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# Sum and Difference Frequency Generation

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Acronym: SFG, DFG

Definition: nonlinear processes generating beams with the sum or difference of the frequencies of the input beams

German: Summen- und Differenzfrequenzerzeugung

Crystal materials lacking inversion symmetry can exhibit a so-called χ(2) nonlinearity. In such nonlinear crystal materials, sum frequency generation (SFG) or difference frequency generation (DFG) can occur, where two pump beams generate another beam with the sum or difference of the optical frequencies of the pump beams. Such a process requires phase matching to be efficient, and usually there is no simultaneous phase matching for sum and difference frequency generation. A sum frequency mixer is sometimes called a FASOR (Frequency Addition Source of Optical Radiation).

A special case is sum frequency generation with an original pump wave and a frequency-doubled part of it, effectively leading to frequency tripling. Such a cascaded process can be much more efficient than direct frequency tripling on the basis of a χ(3) nonlinearity.

## Typical Applications

Some typical applications of sum frequency generation are:

Difference frequency mixing with pump waves of similar frequency can lead to a mixing product with a long wavelength. Some examples are:

• generation of light around 3.3 μm by mixing 1570 nm from a fiber laser and 1064 nm
• generation of light around 4.5 μm by mixing 860 nm from a laser diode and 1064 nm

Such mid-infrared wavelengths are required, e.g., for the spectroscopy of gases.

Difference frequency generation can also be used for generating terahertz waves. For efficient terahertz wave generation, there are special semiconductor-based photomixers, where the terahertz beat note of two similar optical frequencies generates an oscillation of the carrier density in the semiconductor, which is translated into an oscillating current and then into terahertz radiation.

## Insight from a Photon Picture

In a sum frequency mixer, both pump waves experience pump depletion when the signal becomes intense. For efficient conversion, the photon fluxes of both input pump waves should be similar. If one input wave has a lower photon flux, and its power is totally depleted somewhere in the crystal, there can be backconversion during subsequent propagation.

In a difference frequency mixer, the lower-frequency wave is amplified rather than depleted. This is because photons of the beam with highest photon energy (shortest wavelength) are effectively split into two lower-frequency photons, thus adding optical power to both lower-frequency waves. The term parametric amplification emphasizes the aspect of amplification, and the difference frequency mixing product is then called the idler wave.

### Bibliography

 [1] M. Bass et al., “Optical mixing”, Phys. Rev. Lett. 8 (1), 18 (1962) [2] S. Guha and J. Falk, “The effects of focusing in the three-frequency parametric upconverter”, J. Appl. Phys. 51 (1), 50 (1980)

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