Abstract

The boundary-diffraction-wave theory is used to calculate the diffracted field in the shadow boundary region. Discussions are based on expressions derived for a Gaussian beam incident on a circular aperture.

© 1977 Optical Society of America

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References

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  1. A. Rubinowicz, “Die Beugungswelle in der Kirehhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917).
    [Crossref]
  2. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary-diffraction wave,” J. Opt. Soc. Am. 52, 615–625 (part I) and 626–637 (part II) (1962).
    [Crossref]
  3. J. W. Y. Lit and R. Tremblay, “Boundary-diffraction-wave theory of cascaded-apertures diffraction,” J. Opt. Soc. Am. 59, 559–567 (1969).
    [Crossref]
  4. G. Otis, “Application of the boundary-diffraction-wave theory to Gaussian beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
    [Crossref]
  5. G. Otis and J. W. Y. Lit, “Edge-on diffraction of a Gaussian laser beam by a semi-infinite plane,” Appl. Opt. 14, 1156–1160 (1975).
    [Crossref] [PubMed]
  6. E. W. Marchand and E. Wolf, “Diffraction at small apertures black screens,” J. Opt. Soc. Am. 59, 79–90 (1969).
    [Crossref]
  7. On the shadow boundary, the term 1−p⋅ρ̂ in Eq. (2) is equal to zero.
  8. J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
    [Crossref]
  9. W. A. Kleinhaus, “Diffraction from a sequence of apertures and disks,” J. Opt. Soc. Am. 65, 1451–1456 (1975).
    [Crossref]
  10. D. Ludwig, “Uniform asymptotic expansions at a caustic,” Comm. Pure Appl. Math. 19, 215–250 (1966).
    [Crossref]
  11. D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783–807 (1968).
    [Crossref]
  12. S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
    [Crossref]
  13. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 441.
  14. A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965), Chap. 9.
  15. G. N. Watson, Theory of Bessel Functions, (Cambridge U. P., Cambridge, 1952), p. 22.
  16. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 12.
  17. Reference 14, formula 11. 3. 9.
  18. Reference 14, Chap. 7.
  19. Reference 13, Sec. 8. 7.

1975 (2)

1974 (2)

G. Otis, “Application of the boundary-diffraction-wave theory to Gaussian beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
[Crossref]

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

1969 (2)

1968 (1)

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783–807 (1968).
[Crossref]

1966 (1)

D. Ludwig, “Uniform asymptotic expansions at a caustic,” Comm. Pure Appl. Math. 19, 215–250 (1966).
[Crossref]

1962 (1)

1957 (1)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[Crossref]

1917 (1)

A. Rubinowicz, “Die Beugungswelle in der Kirehhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917).
[Crossref]

Abramowitz, A.

A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965), Chap. 9.

Ahluwalia, D. S.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783–807 (1968).
[Crossref]

Boersma, J.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783–807 (1968).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 441.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 12.

Choudhary, S.

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

Felsen, L. B.

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

Keller, J. B.

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[Crossref]

Kleinhaus, W. A.

Lewis, R. M.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783–807 (1968).
[Crossref]

Lit, J. W. Y.

Ludwig, D.

D. Ludwig, “Uniform asymptotic expansions at a caustic,” Comm. Pure Appl. Math. 19, 215–250 (1966).
[Crossref]

Marchand, E. W.

Miyamoto, K.

Otis, G.

Rubinowicz, A.

A. Rubinowicz, “Die Beugungswelle in der Kirehhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917).
[Crossref]

Stegun, I. A.

A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965), Chap. 9.

Tremblay, R.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, (Cambridge U. P., Cambridge, 1952), p. 22.

Wolf, E.

Ann. Phys. (1)

A. Rubinowicz, “Die Beugungswelle in der Kirehhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. 53, 257–278 (1917).
[Crossref]

Appl. Opt. (1)

Comm. Pure Appl. Math. (1)

D. Ludwig, “Uniform asymptotic expansions at a caustic,” Comm. Pure Appl. Math. 19, 215–250 (1966).
[Crossref]

J. Appl. Phys. (1)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[Crossref]

J. Opt. Soc. Am. (5)

Proc. IEEE (1)

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[Crossref]

SIAM J. Appl. Math. (1)

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform asymptotic theory of diffraction by a plane screen,” SIAM J. Appl. Math. 16, 783–807 (1968).
[Crossref]

Other (8)

On the shadow boundary, the term 1−p⋅ρ̂ in Eq. (2) is equal to zero.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 441.

A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965), Chap. 9.

G. N. Watson, Theory of Bessel Functions, (Cambridge U. P., Cambridge, 1952), p. 22.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 12.

Reference 14, formula 11. 3. 9.

Reference 14, Chap. 7.

Reference 13, Sec. 8. 7.

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Figures (1)

FIG. 1
FIG. 1

Defining the geometry of the problem.

Equations (36)

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U D ( P ) = U B ( P ) + U o ( P ) ,
U D ( P ) = U B ( P ) .
U D ( P ) = U B ( P ) + 1 2 U o ( P ) .
U B ( P ) = Γ U ( Q ) exp ( i k ρ ) 4 π ρ p × ρ ̂ 1 p ρ ̂ d l ,
U G ( r , z ) = q ( 0 ) q ( z ) exp [ i k ( z + r 2 2 q ( z ) ) ] ,
1 q ( z ) = 1 R ( z ) + 2 i k w 2 ( z ) .
U B ( x , z ) = a 4 π 0 2 π U G ( a , z 1 ) exp ( i k ρ ) ρ p × ρ ̂ ϕ ̂ 1 p ρ ̂ d ϕ ̂ ,
ρ ̂ = ( x cos ϕ a ) r ̂ + ( z z 1 ) ( x sin ϕ ) ϕ ̂ ρ ,
p = a q ( z 1 ) r ̂ + ( 1 a 2 2 q 2 ( z 1 ) ) .
p × ρ ̂ ϕ ̂ a ρ ( x a cos ϕ 1 z z 1 q ( z 1 ) ) .
ρ z z 1 + x 2 + a 2 2 a x cos ϕ 2 ( z z 1 ) .
U B ( α , z ) = U o ( α , z ) exp ( i y ψ ) S ( α ) ,
α = x / a ,
γ = q ( z ) q ( z 1 ) = 1 + z z 1 q ( z 1 ) = | γ | exp ( i θ ) ,
ψ = k α a 2 / ( z z 1 ) ,
y = 1 2 ( γ α + α γ ) .
S ( α ) = γ 2 π 0 2 π ( α cos ϕ γ ) exp ( i ψ cos ϕ ) α 2 2 α γ cos ϕ + γ 2 d ϕ .
α = | γ | = w ( z ) / w ( z 1 ) ,
w 2 ( u ) = w 0 2 [ 1 + ( λ u / π w 0 2 ) 2 ] .
S ( α = γ ) = 1 2 J o ( ψ ) ,
S ( α < | γ | ) = m = 0 ( i ) m ( α / γ ) m J m ( ψ ) ,
S ( α > | γ | ) = m = 1 ( i ) m ( γ / α ) m J m ( ψ ) .
4 π S ( α ) = 0 2 π exp ( i ψ cos ϕ ) d ϕ + ( y γ α ) 0 2 π exp ( i ψ cos ϕ ) y cos ϕ d ϕ .
S ( α ) = 1 2 J o ( ψ ) + ( y γ α ) S 1 .
S 1 ψ + i y S 1 = i 2 J o ( ψ ) ,
S 1 = 1 2 ( S 1 ( ψ = 0 ) + i 0 ψ J o ( ξ ) exp ( i y ξ ) d ξ ) exp ( i y ψ ) ,
S 1 ( ψ = 0 ) = 0 for α = | γ | , = ( y 2 1 ) 1 / 2 fo α | γ | .
2 S ( α ) = J o ( ψ ) + δ exp ( i y ψ ) + i ψ ( y γ α ) m = 0 J m ( y ψ ) i J m + 1 ( y ψ ) m ! ( 2 m + 1 ) ξ m ,
δ = ± 1 according as α | γ | , = 0 if α = | γ | ,
ξ = ψ ( y 2 1 ) / 2 y .
2 S ( α ) = J o ( ψ ) + [ δ + 2 ( i ξ π ) 1 / 2 m = 0 ( i ξ ) m m ! ( 2 m + 1 ) ] exp ( i y ψ ) .
2 S ( α ) = J o ( ψ ) + { δ ± ( i 1 ) [ C ( 2 ξ π ) 1 / 2 + i S ( 2 ξ π ) 1 / 2 ] } exp ( i y ψ ) ,
U B ( P ) U o ( P ) δ 2 { 1 + ( i 1 ) [ C ( 2 ξ π ) 1 / 2 + i S ( 2 ξ π ) 1 / 2 ] } .
U D ( P ) = 1 2 U o ( P ) [ 1 ( J o ( ψ ) ψ sin θ m = 0 J m ( ψ cos θ ) i J m + 1 ( ψ cos θ ) m ! ( 2 m + 1 ) ζ m ) × exp ( i ψ cos θ ) ] ,
ζ = ξ ( α = | γ | ) = ψ ( sin 2 θ ) / 2 cos θ .
U D ( P ) = 1 2 U o ( P ) [ 1 J o ( ψ ) exp ( i ψ ) ] .