Abstract

We present the approximate polynomial expression for an ellipsoid with rotational symmetry about its major axis, which is on the yz plane and at angle θ with respect to the z axis. These expressions have many possible useful applications in optics as shown. The main optical properties of these types of inclined ellipsoidal surface will be reviewed.

© 2006 Optical Society of America

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References

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  1. D. Malacara-Hernández, "Some parameters and characteristics of an off-axis paraboloid," Opt. Eng. 30, 1277-1281 (1991).
    [CrossRef]
  2. D. Malacara-Hernández and M. Servin, "Aberrations introduced in off-axis testing of spherical surface," Opt. Eng. 35, 3260-3264 (1996).
    [CrossRef]
  3. C. Menchaca and D. Malacara-Hernández, "Directional curvature in a conic surface," Appl Opt. 23, 3258-3261 (1984).
    [CrossRef] [PubMed]
  4. M. E. Mortenson, Geometric Modeling, 2nd ed. (Wiley, 1997), pp. 152-156.
  5. P. C. Gasson, Geometry of Spatial Forms (Ellis Horwood, 1983), pp. 85-87.
  6. D. Malacara-Hernández, Optical Shop Testing (Wiley, 1992), pp. 321-342 and 367-385.
  7. D. Malacara-Hernández and Z. Malacara-Hernández, eds., Handbook of Optical Design (Marcel Dekker, 2004), pp. 270-276.

1996

D. Malacara-Hernández and M. Servin, "Aberrations introduced in off-axis testing of spherical surface," Opt. Eng. 35, 3260-3264 (1996).
[CrossRef]

1991

D. Malacara-Hernández, "Some parameters and characteristics of an off-axis paraboloid," Opt. Eng. 30, 1277-1281 (1991).
[CrossRef]

1984

C. Menchaca and D. Malacara-Hernández, "Directional curvature in a conic surface," Appl Opt. 23, 3258-3261 (1984).
[CrossRef] [PubMed]

Gasson, P. C.

P. C. Gasson, Geometry of Spatial Forms (Ellis Horwood, 1983), pp. 85-87.

Malacara-Hernández, D.

D. Malacara-Hernández and M. Servin, "Aberrations introduced in off-axis testing of spherical surface," Opt. Eng. 35, 3260-3264 (1996).
[CrossRef]

D. Malacara-Hernández, "Some parameters and characteristics of an off-axis paraboloid," Opt. Eng. 30, 1277-1281 (1991).
[CrossRef]

C. Menchaca and D. Malacara-Hernández, "Directional curvature in a conic surface," Appl Opt. 23, 3258-3261 (1984).
[CrossRef] [PubMed]

D. Malacara-Hernández, Optical Shop Testing (Wiley, 1992), pp. 321-342 and 367-385.

D. Malacara-Hernández and Z. Malacara-Hernández, eds., Handbook of Optical Design (Marcel Dekker, 2004), pp. 270-276.

Malacara-Hernández, Z.

D. Malacara-Hernández and Z. Malacara-Hernández, eds., Handbook of Optical Design (Marcel Dekker, 2004), pp. 270-276.

Menchaca, C.

C. Menchaca and D. Malacara-Hernández, "Directional curvature in a conic surface," Appl Opt. 23, 3258-3261 (1984).
[CrossRef] [PubMed]

Mortenson, M. E.

M. E. Mortenson, Geometric Modeling, 2nd ed. (Wiley, 1997), pp. 152-156.

Servin, M.

D. Malacara-Hernández and M. Servin, "Aberrations introduced in off-axis testing of spherical surface," Opt. Eng. 35, 3260-3264 (1996).
[CrossRef]

Appl Opt.

C. Menchaca and D. Malacara-Hernández, "Directional curvature in a conic surface," Appl Opt. 23, 3258-3261 (1984).
[CrossRef] [PubMed]

Opt. Eng.

D. Malacara-Hernández, "Some parameters and characteristics of an off-axis paraboloid," Opt. Eng. 30, 1277-1281 (1991).
[CrossRef]

D. Malacara-Hernández and M. Servin, "Aberrations introduced in off-axis testing of spherical surface," Opt. Eng. 35, 3260-3264 (1996).
[CrossRef]

Other

M. E. Mortenson, Geometric Modeling, 2nd ed. (Wiley, 1997), pp. 152-156.

P. C. Gasson, Geometry of Spatial Forms (Ellis Horwood, 1983), pp. 85-87.

D. Malacara-Hernández, Optical Shop Testing (Wiley, 1992), pp. 321-342 and 367-385.

D. Malacara-Hernández and Z. Malacara-Hernández, eds., Handbook of Optical Design (Marcel Dekker, 2004), pp. 270-276.

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Figures (6)

Fig. 1
Fig. 1

Some orientations for the reflecting ellipsoid: (a) with both the light source and the image displaced along the optical axis, (b) and (c) with both the light source and the image displaced along the optical axis as well as transversally to this axis, (d) with the light source and the image laterally and symmetrically displaced.

Fig. 2
Fig. 2

Tangential and sagittal evolutes for the ellipsoidal reflecting surface.

Fig. 3
Fig. 3

Ronchi test where the ruling is also used as the light source.

Fig. 4
Fig. 4

Definition of the angle of inclination θ and the angle of incident ray α.

Fig. 5
Fig. 5

Different values of the mirror diameter versus defocusing Δl.

Fig. 6
Fig. 6

Different values of the mirror diameter versus lateral displacement Δs of the image.

Tables (1)

Tables Icon

Table 1 Values of the Coefficients a i for θ = 0° and θ = 90°

Equations (62)

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x 2 b 2 + y 2 b 2 + z 2 a 2 = 1.
x 2 b 2 + A ( z z 0 ) 2 + B ( y y 0 ) 2 + C ( y y 0 ) ( z z 0 ) = 1 ,
A = ( sin 2 θ b 2 + cos 2 θ a 2 ) ,
B = ( cos 2 θ b 2 + sin 2 θ a 2 ) ,
C = 2 sin θ cos θ ( 1 b 2 1 a 2 ) .
y 0 = ( a 2 b 2 ) sin θ cos θ a 2 cos 2 θ + b 2 sin 2 θ ,
z 0 = a 2 cos 2 θ + b 2 sin 2 θ .
( a 2 b 2 ) = ( L 2 ) 2 .
z = z 0
C ( y y 0 ) ± ( C 2 4 A B ) ( y y 0 ) 2 + 4 A ( 1 x 2 / b 2 ) 2 A ,
z = ( z 0 + C y 0 2 A ) ( C 2 A ) y ± 1 A x 2 A b 2 ( y y 0 ) 2 A 2 a 2 b 2 .
H 2 = a 2 sin 2 θ + b 2 cos 2 θ ,
z = z 0 + D ( y y 0 ) a b H 2 H 2 ( y y 0 ) 2 H 2 x 2 b 2 ,
D = ( a 2 b 2 ) H 2 sin θ cos θ .
z ( x , y ) = 1 2 ( 2 z x 2 ) 0 , 0 x 2 + 1 2 ( 2 z y 2 ) 0 , 0 y 2 + 1 2 ( 3 z x 2 y ) 0 , 0 x 2 y + 1 6 ( 3 z y 3 ) 0 , 0 y 3 + 1 24 ( 4 z x 4 ) 0 , 0 x 4 + 1 6 ( 4 z x 2 y 2 ) 0 , 0 x 2 y 2 + 1 24 ( 4 z y 4 ) 0 , 0 y 4 .
z ( x , y ) = a 1 x 2 + a 2 y 2 + a 3 x 2 y + a 4 y 3 + a 5 x 4 + a 6 x 2 y 2 + a 7 y 4 ,
a 1 = a 2 b ( H 2 y 0     2 ) 1 / 2 ,
a 2 = a b 2 H 2 ( H 2 y 0     2 ) 1 / 2 [ 1 + y 0     2 ( H 2 y 0     2 ) ] ,
a 3 = a y 0 2 b ( H 2 y 0     2 ) 3 / 2 ,
a 4 = a b y 0 2 H 2 ( H 2 y 0     2 ) 3 / 2 [ 1 + y 0     2 ( H 2 y 0     2 ) ] ,
a 5 = a H 2 8 b 3 ( H 2 y 0     2 ) 3 / 2 ,
a 6 = a 4 b ( H 2 y 0     2 ) 3 / 2 ( 3 y 0     2 H 2 y 0     2 + 1 ) ,
a 7 = a b 8 H 2 ( H 2 y 0     2 ) 3 / 2 [ 5 y 0     4 ( H 2 y 0     2 ) 2 + 6 y 0     2 ( H 2 y 0     2 ) + 1 ] .
W 2 = ( a 2 b 2 ) sin θ cos θ = y 0 z 0 ,
a 1 = z 0 2 b 2 ,
a 2 = z 0     3 2 ( a 2 b 2 + W 4 ) ( 1 + W 4 a 2 b 2 ) ,
a 3 = z 0     2 W 2 2 b 4 a 2 ,
a 4 = W 2 z 0     4 2 a 2 b 2 ( a 2 b 2 + W 4 ) ( 1 + W 4 a 2 b 2 ) ,
a 5 = z 0 ( a 2 b 2 + W 4 ) 8 b 6 a 2 ,
a 6 = z 0     3 4 b 4 a 2 ( 3 W 4 a 2 b 2 + 1 ) ,
a 7 = z 0     5 8 a 2 b 2 ( a 2 b 2 + W 4 ) ( 5 W 8 a 4 b 4 + 6 W 4 a 2 b 2 + 1 ) .
a = r ,
b = r 2 Δ l 2 r Δ l 2 2 r ,
θ = Δ s Δ l .
a 1 = a 2 b 2 + θ 2 4 a ,
a 2 = a 2 b 2 + 3 θ 2 4 a ,
a 3 = θ ( a 2 b 2 ) 2 b 4 ,
a 4 = θ ( a 2 b 2 ) 2 b 4 ,
a 5 = a 8 b 4 + θ 2 ( a 2 b 2 ) 2 8 a b 6 + θ 2 16 a b 2 ,
a 6 = 3 θ ( a 2 b 2 ) 2 4 a b 6 + a 4 b 4 + 3 a θ 2 4 b 2 ,
a 7 = a 8 b 4 [ 6 ( a 2 - b 2 ) θ 2 a 2 b 2 + 1 ] .
c s = 2 a 1 ,
c t = 2 a 2 .
c s     3 = c 2 c t ,
a = [ z 0     2 + ( L 2 ) 2 sin 2 θ ] 1 / 2 ,
b = [ a 2 ( L 2 ) 2 ] 1 / 2 ,
z ref = c ref 2 ( x 2 + y 2 ) + c ref 3 8 ( x 2 + y 2 ) 2 ,
tan θ = Δ s Δ l ,
α = Δ s r .
W ( x , y ) = 2 Δ z = 2 ( z z ref ) ,
Δ z = a 8 b 4 ( b 2 a 2 b 2 ) ( x 2 + y 2 ) 2 ,
a r + L 2 4 r ,
b ( r 2 + L 2 4 ) 1 / 2 .
W ( x , y ) = L 2 16 r 5 ( x 2 + y 2 ) 2 ,
W max = L 2 16 r 5 ( D 2 ) 4 ,
c ref = a 1 + a 2
Δ z = A ( y 2 x 2 ) ,
A = ( a 2 a 1 2 ) .
W ( x , y ) = 2 Δ z = 2 ( z z ref )
W ( x , y ) = ( b 2 a 2 2 a 2 b ) ( y 2 x 2 ) ,
W ( x , y ) = ( Δ s 2 2 r 3 ) ( y 2 x 2 ) ,
W max = ( Δ s 2 2 r 3 ) ( D 2 ) 2 ,

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