=Paper=
{{Paper
|id=Vol-1490/paper6
|storemode=property
|title=Modeling superlattice patterns using the interference of sharp focused spherical waves
|pdfUrl=https://ceur-ws.org/Vol-1490/paper6.pdf
|volume=Vol-1490
}}
==Modeling superlattice patterns using the interference of sharp focused spherical waves==
https://ceur-ws.org/Vol-1490/paper6.pdf
Computer Optics and Nanophotonics
Modeling superlattice patterns using the interference of
sharp focused spherical waves
Fidirko N.S.
Samara State Aerospace University
Abstract. In this paper, modelling of pseudonondiffrational beams forming
superlattice structures in a cross section has been performed. To create such
distributions we suggest using superposition of sharp focused spherical waves.
Thus, we have done simulations for several spherical waves generated by
coherent light sources located on a ring with a certain radius. It is shown that
depending on the configuration of the source field we can achieve different
superlattice patterns in a cross section with a small amount of waves in the
input field. Using more waves, we can obtain Bessel-like beams in the cross
section.
Keywords: interference, optical vortices, sharp focusing, polarization
Citation: Fidirko NS. Modeling superlattice patterns using the interference of
sharp focused spherical waves. Proceedings of Information Technology and
Nanotechnology (ITNT-2015), CEUR Workshop Proceedings, 2015; 1490: 45-
52. DOI: 10.18287/1613-0073-2015-1490-45-52
Introduction
A nondiffracting wave field is comprehended as a monochromatic optical field
whose transverse shape remains invariant in free-space propagation. In 1987, Durnin
proposed that nondiffracting wave fields are exact solutions to the homogeneous
Helmholtz equation [1]; such particular solutions can be described as Bessel functions
and are called nondiffracting Bessel beams. The realizable beams that propagate with
relatively small divergence angles up to a certain range have finite energy and are
known as pseudonondiffracting optical beams. Along with his co-authors, Durnin first
experimentally realized a pseudonondiffracting Bessel beam in a cylindrical
coordinates system [2]. Since then, nondiffracting Bessel beams have been
extensively studied and applied in diverse fields, for example optical manipulation,
the capture of micro particles and optical coherence tomography [3-7].
In recent years, the attention of physicists and mathematicians has been drawn to
two-dimensional nondiffractive superlattice patterns [8]. Besides, realization of such
distributions related to crystals, quasicrystals and other periodic structures has been
actively researched [9-12].
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Fidirko N.S. Modeling superlattice patterns...
A two dimensional distribution made by superposition of several plain lattices is
called a superlattice [8, 12]. In this work, we show an approach to create superlattice
distributions using the interference of sharp focused spherical waves.
Model of sharp focusing
A sharp focused electromagnetic field in the focal area in Cartesian coordinates
can be described with the following equation:
PE ( , )
E( , , z )
2
if 1
H( , , z )
B ( , )T ( )
P
H ( , ) (1)
0 0
exp ik sin cos( ) z cos sin d d ,
where ( , , z ) – cylindrical coordinates in the focal area, (, ) – spherical
angular coordinates of the output pupil of the focusing system, - maximal value of
the azimuth angle, related to the numerical aperture, B(, ) – transmission
function, T () – apodization function (for aplanatic systems it is T () cos ),
k 2 / – wavenumber, – wavelength, f – focal distance. PE (, ) и
PH (, ) – polarization matrixes for electric and magnetic fields respectively:
1 cos2 (cos 1) sin cos (cos 1)
cx ( )
PE ( , ) sin cos (cos 1) 1 sin 2 (cos 1)
c y ( )
; (2)
sin cos sin sin
sin cos (cos 1) 1 cos2 cos 1
cx ( )
PH ( , ) 1 sin 2 cos 1 sin cos (cos 1) . (3)
c y ( )
sin sin sin cos
where cx ( ), c y ( ) is the polarization coefficients of the source field.
For vortex fields B( , ) R exp(im ) , so formula (1) can be reduced to an
equation with one-time integration:
QE (, , )
E(, , z )
H(, , z ) ikf R () T () 1 Q (, , ) sin exp (ikz cos )d. (4)
0 H
where Q E , H (, , ) matrices can be explicitly written for certain types of
polarizations and consist of superposition of Bessel functions of different orders [13-
14].
If all beams are generated by different zones of the optical element supplementing
a lens with a high numerical aperture, the resulting field in the focal area will be a
superposition of the fields established by different zones of the optical element:
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Fidirko N.S. Modeling superlattice patterns...
E( , , z ) Ei ( , , z ). (5)
i
The interference of spherical waves
The next step is to review an opaque diaphragm, imposed on the pupil of a
focusing system with a high numerical aperture. The diaphragm has several small
holes, located evenly on a certain radius from the centre of the diaphragm.
Table 1. – The resulting electric field for the central radius rc 25
N Source field 2
E (latitudinal and longitudinal sections)
3
4
5
Thereby, we gain a system of point light sources. Every source generates a
spherical wave that is being focused on and interferes with waves from other point
sources. Herewith, we can vary the number of point sources and their distance from
the centre of the aperture.
In figure 1, the shapes of the diaphragms generating different number of waves is
evident. In tables 1-5 results of modelling with different parameters are listed.
From the tables above, with the interference of three and four spherical waves in
the cross section of the focal area there are bright light spots located in the lattice
sites; furthermore, the configurations of the lattices can be different. In addition,
longitudinal plane long light channels are formed. The nondiffractional nature of the
beams, the spectrum of which is localized on the ring, has been evident for a long
time. It has been successfully used to create different structures that remain invariant
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Fidirko N.S. Modeling superlattice patterns...
in the longitudinal direction [16-18]. The obtained distributions can be used to create
photonic crystals and plasma channels.
Table 2. – The resulting electric field for the central radius rc 50
N Source field 2
E (latitudinal and longitudinal sections)
3
4
5
With the interference of five or more waves in the cross section, we can see a
superlattice pattern. If the radius of the ring is increased (for comparison see tables 3
and 4) the central spot is decreased, which corresponds to the increase in the
numerical aperture. With the increase in the numerical aperture (radius of the ring),
the interference pattern becomes more complex and different symmetries appear.
а) b) c)
Fig. 1. – Shape of the diaphragm for а) 5 point sources, b) 10 point sources, c) 15 point sources
with a central radius rc 50 , the radius of every point is - rd 2
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Information Technology and Nanotechnology (ITNT-2015)
Computer Optics and Nanophotonics Fidirko N.S. Modeling superlattice patterns...
Table 3. – The resulting electric field for the central radius rc 25
Source field 2
N
E (latitudinal and longitudinal sections)
10
12
15
If we use many point sources so that the ring aperture is tightly filled, Bessel-like
beams begin to form in the cross section. In these situations, the size of the central
spot of the Bessel beam depends on the radius of the ring aperture. It is worth noting
that this approach to generating a Bessel beam is more convenient than creating the
ring aperture.
Conslusion
By varying the number of waves and distance between them one can therefore
obtain a wide range of superlattice patterns in the cross section, which will keep their
structure at a long distance. In this case, the radius of the holes in the diaphragm and
the radius of the ring determine the length of the longitudinal section. Increasing the
size of the holes and the numerical aperture leads to a reduction of the focus depth.
More complex superlattice patterns can be added by increasing the number of
phases of the point sources with special phase elements and by adding polarization to
the focused beam [13, 19-21].
Acknowledgements
This work was financially supported by the Russian Ministry of Education and
Science.
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Table 4. – The resulting electric field for the central radius rc 50
N Source field 2
E (latitudinal and longitudinal sections)
10
12
15
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