Laser beam scanning by rotary mirrors. II. Conic-section scan patterns

Yajun Li

doi:10.1364/AO.34.006417

Applied Optics
Vol. 34,
Issue 28,
pp. 6417-6430
(1995)
•https://doi.org/10.1364/AO.34.006417

Laser beam scanning by rotary mirrors. II. Conic-section scan patterns

Yajun Li

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Yajun Li, "Laser beam scanning by rotary mirrors. II. Conic-section scan patterns," Appl. Opt. 34, 6417-6430 (1995)

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Abstract

Part II of this study is an application of the general theory of Part I to the following scanners: the galvanometer-based scanner, the paddle scanner, and the regular polygon. The scan field produced by these scanners is (or approximates) a circular cone. Therefore the scan pattern on the plane of observation can be one of the following curves, circle, ellipse, parabola, or hyperbola, depending on the position and orientation of the plane. Special topics to be addressed are (1) the effect of input offset, (2) the locus of the instantaneous scan center and the waist of the scan field, (3) the scanning on curved surfaces, and (4) the generalization of the scan-field expression. In Part III, X–Y scanning will be studied.

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	Circular	Elliptic	Parabolic	Hyperbolic
Observation Plane
Center	(0, 0, −P)	(P sin ϕ, 0, − P cos ϕ)	(P cos ϕ, 0, −P sin ϕ)	(P, 0, 0)
ê _X =	(1, 0, 0)	(−cos ϕ, 0, −sin ϕ)	(−sin ϕ, 0, −cos ϕ)	(0, 0, −1)
ê _Y =	(0, 1, 0)	(0, 1, 0)	(0, 1, 0)	(0, 1, 0)
ê _Z =	(0, 0, −1)	(sin ϕ, 0, −cos ϕ)	(cos ϕ, 0, − sin ϕ)	(1, 0, 0)
Scan Vector
R/P =	(tan ϕ cos 2θ, tan ϕ sin 2θ, −1)	$\frac{1}{D_{3}} (sin ϕ cos 2 θ, sin ϕ sin 2 θ - cos ϕ)$	$\frac{1}{D_{4}} (sin ϕ cos 2 θ, sin ϕ sin 2 θ, - cos ϕ)$	(1, tan 2θ, −cot ϕ/cos 2θ)
		D ₃ = cos² ϕ + sin² ϕ cos 2θ	D ₄ = sin 2ϕ cos² θ
Scan Pattern
X/P =	(tan ϕ)cos 2θ	$\frac{1}{D_{3}} sin 2 ϕ {sin}^{2} θ$	cot 2θ + (tan² θ/sin 2ϕ)	cot ϕ/cos 2θ
Y/P =	(tan ϕ)sin 2θ	$\frac{1}{D_{3}} sin ϕ sin 2 θ$	tan θ/cos ϕ	tan 2θ
	X ² + Y ² = ( P tan ϕ)²	(2X − P tan 2ϕ)² + (2Y /tan ϕ)² = (P tan 2ϕ)²	X = (cot ϕ) Y ²/2P + cot 2ϕ	X ² − (Y cot ϕ)² = (P cot ϕ)²
SLB
SLB =	(tan θm)/2	(cos ϕ tan θ _m )/2	tan θ _m /(4 sin ϕ)	(cot ϕ tan θ _m )/2
Scan Linearity
LIN =	100%	See Fig. 3(a)	See Fig. 3(b)	See Fig. 3(c)

Number of Facets M	ψ (deg)	θ₁ (deg)	θ₂ (deg)	DF ₁/D _p	DF ₂/D _p	(DF ₁ + DF ₂)/D _p
4	0	−45.0	45.0	0.20711	0	0.20711
	45.0	−30.0	60.0	0.14943	0.04184	0.19127
	80.0	−9.1	80.9	0.04046	0.18088	0.22133
6	0	−30.0	30.0	0.07735	0	0.07735
	30.0	−25.7	34.3	0.10431	0.01769	0.12199
	60.0	−18.6	41.4	0.08463	0.08413	0.16876
10	0	−18.0	18.0	0.02573	0	0.02573
	18.0	−17.1	18.9	0.04519	0.00623	0.05142
	36.0	−16.0	20.0	0.05755	0.02589	0.08343
12	0	−15.0	15.0	0.01764	0	0.01764
	15.0	−14.5	15.5	0.03228	0.00432	0.03659
	30.0	−13.9	16.1	0.04345	0.01769	0.06114
15	0	−12.0	12.0	0.01117	0	0.01117
	12.0	−11.7	12.3	0.02112	0.00275	0.02388
	24.0	−11.4	12.6	0.02964	0.01118	0.04082
18	0	−10.0	10.0	0.00711	0	0.00771
	10.0	−9.8	10.2	0.01484	0.00191	0.01675
	20.0	−9.7	10.3	0.02128	0.00772	0.02899
20	0	−9.00	9.00	0.00623	0	0.00623
	9.0	−8.89	9.11	0.01208	0.00155	0.01363
	18.0	−8.77	9.23	0.01748	0.00623	0.02371
24	0	−7.50	7.50	0.00431	0	0.00431
	7.5	−7.44	7.56	0.00844	0.00107	0.00952
	15.0	−7.37	7.63	0.01235	0.00432	0.01667
25	0	−7.20	7.20	0.00397	0	0.00397
	7.2	−7.14	7.26	0.00779	0.00099	0.00878
	14.4	−7.08	7.32	0.01142	0.00397	0.01540
36	0	−5.00	5.00	0.00191	0	0.00191
	5.0	−4.98	5.02	0.00378	0.00048	0.00426
	10.0	−4.96	5.04	0.00561	0.00191	0.00752
48	0	−3.75	3.75	0.00107	0	0.00107
	3.75	−3.74	3.76	0.00213	0.00027	0.00240
	7.50	−3.73	3.77	0.00318	0.00107	0.00425
72	0	−2.50	2.50	0.00048	0	0.00048
	2.5	−2.50	2.50	0.00095	0.00012	0.00107
	5.0	−2.50	2.50	0.00142	0.00048	0.00190

	Circular	Elliptic	Parabolic	Hyperbolic
Observation Plane
Center	(0, 0, −P)	(P sin ϕ, 0, − P cos ϕ)	(P cos ϕ, 0, −P sin ϕ)	(P, 0, 0)
ê _X =	(1, 0, 0)	(−cos ϕ, 0, −sin ϕ)	(−sin ϕ, 0, −cos ϕ)	(0, 0, −1)
ê _Y =	(0, 1, 0)	(0, 1, 0)	(0, 1, 0)	(0, 1, 0)
ê _Z =	(0, 0, −1)	(sin ϕ, 0, −cos ϕ)	(cos ϕ, 0, − sin ϕ)	(1, 0, 0)
Scan Vector
R/P =	(tan ϕ cos 2θ, tan ϕ sin 2θ, −1)	$\frac{1}{D_{3}} (sin ϕ cos 2 θ, sin ϕ sin 2 θ - cos ϕ)$	$\frac{1}{D_{4}} (sin ϕ cos 2 θ, sin ϕ sin 2 θ, - cos ϕ)$	(1, tan 2θ, −cot ϕ/cos 2θ)
		D ₃ = cos² ϕ + sin² ϕ cos 2θ	D ₄ = sin 2ϕ cos² θ
Scan Pattern
X/P =	(tan ϕ)cos 2θ	$\frac{1}{D_{3}} sin 2 ϕ {sin}^{2} θ$	cot 2θ + (tan² θ/sin 2ϕ)	cot ϕ/cos 2θ
Y/P =	(tan ϕ)sin 2θ	$\frac{1}{D_{3}} sin ϕ sin 2 θ$	tan θ/cos ϕ	tan 2θ
	X ² + Y ² = ( P tan ϕ)²	(2X − P tan 2ϕ)² + (2Y /tan ϕ)² = (P tan 2ϕ)²	X = (cot ϕ) Y ²/2P + cot 2ϕ	X ² − (Y cot ϕ)² = (P cot ϕ)²
SLB
SLB =	(tan θm)/2	(cos ϕ tan θ _m )/2	tan θ _m /(4 sin ϕ)	(cot ϕ tan θ _m )/2
Scan Linearity
LIN =	100%	See Fig. 3(a)	See Fig. 3(b)	See Fig. 3(c)

Number of Facets M	ψ (deg)	θ₁ (deg)	θ₂ (deg)	DF ₁/D _p	DF ₂/D _p	(DF ₁ + DF ₂)/D _p
4	0	−45.0	45.0	0.20711	0	0.20711
	45.0	−30.0	60.0	0.14943	0.04184	0.19127
	80.0	−9.1	80.9	0.04046	0.18088	0.22133
6	0	−30.0	30.0	0.07735	0	0.07735
	30.0	−25.7	34.3	0.10431	0.01769	0.12199
	60.0	−18.6	41.4	0.08463	0.08413	0.16876
10	0	−18.0	18.0	0.02573	0	0.02573
	18.0	−17.1	18.9	0.04519	0.00623	0.05142
	36.0	−16.0	20.0	0.05755	0.02589	0.08343
12	0	−15.0	15.0	0.01764	0	0.01764
	15.0	−14.5	15.5	0.03228	0.00432	0.03659
	30.0	−13.9	16.1	0.04345	0.01769	0.06114
15	0	−12.0	12.0	0.01117	0	0.01117
	12.0	−11.7	12.3	0.02112	0.00275	0.02388
	24.0	−11.4	12.6	0.02964	0.01118	0.04082
18	0	−10.0	10.0	0.00711	0	0.00771
	10.0	−9.8	10.2	0.01484	0.00191	0.01675
	20.0	−9.7	10.3	0.02128	0.00772	0.02899
20	0	−9.00	9.00	0.00623	0	0.00623
	9.0	−8.89	9.11	0.01208	0.00155	0.01363
	18.0	−8.77	9.23	0.01748	0.00623	0.02371
24	0	−7.50	7.50	0.00431	0	0.00431
	7.5	−7.44	7.56	0.00844	0.00107	0.00952
	15.0	−7.37	7.63	0.01235	0.00432	0.01667
25	0	−7.20	7.20	0.00397	0	0.00397
	7.2	−7.14	7.26	0.00779	0.00099	0.00878
	14.4	−7.08	7.32	0.01142	0.00397	0.01540
36	0	−5.00	5.00	0.00191	0	0.00191
	5.0	−4.98	5.02	0.00378	0.00048	0.00426
	10.0	−4.96	5.04	0.00561	0.00191	0.00752
48	0	−3.75	3.75	0.00107	0	0.00107
	3.75	−3.74	3.76	0.00213	0.00027	0.00240
	7.50	−3.73	3.77	0.00318	0.00107	0.00425
72	0	−2.50	2.50	0.00048	0	0.00048
	2.5	−2.50	2.50	0.00095	0.00012	0.00107
	5.0	−2.50	2.50	0.00142	0.00048	0.00190

Abstract

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Figures (23)

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Equations (46)

Applied Optics