Sum and Difference Frequency Generation

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.

Sum and difference frequency generation can be implemented in different device configurations, depending on the required conversion efficiency and power levels:

• Single-pass bulk devices involve a nonlinear crystal through which the two input beams pass once. This is simple and robust, suitable for high-peak-power pulsed lasers where the high optical intensity ensures sufficient conversion efficiency.
• Waveguide devices confine the light in a small cross-section over a longer distance, maintaining high intensity. This allows for efficient conversion even with continuous-wave (CW) or low-power pulsed sources. Common materials for waveguides include lithium niobate (often periodically poled, PPLN) and gallium arsenide.
• Resonant enhancement cavities can be used to drastically increase the circulating power of one or both input beams (or resonating the output), thereby enhancing the conversion efficiency for CW sources without using waveguides.

Some typical applications of sum frequency generation are:

• generation of red light (→ red lasers), e.g. by mixing the outputs of a 1064-nm Nd:YAG laser and a 1535-nm fiber laser, resulting in an output at 628 nm
• generation of ultraviolet light, e.g. by mixing the output of a 1064-nm Nd:YAG laser with frequency-doubled light at 532 nm, resulting in 355-nm UV light

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.

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, 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.