Compound prism design principles, II: triplet and Janssen prisms

Nathan Hagen; Tomasz S. Tkaczyk

doi:10.1364/AO.50.005012

Applied Optics
Vol. 50,
Issue 25,
pp. 5012-5022
(2011)
•https://doi.org/10.1364/AO.50.005012

Compound prism design principles, II: triplet and Janssen prisms

Nathan Hagen and Tomasz S. Tkaczyk

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Nathan Hagen and Tomasz S. Tkaczyk, "Compound prism design principles, II: triplet and Janssen prisms," Appl. Opt. 50, 5012-5022 (2011)

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Abstract

Continuing the work of the first paper in this series [Appl. Opt. 50, 4998–5011 (2011)APOPAI0003-6935], we extend our design methods to compound prisms composed of three independent elements. The increased degrees of freedom of these asymmetric prisms allow designers to achieve greatly improved dispersion linearity. They also, however, require a more careful tailoring of the merit function to achieve design targets, and so we present several new operands for manipulating the compound prisms’ design algorithm. We show that with asymmetric triplet prisms, one can linearize the angular dispersion such that the spectral sampling rate varies by no more than 4% across the entire visible spectral range. Doing this, however, requires large prisms and causes beam compression. By adding a beam compression penalty to the merit function, we show that one can compromise between dispersion linearity and beam compression in order to produce practical systems. For prisms that do not deviate the beam, we show that Janssen prisms provide a form that maintains the degrees of freedom of the triplet and that are capable of up to $32 °$ of dispersion across the visible spectral range. Finally, in order to showcase some of the design flexibility of three-element prisms, we also show how to design for higher-order spectral dispersion to create a two-dimensional spectrum.

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	Glass 1	Glass 2	Glass 3	$α_{1}$ (deg)	$α_{2}$ (deg)	$α_{3}$ (deg)	$\bar{δ}$ (deg)	Δ (deg)	NL ( $\times 10^{4}$ )	SSR	K ^b
$merit func = linearity$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 1 °$
1a	LASF41	KZFS4	LITHQ	46.11	$- 149.52$	100.18	0.000	1.000	0.095	1.04	0.04
2a	LAF34	KZFS2	CAF2	44.63	$- 146.85$	98.09	0.000	1.000	0.158	1.08	0.11
3a	LASF31A	KZFS4	LITHQ	42.28	$- 145.71$	107.20	0.000	1.000	0.164	1.09	0.06
4a	LASF31A	KZFS4	CAF2	43.79	$- 141.50$	97.84	$- 0.012$	0.988	0.232	1.13	0.12
5a	LASF31A	KZFS11	K7	44.11	$- 150.57$	90.21	0.000	1.000	0.242	1.12	0.07
$merit func = linearity$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 4 °$
1b	LAK34	F5	CAF2	$- 80.90$	33.52	68.39	0.005	3.992	0.281	1.13	0.20
2b	LAK33A	BASF2	CAF2	$- 79.70$	30.10	70.11	0.000	4.000	0.283	1.48	0.21
3b	LASF43	SF1	CAF2	$- 81.83$	40.17	63.90	$- 0.015$	3.992	0.303	1.37	0.24
4b	LASF43	SF1	PK52A	$- 84.18$	38.35	64.80	0.004	3.997	0.317	1.40	0.25
5b	LASF9	SF57	CAF2	$- 85.55$	41.09	63.86	0.003	4.000	0.412	1.36	0.27
$merit func = thinness$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 4 °$ ^c
1c	CAF2	SF56A	LITHQ	57.29	$- 61.66$	57.28	0.014	4.085	2.519	9.08	1.00
2c	CAF2	SF4	CAF2	57.29	$- 62.34$	57.35	$- 0.005$	3.907	2.271	8.26	1.00
3c	LITHQ	SF11	LITHQ	57.29	$- 63.22$	57.29	0.012	4.070	2.710	10.05	1.00
4c	LITHQ	SF56A	LITHQ	57.29	$- 63.25$	57.29	$- 0.003$	3.929	2.503	9.74	1.00
5c	LITHQ	SF14	CAF2	57.29	$- 63.40$	57.29	0.006	4.017	2.556	9.26	1.00

	Glass 1	Glass 2	Glass 3	$α_{1}$ (deg)	$α_{2}$ (deg)	$α_{3}$ (deg)	$\bar{δ}$ (deg)	Δ (deg)	NL ( $\times 10^{4}$ )	SSR	K
$merit func = M_{nl -K}$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 1 °$
1a	CAF2	LASF47	SF2	87.03	$- 103.46$	92.01	0.000	1.000	0.177	2.53	1.00
2a	CAF2	LASF44	F5	91.03	$- 101.29$	93.27	0.000	1.000	0.179	2.63	1.00
3a	CAF2	LASF46A	SF56A	84.61	$- 104.96$	93.17	0.000	1.000	0.181	2.60	1.00
4a	CAF2	LAF33	F2	89.08	$- 104.36$	93.11	0.000	1.000	0.192	2.81	1.00
5a	CAF2	LAF35	BAF4	86.18	$- 107.24$	92.22	0.000	1.000	0.192	2.82	1.00
$merit func = M_{nl -K}$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 4 °$
1b	CAF2	LAK33A	CAF2	94.94	$- 94.30$	94.94	0.000	3.984	1.762	5.74	1.00
2b	CAF2	LAK33A	PK52A	94.14	$- 99.17$	94.92	$- 0.004$	3.903	1.766	5.91	0.99
3b	CAF2	LASF31A	BAF4	92.84	$- 93.87$	92.91	0.000	3.982	1.785	6.40	1.00
4b	CAF2	LAK33A	FK51A	94.58	$- 98.43$	94.90	$- 0.001$	3.963	1.792	5.93	1.00
5b	CAF2	LAF34	CAF2	91.94	$- 90.43$	91.94	0.000	3.983	1.806	5.97	1.00
$merit func = M_{nl -K}$ , $λ range = 475 - 625 nm$ , $Δ^{*} = 4 °$
1c	CAF2	LASF40	CAF2	99.51	$- 89.98$	99.62	$- 0.001$	3.967	0.736	2.56	1.00
2c	CAF2	LASF46A	CAF2	92.60	$- 80.23$	92.60	$- 0.000$	3.995	0.768	2.63	1.00
3c	CAF2	LASF9	CAF2	94.28	$- 85.24$	94.28	0.000	3.994	0.773	2.65	1.00
4c	CAF2	LASF45	PK52A	96.57	$- 95.55$	96.61	$- 0.001$	3.938	0.777	2.69	0.99
5c	CAF2	LASF45	FK51A	96.92	$- 94.93$	96.80	$- 0.001$	3.966	0.780	2.69	1.00

	Glass 1	Glass 2	Glass 3	$α_{1}$ (deg)	$α_{2}$ (deg)	$α_{3}$ (deg)	$\bar{δ}$ (deg)	Δ (deg)	NL ( $\times 10^{4}$ )	SSR	K
$merit func. = linearity$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 1 °$
1a	CAF2	LASF40	LAK8	90.57	12.48	$- 129.12$	0.000	0.998	0.131	1.94	1.00
2a	CAF2	BASF2	LAK14	88.88	10.85	$- 116.08$	$- 0.001$	0.997	0.146	1.91	1.00
3a	CAF2	SF67	KZFS2	79.35	2.88	$- 122.58$	0.000	0.998	0.147	2.07	1.00
4a	CAF2	LASF40	LAF34	88.58	17.95	$- 129.78$	0.000	0.998	0.151	2.02	1.00
5a	PK52A	LAFN7	LAK33A	89.95	10.83	$- 122.57$	0.000	0.997	0.156	2.29	1.00
$merit func. = linearity$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 4 °$
1b	CAF2	SF6	LASF31A	101.78	13.05	$- 110.29$	0.001	3.998	0.193	3.38	1.00
2b	CAF2	LASF9	LASF31A	99.88	20.81	$- 125.26$	0.000	3.999	0.205	3.56	1.00
3b	CAF2	SF14	LASF40	99.57	17.87	$- 120.36$	0.002	3.996	0.223	3.76	1.00
4b	CAF2	SF66	LASF41	98.86	5.18	$- 101.17$	0.000	3.998	0.241	4.07	1.00
5b	CAF2	SF4	LASF31A	103.08	17.07	$- 114.70$	0.000	4.000	0.242	2.82	1.00
$merit func. = linearity$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 8 °$
1c	CAF2	SF8	LASF31A	103.57	14.59	$- 108.33$	0.005	7.958	2.615	4.55	1.00
2c	FK51A	LASF44	BAF4	102.02	$- 101.17$	128.30	0.003	7.971	2.882	4.90	1.00
3c	CAF2	LASF44	BAF4	102.53	$- 90.30$	105.53	0.003	7.971	2.918	4.84	1.00
4c	PK52A	LASF31A	BASF2	98.79	$- 89.80$	106.15	$- 0.010$	7.949	2.950	5.18	1.00
5c	PK51	LASF31A	BASF2	98.96	$- 93.75$	111.46	0.000	7.954	2.981	5.24	1.00

	Glass 1	Glass 2	Glass 3	$α_{1}$ (deg)	$α_{2}$ (deg)	$α_{3}$ (deg)	$\bar{δ}$ (deg)	Δ (deg)	$Δ_{1}$ (deg)	$Δ_{2} (\times 1000)$	$Δ_{3} (\times 1000)$
$merit func = M_{2}$ , $λ range = 400 - 700 nm$ , $δ^{*} = 45 °$
1a	LASF43	LASF31A	FK51A	84.86	$- 92.21$	84.33	44.997	0.099	0.003	$- 0.451$	$- 0.054$
2a	KZFS5	LASF31A	PK51	92.65	$- 82.46$	78.47	44.998	0.099	0.015	$- 0.426$	$- 0.386$
3a	BAF52	LASF31A	BAK4	95.73	$- 78.41$	72.93	44.997	0.099	0.020	$- 0.424$	$- 0.483$
4a	BAF52	LASF31A	PSK3	94.66	$- 72.26$	66.79	44.996	0.097	0.025	$- 0.408$	$- 0.565$
5a	BALF5	LASF31A	BAF52	98.82	$- 76.41$	71.45	44.998	0.098	0.021	$- 0.408$	$- 0.312$
$merit func = M_{2}$ , $λ range = 350 - 1000 nm$ , $δ^{*} = 45 °$
1b	K10	KZFS2	SRF2	73.26	$- 96.86$	91.58	43.981	0.962	$- 0.167$	17.246	2.290
2b	K10	BAF2	SK11	$- 44.31$	121.78	$- 19.04$	44.370	0.896	$- 0.194$	16.394	2.682
3b	F_SILICA	CAF2	BAK1	$- 44.94$	116.01	$- 13.27$	43.352	0.919	$- 0.219$	16.000	2.577
4b	SF6	PBF2	LAK21	$- 52.83$	96.02	$- 86.01$	44.759	0.967	$- 0.227$	15.644	4.190
5b	K10	KZFS2	CAF2	$- 72.47$	94.54	$- 90.26$	44.199	0.856	$- 0.074$	15.642	1.578

	Glass 1	Glass 2	Glass 3	$α_{1}$ (deg)	$α_{2}$ (deg)	$α_{3}$ (deg)	$\bar{δ}$ (deg)	Δ (deg)	NL ( $\times 10^{4}$ )	SSR	K ^b
$merit func = linearity$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 1 °$
1a	LASF41	KZFS4	LITHQ	46.11	$- 149.52$	100.18	0.000	1.000	0.095	1.04	0.04
2a	LAF34	KZFS2	CAF2	44.63	$- 146.85$	98.09	0.000	1.000	0.158	1.08	0.11
3a	LASF31A	KZFS4	LITHQ	42.28	$- 145.71$	107.20	0.000	1.000	0.164	1.09	0.06
4a	LASF31A	KZFS4	CAF2	43.79	$- 141.50$	97.84	$- 0.012$	0.988	0.232	1.13	0.12
5a	LASF31A	KZFS11	K7	44.11	$- 150.57$	90.21	0.000	1.000	0.242	1.12	0.07
$merit func = linearity$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 4 °$
1b	LAK34	F5	CAF2	$- 80.90$	33.52	68.39	0.005	3.992	0.281	1.13	0.20
2b	LAK33A	BASF2	CAF2	$- 79.70$	30.10	70.11	0.000	4.000	0.283	1.48	0.21
3b	LASF43	SF1	CAF2	$- 81.83$	40.17	63.90	$- 0.015$	3.992	0.303	1.37	0.24
4b	LASF43	SF1	PK52A	$- 84.18$	38.35	64.80	0.004	3.997	0.317	1.40	0.25
5b	LASF9	SF57	CAF2	$- 85.55$	41.09	63.86	0.003	4.000	0.412	1.36	0.27
$merit func = thinness$ , $λ range = 400 - 700 nm$ , $Δ^{*} = 4 °$ ^c
1c	CAF2	SF56A	LITHQ	57.29	$- 61.66$	57.28	0.014	4.085	2.519	9.08	1.00
2c	CAF2	SF4	CAF2	57.29	$- 62.34$	57.35	$- 0.005$	3.907	2.271	8.26	1.00
3c	LITHQ	SF11	LITHQ	57.29	$- 63.22$	57.29	0.012	4.070	2.710	10.05	1.00
4c	LITHQ	SF56A	LITHQ	57.29	$- 63.25$	57.29	$- 0.003$	3.929	2.503	9.74	1.00
5c	LITHQ	SF14	CAF2	57.29	$- 63.40$	57.29	0.006	4.017	2.556	9.26	1.00

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Tables (4)

Equations (21)

Applied Optics