# Derivation of the vectorial wave equation from a variational point of view

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750 J. Opt. Soc. Am. A/Vol. 8, No. 5/May 1991

Derivation of the vectorial wave equation from a variationalpoint of view

Thomas Martini JorgensenRiso National Laboratory, DK-4000 Roskilde, Denmark

Received May 2, 1990; accepted January 4, 1991

The electromagnetic field in a linear, inhomogeneous, isotropic, and nonmagnetic dielectric is considered. It isshown that the vectorial wave equation for the electric field can apparently be derived on the basis of a principle ofminimum divergence of the field. It is argued that this principle is preferable to the recently suggested method ofobtaining the scalar wave equation by a principle of minimum Fisher information.

1. INTRODUCTIONThe vectorial wave equation' for the electric field, which isthe basis for most classical optics, is normally derived fromMaxwell's equations. This is done by combining the twoequations involving the curl of the electric vector field E andthe curl of the magnetic vector field H. It follows that, withthis procedure, the magnetic field has to be introduced whenderiving the wave equation for the electric field.

The main purpose of the present paper is to show that,when considering a linear, inhomogeneous, isotropic, andnonmagnetic dielectric, the vectorial wave equation describ-ing the electric wave field can apparently be derived on thebasis of a simple variational principle. The principle is thatthe wave field obeys minimum divergence in the averagesense under the appropriate physical constraint. It appearsthat in this way the wave equation can be derived withoutintroducing the magnetic field.

A field that has zero divergence is said to be transverse.Thus the principle shows that the electric wave field will beas transverse as possible, i.e., as is permitted by the con-straint.

The constraint to be used is based on a Lorentz invariantexpression connecting the energy and the momentum foreach of the photons in the field. Because the energy and themomentum of each photon are closely related to the fre-quencies and wavelengths of the corresponding field, respec-tively, the constraint also implies a relation between thespatial and the temporal frequencies of the electric wavefield.

The mathematical formulation of the variational principleis presented in Section 2. In Section 3 it will be shown howthe solution of this variational problem leads to the vectorialwave equation.

Recently it has been suggested that the scalar approxima-tion of the vectorial wave equation together with Schr6ding-er's equation can be established on the basis of a principle ofminimum Fisher information. 2 4 With respect to such aprinciple, nature should produce as great an uncertainty aspossible under the given physical constraints. However, inSection 4 it is argued that the present suggestion is moreappropriate than the use of minimum Fisher information.

2. FORMULATION OF THE VARIATIONALPROBLEMA volume V characterized by a refractive-index distributionn(x, y, z) is considered. This distribution implies a physicalconstraint on the electric vector field E(x, y, z, t) in thevolume outside which the field is assumed to be zero. Thequestion is now how this constraint can be formulated.First we note the following Lorentz invariant expression,which is valid for a particle in free space 5 :

&2 - 2 2 2

2 p mOCO ,Co(1)

where mO is the rest mass, co the velocity of light in freespace, p the momentum vector, and 6 the total energy of theparticle. The particle to be considered in the present case isa photon. A single nonlocalizable photon is represented bya homogeneous plane wave, that is, one whose equiampli-tude and equiphase planes coincide. For such a photon, mO= 0, 6 = hco, and p = hk, where w is the angular frequency, kis the wave vector, and h = h/2ir with Planck's constantdenoted by h. The following constraint then applies:

h 2W2 h2k2-2kY2 = 2 k W2 + 2) 2 2_ 2__ _ _ _ _ 2

(2)where k, k, and kz are the x, y, and z components of thewave vector k. Equation (2) requires a phase velocity that isequal to the prescribed value of the light velocity. Equation(2) is known as the dispersion relation for vacuum.

Considering the electric field in the volume V, where thelight velocity is permitted to be space dependent, we mustanalogously require that the phase velocities of the planewaves constituting the field be in accordance with the refrac-tive-index distribution. Since there will in general exist acontinuum of spatial and temporal frequencies, we mustincorporate the weight with which the different frequenciesenter the field. The weight functions are given by the fourpower spectral density functions.(three spatial and one tem-poral).

Because a given k vector is related to all space, it will

0740-3232/91/050750-05$05.00 1991 Optical Society of America

Thomas Martini Jorgensen

Vol. 8, No. 5/May 1991/J. Opt. Soc. Am. A 751

generally make no sense claiming that its k2 value shouldequal a specific value of w2n2(x, y, Z)/Co 2 , since this value is ingeneral dependent on the spatial coordinates. To obtain auseful constraint in the case of an inhomogeneous medium,we therefore have to replace the four terms in Eq. (2) withweighted sums, in this way demanding that Eq. (2) be ful-filled in an average sense. With the scalar component de-noted by E, we get

n2(x, y, Z) ddxdydz-f kx 2IE I2dkxdydzdt

- | k 2 E1y2 dkydxdzdt - J kz 2 EzI 2 dkzdxdydt = 0, (3)where Et(cw, x, y, z), Bx(kx, y, z, t), Ey(ky, x, z, t), and Ez(kz x,y, t) denote the Fourier transforms of the scalar field E(x, y,z, t) with respect to t, x, y, and z, respectively. By inverseFourier transformations and the use of Parseval's theorem,Eq. (3) can be reformulated in the following form:JI n2(x, yz) [aE(x, y, z, t)]2 [aE(x, y, z, t)12

f CO2 I at I L ax I

_ [aE(x, y, z, t)]2 _ [OE(x y, Z t) ]2dxdydzdt = 0. (4)

The integration is performed over the volume and time peri-od with E 0. In Eq. (4) it is assumed that E(x, y, z, t) is areal function. Otherwise the square of the four derivativesshould be replaced with products of the derivatives and theappropriate complex conjugates. The assumption is actual-ly no limitation because a physical solution must be real. Ifthe scalar components were permitted to be complex, thesymmetry of both Eq. (4) and the variational principle pre-sented below would simply imply that both the real compo-nent and the imaginary component of the vector field Esatisfy the vectorial wave equation.

It is emphasized that stationary fields, corresponding toCoulomb forces, are not included in the present analysis.This is because no photons are related to such fields.

Equation (3) above is based on the assumption that thevelocity of a photon in a volume element with the refractiveindex n1 is given by co/ni. However, to incorporate thepossibility of evanescent waves,6 we must in general permitthe phase velocities of the plane waves to be smaller, accord-ing to the following consideration. Imagine a plane wavewith frequency w and k vector k1 propagating from a semi-infinite layer with index n1 toward another semi-infinitelayer with index n2 (n1 > n2). The demand for continuity ofthe wave fronts along the boundary layer implies that thetemporal frequency must be the same in both layers and thatthe tangential k component is the same on both sides of thelayer. If the angle of incidence exceeds the critical angle(corresponding to total reflection), Iktangl will exceed wn 2/co.Therefore the superposition of plane waves constituting thefield in the layer with index n2 is forced to have phasevelocities smaller than co/n2. It follows that the left-handside of Eq. (4) should be permitted to be less than zero.

Considering the vector field E(x, y, z, t) = [E(')(x, y, z, t),E(2)(X, y, z, t), E(3)(x, y, z, t)], we find that each of its compo-nents must satisfy the inequality corresponding to Eq. (4).Thus the field E must comply with the following constraint:

y , ) E a 2 - a ( 2J I i [ at] i [ a x i )] 2[F,1 a dxdydzdt _< 0,Lay - L aZ

(5)

where i takes the values 1, 2, and 3.In Section 3 it will be shown that E with great probability

obeys the following variational principle (remember that Edescribes the wave field alone, i.e., not the stationary partarising from the charges in the considered volume):

In a linear, inhomogeneous, isotropic, and nonmagneticdielectric, f(V * E)2dxdydzdt is minimum under theconstraint given by relation (5), where the integration isperformed over the volume and time period with E ;4 0.

3. SOLVING THE VARIATIONAL PROBLEMA necessary condition for the solution to a variational prob-lem is that the corresponding Euler equation7 be satisfied.When a variational problem is to be solved with respect toequality constraints, Lagrange's method of multipliers7 isused. The solution to Euler's equation then depends on theLagrange multipliers. The corresponding degrees of free-dom make it possible to adjust the solution so that theconstraints can be fulfilled.

In the present case we note that the constraints are givenby three inequalities. However, each inequality can simplybe interpreted as a continuum of possible equality con-straints, which all are legal choices. Among these possibili-ties the particular equality constraint that minimizesS (V - E)2dxdydzdt is selected. That is, having solved theEuler equation, we choose the particular values of the La-grange multipliers (among their legal values) that minimizethe square divergence in the average sense.

With the definition

al) aE, ) aE(l ) ME(1 ) aE(3)1(Vax ay a ' at at at

n 2 (X, y, Z) aE~)12 [aE) 1(V -E)' + 2 I _' i1 I Co at J ax

E 1 1 - E 121ay [ a z Ij|(6)

where Ai are Lagrange multipliers, the Euler equations be-come

JEW - ax KE ax] -y [KEU)l/ay] - a KEaPlaz]

-dt [aKEat I = 0, (7)

where the superscript (j) takes the values j = 1, 2, 3 and, e.g.,JaE(a)Iat denotes the partial derivative of J with respect toaE(')/at. After a few straightforward calculations, the fol-lowing equations are obtained from Eqs. (6) and (7):

Thomas Martini Jorgensen

752 J. Opt. Soc. Am. A/Vol. 8, No. 5/May 1991

d -I + +E(2) I(3)ax Lax ay az J

a2E(l) + O2E1 ) + 2E(l) _ n2 (X, y, Z) 2E(1)-02 aY2 0z2 c 2 t2 (8a)

d9 + aE(2+ IE lay Lax ay Oz J

[O2E(2 ) O2 (2) O2E( 2 ) n 2 (x, y, z) 2 E(2)1A2 2 + Oy + 2z2 - c02 8bt

a [E(l) E(2 ) +E (3)z L ax ay Oz J

= [ 2E(3) 2E 3) 2E(3)_ n2(x, y, z) a2 E(3)t3[O 2 + Oy2 + az2 - c 2 Ot2 J (8

Since the medium is considered to be isotropic, the threecomponents of the E vector must enter into the resultingequations in exactly the same way. We therefore concludethat l = 2 = 3 . Equations (8a)-(8c) can now bewritten as

F2n(x, y, Z)12 O2E1V(V E) = H{VE-[ C a9t2 (9)

(In the above analysis it has tacitly been assumed that theconsidered medium is nondispersive. However, dispersioncan be included by considering the case of monochromaticlight, which is no limitation since an electric field can alwaysbe written as a continuous superposition of monochromaticwaves. However, dispersion can be incorporated correctlyonly by proper inclusion of causality. But this is beyond thescope of the present paper.)

If A can be shown always to equal one, Eq. (9) is recognizedas the vectorial wave equation. The procedure of finding ,Ahas been outlined above just before Eq. (6)]. The con-straints given by relation (5) imply that A must be positive.This is shown in Appendix A. It may, however, at first sightbe hard to believe that, no matter what the boundary andinitial conditions should be and for all possible distributionsof the refractive index, u would appear always to equal one.Although we cannot prove that this is actually the case, weare able to show that it is likely that 1A would always take thisvalue. To show that, we consider (with no loss of generality)the case of a monochromatic field with angular frequency w.Equation (9) then becomes

V(V E)= 4V2E + [n(xY.Z) ojE}. (10)

From this equation, we obtainCO2

E = 2[V(V E) -,OV2E]. (11)n 2co2A

By taking the divergence on both sides of Eq. (11), we arriveat the following expression:

V E= c V [V(V E) V2E] -E (), (12)nfXM nf2

From Eq. (12), we get

(V E)2 dxdydzdt = 4 V [V(V E)-sV 2 Elf 2

+ [E. V(n2)] 2 2cO2 [E V(n 2 )]

-[V(V E) - MV2E])dxdydzdt. (13)

Noting the identity V [V(V -E) - V 2 E] = V * [V X (V X E)]= 0 (the divergence of a rotor is always zero), we see that, =1 will always make the first and the last term on the right-hand side of Eq. (13) equal to zero, that is, independent ofboundary and initial conditions and independent of the re-fractive-index distribution. It is therefore likely that thevalue of A that minimizes the square of the divergence in theaverage sense always takes the value one.

As a result of the considerations above let us now assumethat,4 = 1 is always the optimum choice for the Lagrangemultiplier (optimum in the sense that every other choice forA would make the average value of the square divergencelarger). We then still have to know whether the solution toEq. (9) constrained by the equality corresponding to M = 1furnishes a minimum, a maximum, or a point of inflection.In principle it is therefore necessary to consider the second'variation. However, this would require cumbersome calcu-lations in the present case because the variational probleminvolves four independent and three dependent variables.Instead, the special case of a bounded medium with constantpermittivity is considered. Then a superposition of planewaves, polarized orthogonally to their k vectors, which com-plies with the constraint given in relation (5) and which isalso a solution to Eq. (9), can be found. In this case thesquare value of the divergence of the field is zero, and this iscertainly a minimum value. It is therefore tempting tobelieve that the solutions to Eq. (9) will in general furnish aminimum.

We note that Eq. (9) is derived without using Maxwell'sequations and even without introducing the magnetic field.In this connection it should be emphasized that the mediumis assumed to be nonmagnetic.

It is relevant to discuss the variational principle in connec-tion with Maxwell's equations. First a homogeneous andisotropic medium is considered. In this case the use of theCoulomb gauge makes it possible to divide Maxwell's equa-tions into two parts8 : one part that describes the longitudi-nal field EL (V X EL = 0) and another that describes thetransverse field ET (V ET = 0). The longitudinal field ischaracterized as stationary since it originates from thecharges in the considered volume, whereas the transversepart describes electromagnetic waves. It is noted that thedivergence of the electric wave field in this case takes theminimum possible value.

In the case of an inhomogeneous but isotropic medium it isnot possible to divide the equations in a similar manner.Therefore a part of the electric wave field will be longitudi-nal. However, if the variational principle above applies, itshows that the electric wave field will be as transverse aspossible, since the divergence will be as small as possible.

Thomas Martini Jorgensen

Vol. 8, No. 5/May 1991/J....

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