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# Sum and Difference Frequency Generation

Author: the photonics expert

Acronyms: SFG, DFG

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

More general term: nonlinear frequency conversion

Crystal materials lacking inversion symmetry can exhibit a so-called <$\chi^{(2)}$> nonlinearity. In such nonlinear crystal materials, sum frequency generation ({SFG=sum frequency generation}) or difference frequency generation ({DFG=difference frequency generation}) can occur, where two pump beams generate another beam with the sum or difference of the optical frequencies of the pump beams.

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 <$\chi^{(3)}$> nonlinearity.

Sum or difference frequency generation processes require phase matching to be efficient. Usually there is no simultaneous phase matching for both processes, so that only one of them can take place.

## 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 laser 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. That physical mechanism is substantially different from the common one based on a <$\chi^{(2)}$> nonlinearity.

## Wavelength Calculations for SFG and DFG

 Input wavelength 1: Input wavelength 2: SFG output wavelength: calc DFG output wavelength: calc

Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

## Insight from a Photon Picture

### Sum Frequency Generation

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.

### Difference Frequency Generation

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.

## Carrier–Envelope Offset Frequencies

For operation with trains of ultrashort pulses, the carrier–envelope offset frequency (CEO frequency) of the output of a sum or difference frequency mixer is essentially the sum or difference, respectively, of those frequencies for the input. (The result may have to be corrected by subtracting the line spacing, which is identical to the pulse repetition rate, in order to get back to the interval from zero to the line spacing.)

It is interesting to consider what happens if difference frequency generation is applied to the low- and high-frequency components of a broadband frequency comb, which can be generated e.g. with a femtosecond laser, possibly followed by an optical fiber for supercontinuum generation. The CEO frequency of the output is then the difference between two identical frequencies, i.e., zero. This implies that the carrier–envelope offset phase is temporally constant. (In practice, it may still exhibit some drift, but only with a quite limited range.) This principle is realized in some devices for obtaining a more or less constant CEO phase without employing active stabilization methods.

## More to Learn

Encyclopedia articles:

## Suppliers

The RP Photonics Buyer's Guide contains 12 suppliers for sum and difference frequency generators. Among them:

### Bibliography

 [1] M. Bass et al., “Optical mixing”, Phys. Rev. Lett. 8 (1), 18 (1962); https://doi.org/10.1103/PhysRevLett.8.18 [2] S. Guha and J. Falk, “The effects of focusing in the three-frequency parametric upconverter”, J. Appl. Phys. 51 (1), 50 (1980); https://doi.org/10.1063/1.327353 [3] X. Chen et al., “5.32 W ultraviolet laser generation at 266 nm using sum-frequency method with CsB3O5 crystal”, Opt. Express 31 (2), 802 (2023); https://doi.org/10.1364/OE.474095

## Questions and Comments from Users

2023-05-31

When two optical waves are mixed in a medium with χ(2) nonlinearity, the two processes, beating and DFG, may happen simultaneously, correct? If so, a new frequency component corresponding to the difference of the two waves will be generated, and shown on an optical spectrum analyzer. Then how can I tell which process is responsible for this new frequency component? And what is the main advantages of the DFG process considering that the new component can be generated from the more common and easier beating process?