Page 337 - Tunable Lasers Handbook

7 Optical Parametric OsciIIators 297

2. PARAMETRIC INTERACTIONS
Optical parametric oscillators and amplifiers can be created bir using the fre-
quency mixing properties in nonlinear crystals. Nonlinearity in crystals can be
characterized through a set of nonlinear coefficients. In general. the polarization
of a crystal can be expanded in a power series of the applied electric field. For
most materials, the components of polarization vector PI are linearly related to
the components of the applied electric field vector El. Subscripts refer to the vec-
tor components of the polarization and the electric field and are usually
expressed in Cartesian coordinates. Nonlinear crystals have a significant non-
linear response to the electric field which can be described by

where E~ is the permittivity of free space, dlJ are components of a 3 x 6 tensor,
and (EE), is the product of the applied electric fields creating the nonlinear
polarization. Because the polarization depends on the product of the applied
electric fields. frequency mixing can occur. That is, the product of the two elec-
tric fields will contain terms at both sum and difference frequencies. Sum and
difference frequencies are obtained by expanding the product of two sine waves
using trigonometric identities. Optical parametric oscillators use this effect to
generate new frequencies or wavelengths from the pump.
Components of the nonlinear tensor depend on the symmetry Df the nonlin-
ear crystal. For a nonlinear crystal with very low symmetry, all IS components
of the nonlinear tensor may exist. However, in general, crystal symmetry mini-
mizes the number of independent components. Depending on the symmetry,
some of the components are zero while other components may be simply related
to each other. For example, some components may be equal to a given compo-
nent or equal to the negative of a given component. Which components exist
depends on the point group of the nonlinear crystal. Given the point group, the
nonzero components and the relations between them can be determined by refer-
ring to tables [9].
To satisfy conservation of momentum, the nonlinear interaction usually
occurs in a birefringent crystal. Over the range of transparency. the refractive
index of a crystal is usually a monotonically decreasing function of wavelength,
If this is thLe case, the crystal is said to have noma1 dispersion. Thus. in
isotropic materials where there is only one refractive index, conservation of
momenturn (cannot be satisfied. To satisfy conservation of momentum. a bire-
fringent noiidinear crystal is utilized since, in these crystals. two indices of
refraction are available,
In birefringent crystals the refractive index depends on the polarization as
well as the direction of propagation. In uniaxial birefringent crystals, at a given
wavelength, the two refractive indices are given by [ 101