Semiconductor laser theory

Semiconductor lasers (635nm, 520nm, 445nm)
Semiconductor lasers (660nm, 532nm, 405nm)

Semiconductor lasers or laser diodes play an important part in our everyday lives by providing cheap and compact-size lasers. They consist of complex multi-layer structures requiring nanometer scale accuracy and an elaborate design. Their theoretical description is important not only from a fundamental point of view, but also in order to generate new and improved designs. The description can be done at various levels of accuracy and effort, resulting in different levels of understanding. It is common to all systems that the laser is an inverted carrier density system. The carrier inversion results in an electromagnetic polarization which drives an electric field E(t). In most cases, the electric field is confined in a resonator, the properties of which are also important factors for laser performance.

Gain medium

Comparison of gain and absorption calculated in Hartree–Fock approximation (dotted line) and fully taking into account collision terms (solid line). The sample is a Ga(AsSb) quantum well surrounded by GaAs spacers. For the top figure, a density of 1.3 x 1012 cm−2 was used which is well above lasing threshold. For the bottom figure, the carrier density is negligible. The differences in lineshape are obvious especially for the lasing structure. The Hartree–Fock approximation leads to absorption below the bandgap (below about 0.94 eV), which is a natural consequence of the relaxation time approximation, but is completely unphysical. For the low density case, the T2-time approximation also leads to extended tails.

In semiconductor laser theory, the optical gain is produced in a semiconductor material. The choice of material depends on the desired wavelength and properties such as modulation speed. It may be a bulk semiconductor, but more often a quantum heterostructure. Pumping may be electrically or optically (disk laser). All these structures can be described in a common framework and in differing levels of complexity and accuracy.[1]

Light is generated in a semiconductor laser by radiative recombination of electrons and holes. In order to generate more light by stimulated emission than is lost by absorption, the system has to be inverted, see the article on lasers. A laser is, thus, always a high carrier density system that entails many-body interactions. These cannot be taken into account exactly because of the high number of particles involved. Various approximations can be made:

The above-mentioned models yield the polarization of the gain medium. From this, the absorption \alpha or gain g may be calculated via

Optical absorption


 \alpha(\hbar\omega)=-g(\hbar\omega)=-\frac{\omega}{n_\mathrm{b} c}\operatorname{Im}\left[\frac{P(\omega)}{4\pi\epsilon_0\epsilon E(\omega)}\right]\,,

where \hbar\omega denotes the photon energy, n_\mathrm{b} is the background refractive index, c is the vacuum speed of light, \epsilon_0 and \epsilon are the vacuum permittivity and background dielectric constant, respectively, and E(\omega) is the electric field present in the gain medium. "\operatorname{Im}" denotes the imaginary part of the quantity in brackets. The above formula can be derived from Maxwell's equations.[3]

The figure shows a comparison of the calculated absorption spectra for high density where absorption becomes negative (gain) and low density absorption for the two latter theoretical approaches discussed. The differences in lineshape for the two theoretical approaches are obvious especially for the high carrier density case which applies to a laser system. The Hartree–Fock approximation leads to absorption below the bandgap (below about 0.94 eV), which is a natural consequence of the relaxation time approximation, but is completely unphysical. For the low density case, the T2-time approximation also overestimates the strength of the tails.

Laser resonator

A resonator is usually part of a semiconductor laser. Its effects have to be taken into account in the calculation. Therefore, the eigenmode expansion of the electric field is done not in plane waves but in the eigenmodes of the resonator which may be calculated, e.g., via the transfer-matrix method in planar geometries; more complicated geometries often require the use of full Maxwell-equations solvers (finite-difference time-domain method). In the laser diode rate equations, the photon life time \tau_p enters instead of the resonator eigenmodes. In this approximative approach, \tau_p may be calculated from the resonance mode[6] and is roughly proportional to the strength of the mode within the cavity. Fully microscopic modeling of laser emission can be performed with the semiconductor luminescence equations[7] where the light modes enter as an input. This approach includes many-body interactions and correlation effects systematically, including correlations between quantized light and the excitations of the semiconductor. Such investigations can be extended to studying new intriguing effects emerging in semiconductor quantum optics.

See also

References

  1. Chow, W. W.; Koch, S. W. (2011). Semiconductor-Laser fundamentals. Springer. ISBN 978-3540641667
  2. Lindberg, M.; Koch, S. (1988). "Effective Bloch equations for semiconductors". Physical Review B 38 (5): 3342–3350. doi:10.1103/PhysRevB.38.3342
  3. 1 2 3 Haug, H.; Koch, S. W. (2009). Quantum Theory of the Optical and Electronic Properties of Semiconductors (5th ed.). World Scientific. p. 216. ISBN 9812838848
  4. Haug, H.; Schmitt-Rink, S. (1984). "Electron theory of the optical properties of laser-excited semiconductors". Progress in Quantum Electronics 9 (1): 3–100. doi:10.1016/0079-6727(84)90026-0
  5. Hader, J.; Moloney, J. V.; Koch, S. W.; Chow, W. W. (2003). "Microscopic modeling of gain and luminescence in semiconductors". IEEE J. Sel. Top. Quant. Electron. 9 (3): 688–697. doi:10.1109/JSTQE.2003.818342
  6. Smith, F. (1960). "Lifetime Matrix in Collision Theory". Physical Review 118 (1): 349–356. doi:10.1103/PhysRev.118.349
  7. Kira, M.; Koch, S. W. (2011). Semiconductor Quantum Optics. Cambridge University Press. ISBN 978-0521875097

Further reading

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