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Intensity and phase distribution in deep Fresnel diffraction region generated by multi-pinhole aperture
Optik 125 (2014) 6329–6334
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Intensity and phase distribution in deep Fresnel diffraction region generated by multi-pinhole aperture Man Liu a,b,∗ a b
School of Science, Qilu University of Technology, Jinan 250353, China College of Physics and Electronics, Shandong Normal University, Jinan 250014, China
a r t i c l e
i n f o
Article history: Received 15 November 2013 Accepted 17 June 2014 PACS: 42.25.Fx 42.30.Ms Keywords: Intensity distribution Phase distribution The zero-contour of the real and imaginary parts
a b s t r a c t The Kirchhoff diffraction theory is applied to the multi-pinhole aperture diffraction screens, and the intensity, the zero-contour of the real and imaginary parts of complex amplitude and the phase distribution in deep Fresnel diffraction region are simulated. It is found that the number of bright spots, the zero-contours of the real and imaginary parts of complex amplitude and the phase singularities are all related to number of pinholes in diffraction screens. The brightness of bright spots in center of each pattern would become larger with increasing number of pinholes. In addition, there are many lines, on which the intensity value is close to 0. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction In interference field, there are isolated points of vanishing intensity at which the phase is undefined [1]. From phase singularity, equi-phase lines radiate outward in a star-like fashion lead to the phase spiral distribution, so that these phase singularities are also called optical vortices [1,2]. The sign of the phase singularity is positive if the phase circulates counterclockwise and negative if the phase circulates clockwise [3]. Vortices in light fields owing to potential applications have driven a surge of interest in recent years. For example, in selfdefocusing media, an optical vortex induces a waveguide, which may be used for optical switching [4]. In linear optics, optical vortices have been already utilized as “Optical Tweezers” to manipulate micrometer-sized particles such as cells [5,6]. Simpson et al. [7] achieved rotation of the particle using optical vortex, and this technology is called “Optical Spanners”. Optical vortices may also be useful for optical limiting purposes or for optical data storage [8]. The vortex characteristics are common to all wave fields, so understanding the optical vortices can help to understand the vortex characteristics in other physical systems.
∗ Correspondence to: School of Science, Qilu University of Technology, Jinan 250353, China. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2014.06.154 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Recently, some detailed studies of the phase singularities in far-field diffraction region are reported [9,10]. The phase singularities produced by multi-aperture diffraction screens have the complex structure produced by the optical wave interference in the far fields [11,12]. This paper discusses the intensity and the phase singularities distribution generated by multi-pinhole aperture in deep Fresnel diffraction region. The Kirchhoff diffraction theory is applied to the mutli-pinhole aperture diffraction screens. The intensity, the zero-contour of the real and imaginary parts and the phase distribution in deep Fresnel diffraction region are shown. The light intensity and the zero-contour of the real and imaginary parts of complex amplitude and the distribution of phase singularities are studied in detail. It is found that, for an even number of pinholes, the bright spots in diffractive field show central symmetry distribution, and for an odd number of pinholes, the intensity distribution patterns are mirrored in the x-axis. When the observation plane close to the diffraction screens, the zero-points of light intensity can form line segment, on which the eccentricities of the light intensity isoline are close or equal to 1, and the intensity changes very fast on both sides of the zero-line of light intensity. The zerocontours of the real and imaginary parts of complex amplitude are closed curves, so that the number of intersection points of the zero-contour is even, and positive and negative singularities are equal. The lines of zero-intensity change shorter and shorter with the increase in the pinhole. Therefore, the phase singularities in
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M. Liu / Optik 125 (2014) 6329–6334
with
c =1−
∂h(x0 , y0 ) ∂x0
2
−
∂h(x0 , y0 ) ∂y0
d = (h(x0 , y0 ) − z)c − (x0 − x)
2 (6)
∂h(x0 , y0 ) ∂h(x0 , y0 ) − (y0 − y) ∂x0 ∂y0
(7)
where T (x0 , y0 ) is the aperture function, which satisfies
Fig. 1. The geometry of mutli-pinhole aperture diffraction screen and interference field.
central region of light fields change from special phase singularity to the conventional phase singularity, when the number of pinhole is equal to 20, near the center of interference field, the positive and negative phase singularities are alternately appear and form a phase vortex chain. The results of this paper may be important to study the characteristics of intensity and phase in deep Fresnel diffraction region. 2. The calculation methods of interference fields To present the numerical simulating method of the intensity, the zero-contour of the real and imaginary parts and the phase distribution in deep Fresnel diffraction region, we consider the light diffraction geometry shown in Fig. 1. The diffraction screen is square, blocked with a mutli-pinhole opaque plate. The radius of each pinhole is equal. The pinholes are uniformly distributed in a circle of radius R in the x0 y0 -plane, the pinholes are indicated by open dots and the angular coordinate of the nth pinhole is ±n = 2n/N (n = 0, 1, 3, . . ., N − 1). Observation planes parallel to the diffraction screens. Here z is the distance from the diffraction screen to the observation plane. A parallel laser beam with wavelength and unity amplitude illuminates a multi-pinhole diffraction screen in the x0 y0 -plane. With the unimportant constant factor dropped, the complex amplitude of interference field after diffraction screen can be expressed as U0 (x0 , y0 ) = exp[ik(n − 1)h(x0 , y0 )]
(1)
where h(x0 , y0 ) is the thickness of diffraction screen, n is the refractive index of the glass and k = 2/ is the wavenumber. By using the Kirchhoff diffraction theory, we write the complex amplitude U of interference field at an arbitrary point in the observation plane as U(x, y) =
1 4
where
G G
∂U0 (x0 , y0 ) ∂G − U0 (x0 , y0 ) ∂e0 ∂e0 is
Green’s
dx0 dy0
(2) r=
function,
2 1/2
{(x − x0 )2 + (y − y0 )2 + [z − h(x0 , y0 )] } is the length between an arbitrary point in the observation plane and an arbitrary point in the diffraction screen, e0 is unit vector of each point in the x0 y0 -plane along the z-axis, namely e0 = cos ˛៝i + cos ˇ៝j + cos k៝ =
2 1/2
2
= [1 + (∂h/∂x0 ) + (∂h/∂y0 ) ] ∂ 1 = (e0 • ∇ ) = ∂e0
1 ∂h ∂h ៝ ៝ ៝ i− j+k −
∂x0
(3)
∂y0
, and partial derivative
∂h ∂ ∂h ∂ ∂ − · − · + ∂x0 ∂x0 ∂y0 ∂y0 ∂h
(4)
After a simple derivation, Eq. (2) can be written as U(x, y) =
1 4
T (x0 , y0 )
T (x0 , y0 ) =
×
2
(x0 − a)2 + (y0 − b) ≤ 8.0 m
0≤
0,
otherwise
(8)
The complex amplitude of light field in two dimensions can be represented as U(x, y) = exp[i(x, y)] = (x, y) + i (x, y)
(9)
where ≥0 is the amplitude, (x, y) is the phase with the range of (−, ] and (x, y) and (x, y) are the real and imaginary parts of complex amplitude, respectively. The phase is undefined, when the amplitude is equal to 0, and the phase singularities occur at the intersection of the real and imaginary parts of the field. Around a phase singularity, the phase changes by an integer multiple of 2, for a closed loop c enclosing a phase singularity yields an integer multiple of 2,
=
∇ · d = 2S
d = c
(10)
c
where S is an integer which is known as the topological charge, or strength, of the phase singularities, it can be defined as +1 (−1) if the phase value increases by 2 in the counterclockwise (clockwise) direction around the phase vortex. We simulate the intensity, phase and the zero-contour of the real and imaginary parts of complex amplitude distribution of interference field produced by mutli-pinhole aperture diffraction screens in deep Fresnel diffraction region, respectively. The distance z is set at 20.0 m. The wavelength of the illuminating light is set at 0.6328 m. The refractive index n is set at 1.532. The range of the generated surface is 60.0 m × 60.0 m with 301 × 301 sampling points. For N is equal to 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively. The radius of circle is 22.0 m and the radius of pinhole is set at 8.0 m. The range of the observation plane is 12.0 m × 12.0 m with 301 × 301 sampling points. Using these parameters and Eqs. (5) and (9) for the numerical calculations, we can obtain the discrete numerical distributions of the real and imaginary parts of interference field in deep Fresnel diffraction region. Subsequently, the intensity I(x, y) is calculated by using I(x, y) = U(x, y)U ∗ (x, y). Fig. 2a–i shows the intensity distribution patterns produced by the mutli-pinhole apertures diffraction screens with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively. The grayscale intensity distribution patterns are plotted using arbitrary units. The grayscale level number is 32. The value changes from minimum to maximum presenting from dark to light. The size of observation plane is much smaller than the size of diffraction screen. Therefore, the light intensity on the observation plane is the intervention results of the diffraction light from each pinhole in the diffraction plane, there is no transmitted light. From Fig. 2, we can see that the bright spots show central symmetry distribution, if the number of pinhole is equal to an even number, the bright spots are mirrored in the x-axis for an odd number of pinholes, which show a regular arrangement. Fist, the number of bright spot will increase with the number of pinholes N increase, then decrease. In Fig. 2a, we can see that the diffraction
i2(n − 1)c
11 2 [(n − 1)h(x0 , y0 ) + r] exp i r
1,
−
d d + 2∞ r r
dx0 dy0
(5)
M. Liu / Optik 125 (2014) 6329–6334
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Fig. 2. The intensity distribution patterns behind multi-pinhole aperture in deep Fresnel diffraction region for N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively, illuminated by a plane wave.
stripes appear, and symmetry distribution with respect to x-axis and y-axis. In Fig. 2b–i, there are many bright spots, which regular arrangement, the bright spots show uniform distribution, and the bright spots are arranged in different shapes correspond with the number of pinholes in diffraction screen. The bright spots distributed uniformly around the center with the increase in the number of pinholes, the brightness of the ring around the center gets progressively increase form the innermost to the outermost rings. The number of bright spots near the center is equal to the number of pinholes in diffraction screens. In order to analyze the phase distribution of interference field generated by multi-pinhole aperture diffraction screens in deep Fresnel diffraction region, the zero-contour of the real and imaginary parts of complex amplitude are shown in Fig. 3a–i with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, which corresponding to Fig. 2a–i, respectively. The black solid line and dashed lines give the zero-contour of the real and imaginary parts of complex amplitude, respectively. The image sizes are 12.0 m × 12.0 m. In Fig. 3a (for N = 2), we can see that the zero-contour of the real and imaginary parts of complex amplitude, i.e., black solid and the dashed lines have almost equal intervals and they are almost parallel to each other in the horizontal direction, and symmetry distribution with respect to x-axis and y-axis. Except N = 2, by comparing Fig. 3b and d, we can see that the patterns of the real and imaginary parts of complex amplitude are mirrored in the x-axis, respectively. From Fig. 3c, e–i, we may see that the zero-contours of the real and imaginary parts of complex amplitude of interference field in deep Fresnel diffraction region are closed curves. Therefore, the number of intersection points of real and imaginary parts is even, namely the number of phase singularities is even, the number of positive and negative vortices are equal. So, the algebraic sum of topological charge in the whole pattern is zero. In addition, the lines of zero-intensity change shorter and shorter and the number of intersection points of real and imaginary parts become lager with the increase in the pinhole.
3. The phase distribution and topological charges of special phase singularities It would be interesting to know the phase distribution in deep Fresnel diffraction region near the multi-pinhole aperture. Fig. 4a–i shows that the color phase distribution maps produced by the multi-pinhole aperture with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively, in correspondence with the images in Figs. 2 and 3, where the sizes of each image are also 12.0 m × 12.0 m with the phase ranging from − to in eight color levels, namely the black lines indicate phase contours separated by /4. One can see form Fig. 4a that the phase distributions appear striped structures in y direction. From Fig. 4b and d, we can see that the phase singularities are mirrored in the x-axis. From Fig. 4c, e–i, we may see that the positive and negative phase singularities are alternately appear and form a series ring, we call this “phase singularity chains”. In each phase distribution pattern, form the innermost to the outermost “phase singularity chains”, the number of positive and negative phase singularities is equal to each other on each chain, and this is equal to the number of pinholes in the diffraction screens. As the number of pinholes increases, this phenomenon is more obvious. The phase singularities are uniformly distributed on the phase singularity chains. 4. The characteristics of intensity and phase distribution near the zero-lines of light intensity In the following, we analyze the characteristics of intensity distribution near the zero-lines of light intensity. We enlarge the squared regions in Fig. 2c, and it is shown in Fig. 5a. In Fig. 5a, around one circular bright spot, there are four other circular bright spots and four other stilliform bright spots adjacent to it. If the maximum light intensity in Fig. 5a is set at 3.0, we can obtain Fig. 5b. In Fig. 5b, the light intensity on the black line segments is close to 0, we call them “the zero-line of light intensity”.
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M. Liu / Optik 125 (2014) 6329–6334
Fig. 3. Distributions of the zero-contour of the real and imaginary parts of complex amplitude. The black solid lines and dashed lines are, respectively, the zero-contour of the real and imaginary parts. (a–i) For N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively.
In order to observe clearly the light intensity distribution near the zero-lines of light intensity, we use three-dimensional images to display the light intensity distribution near the zero-lines of light intensity (the squared regions in Fig. 5b) and show it in Fig. 5c. From
Fig. 5c, we can see that the light intensity changes rapidly on both sides of the zero-line of intensity. Similarly, we enlarge the squared regions in Fig. 2f, and it is shown in Fig. 6a.
Fig. 4. The phase distribution patterns behind multi-pinhole aperture in deep Fresnel diffraction region with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively, illuminated by a plane wave.
M. Liu / Optik 125 (2014) 6329–6334
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Fig. 5. Detailed local intensity distribution patterns, the range of (a) and (b) is equal, (c) is three-dimensional images corresponding to the zero-line in (b).
Fig. 6. Detailed local intensity distribution patterns, the range of (a) and (b) is equal, (c) is three-dimensional images corresponding to the zero-line in (b).
Table 1 The eccentricities of the elliptical intensity-contour. x (m) y (m) ε
–1.82 –2.35 0.9984
–1.82 –2.33 0.9993
–1.80 –2.32 0.9998
–1.80 –2.30 0.9999
–1.80 –2.28 0.9996
–1.78 –2.27 0.9987
–1.78 –2.25 0.9969
–1.77 –2.23 0.9947
–1.35 –1.90 0.9359
–1.34 –1.82 0.9997
–1.34 –1.79 0.9999
–1.35 –1.75 0.9983
–1.35 –1.60 0.9986
–1.36 –1.55 0.9998
Table 2 The eccentricities of the elliptical intensity-contour. x (m) y (m) ε
–1.38 –1.94 0.9993
–1.37 –1.92 0.9998
When the maximum light intensity in Fig. 6a is also set at 3.0, we can obtain Fig. 6b. In Fig. 6b, the zero-line of light intensity still exists. Fig. 6c shows the three-dimensional intensity distribution near the zero-lines of light intensity, e.g., the squared region of zerolines of light intensity in Fig. 6b and show it in Fig. 6c. In Fig. 6c, we can see that the light intensity on both sides of the zero-line of intensity still changes rapidly, this phenomenon is similar to the one in Fig. 5c. Further simulations have shown that the zero-line of light intensity will become more shorter with increasing number of pinholes. In order to further analyze the phase distribution near the zerolines of light intensity, the expression of eccentricity is given as follows [9]:
1/4 2 1 2 ε= √ ([(∇ ) + (∇ )2 ] − 4ω) 2ω
2
2
(∇ ) + (∇ ) −
2
light intensity rapid change in vertical direction of zero-line of light intensity, while slow change along the direction of zero-line of light intensity. Finally, we obtain the following results: The bright spots of interference fields generated by multi-pinhole apertures diffraction screens in deep Fresnel diffraction region show symmetry distribution. When the observation plane close to the diffraction screen, the eccentricities of the elliptical intensity-contour are close or equal to 1, the light intensity changes rapidly on both sides of the zero-line of intensity. The number of positive and negative phase singularities is equal to each other on each chain and equal to the number of pinholes. For an odd number of multi-pinholes, the patterns of intensity distribution, phase distribution and the zero field 2
[(∇ ) + (∇ )2 ] − 4ω2
(11)
where ε is eccentricity of the elliptical intensity contours, ω ≡ ˝, ˝ = ∇ × ∇ . Using Eq. (11) for the numerical calculations, we can obtain the eccentricities of intensity-contour on the zero-line of light intensity. The results are shown in Tables 1 and 2, respectively. The coordinate points in Table 1 are the points on the zero-line of light intensity in Fig. 5c. The coordinate points in Table 2 correspond to the points on the zero-line of light intensity in Fig. 6c, where x (m) and y (m) are coordinates. From Tables 1 and 2, we can see that the eccentricities are close to 1, showing that the intensity-contours near the zero-lines of light intensity appear very flat ellipse or parabola. Note that the typical structure of phase distribution is strongly anisotropic. The
lines of the real and imaginary part of complex amplitude are all symmetric in the x-axis; for an even number of multi-pinholes, the patterns of intensity distribution, phase distribution and the zero field lines of the real and imaginary part of complex amplitude show central symmetric distribution.
5. Conclusions Based on the diffraction theory of Kirchhoff, in this paper, the intensity, the phase and the zero-contour generated by mutlipinhole aperture diffraction screens in deep Fresnel diffraction
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region are studied in detail and we find that the intensity, phase and the zero-contour of the real and imaginary parts of complex amplitude are all related to the number of pinholes. The eccentricities of the zero-line of light intensity are rather large, showing that the typical phase structure is strongly anisotropic. These results would be helpful for study the phase singularity in near field. References [1] J.F. Nye, M.V. Berry, Dislocation in wave trains, Proc. R. Soc. Lond. Ser. A 336 (1974) 165–190. [2] M. Harris, Light-field fluctuations in space and time, Contemp. Phys. 36 (1995) 215–233. [3] N. Shvartsman, I. Freund, Vortices in random wave fields: nearest neighbor anti-correlations, Phys. Rev. Lett. 72 (1994) 1008–1011. [4] G.A. Swartzlander, D.L. Drugan, N. Hallak, M.O. Freeman, C.T. Law, Optical transistor effect using an optical vortex soliton, Laser Phys. 5 (1995) 704–709.
[5] O. Otto, C. Gutsche, F. Kremer, U.F. Keyser, Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis, Rev. Sci. Instrum. 79 (2008) 23710–23716. [6] T.A. Wood, G.S. Roberts, S. Eaimkhong, P. Bartlett, Characterization of microparticles with driven optical tweezers, Faraday Discuss. 137 (2007) 319–333. [7] N.B. Simpson, K. Dholakia, L. Allen, M.J. Padgett, Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner, Opt. Lett. 22 (1997) 52–54. [8] G.A. Swartzlander, Jr., A.M. Deykoon, D.W. Jackson, C.T. Law, Materials Research Society Symposia Proceedings, Materials for Optical Limiting II, 1998. [9] M.V. Berry, M.R. Dennis, Phase singularities in isotropic random waves, Proc. R. Soc. Lond. A 456 (2000) 2059–2079. [10] M.R. Dennis, Y.S. Kivshar, M.S. Soskin, G.A. Swartzlander, Singular optics: more ado about nothing, J. Opt. A: Pure Appl. Opt. 11 (2009) 90201–90203. [11] J. Masajada, B. Dubik, Optical vortex generation by three plane wave interference, Opt. Commun. 198 (2001) 21–27. [12] R. Gary, M.P. Dauid, Phase vortices from a Young’s three-pinhole interferometer, Phys. Rev. E 75 (2007) 66613–66710.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Intensity and phase distribution in deep Fresnel diffraction region generated by multi-pinhole aperture Man Liu a,b,∗ a b
School of Science, Qilu University of Technology, Jinan 250353, China College of Physics and Electronics, Shandong Normal University, Jinan 250014, China
a r t i c l e
i n f o
Article history: Received 15 November 2013 Accepted 17 June 2014 PACS: 42.25.Fx 42.30.Ms Keywords: Intensity distribution Phase distribution The zero-contour of the real and imaginary parts
a b s t r a c t The Kirchhoff diffraction theory is applied to the multi-pinhole aperture diffraction screens, and the intensity, the zero-contour of the real and imaginary parts of complex amplitude and the phase distribution in deep Fresnel diffraction region are simulated. It is found that the number of bright spots, the zero-contours of the real and imaginary parts of complex amplitude and the phase singularities are all related to number of pinholes in diffraction screens. The brightness of bright spots in center of each pattern would become larger with increasing number of pinholes. In addition, there are many lines, on which the intensity value is close to 0. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction In interference field, there are isolated points of vanishing intensity at which the phase is undefined [1]. From phase singularity, equi-phase lines radiate outward in a star-like fashion lead to the phase spiral distribution, so that these phase singularities are also called optical vortices [1,2]. The sign of the phase singularity is positive if the phase circulates counterclockwise and negative if the phase circulates clockwise [3]. Vortices in light fields owing to potential applications have driven a surge of interest in recent years. For example, in selfdefocusing media, an optical vortex induces a waveguide, which may be used for optical switching [4]. In linear optics, optical vortices have been already utilized as “Optical Tweezers” to manipulate micrometer-sized particles such as cells [5,6]. Simpson et al. [7] achieved rotation of the particle using optical vortex, and this technology is called “Optical Spanners”. Optical vortices may also be useful for optical limiting purposes or for optical data storage [8]. The vortex characteristics are common to all wave fields, so understanding the optical vortices can help to understand the vortex characteristics in other physical systems.
∗ Correspondence to: School of Science, Qilu University of Technology, Jinan 250353, China. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2014.06.154 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Recently, some detailed studies of the phase singularities in far-field diffraction region are reported [9,10]. The phase singularities produced by multi-aperture diffraction screens have the complex structure produced by the optical wave interference in the far fields [11,12]. This paper discusses the intensity and the phase singularities distribution generated by multi-pinhole aperture in deep Fresnel diffraction region. The Kirchhoff diffraction theory is applied to the mutli-pinhole aperture diffraction screens. The intensity, the zero-contour of the real and imaginary parts and the phase distribution in deep Fresnel diffraction region are shown. The light intensity and the zero-contour of the real and imaginary parts of complex amplitude and the distribution of phase singularities are studied in detail. It is found that, for an even number of pinholes, the bright spots in diffractive field show central symmetry distribution, and for an odd number of pinholes, the intensity distribution patterns are mirrored in the x-axis. When the observation plane close to the diffraction screens, the zero-points of light intensity can form line segment, on which the eccentricities of the light intensity isoline are close or equal to 1, and the intensity changes very fast on both sides of the zero-line of light intensity. The zerocontours of the real and imaginary parts of complex amplitude are closed curves, so that the number of intersection points of the zero-contour is even, and positive and negative singularities are equal. The lines of zero-intensity change shorter and shorter with the increase in the pinhole. Therefore, the phase singularities in
6330
M. Liu / Optik 125 (2014) 6329–6334
with
c =1−
∂h(x0 , y0 ) ∂x0
2
−
∂h(x0 , y0 ) ∂y0
d = (h(x0 , y0 ) − z)c − (x0 − x)
2 (6)
∂h(x0 , y0 ) ∂h(x0 , y0 ) − (y0 − y) ∂x0 ∂y0
(7)
where T (x0 , y0 ) is the aperture function, which satisfies
Fig. 1. The geometry of mutli-pinhole aperture diffraction screen and interference field.
central region of light fields change from special phase singularity to the conventional phase singularity, when the number of pinhole is equal to 20, near the center of interference field, the positive and negative phase singularities are alternately appear and form a phase vortex chain. The results of this paper may be important to study the characteristics of intensity and phase in deep Fresnel diffraction region. 2. The calculation methods of interference fields To present the numerical simulating method of the intensity, the zero-contour of the real and imaginary parts and the phase distribution in deep Fresnel diffraction region, we consider the light diffraction geometry shown in Fig. 1. The diffraction screen is square, blocked with a mutli-pinhole opaque plate. The radius of each pinhole is equal. The pinholes are uniformly distributed in a circle of radius R in the x0 y0 -plane, the pinholes are indicated by open dots and the angular coordinate of the nth pinhole is ±n = 2n/N (n = 0, 1, 3, . . ., N − 1). Observation planes parallel to the diffraction screens. Here z is the distance from the diffraction screen to the observation plane. A parallel laser beam with wavelength and unity amplitude illuminates a multi-pinhole diffraction screen in the x0 y0 -plane. With the unimportant constant factor dropped, the complex amplitude of interference field after diffraction screen can be expressed as U0 (x0 , y0 ) = exp[ik(n − 1)h(x0 , y0 )]
(1)
where h(x0 , y0 ) is the thickness of diffraction screen, n is the refractive index of the glass and k = 2/ is the wavenumber. By using the Kirchhoff diffraction theory, we write the complex amplitude U of interference field at an arbitrary point in the observation plane as U(x, y) =
1 4
where
G G
∂U0 (x0 , y0 ) ∂G − U0 (x0 , y0 ) ∂e0 ∂e0 is
Green’s
dx0 dy0
(2) r=
function,
2 1/2
{(x − x0 )2 + (y − y0 )2 + [z − h(x0 , y0 )] } is the length between an arbitrary point in the observation plane and an arbitrary point in the diffraction screen, e0 is unit vector of each point in the x0 y0 -plane along the z-axis, namely e0 = cos ˛៝i + cos ˇ៝j + cos k៝ =
2 1/2
2
= [1 + (∂h/∂x0 ) + (∂h/∂y0 ) ] ∂ 1 = (e0 • ∇ ) = ∂e0
1 ∂h ∂h ៝ ៝ ៝ i− j+k −
∂x0
(3)
∂y0
, and partial derivative
∂h ∂ ∂h ∂ ∂ − · − · + ∂x0 ∂x0 ∂y0 ∂y0 ∂h
(4)
After a simple derivation, Eq. (2) can be written as U(x, y) =
1 4
T (x0 , y0 )
T (x0 , y0 ) =
×
2
(x0 − a)2 + (y0 − b) ≤ 8.0 m
0≤
0,
otherwise
(8)
The complex amplitude of light field in two dimensions can be represented as U(x, y) = exp[i(x, y)] = (x, y) + i (x, y)
(9)
where ≥0 is the amplitude, (x, y) is the phase with the range of (−, ] and (x, y) and (x, y) are the real and imaginary parts of complex amplitude, respectively. The phase is undefined, when the amplitude is equal to 0, and the phase singularities occur at the intersection of the real and imaginary parts of the field. Around a phase singularity, the phase changes by an integer multiple of 2, for a closed loop c enclosing a phase singularity yields an integer multiple of 2,
=
∇ · d = 2S
d = c
(10)
c
where S is an integer which is known as the topological charge, or strength, of the phase singularities, it can be defined as +1 (−1) if the phase value increases by 2 in the counterclockwise (clockwise) direction around the phase vortex. We simulate the intensity, phase and the zero-contour of the real and imaginary parts of complex amplitude distribution of interference field produced by mutli-pinhole aperture diffraction screens in deep Fresnel diffraction region, respectively. The distance z is set at 20.0 m. The wavelength of the illuminating light is set at 0.6328 m. The refractive index n is set at 1.532. The range of the generated surface is 60.0 m × 60.0 m with 301 × 301 sampling points. For N is equal to 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively. The radius of circle is 22.0 m and the radius of pinhole is set at 8.0 m. The range of the observation plane is 12.0 m × 12.0 m with 301 × 301 sampling points. Using these parameters and Eqs. (5) and (9) for the numerical calculations, we can obtain the discrete numerical distributions of the real and imaginary parts of interference field in deep Fresnel diffraction region. Subsequently, the intensity I(x, y) is calculated by using I(x, y) = U(x, y)U ∗ (x, y). Fig. 2a–i shows the intensity distribution patterns produced by the mutli-pinhole apertures diffraction screens with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively. The grayscale intensity distribution patterns are plotted using arbitrary units. The grayscale level number is 32. The value changes from minimum to maximum presenting from dark to light. The size of observation plane is much smaller than the size of diffraction screen. Therefore, the light intensity on the observation plane is the intervention results of the diffraction light from each pinhole in the diffraction plane, there is no transmitted light. From Fig. 2, we can see that the bright spots show central symmetry distribution, if the number of pinhole is equal to an even number, the bright spots are mirrored in the x-axis for an odd number of pinholes, which show a regular arrangement. Fist, the number of bright spot will increase with the number of pinholes N increase, then decrease. In Fig. 2a, we can see that the diffraction
i2(n − 1)c
11 2 [(n − 1)h(x0 , y0 ) + r] exp i r
1,
−
d d + 2∞ r r
dx0 dy0
(5)
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Fig. 2. The intensity distribution patterns behind multi-pinhole aperture in deep Fresnel diffraction region for N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively, illuminated by a plane wave.
stripes appear, and symmetry distribution with respect to x-axis and y-axis. In Fig. 2b–i, there are many bright spots, which regular arrangement, the bright spots show uniform distribution, and the bright spots are arranged in different shapes correspond with the number of pinholes in diffraction screen. The bright spots distributed uniformly around the center with the increase in the number of pinholes, the brightness of the ring around the center gets progressively increase form the innermost to the outermost rings. The number of bright spots near the center is equal to the number of pinholes in diffraction screens. In order to analyze the phase distribution of interference field generated by multi-pinhole aperture diffraction screens in deep Fresnel diffraction region, the zero-contour of the real and imaginary parts of complex amplitude are shown in Fig. 3a–i with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, which corresponding to Fig. 2a–i, respectively. The black solid line and dashed lines give the zero-contour of the real and imaginary parts of complex amplitude, respectively. The image sizes are 12.0 m × 12.0 m. In Fig. 3a (for N = 2), we can see that the zero-contour of the real and imaginary parts of complex amplitude, i.e., black solid and the dashed lines have almost equal intervals and they are almost parallel to each other in the horizontal direction, and symmetry distribution with respect to x-axis and y-axis. Except N = 2, by comparing Fig. 3b and d, we can see that the patterns of the real and imaginary parts of complex amplitude are mirrored in the x-axis, respectively. From Fig. 3c, e–i, we may see that the zero-contours of the real and imaginary parts of complex amplitude of interference field in deep Fresnel diffraction region are closed curves. Therefore, the number of intersection points of real and imaginary parts is even, namely the number of phase singularities is even, the number of positive and negative vortices are equal. So, the algebraic sum of topological charge in the whole pattern is zero. In addition, the lines of zero-intensity change shorter and shorter and the number of intersection points of real and imaginary parts become lager with the increase in the pinhole.
3. The phase distribution and topological charges of special phase singularities It would be interesting to know the phase distribution in deep Fresnel diffraction region near the multi-pinhole aperture. Fig. 4a–i shows that the color phase distribution maps produced by the multi-pinhole aperture with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively, in correspondence with the images in Figs. 2 and 3, where the sizes of each image are also 12.0 m × 12.0 m with the phase ranging from − to in eight color levels, namely the black lines indicate phase contours separated by /4. One can see form Fig. 4a that the phase distributions appear striped structures in y direction. From Fig. 4b and d, we can see that the phase singularities are mirrored in the x-axis. From Fig. 4c, e–i, we may see that the positive and negative phase singularities are alternately appear and form a series ring, we call this “phase singularity chains”. In each phase distribution pattern, form the innermost to the outermost “phase singularity chains”, the number of positive and negative phase singularities is equal to each other on each chain, and this is equal to the number of pinholes in the diffraction screens. As the number of pinholes increases, this phenomenon is more obvious. The phase singularities are uniformly distributed on the phase singularity chains. 4. The characteristics of intensity and phase distribution near the zero-lines of light intensity In the following, we analyze the characteristics of intensity distribution near the zero-lines of light intensity. We enlarge the squared regions in Fig. 2c, and it is shown in Fig. 5a. In Fig. 5a, around one circular bright spot, there are four other circular bright spots and four other stilliform bright spots adjacent to it. If the maximum light intensity in Fig. 5a is set at 3.0, we can obtain Fig. 5b. In Fig. 5b, the light intensity on the black line segments is close to 0, we call them “the zero-line of light intensity”.
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Fig. 3. Distributions of the zero-contour of the real and imaginary parts of complex amplitude. The black solid lines and dashed lines are, respectively, the zero-contour of the real and imaginary parts. (a–i) For N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively.
In order to observe clearly the light intensity distribution near the zero-lines of light intensity, we use three-dimensional images to display the light intensity distribution near the zero-lines of light intensity (the squared regions in Fig. 5b) and show it in Fig. 5c. From
Fig. 5c, we can see that the light intensity changes rapidly on both sides of the zero-line of intensity. Similarly, we enlarge the squared regions in Fig. 2f, and it is shown in Fig. 6a.
Fig. 4. The phase distribution patterns behind multi-pinhole aperture in deep Fresnel diffraction region with N = 2, 3, 4, 5, 6, 8, 10, 16 and 20, respectively, illuminated by a plane wave.
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Fig. 5. Detailed local intensity distribution patterns, the range of (a) and (b) is equal, (c) is three-dimensional images corresponding to the zero-line in (b).
Fig. 6. Detailed local intensity distribution patterns, the range of (a) and (b) is equal, (c) is three-dimensional images corresponding to the zero-line in (b).
Table 1 The eccentricities of the elliptical intensity-contour. x (m) y (m) ε
–1.82 –2.35 0.9984
–1.82 –2.33 0.9993
–1.80 –2.32 0.9998
–1.80 –2.30 0.9999
–1.80 –2.28 0.9996
–1.78 –2.27 0.9987
–1.78 –2.25 0.9969
–1.77 –2.23 0.9947
–1.35 –1.90 0.9359
–1.34 –1.82 0.9997
–1.34 –1.79 0.9999
–1.35 –1.75 0.9983
–1.35 –1.60 0.9986
–1.36 –1.55 0.9998
Table 2 The eccentricities of the elliptical intensity-contour. x (m) y (m) ε
–1.38 –1.94 0.9993
–1.37 –1.92 0.9998
When the maximum light intensity in Fig. 6a is also set at 3.0, we can obtain Fig. 6b. In Fig. 6b, the zero-line of light intensity still exists. Fig. 6c shows the three-dimensional intensity distribution near the zero-lines of light intensity, e.g., the squared region of zerolines of light intensity in Fig. 6b and show it in Fig. 6c. In Fig. 6c, we can see that the light intensity on both sides of the zero-line of intensity still changes rapidly, this phenomenon is similar to the one in Fig. 5c. Further simulations have shown that the zero-line of light intensity will become more shorter with increasing number of pinholes. In order to further analyze the phase distribution near the zerolines of light intensity, the expression of eccentricity is given as follows [9]:
1/4 2 1 2 ε= √ ([(∇ ) + (∇ )2 ] − 4ω) 2ω
2
2
(∇ ) + (∇ ) −
2
light intensity rapid change in vertical direction of zero-line of light intensity, while slow change along the direction of zero-line of light intensity. Finally, we obtain the following results: The bright spots of interference fields generated by multi-pinhole apertures diffraction screens in deep Fresnel diffraction region show symmetry distribution. When the observation plane close to the diffraction screen, the eccentricities of the elliptical intensity-contour are close or equal to 1, the light intensity changes rapidly on both sides of the zero-line of intensity. The number of positive and negative phase singularities is equal to each other on each chain and equal to the number of pinholes. For an odd number of multi-pinholes, the patterns of intensity distribution, phase distribution and the zero field 2
[(∇ ) + (∇ )2 ] − 4ω2
(11)
where ε is eccentricity of the elliptical intensity contours, ω ≡ ˝, ˝ = ∇ × ∇ . Using Eq. (11) for the numerical calculations, we can obtain the eccentricities of intensity-contour on the zero-line of light intensity. The results are shown in Tables 1 and 2, respectively. The coordinate points in Table 1 are the points on the zero-line of light intensity in Fig. 5c. The coordinate points in Table 2 correspond to the points on the zero-line of light intensity in Fig. 6c, where x (m) and y (m) are coordinates. From Tables 1 and 2, we can see that the eccentricities are close to 1, showing that the intensity-contours near the zero-lines of light intensity appear very flat ellipse or parabola. Note that the typical structure of phase distribution is strongly anisotropic. The
lines of the real and imaginary part of complex amplitude are all symmetric in the x-axis; for an even number of multi-pinholes, the patterns of intensity distribution, phase distribution and the zero field lines of the real and imaginary part of complex amplitude show central symmetric distribution.
5. Conclusions Based on the diffraction theory of Kirchhoff, in this paper, the intensity, the phase and the zero-contour generated by mutlipinhole aperture diffraction screens in deep Fresnel diffraction
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region are studied in detail and we find that the intensity, phase and the zero-contour of the real and imaginary parts of complex amplitude are all related to the number of pinholes. The eccentricities of the zero-line of light intensity are rather large, showing that the typical phase structure is strongly anisotropic. These results would be helpful for study the phase singularity in near field. References [1] J.F. Nye, M.V. Berry, Dislocation in wave trains, Proc. R. Soc. Lond. Ser. A 336 (1974) 165–190. [2] M. Harris, Light-field fluctuations in space and time, Contemp. Phys. 36 (1995) 215–233. [3] N. Shvartsman, I. Freund, Vortices in random wave fields: nearest neighbor anti-correlations, Phys. Rev. Lett. 72 (1994) 1008–1011. [4] G.A. Swartzlander, D.L. Drugan, N. Hallak, M.O. Freeman, C.T. Law, Optical transistor effect using an optical vortex soliton, Laser Phys. 5 (1995) 704–709.
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