Exact analytical solution of the generalized Luneburg lens problem

Jacek Sochacki

doi:10.1364/JOSA.73.000789

Journal of the Optical Society of America
Vol. 73,
Issue 6,
pp. 789-795
(1983)
•https://doi.org/10.1364/JOSA.73.000789

Exact analytical solution of the generalized Luneburg lens problem

Jacek Sochacki

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Jacek Sochacki, "Exact analytical solution of the generalized Luneburg lens problem," J. Opt. Soc. Am. 73, 789-795 (1983)

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Abstract

An analytical solution of the Luneburg integral equation is derived that provides a transcendental relationship for the refractive-index distribution. Using this expression, the index profiles for particular generalized Luneburg lenses with one and two finite image distances are calculated.

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ρ	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)

	f = 4		f = 6		f = 7		f = 8		f = 10

0.00	1.0831339	0.0000000	1.0545711	0.0000000	1.0465768	0.0000000	1.0406270	0.0000000	1.0323613	0.0000000
0.05	1.0830270	0.0461669	1.0545011	0.0474157	1.0465173	0.0477775	1.0405753	0.0480503	1.0323202	0.0484345
0.10	1.0827062	0.0923611	1.0542916	0.0948504	1.0463388	0.0955713	1.0404197	0.0961150	1.0321968	0.0968807
0.15	1.0821689	0.1386105	1.0539410	0.1423229	1.0460400	0.1433979	1.0401594	0.1442086	1.0319900	0.1453502
0.20	1.0814112	0.1849435	1.0534460	0.1898531	1.0456186	0.1912743	1.0397927	0.1923460	1.0316985	0.1938550
0.25	1.0804272	0.2313899	1.0528035	0.2374612	1.0450712	0.2392181	1.0393160	0.2405428	1.0313200	0.2424077
0.30	1.0792088	0.2779814	1.0520079	0.2851689	1.0443937	0.2872479	1.0387259	0.2888153	1.0308510	0.2910216
0.35	1.0777459	0.3247518	1.0510525	0.3329995	1.0435799	0.3353839	1.0380172	0.3371812	1.0302878	0.3397108
0.40	1.0760251	0.3717385	1.0499285	0.3809783	1.0426224	0.3836480	1.0371834	0.3856598	1.0296252	0.3884908
0.45	1.0740295	0.4189829	1.0486249	0.4291334	1.0415119	0.4320641	1.0362162	0.4342722	1.0288566	0.4373787
0.50	1.0717375	0.4665321	1.0471276	0.4774967	1.0402361	0.4806601	1.0351051	0.4830427	1.0279735	0.4863938
0.55	1.0691214	0.5144411	1.0454182	0.5261051	1.0387796	0.5294674	1.0338364	0.5319990	1.0269649	0.5355587
0.60	1.0661447	0.5627753	1.0434729	0.5750029	1.0371218	0.5785241	1.0323922	0.5811744	1.0258167	0.5848998
0.65	1.0627590	0.6116156	1.0412597	0.6242438	1.0352354	0.6278764	1.0307486	0.6306095	1.0245097	0.6344497
0.70	1.0588974	0.6610649	1.0387342	0.6738970	1.0330828	0.6775836	1.0288728	0.6803561	1.0230177	0.6842500
0.75	1.0544635	0.7112621	1.0358333	0.7240547	1.0306093	0.7277248	1.0267171	0.7304835	1.0213027	0.7343562
0.80	1.0493101	0.7624056	1.0324595	0.7748487	1.0277322	0.7784128	1.0242090	0.7810905	1.0193068	0.7848470
0.85	1.0431911	0.8148075	1.0284502	0.8264862	1.0243119	0.8298235	1.0212269	0.8323321	1.0169326	0.8358469
0.90	1.0356307	0.8690356	1.0234907	0.8793435	1.0200793	0.8822843	1.0175351	0.8844903	1.0139920	0.8875809
0.91	1.0338664	0.8801910	1.0223324	0.8901214	1.0190905	0.8929530	1.0166724	0.8950769	1.0133045	0.8980517
0.92	1.0319881	0.8914831	1.0210987	0.9009902	1.0180371	0.9036998	1.0157533	0.9057316	1.0125722	0.9085770
0.93	1.0299745	0.9029349	1.0197756	0.9119652	1.0169073	0.9145376	1.0147675	0.9164660	1.0117864	0.9191663
0.94	1.0277969	0.9145775	1.0183441	0.9230671	1.0156848	0.9254839	1.0137006	0.9272954	1.0109360	0.9298313
0.95	1.0254142	0.9264548	1.0167770	0.9343248	1.0143463	0.9365637	1.0125323	0.9382416	1.0100045	0.9405897
0.96	1.0227629	0.9386339	1.0150322	0.9457827	1.0128556	0.9478152	1.0112311	0.9493378	1.0089668	0.9514682
0.97	1.0197369	0.9512257	1.0130394	0.9575145	1.0111527	0.9593011	1.0097444	0.9606391	1.0077808	0.9625108
0.98	1.0161295	0.9644440	1.0106616	0.9696617	1.0091205	0.9711426	1.0079697	0.9722514	1.0063647	0.9738019
0.99	1.0114090	0.9788325	1.0075468	0.9825846	1.0064572	0.9836483	1.0056433	0.9844444	1.0045078	0.9855572
1.00	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000

ρ	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)

	f₁ = 2, f₂ = 3		f₁ = 3, f₂ = 4		f₁ = 4, f₂ = 10

0.00	1.3077550	0.0000000	1.2051903	0.0000000	1.1181855	0.0000000
0.05	1.3073351	0.0382457	1.2049142	0.0414967	1.1180306	0.0447214
0.10	1.3060747	0.0765652	1.2040856	0.0830505	1.1175658	0.0894801
0.15	1.3039645	0.1150338	1.2026984	0.1247195	1.1167864	0.1343138
0.20	1.3009902	0.1537290	1.2007433	0.1665634	1.1156903	0.1792612
0.25	1.2971316	0.1927329	1.1982064	0.2086451	1.1142661	0.2243629
0.30	1.2923603	0.2321334	1.1950690	0.2510315	1.1125034	0.2696620
0.35	1.2866398	0.2720264	1.1913070	0.2937949	1.1103884	0.3152049
0.40	1.2799232	0.3125187	1.1868890	0.3370154	1.1079025	0.3610426
0.45	1.2721513	0.3537315	1.1817753	0.3807830	1.1050223	0.4072315
0.50	1.2632480	0.3958050	1.1759153	0.4252006	1.1017177	0.4538367
0.55	1.2531158	0.4389059	1.1692437	0.4703894	1.0979501	0.5009335
0.60	1.2416281	0.4832364	1.1616749	0.5164956	1.0936690	0.5486120
0.65	1.2286153	0.5290508	1.1530956	0.5636999	1.0888069	0.5969837
0.70	1.2138450	0.5766798	1.1433482	0.6122368	1.0832707	0.6461911
0.75	1.1969830	0.6265752	1.1322070	0.6624230	1.0769264	0.6964263
0.80	1.1775209	0.6793934	1.1193268	0.7147152	1.0695689	0.7479648
0.85	1.1546102	0.7361791	1.1041304	0.7698365	1.0608550	0.8012403
0.90	1.1266158	0.7988526	1.0855023	0.8291092	1.0501212	0.8570439
0.91	1.1201353	0.8124017	1.0811791	0.8416736	1.0476214	0.8686344
0.92	1.1132576	0.8264034	1.0765862	0.8545529	1.0449624	0.8804144
0.93	1.1059111	0.8409355	1.0716745	0.8678007	1.0421141	0.8924166
0.94	1.0979968	0.8561044	1.0663760	0.8814901	1.0390368	0.9046840
0.95	1.0893738	0.8720605	1.0605945	0.8957240	1.0356729	0.9172780
0.96	1.0798253	0.8890326	1.0541812	0.9106593	1.0319338	0.9302922
0.97	1.0689883	0.9073999	1.0468872	0.9265563	1.0276712	0.9438815
0.98	1.0561564	0.9278928	1.0382281	0.9439158	1.0225968	0.9583444
0.99	1.0395116	0.9523702	1.0269570	0.9640130	1.0159682	0.9744398
1.00	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000

ρ	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)

	f = 4		f = 6		f = 7		f = 8		f = 10

0.00	1.0831339	0.0000000	1.0545711	0.0000000	1.0465768	0.0000000	1.0406270	0.0000000	1.0323613	0.0000000
0.05	1.0830270	0.0461669	1.0545011	0.0474157	1.0465173	0.0477775	1.0405753	0.0480503	1.0323202	0.0484345
0.10	1.0827062	0.0923611	1.0542916	0.0948504	1.0463388	0.0955713	1.0404197	0.0961150	1.0321968	0.0968807
0.15	1.0821689	0.1386105	1.0539410	0.1423229	1.0460400	0.1433979	1.0401594	0.1442086	1.0319900	0.1453502
0.20	1.0814112	0.1849435	1.0534460	0.1898531	1.0456186	0.1912743	1.0397927	0.1923460	1.0316985	0.1938550
0.25	1.0804272	0.2313899	1.0528035	0.2374612	1.0450712	0.2392181	1.0393160	0.2405428	1.0313200	0.2424077
0.30	1.0792088	0.2779814	1.0520079	0.2851689	1.0443937	0.2872479	1.0387259	0.2888153	1.0308510	0.2910216
0.35	1.0777459	0.3247518	1.0510525	0.3329995	1.0435799	0.3353839	1.0380172	0.3371812	1.0302878	0.3397108
0.40	1.0760251	0.3717385	1.0499285	0.3809783	1.0426224	0.3836480	1.0371834	0.3856598	1.0296252	0.3884908
0.45	1.0740295	0.4189829	1.0486249	0.4291334	1.0415119	0.4320641	1.0362162	0.4342722	1.0288566	0.4373787
0.50	1.0717375	0.4665321	1.0471276	0.4774967	1.0402361	0.4806601	1.0351051	0.4830427	1.0279735	0.4863938
0.55	1.0691214	0.5144411	1.0454182	0.5261051	1.0387796	0.5294674	1.0338364	0.5319990	1.0269649	0.5355587
0.60	1.0661447	0.5627753	1.0434729	0.5750029	1.0371218	0.5785241	1.0323922	0.5811744	1.0258167	0.5848998
0.65	1.0627590	0.6116156	1.0412597	0.6242438	1.0352354	0.6278764	1.0307486	0.6306095	1.0245097	0.6344497
0.70	1.0588974	0.6610649	1.0387342	0.6738970	1.0330828	0.6775836	1.0288728	0.6803561	1.0230177	0.6842500
0.75	1.0544635	0.7112621	1.0358333	0.7240547	1.0306093	0.7277248	1.0267171	0.7304835	1.0213027	0.7343562
0.80	1.0493101	0.7624056	1.0324595	0.7748487	1.0277322	0.7784128	1.0242090	0.7810905	1.0193068	0.7848470
0.85	1.0431911	0.8148075	1.0284502	0.8264862	1.0243119	0.8298235	1.0212269	0.8323321	1.0169326	0.8358469
0.90	1.0356307	0.8690356	1.0234907	0.8793435	1.0200793	0.8822843	1.0175351	0.8844903	1.0139920	0.8875809
0.91	1.0338664	0.8801910	1.0223324	0.8901214	1.0190905	0.8929530	1.0166724	0.8950769	1.0133045	0.8980517
0.92	1.0319881	0.8914831	1.0210987	0.9009902	1.0180371	0.9036998	1.0157533	0.9057316	1.0125722	0.9085770
0.93	1.0299745	0.9029349	1.0197756	0.9119652	1.0169073	0.9145376	1.0147675	0.9164660	1.0117864	0.9191663
0.94	1.0277969	0.9145775	1.0183441	0.9230671	1.0156848	0.9254839	1.0137006	0.9272954	1.0109360	0.9298313
0.95	1.0254142	0.9264548	1.0167770	0.9343248	1.0143463	0.9365637	1.0125323	0.9382416	1.0100045	0.9405897
0.96	1.0227629	0.9386339	1.0150322	0.9457827	1.0128556	0.9478152	1.0112311	0.9493378	1.0089668	0.9514682
0.97	1.0197369	0.9512257	1.0130394	0.9575145	1.0111527	0.9593011	1.0097444	0.9606391	1.0077808	0.9625108
0.98	1.0161295	0.9644440	1.0106616	0.9696617	1.0091205	0.9711426	1.0079697	0.9722514	1.0063647	0.9738019
0.99	1.0114090	0.9788325	1.0075468	0.9825846	1.0064572	0.9836483	1.0056433	0.9844444	1.0045078	0.9855572
1.00	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000

ρ	n(ρ)	r(ρ)	n(ρ)	r(ρ)	n(ρ)	r(ρ)

	f₁ = 2, f₂ = 3		f₁ = 3, f₂ = 4		f₁ = 4, f₂ = 10

0.00	1.3077550	0.0000000	1.2051903	0.0000000	1.1181855	0.0000000
0.05	1.3073351	0.0382457	1.2049142	0.0414967	1.1180306	0.0447214
0.10	1.3060747	0.0765652	1.2040856	0.0830505	1.1175658	0.0894801
0.15	1.3039645	0.1150338	1.2026984	0.1247195	1.1167864	0.1343138
0.20	1.3009902	0.1537290	1.2007433	0.1665634	1.1156903	0.1792612
0.25	1.2971316	0.1927329	1.1982064	0.2086451	1.1142661	0.2243629
0.30	1.2923603	0.2321334	1.1950690	0.2510315	1.1125034	0.2696620
0.35	1.2866398	0.2720264	1.1913070	0.2937949	1.1103884	0.3152049
0.40	1.2799232	0.3125187	1.1868890	0.3370154	1.1079025	0.3610426
0.45	1.2721513	0.3537315	1.1817753	0.3807830	1.1050223	0.4072315
0.50	1.2632480	0.3958050	1.1759153	0.4252006	1.1017177	0.4538367
0.55	1.2531158	0.4389059	1.1692437	0.4703894	1.0979501	0.5009335
0.60	1.2416281	0.4832364	1.1616749	0.5164956	1.0936690	0.5486120
0.65	1.2286153	0.5290508	1.1530956	0.5636999	1.0888069	0.5969837
0.70	1.2138450	0.5766798	1.1433482	0.6122368	1.0832707	0.6461911
0.75	1.1969830	0.6265752	1.1322070	0.6624230	1.0769264	0.6964263
0.80	1.1775209	0.6793934	1.1193268	0.7147152	1.0695689	0.7479648
0.85	1.1546102	0.7361791	1.1041304	0.7698365	1.0608550	0.8012403
0.90	1.1266158	0.7988526	1.0855023	0.8291092	1.0501212	0.8570439
0.91	1.1201353	0.8124017	1.0811791	0.8416736	1.0476214	0.8686344
0.92	1.1132576	0.8264034	1.0765862	0.8545529	1.0449624	0.8804144
0.93	1.1059111	0.8409355	1.0716745	0.8678007	1.0421141	0.8924166
0.94	1.0979968	0.8561044	1.0663760	0.8814901	1.0390368	0.9046840
0.95	1.0893738	0.8720605	1.0605945	0.8957240	1.0356729	0.9172780
0.96	1.0798253	0.8890326	1.0541812	0.9106593	1.0319338	0.9302922
0.97	1.0689883	0.9073999	1.0468872	0.9265563	1.0276712	0.9438815
0.98	1.0561564	0.9278928	1.0382281	0.9439158	1.0225968	0.9583444
0.99	1.0395116	0.9523702	1.0269570	0.9640130	1.0159682	0.9744398
1.00	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000	1.0000000

Abstract

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Figures (2)

Tables (2)

Equations (44)

Journal of the Optical Society of America