Abstract

The paper presents an asymptotic expression of relative intensity distribution in a Fresnel diffraction pattern at an opaque straight strip illuminated with a spherical wave. The asymptotic expression is used in an analysis showing an area of validity where the asymptotic expression reduces to an asymptotic expression of relative intensity distribution in a Fresnel diffraction at a half plane. The area of validity is defined through width of the geometrical shadow in a Fresnel diffraction pattern at an opaque straight strip and distance of a point under study to the center of the Fresnel diffraction pattern. Within this area, relative intensity in the Fresnel diffraction pattern at an opaque straight strip shows sinusoidal behavior, which can be used in easy location of maxima or minima of the relative intensity. The result of the analysis is supported by experiments realized in the area of validity and outside it.

© 2012 Optical Society of America

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References

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  1. J. Komrska, “Intensity and phase in Fresnel diffraction by a plane screen consisting of parallel strips,” J. Mod. Opt. 14, 127–146 (1967).
  2. F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
    [CrossRef]
  3. E. A. Rueda, F. F. Medina, and J. F. Barrera, “Diffraction criterion for a slit under spherical illumination,” Opt. Commun. 274, 32–36 (2007).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, 1959).
  5. M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).
  6. K. D. Mielenz, “Computation of Fresnel Integrals,” J. Res. Natl. Inst. Stand. Technol. 102, 363–365 (1997).
    [CrossRef]
  7. K. M. Abedin, M. R. Islam, and A. F. M. Y. Haider, “Computer simulation of Fresnel diffraction from rectangular apertures and obstacles using the Fresnel integrals approach,” Opt. Laser Technol. 39, 237–246 (2007).
    [CrossRef]
  8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, eds. (Academic, 2000).
  9. V. M. Serdyuk, “Exact solutions for electromagnetic wave diffraction by a slot and strip,” Int. J. Electron. Commun. (AEÜ) 65, 182–189 (2011).
  10. P. Wolfe, “The diffraction of waves by slits and strips,” SIAM J. Appl. Math. 19, 20–32 (1970).
    [CrossRef]
  11. M. Glass, “Diffraction of a Gaussian beam around a strip mask,” Appl. Opt. 37, 2550–2562 (1998).
    [CrossRef]
  12. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

2011

V. M. Serdyuk, “Exact solutions for electromagnetic wave diffraction by a slot and strip,” Int. J. Electron. Commun. (AEÜ) 65, 182–189 (2011).

2007

K. M. Abedin, M. R. Islam, and A. F. M. Y. Haider, “Computer simulation of Fresnel diffraction from rectangular apertures and obstacles using the Fresnel integrals approach,” Opt. Laser Technol. 39, 237–246 (2007).
[CrossRef]

E. A. Rueda, F. F. Medina, and J. F. Barrera, “Diffraction criterion for a slit under spherical illumination,” Opt. Commun. 274, 32–36 (2007).
[CrossRef]

1998

1997

K. D. Mielenz, “Computation of Fresnel Integrals,” J. Res. Natl. Inst. Stand. Technol. 102, 363–365 (1997).
[CrossRef]

F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
[CrossRef]

1970

P. Wolfe, “The diffraction of waves by slits and strips,” SIAM J. Appl. Math. 19, 20–32 (1970).
[CrossRef]

1967

J. Komrska, “Intensity and phase in Fresnel diffraction by a plane screen consisting of parallel strips,” J. Mod. Opt. 14, 127–146 (1967).

Abedin, K. M.

K. M. Abedin, M. R. Islam, and A. F. M. Y. Haider, “Computer simulation of Fresnel diffraction from rectangular apertures and obstacles using the Fresnel integrals approach,” Opt. Laser Technol. 39, 237–246 (2007).
[CrossRef]

Barrera, J. F.

E. A. Rueda, F. F. Medina, and J. F. Barrera, “Diffraction criterion for a slit under spherical illumination,” Opt. Commun. 274, 32–36 (2007).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, 1959).

Glass, M.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, eds. (Academic, 2000).

Haider, A. F. M. Y.

K. M. Abedin, M. R. Islam, and A. F. M. Y. Haider, “Computer simulation of Fresnel diffraction from rectangular apertures and obstacles using the Fresnel integrals approach,” Opt. Laser Technol. 39, 237–246 (2007).
[CrossRef]

Islam, M. R.

K. M. Abedin, M. R. Islam, and A. F. M. Y. Haider, “Computer simulation of Fresnel diffraction from rectangular apertures and obstacles using the Fresnel integrals approach,” Opt. Laser Technol. 39, 237–246 (2007).
[CrossRef]

Komrska, J.

J. Komrska, “Intensity and phase in Fresnel diffraction by a plane screen consisting of parallel strips,” J. Mod. Opt. 14, 127–146 (1967).

Maosheng, C.

F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
[CrossRef]

Medina, F. F.

E. A. Rueda, F. F. Medina, and J. F. Barrera, “Diffraction criterion for a slit under spherical illumination,” Opt. Commun. 274, 32–36 (2007).
[CrossRef]

Mielenz, K. D.

K. D. Mielenz, “Computation of Fresnel Integrals,” J. Res. Natl. Inst. Stand. Technol. 102, 363–365 (1997).
[CrossRef]

Rueda, E. A.

E. A. Rueda, F. F. Medina, and J. F. Barrera, “Diffraction criterion for a slit under spherical illumination,” Opt. Commun. 274, 32–36 (2007).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, eds. (Academic, 2000).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

Serdyuk, V. M.

V. M. Serdyuk, “Exact solutions for electromagnetic wave diffraction by a slot and strip,” Int. J. Electron. Commun. (AEÜ) 65, 182–189 (2011).

Shiming, Q.

F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, 1959).

Wolfe, P.

P. Wolfe, “The diffraction of waves by slits and strips,” SIAM J. Appl. Math. 19, 20–32 (1970).
[CrossRef]

Xiaoyong, F.

F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
[CrossRef]

Yan, Z.

F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
[CrossRef]

Yongzhi, B.

F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
[CrossRef]

Appl. Opt.

Int. J. Electron. Commun. (AEÜ)

V. M. Serdyuk, “Exact solutions for electromagnetic wave diffraction by a slot and strip,” Int. J. Electron. Commun. (AEÜ) 65, 182–189 (2011).

J. Mod. Opt.

J. Komrska, “Intensity and phase in Fresnel diffraction by a plane screen consisting of parallel strips,” J. Mod. Opt. 14, 127–146 (1967).

J. Res. Natl. Inst. Stand. Technol.

K. D. Mielenz, “Computation of Fresnel Integrals,” J. Res. Natl. Inst. Stand. Technol. 102, 363–365 (1997).
[CrossRef]

Opt. Commun.

E. A. Rueda, F. F. Medina, and J. F. Barrera, “Diffraction criterion for a slit under spherical illumination,” Opt. Commun. 274, 32–36 (2007).
[CrossRef]

Opt. Laser Technol.

F. Xiaoyong, C. Maosheng, Z. Yan, B. Yongzhi, and Q. Shiming, “Fine structure in Fresnel diffraction patterns and its application in optical measurement,” Opt. Laser Technol. 29, 383–387 (1997).
[CrossRef]

K. M. Abedin, M. R. Islam, and A. F. M. Y. Haider, “Computer simulation of Fresnel diffraction from rectangular apertures and obstacles using the Fresnel integrals approach,” Opt. Laser Technol. 39, 237–246 (2007).
[CrossRef]

SIAM J. Appl. Math.

P. Wolfe, “The diffraction of waves by slits and strips,” SIAM J. Appl. Math. 19, 20–32 (1970).
[CrossRef]

Other

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, eds. (Academic, 2000).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, 1959).

M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Coordinate system for observation of Fresnel diffraction at an opaque strip. A spherical wave originating from a point source (x0, y0, z0) and diffracted at the strip located in the plane (xp, yp, 0) produces a diffraction pattern in the plane (x, y, z). (b) Relative intensity in Fresnel diffraction at an opaque strip (solid line) and its asymptotic equivalent (dotted line) evaluated for Δν=10. Dashed lines indicate validity limits of an asymptotic expression of relative intensity in Fresnel diffraction at an opaque strip. (c) Difference between relative intensity in Fresnel diffraction at an opaque strip and its asymptotic equivalent within the range ν6,8.

Fig. 2.
Fig. 2.

Term I2 (a) and I3 (b) of Eq. (18) plotted as a function of ν for different values of Δν.

Fig. 3.
Fig. 3.

(a) Behavior of the border of an area (dashed line) where the quotient (I1+I2)/I3>100 (gray area above the dashed line). (b) Term I46=I4+I5+I6 of Eq. (18) plotted as a function of ν for different values of Δν.

Fig. 4.
Fig. 4.

(a) Quotient (I1+I2)/I46 plotted as a function of ν for different values of Δν. (b) Difference νsνa of the beginning of an interval of ν in which (I1+I2)/I46>100 and limit of validity νa=2+Δν/2 plotted as a function of Δν. (c) Comparison of borders delimiting areas where (I1+I2)/I3>100 and |(I1+I2)/I3|>100 (dashed line), (I1+I2)/(I3+I46)>100 and |(I1+I2)/(I3+I46)|>100 (solid line). The dotted line connects points where νΔν/2=2.

Fig. 5.
Fig. 5.

Plots showing the regions in which |(I1+I2)/(I3+I46)|>100 and νΔν/22. (a) White spaces represent regions where |(I1+I2)/(I3+I46)|100. (b) The dark gray area is a result of high frequency alternation of the both regions. The light gray area represents the region in which (I1+I2)/(I3+I46)>100.

Fig. 6.
Fig. 6.

Fresnel diffraction patterns detected in the regions where |(I1+I2)/(I3+I46)|100 (a, b) and (I1+I2)/(I3+I46)>100 (c, d). (a) A Fresnel diffraction pattern by a metal strip of width xp2xp1=1.8·103m (parameters of the experimental setup according to Fig. 1(a): z0=784·103m, z=884·103m, Δν=4.96). (b) A Fresnel diffraction pattern by a half plane (parameters of the experimental setup according to Fig. 1(a): z0=784·103m, z=884·103m). (c) A Fresnel diffraction pattern by a metal strip of width xp2xp1=6·103m (parameters of the experimental setup according to Fig. 1(a): z0=7·103m, z=298·103m, Δν=128.98). (d) A Fresnel diffraction pattern by a half plane (parameters of the experimental setup according to Fig. 1(a): z0=7·103m, z=298·103m).

Fig. 7.
Fig. 7.

Behaviors of normalized relative intensity along the center rows of images of diffraction patterns in Fig. 6(a) (a), Fig. 6(b) (b), Fig. 6(c) (c), and Fig. 6(d) (d) as a function of pixel’s coordinate measured to the origin of an active area of used digital image sensor.

Equations (19)

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f(xp,yp)=exp(ikrpo)rpop(xp,yp),
p(xp,yp)={1;xp<xp1andxp>xp2,<yp<,0;xp1xpxp2,<yp<.
U(x,y,z)=iλzexp(ikz)f(xp,yp)exp(ik(xxp)2+(yyp)22z)dxpdyp.
U(x,y,z)=Us(x,y,z)(1i12ν1ν2exp(iπν22)),
Us(x,y,z)=exp(ik(z+zo))z+zoexp(ik(xxo)2+(yyo)22(z+zo))
νm=[kπ(1zo+1z)]1/2(xoz+xzozo+zxpm);m=1,2.
C(ν)=0νcos(πx22)dx,
S(ν)=0νsin(πx22)dx,
I(ν,Δν)=12{[1C(ν+Δν2)+C(νΔν2)]2+[1S(ν+Δν2)+S(νΔν2)]2},
Δν=ν1ν2,
ν=ν1+ν22.
p(xp,yp)={1;xp>xp2,<yp<,0;<xpxp2,<yp<.
I(ν2)=12{[12+C(ν2)]2+[12+S(ν2)]2}.
C(x)12+1πxsin(π2x2),
S(x)121πxcos(π2x2),
cos(πνΔν)=cos(π2(νΔν2)2)cos(π2(ν+Δν2)2)+sin(π2(νΔν2)2)sin(π2(ν+Δν2)2)
2sin(π2(x212))=sin(π2x2)cos(π2x2),
I(ν,Δν)=12{2+22π(νΔν2)sin[π2(νΔν2)2π4]22π(ν+Δν2)sin[π2(ν+Δν2)2π4]2cos(πνΔν)π2(νΔν2)(ν+Δν2)+1π2(νΔν2)2+1π2(ν+Δν2)2}.
I(ν,Δν)=12{2+22π(νΔν2)sin[π2(νΔν2)2π4]};ν2+Δν2.

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