=Paper=
{{Paper
|id=Vol-1490/paper13
|storemode=property
|title=Modeling of propagation of optical signals in gradient index media based on fractional Fourier transform
|pdfUrl=https://ceur-ws.org/Vol-1490/paper13.pdf
|volume=Vol-1490
}}
==Modeling of propagation of optical signals in gradient index media based on fractional Fourier transform==
https://ceur-ws.org/Vol-1490/paper13.pdf
Computer Optics and Nanophotonics
Modeling of propagation of optical signals in gradient
index media based on fractional Fourier transform
Zubtsov R.O.,
Samara State Aerospace University
Kirilenko M.S.
Samara State Aerospace University,
Image Processing Systems Institute, Russian Academy of Sciences
Abstract. This research has simulated the propagation of the light beams
through the quadratic index media. Five methods of simulation were considered
and the propagating beams corresponding to different input signals such as
Airy-Gaussian beams, rectangular function pulses, triangular function pulses,
cosine function signals and finite eigenfunctions were demonstrated.
Keywords: gradient index media, fractional Fourier transform, Airy-Gaussian
beams, Hermite-Gaussian modes, eigenfunctions
Citation: Zubtsov RO, Kirilenko MS. Modeling of propagation of optical
signals in gradient index media based on fractional Fourier transform.
Proceedings of Information Technology and Nanotechnology (ITNT-2015),
CEUR Workshop Proceedings, 2015; 1490: 105-111. DOI: 10.18287/1613-
0073-2015-1490-105-111
Introduction
Fractional Fourier transform (FrFT) is a set of linear transformations that
generalize Fourier transform. Fourier transform is generally interpreted as a
convention of the time domain of the signal to its frequency domain.
The canonical FrFT was considered [1] as the Fourier transform of -order, where
is the real value. We can likewise define the FrFT as the operation of the frequency-
time distribution (Wigner distribution function) rotation at a certain angle [2].
Originally, FrFT was used in quantum mechanics; however, recently it has
increasingly become a focus of opticists. As a result, extensive research involving its
properties, optical realization and potentiality opportunities in optic applications has
been performed. Thus, currently FrFT is actively used in optical image processing [3].
Moreover, the fractioning of some transformation provides a new degree of freedom
(fraction order) that can be used for a more complete description of the object (signal)
or as an additional encoding parameter.
FrFT is used in differential equation solving, in quantum mechanics and quantum
optics, in optical theory of diffraction, in optical system and optical signal processing
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descriptions including the application of frequency filters, time filtration and
multiplexing, as well as in pattern recognition, in wavelet-transformations, in
operations with chirp-functions, in encryption, for neural network creation and other
applications. A more detailed review of FrFT can be found in the paper [3] produced
by T. Alieva et al.
The modular lens system and the system of several spherical and/or cylindrical
lenses are among the methods of FrFT optical realization [4-8]. Some of these
systems (especially cylindrical lenses) are used for astigmatic transformation in order
to form vortex beams [5, 8-10].
One of application of FrFT is the description of laser beam propagation in gradient
index media [11, 12].
In this work, we use one-dimensional FrFT to model optical signal propagation in
optical waveguides with parabolic dependence of the refractive index. The
eigenfunctions of the transforms are Hermite-Gaussian modes [1].
During the modelling process special attention is given to Airy-Gaussian beams,
which carry finite power and keep the properties of non-diffracted propagation in a
partial area. They can be experimentally realized with a particularly good
approximation [13-16].
1. General theory
The light beam propagation through the ABCD-system in one-dimensional cases
is described by the Huygens integral:
ik 2
U 2 ( x2 )
k
i 2B
2
U1 ( x1 )exp 2B Ax1 2x1x2 Dx2 dx1 . (1)
where k 2 / , U1 ( x1 ) is input field, U 2 ( x2 ) is output field.
In gradient index media with the refraction index n n0 1 x 2 / 2a 2 , the matrix
of ABCD-system is (beam propagation from z1 0 to z2 z ) [13, 14, 17, 18]:
A B cos( z / a) a sin( z / a)
C D sin( z / a) / a cos( z / a) . (2)
With this type of matrix the integral (1) turns into FrFT.
The complete set of FrFT eigenfunctions is the following set of Hermite-Gaussian
functions:
n ( x) ein / 2 n ( x),
2 x e
21/ 4 x 2
(3)
n ( x) Hn ,
n
2 n!
where H n ( x) – a Hermite polynomial of order n:
d x2
H n ( x) (1)n e x
2
e . (4)
dx n
If the FrFT has finite integration limits (in other words the input beam is limited),
its eigenfunctions are somewhat different from the Hermite-Gaussian modes [19].
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The propagation of a light beam through the gradient index media was modeled.
The Airy-Gaussian beams are:
U1 ( x1 ; 1 , 1 , S1 , q1 )
x x S 2 ikx 2
Ai 1 1 exp iS1 1 1 i 1 1 , (5)
1 1 3 2q1
1 , 1 , S1 , q1 C.
Furthermore, the cosine distribution function is:
f ( x) A cos(x ),
(6)
A, , R.
In addition, the rectangular function, triangle function and eigenfunctions have
been simulated.
2. Simulation
Method 1: direct numerical calculation of FrFT by definition.
The integral (1) is rarely solved analytically, so we use numerical computations. If
typical methods are use, the numerical calculations of quadratic exponents require a
very large number of sampling points because of rapid oscillations in the kernel. The
problem is especially pronounced if is close to 0 or ±2. We assume that functions
and their Fourier transformations are limited (they are not equal to zero in finite
intervals) and this difficulty can be avoided. If 0.5 ≤ ≤ 1.5 or 2.5 ≤ ≤ 3.5, we can
directly calculate the integral. If -0.5 < < 0.5 or 1.5 < < 2.5, we can use the
property of additivity: 11 , where in the transformation of -1 order may be
found immediately.
For the integration we use Simpson's rule (n is even):
b
a f ( x)dx 3
h n/ 2
j 1
f ( x2 j 2 ) 4 f ( x2 j 1 ) f ( x2 j ) ,
(7)
x j a jh, j 0, n 1, h (b a) / n.
Although this method of FrFT calculation can produce accurate results, it operates
slowly and has a computational complexity O(N2) [20].
Method 2: Fast FrFT.
FrFT is a special case of the more general transform class sometimes known as
linear canonical transformations or quadratic-phase transformations. The elements of
this class can be decomposed to a sequence of simple operations such as chirp-
multiplication, chirp-convolution, scaling and typical Fourier transform. There are
two different decompositions demonstrated here, leading to different algorithms.
Method 3: the decomposition into chirp-multiplication, chirp-convolution and
another single chirp-multiplication sequence.
In this approach, we assume that -1 ≤ ≤ 1. The transformation (1) can be written
as:
i 2
f ( x) exp x tan g '( x) , (8)
a 2
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i
g '( x) A exp ( x x ')2 g ( x ')dx ' , (9)
a
i 2
g ( x) exp x tan f ( x) , (10)
a 2
z
where g ( x), g '( x) represents an intermediate result, , csc , A .
a ia
Method 4: another decomposition involving the Whittaker-Shannon interpolation
formula (sinc-interpolation). The defining expression of FrFT can be written as:
i
f ( x) A exp x 2
a
(11)
i 2 i 2
exp xx ' exp x ' f ( x ')dx '.
a a
i
The function exp x '2 f ( x ') can be represented by the Shannon
a
interpolation formula:
i n 2
N
exp ix '2 f ( x ') exp
N a 2x
, (12)
n n
f sinc 2x x ' 2x .
2x
We substitute (12) in (11), change sequence of integration and summation and use
certain algebraic manipulations to obtain the following expression:
m A i m
2
f
2x 2x
exp ( )
a 2x
(13)
N i m n 2 i n n
2
exp ( ) f .
N a 2x a 2x 2x
There are other methods, for example those described in [21], however they are
not well suited for plotting images on a plane. The modeling results are demonstrated
in Figures 1-6.
Fig. 1. – The propagation of Airy-Gaussian beams (version 1)
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Fig. 2. – The propagation of Airy-Gaussian beams (version 2)
Fig. 3. – The propagation of cosine function signals
Fig. 4. – The propagation of rectangular pulses
Fig. 5. – The propagation of triangular pulses
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Fig. 6. – The propagation of eigenfunctions
Acknowledgments
This work was financially supported by the Russian Ministry of Education and
Science and by the Russian Foundation for Basic Research (grant 13-07-00266).
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