Abstract

Peculiarities of high-frequency filtering (contouring) of images of symmetric thick (extended along the optical axis) edge are investigated in analytical form by the Kirchhoff–Fresnel approximation. They are based on a model of equivalent diaphragms (transparencies) earlier proposed by the authors [J. Opt. Soc. Am. 71, 483 (1981)]. Expressions for the fields in special cases of perfect absorption and reflection are derived. For the case of slight volumetric effects, a structure of the contoured images of the front and back faces is shown to be similar to a structure of the contoured image of the plane edge. But the influence of the extension causes a shift of the basic minimum position proportional to the size of the Fresnel zone. When volumetric effects are significant, the profile of the filtered image of the front face has a single maximum (instead of a double one). Here the structure of the filtered image of the back face depends to a great extent on the reflection properties of the inner surface, changing from the form with a double maximum to the form with a single one. There is agreement between theoretical and experimental results.

© 1998 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  2. H. Honl, M. Maue, K. Westphahl, Theorie der Beuding, (Springer-Verlag, Berlin, 1961).
  3. V. A. Borovikov, B. E. Kinber, Geometricheskaya Teoriya Diffraktsii (Geometrical Diffraction Theory) (Svyaz, Moscow, 1978).
  4. Yu. V. Chugui, V. P. Koronkevich, B. E. Krivenkov, S. V. Mikhlyaev, “Quasi-geometrical method for Fraunhofer diffraction calculation for three-dimensional bodies,” J. Opt. Soc. Am. 71, 483–489 (1981).
    [CrossRef]
  5. Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by volumetric bodies of constant thickness,” J. Opt. Soc. Am. A 6, 617–626 (1989).
    [CrossRef]
  6. Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by reflecting volumetric bodies of constant thickness,” Optoelectron. Instrum. Data Process. 4, 113–118 (1991).
  7. K. G. Birch, “A spatial frequency filter to remove zero frequency,” Opt. Acta 15, 113–127 (1968).
  8. S. V. Gupta, D. Sen, “Diffrimoscopic image formation under partially coherent illumination (straight edge),” Opt. Acta 18, 507–521 (1971).
    [CrossRef]
  9. P. J. S. Hutzler, “Spatial frequency filtering and its application to microscopy,” Appl. Opt. 16, 2264–2272 (1977).
    [CrossRef] [PubMed]
  10. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  11. Yu. V. Chugui, B. E. Krivenkov, “Investigation of two-dimensional binary edge contouring of opaque objects,” Opt. Acta 28, 157–167 (1981).
    [CrossRef]
  12. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).
  13. A. Vašiček, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  14. G. D. Dew, “The application of spatial filtering techniques to profile inspection and an associated interference phenomenon,” Opt. Acta 17, 237–257 (1970).
    [CrossRef]

1991 (1)

Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by reflecting volumetric bodies of constant thickness,” Optoelectron. Instrum. Data Process. 4, 113–118 (1991).

1989 (1)

1981 (2)

1977 (1)

1971 (1)

S. V. Gupta, D. Sen, “Diffrimoscopic image formation under partially coherent illumination (straight edge),” Opt. Acta 18, 507–521 (1971).
[CrossRef]

1970 (1)

G. D. Dew, “The application of spatial filtering techniques to profile inspection and an associated interference phenomenon,” Opt. Acta 17, 237–257 (1970).
[CrossRef]

1968 (1)

K. G. Birch, “A spatial frequency filter to remove zero frequency,” Opt. Acta 15, 113–127 (1968).

Birch, K. G.

K. G. Birch, “A spatial frequency filter to remove zero frequency,” Opt. Acta 15, 113–127 (1968).

Born, M.

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

Borovikov, V. A.

V. A. Borovikov, B. E. Kinber, Geometricheskaya Teoriya Diffraktsii (Geometrical Diffraction Theory) (Svyaz, Moscow, 1978).

Chugui, Yu. V.

Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by reflecting volumetric bodies of constant thickness,” Optoelectron. Instrum. Data Process. 4, 113–118 (1991).

Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by volumetric bodies of constant thickness,” J. Opt. Soc. Am. A 6, 617–626 (1989).
[CrossRef]

Yu. V. Chugui, V. P. Koronkevich, B. E. Krivenkov, S. V. Mikhlyaev, “Quasi-geometrical method for Fraunhofer diffraction calculation for three-dimensional bodies,” J. Opt. Soc. Am. 71, 483–489 (1981).
[CrossRef]

Yu. V. Chugui, B. E. Krivenkov, “Investigation of two-dimensional binary edge contouring of opaque objects,” Opt. Acta 28, 157–167 (1981).
[CrossRef]

Dew, G. D.

G. D. Dew, “The application of spatial filtering techniques to profile inspection and an associated interference phenomenon,” Opt. Acta 17, 237–257 (1970).
[CrossRef]

Gupta, S. V.

S. V. Gupta, D. Sen, “Diffrimoscopic image formation under partially coherent illumination (straight edge),” Opt. Acta 18, 507–521 (1971).
[CrossRef]

Honl, H.

H. Honl, M. Maue, K. Westphahl, Theorie der Beuding, (Springer-Verlag, Berlin, 1961).

Hutzler, P. J. S.

Kinber, B. E.

V. A. Borovikov, B. E. Kinber, Geometricheskaya Teoriya Diffraktsii (Geometrical Diffraction Theory) (Svyaz, Moscow, 1978).

Koronkevich, V. P.

Krivenkov, B. E.

Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by reflecting volumetric bodies of constant thickness,” Optoelectron. Instrum. Data Process. 4, 113–118 (1991).

Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by volumetric bodies of constant thickness,” J. Opt. Soc. Am. A 6, 617–626 (1989).
[CrossRef]

Yu. V. Chugui, V. P. Koronkevich, B. E. Krivenkov, S. V. Mikhlyaev, “Quasi-geometrical method for Fraunhofer diffraction calculation for three-dimensional bodies,” J. Opt. Soc. Am. 71, 483–489 (1981).
[CrossRef]

Yu. V. Chugui, B. E. Krivenkov, “Investigation of two-dimensional binary edge contouring of opaque objects,” Opt. Acta 28, 157–167 (1981).
[CrossRef]

Maue, M.

H. Honl, M. Maue, K. Westphahl, Theorie der Beuding, (Springer-Verlag, Berlin, 1961).

Mikhlyaev, S. V.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Sen, D.

S. V. Gupta, D. Sen, “Diffrimoscopic image formation under partially coherent illumination (straight edge),” Opt. Acta 18, 507–521 (1971).
[CrossRef]

Vašicek, A.

A. Vašiček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Westphahl, K.

H. Honl, M. Maue, K. Westphahl, Theorie der Beuding, (Springer-Verlag, Berlin, 1961).

Wolf, E.

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

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (4)

K. G. Birch, “A spatial frequency filter to remove zero frequency,” Opt. Acta 15, 113–127 (1968).

S. V. Gupta, D. Sen, “Diffrimoscopic image formation under partially coherent illumination (straight edge),” Opt. Acta 18, 507–521 (1971).
[CrossRef]

Yu. V. Chugui, B. E. Krivenkov, “Investigation of two-dimensional binary edge contouring of opaque objects,” Opt. Acta 28, 157–167 (1981).
[CrossRef]

G. D. Dew, “The application of spatial filtering techniques to profile inspection and an associated interference phenomenon,” Opt. Acta 17, 237–257 (1970).
[CrossRef]

Optoelectron. Instrum. Data Process. (1)

Yu. V. Chugui, B. E. Krivenkov, “Fraunhofer diffraction by reflecting volumetric bodies of constant thickness,” Optoelectron. Instrum. Data Process. 4, 113–118 (1991).

Other (6)

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).

A. Vašiček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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

H. Honl, M. Maue, K. Westphahl, Theorie der Beuding, (Springer-Verlag, Berlin, 1961).

V. A. Borovikov, B. E. Kinber, Geometricheskaya Teoriya Diffraktsii (Geometrical Diffraction Theory) (Svyaz, Moscow, 1978).

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Figures (13)

Fig. 1
Fig. 1

Coherent optical system for the spatial filtering of images: ABCD is the 3-D edge, and O1 and O2 are the objectives of direct and inverse Fourier transformation.

Fig. 2
Fig. 2

Representation of an absorbing thick symmetric edge shown by the model of equivalent diaphragms: (a) perfectly absorbing 3-D edge, (b) model of (a) in the form of equivalent transparencies, (c) generalized point model for forming a Fraunhofer diffraction pattern (Fourier spectrum).

Fig. 3
Fig. 3

Representation of the reflecting thick symmetric edge shown by the model of equivalent diaphragms: (a) perfectly reflecting 3-D edge, (b) model of (a) in the from of equivalent transparencies, (c) generalized point model.

Fig. 4
Fig. 4

Intensity distribution in the contoured image of the front face of the perfectly absorbing 3-D edge when volumetric effects are slight: N=1.5 and 4.

Fig. 5
Fig. 5

Intensity profile of the filtered image of the front face of the perfectly absorbing 3-D edge when volumetric effects are significant: N=0.4 and 0.2.

Fig. 6
Fig. 6

Intensity distribution in the contoured image of the backface of the perfectly absorbing 3-D edge when volumetric effects are significant: N=0.15 and 0.7.

Fig. 7
Fig. 7

Intensity profile of the filtered image of the front face of the perfectly reflecting 3-D edge when volumetric effects are slight: N=1.8 and 4.

Fig. 8
Fig. 8

Difference between the shift of the coordinate of the basic minimum of the contoured image (with respect to intensity) of the front face of the perfectly reflecting 3-D edge and linear approximation on N.

Fig. 9
Fig. 9

Intensity profile of the filtered image of the back face of the perfectly reflecting 3-D edge when volumetric effects are slight: N=1.8 and 4.

Fig. 10
Fig. 10

Intensity distribution in the contoured image of the back face of the reflecting edge in the case of significant volumetric effects: N=0.2 and 0.5.

Fig. 11
Fig. 11

Schematic diagram of the experimental coherent optical device for HF filtering images of the 3-D edge: 1, He–Ne laser; 2, collimator with pinhole; 3, tested object in the form of a parallelepiped; 4, projection objective; 5, contouring spatial filter of the binary type; 6, photodiode line sensor; 7, pc with adapter.

Fig. 12
Fig. 12

Experimental results of the HF filtering of the absorbing 3-D edge images in a coherent optical system (d=1 mm; N=1.6).

Fig. 13
Fig. 13

Comparison of the theory and the experimental data for the reflecting 3-D edge (d=1 mm): (a) front face, N=1.6; (b) back face, N=1.6; (c) front face, N=0.6; (d) back face, N=0.6.

Equations (50)

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H(ω)=ωω02 rectω2ω0,
h0(x)=ω0πsinc(ω0x),
fˆ(x)=1πω0d2dx2f(x)sin(ω0x)ω0x=1πω0ddxsin(ω0x)ω0x=1πω0sinc(ω0x),
fa(x)=Y(x),ga(x1)=Y(x1).
Fa(θ)=πδ(kθ)+Y˜(dθ)jkθ+12jkθexpj kdθ22,
Y˜(x)=exp(-jπ/4)λd-x expj kξ22ddξ.
fˆa(x)=0.5Y(x)+j2πY(x)exp(-jkx2/2d)x+0.5Y˜*(x),
gˆa(x1)=Y˜(x1)Y(x1).
fr(x)=sign(x),gr(x1)=Y(x1),
Fr(θ)=πδ(kθ)+2Y˜(dθ)jkθ.
fˆr(x)=F-1{Fr(θ)}=0.5Y(x)+jπY(x)exp(-jkx2/2d)x.
gˆr(x1)=sign˜(x1)Y(x1),
f^a(x)=j2πx2ω02[Y˜(-x+dθ0)-Y˜(-x-dθ0)]×exp-j kx22d1+j kx2d-2 sin(ω0x)λd×expj kdθ022-j π4+Y˜(dθ0)exp(jω0x)-Y˜(-dθ0)exp(-jω0x)-jω0x[Y˜(-dθ0)×exp(-jω0x)+Y˜(dθ0)exp(jω0x)]+sin(ω0x)λdω02expj kdθ022+πxω02(λd)3/2[Y˜(x+dθ0)-Y˜(x-dθ0)]×exp-j kx2d-j π4.
f^ a(x)=1πω0sinc(ω0x)-exp(jπ/4)λd2π2ω0sinc(ω0x).
xmin=λd22π,
f^ a(x)=exp(jω0x/2)πexp(jω0x/2)2ω0x-sin(ω0x/2)(ω0x)2.
g^ a(x1)=sinc(ω0x1)2πω0+exp(-jπ/4)πω0λd×sinc(ω0x1)+2ω0exp(jπ/4)(λd)3/2×x1Y(x1)expj kx122dsinc(ω0x1).
g^ a(x1)=12πω0sinc(ω0x1)+exp(-jπ/4)πω0λdsinc(ω0x1).
x1,min=32ω02λd=3N2πω0.
f^r(x)=jπx2ω02[Y˜(-x+dθ0)-Y˜(-x-dθ0)]×exp-j kx22d1+j kx2d-2 sin(ω0x)λd×expj kdθ022-j π4+Y˜(dθ0)exp(jω0x)-Y˜(-dθ0)exp(-jω0x)-jω0x[Y˜(-dθ0)exp(-jω0x)+Y˜(dθ0)exp(jω0x)].
f^r(x)=1πω0sinc(ω0x)-exp(jπ/4)λdπ2ω0sinc(ω0x).
xmin=λd2π.
f^r(x)=exp(jω0x/2)πexp(jω0x/2)ω0x-2 sin(ω0x/2)(ω0x)2.
g^r(x1)=1πω0sinc(ω0x1)-exp(jπ/4)λdπ2ω0sinc(ω0x1)-λd4π2ω0sinc(ω0x1).
x1,min=102λd20π+3λdω02.
g^r(x1)=2 exp(-jπ/4)πω0λdsinc(ω0x1)
f^a(x)=12δ(x)+jπxexp-j kx22dsin(ω0x)πx+exp(jπ/4)2λdexp-j kx22dsin(ω0x)πx.
δ(x)+jπxexp-j kx22dsin(ω0x)πx
=12π -2Y˜(dθ)rectω2ω0exp(jωx)dω.
Fn(θ)=-Y(x)Y˜(x-L+dθ)exp(-jωx)dx=πδ(θ)+1jkθY˜(dθ-L)+expj kdθ22-jωLY˜(L),
p(x)q˜(x+dθ)exp(-jωx)dx
=expj kdθ22p˜(x)q(x)exp(-jωx)dx,
Fn(θ)=expj kdθ22-Y˜(x)Y(x-L)exp(-jωx)dx.
-Y˜(dθ)Y(ω-ω0)exp(-jωx)dω
=πδ(x)-jxY˜(-x-dθ0)exp-j kx22d+Y˜(dθ0)exp(jω0x).
δ(x)+jπxexp-j kx22dsin(ω0x)πx=jπxω02[Y˜(-x+dθ0)-Y˜(-x-dθ0)]×exp-j kx22d+exp(-jω0x)Y˜(-dθ0)-exp(jω0x)Y˜(dθ0).
exp-j kx22dsin(ω0x)πx
=exp(-jπ/4)λd12π- expj kdθ22×rectω2ω0exp(jωx)dω=exp-j kx22d[Y˜(x+dθ0)-Y˜(x-dθ0)].
f^r(x)=j2πxω02[Y˜(-x+dθ0)-Y˜(-x-dθ0)]×exp-j kx22d+Y˜(-dθ0)exp(-jω0x)-Y˜(dθ0)exp(jω0x)+exp(jπ/4)2ω02λd[Y˜(x+dθ0)-Y˜(x-dθ0)]exp-j kx22d
f^ a(x)=j2πx2ω02[Y˜(-x+dθ0)-Y˜(-x-dθ0)]×exp-j kx22d1+j kx2d-2 sin(ω0x)λd×expj kdθ022-j π4+Y˜(dθ0)exp(jω0x)-Y˜(-dθ0)exp(-jω0x)-jω0x[Y˜(-dθ0)×exp(-jω0x)+Y˜(dθ0)exp(jω0x)]+sin(ω0x)λdω02expj kdθ022+πxω02(λd)3/2×[Y˜(x+dθ0)-Y˜(x-dθ0)]×exp-j kx22d-j π4.
f^r(x)=1πω0ddxδ(x)+jπxexp-j kx22dsinc(ω0x)=jπx2ω02[Y˜(-x+dθ0)-Y˜(-x-dθ0)]×exp-j kx22d1+j kx2d-2 sin(ω0x)λd×expj kdθ022-j π4+Y˜(dθ0)exp(jω0x)-Y˜(-dθ0)exp(-jω0x)-jω0x[Y˜(-dθ0)×exp(-jω0x)+Y˜(dθ0)exp(jω0x)].
gˆr(x1)=2gˆa(x1)-sin(ω0x)πx.
H(ω)=rectω2ω0.
fˆa(x)=Y(x)-exp(jπ/4)λd2πδ(x)+λd8πδ(x).
gˆa(x1)=Y(x1)-exp(jπ/4)λd2πδ(x1)-λd8πδ(x1)(whenN1).
gˆa(x1)=Y(x1)12+exp(-jπ/4)λdx1(whenN1).
gˆa(x1)=Y(x1)sin(ω0x)πx+exp(-jπ/4)ω0λd2π2sinc(ω0x1)-λdω08π2sinc(ω0x1)(whenN1),
gˆa(x1)=Y(x1)sin(ω0x)πx12+exp(-jπ/4)λdx1+exp(jπ/4)cos(ω0x1)ω02λd(whenN1).
g^r(x1)=1πω0sinc(ω0x1)-exp(jπ/4)λdπ2ω0sinc(ω0x1)-λd4π2ω0sinc(ω0x1)(whenN1),
g^r(x1)=2 exp(-jπ/4)πω0λdsinc(ω0x1)(whenN1).

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