Editing Mode locking (section)

==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 (waves)|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 [[Beat (acoustics)|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 {{Math|1=''&tau;'' = 2''L''/''c''}}, where {{Mvar|&tau;}} 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, {{Math|1=&Delta;''&nu;'' = 1/''&tau;''}}.

The duration of each pulse of light is determined by the number of modes oscillating in phase (in a real laser, it is not necessarily true that all of the laser's modes are phase-locked). If there are {{Mvar|N}} modes locked with a frequency separation {{Math|&Delta;''&nu;''}}, then the overall mode-locked bandwidth is {{Math|''N''&Delta;''&nu;''}}, 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 function|Gaussian]] temporal shape, the minimum possible pulse duration {{Math|&Delta;''t''}} is given by

: <math>\Delta t = \frac{0.441}{N \, \Delta\nu}.</math>

The value 0.441 is known as the "[[Bandwidth-limited pulse|time–bandwidth product]]" of the pulse and varies depending on the pulse shape. For [[ultrashort pulse|ultrashort-pulse]] lasers, a [[Hyperbolic function|hyperbolic-secant]]-squared (sech<sup>2</sup>) 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&nbsp;GHz bandwidth, the shortest Gaussian pulse consistent with this spectral width is around 300&nbsp;picoseconds; for the 128&nbsp;THz bandwidth Ti:sapphire laser, this spectral width corresponds to a pulse of only 3.4&nbsp;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 (optics)|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 (frequency or phase). 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 interferometer|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 integral 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|Lorentzian function]] with a full-width half-maximum line width

: <math>\Delta\nu_c = \frac{ \Delta\nu_{\text{FSR}}}{\mathcal{F}},</math>

where {{Math|1=&Delta;''&nu;''<sub>FSR</sub> = ''c''/2''L''}} is the frequency difference between adjacent resonances (i.e. the free spectral range) and {{Mvar|{{mathcal|F}}}} is the [[Fabry–Pérot interferometer|finesse]],

:<math>\mathcal{F}=\frac{ \pi R^{\frac{1}{2}} }{ 1-R },</math>

where {{Mvar|R}} 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.