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STUDY ON THE MAINARDI BEAM THROUGH THE FRACTIONAL FOURIER TRANSFORMS SYSTEM
F. Habibi1, M. Moradi2, A. Ansari3 1 Department of Physics, Faculty of Sciences, Shahrekord University, Shahrekord, Iran,
2 Department of Physics, Photonic Research Group, Shahrekord University, Shahrekord, Iran, 3 Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Iran
Abstract
In this paper, we introduced the Mainardi beam and indicated its attributes under the Fractional Fourier transform for power variations of Fractional Fourier transform. The results represent that the behavior of the Mainardi beam is similar to that of the Airy beam. The obtained formula is a very powerful tool to describe propagation of a Mainardi beam through the FFT and the FrFT systems. An analytical expression of the Mainardi beam passing through an Fractional Fourier transform system presented. The influences of the Fractional Fourier transform, rational order of the Mittag-Leffler function (Fourier transform of the Mainardi function) on the normalized intensity distribution and characteristics of the Mainardi beam in the Fractional Fourier transform system examined. Power of the Fractional Fourier transform (p) and rational order of the Mittag-Leffler function (q) control characteristics of the Mainardi beam such as effective beam size, number, width, height, and orientation of the beam spot.
Keywords: Wright function, Mainardi function, Mittag-Leffler function, Airy beam, Fractional Fourier transform.
Citation: Habibi F, Moradi M, Ansari A. Study on the Mainardi beam through the fractional Fourier transforms system. Computer Optics 2018; 42(5): 751-757. DOI: 10.18287/2412-61792018-42-5-751-757.
Introduction
The Fractional Fourier transform (FrFT), as the generalization of a conventional Fourier transform has been widely studied [1- 4]. The FrFT properties of all kinds of laser beams have been investigated with considerable interest [5- 9]. The FrFT of partially coherent beams has been studied, based on the mutual intensity function [10], the Wigner distribution function [11] and tensor method [12]. Cai et al. investigated the FrFT of partially coherent and partially polarized Gaussian model beams using the tensor method [13]. The FrFT pulses were investigated by Dragoman et al. [14]. The experimental results indicated that the source coherence of the partially coherent beam had an influence on the intensity of the Gaussian Schell model beams in the FrFT plane [15]. The FrFT of Airy beams has been investigated [16, 17]. Theoretical introduction of the Olver beams has been done [18]. However, the FrFT of the Mainardi beam has not been investigated, and doing that is the aim of this paper. Based on the generalized Fresnel integral, an analytical expression for the FrFT of the Mainardi beam is obtained, and its properties illustrated by numerical examples. The Wright function, which we denote by WX, ^(z), is so named in honor of E. Maitland Wright, the eminent British mathematician, who introduced and investigated this function in a series of notes starting from 1933 in the theory of partitions. The function is defined by the series representation, convergent in the whole complex plane. WX,^(z) distinguishes the Wright functions of the first kind (X > 0) and the second kind (-1 < X < 0). In fact, functions Fv (z) and Mv (z) are a particular cases of the Wright function of the second kind wx, n(z), i.e.,[18].
(z) = £
0 n!r(Xn + ||) (-z)n
F (z) = £
„=1 n !r(-vn)
1 ™ (_z)n-i = — V-r(vn +1) sin (nvn),
n n=i n!
(_ z)n
m v (z) = Y-
n=i n!r(-vn + (1 -v))
1 ™ (—z)"-1
= — Y-r(vn) sin (nvn).
nn=i (n — 1)!
A noteworthy particular case is
MV2( z) = -^exp(—z 7 4), Vtc
M1/3(z) = 32/3Ai(z/31/3).
(1)
(2)
(3)
(4)
(5)
M denotes the Mainardi function that it is here, like an Airy function. Ai denotes the Airy function that writes as follows: [20]
Ai (z) = -!-j cos (+1)dt.
(6)
We exert Mainardi function in calculation and descript it. Mainardi beam in the Cartesian space as follows.
E (x, y) = MVq exp MVq exp . (7)
The Mainardi beam are described in details with the number, height, and width of the peaks that depend on the factors x0 and a. Myq(z) satisfies the differential equation of order q-1 [19].
n
z
n=
M v, (z) = £-
(-z)n
n!r
—n | +
q
1 ^ (-Z)n-' ( 1
= -Y-—-—rl -n |sin
n n=v (n-1)! I q ) V q
1 q
w
)
(8)
Mainardi beam in the FFT and the FrFT systems
The Fresnel integral is tool for describing of the beam propagation in free space (FFT system).
Fresnel integral of a Mainardi beam is as follows [20].
E (^ z)=inz ftexp Vi 2z (((i- x )2 +( - y )2))
¡J exp
ik (ax ia2 z ay
-exp I —+-+— +
2nz V xo 2kx0 y0
2ky0
(
< J exp
2 z
yi -l y
yo k
2
M i [A | dyi q V yo
Mi | — |MJA I expI — + ayv| dxidyi =
xo ) q V yo ) I xo yo
iaz xo k
2
M, I ^ | dxi:
q V xo ,
(9)
Since the relationship between the primary and last page variables is determined by using Fourier transform and the convolution [21], we note the convolution of two functions, f (t) and f2 (t).
f1(T) ® f (t) =
—t2 l=j f1(x,)/2(t-x,)dx,.( ) v 2 z
= Mi- I® exp [— x2 )= J f (x,) f (t - x,)dx,.
Where t = x - iaz/x0 k or t = y - iaz/y0k. The convolution theorem of the Fourier transform has the following property [20].
00
fi (t) ® f (t) = j fi (|)f (I) exp (-i|T) d|. (11)
We must introduce a new function for the Fourier of f1(T). The familiar Mittag-Leffler function Ea(z) introduced by the Mittag-Leffler [22] and its generalization E a (z) introduced by Wiman [23] are defined by the following formula:
Ea (z) = xn=o znl r(an +1) = Ea,i(z).
(12)
As well-known in probability theory, the Fourier transform of a density provides the so-called characteristic function. In this case we have:
F [1/2 M v(|x|)] =
ico
^ cos (kx)Mv (x) dx = E2v (-k2).
(13)
Where Mv(z) denotes the Mainardi function and Ev(z) denotes the Fourier transform of the Mainardi function (Mittag-Leffler function). Now, we can indicate the Fourier of f1(T).
I"
f (0 = 1
(qn)!'
(14)
By Eq (6) and by a simple mathematical calculation of Eq (13) and Eq (14), we can demonstrate:
M 1/3(z) x J cos (|3/3 + z§)d§,
(15)
M1/5(z) x J cos (|5/5 + z§)d§, M1/7 x J cos /7 + z§)d§,,
(16) (17)
M1/q xJcos(|q/q + z|)d| (q is odd). (18)
We found that our calculations are similar to the previous calculations of the Airy beam [21]. Therefore, Eq. (9) is found to be
E (x, y, z) =
Ä exp [ax I exp lay 2nz V xo) V yo
iz£2 az£
——-i^x + —-2k xo k
d^)
(19)
V"=0 (qn)!]e p( 2k Iy y0k)dI.
The FrFT of various kinds of laser beams have been widely investigated. The Wigner distribution function is rotated by an angle of 9 = pn/2 where p is the power of FrFT. The optical system for performing the FrFT is shown in Figure 1.
Lohmann I system
"l r2
Lohmann II system
"1 "2
Fig. 1. Optical system for the FrFT: Lohmann I system (a), Lohmann II system (b). Numerical calculations
Figure 1a denotes the Lohmann I system and Figure 1b the Lohmann II system; f is the standard focal length. In Figure 1a, the focus of the lens is f /sin 9. The Lohmann I and Lohmann II optical systems are equivalent and are described by the following transfer matrix [5, 16]:
n=o
R =
' A B C D
' cos 9 f sin 9^ sin 9
f
cos 9
(20)
Mainardi beam passing from a Lohmann optical system, obeys the well-known Collins integral formula:
E (x, y, z) = exp (ikz) = iXB
ikD (x2 + y2)
<P M,
J -» —
exp 2B
x, A ax, A
— 1exp| 1e
xo J xo J
y, V a, Y
— 1expI — 1e
yo J Jo J
exp
ik 2B
ik 2B
(Ax,2 - 2x,x) (Ay,2 - 2yy)
dx, x dyi.
(21)
We note that the convolution of the FrFT system, are /¡(t) and /2(t). Therefore, Eq (21) rewritten as follows:
, . exp (ikz)
E (x, y, z ) =-A— exP
ax ay exp I-
Axo Ayo
ikD (x2 + y2)
2B
exp
-ikx2
2BA 2BA
iky2 ia2 B ia2 B - +-+ •
2 Akxo
2 Aky02
(22)
il§(qn)\
/ » ■di
\(qn)\
exp(-
exp
- +
iaB
A kAx0
y iaB
B2
IkÄ
2kA {A kAy0
d^x A A
(1)
2.0 1.6 1.2 0.8 0.4 0
8
-8
I(x) X{mm)
8 -6 -4 -2 0 2 4 6 8
In the following, the properties of Mainardi beams in the FFT and the FrFT systems derived in Section 1 for the parameters are chosen as X = 0.53 |im, /= 1000 mm, a = 0.05 and x0 = y0 = 1 mm. Figure 2 represents the normalized intensity distribution of a Mainardi beam in the FFT system depending on the parameters q, a, x0, and y0. 2.01
1.6
1.2
1.0 0.5 0
0.8
0.4
I(x)
J ^V^ X(mm)
-8
(2) (a) -8
8 (b) -8 8 (c) -8
Fig. 2. Normalized intensity in the FFT system; (1) the x-direction of a Mainardi beam and (2) Intensity graph of a Mainardi beam; (a) q = 3, (b) q = 5 and (c) q = 7
Now, we investigate the influence of the parameter q on the normalized intensity distribution. Using Eq (19), we simulate a Mainardi beam in the FFT system. In Figure 2, with increase in the value of q, the number of lateral side lobes decreases, the main peak expands, and the heights of other peaks decrease.
We investigate the influences of the parameters q and p on the contour graph of the normalized intensity distribution for a Mainardi beam in the FrFT system Using Eq (22). Figure 3 represents the contour graph of the normalized intensity distribution of a Mainardi beam in the FrFT system depending on the parameters p, q, a, x0, and y0. Now, we investigate the influence of the parameters q and p on the normalized intensity distribution.
The variation of the intensity distribution with the fractional power is periodic, and the period is 2. When p < 1, the lateral side lobes are at the left side. In this case, moreover, the spot size of the Mainardi beam decreases with an increase in the value of p and with an increase in the value of q, the number of lateral lobes decreases and the main peak expands and heights of the other peaks decrease. When 1 < p < 2, the lateral side lobes are located on the right side. In this case, however, the spot size of the Mainardi beam increases with an increase in the value of p, and, with an increase in the value of q, the number of lateral lobes decreases and the main peak expands and heights of other peaks decrease. This dramatic phenomenon can be interpreted as follows: when 1 < p < 2, A = cos(pn/2) has a negative value, and B =fsin(pn /2) has a positive value.
X
1
q
-8
(3) -8 8
Y(mm) ' 1.0
0.5
. | It 1°
• •
A-(»mm)
8
Y(mm) 1.0
0.5
II« 1°
X(mm)
8 S
Y(mm) 1.0
0.5
ii* 1«
X{mm)
■8 8
(4)
-8
Y(mm)
• III
X(mm)
1.0 0.5 0
Y(mm)
« * • i
• III
X(mm)
1.0 0.5 0
-8
1.0 0.5 0
-8
Y(mm)
m
X(mm)
1.0 0.5 10
(5) -8 8 -8 8 -8 8
(a) (b) (c)
Fig. 3. Intensity graph in the FrFT system at: (1) p = 0.3 (2) p = 0.5 (3) p = 0.7 (4) p = 1.3 (5) p = 1.5 (6) p = 1.7
and with; (a) q = 3, (b) q = 5, (c) q = 7
Y(mm)
S • •.
• II
X(mm)
1.0 0.5 0
Y(mm) 1.0 8 Y(mm)
0.5
• •
X(mm) -8 X(mm)
1.0 0.5 0
(6)
-8
(a)
(b)
Continuation of Fig. 3
(c)
Figures 4- 6 show the normalized intensities in the x-direction of a Mainardi beam with a different p in the FrFT system for a = 0.05, x0 = y0 = 1 mm, q = 3, 5, and 7. If the power of FrFTs are the values p = 0.3 or p = 1.7, the normalized intensities have the same form; in p = 0.5 or p = 1.5, the normalized intensities also have the same 2.0
form; and in p = 0.7 or p = 1.3 the normalized intensities have the same form, too, but when 1 < p < 2, the spot size of the Mainardi beam rotates compared to that of the p < 1. When p < 1 and q = 3, the spot size of the Mainardi beam in the FrFT system is similar to that of the Mainardi beam in the FFT system.
1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0
I(x) ...........9=3 ----9=5 9~7
j i\\ If -:l\ A A : : 1 A
A ; Iixi 1 \\ ¿SLtixJ -A X(mm)
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0
I(x) ...........9=3; ----q = 5 9-7
X(mm)
8 -6 -4 -2 0 (b) 2 4 6 8
I(x) ...........1=3. ----q = 5 9-7
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0
I(x) ...........q = 3
----1=5
-q = 7
l
^Jil 1 X(mm)
-8 -6 -4 -2 0
4 6 8
(c)
I(x) ...........q = 3c
----q = 5r
-q = 7
ß\
/'■ 1
1; \
In ' ::
Si I / ■ \\Xjjnm)
■ r
8 -6 -4 -2 0
4 6 8
f
Fig. 4. Normalized intensity at q = 3 (dot line), q = 5 (thin line), and q = 7 (thick line) for (a)p = 0.3, (b)p = 0.5, (c)p = 0.7, (d)p = 1.3, (e)p = 1.5, and (f )p = 1.7
Figure 4 shows the normalized intensities in the x-di-rection of a Mainardi beam with a different q in the FrFT system for a = 0.05, x0 = y0 = 1 mm, q = 3, 5, and 7. If p < 1, the spot beam size in the x-direction decreases with an increase in the power of the FrFT. For 1 < p < 2, the spot beam size in the x-direction increases with an increasing power of FrFT.
Figure 5 shows the normalized intensities in the x-di-rection of a Mainardi beam with different q in the FrFT
system for a = 0.05, x0 = y0 = 1 mm, p = 0.3, 0.5, and 0.7. The effective beam size in the x-direction increases with an increase in the rational order of Mittag-Leffler function.
Figure 6 shows the normalized intensities in the x-direction of a Mainardi beam with different q in the FrFT system for a = 0.05, x0 = y0 = 1 mm, p = 1.3, 1.5, and 1.7. The effective beam size in the x-direction increases with an increase in the rational order of the Mittag-Leffler function.
-8 -6 -4 -2 0 2 4 6 8
(a)
Fig. 5. Normalized intensity for p=0.3 (dot line), 0.5 (thin line), and 0.7 (thick line); (a) q = 3, (b) q = 5, (c) q = 7
.8 -6 -4 -2 0 2 4 6
(a)
Fig. 6. Normalized intensity for p=1.3 (dot line), 1.5 (thin line), and 1.7 (thick line); (a) q = 3, (b) q = 5, (c) q = 7
Conclusion
The analytical formulas of a Mainardi beam passing from the FFT and the FrFT systems calculate by using generalized Fresnel and Collin integrals. The obtained formula is a very powerful tool to describe propagation of a Mainardi beam through the FFT and the FrFT systems. The behavior of the Mainardi beam is similar to an Airy beam in the specific case in which the of Mainardi beam q have values 3, 5, and 7. We derived an analytical expression of a Mainardi beam passing through an FrFT system. We graphically illustrated the normalized intensity distribution of a Mainardi beam in the FrFT system, and we discussed the influences of the different q and p, on the normalized intensity distribution and characteristics of the beam. Powers of the FrFT affects spot size of a Mainardi beam, and controls the orientation of the Mainardi beam spot. When p < 1, the effective beam size decreases with an increasing value of p, when 1 < p < 2 the effective beam size increases with an increasing value of p. The rational order of the Mittag -Leffler function influences the number of lateral side lobes of a Mainardi beam in the FrFT system. With an increase in the value of q, the main peak expands and height of the other peaks decrease. Therefore, the power of the FrFT (p) and the rational order of the Mittag-Leffler function (q) control the characteristics of a Mainardi beam such as the effective beam size, number, width, height, and orientation of the beam spots.
References
[1] Dai HT, Liu YJ, Luo D, Sun XW. Propagation dynamics of an Airy beam. Opt Lett 2010; 35(23): 4075-4077. DOI: 10.1364/0L.35.004075.
[2] Tang B, Jiang Ch, Zhu H. Fractional Fourier transform for confluent hypergeometric beams. Phys Lett A 2012; 376(38-39): 2627-2631. DOI: 10.1016/j.physleta.2012.07.017.
[3] Mendlovic D, Ozaktas HM. Fractional Fourier transforms and their optical implementation: I. J Opt Soc Am A 1993; 10(9): 1875-1881. DOI: 10.1364/J0SAA.10.001875.
[4] Ozaktas HM, Mendlovic D. Fractional Fourier transforms and their optical implementation. II. J Opt Soc Am A 1993; 10(12): 2522-2531. DOI: 10.1364/JOSAA.10.002522.
[5] Lohmann AW. Image rotation, Wigner rotation, and the fractional Fourier transform. J Opt Soc Am A 1993; 10(10): 2181-2186. DOI: 10.1364/JOSAA.10.002181.
[6] Cai Y, Lin Q. Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam. J Opt Soc Am A 2008; 25(8): 2001-2010. DOI: 10.1364/JOSAA.25.002001.
[7] Du X, Zhao D. Fractional Fourier transform of truncated elliptical Gaussian beams. Appl Opt 2006; 45(36): 90499052. DOI: 10.1364/AO.45.009049.
[8] Zhou G. Fractional Fourier transform of a higher-order cosh-Gaussian beam. J Mod Opt 2009; 56(7): 886-892. DOI: 10.1080/09500340902783816.
[9] Gao Y-Q, Zhu B-Q, Liu D-Z, Liu X-F, Lin Z-Q. Characteristics of beam aligmenet in a high power four-pass laser amplifier. Appl Opt 2009; 48(8): 1591-1597. DOI: 10.1364/AO.48.001591.
[10] Erden MF, Ozaktas HM, Mendlovic D. Propagation of mutual intensity expressed in terms of the fractional Fourier transform. J Opt Soc Am A 1996; 13(5): 10681071. DOI: 10.1364/JOSAA.13.001068.
[11] Yoshimura H, Iwai T. Properties of the Gaussian schell model source field in a fractional Fourier plane. J Opt Soc Am A 1997; 14(12): 3388-3393. DOI: 10.1364/JOSAA.14.003388.
[12] Lin Q, Cai Y. Tensor ABCD law for partially coherent twisted anisotropic Gaussian-schell model beams. Opt Lett 2002; 27(4): 216-218. DOI: 10.1364/OL.27.000216.
[13] Cai Y, Ge D, Lin Q. Fractional Fourier transform for partially coherent and partially polarized Gaussian-schell model beams. J Opt A 2003; 5(5): 453-459. DOI: 10.1088/1464-4258/5/5/304.
[14] Dragoman D, Dragoman M, Brenner K-H. Variant fractional Fourier transformer for optical pulses. Opt Lett 1999; 24(14): 933-935. DOI: 10.1364/OL.24.000933.
[15] Wang F, Cai Y. Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics. J Opt Soc Am A 2007; 24(7): 1937-1944. DOI: 10.1364/JOSAA.24.001937.
[16] Zhou G, Chen R, Chu X. Fractional Fourier transform of Airy beams. Appl Phys B 2012; 109(4): 549-556. DOI: 10.1007/s00340-012-5117-3.
[17] Khonina SN, Ustinov AV. Fractional Airy beams. J Opt Soc Am A 2017; 34(11): 1991-1999. DOI: 10.1364/JOSAA.34.001991.
[18] Belafhal L, Ez-Zariy L, Hennani S, Nebdi H. Theoretical introduction and generation method of a novel nondiffracting waves: Olver beams. Opt Photonics J 2015; 5(7): 234-246. DOI: 10.4236/opj.2015.57023.
[19] Mainardi F. Fractional calculus and waves in linear viscoelasticity. London: Imperial College Press; 2010. ISBN: 978-1-84816-329-4.
[20] Morris JE, Mazilu M, Baumgartl J, Cizmâr T, Dholakia K. Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength. Opt Express 2009; 17(15): 13236-13245. DOI: 10.1364/OE.17.013236.
[21] Gradshteyn IS, Ryzhik IM. Tables of integrals, series and products. New York: Academic Press; 1980. ISBN: 978-012-294760-5.
[22] Mittag-Leffler G. Sur la Nouvelle fonction Ea(x). Comptes Rendus de l'Académie des Sciences - Series I. Mathematics 1903; 137: 554-558.
[23] Wiman A. Über den Fundamentalsatz in der Teorie der Funktionen Ea(x). Acta Math 1905; 29(1): 191-201.
Author's information
Forouzan Habibi (b. 1990) received Master's degree in Atomic and Molecular Physics in Iran State SKU (2017). She is researcher and programmer. Scientific interests: singular optics, Optical systems, atmospheric turbulence, special functions. E-mail: _ [email protected] .
Mohammad Moradi (b. 1965) received his B. Sc. in Physics in 1989 from Isfahan University, Isfahan, Iran; M. Sc. in Atomic Physics in 1992 from Polytechnic Tehran University, Iran and PhD in Atomic Physics, Optics & Laser from Moscow State University in 2005, Russia. He is currently engaged as an Assistant Professor of Physics in Department of Physics, Shahrekord University, Shahrekord, Iran. E-mail: [email protected] .
Alireza Ansari is associate professor and academic member at department of applied mathematics, faculty of mathematical sciences, Shahrekord University. He received his Ph.D. degree from University of Guilan in year 2011. His research areas include fractional differential equation, integral transforms and special functions. E-mail: ansari. alireza@sci. sku. ac.ir .
Received May 28, 2018. The final version - October 3, 2018.
