Abstract
The present chapter deals with the semiclasscial theory of the laser as developed by Lamb (1964). In this analysis, we will treat the electromagnetic field classically with the help of Maxwell’s equations and the atom will be treated using quantum mechanics. We will consider a collection of two-level atoms placed inside an optical resonator. The electromagnetic field of the cavity mode produces a macroscopic polarization of the medium. This macroscopic polarization is calculated using quantum mechanics.
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Notes
- 1.
Because of the losses, the field in the cavity decays with time as \(\exp \left( { - {{\Omega _n t} \mathord{\left/ {\vphantom {{\Omega _n t} {2Q_n }}} \right. \kern-\nulldelimiterspace} {2Q_n }}} \right)\) and hence the energy decays as \(\exp \left( { - {{\Omega _n t} \mathord{\left/ {\vphantom {{\Omega _n t} {Q_n }}} \right. \kern-\nulldelimiterspace} {Q_n }}} \right)\).Thus, the energy decays to \({1 \mathord{\left/ {\vphantom {1 e}} \right. \kern-\nulldelimiterspace} e}\) of the value at \(t = 0\) in a time \(t_\textrm{c} = {{Q_n } \mathord{\left/ {\vphantom {{Q_n } {\Omega _n }}} \right. \kern-\nulldelimiterspace} {\Omega _n }}\) which is referred to as the cavity lifetime (see also Section 7.4).
- 2.
Actually we have equated each Fourier component; this follows immediately by multiplying Eq. (6.32) by \(\sin K_m z\) and integrating from 0 to L.
- 3.
We will show in Section 6.3 that \(\chi^{\prime\prime}_n \) is negative for a medium with a population inversion.
- 4.
The four quantities \(\rho _{11} \), \(\rho _{12} \), \(\rho _{21} \), and \(\rho _{22} \) form the elements of what is known as the density matrix \(\rho \).
- 5.
This will be justified in Section 6.3.2
- 6.
- 7.
- 8.
Notice that for \(\omega _n \ne \omega _{21} \), i.e., for a mode shifted away from resonance, the value of \(N_t \) increases with increase in the value of \(\left| {\omega _n - \omega _{21} } \right|\).
- 9.
It represents the ratio of the cavity bandwidth to the natural linewidth.
- 10.
See also Section 5.5, where we showed that on a steady-state basis the inversion can never exceed the threshold value.
- 11.
Use has been made of Eq. (6.92)
References
Lamb, W. E. (1964), Theory of an optical maser, Phys. Rev. 134, A1429.
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Thyagarajan, K., Ghatak, A. (2011). Semiclassical Theory of the Laser. In: Lasers. Graduate Texts in Physics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-6442-7_6
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