=Paper=
{{Paper
|id=Vol-1490/paper18
|storemode=property
|title=Asymptotic research in computer optics
|pdfUrl=https://ceur-ws.org/Vol-1490/paper18.pdf
|volume=Vol-1490
}}
==Asymptotic research in computer optics==
https://ceur-ws.org/Vol-1490/paper18.pdf
Computer Optics and Nanophotonics
Asymptotic research in computer optics
Kazanskiy N.L.
Image Processing Systems Institute, Russian Academy of Sciences,
Samara State Aerospace University
Abstract. I give an overview of the methods and possibilities of the asymptotic
studies for solving the computer optics. I analyze the relevance of the use of the
results in the design of diffractive optical elements for laser material processing.
I discuss the prospects of the developed approaches for the study of the
components of diffractive nanophotonics.
Keywords: Asymptotic method, diffraction theory, scalar approximation,
electromagnetic theory, computer optics, laser technology, diffractive
nanophotonics
Citation: Kazanskiy N.L. Asymptotic research in computer optics. Proceedings
of Information Technology and Nanotechnology (ITNT-2015), CEUR
Workshop Proceedings, 2015; 1490: 151-161. DOI: 10.18287/1613-0073-2015-
1490-151-161
Introduction
Asymptotic methods have always been in the focus of scientists-opticians [1-3].
These methods have evident interest in recent years [4-11]. Scientists working in the
field of diffractive computer optics, also actively used the opportunities provided by
asymptotic methods [12-17]. Asymptotic methods are especially good in the study of
such class of diffractive optical elements (DOEs), as focusators of laser radiation [14-
17]. In the paper I give an overview of the methods and possibilities of the asymptotic
studies for solving the computer optics. In particular, I analyze the relevance of the
use of the results in the design of diffractive optical elements for laser material
processing.
1. Focusators research
For creating a new focusator we have several important steps: obtaining a phase
function; study of the phase function; choice of sampling parameters and method for
manufacturing diffractive microrelief; calculation and production of focusator;
experimental study of the microrelief and output parameters of focusator. To study the
phase function of focusator scientists use analytical calculation of the diffraction
patterns of the focused radiation. This calculation must take into account the finite
size and specific physical parameters of created focusator [12-17]. Typically, the
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geometric optics approximation is used to calculate the phase function of focusator.
Diffractive analysis allows us to explore the limits of this approach. It allowed us to
identify the initial values of the physical parameters under which the distortion of the
focus area began. This analysis allows us to identify possible errors in the analytical
solution of the inverse problem of the diffraction theory.
However, we can carry out diffractive analyzes only for simple phase functions,
axially symmetric illuminating beams and focus areas - such as the ring [12], a set of
points [18-20], longitudinal [21-24] or cross [15-17, 25-26] segments. In some cases,
the analytical study can provide diffractive corrections to the phase function of
focusator [14]. Unfortunately, in the framework of the analytical study, we cannot
take into account the effect of sampling and quantization of the phase function
focusator arising during the manufacture of the DOE. However asymptotic analysis is
an important stage of research and complements the capabilities of the computational
experiment [21-28].
Unfortunately, asymptotic methods do not allow us to analyze different methods
for manufacturing micro-relief of diffractive optical elements [29-36]. Such a study is
necessary to select the most appropriate technology for manufacturing DOEs intended
for solving a particular problem.
As an example, I cite the results of the asymptotic study of the geometric-optical
focusator, concentrated laser beam into the ring [12, 21]. Fig. 1 shows the results of
the asymptotic calculation of the intensity distribution in the focal plane of the
geometric-optical focusator into the ring with the following parameters: focal length f
= 750 mm; focusator diameter 2R = 25.6 mm; the wavelength of focused beam λ =
0.6328 µm; focal ring diameter r0 = 0.1 mm.
Fig. 1. – The intensity distribution I(x,y) in the focal plane of the focusator into the ring
Asymptotic calculation shows that the diffraction width of the focus ring is
comparable to the radius of the ring for focusator with these parameters. As a result,
the ring begins to merge with the central spot. The asymptotic calculation shows that
such a draining does not occur when the radius of the focal ring in several (and
furthermore many) times larger than 0.1 mm. So we clearly see the limitations of the
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methods of geometrical optics in calculating the phase function of focusator. The
results can also be used in the study of other types of DOEs [37-45] and to focus the
surface electromagnetic waves [46-48].
2. Temperature calculation
Asymptotic analysis allows us to optimize the phase function of focusator for use
in a specific laser processing technology of a given type of material [49-55].
Focusators have broad prospects for application in a variety of laser materials
processing technologies [49-55]: hardening, cutting, welding, drilling, branding, etc.
Therefore, the problem of investigating the temperature characteristics of the laser
effects produced by focusators is very important. Known focusators form a
predetermined intensity distribution in some areas. Laser technology requires forming
a desired temperature distribution on the object surface intended for processing. In
[26], we conducted an analysis of the temperature distribution formed by focusator
into segment (focusator focuses laser light into a line segment located in the focal
plane). Asymptotic approach allowed us to obtain a phase function of focusator
focusing laser beam into the line segment with a predetermined temperature profile.
For example, in [26] we calculated optical element for focusing the circular (and ring)
beam into the line segment with a constant temperature distribution. Fig. 2 shows the
simulation results for these focusators. Fig. 2a shows a normalized graph of the
calculation of temperature distribution along the focal segment for thermal focusator
focusing uniform beam of circular cross section of radius R = 5 mm with the
following parameters: the wavelength of the focused radiation λ = 1.06 µm; focal
length f = 100 mm; length of focal segment 2d = 2 mm; (4at0)½ = 20 µm (here a is
thermal diffusivity, t0 is the duration of the laser action).
Fig. 2b shows the calculated normalized graph of the temperature distribution
along the focal segment for the heat focusator focusing uniform beam of annular cross
section with radii R1=3 mm and R2=5 mm with the following parameters: the
wavelength of the focused radiation λ=1.06 µm; focal length f=400 mm; length of
focal segment 2d=8 mm; (4at0)½ = 0.2 mm (here used value for thermal diffusivity of
the steel a=12 mm/s2). We can interpret the data in Fig. 2 as a result of forming by the
heat focusator a constant temperature profile on the steel surface by the end of the
laser pulse duration t0 = 0.001s. For investigated focusators standard deviation from
the constant temperature is 8.8% (for the illuminating beam of circular cross-section,
Fig. 2a) and 12.2% (for the illuminating beam of the annular cross-section, Fig. 2b). It
is about two times better than using geometrical optics focusators.
3. Electromagnetic theory
In recent years, we are actively developing new asymptotic methods within the
electromagnetic theory for calculating the field generated by DOE [56-65].
For example, in [57] we presented an asymptotic method for solving problems of
diffraction on the diffractive microrelief. This method combines the geometric-optical
approach and solution to the problem of diffraction by a periodic structure with a
period comparable to the wavelength. We solved the problem of diffraction by a
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standard quasi-periodic structure that combines the functions of a beam splitter and a
diffraction lens. On the basis of the standard solution of the problem we got a simple
expression for the field in a plane adjacent to the DOE. The resulting expression
allows us to estimate the distribution of the field at the output of the DOE without
resorting to complex computational methods.
T
1,0
0,5
a) -1,10 -0,55 0 0,55 1,10 x/d
T
1,0
0,5
b) -1,10 -0,55 0 0,55 1,10 x/d
Fig. 2. – The temperature distribution T on the focal segment for thermal focusators: a)
uniform illuminating beam with a circular cross-section; b) uniform illuminating beam with an
annular cross-section
We have obtained the results of calculation for the intensity distributions of
electromagnetic radiation in the focal plane of the focusator into ring for various
combinations of system parameters. Calculation of the field in the focal region, we
carried out on the basis of the distribution of the field at the output of the DOE
calculated within the electromagnetic theory [57]. Further, the field in the focal plane,
we calculated using the propagator, described in [58], on the base of the field at the
output of the DOE.
Fig. 3 shows an example for calculation of fields generated by focusators into the
ring for the values shown in Table. 1 (all dimensions are in microns). For small
relationship σ/f (σ is parameter of an illuminating Gaussian beam, f is the focal
length), the intensity distribution in the focal plane of focusator into the ring is close
to the intensity distributions obtained [12, 21] in the framework of scalar
approximation. In this case, the energy distribution has good axial symmetry. The
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symmetry is improved in the case of increasing the focal length. The asymmetry in
the energy distribution along the ring appears at the increase in the ratio σ/f. The
presence of asymmetry is due to the following factors:
presence of linear polarization of the incident wave destroys radial symmetry, since
the electric fields from different points of the focusators come at different angles in
different points of the focal plane;
in the case of linear polarization of the incident wave the diffractive coefficient
depends on the direction of the local grating, it appears with increasing ratio σ/f.
Table 1. The parameters of focusators
Parameter Value (option Value (option
1), µm 2), µm
Wavelength λ 1 0.1
Parameter of 50 50
Gaussian beam σ
The distance from 1000 100
the optical element
to the observation
plane
Focal length f 1000 100
Dimensions of the 500×500 500×500
optical element
Uneven intensity of light in the observation plane of focusator into ring caused
unevenness coefficient values in transmission (reflection) of the E- and the H-
polarization depending on the current value of the period of the band structure
(diffraction grating).
Conclusion
In recent years, scientists were actively developing asymptotic methods in the
frame of the electromagnetic theory for calculation of the field formed by the DOEs
[56-64]. We use this methods not only for the study of diffractive optical elements (in
particular, of short-focus DOEs), but also for the study of nanophotonics components
[66-74], for the design of equipment for hyperspectral remote sensing [75-79] and
solving other urgent tasks of diffractive nanophotonics [80].
Acknowledgements
This work was supported by RFBR grants 14-07-00339, 14-07-97008, and
fundamental research programs of the Nanotechnology and Information Technology
Department of the Russian Academy of Sciences.
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Fig. 3. – The calculated intensity distribution in the focal planes of focusators into the ring
with the parameters given in the Table 1 (option 1 - the top; option 2 - the bottom)
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