24 Electro Optic Devices

Dr. Ayushi Paliwal and Dr. Monika Tomar

1. Introduction

Information processing is another field where optics plays a crucial role and has certain advantages with respect to electronic computing, such as massive parallelism, continuous data operation, direct data acquisition process, implementation of fuzzy logic can be implemented etc. External optical modulators are the key components of optical communication which are extensively used in dense wavelength division multiplexing (WDM) systems. Numerous efforts have been made on the integration of lasers with modulators for efficient optical communication. Another advantage of optical modulators is their ability to make good use of the wide bandwidth of optical fibers. The Mach–Zehnder is the most commercially used modulator because of low operating voltage and wide bandwidth, but has disadvantages of large insertion loss and high price. Various other types of modulators have been developed using the electroabsorption modulator, acousto-optic (AO) modulator, electro-optic (EO) modulator, cavity damping based modulator and photoelastic effect based modulator. EO modulators are preferred because of fast response time and feasibility to realize in either reflection or transmission geometry. But, much attention has been paid to EO modulators with refection geometry in the past few decades. EO modulators are based on the EO effect which corresponds to the change in the refractive index of an anisotropic material with applied electric field.

2. Theory of Electro optic effect 2.1 Anisotropy

Anisotropy is the property of being directionally dependent in a crystal. It can be defined as a difference in a material’s physical or mechanical properties (absorbance, refractive index, conductivity, etc.) when measured along different axes. An example of anisotropy is the light coming through a polarizer. In an anisotropic crystal, the refractive index is different in different crystal directions. The optically anisotropic crystals can be classified as uniaxial and optically biaxial crystals. Considering, the coordinate system to be along the three principal axes [along X, Y and Z directions] of a crystal, then we have the following relations:

• X = nx2  , εy = ny2  , εz = nz2 (1)

The index ellipsoid equation describing the optical anisotropy of a crystal can be written as

In an uniaxial crystal, we have nx = ny = no and nz = ne, where no and ne are the ordinary and extraordinary refractive indices, respectively. The refractive index along the optic axis corresponds to the extraordinary index, ne, and the refractive index perpendicular to the optic axis corresponds to the ordinary index, no (Figure 1). The existence of two rays with different indices of refraction is called Optical Birefringence. The birefringence is usually defined as

Δn = ne – no (3)

Since the value of ne may be either higher or lower than no, birefringence may take positive or negative values. If Δn > 0, the crystal is said to be positive, and if Δn < 0, it is said to be negative.

2.2 Permittivity tensor

The modulation of the properties of light propagating in an optically anisotropic crystal is the fundamental principle for the development of electro-optic (EO) device. Permittivity of the material can be characterized by its optical and dielectric properties and is an important parameter for the description of EO effect and, therefore, its origin is explained below. Polarization is an important phenomena while studying the EO effect where reallocation of both bound and free charges in response to an electric field within the material takes place. In an isotropic medium, the induced polarization P and the electric field E, are related by a scalar factor χ called susceptibility. The relation is given by = 0 (4) where 0stands for the permittivity of vacuum. In an anisotropic material, the induced polarisation depends on the direction of the electric field vector and it becomes necessary to define the susceptibility as a second rank tensor. Each of the polarization components Pi is dependent on all three components of the electric field Et, and can be written using Einstein notation as
= 0i,j =1,2,3 (5) The susceptibility tensor has clear physical origin and can be used to define the permittivity tensor ij as follows = = 0(1 + ) (6) The permittivity tensor relates the electric field vector E and the electric displacement vector D = (7) and thus defines the dielectric properties of the crystal. If the medium is homogenous, non-absorbing and magnetically isotropic the permittivity tensor can be shown to be symmetric using the principle of the conservation of EM field energy. The same analysis, applied to a non-absorbing optically active medium, shows more general result = ∗ (8) where ij is a complex conjugate of the permittivity ∗. The symmetrical property of the permittivity tensor makes it possible to find a coordinate system in which the tensor has only its diagonal elements. The axes of such a coordinate system are called the principal dielectric coordinate axes and are of great importance for the description of the electro-optic effect.

2.3 Electro-optic effect

Electro optic effect can be defined as the change in the refractive index due to ion displacement in a ferroelectric crystal with application of an external electric field. It is a nonlinear optic effect exhibited by ferroelectric materials when an electric field is applied leading to the redistribution of charges in the crystal. Macroscopically, EO effect can be described in terms of power series expansion of the polarization, ⃗ as

The above equation describes the linear optical properties of a medium and the application of a static electric field to a second-order non-linear optical (NLO) material is represented by the term in χ(2) in equation 9 where change in the refractive index of the material is proportional to the field referred to as linear EO effect which is also defined as the Pockels effect. Pockels effect describes a linear relationship between the induced change in birefringence (Δn) and the electric field (E). Similarly, the term χ(3) represents the quadratic variation of change in refractive index with applied field defined as Kerr effect. The linear EO effect can be reflected as the change in rotation and deformation of the index ellipsoid resulting in the variation in 1/ nx2, 1/ ny2, and 1/ nz2. Due to the rotation of the ellipsoid, cross-terms have to be included. Due to the presence of an electric field, the equation for the index ellipsoid is given by

where, the parameters Bij are functions of the electric field E. In order to couple the six constants Bij to three components of Ei, 18 coefficients in the form of a 6 × 3 matrix are needed

It is often possible to avoid the complications of the cross-terms by applying the external field parallel to one of the main orientations of the crystal and by choosing the corresponding polarization of the light. Applying the electric field along the c-axis of the LiNbO3 crystal (E = (0, 0, E)), nx and ny are identical to no, while nz = ne propagates the extraordinary beam. Equations (11) and (12) reduce to:

3. Electro optic modulators: Principle of operation

Integrated optical modulator is a device in which the light waves are confined to optical waveguides. The propagation of light waves is influenced by the change in the refractive index of the guided region with the applied electric field, σo through the EO effect of the medium. EO modulators are optical devices that can modulate a beam of light, which can be used to transmit information. Because of the high frequencies of the optical waves, very high modulation bandwidths are possible with light as a carrier of information. The optical beam may be modulated in phase, frequency, polarization, amplitude and direction. The phase and amplitude modulators are attractive due to simple design.

In case of long distance optical transmission, the wave form is susceptible to degradation due to nonlinear effect such as self-phase modulation, and so on. A phase modulator can be used to alter the phase of the carrier to compensate for the degradation. A phase modulator is a device that manipulates the phase of optical carrier signals under the influence of applied electric field. When voltage is not applied, a number of periods of light waves, n, exist in a certain path length. When voltage is applied to the rf-electrode, one or a fraction of one period of a wave is added, which now implies (n+1) waves exist in the same length. In this case, the phase has been changed by 2π and in this condition, the half voltage is called the driving voltage.

Phase modulators involve variation of the phase of the carrier wave. The change in refractive index (∆n) of a biased EO material will depend on effective EO coefficient rc (dependent on the orientation of the material and light polarization) and the magnitude of the applied field V(t) as

Amplitude (intensity) modulation can be achieved, for example by putting a phase modulator in one branch of the interferometer, most commonly a Mach-Zehnder interferometer where a beam splitter is used to divide the laser light into two beams one of which passes through the phase EO modulator. The two beams are again combined with another beam splitter. If the incoming beam has an amplitude A and the beam splitters are dividing a wave into two equal parts, the outgoing field is the sum of two interfering beams

The resulting transmittance of the interferometer T = / is a function of V(t). The half wave voltage is the applied voltage needed to change the transmittance from 0 to 1 i.e to operate the device in a limited region around T(V) = 0.5 (can be adjusted by changing the length of one arm of the interferometer), and a linear intensity modulator is obtained. The modulated transmitted light intensity is directly proportional to the applied electric signal V(t) as

The half wave voltage , (equation 18) is an important parameter for device applications. It is minimized for each device geometry and carrier wavelength by large 3 , which is the figure-of-merit for EO material. On the other hand can be reduced by optimizing the geometry i.e. long propagation distance l and short electrode distance d greatly reduce . 

4. Various method to study EO effect

There are several established methods to measure the EO coefficients of crystals; which include Mach-Zehnder interferometry, Senarmont method, ATR method, and measurement of transmittance change. Mach-Zehnder interferometry is used to observe the interference between split light beams, where an electric field is applied to one of the beam paths. However, it is troublesome to fabricate waveguides through patterning and etching when applying this technique to thin films. The Senarmont method is based on measuring electric-field-induced phase retardation (birefringence). Enhanced sensitivity can be achieved by modulating incident light with lock-in detection. The substrate must be transparent and measurement is done only for alternate changing electric fields. As the ATR method employs a prism-coupling configuration, the sample is slightly damaged when the prism is clamped. Another shortcoming common to the above techniques is that the EO coefficient is obtained at a single wavelength of laser light. Senarmont compensator method, which is a birefringence-based method, can measure a small difference between the ordinary and extraordinary refractive indices rather than measuring the change of individual refractive index. The birefringence in a crystal can be detected by placing the sample between a pair of crossed polarizers to convert the birefringence induced light polarization change to the output light intensity change. Therefore, any birefringence change caused by modulating the refractive index can be measured by recording the output intensity.

5. Summary

• Theory of electro optic effect and Electro optic effect in lithium niobate were discussed in detail
• Electro optic modulators and their principle of operation was discussed.
• Different methods to study EO effect