Difference Frequency Generation

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The generation of a photon at frequency $ω 3$ when photons at frequencies $ω 1$ and $ω 2$ are incident on a crystal with an appreciable second-order susceptibility $χ (2)$ , such that $ω 3 = ω 1 - ω 2$ (assuming $ω 1 > ω 2$ ). In order for energy conservation to hold, this additionally implies that, for every photon generated at the difference frequency $ω 3$ , a photon at $ω 2$ must also be created, while a photon at the higher frequency $ω 1$ must be annihilated. In addition, for the process to occur with an appreciable efficiency of frequency conversion, phase-matching must occur, so that $\mathbf{k}_3=\mathbf{k}_{1}-\mathbf{k}_{2}$ .

In many situations, the field at $ω 1$ is an intense pump field, while the field at $ω 2$ is a weak signal field. Difference-frequency generation yields amplification of the $ω 2$ field (along with generation of another field at $ω 3$ , commonly called the idler field). Thus, this process is termed parametric amplification; when $ω 2 = ω 3$ , the device created is called a Degenerate Parametric Amplifier.

If this process occurs within an Optical Cavity with resonance frequency $ω 2$ , the device is called an Optical Parametric Oscillator, or OPO. If the cavity has modes at both $ω 2$ and $ω 3$ , the device is a Doubly Resonant Optical Parametric Oscillator, and if $ω 2 = ω 3$ , it is a Degenerate Optical Parametric Oscillator. OPOs are often used to generate light that is red-detuned from some available laser source (for example, to generate infrared light using a Titanium-Sapphire laser as a pump).

Difference-frequency generation can also occur in the absence of an applied signal field; under these circumstances, it is called Spontaneous Parametric Fluorescence, or Parametric Down-conversion. In this case, the frequencies of the generated photons are determined by the phase-matching condition for the particular crystal orientation used. Parametric down-conversion is one way to generate entangled photons.