Abstract

Generalized Coddington equations allow the optical properties of an arbitrarily oriented incoming astigmatic ray bundle to be found following refraction by an arbitrary surface. Generalized Coddington equations are developed using the abstract concept of vergence and refraction operators. After suitable incoming vergence and refraction operators have been formed, these operators are re-expressed in a common coordinate system via similarity transformations created from the series of space rotations necessary to align the coordinate systems. The transformed operators are then added together to produce the vergence operator of the refracted ray bundle. When properly applied, these generalized Coddington equations may be used with complex wavefronts and complex refracting surfaces if local surface curvature properties are known for both where the two intersect. The generalized Coddington equations are given in matrix form so that they may be easily implemented.

© 2006 Optical Society of America

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References

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  1. R. Kingslake, 'Who discovered Coddington's equations?' Opt. Photon. News May 1994, pp. 20-23.
    [CrossRef]
  2. D. G. Burkhard and D. L. Shealy, 'Simplified formula for the illuminance in an optical system,' Appl. Opt. 20, 897-909 (1981).
    [CrossRef] [PubMed]
  3. J. E. A. Landgrave and J. R. Moya-Cessa, 'Generalized Coddington equation in ophthalmic lens design,' J. Opt. Soc. Am. A 13, 1637-1644 (1996).
    [CrossRef]
  4. H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, 1950), p. 113.
  5. H. W. Guggenhiemer, Differential Geometry (Dover, 1963).
  6. C. E. Campbell, 'The Refractive Group,' Optom. Vision Sci. 74, 381-387 (1997).
    [CrossRef]
  7. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972), pp. 149-165.

1997 (1)

C. E. Campbell, 'The Refractive Group,' Optom. Vision Sci. 74, 381-387 (1997).
[CrossRef]

1996 (1)

1981 (1)

Burkhard, D. G.

Campbell, C. E.

C. E. Campbell, 'The Refractive Group,' Optom. Vision Sci. 74, 381-387 (1997).
[CrossRef]

Guggenhiemer, H. W.

H. W. Guggenhiemer, Differential Geometry (Dover, 1963).

Kingslake, R.

R. Kingslake, 'Who discovered Coddington's equations?' Opt. Photon. News May 1994, pp. 20-23.
[CrossRef]

Landgrave, J. E. A.

Moya-Cessa, J. R.

Shealy, D. L.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972), pp. 149-165.

Weyl, H.

H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, 1950), p. 113.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Photon. News (1)

R. Kingslake, 'Who discovered Coddington's equations?' Opt. Photon. News May 1994, pp. 20-23.
[CrossRef]

Optom. Vision Sci. (1)

C. E. Campbell, 'The Refractive Group,' Optom. Vision Sci. 74, 381-387 (1997).
[CrossRef]

Other (3)

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972), pp. 149-165.

H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, 1950), p. 113.

H. W. Guggenhiemer, Differential Geometry (Dover, 1963).

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Equations (63)

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n [ V ] ,
n [ V ] = n [ R ] [ V ] [ R ] 1 .
[ H ] = [ 2 x 2 2 x y 2 x y 2 y 2 ] ,
[ K ] = [ H ] S ( x , y ) = [ 2 x 2 2 x y 2 x y 2 y 2 ] S ( x , y ) = [ 2 S ( x , y ) x 2 2 S ( x , y ) x y 2 S ( x , y ) x y 2 S ( x , y ) y 2 ] .
M = 1 2 ( 2 S x 2 + 2 S y 2 ) , C + = 1 2 ( 2 S x 2 2 S y 2 ) , C × = 2 S x y .
[ K ] = [ M + C + C × C × M C + ] .
[ K ] = M [ 1 0 0 1 ] + C + [ 1 0 0 1 ] + C × [ 0 1 1 0 ] .
opl i n = z s n i n z i n = n i n z s z i n ,
opl r e f = z s n r e f z i r e f = n r e f z s z r e f .
Γ = n r e f z s z r e f n i n z s z i n .
z s z i n = cos ϕ i n , z s z r e f = cos ϕ r e f ,
Γ = n r e f cos ϕ r e f n i n cos ϕ i n .
[ F ] = Γ [ K ] = ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ M + C + C × C × M C + ] .
[ R ( θ ) ] = [ cos θ sin θ 0 sin θ cos θ 0 0 0 1 ] ,
[ R ( θ ) ] = [ cos θ sin θ sin θ cos θ ]
[ R ( θ ) ] 1 = [ cos θ sin θ sin θ cos θ ] .
n [ V ] = n [ R ( θ ) ] [ V ] [ R ( θ ) ] 1 ,
[ R ( ϕ ) ] = [ 1 0 0 0 cos ϕ sin ϕ 0 sin ϕ cos ϕ ]
[ R ( ϕ ) ] 1 = [ 1 0 0 0 cos ϕ sin ϕ 0 sin ϕ cos ϕ ] .
[ R ( ϕ ) ] = [ 1 0 0 cos ϕ ] ,
n [ V ] = n [ R ( ϕ ) ] [ V ] [ R ( ϕ ) ] = n [ R ( ϕ ) ] [ R ( θ ) ] [ V ] [ R ( θ ) ] 1 [ R ( ϕ ) ] .
[ R ( ϕ ) ] [ R ( θ ) ] = [ 1 0 0 cos ϕ ] [ cos θ sin θ sin θ cos θ ] = [ cos θ sin θ sin θ cos ϕ cos θ cos ϕ ] = [ R ( ϕ , θ ) ] ,
[ R ( θ ) ] 1 [ R ( ϕ ) ] = [ cos θ sin θ sin θ cos θ ] [ 1 0 0 cos ϕ ] = [ cos θ sin θ cos ϕ sin θ cos θ cos ϕ ] = [ R ( ϕ , θ ) ] ( 1 ) .
n [ V ] = n [ cos θ sin θ sin θ cos ϕ cos θ cos ϕ ] [ V ] [ cos θ sin θ cos ϕ sin θ cos θ cos ϕ ] .
S E = S + C 2 ,
C + = ( C 2 ) cos ( 2 A ) ,
C × = ( C 2 ) sin ( 2 A ) ,
n [ V ] = n [ S E + ( C + ) C × C × S E ( C + ) ] .
n [ V ] = n [ cos θ sin θ sin θ cos ϕ cos θ cos ϕ ] [ S E + C + C × C × S E C + ] [ cos θ sin θ cos ϕ sin θ cos θ cos ϕ ] .
n r e f [ V r e f ] = ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ K ] + n i n [ V i n ] .
n r e f [ R ( θ r e f , ϕ r e f ) ] [ V r e f ] [ R ( θ r e f , ϕ r e f ) ] ( 1 ) = ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ R ( θ s ) ] [ K ] [ R ( θ s ) ] 1 + n i n [ R ( θ i n , ϕ i n ) ] [ V i n ] [ R ( θ i n , ϕ i n ) ] ( 1 ) .
n r e f [ cos θ r e f sin θ r e f sin θ r e f cos ϕ r e f cos θ r e f cos ϕ r e f ] [ S E r e f + ( C + ) r e f ( C × ) r e f ( C × ) r e f S E r e f ( C + ) r e f ] [ cos θ r e f sin θ r e f cos ϕ r e f sin θ r e f cos θ r e f cos ϕ r e f ] = ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ cos θ s sin θ s sin θ s cos θ s ] [ M + C + C × C × M C + ] [ cos θ s sin θ s sin θ s cos θ s ] + n i n [ cos θ i n sin θ i n sin θ i n cos ϕ i n cos θ i n cos ϕ i n ] [ S E i n + ( C + ) i n ( C × ) i n ( C × ) i n S E i n ( C + ) i n ] [ cos θ i n sin θ i n cos ϕ i n sin θ i n cos θ i n cos ϕ i n ] .
[ R ( ϕ r e f , θ r e f ) ] 1 = [ cos θ r e f sin θ r e f cos ϕ r e f sin θ r e f cos θ r e f cos ϕ r e f ] ,
[ [ R ( ϕ r e f , θ r e f ) ] ( 1 ) ] 1 = [ cos θ r e f sin θ r e f sin θ r e f cos ϕ r e f cos θ r e f cos ϕ r e f ] .
n r e f [ V r e f ] = [ R ( θ r e f , ϕ r e f ) ] 1 { ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ R ( θ s ) ] [ K ] [ R ( θ s ) ] 1 + n i n [ R ( θ i n , ϕ i n ) ] [ V i n ] [ R ( θ i n , ϕ i n ) ] ( 1 ) } [ [ R ( θ r e f , ϕ r e f ) ] ( 1 ) ] 1 .
cos θ i n = x i n x c o m x i n x c o m = ( I × y ) ( I × N ) ( I × y ) ( I × N ) .
cos θ i n = ( I I ) ( N I ) ( I N ) ( y N ) ( I × y ) ( I N ) .
( I I ) = 1 ,
( N I ) = cos ϕ i n ,
I × N = sin ϕ i n ,
I × y = ( I x 2 + I z 2 ) 1 2 ( I x and I y are the x and z components of I ) ,
( I y ) = I y ( the y component of I ) ,
( N y ) = N y ( the y component of N ) ,
cos θ i n = N y I y cos ϕ i n ( I x 2 + I z 2 ) 1 2 sin ϕ i n .
cos θ s = x s x c o m x s x c o m = ( N × y ) ( I × N ) ( N × y ) ( I × N ) .
cos θ s = ( N I ) ( N y ) ( N N ) ( y I ) ( N × y ) ( I × N ) ,
cos θ s = cos ϕ i n N y I y ( N x 2 + N z 2 ) 1 2 sin ϕ i n .
cos θ r e f = x r e f x c o m x r e f x c o m = ( R × y ) ( I N ) ( R × y ) ( I × N ) .
cos θ r e f = ( R I ) ( N y ) ( R N ) ( y I ) ( R × y ) ( I × N ) .
R = ( n i n n r e f ) I + k N ,
k = ( 1 ( n i n n r e f ) 2 sin 2 ϕ i n ) 1 2 n i n n r e f cos ϕ i n = cos ϕ r e f n i n n r e f cos ϕ i n = Γ n r e f ,
sin ϕ r e f = ( n i n n r e f ) sin ϕ i n ,
R = n i n I + Γ N n r e f = ( n i n I x + Γ N x n r e f , n i n I y + Γ N y n r e f , n i n I z + Γ N z n r e f ) .
cos θ r e f = ( n i n I x + Γ N x ) ( I x N y I y N x ) + ( n i n I z + Γ N z ) ( I z N y I y N z ) ( ( n i n I x + Γ N x ) 2 + ( n i n I z + Γ N z ) 2 ) 1 2 sin ϕ i n .
n r e f [ 1 0 0 cos ϕ r e f ] [ S E r e f + ( C + ) r e f ( C × ) r e f ( C × ) r e f S E r e f ( C + ) r e f ] [ 1 0 0 cos ϕ r e f ] = ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ M 0 0 M ] + n i n [ 1 0 0 cos ϕ i n ] [ S E i n 0 0 S E i n ] [ 1 0 0 cos ϕ i n ] .
n r e f [ 1 0 0 cos ϕ r e f ] [ S E r e f + ( C + ) r e f 0 0 S E r e f ( C + ) r e f ] [ 1 0 0 cos ϕ r e f ] = ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ M 0 0 M ] + n i n [ 1 0 0 cos ϕ i n ] [ S E i n 0 0 S E i n ] [ 1 0 0 cos ϕ i n ] .
n r e f [ S E r e f + ( C + ) r e f 0 0 ( S E r e f ( C + ) r e f ) cos 2 ϕ r e f ] = ( n r e f cos ϕ r e f n i n cos ϕ i n ) [ M 0 0 M ] + n i n [ S E i n 0 0 S E i n cos 2 ϕ i n ] .
n r e f ( S E r e f + ( C + ) r e f ) = ( n r e f cos ϕ r e f n i n cos ϕ i n ) M S E i n ,
n r e f cos 2 ϕ r e f ( S E r e f ( C + ) r e f ) = ( n r e f cos ϕ r e f n i n cos ϕ i n ) M n i n cos 2 ϕ i n S E i n .
n s = ( n cos ϕ n cos ϕ ) r + n o ,
n cos 2 ϕ t = ( n cos ϕ n cos ϕ ) r + n cos 2 ϕ o ,
n s = ( n cos ϕ n cos ϕ ) r + n s ,
n cos 2 ϕ t = ( n cos ϕ n cos ϕ ) r + n cos 2 ϕ t ,

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