PHYS4121: Laser Physics Resources

This web app illustrates the concept of spectral hole burning (Lecture 7) in an inhomogeneously broadened laser cavity containing cavity modes. The leftmost plot below shows a large Gaussian lineshape due to inhomogeneous Doppler broadening in the cavity. The holes in the Gaussian coincide with the oscillation frequencies of cavity modes. At these frequencies the round trip gain 2$gL$ (where the gain coefficient $g$ = $g(\omega)$ and $L$ is the cavity length) reaches a constant value and is equal to the threshold value $\delta_c$ (the round trip loss). This occurs because the intensity has reached a steady state value and so there can be no net gain around the cavity (see Lectures 7-8). The radiation of each mode only interacts with the subset of atoms that are travelling with the correct velocity to be in resonance with the radiation, thus every mode with a value of $g_0(\omega)$ above the saturation value contributes to the output intensity. There are also holes present at the image frequencies (which are mirrors of the cavity modes) due to the fact that for each mode there are two classes of atoms because each standing wave in the cavity is composed of two oppositely directed travelling waves. Both of these travelling waves lase at the same mode frequency, so the image frequencies do not appear in the output power spectrum on the right.

The overall Gaussian shape is calculated using the inhomogenously broadened lineshape given in Lecture 4: $$ L_I(\omega-\omega_{12}) = \frac{2}{\Delta\omega_I} \left({\mathrm{ln}2}\over\pi\right)^{\frac{1}{2}} \mathrm{exp}\left[-4\mathrm{ln}2\left(\frac{\omega-\omega_{12}}{\Delta\omega_I}\right)^2\right], $$ Here $\omega-\omega_{12}$ is the detuning from the laser transition frequency $\omega_{12}$ and the Doppler width $\omega_I = 7.16 \times 10^{-17} \omega_{12} \left(\frac{T}{M_A}\right)^{\frac{1}{2}} $, $T$ is the discharge temperature and $M_A$ is the relative atomic mass of the lasing atom. These values are characterised by the atom that is used, and this example utilises those of the Ar$^+$ ion.

The holes are Lorentzian shaped as they are produced by inhomogeneously broadened lineshapes with a linewidth $\Gamma_{21}$ characterised by the atom's laser transition. The Lorentzian lineshape is calculated using the following equation from Lecture 4: $$ L_H(\omega-\omega_{12}) = \frac{\Gamma_{12}/2\pi}{(\omega-\omega_{12})^2 + \Gamma_{12}^2/4} $$ The round trip gain without the holes $2g_0(\omega-\omega_{12})L$ is assumed to be proportional to $L_I$ for the purposes of this code.
In the code, for each cavity mode (and image frequency) where the gaussian lineshape is above the saturation value, a Lorentzian centred on the mode frequency is scaled to the height of the difference between the round trip gain and the round trip loss and subtracted from the gaussian. Once all the holes have been subtracted it gives $2g_{\mathrm{th}}(\omega-\omega_{12})L$, the round trip gain which is what is plotted on the left.
The spacing of the cavity modes is the free spectral range $\omega_{FSR} = 2\pi \left(\frac{c}{2L}\right)$ where $L$ is the cavity length which can be varied below. The value of fractional round trip loss and the hole saturation is (Lecture 6) $\delta_c = \mathrm{ln}\left(\frac{1}{R_1R_2}\right) + 2{\alpha}L $, where $R_1$ and $R_2$ are the reflectivities of the two mirrors on either end of the cavity (in this particluar case, both kept constant at 0.99), and $\alpha$ is the total absorption and scattering losses in the cavity per unit length (can be varied below).

The calculation of the output power spectrum utilises the fact that $P_{\mathrm{out}}$ is proportional to the steady state intensity in the cavity $I_{\mathrm{SS}}$ which is itself proportional to $\left[\frac{g_0}{g_{\mathrm{th}}} - 1\right] $ (Lecture 8).