Injection Locking
Locking range of an injection-locked laser from the injection ratio, coupling rate and linewidth enhancement factor α: Adler's symmetric range and the asymmetric semiconductor-laser range, with a locking map.
Coupling rate from the slave cavity
k_c = (1 − R)/(τ_rt√R) = 2π × 24.51 GHz, τ_rt = 2n_gL/c = 7.805 ps.
Locking range
-3.466 to +2.451 GHz
width 5.917 GHz
Injection ratio ρ
-20.00 dB
P_inj/P = 0.01000; √ρ = 0.1000
Adler range (α = 0)
±2.451 GHz
α widens the red side by √(1+α²) = 1.414
At this detuning
In static range
a locked steady state exists; it can still be dynamically unstable
A weak master field injected into a free-running slave laser pulls its frequency onto the master's when the detuning Δf = f_master − f_slave lies inside −(k_c/2π)√ρ√(1 + α²) ≤ Δf ≤ (k_c/2π)√ρ, with ρ = P_inj/P. For α = 0 (gas and solid-state lasers) this is Adler's symmetric range. In a semiconductor laser the gain and the refractive index change together, which stretches the range towards negative detuning: Berger et al. (1991) measured α ≈ 1 in 0.98 µm InGaAs/GaAs quantum-well lasers (the default), against ≈ 3.5 for InGaAsP/InP quantum wells and ≈ 5.7 for InGaAsP/InP double heterostructures. The coupling rate k_c is an input. The estimate above, k_c = (1 − R)/(τ_rt√R) (Kondratiev et al. 2023, Eq. 4), sets the default for a 300 µm cleaved InGaAs/GaAs diode (R 0.32, n_g 3.9 from Wenzel et al. 2021); other authors define it with different factors of order unity. It pairs with ρ = P_inj/P when both powers are taken outside the injection facet; entering the total output of a symmetric slave halves ρ. The static range is not the stable band: relaxation-oscillation dynamics (a Hopf bifurcation, set by the relaxation frequency and damping, which are not inputs here) can make the lock unstable well inside its positive edge, and at very weak injection the stable band can reach slightly past it. The negative edge, a saddle-node, holds.
Locking map
- Upper edge k_c√ρ
- Lower edge −k_c√ρ√(1+α²)
- Adler lower edge (α = 0)
- This detuning
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Static locking range −k_c√ρ√(1 + α²) ≤ Δω ≤ k_c√ρ of the single-mode rate equations, weak injection. The coupling rate k_c is an input; the page estimates it as (1 − R)/(τ_rt√R) (Kondratiev et al. 2023), and other authors' expressions differ by factors of order unity. Defaults: a 300 µm strained InGaAs/GaAs quantum-well diode at 0.98 µm (R 0.32, n_g 3.9, k_c/2π 24.51 GHz) with the α ≈ 1 Berger et al. measured below threshold (Banerjee et al.: 0.34–0.54 in a similar well; unstrained GaAs and InGaAsP lasers have larger α). The dynamically stable band (Hopf edge, set by the relaxation frequency and damping) is not computed: it can end well inside the positive edge or, at very weak injection, reach past it.
- Adler R. (1946). A study of locking phenomena in oscillators. Proc. IRE 34, 351. doi:10.1109/JRPROC.1946.229930
- Lang R. (1982). Injection locking properties of a semiconductor laser. IEEE J. Quantum Electron. 18, 976. doi:10.1109/JQE.1982.1071632
- Mogensen F., Olesen H., Jacobsen G. (1985). Locking conditions and stability properties for a semiconductor laser with external light injection. IEEE J. Quantum Electron. 21, 784. doi:10.1109/JQE.1985.1072760
- Kondratiev N. M. et al. (2023). Recent advances in laser self-injection locking to high-Q microresonators. Front. Phys. 18, 21305. Eq. (4): coupling rate κ = (1 − R_o²)/(τ R_o), R_o the amplitude reflectivity. doi:10.1007/s11467-022-1245-3
- Berger P. R., Dutta N. K., Lopata J., Chand N. (1991). In0.2Ga0.8As/GaAs strained quantum-well lasers. CLEO 1991, paper CTuA7. Measured α ≈ 1.0 at 0.98 µm (≈ 3.5 for InGaAsP/InP quantum wells). opg.optica.org
- Banerjee S., Srivastava A. K., Chand N. (1991). Measurement of the linewidth enhancement factor in strained InGaAs/GaAs single quantum well lasers. Appl. Phys. Lett. 58, 2198. α 0.54, down to 0.34 at lower current, below threshold at 0.97 µm. doi:10.1063/1.104925
- Henry C. H., Olsson N. A., Dutta N. K. (1985). Locking range and stability of injection locked 1.54 µm InGaAsP semiconductor lasers. IEEE J. Quantum Electron. 21, 1152. Range −ω₁√(1 + α²) to ω₁, unstable at the high-frequency end; α 6 ± 1. doi:10.1109/JQE.1985.1072787
- Wenzel H., Kantner M., Radziunas M., Bandelow U. (2021). Semiconductor Laser Linewidth Theory Revisited. Appl. Sci. 11, 6004. Table I: group index 3.9. doi:10.3390/app11136004