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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 23 — Nov. 17, 2003
  • pp: 3128–3135
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Exact description of free electromagnetic wave fields in terms of rays

Miguel A. Alonso  »View Author Affiliations


Optics Express, Vol. 11, Issue 23, pp. 3128-3135 (2003)
http://dx.doi.org/10.1364/OE.11.003128


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Abstract

A set of representations are proposed for vector electromagnetic waves in free space (or homogeneous media) of any state of coherence. These representations take the form of a radiance, i.e., the weight of a ray. The intensity, energy density or Poynting vector at any point correspond simply to the sum of the weights of all the rays through that point. This formalism is valid even for fields with components traveling in all directions, but without evanescent components.

© 2003 Optical Society of America

1. Introduction

The objective of this work is to show that a general traveling monochromatic or partially coherent electromagnetic vector field in free space (or homogeneous medium) can be represented exactly in terms of rays. In this representation, a weight is assigned to each ray, such that the integral of these weights for all the rays that go through a point gives exactly a physically meaningful field property, like the electric energy density (or optical intensity), the magnetic energy density, the total energy density, or the Poynting vector. The framework proposed here works for any numerical aperture, even for fields composed of plane waves traveling in all directions. The conservation of the weights along each ray is exact, regardless of wavelength or state of coherence; the only condition is the absence of evanescent wave components. Additionally, it is shown that when the fields are composed of poorly correlated plane waves, the different weight definitions become equivalent and non-negative, and can be associated with the radiance.

2. Electromagnetic waves

Let us start by writing Maxwell’s equations for a field in free space

·E(r,t)=0,
(1a)
·B(r,t)=0,
(1b)
×E(r,t)=Bt(r,t),
(1c)
×B(r,t)=1c2Et(r,t).
(1d)

The electric energy density (or intensity), magnetic energy density, and Poynting vector are defined as

EE(r,t)=ε08πE(r,t)2,
(2a)
EM(r,t)=18πμ0B(r,t)2,
(2b)
S(r,t)=14πμ0Re[E*(r,t)×B(r,t)].
(2c)

These quantities satisfy the continuity equation:

t[EE(r,t)+EM(r,t)]=·S(r,t).
(3)

From this equation, the Poynting vector is interpreted as an energy flux density. Of course, we can add to it the curl of any vector function, without causing Eq. (3) to change. This means that there is ambiguity in the definition of an electromagnetic flux vector, the standard Poynting vector being just one choice.

Let us first assume that the field is monochromatic, so the electric and magnetic vectors can be written as E(r, t)=U(r) exp(-iωt) and B(r, t)=V(r) exp(-iωt), where ω is the angular frequency. Equations (1) then become

·U(r)=0,
(4a)
·V(r)=0,
(4b)
×U(r)=iωV(r),
(4c)
×V(r)=iωc2U(r).
(4d)

and the corresponding physical properties in Eqs. (2) are

EE(r)=ε08πU(r)2,
(5a)
EM(r)=18πμ0V(r)2,
(5b)
S(r)=14πμ0Re[U*(r)×V(r)].
(5c)

When there are no evanescent components, a general solution to Maxwell’s equations (4) for the electric field is a superposition of linearly-polarized plane waves of the form

U(r)=p=±14πφ(u,p)w(u,pπ4)exp(iku·r)dΩ,
(6)

where φ(u, p) is the complex amplitude or angular spectrum of a wave traveling in the direction of the unit vector u and polarized linearly in the direction w(u,pπ4) . Here, the unit vector w(u,θ) is defined to be perpendicular to u and at an angle θ from some reference direction, as shown in Fig. 1. A convention must be chosen for assigning the reference directions for each u. The choice of this convention, however, does not affect the results of this paper. It follows from Eq. (4c) that the magnetic field is then

V(r)=1cp=±14πφ(u,p)w(u,pπ4+π2)exp(iku·r)dΩ,
(7)

where we used the fact that u×w(u,θ)=w(u,θ+π2) .

Fig. 1. The vector w is constrained to the plane perpendicular to the unit vector u specified as its first argument. As θ increases, w rotates around u.

When the field is partially coherent, one must use the cross-spectral density tensor to describe the electric field. This tensor can also be expanded terms of linearly-polarized plane waves:

WE(r1,r2)=U*(r1)U(r2)
=p1,p2=±14πΞ(u1,p1,u2,p2)w(u1,p1π4)w(u2,p2π4)exp[ik(u2·r2u1·r1)]dΩ1dΩ2,
(8)

where Ξ(u 1, p 1, u 2, p 2)=〈φ*(u 1, p 1)φ(u 2, p 2)〉 is the angular correlation function, which can be interpreted as a measure of the coherence between the plane wave components in the u 1 and u 2 directions, with linear polarizations specified by p 1 and p 2, respectively. The angular correlation function is Hermitian and positive definite, so Ξ(u 1, p 1, u 2, p 2)=Ξ*(u 1, p 1, u 2 p 2) and Ξ(u, p, u, p)≥0. The cross-spectral density tensor for the magnetic field results from simply rotating the vectors w in Eq. (8) by π/2, as in Eq. (7).

3. Rays and the radiance

Compared to the electromagnetic description in Section 2, the radiometric (ray-based) description of an optical field is very simple. The main quantity that describes the field is the radiance B(r,u), which is the weight of a ray (a straight line in free space) that goes through a point r and travels in the direction of the unit vector u. The condition satisfied by the radiance is its conservation along rays. That is, its derivative in the u direction vanishes exactly:

u·B(r,u)=0.
(9)

The radiometric analogues of the electromagnetic quantities in Eqs. (5) are the intensity and flux density given by

I(r)=4πB(r,u)dΩ,
(10a)
F(r)=c4πuB(r,u)dΩ.
(10b)

The factor of c is included in Eq. (10b) for dimensional reasons.

The simplicity of this model is usually regarded as a consequence of considering one or several limiting conditions and approximations, like small wavelength, small coherence length, quasi-homogeneous fields, scalar fields, paraxial or metaxial propagation, etc. The goal of what follows is to express the electromagnetic field exactly in terms of rays, i.e. as a radiance which is exactly conserved along rays, and that presents projection properties like the ones presented in Eqs. (10a) and (10b), regardless of wavelength, coherence properties, or numerical aperture.

4. Electric and magnetic radiance analogues

Let us start by trying to write the electric energy density in a form similar to Eq. (10a). By substituting Eq. (6) into Eq. (5a), we get

EE(r)=ε08πU*(r)·U(r)
=ε08πp1,p2=±14πφ*(u1,p1)φ(u2,p2)w(u1,p1π4)·w(u2,p2π4)exp[ik(u2u1)·r]dΩ1dΩ2.
(11)

More generally, for a partially coherent field, this energy density is given by the trace of the electric cross-spectral density tensor

EE(r)=ε08πtr[WE(r,r)]
=ε08πp1,p2=±14πΞ(u1,p1,u2,p2)w(u1,p1π4)·w(u2,p2π4)exp[ik(u2u1)·r]dΩ1dΩ2.
(12)
Fig. 2. Change of variables described in Eq. (13). Here, the unit vector u bisects the unit vectors u 1 and u 2, α is the angle between these two vectors, and w is a unit vector perpendicular to u and coplanar with u, u 1, and u 2.

The key step of what follows is the following change of variables (illustrated in Fig. 2):

u21=ucosα2w(u,θ)sinα2,
(13)

where u is integrated over all directions, 0≤απ, 0≤θ<2π, and dΩ1 dΩ2=sinα dα dΩ. By using these variables and inverting the order of integration and summation, Eq. (12) can be written in a form identical to Eq. (10a):

EE(r)=4πBE(r,u)dΩ,
(14)

where the electric radiance analogue B E is defined as

BE(r,u)=ε08πp1,p2=±10π2πΞ[ucosα2w(u,θ)sinα2,p1,ucosα2+w(u,θ)sinα2,p2]
×w[ucosα2w(u,θ)sinα2,p1π4]·w[ucosα2+w(u,θ)sinα2,p2π4]
×exp[2ikr·w(u,θ)sinα2]sinαdθdα.
(15)

It is trivial to see that, since in this definition r appears exclusively in an inner product with a vector perpendicular to u, the transport equation in Eq. (9) is satisfied exactly:

u·BE(r,u)=0.
(16)

This function therefore satisfies the main properties of the radiance, for any wavelength and state of coherence.

An analogous definition is associated with the magnetic energy density:

BM(r,u)=ε08πp1,p2=±10π2πΞ[ucosα2w(u,θ)sinα2,p1,ucosα2+w(u,θ)sinα2,p2]
×w[ucosα2w(u,θ)sinα2,p1π4+π2]·w[ucosα2+w(u,θ)sinα2,p2π4+π2]
×exp[2ikr·w(u,θ)sinα2]sinαdθdα.
(17)

such that

u·BM(r,u)=0.
(18)

Of course, a definition corresponding to the total energy density would be given by the sum of B E and B M.

5. Radiance analogue for the Poynting vector

Finally, we look for a radiance analogue coupled to the Poynting vector by a relation of the form in Eq. (10b). The substitution of Eqs. (6) and (7) in (5c) gives

S(r)=18πμ0[U*(r)×V(r)V*(r)×U(r)]
=18πμ0cp1,p2=±14πφ*(u1,p1)φ(u2,p2)exp[ik(u2u1)·r]
×[w(u1,p1π4)×w(u2,p2π4+π2)w(u1,p1π4+π2)×w(u2,p2π4)]dΩ1dΩ2.
(19)

By using w(uj,pjπ4+π2)=uj×w(uj,pjπ4) , this expression can be rewritten as

S(r)=SR(r)+SV(r),
(20)

where

SR(r)=18πμ0cp1,p2=±14πφ*(u1,p1)φ(u2,p2)exp[ik(u2u1)·r]
×(u1+u2)[w(u1,p1π4)·w(u2,p2π4)]dΩ1dΩ2,
(21a)
SV(r)=18πμ0cp1,p2=±14πφ*(u1,p1)φ(u2,p2)exp[ik(u2u1)·r]
×{w(u1,p1π4)[u1·w(u2,p2π4)]+w(u2,p2π4)[u2·w(u1,p1π4)]}dΩ1dΩ2.
(21b)

The corresponding expressions for a partially coherent field result, of course, by making the substitution φ*(u 1, p 1)φ(u 2, p 2)→Ξ(u 1, p 1, u 2, p 2). Notice that the integrand of S R (where R stands for radiometric) in Eq. (21a) points in the direction bisecting the two specified plane wave directions. The integrand of S V (where V stands for vector) in Eq. (21b), on the other hand, points in a less intuitive direction: while it is also always perpendicular to u 2-u 1, it is not constrained to the plane that contains u 1 and u 2.

By using the change of variables in Eq. (13), we find that S R can be written in the form of Eq. (10b):

SR(r)=c4πuBR(r,u)dΩ,
(22)

where

BR(r,u)=ε04πp1,p2=±10π2πΞ[ucosα2w(u,θ)sinα2,p1,ucosα2+w(u,θ)sinα2,p2]
×w[ucosα2w(u,θ)sinα2,p1π4]·w[ucosα2+w(u,θ)sinα2,p2π4]
×exp[2ikr·w(u,θ)sinα2]cosα2sinαdθdα.
(23)

The radiometric-like expression for S V, on the other hand, must be written in the form of Eq. (10a), in terms of a vector radiance:

SV(r)=4πBV(r,u)dΩ,
(24)

where

BV(r,u)=ε04πp1,p2=±10π2πΞ[ucosα2w(u,θ)sinα2,p1,ucosα2+w(u,θ)sinα2,p2]
×(w[ucosα2w(u,θ)sinα2,p1π4]{u·w[ucosα2+w(u,θ)sinα2,p2π4]}
+w[ucosα2+w(u,θ)sinα2,p2π4]{u·w[ucosα2w(u,θ)sinα2,p1π4]})
×exp[2ikr·w(u,θ)sinα2]cosα2sinαdθdα.
(25)

Again, it is easy to show that

u·BR(r,u)=0,
(26a)
(u·)BV(r,u)=0,
(26b)

so both radiance analogues are constant along rays.

Luckily, the perhaps unsatisfactory vector radiance analogue in Eq. (25) can be made unnecessary by modifying the definition of the Poynting vector as commented after Eq. (3). Notice that, for a coherent field, Eq. (21b) can be written as

SV(r)=×p1,p2=±14πiφ*(u1,p1)φ(u2,p2)8πkμ0cexp[ik(u2u1)·r]
×[w(u1,p1π4)×w(u2,p2π4)]dΩ1dΩ2
=i8πkμ0×[U*(r)×U(r)]=×{Re[U(r)]×Im[U(r)]}16πkμ0.
(27)

(For a partially coherent field, we simply consider the expected value or ensemble average of these quantities.) Since this component of the Poynting vector is exactly a curl, we can always drop it and use S R alone as a meaningful measure of the local flux. In this case, Eq. (22) tells us that S=S R can be written exactly in the form of Eq. (10b). This flux definition is at least as meaningful as the standard Poynting vector in Eq. (5c). Care must be taken in making such a claim, nevertheless, since one can argue that ∇·S R vanishes too [because we are considering stationary fields, so the left-hand side of Eq. (3) is zero], and could also, in principle, be written as a curl, although the expression is probably not as simple as that in Eq. (27).

6. Radiometric limit

As discussed above, the radiance analogues proposed here satisfy most of the defining properties of the radiance for any wavelength, state of coherence, and direction of propagation. These properties are their exact conservation along rays, and the fact that their angular projections are tied to physically meaningful quantities. The only possible objections to identifying these definitions with the phenomenological radiance are: 1) that there are several analogues instead of just one, and 2) that all these analogues can be negative for some rays. It is easy to show that such conceptual problems disappear when we consider a field where the different plane wave components are poorly correlated, so Ξ(u 1, p 1, u 2, p 2) is insignificant except when the angle α between u 1 and u 2 is small. It then follows that B EB MB R/2≳0 for all r and u. These functions can therefore be identified as the radiance. In fact, even if we insisted on using the full expression for the Poynting vector including S V, it is easy to see that for small α the integrand in Eq. (25) is much smaller than those of Eqs. (15), (17), and (23), due to the inner products of nearly perpendicular unit vectors. This means that, for fields with small angular coherence, S R dominates over S V, and the vector radiance becomes unnecessary.

7. Concluding remarks

The functions proposed in this work are the analogues for vector fields of generalized radiances for scalar fields proposed earlier [7

7. M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A. 18, 910–918 (2001). [CrossRef]

,8

8. M. A. Alonso, “Radiometry and wide-angle wave fields. III. Partial coherence,” J. Opt. Soc. Am. A. 18, 2502–2511 (2001). [CrossRef]

,9

9. One of the angle-impact Wigner functions in Ref. 4 was found independently in: C.J.R. Sheppard and K.G. Larkin, “Wigner function for highly convergent three-dimensional wave fields,” Opt. Lett.26, 968–970 (2001). [CrossRef]

]. The exact conservation along rays of these functions, together with the physical significance of their angular projections, make them potentially useful tools in the description of wave propagation in homogeneous media: this ray-based description is sufficient in applications where the local effects of polarization are not essential and all that is needed is the electric energy density (or optical intensity), total energy density, or Poynting vector. One can, for example, define analogues of the angle-impact Wigner functions in [8

8. M. A. Alonso, “Radiometry and wide-angle wave fields. III. Partial coherence,” J. Opt. Soc. Am. A. 18, 2502–2511 (2001). [CrossRef]

]. With these, numerically efficient computations of the propagation of partially coherent fields can be performed. It was shown elsewhere [10

10. L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101–112 (2002). [CrossRef]

] for the scalar case that this type of scheme can reduce the computation time considerably. In cases when the local polarization is important, it might be possible to define angle-impact Wigner functions whose arguments also involve polarization.

In the current manuscript, we chose to expand a general monochromatic traveling field in terms of plane waves with linear polarization. As mentioned earlier, the choice of a convention for the reference polarization corresponding to different directions of propagation, although possibly problematic at points, does not affect the validity of the results found here. If circular polarization were used instead, the problem would reduce to choosing a reference phase for each direction of propagation, and this could bring simplifications in the applications for propagation problems.

The functions defined here are valid for homogeneous media away from sources. Several generalizations could be considered, however, like the radiance analogues for homogeneous anisotropic or active media (which also support plane waves). In fact, the determination of the boundary conditions (at least as an asymptotic approximation) that these functions satisfy at the interface between two homogeneous media would allow us to use this framework for modeling the propagation of partially-coherent vector fields through optical systems of arbitrary numerical aperture. Such generalizations are the subject of further research, and are beyond the scope of this paper.

Acknowledgments

I would like to acknowledge the support of The Institute of Optics of the University of Rochester, as well as of the Dirección General de Apoyo al Personal Académico of the Universidad Nacional Autónoma de México (DGAPA-UNAM) through the Grant IN112300 Optica Matemática. Thanks also go to the Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, where part of this work was carried out, and to an anonymous reviewer who brought Refs. [5

5. E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976). [CrossRef]

] and [6

6. M.S. Zubairy and E. Wolf, “Exact equations for radiative transfer of energy and momentum in free electromagnetic fields,” Opt. Commun. 20, 321–324 (1977). [CrossRef]

] to my attention.

References and links

1.

A.T. Friberg (volume editor), Selected Papers on Coherence and Radiometry (SPIE Optical Engineering Press, Milestone Series vol. MS69, Bellingham, 1993).

2.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.

3.

L.A. Apresyan and Yu. A Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Sov. Phys. Usp. 27, 301–313 (1984). [CrossRef]

4.

Yu. A. Kravtsov and L.A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. Vol. XXXVI (North Holland, New York, 1996) pp. 179–244.

5.

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976). [CrossRef]

6.

M.S. Zubairy and E. Wolf, “Exact equations for radiative transfer of energy and momentum in free electromagnetic fields,” Opt. Commun. 20, 321–324 (1977). [CrossRef]

7.

M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A. 18, 910–918 (2001). [CrossRef]

8.

M. A. Alonso, “Radiometry and wide-angle wave fields. III. Partial coherence,” J. Opt. Soc. Am. A. 18, 2502–2511 (2001). [CrossRef]

9.

One of the angle-impact Wigner functions in Ref. 4 was found independently in: C.J.R. Sheppard and K.G. Larkin, “Wigner function for highly convergent three-dimensional wave fields,” Opt. Lett.26, 968–970 (2001). [CrossRef]

10.

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101–112 (2002). [CrossRef]

OCIS Codes
(030.5620) Coherence and statistical optics : Radiative transfer
(080.0080) Geometric optics : Geometric optics
(260.2110) Physical optics : Electromagnetic optics
(350.7420) Other areas of optics : Waves

ToC Category:
Research Papers

History
Original Manuscript: October 15, 2003
Revised Manuscript: November 7, 2003
Published: November 17, 2003

Citation
Miguel Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128-3135 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3128


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References

  1. A.T. Friberg (volume editor), Selected Papers on Coherence and Radiometry (SPIE Optical Engineering Press, Milestone Series vol. MS69, Bellingham, 1993).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287�??307.
  3. L.A. Apresyan and Yu. A Kravtsov, �??Photometry and coherence: wave aspects of the theory of radiation transport,�?? Sov. Phys. Usp. 27, 301-313 (1984). [CrossRef]
  4. Yu. A. Kravtsov and L.A. Apresyan, �??Radiative transfer: new aspects of the old theory,�?? in Progress in Optics, E. Wolf, ed. Vol. XXXVI (North Holland, New York, 1996) pp. 179-244.
  5. E. Wolf, �??New theory of radiative energy transfer in free electromagnetic fields,�?? Phys. Rev. D 13, 869-886 (1976). [CrossRef]
  6. M.S. Zubairy and E. Wolf, �??Exact equations for radiative transfer of energy and momentum in free electromagnetic fields,�?? Opt. Commun. 20, 321-324 (1977). [CrossRef]
  7. M. A. Alonso, "Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions," J. Opt. Soc. Am. A. 18, 910-918 (2001). [CrossRef]
  8. M. A. Alonso, "Radiometry and wide-angle wave fields. III. Partial coherence,�?? J. Opt. Soc. Am. A. 18, 2502-2511 (2001) [CrossRef]
  9. One of the angle-impact Wigner functions in Ref. 4 was found independently in: C.J.R. Sheppard and K.G. Larkin, �??Wigner function for highly convergent three-dimensional wave fields,�?? Opt. Lett. 26, 968-970 (2001). [CrossRef]
  10. L. E. Vicent and M. A. Alonso, "Generalized radiometry as a tool for the propagation of partially coherent fields,�?? Opt. Commun. 207, 101-112 (2002). [CrossRef]

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