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 fullwave solutions that reduce to the paraxial beams in the appropriate limit. Complex Space Source Theory of Spatially Localized Electromagnetic Waves treats the exact fullwave 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: LaguerreGauss 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; HermiteGauss wave; Green function; Helmholtz equation; paraxial equation; BesselGauss 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
 eISBN: 9781613531945
 Page count: 243
 Format: PDF

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 timeaveraged radiative power. The reactive power of the paraxial beam is found to vanish.

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 timeaveraged 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 timeaveraged 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 kw_{0} 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 kw_{0} is increased.

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.

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 timeaveraged 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.

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.

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 fullwave generalizations of the fundamental Gaussian beam is obtained [2].

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 [15]. 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 kw_{0} 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 fullwave 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  ib_{t}, where the length parameter b_{t} is in the range 0 ≤ b_{t} ≤ b and b is the Rayleigh distance. The limiting case of b_{t}=b = 1 is identical to the fullwave 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 b_{t}/b and kw_{0} is examined.

8 Two higherorder full Gaussian waves
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A treatment of the higherorder hollow Gaussian beam is presented. The required higherorder point source in the complex space is introduced. The higherorder 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 higherorder hollow full Gaussian wave. The coshGauss paraxial beam is deduced by the twodimensional Fourier transform technique. The required array of point sources situated on the corners of a square in the complex space is introduced. The fullwave generalization of the coshGauss beam is determined. The basic full Gaussian wave is obtained as a special case of the full coshGauss wave as the side of the square reduces to zero.

9 Basic full complexargument LaguerreGauss wave
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The paraxial wave equation in the cylindrical coordinate system has a series of higherorder solutions known as the complexargument LaguerreGauss beams [1,2]. This series of eigenfunctions is a complete set. These higherorder Gaussian beams are characterized by the radial mode number n and the azimuthal mode number m. The fundamental Gaussian beam is the lowestorder (n = 0, m = 0) mode in this set. The higherorder hollow Gaussian beams discussed in Chapter 8 correspond to n = 0. In this chapter, a treatment of the complexargument LaguerreGauss beams is presented. As for the fundamental Gaussian beam, for the complexargument LaguerreGauss beams also, the reactive power vanishes. The higherorder point source in the complex space required for the fullwave generalization of the complexargument LaguerreGauss beams is deduced. The basic full complexargument LaguerreGauss wave generated by the complex space source is determined [3,4]. The real and reactive powers of the basic full higherorder 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 kw_{0} is increased. For a fixed kw_{0}, in general, the real power decreases as the mode order increases.

10 Basic full realargument LaguerreGauss wave
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The paraxial wave equation in the cylindrical coordinate system, in the same manner as the complexargument LaguerreGauss beams, has another series of higherorder solutions known as the realargument LaguerreGauss beams [1]. These higherorder 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 lowestorder (n = 0, m = 0) mode in this set. For n = 0, the realargument LaguerreGauss beams become identical to the complexargument LaguerreGauss beams. Therefore, for the realargument LaguerreGauss beams as well, the higherorder hollow Gaussian beams treated in Chapter 8 correspond to n = 0. In this chapter, the various aspects of the realargument LaguerreGauss beams are discussed. As for the complexargument LaguerreGauss beams, as well as for the realargument LaguerreGauss beams, the reactive power is zero. The source in the complex space required for the fullwave generalization of the realargument LaguerreGauss beams is derived. The basic full realargument LaguerreGauss wave generated by the complex space source is determined. The real and the reactive powers of the basic full higherorder wave are obtained. The source in the complex space is shown to be a series of higherorder point sources that are similar to those obtained for the complexargument LaguerreGauss beams. Consequently, as is to be expected, the reactive power is infinite, as for the basic full complexargument LaguerreGauss wave. The general characteristics of the real power are investigated. The real power increases, approaching the limiting value of the paraxial beam as kw_{0} is increased. For sufficiently large and fixed kw_{0}, in general, the real power decreases as the mode order increases.

11 Basic full complexargument HermiteGauss wave
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The paraxial wave equation in the Cartesian coordinate system has a series of higherorder solutions known as the complexargument HermiteGauss beams. This series of eigenfunctions is a complete set. These higherorder Gaussian beams are described by the mode numbers m and n in the x and y directions, respectively. The fundamental Gaussian beam is the lowestorder (m = 0, n = 0) mode in this set. In this chapter, a treatment of the complexargument HermiteGauss beams is presented. As for the other previously introduced paraxial beams, the reactive power of the complexargument HermiteGauss beams vanishes. The higherorder point source in the complex space required for the fullwave generalization of the complexargument HermiteGauss beams is derived. The basic full complexargument HermiteGauss wave generated by the complex space source is determined. The real and reactive powers of the basic full complexargument HermiteGauss 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 kw_{0} is increased. For a fixed kw_{0}, the real power decreases as the mode order increases.

12 Basic full realargument HermiteGauss wave
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Kogelnik and Li (1966) and Marcuse (1972) introduced the realargument HermiteGauss beams in connection with laser beams and resonators. The realargument HermiteGauss beams are another series of higherorder 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 lowestorder, (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 realargument HermiteGauss beams are treated. As for the other paraxial beams, the reactive power of the realargument HermiteGauss beams vanishes. The source in the complex space required for the fullwave generalization of the realargument HermiteGauss beams is obtained. The basic full realargument HermiteGauss wave produced by the complex space source is derived. The real and the reactive powers of the basic full realargument HermiteGauss 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 kw_{0} is increased, reaching the limiting value of the paraxial beam. For a fixed kw_{0}, the real power decreases as the mode orders increase.

13 Basic full modified BesselGauss wave
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For a modified BesselGauss 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 BesselGauss beam was introduced and its basic fullwave generalization was treated with particular emphasis on its spreading properties on propagation (Seshadri, 2007). One form of the electromagnetic modified BesselGauss beams, namely the transverse magnetic (TM) modified BesselGauss beam, was investigated and its basic fullwave generalization was obtained (Seshadri, 2008). To generate the TM modified BesselGauss 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 BesselGauss beam and a wave are treated. To produce the linearly polarized modified BesselGauss beam and a wave, a single component of the magnetic vector potential perpendicular to the direction of propagation is required.

14 Partially coherent and partially incoherent full Gaussian wave
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Wolf et al. [57] 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 T_{w}, but the source amplitude is essentially a constant on the time scale of T_{w}; but on a longer time scale T_{f} , on the order of nearly thousands of T_{w}, 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 (b_{t}/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.

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 GutierrezVega [8] have analyzed the generalized AiryGauss 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 “finiteenergy”(modified) fundamental Airy beam are discussed. The fundamental Airy beam is generalized to obtain the fullwave 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.

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 threedimensional scalar Green's function, Fourier transform of scalar Green's function, and Bessel transform of scalar Green's function.

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.

Appendix D: Electromagnetic fields
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This paper discusses the following topics, Poynting vector and generated power, and vector potentials.

Appendix E: Airy integral
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Back Matter
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