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CHAPTER 1
Introduction to Nonlinear Optics
The advent of laser in 1960 was the most important event in the history of modern optics. As an intense coherent light, the laser radiation exhibits the properties of extremely high monochromatic brightness and photon degeneracy. Based on the interaction of laser radiation with matter, a great number of new optical effects and phenomena have been discovered and exploited, which are the major subjects of nonlinear optics. In this new branch of optics, two parallel theories, semiclassical theory and quantum electrodynamic theory can be complementally adopted to explain and describe all of these new optical effects and phenomena.
1.1Definition of Nonlinear Optics
As a new branch of optical physics, nonlinear optics is a study that deals mainly with various new optical effects and novel phenomena arising from the interactions of intense coherent optical radiation with matter. There is a historical reason why this new branch of optical physics is termed ‘nonlinear optics’.
1.1.1Linear optics
Before the 1960s, in the areas of conventional optics many basic mathematical equations or formulae manifested a linear feature. To show this linear feature of conventional optics, we shall consider the following three examples.
First, in order to interpret the refraction, reflection, dispersion, scattering, as well as birefringence of light propagation in a medium, we should consider an important physical quantity, the electric polarization induced in the medium. In the regime of conventional optics, the electric polarization vector P of a medium is simply assumed to be linearly proportional to the electric field strength E of an applied optical wave field, i.e.,
where ε0 is the free-space permittivity and χ is the susceptibility of a given medium. Based on this linear assumption, Maxwell’s equations lead to a set of linear differential equations in which only the terms proportional to the first power of the field E are involved. As a result, there is no coupling between different light beams or between different monochromatic components when they pass through a medium. In other words, if there are several monochromatic optical waves of different frequencies passing through a medium simultaneously, no coherent radiation at any new frequency will be generated.
Second, in conventional optics the attenuation of an optical beam propagating in an absorptive medium can be described as
where I is the beam intensity, z is the variable along the propagation direction, and α is the absorption coefficient of a given medium. The physical meaning of Eq. (1.1.2) is that the decrease of the beam intensity in a unit propagation length is linearly proportional to the local intensity itself. From Eq. (1.1.2) we obtain a very familiar exponential attenuation expression
This expression implies that for a given propagation length of z = l, the transmitted intensity I(l) is linearly proportional to the initial intensity of I = I(0).
The third example is related to a Fabry–Perot (F–P) interferometer that plays a vital role in modern optics. The transmission T of this device is determined by1
Here, F is a constant determined by the reflectivity of the two mirrors of the interferometer, and δ is a phase-shift factor determined by
where λ is the wavelength of the incident beam, L is the spacing between the two mirrors, θ is the angle between the beam and the normal of the mirrors, and finally no is the refractive index of the medium inside the F–P cavity. In the regime of conventional optics, n0 is a constant for a given wavelength and independent of the incident beam intensity. Therefore, the transmission T of the whole device is also a constant for given values of λ, θ, and L. In this case the transmitted intensity It is linearly proportional to the incident intensity I0, i.e.,
So far, we have given three examples that manifest a simple linear relation as shown by Eqs. (1.1.1), (1.1.2), and (1.1.6), respectively. These simple linear assumptions or conclusions given by the conventional optics were widely accepted, as well as verified by most experimental observations and measurements based on the use of ordinary light sources. However, these situations have been changed radically since the beginning of the 1960s.
1.1.2Nonlinear optics
The conceptual design of laser was published in 1958,2 and the first laser device (a pulsed ruby laser) was demonstrated in 1960.3 Shortly after the advent of laser, researchers soon found that when an intense laser beam interacted with certain types of optical media, those simple previous linear assumptions or conclusions were no longer adequate. For the sake of clarity, we shall stay with our three examples and show why some higher-order approximations must be employed when an intense laser field interacts with an optical medium.
The first breakthrough was achieved in 1961 when a pulsed ruby laser beam was sent into a piezoelectric crystal. In this case, researchers for the first time observed the second-harmonic generation at an optical frequency.4 Shortly after this discovery, several other coherent optical frequency-mixing effects (such as optical sum-frequency effect,5 optical third-harmonic effect,6 optical rectification effect,7 optical difference-frequency effect,8,9 and optical parametric amplification/oscillation as well10,11) were sequentially observed. Only since that time, researchers had realized that all these new effects could be reasonably explained by replacing the linear term on the right-hand side of Eq. (1.1.1) by a power series
Here, χ(1), χ(2), and χ(3) are the first-order (linear), second-order (nonlinear), and third-order (nonlinear) susceptibility and so on. They are material coefficients and in general are tensors. Substituting Eq. (1.1.7) into Maxwell’s equations leads to a set of nonlinear differential equations that involve high-order power terms of optical electric field strength; these terms are responsible for various observed coherent optical frequency-mixing effects.12
In the same period, researchers also found that the depletion behavior of an intense laser beam propagating in an absorptive optical medium did not follow the description indicated by Eq. (1.1.2) or Eq. (1.1.3). For instance, in a one-photon absorptive medium, if the intensity of the incident beam is high enough, the attenuation coefficient α is no longer a constant and may become a variable that depends on the incident intensity. Therefore, the exponential attenuation formula like Eq. (1.1.3) cannot be applied and the linear relationship between I(z = l) and I(0) does not hold. In this case, either a saturable absorption or a reverse-saturable absorption effect may take place. Moreover, if there is a two-photon absorption process involved in the medium, the attenuation of an intense incident beam should be described as
where β is the two-photon absorption coefficient of a given medium. In more general cases, if we further extend our consideration to include multi-photon (three-photon or more) absorption processes, Eq. (1.1.8) should be generalized to the following form:
Here, γ is the three-photon absorption coefficient and so on.
Now, let us return to the transmission behavior of the F–P device under the action of an intense laser beam. In this case, Eq. (1.1.6) is no longer applicable. According to the linear electric polarization expressed by Eq. (1.1.1), the refractive index of the medium at a given wavelength should be a constant independent of the light intensity. However, based on the more general assumption expressed by Eq. (1.1.7), the refractive index for centrosymmetric or isotropic media can be written as (see Chapter 5)
where the first term n0 is the linear refractive index independent of the beam intensity, the second term corresponds to an additional nonlinear refractive index contribution proportional to the beam intensity, and n2 is a proportionality coefficient. When the beam intensity is quite low, the second ...
