Abstract

We used the stationary-phase approximation to derive an approximate kernel for the propagation of a monochromatic wave specified on a curved source plane.

© 1988 Optical Society of America

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References

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  1. A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 34.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 42.
  3. A. S. Marathay, “Diffraction of light from an aperture in a spherical surface,” J. Opt. Soc. Am. 65, 909–913 (1975).
    [CrossRef]
  4. E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), App. III.
  5. A. Walther, “Lenses, wave optics, and eikonal functions,” J. Opt. Soc. Am. 59, 1325–1333 (1969).
    [CrossRef]

1975 (1)

1969 (1)

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 42.

Marathay, A. S.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 34.

Walther, A.

Wolf, E.

E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), App. III.

J. Opt. Soc. Am. (2)

Other (3)

E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), App. III.

A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 34.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 42.

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Figures (2)

Fig. 1
Fig. 1

Wave propagation from a curved surface.

Fig. 2
Fig. 2

Coordinate system used in the error analysis.

Equations (21)

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2 u = k 2 u = 0 ,
K R S ( x , y , x , y , z ) = 1 2 π ( i k r 1 r 2 ) exp ikr ,
r = [ ( x x ) 2 + ( y y ) 2 + ( z z ) 2 ] 1 / 2 .
K ( u , υ , x , y , z ) .
K RS ( x , y , x , y , z ) K ( u , υ , x , y , z ) d u d υ = K RS ( x , y , x , y , z ) .
K ( u , υ , x , y , z ) = A ( u , υ , x , y , z ) exp i k r ,
r = [ ( x x ) 2 + ( y + y ) 2 + ( z z ) 2 ] 1 / 2 ,
1 2 π z r i k r A exp i k ( r + r ) d u d υ = 1 2 π z r exp i k r .
r + r = | r 0 n ˆ + w | + | r 0 n ˆ w | ,
r + r = [ r 0 2 + 2 r 0 n ˆ w + w w ] 1 / 2 + [ r 0 2 2 r 0 n ˆ w + w w ] 1 / 2 .
r + r = r 0 + r 0 + 1 2 r 0 + r 0 r 0 r 0 [ w w ( n ˆ w ) 2 ] .
w = u â + υ b ˆ .
r + r = r 0 + 1 2 r 0 r 0 r 0 ( υ 2 + u 2 cos 2 ψ ) ,
1 2 π z 0 r 0 i k r 0 A 2 π i k r 0 r 0 r 0 cos ψ exp i k r 0 = 1 2 π z r 0 exp i k r 0 .
A = 1 2 π i k r 0 cos ψ .
r + r = [ ( r 0 + ζ ) 2 + ξ 2 + η 2 ] 1 / 2 + [ ( r 0 ζ ) 2 + ξ 2 + η 2 ] 1 / 2 .
r + r = r 0 + ζ + [ ( r 0 ζ ) 2 + ξ 2 + η 2 ] 1 / 2 .
T 1 = ( ξ 2 + η 2 ) / 2 r 0 .
T 2 = ( ξ 2 + η 2 ) ζ c / 2 r 0 2 .
( ξ 2 + η 2 ) / 2 r 0 = N λ , ( ξ 2 + η 2 ) ζ c / 2 r 0 2 < λ .
2 ζ c / ( ξ 2 + η 2 ) < / N 2 λ .

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