## Exact description of free electromagnetic wave fields in terms of rays

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

## 2. Electromagnetic waves

**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

*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**,

*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

**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

*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:

*c*is included in Eq. (10b) for dimensional reasons.

## 4. Electric and magnetic radiance analogues

**u**is integrated over all directions, 0≤

*α*≤

*π*, 0≤

*θ*<2

*π*, and

*d*Ω

_{1}

*d*Ω

_{2}=sin

*α dα d*Ω

*dθ*. By using these variables and inverting the order of integration and summation, Eq. (12) can be written in a form identical to Eq. (10a):

*B*

_{E}is defined as

**r**appears exclusively in an inner product with a vector perpendicular to

**u**, the transport equation in Eq. (9) is satisfied exactly:

*B*

_{E}and

*B*

_{M}.

## 5. Radiance analogue for the Poynting vector

*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}.

**S**

_{R}can be written in the form of Eq. (10b):

**S**

_{V}, on the other hand, must be written in the form of Eq. (10a), in terms of a vector radiance:

**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

## 7. Concluding remarks

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]

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

## Acknowledgments

*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]

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]

## References and links

1. | A.T. Friberg (volume editor), |

2. | L. Mandel and E. Wolf, |

3. | L.A. Apresyan and Yu. A Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Sov. Phys. Usp. |

4. | Yu. A. Kravtsov and L.A. Apresyan, “Radiative transfer: new aspects of the old theory,” in |

5. | E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D |

6. | M.S. Zubairy and E. Wolf, “Exact equations for radiative transfer of energy and momentum in free electromagnetic fields,” Opt. Commun. |

7. | M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A. |

8. | M. A. Alonso, “Radiometry and wide-angle wave fields. III. Partial coherence,” J. Opt. Soc. Am. A. |

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. |

**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

- A.T. Friberg (volume editor), Selected Papers on Coherence and Radiometry (SPIE Optical Engineering Press, Milestone Series vol. MS69, Bellingham, 1993).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287�??307.
- 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]
- 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.
- E. Wolf, �??New theory of radiative energy transfer in free electromagnetic fields,�?? Phys. Rev. D 13, 869-886 (1976). [CrossRef]
- 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]
- 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]
- M. A. Alonso, "Radiometry and wide-angle wave fields. III. Partial coherence,�?? J. Opt. Soc. Am. A. 18, 2502-2511 (2001) [CrossRef]
- 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]
- 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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