Modelocking

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 (10-12s) or femtoseconds (10-15s).

The basis of the technique is to induce a fixed phase relationship between the modes of the laser's resonant cavity. The laser is then said to be phase-locked or mode-locked. Interference between these modes causes the laser light to be produced as a train of pulses. Depending on the properties of the laser, these pulses may be of extremely brief duration, as short as a few femtoseconds.

Laser cavity modes
Although laser light is perhaps the purest form of light, it is not of a single, pure frequency or wavelength. All lasers produce light over some natural bandwidth or range of frequencies. A laser's bandwidth of operation is determined primarily by the gain medium that the laser is constructed from, and the range of frequencies that a laser may operate over is known as the gain bandwidth. For example, a typical helium-neon (HeNe) gas laser has a gain bandwidth of approximately 1.5 GHz (a wavelength range of about 0.002 nm at a central wavelength of 633 nm), whereas a titanium-doped sapphire (Ti:Sapphire) solid-state laser has a bandwidth of about 128 THz (a 300 nm wavelength range centred around 800 nm).

The second factor that determines a laser's emission frequencies is the optical cavity or resonant cavity of the laser. In the simplest case, this consists of two plane (flat) mirrors facing each other, surrounding the gain medium of the laser (this arrangement is known as a Fabry-Perot cavity). Since light is a wave, when bouncing between the mirrors of the cavity the light will constructively and destructively interfere with itself, leading to the formation of standing waves between the mirrors.



These standing waves form a discrete set of frequencies, known as the longitudinal modes of the cavity. These modes are the only frequencies of light which are self-regenerating and allowed to oscillate by the resonant cavity; all other frequencies of light are suppressed by destructive interference. For a simple plane-mirror cavity, the allowed modes are those for which the separation distance of the mirrors L is an exact multiple of half the wavelength of the light λ, such that L = q λ/2, when q is an integer known as the mode order.

In practice, the separation distance of the mirrors L is usually much greater than the wavelength of light λ, so the relevant values of q are large (around 105 to 106). Of more interest is the frequency separation between any two adjacent modes q and q+1; this is given (for an empty linear resonator of length L) by Δν:
 * $$\Delta\nu = \frac{c}{2L}$$

where c is the speed of light (≈3×108 m·s−1).

Using the above equation, a small laser with a mirror separation of 30 cm has a frequency separation between longitudinal modes of 0.5 GHz. Thus for the two lasers referenced above, with a 30 cm cavity the 1.5 GHz bandwidth of the HeNe laser would support up to 3 longitudinal modes, whereas the 128 THz bandwidth of the Ti:sapphire laser could support approximately 250,000 modes. When more than one longitudinal modes are excited, the laser is said to be in multi-mode operation. When only one longitudinal mode is excited, the laser is said to be in single-mode operation.

Each individual longitudinal mode has itself some bandwidth or narrow range of frequencies over which it operates, but typically this bandwidth, determined by the finesse of the cavity (see Fabry-Perot interferometer), is much smaller than the inter-mode frequency separation.

Mode-locking theory
In a simple laser, each of these modes will oscillate independently, with no fixed relationship between each other, in essence like a set of independent lasers all emitting light at slightly different frequencies. The individual phase of the light waves in each mode is not fixed, and may vary randomly due to such things as thermal changes in materials of the laser. In lasers with only a few oscillating modes, interference between the modes can cause beating effects in the laser output, leading to random fluctuations in intensity; in lasers with many thousands of modes, these interference effects tend to average to a near-constant output intensity, and the laser operation is known as a c.w. or continuous wave.

If instead of oscillating independently, each mode operates with a fixed phase between it and the other modes, the laser output behaves quite differently. Instead of a random or constant output intensity, the modes of the laser will periodically all constructively interfere with one another, producing an intense burst or pulse of light. Such a laser is said to be mode-locked or phase-locked. These pulses occur separated in time by τ = 2L/c, which is the time taken for the light to make exactly one round trip of the laser cavity. This time corresponds to a frequency exactly equal to the mode spacing of the laser, Δν = 1/τ.

The duration of each pulse of light is determined by the number of modes which are oscillating in phase (in a real laser, it is not necessarily true that all of the laser's modes will be phase-locked). If there are N modes locked with a frequency separation Δν, the overall modelocked bandwidth is NΔν, and the wider this bandwidth, the shorter the pulse duration from the laser. In practice, the actual pulse duration is determined by the shape of each pulse, which is in turn determined by the exact amplitude and phase relationship of each longitudinal mode. For example, for a laser producing pulses with a Gaussian temporal shape, the minimum possible pulse duration Δt is given by


 * $$ \Delta t = \frac{0.44}{N \Delta \nu}.$$

The value 0.44 is known as the time-bandwidth product of the pulse, and varies depending on the pulse shape. For ultrashort pulse lasers, a hyperbolic-secant-squared (sech2) pulse shape is often assumed, giving a time-bandwidth product of 0.315.

Using this equation, we can calculate the minimum pulse duration which can be produced by a laser. For the HeNe laser with a 1.5 GHz bandwidth, the shortest Gaussian pulse which can be produced would be around 300 picoseconds; for the 128 THz bandwidth Ti:sapphire laser, this duration would be only 3.4 femtoseconds. These values represent the shortest possible Gaussian pulses supported by the laser's bandwidth; in a real mode-locked laser, the actual pulse duration depends on many other factors, such as the actual pulse shape, and the overall dispersion of the cavity.

Mode-locking methods
Methods for producing mode-locking in a laser may be classified as either active or passive. Active methods typically involve using an external signal to induce a modulation of the intra-cavity light. Passive methods do not use an external signal, but rely on placing some element into the laser cavity which causes self-modulation of the light.

Active mode-locking
The most common active mode-locking technique places a standing wave acousto-optic modulator into the laser cavity. When driven with an electrical signal, this produces a sinusoidal amplitude modulation of the light in the cavity. Considering this in the frequency domain, if a mode has optical frequency ν, and is amplitude-modulated at a frequency f, the resulting signal has sidebands at optical frequencies ν-f and ν+f. If the modulator is driven at the same frequency as the cavity-mode spacing Δν, then these sidebands correspond to the two cavity modes adjacent to the original mode. Since the sidebands are driven in-phase, the central mode and the adjacent modes will be phase-locked together. Further operation of the modulator on the sidebands produces phase-locking of the ν-2f and ν+2f modes, and so on until all modes in the gain bandwidth are locked. As said above, typical lasers are multi-mode and not seeded by a root mode. So multiple modes need to work out which phase to use. In a passive cavity with this locking applied there is no way to dump the entropy given by the original independent phases. This locking is better described as a coupling, leading to a complicated behavior and not clean pulses. Only because of the dissipative nature of the amplitude modulation the coupling is also dissipative, phase modulation would not work.

This process can also be considered in the time domain. The amplitude modulator acts as a weak shutter to the light bouncing between the mirrors of the cavity, attenuating the light when it is "closed", and letting it through when it is "open". If the modulation rate f is synchronised to the cavity round-trip time τ, then a single pulse of light will bounce back and forth in the cavity. The actual strength of the modulation does not have to be large; a modulator that attenuates 1% of the light when "closed" will mode-lock a laser, since the same part of the light is repeatedly attenuated as it traverses the cavity.

Related to this amplitude modulation (AM) active mode-locking is frequency modulation (FM) mode-locking, which uses a modulator device based on the electro-optic effect. This device, when placed in a laser cavity and driven with an electrical signal, induces a small, sinusoidally varying frequency shift in the light passing through it. If the frequency of modulation is matched to the round-trip time of the cavity, then some light in the cavity sees repeated up-shifts in frequency, and some repeated down-shifts. After many repetitions, the up-shifted and down-shifted light is swept out of the gain bandwidth of the laser. The only light which is unaffected is that which passes through the modulator when the induced frequency shift is zero, which forms a narrow pulse of light.

The third method of active mode-locking is synchronous mode-locking, or synchronous pumping. In this, the pump source (energy source) for the laser is itself modulated, effectively turning the laser on and off to produce pulses. Typically, the pump source is itself another mode-locked laser. This technique requires accurately matching the cavity lengths of the pump laser and the driven laser.

Passive mode-locking
Passive mode-locking techniques are those that do not require a signal external to the laser (such as the driving signal of a modulator) to produce pulses. Rather, they use the light in the cavity to cause a change in some intracavity element, which will then itself produce a change in the intracavity light. The most common type of device which will do this is a saturable absorber.

A saturable absorber is an optical device that exhibits an intensity-dependent transmission. What this means is that the device behaves differently depending on the intensity of the light passing through it. For passive mode-locking, ideally a saturable absorber will selectively absorb low-intensity light, and transmit light which is of sufficiently high intensity.

When placed in a laser cavity, a saturable absorber will attenuate low-intensity constant wave light (pulse wings). However, because of the somewhat random intensity fluctuations experienced by an un-modelocked laser, any random, intense spike will be transmitted preferentially by the saturable absorber. As the light in the cavity oscillates, this process repeats, leading to the selective amplification of the high-intensity spikes, and the absorption of the low-intensity light. After many round trips, this leads to a train of pulses and mode-locking of the laser.

Considering this in the frequency domain, if a mode has optical frequency ν, and is amplitude-modulated at a frequency n f, the resulting signal has sidebands at optical frequencies ν - n f and ν + n f and enables much stronger mode-locking for shorter pulses and more stability than active modelocking, but has startup problems.

Saturable absorbers are commonly liquid organic dyes, but they can also be made from doped crystals and semiconductors. Semiconductor absorbers tend to exhibit very fast response times (~100 fs), which is one of the factors that determines the final duration of the pulses in a passively mode-locked laser. In a colliding-pulse mode-locked laser the absorber steepens the leading edge while the lasing medium steepens the trailing edge of the pulse.

There are also passive mode-locking schemes that do not rely on materials that directly display an intensity dependent absorption. In these methods, nonlinear optical effects in intra-cavity components are used to provide a method of selectively amplifying high-intensity light in the cavity, and attenuation of low-intensity light. One of the most successful schemes is called Kerr-lens mode-locking (KLM), also sometimes called "self mode-locking". This uses a nonlinear optical process, the optical Kerr effect, which results in high-intensity light being focussed differently than low-intensity light. By careful arrangement of an aperture in the laser cavity, this effect can be exploited to produce the equivalent of an ultra-fast response time saturable absorber.

Practical mode-locked lasers
In practice, a number of design considerations affect the performance of a mode-locked laser. The most important are the overall dispersion of the laser's optical resonator, which can be controlled with a prism compressor or some dispersive mirrors placed in the cavity, and optical nonlinearities. For excessive net group delay dispersion (GDD) of the laser cavity, the phase of the cavity modes can not be locked over a large bandwidth, and it will be difficult to obtain very short pulses. For a suitable combination of negative (anomalous) net GDD with the Kerr nonlinearity, soliton-like interactions may stabilize the mode-locking and help to generate shorter pulses. The shortest possible pulse duration is usually accomplished either for zero dispersion (without nonlinearities) or for some slightly negative (anomalous) dispersion (exploiting the soliton mechanism).

The shortest directly produced optical pulses are generally produced by Kerr-lens mode-locked Ti-sapphire lasers, and are around 5 femtoseconds long. Alternatively, amplified pulses of a similar duration are created through the compression of longer (e.g. 30 fs) pulses via self-phase modulation in a hollow core fibre or during filamentation. However, the minimum pulse duration is limited by the period of the carrier frequency (which is about 2.7 fs for Ti:S systems), therefore shorter pulses require moving to shorter wavelengths. Some advanced techniques (involving high harmonic generation with amplified femtosecond laser pulses) can be used to produce optical features with durations as short as 100 attoseconds in the extreme ultraviolet spectral region (i.e. <30 nm). Other achievements, important particularly for laser applications, concern the development of mode-locked lasers which can be pumped with laser diodes, can generate very high average output powers (tens of watts) in sub-picosecond pulses, or generate pulse trains with extremely high repetition rates of many GHz.

Pulse durations less than approximately 100 fs are too short to be directly measured using optoelectronic techniques (i.e. photodiodes), and so indirect methods such as autocorrelation, Frequency-resolved optical gating, Spectral phase interferometry for direct electric-field reconstruction or Multi-photon Intrapulse Interference Phase Scan are used.

Neodymium as a dopant has a lot of laser lines. A group of 8 lines lies between 1050 nm and 1080 nm. If the host material is YAG, these lines have width of 0.5 nm. If the host is YLF, the lines have width of 1 nm. If the host is glass, the lines merge into one continuum of 38 nm FWHM (Broad-spectrum neodymium-doped laser glasses for high-energy chirped-pulse amplification; Greg R. Hays, Erhard W. Gaul, Mikael D. Martinez, and Todd Ditmire; Applied Optics, Vol. 46, Issue 21, pp. 4813-4819 ). Single stack laser mirrors have high reflectivity within a 30 nm range. For the crystal host-materials an etalon selects the main laser line and a Gires-Tournois-Interometer compensates the GVD. For the glass host the same techniques as for Ti:Sa lasers can be used. In the crystal hosts the gain bandwidth allows for pulses of about 1 ps. Gain-loss optimization between the gain medium and the typical 10000 ps modulation period of the AOM then leads to 100 ps pulses at 100 % modulation depth. The round-trip frequency of the cavity must be adjusted with 1 in 1E6 precision onto the AOM standing wave frequency(, , REVIEW OF SCIENTIFIC INSTRUMENTS 78, 013105 2007 ). For this the diffracted light from the AOM is detected by a photodiode, and minimized. Depending on the sign of the detuning the relative phase between the diffracted signal and the reflection on the surface of the AOM changes and can be detected by the interference of the two on a small photodiode array. Any drift in this interferometer leads to drift of the phase, while the cavity length drift leads to amplitude drift with π jumps of the phase.

Applications

 * Photochemistry. The short pulses can be used to probe the process of the reaction at a very high temporal resolution, allowing the detection of short-lived intermediate molecules. This method is particularly useful in biochemistry, where it is used to analyse details of protein folding and function. See femtochemistry.
 * Nuclear fusion. (inertial confinement fusion).
 * Nonlinear optics, such as second-harmonic generation, parametric down-conversion, optical parametric oscillators, and generation of Terahertz radiation
 * Optical Data Storage uses lasers, and the emerging technology of 3D optical data storage generally relies on nonlinear photochemistry. For this reason, many examples use modelocked lasers, since they can offer a very high repetition rate of ultrashort pulses.
 * Femtosecond laser micromachining - The short pulses can be used to micromachine in many types of materials. The Handbook of Femtosecond Micromachining (link following) gives additional information.
 * Two-photon microscopy
 * Corneal Surgery. Femtosecond lasers can create bubbles in cornea, if multiple bubbles are created in a planar fashion parallel to the corneal surface then the tissue separates at this plane and a flap like the one in LASIK is formed (Intralase: Intralasik or SBK (Sub Bowman Keratomileusis) if the flap thickness is equal or less than 100 microns). If done in multiple layers a piece of corneal tissue between these layers can be removed (Visumax: FLEX Femtosecond Lenticle Extraction).