Accurate calculation of diffraction-limited encircled and ensquared energy

Torben B. Andersen

doi:10.1364/AO.54.007525

Applied Optics
Vol. 54,
Issue 25,
pp. 7525-7533
(2015)
•https://doi.org/10.1364/AO.54.007525

Accurate calculation of diffraction-limited encircled and ensquared energy

Torben B. Andersen

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Torben B. Andersen, "Accurate calculation of diffraction-limited encircled and ensquared energy," Appl. Opt. 54, 7525-7533 (2015)

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Abstract

Mathematical properties of the encircled and ensquared energy functions for the diffraction-limited point-spread function (PSF) are presented. These include power series and a set of linear differential equations that facilitate the accurate calculation of these functions. Asymptotic expressions are derived that provide very accurate estimates for the relative amount of energy in the diffraction PSF that fall outside a square or rectangular large detector. Tables with accurate values of the encircled and ensquared energy functions are also presented.

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Equations (46)

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$p$	$E_{c}$	$E_{s}$	$p$	$E_{c}$	$E_{s}$	$p$	$E_{c}$	$E_{s}$	$p$	$E_{c}$	$E_{s}$
0.0	0.0000000000	0.0000000000	5.0	0.8611512954	0.8829572055	30	0.9784387565	0.9808569938	80	0.9919936098	0.9928300563
0.1	0.0024968772	0.0031777989	5.5	0.8833731065	0.8984103943	31	0.9796822570	0.9815750006	82	0.9922743344	0.9930110221
0.2	0.0099501386	0.0126478421	6.0	0.9007520491	0.9078882559	32	0.9802272106	0.9820670730	84	0.9924175357	0.9931800303
0.3	0.0222484508	0.0282219098	6.5	0.9086836501	0.9130170913	33	0.9804141025	0.9825874649	86	0.9925670436	0.9933299203
0.4	0.0392088270	0.0495923702	7.0	0.9099305026	0.9167699371	34	0.9813932161	0.9831924007	88	0.9928065755	0.9934899571
0.5	0.0605804153	0.0763410538	7.5	0.9107710495	0.9212972094	35	0.9819749698	0.9836175385	90	0.9929027332	0.9936320914
0.6	0.0860496724	0.1079512299	8.0	0.9154817851	0.9270515299	36	0.9820773304	0.9840385697	92	0.9930721270	0.9937662812
0.7	0.1152467960	0.1438222636	8.5	0.9236454921	0.9329946732	37	0.9828308624	0.9845414304	94	0.9932589975	0.9939066005
0.8	0.1477532616	0.1832864353	9.0	0.9316619661	0.9376785009	38	0.9834278956	0.9849249928	96	0.9933361332	0.9940279322
0.9	0.1831102815	0.2256273400	9.5	0.9363854242	0.9405423382	39	0.9834962787	0.9852653604	98	0.9935152249	0.9941498655
1.0	0.2208279825	0.2700992347	10.0	0.9376257201	0.9423472394	40	0.9840600713	0.9856914929	100	0.9936491605	0.9942717986
1.1	0.2603950850	0.3159466805	10.5	0.9377803073	0.9443562465	41	0.9846516940	0.9860357408	102	0.9937286072	0.9943778264
1.2	0.3012888547	0.3624238335	11.0	0.9394408391	0.9471478529	42	0.9847194101	0.9863206428	104	0.9939019689	0.9944891991
1.3	0.3429850947	0.4088127639	11.5	0.9432661489	0.9502633660	43	0.9851269204	0.9866813154	106	0.9939923915	0.9945947714
1.4	0.3849679479	0.4544402376	12.0	0.9477971211	0.9528783709	44	0.9856947573	0.9869916954	108	0.9940857600	0.9946899232
1.5	0.4267392876	0.4986924680	12.5	0.9510401849	0.9546385801	45	0.9857823780	0.9872383868	110	0.9942386960	0.9947912495
1.6	0.4678274887	0.5410274321	13.0	0.9522369597	0.9558547694	46	0.9860647621	0.9875408370	112	0.9943009687	0.9948829006
1.7	0.5077953889	0.5809844529	13.5	0.9523319099	0.9570797261	47	0.9865936226	0.9878257374	114	0.9944099228	0.9949700994
1.8	0.5462472752	0.6181908544	14.0	0.9529449339	0.9586224824	48	0.9867121515	0.9880400370	116	0.9945328294	0.9950613479
1.9	0.5828347564	0.6523656153	14.5	0.9549209387	0.9603925276	49	0.9868980924	0.9882983912	118	0.9945836261	0.9951417568
2.0	0.6172614151	0.6833200551	15.0	0.9577299977	0.9620642490	50	0.9873764275	0.9885560909	120	0.9947020339	0.9952226639
2.1	0.6492861625	0.7109556962	15.5	0.9601084172	0.9633617335	51	0.9875296867	0.9887490818	122	0.9947924712	0.9953040283
2.2	0.6787252573	0.7352595368	16.0	0.9612386646	0.9642709566	52	0.9876451124	0.9889691015	124	0.9948455906	0.9953759413
2.3	0.7054529809	0.7562970567	16.5	0.9614013973	0.9650415805	53	0.9880651668	0.9892024257	126	0.9949631300	0.9954512965
2.4	0.7294009983	0.7742033379	17.0	0.9616103985	0.9659657900	54	0.9882516944	0.9893797651	128	0.9950255591	0.9955234451
2.5	0.7505564635	0.7891727315	17.5	0.9626621594	0.9671171865	55	0.9883194821	0.9895671049	130	0.9950891604	0.9955891892
2.6	0.7689589607	0.8014475283	18.0	0.9644795455	0.9683005025	56	0.9886772112	0.9897795341	132	0.9951952572	0.9956590125
2.7	0.7846963989	0.8113060971	18.5	0.9662788204	0.9692624077	57	0.9888918298	0.9899424945	134	0.9952389464	0.9957229895
2.8	0.7978999994	0.8190509425	19.0	0.9673270149	0.9699396164	58	0.9889315590	0.9901057575	136	0.9953146763	0.9957841713
2.9	0.8087385363	0.8249971058	19.5	0.9675754565	0.9704947107	59	0.9892263597	0.9902960133	138	0.9954017452	0.9958481937
3.0	0.8174120034	0.8294612863	20.0	0.9676360950	0.9711349190	60	0.9894614997	0.9904494308	140	0.9954377066	0.9959053919
3.1	0.8241448885	0.8327520049	20.5	0.9681873497	0.9719241402	61	0.9894892932	0.9905917907	142	0.9955216385	0.9959629663
3.2	0.8291792423	0.8351610663	21.0	0.9693798238	0.9727595037	62	0.9897235910	0.9907624940	144	0.9955869283	0.9960210854
3.3	0.8327677256	0.8369565030	21.5	0.9707571714	0.9734913735	63	0.9899704123	0.9909073436	146	0.9956248471	0.9960730675
3.4	0.8351668123	0.8383771131	22.0	0.9717125870	0.9740523514	64	0.9899988884	0.9910329530	148	0.9957097111	0.9961274069
3.5	0.8366303163	0.8396286307	22.5	0.9720348525	0.9744976669	65	0.9901776208	0.9911862464	150	0.9957555084	0.9961798126
3.6	0.8374033927	0.8408815006	23.0	0.9720603117	0.9749491450	66	0.9904269500	0.9913222273	152	0.9958014795	0.9962279489
3.7	0.8377171465	0.8422701677	23.5	0.9723327467	0.9754942321	67	0.9904652946	0.9914361053	154	0.9958794049	0.9962789448
3.8	0.8377839585	0.8438937407	24.0	0.9731104307	0.9761149849	68	0.9905953271	0.9915718964	156	0.9959118053	0.9963261289
3.9	0.8377936161	0.8458178488	24.5	0.9741642962	0.9767066827	69	0.9908384212	0.9917010273	158	0.9959673600	0.9963714158
4.0	0.8379103073	0.8480774811	25.0	0.9750200214	0.9771762445	70	0.9908925777	0.9918046072	160	0.9960323352	0.9964187902
4.1	0.8382705142	0.8506805809	25.5	0.9753960739	0.9775305309	71	0.9909820899	0.9919256682	162	0.9960591275	0.9964615638
4.2	0.8389818102	0.8536121623	26.0	0.9754378395	0.9778662148	72	0.9912112285	0.9920472962	164	0.9961216284	0.9965046152
4.3	0.8401225458	0.8568387202	26.5	0.9755579630	0.9782726938	73	0.9912841952	0.9921432067	166	0.9961710642	0.9965481904
4.4	0.8417423774	0.8603127220	27.0	0.9760532264	0.9787499507	74	0.9913420700	0.9922512765	168	0.9961994318	0.9965875210
4.5	0.8438635767	0.8639769904	27.5	0.9768518598	0.9792203869	75	0.9915509805	0.9923649358	170	0.9962635392	0.9966285407
4.6	0.8464830359	0.8677688136	28.0	0.9776043607	0.9796102460	76	0.9916431976	0.9924556489	172	0.9962986437	0.9966683215
4.7	0.8495748688	0.8716236566	28.5	0.9780130491	0.9799137752	77	0.9916784443	0.9925514723	174	0.9963333341	0.9967050850
4.8	0.8530934967	0.8754783761	29.0	0.9780926596	0.9801876277	78	0.9918625679	0.9926581436	176	0.9963930027	0.9967439447
4.9	0.8569770980	0.8792738803	29.5	0.9781365948	0.9804955666	79	0.9919723713	0.9927438625	178	0.9964180307	0.9967801792
5.0	0.8611512954	0.8829572055	30.0	0.9784387565	0.9808569938	80	0.9919936098	0.9928300563	180	0.9964604359	0.9968150506

Abstract

Cited By

Figures (7)

Tables (1)

Equations (46)

Applied Optics