Analytic eigenmode solution for the self-filtering unstable resonator

William P. Latham; Michael L. Tilton; Thomas R. Ferguson

doi:10.1364/AO.31.001948

Applied Optics
Vol. 31,
Issue 12,
pp. 1948-1959
(1992)
•https://doi.org/10.1364/AO.31.001948

Analytic eigenmode solution for the self-filtering unstable resonator

William P. Latham, Michael L. Tilton, and Thomas R. Ferguson

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William P. Latham, Michael L. Tilton, and Thomas R. Ferguson, "Analytic eigenmode solution for the self-filtering unstable resonator," Appl. Opt. 31, 1948-1959 (1992)

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Abstract

An analytic solution for the bare cavity eigenmodes of a nonsymmetric self-filtering unstable resonator is obtained by a modal expansion in prolate functions, which is a complete and orthogonal set of eigenmodes for a symmetric confocal stable resonator. An accurate representation within the aperture is shown to require only three terms. An efficient use of Gaussian quadrature for the various calculations is described.

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Systems	Γ_ℓ,_k	C₁	C₂	B_ℓK_ℓ(C₁, C₂, r, r′, r″)dA	s	t
Linear	Γ_ℓ × Γ_k	$\frac{2 π a^{2}}{λ f_{1}} = π$	$\frac{2 π a^{2}}{λ f_{2}} = \frac{π}{∣ M ∣}$	$i \frac{\sqrt{C_{1} C_{2}}}{2 π} exp (i C_{2} x x^{'}) exp (i C_{1} x^{'} x^{″}) d x^{'} d x^{″}$	−1	1
Radial	Γ_ℓ,_k	$\frac{2 π a^{2}}{λ f_{1}} = 1.22 π$	$\frac{2 π a^{2}}{λ f_{2}} = \frac{1.22 π}{∣ M ∣}$	(−1)^ℓ+1C₁C₂J_ℓ(C₂rr′) J_ℓ(C₁r′r″)r′r″dr′dr″	0	1

Systems	γ_ℓ,_k	b_ℓ	c	k_ℓ(c, r, r′)	dL	s	t	d_k_,ℓ
Linear	β_k	$\sqrt{i}$	$\frac{2 π a^{2}}{λ f}$	$\sqrt{\frac{c}{2 π}} exp (i c x x^{'})$	dx′	−1	1	1
Radial	γ_ℓ,_k	i^ℓ+1	$\frac{2 π a^{2}}{λ f}$	cJ_ℓ(crr′)	r′dr′	0	1	1

Systems	D_ℓ,_k	S_m_,_p
Linear	${(\frac{i C_{2}}{2 π})}^{1 / 2}$	$i {(- 1)}^{p} β_{m} \sum_{j = 1}^{N} \sum_{n = 1}^{N} w_{j} w_{n} f_{m} (x_{j}^{'}) f_{p} (x_{n}) exp (i C_{2} x_{j}^{'} x_{n})$
Radial	C₂(−1)^ℓ+1	$γ_{l, m} \sum_{j = 1}^{N} \sum_{n = 1}^{N} {\tilde{w}}_{j} {\tilde{w}}_{n} f_{l, m} (r_{j}^{'}) f_{l, p} (r_{n}) J_{l} (C_{2} r_{j}^{'} r_{n})$

	Linear Eigenvalues

k	Prolates β_k	M = −2 SFUR Γ_k	M = −5 SFUR Γk
0	0.99048	0.80201	0.52416
1	0.86581	0.37333	0.10362
2	0.49355	0.05635	0.00618
3	0.15699	0.00239	9.95 × 10⁻⁵
4	0.03265	4.86 × 10⁻⁵	7.95 × 10⁻⁷
5	5.24 × 10⁻³	6.15 × 10⁻⁷	4.00 × 10⁻⁹
6	6.93 × 10⁻⁴	5.35 × 10⁻⁹	1.39 × 10⁻¹¹
7	7.83 × 10⁻⁵	3.41 × 10⁻¹¹	3.50 × 10⁻¹⁴

	Linear Expansion Coefficients (M = −2)

	Even SFUR Eigenmodes				Even SFUR Adjoint Modes

n	k =	0	2	4	0	2	4
0		0.96544	0.13187	0.00078	0.99110	0.26017	0.02353
2		0.26027	0.98213	0.00922	0.13314	0.96552	0.13846
4		0.01345	0.13422	0.99634	0.00046	0.00873	0.99009
6		0.00015	0.00360	0.08494	1 × 10⁻⁷	5 × 10⁻⁶	0.00179
8		4 × 10⁻⁷	0.00003	0.00197	3 × 10⁻¹²	5 × 10⁻¹⁰	5 × 10⁻⁷
	Odd SFUR Eigenmodes				Odd SFUR Adjoint Modes

n	k =	1	3	5	1	3	5

1		0.97878	0.03770	0.00009	0.99928	0.20478	0.01545
3		0.20480	0.99387	0.00350	0.03791	0.97880	0.10451
5		0.00632	0.10387	0.99742	0.00004	0.00341	0.99440
7		0.00005	0.00261	0.07170	5 × 10⁻⁹	1 × 10⁻⁶	0.00107
9		3 × 10⁻⁸	0.00002	0.00153	3 × 10⁻¹⁴	1 × 10⁻¹⁰	2 × 10⁻⁷

	Linear Expansion Coefficients (M = −5)

	Even SFUR Eigenmodes				Even SFUR Adjoint Modes

n	k =	0	2	4	0	2	4
0		0.94884	0.16081	0.00113	0.98660	0.31478	0.03396
2		0.31494	0.97305	0.01153	0.16318	0.94911	0.17265
4		0.02250	0.16513	0.99410	0.00077	0.01066	0.98439
6		0.00046	0.00662	0.10780	3 × 10⁻⁷	9 × 10⁻⁶	0.00227
8		5 × 10⁻⁶	0.00011	0.00356	4 × 10⁻¹¹	2 × 10⁻⁹	8 × 10⁻⁷
	Odd SFUR Eigenmodes				Odd SFUR Adjoint Modes

n	k =	1	3	5	1	3	5

1		0.96759	0.04680	0.00013	0.99888	0.25220	0.02214
3		0.25223	0.99029	0.00442	0.04722	0.96767	0.13192
5		0.01196	0.13078	0.99581	0.00007	0.00426	0.99101
7		0.00022	0.00477	0.09127	2 × 10⁻⁸	2 × 10⁻⁶	0.00136
9		2 × 10⁻⁶	0.00008	0.00273	2 × 10⁻¹²	3 × 10⁻¹⁰	4 × 10⁻⁷

	Circular Eigenvalues

k	Prolates γ_0,_k	M = −2 SFUR Γ_0,_k	M = −5 SFUR Γ_0,_k
0	0.98350	0.64768	0.27715
1	0.47184	0.03867	0.00274
2	0.03516	4.03 × 10⁻⁵	4.18 × 10⁻⁷
3	9.36 × 10⁻⁴	6.93 × 10⁻⁹	1.14 × 10⁻¹¹
4	1.36 × 10⁻⁵	3.67 × 10⁻¹³	3.20 × 100⁻¹⁵

	Circular Expansion Coefficients (M = −2)

	SFUR Eigenmodes				SFUR Adjoint Modes

n	k =	0	1	2	0	1	2
0		0.94038	0.16794	0.00144	0.98535	0.33910	0.03987
1		0.33930	0.97107	0.01344	0.17057	0.94067	0.17858
2		0.02350	0.16960	0.99316	0.00088	0.01224	0.98311
3		0.00040	0.00655	0.11595	4 × 10⁻⁷	0.00001	0.00306
4		2 × 10⁻⁶	0.00009	0.00386	3 × 10⁻¹¹	3 × 10⁻⁹	1 × 10⁻⁶
5		3 × 10⁻¹⁰	5 × 10⁻⁷	0.00005	4 × 10⁻¹⁷	1 × 10⁻¹³	2 × 10⁻¹⁰
6		4 × 10⁻¹¹	1 × 10⁻¹¹	4 × 10⁻⁷	4 × 10⁻²⁰	2 × 10⁻²⁰	8 × 10⁻¹⁵

	Circular Expansion Coefficients (M = −5)

	SFUR Eigenmodes				SFUR Adjoint Modes

n	k =	0	1	2	0	1	2
0		0.91420	0.20243	0.00208	0.97831	0.40324	0.05732
1		0.40350	0.95740	0.01674	0.20716	0.91498	0.22117
2		0.03785	0.20563	0.98902	0.00145	0.01464	0.97354
3		0.00112	0.01152	0.14662	1 × 10⁻⁶	0.00002	0.00384
4		0.00002	0.00029	0.00686	3 × 10⁻¹⁰	8 × 10⁻⁹	3 × 10⁻⁶
5		2 × 10⁻⁷	4 × 10⁻⁶	0.00015	2 × 10⁻¹⁴	1 × 10⁻¹²	5 × 10⁻¹⁰
6		4 × 10⁻¹¹	1 × 10⁻¹¹	4 × 10⁻⁷	4 × 10⁻²⁰	2 × 10⁻²⁰	8 × 10⁻¹⁵

Sum of Squares of Coefficients [see Eqs. (D1) and (D2)]

Number of Terms in Series	M = −2 Linear SFUR Refer to Table V		M = −5 Circular SFUR Refer to Table IX

	Eigenmode	Adjoint	Eigenmode	Adjoint
1	0.93208	0.98227	0.83575	0.95708
2	0.99982	0.99999	0.99857	0.99999
3	0.99999	1.00000	0.99999	1.00000
4	1.00000	1.00000	1.00000	1.00000
5	1.00000	1.00000	1.00000	1.00000

Abstract

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Figures (5)

Tables (10)

Equations (41)

Applied Optics