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© Dr. Rüdiger Paschotta

Last update: 2008-03-28

Definition: a process of optical amplification based on a parametric nonlinearity

Parametric amplification is a phenomenon where a signal can be amplified using a parametric nonlinearity and a pump wave. This articles focuses on optical amplification, even though there are also electronic parametric amplifiers, used e.g. for microwaves. For optical parametric amplifiers, either the χ⁽²⁾ nonlinearity of certain nonlinear crystal materials or the χ⁽³⁾ nonlinearity e.g. of an optical fiber [7] can be utilized.

This article discusses the physical mechanism of parametric amplification based on a χ⁽²⁾ nonlinearity. It is important to distinguish between degenerate and nondegenerate amplification, because these lead to very different features.

Nondegenerate Parametric Amplification

In the nondegenerate case, there is an interaction between three distinct light waves, the angular frequencies of which are ω₁, ω₂, and ω₃ (with the indices of the order of the frequency values). For reasons of energy conservation, the relation ω₃ = ω₁ + ω₂ must hold. The pump wave has the frequency ω₃, a signal wave ω₂, and ω₁ is called the idler frequency. (Exchanging ω₁ and ω₂ would not lead to essential changes of the analysis.) For a simple case with plane waves or with weakly focused collinear Gaussian beams, the nonlinear interaction of the waves with complex amplitudes A₁ to A₃ can be described with the equations

where

is the phase mismatch, calculated from the wavenumbers, z is the coordinate of the propagation direction, and κ is a coupling constant proportional to the nonlinear coefficient of the material. A more symmetric set of equations is sometimes used, which can be obtained from the equations above by replacing the amplitude A₃ with i A₃.

The simplest situation is that with a zero phase mismatch. When there is no idler input (A₁ = 0 at the beginning of the crystal), in the first order there is no signal amplification nor any pump depletion, but a buildup of an idler wave. If the initial A₃ and A₂ are real, a real amplitude of the idler will be built up. (This process may be called difference frequency generation.) In the following, this leads to a growth of A₂, i.e., to signal amplification, and later on to the depletion of the pump wave.

It is instructive to consider that the phase of the initial signal is now rotated by e.g. 30°. This will lead to a rotation of the idler phase by -30°. Consequently, the field contributions added to the signal are also rotated by +30°, so that the signal is amplified as before. This shows that the amplification is phase-insensitive (i.e., independent of the signal phase) in this nondegenerate case (with a separate signal and idler).

In another case, the pump phase is rotated by 30°. This will also rotate the idler by 30°. Both phase changes cancel in the contribution added to the signal, so the signal is again amplified, and only the idler phase is changed.

When there is a phase mismatch (i.e., a nonzero Δk), the relative phases of the waves will change during propagation, so that after some distance the power in signal and idler can be converted back toward the pump. Therefore, phase matching, which can be achieved only in a limited frequency range, is important for efficient amplification. Under certain special conditions, a very large phase-matching bandwidth may be achieved.

The equations also show that the signal amplification is reduced in a medium with strong absorption for the idler wave, which keeps the idler amplitude small. Therefore, the transparency range of the nonlinear crystal has to be large enough to keep the idler wave in the low-loss window, even if one is not interested in the idler wave itself – otherwise, the parametric gain is strongly reduced.

The used equations are based on classical physics. In a quantum-mechanical picture, there are processes where pump photons are converted into signal-idler photon pairs. These are strongly correlated, leading to nonclassical statistics.

Degenerate Parametric Amplification

In the rarely used degenerate case, signal and idler waves are identical not only in frequency but also in polarization, i.e., they are indistinguishable. There is thus only a signal amplitude A₁ and a pump amplitude A₂, but no idler. The signal wavelength is then exactly twice the pump wavelength. The interaction is described with the equations

with

Here, the amplification is phase-sensitive. For example, signal amplification occurs (for zero phase mismatch) if signal and pump amplitudes are real and positive, or there is signal deamplification when the sign of the pump amplitude is changed. The direction of energy transfer is governed by the complex phase of the term A₁² A₂^*.

Phase-sensitive amplification provides a mechanism for producing so-called squeezed states of light, and also in principle it allows to avoid excess amplifier noise. However, the need to maintain a fixed phase relationship between pump and signal makes this kind of optical amplification too inconvenient e.g. for use in optical fiber communications.

Bibliography

See also: optical parametric amplifiers, optical parametric chirped-pulse amplification, optical parametric oscillators, optical parametric generators, amplifiers, amplifier noise, nonlinearities, squeezed states of light, Spotlight article 2007-12-11

Category: nonlinear optics