2011-8-29Department of PhysicsSoutheast UniversityLecture Six: Derivative of The Wave Equation for Nonlinear Optical Media The forms of the linear and nonlinear susceptibility tensors are constrained by the symmetry properties of the optical medium.1. Spatial Symmetry of the Nonlinear MediumBy means of the mathematical method known as group theory, crystallographers have found that all crystals can be classified as belonging to one of 32 possible crystal classes depending on what is called the point group symmetry of the crystal.biaxial crystals. uniaxial crystals. Any additional symmetry property of a nonlinear optical medium can impose additional restrictions on the form of the nonlinear susceptibility tensor.By explicit consideration of the symmetries of each of the 32 crystal classes, one can determine the allowed form of the susceptibility tensor for crystals of that class.Since 11 of the 32 crystal classes possess inversion symmetry, this rule is very powerful, as it immediately eliminates all crystals belonging to these classes from consideration for second-order nonlinear optical interactions.Example 1: Derivative of contracted matrix in Page 46(1) First, we know from the Table 1.5.2 that the class 3m corresponds to a uniaxial crystal belonging to the trigonal crystal system, which has the following nonvanishing tensor components:(2) Then, using the prescription of contracted notation, we have(3) (4) (5) with this condition (3) becomes(6) which is just the same matrix form given in page 46.2. The Wave Equation for Nonlinear Optical Media In the section, we examine how Maxwells equations describe the generation of new components of the field, and more generally we see how the various frequency components of the field become coupled by the nonlinear interaction.As an example, we consider the SFG and show how a new frequency component can be generated, see Figure 2.1.1. Let us now consider the form of the wave equation for the propagation of light through a nonlinear optical medium. We begin with Maxwells equations, which we write in SI units in the form(7) We are primarily interested in the solution of these equations in regions of space that contain no free charges, so thatand that contain no free currents, so that(9) (8) We also assume that the material is nonmagnetic, so that(10) (11) We now proceed to derive the optical wave equation in the usual manner. One can easily find that (12) (13) This is the most general form of the wave equation in nonlinear optics. We can simplify it by using the identity(14) Fortunately, in nonlinear optics the first term can usually be dropped for cases of interest,especially when the slowly varying amplitude approximation is valid(15) (16) (17) By introducing a dimensionless, relative dielectric tensor (18) (19) For the case of an isotropic, dispersionless material, we have Hence the wave equation takes the formFor the case of a dispersive but isotropic medium, we must consider each frequency component of the field separately.We represent the electric, linear displacement, and polarization fields as the sums of their various frequency components:(20) where we represent each frequency component in terms of its complex, slowly varying amplitude as(21) (22) When Eqs. (20) through (22) are introduced into Eq. (18), weobtain a wave equation analogous to (19) that is valid for each frequency component of the field:(23) Later, we will use Eq. (23) to study in detail the nonlinear optical processes including SHG, SFG, DFG, and so on.逆水行舟用力撑逆水行舟用力撑, ,一篙松劲退千寻一篙松劲退千寻; ;古云此日足可惜古云此日足可惜, ,吾辈更应惜秒阴。吾辈更应惜秒阴。 董比武董比武 Exercises(作业):