Complex Space Source Theory of Spatially Localized Electromagnetic Waves
This book begins with an essential background discussion of the many applications and drawbacks for paraxial beams, which is required in the treatment of the complex space theory of spatially localized electromagnetic waves. The author highlights that there is a need obtain exact full-wave solutions that reduce to the paraxial beams in the appropriate limit. Complex Space Source Theory of Spatially Localized Electromagnetic Waves treats the exact full-wave generalizations of all the basic types of paraxial beam solutions. These are developed by the use of Fourier and Bessel transform techniques and the complex space source theory of spatially localized electromagnetic waves is integrated as a branch of Fourier optics. Two major steps in the theory are described as: 1) the systematic derivation of the appropriate virtual source in the complex space that produces the required full wave from the paraxial beam solution; and 2) the determination of the actual secondary source in the physical space that is equivalent to the virtual source in the complex space. Complex Space Source Theory of Spatially Localized Electromagnetic Waves introduces and carefully explains original analytical techniques; includes a treatment of partially coherent and partially incoherent waves; provides treatment of the newly developing area of Airy beams and waves; develops complex space source theory as a branch of Fourier Optics.
Inspec keywords: electromagnetic fields; integral equations; Green's function methods; electromagnetic waves; Helmholtz equations
Other keywords: Laguerre-Gauss wave; full Gaussian wave; spatially localized electromagnetic wave; electromagnetic fields; complex space source theory; Gaussian beam; complex source point theory; point current source origin; Airy integral; Airy beam; Hermite-Gauss wave; Green function; Helmholtz equation; paraxial equation; Bessel-Gauss wave; An integral
Subjects: Electromagnetic waves: theory; Mathematical analysis; Electromagnetic waves, antennas and propagation; Electromagnetic fields; Function theory, analysis; General electrical engineering topics; Integral equations (numerical analysis); Textbooks
- Book DOI: 10.1049/SBEW518E
- Chapter DOI: 10.1049/SBEW518E
- ISBN: 9781613531938
- e-ISBN: 9781613531945
- Page count: 243
- Format: PDF
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Front Matter
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1 Fundamental Gaussian beam
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A plane wave has a unique propagation direction and is not physically realizable since infinite energy is required for its launching. Nearly plane waves or beams are formed by a group of plane waves having a narrow range of propagation directions about a specified direction. A general electromagnetic field is constructed from a single component of magnetic vector potential and a single component of electric vector potential, both in the same direction. The vector potential associated with an electromagnetic beam is separated into a rapidly varying phase and a slowly varying amplitude. The slowly varying amplitude satisfies the paraxial wave equation. For an input distribution having a simple Gaussian profile with circular cross section, the paraxial wave equation is solved to obtain the vector potential. For the fundamental electromagnetic Gaussian beam, the fields are evaluated and the characteristics of the radiation intensity distribution are described. The outward propagations in the +z direction in the space 0 < z < ∞ and in the -z direction in the space -∞ < z < 0 are considered. The secondary source is concentrated on the boundary plane z = 0. The source current density is obtained and the complex power is determined. The time average of the real power is equal to the time-averaged radiative power. The reactive power of the paraxial beam is found to vanish.
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2 Fundamental Gaussian wave
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The secondary source for the approximate paraxial beams and the exact full waves is a current sheet that is situated on the plane z = 0. The beams and the waves generated by the secondary source propagate out in the +z direction in the space 0 < z < ∞ and in the -z direction in the space -∞ < z < 0. The response of the electric current source given by Eq. (1.24) obtained in the paraxial approximation is the fundamental Gaussian beam. The same current source for the full Helmholtz wave equation yields the fundamental Gaussian wave. For the electric current source given by Eq. (1.24), the Helmholtz wave equation is solved to obtain the exact vector potential. The electromagnetic fields are derived, the radiation intensity distribution is determined, and its characteristics are analyzed. The time-averaged power transported by the fundamental Gaussian wave in the ±z direction is obtained. The complex power is evaluated and from there the reactive power is found. The time-averaged power carried by the fundamental Gaussian wave in the ±z direction increases, reaches a maximum greater than 1, decreases, and approaches the value of 1 corresponding to the fundamental Gaussian beam as the parameter kw0 is increased. The reactive power does not vanish for the fundamental Gaussian wave. The reactive power decreases, reaches zero, decreases further, reaches a minimum, then increases and approaches zero (that is, the limiting value for the corresponding fundamental Gaussian beam) as kw0 is increased.
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3 Origin of point current source in complex space
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The concept of a virtual source, introduced by Deschamps, has some novel features. The usual electromagnetic field theory is not valid in the region where the virtual source is located. The field generated by the virtual source in the complex space reproduces the field of a paraxial scalar fundamental Gaussian beam to within an amplitude term in the real space. This novel virtual point source is now introduced.
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4 Basic full Gaussian wave
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The field of a point source of suitable strength with the location coordinates in the complex space reproduces the fundamental Gaussian beam in the paraxial approximation. The resulting full wave is designated the basic full Gaussian wave. The vector potential that generates the basic full Gaussian wave is obtained. The resulting electromagnetic fields are found, the radiation intensity distribution is determined, and its characteristics are described. The time-averaged power transported by the basic full Gaussian wave in the +z and the -z directions is obtained. This power increases monotonically and approaches the limiting value of the fundamental Gaussian beam as kw0 is increased. The surface electric current density on the secondary source plane z = 0 is deduced. In the paraxial approximation, this electric current density reduces to the source electric current density that generates the fundamental Gaussian beam. In the paraxial approximation, both the fundamental Gaussian wave and the basic full Gaussian wave reduce to the same fundamental Gaussian beam. The complex power is evaluated and the reactive power is determined. For the basic full Gaussian wave, the reactive power is infinite. In contrast, for the corresponding paraxial beam, namely the fundamental Gaussian beam, the reactive power vanishes.
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5 Complex source point theory
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Deschamps (1971) and Felsen (1976) introduced the scalar point source with the location coordinates in the complex space for obtaining the exact full wave corresponding to the fundamental Gaussian beam. Postulation of the required source in the complex space for paraxial beams other than the fundamental Gaussian beam is difficult. For the purpose of generalizing the treatments of Deschamps and Felsen to any paraxial beam, a method is developed for deducing the required source with the location coordinates in the complex space (Seshadri, 2009). In this chapter, for the fundamental Gaussian beam, such a method is presented, and the location and the strength of the required point source in the complex space are derived.
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6 Extended full Gaussian wave
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The fundamental Gaussian light beam has no reactive power [1]. For the basic full Gaussian wave, the reactive power is infinite. In order to obtain finite reactive power, a full Gaussian wave generated by a distribution of electric current rather than a point source in the complex space is necessary. A class of such extended full-wave generalizations of the fundamental Gaussian beam is obtained [2].
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7 Cylindrically symmetric transverse magnetic full Gaussian wave
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The fundamental Gaussian beam and wave are generated by the magnetic/electric vector potential, oriented perpendicular to the direction of propagation. If the vector potential is oriented parallel to the propagation direction and is cylindrically symmetrical, transverse magnetic (TM) and transverse electric (TE) beams and waves are excited [1-5]. The lowest order solution is used for the vector potential and is the same as for the fundamental Gaussian beam and wave. The resulting electromagnetic fields are one order higher, which is one order of magnitude in kw0 smaller, than those for the fundamental Gaussian beam. For the TM paraxial beam, the electric field component Er(r, z) is discontinuous across the secondary source plane z = 0, resulting in an azimuthally directed magnetic current sheet on the secondary source plane. Therefore, the problem of the cylindrically symmetric TM Gaussian paraxial beam and full wave is formulated directly in terms of the electromagnetic fields and the source current density. The solution is first obtained in the paraxial approximation. The electromagnetic fields are determined and the characteristics of the real power, the radiation intensity distribution, and the reactive power are obtained. The reactive power of the TM Gaussian beam vanishes.The full-wave generalization of the cylindrically symmetric TM paraxial beam is carried out. The procedure involves the analytic continuation of the asymptotic (IzI → ∞) field from IzI to IzI - ibt, where the length parameter bt is in the range 0 ≤ bt ≤ b and b is the Rayleigh distance. The limiting case of bt=b = 1 is identical to the full-wave treatment introduced by Deschamps [6] and Felsen [7] for the fundamental Gaussian beam [4]. For the cylindrically symmetric TM full Gaussian wave, the required magnetic current density is deduced and the relevant components of the generated electromagnetic fields are determined. The characteristics of the real power, the reactive power, and the radiation intensity distribution are analyzed. The reactive power of the cylindrically symmetric TM full Gaussian wave does not vanish. The dependence of the characteristics of the various physical quantities on bt/b and kw0 is examined.
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8 Two higher-order full Gaussian waves
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A treatment of the higher-order hollow Gaussian beam is presented. The required higher-order point source in the complex space is introduced. The higher-order hollow full Gaussian wave generated by the complex space source is derived. The basic full Gaussian wave is obtained as a special case of the higher-order hollow full Gaussian wave. The cosh-Gauss paraxial beam is deduced by the two-dimensional Fourier transform technique. The required array of point sources situated on the corners of a square in the complex space is introduced. The full-wave generalization of the cosh-Gauss beam is determined. The basic full Gaussian wave is obtained as a special case of the full cosh-Gauss wave as the side of the square reduces to zero.
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9 Basic full complex-argument Laguerre-Gauss wave
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The paraxial wave equation in the cylindrical coordinate system has a series of higher-order solutions known as the complex-argument Laguerre-Gauss beams [1,2]. This series of eigenfunctions is a complete set. These higher-order Gaussian beams are characterized by the radial mode number n and the azimuthal mode number m. The fundamental Gaussian beam is the lowest-order (n = 0, m = 0) mode in this set. The higher-order hollow Gaussian beams discussed in Chapter 8 correspond to n = 0. In this chapter, a treatment of the complex-argument Laguerre-Gauss beams is presented. As for the fundamental Gaussian beam, for the complex-argument Laguerre-Gauss beams also, the reactive power vanishes. The higher-order point source in the complex space required for the full-wave generalization of the complex-argument Laguerre-Gauss beams is deduced. The basic full complex-argument Laguerre-Gauss wave generated by the complex space source is determined [3,4]. The real and reactive powers of the basic full higher-order wave are evaluated. As expected, the reactive power is infinite, as for the basic full Gaussian wave. The general characteristics of the real power are examined. The real power increases monotonically, approaching the limiting value of the paraxial beam as kw0 is increased. For a fixed kw0, in general, the real power decreases as the mode order increases.
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10 Basic full real-argument Laguerre-Gauss wave
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The paraxial wave equation in the cylindrical coordinate system, in the same manner as the complex-argument Laguerre-Gauss beams, has another series of higher-order solutions known as the real-argument Laguerre-Gauss beams [1]. These higher-order Gaussian beams, which form a complete set of eigenfunctions, are characterized by the radial mode number n and the azimuthal mode number m. The fundamental Gaussian beam is the lowest-order (n = 0, m = 0) mode in this set. For n = 0, the real-argument Laguerre-Gauss beams become identical to the complex-argument Laguerre-Gauss beams. Therefore, for the realargument Laguerre-Gauss beams as well, the higher-order hollow Gaussian beams treated in Chapter 8 correspond to n = 0. In this chapter, the various aspects of the real-argument Laguerre-Gauss beams are discussed. As for the complex-argument Laguerre-Gauss beams, as well as for the real-argument Laguerre-Gauss beams, the reactive power is zero. The source in the complex space required for the full-wave generalization of the real-argument Laguerre-Gauss beams is derived. The basic full real-argument Laguerre-Gauss wave generated by the complex space source is determined. The real and the reactive powers of the basic full higher-order wave are obtained. The source in the complex space is shown to be a series of higher-order point sources that are similar to those obtained for the complex-argument Laguerre-Gauss beams. Consequently, as is to be expected, the reactive power is infinite, as for the basic full complex-argument Laguerre-Gauss wave. The general characteristics of the real power are investigated. The real power increases, approaching the limiting value of the paraxial beam as kw0 is increased. For sufficiently large and fixed kw0, in general, the real power decreases as the mode order increases.
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11 Basic full complex-argument Hermite-Gauss wave
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The paraxial wave equation in the Cartesian coordinate system has a series of higher-order solutions known as the complex-argument Hermite-Gauss beams. This series of eigenfunctions is a complete set. These higher-order Gaussian beams are described by the mode numbers m and n in the x and y directions, respectively. The fundamental Gaussian beam is the lowest-order (m = 0, n = 0) mode in this set. In this chapter, a treatment of the complex-argument Hermite-Gauss beams is presented. As for the other previously introduced paraxial beams, the reactive power of the complex-argument Hermite-Gauss beams vanishes. The higher-order point source in the complex space required for the full-wave generalization of the complex-argument Hermite-Gauss beams is derived. The basic full complex-argument Hermite-Gauss wave generated by the complex space source is determined. The real and reactive powers of the basic full complex-argument Hermite-Gauss waves are evaluated. As for the other basic full Gaussian waves, the reactive power is infinite. The general characteristics of the real power are investigated. The real power increases, approaching the limiting value of the paraxial beam as kw0 is increased. For a fixed kw0, the real power decreases as the mode order increases.
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12 Basic full real-argument Hermite-Gauss wave
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Kogelnik and Li (1966) and Marcuse (1972) introduced the real-argument Hermite-Gauss beams in connection with laser beams and resonators. The real-argument Hermite-Gauss beams are another series of higher-order solutions to the paraxial wave equation in the Cartesian coordinate system. This series of eigenfunctions forms an orthonormal set. The fundamental Gaussian beam is the lowest-order, (m, n) = (0, 0), mode in this set, where m and n are the mode numbers in the x and y directions, respectively. In this chapter, the real-argument Hermite-Gauss beams are treated. As for the other paraxial beams, the reactive power of the real-argument Hermite-Gauss beams vanishes. The source in the complex space required for the full-wave generalization of the real-argument Hermite-Gauss beams is obtained. The basic full real-argument Hermite-Gauss wave produced by the complex space source is derived. The real and the reactive powers of the basic full real-argument Hermite-Gauss waves are evaluated. As for the other basic full Gaussian waves, the reactive power is infinite. The characteristics of the real power are discussed. The real power increases as kw0 is increased, reaching the limiting value of the paraxial beam. For a fixed kw0, the real power decreases as the mode orders increase.
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13 Basic full modified Bessel-Gauss wave
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For a modified Bessel-Gauss beam, the modulating function for the Gaussian is a Bessel function of imaginary argument or a modified Bessel function of real argument. The scalar modified Bessel-Gauss beam was introduced and its basic full-wave generalization was treated with particular emphasis on its spreading properties on propagation (Seshadri, 2007). One form of the electromagnetic modified Bessel-Gauss beams, namely the transverse magnetic (TM) modified Bessel-Gauss beam, was investigated and its basic full-wave generalization was obtained (Seshadri, 2008). To generate the TM modified Bessel-Gauss beam and wave, a single component of the magnetic vector potential in the direction of propagation was used. In this chapter, a linearly polarized electromagnetic modified Bessel-Gauss beam and a wave are treated. To produce the linearly polarized modified Bessel-Gauss beam and a wave, a single component of the magnetic vector potential perpendicular to the direction of propagation is required.
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14 Partially coherent and partially incoherent full Gaussian wave
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Wolf et al. [5-7] introduced the idea of planar secondary sources for the treatment of partially coherent light beams. For the extended full Gaussian waves, only the fully coherent waves in which the wave amplitude remains a constant in time were considered [3]. For the partially coherent beams, the wave period is the same, namely Tw, but the source amplitude is essentially a constant on the time scale of Tw; but on a longer time scale Tf , on the order of nearly thousands of Tw, the amplitude changes in a random manner [8]. A majority of treatments of the partially coherent beams are restricted to the paraxial beams for which the beam waist is large compared to the wavelength. There is a need for the treatment of partially coherent, spatially localized electromagnetic waves extended beyond the paraxial approximation to the full waves governed by Maxwell's equations. An analysis of the partially coherent, spatially localized electromagnetic waves was presented for the fundamental Gaussian wave [9,10]. This is a special case (bt/b 1/4 0) for which the virtual source becomes identical to the actual secondary source in the physical space. In this chapter, the treatment of partially coherent, spatially localized electromagnetic waves is enlarged in scope to include the extended full Gaussian waves for which the virtual source is in the complex space requiring a different formulation.
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15 Airy beams and waves
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The Airy functions were analyzed by Kalnins and Miller [1], and the Airy beams were introduced by Berry and Balazs [2] in the context of quantum mechanics. There are other investigations into the properties of the Airy wave packets [3,4]. The original Airy beam cannot be physically realized because infinite power is required to excite the beam. Siviloglou and Christodoulides [5] have introduced a physically realizable Airy beam, and the various properties of this beam have been investigated [6,7]. Bandres and Gutierrez-Vega [8] have analyzed the generalized Airy-Gauss beams that are also physically realizable. All these investigations pertain only to the beams that satisfy the paraxial wave equation. Yan, Yao, Lei, Dan, Yang, and Gao [9], using the virtual source method, have extended the analysis to full Airy waves governed by the exact Helmholtz equation. In this chapter, some aspects of Airy beams and waves are treated. The fundamental Airy beam and the “finite-energy”(modified) fundamental Airy beam are discussed. The fundamental Airy beam is generalized to obtain the full-wave solution, namely the fundamental Airy wave. For the fundamental Airy wave, the radiation intensity distribution is found to be the same as that for a point electric dipole situated at the origin and oriented normally to the propagation direction. A treatment of the basic full modified Airy wave by the use of the complex space source theory is provided. The propagation characteristics of the basic full modified Airy wave are found to be the same as those for the basic full Gaussian wave, provided that for the former an equivalent waist and an equivalent Rayleigh distance are introduced.
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Appendix A: Green's function for the Helmholtz equation
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This paper is an appendix to the book "Complex space source theory of spatially localized electromagnetic waves". It provides a background on three-dimensional scalar Green's function, Fourier transform of scalar Green's function, and Bessel transform of scalar Green's function.
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Appendix B: An integral
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Appendix C: Green's function for the paraxial equation
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There is a difference in the definitions of the Green's functions for the Helmholtz equation and the paraxial equation. This difference is removed by the introduction of an appropriate excitation constant for the solution obtained from the Helmholtz equation. Then, the paraxial approximation of the exact solution found from the Helmholtz equation is the same as the paraxial approximation deduced from the paraxial equation.
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Appendix D: Electromagnetic fields
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This paper discusses the following topics, Poynting vector and generated power, and vector potentials.
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Appendix E: Airy integral
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Back Matter
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