Field theory of monochromatic optical beams. I

We study monochromatic, scalar solutions of the Helmholtz and paraxial wave equations from a field-theoretic point of view. We introduce appropriate time-independent Lagrangian densities for which the Euler-Lagrange equations reproduces either Helmholtz and paraxial wave equations with the $z$-coordinate, associated with the main direction of propagation of the fields, playing the same role of time in standard Lagrangian theory. For both Helmholtz and paraxial scalar fields, we calculate the canonical energy-momentum tensor and determine the continuity equations relating"energy"and"momentum"of the fields. Eventually, the reduction of the Helmholtz wave equation to a useful first-order Dirac form, is presented. This work sheds some light on the intriguing and not so acknowledged connections between angular spectrum representation of optical wavefields, cosmological models and physics of black holes.


Introduction
Light is an electromagnetic phenomenon which can be described by a field theory governed by Maxwell's equations. These are a set of first-order partial differential equations that relates electric and magnetic vector fields each other and to possible sources. However, in many practical instances, a vector field representation of light appears redundant and a simpler scalar field description results appropriate. This is not the case for structured-light beams with nontrivial polarization patterns as the ones considered, for example, in [1]. In this paper, we will not consider such a kind of light. Thus, when a scalar description is appropriate, according to the characteristics of the phenomenon under investigation, monochromatic light propagating in free space can be described either by a field y y = z x y z x, , , where here and hereafter we use the notation = Î x y x , 2 ( ) . In the appendix XI of their book 'Principles of Optics', Born and Wolf derive the energy conservation law for a real, time-dependent scalar wavefield V t r, r ( ) ( ) in free space, with = Î x y z r , , 3 ( ) [2]. Because of the explicit time dependence of V t r, r ( ) ( ) , a continuity equation expressing the local energy conservation law could be deduced from the Lagrangian form of field equations. For the case of a monochromatic field of frequency ω, Born and Wolf first rewrite the field as the real part of a time-harmonic complex amplitude, that is Re , . Then, they take the time averages of the energy density and the energy flux vector to obtain conservation laws involving only the time-independent complex field U(r, ω).
In this work, we pursue the same goal of Born and Wolf, yet following an entirely different and new approach. Instead of considering time-dependent monochromatic fields and erasing such dependence via time averages, we develop an ab initio time-independent field theory taking the monochromatic Helmholtz and PWEs (1) and (2) as the central points around which we build a time-independent Lagrangian field theory [3]. The idea is to deal with action functionals of ( ) where z 1 and z 2 are the limits of integration for the variable z which is associated with the main propagation direction of the field, = d dx dy dz r is the volume measure and  denotes the Lagrange density (Lagrangian, for short). We require S to be stationary for arbitrary variations of the field quantities that vanish at the end points, namely in order to infer the Euler-Lagrange equations reproducing (1) and (2). Thus, in our nonstandard approach, propagation along the z-axis of a time-independent field obeying either HWE or PWE, is formally described in the same manner the time evolution of a time-dependent field is depicted in the standard Lagrangian formalism.

Nonstandard Lagrangian formalism for Helmholtz fields
In this section, we discuss the classical mechanics of a complex scalar field y y = z x, ( ), which is a solution of the HWE (1) where the Laplacian ∂ 2 in the three-dimensional Euclidean space  3 is written as (throughout this paper, we use Einstein's summation convention) where a point in  3 is labeled by the three coordinates x μ , with x 3 =z the longitudinal coordinate and x k , k=1, 2 the transverse coordinates. The two-dimensional gradient of a scalar function f x y z , , ( ) is denoted f and is defined as where  1 and  2 are the orthogonal unit vectors pointing in the x and y Cartesian coordinate directions, respectively. From now on, Greek indexes μ, ν, α, β, K, run from 1 to 3, while Latin indexes i, j, k, l, m, n, K, take the values 1 and 2. When the complex scalar field ψ has two independent real components y 1 and ψ 2 , we may put and regard to ψ and ψ * (instead of ψ 1 and ψ 2 ) as independent fields. In this case, we expect that the two Euler-Lagrange equations Elementary examples thereof are given by plane wave fields , and by Bessel fields where J 0 (z) denotes the zeroth-order Bessel function of first kind. The 'frequency' ζ can be either real and positive, or purely imaginary, namely z z = -* and ζ 2 <0. This can be seen multiplying equation (21) by j x *( ) and integrating over the xy-plane, thus obtaining [4] ò ò ò is the surface element and we assumed that the field j x ( ) vanishes for  ¥ x y , in order to neglect surface terms. From equation (24), it follows that either This relation imposes a constraint upon the Fourier spectrum is not entirely contained within the circle of equation 2 , the field develops purely imaginary frequencies. This fact will have profound consequences upon the quantization of y z x, ( ). Since the right side of equation (24) is always real, it follows that (21), we see that j x ( ) and j x *( ) satisfy the same equation. Therefore, if j x ( ) is a given solution of equation (21), then j x *( ) is also a solution. We conclude this part by noticing that, irrespective of the either positive or purely imaginary value taken by ζ, there are four linearly independent separable solutions of equation (12), namely

Using this result in equation
and the y -ʼs solutions are exponentially growing as z increases. Therefore they represent physically acceptable solutions only for z<0. Vice versa, the ψ + ʼs solutions are exponentially decaying and physically acceptable only for z>0. For fields associated to optical beams, ψ + (ψ − ) and y + * (y -*) are called evanescent waves when z z = i| | and z>0 (z<0).

Symmetries and conservation laws
In this part, we discuss the symmetries of the Lagrangian (16). A different discussion can be found in [5]. To begin with, let us note that such Lagrangian is manifestly invariant under the transformation where Λ is a real constant. From Noether's theorem, it follows that there exists a current y y y y y y y y which has a vanishing three-divergence [6-8] Integrating both sides of this equation over all the xy-plane, we obtain where the right side amounts to the two-dimensional integral of a two-divergence and then vanishes for fields localized within a finite region of the xy-plane. This equation states that during propagation of a monochromatic optical field along the z-axis, the 'charge'  defined as ò y y y y It is instructive to evaluate  for the four fundamental solutions (29). A straightforward calculation gives At first sight, the charges   seems to depend on z, thus contradicting the conservation law (36). However, one should remember that there are only two possibilities for ζ: either ) denotes the full three-momentum of the beam. Therefore, as the beam propagates along the z-axis, the 'energy' H and the two components of the linear momentum P remain constant as a consequence of equation (44), As will be shown later, the 'energy' H coincides with the Hamiltonian of the system. However, the expression in equation (45) is not manifestly positive semidefinite, as a physically realizable Hamiltonian should be, because of the negative 'kinetic energy' term y  -2 | | . We will discuss this point at length later, when proceeding with the quantization of the field. For the moment, we verify that H is actually positive semidefinite for the four fundamental solutions (29). After a straightforward calculation, one finds where C is given again by equation (42). This nice result shows that the evanescent waves do not contribute to the total energy of the field. In a similar manner, we can now calculate P and the outcome is The two terms at the right side of this equation can be discarded under the assumption that the fields and their derivative fall off sufficiently fast at infinity. Thus, we recover the well-known conservation law [6]: However, it should be reminded that particular care must be taken when handling equation (58) because of the risk of improper manipulation of the surface terms [10]. To proceed further, it is useful to define the conserved angular momentum tensor as which reproduce the laws of rays propagation in geometrical optics [11].
It is enlightening to calculate explicitly J z for the fields (29). After a lengthy but straightforward calculation, one finds ) for z z = * and it is equal to zero for z z = -*. Once again, the 'unphysical' evanescent waves generated by the angular spectrum representation do not carry angular momentum. From equation (70), it follows that J z is conserved along with propagation because z z ¶ ¶ µ = J z Re Im 0 z , the latter equality being a consequence of the fact that ζ is either real or purely imaginary. The conjugate fields y  * produce the same J z . One more continuity equation may be derived by rewriting equation (56) This quantity differs from the homonym one introduced by Barnett [9] in that l i L is time-independent. In conclusion, in this section we have shown that all the relevant physical quantities as energy, linear and angular momenta vanishes for evanescent fields. Thus, the latter appears more as virtual fields that do not correspond to any real physical field. However, as we shall see later, they are necessary to preserve unitarity [12].

Hamiltonian formalism
Propaedeutical to the quantization procedure, which will be carried out in part II, is the introduction of the Hamiltonian formalism for equation (1). The procedure for passing from the Lagrangian to the Hamiltonian representation of the field is standard [6]. First, we write down our Lagrangian (16) as where, for the sake of clarity, we omitted the subscript 'HWE'. Then, we determine the fields P z x, ( ) and P z x, *( ) canonically conjugate to y z x, ( ) and y z x, *( ), respectively which shows that actually = 33 H T , as previously stated. The reader familiar with quantum field theory of tachyons, will appreciate the similarity between equation (77) and the Hamiltonian of a Klein-Gordon field with purely imaginary mass [13,14]. Now we are going to show that the canonical Hamiltonian is naturally partitioned in a 'propagating' and an 'evanescent' part. Interestingly, the same phenomenon manifests in the quantization of scalar fields near rapidly rotating stars [4] and in cosmological models of universes with unstable modes [15]. In the latter case, the propagating and evanescent parts are quite suggestively dubbed 'light' and 'dark' components of the Hamiltonian, respectively [16].
At any position z the fields P z x, ( ) and y z x, ( ) can be expanded in terms of the Fourier transform representations: where the complex amplitudes Q z p, ( ) and P z p, ( ) are zdependent. The minus sign in the exponential in equation (80) is not a typo. Substituting equations (79), (80) into (78) we obtain, after some manipulation  ( ) ( ) ( ) ( ) The appearance of the two functions Y i instead of the unique original one ψ, may be hardly surprising as clearly explained by Messiah [19]. Indeed, the solution of a second-order differential equation with respect to z, as the HWE is, requires the knowledge of both y z x, ( ) and y ¶ z x, z ( ) evaluated at the initial position z=0. Therefore, converting the second-order HWE to a first-order Dirac form without loosing information, necessarily introduces a two-component wavefield.
In order to find a Lagrangian for equation (116) it is convenient first to rewrite it in the more suggestive form

Conclusions
In this first part of our two-part work, we have mainly developed a Lagrangian theory for scalar, monochromatic optical fields. Our approach is unconventional in that we consider the coordinate z of a Cartesian reference frame attached to a beam of light and parallel to the propagation direction of the latter, in the same way the time t is considered in ordinary field theory.
In this way, we are able to determine exact conservation laws for several physical quantities and we can highlight the different roles that propagating and evanescent waves have in the description of optical beams. Moreover, this theory also serves to establish the basic formalism before proceeding with the 'phenomenological' quantization of both Helmholtz and PWEs, which will be presented in part II.