.6 is plotted as a function of the effective refractive index defined as λ β. The figure shows five locations where the aplitude of the deterinant drops draatically. At these locations, the value of β is the closest to the real part of the actual coplex root. To find the coplex value of the propagation constant, we used the false position ethod. First, we define two initial points around the real value of β that 7

47 iniizes the deterinant M β MCF. Using these values, the coplex value of β can be obtained through this recursive forula β β β β M MCF β M MCF β M MCF β In equation 3.30, the initial value of β and β are coplex. The nuber of these zeros corresponds to the total nuber of non-degenerate odes Nuerical Exaple: Super Modes in Multicore fiber As a nuerical exaple, we solved equation 3.7 for a seven-core ulticore fiber with core to core spacing of 0.5 µ and core radii of 3.5 µ. The core aterial is considered to be doped fused silica and the cladding is pure fused silica. The operating wavelength is.33 µ. This configuration gives a v nuber of.64 for each fiber. Figure 3.7 shows the intensity distributions of the guided superodes. 8

48 Figure 3. 7: The fundaental superodes for seven core fiber of fused silica cladding and doped fused silica cores at.33 µ operating wavelength. The spacing between the cores is 0.5 µ and core radii of 3.5 µ Hollow waveguide There are two ain structural differences between hollow and solid waveguides. The first is that in hollow waveguides, the solid core is replaced by air or gas. The second is that the cladding of the waveguide is frequently opaque at the operation wavelength. Most of the energy being guided by the hollow waveguide is confined in the core region. The reflectivity of the wall is the ost significant paraeter deterining the perforance of a waveguide. A hollow waveguide reflecting wall surface can be dielectric, etal or a ore coplex dielectric-coated etal structure used to axiize the reflection. The 9

49 losses in these structures are due to the less than perfect reflections of the propagating energy at the wall/core interface. Due to the large nuber of reflections fro the wall, any sall reflection loss will be greatly aplified [3] and [4] Modal calculations in cylindrical hollow-glass waveguide HGW Here we are presenting an approxiate odal solution of the hollow-glass waveguide structure depicted in Figure 3.8. In this solution we assue that the etallic layer is a perfect conductor and we neglect the thin dielectric layer. Thus the proble is siplified to a etallic cylindrical waveguide as presented in Figure 3.9. Glass n g Ag n Ag AgI n AgI Plastic n p i AgI Thickness d AgI Ag Thickness d Ag Glass wall thickness d g Plastic coating thinckness d p.5 µ 0.65 µ 50 µ 65 µ Figure 3. 8: Dielectric coated etallic waveguide structure and aterials paraeters at.55 µ wavelength. 30

50 Figure 3. 9: Siplified representation of the HGW by a etallic cylindrical waveguide. The odes that can be supported in such structure are transverse electric TE z and transverse agnetic TM z odes [5]. The TE z odes can be derived using the following vector potential representations,, 0 z r F a z z θ F A. 3.3 However, the vector potential F ust satisfy the wave propagation equation z z z z z F z F F r r F r r F β θ

51 The solution of equation 3.3 for the geoetry in Figure 3.9, using the constraint that the field should be finite everywhere, is deonstrated in this for F iβ z z β r [ C cos θ + D θ ] e z r, θ, z An J r sin In equation 3.33, C and D are constants. However, takes only discrete values under the boundary condition that the fields ust repeat every π radians in θ as shown below 0,,, In equation 3.33, β r and β z are the transverse and longitudinal propagation constants of the guided ode. They are related through the following equation z + β r β β To obtain values of these propagation constants for the different guided odes, we need to apply the following boundary condition Eθ r a, θ, z

52 In ters of the potential vector F z, this boundary condition can be written as E Fz An iβ z z r a, θ, z β r J β ra [ C cos θ + D sin θ ] e ε r ε θ which is only satisfied provided that χ n J β ra 0 β r, 3.38 a where χ n is the n th zero of the derivative of the Bessel function J of the first kind of order. Using equation 3.35, the longitudinal propagation constant β z is χ n β z β n a Finally, the electric field coponents can be written as [5] E E F r, θ, z ε r A ε iβ z z β r [ C cos θ + D sin θ ] e 0 z n θ β r J r Fz r, θ, z A εr θ εr iβ z z β r [ C sin θ + D cos θ ] e 0 r n J r, E z r, θ, z

53 The TM z odes can be derived in a siilar anner by letting A a F 0 z Az r, θ, z Again, the potential A z has to satisfy the wave propagation equation. Applying the boundary conditions that the tangential field coponents have to be zero at ra, the field has to be finite everywhere, and the field has to repeat every π radian cahnge of θ, we obtain the following solutions of the propagation constants χ n β r a β z n β χ a n, 3.44 where χ n is the n th zero of the Bessel function J of the first kind of order. The electric field coponents can be written as [5] E E j A β β sin ωµε r z ωµε z r z r r, θ, z Bn J r iβ z z β r [ C cos θ + D θ ] e j A β cos ωµεr θ z ωµεr z z θ r, θ, z Bn J r iβ z z β r [ C sin θ + D θ ] e E j β ωµεr z sin ωµε r z r, θ, z + β Az jbn J r iβ z z β r [ C cos θ + D θ ] e

54 Using the above procedures and for the operating wavelength of.55 µ, the transverse field coponents of the electric field as well as the polarization distributions of the TE 0 and TE 0 are depicted in Figure a b Figure 3. 0: The a E x and b E y field distribution for the TE0 ode. c The electric field polarization of TM0 and d TE0 respectively. 35

55 3.3.. Loss echanis in hollow waveguide This section derives a first order forulation of the hollow waveguide losses through Fresnel refection calculation at the core/wall interface. Figure 3.8 presents the structure and aterial properties of a dielectric-coated circular hollow waveguide structure. Using the ray optics approach, any guided ode can be represented by an angle, α, and a polarization status. As depicted in Figure 3. the angle θ is the copleent of α. The reflection loss at core/wall interface can be calculated using Fresnel reflection coefficient for the ulti layer structure shown in Figure 3.. Figure 3. : The optical path of a guided ode in the fiber. For any guided ode the ode angle, α, can be calculated as follows 36

56 α n β r n β z n tan, 3.47 and the incident angle θ n is θ π n α n The Fresnel reflection of the ultilayer structure presented in Figure 3. can be calculated as follows 0, air Z0 Z 0, air Γ, 3.49 Z0 + Z where o, AgI AgI konagi cos θ AgI d AgI k n cos θ d Z d AgI jz o, tan Z0 Z o, AgI, 3.50 Z jz z tan o AgI AgI AgI Z d Z d Z d AgI Au glass o, Au Au kon Au cos θ Au d Au k n cos θ d Z d Au jz o, Au tan Z o, Au, 3.5 Z jz z tan o, glass o Au Au Au kong cos θ glass d glass k n cos θ d Z d glass jz o, tan Z o, glass, 3.5 Z jz d tan o, plastic plastic o g glass glass kon p cos θ p d plastic k n cos θ d Z d plastic jz o, p tan Z o, plastic Z jz d tan o p p plastic 37

57 In equations 3.49 through 3.50 θ h θ sin sin, 3.54 nh and for the TE case, Z ωµ o o, h ko cos θ h However, for the TM case Z o, h θ ko cos ωε n o h h The total transitted power is approxiated by P N out Pin * Γ In equation 3.57, N is the nuber of reflections of the cladding for the specific ode. The nuber of reflections is defined as 38

58 N L, 3.58 Λ where Λ is the spacing between two consecutive reflections as depicted in Figure 3.9. It is defined as a β β z Λ r Figure 3. : Fresnel reflection fro the ultilayer fiber cladding Using equation 3.57, the transitted power through a 300 µ bore diaeter fiber with the paraeters entioned in Figure 3.0 above is depicted in Figure 3.3 as a function 39

59 of the ode angle for both TE and TM polarizations. The figure shows that TE polarization suffers fro less power loss than the TM case. Though, the difference in transittance is very sall, around 0. db. Thus, a ode of electric field that keeps a TE polarization around the boundary will suffer the least aount of power loss. In the previous analysis, we considered the transission losses as for the TE and TM odes. However, when guiding the light inside the fiber, HE is the ost doinant ode and hence the transission loss is ainly related to this ode. The attenuation coefficient of the power, α, for this ode can be expressed as [7] 8U o o 3 koa { zte ytm α k Re + } Figure 3. 3: Transission of a 300 µ bore diaeter hollow waveguide. 40

60 In equation 3.60, U o is the first zero point.404 of the Bessel function J o U o. The noralized TE ipedance and TM adittance are defined as [8] z TE x x zc + i zl tan zl, 3.6 z + i z tan l c and y TE x x yc + i yl tan yl, 3.6 y + i y tan l c where, the noralized ipedance and adittance of the dielectric layer z l and y l and etal are defined as n and y n n z 3.63 l Ag l Ag Ag n and y n n z, 3.64 c AgI c AgI AgI and x n k AgI o d AgI Another loss echanis in the hollow waveguide is the bending loss. Bending the fiber causes the excitation of higher-order odes with larger angles. Thus, the losses increase as depicted in Figure 3.3. In addition, bending the fiber causes coupling to TM odes 4

61 which suffer of high attenuation. While bent, the laser bea can travel in the fiber in two possible odes as depicted in figures 3.4,a and b. In the first one, the light reflects fro both sides, while for the second case the light collides with one side only. The second case is referred to as whispering gallery ode [8]. Figure 3.4,c shows the geoetry of the bent fiber and the ray paths for both odes of propagation. Using this geoetry, the copleent ode angles, θ and θ can be calculated as follows [0] R θ sin, cosα R ± a 3.66 where R is the radius of curvature and a is the bore radius as can be seen in Figure 3.4,c. Figure 3. 4: The two odes of ray and angles corresponding to: a reflection fro both walls of the fiber and b colliding with one side only. c The geoetry used to calculate the angles for both odes. 4

62 The two odes of propagation are separated when θ 90 o. Thus, the first ode of propagation exists when R a α < R As can be seen in equation 3.67, the type of propagation depends on the ode angle and the radius of curvature. This is an iportant factor to be taken into consideration while calculating the bending loss for a particular ode in addition to the state of polarization of that ode. In this section, we will copare the bending loss calculations for the TE 0 and TM 0 odes respectively. The polarization distributions of these two odes are shown in Figure 3.0. Considering a plane of syetry along this fiber, the electric field of the TE 0 is noral to this plane. However, for the TM 0 ode the electric field lies in the plane of syetry. In other words, the TE 0 ode aintains a TE polarization around the inner wall of the fiber. Conversely, the TM 0 can be considered to aintain a TM state of polarization around the walls of the fiber. Using these considerations, the bending losses of both odes can be calculated by calculating the Fresnel reflections fro the inner wall for the TE and TM cases. The angles of incident are calculated fro the ode angle α and the radius of curvature as represented in equation

63 Nuerical Exaple: calculations of the bending loss for the TE 0 and TM 0 odes in a 300 µ core HGW at 980 n To deterine the ode angle for each ode, we first need to copute the transverse and longitudinal propagation constants, β r and β z, through equation 3.39, 3.40 and For the TE 0 ode β β r r π Thus, the ode angle is α TE tan rad Siilarly, for the TM 0 ode β β r r π

64 Thus, the ode angle is α TM tan rad. 3.7 In a bent fiber, light can propagate in two different odes as can be seen in figures 3.4, a and b. Equation 3.67 presents a condition over the ode angle for the first ode of propagation to exist. This equation can be rewritten in the following for a R >. 3.7 α As can be seen by equations 3.70 and 3.7, the ode angles are very sall copared to unity. Thus, for the whispering gallery ode to exist the radius of curvature has to be as sall as the bore radius of the fiber, which is an extree case. So, the light propagates in the first ode as depicted in Figure 4.a. As both ode angles, α TE and α TM, are very sall, the cosine of these angles can be set to unity. Therefore, equation 3.66 can be siplified to R θ, sin R ± a 45

65 Using equation 3.73 and equations 3.49 through 3.59, the noralized transitted power versus the bending curvature is depicted in Figure 3.5 for both the TE 0 and TM 0 odes. As predicted, the TM 0 suffers of a very high bending loss copared to the TE 0 ode. Figure 3. 5: Transittance of the TE 0 and TM 0 odes versus the curvature of the fiber /R for a bore 46

66 CHAPTER FOUR: DESIGN OF OPTICAL COUPLING METHODS The previous section deonstrated different odeling techniques for ultiode fiber, ulticore fiber and hollow cylindrical waveguide. Using these techniques, we investigated the properties of the guided odes in these specialty fibers. Aong those odes, certain higher order odes exhibit interesting coplex field distributions which can be used to overcoe the liitations of these light delivery systes as explained in chapters one and three. In order to benefit fro such field properties we need to selectively couple to those odes. This section deals with the design of novel coupling schees that axiize the power coupled to a certain ode or set of odes in these fibers for different applications. 4.. Single ode fiber to ultiode fiber direct coupling Multiode interference and re-iaging conditions [9] provide the basis for the operation of the wavelength tunable lens. In the past, ultiode interference theory has been utilized in the design and fabrication of devices such as odulators [30] and Mach- Zender switches [3]. For an input field centered syetrically on the optical axis of a ultiode fiber, ultiode interference results in periodic longitudinal locations within the fiber where the source field is duplicated. Characterizing the re-iaging effect in a ultiode fiber, it is possible to design and fabricate a device that provides a echanis 47

67 to overcoe the natural tendency for the light to diverge when exiting the fiber. Traditionally, when fiber based devices are designed and fabricated, it is assued that colliating optics have been incorporated to cobat the natural divergence of the exiting bea [3]. In fact, soe approaches actually involve fabricating lenses on the ends of the fiber [33]. The device we present essentially perfors a lensing operation on the input field incident to the ultiode fiber, condensing the light to wavelength dependent longitudinal locations outside the fiber. By analogy to the traditional lens, we will call these locations outside the ultiode fiber MMF where the field condenses focal planes using our virtual lens odel. These locations and associated spot sizes are a function of the core diaeter of the ultiode fiber, the aterial properties of the fiber, and the wavelength of the source. We present a ethod to theoretically predict the shift of the focal plane as a function of wavelength in coparison to experientally observed behavior and results obtained through FD-BPM siulations. These FD-BPM siulations are also used to verify the validity of the odel we present for the deterination of the bea spot size observed at the wavelength dependent locations where the output bea is condensed Multi ode interference The basis for this work lies in the concept of re-iaging based on the ultiode interference effect associated with ultiode waveguides as was done for the case of a 48

68 planar waveguide [9]. The re-iaging distance is where the input source is replicated in both aplitude and phase. In contrast to the planar waveguide approach, a re-iaging condition is established for a circular ultiode waveguide. Moreover, fractional planes are defined where light appears to be concentrated see Figures 4.,a, 4.,b. The specific planes where field concentration and re-iaging occur are deterined through use of analytical approxiations that are derived on the preise that the input light source is provided by a single ode fiber fusion spliced to a section of MMF. Figure 4. : Geoetry of the device, SMF fusion spliced to a MMF. b Aplitude distribution of the light inside the MMF for the fibers paraeters entioned in table. 49

69 4... SMF to MMF Coupling In order to deterine the longitudinal re-iaging locations inside a MMF, with an input field provided fro a single ode fiber SMF fusion spliced directly to the MMF, it is necessary to first deterine which odes are excited in the MMF based on the SMF source. This ay be derived by the coupling efficiency associated with the propagating odes within the MMF. Using the linearly polarized ode approxiation, the input field provided fro the SMF, E s r can be approxiated by a Gaussian bea as follows E s r / w β g i oz r e e xˆ, 4. where β o is the longitudinal propagation constant for the SMF guided ode, LP 0. The half width half ax HWHM spot size, w g, of the Gaussian bea can be deterined epirically [36], based on the radius of the SMF, a, and the V-nuber, V π a λ n core n clad, where n core and n clad are the core and cladding refractive indices respectively, as V a w g V. 4. ln This input field excites a specific nuber of guided odes inside the MMF. These 50

70 odes, together with the radiating odes, for an orthogonal set [39]. Neglecting the radiating odes effect, the field distribution at any location across the MMF can be written as a series expansion of the guided odes as depicted in equation 4.3 M M N v v v z r c z r E,,,,,, θ ψ θ. 4.3 In equation 4.3 the suffixes ν and are the indices for the guided radial and aziuthal coponents respectively. N and M+ are the total nuber of radial and aziuthal guided odes respectively. The vector,,, z r v θ ψ represents the coplex field aplitude of the guided ode. Applying the separation of variables and solving Maxwell equations inside the MMF for linearly polarized odes,,,, z r v θ ψ can be written as follows [35] > a r x e a r w K d a r x e a r u J c z r z i v v z i v v v v v ˆ cos ˆ cos,,,,,,,,, β β θ θ θ ψ, 4.4 where u v,, and w v, are the noralized transverse propagation constant inside the core, and in the cladding respectively. The sybol β v, corresponds to the longitudinal propagation constant for this ode and a is the radius of the MMF. The sybols J and 5

71 K are the th order Bessel and odified Bessel functions respectively. The noralized transverse wave nuber, u ν and w ν, are defined as u ν, a ko ncore ν, and wν, a βν, ko nclad β, 4.5 where, n core and n clad are the core and cladding refractive indices respectively, and k o is π the wave nuber in free space,. The coefficients cv, and d v, are constants. A direct λ relation is obtained between these constants by applying the continuity of the tangential field coponents at the core cladding interface as follows J u. 4.6 v, d v, cv, K wv, Thus equation 4.4 can be incorporated into equation 4.3 to represent the radial and aziuthal odes within the MMF. At z 0, the left hand side of equation 4.3 should equal the input field, E s r, as shown in the geoetry of Figure 4.,a. In other words, at this position, the input field is projected onto an orthogonal set of the transverse field coponents of the guided and leaky odes inside the MMF with different weights. These weights are referred to as the ode excitation coefficients of these odes. Here, we will neglect the influence of the leaky odes inside the MMF, and hence in equation 4.4 the constant c ν, is the field excitation coefficient of the LP ν, ode. Since of the input field 5

72 defined by equation 4. does not contain any aziuthal coponents, the field distribution inside the MMF should be radially syetric. Thus, the excitation coefficient vanishes for values of greater than 0. This constraint siplifies the representation for the field within the MMF into a su of radial odes as follows N r cv,0 J 0 uv,0 xˆ r a v a Es r,0 N. 4.7 r > d v,0k 0 wv,0 xˆ r a v a For siplicity we will oit the suffix for the aziuthal odes and write the excitation coefficients, the noralized traverse propagation constant and the longitudinal propagation constant as c v, d v, u v, and β v respectively. To calculate the field excitation coefficients, we will use the power coupling coefficient, η v. The power coupling coefficient deterines the aount of the input power that couples to each specific ode in the MMF. In ters of an overlap integral in cylindrical coordinates η ν can be deterined using the following equation r 0 η. 4.8 v 0 E s E s r rdr ψ r rdr v 0 ψ r v rdr 53

73 \The function ψ ν r represents the field distribution of the ν th guided radial ode. Using the fact that the input field is uch saller than the actual core diaeter of the MMF itself, we can neglect the extension of the ode field in the cladding region in the nuerator of equation 4.6. Using this assuption, the field excitation coefficient, c ν, can be related to the power coupling efficiency,η ν, through ν ν η c since the integration in the nuerator in equation 4.8 is the cross correlation between the input and the core fields. In this case the overlap integral can be written as exp exp rdr a r w K w K u J rdr a r u J rdr w r rdrd a r u J w r a a v v g g v ν ν ν θ η. 4.9 Using Hankel Transfor properties, an analytical solution can be deduced as a function of the guided odes and the ters dependant on the physical paraeters of the fibers used exp v v v v v v v g g v w K w K w K u J u J u J u a w a w + + η

74 Fro the asyptotic forulation for the roots of the 0 th order Bessel function [39], the noralized transverse wave nubers can be written as u v v, 4. π π w v V v, 4. Figure 4. : Coupling efficiency as function of ode nuber for MMF core radii of 55

75 5.5µ and 9.5µ respectively In equation 4., V is the V nuber of the ultiode fiber. Thus, equation 4.8 can be odified into the following for + + exp π π π π π π ω ω η v J v V K v V K v J v J v a a v. 4.3 Using the equation above, the power coupling coefficient is calculated for all the allowed radial odes inside two MMF have the sae aterial properties as in table 4. but two different radii: 5.5 µ and 9.5 µ as depicted in Figure 4.. The ost pronounced feature in the graph is the presence of a peak coupling efficiency associated with a specific ode nuber. This feature is the key for predicting the re-iaging and the fractional planes where light is concentrated as will be presented in the next section. 56

76 Table 3. Optical Fiber Characteristics FIBER TYPE CORE RADIUS CLADDING CORE REF. INDEX RADIUS SMF-8 4.5µ 6.5µ.4505 Step Index MMF 5.5µ 6.5µ Re-iaging Conditions In this section we present an analytical forulation of the location of the re-iaging and the fractional planes where light is concentrated using the fact that the power coupling efficiency is axiu for a specific ode nuber. In order to deterine which radial guided ode has the highest coupling efficiency associated with it, it is necessary to look at the derivative of the coupling coefficient with respect to the ode nuber. Using the forulation for the coupling efficiency depicted in equation 4.3, the ode nuber associated with peak coupling efficiency is deterined by taking the derivative with respect to the ode nuber and equating it to zero as follow η v v

77 We introduce a paraeter D ν defined as π π π π π v J v V K v V K v J v J D v. 4.5 This expression can be siplified as π π π π v V K v V K v J v J D v. 4.6 The derivative of D ν with respect toν, D ν, is defined as 58

78 π π π π π π π π π π π π π π π π π π π π π v V v V K v V K v V K v J v V v V K v V K v V K v J v V v V K v V K v J v J v J v J v D D v v 4.7 The derivative of the coupling efficiency with respect to the ode nuber can thus be expressed in ters of D ν, the ode nuber, spot size of the input field, and the radius of the MMF core as follows exp exp 0 v v v v D D v a a D v a a v v π ω ω π ω ω π π η 4.8 The solution of the above equation result in the value of νν p that axiizes the 59

79 coupling coefficient. This ode is, in general, the ost doinant radial ode in the MMF. In order for light within the MMF to be concentrated to on axis location, it is necessary for the phase difference between the peak ode, ν p, and the previous ode to equal an integer ultiple of π. Moreover, we predict that the re-iaging condition should be a special case of this criteria, or ore specifically, a specific integer ultiple of the distance characteristic of a location where field condensation occurs. In the coing paragraphs we will evaluate the validity of the previous stateents. Under the asyptotic forulation given by reference [4.], the difference in the longitudinal propagation constants between two radial odes, ν and ν, can be expressed as follows u u β ν β ν, 4.9 k ν ν oa n core where u ν and u ν are provided in equation 4. for the asyptotic forulation for the roots of the 0 th order Bessel function. Considering the two odes ν p and ν p -, the phase difference between these two odes can thus be expressed as in the following equation. π 4v p 3 β z z β ν p v p k a n o core 60

80 At the following longitudinal location inside the MMF along the optical axis, z, z L p 8ka ncore L π 4v 3 p 8ka ncore π 4v 3 p p,,,3, the phase difference becoes an integer ultiple of π. L p corresponds to the location where the phase difference between the two odes equals π. Recalling Figure 4.,b, the upper part shows the Intensity distribution across the MMF calculated using equation 4.3 for the fiber properties entioned in table 4.. The lower part of the figure shows the average intensity distribution around the central region noralized to the input intensity. In this portion of this graph, we see pronounced axia at 0,, 3, 0, and 3. Beside these locations, there are several axia approxiately corresponding to the values of, 5, 8 and. This serves to justify the stateent ade previously that there are several locations, z, corresponding to local axia along the axis of the ultiode fiber where field condensation occurs. Although these locations ight not be explicit re-iaging locations of the input field, they do correspond to positions where condensing of power along the optical axis occurs. The explicit re-iaging location where the source input field is duplicated is derived fro the representation of the field given in equation 4.3. Looking at the coplex field vector contained in this forulation, the phase ter can be anipulated by 6

81 factoring out the phase ter characteristic of the radial ode that has a axiu coupling associated to it, iβ z e υ p. By doing so, the re-iaging distance can be deterined by looking at the resulting phase difference ter, βν-β νp z. The re-iaging distance is defined as the distance, z re-iaging, that corresponds to when this phase difference between these two guided radial odes equaling an integer ultiple of π. Therefore, under the asyptotic assuption for the lateral propagation constants, the re-iaging distance can be calculated by forulating an expression for the phase difference between the ν th and ν p odes as done in equation 4.9. β β ν v p π z [ ν ν + ν ν ] 4n core ka π p, p : Integer p p z 4. Thus, the distance that the two odes ust propagate to satisfy the re-iaging conditions can be expressed as z 8ncoreka iaging. 4.3 π re In coparing equations 4., 4., and 4.3, we can see that z re-iaging is indeed an integer ultiple of L p when expressed in the following anner 6

82 z 4v 3 L. 4.4 re iaging p p Having deterined the specific planes inside the MMF where field condensation and reiaging occurs, it is now necessary to present the use of this device to condense the light exiting the end facet of the MMF. In the next section, we will deonstrate a novel and siple technique that uses the phenoenon discussed above to create a fiber based wavelength tunable condensing lens Fiber based wavelength tunable condensing lens To force light exiting the MMF to converge towards the optical axis, the actual length of the MMF, L MMF, has to be slightly less than the length specified by equation 4. for a chosen integer value of. By doing so, it is possible to deterine the locations along the optical axis where field condensation will occur. Phase difference relations can be used to deterine these locations by taking into account the propagation within the MMF and the propagation in free space to the locations outside the MMF where the field condensation is observed. The following equation is derived fro this phase difference relation keeping in ind that the propagation constants are wavelength dependent 63

83 z out π L β MMF out, vp β β β νp out, vp vp. 4.5 Fro continuity of the tangential coponent of the wave vector, the associated longitudinal propagation constant of a particular ode after exiting the MMF can be deterined using the forula below in ters of the longitudinal propagation constant inside the MMF n β β k out, ν o core ν In forulating equation 4., the phase ter associated with the ode that has a axiu coupling was factored out of the field representation in equation 4.3 resulting in a particular phase difference ter. In order to deterine a st order forulation for equation 4.5, the sae phase ter factoring approach can be adopted, only for approxiation purposes the phase ter associated with the fundaental ode is factored out. In doing so, the sae asyptotic approxiations ade previously can be applied to the β -β ν ter contained within the phase difference expression for the field as done in equation 4.. β ν can be isolated fro this relationship and substituted back into equation 4.. Thus, the longitudinal propagation constant of a particular ode after exiting the MMF into free-space can be forulated as follows 64

84 β out, ν k n core k o β + β ν + ν π ν + ν z reiaging + z reiaging π. 4.7 Assuing that the fundaental longitudinal propagation constant can be approxiated as k o n core and that the last squared ter in equation 4.7 is uch saller than the other ters under the square root, this equation can be siplified into the following equation β out, ν k o k o n n core core ν + ν reiaging ν + ν z z reiaging π π. 4.8 This approxiate forulation for the longitudinal propagation constants outside the MMF can be used to for the expression for the β out, vp βout, vp ter included in equation 4.5. Along with the expression for β ν p β in equation 4.0, equation vp 4.5 can be anipulated into the following equation for the locations along the optical axis where field condensation of the light exiting the MMF occurs z out π L n core z 4v p 3 π zreiaging λ v 3 π MMF p reiaging λ 65

85 Using the relation defined for L p in equation 4., the above equation can be rewritten into a ore copact for in ters of L p, the actual length of the MMF used, and the core refractive index z out Lp LMMF n core In ters of the fiber paraeters and the wavelength, equation 4.30 can be written as follows z out 4v 6a MMF. 4.3 p 3λ L n core Due to the finite wavelength range that the source can be tuned, we operate over a specific range of wavelengths around a central wavelength, λ o. Assuing that the value of ν p doesn t change within this range of wavelength, equation 4.3 is expanded around the central wavelength using a Taylor expansion as follows z 3a L MMF 6a λ O λ v p 3 λo n core 4v p 3 λo out + 66

86 It is iportant to look at how z out changes as certain syste paraeters are varied. In particular, it is clear that this output distance exhibits inherent wavelength dependence.. Using the forulation above, the derivative of z out relative to the wavelength deonstrates that it is approxiately constant with respect to the varying wavelength as follows z out λ 6a ν 3 λ p o Another influencing syste paraeter is the radius of the MMF, a, because the peak ode index ν p, and L MMF depend strongly on the MMF radius. For a specific wavelength, λ o, z out changes considerably when the MMF radius is varied as deonstrated in the following equation zout 3a 64a ν p + a 4ν 3 λ 4ν 3 λ a n p o p o core L a MMF As the radius of the MMF changes, ν p changes according to equation 4.4. That changes the location of the working axiu, and thus the proper MMF length, L MMF. Using the fiber aterial paraeters in table 4., the change of L MMF as a function of the core radius is depicted in Figure 4.3. L MMF is assued to be 0.95 of the length corresponding to the peak at ν p -L p. 67

87 Figure 4. 3: Change of L MMF versus the MMF radius a. L MMF is considered to be 0.95 of the re-iaging distance inside the MMF. Now that the locations outside the MMF where field condensation will occur have been identified, in order to estiate the spot size at the focal plane, we need to calculate the intensity distribution outside the MMF. The series expansion in equation 4.3 can be used with a odification in the phase ter to copensate for the free space propagation. In addition, we can neglect the field expansion in the cladding region as ost of the power is concentrated in the central region. Thus, the output field can be written as 68

88 E r +, 4.35 N i βvlmmf + β,, vt out, 0 ou z out r LMMF zout ηv J uv e v a where β out,ν is the longitudinal propagation constant for the ν th order ode in air as defined in equation 4.8. In this section, we have deonstrated a novel application of selective ode excitation without using any additional optical eleents. By direct coupling of the light fro a SMF to MMF, only radial odes are exited in the MMF. Properly selecting the length of the MMF, the device can be utilized as a wavelength tunable condensing lens. However, for other type of applications, we ight need to add extra optical eleents. For instance, selective excitation of certain higher order odes in few odes fiber requires at least one phase eleent to atch the phase profile of the desired ode. In the next section, we will explain in details the design and analysis of single phase eleent for selective excitation of the LP and LP odes in few odes step index fiber. 4.. Single Phase Eleents Figure 3. shows the first three higher order odes in few odes fiber. The phase profiles of these odes are orthogonal to each other. Consider two functions with radial syetric aplitudes, and their phase profiles atch two different odes. Although the aplitudes ight not be orthogonal to each other, the cross correlation 69

89 between these two functions vanishes. However, this integral is axiized when the two phases atch. Thus, to selectively couple to a certain linearly polarized ode, LP ν,, the phase field at the input facet of the MMF has to atch that ode. This can be achieved through placing a phase eleent with a phase profile equivalent to that of the desired ode in the path of the input light. In spite of the fact that one ode has been selectively excited inside the MMF its coupling efficiency ight not necessary reach 00%. The coupling efficiency to any particular ode can be calculated through the overlap integral represented by equation 3.5 where, E s r,θ is the field profile at the input facet of the fiber. For a step index fiber of core radius a and a working wavelength λ the field distribution of the guided linearly polarized odes can be represented as in 4.4 and 4.6 [4] ψ ν, c r, θ, z cν, ν, J J K u u ν, w ν, ν, r iβν, z cos θ e a r K wν, cos θ e a iβν, z r a r > a In equation 4.36, the suffixes ν and are the indices for the guided radial and aziuthal coponents respectively, and c v, is the excitation coefficient of the LP v, ode as entioned in the previous section. It is defined as the square root of the power coupling coefficient, η ν,, represented by the following overlap integral 70

90 π E r, θ ψ in ν, 0 0 ην, π π Ein r, θ rdrdθ r, θ,0 ν, rdrdθ, 4.37 ψ r, θ,0 rdrdθ where, E in r,θ is the field profile at the input facet of the fiber. Figure 3. depicts the aplitude and phase profiles of the first few linearly polarized odes. Upon closer inspection of the phase profiles of these odes, one notices that they are orthogonal to each other. Thus, in order to selectively excite one particular ode, it is sufficient to atch the phase profile of that specific ode. This can be achieved through phase odulation of the input field by placing a proper phase eleent in its path as shown in Figure 4.4. In this setup, the first lens colliates the light out of the single ode fiber SMF, and the second lens focuses the phase odulated light to the input facet of a larger core fiber that sustains ore than one ode. We will refer to this fiber as a large core fiber hence forth. 7

91 Figure 4. 4: The coupling schee showing the phase eleent placed in the path of the input bea. Proper selection of the focal length of the focusing lens has a great ipact on achieving high coupling efficiency to the desired ode as we will present later in this section. For the coupling schee presented in Figure 4.4, the light out of the SMF is assued to be Gaussian with a bea waist defined as [39] V +.879V w S as S S In equation 4.38, a S and V S are the SMF radius and it s V-nuber respectively. The field iediately after the colliating lens has a bea waist of 7

92 w g w S Lλ + πws Assuing perfect colliation, the bea waist at the phase eleent will be w g as well. In this section we present two coupling eleents with phase profiles atch that of the LP and LP respectively. The phase coupling whose a phase profile identical to that of the LP ode has a transittance, Tx,y, which can be represented by a sign function as follows T x, y sign, 4.36 x where x > 0 sign x 0 x x < 0 The field distribution at the input of the fiber can be calculated using Fresnel approxiation as follows E in x + y x, y sign x Exp i f λ w x y g Exp iπ η x + η y dx dy x y Equation 4.38 can be siplified as 73

93 f y f x w Exp w Exp f i w y x E y x x x g y g g in λ η λ η η η π η π λ and where,, 4.39 In equation 4.39, λ is the working wave length and is the convolution operator. The convolution in equation 4.39 can be written as dt t e w Exp x w t x x g g η η η π π 4.40 This integration can be split in two parts as follows dt t e dt t e dt t e dt t e w Exp x w t x w t x w t x w t x x g g g g g η η η η η η π π π π π. 4.4 Knowing that [4] 74

94 + 0 0 ax Ei dt e e dt t x e x a t ax at π, 4.4 where Ei is the exponential integral defined as x t dt t e x Ei, 4.43 equation 4.4 can be represented as x g w t w x g w t w x x g w Ei dt e e w Ei dt e e w Exp x g x g x g x g η π π η π π η η π η π η π η π η π 4.44 This is siplified to x g w x x g w e w Exp x g η π π η η π η π erfi, 4.45 where erfiz referred to as iaginary error function is defined as 75

95 z t dt e z 0 erfi π Thus, the field in equation 4.39 can be written as x g w g y x in w e f i w E y x g η π λ π η η η η π erfi, Substituting η x and η y in ters of x and y, equation 4.47 is written as g G G w y x g in w f w w x e f w i r E G π λ λ π θ erfi, In equation 4.48, f represents the effective focal length of the second lens in Figure 4.4, and k is the wave nuber of the light, λ π. Notice that we used the Cartesian coordinates to represent both the phase transittance of the coupling eleent and the field at the input of the large core fiber. However, it is ore convenient to use polar coordinates in this cylindrical syetric structure. Defining the following noralized polar coordinates 76

96 x + y ρ wg, 4.49 θ tan y x equation 4.48 can be written as follows iπwg ρ θ ρ E, in e erfi ρ cosθ λf The aplitude and phase distributions of E in are depicted in Figure 4.5. Notice that the phase profile atches that of the LP ode presented in Figure 3,e. However, the aplitude differs fro that presented in Figure 3.b. This aplitude isatch reduces the coupling efficiency to the LP ode. Figure 4. 5: The calculated a Aplitude and b phase profiles of the field at the input facet of the larger core fiber. 77

97 The coupling efficiency is calculated through the overlap integration in equation Equation 4.37 can be written as D D N f η, 4.5 where + π ρ π ρ θ ρ ρ ρ θ ρ θ ρ ρ ρ θ ρ cos erfi cos erfi G G w a w a f d d c K w K u J e d d b J e N, 4.5 π ρ θ ρ ρ θ ρ 0 0 cos erfi d d e D 4.53 and + π π θ ρ ρ ρ θ ρ ρ ρ G G w a w a d d c K w K u J d d b J D In equations 4.5 through 4.54, b and c are defined as 78

98 a w w c a w u b G G Equation 4.5 can be siplified by expanding the error function in a polynoial series around zero as + + 0! erfi n n n n z z π Using this expansion, Equation 4.5 can be written as cos! n n n f d e d n n N ρ ρ ρ ψ θ θ π ρ π, 4.57 where, ρ ψ is the LP ode radial dependant ter defined as > G G w a c K w K u J w a b J ρ ρ ρ ρ ρ ψ

99 A closed for solution nuerator, N f, is obtained by, first, carrying out the integration over θ as shown in equation It has the following closed for solution π 0 cos 5 Γ / Γ + n n+ θ dθ n Γ + n n!n + In the second integration, a closed for solution is obtained by approxiating ψ ρ by a first order Herite-Gaussian as follows [9] ψ ρ p ρ e pρ, 4.60 where p and p are constants. The values of these constants can be obtained by applying two constraints. First, the areas under both curves are equal, ψ ρ ρ ρ ρ p d p e dρ Applying this constraint, we obtain the following linear relation between the two constants 80

100 p p ψ ρ dρ 0 The integration in the denoinator has the following solution [4] Thus where 0 c a w u w g J ψ + ρ dρ c J bρ dρ A K 0 K a J 0 u J u + w b K w K wg 0 c ρ w dρ, 4.64 p h p, 4.65 J 0 u J u + b K w w K0 h A 4.66 c In the second constraint, the value of p is obtained by iniizing the profile aplitude error, δ R, defined as δr ρ pρ ψ h p ρ e

101 Several optiization techniques can be used to solve for the p coefficient. In this work we used the least square error ethod. In this ethod p is represented as a cobination of two parts, p ~ p + p In equation 4.68, ~ p is assued to result in zero error. Using this assuption and using the first order expansion of the Herite-Gaussian function around ~ p, equation 4.67 can be written as p 3 ρ p ρ hρ e h p ρ e p δ R By discretizingρ, equation 4.69 can be written in a vector for as follows R M p In equation 4.70, vectors R, and M represent the error, the ter between brackets on the right hand side in equation The value of p is obtained as follows 8

102 T M R T M M p. 4.7 The initial value of p is then updated p new old p p. 4.7 These steps are then repeated till p converges. Using this representation, the integration over ρ can be written as 0 ρ n+ n+ 3 p + ρ e ψ ρ ρ dρ ρ e dρ Using the following relation µ / µ σt σ + t e dt Γ 0 µ, 4.74 equation 4.73 has the following solution n+ ρ n+ p + + µ e ψ ρ ρ dρ Γ

103 Substituting equations 4.59 and 4.75 in 4.75, we obtain N f p π n + n! p + + n n n 0 n! 7.76 In the denoinator, the first integral, D, is very difficult to be solved analytically. However, neglecting the Fresnel losses fro the phase eleent and the lens, we can apply Parseval s theory since E in is proportional to the Fourier transfor of the input Gaussian bea phase odulated by the coupling eleent. Thus, D πwg λ f The Second integral in the denoinator part of equation 4.5 has the following solution where D πa H u,, 7.78 w J [ ] u J u J 0 u J u + K w [ K w K w K ] H u, w 0 w

104 Substituting equation 4.76, 4.77 and 4.78 into 4.5 and we obtain the following expression for the coupling efficiency η n + π n! n λ f p + p + n n 0 n! π w a H u, w g Figure 4. 6: The coupling efficiency to the LP ode as a function of the nuber of expansions. 85

105 Figure 4. 7: Coupling Efficiency vs. the effective focal length of the few odes fiber objective using exact nuerical integration in equation 4.4 and the analytical solution in equation This expression is represented in ters of an infinite series. However, a few expansion ters ight only be needed to obtain a precise value of η. Figure 4.6 depicts the coupling efficiency as a function of the nuber of expansion ters. In these calculations we consider an effective focal length, f, of 8 for the focusing lens. The figure shows that by including ore than 6 expansion ters into the efficiency calculation, the 86

106 calculated value of η converges to a stable value. Therefore, in the rest of this work we use 6 expansion ters to evaluate η when using the expression in To predict the accuracy of this expression, we copare the exact nuerical calculation of the overlap integral in 4.37 and equation 4.80 as a function of f. These results are depicted in Figure 4.7. In the nuerical calculations, E in is coputed through the Fresnel propagation of the input Gaussian bea phase odulated by the coupling eleent through the second lens. The figure shows a good agreeent between both cases. However, the slight difference in the graphs is ainly due to the first order Herite-Gaussian approxiation as presented in equation Additionally, the graphs show that the coupling efficiency is axiized around f 8. This sees contradictory with equation 4.80 due to the presence of f in the nuerator. The explanation for the axiization of η at a specific effective focal length value can be traced to the forulation of p and p. However, constants p and p iplicitly depend on f through b, c, and w g as presented in equations 4.55, and Figure 4.8 shows the values of p and p as a function of f. Notice that both p and p increases with f. Siilarly, the transittance of the phase eleent required to excite the LP is T x, y sign x sign, 4.8 y and the resulting field distribution at the input facet of the fiber is 87

107 π wg Ein r, θ erfi ρ sinθ erfi ρ cosθ iλf Figure 4. 8: Dependence on values of p and p on effective focal length showing increasing relationship for both paraeters. Again, the phase profile of this field atches that of the LP while the aplitude differs. For the used focal length of 9, the calculated coupling efficiencies using equation 4.8 is 8.47% while it is 83.4% using the nuerical integration. Using equation 4.4 the coupling efficiency for the LP ode is about 60%. 88

108 4.3. Dual eleents Dual eleents are two phase eleents aligned on the front and back sides of a substrate aterial such as fused silica wafer as depicted in Figure 4.9. The coupling schee is depicted in Figure 4.0. In this schee, the first eleent odulates the aplitude of the incident bea and the second one fixes the phase profile. However, designing the first phase is the ain challenge. This section deonstrates two design techniques. The least square error ethod and Method of projection MOP. In the first ethod, the phase surface of the first eleent is fitted to a two-diensional polynoial expansion, while within the MOP ethod the phase of the first eleent is nuerically calculated. Figure 4. 9: Dual eleent structure. 89

109 Figure 4. 0: Dual eleents coupling schee Least Square Error ethod In this ethod, the phase surface of the first eleent is represented by a two diensional polynoial expansion as depicted in equation h k ah k x y, h 0,,.., N and k φ x, y, 0,,..., M 4.84 h k Considering an input Gaussian bea, E g x,y, of specific bea width, w g, the goal is to iniize the difference between the intensity distributions of the output field at a certain observation plane, Ia h,k,x,y, and the target intensity, I o x,y. 90

110 di x, y I x, y I a,, x, y 4.85 o h k The second eleent corrects for the phase of the field at the observation plane. Figure 4. depicts a scheatic diagra of the ethod. Initial guess of the polynoial coefficient Expiφ Second eleent Fresnel propagation E Expiφ Target Field Calculate new coefficients Back Fresnel propagation di E o - E E o Expiφ o Figure 4. : Scheatic diagra of the diffractive eleent design ethod. The field distribution at the observation plane using the above schee can be written as ik x + y h k E ah k x y Eg x y Exp Expi ah k x y z,,,,, x y λ h k dxdy,

111 where λ, k, and z are the working wavelength, the wave nuber and the propagation distance respectively. The intensity distribution at this plane is I ah, k, x, y E ah, k, x, y E ah, k, x, y Using equation 4.86 and representing the coefficient a h,k as a suation of the desired value, a ~ h, k, and a displaceent of ah,k, equation 4.85 can be written as I ah, k, x, y di x, y ah, k a h k h, k Writing equation 4.88 in a atrix for, one obtains the following expression di M a In equation 4.89, atrices di, M, and a are defined in appendix A. Multiplying both sides by the transpose of atrix M, equation 4.89 can be written as T T M di M M a a can be calculated by solving the set of linear equations represented in Updating the value of a and substituting this new value in equation 4.89, one gets a new value of di. Subsequently, this process repeats till the displaceent coefficient reaches a very sall value. However, the convergence of this schee is critically dependant on the initial guess. 9

112 Least Square Error for Designing a Dual Eleent for Selective Excitation of The LP Mode Using this technique the calculated coupling efficiency to LP reaches 94% of the total input power copared to 60% using phase atching only with a wavelength of 633 n and Gaussian bea of 50 µ waist. Figure 4. depicts the two phase eleents and the aplitude profile of the field just behind the second phase eleent and at the input facet of the large core fiber. Figure 4. : a The first phase eleent, b the second phase eleent c phase c and intensity d distributions of the output field at the input facet of the fiber. 93

113 4.3.. Method of Projection MOP MOP is an iterative technique, where the first phase eleent is initially set to have a rando phase. Applying a two diensional Fresnel propagator, the coplex field distribution is coputed at the second phase eleent, then its intensity is set to the target field intensity. The second phase eleent is updated by the phase difference between the coputed field at the surface of this phase eleent and the desired phase. Figure 4.3 depicts a scheatic diagra of the MOP. Initial guess of the phase φ φ o -φ Expiφ Fresnel propagation E Expiφ Target Field E Expiφ Back Fresnel propgation E o Expiφ E o Expiφ o Figure 4. 3: Scheatic diagra of MOP. 94

114 MOP for Designing a Dual Eleent for Selective Excitation of a High order super ode in seven core fiber Figures 4.4, c and 4.4, d depict two phase eleents designed to atch both aplitude and phase of the last fundaental superode in the seven core as depicted in Figure 3.7. Figure 4. 4: a aplitude and b phase profiles of the field at the input facet of the seven core fiber when using the dual phase eleents c and d. 95

115 Using the coupling schee depicted in Figure 4.0 and considering a 633 n source, the calculated aplitude and phase profiles of the field at the input facet of the ulticore fiber are depicted in figures 4.3, a and b. Using the overlap integral in 4.7, the estiated axiu coupling efficiency is about 98% when both L and f distances in Figure 4.0 are set to. 4.4 Sub-wavelength periodical structure for coupling to hollow waveguide. In chapter three, we presented first order calculations of the transission and bending losses inside the cylindrical hollow waveguides based on Fresnel reflections. These calculations showed that the TE 0 ode suffers the least aount of losses as depicted in figures 3.3 and 3.5. Coupling the input light to this ode, the perforance of the light delivery syste can be draatically iproved. To selectively excite this ode, atching the phase and aplitude profiles of the TE 0 ode only will not be sufficient due to the degeneracy between this ode and the TM 0 ode. Thus, it is desired to design an optical coponent that converts an incident linearly polarized light into a rotating one siilar to that depicted in Figure 3.0, d. In order to design this optical coponent, we first need to review an iportant property of birefringent crystals. For a birefringent crystal, the transittance of this crystal for an incident linearly polarized light of angle ψ relative to the crystal axis fast axis with a phase retardation of π can be written as [44] 96

116 i cosψ x, y T x, y. 4.9 i sinψ x, y Thus, it is possible to obtain the desired output polarization distribution if the axis of the crystal varies in space. Realization of such coponent is alost ipossible using traditional bierefringent crystals. However, artificial birefringent crystals can be fabricated using subwavelength gratings [45]. Figure 4. 5: Sub-wavelength grating structure that fors an artificial birefringent crystal. Figure 4.5 depicts the subwavelength grating structure. In this figure, ε, and ε are the parallel and noral effective perittivities of the grating. ε s and Λ are the substrate perittivity and the grating period. The birefringence of such structure can be calculated 97

117 using effective index ethod [46]. For the configurations shown in Figure 4., the calculated perittivities are Λ + o o o s f f ε ε ε ε λ π ε ε 0 3, 4.9 and Λ + s o o o o o s f f ε ε ε ε ε ε ε λ π ε ε In equations 4.9 and 4.93 o s o s f f f f ε ε ε ε ε ε , 4.94 where f is the in plane filling ratio. Using the effective refractive index calculations entioned above, the required depth of the grating to have π retardation is ε ε λ d

118 4.4.. Polarization converter eleent for coupling to TE 0 ode. For fused silica substrate, operation wavelength λ0.98 µ, period of λ/ and filling factor of 0.5, the calculated depth is about 6.84 µ. In a siilar way, the calculated depth for the working wavelength of.55µ is.06 µ. Figure 4.6 depicts the proposed sub-wavelength structure. Figure 4. 6: The proposed design of polarization converter eleent to couple the light at the input of the hollow waveguide to the TE 0 ode. In this design, the working space is divided into sectors. Each section represents a sub-wavelength grating with grating vector rotated by π/6 relative to the next sector. Thus each sector rotates the incident linearly polarization by ψ, where ψ is the 99

119 angle of polarization of the incident light relative to the grating vector. For an incident y polarized light the output polarization status will result in the field distribution and polarization status depicted in Figure 4.7 below. This field, to great extend, atches the desired TE 0 ode. On the other hand, for an incident x polarized light the output field will be siilar to that of TM 0 ode. E x E y a b Figure 4. 7: Calculated a Field and b polarization distributions of the output of the polarization converter eleent. This section presented the design of novel optical eleents that couple the light to the desired guided odes in different specialty waveguides. However, fabricating soe of these eleents is challenging. The next section suarizes several fabrication techniques and procedures to realize these eleents. 00

120 CHAPTER FIVE: FABRICATION TECHNIQUES Lithography is a key process technology for fabricating icro-optical eleents and is used to copy a aster pattern into solid aterial such as silicon, glass or GaAs. This process has two ajor steps. First, a pattern is written into a resist aterial by exposing it to light photolithography or an electron bea e-bea lithography and developing in order to reove resist in selected areas. Second, the pattern is transfer etched into a substrate aterial using an etching technique. The substrate is etched where it is not protected by the resist aterial. Photolithography is ost widely used. In the next section, photolithography using a contact aligner and a stepper syste is introduced. Both require a ask to control which portions of the resist are exposed, and hence the pattern. The photoask is typically an optically flat and transparent glass or quartz with a etal absorber pattern e.g. 80 n of chroiu. [47] E-bea lithography EBL does not require a ask because the pattern is written by steering a narrow bea of electrons to write the pattern into the resist. In section three, fabrication of the polarization converter eleent using EBL is discussed. Section four introduces the use of the focused ion bea FIB syste for integration of icro optical eleents into real devices. The ain advantage of FIB over lithography is that devices can be ade in a single process step without a ask. 0

121 5.. Photolithography Figure 5. shows two ajor photolithography systes: contact aligners and steppers. In contact aligner systes, the photo-ask is usually in direct contact with the photoresistcoated surface during exposure as depicted in Fig. 5..a. Ultraviolet light passes through a ask where it is selectively blocked by the etalized patterns. In this anner, the photoresist is exposed with the sae pattern as the ask. Figure 5. : The two ajor photolithography systes: a Contact aligners and b 0

122 steppers. For the research described in this paper, the pattern on the photo-ask was generated using EBL because it could produce higher resolution patterns than photolithography alone. The transparent pattern on the ask was transferred to the photoresist coated on the wafer surface and then to the substrate using etching techniques. This procedure results in a : iaging of the entire ask onto the wafer. Contact asks degrade fast through wear. Defects resulting fro hard contact on both the photoresist and the wafer ake this ethod unsuitable for very large scale integration anufacturing. However, this ethod is still coonly used in research and developent as well as ask aking [47]. A ore robust ethod uses an optical stepper where the photo-ask does not contact the substrate, but is iaged onto the photoresist fro soe distance away. Basically, the optical stepper is an iaging syste where the photo-ask is in an object plane and the photo-resist layer on the wafer is in the iage plane [48-49]. The scale of the pattern is often reduced by the iaging syste. A 5: reduction is coon. The photo-ask is illuinated by a UV source through condenser optics during exposure. This light incident on the photo-ask is well colliated and the effective source of UV light is iaged onto the pupil plane inside the stepper. Figure 5.b shows a conceptual diagra of a typical optical stepper syste. With a deep UV light source, resolution of µ can be achieved, depth of focus of ±6 µ is possible, and an overlay accuracy of ±0.5 µ has been realized []. 03

123 5... Exaple of Photolithography: Fabrication of Single phase eleents Figure 5 depicts Zygo profiles of the two phase eleents presented in section 4.. These eleents were fabricated in Shiply PR805 photoresist using a stepper syste. The refractive index of the photoresist is.6406 at the working wavelength of 633 n. To achieve a π phase shift, the step height was set to 494 n. Figure 5. : a and b Zygo iages of the first and second phase eleents for coupling to LP and LP odes respectively, fabricated in PR805 photoresist 5.. Electron Bea Lithography EBL is a high resolution pattering technique in which high-energy electrons 0 to 00 KeV are focused into a narrow bea that exposes electron sensitive resist. Unlike photolithography, EBL does not liit feature resolution by diffraction because the 04

124 quantu echanical wavelength of the high energy electron bea is exceedingly sall [47]. On the other hand, resolution of the EBL syste is affected by scattering of electrons fro the resist and substrate. Back scattering of electrons exposes the resist over an area slightly greater than the spot size. This phenoena anifests itself in sall variations in the geoetry of exposed structures. The effect is ore pronounced fy sall structures 5... Introduction to E-Bea Lithography EBL is a high-resolution and askless pattering technique. Electrons eitted fro a source are focused onto a substrate through an iaging colun. The focused bea is scanned over the substrate using deflectors as depicted in Fig A coputer controls the bea blanker, the deflector, and the oving stage according to received ask data. Total writing tie depends on total exposed area and required dose to fully expose the e-bea resist. At the end of the writing process, a pattern is written into the e-bea resist. E-bea resists are produced for direct writing applications. Bobardent of polyers by electrons causes bond breaking. A subsequent cheical developing process is required. For positive resists, exposes areas are reoved during developing. For negative resists, exposed resist reains after developing. In either case, the reaining resist serves as a ask during a subsequent etching process. Either wet or dry etching can be used to transfer etch into the substrate. 05

125 Figure 5. 3: Scheatic diagra of e-bea syste Fabrication of Polarization Converter Eleent As presented in section 4., the polarization converter eleent is ade of sub-wavelength gratings with a µ period. Each grating is oriented such that the grating vector is rotated by π/6 relative the previous one. This structure has features as sall as 375 n with a depth of µ in order to achieve π phase retardation when using fused silica as a substrate. In contrast, only 996 n depth is required when using GaAs. 06

126 In the next sections, different fabrication procedures are used to realize this structure in both fused silica and GaAs. First, e-bea resist ZERP 50 was used as a ask for etching into a GaAs substrate where an etching selectivity ore than : was achieved. This selectivity is sufficient for GaAs, but not for fused silica that requires up to :. For fused silica, a hard etal ask ust be used.. The hard ask is fabricated using a lift off process and a bi-layer PMMA resist. For both techniques, the pattern ust be properly transferred to the resist. This requires selecting an accurate dose while writing the pattern. Quality of the hard ask, and eventually the etched pattern, is deterined by the surface condition in the developed regions and the aount of resist left after developing. The next section discusses pattern transfer writing in ZEP50 resist using EBL Writing the pattern into the e-bea resist The desired pattern was the first generated using our GDS aster software. The pattern was fractured using a Linux based coercial software, CATS, and written to the resist using the Lieca EBPG E-Bea at 50 KeV. Different e-bea resists have different polarity, sensitivity, resolution, and etch resistance. PMMA and ZEP50 are exaples of positive e-bea resists. Although both resists have high resolution, ZEP50 has higher sensitivity and higher dry etch resistance. This reduces the e-bea writing tie when using ZEP50 copared to PMMA. In addition, ZEP50 holds for a longer 07

127 tie when etching chroe. Finally, ZEP50 resist can produce an under-cut profile siilar to PMMA as depicted in Fig This profile is favorable for the lift-off process. In this work, ZEP50 was used as a ask for etching in GaAs and to generate a hard ask in chroe. Figure 5. 4: ZEP50 resist profile. Using ZEP50, proper writing dose is required to generate a pattern with the required diensions. High doses will result in washed out patterns, while low doses will create under exposed patterns. In addition, the dose is a function of the feature size. To 08

128 deterine the proper dose, variable period gratings were fabricated with a 50:50 duty cycle. The periods varied fro 00 n to 3.95 µ as shown in Fig. 5.5a. Figure 5. 5: a the GDS pattern of gratings of varying periods b the pattern written in the e-bea resist. Using this pattern, a dose atrix was written in ZEP 50 e-bea resist with a dose varying fro 50 µc/c to 300 µc/c. The resist was spun at 800 rp on a four inch fused silica wafer coated with 300 n of chroe. The spun resist layer was 500 n thick. After writing the pattern, the wafer is developed for 90 seconds using ZEP-D 09

129 developer. The wafer was rinsed in isopropanol for 0 seconds iediately after developing. Figure 5.5b shows an optical icroscope iage of the pattern after developing. This pattern was written using a dose of 90 µc/c. Observing the quality of the written patterns under the optical icroscope, the dose ap in Fig. 5.6 was generated Figure 5. 6: Map of pattern quality in ters of dose values and feature size. Figure 5.6 shows a 90 µc/c dose is optiu for features as sall as 00 n. Saller features can be written with a dose between 70 µc/c and 90 µc/c. For the 0

130 patterns written at 90 µc/c, SEM iages were taken after sputtering a 00 angstro thick fil of gold-palladiu on the developed resist. 00 n 350 n 450 n 750 n Figure 5. 7: SEM iages of the varying period gratings using 90 µc/c after developing and coating with 00 angstro layer of AuPd.

131 Figure 5.7 depicts four SEM iages of different grating periods, 00n, 350 n, 450 n and 750 n. For the 00 n period patterns, the irregularities of the lines are iaging artifacts due to charging of the substrate. In this figure, resist residue reains in the developed areas and ay degrade the quality of the hard ask and hence the final etched pattern. The residue can be reoved by oxygen plasa etching. In this process we used 40 scc of oxygen with an ICP power of 400 W and RIE power of 40 W. The easured etch rate of ZEP50 was 0 n/sec. Thus, five seconds of oxygen plasa etching reoved 50 n of resist. Figure 5. 8: The polarization converter pattern transferred to the ZEP50 e-bea resist. As entioned in chapter four, the polarization converter eleent consists of

132 sectors of subwavelength gratings; each rotated by π/6 relative to the next sector. The gratings were designed for an operation wavelength of.55 µ. Each grating has a period of λ/ and filling factor of 0.5. Thus, the iniu feature of this structure is 375 n. Fro the e-bea resist analysis depicted in Fig. 5.6, the proper dose to write this structure was 90 µc/c. Using this dose, Fig. 5.8 shows an SEM iage of the pattern generated in ZEP 50 after developing, plasa etching for 5 seconds, and coating with a thin layer of chroe to prevent the charging when using the SEM. This figure shows the pattern was properly transferred in the resist. The next step was to transfer etch this pattern into the substrate aterial Transfer Etching into Galliu Arsenide First, ZEP 50 was spin-coated directly onto a GaAs wafer. Second, a pattern of gratings was written into the resist. The gratings had a 50% duty cycle with periods varying fro 400 n to 600 n in 00 n increents. Third, the resist was developed using ZEP-RD for 90 seconds. Using the resist as a ask, the pattern was transfer etched in the GaAs substrate using a BCl 3 /Ar plasa etch for two inutes. In this process, 0 scc of BCl 3, 0 scc of Ar, and 5 scc of N was used at a pressure of 0 T. The RIE power was 60 W and the ICP power was 500 W. The profiles of the etched pattern were iaged using a SEM and are depicted in Fig Saller period gratings had saller depth than the larger period gratings 79 n for 600 n period grating copared to 3

133 799 n at 400 n period grating as shown in Fig. 5.9a and 5.9d respectively. This was due to a icro loading effect. Resist at the saller period gratings etched faster than the larger period gratings. The eroded patterns in Fig. 5.9 were caused by resist being copletely etched and no longer asking the GaAs. It is therefore necessary to increase etching selectivity and reduce the resist etch rate. The first was achieved through reducing RIE power, increasing ICP power, and increasing pressure. The resist etching rate was decreased by reducing the flow rate of the BCl 3 gas. Table 5. shows the different recipes used, the resulting etch rates, and etch selectivities. 4

134 Figure 5. 9: Cross section SEM iages of gratings with different periods etched into GaAs using 500 n of ZEP50 as a ask and BCl 3 /Ar/N plasa for two inutes. Grating periods are a 600 n, b 800 n, c 000 n, and d 400 n. 5

135 Table 5. GaAs dry etching process when using ZEP 50 as a ask. Process BCL 3 scc Ar scc Pressure T RIE W ICP W Etch rate n/in Selectivity : : : : : Figure 5. 0: SEM iage of the pattern transfer etched in GaAs using ZEP 50 as a ask. 6

136 Figure 5. : SEM cross sectional iage of the polarization converter pattern transfer etched in the GaAs substrate using a process for 4 inutes, b process 3 for 5 inutes, c process 5 for 6 inutes, and d process 4 for 7 inutes. 7

137 Table 5. shows dry etching recipies using BCl 3 /Ar plasa. It is clear fro this Table, increasing the ICP fro 500 W to 800 W iproved the resist:gaas etch selectivity fro : to :. Increasing the pressure fro 0 T to 5 T, however, increases etch rate fro 40 n/in to 70 n/in. Figure 5.0 shows a top view iage of the polarization converter eleent transfer etched into a GaAs substrate taken by SEM. As can be seen, the pattern was properly transferred into the substrate. The profile of this eleent can be inspected by cleaving through it. Figure 5. shows profiles of soe eleents fabricated using different process fro Table 5.. The icro loading effect is apparent in Fig. 5.d, when using the process 5 for seven inutes Focused ion bea syste for prototyping icro-optical eleents A FIB is often used in saple preparation for characterization easureents using a Transission Electron Microscope TEM or Scanning Transission Electron Microscope STEM. Others have deonstrated the use of a FIB for icro/nano fabrication on flat surfaces of optical coponents such as gratings, photonic crystals, and icro lenses [50-54]. The ain liitation of prototyping and fabrication with FIB is cost for ass production when copared to the available lithography tools such as EBL and photolithography. Moreover, FIB does not iprove the resolution or surface quality over these techniques. Fabrication tie and coplexity becoe iportant issues as the device size increases. On the other hand, the FIB is a copetent tool for rapid prototyping of 8

138 new devices, since it is a askless fabrication technique and can be used on a single chip with sall field sizes [50]. Figure 5. : a Representation of the FIB as an iaging syste. b Scheatic picture of a typical FIB syste. A FIB workstation takes charged particles fro a source, focuses the into a bea through electrostatic lenses, and scans across sall areas of the saple using deflection plates or scan coils. The charged particles ions are coposed of a liquid 9

139 etal ion source LMIS, Ga +, where ions are produced through field evaporation by applying a high voltage on the eitter. The eitter is a highly sharpened etallic needle. As liquid etal flows through the eitter, an electric field aintains the conical shape of the liquid. The conical shape provides a sall end radius, typically in the range of 5 n [55] Focused Ion Bea as an iaging syste Focused ion bea is a point to point iaging syste, where the shape and the size of the ion source at the object plane is transferred to the iage plane as illustrated in Fig. 5.a. To transfer the iage of the source to the substrate, an electrostatic lens is used. In coparison to an optical lens that consists of a transparent aterial of refractive index greater than unity, the electrostatic lens consists of an electrostatic field generated by a set of electrodes and insulators separating the. It is very difficult to calculate the refractive index of the electrostatic lens as the geoetry of the electrostatic field depends on the potential distribution generated by the electrodes. Instead, equations of otion of ions passing through the lens can be written using the paraxial approxiation as follows [56]. φ & r qr 5. 0

140 qr φ & z qφ +, 5. 4 where q is ion charge, φ is potential, is ion ass, φ and φ are the first and second derivatives of the potential with respect to r. Equation shows the second-order derivative of the radial position, r, is linearly proportional to r. This eans ions located farther fro the axis accelerate faster towards that axis. Thus the focal point is independent of r, which iplies lens action. As in any iaging syste, resolution is very iportant. In the next part we will briefly discuss spot size liitations of the FIB syste Spot size liitation The focused bea size of any FIB syste depends on the presence of iaging aberrations and the Coulob effects in the bea. Unfortunately, electrostatic lenses suffer fro any aberrations known to optics plus ore due to the anisotropic properties of the lens. As an LMIS is essentially a point source, the principal aberrations of interest are spherical and chroatic aberrations. It was assued the current distribution was Gaussian at the source plane and that the source was well aligned on axis. Aberrations that depend on r can then be neglected. Given the source diaeter d g, the spot diaeter at the iage plane is

141 d d + d + d. 5.3 g c s In this equation d s and d c are the spherical and chroatic aberration disks diaeters. Equation 5.3 can be written in ters of the ion energy U, half width full odulation of the ion energy distribution U, aperture angle α, and spherical and chroatic aberration coefficients C so and C co, as follows [56]. 3 C U d d g + Ccoα + soα 5.4 U Equation 5.4 shows saller spot size at the iage plane can be achieved using saller aperture angle, α. Reducing aperture angle, however, eans reducing aperture size. This eventually reduces bea current at the saple/iage plane. Fro the above arguents, a tradeoff is required to achieve the desired spot size and current value at the saple according to each specific application of the FIB syste. Table 5. shows soe typical values of d co and d so for a LMIS focusing colun for different aperture angles with paraeters C so 0 3 c, C co 0 c, U 5 ev, U 0 4 ev, and d g 50n [56]. The above text discussed the FIB as a syste that iages the source to the substrate. Liitations of spot size or agnification of the source was also discussed. It is also iportant to highlight soe theory on the interaction between the ion bea and the solid

142 substrate. The following section discusses the basics of ion-solid interaction and its iportance for icro-fabrication. Table 5. Typical values for the spherical and chroatic aberrations disks diaeters for a LMIS focusing colun with the paraeters entioned in the above paragraph. α rad d s µ d c µ x x x0 -.4x x0-5.0x Interaction of ions with the substrate Ion-solid interaction is substantially different than electron-solid interaction due to 3

143 intrinsically large difference in ass of the ipinging particles. There are two basic classes of interaction between ions and solids. These are elastic and inelastic interactions. Elastic interactions cause defects through sputtering fro displaced atos in the lattice. Inelastic interactions produce other fors of energy such as secondary eitted electrons or x-rays. With elastic interactions, ipinging ions with a specific energy can sputter the aterial atos, on a very sall scale, and peranently change the surface cheistry of the substrate. By controlling the locations and depth of aterial to be reoved, extreely sall patterns can be directly illed in the substrate. The nuber of target atos sputtered per priary ion is a very iportant quantity called the sputtering yield. This quantity strongly depends on the substrate aterial and ion source. The total sputtered volue of the aterial per unit tie per unit incident bea current in na is called illing rate. For a structure of certain volue, the total illing tie can be estiated as total tie 3 Volue µ 3 illing rate µ s na Bea current na 5.5 Table 5.3 shows typical values of illing sputtering rate and total tie for three substrate types: Si, SiO, and GaAs considering Ga + LMIS with a potential of 30 KeV, bea current of 000 pa, and a total illed volue of µ 3 [57]. 4

144 Table 5. 3 Typical values of the sputtering rate and total illing tie for different substrates with the ion bea paraeters entioned above. Substrate Sputtering rate Milling tie sec Si SiO GaAs The inelastic process, specifically the production of secondary electrons, is very useful for constructing an iage of the illed structure. This can be accoplished by collecting the secondary eitted electrons while scanning across the saple. The inelastic process, specifically the production of secondary electrons, is very useful for constructing an iage of the illed structure. This can be accoplished by collecting the secondary eitted electrons while scanning across the saple FIB as a tool for icro-fabrication Figure 5.b shows a scheatic diagra of a typical FIB syste. The three ain coponents of the syste are the source object, the iaging optics, and the substrate 5

145 chaber iage plane. High voltage is applied to the eitter a voltage difference between the suppressor and extractor to extract ions out of the source typically 30 kv. The bea acceptance aperture BAA controls bea current in the colun, and therefore the aplitude of syste aberrations in Eq The first lens colliates the bea and the second lens serves as the iaging lens. The autoatic variable aperture AVA can be set for the desired current to the saple. The saple holder is placed on a oving stage that can be autoated. With the appropriate background, FIB for icro-fabrication can now be discussed. First, feature size liits bea current because the iaged spot size is proportional to acceptance aperture size as in Equation 5.4. Reducing bea current will reduce syste aberrations, but will require a longer illing tie as quantified in Equation 5.5. Given bea current and total illing volue, total illing tie can be estiated using Equation 5.5. Patterns can be defined in two forats: basic geoetric eleents or strea of points. For the first forat, patterns are described as a cobination of basic geoetric eleents. Each eleent is illed to a specific width such that the final pattern is achieved. To ill one eleent, the bea deflector controls the trajectory of the ion bea to fill the whole area of this pattern 6

146 Figure 5. 3: FIB syste as a tool for icro fabrication In the second forat, the pattern is defined as a strea of points that ust each be illed for a specific tie. In this forat, the ion bea is raster scanned over the allocated spots in the pattern. Once the pattern is generated, the location of the pattern to be illed is anually selected and aligned. This ethod is convenient for applications that do not require high precision. To autoate the illing process, a script can be written that contains a flow of coands to be executed until the final pattern is illed. Figure 5.3 shows a scheatic diagra of the FIB as a tool for icro fabrication. This section discussed the basics of icro-fabrication and iaging using a FIB syste. The following section discusses a illing process that allows arbitrarily shaped refractive icro-optical eleents to be sculpted. 7

147 Subtractive Milling Process Subtractive illing relies on quantization of the optical eleent surface into discrete heights. Each height can be represented by basic geoetrical shapes or a ore coplicated strea file. In this work, we present the fabrication of radially syetric diffractive optical eleents. For these structures, the basic geoetrical shapes are rings and circles. The technique ay be applied to structures that are not radially syetric. The depth profile of a spherical lens or irror of radius R is h R R W /, 5.6 where W is the diaeter of the icro-optic eleent. Figure 5.4a shows the quantization of a irror surface into N levels of equal spacing δh h/n. The radius of each level is calculated fro equation 5.. ρ i R R N i δ i, i0,,., N- 7 8

148 Figure 5. 4: a Quantization of the spherical irror b subtractive illing technique; N concentric circles are illed sequentially starting by the largest one results in a ultilevel spherical irror To fabricate a spherical irror, a script was written to calculate the radii of N concentric circles according to Eq As shown in Fig. 5.4b, each circle is illed to a depth of δh starting with the largest radius, ρ o W/. In the case of a positive focal length 9

149 icro-lens, the script generates N concentric rings of inner radii calculated fro Eq. 5.7 and outer radii that are greater than or equal to W/. The inner radii ust be less than W/. The fabrication procedure starts by illing the ring of the sallest inner radius, ρ N Rδh δh to a depth of δh, followed by the next saller ring, and so on until the irror is copletely illed into the substrate as shown in Fig. 5.4b Integrated Mirror Exaple As an exaple of the subtractive illing technique, a icro-optic eleent was integrated onto the surface of a silicon v-groove irror. One advantages of FIB illing is the ability to ill into tilted surfaces. The v-groove was fabricated by wet etching and was designed to passively align an optical fiber as shown in Fig. 5.5a. A silicon wafer was coated with a layer of SiN and patterned using a g-line stepper with 5X reduction. The patterned areas were dry etched in an RIE plasather for -4 inutes to reove the SiN. The saple was cheically etched in a KOH cheical bath at 90 o C for five hours. This resulted in 54.7 o tilted walls corresponding to the crystal lattice planes of the silicon wafer. Figure 5.5b shows the geoetry of the optical eleent that was fabricated. As illustrated in the figure, the distance between the fiber and the irror, r, defines the location of the iage point. Due to the finite size of the irror aperture, the axiu distance that the fiber can be placed in front of the irror is restricted to a few hundred 30

150 icrons. For a v-groove with wall angle θ, the location of the fiber center relative to the surface of the v-groove l t can be calculated fro the geoetry shown in Fig. 5.5a as follows l t r D tan θ 5.8 cos θ where D is the half width of the v-groove. For the v-groove to include the full cladding region of the fiber, l t ust be greater than r, the radius of the cladding. cos θ + D r 5.9 sin θ 3

151 Figure 5. 5: a Geoetry of the fiber inside the v-groove. b The location of the fiber relative to the integrated icro lens. For a specific v-groove of width D and wall tilt angle θ, the lens ust be fabricated so the center is located at a distance l t under the v-groove surface. For this work, W340 µ, D70 µ, and r6.5 µ. Using Eq. 5.4, the height of the center of the irror should be 7.65 µ. The irror diaeter liited the axiu distance the fiber can be oved, r, while the bea was still within the irror as shown in Fig. 5.5b. The axiu distance, r ax, was calculated using Gaussian bea propagation presented in Eq W W r z cos ax o θ z w o o π wo 5.0 λ where w o was the ode field radius of the single ode fiber. The location of the iniu bea waist, r, at the iage location P is defined as r R R r R + 5. R r + z o 3

152 where, R is the radius of curvature of the irror as shown in Fig. 5.5b. Figure 5. 6: SEM iage of the icro spherical irror fabricated on the wall of the v- groove. 33

153 CHAPTER SIX: EXPERIMENTAL VERIFICATION OF SELECTIVE COUPLING SCHEMES In this chapter we deonstrate the experiental analysis and characterization of the different coupling schees presented in chapter five. First, we present, in detail, the experiental results and characterization of a MMF condensing lens. A SMF is fusion spliced to a MMF of a specific length, and then we recorded the intensity distributions of the back reflected light for different wavelengths incident on a planar irror using a tunable laser diode and a CCD caera. The subsequent analysis showed good atching to the theoretical predictions ade. In the second section, we show the experiental analysis of the fabricated phase eleent for coupling to the LP ode. The results exhibited a great agreeent with the theoretical analysis. The third section deals with the hollow-glass-waveguide HGW characterization and analysis. We characterized the transission and bending losses in a eter long hollow glass waveguide and copared the easureents with the first order calculations presented in chapter five. Additionally, we easured the optical functionality of the fabricated polarization converter eleent as well as its influence on the bending loss in the hollow waveguide. In the last section we analyze the icro lens fabricated on the side wall of a fiber v-groove using the FIB syste. 34

154 6.. MMF as a Condensing Lens The tunable fiber optic lens described here is unique with respect to the relative siplicity in fabrication and the ability to significantly change the focal length. It consists of a ultiode fiber Thor Labs AFS05/5 fusion spliced onto a single ode fiber Corning SMF-8 attached to a tunable laser source via an 80/0 splitter, which allows for detection of any returning signal. Agilent 8635A InGaAs power sensor and 8640A tunable laser odules were used in conjunction with the 864A Lightwave easureent syste to ake the easureents presented in this paper. For the fiber paraeters cited in table 3., the ode associated with axiu coupling is ν p 4. This can be observed looking at Figure 4. which shows the relationship between the coupling efficiency and the ode nuber for MMF core radii of 5.5µ and 9.5µ respectively. Using these paraeters at a wavelength of.55µ results in L p Fro equation 4., the re-iaging location is found to be z 3 L Using these values, Figure 6. is constructed in order reiaging p to represent the intensity distribution along the axis of the ultiode fiber with the length in the z-direction noralized to L p. As entioned in the theory section, the ost significant feature is the presence of very distinguished axia at locations such as 3L p, and 0L p. The length of the MMF spliced onto the SMF can be chosen with respect to these distinguished axia observed inside the MMF. 35

155 Figure 6. : Power, noralized to the input power, across an area equals to the SMF core along the MMF By cleaving the ultiode fiber at a length slightly less than the length specified at any one of these locations, the light exiting the fiber converges to an on-axis location in the air outside the fiber as depicted in Figure 6.. In this case, the fiber end facet is located at a longitudinal location that corresponds to a length slightly less than that defined for the 3 case in equation 4.9. If a irror is placed at this sae plane, the coupling of the light reflected back through the fiber would be a axiu copared to any other location in the near vicinity which indicates the location of the focal plane. Thus, the shift in the focal plane can be experientally easured by detecting the power reflected fro a irror placed in front of the end facet of the MMF while sweeping the wavelength. 36

156 Figure 6. : Calculation of intensity inside ulti-ode fiber λ.55 µ using equations 5 and 5 for the field inside and outside the MMF. The MMF is polished slightly less than 0 corresponds to 3L p. [SMF of 4.5 µ core radius and MMF core radius of 5.5 µ.] To verify the validity of choosing the a length of MMF less than the 3 case, a scalar cylindrically syetric FD-BPM siulation was perfored for a single ode fiber and step index ultiode fibers using the paraeters depicted in table 6.. Figure 5 shows specifically the field convergence at longitudinal on axis locations after exiting the end facet of the MMF. In order to accurately copare experiental and theoretical results, the length of the MMF fusion spliced onto the SMF ust be accurately known. The process used to fabricate this device inherently provides this inforation and a procedure to repeatedly 37

157 cleave the sae length of MMF onto the SMF each tie a device is fabricated. Using the scale on the fiber cleaver, a MMF splice length of was produced to be fusion spliced to the SMF. Once this was done, a cleave is ade fro the end facet of the MMF, leaving a length less than 0 long MMF spliced to the SMF. This length is conveniently slightly less than the 3 case using the forulation of equation 4.. The actual length of the MMF can be ore closely deterined using a icroscope equipped with a CCD caera. Once the fusion splice interface is found, the calibrated caera software to observe icroscope iages can be used to provide the distance fro this interface to the end facet of the MMF. The bare fiber device was then appropriately connectorized into an FC ferrule with a stereo icroscope used to verify that the MMF only barely protruded through the end facet of the FC ferrule. A 0.5 µ polishing disk and colloidal slurry were then used to hand polish the end facet of the MMF until the surface was of optical quality as depicted in Figure 6.3. The single ode fiber and step index ultiode fiber used have paraeters defined in table 6.. As done in [6.], sweeping the wavelength of the source into the SMF we observed the reflected power that results for irror placeent at specific longitudinal displaceents fro the end facet of the MMF. Figure 6.4 depicts the experiental setup used for these easureents. 38

158 Figure 6. 3: Device connectorized in FC connector and 0x icroscope iage of end facet of MMF quality Figure 6. 4: the syste setup showing the SMF fused splice to MMF. The ost notable feature for each wavefor associated with a specific longitudinal displaceent is the drop in output power around 5 db seen at a 39

159 particular wavelength over the µ wavelength range. This drop corresponds to a plane located behind the focal plane at this wavelength as depicted in Figure 6.5. The shift over this wavelength range follows a linear relation that can be expressed as [ µ ] 634. z out λ 4 µ 6., exp + This drop in power can be explained by carefully observing the intensity distribution in Figure 4., b specifically at the locations z 3L p and 0L p. Both locations correspond to local axia along the MMF axis. For the first location, the light destructively interferes iediately after it condenses, while in the second location the destructive interference occurs before the light convergences. Link Loss vs. Wavelength Link Loss db Wavelength n Start 40u 80u 0u 60u 00u Figure 6. 5: The experientally easured back-coupled power versus the wavelength. 40

160 As the MMF is cleaved to a length slightly shorter than 3L p the drop in power is the ost notable feature. Figure 6. shows the siulated field distribution in a region of behind the MMF end facet using the odes expansion expressions in 4.3 and To copare these experiental results with the first order approxiation the expression in equation 4.3 is used to calculate the location of the focal plane over this particular wavelength range. [ µ ] z out 408 λ µ 6. Equations 6. and 6. deonstrate a correlation between the experiental easureents and the predicted linear relation in equation On the other hand, there is an error of 7% in the slope calculated using our first order approxiation. This error is predictable due to the approxiations carried while driving the expression presented by equation In addition, a cylindrical syetric FD-BPM is used to siulate the experiental setup. 4

161 Figure 6. 6: FD-BPM, experiental easureents, and first order results for focal point location as a function of wavelength The power that couples back to the SMF is calculated for four different wavelengths. Figure 6.6 depicts the results obtained using the FD-BPM in addition to the experiental and the first order approxiation results. The figure shows alost a constant shift of 00µ between the first order approxiation and both the experiental and the FD-BPM results. This shift is due to the presence of error of about 3% in calculating the longitudinal propagation constant using the asyptotic assuption in 4

162 coparison to the exact values. This results in inaccuracy of the value of L p and thus the calculated z out. In addition, by rounding of the value of ν p to the nearest integer, an additional error in calculating L p is introduced. Thus, the first order approxiations are seen to be able to predict the location where the light condenses outside the MMF as a function of wavelength with an acceptable error of 7%. This error is not as significant when estiating the change of the spot size at these planes over the operating range of wavelength using st order approxiations in coparison to coplete FD-BPM siulations. The ode series expansion expression presented in equation 4.35 is used to calculate the intensity distribution outside the MMF for four wavelengths:.44,.48,.5 and.56 µ as depicted in Figure

163 Figure 6. 7: Siulated field distribution out of the MMF facet for four different wavelengths deonstrating wavelength dependence of focal position. The HWHM spot sizes at the focal planes are depicted in Figure 6.8 together with the experientally easured data and the results fro FD-BPM siulations. Looking at the curves in Figure 6.8, it is apparent that the spot size slightly decreases as the wavelength is increased. An interesting property of this device is the size of the bea at the focus location associated with a particular wavelength. To enhance the idea of the 44

164 actual size of a bea at a focus location, the experientally easured bea is depicted in Figure 6.9, overlaid on the actual size of the MMF end facet. Figure 6. 8: Spot size HWHM at focal location as a function of wavelength for BPM, first order solutions, and experiental results. 45

165 Figure 6. 9: Depiction of exiting bea at focus location overlaid on actual size of MMF end facet. 6.. Excitation of LP and LP in Large Core Fiber Using Single Phase Eleent In our experiental setup, a HeNe laser source is coupled to a single ode fiber and the output is then colliated using a 0x objective lens. The phase odulated light is then 46

166 coupled into a larger core fiber which sustains a few odes using a 0x objective lens with effective focal length of 9. The fiber used is Corning SMF 8 which is single ode at 550 n, and supports four odes at a wavelength of 63.8 n. Without any phase eleents, we axiized the light coupled to the LP 0 to guarantee iniu tilt and shift in the input bea. The phase eleents shown in Figure 5., a and b were fabricated in Shiply PR805 photo resist using a stepper syste. The refractive index of the photoresist is.6406 at the working wavelength. To achieve a π phase shift the step height was set to 494 n. For the eleent depicted in Figure 5., a, the resulting far field intensity distribution at the output of the fiber is depicted in Figure 6.9, a. The figure shows that the light is selectively coupled to the LP ode. Measuring the output power, 8.4% of the power at the input facet of the large core fiber is coupled to the LP ode. This value is very close to the calculated coupling efficiency of 83.7% using equation Nevertheless, only 60% coupling efficiency was achieved when using the second phase eleent to selectively excite the LP ode. The far field distribution after the large core fiber is presented in Figure 6.0, b. However, in both cases the cross talk is iniized when properly aligning the input light and the phase eleent to be on axis with the few odes fiber. 47

167 Figure 6. 0: Observed far field intensity distributions of the light at the out of the fiber using the a first phase eleent and b the second phase eleent. 48

168 6.3. Nuerical analysis of the selective excitation of the highest order superode in seven core fiber In this section, we calculate the effect of the tilt and shift of the input bea on the coupling efficiency to the highest order superode. Figure 6. depicts the change of the coupling efficiency when tilting and shifting the input bea. One notices that the efficiency is ore sensitive to tilting than shifting. Thus, a very precise alignent is required to achieve high coupling efficiency. Figure 6. : Coupling efficiency to the last superode in seven core fiber against the tilt 49

169 and shift in the input bea Selective Coupling to TE 0 ode in HGW In chapters four and five, we have illustrated the design and fabrication of polarization converter eleent to selective couple to the TE 0 ode in order to iniize the bending loss in hollow waveguide. Here, we first characterize the transittance and bending losses in glass hollow waveguide, and then study the optical behavior of the fabricated polarization converter eleent and its effect on the transission and bending losses in the fiber Measureent of the Transission and Bending Loss in HGW Figure 6. depicts an SEM iage of the HGW we used in our experient. In any HGW, the thickness and refractive index of the dielectric and etal layers essentially affect the bending and transission losses of this waveguide. As can be seen in Figure 6., the dielectric layer AgI has is. µ thick, while the thickness of the etal Silver layer is about 65 n. At a wavelength of.55 µ, the refractive indexes of the AgI and silver layers are.38 and.9+i6.9 respectively. To easure the transittance and bending losses we used a 9.5 c long piece of this HGW as depicted in Figure

170 Figure 6. : SEM iages of the HGW showing the dielectric AgI and the etal Ag layers. 5

171 Figure 6. 3: Experiental setup for easuring transission and bending losses in HGW. In this setup, we used a tunable laser diode source with a wavelength range between.540 and.565 µ. The source is pigtailed to a SMF at.55 µ with a colliating lens at the output of this fiber. A linear polarizer is placed in front of the colliator in order to fix the polarization of the input light to be perpendicular to the plane of incidence. The oving irror slides across the optical path and reflects the light towards an IR CCD caera where a far field iage is fored. These irror and caera are iportant for rough alignent as well as for capturing iages of the far field intensity when using the optical eleent. When the irror is not crossing the optical path, a focusing lens 0 c focal length is used to axiize the coupling to the HGW. First, we aligned the light out of the colliator to the center of the HGW when it is straight no bending through 5

172 three diensional stage in addition to tilt and rotation stages. Alignent is achieved by axiizing power read by the detector. The aount of power easured directly out of the colliator is about.6 W. However, the easured transitted power in the fiber, when it is straight, is.4 W. Hence, the transittance of this HGW is and the power attenuation coefficient is about Figure 6. 4: The easured bending loss in HGW circles represent three different easureent and the dashed line is the average and the calculated bending loss using equation

173 Using equation 3.60 and the HGW paraeters defined in Figure 3.8, the calculated power attenuation coefficient is which is close to the easured one. Bending the fiber with different radius of curvatures, we recorded the readings of the detector behind the fiber as shown in the setup depicted by Figure 6.3. Figure 6.4 shows the easured bending loss in the HGW at.55 µ as well as the calculated bending loss using equation Characterization of the Fabricated Polarization Conversion Eleent In chapters four and five, we deonstrated the design and fabrication of a subwavelength structure phase eleent that converts the linearly polarized input light into a rotating one for selectively excitation of the TE 0 ode inside HGW. In this section, we are going to test the optical perforance of the eleent fabricated in GaAs substrate. Additionally, we will show that by placing this eleent in the path of the input light, the transittance of the HGW iproves considerably. As presented in chapter five, the polarization converter eleent is fabricated in a GaAs substrate which has a high refractive index n GaAs at λ.55 µ. Hence, the axiu transittance through the substrate itself is calculated through Fresnel reflection as follows R T,

174 where n R + n GaAs GaAs Thus, the axiu transittance is However, the ultiple reflections fro the two sides of the wafer Fabry Perot cause the change of the transittance with the wavelength as depicted in Figure 6.5. As can be seen in the figure, the axiu transittance is about which is very close to the calculated one. Figure 6. 5: Transittance through the GaAs wafer as a function of wavelength. 55

175 To characterize the perforance of this eleent, we placed it in the setup depicted in Figure 6.6 below. Figure 6. 6: Experiental setup for testing the perforance of the polarization converter eleent. In this setup, the Source is pigtailed to a SMF at.55 µ with a colliating lens at the end of this fiber. The first polarizer fixes the polarization of the light to be parallel to the table. However, the second polarizer is used to analyze the polarization distribution of the light out of the optical eleent. This can be done by rotating the polarizer and recording the intensity distributions for different rotation angles using the iaging lens and the IR CCD caera. Using this setup, we first recorded the far field pattern without the second polarizer as depicted in Figure 6.7. As can be seen, the intensity distribution is a donut shape which atches the predicted one for the TE 0 ode. 56

176 Figure 6. 7: The far field intensity distribution of the light out of the polarization converter eleent. 57

177 Figure 6. 8: Far field intensity distributions for different polarization angles for the light out of the polarization converter eleent. 58

178 As can be seen in Figure 6.8, when rotating the second polarizer the recorded intensity pattern is, basically, a section of the field out of the polarization converter eleent. This section rotates correspondingly with the polarized. That indicates that the polarization distribution of the output field is siilar to that in Figure 4.3, b. Figure 6. 9: Measured polarization distribution of the output field. The polarization distribution of the output field can be constructed fro the recorder 59