Spatial Coherence Manipulation on the Statistical Photonic Platform

: Coherence, like amplitude, polarization and phase, is a fundamental characteristic of the light fields and is dominated by the statistical optical property. Generally, accurate coherence manipulation is challenging since coherence as a statistical quantity requires the combination of various bulky optical components and fast tuning of optical media. Spatial coherence as another pivotal optical dimension still has not been significantly manipulated on the photonic platform. Here, we theoretically and


Introduction
Coherence is one of the most important concepts in optics and is strongly related to the ability of light to exhibit interference effects. The statistical optical property as a fundamental characteristic dominates the coherence of electromagnetic waves (EMWs) 1,2 , which has boosted the development of lasers, precision measurements, and information transmission. The beams with partial coherence have been applied in various fields, such as plasma-instability suppression 3 , varying polarization modes 4 , structuring random solitons 5 , ghost imaging [5][6][7] and optical communication, which can significantly reduce the scintillation and beam wandering caused by turbulent atmosphere 8 . However, the generation of partially coherent beams traditionally requires a combination of various optical components such as lenses, rotating ground glass, and a spatial light modulator with specific distances in between 9,10 , which is complex, bulky, and is challenging to be integrated on a single photonic platform. The degree of coherence (DOC) is also challenging to be accurately manipulated due to the inevitable statistical roughness of the ground glass or other disordered media.
In the past decade, metasurfaces have attracted much attention due to their great potentials in integrated optical systems and advanced photonics. By tailoring the geometric parameters and spatial symmetry of the sub-wavelength metallic or dielectric building blocks, versatile local resonances such as waveguide modes 11 , Fano resonances 12,13 , and bound states in the continuum [14][15][16] can be generated, which strikingly boosts the light-matter interactions between EMWs and the nano-structures.
Such resonances locally introduce abrupt changes in optical components in an ultra-thin plane, leading to significant wave-front [17][18][19] and polarization [20][21][22] manipulation of EMWs. Although several optical dimensional manipulations have been systematically investigated, such as amplitude, polarization, and phase of EMWs [23][24][25][26][27] , which have been applied in sub-resolution focusing 28 , arbitrary spin-to-orbital angular momentum conversion 29 , and information photonics 26 , the spatial coherence as another pivotal optical dimension has not been significantly manipulated, especially in the photonic platform. Generally, coherence manipulation is challenging since coherence as a statistical quantity requires either fast tuning of optical media or fabricating numerous different samples. Recently, researchers engineered the individual input-output responses with disordered metasurfaces to realize the wavefront shaping 30 . However, the wide range of statistical optical properties and their applications on the photonic platform still have not been revealed and quantitatively studied.
Here, we introduce a strategy with statistical metasurfaces to accurately manipulate the spatial coherence of EMWs by projecting a temporal random phase distribution onto the wave-front. The statistical photonic properties such as information entropy are quantitatively demonstrated and the partially coherent vortex beams with the pre-defined DOC can be successfully obtained on the statistical photonic platform. The experimental results meet well with the theoretical predictions.
The proposed strategy can also be easily extended to manipulate the spatial coherence of other special beams such as partially coherent vortex beam generations. Our approach paves the way for a new direction in statistical photonic manipulation and provides a routine to apply partially coherent beams in the information photonic systems.

Concept of the statistical metasurfaces.
To quantify the statistical properties on the photonic platform, we start from the harmonic formula of the electric fields of EMWs: , where A(r) and j (r, t) denote the amplitude and phase of the electric fields at an arbitrary point r, respectively. Generally, the statistical behavior of j (r, t) decides the coherence of the EMWs, which can be described by the mutual coherence function (MCF) in the space-time domain 31 : , where t = t2 -t1, the asterisk denotes the complex conjugate and the angle brackets denote an ensemble average over the fluctuating field. By substituting Eq. (1) to Eq.
(2), the MCF can be express as: , where we assume A(r) does not change with r for convenience both in theory and experiments. Specifically, for narrowband or quasi-monochromatic EMWs the longitudinal coherent length of the light is much greater than the maximum pathlength difference between r1 and r2 to the observation point. Thus, the MCF is a slow varying function in the t domain, and can be simplified to a mutual intensity function G(r1, r2) = G(r1, r2, t = 0). Accordingly, the degree of the spatial coherence of a scalar partially coherent beam takes the form: . 11 22 ,, , ,   Figure 2d shows the calculated information entropy with increasing of the disorder of the phase distribution. The information entropy is calculated through: , (6) where N is the number to evenly segment the phase interval 0-2p, is the possibility of the phase located in the j-th interval. Compared with conventional information entropy using 2 as the base of the logarithm 36 , here we employ N as the base to obtain convergent results, which changes the unit system in definition (see The speckle can be viewed as a coherent superposition of the incident light, and the light coming from the speckle should also be coherent 39 . In contrast, the light coming from different light speckles is separated by different superposition processes, such as the low-intensity areas between the speckles resulting from statistically averaged superposition. Thus, the size of the speckles can sever as a significant indicator of the coherence of the beam. The smaller the speckle is, the lower the spatial coherence will be. Figure 3g illustrates the detailed experimental setup for capturing the aforementioned instantaneous intensity distributions in Fig. 3a-e. Spatial coherence manipulation of the statistical metasurfaces. The output light distributions captured by the CCD camera can be statistically analyzed to measure the normalized fourth-order correlation function (FOCF) 40 , which is defined as: , where the angle brackets denote the time average and I (r, t) is the instantaneous intensity distribution. Since the probability distribution of the random phase embedded in the sample is spatially uniform, the random process is ergodic while scanning the sample. Thus, one can use the time average over the correlation of the sequence of instantaneous intensity distributions between r1 and r2 to measure the spatial coherence width of the fields, instead of employing an ensemble average.
Particularly, the normalized FOCF is linearly dependent on the square of the modulus of DOC function provided that the random process obeys the Gaussian statistics, i.e., .
In the experiment, a series of pictures were captured by a CCD camera during the scanning process, and each picture shows one instantaneous intensity distribution of ( ) ( ) ( ) ( ) ( ) 12 (2) 12 12 ,, , where and denote the average intensity at point r1 and r2, respectively, which are given by: .
The recorded pictures were post-processed through MATLAB to calculate the FOCF using Eq. (9). We took M = 1000 pictures of the instantaneous intensity to achieve convergent results of FOCF, which can determine the spatial coherence width of the output beam.
where Ai is the complex envelope of the incident light, t0 is the average time delay associated with the metasurface. Thus, the mutual intensity of the transmitted light is: (12) Accordingly, the general relationship between the incident and transmitted mutual intensity is: , (13) where Gi (x1, y1; x2, y2) is the mutual intensity of the incident light. Specifically, when T (x, y) is a unitary phase mask, the DOC of the transmitted beam is the same as that of the incident beam. Thus, we can load a unitary phase mask onto the generated partially coherent beam, and obtain a special beam with the same spatial coherence.
To demonstrate the spatial coherence manipulation of the special beams, we employed a vortex phase plate on the transmission path of the generated partially coherent beam (Fig. 5). The calculated intensity distributions of the partially coherent vortex beams with different coherences are shown in Fig. 5a and the corresponding measured results captured by the CCD camera are shown in Fig. 5b. The vortex phase plate adds an additional wavefront to the partially coherent beam generated by the statistical metasurface, which will not alter the beam width and spatial coherence. The central dark spot of the doughnut intensity profile for the partially coherent vortex beam phenomenologically becomes smaller and weaker with decreasing of the spatial coherence resulting from increasing of the disorder of the random phase distribution.
Similar operations can also be applied in phase-only partially coherent hologram generation.

Discussion
Traditionally the DOC of the beam has to be measured before further applications, while in our strategy the DOC of the generated beam can be predefined and accurately manipulated. It is noteworthy that the limitation of phase-only-based partially coherent special beam generation is not fundamental, the transmission screen T(x, y) with arbitrary amplitude and phase distribution can also be applied. The difference is that the coherence distribution will change when loading the mask T(x, y), but the DOC of such special beams still can be determined through Eq. (13). The proposed statistical scheme simplifies the experimental setups for the spatial coherence manipulation of light, and provides the possibilities for further statistically characteristics and applications on an ultracompact photonic platform. Compared with traditional methods to control the spatial coherence of light by employing bulky disordered media, our approach also significantly reduces the energy loss of the incident light, which further promotes the applications of partially coherent beams.
Although we employ a mechanical method to modulate the random phase distributions at the seconds level, the tuning speed limitation is not fundamental. For example, the spatial coherence manipulation can be real-time modulation by adopting a micro-electromechanical system 42 or employing structured light 43 to scan the sample.
In summary, we propose a design strategy to accurately manipulate the spatial coherence of EMWs by loading a temporal random phase distribution onto the wavefront on the statistical photonic platform. The proposed statistical metasurface consists of high-efficiency dielectric nanofins that can locally manipulate the correlation between different locations of EMWs, leading to partially coherent light with a pre-defined DOC. We demonstrate for the first time that the output beam can be continuously modulated from fully coherent to incoherent on the statistical photonic platform. We also demonstrate the statistical properties of the metasurface such as information entropy and apply this design strategy to partially coherent vortex beams generations, and the experimental results meet well with the theoretical predictions. This strategy not only significantly simplifies the experimental setups to realize partially coherent beams, but also enables pre-designed and accurate manipulation of the spatial coherence for different kinds of light beams. Our approach paves the way to generate partially coherent beams with extraordinary correlation functions on the statistical photonic platforms, which can boost the applications in information photonics such as turbulent information transmission and information retrieval in disordered or perturbative media. was performed to blank-etch the TiO2 layer until the ZEP 520A resist was exposed.

Methods
Here the etching conditions were 20 sccm of CHF3 at 50 W bias power/500 W induction power at an operating pressure of 10 mTorr, which resulted in a TiO2 etching rate of ~20 nm per minute. Finally, O2 plasma with small addition of CHF3 was used to fully remove the remaining ZEP resist. Data availability. The data that support the finding of this study are available from the corresponding author upon request.