Propagation of cross-spectral densities from spherical sources
R. Borghi,1F. Gori,2,* O. Korotkova,3and M. Santarsiero2
1Dipartimento di Elettronica Applicata, Università degli Studi “Roma Tre”, Via della Vasca Navale 84, I-00146 Rome, Italy
2Dipartimento di Fisica, Università degli Studi “Roma Tre”, Via della Vasca Navale 84, I-00146 Rome, Italy
3Department of Physics, University of Miami, 1320 Campo Sano Drive, Coral Gables, Florida 33146, USA
*Corresponding author: gori@uniroma3.it Received June 14, 2012; accepted June 26, 2012;
posted June 28, 2012 (Doc. ID 170605); published July 25, 2012
We show that the cross-spectral density in the far zone of a homogeneous spherical source can be described as a low- pass filtered version of that existing across the source surface. We prove that, to an excellent approximation, the corresponding filter with respect to a (normalized) spatial frequency ξ has a functional structure of the form !!!!!!!!!!!!!
1 − ξ2
p ,
for 0 ≤ ξ ≤ 1. The cases of spatially incoherent and Lambertian sources are treated as significant examples. © 2012 Optical Society of America
OCIS codes: 030.6600, 030.1640, 350.5500.
Not long ago, Agarwal et al. [1] presented an investigation about the cross-spectral density (CSD) [2] of sunlight starting from the remark that previous evaluations were based on the hypothesis that the Earth could be assumed to be in the far zone of the Sun. Using the traditional cri- terion for evaluating the far zone distance, such a hypoth- esis is not even approximately valid. The authors of [1]
devised a method, without the above far zone hypothesis, for checking the validity of previous results through a di- rect numerical evaluation of the CSD. To this end, they referred to a stochastic spherical source and expressed the field produced outside the source in a single realiza- tion as a series, with perfectly uncorrelated random coefficients (delta-correlation hypothesis), of spherical harmonics multiplied by outgoing spherical Hankel func- tions [3]. They found through numerical evaluations that the CSD tends very rapidly, namely within a very few wavelengths from the source surface, to a limiting func- tional form, which turned out to be approximately equal to that produced by a spatially incoherent circular source with uniform intensity. Shortly after, it was shown [4]
that the condition characterizing the far zone in the case of partially coherent sources is much less restrictive than in the typical case of coherent sources, which partly jus- tifies the results of [1]. Moreover, the modal analysis for partially coherent sources [2] has recently been extended to homogeneous spherical sources [5]. This allows us to consider the problem introduced in [1] within a more general treatment, in which the choice of the correlation function across the source surface is not limited to be of delta-correlated type.
The aim of this Letter is to establish a general frame- work to be used for studying the free-space propagation of the CSD from partially coherent homogeneous sphe- rical scalar sources. In particular, we show that the effect of the propagation on the CSD can, to an excellent ap- proximation, be thought of as due to a suitable filtering process. To illustrate the feasibility of the proposed ap- proach, two examples, which are of particular signifi- cance, are considered: delta-correlated and Lambertian sources. In both cases the present approach gives accu- rate approximations of the propagated CSD that show an
excellent agreement with the results based on the direct numerical evaluations of the series solution.
We suppose the CSD at two points on the surface of a homogeneous spherical source [5] depends only on the angle, say φ, between the radii that specify the points.
A key ingredient in this type of problem is the use of ex- pansions into a series of Legendre polynomials [3] having cos φ as argument. Accordingly, we shall think of the CSD at the surface of the source, say W0, as a function of cos φ. On the other hand, the natural variable remains φ, and this is to be kept in mind when interpreting the formulas we are going to derive. Let us write W0 in the form
W0!ˆr1; ˆr2" # S0g0!cos φ"; (1) where cos φ # ˆr1· ˆr2, ˆr1 and ˆr2 being unit vectors. The functions S0 and g0 denote the spectral density and the spectral degree of coherence, respectively [2]. The explicit dependence of S0 and g0 on the temporal fre- quency will be omitted throughout the Letter. As shown in [5], the spectral degree of coherence across the source can be expanded in the following series:
g0!cos φ" #X∞
l#0
"
l $1 2
#
BlPl!cos φ"; (2)
where Pl!·" denotes the lth-order Legendre polynomial [3] and the coefficients Bl, namely,
Bl# Z1
−1
g0!ξ"Pl!ξ"dξ; l # 0; 1; …; (3) uniquely specify the spectral degree of coherence across the source. At a pair of points !r1; r2" outside the source, the CSD is easily found to be given by the following expansion [1]:
W!r1; r2" # S0
X∞
l#0
"
l $1 2
#
Blh%l!kr1"hl!kr2"
jhl!ka"j2 Pl!cos φ";
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0146-9592/12/153183-03$15.00/0 © 2012 Optical Society of America
where a denotes the radius of the source, hl!·" denotes the (outgoing) spherical Hankel function [3], and k # 2π ∕ λ, λ being the wavelength. When r1 and r2 are in the far zone, the propagated CSD expressed by Eq. (4) is well approximated by replacing the spherical Hankel functions in the numerator of the fraction by the leading terms of their asymptotic expansions (see for- mula 10.52.4 of [3]), so that Eq. (4) becomes
W!r1; r2" ∼ S0exp&i!kr2− kr1"' r1r2 g
∞!cos φ"; (5) where the (unnormalized) angular correlation function g∞ is given by
g∞!cos φ" #X∞
l#0
"
l $1 2
# Bl
jhl!ka"j2Pl!cos φ": (6) On comparing Eqs. (2) and (6), it is seen that the main effect of propagation consists in multiplying the source coefficients Bl by the factors 1∕ jhl!ka"j2 (l # 0; 1; 2; …), and this suggests a possible way of interpreting the pro- pagation process of the CSD in terms of a suitable filter- ing action. To this aim, we start from formula 8.714.1 of [6], which, after simple algebra, leads to the following ex- act integral representation of Legendre polynomials:
Pl!cos φ" #2φ π
Z1
0
cosh$
l $12
%φti
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
2!cos φt − cos φ"
p dt: (7)
For sufficiently small values of φ, the inverse square root factor is smooth enough within the integration interval [0,1] to be expanded as a truncated Taylor series with respect to the angle φ such that !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
2!cos φt − cos φ"
p
≃ φ !!!!!!!!!!!!
1− t2
p , and, thanks to formula 8.411.8 of [6], we have
Pl!cos φ" ≃ J0
&"
l $1 2
# φ
'
; (8)
where Jn!·" denotes the nth-order Bessel function of the first kind [3].
Equation (8) is the key to understanding the filtering interpretation of the CSD propagation. First of all, on sub- stituting from Eq. (8) into Eq. (6), we have
g∞!cos φ" ≃X∞
l#0
"
l $1 2
# Bl jhl!ka"j2J0
&"
l $1 2
# φ
' : (9)
Then, we note that the squared modulus of hl!ka" can be approximated according to the following expression (see formula 8.479 of [6]):
1
!ka"2jhl!ka"j2≃
!!!!!!!!!!!!!!!!!!!!!!!!!!!
1−"l $12 ka
#2
s
; ka≫ 1; (10)
which is valid for 0≤ l < ka. Figure1shows the behavior, versus the index l, of both sides of Eq. (10) written for ka # 100. Numerical evaluations indicate that the approximation furnished by Eq. (10) becomes better
and better upon increasing ka. Upon substituting from Eq. (10) into Eq. (9), the function g∞is expressed through the following finite expansion:
g∞!cos φ" ≃ !ka"2
×X&ka'
l#0
"
l $1 2
# Bl
!!!!!!!!!!!!!!!!!!!!!!!!!!!
1−"l $12
ka
#2
s
J0&"l $12
ka
# kaφ
'
; (11)
which can be approximately evaluated, in the limit of large values of ka, simply by replacing the sum over l by an integral with respect to the dimensionless variable ξ # !l $ 1 ∕ 2" ∕ ka [7], thus obtaining
g∞!cos φ" ∝ Z1
0 B!ξ" !!!!!!!!!!!!!
1− ξ2
q J0!kaφξ"ξdξ; (12)
where B!ξ" is the function obtained from the set of Bl
coefficients through interpolation at the nodes
Bl# B"l $ 1∕ 2 ka
#
: (13)
Equations (12) and (13) constitute the main result of this Letter and provide the filtering interpretation of the CSD propagation process promised at the beginning.
To this end, it is sufficient to read Eq. (12) as the Hankel transform of the radial function B!ξ"F!ξ", where
F!ξ" # !!!!!!!!!!!!!
1− ξ2
q circ!ξ"; (14)
with circ!·" representing the characteristic function of the unitary circle. Moreover, from the interpolation rule in Eq. (13) it clearly appears that ξ can be interpreted as a sort of“spatial” frequency and, together with the ortho- gonal expansion in Eq. (6), B!ξ" as the “spectrum” asso- ciated to the g0. In this way the free-propagation process in the far zone simply corresponds to filtering the degree of coherence across the source via the low-pass function given in Eq. (14).
It should be noted that Eq. (12) does not provide only a physically insightful way to interpret the CSD propaga- tion. It also gives a numerical tool to achieve accurate analytical estimates of the angular degree of coherence g∞. To show this, in this Letter two significant examples of partially coherent sources will be considered. The first example is that of an incoherent source for which the
0 20 40 60 80 100
0.0 0.2 0.4 0.6 0.8 1.0
l
Fig. 1. Behavior, as a function of l, of the l.h.s. (dots) and of the r.h.s. (solid curve) of Eq. (10), for ka # 100.
3184 OPTICS LETTERS / Vol. 37, No. 15 / August 1, 2012
“spectrum” Blturns out to be flat with respect to the in- dex l; i.e., B does not depend on ξ. Upon using formula 6.567.1 of [6] from Eq. (12), we have at once
g∞!cos φ" ≈3j1!kaφ"
kaφ ; (15)
where jn!·" denotes the nth-order spherical Bessel func- tion of the first kind [3] and the factor 3 has been intro- duced to normalize the function to unity. In Fig.2, the graph of the r.h.s. of Eq. (15) is drawn (full line) together with points (black circles) evaluated by summing the ser- ies in Eq. (4). The numerical value ka # 100 was chosen, while kr1# kr2# 1000. Differences between the full line and the circle centers are barely discernible and become inappreciable at the graphical level when ka grows further. As a second fundamental test case, consider a so-called Lambertian [2] spherical source, whose CSD is given by
W0!ˆr1; ˆr2" # S0sin!kajˆr1− ˆr2j"
kajˆr1− ˆr2j : (16) This is the celebrated correlation function of black body radiation [8]. The modal structure of the source in Eq. (16) has recently been investigated in [5], where it has been shown that the spectrum Bl is proportional to j2l!ka". Moreover, upon using formulas 10.18.4 and 10.18.18 of [3], after some algebra it is possible to recast Eq. (12), in the limit of ka≫ 1, as
g∞ ∝ Z 1
0
1− sin!kaπξ − 2ka"
2 J0!kaφξ"ξdξ; (17) and to evaluate the integral through standard asymptotic techniques [7], thus obtaining
g∞!cos φ" ≈2J1!kaφ"
kaφ ; (18)
where use has been made of formula 6.651.1 of [6].
Figure2shows the behavior, as a function of kaφ, of the spectral degree of coherence (suitably normalized) nu- merically evaluated by summing the series in Eq. (4) for a Lambertian source (open circles) having ka # 100, at kr1# kr2# 1000, together with the analytical estimates provided by Eq. (18) (dotted curve). Even in this case, the differences between the two predictions are very small and decrease further upon increasing ka.
The free-propagation problem of the partially coherent radiation emitted from quasi-monochromatic homoge- neous scalar spherical sources has been dealt with. In particular, a physically insightful filtering interpretation of the propagation effect on the CSD associated to the source has been proposed and numerically validated on the field emitted by delta-correlated and Lambertian sources. We believe that the proposed approach could be suitably extended to other important classes of partially coherent sources and also to include in the above filter- ing picture the vectorial character of electromagnetic fields emitted by spherical sources [9].
References
1. G. S. Agarwal, G. Gbur, and E. Wolf, Opt. Lett. 29, 459 (2004).
2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
3. Digital Library of Mathematical Functions. Release date 2012-03-23. National Institute of Standards and Technology, fromhttp://dlmf.nist.gov/.
4. F. Gori, Opt. Lett.30, 2840 (2005).
5. F. Gori and O. Korotkova, Opt. Commun.282, 3859 (2009).
6. I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, 1994).
7. C. M. Bender and S. A. Orszag, Advanced Mathematical Method for Scientists and Engineers (McGraw-Hill, 1978).
8. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).
9. T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A.
Friberg, Phys. Rev. E71, 036618 (2005).
0 2 4 6 8 10
0.2 0.0 0.2 0.4 0.6 0.8 1.0
ka Fig. 2. Behavior, as a function of the kaφ, of the spectral de- gree of coherence for a δ-correlated (dots) and for a Lambertian (open circles) source, together with the theoretical estimates provided by Eqs. (15) (solid curve) and (17) (dotted curve), respectively. ka # 100 and kr1# kr2# 1000.
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