=Paper=
{{Paper
|id=Vol-2667/paper71
|storemode=property
|title=Optimal orthogonal bases in optical applications
|pdfUrl=https://ceur-ws.org/Vol-2667/paper71.pdf
|volume=Vol-2667
|authors=Vladimir Andreev,Anton Bourdine,Vladimir Burdin
}}
==Optimal orthogonal bases in optical applications ==
https://ceur-ws.org/Vol-2667/paper71.pdf
Optimal orthogonal bases in optical applications
Vladimir Andreev Anton Bourdine Vladimir Burdin
Povolzhskiy State University of Povolzhskiy State University of Povolzhskiy State University of
Telecommunications and Informatics Telecommunications and Informatics; Telecommunications and Informatics
Samara, Russia Samara, Russia Samara, Russia
JSC “Scientific Production Association burdin-va@psuti.ru
State Optical Institute Named after S.I.
Vavilov
St. Petersburg, Russia
bourdine@yandex.ru
Abstract—The tools of diffraction optics allow to implement often, these waves also correspond to the eigenfunctions of
in optics a wide range of mathematical functions useful for optical fibers with a constant refractive index, i.e. to Bessel
various applications. The orthogonal bases are of particular modes. However, it is not so easy to perform optical
interest as they are optimal in terms of representation and expansion by the basis of conical waves. A zero-order Bessel
transmission of optical information. The scientific school of
beam can be formed using a glass cone (refractive axicon)
Professor Viktor A. Soifer, Academician of the Russian
Academy of Sciences, pays considerable attention to addressing [3], but the generation of high-order Bessel modes required
the problems in this area. The following problems have been the development of fundamentally different optical elements,
solved successfully: optical multiplexing - demultiplexing of which can be referred to with the concept of “Bessel optics”
various laser beams for modal compaction of communication [4]. Hermite-Gaussian modes and Laguerre-Gaussian modes,
channels, numerical and optical implementation of the which are the eigenfunctions of gradient media, are used
Karhunen-Loeve expansion for the investigation of the stability widely in the theory of resonators, gradient waveguides, and
of vortex beams propagation in a medium with random paraxial optical systems [5]. When analyzing wavefront
fluctuations, and the use of eigenfunctions of bounded optical aberrations, the Zernike basis is used [6]. Generation, as well
systems for signal transmission with less distortion. The results
as optical decomposition by such bases, became possible
achieved in the development of new optical devices can serve as
the basis for the advanced information technologies. only after the development of diffractive optical elements
(DOEs). In the works of A.M. Prokhorov, I.N. Sisakyan,
Keywords—diffraction optics, mathematical functions, V.A. Soifer et al. [4, 7–10] it was proposed to synthesize
orthogonal bases, optical information, scientific school, laser optical elements - “modans” that generate and select
beam, Karhunen-Loeve expansion, vortex beams
individual laser radiation modes. A similar statement of the
problem was contained in the article A.W. Lohmann, G.K.
Grau et al. [11] published a year after the publication of
I. INTRODUCTION M.A. Golub, A.M. Prokhorov, I.N. Sisakyan and V.A. Soifer
In information theory, the optimal representation of a [6]. These pioneering works were developed further at the
certain signal [1–2] means choosing an orthogonal basis with
scientific school of Professor Viktor A. Soifer, Academician
the minimal number of coefficients of expansion by the basis
of the Russian Academy of Sciences [12].
functions. In optical applications, special attention is paid to
The group of Prof. V.V. Kotlyar calculated, and then
the bases representing the solution of a differential or
produced in collaboration with Prof. S.N. Khonina and the
integral operator of propagation through a specific optical group of Prof. J. Turunen (University of Joensuu, Finland)
medium or system. As a rule, these are laser radiation the DOEs that enable the formation of multimode laser
modes. In addition, the bases that are optimal in terms of the
beams with the pre-defined self-reproduction properties [13–
presentation and transmission of optical information are of
18].
particular interest. For example, the Karhunen-Loeve basis,
which provides the minimum number of expansion terms in III. KARHUNEN-LOEVE BASIS
the representation of a random signal, as well as In addition to the bases listed above, other optimal bases
eigenfunctions of bounded optical systems, the matching are known that have no analytical representation. They are
with which ensures the transmission of a signal with less usually associated with additional conditions or restrictions
distortion. Such complex basis functions, which sometimes imposed on optical systems or the optical signal.
even have no analytical representation, can be implemented In the statistical approach to the description of signals,
in optics only by using the tools of diffraction optics. The the optimal basis for representing particular realizations of
scientific school of Professor Viktor A. Soifer, Academician random signals is the Karhunen-Loeve basis (KL) [19], in
of the Russian Academy of Sciences made a great which the error rate averaged over the ensemble of
contribution to the development of theoretical foundations implementations is minimal. That is, the KL expansion
and methods of diffraction optics. This article provides a provides the minimum number of terms among all possible
brief overview of the achievements of the scientific school expansions in the representation of a random signal for a
related to the formation and analysis of optical signals based given mean square error [20]. This property is relevant for
on optimal orthogonal bases. various applications: from recognition problems to the
problem of increasing the stability of optical signal
II. LASER RADIATION MODES
transmission under atmospheric turbulence [21–26].
The plane wave basis is well known in optics, its
At the beginning of the 1990s, the problem of calculating
spectrum can be generated in the focal plane of a lens. Along
the KL basis for the exponential cosine correlation function
with the plane waves, expansion in conical waves is used
[27] was successfully solved at the Image Processing
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)
Data Science
Systems Institute of Russian Academy of Sciences (IPSI coordinate system for a finite (space-limited) Fourier
RAS), and then the possibility of its optical realization was transform correspond to prolate angular spheroidal functions.
studied [28, 29] in order to form decorrelated criteria of Spheroidal functions are also eigenfunctions for a two-lens
optical signals. In recognition of these results V.A. Soifer system, in which an additional restriction appears in the
and S.N. Khonina received the First Prize of the German plane of the spatial spectrum [48–49].
Society for the Advancement of Applied Informatics for the Another attractive feature of the communication mode
best scientific work in the field of image processing and method is that it simplifies free space diffraction to ordinary
pattern recognition in 1993. mathematical multiplication, thereby making it an interesting
It becomes more and more urgent to tackle the issues tool for propagating waves and synthesizing fields [64]. To
related to the transmission of an optical signal over implement this approach, methods of calculating DOEs
significant distances in free space, use of optical radiation correlated with PSWFs [65] were used at IPSI RAS. The
for sensing the Earth’s surface, determining environmental possibility of optical generation of an arbitrary superposition
parameters, location and navigation [30–32]. The use of of spheroidal functions allows to form optical fields passing
optical radiation for these applications requires to take into through the corresponding optical systems without distortion
account the effect of atmospheric turbulence [33–35]. [66–68].
Therefore, a lot of efforts are aimed at finding the possibility The theory of communication modes (or eigenfunctions
to overcome the negative impact of turbulence of the of optical operators) is applicable to arbitrary optical systems
medium. An overview of the current situation in this area and electromagnetic waves [69–74].
can be found in the joint publication of the researchers from A particular type of optical system is optical fiber. The
IPSI RAS, Samara University and University of Miami [36]. current level of use of optical fiber for transmitting
Also, at the initiative of Academician V.A. Soifer, numerical information over time and frequency channels tends to the
and experimental studies were performed on the resistance limit of bandwidth. An additional increase in the number of
of vortex beams to random fluctuations of the optical information channels is possible on the mode division
medium [37–42]. multiplexing (MDM) [10, 75]. This technology includes the
In order to analyze the ability of certain beams to transmission of information in various transverse modes on a
maintain information stability (for example, the orbital single physical medium - optical fiber. The transmitted
angular momentum) under the influence of random information can be contained in the mode structure and in
fluctuations of the optical medium, numerical simulation or the energy component carried by each mode in the laser
laboratory experiments with turbulence simulators are used, beam individually. Moreover, multiplexing based on vortex
including diffusers, scattering screens and turbulence cells beams associated with the orbital angular momentum is of
[43–44]. The synthesis of such a simulator of turbulence can the greatest interest [76–78]. For mode channel multiplexing
be implemented using the KL expansion for the given based on the orbital angular momentum in real (bounded)
correlation operators based on a search for the fibers, it becomes necessary to calculate vortex
eigenfunctions of these operators [45–46]. eigenfunctions [78]. The propagation of an optical signal
through multi-lens optical systems and gradient waveguides
IV. THE BASIS OF PROLATE SPHEROIDAL
is well described by the fractional Fourier transform [79–84].
WAVE FUNCTIONS
Spatial constraint inevitably leads to the necessity to
When analyzing and compensating the atmospheric
consider spatially bounded propagation operators and
distortions, not only the KL expansion, but also the basis of
calculate the corresponding eigenfunctions to simulate the
prolate spheroidal wave functions (PSWFs) is used [47].
propagation of an optical signal [85–86]. This approach
According to the Fourier transform theory, a signal cannot
allows both to understand the nature of optical signal
be sharply bounded both in the object domain and in the
distortions, and to form an approximation of the initial signal
spatial frequency band, but by using the PSWFs it is possible
through decomposition by eigenfunctions of the lens system
to provide the best field concentration in the object and
by analogy with the PSWFs. When forming such an
spatial-frequency domains simultaneously [48–49]. PSWFs
approximation, a compromise can be observed between the
are also used in various applications: in the theory of antenna
accuracy of the approximation and the ability to transmit
synthesis, in image-based reconstruction of objects, for
signal without distortion.
superresolution, in the theory of resonators, in digital
filtering [50–55]. At the beginning of 2000s, under the VI. CONCLUSION
direction of V.A. Soifer a new stable method was developed Modern computing resources provide the possibility to
at IPSI RAS for calculating the eigenvalues of the zero-order calculate the eigenfunctions of fairly complex operators,
PSWFs for arbitrary parameter values [56], as well as for including those describing near-field optics and scanning
approximating the eigenfunctions by finite series [57–58]. optical systems [87–88], thus the diffraction optics tools
Later, on the basis of the developed algorithms, the allow to implement these complex expansions in optics. In
possibilities of applying the PSWF basis to the problems of this area, the academic school of Academician V.A. Soifer
forming non-diffraction beams [59–60] and increasing the has been at the level of world priorities for several decades
resolution of imaging systems [61] were investigated. [10, 89–93], creating new optical devices and forming
advanced information technologies on this basis [94–97].
V. COMMUNICATION MODES
The basis of spheroidal functions is closely related to the ACKNOWLEDGEMENTS
concept of communication modes [62–63], which are the This work was performed with financial support from
eigenfunctions of some optical propagation operator. In RFBR, DST, NSFC and NRF foundations in accordance
particular, the communication modes in the Cartesian with research project 19-57-80006 BRICS_t.
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 324
Data Science
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