With the maturing of liquid crystal technology, liquid crystal variable retarder (LCVR) has been widely used in optical systems. In practice, it is of great importance to characterize the polarization properties of the LCVR for its control and applications to accurately modulate the polarization state of the light in the optical systems. In this paper, the Mueller matrix ellipsometry (MME) is applied to comprehensively characterize the polarization properties including the retardance and the fast axis azimuth of the LCVR versus the driving voltage, the wavelength, as well as the incidence and the azimuth of the light. An analytical model is constructed to describe the LCVR based on the Mueller matrix calculus, in which the influences of the incidence and the azimuth of the light are considered. A house-developed spectroscopic Mueller matrix ellipsometer is employed to perform the experiments. Simulations and experiments about a commercial LCVR product are presented and discussed to verify the proposed method.
Polarization distortion is a phenomenon which the polarization state of output light deviates from the theoretical expectation. Due to the design defects and process limitations, polarization distortion in beam splitter is inevitable, which results in the significant errors in the optical systems. A theoretical analysis method based on Mueller matrix is proposed for characterizing the beam splitter. In the propose approach, polarization distortion in the beam splitter including depolarization, linear and circular birefringence, and linear diattenuation, circular dichroism have been considered. With the proposed method, we can characterize the beam splitters and extract the related effective optical parameters of polarization distortion. The Mueller matrices of two different commonly used beam splitters measured by a commercial Mueller matrix ellipsometer (MME) are consistently fitted by the proposed method and the residual errors have shown the improvement compared to the conventional methods.
KEYWORDS: Optical lithography, Lithography, Double patterning technology, Photomasks, Very large scale integration, Process control, Manufacturing, Manufacturing equipment, Integrated circuits, Integrated optics, Current controlled current source
Triple patterning (TP) lithography becomes a feasible technology for manufacturing as the feature size further scale down to sub 14/10 nm. In TP, a layout is decomposed into three masks followed with exposures and etches/freezing processes respectively. Previous works mostly focus on layout decomposition with minimal conflicts and stitches simultaneously. However, since any existence of native conflict will result in layout re-design/modification and reperforming the time-consuming decomposition, the effective method that can be aware of native conflicts (NCs) in layout is desirable. In this paper, a bin-based library matching method is proposed for NCs detection and layout decomposition. First, a layout is divided into bins and the corresponding conflict graph in each bin is constructed. Then, we match the conflict graph in a prebuilt colored library, and as a result the NCs can be located and highlighted quickly.
The diagnosis and control of the polarization aberrations is one of the main concerns in a hyper numerical aperture (NA) lithography system. Waveplates are basic and indispensable optical components in the polarimetric diagnosis tools for the immersion lithography system. The retardance of a birefringent waveplate is highly sensitive to the incident angle of the light, which makes the conventional waveplate not suitable to be applied in the polarimetric diagnosis for the immersion lithography system with a hyper NA. In this paper, we propose a method for the optimal design of a wideview- angle waveplate by combining two positive waveplates made from magnesium fluoride (MgF2) and two negative waveplates made from sapphire using the simulated annealing algorithm. Theoretical derivations and numerical simulations are performed and the results demonstrate that the maximum variation in the retardance of the optimally designed wide-view-angle waveplate is less than ± 0.35° for a wide-view-angle range of ± 20°.
Mueller matrix ellipsometry has been demonstrated as a powerful tool for nanostructure metrology in high-volume manufacturing. Many factors may induce depolarization effect in the Mueller matrix measurement, and consequently, may lead to accuracy loss in the nanostructure metrology. In this paper, we propose to apply a Mueller matrix decomposition method for the Mueller matrix measurement to separate the depolarization effect caused by the MME system. The method is based on the polar decomposition by decomposing the measured depolarizing Mueller matrix into a sequence of three matrices corresponding to a diattenuator followed by a retarder and a depolarizer. Since the depolarization effects will be only reflected in the depolarizer matrix, the other two matrices are used to extract the structure parameters of the measured sample. Experiments performed on a one-dimensional silicon grating structure with an in-house developed MME layout have demonstrated that the proposed method achieves a higher accuracy in the nanostructure metrology.
Dual-rotating compensator Mueller matrix ellipsometer (DRC-MME) has been designed and applied as a powerful tool for the characterization of thin films and nanostructures. The compensators are indispensable optical components and their performances affect the precision and accuracy of DRC-MME significantly. Biplates made of birefringent crystals are commonly used compensators in the DRC-MME, and their optical axes invariably have tilt errors due to imperfect fabrication and improper installation in practice. The axis tilt error between the rotation axis and the light beam will lead to a continuous vibration in the retardance of the rotating biplate, which further results in significant measurement errors in the Mueller matrix. In this paper, we propose a simple but valid formula for the retardance calculation under arbitrary tilt angle and azimuth angle to analyze the axis tilt errors in biplates. We further study the relations between the measurement errors in the Mueller matrix and the biplate axis tilt through simulations and experiments. We find that the axis tilt errors mainly affect the cross-talk from linear polarization to circular polarization and vice versa. In addition, the measurement errors in Mueller matrix increase acceleratively with the axis tilt errors in biplates, and the optimal retardance for reducing these errors is about 80°. This work can be expected to provide some guidences for the selection, installation and commissioning of the biplate compensator in DRC-MME design.
In this paper, we presented a microstrip patch antenna sensor that is capable to measure the shear and normal deformations simultaneously. The sensor was designed based on the electromagnetic interference between a microstrip patch antenna and a metallic reflector separated by a distance. By placing the reflector on top of the patch antenna, the electromagnetic wave radiated by the patch antenna is reflected by the reflector and interferes with the electromagnetic field of the radiation patch, which in turn changes the antenna resonant frequencies. Since the antenna resonant frequencies are related to the lateral and vertical positions of the metallic reflector, the shear force and normal pressure that shift the reflector laterally and vertically can be detected by monitoring the antenna resonant frequencies. The numerical simulation and experimental measurements were carried out to evaluate the relationship between the antenna resonant frequencies and the shear and normal displacements. A data processing scheme was developed to inversely determine the shear and normal displacements from the antenna resonant frequencies.
KEYWORDS: Inverse optics, Scatterometry, Inverse problems, Optics manufacturing, Transmission electron microscopy, Error analysis, Integrated optics, Nanostructures, Process control, Chemical elements
Optical scatterometry is widely used in the process control of integrated circuits (IC) manufacturing due to its inherent
advantages such as nondestruction, high sampling rate, large aerial coverage and low-cost. However, in the conventional
inverse problem solvent of optical scatterometry, the measurement errors are usually excessively simplified as normally
distributed errors, which deviate from the actual complex ones. In this work, we will demonstrate that there exist typical
outlying measurement errors in the measurement signature, and these outlying measurement errors will notably affect the
result of each iteration step in the conventional Gauss-Newton (GN) method. By performing a method based on the
principle of least trimmed squared estimator (LTS) regression instead of each GN iteration step, the higher measurement
accuracy of a nanostructure can be achieved. The remarkably improved reconstruction of a deep-etched multilayer
grating has demonstrated the feasibility of the proposed method.
In this paper, we studied the microstrip patch antenna for the purpose of temperature sensing. The relationship between the antenna resonant frequency shift and temperature variation is first derived based on the transmission line model. A substrate material was selected to achieve a linear sensor response. Temperature chamber tests on patch antenna sensors bonded to three different test samples were carried out. Preliminary experimental results indicated a linear relation between the normalized antenna resonant frequency changes and temperature variations. However, a large discrepancy between the measured and predicted sensitivities was observed, which indicated that the thermal strain might have a significant influence on the dielectric constant of the substrate. To account for this effect, we introduced a strain coefficient of dielectric constant to quantify the effect of strain on the dielectric constant. With the modified theoretical predictions, the errors between the measurements and predictions were within the systematic error of the reference thermocouple, which validates the feasibility of using a microstrip patch antenna for temperature sensing.
Due to the rich information provided by the Mueller matrices when the most general conical diffraction configuration is considered, the Mueller matrix polarimetry has demonstrated a great potential in semiconductor manufacturing. As the configurations of the incidence and azimuthal angles have different influences on the measurement accuracy, it is necessary to select an optimal one among the multitude of possible options. We introduce the norm of a configuration error propagating matrix to assess the measurement accuracy for different measurement configurations. The optimal configuration for a Si grating sample was achieved by minimizing the norm of the configuration error propagating matrix. Experimental results show the agreement between the theoretically predicted optimal configuration and the experimental exhibited one obtained by using a dual-rotating compensator Mueller matrix polarimeter and thus demonstrated the validity of the proposed optimization method.
As a non-imaging optical measurement technique, spectroscopic Mueller matrix polarimetry (MMP) has been introduced for critical dimension (CD) and overlay metrology with recent great success. Due to the additional information provided by the Mueller matrices when the most general conical diffraction configuration is considered, MMP has demonstrated a great potential in semiconductor manufacturing. In order to make full use of the additional information provided by the Mueller matrices, it is of great importance for MMP to optimize the measurement configuration. In this paper, we introduce the norm of a configuration error propagating matrix as the cost function to optimize the measurement configuration for spectroscopic MMP with the aim of finding an optimal combination of fixed incidence and azimuthal angles, which provides higher measurement accuracy. The optimal measurement configuration can be achieved by minimizing the norm of the configuration error propagating matrix in the available ranges of incidence and azimuthal angles. Experiments performed on a silicon grating with a dual-rotating compensator Mueller matrix polarimeter have demonstrated the validity of the proposed measurement configuration optimization method.
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