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
\end{displaymath} (1)

The field autocorrelation function $\Phi(\tau)$ is given by
\Phi(\tau) = \langle E^*(t+\tau)E(t) \rangle
\end{displaymath} (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))}
\end{displaymath} (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
\end{displaymath} (4)

The following notation will be used from hereon to simplify expressions:
\Delta \phi(t,\tau) = \delta \phi(t+\tau) - \delta \phi(t)
\end{displaymath} (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
\end{displaymath} (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}
\end{displaymath} (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)
\end{displaymath} (8)

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

\delta = \tan^{-1}\left( {\Gamma_R \over \Omega_R} \right)
\end{displaymath} (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)}
\end{displaymath} (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].


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

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

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.