Wave coupling theory of nonlocal linear electro-optic effect in a linear absorbent medium

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 Chi((1)). When linear-absorption coefficients alpha(11) and alpha(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 alpha(11) = alpha(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 Chi((2)). More interestingly, when f alpha(1) not equal alpha(22), 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 Chi((2)) to judge whether Chi((2)) has a nonlocal response. Finally, in the case of alpha(11) not equal alpha(22), 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 Chi((1)) and Chi((2)) and the absorption coefficients alpha(11) and alpha(22)