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, XY scanning will be studied.

© 1995 Optical Society of America

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  1. Y. Li, J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 28, 6403–6416 (1995).
    [CrossRef]
  2. L. Beiser, Laser Scanning Notebook (SPIE Press, The International Society for Optical Engineering, Bellingham, Wash., 1992).
  3. L. Beiser, R. B. Johnson, “Scanners,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 9.
  4. J. I. Montagu, “Galvanometric and resonant low-inertia scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 525–613.
  5. R. J. Sherman, “Beam-deflecting systems and their uses,” in The Photonics Design and Applications Handbook, P. L. Jacobs, ed. (Laurin, Pittsfield, Mass., 1990), Book 3, pp. H173–H175.
  6. L. Beiser, “Laser scanning systems,” in Laser Applications, M. Ross, ed. (Academic, New York, 1974), Vol. 2, pp. 53–159.
  7. G. Marshall, “Scanning devices and systems,” in Applied Optics and Optical Engineering, R. Kingslake, B. J. Thompson, eds. (Academic, New York, 1980), Vol. 6, pp. 203–262.
  8. S. Reich, “The use of electro-mechanical mirror scanning devices,” in Laser Scanning Components and Techniques: Design Considerations/Trends, L. Beiser, G. F. Marshall, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 84, 47–56 (1976). Reprinted in Selected Papers on Laser Scanning and Recording, SPIE Milestone Series378, L. Beiser, ed. (1985), pp. 229–238.
  9. D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), Chap. 8, pp. 270–312.
  10. R. E. Hopkins, D. Stephenson, “Optical systems for laser scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 27–81.
  11. R. J. Sherman, “Polygonal scanners: applications, performance, and design,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 351–408.
  12. L. Beiser, “Design equations for a polygon laser scanner,” in Beam Deflection and Scanning Technologies, L. Beiser, G. F. Marshall, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1454, 60–66 (1991).
  13. K. O. G. Varughese, K. Siva, Rama Krishna, “Flattening the field of postobjective scanners by optimum choice and positioning of polygons,” Appl. Opt. 32, 1104–1108 (1993).
    [CrossRef] [PubMed]

1995 (1)

Y. Li, J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 28, 6403–6416 (1995).
[CrossRef]

1993 (1)

Beiser, L.

L. Beiser, Laser Scanning Notebook (SPIE Press, The International Society for Optical Engineering, Bellingham, Wash., 1992).

L. Beiser, R. B. Johnson, “Scanners,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 9.

L. Beiser, “Laser scanning systems,” in Laser Applications, M. Ross, ed. (Academic, New York, 1974), Vol. 2, pp. 53–159.

L. Beiser, “Design equations for a polygon laser scanner,” in Beam Deflection and Scanning Technologies, L. Beiser, G. F. Marshall, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1454, 60–66 (1991).

Hopkins, R. E.

R. E. Hopkins, D. Stephenson, “Optical systems for laser scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 27–81.

Johnson, R. B.

L. Beiser, R. B. Johnson, “Scanners,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 9.

Katz, J.

Y. Li, J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 28, 6403–6416 (1995).
[CrossRef]

Krishna, Rama

Li, Y.

Y. Li, J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 28, 6403–6416 (1995).
[CrossRef]

Marshall, G.

G. Marshall, “Scanning devices and systems,” in Applied Optics and Optical Engineering, R. Kingslake, B. J. Thompson, eds. (Academic, New York, 1980), Vol. 6, pp. 203–262.

Montagu, J. I.

J. I. Montagu, “Galvanometric and resonant low-inertia scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 525–613.

O’Shea, D. C.

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), Chap. 8, pp. 270–312.

Reich, S.

S. Reich, “The use of electro-mechanical mirror scanning devices,” in Laser Scanning Components and Techniques: Design Considerations/Trends, L. Beiser, G. F. Marshall, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 84, 47–56 (1976). Reprinted in Selected Papers on Laser Scanning and Recording, SPIE Milestone Series378, L. Beiser, ed. (1985), pp. 229–238.

Sherman, R. J.

R. J. Sherman, “Beam-deflecting systems and their uses,” in The Photonics Design and Applications Handbook, P. L. Jacobs, ed. (Laurin, Pittsfield, Mass., 1990), Book 3, pp. H173–H175.

R. J. Sherman, “Polygonal scanners: applications, performance, and design,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 351–408.

Siva, K.

Stephenson, D.

R. E. Hopkins, D. Stephenson, “Optical systems for laser scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 27–81.

Varughese, K. O. G.

Appl. Opt. (2)

Other (11)

L. Beiser, Laser Scanning Notebook (SPIE Press, The International Society for Optical Engineering, Bellingham, Wash., 1992).

L. Beiser, R. B. Johnson, “Scanners,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 9.

J. I. Montagu, “Galvanometric and resonant low-inertia scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 525–613.

R. J. Sherman, “Beam-deflecting systems and their uses,” in The Photonics Design and Applications Handbook, P. L. Jacobs, ed. (Laurin, Pittsfield, Mass., 1990), Book 3, pp. H173–H175.

L. Beiser, “Laser scanning systems,” in Laser Applications, M. Ross, ed. (Academic, New York, 1974), Vol. 2, pp. 53–159.

G. Marshall, “Scanning devices and systems,” in Applied Optics and Optical Engineering, R. Kingslake, B. J. Thompson, eds. (Academic, New York, 1980), Vol. 6, pp. 203–262.

S. Reich, “The use of electro-mechanical mirror scanning devices,” in Laser Scanning Components and Techniques: Design Considerations/Trends, L. Beiser, G. F. Marshall, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 84, 47–56 (1976). Reprinted in Selected Papers on Laser Scanning and Recording, SPIE Milestone Series378, L. Beiser, ed. (1985), pp. 229–238.

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), Chap. 8, pp. 270–312.

R. E. Hopkins, D. Stephenson, “Optical systems for laser scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 27–81.

R. J. Sherman, “Polygonal scanners: applications, performance, and design,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 351–408.

L. Beiser, “Design equations for a polygon laser scanner,” in Beam Deflection and Scanning Technologies, L. Beiser, G. F. Marshall, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1454, 60–66 (1991).

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Figures (23)

Fig. 1
Fig. 1

Galvanometer-based scanner showing (a) the scanner with the beam incident at an angle ϕ to the axis of rotation and (b) the components in the scanning system, with the incident beam in the xz plane.

Fig. 2
Fig. 2

Galvanometer-based scanner showing graphs of (a) beam deflection angle DA plotted versus the mirror rotation angle θ and (b) the ATF plotted versus θ.

Fig. 3
Fig. 3

LIN in a conic-section scan pattern produced by a galvanometer-based scanner showing (a) elliptical, (b) parabolic, and (c) hyperbolic scanning plotted against the maximum angle of rotation.

Fig. 4
Fig. 4

Optical arrangements for a galvanometer-based scanning system where Q O is the object point, Q1 is the incident point, Q I is the image point, and Q2 is the center of the scanning spot. The configurations show (a) the effect of moving Q O in the radial direction, (b) the effect of moving Q O in the tangential direction, (c) a straight line scan produced by an angle of incidence normal to the axis of rotation, and (d) a hyperbolic scan pattern on an observation plane parallel to the axis of rotation.

Fig. 5
Fig. 5

Scan pattern produced by a galvanometer-based scanner showing enlargement and rotation of the scanning spot across a circular path when ϕ = 45°. The cross section of the incident beam is (a) circular and (b) elliptical.

Fig. 6
Fig. 6

(a) Schematic diagram of a galvanometer-based scanner with input offset. (b) The one-sheet hyperboloid of revolution represents the general structure of the scan field produced by a galvanometer-based scanner.

Fig. 7
Fig. 7

Paddle scanner showing (a) the scanner notation in which the single stationary scan center is only in the sense of first approximation (see text) and (b) the optical arrangement.

Fig. 8
Fig. 8

Linearity of the scan pattern produced by a paddle scanner.

Fig. 9
Fig. 9

Schematic diagram of a paddle scanner showing (a) incidence and reflection angles and the waist of the scan field and (b) the locus of the scan center and a graphic analysis of the waist of the scan field.

Fig. 10
Fig. 10

(a) Regular polygon scanner, and (b) an optical schematic diagram of the polygon facet being illuminated in a scanning system.

Fig. 11
Fig. 11

Cross-sectional view of a polygon scanner showing three special beams at the start (the downdeflected beam), the neutral position, and the end of a scan (the updeflected beam).

Fig. 12
Fig. 12

Cross section of a polygon scanner showing (a) various angles when m p cos(Θ p /2) > 1 and Δθ < 0, (b) various angles when m p cos(Θ p /2) < 1 and Δθ > 0, and (c) various angles when m p cos(Θ p /2) = 1 and Δθ = 0.

Fig. 13
Fig. 13

X/P) versus θ for a polygon scanner when the incident plane is perpendicular to the rotation axis and (a) ψ = 0° and (b) ψ = 30°.

Fig. 14
Fig. 14

Linearity of the straight-line scan produced by a polygon scanner when the incident plane is perpendicular to the rotation axis and ψ = 0°.

Fig. 15
Fig. 15

Locus of the scan center (open circles) for a regular polygon of (a) M = 6 sides and (b) M = 10 sides when m p = 1.

Fig. 16
Fig. 16

Various quantities related to the depth of a scan field for a polygon scanner (see text).

Fig. 17
Fig. 17

Optical schematic showing a polygon scanner with an input offset and normally incident light.

Fig. 18
Fig. 18

Optical schematic showing a polygon scanner with an input offset and non-normally incident light.

Fig. 19
Fig. 19

Scan patterns produced by a polygon scanner with non-normally incident light for two different ratios of (D p /R 2). Note that the actual scan pattern includes only part of the solid curve (see text). It is very close to a circle of radius (R 2 + D p ).

Fig. 20
Fig. 20

Angle Θ of the scan field versus the rotation angle θ for different values of (D p /R 2) and numbers of mirror facets.

Fig. 21
Fig. 21

Optical schematic showing an inverted regular polygon scanner.

Fig. 22
Fig. 22

Optical schematic for a galvanometer-based scanner with non-normally incident light projecting a scan pattern on a cylindrical surface.

Fig. 23
Fig. 23

Scan patterns produced by a galvanometer-based scanner with non-normally incident light projecting onto cylindrical surfaces with different radii of curvature R cy .

Tables (2)

Tables Icon

Table 1 Relationships for Conic-Section Scan Patternsa

Tables Icon

Table 2 Displacement of the Scan Center along the Optical Axis of the Scan Lens when mp = 1a

Equations (46)

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s ^ ( r ) = ( sin ϕ cos 2 θ , sin ϕ sin 2 θ , - cos ϕ ) .
DA = arccos [ s ^ ( r ) · e ^ ref ] = arccos ( 1 - 2 sin 2 ϕ sin 2 θ ) .
ATF ~ lim θ 0 DA θ = 2 sin ϕ .
U = arccos [ s ^ ( r ) · ( - k ^ ) ] = arccos ( cos ϕ ) = ϕ .
z = - ( cot ϕ ) ( x 2 + y 2 ) 1 / 2 .
R 2 = P tan ϕ .
D X = w 2 [ sin 2 2 θ + ( cos 2 θ / cos ϕ ) 2 ] 1 / 2 ,
D Y = w 2 [ cos 2 2 θ + ( sin 2 θ / cos ϕ ) 2 ] 1 / 2 ,
r 0 = ( 0 , d 1 , d 2 ) .
z = - ( cot ϕ ) ( x 2 + y 2 - d 1 2 ) 1 / 2 - d 2 .
X 2 + Y 2 = R 4 2 ,
X = R 4 cos ( 2 θ + θ 0 ) ,             Y = R 4 sin ( 2 θ + θ 0 ) ,
r 0 = d ( 0 , 1 , 0 ) .
r 1 = d ( - tan θ , 1 , 0 ) ,
y = x tan 2 θ + d / cos 2 θ ,             z = 0.
LIN = 200 × cos 2 2 θ m 1 + cos 2 2 θ m + ( d / P ) sin 2 θ m ( % ) ,
y = x tan 2 ( θ + δ θ ) + d / cos 2 ( θ + δ θ ) .
x = - d sin 2 θ ,             y = d cos 2 θ .
θ = θ 0 + θ m cos τ ,
W S F = Q 2 Q 2 ¯ = ( d / cos 2 θ m ) - d = 2 d sin 2 θ m / cos 2 θ m .
- θ 1 < θ < θ 2 ,
θ 1 = ( Θ p / 2 ) - Δ θ ,             θ 2 = ( Θ p / 2 ) + Δ θ
Δ θ = ψ - arcsin [ ( 2 x 0 / D t ) sin ψ ] = ψ - arcsin [ m p sin ψ cos ( Θ p / 2 ) ] ,
n ^ = ( cos θ , sin θ , 0 ) .
r 1 = D p 2 [ m p + 1 - m p cos θ cos ( θ - ψ ) cos ψ , 1 - m p cos θ cos ( θ - ψ ) sin ψ , 0 ] .
s ^ ( r ) = [ cos ( 2 θ - ψ ) , sin ( 2 θ - ψ ) , 0 ] .
X = P tan ( 2 θ - ψ - ψ 0 ) - D p sin ( θ - ψ ) + ( m p / 2 ) sin ψ cos ( 2 θ - ψ - ψ 0 ) , Y = 0 ,
x = D p { cos θ + [ sin ( θ - ψ ) + m p sin ψ ] sin ( 2 θ - ψ ) } / 2 , y = D p { sin θ - [ sin ( θ - ψ ) + m p sin ψ ] cos ( 2 θ - ψ ) } / 2.
D F 1 = D p 4 cos ψ | 1 - 1 cos θ [ 1 + sin ( θ - 2 ψ ) tan ( θ / 2 ) + ( m p - 1 ) sin 2 ψ sin θ ] | ,
sin 3 θ ( m p - cos θ ) cos 2 θ - ( m p - 1 ) = tan ψ .
ψ 0 ,             x 0 ,             x 0 sin ψ - d .
X = P tan ( 2 θ - ψ 0 ) - ( D p sin θ - d ) / cos ( 2 θ - ψ 0 ) , Y = 0 ,
x = D p ( 1 - 0.5 cos 2 θ ) cos θ - d sin 2 θ , y = D p sin 3 θ + d cos 2 θ ,
n ^ = ( cos θ , sin θ , 0 ) .
r 1 = D p 2 cos θ ( 1 , 0 , cot ϕ ) .
s ^ ( r ) = ( sin ϕ cos 2 θ , sin ϕ sin 2 θ , - cos ϕ ) .
X = R 2 cos 2 θ + D p cos θ , Y = R 2 sin 2 θ + D p sin θ ,
X 2 + Y 2 = R 2 2 + D p ( D p + 2 R 2 cos θ )
X 2 + Y 2 = ( R 2 + D p ) 2 - 4 R 2 D p sin 2 ( θ / 2 ) .
X 2 + Y 2 ( R 2 + D p ) 2 .
X = R 2 cos 2 θ - D p cos θ , Y = R 2 sin 2 θ - D p sin θ .
X 2 + Y 2 ( R 2 - D p ) 2 .
R 3 = ( P + R s p ) sin ϕ × { cos ϕ - [ R s p 2 / ( P + R s p ) 2 - sin 2 ϕ ] 1 / 2 } .
sin ϕ < R s p / ( P + R s p ) .
X = R 2 cos 2 θ 1 + tan 2 ϕ sin 2 2 θ { ( 1 + R c y P ) - [ ( R c y P ) 2 - ( 1 + 2 R c y P ) tan 2 ϕ sin 2 2 θ ] 1 / 2 } , Y 2 + ( Z + R c y ) 2 = R c y 2 .
tan ϕ < R c y / ( P sin 2 θ m ) ( P + 2 R c y ) 1 / 2 ,

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