Optical Parametric Oscillator

a source of coherent optical radiation. The basic element of an optical parametric oscillator is a nonlinear crystal in which a high-power light wave of fixed frequency parametrically excites light waves of a lower frequency. The frequencies of the parametrically excited waves are determined by the dispersion of the light in the crystal. By changing the dispersion of the medium, that is, the value of n, the frequency of the waves radiated by the optical parametric oscillator can be controlled.

Optical parametric oscillators were proposed in 1962 by S. A. Akhmanov and R. V. Khokhlov of the USSR. The first such generators were constructed in 1965 by Giordmaine and Miller in the USA and somewhat later by Akhmanov, Khokhlov, and co-workers. When a high-intensity light wave (the pump wave) propagates in a crystal, it modulates the crystal’s dielectric constant ∊. The electric field of the pump wave is

E_p = E_p0 sin(ω_pt - k_px + φ_p

where k_p = ω_p/v_p is the wave number, φ_p is the initial phase, t is time, and x is the coordinate of a point in the crystal. The dielectric constant ∊ varies in accordance with the law for a traveling wave: ∊ = ∊₀[1 + m sin (ω_p t + k_p x + φ_p], where

m = 4πXE_p0/∊₀

is called the depth of modulation of the dielectric constant, in which X is a quantity that characterizes the nonlinear properties of the crystal.

At the entrance face (x= 0) of the crystal in which the dielectric constant ∊ is varying with time, electromagnetic oscillations may be excited at the frequencies ω₁ and ω₂ and the phases φ₁ and φ₂. The relationships between these quantities are given by the equations ω_ι + ω₂ = ω_p and φ₁ + φ₂ = φ_p, which are like the equations for the parametric excitation of oscillations in a two-circuit system. The oscillations at the frequencies

Figure 1. Relationship between the index of refraction η and the wave frequency ω for normal dispersion

ω₁ and ω₂ are propagated inside the crystal as two light waves. The pump wave imparts energy to the waves along the entire propagation path if the following relationship between the phases holds:

(1) φ_p (x) = φ₁ (x) + φ₂ (x)+ π/2

This corresponds to the conditions for phase matching

(2) k₁ + k₂ = k_p

Figure 2. (a) The phase-matching condition in a nonlinear crystal, (b) change in length of the wave vector k_p of the extraordinary wave and k₁ and k₂ of the ordinary generated waves when the crystal is rotated, (c) dependence of the frequencies ω₁ and ω₂ of the generated waves on θ; (θ) angle between the optic axis of the crystal and the pump beam, (θ_x) phase-matching direction

Equation (2) indicates that the wave vectors k_p of the pump wave and k₁ and k₂ of the excited waves form a closed triangle. It follows from (2) that the condition for the crystal’s indexes of refraction at the frequencies ω_p, ω₁, and ω₂ is n (ω_p) ≥ n (ω₂) + [n(ω₁) – n (ω₂)]ω₁/ω_p

With phase matching, the amplitudes of the excited waves continuously increase as the waves propagate in the crystal:

where δ is the attenuation coefficient of the waves in the ordinary, or linear, medium. Clearly, parametric excitation occurs if the pump field exceeds the threshold:

In a medium having normal dispersion, where the index of refraction n increases with the frequency ω, phase-matched interaction of the waves is unrealizable (Figure 1). In anisotropic crystals, however, where two types of waves, ordinary and extraordinary, can propagate, the phase-matching condition can be fulfilled if not only the frequency dependence of the index of refraction but also the index’s dependence on the polarization of the wave and on the direction of propagation is made use of. For example, in a negative uniaxial crystal the index of refraction of the ordinary wave n₀ is greater than the index of refraction of the extraordinary wave n_e, which depends on the direction of propagation of the wave with respect to the crystal’s optic axis. If the wave vectors are parallel to one another, the phase-matching condition is satisfied in the specific direction for which

(4)

2n_e (ω_p, θ_s) = n_o (ω₁) + n_o (ω_p ω₁)

2n_e (ω_p, θ_s) = n_o (ω₂) + n_o (ω_p – ω₂)

The angle θ_s with respect to the optic axis is called the phase-matching angle and is a function of the pump frequency and the frequency of one of the excited waves. By varying the direction of propagation of the pump wave relative to the optic axis through rotation of the crystal, the frequency of the optical parametric oscillator can be tuned smoothly (Figure 2). Other methods of frequency tuning exist; they involve the dependence of the index of refraction n on such factors as the temperature and an external electric field.

To increase the power of the optical parametric oscillator, the crystal is positioned inside an open resonator so that the waves traverse the crystal many times during the action of the pump —that is, the effective length of the crystal is increased (Figure 3). The frequency tuning of such a resonator-type optical parametric oscillator proceeds in small steps determined by the difference between the frequencies corresponding to the longitudinal modes of the resonator. Smooth tuning can be obtained by combining rotation of the crystal with variation of the resonator parameters.

Optical parametric oscillators are manufactured in many countries. The radiation of a laser (pulsed and continuous types) or of the laser’s optical harmonics serves as the pump source. Existing parametric generators cover the wavelength range from 0.5 to 4 microns. Oscillators are being developed that are tunable in the wavelength region of 10 to 15 microns. Some oscillators provide frequency tuning within 10 percent of ω_p. The unique characteristics of optical parametric oscillators—coherence of

Figure 3. A nonlinear crystal positioned in an optical resonator: (M₁) and (M₂) mirrors forming the resonator

frequency tuning—make such a generator a basic instrument for radiation, narrowness of spectral lines, high power, and smooth spectroscopic research (for example, time-resolved spectroscopy). These characteristics also permit optical parametric oscillators to be used for selective action on a substance, especially on biological objects.