Abstract

The possibility of applying an ordinary quasi-geometrical technique for Fraunhofer diffraction calculations on absolutely absorbing three-dimensional (3-D) bodies of constant thickness is investigated. It is shown that such an application can lead to results that are inadequate for finding the physical diffraction pattern of 3-D bodies. A modified version of the technique is suggested that, to a greater degree, takes into account secondary diffraction and thus permits a more exact presentation of characteristic features of the light diffracted by 3-D bodies. Some examples of this approach applied to the light-diffraction analysis of simple 3-D bodies are given. It is shown experimentally and with calculations that this approach permits description of the diffraction effects of 3-D bodies of the class mentioned in rather simple form and with good accuracy.

© 1981 Optical Society of America

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References

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  1. R. M. Bytchkov, V. P. Koronkevitch, and Yu. V. Chugui, “Threaded article parameter measurement by spatial spectra anaylsis,” Appl. Opt. 18, 197–200 (1979).
    [CrossRef] [PubMed]
  2. M. Kochsiek, H. Kunzmann, and J. Tantau, “Anwendung Beugungsoptischer Methoden zur Messung der Durchmesser von kleinen Wellen,” PTB-Mitt. 87, 279–282 (1977).
  3. S. Schmidt, “Ein Beitrag zur Ekrlärung der Lichtbeugung am metallischen Kriszylinder,” PTB-Mitt. 86, 239–247 (1976).
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1965).
  5. H. Hönl, A. W. Maue, and K. Westpfahl, Theorie der Beugung (Springer-Verlag, Berlin, 1961).
  6. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1961).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data-processing systems,” Proc. IEEE 54, 1055–1063 (1966).
    [CrossRef]
  9. G. Harburn, J. K. Ranniko, and R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik 48, 321–328 (1977).
  10. H. Arsenault and N. Brousseau, “Space variance in quasi-linear coherent optical processing,” J. Opt. Soc. Am. 63, 555–558 (1973).
    [CrossRef]
  11. It is obvious that the quasi-geometric approach application is right when a true Fresnel image of the back face reduced to the input plane differs only slightly from the initial one g(x0). For this it is necessary that the Fresnel zone size ∊=λd be much less than the characteristic size D of the binary function g(x0), i.e., D≫λd.

1979 (1)

1977 (2)

M. Kochsiek, H. Kunzmann, and J. Tantau, “Anwendung Beugungsoptischer Methoden zur Messung der Durchmesser von kleinen Wellen,” PTB-Mitt. 87, 279–282 (1977).

G. Harburn, J. K. Ranniko, and R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik 48, 321–328 (1977).

1976 (1)

S. Schmidt, “Ein Beitrag zur Ekrlärung der Lichtbeugung am metallischen Kriszylinder,” PTB-Mitt. 86, 239–247 (1976).

1973 (1)

1966 (1)

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data-processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[CrossRef]

1961 (1)

Arsenault, H.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1965).

Brousseau, N.

Bytchkov, R. M.

Chugui, Yu. V.

Goodman, J. W.

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

Harburn, G.

G. Harburn, J. K. Ranniko, and R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik 48, 321–328 (1977).

Hönl, H.

H. Hönl, A. W. Maue, and K. Westpfahl, Theorie der Beugung (Springer-Verlag, Berlin, 1961).

Keller, J. B.

Kochsiek, M.

M. Kochsiek, H. Kunzmann, and J. Tantau, “Anwendung Beugungsoptischer Methoden zur Messung der Durchmesser von kleinen Wellen,” PTB-Mitt. 87, 279–282 (1977).

Koronkevitch, V. P.

Kunzmann, H.

M. Kochsiek, H. Kunzmann, and J. Tantau, “Anwendung Beugungsoptischer Methoden zur Messung der Durchmesser von kleinen Wellen,” PTB-Mitt. 87, 279–282 (1977).

Maue, A. W.

H. Hönl, A. W. Maue, and K. Westpfahl, Theorie der Beugung (Springer-Verlag, Berlin, 1961).

Ranniko, J. K.

G. Harburn, J. K. Ranniko, and R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik 48, 321–328 (1977).

Schmidt, S.

S. Schmidt, “Ein Beitrag zur Ekrlärung der Lichtbeugung am metallischen Kriszylinder,” PTB-Mitt. 86, 239–247 (1976).

Tantau, J.

M. Kochsiek, H. Kunzmann, and J. Tantau, “Anwendung Beugungsoptischer Methoden zur Messung der Durchmesser von kleinen Wellen,” PTB-Mitt. 87, 279–282 (1977).

Vander Lugt, A.

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data-processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[CrossRef]

Westpfahl, K.

H. Hönl, A. W. Maue, and K. Westpfahl, Theorie der Beugung (Springer-Verlag, Berlin, 1961).

Williams, R. P.

G. Harburn, J. K. Ranniko, and R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik 48, 321–328 (1977).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1965).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Optik (1)

G. Harburn, J. K. Ranniko, and R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik 48, 321–328 (1977).

Proc. IEEE (1)

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data-processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[CrossRef]

PTB-Mitt. (2)

M. Kochsiek, H. Kunzmann, and J. Tantau, “Anwendung Beugungsoptischer Methoden zur Messung der Durchmesser von kleinen Wellen,” PTB-Mitt. 87, 279–282 (1977).

S. Schmidt, “Ein Beitrag zur Ekrlärung der Lichtbeugung am metallischen Kriszylinder,” PTB-Mitt. 86, 239–247 (1976).

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1965).

H. Hönl, A. W. Maue, and K. Westpfahl, Theorie der Beugung (Springer-Verlag, Berlin, 1961).

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

It is obvious that the quasi-geometric approach application is right when a true Fresnel image of the back face reduced to the input plane differs only slightly from the initial one g(x0). For this it is necessary that the Fresnel zone size ∊=λd be much less than the characteristic size D of the binary function g(x0), i.e., D≫λd.

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Figures (12)

Fig. 1
Fig. 1

Optical Fourier-transform unit. 1 is an absolutely absorbing object of constant thickness d (its section); 2 is a Fourier-transform objective.

Fig. 2
Fig. 2

Imaging system. P and P0 are the planes corresponding to front and back object faces; P1 is a frequency plane; P′ and P′0 are image planes conjugate to planes P and P0; P″0 is a plane of the image g(x0) obtained with the QG approach.

Fig. 3
Fig. 3

Production of 3-D edge spectrum (a) with the QG approach and (b) its equivalent model.

Fig. 4
Fig. 4

Curves of the modulus (solid line) and phase (broken line) of the function Y ˜(x).

Fig. 5
Fig. 5

Curves of the function Y(x) Y ˜(x + a) for (a) a > 0 and (b) a < 0.

Fig. 6
Fig. 6

Objects: 3-D edge with (a) a limiting diaphragm and (b) a biplanar slit.

Fig. 7
Fig. 7

(a) Curves of the dependences of IFR (u) (solid lines) and IMQG (u) (broken lines), (b) curve of the dependence of IQG (u).

Fig. 8
Fig. 8

Spectra of the object simulating a 3-D edge with a limiting aperture for various values of d.

Fig. 9
Fig. 9

Curves of the dependence of IFR (u) for (a) d = 3 mm and (b) d = 7 mm.

Fig. 10
Fig. 10

Light diffraction by an object in the form of a gauge block: (a) experimental scheme, (b) photograph of direct and reflected spectra from the gauge block.

Fig. 11
Fig. 11

Equivalent scheme for producing the gauge block spectra: (a) a direct one for u < u1, (b) a direct one for u > u1, and (c) a reflected one.

Fig. 12
Fig. 12

Curves of dependence of position of minima of (a) direct and (b) reflected spectra of a gauge block. Solid lines are theoretical results obtained with the MQG approach; circles are experimental data; broken lines and crosses are calculated and experimental results, respectively, for the plane slit spectrum d = 0.

Equations (35)

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F Q G ( u ) = ( P ) f ( x ) g ( x + u d / F ) e - j ( k u x / F ) d x ,
F Q G ( u ) = - f ( x ) e - j ( k u x / F ) d x { f ( x ) } .
F Q G ( u ) = - g ( x + u d / F ) e - j ( k u x / F ) d x e j ( k u 2 d / F 2 ) { g ( x ) } .
h ( x , u ) = e - j ( k u x / F ) g ( x + u d / F )
f ( x ) = Y ( x ) ,             g ( x 0 ) = Y ( x 0 ) ,
Y ( x ) = { 1 for x 0 0 for x < 0 .
F Q G ( u ) = { { Y ( x ) } for u > 0 { Y ( x ) } e j ( k u 2 d / F 2 ) for u < 0 .
F ( θ ) e j k θ x Y ( x )
Δ = θ d = u d / F .
F F R ( u ) = e j ( k u 2 d / 2 F 2 ) { [ f ( x 0 ) e j ( k x 0 2 / 2 d ) ] g ( x 0 ) } ,
F F R ( u ) = - f ( x ) g ˜ ( x + u d / F ) e - j ( k u x / F ) d x ,
g ˜ ( x ) - g ( x 0 ) e j [ k ( x - x 0 ) 2 / 2 d ] d x 0
f ( x ) = Y ( x ) ,             g ( x 0 ) = Y ( x 0 - A ) ,
F F R ( u ) = - Y ( x ) Y ˜ ( x - A + u d / F ) e - j ( k u x / F ) d x ,
Y ˜ ( x ) = - Y ( x 0 ) e j [ k ( x - x 0 ) 2 / 2 d ] d x 0
Φ ( x ) = 0 x exp ( j π t 2 ) d t
Y ˜ ( x ) = λ d [ Φ ( ) + Φ ( x / λ d ) ] .
Y ( x ) Y ˜ ( x + a ) Y ( x ) ,
Y ( x ) Y ˜ ( x + a ) Y ˜ ( x + a ) .
F F R ( u ) - Y ( x ) e - j ( k u x / F ) d x = { Y ( x ) } ,
F F R ( u ) - Y ˜ ( x - A + u d / F ) e - j ( k u x / F ) d x = e - j ( k u A / F ) e j ( k u 2 d / F 2 ) { Y ˜ ( x ) } .
{ Y ˜ ( x ) } = e - j ( k u 2 d / 2 F 2 ) { Y ( x ) } ,
F F R ( u ) e - j ( k u A / F ) e j ( k u 2 d / 2 F 2 ) { Y ( x ) } .
F F R ( u ) = { { Y ( x ) } for u > u 0 e - j ( k u x / F ) e j ( k u 2 d / 2 F 2 ) { Y ( x ) } for u < u 0 .
F Q G ( u ) = { { Y ( x ) } for u > u 0 e - j ( k u x / F ) e j ( k u 2 d / F 2 ) { Y ( x ) } for u < u 0 .
F M Q G ( u ) = - f ( x ) g ( x + u d / 2 F ) e - j ( k u x / F ) d x .
f ( x ) = Rect ( x / 2 A ) ,             g ( x 0 ) = Y ( x 0 + A ) .
F M Q G ( u ) = { sin ( k u A / F ) / u for u > 0 e j ( k u 2 d / 4 F 2 ) sin [ k u A F ( 1 + u d 4 A F ) ] / u for - u 0 < u < 0 0 for u < - u 0 ,
f ( x ) = Y ( A - x ) ,             g ( x 0 ) = Y ( A + x 0 ) .
F M Q G ( u ) = { e j ( k u 2 d / 4 F 2 ) sin { k u A / F [ 1 + ( u d / 4 A F ) ] } / u for u > - u 0 0 for u < - u 0 ,
F 0 ( u ) = sin ( k u A F ) / u .
ũ M Q G ( n ) = ( 2 A F / d ) [ 1 + n ( λ d / 2 A 2 ) - 1 ] .
ũ 0 ( n ) = n ( λ F / 2 A ) ,
f ( x ) = Y ( x + h cos α ) , g ( x 0 ) = Y ( - x 0 - l sin α ) for u < F tan α , f ( x ) = Y ( - x ) , g ( x 0 ) = Y ( x 0 + h cos α - l sin α ) for u > F tan α .
f ( x ) = Y ( - x ) ,             g ( x 0 ) = Y ( x 0 - l sin α ) ,