Stability and beam divergence of multimode lasers with internal variable lenses

H. P. Kortz; R. Iffländer; H. Weber

doi:10.1364/AO.20.004124

Applied Optics
Vol. 20,
Issue 23,
pp. 4124-4134
(1981)
•https://doi.org/10.1364/AO.20.004124

Stability and beam divergence of multimode lasers with internal variable lenses

H. P. Kortz, R. Iffländer, and H. Weber

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H. P. Kortz, R. Iffländer, and H. Weber, "Stability and beam divergence of multimode lasers with internal variable lenses," Appl. Opt. 20, 4124-4134 (1981)

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Abstract

A resonator with an internal lens can be characterized by its equivalent g-parameters which give the regions of stability if the lens has variable focal length. However, other properties are not given by this equivalence and have to be investigated by means of the resonator matrix method. Beam diameter, divergence angle, number of transversal modes, positions of beam waists, stability, and the correlations of these parameters with position and focal length of the internal lens are discussed. Regions of low sensitivity to variations of the focal length are evaluated and compared with experimental results. For some special resonator configurations numerical results are presented.

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Type	Instability	Critical value of refractive power	Critical value of g*-parameter
I	$g_{1}^{} \cdot g_{2}^{} = 1$	$D_{I} = - \frac{1}{ρ_{1} - d_{1}} - \frac{1}{ρ_{2} - d_{2}}$	$g_{1, I}^{*} = \frac{1 - d_{1} / ρ_{1}}{1 - d_{2} / ρ_{2}}$	$g_{2, I}^{*} = \frac{1 - d_{2} / ρ_{2}}{1 - d_{1} / ρ_{1}}$
	$g_{i}^{*} > 0$
II	$g_{1}^{} \cdot g_{2}^{} = 0$	$D_{II} = \frac{1}{d_{2}} - \frac{1}{ρ_{1} - d_{1}}$	$g_{1, II}^{*} = 0$	$g_{2, II}^{*} = g_{2} - g_{1} \cdot \frac{d_{1}}{d_{2}} \frac{1 - d_{2} / ρ_{2}}{1 - d_{1} / ρ_{1}}$
III	$g_{1}^{} \cdot g_{2}^{} = 0$	$D_{III} = \frac{1}{d_{1}} - \frac{1}{ρ_{2} - d_{2}}$	$g_{1, III}^{*} = g_{1} - g_{2} \cdot \frac{d_{2}}{d_{1}} \cdot \frac{1 - d_{1} / ρ_{1}}{1 - d_{2} / ρ_{2}}$	$g_{2, III}^{*} = 0$
IV	$g_{1}^{} \cdot g_{2}^{} = 1$	$D_{IV} = \frac{1}{d_{1}} + \frac{1}{d_{2}}$	$g_{1, IV}^{*} = - d_{2} / d_{1}$	$g_{2, IV}^{*} = - d_{1} / d_{2}$
	$g_{i}^{*} < 0$

				L₀₁			L₀₂			Θ₀₁			Θ₀₂

D	w₁	w₂	w₀_L	sph	pl	pl/sph	sph	pl	pl/sph	sph	pl	pl/sph	sph	pl	pl/sph
D_I	∞	∞	∞	ρ₁	0	0	ρ₂	0	ρ₂	π	0	0	π	0	π
D_II	∞	0	∞	ρ₁	0	0	0	0	0	π	0	0	π	π	π
D_III	0	∞	∞	0	0	0	ρ₂	0	ρ₂	π	π	π	π	0	π
D_IV	0	0	∞	0	0	0	0	0	0	π	π	π	π	π	π

Rod	Type	R/mm	l/mm	P_0/W	P_c_/_k_W	α
1	Schott LG 703	3	180	1050	8.8	0.86
2	Schott LG 703	4	180	1300	10.9	0.86
3	Schott LG 706	3	180	720	6.0	0.86
4	Schott LG 706	4	180	1300	10.9	0.86
5	Schott LG —	4	180	560	4.7	0.94
6	Schott LG —	3	100	440	6.6	0.94
7	Hoya LHG 8	3	100	730	11.1	1.54
8	Kigre Q 98	3	100	630	9.5	—
9	Schott LG 706	3	100	330	5.0	—
10	YAG	3	90	1440	28.8	1.0

Type	Instability	Critical value of refractive power	Critical value of g*-parameter
I	$g_{1}^{} \cdot g_{2}^{} = 1$	$D_{I} = - \frac{1}{ρ_{1} - d_{1}} - \frac{1}{ρ_{2} - d_{2}}$	$g_{1, I}^{*} = \frac{1 - d_{1} / ρ_{1}}{1 - d_{2} / ρ_{2}}$	$g_{2, I}^{*} = \frac{1 - d_{2} / ρ_{2}}{1 - d_{1} / ρ_{1}}$
	$g_{i}^{*} > 0$
II	$g_{1}^{} \cdot g_{2}^{} = 0$	$D_{II} = \frac{1}{d_{2}} - \frac{1}{ρ_{1} - d_{1}}$	$g_{1, II}^{*} = 0$	$g_{2, II}^{*} = g_{2} - g_{1} \cdot \frac{d_{1}}{d_{2}} \frac{1 - d_{2} / ρ_{2}}{1 - d_{1} / ρ_{1}}$
III	$g_{1}^{} \cdot g_{2}^{} = 0$	$D_{III} = \frac{1}{d_{1}} - \frac{1}{ρ_{2} - d_{2}}$	$g_{1, III}^{*} = g_{1} - g_{2} \cdot \frac{d_{2}}{d_{1}} \cdot \frac{1 - d_{1} / ρ_{1}}{1 - d_{2} / ρ_{2}}$	$g_{2, III}^{*} = 0$
IV	$g_{1}^{} \cdot g_{2}^{} = 1$	$D_{IV} = \frac{1}{d_{1}} + \frac{1}{d_{2}}$	$g_{1, IV}^{*} = - d_{2} / d_{1}$	$g_{2, IV}^{*} = - d_{1} / d_{2}$
	$g_{i}^{*} < 0$

				L₀₁			L₀₂			Θ₀₁			Θ₀₂

D	w₁	w₂	w₀_L	sph	pl	pl/sph	sph	pl	pl/sph	sph	pl	pl/sph	sph	pl	pl/sph
D_I	∞	∞	∞	ρ₁	0	0	ρ₂	0	ρ₂	π	0	0	π	0	π
D_II	∞	0	∞	ρ₁	0	0	0	0	0	π	0	0	π	π	π
D_III	0	∞	∞	0	0	0	ρ₂	0	ρ₂	π	π	π	π	0	π
D_IV	0	0	∞	0	0	0	0	0	0	π	π	π	π	π	π

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Equations (55)

Applied Optics