Feb 6, 2017 - Geometrical optics (GO) is a reduced model of wave dynamics [1, 2] that is widely used in many contexts rang- ing from quantum dynamics ...

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arXiv:1612.06184v2 [physics.plasm-ph] 6 Feb 2017

1

Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA 2 Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA (Dated: February 7, 2017)

Even when neglecting diffraction effects, the well-known equations of geometrical optics (GO) are not entirely accurate. Traditional GO treats wave rays as classical particles, which are completely described by their coordinates and momenta, but vector-wave rays have another degree of freedom, namely, their polarization. The polarization degree of freedom manifests itself as an effective (classical) “wave spin” that can be assigned to rays and can affect the wave dynamics accordingly. A well-known manifestation of polarization dynamics is mode conversion, which is the linear exchange of quanta between different wave modes and can be interpreted as a rotation of the wave spin. Another, less-known polarization effect is the polarization-driven bending of ray trajectories. This work presents an extension and reformulation of GO as a first-principle Lagrangian theory, whose effective-gauge Hamiltonian governs the aforementioned polarization phenomena simultaneously. As an example, the theory is applied to describe the polarization-driven divergence of right-hand and left-hand circularly polarized electromagnetic waves in weakly magnetized plasma.

I.

INTRODUCTION A.

Motivation

Geometrical optics (GO) is a reduced model of wave dynamics [1, 2] that is widely used in many contexts ranging from quantum dynamics to electromagnetic (EM), acoustic, and gravitational phenomena [3–5]. Mathematically, GO is an asymptotic theory with respect to a small parameter that is a ratio of the wave relevant characteristic period (temporal or spatial) to the inhomogeneity scale of the underlying medium. Practical applications of GO are traditionally restricted to the lowest-order theory, where each wave is basically approximated with a local eigenmode of the underlying medium at each given spacetime location. Then, the wave dynamics is entirely determined by a single branch of the local dispersion relation. However, this approximation is not entirely accurate, even when diffraction is neglected. If a dispersion relation has more than one branch, i.e., a vector wave with more than one polarization at a given location, then the interaction between these branches can give rise to important polarization effects that are missed in the traditional lowest-order GO. One interesting manifestation of such polarization effects is the polarization-driven bending of ray trajectories. At the present moment, it is known primarily in two contexts. One is quantum mechanics, where polarization effects manifest as the Berry phase [6] and the associated Stern-Gerlach force experienced by vector particles, i.e., quantum particles with spin. Another one is optics, where a related effect has been known as the Hall effect of light; namely, even in an isotropic dielectric, rays propagate somewhat differently depending on polarization if the dielectric is inhomogeneous (see, c.f. Refs. [7–11]). But the same effect can also be anticipated for waves in plasmas, e.g., radiofrequency (RF) waves in tokamaks. In fact, since for RF waves in laboratory plasma is typ-

ically larger than that for quantum and optical waves, the polarization-driven bending of ray trajectories in this case can be more important and perhaps should be taken into account in practical ray-tracing simulations. However, ad hoc theories of polarization effects available from optics are inapplicable to plasma waves, which have more complicated dispersion and thus require more fundamental approaches. Thus, a different theory is needed that would allow the calculation of the polarization-bending of the ray trajectories for plasma waves and, also more broadly, waves in general linear media. Relevant work was done in Refs. [12, 13], where a systematic procedure was proposed to asymptotically diagonalize the dispersion operator for linear vector waves. Polarization effects emerge as O() corrections to the GO dispersion relation. However, this approach excludes mode conversion, i.e., the linear exchange of quanta between different branches of the local dispersion relation. Since the group velocities of the different branches eventually separate, mode conversion is typically followed by ray splitting and, in this particular context, was studied extensively (see, e.g., Refs. [14–20]). However, these works considered wave modes that are resonant in small, localized regions of phase space. Hence, the nonadiabatic dynamics was formulated as an asymptotic scattering problem between two wave modes, so the polarizationdriven bending of ray trajectories was not included. The main message of this work is that mode conversion and the polarization-driven bending of ray trajectories are two sides of the same coin and can be considered simultaneously within a unified theory. The first general theory that captures them both simultaneously was proposed in Ref. [21]. This theory was successfully benchmarked [22] against previous theories describing the Hall effect of light [7–11]. However, the formulation in Ref. [21] is still limited since it requires that the wave equation be brought to a certain (multisymplectic) form resembling the Dirac equation. Although any nondissipative vector wave allows for such representation in prin-

2 ciple [21, 23], casting the wave dynamics into the specific framework adopted in Ref. [21] can be complicated. Thus, practical applications require a more flexible formulation that do not rely on this specific framework. Here we propose such a theory. In addition to generalizing the results of Ref. [21], we also introduce, in a unified context and an instructive manner, some of the related advances that were made recently in Refs. [24–26]. It is expected that the comprehensive analysis presented in this work will facilitate future practical implementations of the proposed theory, particularly in improving ray-tracing simulations.

B.

Outline

We consider general linear nondissipative waves determined by some Hermitian dispersion operator. Using the Feynman reparameterization and the Weyl calculus, we obtain a reduced Lagrangian for such waves. In contrast with the traditional GO Lagrangian, which has an accuracy of O(0 ) in the GO parameter , our Lagrangian is O(1 )-accurate, so it captures polarization effects. As an example, we apply the formulation to study polarization effects on the propagation of EM waves propagating in weakly magnetized plasma. (The case of strongly magnetized plasma will be discussed in a separate paper.) The advantages of our theory are as follows. (i) The theory is derived in a variational form, so the resulting equations are manifestly conservative. (ii) Through the use of the Feymann reparameterization, we can obtain the dynamics of continuous waves and of their rays directly from a variational principle. (iii) The theory assumes no specific wave equation, so quantum spin effects and classical polarization effects can be studied on the same footing. (iv) Moreover, a related formalism [21] is applicable to develop new reduced theories for relativistic spinning particles [21, 24]. This paper is organized as follows. In Sec. II, the basic notation is defined. In Sec. III, the variational formalism used to describe vector waves is presented. In Sec. IV, a general procedure is proposed to block-diagonalize the wave dispersion operator. In Sec. V, we reparametrize the wave action to facilitate asymptotic analysis. In Sec. VI, the leading order GO approximation is discussed. In Sec. VII, the more accurate model that includes polarization effects is discussed. In Sec. VIII, the theory is applied to describe polarization effects on the propagation of EM waves in weakly magnetized plasma. In Sec. IX, our main results are summarized. Finally, Appendix A presents a brief introduction to the Weyl symbol calculus.

mitian conjugate.” The identity N × N matrix is denoted by IN . The Minkowski metric is adopted with signature (+, −, −, −). Greek indices span from 0 to 3 and refer to spacetime coordinates xµ = (x0 , x) with x0 corresponding to the time variable t. Also, partial derivatives on spacetime will be denoted by ∂x , where . the individual components are ∂µ = ∂/∂xµ = (∂t , ∇) . 4 3 and d x = dt d x. Latin indices span from 1 to 3 and denote the spatial variables, i.e., x = (x1 , x2 , x3 ) and . ∂i = ∂/∂xi . Summation over repeated indexes is assumed. In particular, for arbitrary four-vectors a and b, . we have a · b = aµ bµ = a0 b0 − a · b. In Euler-Lagrange equations (ELEs), the notation “δa :” means that the corresponding equation is obtained by extremizing the action integral with respect to a. III.

BASIC EQUATIONS

A.

Wave action principle

The dynamics of any nondissipative linear wave can be described by the least action principle, δS = 0, where the real action S is bilinear in the wave field [23]. We represent a wave field, either classical or quantum, as a complex-valued vector Ψ(x). We allow this vector field ¯ . In the to have an arbitrary number of components N absence of parametric resonances [27], the action can be written in the form [28] Z . S = d4 x d4 x0 Ψ† (x)D(x, x0 )Ψ(x0 ), (1) ¯ ×N ¯ Hermitian matrix kernel [D(x, x0 ) = where D is a N † 0 D (x , x)] that describes the underlying medium. Varying S with respect to Ψ† leads to Z δΨ† : 0 = d4 x0 D(x, x0 )Ψ(x0 ). (2) Similarly, varying with respect to Ψ gives the equation adjoint to Eq. (2), which we do not need to discuss. It is convenient to describe the wave Ψ(x) as an abstract vector |Ψi in the Hilbert space of wave states with inner product [23, 29] Z . hΥ|Ψi = d4 x Υ† (x)Ψ(x). (3) In this representation, Ψ(x) = hx|Ψi, where |xi are the eigenstates of the coordinate operator x ˆ such that hx|ˆ xµ |x0 i = xµ hx|x0 i = xµ δ 4 (x − x0 ). We also introduce the momentum (wavevector) operator pˆ such that hx|ˆ pµ |x0 i = i∂[δ 4 (x − x0 )]/∂xµ in the x-representation [30]. Thus, the action (1) can be rewritten as ˆ S = hΨ|D|Ψi ,

II.

NOTATION

The following notation is used throughout the pa. per. The symbol “=” denotes definitions, “c. c.” denotes “complex conjugate,” and “h. c.” denotes “Her-

(4)

ˆ is the Hermitian dispersion operator such that where D ˆ 0 i. Treating hΨ| and |Ψi as indepenD(x, x0 ) = hx|D|x dent variables [23] and varying the action (4) gives δ hΨ| :

ˆ |Ψi = 0 D

(5)

3 (plus the conjugate equation), which is the generalized vector form of Eq. (2). Specifically, Eq. (2) is obtained by projecting Eq. (5) with hx| and using the fact that the R operator d4 x |xi hx| = ˆ 1 is an identity operator.

B.

IV.

EIGENMODE REPRESENTATION A.

Variable transformation

We introduce a unitary τ -independent transformation ˆ that maps |Ψi to some N ¯ -dimensional abstract vector Q ¯ yet to be defined: |ψi

Extended wave function

ˆ |ψi ¯ . |Ψi = Q As shown in Refs. [21, 23], reduced models of wave propagation are convenient to develop when the action is of the symplectic form; namely, . ˆ Ssymplectic = hΨ|(ˆ p0 IN¯ − H)|Ψi ,

(6)

where pˆ0 = i∂t (in the x-representation) and “the wave ˆ = H(ˆ ˆ ) is some Hermitian operator Hamiltonian” H x, p that is local in time, i.e., commutes with tˆ. (For extended discussions, see Refs. [23, 31].) In order to cast the general action (4) into the symplectic form (6), let us perform the so-called Feynman reparameterization [32, 33] that lifts the wave dynamics governed by Eq. (4) from R4 to R5 . Specifically, we let the wave field depend on spacetime and on some parameter τ so that Ψ(τ, x) = hx|Ψ(τ )i. Note that |Ψ(τ )i belongs to the same Hilbert space defined in Sec. III A. Thus, theR inner product remains the same; i.e., hΥ(τ 0 )|Ψ(τ )i = d4 x Υ† (τ 0 , x)Ψ(τ, x). We consider the following “extended” action: . SX =

(10)

Inserting Eq. (10) into Eqs. (8) leads to ¯ τ ψi ¯ − c. c. , Lτ = −(i/2) hψ|∂ ˆ eff |ψi ¯D ¯ , LD = hψ|

(11a) (11b)

. ˆ† ˆ ˆ ˆ eff = where D Q DQ. In what follows, we seek to conˆ ˆ eff is expressed in a struct Q such that the operator D block-diagonal form. The procedure used is identical to that given in Refs. [12, 13]. However, in order to account ˆ eff will be made only blockfor resonant-mode coupling, D diagonal, instead of fully diagonal as in Refs. [12, 13]. B.

Weyl representation

Let us consider Eq. (11b) in the Weyl representation. (Readers who are not familiar with the Weyl calculus are encouraged to read Appendix A before continuing further.) In this representation, LD is written as [28] Z LD = Tr d4 x d4 p Deff (x, p)W (τ, x, p), (12)

Z dτ L,

(7)

. where L = Lτ + LD , . Lτ = −(i/2) [ hΨ(τ )|∂τ Ψ(τ )i − c. c. ] , . ˆ LD = hΨ(τ )|D|Ψ(τ )i ,

(8a) (8b)

where ‘Tr’ represents the matrix trace. The Wigner ten¯ is defined as sor W (τ, x, p) corresponding to |ψi Z d4 s ip·s s s . W mn (τ, x, p) = e hx + |ψ¯m i hψ¯n |x − i , (2π)4 2 2 (13) and Deff (x, p) is the Weyl symbol [Eq. (A1)] correspondˆ eff . It can be written explicitly as ing to the operator D

and ∂τ Ψ(τ, x) = hx|∂τ Ψ(τ )i. Note that the Lagrangian L is local in the parameter τ ; i.e., the abstract vectors are all evaluated at τ . From hereon, all fields will be evaluated at τ , and we will avoid mentioning the dependence of |Ψi on τ explicitly. The ELE corresponding to the action SX is given by

where ‘?’ is the Moyal product [Eq. (A6)] and D(x, p), Q(x, p), and [Q† ](x, p) are the Weyl symbols correspondˆ Q, ˆ and Q ˆ † , respectively. Also, the Weyl repreing to D, ˆ†Q ˆ = ˆIN¯ , is sentation of the unitary condition, Q

ˆ |Ψi . i∂τ |Ψi = D

[Q† ](x, p) ? Q(x, p) = IN¯ ,

Deff (x, p) = [Q† ](x, p) ? D(x, p) ? Q(x, p),

(9)

Note that Eq. (9) can be interpreted as a vector Schr¨ odinger equation in the extended variable space, ˆ acts as the Hamiltonian operator. The dynamwhere D ics of the original system described by Eq. (5) is a special case of the dynamics governed by Eq. (9), which corresponds to a steady state with respect to the parameter τ ; i.e., ∂τ Ψ = 0. The advantage of the representation (7) is that the action has the manifestly symplectic form, so we can proceed as follows.

(14)

(15)

which will be used below. C.

Eigenmode representation

Let us assume that the symbols Deff and Q can be expanded in powers of the GO parameter 1 1 = max , 1, (16) ωT |k|`

4 where ω and |k| are understood as the characteristic wave frequency and wave number, respectively. Also, T and ` are the characteristic time and length scales of the background medium, correspondingly. Hence, we write Deff (x, p) = Λ(x, p) + U (x, p) + O(2 ), 2

Q(x, p) = Q0 (x, p) + Q1 (x, p) + O( ),

(17a)

[Q†0 ](x, p)Q0 (x, p)

= IN¯ .

¯

Λ(x, p) = diag [ λ(1) (x, p), ... , λ(N ) (x, p) ].

(18) (19)

ˆ By properties of the Weyl transformation, the fact that D is a Hermitian operator ensures that D(x, p) is a Hermi¯ orthonormal eigenvectian matrix. Hence, D(x, p) has N tors eq (x, p), which correspond to some real eigenvalues λ(q) (x, p). Let us construct Q0 (x, p) out of these eigenvectors so that Q0 (x, p) = e1 (x, p), ... , eN¯ (x, p) , (20) where the individual eq form the columns of Q0 . From Eq. (19), we then find [Q†0 ](x, p) = Q†0 (x, p). Hence, the

(21)

To the next order in , Eq. (15) reads as follows:

(17b)

¯ ×N ¯ matrices of order unity. where (Λ, U, Q0 , Q1 ) are N To the lowest-order in , the Moyal products in Eqs. (14) and (15) reduce to ordinary products, so Λ(x, p) = [Q†0 ](x, p)D(x, p)Q0 (x, p),

matrix Λ has the following diagonal form:

Q†0 Q1 + [Q†1 ]Q0 + (i/2){Q†0 , Q0 } = 0.

(22)

Here we assumed that term that involves the Poisson bracket {Q†0 , Q0 }, which arises from the expansion of the Moyal star product [Eq. (A9)], is of the first order in . Following Ref. [12], we let Q1 = Q0 (A + iG) and ¯ ×N ¯ Hermi[Q†1 ] = Q†1 , where A(x, p) and G(x, p) are N tian matrices. Then, Eq. (22) gives A(x, p) = −(i/4){Q†0 , Q0 }.

(23)

In order to determine G(x, p), we write Eq. (14) to the first order in . Introducing the bracket ∂A ∂B . ∂A ∂B {A, B}C = − C C ∂pµ ∂xµ ∂xµ ∂pµ

(24)

and noting that DQ0 = Q0 Λ, we obtain [12]

U (x, p) = Q†1 DQ0 + Q†0 DQ1 + (i/2){Q†0 D, Q0 } + (i/2){Q†0 , D}Q0 = (A − iG)Q†0 DQ0 + Q†0 DQ0 (A + iG) + (i/2){Q†0 D, Q0 } + (i/2){Q†0 , D}Q0 = AΛ − iGΛ + ΛA + iΛG + (i/2){Q†0 D, Q0 } + (i/2){Q†0 , DQ0 } − (i/2){Q†0 , Q0 }D = iΛG − iGΛ − (i/4){Q†0 , Q0 }Λ − (i/4)Λ{Q†0 , Q0 } + (i/2){ΛQ†0 , Q0 } + (i/2){Q†0 , Q0 Λ} − (i/2){Q†0 , Q0 }D = i(ΛG − GΛ + δU ), (25) ¯ ×N ¯ matrix given by where δU (x, p) is a N δU (x, p) = (1/4){Q†0 , Q0 }Λ + (1/4)Λ{Q†0 , Q0 } + (1/2){Λ, Q0 }Q† + (1/2){Q†0 , Λ}Q0 − (1/2){Q†0 , Q0 }D .

(26)

0

Since (ΛG − GΛ)mn = Gmn [λ(m) − λ(n) ] (no summation is assumed here over the repeating indices), one can diagonalize U by adopting Gmn = δU mn /[λ(n) − λ(m) ], as done in Refs. [12, 13]. However, this method is applicable only when |λ(m) − λ(n) | & O(1); otherwise, when |λ(m) −λ(n) | ' O(), G & O(−1 ) so Q1 & O(1), which is in violation of the assumed ordering in Eq. (17b). Hence, instead of diagonalizing U , we propose to only blockdiagonalize U as follows. When |λ(m) − λ(n) | & O(1), we choose the off-components of Gmn so that U mn = 0. (We call such modes nonresonant.) When |λ(m) −λ(n) | ' O(), we let Gmn = 0. (We call such modes resonant.) By following this prescription and permutating the ma-

trix rows, we obtain U in the following form: U1 (x, p) 0 ... 0 0 U2 (x, p) . . . 0 U (x, p) = , .. .. . . . . . . . . 0 0 0 UJ (x, p)

(27)

where Uj (x, p) are nj × nj Hermitian matrices and J is PJ ¯ . Note that, the total number of blocks, so j=1 nj = N in the particular case where only nonresonant modes are present, U (x, p) is diagonal, and one recovers the results obtained in Refs. [12, 13]. Since the matrix Deff ≈ Λ+U is made block-diagonal, the Lagrangian (12) is unaffected by the matrix elements

5 W mn with indices (m, n) such that U mn = 0. Thus, without loss of generality, we can write LD =

J X

Z Tr

d4 x d4 p [[Deff ]]j [[W ]]j ,

(28)

j=1

where [[Deff ]]j ' [[Λ + U ]]j and [[W ]]j represent the jth matrix block of Λ + U and W , respectively. Hence, nonresonant eigenmodes are decoupled while resonant eigenmodes that belong to the same block remain coupled.

V.

REDUCED ACTION A.

Basic equations

Now that blocks of mutually nonresonant modes are decoupled, let us focus on the dynamics of modes within a single block of some size N . Hence, the block index will be dropped, and we adopt Z . Lτ = −(i/2) d4 x ψ † (∂τ ψ) − (∂τ ψ † )ψ , (29a) Z . LD = Tr d4 x d4 p [[Λ + U ]] [[W ]]. (29b) Here ψ is a complex-valued function with N components, and [[W ]] is the N × N Wigner tensor with elements Z s s d4 s ip·s m e hx + |ψ m i hψn |x − i . [[W ]] n (τ, x, p) = (2π)4 2 2 (30) Since we consider the coupled dynamics of some N resonant modes, only N columns of Q0 actually contribute to [[Deff ]]. For clarity, let us denote the resonant eigenmodes as eq with indices q = 1, ..., N . Then, in order to calculate [[U ]], one can use Eq. (25). After block¯ × N matrix diagonalizing U and introducing the N Ξ(x, p) = [e1 (x, p), ... , eN (x, p)],

(31)

one obtains i i i [[U ]] = {Ξ† , Ξ}Λ + Λ{Ξ† , Ξ} + {Λ, Ξ}Ξ† 4 4 2 i † i † + {Ξ , Λ}Ξ − {Ξ , Ξ}D , (32) 2 2 which is a N × N Hermitian matrix. Furthermore, it is convenient to split [[Deff ]] as follows: [[Deff ]] = λIN + U,

(33)

. where λ = N −1 Tr [[Deff ]] is the average of the eigenval. ues of [[Deff ]] and U = [[Deff ]] − λIN¯ is the remaining traceless part of [[Deff ]]. In the special case when all λ(q) within the block are identical and [[U ]] is traceless, then Λ = λIN , and

U = [[U ]]. We call such modes degenerate. Then, the expression (32) for [[U ]] simplifies, and one obtains i i i U(x, p) = {Ξ† , Ξ}λ + λ{Ξ† , Ξ} + {λ, Ξ}Ξ† 4 4 2 i i + {Ξ† , λ}Ξ − {Ξ† , Ξ}D 2 2 1 1 = − Ξ† {λ, Ξ} − {Ξ† , λ}Ξ 2i 2i 1 1 + {Ξ† , Ξ}D − {Ξ† , Ξ}λ 2i 2i = − Ξ† {λ, Ξ} A + (∂p Ξ† )(D − λIN )(∂x Ξ) A , (34) where we used the bracket introduced in Eq. (24) and the subscript ‘A’ denotes “anti-Hermitian part;” i.e., for any . matrix M , then MA = (M − M † )/(2i). The expression in Eq. (34) can also be written more explicitly as ∂λ ∂Ξ ∂λ † ∂Ξ U(x, p) = − + Ξ† µ Ξ ∂pµ ∂x A ∂xµ ∂pµ A † ∂Ξ ∂Ξ + (35) (D − λIN ) µ . ∂pµ ∂x A Examples of physical systems, where these simplified formulas are applicable, include spin-1/2 particles [21, 24] and EM waves propagating in isotropic dielectrics [22].

B.

Parameterization of the action

In order to derive the corresponding ELEs, let us adopt the following parameterization: p ψ(τ, x) = I(τ, x) z(τ, x) eiθ(τ,x) . (36) Here θ(τ, x) is a real variable that serves as the rapid phase common for all N modes (remember that all modes within the block of interest are approximately resonant to each other). Also, I(τ, x) is a real function, and z(τ, x) is a N -dimensional complex unit vector (z † z = 1), whose components describe the amount of quanta in the corresponding modes. (Since we parameterize the N -dimensional complex vector ψ by the N -dimensional complex vector z plus two independent real functions θ and I, not all components of z are truly independent. For an extended discussion, see Ref. [21].) After substituting the ansatz (36) into Eq. (29a), the Lagrangian Lτ is given by Z Lτ = d4 x I ∂τ θ − (i/2)(z † ∂τ z − c. c.) . (37) (Here we formally introduce to denote that z is a slowlyvarying quantity; however, this ordering parameter will be removed later.) Now, we calculate the Wigner tensor (30). Substituting Eq. (36) into Eq. (30), we obtain

6

p d4 s p I(x + s/2, τ )z m (x + s/2, τ )eiθ(x+s/2,τ ) I(x − s/2, τ )zn∗ (x − s/2, τ )e−iθ(x−s/2,τ ) eip·s 4 (2π) Z d4 s = I(τ, x)z m (τ, x)zn∗ (τ, x)ei(p−k)·s (2π)4 " √ # √ Z ∂( I z m ) √ ∗ √ m ∂( I zn∗ ) d4 s sµ + · Izn − Iz ei(p−k)·s + O(2 ) (2π)4 2 ∂xµ ∂xµ " √ √ ∗ # m √ 4 √ i I z ) ∂( I zn ) ∂[δ (p − k)] ∂( = I(τ, x) z m (τ, x)zn∗ (τ, x) δ 4 (p − k) − Izn∗ − Iz m + O(2 ), 2 ∂pµ ∂xµ ∂xµ (38)

[[W ]]mn (τ, x, p) =

Z

. where we introduced the four-wavevector kµ (τ, x) = −∂µ θ(τ, x) = (ω, −k), which is considered a slow function. µ [Accordingly, the contravariant representation is k (x, τ ) = (ω, k).] Inserting Eq. (38) into Eq. (29b) and integrating over the momentum coordinate, we obtain Z LD = d4 x d4 p (λ[[W ]]mm + U mn [[W ]]nm ) # " √ Z Z i ∂[δ 4 (p − k)] ∂( I z m ) √ ∗ 4 † † 4 4 = d x I λ(x, k)z z + z U(x, k)z − d xd pλ Izm − c. c. + O(2 ) 2 ∂pµ ∂xµ Z Z i 4 µ † ∂z 4 † d x I v (τ, x) z − c. c. + O(2 ), (39) = d x I λ(x, k) + z U(x, k)z − 2 ∂xµ

where we integrated by parts and used z † z = 1. Here ∂λ(x, p) . µ v (τ, x) = − (40) ∂pµ p=k(τ,x) is the zeroth-order (in ) group velocity of the wave. We then introduce the convective derivative d . ∂ ∂ = + v µ (τ, x) µ . dτ ∂τ ∂x

VI.

TRADITIONAL GEOMETRICAL OPTICS A.

Continuous wave model

To lowest order in , the Lagrangian (42) can be approximated simply with Z . LGO = d4 x I [ ∂τ θ + λ(x, k) ] , (43)

(41)

Summing Eqs. (37) and (39), we obtain the action R S = dτ L + O(2 ), where the Lagrangian is given by Z 4 L = d x I ∂τ θ + λ(x, k) dz † i † dz † z − z + z U(x, k)z . (42) − 2 dτ dτ Equation (42), along with the definitions in Eqs. (31)(33), (40), and (41), is the main result of this work. The first line on the right-hand side of Eq. (42) represents the lowest-order GO Lagrangian. The terms in the second line of Eq. (42) are O() and introduce polarization effects. [Importantly, diffraction terms would be O(2 ) and thus are safe to neglect in our first-order theory.] In what follows, we discuss the consequences of this theory and provide an example, where we apply the theory to study polarization effects on EM waves in weakly magnetized plasmas.

which one may interpret as a Hayes-type representation [34] of the GO wave Lagrangian in the extended (τ, x) space. This Lagrangian is parameterized by just two functions, the rapid phase θ and the totalR action density I. Thus, varying the action SGO = dτ LGO , we obtain the following ELEs: δθ : ∂τ I + ∂µ (v µ I) = 0, δI : ∂τ θ + λ(x, k) = 0,

(44a) (44b)

where v µ (τ, x) is the GO four-group-velocity (40). As mentioned in Sec. III, the dynamics of the physical wave propagating in spacetime is obtained by adopting ∂τ Ψ = 0, which also corresponds to ∂τ I = ∂τ θ = 0. Hence, Eqs. (44) become ∂ ∂λ ∂λ − I +∇· I = 0, (45a) ∂t ∂ω ∂k λ(x, k) = 0. (45b) Equation (45a) is the action conservation theorem, or the photon conservation theorem. Equation (45b) is the local

7 dispersion relation. For an in-depth discussion of these equations, see, e.g., Refs. [1, 4]. B.

Point-particle model

The ray equations corresponding to the above field equations can be obtained as the point-particle limit. In this limit, I can be approximated with a delta function I(τ, x) = I0 δ 4 (x − X(τ )).

. where Pµ (τ ) = −∂µ θ(τ, X(τ )). Similarly, Z d4 x δ 4 (x − X(τ ))λ(x, −∂θ) = λ(X(τ ), P (τ )). (48)

(49)

This is a covariant action, where X(τ ) and P (τ ) serve as canonical coordinates and canonical momenta, respectively. Treating X and P as independent variables leads to ELEs matching Hamilton’s covariant equations µ

∂λ dX =− , (50a) dτ ∂Pµ dPµ ∂λ δX µ : = . (50b) dτ ∂X µ These are the commonly known ray equations; for instance, see Ref. [1]. They can also be written as δPµ :

dX 0 ∂λ =− , dτ ∂P0 dP 0 ∂λ = , dτ ∂X 0

dX ∂λ = , dτ ∂P dP ∂λ =− . dτ ∂X

λ(X, P ) = 0.

(51)

(46)

Here I0 denotes the total action, which is conserved according to Eq. (45a). The value of I0 is not essential below so we adopt I0 = 1 for brevity. In this representation, the wave packet is located at the position X(τ ) in space-time, and the independent parameter is τ . [This means that at a given τ , the wave packet is located at the spatial point X(τ ) at time t(τ ).] When inserting Eq. (46) into Eq. (43), the first term in the action gives the following: Z dτ d4 x I ∂τ θ Z = dτ d4 x δ 4 (x − X(τ )) ∂τ θ(τ, x) Z = − dτ d4 x θ(τ, x)[∂τ δ 4 (x − X(τ ))] Z = dτ d4 x θ(τ, x)[X˙ µ (τ )∂µ δ 4 (x − X(τ ))] Z = − dτ d4 x ∂µ θ(τ, x)X˙ µ (τ )δ 4 (x − X(τ )) Z = dτ Pµ (τ )X˙ µ (τ ), (47)

Thus, the point-particle action is expressed as Z h i ˙ ) + λ(X, P ) . SGO = dτ P (τ ) · X(τ

Note that the first term in the integrand in Eq. (49) represents the symplectic part of the canonical phasespace Lagrangian, and the second term represents the Hamiltonian part. Since the Hamiltonian part λ(X, P ) does not depend explicitly on τ , then dλ(X, P )/dτ = 0 along the ray trajectories. Thus, the ray dynamics lies on the dispersion manifold defined by

As a reminder, λ(x, p) is defined as the average eigenvalue . of the resonant block, i.e., λ = N −1 Tr [[Deff ]]. The GO action (49) is only accurate to lowest order in ; hence, one can approximate λ(x, p) ' λ(n) (x, p), where λ(n) is any particular resonant eigenvalue. This occurs because the resonant eigenvalues differ by O() and because the polarization coupling is also O().

VII.

EXTENDED GEOMETRICAL OPTICS

In this section, we explore the polarization effects determined by the Lagrangian (42). For the sake of conciseness, we only discuss the point-particle ray dynamics. For an overview of the continuous-wave model, see Ref. [21].

A.

Point-particle model

The ray equations with polarization effects included can be obtained as a point-particle limit of the Lagrangian (42). As in Sec. VI B, we approximate the wave packet to a single point in spacetime [Eq. (46)]. As shown in Refs. [21, 25], the Lagrangian (42) can be replaced by a point-particle Lagrangian so the action is dτ P · X˙ − (i/2)(Z † Z˙ − Z˙ † Z) + λ(X, P ) + Z † U(X, P )Z ,

Z SXGO =

(52)

. where Z(τ ) = z(τ, X(τ )) is the point-particle polarization vector and we dropped the GO ordering parameter . In the complex representation, Z and Z † are canonical conjugate, and Z † (τ )Z(τ ) = 1.

(53)

Even though the components of Z are not independent by definition (Sec. V B), it can be shown [21] that treating them as independent in this point-particle model leads to correct results provided that the initial conditions satisfy Eq. (53). Hence, the independent variables in SXGO are

8 (X, P, Z, Z † ), and the corresponding ELEs are δPµ : δX µ : δZ † : δZ :

dX µ ∂λ ∂U =− − Z† Z, dτ ∂Pµ ∂Pµ ∂U ∂λ dPµ + Z† Z, = dτ ∂X µ ∂X µ dZ = −iUZ, dτ † dZ = iZ † U. dτ

(54a) (54b) (54c) (54d)

Together with Eqs. (31)-(33), Eqs. (54) form a complete set of equations. The first terms on the right-hand side of Eqs. (54a) and (54b) describe the ray dynamics in the GO limit. The second terms describe the coupling to the mode polarization. Equations (54c) and (54d) describe the wave-polarization dynamics. As in Sec. VI B, the Hamiltonian part of Eq. (52) is constant along the ray trajectories. As before, the ray dynamics lies on the dispersion manifold defined by setting the Hamiltonian part to zero; i.e., †

λ(X, P ) + Z U(X, P )Z = 0.

(55) . As a reminder, λ(x, p) is defined through Eq. (33) as λ = . −1 N Tr [[Deff ]], and U = [[Deff ]] − λIN¯ is the remaining traceless part of [[Deff ]]. B.

Precession of the wave spin

Let us also describe the rotation of Z(τ ) as follows. Since U(X, P ) is a traceless Hermitian N × N matrix, it can be decomposed into a linear combination of N 2 − 1 generators Tu of SU(N ), which are traceless Hermitian matrices, with some real coefficients −W u [35]: U =−

2 NX −1

Tu W u ≡ −T · W.

(56)

u=1

Then, we introduce the (N 2 − 1)-dimensional vector . S(τ ) = Z † (τ )T Z(τ ) (57) so that Z † UZ = −S·W. The components of S(τ ) satisfy the following equation: dτ Sw = Z † Tw (dτ Z) + (dτ Z † )Tw Z = iZ † [U, Tw ]Z = −i Z [Tu , Tw ]ZW

u

= fuwv (Z † T v Z)W u = fwvu S v W u ,

d S = S ∗ W, dτ

(58)

where fabc are structure constants. They are defined via [Ta , Tb ] = ifabc T c so that the structure constants fabc are antisymmetric in all indices [35].

(59)

which is understood as a generalized precession equation. In the particular case when S is conserved (we call such waves “pure states”), then Eqs. (54a), (54b), and (59) form a closed set of equations, and λ − S · W serves as an effective scalar Hamiltonian. The dynamics of Z and Z † does not need to be resolved in this case, so one can rewrite SXGO as a functional of (X, P ) alone: Z SXGO = dτ [P · X˙ + λ(X, P ) − S · W(X, P )]. (60) An example of the dynamics described by such action will be discussed in Sec. VIII E. A more general case is when S is close to some eigenvector w of W that corresponds to some nondegenerate eigenvalue Ωw . If Ωw is large enough, then S(τ ) will remain close to w(τ ) and will only experience smallamplitude oscillations. These oscillations can be understood as a generalized zitterbewegung effect [38], and they ˙ becomes zero. In this are transient, i.e., vanish when W regime, no mode conversion occurs at τ → ∞. In contrast, if Ωw is not large enough, the change of S governed by Eq. (59) is not necessarily negligible. This corresponds to mode conversion and causes ray splitting at τ → ∞ (see, e.g., Refs. [14–20]). This is discussed below.

C.

= iZ † UTw Z − iZ † Tw UZ †

For example, consider the case when only two waves are resonant. Then, N 2 − 1 = 3, T v are the three Pauli matrices divided by two (so |S|2 = 1/2), and fwuv is the Levi-Civita symbol, so fwvu S v W u = (S × W)w . For a Dirac electron, which is a special case, such S is recognized as the spin vector undergoing the well known precession equation, dτ S = S × W [21]. In optics, this is an equation for the Stokes vector that was derived earlier to characterize the polarization of transverse EM waves in certain simple media [7, 36, 37]. Hence, it is convenient to extend this quantum terminology also to N resonant waves. We will call the corresponding (N 2 − 1)-dimensional vector S a generalized “wave-spin” vector and express fwvu S v W u symbolically as (S∗W)w , where ‘∗’ can be viewed as a generalized vector product. Notably, using the concept of spin vector S, one can rewrite Eqs. (54c) and (54d) as follows:

Mode conversion as a form of spin precession

Equation (54c) [and thus Eq. (59)] can also describe mode conversion as it is understood in Refs. [14–20]. This is shown as follows. Let us consider the resonant interaction between two modes as an example; then, U is a 2 × 2 matrix. From Eq. (33), U is Hermitian and traceless and can be represented as ∆λ/2 U12 U(X, P ) = , (61) ∗ U12 −∆λ/2

9 . where ∆λ(X, P ) = [λ(1) −λ(2) ]/2+(U11 −U22 )/2 and the coefficient U12 determines the mode coupling. Suppose that, absent coupling (U12 = 0), the dispersion curves of two modes cross at some point (X∗ , P∗ ). Suppose also that ∆λ(τ ) = ∆λ(X(τ ), P (τ )) changes along the ray trajectory approximately linearly in τ . Then, ∆λ ≈ ατ , where α is some constant coefficient and we chose the origin on the time axis such that ∆λ(τ = 0) = 0 for sim. plicity. Similarly, U12 (τ ) = U12 (X(τ ), P (τ )) ' β + γτ , where β and γ are some constants. Assuming β is sufficiently large, we neglect the term γτ for it only causes a correction to the dominant effect. Thus, near the modeconversion region, Eq. (54c) is approximately written as d Z1 ατ /2 β Z1 i = . (62) β ∗ −ατ /2 Z2 dτ Z2 Equation (62) is the well-known equation for mode conversion that was studied by Zener in Ref. [39]. After eliminating Z2 , the governing equation for Z1 is α α2 τ 2 Z¨1 (τ ) + |β|2 + i + Z1 (τ ) = 0. (63) 2 4 . √ . Letting w = τ α eiπ/4 and n = −i|β|2 /α, the equation above can be written as a Weber equation 1 w2 00 Z1 (w) + n + − Z1 (w) = 0, (64) 2 4 whose solutions are the parabolic cylinder functions Dn (w). In Refs. [17, 39], the matrix connecting the waves entering and exiting the resonance are obtained by analyzing asymptotics of Dn (w). Specifically, Z1,out T −C ∗ Z1,in = , (65) Z2,out C T Z2,in where

√

2πη , (66) ηΓ(−i|η|2 ) √ . where Γ is the Gamma function and η = β/ α. The transmission and conversion coefficients for the wave quanta are, correspondingly, 2

T = exp(−π|η| ),

C=−

|T | = exp(−2π|β|2 /|α|),

(67)

|C|2 = 1 − |T |2 .

(68)

(Also see Ref. [16] for a somewhat different approach leading to the same answer.) This calculation shows that mode conversion, in the way as commonly described in literature [14–20], is nothing but a manifestation of the wave-spin precession described by Eqs. (54c) and (59). Note that the present point-particle model cannot capture ray-splitting because it introduces only one ray for the whole field. However, this theory does predict the transfer of wave quanta, which is a prerequisite for ray-splitting. For a complete analysis on ray-splitting mode conversion, please refer to Refs. [1, 18, 19].

VIII.

DISCUSSION: WAVES IN WEAKLY MAGNETIZED PLASMAS

A simplified form of the theory above was applied to describe spin-1/2 particles [21, 24] and waves in isotropic dielectrics [22]. Here we present another example of its application, namely, EM waves in weakly magnetized cold plasmas. (The case of strongly magnetized plasmas will be discussed in a separate paper.) We assume that the plasma response is determined by particles of just one type, e.g., electrons. The generalization to multicomponent plasma is straightforward to do.

A.

Dispersion operator

The linearized equations of motion are [40] ∂t v = (q/m)E + (q/mc)v × B0 , ∂t E = −4πqn0 v + c∇ × B, ∂t B = −c∇ × E.

(69a) (69b) (69c)

Here q, m, n0 (x), and v(t, x) are the particle charge, mass, unperturbed background density, and flow velocity, respectively. Also, E(t, x) denotes the perturbation electric field, B(t, x) is the perturbation magnetic field, B0 (x) is the background magnetic field, and c is the speed of light. We introduce a re-scaled velocity field . ¯ (t, x) = v(t, x)[4πn0 (x)m]1/2 , so v ¯ = ωp E + v ¯ × Ω, ∂t v ¯ + c∇ × B, ∂t E = −ωp v ∂t B = −c∇ × E,

(70a) (70b) (70c)

. where ωp (x) = [4πq 2 n0 (x)/m]1/2 is the plasma frequency . and Ω(x) = qB0 (x)/(mc) is the gyrofrequency. Let us write Eqs. (70) using the abstract Hilbert space notation. Let |vi be a state vector representing the velocity field such that v(x) = hx|vi. Likewise, we introduce |Ei and |Bi as the state vectors of E(x) and B(x), respectively. Then, Eqs. (70) can be written as follows:

ˆ |¯ pˆ0 |¯ vi = iˆ ωp |Ei − (α · Ω) vi , ˆ ) |Bi , pˆ0 |Ei = −iˆ ωp |¯ vi + ic(α · p ˆ ) |Ei , pˆ0 |Bi = −ic(α · p

(71a) (71b) (71c)

. . ˆ = where ω ˆ p = ωp (ˆ x) and Ω Ω(ˆ x). (As a reminder, ˆ = −i∇ are the components of the fourpˆ0 = i∂t and p . momentum operator in the x-representation.) Also, α =

10 α1 , α2 , α3 are 3 × 3 Hermitian matrices [41] 0 0 0 . α1 = 0 0 −i , 0 i 0 0 0 i . α 2 = 0 0 0 , −i 0 0 0 −i 0 . α 3 = i 0 0 . 0 0 0

(72a)

(72b)

(72c)

These matrices serve as generators for the vector product. Namely, for any two column vectors A and B, one has (α · A)B = iA × B, T j

For the sake of simplicity, we consider the case of a wave propagating in a weakly magnetized plasma. (The general case will be described in a separate paper.) Thus, supposing that the typical wave frequency is much larger than the gyrofrequency (ω ∼ p0 Ω), we expand the dispersion symbol (78) in powers of Ω/p0 : D ' D0 + D1 + O(Ω2 /p20 ),

(79)

D0 (x, p) = −p20 + (α · p)2 + ωp2 (x),

(80a)

where

D1 (x, p0 ) =

· Ω)/p0 .

(80b)

(73a) j

A α B = −i(A × B) ,

(73b)

where the superscript ‘T’ denotes the matrix transpose. The next step is to construct a dispersion operator for the electric field state |Ei. Starting from Eq. (71a), we solve for the velocity field in terms of the electric field. Hence, we formally obtain the following: ˆ −1 |Ei |¯ vi = iˆ ωp (ˆ p0 I3 + α · Ω) # " ˆ ˆ 2 α·Ω (α · Ω) 1 − 2 |Ei , + = iˆ ωp ˆ2 ˆ 2) pˆ0 pˆ0 − Ω pˆ0 (ˆ p20 − Ω

To simplify the following calculation, we assume that D1 ∼ O(Ω/p0 ) is comparable in magnitude to the GO parameter , but this is not essential. Hence, we will consider D1 as a perturbation only. Following Sec. IV C, the next step is to identify the eigenvalues and eigenmodes of the dispersion symbol D0 (x, p). The corresponding eigenvalues are λ(1) (x, p) = −p · p + ωp2 (x),

(74)

. ˆ where Ω = |Ω(ˆ x)|. Similarly, we obtain ˆ )ˆ |Bi = −ic(α · p p−1 |Ei from Eq. (71c). Substitut0 ing these results into Eq. (71b), we obtain ˆ |Ei = 0, D

(75)

where ˆ 2 ˆ ω ˆ p2 (α · Ω) ω ˆ p2 pˆ0 (α · Ω) . ˆ = ˆ )2 + ω + (76) D −ˆ p20 +(α· p ˆ p2 − ˆ2 ˆ2 pˆ20 − Ω pˆ20 − Ω serves as the dispersion operator for |Ei. (For convenience, we let c = 1.) Since ωp (x) and Ω(x) are indeˆ so pendent of time, then pˆ0 commutes with ω ˆ p and Ω, ˆ is manifestly Hermitian. The corresponding action (4) D ˆ for the electric field is S = hE|D|Ei, and the extended action (7) is Z i . ˆ SX = dτ − (hE|∂τ Ei − c. c.) + hE|D|Ei . (77) 2 ¯ = 3. Note that E is a three-dimensional vector field, so N

(2)

(81a)

ωp2 (x),

(81b)

λ(3) (x, p0 ) = −p20 + ωp2 (x),

(81c)

λ

(x, p) = −p · p +

where p · p = p20 − p2 . These eigenvalues correspond to the dispersion relations of two transverse EM waves and of longitudinal Langmuir oscillations, respectively. The matrix Q0 defined in Eq. (20) is given by Q0 (p) = [ e1 (p), e2 (p), ep (p) ],

(82)

where e1 (p) and e2 (p) are any two orthonormal vectors . in the plane normal to ep (p) = p/|p|. A right-hand convention is adopted such that e1 × e2 = ep . One can easily verify that these vectors are indeed eigenvectors of D0 (x, p). For example, D0 e1 = [−p20 + (α · p)(α · p) + ωp2 ] e1 = (−p20 + ωp2 ) e1 − p × (p × e1 ) = (−p20 + p2 + ωp2 ) e1 = λ(1) e1 ,

B.

−ωp2 (x)(α

(83)

EM waves in weakly magnetized plasma

We now follow the procedure given in Secs. IV and V to block-diagonalize the dispersion operator. The Weyl ˆ is symbol of D 2

2

2

ωp p0 (α · Ω) ωp (α · Ω) . D = −p20 +(α·p)2 +ωp2 − + 2 . (78) p20 − Ω2 p0 − Ω2

where Eq. (73a) was used. Similar calculations follow for the other two eigenmodes e2 and ep . We now analyze the dynamics of the transverse EM waves. From Sec. V, the eigenvalue is λ(x, p) = −p · p + ωp2 (x), and Ξ(p) = [e1 (p), e2 (p) ] is a 3 × 2 matrix. Since Ξ(p) only depends on the spatial momentum coordinate, then the polarization-coupling

11 Hamiltonian U in Eq. (35) is given by

∂Ξ Ξ ∂pµ A ∂λ † ∂Ξ =− · Ξ ∂x ∂p 1 A ∂ ∂λ e e e · =− e2 ∂p 1 2 A ∂x ! ∂ ∂ e1 e1 ∂p e2 e1 ∂p ∂λ =− · ∂ ∂ e2 ∂p e1 e2 ∂p e2 ∂x A ! 1 ∂ 0 e ∂p e2 1 ∂λ =− · , ∂ −e1 ∂p e2 0 i ∂x

∂λ U(x, p) = ∂xµ

on the matrix D1 (x, p0 ) is given by ωp2 e1 † Ξ D1 Ξ = − (α · Ω) e1 e2 p0 e2 ωp2 e1 (α · Ω)e1 e1 (α · Ω)e2 =− p0 e2 (α · Ω)e1 e2 (α · Ω)e2 ωp2 0 ep · Ω =i 0 p0 −ep · Ω

†

=−

ωp2 (ep · Ω)σy , p0

where we used Eq. (73b). (84) C.

q

where e is a row of those Eq. (84)

q

δrq .

(85)

Now, let us discuss the point-particle ray dynamics. Following Sec. VII, we substitute λ(x, p) = −p·p+ωp2 (x), Eq. (85), and Eq. (91) into Eq. (52). We then obtain the point-particle action Z SXGO = dτ P · X˙ − (i/2)(Z † Z˙ − Z˙ † Z) − P · P + ωp2 (X) + Σ(X, P )Z † σy Z , (92)

(86)

where the polarization-coupling matrix is given by

where σy is the y-component of the Pauli matrices σy =

0 −i i 0

ωp2 . Σ(x, p) = −(∇ωp2 ) · F − (ep · Ω) p0

and F(p) is a vector with components given by ∂ . e2 . F(p) = e1 ∂p

(87)

For example, one may choose √px pz p − √ 2y 2 p p2x +p2y px +py p p . . √ px √y z e1 (p) = p √p2x +p2y , e2 (p) = p2x +p2y , (88) p2 +p2 0 − xp y

(89)

T . where p⊥ = px , py , 0 ; or, more explicitly, py pz −px . F(p) = |p||p⊥ |2 0

(90)

(The specific choice of e1 and e2 does not affect the resulting equations within the accuracy of the present theory. For more details, see Sec. VIII G.) Returning to the perturbation caused by the background magnetic field, the projection of the eigenmodes

(93)

and Z(τ ) is a complex-valued vector with two components that describe the degree of polarization along the vectors e1 and e2 . It is normalized such that Z † Z = 1. In the action (92), the two polarization modes are coupled through the Pauli matrix σy . However, these modes can be decoupled when using the basis of circularly polarized modes. We introduce the variable transformation Z(τ ) = R Γ(τ ),

(94)

where . 1 R= √ 2

so that p ×p F(p) = ⊥ , |p||p⊥ |2

Ray dynamics

q

is the dual to eq , so e er = (Specifically, e vector, whose elements are complex-conjugate of eq .) Since ∇λ = ∇ωp2 , we can also write in the form U(x, p) = −σy (∇ωp2 ) · F,

(91)

1 1 i −i

(95)

and Γ(τ ) is a new vector with components denoted as . Γ+ Γ(τ ) = . (96) Γ− Inserting Eq. (94) into the action (92) leads to Z h SXGO = dτ P · X˙ − (i/2)(Γ† Γ˙ − Γ˙ † Γ) −P · P + ωp2 (X) + Σ(X, P )Γ† σz Γ , (97) where σz is another Pauli matrix, 1 0 σz = . 0 −1

(98)

12

FIG. 1: Comparison between ray trajectories calculated using the equations of traditional GO [Eqs. (109), dashed line] and extended GO [Eqs. (104)]. The blue and red lines represent the ray trajectories for the right-hand and left-hand polarized rays, respectively. For simplicity, nonmanetized plasma is considered, so the Faraday effect is absent. The plasma frequency is given by ωp2 (x) = y 2 + z 2 . The initial location of the ray trajectories is X0 = (0, 1, 0), and the initial momentum is P0 = (5, 0, 1). (The units are arbitrary, since the figure is a general illustration only.) For this simulation, the GO parameter is roughly ∼ 1/ |P0 | ∼ 0.2. Due to the radial gradient in the plasma frequency, the wave rays follow helical trajectories along the x axis.

Here Γ± (τ ) represent the wave quanta belonging to the right-hand and left-hand circularly polarized modes, respectively (as defined from the point of view of the source). Also, Γ is normalized such that Γ† Γ = 1. Treating X(τ ), P (τ ), Γ(τ ), and Γ† (τ ) as independent variables, we obtain the following ELEs: δPµ : δX µ : δΓ† : δΓ :

dX µ ∂Σ = 2P µ − Γσz Γ, dτ ∂Pµ ∂ωp2 ∂Σ dPµ = + Γσz Γ, dτ ∂X µ ∂X µ dΓ = −iΣσz Γ, dτ dΓ† = iΓΣσz . dτ

(99a) (99b) (99c) (99d)

Together with Eq. (93), Eqs. (99) form a complete set of equations. The first terms on the right-hand side of Eqs. (99a) and (99b) describe the ray dynamics in the GO limit. The second terms describe the coupling of the mode polarization and the ray curvature.

D.

Restating the Faraday effect

To better understand the polarization equations, let us rewrite Eq. (99c) as an equation in the basis of linearly polarized modes: Z˙ = RΓ˙ = −iΣRσz Γ = −iΣ(Rσz R−1 )Z = −iΣσy Z. (100) [This equation could also be obtained if the ray equations were derived directly from the action (92).] Since Σ is a scalar and σy is constant, this can be readily integrated,

yielding [42] Z(τ ) = exp(−iΘσy )Z0 = (I2 cos Θ − iσy sin Θ)Z0 , (101) R . τ where Θ(τ ) = 0 dτ 0 Σ(X(τ 0 ), P (τ 0 )) is the polarization . precession angle and Z0 = Z(τ = 0). This result can be also be expressed explicitly as follows: cos Θ − sin Θ Z(τ ) = Z0 . (102) sin Θ cos Θ It is seen that the polarization of the EM field rotates at the rate Σ(t) in the reference frame defined by the basis vectors (e1 , e2 ). The first term in Eq. (93) is identified as the rate of change of the wave Berry phase [6]. (In optics, the rotation of the polarization plane caused by the Berry phase is also known as the Rytov rotation [37, 43, 44].) The second term in Eq. (93) is identified as the rate of change due to Faraday rotation. E.

Dynamics of pure states

If a ray corresponds to a strictly circular polarization such thatR σz Γ = ±Γ, the action (97) can be simplified to SXGO = dτ L± , where the Lagrangian is given by L± = P · X˙ − P · P + ωp2 (X) ± Σ(X, P ).

(103)

Here the Lagrangian L± governs the propagation of righthand and left-hand polarization modes, respectively. The corresponding ELEs are δPµ :

∂Σ dX µ = 2P µ ∓ , dτ ∂Pµ

(104a)

δX µ :

∂ωp2 dPµ ∂Σ = ± , µ dτ ∂X ∂X µ

(104b)

13 or in terms of spacetime components, dX 0 ∂Σ = 2P 0 ∓ , dτ ∂P0 dP0 = 0, dτ

G.

dX ∂Σ = 2P ± , dτ ∂P ∂ωp2 dP ∂Σ =− ∓ . dτ ∂X ∂X

The first terms on the right-hand side of Eqs. (104) describe the ray dynamcis in the GO limit. The second terms describe the coupling of the mode polarization and the ray curvature. They are also responsible for the polarization-driven bending of ray trajectories. As shown, P0 remains constant because the background medium is time independent. In order to obtain the value of P0 , we note that the ray Hamiltonian H± (X, P ) = −P · P + ωp2 (X) ± Σ(X, P )

and substituting them into Eq. (110), we obtain

(107)

Numerical simulations

To illustrate the polarization-driven divergence of the ray trajectories, Fig. 1 shows the ray trajectories for a right-polarized and left-polarized waves using the Lagrangian (103). For completeness, we also show the calculated ray trajectory as determined by the lowest-order GO ray Lagrangian LGO = P · X˙ − P · P + ωp2 (X),

(108)

which does not account for polarization effects. As anticipated, the ray trajectories predicted by the Lagrangian (103) differ noticeably from the “spinless” ray trajectory predicted by Eq. (108); namely; µ

δPµ : δX µ :

dX = 2P µ , dτ ∂ωp2 dPµ = . dτ ∂X µ

ωp2 (X) [ep (P) · Ω(X)] P0 ωp2 (X) ' P · X˙ − P · P + ωp2 (X ∓ F) ∓ [ep (P) · Ω(X)], P0 (110) ∓ (∇ωp2 ) · F ∓

(106)

. where ω(X, P) = (P2 + ωp2 )1/2 is the wave frequency in . the GO limit and P∗µ (X, P) = (ω(X, P), P).

F.

L± = P · X˙ − P · P + ωp2 (X)

where we assumed that ωp2 (x) is smooth and neglected terms of O(2 ) as usual. Introducing the variables . . xµ (τ ) = (X 0 , X ∓ F(P)), pµ (τ ) = Pµ (111)

Setting the Hamiltonian equal to zero, we use Eq. (105) to determine P0 . One finds P0 ' ω(X, P) ± Σ(X, P∗ )/[2ω(X, P)],

It is possible to obtain an alternative, noncanonical representation of the ray Lagrangian (103) that is invariant with respect to the choice of F(p) for pure states and explicitly shows the so-called Berry connection. Starting from Eq. (103) and substituting Eq. (93), we can write

(105)

is independent of τ , so one can readily verify that H± (X(τ ), P (τ )) = constant.

Noncanonical representation and the Berry connection

(109a) (109b)

This divergence along the x-axis is driven by polarization effects. For EM waves propagating in isotropic nonbirefringent dielectrics, this effect is called the Hall effect of light in the optics literature [7].

L± ' p · x˙ − p · p + ωp2 (x) ωp2 (x) [ep (p) · Ω(x)] p0 = p · x˙ − p · p + ωp2 (x) ˙ ∓ ∓p·F

± p˙ · F(p) ∓

ωp2 (x) [ep (p) · Ω(x)], p0

(112)

where we dropped a perfect time derivative. We also approximated X ' x in the Faraday rotation term in Eq. (112) since it is already O(). Note that |x − X| is of the order of the wavelength, i.e., small enough to make x and X equally physical as measures of the ray location. The term p˙ · F(p) is known as the Berry connection term [8]. It is to be noted that adding ∂p χ(p) to F(p), where χ(p) is an arbitrary scalar function, changes L± by a perfect derivative and does not affect the equations of motion. The ELEs corresponding to the Lagrangian (112) are given by ωp2 dx0 = 2p0 ∓ 2 (ep · Ω), dτ p0 ωp2 ∂ dx = 2p ± p˙ × (∇p × F) ∓ (ep · Ω), dτ p0 ∂p dp0 = 0, dτ " # ∂ωp2 dp ∂ ωp2 =− ± (ep · Ω) . dτ ∂x ∂x p0

(113a) (113b) (113c) (113d)

These equations are equivalent to Eqs. (104) within the accuracy of the theory. Substituting Eq. (90), we can also write Eq. (113b) as dx p˙ × p ωp2 ∂ = 2p ± ∓ (ep · Ω). dτ |p|3 p0 ∂p

(114)

14 Hence, with the use of the noncanonical coordinates (x, p), the equations of motion no longer depend on the specific choice of F(p); i.e., they are invariant with respect to the choice (88) of vectors e1 and e2 . Note that the same equations could be obtained directly from the point-particle limit of Eq. (84), if one substitutes ˙ For an extended discussion of pure states −∇λ = p. governed by noncanonical Lagrangians, see Ref. [12].

IX.

CONCLUSIONS

Even when neglecting diffraction, the well-known equations of geometrical optics (GO) are not entirely accurate. Traditional GO treats wave rays as classical particles, which are completely described by their position and momentum coordinates. However, vector waves have another degree of freedom, namely, their polarization. Polarization dynamics are manifested in two forms: (i) mode conversion, which is the transfer of wave quanta between resonant eigenmodes and can be understood as the precession of the wave spin, and (ii) polarization-driven bending of ray trajectories, which refers to deviations of the GO ray trajectories arising from first-order corrections to the GO dispersion relation. They are easily understood by drawing parallels with quantum mechanics, where similar effects (yet involving ~) are known as spin rotation and spin-orbital coupling. In this work, we propose a first-principle variational formulation that captures both types of polarizationrelated effects simultaneously. We consider general linear nondissipative waves, whose dynamics are determined by ˆ Using the Feynman repasome dispersion operator D. rameterization and the Weyl calculus, we obtain a reduced Lagrangian model for such general waves. In contrast with the traditional GO Lagrangian, which is O(0 )accurate in the GO parameter , our Lagrangian is O()accurate. In our procedure, polarization effects are contained in the O() corrections to the GO Lagrangian. These corrections may be especially significant for modeling RF waves in laboratory plasmas because such waves can have not-too-small (as opposed, for instance, to quantum particles whose spin effects are typically weak). As an example, we apply the formulation to study the polarization-driven divergence of RF waves propagating in weakly magnetized plasma. Assessing the importance of polarization effects on waves propagating in strongly magnetized plasma will be discussed in a separate paper. Likewise, the method of including dissipation [26] in the above theory will also be described separately. This work was supported by the U.S. DOE through Contract No. DE-AC02-09CH11466, by the NNSA SSAA Program through DOE Research Grant No. DENA0002948, and by the U.S. DOD NDSEG Fellowship through Contract No. 32-CFR-168a.

Appendix A: The Weyl transform

This appendix summarizes our conventions for the Weyl transform. (For more information, see the excellent reviews in Refs. [1, 45–47].) The Weyl symbol A(x, p) of any given operator Aˆ is defined as Z . ˆ − s/2i , A(x, p) = d4 s eip·s hx + s/2|A|x (A1) where p·s = p0 s0 −p·s and the integrals span over R4 . We shall refer to this description of the operators as a phasespace representation since the symbols (A1) are functions of the eight-dimensional phase space. Conversely, the inverse Weyl transformation is given by Z 4 4 4 d x d p d s ip·s/ ˆ A= e A(x, p) |x − s/2i hx + s/2| . (2π)4 (A2) ˆ 0 i can be expressed as Hence, A(x, x0 ) = hx|A|x Z d4 p ip·(x0 −x) x + x0 0 A(x, x ) = e A ,p . (A3) (2π)4 2 In the following, we outline a number of useful properties of the Weyl transform. . R 4 ˆ the trace Tr[A] ˆ = ˆ • For any operator A, d x hx|A|xi can be expressed as Z 4 4 d xd p ˆ = Tr[A] A(x, p). (A4) (2π)4 ˆ then A† (x, p) is the • If A(x, p) is the Weyl symbol of A, † Weyl symbol of Aˆ . As a corollary, the Weyl symbol of a Hermitian operator is real. ˆ = AˆB, ˆ the corresponding Weyl symbols sat• For any C isfy [48, 49] C(x, p) = A(x, p) ? B(x, p).

(A5)

Here ‘?’ refers to the Moyal product, which is given by ˆ . A(x, p) ? B(x, p) = A(x, p)eiL/2 B(x, p),

and Lˆ is the Janus operator − − → ← − → − . ← Lˆ = ∂p · ∂x − ∂x · ∂p = {·, ·}.

(A6)

(A7)

The arrows indicate the direction in which the derivaˆ = {A, B} is the canonical Poisson tives act, and ALB bracket in the eight-dimensional phase space, namely, ← − → − ← − → − ← − → − ← − → − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ˆ L= 0 0 − + · − · . (A8) ∂p ∂x ∂x0 ∂p0 ∂x ∂p ∂p ∂x ˆ is small, one can use the following Provided that ALB asymptotic expansion of the Moyal product: i A ? B ' A B + {A, B}. 2

(A9)

15 • The Moyal product is associative; i.e., A ? B ? C = (A ? B) ? C = A ? (B ? C).

For any two functions f and g, one has (A10)

• Now we tabulate some Weyl transforms of various operators. (We use a two-sided arrow to show the correspondence with the Weyl transform.) First of all, the Weyl transforms of the identity, position, and momentum operators are given by ˆ 1 ⇔ 1,

x ˆ µ ⇔ xµ ,

pˆµ ⇔ pµ .

f (ˆ x) ⇔ f (x),

g(ˆ p) ⇔ g(p).

(A12)

Similarly, using Eq. (A6), one has pˆµ f (ˆ x) ⇔ pµ f (x) + (i/2)∂µ f (x), f (ˆ x)ˆ pµ ⇔ pµ f (x) − (i/2)∂µ f (x).

(A13) (A14)

(A11)

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