We can find that the intensity is N2 times the intensity per mode when ωct/2 is an integral multiple of π. Therefore the average power of the laser is larger than the power per mode N times. And thus we obtain the (approximate) relation that the peak power is N times the average.
peak ave
P = N × P
(2.25) The pulse width is estimated by using
P
peak⋅ ∆ t
p≈ P
ave⋅ τ
RT (2.26) With substituting the average power and the number N of oscillating modes locked together which is approximately the line width Δν divided by the cavity mode spacing c/2d, the From the above equations, we can find that by using the technology of mode-locking a very giant and sharp pulse of high peak power, narrow pulse-width and high repetition rate can be produced. Usually, in order to achieve mode-locking, a thin shutter inside the laser resonator as a form of loss modulation with narrow opening time for every periodic duration can be used.2-3-2 Amplitude modulation
Amplitude modulation mode-locking is one of the most used methods for active mode locking. The main mechanism of amplitude modulation is directly modulating the optical amplitude or loss to actively generate short pulse trains with high repetition rate. It will offer the periodic loss like sinusoidal signal in the time domain so that only the pulses which pass
through the modulator at the lowest loss will exist.
Fig. 2.10 The amplitude modulation or loss modulation of active mode-locking process.
We usually introduce the amplitude modulator such as acousto-optic modulator and electro-optic modulator into the cavity which periodically varies the intracavity loss. All the lights in the cavity will experience a net loss larger than the net gain, which block the lasing mechanism, except those which pass through the modulator around the lowest loss point, which will generate the laser pulse at this moment. The modulation period has to be matched to the cavity round-trip time, and then the laser beam which is incident at particular point in the modulation cycle will be incident at the same point after one round trip of the cavity so that the phase of each lasing mode will be fixed. Fig. 2.10 shows the amplitude modulation or loss modulation of active mode-locking process in the time domain.
As mentioned in the theory of mode locking, a laser consist of a gain medium inside two reflected mirrors and the distance between the two reflected mirrors which is the cavity round trip will decide the longitudinal modes separated in frequency by ωc= 2π(c/2d)=2π/τRT, where τRT is the round-trip time. If the gain is larger than the loss for these longitudinal modes, these longitudinal modes will all be lasing. But the phase between these longitudinal modes is random and not related to each other. Placing the amplitude modulator near one of the mirrors, and a cosinusoidal signal of the modulation at the frequency Ω=Nωc will be used to generates sidebands at ω0±Ω, which the frequency of the central mode is ω0. These matched modes lasing in the same round trip cycle will have the fixed phase condition so that they will interference to each other to generate an extremely giant and short pulse.
Now we consider the mathematical theory of amplitude modulation. The signal of the
Where we define A is a slowly varying field envelope, that is already normalized to the total power flow in the beam, TR =2L/Vg,0 is dispersion coefficients per resonator round-trip, g(T) is the gain and Dg is the curvature of the gain at the maximum of the Lorentzian lineshape (is also the gain dispersion).
The first and second terms of the Master equation are the interaction by gain. The third and fourth terms of the equation are the interaction by loss and modulator. We fix the gain in Eq. at its stationary value because it might be constant in time domain, and the equation can be solved by separation of variables. The pulses will have a width much shorter than the round-trip time TR, and they will be located in the minimum of the amplitude modulation where the cosine-function can be approximated by a parabola function and the equation will
become a linear P.D.E. The differential equation corresponds to the Schrödinger-Operator of the harmonic oscillator problem. Therefore, we can calculate the eigen functions of this operator which are the Hermite-Gaussians
where τa defines the width of the Gaussian.
Thus active mode-locking without detuning between resonator round-trip time and modulator period leads to Gaussian steady state pulses with a FWHM pulse width
∆ t
FWHM= 2 (2 ) 1.66 In τ
a= τ
a (2.33) And the spectrum of the Gaussian pulse is obtained by
Therefore, the time-bandwidth product of the Gaussian pulse is
∆ t
FWHM⋅ ∆ f
FWHM= 0.44
(2.36) The stationary pulse of the mode-locked laser is achieved when two effects which the generation of the pulse are due to the parabolic amplitude modulation in the time domain and the parabolic filtering due to the gain in the frequency domain corresponding to pulse shortening and pulse stretching balance. Actively mode-locking typically by amplitude modulation only results in pulse width in the range of several tens picoseconds because the pulse width does only scale with the inverse square root of the gain bandwidth and external modulation is limited to electronic speed.In the frequency domain, the amplitude modulator transfers energy from each mode to its neighboring mode. In other words, it redistributes energy from the center mode to the wings of the spectrum. This process seeds and injection will lock neighboring modes. We can see the phenomenon of neighboring modes locking from the below equation which is the fourth term of above master equation So as we have mentioned before, if the modulation frequency is the same or integer times to the cavity round-trip frequency. The sidebands generated from each running mode are injected into the neighboring modes which lead to synchronization and locking of neighboring modes, i.e. mode-locking.
References
[2.1] Joseph T. Verdeyen, “Laser Electronics,” third edition, Prentice Hall Englewood Cliffs, New Jersey 07632
[2.2] Amnon Yariv, Pochi Yen, “Photonics,” sixth edition, New York Oxford university press 2007.
[2.3] B. E. A. Saleh and M. C. Teich, “Fundamentals of Photonics,” John Wiley & Sons, New York _1991_.
[2.4] Y. Wang and Chang-Quing Xu, “Actively Q-switched fiber lasers: Switching dynamics and nonlinear processes,” Progress in Quantum Electronics 31, 131–216 (2007)
[2.5] Y. Wang, A. Martinez-Rios and Hong Po, “Analysis of a Q-switched ytterbium-doped double-clad fiber laser with simultaneous mode locking,” Opt. Commun. 224, 113-123 (2003).
[2.6] P. K. Das, C. M. DeCusatis, “Acoustio-Optic Signal Processing: Fundamentals&
Applications,” Artech House, Boston‧London.
[2.7] Herman A. Haus, “Mode-locking of Lasers”, IEEE J.on Selected topics in Quant.
Electron. 6, 1773(2000)
[2.8] Dirk J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser-Part I:Theory” IEEE J. Quantum Electronics, vol. 6, no. 11, pp. 694-708 (1970)