Calculation of prolate functions for optical analysis

William P. Latham; Michael L. Tilton

doi:10.1364/AO.26.002653

Applied Optics
Vol. 26,
Issue 13,
pp. 2653-2658
(1987)
•https://doi.org/10.1364/AO.26.002653

Calculation of prolate functions for optical analysis

William P. Latham and Michael L. Tilton

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William P. Latham and Michael L. Tilton, "Calculation of prolate functions for optical analysis," Appl. Opt. 26, 2653-2658 (1987)

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Abstract

The linear and circular prolate functions are basis sets for the bare cavity eigenmodes of stable and unstable laser resonators having rectangular or circular symmetry. Early calculations of these functions used power series expansions. A matrix technique which was first used by Sanderson and Streifer is used here to demonstrate rapid and accurate numerical calculations of prolate functions.

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Eigenfunction	β_k	a	b	K(c,x,y)	d_k
Circular Prolates, ϕ_ℓk(x)	γ_ℓ,k	0	1	√cxy J_ℓ(cxy)	1
Linear Prolates, ψ_k(x)	λ_k	−1	1	exp(icxy)	λ_k

	N = 6	N = 10	N = 15	N = 24	N = 40	N = 80

	.88056058	.88056076	.88056076	.88056076	.88056076	.88056076
	.35564125	.35564095	.35564095	.35564098	.35564092	.35564098
	.35867363 E–1	.35867721 E–1	.35867706 E–1	.35867732 E–1	.35867706 E–1	.35867766 E–1
	.11524374 E–2	.11522414 E–2	.11522460 E–2	.11522381 E–2	.11522311 E–2	.11522470 E–2
	.18815203 E–4	.18889350 E–4	.18907645 E–4	.18883109 E–4	.18871044 E–4	.18873188 E–4
	.21277961 E–6	.19506624 E–6	.19239708 E–6	.20387050 E–6	.19744330 E–6	.20130742 E–6
	-----------	.35828301 E–7	.50898183 E–7	.47300404 E–7	.23037570 E–7	.48458858 E–7
	-----------	.11745461 E–7	.12596061 E–7	.15218891 E–7	.23037568 E–7	.34920145 E–7
	-----------	.88052969 E–8	.12596056 E–7	.15218887 E–7	.16142064 E–7	.34920131 E–7

CPU (min)	.92	1.02	1.03	1.05	1.18	2.35

	N = 6	N = 10	N = 15	N = 24	N = 40	N = 80

	1.4921626	.99999499	.99999869	.99999869	.99999887	.99999875
	1.0088998	.99994719	.99987978	.99987996	.99987984	.99987942
	.97439581	.99659437	.99700540	.99700564	.99700534	.99700540
	.96274185	.96208018	.96054637	.96054661	.96054661	.96054665
	.33395001	.74484408	.74790347	.74790347	.74790353	.74790347
	.32081228	.32350591	.32027680	.32027689	.32027698	.32027701
	-------	.58974341 E–1	.60784385 E–1	.60784586 E–1	.60784388 E–1	.60784504 E–1
	-------	.66820979 E–2	.61262734 E–2	.61263009 E–2	.61262450 E–2	.61262962 E–2
	-------	.30792027 E–3	.41831171 E–3	.41823607 E–3	.41827830 E–3	.41819745 E–3

CPU (min)	.95	1.07	1.15	1.18	1.23	2.56

	N = 6	N = 10	N = 15	N = 24	N = 40	N = 80

	.62963057	.62963045	.62963057	.62963057	.62963045	.62963074
	.67214449 E–2	.67214430 E–2	.67214449 E–2	.67214458 E–2	.67214528 E–2	.67214477 E–2
	.19236634 E–5	.19234935 E–5	.19234885 E–5	.19235304 E–5	.19235479 E–5	.19235222 E–5
	.98847111 E–10	.98383843 E–10	.98160910 E–10	.98116265 E–10	.98574038 E–10	.98131871 E–10
	.95222491 E–14	.20774031 E–14	.45375614 E–14	.26301794 E–14	.22449425 E–14	.20574468 E–14

CPU (min)	.95	.97	.98	1.07	1.32	2.85

	N = 6	N = 10	N = 15	N = 24	N = 40	N = 80

	1.0000442	.99999964	1.0000000	1.0000001	1.0000001	.99999964
	.99258143	.99947798	.99947894	.99947774	.99947810	.99947798
	.90362161	.92600876	.92600965	.92601001	.92601001	.92600942
	.25408736	.24212666	.24212578	.24214584	.24212569	.24214566
	.73272777 E–2	.45387885 E–2	.45381812 E–2	.45381789 E–2	.45381733 E–2	.45381780 E–2
	.72985475 E–8	.20744079 E–4	.20703239 E–4	.20703352 E–4	.20703672 E–4	.20703957 E–4
	-----------	.40768075 E–7	.41697643 E–7	.41729450 E–7	.41720561 E–7	.41739558 E–7
	-----------	.41771950 E–10	.43969764 E–10	.43214661 E–10	.44285253 E–10	.43926265 E–10

CPU (min)	.92	.93	.97	1.03	1.28	2.95

CIRCULAR PROLATE FUNCTIONS
ℓ = 0		C = 10.000
NPOINTS = 10
1.0	0.21E–08	−.33E–05	−.12E–06	−.91E–07
0.21E–08	1.0	−.17E–06	−.32E–06	−.75E–07
−.33E–05	−.17E–06	1.0	0.87E–07	−.21E–06
−.12E–06	−.32E–06	0.87E–07	1.0	0.19E–06
−.91E–07	−.75E–07	−.21E–06	0.19E–06	1.0

Abstract

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Tables (6)

Equations (3)

Applied Optics