Frequency locking in an index-guided semiconductor laser array

Weng W. Chow

doi:10.1364/JOSAB.3.000833

Journal of the Optical Society of America B
Vol. 3,
Issue 6,
pp. 833-836
(1986)
•https://doi.org/10.1364/JOSAB.3.000833

Frequency locking in an index-guided semiconductor laser array

Weng W. Chow

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Weng W. Chow, "Frequency locking in an index-guided semiconductor laser array," J. Opt. Soc. Am. B 3, 833-836 (1986)

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Abstract

A theory is developed for investigating frequency locking in an index-guided semiconductor laser array. This theory, which is based on the expansion of the laser field in terms of the passive array eigenmodes, and on the use of semiclassical laser theory to treat the interaction of these modes among themselves, is valid for all values of coupling between lasers and for all array sizes. To illustrate the application of this theory, we use it to determine the lockband of a two-laser array with dissimilar waveguides.

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Coefficient	Physical Content
$β_{n} = \frac{F_{n}}{N} \frac{γ_{c υ}}{γ} \int d^{3} r n^{2} (x, y) {u_{n}}^{4} (r) \int_{- (ν_{n} - ν_{g}) / γ}^{\infty} d u {(u + \frac{ν_{n} - ν_{g}}{γ})}^{1 / 2} L^{2} (u) η (r, u)$	Self-saturation
$ρ_{n} = \frac{F_{n}}{N} \frac{γ_{c υ}}{γ} \int d^{3} r n^{2} (x, y) {u_{n}}^{4} (r) \int_{- (ν_{n} - ν_{g}) / γ}^{\infty} d u {(u + \frac{ν_{n} - ν_{g}}{γ})}^{1 / 2} u L^{2} (u) η (r, u)$	Self-mode pushing
$Ө_{n m} = \frac{F_{n}}{N} \frac{γ_{c υ}}{γ} \int d^{3} r n^{2} (x, y) {u_{n}}^{2} (r) {u_{m}}^{2} (r) \int_{- (ν_{n} - ν_{g}) / γ}^{\infty} d u {(u + \frac{ν_{n} - ν_{g}}{γ})}^{1 / 2} η (r, u) [L (u + \frac{ν_{n} - ν_{m}}{γ}) L (u) + Re (p)]$	Cross mode saturation
$τ_{n m} = \frac{F_{n}}{N} \frac{γ_{c υ}}{γ} \int d^{3} r n^{2} (x, y) {u_{n}}^{2} (r) {u_{m}}^{2} (r) \int_{- (ν_{n} - ν_{g}) / γ}^{\infty} d u {(u + \frac{ν_{n} - ν_{g}}{γ})}^{1 / 2} η (r, u) [u L (u + \frac{ν_{n} - ν_{m}}{γ}) L (u) - Im (p)]$	Cross mode pushing
$p = [γ_{υ} D (\frac{ν_{m} - ν_{n}}{γ_{c}}) + γ_{c} D (\frac{ν_{m} - ν_{n}}{γ_{υ}})] [D (\frac{ν_{m} - ν_{n}}{γ} - u) + D (u)] D (u) / (4 γ_{c υ})$	Population pulsation contribution
$\begin{array}{l} Λ_{n} = & [(Ө_{n m n n} + Ө_{n n n m}) {(I_{n} I_{m})}^{1 / 2} + Ө_{n m m m} {I_{m}}^{3 / 2} / {I_{n}}^{1 / 2}] exp [i (ψ_{n} - ψ_{m})] + Ө_{n n m n} {(I_{n} I_{m})}^{1 / 2} exp [- i (ψ_{n} - ψ_{m})] \\ + Ө_{n m n m} I_{m} exp [2 i (ψ_{n} - ψ_{m})] \end{array}$	Third-order frequency locking

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Journal of the Optical Society of America B