Nonlinear theory of semiconductor lasers

Laser theory of Fabry-Perot (FP) semiconductor lasers proves to be nonlinear, since the gain,[1][2] the refractive index[3] and the loss coefficient[4] are the functions of energy flux.

The nonlinear theory[2] made it possible to explain a number of experiments some of which could not even be explained (for example, natural linewidth), much less modeled, on the basis of other theoretical models; this suggests that the nonlinear theory developed is a new paradigm of the laser theory.

we have defined η as a specific gain factor; σ is specific conductivity that describes incoherent losses (for example, on free electrons).

where S is Poynting vector; V=sz, 0

The formulas for the line shape in FP and in DFB lasers were derived.

These formulas for the line shape are similar and have the following form:

The theory of natural linewidth in semiconductor lasers has an independent significance.

Using the density matrix equations with relaxation, the following derivations have been made: Einstein’s spectral coefficient in a semiconductor laser and, accordingly, Einstein’s coefficient;[1][2][10] formula for the saturation effect in a semiconductor laser was derived; it was shown that the saturation effect in a semiconductor laser is small.

[1][2] Gain in a semiconductor laser has been derived using the density matrix equations with relaxation.

[1][2] It has been found that Fabry-Perot laser gain depends on energy flux, and this determines the ‘basic nonlinear effect’ in a semiconductor laser

is Einstein coefficient for induced transition between the two energy levels when exposed to a narrow-band wave is written in the following form:[2][10]

[1][2] Necessary conditions for induced radiation are determined by the requirement for the gain to be greater than zero.

Necessary condition for induced radiation of the 1st kind formulated by Bernard and Duraffourg[2][11] is that the population of levels located above becomes more than the population of levels located below

The necessary condition of induced radiation of the 2nd kind formulated by Noppe[1][2] is that:

The necessary condition of induced radiation of the 2nd kind allows formulating the basic restriction of laser capacity,[1][2] which has been confirmed experimentally:

Based on the developed theory, experimental output characteristics have been simulated: natural linewidth (see simulation in,[2][6]) (see Fig.2), experimental watt - ampere characteristics[1][2][11] (see Fig.4) and dependence of experimental output radiation line-length on the current in Fabry-Perot semiconductor injection lasers,[1][2] (see Fig.3), as well as linewidth in DFB lasers (see simulation in,[7][8]).

Created theory makes it possible to simulate the majority published experiments on the measurement of the natural linewidth in Fabry-Perot lasers and distributed feedback DFB lasers[2][6][7][8][9][12] with help of two methods (using (13) and (15)).

Based on the formula derived for the line shape,[2][6] 12 experiments on measuring the natural linewidth in Fabry-Perot lasers (for example see Fig.2) and 15 experiments in DFB lasers[2][9] have been simulated.

Based on the formula derived for the natural linewidth,[2][6][8] 15 experiments on measuring the natural linewidth in Fabry-Perot lasers[2][6] and 15 experiments in DFB lasers[2][9] have been simulated.

The derived formula for line shape of radiation (of FP lasers[2][6][12] and DFB lasers[2][7]) is distinguished from Lorentz line formula.

Based on the developed theory, experimental output characteristics have been simulated: natural linewidth (see simulation in,[5][7]), experimental watt - ampere characteristics[10] (see Fig.4), and dependence of experimental output radiation line-length on the current in Fabry-Perot semiconductor injection lasers[13] (see Fig.3), as well as linewidth in DFB lasers (see simulation in,[2][9]).

On the basis of the nonlinear theory, recommendations have been made for the development of lasers with smaller natural linewidth and lasers with higher output power.

[1][2] Based on the solution of the density matrix equations, Einstein coefficient for induced transition has been derived; it has been shown that the saturation effect is small for semiconductor lasers.

[1][2] The formula of gain depending on the energy flux has been derived; it is the basic nonlinear effect in a laser.

We derived the gain g for a Fabry-Perot semiconductor laser based on the density matrix equations and expressions for the natural linewidth.

The resulting dependence of g on the energy flux has been called the main nonlinear effect in semiconductor lasers;[1][2] derivation of this relation formula is presented in.

[1][2] Experimental wavelength shift versus normalized current (J/Jth), and the output power versus current have been simulated for a high-power laser with a quantum well of intrinsic semiconductor.

Broadening of the states density due to different effects has been taken into consideration.

The nonlinear theory made it possible to explain a number of experiments some of which could not even be explained (for example, natural linewidth), much less modeled, on the basis of other theoretical models; this suggests that the nonlinear theory developed is a new paradigm of the laser theory.

Due to the nonlinear theory development, recommendations can be given for creating lasers with smaller natural linewidth, and lasers with higher output power.

Figure.1. Functions and versus energy flux I for two sets of characteristic parameters. [ 1 ] [ 2 ]
Figure.2. Simulating experimental curve [ 2 ] [ 14 ] of the natural linewidth of Fabry-Perot semiconductor lasers as functions of inverse output power Δν e (1/P ) (Ke=14) by theoretical curve Δνe(1/P ) [ 2 ] [ 6 ] (K t =14).
Figure 3. Wavelength shift Δλ (theoretical [ 1 ] [ 2 ] and experimental [ 1 ] [ 2 ] [ 15 ] ) versus normalized current (J/Jth)
Figure 4. Experimental [ 11 ] and theoretical [ 1 ] [ 2 ] output power versus current for a powerful laser.