Design of Plane Mirror Systems*

David A. Berkowitz

doi:10.1364/JOSA.55.001464

Journal of the Optical Society of America
Vol. 55,
Issue 11,
pp. 1464-1467
(1965)
•https://doi.org/10.1364/JOSA.55.001464

Design of Plane Mirror Systems

David A. Berkowitz

Not Accessible

Your library or personal account may give you access

Get PDF
Email
Share
Get Citation
Copy Citation Text
David A. Berkowitz, "Design of Plane Mirror Systems*," J. Opt. Soc. Am. 55, 1464-1467 (1965)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Abstract

It is shown that the rotation of an optical image by a system of plane mirrors relates directly to the angles between the successive planes into which the optical axis reflects. This permits simpler visualization of mirror systems and rapid design of such systems for particular image rotations. Tables of the rotation are presented in terms of angles easily measured and specified in the laboratory.

Full Article | PDF Article

More Like This

Small Image Shifts in Plane Mirror Systems

Sten Walles
Appl. Opt. 4(6) 737-742 (1965)

Analysis and Design of Plane-Mirror Systems

R. J. Pegis and M. M. Rao
Appl. Opt. 2(12) 1271-1274 (1963)

Scanning, Coherent Light Beams—Plane-Wave Expansions and Interactions with Dispersive Systems*

Casper W. Barnes
J. Opt. Soc. Am. 55(4) 382-391 (1965)

Previous Article Next Article

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Figures (3)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Tables (3)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (7)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Name of vector	Reference plane	Angle between vector and the reference plane
object	x–z plane	0
object	P₀₁	90°−β
first image	P₀₁	90°+β
first image	P₁₂	90°+β−A
second image	P₁₂	90°−β+A
second image	P₂₃	90°−β+A−B
third image	P₂₃	90°+β−A+B

		ϕ=15		ϕ=30		ϕ=45		ϕ=60		ϕ=75		ϕ=90
α, (−α)	θ, (180-θ)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)
0	15	3.97	164.50	8.50	162.81	14.51	159.25	24.15	151.81	44.01	134.01	90.00	90.00
	30	7.63	149.13	16.10	146.31	26.57	140.77	40 89	130.89	61.81	114.15	90.00	90.00
	45	10.73	134.01	22.21	130.89	35.26	125.26	50.77	116.57	69.25	104.51	90.00	90.00
	60	13.06	119.15	26.57	116.57	40.89	112.21	56.31	106.10	72.81	98.50	90.00	90.00
	75	14.51	104.51	29.15	103.06	44.01	100.73	59.13	97.63	74.50	93.97	90.00	90.00
	90	15.00	90.00	30.00	90.00	45.00	90.00	60.00	90.00	75.00	90.00	90.00	90.00
	105	14.51	75.49	29.15	76.94	44.01	79.27	59.13	82.37	74.50	86.03	90.00	90.00
	120	13.06	60.85	26.57	63.44	40.89	67.79	56.31	73.90	72.81	81.50	90.00	90.00
	135	10.73	45.99	22.21	49.11	35.26	54.74	50.77	63.44	69.25	75.49	90.00	90.00
	150	7.63	30.87	16.10	33.69	26.57	39.23	40.89	49.11	61.81	65.85	90.00	90.00
	165	3.97	15.50	8.50	17.19	14.51	20.75	24.15	28.19	44.01	45.99	90.00	90.00
15	15	4.41	163.32	10.31	159.78	19.88	152.24	40.07	134.11	88.05	88.05	134.01	44.01
	30	8.41	146.91	19.04	140.94	33.98	130.23	56.29	111.90	86.03	86.03	114.15	61.81
	45	11.68	130.95	25.39	124.08	42.09	113.71	62.08	99.75	83.88	83.88	104.51	69.25
	60	13.99	115.53	29.29	109.05	45.99	100.73	63.69	91.19	81.50	81.50	98.50	72.81
	75	15.27	100.59	31.02	95.33	47.06	89.62	63.07	83.93	78.77	78.77	93.97	74.50
	90	15.50	86.03	30.87	82.37	45.99	79.27	60.85	76.94	75.49	75.49	90.00	75.00
	105	14.74	71.71	29.04	69.66	43.08	68.91	57.11	69.48	71.36	71.36	86.03	74.50
	120	13.06	57.47	25.66	56.79	38.33	57.91	51.57	60.90	65.85	65.85	81.50	72.81
	135	10.57	43.22	20.86	43.45	31.61	45.72	43.67	50.37	58.00	58.00	75.49	69.25
	150	7.44	28.90	14.77	29.49	22.79	31.95	32.60	36.94	45.99	45.99	65.85	61.81
	165	3.84	14.48	7.67	14.93	12.02	16.52	17.77	19.90	26.96	26.96	45.99	44.01
30	15	5.38	161.72	14.28	154.71	34.05	136.71	86.23	86.23	134.11	40.07	151.81	24.15
	30	10.13	143.96	25.11	132.70	49.11	112.21	82.37	82.37	111.90	56.29	130.89	40.89
	45	13.80	127.04	31.67	114.60	54.07	97.39	78.30	78.30	99.75	62.08	116.57	50.77
	60	16.19	111.06	34.72	99.46	54.58	86.39	73.90	73.90	91.19	63.69	106.10	56.31
	75	17.29	95.94	35.17	86.18	52.67	76.87	69.01	69.01	83.93	63.07	97.63	59.13
	90	17.19	81.50	33.69	73.90	49.11	67.79	63.44	63.44	76.94	60.85	90.00	60.00
	105	16.03	67.53	30.66	62.04	44.14	58.53	56.91	56.91	69.48	57.11	82.37	59.13
	120	13.97	53.86	26.33	50.19	37.81	48.66	49.11	49.11	60.90	51.57	73.90	56.31
	135	11.16	40.35	20.88	38.12	30.11	37.90	39.64	39.64	50.37	43.67	63.44	50.77
	150	7.77	26.89	14.50	25.71	21.05	26.10	28.19	28.19	36.94	32.60	49.11	40.89
	165	3.99	13.45	7.44	12.95	10.86	13.34	14.75	14.75	19.90	17.77	28.19	24.15
45	15	7.54	159.00	25.54	142.44	84.68	84.68	136.71	34.05	152.33	19.88	159.25	14.51
	30	13.86	139.12	39.23	116.57	79.27	79.27	112.21	49.11	130.23	33.98	140.77	26.57
	45	18.29	120.86	44.29	99.03	73.68	73.68	97.39	54.07	113.71	42.09	125.26	35.26
	60	20.75	104.51	44.83	85.58	67.79	67.79	86.39	54.58	100.73	45.99	112.21	40.89
	75	21.47	89.48	42.84	74.06	61.52	61.52	76.87	52.67	89.62	47.06	100.73	44.01
	90	20.75	75.49	39.23	63.44	54.74	54.74	67.79	49.11	79.27	45.99	90.00	45.00
	105	18.90	62.22	34.46	53.14	47.34	47.34	58.53	44.14	68.91	43.08	79.27	44.01
	120	16.14	49.42	28.75	42.87	39.23	39.23	48.66	37.81	57.91	38.33	67.79	40.89
	135	12.70	36.90	22.29	32.44	30.36	30.36	37.90	30.11	45.72	31.61	54.74	35.26
	150	8.74	24.54	15.23	21.80	20.75	20.75	26.10	21.05	31.95	22.79	39.23	26.57
	165	4.46	12.25	7.73	10.96	10.55	10.55	13.34	10.86	16.52	12.02	20.75	14.51
60	15	14.11	151.90	83.50	83.50	142.44	25.54	154.71	14.28	159.78	10.31	162.81	8.50
	30	24.13	127.83	76.94	76.94	116.57	39.23	132.70	25.11	140.94	19.04	146.31	16.10
	45	29.43	108.36	70.27	70.27	99.03	44.29	114.60	31.67	124.08	25.39	130.89	22.21
	60	31.15	92.29	63.44	63.44	85.58	44.83	99.46	34.72	109.05	29.29	116.57	26.57
	75	30.47	78.40	56.40	56.40	74.06	42.84	86.18	35.17	95.33	31.02	103.06	29.15
	90	28.19	65.85	49.11	49.11	63.44	39.23	73.90	33.69	82.37	30.87	90.00	30.00
	105	24.80	54.14	41.54	41.54	53.14	34.46	62.04	30.66	69.66	29.04	76.94	29.15
	120	20.64	42.92	33.69	33.69	42.87	28.75	50.19	26.33	56.79	25.66	63.44	26.57
	135	15.92	32.01	25.56	25.56	32.44	22.29	38.12	20.88	43.45	20.86	49.11	22.21
	150	10.82	21.26	17.19	17.19	21.80	15.23	25.71	14.50	29.49	14.77	33.69	16.10
	165	5.47	10.61	8.64	8.64	10.96	7.73	12.95	7.44	14.93	7.67	17.19	8.50
75	15	82.75	82.75	151.90	14.11	159.00	7.54	161.72	5.38	163.32	4.11	164.50	3.97
	30	75.49	75.49	127.83	24.13	139.12	13.86	143.96	10.13	146.91	8.41	149.13	7.63
	45	68.19	68.19	108.36	29.43	120.96	18.29	127.04	13.80	130.95	11.68	134.01	10.73
	60	60.85	60.85	92.29	31.15	104.51	20.75	111.06	16.19	115.53	13.99	119.14	13.06
	75	53.46	53.46	78.40	30.47	89.48	21.47	95.94	17.29	100.59	15.27	104.51	14.51
	90	45.99	45.99	65.85	28.19	75.49	20.75	81.50	17.19	86.03	15.50	90.00	15.00
	105	38.46	38.46	54.14	24.80	62.22	18.90	67.53	16.03	71.71	14.74	75.49	14.51
	120	30.87	30.87	42.92	20.64	49.42	16.14	53.86	13.97	57.47	13.06	60.85	13.06
	135	23.21	23.21	32.01	15.92	36.90	12.70	40.35	11.16	43.22	10.57	45.99	10.73
	150	15.50	15.50	21.26	10.82	24.54	8.74	26.89	7.77	28.90	7.44	30.87	7.63
	165	7.76	7.76	10.61	5.47	12.25	4.46	13.45	3.99	14.48	3.84	15.50	3.97

Name of vector	Reference plane	Angle between vector and the reference plane
object	x₁–z₁ plane	0
object	P₀₁	90°−β₁
first image	P₀₁	90°+β₁
first image	Q₁₂	90°+β₁−A
first image	P₁₂	90°+β₁−A−B
final image	P₁₂	90°−β₁+A+B
final image	y₁–z₁ plane	90°−β₁+A+B+90°−β₂

Name of vector	Reference plane	Angle between vector and the reference plane
object	x–z plane	0
object	P₀₁	90°−β
first image	P₀₁	90°+β
first image	P₁₂	90°+β−A
second image	P₁₂	90°−β+A
second image	P₂₃	90°−β+A−B
third image	P₂₃	90°+β−A+B

		ϕ=15		ϕ=30		ϕ=45		ϕ=60		ϕ=75		ϕ=90
α, (−α)	θ, (180-θ)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)	γ	A, (180-A)
0	15	3.97	164.50	8.50	162.81	14.51	159.25	24.15	151.81	44.01	134.01	90.00	90.00
	30	7.63	149.13	16.10	146.31	26.57	140.77	40 89	130.89	61.81	114.15	90.00	90.00
	45	10.73	134.01	22.21	130.89	35.26	125.26	50.77	116.57	69.25	104.51	90.00	90.00
	60	13.06	119.15	26.57	116.57	40.89	112.21	56.31	106.10	72.81	98.50	90.00	90.00
	75	14.51	104.51	29.15	103.06	44.01	100.73	59.13	97.63	74.50	93.97	90.00	90.00
	90	15.00	90.00	30.00	90.00	45.00	90.00	60.00	90.00	75.00	90.00	90.00	90.00
	105	14.51	75.49	29.15	76.94	44.01	79.27	59.13	82.37	74.50	86.03	90.00	90.00
	120	13.06	60.85	26.57	63.44	40.89	67.79	56.31	73.90	72.81	81.50	90.00	90.00
	135	10.73	45.99	22.21	49.11	35.26	54.74	50.77	63.44	69.25	75.49	90.00	90.00
	150	7.63	30.87	16.10	33.69	26.57	39.23	40.89	49.11	61.81	65.85	90.00	90.00
	165	3.97	15.50	8.50	17.19	14.51	20.75	24.15	28.19	44.01	45.99	90.00	90.00
15	15	4.41	163.32	10.31	159.78	19.88	152.24	40.07	134.11	88.05	88.05	134.01	44.01
	30	8.41	146.91	19.04	140.94	33.98	130.23	56.29	111.90	86.03	86.03	114.15	61.81
	45	11.68	130.95	25.39	124.08	42.09	113.71	62.08	99.75	83.88	83.88	104.51	69.25
	60	13.99	115.53	29.29	109.05	45.99	100.73	63.69	91.19	81.50	81.50	98.50	72.81
	75	15.27	100.59	31.02	95.33	47.06	89.62	63.07	83.93	78.77	78.77	93.97	74.50
	90	15.50	86.03	30.87	82.37	45.99	79.27	60.85	76.94	75.49	75.49	90.00	75.00
	105	14.74	71.71	29.04	69.66	43.08	68.91	57.11	69.48	71.36	71.36	86.03	74.50
	120	13.06	57.47	25.66	56.79	38.33	57.91	51.57	60.90	65.85	65.85	81.50	72.81
	135	10.57	43.22	20.86	43.45	31.61	45.72	43.67	50.37	58.00	58.00	75.49	69.25
	150	7.44	28.90	14.77	29.49	22.79	31.95	32.60	36.94	45.99	45.99	65.85	61.81
	165	3.84	14.48	7.67	14.93	12.02	16.52	17.77	19.90	26.96	26.96	45.99	44.01
30	15	5.38	161.72	14.28	154.71	34.05	136.71	86.23	86.23	134.11	40.07	151.81	24.15
	30	10.13	143.96	25.11	132.70	49.11	112.21	82.37	82.37	111.90	56.29	130.89	40.89
	45	13.80	127.04	31.67	114.60	54.07	97.39	78.30	78.30	99.75	62.08	116.57	50.77
	60	16.19	111.06	34.72	99.46	54.58	86.39	73.90	73.90	91.19	63.69	106.10	56.31
	75	17.29	95.94	35.17	86.18	52.67	76.87	69.01	69.01	83.93	63.07	97.63	59.13
	90	17.19	81.50	33.69	73.90	49.11	67.79	63.44	63.44	76.94	60.85	90.00	60.00
	105	16.03	67.53	30.66	62.04	44.14	58.53	56.91	56.91	69.48	57.11	82.37	59.13
	120	13.97	53.86	26.33	50.19	37.81	48.66	49.11	49.11	60.90	51.57	73.90	56.31
	135	11.16	40.35	20.88	38.12	30.11	37.90	39.64	39.64	50.37	43.67	63.44	50.77
	150	7.77	26.89	14.50	25.71	21.05	26.10	28.19	28.19	36.94	32.60	49.11	40.89
	165	3.99	13.45	7.44	12.95	10.86	13.34	14.75	14.75	19.90	17.77	28.19	24.15
45	15	7.54	159.00	25.54	142.44	84.68	84.68	136.71	34.05	152.33	19.88	159.25	14.51
	30	13.86	139.12	39.23	116.57	79.27	79.27	112.21	49.11	130.23	33.98	140.77	26.57
	45	18.29	120.86	44.29	99.03	73.68	73.68	97.39	54.07	113.71	42.09	125.26	35.26
	60	20.75	104.51	44.83	85.58	67.79	67.79	86.39	54.58	100.73	45.99	112.21	40.89
	75	21.47	89.48	42.84	74.06	61.52	61.52	76.87	52.67	89.62	47.06	100.73	44.01
	90	20.75	75.49	39.23	63.44	54.74	54.74	67.79	49.11	79.27	45.99	90.00	45.00
	105	18.90	62.22	34.46	53.14	47.34	47.34	58.53	44.14	68.91	43.08	79.27	44.01
	120	16.14	49.42	28.75	42.87	39.23	39.23	48.66	37.81	57.91	38.33	67.79	40.89
	135	12.70	36.90	22.29	32.44	30.36	30.36	37.90	30.11	45.72	31.61	54.74	35.26
	150	8.74	24.54	15.23	21.80	20.75	20.75	26.10	21.05	31.95	22.79	39.23	26.57
	165	4.46	12.25	7.73	10.96	10.55	10.55	13.34	10.86	16.52	12.02	20.75	14.51
60	15	14.11	151.90	83.50	83.50	142.44	25.54	154.71	14.28	159.78	10.31	162.81	8.50
	30	24.13	127.83	76.94	76.94	116.57	39.23	132.70	25.11	140.94	19.04	146.31	16.10
	45	29.43	108.36	70.27	70.27	99.03	44.29	114.60	31.67	124.08	25.39	130.89	22.21
	60	31.15	92.29	63.44	63.44	85.58	44.83	99.46	34.72	109.05	29.29	116.57	26.57
	75	30.47	78.40	56.40	56.40	74.06	42.84	86.18	35.17	95.33	31.02	103.06	29.15
	90	28.19	65.85	49.11	49.11	63.44	39.23	73.90	33.69	82.37	30.87	90.00	30.00
	105	24.80	54.14	41.54	41.54	53.14	34.46	62.04	30.66	69.66	29.04	76.94	29.15
	120	20.64	42.92	33.69	33.69	42.87	28.75	50.19	26.33	56.79	25.66	63.44	26.57
	135	15.92	32.01	25.56	25.56	32.44	22.29	38.12	20.88	43.45	20.86	49.11	22.21
	150	10.82	21.26	17.19	17.19	21.80	15.23	25.71	14.50	29.49	14.77	33.69	16.10
	165	5.47	10.61	8.64	8.64	10.96	7.73	12.95	7.44	14.93	7.67	17.19	8.50
75	15	82.75	82.75	151.90	14.11	159.00	7.54	161.72	5.38	163.32	4.11	164.50	3.97
	30	75.49	75.49	127.83	24.13	139.12	13.86	143.96	10.13	146.91	8.41	149.13	7.63
	45	68.19	68.19	108.36	29.43	120.96	18.29	127.04	13.80	130.95	11.68	134.01	10.73
	60	60.85	60.85	92.29	31.15	104.51	20.75	111.06	16.19	115.53	13.99	119.14	13.06
	75	53.46	53.46	78.40	30.47	89.48	21.47	95.94	17.29	100.59	15.27	104.51	14.51
	90	45.99	45.99	65.85	28.19	75.49	20.75	81.50	17.19	86.03	15.50	90.00	15.00
	105	38.46	38.46	54.14	24.80	62.22	18.90	67.53	16.03	71.71	14.74	75.49	14.51
	120	30.87	30.87	42.92	20.64	49.42	16.14	53.86	13.97	57.47	13.06	60.85	13.06
	135	23.21	23.21	32.01	15.92	36.90	12.70	40.35	11.16	43.22	10.57	45.99	10.73
	150	15.50	15.50	21.26	10.82	24.54	8.74	26.89	7.77	28.90	7.44	30.87	7.63
	165	7.76	7.76	10.61	5.47	12.25	4.46	13.45	3.99	14.48	3.84	15.50	3.97

Abstract

Cited By

Figures (3)

Tables (3)

Equations (7)

Journal of the Optical Society of America