先進光纖光柵之製作與特性量測

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(1)國立交通大學光電工程研究所博士論文. Fabrication and Characterization of Advanced Fiber Bragg Gratings 先進光纖光柵之製作與特性量測. 研究生：莊凱評指導教授：賴暎杰博士. 中華民國九十三年六月.

(2) Fabrication and Characterization of Advanced Fiber Bragg Gratings 先進光纖光柵之製作與特性量測. 研究生：莊凱評指導教授：賴暎杰博士. Student : Kai-Ping Chuang Advisor : Dr. Yinchieh Lai. 國立交通大學電機資訊學院光電工程研究所博士論文 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in The Institute of Electro-Optical Engineering National Chiao-Tung University Hsin-Chu, Taiwan, R.O.C.. 中華民國九十三年六月.

(3) 先進光纖光柵之製作與特性量測. 研究生：莊凱評. 指導教授：賴暎杰博士. 國立交通大學電機資訊學院光電工程研究所. 摘要. 隨著光纖光柵在光纖通訊及光纖感測領域被發現有越來越多的應用，致力於改善光纖光柵的製作與特性量測對於製作具有特殊光學特性的濾波元件而言就顯得非常重要。在本論文中，我們利用逆散射之剝層（layer-peeling）法來設計先進的光纖光柵結構，並根據實驗的實際製作架構，使用最小方差逼近（least-square fitting）法來找出最佳的光纖光柵曝照參數。在製作具有複雜結構之先進光纖光柵方面，我們提出了三個新的製作光纖光柵的方法。第一個方法我們稱之為偏振控制雙光束干涉法，第二個方法稱之為偏振控制相位光罩法，這兩個製作方法目的都是利用曝照光束偏振的特性，在製作光纖光柵逐段曝照過程中既可以任意控制光柵折射率（ac-index）的大小，同時又可以保持光柵不同位置的平均折射率（dc-index）為定值。藉著利用這些方法，具有低損耗及良好頻譜響應的濾波元件可以順利的被完成。第三個方法是干涉式側向繞射光柵位移監控的技術，我們提出這個光學監控方法來製作具有較長長度的光纖光柵元件。我們成功完成一些實驗的例子來證明這些方法的可行性，也利用 LabView 的自動控制 I.

(4) 軟體來建立自動化的光柵曝照系統，藉此可以提升光纖光柵製作過程的準確性及重複性。在光纖光柵特性量測方面，我們發展光纖式麥克森干涉儀（ the balanced Michelson interferometer）的方法及側向繞射（side-diffraction）量測技術來分析光纖光柵完整的特性，包括光柵的複數反射係數及複數耦合係數等，這些係數代表著光纖光柵所有的光學和結構參數。當我們再結合逆散射之剝層法反推時，光纖光柵不同位置的平均折射率變化也可以求得。對未來先進光纖光柵製程的改善工作，這些特性的分析方法希望能夠有所幫助。. II.

(5) Fabrication and Characterization of Advanced Fiber Bragg Gratings Student: Kai-Ping Chuang. Advisor: Dr. Yinchieh Lai. Institute of Electro-Optical Engineering College of Electrical Engineering and Computer Science National Chiao-Tung University. ABSTRACT. When the FBG devices begin to find a lot of applications in fiber communication and fiber sensing, it also becomes more important to further improve the FBG fabrication and characterization techniques for achieving more complex optical filter properties. In this thesis, in order to design the advanced fiber grating structures, the layer peeling inverse design method combined with the least square fitting method is developed. Based on these synthesis methods, the best experimental parameters for our sequential writing setup can be found. For the fabrication of advanced fiber grating structures, we have also proposed three new exposure methods. One is the two-beam interferometer method with the polarization control, and another is the phase mask method with the polarization control. Both of them have the same purpose for achieving a controllable ac-index profile in a single scan, with the dc-index profile being kept constant during the scan. The spectral shapes with a steep edge, very low sidelobes, a flat top, and very little ripples can be achieved by the use of these methods. The third method is the interferometric side-diffraction position monitoring technique for writing long fiber Bragg gratings. Some examples are presented to demonstrate the feasibility of these. III.

(6) methods. The automation of the exposure system has also been setup for enhancing the accuracy and repetibility of the fabrication process. In order to determine the complete characteristics of the fiber gratings, we develop the balanced Michelson interferometer method and the side-diffraction technique. The characteristics to be determined include the measurement of the complex reflection coefficient and the complex coupling coefficient of the grating. When the discrete layer peeling method is combined with these two measurement methods, we can get all of the information about the grating structure including the dc-index change. In addition, this analysis process may also be helpful for improving the fabrication processes of advanced FBGs.. IV.

(7) Acknowledgement 在這四年的攻讀博士生涯裏，實驗設備的從無到有以及到論文期刊的發表，指導教授賴暎杰博士的悉心教導與支持給了我在研究過程中最大的幫助。記得在博二到博三的這段求學期間，因為新的實驗架構遲遲無法完成，以致於無法順利的發表文章，在心裡有點急的情況下求助於老師，記得當時老師的一句話“只要一天比一天實驗有所進展，就很好了。“，也是因為抱持著這樣的信念，能讓自己一步一步的完成博士學業。在此，由衷的感謝賴教授的細心教導。光纖光柵實驗室的建立過程中，也多虧了所辦湯敏雄先生及實驗室學弟妹的幫忙，尤其是學弟妹孟璋、易霖、慧萍、夙鴻及淑慧在實驗上的幫助，實驗室才能夠建立的如此完善，在此一併致謝。此外，要特別感謝澄鈴學姊、許立根博士、瑞光、維巍及桂珠同學，除了在學業的討論上給了很大的幫助外，也給了我博士求學生涯中多采多姿的生活。家，可以說是我的最後精神堡壘，非常感激我的父母和弟弟這四年來的全力支持，讓我無後顧之憂，得以專心的取得博士學位。女朋友虹雯的鼓勵與陪伴，使我這一路走來無怨無悔，伴我渡過最困難的時期。在此，將此論文獻給所有關心我的人。. V.

(8) Contents Chinese Abstract………………………………………………………Ⅰ English Abstract………………………………………………………Ⅲ Acknowledgement…………………………………………………….Ⅴ Contents……………………………………………………………….Ⅵ List of Figures…………………………………………………………Ⅷ List of Acronyms……………………………………………………..ⅩⅣ List of Symbols……………………………………………………….ⅩⅤ. Chapter 1. Introduction 1.1 Brief Review of Fiber Bragg Gratings……………………1 1.2 Motivation and Approach of the Research………………..3 1.3 Organization of the Dissertation…………………………..5. Chapter 2. Synthesis of Fiber Bragg Gratings 2.1 Introduction……………………………………………..12 2.2 Fundamentals of Fiber Bragg Gratings………………....12 2.3 Different Types of Fiber Bragg Gratings……………….14 2.3.1. Uniform Fiber Bragg Gratings……………………………15. 2.3.2. Phase-shifted Fiber Bragg Gratings……………………….15. 2.3.3. Apodized Fiber Bragg Gratings…………………………...16. 2.3.4. Chirped Fiber Bragg Gratings……………………………..16. 2.4 Coupled-Mode Theory………………………….……….17 2.5 Discrete Layer-Peeling Method………………………....18 2.6 Least Square Method…………………………………….21. VI.

(9) 2.7 Summary…………………………………………….….22. Chapter 3. Characterization of Fiber Bragg Gratings 3.1 Introduction……………………………………………...34 3.2 The Balanced Michelson Interferometer Method……….34 3.3 The Side-Diffraction Technique…………………………36 3.4 Characterization of Fiber Gratings………………………39 3.5 Summary………………………………………………...43. Chapter 4. Fabrication of Advanced Fiber Bragg Gratings 4.1 Introduction……………………………………………...67 4.2 Setup of the Exposure System…………………………..68 4.3 Two-Beam Interferometer Method with Polarization Control……………………………………………………70 4.4 Phase Mask Method with Polarization Control…………75 4.5 Interferometric Side-Diffraction Position Monitoring Technique for Writing Long Fiber Bragg Gratings………80 4.6 Summary………………………………………………...85. Chapter 5. Conclusions and Future Work 5.1 Conclusions…………………………………………….124 5.2 Future Work………………………………………….....126. List of Publications……………………………………..…………127. VII.

(10) List of Figures Fig.2.1. The refractive-index modulation along the length of waveguides ………………………………………………………….……26. Fig.2.2. The common types of fiber Bragg gratings; (a) uniform FBGs, (b) phase-shift FBGs, (c, d) apodized FBGs, and (e) chirped FBGs..…..……………………………………………………27. Fig.2.3 (a) The spectra of three Bragg gratings with different coupling constants of 2, 4, and 8.; (b) time delay of the same gratings as in Fig. 2.3(a)…………………………………………………28 Fig.2.4. A π-phase shift uniform fiber Bragg grating. (the parameters are kL=1.6 and L=5 mm)……..……………………………..29. Fig.2.5 (a) A Gaussian apodized FBG with non-constant dc index; (b) a pure apodization FBG with a Gaussian index profile……….30 Fig.2.6. A chirped FBG with the grating length = 30 mm, chirped rate = 0.1667 nm/cm, and a tanh index profile. (a) The reflection and transmission spectra; (b) the phase and time delay………….31. Fig.2.7. The description of the inverse and the forward methods for the synthesis and analysis of fiber gratings……………………...32. Fig.2.8. A discretized model of the LP method……………………….33. Fig.3.1. Fabry-Perot cavity consisting of a fiber Bragg grating and a reference reflector……………………………………………47. Fig.3.2. Experimental setup of the balanced Michelson interferometer.48. Fig.3.3. The interference Iinter(λ) of combined signals……………….49. Fig.3.4. The inverse Fourier transform of Iinter.(λ) …..…………….…49. VIII.

(11) Fig.3.5. The algorithm for finding the phase spectrum of the fiber grating………………………………………………………..50. Fig.3.6 (a)The measured interference pattern Iinter(λ), (b)The calculated phase spectrumψ(δ). (cited from [12])…………………….51 Fig.3.7 (a) Measured results using a commercial equipment “Advantest Q7760”, and (b) our measuring method. (cited from [12])….52 Fig.3.8. The side-diffraction setup for measuring index modulation: M, reflection mirror; SL, spherical lens………………………….53. Fig.3.9. The side-diffraction interference method for measuring grating period change: BS, beam splitter; M, reflection mirror; SL, spherical lens………………………………………………...54. Fig.3.10 The improved side-diffraction interference method for measuring the grating period variation and the ac-index modulation…………………………………………………..55 Fig.3.11. Flow chart of the characterization method of fiber gratings.56. Fig.3.12 The balanced Michelson interferometer setup used to measure the highly reflecting fiber gratings………………………….57 Fig.3.13 The reflection spectrum and interference pattern by the use of the balanced Michelson interferometer method…………….58 Fig.3.14 The calculated phase information of the grating……………..59 Fig.3.15 (a) The Ac-index modulation from the DLP is compared with the side-diffraction method. (b) The Ac-index modulation from the DLP is compared with the side-diffraction interference method……………………………………………………….60 Fig.3.16 The spatial grating phase is calculated from DLP method……61 IX.

(12) Fig.3.17 The reflection spectra of the target and the calculated result from DLP method………………………………………………….62 Fig.3.18 The group time delays of the target and the calculated result from DLP method……………………………………………63 Fig.3.19 The CCD interference period corresponds to the relative grating period…………………………………………………………64 Fig.3.20 (a) dc-index amplitude ∆ndc(z) = 0…………………………….65 Fig.3.20 (b) dc-index amplitude ∆ndc(z) = ac-index amplitude ∆nac(z) * 0.5…………………………………………………………….65 Fig.3.20 (c) dc-index amplitude ∆ndc(z) = ac-index amplitude ∆nac(z) * 1 ………………………………………………………………..66 Fig.3.20 (d) dc-index amplitude ∆ndc(z) = ac-index amplitude ∆nac(z) * 1.3…………………………………………………………….66 Fig.4.1. An auto-controlled FBG exposure system based on the two exposure methods of a phase mask and a two-beam interferometer………………………………………………...89. Fig.4.2. The way to flatten the dc-index of the fiber grating………….90. Fig.4.3. The reflection and transmission spectra of the four channel FBG filter…………………………………………………………..91. Fig.4.4. Narrow-band filter by 100 sequences of step writing………...92. Fig.4.5 (a) The reflection and transmission spectra, and (b) the phase and time delay of the simulated chirped FBG………………..93 Fig.4.6. The experimental result of a 10cm long chirped FBG………..94. Fig.4.7. The Labview program for fabricating FBG…………………..95. Fig.4.8. Schematic diagram of fiber Bragg grating fabrication system by. X.

(13) using two-beam interferometer method with polarization control………………………………………………………..96 Fig.4.9. The refractive index profile at each sequential position of fiber ……………………………………………………………….97. Fig.4.10 (a) The relation of the half-wave plate angle and the refractive index modulation amplitude along the FBG position and (b) experimental and theoretical results for the refractive index modulation versus the rotation angle of the half-wave plate…98 Fig.4.11 The non-linear effect of refractive index change depends on a power law…………………………………………………….99 Fig.4.12 (a) Reflection spectra of the cos2 pure apodized (experiment) and ordinarily apodized (simulation) FBGs and (b) reflection and transmission spectra of a π-phase-shifted FBG…………….100 Fig.4.13 The reflection spectrum of pure gaussion-apodization FBG influences from the UV-induced birefringence……………..101 Fig.4.14 The Labview program is used to control the two-beam interferometer method with the polarization control……….102 Fig.4.15 Experimental setup for writing complex fiber grating structures: s and f, slow and fast axes of half-wave plate………………103 Fig.4.16 A constant dc-index structure and its reflection spectrum……104 Fig.4.17 A non-constant dc-index structure and its reflection spectrum105 Fig.4.18 A non-constant dc-index structure and its reflection spectrum106 Fig.4.19 The way of achieving a constant dc refractive index change...107 Fig.4.20 The reflection spectra of a pure apodized FBG and an ordinary apodized FBG with non-constant dc index………………….108. XI.

(14) Fig.4.21 Plots of the coupling coefficient reconstructed from a dispersionless reflection spectrum…………………………..109 Fig.4.22 Target and calculated reflection and group delay spectra from DLP…………………………………………………………110 Fig.4.23 The normalized refractive-index profile and phase of the designed dispersionless FBG……………………………….111 Fig.4.24 The spectral response calculated from DLP before and after using the least square fitting………………………………..112 Fig.4.25 (a) The measured reflection and time delay spectra of the dispersionless FBG and (b) the simulated reflection and time delay spectra of the “standard” Gaussian-apodized FBG…..113 Fig.4.26 The PMD and CD measurement of the dispersionless FBG…114 Fig.4.27 The calculated birefringence from PMD and CD by equation (4.3)………………………………………………………....115 Fig.4.28 The Labview program is used to fabricate FBG by use of the phase mask method with the polarization control…………..116 Fig.4.29 Traditional position monitoring method……………………...117 Fig.4.30 Experimental setup for fabricating and monitoring fiber grating: SL, spherical lens; M, reflecting mirror; BS, beam splitter; PBS, polarization beam splitter; HWP, half-wave plate; BC, beam combiner; θ1, the angle of the FBG writing beams; θ2, the input angle of the probe beam; θ3, the interfering angle of the probe and the reference beams……………………………...118 Fig.4.31 The Labview program of side-diffraction monitoring method for fabricating long FBG……………………………………….119 XII.

(15) Fig.4.32 Algorithm for calculating the phase of interference pattern….120 Fig.4.33 A typical experimental interference pattern and the calculated phase distribution……………………………………………121 Fig.4.34 The overlap-step-scan exposure method……………………..122 Fig.4.35 The reflection and transmission spectra of a 75-mm long gaussion apodized FBG with a constant dc refractive index along the whole grating……………………………………..123. XIII.

(16) List of Acronyms Acronyms. Descriptions. FBG. Fiber Bragg Grating. GLM. Gel’fand-Levitan-Marchenko. LP. Layer Peeling. GA. Genetic Algorithms. EA. Evolutionary Algorithms. DLP. Discrete Layer Peeling. LS. Least Square. DFB. Distributed Feedback. FP. Fabry-Perot. OSA. Optical Spectrum Analyzer. PZT. Piezoelectric Translator. XIV.

(17) List of Symbols Symbols. Descriptions. λB. Bragg wavelength. neff. effective modal index. Λ. grating period. δ. wavenumber detuning. β. propagation constant. θ (z ). spatial grating phase. φ (λ ). spectrum phase. ∆n(z ). refractive index change. ∆nac ( z ). ac refractive-index change. ∆ndc ( z ). dc refractive-index change. ∆Λ. grating period change. n ( x, y ). the index change of the unperturbed fiber. n ( x, y , z ). the index change of the perturbed fiber. u( z). the envelope of the forward field. v( z ). the envelope of the backward field. r. reflection coefficient. t. transmission coefficient. ρj. complex reflection coefficient of the j-th section. q( z ). complex coupling coefficient. τg. group delay time. k. grating vector. h. impulse response XV.

(18) ∆. grating section distance. Am(z). the refractive index envelope of the m-th small gaussion beam. ws. the width of the gaussion beam. Pd. the first-order diffraction power]. XVI.

(19) Chapter 1. Introduction. 1.1 Brief Review of Fiber Bragg Gratings. A fiber Bragg grating (FBG) is a section of the optical fiber in which the refractive index of the core is perturbed to form a periodic index modulation profile. When the reflection from a period of the grating is in phase with that from the next period, maximum mode coupling or reflection occurs and the Bragg condition is fulfilled:. λ B = 2neff Λ. (1.1). Here λ B is the Bragg wavelength, neff is the effective modal index and Λ is the perturbation period. The formation of permanent gratings by the photosensitivity in an optical fiber was first demonstrated by Hill et al. [1] in 1978. Here photosensitivity means that the exposure of UV lights will lead to a rise in the refractive index of certain doped glasses. Fiber gratings are usually fabricated by a variant of the transverse holographic method first proposed by Meltz et al. [2] in 1989. Afterward, the phase mask technique [3-5] has been widely used recently, which has the advantages of less-stringent requirements on the UV source, simplifying the manufacturing process and yet yielding fiber gratings with high performance. In order to write long, complex fiber gratings with advanced characteristics, the schemes of sequential writing have been proposed [6-8]. Because a fiber grating can be designed to have an almost arbitrary and complicated reflection response, it has a variety of applications as are well described by Hill and Meltz [9]. For telecommunication, the two most promising applications have been the wavelength selective devices [10-12] and the dispersion compensation devices [13-14]. FBGs have also. 1.

(20) become popular as sensing devices in applications ranging from structural monitoring to chemical sensing [15-17]. Another noteworthy application of fiber gratings is to be used as the narrowband reflectors for fiber lasers [18-20]. In order to adopt fiber gratings for various applications, it is important to have tools for synthesis, fabrication and characterization of special fiber grating devices. The most common mathematical model that describes wave propagation in fiber gratings is the coupled-mode theory [21]. The synthesis problem of gratings amounts to finding the grating structure (grating amplitude and phase or equivalently the coupling constant) from a pre-specified, complex reflection or transmission spectrum. These popular inverse algorithms include Gel’fand-Levitan-Marchenko (GLM) inverse-scattering method [22-23], Layer-Peeling (LP) inverse-scattering algorithms [24-26], Genetic Algorithm (GA) method [27] and Evolutionary Algorithms (EA) [28-29]. The increasing complexity and more-demanding specifications of fiber gratings for these applications require increasingly precise measurement of the grating parameters. Characterization of FBGs means to determine the complex reflection coefficient or the complex coupling coefficient of the grating. Indirect methods for characterizing fiber-grating profiles based on the measured reflection and transmission spectra can be achieved by the inversion techniques [22-29]. Although in principle such techniques are very powerful, their inherent limitations (due to noise, for example) as well as the demonstration of their practical application have yet to be shown. In addition, such methods are not necessarily efficient in determining the precise nature and the location of the defects in the written grating structure. The direct methods are usually based on interferometry [30-33], side-diffraction [34-36] or heat-scan method [37]. In Ref.31, a simple interferometric approach was used to characterize the weak fiber gratings. The group delay of the grating (or grating phase) can be obtained from measurement of the interference fringes from the grating and a reference reflector. By. 2.

(21) using the side-diffraction or heat-scan technique, the grating index amplitude and period change can be obtained. One can also combine the inverse algorithms methods with the direct characterization methods to find the complex coupling coefficient of the grating [38]. With these powerful tools, the design, fabrication and characterization of advanced fiber gratings can be achieved for various practical applications.. 1.2 Motivation and Approach of the Research. Advanced fiber Bragg gratings with special structures have become popular for various practical applications in modern fiber communication systems. Design, fabrication and characterization techniques of fiber gratings must be developed to meet the stringent requirements of fiber communication. In this work, for the design of advanced fiber gratings, the problem will be treated by the layer-peeling inverse synthesis method. The method is fast and accurate, and amounts to finding the grating structure (grating amplitude and phase) from a specified, complex spectrum. In this respect we will refer to the results achieved by Skaar et. al. [26]. He simplified the discrete layer-peeling (DLP) synthesis method proposed by Feced et. al. [24] to improve its clarity and efficiency. In order to meet the requirement of practical fabrication, the Least Square (LS) method [39] has been utilized to find the best experimental parameters for our sequential writing setup. In particular, the FBGs can act as the channel multiplexer-demultiplexers or the dispersion compensators in dense wavelength division multiplexing (DWDM) systems, in which the requirements for the optical filter properties are very stringent. The spectral shape of these optical filters needs to have a steep edge, very low sidelobes, and a flat top with very little ripples. For FBG filters to meet these requirements, it is important to apodize the FBG structure such that its dc-index change remains a constant across the whole grating (pure. 3.

(22) apodization). For more advanced FBGs like dispersionless FBGs, multiple π-phase shifts are also required. After we find the structure parameters of these fiber gratings from the DLP technique, the practical fabrication method for implementing these structures will become the most important issue. In the literature, several fabrication methods for these so-called pure apodized and/or phase-shifted FBGs have been developed. The double-UV exposure methods [40-42] need a two-stage exposure process and the variable-diffraction-efficiency phase mask method [4] needs a special designed mask. The phase mask dithering method [5] adopts mask dithering to achieve a constant dc refractive-index change during a single scan, but it will more easily induce extra vibration for the interferometric control of the relative position between the fiber and the phase mask. Recently it was shown that the apodized FBGs with constant average index and phase shifts can be fabricated by using a phase mask polarization control method [6]. However, in this method the FBG length is limited by the size of the phase mask and the precision adjustment of the fiber-to-phase-mask distance and the polarizer alignment are needed. In order to practically achieve the purposes of pure apodization index profiles, multiple π-phase shifts and long fiber grating length, in this work we propose and demonstrate three new methods which are capable of fabricating FBGs with these complex structures. The first method is based on the two-beam interferometer technique with the polarization control on one of the interfering beams. The second method is also based on the polarization control of the exposure UV beam, and can be combined with the sequential writing setup based on either the two-beam interference or the phase-mask approach to produce pure apodized complicated FBGs in a single scan. The third method is based on the side-diffraction interference technique, which is capable of monitoring the fiber grating position without the accumulative error due to the drift of the interferometer and inaccurate grating period. The increasing complexity and more-demanding specifications of fiber gratings for. 4.

(23) various applications require increasingly precise measurement of the grating parameters. For the characteristics measurement of fiber gratings, we develop two direct methods to measure the important grating parameters including the grating phase (ψ) , the refractive index change ( ∆n ) and the grating period change ( ∆Λ ). The two methods are the balanced Michelson interferometer technique for measuring the spectrum phase information and the side-diffraction technique for measuring the refractive index change and the grating period change. These methods are efficient in determining the precise nature and the location of the defects in the written grating structure. The accurate measurement of grating parameters is useful for the design and fabrication of FBGs. The causal relationship among characterization, design and fabrication will be helpful for finding the system error of the FBG exposure setup. Eventually this system error may also be taken into account in the design of the FBG structure from the beginning [43].. 1.3 Organization of the Dissertation. The main organization of this thesis can be divided into three parts, which include the theoretical analyses, the characterization methods and the fabrication methods of the fiber gratings. In the introductory chapter, we describe the history of the fiber gratings and the purposes of this thesis research. It contains the review of the previous works and the challenges of the proposed methods. The chapter 2 describes the synthesis of fiber gratings, which include a short introduction on various types of fiber gratings and the developed methods for the design of complex fiber grating structures. In chapter 3, we improve the measurement methods based on the balanced Michelson interferometer method and the side-diffraction technique for the characterization of fiber gratings. The complex reflection spectrum and the complex coupling coefficient can be obtained by using the proposed. 5.

(24) analysis process. In chapter 4, we setup the auto-controlled fabrication system and propose three new fabrication methods to actually fabricate advanced fiber gratings with the desired complex grating structures. The proposed methods can be based on the phase mask or the two-beam interferometer setups to write long, complex fiber gratings. We also develop the Labview program to control the fabrication process. So that the accuracy and repetibility of the fabrication process can be enhanced. Finally, in chapter 5 we draw a conclusion of this research and provide directions for performing further research related to this topic.. 6.

(25) References for Chapter 1:. [1] K. O. Hill, Y. Fujii, D. C. Johnsen, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguide: Application to reflection filter fabrication,” Appl. Phys. Lett., vol. 32, pp.647-649, 1978. [2] G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett., vol. 14, pp. 823-825, 1989. [3] K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett., vol. 62, pp.1035-1037, 1993. [4] J. Albert, K. O. Hill, B. Malo, S. Theriault, F. Bilodeau, D. C. Johnson, and L. E. Erickson, “Apodisation of the spectral response of fiber Bragg gratings using a phase mask with variable. diffraction efficiency,” Electron. Lett., vol. 31, pp. 222-223, 1995.. [5] W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving-scanning beam technique,” Opt. Lett., vol. 20, pp. 2051-2053, 1995. [6] J. B. Jensen, M. Plougmann, H. –J. Deyerl, P. Varming, J. Hubner, and M. Kristensen, “Polarization control method for ultraviolet writing of advanced Bragg gratings,” Opt. Lett., vol. 27, pp. 1004-1006, 2002. [7] A. Asseh, H. Storoy, B. E. Sahlgren, S. Sandgren, and R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles.” J.Lightwave Tech., vol. 15, pp. 1419-1423, 1997. [8] I. Petermann, B. Sahlgren, S. Helmfrid, A. T. Friberg, and P.-Y. Fonjallaz, “Fabrication of advanced fiber Bragg gratings by use of sequential writing with a continuous-wave ultraviolet laser source,” Appl. Opt., vol. 41, pp. 1051-1056, 2002. [9] K. O. Hill, G. Meltz, “Fiber Bragg Grating Technology: Fundamentals and Overview,” J.. 7.

(26) Lightwave Tech., vol. 15, pp. 1263-1276, 1997. [10] I. Baumann, J. Seifert, W. Nowak, and M. Sauer, “Compact all-fiber add-drop multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett., vol. 8, pp. 1331-1333, 1996. [11] M. Ibsen, R. Feced, P. Petropoulos and M. N. Zervas, “99.9% reflectivity dispersion-less square-filter fibre Bragg gratings for high speed DWDM networks,” Optical Fiber Communication Conference, pp. 230-232, 2000. [12] M. Ibsen, “Advanced fibre Bragg grating design and technology,” Optical Communication, ECOC’01 27th European Conference, vol. 5, pp. 181-182, 2001. [13] J. A. R. Williams, K. S. I. Bennion, and N. J. Doran, ”Fiber dispersion compensation using a chirped in-fiber Bragg grating,” Electron. Lett., vol. 30, pp. 985-987, 1997. [14] W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled fiber grating based dispersion slope compensator,” IEEE Photon. Technol. Lett., vol. 11, pp. 1280–1282, Oct. 1999. [15] A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Tech., vol. 15, pp. 1442-1463, 1997. [16] Youlong Yu, Hwayaw Tam, Wenghong Chung, and Muhtesem Suleyman Demokan, “ Fiber Bragg grating sensor for simultaneous measurement of displacement and temperature,” Opt. Lett., vol. 25, pp. 1141-1143, 2000. [17] Sarfraz Khaliq, Stephen W. James, and Ralph P. Tatam, “Fiber-optic liquid-level sensor using a long-period grating,” Opt. Lett., vol. 26, pp. 1224-1226, 2001. [18] J. T. Kringlebotn, J. L. Archambault, L. Reekie, and D. N. Payne, “Er3+:Yb3+-codoped fiber distributed-feedback laser,” Opt. Lett. 19, pp. 2101-2103, 1994. [19] W. H. Loh, B. N. Samson, L. Dong, G. J. Cowle, and K. Hsu, “High Performance Single Frequency Fiber Grating-Based Erbium:Ytterbium-Codoped Fiber Lasers,” J. Lightwave. 8.

(27) Tech., vol. 16, pp. 114-118, 1998. [20] M. Ibsen, S. Y. Set, G. S. Goh, and K. Kikuchi, “Broad-Band Continuously Tunable All-Fiber DFB Lasers,” IEEE Photon. Technol. Lett., vol. 14, pp. 21-23, 2002. [21] T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Tech., vol. 15, pp.1277-1294, 1997. [22] G.-W. Chern and L. A. Wang, “Analysis and design of almost-periodic vertical-grating-assisted codirectional coupler filters with nonuniform duty ratios,” Appl. Opt., vol. 39, pp. 4926-4937, 2000. [23] G. H. Song, “Toward the ideal codirectional Bragg filter with an acoustooptic-filter design,” J. Lightwave Tech., vol. 13, pp.470-480, 1995. [24] R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron., vol. 35, pp. 1105-1115, 1999. [25] L. Poladian, “Simple grating synthesis algorithm,” Opt. Lett., vol. 25, pp. 787-789, 2000. [26] J. Skaar, L. Wang, and T. Erdogan , “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron., vol. 37, pp.165-173, 2001. [27] J. Skaar and K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Tech., vol. 16, pp. 1928-1932, 1998. [28] Thomas Back, “Evolutionary Algorithms in Theory and Practice,” Oxford, New York, 1996. [29] Cheng-Ling Lee and Yinchieh Lai, “Evolutionary programming synthesis of optimal long-period fiber Bragg filters for EDFA gain flattening,” IEEE Photon. Technol. Lett., vol. 14, pp. 1557-1559, 2002. [30] Mark Froggatt, “Distributed measurement of the complex modulation of a photoinduced Bragg grating in an optical fiber,” Appl. Opt., vol. 35, pp. 5162-5164, 1996. [31] J. Skaar, “Measuring the group delay of fiber Bragg gratings by use of end-reflection. 9.

(28) interference”, Opt. Lett., vol. 24, pp.1020-1022, 1999. [32] E. Ingemar Petermann, Johannes Skaar, Bengt E. Sahlgren, Raoul A. H. Stubbe, and Ari T. Friberg, “Characterization of Fiber Bragg Gratings by Use of Optical Coherence-Domain Reflectometry,” J. Lightwave Tech., vol. 17, pp. 2371-2378, 1999. [33] Shay Keren and Moshe Horowitz, “Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses,” Opt. Lett., vol. 26, pp. 328-330, 2001. [34] L. M. Baskin et al., “Accurate Characterization of Fiber Bragg Grating Index Modulation by Side-Diffraction Technique”, IEEE Photon. Technol. Lett., vol. 15, pp. 449-451, 2003. [35] F. El-Diasty, A. Heaney, and T. Erdogan, “Analysis of fiber grating by a side-diffraction interference technique,” Appl. Opt., vol. 40, pp. 890–896, 2001. [36] P. A. Krug, R. Stolte, and R. Ulrich, “Measurement of index modulation along an optical fiber Bragg grating,” Opt. Lett., vol. 20, pp. 1767-1769, 1995. [37] S. Sandgren, B. Sahlgren, A. Asseh, W. Margulis, F. Lrurell, R. Stubbe, and A. Lidagrd, “Characterisation of Bragg gratings in fibres with the heat-scan technique,” Electron. Lett., vol. 31, pp. 665-666, 1995. [38] Ph. Giaccari, H. G. Limberger, and R. P. Salathé, “Local coupling-coefficient characterization in fiber Bragg gratings,” Opt. Lett., vol. 28, pp. 598-600, 2003. [39] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, New York, 1992). [40] B. Malo, S. Theriault, D.C. Johnson, F. Bilodeau, J. Albert, and K.O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprinted using a phase mask,” Electron. Lett., vol. 31, pp. 223-225, 1995. [41] H. Singh, and M. Zippin, “Apodized fiber Bragg gratings for DWDM applications using uniform phase mask,” in Proceedings of the 24th European Conf. on Optical Communication (Lerko Print S.A., Madrid, Spain), vol. 1, pp. 189-190, 1998. [42] C. Yang, and Y. Lai, “Apodised fiber Bragg gratings fabricated with uniform phase mask. 10.

(29) using low cost apparatus,” Electron. Lett., vol. 36, pp. 655-657, 2000. [43] Alexander V. Buryak and Dmitrii Yu. Stepanov, “Correction of systematic errors in the fabrication of fiber Bragg gratings,” Opt. Lett., vol. 27, pp. 1099-1101, 2002.. 11.

(30) Chapter 2. Synthesis of Fiber Bragg Gratings. 2.1 Introduction. The synthesis of fiber Bragg gratings amounts to finding the grating structure from a specified, complex spectrum. The purpose of synthesis is helpful for finding the optimal experimental parameters and for the characterization of already fabricated gratings. The objective of this chapter is to understand the characteristics of fiber gratings and then to use the layer-peeling method to find the grating structure for the required filter properties. The content of this chapter is arranged as follows: In section 2.2, we introduce the fundamentals of fiber Bragg gratings which include a short introduction on the grating structure and the photosensitivity of a fiber. In section 2.3, various types of fiber Bragg gratings are discussed, which include uniform FBGs, phase-shifted FBGs, apodized FBGs and chirped FBGs. The theoretic model and the synthesis method of fiber gratings are described in section 2.4 and 2.5. In order to practically implement the ideal grating structure constructed from the layer-pealing method by the sequential writing setups, we use the least square method to find the experimental parameters and the procedure is described in section 2.6. Finally, a summary for this chapter is given in section 2.7.. 2.2 Fundamentals of Fiber Bragg Gratings. Fiber Bragg gratings have almost periodic structures consisting of a variation of the refractive index along the propagation direction, as shown in Fig. 2.1. We assume that the unperturbed fiber has a refractive index profile n( x, y ) and the perturbed fiber has the z-dependent index. 12.

(31) n( x, y, z ) . The z-dependence of the index perturbation is approximately quasi-sinusoidal in the sense that it can be written as 2. n 2 − n = ∆ε r ,ac ( z ) cos(. 2π z + θ ( z )) + ∆ε r ,dc ( z ) Λ. (2.1). where Λ is a chosen design period so that θ(z) becomes a slowly varying function of z when compared to the grating period Λ. The functions ∆ε r ,ac ( z ) and ∆ε r ,dc ( z ) are real functions and are slowly varying. They also satisfy. ∆ε r ,ac ( z ) << nco2 ,. ∆ε r ,dc ( z ) << nco2. (2.2). Note that since the index perturbation is small, (2.1) can be written as ∆n( z ) = n − n = ∆nac ( z ) cos(. 2π z + θ ( z )) + ∆ndc ( z ) Λ. (2.3). where ∆nac (z ) and ∆n dc (z ) are the “ac” and “dc” index changes, respectively. We have 2. also used the approximation n 2 − n ≈ 2nco (n − n) . θ(z) is the spatial grating phase. One can also say that (n+ ∆n dc (z ) ) is the effective refractive index neff. The typical index modulation values of a fiber grating are about 10-6~10-3, which are dependent on the dopants in the fiber and the UV-exposure condition. For commercial photosensitive fibers, germanium and boron co-doped fibers have been demonstrated to be excellent for increasing the photosensitivity of the silica fibers. Some examples are like FberCore PS-1500 and 3M GF-3 fibers [1]. The high-pressure hydrogen loading of the fiber also gives great help for increasing the photosensitivity to reduce the exposure time [1-2]. Other schemes to increase photosensitivity have also been suggested, including the codoping of the perform core with materials such as aluminum [3], tin [4], and phosphorus [5]. Another important property of the photo-induce refractive index change is its anisotropy. This characteristic property can be easily observed by irradiating the fiber from the side with the UV light polarized perpendicular to the fiber axis. The anisotropy in the photo-induce. 13.

(32) refractive index change results in the birefringence of the fiber. The magnitude of the birefringence proportionally to the total induced index change was measured to be as smaller as 0.2% and as larger as 8% of the total index change [6]. The effect is useful for fabricating polarization mode converting devices or rock filters [7]. The magnitude of the refractive index change also depends on the UV-exposure condition, such as the wavelength, intensity, and total dosage of the irradiating lights. Various continuous wave (CW) and pulsed laser light sources with the wavelength ranging from the visible to the UV have been used to photo-induce the refractive index changes in the optical fiber [8-10]. In practice, the most commonly used light sources are KrF and ArF excimer lasers that generate 248 and 193 nm optical pulses. In this thesis, we use CW 244 nm Argon Ion SHG UV laser source as the exposure light.. 2.3 Types of Fiber Bragg Gratings. The optical properties of a fiber grating are determined by the variation of refractive index. ∆n(z ) along the fiber axis z. In equation (2.3), the parameters of ∆n(z ) are what can be controlled in a fiber grating. They include the ac-index change ∆nac (z ) , the dc-index change ∆n dc (z ) (or effective index neff), the grating period Λ, and the grating phase θ(z). Fig. 2.2. illustrates the common types of fiber Bragg gratings which include uniform FBGs, phase-shift FBGs, apodized FBGs, and chirped FBGs. The discussion of this section will describe the difference among these grating types and the important parameters for the design and fabrication of these FBGs.. 14.

(33) 2.3.1 Uniform Fiber Bragg Gratings. The variation of refractive index ∆n(z ) for a uniform FBG is shown in Fig. 2.2(a). For this grating structure, the index modulation ∆nac (z ) and the effective index neff are keeping constant along the grating length. The grating period is also fixed along the whole grating. The quantities of interest are the reflection spectrum and the time delay. Fig. 2.3(a) shows the spectra of three Bragg gratings with different coupling constants of 2, 4, and 8. These gratings have the fixed finite grating length, but with different index modulation amplitudes. One can see that a uniform period and index-modulation grating will produce side lobes at the shorter and longer wavelength bands. The time delay spectra of these gratings are shown in Fig. 2.3(b). They are calculated by. τg = −. λ2 dφ (λ ) 2πc dλ. (2.4). Near the edge of the band stop, strong dispersion can be seen with the increasing strength of the grating. At the center of the stop band, the time delay is minimum.. 2.3.2 Phase-shifted Fiber Bragg Gratings. For many applications, the transmission characteristics of a fiber Bragg grating attracted more attention. For examples, the applications of distributed feedback (DFB) fiber lasers [11-12] and superstructure band-pass filters [13] are becoming more popular. The single λ / 4 (or π) phase-shifted DFB structure has a pass-band in the middle of the stop band as shown in Fig. 2.2(b). It is like a Fabry-Perot (FP) resonator, which works in the same way as a bulk FP interferometer expect that the grating is a narrow-band distributed reflector. The example spectra of a grating with a π-phase shift in the middle of the grating length is shown in Fig. 2.4 the grating has a uniform index shape with a grating length of 5 mm and the coupling. 15.

(34) coefficient of 1.6. There is a transmission peak appeared in the middle of the stop band.. 2.3.3 Apodized Fiber Bragg Gratings. “Apodization” is a word often encountered in filter design. Its meaning can be described as designing a special index envelope shape to improve the spectral properties. The commonly used apodization functions are as follows: Gaussian, Tanh, Sinc, and Raised cosine etc.. The apodization can be achieved through controlling the parameters of ∆nac (z ) in equation (2.3). The beneficial effects of apodization are not only in the smoothness of the reflection spectrum, but also in the dispersion characteristics. One simple method to apodize a FBG is to make the profile of ∆nac (z ) to be one of the apodization functions. In this case due to the usual fabrication procedure, the value of ∆n dc (z ) may not be constant along the whole grating as shown in Fig. 2.2(c). The reflection spectrum of this grating has large side lobes in the shorter wavelength band, which is shown in Fig. 2.5(a). This kind of filters cannot be used in the DWDM systems which require a steep edge, very low sidelobes, and a flat top with very little ripples. In order to meet the requirement, the dc-index change ∆n dc (z ) must be kept constant throughout the grating when the ac-index change ∆nac (z ) is apodized. This type of gratings are called the pure (or true) apodized FBGs. The Gaussian apodized grating index profile is shown in Fig. 2.2(d), and the corresponding reflection and transmission spectra are shown in Fig. 2.5(b). One can see that there are very low side lobes on both sides of the central wavelength.. 2.3.4 Chirped Fiber Bragg Gratings. “Chirp” is the high-pitched varying sound emitted by certain birds and bats. Gratings that. 16.

(35) have a non-uniform period along the whole grating are called to be chirped. Chirped gratings have many different forms which may be linear [14], quadratic [15], or random along the grating [16]. Besides, the most important property of chirped gratings is the phase information, which can be contributed from the dc-index change, phase shift and period change terms. The chirped FBGs can be fabricated by controlling one of these parameter terms. In general, the period-chirped FBG has the structure of different period along the grating, which is shown in Fig. 2.2(e). The spectra of one example chirped FBG is shown in Fig. 2.6. The simulation is done with the grating length = 30 mm, chirped rate = 0.1667 nm/cm, and a tanh index profile. Fig. 2.6(a) shows the reflection and transmission spectra, and the Fig. 2.6(b) shows the phase and time delay spectra. Chirped gratings have many applications. In particular, the linearly chirped grating can act as the dispersion compensating device. The other applications include the chirped pulse amplification [17], amplifier gain flattening [18], and optical sensing [19].. 2.4 Coupled-Mode Theory. The relation between the spectral response of a fiber grating and the corresponding grating structure is usually calculated by the coupled-mode theory. The fiber is assumed to be lossless and single-mode in the wavelength range of interest. In other words, we consider only one forward and one backward propagating modes. Moreover, the fiber is assumed to be weakly guiding. That is the difference between the refractive indices in the core and the cladding is very small. Under this assumption the electric and magnetic fields are approximately transverse to the fiber axis, and one can ignore all the polarization effects due to the fiber structure and consider solely the scalar wave equation [20]. The coupled-mode equations can be written as. 17.

(36) du = + iδ u + q ( z ) v dz dv = −iδv + q ∗ ( z )u dz. (2.5). In these equations, we ignore the terms that are rapidly oscillating since they contribute little to the growth and decay of the modes. The fields of u and v correspond to the forward and backward propagating amplitudes. In (2.5) the wavenumber detuning δ=β-π/Λ and the coupling coefficient q of the grating is given by : z. q( z ) = iκ ( z ) exp(−2i ∫ σ ( z ' )dz '). (2.6). 0. where κ (z ) is a complex, slowly varying function of z and σ (z ' ) is a real, slowly varying function that accounts for the dc index variation from ∆ε r ,dc ( z ) . Here q(z) is related to the index modulation by : q( z ) =. ηπ∆nac ( z ) λ z. π. 0. 2. arg q ( z ) = θ ( z ) − 2ηk ∫ ∆ndc ( z ' )dz ' +. (2.7). The modulus of q is proportional to the ac-index modulation amplitude. The termθ(z) is the spatial grating phase and the integral term in (2.7) is the modification to the spatial phase due to the increased dc index.. 2.5 Discrete Layer-Peeling Method. This layer-peeling method for synthesis of fiber gratings comes from an inherently discrete model [21]. It was first developed by geophysicists like Goupillaud and Robinson, and was extended by Bruckstein et. al. [22,23]. In this thesis, the described discrete layer-peeling (DLP) method was developed by J. Skaar et. al. [24]. This method is an inverse method for finding the structure parameters of the complex coupling coefficient of a fiber grating from the. 18.

(37) complex reflection spectrum. The description of the inverse and the forward methods for the synthesis of fiber gratings. are summarized in Fig. 2.7. The discretized LP method can be described as shown in Fig. 2.8. The starting point for the numerical modeling of FBGs is based on the transfer matrix formulation, which connects the field at point z+Δ with the field at z by the following relation : q δ   cosh (γ∆ ) + i sinh (γ∆ ) sinh (γ∆ )   γ γ u(z, δ ) u ( z , δ ) u ( z + ∆, δ )    = [T ] =    v ( z + ∆, δ )   *   v(z, δ ) q δ  v( z, δ )    ( ) ( ) ( ) sinh cosh sinh ∆ ∆ − ∆ i γ γ γ   γ  γ. (2.8). Here u ( z , δ ) and v( z , δ ) are the slowly varying amplitudes of the forward and backward propagating fields, and γ 2 = q − δ 2 , where q is the coupling coefficient and δ = β − β B 2. is the wavenumber detuning compared to a Bragg design wavenumber β B . We refer to (2.8) as the piecewise uniform model since the grating is considered uniform in the interval [z, z+ Δ]. The matrix T can be replaced by the product of two transfer matrixes T∆ * Tρ . Here. 0 exp(iδ∆)  T∆ =  0 exp(−iδ∆) . (2.9). is the pure propagation transfer matrix obtained by letting q Æ 0, and. (. Tρ = 1 − ρ j. ). 2 −1 / 2.  1  − ρ j. − ρ ∗j   1 . (2.10). is the discrete reflector matrix obtained by letting q Æ ∞ and holding qΔ constant. The discrete reflection coefficient is given by. ρ. j. = − tanh. (q. j. ∆. )q. q *j. .. (2.11). j. It is straightforward to show that the result of transferring the fields using T∆ ⋅ Tρ is equivalent to the recursive formulation given below. 19.

(38) r j +1 (δ ) = exp(− 2iδ∆ ) ⋅. r j (δ ) − ρ j. (2.12). 1 − ρ *j r j (δ ). Here r j (δ ) = v j (δ ) / u j (δ ) . To obtain an explicit expression for the determination of ρ1 by the inverse Fourier transform, we note that the spectrum r1(δ) can be written as a discrete-time Fourier transform of the impulse response h1(τ), ∞. r1 (δ ) = ∑ h1 (τ ) exp(iδτ 2∆ ). (2.13). τ =0. Since the impulse response forτ=0 is the same as if only the first reflector is present, we can see thatρ1 is simply the zeroth order Fourier coefficient of the series in (2.13) :. ρ1 = h1 (0 ) =. ∆. π 2∆. π. 1 −π 2 ∆. ∫ r (δ )dδ. (2.14). For numerical implementation, the spectral dependence must also be discretized, and hence the calculation ofρ1 by the inverse Fourier transform of r1(δ) can be achieved by the discrete Fourier transform. ρ1 =. 1 M. M. ∑ r (m) m =1. (2.15). 1. Here r1(m) denotes a discretized version of the spectrum r1(δ) in the range. δ ≤ π 2∆ , and. M is the number of wavelengths in the spectrum. After determiningρj , we can get the complex coupling constant q(z) by  1  1 − ρ j  q j = ln 2∆  1 + ρ j    arg q j = − arg ρ j. 20.    . (2.16).

(39) 2.6 The Least Square Method. From the DLP method, the normalized refractive-index modulation amplitude and phase profiles can be reconstructed from the target spectrum. In order to realize this complicated FBG by the proposed fabrication methods, it is important to include the overlap-step-scan effects in the design [25]. We use the least square (LS) fitting method [26] to find the best experimental exposure parameters. In our experiment we will use a small gaussion beam to fabricate fiber gratings by an overlap-step-scan exposure method. It is assumed that Aid(z) is the refractive index envelope reconstructed from the Layer-Peeling method and Am(z) is the refractive index envelope of the m-th small gaussion beam. Therefore the following merit function can be used for determining the optimum parameters of the overlap-step-scan exposure. 2.   σ ({C m }) = ∫  Aid ( z ) − ∑ Am ( z ) dz m  . (2.17). where Cm represents the amplitude of the m-th small gaussion beam, and Am(z) can be expressed as. (. Am (z ) = C m ⋅ exp − ( z − z m ) ws 2. 2. ). (2.18). In the above expression ws is the width of the gaussion beam and zm is the central position of the m-th exposure gaussion beam. We want to let. ∑A. m. ( z ) be as close to Aid(z) as possible.. m. This can be achieved by the least square method, which solves the following equations to find the optimum values.. (z − z m ) ∂σ   = -2∫ A id ( z ) − ∑ Am ( z ) ⋅ exp(− )dz = 0 ∂C m ws m   2. 21. (2.19).

(40) 2.7 Summary. In this chapter we have described the characteristics of fiber Bragg gratings with various types of refractive index profiles. It is helpful for the design of various complex FBG structures. The discrete layer-peeling method has been adopted for the design of advanced fiber gratings. We also combine this synthesis method with the least square method to find the best experimental parameters for sequential UV exposure. These developed methods will be applied to the design of advanced FBG structures and can be realized by using the proposed fabrication methods to be described in the following chapter.. 22.

(41) References for Chapter 2:. [1] P. L. Swart, M. G. Shlysgin, A. A. Chtcherbakov, and V. V. Spirin, “Photosensitivity measurement in optical fibre with Bragg grating interferometers,” Electron. Lett., vol. 38, pp. 1508-1510, 2002. [2] P. J. Lemaire, R. M. Atkins, V. Mizrahi, and W. A. Reced, ”High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett., vol. 29, pp. 1191-1193, 1993. [3] G. Meltz and W. W. Morey, SPIE, vol. 1515, pp. 185, 1991. [4] L. Dong, J. L. Cruz, L. Reekie, M. G. Xu and D. N. Payne, “Enhanced photosensitivity in tin-codoped germanosilicate optical fibers,” IEEE Photon. Technol. Lett.,vol. 7, pp. 1048-1050, 1995. [5] Thomas A. Strasser, “photosensitivity in phosphorus fibers,” in OFC’96, invited TuO1. [6] T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B, vol. 11, pp. 2100-2105, 1994. [7] K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: Application to the external writing of rocking filters,” Electron. Lett., vol. 27, pp. 1548–1550, 1991. [8] K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: Application to reflection filter fabrication,” Appl. Phys. Lett., vol. 32, pp. 647–649, 1978. [9] C. G. Askins, T.-E. Tsai, G. M. Williams, M. A. Putnam, M. Bashkansky, and E. J. Friebele, “Fiber Bragg reflectors prepared by a single excimer pulse,” Opt. Lett., vol. 17, pp. 833-835, 1992. [10] D. C. Psaila and C. Martijn de Sterke, “Fabrication of rocking filters at 193 nm,” Opt.. 23.

(42) Lett., vol. 21, pp. 1550-1552, 1996. [11] H. A. Haus and C. V. Shank, “Anti-symmetric type of distributed feedback lasers,” J. Quantum Electron, vol. 12, pp. 532-539, 1976. [12] R. Kashyap, P. E. McKee, and D. Armes, “UV written reflection grating structures in photosensitive optical fibres using phase-shifted phase-mask,” Electron. Lett., vol. 30, pp. 1977-1978, 1994. [13] L. Wei and W. Y. Lit, “Phase-shifted Bragg grating filters with symmetrical structures,” IEEE J. Lightwave Tech., vol. 15, pp. 1405-1410, 1997. [14] K. C. Byron, K. Sugden, T. Bircheno, and I. Bennion, “Fabrication of chirped Bragg gratings in photosensitive fibre,” Electron. Lett. vol. 29, pp. 1659, 1993. [15] B. Eggleton, P. A. Krug, and L. Poladin, “Dispersion compensation by using Bragg grating filters with self induced chirp,” in Tech. Digest of Opt. Fib. Comm. Conf., OFC’94, pp. 227. [16] F. Ouellette, “The effect of profile noise on the spectral response of fiber grating,” in Bragg Gratings, Photosensitivity, Poling in Glass Fibers and Waveguides: Applications and Fundamentals, vol. 17, OSA Technical Digest Series (Optical Society of America, Washington, DC, 1997), paper BMG13, pp. 222-224. [17] A. Boskovic, M. J. Guy, S. V. Chernikov, J. R. Taylor, and R. Kashyap, “All-fiber diode pumped, femtosecond chirped pulse amplification system,” Electron. Lett., vol. 31, pp. 877-879, 1995. [18] R. Kashyap, R. Wyatt, and P. F. McKee, “Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings,” Electron. Lett., vol. 29, pp. 1025, 1993. [19] A. D. Kersey and M. A. Davis, “Interferometric fiber sensor with a chirped grating distributed sensor element,” Proc. OFS’94, pp. 319-322, Glasgow UK, 1994.. 24.

(43) [20] A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983). [21] R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum El., vol. 35, pp. 1105-1115, 1999. [22] A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev., vol. 29, pp. 359-389, 1987. [23] A. M. Bruckstein, B. C. Levy and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math, vol. 45, pp. 312-335, 1995. [24] J. Skaar, L. Wang, and T. Erdogan , “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron., vol. 37, pp.165-173, 2001. [25] L. G. Sheu, K. P. Chuang, and Y. Lai, “Fiber Bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photon. Technol. Lett., vol. 15, pp. 939-941, 2003. [26] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, New York, 1992).. 25.

(44) Fig. 2.1 The refractive-index modulation along the length of waveguides.. 26.

(45) Fig. 2.2 The common types of fiber Bragg gratings; (a) uniform FBGs, (b) phase-shift FBGs, (c, d) apodized FBGs, and (e) chirped FBGs.. 27.

(46) 5 abs(qL)=2 abs(qL)=4 abs(qL)=8. Reflectivity (dB). 0 -5 -10 -15 -20 -25 1549.6. 1549.8. 1550.0. 1550.2. 1550.4. Wavelength (nm) (a). 600 abs(qL)=2 abs(qL)=4 abs(qL)=8. Time Delay (ps). 400. 200. 0. -200 1549.6. 1549.8. 1550.0. 1550.2. 1550.4. Wavelength (nm) (b). Fig. 2.3 (a) The spectra of three Bragg gratings with different coupling constants of 2, 4, and 8.; (b) time delay of the same gratings as in Fig. 2.3(a).. 28.

(47) Reflectivity (dB). 5. 0. 0. -5. -5. -10. -10. -15. -15. -20. -20. -25. -25. Transmission (dB). 5. -30. -30 1549.2. 1549.6. 1550.0. 1550.4. 1550.8. Wavelength (nm). Fig. 2.4 A π-phase shift uniform fiber Bragg grating. (the parameters are abs(qL)=1.6 and L=5 mm). 29.

(48) Reflectivity (dB). 5. 0. 0. -5. -5. -10. -10. -15. -15. -20. -20. -25. -25. -30. -30. -35. -35. -40 1549.2. 1549.6. 1550.0. 1550.4. Transmission (dB). 5. -40 1550.8. 5. 5. 0. 0. -5. -5. -10. -10. -15. -15. -20. -20. -25. -25. -30. -30. -35. -35. -40 1549.2. 1549.6. 1550.0. 1550.4. Transmission (dB). Reflectivity (dB). Wavelength (nm) (a). -40 1550.8. Wavelength (nm) (b). Fig. 2.5 (a) A Gaussian apodized FBG with non-constant dc index; (b) a pure apodization FBG with a Gaussian index profile.. 30.

(49) Reflectivity (dB). 5. 0. 0. -5. -5. -10. -10. -15. -15. -20. -20. -25. -25. -30. -30. -35. -35. -40 1548.5. 1549.0. 1549.5. 1550.0. 1550.5. 1551.0. Transmission (dB). 5. -40 1551.5. Wavelength (nm) (a). 600. Phase Time Delay. 500. 200 100 0. Phase (rad). -100 300. -200. 200. -300. 100. -400. Time Delay (ps). 400. -500 0 1548.5. 1549.0. 1549.5. 1550.0. 1550.5. 1551.0. -600 1551.5. Wavelength (nm) (b). Fig. 2.6 A chirped FBG with the grating length = 30 mm, chirped rate = 0.1667 nm/cm, and a tanh index profile. (a) The reflection and transmission spectra; (b) the phase and time delay.. 31.

(50) inverse. coupling coefficient q(z) ( index ∆ n and phase θ ). reflection coefficient r(λ) or t(λ). forward. Fig. 2.7 The description of the inverse and the forward methods for the synthesis and analysis of fiber gratings.. 32.

(51) Optical Waveguide. v1 u1. ρ1 v2 ρ2 u2. section 1. ρ3 z section 2. Fig. 2.8 A discretized model of the LP method.. 33.

(52) Chapter 3. Characterization of Fiber Bragg Gratings. 3.1 Introduction. The characterization of a fiber Bragg grating involves the determination of the complex reflection coefficient or the complex coupling coefficient of the grating. The mostly direct method for finding FBG reflection and transmission spectra is by using the optical spectrum analyzer (OSA). This measurement can help us to find the performances of a fiber grating initially and is not enough to get the grating structures for improving the design and fabrication of advanced fiber gratings. Accurate characterization of the amplitude (refractive-index modulation) and phase (dispersion) of FBGs are needed for many applications. In the literature, there have been a lot of different approaches based on the interferometry [1-4], side-scattering [5-7], heat-scan [8-9] or phase modulation methods [10-11]. In this chapter, we describe two measurement methods that we developed for measuring the spectral phase, index amplitude and period change of the FBGs. The two methods are based on the balanced Michelson interferometer technique for measuring the spectral phase information and the side-diffraction technique for measuring the refractive index change and the grating period change.. 3.2 The Balanced Michelson Interferometer Method. The characterization of fiber Bragg gratings is very important for their applications in high-bit rate fiber communication systems. It is not easy to measure the phase information of a fiber grating. In reference 4, Skaar used a lossless Fabry-Perot-like system consisting of two reflectors, where one reflector has known characteristics, and the other has not. The first. 34.

(53) reflector is a FBG with unknown characteristics, and the other reflector is quite broad-band when compared to the FBG as shown in Fig. 3.1. They obtained the complex reflection spectrum of the FBG, or equivalently the reflectivity and group delay spectra, from the measurement of the reflectivity of the Fabry-Perot structure. In this paper, we analyze a similar structure, namely a balanced Michelson interferometer consisting of two general reflectors, where one of them is the FBG to be characterized, and the other is a reference reflector [12]. The experimental setup is shown in Fig. 3.2 and can be analyzed as follows. In the FBG path, we obtain the path 1 signal 2π ~ E1 (λ ) = E1 (λ ) ⋅ exp(i L1 + iφ (λ )). (3.1). λ. where φ (λ ) is the phase information of the FBG, and L1 is the path length from coupler. In the reference path, we obtain the path 2 signal 2π ~ E2 (λ ) = E2 (λ ) ⋅ exp(i L2 ). (3.2). λ. where L2 is the path length from coupler. When these two signal are combined at the output port of the coupler which is connected to an Optical Spectrum Analyzer (OSA), the measured output signal spectrum can be expressed as I int er (λ ) = I grating + I reference + 2 ⋅ I grating ⋅ I reference ⋅ cos( ~ where l = L1 − L2 , I grating = E1. 2. 2π. λ. l + φ (λ )). (3.3). ~ 2 and I reference = E 2 .. From equation (3.3) the interference spectrum Iinter(λ) will look like that in Fig. 3.3. One can take the inverse Fourier transform of Iinter(λ) to get the result shown in Fig. 3.4. By multiplying the result by a filtering window function centered at the peak corresponding to the angular frequency +τand then taking a Fourier transform back to the spectral domain, we can get the phase response φ (λ ) from the result and the group delay τg=d φ /dω can also be calculated. The whole algorithm is summarized in Fig. 3.5. In order to test our algorithm, we analyze a structure consisting of a gaussion apodized,. 35.

(54) linear chirped FBG with the length L=5mm, the chirp rate = 0.5 (nm/cm), and the maximum grating reflection = 96.84%. The measured interference pattern Iinter(λ) is shown in Fig. 3.6(a). Fig. 3.6(b) shows the result of the calculated phase spectrum φ (λ ) . We have also calculated the group delay time τg=d φ /dω from the result of the phase response φ (λ ) . For comparison, we experimentally measure the group delay of this grating sample using a commercial dispersion measurement equipment (“Advantest Q7760”). The measured result (Fig. 3.7(a)) is compared with the result from our measuring method (Fig. 3.7(b)). One can see that we can get good measured results and the measurement resolution is about 5 ps, which is probably limited by the resolution of the OSA as well as the environmental vibration. The presented method has been carried out experimentally, and has been shown to be easy and accurate for the measurement of the grating phase. Some of the fabricated advanced fiber Bragg gratings shown in the next chapter will be measured by using this method.. 3.3 The Side-Diffraction Technique. The spatial quality of a FBG will strongly affect the performance of the FBG device. By using the side-diffraction technique, we can directly measure the refractive index modulation and the grating period profile. In this section we describe how this side-diffraction method can be used to get the information of grating index and period changes. And we also proposed an improved method for increasing the measuring resolution when compared with the prior methods in the literature. The side-diffraction technique can be divided into two categories of without/with interference as shown in Fig. 3.8 and Fig. 3.9. First, a side-diffraction method for measuring the refractive index modulation is shown in Fig. 3.8. A He-Ne laser as a probe beam is. 36.