Abstract

A paraxial model describing the astigmatism generated by a plane-parallel plate is derived. This model fits the framework of the 4×4 matrix formalism that Arsenault used to describe cylindrical lenses. The framework including this new model is used to build a compact system description of a plane-parallel plate combined with a cylindrical lens, from which several imaging properties are derived. Calculation results are compared with ray-trace simulation results and measurements. Both the ray-trace and the experimental results are in excellent agreement with the calculated results.

© 2007 Optical Society of America

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References

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  1. J. Braat, "Analytical expressions for the aberration coefficients of a tilted plane parallel plate," Appl. Opt. 36, 8459-8466 (1997).
    [CrossRef]
  2. H. H. Arsenault, "The rotation of light fans by cylindrical lenses," Optics Commun. 31, 275-278 (1979).
    [CrossRef]
  3. H. H. Arsenault, "A matrix representation for non-symmetrical optical systems," J. Opt. 11, 87-91 (1980).
    [CrossRef]
  4. J. Schleipen, B. Hendriks, and S. Stallinga, "Optical heads," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 1666-1693.
  5. Eric W. Weisstein, "Torus," from MathWorld-A Wolfram Web Resource, http://mathworld.wolfram.com/Torus.html.
  6. The ray-trace program used zemax, www.zemax.com.

1997

1980

H. H. Arsenault, "A matrix representation for non-symmetrical optical systems," J. Opt. 11, 87-91 (1980).
[CrossRef]

1979

H. H. Arsenault, "The rotation of light fans by cylindrical lenses," Optics Commun. 31, 275-278 (1979).
[CrossRef]

Arsenault, H. H.

H. H. Arsenault, "A matrix representation for non-symmetrical optical systems," J. Opt. 11, 87-91 (1980).
[CrossRef]

H. H. Arsenault, "The rotation of light fans by cylindrical lenses," Optics Commun. 31, 275-278 (1979).
[CrossRef]

Braat, J.

Hendriks, B.

J. Schleipen, B. Hendriks, and S. Stallinga, "Optical heads," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 1666-1693.

Schleipen, J.

J. Schleipen, B. Hendriks, and S. Stallinga, "Optical heads," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 1666-1693.

Stallinga, S.

J. Schleipen, B. Hendriks, and S. Stallinga, "Optical heads," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 1666-1693.

Weisstein, Eric W.

Eric W. Weisstein, "Torus," from MathWorld-A Wolfram Web Resource, http://mathworld.wolfram.com/Torus.html.

Appl. Opt.

J. Opt.

H. H. Arsenault, "A matrix representation for non-symmetrical optical systems," J. Opt. 11, 87-91 (1980).
[CrossRef]

Optics Commun.

H. H. Arsenault, "The rotation of light fans by cylindrical lenses," Optics Commun. 31, 275-278 (1979).
[CrossRef]

Other

J. Schleipen, B. Hendriks, and S. Stallinga, "Optical heads," in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 1666-1693.

Eric W. Weisstein, "Torus," from MathWorld-A Wolfram Web Resource, http://mathworld.wolfram.com/Torus.html.

The ray-trace program used zemax, www.zemax.com.

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Figures (7)

Fig. 1
Fig. 1

Refraction of the chief ray and an arbitrary ray by a surface drawn in the meridional plane. The difference between the propagation vectors of both rays equals δ k .

Fig. 2
Fig. 2

Translation of the chief ray and an arbitrary ray over a distance d drawn in the meridional plane.

Fig. 3
Fig. 3

Combination of a PPP with an anamorphic lens consisting of a cylindrical and a spherical surface. The orientation of the cylindrical axis is at an arbitrary angle γ with the x axis.

Fig. 4
Fig. 4

Ray fans of a converging astigmatic beam with its first focal line in the x direction and hence the second focal line in the y direction. The first and second focal lines are indicated by the near and far gray rectangles, respectively. (a) A ray fan lying in the xz plane converges to a single focus at the second focal line. The ray fan shown in (c) is lying in the yz-plane and its rays come to focus at the first focal line. For a ray fan outside the xz and yz planes, as in (b), the rays do not come to a focus. This fan however rotates such that all its rays pass through both focal lines. Arsenault has described this rotation of ray fans earlier (Ref. [2]).

Fig. 5
Fig. 5

Comparison between calculated (a) and simulated (b) results for the system shown in Fig. 3. The thin circles have the same area as the calculated spot of least confusion.

Fig. 6
Fig. 6

Photo and outline of the experimental setup.

Fig. 7
Fig. 7

Comparison between measured and calculated results.

Equations (143)

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4 × 4
4 × 4
4 × 4
| k | = 1
n k = [ μ x , μ y , ( n 2 μ 2 ) 1 / 2 ] ,
μ = ( μ x 2 + μ y 2 ) 1 / 2
k + δ k
| k + δ k | = 1
| δ k | 1
n ( k + δ k ) = ( μ x + δ μ x , μ y + δ μ y , ( n 2 ( μ + δ μ ) 2 ) ) 1 / 2 ,
n δ k = ( δ μ x , δ μ y , μ · δ μ n 2 μ 2 ) .
z = f ( x , y )
z 1 2 i j p i j r i r j = 1 2 p x x x 2 + ( p x y + p y x ) x y + p y y y 2 .
r = ( x , y )
a = ( a x , a y , [ 1 a x 2 a y 2 ) 1 / 2 ]
a x = z x = p x x x + p x y y ,
a y = z y = p y x x + p y y y .
p x y
p y x
| a x | , | a y | 1
a ( a x , a y , 1 )
n ( k + δ k ) × a = n ( k + δ k ) × a .
μ x = μ x
μ y = μ y
μ = μ
δ μ x n 2 μ 2 a x = δ μ x ] n 2 μ 2 a x ,
δ μ y n 2 μ 2 a y = δ μ y n 2 μ 2 a y .
r = r
δ μ = δ μ ( n 2 μ 2 n 2 μ 2 ) × [ p x x p y x p x y p y y ] r δ μ + P r ,
[ r δ μ ] = [ 1 P 0 1 ] [ r δ μ ] = [ L ] [ r δ μ ] ,
2 × 2
2 × 2
z = ± ( R ± r ) ± ( R ± r 2 y 2 ) 2 x 2 .
z = 0
x , y = 0
z x 2 2 ( R + r ) + y 2 2 r + O ( x , y ) 3 R r .
1 / R x p x x = 1 / ( R + r )
1 / R y p y y = 1 / r
p x y = p y x = 0
( μ 1 )
P t o r o i d a l = ( n n ) [ 1 / R x 0 0 1 / R y ] ,
R x R y
R x = R y R sph
P s p h e r i c a l = n n R sph [ 1 0 0 1 ] K [ 1 0 0 1 ] ,
m - 1
R
R y R cyl
P cylindrical = n n R cyl [ 0 0 0 1 ] = K [ 0 0 0 1 ] .
K = 1 / f
r r = d ( [ k + δ k ] [ k + δ k ] z [ k ] [ k ] z ) = d ( μ + δ μ n 2 ( μ + δ μ ) 2 μ n 2 μ 2 ) .
δ μ
r r = d ( δ μ n 2 μ 2 + ( μ · δ μ ) μ ( n 2 μ 2 ) 3 / 2 ) = d ( n 2 μ 2 ) 3 / 2 × [ n 2 μ y 2 μ x μ y μ x μ y n 2 μ x 2 ] [ δ μ x δ μ y ] D δ μ ,
[ r δ μ ] = [ 1 0 D 1 ] [ r δ μ ] = [ T ] [ r δ μ ] .
D = d n ( 1 0 0 1 ) .
μ = ( sin   β , 0 )
n δ k = ( δ μ x , δ μ y , μ · δ μ 1 μ 2 ) = ( δ μ x , δ μ y , μ x δ μ x 1 μ x 2 ) .
R = [ cos β 0 sin β 0 1 0 sin β 0 cos β ] = [ 1 μ x 2 0 μ x 0 1 0 μ x 0 1 μ x 2 ] ,
R n δ k = ( δ μ x 1 μ x 2 , δ μ y , 0 ) .
( ξ , y , ζ )
[ ( r r ) ξ ( r r ) ζ 0 ] = R [ D 11 D 21 0 D 12 D 22 0 0 0 0 ] R 1 [ δ μ ξ δ μ ζ 0 ] ,
D = d ( n 2 μ x 2 ) 3 / 2 [ n 2 ( 1 μ x 2 ) 0 0 n 2 μ x 2 ] = d ( n 2 sin 2 β ) 3 / 2 [ n 2 ( 1 sin 2 β ) 0 0 n 2 sin 2 β ] .
D = d n 2 sin 2 β [ 1 0 0 1 ] d ( n 2 1 ) sin 2 β ( n 2 sin 2 β ) 3 / 2 [ 1 0 0 0 ] .
A D = ( n 2 1 ) sin 2 β ( n 2 sin 2 β ) 3 / 2 d = 2 W 22 N A 2 λ ,
W 22
D p p p A D [ 1 0 0 0 ] .
n 2
R cyl
R s p h
n 3
s 0
s 1
[ r δ μ ] = [ T ( s 1 ) ] [ L s p h ] [ T ( t 2 n 3 ) ] [ R ( γ ) ] [ L c y l ] [ R ( γ ) ] - 1 × [ T ( s 0 ) ] [ T p p p ] [ r δ μ ] M [ r δ μ ] [ I K J L ] [ r δ μ ] .
L sph
L cyl
4 × 4
r = I · r + J · δ μ
r
δ μ
J = 0
J = 0
J · δ μ = 0
| J | = 0
| J | = 0
s 1
0 = a s 1 2 + b s 1 + c ,
s 1 = b 2 a ( b 2 a ) 2 c a ,
A D syst = 2 ( b 2 a ) 2 c a .
r = I · r + J · δ μ
( ρ , ϕ )
( δ μ x , δ μ y )
( x y ) = I · r + J ( ρ   cos   ϕ ρ   sin   ϕ ) = I · r + ρ ( J 11   cos   ϕ + J 12   sin   ϕ J 21   cos   ϕ + J 22   sin   ϕ ) .
I · r
( x y ) = I · r + ( a   cos   φ   cos ( t Δ ) b   sin   φ   sin ( t Δ ) a   sin   φ  cos ( t Δ ) + b  cos   φ   sin ( t Δ ) ) ,
{ ρ · J 11 = a   cos   φ · sin   Δ b   sin   φ · cos   Δ ρ · J 12 = a   cos   φ · cos   Δ + b   sin   φ · sin   Δ ρ · J 21 = a   sin   φ · sin   Δ + b   cos   φ · cos   Δ ρ · J 22 = a   sin   φ · cos   Δ b   cos   φ · sin   Δ .
b = 0
tan φ = J 21 + J 22 J 11 + J 12 .
x
a = b
( a b ) 2
( a b ) 2 = ( J 12 J 21 ) 2 + ( J 11 + J 22 ) 2 .
s 1
0 = a s 1 2 + b s 1 + c ,
s 1 = b 2 a .
a = b
J 11 = J 22
J 12 = J 22
Φ = 2 a b = 2 ρ J 12 J 21 J 11 J 22 .
n 1 = 1.0
n 2 = 1.520241
n 3 = 1.579827
β = 45 °
t 1 = 4.0   mm
t 2 = 0.9   mm
R cyl = 12.0 mm
R sph = 5.8   mm
s 0 = 3.3   mm
NA = 0.0874
γ = 45 °
s 1 = 4.13693
5.76945   mm
16.5 °
1 9 5 . 3 μ m
1 9 4 . 1 μ m
f 1 = 120   mm
f 2 = 140   mm
f cyl = 980   mm
γ 1 st
0.02   mm
γ 1 st
γ 1 st
f cyl
γ 1st
4 × 4
2 × 2
J = { D ( s 1 ) · P s p h + 1 } · { D ( s 0 + t 2 n 3 A D p p p ) + D ( t 2 n 3 ) · R ( γ ) · P c y l · R - 1 ( γ ) · D ( s 0 A D p p p ) } + D ( s 1 ) { R ( γ ) P c y l R - 1 ( γ ) D ( s 0 A D p p p ) + 1 } . 
D ( s 0 ) + D p p p
D ( s 0 A D p p p )
{ J 11 = s 1 + ( 1 s 1 K s p h ) ( s 0 A D p p p + t 2 / n 3 ) [ ( 1 s 1 K s p h ) t 2 / n 3 + s 1 ] ( s 0 A D ) K c y l sin 2 γ J 12 = [ s 1 ( 1 K s p h t 2 / n 3 ) + t 2 / n 3 ] s 0 K c y l   sin   γ   cos   γ J 21 = [ s 1 ( 1 K s p h t 2 / n 3 ) + t 2 / n 3 ] ( s 0 A D p p p ) K c y l   sin   γ   cos   γ J 22 = s 1 + ( 1 s 1 K s p h ) ( s 0 + t 2 / n 3 ) [ ( 1 s 1 K s p h ) t 2 / n 3 + s 1 ] s 0 K c y l cos 2 γ .
| J | = ( J 11 J 22 ) ( J 12 J 21 )
a = 1 + K s p h 2 ( s 0 + t 2 / n 3 ) ( s 0 + t 2 / n 3 A D p p p ) K s p h [ 2 ( s 0 + t 2 / n 3 ) A D p p p ] K s p h K c y l ( K s p h t 2 / n 3 1 ) ( s 0 ( s 0 + t 2 / n 3 A D p p p ) A D ( t 2 / n 3 ) sin 2 γ ) + K c y l ( K s p h t 2 / n 3 1 ) ( s 0 A D p p p sin 2 γ ) ,
b = 2 ( s 0 + t 2 / n 3 ) A D p p p 2 K s p h ( s 0 + t 2 / n 3 ) × ( s 0 + t 2 / n 3 A D p p p ) + K c y l ( 2 K s p h t 2 / n 3 1 ) × [ s 0 ( s 0 + t 2 / n 3 A D p p p ) A D p p p ( t 2 / n 3 ) sin 2 γ ] K c y l ( t 2 / n 3 ) ( s 0 A D p p p sin 2 γ ) ,
c = ( s 0 + t 2 / n 3 ) ( s 0 + t 2 / n 3 A D p p p ) K c y l ( t 2 / n 3 ) [ s 0 ( s 0 + t 2 / n 3 A D p p p ) A D p p p ( t 2 / n 3 ) sin 2 γ ] ,
δ k

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