國立交通大學
國立交通大學
國立交通大學
國立交通大學
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顯示科技
顯示科技
顯示科技研究所
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碩士學位論文
碩士學位論文
碩士學位論文
碩士學位論文
藉由
藉由
藉由
藉由二氧化碳雷射加熱
二氧化碳雷射加熱
二氧化碳雷射加熱與張力
二氧化碳雷射加熱
與張力
與張力
與張力拉伸
拉伸
拉伸
拉伸產生
產生
產生
產生
長週期光纖
長週期光纖
長週期光纖
長週期光纖光柵之研究
光柵之研究
光柵之研究
光柵之研究
Periodically Tapered Long-Period Fiber
Gratings by CO
2
Laser Heating and Tension
Stretching
研究生
研究生
研究生
研究生：
：
：
：呂柏萱
呂柏萱
呂柏萱
呂柏萱
指導教授
指導教授
指導教授
指導教授：
：
：
：賴暎杰
賴暎杰
賴暎杰
賴暎杰
中華民國九十
中華民國九十
中華民國九十
中華民國九十九
九
九
九年六月
年六月
年六月
年六月
Periodically Tapered Long-Period Fiber Gratings by CO
2Laser Heating and Tension Stretching
研 究 生：呂柏萱 Student：Po-Hsuan Lu
指導教授：賴暎杰 老師 Advisor：Yin-Chieh Lai
國 立 交 通 大 學
顯 示 科 技 研 究 所
碩 士 論 文
A Thesis
Submitted to Display Institute
College of Electrical Engineering and Computer Science National Chiao Tung University
in Partial Fulfillment of the Requirements for the Degree of Master in Display
June 2010
Hsinchu, Taiwan, Republic of China
i
摘
摘
摘
摘
要
要
要
要
論文名稱：藉由二氧化碳雷射加熱與張力拉伸製作長週期光纖光柵校所別：國立交通大學顯示科技研究所 頁數：1 頁 畢業時間：九十八學年度第二學期 學位：碩士 研究生：呂柏萱 指導教授：賴暎杰 老師 關鍵詞：光纖濾波器、光纖感測器、熔拉、長週期光纖光柵 在本論文中，我們提出一種週期性熔錐形狀的長週期光纖光柵及製作方法。 有別於以往二氧化碳雷射產生不對稱的長週期光纖光柵，我們產生的長週期光纖 光柵是對稱的結構，而這個對稱的結構是藉由熔拉所產生。我們並藉由解三層結 構光纖模態的數值模擬得到和實驗相吻合的結論。我們發現當週期性熔錐形狀明 顯時，其對溫度的靈敏度升高。而將具週期性熔錐形狀光纖光柵置於較高折射率 的介質中時，其對溫度的靈敏度亦升高。 具週期性熔錐形狀光纖光柵的製作方法如下：首先將高功率的二氧化碳雷射 經過柱狀透鏡聚焦後加熱光纖，此光纖的一端固定而另一端則懸掛重物(重物即 為光纖所受到的張力)，熔融的光纖經張力拉伸後產生熔拉的形狀。將上述光纖 裝置於一個移動平台上，並用一個開關去控制二氧化碳雷射的加熱的時間。我 們利用 Labview 進行自動控制，藉由控制開關打開的時間可以決定加熱的時間， 藉由移動平台的移動的距離可以決定光柵的週期。此對稱光纖光柵對溫度的靈 敏度(週期為 700 μm 且拉伸後的直徑 100 μm 的樣本)測得為 0.238 nm/˚C，比 傳統的 0.059 nm/˚C 改善很多。
ii
ABSTRACT
Title：Periodically Tapered Long-Period Fiber Gratings by CO2 Laser Heating
and Tension Stretching Pages：1 Page
School：National Chiao Tung University Department：Display Institute
Time：June, 2010 Degree：Master
Researcher：Po-Hsuan Lu Advisor：Prof. Yin-Chieh Lai
Keywords： Fiber filter、Fiber sensor、Taper、Long period fiber grating
In this thesis, we demonstrate a periodically tapered long period fiber grating and its fabrication method. Not as the same as the asymmetric long period fiber gratings produced before, we have produced devices with a symmetric structure tapered by a CO2 Laser. The produced LPFG with deep corrugated
surface structure and immersed in higher index media results in high temperature sensitivity. The numerical results performed by solving waveguiding modes in three-layer optical fibers agree with the experimental observation.
The procedure of making a periodically tapered long period fiber grating is as following: First, let the high power CO2 laser beam pass through the
cylindrical lens and focused on the fiber, which is fixed at one end and tagged with a weight at the other end ( the stress of the fiber equals the weight ). Then, the melting fiber will be extruded and shaped like a tapered fiber with laser heating. The fiber is put on a translation stage and a shutter is used to control
iii
the heating time of the CO2 laser. We use the Labview software to control the
translation stage and the shutter automatically so that we can set the parameters like the heating time and period easily. This kind of symmetric long period fiber gratings is more sensitive to temperature. We have measured the temperature sensitivity 0.238 nm/˚C for one of our LPFG samples, which is much better than the temperature sensitivity of traditional long period fiber gratings (0.059 nm/˚C).
iv
ACKNOWLEDGEMENT
研究所這兩年我非常感謝賴暎杰老師的指導，老師在思考問題和解決問題的 能力上對我幫助甚多，而老師溫和的個性和豐富的學養是我學習的典範。 我要特別感謝的是徐桂珠學姐對我論文研究上的指導，讓我的研究有方向並 能順利進行。也恭喜學姐在去年順利地榮升中央大學光電所的助理教授！ 此外感謝郭立強學長提供儀器，讓我能很快地檢視我實驗成果的好壞；感謝 鞠曉山學長的帶領，我們實驗室才能維持得這麼好；感謝張宏傑學長、莊佩蓁學 姐、劉秀鳳學姐、鍾佩芳學姐、池昱勳學長、顏子翔學長對我的幫助，讓我很順 利地融入實驗室，成為實驗室的一分子；感謝我的同學家豪、柏崴，一起在課業 上的互相幫助，今天才能一起順利畢業；也感謝學弟力行、聖閔、學妹姿媛，你 們都是實驗室的好幫手，未來實驗室就要靠你們共同努力了。 在實驗室的這段期間有很多美好的回憶，大家一起出遊、一起玩牌、一起慶 生、一起打羽球、一起看電影，雖然只是短短的兩年，可是我一輩子也忘不了。 最後我要感謝我的家人和朋友，他們給我很多支持和鼓勵，讓我能面對許多挫折 和挑戰，謝謝！v
CONTENTS
ABSTRACT( in Chinese)……….………i
ABSTRACT (in English)...ii
ACKNOWLEDGEMENT………iv
CONTENTS………..v
LIST OF FIGURES………vii
Chapter 1：Introduction………..……….1
1.1 Introduction of fiber gratings...………1
1.1-1 Various fabrication methods………1
1.1-2 Characteristics………...3
1.1-3 Applications……… 4
1.2 Motivation of the thesis ………5
1.2-1 Previous work………5
1.2-2 Motivation………...6
1.3 Structure of the thesis………..6
1.4 References………6
Chapter 2：Theory………10
2.1 Phase-matching condition………10
2.2 Coupled-mode theory………12
2.3 Wave guiding condition……….14
2.4 Dispersion mechanism ……….17
2.4-1 Material dispersion……….17
2.4-2 Waveguide dispersion………19
2.5 References………..19
vi
3.1 Experimental setup……….21
3.1-1 Instruments utilized in fabricating LPFG……….21
3.1-2 Fabrication process………23
3.2 Results and discussion ……….25
3.2-1 The surface structure observed by CCD camera…….…..25
3.2-2 The experiment of measurement………..………27
3.2-3 The transmission spectrum of LPFGs………..………28
3.2-4 Discussion...32
3.3 Simulation………33
3.3-1 Assumption……….33
3.3-2 Simulation method………..34
3.3-3 Simulation results………35
3.4 Possible improvement of the fabrication process………..40
Chapter 4：Conclusions and future work……….42
4.1 Conclusions……….42
vii
LIST OF FIGURES
Fig. 1.1 Two coherent UV beams produce an interference pattern in the fiber..
…...………..……….………….……...2
Fig. 1.2 The fiber grating is fabricated by a UV beam incident to a phase mask (PM). The period of the fiber grating is half of the PM period………3
Fig. 1.3 Conventional reflection spectrum of FBGs………...…...………4
Fig. 1.4 Conventional transmission spectrum of LPFGs………..…...4
Fig. 2.1 A light-wave diffracted by a fiber grating………..…………11
Fig. 2.2 The schematic diagram of phase-matching condition……...…………12
Fig. 2.3 Step profile of three layer fiber………..…………14
Fig. 2.4 Sellmeier and Cauchy’s refractive index dispersion curves…...……..18
Fig. 3.1 Schematic diagram of the experimental setup for fabricating LPFG… ……….………24
Fig. 3.2 The photography of the experimental setup for fabricating LPFG (a) the overall image (b) the fiber on the fiber holder with one end tagged with a weight.……….………24
Fig. 3.3 shows the CCD camera we used……….………25
Fig. 3.4 Optical micrograph of the periodically tapered LPFGs with (a) shallow (waist diameter around 104.4 µm) and (b) deep (waist diameter around 65.3 µm) surface corrugation……….…….…….26
Fig. 3.5 Optical micrograph of the periodically tapered LPFGs with grating periods of (a) 150 µm (b) 200 µm (c) and 300 µm………....………26
Fig. 3.6 The instruments of tunable laser and power sensor…...…….……….27
viii
Fig. 3.8 The spectral transmission of sample1in the air with tuning temperature ranged from 20 ℃ to 55.5 ℃………28 Fig. 3.9 The relationship of resonance wavelength shift and resonance dip to the environmental temperature for sample1 in the air………....……….………29 Fig. 3.10 The spectral transmission of sample1 under water with tuning temperature ranged from 40.1 ℃ to 54.4 ℃……….……….30 Fig. 3.11 The relationship of resonance wavelength shift and resonance dip to the environmental temperature for sample1 under water……....………...……30 Fig. 3.12 The spectral transmission of sample2 in the air with tuning temperature ranged from 30 ℃ to 45 ℃………....……….31 Fig. 3.13 The relationship of resonance wavelength shift and resonance dip to the environmental temperature……….……..….………..32 Fig. 3.14 The logarithm of the module by changing the effective refractive index……….………..………34 Fig. 3.15 The coupling wavelength with grating period Λ = 500µm and air as the surrounding medium, (a) rco = 4.1µm, (b) rco = (4.1×0.95) µm..…..……….37
Fig. 3.16 The coupling wavelength with grating period Λ = 500 µm and the change of core index ∆nco = -5×10-4...………...………..38
Fig. 3.17 The coupling wavelength with grating period Λ = 700 µm, rco =
(4.1×0.9) µm and ∆nco = -5×10-4 ……….…………39
Fig. 3.18 The coupling wavelength with grating period Λ = 700 µm, rco =
- 1 -
Chapter 1
Introduction
1.1 Introduction of Fiber Gratings
Fiber gratings are fiber devices that have already found many applications in optical communication and optical sensing. They can be classified according to the grating periods into two types, fiber Bragg gratings (also called reflection and short-period fiber gratings, denoted as FBGs) and long-period fiber gratings (also called transmission gratings, denoted as LPFGs). The periods of FBGs are normally under one micron meter while the periods of LPFGs are usually several hundred micron meters. In this thesis, we use a CO2 laser to
fabricate LPFGs. Because the focused CO2 laser beam size is much larger
than one micron meter, we can’t fabricate FBGs using our experimental setup. However, the CO2 laser is quite suitable for fabricating new kinds of LPFGs.
1.1-1 Various Fabrication Methods
Fiber gratings are first produced by exposing an optical fiber to ultra-violet (UV) lights that perturb the refractive index of the core to form a periodic index modulation profile. The photosensitive phenomenon in optical fibers was first demonstrated by Hill el al. in 1978 [1-1]. Here photosensitive means that the refractive index of certain doped glass can be raised by the exposure of UV lights. The magnitude of the refractive index change depends on the exposure time and dopant density (such as Germania) in the fiber and is typically between 10-6 and 10-3. The dopants are usually doped only in the core of fibers, so that the UV induced index change is only in the core, not in the cladding of
fibers. By using the techniques such as hydrogen loading, an index change as high as 10-2 can be obtained.
Since the period of fiber Bragg gratings the only way to fabricate
interferometer or a phase mask
holographic method was first proposed by Meltz et al [1
interference pattern is imposed on the fiber from the side as shown in 1.1. After a few years, the phase mask technique
1.2 became widely used to simplify the fabricat easily produce fiber gratings with high performance.
and complex fiber gratings with advanced characteristics, the systems of sequential writing have also been prop
Fig. 1.1 Two coherent UV beams produce an interference pattern in the fiber. The period of the grating depends on the angle
- 2 -
fibers. By using the techniques such as hydrogen loading, an index change as can be obtained.
fiber Bragg gratings are usually under one micron
the only way to fabricate such a short period grating is by utilizing either an interferometer or a phase mask to perform UV exposure. The transverse first proposed by Meltz et al [1-2] in 1989. The UV interference pattern is imposed on the fiber from the side as shown in
the phase mask technique [1-3]-[1-5] as shown in me widely used to simplify the fabrication procedure and
produce fiber gratings with high performance. In order to fabricate long and complex fiber gratings with advanced characteristics, the systems of sequential writing have also been proposed and demonstrated [1-6]-[1
1.1 Two coherent UV beams produce an interference pattern in the fiber. The period of the grating depends on the angle θ of the two incident beams. fibers. By using the techniques such as hydrogen loading, an index change as
micron meter, either an he transverse 2] in 1989. The UV interference pattern is imposed on the fiber from the side as shown in figure as shown in figure procedure and to more In order to fabricate long and complex fiber gratings with advanced characteristics, the systems of
[1-8].
1.1 Two coherent UV beams produce an interference pattern in the fiber. of the two incident beams.
Fig. 1.2 The fiber grating is fabricated by a UV beam incident to a phase mask (PM). The period of the fiber grating is half of the PM period.
On the contrary, LPFGs one of the methods. In addition
by thermally induced methods such as the use of a CO discharge with a conventional splicer
more available than UV photo
1.1-2 Characteristics
FBGs couple the fundamental core mode to the traveling direction. To inspect
reflection spectrum while the cladding mode which can
Figure1.3 shows the conventional reflection spectrum of FBGs. Fig shows the conventional transmission spectrum of LPFGs.
- 3 -
The fiber grating is fabricated by a UV beam incident to a phase mask The period of the fiber grating is half of the PM period.
have larger period than FBGs. UV illumination is just In addition, LPFGs have also been designed and realized by thermally induced methods such as the use of a CO2 laser [1-9]
with a conventional splicer [1-10]. These methods are cheaper and more available than UV photo-imprinting method.
Characteristics
s couple the fundamental core mode to the mode of the To inspect a fiber Bragg grating, we can observe
the LPFGs only the core couple to the co-propagated can only be observed in the transmission spectrum. 1.3 shows the conventional reflection spectrum of FBGs. Fig
shows the conventional transmission spectrum of LPFGs.
The fiber grating is fabricated by a UV beam incident to a phase mask
UV illumination is just been designed and realized 9] and arc These methods are cheaper and
mode of the reverse fiber Bragg grating, we can observe the propagated transmission spectrum. 1.3 shows the conventional reflection spectrum of FBGs. Figure1.4
- 4 -
Fig. 1.3 Conventional reflection spectrum of FBGs.
Fig. 1.4 Conventional transmission spectrum of LPFGs.
1.1-3 Applications
Long-period fiber gratings have been developed into a crucial optical component in many applications. They are used in optical communication
- 5 -
systems such as low-loss in-fiber band rejection filters [1-11], gain-flattening devices in erbium-doped fiber amplifiers (EDFAs) [1-12], highly sensitive optical fiber sensors, selective mode converters, and polarization-selective devices. They may provide options to the network designers that should influence, for example, the deployment of wavelength-division-multiplexed (WDM) systems, channel selection, and deployment of transmitters in the upstream path in a network, and should make routing viable.
In sensing applications, the conventional LPFGs have limited wavelength sensitivity of temperature and refractive index due to the tight mode field confinement. Numerous kinds of LPFGs have been proposed to enhance the sensitivity, including the short-grating period [1-13], ultra-thin cladding layer [1-14], asymmetric CO2 laser ablation [1-15]-[1-17], surface-corrugated fiber
[1-18].
1.2 Motivation of the thesis
1.2-1 Previous Work
In the previous work, our lab had produced two kinds of asymmetric long-period fiber gratings. Our first kind of long-period fiber grating is with surface corrugated structure which was ablated by focused CO2 laser[1-18].
Our second kind of long-period fiber gratings is produced by using a metal grating side-contacted on tapered fiber[1-19]. The lack of structure symmetry on the fiber cross section may cause asymmetry mode profiles and complicated mode coupling, and exhibit polarization-dependent losses. In particular, since the asymmetric structure is polarization dependent, this may cause some problems the resonance dips not sharp enough for practical
- 6 - sensing applications.
1.2-2 Motivation
We want to make a highly symmetric long-period fiber grating which is theoretically polarization independent. So we choose a cylindrical lens instead of a circular lens to assure the heating area is more symmetrical along with respect to the fiber axis. Since it has the waveguide structure to let the cladding modes spread more out of the fiber, we predict it can be highly sensitive to the environmental fluctuations. This means that it can be utilized as a sensor device for different applications.
1.3 Structure of the thesis
The present thesis comprises four chapters. Chapter 1 is an introductory chapter consisted of an introduction to fiber gratings and the motivation of our research. Chapter 2 describes the principles and analysis of our research work. We use a three-layer waveguide to calculate the mode refractive indices, which can be used to predict the coupling wavelength of the LPFG. Chapter 3 describes our experimental procedures, and shows the measurement results. Finally, chapter 4 gives the conclusions and discusses possible future works.
1.4References
[1-1] K. O. Hill, Y. Fujii, D. C. Johnson, and B. Kawasaki, “Photosensitivity in Optical Fiber Waveguides : Application to Reflection Fiber Fabrication, ” Appl. Phys. Lett. 32(10), p647(1978).
[1-2] G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method, ” Opt. Lett. 14, 823-825
- 7 - (1989).
[1-3]M. Gagné, L. Bojor, R. Maciejko, and R. Kashyap, “Novel custom fiber Bragg grating fabrication technique based on push-pull phase shifting interferometry, ” Opt. Express 16, 21550-21557 (2008).
[1-4]J. Albert, K. O. Hill, B. Malo, S. The´riault, F. Bilodeau, D.C. Johnson and L.E. Erickson, “Apodization of the spectral response of fiber Bragg gratings using a phase mask with variable diffraction efficiency,” Electron. Lett. 31, 222, (1995).
[1-5]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 fiber-scanning beam technique, ” Opt. Lett. 20, 2051-2053 (1995).
[1-6]J. B. Jensen, N. Plougmann, H.-J. Deyerl, P. Varming, J. Hubner, and M. Kristensen, “Polarization control method for ultraviolet writing of advanced Bragg gratings, ”Opt. Lett. 27, 1004-1006 (2002).
[1-7] A. Asseh, H. Storoy, Bengt E. Sahlgren, S. Sandgren, and Raoul A. H. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” Journal of Lightwave Technology, vol. 15, pp.1419-1423, (1997).
[1-8] 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. 41, 1051-1056 (2002).
[1-9]D. D. Davis, T. K. Gaylord, E. N. Glytsis, S. G. Kosinski, S. C. Mettler, and A. M. Vengsarkar, “Long period fiber grating fabrication with focused CO2 laser beams,” Electron. Lett. 34, 302-303 (1998).
- 8 -
“Investigation of the Writing Mechanism of Electric-Arc-Induced Long-Period Fiber Gratings,” Appl. Opt. 42, 3776-3779 (2003).
[1-11] A.M. Vengsarkar, P.J. Lemaire, J.B. Judkins, V. Bhatina, T. Erdogan and J.E. Sipe, “Long-period fiber gratings as band-rejection filter, ” IEEE J. Lightwave Technol. 14, 58–65(1996).
[1-12] A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano and, C. R. Davidson, “Long- period fiber-grating-based gain equalizers, ” Opt. Lett. 21, 336-338 (1996).
[1-13] H. J. Patrick, A. D. Kersey, and F. Bucholtz, “Analysis of the Response of Long Period Fiber Gratings to External Index of Refraction, ” J. Lightwave Technol. 16, 1606- (1998).
[1-14] K.-W. Chung and S. Yin, “Analysis of a widely tunable long-period grating by use of an ultrathin cladding layer and higher-order cladding mode coupling, ” Opt. Lett. 29, 812-814 (2004).
[1-15] M. Yan, S. Luo, L. Zhan, Z. Zhang, and Y. Xia, “Triple-wavelength switchable Erbium doped fiber laser with cascaded asymmetric exposure long-period fiber gratings, ”Opt. Express 15, 3685-3691 (2007)
[1-16]Y. Rao and T. Zhu, “ A Highly Sensitive Fiber-Optic Refractive Index Sensor Based on an Edge-Written Long-Period Fiber Grating, ” in Nonlinear Photonics, OSA Technical Digest (CD) ,paper JWA53 (Optical Society of America,( 2007).
[1-17] N.-K. Chen, D.-Y. Hsu, and S. Chi, “Widely tunable asymmetric long-period fiber grating with high sensitivity using optical polymer on laser-ablated cladding, ” Opt. Lett. 32, 2082-2084 (2007).
[1-18]W. Ding and S. R. Andrews, “Modal coupling in surface-corrugated long-period-grating fiber tapers, ” Opt. Lett. 33, 717-719 (2008).
- 9 -
[1-19] K.-C. Hsu, N.-K. Chen, C.-L. Lee, Y.-S. Chih, P.-J. Jhuang, Y. Lai, and C. Lin, “Spectral Response of Long-Period Fiber Grating Based on Tapered Fiber With Side-Contacted Metal Grating, ” J. Lightwave Technol. 28, 1057-1063 (2010).
- 10 -
Chapter2
Theory
In this chapter we will introduce the phase-matching condition and coupled-mode theory at first. Then we use a uniform waveguide model, which contained three layers of media, including core, cladding, and surrounding medium, to derive boundary conditions at the interfaces. With the boundary conditions, we can get the propagating conditions of the waveguide. By solving the waveguide propagating conditions, we can calculate the effect refractive indices of the fundamental core mode and higher order cladding modes propagating in the waveguide with different incident wavelengths. Combining the phase-matching condition and the effective refractive indices calculated by the model, we can figure out how resonance wavelength change as the temperature increases or the surrounding medium is changed. In order to simulate broad band transmission spectrum of LPFGs, we should also consider the material dispersion. So we will also introduce the dispersion mechanism at last.
2.1 Phase-matching condition
Figure 2.1 shows a light-wave diffracted by a fiber grating. A light-wave incident into the grating at an angle θ1 can be described by the grating
equation
nsinθ nsinθ m (2.1)
where θ2 is the angle of the diffracted wave and the integer m is the diffraction
Fig. 2.1 A light
Equation (2.1) is able to determine the wavelength at which a fiber grating can most efficiently couple lights between two modes. We can even know the coupling direction by θ2.
For Bragg gratings, θ2 is negative, and the coupling occurs between modes
traveling in opposite directions; for transmission gratings,
coupling is between modes traveling in the same direction. Since the mode propagation constant β is simply
rewrite for guided modes as β β m
Λ .
For example, first-order reflective diffraction Bragg grating, m=-1. Positive and negative
propagate in the +z and –z direction, respectively. The physical meaning is the phase-matching condition, and the
2.2.
- 11 -
Fig. 2.1 A light-wave diffracted by a fiber grating.
Equation (2.1) is able to determine the wavelength at which a fiber grating can most efficiently couple lights between two modes. We can even know the
is negative, and the coupling occurs between modes ling in opposite directions; for transmission gratings, θ2 is positive, the
coupling is between modes traveling in the same direction. Since the mode is simply β = (2π/λ)neff , where neff = ncosinθ, we may
rewrite for guided modes as
. order reflective diffraction usually dominates in a fiber 1. Positive and negative β values describe modes that z direction, respectively. The physical meaning is the matching condition, and the schematic diagram can be shown as Equation (2.1) is able to determine the wavelength at which a fiber grating can most efficiently couple lights between two modes. We can even know the
is negative, and the coupling occurs between modes is positive, the coupling is between modes traveling in the same direction. Since the mode , we may
(2.2) usually dominates in a fiber values describe modes that z direction, respectively. The physical meaning is the can be shown as figure
- 12 -
Fig. 2.2 The schematic diagram of phase-matching condition.
By using equation (2.2) and recognizing β2<0, we find that the resonant
wavelength for the reflection of a mode of index neff,1 into another mode of
index neff,2 is
mλ n , n ,Λ. (2.3)
If the two modes are identical, i.e. n , n , n , we get the familiar result for Bragg reflection：
mλ 2n Λ. (2.4)
When light is diffracted by a transmission grating, the first mode is a core mode while the second one is a cladding mode. Since here β2>0, the resonant
wavelength for a transmission grating is
mλ n , n ,Λ. (2.5)
2.2 Coupled-mode theory
In the following derivation, for simplicity a perturbation applied to the core refractive index nco will be described by [9]
∆nz ∆nz1 ν cos πΛ z z ! (2.6)
- 13 -
ν is the fringe visibility of index change, Λ is the grating period, and z describes the grating phase.
Since the long-period fiber grating couples the forward propagating guided core mode of amplitude Az to one of the co-propagating cladding modes of amplitude Az, the coupled-mode equations of long-period fiber grating can be written as
#A%&'
#' iσ(Az iκzA)z (2.7) #A%*'
#' iσ(A)z iκ+zA)z (2.8)
where the amplitudes Az and A)z are Az , A
z exp 0σ112σ 33' exp iδz 5 (2.9)
A)z , A
z exp 0σ112σ 33' exp iδz 5 (2.10)
, and σ, σ are dc coupling coefficients defined in
σ67z ω8%&z9 ε∆n = ;;;;;<6:x, y · ε;;;;<@:+x, ydxdy (2.11) The mode coupling coefficient functions are defined as
κ67z νσ67z k,j=1,2 (2.12) where ε;;;;<x, y are transverse mode fields. When j is equal to 1, it refers to the @: core mode. When j is equal to 2, it describes the cladding mode.
κ κ κ+ is the ac cross-coupling coefficient and σ( is the dc self-coupling coefficient defined as
σ ( , δ σ11Bσ33 #5 #'. (2.13)
Here the detuning, which is assumed to be constant along z, is δ, (β β) π Λ π∆n λ λD. (2.14)
cladding modes, λD, ∆n
weak grating, ∆n is the difference of the core and the cladding effective indices, and Λ is the period of the long
initial conditions A0
solutions can be found. The core mode and the cladding mode transmission coefficients for uniform long
T
G
A %&L A%&IG
cos
T
)G
A %*L A%&IG
κ3 κ32σ(2.3 Wave guiding conditions
Figure 2.3 depicts the fiber structure considered
including core, cladding, and the surrounding medium of the fiber.
Fig. 2.3 Step profile of three layer
- 14 -
Λ is the design wavelength for an infinitesimally
is the difference of the core and the cladding effective is the period of the long-period fiber grating. When appropriate 1 and A)0 0 are specified, closed form
solutions can be found. The core mode and the cladding mode transmission coefficients for uniform long-period fiber gratings can be shown to be
cos
JKκ
σ(
zL
σ(3 κ32σ(3sin
JKκ
3 σ(3sin
JKκ
σ(
zL
Wave guiding conditions
2.3 depicts the fiber structure considered as a three-layer waveguide including core, cladding, and the surrounding medium of the fiber.
Fig. 2.3 Step profile of three layer fiber.
is the design wavelength for an infinitesimally is the difference of the core and the cladding effective period fiber grating. When appropriate are specified, closed form solutions can be found. The core mode and the cladding mode transmission
σ(
zL
(2.15)(2.16)
- 15 -
The modes of such a straight fiber are approximations to the solutions of Maxwell’s equations within local regions of the LPFGs. The variables r0 and r1
are the radius of the core and cladding. The core with constitutive parameters µ0 and ε0 is embedded in the cladding with µ1 andε1, which is surrounded by
the external medium withµ2 andε2. The field distributions and propagation
constants of the modes can be obtained by solving the appropriate boundary-value problem of the wave-guiding equations. The wave-guiding equation for longitudinal components of the fields ( Ez and Hz) in the cylindrical
polar coordinates ( r, φ, z) with the z-axis along the axis of the fiber is given by =M=M JrM=ML =MφM33 k= PEH''S 0 (2.17)
where k= ωεµ k'.
Inside the core, solutions for E' and H' are Bessel functions, which are given by
E' AJUk=r cosm exp ik'z (2.18)
H' BJUk=r sinm exp ik'z (2.19)
with k' k=I ωεIµI.
In the cladding region, solutions for E' and H' are
E' CJUk=r cosm expik'z DYUk=r cosm exp ik'z (2.20)
H' FJUk=r sinm expik'z GYUk=r sinm exp ik'z (2.21)
with k' k= ωεµ.
In the surrounding, the field associated with the guiding wave is evanescent in the radial direction. By letting k= iα=, the solutions in the surrounding are E' IHU)iα=r cosm expik'z (2.22)
H' JHU)iα=r sinm expik'z (2.23)
- 16 -
In the aforementioned solutions, JU and YU are the Bessel functions of the first and second kind, respectively, and HU) is the Hankel function. There are eight coefficients A, B, C, D, F, G, I, and J, and to be solved by boundary conditions at interfaces between different dielectric media. The following four equations at r rI are obtained:
AJU k=]rI CJU k=1rI DYU k=1rI 0 (2.24) BJU k=]rI FJU k=1rI GYU k=1rI 0 (2.25) A ωε] 6^]=]JU ′ k =]rI B U6_ 6^]3 =]3JU k=1rI C ωε1 6^1=]JU ′ k =1rI D ωε1 6^1=]YU ′ k =1rI F U6_ 6^13 =]3JU k=1rI G U6_ 6^13 =]3YU k=1rI 0 (2.26) AU6_ 6^]3 = ] 3JU k=]rI B ωµ] 6^]=]JU ′ k =]rI C U6_ 6^13 = ] 3JU k=1rI D U6_ 6^13= ] 3YU k=1rI F ωµ1 6^1=]JU ′ k =1rI G ωµ1 6^1=]YU ′ k =1rI 0 (2.27)
and the other four are obtained from boundary conditions at r r, as CJU k=1r DYU k=1r IHU iα=3r 0, (2.28) FJU k=1r GYU k=1r JHU iα=3r 0, (2.29) C ωε1 6^1=1JU ′ k =1r D ωε1 6^1=1YU ′ k =1r F U6_ 6^13 = 1 3JU k=1r G U6_ 6^13 = 1 3YU k=1r I0ωε3 α^3=1HU ′ iα =3r J U6_ α^33 =13HU iα=3r 0 (2.30) CU6_ 6^13 =13JU k=1r D U6_ 6^13 =13YU k=1r F ωµ1 6^13 =13JU ′ k =1r G ωµ1 63^1=13YU ′ k =1r I U6_ α^33=13HU ′ iα =3r J 0ωµ3 α^3=1HU ′ iα =3r 0 (2.31)
Equations (2.24)–(2.31) resulted from the continuity of Ez, Hz, and Eφ, Hφ
give rise to the guidance conditions for the waveguide modes. Since the eight equations are a set of homogeneous linear equations, the determinant of the coefficient matrix has to be zero in order to have nonzero solutions. This condition is the final characteristic equation for determining the dispersion
- 17 - relation of the modes.
2.4 Dispersion Mechanism
The dispersion property is important for determining the spectral characteristics of a fiber filter component with its evanescent wave interacted with the environment. The causes of dispersion can be divided into two categories, material and waveguide dispersion. In general, the material dispersion comes from the material dispersion properties of the fiber material and the external medium while the waveguide dispersion results from the waveguiding fiber structure. For the filter fiber devices studied here, the relative refractive index difference of the fiber waveguide changes with the variation of wavelength through the material and waveguide dispersion. Therefore, the spectral responses will also vary accordingly.
These influences are important when a wide range of wavelengths are considered.
2.4-1 Material Dispersion
The material dispersion is defined as the variation of refractive index with the wavelength of light. The material dispersion of glass can be expressed by Sellmeier equation. The formula is valid at frequencies far from material resonance:
n
1 ∑
Aaλ3 λ3Bλa3 b7c (2.32) The summation is taken over all the resonance wavelengths. For silica, a three-term Sellmeier equation is typically used, accounting for the resonances in the ultraviolet and infrared:
- 18 -
n
1
A1λ3 λ3Bλ13 A3λ3 λ3Bλ33 Adλ3 λ3Bλd3 (2.33) Equation (2.42) is plotted in figure 2.4 for pure silica and for 4.1% Germania-doped silica, using the empirically determined Sellmeier coefficients given in Table 2.1[2.1-2.2]. Material A1 λ(µm) A2 λ(µm) A3 λ(µm) SiO2 [1.17] 0.6961663 0.0684043 0.4079426 0.1162414 0.8974794 9.896161 GeO2 [1.18] 0.68671749 0.072675189 0.43481505 0.11514351 0.89656582 10.002398Table 2.1 Sellmeier parameters used in Fig. 2-4
- 19 -
2.4-2 Waveguide Dispersion
In contrast to material dispersion, the waveguide dispersion due to the waveguide fiber structure doesn’t have apparent influence on the spectral characteristics. In fused-tapered fibers, the length, diameter, strain, bending, and dopant diffusion at the tapered region are the key issues for waveguide dispersion. In general, the variation of these parameters will mainly vary the dispersion slope of the guiding modes. Usually, people discussed the influences of waveguide dispersion on fiber devices based on special fibers [2.3-2.6].
However, the waveguide structure can also be locally changed to alter the waveguide dispersion for light propagation and the spectral responses of the fiber devices by using standard fibers [2.7, 2.8]. The electromagnetic field distribution is locally modified to vary the dispersion characteristics or to excite the higher-order modes [2.9]. Therefore, novel fiber devices can be achieved based on the variation of waveguide structure of the standard fiber in a small local area. In this dissertation, since the dispersion measurement instruments are not available in these cases, the influences of the variation of the waveguide structure on the spectral characteristics are studied instead.
2.5 Reference
[2.1] H. Malitson, “Interspecimen Comparison of the Refractive Index of Fused Silica.” Journal of the Optical Society of America, vol. 55, pp. 1205-1209, 1965. [2.2] W. Fleming, “Material Dispersion in Lightguide Glasses,” Electronics Letters,vol. 14, pp. 326-328, 1978.
- 20 - directional
couplers of dissimilar single-mode fibers,” J. Lightwave Technol. LT-5, 1196-1198 (1987).
[2.4] J. Chung and A. Safaai-Jazi, “Narrow-band spectral filters made of W-index and step-index fibers,” J. Lightwave Technol. 10, 42-45 (1992). [2.5] Monerie, “Propagation in doubly clad single-mode fibers,” IEEE J. Quantum Electron. QE-18, 535-542 (1982)
[2.6] A. Arbore, “Application of fundamental-mode cutoff for novel amplifiers and lasers,” in Optical Fiber Communication Conference OFC’05 (Optical Society of America, Washington, D.C., 2005), paper OFB4.
[2.7]K. Chen, S. Chi, S. M. Tseng, and Y. Lai, “Wavelength-tunable fiber codirectional coupler filter based on asymmetric side-polished fiber coupler with local dispersive intermediate layer,” submitted to ECOC 2006 conference. [2.8] K. Chen and S. Chi, “Influence of a holey cladding structure on the spectral characteristics of side-polished endlessly single-mode photonic crystal fibers,” accepted by Opt. Lett. (2006).
[2.9] K. Furusawa, A. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, Optics Express 9, 714 (2001).
- 21 -
Chapter3
Experiment setup and results
In this chapter, we will introduce the instruments used in our experimental setup first. Then, we will describe the steps of producing a LPFG sample. In the experimental results, we will show the surface structure of the sample with high resolution CCD camera and the transmission spectrum measured by a tunable laser. We use a temperature controller to measure the temperature sensitivity of the LPFG samples. Some samples are also put into the higher index surrounding media like water and ethanol for studying their properties in different operating regimes. At last, we will discuss the effects of periodic surface corrugation for a single mode fiber filter (SPF) by experimental results and simulation.
3.1 Experimental setup
3.1-1 Instruments utilized in fabricating LPFG
We fabricate the studied long-period fiber gratings in use of the point-by-point writing method by CO2 laser. In the following, we introduce the
used fibers and instruments for the fabrication:
1. Fibers
The fiber we use is the standard telecommunication single-mode step-index fiber (SMF-28) from Corning Corporate. The diameters of cladding and core are 125/8.2 µm respectively. The numerical aperture of fiber is 0.13. It is important to mention that this fiber is inexpensive and readily available.
- 22 -
2. Laser source
The instrument of our laser source is Synrad 48 series CO2 laser. The beam
diameter is 3.5 mm and the center wavelength is 10.6 µm. The maximum power is about 25 watts, but the stability (cold start) is ±5%.
Such high power laser can even heat glass to melting state. So we should be careful when operating this instrument. Before we start to run CO2 laser, we
should put on a pair of CO2-laser-protecting glasses for safe reasons. Then we
turn on the cooling first. When the cooling is working at the appropriate temperature, then we can turn on the laser.
3. Mirrors
These reflective mirrors are utilized in the wavelength of infrared ray including the wavelength of CO2 laser. We use mirrors to control the laser
beam propagation, and most important of all let the beam propagate horizontally to avoid the beam reflecting into our eyes.
4. Shutter
We utilize a relay and the aluminum block to form a basic shutter. The relay is controlled by the voltage of power supply. When the power supply is open (12 V), the shutter will be closed; when the power is closed (0 V), the shutter will be opened. So we can control the power supply with a transistor which can be connected with the port of computer by some electric circuit design.
5. Cylindrical lens
The cylindrical lens we used is from Ronar-Smith with its efficient focusing length 5.08 mm. We can focus the laser beam to about 200 µm in the z-axis. This scale is small enough for producing LPFG with period larger than 200 µm.
6. Translation Stage
- 23 -
movement 33 nm. It can be easily connected to the computer with RS-232 port and controlled automatically.
7. Computer Program
We need to fabricate LPFG automatically by computer programs to avoid the human errors. We both control the shutter and the translation by program Labview6.1.
3.1-2 Fabrication process
Figure 3.1 shows the schematic diagram of the experimental setup. Figure 3.2 shows the photography of the experimental setup (a) the overall image (b) the fiber on the fiber holder with one end tagged with a weight. First, we turn on the power of CO2 laser and use a rock to block the beam. We will wait about
ten minutes to let the power of CO2 laser stable. In the mean time, we remove
the jacket from a single mode fiber over a section about several centimeters and clean the section with ethanol. After we have cleaned the fiber, we put it on the fiber holder which is set on the translation stage. The fiber is only fixed one end on the fiber holder and the other end is tagged with a weight about 10 grams to provide a constant tension on fiber. Then we set the parameters such as the time that the shutter opens each time and the length that the translation stage moves each time. After we have set the parameters and remove the rock, we could run the programs. When the shutter is open, CO2 laser can write on
Fig. 3.1 Schematic diagram of the
Fig. 3.2 The photography
the overall image (b) the fiber on the fiber holder with one end tagged with a
- 24 -
Fig. 3.1 Schematic diagram of the experimental setup for fabricating LPFG.
(a)
(b)
photography of the experimental setup for fabricating LPFG (a) the overall image (b) the fiber on the fiber holder with one end tagged with a
weight.
experimental setup for fabricating LPFG.
of the experimental setup for fabricating LPFG (a) the overall image (b) the fiber on the fiber holder with one end tagged with a
- 25 -
3.2 Results and discussion
3.2-1 The surface structure observed by CCD camera
In our experiment, there are four main factors that will affect the characteristics of fabricated LPFGs, including the laser power, shutter open time, distance translation stage move each time, weight tagged on the fiber, number of the written period. If the open time of shutter is longer or the laser power is high, the higher energy heats the fiber. The heating time or the laser power for writing a fiber is related to the depth of corrugated surface structure. The distance of the translation stage moved each time determines the grating period. The tagged weight on the fiber reduces the heating time to have the same corrugated surface structure and also changes the core refractive index by residual stress relaxation. The number of written period determines the coupling length.After we made our sample with different parameters mentioned above, we observe the surface structure by CCD camera. Figure 3.3 shows the CCD camera we use. Figure 3.4 (a) and (b) show the surface structure of LPFGs with high resolution so that we could only observe single point. Figure 3.5 (a), (b), and (c) show the surface structure of LPFGs with low resolution, and we could see the period of the grating we made.
Fig. 3.3 shows t
Fig. 3.4 Optical micrograph of the periodically tapered LPFGs with (a) shallow (waist diameter around 104.4 µ
surface corrugation.
Fig. 3.5 Optical micrograph of the periodically periods of (a) 150
- 26 -
Fig. 3.3 shows the CCD camera we used.
Optical micrograph of the periodically tapered LPFGs with (a) shallow (waist diameter around 104.4 µm) and (b) deep (waist diameter around 65.3
surface corrugation.
Optical micrograph of the periodically tapered LPFGs with grating periods of (a) 150 µm (b) 200 µm (c) and 300 µm.
Optical micrograph of the periodically tapered LPFGs with (a) shallow m) and (b) deep (waist diameter around 65.3 µm)
- 27 -
3.2-2 The experiment of measurement
In our measurement, a tunable laser source from Agilent 8164A spanning from 1500 nm to 1640 nm was launched into the measured LPFG sample for performing measurement. In order to measure the sensitivity to the temperature, the LPFG sample was heated by a temperature controller to stabilize the temperature within 0.1 ℃. Also, we change the surrounding medium of different refractive index to see the effect on the transmission spectrum. We use a power sensor Agilent 81634B to observe the spectrum. Figure 3.6 shows the instruments, tunable laser and power sensor. Figure 3.7 demonstrates the transmission spectrum of the background light source.
Fig. 3.6 The instruments of tunable laser and power sensor.
- 28 -
The power of background is ranged from -0.5 dBm to -1 dBm. We should subtract the background light source when we analyze the transmission spectrum of our LPFG samples.
3.2-3 The transmission spectrum of LPFGs
In the following, we will show two transmission spectrums of LPFG samples, sample1 with shallow surface corrugated structure (about 10% surface depth) and sample2 with deep surface corrugated structure (about 20% surface depth). The period and number of periods are the same for these two samples. Their period is 700µm and period number is 20 points.
We will examine the sample1 first. Figure 3.8 shows the transmission spectrum of sample1 in the air with different temperatures.
Fig. 3.8 The spectral transmission of sample1in the air with tuning temperature ranged from 20 ℃ to 55.5 ℃.
As we can see that there are two resonance dips in figure 3.8. The insertion loss is about -3 dB, but the sensitivity for resonance dip near 1565nm is
1505 1540 1575 1610 1645 -18 -15 -12 -9 -6 -3 0 0.168nm / ℃
T
ra
n
s
m
is
s
io
n
(d
B
)
Wavelength(nm)
20.7℃ 25℃ 30℃ 35.4℃ 40.5℃ 46℃ 51℃ 55.5℃ 0.146nm / ℃- 29 -
around 0.146 nm/℃ and for resonance dip near 1625 nm is around 0.168 nm/℃. Both of them are much larger than conventional sensitivity of LPFG, which is usually 0.059 nm/℃. The depths of resonance dips are ranged from -9.5 dB to -14 dB.
We can observe that the resonance wavelength shift is almost linearly positive-dependent with the temperature of the environment while the resonance dip (dB) is linearly negative-dependent with the temperature ranged from 20 ℃ to 45 ℃. Figure 3.9 shows the relationship of resonance wavelength shift and resonance dip to the environmental temperature.
Fig. 3.9 The relationship of resonance wavelength shift and resonance dip to the environmental temperature for sample1 in the air.
In the following, we similarly show the transmission spectrum as the environmental medium changed. When we put sample1 under water, and we could see the transmission spectrum has much larger resonance wavelength shift and deeper resonance depth compared with sample1 in the air.
Figure 3.10 shows the transmission spectrum of sample1 under water with
20 25 30 35 40 45 50 55 1568 1569 1570 1571 1572 1573 Wavelength Resonace dip
Temperature (
℃
)
W
a
v
e
le
n
g
th
(
n
m
)
Equation y = a + b*x Value wavelengt Intercep 1565.86 wavelengt Slope 0.14686 -14 -13 -12 -11 -10 -9R
e
s
o
n
a
c
e
d
ip
(d
B
)
- 30 -
different temperature. Figure 3.11 shows the relationship of resonance wavelength shift and resonance dip to the environmental temperature.
Fig. 3.10 The spectral transmission of sample1 under water with tuning temperature ranged from 40.1 ℃ to 54.4 ℃.
Fig. 3.11 The relationship of resonance wavelength shift and resonance dip to the environmental temperature for sample1 under water.
1505 1540 1575 1610 1645 -35 -30 -25 -20 -15 -10 -5 0 0.49nm / ° C
Tr
a
n
s
m
is
s
io
n
L
o
s
s
(
d
B
)
Wavelength (nm)
40.1° C 44.6° C 48.7° C 54.4° C 0.83nm /° C 39 42 45 48 51 54 1554 1556 1558 1560 1562 1564 1566 1568 Wavelength Resonance dipTemperature (
℃
)
Wave
len
gth (nm)
Equation y = a + b*x Value wavelengt Interce 1521.70 wavelengt Slope 0.81688 -27 -26 -25 -24 -23 -22R
eson
ance
d
ip (dB)
- 31 -
We then show the transmission spectrum of sample2 in the figure 3.12 and the relationship of resonance wavelength shift and resonance dip to the environmental temperature in the figure 3.12. As we can see that the insertion loss -18 dB is much larger than sample1 and the spectrum is not perfectly smooth. However, the sensitivity of the dip near 1510 nm is about 0.238 nm/℃ and is more sensitive than sample1. This higher sensitivity is probably due to the deeper surface corrugated structure. In figure 3.13, we show the relationship of resonance wavelength shift and resonance dip with respect to the environmental temperature.
Fig. 3.12 The spectral transmission of sample2 in the air with tuning temperature ranged from 30 ℃ to 45 ℃.
1500
1530
1560
1590
1620
1650
-40
-35
-30
-25
-20
T
ra
n
s
m
is
s
io
n
(d
B
)
Wavelength (nm)
30℃
34℃
40℃
45℃
0.24nm/℃
- 32 -
Fig. 3.13 The relationship of resonance wavelength shift and resonance dip to the environmental temperature.
3.2-4 Discussion
We have observed that the coupling wavelength increases as the temperature increases. This is due to the change of the phase-matching condition, since the effective refractive indices of the core and cladding modes are related to the temperature and the grating period is also related to the linear thermal expansion of the single mode fiber.
The temperature sensitivity of LPFGs has been shown to be related to the environmental medium. The results show that the sensitivity of LPFG is better under water than in the air. This can be explained more fraction of field will leak to the environmental medium when the environmental refractive index is closer to the cladding refractive index. Since more fraction of field is in the environmental medium, the influence of the surrounding medium to the field is
30 33 36 39 42 45 1510 1511 1512 1513 1514 Wavelength Resonance dip
Temperature (
℃
)
W
a
v
e
le
n
g
th
(n
m
)
Equation y = a + b*x Value Wavelength Intercept 1502.7039 Wavelength Slope 0.24673 -18 -17 -16 -15R
e
s
o
n
a
n
c
e
d
ip
(d
B
)
- 33 -
larger. Based on the same argument, we can also explain why the sensitivity increases with deeper depth of corrugated structure. More fraction of field will leak to the environmental medium when the corrugation depth is deeper.
We have also observed that the coupling wavelength is shorter as the corrugated surface is deeper by comparing figure 3.8 with figure 3.12 and the coupling wavelength is shorter as the surrounding refractive index increases by comparing figure 3.8 and figure 3.10. According to Equation (2.5), the resonance wavelength is related to the effective index difference between fundamental mode and cladding modes, and also related to the grating period. Since the grating period is fixed for both samples, thus the decreasing resonance wavelength with increasing corrugated surface and index of surrounding media may properly due to the decreasing effective index difference of two coupling modes. In order to clarify the effective index of fiber modes, numerical simulation is performed to analyze the dispersion relation of fiber modes for specific grating periods.
3.3 Simulation
3.3-1 Assumption
We have observed the LPFG samples by a CCD camera, so the images will be limited by the resolution and aberration of the CCD camera. We can tell the diameter value of cladding by the CCD camera, but we can’t tell the diameter value of core. Therefore we will assume the ratio of core diameter and cladding diameter is the same with the original SMF-28 fiber, so we can predict the diameters of core in the tapered regions.
fiber model. We change the parameters such as
and environmental medium in our simulation. Since we only consider the modes traveling in straight
3.3-2 Simulation method
We use Mathematica6.0 as our simulation tool. The
propagating in the model have been derived in equations (2.24) to (2.31), and these equations can be written as a matrix form. Since the equations are all equal to zero, the matrix is a homogeneous matrix.
We can find out effective refr
following steps. We use a module to calculate the determinant of the matrix and return the determinant value under particular wavelength. Plot the logarithm of the module by changing the effective refractive index
figure 3.14.
Fig. 3.14 The logarithm of the module by changing the effective refractive
- 34 -
model. We change the parameters such as the indices of core, cladding, and environmental medium in our simulation. Since we only consider the
straight waveguides, we can only tell the trend of the dips.
2 Simulation method
We use Mathematica6.0 as our simulation tool. The conditions for wave propagating in the model have been derived in equations (2.24) to (2.31), and these equations can be written as a matrix form. Since the equations are all equal to zero, the matrix is a homogeneous matrix.
We can find out effective refractive indices of core and cladding by the following steps. We use a module to calculate the determinant of the matrix and return the determinant value under particular wavelength. Plot the logarithm of the module by changing the effective refractive index shown in
Fig. 3.14 The logarithm of the module by changing the effective refractive index.
indices of core, cladding, and environmental medium in our simulation. Since we only consider the , we can only tell the trend of the dips.
conditions for wave propagating in the model have been derived in equations (2.24) to (2.31), and these equations can be written as a matrix form. Since the equations are all
active indices of core and cladding by the following steps. We use a module to calculate the determinant of the matrix and return the determinant value under particular wavelength. Plot the shown in
- 35 -
The logarithm of roots will form a discontinuous point in the plot, and we can find out the approximate value of roots by the plot. Then we get the more accurate value of roots by Newton’s method, which is the built-in command in Mathematica6.0. These roots represent the effective refractive indices of mode propagating in the model, including the core mode and cladding modes. Since the fiber we use is a single mode fiber, there should be only one core mode. In figure 3.14, we can see that there are one core mode and many cladding modes, and the effective refractive index of core mode is much higher than the cladding modes.
As stated in chapter 2, the phase-matching condition describes that
Λ
8eff,%&^eB8eff,%*ghhijk
λ .
We plot 1/Λ and n ,= n ,)m##08n/λ simultaneously and the coupling wavelength can be determined by the intersection points of 1/Λ and n ,= n ,)m##08n/λ. In order to plot n ,= n ,)m##08n/λ, we need to
find n ,= and n ,)m##08n under different wavelengths.
3.3-3 Simulation results
Simulation1
We simulate the transmission spectra with different diameters of core and cladding. Since we assume the diameters of core and cladding are in the same ratio, we denote only one of them. Figure 3.15 shows the coupling wavelength with the grating period Λ = 500 µm, 510 µm, and the air as the surrounding medium, but with different radii of core (denoted as rco) (a) rco = 4.1 µm, (b) rco
= (4.1×0.95) µm.
The slashes are n ,= n ,)m##08n/λ for different cladding modes. M11-M12 means the fundamental core mode coupling to the first cladding mode,
- 36 -
M11-M13 means the fundamental core mode coupling to the second cladding
mode, and so on. The horizontal line is 1/Λ with Λ 500µm and Λ 510µm.
The intersection points are the coupling wavelengths. In figure 3.15 (a), we can observe that the wavelength of intersection points is longer when the grating period increases. The simulation result can be used to explain the red shift of coupling wavelength when the environmental temperature increases in the air.
Compare figure 3.15 (a) and (b), one can observe the coupling wavelength is shorter as the diameter of core decreases. These simulation results verify that the index difference of the two coupling modes at specific wavelength decreases with deeper surface corrugated surface, which coincides with the experimental results shown in figure 3.8 and figure 3.12.
- 37 - (a)
(b)
Fig. 3.15 The coupling wavelength with grating period Λ = 500µm and air as the surrounding medium, (a) rco = 4.1 µm, (b) rco = (4.1×0.95) µm.
1540
1610
0.0015
0.0020
0.0025
0.0030
(N
co
-N
cl
)/
λ
(1
/µ
m
)
Wavelength (nm)
M11-M12 M11-M13 M11-M14 M11-M15 M11-M16 M11-M17 M11-M18 M11-M19 500µ m 510µ m1540
1610
0.0015
0.0020
0.0025
0.0030
(N
co
-N
cl
)/
λ
(
1
/µ
m
)
Wavelength (nm)
M11-M12 M11-M13 M11-M14 M11-M15 M11-M16 M11-M17 M11-M18 M11-M19 500µ m 510µ m- 38 -
Simulation2
In the above simulation, we consider the changing diameters of core and cladding. In the following, we subtract the index of core a small number to study the corresponding changes.
Figure 3.16 shows the coupling wavelength with the grating period Λ = 500 µm and the core index change ∆nco = -5×10-4. Comparing figure 3.16 and
figure 3.15 (a), the coupling wavelength is shorter as the core index decreases.
Fig. 3.16 The coupling wavelength with grating period Λ = 500 µm and the change of core index ∆nco = -5×10-4.
Simulation3
In this simulation we consider both the parameters of the diameter and core index. We set the diameter of core to be 90﹪of SMF-28 and the change of
1540
1610
0.0015
0.0020
0.0025
0.0030
(N
c
o
-N
c
l)/
λ
(1
/µ
m
)
Wavelength (nm)
M11-M12 M11-M13 M11-M14 M11-M15 M11-M16 M11-M17 M11-M18 M11-M19 500µ m- 39 -
core index ∆nco = -5×10-4. Figure 3.17 shows the simulation results with grating
period Λ = 700µm.
Fig. 3.17 The coupling wavelength with grating period Λ = 700µm, rco = (4.1×0.9) µm and ∆nco= -5×10-4.
In figure 3.17, the coupling wavelengths are near 1580nm and 1620nm. It’s quite similar to sample1 even though we use such a simple model.
Next, we set the diameter of core to be 80﹪of SMF-28 and the change of core index ∆nco = -5×10-4. Figure 3.18 in the next page shows the results with
the grating period Λ = 700µm. It’s also quite similar to the experimental results of sample2.
1540
1610
0.0010
0.0015
0.0020
0.0025
0.0030
(N
c
o
-N
c
l)/
λ
(1
/µ
m
)
Wavelength (nm)
M11-M12 M11-M13 M11-M14 M11-M15 M11-M16 M11-M17 M11-M18 M11-M19 700µ m- 40 -
Fig. 3.18 The coupling wavelength with grating periodΛ = 700µm, rco = (4.1×0.8) µm and ∆nco = -5×10-4.
3.4 Possible improvement of the fabrication
process
The obtained results could be better if we could make some improvements in our fabrication method. First, the period is not very good under CCD camera and some of the points are deeper and some of them are shallower. The non-uniformity may cause large insertion loss in the transmission spectrum.
There are two reasons for causing the non-uniformity of the LPFGs: the instability of CO2 laser and the rough design of shutter. The CO2 laser we use
has ±5% error, but we can open air conditioners to help laser more stable. The shutter is designed by a delay, which has problems when the switching time interval is short. When the heating time is short, this shutter is not accurate
1540
1610
0.0010
0.0015
0.0020
0.0025
0.0030
(N
c
o
-N
c
l)
/λ
(
1
/µ
m
)
Wavelength (nm)
M11-M12 M11-M13 M11-M14 M11-M15 M11-M16 M11-M17 M11-M18 M11-M19 700µ m- 41 -
enough. Also, I have found that the transmission power passing through the shutter is not equal each time. This may be the main reason that was causing the non-uniformity. Better results can be expected by using a better shutter with short response time and the same transmission power.
