In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω 0 can generate harmonic components with central frequencies 2 ω 0 , 3 ω 0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω 1 and ω 2 , with ω 1 > ω 2 , can generate components with central frequency ω 1 + ω 2 (sum-frequency generation) and ω 1 − ω 2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω 0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities. The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.
In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω 0 can generate harmonic components with central frequencies 2 ω 0 , 3 ω 0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω 1 and ω 2 , with ω 1 > ω 2 , can generate components with central frequency ω 1 + ω 2 (sum-frequency generation) and ω 1 − ω 2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω 0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities. The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.
Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam. Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.
In the last chapter, we examined the effect of cubic nonlinearity on selfrefraction of an optical beam. We have also seen that the presence of a quadratic nonlinearity can also contribute to change in the profile of a Gaussian beam in a crystal. In this chapter we will see how the effect of nonlinearity, alongwith the proper feedback, can give rise to optical bistability and hysteresis. This is not unexpected as hysteresis and bistability are also observed in nonlinear electronic circuits with feedback, such as in the Schmitt trigger, as well as in hybrid optical devices, such as an acoustooptic device with feedback (Banerjee and Poon, 1991). In what follows, we will examine two different types of optically bistable devices based on optical nonlinearities. The first is a nonlinear ring resonator comprising a two-level gain medium as the nonlinear element. We will treat both absorptive and dispersive bistability. Absorptive bistability is the case when the incident optical frequency is close to or equal to the transition frequency of the atoms from one level to another. In this case, the absorption coefficient becomes a nonlinear function of the incident intensity. On the other hand, if the frequencies are far apart, the medium behaves like a Kerr-type material and the system exhibits what is called dispersive bistability. In this case the material can be modeled by an effective n2 or v (3), meaning that the refractive index becomes a nonlinear function of the optical intensity. The second configurationwewill examine is a linear-nonlinear interface. In this case, the interface provides the feedback due to refractive index mismatch. In fact, the reflection coefficient (as well as the transmission coefficient) is modified due to the nonlinearity, which, as we shall see later, is responsible for demonstrating optical switching and bistability.
Introduction In 1961, just a few months after Maiman invented the ruby laser, Franken focused the pulses emitted from such a laser, of wavelength 694 nm, on a quartz plate, and examined the spectrum of the light transmitted using a simple prism (see Figure 7.1). He thus discovered that ultra-violet light of wavelength 347 nm was emerging from the quartz plate. Clearly, as it propagated through the quartz, the light of frequency ω had generated the second harmonic, of frequency 2ω. It thus transpires that in optics, as in any other part of physics, a system subjected to a strong enough sinusoidal excitation will leave the linear response regime. Nonlinearities cause harmonics of the excitation frequency to appear. But what intensity is needed before nonlinear effects will appear? One might think that a natural scale would be the electric field of the nucleus at the location of an atomic electron. In the case of the hydrogen atom in its ground state, this field is about, or 3 × 10 11 V.m −1 . (Here e is the charge of the electron, and a 0 the Bohr radius, of the order of 5 × 10 −11 m). In fact, experiment shows that, in the transparency zone of a dielectric material like quartz, a field of just 10 7 V.m −1 (corresponding to a light intensity of 2.5 kW.cm −2 ) is sufficient for nonlinear effects to appear perturbatively.
In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11. Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states. In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2. In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations. A laser with an injected signal From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element. In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.
In earlier chapters we have considered in detail the susceptibility formalism of nonlinear optics, which is perhaps the most familiar approach and has a wide range of application. Starting from the constitutive relations of Chapter 2, the susceptibility formalism is quite general. In many practical applications, a particular phenomenon can be described accurately by a single order of nonlinearity, and the susceptibility then provides a useful and convenient description. However, this is not always the case. Some of the most interesting phenomena in nonlinear optics involve close resonance with the transition frequencies of the medium, and perhaps also the use of very intense optical fields. As remarked in §4.5, in these circumstances the resonant susceptibilities display mathematical divergences which are clearly unphysical. Strictly, these divergences occur only because higher-order nonlinearities have been neglected. Successive orders of nonlinearity take into account such effects as saturation, power broadening and level shifts (optical Stark effect). For intense fields or very close resonance, the contributions from several orders of nonlinearity may be comparable in magnitude. Therefore, despite its generality, the susceptibility formalism does not necessarily provide the most practical approach for the description of resonant processes. This chapter is concerned with the problem of deriving alternative and more manageable descriptions of resonant nonlinear processes, illustrated by examples. For much of the chapter, the nonlinear medium is treated as a two-level system (described in §6.2).
The colored rings shown in figure 1 are a spectacular manifestation of one type of nonlinear optical frequency conversion: parametric amplification of quantum noise. The amplification in this case is produced by the propagation of an intense pulse of ultraviolet radiation through a crystal of barium borate. When intense electromagnetic radiation propagates through such a material, whose polarization response at optical frequencies manifests a strongly nonlinear dependence on electric field amplitude, nonlinear mixing of the input radiation generates new spectral components. We can exploit this phenomenon to generate coherent radiation at frequencies for which we have no convenient laser sources.