Polarization in Maxwell optics

Accepted Manuscript Title: Polarization in Maxwell Optics Author: Sameen Ahmed Khan PII: DOI: Reference:

S0030-4026(16)31470-X http://dx.doi.org/doi:10.1016/j.ijleo.2016.11.134 IJLEO 58550

To appear in: Received date: Accepted date:

6-10-2016 25-11-2016

Please cite this article as: Sameen Ahmed Khan, Polarization in Maxwell Optics, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.134 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Sameen Ahmed Khan1 Department of Mathematics and Sciences College of Arts and Applied Sciences (CAAS) Dhofar University Post Box No. 2509, Postal Code: 211 Salalah, Sultanate of Oman.

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Polarization in Maxwell Optics

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Abstract

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A unified formalism of light beam optics and light polarization is presented. The starting point of our formalism is an exact eightdimensional matrix-representation of Maxwells equations in an inhomogeneous medium, which is presented in detail. The beam-optical Hamiltonians are derived without any specifications on the varying refractive index. The new formalism generalizes the traditional and non-traditional treatments of Helmholtz optics. As for the light polarization, the elegant Mukunda-Simon-Sudarshan rule for transition from scalar optics to vector wave optics is obtained as the paraxial limit of the general formalism presented here. The new formalism is a suitable candidate to extend the traditional theory of polarization beyond the paraxial approximation. The unified formalism light beam-optics and light polarization advances the Hamilton’s opticalmechanical analogy into the wavelength-dependent regime.

PACS: 02, 42.15.-i, 42.90.+m, 42.15.Fr, 42.25, 42.25.Ja, 42.25.-p Keywords and phrases: Matrix representation of Maxwell’s equations, Vector wave optics, Maxwell optics, wavelength-dependent effects, FoldyWouthuysen transformation; Beam propagation, Hamiltonian description, Aberrations, Polarization.

1

E-mail address: [email protected] URL: http://orcid.org/0000-0003-1264-2302.

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Introduction

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1

Sameen Ahmed Khan

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It is natural to develop a beam optics formalism based on Maxwell’s theory of electromagnetism, as it is the correct theory of light. This is done conveniently using the Helmholtz equation; but the resulting beam-optical Hamiltonian is no different from the purely geometric approach of the Fermat’s principle of least time [1]. There are two sources of this equivalence. Firstly, the Helmholtz equation is approximate as its derivation from Maxwell’s equations does not take into account the spatial and temporal derivatives of the permittivity and permeability of the medium. Secondly, there is the issue of the “square-root” of the Helmholtz operator, in the derivation of the beamoptical Hamiltonian. It is possible to circumvent the square-root procedure, by virtue of the mathematical similarities between the Helmholtz equation and the Klein-Gordon equation [2]-[5]. The fact that the Helmholtz equation is approximate remains unaddressed irrespective of the machinery employed. Any formalism using the Helmholtz equation as a base is definitely an approximate, even though it suffices in many situations. The absence of the wavelength λ is another issue in beam optics, which requires a critical approach. The polarization of light is done separately using very diverse techniques such as group theory [6]-[11]. Consequently, the question of the role of light polarization in beam optics remains open. We present here a formalism firmly based on Maxwell’s equations, which leads to a unified formalism of beam optics and light polarization. The formalism also leads to the λ-dependent modifications of the beam optics. This is so in the paraxial regime as well as in the modifications of the aberrating coefficients at each order. In Section 2, we present the exact matrix representation of Maxwell’s equations. In Section 3, we develop the matrix formulation of Maxwell optics by deriving the general expressions for the Hamiltonians, without making any specific assumptions on the form of the varying refractive index. In Section 4, we address the polarization by deriving the Mukunda-SimonSudarshan (MSS) theory for the transition from scalar optics to vector wave optics for paraxial systems. In the same section the MSS-rule is applied to the Gaussian beams. Section 5, our final section, has the concluding discussions.

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Polarization in Maxwell Optics

Matrix representation of Maxwell’s equations

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2

3

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Any attempt to deal with Maxwell’s equations directly in their usual form is bound to be limited by approximations. Helmholtz equation is one of the most widely used approximations. A possible way to overcome this situation is to express Maxwell’s equations in a matrix form: a single matrix equation containing all the four Maxwell’s equations. So, we require an exact matrix representation of Maxwell’s equations for a medium with varying refractive index. This is accomplished only when we pay due regards to the spatial and temporal variations of the permittivity (r, t) and permeability µ(r, t). The available matrix representations do not meet this crucial requirement completely! The possible reasons are perhaps that the various matrix representations were derived with very different motivations, for instance: spinor analysis or seeking symmetries and the photon wave function [12][28]. Hence, an exact matrix representation of Maxwell’s equations using eight-dimensional matrices was specially developed to meet the stringent requirements of the Maxwell optics [29]. The beam-optical Hamiltonian derived using the eight-dimensional matrix representation is exact. Moreover, the derived beam-optical Hamiltonian has a mathematical correspondence with the Dirac equation. The mathematical correspondence ensures the applicability of the rich and powerful machinery of relativistic quantum mechanics, particularly the Foldy-Wouthuysen transformation technique [30]-[31]. From the onset, let us borne in mind that the six-dimensional electromagnetic field governing optics and the four-dimensional Dirac field are two completely different physical objects. Let us note, that the Foldy-Wouthuysen transformation was historically devised for the Dirac equation (the equation for spin1/2 particles such as the electron). But it can also be applied to a specific class of matrix equations. Our matrix representation of Maxwell’s equations falls within the special category of matrix equations [30]-[31]. The extreme similarities in the underlying mathematical structures of the two systems ensures that we can use the powerful machinery of relativistic quantum mechanics. The use of quantum methodologies (particularly, the FoldyWouthuysen transformation technique) has served as a powerful analytic tool leading to well-established results and even in predicting new affects. The matrix formulation of Maxwell optics was decisively influenced by the ‘quantum theory of charged-particle beam optics’ [32]-[48].

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Sameen Ahmed Khan

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∇ · D (r, t) = ρ , ∂ ∇ × H (r, t) − D (r, t) = J , ∂t ∂ ∇ × E (r, t) + B (r, t) = 0 , ∂t ∇ · B (r, t) = 0 .

ip t

In an inhomogeneous medium, Maxwell’s equations [49]-[50] in presence of sources and currents are

(1)

an

In the present study of optics, we assume that the media is linear implying D = E, and B = µH. Here,  = (r, t) is the permittivity and µ = µ(r, t) is the permeability of the medium. The speed of light in this medium is q v(r, t) = 1/ (r, t)µ(r, t). In optics, the chief quantity is the refractive q

M

index of the medium, n(r, t) = c/v(r, t) = c (r, t)µ(r, t). The quantity q

Ac ce p

te

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h(r, t) = µ(r, t)/(r, t) has the dimensions of resistance. In the present study, it is realistic to set aside the resistance function. So, we shall use the ‘basic optical quantity’, the refractive index n(r, t) and the ignorable quantity, the resistance h(r, t) in place of µ(r, t) and (r, t) in the matrix representation of Maxwell’s equations. This switchover leads to the physical basis of the approximations. This will be demonstrated through our exact matrix representation of Maxwell’s equations. In terms of these quantities  = 1/hv = n/ch and µ = h/v = nh/c. In our matrix representation, we shall be using the Riemann-Silberstein complex vector [51, 52] defined by 



1 1 1 D (r, t) ±i q B (r, t) F ± (r, t) = √  q 2 (r) µ(r) 



1 q 1 = √  (r)E (r, t) ±i q B (r, t) . 2 µ(r)

(2)

The Riemann-Silberstein complex vector, F (r, t) can also be derived from the potential Z (r, t), (

i ∂ F (r, t) = ∇ × Z (r, t) + ∇ × Z (r, t) v ∂t

)

.

(3)

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Polarization in Maxwell Optics

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(

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Z (r, t) is the superpotential and is commonly known as the polarization potential or the Hertz Vector [50]. This further leads to the wave-equation 1 ∂2 ∇ − 2 2 Z (r, t) = 0 . v ∂t )

2

(4)

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The Riemann-Silberstein vector has interesting transformation properties [53]. There are certain advantages in building the matrix-representations of Maxwell’s equations from the Riemann-Silberstein vector (see [26]-[28], [54]-[57]). Riemann-Silberstein vector can be used to express many of the quantities associated with the electromagnetic field 1 E×B µ

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Poynting Vector : S =



= −iv F † × F

 !

M

1 1 E · E + B · B Energy Density : u = 2 µ

te

d

Momentum Density : pEB

Ac ce p

Angular Momentum Density : LEB

= F† · F =  (E × B)  i = − F† × F v =  {r × (E × B)}  o in = − r × F† × F , v

(5)

where ‘† ’ is the hermitian conjugate. The other quantities are 1Z 3 Total Energy : E = dr 2 =

Z

Total Momentum : P = 

n

(

1 E · E + B · B µ

d3 r F † · F Z

)

o

d3 r {E × B}

o iZ 3 n † d r F ×F Zv

= − Total Angular Momentum : M = 

d3 r {r × (E × B)}

 o iZ 3 n = − d r r × F† × F v

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Sameen Ahmed Khan

Z

=

(

1 r E · E + B · B µ

n 

d3 r r F † · F

o

!)

ip t

1Z 3 Moment of Energy : N = dr 2

.

(6)

 

 Ψ± (r, t) = 

−Fx± ± iFy± Fz± Fz± Fx± ± iFy±



an



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In this form, the aforementioned quantities look like the quantum-mechanical expectation values! The Riemann-Silberstein vector as a possible candidate for the photon wave function has been advocated [26]-[28]. In order to derive the matrix representation, we define

   

.

(7)

M

The sources are incorporated through the following vectors 

1 √ 2

   

d

W± =

!

−Jx ± iJy Jz − vρ Jz + vρ Jx ± iJy

    

.

(8)

"

0 1l 1l 0

#

,

Ac ce p

Mx =

te

The various matrices arising in our matrix representation are

"

Σ =

σ 0 0 σ

"

My =

#

"

,

α=

#

0 −i1l i1l 0 0 σ σ 0

"

,

Mz = β =

#

"

,

I=

1l 0 0 1l

1l 0 0 −1l

#

,

#

,

(9)

where 1l is the 2 × 2 unit matrix and σ are the Pauli triplets "

σ =

σx =

0 1 1 0

#

"

, σy =

0 −i i 0

#

"

, σz =

1 0 0 −1

#!

.

(10)

The following two coupling functions naturally occur in our matrix representation 1 1 1 ∇v(r, t) = ∇ {ln v(r, t)} = − ∇ {ln n(r, t)} 2v(r, t) 2 2 1 1 w(r, t) = ∇h(r, t) = ∇ {ln h(r, t)} . (11) 2h(r, t) 2 u(r, t) =

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Polarization in Maxwell Optics

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#"

#

"

= −v(r, t) "

−

"

#"

v(r, ˙ t) I 0 Ψ+ Ψ+ − Ψ− Ψ− 2v(r, t) 0 I " #" # ˙ h(r, t) 0 iβαy Ψ+ + Ψ− 0 2h(r, t) iβαy

I 0 0 I

{M · ∇ + Σ · u} −iβ (Σ∗ · w) αy #"

W+ W−

#

cr

I 0 0 I

−iβ (Σ · w) αy {M ∗ · ∇ + Σ∗ · u}

#"

us

"

#

,

Ψ+ Ψ−

#

(12)

an

∂ ∂t

ip t

As we shall see shortly, the coupling functions shall facilitate the approximations. In the end, the exact matrix representation is

3

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te

d

M

where ‘∗ ’ stands for complex-conjugation and the time derivatives are v˙ = ∂v/∂t and h˙ = ∂h/∂t. The matrix representation has thirteen 8 × 8 matrices, of which ten are hermitian. The remaining three matrices, which contain w(r, t) are antihermitian. In our matrix representation of Maxwell’s equations, the refractive index is the dominant function and the resistance function occurs very weakly. This vindicates our choice of using the refractive index and resistance function over the permittivity and permeability of the inhomogeneous medium respectively. By approximating w(r, t) to zero, we can reduce the representation from eight to four dimensions. This is precisely, the methodology, we follow in developing the matrix formalism of Maxwell optics. It is fascinating to note, that Maxwell’s equations can be obtained from Fermat’s principle of geometrical optics through the process of ‘wavization’, in analogy to the quantization of classical mechanics [58]. Matrix methods have been used in other areas of optics and provide great advantages of simplifying and presenting the equations of optics [59].

Matrix formulation of Maxwell optics

Equipped with the exact matrix representation of Maxwell’s equations, we are now in a good position to develop the Maxwell optics. A comprehensive account of the beam optics within the Maxwell framework is to be found in [60]-[62]. In the optical studies, the charge distribution, ρ = 0 and the

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Sameen Ahmed Khan

sources, W ± = 0. So, the very general expression in Eq. (12) simplify to "

Ψ+ Ψ−

#

"

= −v(r)

{M · ∇ + Σ · u} −iβ (Σ∗ · w) αy

−iβ (Σ · w) αy {M ∗ · ∇ + Σ∗ · u}

#"

Ψ+ Ψ−

#

.(13)

ip t

∂ ∂t

cr

We are studying monochromatic quasiparaxial beams. So, the electromagnetic fields can be written as

us

E(r, t) = E(r) exp(−iωt) , B(r, t) = B(r) exp(−iωt) .

an

This leads to

(14)

Ψ± (r, t) = ψ ± (r) exp(−iωt) ,

ω > 0.

(15)

Mz 0 0 Mz

"

"

ψ+ ψ−

ψ+ ψ−

#

{M ⊥ · ∇⊥ + Σ · u} −iβ (Σ∗ · w) αy

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−

"

#

∂ ∂z

te

ω =i v(r)

#

d

"

M

Assuming the optical axis to be the z-axis, we rearrange the terms in Eq. (13) as

−iβ (Σ · w) αy − {M ∗⊥ · ∇⊥ + Σ∗ · u}

#"

#

ψ+ (16) . ψ−

Now, the process of ‘wavization’ [63] is introduced using the customary Schr¨odinger replacements b⊥ , −i–λ∇⊥ −→ p – ∂ −→ pz , −iλ ∂z

(17)

where the reduced wavelength –λ = λ/2π = c/ω is the analogue of h ¯ = h/2π, which is the reduced Planck’s constant. We note, that the spatial variable ‘z’ along the optic axis (z-axis) has the ‘role of time’. In quantum mechanics, the most basic commutator is [p, q] = −i¯ h. The analogue of this relation in wave optics is the commutator [p, q] = pq − qp = −i–λ. The spatially b ⊥ = −i∇⊥ need varying refractive index n(r) and the transverse momenta p not commute. It is precisely this non-commutativity, which leads to the

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Polarization in Maxwell Optics

9

wavelength-dependent affects. Multiplying on both sides of the Eq. (16) from the left with #−1

"

= (i–λ)

β 0 0 β

#

ip t

Mz 0 0 Mz

,

(18)

we obtain "

ψ + (r ⊥ , z) ψ − (r ⊥ , z)

#

"

=

b H

g

ψ + (r ⊥ , z) ψ − (r ⊥ , z)

where b H g = −n0

β 0 0 −β

# b + Ebg + O g "

Eb

g

= − (n (r) − n0 )

β 0 0 β

#

βg

(19)

β {M ⊥ · p⊥ − i–λΣ · u} 0 + 0 β {M ∗⊥ · p⊥ − i–λΣ∗ · u} " # – (Σ · w) αy 0 −λ = . (20) −–λ (Σ∗ · w) αy 0 #

d

M

"

b O g

,

an

"

#

us

∂ i–λ ∂z

cr

"

(i–λ)

Ac ce p

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Equation (20) is the fundamental beam-optical equation, where ‘g’ stands for grand, signifying the eight dimensions and βg =

"

I 0 0 −I

#

.

(21)

The beam-optical Hamiltonian in Eq. (20) is exact and in total mathematical correspondence with the Dirac equation. The first approximation is made on the physical ground that the logarithmic divergence of the resistance function, which is the weak coupling between the upper components Ψ+ and the lower components Ψ− is ignorable. The physically reasonable approximation, w ≈ 0 leads to two equivalent and independent matrix equations in four dimensions. This totally justifies our usage of n(r, t) and h(r, t) in preference to (r, t) and µ(r, t). We omit ‘+ ’ throughout leading and the resulting four-dimensional beamoptical Hamiltonian i–λ

∂ b (r) ψ (r) = Hψ ∂z

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Sameen Ahmed Khan

(22)

ip t

b = −n β + Eb + O b H 0 Eb = − (n (r) − n0 ) β − i–λβΣ · u b = i (M p − M p ) = β (M · p O y x x y ⊥ b ⊥) .

!

"

= out

A B C D

#

r⊥ p⊥

!

an

r⊥ p⊥

us

cr

The original boundary value problem in Maxwell’s equations in Eq. (1) has been replaced by a first-order initial value problem in Eq. (22), by employing matrix methods. This switchover results in the powerful Fourier optic approach and enables many applications [64]. In optics, a first-order system relates the ray parameters in the input and output planes as .

(23)

in

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d

M

This is known as the Kogelnik’s ABCD-law [65]. When we take the square of the Hamiltonian in Eq. (22), it is found that it differs substantially from the squares of the Hamiltonians in the squareroot approach [66]-[68] and the non-traditional prescription of Helmholtz optics [2]-[4]. It is precisely the same type of difference, which exists in quantum mechanics. There too, the square of the Dirac Hamiltonian has terms such as the Pauli term −¯ hqΣ·B, which couples the spin to the magnetic field. Such terms are absent in the Klein-Gordon case. These extra terms are the source of the wavelength-dependent modifications of scalar beam optics in our formalism of Maxwell optics. Importantly, the beam-optical Hamiltonian in Eq. (22) has the mathematical correspondence with the Dirac equation. So we are able to employ the quantum methodologies, particularly the FoldyWouthuysen transformation technique [30, 31]. The first iteration in the Foldy-Wouthuysen expansion gives the beam-optical Hamiltonian to order b 2⊥ /n20 ), (p i–λ

∂ c(2) |ψi , |ψi = H ∂z 1 b2 βO , 2n0   1 2 b = − n (r) − p β − i–λβΣ · u . 2n0 ⊥

c(2) = −n β + Eb − H 0

(24)

b ⊥ )2 = This is the paraxial Hamiltonian. We have used the identities, (M ⊥ · p b 2 = −p b 2⊥ and O b 2⊥ . As we are interested in a beam propagating in the +z p direction, matrix β has to be omitted while computing the transfer maps.

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Polarization in Maxwell Optics

11

∂ c(4) |ψi , |ψi = H ∂z 1 b2 βO 2n0

"

h i 1 ∂ b b b Eb + i– − 2 O, O, λ O 8n0 ∂z

1 b4 + β O 3  8n0

h

i b Eb + i– O, λ

(

!#

∂ b O ∂z

!2  

,



an

+

 

cr

c(4) = −n β + Eb − H 0

us

i–λ

ip t

The second iteration in the Foldy-Wouthuysen expansion gives the leading-order aberrating Hamiltonian describing the third-order aberrations. c(4) to order (p b 2⊥ /n20 )2 and is This is the Hamiltonian, H

)

M

1 2 1 4 b⊥ − b = − n(r) − p p 2n0 8n30 ⊥ 1 h b bi2 1 h b h b bii O, E + 3 β O, − 2 O, E + ··· . 8n0 8n0

(25)

4

Ac ce p

te

d

The first parenthesis contains the expressions obtained in the traditional prescriptions. Rest of the terms are wavelength-dependent. The Hamiltonians in our matrix-based formalism of Maxwell optics intrinsically contain wavelength-dependent parts. Consequently, these intrinsic wavelength-dependent contributions modify the paraxial behaviour as well as the aberration coefficients (at each order). The ‘Lie algebraic treatment of light beam optics’ [66]-[69] is completely obtained from our exact formalism in the limit –λ −→ 0, which we call as the traditional limit of our formalism. This is very similar or analogous to the classical limit obtained by taking h ¯ −→ 0 in quantum prescriptions.

Polarization

In the preceding section, we saw the development of beam optics using a matrix formulation of Maxwell’s equations. The scalar optics was generalized to wave optics by Mukunda, Simon and Sudarshan (MSS) for optical systems in the paraxial regime. The MSS-theory for transition from scalar optics to vector optics is based on a group theoretic approach and takes care of the polarization. The matrix formulation of Maxwell optics, we have developed describes both beam optics and polarization. The MSS-theory is obtained

Page 11 of 41

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Sameen Ahmed Khan

an

us

cr

ip t

as a leading order approximation of our formalism. Our formalism is an appropriate candidate to generalize the MSS-theory way beyond the paraxial regime [70]. In this section, we shall derive the elegant MSS-rule for transition from scalar optics to vector wave optics. We shall then apply the MSS-rule to the Gaussian beams. The beam-optical Hamiltonian in our formalism is exact and does not have any specifications on the spatially varying refractive index. In the initial studies of polarization, it suffices to focus on the restricted case of a constant refractive index. The MSS-rule is obtained even with this simplification! Consequently, in this section, we shall confine our derivations to the case of a constant refractive index. When the refractive index has a constant value, n (r) = nc , the beam-optical Hamiltonian in Eq. (22) reduces to b b ⊥) , H c = −nc β + β (M ⊥ · p = −nc β + i (My px − Mx py ) .

(26)

M

The Dirac Hamiltonian can be exactly diagonalized for a free particle. Similar is the situation, when the refractive index is a constant. The required transform in the Foldy-Wouthuysen procedure is (nc + Pz ) − iβ (My px − Mx py ) q

ψ,

(27)

2Pz (nc + Pz )

te

d

ψ −→ ψ0 = T + ψ = q

Ac ce p

b 2⊥ . This transforms the beam-optical equation in where Pz = + n2c − p Eq. (26) to the exactly diagonalized form

i–λ

n o1 ∂ b 2⊥ 2 ψ0 , ψ0 = −β n2c − p ∂z

(28)

b 2⊥ /n2c ), The square-roots in Eq. (27) are to expanded in a power series, in (p to the required degree of accuracy. From Eq. (28), we conclude that the components of ψ0 act as ‘scalars’ as the beam propagates along the z-axis. In scalar optics, it is assumed that the components of E and B, which act as scalars; this is not case. It is the components of the defined vector ψ0 in Eq. (2) and Eq. (27), which behave as scalars. In the input free-space region, Eq. (28) is

i–λ

∂ ψ0(in) = −βPz ψ0(in) , ∂z (nc + Pz ) − iβ (My px − Mx py ) q ψ0(in) = T + ψ(in) = ψ(in) . 2Pz (nc + Pz )

(29)

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Polarization in Maxwell Optics

13

The input beam is in the forward direction along the z-axis (optic axis). So, ∂ ψ(in) = −Pz ψ(in) , ∂z

(30)

ip t

i–λ

βψ0(in) = ψ0(in) =

ψ0u(in) ψ0L(in)

#

ψ0L(in) = 0 .

an

Consequently, the lower components of ψ0(in)

,

us

"

cr

We note that px , py and ∂/∂z commute with each other and also with β. So, Pz commutes with β. We also note, that ∂/∂z, Pz , T + and β commute among themselves. The mutual consistency of the Eq. (27) and Eq. (30) requires that (31)

(32)

d

M

The vanishing of the lower part has consequences. The z-evolution described by Eq. (28) for ψ0 does not couple the upper components of ψ0 with the lower components of ψ0 . So, the lower components of ψ0 (z) are identically zero for all values of z, as the beam propagates. So, in all generality, ψ0L (z) = 0 .

(33)

n o1 ∂ c b 2⊥ 2 ψ0u = Hψ ψ0u = − n2 − p 0u , ∂z

Ac ce p

i–λ

te

The z-evolution of the two-component upper part ψ0u (z) is given by (34)

The beam is now completely characterized jointly by the constraints in Eq. (33) and the z-evolution in Eq. (34). The constraints in Eq. (33) has interesting implications on the beam fields. In the region of free space, we have 

ψ0 = T + ψ = q

1

2Pz (nc + Pz )

   

(nc + Pz ) (−Fx + iFy ) + (−px + ipy ) Fz (nc + Pz ) Fz + (−px + ipy ) (Fx + iFy ) (−px − ipy ) (−Fx + iFy ) + (nc + Pz ) Fz (−px − ipy ) Fz + (nc + Pz ) (Fx + iFy )

    

.(35)

The constraints, ψ0L (z) = 0 in Eq. (33) in the region of free space translates to the following relations i–λ Ez = {(∇⊥ · E ⊥ ) + v (∇ × B)z } , (nc + Pz )

Page 13 of 41

14

Sameen Ahmed Khan i–λ 1 = + (∇⊥ · B ⊥ ) − (∇ × E)z , (nc + Pz ) v ( ) 1 i–λ 1 ∂ ∂ = − Ey − Ez + Bz , v (nc + Pz ) v ∂y ∂x ( ) 1 i–λ 1 ∂ ∂ = + Ex + Ez − Bz . v (nc + Pz ) v ∂x ∂y 

Bz Bx

(36)

cr

By

ip t



M

an

us

These relations hold for any beam moving in the direction of +z-axis. The aforementioned relations in Eq. (36) on the fields are consistent with the results of Mukunda, Simon and Sudarshan, which were obtained using a group theoretical analysis [6]-[11]. Let us now examine the implications of the constraints in Eq. (33) on the z-evolution of the beam fields (E, B). The values of the beam fields in the output plane at z 00 and the input plane at z 0 is formally given by integrating the Hamiltonian in Eq. (28) i Z z00 c dz H0 (z) ψ0u (z ) = ℘ exp − – λ z0 (

00

"

#)

ψ0 (z 0 ) ,

d

b (z 00 , z 0 )ψ (z 0 ) , = G 0 0u   i = exp − – φ(z 00 , z 0 ) ψ0u (z 0 ) , λ

te

(37)

Ac ce p

where ℘ signifies the z-ordering [71]-[74]. We write the z-evolution of the beam fields as " √ # " √ # E E 00 00 0 b (z ) = G0 (z , z ) √1 B (z 0 ) . (38) √1 B µ µ

Now, using Eq. (38) along with the conditions on the fields in Eq. (36), we express the output fields (E(z 00 ), B(z 00 )), in terms of the input fields (E(z 0 ), B(z 0 )). After a straightforward algebra, we obtain the relations given below (

)

1 ∂φ ∂φ Ex (z ) = Ez (z 0 ) − vBz (z 0 ) , 0 Ex (z 0 nc + P z ∂x ∂y ( ) 1 ∂φ ∂φ b E (z 0 ) − G b Ey (z 00 ) = G Ez (z 0 ) + vBz (z 0 ) , 0 y 0 nc + Pz ∂y ∂x 00

b G

0

b )−G

b ∇ · E (z 0 ) Ez (z 00 ) = G 0 ⊥ ⊥

Page 14 of 41

Polarization in Maxwell Optics (

∂φ ∂φ ∂φ ∂φ Ex (z ) + Ey (z 0 ) + v By (z 0 ) − Bx (z 0 ) 0 ∂x ∂y ∂x ∂y ( ) 1 ∂φ ∂φ 1 b B (z 0 ) − G b Ez (z 0 ) + Bz (z 0 ) , Bx (z 00 ) = G 0 x 0 nc + Pz v ∂y ∂x ( ) 1 1 00 0 0 ∂φ 0 ∂φ b b By (z ) = G0 By (z ) + G0 Ez (z ) − Bz (z ) , nc + Pz v ∂x ∂y

!)

0

,

cr

ip t

b +G

1 n c + Pz

15

b B (z 0 ) Bz (z 00 ) = G 0 z 0

1 n c + Pz

(

!)

∂φ ∂φ 1 ∂φ ∂φ Bx (z ) + By (z 0 ) − Ey (z 0 ) − Ex (z 0 ) (39). ∂x ∂y v ∂x ∂y

us

b +G

0

M

  √   Ey    √  Ez    b (z 00 , z 0 )M  G 0  √1 B  µ x     1  √µ By   

d

=

Ac ce p

√1 Bz µ

           00  (z )          

te

  √   Ey    √  Ez      √1 B  µ x     1  √µ By   

an

The aforementioned six relations for the fields in (39) are summarized in the matrix below  √  √   Ex Ex

√1 Bz µ

           0  (z ) ,          

(40)

where



M =

1

0

     0        1 ∂φ  2nc ∂x      0      0       1 2nc

∂φ ∂y

1

1 2nc



∂φ ∂y

− 2n1 c



− 2n1 c





− 2n1 c

0

1 2nc



∂φ ∂x





∂φ ∂y







0

0

∂φ ∂y

∂φ ∂x





1 2nc

∂φ ∂y



1 2nc



1

0

0

1 ∂φ ∂x



1 2nc





∂φ ∂y

 

      − 2n1 c ∂φ  ∂x     0   . (41)     ∂φ 1 − 2nc ∂x         − 2n1 c ∂φ  ∂y    

0





1 2nc

0

0 − 2n1 c

1

0

− 2n1 c

∂φ ∂x

∂φ ∂x



∂φ ∂y



1

Page 15 of 41

16

Sameen Ahmed Khan

ip t

We have kept only the leading order terms in the matrix M in Eq. (41). This is in view of the paraxial approximation, when approximating 1 1 1 2 1 b⊥ + . . . ≈ = + 3p . nc + Pz 2nc 8nc 2nc

"





0 0 0  =   0 0 −i  , 0 i 0

"

1 S1 S2 , Gy = 2 −S2 S1   0 0 i  S2 =   0 0 0 . −i 0 0

#

,

an

S1

#

−S2 S1 −S1 −S2

us

Gx

1 = 2

cr

We follow the notation in [6] and have

(42)

(43)

The matrices Gx and Gy satisfy the identities G3x = G3y = 0 ,

M

G2y = −G2x ,

Gx Gy − Gy Gx = 0 .

(44)

Let φ(r ⊥ ) be any general polynomial in r ⊥ , then to the leading order in –λ i M ≈ exp − – {φ (Q⊥ ) − φ (r ⊥ )} , λ –λ Q⊥ = r ⊥ + G⊥ . n0 

Ac ce p

te

d



This enables us to express Eq. (38) as # # " √ " √ E E 00 00 0 b (p (z ) = G (z 0 ) . 0 b ⊥ , r ⊥ → Q⊥ ; z , z ) √1 B √1 B µ µ

(45)

(46)

Equation (46) is the elegant Mukunda-Simon-Sudarshan rule for the transition from the scalar theory of optics to the vector theory of wave optics. b (r , p Whenever, an operator G 0 ⊥ b ⊥ ) is used to describe any ideal linear optical system in scalar optics b (r , p ψ(out) = G 0 ⊥ b ⊥ ) ψ(in) ,

(47)

then the same system is described completely in vector wave optics by a b (r , p b ⊥ ) readily obtained from G matrix function of (r ⊥ , p 0 ⊥ b ⊥ ) by the simple substitution r ⊥ → Q⊥ .

Page 16 of 41

Polarization in Maxwell Optics

17

4.1

us

cr

ip t

As mentioned earlier, our formalism is built from an exact matrix representation and works for beam optics as well as polarization. Moreover, the MSS-rule is readily obtained within the paraxial approximation and the simplification of a constant refractive index [70]. So, our formalism of Maxwell optics is an appropriate candidate for generalizing the MSS-theory far beyond the paraxial approximation.

Gaussian beams in Maxwell optics

s

"

1 2 1 exp − 2 r 2⊥ π σ0 σ0

#"

# √ E . √1 B µ

(48)

d

F (r ⊥ ; 0) =

M

an

As a first application of the Mukunda-Simon-Sudarshan-rule, let us consider the Gaussian beams. The beam is propagating along the z-axis, with its waist positioned at the transverse plane at z = 0. The field in the waist plane is

te

We note, that k = 1/–λ. The complex radius of curvature, q(z) is related to the radius of curvature of the phase front, R(z) as

Ac ce p

1 2i 1 = + 2 , q(z) R(z) kσ (z)

(49)

where σ(z) is the beam width. Let us consider a plane wave polarized in the x-direction (without any loss of generality and n0 = 1). Then the column " √ # E vector √1 B for the fields simplifies to µ

F (r ⊥ ) =

          

√ Ex0 0 0 0 √1 By0 µ 0

          

.

(50)

Page 17 of 41

18

Sameen Ahmed Khan



2 1 exp π σ0

  #  1  − 2 (r ⊥ + –λG⊥ )2   σ0   

          

.

(51)

.

(52)

an

us

F (r ⊥ ; 0) =

"

√ Ex0 0 0 0 √1 By0 µ 0

cr

s

ip t

Using the MSS-rule, the fields in the plane at z = 0 are

M

The fields at a later transverse plane at z = z 00 are

2 1 ik exp π σ(z) 2q

  #  (r ⊥ + –λG⊥ )2      

Ac ce p

te

F (r ⊥ ; z 00 ) =

"

d

s



√

Ex0 0 0 0 √1 By0 µ 0

          

Using the properties of the matrices Gx and Gy in Eq. (44), the exponential in Eq. (52) is readily obtained in a closed form. So, the expression in Eq. (52) simplifies to

s

F (r ⊥ ; z 00 ) =



2 1 exp π σ(z)

"

    ik 2 r⊥ M    2q(z)    #

√

Ex0 0 0 0 √1 By0 µ 0

          

,

(53)

Page 18 of 41

Polarization in Maxwell Optics

19

where M is the matrix

1−

xy 4q 2

x 2q

y 2 −x2 8q 2

y 2 −x2 8q 2

y 2q

y 2 −x2 8q 2

xy − 4q 2

x − 2q

y − 2q

1

y 2q

x − 2q

xy − 4q 2

2 2 − y 8q−x2

y 2q

y 2 −x2 8q 2

xy − 4q 2

xy 4q 2

x − 2q

xy − 4q 2

x 2q

0

x − 2q

2

− y 8q−x2 y − 2q

2

1+

y − 2q x 2q



ip t

xy − 4q 2

xy − 4q 2

                       

cr

y 2 −x2 8q 2

0

x 2q

.

(54)

us

M=

                       

1+

1−

y 2 −x2 8q 2

an



y − 2q

y 2q

1

s

"

#

2 1 ik 2 exp r . π σ(z) 2q(z) ⊥

(55)

d

Ω(r ⊥ ; z) =

M

In order to express the fields in component form, we shall use the field amplitudes given by

te

In term of components, we have

y 2 − x2 √ 1 Ex0 + √ By0 Ω(r ⊥ ; z) Ex0 + 2 8q µ " !# xy √ 1 Ω(r ⊥ ; z) − 2 Ex0 + √ By0 4q µ " !# x √ 1 Ω(r ⊥ ; z) − Ex0 + √ By0 2q µ !# " 1 xy √ Ex0 + √ By0 Ω(r ⊥ ; z) − 2 4q µ " ! # 2 2 √ y −x 1 1 Ω(r ⊥ ; z) − Ex0 + √ By0 + √ By0 8q 2 µ µ " !# y √ 1 Ω(r ⊥ ; z) − Ex0 + √ By0 . (56) 2q µ √

"

Ac ce p

Ex (r ⊥ ; z) =

!#

Ey (r ⊥ ; z) = Ez (r ⊥ ; z) =

Bx (r ⊥ ; z) =

By (r ⊥ ; z) =

Bz (r ⊥ ; z) =

We note, that Ey (r ⊥ ; z) = cBx (r ⊥ ; z). Eq. (56) has the electric and magnetic fields of a Gaussian beam in Maxwell optics. It is to be noted, that the

Page 19 of 41

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Sameen Ahmed Khan

5

Concluding remarks

us

cr

ip t

electric field E(r ⊥ ; z) has a component Ez (r ⊥ ; z) along the beam axis. Additionally, Ey (r ⊥ ; z) and Bx (r ⊥ ; z) have the cross-polarization components. These components vanish only along the two planes defined by x = 0 and y = 0 respectively. It is to be noted, that the intersection of these two planes is the z-axis, which is the optics axis. The MSS-rule leading to the Gaussian Maxwell beams was derived using a group theoretical analysis [6]-[11]. Here, the same rule is obtained using an entirely different framework.

Ac ce p

te

d

M

an

It was noted, that even though Maxwell’s equations are linear, it is difficult to use them directly as they are constrained and coupled. So, it is useful to express Maxwell’s equations in a matrix form. The matrix representation of Maxwell’s equations presented here is exact and has a Dirac-like form [29]. Consequently, it is possible to employ quantum methodologies, in particular the Foldy-Wouthuysen transformation technique, to develop a formalism of Maxwell optics. In the limit of low wavelength, Maxwell optics (vector optics) presented here reproduce the ‘Lie algebraic formalism of light beam optics’ [66]-[69]. The matrix representation of Maxwell’s equations is central to our formalism. So, we have presented the derivation of the required matrix representation of Maxwell’s equations in detail [29]. The Foldy-Wouthuysen transformation is vital to our matrix-based formalism of Maxwell optics. Hence, we have outlined the essential aspects of the FoldyWouthuysen transformation and its applications in optics [30, 31]. Our formalism of Maxwell optics leads to a unified theory of beam optics and light polarization. The beam-optics is based on the exact Hamiltonians. Where as in the case of polarization the elegant Mukunda-Simon-Sudarshan rule for transition from scalar to vector optics was derived from the unified formalism [70]. In fact, the derivation of the MSS-rule is based on the paraxial limit of our formalism, that too with the idealization of a constant refractive index. In the limit of low wavelength, our formalism leads to the ‘Lie algebraic treatment of light beam optics’, which is a cornerstone of optics. With this established result, along with the MSS-rule, the new formalism is the ideal candidate to generalize the MSS theory beyond the paraxial approximation taking into account the variation of the refractive index. The unified formalism light beam-optics and light polarization advances the Hamilton’s optical-mechanical in the wavelength-dependent regime [75, 76].

Page 20 of 41

Polarization in Maxwell Optics

21

Ac ce p

te

d

M

an

us

cr

ip t

The matrix-based formalism of the Maxwell optics is based on the techniques developed in the quantum theory of charged-particle beam optics. So, a few remarks on the quantum formalisms are appropriate. The quantum formalism of charged-particle beam optics had its origins in the question, where is the h ¯ in charged-particle beam optics? This question has been addressed in substantial detail by developing a quantum theory of charged-particle beam optics. In 1989, Jagannathan, Mukunda, Simon and Sudarshan worked out the focusing action of a magnetic electron lens using the Dirac equation [32][34]. This gave birth to the quantum theory of charged-particle beam optics. In the quantum theories of charged-particle beam optics, the h ¯ occurs through – ¯ /p0 , where p0 is the design or avthe reduced de Broglie wavelength λ0 = h erage momentum of the beam particles. In the limit –λ0 −→ 0 the quantum prescriptions lead to the well-known ‘Lie algebraic formalism of chargedparticle beam optics’ [77]-[86]. Following Jagannathan’s 1990 blueprint, the quantum theory of aberrations was developed using the Klein-Gordon equation [87, 88]. A unified theory of the beam-optics and the spin dynamics of a Dirac particle beam was also presented, considering the anomalous magnetic moment [35]. The paraxial limit of this approach leads to the beam-optical version of the Thomas-Bargmann-Michel-Telegdi (Thomas-BMT) equation. A detailed account of these historical developments is available in the encyclopedic works of Hawkes and Kasper [89]. Thereafter, the field of quantum approaches to beam optics has been under active research [36]-[48]. A diffractive quantum limit has been also considered for particle beams [90]. There have been a series of meetings on the emerging field of QABP: the Quantum Aspects of Beam Physics (see the proceedings [91]-[93]). Quantum methodologies are a powerful tool to handle a variety of systems. Quantum methodologies have been also used to study multi-particle effects in charged-particle beam optics. In passing, we note the models developed by Fedele et al. (the thermal wave model [94]-[98]) and Cufaro Petroni et al. (the stochastic collective dynamical model [99]) for treating the beam phenomenologically as a quasiclassical many-body system. In these models, the basic equation is a Schr¨odinger-like equation with the beam emittance playing the role of h ¯ , the Planck’s constant. The quantum-like approach has been also used to develop a Diffraction Model for the beam halo [100]-[103]. Few remarks on the Foldy-Wouthuysen are appropriate. The FoldyWouthuysen transformation was originally constructed for the Dirac equation, which is for the spin-1/2 particles [104]-[113]. Based on its success, the FW transformation technique was extended to the equations for other

Page 21 of 41

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Sameen Ahmed Khan

Ac ce p

te

d

M

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spins [114]-[116]. It was soon seen, that the FW technique works for a class of equations having a certain mathematical structure. So, the equations on which the FW is employed need not be even quantum mechanical! Consequently, the Foldy-Wouthuysen technique has been successfully used in diverse systems including: atomic systems [117, 118]; synchrotron radiation [119]; derivation of the Bloch equation for polarized beams [120]; among others. It has found applications even in acoustics [121]-[126] and ocean acoustics [127]. The Helmholtz equation governing scalar optics is mathematically very similar to the Klein-Gordon equation for a spin-0 relativistic particle. So, it is possible to employ quantum methodologies in order to develop an alternate prescription of Helmholtz optics. This requires the linearization of the Helmholtz wave equation using a Feshbach-Villars-like procedure used for linearizing the Klein-Gordon equation [128, 129]. The resulting non-traditional formalism of Helmholtz optics [2]-[4] has wavelength-dependent modifications of the traditional square-root approaches. In the context of Helmholtz optics, we note that the square-root operators have been extensively examined [130]. A detailed account of the Foldy-Wouthuysen transform and its applications in optics is available in [30, 31]. The proposal to use the Foldy-Wouthuysen procedure for the Helmholtz equation was first mentioned as a comment by Fishman and McCoy [131]. The same idea was independently outlined by Jagannathan and Khan (see, p. 277 in [88]). The quantum methodologies and the Foldy-Wouthuysen transformation technique in particular, are shedding new facets of the analogies between the light beam optics and the charged-particle beam optics [75, 76]. If not for the quantum methodologies, there would be no unified formalism of light beam optics and light polarization [132]. The analogy between geometrical light optics and the classical theories of charged-particle beam optics has been known and exploited for a very long time (for comprehensive and historical accounts see [89] and [133]-[135]). Now, it is possible to see the Hamilton’s optical-mechanical analogy in the wavelength-dependent regime [75, 76]. Usually, Ren´e Descartes (1596-1650) is credited as the originator of the analogy. The analogy has a much older history and its beginning can be attributed to Ibn al-Haytham (0965-1037, Latinized name: Alhazen) [136]-[142]. But it was William Rowan Hamilton (1805-1865), who in 1831, closely examined the trajectories of material particles in various potential fields and compared them with the paths of rays of light in media with spatially varying refractive index. Thus it was Hamilton,

Page 22 of 41

Polarization in Maxwell Optics

23

Acknowledgments

cr

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who laid a rigorous mathematical foundation to the optical-mechanical analogy. Some historical aspects of the analogy were also addressed during the 2015 International Year of Light and Light-based Technologies [143]-[149].

M

an

us

I am indebted to my Doctoral Thesis Advisor, Professor Ramaswamy Jagannathan (Institute of Mathematical Sciences, IMSc, Chennai, India) for all my training in the emerging field of quantum mechanics of charged-particle beam optics. He elegantly supervised my doctoral thesis. He gave the brilliant suggestion to use the Foldy-Wouthuysen transformation technique to investigate the scalar optics (Helmholtz optics). Later he extended the suggestion to develop a matrix formalism of the Maxwell optics.

References

te

d

[1] V. Lakshminarayanan, Ajoy Ghatak and K. Thyagarajan, Lagrangian Optics, Springer, 2002, http://dx.doi.org/10.1007/ 978-1-4615-1711-5.

Ac ce p

[2] Sameen Ahmed Khan, Ramaswamy Jagannathan, and Rajiah Simon, Foldy-Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams, E-Print arXiv: arXiv:physics/0209082 [physics.optics]; http://arXiv.org/ abs/physics/0209082/ (2002). (communicated). [3] Sameen Ahmed Khan, Wavelength-dependent modifications in Helmholtz Optics, Int. J. Theor. Phys. 44(1) (2005) 95125, http://dx.doi.org/10.1007/s10773-005-1488-0; EPrint arXiv: arXiv:physics/0210001 [physics.optics]; http: //arxiv.org/abs/physics/0210001. [4] Sameen Ahmed Khan, Quantum Methodologies in Helmholtz Optics. Optik-International Journal for Light and Electron Optics, 127(20) (2016) 9798-9809, http://dx.doi.org/10.1016/j.ijleo.2016.07. 071.

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[5] Sameen Ahmed Khan, Linearization of Wave Equations, (communicated).

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[6] N. Mukunda, R. Simon and E.C.G. Sudarshan, Paraxial-wave optics and relativistic front description I: the scalar theory, Phys. Rev. A 28 (1983) 2921-2932, http://dx.doi.org/10.1103/PhysRevA.28.2921.

us

[7] N. Mukunda, R. Simon and E.C.G. Sudarshan, Paraxial-wave optics and relativistic front description II: the vector theory, Phys. Rev. A 28 (1983) 2933-2942, http://dx.doi.org/10.1103/PhysRevA.28.2933.

an

[8] N. Mukunda, R. Simon and E.C.G. Sudarshan, Fourier optics for the Maxwell field: formalism and applications, J. Opt. Soc. Am. A 2(3) (1985) 416-426, http://dx.doi.org/10.1364/JOSAA.2.000416.

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[9] N. Mukunda, R. Simon, and E.C.G. Sudarshan, Paraxial Maxwell beams: transformation by general linear optical systems, J. Opt. Soc. Am. A 2(8) (1985) 1291-1296; http://dx.doi.org/10.1364/JOSAA. 2.001291.

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[10] R. Simon, E.C.G. Sudarshan, N. Mukunda, Gaussian-Maxwell beams, J. Opt. Soc. Am. A 3(4) (1986) 536-540; http://dx.doi.org/10.1364/ JOSAA.3.000536. [11] R. Simon, E.C.G. Sudarshan and N. Mukunda, Cross polarization in laser beams, Appl. Opt. 26(9) (1987) 1589-1593, http://dx.doi.org/ 10.1364/AO.26.001589. [12] O. Laporte, G.E. Uhlenbeck, Applications of spinor analysis to the Maxwell and Dirac Equations, Phys. Rev., 37 (1931) 1380-1397. http: //dx.doi.org/10.1103/PhysRev.37.1380. [13] J.R. Oppenheimer, Note on Light Quanta and the Electromagnetic Field, Phys. Rev. 38, 725-746 (1931). http://dx.doi.org/10.1103/ PhysRev.38.725. [14] E. Moses, Solutions of Maxwell’s equations in terms of a spinor notation: the direct and inverse problems, Phys. Rev., 113(6) (1959) 1670-1679. http://dx.doi.org/10.1103/PhysRev.113.1670.

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[15] E. Majorana, quoted after R. Mignani, E. Recami, and M. Baldo, About a Diraclike Equation for the Photon, According to Ettore Majorana, Lett. Nuovo Cimento 11(12), 568-572 (1974). http://dx.doi.org/10. 1007/BF02812391.

cr

[16] T. Ohmura, A New Formulation on the Electromagnetic Field, Prog. Theor. Phys. 16(6), 684-685 (1956). http://dx.doi.org/10.1143/ PTP.16.684.

an

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[17] R.H. Good Jr., Particle Aspect of the Electromagnetic Field Equations, Phys. Rev. 105, 1914-1919 (1957). http://dx.doi.org/10. 1103/PhysRev.105.1914.

M

[18] J.S. Lomont, Dirac-Like Wave Equations for Particles of Zero Rest Mass and Their Quantization, Phys. Rev. 111, 1710-1716 (1958). http://dx. doi.org/10.1103/PhysRev.111.1710.

d

[19] M. Sachs and S.L. Schwebel, On Covariant Formulations of the MaxwellLorentz Theory of Electromagnetism, J. Math. Phys. 3(5), 843-848 (1962). http://dx.doi.org/10.1063/1.1724297.

te

[20] E. Giannetto, A Majorana-Oppenheimer formulation of quantum electrodynamics, Lett. Nuovo Cim. 44(3), 140-144 (1985). http://dx.doi. org/10.1007/BF02746912.

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[21] W.H. Inskeep, On Electromagnetic Spinors and Quantum Theory, Zeitschrift fr Naturforschung 43A, 695-696 (1988). http://dx.doi. org/10.1515/zna-1988-0715. [22] V.V. Dvoeglazov, Electrodynamics with Weinberg’s photons, Hadronic Journal. 16, 423-428 (1993). E-Print arXiv: http://arxiv.org/abs/ hep-th/9306108. [23] S. Esposito, Covariant Majorana formulation of electrodynamics, Found. Phys. 28(2), 231-244 (1998). http://dx.doi.org/10.1023/ A:1018752803368. E-Print arXiv: http://arxiv.org/abs/hep-th/ 9704144. [24] T. Ivez´c Lorentz Invariant Majorana Formulation of the Field Equations and Dirac-like Equation for the Free Photon, Electronic Journal of Theoretical Physics 3(10), 131-142 (2006). http://www.ejtp.com/

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articles/ejtpv3i10p131.pdf. E-Print arXiv: http://arxiv.org/ abs/physics/0605030.

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[25] P.J. Mohr, Solutions of the Maxwell equations and photon wave functions, Annals of Physics, 325(3) (2010) 60763. dx.doi.org/10.1016/ j.aop.2009.11.007.

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[26] I. Bialynicki-Birula, On the wave function of the photon, Acta Physica Polonica A 86, 97-116 (1994). http://przyrbwn.icm.edu.pl/APP/ ABSTR/86/a86-1-8.html.

an

[27] I. Bialynicki-Birula, The Photon Wave Function, in: J.H. Eberly, L. Mandel, E. Wolf (Eds.), Coherence and Quantum Optics VII, Plenum Press, New York, 1996, pp. 313-322. http://dx.doi.org/10.1007/ 978-1-4757-9742-8_38.

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[28] I. Bialynicki-Birula, Photon wave function, in: Progress in Optics, vol. XXXVI, E. Wolf (Ed.), Elsevier, Amsterdam, 1996, pp. 245-294. http: //dx.doi.org/10.1016/S0079-6638(08)70316-0.

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