Abstract

An efficient two-dimensional matrix method is presented that facilitates the design of optical systems with tilted surfaces for which the requirement or knowledge of the orientation of the image plane is necessary, i.e., for which a generalized Scheimpflug condition is needed. In more general terms, the method results in imaging properties of second-order expansion, but the method is linear. Therefore the complexity of the design process is considerably reduced. The strength of the design method is demonstrated in detail for a novel application in which a reflective optical system of several surfaces is required for rotationally symmetric triangulation.

© 2005 Optical Society of America

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References

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  1. J. M. Howard, B. D. Stone, “Imaging with four spherical mirrors,” Appl. Opt. 39, 3232–3242 (2000).
    [CrossRef]
  2. B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
    [CrossRef]
  3. J. M. Howard, B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045–3056 (1998).
    [CrossRef]
  4. B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
    [CrossRef]
  5. L. H. J. F. Beckmann, D. Ehrlichmann, “Three- mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.
  6. D. Korsch, Reflective Optics (Academic, San Diego, Calif., 1991).

2000 (1)

1998 (1)

1994 (1)

1992 (1)

Beckmann, L. H. J. F.

L. H. J. F. Beckmann, D. Ehrlichmann, “Three- mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.

Ehrlichmann, D.

L. H. J. F. Beckmann, D. Ehrlichmann, “Three- mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.

Forbes, G. W.

Howard, J. M.

Korsch, D.

D. Korsch, Reflective Optics (Academic, San Diego, Calif., 1991).

Stone, B. D.

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

Other (2)

L. H. J. F. Beckmann, D. Ehrlichmann, “Three- mirror off-axis systems for laser applications,” in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 340–348.

D. Korsch, Reflective Optics (Academic, San Diego, Calif., 1991).

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Figures (7)

Fig. 1
Fig. 1

Basal ray and a parabasal ray going through a tilted surface. The incident angle of the base ray is γ, and the radius of the surface is r.

Fig. 2
Fig. 2

(a) Notation for tilted reflecting surfaces. (b) The same sign convention is applied to centered systems.

Fig. 3
Fig. 3

(a) Classical triangulation sensor. The measurement result at gaps or curved objects is dependent on the angular orientation φ of the sensor. (b) Imaging requirements for rotationally symmetric triangulation. The optical system is actually rotated around the illumination axis.

Fig. 4
Fig. 4

Solution with one mirror with α = 40 ° . The stop is located at the mirror. The following numerical values result for this case:  γ = 37.5 ° , β = 0.71 , and α = 24.9 ° . The rays are traced by exact ray tracing.

Fig. 5
Fig. 5

Basic layout of an optical system with two mirrors. Only the basal ray is shown. The distance between the two mirrors (along the basal ray) is denoted by d, which is always positive.

Fig. 6
Fig. 6

Optical design examples for rotationally symmetric triangulation systems with two mirrors. Scaling for all layouts is the same.

Fig. 7
Fig. 7

(a) Mock-up of the rotationally symmetric triangulation sensor. In the middle, the optics is clearly visible. Above, the complementary metal-oxide semiconductor detector with its adjustment can be seen. The object is located at the bottom. The laser-diode system is located behind the detector (not visible). (b) An image from the detector for a perfect scattering object. The disjunction of the ring is due to a mechanical mount for a small mirror between the optical system and the detector, which is visible in (a). This mirror redirects the illumination beam.

Equations (40)

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β t = d l ( t ) d l ( t ) = d h t ; O d h t ; O = r n cos γ cos γ a ( t ) ( n cos γ n cos γ ) + r n ( cos γ ) 2 = n cos γ n cos γ a ( t ) a ( t ) .
n a ( t ) cos γ cos γ n a ( t ) cos γ cos γ = n cos γ n cos γ r cos γ cos γ .
d l ( t ) = β t d l ( t ) .
d a ( t ) = n n β t 2 d a ( t ) β t sin γ β t n ( n cos γ + 2 n cos γ ) n ( n cos γ + 2 n cos γ ) n n cos γ cos γ d l ( t ) .
cot α ( t ) = n n β t cot α ( t ) + sin γ β t n ( n cos γ + 2 n cos γ ) n ( n cos γ + 2 n cos γ ) n n cos γ cos γ ,
m ( t ) = sin α ( t ) sin α ( t ) β t .
( d l ( t ) d a ( t ) ) = [ β t 0 β t sin γ β t n ( n cos γ + 2 n cos γ ) n ( n cos γ + 2 n cos γ ) n n cos γ cos γ n n β t 2 ] ( d l ( t ) d a ( t ) ) .
β s = d h s ; O d h s ; O = r n a ( s ) ( n cos γ n cos γ ) + r n = n n a ( s ) a ( s ) ,
n a ( s ) n a ( s ) = n cos γ n cos γ r .
d l ( s ) = β s cos γ cos γ d l ( s ) ,
d a ( s ) = n n β s 2 d a ( s ) β s sin γ β s 1 cos γ d l ( s ) .
cot α ( s ) = n cos γ n cos γ β s cot α ( s ) + ( β s 1 ) tan γ .
( d l ( s ) d a ( s ) ) = [ β s cos γ cos γ 0 β s sin γ β s 1 cos γ n n β s 2 ] ( d l ( s ) d a ( s ) ) .
β t = r cos γ 2 a ( t ) r cos γ = a ( t ) a ( t ) , n a ( t ) n a ( t ) = 2 n r cos γ ,
β s = r r 2 a ( s ) cos γ = a ( s ) a ( s ) , n a ( s ) n a ( s ) = 2 n cos γ r .
( d l ( t ) d a ( t ) ) = [ β t 0 β t ( β t + 1 ) tan γ β t 2 ] ( d l ( t ) d a ( t ) ) ,
( d l ( s ) d a ( s ) ) = [ β s 0 β s ( β s 1 ) tan γ β s 2 ] ( d l ( s ) d a ( s ) ) .
cot α ( s ) = β s cot α ( s ) + ( β s 1 ) tan γ ,
cot α ( t ) = β t cot α ( t ) + ( β t + 1 ) tan γ .
α = π 2 + ( π + α + 2 γ ) = α + 2 γ + π 2 ,
a sin α = a sin ( π 2 + α )
a a = sin α sin ( π 2 + α ) = sin α sin ( π + α + 2 γ ) = sin α sin ( α + 2 γ ) .
cot α = β cot α + tan γ ( β + 1 ) .
β = sin α sin ( α + 2 γ ) .
0 = tan ( α + 2 γ ) + sin α sin ( α + 2 γ ) cot α + tan γ [ sin α sin ( α + 2 γ ) + 1 ] .
k = 1 + 4 β ( 1 + β ) 2 ( cos γ ) 2 .
r = a 2 β 1 + β 1 cos γ .
r 0 = r ( cos γ ) 3 .
a long axis = r 0 1 + k , b short axis = r 0 1 + k 1 + k .
2 π = α 2 γ 1 + ( 2 π 2 γ 2 ) + ( π 2 + α )
α = α + 2 ( γ 1 + γ 2 ) π 2 for β > 0 .
α = α + 2 ( γ 1 + γ 2 ) + π 2 for β < 0 .
cot α = cot [ α + 2 ( γ 1 + γ 2 ) β β π 2 ] = tan [ α + 2 ( γ 1 + γ 2 ) ] ,
a 2 cos ( α ) + d cos ( α + 2 γ 1 + π 2 ) = a 1 sin ( α ) for β > 0 .
a 2 = β 2 a 2 = β 2 ( d + a 1 ) = β 2 d + β 2 β 1 a 1 = β 2 d + β a 1 .
β 2 = sin ( α + 2 γ 1 ) + a 1 d sin α sin [ α + 2 ( γ 1 + γ 2 ) ] a 1 d β .
( d l d a ) = [ β 2 0 β 2 tan γ 2 ( β 2 + 1 ) β 2 2 ] [ β 1 0 β 1 tan γ 1 ( β 1 + 1 ) β 1 2 ] ( d l d a ) ,
( d l d a ) = [ β 0 β [ tan γ 2 ( β 2 + 1 ) + tan γ 1 ( β + β 2 ) ] β 2 ] ( d l d a ) ,
cot α = β cot α + tan γ 2 ( β 2 + 1 ) + tan γ 1 ( β + β 2 ) .
0 = tan [ α + 2 ( γ 1 + γ 2 ) ] + β cot α + β tan γ 1 + tan γ 2 + { sin ( α + 2 γ 1 ) + a 1 d sin α sin [ α + 2 ( γ 1 + γ 2 ) ] a 1 d β } ( tan γ 1 + tan γ 2 ) .

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