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Being an important optical phenomenon, the linear electro-optic effect has diverse applications in the optical modulation and optical switching. The refractive index ellipsoid theory has been widely used to study the linear electro-optic effect for a long time. Despite of its visualization such a theory has limitations and cannot deal with a lot of cases in which the linear absorption cannot be neglected, or the electric displacement vector has a nonlocal response to electric field, etc. To overcome such shortcomings, in 2001 a wave coupling theory of linear electro-optic effect was developed by She and Lee (She W, Lee W 2001 Opt. Commun. 195 303). And in 2016 we generalized this wave coupling theory to the treatment of nonlocal linear electro-optic effect in which the displacement vector has a nonlocal response to electric field. In this paper, we use this wave-coupling theory to investigate how the linear absorption influences the linear electro-optic effect in a nonlocal medium. Starting from Maxwell's equations and considering the linear absorption and the nonlocality of the susceptibility tensors, we obtain two coupling equations for two orthogonal linear polarized waves and also analytical solutions of the resulting equations, which can be used to describe the nonlocal linear electro-optic effect for a light beam propagating along any direction, with an external direct current electric field applied along an arbitrary direction in a linear absorbent crystal. With such solutions, we study the influences of the linear absorption on the phase, amplitude, shape of the output beam, as well as the half-wave voltage and the extinction ratio of electro-optic modulation. The results show that no matter whether there exists linear absorption, the Rayleigh distance of the Gaussian beam in the crystal will be shortened as a result of the nonlocality of (1). When linear-absorption coefficients 11 and 22 are equal, the linear absorption damps equally the amplitudes of the two polarized output beams with keeping their phases and shapes unchanged. So in the case of 11=22, just as in a lossless medium, the phenomenon that the output beam is no longer a Gaussian beam in an electro-optic amplitude modulation scheme can be considered as a possible signal of the nonlocal response of (2). More interestingly, when 1122, the linear absorption not only reduces the amplitudes of output beams, but also changes their phases and shapes. In such a case one need to measure the nonlocal characteristic length of (2) to judge whether (2) has a nonlocal response. Finally, in the case of 1122, as a result of linear absorption, the extinction ratio is reduced, but the half-wave voltage keeps nearly unchanged in an electro-optic amplitude modulation scheme. Besides the discussion on the influence of the linear absorption, we also make suggestions of how to measure the nonlocal characteristic lengths of (1) and (2) and the absorption coefficients 11 and 22.
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Keywords:
- linear electro-optic effect /
- nonlocal /
- linear absorption /
- electro-optic modulation
[1] Yariv A 1988 Quantum Electronics (3rd Ed.) (New York: Wiley) pp1-17, 105-135
[2] van der Valk N C J, Wenckebach T, Planken P C M 2004 J. Opt. Soc. Am. B 21 622
[3] Chen L, Xu Q, Wood M G, Reano R M 2014 Optica 1 112
[4] Borshch V, Shiyanovskii S V, Li B X, Lavrentovich O D 2014 Phys. Rev. E 90 062504
[5] Zhang X, Chung C, Hosseini A, Subbaraman H, Luo J, Jen A K, Nelson R L, Lee C Y, Chen R T 2015 J. Lightwave Technol. 34 2941
[6] Wang J, Li P, Weng S, Wu L, Ning T, Li J 2016 Chin. Opt. Let. 14 100603
[7] Liu K, Shi J, Zhou Z, Chen X 2009 Opt. Commun. 282 1207
[8] Gobert O, Paul P M, Hergott J F, Tcherbakoff O, Lepetit F, Oliveira P D, Viala F, Comte M 2011 Opt. Express 19 5410
[9] Qian S X, Wang G M 2002 Nonlinear Optics (1st Ed.) (Shanghai: Fudan University Press) pp334-343 (in Chinese) [钱士雄, 王恭明 2002 非线性光学 (第一版) (上海: 复旦大学出版社) 第334343页]
[10] Yariv A 1973 IEEE J. Quantum Elect. 9 919
[11] Nelson D F 1975 J. Opt. Soc. Am. 65 1144
[12] Gunning M J, Raab R E 1998 Appl. Opt. 37 8438
[13] She W, Lee W 2001 Opt. Commun. 195 303
[14] Wu D D, Chen H B, She W L, Lee W K 2005 J. Opt. Soc. Am. B 22 2366
[15] Zheng G, She W 2006 Opt. Commun. 268 323
[16] Zheng G, Wang H, She W 2006 Opt. Express 14 5535
[17] Chen L, Zheng G, Xu J, Zhang B, She W 2006 Opt. Lett. 31 3474
[18] Huang D, She W 2007 Opt. Express 15 8275
[19] Tang H, Chen L, She W 2010 Opt. Express 18 25000
[20] Wu D, She W 2016 Opt. Express 24 2867
[21] Wang G, Wang R 2013 Appl. Phys. Lett. 102 021906
[22] Guo Q, Luo B, Yi F, Chi S, Xie Y 2004 Phys. Rev. E 69 016602
[23] Krolikowski W, Bang O, Nikolov N I, Neshev D, Wyller J, Rasmussen J J, Edmundson D 2004 J. Opt. Soc. Am. B 6 S288
[24] Haus H A 1984 Waves and Fields in Optoelectronics (1st Ed.) (New Jersey: Prentice-Hall) pp81-157
[25] Mitrofanov O, Gasparyan A, Pfeiffer L N, West K W 2005 Appl. Phys. Lett. 86 202103
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[1] Yariv A 1988 Quantum Electronics (3rd Ed.) (New York: Wiley) pp1-17, 105-135
[2] van der Valk N C J, Wenckebach T, Planken P C M 2004 J. Opt. Soc. Am. B 21 622
[3] Chen L, Xu Q, Wood M G, Reano R M 2014 Optica 1 112
[4] Borshch V, Shiyanovskii S V, Li B X, Lavrentovich O D 2014 Phys. Rev. E 90 062504
[5] Zhang X, Chung C, Hosseini A, Subbaraman H, Luo J, Jen A K, Nelson R L, Lee C Y, Chen R T 2015 J. Lightwave Technol. 34 2941
[6] Wang J, Li P, Weng S, Wu L, Ning T, Li J 2016 Chin. Opt. Let. 14 100603
[7] Liu K, Shi J, Zhou Z, Chen X 2009 Opt. Commun. 282 1207
[8] Gobert O, Paul P M, Hergott J F, Tcherbakoff O, Lepetit F, Oliveira P D, Viala F, Comte M 2011 Opt. Express 19 5410
[9] Qian S X, Wang G M 2002 Nonlinear Optics (1st Ed.) (Shanghai: Fudan University Press) pp334-343 (in Chinese) [钱士雄, 王恭明 2002 非线性光学 (第一版) (上海: 复旦大学出版社) 第334343页]
[10] Yariv A 1973 IEEE J. Quantum Elect. 9 919
[11] Nelson D F 1975 J. Opt. Soc. Am. 65 1144
[12] Gunning M J, Raab R E 1998 Appl. Opt. 37 8438
[13] She W, Lee W 2001 Opt. Commun. 195 303
[14] Wu D D, Chen H B, She W L, Lee W K 2005 J. Opt. Soc. Am. B 22 2366
[15] Zheng G, She W 2006 Opt. Commun. 268 323
[16] Zheng G, Wang H, She W 2006 Opt. Express 14 5535
[17] Chen L, Zheng G, Xu J, Zhang B, She W 2006 Opt. Lett. 31 3474
[18] Huang D, She W 2007 Opt. Express 15 8275
[19] Tang H, Chen L, She W 2010 Opt. Express 18 25000
[20] Wu D, She W 2016 Opt. Express 24 2867
[21] Wang G, Wang R 2013 Appl. Phys. Lett. 102 021906
[22] Guo Q, Luo B, Yi F, Chi S, Xie Y 2004 Phys. Rev. E 69 016602
[23] Krolikowski W, Bang O, Nikolov N I, Neshev D, Wyller J, Rasmussen J J, Edmundson D 2004 J. Opt. Soc. Am. B 6 S288
[24] Haus H A 1984 Waves and Fields in Optoelectronics (1st Ed.) (New Jersey: Prentice-Hall) pp81-157
[25] Mitrofanov O, Gasparyan A, Pfeiffer L N, West K W 2005 Appl. Phys. Lett. 86 202103
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