Near-field effects on partially coherent light scattered by an aperture

We investigate how the near field affects partially coherent light scattered from an aperture in an opaque screen. Prior work on this subject has focused on the role of surface plasmons, and how they affect spatial coherence is well documented. Here, we consider other near-field effects that might impact spatial coherence. We do this by examining the statistics of the near-zone field scattered from an aperture in a perfect electric conductor plane—a structure that does not support surface plasmons. We derive the near-field statistics (in particular, cross-spectral density functions) by applying electromagnetic equivalence theorems and the Method of Moments. We find, even in the absence of surface plasmons, that near-field physics can affect the coherence of the scattered field. The analysis and findings presented herein complement the existing coherence-related surface plasmons literature, and could find use in the design of photonic devices built to engineer spatial coherence.


Introduction
Motivated by the discovery of directional thermal emission(DTE), near-field effects on coherence have been studied for the past 30 years. In 1988, the first experimental results of directional emission from a heated grating were published [1,2]. Since that time, DTE has been linked both theoretically and experimentally to the excitation of surface plasmons [3][4][5][6], and recently, fabricated structures designed to directionally emit thermal light at specific wavelengths have been demonstrated [7][8][9][10].
Directionality of a source implies spatial coherence; the existence of DTE proves that near-field physics can have a significant impact on source coherence. In the case of DTE, this impact is profound in that an initially incoherent source (a thermal source) is transformed into a partially coherent one by the excitation of surface plasmons interacting with wavelength-scale structures.
Realizing the potential of this transformation, several authors have proposed using surface plasmons and wavelength-scale structures to precisely control spatial coherence for reasons other than just spectral or directional emission [11][12][13][14][15][16][17]. These plasmonic methods would permit coherence manipulation at the nano-or micrometer scale-coherence control on a microchip-and be superior in terms of size, weight, power, and complexity to the more traditional approaches for coherence control [18][19][20][21][22][23]. Even at longer wavelengths, where true surface plasmons are not excited, researchers have fabricated materials to excite so-called 'spoof plasmons' [24][25][26][27]. Although these spoof plasmons have not been used to affect spatial coherence to date, in principal, they could be used for that purpose.
With the link between surface plasmons and coherence well established, we seek to answer whether there are other near-field phenomena (perhaps secondary or tertiary effects) that can impact spatial coherence. We do this by examining the scattering of a partially coherent wave from an aperture in an infinite perfect electric conductor (PEC) plane-a structure that does not support surface plasmons. We simplify the analysis and computational burden by assuming a two-dimensional(2D), z-invariant geometry; applying our results to a three-dimensional(3D) aperture geometry is straightforward.
In this paper, we derive the transverse magnetic (TM) and transverse electric (TE) expressions for the crossspectral density (CSD) function [18,28] in the aperture using electromagnetic (EM) theory and the commonly used physical optics(PO) approximation [29][30][31]. In the former case, we apply Love's equivalence theorem and image theory [29][30][31], and enforce the continuity of transverse aperture fields to derive a magnetic field integral equation (MFIE) for the unknown equivalent magnetic current (proportional to the electric field in the aperture). Then, using the Method of Moments(MoM) [30,32,33], we numerically solve for the CSD of this current. We compare and contrast the exact EM and PO normalized CSD functions, also known as the spectral degrees of coherence (SDoCs) [18,28], and discuss their physical significance.
We find that, even in the absence of surface plasmons, the near field does affect spatial coherence, and its impact on the far-zone intensity (or equivalently, the spectral density when working in the space-frequency domain [28]) is most pronounced at large, nonparaxial, scattering angles. We also find, even for relatively large apertures and depending on the incident field's SDoC, that near-field effects can still be observed.
The analysis and findings presented in this paper complement the existing near-field coherence research involving surface plasmons, and will be useful in future photonic devices that manipulate or control spatial coherence.
2. Analysis 2.1. Aperture geometry and assumptions Figure 1 depicts the scattering geometry considered in this paper. A stochastic field, incident from the left, illuminates a slit of width d in an infinitesimally thin PEC (s  ¥) plane immersed in a linear, homogeneous, and isotropic medium with permittivity ε and permeability μ. We assume that the scatterer and the incident field are z-invariant. As a result, Maxwell's equations decouple into independent TM and TE sets, and the general 3D problem becomes 2D. We observe the resulting scattered field in the slit at y=0 and in the far-zone as a function of f.
We note that the incident field shown in figure 1 should be interpreted as a single realization of a whole ensemble of fields, and hence, the scattered field, resulting from that incident field, should be interpreted in the same manner. We infer physical behavior of both the incident and scattered fields from statistical moments taken over the whole ensemble of possible fields.
Since we are concerned with coherence, we require only the first and second moments of the fields, better known as the mean and CSD function, respectively. To find these moments for the scattered field, we must know the incident field's mean and CSD function. We assume that the former is zero and hence, the mean scattered field is also zero. For the latter, we assume a Schell-model [18,28,34] form for the incident field CSD function, and further, that the mean intensity of the incident field fully and uniformly illuminates the slit. These two assumptions result in a CSD function of the form where α=x, z for TM and TE polarizations, respectively, I 0 is the average intensity, and γ is the SDoC [18,28,34].

MFIEs
Returning briefly to figure 1, by applying Love's equivalence theorem and image theory [29][30][31], we transform the aperture geometry into an equivalent one, in which a magnetic current  radiates in unbounded space. This current is unknown, and we derive an equation for  in terms of the incident field by enforcing the continuity of the transverse magnetic field in the slit, namely, where n is the unit normal to the slit (ŷ here), k w me = is the wavenumber, ω is the radian frequency, and ∇is the del operator.
The remaining symbols in equation (2) are H inc , which is the incident magnetic field, and F , which is the electric vector potential [29,30]. F contains  and is given by the integral expression Note that  is related to the electric field in the slit by We will return to this simple relation when we present the PO solutions.
For TM polarization, H inc and  are z directed, and equation (2) simplifies to In the case of TE polarization, H inc and  are x directed, and equation (2) becomes Note that the condition on x specified in equation (2) applies to these equations as well. We solve these two MFIEs for their respective  in the next section.

MoM solutions
The first step in the MoM is to expand  in a set of basis functions. These basis functions should match, as closely as possible, the physical behavior of the current. After expansion, the resulting integral equation is 'tested' by taking the inner product of both sides of the equation with another set of functions known as testing functions. For accurate results, the combination of basis and testing functions must overcome the source-point singularity in the Green's function (Hankel function) and make the expressions on the right-hand sides of equations (5) and (6) integrable. We begin with the TM polarization MFIE given in equation (5). Being in the z direction, TM  goes like x 1 at the slit edges [35,36], and the Hankel function, at the source-point singularity, goes like x ln ( ). Taking these two factors into account, we choose rectangular basis and delta testing functions to discretize equation (5) in a procedure called point matching [30,33,37]. Equation (5) becomes where m i TM is the unknown complex weight of the ith basis function, x i is the center of the ith rectangular basis function, w is the width of a basis function (assumed to have equal widths), x j is the location of the jth delta testing function, and x rect( ) is [38] x x x rect Note that the delta testing functions are located at the centers of the rectangular basis functions. We now proceed to the TE polarization MFIE given in equation (6). The x-directed TE  must go to zero at the slit edges and does so like x [35,36]. Although the Hankel function at the source-point singularity behaves like x ln ( ), the derivatives on the outside of the integral require a higher-order combination of basis and testing functions than in the TM case to make the expression integrable.
Using the symmetry of the Green's function and some basic calculus, we redistribute one of the derivatives to operate on TE  yielding We now choose triangular basis functions that span two discretized aperture segments, and rectangular testing functions that extend from the center of one segment to the center of an adjacent segment [33]. Note by choosing triangular basis functions, we ensure that TE  goes to zero at the slit edges. Substituting in these functions and simplifying produces The final form for both the TM and TE discretized MFIEs-equations (7) and (10), respectively-is Here, b is a vector containing the incident field in the aperture, m is a vector containing the unknown basis function weights and physically represents the magnetic current, and Z is the impedance matrix, which physically models how the field radiated from a current element affects or couples to every other element. Since Z contains all the near-zone interactions of the aperture field, it is instrumental in determining how and to what extent the near field affects spatial coherence. We will return to Z in a later section. Lastly, the far-zone electric field radiated by these currents is

PO solutions
Here, we briefly present the PO solutions [29][30][31]38] to the aperture problem depicted in figure 1. We do so because the PO approximation is commonly used in Fourier and statistical optics to predict the scattering (or diffraction patterns) from apertures in opaque screens [34,38]. It is well known that the PO approximation returns inaccurate results for wavelength-scale apertures [29][30][31]. Later, we show that it also returns inaccurate results for spatially incoherent incident fields, regardless of aperture size. Returning to equation (4), the PO solution assumes that the aperture field is approximately the incident field, namely, 2.5. CSD functions, SDoCs, and far-zone spectral densities In this section, we derive relations for the CSD functions and SDoCs of the near-zone MoM and PO fields. These second-order statistical moments show how the near-zone affects the coherence of the scattered field. To examine how these near-field coherence effects impact the far-zone, we also find the MoM and PO far-zone spectral densities (SDs). Although we do find relations for the MoM-based field statistics, these equations are in terms of the impedance matrix Z and therefore, numerical in nature. Referring back to equation (12), we easily find the currents by The CSD function of m is found by multiplying (on the right) both sides of equation (16) by m † , where †is the conjugate transpose, and then taking the expectation (ensemble average), namely, where the moment bb á ñ † is the discretized CSD function of the incident field and equal to ( ) † where X diag( )returns the diagonal elements of X as a vector. The MoM far-zone SD is found by taking the expectation of the magnitude squared of equation (13) where the moments in equation (20) are the TM and TE current CSD functions given in equation (17).
For the PO approximation, the CSD function of the currents is found by taking the autocorrelation of equation (14), which is trivially equal to equation (1); the SDoC of the PO currents is therefore γ. We derive the PO far-zone SD by computing the second moment of equation (15), namely, The bracketed function in the integrand is the Fourier transform of the aperture autocorrelation function . The far-zone PO SD is therefore proportional to the convolution of the Fourier transforms of  and γ, and physically is the coherent diffraction pattern filtered or smoothed by g [34,39,40].

Discussion
Having derived the requisite statistical moments, we now discuss their physical significance. Referring back to equations (17) and (19), we first note that in general, the CSD function and SDoC of the currents-or equivalently, the near field via equation (4)-are not equal to those of the incident field. This stands in contrast to the PO CSD function and SDoC, which are equal to those of the incident field. These statements are true regardless of aperture size. The fact that the MoM CSD function and SDoC differ from the commonly used PO statistics is not that surprising-even in the absence of surface plasmons. What truly matters is whether these differences give rise to observable effects. To begin addressing this, we start with an approximate, analytically simple model for a spatially incoherent incident field, i.e., is the Dirac delta function [28,34]. The discretized incident field CSD function bb á ñ † , given in equation (18), is proportional to the identity matrix I, and W m simplifies to The PO CSD function is equal to equation (24). We note that the incoherent CSD function in equation (24) cannot be represented exactly by the discrete bb á ñ † , which has an equivalent width equal to that of a testing function, or w. Figure 2 shows the TM and TE SDoC magnitudes corresponding to the discrete CSD function in equation (25) for several values of the aperture width d. The figure is organized into two columns, where the first column shows the TM m g | |and the second column shows the TE m g | |. The first 3 rows of figure 2 show the full 2D ; m g | | row 4 shows the corresponding 1D cuts through m g | |with x 2 =0. To compute Z, we discretized the TM and TE aperture currents using w=λ/50 segments.
It is quite clear from figure 2 that the impact of the near field on coherence is most pronounced in the TE polarization. This is due to the direction of the TE current or aperture field, which is oriented normal to the aperture edge. By visualizing the TE current as numerous tiny magnetic dipole antennas placed end to end, we see physically that this results in significant near-field coupling among 'adjacent' dipoles. Figure 2(h) shows that the widths of the SDoC magnitudes (distances between the first zeros) are λ/2. We note, however, that there is significant residual coherence for separations beyond λ/2 extending several λ as seen in figures 2(d) and(f).
On the other hand, the near zone has little affect on TM fields. The TM current is oriented parallel to the aperture edge in the infinite, or invariant direction. In this case, the tiny magnetic dipole antennas are placed side by side, resulting in negligible coupling between nearby antennas. Figure 2(g) supports this physical picture, showing that the widths of the SDoC magnitudes are approximately λ/10 and independent of d.
These findings are easily generalized to 3D problems. For instance, assuming a d x ×d y rectangular aperture in the x-y plane, the 'TM polarization' scenario applies to the x-and y-directed currents parallel to the aperture edges at y=±d y /2 and x=±d x /2, respectively. The TE scenario applies to the x-and y-directed currents normal to the aperture edges at x=±d x /2 and y=±d y /2, respectively. Likewise, for a circular aperture in the x-y plane, the TM and TE scenarios apply to the fand ρ-directed currents, respectively.
To investigate how the near-field coherence phenomena shown in figure 2 impact the far zone, figure 3 shows the MoM and PO far-zone SDs with an incoherent incident field. Like figures 2, 3 is organized into left and right columns, which show the TM and TE SDs, respectively. Row 1 shows S MoM and S PO on a Cartesian grid; row 2 shows S MoM and S PO on a more physical, polar grid. All SDs are plotted in decibels (dB).
Here, we see differences in the MoM and PO far-zone SDs for both polarizations. Because the PO current CSD function mirrors that of the incident field, S PO is featureless showing no sign of the aperture. In the TE case, it varies with f because of the sin 2 f in equation (22). Although S MoM and S PO differ for both polarizations, the differences are much more pronounced in the TE case, especially at large observation angles where we observe directional scattering. We also see partial constructive and destructive interference (ripples) in S MoM in figure 3(b). Since the incident field is incoherent, these telltale signs of (partial) spatial coherence must be due to near-field effects (see figure 2). We do not observe any interference phenomena in S PO or TM S MoM . Figures 2 and 3 show unequivocally that the near-field most strongly affects coherence in the TE polarization. We therefore focus on the TE case for the remainder of this paper.
We now consider a more realistic incident field CSD function than equation (24), namely, where β>0 is the coherence width (in units of waves) of the incident field. We note that the above CSD function with β=1 is consistent with those of blackbody or Lambertian radiators [28], and therefore, could be considered a 'natural' spatially incoherent source. Figure 4 shows the normalized TE near-field SDs (second moment of the currents), the 1D cuts through the SDoC magnitudes (x 2 =0), and the corresponding S MoM and S PO in dB on polar grids for different β. We assumed d=10λ and discretized the TE current using w=λ/50 segments.
The results in figure 4 show, not surprisingly, that the PO approximation improves (agrees better with the corresponding MoM result) as β increases, or equivalently, as the incident field becomes more spatially  (26)), the 'side lobes' are significantly higher for the MoM m g | |. This physically means that the scattered field is spatially more coherent than predicted by the commonly used PO approximation.
Lastly, figures 4(e) and (f) show the far-zone SD results, where, again, we observe significant differences between S MoM and S PO when β=1. Like the SD results in figure 3, these differences are most noticeable at large scattering angles.
We note here that d=10λ, which is large enough for the PO approximation to be accurate (see the β=20 results in figure 4(f)). The β=1 results in figure 4 show that the PO approximation gives nonphysical results for incoherent incident fields. This finding is independent of aperture size and a key result in this paper.

Young's experiment
In the previous section, we showed and explained that because of the direction of the TE current, significant near-field coupling occurs, generally resulting in an increase in spatial coherence. Here, we investigate this phenomenon further using a two-slit geometry, i.e., Thomas Young's experiment [28,34]. The purpose is to observe interference of the light emitted from the two slits when illuminated by a spatially incoherent TE field. The two-slit geometry is similar to that depicted in figure 1-the difference being that there are now two apertures (of width d) centered on the origin and spaced D apart (when measured from center to center).
The MoM CSD and SDoC are still, in general, given by equations (17) and (19); however, the TE MFIE is now a system of coupled MFIEs in terms of the currents in slits1 and2: The same triangular basis and rectangular testing functions that were used in the single aperture problem are used here as well. Since the analysis is very similar to that presented in section 2.3, we omit the details for the sake of brevity. The MoM far-  Figure 6 shows the corresponding S MoM and S PO , plus their incoherent and coherent contributions. Here, we assumed the slits were d=5λ, the currents were discretized using w=λ/50 segments, and the incident field CSD function was given by equation (26) with β=1.
The SDoC magnitudes in figure 5 are block 2×2, with the off-diagonal blocks physically showing how much one slit coherently couples to the other. As physically expected, the amplitudes of these blocks decrease as D increases. In addition, one notices that the off-diagonal block amplitudes for the MoM m g | |are significantly higher than the corresponding PO SDoC magnitudes, which are very weak. This means that, although illuminated by a spatially incoherent field, the light emitted from the two slits is actually partially coherent, and near-field physics are responsible for the transformation. Figure 6 shows the extent to which coherent near-field coupling is observable in the far zone. For all values of D, there is a coherent contribution to S MoM which causes observable (in some cases, minor) differences between S MoM and S MoM,i located predominately at large scattering angles. As physically expected, these differences are most noticeable when D is the smallest (see figure 6(a)). We observe no discernible differences between S PO and S PO,i .

Conclusion
In this paper, we studied how the near field affects spatial coherence considering phenomena other than surface plasmons, whose role is well documented. We derived the second-order statistical moments of the field (CSD functions, SDoCs, and far-zone SDs) scattered from an aperture in an infinite PEC screen (a structure that does not support surface plasmons) using both rigorous EM theory and the commonly used PO approximation. We compared and contrasted the EM and PO field statistics and discussed their physical significance at length.
From our analysis, we concluded that the near field does impact spatial coherence, even in the absence of surface plasmons, and the effects are observable. By examining the far-zone SDs and assuming a spatially incoherent incident field, we found that near-field coherence effects were most noticeable at large, nonparaxial scattering angles, where significant differences were noted between the EM and PO SDs. Most interestingly, we found that this held true even for optically (or electrically) large apertures, where PO was assumed to be accurate.
To further examine these effects, we considered the near-and far-zone field statistics of Young's classic, twoslit experiment. Even with an incoherent incident field and large, widely separated apertures, we observed interference of the light emitted from the slits implying that near-field physics had increased the spatial coherence of the field.
The analysis and findings presented in this work complement the prior theoretical and experimental coherence research involving surface plasmons, and could find use in future, compact, low-power devices designed to manipulate or control spatial coherence.