Spectroscope Slit Images in Partially Coherent Light

Klaus D. Mielenz

doi:10.1364/JOSA.57.000066

Journal of the Optical Society of America
Vol. 57,
Issue 1,
pp. 66-74
(1967)
•https://doi.org/10.1364/JOSA.57.000066

Spectroscope Slit Images in Partially Coherent Light

Klaus D. Mielenz

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Klaus D. Mielenz, "Spectroscope Slit Images in Partially Coherent Light," J. Opt. Soc. Am. 57, 66-74 (1967)

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Abstract

The apparatus function of prism or grating spectroscopes with finite slits is derived for the general case of partially coherent illumination of the slit. Numerical results are presented for the border cases of coherent and incoherent slit illumination, as well as for full and partial illumination of the spectroscope aperture.

The dependence of line intensity and line width on slit width, as well as the optimum slit width, are determined for each of these cases. It is shown that the performance of the spectroscope reaches an optimum when the aperture is at least fully illuminated.

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v₀/π	A₀^inc	(v₀′/π)^inc	A₀^coh	$A_{max}^{coh}$	(v₀′/π)^coh
0.00	0.00000	0.500	0.00000	0.00000	0.500
0.25	0.46737	0.530	0.23346	0.23346	0.518
0.50	0.77369	0.642	0.76153	0.76153	0.533
0.75	0.88883	0.861	1.22669	1.22669	0.589
1.00	0.90282	1.111	1.38999	1.38999	0.710
1.50	0.93109	1.607	1.04842	1.08348	1.366
2.00	0.94994	2.104	0.81509	1.26019	1.760
3.00	0.96641	3.101	1.13675	1.23121	2.804
4.00	0.97475	4.100	0.90238	1.21861	3.782

v′/π	Incoherent (q=∞)	Semicoherent (q=1)	Coherent (q=0)
0.00	1.0000	1.0000	1.0000
0.25	0.8277	0.8162	0.8155
0.50	0.4509	0.4194	0.4175
0.75	0.1381	0.1028	0.1005
1.00	0.0220	0.0019	0.0004
1.25	0.0302	0.0262	0.0257
1.50	0.0404	0.0399	0.0398
1.75	0.0202	0.0160	0.0157
2.00	0.0051	0.0004	0.0000

v₀/π	A₀	v₀′/π
0.00	0.00000	0.500
0.25	0.21863	0.509
0.50	0.60599	0.555
0.75	0.82386	0.703
1.00	0.86681	0.959
1.50	0.88773	1.454
2.00	0.92909	1.945
3.00	0.95176	2.940
4.00	0.96346	3.938

v₀/π	A₀	v₀′/π
0.6	0.81622	0.577
0.7	0.92686	0.625
0.8	0.99721	0.694

q	v₀^opt	v₀′(v₀^opt)	Q(v₀^opt)
0.00	0.83π	0.62π	0.00
0.75	0.70π	0.63π	3.49
1.00	0.65π	0.63π	3.80
∞	0.50π	0.63π	3.80

v₀/π	A/A₀ for q=∞

v′/π	0.25	0.5	0.75	1.0	1.5	2.0	3.0	4.0
0.00	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000
0.25	0.82771	0.87644	0.94311	0.99590	0.99698	0.99954	0.99987	0.99995
0.50	0.45088	0.58345	0.77450	0.94414	0.99494	0.99493	0.99859	0.99942
0.75	0.13814	0.28567	0.52378	0.78338	0.98846	0.98673	0.99646	0.99856
1.00	0.02203	0.10172	0.26989	0.52609	0.93055	0.98387	0.99579	0.99830
1.25	0.03024	0.03769	0.09914	0.26833	0.76932	0.97698	0.99462	0.99788
1.50	0.04058	0.03045	0.03546	0.10271	0.51897	0.91828	0.98743	0.99549
1.75	0.02016	0.02744	0.02744	0.03169	0.04232	0.75873	0.97722	0.99243
2.00	0.00561	0.01838	0.03141	0.03522	0.10592	0.51306	0.97402	0.99158
2.25	0.00986	0.01170	0.01987	0.03207	0.04580	0.26734	0.96695	0.99026
2.50		0.01064	0.01135	0.02207	0.03862	0.10724	0.90813	0.98255
2.75			0.01181	0.01494	0.03537	0.04766	0.75032	0.97192
3.00				0.01374	0.02488	0.04050	0.50865	0.96864
3.25					0.01731	0.03721	0.26697	0.96155
3.50					0.01623	0.02656	0.10901	0.90289
3.75						0.01880	0.04994	0.74612
4.00						0.01747	0.04280	0.50646
4.25							0.03948	0.26679
4.50							0.02866	0.10997
4.75							0.02072	0.05120
5.00							0.01935	0.04408
5.25								0.04075
5.50								0.02987
5.75								0.02187
6.00								0.02084

v₀/π	A/A₀ for q=0

v′/π	0.25	0.5	0.75	1.0	1.5	2.0	3.0	4.0
0.00	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000
0.25	0.81547	0.83072	0.85784	0.89879	1.01495	1.06735	0.96311	1.03155
0.50	0.41747	0.45632	0.52898	0.64695	1.03344	1.24458	0.87575	1.11033
0.75	0.10048	0.13646	0.21367	0.36184	0.99023	1.44847	0.79032	1.19326
1.00	0.00044	0.00751	0.04164	0.14660	0.82774	1.54608	0.75494	1.22922
1.25	0.02575	0.00975	0.00019	0.03595	0.55705	1.42437	0.79342	1.19166
1.50	0.03976	0.02504	0.00641	0.00250	0.27107	1.08139	0.89811	1.09995
1.75	0.01570	0.01282	0.00724	0.00070	0.07592	0.64362	1.02212	1.00735
2.00	0.00003	0.00037	0.00139	0.00229	0.00418	0.27678	1.08309	0.96800
2.25	0.00823	0.00398	0.00036	0.00117	0.00482	0.07097	1.00367	1.01199
2.50		0.00876	0.00257	0.00006	0.01251	0.00474	0.77296	1.13316
2.75			0.00213	0.00016	0.00639	0.00194	0.46864	1.27830
3.00				0.00040	0.00017	0.00591	0.20540	1.35043
3.25					0.00205	0.00302	0.05342	1.25469
3.50					0.00450	0.00015	0.00348	0.97272
3.75						0.00053	0.00168	0.59501
4.00						0.00124	0.00496	0.26323
4.25							0.00253	0.06894
4.50							0.00011	0.00443
4.75							0.00049	0.00234
5.00							0.00114	0.00679
5.25								0.00347
5.50								0.00015
5.75								0.00071
6.00								0.00164

v₀/π	A/A₀ for q=1

v′/π	0.25	0.5	0.75	1.0	1.5	2.0	3.0	4.0
0.00	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000
0.25	0.81618	0.84110	0.89880	0.97384	1.00435	0.99396	0.99726	0.99842
0.50	0.41942	0.48521	0.64460	0.86507	1.01105	0.98112	0.99267	0.99609
0.75	0.10276	0.17049	0.35383	0.64231	0.98693	0.97476	0.99218	0.99636
1.00	0.00186	0.02932	0.13655	0.35947	0.87284	0.97788	0.99412	0.99760
1.25	0.02619	0.01680	0.03421	0.13249	0.64256	0.95558	0.98880	0.99516
1.50	0.03992	0.02709	0.01432	0.02932	0.35850	0.84505	0.97367	0.98956
1.75	0.01603	0.01685	0.01908	0.01723	0.13420	0.62101	0.96288	0.98704
2.00	0.00037	0.00489	0.01634	0.02375	0.03222	0.34633	0.96414	0.98816
2.25	0.00838	0.00606	0.00821	0.01634	0.01941	0.13063	0.94301	0.98314
2.50		0.00949	0.00519	0.00689	0.02547	0.03262	0.83448	0.96773
2.75			0.00671	0.00687	0.01824	0.01999	0.61296	0.95588
3.00				0.00915	0.00868	0.02565	0.34195	0.95657
3.25					0.00823	0.01876	0.12992	0.93597
3.50					0.01034	0.00948	0.03371	0.82868
3.75						0.00882	0.02189	0.60879
4.00						0.01076	0.02615	0.33977
4.25							0.01981	0.12954
4.50							0.01063	0.03424
4.75							0.00977	0.02168
5.00							0.01157	0.02702
5.25								0.02039
5.50								0.01128
5.75								0.01033
6.00								0.01207

v₀/π	A/A₀ for q=∞

v′/π	0.25	0.5	0.75	1.0	1.5	2.0	3.0	4.0
0.00	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000
0.25	0.82771	0.87644	0.94311	0.99590	0.99698	0.99954	0.99987	0.99995
0.50	0.45088	0.58345	0.77450	0.94414	0.99494	0.99493	0.99859	0.99942
0.75	0.13814	0.28567	0.52378	0.78338	0.98846	0.98673	0.99646	0.99856
1.00	0.02203	0.10172	0.26989	0.52609	0.93055	0.98387	0.99579	0.99830
1.25	0.03024	0.03769	0.09914	0.26833	0.76932	0.97698	0.99462	0.99788
1.50	0.04058	0.03045	0.03546	0.10271	0.51897	0.91828	0.98743	0.99549
1.75	0.02016	0.02744	0.02744	0.03169	0.04232	0.75873	0.97722	0.99243
2.00	0.00561	0.01838	0.03141	0.03522	0.10592	0.51306	0.97402	0.99158
2.25	0.00986	0.01170	0.01987	0.03207	0.04580	0.26734	0.96695	0.99026
2.50		0.01064	0.01135	0.02207	0.03862	0.10724	0.90813	0.98255
2.75			0.01181	0.01494	0.03537	0.04766	0.75032	0.97192
3.00				0.01374	0.02488	0.04050	0.50865	0.96864
3.25					0.01731	0.03721	0.26697	0.96155
3.50					0.01623	0.02656	0.10901	0.90289
3.75						0.01880	0.04994	0.74612
4.00						0.01747	0.04280	0.50646
4.25							0.03948	0.26679
4.50							0.02866	0.10997
4.75							0.02072	0.05120
5.00							0.01935	0.04408
5.25								0.04075
5.50								0.02987
5.75								0.02187
6.00								0.02084

Abstract

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Tables (8)

Equations (62)

Journal of the Optical Society of America