REVIEW

Review of thermal infrared polarimetry, I: theory
Nathan Hagen *Utsunomiya University, Department of Optical Engineering, Utsunomiya, Japan

Abstract. We review the physics underlying thermal infrared emission, focusing on the processes that generate polarization. We also discuss a number of features of the theory that are
not widely discussed in the literature, including the Straubel theorem for absorbing media, themixed Poynting vector, and the emission of nonisothermal media. Using a model for emission
from nonisothermal media, we present a quantitative discussion of the conditions under which
Kirchhoff’s law is valid. Using this model, we show how it is possible for many blackbody simulators and infrared camera shutters to produce emission very close to true blackbodies. © 2022
Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.61.7.070902]

Keywords: infrared; polarization; emissivity; blackbody; Kirchhoff’s law; Straubel’s theorem.
Paper 20220227V received Mar. 8, 2022; accepted for publication Jul. 5, 2022; published online
Jul. 26, 2022.

1 Introduction
Although research in infrared polarization has been pursued at least since Millikan’s studies on
emission polarization in 1895,1,2 infrared polarimetry has never reached widespread use. As aresult, there are many polarimetric features specific to infrared wavelengths that are not widely
known, despite their fascinating physics. This review aims to survey the physics underlying
infrared polarimetry and to explore a few of the less common features of the theory.
In the infrared polarization literature, writers often refer to the thermal emission of surfaces, so
it is easy to slip into thinking of infrared light emission as a surface phenomenon rather thana volumetric one. A search through the literature encounters “surface emissivity” as a common
phrase, and Planck’s blackbody law, after all, is generally expressed in terms of surfaces andsurface temperatures, rather than volumes and volumetric temperature distributions. Nevertheless,emission is a volumetric property, much like absorption but played out in reverse. For opaquematerials, in which absorption completely attenuates incident light after passing only microns
deep inside the material, this volumetric behavior is not obvious. For semitransparent (diathermous) materials, however, understanding the volumetric behavior is essential to accurately
modeling their emission properties.3,4
In Sec. 2, we model the thermal light emission that constantly occurs inside every materialand show that the surface emission that we see is simply the portion of that internal light thatreaches the surface, transmits, and refracts into the air.3,5 This model holds even for highly reflective materials such as metals, and we find that the apparent skin-like quality of emission derives
from the fact that so many materials in the infrared have strong absorption, so only the layers
close to the surface are able to participate in generating the emitted light that we see. In Sec. 6, we
also show that emitted light is polarized by the same Fresnel reflection and transmission effects
that generate polarization from smooth visible-light surfaces. Because emission polarization is
actually antagonistic to reflection polarization, the combination of emitted and reflected lightoften causes most of the polarimetric signature to cancel, producing the characteristically low
degree of polarization values that are typically seen in this regime.6 In outdoor environments,
this cancellation can become small for imaging quite hot surfaces and blue sky (very cold)
backgrounds.
The advantage of going carefully through the steps of deriving emission polarization fromfirst principles shows that Kirchhoff’s law, which is often presented as an empirical formula, canbe derived from basic physics. It also allows us to quantify the range inside which Kirchhoff’s*Address all correspondence to Nathan Hagen, nh@hagenlab.org

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July 2022 • Vol. 61(7)Hagen: Review of thermal infrared polarimetry, I: theory

law is valid (Sec. 7) and to demonstrate that, although thermal emission may be a totally incoherent process, interference effects still play a role in determining emission properties when thinfilms (with thicknesses on the order of a wavelength) are present or when using high-resolution
spatial/spectral detection.
In addition to improving understanding of the theory, some motivations for pursuing thiswork are to resolve some questions that are not easily clarified by a casual read through theliterature. These questions include the following:
1. If a surface gets wet, with a very thin layer of water, how is its emissivity effected?
How thick does the water coating have to be to dominate the surface underneath?
2. Anodized aluminum parts are ubiquitous in optical systems, but if we anodize an aluminum surface, how does the process change its emissivity?
3. How should we model the emissivity of a thin (partially transparent) plastic film?
4. If we paint a surface, how thick should the paint be to obscure the underlying surface?
For that matter, why do painted shutters work so well as simulated blackbodies?
In the discussion below, we survey the theoretical tools needed to resolve these questions,though the discussion of anodize layers, and many of the details on paint layers are postponed
until Part II of the review. To the extent possible, we aim to provide quantitative answers to thesequestions and bring together into one place the various work scattered through the literature onthese topics. Part II of the review uses these tools to guide analysis of infrared imaging polarization measurements.

2 Calculating Emitted Surface Radiance with the Emission Density, jλ
The first widely known attempt to apply the tools of radiance transfer to the problem of volumetric emission from solids is that of McMahon in 1950.3 This work was later generalized by
Gardon in 1956 to be applied to non-normal angles of emission.4 Despite its beautiful physics,
the resulting model is not well known in the infrared imaging community even though it is
widely used in the heat-transfer literature. In the discussion below, we modify the conventional
derivation by centering the calculations around radiance rather than optical power. This has theadvantage of simplifying expressions while also modernizing the units, so that they are morefamiliar to the modern reader.
The calculation of the emissivity of an object can be quite involved if presented all at once, so
we start with a simple setup and add complexity to the model in stages. At the first stage of thecalculation, we aim to calculate the normal-angle emittance of a material with known refractive~
index nðλÞ and absorption index kðλÞ, i.e., a complex refractive index of nðλÞ
¼ nðλÞ þ ikðλÞ.
The electric field of a plane wave propagating along z is then written as
EQ-TARGET;temp:intralink-;e001;116;275

E ¼ A expð−i2π ñðλÞz∕λÞ expðiωtÞ;(1)

E ¼ Ae−i2πnðλÞz∕λ e−αðλÞz∕2 eiωt ;(2)

orEQ-TARGET;temp:intralink-;e002;116;232

where λ is the wavelength in vacuum and αðλÞ ¼ 4πkðλÞ∕λ is the intensity absorption coefficient.
In the discussion below, most variables will have a wavelength dependence. Writing an
explicit wavelength dependence, such as αðλÞ, for each variable quickly becomes tedious andmakes the equations more difficult to read. As a result, we generally drop the wavelength
dependence from intermediate results and include the explicit dependence when it helps toclarify context. In addition, rather than “emitting medium,” we generally refer to the region fromwhich we are calculating the emission as “the solid.” This simplifies the nomenclature, but readers should note that it can refer to liquids and gases as well, without difficulty.
We start by postulating that the medium is uniform in temperature and composition and has
negligible scattering and that each point within the solid can be characterized as emitting isotropic thermal radiation. For highly opaque materials, radiation emitted from within the volume
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(a)

(b)

(c)

Fig. 1 A plane-parallel slab of partially transparent emitting medium: “the solid.” (a) Radiation isemitted from volume d V toward surface element d A, along the ray at angle θ to the surface normal. (b) Geometry of the emission solid angle. (c) Each ray splits into transmitted and reflectedcomponents at the top and bottom surfaces. Summing over these secondary rays gives the total
light exiting the surface. (c) The red rays are emitted from d V in an initially upward direction,
whereas the blue rays are emitted in an initially downward direction.

is quickly absorbed by the surrounding matter and does not reach the surface. For semitransparent materials, however, this volumetric emission j can reach the surface, so calculating thelight emitted by the entire solid requires integrating j over the object’s volume. The setup forthis calculation is illustrated in Fig. 1.
Figure 1 shows the radiation emitted from an element of volume dV in the interior of the solidthat reaches a small area element dA of the surface, as originally constructed by Gardon.4 From
this geometry, Gardon develops an integral over dV and the differential solid angle dω, taking
advantage of the conical symmetry of the problem to define dV as a differential sector of aconical annulus. In this way, any region of the solid lying inside the region bounded by theblue lines [with differential angles dθ and dϕ, as indicated in Fig. 1(b)], produces radiation
propagating within the same element of the solid angle. The development of the integral is shownin Sec. 10, the end result of which is a simple one-dimensional (1D) integral along the ray
between dV and dA.
The quantity jðλ; TÞ is the material’s spectral emission density, at a given temperature T,
expressed in units of Wm−3 sr−1 μm−1 —spectral radiance per unit length. Thus, we can expectthat integrating jðλ; TÞ over a length will lead to the familiar spectral radiance LðλÞ given by
the Planck blackbody equation, viz.,Lbb ðλÞ ¼

EQ-TARGET;temp:intralink-;e003;116;348

2hc2
1; ½Wm−3 sr−1 μm−1 ;n2 λ5 ehc∕ðnλkB TÞ − 1(3)for Planck constant h, Boltzmann constant kB , and the refractive index of the medium n.
The “bb” subscript on L indicates that the radiance corresponds to a blackbody.
In the plane parallel medium shown in Fig. 1, the radiance j is generated in dV, lying at a
depth of z below the surface. As it propagates, the emitted light is attenuated by absorption, so
only the fraction expð−αz∕ cos θÞ arrives at the surface. Therefore, the element of radiance dL
arriving at the surface from this layer at depth z isdL ¼ je−αz∕ cos θ dz:

EQ-TARGET;temp:intralink-;e004;116;232(4)

By integrating this over the thickness of the slab, from z ¼ 0 to z ¼ b, we obtain the total spectral radiance arriving at the surface as
Z b
j
(5)
L¼j
e−αz∕ cos θ dz ¼ ð1 − e−αb∕ cos θ Þ cos θ:α0EQ-TARGET;temp:intralink-;e005;116;176

We simplify this asLðλ; θÞ ¼

EQ-TARGET;temp:intralink-;e006;116;121jðλÞ½1 − βðλ; θÞ cos θ;
αðλÞ

(6)

where we define
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EQ-TARGET;temp:intralink-;e007;116;735

β ¼ expð−αb∕ cos θÞ;(7)

as the bulk internal transmission of the entire slab, and we have assumed that the emission
density is uniform throughout the volume. For inhomogeneous materials, or for materials witha temperature that varies throughout the volume, the depth dependence of j prevents it frombeing pulled out of the integral as a constant.
So far we have ignored any secondary reflections inside the volume: the portion of light thatis not transmitted through the upper surface is instead reflected downward, and this radiance isρð1 − βÞðj∕αÞ cos θ. After traversing the volume of the slab (and attenuating due to absorption),
it reaches the far side of the slab, where it reflects again. Traversing the slab one more time brings
it back to the upper surface, where again a portion of the radiation transmits through the surface.
This second component is therefore
ρ2 β2 τð1 − βÞ

EQ-TARGET;temp:intralink-;e008;116;594

jcos θ;
α(8)where ρ2 is the two internal reflections, β2 is the attenuation along that doubled path, and τ is thetransmission through the surface. This process continues to occur, giving another contribution of
ρ4 β4 τð1 − βÞ

EQ-TARGET;temp:intralink-;e009;116;528

jcos θ;α(9)and other higher-order terms. Summing all of these together gives the total upwelling radiancereaching the inside of the surface asL↑ ¼ ð1 − βÞ

EQ-TARGET;temp:intralink-;e010;116;462

∞
X
j
cos θ
ρ2n β2n :
α
n¼0

(10)

So far, we have only considered upwelling emission from the layers inside the slab. Because
any thermal emitted radiance is isotropic, there is an equal amount emitted in the downward
direction. For this portion, light reaching the upper surface of the slab must first undergo one
reflection and suffer attenuation from traversing the volume of the medium, so the first portion ofradiance reaching the upper surface isρβτð1 − βÞ

EQ-TARGET;temp:intralink-;e011;116;353

jcos θ:
α(11)

Likewise summing all contributions from multiple reflections, we obtain the total downwellingradiance reaching the inside of the top surface asL↓ ¼ ð1 − βÞ

EQ-TARGET;temp:intralink-;e012;116;288

∞
X
j
cos θ
ρ2nþ1 β2nþ1 :
α
n¼0

(12)

The total radiance reaching the underside of the surface is the sum of the upwelling anddownwelling terms


∞
X
j
j 1−β
n
n
L ¼ L↑ þ L↓ ¼ ð1 − βÞ cos θρ β ¼
cos θ:(13)αα 1 − ρβn¼0
EQ-TARGET;temp:intralink-;e013;116;215

If the surface transmittance is τðλÞ, the spectral radiance L 0 ðλÞ that passes through the surface isj ð1 − βÞL0 ¼ τ
cos θ:(14)
α 1 − ρβ
EQ-TARGET;temp:intralink-;e014;116;154

Having reached this point, we do not yet know how this emitted radiance L 0 relates to thestandard method of representing surface-emitted radiance, viz.,Lobject ¼ ϵLbb :EQ-TARGET;temp:intralink-;e015;116;84

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If we consider a material that is highly opaque (large k), then we know that the bulk transmissionof the material is approximately zero (β ¼ 0). In this situation, Kirchhoff’s law is unambiguous,
giving ϵ ¼ 1 − ρ. In our model of emission from the solid Eq. (14), we use Kirchhoff’s law toexpress the surface transmittance as τ ¼ 1 − ρ. Thus, if we equate our calculated emitted radiance L 0 Eq. (14) with the familiar form of the object radiance Eq. (15), we have
ð1 − ρÞ

EQ-TARGET;temp:intralink-;e016;116;675

jcos θ ¼ ð1 − ρÞLbb :
α

(16)

jðλÞ
¼ Lbb ðλÞ:
αðλÞ

(17)

At normal incidence, we conclude that

EQ-TARGET;temp:intralink-;e017;116;622

This means that the emission density divided by the absorption coefficient is equal to the blackbody radiance for the same conditions of wavelength and temperature.

3 Straubel Theorem for Radiance Scaling Across an Interface
So far we have obtained the expression for the emitted radiance L2 in terms of the internal angle
θ1 but not yet in terms of the external angle θ2 . (Subscript 1 refers to quantities inside the solid,
and subscript 2 refers to quantities in the external medium—usually air.) For this, we need toapply the angle-dependent refraction effects at the surface, making use of a well-known theorem
in radiometry governing the scaling of the solid angle Ω across an interface [shown in (Fig. 2)],
apparently first discovered by Straubel in 1902.7,8 To start, from Snell’s law we know thatn1 sin θ1 ¼ n2 sin θ2 . Taking the derivative of both sides with respect to θ2 , this becomes

EQ-TARGET;temp:intralink-;e018;116;427

n1 cos θ1

dθ1¼ n2 cos θ2 ;
dθ2

(18)

which leads to

EQ-TARGET;temp:intralink-;e019;116;371dθ1 n2 cos θ2¼:dθ2 n1 cos θ1(19)

Using the definition of the differential solid angle, dΩ1 ¼ sin θ1 dθ1 dϕ, we also writedΩ1 ¼ sin θ1 dθ1 dϕ ¼

EQ-TARGET;temp:intralink-;e020;116;316n2
sin θ2 dθ1 dϕ;
n1

(20)

so thatFig. 2 Refraction of a cone of light, illustrating the geometry of Straubel’s theorem.
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EQ-TARGET;temp:intralink-;e021;116;735n2
dΩ1 n1 sin θ2 dθ1 dϕ n2 dθ1¼¼:sin θ2 dθ2 dϕdΩ2
n1 dθ2(21)

Substituting Eq. (19) into this, we obtain

EQ-TARGET;temp:intralink-;e022;116;686dΩ1 n22 cos θ2¼;dΩ2 n21 cos θ1(22)which is known as Straubel’s theorem.
If we ignore for a moment the reflection losses on passing through a smooth interface, thenthe power entering through an element of solid angle dΩ1 is equal to the power exiting theinterface through dΩ2 :L2 AdΩ2 ¼ L1 AdΩ1 :(23)

EQ-TARGET;temp:intralink-;e023;116;593

Substituting Eq. (22) into this and incorporate surface transmission loss τ, we haven2 τð1 − βÞL2 ¼ 22
Lbb cos θ2 :n1 1 − ρβ
EQ-TARGET;temp:intralink-;e024;116;550(24)This result, in which the emitted spectral radiance is proportional to the cosine of the exitanceangle θ2 , is commonly described as Lambert’s cosine law.
As we will see in Sec. 4 below, since the derivation of Straubel’s theorem assumes dielectriconly media, the theorem cannot hold in general for absorbing-material interfaces, yet how thetheorem should be expressed in the more general case of absorbing media has apparently not
been discussed in the literature. The silence on the subject is likely because media with significant k are generally opaque, so they have little transmitted light to measure, making this a case oflimited utility for most situations. Likewise, media with small k transmit sufficient light formeasurement, but the small value of k minimizes the effect on the refraction angle.
For intermediate cases, we need to use an expression giving the dependence of the refraction
angle on the solid’s absorption index [given in Eq. (26)].

4 Generalized Snell’s Law
When working with absorbing materials, Snell’s law is used directly with the complex index
n~ ¼ n − ik:
EQ-TARGET;temp:intralink-;e025;116;305

sin θ̃2 ¼

ñ1
sin θ̃1 ;
ñ2

(25)

where we have explicitly labeled each of the complex variables with a tilde. The meaning of acomplex-valued angle θ~ 2 may be unclear, but it is convenient for the purposes of calculating theFresnel reflection coefficients ρ. However, to get the correct result for emitted light from the solid
refracting into an ambient lossless medium (such as air), we can get a better feel for how thecomplex variables affect the refraction angles by working entirely in terms of real-valued geometry. For an interface between an absorbing emitter (the solid) and a nonabsorbing medium(the ambient air), we have n~ 2 ¼ n2 and n~ 1 ¼ n1 − ik1 . For this case, the expression relating
the real-valued angle inside the solid to the transmission angle in the ambient is given asn1
sin θ2 ¼
sin θ1 F−1∕2 ;(26)
n2
EQ-TARGET;temp:intralink-;e026;116;164

where the Snell’s law correction factor F is given by Bell et al. as9
F¼

EQ-TARGET;temp:intralink-;e027;116;107

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1 −a þ ða2 þ 4½n21 k21 − n22 ðn21 − k21 Þsin2 θ2 Þ1∕2;
2
k21 − n22 ½1 − ðk21 ∕n21 Þsin2 θ2070902-6

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(a)

(b)

Fig. 3 (a) Curves for θ2 versus θ1 refraction going from air ðn2 ; k 2 Þ ¼ ð1;0Þ into an absorbing
medium ðn1 ; k 1 Þ. (b) Curves for θ1 versus θ2 going from an absorbing medium ðn1 ; k 1 Þ into air.
For both plots, the solid lines give the generalized Snell’s law Eq. (26), and the dashed lines show
the standard Snell’s law using the real-valued refractive indices alone. Note that the material
constants are taken for λ ¼ 10 μm, except for the alumina curve indicated at 12.2 μm. Refractive
indices for each material are obtained from the following references: Al2 O3 ,10 H2 O,11 SiO2 ,12graphite,13 copper,13 paint,14 and aluminum.15

where a ¼ n21 − k21 þ n22 sin2 θ2 and is limited to cases in which the ambient medium is nonabsorbing (k2 ¼ 0).
In Fig. 3, a set of curves show the refraction angles given by Eq. (26), drawn together with asecond set of dashed curves that are obtained by taking the real parts of n~ 1 and n~ 2 and calculatedusing the standard Snell’s law. We can see that, for angles close to normal (small θ1 ; θ2 ), thestandard Snell’s law actually provides a good approximation to the generalized angle calculation.
We can also see this mathematically, by taking the approximation n2 sin θ2 ≪ fn1 ; k1 g in
Eq. (27), from which we obtain F ≈ 1 (where fn; kg means “both of n and k”). The approximation is not valid when k1 is comparable to or larger than n1 .
If we want to know the incident angle θ1 from within the solid for a given emission angle θ2 ,we use Eq. (26) in reverse. Although this produces a transcendental function, this representationof the Snell correction factor is easy to apply due to the smooth monotonic relationship between
θ1 and θ2 [see Fig. 3(b)]. Unfortunately, however, Eq. (27) does not provide us with the complex
form of the angle inside the solid, θ~ 1 , which we need in Sec. 5. The easiest way to get thecomplex θ~ 1 is to work out Snell’s law Eq. (25) backward, either using a nonlinear solver orby interpolation.

5 Polarization and the Complex Fresnel Equations
So far, all of our expressions for the emitted radiance and emissivity of the solid have been
written in terms of the Fresnel intensity coefficients for surface transmittance τ and reflectivity
ρ. The Fresnel coefficients are more often written in terms of complex amplitudes t and r, fromwhich the intensity coefficients are derived as ρ ¼ rr and a similar expression for the transmission. (The  superscript indicates a complex conjugate.) The most general form of the complex-valued Fresnel reflection and transmission, written for p-polarized light (electric field
vibration in the plane of incidence at the surface) and for s-polarized light (electric field vibration
perpendicular to the plane of incidence), are written as16,17r̃p ¼

EQ-TARGET;temp:intralink-;e028;116;195ñ2 cos θ̃1 − ñ1 cos θ̃2;ñ2 cos θ̃1 þ ñ1 cos θ̃2r̃s ¼

ñ1 cos θ̃1 − ñ2 cos θ̃2;ñ1 cos θ̃1 þ ñ2 cos θ̃2(28)

and the corresponding surface transmission amplitude coefficients are,17t̃p ¼

EQ-TARGET;temp:intralink-;e029;116;1372ñ1 cos θ̃1;ñ2 cos θ̃1 þ ñ1 cos θ̃2t̃s ¼

2ñ1 cos θ̃1:ñ1 cos θ̃1 þ ñ2 cos θ̃2(29)

Taking the magnitude squared of r and t, respectively, we obtain the intensity reflectivity andtransmittance as18–21
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July 2022 • Vol. 61(7)Hagen: Review of thermal infrared polarimetry, I: theoryρp ¼ r̃p r̃p ;ρs ¼ r̃s r̃s ;Refñ2 cos θ̃2 g;
Refñ1 cos θ̃1 g

τs ¼ ðt̃s t̃s Þ

EQ-TARGET;temp:intralink-;e030;116;735

(30)

and
τp ¼ ðt̃p t̃p Þ

EQ-TARGET;temp:intralink-;e031;116;701

Refñ2 cos θ̃2 g:
Refñ1 cos θ̃1 g(31)

The fraction at the rightside of τp and τs in Eq. (31) expresses the change in the beam crosssectional area upon refraction.
For unpolarized light, it is common to define an unpolarized average reflectivity ρa given as1
ρa ¼ ðρp þ ρs Þ:
2

EQ-TARGET;temp:intralink-;e032;116;617(32)

Because thermally emitted light from thick media is typically incoherent, the phase relationshipbetween s and p reflected light can be considered random, that is, emitted radiance is generally
partially linearly polarized, but not elliptically polarized, for thick media.
An example set of reflectivity curves is shown in Fig. 4, at λ ¼ 10 μm, for the cases of lightincident from air and reflecting from an absorbing medium. Thus, in this case, ñ1 ¼ n1 andθ~ 1 ¼ θ1 . Note that, although the water reflectivity curve looks like a curve for a nonabsorbing
interface, it does not have a true Brewster angle.22 Instead, it has a pseudo-Brewster angle(for λ ¼ 10 μm) at an external angle of incidence of θ2 ¼ 50.6 deg, where ρp ¼ 4.2 × 10−4 .
A difficulty with using the complex Fresnel equations in our situation is that the radiancetransfer approach allows us to know θ1 inside the solid, but it does not tell us the corresponding
value for θ~ 1. If we naively attempt to use θ1 in place of θ~ 1 in Eqs. (28) and (29), we will obtain
unphysical reflection and transmission coefficient values. Thus, if we invert Eq. (27) for thegeneralized Snell’s law, we can calculate the correct angle of emission from the solid. Instead,
we must invert Eq. (25) to obtain the corresponding complex angle θ~ 1 for use in the Fresnel
equations.
Another phenomenon that may be unfamiliar to many is that, for absorbing incident media,
interference between the incident wave and reflected wave causes nodes in the intensity distribution near the material interface, and this alters the behavior at the interface.23,24 As a result, theenergy conservation requires the expressionτ þ ρ þ μ ¼ 1;

EQ-TARGET;temp:intralink-;e033;116;340(33)

where μ can be positive or negative. This extra μ term, which has been termed the “mixed
Poynting vector,” becomes negative when a maximum node (intensity peak) occurs near the

Fig. 4 Fresnel surface reflectivity Eq. (30) as a function of angle of incidence, for light incident fromair onto five different polished materials, at 10 μm wavelength. Solid curves: s-pol, dashed curves:
p-pol. Refractive indices for each material are obtained from the following references: Al2 O3 ,10
H2 O,11 SiO2 ,12 graphite,13 and copper.13
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interface itself and positive when a minimum node (intensity valley) occurs near the interface,causing a small increase or decrease in the amount of transmitted light depending on whether thenode at the interface is a valley or a peak. This extra term is written as25,26
μp ¼ ðr̃p − r̃p Þ

EQ-TARGET;temp:intralink-;e034;116;699

ñ1 cos θ̃1 − ñ1 cos θ̃1;ñ1 cos θ̃1 þ ñ1 cos θ̃1

μs ¼ ðr̃s − r̃s Þñ cos θ̃1 − ñ1 cos θ̃1;ñ1 cos θ̃1 þ ñ1 cos θ̃1(34)

which at normal incidence simplifies to27

EQ-TARGET;temp:intralink-;e035;116;640μθ1 ¼0 ¼

4k1 ðn1 k2 − n2 k1 Þ:
n1 jñ1 þ ñ2 j2(35)

Researchers who discuss the mixed Poynting vector almost universally assert that μ is
“small” and therefore can be neglected.28–31 However, if one proceeds to substitute n; k values
for example materials into Eq. (34), one quickly finds that μ can be surprisingly large in thethermal infrared domain for a wide class of materials. It is quite common for metals to haven < 1, and in this regime, it is not unusual to have jμj > 1 (i.e., transmission >200%) at higher
angles of incidence, as shown in Fig. 5. Thus, what is missing from the literature is a descriptionof the correct context for calling μ “small.”Figure 5 shows example curves of μðθÞ for light transmitted across an interface from anabsorbing medium (in this case, silica and alumina) into air. For silica, we see that μ is small(jμj < 0.019), but it seems careless to refer to even a 2% change as negligible, especially as it
occurs at normal incidence and not just at large angles of incidence. For alumina, the larger
ratio between the absorption and refractive indices means that μ plays a larger role. Althoughμ ¼ −0.011 at normal incidence, it reaches μp ¼ −1.14 and μs ¼ −2.62 at an angle of incidence
θ 0 ¼ 79 deg. This does not violate Eq. (33) because we also find that τp ¼ 1.75 and τs ¼ 3.08
there.
From these results, we can conclude that the nonlinear interaction between the incident wave
and reflective wave at the interior boundary of the solid causes an enhancement (or in other cases,
diminishment) of the internally reflected wave intensity. Because the absorption coefficient ofthese materials is large, this enhancement will be small in comparison with the absorption experienced by the wave as it propagates back toward the interior of the solid. For all but the thinnestof films, this likely makes the effect difficult to observe using the standard experimental
approach of taking the ratio between the incident light and the transmitted light. However,semitransparent (and thus extremely thin) metal coatings on mylar films may provide the rightcombination of circumstances to observe this effect.
However, as we see in Eq. (36), the surface transmittance is proportional to the object’s
emittance. This means that any impact that the mixed Poynting vector has on the surface transmittance should be observable as a change in emittance. Either this impact has not been observed
before, due to the limited range of wavelengths and angles over which it generally occurs,
(a)

(b)

Fig. 5 Fresnel intensity reflection ρðθÞ, transmission τðθÞ, and modulation μðθÞ terms, calculated
for light emitted into air from within (a) silica and (b) alumina, using Eqs. (30), (31), and (34).
The refractive indices used here are the same as in Fig. 4.
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6 Emissivity
The emissivity is defined as the emitted radiance of a body in proportion to the emitted radianceof an equivalent (same temperature and same angle) blackbody emitter, so with Eq. (24) ourradiative model gives the spectral emissivity asϵm ðλ; θ2 Þ ¼

EQ-TARGET;temp:intralink-;e036;116;637

L2;m ðλ; θ1 Þ
L2;m ðλ; θ1 Þ¼
¼ τm ðλ; θ2 Þ½1 − βðλ; θ2 Þ:
Lbb ðλ; θ1 Þ
Lbb ðλÞ cos θ2(36)Here, “m” is used to index the polarization state as either p or s. When the solid is optically thick,so that as β → 0, we can see that this equation simplifies to ϵm ¼ τm —the emissivity of theobject is equal to the object surface transmittance. This is from the perspective of light generated
from inside the solid, refracting through the surface. On the other hand, we can look from theperspective of light in air incident on the surface of the solid. For an opaque solid, all lighttransmitted through the surface is absorbed, so this is also equivalent to ϵm ¼ αm , which is thebasic expression of the Kirchhoff law. Figure 6 shows curves obtained from Eq. (36) for water,
Al2 O3 (alumina), SiO2 (silica), graphite, and copper.
At normal angles, when the ambient medium is air (n2 ¼ 1, k2 ¼ 0), the emissivity Eq. (36)simplifies to
ϵ¼

EQ-TARGET;temp:intralink-;e037;116;472

4n1;
ðn1 þ 1Þ2 þ k21(37)

which is shown in Fig. 7. The curve for k1 ¼ 0 may appear to be nonphysical because a nonabsorbing medium cannot be an emitter, but this result can be taken as an asymptotic limit of a
medium in which k1 is brought closer to zero, whereas the medium itself becomes semi-infinite
in extent. If the limit is taken in this way, k1 can be brought arbitrarily close to zero, if not quite
all of the way to zero.
From Eq. (27), we can calculate the incident angle θ1 from the emission angle θ2 , and by
inverting Eq. (25) we can calculate the complex angle θ~ 1 that is needed for the general Fresnel
equations Eq. (31). Because β ¼ expð−αb∕ cos θ1 Þ is a function of the material thickness b, wefind that, for small values of thickness, ε increases approximately linearly with b, but that as b
becomes large, β approaches zero, and the emissivity approaches an asymptotic value. Following
Edalatpour, we can define a “critical thickness” bc at which the emissivity reaches 99% of its
asymptotic value.33 Taking the logarithm of Eq. (7)Fig. 6 Fresnel emissivity Eq. (36) as a function of angle of emission, for light emitted into air, fromair onto five different polished materials, at 10 μm wavelength. Each material is assumed to be
thick enough that β ≈ 0. Solid curves: s-pol, dashed curves: p-pol. The refractive indices used here
are the same as in Fig. 4.
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Fig. 7 Normal-angle emissivity of light emitted into air, for various ðn; k Þ. (Adapted from Ref. 32.)bc ðλÞ ¼ 4.6

EQ-TARGET;temp:intralink-;e038;116;545

λ cos θ1;
4πk1 ðλÞ(38)

for 4.6 ≈ lnð0.01Þ. In the thermal infrared (3 to 15 μm), typical values for k1 in metals are 30 to
90,34 giving critical thicknesses of 24 to 360 nm (i.e., less than a micron). Many nonmetal
absorbing materials have 0.05 < k1 < 5,35 giving bc ¼ 0.44 to 220 μm (i.e., well under 1 mm).
In the outdoors, the surfaces exposed to water tend to be hydrophilic, so water tends to completely wet the surface as a film rather than drawing together into beads. This includes mostglasses and ceramics and the most common metals (aluminum, steel, copper, etc.).36 After being
saturated with rainwater and allowed to drain, many surfaces tend to retain a thin water filmof roughly 3 μm thickness.37 Because water at λ ¼ 4 μm and 10 μm has absorption indicesof k1 ¼ 0.0046 and 0.051, the critical thicknesses at which the water completely obscures theemission and reflection of the underlying surface is about bc ¼ 0.8 mm. This thickness is only
achieved as a layer fully covering a surface when the water cannot fully drain, so we expect thatmost wet surfaces are not fully obscured by their overlying water films.

6.1 Emission from Films and Coatings
Although Eq. (36) allows us to calculate the emissivity at any angle of incidence, for a homogeneous material, many materials naturally form a surface oxide layer upon exposure to air. Purealuminum, for example, produces a natural amorphous oxide layer that is typically 2 to 4 nm
thick.38 Naturally, the question arises about how these surface layers modify the appearance of
these materials in the infrared. For nontransparent coatings, the object will take on the opticalproperties of the coating material if the layer is very thick. Conversely, it will maintain the opticalproperties of the substrate if the coating is quite thin. To define “thick” and “thin” for this usage,
we once again make use of the critical thickness bc of the coating material to make a good
approximation.
References 39 to 41 develop a model for going beyond the approximate answer that wesuggest here, for the case of transparent surface layers. For many coating layers, the coherence
length of blackbody spectra exceeds the coating thicknesses. As a result, even though the emission of the substrate is incoherent, interference effects produced by multiple reflections within
the surface layer will modify the coating optical properties. Modeling this requires the use of acoherent-light model that tracks the phase of a light wave passing through the coating, closely
resembling the approach typically found in thin film optical models at shorter wavelengths. The
resulting model accurately reproduces the modulations seen when monitoring the emissivity ofa film as it is being grown on top of a substrate (as in Fig. 8).
A general approach that allows for the thickness of the film to be of the same order as thecoherence length of the emitted light requires use of a partial coherence theory of thin film
optics.42,43 Such a partial coherence theory allows one to go smoothly between the coherent andincoherent extremes. We can also note that, although thermal emission is incoherent (has a very
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Fig. 8 Oscillations in the normal-angle emittance of a surface film as a function of the film thickness, shown for a diamond film on a silicon substrate, measured for λ ¼ 2.3 μm at a temperatureof 1073 K. (Data obtained from Ref. 33).

short coherence length), the appropriate coherence length to use when modeling a measurement
is also determined by the angular size of the detection aperture and the spectral bandwidth ofdetection, rather than solely by the size and spectrum of the source alone. For small solid angles
and narrow spectral bandwidths, even thermal light becomes coherent.
Although a model of a transparent coating on an absorbing substrate is quite useful, transparent coatings are somewhat rare among materials in the thermal infrared. Most coatings, suchas the natural oxide coatings that form on many metals upon exposure to air, are partially transparent and partially absorbing/emitting. To model this more general case, it is necessary toaugment the coating model to include not only interference of partial reflections but also
absorption and incoherent emission within the film. This is the approach taken in Ref. 33, whichmodels the effect of growing a thin metal film or an absorbing dielectric film on top of anabsorbing substrate. As we should expect, the model shows a continuous transition from theoptical properties of the substrate to those of the film as the film increases in thickness. Thechange is most rapid when the film is at its thinnest and asymptotically approaches its final
value when the film thickness exceeds bc (see Fig. 8). As with the transparent film, we alsofind oscillations in the emissivity curve as a function of thickness or wavelength.

7 Refining Kirchhoff’s Law
The basic form of Kirchhoff’s law expresses the equivalence of emission and absorption:
ϵm ðθ; ϕ; λÞ ¼ βm ðθ; ϕ; λÞ for polarization state m. In most of the literature, the conservationof energy for the interaction of light with media is invoked to say that probabilities of reflection,transmission, and absorption sum to 1: ρ∘m þ τ∘m þ β∘m ¼ 1. Using the equivalence property ofemission and absorption, this leads to the form of Kirchhoff’s law most commonly used inthermal imaging:ϵ∘m ¼ 1 − ρ∘m − τ∘m :EQ-TARGET;temp:intralink-;e039;116;232(39)

The superscript ∘ is used to indicate that these are properties of the object (apparent properties)
and not just a single surface (surface properties). For opaque media, τ∘ ¼ 0, in which case
Eq. (39) provides a simple method for obtaining the emissivity from the reflectivity ρ∘.
In the literature, Kirchhoff’s law is more often written as ϵm ¼ 1 − ρm − τm , i.e., with theFresnel coefficients ρ and τ rather than the apparent object reflectivity and transmittance. Doingso, however, ignores the effect of internal reflections within the object and internal scattering, so
it is strictly only true for either opaque τ∘ ¼ 0 or diaphanous ρ∘ media. In addition, there is also acommon assumption that, because the interface is an infinitesimally thin region, no absorption ispossible at the surface. If true, this would allow the surface transmittance to be calculated
from the complement of the reflectivity when the emissivity is negligible, τ ¼ 1 − ρ. However,for absorbing media in which the waves are incident from within an absorbing medium,
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we have seen that there can also exist a small amount of (positive or negative) “apparent surface
absorption” μ.
A weakness of using the Kirchhoff law for deriving ϵ from ρ is that its validity can be unclear
for inhomogeneous media and particularly for nonisothermal media. The direct approach ofmodeling the emissivity from the spectral emission density jðλÞ provides better insight into thephysical behavior.
Using the incoherent sums of partial waves within the solid, McMahon derived expressions
for the normal-angle emissivity, reflectivity, and transmittance as39m ð1−βÞ>ϵm ðλ; θ1 Þ ¼ τ1−ρ
>
β
>
>
hm
i
=
2 ð1−ρ Þ2βρ∘m ðλ; θ1 Þ ¼ ρm 1 þ 1−ρ 2mβ2
;
(40)
m
>
h
i
>
>
2
>
ð1−ρm Þ;τ∘m ðλ; θ1 Þ ¼ βρm 1−ρ2 β2
EQ-TARGET;temp:intralink-;e040;116;640mwhere ρ∘ and τ∘ are the apparent spectral reflectivity and spectral transmittance of the object,
respectively, and ρ and τ retain their meaning as the Fresnel intensity reflectivity and transmittance of the interface between the solid and the ambient medium, respectively. The wavelengthand angle dependence of ρ and β were suppressed for clarity. Adding all three together, withsome algebra we obtain the Kirchhoff law
ϵm ðλ; θ1 Þ þ ρ∘m ðλ; θ1 Þ þ τ∘m ðλ; θ1 Þ ¼ 1:

EQ-TARGET;temp:intralink-;e041;116;498(41)

Thus, for a homogeneous and isothermal medium, we have been able to derive Kirchhoff’s
law as a consequence of the Fresnel equations, for all angles and all polarization states. One
difference in Eq. (40) from McMahon’s results is that the expression for ϵm uses τm in placeof McMahon’s ð1 − ρm Þ. These are usually asserted to be equivalent, but we have seen from
Eq. (33) that, although they are typically close, they are not exactly equivalent due to the presence of the mixed Poynting vector term μ. Thus, it seems that the mixed Poynting vector may
point to a deviation from Kirchhoff’s law, for situations in which μ is significant.

7.1 Nonisothermal MediaTreatments of Kirchhoff’s law generally state that the law is valid when the emitting object is in
local thermodynamic equilibrium (LTE) and may further add that this is not satisfied when an
object is not isothermal. Because LTE is never exactly satisfied in practice, it is even more important to be able to say quantitatively to what degree a given situation permits the law to provide
accurate results, that is, rather than evaluating a situation as valid or invalid for applying the law,
we should instead express how closely we can expect the law to hold. First, we start with aheuristic analysis to gain insight into the shape of the problem before going on to developing
the path integral for the nonisothermal case.
We can start with an example of a painted metal plate that is exposed to the open air and iscooled by air convection currents and radiative losses. The paint coating has an absorption index
of k1 ¼ 0.47 at wavelength λ ¼ 10 μm. In this case, the skin depth of the coating is α−1 ¼λ4πk1 ¼ 1.7 μm. Regardless of the status of Kirchhoff’s law, we can expect that most of the emission from the solid will take place within this narrow region just below the surface. If the paintlayer is nonisothermal, then any thermal gradient will have to be significant within this skindepth. Because most paint coatings are 5 to 200 μm thick, this narrow region is the outermost
surface of the paint. If the temperature difference between the warm plate (the “substrate” of thecoating) and the surface of the coating is 5°C and the coating is 100 μm thick, then the temperature delta across the skin depth is only 5 K × ð0.7 μm∕100 μmÞ ¼ 35 mK. This gives the
maximum possible effect that the thermal gradient within this coating can have on the emissivity,
making the surface appear as if it were 35 mK warmer than it really is. In practice, the emission
by deeper layers is attenuated by absorption before reaching the surface, so the change in emission is even smaller than this and therefore below the noise level of most thermal infrared
imagers. With this result in mind, we can say that painted metal surfaces used as blackbody
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simulators will likely have an apparent radiative temperature that differs by less than ∼35 mKfrom the temperature of the underlying metal substrate as long as the high-emissivity coating
layer is kept to thicknesses near 10 μm.
To make this more quantitative, we develop a model for the emission of a nonisothermal
solid. The spectral emission density j becomes a function of z due to the linear depth-dependenceof temperature within the solid: T ¼ T s þ γz for surface temperature T s and temperature gradientγ (in units of K∕μm), so
jðλÞ ¼

EQ-TARGET;temp:intralink-;e042;116;651

2hc2
αðλÞ
:
λ5 exp½hc∕λkB ðT s þ γzÞ − 1(42)Inserting this into Eq. (4) gives the element of radiance dL emitted from within a layer ofthickness dz (see Fig. 1). Integrating this over the thickness of the solid, from z ¼ 0 to
z ¼ h, we obtain the spectral radiance arriving at the surface asZ
2hc2 α be−αz∕ cos θ
dz:
(43)
L¼
λ5
0 exp½hc∕λkB ðT s þ γzÞ − 1
EQ-TARGET;temp:intralink-;e043;116;570

Although this is a complicated integral, the expressionfðzÞ ¼ ðexp½hc∕λkB ðT s þ γzÞ − 1Þ−1

(44)

EQ-TARGET;temp:intralink-;e044;116;513

is close to linear when the gradient term is small in comparison with the surface temperature,γz ≪ T s . If we take the Taylor series expansion of this expression about the point γ ¼ 0, then we
obtain 
1aγzea∕T sfðzÞ ¼þ :::(45)
þ
T 2s ðea∕T s − 1Þ2ea∕T s − 1
EQ-TARGET;temp:intralink-;e045;116;445

for a ¼ hc∕λkB. Putting this expansion into Eq. (43), we obtain




γhc
cos θ
αbβ
L ¼ Liso 1 þ;1−cos θð1 − βÞ
αλkB T 2s 1 − e−a∕T s
EQ-TARGET;temp:intralink-;e046;116;387(46)

where Liso is the radiance obtained for an isothermal solid at T s . When the solid is semi-infinite
in extent (b → ∞), then this simplifies to
L ¼ Liso ½1 þ M;EQ-TARGET;temp:intralink-;e047;116;318

(47)

where

MðθÞ ¼

EQ-TARGET;temp:intralink-;e048;116;275


γhc
cos θ
αλkB T 2s 1 − e−hc∕ðλkB T s Þ(48)

is the nonisothermal modifier term. Although following a different derivation, this last result
agrees with that given by Ref. 44. We can note that M ∝ 1∕α, so that materials with large absorption index k tend to have values closely agreeing with Liso unless the thermal gradient is extreme.
Figure 9 shows example curves of MðθÞ for different paint coating thicknesses from 2 to200 μm and a temperature difference ΔT ¼ 4°C between the two sides of the coating (the uppersurface exposed to air and the lower surface touching the substrate). For very thin layers of paintthat still have a thermal delta of 4°C, we see that the nonisothermal effect on the emittance canreach as high as 10%. However, for more ordinary thermal deltas, the value of M quickly goes
below our ability to measure.
Because there is a wide variety of base media and additives in paints, there is a corresponding
wide range of refractive and absorption indices. Oppenheim gives a representative measurement of ðn1 ; k1 Þ ¼ ð1.52; 0.46Þ, whereas Richmond gives ðn1 ; k1 Þ ¼ ð1.4; 0.01Þ, and Tyo finds
ðn1 ; k1 Þ ¼ ð1.6; 0.1Þ.6,14,45 These values fall within the range of our own experimental measurements, in which the high-k paints generally contain a large number of pigment particles, making
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Fig. 9 Emission modifier M, calculated from Eq. (48) for different thicknesses b of paint coatings,
assuming ðn 1 ; k 1 Þ ¼ ð1.52; 0.1Þ, and a temperature difference of 4°C across the coating.

them appear black in the visible spectrum. The paint layers simulated in Fig. 9 assume intermediate values of ðn1 ; k1 Þ ¼ ð1.52; 0.1Þ.
In addition to the requirement that the emitting object be isothermal, one sometimes encounters the statement that the object must also be homogeneous for Kirchhoff’s law to hold.41
However, it is clear that inhomogeneous objects do not by themselves cause a violation of theequilibrium condition for Kirchhoff’s law because Kirchhoff’s law has been applied successfully
to multilayer films and to composite materials such as concrete.

7.2 Nonisothermal Behavior in Paints and Glasses
Figure 9 shows that a 4°C temperature difference across any paint coating more than 5-μm thickproduces such a small change in emissivity that it will be difficult to measure. Although thissimulation assumes a temperature delta of 4°C, the author’s experience with thin paint coatings
in the lab indicate that deltas of 0.2°C to 0.5°C are more likely (using contact thermisters tomeasure substrate and coating surface temperature). As a result, for blackbody simulator paintcoatings, we should be able to neglect the effect of deviating from isothermal sources.
If the thermal gradient becomes steeper and the absorption coefficient k becomes smaller(so that deeper regions participate in emission), the change in emissivity can become significant.
For example, for a coating layer with k ¼ 0.009, a temperature delta of 2°C across a layer of
5 μm thickness (i.e., a gradient of γ ¼ 0.4 K∕μm) produces a 1.5% change in emissivity at
normal incidence, falling off to 1.2% near the grazing incidence.
We can see from Eq. (48) that M is proportional to the thermal gradient γ and inversely
proportional to the absorption index k. Thus, M increases linearly with the thermal change
ΔT across the emitter’s skin depth. If we scale M by k∕γ, then we can use the same curve forall values of k, ΔT, and b, as shown in Fig. 10. The proportionality M ∝ γ∕k also shows why

Fig. 10 If the refractive index is held constant, the scaled modifier Mk ∕γ produces one curve forall different absorption indices k and different thermal gradients γ (i.e., all of the curves of Fig. 9).
The curves shown here are calculated from Eq. (48), with the k -values included in the curve labels
for ease of reference.
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metals present no difficulty for Kirchhoff’s law—the large k value and low γ of most metals
make M negligible in almost all circumstances.
Using the curves shown in Fig. 10, we can interpret the results in the following way. The blue
curve shows that, if there is a 4°C temperature difference between the substrate and the surface ofthe coating layer, for the coating layer thickness of 10 μm, then the deviation of emission fromthe isothermal case will be about 5%. One can likely achieve these kinds of conditions by heating
the substrate to a high temperature (such as 100°C using the painted kitchen pot shown in Fig. 3of Part II of this review) while cooling the surface with a fan blowing cool air. This illustrates theconditions under which the nonisothermal effect becomes important.
In the case of silica (SiO2 ) and alumina (Al2 O3 ) glasses, the emission skin depth at
λ ¼ 10 μm is 2.0 and 0.9 μm, respectively. For alumina, however, 10 μm happens to be close
to an absorption peak, so it is not necessarily representative of the material’s overall behavior in
the thermal infrared. In the midwave infrared (MWIR) (3 to 5 μm), for example, the absorption
coefficient for silica drops to k ¼ 0.00035—three orders of magnitude lower than the value at
10 μm. Similarly, the absorption index for alumina drops to 0.0018 and for paint (if we takepolymethyl methacrylate as representative of a plastic-based paint) to 0.00065. Thus, in the
MWIR at λ ¼ 4 μm, the skin depths of silica, alumina, and acrylic-base paint become 2.3,
0.5, and 1.2 mm, respectively. The lower absorption coefficients in the MWIR versus longwave
infrared (LWIR) mean that nonisothermal effects will be more likely in the midwave regime.
Although most natural materials, under normal circumstances, do no deviate significantly
from Kirchhoff’s law, one category of natural materials that consistently exhibits significant
departures from Kirchhoff’s law is layered fine particulate such as quartz sand powder.46
Reference 46 measured up to 6% deviations from Kirchhoff’s law for dry unpacked quartz dust.
The air gaps between particles reduce the thermal conductivity of the material while preventing
convective cooling; at the same time, this allows for sunlight to penetrate deeply and heat up theunderlying particle layers, producing a large thermal gradient. Another material having similar
behavior is lunar soil.47,48 One might expect fresh snow to have similar properties, due to having
a similar physical structure, but experimental measurements have shown that snow exhibits aclose correspondence to Kirchhoff’s law.49
One can of course envisage cases in which nonisothermal behavior has a predominant effecton emission. For example, most “transparent” materials in the infrared actually have a small
amount of absorption present. We can imagine a thick slab of such a medium, in which theslab thickness is larger than the material’s thermal penetration depth, but much shorter than the
optical absorption depth. Within such a thick slab, emission can take place over a large range of
distances, allowing the various emitting layers to have large differences in temperature.

7.3 Kirchhoff’s Law for Gases
When modeling gases in the thermal infrared, it is common to write Kirchhoff’s law simply as
absorption equals emission: ð1 − βÞ ¼ ϵ. Although one can interpret this as equivalent to
Eq. (41) when the reflectivity is zero and the transmission is ð1 − βÞ, equating absorption andemission is actually the most fundamental form of Kirchhoff’s law.50 As a result, it is not necessary to know the scattering (the diffuse reflection) by the gas cloud to calculate the emission.
Instead, one needs to know only the gas temperature T gas and its absorbance χ along the relevantline of sight. For a gas cloud of thickness l along the line of sight z, the absorbance is51
 

Z l
χ gas ðλÞ ¼ 1 − exp − σðλÞcðzÞdz ;(49)
EQ-TARGET;temp:intralink-;e049;116;1940

for gas concentration (i.e., number density) c and absorption cross-section σ. Equating theemissivity with the absorbance χ, the emitted radiance of a gas at uniform temperature is52
Lgas ðλÞ ¼ χ gas ðλÞLbb ðλ; T gas Þ:

EQ-TARGET;temp:intralink-;e050;116;125(50)

When the gas is nonuniform in temperature, however, one must divide the gas cloud into a
series of separate parcels, each with their own locally uniform temperature and their own local
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absorption value (for light passing from one side to the other side of the parcel along the line ofsight).53–55 Although this substantially increases the complexity of the calculation, it is a straightforward process to model. In the unlikely case of a linear thermal gradient along the line of sight
through the gas cloud, one reaches a result analogous to that of the linear gradient in a solid given
in Eq. (46). What is different from solids is that most gases are weak absorbers/emitters, so thatlight can propagate through a large depth of gas before extinction. In addition, the lack of direct
thermal conduction means that large differences in temperature between parcels of gas are
possible. As a result, modeling a gas cloud as an isothermal body with uniform absorbance/
emissivity via Kirchhoff’s law is often subject to larger errors than in the case of solids.

8 Rough Surfaces and Scattering MediaSo far we have considered only interfaces with smooth boundaries. When the boundaries arerough, incident light from one angle is scattered into a range of transmission and reflection
angles, requiring a modification to the emissivity model that we have used so far. Jordan et al.
provide an approach for doing this by modeling the orientation angle of the interface as comprising a probability distribution over angles.18,56,57 In general, we can expect that, as the surface
roughness becomes large, multiple reflections among surface facets increase, causing a reduction
in reflectivity and a consequent increase in the emissivity.14 The effect on emission polarization
is more complex. Following Chang et al., we can model the emitting surface roughness as consisting of triangular corrugations of a given angle ϕ (see Fig. 11). From the standpoint of theviewer, the emission angle is θ 0 , but the Fresnel equations for emission use the local surface
normal to calculate the polarization properties. Thus, if the corrugations are ordered and not
randomly oriented, this allows for polarized emission even at θ 0 ¼ 0, as shown in Fig. 11.
For emission angles above the local tilt ϕ, the surface emission is affected by shadowing, inwhich part of the facet emits directly into the external medium, but emission from the remainderof the facet is reflected from the opposite face, inducing a change in polarization onreflection.58,59 Although this shadowing effect makes analytical analysis of the problem difficult,numerical simulations have shown that the corrugation tends to increase overall emissivity, but
the shadowing effect tends to reduce the degree of polarization in the emitted light.18
Goldstein and Chenault show polarimetric measurements of reflection from various paints
for a range of angles of incidence.60,61 Although the measurements were taken in the visible andnear-infrared regions, they also contained very useful tables of the RMS surface roughness foreach paint type, showing that the RMS roughness ranges from 50 nm for glossy paints to as
much as 3 μm for matte paints.62

Fig. 11 Normalized linear polarization component of thermal emission from a corrugated surface
with tilt angles of 30 deg and 60 deg. The behavior shown here is for out-of-plane viewing angles,
at which the shadowing effect is irrelevant but reflection plays a role. (Inset) The emission geometry for a corrugated surface with facet tilts of ϕ and indicating the shadowed region that occurs foran emission angle of θ 0 . Light emitted from the shadowed region of the surface experiences reflection from the opposing facet. (Data for the curves shown here were obtained from Ref. 18.).
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Fig. 12 Kubelka–Munk model of light propagation through a strongly scattering medium, showing
upwelling flux I and a downwelling flux J in red and blue, respectively. The three reflectivities ρe , ρi ,and ρs represent the hemispherical-average reflectivities for light reflecting at the boundary
between two media.

Germer considers the effect of surface microroughness—surface scattering features that aresmall with respect to the light wavelength—and its effect on optical scattering and observed
polarization.63,64 Although Germer’s simulations provide only for the case of visible light, theoverall approach is equally applicable to infrared thermal emission and reflection and indicate
that, because microroughness and subsurface defects have a somewhat different angular dependence of polarization, the two effects can be separated from one another with careful measurements. When a material is itself inhomogeneous, so that light propagating through the medium
experiences a significant amount of scatter as it traverses the volume, the reflectivity andemissivity values calculated using a homogeneous material model are no longer reliable. One
approach to take in this case is to form a Monte Carlo model of the rays propagating through themedia, but this does not provide much physical insight to apply across various different media.
When the scattering is large, however, we can take the diffuse light approximation, in which case
we can apply the Kubelka–Munk approach to calculating the reflectivity.65 In this approach, lightis assumed to be evenly distributed in angle, so the calculated reflectivity is angle independent.
This allows the problem to be simplified into a one-dimensional differential equation. Making
use of the Kirchhoff law, we are then able to infer the emissivity from ϵ ¼ 1 − ρ.
In the diffuse light approach, we imagine the light as being composed of two streams, an
upwelling flux I and a downwelling flux J (shown as red and blue, respectively, in Fig. 12). Eachstream is subjected to absorption and scattering effects, for which we use coefficients K and S.
If we examine the behavior of the two light streams as they traverse a layer of differential thickness dz inside the medium, we see that upwelling flux I suffers from absorption loss (−IKdz)
and scattering loss (−ISdz) upon traversing the layer, but gains from backscattered downwelling
light (þJSdz). The downwelling flux J likewise suffers from absorption and scattering but gains
from backscattered upwelling light. As a result, the differential changes in fluxes across the layer,
defined as

EQ-TARGET;temp:intralink-;e051;116;227dI Iðz þ dzÞ − IðzÞ
¼
;
dz
dz

dJ JðzÞ − Jðz þ dzÞ
¼
;
dz
dz

(51)

are given by the pair of coupled differential equations
EQ-TARGET;temp:intralink-;e052;116;172dI¼ −KI − SI þ SJ;
dz

dJ¼ þKJ þ SJ − SI:
dz

(52)

The solutions for these equations are45
I ¼ I 1 ð1 − BÞeAz þ I 2 ð1 þ BÞe−Az
EQ-TARGET;temp:intralink-;e053;116;118

J ¼ I 1 ð1 þ BÞeAz þ I 2 ð1 − BÞe−Az


(53)

for
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A¼

EQ-TARGET;temp:intralink-;e054;116;735

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
KðK þ 2SÞ;B¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
K∕ðK þ 2SÞ;(54)

and I 1 , I 2 are constants to be determined by the boundary conditions of the problem.
Although the K and S coefficients are intended as experimentally determined parameters,
researchers have established that in many cases we can approximate the absorption coefficientas K ≈ 2α and the backscattering coefficient as S ≈ ση, where we model the medium ascomposed of discrete scatterers of scattering cross-section σ and number density η.66,67 A moredetailed calculation of K and S that takes into account specific material properties is presented inRefs. 68 and 69.
The steps toward calculating I 1 and I 2 are given in detail in Ref. 45, so we skip to thesolution:ρ ¼ ρe þ ð1 − ρe Þð1 − ρi Þ

EQ-TARGET;temp:intralink-;e055;116;603

ð1 − BÞMeAh − ð1 þ BÞOe−Ah;
MNeAh − OPe−Ah

(55)

where
EQ-TARGET;temp:intralink-;e056;116;548

M ¼ ð1 þ BÞ − ρs ð1 − BÞ;(56)

N ¼ ð1 þ BÞ − ρi ð1 − BÞ;(57)

O ¼ ð1 − BÞ − ρs ð1 þ BÞ;(58)

P ¼ ð1 − BÞ − ρi ð1 þ BÞ;(59)

EQ-TARGET;temp:intralink-;e057;116;505

EQ-TARGET;temp:intralink-;e058;116;483

EQ-TARGET;temp:intralink-;e059;116;461

and ρe is the “external” reflectivity for light incident from the ambient medium onto the surfaceof the coating layer; ρi is the “internal” reflectivity for light incident on the surface of the coating
layer from inside the coating; and ρs is the “substrate” reflectivity for light incident from thecoating onto the substrate. The substrate itself is assumed to be fully opaque. Each of these threereflectivities ρe , ρi , and ρs are hemispherical average reflectivities obtained from the Fresnel
equations. In the solution given in Eq. (55), the material reflectivity ρ is the diffuse reflectance,
so it is independent of angle or polarization.
For a direct model of scattering media that does not derive the emissivity from applying
Kirchhoff’s law to the reflectivity, it becomes necessary to account for the emission densityjðλ; TÞ within the medium itself. Xu and Shen show that this adds a new source term to eachof the equations in Eq. (53), requiring modification of the solution.70

9 Conclusion
One of the primary aims of this review was to gather together material illustrating how the elements in the theory of emission polarization can be derived from the Fresnel laws and volumetricblackbody radiation inside absorbing materials. We also considered the generalization of theStraubel theorem to absorbing materials, quantified the requirements for Kirchhoff’s law to hold,and discussed the potential role of the mixed Poynting vector on emission. The physics of thermal emission is likely one of the few places in which the mixed Poynting vector may play anontrivial role in practical measurement situations. Along the way through our survey, we also
provided some of the answers to the questions that originally motivated gathering together thetheory in one place. (Other answers await development until Part II of this review.) For example,
we found that a layer of water fully obscures its underlying substrate when 1 mm thick or more.
For hydrophilic surfaces in which water spreads widely without collecting into beads, water only
forms layers this thick when it cannot drain. Wet surfaces that can drain retain only very thinlayers of water, in which case the emission/reflection from the surface will be predominantly
from the substrate and not the water layer.
We also found that high-emissivity paints can be considered to fully obscure their substrates
when over about 10 μm thick, though this depends a lot on the type and density of colorants
used in the paint. Surprisingly, a fully obscuring paint at visible wavelengths is generally a good
guide that the paint will also be fully obscuring (or nearly so) in the thermal infrared as well.
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There is, however, some concern about the use of paints for blackbody simulators. If the heater
underneath the paint layer is pushed to a high temperature, so that the thermal gradient acrossthe paint becomes large, this gradient can cause measurable deviations from Kirchhoff’s law.
Finally, the McMahon–Gardon model of emission (Sec. 2) shows that the emission from athin (partially transparent) plastic film is not simple. Most plastics have a complex absorption
spectrum in the thermal infrared, particularly in the LWIR. Although some wavelengths arehighly transparent, others are significantly absorbing, making their spectral dependence a major
factor. In addition, any experimental measurements are prone to differences in composition andany coloring agents (such as carbon particles) or impurities. For example, clear plastic bags
are more transparent in the infrared than black plastic bags because the latter contain a large
concentration of absorbing carbon particles.

10 Appendix A: Gardon’s Integral
In Sec. 2, we said that a 1D integral along the ray is sufficient to model the emissivity from asolid slab. We can confirm this with a rigorous treatment of the differential solid angle elements
and the differential surface area element.
With the setup shown in Fig. 1, the volume element dV is part of an annular ring of radius r,with radial thickness dr and axial thickness dz. From the center of the ring, the elemental volume
dV subtends an angle of dϕ. In cylindrical polar coordinates ðr; ϕ; zÞ, we write the elemental
volume asdV ¼ rdrdϕdz:(60)

EQ-TARGET;temp:intralink-;e060;116;477

However, a more convenient representation is to rewrite this in terms of θ and dθ rather thanr and dr. Using the triangle formed by r, z, and θ,
r ¼ z tan θ:(61)

EQ-TARGET;temp:intralink-;e061;116;422

The geometry shown in Fig. 1 gives
EQ-TARGET;temp:intralink-;e062;116;379dθ ¼ ds∕l ¼

dr cos θ;z∕ cos θ(62)

so
dr ¼ ðz cos2 θÞdθ;(63)

rdr ¼ z2 ðtan θ∕cos2 θÞdθ:(64)

EQ-TARGET;temp:intralink-;e063;116;323

andEQ-TARGET;temp:intralink-;e064;116;279

Therefore, in terms of the more convenient coordinates ðz; ϕ; θÞ, the volume element is
dV ¼ z2

EQ-TARGET;temp:intralink-;e065;116;236

sin θdz dϕ dθ:cos3 θ

(65)

Again making use of the geometrical relationships shown in Fig. 1, we write the solid angle
element dω, which is the solid angle subtended by surface area element dA from dV, asdω ¼

EQ-TARGET;temp:intralink-;e066;116;169

projected area dA cos θ dA cos θ
cos3 θ
¼¼ 2¼ dA 2 :2
2
2
distancelz ∕cos θz(66)

Similarly, for the solid angle element dΩ defined within the blue lines drawn in Fig. 1(b)representing the solid angle subtended by dV from dA, we have
EQ-TARGET;temp:intralink-;e067;116;100

Optical EngineeringdΩ ¼

projected area cos θrdr dϕ¼¼ sin θdθ dϕ:distance2
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The element of volume dV emits power dP in the direction of dA:
dP ¼ jdVdω:EQ-TARGET;temp:intralink-;e068;116;723(68)

The power arriving at dA from dV is less than this because of absorption along the path.
Including path absorption, we havedP ¼ je−kz∕ cos θ dVdω:EQ-TARGET;temp:intralink-;e069;116;668(69)

The corresponding elemental radiance reaching the surface from inside the solid is given by
the power dP per units area dA per unit solid angle dΩ:dL ¼ dP∕ðdA dΩÞ:

EQ-TARGET;temp:intralink-;e070;116;613(70)

With this expression, we can solve for the radiance arriving at dA from within the solid byintegrating over all layers:
Z ∞ −kz∕ cos θ
Z ∞
Z z¼∞
dP
je
dV dω
dz ¼
dz ¼ jL¼
e−kz∕ cos θ dz:(71)
dA½sin θdθ dϕ
z¼0 dAdΩ
00
EQ-TARGET;temp:intralink-;e071;116;558

Acknowledgments
The author is grateful to Photron Inc. for allowing the use of their FLIR x6900sc InSb camerain the MWIR polarimetry experiments. The author would also like to thank Shuhei Shibata forhelp with conducting the FLIR A35 LWIR polarimetry experiments. A portion of this research
was funded by a grant from Denso Corp. and by JSPS KAKENHI (Grant No. JP 20K04516).

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