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Haus Master Equation

Definition: an analytical equation describing the evolution of ultrashort pulses in a laser resonator

Categories: methods, light pulses

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The Master equation as introduced by Hermann A. Haus [1] is the central piece of a physical model which can be used to describe the pulse evolution in the resonator of a mode-locked laser. The basic ideas underlying the model are the following:

form of Haus Master equation

As an example, consider the case of an actively mode-locked laser with intracavity dispersion and a Kerr nonlinearity. In that case, the Haus Master equation reads

form of Haus Master equation

where the terms on the right-hand side describe (in that order) the constant part of the laser gain, the frequency dependence of the gain and the second-order dispersion, the resonator losses, the time-dependent modulator loss (approximated for small time arguments), and the Kerr nonlinearity. This is a generalized Landau–Ginzberg equation. The Master equation can be seen as a generalization of the nonlinear Schrödinger equation, as is often used for the study of soliton pulse phenomena.

Approximations

It is important to recognize which kind of approximations are usually required:

Applications

In not too complicated situations, analytical solutions of the Haus Master equation are known for the steady state (as reached after many resonator round trips). For example, this is the case for the actively mode-locked laser without dispersion and nonlinearities. The result is that the pulse shape is Gaussian, and the calculated pulse duration exactly agrees with that of the earlier Kuizenga–Siegman theory. There are also higher-order solutions described with Hermite–Gaussian functions, but these experience lower net gains per resonator round trip, and are thus not observed.

In more difficult cases, approximated results can be obtained by using a simple ansatz for the function A(t), e.g. corresponding to sech2-shaped pulses (possibly with a chirp). This leads to equations which allow one e.g. to calculate pulse duration, chirp, and spectral bandwidth as functions of the laser parameters.

In still more complicated cases, analytical solutions usually require approximations, the validity conditions of which are often not well satisfied in realistic situations. In particular, the time-dependent loss caused by a slow saturable absorber is difficult to treat, as it depends on the optical intensity at earlier times and thus introduces an integral on the right-hand side of the Haus Master equation. Another issue is the stability of solutions; there are cases where the theoretical solution is dynamically unstable and can thus never be observed in practice. For these reasons, numerical algorithms are often used to calculate the steady-state pulse profile. However, this approach removes the ability to obtain analytical equations as results, which can be more helpful in recognizing relations between parameters. There is then usually no more advantage over using straightforward pulse propagation models, which are less dependent on various approximations (see above), and thus able to describe reliably a wider range of phenomena. Also, such models are not more complex to implement and validate.

Another use of the Master equation is to derive dynamic equations (coupled differential equations) for a limited number of pulse parameters, such as pulse energy, duration, chirp, and temporal position, using the moment method [4]. The emphasis is then not necessarily on the steady state, but on the evolution of pulse parameters, which can be used, e.g., to investigate noise properties.

In conclusion, the Haus Master equation can be considered as a useful tool mainly for the study of simple situations, where analytical solutions can be obtained, and as the basis of some dynamic models, whereas simple pulse propagation models (treating different effects on the pulses sequentially) are usually more appropriate for more complex situations, particularly for passively mode-locked lasers.

Bibliography

[1]H. A. Haus et al., “Structures for additive pulse mode locking”, J. Opt. Soc. Am. B 8 (10), 2068 (1991)
[2]A. M. Dunlop et al., “Pulse shapes and stability in Kerr and Active Mode-Locking (KAML)”, Opt. Express 2 (5), 204 (1998) (extended equations, including the transverse spatial dimensions)
[3]H. A. Haus, “Mode-locking of lasers”, IEEE J. Sel. Top. Quantum Electron. 6 (6), 1173 (2000)
[4]N. G. Usechak and G. P. Agrawal, “Rate-equation approach for frequency-modulation mode locking using the moment method”, J. Opt. Soc. Am. B 22 (12), 2570 (2005)

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See also: pulse propagation modeling, mode locking, ultrashort pulses, Kuizenga–Siegman theory
and other articles in the categories methods, light pulses

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