# Mode-Locked Fiber Lasers

As discussed with some detailed in the “Lasers” portion of this website, there are two main methods to obtain short optical pulses: Q-switching and mode-locking. Q-switching was achieved first with fiber lasers, and some of the first experiments used an intracavity acousto-optic modulator to carve out pulses with Er- and Nd-doped fiber lasers. For applications that do not require short pulses, Q-switching is still a useful technique as it can produce relatively high peak powers over a tunable wavelength range. However, Q-switched pulses are limited to pulse durations closer to 100 ns, which does not suffice in many ultrafast experiments. As such, mode locking is used much more in practice for high peak power, ultrafast optical science applications.

PHYSICS OF MODE-LOCKING

Mode-locking involves the “locking” of multiple longitudinal cavity modes with respect to one another by synchronizing the phase between the modes to a constant value. To do our analysis, we will start with the operation of a continuous wave fiber laser.

If there are no frequency-selection components in a cavity, there exist many modes simultaneously oscillating the laser cavity. The frequency spacing is determined by the length, given as:

$$\Delta\nu = \frac{c}{L_{opt}}$$

where $L_{opt}$ is the optical length for one cavity round trip. The optical length is determined by the type of cavity; for loop mirror cavities it would be $L_{opt} = n_{eff}L$ and for a Fabry-Perot cavity it would be $L_{opt} = 2n_{eff}L$. The modes that a cavity can support are those whose cavity length is an exact half multiple of the wavelength, i.e.

$$L_{cav} = q\lambda/2$$

where “q” is the mode order. For very large cavities ($L >> \lambda/2$), then many modes would be supported, but the number of modes that actually oscillate depends on the gain bandwidth of the laser medium. If the gain bandwidth is large compared to the longitudinal mode spacing, then there would be thousands of modes that could fit into the gain profile of the medium. For example, a Ti:Sapphire laser with a 30 cm resonator would have on the order of 250,000 modes. The total optical field can then be written as:

$$E(t) = \sum_{m=-M}^{M}E_me^{i\phi_m-i\omega_m t}$$

where “m” stands for the “mth” mode, such that 2M + 1 modes represents the total number of modes. In the case without mode locking such that all of the modes oscillate independently of each other with no definite phase relationship, the interference terms that would arise when taking the magnitude would go away, making the intensity time-independent. This would be the case for the cw-operation of a multimode laser.

Now, with mode locking, there is a time dependence. This occurs because with mode locking, the phases of the longitudinal modes are fixed such that the phase difference between any two neighboring modes is given by a constant value:

$$\phi_m-\phi_{m-1} = \phi$$

We can note that $\phi_m = m\phi + \phi_0$ to see how we might have mode-locked modes. If there is an initial phase difference to the modes $\phi_0$, we can establish a fixed phase between modes by adding an “ordered” phase to each additional mode; for example, the $m=1$ mode would have a phase given by $\phi_1 = \phi + \phi_0$, $m=2$ modes would have a phase given by $\phi_2 = 2\phi + \phi_0$ which differs from $\phi_1$ by $\phi$, etc. The mode frequency can then be given by $\omega_m = \omega_0 + 2m\pi\Delta\nu$. If we now use these relations in our summation and assume that all of the modes have the same amplitude $E_0$, we can solve for the intensity summation analytically:

$$|E(t)|^2 = \frac{sin^2[(2M+1)\pi\Delta\nu t + \phi/2]}{sin^2(\pi\Delta\nu t + \phi/2)}E_0^2$$

We can see here how we would obtain ultrashort pulses. As we know from previous sections, the broader the bandwidth means the smaller the possible pulse duration. As we add more and more pulses together, we will have a broader and broader pulse bandwidth for an effectively single pulse. The exact bandwidth of the pulse would be given by $N\Delta\nu$, where then the duration is limited by the time bandwidth product:

$$\Delta{t} = \frac{0.441}{N\Delta\nu}$$

ACTIVE MODE LOCKING

Active mode-locking refers to the active modulation of either the amplitude or frequency of an intracavity optical field. Amplitude modulation is referred to as AM mode locking, while frequency modulation is referred to as FM. If these sound like familiar abbreviations, that’s because they should! AM and FM radio stations broadcast their sounds in exactly the same way, just via modulation of radio waves vs visible or infrared waves in a laser.

We can understand active mode locking by considering a modulator (acousto-optic or electro-optic, for example) that modulates the light field with a frequency $f_m$. If the mode spacing is $\Delta\nu$, then the modulated field would produce sidebands at $\nu+f_m$ and $\nu-f_m$. We can see this mathematically by considering that the field we are modulating is a wave oscillating at a frequency $cos(A)$. If this is then modulated by another wave (here acousto-optic or electro-optic) oscillating at $cos(B)$, the net result is given as:

$$cos(A)*cos(B) = \frac{1}{2}cos(A+B) + \frac{1}{2}cos(A-B)$$

which we obtained from a simple trigonometric identity. These halved terms are the sidebands, and they clearly carry the modulation data. Now, for the case of mode-locking, if the modulation frequency occurs at the same rate as the mode spacing, i.e. $f_m = \Delta\nu$, then the two adjacent modes would oscillate in phase with the original modulated mode. If the modulation then continues for further modes, then eventually all of the modes will be oscillating in phase. Hence, this gives us frequency modulation for mode-locking!

We can also consider this in the time domain (amplitude modulation) by looking at the following image.

Here we can see that the cavity loss is modulated such that the minima of the cavity loss correspond to the peaks of the signal intensity.

XPM-INDUCED MODE LOCKING

Although electro-optic modulators are useful in creating short pulses, they introduce a non-fiber element to the cavity design which can be undesirable for all-fiber cavities.