Frequency comb

Last updated

A frequency comb or spectral comb is a spectrum made of discrete and regularly spaced spectral lines. In optics, a frequency comb can be generated by certain laser sources.

Contents

A number of mechanisms exist for obtaining an optical frequency comb, including periodic modulation (in amplitude and/or phase) of a continuous-wave laser, four-wave mixing in nonlinear media, or stabilization of the pulse train generated by a mode-locked laser. Much work has been devoted to this last mechanism, which was developed around the turn of the 21st century and ultimately led to one half of the Nobel Prize in Physics being shared by John L. Hall and Theodor W. Hänsch in 2005. [1] [2] [3]

The frequency domain representation of a perfect frequency comb is like a Dirac comb, a series of delta functions spaced according to

where is an integer, is the comb tooth spacing (equal to the mode-locked laser's repetition rate or, alternatively, the modulation frequency), and is the carrier offset frequency, which is less than .

Combs spanning an octave in frequency (i.e., a factor of two) can be used to directly measure (and correct for drifts in) . Thus, octave-spanning combs can be used to steer a piezoelectric mirror within a carrier–envelope phase-correcting feedback loop. Any mechanism by which the combs' two degrees of freedom ( and ) are stabilized generates a comb that is useful for mapping optical frequencies into the radio frequency for the direct measurement of optical frequency.

An ultrashort pulse of light in the time domain. The electric field is a sinusoid with a Gaussian envelope. The pulse length is on the order of a few 100 fs Gaussian wave packet.svg
An ultrashort pulse of light in the time domain. The electric field is a sinusoid with a Gaussian envelope. The pulse length is on the order of a few 100  fs

Generation

Using a mode-locked laser

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T; the Fourier transform of a time-domain Dirac comb is a Dirac comb in the frequency domain. Dirac comb.svg
A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T; the Fourier transform of a time-domain Dirac comb is a Dirac comb in the frequency domain.

The most popular way of generating a frequency comb is with a mode-locked laser. Such lasers produce a series of optical pulses separated in time by the round-trip time of the laser cavity. The spectrum of such a pulse train approximates a series of Dirac delta functions separated by the repetition rate (the inverse of the round-trip time) of the laser. This series of sharp spectral lines is called a frequency comb or a frequency Dirac comb.

The most common lasers used for frequency-comb generation are Ti:sapphire solid-state lasers or Er:fiber lasers [4] with repetition rates typically between 100 MHz and 1 GHz [5] or even going as high as 10 GHz. [6]

Using four-wave mixing

Four-wave mixing is a process where intense light at three frequencies interact to produce light at a fourth frequency . If the three frequencies are part of a perfectly spaced frequency comb, then the fourth frequency is mathematically required to be part of the same comb as well.

Starting with intense light at two or more equally spaced frequencies, this process can generate light at more and more different equally spaced frequencies. For example, if there are a lot of photons at two frequencies , four-wave mixing could generate light at the new frequency . This new frequency would get gradually more intense, and light can subsequently cascade to more and more new frequencies on the same comb.

Therefore, a conceptually simple way to make an optical frequency comb is to take two high-power lasers of slightly different frequency and shine them simultaneously through a photonic-crystal fiber. This creates a frequency comb by four-wave mixing as described above. [7] [8]

In microresonators

An alternative variation of four-wave-mixing-based frequency combs is known as Kerr frequency comb. Here, a single laser is coupled into a microresonator (such as a microscopic glass disk that has whispering-gallery modes). This kind of structure naturally has a series of resonant modes with approximately equally spaced frequencies (similar to a Fabry–Pérot interferometer). Unfortunately the resonant modes are not exactly equally spaced due to dispersion. Nevertheless, the four-wave mixing effect above can create and stabilize a perfect frequency comb in such a structure. [9] Basically, the system generates a perfect comb that overlaps the resonant modes as much as possible. In fact, nonlinear effects can shift the resonant modes to improve the overlap with the perfect comb even more. (The resonant mode frequencies depend on refractive index, which is altered by the optical Kerr effect.)

In the time domain, while mode-locked lasers almost always emit a series of short pulses, Kerr frequency combs generally do not. [10] However, a special sub-type of Kerr frequency comb, in which a "cavity soliton" forms in the microresonator, does emit a series of pulses. [11]

Using electro-optic modulation of a continuous-wave laser

An optical frequency comb can be generated by modulating the amplitude and/or phase of a continuous-wave laser with an external modulator driven by a radio-frequency source. [12] In this manner, the frequency comb is centered around the optical frequency provided by the continuous-wave laser and the modulation frequency or repetition rate is given by the external radio-frequency source. The advantage of this method is that it can reach much higher repetition rates (>10 GHz) than with mode-locked lasers and the two degrees of freedom of the comb can be set independently. [13] The number of lines is lower than with a mode-locked laser (typically a few tens), but the bandwidth can be significantly broadened with nonlinear fibers. [14] This type of optical frequency comb is usually called electrooptic frequency comb. [15] The first schemes used a phase modulator inside an integrated Fabry–Perot cavity, [16] but with advances in electro-optic modulators new arrangements are possible.

Low-frequency combs using electronics

A purely electronic device which generates a series of pulses, also generates a frequency comb. These are produced for electronic sampling oscilloscopes, but also used for frequency comparison of microwaves, because they reach up to 1 THz. Since they include 0 Hz, they do not need the tricks which make up the rest of this article.

Widening to one octave

For many applications, the comb must be widened to at least an octave:[ citation needed ] that is, the highest frequency in the spectrum must be at least twice the lowest frequency. One of three techniques may be used:

These processes generate new frequencies on the same comb for similar reasons as discussed above.

Carrier–envelope offset measurement

Difference between group and phase velocity leading to carrier-envelope offset Wave opposite-group-phase-velocity.gif
Difference between group and phase velocity leading to carrier–envelope offset

An increasing offset between the optical phase and the maximum of the wave envelope of an optical pulse can be seen on the right. Each line is displaced from a harmonic of the repetition rate by the carrier–envelope offset frequency. The carrier–envelope offset frequency is the rate at which the peak of the carrier frequency slips from the peak of the pulse envelope on a pulse-to-pulse basis.

Measurement of the carrier–envelope offset frequency is usually done with a self-referencing technique, in which the phase of one part of the spectrum is compared to its harmonic. Different possible approaches for carrier–envelope offset phase control were proposed in 1999. [17] The two simplest approaches, which require only one nonlinear optical process, are described in the following.

In the "f − 2f" technique, light at the lower-energy side of the broadened spectrum is doubled using second-harmonic generation (SHG) in a nonlinear crystal, and a heterodyne beat is generated between that and light at the same wavelength on the upper-energy side of the spectrum. This beat signal, detectable with a photodiode, [18] includes a difference-frequency component, which is the carrier–envelope offset frequency.

Alternatively, difference-frequency generation (DFG) can be used. From light of opposite ends of the broadened spectrum the difference frequency is generated in a nonlinear crystal, and a heterodyne beat between this mixing product and light at the same wavelength of the original spectrum is measured. This beat frequency, detectable with a photodiode, is the carrier–envelope offset frequency.

Because the phase is measured directly, and not the frequency, it is possible to set the frequency to zero and additionally lock the phase, but because the intensity of the laser and this detector is not very stable, and because the whole spectrum beats in phase, [19] one has to lock the phase on a fraction of the repetition rate.

Carrier–envelope offset control

In the absence of active stabilization, the repetition rate and carrier–envelope offset frequency would be free to drift. They vary with changes in the cavity length, refractive index of laser optics, and nonlinear effects such as the Kerr effect. The repetition rate can be stabilized using a piezoelectric transducer, which moves a mirror to change the cavity length.

In Ti:sapphire lasers using prisms for dispersion control, the carrier–envelope offset frequency can be controlled by tilting the high reflector mirror at the end of the prism pair. This can be done using piezoelectric transducers.

In high repetition rate Ti:sapphire ring lasers, which often use double-chirped mirrors to control dispersion, modulation of the pump power using an acousto-optic modulator is often used to control the offset frequency. The phase slip depends strongly on the Kerr effect, and by changing the pump power one changes the peak intensity of the laser pulse and thus the size of the Kerr phase shift. This shift is far smaller than 6 rad, so an additional device for coarse adjustment is needed. A pair of wedges, one moving in or out of the intra-cavity laser beam can be used for this purpose.

The breakthrough which led to a practical frequency comb was the development of technology for stabilizing the carrier–envelope offset frequency.

An alternative to stabilizing the carrier–envelope offset frequency is to cancel it completely by use of difference frequency generation (DFG). If the difference frequency of light of opposite ends of a broadened spectrum is generated in a nonlinear crystal, the resulting frequency comb is carrier–envelope offset-free since the two spectral parts contributing to the DFG share the same carrier–envelope offset frequency (CEO frequency). This was first proposed in 1999 [17] and demonstrated in 2011 using an erbium fiber frequency comb at the telecom wavelength. [20] This simple approach has the advantage that no electronic feedback loop is needed as in conventional stabilization techniques. It promises to be more robust and stable against environmental perturbations. [21] [22]

Applications

Spectrum of the light from the two-laser frequency combs installed on the High Accuracy Radial Velocity Planet Searcher. Laser frequency comb installed on HARPS.jpg
Spectrum of the light from the two-laser frequency combs installed on the High Accuracy Radial Velocity Planet Searcher.

A frequency comb allows a direct link from radio frequency standards to optical frequencies. Current frequency standards such as atomic clocks operate in the microwave region of the spectrum, and the frequency comb brings the accuracy of such clocks into the optical part of the electromagnetic spectrum. A simple electronic feedback loop can lock the repetition rate to a frequency standard.

There are two distinct applications of this technique. One is the optical clock , where an optical frequency is overlapped with a single tooth of the comb on a photodiode, and a radio frequency is compared to the beat signal, the repetition rate, and the CEO-frequency (carrier–envelope offset). Applications for the frequency-comb technique include optical metrology, frequency-chain generation, optical atomic clocks, high-precision spectroscopy, and more precise GPS technology. [24]

Illustration showing how trace gases are detected in the field using a mobile dual-frequency comb laser spectrometer. The spectrometer sits in the center of a circle which is ringed with retroreflecting mirrors. Laser light from the spectrometer (yellow line) passes through a gas cloud, strikes the retroreflector and is returned directly to its point of origin. The data collected are used to identify leaking trace gases (including methane), as well leak locations and their emission rates. Dual-Comb Spectroscopy Detection of Trace Gases in the Field.jpg
Illustration showing how trace gases are detected in the field using a mobile dual-frequency comb laser spectrometer. The spectrometer sits in the center of a circle which is ringed with retroreflecting mirrors. Laser light from the spectrometer (yellow line) passes through a gas cloud, strikes the retroreflector and is returned directly to its point of origin. The data collected are used to identify leaking trace gases (including methane), as well leak locations and their emission rates.

The other is doing experiments with few-cycle pulses, like above-threshold ionization, attosecond pulses, highly efficient nonlinear optics or high-harmonics generation. These can be single pulses, so that no comb exists, and therefore it is not possible to define a carrier–envelope offset frequency, rather the carrier–envelope offset phase is important. A second photodiode can be added to the setup to gather phase and amplitude in a single shot, or difference-frequency generation can be used to even lock the offset on a single-shot basis, albeit with low power efficiency.

Without an actual comb one can look at the phase vs frequency. Without a carrier–envelope offset all frequencies are cosines. This means that all frequencies have the phase zero. The time origin is arbitrary. If a pulse comes at later times, the phase increases linearly with frequency, but still the zero-frequency phase is zero. This phase at zero frequency is the carrier–envelope offset. The second harmonic not only has twice the frequency, but also twice the phase. Thus for a pulse with zero offset the second harmonic of the low-frequency tail is in phase with the fundamental of the high-frequency tail, and otherwise it is not. Spectral phase interferometry for direct electric-field reconstruction (SPIDER) measures how the phase increases with frequency, but it cannot determine the offset, so the name “electric field reconstruction” is a bit misleading.

In recent years, the frequency comb has been garnering interest for astro-comb applications, extending the use of the technique as a spectrographic observational tool in astronomy.

There are other applications that do not need to lock the carrier–envelope offset frequency to a radio-frequency signal. [25] These include, among others, optical communications, [26] the synthesis of optical arbitrary waveforms, [27] spectroscopy (especially dual-comb spectroscopy) [28] or radio-frequency photonics. [13]

On the other hand, optical frequency combs have found new applications in remote sensing. Ranging lidars based on dual comb spectroscopy have been developed, enabling high-resolution range measurements at fast update rates [29] . Optical frequency combs can also be utilized to measure greenhouse gas emissions with great precision. For instance, in 2019, scientists at NIST employed spectroscopy to quantify methane emissions from oil and gas fields [30] . More recently, a greenhouse gas lidar based on electro-optic combs has been successfully demonstrated [31] .

History

Theodor W. Hänsch and John L. Hall shared half of the 2005 Nobel Prize in Physics for contributions to the development of laser-based precision spectroscopy, including the optical frequency-comb technique. The other half of the prize was awarded to Roy Glauber.

Also in 2005, the femtosecond comb technique was extended to the extreme ultraviolet range, enabling frequency metrology in that region of the spectrum. [32] [33] [34] [35]

See also

Related Research Articles

<span class="mw-page-title-main">Nonlinear optics</span> Branch of physics

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >108 V/m and thus comparable to the atomic electric field of ~1011 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

Mode locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or femtoseconds (10−15 s). A laser operated in this way is sometimes referred to as a femtosecond laser, for example, in modern refractive surgery. The basis of the technique is to induce a fixed phase relationship between the longitudinal modes of the laser's resonant cavity. Constructive interference between these modes can cause the laser light to be produced as a train of pulses. The laser is then said to be "phase-locked" or "mode-locked".

<span class="mw-page-title-main">Ti-sapphire laser</span> Type of laser

Ti:sapphire lasers (also known as Ti:Al2O3 lasers, titanium-sapphire lasers, or Ti:sapphs) are tunable lasers which emit red and near-infrared light in the range from 650 to 1100 nanometers. These lasers are mainly used in scientific research because of their tunability and their ability to generate ultrashort pulses thanks to its broad light emission spectrum. Lasers based on Ti:sapphire were first constructed and invented in June 1982 by Peter Moulton at the MIT Lincoln Laboratory.

In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.

<span class="mw-page-title-main">Terahertz time-domain spectroscopy</span>

In physics, terahertz time-domain spectroscopy (THz-TDS) is a spectroscopic technique in which the properties of matter are probed with short pulses of terahertz radiation. The generation and detection scheme is sensitive to the sample's effect on both the amplitude and the phase of the terahertz radiation.

<span class="mw-page-title-main">Optical parametric oscillator</span>

An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave with frequency into two output waves of lower frequency by means of second-order nonlinear optical interaction. The sum of the output waves' frequencies is equal to the input wave frequency: . For historical reasons, the two output waves are called "signal" and "idler", where the output wave with higher frequency is the "signal". A special case is the degenerate OPO, when the output frequency is one-half the pump frequency, , which can result in half-harmonic generation when signal and idler have the same polarization.

Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.

<span class="mw-page-title-main">Theodor W. Hänsch</span> German physicist and nobel laureate

Theodor Wolfgang Hänsch is a German physicist. He received one-third of the 2005 Nobel Prize in Physics for "contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique", sharing the prize with John L. Hall and Roy J. Glauber.

<span class="mw-page-title-main">Silicon photonics</span> Photonic systems which use silicon as an optical medium

Silicon photonics is the study and application of photonic systems which use silicon as an optical medium. The silicon is usually patterned with sub-micrometre precision, into microphotonic components. These operate in the infrared, most commonly at the 1.55 micrometre wavelength used by most fiber optic telecommunication systems. The silicon typically lies on top of a layer of silica in what is known as silicon on insulator (SOI).

<span class="mw-page-title-main">Supercontinuum</span>

In optics, a supercontinuum is formed when a collection of nonlinear processes act together upon a pump beam in order to cause severe spectral broadening of the original pump beam, for example using a microstructured optical fiber. The result is a smooth spectral continuum. There is no consensus on how much broadening constitutes a supercontinuum; however researchers have published work claiming as little as 60 nm of broadening as a supercontinuum. There is also no agreement on the spectral flatness required to define the bandwidth of the source, with authors using anything from 5 dB to 40 dB or more. In addition the term supercontinuum itself did not gain widespread acceptance until this century, with many authors using alternative phrases to describe their continua during the 1970s, 1980s and 1990s.

In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and dispersive effects in the medium. There are two main kinds of solitons:

The Mamyshev 2R regenerator is an all-optical regenerator used in optical communications. In 1998, Pavel V. Mamyshev of Bell Labs proposed and patented the use of the self-phase modulation (SPM) for single channel optical pulse reshaping and re-amplification. More recent applications target the field of ultrashort high peak-power pulse generation.

Time stretch dispersive Fourier transform (TS-DFT), otherwise known as time-stretch transform (TST), temporal Fourier transform or photonic time-stretch (PTS) is a spectroscopy technique that uses optical dispersion instead of a grating or prism to separate the light wavelengths and analyze the optical spectrum in real-time. It employs group-velocity dispersion (GVD) to transform the spectrum of a broadband optical pulse into a time stretched temporal waveform. It is used to perform Fourier transformation on an optical signal on a single shot basis and at high frame rates for real-time analysis of fast dynamic processes. It replaces a diffraction grating and detector array with a dispersive fiber and single-pixel detector, enabling ultrafast real-time spectroscopy and imaging. Its nonuniform variant, warped-stretch transform, realized with nonlinear group delay, offers variable-rate spectral domain sampling, as well as the ability to engineer the time-bandwidth product of the signal's envelope to match that of the data acquisition systems acting as an information gearbox.

<span class="mw-page-title-main">Optical rogue waves</span>

Optical rogue waves are rare pulses of light analogous to rogue or freak ocean waves. The term optical rogue waves was coined to describe rare pulses of broadband light arising during the process of supercontinuum generation—a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input waveform—in nonlinear optical fiber. In this context, optical rogue waves are characterized by an anomalous surplus in energy at particular wavelengths or an unexpected peak power. These anomalous events have been shown to follow heavy-tailed statistics, also known as L-shaped statistics, fat-tailed statistics, or extreme-value statistics. These probability distributions are characterized by long tails: large outliers occur rarely, yet much more frequently than expected from Gaussian statistics and intuition. Such distributions also describe the probabilities of freak ocean waves and various phenomena in both the man-made and natural worlds. Despite their infrequency, rare events wield significant influence in many systems. Aside from the statistical similarities, light waves traveling in optical fibers are known to obey the similar mathematics as water waves traveling in the open ocean, supporting the analogy between oceanic rogue waves and their optical counterparts. More generally, research has exposed a number of different analogies between extreme events in optics and hydrodynamic systems. A key practical difference is that most optical experiments can be done with a table-top apparatus, offer a high degree of experimental control, and allow data to be acquired extremely rapidly. Consequently, optical rogue waves are attractive for experimental and theoretical research and have become a highly studied phenomenon. The particulars of the analogy between extreme waves in optics and hydrodynamics may vary depending on the context, but the existence of rare events and extreme statistics in wave-related phenomena are common ground.

The carrier-envelope phase (CEP) or carrier-envelope offset (CEO) phase is an important feature of an ultrashort laser pulse and gains significance with decreasing pulse duration, in a regime where the pulse consists of a few wavelengths. Physical effects depending on the carrier-envelope phase fall into the category of highly nonlinear optics.

Vernier spectroscopy is a type of cavity enhanced laser absorption spectroscopy that is especially sensitive to trace gases. The method uses a frequency comb laser combined with a high finesse optical cavity to produce an absorption spectrum in a highly parallel manner. The method is also capable of detecting trace gases in very low concentration due to the enhancement effect of the optical resonator on the effective optical path length.

<span class="mw-page-title-main">Ursula Keller</span> Swiss physicist

Ursula Keller is a Swiss physicist. She has been a physics professor at the ETH Zurich, Switzerland since 2003 with a speciality in ultra-fast laser technology, an inventor and the winner of the 2018 European Inventor Award by the European Patent Office.

Kerr frequency combs are optical frequency combs which are generated from a continuous wave pump laser by the Kerr nonlinearity. This coherent conversion of the pump laser to a frequency comb takes place inside an optical resonator which is typically of micrometer to millimeter in size and is therefore termed a microresonator. The coherent generation of the frequency comb from a continuous wave laser with the optical nonlinearity as a gain sets Kerr frequency combs apart from today's most common optical frequency combs. These frequency combs are generated by mode-locked lasers where the dominating gain stems from a conventional laser gain medium, which is pumped incoherently. Because Kerr frequency combs only rely on the nonlinear properties of the medium inside the microresonator and do not require a broadband laser gain medium, broad Kerr frequency combs can in principle be generated around any pump frequency.

The numerical models of lasers and the most of nonlinear optical systems stem from Maxwell–Bloch equations (MBE). This full set of Partial Differential Equations includes Maxwell equations for electromagnetic field and semiclassical equations of the two-level atoms. For this reason the simplified theoretical approaches were developed for numerical simulation of laser beams formation and their propagation since the early years of laser era. The Slowly varying envelope approximation of MBE follows from the standard nonlinear wave equation with nonlinear polarization as a source:

Spectral interferometry (SI) or frequency-domain interferometry is a linear technique used to measure optical pulses, with the condition that a reference pulse that was previously characterized is available. This technique provides information about the intensity and phase of the pulses. SI was first proposed by Claude Froehly and coworkers in the 1970s.

References

  1. ↑ Hall, John L. (2006). "Nobel Lecture: Defining and measuring optical frequencies". Reviews of Modern Physics. 78 (4): 1279–1295. Bibcode:2006RvMP...78.1279H. doi: 10.1103/revmodphys.78.1279 .
  2. ↑ Hänsch, Theodor W. (2006). "Nobel Lecture: Passion for precision". Reviews of Modern Physics. 78 (4): 1297–1309. Bibcode:2006RvMP...78.1297H. CiteSeerX   10.1.1.208.7371 . doi:10.1103/revmodphys.78.1297.
  3. ↑ "The Nobel Prize in Physics 2005". www.nobelprize.org. Retrieved 2017-11-16.
  4. ↑ Adler, Florian; Moutzouris, Konstantinos; Leitenstorfer, Alfred; Schnatz, Harald; Lipphardt, Burghard; Grosche, Gesine; Tauser, Florian (2004-11-29). "Phase-locked two-branch erbium-doped fiber laser system for long-term precision measurements of optical frequencies". Optics Express. 12 (24): 5872–80. Bibcode:2004OExpr..12.5872A. doi: 10.1364/OPEX.12.005872 . ISSN   1094-4087. PMID   19488226.
  5. ↑ Ma, Long-Sheng; Bi, Zhiyi; Bartels, Albrecht; et al. (2004). "Optical Frequency Synthesis and Comparison with Uncertainty at the 10−19 Level" (PDF). Science. 303 (5665): 1843–1845. Bibcode:2004Sci...303.1843M. doi:10.1126/science.1095092. PMID   15031498. S2CID   15978159.
  6. ↑ Bartels, Albrecht (14 July 2009). "10-GHz Self-Referenced Optical Frequency Comb". Science. 326 (5953): 681. Bibcode:2009Sci...326..681B. CiteSeerX   10.1.1.668.1986 . doi:10.1126/science.1179112. PMID   19900924. S2CID   30199867.
  7. ↑ Boggio, J. C.; Moro, S.; Windmiller, J. R.; Zlatanovic, S.; Myslivets, E.; Alic, N.; Radic, S. (2009). "Optical frequency comb generated by four-wave mixing in highly nonlinear fibers". Cleo/Qels 2009: 1–2.
  8. ↑ Sefler, G.A.; Kitayama, K. (1998). "Frequency comb generation by four-wave mixing and the role of fiber dispersion". Journal of Lightwave Technology . 16 (9): 1596–1605. Bibcode:1998JLwT...16.1596S. doi:10.1109/50.712242.
  9. ↑ P. Del'Haye; A. Schliesser; O. Arcizet; T. Wilken; R. Holzwarth; T. J. Kippenberg (2007). "Optical frequency comb generation from a monolithic microresonator". Nature. 450 (7173): 1214–1217. arXiv: 0708.0611 . Bibcode:2007Natur.450.1214D. doi:10.1038/nature06401. PMID   18097405. S2CID   4426096.
  10. ↑ Jérôme Faist; et al. (2016). "Quantum Cascade Laser Frequency Combs". Nanophotonics. 5 (2): 272. arXiv: 1510.09075 . Bibcode:2016Nanop...5...15F. doi:10.1515/nanoph-2016-0015. S2CID   119189132. "In contrast to mode-locked lasers, microresonator-based frequency combs (also called Kerr combs) can exhibit complex phase relations between modes that do not correspond to the emission of single pulses while remaining highly coherent [8]."
  11. ↑ Andrew M. Weiner (2017). "Frequency combs: Cavity solitons come of age". Nature Photonics. 11 (9): 533–535. doi:10.1038/nphoton.2017.149. S2CID   126121332.
  12. ↑ Murata, H.; Morimoto, A.; Kobayashi, T.; Yamamoto, S. (2000-11-01). "Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators". IEEE Journal of Selected Topics in Quantum Electronics. 6 (6): 1325–1331. Bibcode:2000IJSTQ...6.1325M. doi:10.1109/2944.902186. ISSN   1077-260X. S2CID   41791989.
  13. 1 2 Torres-Company, Victor; Weiner, Andrew M. (May 2017). "Optical frequency comb technology for ultra-broadband radio-frequency photonics". Laser & Photonics Reviews. 8 (3): 368–393. arXiv: 1403.2776 . doi:10.1002/lpor.201300126. S2CID   33427587.
  14. ↑ Wu, Rui; Torres-Company, Victor; Leaird, Daniel E.; Weiner, Andrew M. (2013-03-11). "Supercontinuum-based 10-GHz flat-topped optical frequency comb generation". Optics Express. 21 (5): 6045–6052. Bibcode:2013OExpr..21.6045W. doi: 10.1364/OE.21.006045 . ISSN   1094-4087. PMID   23482172.
  15. ↑ Metcalf, A. J.; Torres-Company, V.; Leaird, D. E.; Weiner, A. M. (2013-11-01). "High-Power Broadly Tunable Electrooptic Frequency Comb Generator". IEEE Journal of Selected Topics in Quantum Electronics. 19 (6): 231–236. Bibcode:2013IJSTQ..19..231M. doi:10.1109/JSTQE.2013.2268384. ISSN   1077-260X. S2CID   37911312.
  16. ↑ Kobayashi, T.; Sueta, T.; Cho, Y.; Matsuo, Y. (1972-10-15). "High‐repetition‐rate optical pulse generator using a Fabry‐Perot electro‐optic modulator". Applied Physics Letters. 21 (8): 341–343. Bibcode:1972ApPhL..21..341K. doi:10.1063/1.1654403. ISSN   0003-6951.
  17. 1 2 H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, U. Keller (1999). "Carrier–envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation", Appl. Phys. B 69, 327.
  18. ↑ Hu, Yue (15 March 2017). "Computational Study of Amplitude-to-Phase Conversion in a Modified Unitraveling Carrier Photodetector". IEEE Photonics Journal. 9 (2): 2682251. arXiv: 1702.07732 . Bibcode:2017IPhoJ...982251H. doi:10.1109/JPHOT.2017.2682251. S2CID   19450831.
  19. ↑ "Archived copy" (PDF). www.attoworld.de. Archived from the original (PDF) on 7 October 2007. Retrieved 17 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)
  20. ↑ G. Krauss, D. Fehrenbacher, D. Brida, C. Riek, A. Sell, R. Huber, A. Leitenstorfer (2011). "All-passive phase locking of a compact Er:fiber laser system", Opt. Lett., 36, 540.
  21. ↑ T. Fuji, A. Apolonski, F. Krausz (2004). "Self-stabilization of carrier–envelope offset phase by use of difference-frequency generation", Opt. Lett., 29, 632.
  22. ↑ M. Zimmermann, C. Gohle, R. Holzwarth, T. Udem, T.W. Hänsch (2004). "Optical clockwork with an offset-free difference-frequency comb: accuracy of sum- and difference-frequency generation", Opt. Lett., 29, 310.
  23. ↑ "HARPS Laser Frequency Comb Commissioned". European Southern Observatory. 22 May 2015.
  24. ↑ Optical frequency comb for dimensional metrology, atomic and molecular spectroscopy, and precise time keeping Archived 2013-06-27 at the Wayback Machine
  25. ↑ Newbury, Nathan R. (2011). "Searching for applications with a fine-tooth comb". Nature Photonics. 5 (4): 186–188. Bibcode:2011NaPho...5..186N. doi:10.1038/nphoton.2011.38.
  26. ↑ Temprana, E.; Myslivets, E.; Kuo, B. P.-P.; Liu, L.; Ataie, V.; Alic, N.; Radic, S. (2015-06-26). "Overcoming Kerr-induced capacity limit in optical fiber transmission". Science. 348 (6242): 1445–1448. Bibcode:2015Sci...348.1445T. doi:10.1126/science.aab1781. ISSN   0036-8075. PMID   26113716. S2CID   41906650.
  27. ↑ Cundiff, Steven T.; Weiner, Andrew M. (2010). "Optical arbitrary waveform generation". Nature Photonics. 4 (11): 760–766. Bibcode:2010NaPho...4..760C. doi:10.1038/nphoton.2010.196.
  28. ↑ Coddington, Ian; Newbury, Nathan; Swann, William (2016-04-20). "Dual-comb spectroscopy". Optica. 3 (4): 414–426. Bibcode:2016Optic...3..414C. doi: 10.1364/OPTICA.3.000414 . ISSN   2334-2536. PMC   8201420 . PMID   34131580.
  29. ↑ Nürnberg, Jacob; Willenberg, Benjamin; Phillips, Christopher R.; Keller, Ursula (2021-08-02). "Dual-comb ranging with frequency combs from single cavity free-running laser oscillators". Optics Express. 29 (16): 24910. doi:10.1364/OE.428051. ISSN   1094-4087.
  30. ↑ robin.materese@nist.gov (2009-12-31). "Optical Frequency Combs". NIST. Retrieved 2022-02-16.
  31. ↑ "Optica Publishing Group". opg.optica.org. doi:10.1364/oe.515543 . Retrieved 2024-04-02.
  32. ↑ Jones, R. Jason; Moll, Kevin D.; Thorpe, Michael J.; Ye, Jun (20 May 2005), "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity" (PDF), Physical Review Letters , 94 (19): 193201, Bibcode:2005PhRvL..94s3201J, doi:10.1103/PhysRevLett.94.193201, PMID   16090171 , retrieved 2014-07-31
  33. ↑ Gohle, Christoph; Udem, Thomas; Herrmann, Maximilian; Rauschenberger, Jens; Holzwarth, Ronald; Schuessler, Hans A.; Krausz, Ferenc; Hänsch, Theodor W. (2005), "A frequency comb in the extreme ultraviolet", Nature , 436 (14 July 2005): 234–237, Bibcode:2005Natur.436..234G, doi:10.1038/nature03851, PMID   16015324, S2CID   1029631
  34. ↑ Kandula, Dominik Z.; Gohle, Christoph; Pinkert, Tjeerd J.; Ubachs, Wim; Eikema, Kjeld S.E. (2 August 2010). "Extreme ultraviolet frequency comb metrology". Physical Review Letters. 105 (6): 063001. arXiv: 1004.5110 . Bibcode:2010PhRvL.105f3001K. doi:10.1103/PhysRevLett.105.063001. PMID   20867977. S2CID   2499460.
  35. ↑ Cingöz, Arman; Yost, Dylan C.; Allison, Thomas K.; Ruehl, Axel; Fermann, Martin E.; Hartl, Ingmar; Ye, Jun (2 February 2012), "Direct frequency comb spectroscopy in the extreme ultraviolet", Nature, 482 (7383): 68–71, arXiv: 1109.1871 , Bibcode:2012Natur.482...68C, doi:10.1038/nature10711, PMID   22297971, S2CID   1630174

Further reading

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.