13.1   INTRODUCTION

Who has not enjoyed standing on a beach caressed by the breaking waves? Surface
waves usually originate from wind blowing over the water surface. As the waves
approach the sloping beach, their amplitude increases and they slow down. Unlike
the waves that we discussed in Chapter 2, the speed of these waves depends on
the amplitude. Thus, the larger-amplitude portion of the wave travels faster than the
smaller-amplitude portion, and eventually the wave breaks (Fig. 13.1). The dependence
of the wave speed on the amplitude of the wave is a characteristic nonlinear
phenomenon.

In the case of light waves, when we are dealing with light from ordinary sources
with small powers, the propagation of lightwaves in any medium is linear. This means
that the propagation effects of a light wave are independent of the intensity of the light
wave. The characteristics of the emerging light beam such as frequency, phase, and
wave shape, remain unchanged as we change its intensity. Apart from this, when two
light beams propagating through a medium intersect, each beam emerges without any
modification brought about by the other beam. Thus, referring to Fig. 13.2, we see
that there would be no change in the output beam 1 whether or not beam 2 is present,
and conversely. Thus, one light beam does not modify any of the properties of another
light beam, even if they cross each other. This happens whenever the intensities of
the two light beams are small, and this is referred to as linear optics.

When the light intensity becomes large, the electric field associated with the light
beam can modify the property of the medium to such an extent that it can then
affect its own propagation as well as that of other beams crossing it. For example,
a red beam entering an appropriate crystal can get converted partly to a blue output
beam under certain conditions (Fig. 13.3). This happens due to a nonlinear effect
(called second harmonic generation) in which a light beam of frequency f creates a
beam having double its frequency, 2f (half the wavelength). Similarly, the refractive
index of the medium can get changed due to the large intensity of the beam; this
change of refractive index would in turn change the phase with which a light wave
emerges from a medium. Such effects, referred to as nonlinear optical effects, have

become very important since discovery of the laser. The nonlinear optical effects
are almost instantaneous: that is, any change in intensity of the beam leads almost
instantaneously to a change in the medium property, and the medium relaxes to its
original state within times less than a few femtoseconds (1 femtosecond = 10^-15 s).