Mode Locking

Q-switching produces very high energy pulses but the pulse durations are typi- cally in the nanosecond regime. In order to produce ultrashort pulses of durations in picoseconds or shorter, the technique most commonly used is mode locking. In order to understand mode locking let us first consider the formation of beats when two closely lying sound waves interfere with each other. In this case we hear beats due to the fact that the two sound waves (each of constant intensity) being of slightly different frequency will get into and out of phase periodically (see Fig.7.16). When they are in phase then the two waves add constructively to produce a larger inten- sity. When they are out of phase, then they will destructively interfere to produce no sound. Hence in such a case we hear a waxing and waning of sound waves and call them as beats.

Fig. 7.16 The top curve is obtained by adding the lower two sinusoidal variations and corresponds to beats as observed when two sound waves at closely lying frequencies interfere with each other

Mode locking is very similar to beating except that instead of just two waves now we are dealing with a large number of closely lying frequencies of light. Thus we expect beating between the waves; of course this beating will be in terms of intensity of light rather than intensity of sound. In order to understand mode locking, we first consider a laser oscillating in many frequencies simultaneously. Usually these waves at different frequencies are not correlated and oscillate almost independently of each other, i.e., there is no fixed-phase relationship between the different frequencies. In this case the output consists of a sum of these waves with no correlation among them. When this happens the output is almost the sum of the intensities of each individual mode and we get an output beam having random fluctuations in intensity.

In Fig.7.17we have plotted the output intensity variation with time obtained as a sum of eight different equally spaced frequencies but with random phases. It can be seen that the output intensity varies randomly with time resembling noise.

Now, if we can lock the phases of each of the oscillating modes, for example bring them all in phase at any time and maintain this phase relationship, then just

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Fig. 7.17 The intensity variation obtained by adding eight equally spaced frequencies with random phases. The intensity variation is noise like

like in the case of beats, once in a while the waves will have their crests and troughs coinciding to give a very large output and at other times the crests and troughs will not be overlapping and thus giving a much smaller intensity (see Fig.7.18). In such a case the output from the laser would be a repetitive series of pulses of light and such a pulse train is called mode-locked pulse train and this phenomenon is called mode locking. Figure7.19shows the output intensity variation with time corresponding to the same set of frequencies as used to plot Fig.7.17, but now the different waves have the same initial phase. In this case the output intensity consists of a periodic series of pulses with intensity levels much higher than obtained with random phases.

The peak intensity in this case is higher than the average intensity in the earlier case by the number of modes beating with each other. Also the pulse width is inversely proportional to the number of frequencies.

Fig. 7.18 Figure showing interference between four closely lying and equally spaced frequencies and which are in phase at the beginning and retain a constant phase relationship. Note that the waves add constructively periodically

7.7 Pulsed Operation of Lasers 175

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Fig. 7.19 The output intensity variation for the same situation as in Fig.7.17but now the waves are having the same phase at a given instant of time and they maintain their phase relationship.

Note that the resultant intensity peaks periodically

If the number of waves interfering becomes very large, e.g., a hundred or so, then the peak intensity can be very high and the pulse widths can be very small.

Figure7.20 shows the output mode-locked pulse train coming out of a titanium sapphire laser. Such mode-locked pulse train can be very short in duration (in picoseconds) and have a number of applications.

Mode locking is very similar to the case of diffraction of light from a grating.

In this case the constructive interference among the waves diffracted from differ- ent slits appears at specific angles and at other angular positions, the waves almost cancel each other. The angular width of any of the diffracted order depends on the number of slits in the grating similar to the temporal width of the mode-locked pulse train depending on the number of modes that are locked in phase.

In order to understand the concept of mode locking, we consider a laser formed by a pair of mirrors separated by a distance d. If the bandwidth over which gain exceeds losses in the cavity isν(see Fig.7.21), then, since the intermodal spacing is c/2n0d, the laser will oscillate simultaneously in a large number of frequencies.

The number of oscillating modes will be approximately (assuming that the laser is oscillating only in the fundamental transverse mode)

Fig. 7.20 Mode-locked pulse train from a titanium sapphire laser. Each division

corresponds to 2 ns (Adapted from French et al. (1990) © 1990 OSA)