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

Full Article  |  PDF Article

References

  • View by:
  • |

  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]

J. Opt. Soc. Am. A.

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]

Opt. Commun.

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]

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]

Opt. Lett.

Phys. Rev. D

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

Progress in Optics

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.

Sov. Phys. Usp.

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]

Other

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.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
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.

Fig. 2.
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.

Equations (58)

Equations on this page are rendered with MathJax. Learn more.

· E ( r , t ) = 0 ,
· B ( r , t ) = 0 ,
× E ( r , t ) = B t ( r , t ) ,
× B ( r , t ) = 1 c 2 E t ( r , t ) .
E E ( r , t ) = ε 0 8 π E ( r , t ) 2 ,
E M ( r , t ) = 1 8 π μ 0 B ( r , t ) 2 ,
S ( r , t ) = 1 4 π μ 0 Re [ E * ( r , t ) × B ( r , t ) ] .
t [ E E ( r , t ) + E M ( r , t ) ] = · S ( r , t ) .
· U ( r ) = 0 ,
· V ( r ) = 0 ,
× U ( r ) = i ω V ( r ) ,
× V ( r ) = i ω c 2 U ( r ) .
E E ( r ) = ε 0 8 π U ( r ) 2 ,
E M ( r ) = 1 8 π μ 0 V ( r ) 2 ,
S ( r ) = 1 4 π μ 0 Re [ U * ( r ) × V ( r ) ] .
U ( r ) = p = ± 1 4 π φ ( u , p ) w ( u , p π 4 ) exp ( i k u · r ) d Ω ,
V ( r ) = 1 c p = ± 1 4 π φ ( u , p ) w ( u , p π 4 + π 2 ) exp ( i k u · r ) d Ω ,
W E ( r 1 , r 2 ) = U * ( r 1 ) U ( r 2 )
= p 1 , p 2 = ± 1 4 π Ξ ( u 1 , p 1 , u 2 , p 2 ) w ( u 1 , p 1 π 4 ) w ( u 2 , p 2 π 4 ) exp [ ik ( u 2 · r 2 u 1 · r 1 ) ] d Ω 1 d Ω 2 ,
u · B ( r , u ) = 0 .
I ( r ) = 4 π B ( r , u ) d Ω ,
F ( r ) = c 4 π u B ( r , u ) d Ω .
E E ( r ) = ε 0 8 π U * ( r ) · U ( r )
= ε 0 8 π p 1 , p 2 = ± 1 4 π φ * ( u 1 , p 1 ) φ ( u 2 , p 2 ) w ( u 1 , p 1 π 4 ) · w ( u 2 , p 2 π 4 ) exp [ ik ( u 2 u 1 ) · r ] d Ω 1 d Ω 2 .
E E ( r ) = ε 0 8 π tr [ W E ( r , r ) ]
= ε 0 8 π p 1 , p 2 = ± 1 4 π Ξ ( u 1 , p 1 , u 2 , p 2 ) w ( u 1 , p 1 π 4 ) · w ( u 2 , p 2 π 4 ) exp [ ik ( u 2 u 1 ) · r ] d Ω 1 d Ω 2 .
u 2 1 = u cos α 2 w ( u , θ ) sin α 2 ,
E E ( r ) = 4 π B E ( r , u ) d Ω ,
B E ( r , u ) = ε 0 8 π p 1 , p 2 = ± 1 0 π 2 π Ξ [ u cos α 2 w ( u , θ ) sin α 2 , p 1 , u cos α 2 + w ( u , θ ) sin α 2 , p 2 ]
× w [ u cos α 2 w ( u , θ ) sin α 2 , p 1 π 4 ] · w [ u cos α 2 + w ( u , θ ) sin α 2 , p 2 π 4 ]
× exp [ 2 ik r · w ( u , θ ) sin α 2 ] sin α d θ d α .
u · B E ( r , u ) = 0 .
B M ( r , u ) = ε 0 8 π p 1 , p 2 = ± 1 0 π 2 π Ξ [ u cos α 2 w ( u , θ ) sin α 2 , p 1 , u cos α 2 + w ( u , θ ) sin α 2 , p 2 ]
× w [ u cos α 2 w ( u , θ ) sin α 2 , p 1 π 4 + π 2 ] · w [ u cos α 2 + w ( u , θ ) sin α 2 , p 2 π 4 + π 2 ]
× exp [ 2 ik r · w ( u , θ ) sin α 2 ] sin α d θ d α .
u · B M ( r , u ) = 0 .
S ( r ) = 1 8 π μ 0 [ U * ( r ) × V ( r ) V * ( r ) × U ( r ) ]
= 1 8 π μ 0 c p 1 , p 2 = ± 1 4 π φ * ( u 1 , p 1 ) φ ( u 2 , p 2 ) exp [ ik ( u 2 u 1 ) · r ]
× [ w ( u 1 , p 1 π 4 ) × w ( u 2 , p 2 π 4 + π 2 ) w ( u 1 , p 1 π 4 + π 2 ) × w ( u 2 , p 2 π 4 ) ] d Ω 1 d Ω 2 .
S ( r ) = S R ( r ) + S V ( r ) ,
S R ( r ) = 1 8 π μ 0 c p 1 , p 2 = ± 1 4 π φ * ( u 1 , p 1 ) φ ( u 2 , p 2 ) exp [ ik ( u 2 u 1 ) · r ]
× ( u 1 + u 2 ) [ w ( u 1 , p 1 π 4 ) · w ( u 2 , p 2 π 4 ) ] d Ω 1 d Ω 2 ,
S V ( r ) = 1 8 π μ 0 c p 1 , p 2 = ± 1 4 π φ * ( u 1 , p 1 ) φ ( u 2 , p 2 ) exp [ ik ( u 2 u 1 ) · r ]
× { w ( u 1 , p 1 π 4 ) [ u 1 · w ( u 2 , p 2 π 4 ) ] + w ( u 2 , p 2 π 4 ) [ u 2 · w ( u 1 , p 1 π 4 ) ] } d Ω 1 d Ω 2 .
S R ( r ) = c 4 π u B R ( r , u ) d Ω ,
B R ( r , u ) = ε 0 4 π p 1 , p 2 = ± 1 0 π 2 π Ξ [ u cos α 2 w ( u , θ ) sin α 2 , p 1 , u cos α 2 + w ( u , θ ) sin α 2 , p 2 ]
× w [ u cos α 2 w ( u , θ ) sin α 2 , p 1 π 4 ] · w [ u cos α 2 + w ( u , θ ) sin α 2 , p 2 π 4 ]
× exp [ 2 ik r · w ( u , θ ) sin α 2 ] cos α 2 sin α d θ d α .
S V ( r ) = 4 π B V ( r , u ) d Ω ,
B V ( r , u ) = ε 0 4 π p 1 , p 2 = ± 1 0 π 2 π Ξ [ u cos α 2 w ( u , θ ) sin α 2 , p 1 , u cos α 2 + w ( u , θ ) sin α 2 , p 2 ]
× ( w [ u cos α 2 w ( u , θ ) sin α 2 , p 1 π 4 ] { u · w [ u cos α 2 + w ( u , θ ) sin α 2 , p 2 π 4 ] }
+ w [ u cos α 2 + w ( u , θ ) sin α 2 , p 2 π 4 ] { u · w [ u cos α 2 w ( u , θ ) sin α 2 , p 1 π 4 ] } )
× exp [ 2 ik r · w ( u , θ ) sin α 2 ] cos α 2 sin α d θ d α .
u · B R ( r , u ) = 0 ,
( u · ) B V ( r , u ) = 0 ,
S V ( r ) = × p 1 , p 2 = ± 1 4 π i φ * ( u 1 , p 1 ) φ ( u 2 , p 2 ) 8 π k μ 0 c exp [ ik ( u 2 u 1 ) · r ]
× [ w ( u 1 , p 1 π 4 ) × w ( u 2 , p 2 π 4 ) ] d Ω 1 d Ω 2
= i 8 π k μ 0 × [ U * ( r ) × U ( r ) ] = × { Re [ U ( r ) ] × Im [ U ( r ) ] } 16 π k μ 0 .

Metrics