Laser mode locking

Laser sources of coherent light are so ubiquitous nowadays, that they are considered one of the important inventions of the 20th century. The simplest laser consists of an optical cavity and a stimulated emission amplifier, which by its feedback action generates nearly monochromatic light. However, most actual lasers emit light of a broad spectral width, making them multimode; that is, the spectrum consists of several, often many, nearly discrete modes. Since the laser mode frequencies are almost equally spaced, the spectrum is sometimes referred to as a ‘frequency comb’.

The coherence of the laser light implies that a well-defined phase can be attributed to the light mode amplitudes. Still, the inevitable presence of noise generates a phase diffusion, which ultimately limits the coherence of the light. Light emitted by spontaneous emission from the amplifier is a fundamental noise source. It follows that in a free-running multimode laser the phases of the lasing modes are random. In this case the intensity of the emitted light is constant up to small fluctuations, as shown in figure 1, and the laser is said to operate in continuous wave (cw) mode, or sometimes quasi-cw to distinguish it from single mode operation.

Rather than being a drawback, the spectral width of the laser light can be harnessed when the phases of the light modes can be controlled. In particular, if the phases are aligned, the light in the cavity is compressed into short pulses, and the output light is a pulse train whose repetition rate is the cavity round-trip time. In this case the laser is said to be mode locked.

Mode locking is an important technological tool and its history is almost as long as that of that of the laser itself [1]. This article focuses on a particular method of achieving mode locking and pulses—passive mode locking. Passive mode locking is obtained by introducing a saturable absorber into the laser cavity that is a nonlinear absorber that becomes more transparent as the intensity of light increases. In the presence of a pulse the saturable absorber creates an instantaneous profile of net gain, which is positive near the peak of pulse and negative at its tails, as shown in figure 2. As the picture suggests, the saturable absorber action further compresses the pulse, and after many passes the result is a short pulse whose bandwidth is limited by that of the amplifier gain and factors like the spectral response of the cavity mirrors.

Saturable absorbers were initially designed as ‘reverse amplifiers’, that is passive resonant media that can absorb a certain amount of light before they saturate. However, the slow relaxation times of the media, typically much longer than optical time scales, limited the shortness of the pulses. For this reason they are mostly replaced nowadays by ‘fast saturable absorbers’, which often take advantage of the Kerr effect, the nonlinear dependence of the index of refraction on intensity, and convert it to nonlinear absorption. A class of ‘saturable absorbers’ of this type operate by passing the light in an interferometer where the two arms experience different Kerr nonlinearities, as shown in figure 3; this method is called additive-pulse mode locking. A similar method, the Kerr-Lens mode locking, is used to mode lock the Ti:Sapphire laser, whose very broad gain bandwidth makes it a favorite source of ultrashort pulses. Elaborate fine tuning has allowed the formation of pulses shorter than 5 fs (5.10^-15sec) in Ti:Sapphire, less than 2 optical periods [2].

Figure 1. A schematic representation of the optical wave form of a multimode laser operating in cw (top) and when it is mode locked (bottom). The graphs on the right represent the output waveform—a nearly constant intensity for the cw, and a train of pulses in mode locking. The graphs on the left represent the amplitudes of the lasing modes - cw is disordered, while mode locking happens when the phases are nearly aligned, like the spins in a ferromagnet.

The subject of this article is the physical process of mode locking, in particular its interplay with the dephasing action of noise, but to complete this introduction, here is a brief and partial list of common applications for femtosecond laser pulses:

• Like the flash of a camera, the ultrashort pulses enable the probing of ultrafast processes. In this manner it has become possible to observe in real time chemical reactions and electronic processes like magnetic relaxation. Often these experiments are carried out by first exciting the system with a strong pump pulse, and then observing it with a weak probe pulse.
• The frequency ‘combs’ of the high-bandwidth mode locked lasers are a source of extremely stable oscillators. Beating different modes of the comb produces microwave signals whose quality competes with and surpasses standard atomic clocks. This work has recently been awarded the Nobel prize.

Figure 2. The net gain is positive for the bright parts of the waveform and negative for the dark parts. The outcome of each pass the saturable absorber-amplifier combination is a sharpening shortening of the pulse.

• When the light is compressed into shorter and shorter time intervals, its peak intensity rises accordingly. The pulses can then be further amplified to become ultra-high intensity light ‘bullets’, that can be used, for example, to generate entangled photons by parametric down-conversion, or to create self-focusing air plasma filaments.
• The femtosecond pulses are used as a stepping stone for the creation of even shorter ‘attosecond’ pulses. Coherent extreme ultra-violet sources are produced by a nonresonant ionization of Helium using strong ultrashort pulses. The recombination process produces attosecond pulses from the high harmonics of the source.

Phase transitions and critical phenomena

The mode locking process can be thought of as follows: The amplitudes of the laser modes form an array of random phase complex numbers, or planar arrows with a random direction, disordered because of the presence of noise. Mode locking is obtained by an ordering agent (the saturable absorber) that imposes an ordering interaction between the modes.

This scenario is quite similar to the ferromagnetic ordering of planar spins (also called XY spins), where the ferromagnatic interaction has to overcome the disordering thermal noise. Statistical light-mode dynamics (SLD) utilizes this analogy to show that mode locking is a thermodynamic phase transition, although the laser system is far from equilibrium.

Figure 3. A possible configuration for a nonlinear ‘absorber’ in an additive pulse mode locked laser. The light in one arm of a Mach-Zehnder interferometer is passed through a Kerr medium, and receives an intensity-dependent phase shift. As a consequence the output splitting at the final beam splitter depends on the intensity. One of the output beams is coupled back to the laser and the other is discarded.

The next section is devoted to a detailed exposition of SLD. For the sake of completeness a brief review is given here of the ideas from phase transition theory that are going to be used, using the Ising model of magnetism as an example [3]. Ising modeled the magnet as a set of ‘spins’, classical variables s_n that can only take the values ±1. The partition function at temperature T, Z = å_{s} exp(−H({s})/T ), is obtained as usual as a sum over all the possible configurations of the system. Assuming that spin n has a magnetic moment µ_n and that the exchange interaction coefficient between spins n and m is J_nm, the energy of a spin configuration is H({s}) = å_nµ_ns_nB − å_nJ_nms_ns_m, where B is the external magnetic field.

In most cases, when the interaction is ferromagnetic J_nm > 0, there is a threshold temperature T_c below which the system magnetizes spontaneously, that is, the mean magnetization M = {s} is nonzero even without external magnetic field—this is the phenomenon of ferromagnetism. The sign of M depends on whether the external field approaches zero from positive or negative direction, and if the external is changed continuously across zero the magnetization changes its sign discontinuously. Such a discontinuous dependence of thermodynamic quantities on external conditions is a first order phase transition like the boiling of a liquid. On the other hand, the transition from zero to nonzero spontaneous magnetization at T_c is continuous: M is a continuous function of the temperature, although not a smooth one. Continuous phase transitions are usually accompanied by critical phenomena that include the divergence of response coefficients and of fluctuations. For this reason T_c is called the critical temperature. The singularities that occur at the critical point or near it are characterized by a set of critical exponents, which have the appealing property of universality. Namely, they depend on gross features of the system rather than on its microscopic details, and in this way allow quantitative comparison between experiments and theory.

Calculating the partition function or thermodynamic quantities directly is usually a difficult task, even for simplified models like the Ising model. Fortunately there is an approximation method of wide applicability—mean field theory. There are many approaches to mean field theory, and the different formulations can sometime seem completely unrelated. The common theme of the various mean field theories is that in one way or another they neglect correlations, and replace a fluctuating quantity (or quantities) by a mean one, which has to be determined self-consistently. For example, in the Ising model the mean field approximation consists of replacing the exchange interaction of two spins by an interaction of a spin and a mean field {s}. One can then use this approximation to calculate the mean magnetization, obtaining an equation for M. When the solution of this equation is nonzero spontaneous magnetization has occurred. Mean field theory is usually a good indication of the general structure of the phase diagram, and approximates thermodynamic quantities to within an order of magnitude, but cannot be expected to give precise predictions, unless some other conditions are obeyed.

In general mean field theory becomes better when the dimensionality of the system increases. Below the upper critical dimension, the mean field approximation is good in general but breaks down near the critical points, while below the lower critical dimension the interaction is not strong enough to impose an ordering transition, and the mean field approximation is invalid. In the presence of long range interaction the accuracy of mean field theory improves and it can even become exact. SLD of passive mode locking is another such case, to which we now turn.

Statistical light-mode dynamics

The quantitative theory of passive mode locking was pioneered by Herman Haus in 1975 [4]. Haus realized that a first-principles modeling of the mode locking process is not necessary. A first-principles analysis of a single mode laser is difficult, and adding to it a nonlinear absorptive interaction between different modes makes it intractable. Haus’ method applies the idea of Landau (although probably independently) of studying effective dynamics. Specifically one studies the time evolution of the complex envelope of the optical waveform in cavity round-trip intervals, under the assumption that the waveform changes only slightly during each such interval. That is, the waveform dynamics occurs on two widely different time scales: The time variable t describes the waveform on scales much shorter than the round-trip time. The waveform y(t) can change from a nearly constant shape in cw operation to a sharply peaked one after mode locking has occurred. The slow variable t describes the evolution of y(t) on time scales much longer than the cavity round-trip time.

The Haus model describes of the effective dynamics for y(t,t) resulting from the action of the amplifier and the saturable absorber. The action of the saturable absorber is modeled using a transmissivity function s(|y|²). Fast saturable absorption is assumed in that s depends on the instantaneous intensity |y|²; when the intensity is not too large s can be approximated linearly s » s₀ + a|y|². The modeling of the amplifier is somewhat more elaborate: It includes the differential overall net gain g at the band center, and the curvature b of spectral response of the amplifier. Together, these yield a partial differential equation for y of Ginzburg-Landau type,

(s₀ has been absorbed into g). The net gain g is not constant, but changes slowly, and depends on the total optical energy in the cavity P = ∫|y|²d t rather than on the instantaneous power. The Haus effective equation, often called the mode locking master equation, predicts the formation of pulses with exponentially decaying tails, as actually observed in passively mode locked lasers. It should be remarked that in most cases the waveform dynamics also includes dispersive effects, most importantly chromatic dispersion and Kerr nonlinearity. These phenomena are usually included in the master equation, but are omitted here to avoid obscuring the essential physics.

Around 1990 it was realized that there is a deficiency in the Haus theory of mode locking, specifically in its description of the onset of mode locking. According to the master equation the cw mode operation is absolutely unstable against infinitesimal perturbations, but in practice many lasers continue to operate without mode locking until an external ‘morning kick’ drives them into the mode locked state. This self-starting problem, as it came to be called, was studied by several authors. The self-starting problem was related to noise-induced decoherence that inhibits the coherent process of passive mode locking, but a quantitative theory of this mechanism was achieved only with SLD.

An independent step toward the development of SLD was taken in 1993 when Haus and Mecozzi [5] studied directly the role of noise in the effective dynamics. A simple but adequate for many purposes way of taking noise into account is to add a random term h(t,t) to the right-hand side of the master equation. Haus and Mecozzi used perturbation theory to calculate the fluctuations of the pulses in mode locked laser. However, noise has a nonperturbative consequence in many body systems - it introduces entropy.

At this point SLD becomes essential. Statistical mechanics provides precisely the needed ideas and tools to understand the interplay between the ordering interaction of the saturable absorber, and the disordering noise. There are two important technical properties that make the quantitative analysis of the SLD of passive mode locking easier than most notoriously difficult strongly interacting far from equilibrium systems. However, before describing them, I would like to point out the main conclusions of SLD with regards to the onset and stability of mode locking.

Firstly, in the presence of noise one should distinguish between dynamical stability and statistical stability. Just like the solid state is dynamically stable but the vapor phase is statistically stable at high temperatures, the cw state is statistically stable when the cavity noise is large despite being dynamically unstable. The waveform of a mode locked laser has a very short bright interval - the pulse, and the rest is ‘dark’; nonetheless, the dark waveform carries a significant part of the cavity power. Since in SLD an increase in the noise is equivalent to a decrease in the intracavity power, it follows immediately that there is a minimum threshold power required to sustain mode locked operation of laser [6]. Below this threshold the pulses simply ‘melt’. The onset of mode locking at the threshold ‘temperature’ is thus a first order ordering phase transition.

Figure 4. The mean-field free energy as a function of pulse power for a set of different ‘temperatures’. The minimum at zero is the cw configuration. At low ‘temperature’, that is weak noise or strong cavity power, there is a second minimum at a finite pulse power, which signifies the mode locked configuration. The minimum with lower energy is the statistically stable one, and the other is metastable.

Secondly, in addition to the statistical steady state configuration, there may exist long-lived metastable states, similar to the existence of supercooled liquids. Because of the nonlinear character of the saturable absorber interaction, the cw state is always metastable, no matter how weak the noise or how strong the laser power. This is the reason behind the self-starting problem. In order to achieve mode locking the laser must cross an activation barrier, either by an external perturbation, or through a rare noise activation. This point has been recently demonstrated when the exponential tails of the self-start time were measured [7]

Consider now the case of the effective dynamics described by equation (1) above with additive gaussian white noise with correlation function {h(t,t)^* h(t',t')} = 2Td(t − t') d(t − t') (the star superscript denotes complex conjugation). The parameter T that measures the noise strength has the same role in SLD as temperature in equilibrium statistical mechanics. The simplifying feature of the dynamics of the Haus model is that it is derived from the functional H = ∫dt (−a|y|⁴ + β|y'|² )+u(P), where −u'(P) is equal to the gain function g(P). This property ensures that the statistical steady state of the laser waveform obeys detailed balance and has the Gibbs-like probability distribution function r = exp(−H/T ) [3].

Figure 5. The solid lines shows the pulse power in the Haus effective model as a fraction of the total intracavity power for different values of the ‘inverse temperature’ αP2/T . It is a typical first order phase transition, exhibiting a jump from cw to mode locking at αP2/T = 9 and metastable states, shown as broken lines.

At this point the problem has been mapped to one of equilibrium statistical physics, and we can proceed directly to calculate the free energy F by mean field theory. In SLD the mean field approximation neglects correlations between the pulse and the ‘dark’ part of the wave form. The free energy then becomes a function only of the fraction y of the power contained in the pulse as shown figure 4. The mean field free energy has the typical behavior of a first order phase transition. Below the transition ‘temperature’ (that is, when noise is weak or the laser power is strong enough), its global minimum occurs at a positive y, indicating mode locking, while above the transition the minimum jumps to y = 0, showing that the statistically stable state is the cw. The existence of metastable states is manifest by the fact that there is a range of ‘temperatures’ with two minima, which exchange stability at the mode-locking threshold.

It is a fortunate property of SLD that, like mechanical or magnetic systems with long-range interactions, the mean field theory is actually exact in the thermodynamic limit - which in this case means when the number of active laser modes becomes very large [6]. It is therefore possible to predict precisely the transition point, which occurs when αP²/T = 9, above which the mode locking is stable, while the pulse remains metastable as long as αP²/T > 8. The change in the pulse power as this parameter is varied is shown in figure 5.

Multipulse mode locking

The accompanying video clips show the of oscilloscope signal at the output of a mode locked fiber laser from the electro-optics lab of the department of electrical engineering of the Technion, taken at different increasing and decreasing noise level.

Figure 6. The oscillatory transmissivity function s(x)—the total net gain experienced by a pulse of power x in an interferometric implementation of a ‘saturable absorber’. When the pulse power reaches the first minimum it is liable to break down into two weaker pulses. In the mode locked steady state the pulse powers are confined to the interval shown on the graph by bold black.

Evidently, as the noise decreases, the mode locking action results in the formation of several pulses, rather than a stronger single pulse. It should be stressed that the output is one of several pulses per round-trip, rather than a repeated single pulse. The phenomenon of multipulse mode locking has been observed often in recent years, and can be understood by reconsidering the saturable absorption in the Haus effective model [8]. When Haus first formulated his equation, the mode locked pulses were much less short than those available in modern lasers, and accordingly their peak power was much lower. Under those circumstances it was reasonable to approximate the saturable absorber transmissivity function by its weak field Taylor expansion. When the pulse peak power reaches higher values, on the other hand, the transmissivity function saturates, and often even decreases, as shown in figure 6. The single pulse configuration then becomes unstable, and the pulse breaks into two or more pulses.

In SLD the multipulse mode locking acquires the fascinating significance of a cascade of first order phase transitions, where each phase is labeled by the number of pulses [9]. Replacing the term a|y|²y in the master equation by s|y|²y with an appropriate transmissivity function s, the phenomenology of multipulse mode locking is satisfactorily described. Figure 7 shows the phase diagram, which specifies the number of pulses as a function of the intracavity power and noise strength, as measured experimentally, and as calculated by SLD. It follows from the SLD analysis that when the number of pulses is large, the power of each pulse approaches a constant value, again in accordance with the experimental observations. SLD can also be used to calculate the hysteresis curves that accompany the phase transitions [9].

Figure 7. Theoretical (left) and experimental multipulse mode locking phase diagrams. Each phase is labeled by the number of pulses, where zero pulses stand for the cw phase.

Critical phenomena in light

When the pressure of a liquid gas system is increased, the boiling point (coexistence temperature between liquid and vapor) typically increases, but at the same time the difference between the pressure of the liquid and the gas phases decreases. This trend continues until the liquid and the gas phases become indistinguishable at the critical point. The SLD analog of a pressure increase is the injection of an external pulse [10]. When the injected pulse has the correct parameters, it acts as a seed for the mode locking action, and lowers the threshold power. In fact, the ‘mode locking’ phase transition becomes then a transition between a weak pulse and a strong pulse. As in the liquid-gas transition, stronger injection implies a smaller jump in the pulse power, until the two pulses become indistinguishable at the critical point.

The experimental realization of this idea was a difficult task, because of the high degree of compatibility required between the external and internal pulses. This was finally achieved by using the internal pulse itself as source for generating the external pulse [10]. The experimental graph of the pulse power as a function of total power and seeding power shown in figure 8 exhibits the familiar structure of the equation of state near a critical point, obtained for the first time in a light-mode system. An important diagnostic for the critical phenomena is the set of critical exponents a, b, and g, describing the singularities of the pulse power as a function of the seeding on and off the coexistence curve, and the divergence of the response coefficient as the critical point is approached (respectively). As shown in figure 9, the exponents were measured and within small experimental errors the classical values b = 1/2, g = 1, and d = 3 were obtained. This is in perfect agreement with the prediction of the mean field theory that as explained above, has been shown to be exact in SLD.

Figure 8. Theoretical (left) and experimental surface graphs of the pulse magnitude as a function of the external seeding and total cavity power. The discontinuity occurs at the coexistence line that terminates at the critical point.

Figure 9. Experimental measurement of the critical exponents. T stands for the ‘tempearture’, h the external seeding, x the pulse magnitude, and the subscript c labels the values at the critical point.

Conclusions and outlook

The key idea of SLD is that weak cavity noise can become important because, as it is coupled to a very large number of modes, it carries significant entropy. The complex mode dynamics of the laser has to be therefore studied statistically. This realization has led to identification of several familiar thermodynamic phenomena, like phase transitions and barrier crossing in standard mode locked laser systems, and to the introduction of new concepts like critical phenomena into this field. Similar ideas stand behind the application of glass physics to the theory of random lasers [11].

On the other hand, noise-induced fluctuations have long been studied, usually in an attempt to reduce them. Common examples are jitter - the timing fluctuations of pulses, and the line width and stability of frequency combs. These phenomena were studied theoretically using perturbation theory, as is reasonable since the noise is weak.

However, full account of noise must include entropy nonperturbatively. The future inclusion of entropy in the study of fluctuations will improve the understanding of this important subject.

References

[1] H. A. Haus, “Mode-Locking of Lasers”, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000)

[2] T. R. Schibli et al., “Toward single-cycle laser systems”, IEEE J. Sel. Top. Quantum Electron. 9, 990 (2003)

[3] H. E. Stanley Introduction to Phase Transitions and Critical Phenomena, (Oxford university press, 1987)

[4] H. A. Haus, “Theory of mode locking with a fast saturable absorber”, J. Appl. Phys., 46, 3049 (1975).

[5] H. A. Haus and A. Mecozzi,“Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29 983 (1993).

[6] A. Gordon and B. Fischer, “Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers”, Phys. Rev. Lett. 89, 103901, (2002);

Gat O, Gordon A, and Fischer B, “Light-mode locking: a new class of solvable statistical physics systems”, New J. Phys. 7, 151 (2005); M.

Katz, A. Gordon, O. Gat, and B. Fischer, ‘Statistical theory of passive mode locking with general dispersion and Kerr effect,’ Phys. Rev. Lett., 97, 113902 (2006).

[7] Vodonos B, Bekker A, Smulakovsky V, Gordon A, Berger N, Gat O, and Fischer B, “Experimental study of the stochastic process of the pulsation self-starting in passive mode- locking’, Opt. Lett. 30, 2787–2789 (2005)”; A. Gordon, O. Gat, B. Fischer, and F. X. K.artner, “Self-starting of passive mode locking,” Opt. Express, 14, 11142 (2006).

[8] N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton Solutions of the Complex Ginzburg-Landau Equation”, Phys. Rev. Lett. 79, 4047 (1997).

[9] Vodonos B, Weill R, Gordon A, Bekker A, Smulakovsky V, Gat O and Fischer B, “Formation and annihilation of laser light pulse quanta in thermodynamic-like pathway”, Phys. Rev. Lett. 93,153901 (2004); R Weill, B Vodonos,

A Gordon, O Gat, and B Fischer, “Statistical light-mode dynamics of multipulse passive mode locking,” Phys. Rev. E, 76, 031112 (2007).

[10] Weill R, Rosen A, Gordon A, Gat O and Fischer B, “Critical Behavior of Light”, Phys. Rev. Lett., 95, 013903 (2005).

[11] L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, “ Glassy Behavior of Light”, Phys. Rev. Lett. 96, 065702 (2006).