## Abstract

Spatial light modulators (SLMs) support flexible system concepts in modern optics and especially phase-only SLMs such as micromirror arrays (MMAs) appear attractive for many applications. In order to achieve a precise phase modulation, which is crucial for optical performance, careful characterization and calibration of SLM devices is required. We examine an intensity-based measurement concept, which promises distinct advantages by means of a spatially resolved scatter measurement that is combined with the MMA’s diffractive principle. Measurements yield quantitative results, which are consistent with measurements of micromirror roughness components, by white-light interferometry. They reveal relative scatter as low as ${10}^{-4}$, which corresponds to contrast ratios up to 10,000. The potential of the technique to resolve phase changes in the subnanometer range is experimentally demonstrated.

© 2016 Optical Society of America

## 1. INTRODUCTION

Spatial light modulators (SLMs) offer the programmable, spatial control of light (amplitude, phase, polarization, etc.) and have become a versatile instrument in modern optics, e.g., for pattern generation in photolithography [1–4], wavefront control in adaptive optics systems [5,6], or the generation of computer-generated holograms for optical tweezers [7,8]. As opposed to amplitude modulation, where energy is lost by (partially) blocking light, phase modulation offers the capability to redistribute coherent light without losses caused by the modulation principle itself. Therefore, phase-only SLMs are very attractive in scenarios that benefit from a high optical efficiency, e.g., high-power laser beam shaping or holographic optical tweezers. Furthermore, pure phase modulation is desired in applications such as aberration correction or wavefront shaping. Liquid-crystal (LC) SLMs typically provide coupled amplitude and phase modulation, and strong efforts have been dedicated to the development of phase-only LC SLMs [9–11]. The phase accuracy is a key factor for many application-relevant parameters, e.g., the contrast ratio and gray-level performance in projection applications or the diffraction efficiency in holography. Consequently, special focus is being directed to the accurate calibration of the phase modulation, which has become a burgeoning field of active research [12–16].

We study micromirror arrays (MMA, Fig. 1), which are SLMs that provide pure phase modulation due to their physical working principle, i.e., the reflection of light from a programmable surface topography. Compared to conventional SLMs like digital micromirror devices and LC SLMs, their main features are high precision, polarization independence, analog phase control, high-speed capabilities in the $1\dots 1000\text{\hspace{0.17em}}\mathrm{kHz}$ range, and a broad spectrum of usable wavelengths from the deep UV to the infrared. From an SLM developer’s perspective, two strategic directions are pursued to efficiently achieve a high phase accuracy. On the one hand, the technology is continuously improved by process innovations [17–19] and, on the other hand, precise characterization techniques are developed to better understand and optimize the individual device performance through calibration [20,21]. Because the surface topography is the most important factor for the optical performance of an MMA, optical profiling techniques such as white-light interferometric (WLI) microscopy are classical candidates for a variety of MMA inspection, characterization, and optimization tasks. For example, a WLI calibration routine provides precise control of the micromirror deflections with an uncertainty of only a few nanometers corresponding to about $\lambda /100$ for visible light [20]. The ultimate goal of all these characterizations and calibration efforts is to deliver the SLM with the best possible phase accuracy to support an out-of-the-box utilization with minimal or no application-specific calibration. Naturally, the better the calibration the more applications of precision optics might be addressed in this manner. For example, in order to generate 256 intensity levels of the zero diffraction order of an MMA, which might be used to generate gray scale patterns [22], the micromirror deflection has to be set with accuracy in the order of $\lambda /1000$. In the optical domain this corresponds to subnanometer deflection accuracy, which is very challenging to measure with interferometric microscopy due to the vibration and temperature related noise level in a standard lab environment.

Looking for a potential alternative we started to study an intensity-based measurement approach where the phase modulation of the MMA is indirectly measured by the intensity of scattered light. Whereas mechanical vibrations, thermal fluctuations, and aberrations limit the phase resolution practically achievable in a two-beam interferometric microscope, the intensity-based approach is generally less sensitive to these environmental conditions. Consequently, it should offer better scalability, i.e., the possibility to improve the sensitivity to subtle phase changes by increasing the dynamic range of the intensity measurement, e.g., with the help of filters. Moreover, in a scatter-based measurement even laterally very small surface features contribute to the measurement signal and might be extracted by appropriate measurement and analysis. In a WLI microscope, on the other hand, the magnification has to be increased to resolve finer surface features and this leads to longer measurement times due to the smaller field of view. Because of these potential advantages it appears very attractive to study an intensity-based scatter approach as a complementary tool for MMA characterization.

In this paper we focus on a new scatter technique that is specialized for diffractive MMAs. It is partially based on our previous work on MMA contrast [23]. Here we present for the first time a self-contained description of spatially resolved scatter. The diffractive working principle of MMAs provides an elegant way to extract scatter information and is combined with a measurement design that provides spatial information.

After a brief review of some basic optical properties of MMAs, we illustrate our measurement concept in the next section and describe its implementation in Section 3. First results are discussed in Section 4 where we show that spectral scatter measurements are consistent with estimations based on surface roughness parameters obtained by WLI measurements.

## 2. MEASUREMENT CONCEPT

Light scatter from structured surfaces carries valuable information about topographic properties. Scatter measurements therefore are used as fast, noncontact, versatile instruments in both scientific and industrial environments, e.g., in the optics, semiconductor, and paper industries for surface roughness measurements or the detection of particles and contaminations. Since the phase modulation of diffractive MMAs is primarily determined by their surface topography, the study of surface scatter appears as a suitable approach for MMA characterization [24]. Nonetheless, the application of scatter techniques to diffractive MMAs is not straightforward since samples of scatter measurements typically are supposed to be sufficiently smooth. MMAs, however, consist of a regular grid of micron-sized elements with abrupt surface-height changes at the micromirror edges, which lead to strong diffraction effects. Additionally, a high spatial resolution is required in order to investigate varying characteristics across the MMA surface. Typical scatter measurements, however, do not provide spatial information. The two main challenges for a measurement of scatter from diffractive MMAs are thus (a) the separation of scatter and diffraction effects, and (b) the recovery of spatial scatter information.

#### A. Optical Properties of Diffractive MMAs

MMAs act as phase gratings where monochromatic light is being diffracted into distinct directions that are determined by the mirror pitch and the wavelength according to the grating equation [25]. The intensity in these so-called diffraction orders is mainly determined by the deflection state of the micromirrors, which here is supposed to be the same for the whole MMA. The deflection $d$ is defined as the distance of the micromirror edge, which is parallel to the rotation axis, to the plane of the array (see inset of Fig. 2). To illustrate the relationship between deflection and diffraction, the intensity in the zero diffraction order is shown as a function of the deflection in Fig. 2 for an ideal MMA. Here ideal means that the MMA consists of perfectly flat, smooth micromirrors with exact deflections, and that it extends infinitely with a 100% fill factor. We want to highlight two important cases for the following discussion. When all micromirrors are not tilted ($d=0$), the MMA basically acts as a plane mirror and reflects the maximum of light in the specular direction, i.e., the zero diffraction order. If all micromirrors are deflected to a multiple of a quarter of the wavelength ($d=n\lambda /4$), the MMA forms a blazed grating and diffracts all the light into a single higher-diffraction order yielding zero intensity in the specular direction.

#### B. Separation of Grating Diffraction and Scatter Contributions

For real MMAs any deviation from the ideal phase profile such as introduced by micromirror surface corrugation results in light scatter (Fig. 3), which superimposes the diffraction pattern of the ideal MMA. For clarity from here on we use the terms diffraction and scatter to distinguish between the diffraction contributions of an ideal MMA (as defined in the previous section) and the additional scatter contributions due to, e.g., surface imperfections, respectively. This choice of terms reflects our experimental interest to investigate the deviation of a real MMA’s phase modulation from that of an ideal one.

The general idea to realize this separation in practice is to choose appropriate micromirror tilt angles to suppress the diffraction into certain directions. For example, the diffraction contribution in the specular direction (zero diffraction order) vanishes when the MMA fulfills the blaze condition $d=n\lambda /4$ as discussed above (see Fig. 2).

#### C. Principle of Spatially Resolved Scatter Measurements

Typical scatter measurements, e.g., angle-resolved scattering (ARS), do not provide spatial resolution. To review the physical reason for this, let us take a look at the geometry of an ARS measurement, which is sketched in Fig. 4(a). The power $d{P}_{s}$ that is scattered into a solid angle ${d\mathrm{\Omega}}_{s}$ is measured for different incident and scatter angles ${\theta}_{i}$, ${\theta}_{s}$ and yields the ARS after normalization to ${d\mathrm{\Omega}}_{s}$ and the incident power ${P}_{i}$. Generally, the surface profile can be represented as a sum of different spatial-frequency terms, each corresponding to a certain scatter angle. The basic idea of ARS measurements is to investigate the scattered light in a large range of angles to extract information about the profile. For smooth, clean, front-surface-reflective surfaces the so-called power spectral density (PSD) can be derived from the ARS and allows deep insight into statistical surface properties, e.g., the rms roughness or rms slope [24]. More specifically, the PSD is defined as the squared absolute value of the Fourier transform (FT) of the sample’s surface profile. As such, it represents the spatial-frequency spectrum of the surface profile. We may therefore say that an ARS measurement takes place in spatial-frequency space. It follows from the basic properties of the FT that a lateral shift of the surface profile will appear only as an additional phase factor [26]. This phase factor is lost in the PSD by taking the absolute value of the FT. Therefore, the PSD is invariant under arbitrary shifts in the surface profile so that surface features cannot be localized, i.e., there is no spatial resolution.

One way to recover the spatial information that is contained in the phase of the scattered light field is to optically perform another FT. This can be easily achieved by a focusing lens or mirror, which is well known to realize an FT between its conjugate focal planes [25]. Two lenses in a $4f$ arrangement may be used to provide the required transformations into spatial-frequency space and back into position space, i.e., realizing an imaging setup [Fig. 4(b)]. However, the surface profile information is then encoded in the phase of the light field and again is lost in the measurement process just as the spatial resolution in an ARS measurement.

It seems that we can measure only surface profile statistics without spatial resolution (ARS, PSD) or do imaging without access to profile information. However, we might be able to gain some insight into the scattering behavior of the MMA while maintaining spatial resolution. To this end we introduce a so-called Fourier aperture that is positioned in the intermediate Fourier plane, i.e., in spatial-frequency space. It serves as a spatial-frequency low-pass filter and by blocking the higher diffraction orders it enables the separation of scatter and diffraction as discussed in the previous subsection. The light scattered from a point on the MMA into the solid angle $\mathrm{\Omega}$ is imaged onto a detector, which provides a trade-off between positional and spatial-frequency information. A smaller Fourier aperture decreases the spatial resolution of the image but also narrows the spatial-frequency range over which the scatter is integrated, i.e., the spatial-frequency resolution is increased and vice versa. For the extreme case of a very small aperture, the spatial resolution vanishes and a very high spatial-frequency resolution is achieved like in a typical scatter measurement. In the other extreme, a very large Fourier aperture collects a broad spectrum of spatial frequencies including higher diffraction orders and provides a high spatial resolution.

In principle, the scatter could be measured in any direction by realigning the detection optics and choosing a suitable micromirror deflection to minimize diffraction in the chosen direction. The zero diffraction order offers the unique advantage of being achromatic and thus enabling multispectral measurements without any change in the geometry of the setup. Furthermore, in this configuration it is easy to perform relative measurements by using the MMA with all micromirrors set to zero deflection as a plane mirror reference (refer to Section 2A and Fig. 2). The normalization to such a reference measurement eliminates the influence of illumination intensity, exposure time, fill factor, overall MMA reflectivity, and camera sensitivity.

Let the MMA state be designated by the deflections $\mathit{d}=({d}_{1},\dots ,{d}_{n})$ of its $n$ micromirrors and $\mathrm{\Delta}P(x,y;\mathit{d})$ be the optical power that is scattered into the solid angle $\mathrm{\Omega}$ and imaged to the point $(x,y)$ in the image plane. Let $\widehat{\mathit{d}}$ be the MMA state to be investigated and $\mathit{d}=0=(0,\dots ,0)$ the zero tilt state, i.e., the MMA acting nearly as a plane mirror. With the latter state used as a reference, we define the relative scatter ${S}_{\text{rel}}(x,y;\widehat{\mathit{d}})$ as

So far we assumed $\widehat{\mathit{d}}=(\lambda /4,\dots ,\lambda /4)$ to result in zero intensity in the zero diffraction order leaving only the scatter to be measured. If the deflection could be set exactly, such a measurement would comprise all deviations from the ideal MMA as defined above. In practice there are two reasons for a slightly different approach. First, the deflection can only be set to a certain precision and is known to exhibit certain systematic uncertainties. Second, asymmetric scatter contributions can actually be compensated to some extent by the micromirror deflection because of the MMA’s diffractive working principle, i.e., the coherent addition of all diffraction and scatter contributions. Conversely, the symmetric scatter contributions, which cannot be compensated, are the most relevant for applications so we choose $\widehat{\mathit{d}}$ as the MMA state in which the combined diffraction and scatter (into the zero order) have been minimized to isolate mostly the symmetric scatter contributions. The minimization procedure and the image processing involved in the measurement of ${S}_{\text{rel}}(x,y;\mathit{d})$ are discussed in Sections 3C and 3E, respectively.

## 3. EXPERIMENTAL WORK

#### A. System Setup

The experiment is based upon an existing characterization setup for diffractive MMAs (Fig. 5) [20–22]. Five laser sources provide different illumination wavelengths from the deep ultraviolet to the near infrared (NIR): (i) a KrF excimer laser “MLI-1000LC” by MLase AG at 248 nm (10 W average power, 1 kHz repetition rate, 10 ns pulse width, $3\text{\hspace{0.17em}}\mathrm{mm}\times 6\text{\hspace{0.17em}}\mathrm{mm}$ beam exit); (ii) three customer-specific laser diode modules “51nano” by Schäfter $+$ Kirchhoff GmbH at 405, 680, and 830 nm and all modules have 1 mW maximum power, temperature and power stabilized, single-mode fiber exit, and collimator with Gaussian beam profile, $\text{diameter}\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\mathrm{mm}$. The electronics have been specifically tuned to provide a modulation up to 100 kHz and lowest residual glow (ratio of max/min intensity modulation $>{10}^{5}$) to support stroboscopic scattering experiments with pulsed laser operation; (iii) a diode-pumped solid-state laser module “Flexpoint” by Laser2000 GmbH at 532 nm (1 mW). The pulsed operation of the sources together with synchronization to the MMA actuation cycle enables stroboscopic experiments (see also Section 3E). Only one source at a time is operated to provide a monochromatic illumination of the MMA. A multispectral device characterization therefore comprises a sequence of experiments for the wavelengths of interest.

The laser beams are individually expanded by variable zoom optics (6 mm entrance aperture, $1x-8x$ magnification, Sill Optics GmbH) and the beam profile of the excimer laser is symmetrized with an anamorphic prism pair (CVI Laser Optics). Additionally, a rotating diffuser (SÜSS MicroTec AG) is placed in front of each zoom to reduce interference effects and speckle. The beams of all sources pass a cascade of custom-made dichroic mirrors (mso jena Mikroschichtoptik GmbH, now Optics Balzers Jena GmbH), which form a beam combiner for an efficient overlay of all s-polarized sources.

An achromatic beam homogenization has been achieved with a fly’s eye condenser [28] in a reflective layout. Two faceted mirror arrays ($30\times 30$ square facets, 1 mm pitch, $f=100\text{\hspace{0.17em}}\mathrm{mm}$, quartz substrate etched by advanced microoptic systems GmbH, Al-coated in-house) and a spherical integrating mirror ($f=2\text{\hspace{0.17em}}\mathrm{m}$) form a reflective Köhler integrator in an imaging arrangement to produce a square illumination profile at the MMA sample. The MMA in its ceramic packaging (Fig. 1) is mounted on an XY stage with tip/tilt adjustment capabilities (OWIS GmbH). It is controlled by an address electronics developed in-house, which provides a voltage resolution of 10 bit.

Only a single spherical mirror ($f=300\text{\hspace{0.17em}}\mathrm{mm}$) is utilized to image the MMA onto a camera. This layout features three characteristics. (i) It magnifies the active MMA area ($4.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}\times 4.1\text{\hspace{0.17em}}\mathrm{mm}$) to fill the camera sensor, (ii) it creates an intermediate focus where a motorized iris diaphragm (OWIS GmbH) is placed as a Fourier aperture, and (iii) it enables diffraction-limited imaging of the MMA with single-mirror resolution for all laser wavelengths on the camera with virtually no distortions. This last feature is not easy to achieve with a spherical mirror, which usually creates significant aberrations at larger incidence angles. An optical simulation conducted in-house showed that $<6\xb0$ angular incidence may effectively avoid any significant distortion if only one mirror is used for imaging. In our setup we realized an incidence angle below 1.5° that enables diffraction limited imaging for all illumination wavelengths up to the NIR. A second spherical mirror, which would be necessary for a standard reflective implementation of a $4f$ setup as shown in Fig. 4(b), significantly increases the aberrations caused by an oblique incidence and would set a spectral limit below 500 nm.

As an image sensor we use a charge-coupled device (CCD) camera with a 16 bit analog-to-digital (A/D) converter (Finger Lake Instruments ML-1603-2-LUM with Kodak KAF-1603 sensor, $1536\times 1024\text{\hspace{0.17em}}\text{pixels}$, 9 μm pitch, and Lumogen-coating). The illumination power can be adjusted by neutral density filters to reposition the signal within the camera’s dynamic range, which allows effective increase of the dynamic range by choosing different illumination intensities for the scatter and normalization measurements.

The key tasks that were necessary to develop a spatially resolved scatter measurement are a careful consideration of the CCD linearity, a systematic stray-light reduction, the fine tuning of the micromirror deflections, and the development of measurement and image processing routines. Apart from standard techniques to control stray light, such as the strategic placement of baffles, we found that the introduction of an adjustable slit right in front of the MMA was a decisive measure. In this way, the illumination can be tailored to the active micromirror area to avoid scatter from surrounding passive device structures, which easily couple into the imaging path.

#### B. CCD Linearity

Since we want to measure both low and high light intensities to calculate the relative scatter of the MMA, the camera linearity, i.e., the (ideally linear) relationship between the number of incident photons ${N}_{q}$ and the measurement signal $N$, should be characterized in a reasonable dynamic range of at least three to four orders of magnitude.

Modern scientific CCD cameras, whose A/D conversion electronics are attuned to the linear part of the CCD transfer curve to avoid the saturation region close to the well depth, are generally considered to be highly linear devices. The linearity stated by camera manufacturers is typically a number given in percent, e.g., $>99\%$. This would suggest that the error due to nonlinearity is at most 1% for any measurement. However, since this measure of linearity is often derived from the maximal deviation from a linear camera response divided by the highest signal [29], it is in this case biased toward high signal levels, and the impact of the detector nonlinearity on measurements at low light levels can hardly be estimated. Therefore we decided to individually characterize our camera’s linearity. In the following discussion we use the term analog-to-digital units (ADU) for the output units of the CCD camera.

Let us consider the case of monochromatic radiation impinging on a CCD camera, which corresponds to our experimental conditions. In this case the camera signal $N$ can be written as the number of incident photons ${N}_{\mathrm{q}}$ multiplied by a conversion factor $C$, which comprises the wavelength-dependent quantum efficiency (converted electrons per incident photon) and the overall gain of the CCD readout electronics (converted electrons per ADU):

For a linear detector, $C$ is by definition independent of the number of incident photons ${N}_{\mathrm{q}}$. For a real, nonlinear detector, however, it might depend on both wavelength $\lambda $ and the signal level $N$ (or equivalently on ${N}_{\mathrm{q}}$). If we assume that these two variables are separable, i.e., $C(N,\lambda )={C}_{\mathrm{N}}(N){C}_{\mathrm{\lambda}}(\lambda )$, we may write

To measure the relative gain the MMA is replaced by a mirror and the laser sources are operated in continuous-wave mode to provide a constant photon flux at the camera. A series of measurements with varying exposure times is then performed to acquire signals across the dynamic range of the camera, and, after a bias correction, a linear fit of the data provides the signal ${N}_{\text{lin}}$, which would be expected from a linear detector. Each (bias corrected) measured signal $N$ is then divided by the corresponding ${N}_{\text{lin}}$ value according to Eq. (4).

We measured the relative gain of our camera by performing such measurements for different wavelengths as shown in Fig. 6. The measurements clearly show that the relative gain is independent of the illumination wavelength, which confirms the assumption from above that the dependencies of wavelength and signal level are separable. Physically this is plausible because each photon of the same wavelength should have the same probability to create an electron in the CCD regardless of the photon flux. On the other hand the wavelength independency shows that the nonlinearity is introduced after the photon-electron conversion, i.e., within the readout electronics.

The relative gain of our camera decreases monotonically toward lower signal levels. It is $>0.95$ above 3000 ADU corresponding to a nonlinearity error $<5\%$ but drops to about 0.75 for the lowest measured signal levels around 50 ADU. This means that the camera only displays about 75% of the true signal at low light levels. We found the relative gain to be reproducible in repeated measurements and since it depends only on the signal level, a fit of the measured relative gain data ${C}_{\mathrm{N}}(N)$ (solid line in Fig. 6) can be used to calculate a nonlinearity-corrected signal ${N}_{\text{lin}}=N/{C}_{\mathrm{N}}(N)$ for any measured signal $N$ according to Eq. (4).

#### C. Intensity-based MMA Calibration

We want to measure the relative scatter ${S}_{\text{rel}}(x,y;\widehat{\mathit{d}})$ across the MMA area for the MMA state $\widehat{\mathit{d}}=({\widehat{d}}_{1},\dots ,{\widehat{d}}_{n})$ in which diffraction into the zero order has been minimized as outlined in Section 2C. To find the optimal individual micromirror deflections ${\widehat{d}}_{1},\dots ,{\widehat{d}}_{n}$ we directly utilize the projected intensity in the image plane that is to be minimized and hence call this process an intensity-based MMA calibration. More specifically, we measure the optical response of each micromirror for different deflection settings to determine the optimal MMA state $\widehat{\mathit{d}}$.

One starting point for this approach is a profilometric calibration of the micromirror deflections by WLI, which is based on the measurement of several surface profiles related to different mirror-deflection states. After extracting the individual mirror deflections with a pattern matching algorithm, the relation between address voltage and deflection for each mirror in the array is described individually by a polynomial function. By utilizing these characteristic curves the deflections can be set with an accuracy of a few nanometers [20,21], which has been found to, however, still provide good potential for contrast optimization.

Essential for an intensity-based calibration is also the localization of the optical response of each micromirror. This is provided by a geometric calibration of the imaging system (next subsection). For the following let us assume that we know the position of each micromirror (more precisely, its optical response) in the recorded camera image.

The micromirrors’ optical responses do slightly overlap in the image plane because spatial frequencies higher than about half the micromirror pitch frequency are blocked by the Fourier aperture. Nevertheless we may assume that the intensity at a certain location in the image plane is dominated by the deflection state of the corresponding micromirror. This assumption is found to be valid when the deflections are nearly the same between adjacent micromirrors, which is provided by the WLI calibration.

The deflections for all micromirrors might then be optimized in parallel. For a single measurement all mirrors are set to the same deflection $d$, i.e., the MMA state is $\mathit{d}=({d}_{1}=d,\dots ,{d}_{n}=d)$, and the intensities ${N}_{i}(d)$ at all micromirror image positions $i=1,\dots ,n$ are evaluated. Such intensity measurements are then performed for a range of deflections $d={d}^{(1)},\dots ,{d}^{(p)}$, as sketched in Fig. 7(a). Finally, for each micromirror $i$ the deflection ${\widehat{d}}_{i}$ that yields the lowest intensity is extracted,

These deflections are combined into the calibrated MMA state $\widehat{\mathit{d}}=({\widehat{d}}_{1},\dots ,{\widehat{d}}_{n})$, which we also call a deflection map [Fig. 7(b)].

#### D. Geometric Calibration of the Imaging System

In our setup a CCD camera samples the two-dimensional intensity distribution in the image plane and the acquired data is output as an image whose pixels correspond to the CCD’s photosites. The prerequisite for any optical calibration is to know the position of each micromirror’s optical response in such an image. More specifically, since both the MMA and the camera image have intrinsic coordinate systems, this means finding a coordinate transformation, i.e., a bijective mapping between the MMA and the camera image coordinate systems. We refer to this process as geometric calibration.

The MMA consists of $256\times 256$ micromirrors arranged in a square grid, which suggests a Cartesian coordinate system with physically defined row ($r$) and column ($c$) axes. Cardinal directions are technologically defined so that the row coordinate increases from north to south and the column coordinate from west to east as shown in Fig. 8(a). The camera image consists of a square grid of pixels for which we use the standard (Cartesian) coordinate system for images: The first coordinate ($x$) increases from left to right and the second ($y$) from top to bottom. The relation of both coordinate systems in the current setup is shown in Fig. 8(a).

Since the light field sampled by the CCD chip is an image of the MMA generated by low-NA optics, we start with the assumption that the image is free of distortions. The mapping from MMA coordinates $(r,c)$ to image coordinates $(x,y)$ can then be described by a simple combination of translation, rotation, and scaling. (Both coordinate systems are left-handed, so there is no need for a reflection.) Using vector notation this may by concisely written as

To find the parameter set $\widehat{p}$, which provides the best mapping in some sense, we generate a number of reference points $i=1,\dots ,n$ at known MMA positions ${(r,c)}_{i}$ and measure the corresponding coordinates ${(x,y)}_{i}$ in the camera image. More specifically, the whole MMA is set to zero deflection except for certain micromirrors that are deflected to about $\lambda /4$, which generate a bright background with dark spots at the positions of the deflected mirrors that are detected and localized by image processing software (LabVIEW Vision).

The above transformation model, Eq. (6), is then fitted to these reference points as follows. For a certain parameter set $p$ the image positions ${(x,y)}_{i}^{p}={T}_{p}[{(r,c)}_{i}]$ of all reference points are calculated and compared with the measured positions ${(x,y)}_{i}$. The sum of squares of the Euclidian distances between the measured and calculated spot positions serves as the loss function

with ${\Vert (a,b)\Vert}^{2}={a}^{2}+{b}^{2}$. We use the Levenberg–Marquardt method to find the optimal parameter set $\widehat{p}$ for which $L(p)$ becomes minimal. The coordinate transformation between the MMA and image coordinate systems is then given by ${T}_{\widehat{p}}$ according to Eq. (6).For practical reasons we chose to perform the geometric calibration in two steps: First, the transformation parameters are coarsely determined by utilizing specific patterns to detect and localize the corners of the MMA as shown in Fig. 9(a). Second, a grid of reference points is used as described above to yield precise transformation parameter values [Figs. 9(b) and 9(c)].

The first step serves to quickly gain an estimate at which image position, the spot corresponding to a certain deflected micromirror, will appear. This eliminates the need to find the correspondence of deflected micromirrors and detected spots and offers a speed advantage because a small bounding box can be defined around each estimated spot position as the search area for the spot finding algorithm. Furthermore, spots whose measured positions are too far off the initial estimate, which indicates an inaccurate spot detection, e.g., because of close-by defective mirrors or particles, can be filtered out to improve the robustness of the calibration procedure.

The aim of the second calibration step is to obtain precise transformation parameters, but it also provides information to assess the validity of the assumption that there are no significant image distortions. For this purpose we use the distances between the measured spot positions ${(x,y)}_{i}$ and the positions calculated with the optimal transformation ${(x,y)}_{i}^{\widehat{p}}={T}_{\widehat{p}}[{(r,c)}_{i}]$ [see Fig. 9(c)]. A simple measure is the mean absolute deviation $1/n\times \sum _{i=1}^{n}\Vert {(x,y)}_{i}-{(x,y)}_{i}^{\widehat{p}}\Vert $, which is typically about 0.2 pixels or about 1/10 of the micromirror pitch. This indicates that the mapping between the MMA and the camera image can be well described by the distortion-free model given in Eq. (6). Furthermore, the deviations across the MMA area show no signs of systematic patterns like an increase of the deviations with the distance to the optical axis, which would be expected for instance in the presence of radial distortion. The setup thus appears to be free from any practically relevant distortions.

#### E. Measurement Automation and Image Processing

The diffractive MMAs have been designed for a highly dynamic pulsed operation, which means that the micromirrors are deflected for a fixed time. Consequently, the laser sources have to be synchronized with the MMA actuation cycle so that the MMA is illuminated precisely when the micromirrors are deflected. Within one camera exposure, which is typically about 200 to 1000 ms, several laser pulses are usually integrated. A self-written LabVIEW program, running on a PC, controls this pulsed operation and synchronization of the MMA, the laser sources, and the CCD camera and is also used to perform automated measurement and analysis tasks.

We use trigger inputs to the laser sources to dynamically switch them on and off. The trigger rate of the MMA we use can be as high as 1000 Hz. The present experiment comprises different trigger rates ranging from 2.5 up to 1000 Hz. For example, in one trigger scheme we use a deflection time of 1150 μs and a laser pulse width of 850 μs at a repetition rate of 400 Hz.

To perform a relative scatter measurement according to Eq. (1) for a particular illumination wavelength, a set of so-called scatter images is acquired with the deflections calibrated toward minimal intensity as described in Section 3C and a set of so-called reference images with the deflections set to zero, i.e., the MMA acting basically as a plane mirror. Both image sets are fed to an image processing routine (Fig. 10).

(1) An average image is calculated from $N$ exposures to average out the jitter in the exposure time, which is introduced by the camera’s mechanical shutter, and to reduce the noise by a factor of $1/\surd N$. (2) A bias image, i.e., the A/D converter offset, is then subtracted to get physically meaningful measurement values. (3) A spatial averaging over an $M\times M$ neighborhood around each pixel further reduces the noise by a factor of $1/M$ and inherently reduces the spatial resolution. Nevertheless, with a micromirror pitch of 16 μm the image consists of about $3.5\times 3.5$ pixels per micromirror in the current configuration, and since the spatial resolution is limited by the Fourier aperture to less than the mirror pitch, no physical information is lost, e.g., by $3\times 3$ spatial averaging. If the signal level is very low, negative pixel values that cannot be interpreted physically will occur due to pixel noise and lead to artifacts in the subsequent analysis. The main purpose of the initial averaging steps is therefore to minimize the number of these negative pixel values. Since the averaging steps are linear operations, the order of the first three processing steps is arbitrary but the chosen one minimizes the number of necessary calculation steps. (4) The nonlinearity correction is straightforwardly performed by dividing each pixel value by the respective relative gain (see Fig. 6 in Section 3B). Since the relative gain is not defined for negative values, the correction cannot be performed for the aforementioned pixels with negative values, resulting in undefined pixel values (NaN). (5) These NaNs are replaced by the median of the surrounding pixels to enable further processing. (6) An optional, additional spatial averaging further reduces image noise.

This image processing routine was initially implemented as a macro for the free and open-source Fiji image processing package (Fiji Is Just ImageJ) [31] but was also implemented in LabVIEW recently. After both sets of exposures are processed, the scatter image is divided by the reference image to yield the relative scatter ${S}_{\text{rel}}(x,y)$ across the MMA, which we call a scatter map.

## 4. EXPERIMENTAL RESULTS AND DISCUSSION

Spectral-scatter measurements were performed on several MMAs. The data shows generally a very low relative scatter level in the range of ${10}^{-2}-{10}^{-3}$, reaching as low as ${10}^{-4}$ and below for some devices and wavelengths. From an application perspective, this corresponds to global contrast ratios up to 10,000 and above, since the contrast ratio is simply the reciprocal of the relative scatter as defined in Eq. (1) [23]. As an example the scatter map of a device with a particularly low scatter is shown in Fig. 11(a). This very low scatter is not only an interesting device property because of the extraordinary contrast potential but is among the highest that has been reported for SLMs to the best of our knowledge. It also demonstrates the suitable dynamic range of our measurement setup of at least four orders of magnitude. Scatter measurement on the same device with deflection offsets of $+0.5$ and $+1.0\text{\hspace{0.17em}}\mathrm{nm}$ with respect to the calibrated state were performed. This deflection offset is based on the deflection curves obtained by WLI calibration. The seemingly small changes in the deflection, i.e., the phase modulation, result in an increase of the measurement signal by 50% ($+0.5\text{\hspace{0.17em}}\mathrm{nm}$) and 300% ($+1.0\text{\hspace{0.17em}}\mathrm{nm})$ in the central region where the scatter is lowest [Fig. 11(b)]. Here we intentionally deviated from the condition of minimized intensity and the measurement signal clearly incorporates a significant contribution of grating diffraction. On the one hand, this result underlines the accuracy with which the phase modulation can be adjusted by a combination of WLI and intensity calibration. On the other hand it also demonstrates the potential of the technique to resolve phase changes in the subnanometer range. In comparison, such sensitivity would be quite difficult to reach with an interferometric microscope in standard laboratory conditions, which is a main reason why we started the investigation of complementary techniques with better scalability. This first result thus seems promising for the further development of the intensity-based measurement concept for the precise investigation of the phase modulation of MMAs.

To check the consistency of our measurement results we estimated the relative scatter based on the micromirror surface roughness, which is known to be a determining factor for the MMA contrast [32]. In general, surface corrugation causes surface scattering, which scales inversely with the wavelength squared [24]. However, as a central result of the studies performed earlier, only a single symmetric spatial-frequency component of the surface corrugation remains in the specular direction after consideration of the diffraction from the regular grid of micromirrors. Here this spatial-frequency component is represented by the surface parameter ${a}_{1}$, which corresponds to the peak-to-valley amplitude of the symmetric surface corrugation at the mirror pitch frequency. For the case of an MMA with surface corrugation, the relative scatter ${S}_{\text{rel}}$ into the specular direction can be estimated as [32]

The parameter ${a}_{1}$ is routinely measured for each MMA device at several positions by WLI and might therefore serve as a starting point for a quantitative consistency check. Since the measurement setup was built with multispectral measurement capabilities, we can vary both ${a}_{1}$ and $\lambda $ in Eq. (7) by choosing MMA devices with different surface roughness and by switching between the five laser sources in the range 248–830 nm, respectively.

The results of spectral-scatter measurements for two typical MMA devices and the estimations calculated from the respective values of ${a}_{1}$ with Eq. (7) are shown in Fig. 12. For these measurements the scatter map was derived from scans over different deflection settings instead of the individually optimized actuation of all micromirrors. It could be shown, however, that this method produces equivalent results. Sample A has a very shallow surface corrugation (${a}_{1}$) of about 2 nm resulting in scatter as low as ${10}^{-4}$ at higher wavelengths while sample $B$ has stronger surface corrugation around 6.6 nm leading to relative scatter from ${10}^{-2}$ to $5\times {10}^{-4}$ over the spectral range. This shows that the very low relative scatter measured for some devices is indeed to be expected with the current, excellent state of the MMA technology in terms of micromirror roughness. Generally, the measured data fits well to the estimations, which underpins the plausibility of the present data. Latest experiments indicate that additional factors also come into play for distinct samples, which will be subject for future analysis together with an investigation of the spatial variation of the scatter across the MMA.

Another conclusion that might be drawn from the dominant influence of the surface roughness is the effectiveness of the employed calibration procedures. In the current approach, the intensity is minimized in the image plane. In this case, the micromirror surface roughness or more precisely the symmetric roughness term at the mirror pitch (spatial) frequency cannot be compensated by an appropriate deflection. Conversely, however, we can conclude that other device features such as hinge stiffness, pre-deflection, or the electrode-mirror distance are effectively compensated for by the combination of WLI and intensity calibration.

## 5. SUMMARY

For the characterization of optical MMA properties and their calibration toward a precise phase modulation, an intensity-based measurement concept appears promising. It might complement the successful WLI tools due to its scalability in terms of phase sensitivity. To this end we implemented a spatially resolved measurement of scattered light from diffractive MMAs. The setup consists of a multispectral zero-order imaging system with a Fourier aperture to block higher diffraction orders. The MMA is driven as a blazed grating in order to minimize the zero-order diffraction and to focus on surface roughness contributions and deflection variations. The MMA state with non-tilted micromirrors serves as a reference to define the so-called relative scatter. Through stray-light minimization and appropriate image processing, a dynamic range of at least four orders of magnitude has been reached. Measurements reveal very low relative scatter down to ${10}^{-4}$ and below in selected samples and are consistent with calculations from micromirror roughness parameters obtained by WLI. From an application perspective, this corresponds to global contrast ratios of 10,000 and above, which is one of the highest values reported for SLMs to the best of our knowledge. The dominant contribution of the micromirror surface roughness to the measured scatter signal shows that the employed WLI and intensity calibration is effective. The first experiments clearly show for the first time that spatially resolved scatter techniques can also be successfully applied to diffractive MMAs with subnanometer sensitivity. Together with the theoretical consistency of the measurement data this provides a very promising start of a new intensity-based measurement concept for the precise phase analysis and calibration of diffractive MMAs. One of our next goals is, e.g., the calibration of MMAs toward 256 gray levels.

## Funding

Fraunhofer-Gesellschaft (Germany); Agence Nationale de la Recherche (ANR, France), Federal Ministry of Education and Research (BMBF, Germany).

## Acknowledgment

The MMA devices used in the present work have been developed in the
Joint-Programme Inter Carnot Fraunhofer PICF 2011, “Micromirror
Enhanced Microscopic Imaging for high-speed angular and spatial light
control in spectral Optogenetics & Photomanipulation in biological
applications” (MEMI-OP) and the support by the *Agence
National de Recherche* (ANR, France) and the *Federal
Ministry of Education and Research* (BMBF, Germany) is
gratefully acknowledged. The authors thank all colleagues at the
Fraunhofer IPMS who have contributed to the achieved results by technical
assistance and helpful discussions, and the Fraunhofer–Gesellschaft for
the financial support. C. S. thanks Prof. Harald Schenk from the
Brandenburg University of Technology Cottbus-Senftenberg for the guidance
and supervision of his Ph.D. work, Uwe Scadock for his invaluable
technical assistance, Christoph Skupsch for the many fruitful discussions
and stimulating comments, and Christopher Bunce for the continuous help
with the ins and outs of LabVIEW. J. H. gratefully acknowledges the
inspiring and helpful talks with colleagues of Mycronic AB about
diffractive micromirror arrays.

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