Abstract

We propose the design of tilted null screens for testing off-axis segments of conic surfaces. The tilt allows us to control the size of the screen and the sensitivity of the test. For positive tilt angles the sensitivity is increased while the size of the screen is reduced in the sagittal caustic region and vice versa in the tangential caustic region. Further analysis and preliminary experimental results are presented for a fast off-axis concave parabolic mirror with an elliptical aperture. An offset distance of XC=25.4  mm yields radius of curvature at the vertex R=20.4  mm; major axis of the mirror DM=49.4  mm; and minor axis Dm=29.5  mm.

© 2006 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, 1978), pp. 283 and 323.
  2. A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nunez, " Ronchi and Hartmann tests with the same mathematical theory," Appl. Opt. 31, 2370- 2376 ( 1992).
    [CrossRef] [PubMed]
  3. A. B. Meinel and M. P. Meinel, " Optical testing of off-axis parabolic segments without auxiliary optical elements," Opt. Eng. 28, 071- 075 ( 1989).
  4. R. Díaz-Uribe, " Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics, the Large Millimeter Telecope," Appl. Opt. 39, 2790- 2804 ( 2000).
    [CrossRef]
  5. R. Díaz-Uribe and M. Campos-Garcia, " Null screen testing of fast convex aspheric surfaces," Appl. Opt. 39, 2670- 2677 ( 2000).
    [CrossRef]
  6. M. Campos-García, R. Díaz-Uribe, and F. Granados-Agustín, " Testing fast aspheric convex surfaces with a linear array of sources," Appl. Opt. 43, 6255- 6264 ( 2005)
    [CrossRef]
  7. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chaps. 9 and 10.
  8. C. Menchaca and D. Malacara, " Directional curvatures in a conic mirror," Appl. Opt. 23, 3258- 3259 ( 1984).
    [CrossRef] [PubMed]
  9. M. Avendan̄o-Alejo, M. Campos-Garcia, and R. Díaz-Uribe, " Testing a fast off-axis parabolic mirror using tilted null-screens," in Eighth International Symposium on Laser Metrology, R.Rodriguez-Vera and F.Mendoza-Santoyo, eds., Proc. SPIE 5776, 553- 560 ( 2005).
  10. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer 2000), pp. 480- 502.

2005

2000

1992

1989

A. B. Meinel and M. P. Meinel, " Optical testing of off-axis parabolic segments without auxiliary optical elements," Opt. Eng. 28, 071- 075 ( 1989).

1984

Arnold, V. I.

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer 2000), pp. 480- 502.

Avendan¯o-Alejo, M.

M. Avendan̄o-Alejo, M. Campos-Garcia, and R. Díaz-Uribe, " Testing a fast off-axis parabolic mirror using tilted null-screens," in Eighth International Symposium on Laser Metrology, R.Rodriguez-Vera and F.Mendoza-Santoyo, eds., Proc. SPIE 5776, 553- 560 ( 2005).

Campos-Garcia, M.

R. Díaz-Uribe and M. Campos-Garcia, " Null screen testing of fast convex aspheric surfaces," Appl. Opt. 39, 2670- 2677 ( 2000).
[CrossRef]

M. Avendan̄o-Alejo, M. Campos-Garcia, and R. Díaz-Uribe, " Testing a fast off-axis parabolic mirror using tilted null-screens," in Eighth International Symposium on Laser Metrology, R.Rodriguez-Vera and F.Mendoza-Santoyo, eds., Proc. SPIE 5776, 553- 560 ( 2005).

Campos-García, M.

Cardona-Nunez, O.

Cordero-Dávila, A.

Cornejo-Rodriguez, A.

Díaz-Uribe, R.

Granados-Agustín, F.

Malacara, D.

Meinel, A. B.

A. B. Meinel and M. P. Meinel, " Optical testing of off-axis parabolic segments without auxiliary optical elements," Opt. Eng. 28, 071- 075 ( 1989).

Meinel, M. P.

A. B. Meinel and M. P. Meinel, " Optical testing of off-axis parabolic segments without auxiliary optical elements," Opt. Eng. 28, 071- 075 ( 1989).

Menchaca, C.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chaps. 9 and 10.

Appl. Opt.

Opt. Eng.

A. B. Meinel and M. P. Meinel, " Optical testing of off-axis parabolic segments without auxiliary optical elements," Opt. Eng. 28, 071- 075 ( 1989).

Other

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chaps. 9 and 10.

M. Avendan̄o-Alejo, M. Campos-Garcia, and R. Díaz-Uribe, " Testing a fast off-axis parabolic mirror using tilted null-screens," in Eighth International Symposium on Laser Metrology, R.Rodriguez-Vera and F.Mendoza-Santoyo, eds., Proc. SPIE 5776, 553- 560 ( 2005).

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer 2000), pp. 480- 502.

D. Malacara, Optical Shop Testing (Wiley, 1978), pp. 283 and 323.

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Figures (13)

Fig. 1
Fig. 1

Schematic of the null screen design. Dc is the length of the mirror and L is the size of the CCD sensor.

Fig. 2
Fig. 2

(Color online) Tangential and sagittal normal caustic surfaces for the off-axis parabolic mirror under test and possible orientations of the null screen in sagittal and tangential regions.

Fig. 3
Fig. 3

Design of the tilted null screen in two regions: NRS, normal to the reflecting surface.

Fig. 4
Fig. 4

Singularities in the design of the null screens that occur when the null screen plane is placed between the sagittal and tangential normal caustic surfaces.

Fig. 5
Fig. 5

Evolution of the null screens near the sagittal normal caustic as a function of tilt angle; the position of the pinhole is fixed.

Fig. 6
Fig. 6

(Color online) Dimensions for the X and Y coordinates for the null screen placed near the sagittal normal caustic as a function of tilt angle φ.

Fig. 7
Fig. 7

Evolution of null screens near the tangential normal caustic as a function of the tilt angle; the position of the pinhole is fixed.

Fig. 8
Fig. 8

Dimensions of the X and Y coordinates for the null screen placed near the sagittal normal caustic as a function of tilt angle φ.

Fig. 9
Fig. 9

Back ray tracing for an off-axis parabolic mirror test. We are assuming only a tiny variation in the conic constant.

Fig. 10
Fig. 10

Differences between ideal back ray tracing and a deformed surface under test with a slight variation of conic constant Δk = −0.001 in both regions near either sagittal or tangential normal caustics.

Fig. 11
Fig. 11

Simulated sensitivity of the test for the null screen placed near both sagittal and tangential normal caustics as a function of the angle.

Fig. 12
Fig. 12

Simulated sensitivity of the test for the null screen placed near both sagittal and tangential normal caustics as a function of size of the X and Y coordinates.

Fig. 13
Fig. 13

Preliminary test of the off-axis parabolic mirror for two different tilt null screens.

Equations (61)

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X C = 25.4   mm
R = 20.4   mm
D M = 49.4   mm
D m = 29.5   mm
( F > 1 )
( x 0 , y 0 , b )
x 0   and   y 0
L × L
( 0 , 0 , a )
z ( x , y ) = D B + B 2 A D ,
A = c ( 1 + k cos 2 θ ) ,
B = 1 ( 1 + k sin 2 θ ) 1 / 2 c k x cos θ sin θ ,
D = c ( 1 + k sin 2 θ ) x 2 + c y 2 ,
tan θ = c x c [ 1 ( k + 1 ) c 2 x c 2 ] 1 / 2 ,
x x 0 = z ( x , y ) a b a = y y 0 .
R ^ = I ^ 2 ( I ^ N ^ ) N ^ = ( R x , R y , R z ) ,
I ^
N ^
N ^ = ( z x , z y , 1 ) ( z x 2 + z y 2 + 1 ) 1 / 2 = ( z / x , z / y , 1 ) [ ( z / x ) 2 + ( z / y ) 2 + 1 ] 1 / 2 .
A p X + B P Y + C p Z = D p
( A p , B p , C p )
D p
D p / C p
X = x + R x M a , Y = y + R y M a , Z = z + R z M a ,
M a = [ D p ( A p x + B p y + C p z ) A p R x + B p R y + C p R z ] .
A p = B p = 0
C p = 1
D p = a
D p = a
D p
f t = R t 2 = { [ 1 k c 2 ( x cos θ z sin θ + x c ) 2 + y 2 ] 1 / 2 } 3 c ,
f s = R s 2 = [ 1 k c 2 ( x cos θ z sin θ + x c ) 2 + y 2 ] 1 / 2 c .
f t   and   f s
f s 3 = f t R 2
x c = 25.4   mm
0.049 mm 1
a = 20   mm
b = 21.8   mm
L = 4.4   mm
D c = 49.4   mm
B p = 0
A p X + C p Z = D p
D p = a
A p   and   C p
φ = arctan ( C p / A p )
φ m s = 17.5 ° )
a = 205   mm
b = 223 mm
φ > 45 °
( φ = 0 ° )
ρ = 1 N i = 1 N j = 1 N ( d x i j 2 + d y i j 2 ) 1 / 2 .
Δ k = 0.001
φ = 1 0 °
φ = 0 °
Δ k
φ m = 17.5 °
φ m
φ m
φ m
20   μm
φ = 30 ° , 45 °

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