Does a theory of semiconducting laser line width exist? B. Spivak UW, S. Luryi SUNY

A cartoon picture of a laser Active medium  >0 Semi-transparent mirrors Pumping n1n1 n2n2 A requirement for lasing : the inverse population n=n 1 -n 2 >0 laser light

Characteristic parameters: Laser line width:  Spectral width of a cavity :  C Uniform broadening of electron eigenstates 1/   Inverse life time with respect to photon 1/  ph radiation The width of the excitation spectrum  I The spectral width of the amplification  The frequency of Rabi oscillations  R (I)  C  I,    ph  h  R << h  

An example: semiconducting injection lasers electrons holes Why the laser’s line width is so narrow ? II holes electrons   P N light

A very simple theory (model ?) of laser

lasing mode Z is a complex number is an eigenfunction of Maxwell equations with appropriate boundary conditions at the cavity mirrors

(Strictly speaking incorrect) rate equations  description of the laser kinetics N+1 >>1 N~|Z |2 is the number of photons, n=n 1 –n 2 is the electron population difference,  characterizes loss of photons as they leave the cavity through mirrors, I is the injection intensity,  nr is a characteristic time of non- radiative recombination

relaxation oscillations of the laser intensity t N(t) Perturbations of the number of the photons (or |Z|) decay in time, and N(t) approaches it’s equilibrium value

Intensity fluctuations: one can introduce delta-correlated in time random Langevin sources in the rate equations N+1

Full statistics of the intensity fluctuation (C.H. Henry, P.S. Henry, M. Lax, 1984) The frequency of the relaxation oscillations is smaller than the spontaneous emission rate.

Schawlow-Townes theory of laser’s line width Z   an(    Z<<Z a photon  Z 

Questions: 1.What is the frequency interval in which spontaneous emission of photon determines the value of  2.More importantly: Is it correct that at the mean field level (before the spontaneous emission is taken into account) a single frequency generation takes place? If not, then ……. 3.What is the relation of the problem with the problem of turbulent plasma?

Another question which will help us to understand what the problem is: What is the number of lasing modes, N, and how does N depend on I? In the case of semiconductor lasers nobody really knows for sure, but it looks like if no precautions are taken (no distributed feedback) the number of modes first increases with the injection intensity I (N<100) and then, at larger I, it decreases with I.

A simple (and probably, not entirely correct) model.    This model can explain the increase of N with I, but can not explain its subsequent decrease with I electrons I st [n  ] is a scattering integral describing electron and electron- phonon scattering which conserve the total number of electrons And holes and redistribute them in energy. ( The characteristic rate is 1/   

Back to the problem of laser line width    Does the line width  increase with the injection intensity ? Solution of the kinetic equations gives us region of applicability of the kinetic equation :   electrons

Why the laser line width is so narrow? Because   is short? What is wrong with previous arguments?  h/    The electron distribution function should be calculated with a precision better than the broadening of the electron levels. This poses a very difficult theoretical problem, which does not have precedents in the kinetic theory 

a. The limit    corresponds to the model of two level system where at the mean field level  (Is the semiconducting laser line width narrow because   is short?) b. Any way in the framework of the model the line width increases with I ! Ignoring this fact we can write

#2 A spatial hole burning n(r),  (r) leads to both competition between modes, and to a competition between harmonics with different frequencies within one mode.

#3 Beating in time: Consider for example two lines (R. Kazarinov, C. Henry) There is no stationary solution of this problem! n(t) depends on time, which leads to a competition between modes and a competition of EMF oscillations at different frequencies with a mode

  cc    linear analysis of Maxwell equations at  :  c Re Im At I>I c there are no stationary solutions of the problem !

Theory is not that impressive What about experiment ?

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, M. Aoki, IEEE J. Quantum Electron. QE-27, 1782 (1991) Experiments are not that impressive either

Other side of the problem: A model of two level system couples with EMF This system of equations is equivalent to that exhibiting chaos and the Lorenz strange attractor! (Haken)

Conclusion: We do not know why semiconducting laser line widths are narrow and which factors determine it