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{{Short description|Nontraditional laser theory of Fabry-Perot semiconductor lasers}}
{{Research paper|date=November 2016}}

[[Semiconductor laser theory|Laser theory]] of Fabry-Perot (FP) semiconductor [[lasers]] proves to be nonlinear, since the [[Gain (laser)|gain]],<ref name="a1">Noppe M G On Nonlinear Theory for Semiconductor Lasers. 2016 Laser Phys. 26055004 (doi:10.1088/1054-660X/26/5/055004)</ref><ref name="a4" /> the [[refractive index]]<ref name="a2">Partovi and E.M.Garmire, J. Appl.Phys, 69, 6885 (1991).</ref> and the loss coefficient<ref name="a3">Said A A et al. Opt. Soc. Am. B 1992 9 405</ref> are the functions of [[energy flux]]. The nonlinear theory<ref name="a4">Noppe M.G.  “Fundamentals of nonlinear theory for semiconductor lasers” (Publishing House SB RAS, 2016.  Novosibirsk, 2016). (To purchase a monograph, please use the following link: [https://www.facebook.com/nonlineartheorysemiconductorlasers "Fundamentals of nonlinear theory for semiconductor lasers"])</ref> made it possible to explain a number of experiments some of which could not even be explained (for example, natural [[Laser linewidth|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.

==Equations in the gain medium==
[[Maxwell's equations]] describe the field for passive medium and cannot be used in describing the field in laser and [[quantum amplifier]]. Phenomenological equations are derived for electromagnetic field in the [[gain medium]], i.e. Maxwell's equations for the gain medium, and [[Poynting's theorem]] for these equations.<ref name="a1" /><ref name="a4" /><ref name="a5">Noppe M G On nonlinear refraction in semiconductor lasers; simulation of experiment, J. Mod. Opt. 2004 51 153</ref> Maxwell's equations in the gain medium are used to obtain equations for energy flux, and to describe nonlinear phase effect.<ref name="a1" /><ref name="a4" /><ref name="a5" />
      
<math>(1)\quad rot\vec H =(\sigma-\eta)\vec E+{\partial \vec D \over \partial t},</math><br/> we have defined η as a specific gain factor; σ is specific [[Electrical resistivity and conductivity|conductivity]] that describes incoherent losses (for example, on free electrons). Other Maxwell's equations are used unchanged.<br/>
<math>(2)\quad rot\vec E = -{\partial B \over \partial t},  divD = 0, divB = 0,</math><br/>
<math>(3)\quad D=\epsilon'\epsilon_0E,  B=\mu_0\mu H,</math><br/>
Poynting theorem follows from (1)-(3):<br/>
<math>(4)\quad \int_V \eta|E(\omega)|^2\, dv=\int_V\sigma|E(\omega)|^2\, dv+\oint_{S}(Sn)ds,</math>

where S is [[Poynting vector]]; V=sz, 0 <z<L, where s is cross section (to axis z) of [[active laser medium]].<br/>Equations for energy flux follow from (4):<br/>
<math>(5)\quad \frac{dI^+}{dz}=(\Gamma g(I^+)-\alpha (I^+))I^+,</math><br/>
<math>(6)\quad \frac{-dI^-}{dz}=(\Gamma g(I^-)-\alpha (I^-))I^-,</math><br/>
where<br/>
<math>(7)\quad \alpha (I)=(1-\Gamma )\alpha_{out}+\Gamma \alpha_{in}+\alpha_x+\alpha_{2p}(I),</math><br/>
where <math>I</math> is the energy flux; <math>s</math> is sectional area of the active zone of the laser; <math>\Gamma</math> is confinement factor; <math>\alpha_{in}</math> is absorption factor in active zone; <math>\alpha_{out}</math> is absorption factor outside active zone; <math>\alpha_x</math> is losses due to [[incoherent scattering]]; <math>\alpha_{2p}(I)</math> is two-photon absorption factor;<ref name="a4" /><ref name="a3" /> and <math>\alpha_{2p}(I)=\beta\cdot I</math>).

==Formulas for the line shape and natural linewidth==
Theory of natural linewidth in semiconductor lasers has been developed, it follows that refractive index n in FP lasers<ref name="a2" /><ref name="a5" /> and effective refractive index n<sub>ef</sub> in [[Distributed feedback laser|Distributed FeedBack]] (DFB) lasers<ref name="a5" /><ref name="a6">Noppe M G, The Natural Linewidth of Fabry-Perot Semiconductor Lasers, Laser Phys., 24, 125006 (2014). DOI:10.1088/1054-660X/24/12/125006</ref> are the functions of E:<br/>
<math>(8)\quad n=n_0+n_1(\omega)E+n_2(\omega)E^2,</math><br/>
<math>(9)\quad n=n_{ef}^{0}+n_{ef}^{1}E+n_{ef}^{2}E^2,</math><br/>
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:<br/>
<math>(10)\quad L(v)=2\int_{0}^{\infty}\cos (2\pi v\tau)exp(A_{pa}|\tau|-B\tau^2)d\tau,</math><br/>
where <math>v=\tfrac{\omega - \omega L}{2\pi};\quad \omega L</math> is laser generation frequency;

<math>(11)\quad A_{pa}=\frac{D_0}{P},\quad B=(D_1+D_2P^{(1/2)})^2,</math><br/>
where <math>D_0, D_1</math>, and <math>D_2</math> have different form for FP and for DFB lasers.<ref name="a4" /><ref name="a6" /><ref name="a7">Noppe M G. On natural linewidth of distributed feedback lasers; simulation of experiments. In Proc. XII Intern. Confer.  (APEIE – 2014)- v.1, pp. 456 - 460)</ref><ref name="a8">Noppe M G. On formula for natural linewidth in Fabry-Perot lasers; simulation of experiments In Proc. XII Intern.  Confer. (APEIE – 2014)- v.1, pp. 472 - 477)</ref><ref name="a9">Noppe M G On formula for natural linewidth in distributed feedback lasers; simulation of experiments. In Proc. XII Intern. Confer. (APEIE – 2014)- v.1, pp. 461 -467</ref>
Let us write the natural linewidth Δν<ref name="a4" /><ref name="a8" /><ref name="a9" /><br/>
<math>(12)\quad \Delta v=\Delta v^{(c)}T\left (\left (mW\right ) \frac{A_{pa}}{P}+B \right ) F_N\left (\frac{P^{(c)}}{P},k \right ),</math><br/>
where <math>T\left (\left (mW\right ) \frac{A_{pa}}{P}+B \right )</math> is the bridge function;<ref name="a4" /><ref name="a8" /><ref name="a9" /> <math>\Delta v^{(c)}</math> and <math>P^{(c)}</math> are characteristic linewidth and characteristic laser power; k is characteristic parameter of laser nonlinearity; q is non-dimensional inverse power:

<math>(13)\quad F_N(q,k)=1/\int_{0}^{\infty}exp(-xq-x^2(k/q^{1/2}-1)^2)dx</math>

The theory of natural linewidth in semiconductor lasers has an independent significance. At the same time, the developed theory is an integral part of the nonlinear theory of lasers, and its concepts and the introduced characteristic parameters are used in all parts of the nonlinear theory.

==Gain in a semiconductor laser==
Using the [[density matrix]] equations with relaxation, the following derivations have been made: Einstein’s spectral coefficient in a semiconductor laser and, accordingly, [[Einstein coefficients|Einstein’s coefficient]];<ref name="a1" /><ref name="a4" /><ref name="a11">Noppe M. G. The Resonance and Nonresonance Coefficients of Stimulated Transitions for a System with Relaxation, Technical Physics Letters 2000, V. 26, 10-11</ref> formula for the saturation effect in a semiconductor laser was derived; it was shown that the saturation effect in a semiconductor laser is small.<ref name="a1" /><ref name="a4" /> Gain in a semiconductor laser has been derived using the density matrix equations with relaxation.<ref name="a1" /><ref name="a4" /> It has been found that Fabry-Perot laser gain depends on energy flux, and this determines the ‘basic nonlinear effect’ in a semiconductor laser

<math>(14)\quad g=g(I)=R\Delta \omega_e/(1-I(\delta \Delta\omega_e/\delta I)/\Delta\omega_e),</math>

where<br/>
<math>(15)\quad R=(\hbar\omega n_g/c)B_{21}(f_2-f_1)S_P(\omega).</math>

where <math>B_{21}</math> is Einstein coefficient for induced transition between the two energy levels when exposed to a narrow-band wave is written in the following form:<ref name="a4" /><ref name="a11" /><br/>
<math>(16)\quad B_{21}=(\hat{H}_{21}/\hbar)^2/(2\Gamma_0)</math>
where <math>\Delta \omega_e</math> is effective natural linewidth; <math>I</math> is the energy flux; <math>S_P(\omega)</math> is spectral density of transitions.

===Necessary condition for induced radiation of the 1st  kind===
Necessary conditions for induced radiation of the 1st and 2nd kind have been defined in.<ref name="a1" /><ref name="a4" /> 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<ref name="a4" /><ref name="a12">Andreev, A.Yu., et al. Semiconductors, 2009,43 543-547</ref> is that the population of levels located above becomes more than the population of levels located below

<math>(17)\quad f_2(E_2)>f_1(E_1).</math>

===Necessary condition for induced radiation of the 2nd  kind===
The necessary condition of induced radiation of the 2nd kind formulated by Noppe<ref name="a1" /><ref name="a4" /> is that:

<math>(18)\quad \Delta \omega_e/(1-I(\delta \Delta \omega_e/\delta I)/\Delta \omega_e)>0.</math>

[[File:Versus_energy_flux_I_for_two_sets_of_characteristic_parameters.jpg|thumb|center|upright=2.0|Figure.1. Functions <math>g_1(I)</math> and <math>g_2(I)</math> versus energy flux I for two sets of characteristic parameters.<ref name="a1" /><ref name="a4" />]]

The necessary condition of induced radiation of the 2nd kind allows formulating the basic restriction of laser capacity,<ref name="a1" /><ref name="a4" /> which has been confirmed experimentally:

<math>(19)\quad I<I(M),</math>

where <math>I</math> is energy flux; <math>I(M)</math> is the characteristic parameter of ultimate power. Figure 1 shows the function <math>g(I)</math> for two sets of characteristic parameters.

==Simulation of experiments==
4.1. Maxwell's equations in the gain medium are used to obtain equations for energy flux.<ref name="a1" /><ref name="a4" /><ref name="a5" /> Nonlinear phase effect has been described and simulated,<ref name="a1" /><ref name="a4" /> using the nonlinearity of refractive index.<ref name="a2" /> (see Fig.3).<br/>
<br/>
4.2. Based on the developed theory, experimental output characteristics have been simulated: natural linewidth (see simulation in,<ref name="a4" /><ref name="a6" />) (see Fig.2), experimental watt - ampere characteristics<ref name="a1" /><ref name="a4" /><ref name="a12" /> (see Fig.4) and dependence of experimental output radiation line-length on the current in Fabry-Perot semiconductor injection lasers,<ref name="a1" /><ref name="a4" /> (see Fig.3), as well as linewidth in DFB lasers (see simulation in,<ref name="a7" /><ref name="a8" />). 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<ref name="a4" /><ref name="a6" /><ref name="a7" /><ref name="a8" /><ref name="a9" /><ref name="a10">Noppe M.G. On the Line Form and Natural Linewidth; Simulation and Interpretation of Experiments. In Proc. XII Intern. Confer. (NUSOD-2012), 123.</ref> with help of two methods (using (13) and (15)). Based on the formula derived for the line shape,<ref name="a4" /><ref name="a6" /> 12 experiments on measuring the natural linewidth in Fabry-Perot lasers (for example see Fig.2) and 15 experiments in DFB lasers<ref name="a4" /><ref name="a9" /> have been simulated. Based on the formula derived for the natural linewidth,<ref name="a4" /><ref name="a6" /><ref name="a8" /> 15 experiments on measuring the natural linewidth in Fabry-Perot lasers<ref name="a4" /><ref name="a6" /> and 15 experiments in DFB lasers<ref name="a4" /><ref name="a9" /> have been simulated. The derived formula for line shape of radiation (of FP lasers<ref name="a4" /><ref name="a6" /><ref name="a10" /> and DFB lasers<ref name="a4" /><ref name="a7" />) is distinguished from Lorentz line formula.<br/>
<br/>
4.3. Based on the developed theory, experimental output characteristics have been simulated: natural linewidth (see simulation in,<ref name="a5" /><ref name="a7" />), experimental watt - ampere characteristics<ref name="a11" /> (see Fig.4), and dependence of experimental output radiation line-length on the current in Fabry-Perot semiconductor injection lasers<ref name="a13">Bernard M.G., Duraffourg G. 1961 Phys. Status Solidi 127 699</ref> (see Fig.3), as well as linewidth in DFB lasers (see simulation in,<ref name="a4" /><ref name="a9" />).<br/>
<br/>
4.4. 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.<ref name="a1" /><ref name="a4" />
 
[[File:Simulating experimental curve of the natural linewidth of Fabry-Perot semiconductor lasers.jpg|thumb|center|upright=2.0|Figure.2. Simulating experimental curve<ref name="a4" /><ref name="a15">Elsasser  W., Gobel E.O., Kuhl J., IEEE JQE,1983 19 981</ref> of the natural linewidth of Fabry-Perot semiconductor lasers as functions of inverse output power Δν<sub>e</sub>(1/P ) (Ke=14) by theoretical curve Δνe(1/P )<ref name="a4" /><ref name="a6" /> (K<sub>t</sub>=14).]]

[[File:Wavelength shift Δλ versus normalized current.jpg|thumb|center|upright=2.0|Figure 3. Wavelength shift Δλ (theoretical<ref name="a1" /><ref name="a4" /> and experimental<ref name="a1" /><ref name="a4" /><ref name="a14">Ito M, Kimura T 1980 IEEE J. QE 16 910</ref>) versus normalized current (J/Jth)]]

[[File:Experimental and theoretical output power versus current for a powerful laser.jpg|thumb|center|upright=2.0|Figure 4. Experimental<ref name="a12" /> and theoretical<ref name="a1" /><ref name="a4" /> output power versus current for a powerful laser.
]]

==Conclusion==
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.<ref name="a1" /><ref name="a4" /> The formula of gain depending on the energy flux has been derived; it is the basic nonlinear effect in a laser. It has been stated that the main effect resulting in nonlinearity is the saturation effect.<ref name="a1" /><ref name="a4" /> For semiconductor lasers, the saturation effect is negligible. We derived the gain g for a Fabry-Perot semiconductor laser based on the density matrix equations and expressions for the natural linewidth.<ref name="a1" /><ref name="a4" /> Thus, the linewidth theory<ref name="a4" /><ref name="a8" /><ref name="a9" /> is an integral part of the nonlinear theory. The resulting dependence of g on the energy flux has been called the main nonlinear effect in semiconductor lasers;<ref name="a1" /><ref name="a4" /> derivation of this relation formula is presented in.<ref name="a1" /><ref name="a4" />  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.

==References==
{{Reflist}}

[[Category:Semiconductor lasers]]