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The superposition of the Bessel and mirrored Bessel beams and investigation of their self-healing characteristic
Journal Pre-proof The Superposition of the Bessel and Mirrored Bessel Beams and Investigation of Their Self-Healing Characteristic Fazel Saadati-Sharafeh, Abdollah Borhanifar, Alexey P. Porfirev, Pari Amiri, Ehsan A. Akhlaghi, Svetlana N. Khonina, Yashar Azizian-Kalandaragh
PII:
S0030-4026(19)31956-4
DOI:
https://doi.org/10.1016/j.ijleo.2019.164057
Reference:
IJLEO 164057
To appear in:
Optik
Received Date:
6 November 2019
Accepted Date:
11 December 2019
Please cite this article as: Fazel Saadati-Sharafeh, Abdollah Borhanifar, Alexey P. Porfirev, Pari Amiri, Ehsan A. Akhlaghi, Svetlana N. Khonina, Yashar Azizian-Kalandaragh, The Superposition of the Bessel and Mirrored Bessel Beams and Investigation of Their Self-Healing Characteristic, (2019), doi: https://doi.org/10.1016/j.ijleo.2019.164057
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
The Superposition of the Bessel and Mirrored Bessel Beams and Investigation of Their Self-Healing Characteristic Fazel Saadati-Sharafeh1,2 , Abdollah Borhanifar1 , Alexey P. Porfirev3,4 ,
1 Department
of Mathematics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
of Engineering Sciences, Sabalan University of Advanced Technologies (SUAT), Namin, Iran 3 Samara
National Research University, Samara, Russia
Processing Systems Institute Branch of the Federal Scientific Research Centre ”Crystallography and Photonics”
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4 Image
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2 Department
of
Pari Amiri2,5 , Ehsan A. Akhlaghi6,7 , Svetlana N. Khonina3,4 , Yashar Azizian-Kalandaragh2,8
of Russian Academy of Sciences, Samara, Russia
6 Department
of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
Research Center, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran 8 Department
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7 Optics
of Engineering Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
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5 Department
of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
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October 28, 2019
Abstract
The superposition of the Bessel and Mirrored Bessel beams, and their self-healing characteristic, using the defined similarity function have been investigated, quantitatively. We use the Huygens convolution method to propagate these beams to study their self-healing behavior. Numerical simulations and observational
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results show that the self-healing property of the superposed beams is better than the self-healing property of the Bessel beams.
Keyword : Bessel beams, Mirrored Bessel beams, Huygens convolution method, Self-healing.
1
Introduction
In resent years, theoretical and experimental challenges on the properties of structured light have been described by some mathematical special functions. The generation and propagation of the structured light beams using ∗ Corresponding
author. Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran. E-mail
addresses: [email protected], [email protected], (Tel):+984532324045, (Fax):+984532323611(Yashar Azizian-Kalandaragh).
1
∗
some special functions have been attracted fundamentally and experimentally by several researchers [1, 2, 3, 4, 5, 6]. One of the most important structured light beams is Bessel beam, which is the exact solution of scalar Helmholtz equation in cylindrical coordinates and its amplitude can be describe using Bessel function of the first kind. It is impossible to generate these ideal beams experimentally because they have infinite transverse extent and energy. For this reason, little attention has been paid to the ideal Bessel beams by researchers, but after introducing the non-diffracting beams by Durnin [7], several researchers reported different interesting results about various types of non-diffractive beams. Eliminating the hardships of ideal Bessel beams physical realization is practicable by constructing the efficient approximate Bessel beams, which have finite energy and
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all of their useful properties. The Bessel-Gauss beams that have finite-energy, have been introduced by Gori and Guattari [8]. The BG beams amplitude is expressed by the product of two functions, Gaussian and nth-order first kind Bessel
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functions. These beams carry orbital angular momentum (OAM) and their intensity distribution is radially symmetric. Several studies have been accomplished on the generalizations of BG beams, for example, by the
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product of Gaussian function by modified Bessel function with complex argument, Kotlyar et al. proposed asymmetric Bessel-Gauss beams with integer and fractional OAM [9]. The asymmetry degree of aBG beams is
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a nonnegative real parameter c. If c = 0, then the aBG beam is radially symmetric and by increasing c, the beam symmetry disappears and the shape of the beam will be semi-crescent. Li et al. presented generalized asymmetric Bessel mode with a complex parameter c, which determines the asymmetry degree and orientation
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of the optical crescent [10]. Arlt et al. illustrated a method for creating a high-order Bessel beam by illuminating an axicon with the appropriate Laguerre-Gaussian light beam [11]. Since the vortex beams with orbital angular momentum can improve the communication efficiency, then the high-order Bessel beams is important and
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suitable for atom guiding.
Because of changes in intensity profile and shape of beams after propagation beyond the obstacle, similarity/difference value of these beams with/from the undisturbed propagated beams is one of the researchers’ questions. Essentially, it is required that, we have a criterion to judge about the beams’ properties, especially, the self-healing property. Structured light beam’s self-healing characteristic is one of the most interesting and
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important properties, that describes the beams ability to reform their amplitude at the minimum distance from the obstruction. There are several works about this property of the beams reported by scientists. For example, Almazov and Khonina have been examined the self-healing property of the Bessel and LG beams, experimentally [12]. Chu et al. studied analytically the self-healing characteristic of the Airy beams in free space [13] and then, this property of optical Airy beams demonstrated both theoretically and experimentally in [14, 15]. One of the non-diffracting beams which have self-healing property and has been presented by Ring et al. [16], is the Pearcey beam. Pure Pearcey beams have infinite energy similar to Bessel and Airy beams and so Ring et al. have modulated Pearcey function with a Gaussian function without changing their properties, and then investigated the self-healing property of them analytically and experimentally. Since there are several
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applications for Bessel beams in optical manipulation [11, 17], optical microscopy [18], quantum communication [19] because of their special non-diffractive and self-healing properties, then have been done and presented numerous researches about self-healing property of different kinds of Bessel beams, too. Aiello et al. offered a simple explanation of the self-healing mechanism for the Bessel beams, and obtained the minimum distance of the self-reconstruction of them beyond the obstruction [20]. Fahrbach et al. identified different self-healing measurements of modified Bessel beams relative to a focused Gaussian beam. They illustrated coherent light propagation and scattering through inhomogeneous media by exciting fluorescence in a plane parallel to the propagation axis using a light-sheet-based microscope [18]. There are some other researches about the self-
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healing property of various kinds of Bessel beams such as Bessel-like beams [21], Bessel-Gaussian beams [22], and asymmetric Bessel beams [23, 24]. As our knowledge, there are no reports that present the detailed and quantitatively study of the self-healing property of Bessel beams and superposition of Bessel and mirrored Bessel
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(SBMB) beams.
In this paper, for the first time, in order to further elucidation, we report the high precision self-healing
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characteristic of structured light beams by introducing a new similarity function as a criterion for this property.
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The Bessel beam and SBMB beams are used to investigate and analyze their self-healing behavior, quantitatively.
Quantification of the self-healing property of the beams
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Complex amplitude of the Bessel beams in cylindrical coordinates is described by[25]: [ ] √ En (r, ϕ, z) = exp inϕ + iz k 2 − α2 Jn (αr),
(1)
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where k is the wave number, Jn is the Bessel function of order n, and α = k sin(θ0 ), where θ0 is the angle of a conical wave. Using Dove Prism, makes the Eq. (1) mirrored on x (or y) axis, that is described by: [ ] √ M En (r, ϕ, z) = exp −inϕ + iz k 2 − α2 Jn (αr),
(2)
So, the electrical field and intensity profile of the SBMB beams become: [ √ ] E(r, ϕ, z) = 2 cos(nϕ) exp iz k 2 − α2 Jn (αr),
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(3)
I(r, ϕ, z) = 4 cos2 (nϕ)Jn2 (αr).
Depicting numerical values of the beam’s self-healing, we define the following similarity function S, that
gives the similarity percent of intensity distribution for the original and masked beams, which called A and B, respectively: S(A, B) =
( 1−
√
∥A2 − B 2 ∥ ∥(A + B)2 ∥
) × 100,
(4)
where the matrix power is matrix with the power of all its entries and the matrix norm is the Frobenius norm. In order to exert the average efficiency and to give more weight to bright points’ intensity respect to the dark points’ intensity, we use square of intensity matrix in Eq. (4). 3
3
Simulation and experiment results
In this research, we studied quantitatively the self-healing property of the Bessel beams and the SBMB beams by simulation and experiment, in detail. For this purpose, two circular mask with different sizes of areas 0.22 mm2 (mask 1) and 0.88 mm2 (mask 2) are used for each order of Bessel beams n = 0, 2, and 5. The diameters of these circular masks selected approximately equal to the half and full width of the first ring of the Bessel beam. The similarity of original and masked beams is compared in various propagation distances from the initial plane (z = 0 mm) by using the introduced similarity Eq. (4) and investigated the self-healing behavior of beams. The simulation of the beams propagation is done using Huygens convolution method. The minimum
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propagation distance from the initial plane in which the masked beams reach to maximum similarity with the original beams is called optimum self-healing distance (zopt ) and the plane at this distance is named optimum
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plane. The self-healing behavior of the Bessel and SBMB beams is investigated in 1500 µm×1500 µm area with
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1000 × 1000 pixels sampled array.
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Figure 1: Experimental setup: Laser is a solid-state laser (λ = 532 nm); PH is a pinhole; L1 , L2 , and L3 are lenses with focal lengths f1 = 250 mm, f2 = 500 mm, and f3 = 150 mm, respectively; D1 and D2 are diaphragms; SLM is a spatial light modulator PLUTO VIS (1920 × 1080 pixels, 8 µm pixel pitch); and Cam is
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a video-camera.
Figure 1 shows the experimental setup. A Gaussian beam from a solid state laser (λ = 532 nm) was expanded and collimated using a pinhole (PH) and lens L1 (f1 = 250 mm), and a spatial light modulator SLM (HOLOEYE PLUTO VIS, 1920 × 1080 pixels and 8 µm pixel pitch). A diaphragm D1 was utilized to
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single out the central bright ring from surrounding bright and dark rings resulting from the diffraction by the pinhole. Lenses L2 (f2 = 500 mm) and L3 (f3 = 150 mm) and diaphragm D2 , were used to spatially filter the phase-modulated laser beam. A video-camera (Cam) was used to record the intensity distributions at different distances from the lens L3 . Table 1 depicts simulation results of the intensity distribution of the second-order Bessel beam with mask 2 propagated in 6 arbitrary propagation distances and their similarity values respect to the propagated original beam with no mask. The similarity value of the Bessel beams respect to the original beams in the initial plane z = 0 mm equals 92 percent. It means that, exerting mask 2 caused to 8 percent perturbation on the original beam. After propagation of the beam along the z-axis, the maximum similarity value between masked and unmasked beams is 96.73 percent that happens at the zopt = 30 mm distance from the initial plane. On the 4
Table 1: propagation of the masked Bessel beam with mask 2 and its similarity values respect to the propagated original Bessel beams in different propagation distances
Intensity profile of Bessel beams z = 0 mm
z = 10 mm
z = 20 mm
zopt = 30 mm
z = 40 mm
z = 50 mm
z = 60 mm
92
93.86
95.95
96.73
96.26
93.60
87.96
Similarity(%)
other hand, the difference of the propagated masked beam respect to the propagated original beam, in the
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minimum distance propagation, is 3.27 percent.
In Table 2 the zero-order Bessel beam is investigated by simulation and experiment, in which the original
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and masked intensity distribution is shown in the initial and optimum plane. In addition, the similarity value for each case is presented. The difference of similarities of the original and masked beams with Mask 2 at the
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initial and optimum plane is more than for the case with mask 1. This is due to the size of mask 2 is more than the size of mask 1, therefore, it has affected the whole of first bright ring of Bessel beam. This results in the similarity value for the case mask 2 being less than the mask 1, since the original beam has less disturbance.
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Because the results are the same for the SBMB beams in this section, the related results and images have omitted.
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In this part, it has been investigated quantified self-healing behavior of the masked Bessel and SBMB beams with mask 1 after propagation for orders n = 2 and 5 (Table 3). By increasing the order of Bessel beams, the diameter of central dark region and bright rings of the beam increases, and therefore, the intensity density of
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the bright rings reduces. Thus, for case n = 5, the similarity value in the initial plane is more, while the same mask applied. This is established for the SBMB beams, too. In fact, increasing the order of beams caused the growth of the central dark region diameter and number of petals, and so reduction of the intensity distribution on each petal.
Results show that both of beams have self-healing property observationally, and the self-healing value of
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the SBMB beams is more than the Bessel beams’ self-healing value. Along the propagation of the beams, the self-healing is occurred and the simulation results show that the similarity value in the optimum plane are the
Table 2: propagation of the masked Bessel beams with two mask 1 and mask 2, and their self-healing values Mask 1
Mask 2
Intensity profile of Bessel beams
Similarity(%)
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 30 mm
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 30 mm
82.21
59.25
97.04
67.56
56.69
36.88
96.26
65.32
5
Table 3: propagated Bessel and SBMB beams with mask 1 and their self-healing values Order of beams
n=2
n=5
Intensity profile of Bessel beams z = 0 mm Similarity(%)
95.16
zopt = 30 mm
89.71
97.24
93.10
z = 0 mm 96.83
zopt = 30 mm
94.43
97.41
95.61
Intensity profile of
z = 0 mm 92.57
zopt = 20 mm
84.42
97.08
92.17
z = 0 mm 95.44
zopt = 10 mm
95.52
97.09
95.83
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Similarity(%)
of
SBMB beams
same and reached to the maximum possible value (∼ 97%), whereas experimental results are various. The difference between the simulation and the experiment result is due to the imperfections and deficiencies in the
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experimental work.
In addition, it is observed that by increasing the beams’ order the similarity value at the optimum plane is
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increased from 97.24 (93.10, in the experiment) percent to 97.41 (95.61) percent. Furthermore, the optimum propagation distance for the SBMB beams is less than for the Bessel beams. The remarkable note for these
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two observed effects is that the superposition of the beams and increasing of the beams’ order are two effective factors to improve the self-healing behavior of the beam. This is because the increasing order of the Bessel beam and the superposition of the beams results in a more complex phase distribution and therefore increases
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the rate of phase change in the propagation direction. This accelerates the wave reconstruction process. Table 4: propagated Bessel and SBMB beams with two mask 1 and their self-healing values Order of beams
n=2
n=5
Intensity profile of
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 20 mm
z = 0 mm
z = 0 mm
zopt = 20 mm
zopt = 30 mm
93.05
88.79
97.11
91.92
95.52
93.76
97.22
95.18
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 20 mm
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 10 mm
89.07
81.09
96.99
89.94
93.51
94.55
96.88
94.83
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Bessel beams
Similarity(%)
Intensity profile of SBMB beams
Similarity(%)
The self-healing property of the Bessel and SBMB beams with two circular masks of area 0.22 mm2 is studied, too. Here, results depict the beams’ self-healing behavior is similar to the previous status. Comparing
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the results of two Tables (3), and (4) shows that as expected, the self-healing behavior of the beams is reduced when the number of applied masks increases. According to the Table (4), the SBMB beams have more selfhealing value than the Bessel beams and the optimum propagation distance of the SBMB beams is the same or shorter than the optimum propagation distance of the Bessel beams. Table 5: propagated Bessel and SBMB beams with mask 2 and their self-healing values Order of beams
n=2
n=5
Intensity profile of Bessel beams z = 0 mm
zopt = 30 mm
zopt = 30 mm
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 30 mm
92
87.94
96.73
90.76
94.52
93.80
96.54
94.41
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 20 mm
z = 0 mm
90.46
83.57
96.69
89.68
94.07
Intensity profile of
z = 0 mm
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SBMB beams
Similarity(%)
of
z = 0 mm
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Similarity(%)
95.06
zopt = 30 mm
zopt = 30 mm
96.40
94
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Self-healing property of the beams with mask 2 is shown in Table (5). The comparison of the results in Tables (3) and (5) shows that beams self-healing behavior does not change with increasing the mask size. All of
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the masked beams are reconstructed after propagation and as expected, the amount of the all beams’ similarity value decreased with increasing mask size. It should be noted that the intensity distribution of the SBMB beams in the optimum plane is symmetric respect to the x-axis and uniform vice versa the Bessel beams’ profile. This
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is due to the asymmetric phase structure for the Bessel beams respect to the SBMB beams.
Conclusions
In this paper, the self-healing property of the Bessel beams and the SBMB beams were investigated, observationally and quantitatively. The investigations were done by simulation and experiment. In addition, the new
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similarity function has been introduced which is a good criterion for comparing the similarity of original and masked beams and beams self-healing behavior. The self-healing property of the beams is studied by investigation of the similarity value and the optimum distance propagation of the beams. Results show that the superposed beams have great self-healing property respect to the Bessel beams. In addition, it is shown by increasing the order of the Bessel and SBMB beams the self-healing property is increased. Further, the effect of mask size was examined, which did not have a significant effect as a result of resizing the mask if the number of areas did not change.
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Acknowledgement
The experimental investigation was performed by A. P. Porfirev and was financially supported by the Russian Science Foundation (19-72-00018).
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https://doi.org/10.1103/PhysRevA.91.053840
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PII:
S0030-4026(19)31956-4
DOI:
https://doi.org/10.1016/j.ijleo.2019.164057
Reference:
IJLEO 164057
To appear in:
Optik
Received Date:
6 November 2019
Accepted Date:
11 December 2019
Please cite this article as: Fazel Saadati-Sharafeh, Abdollah Borhanifar, Alexey P. Porfirev, Pari Amiri, Ehsan A. Akhlaghi, Svetlana N. Khonina, Yashar Azizian-Kalandaragh, The Superposition of the Bessel and Mirrored Bessel Beams and Investigation of Their Self-Healing Characteristic, (2019), doi: https://doi.org/10.1016/j.ijleo.2019.164057
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
The Superposition of the Bessel and Mirrored Bessel Beams and Investigation of Their Self-Healing Characteristic Fazel Saadati-Sharafeh1,2 , Abdollah Borhanifar1 , Alexey P. Porfirev3,4 ,
1 Department
of Mathematics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
of Engineering Sciences, Sabalan University of Advanced Technologies (SUAT), Namin, Iran 3 Samara
National Research University, Samara, Russia
Processing Systems Institute Branch of the Federal Scientific Research Centre ”Crystallography and Photonics”
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4 Image
ro
2 Department
of
Pari Amiri2,5 , Ehsan A. Akhlaghi6,7 , Svetlana N. Khonina3,4 , Yashar Azizian-Kalandaragh2,8
of Russian Academy of Sciences, Samara, Russia
6 Department
of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
Research Center, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran 8 Department
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7 Optics
of Engineering Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
re
5 Department
of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
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October 28, 2019
Abstract
The superposition of the Bessel and Mirrored Bessel beams, and their self-healing characteristic, using the defined similarity function have been investigated, quantitatively. We use the Huygens convolution method to propagate these beams to study their self-healing behavior. Numerical simulations and observational
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results show that the self-healing property of the superposed beams is better than the self-healing property of the Bessel beams.
Keyword : Bessel beams, Mirrored Bessel beams, Huygens convolution method, Self-healing.
1
Introduction
In resent years, theoretical and experimental challenges on the properties of structured light have been described by some mathematical special functions. The generation and propagation of the structured light beams using ∗ Corresponding
author. Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran. E-mail
addresses: [email protected], [email protected], (Tel):+984532324045, (Fax):+984532323611(Yashar Azizian-Kalandaragh).
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some special functions have been attracted fundamentally and experimentally by several researchers [1, 2, 3, 4, 5, 6]. One of the most important structured light beams is Bessel beam, which is the exact solution of scalar Helmholtz equation in cylindrical coordinates and its amplitude can be describe using Bessel function of the first kind. It is impossible to generate these ideal beams experimentally because they have infinite transverse extent and energy. For this reason, little attention has been paid to the ideal Bessel beams by researchers, but after introducing the non-diffracting beams by Durnin [7], several researchers reported different interesting results about various types of non-diffractive beams. Eliminating the hardships of ideal Bessel beams physical realization is practicable by constructing the efficient approximate Bessel beams, which have finite energy and
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all of their useful properties. The Bessel-Gauss beams that have finite-energy, have been introduced by Gori and Guattari [8]. The BG beams amplitude is expressed by the product of two functions, Gaussian and nth-order first kind Bessel
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functions. These beams carry orbital angular momentum (OAM) and their intensity distribution is radially symmetric. Several studies have been accomplished on the generalizations of BG beams, for example, by the
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product of Gaussian function by modified Bessel function with complex argument, Kotlyar et al. proposed asymmetric Bessel-Gauss beams with integer and fractional OAM [9]. The asymmetry degree of aBG beams is
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a nonnegative real parameter c. If c = 0, then the aBG beam is radially symmetric and by increasing c, the beam symmetry disappears and the shape of the beam will be semi-crescent. Li et al. presented generalized asymmetric Bessel mode with a complex parameter c, which determines the asymmetry degree and orientation
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of the optical crescent [10]. Arlt et al. illustrated a method for creating a high-order Bessel beam by illuminating an axicon with the appropriate Laguerre-Gaussian light beam [11]. Since the vortex beams with orbital angular momentum can improve the communication efficiency, then the high-order Bessel beams is important and
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suitable for atom guiding.
Because of changes in intensity profile and shape of beams after propagation beyond the obstacle, similarity/difference value of these beams with/from the undisturbed propagated beams is one of the researchers’ questions. Essentially, it is required that, we have a criterion to judge about the beams’ properties, especially, the self-healing property. Structured light beam’s self-healing characteristic is one of the most interesting and
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important properties, that describes the beams ability to reform their amplitude at the minimum distance from the obstruction. There are several works about this property of the beams reported by scientists. For example, Almazov and Khonina have been examined the self-healing property of the Bessel and LG beams, experimentally [12]. Chu et al. studied analytically the self-healing characteristic of the Airy beams in free space [13] and then, this property of optical Airy beams demonstrated both theoretically and experimentally in [14, 15]. One of the non-diffracting beams which have self-healing property and has been presented by Ring et al. [16], is the Pearcey beam. Pure Pearcey beams have infinite energy similar to Bessel and Airy beams and so Ring et al. have modulated Pearcey function with a Gaussian function without changing their properties, and then investigated the self-healing property of them analytically and experimentally. Since there are several
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applications for Bessel beams in optical manipulation [11, 17], optical microscopy [18], quantum communication [19] because of their special non-diffractive and self-healing properties, then have been done and presented numerous researches about self-healing property of different kinds of Bessel beams, too. Aiello et al. offered a simple explanation of the self-healing mechanism for the Bessel beams, and obtained the minimum distance of the self-reconstruction of them beyond the obstruction [20]. Fahrbach et al. identified different self-healing measurements of modified Bessel beams relative to a focused Gaussian beam. They illustrated coherent light propagation and scattering through inhomogeneous media by exciting fluorescence in a plane parallel to the propagation axis using a light-sheet-based microscope [18]. There are some other researches about the self-
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healing property of various kinds of Bessel beams such as Bessel-like beams [21], Bessel-Gaussian beams [22], and asymmetric Bessel beams [23, 24]. As our knowledge, there are no reports that present the detailed and quantitatively study of the self-healing property of Bessel beams and superposition of Bessel and mirrored Bessel
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(SBMB) beams.
In this paper, for the first time, in order to further elucidation, we report the high precision self-healing
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characteristic of structured light beams by introducing a new similarity function as a criterion for this property.
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The Bessel beam and SBMB beams are used to investigate and analyze their self-healing behavior, quantitatively.
Quantification of the self-healing property of the beams
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Complex amplitude of the Bessel beams in cylindrical coordinates is described by[25]: [ ] √ En (r, ϕ, z) = exp inϕ + iz k 2 − α2 Jn (αr),
(1)
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where k is the wave number, Jn is the Bessel function of order n, and α = k sin(θ0 ), where θ0 is the angle of a conical wave. Using Dove Prism, makes the Eq. (1) mirrored on x (or y) axis, that is described by: [ ] √ M En (r, ϕ, z) = exp −inϕ + iz k 2 − α2 Jn (αr),
(2)
So, the electrical field and intensity profile of the SBMB beams become: [ √ ] E(r, ϕ, z) = 2 cos(nϕ) exp iz k 2 − α2 Jn (αr),
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(3)
I(r, ϕ, z) = 4 cos2 (nϕ)Jn2 (αr).
Depicting numerical values of the beam’s self-healing, we define the following similarity function S, that
gives the similarity percent of intensity distribution for the original and masked beams, which called A and B, respectively: S(A, B) =
( 1−
√
∥A2 − B 2 ∥ ∥(A + B)2 ∥
) × 100,
(4)
where the matrix power is matrix with the power of all its entries and the matrix norm is the Frobenius norm. In order to exert the average efficiency and to give more weight to bright points’ intensity respect to the dark points’ intensity, we use square of intensity matrix in Eq. (4). 3
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Simulation and experiment results
In this research, we studied quantitatively the self-healing property of the Bessel beams and the SBMB beams by simulation and experiment, in detail. For this purpose, two circular mask with different sizes of areas 0.22 mm2 (mask 1) and 0.88 mm2 (mask 2) are used for each order of Bessel beams n = 0, 2, and 5. The diameters of these circular masks selected approximately equal to the half and full width of the first ring of the Bessel beam. The similarity of original and masked beams is compared in various propagation distances from the initial plane (z = 0 mm) by using the introduced similarity Eq. (4) and investigated the self-healing behavior of beams. The simulation of the beams propagation is done using Huygens convolution method. The minimum
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propagation distance from the initial plane in which the masked beams reach to maximum similarity with the original beams is called optimum self-healing distance (zopt ) and the plane at this distance is named optimum
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plane. The self-healing behavior of the Bessel and SBMB beams is investigated in 1500 µm×1500 µm area with
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1000 × 1000 pixels sampled array.
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Figure 1: Experimental setup: Laser is a solid-state laser (λ = 532 nm); PH is a pinhole; L1 , L2 , and L3 are lenses with focal lengths f1 = 250 mm, f2 = 500 mm, and f3 = 150 mm, respectively; D1 and D2 are diaphragms; SLM is a spatial light modulator PLUTO VIS (1920 × 1080 pixels, 8 µm pixel pitch); and Cam is
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a video-camera.
Figure 1 shows the experimental setup. A Gaussian beam from a solid state laser (λ = 532 nm) was expanded and collimated using a pinhole (PH) and lens L1 (f1 = 250 mm), and a spatial light modulator SLM (HOLOEYE PLUTO VIS, 1920 × 1080 pixels and 8 µm pixel pitch). A diaphragm D1 was utilized to
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single out the central bright ring from surrounding bright and dark rings resulting from the diffraction by the pinhole. Lenses L2 (f2 = 500 mm) and L3 (f3 = 150 mm) and diaphragm D2 , were used to spatially filter the phase-modulated laser beam. A video-camera (Cam) was used to record the intensity distributions at different distances from the lens L3 . Table 1 depicts simulation results of the intensity distribution of the second-order Bessel beam with mask 2 propagated in 6 arbitrary propagation distances and their similarity values respect to the propagated original beam with no mask. The similarity value of the Bessel beams respect to the original beams in the initial plane z = 0 mm equals 92 percent. It means that, exerting mask 2 caused to 8 percent perturbation on the original beam. After propagation of the beam along the z-axis, the maximum similarity value between masked and unmasked beams is 96.73 percent that happens at the zopt = 30 mm distance from the initial plane. On the 4
Table 1: propagation of the masked Bessel beam with mask 2 and its similarity values respect to the propagated original Bessel beams in different propagation distances
Intensity profile of Bessel beams z = 0 mm
z = 10 mm
z = 20 mm
zopt = 30 mm
z = 40 mm
z = 50 mm
z = 60 mm
92
93.86
95.95
96.73
96.26
93.60
87.96
Similarity(%)
other hand, the difference of the propagated masked beam respect to the propagated original beam, in the
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minimum distance propagation, is 3.27 percent.
In Table 2 the zero-order Bessel beam is investigated by simulation and experiment, in which the original
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and masked intensity distribution is shown in the initial and optimum plane. In addition, the similarity value for each case is presented. The difference of similarities of the original and masked beams with Mask 2 at the
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initial and optimum plane is more than for the case with mask 1. This is due to the size of mask 2 is more than the size of mask 1, therefore, it has affected the whole of first bright ring of Bessel beam. This results in the similarity value for the case mask 2 being less than the mask 1, since the original beam has less disturbance.
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Because the results are the same for the SBMB beams in this section, the related results and images have omitted.
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In this part, it has been investigated quantified self-healing behavior of the masked Bessel and SBMB beams with mask 1 after propagation for orders n = 2 and 5 (Table 3). By increasing the order of Bessel beams, the diameter of central dark region and bright rings of the beam increases, and therefore, the intensity density of
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the bright rings reduces. Thus, for case n = 5, the similarity value in the initial plane is more, while the same mask applied. This is established for the SBMB beams, too. In fact, increasing the order of beams caused the growth of the central dark region diameter and number of petals, and so reduction of the intensity distribution on each petal.
Results show that both of beams have self-healing property observationally, and the self-healing value of
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the SBMB beams is more than the Bessel beams’ self-healing value. Along the propagation of the beams, the self-healing is occurred and the simulation results show that the similarity value in the optimum plane are the
Table 2: propagation of the masked Bessel beams with two mask 1 and mask 2, and their self-healing values Mask 1
Mask 2
Intensity profile of Bessel beams
Similarity(%)
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 30 mm
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 30 mm
82.21
59.25
97.04
67.56
56.69
36.88
96.26
65.32
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Table 3: propagated Bessel and SBMB beams with mask 1 and their self-healing values Order of beams
n=2
n=5
Intensity profile of Bessel beams z = 0 mm Similarity(%)
95.16
zopt = 30 mm
89.71
97.24
93.10
z = 0 mm 96.83
zopt = 30 mm
94.43
97.41
95.61
Intensity profile of
z = 0 mm 92.57
zopt = 20 mm
84.42
97.08
92.17
z = 0 mm 95.44
zopt = 10 mm
95.52
97.09
95.83
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Similarity(%)
of
SBMB beams
same and reached to the maximum possible value (∼ 97%), whereas experimental results are various. The difference between the simulation and the experiment result is due to the imperfections and deficiencies in the
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experimental work.
In addition, it is observed that by increasing the beams’ order the similarity value at the optimum plane is
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increased from 97.24 (93.10, in the experiment) percent to 97.41 (95.61) percent. Furthermore, the optimum propagation distance for the SBMB beams is less than for the Bessel beams. The remarkable note for these
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two observed effects is that the superposition of the beams and increasing of the beams’ order are two effective factors to improve the self-healing behavior of the beam. This is because the increasing order of the Bessel beam and the superposition of the beams results in a more complex phase distribution and therefore increases
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the rate of phase change in the propagation direction. This accelerates the wave reconstruction process. Table 4: propagated Bessel and SBMB beams with two mask 1 and their self-healing values Order of beams
n=2
n=5
Intensity profile of
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 20 mm
z = 0 mm
z = 0 mm
zopt = 20 mm
zopt = 30 mm
93.05
88.79
97.11
91.92
95.52
93.76
97.22
95.18
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 20 mm
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 10 mm
89.07
81.09
96.99
89.94
93.51
94.55
96.88
94.83
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Bessel beams
Similarity(%)
Intensity profile of SBMB beams
Similarity(%)
The self-healing property of the Bessel and SBMB beams with two circular masks of area 0.22 mm2 is studied, too. Here, results depict the beams’ self-healing behavior is similar to the previous status. Comparing
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the results of two Tables (3), and (4) shows that as expected, the self-healing behavior of the beams is reduced when the number of applied masks increases. According to the Table (4), the SBMB beams have more selfhealing value than the Bessel beams and the optimum propagation distance of the SBMB beams is the same or shorter than the optimum propagation distance of the Bessel beams. Table 5: propagated Bessel and SBMB beams with mask 2 and their self-healing values Order of beams
n=2
n=5
Intensity profile of Bessel beams z = 0 mm
zopt = 30 mm
zopt = 30 mm
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 30 mm
92
87.94
96.73
90.76
94.52
93.80
96.54
94.41
z = 0 mm
z = 0 mm
zopt = 30 mm
zopt = 20 mm
z = 0 mm
90.46
83.57
96.69
89.68
94.07
Intensity profile of
z = 0 mm
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SBMB beams
Similarity(%)
of
z = 0 mm
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Similarity(%)
95.06
zopt = 30 mm
zopt = 30 mm
96.40
94
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Self-healing property of the beams with mask 2 is shown in Table (5). The comparison of the results in Tables (3) and (5) shows that beams self-healing behavior does not change with increasing the mask size. All of
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the masked beams are reconstructed after propagation and as expected, the amount of the all beams’ similarity value decreased with increasing mask size. It should be noted that the intensity distribution of the SBMB beams in the optimum plane is symmetric respect to the x-axis and uniform vice versa the Bessel beams’ profile. This
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is due to the asymmetric phase structure for the Bessel beams respect to the SBMB beams.
Conclusions
In this paper, the self-healing property of the Bessel beams and the SBMB beams were investigated, observationally and quantitatively. The investigations were done by simulation and experiment. In addition, the new
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similarity function has been introduced which is a good criterion for comparing the similarity of original and masked beams and beams self-healing behavior. The self-healing property of the beams is studied by investigation of the similarity value and the optimum distance propagation of the beams. Results show that the superposed beams have great self-healing property respect to the Bessel beams. In addition, it is shown by increasing the order of the Bessel and SBMB beams the self-healing property is increased. Further, the effect of mask size was examined, which did not have a significant effect as a result of resizing the mask if the number of areas did not change.
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Acknowledgement
The experimental investigation was performed by A. P. Porfirev and was financially supported by the Russian Science Foundation (19-72-00018).
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