Abstract

Rectilinear propagation of light rays in homogeneous isotropic media makes it possible for optical generation of ruled surfaces as the ray is deflected by a rotatable mirror. Scan patterns on a plane or curved surface are merely curves on the ruled surface. Based on this understanding, structures of the scan fields produced by mirror- scanning devices of different configurations are investigated in terms of differential geometry. Expressions of the first and second fundamental coefficients and the first and second Gauss differential forms are given for an investigation of the intrinsic properties of the optically generated ruled surfaces. The Plücker ruled conoid is then generalized for mathematical modeling of the scan fields produced by single-mirror scanning devices of different configurations. Part II will be devoted to a study of multi-mirror scanning systems for optical generation of well-known ruled surfaces such as helicoids and hyperbolic paraboloids.

© 2011 Optical Society of America

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References

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  1. D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.
  2. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”
  3. V. I. Smirnov, A Course of Higher Mathemantics II: Advanced Calculus, trans. by D. E. Brown, trans. ed. I. N. Sneddon (Pergamon, 1964), Chap. V, “Foundations of Differential Geometry.”
  4. Y. Li, “Laser beam scanning by rotary mirrors. II. Conic-section scan patterns,” Appl. Opt. 34, 6417–6430 (1995).
    [CrossRef] [PubMed]
  5. Y. Li and J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror scanning devices,” Appl. Opt. 34, 6403–6416 (1995).
    [CrossRef] [PubMed]
  6. Y. Li and J. Katz, “Asymmetric distribution of the scanned field of a rotating reflective polygon,” Appl. Opt. 36, 342–352 (1997).
    [CrossRef] [PubMed]
  7. Y. Li, “Single-mirror beam steering system: analysis and synthesis of high-order conic-section scan patterns,” Appl. Opt. 47, 386–397 (2008).
    [CrossRef] [PubMed]
  8. Y. Li, “Beam deflection and scanning by two-mirror and two-axis systems of different architectures: a unified approach,” Appl. Opt. 47, 5976–5984 (2008).
    [CrossRef] [PubMed]
  9. R. J. Sherman, “Polygon scanners: applications, performance and design,” in Optical Scanning, G.F.Marshall, ed. (Dekker, 1991), pp. 351–406.
  10. W. L. Edge, The Theory of Ruled Surfaces, 1st ed. (Cambridge University, 1931), Sec. 3.2.2.

2008

1997

1995

Abbena, E.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

Cohn-Vossen, S.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.

Edge, W. L.

W. L. Edge, The Theory of Ruled Surfaces, 1st ed. (Cambridge University, 1931), Sec. 3.2.2.

Gray, A.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

Hilbert, D.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.

Katz, J.

Li, Y.

Salamon, S.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

Sherman, R. J.

R. J. Sherman, “Polygon scanners: applications, performance and design,” in Optical Scanning, G.F.Marshall, ed. (Dekker, 1991), pp. 351–406.

Smirnov, V. I.

V. I. Smirnov, A Course of Higher Mathemantics II: Advanced Calculus, trans. by D. E. Brown, trans. ed. I. N. Sneddon (Pergamon, 1964), Chap. V, “Foundations of Differential Geometry.”

Appl. Opt.

Other

R. J. Sherman, “Polygon scanners: applications, performance and design,” in Optical Scanning, G.F.Marshall, ed. (Dekker, 1991), pp. 351–406.

W. L. Edge, The Theory of Ruled Surfaces, 1st ed. (Cambridge University, 1931), Sec. 3.2.2.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

V. I. Smirnov, A Course of Higher Mathemantics II: Advanced Calculus, trans. by D. E. Brown, trans. ed. I. N. Sneddon (Pergamon, 1964), Chap. V, “Foundations of Differential Geometry.”

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Figures (6)

Fig. 1
Fig. 1

(a) Geometry of prismatic polygon x 0 is the radius of the circle inscribed by the polygon; (b) Polygon facet under illumination and illustration of notation.

Fig. 2
Fig. 2

Contour line maps of the normalized Gaussian curvature K and mean curvature H of the ruled surface optically generated by (a) a galvo scanner when input offset x 0 = 2 y 0 , (b) a cantilevered scanner when the mirror is inclined at angle γ = 45 ° and the incident ray inclined at angle ϕ = 10 ° , both measured from the axis of rotation.

Fig. 3
Fig. 3

(a) Schematic diagram illustrating the ruled surface generated by a rotating polygon. Base curve of the ruled surface is a line segment bounded by two distinct endpoints Q 1 and Q 1 on incident ray. (b) Generating the Plücker conoid by a straight line connected orthogonally to the z axis, rotating it about the axis, and moving it straight up-and-down between Q 1 and Q 1 .

Fig. 4
Fig. 4

Ruled surfaces generated by a galvo scanner. (a) The com ponents in the scanning system; (b) The one-sheet hyperboloid of revolution as a doubly ruled surface.

Fig. 5
Fig. 5

(a) Pyramidal polygon. (b) Cantilevered scanner as a special case of pyramidal polygon when the number of facets reduces to one. (c) Illustration of notation.

Fig. 6
Fig. 6

Diagram illustrating the ruled surface generated by a cantilevered scanner: when the incident ray is parallel to the spin axis but at a distance x 0 away from the axis and mirror inclination angle 0 ° < γ < 90 ° .

Equations (48)

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r · n ^ ( θ ) = p ,
s ^ ( i ) × ( r r 0 ) = 0 .
r 1 ( θ ) = r 0 + p n ^ ( θ ) · r 0 n ^ ( θ ) · s ^ ( i ) s ^ ( i ) .
s ^ ( r ) = s ^ ( r ) ( θ ) = [ m ( r ) ( θ ) , n ( r ) ( θ ) , l ( r ) ( θ ) ] = r r 1 ( θ ) | r r 1 ( θ ) | .
s ^ ( r ) ( θ ) = s ^ ( i ) 2 [ s ^ ( i ) · n ^ ( θ ) ] n ^ ( θ ) .
r ( θ , ρ ) = r 1 ( θ ) + ρ s ^ ( r ) ( θ ) ,
x x 1 ( θ ) m ( r ) ( θ ) = y y 1 ( θ ) n ( r ) ( θ ) = z z 1 ( θ ) l ( r ) ( θ ) = ρ .
[ z z 1 ( θ ) ] 2 = [ l ( r ) ( θ ) ] 2 1 [ l ( r ) ( θ ) ] 2 { [ x x 1 ( θ ) ] 2 + [ y y 1 ( θ ) ] 2 } .
I = ( d s ) 2 = E ( d ρ ) 2 + 2 F d ρ d θ + G ( d θ ) 2 ,
E = r ρ r ρ , F = r ρ r θ and G = r θ r θ .
E = 1 , F = [ ( n ^ · r 0 ) ( n ^ · s ^ ( i ) ) + ( n ^ · s ^ ( i ) ) ( p n ^ · r 0 ) ] [ 2 1 ( n ^ · s ^ ( i ) ) 2 ] ,
G = { 1 ( n ^ · s ^ ( i ) ) 2 [ n ^ · r 0 + n ^ · s ^ ( i ) n ^ · s ^ ( i ) ( p n ^ · r 0 ) ] 2 + 8 ρ ( n ^ · s ^ ( i ) ) [ n ^ · r 0 + n ^ · s ^ ( i ) n ^ · s ^ ( i ) ( p n ^ · r 0 ) ] + 4 ρ 2 [ ( s ^ ( i ) · n ^ ) 2 | n ^ | 2 + ( s ^ ( i ) · n ^ ) 2 ] } .
II = L ( d ρ ) 2 + 2 M d ρ d θ + N ( d θ ) 2 ,
L = r ρ ρ ( r ρ × r θ ) E G F 2 , M = r ρ θ ( r ρ × r θ ) E G F 2 , and N = r θ θ ( r ρ × r θ ) E G F 2 ,
L = 0 , M = 4 G F 2 [ ( n ^ · r 0 ) ( n ^ · s ^ ( i ) ) + ( n ^ · s ^ ( i ) ) ( p n ^ · r 0 ) ] [ s ^ ( i ) ( n ^ × n ^ ) ] ,
N = 4 ρ G F 2 ( { [ n ^ · ( r 0 + ρ s ^ ( i ) ) ] ( n ^ · s ^ ( i ) ) 2 [ n ^ · ( 2 r 0 + ρ s ^ ( i ) ) ] ( n ^ · s ^ ( i ) ) + [ ( n ^ · s ^ ( i ) ) 4 ( n ^ · s ^ ( i ) ) 2 ( n ^ · s ^ ( i ) ) ] ( p n ^ · r 0 ) } [ s ^ ( i ) ( n ^ × n ^ ) ] { ( n ^ · s ^ ( i ) ) [ n ^ · ( r 0 + ρ s ^ ( i ) ) ] + ( n ^ · s ^ ( i ) ) ( p n ^ · r 0 ) } [ s ^ ( i ) ( n ^ × n ^ ) ] ρ ( n ^ · s ^ ( i ) ) 2 { n ^ [ ( s ^ ( i ) 2 ( n ^ · s ^ ( i ) ) n ^ ) × n ^ ] } ) .
K = L N M 2 E G F 12 2 = M 2 G F 12 2 and H = E N 2 F M + G L 2 ( E G F 2 ) = N 2 F M 2 ( G F 2 ) ,
n ^ = ( cos θ , sin θ , 0 ) ,
r 0 = ( x 0 , y 0 , z 0 ) ,
s ^ ( i ) = ( sin ϕ , 0 , cos ϕ ) ,
r 1 = ( x 1 , y 1 , z 1 ) = ( x 0 y 0 sin θ cos θ , y 0 , x 0 ( 1 cos θ ) y 0 sin θ tan ϕ cos θ + z 0 ) ,
s ^ ( r ) = ( sin ϕ cos 2 θ , sin ϕ sin 2 θ , cos ϕ ) .
r ( θ , ρ ) = ( x 1 , y 1 , z 1 ) + ρ ( sin ϕ cos 2 θ , sin ϕ sin 2 θ , cos ϕ ) = [ x 1 , y 0 , ( x 1 x 0 cot ϕ + z 0 ) ] + ρ ( sin ϕ cos 2 θ , sin ϕ sin 2 θ , cos ϕ ) ,
{ [ ( z z 0 ) tan ϕ + x 0 ] 2 ( x 2 + y 2 ) + y 0 2 } 2 = 4 x 0 2 { [ ( z z 0 ) tan ϕ ( x x 0 ) ] 2 + ( y y 0 ) 2 } .
f 1 2 ( x , y , z ) = f 1 ( x , y , z ) f 2 ( x , y , z ) ,
( z z 0 ) 2 tan 2 ϕ ( x 2 + y 2 ) + y 0 2 = 0 .
x 2 ( y 0 tan ϕ ) 2 + y 2 ( y 0 tan ϕ ) 2 ( z z 0 ) 2 y 0 2 = 1 .
n ^ = ( sin γ cos θ , sin γ sin θ , cos γ ) ,
r 0 = ( x 0 , 0 , 0 ) .
r 1 = ( x 1 , y 1 , z 1 ) = x 0 ( 1 , 0 , tan γ cos θ ) 1 + tan γ tan ϕ cos θ ,
s ^ 1 ( r ) = ( m 1 ( r ) , n 1 ( r ) , l 1 ( r ) ) = ( a cos 2 θ + b cos θ d 1 , a sin 2 θ + b sin θ , c cos θ + d 2 ) ,
r ( θ , ρ ) = ( x 1 , y 1 , z 1 ) + ρ ( m 1 ( r ) , n 1 ( r ) , l 1 ( r ) ) = x 0 ( 1 , 0 , tan γ cos θ ) 1 + tan γ tan ϕ cos θ + ρ ( a cos 2 θ + b cos θ d 1 , a sin 2 θ + b sin θ , c cos θ + d 2 ) .
f 1 2 ( x , y , z ) = f 1 ( x , y , z ) f 2 ( x , y , z ) ,
f 1 ( x , y , z ) = { r 2 [ 2 ( x x 0 ) cos 2 γ cos 2 γ tan 2 ϕ + 2 z cos 2 γ tan 3 ϕ + ( z tan ϕ + x 0 ) cos 2 2 γ ] z 2 [ ( z tan ϕ + x 0 ) ( 1 + tan 2 ϕ ) 2 x 0 cos 2 γ tan 2 ϕ ] + x 0 2 [ 2 ( z sin 2 γ tan ϕ x cos 2 γ ) cos 2 γ + x 0 cos 2 γ z tan ϕ ] } ,
f 2 ( x , y , z ) = cos γ { r 2 [ ( x tan ϕ cos 2 γ z sin 2 γ ) tan ϕ + x 0 cos 2 γ ] tan ϕ z 2 ( x 0 tan ϕ ) x 0 2 ( x tan ϕ cos 2 γ + z sin 2 γ ) } ,
f 3 ( x , y , z ) = 4 cos γ { r 2 [ ( x tan ϕ + z ) cos 2 γ + 2 ( z tan ϕ x 0 cos 2 γ ) tan ϕ ] cos 2 γ + z 2 [ ( x tan ϕ z ) ( 1 + tan 2 ϕ ) 4 x 0 sin 2 γ tan ϕ ] + x 0 2 [ ( x tan ϕ z ) cos 2 2 γ + 2 z cos 2 γ ] 2 x z x 0 ( 1 + tan 2 ϕ ) cos 2 γ } ,
[ ( x x 0 ) ( x x 0 cos 2 γ ) + y 2 ] 2 = tan 2 2 γ [ ( x x 0 ) 2 + y 2 ] z 2 .
[ r 2 ( sin 2 ϕ + cos 2 γ ) z 2 ] 2 = 4 ( x sin ϕ cos 2 γ z cos ϕ sin 2 γ ) 2 r 2 .
[ r 2 tan 2 ϕ ( z tan ϕ + x 0 ) 2 z 2 + z x 0 ( 1 cot 2 ϕ ) tan ϕ ] 2 = [ ( r 2 tan 2 ϕ x 0 2 ) ( x tan ϕ z ) 2 z x 0 ( z tan ϕ + ε ) ] × [ ( x tan ϕ z ) ( 1 + cot 2 ϕ ) 2 x 0 cot ϕ ] .
r ( θ , ρ ) = 2 d ( 0 , 0 , sin q θ ) + ρ ( cos θ , sin θ , 0 ) ,
r ( θ , ρ ) = [ x 1 ( θ ) , y 1 ( θ ) , z 1 ( θ ) ] + ρ [ m ( θ ) , n ( θ ) , l ( θ ) ] ,
E = 1 , F = x 0 sin θ y 0 sin ϕ cos 2 θ ( sin 2 ϕ cos 2 θ cos 2 ϕ ) , G = ( x 0 sin θ y 0 sin ϕ cos 2 θ ρ sin ϕ sin 2 θ ) 2 + 4 ρ 2 sin 2 ϕ ( 1 sin 2 ϕ sin 2 2 θ ) } .
E = 1 , F = x 0 g ( θ ) ( sin 2 γ sin ϕ tan ϕ cos 2 θ + 2 sin 2 γ sin ϕ cos θ cos 2 γ sin ϕ tan ϕ + cos 2 γ cos ϕ ) , G = x 0 2 ( 1 + tan 2 ϕ ) g 2 ( θ ) 8 x 0 r g ( θ ) sin γ sin ϕ sin θ ( sin γ tan ϕ cos θ + cos γ ) + 4 r 2 sin 2 γ ( sin 2 γ sin 2 ϕ + cos 2 γ cos 2 γ sin 2 ϕ cos 2 θ + sin 2 γ sin ϕ cos ϕ cos θ ) } ,
L = 0 , M = 2 ( x 0 sin θ y 0 ) sin 2 ϕ E F G 2 , N = 2 r sin 2 ϕ E F G 2 [ x 0 + ( 3 x 0 sin θ 4 y 0 ) sin θ cos θ 2 r sin ϕ ] .
L = 0 , M = 4 x 0 sin γ tan γ sin θ ( 1 + tan γ tan ϕ cos θ ) 2 E F G 2 × { sin 2 γ cos γ sin 2 ϕ tan ϕ cos 3 θ + sin γ sin 2 ϕ ( cos 2 γ + cos 2 γ ) cos 2 θ + cos γ sin ϕ cos ϕ ( cos 2 γ sin 2 γ ) cos θ sin γ cos 2 γ ( cos 2 ϕ + sin 2 ϕ ) } , N = 1 E F G 2 { r x 0 tan γ ( 1 + tan γ tan ϕ cos θ ) 2 [ N 1 sin 2 θ + cos θ + tan γ tan ϕ ( 1 + sin 2 θ ) 1 + tan γ tan ϕ cos θ N 2 ] + r 2 N 3 } .
N 1 = [ 6 sin 2 γ sin 2 γ sin 2 ϕ tan ϕ cos 2 θ + 8 sin 2 γ sin 2 ϕ ( cos 2 γ + cos 2 γ ) cos θ + sin 2 γ sin 2 ϕ ( cos 2 γ sin 2 γ ) ] ,
N 2 = [ 2 ( sin 2 γ sin 2 γ sin 2 ϕ tan ϕ ) cos 3 θ + 4 sin 2 γ ( cos 2 γ + cos 2 γ ) sin 2 ϕ cos 2 θ + 2 sin 2 γ ( cos 2 γ sin 2 γ ) sin ϕ cos ϕ cos θ sin 2 2 γ cos 2 ϕ ] ,
N 3 = { sin 3 2 γ sin 3 ϕ cos 3 θ + 3 sin 2 2 γ sin 2 γ sin 2 ϕ sin ϕ cos 2 θ + 6 sin 2 γ sin 2 γ sin ϕ ( sin 2 ϕ + cos 2 γ cos 2 ϕ ) cos θ + 2 sin 2 γ cos ϕ [ ( 3 cos 2 γ 1 ) sin 2 ϕ + 2 cos 2 γ cos 2 γ cos 2 ϕ ] } .

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