Chapter 1 The Nonlinear Optical Susceptibility

1.1. Introduction to Nonlinear Optics

Nonlinear optics is the study of phenomena that occur as a consequence of the modification of the optical properties of a material system by the presence of light. Typically, only laser light is sufficiently intense to modify the optical properties of a material system. The beginning of the field of nonlinear optics is often taken to be the discovery of second-harmonic generation by Franken et al. (1961), shortly after the demonstration of the first working laser by Maiman in 1960.* Nonlinear optical phenomena are “nonlinear” in the sense that they occur when the response of a material system to an applied optical field depends in a nonlinear manner on the strength of the optical field. For example, second-harmonic generation occurs as a result of the part of the atomic response that scales quadratically with the strength of the applied optical field. Consequently, the intensity of the light generated at the second-harmonic frequency tends to increase as the square of the intensity of the applied laser light.

In order to describe more precisely what we mean by an optical nonlinear-ity, let us consider how the dipole moment per unit volume, or polarization (t), of a material system depends on the strength (t) of an applied optical field.* In the case of conventional (i.e., linear) optics, the induced polarization depends linearly on the electric field strength in a manner that can often be described by the relationship

where the constant of proportionality χ(1) is known as the linear susceptibility and ε0 is the permittivity of free space. In nonlinear optics, the optical response can often be described by generalizing Eq. (1.1.1) by expressing the polarization (t) as a power series in the field strength (t) as

The quantities χ(2) and χ(3) are known as the second- and third-order nonlinear optical susceptibilities, respectively. For simplicity, we have taken the fields (t) and (t) to be scalar quantities in writing Eqs. (1.1.1) and (1.1.2). In Section 1.3 we show how to treat the vector nature of the fields; in such a case χ(1) becomes a second-rank tensor, χ(2) becomes a third-rank tensor, and so on. In writing Eqs. (1.1.1) and (1.1.2) in the forms shown, we have also assumed that the polarization at time t depends only on the instantaneous value of the electric field strength. The assumption that the medium responds instantaneously also implies (through the Kramers-Kronig relations†) that the medium must be lossless and dispersionless. We shall see in Section 1.3 how to generalize these equations for the case of a medium with dispersion and loss. In general, the nonlinear susceptibilities depend on the frequencies of the applied fields, but under our present assumption of instantaneous response, we take them to be constants.

We shall refer to as the second-order nonlinear polarization and to as the third-order nonlinear polarization. We shall see later in this section that physical processes that occur as a result of the second-order polarization (2) tend to be distinct from those that occur as a result of the third-order polarization (3). In addition, we shall show in Section 1.5 that second-order nonlinear optical interactions can occur only in noncentrosymmetric crystals—that is, in crystals that do not display inversion symmetry. Since liquids, gases, amorphous solids (such as glass), and even many crystals display inversion symmetry, χ(2) vanishes identically for such media, and consequently such materials cannot produce second-order nonlinear optical interactions. On the other hand, third-order nonlinear optical interactions (i.e., those described by a χ(3) susceptibility) can occur for both centrosymmetric and noncentrosymmetric media.

We shall see in later sections of this book how to calculate the values of the nonlinear susceptibilities for various physical mechanisms that lead to optical nonlinearities. For the present, we shall make a simple order-of-magnitude estimate of the size of these quantities for the common case in which the non-linearity is electronic in origin (see, for instance, Armstrong et al., 1962). One might expect that the lowest-order correction term (2) would be comparable to the linear response (1) when the amplitude of the applied field is of the order of the characteristic atomic electric field strength , where —e is the charge of the electron and is the Bohr radius of the hydrogen atom (here is Planck’s constant divided by 2π, and m is the mass of the electron). Numerically, we find that Eat = 5.14 × 1011 V/m.* We thus expect that under conditions of nonresonant excitation the second-order susceptibility χ(2) will be of the order of χ(1)/Eat. For condensed matter χ(1) is of the order of unity, and we hence expect that χ(2) will be of the order of 1/Eat, or that

Similarly, we expect χ(3) to be of the order of χ(1)/E2at, which for condensed matter is of the order of

These predictions are in fact quite accurate, as one can see by comparing tnese values with actual measured values of χ(2) (see, for instance, Table 1.5.3) and χ(3) (see, for instance, Table 4.3.1).

For certain purposes, it is useful to express the second- and third-order susceptibilities in terms of fundamental physical constants. As just noted, for condensed matter χ(1) is of the order of unity. This result can be justified either as an empirical fact or can be justified more rigorously by noting that χ(1) is the product of atomic number density and atomic polarizability. The number density N of condensed matter is of the order of (a0)−3, and the nonresonant polarizability is of the order of (a0)3. We thus deduce that χ(1) is of the order of unity. We then find that χ(2) (4π ε0)34/m2e5 and χ(3) (4π ε0)68/m4e10. See Boyd (1999) for further details.

The most usual procedure for describing nonlinear optical phenomena is based on expressing the polarization (t) in terms of the applied electric field strength (t), as we have done inEq. (1.1.2). The reason why the polarization plays a key role in the description of nonlinear optical phenomena is that a time-varying polarization can act as the source of new components of the electromagnetic field. For example, we shall see in Section 2.1 that the wave equation in nonlinear optical media often has the form

where n is the usual linear refractive index and c is the speed of light in vacuum. We can interpret this expression as an inhomogeneous wave equation in which the polarization NL associated with the nonlinear response drives the electric field . Since ∂2NL/∂t2 is a measure of the acceleration of the charges that constitute the medium, this equation is consistent with Larmor’s theorem of electromagnetism which states that accelerated charges generate electromagnetic radiation.

It should be noted that the power series expansion expressed by Eq. (1.1.2) need not necessarily converge. In such circumstances the relationship between the material response and the applied electric field amplitude must be expressed using different procedures. One such circumstance is that of resonant excitation of an atomic system, in which case an appreciable fraction of the atoms can be removed from the ground state. Saturation effects of this sort can be described by procedures developed in Chapter 6. Even under nonreso-nant conditions, Eq. (1.1.2) loses its validity if the applied laser field strength becomes comparable to the characteristic atomic field strength Eat, because of strong photoionization that can occur under these conditions. For future reference, we note that the laser intensity associated with a peak field strength of Eat is given by

We shall see later in this book (see especially Chapter 13) how nonlinear optical processes display qualitatively distinct features when excited by such super-intense fields.

1.2. Descriptions of Nonlinear Optical Processes

In the present section, we present brief qualitative descriptions of a number of nonlinear optical processes. In addition, for those processes that can occur in a lossless medium, we indicate how they can be described in terms of the nonlinear contributions to the polarization described by Eq. (1.1.2).* Our motivation is to provide an indication of the variety of nonlinear optical phenomena that can occur. These interactions are described in greater detail in later sections of this book. In this section we also introduce some notational conventions and some of the basic concepts of nonlinear optics.

1.2.1 Second-Harmonic Generation

As an example of a nonlinear optical interaction, let us consider the process of second-harmonic generation, which is illustrated schematically in Fig. 1.2.1. Here a laser beam whose electric field strength is represented as

is incident upon a crystal for which the second-order susceptibility χ(2) is nonzero. The nonlinear...