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Frequency Doubling

Author: the photonics expert

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

Alternative term: second-harmonic generation

More general term: nonlinear frequency conversion

More specific terms: resonant frequency doubling, intracavity frequency doubling

Category: article belongs to category nonlinear optics nonlinear optics

DOI: 10.61835/7jb   Cite the article: BibTex plain textHTML

Crystal materials lacking inversion symmetry can exhibit a so-called <$\chi^{(2)}$> 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 vacuum 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 the same or a similar direction.

frequency doubling
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 <$\chi^{(2)}$> 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 that doubled frequency. Due to the requirement of phase-matching (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 becomes substantial: 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 (not depleted) pump intensity:

$${P_2} = \gamma \;P_1^2$$

where the factor <$\gamma$> depends on details such as the effective mode area, the length of the crystal and many crystal properties including its effective nonlinearity.

Once the conversion efficiency and thus pump depletion becomes significant, the further rise of second-harmonic power becomes slower. Of course, the converted power <$P_2$> cannot become larger than the input power <$P_1$>.

Phase Matching

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% [12, 14, 22]. Even values of the order of 90% are possible with flat-top spatial and temporal profiles.

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 <$\chi^{(2)}$> 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, which lead to a stronger nonlinear interaction.

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. That effect is not relevant for nanosecond pulses from Q-switched lasers, but in that case there can still be some change of pulse duration; often, the frequency-doubled pulses are somewhat shorter than the pump pulses.

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 resonators.

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 greater propagation 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.

For example, 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.

Unfortunately, frequency doubling in waveguides involves various disadvantages, which limit its usefulness in many cases:

  • Waveguides requires special fabrication techniques, which are not well established for all materials. (Particularly well developed are waveguides in LiNbO3 and LiTaO3.)
  • It is necessary to efficiently couple the pump light into the waveguide. This introduces coupling losses and tight alignment tolerances.
  • Angle tuning is not possible with a waveguide.
  • Waveguides typically exhibit higher propagation losses.

For those reasons, the use of waveguides for frequency doubling is not very widespread.

Generating Short Wavelengths

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

Frequency-doubled neodymium-based lasers have largely replaced large argon ion lasers, since they reach similar or better performance 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. However, efficiencies around 80% are achievable in special cases [28].

Design of a Frequency Doubler

For the design of a frequency doubler, a number of non-trivial aspects have to be considered:

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.

Second-harmonic Imaging

Frequency doubling is also sometimes used as a method of imagingsecond-harmonic microscopy, applied e.g. to nonlinear crystals [7] or to biological samples [10]. One may, for example, measure the amount of frequency-doubled light as a function of the position of a tight beam focus sent to the sample. Here, one may also vary the polarization direction of the applied beam.

More to Learn

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The RP Photonics Buyer's Guide contains 38 suppliers for frequency doubling devices. Among them:


[1]P. A. Franken et al., “Generation of optical harmonics”, Phys. Rev. Lett. 7 (4), 118 (1961); https://doi.org/10.1103/PhysRevLett.7.118 (first report of second-harmonic generation, not yet phase-matched)
[2]A. Savage and R. C. Miller, “Measurements of second harmonic generation of the ruby laser line in piezoelectric crystals”, Appl. Opt. 1 (5), 661 (1962); https://doi.org/10.1364/AO.1.000661
[3]A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing”, IEEE J. Quantum Electron. 2 (6), 109 (1966); https://doi.org/10.1109/JQE.1966.1074007
[4]W. J. Kozlovsky et al., “Efficient second harmonic generation of a diode-laser pumped cw Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities”, IEEE J. Quantum Electron. 24, 913 (1988); https://doi.org/10.1109/3.211
[5]G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams”, J. Appl. Phys. 39 (8), 3597 (1968) (a seminal work with a comprehensive quantitative discussion)
[6]S. Fine and W. P. Hansen, “Optical second harmonic generation in biological systems”, Appl. Opt. 10 (10), 2350 (1971); https://doi.org/10.1364/AO.10.002350
[7]R. Hellwarth and P. Christensen, “Nonlinear optical microscopic examination of structure in polycrystalline ZnSe”, Opt. Commun. 12 (3), 319 (1974); https://doi.org/10.1016/0030-4018(74)90024-8
[8]J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation”, IEEE J. Quantum Electron. 20 (10), 1178 (1984); https://doi.org/10.1109/JQE.1984.1072294
[9]K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4”, IEEE J. Quantum Electron. 22 (7), 1013 (1986); https://doi.org/10.1109/JQE.1986.1073097
[10]I. Freund and M. Deutsch, “Second-harmonic microscopy of biological tissue”, Opt. Lett. 11 (2), 94 (1986); https://doi.org/10.1364/OL.11.000094
[11]Y. B. Band et al., “Spectrum of second-harmonic generation for multimode fields”, Phys. Rev. A 42 (3), 1515 (1990); https://doi.org/10.1103/PhysRevA.42.1515
[12]Z. Y. Ou et al., “85% efficiency for cw frequency doubling from 1.08 to 0.54 μm”, Opt. Lett. 17 (9), 640 (1992); https://doi.org/10.1364/OL.17.000640
[13]R. Paschotta et al., “Nonlinear mode coupling in doubly-resonant frequency doublers”, Appl. Phys. B 58, 117 (1994); https://doi.org/10.1007/BF01082345
[14]R. Paschotta et al., “82% efficient continuous-wave frequency doubling of 1.06 μm with a monolithic MgO:LiNbO3 resonator”, Opt. Lett. 19 (17), 1325 (1994); https://doi.org/10.1364/OL.19.001325
[15]V. Pruneri et al., “49 mW of cw blue light generated by first-order quasi-phase-matched frequency doubling of a diode-pumped 946-nm Nd:YAG laser”, Opt. Lett. 20 (23), 2375 (1995); https://doi.org/10.1364/OL.20.002375
[16]R. Wynands et al., “How accurate is optical second-harmonic generation?”, Opt. Lett. 20 (10), 1095 (1995); https://doi.org/10.1364/OL.20.001095
[17]J.-P. Meyn et al., “Tunable ultraviolet radiation by second-harmonic generation in periodically poled lithium tantalate”, Opt. Lett. 22 (16), 1214 (1997); https://doi.org/10.1364/OL.22.001214
[18]J. Webjorn et al., “Visible laser sources based on frequency doubling in nonlinear waveguides”, IEEE J. Quantum Electron. 33 (10), 1673 (1997); https://doi.org/10.1109/3.631263
[19]W. J. Alford and A. V. Smith, “Frequency-doubling broadband light in multiple crystals”, J. Opt. Soc. Am. B 18 (4), 515 (2001); https://doi.org/10.1364/JOSAB.18.000515
[20]Q. H. Xue et al., “High-power efficient diode-pumped Nd:YVO4/LiB3O5 457 nm blue laser with 4.6 W of output power”, Opt. Lett. 31 (8), 1070 (2006); https://doi.org/10.1364/OL.31.001070
[21]J. Burghoff et al., “Efficient frequency doubling in femtosecond laser-written waveguides in lithium niobate”, Appl. Phys. Lett. 89, 081108 (2006); https://doi.org/10.1063/1.2338532
[22]T. Südmeyer et al., “Efficient 2nd and 4th harmonic generation of a single-frequency, continuous-wave fiber amplifier”, Opt. Express 16 (3), 1546 (2008); https://doi.org/10.1364/OE.16.001546
[23]A. Canagasabey et al., “High-average-power second-harmonic generation from periodically poled silica fibers”, Opt. Lett. 34 (16), 2483 (2009); https://doi.org/10.1364/OL.34.002483
[24]C. Stolzenburg et al., “700 W intracavity-frequency doubled Yb:YAG thin-disk laser at 100 kHz repetition rate”, Proc. SPIE 7578, 75780A (2010); https://doi.org/10.1117/12.840875
[25]M. Galli et al., “Generation of deep ultraviolet sub-2-fs pulses”, Opt. Lett. 44 (6), 1308 (2019); https://doi.org/10.1364/OL.44.001308
[26]H. Chi et al., “Demonstration of a kilowatt average power, 1 J, green laser”, Opt. Lett. 45 (24), 6803 (2020); https://doi.org/10.1364/OL.412975
[27]J. P. Phillips et al., “Second and third harmonic conversion of a kilowatt average power, 100-J-level diode pumped Yb:YAG laser in large aperture LBO”, Opt. Lett. 46 (8), 1808 (2021); https://doi.org/10.1364/OL.419861
[28]C. Aparajit et al., “Efficient second-harmonic generation of a high-energy, femtosecond laser pulse in a lithium triborate crystal”, Opt. Lett. 46 (15), 3540 (2021); https://doi.org/10.1364/OL.423725
[29]A. Cifuentes et al., “Polarization-resolved second-harmonic generation imaging through a multimode fiber”, Optica 8 (8), 1065 (2021); https://doi.org/10.1364/OPTICA.430295

(Suggest additional literature!)

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Questions and Comments from Users


Does second-harmonic generation double the linewidth or frequency noise of a laser?

The author's answer:

Let us assume that we are dealing with a single-frequency laser. Its phase fluctuations are doubled in the harmonic beam, which implies that the power spectral density of the phase noise is increased fourfold. The linewidth would then also be doubled, if you simply define it as the r.m.s. value of the instantaneous optical frequency. However, it is more common to take the width of the optical spectrum, and that is more difficult to calculate.


What is the mechanism that causes the decrease of the SHG pulse duration compared to the fundamental?

The author's answer:

In simple situations (without significant group velocity mismatch, chromatic dispersion etc.), the conversion efficiency is higher at the pulse peak than in the wings. Consider e.g. a Gaussian pulse in the limit of low conversion (weak pump depletion): the intensity vs. time of the converted pulse is the pump intensity squared, which results in a Gaussian with a duration which is some 30% smaller.

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