Spectral approach to optical resonator theory

M. D. Feit; J. A. Fleck

doi:10.1364/AO.20.002843

Applied Optics
Vol. 20,
Issue 16,
pp. 2843-2851
(1981)
•https://doi.org/10.1364/AO.20.002843

Spectral approach to optical resonator theory

M. D. Feit and J. A. Fleck

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M. D. Feit and J. A. Fleck, "Spectral approach to optical resonator theory," Appl. Opt. 20, 2843-2851 (1981)

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Abstract

A new computational method for unloaded optical resonators is developed based on the discrete Fourier analysis of information generated by repeated iterations of the optical field corresponding to transits between reflectors. The method is a straightforward extension of the propagating beam method developed earlier for optical fibers for extracting modal properties from numerical solutions to the paraxial scalar wave equation. The method requires computation of a field correlation function, whose Fourier transform reveals the eigenmodes as resonant peaks. Analysis of the location and breadth of these peaks determines the resonator eigenvalues. When the eigenvalues are known, additional discrete Fourier transforms of the field are used to generate the mode eigenfunctions. This new method makes possible the unambiguous identification and accurate characterization of the entire spectrum of transverse resonator modes.

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Mode	Propagation constant		Phase shift	Attenuation
n	$- β_{r}^{'} ({cm}^{- 1})$	$- β_{i}^{'} ({cm}^{- 1})$	ϕ/π(rad)	1 − \|γ\|²%
0	1.7167 × 10⁻³	1.6670 × 10⁻⁴	−5.4645 × 10⁻³	0.3328
1	6.8092 × 10⁻³	6.6743 × 10⁻⁴	−2.1674 × 10⁻²	1.3260
2	1.5456 × 10⁻²	1.4690 × 10⁻³	−4.9199 × 10⁻²	2.8953
3	2.7314 × 10⁻²	2.7833 × 10⁻³	−8.6945 × 10⁻²	5.4145
4	4.2842 × 10⁻²	4.0063 × 10⁻³	−1.3637 × 10⁻¹	7.700
5	6.2046 × 10⁻²	6.5831 × 10⁻³	−1.9749 × 10⁻¹	12.3363
6	8.3664 × 10⁻²	8.6719 × 10⁻³	−2.6631 × 10⁻¹	15.9231
7	1.1041 × 10⁻¹	1.0335 × 10⁻²	−3.5143 × 10⁻¹	18.6736
8	1.4131 × 10⁻¹	1.6185 × 10⁻²	−4.4981 × 10⁻¹	27.6533
9	1.7310 × 10⁻¹	2.0627 × 10⁻²	−5.5100 × 10⁻¹	33.803
10	2.0743 × 10⁻¹	2.6576 × 10⁻²	−6.6025 × 10⁻¹	41.2289
11	2.4989 × 10⁻¹	2.4405 × 10⁻²	−7.9540 × 10⁻¹	38.6209
12	2.9480 × 10⁻¹	3.747 × 10⁻²	−9.3840 × 10⁻¹	52.7350

Mode	Fox-Li		Spectral method
	$- β_{r}^{'} ({cm}^{- 1})$	$- β_{i}^{'} ({cm}^{- 1})$	$β_{r}^{'} ({cm}^{- 1})$	$β_{i}^{'} ({cm}^{- 1})$
0	1.71671 × 10⁻³	1.66736 × 10⁻⁴	1.71672 × 10⁻³	1.66695 × 10⁻⁴
1	6.80920 × 10⁻³	6.67427 × 10⁻⁴	6.80919 × 10⁻³	6.67424 × 10⁻⁴

n	0	1	2	3	4	5	6	7
0	1.0000	0.0005	0.0036	0.0024	0.0044	0.0013	0.0143	0.0026
1	0.0005	1.0000	0.0005	0.0464	0.0004	0.0013	0.0005	0.0047
2	0.0036	0.0005	1.0000	0.0025	0.0140	0.0013	0.0147	0.0027
3	0.0024	0.0464	0.0025	1.0000	0.0023	0.0420	0.0024	0.0232
4	0.0044	0.0004	0.0149	0.0023	1.0000	0.0012	0.0337	0.0025
5	0.0013	0.0013	0.0013	0.0420	0.0012	1.0000	0.0012	0.0677
6	0.0143	0.0005	0.0147	0.0924	0.0337	0.0013	1.0000	6.0026
7	0.0026	0.0047	0.0027	0.0232	0.0325	0.0677	0.0026	1.0000

$- β_{r}^{'} ({cm}^{- 1})$	$- β_{i}^{'} ({cm}^{- 1})$
ν = 0
4.1242 × 10⁻²	4.1533 × 10⁻⁴
2.1740 × 10⁻²	2.2424 × 10⁻³
5.3599 × 10⁻²	5.5141 × 10⁻³
9.9366 × 10⁻²	1.0080 × 10⁻²
1.6039 × 10⁻¹	1.7395 × 10⁻²
2.3650 × 10⁻¹	2.5097 × 10⁻²
−2.9848 × 10⁻¹	5.1595 × 10⁻²
−1.9100 × 10⁻¹	1.1756 × 10⁻¹
ν = 1
1.0543 × 10⁻²	1.0270 × 10⁻³
3.5216 × 10⁻²	3.4503 × 10⁻³
7.3943 × 10⁻²	7.6848 × 10⁻³
1.2738 × 10⁻¹	1.4098 × 10⁻²
1.9417 × 10⁻¹	2.0246 × 10⁻²
2.7976 × 10⁻¹	2.8590 × 10⁻²
−2.5825 × 10⁻¹	5.3747 × 10⁻²
ν = 2
1.8913 × 10⁻²	1.9539 × 10⁻³
5.0983 × 10⁻²	5.2836 × 10⁻³
9.6963 × 10⁻²	9.9871 × 10⁻³
1.5781 × 10⁻¹	1.7487 × 10⁻²
2.3302 × 10⁻¹	2.5328 × 10⁻²
−3.0550 × 10⁻¹	3.7678 × 10⁻²
−2.0212 × 10⁻¹	5.0451 × 10⁻²
ν = 3
2.9148 × 10⁻²	2.8329 × 10⁻³
6.8097 × 10⁻²	7.0439 × 10⁻³
1.2152 × 10⁻¹	1.2613 × 10⁻²
1.8794 × 10⁻¹	2.0710 × 10⁻²
2.7322 × 10⁻¹	3.1532 × 10⁻²
−2.6363 × 10⁻¹	4.9628 × 10⁻²
−1.4365 × 10⁻¹	6.7257 × 10⁻²
ν = 4
4.1416 × 10⁻²	4.2449 × 10⁻³
8.7836 × 10⁻²	8.9656 × 10⁻³
1.4849 × 10⁻¹	1.6292 × 10⁻²
2.2340 × 10⁻¹	2.4834 × 10⁻²
−2.1275 × 10⁻¹	5.5121 × 10⁻²
ν = 5
5.5022 × 10⁻²	5.6033 × 10⁻³
1.0927 × 10⁻¹	1.1211 × 10⁻²
1.7585 × 10⁻¹	1.9121 × 10⁻²
2.6092 × 10⁻¹	2.9803 × 10⁻²
−2.7564 × 10⁻¹	4.3704 × 10⁻²
−1.5986 × 10⁻¹	5.5217 × 10⁻²
ν = 6
7.0854 × 10⁻²	7.1140 × 10⁻³
1.3275 × 10⁻¹	1.4368 × 10⁻²
2.0779 × 10⁻¹	2.1288 × 10⁻²
2.9729 × 10⁻¹	3.6532 × 10⁻²
−2.2742 × 10⁻¹	5.0692 × 10⁻³
−1.0087 × 10⁻¹	7.2452 × 10⁻²
ν = 7
8.8293 × 10⁻²	8.9142 × 10⁻³
1.5673 × 10⁻¹	1.6785 × 10⁻²
2.4196 × 10⁻¹	2.7209 × 10⁻²
−2.9373 × 10⁻¹	3.8499 × 10⁻²
−1.7799 × 10⁻¹	5.3324 × 10⁻²

Mode	Propagation constant		Phase shift	Attenuation
n	$- β_{r}^{'} ({cm}^{- 1})$	$- β_{i}^{'} ({cm}^{- 1})$	ϕ/π(rad)	1 − \|γ\|²%
0	1.7167 × 10⁻³	1.6670 × 10⁻⁴	−5.4645 × 10⁻³	0.3328
1	6.8092 × 10⁻³	6.6743 × 10⁻⁴	−2.1674 × 10⁻²	1.3260
2	1.5456 × 10⁻²	1.4690 × 10⁻³	−4.9199 × 10⁻²	2.8953
3	2.7314 × 10⁻²	2.7833 × 10⁻³	−8.6945 × 10⁻²	5.4145
4	4.2842 × 10⁻²	4.0063 × 10⁻³	−1.3637 × 10⁻¹	7.700
5	6.2046 × 10⁻²	6.5831 × 10⁻³	−1.9749 × 10⁻¹	12.3363
6	8.3664 × 10⁻²	8.6719 × 10⁻³	−2.6631 × 10⁻¹	15.9231
7	1.1041 × 10⁻¹	1.0335 × 10⁻²	−3.5143 × 10⁻¹	18.6736
8	1.4131 × 10⁻¹	1.6185 × 10⁻²	−4.4981 × 10⁻¹	27.6533
9	1.7310 × 10⁻¹	2.0627 × 10⁻²	−5.5100 × 10⁻¹	33.803
10	2.0743 × 10⁻¹	2.6576 × 10⁻²	−6.6025 × 10⁻¹	41.2289
11	2.4989 × 10⁻¹	2.4405 × 10⁻²	−7.9540 × 10⁻¹	38.6209
12	2.9480 × 10⁻¹	3.747 × 10⁻²	−9.3840 × 10⁻¹	52.7350

Abstract

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Figures (9)

Tables (4)

Equations (44)

Applied Optics