Abstract

In this paper, an optical design is presented for an anastigmatic telescope with back focal length corrected with exact ray tracing to eliminate spherical, coma, and astigmatism aberrations. The telescope is formed of three conical mirrors, two of them polished on the same substratum. The optical design is divided into three stages: we began the design obtaining the Gaussian parameters in a first-order solution; posteriorly, were obtained analytically the three mirrors’ asphericity in a third-order design. The final design stage consists of the implementation of the Fermat’s principle, the Abbe sine condition, and the Coddington equations for the exact correction for the three aforementioned aberrations.

© 2011 Optical Society of America

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References

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  1. D. Korsch, “Anastigmatic three-mirror telescope,” Appl. Opt. 16, 2074–2077 (1977).
    [CrossRef] [PubMed]
  2. P. N. Robb, “Three-mirror telescopes: design and optimization,” Appl. Opt. 17, 2677–2685 (1978).
    [CrossRef] [PubMed]
  3. L. G. Seppala, “Improved optical design for the large synoptic survey telescope (lsst),” Proc. SPIE 4836, 111–118 (2002).
    [CrossRef]
  4. V. Draganov, “Compact telescope,” U.S. Patent 6,667,831(23 December 2003).
  5. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994), Chap. 1.
  6. W. Welford, Aberrations of Optical Systems, Series on Optics and Optoelectronics, 1st ed. (Taylor and Francis, 1986), Chaps. 6, 4, 8.
  7. M.J.Kidger, ed., Fundamental Optical Design, Vol. PM92 of SPIE Press Monograph Series (SPIE, 2001), pp. 101–164, Chaps. 6, 7.
    [CrossRef]
  8. W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007), Chap. 10.
  9. D. Korsch, Reflective Optics (Academic, 1991), Chap. 9.
  10. W. Cheney and D. Kindcaid, Numerical Mathematics and Computing, 6th ed. (Thomson Brooks/Cole, 2008), Chap. 3.
  11. URL: www.zemax.com.
  12. URL: http://www.ptc.com/products/mathcad/.

2002 (1)

L. G. Seppala, “Improved optical design for the large synoptic survey telescope (lsst),” Proc. SPIE 4836, 111–118 (2002).
[CrossRef]

1978 (1)

1977 (1)

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994), Chap. 1.

Cheney, W.

W. Cheney and D. Kindcaid, Numerical Mathematics and Computing, 6th ed. (Thomson Brooks/Cole, 2008), Chap. 3.

Draganov, V.

V. Draganov, “Compact telescope,” U.S. Patent 6,667,831(23 December 2003).

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994), Chap. 1.

Kindcaid, D.

W. Cheney and D. Kindcaid, Numerical Mathematics and Computing, 6th ed. (Thomson Brooks/Cole, 2008), Chap. 3.

Korsch, D.

Robb, P. N.

Seppala, L. G.

L. G. Seppala, “Improved optical design for the large synoptic survey telescope (lsst),” Proc. SPIE 4836, 111–118 (2002).
[CrossRef]

Smith, W.

W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007), Chap. 10.

Welford, W.

W. Welford, Aberrations of Optical Systems, Series on Optics and Optoelectronics, 1st ed. (Taylor and Francis, 1986), Chaps. 6, 4, 8.

Appl. Opt. (2)

Proc. SPIE (1)

L. G. Seppala, “Improved optical design for the large synoptic survey telescope (lsst),” Proc. SPIE 4836, 111–118 (2002).
[CrossRef]

Other (9)

V. Draganov, “Compact telescope,” U.S. Patent 6,667,831(23 December 2003).

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994), Chap. 1.

W. Welford, Aberrations of Optical Systems, Series on Optics and Optoelectronics, 1st ed. (Taylor and Francis, 1986), Chaps. 6, 4, 8.

M.J.Kidger, ed., Fundamental Optical Design, Vol. PM92 of SPIE Press Monograph Series (SPIE, 2001), pp. 101–164, Chaps. 6, 7.
[CrossRef]

W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007), Chap. 10.

D. Korsch, Reflective Optics (Academic, 1991), Chap. 9.

W. Cheney and D. Kindcaid, Numerical Mathematics and Computing, 6th ed. (Thomson Brooks/Cole, 2008), Chap. 3.

URL: www.zemax.com.

URL: http://www.ptc.com/products/mathcad/.

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Figures (13)

Fig. 1
Fig. 1

Optical system representing the use of a third mirror with power B 2 = 1 / f 2 , where the light is reflected two times.

Fig. 2
Fig. 2

Mirrors diameters and separations obtained by paraxial ray trace. The diameter D 4 is not shown in the figure because it corresponds to the second reflection through the secondary mirror where D 2 has been calculated.

Fig. 3
Fig. 3

Intersection points must be calculated with the exact ray trace for a marginal ray.

Fig. 4
Fig. 4

Image of an object at infinity P s = and P t = is produced at finite positions by the mirrors in the telescope. The final sagittal position can be different from the tangential position because of astigmatism present in the system.

Fig. 5
Fig. 5

System layout.

Fig. 6
Fig. 6

Aberrations obtained with the methodology shown in this work.

Fig. 7
Fig. 7

Sagittal and tangential image planes plotted on the image plane with Petzval curvature.

Fig. 8
Fig. 8

Spot diagram of the perfectly corrected system with an ideal image plane.

Fig. 9
Fig. 9

Fermat algorithm.

Fig. 10
Fig. 10

Esfer algorithm.

Fig. 11
Fig. 11

Coma algorithm.

Fig. 12
Fig. 12

Codd algorithm.

Fig. 13
Fig. 13

Astig algorithm.

Tables (1)

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Table 1 Obtained Parameters

Equations (67)

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( a b c d ) = ( 1 1 f 2 0 1 ) ( 1 0 d 2 1 ) ( 1 1 f 3 0 1 ) ( 1 0 d 2 1 ) ( 1 1 f 2 0 1 ) ( 1 0 d 1 1 ) ( 1 1 f 1 0 1 ) ,
b 1 = F total = f 1 f 2 2 f 3 A + B + C ,
A = f 2 2 ( f 1 d 1 + f 3 ) + d 2 2 ( f 1 d 1 + f 2 ) , B = d 2 ( f 2 ( f 2 + 2 ( f 1 d 1 + f 3 ) ) + 2 f 3 ( f 3 d 1 ) ) , C = 2 f 2 f 3 ( f 1 d 1 ) .
( a b c d ) = ( 1 0 d 2 + d w 1 ) ( a b c d ) .
( a b c d ) = ( a b c d ) ( θ in h ) .
d = 0 ,
f 2 = [ ( d 1 f 1 ) F total ( d 2 + d w ) f 1 ] [ ( d 1 f 1 + d 2 ) F total ( 2 d 2 + d w ) f 1 ] ,
f 3 = F total ( f 1 + d 2 + d w d 1 ) f 1 d 2 2 Q ,
Q = [ F total 2 + ( F total d w ) 2 d 2 d 2 2 d w 2 ] f 1 2 + [ F total 2 d 1 + ( d 1 + d 2 + d w ) d 2 F total ] 2 f 1 + F total 2 d 1 2 .
P c = ( 1 f 1 + 2 f 2 + 1 f 3 ) .
S I = k = 0 n A k 2 h k Δ ( u n ) k ,
S II = k = 0 n A k A ¯ k h k Δ ( u n ) k ,
S III = k = 0 n A ¯ k 2 h k Δ ( u n ) k ,
S IV = k = 0 n H k 2 c k Δ ( 1 n ) k ,
S V = k = 0 n { A ¯ k 3 A k h k Δ ( u n ) k + A ¯ k A k H k 2 c k Δ ( 1 n ) k } .
L = n i l i .
f = r Paraxial 2 = 1 2 c .
S I * = [ S I k = 0 n ( ε k c k 3 h k 4 Δ n k ) ] ,
S II * = [ S II k = 0 n ( ε k c k 3 h k 3 h ¯ k Δ n k ) ] ,
S III * = [ S III k = 0 n ( ε k c k 3 h k 2 h ¯ k 2 Δ n k ) ] ,
S IV * = [ S IV k = 0 n ( ε k c k 3 h k h ¯ k 3 Δ n k ) ] .
A ¯ = H h ( A h E 1 ) ,
E k + 1 E k = d n h k h k + 1 .
S I * = [ S I ( ε 0 α 0 + ε 1 α 1 + ε 2 α 2 + ε 1 α 3 ) ] = 0 ,
S II * = [ S II ( ε 1 β 1 + ε 2 β 2 + ε 1 β 3 ) ] = 0 ,
S III * = [ S III ( ε 1 γ 1 + ε 2 γ 2 + ε 1 γ 3 ) ] = 0 .
ε 0 = 1 α 0 [ S I + S III δ ξ S II δ μ ] ,
ε 1 = S II γ 2 δ S III β 2 δ ,
ε 2 = S III β 1 + β 3 δ S II γ 1 + γ 3 δ ,
δ = β 1 γ 2 β 2 γ 1 β 2 γ 3 + β 3 γ 2 , ξ = α 1 β 2 α 2 β 1 α 2 β 3 + α 3 β 2 , μ = α 1 γ 2 α 2 γ 1 α 2 γ 3 + α 3 γ 2 .
p ( y , z ) 4 = p ( c 1 , c 2 , c 3 , d 1 , d 2 , d 3 , ε 0 , ε 1 , ε 2 , M , N ) .
OP marginal = [ z 0 d 0 ] 2 + y 0 ] 2 + [ z 1 z 0 ] 2 + [ y 1 y 0 ] 2 + [ z 2 z 1 ] 2 + [ y 2 y 1 ] 2 + [ z 3 z 2 ] 2 + [ y 3 y 2 ] 2 + [ z 4 z 3 ] 2 + [ y 4 y 3 ] 2 .
OP axis = d 0 + d 1 + d 2 + d 2 + ( d 2 + d w ) .
Fermat ( ε 0 , ε 1 , ε 2 ) = OP marginal OP axis = 0 .
Abbe ( ε 1 , ε 2 , ε 3 ) = f Paraxial y 0 M 4 = 0 .
r s = 1 c ( 1 ε c 2 y 2 ) ,
r t = 1 c ( 1 ε c 2 y 2 ) 3 .
1 P s = 2 cos I r s 1 P s ,
1 P t = 2 r t cos I 1 P t .
f s final = P s ( ε 1 , ε 2 , ε 3 ) N ,
f t final = P t ( ε 1 , ε 2 , ε 3 ) N .
Coddington ( ε 0 , ε 1 , ε 2 ) = f S final f T final = 0 .
( ε 0 ε 1 ε 2 ) = ( d f 1 d ε 0 d f 1 d ε 1 d f 1 d ε 2 d f 2 d ε 0 d f 2 d ε 1 d f 2 d ε 2 d f 3 d ε 0 d f 3 d ε 1 d f 3 d ε 2 ) 1 ( f 1 f 2 f 3 ) .
d f ( ε ) d ε = lim h 0 f ( ε + h ) f ( ε ) h .
f 1 = Fermat ( ε 0 , ε 1 , ε 2 ) ,
f 2 = Abbe ( ε 0 , ε 1 , ε 2 ) ,
f 3 = Coddington ( ε 0 , ε 1 , ε 2 ) .
z = 1 2 c ( x 2 + y 2 + ε z 2 ) ,
x 0 = x 1 + L N ( d z 1 ) .
y 0 = y 1 + M N ( d z 1 ) .
F = c ( x 0 2 + y 0 2 ) ,
G = N c ( L x 0 + M y 0 ) .
Δ = F G + G 2 c F [ 1 + ( ε 1 ) N 2 ] .
x = x 0 + L Δ .
y = y 0 + M Δ .
z = N Δ .
J = 1 2 c ( ε 1 ) z + c 2 ε ( ε 1 ) z 2 .
c os I = G 2 c F [ 1 + ( ε 1 ) N 2 ] J .
n cos I = n 2 n 2 ( 1 cos I 2 ) .
K = c ( n cos I n c os I ) .
L = 1 n [ n L K x J ] ,
M = 1 n [ n M K y J ] ,
N = 1 n [ n N + ( 1 z ε c ) K J c ] .
x i = x ( x i 1 , y i 1 , z i 1 , c i , d i , ε i , L i , M i , N i ) ,
y i = y ( x i 1 , y i 1 , z i 1 , c i , d i , ε i , L i , M i , N i ) ,
z i = z ( x i 1 , y i 1 , z i 1 , c i , d i , ε i , L i , M i , N i ) .
f 1 ( x 1 , x 2 , x 3 ) = Esfer ( c 1 , c 2 , c 3 , x 1 , x 2 , x 3 , d 1 , d 2 , H t , L ) , f 2 ( x 1 , x 2 , x 3 ) = Coma ( c 1 , c 2 , c 3 , x 1 , x 2 , x 3 , d 1 , d 2 , H t , f ) , f 3 ( x 1 , x 2 , x 3 ) = Astig ( c 1 , c 2 , c 3 , x 1 , x 2 , x 3 , d 1 , d 2 , θ ) .

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