Abstract

Space and surface 2-D scans produced by single or twin (deflecting) mirror scanning systems are studied. The geometrical and temporal distortions they may exhibit are analyzed. A new type of scanning system is proposed, free of geometrical distortion and capable of covering any desired region in space, from the equator to the pole of its attached spherical coordinates. The drive functions required for obtaining temporally uniform space and (cylindrical) surface scans are established.

© 1983 Optical Society of America

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References

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  1. W. J. Alford, R. D. Vanderneut, V. J. Zaleckas, Proc. IEEE 70, 641 (1982).
    [CrossRef]
  2. J. C. Urbach, T. S. Fisli, G. K. Starkweather, Proc. IEEE 70, 597 (1982).
    [CrossRef]
  3. R. H. Webb, G. W. Hughes, O. Pomerantzeff, Appl. Opt. 19, 2991 (1980).
    [CrossRef] [PubMed]
  4. J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
    [CrossRef]

1982 (2)

W. J. Alford, R. D. Vanderneut, V. J. Zaleckas, Proc. IEEE 70, 641 (1982).
[CrossRef]

J. C. Urbach, T. S. Fisli, G. K. Starkweather, Proc. IEEE 70, 597 (1982).
[CrossRef]

1980 (2)

R. H. Webb, G. W. Hughes, O. Pomerantzeff, Appl. Opt. 19, 2991 (1980).
[CrossRef] [PubMed]

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Alford, W. J.

W. J. Alford, R. D. Vanderneut, V. J. Zaleckas, Proc. IEEE 70, 641 (1982).
[CrossRef]

Broekhuyzen, P.

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Fisli, T. S.

J. C. Urbach, T. S. Fisli, G. K. Starkweather, Proc. IEEE 70, 597 (1982).
[CrossRef]

Hughes, G. W.

Molenaar, J.

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Nivard, J.

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Pomerantzeff, O.

Schreurs, A. W.

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Starkweather, G. K.

J. C. Urbach, T. S. Fisli, G. K. Starkweather, Proc. IEEE 70, 597 (1982).
[CrossRef]

Urbach, J. C.

J. C. Urbach, T. S. Fisli, G. K. Starkweather, Proc. IEEE 70, 597 (1982).
[CrossRef]

Van de Grind, W. A.

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Vanderneut, R. D.

W. J. Alford, R. D. Vanderneut, V. J. Zaleckas, Proc. IEEE 70, 641 (1982).
[CrossRef]

Voorhorst, R.

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Webb, R. H.

Zaleckas, V. J.

W. J. Alford, R. D. Vanderneut, V. J. Zaleckas, Proc. IEEE 70, 641 (1982).
[CrossRef]

Appl. Opt. (1)

Pfluegers Arch. (1)

J. Molenaar, R. Voorhorst, A. W. Schreurs, P. Broekhuyzen, J. Nivard, W. A. Van de Grind. Pfluegers Arch. 383, 173 (1980).
[CrossRef]

Proc. IEEE (2)

W. J. Alford, R. D. Vanderneut, V. J. Zaleckas, Proc. IEEE 70, 641 (1982).
[CrossRef]

J. C. Urbach, T. S. Fisli, G. K. Starkweather, Proc. IEEE 70, 597 (1982).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

General scheme of light reflection.

Fig. 2
Fig. 2

Notations for single mirror systems.

Fig. 3
Fig. 3

General scheme of twin mirror systems.

Fig. 4
Fig. 4

Notations for twin mirror systems.

Fig. 5
Fig. 5

Planar surface scan of a single mirror system with δ = 0 and i = 0. Plotted lines are for 2α and 2β values ranging from −30° to +30° in steps of 5°.

Fig. 6
Fig. 6

Planar surface scan of a single mirror system with δ = π/4 and i = π/2. Note that the β deflection is not doubled.

Fig. 7
Fig. 7

Planar surface scan of a twin mirror system with δ = 0 and i =0.

Fig. 8
Fig. 8

General scheme of a cylindrical surface scan with δ = 0 while i and β0 can take any arbitrary value from −π/2 to + π/2.

Fig. 9
Fig. 9

Particular positions of (a) single mirror systems and (b) twin mirror systems in which the actual exiting light ray Δ2 is reflected perpendicular to the β-scan axis.

Fig. 10
Fig. 10

Addition of a compensating rotation α′ along the z axis of mirror M2 in twin mirror systems for eliminating geometrical distortion. This new mount offers two advantages: (i) the pole is easily scanned, (ii) an open setup (the incoming and exiting light beams stand at right angles).

Fig. 11
Fig. 11

Demonstration of the surface scan achieved with a twin mirror system in which M2 (right-hand side) undergoes—in addition to the usual β scan—a compensating rotation α′.

Fig. 12
Fig. 12

Sweep velocities over a period for (a) a sawtooth drive and (b) a sine wave drive.

Tables (1)

Tables Icon

Table I α Values Yielding Nondistorted β Scans

Equations (42)

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Δ 2 = 2 ( Δ 1 N ) N Δ 1 ,
N = { cos δ cos β cos α sin δ sin α cos δ cos β sin α + sin δ cos δ cos δ sin β .
N = { cos δ cos β sin δ cos δ sin β ,
Δ 1 = { cos ( i + α ) sin ( i + α ) 0 ;
{ tan θ = υ 2 u 2 , tan ϕ = w 2 ( 1 w 2 2 ) 1 / 2 .
{ p = f ( α , β ) , q = g ( α , β ) .
{ p = f ( α ) , q = g ( β ) ,
{ p = υ 1 t T 1 + p 0 , 0 < t < T 1 , q = υ 2 t T 2 + q 0 0 < t < T 2 ,
Δ 2 = { u 2 = 2 cos 2 α cos 2 β 1 , υ 2 = cos 2 β sin 2 α , w 2 = cos α sin 2 β ,
{ tan θ = sin 2 α cos 2 β 2 cos 2 α cos 2 β 1 , tan ϕ = cos α sin 2 β ( 1 cos 2 α sin 2 2 β ) 1 / 2 .
{ Y = tan θ X 0 = υ 2 u 2 X 0 , Z = tan ϕ cos θ X 0 = w 2 u 2 X 0 .
{ Y = cos 2 β sin 2 α 2 cos 2 α cos 2 β 1 X 0 , Z = cos α sin 2 β 2 cos 2 α cos 2 β 1 X 0 .
{ θ = 2 α , ϕ = 2 β ,
{ Y = 2 α X 0 , Z = 2 β X 0 .
{ tan θ = cos β sin 2 α sin 2 β sin 2 α cos β cos 2 α 1 / 2 sin 2 α sin 2 β , tan ϕ = cos α sin β + 1 / 2 sin α sin 2 β [ 1 ( cos α sin β + 1 / 2 sin α sin 2 β ) 2 ] 1 / 2 ,
{ Y = cos β sin 2 α sin 2 α sin 2 β cos β cos 2 α 1 / 2 sin 2 α sin 2 β X 0 , Z = cos α sin β + 1 / 2 sin α sin 2 β cos β cos 2 α 1 / 2 sin 2 α sin 2 β X 0 .
{ tan ϕ = tan γ , Z = tan γ X 0 ,
sin ( α M 2 ) = 1 g R sin ( 2 α M 1 ) .
Δ 2 = { cos 2 β cos α sin α sin 2 β cos α .
{ Y = tan α cos 2 β X 0 , Z = tan 2 β X 0 .
x = X 0 cos 2 β ;
{ Y = tan α X 0 , Z = sin 2 β X 0 ,
{ tan θ = u 2 w 2 , tan ϕ = υ 2 ( 1 υ 2 2 ) 1 / 2 .
{ ϕ = π 2 2 β , ϕ = α .
{ Y = tan ( α + i ) X 0 , Z = sin 2 ( β + β 0 ) X 0 ,
{ θ = π 2 2 ( β + β 0 ) , ϕ = ( α + i ) .
{ tan θ = sin α cos β cos α cos β + sin α sin 2 β , tan ϕ = cos α sin β 1 / 2 sin α sin 2 β [ 1 ( cos α sin β 1 / 2 sin α sin 2 β ) 2 ] 1 / 2 .
{ Y = sin α cos β cos α cos β + sin α sin 2 β X 0 , Z = cos α sin β 1 / 2 sin α sin 2 β cos α cos β + sin α sin 2 β .
i = 2 δ + α .
i = 2 δ α ;
α α = α 0 ,
{ θ = α = 2 α M 1 , ϕ = 2 β ,
{ Y = tan α X 0 , Z = tan 2 β cos α X 0 .
{ Y = sin α X 0 , Z = tan 2 β X 0 .
{ d 1 = tan 2 β X 0 , d 2 = ( α ) rad X 0 .
Δ Z = ( 1 cos α 1 ) tan 2 β X 0 .
{ d ( d 1 ) dt = d ( d 1 ) d β d β dt , d ( d 2 ) dt = d ( d 2 ) d α d α dt .
{ d ( d 1 ) d β = 2 X 0 cos 2 2 β , d ( d 2 ) d α = X 0 .
{ β ( t ) = β m ( 2 t T 1 ) for a sawtooth drive , β ( t ) = β m sin ( π T t π 2 ) for a sine wave drive ,
d ( d 1 ) dt = 4 β m X 0 T { cos 2 [ 2 β m ( 2 t T 1 ) ] } 1
d ( d 1 ) dt = 2 π β m X 0 T cos ( π t T π 2 ) cos 2 [ 2 β m sin ( π t T π 2 ) ]
β ( t ) = 4 π β m arctan ( 2 t T 1 ) .

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