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Master’s thesis May 2019
Department of Physics and Mathematics University of Eastern Finland
Blackbody’s Far-Field Coherence
Mohammad AL Lakki
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Mohammad AL Lakki Blackbody’s Far-Field Coherence, 50 pages
University of Eastern Finland
Master’s Degree Programme in Photonics
Supervisors Prof. Tero Setälä
Prof. Ari Friberg
Abstract
Coherence properties of blackbody radiation in the far field is described in the frequency domain within the classical theory of electromagnetic coherence. The far-field cross-spectral density and its degree of coherence are analytically calculated at any two points using Luneburg equations. The re- sults are compared to those given in the literature, where each field component is propagated sepa- rately. At large polar angles of one of the observation points in the far field, the relative difference between the electromagnetic degrees of coherence (using both methods) becomes significant. The paraxial approximation of our expressions, the polarization of the field, and the radiant intensity is found to coincide with the previously published ones.
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Preface
To a tender Sequoia called Amina, Thank you for blessing me.
To a love seedling called Susu,
Thank you for standing by my side during the long library nights, and for much more.
To a Gazan Child called hope,
To an endangered Orangutan called hope,
To a Hainbuchen tree in Hambacher Forest called hope, Thank you for giving me the strength.
To my esteemed teachers and supervisors at UEF,
Thank you for teaching me and Thank you for your patience. Much obliged.
I would like to express a special gratitude to my supervisor Tero Setälä who was extremely patient with his observations and corrections. I am indebted with the successful completion of this work to his guidance and support.
10.05.2019 Mohammad AL Lakki
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Table of Contents
TABLE OF CONTENTS 4
1 INTRODUCTION 5
2 ELECTROMAGNETIC THEORY 7
2.1 Maxwell’s Equations 8
2.2 The Wave Equation 10
2.3 Solving the Wave Equation 11
2.4 Rayleigh Diffraction Formula 13
3 BLACKBODY RADIATION 18
4 SECOND ORDER COHERENCE THEORY 20
4.1 Stationary and Ergodicity 21
4.2 Coherence functions 23
4.3 Electromagnetic Coherence 27
5 BLACKBODY COHERENCE IN THE CAVITY AND IN THE APERTURE PLANE 30
6 BLACKBODY COHERENCE IN THE FAR FIELD 37
7 CONCLUSIONS 48
8 REFERENCES 49
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Chapter I 1 Introduction
The study of electromagnetic coherence forms one of the major pillar on which the edifice of optics rests. Until this day, nascent technologies and new techniques are derived from a thorough under- standing of the coherence and polarization of light fields. In this work, we focus on applying the second-order coherence theory to blackbody radiation (BBR) in the far-field to uncover its properties and behavior. It is not an exaggeration to say that historically the study of radiation emitted from a blackbody, after what was dubbed as the ultraviolet catastrophe, spurred the herald of modern physics and quantum mechanics in the early 20th century. The spectral energy density was analytically de- scribed through the introduction of what later became known as the photon [1]. In the 1960s, BBR coherence properties were discussed first by Bourret [2], Kano and Wolf [3], and Mehta and Wolf [4]. In their work, the BBR’s mutual coherence function, cross-spectral density, and their generalized degree of coherence inside the cavity was derived. In the 1990s, Gori et al. [5] introduced the super- position of plane waves model to describe the scalar-field correlations for statistically homogeneous and isotropic fields. Soon after, James [6] described the BBR’s polarization and intensity in the par- axial region of the far field. In the early 2000s, Setälä et al. [7] expanded the model to include elec- tromagnetic fields, and rederived the expressions of cross-spectral density, and its generalized degree of coherence for the BBR inside the cavity. Later on Lahiri and Wolf [8] derived the cross-spectral density in the paraxial region of the far field, and most recently Setälä et al. [9] in an all-encompassing paper described the coherence properties of BBR in the cavity, aperture plane, and in the whole far field. We carry on the work from there.
In [9], the 3 × 3 matrix of the cross-spectral density, CSD, of the far field was calculated by propa- gating the 3 × 3 aperture cross-spectral density using the Rayleigh diffraction formula for each field component. This approach led to a far-field CSD that is not transverse . We followed similar footsteps, however we rederived the expression for the CSD in the far field by propagating the field components
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using Luneburg method. We then calculated the electromagnetic degree of coherence from the ob- tained CSD and compared it to the one obtained in [9]. The polarization properties, the paraxial ap- proximation, and the radiant intensity were also obtained and compared to the results in [6,8,9].
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Chapter II 2 Electromagnetic theory
A positively charged body attracts a negatively charged one and repels a positively charged body by means of an electric force that obeys an inverse square law, Coulomb’s law. Together with the mag- netic force observed between magnets or current carrying circuit elements, due to Ampere’s law and Lorentz force, these forces (electric and magnetic) act between bodies at a distance from one another and thus like the gravitational force they fit into what was known as the “action at a distance” concept.
Faraday noticed that placing a dielectric between the two interacting charges would reduce the force between them. He also discovered electromagnetic induction, by which a current can be induced in a loop when it experiences a varying magnetic flux. Faraday therefore concluded that this action at a distance is merely a result of a field that exist and propagate between the interacting charges. We can thus decouple the interacting charges from the field between them and give the field a meaning, its own description and laws. However, it was James Clerk Maxwell who gave a full understanding and a mathematical formalism of the nature of this field, electromagnetic field, which permeates the space between the interacting charges and carries momentum and energy. As a consequence of Maxwell’s work, we know that the interaction between the charges is not instantaneous but delayed by a time equal to that taken by light to move between the bodies, as can be verified now by a simple radio communication between a transmitting antenna and a receiving one. Maxwell’s formalism, together with his theoretical prediction of a displacement current, which paved the way for producing high frequency radio waves by Hertz, [10-12], established the basis of the classical electromagnetic theory of light.
It should be noted here that the field on its own is not an observable quantity, we use it in this work and it is understood as an abstract vector quantity that exists at different points in space and time. We can measure its effects, the energy and momentum it carries, among other observables. Maxwell pre- sented the quantities as field values in an elastic solid. The equations, on their own, are validated by numerous experiments. However, the field quantities do not require a medium to exist.
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2.1 Maxwell’s Equations
We now describe the electric and magnetic field that permeates the space and time. It turns out that it is very complex and futile to give out formulas of the electromagnetic field due to a certain distri- bution of charge sources or currents. What we can do with ease is to give the laws which determine the electromagnetic field at a point from the values of the field at a neighboring point and thus solve any particular problem starting from the given boundary values (boundary conditions) [12]. This means utilizing differential equations. Below we list all of these differential equations, the esteemed Maxwell’s equations [10-12].
The first law is a reiteration of Gauss’s law which states that the sum of the normal components of the electric field, E, at all points making up an arbitrary closed surface is proportional to the charge, 𝑄 (free and bound), enclosed within that surface,
∯ 𝐄 ∙ 𝐧 𝑑𝑠 = 𝑄
𝜀𝑜 , (2.1)
and using the divergence theorem, this can be alternatively written as
∇ ∙ 𝐄 = 𝜌
𝜀𝑜 , (2.2)
where 𝜌 is the charge density and 𝜀𝑜 is the permittivity of space. The properties of any medium is thus included in the charge density which is a combination of a free charge density, 𝜌𝑓, and a bound charge density, a result of non-zero polarization, P (a consequence of an induced electric dipole mo- ment density within the material).
Using the electric flux density vector 𝐃
𝐃 = 𝜀𝑜𝐄 + 𝐏 , (2.3)
Maxwell’s first equation take the form
∇ ∙ 𝐃 = 𝜌𝑓 . (2.4)
The second law was a result of the discovery of electromagnetic induction by Michael Faraday where he demonstrated that a varying magnetic flux creates a force on static electric charges, meaning cre- ating an electric field. Maxwell wrote the differential form which states that the curl of an electric
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field vector at a point is proportional to the time derivative of the magnetic flux density 𝐁 at that point, where the negative sign accounts for Lenz’s law,
∇ × 𝐄 = −𝜕𝐁
𝜕𝑡 . (2.5)
The third law states that there are no magnetic monopoles
∇ ∙ 𝐁 = 0 . (2.6)
The fourth law is a combination of Ampere’s law, steady current producing a magnetic field, and Maxwell’s keen insight that as a varying magnetic field produces an electric field, so does a varying electric field produce a magnetic field. If magnetic fields are seen as results of currents, then the varying electric field is said to constitute a displacement current. In differential form, the law takes the form:
∇ × 𝐁 = 𝜇𝑜𝐉 + 𝜇𝑜𝜀𝑜𝜕𝐄
𝜕𝑡 , (2.7)
where 𝜇𝑜 is the free space permeability, and 𝐉 is the total current density.
However, and similarly to the discussion above, the magnetic field can be a result of free currents and bound currents which are due to the magnetization properties of the material. Rewriting the equation above in terms of free currents, 𝐉𝑓, is accomplished by introducing the magnetic field strength
𝐇 = 1
𝜇𝑜𝐁 − 𝑴 , (2.8)
where M, a material property, is the magnetization induced in the material in the presence of magnetic fields. Applying the curl to Eq. (2.8) leads to
∇ × 𝐇 = 𝐉𝑓+𝜕𝐃
𝜕𝑡 , (2.9)
where 𝐉 = 𝐉𝑓− 𝜕𝑷/𝜕𝑡 − ∇ × 𝐌 .
The displacement current of Maxwell is of enormous importance. If a varying magnetic field produces an electric field and a varying electric field produces a magnetic one. Then one can intuitively think that once we start the vibrations of the field, the field can “propel itself” and propagate. We show this
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below, and note that the field will propagate at a speed that is determined by the electric and magnetic properties of the medium and which account to the speed of light.
2.2 The Wave Equation
We now discuss the propagation of electromagnetic fields in mediums with the following properties:
1. Linear: The relation between the polarization P (response) and the external field E (stimulus) takes the form
𝐏 = 𝜀𝑜𝜒𝑒𝚬 ,
where 𝜒𝑒 is the electric susceptibility of the medium. In this case the response to a stimulus follows the superposition principle. Using this expression in Eq. (2.3), the relationship be- tween the electric flux density vector and the electric field becomes
𝐃 = ε 𝐄 , where ε is the permittivity.
2. Isotropic: Material properties are independent of stimulus orientation, i.e., ε is a scalar.
3. Nondispersive: ε is independent of frequency.
4. Nonmagnetic: Materials with no permanent magnetization, i.e. excluding ferromagnetism 𝐌 = 𝜒𝑚𝐇 , 𝐁 = μ 𝐇 ,
where 𝜒𝑚 is the magnetic susceptibility and μ is the permeability.
5. Homogeneous: Material’s permittivity is constant, ∇𝜀 = 0 . 6. Absence of free charge carriers: 𝐽𝑓 = 𝜌𝑓 = 0 .
If we apply the curl on Eq. (2.5), followed by the identity
∇ × ∇ × 𝐕 = ∇ (∇ ∙ 𝐕) − ∇2𝐕 ,
and Eqs. (2.3), (2.4), (2.7), and (2.8) for a medium with the properties discussed above, we arrive at the equation
∇2𝐄 − 𝜀𝜇𝜕2𝐄
𝜕𝑡2 = 0 , (2.10)
with
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𝜀𝜇 = 1
𝑐2 , (2.11)
and where c is verified to be the speed of light.
From Eq. (2.10) we see that in the mediums described above, the electric field propagates as a wave with a speed equal to that of light in that medium. Similar discussion applies to the magnetic field and we end up with the same wave equation describing its propagation.
Note that, only under the above-mentioned constraints, each electric field component satisfies the wave equation (scalar approach). This is a great simplification since the field can be studied now in a scalar way and we do not have to worry about its vectorial nature. We simply study each component on its own. However these equations have no way of expressing the values of the fields at different points in space and time, they describe how the field changes. To calculate the field values one need to use the boundary conditions; the value of the field or its derivative at a given surface. The boundary conditions may be introduced by an aperture plane, which is placed within the medium of propaga- tion. This will drift the wave description from the above-described constrained one. The boundary conditions will couple the E and H values, as well as the different E components [13]. The scalar approach will still produce meaningful and accurate results under the condition that the size of the aperture is significantly greater than the wavelength of the field.
2.3 Solving the Wave Equation
Equation (2.10) is a linear and homogeneous second order partial differential equation, which can be solved using the separation of variables method, the Fourier method. We thus invoke a trial form
𝐄(𝐫, 𝑡) = 𝐑(𝐫)𝑇(𝑡). (2.12)
Substituting Eq. (2.12) in Eq. (2.10) leads to 𝑇∇2𝐑 − 𝜀𝜇𝐑𝜕2𝑇
𝜕𝑡2 = 0 . (2.13)
Since R and T depend on different variables, this is possible only if
𝜕2𝑇
𝑇𝜕𝑡2 = 𝑐𝑠𝑡 . (2.14)
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A positive value of the constant in Eq. (2.14) will lead to an exponentially increasing or decaying time dependency in the solution. We can now see the harmonic time dependency if we call the con- stant −𝜔2 and thus obtaining
𝐓(𝑡) = 𝐴(𝜔)𝑒−𝑖𝜔𝑡 , (2.15)
where 𝐴(𝜔) is a time independent complex number and 𝜔 now plays the role of the angular fre- quency. The solution of Eq. (2.10) therefore assumes the form
𝐄(𝐫, 𝑡) = 𝐑(𝐫) 𝐴(𝜔)𝑒−𝑖𝜔𝑡 = 𝐄(𝐫, 𝜔) 𝑒−𝑖𝜔𝑡 , (2.16) where E(r, 𝜔) is called the complex field amplitude for a particular 𝜔. Substituting Eq. (2.16) in Eq.
(210) yields
∇2𝐄 +ω2
c2 𝐄 = 0 , (2.17)
which is called the Helmholtz equation.
There is nothing special about any particular frequency and the solution to the wave equation is con- structed by superposing all possible solutions together,
𝐄(𝐫, t) = ∫ 𝐄(𝐫, 𝜔)𝑒−𝑖𝜔𝑡
∞
−∞
𝑑𝜔. (2.18)
In this general description, the complex analytic signal, [17], is defined by excluding the negative frequency components, whose amplitudes carry no new information, as 𝐄(𝐫, −𝜔) = 𝐄(𝐫, 𝜔) . We can then proceed with solving the Helmholtz equation and here we present the solution as a superpo- sition of plane waves, each of which is easily seen to satisfy the Helmholtz equation
𝐄(𝐫, 𝜔) = ∫ 𝐄(𝐊, 𝜔)𝑒±𝑖𝐊∙𝐫
∞
−∞
𝑑3𝐊 ,
(2.19) where 𝐊 = 𝑘𝑥𝐢 + 𝑘𝑦𝐣 + 𝑘𝑧𝐤 and 𝑘 = |𝐊| = 𝜔/c .
Maxwell’s first equation, Eq. (2.4), combined with the introduced medium constraints in Sec. 2.2 lead to
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∇ ∙ 𝐄(𝐫, 𝑡) = 0 . (2.20)
Inserting Eq. (2.18) in Eq. (2.20) leads to ∇ ∙ 𝐄(𝐫, 𝜔) = 0, which when Eq. (2.19) is used results in
± 𝐊 ∙ 𝐄(𝐊, 𝜔) = 0 . (2.21)
This means that for a plane wave, the electric field vibrations are in a plane perpendicular to the propagation direction. The same can be said for the magnetic field whose discussion we did not in- clude. We hence talk about transverse electromagnetic wave under the above described material con- straints.
2.4 Rayleigh Diffraction Formula
In an integral form, we now present the general solution of the scalar monochromatic wave equation under a set of boundary conditions which are imposed by an infinite opaque surface with a plane aperture in the middle [13].
Green’s theorem is an integral theorem that relates the surface and volume integral of two scalar complex functions. It is expressed as
∭(𝐸∇2𝐺 − 𝐺∇2𝐸)d𝑉 = ∬ (𝐸𝜕𝐺
𝜕𝑛 − 𝐺𝜕𝐸
𝜕𝑛)
𝑆 𝑉
𝑑𝑠 , (2.22)
where E and G are any single valued complex valued functions which are continuous and differenti- able along the domain of interest, 𝜕/𝜕𝑛 is the directional derivative along a unit normal vector n that is pointing out of the surface S which is enclosing a volume V,
𝜕𝐹
𝜕𝑛 = ∇𝑉 ∙ 𝐧 , (2.23)
for any scalar complex function F.
2.4.1 Integral Theorem of Helmholtz and Kirchhoff
The function G is a known auxiliary complex function which can be selected, following Kirchhoff and Helmholtz, in such a way as to make the volume integral vanish. E is interpreted as our scalar electric field which satisfies the homogenous Helmholtz equation,
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∇2𝐸 + 𝑘2𝐸 = 0 . (2.24)
The function G in Eq. (2.22) is chosen as a spherical wave solution to the Helmholtz equation out of a point p in a region of volume V and surrounded by a closed surface S as shown in Fig. 2.1,
𝐺 = 𝑒𝑖𝑘𝑟 𝑟 ,
∇2𝐺 + 𝑘2𝐺 = 0 .
(2.25)
If we use Eqs. (2.24), and (2.25) in Eq. (2.22), we arrive at ∬ (𝐸𝜕𝐺
𝜕𝑛 − 𝐺𝜕𝐸
𝜕𝑛)
𝑆
𝑑𝑠 = 0 . (2.26)
It is easily noticed that the singularity of G at 𝐏 means that it does not satisfy the required continuity condition within the volume V. This problem is solved by enclosing the singularity with a small sphere of radius 𝜀 which tends to zero (see Fig. 2.1). The volume of interest of Eq. (2.2.2) is now between the small sphere and the surface S. Thus, the enclosing surface 𝑆′ is now formed of S and the boundary of the small sphere, 𝑆𝜀
𝑆′ = 𝑆 + 𝑆𝜀 . (2.27)
and Eq. (2.26) becomes,
∬ (𝐸𝜕𝐺
𝜕𝑛 − 𝐺𝜕𝐸
𝜕𝑛 𝑑𝑠)
𝑆′
= 0 . (2.28)
Figure 2.1 A schematic diagram showing the observation point 𝐏 enclosed by two surfaces: a small one, 𝑺𝜺, with an infinitesimally small radius 𝜺 and a bigger one, S, with radius r.
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Expanding Eq. (2.28) using Eq. (2.27) yields,
∬ (𝐸𝜕𝐺
𝜕𝑛 − 𝐺𝜕𝐸
𝜕𝑛)
𝑆
𝑑𝑠 = ∬ (𝐺𝜕𝐸
𝜕𝑛− 𝐸𝜕𝐺
𝜕𝑛)
𝑆𝜀
𝑑𝑠 . (2.29)
Using
𝜕𝐺
𝜕𝑛 = ∇𝐺 ∙ 𝐧 = 𝑒𝑖𝑘𝑟 𝑟 (1
𝑟− 𝑖𝑘) , (2.30)
obtained from the definition of G in Eq. (2.25), implies
∬ (𝐺𝜕𝐸
𝜕𝑛− 𝐸𝜕𝐺
𝜕𝑛)
𝑆𝜀
𝑑𝑠 = 4𝜋𝜀2[𝑒𝑖𝑘𝜀 𝜀
𝜕𝐸
𝜕𝑛− 𝐸 𝑒𝑖𝑘𝜀 𝜀 (1
𝑟− 𝑖𝑘)]. (2.31)
If the limit 𝜀 ⟶ 0 of Eq. (2.31) is substituted in Eq. (2.29), we arrive at 𝐸P = − 1
4𝜋∬ (𝐸𝜕𝐺
𝜕𝑛− 𝐺𝜕𝐸
𝜕𝑛)
𝑆
𝑑𝑠 , (2.32)
which is an expression for the field at point 𝐏 in terms of the field and its derivative on a boundary enclosing 𝐏.
2.4.2 Rayleigh Sommerfeld diffraction formula
For an observation point behind an aperture plane (see Fig. 2.2), we divide S into two surfaces, a planar surface on the shadow of the aperture plane, 𝑆′, and a circular one, S, with radius R, taking the
Figure 2.2 A schematic showing an observation point 𝐏 enclosed by the surfaces 𝑺′, which in- cludes the aperture A, and S with radius R.
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observation point 𝐏 as the center. The radius R is taken to be orders of magnitude larger than the wavelength.
From Eq. (2.32), the field at point 𝐏 will be given by, 𝐸p = − 1
4𝜋 ∬ 𝐸𝜕𝐺
𝑆′+𝑆 𝜕𝑛
− 𝐺𝜕𝐸
𝜕𝑛 𝑑𝑠 , (2.33)
with
∬ (𝐸𝜕𝐺
𝜕𝑛 − 𝐺𝜕𝐸
𝜕𝑛)
𝑆
𝑑𝑠 = 0 , (2.34)
under the Sommerfeld radiation condition for large values of R [13]; this is true for diverging waves at S where the field, E, is vanishing at a rate greater or equal to 1/𝑅2.
The integral in Eq. (2.33) can be calculated along 𝑆′ by using the two Kirchhoff’s boundary condi- tions:
1. The field and its differential in the aperture plane are unperturbed by the presence of the screen (opaque wall).
2. The field and its differential are zero outside of the aperture.
Thus,
𝐸p = − 1
4𝜋∬ (𝐸𝜕𝐺
𝜕𝑛− 𝐺𝜕𝐸
𝜕𝑛)
A
𝑑𝑠 . (2.35)
where A indicates integration over the aperture only.
Clearly, the above assumptions cannot coexist. For they mean that the field must be zero everywhere as well. However, under the condition of the aperture being much larger than the wavelength, it was found that the Helmholtz Kirchhoff integral together with Kirchhoff’s boundary conditions produce a reliable solution [13]. Sommerfeld worked around this problem and reduced the Kirchhoff’s bound- ary conditions to one, by using another auxiliary function 𝐺′, a superposition of two spherical waves.
One diverging from the observation point, and its mirror image with a 𝜋 phase shift between them.
This will cancel the integrand which depend on the differential of the field since 𝐺′ will be zero on the surface 𝑆′.
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Using Sommerfeld’s auxiliary function 𝐺′, the field at the observation point will be given by, 𝐸p= − 1
2𝜋∬ 𝐸𝜕𝐺
𝜕𝑛
𝐴
𝑑𝑠 . (2.36)
This is known as the first Rayleigh-Sommerfeld solution, and it is obtained after using 𝐺′ in Eq. (2.35) [13].
2.4.3 Luneburg Method
Treating light field as a scalar quantity, using Green’s theorem, and with the help of Kirchhoff bound- ary conditions, the field at a point far behind the aperture plane can be calculated using the first Rayleigh-Sommerfeld solution. However when we take into account the vectorial nature of light, we shall calculate the x and y components of the field using Eq. (2.36), and calculate the z component from the first Maxwell equation, the Luneburg approach.
Taking the aperture plane to lay in the x-y plane. The x and y field components at a point far from the aperture become,
𝐸𝑥 = − 1
2𝜋∫ 𝐸𝑥𝜕𝐺
𝐴 𝜕𝑧
𝑑𝑠 ,
𝐸𝑦 = − 1
2𝜋∫ 𝐸𝑦𝜕𝐺
𝐴 𝜕𝑧
𝑑𝑠 . (2.37)
Inserting Eqs. (2.37) in Eq. (2.20) and after interchanging the order of integration and differentiation, we arrive at
𝐸𝑧 = 1
2𝜋(∫ 𝐸𝑥𝜕𝐺
𝐴 𝜕𝑥
𝑑𝑠 + ∫ 𝐸𝑦𝜕𝐺
𝐴 𝜕𝑦
) 𝑑𝑠 . (2.38)
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Chapter III 3 Blackbody radiation
Accelerating charges produce electromagnetic radiation. A body at a certain temperature T will thus radiate as a consequence of its accelerating subatomic particles. This is known as thermal radiation.
A blackbody will absorb all radiation falling on it and emit all radiation as a sole result of its uniform and fixed temperature. It is thus in thermodynamic equilibrium with its surrounding. It is often mod- eled by a small opening in a large empty cavity [14]. As a result, a blackbody has a spectral energy density, 𝜌(𝜈) where 𝜈 is the frequency, with a unique characteristic, irrespective of the shape of the body and defined by the body’s temperature alone. Fig. 3.1. shows blackbody’s spectral energy den- sity at different temperatures.
Figure 3.1 Blackbody's spectral energy density for: T = 1500 K (blue), T = 3000 K (red), and T=5000 K (green)
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The distributions depicted in Fig. 3.1, were historically obtained experimentally in 1897/98 by the means of a small hole in an oven, a grating to radially separate different emitted frequencies, and detectors for measuring the spectral energy density [15]. The inversely proportional relationship be- tween the peak wavelength and the temperature was given by Wien’s law but the formula describing the spectral energy density could not be obtained at that time for it required the quantization of the electromagnetic field energy as summarized below.
The blackbody walls are considered to be perfectly conducting. Therefore any electromagnetic field inside the body must vanish at the walls due to the boundary conditions, creating standing modes with nodes at the walls. This limits the wavelengths (and thus the frequencies) to certain permissible val- ues. For a rectangular shaped cavity, we can count the possible standing modes within a unit volume in the K-space between frequencies 𝜈, and 𝜈 + d𝜈 considering the 2 polarization directions for each K value. After multiplying that number by the average energy per mode, we will get 𝜌(𝜈). A detailed description is given in [16]. Classically, we would expect all energy values to be permissible for every mode and using Boltzmann statistics we can calculate the average energy per mode to be 𝑘𝐵T, with 𝑘𝐵 being the Boltzmann constant. This can be directly seen by regarding each mode as a harmonic oscillator with two degrees of freedom. This approach will lead to the ultraviolet catastrophe, meaning while it accounts for the spectral energy density for longer wavelengths, however its value continues to quadratically increase with increasing frequency. A result which is obviously irrational as it implies infinite energy density. If the energy between the walls and the field can only be exchanged in an indivisible quanta of energy - later known to be photons - whose values are directly proportional to the frequency of the mode, then using Boltzmann statistics on what is now a discrete set of permissible energies (𝑛ℎ𝜈 with n being an integer) will result in an average energy per mode which will lead to a description of 𝜌(𝜈) which agrees with the experimental results. Max Plank discovered this approach [1], although he did not give an interpretation to the quantized permissible energy values, and the description of the spectral energy is now known as Plank formula.
𝜌(𝜈) =8𝜋 𝜈2 𝑐3
ℎ𝜈
exp(ℎ𝜈/𝑘𝐵𝑇) − 1 , (3.1)
where ℎ is Planck’s constant, and c is the speed of light in vacuum.
20
Chapter IV 4 Second order coherence theory
Under a certain set of initial electromagnetic field values (and their derivatives), Maxwell’s equations will yield deterministic values of the field at any later time and position. Certainty, however, is not a characteristic of real physical sources. Even the most carefully designed and established laser light carries an inherent uncertainty, however small, due to the unpredictable nature of spontaneous emis- sions and setup vibrations. This inherent uncertainty of the dynamics of the source, yields fluctuations in the field values within a very short time interval, usually shorter than the detection time in experi- ments that we might carry. Hence, what we are measuring are averages and to comprehend any result, like the behavior of electromagnetic (EM) fields or their characteristics, we need to study the field through statistical analysis. For example, different relationships between the random field fluctua- tions at two different points in time or space (or both) control the interference effects and express themselves in the visibility of the interference pattern produced in the Michelson or Young’s inter- ference experiments. Coherence theory infuses the mathematical framework of probability and sta- tistics, like correlation functions, to quantify the fluctuating fields and relate them to measured results.
It discusses the fluctuating field properties and the way they propagate. It is thus the study of the measurable aspects of light, and how measurements are related to light’s nature. Like how statistical mechanics bridges the macroscopic observable properties of matter to its microscopic nature. In the breadth of coherence theory, EM fields are described by stochastic processes, a set of variables or signals (ensembles). In this thesis, we are concerned with measurements that correlate the field at two space-frequency points (𝐫, ω), therefore we are limited to the second order coherence theory.
21
4.1 Stationary and Ergodicity
A stochastic process, like the EM field emanating out of a blackbody aperture, is described by an ensemble of realizations. Thus at each point in time or space, the field has a probability to take a certain value. Different realizations reflect the different possible values. Another way of stating this:
as much as the experimenter is concerned, the source does not yield a deterministic single signal but rather a set of probability densities. As an example and merely for simplicity, if we consider a linearly polarized field, then some of the electric field realizations, E (which can be represented by a scalar value in this case) will progress in time like in Fig. 4.1.
The realizations are actually infinite in number and time, and we considered a quasimonochromatic light in Fig. 4.1. This means that the field can be written as a superposition of monochromatic waves with frequency bandwidth, ∆𝜐, that is significantly less than the average frequency, 𝜐̅; ∆𝜐 ≪ 𝜐̅.
Fig. 4.1 Different realizations of a quasimonochromatic linearly polarized electromagnetic field.
22
The realizations in Fig. 4.1 reflect the nature of the source’s fluctuation and thus their shape is defined by the probability densities, 𝑝(𝐸1, 𝑡1), 𝑝(𝐸1, 𝐸2, 𝑡1, 𝑡2), … . Their interpretations are: 𝑝(𝐸1, 𝑡1)d2𝐸1 is the probability that the field will take a value within the area element d2𝐸1 around 𝐸1 at time 𝑡1. In addition, 𝑝(𝐸1, 𝐸2, t1, t2)𝑑2𝐸1𝑑2𝐸2 is the probability that the field will take a value around 𝐸1 and within 𝑑2𝐸1 at time 𝑡1, and a value around 𝐸2 and within 𝑑2𝐸2 at time 𝑡2.
The intensity of the field for example is thus given at each time by
𝐼(𝑡) = 〈𝐸∗(𝑡)𝐸(𝑡)〉 = ∫|𝐸|2𝑝(𝐸, 𝑡)𝑑2𝐸 , (4.1) and the correlation between the field values at two instants of time is given by
Γ(𝐸1, 𝐸2, 𝑡1, 𝑡2) = 〈𝐸1∗(𝑡1)𝐸2(𝑡2)〉 = ∫ 𝐸1∗(𝑡1)𝐸2(𝑡2)𝑃(𝐸1, 𝐸2, 𝑡1, 𝑡2)𝑑2𝐸1𝑑2𝐸2 , (4.2) where in Eqs. (4.1), and (4.2), <> is called the ensemble average and ∗ denotes complex conjugation.
When the probability densities that characterize the fluctuations in the field values are independent of any particular value of time, the process is called stationary, [17]. In this case, the ensemble average of any field quantity is independent of time,
𝑝(𝐸1, 𝑡1) = 𝑃(𝐸1, 𝑡2) = ⋯ = 𝑃(𝐸1, 𝑡𝑛) , 𝑝(𝐸1, 𝐸2, 𝑡1, 𝑡2) = 𝑃(𝐸1, 𝐸2, 𝑡, 𝑡 + 𝜏) ,
where 𝜏 = t2− t1. As a corollary, this means that an important parameter of the field, the intensity, is constant and does not change with time. It also means that the correlation function between two instants of time is only dependent on the time interval between them
𝐼(𝑡1) = 𝐼(𝑡2) = ⋯ = 𝐼(𝑡𝑛) = 𝐼 ,
Γ(𝑡1, 𝑡2) = Γ(𝜏) .
In applications, this implies that if we pass the light through a Michelson interferometer, then the average pattern that we obtain on the screen is steady, or stationary.
However, calculating ensemble averages is no mean feat. For fields emanating from a black body, which is our interest, the ensemble average is equal to the time average. Processes that obey such a rule are called ergodic, [17], and under this condition
23
𝐼(𝑡) = ∫|𝐸(𝑡)|2𝑑𝑡 , (4.3)
and the two-time correlation function becomes
𝛤(𝜏) = ∫ 𝐸1∗(𝑡)E2(𝑡 + 𝜏)𝑑𝑡 . (4.4)
A stationary process whose autocorrelation function goes to zero after a finite time interval is ergodic [20]. Under this property, a single realization carries all the measurable information and can be viewed as a concatenation of a set of finite stationary realizations [17].
4.2 Coherence functions
We now introduce four coherence functions: the two-time mutual coherence function, its complex degree of temporal coherence, the two-frequency cross-spectral density function, and its complex degree of spectral coherence [18]. These functions lie at the center of the second order coherence theory as they interpret the measured observables, like intensity on a screen, in accordance with the field fluctuations.
4.2.1 Mutual Coherence Function
Let 𝐸(𝐫1, 𝑡) and 𝐸(𝐫2, 𝑡) describe the complex analytic signals for the fields in apertures at points defined by position vectors 𝐫1, and 𝐫2 in a double-slit experiment as shown in Fig. 4.2. We are as- suming a scalar representation of the field and defer the vector properties for the later section when we discuss polarization. The fields then propagate, overlap, and interfere on the detector at point r.
Notice that the fields from 𝐫1, and 𝐫2 cover different path lengths (x represent the path length differ- ence) to reach different points on the screen, which in turn accounts for the time delay 𝜏 between the fields,
𝜏 = 𝑥
𝑐 . (4.5)
24
Figure. 4.2 Schematic of young's double slit experiment. Vectors 𝐫𝟏, 𝐫𝟐, and r represent the posi- tions of the apertures and detection point, respectively. While x represent the path difference be-
tween the aperture positions and the observation point.
The intensity variation on the screen will now tell us about the correlation [defined in Eq. (4.4)]
between the fields at the points 𝐫1, and 𝐫2when a time delay 𝜏 is introduced between them.
The intensity at point r on the screen will be given by [20]:
𝐼(𝐫) = 𝐼(1)(𝐫) + 𝐼(2)(𝐫) + 2√𝐼(1)𝐼(2)ℜ{γ(𝐫1, 𝐫2, 𝜏)} , (4.6)
where 𝐼(1)(𝐫) is the intensity at point r if only the slit at 𝐫1 was open, 𝐼(2)(𝐫, 𝑡) is the intensity at point r if only the slit at 𝐫2 was open, and γ(𝐫1, 𝐫2, 𝜏) is the normalized autocorrelation function, Γ(𝐫1, 𝐫2, τ),
𝛾(𝐫1, 𝐫𝟐, τ) = 𝛤(𝐫1, 𝐫𝟐, 𝜏)
√𝐼(𝐫1)𝐼(𝐫2) . (4.7)
In the language of coherence theory, 𝛤(𝐫1, 𝐫𝟐, τ) is called the mutual coherence function, MCF, and 𝛾(𝐫1, 𝐫𝟐, τ) is called the complex degree of coherence. It can be seen from Eqs. (4.6) and (4.7), how the correlation between field values manifest themselves on the detected intensity pattern. Absence of correlation between field values for instance will result in the absence of any interference.
We can quantify the strength of the interference pattern by calculating the visibility at any point r, using the definition
𝑉(𝐫) =𝐼max− 𝐼min
𝐼max+ 𝐼min , (4.8)
r
25
where 𝐼max is the closest maximum intensity to a point defined by the positon vector r on the screen.
This is the case when ℜ{γ(𝐫1, 𝐫𝟐, 𝜏)} = |γ(𝐫1, 𝐫2, 𝜏)|, the magnitude of the complex degree of coher- ence and 𝐼minis the minimum intensity in the vicinity of r, ℜ{γ(𝐫1, 𝐫𝟐, 𝜏)} = −|γ(𝐫1, 𝐫𝟐, 𝜏)|. Using Eqs. (4.7) and (4.5), we can write the visibility as
𝑉(𝐫) = 2 [𝜂(𝐫) + 1 𝜂(𝐫)]
−1
|γ(𝐫1, 𝐫2, 𝜏)| , (4.9)
with 𝜂(𝐫) = √𝐼(1)/𝐼(2) . Under the condition that 𝐼(1)= 𝐼(2), the magnitude of the complex degree of coherence becomes a direct measure of the visibility. When γ(𝜏) has a magnitude 1, it implies com- plete coherence and the electric fields are varying with complete unison, leading to a distinctive in- terference pattern. When γ has a magnitude less than one and above zero, it leads to a less visible interference pattern. A γ of magnitude zero leads to a washed out pattern, meaning no interference can last long enough to be detected on the screen. For nonharmonic fields, as is always the case in practice, larger path difference x, i.e. longer 𝜏, implies smaller γ. This is because of the shift in the maxima of each monochromatic component on the detecting screen. The width of the absolute value of γ is denoted as the coherence time [20].
4.2.2 Cross-spectral Density
Using the Fourier analysis, we know that fields - that satisfy the Dirichlet conditions: being absolutely integrable, have a finite number of maxima and minima within any time interval, and have a finite number of finite discontinuities in any time interval - can be seen as a superposition of harmonic functions with different amplitudes and phases as represented below,
𝐸(𝐫, 𝑡) = ∫ 𝐸(𝐫, 𝜔) exp(−𝑖𝜔𝑡) 𝑑𝜔
∞ 0
,
𝐸(𝐫, 𝜔) = 1
2𝜋∫ 𝐸(𝐫, 𝑡) exp(−𝑖𝜔𝑡) 𝑑𝑡 .
∞
−∞
(4.10)
where 𝐸(𝐫, 𝜔), the field’s Fourier transform, defines the space dependent part of a monochromatic component of the field.
Analogously to the MCF we now define, in the frequency space, another coherence function of the field that plays an important role in the scope of the second-order coherence theory, the cross-spectral density function, CSD,
𝑊(𝐫𝟏, 𝐫𝟐, 𝜔) = 〈𝐸∗(𝐫1, 𝜔)𝐸(𝐫2, 𝜔)〉 . (4.11)
26
The CSD as defined above describes the correlations of the spectral components of the field at two positions 𝐫1 and 𝐫2 as an ensemble average. One advantage of studying it in contrast to the MCF, is that its propagation laws in dispersive media are easily governed.
When 𝐫1 = 𝐫2 = 𝐫, 𝑊(𝐫𝟏, 𝐫𝟐, 𝜔) = 𝑆(𝐫, 𝜔) = 〈|𝐸(𝐫, 𝜔)|2〉 = 〈𝐸∗(𝜔)𝐸(𝜔)〉 and we call it the spec- tral density (also power spectrum or spectrum). Since the energy of the field is carried by all its monochromatic parts, 𝑆(𝐫, 𝜔)𝑑𝜔 is the energy density carried by the components lying between 𝜔 and 𝜔 + 𝑑𝜔. This leads us to normalize the CSD, and thus define the spectral degree of coherence as
𝜇(𝐫𝟏, 𝐫𝟐, 𝜔) = 𝑊(𝐫1, 𝐫2, 𝜔)
√𝑆(𝐫1, 𝜔)𝑆(𝐫2, 𝜔) , (4.12) with an absolute value that ranges between 0 and 1.
In the double-slit experiment, described in Sec. 4.2.1, the spectral degree of coherence between the slits’ positions will affect the total spectral density on the observation plane
𝑆(𝐫, 𝜔) = 𝑆(1)(𝐫, 𝜔) + 𝑆(2)(𝐫, 𝜔) + 2√𝑆(1)𝑆(2)ℜ {𝜇(𝐫1, 𝐫2, 𝜔)𝑒−𝑖2𝜋𝜔𝑥𝑐 } . (4.13)
𝑆(1)(𝐫, 𝜔) and 𝑆(2)(𝐫, 𝜔) are the spectral densities at point r due to slits 1 and 2, respectively. Thus the interference effects manifest themselves in the frequency domain as well. Practically this can be seen through the addition of filters on the slits’ plane [20].
Stationary functions do not obey the Dirichlet conditions, however the power spectrum and cross- spectral density can still be defined through the Wiener-Khinchine theorem [20],
𝑊(𝐫1, 𝐫2, 𝜔) = 1
2𝜋 ∫ 𝛤(𝐫1, 𝐫2, 𝑡)𝑒𝑖𝜔𝑡𝑑𝑡 .
∞
−∞
(4.14)
The theorem states that cross-spectral density and mutual coherence functions are Fourier transforms of each other. Equation (4.10) is no longer valid though, since the Fourier transform of a stationary function does not exist. Using the coherent mode representation [21] of the cross-spectral density, 𝑊(𝐫1, 𝐫𝟐, 𝜔) in Eq. (4.10) retains the meaning of a correlation function of an ensemble of harmonic realizations that represent the field. The only difference is that for stationary fields, E(r,𝜔) and 𝐸(𝐫, 𝑡) are not Fourier transform pairs, however the measurable quantities, MCF and CSD, are [21].
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4.3 Electromagnetic Coherence
Electromagnetic fields are not scalar in nature, which means that results of an experiment like the double slit, the interaction of light and matter, or the propagation of light, are dependent on the vec- torial nature of light. For instance the superposition of two fully coherent fields with electric fields in orthogonal directions will not yield any intensity interference pattern; Fresnel-Arago second law [22].
Thus, to fully understand the behavior of the EM field in any experiment, its vectorial nature must thus be taken into account. Hence, we now study the degree of polarization of the field and extend the definitions of the coherence functions.
4.3.1 Degree of Polarization
Given at point r, a harmonic component of a planar EM field, meaning the electric field vibrations take place in a plane perpendicular to the direction of propagation, which we can attribute it to the z- axis. We now examine the correlation between the different components of the field at the same point and time. This is given by the 2 × 2 correlation matrix, known as the polarization matrix, J,
𝐉 = (〈𝐸𝑥∗(𝑡)𝐸𝑥(𝑡)〉 〈𝐸𝑥∗(𝑡)𝐸𝑦(𝑡)〉
〈𝐸𝑦∗(𝑡)𝐸𝑥(𝑡)〉 〈𝐸𝑦∗(𝑡)𝐸𝑦(𝑡)〉) . (4.15)
where 𝐸𝑥(𝑡) and 𝐸𝑦(𝑡) are the field components in the 𝑥 and y directions.
The polarization matrix, J, has the following properties:
a. J is Hermitian, since J𝑥𝑦 = J𝑦𝑥∗ follows from (〈𝐸𝑥∗(𝑡)𝐸𝑦(𝑡)〉)∗ = 〈𝐸𝑦∗(𝑡)𝐸𝑥(𝑡)〉 . b. The trace of the polarization matrix, Tr[𝐉(𝐫)], is equal to the intensity of the field at r.
c. the degree of correlation between the field components which is obtained by normalizing the off-diagonal elements, 𝑗𝑥𝑦, becomes
𝑗𝑥𝑦= 𝐽𝑥𝑦
√𝐽𝑥𝑥𝐽𝑦𝑦 .
Based on the polarization matrix, specifically the off diagonal elements, we distinguish three field types: completely polarized, unpolarized, and partially polarized. The field is completely polarized when the x and y components are fully correlated. In this case |𝑗𝑥𝑦| equals 1. This will be true regard- less of our choice of the coordinate system (x and y). The field is unpolarized or natural when |𝑗𝑥𝑦| is equal to zero, and 𝐽𝑥𝑥 = 𝐽𝑦𝑦. In this case, the field is rotating erratically on a plane perpendicular to the direction of propagation. Introducing any kind of polarizing system onto this field will decrease
28
the field’s intensity. When the value of |𝑗𝑥𝑦| is greater than zero and less than one, the field is said to be partially polarized.
The polarization matrix can be written as a sum of two matrices, one corresponding to a fully polar- ized field and the other to an unpolarized one [20]. The ratio of the intensities of the polarized to the total part of the field is known as the degree of polarization,
𝒫(𝐫) = √1 −4Det[𝐉(𝐫)]
Tr[𝐉(𝐫)]2 . (4.16)
where Det, and Tr indicate the determinant and the trace respectively.
4.3.2 Electromagnetic Degree of Coherence
With the vector nature of light accounted for, the observable field values need to be amended in such a way as to include all the field components and correlations between them. For this case, we now extend the mutual coherence and cross-spectral density functions into a mutual coherence matrix 𝚪(r1, 𝐫2, 𝜏), written as
and the cross-spectral density matrix
where (𝑖, 𝑗) = (𝑥, 𝑦, 𝑧).
Taking into account that orthogonal components of the field do not produce any intensity interference pattern, like in the classic Young’s slit experiment (Sec. 4.2.1), it might be tempting to simply gener- alize the degree of coherence from the scalar representation by taking into consideration correlations between parallel components. In the time domain, and for the interference experiment of Fig. 4.2, the generalized complex degree of coherence, 𝛶′(𝐫1, 𝐫𝟐, 𝜏), becomes
𝛶′(𝐫1, 𝐫2, 𝜏) = Tr[𝚪(𝐫1, 𝐫2, 𝜏)]
√𝐼(𝐫1)𝐼(𝐫2) , (4.19)
where 𝐼(𝐫1) = Tr[Γ(𝐫1, 𝐫1, 0)] is the intensity at 𝐫1 and similarly 𝐼(𝐫2) = Tr[Γ(𝐫2, 𝐫2, 0)] is the inten- sity at 𝐫2.
𝚪(𝐫1, 𝐫2, 𝜏) = |𝛤𝑖𝑗(𝐫1, 𝐫2, 𝜏)| = |〈𝐸𝑖∗(𝐫1, 𝑡)𝐸𝑗(𝐫2, 𝑡 + 𝜏)〉|, (4.17)
𝐖(𝐫1, 𝐫2, 𝜔) = |𝑊𝑖𝑗(𝐫1, 𝐫2, 𝜔)| = |〈𝐸𝑖∗(𝐫1, 𝜔)𝐸𝑗(𝐫2, 𝜔)〉| , (4.18)
29
Using this simple generalization, we get a complex degree of coherence which accounts for the visi- bility of the fringes in intensity modulation experiments. However, it does not account for correlations that might exist between the normal components of the field which can be fully correlated in a statis- tical sense, and shows as polarization interference. For example, if we consider two fields that are fully correlated yet normal to each other in the double slit experiment, their intensity will not be modulated and visibility will be zero. However, rotating the field in one of the slits by 90 degrees will reveal a visibility equal to 1. In addition to that, Eq. (4.19) is dependent on the choice of the coordinate system and there are cases when it is easier to work with curvilinear coordinate system rather than Cartesian and we expect the complex degree of coherence to hold to its value in such a case [18]. To this end, a new degree of coherence is defined. It lacks the shortcomings of the earlier definition by taking into account correlations between all field components.
The electromagnetic degree of coherence, Υ(𝐫1, 𝐫2, 𝜏), is defined as
𝛶2(𝐫1, 𝐫2, 𝜏) = ∑ |𝛤𝑖,𝑗 ij(𝐫1, 𝐫2, 𝜏)|2
∑ 𝛤𝑖,𝑗 𝑖,𝑖(𝐫1, 𝐫1, 0)𝛤𝑗,𝑗(𝐫2, 𝐫2, 0) , (4.20) or in other notation
𝛶2(𝐫1, 𝐫𝟐, 𝜏) = Tr[𝚪†(𝐫1, 𝐫2, 𝜏)𝚪(𝐫1, 𝐫2, 𝜏)]
Tr[𝚪(𝐫1, 𝐫1, 0)]Tr[𝚪(𝐫2, 𝐫2, 0)] . (4.21)
In the new definition 𝛶 is real, in contrast to the complex value in the scalar definition and its maxi- mum is unity when all the field components are fully correlated [18].
Analogously to the time domain, in the spectral domain, the definition of the complex degree of coherence becomes,
𝜇2(𝐫1, 𝐫2, 𝜔) = Tr[𝐖†(𝐫1, 𝐫2, 𝜔)𝐖(𝐫1, 𝐫2, 𝜔)]
Tr[𝐖(𝐫1, 𝐫1, 𝜔)]Tr[𝐖(𝐫2, 𝐫2, 𝜔)] . (4.22)
This expression is also real and like its time equivalent, it contains information about correlations of all field components [23]
.
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Chapter V 5 BlackBody Coherence in the cavity and in the aperture plane
In this Chapter, we discuss the coherence properties of radiation inside the cavity of a blackbody and in its aperture plane. As we have already noted in Chap. III, the electromagnetic field is completely independent of the cavity shape, and the sole definer of it is the temperature. Blackbody radiation has the following properties:
1. Stationary: as defined in Chap. IV.
2. Statistically homogeneous: Meaning stationary in space. Any measurable observable at two space points (𝐫a and 𝐫b) has its value dependent only on the separation between the points (𝐫 = 𝐫2− 𝐫1). For example, the electromagnetic degree of coherence at two points defined by position vectors 𝐫a, and 𝐫b, can be given as
𝜇(𝐫a, 𝐫b, 𝜔) = 𝜇 (𝐫b− 𝐫a, 𝜔)
3. Statistically isotropic: Meaning all directions at 𝐫 are equivalent and thus measurable observables depend on 𝑟 = |𝐫b− 𝐫a| only. For example
𝜇(𝐫a, 𝐫b, 𝜔) = 𝜇(𝑟, 𝜔 ).
In [24] and in a quantum mechanical description of the blackbody radiation, these properties are described as a consequence of the commuting relationships:
1. Between the density matrix of the blackbody radiation and the generator of time translation (the hamiltonian operator) for property 1,
2. Between the density matrix and the generator of space translation (the linear momentum operator) for property 2, and
3. Between the density matrix and the generator of rotation (the angular momentum operator) for property 3.
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Starting from these field properties, the theory of isotropic turbulence of incompressible fluids, the Wiener-Khinchin theorem, and Planck’s radiation law, Bourret [2] derived an expression for the time correlation function of the field. That was the MCF for the scalar field as described in Chap. IV but taking real field values rather than complex ones. Kano and Wolf [3] extended Bourret solution by calculating the MCF using the complex analytic signal representation of the field. Later on, the cross- spectral density matrix was analytically calculated by Mehta and Wolf [4] using the Fourier transform of the already obtained MCF [3].
The coherence functions of blackbody radiation can also be obtained using the following model of the field inside the cavity, the plane wave model which is represented schematically in Fig. 5.1. In it, the electromagnetic field inside the cavity at each point is considered as a superposition of plane waves which are spanning all directions,
𝐄(𝐫, 𝜔) = ∫ 𝐄(𝐫, 𝜔; 𝑢̂) 𝑑Ω , (5.1)
where 𝑑Ω is a differential solid angle, 𝐄(𝐫, 𝜔) is a realization of angular frequency 𝜔 that repre- sents the field in an ensemble at point r in the cavity, and 𝐄(𝐫, 𝜔; 𝑢̂) is a plane wave component of that realization which propagates in a direction represented by the unit vector 𝑢̂,
𝑢̂ = sin𝜃cos𝜙 𝑖̂ + sin𝜃sin𝜙 𝑗̂ + cos𝜃 𝑘̂ . (5.2) These plane waves are how the far field, originating from the spontaneously emitting atoms of the cavity walls, appear at the observation point. Due to the isotropy, independency, and randomness of the emission processes, the waves have the same intensity in all directions and are angularly uncor- related and unpolarized [5,7]. The model was hypothesized for scalar fields in [5], and in [7] polari- zation properties were considered. Given our interest, in this thesis we will be proceeding with the latter case.
If the plane perpendicular to the propagation direction is specified by the two unit vectors 𝑠̂ and 𝑝̂, representing the s and p polarization directions and which are defined with respect to the Cartesian coordinate system as,
𝑠̂ = 𝑢̂𝑧× 𝑢̂
|𝑢̂𝑧× 𝑢̂| and 𝑝̂ = 𝑠̂ × 𝑢̂ , (5.3) then the plane waves in Eq. (5.1) can be written as
32
𝐄(𝐫, 𝜔; 𝑢̂) = 𝐸𝑠(𝐫, 𝜔; 𝑢̂)𝑠̂ + 𝐸𝑝(𝐫, 𝜔; 𝑢̂)𝑝̂ , (5.4) where 𝐸𝑠 and 𝐸𝑝 represent the plane wave components in the aforementioned directions. For the plane wave propagating in the direction given by the unit vector 𝑢̂, the s and p polarized compo- nents at position r can be written as
𝐸𝑠(𝐫, 𝜔; 𝑢̂) = 𝐸𝑠(𝑢̂, 𝜔) exp(𝑖𝑘𝑢̂ ∙ 𝐫) 𝐸𝑝(𝐫, 𝜔; 𝑢̂) = 𝐸𝑝(𝑢̂, 𝜔) exp(𝑖𝑘𝑢̂ ∙ 𝐫) ,
(5.5)
where 𝐸𝑠(𝑢̂, 𝜔), and 𝐸𝑝(𝑢̂, 𝜔) represent the values of these components at the origin, r = 0 and 𝑘 =
|𝑘 = 𝜔/𝑐| = 2𝜋/𝜆.
The cross-spectral density, Eq. (4.18), between two points 𝐫aand 𝐫b is thus
𝐖(𝐫a, 𝐫b, 𝜔) = 〈𝐄∗(𝐫a, 𝜔)𝐄T(𝐫b, 𝜔)〉 . (5.6) Using Eqs. (5.1) and (5.4) in Eq. (5.6) yields,
Figure 5. 1 A schematic of the plane wave model for the field at a point denoted by position vector r, and the polar and azimuthal angles 𝜽 and 𝝋, respectively. Some of the plane waves are shown and their direction is indicated by the unit vector u. The s and p polarization direction are also noted
for two of the unit vectors u.
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𝐖(𝐫a, 𝐫b, 𝜔) = 〈∫[𝐸𝑠∗(𝐫a, 𝜔; 𝑢̂1)𝑠̂1+ 𝐸𝑝∗(𝐫a, 𝜔; 𝑢̂1)𝑝̂1] 𝑑Ω1
∙ ∫[𝐸𝑠(𝐫b, 𝜔; 𝑢̂2)𝑠̂2T + 𝐸𝑝(𝐫b, 𝜔; 𝑢̂2)𝑝̂2T] 𝑑Ω2 〉 .
(5.7)
After changing the integration and averaging operation order, and having described the plane waves as uncorrelated and unpolarized with equal amplitudes in all directions,
〈𝐸𝑖∗(𝐫a, 𝜔; 𝑢̂1) ∙ 𝐸𝑗(𝐫b, 𝜔; 𝑢̂2)〉 = { 0 for 𝑖 ≠ 𝑗
𝑎0(𝜔) exp[𝑖𝑘(𝑢̂2. 𝐫b− 𝑢̂1. 𝐫a)] 𝛿(𝑢̂1− 𝑢̂2)for 𝑖 = 𝑗 (5.8)
where 𝑖, 𝑗 ∈ (𝑠, 𝑝) and 𝑎0(𝜔) is a direction independent and frequency dependent constant. The cross- spectral density matrix of Eq. (5.7) is expressed as
Evaluating one of the integrals in Eq. (5.9) yields,
𝐖(𝐫a, 𝐫b, 𝜔) = 𝑎0(𝜔) ∫(𝑠̂𝑠̂𝑇+ 𝑝̂𝑝̂𝑇) exp(𝑖𝑘𝑢̂. 𝐑) 𝑑Ω
4𝜋
, (5.10)
where 𝐑 = 𝐫b− 𝐫a. We can now proceed by expressing the 𝑠̂ and 𝑝̂ vectors in Cartesian coordinates and then calculate the integral analytically. However inside the blackbody cavity, it is easier to cal- culate the integral in Eq. (5.10) by first using the formula [7]
𝕌3 = 𝑢̂𝑢̂T+ 𝑠̂𝑠̂T+ 𝑝̂𝑝̂T , (5.11) where 𝕌3 is a 3 × 3 identity matrix. This formula was verified by expressing the 𝑠̂ and 𝑝̂ vectors in the Cartesian coordinate space. Using Eq. (5.11) in Eq. (5.10) gives
𝐖(𝐫a, 𝐫b, 𝜔) = 𝑎0(𝜔) ∫(𝕌3− 𝑢̂𝑢̂𝑇) exp(𝑖𝑘𝑢̂. 𝐑) 𝑑Ω
4𝜋
(5.12) 𝐖(𝐫a, 𝐫b, 𝜔) = ∬ 𝑎0(𝜔) (𝑠̂1𝑠̂2T+ 𝑝̂1𝑝̂2T) exp[𝑖𝑘(𝑢̂2∙ 𝐫b− 𝑢̂1∙ 𝐫a)] 𝛿(𝑢̂1− 𝑢̂2)𝑑Ω1𝑑Ω2 .
4𝜋
(5.9)
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In Eq. (5.12) 𝑢̂𝑢̂Tabove can be replaced ∇∇T,where ∇ is given by
∇ = 𝜕
𝜕𝑅𝑥𝑖̂ + 𝜕
𝜕𝑅𝑦𝑗̂ + 𝜕
𝜕𝑅𝑧𝑘̂ , leading to a cross-spectral density
𝐖(𝐫a, 𝐫b, 𝜔) = 𝑎0(𝜔) (𝕌3+ 1
𝑘2∇∇T) ∫ exp(𝑖𝑘𝑢̂ ∙ 𝐑) 𝑑Ω
4𝜋
. (5.13)
We can now calculate the integral over the full solid angle (4π) by aligning the R vector with the z axis, using change of variables, and Euler’s formula to get
∫ exp(𝑖𝑘𝑢̂. 𝐑) 𝑑Ω = 4𝜋 sin(𝑘𝑅)
𝑘R = 4𝜋𝑗0(𝑘𝑅) , (5.14)
where 𝑗0(𝑘𝑅) is the spherical Bessel function of order zero and 𝑅 = |𝐑|.
We then calculate the derivates in Eq. (5.13) by using the chain rule
𝜕
𝜕𝑅𝑥 = 𝜕𝑅
𝜕𝑅𝑥
𝜕
𝜕𝑅 = 𝑅𝑥 𝑅
𝜕
𝜕𝑅 , and the following property of spherical Bessel functions
𝑑𝑗𝑚(𝑥) 𝑑𝑥 = 𝑚
𝑥 𝑗𝑚(𝑥) − 𝑗𝑚+1(𝑥) ,
where m is the order of the spherical Bessel function. The cross-spectral density in Eq. (5.13) is cal- culated to be,
W(𝐫a, 𝐫b, 𝜔) = 4𝜋𝑎0(𝜔) {[𝑗0(𝑘𝑅) − 𝑗1(𝑘𝑅)
𝑘𝑅 ] 𝕌3+ 𝑗2(𝑘𝑅)𝐑̂𝐑̂T} , (5.15) which is equivalent to the expression derived in [4] when 4𝑎0(𝜔) is equal to Planck’s law, Eq. (3.1).
Using Eq. (4.22), we can now calculate the EM degree of coherence of the field inside the cavity [25],
