Summary: This in-depth article on mode locking of lasers explains
- the term mode locking and its origin,
- how mode locking with different techniques works, including active, passive and hybrid mode locking,
- continuous vs. synchronous laser pumping,
- technical issues in operation regimes with low and high pulse repetition rates,
- the formation of optical frequency combs, and
- typical instabilities of mode-locked lasers.
Mode locking  (sometimes written as modelocking) is a method (or actually a group of methods) to obtain ultrashort pulses from lasers, which are then called mode-locked lasers. In the mode-locked operation regime, typically a single picosecond or femtosecond pulse circulates in the laser resonator, or sometimes multiple equidistant pulses (→ harmonic mode locking). Each time a pulse hits the output coupler, one obtain a pulse in the laser output. A mode-locked laser thus emits a regular pulse train (see Figure 1). The pulse repetition rate is the inverse of the round-trip time in the laser resonator, or an integer multiple of it in the case of harmonic mode locking.
The gain medium compensates for losses, and the saturable absorber mirror (SA) enforces pulse generation. Each time the circulating pulse hits the output coupler mirror (OC), a pulse is emitted in the output.
Typically, the pulse duration is between 30 fs and 30 ps, in extreme cases down to ≈ 5 fs; in most cases it is orders of magnitude shorter than the pulse spacing. Therefore, the peak power of a mode-locked laser can be orders of magnitude higher than the average power.
Mode locking is achieved by using some kind of mode locking device (mode locker) within the laser resonator – either an active element (an optical modulator) or a nonlinear passive element (a saturable absorber). Without such a mode locking device, the laser would usually emit light in continuous-wave operation, although there are some cases where intracavity nonlinearities lead to some kind of pulse generation.
The steady state of a mode-locked laser means that the pulse parameters (pulse energy, pulse duration, chirp, spectral bandwidth etc.) are all unchanged after each completed round trip, although they may vary substantially within each roundtrip. This implies that the various effects influencing the circulating pulse (e.g. laser gain and propagation losses, nonlinearities and chromatic dispersion) must be in a balance such that they cancel each other after each complete round trip. In simple cases such as soliton mode locking, that balance can be well understood, while in more complicated cases (frequently encountered for mode-locked fiber lasers) there is no simple description, and that balance can only be analyzed with numerical simulations. It can even be quite difficult to find a suitable mode-locking regime for such lasers.
This article focuses more on the methods of mode locking and on some fundamental aspects, whereas the article on mode-locked lasers contains more details on different kinds of mode-locked lasers and their performance characteristics.
Active and Passive Mode Locking
Active Mode Locking
Active mode locking involves the periodic modulation of the resonator losses or alternatively of the round-trip phase change. That can be achieved in different ways, for example
- with an acousto-optic,
- an electro-optic modulator,
- a Mach–Zehnder integrated-optic modulator, or
- a semiconductor electroabsorption modulator.
If the modulation is synchronized with the resonator round trips, this can lead to the generation of ultrashort pulses, usually with picosecond pulse durations. In most but not all cases, the pulse duration achieved is governed by a balance of pulse shortening through the modulator and pulse broadening via other effects, such as the limited gain bandwidth.
More details can be found in the articles on active mode locking.
Passive Mode Locking
Passive mode locking (with a saturable absorber) allows the generation of much shorter (femtosecond) pulses, basically because a saturable absorber, driven by already short pulses, can modulate the resonator losses much faster than an electronic modulator: the shorter the pulse becomes, the faster the loss modulation, provided that the absorber has a sufficiently short recovery time. The pulse duration can be even well below the recovery time of the absorber. In some cases, reliable self-starting mode locking is not achieved.
More details can be found in the articles on passive mode locking.
Hybrid Mode Locking
In some lasers (particularly in mode-locked diode lasers), active and passive mode locking are simultaneously applied. Such hybrid mode-locked lasers combine some key advantages, such as an externally controlled pulse repetition rate, fairly short pulses and reliable starting of mode-locked operation.
There are also situations where an actively mode-locked laser exhibits a slight contribution from a passive mode locking mechanism, even if the laser was not designed for that. For example, in a bulk laser there may be a Kerr lens effect leading to Kerr lens mode locking.
Continuous Versus Synchronous Pumping
Most mode-locked lasers are continuously pumped, often with a laser diode (→ diode-pumped lasers). The pump source then continuously supplies energy to the gain medium, while the circulating pulse extracts energy in regular time intervals. In most cases, the pulse spacing is very short compared with the upper-state lifetime, and the intracavity pulse energy is far below the saturation energy, so that there is negligible gain saturation during each single round-trip; only the combined effect of many round trips is substantial. During one round trip (and during each pulse), the laser gain then remains nearly constant.
Even for some gain media with a fairly short upper-state lifetime, e.g. in dye lasers or some semiconductor lasers (e.g. VECSELs), continuous pumping is possible, provided that the pulse repetition rate is not too low. If that condition can not be fulfilled, so that gain saturation by a single pulse becomes strong (which can make the pulse formation unstable), synchronous pumping can be a solution. This, however, requires another mode-locked laser as the pump source.
High and Low Pulse Repetition Rates
High repetition rate pulse trains are sometimes obtained with harmonic mode locking, where multiple pulses are circulating in the laser resonator with a constant temporal spacing. This allows the generation of multi-gigahertz pulse trains even with fiber lasers, which typically have fundamental round-trip frequencies of only some tens of megahertz. Note that fiber lasers cannot have arbitrarily short resonators due to the limited gain and pump absorption per unit length.
For high pulse repetition rates with fundamental mode locking, i.e., without harmonic mode locking, very short laser resonators are required. If Q-switching instabilities are avoided with a proper design, such lasers can be very simple, stable and compact.
Due to the high pulse repetition rate, the pulse energies obtained from mode-locked lasers are fairly limited – normally nanojoules or picojoules. Higher pulse energies combined with lower repetition rates can be obtained with cavity-dumped mode-locked lasers and particularly with regenerative amplifiers.
The Optical Spectrum: Frequency Combs
The optical spectrum of a pulse train generated in a mode-locked laser is not smooth (as for a single pulse), but rather consists of discrete lines with an exactly constant spacing which equals the pulse repetition rate (→ frequency combs). This is true although the resonance frequencies of the resonator modes are usually not exactly equidistant due to the effect of chromatic dispersion, e.g. in the gain medium: the mode-locking mechanism forces the laser to emit frequencies which can to some extent deviate from the frequencies of the resonator modes. These frequency deviations may not be arbitrarily high, and therefore the generation of broadband spectra is usually possible only if the resonator dispersion is sufficiently small, so that the resonator mode frequencies are approximately equidistant. In the time domain, that condition can be understood via the temporal broadening of pulses caused by dispersion, which must be compensated by the mode-locking mechanism. Note, however, that optical nonlinearities often also play an important role, so that considering the “cold cavity modes” misses an important part of the physical mechanisms.
Origin of the Term “Mode Locking”
The term mode locking originates from a description in the frequency domain: a short pulse is formed in the laser resonator when a fixed phase relationship is achieved between its longitudinal modes, or more precisely, between the lines in the spectrum of the laser output. However, the basic mechanisms leading to mode locking can usually be much more easily understood in the time domain. Also, the concept of resonator modes becomes questionable under the influence of strong optical nonlinearities. Strictly speaking, it is not even advisable to use the term modes for the lines in the spectrum of a mode-locked laser, even though these lines are related to the resonator modes.
It is instructive to consider the synthesis of a periodic pulse train by superposition of sinusoidal oscillations (Figure 4) with equally spaced frequencies, corresponding to different axial resonator modes in a mode-locked laser. The larger the number of frequency components involved, the shorter can be the duration of the generated pulses relative to the round-trip time.
The vertical lines indicate points in time where all the oscillations add up in phase.
An important aspect is that there must be a fixed phase relationship between these modes. This is illustrated in Figure 5: the blue curve shows a pulse train with a fixed phase relationship, so that at regular temporal positions (e.g. at <$t = 0$>) the electric fields of all frequency components add up to a maximum of the total field strength. The red curve shows the electric field for the same strength of all frequency components, but with random relative phases.
The explanations and diagrams above are actually simplified in an important detail: it has been assumed that all involved optical frequencies are integer multiples of the pulse repetition frequency. In reality, it is generally such that every frequency is in addition shifted by a certain amount, called the carrier–envelope offset frequency. One then still gets a periodic pulse train in terms of intensity, but a change of the optical phase at the pulse maximum which evolves continuously. See the article on carrier–envelope offset for more details.
Instabilities and Noise of Mode-locked Lasers
The various mechanisms for mode locking can exhibit various kinds of instabilities, which may prevent proper mode locking. For example, a laser may generate bunches of pulses (instead of single pulses), pulses with unstable energy (→ Q-switching instabilities), pulses which fall apart after some time and are later replaced with new pulses, or pulses which are accompanied by a noise background. Different kinds of modeling can be very helpful in tracking down such problems and to find appropriate remedies.
Even without particular instabilities, the output of a mode-locked laser contains various kinds of noise; the timing jitter (random fluctuations of the pulse positions) is often of special interest. Other types of noise include phase noise, intensity noise, and fluctuations of other parameters such as the pulse duration, chirp, and optical center frequency. The phase noise results in a finite width of the lines in the generated frequency comb. There are interesting relations between different types of noise in mode-locked lasers .
Note that the article on mode-locked lasers contains some more details on different kinds of mode-locked lasers, and more details concerning mode-locking techniques can be found in the articles on active and passive mode locking.
Mode Locking Diagnostics
If a mode-locked laser is not designed properly, or has a problem e.g. related to misalignment or a defect component (such as a saturable absorber), it may not work properly. For example, it may exhibit some instabilities as explained above, or may not at all produce ultrashort pulses.
For checking proper mode locking, i.e., excluding incomplete or partial mode locking, one may use multiple methods:
- With an autocorrelator, one can check the pulse duration.
- Although that is generally a good test, one may overlook certain problems. For example, if the laser emits bunches of pulses rather than a single pulse per round trip as intended, one may not see that in an autocorrelation if the displayed range is smaller than the pulse spacing. Further, some temporally very long background, which would not occur for proper mode locking, may be overlooked if its power is much smaller than the peak power of the pulses – although the total energy in that background may be substantial or even dominating.
- Intensity autocorrelation traces tend to show much less structure than the actual intensity profiles of the pulses – also in cases with substantial pulse fluctuations.
- Also note that one if one sees a very narrow peak at the center of the autocorrelation, this may be an artifact called the coherent spike. It may be generated if the pulses have a varying structure in their wings which largely suppresses the autocorrelation signal in the wings. One should not misinterpret such an observation as an indication for a very short pulse (possibly on top of a longer one), but rather as evidence for some problem.
- When an autocorrelator is not available, at least some indications may be obtained by sending the pulses through a nonlinear crystal which is phase-matched for frequency doubling. Generally, the frequency doubling efficiency will be far higher in the mode-locked state, where the peak power gets far higher than the average power. However, one gets substantially less information than from an autocorrelation.
- One can send the pulses (possibly after some attenuation) to a fast photodiode and analyze the electronic output of that with an RF spectrum analyzer. (The measurement bandwidth is usually insufficient for measuring the pulse duration, but should be at least several times the pulse repetition rate.) For proper mode locking, in the RF spectrum one will see a peak corresponding to the pulse repetition rate and further peaks which are exact harmonics of that – with a decay of peak height related to the limited detection bandwidth. Each peak should be very sharp and stable. Fluctuations of those peaks or additional satellite peaks may provide evidence for mode-locking instabilities, but if those are not seen, that is not yet a firm proof of proper mode locking. Q-switched mode locking leads to relatively low frequency peaks (far below the pulse repetition rate), which may or may not be stable.
- Note that even in continuous-wave operation one obtains peaks at harmonic of the round-trip frequency as beat notes of the laser resonator's modes. These, however, are often not very stable and may exhibit small irregular structures when observed with high enough frequency resolution.
- The optical spectrum can be measured with an optical spectrum analyzer. It may give substantial additional information. For proper mode locking, the spectrum should be stable and smooth, and it should have a width which is compatible with the measured pulse duration.
- A substantial optical bandwidth is not itself a proof of mode locking. It may, for example, result even in continuous-wave operation from inhomogeneous saturation of the laser gain.
- More detailed pulse diagnostics (even with a full analysis of the electric field vs. time) are possible with additional instruments, e.g. for FROG or SPIDER measurements. See the article on pulse characterization for more details.
While various observations can one clearly provide evidence for non-proper mode locking, there are cases where problems are not seen despite using multiple characterization methods.
Mode Locking Devices
Optimum mode-locked performance depends on a well worked-out laser design, including the choice of a suitable mode locking device, be it some kind of modulator or a saturable absorber. The article on mode locking devices gives more details.
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|||R. Paschotta, case study on active mode locking|
|||R. Paschotta, case study on passive mode locking|
See also: active mode locking, passive mode locking, mode-locked lasers, mode locking devices, soliton mode locking, harmonic mode locking, cavity dumping, ultrafast lasers, regenerative amplifiers, light pulses, pulse generation, carrier–envelope offset, frequency combs, timing jitter, modes of laser operation, spotlight 2007-10-11
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