Abstract

This paper deals with the reflection and refraction of a gaussian laser beam at a curved interface between media of different refractive indices. The analysis extends beyond the usual case of normal incidence at spherical surfaces to include arbitrary angles of incidence and interfaces of ellipsoidal shape. By matching the transverse variations of optical phase at the interface, equations for the spot sizes and wavefront radii of the beams are obtained. These results have been converted to ray matrix form, which is particularly convenient for analyzing thick lenses or systems of several elements. With these matrices, one can readily design and evaluate optical systems containing such astigmatic elements as tilted spherical or cylindrical lenses and mirrors.

© 1969 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  2. S. A. Collins, Appl. Opt. 3, 1263 (1964).
    [CrossRef]

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1964 (1)

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Figures (2)

Fig. 1
Fig. 1

Coordinate systems for surface and for incident, reflected, and refracted beams.

Fig. 2
Fig. 2

Definition of ellipsoidal surface expanded about point of incidence O. Center of ellipsoid is at O′, with semiaxes A and B.

Tables (1)

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Table I Ray Matrix Elements

Equations (28)

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ũ i ( x i , y i , z i ) = Ã i exp ( j ϕ i [ x i , y i , z i ] ) , ( i = 1 , 2 , 3 ) ,
ϕ i ( x i , y i , z i ) = k i z i + k i 2 ( x i 2 q T i + y i 2 q S i ) ,
k 1 = 2 π n 1 / λ 0 = k 3 , k 2 = 2 π n 2 / λ 0 , 1 / q i 1 / R i j ( λ 0 / π n i w i 2 ) .
( X 2 + Z 2 ) / A 2 + Y 2 / B 2 = 1 .
X = x 1 + A sin Θ , Y = y 1 , Z = z 1 A sin Θ ,
( x 1 + A sin Θ ) 2 + ( A y 1 / B ) 2 + ( z 1 A cos Θ ) 2 = A 2 .
z 1 x 1 tan Θ + x 1 2 2 R T cos 3 Θ + y 1 2 2 R S cos Θ ,
ϕ 1 ( x 1 , y 1 ) = k 1 x 1 tan Θ + k 1 x 1 2 2 ( 1 q T 1 + 1 R T cos 3 Θ ) + k 1 y 1 2 2 ( 1 q S 1 + 1 R S cos Θ ) .
ϕ 1 ( x 1 , y 1 ) = φ 2 ( x 1 , y 1 ) = φ 3 ( x 1 , y 1 ) .
x 3 = x 1 cos 2 Θ z 1 sin 2 Θ , y 3 = y 1 , z 3 = x 1 sin 2 Θ z 1 cos 2 Θ .
φ 3 ( x 1 , y 1 ) = k 1 x 1 tan Θ + k 1 x 1 2 2 ( 1 q T 3 + 1 2 cos 2 Θ R T cos 3 Θ ) + k 1 y 1 2 2 ( 1 q S 3 + 1 2 cos 2 Θ R S cos Θ ) .
1 q T 3 = 1 q T 1 + 2 R T cos Θ ( tangential plane ) , 1 q S 3 = 1 q S 1 + 2 cos Θ R S ( sagittal plane ) .
w T 3 = w T 1 , 1 / R T 3 = 1 / R T 1 + 2 / ( R T cos Θ ) , w S 3 = w S 1 , 1 / R S 3 = 1 / R S 1 + 2 cos Θ / R S .
x 2 = x 1 cos ( Θ Θ ) + z 1 sin ( Θ Θ ) , y 2 = y 1 , z 2 = x 1 sin ( Θ Θ ) + z 1 cos ( Θ Θ ) .
φ 2 ( x 1 , y 1 ) = k 2 x 1 [ tan Θ cos ( Θ Θ ) sin ( Θ Θ ) ] + k 2 x 1 2 2 [ cos 2 Θ cos 2 Θ 1 q T 2 + cos ( Θ Θ ) R T cos 3 Θ ] + k 2 y 1 2 2 [ 1 q S 2 + cos ( Θ Θ ) R S cos Θ ] .
1 q T 1 = cos 2 Θ cos 2 Θ n r q T 2 + n r cos ( Θ Θ ) 1 R T cos 3 Θ ,
1 q S 1 = n r q S 2 + n r cos ( Θ Θ ) 1 R S cos Θ ,
1 q T 2 = ( n r cos 2 Θ n r 2 sin 2 Θ ) 1 q T 1 + n r [ cos Θ ( n r 2 sin 2 Θ ) 1 2 ] R T [ n r 2 sin 2 Θ ] , 1 q S 2 = 1 n r 1 q S 1 + [ cos Θ ( n r 2 sin 2 Θ ) 1 2 ] n r R S .
w T 2 = ( n r 2 sin 2 Θ ) 1 2 n r cos Θ w T 1 , 1 R T 2 = ( n r cos 2 Θ n r 2 sin 2 Θ ) 1 R T 1 + n r [ cos Θ ( n r 2 sin 2 Θ ) 1 2 ] R T [ n r 2 sin 2 Θ ] , w S 2 = w S 1 , 1 R S 2 = 1 n r 1 R S 1 + cos Θ ( n r 2 sin 2 Θ ) 1 2 n r R S .
Ray vector = [ w i w i / R i ] .
[ w i w i / R i ] = [ A B C D ] [ w i w i / R i ]
[ 1 d 0 1 ] ,
( n r 2 sin 2 Θ ) 1 2 n r cos Θ
cos Θ ( n r 2 sin 2 Θ ) 1 2 R T cos Θ ( n r 2 sin 2 Θ ) 1 2
( 1 n r 2 ) ( n r 2 + 1 ) 1 2 R T n r
cos Θ ( n r 2 sin 2 Θ ) 1 2 R S n r
1 n r 2 R S n r ( n r 2 + 1 ) 1 2
cos Θ ( n r 2 sin 2 Θ ) 1 2

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