Deterministic and controllable photonic scattering media via direct laser writing

Photonic scattering materials, such as biological tissue and white paper, are made of randomly positioned nanoscale inhomogeneities in refractive index that lead to multiple scattering of light. Typically these materials, both naturally-occurring or man-made, are formed through self assembly of the scattering inhomogeneities, which imposes challenges in tailoring the disorder and hence the optical properties. Here, We report on the nanofabrication of photonic scattering media using direct laser writing with deterministic design. These deterministic scattering media consist of submicron thick polymer nanorods that are randomly oriented within a cubic volume. We study the total transmission of light as a function of the number density of rods and of the sample thickness to extract the scattering and transport mean free paths using radiative transfer theory. Such photonic scattering media with deterministic and controllable properties are model systems for fundamental light scattering in particular with strong anisotropy and offer new applications in solid-state lighting and photovoltaics.


I. INTRODUCTION
The scattering of light is a familiar physical process that is abundant in everyday life and in nature; manifestations of scattering are the opacity of white paper, of clouds, and of biological tissue [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Light scattering occurs at any interface between different materials and causes a part of the light to deviate from its original path. A medium with a high density of such interfaces can be a strong, multiple-scattering material. The strength of the light transport through and interaction with the scattering medium is quantified through the following length scales. The scattering mean free path sc quantifies the mean distance between subsequent scattering events. The mean distance that the incident light propagates before its direction is scrambled is the transport mean free path tr . The transport mean free path tr sets the typical length scale for a medium to be diffusive (L > tr > λ, where L is the thickness of the scattering medium λ is the wavelength of light in the embedding medium), and is therefore relevant if such a scattering medium is applied in devices such as solar cells and white LEDs. In solar cells, a thin scattering layer on top of the cell increases the absorption of sunlight and hence the cell's efficiency [14]. White LEDs employ a layer of scattering and phosphor particles to control the spectral and spatial distribution of the emission [13,16].
Applications of scattering media often require a specific optical thickness, which is defined as the ratio of the physical thickness L and transport mean free path tr . This ratio defines the scattering regime (ballistic, diffusive, localization [2,5,7]) and is controlled to meet the demands of an application. For example, if there is demand for high opacity diffusing medium, then the physical thickness should exceed the transport mean free path (L/ tr > 1).
Today LED devices have dimensions down to the few µm and thin film solar cells as well.
Therefore, such devices benefit if the scattering mean free path is in the µm regime. One strategy to minimize the scattering mean free path is to increase the number of interfaces in the medium.
A large number of scattering media with well-defined scattering properties, covering a broad range of optical thicknesses have been presented in the literature [17][18][19][20][21][22]. However, the fabrication of those media rely on random processes and only the average values are controlled during fabrication. To know the detailed internal structure of these media requires careful dissection of the media and sub-micron inspection, destroying the medium in the process. On the other hand, a scattering medium with a precisely known internal structure has become feasible as a result of the recent advances in 3D nanofabrication [24][25][26]. An a priori knowledge of the internal structure, without the need of sacrifice of the medium to extract the internal structure, makes it ideal for studies of light propagation that are sensitive to microscopic wave interference of light. In literature, several scattering media with deterministic geometry and scattering properties have been demonstrated. Matoba et al. [27] have created a large-volume deterministic scattering medium with laser micromachining [28,29], but with undefined scattering strength. An alternative nanofabrication technique is direct laser writing (DLW) [25], offering about 1 . . . 10 nm material deposition precision and a feature size down to about 100 nm [24]. The physical principle of DLW is two-photon polymerization to initiate polymerization at a targeted location inside the volume of the photoresist. The advantage of DLW is the freedom to create deterministic and complex 3D geometries. An example of a DLW-nanofabricated photonic structure with disordered geometry is the hyperuniform disordered medium made by Haberko et al. [30][31][32]. In the current work, we report on the fabrication of deterministic scattering media that combine both a large anisotropy and a short scattering mean free path. We measure the transport mean free path of the structures and validate that light scattering in our structures is in the diffusive regime (L/ tr > 1).

A. Direct laser writing
The structures were fabricated with a direct laser writing system [25] (Nanoscribe Professional GT) in the University of Twente Nanolab. The structures were made using polymer photoresist (Nanoscribe IP-G) with a refractive index of 1.51 [33]. The photoresist is a gel and its high viscosity ensures that the features in the structures do not drift during the writing processes, minimizing possible deformation of the structures. The illumination dose was set to be 14% higher than the polymerization threshold (laser beam power 8.90 mW and piezomotors scan speed of 140 µm/s). This results in rods of an average thickness of 500 nm instead of 100 nm that one could reach with a smaller illumination dose. Although the choice of a dose closer to the polymerization threshold would provide finer features [34], the slightly higher dose provides two advantages in the nanofabrication of those media. Firstly, thicker rods provide a more robust mechanical support of the structure; this is important because the design relies on the self support of the whole structure without any additional walls, unlike earlier demonstrations [35]. The second advantage of a higher dose is the reduction of fabrication disorder through the complete cross-linking of polymer chains across the entire volume that improves mechanical stability [36].

B. Designing a deterministic scattering medium
The scattering medium consists of rods that randomly fill a cubic volume. The randomly intersecting polymer rods run from one facet of the cube to another. An important parameter in our fabrication is the filling fraction of polymer, which is set by the number of rods written in the cubic volume. The desired filling fraction is the one that maximizes light scattering.
Given the rod thickness, this is equivalent to maximizing the air-polymer interface. To create a free-standing structure, we require a minimum number of rods that intersect to form a rigid skeleton that does not collapse under its own weight or under capillary forces. If we increase the number of rods above a certain critical value, we start to decrease the air-polymer surface, reducing the scattering strength of our structures. A design example is shown in Fig. 1b). We developed an algorithm based on Jaynes' solution to Bertrand's paradox [37] to ensure an on-average uniform filling of the volume and the random positioning of the rods in the cube. To keep fabrication times acceptable (7 minutes per structure), we created structures with a lateral size of 15 to 20 µm and a height up to 20 µm. For structures wider and taller than 20 µm, we suggest the model to be divided in cubes of (20 µm) 3 to prevent pyramidal distortions, shadowing effect and weight deformations [38,39].
The DLW setup is ideally suited to write rods in a successive manner [40]. After we generate the coordinates of the rods, we sort them such that the rods attached to the substrate are written first, as illustrated in Fig. 1(a). This creates a stable scaffold to which the other rods are attached to. At the end of the writing process, a rigid structure is created to follow the design model as is shown in Fig. 1 In Fig. 2  In order to study the total transmission and extract the transport mean free path, a set of structures is needed with varying height at a given filling fraction [7,41]. The uniform density of the scatterers certifies an on-average uniform filling fraction also in the structures with a reduced height. In addition, we keep the design of the structures constant but tune the height of our structures by writing deeper in the glass. In this way we avoid the transmission noise that may be introduced by different realizations. In order to study the scattering properties of this design, we fabricated two series of structures with lateral dimensions of 15 and 20 µm. The first series are made of 200 up to 2000 rods/(15µm) 3 with a step size of 1 The fill fraction is numerically estimated by first discretizing the design volume into cubic pixels (voxels) with dimensions of (20 nm) 3 and then binarizing the discretized volume into written and unwritten blocks.
The inifinitesimal lines in the design are converted to finite thickness by assuming a laser focus of ellipsoidal volume with a diameter of 500 nm and an elongation of 3.5 times the diameter.

C. Optical setup
The light scattering strength in our samples is quantified by extracting the transport mean free path tr using total transmission measurements [7]. The experimental set-up is The total transmission T z for a sample of thickness L = z, was measured by recording the total power of the light I z that was transmitted through the structure with given thickness z. The reference power I 0 was measured on a bare glass substrate. The ratio of the total power for given thickness z and the reference I 0 is the total transmission of the sample, In a similar manner, the ballistic light attenuation B z is measured as the ratio B z = P z /P 0 with with P z the power of light that propagates through a sample of thickness z. To extract the transport parameters of tr and sc , we model the measured data with the P3 approximation of radiative transfer theory [42,43].

III. RESULTS
We have fabricated 693 structures with various thicknesses and densities of scatterers and employed scanning electron microscope (SEM) to determine their feature sizes. Figure 4 shows a qualitative comparison of the surface features in the designed (a, c) and the fabricated structures (b, d) that highlights a high degree of similarity. The rod thickness was measured to be 524 ± 60 nm. The elongated focus of the laser beam in the resist results in an elliptical cross-section of the rods with the semi-major axis along z. The resulting size asymmetry of the voxel is 3.5. We note that the elongation of the rods depends on its angle with respect to the z-axis [25]. Although the resemblance between the design and real structure is very good and features remain in the same relative positions as seen in Fig. 5, there are a few errors in the fabricated structure, for e.g. missing rods and shrinkage artifacts.
Typically, the very short rods on the cube faces are washed away during development due to insufficient adhesion. The total number of missing rods is small (few dozen) and their contribution to the structure is very small, estimated to be less than 2 vol% and not easy to discriminate from the final structure as shown in Fig .5. In addition, most fabrication errors repeat in different realization of the structure, making it feasible to compare their optical interference properties. Moreover, the SEM images show no significant pyramidal distortions [30], a distortion that DLW structures frequently suffer from [39,40]. Thus we conclude that the chosen height and laser dose were a good choice and proceed to the optical characterization of the structures. transmission decreases with increasing sample thickness, while the highest drop occurs for 400 rods/(15µm) 3 , signifying the smallest value for sc . We used non-linear least squares fitting of the experimental data with an exponential decay function P = P 0 exp(−z/ sc ), finding a scattering mean free path sc = 2.6 ± 1.5µm where the error is the 95% confidence interval. This value confirms that our media are in the multiple-scattering regime since their thickness is at least 5 times the scattering mean free path.
In the color graph of Fig. 6(b) we plot the ballistic light measurements for the series with lateral dimensions of 15 µm. On the x-axis, the thickness of the structure increases, while  . This is sensible since they correspond to the filling fractions that maximizes air-polymer interface [41,44,45], see Fig. 2.
To check the consistency of the transmission data for different linked dimensions at identical filling fractions (≈ 93%), two total transmission series are made: for the 15 µm cubes with 1400 rods and for the 20 µm cubes with 2400 rods. The structures' lateral dimensions and the number of rods differ, but since the filling fractions coincide, the light transport is expected to be similar. This consistency is validated from Fig. 8 and highlights that the design method is effective for the chosen finite lateral dimensions and scatterer geometry.

A. Transport mean free path
We obtain sc and tr from two of the methods making standard approximations. The transport properties of a scattering medium are derived from radiative transfer theory (RTT). Light transport Monte Carlo methods can solve complex geometries and finite boundary conditions [46]. Those quantities for a non-absorbing medium are connected through the anisotropy factor g, as (1 − g) tr = sc . In the case of our DLW structures that are composed from rods with a diameter on the scale of the wavelength, a substantial anisotropy factor is expected [47]. In such a case, there are several approaches that can be used to extract the transport parameters from experimental data [6,48,49]. We decided to make an analysis combining a MC parameter estimation [46,50] and the so called P3 approximation to radiative transfer [16]. In the MC diffusion parameters estimation, one numerically solves radiative transfer theory by considering the geometry of the system for a range of transport parameters that match with experimental observations. The P3 approximation is an approximation to the radiative transport equations that expands the validity to a wider range of anisotropy values compared to the normal P1 approximation. P3 has been shown to be very accurate in transport parameters estimation [16,42]. In the P3 approximation total transmission measurements are combined with the ballistic light measurements to extract the anisotropy factor g and evaluate the transport mean free path tr . Although there is practically no absorption of the photoresist for 633 nm illumination, during the P3 modeling we allowed for an "effective absorption" a . This effective absorption takes into account the scattered light that leaks from the sides of the cubic structures.
In Fig. 10   absorption" length of a =20 ±2 µm. The estimation using the P3 approximation tends to deviate from the data at small sample thicknesses. This is because the accuracy of the P3 approximation is limited for thin samples (L < l tr ). From the P3 calculation, we estimate a transport mean free path tr =26 ±8 µm. The second series of 600 rods interprets reasonable well with sc =2.54 ±0.15 µm, g=0.95 ±0.01 and a =13.1 ±0.4 µm. With the above estimations, the transport mean free path estimation by the P3 appears for the given structure is tr =56 ±11 µm. This rather high value is attributed to the high anisotropy value that is estimated. For comparison we also use a MC approach to extract the sc and g values from the total transmission. The code is scanning a range of possible values of sc between 0.1 and 300µm and g between -1 and 1. The estimations are then modeled with the least-square difference method to the experimental data for the total transmission, to find the best matching quantities [50]. For the two series in Fig. 10 the MC estimates for the 400 rods yield a scattering mean free path value sc =1.8 ±0.5 µm and g=0.85 ±0.04, corresponding to a transport mean free path of 11.8 ±4.5 µm, while for the 600 rods it estimates a scattering  Table I. The MC approach is a numerical solution of RTT which has been validated by statistical iterations and careful parameter scanning for the exact geometry and boundaries effects [51], but the actual microscopic structure of the sample and in particular its anisotropy caused by the laser focus elongation is not taken into account. The P3 result is still an approximate solution for infinite slabs, where we introduced an effective absorption to model light leaking out of the finite structures, all in all, we trust the estimation of the MC better.
In conclusion, our estimation for the transport mean free path is 11.8 ±4.7 µm for 400 rods and 8.3 ±1.2 µm for the 600 rods series. Thus the DLW structures are diffusive since their thickness surpasses the transport mean free path L > tr .  The curves show the calculated P3 to match the measurements, the dashed black curve matches for scattering mean free path of sc =2.64 µm, and the red solid curve corresponds to sc =2.54 µm.

IV. CONCLUSION
We have implemented a DLW method to fabricate small deterministic optical multiple scattering media with optical thickness L tr > 1. The fabrication process allows full control over the position and shape of the scatterers. We show that one can tune the density of scatterers and accordingly vary the transport mean free path. We demonstrate that our best design has a scattering mean free path of sc =1.8 ±0.5 µm. The deterministic nature of fabrication makes the scattering samples extremely interesting for light propagation studies.
This permits validation of various fundamental and applied aspects of light scattering for a given disordered structure, something that was not possible until now for optical frequencies [52]. In the future we want to investigate the reproducibility of the method to study the clonability of multiple scattering media as optical physical unclonable functions [53].

V. ACKNOWLEDGMENTS
We thank Cock Harteveld, Ad Lagendijk, and Matthijs Velsink for discussions and support. This work is financially supported by the Netherlands Organization for Scientific Research (NWO) (Vici), by STW project 11985, and by the FOM program 'Stirring of light'.