Abstract

A theoretical discussion of the problem of diffraction of spherical scalar waves incident upon a thin black infinite half-plane, which makes use of Maggi’s transformation, has previously appeared in the literature. However, previous treatments have been limited solely to the diffracted component. From the results of previous investigators a simple expression for the total energy distribution, which is suitable for computational purposes, is derived. The results are shown to reduce to Kirchhoff’s formulation of the same problem, for field points not far removed from the shadow-boundary-plane. A rapid approximation method, applicable to the cases of plane and cylindrical waves, is also given. Experimental results were obtained from a photometer employing a refrigerated multiplier phototube. It is shown that the theoretical and experimental intensity distributions agree only when the radius of the point source aperture becomes indefinitely small. Tests with metallic and nonmetallic screens indicate that the nature of the edge of the diffracting screen (for points far removed from the screen) is of minor importance. Possibly of somewhat greater significance, but yet minor, is the nature of the body of the diffracting screen, a metallic screen tending to displace the fringes toward the shadow-boundary-plane.

© 1952 Optical Society of America

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References

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  1. F. Kottler, Ann. Physik 70, 405 (1923).
    [Crossref]
  2. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen’s Principle (Oxford University Press, New York, 1950), second edition, p. 95.
  3. A. Rubinowicz, Ann. Physik 53, 257 (1917). In particular, see Eq. 13, p. 273. Rubinowicz derived this equation from Kirchhoff’s formula solely through geometric considerations.
    [Crossref]
  4. See, for example, Max Born, Optik (Julius Springer Verlag, Berlin, 1933) (reprint Edwards Brothers, Inc., Ann Arbor, Michigan, 1943), pp. 190–195.
  5. The table of Fresnel integrals used is that of C. M. Sparrow, Table of Fresnel Integrals (Edwards Brothers, Inc., Ann Arbor, Michigan, 1934). These have been extended by R. T. Birge.
  6. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, p. 365.
  7. See Sommerfeld’s electromagnetic solution in Born’s Optik (reference 4, p. 209).
  8. Hause, Woodward, and McClellan, J. Opt. Soc. Am. 29, 147 (1939). Corrections had to be made to their theoretical curve because of the effects of finite slit width.
    [Crossref]
  9. E. F. Coleman, Electronics 19, 220 (June, 1946).
  10. R. W. Engstrom, J. Opt. Soc. Am. 37, 420 (1947).
    [Crossref]

1947 (1)

1946 (1)

E. F. Coleman, Electronics 19, 220 (June, 1946).

1939 (1)

1923 (1)

F. Kottler, Ann. Physik 70, 405 (1923).
[Crossref]

1917 (1)

A. Rubinowicz, Ann. Physik 53, 257 (1917). In particular, see Eq. 13, p. 273. Rubinowicz derived this equation from Kirchhoff’s formula solely through geometric considerations.
[Crossref]

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen’s Principle (Oxford University Press, New York, 1950), second edition, p. 95.

Birge, R. T.

The table of Fresnel integrals used is that of C. M. Sparrow, Table of Fresnel Integrals (Edwards Brothers, Inc., Ann Arbor, Michigan, 1934). These have been extended by R. T. Birge.

Born,

See Sommerfeld’s electromagnetic solution in Born’s Optik (reference 4, p. 209).

Born, Max

See, for example, Max Born, Optik (Julius Springer Verlag, Berlin, 1933) (reprint Edwards Brothers, Inc., Ann Arbor, Michigan, 1943), pp. 190–195.

Coleman, E. F.

E. F. Coleman, Electronics 19, 220 (June, 1946).

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen’s Principle (Oxford University Press, New York, 1950), second edition, p. 95.

Engstrom, R. W.

Hause,

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, p. 365.

Kottler, F.

F. Kottler, Ann. Physik 70, 405 (1923).
[Crossref]

McClellan,

Rubinowicz, A.

A. Rubinowicz, Ann. Physik 53, 257 (1917). In particular, see Eq. 13, p. 273. Rubinowicz derived this equation from Kirchhoff’s formula solely through geometric considerations.
[Crossref]

Sparrow, C. M.

The table of Fresnel integrals used is that of C. M. Sparrow, Table of Fresnel Integrals (Edwards Brothers, Inc., Ann Arbor, Michigan, 1934). These have been extended by R. T. Birge.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, p. 365.

Woodward,

Ann. Physik (2)

A. Rubinowicz, Ann. Physik 53, 257 (1917). In particular, see Eq. 13, p. 273. Rubinowicz derived this equation from Kirchhoff’s formula solely through geometric considerations.
[Crossref]

F. Kottler, Ann. Physik 70, 405 (1923).
[Crossref]

Electronics (1)

E. F. Coleman, Electronics 19, 220 (June, 1946).

J. Opt. Soc. Am. (2)

Other (5)

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygen’s Principle (Oxford University Press, New York, 1950), second edition, p. 95.

See, for example, Max Born, Optik (Julius Springer Verlag, Berlin, 1933) (reprint Edwards Brothers, Inc., Ann Arbor, Michigan, 1943), pp. 190–195.

The table of Fresnel integrals used is that of C. M. Sparrow, Table of Fresnel Integrals (Edwards Brothers, Inc., Ann Arbor, Michigan, 1934). These have been extended by R. T. Birge.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, p. 365.

See Sommerfeld’s electromagnetic solution in Born’s Optik (reference 4, p. 209).

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Figures (14)

Fig. 1
Fig. 1

Diffraction by an infinite half-plane.

Fig. 2
Fig. 2

General view of apparatus. At left, H-4 transformer and air-cooled light source box with source aperture in center. In the center, refrigerated photometer unit on measuring microscope base. At right, Sorenson regulator (behind) and amplifier unit with movable eyepiece tube to read milliameter. Behind (at left) is Bakelite screen used in the experiments.

Fig. 3
Fig. 3

Photometer and amplifier units open. Electron multiplier and base pulled out of photometer unit, and 5691 tube and resistors for multiplier dynode stages (in front). Adjustable slit has been removed and opened wide. On right, amplifier and milliammeter removed from case.

Fig. 4
Fig. 4

Refrigerator unit.

Fig. 5
Fig. 5

Photomultiplier electrical circuit. (The fine control resistor should be 200 Ω.)

Fig. 6
Fig. 6

Comparison of diffraction pattern of half-plane with light from point source (dark circles) and slit source (open circles).

Fig. 7
Fig. 7

Comparison of theoretical and experimental values for a thin black screen—symmetric case. ρ0=d=715.44 cm, a=0.0105 cm in a 0.002-in. aluminum foil, w=5.662y (cm).

Fig. 8
Fig. 8

Comparison of experiments and theoretical values for a thin black screen, unsymmetrical case. ρ0=1159.4 cm, d=271.5 cm, a=0.033 cm in a 0.002-in. aluminum foil, w=11.7y (cm).

Fig. 9
Fig. 9

Evolution of fringes for large source apertures. The aperture radius a=0.032 cm, 0.0405 cm, and 0.110 cm for Curves A, B, and C, respectively.

Fig. 10
Fig. 10

Evolution of fringes for small source apertures. The aperture radius a=0.010 cm, 0.014 cm, and 0.021 cm, for Curves D, E, and F, respectively.

Fig. 11
Fig. 11

Relative intensity of the first maximum for various point source sizes.

Fig. 12
Fig. 12

Diffraction by a polished copper edge compared to that due to the same edge painted a diffusely reflecting black. The curve connecting the open circles was obtained from the copper edge.

Fig. 13
Fig. 13

Diffraction by copper screen painted black except for copper edge compared to that due to same screen with edge also painted diffusely reflecting black. Screen symmetrically placed and at greater distances than in Fig. 12. The curve connecting the open circles was obtained from the black edge.

Fig. 14
Fig. 14

Diffraction by a specularly reflecting aluminum screen. The solid curve represents diffraction by a specularly reflecting screen. The broken curve shows the results obtained by covering the body of the same screen with a black paper so that only 1 2 mm of the original edge was left visible.

Equations (23)

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y sin α + x cos α = 0 ,             y 0 ,             α π / 2 ,
x = ρ cos ϕ ,             y = ρ sin ϕ ,             z = z .
r 0 = - i ρ 0 + k z , r = - i ρ 1 cos ϕ 1 - j ρ 1 sin ϕ 1 + k ( z - z 1 ) ,             Γ = - k z .
u ( P , t ) = u 0 ( P , t ) + 1 4 π r 0 × r e i k ( c t - r - r 0 ) r r 0 [ r 0 · r + r 0 r ] · d Γ = u 0 ( P , t ) - 1 4 π × - + ρ 0 ρ 1 sin ϕ 1 e i k ( c t - r - r 0 ) r r 0 [ ρ 0 ρ 1 cos ϕ 1 + z ( z - z 1 ) + r r 0 ] d z .
τ = sinh - 1 z / ρ 0 + sinh - 1 ( z - z 1 ) / ρ 1 ,
d τ d z = 1 r 0 + 1 r = R r r 0 ,
cosh τ = r r 0 + z ( z - z 1 ) ρ 0 ρ 1 .
u ( P , t ) = u 0 ( P , t ) - 1 4 π - + e i k ( c t - R ) sin ϕ 1 R ( cosh τ + cos ϕ 1 ) d τ ,
R 2 = ( r 0 + r ) 2 = ρ 0 2 + ρ 1 2 + z 1 2 + 2 ρ 0 ρ 1 cosh τ ,             ϕ < π .
R ( cosh τ + cos ϕ 1 ) = R 1 2 { 4 cos 2 ϕ 1 / 2 + τ 2 ( 1 + 2 ρ 0 ρ 1 R 1 2 cos 2 ϕ 1 / 2 ) } + 0 ( τ 4 ) .
u ( P , t ) = u 0 ( P , t ) - e i k ( c t - R 1 ) R 1 · ψ 2 tan ϕ 1 / 2 2 π 0 exp - i q 2 τ 2 ψ 2 + τ 2 d τ ,
q = + ( k ρ 0 ρ 1 2 R 1 ) 1 2 ,             ψ = 2 cos ϕ 1 / 2 { 1 + ( 2 ρ 0 ρ 1 cos 2 ( ϕ 1 / 2 ) / R 1 2 ) } 1 2 .
c exp - i q 2 z 2 z 2 + ψ 2 d z ,
0 R exp - i q 2 r 2 e - i π / 2 e - i π / 4 r 2 e - i π / 2 + ψ 2 d r + i - π / 4 0 exp - i q 2 R 2 e 2 i θ R e i θ R 2 e 2 i θ + ψ 2 d θ + R 0 exp - i q 2 x 2 x 2 + ψ 2 d x = 0.
0 exp - i q 2 τ 2 τ 2 + ψ 2 d τ = 0 exp - q 2 τ 2 e i π / 4 τ 2 + i ψ 2 d τ = exp i ( q 2 ψ 2 + π / 4 ) 0 d τ q 2 η exp - η 2 ( τ 2 + i ψ 2 ) d η .
0 exp - i q 2 τ 2 τ 2 + ψ 2 d τ = π ψ 2 1 2 exp i ( q 2 ψ 2 + π / 4 ) × [ ( 1 2 - U ( w ) ) - i ( 1 2 - V ( w ) ) ] ,
U ( w ) = 0 w cos π t 2 2 d t ,             V ( w ) = 0 w sin π t 2 2 d t .
u ( P , t ) = u 0 ( P , t ) - exp i [ k ( c t - R 1 ) + π / 4 + π 2 w 2 ] sin ϕ 1 / 2 2 1 2 R 1 { 1 + 2 ρ 0 ρ 1 cos 2 ϕ 1 / 2 R 1 2 } 1 2 × [ ( 1 2 - U ) - i ( 1 2 - V ) ] ,             ϕ π
I I 0 = u ū u 0 ū 0 = 1 2 [ 1 - y 2 ( d 2 + ρ 0 2 + 8 ρ 0 d ) 4 d 2 ( ρ 0 + d ) 2 + 0 ( y 4 ) ] × [ ( 1 2 - U ) 2 + ( 1 2 - V ) 2 ] + [ 1 - 2 1 2 ( 1 - y 2 ( d 2 + ρ 0 2 + 8 ρ 0 d ) 8 d 2 ( ρ 0 + d ) 2 + 0 ( y 4 ) ) × ( ( 1 2 - U ) cos σ + ( 1 2 - V ) sin σ ) ] ,
w = y { 2 ρ 0 λ d ( ρ 0 + d ) } 1 2 · [ 1 - y 2 4 ( 1 2 d 2 + 1 d ( ρ 0 + d ) + ρ 0 d ( ρ 0 + d ) 2 ) + 0 ( y 4 ) ] , σ = π 4 - 3 π y 4 ρ 0 2 4 λ d 2 ( ρ 0 + d ) 3 + 0 ( y 6 ) .
I / I 0 1 2 [ ( 1 2 - U ( w ) ) 2 + ( 1 2 - V ( w ) ) 2 ] + { U ( w ) + V ( w ) } , w y ( 2 ρ 0 λ d ( ρ 0 + d ) ) 1 2 .
( I / I 0 ) K = 1 2 [ ( 1 2 + U ( w ) ) 2 + ( 1 2 + V ( w ) ) 2 ] , w = { 2 λ ( 1 R + 1 R 0 ) } 1 2 X cos δ ,
1 2 [ ( 1 2 - U ) 2 + ( 1 2 - V ) 2 ] + ( U + V ) - 1 2 [ ( 1 2 + U ) 2 + ( 1 2 + V ) 2 ] = 1 2 [ ( 1 2 - U ) 2 - ( 1 2 + U ) 2 ] + 1 2 [ ( 1 2 - V ) 2 - ( 1 2 + V ) 2 ] + ( U + V ) 0.