Research Article

Vol. 33, No. 6 / June 2016 / Journal of the Optical Society of America A

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Polarization oscillations of near-field thermalemission
MANABU MACHIDA,1 EVGENII NARIMANOV,2 AND JOHN C. SCHOTLAND3,*
1

Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA
Department of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47906, USA
3
Department of Mathematics and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA*Corresponding author: schotland@umich.edu
2Received 10 March 2016; accepted 1 April 2016; posted 15 April 2016 (Doc. ID 260780); published 10 May 2016

We consider the polarization of thermal emission in the near field of various materials, including dielectrics andmetallic systems with resonant surface modes. We find that, at thermal equilibrium, the degree of polarization
exhibits spatial oscillations with a period of approximately half the optical wavelength, independent of materialcomposition. This result contrasts with that of Setala et al. [Phys. Rev. Lett. 88, 123902 (2002)], who find monotonic decay of the degree of polarization for systems in local thermal equilibrium. © 2016 Optical Society ofAmerica
OCIS codes: (030.0030) Coherence and statistical optics; (260.5430) Polarization.
http://dx.doi.org/10.1364/JOSAA.33.001071

1. INTRODUCTION
Coherence theory is one of the cornerstones of optical physics[1]. Its goal is to describe the statistical regularities of electromagnetic fields in terms of field correlations and their relation
to measurable quantities. Implicit in the formulation of coherence theory is the notion of a statistical ensemble of random
fields. However, the origin of the randomness of the fields isnot explicitly part of the theory. As a result, the theory of coherence is primarily concerned with the propagation of correlation functions, which has the powerful consequence that its
predictions are, in some sense, independent of the underlying
probability distribution of the fields.
A notable exception to the above remarks is provided by thetheory of thermal emission of radiation [1,2]. In this context,
the currents, which act as sources of the optical field, are taken
to obey the fluctuation-dissipation theorem. Thus, it is possible
to calculate the correlation function of the field in terms of
the statistics of the current. It follows that the emitted field displays temporal and spatial coherence [3]. Moreover, the nearand far-field coherence functions manifest strikingly different
behavior [4–8]. In particular, in materials that support resonantsurface waves, the spectrum of emitted radiation changes qualitatively on propagation, ranging from extreme narrowing at
subwavelength scales to broadband in the far field. Likewise,
the spatial coherence of emitted light is dramatically modified
in the near field, with a coherence length that is much smaller
than the λ∕2 far-field limit of blackbody radiation. In either
case, the alteration in coherence is due to the decay of evanescent modes of the field on propagation.
1084-7529/16/061071-05 Journal © 2016 Optical Society of America

The near-field polarization of thermal emission has also
received attention [9,10]. It was found that, at local thermodynamic equilibrium, the emitted field becomes depolarized,with the degree of polarization decaying monotonically upon
propagation into the far zone. By the term local, we mean thatthe medium is held at a nonzero temperature and radiates intovacuum at zero temperature. In this case, pure thermal emission occurs, with a net heat flux transferred from the material
to vacuum. In contrast, we study the corresponding problem
for systems in thermal equilibrium, where the net flux vanishes. Instead of monotonic decay, we predict that the degreeof polarization exhibits spatial oscillations with a period ofapproximately λ∕2. We illustrate this result for several materials, including lossless dielectrics and metallic systems withresonant surface plasmon modes. We note that the presenceof polarization oscillations was observed numerically byDorofeyev and Vinogradov [11]. However, these authors
did not provide a mathematical analysis of the phenomenon
nor did they predict the universal oscillatory behavior of thepolarization.
The remainder of this paper is organized as follows. In
Section 2, we recall some basic facts from coherence theory.
We also obtain the expression for the degree of polarization in
the half-space geometry at thermal equilibrium. In Section 3,
we calculate the polarization for several materials, elucidate thepresence of oscillations and provide an analysis of the distancedependence of the degree of polarization. Our conclusions arepresented in Section 4, and Appendix A presents details of themathematical analysis.

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2. COHERENCE AND POLARIZATION
The fundamental quantity of coherence theory in the spacefrequency domain is the cross-spectral density W ij, which isdefined byW ij r; r 0 ; ωδω − ω 0   hE i r; ωE j r 0 ; ω 0 i:(1)Here, Er; ω is the electric field at the position r and frequency ω, the presence of the delta function indicates that thefield is taken to be statistically stationary and h  i denotes theensemble average. We note that W ij r; r 0 ; ω for r ≠ r 0 is ameasure of the spatial coherence of the electric field. The degreeof coherence is defined to be [12,13]
TrW r; r 0 ; ωW r 0 ; r; ω:μ2 r; r 0 ; ω TrW r; r; ωTrW r 0 ; r 0 ; ω(2)It can be seen that 0 ≤ μ ≤ 1. The case μ  0 corresponds
to an incoherent field and μ  1 to a coherent field; otherwise,
the field is said to be partially coherent.
There is a fundamental link between polarization and coherence. If we consider W ij r; r 0 ; ω for r  r 0, then W is a measure of the polarization of the field. The degree of polarization
P is defined as [9,10,14,15]

3 TrW 2 r; r; ω 12(3)− :P r; ω 2 Tr2 W r; r; ω 3
It can be shown that and 0 ≤ P ≤ 1. When P  0 the field
is said to be unpolarized, and if P  1 the field is polarized;
otherwise, the field is partially polarized.
We consider a system consisting of two homogeneous halfspaces, which are in thermal equilibrium at a temperature T .
The lower half-space z < 0 is taken to consist of a nonmagnetic
lossy material with a generally complex dielectric permittivity
ϵ1 ω. The upper half-space z > 0 consists of a nonmagnetic
material with a real and frequency-independent permittivity ϵ2.
In this setting, the fluctuation-dissipation theorem can be used
to relate the cross-spectral density to the Green’s tensor G ij bythe formulaℏω
ImG ij r; r 0 ;
W ij r; r 0 ; ω  2πk20 ℏ coth(4)
2k B T
where k0  2π∕λ is the free-space wavenumber and kB is
Boltzmann’s constant [16].
The Green’s tensor obeys the equation∇ × ∇ × Gr; r 0  − k 20 ϵzGr; r 0   4πδr − r 0 I ;whereϵz ϵ1ϵ2ififz < 0;z > 0;

(5)

(6)

and I is the unit tensor. The Green’s tensor also obeys theboundary conditionsẑ × Gr; r 0 jz0  ẑ × Gr; r 0 jz0− ;(7)ẑ × ∇ × Gr; r 0 jz0  ẑ × ∇ × Gr; r 0 jz0− ;(8)which correspond to the continuity of the tangential electricand magnetic fields, where G is interpreted as the electric fieldradiated by a point source.

Research Article

It will prove useful to expand the Green’s tensor into planewaves [17]:Zd2 q iq·ρ−ρ 0 G ij r; r 0  eg ij z; z 0 ; q;
(9)
2π2
whereg ij z; z 0 ; q 

2πi
r ŝ ŝ
k 2z s i j

0r p p̂ i p̂−j e ik2z z z 0p̂ i p̂ j e ik2z z−z  :ŝi ŝ j(10)Here,p̂  ŝ × k̂ ; (11)k  q k 2z ẑ;pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
where k 2z  ϵ2 k 20 − q2 . The Fresnel reflection coefficientsare given byŝ  q̂ × ẑ;k 2z − k1z;
k2z k1zpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
where k 1z  ϵ1 k 20 − q2 .
rs rp 

k2z ϵ1 − k1z;
k 2z ϵ1 k 1z(12)3. APPLICATIONS
We now describe applications of the above results to variousmaterials. In the cases considered, the system has the temperature T  300 K and the upper half-space is taken to havepermittivity ϵ2  1. The optical properties of the materials thatcomprise the lower half-space are given in Table 1 [18]. InFig. 1, we plot the degree of polarization P as a function ofthe distance z from the interface for several materials including
glass, tungsten, silver, and silicon carbide (SiC). We see that thefield is partially polarized near the interface and becomes depolarized at large distances. In Fig. 2, we plot the corresponding
degree of coherence μ as a function of the transverse distanceρ  jr − r 0 j, for points r and r 0 in the plane z  z 0 . As may beexpected, the field is partially coherent near the interface andbecomes incoherent with propagation, consistent with theresults of [4].
Evidently, the behavior of both P and μ depends sensitively
on the dielectric permittivity of the material under investigation. In the case of glass, which is a nonlossy dielectric at
the wavelength λ  500 nm, the field is relatively less polarized. In contrast, silver exhibits a surface plasmon resonanceat λ  620 nm, and the near-field is strongly polarized.
This should be compared with the case of tungsten, which does
not exhibit a plasmon resonance at λ  500 nm, where it canbe seen that P is correspondingly reduced. Finally, we consider
the case of SiC, which, at λ  11.36 μm, supports surfacephonon polariton modes. We see that, as in the example ofsilver, the near-field is strongly polarized.
Table 1. Optical Properties of Considered Materials
MaterialGlass
WAgSiCλϵ500 nm
500 nm
620 nm
11.36 μm

2.25
4.35 18.1i−15.0 1.0i
−12.2 0.71i

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Fig. 1. Degree of polarization of glass, SiC, silver, and tungsten as afunction of distance z from the interface.

Fig. 3. Comparison of the asymptotic formula [Eq. (13)] with theexact result [Eq. (3)] for the degree of polarization.

Fig. 2. Degree of coherence of glass and SiC as a function of transverse distance ρ in the plane z  z 0 at various distances z 0 above theinterface.

A striking feature of Fig. 1 is the oscillatory nature of the
distance-dependence of the degree of polarization. A straightforward asymptotic analysis of the integral Eq. (9) defining
ImG ij shows that for z ≫ λ, the envelope of the oscillation
decays as P ∼ 1∕z. Moreover, if Reϵ1  ≪ −1, it can be seen
thatpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 sin4πz∕λ 2∕ jReϵ1 j
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :P ∼ 2 4πz∕λ 2∕ jReϵ1 j(13)Thus, the period of the oscillations is asymptotically λ∕2,independent of the material. In Fig. 3, we compare the above
asymptotic formula with the exact result obtained from Eq. (3).
As may be expected, there is excellent agreement for SiC andrelatively poor agreement for glass.
It is important to note that polarization oscillations are notpresent for systems in local thermal equilibrium, where P decays monotonically after reaching a maximum at z ≲ λ. [9].
This difference can be explained by the interference between
modes in the upper half-space, a mechanism that is not present
in the calculations presented in [9].

4. DISCUSSION
We have investigated the polarization of near-field thermalemission. We find that, at thermal equilibrium, the degreeof polarization exhibits spatial oscillations with a period ofapproximately λ∕2, independent of material composition.
The amplitude of the oscillations is greatest in systems withresonant surface waves such as surface-plasmons or surfacephonon polaritons.
In future work, we intend to explore the analogy between
spatial correlations of near-field thermal radiation and nearfield speckle patterns produced by volume scattering. It can be
expected that thermal emission would correspond to a speckle
pattern produced in transmission through a scattering medium,
while thermal equilibrium would correspond to a speckle pattern produced by balanced illumination in reflection and transmission. Thus, the results presented herein could be relevant tothe study of polarization in speckle patterns.
APPENDIX A
Here, we derive the asymptotic form of the degree of polarization given in Eq. (13). We assume that the real part of ϵ1 isnegative. We also assume that ϵ1 has a small imaginary part;its real part is a large negative value, and the height z above
the interface is much larger than the wavelength λ. That is,
we suppose the following relations hold:jImϵ1 j ≪ jReϵ1 j;

(A1)

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pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
4πz≪ jReϵ1 j:
λThe assumption Eq. (A2) implies thatpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ζ ≪ jReϵ1 j;1≪whereζ  4πz
λ(A2)

0(A3)

(A4)

Using Eq. (A1) we haveqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
k 1z ∼ k 20 jReϵ1 j q 2 :(A5)

Let us introduce

0 < q < k0 ;(A6)

2itr s ∼ −1 − pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;jReϵ1 jjReϵ1 jt2
1 jReϵ1 jt 2(A7)(A8)

2 acosζ0 tdt
1 at 20   −3   −24πz
4πz
4πz
4πz
2cos
sin
∼ −2λλλλπ− pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :(A15)
2 jReϵ1 jSimilarly, for W 33, we have
Z 1jReϵ1 j1 − t 2 1sin ζ
2 cosζtdt ∼ −jReϵ1 jt
ζ
0 12it
1pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ −;jReϵ1 j jReϵ1 j

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
jReϵ1 j;
2(A16)


2 jReϵ1 j
4πzcostdt
1 − t 2  1 −1 jReϵ1 jt 2λ0  −24πz4πzcos∼ −2λλ2   pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
π jReϵ1 j
4πz4πz −34πz:−sin
22λλλ(A17)(A9)

W 11  W 22(A18)
k0 ℏ2e ℏω∕kB T − 1
 −2
 

4 24πz4πz;cos× − sin ζ − 23 ζλλ∼ 2πk20
Imr p  ∼ 0:πUsing Eqs. (A2) and (A3), we arrive atand for q > k0
Imr s  ∼ 0;

(A14)

andZ 1

Therefore, we obtain for 0 < q < k0rp ∼1πpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;2 jReϵ1 jq > k0 :Noting that jReϵ1 j ≫ 1, we can express k1z as
8hi
1−t 2>
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ< 1 2jReϵ1 j ; 0 < q < k 0 ;ik1z ∼ ik0 jReϵ1 j h>
1 t2: 1 2jReϵ;q > k0 :1 j
12cosζtdt ∼ − sin ζjReϵ1 jt 2
ζ−2Z 1 t 2 −1

1pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :2π jReϵ1 j8 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ>
2>> 1 − kq ;<0t  rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ>>
q 2>− 1;:
k0Z 1

Research Article(A10)

Thus, we obtain
W 11  W 22(A11)k0 ℏk0 ℏ2∼2πk 0 ℏω∕k T
B
2e−1


Z 1
1
2−2×1 tcosζt
1 jReϵ1 jt 20 
2 jReϵ1 j4πzt dt;(A12)
t2 − 12 cos
1 jReϵ1 jtλk0 ℏk0 ℏ2W 33 ∼ 4πk0 ℏω∕k T
B
2e−1Z 1jReϵ1 j1 − t 2  1cosζt
×1 jReϵ1 jt 202 jReϵ1 j4πz
1−costdt:(A13)
1 jReϵ1 jt 2
λTaking Eq. (A3) into account, we can evaluate the integrals for
W 11 and W 22 as follows:k0 ℏk ℏk0 ℏW 33 ∼ 4πk20 ℏω∕k0 TB
2e−1
−2 24πz
4πz:
cos× −23λλFinally, we obtain1 sin ζ P ∼ ;
2 ζ 

(A19)

(A20)

(A21)

which is equivalent to Eq. (13). Evidently, this formula implies
that the origin of polarization oscillations is due to the interplayof the numerator and denominator of 3.
Funding.

National Science Foundation (NSF).

Acknowledgment. We are grateful to Remi Carminati
for valuable discussions. This work was supported by theNSF Center for Photonic and Multiscale Nanomaterials.

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Vol. 33, No. 6 / June 2016 / Journal of the Optical Society of America A

REFERENCES
1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge University, 1995).
2. D. F. V. James and E. Wolf, “Spectral changes produced in Young’sinterference experiment,” Opt. Commun. 81, 150–154 (1991).
3. W. H. Carter and E. Wolf, “Coherence properties of Lambertian andnonLambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
4. R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherenceof thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
5. A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-fieldspectral effects due to electromagnetic surface excitations,” Phys.
Rev. Lett. 85, 1548–1551 (2000).
6. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet,“Surface electromagnetic waves thermally excited: radiative heattransfer, coherence properties and Casimir forces revisited in the nearfield,” Surf. Sci. Rep. 57, 59–112 (2005).
7. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K.
Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation
scanning tunnelling microscopy,” Nature 444, 740–743 (2006).
8. J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y.
Chen, “Coherent emission of light by thermal sources,” Nature 416,
61–64 (2002).

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9. T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in nearfields of thermal sources: effects of surface waves,” Phys. Rev. Lett.
88, 123902 (2002).
10. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree ofpolarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
11. I. A. Dorofeyev and E. A. Vinogradov, “Fluctuating electromagnetic
fields of solids,” Phys. Rep. 504, 75–143 (2011).
12. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence forelectromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
13. J. Tervo, T. Setala, and A. T. Friberg, “Spatial correlations and degreeof polarization in homogeneous electromagnetic fields,” Opt. Lett. 31,2669–2671 (2006).
14. R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt.
Commun. 23, 147–150 (1977).
15. R. Barakat, “Degree of polarization and principal idempotents of coherency matrix,” Opt. Acta 30, 1171–1182 (1983).
16. G. Agarwal, “Quantum electrodynamics in the presence of dielectrics
and conductors. I. Electromagnetic-field response functions and blackbody fluctuations in finite geometries,” Phys. Rev. A 11, 253–264 (1975).
17. J. E. Sipe, “The dipole antenna problem in surface physics: a newapproach,” Surf. Sci. 105, 489–504 (1981).
18. E. W. Palik, Handbook of Optical Constants of Solids (Academic, 1985).