## Abstract

The degree of optical spatial coherence—a fundamental property of light that describes the mutual correlations between fluctuating electromagnetic fields—has been proven challenging to control at the micrometer scale. We use surface plasmon polaritons—evanescent waves excited on both surfaces of a thin metal film—as a means to mix the random fluctuations of the incident electromagnetic fields at the slit locations of a Young’s double-slit interferometer. Strong tunability of the complex degree of spatial coherence of light is achieved by finely varying the separation distance between the two slits. Continuous modulation of the degree of spatial coherence with amplitudes ranging from 0 to 80% allows us to transform totally incoherent incident light into highly coherent light and vice versa. These findings pave the way for alternative methods to engineer flat optical elements with multifunctional capabilities beyond conventional refractive- and diffractive-based photonic metasurfaces.

## INTRODUCTION

Control of light propagation and interaction with matter relies upon knowledge of the amplitude and phase of the electromagnetic fields, as well as their mutual temporal and spatial correlations, which are described by the complex degree of coherence (*1*–*3*). Planar metal and dielectric surfaces with subwavelength features—also known as metasurfaces (*4*–*6*)—have recently been proposed as a way to manipulate light by producing abrupt changes in the local amplitude and phase of the incident electromagnetic fields (*7*). Typically, the incident fields are highly coherent, and the scattered fields originating from the interaction of incident light with the nanostructured surfaces are mutually coherent. Then, optical interference effects produce desired changes in beam directionality, polarization, intensity, phase, and spin (*8*, *9*).

Optical spatial coherence can provide an alternative, powerful tool to control the flow of light with various degrees of coherence beyond conventional wavefront shaping methods offered by photonic metasurfaces. From an application standpoint, control of the degree of spatial coherence of light can lead to higher-resolution speckle-free imaging (*10*, *11*), ultimate capability to shape (*12*) and redirect light beams (*13*–*15*), and enable novel modulation methods for free-space optical communications and interconnects (*16*, *17*). However, full control of spatial coherence at length scales comparable to the wavelength of light has been proven challenging. Although it has been theoretically suggested that surface plasmon polaritons (SPPs)—electromagnetic waves evanescently bound to metal surfaces—can, in principle, modulate the degree of spatial coherence (*18*–*20*), a very few experimental studies have shown this effect, reporting only modest modulation amplitudes (*21*–*23*).

Here, we show, for the first time, large and finely tunable changes in the amplitude and phase of the complex degree of spatial coherence, controllable at the micrometer scale by simply varying the slit-slit separation distance, as well as the wavelength and polarization state of the incident light. For instance, we report how light incident on a Young’s double-slit can be tuned from completely incoherent to partially coherent and vice versa, with degrees of coherence continuously variable from ~0% (totally incoherent) to ~80% (almost fully coherent).

## RESULTS

### Spatial coherence of optical fields

Partially coherent light exiting the two slits placed at points *P*_{1} and *P*_{2} in an opaque film can generate interference fringes when projected onto a far-zone screen, as the result of constructive and destructive interference caused by the difference in the optical paths from each slit to a specific point on the screen (Fig. 1A and fig. S1). The fringe contrast, or visibility, defined as , where *I*_{max} and *I*_{min} are the maximum intensity and minimum intensity, respectively, in the interference pattern (Fig. 1A), can be used to quantitatively describe the complex mutual correlation function μ(*P*_{1}, *P*_{2}) = |μ|*e*^{iφ} between the electromagnetic fields at the two different points where the slits are located. It can be proven that the visibility of the interference fringes is equal to the amplitude of the degree of spatial coherence (*24*), that is, . The phase φ can be inferred by looking at the interference condition at the center of the screen, with constructive (destructive) interference corresponding to φ = 0 (π). Therefore, Young’s double-slits can be used to directly measure and quantify the changes of the degree of coherence of incident light at the slit locations, with μ = 0 (or 1) corresponding to totally incoherent (or coherent) light.

### Transforming incoherent light into coherent light and vice versa

In the absence of SPPs, the degree of coherence of transmitted light is unaffected after interaction with the slitted screen (Fig. 1A), resulting in μ_{in} = μ_{out}. This condition can be realized by using perfectly opaque screens or very large slit-slit separation distances in real metals or when light is linearly polarized with the electric field direction parallel to the long axis of the slits [transverse electric (TE)–polarized].

Under transverse magnetic (TM) illumination (that is, the electric field perpendicular to the long axis of the slits), SPPs are excited and can introduce additional or eliminate existing correlations between the fields at the two slits, which enables strong modulation of the degree of spatial coherence provided that the SPP excitation process is coherent (*25*–*27*). For example, Fig. 1B shows how SPPs can transform partially coherent light incident onto the slits into fully incoherent light leaving the slits. This is evidenced by the disappearance of the interference fringes that are replaced by a broad, featureless intensity distribution on the projection screen caused by the incoherent sum of two single-slit diffraction patterns (Fig. 1B).

Totally incoherent light incident upon the double slits is unable to generate interference fringes in the far field (Fig. 1C, TE polarization, no SPPs). However, SPPs excited at each slit location and propagating toward the other slit can, under proper conditions, mix the initially uncorrelated fields and generate partially coherent light with nonzero degree of coherence at the output of the double-slit. Because of these SPP-induced correlations introduced by the alternative path that light can take in the form of propagating SPPs, totally incoherent light can be transformed into partially coherent light, and as a result, the far-zone fringe visibility is restored (Fig. 1D). Far-zone interference patterns can therefore be used to infer the complex degree of spatial coherence and quantify the changes induced by SPPs (see the Supplementary Materials for more details).

### Experimental setup

Figure 1E shows a schematic of the experimental setup that was specifically designed to measure wavelength-resolved Young’s double-slit interference patterns as a function of slit-slit separation distance *d* and incident polarization. In particular, double-slits etched in a thin silver film were illuminated by linearly polarized broadband light under Köhler illumination (*28*). The objective of an inverted microscope was purposely defocused to project the interference pattern onto the entrance mask of a spectrograph, which disperses the transmitted light and projects the generated interference patterns onto a two-dimensional imaging camera. Representative wavelength-resolved interference patterns emerging from a Young’s double-slit with *d* = 5 μm, illuminated with partially coherent light with two different polarization states (TE and TM; Fig. 1, F and G, respectively), show the marked changes in fringe visibility caused by SPPs. Remarkably, the results reported in the right insets of Fig. 1 (F and G) show—now experimentally—that SPPs can transform incoherent light incident upon the double-slit into partially coherent light at the output of the slitted screen (solid lines; λ_{1} = 581 nm), as attested by the increased fringe visibility when TM-polarized light is used. Vice versa, partially coherent light can be transformed into incoherent light (dashed lines; λ_{2} = 712 nm), as evidenced by the suppressed fringe contrast due to SPP-induced modulation of the mutual correlation function (see Materials and Methods for more details).

### Effects of spatial coherence of incident light

Several wavelength-resolved interference patterns for the same Young’s double-slit with *d* = 5 μm under varying degrees of spatial coherence of the incident light are reported in Fig. 2. Under TE-polarized illumination (no SPPs), the far-zone interference patterns gradually change as a function of incident wavelength λ, at any given Δθ (Fig. 2, A to C). Light with longer wavelength generates interference fringes that are spatially more spread out compared with shorter wavelengths, as the result of the longer optical path difference required to achieve the same phase shift on the projection screen. The fringe visibility is greatly reduced when light with lower degree of spatial coherence (that is, higher subtended illumination angle Δθ) is used. By rotating the polarization state of the incident light by π/2 (that is, from TE to TM), SPPs from both metal/dielectric interfaces can be turned on, and strong modulation of the fringe visibility is observed as the result of additional beatings between the two SPP modes (Fig. 2, D to F). Both the amplitude (related to fringe visibility) and phase (related to either constructive or destructive interference at the screen center) of the degree of spatial coherence are strongly modulated under TM-polarized illumination due to SPP excitation (Fig. 2, E and F) that determines a redistribution of the far-zone light intensity. Both enhancement and suppression of fringe visibility can be achieved relative to the respective interference patterns under TE illumination (Fig. 2, B and C, no SPPs).

### Theory of optical coherence modulation with surface plasmons

An analytical expression for the output fields that includes SPP contributions can be written as *E*_{i}(λ, *d*) = τ*E*_{i,in}(λ) + τβ*E*_{j,in}(λ), where (*i*, *j*) = (1, 2) or (2, 1); *E*_{i,in} is the incident scalar field at one slit location; τ is the transmission coefficient through the slit; λ is the free-space wavelength of light; and , where β_{t} and β_{b} are the SPP coupling coefficients at the top (that is, glass/Ti/Ag) and bottom (that is, Ag/air) interfaces, respectively, and *k*_{SPP,t} and *k*_{SPP,b} are the corresponding SPP complex wave vectors, which account for SPP absorption due to ohmic losses in the metal.

By using the definition of cross-spectral density function , where the angle brackets indicate ensemble averaging, the degree of spatial coherence of light exiting the double-slit can be written as (*19*, *23*)(1)where μ_{in} is the complex degree of spatial coherence of light incident on the double-slit, which is generally treated as an ad hoc tunable parameter (*21*–*23*). Here, μ_{in} is experimentally tuned using a Köhler illumination setup in which light with varying degrees of spatial coherence is generated by changing the subtended illumination angle through a variable condenser aperture. Moreover, a theoretical framework is developed to calculate the far-field intensity distribution determined by light diffraction and interference either with or without SPP contributions, as described in the Supplementary Materials.

### Beatings between surface plasmons excited on both metal/dielectric interfaces

To better quantify and control the modulation of the complex degree of spatial coherence, SPP contributions from both metal/dielectric interfaces need to be considered and carefully tailored. These contributions can be extracted by analyzing the plasmonic interferogram, that is, a plot of the normalized intensity ratio at a given wavelength obtained by normalizing the light intensity transmitted through each double-slit with varying separation distance to the reference intensity transmitted through the corresponding individual slit (*29*). Figure 3A shows a representative plasmonic interferogram measured at λ = 600 nm. Discrete Fourier transform analysis applied to this data set (*28*, *30*) shows the different SPP contributions originating from both Ag/air (black arrows) and glass/Ti/Ag interfaces (red arrows) as distinct intensity peaks in the Fourier power spectrum amplitude (Fig. 3B). This procedure can be extended to all measured wavelengths, and the resulting data sets are plotted as two-dimensional color maps of plasmonic interferograms (Fig. 3C) and Fourier transform power spectra (Fig. 3D), which reveal the energy-momentum dispersion of SPPs existing on both interfaces. By filtering out the higher-order SPP contributions (with *m* > 1), plasmonic interferograms with only first-order SPP contributions can be reconstructed (red line in Fig. 3A), and the corresponding SPP coupling coefficients β_{t} and β_{b} can be extracted from fits to the filtered data (see Materials and Methods).

### Strong modulation of spatial coherence

Far-zone interference patterns as a function of slit-slit separation distance can be constructed by scanning all of the fabricated Young’s double-slit interferometers with varying separation distances under TE-polarized (Fig. 4, A to C) and TM-polarized (Fig. 4, D to F) illumination. The far-zone interference patterns recorded in the presence of SPPs show strong modulation in the fringe contrast and richer intensity features, with overall increased number of observable maxima and minima (for instance, compare Fig. 4F with Fig. 4C). The amplitude and phase of the complex degree of spatial coherence at any given wavelength can be extracted from quantitative inspection of the interference patterns. The results of such an exercise are reported in Fig. 4 (G to L), for both TE and TM illumination, at λ = 600 nm. The fringe visibility recorded under TE-polarized (no SPPs) illumination with Δθ = 3° gradually decreases as a function of increasing slit-slit separation distance (green line in Fig. 4G). Such a change follows the model derived from the van Cittert–Zernike theorem (black line in Fig. 4G). In contrast, the measured visibility under TM illumination (with SPPs) is significantly modulated (blue line in Fig. 4G). Note that for small separation distances (*d* < 5 μm), interference effects caused by beatings of SPPs supported by both metal/dielectric interfaces are visible, as evidenced by the non–single-periodic oscillations of the visibility curve. In contrast, for large separation distances (*d* > 5 μm), the SPPs propagating along the glass/Ti/Ag interface are more strongly attenuated compared to the SPPs along the Ag/air interface and no longer contribute to visibility modulation. According to Eq. 1, under TE illumination, μ_{out} = μ_{in} because SPPs are not excited (that is, β = 0) and cannot modulate fringe visibility. Thus, the experimental visibility values extracted under TE illumination are a direct measure of the mutual correlation function of the incident light at the slit locations. From this and by using the SPP coupling coefficients obtained from plasmonic interferograms as input parameters in Eq. 1, the amplitude and phase of the complex degree of spatial coherence under TM illumination can be theoretically predicted (red lines in Fig. 4, G and J). The theoretical values agree well with the experimental results over the whole range of visibility, wavelength, and slit-slit separation distance (see the Supplementary Materials for more details).

## DISCUSSION

As highlighted in the first inset of Fig. 4I, by finely varying the slit-slit separation distance within a range of only ~ λ_{SPP}/2 ≈ 240 nm, with λ_{SPP} as the average SPP wavelength at λ = 600 nm, the experimental visibility can be varied from ~0 (totally incoherent) to ~0.8 (almost fully coherent) under TM illumination. Furthermore, as shown in the second inset of Fig. 4I, by simply changing the polarization state from TE (green lines; no SPPs) to TM (blue lines; with SPPs), fully incoherent incident fields can be transformed into highly correlated (φ = 0) or anticorrelated (φ = π) fields with . Compared with a previous work (*23*), higher SPP coupling coefficients achieved by tuning the width of each slit enable stronger amplitude modulation of spatial coherence. In addition, the presence of SPPs on both interfaces can also be used to finely modulate the phase of the complex degree of spatial coherence, as shown in Fig. 4 (K and L).

In principle, dense arrays of nanoslits, nanoholes, or other nanostructures in metal films may be designed to modify the spatial coherence of an entire light beam (*20*). As an alternative to SPPs supported by corrugated, lossy metal surfaces, photonic modes in planar waveguides coupled with dielectric metasurfaces may also be used to strongly modify the spatial coherence of incident light, which can lead to beam-transforming optical elements with significantly lower transmission losses.

In summary, we have achieved full control of the complex degree of spatial coherence of light at the micrometer scale using surface plasmon–based interferometers. Strong modulation of the amplitude and phase of the complex degree of coherence is accomplished by engineering the beatings between the surface plasmons with different wave vectors generated at the two metal/dielectric interfaces. These findings can lead to alternative nanoengineered optical flat surfaces that leverage the degree of spatial coherence to achieve ultimate control of light flow, which is beyond conventional refractive- and diffractive-based photonic metasurfaces.

## MATERIALS and METHODS

### Sample fabrication

Electron beam evaporation was used to deposit an approximately 200-nm-thick layer of silver (Ag) on one of the two surfaces of a 1-mm-thick glass slide that was previously coated with a 3-nm-thick titanium (Ti) layer to improve film adhesion. Multiple Young’s double-slit interferometers, consisting of two identical ~200-nm-wide, ~200-nm-deep, and ~15-μm-long slits, with variable slit-slit separation distance, were etched on the silver film using focused ion beam milling (fig. S2). The slit width was chosen to provide comparable transmission intensity under both TE and TM illumination and high SPP coupling coefficients. The separation distance between the two slits ranged from 0.5 to 9.525 μm in incremental steps of 25 nm. In total, 362 Young’s double-slit interferometers were etched on the metal film, evenly distributed in two separate columns. A single column containing 181 identical individual slits was also etched on the same silver film to serve as reference for normalizing light transmission through the double-slits. The columns of single- or double-slits were placed 600 μm apart, and adjacent slits along each column were separated by 25 μm to avoid optical cross-talk during the measurements that could affect the interference patterns and lead to artificially reduced visibility values.

### Optical characterization

Köhler illumination was used to achieve variable spatial coherence illumination of Young’s double-slit interferometers. The setup consisted of a xenon arc lamp, coupled to an inverted microscope, together with a diffuser, an auxiliary lens, a linear polarizer, a 2-mm-wide slit mask, and a condenser lens system (*28*). Each interferometer was illuminated from the glass side—that is, from the glass(1 mm)/Ti(3 nm)/Ag(200 nm) interface—and the transmitted light intensity was collected with a 20× objective (numerical aperture, 0.75) from the opposite side, that is, from the Ag(200 nm)/air interface. The focal plane of the objective lens was located ~49 μm below the metal surface containing the double-slit interferometers to generate far-zone projections of light intensity onto the imaging plane of a charge-coupled device (CCD) camera. More specifically, the interference pattern was projected onto the entrance slit mask of a spectrograph, dispersed by a grating with 150 grooves/mm, and imaged onto a two-dimensional CCD camera to extract the wavelength dependence of light intensity along the horizontal axis and the spatial intensity distribution along the vertical axis (see Fig. 1). This experimental setup allows the detection of wavelength-resolved interference patterns transmitted through each Young’s double-slit interferometer, from which the fringe visibility over several wavelengths can be captured simultaneously in a single CCD image. By scanning all of the double-slit interferometers with different separation distances using an automated microscope stage, 362 wavelength-resolved interference patterns were obtained for each subtended angle and for TE (that is, no SPPs) or TM (that is, with SPPs) polarization states of the incident light (fig. S3). In addition, a full set of light transmission experiment was also performed for all the double-slits and reference single-slits by focusing the microscope objective directly onto the output metal film surface to measure plasmonic interferograms necessary to determine SPP excitation, propagation, and coupling coefficients for both top (glass/Ti/Ag) and bottom (Ag/air) interfaces as a function of incident wavelength. Note that a Nikon Perfect Focus System was used during the scanning process to keep a constant relative distance between the microscope objective and the bottom sample surface (that is, Ag/air interface). All interference patterns (Figs. 1, 2, and 4 and figs. S3 and S6) were normalized to the corresponding mean intensity value at each wavelength to remove the wavelength-dependent intensity variation introduced by the external light source.

### Extraction of SPP coupling coefficients from plasmonic interferograms

The light spectra transmitted through the double-slits (*I*_{DS}) and reference single-slits (*I*_{SS}) were measured by focusing the objective onto the nanoapertures. The scalar field transmitted through the single-slit (*E*_{SS}) and double-slit (*E*_{DS}) can be expressed as follows(2)(3)where the field incident on each slit is assumed to be the same (*E*_{in}), and the two incident fields on the slits can be considered fully coherent up to a 5-μm separation distance under the subtended illumination angle Δθ ≈ 3 ° (corresponding to μ_{in} ≥ 0.7). Accordingly, the normalized intensity ratio of the transmitted intensity through the double-slit and reference single-slit can be written as(4)where the factor 2 in the denominator arises from the fact that the two slits in the Young’s double-slit interferometer can be fully distinguished by the optical setup; therefore, the intensity (rather than the fields) can be summed up when the two slits are directly imaged.

A representative plasmonic interferogram at λ = 600 nm is plotted in Fig. 3A (black line). The wavelength-resolved plasmonic interferograms are shown in Fig. 3C. By applying discrete Fourier transform to plasmonic interferograms, different orders of SPP contributions can be deconvolved (*28*, *30*), as shown by the peaks and bright bands in Fig. 3 (B and D), which are in good agreement with the calculated wave vectors of different orders of SPP (gray lines in Fig. 3D). SPP contributions from both interfaces (black versus red labels) were evident. After filtering out higher-order SPP contributions (that is, with *m* > 1), plasmonic interferograms that only originate from first-order SPPs can be reconstructed (red line in Fig. 3A). By fitting the reconstructed plasmonic interferograms (with *d* ranging from 1 to 5 μm) to Eq. 4, SPP coupling coefficients β_{t}(λ) and β_{b}(λ) can be extracted (see fig. S7). Thus, the modulation of the visibility in Fig. 4 can be theoretically predicted using Eq. 1, provided that the visibility of the incident fields is known.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/10/e1700133/DC1

Supplementary Text

fig. S1. Role of angle of incidence and subtended angle in Young’s double-slit interference patterns under Köhler illumination.

fig. S2. Scanning electron microscopy images of Young’s double-slit interferometers.

fig. S3. Experimental, wavelength-resolved interference patterns for representative Young’s double-slit interferometers and subtended illumination angles at two different polarizations.

fig. S4. Estimate of effective optical width of individual slits.

fig. S5. Comparison between visibility curves obtained using numerical integration and analytical expressions.

fig. S6. Comparison between theoretical and experimental Young’s double-slit interference patterns for micrometer-scale slit-slit separation distances.

fig. S7. Coupling coefficients of SPPs excited along both metal/dielectric interfaces.

fig. S8. Representative visibility curves.

fig. S9. Experimental SPP energy-momentum dispersion curves.

fig. S10. Fourier transform analysis.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**

**Funding:**This work was supported by the NSF (grant no. CMMI1530547).

**Author contributions:**D.L. and D.P. conceived the experiment, discussed and analyzed the data, and wrote the manuscript. D.L. fabricated the sample and performed the experiment. D.P. supervised the project.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2017 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).