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Montgomery rings derivation using angular spectrum approach
Optik 123 (2012) 831–832
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Montgomery rings derivation using angular spectrum approach M. Abolhassani ∗ , N. Kamani Department of Physics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
a r t i c l e
i n f o
Article history: Received 4 January 2011 Accepted 5 June 2011
PACS: 42.25.Fx 42.30.Kq 42.30.Lr
a b s t r a c t Self-imaging objects are not restricted to periodic ones. Montgomery derived the necessary and sufficient conditions in order that a plane object, illuminated by a plane monochromatic wave of normal incidence, images itself. On the basis of the reduced wave equation, he showed that the conditions for this case demand that the spatial frequencies of the self-imaging object should lie on discrete rings. In this paper, the above result has been obtained by using angular spectrum concept. © 2011 Elsevier GmbH. All rights reserved.
Keywords: Self-imaging objects Montgomery rings Angular spectrum
1. Introduction When a periodic object, like a diffraction grating, is illuminated with a monochromatic plane wave, field (and hence intensity) distribution at some distances behind the object is identical to the field just behind it. This effect is called Talbot or self-imaging effect, and discovered by W.F.H. Talbot in 1836. Rayleigh was the first person who explained this experiment based on diffraction. Montgomery [1] showed that lateral periodicity is a sufficient but not necessary condition for the Talbot effect to occur. In fact, he showed that the spatial frequencies of self-imaging objects should be discrete and located on rings, called Montgomery rings. He started his treatment based on the reduced wave equation and longitudinal periodicity of the wave-field function. A detailed explanation of the subject can be seen in [2]. In this paper, it is shown that Montgomery rings can be found easier and more straightforward by using angular spectrum approach.
and A(fX , fY ;z) represent Fourier transforms of these two complex fields, respectively; i.e.
+∞
−∞
+∞
Suppose that a wave is incident on xy plane traveling with a component of propagation in the positive z direction. Let the complex fields on xy plane (z = 0) and on a second, parallel plane a distance z to first plane are U(x, y, 0) and U(x, y, z), respectively. Let A(fX , fY ;0)
+∞
−∞
0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.06.049
(1)
A(fX , fY ; z) exp [i2(fX x + fY y)] dfX dfY .
(2)
+∞
−∞
The two spatial spectra are related to each other as [3]
A(fX , fY ; z) = A(fX , fY ; 0) exp i
2z
1 − 2 fX2 + fY2
.
(3)
Eq. (3) put into Eq. (2) gives
+∞
+∞
U(x, y, z) = −∞
2z A(fX , fY ; 0) exp i
× exp[i2(fX x + fY y)]dfX dfY .
1 − 2
fX2
+ fY2
(4)
If we wish to have a field distribution at the second plane identical to the field at the first one, U(x, y, z) = U(x, y, 0),
∗ Corresponding author. Tel.: +98 861 3130671; fax: +98 861 4173406. E-mail address: [email protected] (M. Abolhassani).
A(fX , fY ; 0) exp[i2(fX x + fY y)]dfX dfY , −∞
U(x, y, z) =
−∞
2. Theoretical approach
U(x, y, 0) =
(5)
the first exponential in Eq. (4) should be equal to one for every spatial frequency making the field function U(x, y, 0). In other words,
832
M. Abolhassani, N. Kamani / Optik 123 (2012) 831–832
each spatial frequency component of this field should satisfy the following condition: 2z or
1 − 2 fX2 + fY2 = 2m, m = 1, 2, 3, . . . ,
f 2 = fX2 + fY2 =
1 m2 − 2 . 2 z
(6)
(7)
This is exactly identical to the result that had been obtained earlier by Montgomery. In the case of a plane object, illuminated by a plane monochromatic wave of normal incidence, the field just behind the object, U(x, y, 0), is equal to its amplitude transmittance function, t(x, y). Therefore in this case U(x, y, z) = t(x, y). This means that the object is self-imaged.
(8)
3. Conclusions We present a different approach for derivation of Montgomery rings by using angular spectrum method. Present approach is simpler and more straightforward in comparison to other method. Probably, this method is more suitable for studying of selfimaging objects and discovery of the ones, other than linear gratings, with relatively easy technology. References [1] W.D. Montgomery, Self-imaging objects of infinite aperture, J. Opt. Soc. Am. 57 (1967) 772–778. [2] K. Patorsky, The self-imaging phenomenon and its applications, in: E. Wolf (Ed.), Progress in Optics, vol. 27, 1989, pp. 1–108. [3] J. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1996.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Montgomery rings derivation using angular spectrum approach M. Abolhassani ∗ , N. Kamani Department of Physics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran
a r t i c l e
i n f o
Article history: Received 4 January 2011 Accepted 5 June 2011
PACS: 42.25.Fx 42.30.Kq 42.30.Lr
a b s t r a c t Self-imaging objects are not restricted to periodic ones. Montgomery derived the necessary and sufficient conditions in order that a plane object, illuminated by a plane monochromatic wave of normal incidence, images itself. On the basis of the reduced wave equation, he showed that the conditions for this case demand that the spatial frequencies of the self-imaging object should lie on discrete rings. In this paper, the above result has been obtained by using angular spectrum concept. © 2011 Elsevier GmbH. All rights reserved.
Keywords: Self-imaging objects Montgomery rings Angular spectrum
1. Introduction When a periodic object, like a diffraction grating, is illuminated with a monochromatic plane wave, field (and hence intensity) distribution at some distances behind the object is identical to the field just behind it. This effect is called Talbot or self-imaging effect, and discovered by W.F.H. Talbot in 1836. Rayleigh was the first person who explained this experiment based on diffraction. Montgomery [1] showed that lateral periodicity is a sufficient but not necessary condition for the Talbot effect to occur. In fact, he showed that the spatial frequencies of self-imaging objects should be discrete and located on rings, called Montgomery rings. He started his treatment based on the reduced wave equation and longitudinal periodicity of the wave-field function. A detailed explanation of the subject can be seen in [2]. In this paper, it is shown that Montgomery rings can be found easier and more straightforward by using angular spectrum approach.
and A(fX , fY ;z) represent Fourier transforms of these two complex fields, respectively; i.e.
+∞
−∞
+∞
Suppose that a wave is incident on xy plane traveling with a component of propagation in the positive z direction. Let the complex fields on xy plane (z = 0) and on a second, parallel plane a distance z to first plane are U(x, y, 0) and U(x, y, z), respectively. Let A(fX , fY ;0)
+∞
−∞
0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.06.049
(1)
A(fX , fY ; z) exp [i2(fX x + fY y)] dfX dfY .
(2)
+∞
−∞
The two spatial spectra are related to each other as [3]
A(fX , fY ; z) = A(fX , fY ; 0) exp i
2z
1 − 2 fX2 + fY2
.
(3)
Eq. (3) put into Eq. (2) gives
+∞
+∞
U(x, y, z) = −∞
2z A(fX , fY ; 0) exp i
× exp[i2(fX x + fY y)]dfX dfY .
1 − 2
fX2
+ fY2
(4)
If we wish to have a field distribution at the second plane identical to the field at the first one, U(x, y, z) = U(x, y, 0),
∗ Corresponding author. Tel.: +98 861 3130671; fax: +98 861 4173406. E-mail address: [email protected] (M. Abolhassani).
A(fX , fY ; 0) exp[i2(fX x + fY y)]dfX dfY , −∞
U(x, y, z) =
−∞
2. Theoretical approach
U(x, y, 0) =
(5)
the first exponential in Eq. (4) should be equal to one for every spatial frequency making the field function U(x, y, 0). In other words,
832
M. Abolhassani, N. Kamani / Optik 123 (2012) 831–832
each spatial frequency component of this field should satisfy the following condition: 2z or
1 − 2 fX2 + fY2 = 2m, m = 1, 2, 3, . . . ,
f 2 = fX2 + fY2 =
1 m2 − 2 . 2 z
(6)
(7)
This is exactly identical to the result that had been obtained earlier by Montgomery. In the case of a plane object, illuminated by a plane monochromatic wave of normal incidence, the field just behind the object, U(x, y, 0), is equal to its amplitude transmittance function, t(x, y). Therefore in this case U(x, y, z) = t(x, y). This means that the object is self-imaged.
(8)
3. Conclusions We present a different approach for derivation of Montgomery rings by using angular spectrum method. Present approach is simpler and more straightforward in comparison to other method. Probably, this method is more suitable for studying of selfimaging objects and discovery of the ones, other than linear gratings, with relatively easy technology. References [1] W.D. Montgomery, Self-imaging objects of infinite aperture, J. Opt. Soc. Am. 57 (1967) 772–778. [2] K. Patorsky, The self-imaging phenomenon and its applications, in: E. Wolf (Ed.), Progress in Optics, vol. 27, 1989, pp. 1–108. [3] J. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1996.