Abstract

The use of automatically computed aberration coefficients of arbitrary order for an exact determination of the surface of best focus of a symmetrical system, by minimizing the gyration radius of the spot diagram, is demonstrated. For the two optical systems studied, a Schmidt camera and a Ritchey-Chrétien telescope, the results from the exact model are found to be significantly different from the paraxial approximation predictions.

© 1981 Optical Society of America

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References

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  1. B. Tatian, J. Opt. Soc. Am. 64, 1083 (1974).
  2. P. N. Robb, J. Opt. Soc. Am. 66, 1037 (1976).
  3. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  4. C. J. Woodruff, Opt. Acta 22, 933 (1975).
  5. P. J. Sands, J. Opt. Soc. Am. 63, 582 (1973).
  6. T. B. Andersen, Appl. Opt. 19, 3800 (1980).
  7. H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.
  8. C. G. Wynne, Astrophys. J. 152, 675 (1968).

1980 (1)

1976 (1)

1975 (1)

C. J. Woodruff, Opt. Acta 22, 933 (1975).

1974 (1)

1973 (1)

1968 (1)

C. G. Wynne, Astrophys. J. 152, 675 (1968).

Andersen, T. B.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Epps, H. W.

H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.

Peters, P. J.

H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.

Robb, P. N.

Sands, P. J.

Tatian, B.

Woodruff, C. J.

C. J. Woodruff, Opt. Acta 22, 933 (1975).

Wynne, C. G.

C. G. Wynne, Astrophys. J. 152, 675 (1968).

Appl. Opt. (1)

Astrophys. J. (1)

C. G. Wynne, Astrophys. J. 152, 675 (1968).

J. Opt. Soc. Am. (3)

Opt. Acta (1)

C. J. Woodruff, Opt. Acta 22, 933 (1975).

Other (2)

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.

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Figures (2)

Fig. 1
Fig. 1

Contour map of the gyration radius kg for the 1.5-m telescope. The ordinate distance is the distance ( x 1 2 + y 1 2 ) 1 / 2 from the axis, and the abscissa is the axial displacement from the Gaussian image plane (GIP). The lowest contour level displayed is 10 μm, and the contour spacing is 10 μm. The dashed curve displays the best focal surface minimizing kg. Note the different scales on abscissa and ordinate axes.

Fig. 2
Fig. 2

Contour map of the gyration radius kg for the telescope with corrector at the 450-nm wavelength. Explanation of curves same as for Fig. 1. The lowest contour level displayed is 10 μm, and the contour spacing is 10 μm.

Tables (3)

Tables Icon

Table I Relative Errors of the Paraxlal-Approxlmation Coefficients

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Table II Optical Focal Surface Functions for the Epps−Peters UV Camera

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Table III Constructional Data for the Danish 1.5-m Ritchey-Chrétien Telescope with Singlet Field Flattener

Equations (42)

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ρ = x 0 2 + y 0 2 , ψ = ξ 0 2 + η 0 2 , κ = x 0 ξ 0 + y 0 η 0 .
[ x 1 y 1 ] = S ( ρ , ψ , κ ) [ x 0 y 0 ] + T ( ρ , ψ , κ ) [ ξ 0 η 0 ] ,
[ ξ 1 η 1 ] = V ( ρ , ψ , κ ) [ x 0 y 0 ] + W ( ρ , ψ , κ ) [ ξ 0 η 0 ] ,
V = V W = W , S = S + V Δ , T = T + W Δ .
[ ξ 0 η 0 ] = 1 D [ x 0 x * y 0 y * ] ,
{ x 1 [ x 0 , y 0 , ( x 0 x * ) / D , ( y 0 y * ) / D ] , y 1 [ x 0 , y 0 , ( x 0 x * ) / D , ( y 0 y * ) / D ] } ,
ρ * = x 0 2 + y 0 2 , ψ * = x * 2 + y * 2 , κ * = x 0 x * + y 0 y *
[ ρ * ψ * κ * ] = [ 1 0 0 1 D 2 2 D 1 0 D ] [ ρ ψ κ ] ,
[ ρ ψ κ ] = [ 1 0 0 D 2 D 2 2 D 2 D 1 0 D 1 ] [ ρ * ψ * κ * ] ,
[ x 1 y 1 ] = S * ( ρ * , ψ * , κ * ) [ x 0 y 0 ] + T * ( ρ * , ψ * , κ * ) [ x * y * ] ,
[ ξ 1 η 1 ] = V * ( ρ * , ψ * , κ * ) [ x 0 y 0 ] + W * ( ρ * , ψ * , κ * ) [ x * y * ] ,
[ S * ( ρ * , ψ * , κ * ) V * ( ρ * , ψ * , κ * ) T * ( ρ * , ψ * , κ * ) W * ( ρ * , ψ * , κ * ) ] = [ 1 D 1 0 D 1 ] [ S ( ρ , ψ , κ ) V ( ρ , ψ , κ ) T ( ρ , ψ , κ ) W ( ρ , ψ , κ ) ] ,
[ x g y g ] = ω [ x 1 y 1 ] d x 0 d y 0 / ω d x 0 d y 0 .
a ω = 1 / ω d x 0 d y 0 = ( π R 2 ) 1
[ x g y g ] = a ω ω S ( ρ , ψ , κ ) [ x 0 y 0 ] d ω + a ω ω T ( ρ , ψ , κ ) [ ξ 0 η 0 ] d ω = a ω ω S ( ρ , ψ , κ ) [ x 0 y 0 ] d ω + a ω ω T ( ρ , ψ , κ ) [ ξ 0 η 0 ] d ω Δ { a ω ω V ( ρ , ψ , κ ) [ x 0 y 0 ] d ω + a ω ω W ( ρ , ψ , κ ) [ ξ 0 η 0 ] d ω } .
a ω ω ρ i ψ j κ k [ x 0 y 0 ] d ω = { ( k + 1 ) ! 2 k ( 2 i + k + 3 ) [ ( k + 1 2 ) ! ] 2 σ i + [ ( k + 1 ) / 2 ] ψ j + [ ( k 1 ) / 2 ] [ ξ 0 η 0 ] , k odd > 0 [ 0 0 ] , k even 0 ,
a ω ω ρ i ψ j κ k [ ξ 0 η 0 ] d ω = { k ! 2 k 1 ( 2 i + k + 2 ) [ ( k 2 ) ! ] 2 σ i + ( k / 2 ) ψ j + ( k / 2 ) [ ξ 0 η 0 ] , k even 0 [ 0 0 ] , k odd > 0 .
a ω ω G ( ρ , ψ , κ ) [ x 0 y 0 ] d ω = ψ 1 a ω ω G ( ρ , ψ , κ ) κ [ ξ 0 η 0 ] d ω .
[ x g y g ] = [ C 1 ( σ , ψ ) + Δ C 2 ( σ , ψ ) ] [ ξ 0 η 0 ] ,
C 1 ( σ , ψ ) = ψ 1 a ω ω S ( ρ , ψ , κ ) κ d ω + a ω ω T ( ρ , ψ , κ ) d ω ,
C 2 ( σ , ψ ) = ψ 1 a ω ω V ( ρ , ψ , κ ) κ d ω + a ω ω W ( ρ , ψ , κ ) d ω ,
k g 2 ( σ , ψ ) = ω [ ( x 1 x g ) 2 + ( y 1 y g ) 2 ] d ω / ω d ω = a ω ω ( x 1 2 + y 1 2 ) d ω ( x g 2 + y g 2 ) = π 0 ( σ , ψ ) + π 1 ( σ , ψ ) Δ + π 2 ( σ , ψ ) Δ 2 ,
π 0 ( σ , ψ ) = a ω ω ( S 2 ρ + T 2 ψ + 2 S T κ ) d ω ψ C 1 ( σ , ψ ) 2 ,
π 1 ( σ , ψ ) = 2 a ω ω [ S V ρ + TW ψ + ( SW + TV ) κ ] d ω 2 ψ C 1 ( σ , ψ ) C 2 ( σ , ψ ) ,
π 2 ( σ , ψ ) = a ω ω ( V 2 ρ + W 2 ψ + 2 V W κ ) d ω ψ C 2 ( σ , ψ ) 2 .
Δ min ( σ , ψ ) = π 1 ( σ , ψ ) 2 π 2 ( σ , ψ ) .
[ x g y g ] min = [ C 1 ( σ , ψ ) + C 2 ( σ , ψ ) Δ min ( σ , ψ ) ] [ ξ 0 η 0 ] M min ( σ , ψ ) [ ξ 0 η 0 ] ,
k g min 2 ( σ , ψ ) = π 0 ( σ , ψ ) + 1 2 π 1 ( σ , ψ ) Δ min ( σ , ψ ) .
V ( ρ , ψ , κ ) V 0000 , W ( ρ , ψ , κ ) W 0000 .
Δ min , par ( σ , ψ ) = 2 V 0000 σ a ω ω ( S ρ + T κ ) d ω ,
[ x g y g ] min , par = [ C 1 ( σ , ψ ) + W 0000 Δ min , par ( σ , ψ ) ] [ ξ 0 η 0 ] = M min , par ( σ , ψ ) ] [ ξ 0 η 0 ] .
G ( ρ , ψ , κ ) = n = 0 j = 0 n k = 0 j G n , n j , j k , k ρ n j ψ j k κ k
F ( σ , ψ ) = n = 0 j = 0 n F n , n j , j σ n j ψ j
σ 1 π 0 ( σ , ψ ) = σ 1 a ω ω S 2 ρ d ω + σ 1 ψ ( a ω ω T 2 d ω + 2 ψ 1 a ω × ω ST κ d ω C 1 2 ) ,
σ 1 π 1 ( σ , ψ ) = 2 σ 1 a ω ω SV ρ d ω + 2 σ 1 ψ [ a ω ω TWd ω + ψ 1 a ω × ω ( SW + TV ) κ d ω C 1 C 2 ] ,
σ 1 π 2 ( σ , ψ ) = σ 1 a ω ω V 2 ρ d ω + σ 1 ψ ( a ω ω W 2 d ω + 2 ψ 1 a ω × ω VW κ d ω C 2 2 ) .
Δ min ( σ , ψ ) = σ 1 π 1 ( σ , ψ ) 2 σ 1 π 2 ( σ , ψ ) .
I 1 : a ω ω d ω , I 2 : ψ 1 a ω ω κ d ω ,
I 3 : σ 1 a ω ω ρ d ω ,
Δ min , par ( σ , ψ ) = 2 V 0000 [ σ 1 a ω ω S ρ d ω + ( σ 1 ψ ) ψ 1 a ω ω T κ d ω ] ,
V 0000 2 Δ min ( σ , ψ ) = 2 3 V 0000 S 1100 σ + V 0000 ( S 1010 + 1 2 T 1001 ) ψ + ( 1 2 V 0000 S 2200 7 18 S 1100 V 1100 ) σ 2 + 1 3 [ V 0000 ( 2 S 2110 + S 2002 + T 2101 ) 2 S 1100 V 1010 2 S 1010 V 1100 + 1 2 T 1100 W 1100 + 5 8 S 1001 V 1001 7 4 S 1100 W 1001 + S 1001 W 1100 + 1 4 T 1100 V 1001 T 1001 V 1100 ] σ ψ + [ V 0000 ( S 2020 + 1 2 T 2011 ) S 1010 V 1010 1 2 S 1010 W 1001 1 2 T 1001 V 1010 ] ψ 2 + .
V 0000 Δ min , par ( σ , ψ ) = 2 3 S 1100 σ + ( S 1010 + 1 2 T 1001 ) ψ + 1 2 S 2200 σ 2 + 1 3 ( 2 S 2110 + S 2002 + T 2101 ) σ ψ + ( S 2020 + 1 2 T 2011 ) ψ 2 .

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