Coupling loss theory of single-mode waveguide resonators

Christopher A. Hill; Denis R. Hall

doi:10.1364/AO.24.001283

Applied Optics
Vol. 24,
Issue 9,
pp. 1283-1290
(1985)
•https://doi.org/10.1364/AO.24.001283

Coupling loss theory of single-mode waveguide resonators

Christopher A. Hill and Denis R. Hall

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Christopher A. Hill and Denis R. Hall, "Coupling loss theory of single-mode waveguide resonators," Appl. Opt. 24, 1283-1290 (1985)

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Abstract

The past dozen years have seen several important publications on mode coupling losses in circular and rectangular waveguide lasers. It is frequently assumed that the laser mode is pure EH₁₁ (quasi-TEM₀₀). We note a flaw in the widely quoted Laguerre-Gaussian mode expansion method as it originally appeared and show how to reconcile it with later results. Also we summarize and try to remove several discrepancies in the published accounts of how the EH₁₁ loss behaves for the popular near-Case I reflector (i.e., a plane mirror placed within a few guide widths of the guide aperture).

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Quantity		Symbol
Halfwidth of square guide		a
Radius of circular guide		a
Separation of guide and mirror		z (or d)
Mirror radius of curvature		R_M (or c)
Gaussian beam parameters		$ω_{0} b = π ω_{0}^{2} / λ$
Dimensionless quantities:
a²/(λz)		N (Fresnel number of reflector)
ka²/R_M		α
z/R_M		β

Authors	Geometry	Method	EH₁₁ coupling loss (%)	Range
Kogelnik23	TEM₀₀ Coupling	Gaussian beam equations	$100 {(\frac{z}{b})}^{2}$	z/b ≪ 1
Abrams10	Circular	Laguerre-Gaussian expansion, ω₀ = 0.6435a	$≃ 70 {(\frac{z}{b})}^{3 / 2}$	$z / b ⩽ 0.4 (from graph)$
Degnan and Hall15	Circular	Diffraction integral, finite aperture	$605 {(\frac{z}{k a^{2}})}^{3 / 2} or$
			$57 {(\frac{z}{b})}^{3 / 2}$ 38.4N^−3/2	$z / (k a^{2}) ⩽ 0.1, z / b ⩽ 0.48$
Henderson12	Square	Hermite-Gaussian expansion, ω₀ = 0.7032a	$80 {(\frac{z}{b})}^{2}$ or $1310 {(\frac{z}{k a^{2}})}^{2}$ , 33N⁻²	$z / b ⩽ 0.2$
Avrillier and Verdonck19	Square	Diffraction integral, infinite aperture	$161.8 {(\frac{z}{k a^{2}})}^{2}$ (see Ref. 24)	$z / (k a^{2}) ⩽ 0.0125$
Boulnois and Agrawal20	Square	Diffraction integral, infinite aperture	≃33N^−3/2 or $64.5 {(\frac{z}{b})}^{3 / 2}, 524 {(\frac{z}{k a^{2}})}^{3 / 2}$	N≫1

Normalized distance	$\frac{z}{k a^{2}}$	$N = \frac{a^{2}}{λ z}$
z/b	$(\frac{z}{k a^{2}}) 2 {(\frac{a}{ω_{0}})}^{2}$	$\frac{N^{- 1}}{π} {(\frac{a}{ω_{0}})}^{2}$
z/b	$= 4.83 (\frac{z}{k a^{2}})$	= 0.769N⁻¹ (circular)
	$= 4.04 (\frac{z}{k a^{2}})$	= 0.644N⁻¹ (square)

Quantity		Symbol
Halfwidth of square guide		a
Radius of circular guide		a
Separation of guide and mirror		z (or d)
Mirror radius of curvature		R_M (or c)
Gaussian beam parameters		$ω_{0} b = π ω_{0}^{2} / λ$
Dimensionless quantities:
a²/(λz)		N (Fresnel number of reflector)
ka²/R_M		α
z/R_M		β

Authors	Geometry	Method	EH₁₁ coupling loss (%)	Range
Kogelnik23	TEM₀₀ Coupling	Gaussian beam equations	$100 {(\frac{z}{b})}^{2}$	z/b ≪ 1
Abrams10	Circular	Laguerre-Gaussian expansion, ω₀ = 0.6435a	$≃ 70 {(\frac{z}{b})}^{3 / 2}$	$z / b ⩽ 0.4 (from graph)$
Degnan and Hall15	Circular	Diffraction integral, finite aperture	$605 {(\frac{z}{k a^{2}})}^{3 / 2} or$
			$57 {(\frac{z}{b})}^{3 / 2}$ 38.4N^−3/2	$z / (k a^{2}) ⩽ 0.1, z / b ⩽ 0.48$
Henderson12	Square	Hermite-Gaussian expansion, ω₀ = 0.7032a	$80 {(\frac{z}{b})}^{2}$ or $1310 {(\frac{z}{k a^{2}})}^{2}$ , 33N⁻²	$z / b ⩽ 0.2$
Avrillier and Verdonck19	Square	Diffraction integral, infinite aperture	$161.8 {(\frac{z}{k a^{2}})}^{2}$ (see Ref. 24)	$z / (k a^{2}) ⩽ 0.0125$
Boulnois and Agrawal20	Square	Diffraction integral, infinite aperture	≃33N^−3/2 or $64.5 {(\frac{z}{b})}^{3 / 2}, 524 {(\frac{z}{k a^{2}})}^{3 / 2}$	N≫1

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Equations (47)

Applied Optics