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Journal of the Optical Society of America B

Analytical solution of amplitude and phase modulation in semiconductor traveling-wave optical amplifiers subject to transient optical signals

Jayanta Sarma, R. Balasubramanyam, K.A. Shore

Published Feb 1, 1992• 1 citation
Publisher

Abstract

We present a formalism and a technique for analyzing the propagation of optical signals through semiconductor traveling-wave amplifiers. The formalism retains, in particular, the spontaneous recombination term, and the newly developed technique permits analytic, real-time solutions to be obtained, so that both the transient and the steady-state regimes are characterized. This end has been achieved by iteratively applying the method of in order to solve the set of coupled differential equations that result from the formulation of the characteristics physical problem. From the real-time solutions so obtained it is also possible easily to recognize the amplitude and phase-modulation effects that occur not only in the steady state but also in the transient regime.

DOI
10.1364/josab.9.000223

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We present a formalism and a technique for analyzing the propagation of optical signals through semiconductor traveling-wave amplifiers. The formalism retains, in particular, the spontaneous recombination term, and the newly developed technique permits analytic, real-time solutions to be obtained, so that both the transient and the steady-state regimes are characterized. This end has been achieved by iteratively applying the method of in order to solve the set of coupled differential equations that result from the formulation of the characteristics physical problem. From the real-time solutions so obtained it is also possible easily to recognize the amplitude and phase-modulation effects that occur not only in the steady state but also in the transient regime.
A simple model of a traveling-wave parametric amplifier is discussed in quantum-mechanical terms. The amplifier consists of a slab of Raman-active material, illuminated on one face by an intense laser beam that serves as the pump mode; an optical-phonon oscillation serves as the idler mode, and the scattered Stokes light occupies the signal mode. With only a few simplifying assumptions, it is possible to solve the equation of motion for the signal field operator exactly. For large times, both an amplified steady signal and the quantum noise contributed by the material assume simple steady states in an amplifier of finite length. The transient and asymptotic behaviors of the signal and noise intensities are found, together with the corresponding first-order correlation functions. The quasiprobability distribution which specifies the complete density operator for the electric field strength of the amplified signal is also derived.
Differential equations which describe the steady state spatial evolution of nonclassical light are established using standard quantum field theoretic techniques. A Schroedinger equation for the state vector of the optical field is derived using the quantum analog of the slowly varying envelope approximation (SVEA). The steady state solutions are those that satisfy the time independent Schroedinger equation. The resulting eigenvalue problem then leads to the spatial propagation equations. For the degenerate parametric amplifier this method shows that the squeezing parameter obey nonlinear differential equations coupled by the amplifier gain and phase mismatch. The solution to these differential equations is equivalent to one obtained from the classical three wave mixing steady state solution to the parametric amplifier with a nondepleted pump.
Nondegenerate four-wave mixing (FWM) of picosecond optical pulses in semiconductor optical amplifiers is investigated theoretically, taking into account various carrier processes, probe depletion, and cross-gain modulation effects. Analytical solutions for greatly simplified coupled-wave mixing equations are obtained from which conditions where probe depletion and cross-gain modulation effects cannot be neglected are found. On the other hand, the frequency detuning dependence of optimum input pump pulse energy and shaping of output conjugate pulses are observed from numerical simulations. In particular, calculations show that the frequency symmetry of conjugate and probe pulses with respect to the pump pulse does not hold for optical pulse FWM. Two-photon absorption and ultrafast nonlinear refraction effects are shown to decrease the FWM conversion efficiency.
Using semiclassical radiation theory, a formalism similar to that used by Lamb in his Theory of an Optical Maser is developed for studying the amplification of vector traveling waves in a laser-type medium. The effect of the medium on the waves is given in terms of space (or time) dependent field amplitudes and phases and a nonlinear index of refraction. With particular emphasis on typical gaseous media, the effects of Doppler broadening are treated in detail for arbitrary ratios of natural to Doppler linewidths. Polarization and propagation vectors in various directions are considered, and the nonlinear effects are found to make an isotropic medium effectively anisotropic. Lowest order nonlinear effects (due to a polarization cubic in the field amplitudes) are studied extensively, and the frequency dependence of several of these processes is presented in graphical form. In particular, the introduction of fields at new frequencies and polarization effects are considered. The characteristics of these nonlinear processes peculiar to Doppler broadened lines are discussed, and the processes are interpreted in terms of saturation and coherent modulation of the population inversion density. Strong nonlinear effects are considered in a more approximate way and are found to consist of saturation of the various linear and nonlinear processes previously considered. These strong nonlinear effects should occur at low enough intensities to be easily observed in practice on a CW basis. With the present formalism, the analytical results of Gordon, White and Rigden regarding gain saturation in laser amplifiers are obtained, and the extension is made to include frequencies away from line center and the effects of multiple spectral components. Again, the introduction of fields at new frequencies is considered in detail. These results are also discussed in terms of saturation and coherent modulation of the populations and hole burning.
A theoretical analysis is presented of two-wave mixing in photorefractive materials in which the complex amplitude of a weak signal beam has slowly varying but otherwise arbitrary spatial dependence and arbitrary time variation. The set of coupled partial differential equations that couple the dynamics of the optical fields to the photorefractive medium is solved analytically to give the complex amplitude of the transmitted signal in the undepleted pump regime. The results are compared with known results for the transient two-wave mixing of plane waves.