Development of a spectroscopic Mueller matrix imaging ellipsometer for nanostructure metrology

Mueller matrix imaging microscope using dual continuously rotating anisotropic mirrors

We demonstrate calibration and operation of a Mueller matrix imaging microscope using dual continuously rotating anisotropic mirrors for polarization state generation and analysis. The mirrors contain highly spatially coherent nanostructure slanted columnar titanium thin films deposited onto optically thick titanium layers on quartz substrates. The first mirror acts as polarization state image generator and the second mirror acts as polarization state image detector. The instrument is calibrated using samples consisting of laterally homogeneous properties such as straight-through-air, a clear aperture linear polarizer, and a clear aperture linear retarder waveplate. Mueller matrix images are determined for spatially varying anisotropic samples consisting of a commercially available (Thorlabs) birefringent resolution target and a spatially patterned titanium slanted columnar thin film deposited onto a glass substrate. Calibration and operation are demonstrated at a single wavelength (530 nm) only, while, in principle, the instrument can operate regardless of wavelength. We refer to this imaging ellipsometry configuration as rotating-anisotropic-mirror-sample-rotating-anisotropic-mirror ellipsometry (RAM-S-RAM-E).

Coaxial spectroscopic imaging ellipsometry for volumetric thickness measurement

We propose spectroscopic imaging ellipsometry that can measure spectral ellipsometric signals in the entire field of view simultaneously without areal scanning or operation of polarization devices. The proposed imaging ellipsometry is configured in a coaxial optical structure so that the high magnification objective lens is applicable and the spatial resolution is highly increased. Without the operation of polarization components and to efficiently obtain the spectral data in the object plane, the ellipsometric parameters are encoded into the high frequency in the spectral domain and are measured by an imaging Michelson interferometer. The volumetric thickness measurement by the proposed method was verified by comparing the thickness results of the SiO₂/Si sample that has four different thicknesses with commercial ellipsometer results.

Co-axial spectroscopic snap-shot ellipsometry for real-time thickness measurements with a small spot size

Spectroscopic snap-shot ellipsometry of co-axial structure is proposed to solve the large spot size and long measurement time issues of the conventional ellipsometer. By modulating the spectroscopic ellipsometry signal into high-frequency at the spectral domain and measuring the spectrum at the point of the back focal plane, the ellipsometry parameters(Δ, ψ) were measured in real-time with small spot size. Detailed analysis, calibration, and optimization process for the proposed methods are presented. The accuracy and precision of the proposed method were confirmed by comparing the thickness measurement result of SiO₂/Si thin-film samples with a commercial ellipsometer.

Angle-resolved spectral reflectometry with a digital light processing projector

We describe a novel approach for angle-resolved spectral reflectometry using a digital light processing (DLP) projector. Here, the DLP generates ring patterned images which are projected on the back focal plane of an objective lens. This way, the proposed method quickly changes the angle of incidence with ease based on the relation between the radius of the back focal plane and the angle of incidence. As a result, a detector captures the intensity of the image plane based on the angular and spectral axis. As the proposed method detects the interesting spot of a sample image, it can easily locate the measurement spot with viewing the full field of view, and the spot size is reduced by adopting the fiber. This method is verified by comparing the measurement output of the thin-film samples with a commercial ellipsometer. The result shows that our the proposed method enables the high accuracy of the thin-film inspection.

Feature extraction on Mueller matrix data for detecting nonporous electrospun fibers based on mutual information

The surface morphology of electrospun fibers largely determines their application scenarios. Conventional scanning electron microscopy is usually used to observe the microstructure of polymer electrospun fibers, which is time consuming and will cause damage to the samples. In this paper, we use backscattering Mueller polarimetry to classify the microstructural features of materials by statistical learning methods. Before feeding the Mueller matrix (MM) data into the classifier, we use a two-stage feature extraction method to find out representative polarization parameters. First, we filter out the irrelevant MM elements according to their characteristic powers measured by mutual information. Then we use Correlation Explanation (CorEx) method to group interdependent elements and extract parameters that represent their relationships in each group. The extracted parameters are evaluated by the random forest classifier in a wrapper forward feature selection way and the results show the effectiveness in classification performance, which also shows the possibility to detect nonporous electrospun fibers automatically in real time.

On the limits of low-numerical-aperture imaging scatterometry

Although imaging scatterometry has been demonstrated to be a powerful technique for characterization of nano-gratings when high lateral resolution is required, some limits of this novel technique are still undisclosed yet, such as the constraint for the imaging numerical aperture (NA), the number of unit cells for accurate grating reconstruction, and the analyzability of image pixels associated with the grating region. To this end, we establish a vectorial image formation (VIF) model for imaging scatterometry based on the finite-difference time-domain (FDTD) method and vectorial diffraction theory. According to the established VIF model and the simulation results of a Si grating sample with finite numbers of unit cells, we find that accurate grating reconstruction by routine RCWA (rigorous coupled-wave analysis) -based data analysis requires an upper limit for the NA of the employed objective. And enough numbers of unit cells are also required to be covered in the illumination spot. Only in these conditions, the zeroth-order diffraction information of the grating under test can be exclusively and completely collected by the imaging system. Moreover, only the image pixels off the edge of the grating region are analyzable by routine RCWA-based data analysis due to the effect of edge scattering. The required number of grating unit cells and the size of the analyzable region are closely related with the imaging NA and the ratio between the illumination spot size and the size of the grating region D/L. Higher imaging NA or smaller D/L typically requires fewer grating unit cells and meanwhile allows a larger analyzable region. The investigation in this paper promises to provide valuable insights into the application of imaging scatterometry.

Metrology of Nanostructures by Tomographic Mueller-Matrix Scatterometry

The development of necessary instrumentation and metrology at the nanoscale, especially fast, low-cost, and nondestructive metrology techniques, is of great significance for the realization of reliable and repeatable nanomanufacturing. In this work, we present the application of a homemade novel optical scatterometer called the tomographic Mueller-matrix scatterometer (TMS), for the measurement of photoresist gratings. The TMS adopts a dual rotating-compensator configuration and illuminates the nanostructure sequentially under test conditions by a plane wave, with varying illumination directions and records. For each illumination direction, the polarized scattered field along various directions of observation can be seen in the form of scattering Mueller matrices. That more scattering information is collected by TMS than conventional optical scatterometry ensures that it achieves better measurement sensitivity and accuracy. We also show the capability of TMS for determining both grating pitch and other structural parameters, which is incapable by current zeroth-order methods such as reflectometry- or ellipsometry-based scatterometry.

Calibration of focusing lens artifacts in a dual rotating-compensator Mueller matrix ellipsometer

A focusing lens consisting of two or more elements is widely used in ellipsometers for spatial resolution. In a typical ellipsometer layout, a lens is placed before the sample to focus light emerging to the sample, and another after the sample to collect the scattered light from the sample. Accurately calibrating the artifacts of the focusing lens is of great importance. In this paper, a method to improve calibration accuracy is proposed. A general analytical model is deduced to describe the artifacts of the focusing lens. This model can be applied to a system model of a dual rotating-compensator Mueller matrix ellipsometer. By adding a uniaxial crystal to calibration samples, with its optical axis neither parallel nor perpendicular to the incident plane, we can better separate artifacts from the lens before and after the sample. The system model also includes depolarization effects due to a finite numerical aperture (NA) that is related to the focusing lens. Similar effects due to the finite spectral bandwidth are also considered. Our simulation's results have validated the proposed method.

Polarimetric learning: a Siamese approach to learning distance metrics of algal Mueller matrix images

Polarimetric measurements are becoming increasingly accurate and fast to perform in modern applications. However, analysis on the polarimetric data usually suffers from its high-dimensional nature spatially, temporally, or spectrally. This paper associates polarimetric techniques with metric learning algorithms, namely, polarimetric learning, by introducing a distance metric learning method called Siamese network that aims to learn good distance metrics of algal Mueller matrix images in low-dimensional feature spaces. As an experimental example, 12,162 Mueller matrix images of eight algal species are measured via a forward Mueller matrix microscope. Eight classical metric learning algorithms, including principle component analysis, multidimensional scaling, isometric feature mapping, t-distributed stochastic neighbor embedding, Laplacian eigenmaps, locally linear embedding, linear discriminant analysis, and metric learning to rank, are considered, by which the algal Mueller matrix images are mapped to two-dimensional (2D) feature spaces with different distance metrics. Support-vector-machine-based holdout sample classification accuracies of the 2D feature vectors are provided in a supervised manner for quantitative comparisons of the low-dimensional distance metrics, including the results of the eight metric learning algorithms and 16 Siamese architectures with varying convolution, inception, and full connection modules. This study shows that the Siamese approach is an effective metric learning algorithm that can adaptively extract features exhibiting empirical correlations with the fast-axis-orientation-dependent and spatially variant algal retardance induced by the algal microstructures.

Anisotropic contrast optical microscope

An optical microscope is described that reveals contrast in the Mueller matrix images of a thin, transparent, or semi-transparent specimen located within an anisotropic object plane (anisotropic filter). The specimen changes the anisotropy of the filter and thereby produces contrast within the Mueller matrix images. Here we use an anisotropic filter composed of a semi-transparent, nanostructured thin film with sub-wavelength thickness placed within the object plane. The sample is illuminated as in common optical microscopy but the light is modulated in its polarization using combinations of linear polarizers and phase plate (compensator) to control and analyze the state of polarization. Direct generalized ellipsometry data analysis approaches permit extraction of fundamental Mueller matrix object plane images dispensing with the need of Fourier expansion methods. Generalized ellipsometry model approaches are used for quantitative image analyses. These images are obtained from sets of multiple images obtained under various polarizer, analyzer, and compensator settings. Up to 16 independent Mueller matrix images can be obtained, while our current setup is limited to 11 images normalized by the unpolarized intensity. We demonstrate the anisotropic contrast optical microscope by measuring lithographically defined micro-patterned anisotropic filters, and we quantify the adsorption of an organic self-assembled monolayer film onto the anisotropic filter. Comparison with an isotropic glass slide demonstrates the image enhancement obtained by our method over microscopy without the use of an anisotropic filter. In our current instrument, we estimate the limit of detection for organic volumetric mass within the object plane of ≈49 fg within ≈7 × 7 μm² object surface area. Compared to a quartz crystal microbalance with dissipation instrumentation, where contemporary limits require a total load of ≈500 pg for detection, the instrumentation demonstrated here improves sensitivity to a total mass required for detection by 4 orders of magnitude. We detail the design and operation principles of the anisotropic contrast optical microscope, and we present further applications to the detection of nanoparticles, to novel approaches for imaging chromatography and to new contrast modalities for observations on living cells.