Field theory of monochromatic optical beams. II Classical and quantum paraxial fields

This work is the second part of an investigation aiming at the study of optical wave equations from a field-theoretic point of view. Here, we study classical and quantum aspects of scalar fields satisfying the paraxial wave equation. First, we determine conservation laws for energy, linear and angular momentum of paraxial fields in a classical context. Then, we proceed with the quantization of the field. Finally, we compare our result with the traditional ones.


I. INTRODUCTION
In this second paper of the series "Field theory of monochromatic optical beams," we continue the investigation of scalar fields obeying either the Helmholtz wave equation (HWE) and the paraxial wave equation (PWE) with x = (x, y) ∈ R 2 . Specifically, this work is devoted to the study of some properties of paraxial fields, in both classical and quantum regimes. The notation that we use here, is the same as established in part I. The three-dimensional gradient is expressed as ∂ψ/∂x µ ≡ ∂ µ ψ = (∇, ∂ z ) ψ, where a point in R 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 ǫ 1 and ǫ 2 are the orthogonal unit vectors pointing in the x and y Cartesian coordinate directions, respectively. Greek indexes µ, ν, α, β, . . . , run from 1 to 3, while Latin indexes i, j, k, l, m, n, . . . , take the values 1 and 2. Moreover, ∂ 2 = ∂ 2 x + ∂ 2 y + ∂ 2 z and ∇ 2 = ∂ 2 x + ∂ 2 y .

II. TWO LAGRANGIANS FOR A PARAXIAL FIELD
Let φ(x, z) be a complex scalar field satisfying the paraxial wave equation, namely which is reminiscent of the Schrödinger equation for a free particle on a plane. A suitable Lagrangian generating Eq. (4) should be bilinear in the field and its derivatives: where the four coefficients A, B, C, D are determined by imposing the fulfillment of the Euler-Lagrange equation A straightforward calculation shows that substituting Eq. (5) into Eq. (6), one obtains Now, requiring the equality between Eq. (4) and Eq. (7) yields the following relations: The equation A − B = i can be satisfied with different choices of A and B. We distinguish between the symmetric choice A = −B = i/2, leading to the Lagrangian L 1 , and the asymmetric choice A = i, B = 0, which generates the Lagrangian L 2 , where and The first Lagrangian L 1 is much more appealing and it is clearly real, while L 2 is not. However, L 1 and L 2 differ by a total z-derivative which does no affect the dynamics: In fact, as we shall see soon, only L 2 leads to the correct equations in the Hamilton form.
A. First Lagrangian: L 1 In this case there are two independent fields Π and Π * canonically conjugate to φ and φ * , respectively, specifically The Hamiltonian density H 1 is defined in the standard way: The total Hamiltonian H 1 is simply where to obtain the last line, integration by part has been used and a surface term has been discarded. Then, the Hamilton equations give It is clear that Eq. (15) does not reproduce correctly Eq. (4) and, therefore, L 1 must be ruled out.

B. Second Lagrangian: L 2
In this case we have As explained in [18,19], since Π 2 (x, z) is simply proportional to the conjugate of φ(x, z), then there are only two independent fields, namely φ(x, z) and Π 2 (x, z). Therefore, the Hamiltonian density is calculated as It should be noticed that the second line of Eq. (17) coincides with the second line of Eq. (13). A straightforward calculation shows that using H 2 , the Hamilton equations give the correct equations of motion: where Therefore, from now on we will consider only L 2 as the "true" Lagrangian for the PWE and we will drop the subscript "2" writing simply L instead of L 2 . The asymmetry of L with respect to the Cartesian coordinates x, y, z, can be made more manifest by rewriting Eq. (10) as where δ µν T ≡ δ µν − δ 3µ δ 3ν is a transverse Kronecker delta, which can also be seen as the coordinate-component of the dyadic ǫ 1 ǫ 1 +ǫ 2 ǫ 2 , namely δ µν T = (ǫ 1 ǫ 1 + ǫ 2 ǫ 2 ) µν . By definition, δ 3ν

III. SYMMETRIES AND CONSERVATION LAWS
The Helmholtz equation does not contain explicitly the three Cartesian coordinates x, y, z. Moreover, the latter enter in a symmetric manner in the differential operator ∂ 2 = ∂ 2 x + ∂ 2 y + ∂ 2 z . This yields to the invariance of the HWE under translations and rotations of the fields [20]. Conversely, due to its first-order form in the z-coordinate, we do not expect to keep rotational invariance around an arbitrary axis for the paraxial wave equation. In order to illustrate the symmetries exhibited by the PWE, let us consider the field φ(x, z) evaluated in the generic point r = (x, z) and imagine to perform an active transformation that converts, via a translation by a = a µ ǫ µ and a three-dimensional rotation by Λ µ ν , the original field φ(x, z) into the new field φ ′ (x, z): Let r ′ = (x ′ , z ′ ) be the point obtained by translating and rotating the original point r = (x, z) by a and Λ, respectively, that is: Then, by definition, the new field φ ′ (x ′ , z ′ ) evaluated at r ′ must take the same value of the original field φ(x, z) evaluated at r, namely where we have used the rightmost relation in Eq. (22). Because of the arbitrariness of the point r ′ , we can drop the prime symbol ( ′ ) and rewrite Eq. (23) as This equation defines the behavior of a scalar field under translations and rotations. Now, suppose that φ(x, z) is a solution of the PWE, namely Then the question is: What are the admissible transformations (a, Λ) such that φ(Λ −1 r − Λ −1 a) is still a solution of the PWE? An instructive and elegant method for answering this question without embarking on calculations of chained partial derivatives, is furnished by the Fourier transform technique. Suppose that the field φ(x, z) = φ(r) can be expressed as a three-dimensional Fourier integral: where (d 3 p) = dp 1 dp 2 dp 3 . Substituting Eq. (26) into Eq. (25) we obtain Thus, the differential equation (25) became an algebraic equation in the amplitude φ(p 1 , p 2 , p 3 ): From this equation it follows that the Fourier amplitude φ(p 1 , p 2 , p 3 ) can be different from zero only when p 3 + (p 2 1 + p 2 2 )/(2k 0 ) = 0. This constraint compels φ(p 1 , p 2 , p 3 ) to have the form where, because of the Dirac delta property x δ(x) = 0, the amplitude ϕ(p 1 , p 2 , p 3 ) can be a completely arbitrary smooth function of (p 1 , p 2 , p 3 ). By definition of Fourier transform and using Eq. (26), we can write Then, defining the new dummy variable q ν as permits us to rewrite Eq. (30) in the form where we have inverted the first expression in Eq. (31) to write p ν = q µ Λ µ ν . To see whether Eq. (32) is a solution of the PWE, we substitute it into Eq. (25) to eventually obtain the algebraic equation where Eq. (29) has been used. Since the displacement vector a does not enter in Eq. (33), it can take any value. However, Eq. (33) put some limitations on the form of the rotation Λ, which must evidently satisfy the relation where C is an irrelevant constant that we arbitrarily fix to C = 1. Equation (34) naturally splits in and The last relation can be simply written as LL T = I 2 , where with L we denoted the 2 × 2 principal submatrix of Λ obtained from the latter deleting the third row an the third column and I 2 is the 2 × 2 identity matrix. The superscript "T " indicates the transpose of the matrix.
To summarize, we have found that the transformations that leave the PWE invariant consist of translations by arbitrary three-dimensional vectors a and of two-dimensional rotations around the z-axis of the form where L : LL T = I 2 denotes an arbitrary 2 × 2 orthogonal matrix.
A. Canonical energy-momentum tensor for the paraxial wave equation Given the paraxial Lagrangian the canonical energy-momentum tensor can be build in the usual manner as Explicitly, we have: where we have used the shorthand φ ,µ = ∂ µ φ. By definition This means that there are a conserved energy H and a conserved transverse linear momentum P defined as where in agreement with Eq. (17), and It should be noticed that the minus sign in the equation above, opposite to the sign of H in Eq. (43), is consistent with the condition implied by the Dirac delta in Eq. (29), because where the equation of motion (25) has been used. Therefore, one can consider the conserved quantities (P, H) ≡ (P 1 , P 2 , P 3 ) as the components of a conserved three-momentum P µ , where P 3 ∼ −H . Since δ ij T = δ ij , the transverse part T ij of T µν is symmetric and can be written as Then, we can construct the conserved tensor density such that Therefore, the quantity is conserved during propagation, namely It is clear that the antisymmetric tensor J ij consists of only one independent parameter, which amounts to the longitudinal component of the orbital angular momentum: where Λ is a real constant. From the Noether's theorem it follows that there exist a conserved current (see, e.g., Ref. [18], p. 46, Eq. (2.83)) where Eq. (38) has been used. The current J µ has a vanishing three-divergence [3][4][5] namely This continuity equation has the same form, when position z is replaced by time t, of the continuity equation for the conservation of probability in the quantum theory of a free twodimensional particle [21]. Moreover, as noticed in [22], Eq. (55) is strictly connected to the Poynting theorem in classical electrodynamics [1]. 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" Q defined as As we will see later, in the quantum version of the theory this charge simply corresponds to the total number of the particles in the field.

IV. QUANTIZATION OF THE PARAXIAL FIELD
The quantum theory of electromagnetic fields in the regime of paraxial propagation, has been accomplished by several authors in the past [23][24][25][26]. In these works, the quantized fields where vector fields obeying Maxwell equations. However, using the full machinery of quantum electrodynamics is not really necessary for many practical applications. Therefore, in the present work we simply aim at quantizing the complex scalar field φ(x, z) satisfying the paraxial wave equation (14). In practice, we will follows basically the same procedure outlined in [18,19], for the quantization of the nonrelativistic Schrödinger equation.
We begin by rewriting the Lagrangian (20) and the canonically conjugate field Π(x, z) associated with φ(x, z): where the constant multiplicative term that we added, does not alter the dynamics of the fields and can be eliminated by absorbing it into the definition of the field: φ → φ/ √ . Moreover, if we multiply both sides of Eq. (58) by the speed of light c and we define the new time-like variable τ = z/c, then the so-obtained Lagrangian becomes identical to the Lagrangian associated to the Schrödinger equation of a particle of mass m = k 0 /c, whose motion is restricted to the plane xy.
Since Π * = 0, there are only two independent fields, either (φ, Π) or (φ, φ * ). We choose the second pair and write the Hamiltonian density (43) as As usual, the total Hamiltonian is obtained integrating H (x, z) over the xy-plane: where we used integration by part to pass from the first to the second line of Eq. (62). At this point, the classical theory is quantized by simply promoting the two classical fields φ(x, z) and φ * (x, z), to the operatorsφ(x, z) andφ † (x, z), respectively, and then postulating the equal-z canonical commutation relations: and The Hamilton equations of motion now become whereĤ is straightforwardly derived from Eq. (62): From Eqs. (63-66) it follows that with our choice the Hamiltonian operatorĤ must have the dimensions of an energy divided by a velocity, therefore cĤ represents a true energy. Substituting Eq. (66) into Eq. (65), one obtains which correctly reproduces the PWE.

Spectrum of the field
In the Fourier representation, the Hamiltonian (70) becomes manifestly z-independent: According to our analysis about the conservation laws associated to the PWE, there must exist a set of three conserved operators {P 1 ,P 2 ,P 3 }, whereP 3 ≡ −Ĥ and namely:P µ ≡ {P 1 ,P 2 ,P 3 } = dp The invariance of these operators with respect to z-propagation, can be proved directly by calculating the commutator where Eq.
is conserved: N ,Ĥ = 0. Moreover, a straightforward calculations shows that P i ,P j ] = 0. From the equations (78), (83) and by using the Campbell-Baker-Hausdorff formula, it is not difficult to prove thatφ Since the four operators {P 1 ,P 2 ,Ĥ,N} commute, they can be simultaneously diagonalized. The procedure to find a complete set of eigenstates of such operators is pretty standard and can be found in many textbooks; therefore now we will only sketch the procedure following Ref. [27]. Let |n ′ be an eigenstate ofN with eigenvalue n ′ : where n ′ is real number, not necessarily integer. Sincê then it follows that This procedure may be iterated. For example, it is not difficult to see that After repeating this procedure n times, we find Since, from the definition (85) it follows thatN is an Hermitean operator positive semidefinite, then in Eq. (96) we must have n ′ − n ≥ 0 for any integer n and any real number n ′ . Therefore, n ′ must be an integer (otherwise the iteration never stops). If in (96) we choose n ′ = n and define the vacuum state |0 as then it follows that the vacuum state does not contains particles: From Eqs. (89,98) it follows thatNâ which is in contradiction with the fact thatN is positive semidefinite. Therefore, it must bê Finally, putting n ′ = 0 in Eq. (95), we obtain which permits us to identify |p 1 , . . . , p n ≡â † (p 1 ) . . .â † (p n )|0 with the state containing n particles. The single-particle state |p =â † (p)|0 is normalized according to where we used Eq. (80) to rewriteâ(p)â † (p ′ ) = δ(p − p ′ ) +â † (p ′ )â(p). From now on, we assume that the vacuum state is normalized, that is 0|0 = 1. It is not difficult to verify that the two-particle state |p 1 , p 2 =â † (p 2 )â † (p 1 )|0 has the expected Bosons symmetry with respect to the exchange of particles: where Eq. (80) has been repeatedly used. This calculation can be straightforwardly generalized to the n-particle states.
where Eq. (80) has been used n times. Now we are equipped to calculatê H|p 1 , . . . , p n = dp η pâ † (p)â(p)|p 1 , . . . , p n Next, an important quantity to calculate is the so-called propagator for the paraxial wave equation. It is evaluated from the two-point correlation function φ (x, z)φ † (x ′ , z ′ ) 0 defined as which coincides with the so-called Fresnel propagator in paraxial optics [28].
As a further step, we introduce the position states |x 1 , . . . , x n ; z defined as whereX n (z) ≡ n k=1φ † (x k , z). Exploiting the fact thatĤ|0 = 0, it is easy to see that these states obey the Schrödinger equation: The wave function associated with the scalar field is given by the scalar product between position |x; z and momentum |p single-particle states, namely: If we denote |x ≡ |x; 0 , then Eq. (110) shows that we have actually recovered the normalized Fourier basis in a two-dimensional space: The action of the field operatorφ(x, z) on the position state |x 1 , . . . , x n ; z is similar to the action of the annihilation operatorâ(p) on the momentum state |p 1 , . . . , p n , which we have seen in Eq. (104). In the present case we havê φ(x, z)|x 1 , . . . , x n ; z =φ(x, z)φ † (x 1 , z) · · ·φ † (x n , z)|0 = δ(x − x 1 )|x 2 , x 3 , . . . , x n + δ(x − x 2 )|x 1 , x 3 , . . . , x n + . . .

B. Mode expansion in different bases
Amongst the solutions of the paraxial wave equations there are the so-called Hermite-Gauss and Laguerre-Gauss modes [29]. Let u a (x, z) be one of such modes, where the shorthand a denotes a multiple index. By definition, These modes form a complete and orthonormal basis, namely a u * a (x, z)u a (y, z) = δ(x − y), dx u * a (x, z)u b (x, z) = δ ab .
The latter relation tells us that the operatorφ † a creates a particle in the paraxial mode u a (x, z) from the vacuum state. The single-particle states associated to different paraxial modes are automatically orthogonal: where Eqs. (117-118) have been used. From these definitions, any other relation may be straightforwardly calculated. and B = iA 0 ω 0 ǫ 3 × n + i k 0 ǫ 3 (n × ∇) 3 ψ(x, z; t) + c.c., where "c.c." stands for "complex conjugate" and n = n 1 ǫ 1 + n 2 ǫ 2 is a two-dimensional transverse unit vector that fixes the polarization of the field [24]. A 0 is a constant real amplitude with the physical dimensions of an electric field. For the quantum fields, the generalization of the equations above is straightforward. For example, the electric field operator will be written asˆ E =ˆ E + +ˆ E − , whereˆ E − is the Hermitean conjugate ofˆ E + . Then, using Eq. (78) we can readily writê This expression is in full agreement with equations (17) and (18) of Ref. [30], when the latter are reduced to the monochromatic case.

V. CONCLUSIONS
In this work we have studied the paraxial wave equation from a field-theoretic point of view. We began writing a Lagrangian apt to yields the Euler-Lagrange equations correctly reproducing the paraxial wave equation. Then, we studied the symmetries of the latter and we deduced several conservation laws. Then, we quantized the fields and calculated the relevant physical observables. Finally, we compared our results with previously established ones finding full agreement.