**Nonlinear Optical Crystal**

The propagation of a wave through a dielectric material is just an interaction (mainly electric field) of the wave and material. This dynamic process can be described by the following two procedures: response of material to wave, and radiation. If the polarization theory is taken, which means that the electric fields in the wave can induce intensity of polarization in the material, it may be considered that when propagating in through the material, the wave polarizes the response, and the polarization intensity produced as the driving source will induce reradiation. This reradiation is just the light wave propagating through the dielectric material. The relationship between the polarization intensity P and the electric field E can be expressed by the following formula (Equation 1):

P = ε0Χ•E

[ ε0-dielectric constant; X-polarizability tensor ]

If the material has linear response to the light, then X is a constant and independent from E. The optical phenomenon generated by this process falls into the linear optical category, i.e. the light’s propagation pattern follows principles of independent propagation and linear superposition.

If the material has nonlinear response to the light, the Equation 1 will changes to Equation 2:

P = ε0Χ(E)•E

X is related to E. The optical phenomenon generated by this process falls into the nonlinear optical category, i.e. the light’s propagation through the dielectric will produce new frequency. Light waves with different frequency will couple each other, which means that the principles of independent propagation and linear superposition can on longer be applied.

Nonlinear crystal is one kind of dielectric material which can generate nonlinear response. Crystal with nonlinear property does not be ready to give nonlinear response at any condition. Intense of the incident beam is one of the key factors of generating nonlinear response. The nonlinear effect can only be acquired by a light with certainly high intensity which helps the crystal to produce high enough intensity of polarization. This light with high intensity is generally a laser.

**Frequency Doubling**

Equation 2 has the following substitution (Equation 3) where Χ^{(2)}(E) and Χ^{(3)}(E) are respectively the second and third rank nonlinear dielectric susceptibility:

P = ε^{0}Χ(E)•E + Χ^{(2)}(E)•E2 + Χ^{(3)}(E)•E3 + …

Equation 3 tells us that, an incident laser beam can cause nonlinear response in nonlinear optical crystal, and thus produce a series of response light. If the nonlinear response is coincident with the phase-matching, i.e. the highest second rank nonlinear polarization has been acquired, then the response light has a frequency double of that of the incident light. This is what so called the optical frequency doubling.

**Optical Parametric Oscillator**

Giving the characteristic frequency of the working substance, a certain sort of laser device can only generate laser beam with certain wave-length. This limited seriously the application of laser. Generally, tunable laser and the nonlinear effect of nonlinear optical crystal can be applied to expand the coverage of laser wave-length. Ordinary laser outputs merely monochromatic light. But if some nonlinear optical crystal with ability of generating optical parametric amplification was introduced into the laser system, the pump light passing the crystal will give two beams named respectively the signal wave and the idler wave. The sum of the frequencies of these two beams equals to the frequency of the pump light, i.e. the optical parametric oscillation.

In the optical resonator, the beams of pump, signal and idler pass to and fro the nonlinear dielectric for many times. Laser oscillation will be formed in the resonator when energy gain is more than energy loss. This is due to the parametric amplification of the signal light and idler light which are formed continuously by the pump beam.

Tuning is the basic character of OPO. If the wave-length of the pump beam is fixed, any subtle change of the refractive index near to the phase-matching condition will lead to wave-length change of the signal and idler light. New condition of phase-matching thus forms. This kind of tuning can be realized by utilizing the connection between the birefraction and clear side angle of the anisotropic crystal, or by changing the temperature. Rapid tuning within limited scope can be realized by the E-O change of refractive index.

OPO has many advantages such as broad tuning range, compact structure, easy to use, high power, broad coverage of output wave-length (from UV to far IR), superior energy conversion efficiency, etc. OPO has been widely applied to many technology fields including new material, biology, chemistry, resonance spectrum, distance measurement, radar, etc. The common nonlinear crystal applied to OPO includes KTP, LBO, BBO, LiNbO_{3}, etc.