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A simple analytical method is proposed to obtain the exact propagation constant and distribution of electric field intensity of optical waveguides with graded refractive index profile. The method is based on the Wenzel-Kramers-Brillouin (WKB) solution, variational method, modified eigen-equations and discretized scalar wave equation for planar optical waveguide. The expressions of the distribution of electric field intensity based on the conventional WKB method, which diverge around the turning point, have been demonstrated to be very exact in the region beyond the turning point where the refractive index profile varies slowly. The proposed method uses the conventional WKB method to calculate the values of electric field intensity at two adjacent positions beyond the turning point and then the electric field intensity profile for the whole region is obtained by making use of the two calculated values. Two simple and explicit formulas are deduced from the discretized scalar wave equation, which provide a relationship among the values of electric field intensity at three adjacent positions. If the effective refractive index of optical waveguide and the refractive index profile for the whole region are known, we can obtain the value of electric field intensity at any position according to the corresponding values at the adjacent positions by using the two formulas aforementioned. By using the two values calculated by WKB method, the electric field intensity profile for the whole region can be determined through the iterative use of the two formulas. The accuracy of the electric field intensity profile determined by the proposed method is greatly dependent on the accuracy of the applied value of the effective refractive index. To achieve exact propagation constant and distribution of electric field intensity, the variational method and modified eigen-equations are employed in the proposed method. Variational method is a very useful method to improve the accuracy of the propagation constant in the analysis of optical waveguide with step-asymmetrical graded refractive index profile. By combining the traditional variational method and calculation of electric field intensity profile by the proposed method, the improved variational method is presented to obtain the exact propagation constant of optical waveguide. The value of propagation constant calculated by WKB method and the corresponding electric intensity field profile determined by the proposed method are chosen as the initial trial value and trial function in the variational method. Propagation constant and the corresponding electric field intensity profile with better accuracy can be achieved by the variational calculation and then are regarded as the new trial value and trial function. By the iterative use of the variational method and calculation of electric field intensity profile by the proposed method at finite times, quite accurate results are obtained. The modified eigen-equations in combination with the proposed method is another approach to calculating accurate propagation constants of optical waveguides with both the step-asymmetrical and symmetrical graded index profile. In comparison with other published methods, the proposed method has the advantages of the simplicity and considerable accuracy.
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Keywords:
- graded refractive index profile /
- optical waveguides /
- Wentzel-Kramers-Brillouin method /
- variational method
[1] Howerton M M, Moeller R P, Greenblatt A S, Krahenbuhl R 2000 IEEE Photon. Technol. Lett. 12 792
[2] Xue T, Yu J, Yang T X, Ni W J, Li S C 2002 Acta Phys. Sin. 51 1521 (in Chinese) [薛挺, 于建, 杨天新, 倪文俊, 李世忱 2002 物理学报 51 1521]
[3] Wang D L, Sun J Q, Wang J 2008 Acta Phys. Sin. 57 252 (in Chinese) [汪大林, 孙军强, 王健 2008 物理学报 57 252]
[4] Wei Z J, Wan W, Wang J D, Liao C J, Liu S H 2011 Acta Phys. Sin. 60 094216 (in Chinese) [魏正军, 万伟, 王金东, 廖常俊, 刘颂豪 2011 物理学报 60 094216]
[5] Camy P, Román J E, Willems F W, Hempstead M, van der Plaats J C, Prel C, Béguin A, Koonen A M J, Wilkinson J S, Lerminiaux C 1996 IEEE Electron. Lett. 32 321
[6] Koshiba M, Suzuki M 1982 \textit IEEE Electron. Lett. 18 579
[7] Lagu R, Ramaswamy R 1986 IEEE J. Quantum Electron. 22 968
[8] Shao G W, Jin G L 2009 Chin. Phys. B 18 1096
[9] Goyal I C, Gallawa R L, Ghatak A K 1991 Opt. Lett. 16 30
[10] Goyal I C, Jindal R, Ghatak A K 1997 IEEE J. Lightwave Technol. 15 2179
[11] Popescu V A 2004 Opt. Commun. 234 177
[12] Popescu V A 2006 Phys. Lett. A 349 220
[13] Gedeon A 1974 Opt. Commun. 12 329
[14] Janta J, \vCtyroky J 1978 Opt. Commun. 25 49
[15] Feng X, Gar L Y 1994 IEEE J. Lightwave Technol. 12 443
[16] Srivastava R, Kao C, Ramaswamy R V 1987 IEEE J. Ligthtwave Technol. 5 1605
[17] Chung M S, Kim C M 2000 IEEE J. Ligthtwave Technol. 18 878
[18] Cao Z Q, Jiang Y, Shen Q S, Dou X M, Chen Y L 1999 J. Opt. Soc. Am. A 16 2209
[19] Zhan L, Cao Z Q 1998 J. Opt. Soc. Am. A 15 713
[20] Zhu H D, Ding Y, Cao Z Q, Shen Q S 2005 Chin. Phys. Lett. 22 1580
[21] Cao X Q, Liu Q, Jiang Y, Shen Q S, Dou X M 2001 J. Opt. Soc. Am. A 18 2161
[22] Eghlidi M H, Mehrany K, Rashidian B 2005 J. Opt. Soc. Am. B 22 1521
[23] Zariean N, Sarrafi P, Mehrany K, Rashidian B 2008 IEEE J. Quantum Electron. 44 324
[24] Henry C H, Verbeek B H 1989 IEEE J. Ligthtwave Technol. 7 308
[25] Wang L, Huang N 1999 IEEE J. Quantum Electron. 35 1351
[26] Ghasemifard F, Shahabadi M 2011 J. Opt. 13 125703
[27] Gric T, Cada M 2015 \textit J. Electromagn. Wave Appl. 29 124
[28] Cao Z Q 2007 \textit Wave Guiding Optics (Beijing: Science Press) p61 (in Chinese) [曹庄琪 2007 导波光学(北京: 科学出版社) 第61页]
[29] Conwell E 1973 Appl. Phys. Lett. 23 328
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[1] Howerton M M, Moeller R P, Greenblatt A S, Krahenbuhl R 2000 IEEE Photon. Technol. Lett. 12 792
[2] Xue T, Yu J, Yang T X, Ni W J, Li S C 2002 Acta Phys. Sin. 51 1521 (in Chinese) [薛挺, 于建, 杨天新, 倪文俊, 李世忱 2002 物理学报 51 1521]
[3] Wang D L, Sun J Q, Wang J 2008 Acta Phys. Sin. 57 252 (in Chinese) [汪大林, 孙军强, 王健 2008 物理学报 57 252]
[4] Wei Z J, Wan W, Wang J D, Liao C J, Liu S H 2011 Acta Phys. Sin. 60 094216 (in Chinese) [魏正军, 万伟, 王金东, 廖常俊, 刘颂豪 2011 物理学报 60 094216]
[5] Camy P, Román J E, Willems F W, Hempstead M, van der Plaats J C, Prel C, Béguin A, Koonen A M J, Wilkinson J S, Lerminiaux C 1996 IEEE Electron. Lett. 32 321
[6] Koshiba M, Suzuki M 1982 \textit IEEE Electron. Lett. 18 579
[7] Lagu R, Ramaswamy R 1986 IEEE J. Quantum Electron. 22 968
[8] Shao G W, Jin G L 2009 Chin. Phys. B 18 1096
[9] Goyal I C, Gallawa R L, Ghatak A K 1991 Opt. Lett. 16 30
[10] Goyal I C, Jindal R, Ghatak A K 1997 IEEE J. Lightwave Technol. 15 2179
[11] Popescu V A 2004 Opt. Commun. 234 177
[12] Popescu V A 2006 Phys. Lett. A 349 220
[13] Gedeon A 1974 Opt. Commun. 12 329
[14] Janta J, \vCtyroky J 1978 Opt. Commun. 25 49
[15] Feng X, Gar L Y 1994 IEEE J. Lightwave Technol. 12 443
[16] Srivastava R, Kao C, Ramaswamy R V 1987 IEEE J. Ligthtwave Technol. 5 1605
[17] Chung M S, Kim C M 2000 IEEE J. Ligthtwave Technol. 18 878
[18] Cao Z Q, Jiang Y, Shen Q S, Dou X M, Chen Y L 1999 J. Opt. Soc. Am. A 16 2209
[19] Zhan L, Cao Z Q 1998 J. Opt. Soc. Am. A 15 713
[20] Zhu H D, Ding Y, Cao Z Q, Shen Q S 2005 Chin. Phys. Lett. 22 1580
[21] Cao X Q, Liu Q, Jiang Y, Shen Q S, Dou X M 2001 J. Opt. Soc. Am. A 18 2161
[22] Eghlidi M H, Mehrany K, Rashidian B 2005 J. Opt. Soc. Am. B 22 1521
[23] Zariean N, Sarrafi P, Mehrany K, Rashidian B 2008 IEEE J. Quantum Electron. 44 324
[24] Henry C H, Verbeek B H 1989 IEEE J. Ligthtwave Technol. 7 308
[25] Wang L, Huang N 1999 IEEE J. Quantum Electron. 35 1351
[26] Ghasemifard F, Shahabadi M 2011 J. Opt. 13 125703
[27] Gric T, Cada M 2015 \textit J. Electromagn. Wave Appl. 29 124
[28] Cao Z Q 2007 \textit Wave Guiding Optics (Beijing: Science Press) p61 (in Chinese) [曹庄琪 2007 导波光学(北京: 科学出版社) 第61页]
[29] Conwell E 1973 Appl. Phys. Lett. 23 328
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