Simulation model of the KLM laser

To simulate a KLM laser we constructed the equivalent four-mirror cavity system

shown in Fig. 3-5. The cavity consists of two flat mirrors M3 and M4, a 98% high

reflection end mirror M3 and a 95% output coupler M4, and a pair of curved mirrors M1

and M2 with the same radii of curvature R = 10 cm. The laser rod also acting as a Kerr

medium with the refractive index n = 1.76 and length l is placed between the curved

mirrors. M2 and M4 form a linear arm with a distance of 75 cm and M1 with M3 at the

same distance, respectively, in a near-symmetric arrangement. The distance r2 from

the curved mirror M1 to one end surface of the Kerr medium is 53.625 mm and r1 from

the other end surface of Kerr medium to M2 is tunable from 53.61 mm to 53.67 mm.

The total length of the resonator L is approximately 160 cm. The laser cavity is

operated at 1/3-degenerate cavity configuration. Here we assume that no dispersion

exists in the system for our numerical model since there are no dispersion components

in Fig. 3-5.

Fig. 3-5 Schematic of an equivalent four-mirror cavity configuration.

In nearly all of the situations of practical interest in mode-locked lasers, the time

variation of mode-locked pulse is still slow compared with the dephasing time in the

saturable absorbing medium; and the saturation behavior of the absorption will be

essentially that of a simple homogeneous atomic transition. Moreover, most lasers

used for technological applications belong to the so-called Class B lasers [5], which

include all solid state, semiconductor, and CO2 lasers. All these devices have in

common the long lifetime of the excited state (relative to both the medium polarization

lifetime and to the photon lifetime in the cavity). Basic rate equation model for a

single longitudinal and transverse-mode Class B laser involves two equations describing

rate of change of field and population inversion [7]. For Kerr-lens mode-locked lasers,

however, the optical Kerr effect plays a role of fast saturable absorber. Hence, we can

describe the nonlinear transition of an optical pulse through Kerr medium with

sufficient accuracy using only a simple rate-equation approach, without going into more

complex resonant-dipole or Rabi-flopping analyses [5].

Let the reference plane be end face I of the crystal. In a thin-slab approximation

[5], which the axial thickness of a short pulse is small compared to the length of a

typical cavity but still very large compared to an optical wavelength, we therefore

numerically simulate this laser system by using Collin’s integral [8] with round-trip

transmission matrix to calculate light field E(r) under cylindrical symmetry, where r is

the radial coordinate and the rate equations as described in our previous work [6],

2 2

with transmission matrix ^{1 1}

1 1

optical field on the reference plane in Fig. 3-5 before and after Huygens diffraction, r′

and r are the corresponding radial coordinates, λ is the wavelength of laser, J0 is the

Bessel function of zero order, d1′ is distance from end face I through the M2 and the M3;

and a is the aperture radius on the reference plane and it must be chosen large enough

with many times of the fundamental mode radius to ensure that the diffraction loss can

be neglected. In order to include the self-focusing effect in active medium, we

modified the equation to describe the light field passing through the gain medium by

adding the nonlinear phase shift, 2 ₂ ( )r π n lI r( )

φ ⁼ λ , which is caused by optical Kerr

effect, in the equation of field evolution:

1 1

of spontaneous emission whose amplitude and phase are given by the spontaneous

decay term in Eq. (3.12) and a random generator, respectively; and n2 is the nonlinear

refractive index. I(r) is the intensity distribution of laser pulse calculated from the

optical field E-m(r) using I(r) = (1/2nε0)|E-m(r)|², where n is refractive index and ε0 is

permittivity of free space. Similar treatment is for the opposite direction propagation.

Note that because the length of gain medium is far smaller than the cavity length the

gain distribution can be regarded as uniform distribution along the propagating direction.

If the thickness ∆z of the pulse is far smaller than the length of gain medium, the pulse

experienced the uniform gain.

The gain coefficient of the successive pass in the gain medium is related as

2

When we considered Eq. (3.12) without self-focusing effect (n2 = 0), Eqs. (3.12) and

(3.13) can use to model the laser dynamics with the beam-propagation dominant as

cavity is far from degeneration but with interplay of beam propagation and gain

dynamics as cavity is tuned toward degeneration [5]. However, if we considered the

self-focusing effect, act as the so-called Kerr lens, it changes the electric field

distribution and shrinks the spot size of the electric field to modify the gain profile and

result in the KLM mode resonates more easily than the CW mode.

Here we used the spontaneous decay rate γa = 3.125 x 10⁵ s^-1 [9], the total density

N0 = 3.3 x 10²⁵ m^-3 [9], the length l = 9 mm, the stimulated-emission cross section σ = 3.0 x 10^-23 m² [10] and the saturation parameter Es = 1.05 x 10⁶ N/C of the active

medium [10]; and the round-trip time ∆t = 10.67 ns that was determined by cavity

length, the photon energy of the pumping laser hνp = 1.53 eV, and pumping beam radius

wp = 15 µm. We have omitted the dispersion of the active medium so that the gain is assumed to be real.

We calculated the laser output power by integrating the intensity distribution of

laser pulse I(r) with respect to the aperture radius a on the reference plane every

roundtrip. The processes repeat in each roundtrip until reach convergence to

continuous-wave steady state for CW laser output. In order to investigate the

cavity-dependent instability, we set the initial values of E(r) to zero, i.e., E-1(r) = 0, and

changed r1 across the point of degeneration to vary the optical field distribution in the

gain medium corresponding to influence optical Kerr effect on laser dynamics for

calculating the output power. All of parameters and variables used in program have

been set double precision.

References

[1] G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 2001).

[2] H. A. Haus, IEEE J. Sel. Top. Quant. 6, 240 (2000).

[3] H. Haken, Synergetics (Springer-Verlag, Berlin, 1978).

[4] J. H. Lin, M. D. Wei, and W. F. Hsieh, J. Opt. Soc. Am. B 18, 1069 (2001).

[5] A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).

[6] C. H. Chen, M. D. Wei, and W. F. Hsieh, J. Opt. Soc. Am. B 18, 1076 (2001).

[7] X. Hachair, S. Barland, J. R. Tredicce, and G. L. Lippi, Appl. Optics. 44, 4761

(2005).

[8] S. A. Collins, J. Opt. Soc. Am. 60, 1168 (1970).

[9] P. F. Moulton, J. Opt. Soc. Am. B 3, 125 (1986).

[10] J. F. Pinto, L. Esterowitz, G. H. Rosenblatt, M. Kokta, and D. Peressini, IEEE J.

Quantum Electron. 30, 2612 (1994).