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708:
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. 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. This type of optical frequency comb is usually called electrooptic frequency comb. The first schemes used a phase modulator inside an integrated FabryâPerot cavity, but with advances in electro-optic modulators new arrangements are possible.
1413:
1404:
carrierâenvelope offset-free since the two spectral parts contributing to the DFG share the same carrierâenvelope offset frequency (CEO frequency). This was first proposed in 1999 and demonstrated in 2011 using an erbium fiber frequency comb at the telecom wavelength. 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.
352:
1490:. These can be single pulses, so that no comb exists, and therefore it is not possible to define a carrierâenvelope offset frequency, rather the carrierâenvelope 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.
383:
2467:
773:
1494:
zero-frequency phase is zero. This phase at zero frequency is the carrierâenvelope 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.
1493:
Without an actual comb one can look at the phase vs frequency. Without a carrierâenvelope 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
1465:
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
1390:
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
707:
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. In this manner, the frequency comb is centered around the optical frequency provided by the continuous-wave laser and the
1520:
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. Optical frequency combs can also be utilized to measure greenhouse gas
788:
Measurement of the carrierâenvelope 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 carrierâenvelope offset phase control were proposed in 1999. The two simplest
784:
of an optical pulse can be seen on the right. Each line is displaced from a harmonic of the repetition rate by the carrierâenvelope offset frequency. The carrierâenvelope offset frequency is the rate at which the peak of the carrier frequency slips from the peak of the pulse envelope on a
1403:
An alternative to stabilizing the carrierâenvelope 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
1359:, 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, one has to lock the phase on a fraction of the repetition rate.
1446:, 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 (carrierâenvelope offset). Applications for the frequency-comb technique include optical
2128:"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 ."
1367:
In the absence of active stabilization, the repetition rate and carrierâenvelope 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
680:. Nevertheless, the four-wave mixing effect above can create and stabilize a perfect frequency comb in such a structure. Basically, the system generates a perfect comb that overlaps the resonant modes as much as possible. In fact,
1436:
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.
1512:
There are other applications that do not need to lock the carrierâenvelope offset frequency to a radio-frequency signal. These include, among others, optical communications, the synthesis of optical arbitrary waveforms,
555:
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
421:
The most common lasers used for frequency-comb generation are Ti:sapphire solid-state lasers or Er:fiber lasers with repetition rates typically between 100 MHz and 1 GHz or even going as high as 10 GHz.
1532:
The frequency comb was proposed in 2000. Before its introduction, the EM spectrum was divided between the electronic/radio frequency range and the optical/laser frequency range. The radio frequency range had accurate
2431:
752:
the second harmonic can be generated in a long crystal so that by consecutive sum frequency generation and difference frequency generation the spectrum of first and second harmonic widens until they overlap.
3103:
Gohle, Christoph; Udem, Thomas; Herrmann, Maximilian; Rauschenberger, Jens; Holzwarth, Ronald; Schuessler, Hans A.; Krausz, Ferenc; HĂ€nsch, Theodor W. (2005), "A frequency comb in the extreme ultraviolet",
691:
In the time domain, while mode-locked lasers almost always emit a series of short pulses, Kerr frequency combs generally do not. However, a special sub-type of Kerr frequency comb, in which a "cavity
1383:
for dispersion control, the carrierâenvelope 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.
1466:
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.
2173:
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".
3217:
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",
1495:
1116:
Alternatively, difference-frequency generation (DFG) can be used. From light at opposite ends of the broadened spectrum the difference frequency is generated in a nonlinear crystal, and a
720:, 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.
939:
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873:
1624:
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
1570:
Before the frequency comb, the only way to bridge the gap were the harmonic frequency chains, which doubles radio frequency in 15 stages, reaching a frequency multiplication of
684:
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
642:
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
484:
3039:
1608:
1565:
410:. 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
637:
2685:
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".
594:
1213:
1182:
1152:
1091:
552:. 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.
346:
319:
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192:
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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
1350:
1111:
165:
2527:
1498:(SPIDER) measures how the phase increases with frequency, but it cannot determine the offset, so the name âelectric field reconstructionâ is a bit misleading.
3156:
Kandula, Dominik Z.; Gohle, Christoph; Pinkert, Tjeerd J.; Ubachs, Wim; Eikema, Kjeld S.E. (2 August 2010). "Extreme ultraviolet frequency comb metrology".
348:) are stabilized generates a comb that is useful for mapping optical frequencies into the radio frequency for the direct measurement of optical frequency.
1631:
Also in 2005, the femtosecond comb technique was extended to the extreme ultraviolet range, enabling frequency metrology in that region of the spectrum.
2384:
Kobayashi, T.; Sueta, T.; Cho, Y.; Matsuo, Y. (1972-10-15). "High-repetition-rate optical pulse generator using a Fabry-Perot electro-optic modulator".
1537:, allowing highly accurate measurements of absolute frequency. The optical range has no such device. The two ranges are separated by a frequency gap of
1784:
Adler, Florian; Moutzouris, Konstantinos; Leitenstorfer, Alfred; Schnatz, Harald; Lipphardt, Burghard; Grosche, Gesine; Tauser, Florian (2004-11-29).
1521:
emissions with great precision. For instance, in 2019, scientists at NIST employed spectroscopy to quantify methane emissions from oil and gas fields
2023:; A. Schliesser; O. Arcizet; T. Wilken; R. Holzwarth; T. J. Kippenberg (2007). "Optical frequency comb generation from a monolithic microresonator".
1386:
In high repetition rate Ti:sapphire ring lasers, which often use double-chirped mirrors to control dispersion, modulation of the pump power using an
54:. 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
1400:
The breakthrough which led to a practical frequency comb was the development of technology for stabilizing the carrierâenvelope offset frequency.
1120:
beat between this mixing product and light at the same wavelength of the original spectrum is measured. This beat frequency, detectable with a
3477:
2634:
811:
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
1417:
639:. This new frequency would get gradually more intense, and light can subsequently cascade to more and more new frequencies on the same comb.
3456:
716:
A purely electronic device which generates a series of pulses, also generates a frequency comb. These are produced for electronic sampling
359:
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
1332:
This avoids the need for frequency doubling at the cost of a second optical mixing step. Again, practical implementation uses a range of
2227:
Torres-Company, Victor; Weiner, Andrew M. (May 2017). "Optical frequency comb technology for ultra-broadband radio-frequency photonics".
3395:
291:
46:
A number of mechanisms exist for obtaining an optical frequency comb, including periodic modulation (in amplitude and/or phase) of a
3080:
2432:"Carrierâenvelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation"
732:: that is, the highest frequency in the spectrum must be at least twice the lowest frequency. One of three techniques may be used:
3040:"Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity"
2536:
1842:
672:). This kind of structure naturally has a series of resonant modes with approximately equally spaced frequencies (similar to a
194:
is the comb tooth spacing (equal to the mode-locked laser's repetition rate or, alternatively, the modulation frequency), and
3507:
3424:
3389:
3467:
1391:
adjustment is needed. A pair of wedges, one moving in or out of the intra-cavity laser beam can be used for this purpose.
3461:
2602:"Optical clockwork with an offset-free difference-frequency comb: accuracy of sum- and difference-frequency generation"
1983:
Sefler, G.A.; Kitayama, K. (1998). "Frequency comb generation by four-wave mixing and the role of fiber dispersion".
1786:"Phase-locked two-branch erbium-doped fiber laser system for long-term precision measurements of optical frequencies"
673:
1610:. However, those were large and expensive to operate. The frequency comb managed to bridge that gap in one stage.
1461:
3457:
Optical frequency comb for dimensional metrology, atomic and molecular spectroscopy, and precise time keeping
2632:
Optical frequency comb for dimensional metrology, atomic and molecular spectroscopy, and precise time keeping
878:
489:
3434:
944:
1275:
87:
28:
2929:
1218:
1007:
821:
1524:. More recently, a greenhouse gas lidar based on electro-optic combs has been successfully demonstrated.
2468:"Computational Study of Amplitude-to-Phase Conversion in a Modified Unitraveling Carrier Photodetector"
1964:
Boggio, J. C.; Moro, S.; Windmiller, J. R.; Zlatanovic, S.; Myslivets, E.; Alic, N.; Radic, S. (2009).
1475:
804:
436:
3492:
3414:
1573:
3356:
2631:
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Telle, H. R.; Steinmeyer, G.; Dunlop, A. E.; Stenger, J.; Sutter, D. H.; Keller, U. (October 1999).
1925:
1737:
1540:
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1760:
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1650:
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in frequency (i.e., a factor of two) can be used to directly measure (and correct for drifts in)
2601:
2584:
2567:
3497:
3351:
2585:"Self-stabilization of carrierâenvelope offset phase by use of difference-frequency generation"
1920:
1732:
740:
643:
55:
3473:
2904:
2846:
NĂŒrnberg, Jacob; Willenberg, Benjamin; Phillips, Christopher R.; Keller, Ursula (2021-08-02).
1187:
789:
approaches, which require only one nonlinear optical process, are described in the following.
3502:
1985:
815:, includes a difference-frequency component, which is the carrierâenvelope offset frequency.
747:
47:
1157:
3343:
3302:
3236:
3175:
3115:
3056:
2996:
2941:
2859:
2792:
2753:
2694:
2659:
2566:
G. Krauss, D. Fehrenbacher, D. Brida, C. Riek, A. Sell, R. Huber, A. Leitenstorfer (2011).
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2182:
2103:
2042:
1994:
1912:
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1724:
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applications, extending the use of the technique as a spectrographic observational tool in
1130:
1069:
729:
411:
387:
324:
297:
258:
252:
224:
197:
170:
2848:"Dual-comb ranging with frequency combs from single cavity free-running laser oscillators"
1613:
1412:
63:
50:, four-wave mixing in nonlinear media, or stabilization of the pulse train generated by a
8:
2847:
1487:
685:
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665:
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3240:
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1998:
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1801:
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3318:
3292:
3260:
3226:
3199:
3165:
3139:
2823:
2744:
Cundiff, Steven T.; Weiner, Andrew M. (2010). "Optical arbitrary waveform generation".
2726:
2508:
2482:
2366:
2254:
2236:
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2155:
2119:
2093:
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2032:
2020:
1946:
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2159:
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1938:
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1815:
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technique, light at the lower-energy side of the broadened spectrum is doubled using
772:
395:
2730:
2618:
2512:
2370:
2272:
Wu, Rui; Torres-Company, Victor; Leaird, Daniel E.; Weiner, Andrew M. (2013-03-11).
2258:
2210:
1963:
1950:
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1885:
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2002:
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681:
430:
399:
356:
70:
3068:
2930:"Greenhouse gas monitoring using an IPDA lidar based on a dual-comb spectrometer"
2638:
1966:"Optical frequency comb generated by four-wave mixing in highly nonlinear fibers"
1425:
2650:
Newbury, Nathan R. (2011). "Searching for applications with a fine-tooth comb".
2504:
2354:
2329:
Metcalf, A. J.; Torres-Company, V.; Leaird, D. E.; Weiner, A. M. (2013-11-01).
2019:
1965:
1746:
1700:
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1380:
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781:
736:
660:
An alternative variation of four-wave-mixing-based frequency combs is known as
78:
32:
3365:
3314:
776:
Difference between group and phase velocity leading to carrierâenvelope offset
3486:
3016:
2881:
2814:
2765:
2714:
2439:
2413:
2362:
2307:
2202:
2151:
2115:
1843:"Optical Frequency Synthesis and Comparison with Uncertainty at the 10 Level"
1819:
1451:
1442:
1429:
1373:
717:
287:
3038:
Jones, R. Jason; Moll, Kevin D.; Thorpe, Michael J.; Ye, Jun (20 May 2005),
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1934:
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3135:
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3024:
2963:
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2832:
2722:
2315:
2274:"Supercontinuum-based 10-GHz flat-topped optical frequency comb generation"
2250:
2084:
JĂ©rĂŽme Faist; et al. (2016). "Quantum
Cascade Laser Frequency Combs".
2062:
1942:
1877:
1827:
1810:
1785:
1645:
1617:
1514:
407:
377:
283:
59:
3451:
2985:"Optical frequency combs: Coherently uniting the electromagnetic spectrum"
2452:
1416:
Spectrum of the light from the two-laser frequency combs installed on the
676:). Unfortunately the resonant modes are not exactly equally spaced due to
2298:
2273:
2138:
Andrew M. Weiner (2017). "Frequency combs: Cavity solitons come of age".
1625:
1621:
1479:
1369:
360:
51:
3248:
3127:
2194:
2054:
646:. This creates a frequency comb by four-wave mixing as described above.
2605:
2588:
2571:
1640:
1502:
1428:
standards to optical frequencies. Current frequency standards such as
1121:
1117:
812:
808:
415:
351:
74:
2954:
2872:
2545:
2405:
2006:
1496:
Spectral phase interferometry for direct electric-field reconstruction
780:
An increasing offset between the optical phase and the maximum of the
1506:
1447:
1433:
2780:
2600:
M. Zimmermann, C. Gohle, R. Holzwarth, T. Udem, T.W. HĂ€nsch (2004).
1501:
In recent years, the frequency comb has been garnering interest for
3297:
2487:
2098:
1783:
1715:
HĂ€nsch, Theodor W. (2006). "Nobel
Lecture: Passion for precision".
3372:
3231:
3170:
2331:"High-Power Broadly Tunable Electrooptic Frequency Comb Generator"
2241:
2037:
1841:
Ma, Long-Sheng; Bi, Zhiyi; Bartels, Albrecht; et al. (2004).
1517:(especially dual-comb spectroscopy) or radio-frequency photonics.
1440:
There are two distinct applications of this technique. One is the
2328:
2845:
3102:
2779:
Coddington, Ian; Newbury, Nathan; Swann, William (2016-04-20).
2172:
728:
For many applications, the comb must be widened to at least an
36:
2568:"All-passive phase locking of a compact Er:fiber laser system"
1376:
transducer, which moves a mirror to change the cavity length.
406:
The most popular way of generating a frequency comb is with a
3279:
2983:
Diddams, Scott A.; Vahala, Kerry; Udem, Thomas (2020-07-17).
596:, four-wave mixing could generate light at the new frequency
382:
40:
2429:
2271:
1676:"Nobel Lecture: Defining and measuring optical frequencies"
3462:
Rulers of Light: Using Lasers to
Measure Distance and Time
2529:
Phase-stabilized
Ultrashort Laser Systems for Spectroscopy
3155:
1455:
703:
Using electro-optic modulation of a continuous-wave laser
3283:; Theodor HĂ€nsch (2019). "Frequency comb spectroscopy".
1982:
739:
generation by strong self-phase modulation in nonlinear
711:
3329:
2902:
2684:
2383:
3216:
2778:
2335:
IEEE Journal of
Selected Topics in Quantum Electronics
2175:
IEEE Journal of
Selected Topics in Quantum Electronics
1066:
In practice, this is not done with a single frequency
767:
433:
is a process where intense light at three frequencies
16:
Laser source with equal intervals of spectral energies
1840:
1576:
1543:
1338:
1278:
1221:
1190:
1160:
1133:
1099:
1072:
1010:
947:
941:, and mixed with light at the very similar frequency
881:
824:
602:
562:
492:
439:
327:
300:
282:. Thus, octave-spanning combs can be used to steer a
261:
227:
200:
173:
153:
90:
73:
representation of a perfect frequency comb is like a
2226:
221:
is the carrier offset frequency, which is less than
398:of a time-domain Dirac comb is a Dirac comb in the
2927:
1602:
1559:
1362:
1344:
1324:
1264:
1207:
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1146:
1105:
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136:
3037:
2982:
1372:. The repetition rate can be stabilized using a
3484:
3412:
2137:
1454:, high-precision spectroscopy, and more precise
486:interact to produce light at a fourth frequency
2928:Patiño Rosas, William; Cézard, Nicolas (2024).
1901:"10-GHz Self-Referenced Optical Frequency Comb"
39:, a frequency comb can be generated by certain
3468:On-chip, electronically tunable frequency comb
2525:
1215:. This is then mixed with light at frequency
2743:
2621:. European Southern Observatory. 22 May 2015.
1418:High Accuracy Radial Velocity Planet Searcher
2083:
1124:, is the carrierâenvelope offset frequency.
371:
3373:John L Hall & Theodor W HĂ€nsch (2004).
1424:A frequency comb allows a direct link from
761:
668:(such as a microscopic glass disk that has
286:within a carrierâenvelope phase-correcting
3470:, article by Leah Burrows | March 18, 2019
3355:
3296:
3230:
3169:
2953:
2871:
2822:
2804:
2619:"HARPS Laser Frequency Comb Commissioned"
2583:T. Fuji, A. Apolonski, F. Krausz (2004).
2544:
2486:
2297:
2240:
2097:
2036:
1924:
1809:
1736:
1699:
1295:
1251:
1194:
983:
920:
854:
756:These processes generate new frequencies
723:
664:. Here, a single laser is coupled into a
425:
120:
3464:by Steven Cundiff in Scientific American
3334:: Femtosecond optical frequency combs".
2425:
2423:
1460:
1411:
771:
381:
350:
290:. Any mechanism by which the combs' two
3447:Attosecond control of optical waveforms
2649:
1898:
1184:is mixed to produce light at frequency
1004:to produces a beat signal at frequency
934:{\displaystyle 2f_{n}=2f_{0}+2n\,f_{r}}
545:{\displaystyle f_{4}=f_{1}+f_{2}-f_{3}}
3485:
2903:robin.materese@nist.gov (2009-12-31).
2537:Ludwig Maximilian University of Munich
2526:Rauschenberger, Jens (24 April 2007).
2466:Hu, Yue; et al. (15 March 2017).
1714:
1450:, frequency-chain generation, optical
997:{\displaystyle f_{2n}=f_{0}+2n\,f_{r}}
746:a Ti:sapphire laser using intracavity
386:A Dirac comb is an infinite series of
31:made of discrete and regularly spaced
3380:. In Jun Ye, Steven T Cundiff (ed.).
3375:"History of optical comb development"
2420:
2222:
2220:
1325:{\displaystyle f_{n}-n\,f_{r}=f_{0}.}
712:Low-frequency combs using electronics
649:
137:{\displaystyle f_{n}=f_{0}+n\,f_{r},}
1673:
1470:The other is doing experiments with
1265:{\displaystyle f_{n}=f_{0}+n\,f_{r}}
1059:{\displaystyle 2f_{n}-f_{2n}=f_{0}.}
868:{\displaystyle f_{n}=f_{0}+n\,f_{r}}
807:(SHG) in a nonlinear crystal, and a
3330:Steven T. Cundiff; Jun Ye (2003). "
1113:values, but the effect is the same
768:Carrierâenvelope offset measurement
13:
3382:Femtosecond optical frequency comb
3273:
2465:
2217:
1899:Bartels, Albrecht (14 July 2009).
14:
3519:
3440:
1761:"The Nobel Prize in Physics 2005"
818:Conceptually, light at frequency
760:for similar reasons as discussed
479:{\displaystyle f_{1},f_{2},f_{3}}
3433:Nobel prize for Physics (2005)
3210:
3149:
3096:
3031:
2976:
2921:
2896:
2839:
2772:
2737:
2678:
2643:
2625:
2611:
2594:
2577:
2560:
2519:
2459:
2377:
2322:
2265:
2166:
2131:
2077:
1986:Journal of Lightwave Technology
1603:{\displaystyle 2^{15}=10^{4.5}}
1407:
1363:Carrierâenvelope offset control
1272:to produce a beat frequency of
695:" forms in the microresonator,
3188:10.1103/PhysRevLett.105.063001
2013:
1976:
1957:
1892:
1834:
1777:
1753:
1708:
1667:
1:
3069:10.1103/PhysRevLett.94.193201
2229:Laser & Photonics Reviews
1661:
1560:{\displaystyle 10^{5}\times }
366:
3508:Spectrum (physical sciences)
1379:In Ti:sapphire lasers using
632:{\displaystyle 2f_{1}-f_{2}}
7:
1634:
1127:Here, light at frequencies
589:{\displaystyle f_{1},f_{2}}
10:
3524:
2505:10.1109/JPHOT.2017.2682251
2355:10.1109/JSTQE.2013.2268384
1747:10.1103/revmodphys.78.1297
1701:10.1103/revmodphys.78.1279
1527:
1476:above-threshold ionization
1393:
1357:phase is measured directly
1352:values, not a single one.
805:second-harmonic generation
674:FabryâPĂ©rot interferometer
653:
375:
3413:Andrew M. Weiner (2009).
3366:10.1103/RevModPhys.75.325
3336:Reviews of Modern Physics
3315:10.1038/s41566-018-0347-5
3114:(14 July 2005): 234â237,
2905:"Optical Frequency Combs"
1717:Reviews of Modern Physics
1680:Reviews of Modern Physics
1488:high-harmonics generation
699:emit a series of pulses.
372:Using a mode-locked laser
2781:"Dual-comb spectroscopy"
2766:10.1038/nphoton.2010.196
2152:10.1038/nphoton.2017.149
2116:10.1515/nanoph-2016-0015
1620:shared half of the 2005
1208:{\displaystyle n\,f_{r}}
670:whispering-gallery modes
3474:Optical Frequency Combs
3158:Physical Review Letters
3048:Physical Review Letters
3009:10.1126/science.aay3676
2806:10.1364/OPTICA.3.000414
2707:10.1126/science.aab1781
2672:10.1038/nphoton.2011.38
2386:Applied Physics Letters
1935:10.1126/science.1179112
1870:10.1126/science.1095092
1651:Bandwidth-limited pulse
1388:acousto-optic modulator
743:or integrated waveguide
390:spaced at intervals of
3452:Femtosecond laser comb
2475:IEEE Photonics Journal
2251:10.1002/lpor.201300126
1811:10.1364/OPEX.12.005872
1674:Hall, John L. (2006).
1604:
1561:
1467:
1421:
1346:
1326:
1266:
1209:
1178:
1177:{\displaystyle f_{2n}}
1148:
1107:
1087:
1060:
998:
935:
869:
785:pulse-to-pulse basis.
777:
741:photonic crystal fiber
724:Widening to one octave
644:photonic-crystal fiber
633:
590:
546:
480:
426:Using four-wave mixing
403:
363:
342:
315:
276:
242:
215:
188:
161:
138:
56:Nobel Prize in Physics
2453:10.1007/s003400050813
1605:
1562:
1464:
1415:
1347:
1327:
1267:
1210:
1179:
1149:
1147:{\displaystyle f_{n}}
1108:
1088:
1086:{\displaystyle f_{n}}
1061:
999:
936:
870:
775:
748:self-phase modulation
634:
591:
547:
481:
412:Dirac delta functions
388:Dirac delta functions
385:
354:
343:
341:{\displaystyle f_{0}}
316:
314:{\displaystyle f_{r}}
277:
275:{\displaystyle f_{0}}
243:
241:{\displaystyle f_{r}}
216:
214:{\displaystyle f_{0}}
189:
187:{\displaystyle f_{r}}
162:
139:
48:continuous-wave laser
2299:10.1364/OE.21.006045
1656:Magneto-optical trap
1574:
1541:
1336:
1276:
1219:
1188:
1158:
1131:
1097:
1093:but with a range of
1070:
1008:
945:
879:
822:
600:
560:
490:
437:
325:
298:
284:piezoelectric mirror
259:
225:
198:
171:
151:
88:
81:spaced according to
3348:2003RvMP...75..325C
3307:2019NaPho..13..146P
3249:10.1038/nature10711
3241:2012Natur.482...68C
3180:2010PhRvL.105f3001K
3128:10.1038/nature03851
3120:2005Natur.436..234G
3061:2005PhRvL..94s3201J
3001:2020Sci...369..367D
2946:2024OExpr..3213614P
2864:2021OExpr..2924910N
2797:2016Optic...3..414C
2758:2010NaPho...4..760C
2699:2015Sci...348.1445T
2693:(6242): 1445â1448.
2664:2011NaPho...5..186N
2497:2017IPhoJ...982251H
2398:1972ApPhL..21..341K
2347:2013IJSTQ..19..231M
2290:2013OExpr..21.6045W
2195:10.1109/2944.902186
2187:2000IJSTQ...6.1325M
2108:2016Nanop...5...15F
2055:10.1038/nature06401
2047:2007Natur.450.1214D
2031:(7173): 1214â1217.
1999:1998JLwT...16.1596S
1917:2009Sci...326..681B
1862:2004Sci...303.1843M
1856:(5665): 1843â1845.
1802:2004OExpr..12.5872A
1729:2006RvMP...78.1297H
1692:2006RvMP...78.1279H
1482:, highly efficient
686:optical Kerr effect
662:Kerr frequency comb
656:Kerr frequency comb
2637:2013-06-27 at the
1765:www.nobelprize.org
1600:
1557:
1535:frequency counters
1468:
1422:
1342:
1322:
1262:
1205:
1174:
1144:
1103:
1083:
1056:
994:
931:
865:
778:
650:In microresonators
629:
586:
542:
476:
404:
364:
338:
311:
292:degrees of freedom
272:
251:Combs spanning an
238:
211:
184:
157:
134:
3426:978-0-471-41539-8
3391:978-0-387-23790-9
2955:10.1364/oe.515543
2873:10.1364/OE.428051
2546:10.5282/edoc.7110
2406:10.1063/1.1654403
2007:10.1109/50.712242
1614:Theodor W. HĂ€nsch
1480:attosecond pulses
1396:phase-locked loop
1345:{\displaystyle n}
1106:{\displaystyle n}
408:mode-locked laser
396:Fourier transform
160:{\displaystyle n}
64:Theodor W. HĂ€nsch
52:mode-locked laser
3515:
3493:Nonlinear optics
3430:
3416:Ultrafast Optics
3409:
3407:
3406:
3400:
3394:. Archived from
3379:
3369:
3359:
3326:
3300:
3285:Nature Photonics
3268:
3267:
3234:
3214:
3208:
3207:
3173:
3153:
3147:
3146:
3100:
3094:
3093:
3092:
3091:
3085:
3079:, archived from
3044:
3035:
3029:
3028:
2980:
2974:
2973:
2971:
2970:
2957:
2925:
2919:
2918:
2916:
2915:
2900:
2894:
2893:
2875:
2843:
2837:
2836:
2826:
2808:
2776:
2770:
2769:
2746:Nature Photonics
2741:
2735:
2734:
2682:
2676:
2675:
2652:Nature Photonics
2647:
2641:
2629:
2623:
2622:
2615:
2609:
2598:
2592:
2581:
2575:
2564:
2558:
2557:
2555:
2553:
2548:
2534:
2523:
2517:
2516:
2490:
2472:
2463:
2457:
2456:
2436:
2427:
2418:
2417:
2381:
2375:
2374:
2326:
2320:
2319:
2301:
2284:(5): 6045â6052.
2269:
2263:
2262:
2244:
2224:
2215:
2214:
2181:(6): 1325â1331.
2170:
2164:
2163:
2140:Nature Photonics
2135:
2129:
2127:
2101:
2081:
2075:
2074:
2040:
2017:
2011:
2010:
1993:(9): 1596â1605.
1980:
1974:
1973:
1961:
1955:
1954:
1928:
1896:
1890:
1889:
1847:
1838:
1832:
1831:
1813:
1781:
1775:
1774:
1772:
1771:
1757:
1751:
1750:
1740:
1723:(4): 1297â1309.
1712:
1706:
1705:
1703:
1686:(4): 1279â1295.
1671:
1609:
1607:
1606:
1601:
1599:
1598:
1586:
1585:
1566:
1564:
1563:
1558:
1553:
1552:
1484:nonlinear optics
1472:few-cycle pulses
1351:
1349:
1348:
1343:
1331:
1329:
1328:
1323:
1318:
1317:
1305:
1304:
1288:
1287:
1271:
1269:
1268:
1263:
1261:
1260:
1244:
1243:
1231:
1230:
1214:
1212:
1211:
1206:
1204:
1203:
1183:
1181:
1180:
1175:
1173:
1172:
1153:
1151:
1150:
1145:
1143:
1142:
1112:
1110:
1109:
1104:
1092:
1090:
1089:
1084:
1082:
1081:
1065:
1063:
1062:
1057:
1052:
1051:
1039:
1038:
1023:
1022:
1003:
1001:
1000:
995:
993:
992:
973:
972:
960:
959:
940:
938:
937:
932:
930:
929:
910:
909:
894:
893:
874:
872:
871:
866:
864:
863:
847:
846:
834:
833:
802:
758:on the same comb
638:
636:
635:
630:
628:
627:
615:
614:
595:
593:
592:
587:
585:
584:
572:
571:
551:
549:
548:
543:
541:
540:
528:
527:
515:
514:
502:
501:
485:
483:
482:
477:
475:
474:
462:
461:
449:
448:
431:Four-wave mixing
400:frequency domain
357:ultrashort pulse
347:
345:
344:
339:
337:
336:
320:
318:
317:
312:
310:
309:
281:
279:
278:
273:
271:
270:
247:
245:
244:
239:
237:
236:
220:
218:
217:
212:
210:
209:
193:
191:
190:
185:
183:
182:
166:
164:
163:
158:
143:
141:
140:
135:
130:
129:
113:
112:
100:
99:
71:frequency domain
58:being shared by
3523:
3522:
3518:
3517:
3516:
3514:
3513:
3512:
3483:
3482:
3476:explanation by
3443:
3427:
3404:
3402:
3398:
3392:
3377:
3357:10.1.1.152.1154
3281:Nathalie Picqué
3276:
3274:Further reading
3271:
3225:(7383): 68â71,
3215:
3211:
3154:
3150:
3101:
3097:
3089:
3087:
3083:
3042:
3036:
3032:
2981:
2977:
2968:
2966:
2926:
2922:
2913:
2911:
2901:
2897:
2844:
2840:
2777:
2773:
2752:(11): 760â766.
2742:
2738:
2683:
2679:
2648:
2644:
2639:Wayback Machine
2630:
2626:
2617:
2616:
2612:
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2132:
2082:
2078:
2018:
2014:
1981:
1977:
1962:
1958:
1926:10.1.1.668.1986
1897:
1893:
1845:
1839:
1835:
1796:(24): 5872â80.
1782:
1778:
1769:
1767:
1759:
1758:
1754:
1738:10.1.1.208.7371
1713:
1709:
1672:
1668:
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1637:
1594:
1590:
1581:
1577:
1575:
1572:
1571:
1548:
1544:
1542:
1539:
1538:
1530:
1432:operate in the
1426:radio frequency
1410:
1398:
1365:
1337:
1334:
1333:
1313:
1309:
1300:
1296:
1283:
1279:
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178:
174:
172:
169:
168:
167:is an integer,
152:
149:
148:
125:
121:
108:
104:
95:
91:
89:
86:
85:
79:delta functions
17:
12:
11:
5:
3521:
3511:
3510:
3505:
3500:
3495:
3481:
3480:
3471:
3465:
3459:
3454:
3449:
3442:
3441:External links
3439:
3438:
3437:
3431:
3425:
3410:
3390:
3370:
3327:
3291:(3): 146â157.
3275:
3272:
3270:
3269:
3209:
3148:
3095:
3055:(19): 193201,
3030:
2975:
2934:Optics Express
2920:
2895:
2852:Optics Express
2838:
2791:(4): 414â426.
2771:
2736:
2677:
2658:(4): 186â188.
2642:
2624:
2610:
2593:
2576:
2559:
2535:(PhD thesis).
2518:
2458:
2447:(4): 327â332.
2419:
2392:(8): 341â343.
2376:
2341:(6): 231â236.
2321:
2278:Optics Express
2264:
2235:(3): 368â393.
2216:
2165:
2146:(9): 533â535.
2130:
2076:
2012:
1975:
1970:Cleo/Qels 2009
1956:
1891:
1833:
1790:Optics Express
1776:
1752:
1707:
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987:
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955:
951:
928:
924:
919:
916:
913:
908:
904:
900:
897:
892:
888:
884:
875:is doubled to
862:
858:
853:
850:
845:
841:
837:
832:
828:
769:
766:
754:
753:
750:
744:
737:supercontinuum
725:
722:
713:
710:
704:
701:
666:microresonator
654:Main article:
651:
648:
626:
622:
618:
613:
609:
605:
583:
579:
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570:
566:
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376:Main article:
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181:
177:
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145:
144:
133:
128:
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119:
116:
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107:
103:
98:
94:
77:, a series of
33:spectral lines
21:frequency comb
15:
9:
6:
4:
3:
2:
3520:
3509:
3506:
3504:
3501:
3499:
3498:Laser science
3496:
3494:
3491:
3490:
3488:
3479:
3475:
3472:
3469:
3466:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3444:
3436:
3435:Press Release
3432:
3428:
3422:
3418:
3417:
3411:
3401:on 2014-12-27
3397:
3393:
3387:
3383:
3376:
3371:
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3345:
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3333:
3328:
3324:
3320:
3316:
3312:
3308:
3304:
3299:
3294:
3290:
3286:
3282:
3278:
3277:
3266:
3262:
3258:
3254:
3250:
3246:
3242:
3238:
3233:
3228:
3224:
3220:
3213:
3205:
3201:
3197:
3193:
3189:
3185:
3181:
3177:
3172:
3167:
3164:(6): 063001.
3163:
3159:
3152:
3145:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3108:
3099:
3086:on 2014-08-12
3082:
3078:
3074:
3070:
3066:
3062:
3058:
3054:
3050:
3049:
3041:
3034:
3026:
3022:
3018:
3014:
3010:
3006:
3002:
2998:
2995:(6501): 367.
2994:
2990:
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2979:
2965:
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2899:
2891:
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2879:
2874:
2869:
2865:
2861:
2858:(16): 24910.
2857:
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2807:
2802:
2798:
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367:Generation
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