Abstract

The vector approach introduced in an early paper for modeling mirror-scanning devices [Y. Li, Appl. Opt. 34, 6417 (1995)] provides the basis of a rigorous study of the scan field generated by a single-mirror beam steering system, in which a hinged movable mirror is able to turn about a fixed pivot point to steering a single laser beam. Because of fewer constraints on mirror angular motion, the system may behave like a true point source for both vector and raster scanning applications. After a summary of the expressions for scan fields generated under different conditions, some fundamental and advanced topics of the single-mirror system are addressed: (1) basic parameters of high-order conic-section scan patterns, (2) scanning spot kinematics, (3) effect of input offset and pixels distortions on two-dimensional images displayed on screens of different formats, (4) mapping and its inverse between the mirror vector space and the scan vector space, and (5) single-mirror beam steering system as a one-element reflective and continuous image zooming device.

© 2008 Optical Society of America

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References

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  1. Y. Li, "Laser beam scanning by rotary mirrors. II. conic-section scan patterns," Appl. Opt. 34, 6417-6430 (1995).
    [CrossRef] [PubMed]
  2. Y. Li and J. Katz, "Laser beam scanning by rotary mirrors. I. modeling mirror scanning devices," Appl. Opt. 34, 6403-6416 (1995).
    [CrossRef] [PubMed]
  3. Y. Li and J. Katz, "Asymmetric distribution of the scanned field of a rotating reflective polygon," Appl. Opt. 36, 342-352 (1997).
    [CrossRef] [PubMed]
  4. J. I. Montagu, "Galvanometric and resonant scanners," in Handbook of Optical and Laser Scanning, G. F. Marshall, ed. (Dekker, 2004), pp. 417-476.
    [CrossRef]
  5. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, "Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II," Appl. Opt. 40, 1021-1028 (2001).
    [CrossRef]
  6. J. Castro-Ramos, O. de I. Prieto, and G. Silva-Ortigoza, "Computation of the disk of least confusion for conic mirrors," Appl. Opt. 43, 6080-6089 (2004).
    [CrossRef] [PubMed]
  7. F. Willmore, D. Barr, and D. Voils, Analytic Geometry: a Vector Approach (Allyn and Bacon, 1971), Chap. V.
  8. T. Smith, "On systems of plane reflecting surfaces," Trans. Opt. Soc. (London) 30, 68-78 (1928).
    [CrossRef]
  9. W. Kaplan, Advanced Calculus, 5th ed. (Addison-Wesley, 2002), pp. 237-241.

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2001

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1995

1928

T. Smith, "On systems of plane reflecting surfaces," Trans. Opt. Soc. (London) 30, 68-78 (1928).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Optical schematic of single-mirror beam steering system, in which the hinged movable mirror is able to turn about a fixed point to steering a single laser beam. The fulcrum for the mirror is selected as the origin of the coordinates ( x , y , z ) system. The tip T of mirror vector n ^ depicts a spatial curve on the surface of the unit sphere when mirror in motion. A single beam with an arbitrary cross section incident to the system in the direction specified by the ray vector s ^ ( i ) = ( 0 , 1 , 0 ) , and beam axis is parallel to the y-axis with offset ( ξ 0 , η 0 ) . The tip Q of the scan vector R depicts the scan pattern on the plane of observation, which is assumed to be normal to the z-axis.

Fig. 2
Fig. 2

First case example for the high-order conic-section scan pattern analysis. (a) Optical schematic of the system. Mirror rotates around the axis in the yz-plane and in the direction specified by the angle β 0 to the y-axis. Mirror vector n ^ sweeps out a circular cone and depicts a circle of radius ρ 0 on the surface of the unit sphere. (b) Scan pattern on the plane of observation. (c) Half-vertex angle of the noncircular high-order cone swept out by the scan vector R.

Fig. 3
Fig. 3

Second case example for the high-order conic-section scan pattern. (a) Optical schematic of the system. Mirror rotates around the axis lying in the yz-plane and in the direction specified by the angle β 0 to the y-axis. Mirror vector n ^ sweeps out a rectangular cone with half-vertex angles Φ 1 × Φ 2 when the mirror in motion. (b) Scan patterns with double pincushion distortions. (c) SLB.

Fig. 4
Fig. 4

Dependence of mirror direction cosines on the location of scanning spot on the plane of observation.

Fig. 5
Fig. 5

Synthesizing a straight scan line along the X-axis of the observation plane. (a) Optical schematic of the system. Mirror rotates around the y-axis. Mirror vector sweeps out a circular cone with half-vertex angles Φ = 45 ° . (b) LIN when the scanning spot moves from | X 0 |   to   + | X 0 | .

Fig. 6
Fig. 6

Synthesizing a circular scan pattern of radius R 0 on the plane of observation. (a) Optical schematic of the system. Mirror rotates around the axis lying in the yz-plane and in the direction specified by the angle β 0 to the y-axis. (b) The tip of mirror vector depicts an ellipse on the surface of the unit sphere when viewed in the direction of the axis of β 0 = 45 ° . (c) Scanning spot kinematics: the instantaneous radian frequency Ω i of the scanning spot moving around the circle when the mirror rotates at a constant speed of radian frequency ω 0 around the axis of β 0 = 45 ° . The instantaneous radian frequency ω i of mirror rotating around the axis of β 0 = 45 ° when the scanning spot moves at a constant speed of radian frequency Ω 0 around a circle of different radii.

Fig. 7
Fig. 7

Synthesizing a raster pattern on the plane of observation. (a) Dimensions of the raster pattern. (b) Optical schematic of the system. The T-curve on the surface of the unit sphere bares resemblance to the raster pattern on the plane of observation.

Fig. 8
Fig. 8

Curves illustrating the effect of input offset on the deformation of scan patterns on the plane of observation.

Fig. 9
Fig. 9

Scan-line broadening. (a) Uniform broadening of the circular scan lines in case of circular beam input. (b) Nonuniform broadening of the circular scan line in case of elliptical beam input.

Fig. 10
Fig. 10

Image pixels distortion. (a) Square pixel distortions on 5:4 display screen. (b) Nonsquare pixel distortions on 16:9 HDTV display screen.

Fig. 11
Fig. 11

Diagram illustrating graphical determination of mirror motion by covering the desirable scan pattern EFGHIJK on a level map.

Tables (1)

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Table 1 Different Forms of Eqs. (2.12) for Scan Pattern Displayed on a Flat-Screen under Different Conditions

Equations (269)

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( x , y , z )
( i ^ , j ^ , k ^ )
( x , y , z )
n ^
( X , Y , Z )
( ξ , η )
s ^ ( i ) = ( 0 , 1 , 0 )
ξ = ξ ( φ )   and   η = η ( φ ) ,
( ξ , η )
( x , z )
r 0 = ( ξ , 0 , η ) .
s ^ ( i ) × ( r r 0 ) = 0 ,
r = ( x , y , z )
n ^ = ( a , b , c ) ,
a = cos α
b = cos β
c = cos γ
α = α ( t ) ,   β = β ( t ) ,   and   γ = γ ( t ) .
n ^
( x , y , z )
n ^ r = 0.
r M = ξ ( 1 , a / b , 0 ) + η ( 0 , c / b , 1 ) .
s ^ ( r )
s ^ ( r ) = s ^ ( i ) 2 [ s ^ ( i ) n ^ ] n ^ .
s ^ ( i ) = ( 0 , 1 , 0 )
s ^ ( r ) = ( u , v , w ) ,
u = 2 a b ,
v = 1 + 2 b 2 ,
w = 2 b c .
s ^ ( r ) × ( R r M ) = 0 ,
r M
r M
R = ( X , Y , P )
R = ( a c ( P η ) + ξ , 1 2 b 2 2 b c ( P η ) + a ξ + c η b , P ) ,
X = a c ( P η ) + ξ
Y = 1 2 b 2 2 b c ( P η ) + a ξ + c η b .
ξ = η = 0
β 0
n ^
ρ 0
n ^
Φ = arctan ( ρ 0 )
ρ 0
n ^
a = sin   Φ   sin   φ ,
b = cos   Φ   cos   β 0 sin   Φ   sin   β 0   cos   φ ,
c = cos   Φ   sin   β 0 + sin   Φ   cos   β 0   cos   φ .
U = arccos ( s ^ ( r ) k ^ ) = arccos ( 2 b c ) ,
Φ 1 × Φ 2
β 0
n ^
a = sin   φ 1 ,
b = cos ( β 0 ± Φ 1 ) cos   φ 1 ,
c = sin ( β 0 ± Φ 1 ) cos   φ 1 ,
φ 1
n ^
n ^
φ 1
± Φ 1
± Φ 2
φ 2
a = ± sin   Φ 2 ,
b = cos ( β 0 φ 2 ) cos   Φ 2 ,
c = sin ( β 0 φ 2 ) cos   Φ 2 ,
Φ 2 = 5 °
Φ 1 = Φ 2 / 2
SLB = 100 Δ H L ( % ) = 100 tan   Φ 2 4   cos ( β 0 + Φ 1 ) ( % ) ,
Δ H
a 2 + b 2 + c 2 = 1 ,
( a , b , c ) = ( u 2 ( 1 + v ) ,   1 + v 2 ,   w 2 ( 1 + v ) ) .
( u , v , w ) = 1 P 2 + X 2 + Y 2 ( X , Y , P )
( a , b , c ) = 1 2 ( P 2 + X 2 ) 1 Y P 2 + X 2 + Y 2 × ( X , Y + P 2 + X 2 + Y 2 , P ) .
a = cos   α
( 1 , + 1 )
b = cos   β
c = cos   γ
( 0 , + 1 )
π / 2
+ π / 2
+ π / 2
n ^
x > 0
y > 0
( R , θ )
( a , b , c ) = 1 2 ( P 2 + R 2 cos 2 θ ) 1 R   sin   θ P 2 + R 2 × ( R   cos   θ , R   sin   θ + P 2 + R 2 , P ) .
Y = 0
a = X 2 ( P 2 + X 2 ) ,
b = 1 2 ,
c = P 2 ( P 2 + X 2 ) ,
b = 1 / 2
n ^
n ^
β = arccos ( 1 / 2 ) = 45 °
1 / 2 = 0.707
cotan φ = a c = X P .
φ = ω t
v = d X d t = ω P sin 2 φ = ω P ( 1 + X 2 P 2 ) ,
X
ω > 0
+ | X 0 |   to   | X 0 |
v max = ω P ( 1 + X 0 2 P 2 ) ,
v min = ω P .
LIN = 100 v min ( v max + v min ) / 2 ( % ) = 100 1 1 + X 0 2 / 2 P 2 ( % ) .
b c = P 2 P 2 + R 0 2 .
( 1 , Φ , φ )
β 0 = 45 °
n ^
Φ = 1 2   arccos [ 1 1 + cos 2 φ ( 2 1 + ( R 0 / P ) 2 sin 2 φ ) ] .
β 0 = 45 °
ε = sin ( Φ φ = 90 ° ) / sin ( Φ φ = 0 ° ) = 2 .
( R 0 , Θ )
Θ = arctan ( Y / X ) .
β 0 = 45 °
Θ = arctan [ 2 + sin   φ   tan   φ   tan   Φ 2  tan   φ ( 1 cos   φ   tan   Φ ) ] ,
ω 0
φ = ω 0 t .
Ω i = d Θ d t = Θ φ d φ d t + Θ Φ Φ φ d φ d t = ω 0 ( Θ φ + Θ Φ Φ φ ) .
Ω i / ω 0
ω 0
Ω 0
ω i / Ω 0
ω i
Ω 0
ω i / Ω 0
Ω i / ω 0
ω i / Ω 0
R 0 / P
AspR = H / W ,
Y / H = m ( X / W ) + y k , ( k = 1 , 2 ,   N )
m = ( 1 ) k 1 / ( AspR × N )
y k = ( 2 k N 1 ) P / N
β 0 = 45 °
n ^
Φ = π 4 1 2   arccos [ 1 1 + ( m   sin   φ y k   cos   φ ) 2 ] .
N = 4
R 0 / P = 0.5
ξ 0 / P = η 0 / P = 0 , 0.01
R 0 / P = 0.5
ξ 0 / P = η 0 / P = 0.1
( ξ 0 , η 0 )
D b
R 0 / P = 0.5
1.12 D b
1.41 D b
n ^
s ^ ( r )
n ^
s ^ ( r )
( a , b , c )
( u , v , w )
s ^ ( r )
n ^
( u , v , w ) = F ( a , b , c )   or   F : ( a , b , c ) ( u , v , w ) .
D a b c
D a b c { a : ( 1 , 1 ) ,   b : ( 0 , 1 ) ,   c : ( 0 , 1 ) } .
R a b c
D a b c
R a b c
D u v w { u : ( 1 , 1 ) ,   v : ( 1 , 1 ) ,   w : ( 0 , 1 ) } .
R u v w
R u v w
R u v w
R a b c
R u v w
R a b c
Φ = c 1
φ = c 2
( a , b , c )
J = ( u , v , w ) ( a , b , c ) = | u a u b u c v a v b v c w a w b w c | = | 2 b 2 a 0 0 4 b 0 0 2 c 2 b | = 16 b 3 ,
u a = u / a
b > 0
J 0
R a b c
R u v w
R a b c
R u v w
F 1 : ( u , v , w ) ( a , b , c ) ,
J = ( a , b , c ) ( u , v , w ) = 1 ( u , v , w ) ( a , b , c ) = 1 16 b 3 .
b > 0
J 0
R u v w
R a b c
( a , b , c )
X κ X
Y κ Y .
( a , b , c )
( a κ , b κ , c κ )
( a κ , b κ , c κ ) = 1 2 ( κ 2 a 2 + c 2 ) 1 + κ ( 1 2 b 2 ) κ 2 + ( 1 κ 2 ) ( 2 b c ) 2
× ( κ a , κ ( 1 2 b 2 ) + κ 2 + ( 1 κ 2 ) ( 2 b c ) 2 2 b , c ) ,
( a κ , b κ , c κ ) ( a , b , c )
κ = 1
θ 1
θ 2
Y 1
Y 2
Φ = const
φ = const
tan   θ 1 = d Y 1 d X = Y b b φ + Y c c φ X a a φ + X c c φ
tan   θ 2 = d Y 2 d X = Y b b Φ + Y c c Φ X a a Φ + X c c Φ ,
a φ = a / φ
X a = X / a
Φ = c 1
φ = c 2
θ 1 θ 2 = ± π 2   or   1 + tan   θ 1   tan   θ 2 = 0 .
( X a a φ + X c c φ ) ( X a a Φ + X c c Φ ) + ( Y b b φ + Y c c φ ) ( Y b b Φ + Y c c Φ ) = 0.
sin   φ G ( Φ , φ ) = 0 ,
G ( Φ , φ ) = cos   Φ   cos   φ { ( b c ) 2 + 4 b 2 c [ c ( 1 2 b 2 ) + 2 b 3 ] } sin   Φ { b 2 c 2 4 b 2 c 2 ( 1 + 2 b 2 ) + 2 2 ( sin   Φ   cos   φ cos   Φ   cos   2 φ ) b 4 c } .
sin   φ = 0 ,   or   G ( Φ , φ ) = 0.
Φ = 15 °
φ = 0 °
( e ^ X , e ^ Y , e ^ Z )
( X , Y , Z )
e ^ X = ( 1 , 0 , 0 ) ,
e ^ Y = ( 0 , 1 , 0 ) ,
e ^ Z = ( 0 , 0 , 1 ) .
e ^ X = ( a X , b X , c X ) ,
e ^ Y = ( a Y , b Y , c Y ) ,
e ^ Z = ( a Z , b Z , c Z ) .
R = P Δ Z [ 2 a c , 1 + 2 b 2 , 2 b c ] + ξ ( a ξ , b ξ , c ξ ) + η ( a η , b η , c η ) ,
( a , b , c )
Δ Z = ( 2 a c ) a Z ( 1 2 b 2 ) b Z + ( 2 b c ) c Z ,
a ξ = 1 2 a ( b a Z a b Z ) ,
b ξ = a ( b a Z a b Z ) ( 1 2 b 2 ) b ,
c ξ = 2 c ( b a Z a b Z ) ,
a η = 2 a ( b c Z c b Z ) ,
b η = c ( b c Z c b Z ) ( 1 2 b 2 ) b ,
c η = 1 2 c ( b c Z c b Z ) .
X = R · e ^ X ,
Y = R · e ^ Y .
X = Δ X Δ Z P + ξ ( a ξ a X + b ξ b X + c ξ c X ) + η ( a η a X + b η b X + c η c X ) ,
Y = Δ Y Δ Z P + ξ ( a ξ a Y + b ξ b Y + c ξ c Y ) + η ( a η a Y + b η b Y + c η c Y ) ,
Δ X
Δ Y
ξ = 0
η = 0
X X S T = a c P
Y Y S T = 1 2 b 2 2 b c P
ξ = ξ 0
η = η 0
X = X S T a c η 0 + ξ 0
Y = Y S T + 1 2 b 2 2 b c η 0 a ξ 0 + c η 0 b
ξ = w ξ   cos   ϕ + ξ 0
η = w η   sin   ϕ + η 0
X = X S T a c ( w η   sin   φ + η 0 ) + ( w ξ   cos   φ + ξ 0 )
Y = Y S T + 1 2 b 2 2 b c ( w η   sin   φ + η 0 ) a ( w ξ   cos   φ + ξ 0 ) + c ( w η   sin   φ + η 0 ) b
   ξ ( w , φ ) = { w   cos   2 φ + ξ 0 when   0 < φ π / 2 w + ξ 0 when   π / 2 < φ < π w   cos   2 φ + ξ 0 when   π < φ 3 π / 2 w + ξ 0 when   3 π / 2 < φ 2 π    a n d    η ( h , φ ) = { h + η 0 when   0 < φ π / 2 h   cos   2 φ + η 0 when   π / 2 < φ < π h + η 0 when   π < φ 3 π / 2 h   cos   2 φ + η 0 when   3 π / 2 < φ 2 π
X = X S T a c η ( h , φ ) + ξ ( w , φ )
Y = Y S T + 1 2 b 2 2 b c η ( h , φ ) a ξ ( w , φ ) + c η ( h , φ ) b
( x , y , z )
n ^
s ^ ( i ) = ( 0 , 1 , 0 )
( ξ 0 , η 0 )
β 0
n ^
ρ 0
β 0
n ^
Φ 1 × Φ 2
Φ = 45 °
| X 0 |   to   + | X 0 |
R 0
β 0
β 0 = 45 °
Ω i
ω 0
β 0 = 45 °
ω i
β 0 = 45 °
Ω 0

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