mode-locked lasers and ultrashort light pulses

Appl. Phys. 2, 281--296 (I973) �9 by Springer-Verlag 1973 Appl ied

Physics

Mode-Locked Lasers and Ultrashort Light Pulses D. yon der Linde Bell Laboratories, Murray Hill, New Jersey 07974

Received 3 September 1973 / Revised 2 October 1973

Abstract. This article reviews some aspects of the generation of very short optical pulses using mode-locked lasers, with the emphasis laid on pulsed laser systems. Active and passive mode-locking is discussed, and the problem of measuring ultrashort light pulses is considered. A survey of some commonly used sources and techniques for the generation of ultrashort light pulses follows. The paper concludes with a short account of limitations in the generation and propagation of these pulses.

Index Headings: Lasers Mode locking - Ultrashort light pulses

The invention of the ruby laser [1] marked the beginning of a rapid development of sources of coherent optical radiation. One major trend in the laser field was concerned with the development of techniques for obtaining very short laser pulses. The generation of laser pulses of about 10-Ss duration by means of laser Q-switching [2] was a first significant step towards shorter pulses. With the introduc- tion of laser mode-locking, pulses much shorter than a nanosecond (10 -9 s) became available I-3]. These subnanosecond and picosecond (10-12s) optical pulses are commonly referred to as ultrashort light pulses. An important aspect of the progress in generation of ultrashort pulses is the increase in peak power made possible by these developments. Picosecond laser pulses with peak power densities of 1017 to 1018 W/cm 2 have been generated in several laboratories. The electric field strength in these pulses reaches values comparable with inneratomic fields, e.g. up to 10 9 to 10 l~ W/cm, and these high fields enabled researchers to observe many new physical phenomena. Ultrashort light pulses have opened up a wide field of new laser applications, which range from basic research in nonlinear optics and picosecond spectroscopy to optical communication and precision optical ranging. Also, powerful ultrashort optical pulses will eventually lead to the achievement of laser action in the X-ray spectral region. Out-

standing among the possible future applications is the prospect of controlling thermonuclear fusion by employing energetic ultrashort laser pulses for heating of matter to thermonuclear temperatures. In many countries large scale research efforts are underway to explore the feasibility of laser fusion. These few remarks may be sufficient to show that the physics of ultrashort laser pulses will remain the subject of intensive study for some time to come.

1. Generation and Measurement of Ultrashort Optical Pulses

The production of very short optical pulses by means of laser mode-locking requires a laser system with an active medium that provides a large frequency bandwidth for light amplification. To obtain light pulses of duration, say, tp, a gain bandwidth of at least Avgai n ~-~ 1/tp is needed. For a discussion of the basic idea of laser mode- locking, let us consider the schematic of a laser, shown in Fig. la. The laser consists of an optical resonator, usually formed by two mirrors, and an active material for light amplification. This simple laser system has a well known set of resonant modes with characteristic resonant frequencies [4]. We restrict ourselves to the discussion of axial modes

282 D. von der Linde

M2 LM

( a )

L

(b)

I

I I

MI

ii C -----~ ~,~-- 8z,' = - -

,, 2L

I I [ I l l l l l l l l l l l l l l l I I I I I l l [ I I I I I I I I ~ Z / I

uo

Fig. 1. a) Schematic of a laser (M~ and Me: mirrors, LM: laser medium), b) Spectrum of the cavity modes

only 1. Then we are left with a set of equally spaced resonant frequencies (Fig. lb). The frequency spacing 5v of the modes is given by

I ( t ) (a )

I 2L I I T = ~ - i i I tf I ~ 1/LXu ,~ ~i ] I I I I

Z (t) (13)

T: b I tp -~ I Igsu ~'!1

I

Fig. 2. Mul t imode laser output for r an d o m phases (a), and for constant phases (b)

C 5 v - 2 L ' (1)

where L is the optical path between the two mirrors forming the resonator. In the presence of strong dispersion the modes are not equally spaced, but for the moment we neglect dispersion. Suppose now that the active material provides gain for modes in a frequency band of width A v centered at the frequency Vo, and that all modes in this band are excited. The total laser output field resulting from the superposition of these modes can be written as

E(z,t)= ~ A, exp -ion, t - +i(p, , (2) n = - N / 2

where A,, q~, and 0 4 = 2u(nfv + Vo) are the amplitudes, phases and resonant frequencies, respectively, of the cavity modes. N + 1 = Av/Sv + 1 is the total number of modes. The mode amplitudes usually form a more or less smooth distribution determined by the frequency variation of the gain. However, there is nothing in the cavity that fixes the mode phases, so far e. Figure 2a

1 Off-axial modes can be suppressed by introducing an aperture into the resonator to make the cavity very lossy for the off-axial (transverse) modes.

Nonl inear mode interaction in the laser medium can lead to "self-locking" of the phases [5].

illustrates the time behavior of the output, when the mode phases are randomly distributed. In this case the output power fluctuates in an irregular manner, and the mean duration tft of the fluctuations is approximately equal to the inverse width of the excited frequency spectrum, tf~ ~- 1/A v. A much more regular picture results if the phases of all modes have the same value, ~o,= q~o. For simplicity, let us also assume that the amplitudes are equal, A, = Ao. Then the sum in (2) is easily calculated, and the output can be written as

I = A0 2 sin(Sc0t/2) - . (3)

This function is shown in Fig. 2b. Now, for constant phases, the output consists of a regular sequence of well-defined pulses. From (3) the following properties of the output pulse train can be readily verified: (i) The spacing in time of the maxima is

T = 1/Sv, (4)

(ii) the peak intensity is (N + 1) times the average intensity,

/peak = (N + 1) ( I ) , (5)

M o d e - L o c k e d Lase r s a n d Ul t rash( ) r t L igh t Pulses 283

(iii) the pulse duration t r is approximately given by

1 1 t p - N a y - Av (6)

A laser forced to maintain a constant frequency spacing of the modes and well-defined mode phases is said to be phase-locked or mode-locked. As in our example, the output of a mode-locked laser consists of a regular pulse train. The pulse width of a perfectly mode-locked laser is given by the inverse width of the oscillating frequency spectrum, and the peak intensity exceeds the average intensity by a factor of T/tp = A v/av. In the following, we shall discuss techniques for locking together the resonator modes with well- defined amplitude and phase relations.

Active Mode-Lockin9

Proper mode coupling can be achieved by inserting into the laser resonator a device for periodic modulation of either the loss or of the refractive index of the cavity. This method is called active mode- locking, because the modulator is driven by a source independent of the laser. There exists a variety of acoustooptic and electrooptic modulation techniques [6]. For the understanding of active mode- locking, it is helpful to consider first the modulation of a monochromatic light wave, Eo(t)=�89 exp(- icoot)+c.c . , by a sinusoidally driven modulator, for example, a loss modulator (amplitude modulation). Let us assume that the time variation of the transmission of the modulator is given by

T(t) = 1 - 5,, [1 - cos(12t + (o)], (7)

where 6,, is the modulation index, [2 and q~ are the modulation frequency and phase, respectively. The modulated wave, E1 = TEo, can be written as

(t) = �89 0 [(1 - am) e-io)o, Ei

+ ~ (e- i[(~'~176 + e-i[(~~176 + c.c. (8)

The modulator generates two new frequency components, COo + Q and COo- f2. The phases of the sidebands are related to the phase of the modulator, e.g., we have + ~ for coo - O, and - cp for coo + O. Consider now repeated modulation by an array of identical modulators, and let u,; assume that the time dependence of the transmission of all these modula-

[ AZ =2 TrC/~I i (0) i -= ~i

EO(t) ~ El(t) R E2 (t) R E3 (t) .-- #

"U ; U U MOD 1 MOD 2 MOD 3

, i ; ,L I, . , l l l l l , , . 0 "~ "(u

~ 1 (b)

~ / , - - L ~ U M 1 MOD M2

Fig. 3. Generation of sidebands by an array of amplitude modulators (a). Schematic of a laser actively mode-locked by internal modulation (b)

tors is given by (7) (see Fig. 3). We also require that the spacing Az of two subsequent modulators satisfies the relation

2xc A z = ~ m, m = 1 , 2 , 3 . . . . . (9)

This condition ensures that the light wave stays in phase with the modulation when travelling from one modulator to the other. Each modulator adds a new pair of sidebands with the proper phase. It is easy to see that the light wave, after having travelled through N modulators, can be represented by an expression of the form

N

EN(t)= ~ a,e-i[(~~176 +c.c. (10) n = - N

Hence, the final field consists of (2N+ 1) equally spaced frequency components, all oscillating in phase. In other words, the output field is "mode- locked", according to our definition, and we end up with an output intensity in the form of a train of pulses similar to the pulse train shown in Fig. 2b. In an actual laser the light travels back and forth between the cavity mirrors and interacts with the same modulator many times. The light stays in phase with the modulation, if the modulation frequency is an integral multiple of the inverse round-trip time:

r ~2/2z~ = m3v = m ~ - ; m = 1, 2, 3 . . . . . (1 l)

284 D. vonde r Linde

The final number of oscillating modes (= number of sidebands) is determined by the spectral range over which sufficient laser gain is available. Usually the 1 number of oscillating modes is small compared with the number of modes contained in the gain bandwidth. For a rigorous treatment of active mode-locking, the coupled equations of motion for a large number of modes have to be considered. This complicated problem can be solved only when simplifying To 1/2 approximations are made (see references given in [6]). To

When the number of relevant modes is very large, it is sometimes easier to study the pulse evolution in the time domain [7]. We shall use the time domain picture in the subsequent discussion of passive mode-locking.

(a}

T s

(b )

Passive Mode-Locking

For passive mode-locking of lasers no external source for driving the modulator is required, because the laser itself provides the necessary modulator control. A widely used technique is passive mode-locking by means of nonlinear absorbers [3] (usually solutions of organic dyes). We begin the discussion of this method by considering the transmission characteristic of a typical saturable absorber (Fig. 4a). At low incident power, the absorber transmission is constant, independent of the incident power. How- ever, at sufficiently high power density the transmission begins to increase, because the intense light partially depletes the ground state of the absorber. The power density necessary to decrease the absorption coefficient to one half of the low power value is called the saturation intensity I S. In organic dyes suitable for mode-locking purposes Is is of the order of 10 6 to several times ] 0 7 W/cm 2. Figure 4b explains the basic mechanism of the generation of short pulses in a laser with a saturable absorber. Consider two light pulses of different peak power I i > 12 - - Is, incident on the saturable absorber. The transmitted pulses differ from the incident pulses in two ways: First, the intensity ratio of the transmitted pulses, (11/I2)tr, exceeds the intensity ratio of the incident pulses, because the stronger pulse sees less absorption than the weaker pulse. Second, the transmitted pulses are both somewhat shortened, because the pulse wings suffer more losses than the pulse peak, as a result of the non-linear absorption characteristic. We have assumed here that the

SATURABLE ABSORBER

II 22

U Fig. 4. Transmission versus incident intensity for a saturable absorber (a). Interaction of short pulses with a saturable absorber (b)

absorber responds to the instantaneous pulse intensity. This is correct only when the recovery time of the absorption, e.g., the excited state lifetime, is short compared to the pulse duration. If this condition is not satisfied, the trailing part of the pulse interacts with absorbing centers already bleached by the pulse front, and the absorber is ineffective for the later parts of the pulse. For example, only the pulse front gets shortened during the interaction with the absorber, and the pulse assumes an asymmetric shape. Let us now discuss the pulse evolution in a laser with a saturable absorber [8]. First we consider a very early stage of laser action, just at the onset of stimulated emission. The phases of the cavity modes are initially distributed in a random way, and the radiation field in the cavity consists of many irregular fluctuations of low power (Fig. 5). As the power builds up, the peak intensity of the biggest fluctuation at some point will become comparable to the saturation intensity of the absorber, and this peak will start bleaching the absorber. In the nonlinear absorption regime the biggest peak always sees a

Mode-Locked Lasers and Ultrashort Light Pulses 285

(a/

M2 SA M1

/ / ' ' L /

I(t)

L/C

Ib)

t Z(t) I / L/C

(c)

t i(t) f / (d)

I / '-. I L/C

Fig. 5. Evolution of short pulses in a laser with saturable absorber. Schematic of a typical cavity configuration (a); low intensity regime, I4~I~, with random fluctuations (b); I~-l,, onset of discrimination of weak peaks (c); final energy distribution (d)

larger net gain than all the rest of the fluctuations, and this peak will grow at the fastest rate. Moreover, the growth rate itself increases because bleaching becomes more effective as the intensity builds up. One finally ends up with one big pulse having collected most of the energy. The laser gain is saturated by this pulse before the other smaller fluctuations have grown to a comparable level, so that the energy carried by the rest of the fluctuations remains very small compared to the energy of the main pulse. Thus the initially chaotic energy distribution is transformed by the saturable absorber into a single, well-defined pulse. The process of selectively amplifying the peak fluctuation and discriminating all weaker spikes corresponds in the. frequency domain to locking of the phases of the cavity modes to a constant value for all modes. Regarding the statistical nature of the initial radiation field, it might be expected that the final output of a passively mode-locked laser still shows some irregular behavior. In fact, there are a number of

imperfections typical of passively mode-locked systems. Imperfect discrimination of the weaker fluctuations leads to a background energy between the main pulses of the train. When more than one fluctuation of the initial field happened to have practically the same size, the absorber cannot discriminate them, and a pulse train consisting of multiple pulses in one round-trip interval will be obtained. Extensive experimental [9-11, 16] and theoretical [8, 12-14] work has dealt with the analysis of various types of imperfect mode-locking. We shall discuss some of these imperfections in more detail in Section 2.

Measurement of Ultrashort Pulses

Mode-locked lasers are capable of generating light pulses of only a few picoseconds in duration. The measurement of these ultrashort light pulses con- stitutes a difficult problem. Until recently the time resolution of commonly used photodetectors and real time oscilloscopes has limited direct measurement to pulses longer than 100 ps. In the last few years, electronic streak cameras with a time resolution of a few picoseconds [15] and less [16] have been developed, and these instruments at the present time represent the most powerful tool for analyzing ultrashort light pulses. However, so far, most of the picosecond measurements have been performed by means of indirect, nonlinear optical techniques such as the measurement of the intensity autocorrelation function. The two photon fluorescence method (TPF) [17] has been the most popular pulse-width measuring technique, because of its apparent simplicity. TPF data require a very careful discussion, and the pulse cannot be reconstructed unambiguously from the measurement of the autocorrelation function. The problem of pulse measurement by means of TPF has been extensively dealt with in the literature [ 14, 18-20]. Here, only a brief outline of the theoretical basis and of the experimental technique will be given. Figure 6 shows a schematic of a typical TPF experiment. The pulse to be analyzed is split into two pulses of equal intensity. Subsequently, the two beams are recombined in a cell containing a fluorescent dye. The lowest excited states of the dye correspond to twice the fundamental frequency of the light pulses, so that the dye cannot be excited by normal single photon absorption. However, at high

286 D. yon der Linde

--~I,. BEAM SPLITTER

Fig. 6. Schematic of TPF experiment. (Dotted area: no pulse overlap. Solid black area: pulse overlap zone)

Fir) r( t)

~,, f 2F~

I (f) F (7-)

(o)

T 1 / 2 ~- tp

(b)

4 ~ ~ tf[

�9 ,, 7-

power density, excitation by simultaneous absorption of two photons takes place. When the two pulses traverse the dye solution, a uniform fluorescent trace is generated in the region where the pulses do not overlap with each other. On the other hand, an enhanced fluorescence is obtained from the area where the two pulses meet. The width of the region of enhanced fluorescence is a measure of the pulse duration. Usually, the fluorescence is photographed with a high resolution camera, and the pulse width is determined from these photographs. Let us now discuss, how the TPF pattern is related to the characteristics of the pulse. It can be shown [ 18] that the fluorescence along the trace is given by

F(z) = 2F0(1 + 2G(Z)(z)/G(2)(O)), (12)

where z = 2z/v o. Fo is the single pass fluorescence of one pulse, z is the distance in the direction of the beams and v o is the group velocity of the pulse in the dye solution. G(2)(z) is the intensity autocorrelation function, defined by

+m

G(2)(z) = ~ J(t + z) J(t) dr. (13) - o o

J(t) is the intensity of the pulse. One can regard G(z)('c)/G(2)(O) a s a measure of the overlap of two identical pulses with a relative delay z. For zero delay the overlap has a maximum, and the fluorescence is 6 times the single pass fluorescence. When is large compared to the pulse width tp, there will be no overlap, and one measures the fluorescence generated independently by the two pulses, i.e., 2Fo.

l(t)

P 11

F(T) (C)

Fig. 7. Shape of TPF traces for various intensity distributions: bell-shaped pulse (a); continuous random fluctuations (b); pulse with amplitude substructure (c)

Figure 7 summarizes the properties of TPF traces for three cases of practical interest. First, the fluorescence is discussed for a simple bell-shaped pulse. As shown above, the peak fluorescence at r = 0 is three times the background fluorescence. The pulse width tp is proportional to the width of the fluorescence maximum, z~:

tp = ]~'~�89 (14)

The proportionality constant 7 depends on the pulse shape (typically 7 - 1.5). Note that the fluorescence peak is always symmetric, even when the pulse shape is asymmetric (G~2)(z) is symmetric by definition). Figure 7b shows the TPF trace for a continuous sequence of random fluctuations. F(z) looks very similar to the pulse case, except for the larger background fluorescence. Now the contrast ratio, R = F(O)/F(oo) is only 1.5, because even for arbitrarily large z there is always a certain amount of overlap between fluctuations travelling in +z and those travelling in - z direction. When a pulse is super- imposed on a certain level of quasi-continuous back-

P 7-


ground radiation, the contrast ratio will have a value between 1.5 and 3, and the amount of background energy can be inferred from the actual calue of R [14]. Finally, Fig. 7c shows a pulse exhibiting a rapidly varying substructure of mean duration t s. Now the TPF pattern has a very sharp peak, with a width related to the width of the substructure. The sharp spike rides on a broad peak, which reflects the total duration tv of the noisy light pulse. The top of the broad structure corresponds to a constrast of two. If there is no background energy, the full contrast at the top of the sharp spike is three, the same as for a smoothly varying pulse (first example: Fig. 7a). By a careful quantitative measurement of TPF traces, one can determine to which of these three categories the actual pulse belongs. The total pulse duration, and, if any, the width of the substructure and the background level can be estimated from quantitative TPF data. However, note that the detailed shape of the pulses cannot be inferred unambiguously from TPF experiments. The same is true for techniques using second harmonic generation [47, 2], which, like TPF, basically measure the intensity autocorrelation function (second-order correlation function). On the other hand, measurements of the approximate pulse shape were made using higher- order correlation functions and correlations of more general type [22]. A very useful technique employing higher-order correlations for pulse analysis is the ultrafast light gate of Duguay et aL [9]. The light gate is basically an electrodeless Kerr cell with a picosecond optical pulse replacing the high voltage trigger pulse of conven- tional Kerr cells. The gate is used as a shutter for ultrafast photography. Exposure times in the picosecond regime are readily obtained when the gate is triggered by a picosecond light pulse. Thus it was possible to take a stop-motion photograph of a frequency doubled picosecond pulse during its flight through a scattering medium [-26]. The analysis shows that the picture obtained in this experiment is related to a fourth-order correlation function, ~j2(t + z) J2(t) dt. On the other hand, when the gate is triggered by the second harmonic pulse and the fundamental pulse is photographed, then the picture is given by ~Ja(t +'c)J(t)dr, which is the function Auston [22] used in his pulse shape measurement. Thus it is possible to obtain a direct photograph of the pulse shape with the-help of the picosecond shutter.

Obviously, in pulse analysis techniques involving only correlation functions of the pulse inten~sity possible frequency modulation of the pulses cannot be seen. The presence of significant frequency modulation can be inferred by comparing the pulse width t v and the inverse frequency width 1/A vp of the pulses. If there is no frequency modulation, the pulse width bandwidth product, trA vv, is approximately unity. Large values of tpAvp indicate frequency modulation of the pulses. An ingenious method for measuring the frequency modulation of ultrashort light pulses has been described by Treacy [23].

2. Survey of Sources of Ultrashort Light Pulses

After the first report of laser mode-locking in 1964 by Hargrove et al. [24], who used acoustooptic loss modulation to mode-lock a He-Ne laser, the power of the mode-locking technique for generation of extremely short optical pulses has been demonstrated for a wide variety of lasers. In this section we review some of the common sources of ultrashort light pulses, and we discuss examples of techniques for further shortening the pulse width, and for increasing the peak power of mode-locked pulses.

Mode-Locked Solid State Lasers

As a typical example of a passively mode-locked solid state laser, we discuss here the neodymium glass laser, which has become one of the most useful sources of powerful picosecond light pulses. Nd- doped glass is a very attractive laser medium for generation of short pulses, because it offers a large bandwidth for laser gain, e.g., 200-300 cm-1, centered in the near infrared, at 2 = 1.06 gin. Passive mode-locking in this spectral region is no problem because very rapidly recovering saturable dyes are available for this wavelength. A typical cavity configuration of a passively mode-locked laser is shown in Fig. 5. The reflectivity of the output mirror is usually about 50 To, and the initial absorption of the saturable dye is adjusted to approximately 25 Too. The active material and the cell containing the saturable dye are placed at the Brewster angle to avoid unwanted mode-selection effects that would narrow the output frequency spectrum and broaden the pulses. Frequently, a dye cell in contact with one of the mirrors is used, because a thin cell separated from the mirror leads to multiple pulses per cavity

288 D. vonde r Linde

Fig. 8. Oscilloscope trace showing the output pulse train from a mode-locked Nd : glass laser

round-trip, with a spacing given by twice the distance of the dye cell from the mirror. Figure 8 shows an output pulse train from a mode- locked Nd: glass laser. The spacing between adjacent pulses of the train is 10 ns, which corresponds to a full round-trip in a cavity of 150 cm in length. The total duration of the pulse train is about 500 ns. Typically, the energy of a single pulse at the train maximum is 1 mJ, in a beam of about 0.1 cm 2 cross section. Reproducible regular pulse trains are obtained only under carefully controlled conditions. For example, a well-defined stable laser cavity is essential, because irreproducible transverse mode distributions lead to irregular pulse trains. Best results are obtained, when the laser is pumped close to threshold and operated at low power. However, the pulse structure can be quite complicated even when the pulse trains look regular on a nanosecond time scale. Experimentally, the following deviations from perfect mode-locking have been observed in passively mode-locked lasers: (i) background energy between the main pulses of the train; (ii) much larger frequency width than the inverse of the pulse duration, indicating a frequency modulation; (iii) variation of the pulse width along the train; (iv) development of an irregular subpicosecond structure. In the last few years the various forms of imperfect mode-locking have been investigated extensively, and the pulse evolution in passively mode-locked lasers is now quite well understood. In the following we briefly summarize some of the main results. It has been confirmed independently by several workers [10, 21, 25] that under good mode-locking conditions the background energy is typically 10% of the total energy of the pulse train, or less. The background consists of weak peaks, randomly distributed over the round-trip time [26, 40]. However, when the

laser is operated at high power, nonlinear losses (see Section 3) can limit further growth of the pulses, while the weak background peaks still see the full laser gain. Under these conditions the background energy can become a sizable fraction of the total energy. Let us now turn to the results concerning the variation of the pulses along the train [9-11, 16, 41]. In the front of the pulse train at low power (100 to 300 MW/cm 2) the pulse duration is about 5 ps (bandwidth limited), and it has been shown that the pulses have a smoothly varying, approximately Gaussian shape at this stage [10]. Later in the train, e.g. at the maximum, the pulses deteriorate. The overall duration increases to more than 10 ps, and a rapidly varying amplitude substructure develops. At the same time a strong frequency modulation is observed, and the pulse width bandwidth product increases to a value of 20 or more. To illustrate the variation of the pulses, a TPF trace and a spectrum is shown in Fig. 9a and 9b for a single pulse selected from the train front and from the end, respectively [10]. Ruby lasers can be mode-locked in the same way as Nd: glass lasers. Because of the narrower gain bandwidth of ruby (~-10cm-~), the pulse duration is longer for ruby, typically 20-30 ps [27]. The gain bandwidth in Nd : YAG lasers of only about 4 cm- 1 restricts the pulse duration to 30-50 ps. YAG lasers can be operated in a cw fashion, and active modulation techniques are well suited for mode-locking the cw YAG laser [28]. Note that pulses of any desired duration (longer than minimum width) can be generated by narrowing the gain bandwidth by means of suitable frequency filters [29]. When only a small fraction of the total gain band is utilized for mode-locking, frequency tuning of the pulses over the gain band is possible. This technique plays an important role in mode-locked organic dye lasers, which will be discussed next.


I

o

t

b-

ol

'1ram

lmm

~ t l psec PULSE

(a)

o

(b)

/ - 500 0

(UO-U)GHz

PULSE SPECTRUM

~-- IOOGHz

50O

Fig. 9. Examples of experimental TPF traces and spectra for single pulses from a Nd: glass laser. Bandwidth-limited pulse (tp A vp-~ 0.5) from train front (a); pulse with amplitude substructure (ts < 1 ps) from the end of the train (b). For both pulses a contrast of R = 3 was measured

Mode-Locked Dye Lasers

Dye lasers are very suitable for generation of ultrashort light pulses because organic dyes provide efficient laser gain over a wide frequency range, typically 1000-3000 cm- 1. This large bandwidth allows significant frequency tuning even for relatively wide-band ultrashort pulses. A variety of suitable dyes is available, and a large spectral range can be covered with the same laser system simply by changing the active laser dye. A very convenient method for mode-locking of dye lasers is direct modulation of the laser gain, and this technique was used in the first successful mode- locked dye laser [30]. In this scheme the output of another mode-locked laser serves to pump the active dye solution which is placed in a separate optical resonator. The pump pulses produce a rapidly varying gain in the dye laser, because the gain is substantial only for a very short time after the absorption of a picosecond pump pulse (excited state lifetimes of laser dyes are typically a few ns). To obtain mode-locking, the round-trip time in the dye laser cavity has to match the repetition time of the pump pulse train. Frequency tuning of the dye laser output can be achieved by varying the concentration of the dye solution, or by introducing frequency selective elements into the cavity, such as a grating reflector or a Fabry-P6rot interferometer. The duration of

the frequency tunable pulses generated in this way is similar to the duration of the pump pulses, or even shorter. A variety of dye lasers have been mode- locked by gain modulation, using either mode- locked ruby or Nd: lasers, or their respective second harmonics [31]. An attractive feature of laser- pumped dye lasers is the following: With the same mode-locked pump source, one can excite two or more different dye lasers at the same time to obtain synchronized ultrashort pulses at different frequencies. Organic dye lasers can also be pumped with fast flashlamps (flash duration ~- 10 gs), and these systems can be passively mode-locked by saturable absorbers [32] in a very similar way as neodymium or ruby lasers. The tuning range for a particular pair of laser dye and saturable absorber is typically a few hundred A, e.g., about 400 • for the rhodamine 6 G laser. With this laser the generation of bandwidth- limited pulses of 3-5 ps has been reported [33] for operation at not too high power. With various combinations of laser dyes and saturable absorbers the spectral region from 5800-7000A has been continuously covered [33]. In passively mode-locked dye lasers the energy of a single pulse in the train is typically 10 - s j (about 10-100 times less than in ruby or Nd:glass lasers). It has been shown that practically all the energy

290 D. vonder Linde

resides in the pulses; background energy is negligibly small [34]. Recently, passive mode-locking of a cw rhodamine 6G dye laser has been demonstrated, and continuous trains of frequency tunable pulses of 1.5-2 ps duration and of 50-100 W peak power were observed [35]. It is very likely that continuous mode- locking can be extended to other types of dye lasers, and tunable cw picosecond dye lasers will certainly make possible many new applications of ultrashort light pulses.

Mode-Locked Gas Lasers

It has been mentioned above that laser mode-locking was demonstrated for the first time with an internally modulated cw He-Ne gas laser. Active locking techniques such as internal loss modulation are particularly well suited for mode-locking of cw gas lasers; passive locking with saturable absorbers is more difficult because of the relatively low power of most cw gas lasers. In general, the gain bandwidth in gas lasers is only about 1-10GHz. Consequently, the duration of mode-locked pulses from gas lasers is limited typically to 10 -9 to 10 -1~ s. For example, mode- locking of the argon ion laser gives pulses of about 200 ps. With an average power of 1 W and a repetition frequency of 100 MHz the peak power is of the order of 50 W for such a laser. Subnanosecond pulses of much higher peak power have been generated recently with pulsed CO2 gas lasers. Laser action at gas pressures exceeding 1 atm is possible when the laser gas mixture (typically CO2:N2:He~_10:10:80) is excited by a pulsed transversal electrical discharge [36]. The line width at atmospheric pressure is several GHz [-37]. The CO2 laser has been mode-locked both by active and passive techniques. With a GaAs crystal as an acoustooptic modulator pulse durations of 0.5 1 ns have been reported [38], and the peak power of the pulses was about 1 MW. Much higher peak power was obtained with passively mode-locked TEA-COz lasers (Transversely Excited Atmospheric pressure), using SF 6 gas as a saturable absorber. With the CO2-SF6 system pulses of 1-5 ns duration with peak power of the order of 108 W were generated [39]. It is interesting to consider the lasing spectrum of COz gas. The lasing band of CO2 at 10.6 gm consists of a series of closely spaced lines with a frequency

spacing of about 60 GHz (-~ 2 cm-1). These lines correspond to the rotational sublevels of the vibra- tional levels involved in the laser transition (00~ ---, 10~ When the gas pressure is increased, the rotational components are broadened. At a pressure of about t0atm the rotational lines overlap, and continuous laser gain is available over a frequency range of about 1 THz (1012 s-1). If efficient modulators can be found, ultrashort pulses of approximately 10-12 s duration are feasible at 10.6 gin. The laser gain and the storable energy for high pressure CO2 lasers compare favorably with solid state lasers (gain: a few 10 -2 cm -1, energy density: - l 0 -2 J/cm3). The efficiency of the CO2 laser is excellent (_~ 10%). Thus high pressure CO2 is a promising laser medium for efficient generation of high power, ultrashort pulses. Another advantage of CO2 gas is that there is no serious material damage problem like, for instance, in a solid state medium at very high power.

Pulse Amplification and Pulse Shortenin 9

Very often it is impossible to lock all the modes within the gain bandwidth of a laser, and the duration of the output pulses is considerably longer than the inverse gain bandwidth. Also, in many cases, lasers have to be operated at relatively low power in order to get the best results. Hence, there is a need for techniques for further shortening the duration and for increasing the power of mode-locked pulses. The power is readily increased by passing the pulses through a laser amplifier. Most effective amplification is achieved when only a single pulse out of the entire output pulse train is selected for amplification. In this case, a very large increase of the pulse power can be obtained because the energy stored in the amplifier medium is supplied just to one pulse instead of being distributed over all pulses of the train. A schematic of a system for selection and amplification of a single ultrashort light pulse is depicted in Fig. 10. For pulse extraction from a mode-locked pulse train, a very fast-responding switching system is needed because of the short repetition time of only about 10- 8 s. The switch consists of a Pockels cell, or a Kerr cell, placed between two crossed polarizing prisms. Properly timed, fast rising high voltage pulses for actuating the electrooptic cell can be generated by means of a spark gap in conjunction with a simple pulse-forming network [43].


. . , , 1 1 1 ~ 1 1 . . . .

INPUT PULSE TRAIN

E.O. CELL

A M P L I F I E R

-.1 , 0 I I - - , H V

SPARK GAP

AMPLIFIED SINGLE PULSE

Fig. 10. Schematic of a system for extraction and amplification for a single pulse

Significant amplification of a single pulse can be achieved with relatively modest effort. For example, a Nd:glass laser amplifier of 30 cm length gives an energy gain of 10-20. Because of the low energy of the input pulses (< 10 -3 J / c m 2) gain saturation is no problem, and one can use multiple transits of the pulse for more efficient extraction of energy from the amplifier. The shape and the duration of the pulse does not change during linear amplification. However, effective reduction of the pulse duration can be obtained by nonlinear amplification in a two- component medium consisting of a laser amplifier and saturable absorbers. Pulse compression in such a system on a nanosecond time scale was studied by Basov et al. [44]. Shortening of picosecond pulses in a two-component medium has been reported by Penzkofer et al. [45]. Figure 11 shows a schematic of their experimental set-up. The arrangement consists of a laser amplifier with a saturable absorber at both ends. Suitable mirrors allow the input pulse to make multiple transits through the amplifier and the absorbers. Pulse shortening in a nonlinear absorber has been discussed in Section ~. Here the same mechanism is operative. One starts with a well defined input pulse extracted from the output pulse train of a mode- locked Nd:laser. The power density of the input pulse matches the saturation intensity of the saturable absorbers, to ensure optimum non-linear interaction. The concentration of the saturable absorber can be chosen to be very high, e.g., To -~ 10 -2, and the pulse

shortening effect in a single transit through one of the absorbers is quite large, e.g. a factor of about 1.5 [45]. During the interaction with the absorber, the peak power of the pulse drops below the saturation intensity. The laser amplifier serves for restoring the pulse power to a value Ipeak --- Ix, so that the pulse matches the optimum nonlinear region of the subsequent absorber. Because of the strong pulse shortening during one absorber transit, only a few complete round-trips in the system are necessary to achieve drastic pulse shortening. Starting with pulses of about 8 ps duration, output pulses of subpicosecond duration have been observed for only four transits. One can also take advantage of the inherent non- linearity of light amplification at high power to achieve pulse shortening. In a laser amplifier, amplification becomes nonlinear when the pulse energy is of the order of hco/2a, because of the onset of gain saturation (co is the optical frequency, a is the stimulated emission cross section). Under certain conditions, the saturation effect can lead to significant pulse sharpening [46]. In the regime of nonlinear amplification the laser gain is saturated during interaction with the leading part of the pulse, and the later parts see greatly reduced amplification. This effect leads to a selective amplification of the pulse front and sharpening of the pulse. Parametric light amplification and stimulated Raman scattering show similar gain nonlinearities, and these effects can also be used for generation of ultrashort pulses. For example, saturation of the Raman gain plays an

I NPUT P U L S E _ _ / f N k ~ -

OUTPUT PULSE

A M P L I F I E R S. ABSORBER

S. ABSORBER Fig. 1 l. Pulse shortening in a two-component system formed by a laser amplifier and two saturable absorbers

292 D. yon der Linde

MODULATOR DISPERSIVE DELAY LINE

I--1

,wO-r

" l - - =tp , . J t AWrn " J't , ~o0_ w

_L___ :/'" I

I (t)

I (t) I ( t)

JL_ "t 5

Fig. t2. Frequency modulation and compression of optical pulses: incident pulse with constant frequency (left); generation of a frequency sweep by the modulator (center); compression of the modulated pulse by the delay line (right)

important role in the evolution of ultrashort pulses in backward stimulated Raman scattering [47], and in the generation of subpicosecond Stokes pulses in Raman lasers pumped by a train of picosecond light pulses [48]. This chapter is concluded with a brief discussion of the compression of frequency-modulated optical pulses. The compression technique allows a drastic shortening of the pulses, in principle without energy losses, so that the peak power increases at the same rate as the pulse width decreases. The idea of pulse compression is explained in Fig. 12. A compression experiment requires two steps. First, a frequency modulation is applied to the pulse, for example, a linear frequency modulation: The frequency is made to increase linearly with time. The total frequency change along the pulse, Avm, is assumed to be large compared to the initial frequency width of the pulse, A vp ~- 1/tp. In the second step, the modulated pulse is sent through a delay line which has a frequency dependent transit time. We assume that the delay time of a wave group depends linearly on the frequency of the group, and that the delay line is adjusted to provide a time delay equal to the input pulse duration tp for wave groups differing in frequency by an amount Av,,. With this adjustment, the trailing edge of the pulse will catch up with the pulse front when the pulses traverse the delay line, and a very short output pulse is obtained. In the ideal case, the

duration of the output pulse is given by the inverse of the applied frequency sweep, (tp)ou t -~ 1/A v,,, and output power is increased by a factor of t /( tp)o,t = A v,,]A vp. The assumption of a linear frequency sweep is of course an idealization. In an actual compression experiment one has usually a more complicated frequency variation. For example, passive frequency modulation by means of the optical Kerr effect [49] gives a frequency variation proportional to the time derivative of the pulse envelope. This situation is indicated by the dotted curve in Fig. 12. It is seen that the frequency variation is approximately linear only close to the pulse maximum; the frequency sweep changes sign in the pulse wings. Consequently, only a fraction of the pulse close to the maximum can be compressed, and the pulse wings are broadened. Experimentally, pulse compression by a factor of 2 was obtained with mode-locked pulses from a He-Ne laser, actively frequency modulated by means of an electrooptic crystal [50]. Treacy made use of the inherent frequency modulation of the pulses in mode-locked Nd:glass lasers to demonstrate pulse compression effects [51]. Pulse compression by a factor of four was reported for mode-locked Nd : laser pulses, passively frequency modulated in CS 2 [52]. It should be noted that compression techniques require a careful matching of the dispersion of the delay line to the amount of frequency modulation


present in the pulses. The higher the desired compression ratio, the narrower are the tolerances for the adjustment of the delay line. The main problem of the compression technique is the generation of a well defined, reproducible frequency modulation.

3. Limitations of Production and Propagation

We turn now to a discussion of some effects which limit the power and the duration of ultrashort light pulses. First, there are limiting effects directly related to the sources or to the method of pulse generation. Second, there are limitations of a more general kind, characteristic of the interaction of very powerful and/or very short pulses with matter. The discussion is restricted to a few examples of practical interest.

Efficiency of Mode-Locking

The pulse duration in mode-locked lasers is ultimately limited by the gain bandwidth of the active material. However, it has been noted above that in most cases the gain bandwidth is only poorly utilized; the duration of the output pulses is usually much longer than the inverse gain bandwidth. The reason for this is in many cases a competition between the modulator, trying to couple together many modes, and opposing mechanisms, tending to narrow the frequency spectrum of the laser output. The final pulse width reflects a trade-off between frequency broadening and narrowing processes. As an example we discuss here ~he spectral narrowing that takes place during the linear amplification regime of a pulsed, passively mode-locked laser, for instance, a Nd : glass laser mode-locked by a saturable absorber. The absorber is effective only at relatively high power, e.g., when the intensity peaks become comparable with the saturation intensity. It takes as much as several thousand round-trips for the radiation to build up to this intensity level. During this long phase of linear amplification, the initially broad frequency spectrum narrows considerably as a result of the natural mode selection (preferential amplification of frequency components close to the gain maximum). In the case of the Nd:glass laser this effect reduces the spectral width from an initial value of about 200 cm- 1 to only a few wavenumbers. Thus at the end of the linear amplification regime the duration of the intensity fluctuations is roughly l0 ps, much longer than at the beginning of laser

action. In the subsequent nonlinear absorption regime the strongest intensity fluctuation is isolated, and the weaker peaks are eliminated, as discussed in Section 1.2. However, the pulse shortening effect of the saturable absorber is too small to compensate the broadening of the fluctuations that had occurred during the linear amplification. This explains why in a pulsed, passively mode-locked Nd:glass laser the final pulse duration is typically 6 ps [10], which corresponds to a band-width of only about 3 cm- 1 much less than the gain bandwidth. Of course, the same effects are operative in other pulsed, passively locked systems. The rate of spectral narrowing is determined by the curvature of the top of the gain curve. With a suitable frequency filter, a gain curve with a very flat top can be produced, and the speed of spectral narrowing is significantly slowed down. Indeed, much shorter pulses (~_ 1.5 ps) are obtained, when the curvature of the top of the gain profile is carefully compensated [53]. The efficiency of saturable absorbers for generation of ultrashort pulses is ultimately limited by the recovery time. To obtain good mode-locking the recovery time has to be of the same order as the desired pulse duration, or shorter. If this condition is not satisfied, the saturable absorber cannot efficiently isolate the strongest intensity fluctuation because the absorber responds only to a mean value of the fluctuations over a time interval of the order of the recovery time. As a result, only groups of fluctuations are isolated [42], with a total width approximately equal to the recovery time. In addi- tion, the probability of multiple pulsing increases [-8], when the recovery is slow. Unfortunately, the usefulness of very fast saturable absorbers is limited by the following consideration. The saturation intensity I s is inversely proportional to the recovery time z. The regime where the saturable absorber is effective shifts to higher intensities as the recovery time decreases. For r = 10 -13 s I s is several times 10 8 W/cm z even for a transition with unity oscillator strength. For such a high value of Is the laser gain saturates before significant absorber bleaching takes place, and mode-locking is not possible.

Broadening of Short Pulses in Dispersive Media

In a medium with a constant refractive index no an optical pulse propagates with a velocity v=c/n o without changing the shape of the envelope.

2 9 4 D. y o n d e r L i n d e

When the frequency dependence of the index is taken into account in first order, the pulse still maintains its shape; but the propagation velocity is now the group velocity:

vo= C(no + Oo ~ ) - t . (15)

This approximation is valid when the frequency spread of the wavepacket is small, e.g., when the pulse is not too short. For very short pulses this approximation breaks down. One finds that the pulse broadens monotonically during propagation the pulse broadening can be described in a general way by the expression

{ 2z62p12] -~ (16) At==Ato[1 + \ (Ato) 2 1 ] '

where A t z and A t o is the r.m.s, width of the pulse of z and z = 0, respectively. The quantity Am is the r.m.s, spectral width, and p = A ~o A t o > �89 is the time- bandwidth product. 52 is a measure of the material dispersion:

5 1 [ dn 09~ d2n) (17) 2 = c ~ - - ~ - o + 2 de) 2 '

Let us consider a specific example. In optical glass 62 is typically 3 x 10-28 sec2/cm. For an initial pulse duration of tp(0) = 10-13 s the pulse width increases by 50%, when the pulse travels only a distance of z = z~r~t = 5 cm in optical glass. For z >> z~,it the pulse duration increases linearly with the travelled distance. Figure 13 illustrates the broadening of a Gaussian pulse in optical glass for different initial widths.

tp(Z)/tp(O)

0.1 ps

10 / p = 1/2

ps p=20

/ ~ o2p~

E I P I I I I t I I ~ z 0 40 20 50 40 50 60 70 80 90 100

(cm)

F ig . 13. B r o a d e n i n g o f a G a u s s i a n s h a p e d o p t i c a l p u l s e in g l a s s

It should be noted that the broadening rate is particularly large when the pulse is not bandwidth- limited. From (16) it is seen that the critical distance for dispersive broadening is proportional to the time-bandwidth product p. Figure 13 clearly demonstrates the difficulty of generating subpicosecond pulses in a glass laser and other laser systems of comparable dispersion; the pulses would spread rapidly when travelling back and forth through the material of the cavity. Two methods for overcoming dispersion broadening are mentioned here. First, the normal dispersion of materials (dn/&o >0) can be compensated by introducing into the laser cavity a system with anomalous dispersion (dn/dco < 0), e.g., optical grat- ings [51, 52] or suitable interferometers [54]. Second, by sending the broadened pulse through a system of opposite dispersion, the original pulse can in principle be restored in a way closely analogous to the compression of a frequency modulated pulse.

Limitation of Peak Power

In this section some limitations of the peak power in the generation and propagation of optical pulses are outlined. When materials are exposed to very intense optical pulses, serious damage and destruction of the material can occur, and these effects set an upper limit on the useful power. Optical damage on material surfaces and of the bulk can result from absorbing impurities, which lead to thermally induced fractures. These damage effects depend on the composition and preparation of the material. On the other hand, damage occurs also in very pure materials. There are indications that electron avalanche breakdown induced by the strong electric field of a powerful optical pulse plays an important role in damage of pure materials [55]. The electron avalanche mechanism predicts that the damage threshold for short optical pulses is governed by the pulse energy. Experimentally, the threshold energy density for transparent optical media is of the order of 103 J/cm z [56]. Since the energy is the significant parameter for damage, the critical power density increases when the pulse duration is decreased. For example, in the nanosecond regime, the maximum tolerable energy limits the power density to approximately several GW/cm 2. Much higher power density is possible from the point of view of damage, when picosecond pulses are used. With the above


value of the threshold energy, the critical power density is estimated to be of the order of 1014 W / c m 2.

Although at first sight this high value of the threshold power density seems to indicate that damage is not a serious problem, damage does occur when self- focussing takes place. Self-focussing of optical beams results from the intensity dependence of the refractive index, which becomes significant at high power densities. The change of refractive index 6n induced by an optical field is given by

6 n : n2 g f f { ) , (18)

where E2(t) denotes the time averaged square of the electric field. The constant n2 :is called the nonlinear refractive index (n 2 > 0). The self-induced change of refractive index leads to a focussing of the beam in a characteristic distance z s [573,

d (no) ~ (19) Zs~- T \ 6 n / '

where d is the beam diameter, and n o is the low power refractive index. As an example, the focussing length in glass (n2 ~ 2 x 10 -18 cm2/v 2 [9]) for an optical pulse of 105~ W/cm z peak intensity is calculated. For a beam diameter of 0.3 c m z s is found to be about 33 cm. Self-focussing leads to extremely high local power density, and the damage threshold can be readily exceeded, even if the intensity in the undistorted beam is much lower than the critical value. In fact, per- manent microscopic damage tracks are commonly observed in the active material of laser amplifiers when long amplifier rods are used. The material damage and the beam distortions that occur when self-focussing is present give rise to severe scattering and diffraction losses during propagation, and to avoid these effects the power density has to be kept below a critical value determined by the length of the material and its nonlinear index. The intensity dependence of the refi'active index affects the pulse propagation also in another way, even if self-focussing and damage is avoided. The self-induced index change gives rise to a change of the pulse phase with time (self-phase modulation). The phase variation can be written [52] as

q~(t)= 2~v~ zn2E2(t), (20) c

where v0 is the optical frequency, and z is the distance travelled by the pulse in the medium. The self-phase modulation leads to broadening of the frequency spectrum. The contribution of the self-phase modulation to the spectral width is approximately given by [9, 52]

(Av)spM_ voz 6n (21) c fp

For example, in a Nd: glass laser, the self-phase modulation leads to a frequency broadening per unit length ofd/dz(Av)spM~- 1.5 x 109 c m -1 s -1 for a 5 ps optical pulse with a peak intensity of 10 9 W / c m 2.

For a laser rod of 20 cm length we get a frequency broadening of 60 GHz (2 cm-1) per round-trip at this intensity. Under these conditions, it takes only a few round-trips in the cavity to produce an amount of frequency broadening much larger than the initial frequency width of the pulse. The pulse duration and the pulse shape are not directly affected by the phase modulation. However, it was pointed out in the previous section that dispersive pulse broadening is particularly strong for pulses with a broad frequency spectrum (p ~> 1). Therefore a strongly phase modulated pulse is rapidly distorted when travelling in a dispersive medium. Cooperation of self-phase modulation and dispersion explains the experimentally observed rapid pulse deterioration along the train in pulsed mode-locked lasers, in particular in Nd : glass lasers, where these effects limit the useful power density to a few hundred MW/cm z. To avoid deteriorated pulses, one has to select pulses from the train that do not exceed this intensity level. For laser amplifiers self- phase modulation can be minimized by keeping the distance travelled in material as short as possible, and by increasing the beam cross section to limit the power density.

References

I. T.H. Maiman: Nature (Lond.) 187, 495 (1960) R.J.Collins, D.F.Nelson, A.L.Schawlow, W.Bond, C.G.B. Garrett, W. Kaiser: Phys. Rev. Letters 5, 303 (t960)

2. F.J.McClung, R.W.Hellworth: J. Appl. Phys. 33, 838 (t962) 3. H.W.Mocker, R.J.Collins: Appl. Phys. Letters 7, 270 (1965)

A. J. DeMaria, D. A. Stetser, H. Heynau: Appl. Phys. Letters 8, 174 (1966)

4. H.Kogelnik, T.Li: Appl. Optics 5, 1550 (1966) 5. M.H.Crowell: IEEE J. Quant. Electr. QE-1, 12 (1965) 6. P.W.Smith: Proc. IEEE 58, 1342 (1970), and references

therein

296 D. vonder Linde

7. D.J. Kuizenga, A.E.Siegman: IEEE J. Quant. Electr. QE-6, 694 (1970)

8. P.G.Kryukov, V.S.Letokhov: IEEE J. Quant. Electr. QE-8, 766 (1972) B. Ya. Zel'dovich, T. I. Kuznetsova: Sov. Phys. Uspekhi 15, 25 (1972)

9. M.A.Duguay, J.W.Hansen, S.L.Shapiro: IEEE J. Quant. Electr. QE-6, 725 (1970)

10. D.vonderLinde: IEEE J. Quant. Electr. QE-8, 328 (1972) 1t. M.C.Richardson: IEEE J. Quant. Electr. QE-9, 768 (1973) 12. V.S.Letokhov: Soy. Phys. JETP 28, 562 (1969) 13. T.I.Kuznetsova: Soy. Phys. JETP 30, 904 (1970) 14. R.H. Picard, P.Schweitzer: Phys. Rev. 1, 1803 (1970) 15. D.J.Bradley, B.Liddy, W.Sibbett, W.E.Sleat: Appl. Phys.

Letters 20, 219 (1972) M. Ya. Schelev, M.C. Richardson, A. J. Alcock: Rev. Sci. Instr. 43, 1819 (1972)

16. N.G. Basov et al., Conference on Nonlinear Optics, Minsk (1972); preprint FIAN No. 82 (1972)

17. J.A. Giordmaine, P.M. Rentzepis, S.L. Shapiro, K.W. Wecht: Appl. Phys. Letters 11, 216 (1967)

18. J.R.Klauder, M.A.Duguay, J.A.Giordmaine, S.L.Shapiro: Appl. Phys. Letters 13, 174 (1968)

19. T.I.Kuznetsova: Sov. Phys. JETP 28, 1303 (1969) 20. H.E.Rowe, T.Li: IEEE J. Quant. Elcetr. QE-6, 49 (1970) 21. J.A.Armstrong: Appl. Phys. Letters 10, 16 (1967)

H.P.Weber: J. Appl. Phys. 38, 231 (1967) 22. D.H.Auston: Appl. Phys. Letters 18, 249 (1971)

D.vonderLinde, A.Laubereau: Opt. Commun. 3, 279 (1971) 23. E.B.Treacy: J. Appl. Phys. 42, 3848 (1971) 24. L.E. Hargrove, R.L. Fork, M.A. Pollack: Appl. Phys. Letters

5, 4 (1964) 25. R.J. Harrach, T.D. MacVicar, G.I. Kachen, L.L.Steinmetz:

Opt. Commun. 5, 175 (1972) 26. M.A.Duguay, A.T.Mattick: Appl. Opt. 10, 2162 (1971) 27. M.W.McGeoch: Opt. Commun. 7, 116 (1973) 28. D. Kuizenga, A.E. Siegman: IEEE J. Quant. Electr. QE-6, 709

(1970) 29. D.vonderLinde, M.Maier, W.Kaiser: Phys. Rev. 178, l[

(1969) 30. W.H. Glenn, M.J. Brienza, A.J. DeMaria: AppI. Phys. Letters

12, 54 (1968) 31. C.V.Shank, E.P.Ippen, in Dye Lasers, ed. by F.P.Schfifer

(Springer Verlag, Berlin, Heidelberg, New York 1973)

32. W.Schmidt, F.P.Sch~ifer: Phys. Letters 26A, 558 (1968) 33. E.G.Arthurs, D.J. Bradley, A. G. Roddie: Appl. Phys. Letters

20, 125 (1972); Appl. Phys. Letters 19, 480 (1971) 34. D.J.Bradley, B.Liddy, A.G.Roddie, W.Sibbett, W.E.Sleat:

Opt. Commun. 3, 426 (1971) 35. E.P.Ippen, C.V.Shank, A.Dienes: Appl. Phys. Letters 21, 348

(1972) 36. A.J.Beaulieu: Appl. Phys. Letters 16, 504 (1970) 37. T.K.McCubbin, R.Darone, J.Sorell: Appl. Phys. Letters 8,

118 (1966) 38. R.L.Abrams, O.R.Wood: Appl. Phys. Letters 19, 518 (1971) 39. R.Fortin, F.Rheault, J. Gilbert, M.Blanchard, J.L.Lacham-

bre: Canad. J. Phys. 51, 414 (1973) 40. A.A.Malyutin, M.Ya.Shchelev: JETP Letters 9, 266 (1969) 41. V.V. Korobkin, A.A. Malyutin, A.M. Prokhorov: JETP Let-

ters 12, 150 (1970) 42. V.I. Malyshev, A. A. Sychev, V. A. Babenkov: JETP Letters 13,

419(1971) 43. D.vonderLinde, O.Bernecker, A.Laubereau: Opt. Commun.

2, 215 (1970) 44. N.G. Basov, P.G. Kryukov, V.S. Letokhov, Yu. A. Matveets:

Sov. Phys. JETP 29, 830 (1969) 45. A. Penzkofer, D.vonderLinde, A.Laubereau, W.Kaiser: Appl.

Phys. Letters 21 (1972) 46. P.G.Kryukov, V.S.Letokhov: Sov. Phys.-Uspekhi 12, 64t

(1970) 47. M.Maier, W.Kaiser, J.A.Giordmaine: Phys. Rev. 177, 580

(1969) 48. M.J.Colles: Appl. Phys. Letters 19, 23 (1971) 49. R.A. Fisher, P.L. Kelley, T. K. Gustafson: Appl. Phys. Letters

14, 140 (1969) 50. M.A.Duguay, J.W.Hansen: Appl. Phys. Letters 14, 14 (1969) 51. E.B.Treacy: Phys. Letters 28A, 34 (1968) 52. A.Laubereau, D.vonderLinde: Z. Naturforsch. 25a, 1626

(1970) 53. D.vonderLinde, K.F.Rodgers: Opt. Commun. 8, 9t (1973) 54. F. Gires, P. Tournois: C. R. Acad. Sci. (Paris) 258, 6112 (1964) 55. D.W.Fradin, M.Bass: Appl. Phys. Letters 22, 206 (1973)

E.Yablonovich, N.Bloembergen: Phys. Rev. Letters 29, 907 (1972)

56. S.A. Belozerov, G.M.Zverev, V.S.Naumov, V.A. Pasbkov: Sov. Phys. JETP 35, 158 (1972)

57. P.L.Kelley: Phys. Rev. Letters 15, 1005 (1965)

mode-locked lasers and ultrashort light pulses

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