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Variational formulation of covariant eikonal theory for vector waves
Volume 120, number 7
PHYSICS LETTERS A
9 March 1987
VARIATIONAL FORMULATION OF COVARIANT EIKONAL THEORY FOR VECTOR WAVES Allan N. KAUFMAN, Huanchun YE and Yukkei HUI Lawrence BerkeleyLaboratory and Physics Department, University of California, Berkeley, CA 94720, USA Received 21 October 1986; accepted for publication 22 December 1986
The eikonal theory ofwave propagation isdeveloped by means ofa Lorentz-covariant variational principle, involving functions defined on the natural eight-dimensional phase space of rays. The wave field is a four-vector representing the electromagnetic potential, while the medium is represented by an anisotropic, dispersive nonuniform dielectric tensor D’~”(k,x). The eikonal expansion yields, to lowest order, the hamiltonian ray equations, which define the lagrangian manifold k(x), and the wave-action conservation law, which determines the wave-amplitudetransport along the rays. The first-ordercontribution to the variational principle yields a concise expression forthe transport ofthe polarization phase. The symmetry between k-space and x-space allows for a simple implementation ofthe Maslov transform, whichavoids the difficulties of caustic singularities.
Wave propagation in various media is often well represented by the eikonal theory, also known as ray optics, geometrical optics, and WKB [1]. In this paper, we present a variational approach. Leading to some new results, as well as concise expressions for old results. While our immediate motivation is electromagnetic wave propagation in magnetized plasma [2], we expect that our methods are applicable to other media, such as elastic waves [3]. Our emphasis here is on the vector character of the wave field [4], which propagates through an anisotropic medium. We also emphasize the phase-space concept, as a way of avoiding the singularities of caustics. In addition, it is convenient to utilize a covariant formulation [5], treating space and time on an equal footing; this is the key to conciseness. We represent the electromagnetic wave field by the four-potential A,~(x),with A~=(A, —0), and x=(x,t). (We set c=1.) The wave equation for linear dissipationless propagation can be expressed as an integral equation of the form d~x”D~(x’,x”)A~(x”) 0 (1)
f
Here D”~(x’,x”) represents the (in general) anisotropic dispersion tensor of the medium, as a twopoint kernel. We consider it as given; methods for its 0375-960 1/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
derivation are discussed in ref. [6]. The variational equation below requires that it be symmetric: D~(x’,x”) =DIw(x~~, x’); asymmetry represents dissipation, which must be treated by other methods. Nonlinearity can also be included, by allowing the medium to respond “ponderomotively” to the wave [6]. We introduce the quadratic action functional: x”)A~(x”). ~ d4x’J d4x” A,~(x’)D’~’(x’,
J
(2) The requirement that S be stationary with respect to arbitrary variations of the four-potential field is equivalent to eq. (1). It is advantageous to introduce the phase-space concept as early as possible. We define the dispersion tensor as a function on phase space (k, x): D/w(k,x)=$d4s1Yw(x~=x+s/2,x~~x_s/2)
x exp ( ik~s). (3) It is thus the Fourier transform of D~(x’,x”) with respect to the two-point separation s=x’ —x”. The wave four vector k= (k, co), together with x= (x, t), coordinatizes the eight-dimensional phase space (k, x) of the rays. Analogously, we introduce the bilinear Wigner tensor: —
—
327
Volume 120, number 7
A~(k,x)
=J
PHYSICS LETTERS A
where we have discarded oscillating terms exp[2i0(x)], which vanishrapidly upon x-integration in (5).
d4sA~(x+s/2)A~(x—s/2)
xexp(—iks),
(4)
which likewise lives on the phase space. In terms of these, the action functional reads S(A)
9 March 1987
(5). Substituting (10) into (8), and performing the kintegration of (7), we obtain the action functions S( a) = f d 4x L (x) in terms of a lagrangian density: 2(x),
JJD~(k,x)A
1~u(k,x), (5) where JJ=Jf d~xd4k/(2m )4 We see that k-space and x-space enter (5) on an equal footing. At each point (k, x) in this space, we represent the tensor D’~’in terms of its local (orthonormal eigenvectors ea and (real) eigenvalues Da: D~Ul~(k, x)—~Da(k,x)e~(k,x)e~*(k,x),
(6)
(11)
L(x)=~Da(k_—80/ôx,x) IaaI where
aa(x)=e~(k=a0/ax,x)a,j(x)
(12)
is the scalar projection of the amplitude on the polarization direction ea. The variation of the action S is now taken with respect to the amplitude a~(x)and the phase 9(x). The former variation yields
where
Da(k=ô0/c9x, x)aa(x)=0, D~(k,x)e~(k,x) =Da(k, x)e~(k,x)
(13)
for each a. For a non-trivial solution, we require = 0 for one polarization, denoted I, and allow
and
Da
e~*(k,x)e~(k,x) =
aa= 0 for the other polarizations a ~ I. Thus we obtain the eikonal equation
(We assume that the eigenvalues are non-degenerate, for the near degenerate case see ref. [7].) Sub-
D 1(k_ô0/19x, x)=0, with the amplitude expressed as
(14)
stituting (6) into (5), we obtain
a,~(x)=e~(x)a~(x).
(15)
~ J$Da(k,X)A~(k,X),
(7)
a
where A~(k,x) =e~*(k,x)A~(k,x)e~(k,x)
The polarization (8)
is the projection of the Wigner tensor on the local polarization ea( k, x), and Da ( k, x) is the scalar dispersion function. We now assume that the wave field A~(x) can be expressed in eikonal form: A~(x)=a,~(x) exp[i0(x)]+c.c., (9) where the phase 0(x) is real and has slowly varying first-derivative k~(x)= a0(x)/ax~,while the amplitude a,~(x)is complex and is slowly varying. When we substitute (9) into (4), we obtain, to lowest order in the eikonal parameter, 4[ô4(k—ô0Iôx)a~(x)a~(x) A~,(k,x) = (2it) +ô4(k+ä0/8x)a~(x)a~(x)], (10) 328
e~(x)=~(k=ô0Iôx,x)
(16)
can absorb the complex phase of the amplitude vector field a~(x),allowing the scalar amplitude A1 (x) to be real and positive. The eikonal equation (14) is solved by Hamilton’s method. In eight-dimensional phase space, the ray equations are t’ ôD dx = 1 dk1, = t9D1 (17) dt~ akM da 9x,, —
~,
—
~.
These hamiltonian equations yield the family of ray orbits [k(c), x(a)]. With appropriate initial condilions, the rays generate a four-dimensional “lagrangian submanifold” k(x) [8]. The phase 0(x) is then obtained by integrating k( a) along ray x( o). On varying S with respect to the phase 0(x), we obtain the conservation law
Volume 120, number 7
PHYSICS LEUERS A
(18) for the wave-action four-flux J~’(x)[91,defined as =
—
ÔL/8k,, =
—
[8D1( k, x)/t9k~](x) a? (x)
(dx~/da) a? (x).
(19) 4(x) is the familiar
The temporal component of P wave-action density:
9 March 1987
determined by (16) up to a complex phase factor. That is, the replacement e~(k,x)—*e~(k,x) x exp [i~(x)] leaves (6) invariant, so that the polarization phase is not as yet determined. Forthe polarization phase, it is necessary to expand the action functional to first order in the eikonal parameter. After some algebra, and using the zeroorder equations, we find the first order lagrangian to be L’(x) =ia?(x) (ej~de
J(x;
t) =
1/da+~ D’~”{e~, ~*}),
(8D1/&v)a?,
while the spatial part ofJ~(x)is the wave-action flux density (8a/ôk)J. Thus (18) reads 8J(x; t)/ôt=—V
(.Jôw/äk),
—
—
020
,
} represents the canonical Poisson bracket:
ôfôg
t3f Og
c9x~äk,,~ On taking the variation of the action functional with respect to the amplitude, we now obtain the additional equation L’ = 0, which yields the desired polarization phase transport: e*. = ~D~”{e~,~*}. (23) ‘ dc Note that this transport relation lives in phase space; —
i.e., it is not necessaryto project onto x-space. On the other hand, the ray divergence equation (21) definitely refers to x-space. We now proceed to the elimination of caustic singularities. Let us again substitute the Wigner tensor (10) into (8), obtaining the scalar field: 4a?(x) A?(k,x)=(2x) x [ô4(k—00/Ox) +o4(k+ 80/Ox)]. (24) Integrating over an element of phase-space volume
daOx’4Ox~= OX”8X~’ 02D, 020 02D, 020 + Ox’4Ok,. OX’4OX” + Ox”Ok,. Ox’4Ox’4
+ Ok,.OkA
{
82D 1
82D,
where {fg}=
this conservation law reflects the invariance of the action functional under a uniform phase shift 0(x) 0(x) + c. On substituting expression (19) for .P’(x) into the conservation law (18), we obtain the amplitude transport equation: 2D 2D 20(x) d ln a 2 8 1(k,x) t9 1 8 4 Ox~ (20) 1= da ôk~8xIL + Ok,. 8/c Ox’ The first term on the right represents the medium nonuniformity, while the second represents the divergence (or convergence) of a ray bundle. The latter quantity is determined, by its transport relation: d
(22)
020 020 Ox~0x’~ OxPOxV’
(see (7)), we have d4xd4kA?(k,x)/(2x)4=a?(x) d4x. (21)
obtained by differentiating (14) twice with respect to x. This nonlinear equation for the ray divergence leads in general to a singularity in a finite distance, i.e., to a caustic. To avoid that singularity, we may use the Maslov transform [8]. Inserting (15) into (9), we have A,,(x) e~(x) a 1 ( x) exp [i0(x)] + c.c., with 0(x) determined by (17), a, (x) determined by 20), and e,. (x)
(25)
We see that A?( k, x) is the wave density in phase space, while a?(x) is the density in x-space. From (24), we see that A? is supported on the lagrangian manifold k_—k(x) =80(x)/Ox, i.e., k, =00(x)/Ox’, and so on. At a caustic, k(x) becomes double-valued, so that the x-representation ofthe eikonal breaks down. The Maslov procedure [8] is a simple way to avoid the caustic singularity. One Fourier transforms the 329
Volume 120, number 7
PHYSICS LETTERS A
field A,.(x) from (x’, x2, x3, x4) space, to (k1, x2, x3, x4) space, where the (x’, k,) pair is chosen to represent the singular direction in 020/Ox’4Ox”. One can now make the eikonal assumption (9) in the new space, and the whole procedure outlined above carries through in the same way. Space does not permit a fuller discussion here. The Maslov transform is a special case of a more general approach, recently developed by Littlejohn [10—12]. His method involves a continuous transformation of coordinates in phase space, and is based on the wave-packet approximation. The variational method discussed here allows for a treatment ofwave angular momentum [13] transport, and its consequences. This work is in progress. Applications of the present formalism are under way. This research was supported by the Office of Basic Energy Sciences of the U.S. Department of Energy under Contract DE-AC0376SF00098. We have benefited from discussions with B. Boghosian, R. Littlejohn, S. McDonald, S. Omohundro, and P. Similon.
330
9 March 1987
References [111. Bernstein and L. Friedland, Geometric optics in spaceand time-varying plasma, in: Handbook of plasma physics, Vol. 1, eds. R. Sagdeev and M. Rosenbluth (North-Holland, Amsterdam, 1983) p. 367. [2] S. McDonald and A. Kaufman, Phys. Rev. A 32 (1985) 1708. [3] J. Achenbach,A. Gautesen and H. McMaken, Ray methods for waves in elastic solids (Pitman, Boston, 1982). [4] H. Berk and D. Pfirsch, J. Math. Phys. 21(1980) 2054. [5] A. Kaufman andD. HoIm, Phys. Lett. A 105 (1984) 277. [6] S. McDonald, C. Grebogi and A. Kaufman, Phys. Lett. A 111 (1985) 19. [7] L. Friedland, Phys. Fluids 28 (1985) 3260. [8] V. Maslov and M. Fedoriuk, Semi-classical approximation in quantum mechanics (Reidel, Dordrecht, 1981). [9] R. Dewar, Astrophys. J. 174 (1972) 301. [10] R. Littlejohn, Phys. Rep. 138 (1986) 193. [11] R. Littlejohn, Phys. Rev. Lett. 54 (1985) 1742. [12] R. Littlejohn, Phys. Rev. Lett. 56 (1986) 2000. [13] D. Soper, Classical field theory (Wiley, New York, 1976) ch. 9.
PHYSICS LETTERS A
9 March 1987
VARIATIONAL FORMULATION OF COVARIANT EIKONAL THEORY FOR VECTOR WAVES Allan N. KAUFMAN, Huanchun YE and Yukkei HUI Lawrence BerkeleyLaboratory and Physics Department, University of California, Berkeley, CA 94720, USA Received 21 October 1986; accepted for publication 22 December 1986
The eikonal theory ofwave propagation isdeveloped by means ofa Lorentz-covariant variational principle, involving functions defined on the natural eight-dimensional phase space of rays. The wave field is a four-vector representing the electromagnetic potential, while the medium is represented by an anisotropic, dispersive nonuniform dielectric tensor D’~”(k,x). The eikonal expansion yields, to lowest order, the hamiltonian ray equations, which define the lagrangian manifold k(x), and the wave-action conservation law, which determines the wave-amplitudetransport along the rays. The first-ordercontribution to the variational principle yields a concise expression forthe transport ofthe polarization phase. The symmetry between k-space and x-space allows for a simple implementation ofthe Maslov transform, whichavoids the difficulties of caustic singularities.
Wave propagation in various media is often well represented by the eikonal theory, also known as ray optics, geometrical optics, and WKB [1]. In this paper, we present a variational approach. Leading to some new results, as well as concise expressions for old results. While our immediate motivation is electromagnetic wave propagation in magnetized plasma [2], we expect that our methods are applicable to other media, such as elastic waves [3]. Our emphasis here is on the vector character of the wave field [4], which propagates through an anisotropic medium. We also emphasize the phase-space concept, as a way of avoiding the singularities of caustics. In addition, it is convenient to utilize a covariant formulation [5], treating space and time on an equal footing; this is the key to conciseness. We represent the electromagnetic wave field by the four-potential A,~(x),with A~=(A, —0), and x=(x,t). (We set c=1.) The wave equation for linear dissipationless propagation can be expressed as an integral equation of the form d~x”D~(x’,x”)A~(x”) 0 (1)
f
Here D”~(x’,x”) represents the (in general) anisotropic dispersion tensor of the medium, as a twopoint kernel. We consider it as given; methods for its 0375-960 1/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
derivation are discussed in ref. [6]. The variational equation below requires that it be symmetric: D~(x’,x”) =DIw(x~~, x’); asymmetry represents dissipation, which must be treated by other methods. Nonlinearity can also be included, by allowing the medium to respond “ponderomotively” to the wave [6]. We introduce the quadratic action functional: x”)A~(x”). ~ d4x’J d4x” A,~(x’)D’~’(x’,
J
(2) The requirement that S be stationary with respect to arbitrary variations of the four-potential field is equivalent to eq. (1). It is advantageous to introduce the phase-space concept as early as possible. We define the dispersion tensor as a function on phase space (k, x): D/w(k,x)=$d4s1Yw(x~=x+s/2,x~~x_s/2)
x exp ( ik~s). (3) It is thus the Fourier transform of D~(x’,x”) with respect to the two-point separation s=x’ —x”. The wave four vector k= (k, co), together with x= (x, t), coordinatizes the eight-dimensional phase space (k, x) of the rays. Analogously, we introduce the bilinear Wigner tensor: —
—
327
Volume 120, number 7
A~(k,x)
=J
PHYSICS LETTERS A
where we have discarded oscillating terms exp[2i0(x)], which vanishrapidly upon x-integration in (5).
d4sA~(x+s/2)A~(x—s/2)
xexp(—iks),
(4)
which likewise lives on the phase space. In terms of these, the action functional reads S(A)
9 March 1987
(5). Substituting (10) into (8), and performing the kintegration of (7), we obtain the action functions S( a) = f d 4x L (x) in terms of a lagrangian density: 2(x),
JJD~(k,x)A
1~u(k,x), (5) where JJ=Jf d~xd4k/(2m )4 We see that k-space and x-space enter (5) on an equal footing. At each point (k, x) in this space, we represent the tensor D’~’in terms of its local (orthonormal eigenvectors ea and (real) eigenvalues Da: D~Ul~(k, x)—~Da(k,x)e~(k,x)e~*(k,x),
(6)
(11)
L(x)=~Da(k_—80/ôx,x) IaaI where
aa(x)=e~(k=a0/ax,x)a,j(x)
(12)
is the scalar projection of the amplitude on the polarization direction ea. The variation of the action S is now taken with respect to the amplitude a~(x)and the phase 9(x). The former variation yields
where
Da(k=ô0/c9x, x)aa(x)=0, D~(k,x)e~(k,x) =Da(k, x)e~(k,x)
(13)
for each a. For a non-trivial solution, we require = 0 for one polarization, denoted I, and allow
and
Da
e~*(k,x)e~(k,x) =
aa= 0 for the other polarizations a ~ I. Thus we obtain the eikonal equation
(We assume that the eigenvalues are non-degenerate, for the near degenerate case see ref. [7].) Sub-
D 1(k_ô0/19x, x)=0, with the amplitude expressed as
(14)
stituting (6) into (5), we obtain
a,~(x)=e~(x)a~(x).
(15)
~ J$Da(k,X)A~(k,X),
(7)
a
where A~(k,x) =e~*(k,x)A~(k,x)e~(k,x)
The polarization (8)
is the projection of the Wigner tensor on the local polarization ea( k, x), and Da ( k, x) is the scalar dispersion function. We now assume that the wave field A~(x) can be expressed in eikonal form: A~(x)=a,~(x) exp[i0(x)]+c.c., (9) where the phase 0(x) is real and has slowly varying first-derivative k~(x)= a0(x)/ax~,while the amplitude a,~(x)is complex and is slowly varying. When we substitute (9) into (4), we obtain, to lowest order in the eikonal parameter, 4[ô4(k—ô0Iôx)a~(x)a~(x) A~,(k,x) = (2it) +ô4(k+ä0/8x)a~(x)a~(x)], (10) 328
e~(x)=~(k=ô0Iôx,x)
(16)
can absorb the complex phase of the amplitude vector field a~(x),allowing the scalar amplitude A1 (x) to be real and positive. The eikonal equation (14) is solved by Hamilton’s method. In eight-dimensional phase space, the ray equations are t’ ôD dx = 1 dk1, = t9D1 (17) dt~ akM da 9x,, —
~,
—
~.
These hamiltonian equations yield the family of ray orbits [k(c), x(a)]. With appropriate initial condilions, the rays generate a four-dimensional “lagrangian submanifold” k(x) [8]. The phase 0(x) is then obtained by integrating k( a) along ray x( o). On varying S with respect to the phase 0(x), we obtain the conservation law
Volume 120, number 7
PHYSICS LEUERS A
(18) for the wave-action four-flux J~’(x)[91,defined as =
—
ÔL/8k,, =
—
[8D1( k, x)/t9k~](x) a? (x)
(dx~/da) a? (x).
(19) 4(x) is the familiar
The temporal component of P wave-action density:
9 March 1987
determined by (16) up to a complex phase factor. That is, the replacement e~(k,x)—*e~(k,x) x exp [i~(x)] leaves (6) invariant, so that the polarization phase is not as yet determined. Forthe polarization phase, it is necessary to expand the action functional to first order in the eikonal parameter. After some algebra, and using the zeroorder equations, we find the first order lagrangian to be L’(x) =ia?(x) (ej~de
J(x;
t) =
1/da+~ D’~”{e~, ~*}),
(8D1/&v)a?,
while the spatial part ofJ~(x)is the wave-action flux density (8a/ôk)J. Thus (18) reads 8J(x; t)/ôt=—V
(.Jôw/äk),
—
—
020
,
} represents the canonical Poisson bracket:
ôfôg
t3f Og
c9x~äk,,~ On taking the variation of the action functional with respect to the amplitude, we now obtain the additional equation L’ = 0, which yields the desired polarization phase transport: e*. = ~D~”{e~,~*}. (23) ‘ dc Note that this transport relation lives in phase space; —
i.e., it is not necessaryto project onto x-space. On the other hand, the ray divergence equation (21) definitely refers to x-space. We now proceed to the elimination of caustic singularities. Let us again substitute the Wigner tensor (10) into (8), obtaining the scalar field: 4a?(x) A?(k,x)=(2x) x [ô4(k—00/Ox) +o4(k+ 80/Ox)]. (24) Integrating over an element of phase-space volume
daOx’4Ox~= OX”8X~’ 02D, 020 02D, 020 + Ox’4Ok,. OX’4OX” + Ox”Ok,. Ox’4Ox’4
+ Ok,.OkA
{
82D 1
82D,
where {fg}=
this conservation law reflects the invariance of the action functional under a uniform phase shift 0(x) 0(x) + c. On substituting expression (19) for .P’(x) into the conservation law (18), we obtain the amplitude transport equation: 2D 2D 20(x) d ln a 2 8 1(k,x) t9 1 8 4 Ox~ (20) 1= da ôk~8xIL + Ok,. 8/c Ox’ The first term on the right represents the medium nonuniformity, while the second represents the divergence (or convergence) of a ray bundle. The latter quantity is determined, by its transport relation: d
(22)
020 020 Ox~0x’~ OxPOxV’
(see (7)), we have d4xd4kA?(k,x)/(2x)4=a?(x) d4x. (21)
obtained by differentiating (14) twice with respect to x. This nonlinear equation for the ray divergence leads in general to a singularity in a finite distance, i.e., to a caustic. To avoid that singularity, we may use the Maslov transform [8]. Inserting (15) into (9), we have A,,(x) e~(x) a 1 ( x) exp [i0(x)] + c.c., with 0(x) determined by (17), a, (x) determined by 20), and e,. (x)
(25)
We see that A?( k, x) is the wave density in phase space, while a?(x) is the density in x-space. From (24), we see that A? is supported on the lagrangian manifold k_—k(x) =80(x)/Ox, i.e., k, =00(x)/Ox’, and so on. At a caustic, k(x) becomes double-valued, so that the x-representation ofthe eikonal breaks down. The Maslov procedure [8] is a simple way to avoid the caustic singularity. One Fourier transforms the 329
Volume 120, number 7
PHYSICS LETTERS A
field A,.(x) from (x’, x2, x3, x4) space, to (k1, x2, x3, x4) space, where the (x’, k,) pair is chosen to represent the singular direction in 020/Ox’4Ox”. One can now make the eikonal assumption (9) in the new space, and the whole procedure outlined above carries through in the same way. Space does not permit a fuller discussion here. The Maslov transform is a special case of a more general approach, recently developed by Littlejohn [10—12]. His method involves a continuous transformation of coordinates in phase space, and is based on the wave-packet approximation. The variational method discussed here allows for a treatment ofwave angular momentum [13] transport, and its consequences. This work is in progress. Applications of the present formalism are under way. This research was supported by the Office of Basic Energy Sciences of the U.S. Department of Energy under Contract DE-AC0376SF00098. We have benefited from discussions with B. Boghosian, R. Littlejohn, S. McDonald, S. Omohundro, and P. Similon.
330
9 March 1987
References [111. Bernstein and L. Friedland, Geometric optics in spaceand time-varying plasma, in: Handbook of plasma physics, Vol. 1, eds. R. Sagdeev and M. Rosenbluth (North-Holland, Amsterdam, 1983) p. 367. [2] S. McDonald and A. Kaufman, Phys. Rev. A 32 (1985) 1708. [3] J. Achenbach,A. Gautesen and H. McMaken, Ray methods for waves in elastic solids (Pitman, Boston, 1982). [4] H. Berk and D. Pfirsch, J. Math. Phys. 21(1980) 2054. [5] A. Kaufman andD. HoIm, Phys. Lett. A 105 (1984) 277. [6] S. McDonald, C. Grebogi and A. Kaufman, Phys. Lett. A 111 (1985) 19. [7] L. Friedland, Phys. Fluids 28 (1985) 3260. [8] V. Maslov and M. Fedoriuk, Semi-classical approximation in quantum mechanics (Reidel, Dordrecht, 1981). [9] R. Dewar, Astrophys. J. 174 (1972) 301. [10] R. Littlejohn, Phys. Rep. 138 (1986) 193. [11] R. Littlejohn, Phys. Rev. Lett. 54 (1985) 1742. [12] R. Littlejohn, Phys. Rev. Lett. 56 (1986) 2000. [13] D. Soper, Classical field theory (Wiley, New York, 1976) ch. 9.