Mode Locking in Lasers

The conventional way of obtaining the ultrashort laser pulses is utilizing the mode-locking phenomena. This basic principle can be understood as follows. The laser radiation contains a number of different frequencies (modes) which differ in frequency by multipoles of $\Delta \nu =c/2L$. Normally, these modes have random phases relative to each other. However, it is possible to lock their phases together. For instance, if the intensity of a monochromatic light of frequency $\nu_0$ is modulation at a frequency $f$, the Fourier analysis shows that besides the carrier frequency, sidebands at frequencies $\nu_0\pm mf$, ($m$=1,2,3, ...) are generated. This intensity modulation can be performed by a Pockels cell.

When the modulator is placed inside the laser resonator and the modulation frequency $f$ is tuned to the mode separation $\Delta\nu$, the carrier and the frequency and the sidebands correspond to possible resonator modes, where laser oscillation is possible within the gain profile of the active media. Since the sidebands are coupled by the modulation function, the oscillating laser modes are also coupled. Let us assume a sinusoidal modulation of the modulator transmission $T=[1+\delta\cos(2\pi f t)]/2$ with the modulator factor $\delta<1$. The amplitude of the $m^{th}$ mode is then

$$A_m(t) = TA_0 cos\omega_mt = \frac{A_0}{2}[1+\delta\cos(2 \pi f t)], cos(\omega_mt)$$ (94)

which can be rewritten as

$$A_m(t) = \frac{A_0}{2}\, cos\omega_mt + \frac{A_0\delta}{4}[\cos(\omega_m+2 \pi f)t + \cos(\omega_m-2 \pi f)t].$$ (95)

In case the modulation frequency $f$ is equal to the mode spacing $\Delta \nu =c/2L$, $\omega_m\pm 2 \pi f = \omega_{m\pm 1}$ and the sidebands amplitudes are generated in the adjacent resonator modes. The phases of these sidebands are determined by that of carrier frequency and by modulation phase. The modulation of these three waves generates new sidebands at $\omega\pm 2\pi f n$, until all modes within the gain profile of the active medium oscillate with mutually coupled phases. Then interference occurs giving a series of sharp peaks. Within the bandwidth $\delta\nu$ of the spectral gain profile the superposition of $2p+1$ modes results in a total amplitude

$$A_m(t) = \sum_{m=-p/2}^{p/2}A_m \cos(\omega_0+2 \pi f m)t.$$ (96)

The resultant total laser intensity $I(t)=A(t)A(t)^*$ then becomes

$$I(t) = A_0^2\,\frac{\sin^2[(2p+1)\pi f t]}{\sin^2[\pi f t]}\,\cos^2\omega_0t.$$ (97)

Eq. (97) represents a periodic function with a period $T=1/\Delta\nu = 2L/c$. The pulse width $\Delta T \approx 1/\delta\nu$ depends on the spectral gain profile $\delta\nu$.

For instance, in dye lasers the spectral width of the gain profile is very large ( $\delta\nu \approx 3\cdot 10^{13} sec^{-1}$), which is equivalent to $\Delta\lambda\approx 30~nm$. Practically, for these lasers the pulse width of $\Delta T \approx 3\cdot 10^{-13} sec^{-1} = 300 fsec$ have been observed.

Even better results (up to several femtoseconds) are obtained for Ti:Sapphire lasers. An additional advantage of the Ti:Sapphire lasers is that the mode locking can be achieved passively by exploring the optical Kerr effect, which arises from a change in refraction index of an active medium when the intense laser pulses are generated. The procedure makes use of the fact that the gain of a frequency component of the radiation is very sensitive to amplification and, once a particular frequency begins to grow, it quickly becomes the dominate one.

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