Advances in Mode-Locked Semiconductor Lasers

CHAPTER

3 Advances in Mode-Locked Semiconductor Lasers E. A. Avrutin* and E. U. Rafailov†

Contents

1. Introduction 2. Mode-Locking Techniques in Laser Diodes: The Main Features 3. Mode-Locking Theory: Recent Progress and the State of the Art 3.1. Self-consistent pulse profile analysis 3.2. Traveling-wave ML models 3.3. Frequency-domain analysis of ML 3.4. Delay-differential model of ML 4. The Main Predictions of Mode-Locked Laser Theory 4.1. Operating regime depending on the operating point 4.2. The main parameters that affect mode-locked laser behaviour 5. Important Tendencies in Optimizing the Mode-Locked Laser Performance 5.1. Achieving a high gain-to-absorber saturation energy ratio 5.2. Improving stability and pulse duration by reducing the SA recovery time 5.3. Increasing the optical power: Broadening the effective modal cross section 5.4. Engineering the bit rate. High power and high bit rate operation. Harmonic ML 5.5. Noise considerations in ML operation

94 96 98 99 100 101 102 112 112 114 118 118 120 121 125 128

* University of York, York, UK {

University of Dundee, Dundee, UK

Semiconductors and Semimetals, Volume 86 ISSN 0080-8784, DOI: 10.1016/B978-0-12-391066-0.00003-4

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2012 Elsevier Inc. All rights reserved.

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6. Novel Mode-Locking Principles 6.1. QD materials 6.2. Femtosecond pulse generation by mode-locked vertical cavity lasers. Coherent population effects as possible saturable absorption mechanism 6.3. Spontaneous ML in single-section lasers 6.4. Minitaturization and integration: Ring and microring resonator cavities 7. Overview of Possible Applications of Mode-Locked Lasers 7.1. Optical and optically assisted communications 7.2. Biophotonics and medical applications 8. Concluding Remarks References

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1. INTRODUCTION In the most general sense, mode locking (ML) is a regime of laser generation whereby the laser emits light in several modes with a constant phase relation between them, that is, with constant and precisely equidistant frequencies. Usually, the term is used more specifically, referring to what is, rigorously speaking, amplitude-modulation (AM) ML, meaning that the relative phases of the (longitudinal) modes may be considered approximately equal. In time domain, this corresponds to the laser’s emitting a train of ultrashort (shorter than the round-trip) optical pulses (see, e.g., Siegman, 1986), at a repetition frequency F near the cavity round-trip frequency or its harmonic: F
Mh vg =2L

(3.1)

vg being the group velocity of light in the laser, L the cavity length, and Mh the harmonic number, or the number of pulses coexisting in the cavity; in the simplest and most usual case, Mh ¼1. In most cases (some important exceptions will be mentioned in Section 6.3), ML does not occur spontaneously and requires a specialist laser construction and/or operating conditions. Specifically, it is usually achieved either by modulation of the laser net gain at a frequency F (Eq. 3.1), known as active ML, or by exploiting nonlinear properties of the medium to shorten the propagating pulse, countering the broadening effects of gain saturation and dispersion; this is known as passive ML. Passive ML, in turn, is usually achieved by introducing a saturable absorber (SA) into the laser cavity. The SA both facilitates a self-starting mechanism for ML and, most importantly, plays a crucial role in

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shortening the duration of the circulating pulses. More recently, saturable refractive index nonlinearities approximately equivalent in their action to saturable absorption have been intensely studied; salient examples are additive pulse ML and Kerr lens ML in solid-state lasers. A combination of active and passive methods of ML is known as hybrid ML; if the external modulation is in the form of short pulses, the corresponding regime is referred to as synchronous ML. Semiconductor lasers (at least in a monolithic configuration) cannot yet directly generate the sub-100fs pulses routinely available from diodepumped crystal-based lasers (Brown et al., 2004; Ell et al., 2001; Innerhofer et al., 2003), but they represent the most compact and efficient sources of picosecond and sub-picosecond pulses. They are directly electrically pumped, and the bias current can be easily adjusted to determine the pulse duration and the optical power, thus offering, to some extent, electrical control of the characteristics of the output pulses. These lasers also offer the best option for the generation of high-repetition rate trains of pulses, owing to their small cavity size L in Eq. (3.1) and hence the large (giga- to terahertz range) values of F. Ultrafast diode lasers have thus been favored over other laser sources for high-frequency applications such as optical data/telecoms. Being much cheaper to fabricate and operate, ultrafast semiconductor lasers also offer the potential for dramatic cost savings in a number of applications that traditionally use solid-state lasers. The deployment of high-performance ultrafast diode lasers could therefore have a significant economic impact, by enabling ultrafast applications to become more profitable, and even facilitate the emergence of new applications. The basic physical mechanisms underlying the generation of short pulses from diode lasers are fundamentally similar to those of other types of lasers, but a number of features are very different. Semiconductors have both a higher gain per unit length and a higher nonlinear refractive index than other gain media. The interaction of the pulse with the gain and the resulting large changes in the nonlinear refractive index lead to significant self-phase modulation, imparting a noticeable chirp to the ML pulses, usually up-chirp in the case of passively mode-locked lasers, which combined with the positive dispersion of the gain material, leads to substantial pulse broadening. This mechanism has been among the limitations in obtaining pulse durations of the order of 100fs directly from the diode lasers, with picosecond pulses being the norm. Furthermore, a strong saturation of the gain also results in stabilization of the pulse energy, which limits the average and peak power to much lower levels than in vibronic lasers. Average output power levels for modelocked laser diodes are usually between 0.1 and 100mW, while peak power levels remain between 10mW and 1W. Only with additional amplification/compression setups, can the peak power reach the kW

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level (Kim et al., 2005). In addition, the typical scales of carrier relaxation times in semiconductor materials are of the order of hundreds of picoseconds, comparable to the ML repetition time, leading to a rich variety of dynamic instabilities in the laser behavior (see, e.g., Avrutin et al., 2000 for an overview). This combination of practical promise and scientific challenge has made ML in semiconductor lasers an important topic of research for more than two decades. From the late 1980s, monolithic and multi-gigahertz constructions, reviewed in an earlier paper (Avrutin et al., 2000), became a research priority. The most recent years have seen considerable progress in both improving the theoretical understanding of ML in semiconductor lasers, and using this understanding to improve their performance in terms of power, pulse duration/chirp, stability, repetition rates accessible, and integrability issues. To this end, new constructions as well as new materials have been proposed. Here, we attempt to summarize both the current theoretical understanding of ML in semiconductor lasers, including the recent theoretical progress (Sections 2–4), and the use of this understanding for improving the laser performance (Sections 4 and 5). The use of more radically novel ML principles is also briefly discussed (Section 6).

2. MODE-LOCKING TECHNIQUES IN LASER DIODES: THE MAIN FEATURES To realize the advantages of mode-locked laser diodes to the full, a variety of ML techniques and device structures have been investigated and optimized (Vasil’ev, 1995). All three main forms of ML – active, passive, and hybrid – have been extensively studied for semiconductor lasers. Purely active ML in a semiconductor laser can be achieved by direct modulation of the gain section current with a frequency very close to the pulse repetition frequency in the cavity, or to a subharmonic of this frequency. Alternatively, an electroabsorption segment (as in Fig. 3.1) of a multi-element device can be modulated to produce the same effect, or a separate modulation section introduced. The main advantages of active ML techniques are the resultant low jitter and the ability to synchronize the laser output with the modulating electrical signal, which is a fundamental attribute for optical transmission and signal-processing applications. However, repetition frequencies in excess of several tens of GHz are not readily obtained through direct modulation of lasers because fast RF modulation, particularly of current, becomes progressively more difficult with increase in frequency. Passive ML of semiconductor lasers typically utilizes a SA region in the laser diode. Upon start-up of laser emission, the laser modes initially oscillate with relative phases that are random; in other words, the

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Reverse bias

Forward bias

er

orb

e

rat

bst

L abs

Su n

L gai L total

FIGURE 3.1 A schematic diagram of a monolithic two-section laser.

radiation pattern consists of irregular bursts. If one of these bursts is energetic enough to provide an fluence of the order of the saturation fluence of the absorber, it will partly bleach the absorption. This means that around the peak of the burst where the intensity is higher, the loss will be smaller, while the low-intensity wings become more attenuated. The pulse generation process is thus initiated by this family of intensity spikes that experience lower losses within the absorber carrier lifetime. The dynamics of absorption and gain play a crucial role in pulse shaping. In steady state, the unsaturated losses are higher than the gain. When the leading edge of the pulse reaches the absorber, the loss saturates more quickly than the gain, which results in a net gain window, as depicted in Fig. 3.2. The absorber then recovers from this state of saturation to the initial state of high loss, thus attenuating the trailing edge of the pulse. It is thus easy to understand why the saturation fluence and the recovery time of the absorber are of primary importance in the formation of modelocked pulses. In practical terms, the SA can be monolithically integrated into a semiconductor laser, by electrically isolating one section of the device (Fig. 3.1). By applying a reverse bias to this section, the carriers photogenerated by the pulses can be more efficiently swept out of the absorber, thus enabling the SA to recover more quickly to its initial state of high loss. An increase in the reverse bias serves to decrease the absorber recovery time, which will have the effect of further shortening the pulses.

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A Loss and gain

Loss Net gain

B

Gain Time

Intensity

Time

FIGURE 3.2 The mechanism of passive mode locking: (A) the loss and gain dynamics that lead to (B) pulse generation.

Alternatively, a SA can also be implemented through ion implantation on one of the facets of the laser, thus increasing the nonradiative recombination (Zarrabi et al., 1991; Deryagin et al., 1994; Delpon et al., 1998). Passive ML provides the shortest pulses achievable by all three techniques and the absence of a RF source simplifies the fabrication and operation considerably (albeit at the expense of somewhat larger pulse jitter – typically 0.1–1ps rather than tens of fs – and RF linewidth than in active or hybrid ML). It also allows for higher pulse repetition rates than those determined solely by the cavity length, by means of harmonic ML (Mh >1 in Eq. (3.1); the means of achieving this will be reviewed in more detail in Section 5.4). Hybrid ML is usually achieved by applying RF modulation to the SA section, which in this case doubles as an electroabsorption modulator (RF modulation of the gain section current is also possible but has been proven to be less efficient). In this case, the pulse generation may be seen as initiated by a modulation provided by the RF signal, while further shaping and shortening is assisted by the SA. This process results in highquality pulses, synchronized with an external source.

3. MODE-LOCKING THEORY: RECENT PROGRESS AND THE STATE OF THE ART As mentioned above, ML in lasers is a result of the balance of several physical processes in semiconductors (nonlinear gain/saturable absorption, modulation, dispersion). It is also influenced by others such as self-phase

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modulation, is noise driven, and is prone to dynamic instabilities. This complex picture makes ML theory and modeling very important for qualitative understanding, and particularly for quantitative analysis and optimization, of the laser behavior. Theoretical analysis of ML in semiconductor lasers has indeed been, and continues to be, an important research subject. Until recently, there were three main types of mode-locked laser models.

3.1. Self-consistent pulse profile analysis Conceptually the simplest, and historically the oldest, models of modelocked lasers are time-domain lumped models. Such models treat a hypothetical ring laser geometry with unidirectional propagation, and assume that the pulsewidth is much smaller than the repetition period. The amplification and gain dispersion/group velocity dispersion (GVD), which in reality are experienced by the pulse simultaneously, may then be approximately treated in two independent stages. This allows the representation of the distributed amplifier by a lumped gain element performing the functions of amplification and self-phase-modulation. Mathematically, this element can be described by a nonlinear integral or ^ acting on the complex pulse shape function integrodifferential operator G (complex slow amplitude) Y(t), t being the local time of the pulse. Separate time scales are introduced explicitly for the pulse (the short time scale) and relaxation period between pulses (the long time scale). A similar ^ is introduced to describe element, described by a nonlinear operator Q, the saturable absorption, and the dispersion (gain/absorption and group ^ Assumvelocity) is introduced as a separate, linear dispersion operator D. ing small gain and saturable absorption per pass (inherited from the nonsemiconductor laser theory, for which the model was originally developed) and neglecting dispersion, the model yields transcendental equations for the case of steady ML pulse stream. The equations allow for quasi-analytical calculation of ML boundaries, and also of the pulse energy (but not duration and peak power, which cannot be calculated without accounting for dispersion). This is known as New’s model of ML (though G. New’s original paper (New, 1974) related to non-semiconductor lasers). A different approach is the one in which dispersion is taken into account, but, in addition to the approximation of the small gain and absorption per pass, also the gain and absorption saturation are taken as ^ Q, ^ and D ^ small. Then, expanding the integral and differential operators G, in series and requiring that both the amplitude and the shape of the pulse is conserved from one repetition period to the next, it is possible to derive a single integrodifferential equation for the pulse profile Y(t; on the short time scale). The equation, known as the master equation of ML, admits an analytical solution of the form

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 t 1þib YðtÞ ¼ Y0 expðiDotÞ cosh tp

(3.2)

known as the self-consistent profile (SCP). The corresponding theoretical approach is known as the SCP, or Haus’s, ML theory; it is based on H. Haus in the work on ML in lasers of an arbitrary type (Haus, 1975) and was later extended to account for large self-phase modulation in diode lasers (Koumans and vanRoijen, 1996; Leegwater, 1996). Substituting the SCP in the ML equation, one obtains three complex, or six real, transcendental algebraic equations (Leegwater, 1996; Koumans and vanRoijen, 1996) for six real variables: pulse amplitude jY0j, duration measure tp, chirp parameter b, optical frequency shift Do¼oo0, repetition period detuning dT, and phase shift arg(Y0) (which is not a measurable parameter, so in reality there are five meaningful equations). These equations, being nonlinear and transcendental, generally speaking, cannot be solved analytically, but still allow for some insight into the interrelation of pulse parameters. The Haus model also allows the conditions for stable ML to be determined; as will be discussed later, these give different values from those obtained using New’s model. Haus’s theory of ML can also be extended to the case of active ML; building on this approach, an analytical theory of phase noise in actively mode-locked lasers has been developed (Rana et al., 2004) allowing for jitter calculations from first principles. Thus, lumped models, particularly the SCP model, allow for considerable analytical progress in analysis of ML, but quantitatively are very far from describing monolithic mode-locked semiconductor lasers accurately. Also, being essentially steady-state, they can show the limits of stability but not describe the dynamics of the unstable case.

3.2. Traveling-wave ML models At the other end of the spectrum of theoretical approaches to the ML laser properties are distributed time-domain, or traveling-wave, models, which treat the propagation of an optical pulse through a waveguide medium with space as well as time resolution. The model starts with decomposing the optical field in the laser into components propagating in the forward and reverse longitudinal direction (say, z) Yðr; tÞ ¼ Fðx; yÞðYf expðibref zÞ þ Yr expðibref zÞexpðioref tÞ

(3.3)

with F being the transverse/lateral waveguide mode profile and oref and bref ¼n(oref)kref ¼n(oref)oref/c being the reference optical frequency and the corresponding wavevector, respectively. This results in equations for slowly varying amplitudes Yf,r:

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@Yf;r 1 @Yf;r  þ ¼ @z vg @t




101

  1 ^g  aint þ ikref D^ nmod Yf;r þ iKrf;fr Yr;f þ Fspont ðz; tÞ 2 mod

(3.4) where ^ gmod is the modal gain, which in general is implemented as an operator to represent gain dispersion, aint, the internal parasitic loss, D^ nmod dynamic correction to the modal refractive index, including the GVD operator if needed. At z values within SA sections, the gain value g is naturally changed to a, a being the saturable absorption coefficient. The coupling coefficients Krf,fr allow for introduction of built-in distributed reflectors or dynamic gratings introduced by standing wave profiles, and the last term is the Langevin noise source that drives the model. The equations are directly solved numerically, in a system with spatially resolved rate equations for the carrier populations in the gain and SA sections which are used to calculate the gain and absorption dynamics. The latter is most often done using phenomenological relations, though an interesting recent development has been an introduction of an efficient, if somewhat simplified, miscroscopically based model of spectral–temporal dynamics of gain and absorption into a travelingwave simulator ( Javaloyes and Balle, 2010). Traveling-wave models are very powerful and general. They combine a large-signal approach with accurate account for spectral features. They have been extensively and successfully used by a number of research groups to analyze and design many edge-emitting laser constructions including mode-locked lasers. They also form the core of several commercial or shareware simulators, some of which have been applied to ML laser design (see Nikolaev et al., 2005 for an overview). The main limitations of traveling-wave models are, first, the absence of any analytically solvable cases, and, second, the relatively high requirements such models pose on the computing time and memory (Nikolaev et al., 2005). The approach is essentially numerical, which limits the physical insight gained from the models.

3.3. Frequency-domain analysis of ML An approach totally alternative (or complementary) to the time-domain analysis of ML is offered by the technique of modal analysis, either static or dynamic (as in Avrutin et al., 2003 and references therein and also in Nomura et al., 2002; Renaudier et al., 2007). In this approach, instead of analysing the pulse shape dynamics, a modal decomposition is used and the dynamics of mode amplitudes and phases followed, with the dynamics of gain and absorption providing the nonlinear coupling terms that

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ensure ML, given the right parameter values. The advantage of the modal expansion is that the time steps can be much longer (and the number of variables can be smaller) than in the traveling-wave model, which makes the modal approach particularly efficient in analyzing, say, long-scale dynamics of external locking of DBR hybridly mode-locked lasers. It also has the logical advantage of describing steady-state ML as a steady-state solution of the model. By using the limiting cases of either only three modes (the minimum number for ML) or a quasi-continuous supermode including a large number of modes, the frequency-domain analysis of ML allows for considerable analytical progress. This included an early but very thorough investigation of linewidth and jitter in both actively and passively mode-locked lasers (Kim and Lau, 1993). However, the frequency-domain approach has significant limitations too, chief of which is the inherent assumption of weak to modest nonlinearity (the model is not large-signal, which can be a significant limitation in treating edge-emitting mode-locked lasers). Thus, time–domain approaches are more widely used. A more detailed review of the modal approach to ML is presented in Avrutin et al. (2000).

3.4. Delay-differential model of ML Arguably the most significant development in ML theory during recent years has been the development of a model intermediate between the traditional lumped models of ML and the traveling-wave models (Vladimirov and Turaev, 2005; Vladimirov et al., 2004). This approach, known as the delay-differential equation (DDE) model or theory, contains a rigorous extension of Haus’s and New’s theories for the case of strong gain and absorption per pass (which is usually the case in semiconductor lasers). Like those traditional ML models, it can provide significant analytical insight, with parameter values more relevant to semiconductor lasers, but unlike those models, it allows also for detailed, self-starting, and very computationally efficient, large-signal numerical analysis. This combination of analytical insight and numerical potential, small- and large-signal possibilities, within a single framework – which is unique among time-domain ML models – makes the DDE model very attractive for analysis of performance trends (if not necessarily numerical parameters) in real lasers. Extensions of the model to specialized constructions such as quantum dot (QD) lasers have also been developed and used successfully (Viktorov et al., 2006); the reader is referred to a recent monograph (Rafailov et al., 2011) for details of QD ML laser theory. As the traditional ML theory, the DDE uses a model of unidirectional ring propagation, giving a gain operator in the form
   ^ ðtÞ ¼ exp 1 1  iaHg GðtÞ YðtÞ (3.5) GY 2

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with aÐHg the Henry linewidth enhancement factor in the amplifier and G (t)¼G g(z,t)dz the total gain integrated over the length of the amplifying region. The SA element is described by a similar element:
 1 ^ (3.6) Qs YðtÞ ¼ exp  ð1  iaHa ÞQs ðtÞ YðtÞ 2 with Qs the saturated absorption and aHa the absorber linewidth enhancement factor. There are, however, significant differences from the traditional models. The DDE model dispenses with the assumption of G, Qs 1 necessary in traditional ML models. It also does not, in its general form, introduce the two separate time scales, neglecting stimulated recombination between the pulses and nonstimulated recombination during them. Instead, ordinary differential equations are written for G, Qs, based on the spatially integrated carrier rate equations. Assuming that the pulse in the unidirectional cavity treated by the model passes the absorber before the amplifier, the equations take the form dGðtÞ PðtÞ G0  GðtÞ ¼ ½expðGðtÞÞ  1expðQðtÞÞ þ dt Ug tg

(3.7)

dQðtÞ Pð t Þ Q 0  Q ¼ ð1  expðQÞÞ þ dt Ua ta

(3.8)

Here, G0 is the unsaturated gain determined by the pumping conditions, Q0 is the unsaturated absorption, P(t)¼vgℏoAXjY(t)j2 is the optical power, tg and ta are the gain and SA recovery times, and Ug ¼

ℏoAXg ; sg

Ua ¼

ℏoAXa sa

(3.9)

are the saturation energies of the amplifier and the SA, with sg ¼dg/dN; sa ¼da/dN being the crosssections (the derivatives by the carrier density N) of the gain and saturable absorption and AXg and AXa the crosssections of the optical beam (mode) in the gain and SA sections. The second difference of DDE from traditional ML theories is in the treatment of the dynamics of ML. As the previous theories, it describes pulse propagation by cascading the operators. However, in previous theories, the dynamics described by an iterative procedure, pffiffiffi is essentially  ^Q ^D ^ Yi ðtÞ, where i is the number of the pulse kG setting Yiþ1 ðtÞ ¼ round-trip (determining the ‘‘slow’’ evolution of the ML pulse), the time t is on the fast time scale commensurate with the pulse duration, and the dimensionless parameter k<1 introduces the total (integrated) unsaturable intensity losses in the cavity, both disspative and outcoupling. The stationary ML equation can then be obtained by writing out the

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condition that the broadening and narrowing cancel each other, and the shape of the pulse is conserved from onep repetition to the next. ffiffiffi ^ ^ ^  period idc Mathematically, this is expressed as kGQD YðtÞ ¼ e Yðt þ dT Þ, where dT is the shift of the pulse, or detuning between the repetition period and the round-trip of the ‘‘cold’’ cavity (or its fraction in case of locking at harmonics of the fundamental frequency), and dc is the optical phase shift induced by the round trip. In between the pulses, on the slow time scale commensurate with the round-trip time, gain and SA are allowed to recover with their characteristic relaxation times, according to the rate equation for carrier density with S¼0. This allows one to calculate the values of gain and saturable absorption at the onset of the pulse, given the pulse energy and repetition period. In DDE, with the two different scales for pulse analysis, abandoned, in general, the iteration procedure is substituted by a delay one. Inpits  general form, this procedure may be written as ffiffiffi ^ most ^D ^ Yðt  TRT Þ, where TRT is again the round-trip of the cold YðtÞ ¼ kGQ cavity, and t is still the local time of the pulse. A particularly useful form of ^ is expanded as a this model, however, is obtained if the dispersion operator D differential one. The authors of Vladimirov and Turaev (2005) and Vladimirov et al. (2004) showed that element with for  a bandwidth-limiting   

^ T ðoÞ ¼ 1= 1  i o  op =g YT (i.e., neglecting a Lorenzian spectrum DY GVD), assuming, without much loss of generality, that the peak gain frequency op coincides with one of the laser resonator modes, and taking it as the reference optical frequency, we can rewrite the equation for Y as YðtÞ ¼ g1

@YðtÞ pffiffiffi ^ ^  þ kGQ Yðt  TRT Þ @t

(3.10)

Equation (3.10), which is a delay-differential one, gives the DDE model its name. Note that the original papers (Vladimirov and Turaev, 2005; Vladimirov et al., 2004) present a slightly more general form of this equation, without assumptions on the peak and reference frequencies. Equations (3.7), (3.8), and (3.10) are a closed system suitable for a detailed numerical simulation of both stationary and dynamic behavior of passive ML. They can also be fairly easily adapted to allow numerical analysis of hybrid ML behavior. As shown in Vladimirov and Turaev (2005), the DDE model also allows for significant analytical progress, similar to one achieved with classical New’s and Haus’s models, but for large single-pass gain and absorption, more relevant for most semiconductor laser constructions than the classical SCP. In the analytical procedure, the slow absorption and gain approximation (neglect of nonstimulated recombination and absorption recovery during the pulse) has to be reintroduced, and the slow (relaxation of gain and absorption between pulses) and fast (evolution during the pulse) stages of laser dynamics are, as in the traditional ML models, treated separately.

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In particular, it is possible to analyze the stability of the solutions by requiring that net gains both immediately before the pulse and immediately after the pulse are smaller than one: G  Q þ ln k < 0 Gþ  Qþ þ ln k < 0

(3.11)

Here G, Q are the total (integrated) gain and absorption immediately before the pulse, and Gþ, Qþ are the values immediately after the pulse. This is a generalization of the condition proposed in traditional ML theory (New, 1974) for the case ln(1/k)1, to the case of an arbitrary k. Further analytical progress is possible in two limiting cases. The first is the case of a model without spectral filtering (g!1), which the authors of (Vladimirov and Turaev, 2005) identified as the generalized New’s model for the arbitrarily large gain and absorption per pass. This theory, like the traditional New’s theory, cannot predict the pulse shape, peak power, or duration. The total pulse energy Up, however, can be estimated approximately from a closed system of five (nonlinear and transcendental) equations for the five unknowns: G, Q, and Up. The solution gives the dependence of pulse energy on the operating point, represented by the unsaturated gain (which is related to pumping current) and absorption (which is related to the reverse bias applied to the absorber). The other fundamental absorber parameter also dependent on the reverse bias, the absorber lifetime, only enters the calculations through the relaxation equation for the absorption between pulses, obtained from Eq. (3.8), and does not influence the results from this model at all if ta TRT (in which case, obviously, Q
Q0). The solution to this nonlinear algebraic equation system can then be substituted into the inequalities (Eq. 3.11) to analyze the stability boundaries of the ML operating range with respect to the leading-edge and tralining-edge instability. The curves, in general, can only be calculated numerically, though some special points can be determined analytically (Vladimirov and Turaev, 2005). In particular, the model confirms the conclusions from the traditional ML analysis regarding the importance of the gain-to-absorber saturation energy ratio s¼Ug/ Ua ¼saAXg/sgAXa. Namely, it shows that the condition s>1 for any range of successful ML to be present, derived in the traditional SCP approach for the case of small gain and loss per period (and thus small dissipative loss), needs to be generalized in the case of arbitrary losses in the cavity as, sk > 1

(3.12)

If treating a more realistic construction, an extra geometric factor could also be required.

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A second case when a full (semi-) analytical solution of the DDE model (with relaxation terms during the pulse neglected) is possible is when the dispersion is taken into account, but the saturation of gain and absorption during the pulse is assumed to be small, as in the Haus model of ML (though the gain and absorption themselves are not necessarily small, unlike the case of the traditional Haus model). The authors of (Vladimirov and Turaev, 2005) called this the generalized Haus model. In this case, a steady-state solution is sought, as in the standard Haus’s model; in our notations, this means Y(tþTRT)¼eidcY(tdT). Then, from Eq. (3.10), @Yðt  dT Þ þ Yðt  dT Þ ¼ FðuðtÞÞYðtÞ (3.13) @t Ðt Ð where u(t)¼1/Ug 1P(t0 )dt0 ¼(vgℏoAX/Ug) t1jY(t0 )j2dt0 is the normalized pulse energy up to point t, and       pffiffiffi (3.14) FðuÞ ¼ kexp GðuÞ 1  iaHg  QðuÞ 1  iaHq  idc g1

is the ‘‘complex net gain,’’ which can be written explicitly, with G(u) and Q(u) obtained from transcendental equations derived (Vladimirov and Turaev, 2005) by integrating Eqs. (3.7) and (3.8) with relaxation terms omitted during the pulse. Next, assuming that the single-pass pulse shift is significantly smaller than the pulse duration and that the saturation of both the gain and absorption during the pulse is weak enough (u(t)
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0.0

-0.5 LN

-q0

-1.0

-1.5 LH -2.0

TN TH

-2.5 0.0

TH

LH TN 0.5

LN 1.0 g0

1.5

2.0

FIGURE 3.3 Stability boundaries of ML with respect to leading (L) and trailing (T) edge instabilities, calculated semi-analytically in traditional (dashed lines) and generalized (solid lines) New’s (subscript N) and Haus’s (subscript H) models. Fully numerical calculation results are shown as dots. In the calculations, s¼25, TRT/ta ¼1.875, ta/tg ¼0.0133, k¼0.1. After Vladimirov and Turaev (2005), reproduced with permission.

procedure. As seen in the figure, standard Haus’s and New’s models disagree widely with numerical simulations in predicting the instability boundaries of ML in a typical diode laser (with the range predicted by New’s model being too wide, and that from Haus’s model, too narrow, as noted also in Dubbeldam et al., 1997). The generalized Haus’s model gives good agreement within its validity limits at low currents/unsaturated gain values, while the generalized New’s model gives very good agreement with numerical simulations at all parameter values (there are some modest deviations, which will be discussed in more detail below). Thus, the large-signal nature of the DDE model is shown to be a very important advantage over the classical ML theories. Apart from allowing some analytical progress in the limiting cases, the DDE model also allows the use of numerical techniques that have been developed for the analysis of DDEs, in particular of numerical packages that allow a full bifurcation analysis of DDEs (Vladimirov and Turaev, 2005). Such a study was indeed performed in Vladimirov and Turaev (2005), comprising the full (in)stability analysis of the stationary solution of the DDE. The stationary solution (the steady-state light– current characteristic of the laser) itself is found by seeking the steadystate light output in the form of Y(t)¼Y0sexp(iDost), which gives an equation for the steady state in a parameteric form. The equation is a

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0 H1

-q0

-1

Stable CW H2

H3

-2

Hq

-3

H4

-4

0

1

2 g0

3

4

FIGURE 3.4 Bifurcation analysis of the steady-state solutions of the DDE model. Parameters used: gta ¼33.3; aHg, a¼0, the rest as in Fig. 3.3. After Vladimirov and Turaev (2005), reproduced with permission.

transcendental trigonometrical one and thus has an infinite set of formal solutions, corresponding to the cavity modes. The steady-state solution, as usual in laser theory, is the one with the lowest value of the threshold gain Gs (Y¼0), in other words, the closest to the peak of the gain spectrum. Figure 3.4, after Vladimirov and Turaev (2005), shows the results of a numerical bifurcation analysis of this solution. The line H1 indicates the Andronov-Hopf bifurcation (transition from a steady state to a periodically oscillating solution with an amplitude smoothly increasing from zero as the controlling parameter, e.g., the unsaturated gain in this case, increases beyond a critical value), corresponding to oscillations at the fundamental ML frequency. ML is predicted at a certain range of conditions (unsaturated gain and absorption) above threshold, whereas at high enough unsaturated gain (or current) and low enough absorption, CW lasing is expected to be stable. The line Hq indicates the Andronov-Hopf bifurcation corresponding to passive Q-switching, or self-sustained-pulsation, instability, which essentially means the well-known relaxation oscillations in the laser being turned from decaying to self-sustained pulsations (SP) by the positive feedback provided by the SA. The frequency of these oscillations is determined mainly by the unsaturated gain, gain cross section, gain relaxation time, and losses in the cavity, and is typically of the order of 1GHz, or about an order of magnitude below the ML

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6

5

|A|2

4

3

2

1

0

0

1

2

3 g0

4

5

6

FIGURE 3.5 Bifurcation diagram obtained by direct numerical implementation of a DDE model. Q0 ¼4, the other parameters as in Fig. 3.3. After Vladimirov and Turaev (2005), reproduced with permission.

frequency. Thus, at low currents and with a high enough saturable absorption in the cavity, the ML pulse train is expected to be modulated by the self-pulsing envelope. The lines Hm, m>1, show the bifurcations corresponding to a solution oscillating at the mth harmonic of the fundamental ML frequency. At high enough values of unsaturated absorption, there are ranges of G0 (or current) in which ML at higher harmonics is predicted to be stable, but ML at fundamental frequency is not. These predictions are confirmed by a full numerical integration of the DDE model (Fig. 3.5), showing the extrema of the laser intensity time dependence calculated for different values of the pumping parameter g0 ¼ (ta/tg)G0. For each unsaturated gain, the initial transient is omitted before the start of registering signals. At low values of g0 (and thus current), the laser exhibits a regime when the ML pulse power is modulated by passive Q-switching envelope, originally with nearly 100% modulation depth (Fig. 3.6A). As the pumping parameter increases, the Q-switching modulation gradually decreases in amplitude, and eventually the modulation regime undergoes the backward bifurcation, moving to a stable ML regime (this corresponds to the border of the trailing-edge instability in Fig. 3.3). Within the area of stable ML, the fundamental round-trip frequency, a train of short pulses is observed as in Fig. 3.7A, with amplitudes increasing with G0. At higher still pumping, the laser dynamics sees areas

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3

A

|A|2

2 1 0

B

|A|2

2 1 0

0

10

20

30

40

50

60

70

t

FIGURE 3.6 Illustration of the aperiodic regimes in Fig. 3.5: combined mode-locking/ Q-switching regime at G0 ¼50 (A) and chaotic pulse competition regime at G0 ¼350 (B). After Vladimirov and Turaev (2005), reproduced with permission.

4

A

|A|2

3 2 1 0 B |A|2

3 2 1 0 C |A|2

3 2 1 0

0

1

2

3 t

4

5

6

7

FIGURE 3.7 Illustration of the periodic regimes in Fig. 3.5: fundamental frequency mode locking at G0 ¼150 (A) and first and second harmonic ML G0 ¼225 (B) and 270 (C). After Vladimirov and Turaev (2005), reproduced with permission.

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of harmonic ML at the second and third harmonic of the fundamental ML frequency Fig. 3.7B and C, separated by narrow areas of unstable operation. Finally, the ML breaks up completely with the onset of chaotic modulation of the pulse power, with multiple pulse trains competing in the cavity, as in Fig. 3.6B (the regimes separating fundamental frequency ML and harmonic ML areas are similar). Eventually, the system undergoes a transition to CW single-frequency operation in agreement with the bifurcation diagrams of Fig. 3.3. An interesting result obtained in Vladimirov and Turaev (2005) is that, while the conditions (Eq. 3.11) of negative net gain before and after the pulse are useful indications of the stability ranges of mode-locked operation, the onset of instabilities in numerical simulations does not coincide with those limits exactly. This may be caused in part by the omission of gain dispersion in the analytical study and in part by the neglect of absorber relaxation during the pulse (though the pulses simulated in Vladimirov and Turaev (2005) were about an order of magnitude shorter than TRT, and several times shorter than ta, which was about ½TRT, so this was probably not a very significant factor). However, there is also a genuine physical reason for the discrepancy, in that not all small fluctuations in an ML laser were found to grow into full-scale instabilities even if a window of positive gain preceded the ML pulse. Instead, stable ML operation was shown to be possible in a range of parameter (unsaturated gain and absorption) values such that before the pulse, the fast absorption had recovered to its unsaturated value, but the slower gain continued recovery, leading to a window of positive net gain preceding the pulse (some previous studies using modifications of Haus’s model for semiconductor lasers indicated the possibility of positive net gain at the trailing edge of a stable ML pulse, see, e.g., Dubbeldam et al., 1997). The possibility of stable ML operation despite a positive net gain window is confirmed by more accurate traveling-wave simulations. One of the consequences of this effect is that the onset of instabilities may be expected to be sensitive to perturbations such as spontaneous noise. The effect of spontaneous emission was indeed studied analytically and numerically in Vladimirov and Turaev (2005), with the noise introduced as a delta-correlated random term in the right-hand-side of Eq. (3.10). It was concluded that, while the onset of Qswitching oscillations (trailing pulse edge instability) is a dynamic process independent of noise, the onset of the chaotic envelope instability (leadingedge instability) is strongly affected by the noise, with an increase in the noise narrowing the window of stable ML. This is fully confirmed by the more complex traveling-wave models described in Section 3.2. The DDE model when used as a numerical tool is not only fully largesignal but also self-starting: it does not, unlike previous lumped timedomain theories, require a trial pulse to start with and can reproduce the emergence of ML pulse train from randomly pulsing light output that is seen as the laser crosses the threshold condition.

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There remain, however, some limitations to linking the DDE approach directly to the performance of a specific practical laser construction. First, the model as studied in Vladimirov and Turaev (2005) does not account for fast absorber saturation and fast gain nonlinearity (though it could be possible to include them, at least in some approximation, and in the QD case, the explicit introduction of fast nonlinearities may not be necessary; instead, separate rate equations for dot and reservoir populations are used in this case). Second, the spatial integration of gain and absorption in DDE is mathematically justified only if both g and a have a simple linear dependence on the carrier densities in the corresponding elements, which is in itself an approximation, or alternatively if G, Q1. Finally, like traditional ML theories, the DDE model studies an artificial unidirectional ring geometry. The latter two assumptions are relatively easily addressed in the case of ML in vertical external cavity surface-emitting lasers (VECSELs; see Section 5.3), which consist of an amplifying (gain) chip and a SESAM (semiconductor saturable absorber mirror) chip, separated by an unguided propagation path (possibly with collimating optics). As both the gain chip and SESAM are very short asymmetric resonators, the lumped-element formalism is a very natural one for their description. Indeed, a special version of DDE, derived independently and presented in a somewhat different form than one of Vladimirov and Turaev (2005), was successfully used to analyze the dynamics of external-cavity VECSELs, with the predicted pulse duration and stability ranges matching the experimentally observed ones not only qualitatively but with a reasonable numerical agreement (Mulet and Balle, 2005). More advanced constructions known as MIXCELs (see Section 5.3), with the QW gain and QD absorber layers located in one chip, could be described by a similar, possibly even somewhat simpler, model, with the single-chip reflectance operator containing the effects of both the gain and the absorption. In the case of edge-emitting lasers, the accuracy of a DDE model is more suspect, and travelingpwave analysis looks more suitable for quantitative analysis of specific laser structures.

4. THE MAIN PREDICTIONS OF MODE-LOCKED LASER THEORY 4.1. Operating regime depending on the operating point The most basic result of all the modern ML theories, confirmed by the experiments, is that the dynamics of notionally mode-locked semiconductor lasers are quite rich and can show, apart from stable ML, a number of other dynamic regimes. Here, we shall briefly discuss the general trends in their dependence on the laser parameters.

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One of the most important features in the dynamic map of operating regimes of a mode-locked laser is the SP, or passive Q-switching, instability at low currents. As shown in Fig. 3.5, produced by the DDE model, the range of current, or unsaturated gain, values in which this regime is observed increases with the amount of saturable absorption in the laser (which, in a given laser construction, either QW or QD, may vary to some extent with reverse bias, due to electroabsorption). The other model parameter affected by the reverse bias is the absorber lifetime ta, which is known to decrease approximately exponentially with the reverse bias in QW materials (see Nikolaev and Avrutin, 2003 and references therein) and to some extent in QDs too (Malins et al., 2006). The dependence of the Q-switching range on ta is not straightforward; the Q-switching range tends to be broadest at a certain absorber recovery time, of the order of the round-trip time, though somewhat longer. At longer ta, the SP range slowly decreases; however, it also decreases as ta is decreased and at ta of a fraction of the round-trip, Qswitching instability can be expected to disappear, leaving a broad area of stable ML. This is illustrated by Fig. 3.8 produced using a traveling-wave model and showing approximate borders of different dynamic regimes for a representative Fabry-Perot laser operating at 40GHz. This means, first, that care needs to be taken when interpreting the bias voltage effects on the performance of either QW or QD mode-locked lasers, as the unsaturated absorption, the saturable absorption cross-section, and the SA recovery time ta are all likely to be affected by voltage variation. The effect on the latter is probably the most significant though, since the dependence of 6

Current limits, I/Ith

5

Mode-locking

Incomplete/ unstable ML + "CW"

4 3 L

2

SP

+M

Self-pulsing

1 10 100 Absorber relaxation time (ps)

FIGURE 3.8 Schematic diagram of regimes in a generic QW mode-locked laser operating at the repetition rate of 40GHz.

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ta on voltage is quite strong (exponential), while the effect on the unsaturated absorption appears, from measured threshold currents, to be more modest. Second, it means that for the same absorber parameters, longer lasers with longer repetition periods are less likely to suffer from the Q-switching instability, which needs to be borne in mind when analyzing the dynamics of QD lasers (due to the relatively low gain, these often have to be quite long if stable operation at the ground level wavelength band is desired). The lower current (or unsaturated gain) limit of the self-pulsing instability may be positioned either below or above the low boundary of ML itself, depending on the gain and absorber saturation energies (s-parameter) and the absorber recovery time. If the boundary for ML is below that for self pulsing (which tends to happen in long lasers, when ta is significantly smaller than TRT but not small enough to completely eliminate self pulsing), then the stable ML range is split in two by the self-pulsing area, with an area of stable ML seen below the Q-switching limit at currents just above threshold. The area is narrow, however, and the pulse powers generated in this regime are typically rather low. If, on the other hand, the boundary for ML is above that for self pulsing (which tends to be the case for shorter lasers or longer absorber relaxation time, when TRT
4.2. The main parameters that affect mode-locked laser behaviour To understand the rationale behind the recent advances in mode-locked laser science and technology, it is instructive to briefly overview the effects of the main laser parameters affecting ML in semiconductor lasers in general, as predicted by the models discussed above and confirmed by experiments.

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The effects of the operating point (gain section bias and SA voltage, determining the unsaturated gain and absorption and the SA time) on the operating regime of the laser have been discussed in the previous section. Furthermore, within the stable ML range, the effect of the operating point on the pulse parameters is as follows. The pulse amplitude grows with pumping current, as illustrated by Fig. 3.9A, calculated by a traveling-wave simulation. Notice that qualitatively, the figure is quite close to the bifurcation behavior seen from A 35

Pulse amplitude (r.u.)

30 25 20 15

Self pulsing/ ML with self pulsing envelope

Stable mode locking

Unstable / chaotic tSA=15 ps

ML with period doubling

10 5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Current I/Ith

Pulse duration (ps)

B 4.0

3.5

3.0

Stable mode locking

2.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Current I/Ith

FIGURE 3.9 Simulated dependence of the pulse amplitude (A) and duration (B) on the pumping current – simulations using a traveling-wave model with typical values for a QW laser. Solid line: obtained by slowly ramping the current; filled circles: time-averaged steady state results for a fixed current.

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Fig. 3.5, except that in the numerical model, and with the different set of parameters, only the second rather than third harmonic operation is predicted. As regards the pulse duration, it has been predicted by early frequency-domain theories (Lau and Paslaski, 1991) to reach a minimum near the area of Q-switching instability; this has been later confirmed by both experiments and time-domain simulations. The pulse duration thus tends to grow with current within the stable ML range above the upper boundary of Q-switching (as seen, e.g., Fig. 3.9B). On the other hand, if an area of stable ML below the Q-switching range is observed (which is sometimes the case in longer resonators), then a decrease of pulse duration with current can be expected within this area. The dependence of the pulse duration on ta (and thus on the absorber bias) within the stability range is shown in Fig. 3.10. As seen in the figure, to achieve stable ML, the absorber relaxation time needs to be below a certain critical value; longer ta produce instabilities. Within the stable ML range, a decrease in ta tends to shorten the pulses, due to both the effects of partial absorber relaxation during the pulse and, probably more significantly, the fact that the slow relaxation of the absorber leads to the absorber being always partially saturated, thus reducing the initial absorption Q. Among the other important parameters that affect the passive ML properties are (i) The s-factor, or the absorber to gain saturation energy ratio. While it is well known that the increase in the absorber to gain saturation energy ratio s facilitates ML, it is not immediately intuitively clear

5.5

Stable mode-locking

ML pulse duration (ps)

5.0 4.5 4.0

Leading edge instabilities

3.5 3.0 2.5 2.0

I/Ith = 5.2

1.5 0

10

20

30 40 50 60 70 80 Absorber recovery time (ps)

90 100

FIGURE 3.10 Typical simulated dependence of pulse duration on the absorber recovery time simulations using a traveling-wave model.

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whether an increased s helps ML stability, as it is known (Kuznetsov, 1985) that the passive Q-switching regime, which is one of the instabilities limiting the ML range, is also facilitated by an increase in s. However, the results from both DDE and travelling wave simulations (Bandelow et al., 2006) show that in fact it is the stable ML range that is increased with s at the expense of the Q-switching/self-pulsing range. (ii) Dispersion of gain and group velocity. Most models of mode-locked laser operation predict that without gain dispersion, stable ML with a finite pulse duration is impossible, and so the gain dispersion, or width of gain curve, represented by the parameter g in the DDE or oL in the traveling-wave model, should play an important role in determining the pulse width and stability. Within the range of gain dispersion typical in mode-locked semiconductor lasers, which usually corresponds to ℏoL of the order of tens of meV (or the wavelength range of tens of nanometers) and does not change too much with operating conditions or construction, gain dispersion is not the most drastic factor limiting the pulse width. However, achieving a broad gain spectrum is still desirable. This may be one of the advantages of QD active media, as discussed below. GVD, like gain dispersion, acts to broaden the pulses in the case of normal dispersion (which is usual in semiconductor lasers). As discussed above, the effect of this parameter is modest in most semiconductor lasers since the pulse durations at which it would become important (100fs) are never achieved; however, with stronger GVD possible in QD lasers, some account for this effect may be necessary. (iii) The gain suppression and absorber compression coefficients. Pulses generated by ML lasers tend to be of picosecond duration. This is below the critical pulsewidth at which the fast gain saturation, rather than the average carrier density dynamics, begins to dominate the pulse amplification and shaping, at least in the gain section; this critical pulse duration has been estimated (Mecozzi and Mork, 1997; Mork and Mecozzi, 1997) to be of the order of tcriteg/(vgsg), eg being the gain compression coefficient. With typical semiconductor parameters, this estimate gives values of the order of 10ps. Thus, the gain compression (and, similarly, absorber compression) effects and the coefficients that describe them (if introduced) may be expected to play a significant part in ML properties. In practice, the effect of nonlinearities is twofold. First, gain compression tends to broaden ML pulses, with absorption compression having the opposite effect. Second, and in some regards more importantly, an increase in gain compression stabilizes ML operation, suppressing the Qswitching instability (the latter can be easily shown by rate equation

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analysis of Q-switched lasers; Avrutin et al., 1991). Again, fast absorber saturation has the opposite effect. (iv) The gain and absorber linewidth enhancement factors. The linewidth enhancement factors have a very modest effect on pulse energy for a given current and absorption, but a more noticeable one on amplitude and duration. They do not significantly affect the onset of the Q-switching instability (the lower current or unsaturated gain limit of ML stability), but have a stronger effect on the upper limit of ML stability associated with the irregular envelope and pulse competition. ML behavior is the most stable when the gain and absorber linewidth enhancement factors are not too different from each other. According to the DDE model predictions, the most stable operating point (which also corresponds to the highest pulse amplitude and lowest duration) is for aHg¼aHa; however, traveling-wave and modal analysis predict that the best quality ML is achieved with aHg>aHa; the discrepancy is likely to be caused by the different geometry of the long amplifier and the shorter absorber. The main parameter determined by the linewidth enhancement factors is the chirp (dynamic shift of the instantaneous frequency) of the pulse. Passively mode-locked pulses tend to be up-chirped (with the instantaneous optical frequency increasing toward the end of the pulse) when the absorber saturation factor aHa is small and the chirp is mainly caused by aHg. With a certain combination of aHg and aHa (typically aHg>aHa), an almost complete compensation of chirp is possible; with aHa >aHg, the pulse is typically down-chirped (Salvatore et al., 1996). As up-chirp is observed more frequently than down-chirp in experiments, one may conclude that typical values of aHa are smaller than aHg. In active ML, downchirp is typically observed, while hybrid ML allows the chirp to be tuned to some extent, and there is typically a combination of bias and current or voltage modulation amplitude for which the chirp is minimized and close to zero, if only in a very narrow range of operating parameters.

5. IMPORTANT TENDENCIES IN OPTIMIZING THE MODE-LOCKED LASER PERFORMANCE 5.1. Achieving a high gain-to-absorber saturation energy ratio The importance of a high gain-to-absorber saturation energy ratio s for short-pulse generation has been semi-empirically understood for some time, and the recent analysis with the DDE model reconfirmed its importance also for maximizing the stable ML range (Vladimirov and Turaev, 2005; Vladimirov et al., 2004). Since, by definition, s¼Ug/Ua ¼saAXg/ sgAXa, there are at least two ways of increasing s:

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(i) In most non-semiconductor lasers, where the absorber and gain regions are discrete elements, the most straightforward way of increasing the s-parameter is to ensure that the light is focused more tightly in the SA than in the gain section so that AXa 1. The authors of Vladimirov and Turaev (2005) and Vladimirov et al. (2004) used their analysis to conclude further that when designing a QW laser for ML purposes, a structure with a smaller number of QWs was preferable to one with a larger number – indeed, the smaller number of QWs means a smaller confinement factor, hence a higher threshold carrier density, hence a smaller dg/dN at threshold due to the sublinear g(N), which in turns gives a higher value of the ratio s. These considerations influenced the choice of structures with just 2–3 QWs for realizing DBR ML lasers capable of generating very stable pulses about 2ps long at 40Gbit/s (Bandelow et al., 2006). It may be argued that the same logic also accounts in part for the success of QD mode-locked lasers, in

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which the dependence of gain on the (total) carrier density in the active layer is even more sublinear than in QWs. However, it has to be borne in mind that the concept of total carrier density is somewhat misleading in QDs; a more accurate picture is given by more complex analysis, considering separately the population of the dots themselves and of the reservoir that supplies them with carriers.

5.2. Improving stability and pulse duration by reducing the SA recovery time As described above (Section 4.1), decreasing the SA recovery time leads to both stabler and shorter ML pulses. Therefore, a number of teams have suggested alternative routes for reducing the time ta: (i) In earlier experiments, heavy ion implantation of the laser facet was used to produce SAs with a recovery time ta 10ps, governed by nonradiative recombination. This enabled some of the first results on mode-locked monolithic semiconductor lasers, with picosecond pulses at both 0.87 and 1.55mm wavelength band (Deryagin et al., 1994; Zarrabi et al., 1991). (ii) In more modern QW laser constructions with a reverse biased SA, the most fundamental of the processes governing ta is the sweepout of photocarriers from the absorber QWs into the waveguide layer by the bias field. In InGaAsP materials, most frequently used in lasers operating at 1.55 mm, the potential well for heavier holes is deeper than that for lighter electrons, making hole sweepout a bottleneck for achieving small ta at sensible values of the SA reverse bias (of the order of a few volts). On the other hand, AlGaInAs quaternaries have a shallower potential well for holes, making for much more efficient sweepout. Absorber recovery times as low as 2.5–3ps at a bias of 4 V, several times shorter than in InGaAsP quaternaries at the same bias voltage, have been measured in such QW heterostructures (Green et al., 2011; Lianping et al., 2011a). Lasers with AlGaInAs quaternaries have been successfully used for ML operation in both a relatively traditional 40GHz construction (Hou et al., 2009) and in more advanced harmonic mode-locked lasers operating at 160þ GHz (Hou et al., 2010a,b; Lianping et al., 2011a), as will be discussed in Section 5.4. The pulse durations obtained were in the subpicosecond range for harmonic operation at 160GHz. (iii) Third, it has been pointed out that faster sweepout may be achieved by engineering the quantum well profile to include steps or oblique rather than vertical walls (Nikolaev and Avrutin, 2004a). At the time of writing, such structures do not appear to have been realized experimentally, but theoretical predictions are encouraging.

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(iv) It is also important to note that, even after carriers have been swept out of the absorbing (active) layer, they need to be efficiently removed from the waveguide layer, to prevent both their diffusion back to the SA and the carrier screening of the electric field applied to the well. In a standard structure, this happens through drift in the electric bias field, with the transport of heavy holes limiting the process speed (Nikolaev and Avrutin, 2004b; Nikolaev et al., 2005). To avoid this, lasers with unitraveling carrier absorbers have been proposed and realized (Scollo et al., 2005, 2009). In these devices, the SA is not a reverse biased section of the same p-i-n heterostructure as the gain section, as is the case in more usual constructions. Instead, the laser heterostructure is etched away in the SA section and a separate structure is grown for the SA, in which light is absorbed in a special pþ-layer. Then, the holes are majority carriers so they are removed from the structure via collective relaxation (a faster process than drift), and it is only the electron drift (faster than hole drift) that limits the absorber recovery speed. The limitation of such structures is the need for a regrowth procedure, which increases the complexity and potentially the cost of the structure. Besides, even very small (<104) residual reflectances between the laser and absorber waveguides can lead to formation of satellite pulses (as in compound cavity lasers discussed in Section 5.4), which was indeed encountered in practice. Satellite-free pulses 0.9ps long at the repetition rate of 40GHz have been achieved in UTC structures, or 0.6-ps pulses with some satellite pulses (Scollo et al., 2005, 2009). (v) Finally, absorber recovery speed is one of the advantages of QD materials (Section 6.1), which have shallower potential wells for both electrons and holes than Quantum Wells, hence faster sweepout. The relatively small total numbers of photogenerated carriers compared to QWs can also partly alleviate any potential problems with carrier screening of the bias field.

5.3. Increasing the optical power: Broadening the effective modal cross section Optimizing the absorber parameters can, as shown above, help reduce the pulse duration, and increasing the stability range helps somewhat extend the achievable power range too. However, nonlinear processes, such as the onset of high-current instabilities, by necessity limit the total average power in standard mode-locked lasers to a few units to tens of milliwatts, depending on the construction and repetition rate. A route to a considerable increase in the output power is suggested by observing that in the theoretical analysis in Section 3, the nonlinearities that limit the achievable

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power (including irregular nonlinearities) respond, not to the absolute power, but to the photon densities Sg,a ¼jYg,aj2 ¼P/(vgℏoAX(g,a)) in the gain and absorption sections of the laser. With an increase in the modal crosssection AXg, the entire map of the dynamic regimes of the laser, including the onset of chaotic instabilities, is scaled toward higher power while keeping the photon density constant. Therefore increasing the modal cross section AXg is a robust way of achieving stable high-power output. As with ordinary CW operating lasers, arguably the main challenge is to achieve this while keeping the laser operating in a single transverselateral mode. (i) One laser construction that realizes this principle, whether operated CW or mode-locked, is a slab-coupled laser waveguide, in which the active layer is positioned at the edge (rather than in the center, as is the case usually) of a waveguide (slab) layer, which is as thick as 3– 4mm in the transverse direction and unbounded laterally. Lateral waveguiding is by means of a relatively shallow ridge, with the width also of a few microns. The combination of the waveguide structure and active layer position ensures laser operation in a single longitudinal-lateral mode which is broad (e.g., in Ahmad and Rana, 2008a, the effective cross-section was AXg ¼AXa
14mm2, about an order higher than in ordinary stripe lasers) and fairly isotropic. By introducing the SA section in the usual way of electrically isolating a section of the cavity, passive ML of a structure of this type was realized (Ahmad and Rana, 2008a,b; Gopinath et al., 2007; Juodawlkis et al., 2011; Plant et al., 2006). Average power value achieved at l¼1.55mm were as high as 210mW with F
4.6GHz (L¼9mm) (Ahmad and Rana, 2008a). The pulses corresponding to the highest average power were not the shortest (25ps in Ahmad and Rana, 2008a), though at different operating conditions, pulses <6 ps long were realized. At l¼890nm, ML with an even higher average power of 489mW at 7.92GHz from a 5-mm device with a 300-mm long SA was demonstrated (Gopinath et al., 2007). By applying fundamental and subharmonic voltage modulation to a modulator section fabricated next to the SA section, the same team that produced some of the passive ML results (Ahmad and Rana, 2008a) reported hybrid ML operation of a similar, slightly shorter (F
5.2GHz) structure, with the average power of 220mW and the pulse duration reduced by about 20% (from 14.5 to 11.7ps in an example quoted) compared to the passive ML case (Ahmad and Rana, 2008b). As expected, hybrid modulation led to reduction of noise. Colliding-pulse ML (CPM) has also been realized in slab-coupled lasers, yielding an average power of up to 240mW per (uncoated) facet (480mW in total) at 8.6GHz, with pulse durations of 8–14ps (11ps at the highest power), at l¼1.55mm ( Juodawlkis et al., 2011).

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The limitation of slab-coupled structures is that the very low confinement factor limits the minimum resonator length needed to achieve lasing and is thus likely to pose an upper limit of at most 10–20GHz on the repetition frequency, unless some form of a harmonic ML construction (Section 5.4) were to be implemented. Theoretical predictions (Avrutin et al., 2010) imply that asymmetric nonbroadened waveguide lasers such as those used for gain-switching (Ryvkin et al., 2011), which share the feature of large modal cross-section with the slab-coupled lasers, can deliver similar powers when operated in a mode-locked regime, though so far there has been no experimental confirmation of this. (ii) As mentioned in Section 5.1, tapered mode-locked lasers also achieve high power by expanding the mode, in this case in the lateral rather than transverse direction, and, importantly, in the gain section only. The added advantage of the simultaneously increased s-parameter (Section 5.1) helps further improving the achievable power by increasing the stable operation range. In the l¼1300nm QD structure analyzed in Nikitichev et al. (2011a), besides the high peak power of 3.6W, the high average power of about 210mW was also achieved; again, as in slab-coupled lasers, this was at somewhat different bias conditions from those for those for shortest and most intense pulses (corresponding to pulses
6ps long at 14.6GHz). Most recently, the laser design was improved further, on the basis of numerical simulations of the dynamic ML regimes and their dependence on the structural parameters. Two designs with different gain and absorber section lengths were proposed. One device design demonstrated the record – high peak power of 17.7W with 1.26ps pulse width (Fig. 3.11) and a second design enabled the generation of a Fourier-limited 672fs pulse width with a peak power of 3.8W (Nikitichev et al., 2011b). A maximum output average power of 288W with 28.7pJ pulse energy was also attained. Quantum-dot tapered gain-guided lasers have therefore shown promising results as high-power ultra-fast and ultra-compact semiconductor-based laser sources. (iii) Finally, in mode-locked VECSELs, the power can be increased by increasing the lateral dimensions of the active area. The highest average output power achieved so far in such structures (and thus in any single ML semiconductor laser) has been 6.4W, in pulses 28ps long at F¼2.5GHz (Rudin et al., 2010). For the time being, this high power is achieved at the expense of sacrificing some of the usual advantages of semiconductor laser sources. VECSELs are external cavity structures and thus need some optical alignment and have a footprint somewhat larger than monolithic ones (although still smaller than the vast majority of non-semiconductor

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B

1.0

QD layers Taper section Absorber section

Intensity (a.u.)

0.8

-40 RF power (dBm)

A

0.6

-45 -50 f = 10 GHz

-55

Lorentzian fit Δt = 2.52 ps

-60 -65 -70

8

10 12 Frequency (GHz)

Δt = ~ 1.26 ps (deconvolved)

0.4 0.2 0.0 -10

-5

0 5 Time delay (ps)

10

FIGURE 3.11 (A) Schematic of a tapered QD laser for ultrahigh power pulse generation; (B) the pulse achieved.

ML sources), and most often use optical rather than injection pumping which is less efficient (the optical to optical efficiency in Rudin et al. (2010) was 17%). However, considerable progress has been achieved toward overcoming these limitations. In particular, the first generation of VECSELs used separate gain chips and SESAMs; the latter could use QWs or QDs layers. Such structures were used for generating average powers as high as 2.1W, with a pulse duration down to 4.7ps at 4 repetition rate at l
960nm (Aschwanden et al., 2005). These lasers required a ‘‘folded cavity’’ with a need for nontrivial optical alignment. The more recent MIXCEL (modelocked integrated external-cavity surface-emitting laser) structures, of which the construction of Rudin et al. (2010) has been the most successful so far, combine a QW active region and a QD SA layer within the same wafer structure, and thus reduce the number of components in the cavity to just two – the MIXCEL chip and the output coupler doubling as the external mirror; this also allows for a straight external cavity, simplifying the optical alignment. The curvature of the output coupler, the cavity length, and the size of the pumping spot allow the size of the waveguide mode to be tuned (in Rudin et al., 2010, the pumping spot and the mode had the radii of 215 and 223mm, respectively, corresponding to a mode cross-section three orders of magnitude greater than in monolithic in-plane structures). A fully vertically, monolithically or hybridly, integrated MICXEL type laser looks foreseeable in future. Current MIXCEL constructions are specifically designed for optical pumping. They contain an intracavity multistack ‘‘pump DBR’’ structure which reflects light at the pump wavelength (808nm in Rudin et al., 2010), but not the signal wavelength (960 nm in Rudin et al., 2010), thus blocking the penetration of pumping light into the

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High-Power ML in Semiconductor Lasers

Material l construction type (mm) ML type

Max Min Max Ppeak tpulse Paver Frep (mw) (GHz) (W) (ps) Reference

Slab coupled

QW

1.55 Passive 210

4.6

Slab coupled

QW

1.55 Hybrid 220

5.2

Slab coupled

QW

0.89 Passive 490

7.92

Slab coupled

QW

1.55 Passive 480 CPM

8.6

Tapered

QD

1.26 Passive

10

VECSEL

6400 2.5

3.95 6

10

8

17.7 1.26 3.8 0.67 28

Ahmad and Rana (2008a) Ahmad and Rana (2008b) Gopinath et al. (2007) Juodawlkis et al. (2011) Nikitichev et al. (2011b) Rudin et al. (2010)

SA but not affecting the cavity structure for the signal light. In principle, electrical isolation of the SA and the active section (e.g., by intracavity electric contacts) could also be possible in vertical cavity structures so in principle electrically pumped MIXCEL type should be possible too. However, due to thermal budget considerations it is not clear how powerful these can be made. Table 3.1 summarizes some of the recent achievements in high-power pulse generation by ML.

5.4. Engineering the bit rate. High power and high bit rate operation. Harmonic ML Depending on the potential application, a very wide range of repetition rates may be required from a ML laser. A number of applications in manufacturing, sensing, and nonlinear imaging involve high output power at moderate (≲1GHz) repetition rates, whereas potential applications such as high-bit-rate communications may benefit from an ML laser

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with a multigigahertz (80, 100, 160, or 640GHz) repetition rate (Avrutin et al., 2000). As mentioned in the previous section, low bit rates with a high power in a ML semiconductor laser are achieved by increasing the cavity length (L in Eq. 3.1), which necessitates using an external cavity construction, whether edge-emitting or VECSEL. This is not a trivial task since, as follows from the theoretical analysis of Section 3, at low bit rates, ML lasers are particularly prone to switching to harmonic ML as the current (and hence power) is increased. Still, ML at repetition rate as low as 0.19 GHz has been recently realized with a QD edge-emitting laser (Cataluna et al., 2011b) in an external cavity configuration (Fig. 3.12). In addition to high-pulse energy, external cavity constructions also have the advantage of low jitter and RF linewidth (down to 30Hz at some operating conditions in Cataluna et al., 2011b). At the other end of the repetition rate scale, multigigahertz repetition rates can be achieved by straightforwardly decreasing L. However, this makes for high-fabrication tolerance, and obtaining substantial ML powers from short resonators may necessitate working high above threshold, making them prone to instabilities and limiting the power. Hence, harmonic ML techniques that use Mh >1 L in Eq. (3.1) have attracted attention. Second harmonic operation at 80GHz (M¼2) has been experimentally observed in standard notionally 40-GHz lasers at some operating conditions (involving high current) in agreement with both DDE and traveling-wave (see Section 3) theoretical simulations (Bandelow et al., 2006). However, stable multigigahertz generation by operation at higher harmonics requires specialist constructions. These fall into two categories. The first is CPM, including multiple (MCPM; Martins et al., 1995; McDougall et al., 1997) and asymmetric (ACPM; Shimizu et al., 1995, 1997) CPM constructions. These achieve ML at the Mhth harmonic by positioning one SA (in CPM or ACPM) or several SAs (in MCPM) at fraction(s) Mh0 /Mh of the laser cavity length, where Mh0
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A QD-ECMLL

TS

Lens

Lens

OI HWP

OC

Gain Absorber

OSA

PC Osc

Autoco

FS SMF

PD

RFSA B -40

RF power (dBm)

-50

-40

281 MHz

1 GHz

-50

-60

-60

-70

-70

-80 -50

-80 -40

238 MHz

750 MHz

-50

-60

-60

-70

-70

-80 -50

375 MHz

-80 -50

191 MHz

-60

-60

-70

-70

-80

-80 0

2

4

6

Frequency (GHz)

8

10

2

4

6

8

10

Frequency (GHz)

FIGURE 3.12 (A) Schematic of an external cavity QD laser configuration; (B) RF spectra for the variable repetition rate from 1GHz to 191MHz from this construction.

of deeply etched slots, either single or multiple in a 1D photonic-bandgap (PBG) mirror arrangement (Yanson et al., 2002). A 608-mm long cavity used ICRs at 1/33 of the cavity lengths, giving F¼2.1THz (similar structures with lower Mh values produced ML at bit rates of the order of hundreds of GHz depending on Mh and L).

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Later (Hou et al., 2010b; Lianping et al., 2011b), the technique was extended to l¼1.55mm by using AlGaInAs quaternary materials, whose short absorber recovery time (see Section 5.2) proved indispensable for high bit rate ML. The ICR slots in these structures were filled with dielectric (rather than air as in Yanson et al., 2002) improving the reliability of the structure. Pulses as short as 0.8–0.9ps at repetition rates of 160– 640GHz have been achieved (Hou et al., 2010b; Lianping et al., 2011b), with the time-bandwidth product of 0.8 indicating moderate chirp. Theoretical predictions (Lianping et al., 2011b) show that the scheme is relatively tolerant to the precision of the ICR position, with harmonic operation (with a somewhat increased pulse duration) maintained for imperfections of the ICR position of up to 10mm. Theoretical analysis conducted in Yanson et al. (2002) implied also that the stability of harmonic ML operation at high Mh numbers would be higher with several weak reflectors than with one strong ICR, which was supported by experimental results (Yanson et al., 2002). This idea has seen its ultimate realization in the so-called discrete mode laser structures. Such structures contain multiple intracavity reflectors, but in the form, not of deep-etched structures, but of simple step discontinuities in the laser waveguide thickness. Taken individually, these steps have weaker reflectances than deeply etched structures; however, they are more production friendly (a single step, unlike an etched slot, does not require high-resolution ionbeam etching). Tailoring the number of steps and the distances between them presents a very powerful tool for engineering the cavity to select either a single mode or a set of a predetermined number of equidistant modes (O’Brien et al., 2010a,b). Originally designed for single-mode operation, such lasers have recently been adapted for harmonic ML (O’Brien et al., 2010a,b). Lasers 875mm long have been reported to provide nearly transform-limited pulses about 2ps long, at F¼100GHz (Mh¼2) (O’Brien et al., 2010a); higher harmonic numbers should be possible as well. In principle, ACPM (or MCPM) and compound cavity techniques can be combined in one structure. Other cavity structures that have been proposed for high-frequency ML, not restricted by the cavity length, are resonators with sample grating reflectors (Kim et al., 1999) and possibly coupled-ring resonators (Agger et al., 2010; Liu et al., 2005) discussed in Section 6.4.

5.5. Noise considerations in ML operation Achieving low-phase noise ML, hence low-timing jitter and RF linewidth, is extremely important for a large range of potential applications. Theoretical work on both active and passive ML, using both time–domain approach (noise-driven Haus model; Rana et al., 2004; Jiang et al., 2001) and frequency-domain approach (Kim and Lau, 1993), has been

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presented and subsequently adapted for interpreting the experiments by a number of authors (see, e.g., Kefelian et al., 2008). Overall, it would appear that routes to jitter and linewidth optimization are somewhat less clear than in the case of more straightforward ML parameters such as duration and power. However, some trends can be identified from the theory. It was noted in all theories, for example, that the noise effects (quantified as either jitter or RF linewidth) are more pronounced in high-repetition frequency ML; in Jiang et al. (2001), the RF linewidth of ML was shown to scale roughly as the square of the pulsewidth-to-round trip ratio. As the RF linewidth of ML is approximately proportional to the Shawlow-Townes linewidth of an individual optical mode ( Jiang et al., 2001), lasers with a low optical linewidth, such as external cavity constructions, can be expected to have a lower ML linewidth and jitter than monolithic ones. Hybrid ML was shown to have a lower phase noise than purely passive ML, with the phase noise spectrum turning from a 1/F2 dependence typical for passive ML to a Lorentzian (Rana et al., 2004) for active/hybrid ML, and the low-frequency phase noise inversely proportional to the modulation strength. It was also shown (Rana et al., 2004) that self-phase modulation leads to excess low-frequency phase noise, hence excess jitter; therefore, achieving transform-limited pulses should also help low-noise operation. This may be the case, for example, in single-section mode-locked lasers (Section 6.3). An even clearer consequence of the conclusions above is that ML constructions such as VECSELs, particularly those with coherent absorbers (Section 6.2), which have a short semiconductor chip in a long external cavity (hence small optical mode linewidth) and produce utlrashort transform-limited pulses at relatively low repetition rates, can be expected to operate with a low jitter and RF linewidth, which is indeed the case. A somewhat more complex situation arises when a monolithic ML laser is placed in an external cavity, creating, in general, a compound cavity arrangement. In this case, both deterioration (e.g., Passerini et al., 2005) and improvement of noise properties has been reported. Experiments (Lin et al., 2010; Solgaard and Lau, 1993) and numerical simulations (Avrutin and Russell, 2009) show that the relation between the optical lengths of the external and intrinsic cavities may be important in determining which is the case at least in some constructions with short (optical length comparable to that of the laser) external cavities. However, with suitable external cavity construction, considerable linewidth and jitter reduction has been proved possible (e.g., from 46 to 1.1kHz and from
300 to
30fs, respectively, in a recent study involving a QD modelocked laser in a long (18m) external cavity (Lin et al., 2011); in Solgaard and Lau, 1993), reduction of the linewidth from 8kHz to 350Hz was reported).

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6. NOVEL MODE-LOCKING PRINCIPLES 6.1. QD materials As seen from Section 5, much of the progress in ML laser technology achieved in the past decade, has been associated with the use of QD mode-locked lasers, which is largely to do with the novel physical properties of these materials (Rafailov et al., 2007, 2011). Ultrafast QD lasers have been considered in detail in a separate monograph (Rafailov et al., 2011); here, we shall reiterate only the main points relevant for ML. QDs are essentially clusters of semiconductor material having all three dimensions of a few nanometers. QDs used in monolithic mode-locked laser structures belong to the category of self-assembled QDs. They have the geometry of a pyramid or a truncated pyramid with a base of 5–10nm and are arranged in layers (typically 1–10) within the structure, with a density of 1010–1011 cm2 per layer. In each dot, electrons and holes see full three-dimensional localization and are thus occupying discrete energy levels rather than continuous bands as in bulk or QW semiconductors. In lasers, the most significant are transitions involving the lowest electron energy state (the ground state) and the next one (the first excited state, sometimes referred to simply as the excited state, which is a few tens of meV higher). There is no direct interaction between carriers in different dots; the (imperfect) thermodynamic equilibrium between them is achieved through the processes of capture of carriers from the reservoir formed by nearby QW and bulk layers into the dots, their possible thermal escape and recapture into different dots. Due to the dispersion of dot size and composition, there is considerable inhomogeneous broadening of optical emission and absorption in dots (about 50–100meV), which largely determines their emission and absorption spectra. These features translate into a number of advantages, of which the following are relevant to ML. First, mode-locked QD lasers share with other QD lasers the advantages of a possible low threshold, reduced amount of amplified spontaneous emission (hence narrow linewidth/low noise), and in particular lower temperature sensitivity. A team headed by one of the authors demonstrated stable passive mode-locked operation of a two-section QD laser over an extended temperature range, at relatively high-output average powers (Cataluna et al., 2006a). We observed very stable modelocked operation from 20 to 70  C, with the corresponding RF spectra exhibiting signal-to-noise ratios well over 20dB and a 3dB-linewidth smaller than 80kHz. The ML regime became less stable only at 80  C, with the 3dB-linewidth increasing to 700kHz, and the signal-to-noise ratio measured as 15dB. No self-pulsations were observed in the entire temperature range. Additionally, to meet the requirements for high-speed

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communications, it is important to investigate the temperature dependence of the pulse duration. For instance, in communication systems with transmission rates of 40Gbit/s or more, the temporal interval between pulses is less than 25ps and so the duration of the optical pulses should be well below this value at any operating temperature. We have shown that, perhaps counterintuitively, the pulse duration and the spectral width decrease significantly as the temperature is increased up to 70  C (Cataluna et al., 2006b). The combination of all these effects resulted in a sevenfold decrease of the time-bandwidth product (the pulses were still highly chirped due to the strong self-phase modulation and dispersion effects in the semiconductor material). The discrete nature of QDs also helps reduce the effects of surface recombination, allowing for higher pulse energies and extending the possibilities of etching in the laser technology (in the context of ML lasers, this helps precise cavity definition and possibly harmonic cavity techniques). Second, the very broad gain spectrum, and stronger spectral hole burning than in other type of lasers (due to inhomogeneous broadening exceeding the inhomogeneous broadening) tend to lead to very broad emission spectrum, which potentially opens a possibility of direct femtosecond pulse generation by ML in monolithic structures. In 2004, a team headed by one of the authors demonstrated for the first time the generation of sub-picosecond pulses directly from a QD laser, the shortest pulse durations being 390fs, without any form of pulse compression (Klopp et al., 2009) from a two-section passively mode-locked QD laser (Fig. 3.13).

2.5 2.0 1.5

Dt ~393 fs

Intensity

Normalised intensity (a. u.)

3.0

Dl~14 nm 1240

1250 1260 1270 Wavelength (nm)

1280

1.0 Experiment Gaussian fit - 550 fs

0.5 0.0 -4

FIGURE 3.13

-3

-2 -1 0 Time delay (ps)

1

2

Optical spectrum and sub-picosecond pulse from a two-section QD laser.

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Owing to the excellent electrical characteristics of the device, it was possible to apply very high values of current and reverse bias (up to 10 V), which provided some latitude for exploring a wider range of these parameters. Mode-locked operation was observed over a broad range of injection currents from above laser threshold (30mA) up to 360mA and over a relatively wide latitude of reverse bias levels on the absorber section (from 4.5 to 10V). The broad spectrum (14nm) and the ultrashort pulse durations measured suggest that the generation of pulses in the sub-100fs domain may yet be possible from relatively simple QD laser configurations. Since then, sub-picosecond pulse durations have also been demonstrated in single-section QD lasers (see Section 6.3). Third, as mentioned above, the role of ultrafast (characteristic time of 50–100fs) carrier collisions that normally establish the quasiequilibrium energy distribution of carriers in a semiconductor laser is taken in QD lasers by the processes of carrier capture and escape, which are 1–2 orders of magnitude slower. This means the corresponding increase in the gain compression in QD lasers by about the same order of magnitude. As discussed in Section 4.2, this suppresses the Q-switching instability and leads to a broader range of ML. Indeed, experimentally, the SSP instability range in QD lasers tends to be considerably narrower than in QW ones with comparable performance. Often it is completely absent. Fourth, a number of advantages are associated with the properties of the QD SAs, chief of which is the short recovery time due to fast carrier sweepout out of QDs (Sections 4.2 and 5.2). Absorber recovery in QDs has been shown experimentally (Malins et al., 2006; Viktorov et al., 2009) and theoretically (Viktorov et al., 2006, 2009) to be a complex process, with at least two time constants involved, and both intralevel transitions (capture and escape of carriers) and field-induced sweepout playing a part (Fig. 3.14A). The capture and escape rates in the absorber section (which have an activation dependence on temperature) have been shown theoretically (Cataluna et al., 2007; Viktorov et al., 2006) to be responsible for a decrease of absorber recovery time with increasing temperature, leading to a decrease in the pulse durations. This fact was verified using ultrafast spectroscopy to probe the absorber recovery time as a function of temperature. Furthermore, p-doping has the potential of enabling an accelerated gain recovery, through a pre-filling of the hole states, as previously demonstrated in high-speed laser modulation experiments (Fathpour et al., 2005). With the intracavity dynamics sped up by temperature, the slowest time constant in the absorber response, associated in this case with the field sweepout, can be brought down to 1–2ps with realistic values of reverse bias (Malins et al., 2006; Fig. 3.14B). This ultrafast SA recovery dynamics is certain to have contributed to the generation of sub-picosecond pulses from QD lasers and was reported by several groups (see, e.g., Thompson et al., 2006; Laemmlin

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A 0.20 Dt1 <1 ps

Intensity (a. u.)

0.15

Dt2 ~100 ps

0.10

0.05

0.00 0

50 100 Time delay (ps)

150

2D Graph 1 B

0.006 0v 2v 4v 6v 8v 10 v

Probe transmission

0.005 0.004 0.003 0.002 0.001 0.000 0

100 Delay (ps)

200

300

FIGURE 3.14 Pump–probe measurements of quantum-dot saturable absorber: (A) passive and (B) with reverse bias. Note the multiple scales and the reduction of the recovery time with reverse bias.

et al., 2006). Indeed, it was confirmed that the pulse duration decreases exponentially with increasing reverse bias (up to 8V) on the SA (Thompson et al., 2006). This decrease was attributed mainly to the corresponding exponential decrease of the absorber recovery time as the reverse bias is increased (Malins et al., 2006; it has to be noted that there are some limits within which this is beneficial; at very high bias, the laser behavior degrades (Fiol et al., 2009) possibly due to strong unsaturable absorption introduced, or to an increased spectral misalignment between gain and absorption bands).

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The possibilities demonstrated so far open the way for the ultimate deployment of ultra-stable, uncooled mode-locked diode lasers incorporating QD materials. Another advantage of QD SAs is the low saturation fluence (effectively, a high absorber cross section at a given wavelength) due to the inhomogeneous nature of absorption saturation. Fifth, QD lasers open a unique opportunity of achieving dual-wavelength lasing, including dual-wavelength ML. It has been observed that laser emission in QD lasers can access the transitions in ground-state (GS), excited-state (ES), or both (Markus et al., 2003). Sub-picosecond gain recovery has been demonstrated for both GS and ES transitions in electrically pumped QD amplifiers (Schneider et al., 2005). It has also demonstrated an optically gain-switched QD laser, where pulses were generated from both GS and ES, and where the ES pulses were shorter than those generated by GS alone (Rafailov et al., 2006). Passive ML via GS (1260nm) or ES (1190nm) in a QD laser demonstrated at repetition frequencies of 21 and 20.5GHz, respectively (Cataluna et al., 2005). The switch between these two states in the ML regime was easily achieved by changing the electrical biasing conditions, thus providing full control of the operating spectral band. It is important to stress that the average power in both operating modes was relatively high and exceeded 25mW. In the range of bias conditions explored in this study, the shortest pulse duration measured for ES transitions was 7ps, where the spectral bandwidth was 5.5nm, at an output power of 23mW (Cataluna et al., 2006b). These pulse durations are similar to those generated by GS ML at the same power level. Although the pulses generated from both GS and ES ML are still far from the transform limit (with a time-bandwidth product exceeding 7), they could be reduced by using external compression techniques. More recently, a dual-wavelength passive ML regime where pulses are generated simultaneously from both ES (l¼1180nm) and GS (l¼1263 nm), in a two-section GaAs-based QD laser (Cataluna et al., 2010). This is the widest spectral separation (83nm) ever observed in a dual-wavelength mode-locked non-vibronic laser (Fig. 3.15). The dual-wavelength ML regime was achieved in a range of bias conditions, which simultaneously satisfied the conditions for achieving ML via GS and ES – for current levels in the gain section between 330 and 430mA, and values of reverse bias between 6 and 10V in the SA region. The excited-state levels have higher degeneracy, and consequently higher saturated gain than the GS. This means that a transition from the GS to the ES can be achieved by increasing loss, which in this case can be manipulated through the increase in reverse bias applied to the SA. The spectral separation between the two modes results in different repetition rates, due to the dispersive nature of the laser semiconductor material, which induces

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A

Intensity (a. u.)

60 50 40 30 20 10 1100

1150 1200 1250 Wavelength (nm)

1300

19 20 21 Repetition frequency (GHz)

22

B -55

RF power (dBm)

-60 -65 -70 -75 -80 18

FIGURE 3.15 (A) Optical spectrum and (B) RF spectrum characteristic of the dualwavelength mode-locked regime in a mode-locked QD laser.

different cavity roundtrip times for the propagation of the two modes. As such, the repetition rates of the generated pulses were 19.7 and 20GHz for the ES (l¼1180nm) and GS (l¼1263nm), respectively. The development of dual- and multiple-wavelength ultrafast lasers is a research area that aims to address a number of important applications such as time-domain spectroscopy, wavelength division multiplexing (WDM), and nonlinear optical frequency conversion. Finally, under certain conditions, QD lasers show weak self-phase modulation and consequently can be expected to produce nearly transformlimited pulses. Unfortunately, this occurs within a relatively narrow current range at low currents (Newell et al., 1999); in general, self-phase modulation of QD lasers has a very complex nature due to a large number of different carrier populations each with its own dynamics, and can be

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stronger than in other types of lasers (Dagens et al., 2005; Kim and Delfyett, 2009; Martinez et al., 2005; Su and Lester, 2005). It has to be noted that most of the QD laser progress, including the ultrashort pulses and high-power operation from ML lasers, has been achieved with InGaAsP/InP QD structures emitting at l¼1.1–1.3mm. For many prospective applications, particularly those in communications, l¼1.5mm is required. QD and quantum dash (QDh) lasers operating at l¼ 1.5mm have been intensely researched, and single-section QD and particularly QDh lasers have shown extremely promising performance (Section 6.3). True QD lasers working at l¼1.5mm with SA-induced ML, on the other hand, do show successful ML operation, but the emission observed, particularly in earlier experiments, tends to show very strong chirp, leading to very elongated pulses (Heck et al., 2009) to the extent that in some experiments no pulse structure could be distinguished without spectral filtering. QDh lasers fair better in this spectral range (Rosales et al., 2011). Further work on l¼1.5mm QD lasers will be useful to optimize their behavior and establish the possible range of their applications.

6.2. Femtosecond pulse generation by mode-locked vertical cavity lasers. Coherent population effects as possible saturable absorption mechanism One of the most important developments in mode-locked semiconductor laser technology in the recent years has been direct generation of ultrashort (tens to a couple of hundreds of femtosecond) pulses that until recently have been the domain of solid-state lasers only. Only one type of semiconductor lasers has been reported so far to produce such pulses, namely vertical external cavity disk lasers, consisting of separate gain and SESAM chips in a folded (V-shaped) external cavity arrangement (Klopp et al., 2009, 2011; Quarterman et al., 2009; Wilcox et al., 2008). Single pulses 260fs long (Wilcox et al., 2008) and 190fs long at l¼1040nm (Klopp et al., 2009) were obtained from such lasers, followed later (Quarterman et al., 2009) by ‘‘pulse molecules’’ consisting of several 60-fs long pulses separated by intervals of about 1ps, with an envelope about 2ps at half maximum; this was the first demonstration of sub-100fs pulses, if not single, from a semiconductor laser. In Klopp et al. (2011), pulses about 110 fs long were reported at a repetition rate of 92GHz, the 18th harmonic of the 29-mm long cavity (see analysis of harmonic ML in Section 3). Careful spectral alignment between gain and absorption spectra (possible in the external cavity construction by tuning the SESAM temperature) was noted by all authors as the necessary condition of this revolutionary performance. Optimization of the effective gain spectrum of the gain chip (including optimizing the layer structure for flattening the

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reflectance spectrum) is also important for the very short pulse durations in question. Significant differences between lasers of this type and other semiconductor lasers are the very small length of the optical path in both the gain and absorber sections and, correspondingly, the low gain and absorption per pass, combined with the relatively broad cross-section which makes it easier to keep the gain and absorption saturation per pulse weak. These features have very probably contributed to the uniquely short pulses generated by lasers of this type (about an order of magnitude shorter than the best values obtained from monolithic constructions at comparable bit rates). However, the qualitatively different pulse parameters and different regimes obtained suggest that different physical mechanisms may be responsible for this femtosecond ML regime as compared to the picosecond regime more usual in semiconductor lasers. Indeed, such short-pulse durations are difficult to reproduce theoretically in the model of slow saturable absorption which usually applies in semiconductor lasers. One of the teams that produced the femtosecond ML pulses attributed their results to coherent effects in the SESAM elements, namely the optical (ac) Stark effect. This effect, similar in nature to self-induced transparency, has the relaxation rate equal to the inverse of the dephasing (coherence decay) time. In semiconductors, this time is determined by carrier–carrier (and carrier–phonon) collisions and is of the order of 50– 100fs. This belongs to the fast, rather than slow, absorber regime, when the SA recovery time is shorter than the pulse duration. Indeed, theoretical analysis (Mihoubi et al., 2008; Wilcox et al., 2008) showed that the pulse duration possible with this mechanism is about twice the dephasing time, which agrees well with the experiments. Further developments in this work can be expected in the near future, including further improvement and optimization of pulse parameters, moving toward simpler structures than the present optically pumped lasers in a folded cavity, and improving the understanding of the underlying physics.

6.3. Spontaneous ML in single-section lasers Most of the tendencies in improving ML pulse properties covered in Section 5 have dealt with evolutionary improvements in the passive ML constructions including a SA. However, in recent years, a truly revolutionary development in ML laser technology occurred, when several teams have observed – and utilized – ML in single-section lasers without SA sections and without any external modulation either. QDh lasers operating at l¼1.56mm, with dash or dash-in-a-well structures, were seen to generate sub-picosecond optical pulses at cavity length corresponding to repetition frequencies ranging from about 10 to 350GHz (Duan et al., 2009; Gosset et al., 2006; Merghem et al., 2009). In a broad

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range of repetition frequencies, transform-limited operation was obtained, with pulses from about 2ps long at 42GHz to 0.56ps at 346GHz. The use of single-section QDh lasers for a wide range of practical applications was successfully demonstrated, showing the maturity of this technology (Duan et al., 2009; Lelarge et al., 2007). All-optical clock recovery at 40Gbit/s was achieved, with jitter removal effectively demonstrated, and subharmonic clock recovery at 40Gbit/s from 80 to 160Gbit/s streams achieved, opening the way for the use of such lasers for optical timedivision multiplexing (OTDM) demultiplexing systems. By beating the modes of QDh lasers, microwave electrical signal was generated, with a RF linewidth in the range of 20–50kHz, depending on the number of dash layers in the structure. Generation of an optical frequency comb at 100GHz, of use for WDM communications, was also successfully shown. ML in QD single-section lasers have also been reported, in a similar (slightly wider) range of repetition frequencies (from 10 to 100GHz in a monolithic construction, or 440GHz with an external element), and pulse durations down to 0.3ps at the repetition rate of 50GHz (Lu et al., 2011; Renaudier et al., 2005; Rosales et al., 2011). Single-section ML has been also reported in QWs (Sato, 2003; Yang, 2011) and even bulk materials (Yang, 2011) showing that the effect is fairly generic. Single-section ML lasers have two considerable advantages over ML lasers with a SA. First, the single contact simplifies the laser operation, and second, the absence of the SA eliminates the risk of the Q-switching instability, which has not been observed in single-section ML. Some of the techniques (harmonic ML, Section 5.4) that are used with SA ML may also be opened to single-section ones. Further optimization of these lasers may partly depend on the establishment of full understanding of their behavior. At the time of writing, a full, universally agreed, theoretical explanation for ML in single-section lasers is still pending, and it is not impossible that different effects can play the main part in different constructions. In the past, some authors (Shore and Yee, 1991; Yee and Shore, 1993) used frequency-domain models with postulated nonlinearity coefficients to show that nonlinearities in single-section semiconductor lasers could lead to a steady-state regime with fixed phases; however, this was predicted to produce, not the AM ML which corresponds to short-pulse emission, but the so-called frequency modulation ML, in which the phases of adjacent modes differ approximately by p, and the outcome is a CW-like regime with periodic carrier frequency oscillation. More recent work (Nomura et al., 2002; Renaudier et al., 2007), also using frequencydomain analysis with microscopically calculated (Nomura et al., 2002) or phenomenological (Renaudier et al., 2007) description of linear and nonlinear gain, predicted a possibility of ML-type signal generation in a

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single-section laser with three modes involved in lasing. The underlying mechanism was mode coupling by population pulsations/four-wave mixing effects as in earlier studies, and for certain cavity lengths and active layer parameters, AM ML was predicted. This appears to agree qualitatively with traveling-wave modeling (Section 3) analysis for the case of a DBR laser without an SA (Bardella and Montrosset, 2005); the authors of Bardella and Montrosset (2005) also identified the role of fourwave mixing in their construction. A full analysis for truly multimode Fabry–Perot construction, either in time or frequency domain, which would be the final proof of the validity of this explanation, has not however been reported, to the best of our knowledge. Yang (2007, 2011) put forward a theory which attributes the (AM) ML in single-section lasers with deeply etched waveguide structures to carrier redistribution in the lateral (in-plane, but perpendicular to wave propagation) direction and the associated variations of the lateral mode profile. In the case of significant scattering losses at the lateral waveguide walls, this redistribution can reduce the losses for high-energy light, producing effective saturable absorption. Stability analysis using a generalization of Haus’s ML theory to the case of multiple spatial modes of an unperturbed waveguide has been performed to justify this explanation, but qualitative analysis of fully developed ML is not yet available. In a separate study, double-lateral modes (both of which were lasing, unlike the case treated in Yang, 2007, 2011) have been seen operating phase-locked in a single-section construction, producing a ML-type regime at a frequency different from the round trip (Enard et al., 2010). Table 3.2 summarizes some of the recent achievements in ulrashort pulse generation by ML; it is seen that most of these are associated with novel ML principles.

6.4. Minitaturization and integration: Ring and microring resonator cavities One important potential application of mode-locked lasers would be as clock generators in future optoelectronic integrated circuits (OEICs). The majority of ML lasers studied so far, however, have been Fabry–Perot constructions with at least one reflector formed by the chip facet, which makes them poorly suitable for integration. Fabry–Perot lasers with etched reflectors such as those discussed in Section 5.4, Bragg lasers with DBR reflectors either side, and ring resonators can all be seen as potential candidates for monolithically integrable ML laser constructions. Ring lasers, in particular, have been attracting attention for a number of years, with the improvement in fabrication technology gradually removing the high scattering losses which reduced the efficiency of early structures (see review in Avrutin et al., 2000) to 15–20% at most. In the last

TABLE 3.2

Short-Pulse Generation by ML in Semiconductor Lasers

Construction

Material type l (mm)

Monolithic QD tandem Tapered QD tandem Single section QDh

ML type

1.26

Passive

1.26

Passive

1.56

Passive self ML

Single section QD

1.54

Passive self ML

VECSEL

QW

1.04

VECSEL

QW

1.03

Passive (coherent SA) Passive (coherent SA)

Chirp properties

Frep (GHz)

Transform limited Transform limited

10

Transform limited Transform limited Transform limited

50 (10–100)

Max Paverage (W) Min tpulse (ps) Reference

288

40 350

92

0.39

Klopp et al. (2009)

0.67

0.19

Nikitichev et al. (2011b) Duan et al. (2009), Gosset et al. (2006), Merghem et al. (2009) Quarterman et al. (2009) Klopp et al. (2009)

0.11

Klopp et al. (2011)

2 0.56

50

0.295

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decade, the technology has been transferred from GaAs/AlGaAs structures to InGaAsP quaternaries (Barbarin et al., 2006), and hybrid integration with silicon-passive Si waveguides also realized (Barbarin et al., 2005; Tahvili et al., 2011), with avoiding internal reflectances at butt joints between different waveguide sections identified as the necessary condition for such integration. The formula (3.1) in a standard ring construction is modified as F
Mhvg/Lring, where Lring is the ring circumference. Moreover, the ring laser is by default a colliding-pulse structure, with clockwise and counterclockwise propagating pulses colliding in the absorber (s). The amplitudes of the two pulses are by default the same, though in an integrated structure are influenced by the relative positions of the SA, amplifier, and output couplers (Tahvili et al., 2011). The ring structure is more amenable to integration than a linear one; however, the length of the ring still remains the necessary limitation to the footprint of the device. The next step toward miniaturized, integrable constructions of mode-locked lasers could be the recently proposed use of coupled microring resonators, possibly embedded in a two-dimensional photonic crystal, for creating utlracompact lasers for active ML (Agger et al., 2010; Liu et al., 2005). Coupled-ring resonators can be viewed as slow-light structures, so could allow for ML at not-too-high (tens of GHz) repetition rates for very small physical footprints of the device. Theoretical analysis using the coupled-mode frequency-domain formalism suggests that such structures may be realizable in practice, though achieving stable ML poses stringent requirements on the fabrication accuracy (Agger et al., 2010). Passive sub-picosecond pulse generation in microring devices has also been predicted theoretically (Gil and Columbo, 2011).

7. OVERVIEW OF POSSIBLE APPLICATIONS OF MODE-LOCKED LASERS Mode-locked semiconductor lasers are well suited for a wide range of applications, including optical fiber communications, optical clock distribution, clock recovery, radio over fiber signal generation, optical sampling of high-speed signals, and metrology (Delfyett et al., 1991; Ohno et al., 2004; Takara, 2001; Vieira et al., 2001). Lasers using novel active media such as QDs and dashes, particularly single-section mode-locked lasers, show particular promise.

7.1. Optical and optically assisted communications Mode-locked lasers and ultrafast semiconductor optical amplifiers are poised to make a large impact on the next generation of optical networks and communication systems. The current InAs QD technology is well

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suited for optical communications using the O-band (1260–1360nm) as defined by the International Telecommunication Union (ITU-T). The O-band coincides with the spectral window of lowest dispersion in optical fibers, and is of particular interest for metropolitan networks. The telecom optical C-band of 1530–1565nm (erbium window) is more often addressed using QW materials (see, e.g., Section 5.2), but the use of QD and QDh active layers is also attracting increasing attention (e.g., Reithmaier et al., 2005), particularly with single-section QDh mode-locked lasers grown on InP as mentioned above (Section 6.3). For OTDM systems, the possibility to generate pulses at very highrepetition rate and with record-low jitter is highly desirable, where a mode-locked laser can be used either as a pulse source or in a clock recovery circuit. In OTDM, different slower data signals are interleaved to form a single faster signal. If a shorter pulse duration is used, the switching window can be narrowed, enabling higher bit rates. Recently, the enormous potential of QD mode-locked lasers for OTDM up to 160 Gbit/s has been demonstrated, by temporally interleaving the split output of a 40GHz hybridly mode-locked QD laser (Schmeckebier et al., 2010). Return-to-zero eye diagrams for transmission rates of 40 and 80 Gbit/s were also presented in this chapter, depicting a clear open-eye. On the other hand, ultrashort pulse sources such as monolithic modelocked lasers can be deployed in WDM systems. In this transmission format, each signal is assigned a different wavelength, enabling the independent propagation down the fiber. Due to the broad bandwidth of QD active layers, QD lasers have a particular advantage here. By engineering the inhomogeneous broadening caused by the size distribution of QDs, a very broad laser emission spectrum can be obtained with nearly uniform intensity distribution (Rafailov et al., 2007). Using slicing techniques, this spectrum can be converted into an array of different equally spaced wavelengths, which is a cost-effective technique to use WDM, without resorting to the use of several laser sources. A uniform 93-channel multiwavelength QD laser has been demonstrated recently (Liu et al., 2007) over a wavelength range from 1638 to 1646nm. The 93 channels were generated directly from the single monolithic chip, without external components or slicing setups, as the channel/mode spacing was simply determined by the length of the cavity (4.5mm). The stability of the laser modes was attributed to the inhomogeneous broadened nature of the gain. With the additional features of low threshold and resilience to temperature, mode-locked lasers, particularly QD and QDh ones, have become suitable candidates for the next generation of telecommunication sources, either fiber-based or free space. In fact, the use of QD lasers in space applications is very promising, as they also exhibit enhanced radiation hardness, when compared to QW materials (Guffarth et al., 2003).

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Due to the cluster-like nature of QD gain material and enhanced carrier confinement, the impact of high-energy radiation on active layers and the consequent generation of defects is less likely to decrease the efficiency of the material, as the carriers are spatially confined. This greatly enhances the reliability of the lasers used in space-borne applications, such as intersatellite communications. Finally, results have been recently reported showing the promise of QD mode-locked lasers for the generation of microwave signals, directly from the intrawaveguide SA (Lin et al., 2009). In this chapter, authors demonstrate a differential efficiency of 33% in optical-to-RF power conversion, while the best extraction efficiency of the SA is shown to be about 86% (for a 10GHz signal). These results could pave the way for a new range of applications where monolithic passively mode-locked lasers could be used as compact RF sources for wireless communications (Kim et al., 2009).

7.2. Biophotonics and medical applications Telecoms and datacoms are two particular niches of applications that semiconductor ultrafast laser diodes have been traditionally designed to address (Avrutin et al., 2000). The ultimate goal is to access applications that have been mainly in the domain of solid-state lasers. Such is the case of biophotonics and medical applications in particular, where compact, rugged, and turnkey sources are crucial for the deployment of sophisticated and noninvasive optical diagnostics and therapeutics. In this respect, compact and simple semiconductor ultrafast sources based on QD materials can offer a number of advantages. Optical coherence tomography (OCT) – a technique that enables imaging with resolutions up to the micrometer level – is one of the medical diagnostics that may benefit in the near future from developments in QD ultrafast lasers. The resolution achieved by this technique is determined by the wavelength and spectral bandwidth of the optical source and it is desirable to achieve as high a bandwidth as possible. The optical source used in OCT should also have a short coherence length. All these requirements can be satisfied by mode-locked lasers, which have been the best performing sources deployed for OCT, in particular Ti:Sapphire lasers (800nm) and Cr:Forsterite and fiber lasers (1300nm; Brezinski and Fujimoto, 1999; Fischer et al., 2005). However, in order to improve the practical prospects of the technique, it is crucial to decrease the footprint and complexity of the laser system. Superluminescent diodes have been used to this end, but power levels can be very low. One possible alternative could be to use mode-locked lasers, particularly those based on QDs. The spectral range that is most routinely accessed with QD lasers (around 1.3mm) can penetrate deeper into biological tissue as it suffers less

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scattering and absorption than at 800nm. In fact, the use of this longer wavelength has allowed imaging depths of 3mm in non-transparent tissues (Brezinski et al., 1996).

8. CONCLUDING REMARKS We have attempted to review the most important developments in ultrashort pulse generation by mode-locked semiconductor lasers. Due to the practical limitations and the increasingly broad nature of the field, the choice of emphasis may have been by necessity subjective; apologies are extended to those authors whose work may not have been given due prominence. The overall conclusion is that mode-locked semiconductor laser technology is now successfully challenging lasers of other type in a wide variety of existing applications, and new applications can emerge, stimulated by high-performance available.

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