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Table Of Integrals Series And Products

An overview of the recent developments in the field of cylindrical vector beams is
provided. As one class of spatially variant polarization, cylindrical vector beams
are the axially symmetric beam solution to the full vector electromagnetic wave
equation. These beams can be generated via different active and passive methods.
Techniques for manipulating these beams while maintaining the polarization symmetry
have also been developed. Their special polarization symmetry gives rise to unique
high-numerical-aperture focusing properties that find important applications in
nanoscale optical imaging and manipulation. The prospects for cylindrical vector
beams and their applications in other fields are also briefly discussed.

Cylindrical vector beams: from mathematical concepts to applications

Optical wireless communication (OWC) refers to transmission in unguided propagation media through the use of optical carriers, i.e., visible, infrared (IR), and ultraviolet (UV) bands. In this survey, we focus on outdoor terrestrial OWC links which operate in near IR band. These are widely referred to as free space optical (FSO) communication in the literature. FSO systems are used for high rate communication between two fixed points over distances up to several kilometers. In comparison to radio-frequency (RF) counterparts, FSO links have a very high optical bandwidth available, allowing much higher data rates. They are appealing for a wide range of applications such as metropolitan area network (MAN) extension, local area network (LAN)-to-LAN connectivity, fiber back-up, backhaul for wireless cellular networks, disaster recovery, high definition TV and medical image/video transmission, wireless video surveillance/monitoring, and quantum key distribution among others. Despite the major advantages of FSO technology and variety of its application areas, its widespread use has been hampered by its rather disappointing link reliability particularly in long ranges due to atmospheric turbulence-induced fading and sensitivity to weather conditions. In the last five years or so, there has been a surge of interest in FSO research to address these major technical challenges. Several innovative physical layer concepts, originally introduced in the context of RF systems, such as multiple-input multiple-output communication, cooperative diversity, and adaptive transmission have been recently explored for the design of next generation FSO systems. In this paper, we present an up-to-date survey on FSO communication systems. The first part describes FSO channel models and transmitter/receiver structures. In the second part, we provide details on information theoretical limits of FSO channels and algorithmic-level system design research activities to approach these limits. Specific topics include advances in modulation, channel coding, spatial/cooperative diversity techniques, adaptive transmission, and hybrid RF/FSO systems.

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Survey on Free Space Optical Communication: A Communication Theory Perspective

Tables of integrals

Underwater wireless information transfer is of great interest to the military, industry, and the scientific community, as it plays an important role in tactical surveillance, pollution monitoring, oil control and maintenance, offshore explorations, climate change monitoring, and oceanography research. In order to facilitate all these activities, there is an increase in the number of unmanned vehicles or devices deployed underwater, which require high bandwidth and high capacity for information transfer underwater. Although tremendous progress has been made in the field of acoustic communication underwater, however, it is limited by bandwidth. All this has led to the proliferation of underwater optical wireless communication (UOWC), as it provides higher data rates than the traditional acoustic communication systems with significantly lower power consumption and simpler computational complexities for short-range wireless links. UOWC has many potential applications ranging from deep oceans to coastal waters. However, the biggest challenge for underwater wireless communication originates from the fundamental characteristics of ocean or sea water; addressing these challenges requires a thorough understanding of complex physio-chemical biological systems. In this paper, the main focus is to understand the feasibility and the reliability of high data rate underwater optical links due to various propagation phenomena that impact the performance of the system. This paper provides an exhaustive overview of recent advances in UOWC. Channel characterization, modulation schemes, coding techniques, and various sources of noise which are specific to UOWC are discussed. This paper not only provides exhaustive research in underwater optical communication but also aims to provide the development of new ideas that would help in the growth of future underwater communication. A hybrid approach to an acousto-optic communication system is presented that complements the existing acoustic system, resulting in high data rates, low latency, and an energy-efficient system.

Underwater Optical Wireless Communication

In a turbulent atmosphere, starting with a cos-Gaussian excitation at the source plane, the average intensity profile at the receiver plane is formulated. This average intensity profile is evaluated against the variations of link lengths, turbulence levels, two frequently used free-space optics wavelengths, and beam displacement parameters. We show that a cos-Gaussian beam, following a natural diffraction, is eventually transformed into a cosh-Gaussian beam. Combining our earlier results with the current findings, we conclude that cos-Gaussian and cosh-Gaussian beams act in a reciprocal manner after propagation in turbulence. The rates (paces) of conversion in the two directions are not the same. Although the conversion of cos-Gaussian beams to cosh-Gaussian beams can happen over a wide range of turbulence levels (low to moderate to high), the conversion of cosh-Gaussian beams to cos-Gaussian beams is pronounced under relatively stronger turbulence conditions. Source and propagation parameters that affect this reciprocity have been analyzed.

Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere.

The scintillation index is formulated for a flat-topped Gaussian beam source in atmospheric turbulence. The variations of the on-axis scintillations at the receiver plane are evaluated versus the link length, the size of the flat-topped Gaussian source, and the wavelength at selected flatness scales. The existing source model that represents the flat-topped Gaussian source as the superposition of Gaussian beams is employed. In the limiting case our solution correctly matches with the known Gaussian beam scintillation index. Our results show that for flat-topped Gaussian beams scintillation is larger than that of the single Gaussian beam scintillation when the source sizes are much smaller than the Fresnel zone. However, this trend is reversed and scintillations become smaller than the Gaussian beam scintillations for flat-topped sources with sizes much larger than the Fresnel zone.

Scintillation index of flat-topped Gaussian beams

Analytical formula is derived for the M2-factor of coherent and partially coherent dark hollow beams (DHB) in turbulent atmosphere based on the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Our numerical results show that the M2- factor of a DHB in turbulent atmosphere increases on propagation, which is much different from its invariant properties in free-space, and is mainly determined by the parameters of the beam and the atmosphere. The relative M2-factor of a DHB increases slower than that of Gaussian and flat-topped beams on propagation, which means a DHB is less affected by the atmospheric turbulence than Gaussian and flat-topped beams. Furthermore, the relative M2-factor of a DHB with lower coherence, longer wavelength and larger dark size is less affected by the atmospheric turbulence. Our results will be useful in long-distance free-space optical communications.

M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere.

The propagation of phase-locked and non-phase-locked laser array beams of radial and rectangular symmetries in a turbulent atmosphere are investigated based on the extended Huygens–Fresnel integral. The beamlet used in our paper for constructing the laser array beams is of elliptical Gaussian mode. Analytical formulae for the average irradiance of phase-locked and non-phase-locked radial and rectangular laser array beams are derived through vector integration and tensor operation. The irradiance properties of these laser array beams in a turbulent atmosphere are studied numerically. It is found that both phase-locked and non-phase-locked radial and rectangular laser array beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The propagation properties are closely related to the parameters of laser array beams and the structure constant of the turbulent atmosphere.

Propagation of laser array beams in a turbulent atmosphere

The source and receiver plane characteristics of flat topped (FT) beam propagating in turbulent atmosphere are investigated. To this end, source size, beam power and M2 factor of source plane FT beam are derived. For a turbulent propagation medium, via Huygens Fresnel diffraction integral, the receiver plane intensity is found. Power captured within an area on the receiver plane is calculated. Kurtosis parameter and beam size variation along the propagation axis are formulated. Graphical outputs are provided displaying the variations of the derived source and receiver plane parameters against the order of flatness and propagation length. Analogous to free space behavior, when propagating in turbulence, the FT beam first will form a circular ring in the center. As the propagation length increases, the circumference of this ring will become narrower, giving rise to a downward peak emerging from the center of the beam, eventually turning the intensity profile into a pure Gaussian shape.

Yahya Baykal

Papers

Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere.

Scintillation index of flat-topped Gaussian beams

M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere.

Propagation of laser array beams in a turbulent atmosphere

Flat topped beams and their characteristics in turbulent media.