## Properties of quasi-homogeneous isotropic electromagnetic sources |

Optics Express, Vol. 20, Issue 22, pp. 24910-24917 (2012)

http://dx.doi.org/10.1364/OE.20.024910

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### Abstract

We study the properties of Quasi-Homogeneous Isotropic Electromagnetic (QuHIEM) Sources, a model for partially-coherent secondary light sources beyond the scalar and paraxial approximations. Our results include polarization properties in the far zone and the realizability condition. We demonstrate these results for sources with a degree of coherence described by Gaussians.

© 2012 OSA

## 1. Introduction

1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. **58**, 1256–1259 (1968). [CrossRef]

*quasi-homogeneous*model, in which the spatial coherence properties are assumed to be independent of the location with the source. In the scalar approximation, the cross-spectral density function at angular frequency

*ω*of the source factorizes as follows (see ref. [4], section 5.3.2, pp.90–95) Here

*ρ*, (

_{i}*i*= 1, 2) are the two-dimensional position vectors of two points within the source plane of a secondary source. Using this model, the radiant intensity of the field in some far-zone point with position vector

**r**=

*r*

**s**(defined with respect to a suitable origin within the source) is given by: where

*k*=

*ω*/

*c*,

*c*being the speed of light,

**s**

_{⊥}is the projection of the unit vector

**s**onto the source plane (

**s**

_{⊥}is treated as a two-dimensional vector), and is the two-dimensional Fourier transform of the spectral degree of coherence. Similarly

*S*̃

^{(0)}(

**f**,

*ω*) is the two-dimensional Fourier transform of the source power spectrum

*S*

^{(0)}(

*ρ*,

*ω*).

6. W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer. **65**, 1067–1071 (1975). [CrossRef]

*μ*

^{(0)}(

*ρ*,

*ω*) = sin(

*kρ*)/

*kρ*. Another important example is the condition for spectral invariance of the radiated field: If the spectral degree of coherence obeys the

*scaling law*,

*μ*

^{(0)}(

*ρ*,

*ω*) =

*h*(

*ωρ*/

*c*), then the radiated field will have the same power spectrum as the source [7

7. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. **56**, 1370–1372 (1986). [CrossRef] [PubMed]

*Wolf effect*, i.e. correlation-induced spectral changes [8

8. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature **326**, 363–365 (1987). [CrossRef]

9. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. **59**, 771–818 (1996). [CrossRef]

## 2. Quasi-homogeneous isotropic electromagnetic sources

*quasi-homogeneous*,

*isotropic electro-magnetic*(QuHIEM) model source, in which the coherence properties of the source, in addition to obeying the spatial invariance inherent in the quasi-homogeneous model, is also isotropic, in the sense that the coherence properties are independent of the orientation of the axes in the source plane. For such a source we have the following cross-spectral density matrix (see ref. [4], Section 9.1): where the isotropic spectral degree of coherence is given by (see ref. [16]): Thus the assumption of isotropy means that the correlation properties of such sources are specified entirely by the two real-valued scalar functions

*A*(

*kρ*) and

*B*(

*kρ*) (rather than the three real and three complex functions needed for more general sources). We have assumed the spectral degree of coherence obeys the scaling law [7

7. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. **56**, 1370–1372 (1986). [CrossRef] [PubMed]

*globally incoherent*, i.e. the coherence area is very small compared to the overall size of the source. Different models, such as the Schell model [17

17. A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. **AP15**187–188 (1967). [CrossRef]

## 3. Radiation, degree of polarization, and the realizability condition for the QuHIEM sources

20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. **53**, 969–978 (2006). [CrossRef]

**s**, so ℰ(

*r*

**s**,

*ω*) is

*not*the electric field. However, the electric field is given in terms of ℰ(

*r*

**s**,

*ω*) as follows: It can be seen that

**s**·

**E**

^{(∞)}(

*r*

**s**,

*ω*) = 0. Note that since

*s*= cos

_{z}*θ*, the inclination factors appear as a direct consequence of the vector diffraction formulation.

**s**in terms of standard angular variables, viz., then it will be useful to write

**E**

^{(∞)}(

*r*

**s**,

*ω*) in terms of the the following two vectors: In that case, eq.(7) may be re-written as follows: where the 2 × 2 matrices

**P**(

*θ*) and

**R**(

*ϕ*) are given by

*J*

_{i,j}(

*r*

**s**,

*ω*), expressed in the {

**ê**

*,*

_{θ}**ê**

*} coordinate system, is then given by a matrix transformation of the 2 × 2 matrix 𝒲*

_{ϕ}_{α,β}(

*r*

**s**,

*r*

**s**,

*ω*), viz.,

*r*

**s**and

*ω*for clarity. That the properties of the far-zone field might be reduced to a single 2 × 2 tensor, rather than a 3 × 3 tensor (as considered in [20

20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. **53**, 969–978 (2006). [CrossRef]

**s**. For plane waves, Gauss’s law, ∇

**·E**(

*r*

**s**,

*ω*) = 0, implies that the field component parallel to

**s**is zero. Thus the use of spherical polar coordinates (rather than the cartesian coordinates used in [20

20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. **53**, 969–978 (2006). [CrossRef]

*(*

_{n}*u*) being the

*n*-th order Bessel function of argument

*u*. Substituting from eq.(17) into eq.(15) and performing the requisite matrix multiplications, we find that the coherency matrix in the {

**ê**

*,*

_{θ}**ê**

*} basis is given by: The Poynting vector of the field in the radiation zone is therefore given by The degree of polarization of the radiated field is*

_{ϕ}*A*(

*kρ*) and

*B*(

*kρ*) have a number of constraints. First, the definition eq.(5) requires that

*B*(0) =

*B*′(0) = 0. Additionally, Gori et al. [21

21. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A **25**, 1016 (2008). [CrossRef]

*realizability condition*, based on the requirement that the cross-spectral tensor is a positive-definite function. The condition entails the following: For our model, this implies: Furthermore, we require that

**J**(

*r*

**ŝ**,

*ω*) is zero when

*θ*=

*π*/2, i.e. there is no radiation parallel to the source plane. This requirement is born of the fact that the solid angle subtended by a planar source is zero when

*θ*=

*π*/2 and hence the radiated intensity should be zero as well. (This seems the usual circumstance in radiometry, although we are not aware of a rigorous proof.) Mathematically, this requirement implies that

*θ*= 0, i.e. when viewing the source along the normal, we find

*I*

_{2}(0) = 0 from the properties of J

_{2}(

*x*); and hence In other words, the field on-axis from an isotropic source must be completely unpolarized. This is a reasonable result, simply because there is nothing to break the isotropic symmetry. Off-axis, however, is symmetry broken, and it is indeed plausible that the

**ê**

*component of the field might be greater than the*

_{θ}**ê**

*component, or vice-versa.*

_{ϕ}## 4. Examples

15. D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. **109**, 209 (1994). [CrossRef]

*A*(

*kρ*) and

*B*(

*kρ*) were given by: It was found that, in this case, the radiated field obeys Lambert’s law, i.e. the normalized amplitude of the Poynting vector 𝒮

^{(∞)}(

*θ*) = S

^{(∞)}(

*r*

**s**,

*ω*)/S

^{(∞)}(

*r*

**ê**

*,*

_{z}*ω*) is given by Further, the degree of polarization P

^{(∞)}(

*θ*) = 0 for all values of

*θ*.

*A*(

*ρ*) and

*B*(

*ρ*) in eq.(5), are given by two Gaussians, defined as follows: where

*k*=

*ω*/

*c*and

*a*and

*b*are dimensionless quantities. In this case, the integrals,

*I*

_{1}and

*I*

_{2}, are given by: If we invoke the criterion eq.(25), which ensures that 𝒮

^{(∞)}(

*π*/2) = 0, we find that

*B*

_{0}=

*a*

^{2}exp(−

*a*

^{2}/2)/2

*b*

^{4}(

*b*

^{2}− 1)exp(−

*b*

^{2}/2), and hence we obtain the following expressions for the radiation pattern and degree of polarization:

^{(∞)}(

*θ*) = 0. First, the formula suggests the criterion

*b*= 1, implying that the coherence length of the second part of the cross-coherency tensor is equal to the wavelength; however the condition eq.(25) cannot be satisfied if

*b*= 1, and hence we discard this possibility. The second criterion is

*a*=

*b*, implying that the two parts of the cross-coherency tensor have the same coherence lengths; in this case, the realizability condition requires that

*a*> 1.

*a*,

*b*→ 0, we would expect the results to be independent of the functional form of

*A*(

*ξ*) and

*B*(

*ξ*). In this case the two expressions become: where and the realizability condition requires that

*μ*≥ 1. In other words, for highly spatially incoherent sources it appears that the degree of polarization will

*always*be directionally dependent in the far zone.

*a*and

*b*(see Fig. 2).

## 5. Conclusions

## Acknowledgments

## References and links

1. | A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. |

2. | E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. |

3. | A. T. Friberg, ed., |

4. | E. Wolf, |

5. | J. H. Lambert, |

6. | W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer. |

7. | E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. |

8. | E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature |

9. | E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. |

10. | D. F. V. James, “Change of polarization of light-beams on propagation in free-space,” J. Opt. Soc. Am. A |

11. | F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics |

12. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

13. | J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A |

14. | O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. |

15. | D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. |

16. | G. K. Batchelor, |

17. | A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. |

18. | G. Toraldo di Francia, |

19. | J. D. Jackson, |

20. | M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt. |

21. | F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(050.1960) Diffraction and gratings : Diffraction theory

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: September 18, 2012

Revised Manuscript: October 11, 2012

Manuscript Accepted: October 11, 2012

Published: October 16, 2012

**Citation**

Asma Al-Qasimi and Daniel F. V. James, "Properties of quasi-homogeneous isotropic electromagnetic sources," Opt. Express **20**, 24910-24917 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24910

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### References

- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am.58, 1256–1259 (1968). [CrossRef]
- E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am.68, 6–17 (1978). [CrossRef]
- A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS 69 (SPIE, 1993).
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
- J. H. Lambert, Photometria sive de mensure et gradibus luminis colorum et umbra (Bassel, 1760).
- W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer.65, 1067–1071 (1975). [CrossRef]
- E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56, 1370–1372 (1986). [CrossRef] [PubMed]
- E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326, 363–365 (1987). [CrossRef]
- E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59, 771–818 (1996). [CrossRef]
- D. F. V. James, “Change of polarization of light-beams on propagation in free-space,” J. Opt. Soc. Am. A111641–1643 (1994). [CrossRef]
- F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998). [CrossRef]
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003). [CrossRef]
- J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A212205–2215 (2004). [CrossRef]
- O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246, 35–43 (2005). [CrossRef]
- D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun.109, 209 (1994). [CrossRef]
- G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Science Classics, 1990).
- A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.AP15187–188 (1967). [CrossRef]
- G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955), pp. 218–221.
- J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 437.
- M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006). [CrossRef]
- F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008). [CrossRef]

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