Mode locking

Mode lockingis a technique inopticsby which alasercan be made to produce pulses of light of extremely short duration, on the order of picoseconds (10⁻¹²s) or femtoseconds (10⁻¹⁵s). A laser operated in this way is sometimes referred to as afemtosecond laser,for example, in modernrefractive surgery.The basis of the technique is to induce a fixedphaserelationship between thelongitudinal modesof the laser'sresonant cavity.Constructive interferencebetween 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

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, purefrequencyorwavelength.All lasers produce light over some naturalbandwidthor range of frequencies. A laser's bandwidth of operation is determined primarily by thegain mediumfrom 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 typicalhelium–neon laserhas a gain bandwidth of about 1.5GHz(a wavelength range of about 0.002nmat 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 theoptical cavity(or resonant cavity) of the laser. In the simplest case, this consists of two plane (flat)mirrorsfacing each other, surrounding the gain medium of the laser (this arrangement is known as aFabry–Pérotcavity). Since light is awave,when bouncing between the mirrors of the cavity, the light constructively and destructivelyinterfereswith itself, leading to the formation ofstanding waves,ormodes,between the mirrors. These standing waves form a discrete set of frequencies, known as thelongitudinal modesof 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 mirrorsLis an exact multiple of half the wavelength of the lightλ,such thatL=qλ/2,whereqis an integer known as the mode order.

In practice,Lis usually much greater thanλ,so the relevant values ofqare large (around 10⁵to 10⁶). Of more interest is the frequency separation between any two adjacent modesqandq+ 1; this is given (for an empty linear resonator of lengthL) by Δν:

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

wherecis thespeed of light(≈ 3×10⁸m/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 theQfactorof the cavity (seeFabry–Pérot interferometer), 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 individualphaseof 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 causebeatingeffects 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 byτ= 2L/c,whereτis 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 areNmodes locked with a frequency separation Δν,the overall mode-locked bandwidth isNΔν,and the wider this bandwidth, the shorter thepulse durationfrom 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 aGaussiantemporal shape, the minimum possible pulse duration Δtis given by

\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. Forultrashort pulselasers, ahyperbolic-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 overalldispersionof 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 or not, a reference is needed. A common way to measure laser frequency is to link it with the geometrical property of an optical cavity. TheFabry- Perot cavityis 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 wavelength of light. Deviation of laser frequency from this condition will decrease frequency transmission. The relation between transmission and frequency deviation is given by aLorentzian Functionwith a full width half maximum line width

\Delta \nu _{c}={\frac {\Delta \nu _{\text{FSR}}}{\mathcal {F}}}.

where $\Delta \nu _{\text{FSR}}={\frac {c}{2L}}$ is the frequency difference between adjacent resonances (i.e. the free spectral range) and ${\mathcal {F}}$ is thefinesse,

{\mathcal {F}}={\frac {\pi R^{\frac {1}{2}}}{1-R}}

where $R$ is the reflectivity of mirrors. As it’s clear from the equation, to obtain a small cavity line width, mirrors must have higher reflectivity. Therefore 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 amodulationof 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 waveelectro-optic modulatorinto the laser cavity. When driven with an electrical signal, this produces a sinusoidalamplitude modulationof the light in the cavity. Considering this in the frequency domain, if a mode has optical frequencyνand is amplitude-modulated at a frequencyf,the resulting signal hassidebandsat optical frequenciesν−fandν+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 theν− 2fandν+ 2fmodes, 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 theentropygiven 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 ratefis synchronised to the cavity round-trip timeτ,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 isfrequency-modulation(FM) mode locking, which uses a modulator device based on theacousto-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 (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. A commonly used device to achieve this is asaturable absorber.

A saturable absorber is an optical device that exhibits an intensity-dependent transmission, meaning that the device behaves differently depending on the intensity of the light passing through it. For passive mode locking, ideally a saturable absorber selectively absorbs low-intensity light, but transmits light of sufficiently high intensity. When placed in a laser cavity, a saturable absorber attenuates low-intensity constant-wave light (pulse wings). However, because of the somewhat random intensity fluctuations experienced by an un-mode-locked laser, any random, intense spike is 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 frequencyνand is amplitude-modulated at a frequencynf,the resulting signal hassidebandsat optical frequenciesν−nfandν+nfand enables much stronger mode locking for shorter pulses and more stability than active mode locking, but has startup problems.

Saturable absorbers are commonly liquidorganicdyes, but they can also be made from dopedcrystalsandsemiconductors.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 acolliding-pulse mode-locked laserthe absorber steepens the leading edge, while thelasing mediumsteepens 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 opticaleffects in intracavity 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 calledKerr-lens mode locking(KLM), also sometimes called "self-mode-locking". This uses a nonlinear optical process, the opticalKerr effect,which results in high-intensity light being focussed differently from 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.

Hybrid modelocking

In some semiconductor lasers a combination of the two above techniques can be used. Using a laser with a saturable absorber and modulating the electrical injection at the same frequency the laser is locked at, the laser can be stabilized by the electrical injection. This has the advantage of stabilizing the phase noise of the laser and can reduce the timing jitter of the pulses from the laser.

Mode locking by residual cavity fields

Coherent phase-information transfer between subsequent laser pulses has also been observed fromnanowire lasers.Here, the phase information has been stored in the residual photon field of coherentRabi oscillationsin the cavity. Such findings open the way to phase locking of light sources integrated onto chip-scale photonic circuits and applications, such as on-chip Ramsey comb spectroscopy.^[1]

Fourier-domain mode locking

Fourier-domain mode locking (FDML) is a laser mode-locking technique that creates a continuous-wave, wavelength-swept light output.^[2]A main application for FDML lasers isoptical coherence tomography.

Practical mode-locked lasers

In practice, a number of design considerations affect the performance of a mode-locked laser. The most important are the overalldispersionof the laser'soptical resonator,which can be controlled with aprism compressoror some dispersive mirrors placed in the cavity, and opticalnonlinearities.For excessive netgroup delay dispersion(GDD) of the laser cavity, thephaseof 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 theKerr 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 byKerr-lens mode-lockedTi-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 byself-phase modulationin 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 (involvinghigh-harmonic generationwith amplified femtosecond laser pulses) can be used to produce optical features with durations as short as 100attosecondsin theextreme ultravioletspectral region (i.e. <30 nm). Other achievements, important particularly forlaser applications,concern the development of mode-locked lasers that can be pumped withlaser 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 usingoptoelectronictechniques (i.e.photodiodes), and so indirect methods, such asautocorrelation,frequency-resolved optical gating,spectral phase interferometry for direct electric-field reconstructionormultiphoton intrapulse interference phase scanare used.

Applications

Nuclear fusion (inertial confinement fusion).
Nonlinear optics, such assecond-harmonic generation,parametric down-conversion,optical parametric oscillators,and generation ofterahertz radiation.
Optical data storage uses lasers, and the emerging technology of3D optical data storagegenerally 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 ofinkjet printers.
Two-photon microscopy.
Corneal surgery (seerefractive surgery). Femtosecond lasers can be used to create bubbles in thecornea.A line of bubbles can be used to create a cut in the cornea, replacing themicrokeratome,e.g. for the creation of a flap inLASIKsurgery (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 assmall 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, formingnanostructures.The immensely increased surface area can absorb virtually all the light that falls on it, thus rendering it deep black. This is one type ofblack gold^[3]^[4]
Photonic sampling, using the high accuracy of lasers over electronic clocks to decrease the sampling error in electronic ADCs.

Locking mechanism of laser cavity

Monochromatic lightis the property of the laser depends on the fundamental working principle of the laser which contains frequency selective elements. For example indiode laser,external mirror resonatorandgratingare 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’.

Error signal generation

The reason for generation to create error signals is to create an electronic signal which is proportional to the laser's deviation from a particular set frequency or phase which is called ‘Lock point’. If the laser frequency is large then the signal is positive, if frequency is very small then the signal is negative. The point where the signal is zero is called a lock point. Laser locking based on an error signal which is a function of frequency is called frequency locking and if the error signal is a function of phase deviation of laser then this locking is referred to as phase locking of laser. If the signal is created using an optical setup involving references like frequency references. Using the reference, the optical signal is directly converted in over frequencies which can be detected directly. The other way is to record the signal using a photodiode or camera and further change this signal electronically.

References

^Mayer, B., et al."Long-term mutual phase locking of picosecond pulse pairs generated by a semiconductor nanowire laser".Nature Communications 8 (2017): 15521.
^R. Huber, M. Wojtkowski, J. G. Fujimoto,"Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography",Opt. Express 14, 3225–3237 (2006).
^"Ultra-Intense Laser Blast Creates True 'Black Metal'".Retrieved2007-11-21.
^Vorobyev, A. Y.; Guo, Chunlei (28 January 2008)."Colorizing metals with femtosecond laser pulses".Applied Physics Letters.92(4): 041914.Bibcode:2008ApPhL..92d1914V.doi:10.1063/1.2834902.

External links

Encyclopedia of laser physics and technology on mode lockingandmode-locked lasers

[nwpl-1] Mayer, B., et al."Long-term mutual phase locking of picosecond pulse pairs generated by a semiconductor nanowire laser".Nature Communications 8 (2017): 15521.

[FDML-2] R. Huber, M. Wojtkowski, J. G. Fujimoto,"Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography",Opt. Express 14, 3225–3237 (2006).

[3] "Ultra-Intense Laser Blast Creates True 'Black Metal'".Retrieved2007-11-21.

[4] Vorobyev, A. Y.; Guo, Chunlei (28 January 2008)."Colorizing metals with femtosecond laser pulses".Applied Physics Letters.92(4): 041914.Bibcode:2008ApPhL..92d1914V.doi:10.1063/1.2834902.

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