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Encyclopedia of Laser Physics and Technology

© Dr. Rüdiger Paschotta

Last update: 2008-05-13

(Acronym: SHG = second-harmonic generation)

Definition: the phenomenon that an input wave in a nonlinear material can generate a wave with twice the optical frequency

Crystal materials lacking inversion symmetry can exhibit a so-called χ⁽²⁾ nonlinearity (→ nonlinear crystal materials). This can give rise to the phenomenon of frequency doubling [1], where an input (pump) wave generates another wave with twice the optical frequency (i.e. half the wavelength) in the medium. This process is also called second-harmonic generation. In most cases, the pump wave is delivered in the form of a laser beam, and the frequency-doubled (second-harmonic) wave is generated in the form of a beam propagating in a similar direction.

Figure 1: A typical configuration for frequency doubling: an infrared input beam at 1064 nm generates a green 532-nm wave during its path through a nonlinear crystal.

The article on nonlinear crystal materials lists a number of crystal materials, many of which are popular for frequency doubling. Examples are lithium niobate (LiNbO₃), potassium titanyl phosphate (KTP = KTiOPO₄), and lithium triborate (LBO = LiB₃O₅).

The Physical Mechanism

The physical mechanism behind frequency doubling can be understood as follows. Due to the χ⁽²⁾ nonlinearity, the fundamental (pump) wave generates a nonlinear polarization wave which oscillates with twice the fundamental frequency. According to Maxwell's equations, this nonlinear polarization wave radiates an electromagnetic field with this doubled frequency. Due to phase matching issues (see below), the generated second-harmonic field propagates dominantly in the direction of the nonlinear polarization wave. The latter also interacts with the fundamental wave, so that the pump wave can be attenuated (→ pump depletion) when the second-harmonic intensity develops: energy is transferred from the pump wave to the second-harmonic wave.

For low pump intensities, the second-harmonic conversion efficiency is small and grows linearly with increasing pump intensity, so that the intensity of the second-harmonic (frequency-doubled) wave grows with the square of the pump intensity:

Once pump depletion becomes significant, the further rise of second-harmonic power becomes slower.

Frequency doubling is a phase-sensitive process which usually requires phase matching to be efficient. This means that the second-harmonic field contributions generated at different locations in the nonlinear crystal coherently add up at the crystal's exit face. With proper phase matching and a pump beam with high intensity, high beam quality, and moderate optical bandwidth, achievable power conversion efficiencies often exceed 50%, in extreme cases even 80% [8, 10, 16]. On the other hand, the conversion efficiencies are typically extremely small when phase matching does not occur. In such cases, the energy transferred by the χ⁽²⁾ nonlinearity quickly oscillates back and forth between pump and second-harmonic wave, rather than consistently going in a certain direction. The lack of phase matching is also the reason why second-harmonic generation is usually not accompanied by other processes such as sum frequency generation of the pump and second-harmonic wave, or second-harmonic generation of the second-harmonic wave itself: phase matching for second-harmonic generation usually does not imply phase matching for the other mentioned processes.

Nonlinear Frequency Conversion of Laser Pulses

High conversion efficiencies can be achieved even with moderate or low average pump powers when the pump light is delivered in the form of pulses, as e.g. generated with a mode-locked or Q-switched laser. This is simply because for a given average power a pulsed laser exhibits higher peak powers. Note, however, that for frequency conversion of ultrashort pulses, the effective interaction length and hence the conversion efficiency can be limited by group velocity mismatch, which causes a temporal walk-off.

Intracavity and Resonant Frequency Conversion

Efficient frequency doubling at moderate powers (e.g. in continuous-wave operation) is often accomplished with intracavity frequency doubling, i.e., by placing the frequency doubler crystal inside a laser resonator, thus exploiting the high intracavity intensity. Yet another technique is to use a resonant enhancement cavity external to the laser (→ resonant frequency doubling). This is possible for single-frequency operation and also with mode-locked lasers, but usually requires active stabilization of one of the involved cavities.

Second-harmonic Generation in Waveguides

Nonlinear waveguides present a way to achieve efficiency frequency doubling at fairly low power levels, i.e., without resorting either to short pulses or to resonant enhancement. They key is that a waveguide makes it possible to maintain a small mode area (and thus high intensities for a given power level) over a longer length than would be possible in a bulk medium, where diffraction would limit the interaction length to something of the order of the Rayleigh length.

In particular, high-quality channel waveguides can be fabricated with different techniques in lithium niobate (LiNbO₃) and lithium tantalate (LiTaO₃), which are nonlinear crystal materials with particularly high nonlinearity. The most important techniques are ion exchange or proton exchange (exposing a small stripe on the crystal surface to a liquid, e.g. benzoic acid) and titanium or zinc indiffusion (by strongly heating a crystal with a narrow stripe of titanium (or zinc) metal deposited on the surface with lithographic techniques). (A variation is vapor phase indiffusion.) Such waveguides can be several centimeters long and can exhibit propagation losses well below 1 dB/cm and second-harmonic conversion efficiencies of more than 100%/W in a 1 cm long device.

Generating Short Wavelengths

Frequency doubling is a frequently used technique for generating light with short wavelengths:

• Green light with wavelength 532 nm can be generated by frequency-doubling the output of a neodymium- or ytterbium-based 1064-nm laser (→ green lasers). Green laser pointers are also usually based on this approach.
• Many blue laser systems are based on a frequency-doubled laser in the 0.9-μm region (e.g. 914 nm from Nd:YVO₄).
• By further frequency doubling (or by sum frequency generation) still shorter wavelengths in the ultraviolet region (→ ultraviolet lasers) become accessible. The main challenges in this spectral region are the limited transparency range and (sometimes) limited durability of nonlinear crystal materials, and the strong chromatic dispersion (sometimes preventing phase matching or at least a large phase-matching bandwidth).

Frequency-doubled neodymium-based lasers can compete with large-sized argon ion lasers in terms of output power and beam quality, whereas having a far higher power efficiency and a longer lifetime.

For frequency doubling of ultrashort pulses, high single-pass conversion efficiencies are difficult to obtain at short wavelengths, because strong group velocity mismatch limits the interaction length, while optical damage limits the applicable optical intensities.

Design of a Frequency Doubler

Years ago, the author (while working on frequency doublers for the generation of squeezed states of light) was told by a colleague that "Any dog can put a frequency doubler in front of a laser". Apparently, this colleague was assuming noncritical phase matching near room temperature – which is generally no option. Also, at least the design of an optimized frequency doubler for a given situation goes well beyond the intellectual capacity of most dogs, as a number of non-trivial aspects have to be considered:

• What nonlinear crystal material to use? Where to get it from with suitable anti-reflection coatings? How much protection will it need? Will it be sufficiently durable?
• Use noncritical or critical phase matching?
• What is the optimum crystal length and pump beam radius, taking into account factors such as beam divergence, spatial walk-off, group velocity mismatch and damage threshold (see the spotlight 2007-09-21)?
• For noncritical phase matching: how good does the spatial homogeneity of the crystal temperature in the oven have to be? Can the coatings withstand the temperature cycles?
• For critical phase matching: how asymmetric will the second-harmonic beam profile be? What is the expected beam quality? Should one use walk-off compensation?
• In the case of a double pass through a crystal: what is the influence of relative phase changes of the fundamental and second-harmonic waves between the two passes?
• If resonant frequency doubling is considered: how to design the enhancement resonator for optimum performance and minimum sensitivity?

For finding the best configuration without costly and time-consuming iterations in the laboratory, it is recommended to carry out a careful design study as the first step.

Bibliography

See also: resonant frequency doubling, intracavity frequency doubling, phase matching, parametric nonlinearities, nonlinear crystal materials, frequency tripling, frequency quadrupling, green lasers, blue lasers, ultraviolet lasers, Spotlight article 2006-08-15, Spotlight article 2006-09-29, Spotlight article 2007-03-05, Spotlight article 2007-09-21, Spotlight article 2007-10-17, Spotlight article 2008-01-27

Category: nonlinear optics

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics Consulting GmbH. Contact this distinguished expert in laser technology, nonlinear optics and fiber optics, and find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, or staff training) could become very valuable for your business!