Abstract
A theoretical analysis of single-mode emission of an injection laser is carried out in the framework of the nonlinear gain method. Nonlinear gain is regarded as the result of interaction of a strong (laser) field with weak fields having symmetrically detuned frequencies (superluminescent emission). The interaction is via beats in the inversion at the difference frequencies and correspondingly via a dynamic phase grating of the complex dielectric constant of the active region. Expressions are obtained for the spectral profile of the optical radiation in a single-mode laser with solitary diode cavity, and also in a laser with external cavity. An expression is obtained for the intensity-fluctuation spectrum in a single-mode laser with external cavity. It is shown that in such a laser the superluminescent modes adjacent to a laser mode are split into two components, each due to such an interaction.
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Abbreviations
- a, b:
-
complex dimensionless coefficients of the eigenvector expansion of the field amplitudes
- c:
-
velocity of light
- D:
-
electric induction
- E0(z):
-
laser-mode field amplitude on the cavity axis
- E1(z):
-
complex amplitude of weak field of the long-wave spectral component
- E2(z):
-
complex amplitude of weak field of the short-wave spectral component
- E(x, y, z, t):
-
total intensity of the light-wave field
- f1, f2, f3, f4 :
-
functions characterizing the Fabry-Perot cavity
- ge :
-
total effective gain in diode cavity, including the mirror losses
- gm :
-
mode gain for the laser field
- g1, g2 :
-
waveguide gains for the long- and short-wave components respectively
- g0 :
-
mode gain, saturated by the spontaneous emission, at laser field frequency
- δg:
-
nonlinear increment to mode gain
- δgsp :
-
laser-mode gain deficit produced by spontaneous emission
- G:
-
local value of the gain
- G0 :
-
laser-field threshold local gain
- h1, h2 :
-
coefficients of expansion of the dipole amplitude in terms of the eigenvectors
- ℏω:
-
photon energy
- I(Ω):
-
spectral density of laser-intensity fluctuations
- J:
-
pumping rate
- ℓ:
-
laser-diode length
- L:
-
length of external part of the cavity
- n:
-
refractive index
- n* :
-
group refractive index
- N0 :
-
threshold density of injected electrons
- N:
-
density of injected electrons
- δN:
-
change of electron density
- p(ω):
-
spectral density of the optical power of the radiation from the effective dipole
- p(ω):
-
spectral density of laser optical power
- P0 :
-
threshold pump power
- q0=q0′+iq0″:
-
complex constant of laser-mode propagation along the cavity axis
- q1=q1′+iq1″:
-
complex propagation constant of the long-wave field component
- q2=q2′+iq2″:
-
complex propagation constant of the short-wave field component
- Δq:
-
change of real part of propagation constant due to change of field frequency by linear dispersion
- δq1, δq2 :
-
increments to field propagation constants at the frequencies ω1 and ω2 with account taken of the gain and of the deviation of the dispersion from linearity
- Q:
-
integral indicative of the effective cross section of the light flux
- r:
-
diode-mirror reflection coefficient
- R:
-
waveguide value of amplitude-phase coupling coefficient
- s(ω):
-
effective-dipole spectral emission density
- sV(ω):
-
spontaneous-sources spectral density averaged over the diode volume
- u0(x, y):
-
transverse distribution of laser-mode field
- u1(x, y):
-
transverse distribution of long-wave field component
- u2(x, y):
-
transverse distribution of short-wave field component
- α:
-
dimensionless coefficient indicative of the contribution of the spontaneous emission to one laser mode
- αm=(1/ℓ)ln(1/r):
-
diode-mirror losses
- γ, θ:
-
dimensionless coefficients of the order of unity, indicative of the dependence of the mode and local gains on the electron density
- Γ:
-
total probability of electron recombination (with allowance for stimulated recombination)
- δ1, δ2, δ3, δ4 :
-
effective complex propagation constants of the interacting fields
- ε0(x,y,ω):
-
complex dielectric constant
- δε:
-
variation of dielectric constant
- η:
-
relative excess of pump above threshold
- ηe :
-
external quantum yield
- κ(ω):
-
spectrally selective feedback coefficient of the external dispersive element of the cavity
- κ0 :
-
value of κ(ω) for ω corresponding to maximum reflection
- λ1, λ2 :
-
eigenvalues of propagation constants
- δλ:
-
variations of propagation-constant eigenvalue
- ρ(ω):
-
spectral amplitude of effective dipole
- ρV(ω):
-
spectral amplitude of spontaneous sources averaged over the diode volume
- τ:
-
electron spontaneous-recombination time
- τp :
-
photon relaxation time
- ψ:
-
phase advance of laser field in the external part of the cavity
- ω0 :
-
laser-field central frequency
- ω1 :
-
frequency of long-wave component of the field
- ω2 :
-
frequency of the short-wave component of the field
- ωg :
-
frequency corresponding to the m-th longitudinal resonance of the laser-diode cavity
- Ω:
-
difference frequency
- Ω0 :
-
relaxation-resonance frequency
- ΩL :
-
spectral detuning of laser frequency from the diode longitudinal-resonance frequency
- ΩR :
-
spectral distance between the laser frequency and the satellite peaks
- Ωt :
-
spectral detuning of selective element from longitudinal resonance
- Ωs :
-
width of selectivity profile of external reflecting element
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Additional information
Optoelectronics Laboratory, Quantum Radiophysics Division, Lebedev Physics Institute, Academy of Sciences of the USSR. Translated from Preprint No. 108 of the Lebedev Physics Institute, Academy of Sciences of the USSR, 1989.
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Bogatov, A.P. Fine structure of the emission spectrum of a single-mode injection laser. Nonlinear gain of active semiconducting medium. J Russ Laser Res 10, 485–510 (1989). https://doi.org/10.1007/BF01442222
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DOI: https://doi.org/10.1007/BF01442222