Mode locking
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 or femtoseconds. 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".
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 from which the laser is constructed, and the range of frequencies over which a laser may operate is known as the gain bandwidth. For example, a typical helium–neon laser has a gain bandwidth of about 1.5 GHz, whereas a titanium-doped sapphire solid-state laser has a bandwidth of about 128 THz.The second factor to determine a laser's emission frequencies is the optical cavity of the laser. In the simplest case, this consists of two plane mirrors facing each other, surrounding the gain medium of the laser. Since light is a wave, when bouncing between the mirrors of the cavity, the light constructively and destructively interferes with itself, leading to the formation of standing waves, or modes, 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 that 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 is an exact multiple of half the wavelength of the light, such that, where is an integer known as the mode order.
In practice, is usually much greater than, so the relevant values of ' are large. Of more interest is the frequency separation between any two adjacent modes ' and ; this is given by, where is the speed of light.
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 mode is 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 some bandwidth or narrow range of frequencies over which it operates, but typically this bandwidth, determined by the Q factor of the cavity, is much smaller than the intermode frequency separation.
Mode-locking theory
In a simple laser, each of these modes oscillates 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 fluctuations in intensity; in lasers with many thousands of modes, these interference effects tend to average to a near-constant output intensity.If instead of oscillating independently, each mode operates with a fixed phase between it and the other modes, then 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, where 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,.
The duration of each pulse of light is determined by the number of modes oscillating in phase. If there are modes locked with a frequency separation, then the overall mode-locked bandwidth is, 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 is given by
The value 0.441 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 pulse shape is often assumed, giving a time–bandwidth product of 0.315.
Using this equation, the minimum pulse duration can be calculated consistent with the measured laser spectral width. For the HeNe laser with a 1.5 GHz bandwidth, the shortest Gaussian pulse consistent with this spectral width is around 300 picoseconds; for the 128 THz bandwidth Ti:sapphire laser, this spectral width corresponds to a pulse of only 3.4 femtoseconds. These values represent the shortest possible Gaussian pulses consistent with 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.
Subsequent modulation could, in principle, shorten the pulse width of such a laser further; however, the measured spectral width would then be correspondingly increased.
Principle of phase and mode locking
There are many ways to lock frequency, but the basic principle is the same, which is based on the feedback loop of the laser system. The starting point of the feedback loop is the quantity that must be stabilized. To check whether frequency changes with time, a reference is needed. A common way to measure laser frequency is to link it with a geometrical property of an optical cavity. The Fabry-Pérot cavity is most commonly used for this purpose, consisting of two parallel mirrors separated by some distance. This method is based on the fact that light can resonate and be transmitted only if the optical path length of a single round trip is an integer multiple of the wavelength of the light. Deviation of a laser's frequency from this condition will decrease transmission of that frequency. The relation between transmission and frequency deviation is given by a Lorentzian function with a full-width half-maximum line widthwhere is the frequency difference between adjacent resonances and is the finesse,
where is the reflectivity of mirrors. Therefore, to obtain a small cavity line width, mirrors must have higher reflectivity, so to reduce the line width of the laser to the lowest extent, a high finesse cavity is required.
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 intracavity 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 electro-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 and is amplitude-modulated at a frequency, then the resulting signal has sidebands at optical frequencies and. If the modulator is driven at the same frequency as the cavity mode spacing, 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 and 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. The coupling is only dissipative because of the dissipative nature of the amplitude modulation; otherwise, the 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 is synchronised to the cavity round-trip time, 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, active mode locking is frequency-modulation mode locking, which uses a modulator device based on the acousto-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 upshifts in frequency, and some repeated downshifts. After many repetitions, the upshifted and downshifted light is swept out of the gain bandwidth of the laser. The only light 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 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.