Abstract

A general diffraction analysis is presented to determine the irradiance distributions produced by the interference of diffracted wavefronts from multiple pupil configurations. Specifically, interference phenomena in the geometrical shadows of the Fresnel region close to the diffracting apertures are considered. The effects of multiple pupil geometry variations and preliminary impact of phase errors are considered. Finally, physical insights are offered for the phenomena of diffractive interference in the geometric shadow.

© 1989 Optical Society of America

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References

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  1. J. E. Harvey, J. L. Forgham, “The Spot of Arago: New Relevance for an Old Phenomena,” Am. J. Phys. 52, 243 (1984) and references therein.
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 370.
  3. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions, ABS Math Series55 (1964).
  4. Ref. 2, p. 440.
  5. J. S. Fender, Ed., “Synthetic Aperture Systems,” Proc. Soc. Photo-Opt. Instrum. Eng.440 (1983), volume dedicated to Synthetic Aperture Systems and references therein.
  6. R. R. Butts, “The Effects of Piston and Tilt Errors on the Performance of a Multiple Aperture Telescope,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 85 (1981).
  7. P. V. Avizonis, J. S. Fender, “Impact of Tilt Errors in Phasing Algorithms,” Opt. Eng., submitted.
  8. Ref. 2, p. 435.
  9. E. Lommel, “Die Beugungserscheinungen geradlinig begrenzter Schirme,” Abh. Bayer. Akad. Wiss. Math. Naturwiss. KI. 15, Abth. 2, 233 (1885).
  10. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922), pp. 537–550.

1984 (1)

J. E. Harvey, J. L. Forgham, “The Spot of Arago: New Relevance for an Old Phenomena,” Am. J. Phys. 52, 243 (1984) and references therein.
[CrossRef]

1981 (1)

R. R. Butts, “The Effects of Piston and Tilt Errors on the Performance of a Multiple Aperture Telescope,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 85 (1981).

1885 (1)

E. Lommel, “Die Beugungserscheinungen geradlinig begrenzter Schirme,” Abh. Bayer. Akad. Wiss. Math. Naturwiss. KI. 15, Abth. 2, 233 (1885).

Avizonis, P. V.

P. V. Avizonis, J. S. Fender, “Impact of Tilt Errors in Phasing Algorithms,” Opt. Eng., submitted.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 370.

Butts, R. R.

R. R. Butts, “The Effects of Piston and Tilt Errors on the Performance of a Multiple Aperture Telescope,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 85 (1981).

Fender, J. S.

P. V. Avizonis, J. S. Fender, “Impact of Tilt Errors in Phasing Algorithms,” Opt. Eng., submitted.

Forgham, J. L.

J. E. Harvey, J. L. Forgham, “The Spot of Arago: New Relevance for an Old Phenomena,” Am. J. Phys. 52, 243 (1984) and references therein.
[CrossRef]

Harvey, J. E.

J. E. Harvey, J. L. Forgham, “The Spot of Arago: New Relevance for an Old Phenomena,” Am. J. Phys. 52, 243 (1984) and references therein.
[CrossRef]

Lommel, E.

E. Lommel, “Die Beugungserscheinungen geradlinig begrenzter Schirme,” Abh. Bayer. Akad. Wiss. Math. Naturwiss. KI. 15, Abth. 2, 233 (1885).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922), pp. 537–550.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 370.

Abh. Bayer. Akad. Wiss. Math. Naturwiss. KI. (1)

E. Lommel, “Die Beugungserscheinungen geradlinig begrenzter Schirme,” Abh. Bayer. Akad. Wiss. Math. Naturwiss. KI. 15, Abth. 2, 233 (1885).

Am. J. Phys. (1)

J. E. Harvey, J. L. Forgham, “The Spot of Arago: New Relevance for an Old Phenomena,” Am. J. Phys. 52, 243 (1984) and references therein.
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. R. Butts, “The Effects of Piston and Tilt Errors on the Performance of a Multiple Aperture Telescope,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 85 (1981).

Other (7)

P. V. Avizonis, J. S. Fender, “Impact of Tilt Errors in Phasing Algorithms,” Opt. Eng., submitted.

Ref. 2, p. 435.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922), pp. 537–550.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 370.

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions, ABS Math Series55 (1964).

Ref. 2, p. 440.

J. S. Fender, Ed., “Synthetic Aperture Systems,” Proc. Soc. Photo-Opt. Instrum. Eng.440 (1983), volume dedicated to Synthetic Aperture Systems and references therein.

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Figures (10)

Fig. 1
Fig. 1

Physical construct of diffractive interference in the geometric shadow (Fresnel region) as produced by two transmitting apertures A and B in the X1 plane at Z = 0, Z being the propagation axis. Transverse diffractive waves and their interference in the geometric shadow are indicated by solid lines at locations Z2 and Z3.

Fig. 2
Fig. 2

Six-pupil Fresnel region interference for N = 21.4 and resultant diffraction field, especially to be noted is the diffractive interference in the geometric shadow between pairs of apertures. Irradiance produced by the interference at the central geometric axis of the dark region was too low to be recorded for the dynamic range of the film.

Fig. 3
Fig. 3

Geometry used to define the parameters.

Fig. 4
Fig. 4

Fresnel interference on-axis (χ = 0) between two wavefronts in the geometric shadow as a function of Fresnel number N which is defined by Eq. (4). The calculation is normalized to the peak intensity in the focal plane, N = 0, and the pupil separation factor s was 1.2.

Fig. 5
Fig. 5

Transverse diffractive interference fringes in the geometric shadow due to two wavefronts as a function of normalized, transverse position χ defined by Eq. (4): (a) Pattern at Fresnel number N = 33.3 representing one of the near-field on-axis peaks of Fig. 2. (b) Pattern at Fresnel number N = 36.3 representing one of the near-field on-axis valleys of Fig. 2.

Fig. 6
Fig. 6

Transverse distribution of diffractive interference fringes in the geometric shadow due to two wavefronts for the parameters of Fig. 5, except with phase errors of Δϕ = π/10 and π/4.

Fig. 7
Fig. 7

Geometry for an array of circular aperture equally separated about the Z axis.

Fig. 8
Fig. 8

On-axis or central intensity for L = 4, n = 1 of radiators in circular symmetry with λf/a2 ~ 1.

Fig. 9
Fig. 9

(a) Transverse intensity distribution for N = 23.75 corresponding to the peak for the on-axis intensity of a circularly symmetric set of four pupils. (b) Transverse intensity distribution for N = 25.1 corresponding to a minimum on-axis intensity for four circularly symmetric radiating pupils. (c) Transverse intensity distribution for N = 30.8 corresponding to a maximum on-axis intensity for four circularly symmetric radiating pupils.

Fig. 10
Fig. 10

Near-field interference pattern contours for the case of N = 23.75, i.e., for which Fig. 9(a) is the transverse cut.

Equations (45)

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U n ( X , Z ) = ( 1 / i λ Z ) X n - a / 2 X n + a / 2 U n ( X n - X n ) × exp [ - ( i k / 2 Z ) ( X - X 1 ) 2 ] d X 1 .
U n ( X 1 - X n ) = A 0 exp ( i k X 1 2 / 2 f ) ,
U n ( X , Z ) = ( a A 0 / 2 i λ Z ) exp ( - i k X 2 / 2 Z ) × - 1 1 exp [ ( - i k / 2 Z ) ( 1 - Z / f ) ( X n + a ξ / 2 ) 2 ] × exp [ ( i k X / Z ) ( X n + a ξ / 2 ) ] d ξ .
U ( X , Z ) = n U n ( X , Z ) .
X = ( 1 - Z / f ) a χ , X n = ± ( 2 n + 1 ) a s / 2 , N = ( k / 2 Z ) ( 1 - Z / f ) a 2 .
U ( χ , N ) = ( a A 0 / 2 i λ Z ) exp [ - i N ( 1 - Z / f ) χ 2 ] × n = 0 N t exp [ - i ( 2 n + 1 ) 2 N s 2 / 4 ] exp [ ± i ( 2 n + 1 ) N s χ ] × - 1 1 exp ( - i N ξ 2 / 4 ) exp ( i N χ ξ ) × exp [ i ( 2 n + 1 ) N s ξ / 2 ] d ξ .
U ( χ , N ) = ( a A 0 / 2 i λ Z ) exp [ - i N ( 1 - Z / f ) χ 2 ] × exp ( - i N s 2 / 4 ) [ F 1 exp ( i N s χ ) + F - 1 exp ( - i N s χ ) ] ,
F 1 = - 1 1 exp ( - i N ξ 2 / 4 ) exp ( - i N s ξ / 2 ) exp ( i N χ ξ ) d ξ , F - 1 = - 1 1 exp ( - i N ξ 2 / 4 ) exp ( i N s χ ξ / 2 ) exp ( i N χ ξ ) d ξ ;
E ( Ω ) = C ( Ω 2 / π ) + i S ( Ω 2 / π ) , E * ( Ω ) = C ( Ω 2 / π ) - i S ( Ω 2 / π ) .
F 1 = 2 π / N exp [ i N ( s 2 - χ ) 2 ] E * [ N ( ξ 2 + s 2 - χ ) ] ξ = - 1 1 , F - 1 = 2 π / N exp [ i N ( s 2 + χ ) 2 ] E * [ N ( ξ 2 - s 2 - χ ) ] ξ = - 1 1 .
U ( χ , N ) / U ( 0 , 0 ) = ( f / 4 Z ) ( 2 π / N ) exp ( i N Z χ 2 / f ) × { E * [ N ( 1 2 + s 2 - χ ) ] + E * [ N ( 1 2 - s 2 + χ ) ] + E * [ N ( 1 2 - s 2 - χ ) ] + E * [ N ( 1 2 + s 2 + χ ) ] } .
I ( χ , N ) = [ Re U ( χ , N ) ] 2 + [ Im U ( χ , N ) ] 2 .
I ( χ , N ) / I ( 0 , 0 ) = ( λ f N - π a 2 8 a 2 N ) ( { C [ 2 N / π ( 1 2 + s 2 - χ ) ] + C [ 2 N / π ( 1 2 + s 2 + χ ) ] + C [ 2 N / π ( 1 2 - s 2 - χ ) ] + C [ 2 N / π ( 1 2 + s 2 + χ ) ] } 2 + { S [ 2 N / π ( 1 2 + s 2 - χ ) ] + S [ 2 N / π ( 1 2 - s 2 + χ ) ] + S [ 2 N / π ( 1 2 + s 2 + χ ) ] + S [ 2 N / π ( 1 2 - s 2 - χ ) ] } 2 ) .
I ( 0 , N ) / I ( 0 , 0 ) = ( N λ f + π a 2 2 a 2 N ) ( { C [ N / 2 π ( 1 + s ) ] + C [ N / 2 π ( 1 - z ) ] } 2 + { S [ 2 N / 2 π ( 1 + s ) ] + S [ 2 N / 2 π ( 1 - s ) ] } 2 ) .
U n ( X 1 - X n ) = A 0 exp [ - i ( ϕ n - k X 1 2 / 2 f ) ] .
U ( χ , N ) / U ( 0 , 0 ) = ( f / 4 Z ) exp [ - i N ( 1 - Z / f ) χ 2 ] × n = 0 N t exp [ - i ( 2 n + 1 ) 2 + N s 2 / 4 ] × exp [ ± i ( 2 n + 1 ) N s χ ] × exp ( - i ϕ n ) - 1 1 exp ( i N ξ 2 / 4 ) exp ( i N χ ξ ) × exp [ i ( 2 n + 1 ) N s ξ / 2 ] d ξ .
U ( χ , N ) / U ( 0 , 0 ) = ( f / 4 Z ) exp [ - i N ( 1 - Z / f ) χ 2 ] × exp ( - i N s 2 / 4 ) { F 1 exp [ i ( N s χ - ϕ 1 ) ] + F - 1 exp [ - i ( N s χ + ϕ - 1 ) ] } ;
U ( χ , N ) / U ( 0 , 0 ) = ( π f / 8 N Z ) exp ( i N Z χ 2 / f ) × ( { E * [ N ( 1 2 + s 2 - χ ) ] + E * [ N ( 1 2 - s 2 + χ ) ] } exp ( - i ϕ 1 ) + { E * [ N ( 1 2 - s 2 - χ ) ] + E * [ N ( 1 2 + s 2 + χ ) ] } exp ( - i ϕ - 1 ) ) .
I ( χ , N ) / I ( 0 , 0 ) = ( λ f N + π a 2 8 a 2 N ) { ( C 1 + C 2 ) 2 + ( S 1 + S 2 ) 2 + ( C 3 + C 4 ) 2 + ( S 3 + S 4 ) 2 + 2 [ ( C 1 + C 2 ) ( C 3 + C 4 ) + ( S 1 + S 2 ) ( S 3 + S 4 ) ] × cos ( Δ ϕ ) + 2 [ ( S 1 + S 2 ) ( C 3 + C 4 ) - ( S 3 + S 4 ) ( C 1 + C 2 ) ] sin ( Δ ϕ ) } ,
C 1 = C [ 2 N / π ( 1 2 + s 2 - χ ) ] ,             S 1 = S [ 2 N / π ( 1 2 + s 2 - χ ) ] , C 2 = C [ 2 N / π ( 1 2 - s 2 + χ ) ] ,             S 2 = S [ 2 N / π ( 1 2 - s 2 + χ ) ] , C 3 = C [ 2 N / π ( 1 2 - s 2 - χ ) ] ,             S 3 = S [ 2 N / π ( 1 2 - s 2 - χ ) ] , C 4 = C [ 2 N / π ( 1 2 + s 2 + χ ) ] ,             S 4 = S [ 2 N / π ( 1 2 + s 2 + χ ) ] .
I ( 0 , N ) / I ( 0 , 0 ) = ( λ f N + π a 2 2 a 2 N ) [ ( C 1 + C 2 ) 2 + ( S 1 + S 2 ) 2 ] cos 2 ( Δ ϕ / 2 ) ,
U n ( X 1 - X n ) = A 0 exp ( i k X 1 2 / 2 f ) exp [ i k α n ( X 1 - X n ) ] ,
U ( X , Z ) / U ( 0 , f ) = ( f / 4 Z ) exp ( - i k X 2 / 2 Z ) n = 0 N t F n F n = exp [ - i ( 2 n + 1 ) 2 N s 2 / 4 ] exp ( i ( 2 n + 1 ) × ( N 1 - Z / f ) ( Z α n / a ) s ] exp [ ± i ( 2 n + 1 ) N s χ ¯ n ] × - 1 1 [ exp ( - i N ξ 2 / 4 ) ] exp [ i ( 2 n + 1 ) N s ξ / 2 ] × exp ( i N χ ¯ n ξ ) d ξ .
X + Z α n = ( 1 - Z / f ) a χ ¯ n ,
I ( χ ¯ , N ) = ( 1 / 16 ) ( f λ N / π a 2 + 1 ) n = 0 N t F n · m = 0 M F m * .
n F n = F 0 + F - 0 = π / 2 N { exp ( i N χ ¯ 0 2 ) exp [ - i ( N 1 - Z / f ) ( Z α 0 a ) s ] × E * [ N ( ξ 2 + s 2 - χ ¯ 0 ) ] · | ξ = - 1 1 + exp ( + i N χ ¯ - 0 2 ) exp [ i ( N 1 - Z / f ) ( Z α - 0 a ) s ] × E * [ N ( ξ 2 - s 2 - χ ¯ - 0 ) ] | ξ = - 1 1 } .
E [ N ( ξ 2 ± s 2 - χ ¯ ± 0 ) ] E [ N ( ξ 2 ± s 0 - χ ) ] .
exp ( i N χ ¯ 0 2 ) exp [ - i ( N 1 - Z / f ) ( Z α 0 a ) s ] exp [ i N Z α 0 a ( 2 X a + Z α 0 a - s ) ] exp ( i N X 2 / α 2 ) , exp ( i N χ ¯ - 0 2 ) exp [ i ( N 1 - Z / f ) ( Z α - 0 a ) s ] exp [ - i N Z α - 0 a ( 2 X a + Z α - 0 a - s ) ] exp ( i N X 2 / α 2 ) .
2 X m / a = ( s - 1 ) .
exp ( i N χ ¯ 0 2 ) exp [ - i ( N 1 - Z / f ) ( Z α 0 a ) s ] exp ( - i N s Z α 0 / a ) × exp ( i N X 2 / α 2 ) , exp ( - i N χ ¯ - 0 2 ) exp [ i ( N 1 - Z / f ) ( Z α - 0 a ) s ] exp ( i N s Z α - 0 / a ) × exp ( i N X 2 / α 2 ) .
n F n = π / 2 N exp ( + i N s 2 / 4 ) exp [ - i N s Z ( α 0 - α - 0 ) / 2 a ] × exp ( i N X 2 / α 2 ) { exp [ - i N s Z ( α 0 + α - 0 ) / 2 a ] × E * [ N ( ξ 2 + s 2 - χ ) ] | ξ = - 1 1 + exp [ i N s Z ( α 0 + α - 0 ) / 2 a ] × E * [ N ( ξ 2 - s 2 - χ ¯ ) ] | ξ = - 1 1 }
I ( χ , N ) / I ( 0 , 0 ) = ( λ f N + π a 2 8 a 2 N ) [ ( C 1 + C 2 ) 2 + ( C 3 + C 4 ) 2 + ( S 1 + S 2 ) 2 + ( S 3 + S 4 ) 2 + 2 [ ( C 1 + C 2 ) ( C 3 + C 4 ) + ( S 1 + S 2 ) ( S 3 + S 4 ) ] × cos ( N s Z Δ α / a ) + 2 [ ( C 1 + C 2 ) ( S 3 + S 4 ) - ( S 1 + S 2 ) ( C 3 + C 4 ) ] sin ( N s Z Δ α / a ) ] .
I ( 0 , N ) / I ( 0 , 0 ) = ( λ f N + π a 2 2 a 2 N ) { ( C 1 + C 2 ) 2 + ( S 1 + S 2 ) 2 } × cos 2 ( N s Z Δ α / 2 a ) .
U n ( X 1 - X n ) = A 0 exp [ + i k X 1 2 / 2 ( f ± Δ f n ) ] ,
U ( r ¯ , Z ) = ( 1 / i λ Z ) n ρ n ρ n + a A n ( ρ ¯ - ρ ¯ n ) exp [ - i k ( r ¯ - ρ ¯ ) 2 / 2 Z ] d 2 ρ ¯ .
( ρ ^ n - r ^ ) · ξ ¯ = ρ ^ n - r ^ ξ cos ( θ - θ n ) ,
U ( r ^ , N ) = ( a 2 A 0 / i λ Z ) exp [ - i N ( 1 - Z / f ) r ^ 2 ] × n exp ( - i N ρ ^ n 2 ) exp ( 2 i N ρ ^ n · r ) × 0 2 π 0 1 exp ( - i N ξ 2 ) exp [ - 2 i N ρ ^ n - r ^ ξ × cos ( θ - θ n ) ] ξ d ξ d θ .
U ( r ^ , N ) / U ( 0 , 0 ) = ( 2 f / N t Z ) exp [ - i N ( 1 - Z / f ) r ^ 2 ] n N t exp ( - i N ρ ^ n 2 ) × exp ( 2 i N ρ ^ n · r ^ ) 0 1 ξ J 0 [ 2 N ρ ^ n - r ^ ξ ] × exp ( - i N ξ 2 ) d ξ .
0 1 ξ J 0 [ 2 N ρ ^ n - r ^ ξ ] exp ( - i N ξ 2 ) d ξ = C ˜ [ 2 N , 2 N ρ ^ n - r ^ ] - i S ˜ [ 2 N , 2 N ρ ^ n - r ^ ] = ( D I ) n ,
C ˜ [ 2 N , 2 N ρ ^ n - r ^ ] = ( 1 / N ) cos ( N ) { ( 1 / ρ ^ n - r ^ ) J 1 ( 2 N ρ ^ n - r ^ ) - ( 1 / ρ ^ n - r ^ ) 3 J 3 ( 2 N ρ ^ n - r ^ ) + } + ( 1 / N ) sin ( N ) { ( 1 / 1 ρ ^ n - r ^ ) 2 J 2 × ( 2 N ρ ^ n - r ^ ) - ( 1 / ρ ^ n - r ^ ) 4 J 4 × ( 2 N ρ ^ n - r ^ ) + }
C ˜ [ 2 N , 2 N ρ ^ n - r ^ ] = ( 1 / N ) { cos ( N ) U ˜ 1 ( 2 N , 2 N ρ ^ n - r ^ ) + sin ( N ) U 2 ( 2 N , 2 N ρ ^ n - r ^ ) } , S ˜ ( 2 N , 2 N ) ρ ^ n - r ^ ) = ( 1 / N ) { sin ( N ) U 1 ( 2 N , 2 N ρ ^ n - r ^ ) - cos ( N ) U 2 ( 2 N , 2 N ρ ^ n - r ^ ) } .
( DI ) n = ( I / N ) exp ( - i N ) [ U 1 ( 2 N , 2 N ρ ^ n - r ^ ) + i U 2 ( 2 N , 2 N ρ ^ n - r ^ ) ] .
U ( r ^ , N ) / U ( 0 , 0 ) = ( 2 / N t ) ( f λ N + π a 2 π a 2 ) exp [ - i N ( 1 - Z / f ) r ^ 2 ] × n N t exp ( - i N ρ ^ n 2 ) exp ( 2 i N ρ ^ n · r ^ ) · ( DI ) n .
U ( r ^ , N ) / U ( 0 , 0 ) = ( 2 / N t ) ( f λ N + π a 2 π a 2 ) exp [ - i N ( 1 - Z / f ) [ × n N t exp ( - i n 2 N s 2 ) l = 1 L exp ( 2 i n s r ^ × cos ( θ - l ψ ) · ( DI ) n , l ,
U ( 0 ^ , N ) / U ( 0 , 0 ) = ( 2 / N t ) ( f λ N + π a 2 π a 2 N ) exp [ - i N ( 1 + s 2 ) × l = 1 4 0 1 ξ J 0 ( 2 N s ξ ) exp ( - i N ξ 2 ) d ξ .

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