Mode locking

Laser mode structure

A mode-locked, fully reflecting cavity supporting the first 30 modes. The upper plot shows the first 8 modes inside the cavity (lines) and the total electric field at various positions inside the cavity (points). The lower plot shows the total electric field inside the cavity.

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 (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 centered at 800 nm).

The second factor to determine 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–Pérot cavity). 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 L is an exact multiple of half the wavelength of the light λ, such that L = qλ/2, where q is an integer known as the mode order.

In practice, L is usually much greater than 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:

${\displaystyle \Delta \nu ={\frac {c}{2L}},}$

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

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 (see Fabry–Pérot interferometer), is much smaller than the intermode frequency separation.

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, 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, where τ 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 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 N modes locked with a frequency separation Δν, the overall mode-locked 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

${\displaystyle \Delta t={\frac {0.441}{N\,\Delta \nu }}.}$

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 (sech²) 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 would be around 300 picoseconds; for the 128 THz bandwidth Ti:sapphire laser, this spectral width would correspond to a pulse of only 3.4 femtoseconds duration. 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.

                                                ∆νC=∆νFSR/ℱ 

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.

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 by 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:sapphire 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 that 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 multiphoton intrapulse interference phase scan are used.

• 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 mode-locked lasers, since they can offer a very high repetition rate of ultrashort pulses.
• Femtosecond laser nanomachining – the short pulses can be used to nanomachine in many types of materials.
• An example of pico- and femtosecond micromachining is drilling the silicon jet surface of inkjet printers.
• Two-photon microscopy.
• Corneal surgery (see refractive surgery). Femtosecond lasers can be used to create bubbles in the cornea. A line of bubbles can be used to create a cut in the cornea, replacing the microkeratome, e.g. for the creation of a flap in LASIK surgery (this is sometimes referred to as Intralasik or all-laser surgery). Bubbles can also be created in multiple layers so that a piece of corneal tissue between these layers can be removed (a procedure known as small incision lenticule extraction).
• A laser technique has been developed that renders the surface of metals deep black. A femtosecond laser pulse deforms the surface of the metal, forming nanostructures. The immensely increased surface area can absorb virtually all the light that falls on it, thus rendering it deep black. This is one type of black gold[3][4]
• Photonic sampling, using the high accuracy of lasers over electronic clocks to decrease the sampling error in electronic ADCs.

Monochromatic light is the property of the laser depends on the fundamental working principle of the laser which contains frequency selective elements. For example in diode laser, external mirror resonator and grating are those elements. With the help of these elements, frequency selection leads to a very narrow spectral emission of light. However, when observed closely, there are frequency fluctuations that occur on different time scales. There can be different reasons for their origin, e.g. fluctuation in input voltage, acoustic vibration or change in pressure and temperature of the surrounding. So, to narrow down these frequency fluctuations, it is necessary to stabilize the phase or frequency of the laser to an external extent. Stabilizing laser property using any external source or external reference is generally called ‘Laser locking’ or simply ‘Locking’.

• Fiber laser
• Disk laser
• Laser construction
• Soliton
• Vector soliton
• Dissipative soliton
• Saturable absorption
• Solid state laser
• Femtotechnology
• Frequency comb
• Ultrafast optics
• Q-switching

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• H. Zhang et al., "Coherent energy exchange between components of a vector soliton in fiber lasers", Optics Express, 16,12618–12623 (2008).
• H. Zhang et al, “Multi-wavelength dissipative soliton operation of an erbium-doped fiber laser”, Optics Express, Vol. 17, Issue 2, pp. 12692–12697
• L.M. Zhao et al, “Polarization rotation locking of vector solitons in a fiber ring laser”, Optics Express, 16,10053–10058 (2008).
• Qiaoliang Bao, Han Zhang, Yu Wang, Zhenhua Ni, Yongli Yan, Ze Xiang Shen, Kian Ping Loh, and Ding Yuan Tang, Advanced Functional Materials,"Atomic layer graphene as saturable absorber for ultrafast pulsed lasers "
• Zhang, H.; et al. (2010). "Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser" (PDF). Applied Physics Letters. 96 (11): 111112. arXiv:1003.0154. Bibcode:2010ApPhL..96k1112Z. doi:10.1063/1.3367743. S2CID 119233725. Archived from the original (PDF) on July 16, 2011.
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• Encyclopedia of laser physics and technology on mode locking and mode-locked lasers