Second-Order Nonlinearities and Optical Rectification

Jacob B. Khurgin    DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING JOHNS HOPKINS UNIVERSITY BALTIMORE, MARYLAND

I Introduction

As stressed throughout these two volumes, using semiconductor materials as nonlinear optical elements allows one fully to bridge the gap between optics and electronics. It opens the possibility for truly integrated optoelectronic circuits in which all major elements — sources and detectors of radiation, electronic signal processing circuitry, and nonlinear optical devices such as switches, modulators, and optical converters — are built from the same family of semiconductor materials, which are compatible with each other. In this chapter, we are interested in the second-order (quadratic) nonlinear phenomena. Being the lowest-order nonlinear optical effects, the second-order nonlinear phenomena are thus the ones observable at the lowest power densities, and their use is, in principle, preferable to the use of third- and higher-order effects. The most ubiquitous “bread-and-butter” optoelectronic materials — the III–V binary semiconductors, their alloys, and quantum-size layered structures (quantum wells and superlattices) — exhibit, as a result of the heteropolar nature of their bonding, a very high second-order susceptibility that is typically at least an order of magnitude higher than those of the competing materials. At the same time, many of the III–V materials (and, similar to them in many ways, the II-VI materials) have direct bandgaps. The values of the bandgaps cover the important range—from as low as 0.5 eV for antimonides to as high as 3.5 eV for nitrides. Furthermore, after enormous effort devoted to the refinement of their growth and fabrication methods, the high-optical-quality and high-damage-threshold III–V semiconductor materials can be routinely grown and formed in the waveguides, making them compatible with laser diodes and with optical fiber technology.

Yet, even with such an impressive array of properties, the semiconductors have not found use in many second-order nonlinear devices owing to their only, but very important, shortcoming: neither the zinc-blende nor hexagonal lattice structures in which all III–V and II-VI semiconductors crystallize have sufficient birefringence to compensate for the large dispersion inherent in these materials. This, until very recently, has made the phasematching necessary for efficient frequency conversion unattainable in semiconductors. Since phasematching is of such paramount importance, we will distinguish in this chapter between two types of second-order nonlinear processes—those that require phasematching (i.e., harmonic generation and sum/difference generation) and those that do not (i.e., linear electro-optic effect and optical rectification, including terahertz oscillation generation).

In the nearly four decades since the inception of nonlinear optics (Armstrong et al, 1962; Ward, 1965; Butcher and McLean, 1963/1964), several theories adequately describing the origin of the second-order non-linearities in solids and the nature of interaction between linear and nonlinear waves have been developed, and several excellent textbooks and monographs on the subject have been published. It is, therefore, beyond the scope of this chapter to consider in great detail all the aspects of second-order nonlinearities, for which the reader is referred to full-scale treatments. Yet, when it comes to the study of the origin of nonlinearity and descriptions of the nonlinear characteristics of the materials, most of the popular texts concentrate on a few well-known examples of ionic crystals, such as KTiPO4 (KTP), LiNbO3, KDHPO4 (KDP), and others. The nonlinearity of these insulating crystals can be treated using a semiclassical model of anharmonic oscillators, and yet, when one considers the nonlinear properties of semiconductors, one has to take into account the dispersion of the bands—a topic not dealt with in most texts. Besides, historically, when the second-order nonlinear phenomena have been mentioned, harmonic generation has been considered first and treated in the most detail. Yet, in recent years, as the new sources of radiation in the blue and ultraviolet (UV) regions of the spectrum have continued to improve, second harmonic generation has taken a back seat to optical parametric generation and difference frequency generation of infrared (IR) and far-IR light. At these frequencies, the nonlinear properties are quite different from those encountered in second harmonic generation, and in our opinion there is a lack of comprehensive literature dealing adequately with this subject. Furthermore, in the last one and a half decades, development of fundamentally new quantum-size semiconductor heterostructures—quantum wells and superlattices— has taken place. The large size (in comparison to the atomic size) of the wavefunctions in these structures, as well as the ability to manipulate the asymmetry of the structure, open the prospects of engineering artificially large nonlinearities. So much progress has taken place in the last few years in the area of quantum-size semiconductor nonlinear structures that there is a need for a book treating this subject.

The purpose of this chapter is to complement the existing literature on second-order nonlinear optics in the areas specific to the semiconductors and their heterostructures. It is structured as follows.

After this brief introduction, we begin in Section II with the specifics of calculating nonlinear optical susceptibilities in the system with nonlocalized states. In order to do this, both classical and quantum theories of χ(2) are reviewed, and a few competing approaches are described. Special attention is given to the consequences of using A · P and E · r interaction Hamiltonians. We then present in some detail the simple bond-polarizability theory of nonresonant χ(2) at frequencies below the bandgap and compare it with the classical estimate and with more detailed full bandstructure calculations.

In Section III, the results of the measurements of χ(2) in bulk semiconductors are described, and comparisons with different theoretical predictions are made. Then the problems associated with phasematching are considered, and a short review of the experimental results in harmonic, difference frequency, and parametric generation in bulk semiconductors is given.

Section IV is devoted to the new and exciting area that has developed in the last decade — nonlinear optics of the semiconductor quantum wells and superlattices, which have large dipole moments and thus very high nonlinearities. The distinction is drawn between two types of processes — band-to-band and intersubband — and the latest experimental results are reviewed for both types of processes.

In Section V, we focus on the resonant or near-resonant second-order properties of semiconductors—optical rectification and THz generation — which, we shall show, require special treatment. We present a model for their calculation in bulk materials and quantum wells and then describe some experimental results.

The last section mentions a number of important topics that were not touched on during this brief review, and presents some conclusions.

II Second-Order Nonlinear Effects in Bulk Semiconductors

1 GENERAL CONSIDERATIONS

We are interested in the responses (linear and nonlinear) of the crystalline materials to the electromagnetic field in the optical range, covering wavelengths ranging from near-IR to near-UV (0.2 µm ≤ λ ≤2 µm). This response is determined by the motion of bound electrons situated in the outer shells of the atoms constituting the crystal. These electrons are called valence electrons. There are two alternative ways of describing the motion of the valence electrons in the solid state. Owing to the presence of a periodic lattice, valence electrons are delocalized and are well described by the extended or Bloch states

k(r)=uk,n(r)eikr

where n is an index of a band, k is a wavevector, and uk,n(r) is a so-called “periodic” atomic wavefunction. This description is the best for evaluating transport properties of the semiconductor, as well as its absorption. The reason behind this is very simple: the transport properties are determined by the relatively few states near the extrema of the bands that have free carriers—electrons near the bottom of the first conduction band and holes near the top of the valence band. Therefore, precise descriptions of these individual states, such as mobility and density of states, are necessary. Similarly, when absorption of the optical radiation is the subject of interest, for the photon of given energy hv, only a pair of states in the conduction and valence bands with energy difference Ek,c — Ek,ν = hν will participate in the absorption process, and a precise description of these states that includes the transition dipole strength and joint density of states must be provided.

When, however, the processes of interest are nonresonant...