We study theoretically the absorbed power by a dielectric sphere when it is illuminated with partially coherent light coming from two pinholes. We present a general theory of Mie scattering of partially coherent light (based on the angular spectrum method); this theory is applied to the aforementioned particular scattering problem which is solved analytically. We found that, if the diameter of the sphere is smaller than the skin depth, the absorbed power by the sphere depends complicatedly on the degree of coherence of light between the pinholes. The absorbed power for coherent illumination can be smaller or greater than that for incoherent light between pinholes, depending on the geometrical configuration. Furthermore, there are particular setups in which the absorbed power is independent of the degree of coherence, despite that the intensity distribution of the electric field inside the sphere depends significantly on the spatial coherence. Hence, by tuning the coherence length between the pinholes, the absorbed power by the sphere can be controlled; if a whispering gallery mode is excited, the absorbed power can be varied over a wide range. Our study might have implications in the understanding of light absorption in photovoltaic nano-devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The scattering of light by particles is a ubiquitous physical phenomenon. The understanding of this effect is crucial for applications such as particle trapping, biosensing, near-field microscopy, biomedical diagnosis, nanoantennas, photovoltaics, atmospheric remote sensing, and meteorological phenomena. Since the seminal work of Mie who established the analytical theory of electromagnetic scattering of a plane wave by a spherical object in the early 20th century , the scattering problem under deterministic or fully coherent illumination has been extensively studied. However, its counterpart, the studies devoted to the scattering of partially coherent light have been limited. Our paper concerns to the latter case.
The influence of transverse spatial coherence on light scattering by particles was studied in , finding that, for a rotational-invariant scatterer (intensity of the incident field across the particle varies slowly), the extinction does not depend on the spatial degree of coherence, whereas the angular pattern of the radiant intensity depends on that. Particularly, for a sphere (Mie scattering), the angular distribution of the radiant intensity can be strongly perturbed by the spatial coherence of a Gaussian Schell-model beam, as demonstrated in ; in a later study by same group , it was shown that the total scattered power is independent of the state of coherence of such a beam. By varying the spatial coherence of a partially coherent beam with a Bessel-type correlation, the direction of the maximum of the radiant intensity can be shifted [5–7]. We mention that [2–7] are restricted to scalar theory. An electromagnetic theory of Mie scattering of partial coherent beams is described in ; the spectral cross-density is represented in terms of the series of coherent modes and vector spherical harmonics and the theory is applied for obtaining the far-field radiated power when a distant blackbody source (Sun) illuminates a sphere, obtaining that the radiant intensity (paraxial limit) is related to integral of the scattering amplitudes (plane wave response) over the solid angle subtended by the source surface. The study  presents an electromagnetic theory of scattering of a partially coherent beam in the far-field by a sphere, showing the influence on the scattering radiant intensity and degree of coherence by the spatial coherence of the incident beam. Vortices in the complex degree of coherence which arise from Mie scattering of a dielectric sphere were examined in [10, 11].
Concerning cylindrical geometry, it was found that, when metallic nanocylinders are illuminated with stochastic light, the coherence and polarization characteristics of near-field are affected by the excitation of plasmonic resonances of such objects . Based on the angular spectrum methodology, the paper  establishes a general theoretical formulation for solving the scattering problem of a 2D-electromagnetic partially coherent beam incident on a circular cylinder, observing that the directionality and strength of the scattering width decrease in the forward region as the spatial coherence length of the incoming beam diminishes. A physical optics approach (suitable for a cylinder with diameter larger than the wavelength) was implemented for the scattering of partially coherent light . The effect of superscattering was analyzed for partially coherent illumination, finding that this effect persists independently of the degree of coherence, but the direction of the maximal scattering cross-section deviates from the forward direction as the degree of coherence decreases .
As seen, most of the aforementioned studies related to scattering of partially coherent light treat effects of spatial coherence on the far-field zone. Our paper focuses on the influence by the spatial coherence on the absorbed power by a dielectric sphere. We consider that the sphere is illuminated with stochastic light coming from two pinholes whose coherence length between these pinholes is varied. Absorption is an essential process in photovoltaic cells in which usually stochastic light is incident on them, thus it is crucial to understand until what extent the coherent properties affect the absorptive properties of these devices. For solving the scattering of an arbitrary partially coherent beam (arising from a secondary source), we provide a general theoretical formalism based on the angular spectrum and vector spherical harmonic expansions of cross-spectral densities. We apply this formalism to the aforementioned case for which a dielectric sphere is illuminated with partially coherent light coming from two pinholes.
Our paper is organized as follows. Section 2 deals with the theoretical framework for describing the scattering of a partially coherent beam by a spherical particle; expression for the absorbed power by a sphere is obtained. In Sec. 3, the theory is applied to the scattering problem of partially coherent light from two pinholes. The results together with their analysis are presented in Sec. 4. Last section is dedicated to conclusions. Four appendices complement the theoretical formulation.
Figure 1(a) illustrates our general setup of study. A partially coherent beam originating from a stochastic secondary planar source irradiates a sphere with radius a. The sphere has dielectric [magnetic] function ϵ2(ω) [µ2(ω)] and it is embedded in an absorptionless medium with dielectric and magnetic functions ϵ1(ω) and µ1(ω) (Im[ϵ1] = Im[µ1] = 0), respectively. A secondary planar source is a stochastic distribution of electromagnetic fields whose statistical properties are known at an arbitrary plane. Furthermore, we assume that the secondary source is stationary in the wide sense, thus the propagating and scattering fields are stationary in the wide sense as well. We set a primed Cartesian coordinate system (x′, y′, z′) in which the secondary planar source is located at plane z′ = 0. The center of the sphere is placed on the z′-axis at coordinate z′ = d (d > 0). Also we establish an unprimed Cartesian coordinate system (x, y, z) whose origin is center of the sphere where the transformation between these Cartesian systems is given by
As consequence, the relations between the unit Cartesian vectors of these coordinate frames are
We consider that E0(r, ω) (ω being the angular frequency) is the incident stochastic electric field in frequency domain. Then, the second order correlations of the Cartesian components of the incident electric field is16], can be determined from the known cross-spectral density at plane z′ = 0, that is, where ; hereafter, subscript n indicates 1 or 2. Then,
is the Fourier spectrum of , namely,
Here, d2κn ≡ dkxndkyn, and where
Our goal is to find, from the incident cross-spectral density densities related to the correlations of the Cartesian components of the stochastic scattering and [ and , respectively], that is,(3)). As we will see, the scattering problem is completely solved if the incident cross-spectral density is expanded into a series of vector spherical harmonics with respect to the center of the sphere as (11) in terms of spherical coordinates of kn, namely kxn = k1 sin αn cos βn, kyn = k1 sin αn sin βn, kzn = k1 cos αn. Then, (11) becomes 17], we found the following multipolar expansion Appendix A. Finally, the weighting coefficients of series (8) are (14) involves only the correlation of the x- and y-components of the electric field; the z-component of the electric field is implicitly included by using the fact that the electric field is divergence free, implying that Ez depends on Ex and Ey .
2.1. Cross-correlations of scattering and internal fields
As mentioned, once the multipolar expansion of the incident cross-spectral density (8) is obtained, the cross-spectral densities involving the scattering and internal fields are straightforwardly determined which are represented in matrix:17], as a direct consequence of the boundary conditions that the electromagnetic fields must fulfill at the interface r = a, it is found that Appendix B); and is the i-th (i = x, y, z) Cartesian component of vector spherical harmonics and , respectively, where is defined as (9), but jl(kr) is replaced by hl(kr) (first-kind Hankel spherical function of l order), whereas is obtained from (10) with .
2.2. Absorbed power by the sphere
The spectral electric intensity inside the sphere is given by(21) can be solved analytically as follows. By using (17), S(r, ω) turns out to be a superposition of any combination of the scalar products of the elements of following left and right columns: Appendix C) and the remaining integral with respect to the radial coordinate is solved by using 
3. Scattering of partially coherent light from two pinholes
A screen is placed at z′ = 0 and it has two pinholes that are located at and with the restriction that ρ01 ≠ ρ02 [see Fig. 1(b)]. As the screen is illuminated, the cross-spectral density tensor of this secondary source at plane z′ = 0 can be expressed as19] Appendix D for the x- and y-components of and , the weighting coefficients (15) of incident beam are obtained analytically. After a lengthy algebra,
We assume that coherence properties of the planar secondary source with two pinholes is described by a Gaussian Shell-model with half-width w0 and coherence length σ0, that is,16].
4. Results and discussion
We consider that the background medium is vacuum (ϵ1 = µ1 = 1) and the angular frequency of incident light corresponds to a free-space wavelength λ1 = 2πc/ω = 500 nm. The sphere is nonmagnetic (µ2 = 1) and has a refractive index of . These optical parameters render a skin depth δs such that k1δs = 1(2Im[n2]) = 50 (Im […] denotes imaginary part). We consider that the pinholes are in the x′-axis and separated symmetrically with respect to the origin of the primed coordinate system, that is, . We assume that, at source (Gaussian Schell-model) plane, light is polarized in the x-direction, thus, and . Moreover, we normalized the absorbed power as
Figure 2 shows the normalized spectral absorbed power as a function of the normalized radius k1a for a sphere-source distance k1d = 50, a half inter-pinhole separation k1x0 = 25, and two states of source coherence (coherent and incoherent); the peaks of the curves correspond to whispering gallery mode (WGM) resonances. It can be noticed that, for a particular value of radius a, the absorbed power can be or or ; this applies to either on- and off-resonance conditions. Hence, the spatial coherence of the source influences the absorbed power by the particle. According to Fig. 2, the perturbation of the absorbed power by state of coherence of the source becomes weaker as the size of the sphere increases, implying the decreasing of the height of resonance peaks. Moreover, we can differentiate two regimes of perturbation by the source degree of coherence which are delimitated by the skin depth: (1) the strong perturbation regime occurs when 2a < δs and (2) weak perturbation regime happens in the limit where 2a is approximately equal or larger than δs.
Now we take a close look at the behavior of absorbed power as the inter-pinhole separation is varied for two cases corresponding to the strong perturbation regime: on-resonance (k1a = 2.8569) and off-resonance (k1a = 3.275); the bottom inset of Fig. 2 shows that these conditions are fulfilled for such radii. In addition, we examine in detail the case when 2a ∼ δs (k1a = 20.125) for which the degree of coherence of the source perturbs slightly the absorbed power (weak perturbation regime); see top inset of Fig. 2.
4.1. On-resonance (2a < δs)
Now we study the resonance case, mentioning that the resonance occurring at k1a = 2.8569 is a transverse magnetic (TM) WGM with angular number l = 4. Figure 3(a) depicts the absorbed spectral power as a function of half inter-pinhole distance k1x0. The incoherent curve is a decreasing function of the inter-pinhole separation as a consequence of the reduction of the intensity of the incident electric field as the pinhole-sphere distance grows. The coherent curve exhibits large oscillations around the incoherent curve. Consequently, there are intervals of the inter-pinhole separation in which the and others where . As expected, the partially coherent curve is bounded between and . Therefore, if the inter-pinhole distance is fixed, it is possible to control the absorbed power by varying the coherence length between the pinholes. Also we can notice that there are specific inter-pinhole distances in which the curves , , and intersect. When this happens, the absorbed power is independent of degree of coherence of the secondary source.
To understand the aforementioned characteristics of the curves of Fig. 3(a), we examine the spectral electric intensity inside the sphere for three inter-pinhole spacings [indicated in Fig. 3(a)]: first case (k1x0 = 22, denominated Ion) in which , the second case (k1x0 = 34.572, denominated IIon) in which , and third case (k1x0 = 44, denominated IIIon) in which . As seen in (21), the spectral absorbed power by the sphere is proportional to the volume integration of the electric intensity inside the particle. The spectral electric intensity is normalized as
Figure 3(b) shows the electric intensity contour plots associated to case Ion for the aforementioned three states of coherence. In this case, it can be seen that the incoherent state allows the excitation of the WGM resonance (xz-plane), whereas the coherent illumination cannot excite the WGM resonance. Consequently, the extension and strength of high intensity regions for the incoherent state are greater than those for the coherent state, explaining . For the partially coherent state, the WGM resonance is excited, but, relative to the incoherent case, the peaks of intensity are lower, implying that .
The electric intensity distributions for case IIon are depicted in Fig. 3(c). As seen, the contour plots for the coherent state exhibit faster spatial variations than those for the partially and incoherent cases, as well as a higher contrast between low and high intensities regions. Even though the electric intensity contour plots depend considerably on the state of coherence, it is surprising that these distributions yield the same power absorbed (actually, as mentioned, the absorbed power is independent of the state of coherence).
Figure 3(d) illustrates the intensity patterns for case IIIon. For this case, we can appreciate that the illumination corresponding to any of the three states of coherence allows the excitation of the WGM resonance. However, the WGM resonance (xz-plane) for the coherent state is sharper than that for the other states (partially coherent and incoherent). Also the high-intensity peaks of the WGM (xz-plane) for the coherent case are larger than those for the partially coherent and incoherent cases. Consequently, .
4.2. Off-resonance (2a < δs)
Now we analyze the off-resonance case where k1a = 3:275. In Fig. 4(a), we plot the normalized absorbed power versus the normalized half inter-pinhole spacing k1x0. Similarly as the on-resonance case, the curve is a decreasing function of k1x0 and undulates around curve . Hence, the behavior for both on- and off-resonance cases is quite alike, that is, there are points for which and vice versa. However, in comparison with the on-resonance case, the deviations of from for the off-resonance case are smaller.
Again, to gain insight of the behavior of plots of Fig. 4(a), we obtain the intensity distribution inside the sphere for three different inter-pinhole distances: , designated as Ioff), , designated as IIoff), and , designated as IIIoff).
Figure 4(b) depicts the normalized electric intensity for case Ioff. The spatial distributions for the coherent state are more complex and have higher contrast than those for the incoherent state. We know that , however, by looking the intensity patterns, we cannot determine which coherence state yields the largest absorbed power. This shows the complicated dependence of the geometrical parameters of the setup and the state of coherence of the source on the absorbed power by the sphere.
For case IIoff, the normalized intensity contour plots are shown in Fig. 4(c). In the same manner as for the resonance case, although the electric intensity contour plots corresponding to the three states of coherence are noticeably different among them, the absorbed power is the same regardless the state of coherence.
The electric intensity contour plots for case IIIoff are displayed in Fig. 4(d). For this case, the intensity patterns among the different states of coherence look similar. The difference is that, as the degree of coherence between the pinholes decreases, the spatial distribution of the electric intensity becomes smoother. Also by visually inspecting the contour plots, it is not possible to discern which state of coherence gives the largest absorbed power.
4.3. Case 2a ∼ δs
When the size of the sphere is around or larger that the skin depth δs, light cannot circulate around the sphere. As a consequence WGM resonances cannot be built up. As mentioned, under this condition, the power absorbed is almost the same regardless of the state of coherence. Figure 5 shows the normalized spectral intensity for k1a = 20.185. This case yields . Contrary to the on- and off-resonance cases, by comparing the contour plots of electric intensity corresponding to different states of coherence, we can observe that they are practically identical, excepting the intensity contour plots in the xy-plane in which slight differences are perceptible. Hence, in this limit, the intensity distributions and the absorbed power are nearly independent of the degree of coherence of the source. The contour plots of Fig. 5 have very intricate patterns because of the size of the particle is larger than the wavelength.
4.4. Additional comments
Transverse electric (TE) WGM resonances can be also excited with the illumination condition and . However, as the geometrical settings are modified, the deviation of curve from for TE resonances is much smaller than that for TM resonances. If light polarization at the source plane is rotated π/2 radians, namely, and ), then the effect would be inverted, that is, oscillations of from for TE-WGM resonances would be larger than those TM-WGM resonances.
We studied the absorbed power by a dielectric sphere when the particle is illuminated with light coming from two pinholes in which the degree of coherence between these pinholes is varied. To describe this effect, based on the angular spectrum methodology, we presented a general theory of Mie scattering of partially coherent light. We distinguished two regimes: (1) the size of the particle is smaller than the skin depth, and (2) the diameter of the sphere is about or larger than the skin depth.
When 2a < δs, we have shown that the absorbed power by a dielectric sphere is influenced by the degree of coherence between the pinholes, as well as the geometrical parameters of the setup. Some conditions yield that the absorbed power for the coherent state of the secondary source is larger than that for the incoherent case, while other conditions render the opposite. It is possible to vary the power absorbed by tuning the coherent length of the source, implying that the modulation of the spatial coherent acts as an agent for controlling the absorbed power; when WGH resonances are excited, the absorbed power can be modified with a large extension. For certain cases, although the electric intensity distribution inside the sphere depends appreciably on the coherence length of the secondary source, the absorbed power is surprisingly independent of last parameter.
In the regime 2a ≳ δs, the absorbed power and the electric intensity inside the sphere is practically independent of the degree of coherence between the pinholes.
As a consequence of the current technological capabilities for manipulating and manufacturing nano-structures, photovoltaic nano-devices with unconventional geometries have been studied; for example, nanowires  and spherical nanoshells . Therefore, our paper might impact in the understanding of absorption of stochastic light in such devices.
A. Explicit factors of (14)
The explicit expressions of factors of (14) are
B. Mie coefficients
The Mie factors are defined as
C. Orthogonality properties of vector spherical harmonics
The orthogonality of the vector spherical harmonics are 
D. Cartesian components of vector spherical harmonics
The x- and y-components of are 
CONACYT (353198) scholarship.
JAG-G acknowledges the financial support by CONACYT through scholarship (No. 353198).
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