Blackbody far-field coherence

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Blackbody far-field coherence

M

^OHAMMAD

A

^{L LAKKI}

,

^1,*

T

^ERO

S

^ETÄLÄ¹ ^AND

A

^RI

T. F

^RIBERG

,

¹

1Institute of Photonics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland

*mohaal@uef.fi

Abstract: We revisit the spectral coherence properties of far-field radiation emanating from an aperture in a blackbody cavity on the basis of Kirchhoff’s boundary conditions. We point out that the far-zone cross-spectral density matrix derived earlier in the literature by separately propagating all three aperture-field components, does not show transversality of the field at nonparaxial directions. This is not the case when Luneburg’s diffraction integrals are applied on the transverse source field components to determine the entire far field. We compare the electromagnetic degrees of coherence for the two methods and show that over important angular separations their values coincide with high accuracy. The results of this work and of others concerning the far-field intensity, polarization, and paraxial angular coherence are in full agreement.

1. Introduction

Blackbody radiation can be described as a three-component electromagnetic field whose tem- poral and spectral coherence matrices have closed-form expressions inside a cavity, within an aperture in the cavity’s wall, and in the aperture far zone [1–3] (for a detailed review, see [4]). In particular, the far field is known to be spectrally unpolarized in every direction and it exhibits a Lambertian intensity distribution [3, 5, 6]. One way to derive the far-field cross-spectral density matrix (CSDM) is by separately propagating all three field components from the aperture to the far zone using the Rayleigh–Sommerfeld diffraction formula [7–9]. This method is valid provided the aperture field is explicitly known [10]. However, this is not often the case and Kirchhoff’s boundary conditions (KBC) are usually employed. While justified and practical in many cases, the KBC constitute an approximation which is inconsistent with Maxwell’s equations.

In this work, we first show that due to the KBC, the blackbody’s far-field CSDM resulting from propagation using the Rayleigh–Sommerfeld formulation does not reflect the transverse nature of the electric field at highly nonparaxial directions [3]. We rederive the far-field CSDM by employing Luneburg’s diffraction integrals [11, 12], which express the entire three-component half-space field in terms of two orthogonal source-field components in the aperture plane and guarantee the transversality of the far field. We remark that the ensuing far-field CSDM is not necessarily accurate either since the KBC are invoked. We further compare the far-field electromagnetic degrees of coherence obtained with the Rayleigh and Luneburg propagation methods. Significant differences are found when the angular separation is large and one of the far-field observation directions is highly nonparaxial. However, over angular separations with a significant degree of coherence, the two methods are found to imply similar results. We emphasize that all the results concerning the far-field intensity, polarization state, and spatial coherence within the paraxial regime obtained in this work and in prior literature using these two methods of propagation, are in complete agreement.

This work is organized as follows. In Sec. II, Luneburg’s diffraction integrals are recalled, while in Sec. III they are applied to compute the far-field CSDM of blackbody radiation originat- ing from an aperture in a cavity. In Sec. IV, the far-field coherence properties are discussed and compared to those obtained with the Rayleigh diffraction method. Section V briefly summarizes the work.

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2. Aperture - Far Field Propagation

Within the scalar framework, the light field away from an aperture in an opaque plane, can be obtained from the Rayleigh–Sommerfeld formulation of diffraction [7, 8]. This approach holds for all three orthogonal (Cartesian) electric components of an electromagnetic wave and the result is correct provided the field is exact at the aperture plane. However, approximating the field values with the KBC, which is often the case, may lead to results that are not consistent with Maxwell’s equations. Specifically, the diffracted electric field may not obey the transversality (divergence) condition and thereby the field in the far zone may not be transverse with respect to the propagation direction. This can be avoided by propagating the field components that are transverse to the aperture normal using the Rayleigh–Sommerfeld diffraction formula and then calculating the longitudinal component from the divergence condition of Maxwell’s equations.

This amounts to the Luneburg approach [11, 12], in which the diffracted transverse components, at an observation pointr, are given by

E_i(r, ω)=−1 2π

∬ _∞

−∞

ikR−1 R³

exp(ikR)zE_i(ρ, ω)d²ρ, (1) withi∈ (x,y), and the longitudinal component is written as

Ez(r, ω)= 1 2π

∬ _∞

−∞

ikR−1 R³

exp(ikR)

(x−ρ_x)Ex(ρ, ω)+(y−ρ_y)Ey(ρ, ω)

d²ρ, (2) for the case of an aperture located in thexyplane (see Fig. 1). Above,Ei(ρ, ω),i∈ (x,y), represents the transverse aperture-field component at frequencyωand spatial positionρ=[ρx, ρy,0]^T, with T denoting the transpose. Henceforth, ρ is reserved for aperture-plane coordinates. In addition,R=|r−ρ|,r=[x,y,z]^T, andk=ω/cis the wavenumber withcbeing the speed of light in vacuum.

Defining a propagation matrix

T(r,ρ, ω)= ikR−1

2πR³

exp(ikR)







−z 0 0

0 −z 0

x−ρx y−ρy 0







, (3)

the Luneburg equations in Eqs. (1) and (2) can be collected as E(r, ω)=

∬ _∞

−∞

T(r,ρ, ω)E(ρ, ω)d²ρ. (4) The fields in Eq. (4) are given by the column vectorsE(r, ω) = [Ex(r, ω),Ey(r, ω),Ez(r, ω)]^T andE(ρ, ω)=[Ex(ρ, ω),Ey(ρ, ω),Ez(ρ, ω)]^T. Note that in the Luneburg method,E(r, ω)is only specified in terms of the aperture-field componentsEx(ρ, ω)andEy(ρ, ω). In the far zone, we may use the approximationR≈r−uˆ·ρ, wherer =|r|and the unit vector ˆu=r/rdetermines the far-zone direction. Hence the propagation matrix in Eq. (3) becomes

T(r,ρ, ω)= ik

2πrexp(ikr)exp(−ikσ·ρ)L(uˆ), (5) whereσis the aperture-plane component of ˆu, see Fig. 1. The matrixL(uˆ)is of the form

L(uˆ)=







−u_z 0 0

0 −u_z 0 u_x u_y 0







, (6)

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and depends solely on the direction ˆu=[uˆx,uˆy,uˆz]^T, which in spherical polar coordinates reads as

ˆ

u=[sinθcosϕ,sinθsinϕ,cosθ]^T. (7) Apart from being real valued,L(uˆ)has the following property which will prove useful later,

ˆ

u^TL(uˆ)=0. (8)

The formalism in Sec. II holds for any three-component electric field in the aperture. We now apply it to the radiation emanating from a blackbody cavity through a circular opening of radius εlocated in thexy-plane as shown in Fig. 1.

Fig. 1. Illustration of the notations.

3. Cross-spectral density matrix of blackbody’s far field

In the context of the second-order coherence theory [9, 13] the cross-spectral density matrix for an electromagnetic field at two spatial pointsr₁andr₂, and at frequencyω, is defined as

W(r₁,r₂, ω)=hE^∗(r₁, ω)E^T(r₂, ω)i, (9) whereE(r, ω)represents a monochromatic realization in an ensemble which describes the field in the space-frequency domain. Further, the asterisk denotes complex conjugation and the angle brackets stand for ensemble averaging. The 3×3 CSDM in the aperture plane of a blackbody cavity has been derived in [3], and the result is

W⁽⁰⁾(ρ₁,ρ₂, ω)=2πa0(ω) j₀(kρ) − j₁(kρ) kρ

U₃+j₂(kρ)ρˆρˆ^T

−iJ₂(kρ) kρ

ˆ

ρuˆ^T_z +uˆzρˆ^T B(ρ₁)B(ρ₂), (10) where the aperture-plane difference vector isρ =ρ₂−ρ₁, andρ=|ρ|, ˆρ=ρ/ρ. In addition, 4a0(ω)coincides with Planck’s spectrum [3,14],U₃is the 3×3 identity matrix, and ˆu_zis the unit vector in thezdirection. The functionsj_i(kρ), withi ∈ (0,1,2), are spherical Bessel functions

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of orderi,J₂(kρ)is the Bessel function of the first kind and order 2, and B(ρ)is a blocking function

B(ρ)=











1, ρ < ε, 0, otherwise.

(11) The values for B(ρ) follow from Kirchoff’s boundary conditions which consider the field as unperturbed in the aperture and zero in the shadow of the blackbody wall at z = 0. This approximation was employed in [3] and is often considered valid for an aperture radius which is much larger than the wavelength [7].

Using Eqs. (4) and (5) in Eq. (9) and after reversing the order of averaging and integration we obtain the CSDM of blackbody radiation in the far zone as

W^(∞)(r₁uˆ₁,r₂uˆ₂, ω)= k²exp[ik(r₂−r₁)]

4π²r₁r₂ L(uˆ₁)

∬ _∞

−∞

W⁽⁰⁾(ρ₁,ρ₂, ω)

×exp[−ik(σ₂·ρ₂−σ₁·ρ₁)]d²ρ₁d²ρ₂L^T(uˆ₂). (12) As shown in [3], on changing to the sum and difference coordinates, the integral in Eq. (12) can be evaluated and thus

W^(∞)(r₁uˆ₁,r₂uˆ₂, ω)=L(uˆ₁)W^(∞)

R (r₁uˆ₁,r₂uˆ₂, ω)

u_1,zu_2,z L^T(uˆ₂), (13) whereW^(∞)

R (r₁uˆ₁,r₂uˆ₂, ω)is the far-field CSDM obtained by propagating each field component according to the Rayleigh–Sommerfeld diffraction integral, hence the subscript R. This CSDM is given as [3]

W^(∞)

R (r₁uˆ₁,r₂uˆ₂, ω)= 2a0(ω)A⁰J1(kσε) kσε

exp[ik(r2−r1)]

r₂r₁ u_1,zu_2,z(1−σ¯²)⁻¹^/²

×h

M+σ¯²uˆzuˆ^T_z − (1−σ¯²)¹^/²(uˆzσ¯^T+σ¯uˆ^T_z)i

, (14)

whereMis

M=U₃−σ¯σ¯^T−uˆzuˆ^T_z, (15) andσ =σ2−σ1,σ =|σ|, ¯σ =(σ1+σ2)/2, ¯σ = |σ¯|. The area of the circular aperture is denoted byA⁰. Using Eqs. (6) and (14) in Eq. (13) results in

W^(∞)(r₁uˆ₁,r₂uˆ₂, ω)= 2a0(ω)A0J₁(kσε) kσε

exp[ik(r₂−r₁)]

r₂r₁ u_1,zu_2,z(1−σ¯²)⁻¹^/²

×

"

M−uˆzσ^T₁

u_1,z M−Mσ2uˆ^T_z u_2,z +

ˆ uzσ^T₁

u_1,z Mσ2uˆ^T_z u_2,z

#

. (16)

The CSDMs of Eqs. (14) and (16) are seen to be Hermitian, as expected, under the interchange of the observation points. In addition, their dependence on the aperture radiusεis identical and results from the Fourier transform of the blocking function, Eq. (11), as shown in [3]. Next we use Eq. (16) to extract information on the coherence properties of blackbody radiation in the far zone in order to compare them with what is obtained from Eq. (14) in [3] and with previously published results [5, 6].

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4. Blackbody’s Far-field coherence properties

By settingr₁ = r₂ in Eq. (16), we obtain the spectral polarization matrix of the blackbody radiation in the far zone

Φ^(∞)

3 (ru, ωˆ )=a₀(ω)A⁰cosθ r²

U₃−uˆuˆ^T

, (17)

with the subscript 3 indicating a 3×3 matrix. SinceΦ^(∞)

3 (ru, ωˆ )uˆ=0, the far field is transverse in every direction. Hence the local electric field can be expressed using thesandppolarization states whose unit vectors, shown in Fig. 1, are defined as

ˆ

s= uˆ_z×uˆ

|uˆz×uˆ| and pˆ=sˆ×u.ˆ (18) Since ˆu,s, and ˆˆ pform an orthonormal vector triad, they obey

U₃=uˆuˆ^T+sˆsˆ^T+pˆpˆ^T. (19) Consequently, in the localspframe the field can be described in terms of the 2×2 polarization matrix

Φ^(∞)

2 (ru, ωˆ )= a₀(ω)A0cosθ

r² U₂, (20)

whereU₂ is the 2×2 identity matrix. The degree of polarization can be calculated from the expression [9]

P²(ru, ωˆ )=1−4det[Φ^(∞)

2 (ru, ωˆ )]

tr[Φ^(∞)

2 (ru, ωˆ )] , (21)

givingP=0, thereby confirming that the electric field is unpolarized in every direction as found earlier [5]. The radiant intensity, i.e., the power radiated from the aperture plane per unit solid angle in a certain direction, is defined by

I(ru, ωˆ )= lim

r→∞r²tr[Φ^(∞)

2 (ru, ωˆ )]. (22)

Substituting Eq. (20) into Eq. (22) results in

I(ru, ωˆ )=2a0(ω)A0cosθ, (23) indicating that the far field obeys Lambert’s cosine law [5]. Assuming equal distances from the aperture,r₁=r₂ =r, the paraxial approximation (σ1 =σ2≈0) of the CSDM in Eq. (16) takes the form

W^(∞)(ruˆ₁,ruˆ₂, ω) ≈ 2a0(ω)A0J₁(kσε)

kσεr² (U₃−uˆzuˆ^T_z), (24) which corresponds to a transversal field as can be verified by multiplying the CSDM with ˆuz. We point out that the above results on far-field polarization, radiant intensity, and paraxial spatial coherence are also obtained from the CSDM of Eq. (14), derived from the Rayleigh–Sommerfeld diffraction integrals [3]. In addition, they have been found earlier by employing the Luneburg method [5, 6].

Next we show that the far-field CSDMs and the related degrees of coherence can differ significantly at nonparaxial directions. In particular,W^(∞)(r₁uˆ₁,r₂uˆ₂, ω)obeys the transversality

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requirement everywhere in the far zone, as can be verified directly by using Eqs. (8) and (13) in the products

W^(∞)(r₁uˆ₁,r₂uˆ₂, ω)uˆ₂=uˆ^T₁ W^(∞)(r₁uˆ₁,r₂uˆ₂, ω)=0. (25) However, the same does not hold for W^(∞)

R (r₁uˆ₁,r₂uˆ₂, ω) in Eq. (14) at large values of the angleθ. In addition, it is straightforward to see from Eqs. (6) and (13) thatW(∞)(r1uˆ1,r2uˆ2, ω) andW^(∞)

R (r₁uˆ₁,r₂uˆ₂, ω)differ only in the matrix elements involving thezcomponent. These elements can as well be directly obtained by using the transversality requirement, Eq. (25), and the elements ofW^(∞)

R (r₁uˆ₁,r₂uˆ₂, ω)involving thexandyfield components.

The degree of coherence for electromagnetic fields in the spectral domain was introduced in [15] (see also [16]) as the Frobenius norm of the CSDM normalized by the square roots of the spectral densities at the observation points,

µ²(r₁uˆ₁,r₂uˆ₂, ω)= tr[W(∞)†(r1uˆ1,r2uˆ2, ω)W(∞)(r1uˆ1,r2uˆ2, ω)]

trW(∞)(r₁uˆ₁,r₁uˆ₁, ω)trW(∞)(r₂uˆ₂,r₂uˆ₂, ω), (26) where†denotes Hermitian adjoint. For interfering planar fields,µ(r1uˆ1,r2uˆ2, ω)represents the contrasts in the modulations of the Stokes parameters [17]. Substituting Eq. (16) in Eq. (26) results in

µ(r₁uˆ₁,r₂uˆ₂, ω)=ξ(uˆ₁,uˆ₂)

J₁(kεσ) kεσ

cosθ₁cosθ₂ 1−σ¯²

1/2

, (27)

whereξ(uˆ₁,uˆ₂)is given by

ξ²(uˆ₁,uˆ₂)=sec²θ₁+sec²θ₂+(σ¯²−2)

(σ¯ ·σˆ₁)²tan²θ₁+(σ¯ ·σˆ₂)²tan²θ₂+σ¯² +tan²θ₁tan²θ₂[(σ¯ ·σˆ₁)(σ¯ ·σˆ₂) − (σˆ₁·σˆ₂)]², (28) which is a function of the observation points’ directions only. The degree of coherence is thus independent ofr1andr2. For equal polar angles (θ1 =θ2),ξ(uˆ1,uˆ2)depends on the azimuthal angles solely through the difference∆ϕ=ϕ2−ϕ1, as expected due to symmetry and as can be directly seen from Eqs. (27) and (28) using

σ₁·σ₂=sin²θcos∆ϕ, (29)

¯

σ²= (1+cos∆ϕ)sin²θ

2 , (30)

σ=[2(1−cos∆ϕ)]¹^/²sinθ. (31)

In Fig. 2 we plot the electromagnetic degree of coherence for aperture sizekε =100 with respect toθ₁ as the observation points’ polar angles vary along any quartercircle specified by ϕ₁ =ϕ₂. Angleθ₂ is fixed at 0.1π(blue curve), 0.2π(green), 0.3π(black), and 0.4π(red). We find that the angular coherence spreads over larger separations as we move farther away from thezaxis. This is visualized by the increase in the full width at half maximum (FWHM) of the degree of coherence from 0.014π(blue curve) to 0.018π(green), 0.024π(black), and 0.047π (red). Physically this behavior is explained by the smaller (incoherent) source area for larger θvalues. However when the points are encircling thezaxis at polar anglesθ1 =θ2 =θ, the FWHM ofµ(r1uˆ1,r2uˆ2, ω)as a function of the arc length, measured by sinθ∆ϕ, does not change for differentθvalues. This is seen in Fig. 3 which shows the degree of coherence forϕ2=0.4π, andθ=0.1π(red curve), 0.22π(green), and 0.4π(blue) in the case ofkε=100.

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Fig. 2. Behavior of the electromagnetic degree of coherence,µ, as a function ofθ₁ forθ₂=0.1π(blue curve), 0.2π(green), 0.3π(black), and 0.4π(red), whenϕ₁=ϕ₂. The FWHM is indicated by the dashed line.

Fig. 3. Variation of the electromagnetic degree of coherence, µ, as a function of sinθ ϕ₁forϕ₂=0.4πandθ₁ =θ₂ =θ=0.1π(red curve), 0.22π(green), and 0.4π (blue).

In Fig. 4 we exemplify the dependence of the degree of coherenceµ(r₁uˆ₁,r₂uˆ₂, ω)on the hole size. Two aperture radii,ε≈16λ(blue curve) andε≈46λ(red curve), are considered for points in the nonparaxial regime with ϕ1 = ϕ2. As expected, the width of µ(r₁uˆ₁,r₂uˆ₂, ω)increases with decreasing aperture radius. This is also noted in [3], where the degree of coherence was calculated by substituting Eq. (14) in Eq. (26) yielding

µ^(R)(r₁uˆ₁,r₂uˆ₂, ω)=√ 2

J₁(kεσ) kεσ

cosθ₁cosθ₂ 1−σ¯²

1/2

, (32)

which is identical to Eq. (27) with ξ(uˆ₁,uˆ₂) = √

2. We remark that no observable difference is seen between µ(r₁uˆ₁,r₂uˆ₂, ω) and µ⁽^R⁾(r₁uˆ₁,r₂uˆ₂, ω)if the latter is plotted in the cases of Figs. 2–4 (i.e., the respective lines would overlap). For this reason the degrees of coherence obtained from the Rayleigh–Sommerfeld formulation are omitted in the above figures.

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Fig. 4. Far-field electromagnetic degree of coherence as a function ofθ₁, for both far-field points in any plane defined byϕ₁=ϕ₂and withθ₂=0.35π. The blue and red lines correspond toε=16λandε=46λ, respectively.

To compare the electromagnetic degrees of coherenceµ^(R)(r₁uˆ₁,r₂uˆ₂, ω)andµ(r₁uˆ₁,r₂uˆ₂, ω), we use their relative difference,

∆µ(uˆ₁,uˆ₂)=

µ^(R)(r1uˆ1,r2uˆ2, ω) −µ(r1uˆ1,r2uˆ2, ω)

µ(r₁uˆ₁,r₂uˆ₂, ω) . (33) Inserting Eqs. (27) and (32) in Eq. (33) results in

∆µ(uˆ₁,uˆ₂)=1−

√2

ξ(uˆ₁,uˆ₂), (34)

whose behavior as a function of the observation points’ polar angles,θ1 andθ2, is illustrated in Fig. 5 for ϕ1 = ϕ2. We see that∆µ(uˆ₁,uˆ₂)increases with increasing angular separations

∆θ =|θ2−θ1|and equals zero when the two points coincide. The latter property is expected since the polarization matrix is identical for both approaches to calculate the far-field CSDM.

For large∆θ the relative difference is more pronounced when one of the directions is highly nonparaxial. However, in these cases, due to the narrow angular extent of coherence, the degrees of coherence are both small and we conclude that in the regions of significant (nonzero) values they coincide with high accuracy. We finally remark that the CSDM dependence on the aperture size stems from the Fourier transform of the blocking function in both approaches, thus the relative difference in Eq. (34) is independent of the aperture radius.

5. Conclusion

Despite being inconsistent with Maxwell’s equations, Kirchhoff’s boundary conditions are known to produce results in optics that are in excellent agreement with experiments. Nonethe- less, using them to approximate the field values in the blackbody’s aperture plane lead to a nontransversal far field in the nonparaxial directions, when the Rayleigh–Sommerfeld formulation is used to propagate all field components separately [3]. Following Luneburg’s method, which guarantees far-field transversality, we derived in this work the cross-spectral density matrix and the electromagnetic degree of coherence at any two directions in the far zone. The relative difference of the degrees of coherence obtained in the two formulations was shown to

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Fig. 5. Illustration of the relative difference of the degrees of coherence,∆µ, atϕ₁=ϕ₂ as a function of the polar anglesθ₁andθ₂.

be significant at large angular separations and when one of the directions is highly nonparaxial.

However, in these cases the angular coherence is small indicating that within the significant values the two degrees of coherence coincide. Our formalism reproduces the known results for blackbody radiation in the far zone concerning the polarization, intensity distribution, and spatial coherence in the paraxial regime. Even though we focused on blackbody radiation, similar conclusions on the Rayleigh–Sommerfeld and Luneburg methods are expected to hold for other random sources as well, if the Kirchhoff boundary conditions are employed.

Funding

This research was funded by the Academy of Finland (projects 308393, 310511, and 320166).

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