Abstract

The mean intensity of an illuminated mirror with random tilt errors (jitter) is expressed as the transform of the convolution of the pupil function times the tilt-error characteristic function. The result is applied to a mirror with a centered hole.

© 1997 Optical Society of America

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References

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  1. D. A. Holmes, P. V. Avizonis, “Approximate optical system model,” Appl. Opt. 15, 1075–1082 (1976).
    [CrossRef] [PubMed]
  2. R Butts, “Effects of piston and tilt errors on the performance of multiple mirror telescopes,” in Wavefront Distortions in Power Optics, C. A. Klein, ed., Proc. SPIE293, 85–89 (1981).
  3. J. B. Shellan, “Phased-array performance degradation due to mirror misfigures, piston errors, jitter, and polarization errors,” J. Opt. Soc. Am. A 2, 555–567 (1985).
    [CrossRef]
  4. C. B. Hogge, J. F. Schultz, D. B. Mason, W. E. Thompson, “Physical optics of multiaperture systems,” Appl. Opt. 27, 5127–5134 (1988).
    [CrossRef] [PubMed]
  5. P. V. Avizonis, J. S. Fender, R. R. Butts, “Effects of piston and tilt bias errors in multiple wavefront interference,” Opt. Eng. 28, 1260–1266 (1989).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1975).

1989 (1)

P. V. Avizonis, J. S. Fender, R. R. Butts, “Effects of piston and tilt bias errors in multiple wavefront interference,” Opt. Eng. 28, 1260–1266 (1989).
[CrossRef]

1988 (1)

1985 (1)

1976 (1)

Avizonis, P. V.

P. V. Avizonis, J. S. Fender, R. R. Butts, “Effects of piston and tilt bias errors in multiple wavefront interference,” Opt. Eng. 28, 1260–1266 (1989).
[CrossRef]

D. A. Holmes, P. V. Avizonis, “Approximate optical system model,” Appl. Opt. 15, 1075–1082 (1976).
[CrossRef] [PubMed]

Butts, R

R Butts, “Effects of piston and tilt errors on the performance of multiple mirror telescopes,” in Wavefront Distortions in Power Optics, C. A. Klein, ed., Proc. SPIE293, 85–89 (1981).

Butts, R. R.

P. V. Avizonis, J. S. Fender, R. R. Butts, “Effects of piston and tilt bias errors in multiple wavefront interference,” Opt. Eng. 28, 1260–1266 (1989).
[CrossRef]

Fender, J. S.

P. V. Avizonis, J. S. Fender, R. R. Butts, “Effects of piston and tilt bias errors in multiple wavefront interference,” Opt. Eng. 28, 1260–1266 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1975).

Hogge, C. B.

Holmes, D. A.

Mason, D. B.

Schultz, J. F.

Shellan, J. B.

Thompson, W. E.

Appl. Opt. (2)

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

Opt. Eng. (1)

P. V. Avizonis, J. S. Fender, R. R. Butts, “Effects of piston and tilt bias errors in multiple wavefront interference,” Opt. Eng. 28, 1260–1266 (1989).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1975).

R Butts, “Effects of piston and tilt errors on the performance of multiple mirror telescopes,” in Wavefront Distortions in Power Optics, C. A. Klein, ed., Proc. SPIE293, 85–89 (1981).

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Figures (7)

Fig. 1
Fig. 1

Mean intensity versus Θ = kW sin θ for (a) Gaussian and (b) uniform jitter probability densities with a standard deviation σ; α = kW = 106 for β = kWσ = 0, 1, 3; and a square mirror with a side W.

Fig. 2
Fig. 2

Mean intensity versus Θ = kW sin θ for Gaussian and uniform jitter probability densities with a standard deviation σ, α = kW = 106 for β = kWσ = 10, and a square mirror with a side W.

Fig. 3
Fig. 3

Strehl ratio versus β = kWσ for a square mirror with a side W.

Fig. 4
Fig. 4

Mean intensity versus Θ = kD sin θ for a uniform jitter probability density for a circular mirror, diameter D, that has a concentric hole with diameter D/10. The assumed value of kD is 106, and the standard deviations of the jitter are σ = 0, 0.25, 0.50, 1.00, and 2.00.

Fig. 5
Fig. 5

Overlap areas of two circles with concentric holes.

Fig. 6
Fig. 6

Overlap areas of unequal circles, diameters D and d, with centers 2t apart for (1) 2t ≥ (D + d)/2, (2) (D + d)/2 ≥ 2t > (Dd)/2, and (3) (Dd)/2 ≥ 2t ≥ 0.

Fig. 7
Fig. 7

Geometry of the overlap area for the case (D + d)/2 ≥ 2t > (Dd)/2.

Equations (42)

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I ( t x , t y ) = I ( t x , t y ξ , ζ ) f ( ξ , ζ ) d ξ d ζ ,
U ( x , y ) = ( λ z ) - 1 U ( x 1 , y 1 ) exp [ - j ( k / z ) × ( x 1 x + y 1 y ) ] d x 1 d y 1 ,
U ( x , y ξ , ζ ) = ( λ z ) - 1 { U ( x 1 , y 1 ) exp [ j k ( ξ x 1 + ζ y 1 ) ] } × exp [ - ( j k / z ) ( x 1 x + y 1 y ) ] d x 1 d y 1
U ( t x , t y ξ , ζ ) = ( λ z ) - 1 U ( x 1 , y 1 ) × exp { - j k [ ( t x - ξ ) x 1 + ( t y - ζ ) y 1 ] } d x 1 d y 1 = U ( t x - ξ , t y - ζ ) .
I ( t x , t y ξ , ζ ) = I ( t x - ξ , t y - ζ ) ,
I ( t x , t y ) = - - I ( t x - ξ , t y - , ζ ) f ( ξ , ζ ) d ξ d ζ = F - 1 [ F { I ( t x , t y ) } F { f ( t x , t y ) } ] ,
I ( t x , t y ) = F - 1 [ F { I ( t x , t y ) } Φ ( ω x , ω y ) ] .
U ( t x , t y ) = ( P / A λ 2 z 2 ) 1 / 2 F { P ( ω ¯ x , ω ¯ y ) } ,
I ( t x , t y ) = U ( t x , t y ) 2 = ( P / A λ 2 z 2 ) F { P ( ω ¯ x , ω ¯ y ) } F * { P ( ω ¯ x , ω ¯ y ) } .
F { I ( t x , t y ) } = ( P / A λ 2 z 2 ) P ( ω ¯ x , ω ¯ y ) P ( ω ¯ x , ω ¯ y ) ,
I ( t x , t y ) = ( P / A λ 2 z 2 ) F - 1 { [ P ( ω ¯ x , ω ¯ y ) P ( ω ¯ x , ω ¯ y ) ] Φ ( ω x , ω y ) } .
Φ [ k ( x a - x b ) ] = exp [ - ( x a - x b ) 2 k 2 σ 2 / 2 ] ,
Φ [ k ( x a - x b ) ] = sin [ ν k ( x a - x b ) ] / [ ν k ( x a - x b ) ] .
I ( t x , t y ) = F - 1 { Φ ( ω x , ω y ) } = f ( t x , t y ) .
P ( ω ¯ x , ω ¯ y ) P ( ω ¯ x , ω ¯ y ) = W 2 [ 1 - ( k W ) - 1 ω x ] × [ 1 - ( k W ) - 1 ω y ]
I ( t x , t y ) = 4 ( P / λ 2 z 2 ) 0 k W 0 k W ( 1 - ω x / k W ) × ( 1 - ω y / k W ) cos ( ω x t x ) cos ( ω y t y ) × Φ ( ω x , ω y ) d ω x d ω x .
I ( t x , t y ) = 0 1 0 1 ( 1 - ω x ) ( 1 - ω y ) cos ( α ω x x ) cos ( α ω y y ) Φ ( 1 ) ( β ω x , β ω y ) d ω y d ω x [ 0 1 ( 1 - ω ) Φ ( 1 ) ( 0 ) d ω ] 2 ,
{ SR } = I ϕ = θ = 0
{ SR } = [ 2 0 1 ( 1 - ω ) Φ ( 1 ) ( β ω ) d ω ] 2 .
[ P P ] = ( D 2 / 2 ) { cos - 1 ( ρ / k D ) - ( ρ / k D ) × [ 1 - ( ρ / k D ) 2 ] 1 / 2 } ,
I ( t r ) 0 k D { cos - 1 ( ω ρ / k D ) - ( ω ρ / k D ) × [ 1 - ( ω ρ / k D ) 2 ] 1 / 2 } Φ ( ω ρ ) J 0 ( ω ρ t r ) ω ρ d ω ρ .
I ( Θ ) = 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] Φ ( 1 ) ( β ρ ) J 0 ( α ρ Θ ) ρ d ρ 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] ρ d ρ ,
{ SR } = 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] Φ ( 1 ) ( β ρ ) ρ d ρ 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] ρ d ρ ,
C C = ( D / 2 ) 2 { cos - 1 ( ρ / k D ) - ( ρ / k D ) × [ 1 - ( ρ / k D ) 2 ] 1 / 2 }             for k D ρ 0 ,
c c = ( q D / 2 ) 2 { cos - 1 ( ρ / q k D ) - ( ρ / q k D ) × [ 1 - ( ρ / q k D ) 2 ] 1 / 2 }             for q k D ρ 0
- 2 C c = - 2 ( D / 2 ) 2 ( cos - 1 ( w + p / 4 w ) + q 2 cos - 1 [ ( w / q ) - p / 4 q w ] - [ 1 - ( w + p / 4 w ) 2 ] 1 / 2 ( w + p / 4 w ) - q 2 { 1 - [ ( w / q ) - p / 4 q w ] 2 } 1 / 2 × [ ( w / q ) - p / 4 q w ] ) for ( 1 + q ) k D / 2 ρ ( 1 - q ) k D / 2 ,
- 2 C c = - 2 π q 2 ( D / 2 ) 2             for ( 1 - q k ) D / 2 ρ 0.
N I ( Θ ) = 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] [ Φ ( 1 ) ( β ρ ) J 0 ( α ρ Θ ) + q 2 Φ ( 1 ) ( q β ρ ) J 0 ( q α ρ Θ ) ] ρ d ρ - 2 ( 1 - q ) / 2 ( 1 + q ) / 2 ( cos - 1 ( ρ + p / 4 ρ ) + q 2 cos - 1 [ ( ρ / q ) - p / 4 q ρ ] - [ 1 - ( ρ + p / 4 ρ ) 2 ] 1 / 2 ( ρ + p / 4 ρ ) - q 2 { 1 - [ ( ρ / q ) - p / 4 q ρ ] 2 } 1 / 2 × [ ( ρ / q ) - p / 4 q ρ ] ) Φ ( 1 ) ( β ρ ) J 0 ( α ρ Θ ) ρ d ρ - 2 π q 2 0 ( 1 - q ) / 2 Φ ( 1 ) ( β ρ ) J 0 ( α ρ Θ ) ρ d ρ ,
N = 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] [ Φ ( 1 ) ( β ρ ) + q 2 Φ ( 1 ) ( q β ρ ) ρ d ρ - 2 ( 1 - q ) / 2 ( 1 + q ) / 2 ( cos - 1 ( ρ + p / 4 ρ ) + q 2 cos - 1 [ ( ρ / q ) - p / 4 q ρ ] - [ 1 - ( ρ + p / 4 ρ ) 2 ] 1 / 2 ( ρ + p / 4 ρ ) - q 2 { 1 - [ ( ρ / q ) - p / 4 q ρ ] 2 } 1 / 2 [ ( ρ / q ) - p / 4 q ρ ] ) Φ ( 1 ) ( β ρ ) ρ d ρ - 2 π q 2 0 ( 1 - q ) / 2 Φ ( 1 ) ( β ρ ) ρ d ρ ,
{ SR } = N 0 I ( 0 ) ,
I ( 0 ) = 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] [ Φ ( 1 ) ( β ρ ) + q 2 Φ ( 1 ) ( q β ρ ) ] ρ d ρ - 2 ( 1 - q ) / 2 ( 1 + q ) / 2 × ( cos - 1 ( ρ + p / 4 ρ ) + q 2 cos - 1 [ ( ρ / q ) - p / 4 q ρ ] - [ 1 - ( ρ + p / 4 ρ ) 2 ] 1 / 2 ( ρ + p / 4 ρ ) - q 2 { 1 - [ ( ρ / q ) - p / 4 q ρ ] 2 } 1 / 2 [ ( ρ / q ) - p / 4 q ρ ] ) × Φ ( 1 ) ( β ρ ) ρ d ρ - 2 π q 2 0 ( 1 - q ) / 2 Φ ( 1 ) ( β ρ ) ρ d ρ ,
N 0 = 0 1 [ cos - 1 ( ρ ) - ρ ( 1 - ρ 2 ) 1 / 2 ] ( 1 + q 2 ) ρ d ρ - 2 ( 1 - q ) / 2 ( 1 + q ) / 2 ( cos - 1 ( ρ + p / 4 ρ ) + q 2 cos - 1 [ ( ρ / q ) - p / 4 q ρ ] - [ 1 - ( ρ + p / 4 ρ ) 2 ] 1 / 2 ( ρ + p / 4 ρ ) - q 2 { 1 - [ ( ρ / q ) - p / 4 q ρ ] 2 } 1 / 2 [ ( ρ / q ) - p / 4 q ρ ] ) × ρ d ρ - 2 π q 2 0 ( 1 - q ) / 2 ρ d ρ ,
C C - 2 ( C c - c c ) - c c = C C + c c - 2 C C .
C c = ( 1 / 4 ) ( D 2 a d + d 2 a d ) - ( 1 / 2 ) ( D / 2 ) 2 sin ( 2 a D ) - ( 1 / 2 ) ( d / 2 ) 2 sin ( 2 a d ) .
D cos ( a D ) + d cos ( a d ) = 4 t ,
d 2 [ 1 - cos 2 ( a d ) ] = D 2 [ 1 - cos 2 ( a D ) ] ,
cos ( a D ) = ( 2 t / D ) - ( d 2 - D 2 ) / 8 t D ,
cos ( a d ) = ( 2 t / d ) - ( D 2 - d 2 ) / 8 t d ,
C c = ( D / 2 ) 2 cos - 1 [ ( 2 t / D ) - ( d 2 - D 2 ) / 8 t D ] + ( d / 2 ) 2 cos - 1 [ ( 2 t / d ) - ( D 2 - d 2 ) / 8 t d ] - ( D / 2 ) 2 { 1 - [ ( 2 t / D ) - ( d 2 - D 2 ) / 8 t D ] 2 } 1 / 2 × [ ( 2 t / D ) - ( d 2 - D 2 ) / 8 t D ] - ( d / 2 ) 2 × { 1 - [ ( 2 t / d ) - ( D 2 - d 2 ) / 8 t d ] 2 } 1 / 2 [ ( 2 t / d ) - ( D 2 - d 2 ) / 8 t d ] .
C c = ( D / 2 ) 2 ( cos - 1 ( w + p / 4 w ) + q 2 cos - 1 [ ( w / q ) - p / 4 q w ] - [ 1 - ( w + p / 4 w ) 2 ] 1 / 2 ( w + p / 4 w ) - q 2 { 1 - [ ( w / q ) - p / 4 q w ] 2 } 1 / 2 [ ( w / q ) - p / 4 q w ] ) for ( 1 + q ) / 2 w ( 1 - q ) / 2.
C C = ( D / 2 ) 2 [ cos - 1 w - w ( 1 - w 2 ) 1 / 2 ]
C c = π ( q D ) 2 / 4.

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