Model of Diode Laser Spectra

Krishna Myneni

February 9, 2001

According to the Wiener-Khinchine Theorem, the optical spectrum is given by the Fourier transform of the field autocorrelation function [1]

$$S_E(\omega) = \int_{-\infty}^{+\infty}\Phi(\tau)e^{-i\omega\tau}d\tau$$ (1)

The field autocorrelation function $\Phi(\tau)$ is given by

$$\Phi(\tau) = \langle E^*(t+\tau)E(t) \rangle$$ (2)

where the brackets denote averaging over $t$. The time dependent $E$ field of a laser can be expressed as

$$E(t) = \sqrt{P + \delta P(t)}\cdot e^{-i(\omega_0 t + \phi + \delta \phi (t))}$$ (3)

where $P$ and $\phi$ are the steady state power and initial phase values, and $\delta P(t)$ and $\delta \phi (t)$ represent the random power and phase fluctuations inside the laser. In diode lasers spontaneous emission is the source of $\delta P(t)$ and $\delta \phi (t)$. Given suitable random distributions for $\delta P(t)$ and $\delta \phi (t)$, the above equations may be integrated numerically to evaluate the optical spectrum $S_E(\omega)$. In the simplest approximation $\delta P(t)$ is taken to be zero, and thus

$$\Phi(\tau) \approx P e^{i\omega_0\tau}\left\langle e^{i(\delta \phi(t+\tau) - \delta \phi(t))}\right\rangle$$ (4)

The following notation will be used from hereon to simplify expressions:

$$\Delta \phi(t,\tau) = \delta \phi(t+\tau) - \delta \phi(t)$$ (5)

Since spontaneous emission events are random, we can assume $\Delta \phi(t, \tau)$ to be a Gaussian random variable. Then it can be shown by using the result for the mean of the function of a random variable $X$,

$$\left\langle f(X) \right\rangle = \int_{-\infty}^{+\infty}f(x)p(x)dx$$ (6)

where $p(x)$ is the probability density function for $X$, that

$$\left\langle e^{i\Delta \phi(t,\tau)} \right\rangle = e^{-{1 \over 2}\langle (\Delta \phi(t, \tau))^2\rangle}$$ (7)

and the time average $\langle (\Delta \phi(t, \tau))^2 \rangle$ can be evaluated from a transient analysis of the laser rate equations to a sudden change in the field intensity caused by the spontaneous emission event. This yields the following expression in terms of phenomenological parameters [1][2]:

$$\langle (\Delta \phi(t, \tau))^2 \rangle = {R_{sp} \over 2... ...ta) - e^{-\Gamma_R \tau}\cos(\Omega_R \tau - 3\delta)]\right)$$ (8)

where

$$b = {\Omega_R \over \sqrt{\Omega_R^2 + \Gamma_R^2}}$$ (9)

and

$$\delta = \tan^{-1}\left( {\Gamma_R \over \Omega_R} \right)$$ (10)

$\Omega_R = 2\pi \nu_R$, where $\nu_R$ is the relaxation oscillation frequency, $\Gamma_R$ is the damping rate of the relaxation oscillations, $\alpha$ is the linewidth enhancement factor, and $R_{sp}$ is the spontaneous emission rate. Putting the expression for $\langle (\Delta \phi(t, \tau))^2 \rangle$ from equation 8 into equation 7, and substituting the resulting expression into equation 4, the field autocorrelation function in the approximation $\delta P(t) = 0$ is given by

$$\Phi(\tau) \approx Pe^{i\omega_0 \tau}e^{-{R_{sp} \over 4P}... ...a) - e^{-\Gamma_R \tau}\cos(\Omega_R \tau - 3\delta)]\right)}$$ (11)

Equation 11 describes a real autocorrelation function; hence, its Fourier transform will yield a symmetric spectrum. The spectrum is centered at frequency $\omega_0$. Its central peak has a Lorentzian profile and symmetric sidebands are present at $\omega = \pm \Omega_R$. Real diode laser spectra are asymmetric. An analysis of the spectrum including correlated power fluctuations was developed by van Exter, et. al. [3].

Bibliography

1

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, New York: Van Nostrand Reinhold, 1993.

2

C. H. Henry, Theory of the phase noise and power spectrum of a single mode injection laser, IEEE J. Quant. Electron., QE-19, 1391.

3

M. P. van Exter, W. A. Hamel, J. P. Woerdman, and B. R. P. Zeijlmans, Spectral signature of relaxation oscillations in semiconductor lasers, IEEE J. Quant. Electron., 28, 1470.

Last revised on Thursday, 10-Jan-2008 06:44:14 EST
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