Modulators

Abstract

The application of an external field can modify the optical properties of a material. By controlling the strength and frequency of the external field, the amplitude, phase, or polarization of the optical beam can be controlled and modulated. The most important modulation methods are those using electric fields, but both acoustic fields and magnetic fields can be exploited. This chapter describes linear and quadratic electro-optic effects, and explains the most common modulation and switching schemes. Modulation methods based on liquid crystals are also briefly discussed. The final part of the chapter is devoted to the acousto-optic effect and its applications to amplitude modulation and to beam deflection.

Problems

4.1

Consider the electro-optic amplitude modulator of Fig. 4.2, using a lithium niobate crystal. The operating wavelength is $\lambda =1530$ nm, the electro-optic crystal is lithium niobate, the crystal length is $L=6$ cm, the crystal thickness is $d=40~{\upmu }$m, and the output intensity $I_B$ is equal to zero when no voltage is applied to the crystal. (i) Calculate the value of the applied voltage $V_{\pi }$ required to obtain $100\,\%$ transmission at input B; (ii) Calculate the upper limit to the modulation frequency, as determined by the condition that the transit time of the light beam through the crystal be one fifth of the modulation period.

4.2

Consider the electro-optic amplitude modulator of Fig. 4.2, using a lithium tantalate crystal. The operating wavelength is $\lambda =1550$ nm, the crystal length is $L=4$ cm, the crystal thickness is $d=40~{\upmu }$m, and the output intensity $I_B$ is equal to zero when no voltage is applied to the crystal. Assume that the applied voltage V is the sum of a bias voltage $V_o$ plus a signal voltage $V_1$. Calculate: (i) the value of $V_o$ required to bring the modulator in the linear regime; (ii) the upper limit to $V_1$, subject to the condition that the maximum output intensity does not exceed the limit of $75\,\%$ of the input intensity.

4.3

A light beam at $\lambda =1550$ nm, linearly polarized along x, propagates along the optical axis (the z axis) of a KDP crystal. A voltage V is applied to the crystal along the z axis. After crossing the crystal the light beam goes through a sequence of a quarter-wave plate, having the optical axis in the x-z plane forming an angle of $45^{\circ }$ with the x axis, and of a polarizer, with optical axis along y. Calculate the value of V required for a $100\,\%$ transmission of the incident beam through the sequence crystal-plate-polarizer.

4.4

An amplitude modulator based on the electro-optic Kerr effect can be made by putting the Kerr cell in between two crossed polarizers, as in Fig. 4.6. The external electric field $E'$ is applied transversally, and the incident light is linearly polarized at $45^{\circ }$ to the electric field. Calculate the half-wave voltage $V_{\pi /2}$ by assuming that the wavelength of light is 600 nm, the path-length of the light beam inside the Kerr cell is 5 cm, the distance between the two capacitor plates is 2 mm, and the electro-optic material is liquid nitrobenzene ($n_K^e-n_K^o = 2.6 \times 10^{-18}$ m$^2$/V$^2$).

4.5

A light beam at $\lambda =1500$ nm goes through an acousto-optic cell made of flint glass ($us = 3$ km/s, $n = 1.95$) in which an acoustic wave of frequency 200 MHz is propagating. Calculate: (i) the wavelength difference between the diffracted wave and the incident wave; (ii) the deflection angle of the diffracted wave.

4.6

A light beam at $\lambda =650$ nm goes through an acousto-optic cell made of water ($n = 1.33,~p = 0.31,~u_s = 1500$ m/s, $\rho = 1000$ kg/m$^3$) having a square cross-section. Calculate: (i) the frequency of the acoustic wave producing a deflection angle of $2^{\circ }$; (ii) the acoustic power required to transfer all the input power to the diffracted beam.