Abstract

A rigorous vector analysis is performed to investigate the structure of the scan field produced by a rotating polygon, and it is shown that the scan field is asymmetric to the ray reflected by the polygon at a neutral scan position. Some fundamental aspects of the polygon scanning systems are addressed, such as the scan duty cycle, the locus of the scan center, the depth of the scan field, and the off-axis defocus in convergent beam scanning.

© 1997 Optical Society of America

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References

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  1. L. Beiser, Laser Scanning Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1992).
  2. L. Beiser, “Fundamental architecture of optical scanning systems,” Appl. Opt. 34, 7307–7317 (1995).
    [CrossRef] [PubMed]
  3. Y. Li, J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 34, 6403–6416 (1995).
    [CrossRef] [PubMed]
  4. Y. Li, “Laser beam scanning by rotary mirrors. II. Conic-section scan patterns,” Appl. Opt. 34, 6417–6430 (1995).
    [CrossRef] [PubMed]
  5. J. I. Montagu, “Galvanometric and resonant low-inertia scanners,” in Optical Scanning, G. F. Marshall, ed. (Dekker, New York, 1991), pp. 525–613.
  6. R. J. Sherman, “Beam-deflecting systems and their uses,” in The Photonics Design and Applications Handbook, 36th ed. P. L. Jacobs, ed. (Laurin, Pittsfield, Mass., 1990), Book 3, pp. H173–H175.
  7. L. Beiser, “Laser scanning systems,” in Laser Applications, M. Ross, ed. (Academic, New York, 1974), Vol. 2, pp. 53–159.
  8. S. Reich, “The use of electro-mechanical mirror scanning devices,” in Laser Scanning Components and Techniques: Design Considerations/Trends, L. Beiser, G. Marshall, eds., Proc. SPIE84, 47–56 (1976).
  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; R. E. Hopkins, M. J. Buzawa, “Optics for laser scanning,” Opt. Eng. 15, 90–94 (1976).
  11. R. J. Sherman, “Polygonalscanners: 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. SPIE1454, 60–66 (1991).
  13. J. E. Klein, “Geometrical relationships characterizing polygonal scan wheels,” in International LensDesign Conference, R. E. Fischer, ed., Proc. SPIE554, 469–477 (1985).
  14. 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]
  15. L. Rade, B. Westergren, BETA—Mathematics Handbook, 2nd ed. (CRC, Boca Raton, Fla., 1990), p. 137.

1995 (3)

1993 (1)

Beiser, L.

L. Beiser, “Fundamental architecture of optical scanning systems,” Appl. Opt. 34, 7307–7317 (1995).
[CrossRef] [PubMed]

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

L. Beiser, Laser Scanning Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1992).

L. Beiser, “Design equations for a polygon laser scanner,” in Beam Deflection and Scanning Technologies, L. Beiser, G. F. Marshall, eds., Proc. SPIE1454, 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; R. E. Hopkins, M. J. Buzawa, “Optics for laser scanning,” Opt. Eng. 15, 90–94 (1976).

Katz, J.

Klein, J. E.

J. E. Klein, “Geometrical relationships characterizing polygonal scan wheels,” in International LensDesign Conference, R. E. Fischer, ed., Proc. SPIE554, 469–477 (1985).

Li, Y.

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.

Rade, L.

L. Rade, B. Westergren, BETA—Mathematics Handbook, 2nd ed. (CRC, Boca Raton, Fla., 1990), p. 137.

Reich, S.

S. Reich, “The use of electro-mechanical mirror scanning devices,” in Laser Scanning Components and Techniques: Design Considerations/Trends, L. Beiser, G. Marshall, eds., Proc. SPIE84, 47–56 (1976).

Sherman, R. J.

R. J. Sherman, “Beam-deflecting systems and their uses,” in The Photonics Design and Applications Handbook, 36th ed. P. L. Jacobs, ed. (Laurin, Pittsfield, Mass., 1990), Book 3, pp. H173–H175.

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

Siva Rama Krishna, 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; R. E. Hopkins, M. J. Buzawa, “Optics for laser scanning,” Opt. Eng. 15, 90–94 (1976).

Varughese, K. O. G.

Westergren, B.

L. Rade, B. Westergren, BETA—Mathematics Handbook, 2nd ed. (CRC, Boca Raton, Fla., 1990), p. 137.

Appl. Opt. (4)

Other (11)

L. Rade, B. Westergren, BETA—Mathematics Handbook, 2nd ed. (CRC, Boca Raton, Fla., 1990), p. 137.

L. Beiser, Laser Scanning Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1992).

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, 36th ed. 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.

S. Reich, “The use of electro-mechanical mirror scanning devices,” in Laser Scanning Components and Techniques: Design Considerations/Trends, L. Beiser, G. Marshall, eds., Proc. SPIE84, 47–56 (1976).

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. E. Hopkins, M. J. Buzawa, “Optics for laser scanning,” Opt. Eng. 15, 90–94 (1976).

R. J. Sherman, “Polygonalscanners: 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. SPIE1454, 60–66 (1991).

J. E. Klein, “Geometrical relationships characterizing polygonal scan wheels,” in International LensDesign Conference, R. E. Fischer, ed., Proc. SPIE554, 469–477 (1985).

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Figures (15)

Fig. 1
Fig. 1

Schematic diagram of a regular prismatic polygonal scanner, showing the dimension of the polygon, the incident and reflected beams, fixed coordinate system Oxy, moving coordinate system Oxy′, and the unit vectors.

Fig. 2
Fig. 2

Cross-sectional view of a polygon scanner, showing the three special rays at the start, ℛ(-), the neutral position, ℛ0, and the end of a scan, ℛ(+), as well as the location of the observation plane and scan vector rS.

Fig. 3
Fig. 3

Cross-sectional view of a polygonal scanner, showing the locus of the point of reflection when the incident ray (a) intersects the inscribed circle of the polygon, and (b) passes through the annular zone between the circumscribed and the inscribed circles without intersecting the inner one.

Fig. 4
Fig. 4

Diagram for deriving an expression for asymmetrical angle θB.

Fig. 5
Fig. 5

Graph of θB versus ψ for a polygon scanner of arbitrary number of sides with the normalized offset (x0/RT) as a parameter sampled between zero and 2.0.

Fig. 6
Fig. 6

Graph showing (a) the movement of the point of reflection on the polygon facet as a function of rotation angle θ with normalized offset (x0/RF) as a parameter, and (b) the variation of the optical power carried by the output beam.

Fig. 7
Fig. 7

Diagram for deriving an expression for the vignetting effect.

Fig. 8
Fig. 8

Graph for scan duty cycle ηD versus the normalized beam width (2a/W) with number M of polygon facets as a parameter.

Fig. 9
Fig. 9

Graph showing the movement of the point of reflection on the polygon facet as a function of rotation angle θ with angle ψ as a parameter. The point of reflection moves from one facet edge to the other when ψ = 0°, 20°, and 40°; it changes direction at θ = -14.2° and then returns to the starting point when ψ = 60°.

Fig. 10
Fig. 10

(a) One-way movement of the point of reflection on the polygon facet from point Q(-) to point Q(+) when rotation angle θ is between -θ(-) = -23.9° and θ(+) = 28.1°. The incident ray makes an angle ψ = 40° to the x axis and normalized offset x0/RF = 1.2; (b) change of direction at θ = - 14.2° when ψ = 60°.

Fig. 11
Fig. 11

Utilization factor of facet width ηW versus normalized offset x0/RF with ψ as a parameter that is sampled from zero to 60°.

Fig. 12
Fig. 12

Plot of the F - tan θ equation, showing normalized incremental movement ΔX/P versus θ for a polygon scanner when the incident plane is perpendicular to the rotation axis and ψ = 30°. The observation plane is parallel to the axis of rotation and located at a distance P away from the axis.

Fig. 13
Fig. 13

Locus of the scan center in the system of M = 6, ψ = 20° when normalized input offset x0/RF varies from zero to 1.5 to show parallel beam scanning.

Fig. 14
Fig. 14

Graph of the locus of the scan center (open circles) and the depth of the scan field for systems of (a) M = 6, ψ = 0°, (b) M = 6, ψ = 30°, x0 = RF, (c) M = 10, ψ = 55°, x0 = RF.

Fig. 15
Fig. 15

Scanning with a convergent beam, showing the curved scan pattern and the asymptotic circle for the scan pattern.

Equations (44)

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Θ=360°/M.
ŝi=-iˆ cos ψ+jˆ sin ψ,
r-iˆx0×ŝi=0,
nˆ=iˆ cos θ+jˆ sin θ,
r·nˆ=RF,
rQ=iˆx0+RF-x0 cos θcosθ-ψiˆ cos ψ+jˆ sin ψ,
ŝr=ŝi-2ŝi ·nˆnˆ=iˆ cos2θ-ψ+jˆ sin2θ-ψ.
DA=arccosŝr·ŝrθ=0.
r-rQ×ŝr=0.
y=x-x0tan2θ-ψ-2RF-x0 cos θcos2θ-ψsinθ-ψ.
AQ¯=rQ-iˆx0=RF-x0 cos θcosθ-ψ,
BT¯=RT2-x0 sin ψ21/2-RF2-x0 sin ψ21/2  when RF>x0 sin ψ,
BB¯=2RT2-x0 sin ψ21/2  when RT>x0 sin ψ>RF,
BB¯max=2RT2-RF21/2=W  when x0 sin ψ=RF+0+,
-θ(-)θθ(+).
DA(-)=-2θ(-),  DA(+)=2θ(+).
θ(-)=Θ/2-θB,  θ(+)=Θ/2+θB,
AS=DA(+)-DA(-)2Θ=100×θBΘ%.
γ=2ψ-DA(+)=2 arcsinx0/RTsin ψ-Θ.
nˆ=iˆ cos θ+jˆ sin θ,  û=-iˆ sin θ+jˆ cos θ.
y=rQ·û=-RF sinθ-ψ+x0 sin ψsinθ-ψ.
-θ(-)<θ<θ(+).
-θ(-)<θ<θ(+).
-θ(-)<θ<θ(-),  θ(+)<θ<θ(+).
A=a2π-arccosh/a,
h=-RT sinθ-ψ±Θ/2+x0 sin θ.
VE=PP0=Aπa2=1-1πarccosha.
ηD=θ(-)+θ(+)Θ=1-1Θarcsinx0 sin ψ+aRT-arcsinx0 sin ψ-aRT.
ηD=1-2a/Wcos ψ.
ηW=ΔW/W.
ηW=0.51-x0/RF2 sin2 ψ-11/2 cotΘ/2  when x0 sin ψRF,  ηW=1  when x0 sin ψ<RF.
r·êN=P,  êN=iˆ cos ϕ+jˆ sin ϕ,
rS=1cos2θ-ψ-ϕPiˆ cos2θ-ψ+jˆ sin2θ-ψ+2RF sinθ-ψ+x0 sin ψiˆ sin ϕ-jˆ sin ϕ.
X=rS·êX=P tan2θ-ψ-ϕ-2RF sinθ-ψ+x0 sin ψcos2θ-ψ-ϕ,  Y=0,
XP tan2θ-ψ-ϕ
y=x-x0tan2θ+2δ-ψ-2RF-x0 cosθ+δcos2θ+2δ-ψsinθ+δ-ψ.
x=RF cos θ+RF sinθ-ψ+x0 sin ψsin2θ-ψ,  y=RF sin θ-RF sinθ-ψ+x0 sin ψcos2θ-ψ.
θC=ψ-arcsin2x0/3RFsin ψ.
DFC1F1¯.
Θ<2arcsinx0RTsin ψ-arcsin2x03RTsin ψ.
dydx=dy/dθdx/dθ=tan2θ-ψ.
xg=xg-2d cos θ,  yg=yg-2d sin θ.
Rc=2RF2+xg2+yg2-2RFxg3/2RF2+2xg2+yg2-3RFxg.
xc=xg2+2yg2-RFxgRF2+2xg2+yg2-3RFxgRF,  yc=xgyg-RFygRF2+2xg2+yg2-3RFxgRF.

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