Linear and quadratic electrooptic effects in symmetric and asymmetric quantum-well structures

In the first part of the paper, the changes in the complex index of refraction of quantum-well structures proportional to the applied dc electric field (F) and its square are obtained from stationary perturbation theory. Expressions are obtained for the linear and quadratic changes in index and absorption coefficients of asymmetric and symmetric quantum well structures. The triple resonance term does not contribute to the quadratic Kerr effect for a symmetrical well in an expansion about F=0. In the second part of the paper, these results are applied to several types of modulators. First we consider two kinds of silicon/silicon-germanium high-barrier asymmetric quantum well structures at 1.55 μm. For the step well, the dominant triple resonant term of the quadratic Kerr effect adds to the linear electrooptic effect (LEG) to yield an intensity extinction ratio of 15 dB in switching from F=+8 V/μm to -8 V/μm and a very small insertion loss of 0.04 dB, for a modulator length of 200 pm and for narrow linewidths (FWHM=7.6 meV). For the case of asymmetric barriers confining silicon wells, a comparable insertion loss is obtained but with an extinction ratio of 44 dB for FWHM=20 meV, a clearly superior performance. For symmetrical silicon-based wells, the double-resonance modulation, though small at λ=1.55 μm, is appreciable at λ=10.6 μm. For the Kerr effect in symmetrical gallium arsenide wells at λ=10.6 μm reported by Sa´ar et al., we show that it is the double-resonance term which yields the six order-of-magnitude enhancement with respect to bulk and that an even larger enhancement of the Kerr effect can be obtained in an asymmetric GaAs structure together with the LEO