# Pulsed Lasers

This is a continuation from the previous tutorial - **laser power**.

In the CW (continuous wave) operation of a laser, the laser is pumped continuously at a constant pumping level, and the loss of the laser cavity is also kept constant so that the laser has a constant output power when it reaches steady state.

A laser can also be pulsed to deliver short optical pulses at its output. In pulsed operation, the net gain seen by the laser field is not kept constant but is temporally varied by pulse pumping the gain medium and/or by modulating the cavity loss.

Depending on the laser material, the cavity design, and the technique employed for the pulsed operation, laser pulses of temporal pulsewidths ranging from the order of microseconds to the order of femtoseconds with large ranges of pulse repetition rates, from single shots to gigahertz, and pulse energies, from femtojoules per pulse to joules per pulse, can be generated.

Many effective techniques have been developed for the generation of laser pulses. The simplest approach is ** gain switching**, while the most important and most commonly employed techniques are

**and**

*\(Q\) switching***.**

*mode locking*Here we only discuss the basic principles of these techniques.

**Gain Switching**

Gain switching is a technique that is used to generate very short laser pulses through the control of oscillator transients. The concept of gain switching is straightforward: the gain parameter \(\Gamma\mathrm{g}\) of a laser is switched on rapidly above the laser threshold, which is defined by the loss parameter \(\gamma_\text{c}\), by fast, pulsed pumping so that a very short laser pulse is generated through the transient effects of the laser oscillator.

In a gain-switched laser, the gain medium is pumped so fast that the population inversion builds up more rapidly than the photons in the cavity. The gain is thus raised considerably above the threshold before the laser field starts to build up from the initial noise level in the cavity.

The transient effects that follow under the excessively high-gain condition result in the generation of a short, powerful laser pulse.

Because the intracavity laser photon density grows exponentially in time with the net gain, for a gain-switched laser pulse to have a short risetime it is only necessary to create a large excess gain above the threshold before stimulated emission starts to reduce it by depleting the population inversion.

This condition requires hard and fast pumping. In addition, a long lifetime \(\tau_2\) for the upper laser level in comparison to the pump-pulse duration also helps in building up the excess population inversion.

To make sure of a short falltime for the gain-switched pulse, first the gain has to be terminated when the photon density builds up to its peak value, and then the intracavity photons have to be depleted quickly.

The first of these two conditions requires short pumping duration, while the second requires a short photon lifetime, corresponding to a large photon decay rate \(\gamma_\text{c}\).

In addition, if gain saturation occurs, the laser gain can be terminated even more rapidly, thus reducing the pulse falltime substantially. Figure 11-11 illustrates the basic concept of gain switching.

From the above discussion, one can see that the conditions for the generation of very short laser pulses by gain switching are

- A large excess population inversion at the onset of laser oscillation
- A short photon lifetime \(\tau_\text{c}\)
- Sufficient gain saturation after pulse buildup

The technical aspect of gain switching is in the choice of the pump and laser parameters best to satisfy these conditions.

Successful gain switching of a laser can be accomplished by choosing

- A very short and strong pump
- A short laser cavity
- A laser medium that has a low saturation intensity and a long fluorescence lifetime for the upper laser level in comparison to the pump-pulse duration

Physical constraints sometimes make it impossible to fulfill all these requirements. However, it is not necessary to satisfy all of these conditions fully before a short laser pulse can be generated by gain switching.

For example, by pumping the gain medium hard enough, a very short pulse can be generated even when the pump pulse is longer than the fluorescence lifetime of the upper laser level.

Certainly, if the fluorescence lifetime is long, the gain medium can be pumped less hard for a desired pulsewidth, or a shorter pulse can be generated with the same pump.

Therefore, a typical gain-switched laser that generates ultrashort pulses is a laser with a very short cavity pumped by a strong pump pulse that has a temporal duration on the order of or shorter than the fluorescence lifetime of the gain medium.

Short laser pulses are ideal sources for optically pumping a secondary gain-switched laser. This approach has been demonstrated for gain switching in solid-state lasers, dye lasers, and semiconductor lasers.

In gain-switched semiconductor lasers that are electrically pumped, the pump current pulses are typically one order of magnitude shorter than the carrier lifetime of the semiconductor gain medium.

The dynamics of the laser oscillator are solely responsible for the behavior of a gain-switched laser pulse. Therefore, if not well-controlled, many transient phenomena, such as spiking and relaxation oscillation, can take place. The output then consists of a series of spikes if the relaxation oscillation is not damped.

However, by choosing a cavity of a proper photon lifetime and by controlling the level and duration of pumping, a single clean pulse without relaxation oscillation spikes can be generated.

It is then only necessary to pump the laser medium as fast as possible in a cavity as short as possible to generate a very short pulse. No special optical elements are required in the laser cavity.

By repetitively gain switching a laser with a periodic train of pump pulses, a train of regularly spaced, gain-switched pulses can be generated. The gain switching technique has been used to generate single pulses of temporal widths ranging from 1 picosecond in an optically pumped short-cavity GaAs laser to a few hundred nanoseconds in CO_{2} lasers.

To generate an ultrashort laser pulse by gain switching, it is important to have an extremely small cavity lifetime \(\tau_\text{c}\) because the shortest cavity photon decay time is limited by \(\tau_\text{c}\).

Sometimes \(\tau_\text{c}\) can be smaller than the cavity round-trip time \(T\) because of high intracavity loss or high output-coupling loss.

Unless the small \(\tau_\text{c}\) is caused by a high distributed loss, however, the shortest pulse that can be generated by gain switching from a laser with \(\tau_\text{c}\lt{T}\) is limited by \(T\) rather than \(\tau_\text{c}\).

This is because it takes at least one round trip to deplete all the intracavity photons by output coupling through the mirrors.

This limitation applies also to \(Q\) switching, which is also a transient technique. It does not apply to mode locking, which does not reply on transient phenomena to generate ultrashort laser pulses.

Therefore, the cavity length usually imposes a direct physical limitation on the laser dynamics so that the shortest pulsewidth generated by the transient technique of gain switching or \(Q\) switching can only be as short as the cavity round-trip time.

**\(Q\) Switching**

\(Q\) switching is the most widely used technique for the generation of high-intensity giant laser pulses of short duration. Similar to gain switching, \(Q\) switching relies on the transient dynamics of a laser to generate very short pules. However, it does not require extremely fast pumping, as does the technique of gain switching. In fact, it is possible to pump the gain medium continuously while switching the cavity \(Q\) factor repetitively to generate a periodic train of \(Q\)-switched pulses.

The principle of \(Q\) switching is based on delaying the onset of laser oscillation relative to the start of pumping to accumulate a large population inversion.

This task is accomplished by reducing the laser cavity \(Q\) factor in the early stage of pumping to prohibit the depletion of population inversion caused by premature laser oscillation.

Upon reaching a large population inversion, the \(Q\) factor is rapidly increased, resulting in a large excess gain above threshold and a burst of high-intensity short pulse driven by the transient dynamics of the laser.

Because \(Q=\omega/\gamma_\text{c}\) according to (11-28) [refer to the resonant optical cavities tutorial], modulating the cavity \(Q\) factor is equivalent to modulating the cavity loss rate \(\gamma_\text{c}\).

The basic principle of \(Q\) switching is illustrated in Figure 11-12 below. In contrast to gain switching where the cavity loss rate is kept constant, \(\gamma_\text{c}(t)\) for a \(Q\)-switched laser is a time-varying function.

In the ** pumping phase** of a \(Q\)-switched laser, population inversion builds up without being depleted by stimulated emission.

Clearly, if the pump-pulse duration is much shorter than the fluorescence lifetime of the gain medium, gain switching can be very effective, and there is no need for \(Q\) switching. This is the condition discussed above for gain switching.

When the pump pulse is long, the gain grows slowly. However, given sufficient time, the gain can still accumulate to a substantial value if \(\tau_2\) is sufficiently long. This is the situation when \(Q\) switching can be effectively implemented to generate a short pulse.

Therefore, a large \(\tau_2\) is even more desirable for \(Q\) switching than for gain switching.

For most efficient utilization of the pump energy, the pump duration should not be too much longer than \(\tau_2\) although it does not have to be short.

Because of spontaneous relaxation, population inversion, if not depleted by laser oscillation, cannot continue to build up much longer than a period of \(\tau_2\). For a repetitively \(Q\)-switched laser under continuous pumping, this fact means that the overall efficiency of the laser drops when the repetition rate is below \(1/\tau_2\).

The major difference between \(Q\) switching and gain switching is in the pumping phase when \(\gamma_\text{c}(t)\) is kept high in the case of \(Q\) switching. In the lasing phase, gain-switched and \(Q\)-switched lasers are driven by the same transient laser dynamics initiated by the initial excess population inversion. This can be seen by comparing Figure 11-12 to Figure 11-11.

It is clear that the conditions for the generation of very short laser pulses by gain switching discussed earlier apply equally well to \(Q\) switching.

\(Q\) switching differs from gain switching only in the technical aspect of how the large initial excess population inversion is achieved.

Gain switching relies on a very fast and strong pump pulse to achieve a high peak gain before stimulated emission starts.

This condition is not required for \(Q\) switching as the high cavity loss of a \(Q\)-switched laser in the pumping phase prohibits the laser from oscillating. Thus the requirement on the pump for \(Q\) switching is less demanding than that for gain switching.

In comparison to a gain-switched laser, the technical demand in a \(Q\)-switched laser is shifted from the pump to the \(Q\) switch.

In order to generate a very short pulse by \(Q\) switching, it is therefore desirable to have

- An effective \(Q\) switch that switches the cavity \(Q\) very rapidly from a very low value to a high value at the moment the gain reaches a desired high level
- A short laser cavity
- A laser medium that has a low saturation intensity and a long fluorescence lifetime

Similarly to the case of gain switching, these requirements do not have to be fully satisfied, but a \(Q\)-switched laser pulse cannot be shorter than the cavity round-trip time and a laser with a high saturation intensity has to be operated at a high power level.

In the ideal situation, the value of \(\gamma_\text{c}(t)\) is switched abruptly from a high level, \(\gamma_\text{cp}\), for the pumping phase to a low level, \(\gamma_\text{cl}\), for the ** lasing phase**.

In practice, because it takes a time delay, \(t_\text{d}\), shown in Figure 11-12, after the time of switching into the lasing phase for the \(Q\)-switched pulse to build up to a significant level, the condition of ideal \(Q\) switching can be approximated by ** fast \(Q\) switching** where the transition from the pumping phase to the lasing phase is completed within a time duration less than the time delay of the pulse buildup.

In this ideal, or nearly ideal, situation of fast \(Q\) switching, the characteristics of the \(Q\)-switched pulse are completely determined by the initial pumping ratio at the onset of the lasing phase:

\[\tag{11-93}r=\frac{\Gamma\mathrm{g}_\text{i}}{\gamma_\text{cl}}\]

where \(\mathrm{g}_\text{i}\) is the ** initial gain parameter** at the onset of the lasing phase, as illustrated in Figure 11-12.

In a \(Q\)-switched laser, the initial gain parameter \(\mathrm{g}_\text{i}\) for the lasing phase actually shoots over the threshold level defined by \(\gamma_\text{cl}\) because of the \(Q\)-switching action.

Therefore, the parameter \(r\) defined in (11-93) has a somewhat different meaning from that defined in (11-76) [refer to the laser power tutorial].

Clearly, we always have \(r\gt1\) for \(Q\)-switching operation.

The ** peak output power** of a \(Q\)-switched pulse is approximately given by

\[\tag{11-94}P_\text{pk}\approx\frac{\tau_2}{\tau_\text{cl}}(r-\ln{r}-1)P_\text{out}^\text{sat}\]

where \(\tau_\text{cl}=1/\gamma_\text{cl}\) is the photon lifetime in the lasing phase and \(P_\text{out}^\text{sat}=\mathcal{V}_\text{mode}S_\text{sat}h\nu\gamma_\text{out}\) as defined in (11-82) [refer to the laser power tutorial].

For \(1.2\lt{r}\lt5\), the FWHM (full width half maximum) pulsewidth, \(\Delta{t}_\text{ps}\), can be quite accurately approximated by the following formula:

\[\tag{11-95}\Delta{t}_\text{ps}=\frac{2.5}{(r-\ln{r}-1)^{1/2}}\tau_\text{cl}\]

which is obtained by approximate analytical fitting of \(Q\)-switched pulses.

The energy of a \(Q\)-switched pulse can be approximated by

\[\tag{11-96}U\approx{P}_\text{pk}\Delta{t}_\text{ps}\]

Clearly, the larger the value of \(r\), the more dramatic the \(Q\)-switching behavior is, resulting in a more powerful \(Q\)-switched pulse with a higher peak power and a smaller pulsewidth.

Depending on the technique used to modulate the cavity \(Q\) factor, the type of \(Q\) switching can be generally categorized as active or passive.

In active \(Q\) switching, the \(Q\) switch that modulates the cavity \(Q\) is controlled by an externally applied signal. Various techniques have been developed for active \(Q\) switching, including mechanical modulation, electro-optic modulation, acousto-optic modulation, and magneto-optic modulation.

Today, most of the active \(Q\)-switched lasers use electro-optic or acousto-optic modulators, which modulate the cavity \(Q\) by modulating the loss in the cavity. Both of these two types of modulators are controlled by external electronic signals, which have the advantages of stability and flexibility of \(Q\) modulation and ease of synchronization with measurement apparatus.

Typical electro-optic modulators are based on the Pockels effect. They have fast switching times in the nanosecond range with a large \(Q\) modulation and can be controlled with precise timing, but they often require a large switching voltage and are difficult to operate at a high repetition rate.

The acousto-optic modulators are Bragg diffractors driven by an RF signal. They have a slower switching speed and a smaller \(Q\) modulation than electro-optic modulators, thus producing longer pulses. They can be easily operated at a high and variable repetition rate and are primarily used in continuously pumped, repetitively \(Q\)-switched lasers at a repetition rate in the kilohertz range.

High-frequency intracavity electro-optic modulation based on the electroabsorption effect can be applied to a semiconductor laser for the generation of actively \(Q\)-switched picosecond pulses at a high repetition rate in the gigahertz range.

In a passively \(Q\)-switched laser, the \(Q\) switch is typically a nonlinear optical element that changes the cavity loss by responding directly to the intracavity laser intensity.

The most commonly used passive \(Q\) switch is a saturable absorber, the optical properties of which are discussed in the nonlinear optical modulators and switches tutorial. With a proper arrangement, any all-optical switch, such as a Kerr lens, also discussed in the nonlinear optical modulators and switches tutorial, can function as a passive \(Q\) switch.

Passive \(Q\) switching has the advantage of being simple and inexpensive, but the pulses generated by passive \(Q\) switching are often subject to larger intensity fluctuations and timing jitter than those generated by active \(Q\) switching.

Solid-state lasers, such as Nd : YAG, ruby, and Ti : sapphire, are primary candidates for \(Q\) switching because they normally have a long fluorescence lifetime. The pulsewidth of a pulse generated by a \(Q\)-switched solid-state laser is typically in the range of a few nanoseconds to hundreds of nanoseconds.

Many useful laser materials, such as laser dyes and semiconductors, have a very small \(\tau_2\) on the order of nanoseconds or less. They are not easy to \(Q\) switch unless a very efficient pump source and a very fast \(Q\) switch are used. As a consequence, the \(Q\)-switched pulses generated by these lasers are typically on the order of tens or hundreds of picoseconds although pulses as short as a few picoseconds at a repetition rate as high as a few tens of gigahertz have been generated by passive \(Q\) switching of semiconductor lasers.

It is also possible to combine \(Q\) switching with mode locking in a \(Q\)-switched mode-locked laser to generate a train of very short mode-locked pulses under a long \(Q\)-switched envelope.

**Example 11-6**

The characteristics of the Nd : YAG microchip laser described in Examples 11-1 to 11-5 [refer to the resonant optical cavities tutorial, the laser oscillation tutorial, and the laser power tutorial] in ideal \(Q\)-switching or gain-switching operation are considered in this example.

For the gain-switching operation, all of the laser parameters, including those of the cavity and the gain medium, remain the same as those described in Examples 11-1 to 11-5 except that the pump is an optical pulse at the pump wavelength of 808 nm.

For the \(Q\)-switching operation, a \(Q\) switch introduces an additional high loss to the laser in the pumping phase, but the laser parameters in the lasing phase are the same as those for the gain-switching operation. A possible \(Q\)-switching mechanism is passive \(Q\) switching by codoping the Nd : YAG with Cr^{4+} ions as the saturable absorber.

For direct comparison with CW operation, we take the pumping ratio to be \(r=1.85\), as found in Example 11-4 [refer to the laser power tutorial] for a CW output power of 1 mW.

(a) What are the required conditions for the laser to be nearly ideally \(Q\)-switched?

(b) Find the peak power, pulsewidth, and pulse energy of the ideally \(Q\)-switched pulse. Compare the peak power of the \(Q\)-switched pulse to that of the CW power of 1 mW.

(c) What are the required conditions for the laser to be nearly ideally gain switched?

(d) What are the characteristics of the ideally gain-switched pulse?

**(a)**

Because the laser parameters in the lasing phase are the same as those of the CW laser described in Examples 11-1 to 11-5, we have \(\gamma_\text{cl}=\gamma_\text{c}=5.78\times10^8\text{ s}^{-1}\) and \(\tau_\text{cl}=\tau_\text{c}=1.73\text{ ns}\) from Example 11-1 [refer to the resonant optical cavities tutorial].

Two of the three conditions for ideal \(Q\) switching, namely, a short laser cavity and a gain medium with a long fluorescence lifetime and a low saturation intensity, are already met by this laser. Therefore, the only requirement that has to be considered is an effective \(Q\) switch that switches the cavity from a high loss of \(\gamma_\text{cp}\) to a low loss of \(\gamma_\text{cl}\).

First, \(\gamma_\text{cp}\) has to be larger than \(\Gamma\mathrm{g}_\text{i}\), which for a pumping ratio of \(r=1.85\) is \(\Gamma\mathrm{g}_\text{i}=r\gamma_\text{cl}=1.85\gamma_\text{cl}\). Thus, an effective \(Q\) switch has to keep \(\gamma_\text{cp}\gt1.85\gamma_\text{cl}=1.07\times10^9\text{ s}^{-1}\).

Next, the \(Q\) switch has to switch fast enough. Quantitively, the \(Q\) switch has to switch the cavity loss from the high value of \(\gamma_\text{cp}\) to the low value of \(\gamma_\text{cl}\) within a time interval \(\Delta{t}_\text{QS}\) that is shorter than the pulse delay time \(t_\text{d}\), as shown in Figure 11-12, for the process to qualify as fast \(Q\) switching.

The pulse delay time can be estimated by considering the fact that the pulse grows from a seed of spontaneous emission to the saturation photon density exponentially with a rate of \(\Gamma\mathrm{g}_\text{i}-\gamma_\text{cl}\).

The saturation photon density is \(S_\text{sat}\), which has a value of \(S_\text{sat}=8.15\times10^{17}\text{ m}^{-3}\) found in Example 11-4 [refer to the laser power tutorial] for this laser.

The seed of spontaneous emission is one photon per mode, which translates into a spontaneous photon density of \(1/\mathcal{V}_\text{mode}\), with \(\mathcal{V}_\text{mode}=1.57\times10^{-11}\text{ m}^3\), also found in Example 11-4 [refer to the laser power tutorial] for this laser.

If we take \(t_\text{d}\) to be the time it takes the photon density of the oscillating laser mode to grow exponentially with a rate of \(\Gamma\mathrm{g}_\text{i}-\gamma_\text{cl}\) from \(1/\mathcal{V}_\text{mode}\) to \(S_\text{sat}\), then \(t_\text{d}\) can be found as

\[\begin{align}t_\text{d}&=\frac{1}{\Gamma\mathrm{g}_\text{i}-\gamma_\text{cl}}\ln\frac{S_\text{sat}}{1/\mathcal{V}_\text{mode}}=\frac{\tau_\text{cl}}{r-1}\ln(S_\text{sat}\mathcal{V}_\text{mode})\\&=\frac{1.73}{1.85-1}\ln(8.15\times10^{17}\times1.57\times10^{-11})\text{ ns}=33.3\text{ ns}\end{align}\]

Therefore, the requirements for ideal \(Q\) switching of this laser at the given pumping ratio of \(r=1.85\) are \(\gamma_\text{cp}\gt1.07\times10^9\text{ s}^{-1}\) and \(\Delta{t}_\text{QS}\ll33.3\text{ ns}\).

**(b)**

From Example 11-4 [refer to the laser power tutorial], we find that \(P_\text{out}^\text{sat}=1.18\text{ mW}\) and \(\tau_2=240\text{ μs}\) for this laser. Therefore, from (11-94), the peak power of the ideally \(Q\)-switched pulse is

\[\begin{align}P_\text{pk}&\approx\frac{\tau_2}{\tau_\text{cl}}(r-\ln{r}-1)P_\text{out}^\text{sat}\\&=\frac{240\times10^{-6}}{1.73\times10^{-9}}\times(1.85-\ln1.85-1)\times1.18\times10^{-3}\text{ W}=38.4\text{ W}\end{align}\]

Compared to the 1 mW output power of the laser in CW operation, the peak power of this \(Q\)-switched pulse is \(3.84\times10^4\) times higher primarily because of the fact that \(\tau_2\) is five orders of magnitude larger than \(\tau_\text{cl}\). This demonstrates that a gain medium that has a large \(\tau_2\) makes a good \(Q\)-switched laser.

The pulsewidth is found from (11-95) to be

\[\Delta{t}_\text{ps}=\frac{2.5}{(r-\ln{r}-1)^{1/2}}\tau_\text{cl}=\frac{2.5}{(1.85-\ln1.85-1)^{1/2}}\times1.73\text{ ns}=8.93\text{ ns}\]

Compared to the cavity round-trip time of \(T=6.07\text{ ps}\) found in Example 11-1 [refer to the resonant optical cavities tutorial], which sets the ultimate lower limit for the pulsewidth of a \(Q\)-switched pulse, this pulsewidth is quite long. It can be shortened by pumping the laser higher to increase the pumping ratio \(r\) and by using an output-coupling mirror of a lower reflectivity to reduce \(\tau_\text{cl}\).

The pulse energy is simply

\[U\approx{P}_\text{pk}\Delta{t}_\text{ps}=38.4\times8.93\times10^{-9}\text{ J}=343\text{ nJ}\]

This pulse energy is not very high because of the small amount of energy that can be stored in the small gain volume of the microchip laser. To increase the \(Q\)-switched pulse energy, one must increase the volume of the gain medium as well as the mode volume of the oscillating laser field.

**(c)**

Two of the three conditions for ideal gain switching are the same as those for ideal \(Q\) switching, which are already met by this laser. The only condition remaining to be considered is a very short and strong pump for gain switching.

Whether a pump pulse is short or not is relative to the fluorescence lifetime \(\tau_2\) of the gain medium. Because \(\tau_2\) is the relaxation time constant of the excited population in the upper laser level, the pump energy can be efficiently stored in the population inversion of the gain medium if the pump-pulse duration is much smaller than \(\tau_2\).

If \(\tau_2\) is much smaller than the pump-pulse duration, then the pump energy cannot be efficiently stored in the population inversion of the gain medium because population relaxation during the pumping process is significant.

From this discussion, we understand that for ideal gain switching, it is necessary that the pump pulse be much shorter than \(\tau_2\) of the gain medium. It does not have to be extremely short, however. A pump pulse that has a duration of \(\tau_2/10\) is short enough, while one that has a duration of \(\tau_2/100\) is close to an ideal delta pulse pump.

For ideal gain switching of this Nd : YAG laser with \(\tau_2=240\text{ μs}\), we need a short pump pulse of a few microseconds or less in duration that has a sufficiently high energy to pump the laser to the desired pumping ratio of \(r=1.85\) in such a short duration.

**(d)**

The characteristics of an ideally gain-switched pulse are the same as those of an ideally \(Q\)-switched pulse found in (b). The only difference is in the pump pulse. Ideal \(Q\) switching can be accomplished with a relatively long pump pulse so long as the \(Q\) switch satisfies the conditions discussed in (a).

**Mode Locking**

Mode locking is the most important technique for the generation of repetitive, ultrashort laser pulses. The principle of mode locking is very different from those of gain switching and \(Q\) switching in that it is not based on the transient dynamics of a laser. Instead, a mode-locked laser operates in a dynamic steady state.

** A pulsed laser can oscillate in multiple longitudinal modes regardless of whether the gain medium is homogeneously or inhomogeneously broadened**. Mode locking refers to the situation when all of the oscillating longitudinal modes of a laser are locked in phase.

When this phase locking is accomplished, constructive interference of all of the oscillating modes results in a short pulse circulating inside the cavity, which is regeneratively amplified by the gain medium after periodically delivering an output pulse through an output-coupling mirror in each round trip.

The mode-locking operation is accomplished by a nonlinear optical element known as the ** mode locker** that is placed inside the laser cavity, typically near one end of the cavity if the laser has the configuration of a linear cavity.

Viewed in the frequency domain, mode locking is a process that generates a train of short laser pulses by locking multiple longitudinal laser modes in phase. The function of the mode locker in the frequency domain is thus to lock the phases of the oscillating modes together through nonlinear interactions among the mode fields.

In the time domain, the mode-locking process can be understood as a regenerative pulse-generating process by which a short pulse circulating inside the laser cavity is formed when the laser reaches steady state. The action of the mode locker in the time domain resembles that of a pulse-shaping optical shutter that opens periodically in synchronism with the arrival at the mode locker of the laser pulse circulating in the cavity. Consequently, the output of a mode-locked laser is a train of regularly spaced pulses of identical pulse envelope.

The simplest case of multimode oscillation is when there are only two oscillating longitudinal modes of frequencies \(\omega_1\) and \(\omega_2\). Then, the total laser field at a fixed location is

\[\tag{11-97}E(t)=\mathcal{E}_1\text{e}^{\text{i}\varphi_1(t)}\text{e}^{-\text{i}\omega_1t}+\mathcal{E}_2\text{e}^{\text{i}\varphi_2(t)}\text{e}^{-\text{i}\omega_2t}\]

where \(\mathcal{E}_1\) and \(\mathcal{E}_2\) are the magnitudes of the field amplitudes and \(\varphi_1\) and \(\varphi_2\) are the phases.

With all the phase information included in \(\varphi_1\) and \(\varphi_2\), \(\mathcal{E}_1\) and \(\mathcal{E}_2\) are postive, real quantities. The intensity of this laser is given by

\[\tag{11-98}\begin{align}I(t)&=2c\epsilon_0n|E(t)|^2\\&=2c\epsilon_0n\{\mathcal{E}_1^2+\mathcal{E}_2^2+2\mathcal{E}_1\mathcal{E}_2\cos[(\omega_1-\omega_2)t-\varphi_1(t)+\varphi_2(t)] \}\end{align}\]

In general, the phases can vary with time.

If \(\varphi_1(t)\) and \(\varphi_2(t)\) vary randomly with time on a characteristic time scale that is shorter than \(2\pi/(\omega_1-\omega_2)\), the beat note of the two frequencies cannot be observed even with a very fast detector. In this situation, the output of the laser has a constant intensity that is the incoherent sum of the intensities of the individual modes. This situation simply represents the ordinary multimode oscillation of a CW laser.

If \(\varphi_1\) and \(\varphi_2\) are time independent, the laser intensity given in (11-98) becomes periodically modulated with a period of \(2\pi/(\omega_1-\omega_2)\) defined by the beat frequency, as shown in Figure 11-13(a).

The modulation depth of this intensity profile depends on the ratio between \(\mathcal{E}_1\) and \(\mathcal{E}_2\). When \(\mathcal{E}_1=\mathcal{E}_2\), the modulation depth is \(100\%\) with \(I_\text{min}=0\). In this situation, \(I(t)\) resembles a train of periodic "pulses" that have a duty cycle of \(50\%\) and a peak intensity of twice the average intensity. This is simply ** coherent mode beating** and is the best one can do with two oscillating modes.

The periodic intensity profile created by two-mode beating, which is shown in Figure 11-13(a), is certainly far from what we normally expect from a train of mode-locked pulses.

As the number of modes that are locked in phase increases, the characteristics of periodic pulses become increasingly apparent in the output of the laser, as demonstrated in Figure 11-13(b).

At a given pulse repetition rate, we can reduce the pulse duty cyle, by increasing the number of modes. In general, a practical useful mode-locked laser oscillates in a large number of modes.

For a laser of many oscillating modes, we have the following laser field:

\[\tag{11-99}E(t)=\sum_q\mathcal{E}_q\text{e}^{\text{i}\varphi_q}\text{e}^{-\text{i}\omega_qt}\]

where again \(\mathcal{E}_q\) are taken to be positive, real quantities representing the magnitudes of the field amplitudes, and the summation is taken over all of the oscillating modes.

As discussed above, if the phases \(\varphi_q\) vary randomly with time, (11-99) describes the field of a CW multimode laser, which is of no interest here.

For mode locking, we consider the situation when \(\varphi_q\) are time independent. In the case of only two oscillating modes with \(\varphi_1\) and \(\varphi_2\) being time-independent constants, the phase difference \(\varphi_1-\varphi_2\) merely shifts the mode beating pattern with respect to the origin of the time axis and is of no physical significance.

With more than two oscillating modes, however, only one phase can be arbitrary because the relative phases among different modes are significant. Consequently, the temporal characteristics of the combined laser field described in (11-99) depend on the phase relationships among the oscillating modes, as well as on the distribution of the field amplitudes \(\mathcal{E}_q\) and the frequency spacing between neighboring modes.

We consider the situation when the oscillating laser modes are equally spaced with a longitudinal mode spacing of \(\Delta\omega_\text{L}\). The magnitudes and phases of the mode fields are functions of the mode frequencies, but not all of the phases vary with time. Their spectral distribution can be described by a complex spectral envelope function \(\mathcal{E}(\omega)\) through

\[\tag{11-100}\mathcal{E}_q\text{e}^{\text{i}\varphi_q}=\frac{\Delta\omega_\text{L}}{2\pi}\mathcal{E}(\omega_q-\omega_0)\]

For simplicity, we have chosen \(\omega_0\) to be a longitudinal mode frequency near the center of the spectrum. Thus we have \(\omega_q=\omega_0+n\Delta\omega_\text{L}\), and the total field in (11-99) can then be transformed as follows:

\[\tag{11-101}\begin{align}E(t)&=\frac{\Delta\omega_\text{L}}{2\pi}\sum_{q=-\infty}^\infty\mathcal{E}(\omega_q-\omega_0)\text{e}^{-\text{i}\omega_qt}\\&=\frac{\Delta\omega_\text{L}}{2\pi}\text{e}^{-\text{i}\omega_0t}\sum_{n=-\infty}^\infty\mathcal{E}(n\Delta\omega_\text{L})\text{e}^{-\text{i}n\Delta\omega_\text{L}t}\\&=\frac{\Delta\omega_\text{L}}{2\pi}\text{e}^{-\text{i}\omega_0t}\mathcal{F}^{-1}\mathcal{F}\left\{\sum_{n=-\infty}^\infty\mathcal{E}(n\Delta\omega_\text{L})\text{e}^{-\text{i}n\Delta\omega_\text{L}t}\right\}\\&=\Delta\omega_\text{L}\text{e}^{-\text{i}\omega_0t}\mathcal{F}^{-1}\left\{\sum_{n=-\infty}^\infty\mathcal{E}(n\Delta\omega_\text{L})\delta(\omega-n\Delta\omega_\text{L})\right\}\\&=\text{e}^{-\text{i}\omega_0t}\mathcal{F}^{-1}\left\{\mathcal{E}(\omega)\cdot\Delta\omega_\text{L}\sum_{n=-\infty}^\infty\delta(\omega-n\Delta\omega_\text{L})\right\}\\&=\text{e}^{-\text{i}\omega_0t}\mathcal{E}(t)*\sum_{m=-\infty}^\infty\delta(t-mT)\\&=\text{e}^{-\text{i}\omega_0t}\sum_{m=-\infty}^\infty\mathcal{E}(t-mT)\end{align}\]

where \(T=2\pi/\Delta\omega_\text{L}\), and

\[\tag{11-102}\mathcal{E}(t)=\mathcal{F}^{-1}\{\mathcal{E}(\omega)\}=\frac{1}{2\pi}\displaystyle\int\limits_{-\infty}^\infty\mathcal{E}(\omega)\text{e}^{-\text{i}\omega{t}}\text{d}\omega\]

In (10-101) and (10-102), \(\mathcal{F}\{\cdot\}\) means taking the Fourier transform from the time domain to the frequency domain, and \(\mathcal{F}^{-1}\{\cdot\}\) means taking the inverse Fourier transform from the frequency domain to back to the time domain.

The result in (11-101) is obtained under the assumption that the phases \(\varphi_q\) do not vary with time. It shows that when phases \(\varphi_q\) do not vary with time, the total field \(E(t)\) is a periodic function of time with a period \(T\) determined by the mode spacing and a temporal profile \(\mathcal{E}(t)\) determined by the spectral envelope.

Figure 11-14 shows the spectral and temporal characteristics of the field and intensity profiles of a completely mode-locked laser, in which all of the longitudinal modes are locked to the same phase.

The spectral width, \(\Delta\omega_\text{ps}\), of a laser pulse is defined as the FWHM of the spectral intensity distribution, \(I(\omega)\), as shown in Figure 11-14(b). Correspondingly, the temporal pulsewidth, \(\Delta{t}_\text{ps}\), is defined as the FWHM of the temporal intensity profile of an individual pulse, as illustrated in Figure 11-14(d).

Because of the Fourier-transform relationship, given in (11-102), between the temporal field profile, \(\mathcal{E}(t)\), and the spectral field profile, \(\mathcal{E}(\omega)\), the temporal and spectral widths of a pulse are subject to the following relation:

\[\tag{11-103}\Delta\nu_\text{ps}\Delta{t}_\text{ps}\ge{K}\]

where \(\Delta\nu_\text{ps}=\Delta\omega_\text{ps}/2\pi\) and \(K\) is a constant of the order of unity that depends on the pulse shape.

For any pulse with a given pulse shape, the best one can hope for is \(\Delta\nu_\text{ps}\Delta{t}_\text{ps}=K\). When this is accomplished, the pulse is said to be ** Fourier-Transform limited**, or simply

**.**

*transform limited*A transform-limited pulse is one that has the smallest pulsewidth \(\Delta{t}_\text{ps}=K/\Delta\nu_\text{ps}\) for a given pulse spectral width \(\Delta\nu_\text{ps}\).

Two pulse shapes are of most interest for mode-locked lasers. One is the Gaussian pulse, and the other is the \(\text{sech}^2\) pulse.

For the Gaussian pulse, both \(\mathcal{E}(\omega)\) and \(\mathcal{E}(t)\) are Gaussian functions because the Fourier transform of a Gaussian function is another Gaussian function, and both its temporal intensity profile and spectral intensity profile are also Gaussian.

For a \(\text{sech}^2\) pulse, both \(\mathcal{E}(\omega)\) and \(\mathcal{E}(t)\) are its sech functions because the Fourier transform of a sech function is another sech function, and both its temporal intensity profile and its spectral intensity profile are \(\text{sech}^2\) functions.

The transform-limit constants are \(K=2\ln2/\pi=0.4413\) for a Gaussian pulse and \(K=4\ln^2(1+\sqrt{2})/\pi^2=0.3148\) for a \(\text{sech}^2\) pulse.

Actively mode-locked pulses tend to have Gaussian shapes, whereas passively mode-locked pulses often have \(\text{sech}^2\) shapes.

When all of the modes of a laser are locked to a common phase, we can set \(\varphi_q=\varphi_0=0\) because a constant common phase has no physical significance. This is the ideal situation of ** complete mode locking**.

From (11-100), we find that the spectral envelope is a real function when \(\varphi_q=0\). This implies that \(\mathcal{E}(\omega)=\mathcal{E}^*(\omega)\) and

\[\tag{11-104}\begin{align}\mathcal{E}(t)&=\frac{1}{2\pi}\displaystyle\int\limits_{-\infty}^\infty\mathcal{E}(\omega)\text{e}^{-\text{i}\omega{t}}\text{d}\omega\\&=\frac{1}{2\pi}\displaystyle\int\limits_{-\infty}^\infty\mathcal{E}^*(\omega)\text{e}^{-\text{i}\omega{t}}\text{d}\omega\\&=\mathcal{E}^*(-t)\end{align}\]

Therefore, \(I(t)=I(-t)\) if the laser pulse is completely mode locked.

From the above discussions, it can be concluded that ** a completely mode-locked laser pulse has a symmetric temporal intensity profile and is transform limited. It does not necessarily have a symmetric spectral intensity profile, but an asymmetric temporal pulse shape or a deviation from the transform limit signifies incomplete mode locking**.

The reverse is not true, however, because ** a transform-limited pulse is not necessarily completely mode locked**. When the longitudinal laser modes are not locked in phase, pulses can still be formed, but with less than ideal characteristics.

A completely mode-locked pulse, being transform limited, satisfies the condition \(\Delta{t}_\text{ps}\Delta\nu_\text{ps}=K\). Because thenumber of oscillating modes can be estimated with

\[\tag{11-105}N\approx\frac{\Delta\nu_\text{ps}}{\Delta\nu_\text{L}}\]

The temporal width of a mode-locked pulse is inversely proportional to the number of oscillating modes:

\[\tag{11-106}\Delta{t}_\text{ps}\approx\frac{K}{N\Delta\nu_\text{L}}=K\frac{T}{N}\]

At a fixed longitudinal mode spacing \(\Delta\nu_\text{L}\), hence a fixed pulse repetition rate \(f_\text{ps}=1/T\), the pulsewidth can be shortened by increasing the number of oscillating modes.

It is common to expect a pulsewidth that is two to five orders of magnitude smaller than the pulse spacing in a train of mode-locked pulses. The relation in (11-106) indicates that this requires locking of hundreds to hundreds of thousands of oscillating modes.

An inhomogeneously broadened laser naturally oscillates in multiple longitudinal modes. In such a laser, the mode locker only has to lock these modes in phase to produce a train of mode-locked pulses.

However, many mode-locked lasers that produce ultrashort pulses are homogeneously broadened. In the free-running steady state of a homogeneously broadened laser, only one longitudinal mode will oscillate because of homogeneous saturation across the gain medium.

Even though it is possible to force multimode oscillation in a homogeneously broadened laser when it is pulsed, the homogeneously broadened gain medium has a natural tendency to narrow the spectral bandwidth of the oscillating laser field.

Therefore, besides locking the phases of the oscillating laser modes together, the mode locker has the function of expanding the spectral width of the laser pulse to counteract the spectral narrowing effect of the gain medium.

For the pulses generated by a given mode-locked laser, the pulse spectral bandwidth \(\Delta\nu_\text{ps}\) is ultimately limited by the spontaneous linewidth \(\Delta\nu\) of the gain medium because \(\Delta\nu\) sets the limit for the gain bandwidth of the laser.

Therefore, the mode-locked pulses that can be generated from a given laser, regardless of whether it is homogeneously or inhomogeneously broadened, are subject to the following absolute limitation:

\[\tag{11-107}\Delta{t}_\text{ps}\ge\frac{K}{\Delta\nu_\text{ps}}\ge\frac{K}{\Delta\nu}\]

where \(\Delta\nu\) has the values listed in Table 10-1 [refer to the optical transitions for laser amplifiers tutorial] for many representative laser gain media.

For most mode-locked lasers, only a fraction of the laser gain bandwidth is utilized so that \(\Delta\nu_\text{ps}\) is only a fraction of \(\Delta\nu\). This fraction of bandwidth utilization depends on a number of operating parameters, including the modulation strength and the modulation frequency of the mode locker, as well as the type of mode locker used. Increasing this fraction is the key to reducing the temporal pulsewidth of mode-locked pulses.

A continuously mode-locked laser delivers a steady train of short pulses at a constant average output power \(\bar{P}\), while each pulse has a high peak power \(P_\text{pk}\). Effectively, the energy of laser output in each pulse repetition period \(T\) is concentrated within the duration of the pulsewidth \(\Delta{t}_\text{ps}\).

Therefore, the peak power of the pulses is enhanced over the average laser power by a factor of \(T/\Delta{t}_\text{ps}\) in accordance with

\[\tag{11-108}P_\text{pk}=K'\frac{T}{\Delta{t}_\text{ps}}\bar{P}=K'\frac{\bar{P}}{f_\text{ps}\Delta{t}_\text{ps}}=\frac{K'}{K}N\bar{P}\]

where \(K'\) is a constant of the order of unity that depends on the pulse shape. For a Gaussian pulse \(K'=2\sqrt{\ln2}/\sqrt{\pi}=0.9394\). For a \(\text{sech}^2\) pulse, \(K'=\ln(1+\sqrt{2})=0.8814\).

From (11-108), we see that the enhancement of the pulse peak power over the average power is proportional to the number of locked modes.

**Example 11-7**

By properly incorporating a suitable mode locker in the laser cavity, a CW Nd : YAG laser can often be mode locked with little additional loss, thus maintaining average power while delivering a regular train of ultrashort laser pulses.

A mode-locked Nd : YAG laser consists of a Nd : YAG gain medium that has a spontaneous linewidth of \(\Delta\nu=150\text{ GHz}\) in a Fabry-Perot cavity that has a round-trip optical path length of \(l_\text{RT}=2\text{ m}\).

The laser is continuously pumped to have an average output power of \(\bar{P}=2\text{ W}\). The mode locker used in this laser generates pulses of Gaussian temporal and spectral shapes.

(a) What is the repetition rate of the mode-locked pulses? Does it vary with pulsewidth or laser output power?

(b) If transform-limited pulses of \(\Delta{t}_\text{ps}=100\text{ ps}\) are generated, how much of the bandwidth of the gain medium is utilized? How many longitudinal modes should oscillate and be locked to generate such pulses?

(c) What is the peak power of the pulses?

(d) What is the pulsewidth of the shortest pulses that can possibly be generated from this laser? Under what conditions can such pulses be generated?

**(a)**

The pulse repetition rate is determined by the cavity round-trip time \(T\), which in turn is determined by the round-trip optical path length \(l_\text{RT}\). Therefore,

\[f_\text{ps}=\frac{1}{T}=\frac{c}{l_\text{RT}}=\frac{3\times10^8}{2}\text{ Hz}=150\text{ MHz}\]

It does not vary with either pulsewidth or laser output power. We also find from this result \(T=6.7\text{ ns}\).

**(b)**

For transform-limited Gaussian pulses of \(\Delta{t}_\text{ps}=100\text{ ps}\), we have

\[\Delta\nu_\text{ps}=\frac{K}{\Delta{t}_\text{ps}}=\frac{0.4413}{100\times10^{-12}}\text{ Hz}=4.413\text{ GHz}\]

Because \(\Delta\nu=150\text{ GHz}\), we have \(\Delta\nu_\text{ps}/\Delta\nu=4.413/150=2.94\%\). Therefore, only \(2.94\%\) of the bandwidth of the gain medium is used.

The longitudinal mode spacing is simply the same as the pulse repetition rate: \(\Delta\nu_\text{L}=f_\text{ps}=150\text{ MHz}\). The number of oscillating modes that are locked to generate these pulses can be found from (11-105):

\[N=\frac{\Delta\nu_\text{ps}}{\Delta\nu_\text{L}}=\frac{4.413\times10^9}{150\times10^6}=30\]

Only 30 oscillating modes are required because the pulsewidth of \(100\text{ ps}\) is relatively long for mode-locked pulses in a cavity that has a round-trip time of \(T=6.7\text{ ns}\).

**(c)**

The peak power of these Gaussian pulses can be found by using (11-108) with \(K'=0.9394\):

\[P_\text{pk}=K'\frac{T}{\Delta{t}_\text{ps}}\bar{P}=0.9394\times\frac{6.67\times10^{-9}}{100\times10^{-12}}\times2\text{ W}=125\text{ W}\]

This peak power is only about 63 times the average power because of the moderate value of the \(T/\Delta{t}_\text{ps}\) ratio and the correspondingly small number of oscillating modes.

**(d)**

The pulsewidth is ultimately limited by the condition given in (11-107). For Gaussian pulses, \(K=0.4413\). Thus, the shortest pulses that can be generated from this laser have the following pulsewidth:

\[\Delta{t}_\text{ps}^\text{min}=\frac{K}{\Delta\nu}=\frac{0.4413}{150\times10^9}\text{ s}=2.94\text{ ps}\]

Such pulses are generated under the following conditions:

- The entire bandwidth of the laser gain medium is utilized so that \(\Delta\nu_\text{ps}=\Delta\nu\).
- The pulses are transform limited so that \(\Delta{t}_\text{ps}\Delta\nu_\text{ps}=K\).

To utilize the entire bandwidth of the laser gain medium is not a simple matter. Aside from pumping the laser sufficiently to realize its entire potential gain bandwidth, it requires that all optical elements in the laser cavity, including the mirrors and the mode locker, have bandwidths larger than \(\Delta\nu\).

It also requires that the mode-locking mechanism be strong enough to force all modes across the entire bandwidth to oscillate and lock in phase.

The generation of transform-limited pulses is not a trivial matter, either. It requires elimination or compensation of all possible sources of dispersion in the laser while using an effective mode-locking scheme to lock all oscillating modes perfectly in phase.

In comparison to the transient techniques of gain switching and \(Q\) switching, the requirements and optimum conditions for operation of a mode-locked laser that is regeneratively pulses are very different.

Because a ** regeneratively pulsed laser** does not function by control of the laser transient, it does not depend on rapid depletion of intracavity photons to generate a short pulse, as do

**.**

*transiently pulsed lasers*Consequently, it does not require a very short photon lifetime and a correspondingly short laser cavity. On the contrary, it requires a sufficiently long cavity for the pulse to fit in and circulate around.

In general, a pulse generated by the transient technique of gain switching or \(Q\) switching cannot be shorter than the cavity round-trip time. Therefore, it has a spatial span longer than the cavity length.

In contrast, a pulse generated by a regenerative approach such as mode locking always has spatial span much shorter than the cavity length and can have a pulsewidth significantly shorter than the cavity round-trip time.

This comparison is illustrated in Figure 11-15.

In a transiently pulsed laser, photon energy is distributed throughout the laser cavity, and a pulse is generated through fast temporal evolution of this distributed energy. As a result, the laser can be modeled as a lumped device.

In a regeneratively pulsed laser, however, the photon energy is localized and circulates in the cavity. Therefore, a mode-locked laser cannot be modeled as a lumped device.

Another difference between a transiently pulsed laser and a regeneratively pulsed laser is the characteristic requirements of the gain medium. A transiently pulsed laser requires a long fluorescence lifetime and prefers to have it as long as possible.

The fluorescence lifetime \(\tau_2\) varies among different types of regeneratively pulsed lasers, but in general a particularly large \(\tau_2\) is not required.

A ** synchronously pumped laser** can successfully operate on a gain medium that has a very small \(\tau_2\) or even one that has no energy-storage mechanism such as in the case of a synchronously pumped OPO.

In certain mode-locked systems, \(\tau_2\) is preferred to be sufficiently large but not so large as to cause competition between transient oscillation and the buildup of mode-locked pulses.

Therefore, the \(\tau_2\) requirement of a mode-locked laser is a sophisticated issue that depends on the specific type and mechanism of the mode-locking operation.

Although repetitive pulses can also be generated by the repetitive operation of transiently pulsed lasers, a regeneratively pulsed laser offers certain advantages.

The pulses generated from a regeneratively pulsed laser do not have to build up from noise once the laser has reached steady state. In the steady state, the pulses bear no relation to the initial noise form which they have developed. Thus these pulses tend to have much better characteristics than those generated by transient techniques.

The pulses are usually very smooth and maintain a very high degree of coherence from one to another over a long period of time, making them very useful for many time-resolved spectroscopic applications.

They are not affected by the transient effects, such as relaxation oscillations, of transiently pulsed lasers. In a mode-locked laser, the regeneratively generated pulses can be transform limited if all of the oscillating modes are completely locked in phase.

Finally, a regeneratively pulsed laser can generate much shorter pulses at a higher repetition rate than a transiently pulsed laser of the same gain medium can.

Similarly to \(Q\) switching, mode locking can also be either active or passive, depending on the type of mode locker used. In an actively mode-locked laser, operation of the mode locker is controlled by an externally applied signal. In a passively mode-locked laser, the mode locker functions directly in response to the optical field in the laser cavity through a nonlinear optical mechanism. Mode locking can take the form of either periodic loss modulation or periodic gain modulation. The most important mode-locking techniques are illustrated in Figure 11-16.

For active mode locking with loss modulation, the most commonly employed technique is acousto-optic modulation with an externally applied RF signal, as shown in Figure 11-16(a). An acousto-optic modulator used for mode locking is different from one used for \(Q\) switching: an acousto-optic mode locker is a standing-wave Bragg diffractor that is turned on continuously, but an acousto-optic \(Q\) switch is a traveling-wave Bragg diffractor that is turned on and off to switch the cavity \(Q\) between different values.

A very important technique, known as ** synchronous pumping** and illustrated in Figure 11-16(b), for generating ultrashort laser pulses can be considered as active mode locking by gain modulation. For synchronous pumping, the gain medium in the laser cavity is localized and placed near one end of the cavity and is pumped periodically, either optically or electrically, with a train of very short pulses at the same repetition rate as that of the periodic arrival at the gain medium of the pulse circulating inside the laser cavity.

Passive mode locking can be accomplished by using a saturable absorber localized and placed near one end of a linear laser cavity, as shown in Figure 11-16(c). Sometime, a saturable absorber used for passive \(Q\) switching can also be used for passive mode locking, but in general the requirements for passive mode locking are very different from those for passive \(Q\) switching.

Passive mode locking with a saturable absorber can be arranged in a ring configuration shown in Figure 11-16(d) for ** colliding-pulse mode locking**. In this mode-locking scheme, there are two intracavity laser pulses that circulate in opposite directions and collide at the saturable absorber to enhance the pulse-shortening function of the saturable absorber.

Passive mode locking can also be accomplished without the use of a saturable absorber by employing nonlinear refractive index changes through the real part of \(\chi^{(3)}\), or even \(\chi^{(2)}\), of a nonlinear optical element. Two very important concepts belonging to this category are ** additive-pulse mode locking** and

**, which are illustrated in Figures 11-16(e) and (f), respectively.**

*Kerr-lens mode locking*Mode locking has been applied to a wide variety of laser materials to generate laser pulses with pulsewidths ranging from the order of \(10\text{ fs}\) to the order of \(1\text{ ns}\). For a given laser material, passive mode locking typically generates shorter pulses than active mode locking, but active mode locking often produces pulses with less fluctuation and jitter. Some systems combine active mode locking with passive mode locking in a form of hybrid mode locking to realize the advantages of both.

It is also possible to combine mode locking with a transient pulsing technique. In this situation, the laser does not reach a complete steady state. An important example of this possibility is the operation of ** \(Q\)-switched mode-locked lasers** by combining \(Q\) switching and mode locking. Because the transiently \(Q\)-switched pulse has a duration longer than the cavity round-trip time, the result is a finite train of equally spaced mode-locked pulses with unequal amplitudes under a \(Q\)-switched envelope.

**Example 11-8**

Nd : YAG lasers can undertake all modes of laser operation, including CW, gain-switching, \(Q\)-switching, and mode-locking operations, as Examples 11-1 to 11-7 illustrated [refer to the resonant optical cavities tutorial, the laser oscillation tutorial, and the laser power tutorial].

With the exception of synchronous pumping, almost all other mode-locking techniques can be successfully employed to mode lock Nd : YAG lasers, either in a pure form of CW mode locking or in a hybrid form that combines \(Q\) switching with mode locking, or otherwise.

These being the facts, however, the microchip Nd : YAG laser with its cavity parameters described in Examples 11-1 to 11-6 cannot be mode locked by any means. Give quantitative reasons for this problem.

First, consider the fact that the longitudinal mode spacing of this microchip laser is \(\Delta\nu_\text{L}=164.8\text{ GHz}\), as found in Example 11-3 [refer to the laser oscillation tutorial], while the entire linewidth of the Nd : YAG plate used for this laser is only \(\Delta\nu=150\text{ GHz}\). Although a homogeneously broadened laser can oscillate in multiple longitudinal modes when the laser is mode locked, as discussed in this tutorial above, this microchip laser can only oscillate in a single longitudinal mode regardless of how it is being operated because \(\Delta\nu_\text{L}\gt\Delta\nu\), not because it is homogenously broadened. Clearly, there is no possibility of mode locking if a laser can only oscillate in one longitudinal mode.

We can see the problem from another angle in the time domain. According to the illustration in Figure 11-15 (b) and the discussion in the tutorial above, a mode-locked pulse must have a spatial span that is much shorter than the cavity length to allow it to circulate inside the cavity as a regenerative pulse. For a laser with a linear Fabry-Perot cavity such as the microchip laser under consideration, the mode-locked pulse has to fit loosely into the length of the cavity to allow it to circulate inside without wrapping itself up, thus having a pulsewidth that is much shorter than one-half of the cavity round-trip time: \(\Delta{t}_\text{ps}\ll{T/2}\). From Example 11-1 [refer to the resonant optical cavities tutorial], we find that \(T=6.07\text{ ps}\) for this laser. Therefore, any mode-locked pulse that can possibly be generated from this laser has a pulsewidth subject to the limitation \(\Delta{t}_\text{ps}\ll3.03\text{ ps}\). However, the pulsewidth of a mode-locked pulse is also subject to the limitation given in (11-107). With \(\Delta\nu=150\text{ GHz}\), we have \(\Delta{t}_\text{ps}\gt2.94\text{ ps}\) according to the calculation in Example 11-7 above if the pulse has a Gaussian shape. These two conflicting limitations cannot be satisfied simultaneously, thus excluding any possibility of mode locking this laser.

The next tutorial covers the topic of **optical fiber lasers**.