平面光波导衍射场特性分析及其在参数测量中的应用
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摘要
随着信息社会的到来，人们对信息的需求量与日俱增。平面光波导作为集成光路中重要组成元素，对其参数和性能测量显得尤为重要，测量所得的参数和数据可以为集成光路的制作工艺提供依据和参考，从而提高集成光学系统的工作性能。本论文主要围绕着平面光波导折射率测量方法展开讨论，分析了星形耦合器的总耦合效率。本论文从基尔霍夫衍射积分公式以及平面光波导的模场分布出发，导出了平面光波导的远场衍射表达式，分析了低阶模的衍射特征，并给出了它们的衍射场的相对强度分布曲线图，分析了TE基模的衍射场中央亮条纹光强下降到最大值1／e~2随波导归一化频率的变化情况以及TE_1、TE_2、TE_3模极大值对应的空间频率随着归一化频率的变化情况，并用指数函数进行了拟合，拟合的结果跟从衍射表达式数值求解的结果吻合得较好。本文介绍并讨论了高斯光源与光波导之间的耦合、两相邻光波导之间的耦合系数，从空间非接触的两光波导之间耦合效率出发，分析了星形耦合器的耦合效率问题。最后，在传输模近场法的基础上讨论了一种基于远场扫描来测量渐变折射率光波导的方法，分析了衍射场空间角步长和构建波导折射率分布区域之间的关系，讨论和分析了三种有限差分方式的精度问题，重构波导折射率分布应用了利用最小二乘法得到的逆矩阵算法，这种测量折射率分布的方法具有高空间分辨率以及高灵敏度的优点。
With information society on the horizon, the information demand is always increasing dramatically. Planar optical waveguides are the base of the integrated optical circuits, and measurement of their parameters and performance is very imperative for improving the performance of devices, because the data from the measure provide a theoretical base and guidance for the fabrication technique of integrated optical circuits in turn. This paper centers on the measurement of refractive index profile of planar waveguide, and in addition, the total coupling of star coupler is analyzed. Firstly, the far-field diffraction equation of planar waveguide's end-surface is derived from Kirchhoff diffraction integral equation, and its diffractive characteristics are analyzed for the relations between diffraction field and waveguide parameters. Secondly, the coupling efficiency is reviewed and discussed for between the Gaussian beam and planar waveguide, two adjacent planar waveguide and NxN star couplers. Finally, reconstruction of graded refractive index profile of planar wave is presented on the base of inverse matrix algorithm for near-field method, applying discrete Fourier transform and central finite difference method, and herein sampling step of far-field diffraction, inverse Fourier transform and central finite difference method are systematically and detailedly discussed.
With information society on the horizon, the information demand is always increasing dramatically. Planar optical waveguides are the base of the integrated optical circuits, and measurement of their parameters and performance is very imperative for improving the performance of devices, because the data from the measure provide a theoretical base and guidance for the fabrication technique of integrated optical circuits in turn. This paper centers on the measurement of refractive index profile of planar waveguide, and in addition, the total coupling of star coupler is analyzed. Firstly, the far-field diffraction equation of planar waveguide's end-surface is derived from Kirchhoff diffraction integral equation, and its diffractive characteristics are analyzed for the relations between diffraction field and waveguide parameters. Secondly, the coupling efficiency is reviewed and discussed for between the Gaussian beam and planar waveguide, two adjacent planar waveguide and NxN star couplers. Finally, reconstruction of graded refractive index profile of planar wave is presented on the base of inverse matrix algorithm for near-field method, applying discrete Fourier transform and central finite difference method, and herein sampling step of far-field diffraction, inverse Fourier transform and central finite difference method are systematically and detailedly discussed.
引文
1 吴重庆编著．光波导理论[M]．第2版，北京：清华大学出版社，2005．
2 刘增基，周洋溢，胡辽林等编著．光纤通信[M]．西安：西安电子科技大学出版社，2001：1-4
3 丁玥，逆矩阵法处理近场模式测定介质光波导折射率[D]．吉林大学硕士论文，2005：53-69
4 西原浩，春名正光，栖原敏明[日本]著．集成光路[M]．梁瑞林译．北京：科学出版社，2004：1-7
5 Conwell E. M.. Modes in optical waveguides formed by diffusion[J]. Appl. Phys. Lett., 1973, 26(4): 328-329
6 曹庄琪．非均匀平板波导的色散方程[J]．光学学报，1994，14(1)：1223-1226
7 Suematsu Y., Furuya K.. Propagation mode and scattering loss of a two dimensional dielectric waveguide with gradual distribution of refractive index [J], IEEE Trans. Microwave Theory Tech.. 1972, 20(8): 524-530
8 Marcuse D.. TE modes of graded-index slab waveguides [J]. IEEE J. quantum Electron., 1973, 9(10): 1000-1006
9 Hocker G. B., Burns W. K.. Modes in diffused optical waveguides of arbitrary index profile [J]. IEEE J. Quantum Electron., 1975, 11(6): 270-276
10 Shiff L. I.. quantum mechanics[M]. McGow-Hill, 1968:268-279
11 Xiang F, Yip G L. A modified WKB method for the improved phase shift at a turning point [J]. J. of Lightwave Technol., 1994, 12(3): 443-452
12 Wang J. Y., Qiao L. a refined WKB method for symmetric planar waveguides with truncated-index profiles and graded-index profiles [J]. IEEE J. Quant. Electron., 1991, 27(4):878-883
13 曹庄琪．导波光学中的转移矩阵问题[M]．上海：上海交通大学出版社，2000：79-95
14 Liu Q, Cao Z Q, Shen Q S, et al. Analysis of planar waveguides with index discontinuity of index slop discontinuity [J]. Opt. and Quant. Electron., 2001, 33(6): 675-683
15 郭福源，林斌，陈钰清等．平面光波导衍射特性分析[J]．光学学报，2002，22(12)：1513-1517
16 Yoshimura H., Takai N., Asakura T. Far-field analysis of fluctuating supermodes radiated from phase-locked array lasers [J]. IEEE J. of quant. Electron., 1994, 30(2): 222-229
17 Mcgreer K. A. Diffraction from concave gratings in planar waveguides[J]. IEEE Photon. Technol. Lett. 1995, 7(3): 324-326.
18 郭长志．半导体激光模式理论[M]．北京：人民邮电出版社，1989：87-135
19 Bledowski A., Zakowicz W.. radiation properties of a planar dielectric waveguide loaded with conducting-strip diffraction grating [J]. IEEE Trans. On Microwave Theory and Tech., 1997, 45(9): 1637-1640
20 郭福源，林斌，陈钰清．平面光波导基模的衍射特性．激光与红外[J]，2002，32(5)：323-326
21 郭福源，林斌，陈钰清等．介质平面波导TE_0模衍射场的光束参量[J]．光学学报，2003，23(6)：702-706
22 上官成木生，郭福源，卢勇．平面波导TE_0模的远场衍射特性分析[J]．光电子·激光，2004，15(3)：299-302
23 秦秉坤，孙雨南编著．介质光波导及其应用[M]．北京：北京理工大学出版社，1991：177-198
24 White K. J.. Practical application of the refracted near-field technique for measurement of optical fiber refractive index profile [J]. Optical Quantum Electronics, 1979, 11(2): 185-196
25 Ganfield B. K., Vigil S. R., Kuzyk M. G. Refractive and dispersive measurement of polymer optical fibers and fiber perform[C]s. Proc. of SPIE, 1998, 3473:34-42
26 Koike Y, Ishigure T., Nihel E.. High-bandwidth graded-index polymer optical fiber. J. Lightwave Technol., 1995, 13(7): 1475-1489
27 付国兰，曹庄琪，丁渊，等．渐变折射率波导表面折射率的确定[J]．光学学报，2005，25(2)：161-164
28 Marcuse D. Refractive index determination by the focusing method[J]. Appl. Opt., 1979, 18(1):14-22
29 Zheng S. L., Zhang X. M., Chen K. S., et al. Applicability of the focusing method to measure the refractive profile of polymer optical fibers[J]. Acta Photonica Sinica, 2005, 34(8): 1205-1208.
30 Morishita K.. Measurement of refractive-index profile of single-mode fibers by the propagation-mode near-field method [J]. J. Lightwave Technol., 1985, 3(2):244-247
31 郝寅雷，吴亚明．迭代法重建平板光波导折射率分布[J]．光学学报，2005，25(6)：726-730
32 Iga K., Kokubn Y., Yamamoto N. refractive index profile measurement of the focusing fibers by using transverse differential interferometry [J].Trans. IECE Jpn, 1976, 60(76): 1176-1179
33 Morishita K.. Index profiling of three-dimensional optical waveguides by the propagation-mode near-field method[J], J. Lightwave Technol., 1986, 4(8): 1120-1124
34 马卫东，刘青，胡国华等．平面光波导技术及其实用器件[J]．光纤通信技术，2003，7：27-28
35 Khomchenko V. A.. m-lines technique application for studying of optical nonlinearities in thin films ate a low light intensity. Optics Communications, 2002, 201:363-372
36 Puscas N. N., Grobnic D. M., Popescu I. M., et al. Characterization of Er~(3+) doped Ti: LiNbO_3 waveguides: Losses, absorption spectra, and near field measurements [J]. Opt. Eng, 1996, 35:1311-1318
37 Stewart W. J.. Optical fiber and perform profiling technology [J]. IEEE transactions on microwave theory and technique. 30(10): 1439-1454
38 Goodman J. W.. Introduction to Fourier Optics [M]. second edition. Singapore: McGRAW-HILL Companies: 32-61
39 周树同 编译，光纤理论与测量[M]．上海：复旦大学出版社，1988：178-218
40 国分泰雄著．光波工程[M]．王友功译．北京：科学出版社，2002：221-231．
41 Yacoubian A, Lin W, Olson D, et al. electro-optic modulator coupling loss improvement by tapering waveguides and fibers[C]. Aerospace Conference, 2001, IEEE Proceedings, 2001, 3: 1481-1488
42 伍晓明，章燕申．高斯光束-光波导的耦合[J]．光电子·激光，2002，13(2)：162-164
43 Hardy A., Streifer W.. coupled mode theory of parallel waveguides[J]. J. of Lightwave Technol., 1985, 3(5):1135-1146
44 Yuen W. P.. On the different formulations of the coupled-mode theory for parallel dielectric waveguides[J]. J. of Lightwave Technol., 1994, 12(1):82-85
45 韦珏，徐晓峰，耿和平等．平面光波导折射率分布的测定[J]．光子学报．2005，34(4)： 534-536
46 郑志强．功能聚合物光纤及波导放大特性研究[D]．中国科学技术大学博士论文，2004：14-16
47 郭福源．光波导理论及其在波分复用器件中应用的研究[D]．浙江大学博士论文．2003．
48 Gaskill J. D.. Linear system, Fourier Transforms, and Optics[M]. New York: John Wiley & Sons, Inc., 1978:385-390
49 谢建平，明海．近代光学基础[M]．合肥：中国科学技术大学出版社，1989：87-93
50 Dragone C.. Efficient N×N star couplers using Fourier optics. J. Lightwave technol., 1989, 7(3): 479-489
51 邱亮，丁竹，杨祥林．N×N星形平板介质光波导耦合器的理论模型[J]．东南大学学报，1998，28(1)：33-36
52 Say S. G., Irene G. H. O.. Reconstruction of bandlimited signals from irregular sampling[J]. Signal Processing. 1995, 46:315-329
53 Abdul J. J.. The Shannon sampling theorem-its various extensions and applications: a tutorial review[J]. Proceedings of the IEEE, 1977, 65(11): 1565-1596
54 Raghu M., Roni Y.. Reconstruction error characterization and control: a sampling theory approach [J]. IEEE transactions on visualization and computer graphics. 1996, 2 (4): 364-378
55 Urbach H. P.. Generalised sampling theorem for band-limited functions[J]. Mathematical and Computer Modelling, 2003,38:133-140
56 Brooks D., Ruschin S.. Improved near-field method for refractive index measurement of optical waveguides[J]. IEEE Photon. Technol. Lett., 1996, 8:254-256
57 Lin J. H., Chen C. K.. An inverse algorithm to calculate the refractive index profile of periodically segmented waveguides from the measured near-field intensities[J]. J. of Lightwave Technol., 2002, 20(1): 58-64
58 梁昆淼编，刘法，缪国庆修订．数学物理方法[M]．第三版，北京：高等教育出版社，1999：462-467
59 葛德彪，玉波．电磁场时域有限差分方法[M]．西安：西安电子科技大学出版社，2002：8-24
60 Federico C., Francesco S., Ibrahim M., et al. A finite differences method for the reconstruction of refractive index profiles from near-field measurements [J]. J. of Lightwave Technol.. 1998, 16 (7):1348-1353
61 费丰，孙权社．新型单模光纤模场直径测试系统[J]．航空计测技术，2000，20(5)：15-18
2 刘增基，周洋溢，胡辽林等编著．光纤通信[M]．西安：西安电子科技大学出版社，2001：1-4
3 丁玥，逆矩阵法处理近场模式测定介质光波导折射率[D]．吉林大学硕士论文，2005：53-69
4 西原浩，春名正光，栖原敏明[日本]著．集成光路[M]．梁瑞林译．北京：科学出版社，2004：1-7
5 Conwell E. M.. Modes in optical waveguides formed by diffusion[J]. Appl. Phys. Lett., 1973, 26(4): 328-329
6 曹庄琪．非均匀平板波导的色散方程[J]．光学学报，1994，14(1)：1223-1226
7 Suematsu Y., Furuya K.. Propagation mode and scattering loss of a two dimensional dielectric waveguide with gradual distribution of refractive index [J], IEEE Trans. Microwave Theory Tech.. 1972, 20(8): 524-530
8 Marcuse D.. TE modes of graded-index slab waveguides [J]. IEEE J. quantum Electron., 1973, 9(10): 1000-1006
9 Hocker G. B., Burns W. K.. Modes in diffused optical waveguides of arbitrary index profile [J]. IEEE J. Quantum Electron., 1975, 11(6): 270-276
10 Shiff L. I.. quantum mechanics[M]. McGow-Hill, 1968:268-279
11 Xiang F, Yip G L. A modified WKB method for the improved phase shift at a turning point [J]. J. of Lightwave Technol., 1994, 12(3): 443-452
12 Wang J. Y., Qiao L. a refined WKB method for symmetric planar waveguides with truncated-index profiles and graded-index profiles [J]. IEEE J. Quant. Electron., 1991, 27(4):878-883
13 曹庄琪．导波光学中的转移矩阵问题[M]．上海：上海交通大学出版社，2000：79-95
14 Liu Q, Cao Z Q, Shen Q S, et al. Analysis of planar waveguides with index discontinuity of index slop discontinuity [J]. Opt. and Quant. Electron., 2001, 33(6): 675-683
15 郭福源，林斌，陈钰清等．平面光波导衍射特性分析[J]．光学学报，2002，22(12)：1513-1517
16 Yoshimura H., Takai N., Asakura T. Far-field analysis of fluctuating supermodes radiated from phase-locked array lasers [J]. IEEE J. of quant. Electron., 1994, 30(2): 222-229
17 Mcgreer K. A. Diffraction from concave gratings in planar waveguides[J]. IEEE Photon. Technol. Lett. 1995, 7(3): 324-326.
18 郭长志．半导体激光模式理论[M]．北京：人民邮电出版社，1989：87-135
19 Bledowski A., Zakowicz W.. radiation properties of a planar dielectric waveguide loaded with conducting-strip diffraction grating [J]. IEEE Trans. On Microwave Theory and Tech., 1997, 45(9): 1637-1640
20 郭福源，林斌，陈钰清．平面光波导基模的衍射特性．激光与红外[J]，2002，32(5)：323-326
21 郭福源，林斌，陈钰清等．介质平面波导TE_0模衍射场的光束参量[J]．光学学报，2003，23(6)：702-706
22 上官成木生，郭福源，卢勇．平面波导TE_0模的远场衍射特性分析[J]．光电子·激光，2004，15(3)：299-302
23 秦秉坤，孙雨南编著．介质光波导及其应用[M]．北京：北京理工大学出版社，1991：177-198
24 White K. J.. Practical application of the refracted near-field technique for measurement of optical fiber refractive index profile [J]. Optical Quantum Electronics, 1979, 11(2): 185-196
25 Ganfield B. K., Vigil S. R., Kuzyk M. G. Refractive and dispersive measurement of polymer optical fibers and fiber perform[C]s. Proc. of SPIE, 1998, 3473:34-42
26 Koike Y, Ishigure T., Nihel E.. High-bandwidth graded-index polymer optical fiber. J. Lightwave Technol., 1995, 13(7): 1475-1489
27 付国兰，曹庄琪，丁渊，等．渐变折射率波导表面折射率的确定[J]．光学学报，2005，25(2)：161-164
28 Marcuse D. Refractive index determination by the focusing method[J]. Appl. Opt., 1979, 18(1):14-22
29 Zheng S. L., Zhang X. M., Chen K. S., et al. Applicability of the focusing method to measure the refractive profile of polymer optical fibers[J]. Acta Photonica Sinica, 2005, 34(8): 1205-1208.
30 Morishita K.. Measurement of refractive-index profile of single-mode fibers by the propagation-mode near-field method [J]. J. Lightwave Technol., 1985, 3(2):244-247
31 郝寅雷，吴亚明．迭代法重建平板光波导折射率分布[J]．光学学报，2005，25(6)：726-730
32 Iga K., Kokubn Y., Yamamoto N. refractive index profile measurement of the focusing fibers by using transverse differential interferometry [J].Trans. IECE Jpn, 1976, 60(76): 1176-1179
33 Morishita K.. Index profiling of three-dimensional optical waveguides by the propagation-mode near-field method[J], J. Lightwave Technol., 1986, 4(8): 1120-1124
34 马卫东，刘青，胡国华等．平面光波导技术及其实用器件[J]．光纤通信技术，2003，7：27-28
35 Khomchenko V. A.. m-lines technique application for studying of optical nonlinearities in thin films ate a low light intensity. Optics Communications, 2002, 201:363-372
36 Puscas N. N., Grobnic D. M., Popescu I. M., et al. Characterization of Er~(3+) doped Ti: LiNbO_3 waveguides: Losses, absorption spectra, and near field measurements [J]. Opt. Eng, 1996, 35:1311-1318
37 Stewart W. J.. Optical fiber and perform profiling technology [J]. IEEE transactions on microwave theory and technique. 30(10): 1439-1454
38 Goodman J. W.. Introduction to Fourier Optics [M]. second edition. Singapore: McGRAW-HILL Companies: 32-61
39 周树同 编译，光纤理论与测量[M]．上海：复旦大学出版社，1988：178-218
40 国分泰雄著．光波工程[M]．王友功译．北京：科学出版社，2002：221-231．
41 Yacoubian A, Lin W, Olson D, et al. electro-optic modulator coupling loss improvement by tapering waveguides and fibers[C]. Aerospace Conference, 2001, IEEE Proceedings, 2001, 3: 1481-1488
42 伍晓明，章燕申．高斯光束-光波导的耦合[J]．光电子·激光，2002，13(2)：162-164
43 Hardy A., Streifer W.. coupled mode theory of parallel waveguides[J]. J. of Lightwave Technol., 1985, 3(5):1135-1146
44 Yuen W. P.. On the different formulations of the coupled-mode theory for parallel dielectric waveguides[J]. J. of Lightwave Technol., 1994, 12(1):82-85
45 韦珏，徐晓峰，耿和平等．平面光波导折射率分布的测定[J]．光子学报．2005，34(4)： 534-536
46 郑志强．功能聚合物光纤及波导放大特性研究[D]．中国科学技术大学博士论文，2004：14-16
47 郭福源．光波导理论及其在波分复用器件中应用的研究[D]．浙江大学博士论文．2003．
48 Gaskill J. D.. Linear system, Fourier Transforms, and Optics[M]. New York: John Wiley & Sons, Inc., 1978:385-390
49 谢建平，明海．近代光学基础[M]．合肥：中国科学技术大学出版社，1989：87-93
50 Dragone C.. Efficient N×N star couplers using Fourier optics. J. Lightwave technol., 1989, 7(3): 479-489
51 邱亮，丁竹，杨祥林．N×N星形平板介质光波导耦合器的理论模型[J]．东南大学学报，1998，28(1)：33-36
52 Say S. G., Irene G. H. O.. Reconstruction of bandlimited signals from irregular sampling[J]. Signal Processing. 1995, 46:315-329
53 Abdul J. J.. The Shannon sampling theorem-its various extensions and applications: a tutorial review[J]. Proceedings of the IEEE, 1977, 65(11): 1565-1596
54 Raghu M., Roni Y.. Reconstruction error characterization and control: a sampling theory approach [J]. IEEE transactions on visualization and computer graphics. 1996, 2 (4): 364-378
55 Urbach H. P.. Generalised sampling theorem for band-limited functions[J]. Mathematical and Computer Modelling, 2003,38:133-140
56 Brooks D., Ruschin S.. Improved near-field method for refractive index measurement of optical waveguides[J]. IEEE Photon. Technol. Lett., 1996, 8:254-256
57 Lin J. H., Chen C. K.. An inverse algorithm to calculate the refractive index profile of periodically segmented waveguides from the measured near-field intensities[J]. J. of Lightwave Technol., 2002, 20(1): 58-64
58 梁昆淼编，刘法，缪国庆修订．数学物理方法[M]．第三版，北京：高等教育出版社，1999：462-467
59 葛德彪，玉波．电磁场时域有限差分方法[M]．西安：西安电子科技大学出版社，2002：8-24
60 Federico C., Francesco S., Ibrahim M., et al. A finite differences method for the reconstruction of refractive index profiles from near-field measurements [J]. J. of Lightwave Technol.. 1998, 16 (7):1348-1353
61 费丰，孙权社．新型单模光纤模场直径测试系统[J]．航空计测技术，2000，20(5)：15-18