1. INTRODUCTION

Nanostructured surfaces have a large technological and economical importance due to their various application fields. Surface functionalities, such as light management and structural colors, and hydrophobic or hydrophilic properties, are achieved through periodical nano-texturing [1,2]. The requirements of the economically efficient, fast mass-production for large precisely structured surfaces sharpen the demand for in-line metrology to support process development and process control.

The structures of interest in this paper are sinusoidal gratings with both period $\mathrm{\Lambda }$ and peak-to-valley heights $H$ of about 200 nm printed in a roll-to-roll procedure on a transparent foil. Height deviations of more than 10% to the nominal value have to be detected with a method fast enough for the in-line application.

Established microscopy techniques (interference and confocal microscopy, and scanning electron or atomic force microscopy [3–6]) do not meet the time resolution requirements or do not provide the robustness needed for the in-process quality inspection. Non-contact scattered light-based methods are predestined for a fast surface inspection and process control, since the scattered light distributions from illuminated surfaces contain information about the topography, including local defects [7]. However, solving an inverse problem is needed for extracting the structural information of the measurement object on a sub-wavelength scale, which typically means comparing measured with simulated scattering distributions. Therefore, the current angular and spectroscopic scatterometry methods [8] depend on complex mathematical models and experiments for solving both the forward—accurate simulation of the measurement process—and the inverse problem. The measured scattering light information that best matches the calculated scattering behavior is found by a direct optimization or library search [8].

Several methods can be applied for calculating the light scattering, differing in complexity and limitations regarding the computational algorithm and the modeling of the sample. The classical scalar diffraction theory fails for surface structures with curvature radii in the sub-wavelength range [9]. Only rigorous simulation algorithms based on the Maxwell equations are appropriate to perform the computations. The method used in this work for solving the Maxwell equations is the discrete dipole approximation (DDA) algorithm [10–12], which is a powerful method for computing scattering and absorption of electromagnetic waves by particles of arbitrary geometry and composition. Replacing the scatterer with a set of interacting dipoles, the internal field and, consequently, all the measured scattering quantities can be calculated. The equations of the DDA algorithm turn exactly into the integral form of Maxwell’s equations when assuming infinitesimal distances between the dipoles [13]. Depending on the parameters of the addressed scattering problem, e.g., target size, shape, orientation, or refraction index, the simulations can be very demanding in terms of the hardware and computational time. The practical application of the method relies on efficient numerical algorithms. Several codes are available [14] differing in the numerical approach and the intended application. Larger scattering problems require computational acceleration, such as CPU or GPU parallelization [15]. This possibility is included in the open source code Amsterdam DDA (ADDA) [16], which is used for this work.

During the development of the DDA method, extended studies [12,16–18] for the comparison with other methods for solving the scattering problem, starting from the rigorous electromagnetic theory, including the exact methods Mie (sphere) or T-matrix, showed the validity of the approximation and of the chosen numerical algorithms. Algorithms for periodical structures [19] were developed by calculating the far-field scattering for a “target unit cell,” consisting of a discretized period of the structure, and extending it for a one- or two-dimensional infinite structure. However, the standard formalism for finite targets allows considering aperiodic defects of the periodical structure and real topographies extracted from microscopy data, respectively, at the expense of modeling large targets.

Currently used angular and spectroscopic scatterometry methods, mostly for semiconductor applications, are based on laboratory or in situ measurements of the specular reflectivity (0th order diffraction) for determining mean dimensional parameters of the sub-wavelength gratings. Series of measurements at different incident angles or incident wavelengths allow to accurately discriminate the effect of several form parameters and grating periodicity on the optical response [8,20–22]. Scatterometry is a critical technique for the high-volume wafer metrology, used in the process control to monitor relative deviations from the intended grating design. However, the in-line applicability is restricted to processes that allow a time resolution of several milliseconds [23].

For this reason, the aim of this paper is to investigate a grating height measurement approach with a fixed incident angle, a single wavelength, and a single measurement at a sub-wavelength sinusoidal grating by using the first-order diffraction, which provides in-line capability. After a short description of the ADDA simulation algorithm and the measurement principle, the optimal geometrical and optical parameters of the measuring setup and the corresponding calibration function are determined through scattered light simulations in Section 2. The limits of measurability and the measurement uncertainty for the grating height are estimated in Section 3 for different instrumental and simulation parameters. Diffraction measurements are performed offline in a scanning setup to test the sensitivity of the method, and the results are presented in Section 4. Results of reference atomic force microscopy (AFM) measurements are discussed in Section 5 together with the uncertainty budget for the grating height, and a short conclusion finishes the paper.

2. SIMULATION OF LIGHT SCATTERING AND MEASUREMENT PRINCIPLE

A. Discrete Dipole Approximation Algorithm

Initially, the target object was approximated by a similarly shaped ensemble of point dipoles characterized by their polarizations. From the interaction between the dipoles and with the incident field, written as a system of complex linear equations, the polarizations can be numerically computed. Alternatively, DDA formulations can be derived from the integral equation for the electric field, which is discretized by dividing the scatterer into small cubical subvolumes, further called also dipoles. The exciting electric field ${\mathit{E}}_{\mathrm{exc},i}$ at the dipole $i$ with the position ${\mathit{r}}_i$ is a sum of the incident field ${\mathit{E}}_{\mathrm{inc},i}={\mathit{E}}_0\exp (i\mathit{k}·{\mathit{r}}_i-i\omega t)$ and the field due to all other dipoles, but excluding the field of the dipole $i$ itself:

Considering different prescriptions for the polarizability and the interaction term [10,11], the system of linear equations can be inverted numerically, delivering the set of polarization vectors ${\mathit{P}}_i$. The scattered field at each scattering field position $\mathit{r}$ is

ADDA delivers $4\pi$ sr scattering information for two orthogonal incident polarizations. The result for any other incident polarization can be calculated through the Mueller scattering matrix. Therefore, it represents a powerful tool to determine the relevant features of the scattering pattern and the suitable parameters of a measuring method.

B. Simulative Design of the Measuring Method

For sub-wavelength gratings, it is possible to observe the first diffraction maximum at large incidence angles $\alpha$ (Fig. 1) up to a wavelength-to-period ratio $\lambda /\mathrm{\Lambda }$ of 2. However, according to the grating equation $\lambda /\mathrm{\Lambda }=\sin (\mathit{\text{in}}\mathit{\text{cidence angle}})+\sin (\mathit{\text{diffraction angle}})$ for the first order, diffraction measurements of a grating with $\lambda /2$ periodicity imply the experimentally unrealistic incidence and diffraction angles of 90°. In other words, for the gratings that are the focus of this work, with $\mathrm{\Lambda }\sim 204\mathrm{nm}$, the largest laser wavelength that can be used for diffraction measurements is the violet 405 nm. Incidence angles above 60° result in strongly enlarged measuring spot sizes and, implicitly, higher averaging of the measured parameters. However, this is not an impediment for the in-line application. The challenge is to determine the dimensional parameters of the gratings with nm precision from measurements of the position and the intensity of the diffraction maxima. The grating period can be computed straightforward from the grating equation, using the position of the diffraction maxima. However, the influence of the grating height on the diffracted intensities is complexly determined by the optical and geometrical parameters of the measurement setup. Therefore, the best configuration with respect to the incident angle, wavelength, polarization state, and used optical signal has to be found by means of scattering light simulations performed with the ADDA code for various scattering setups and grating parameters.

 

Fig. 1. Schematics of the simulation geometry in the scattering plane.

Download Full Size | PDF

Figure 1 shows the schematics of the simulation geometry in the so-called scattering plane, defined by the incident beam and its projection on the scattering target, in this case, the grating. The scattering (polar) angles $\theta$ are defined with respect to the incident beam. The first-order diffraction maxima are observed in reflection and in transmission, denoted by indices $R$ and $T$, respectively. To avoid confusion, the scattering angles ${\theta }_R$ are set to negative values. In this configuration, the incidence angle $\alpha$ equals the polar angle $\theta$ of the normal to the grating surface. Note that in the 3D view of the setup, the scattering directions are described by the scattering angles $\theta$ and the azimuthal angles $\varphi$. For the scattering plane depicted in Fig. 1, the azimuthal angles are ${\varphi }_R=90°$ and ${\varphi }_T=270°$.

A set of sinusoidal gratings with different $\mathrm{\Lambda }$ between 203 nm and 205 nm and broadly varying heights $H$ was approximated by a regularly spaced dipole model (Fig. 2). Choosing a dipole dimension of 20 nm allows for a good approximation of the structures while keeping the total number of dipoles, implicitly the size of the interaction matrix, under 10 million, which can be computed in several hours by a workstation. For the same reason, the target size was limited to $8\mathrm{\mu m}\times 8\mathrm{\mu m}$, and the substrate was neglected. For the calculations with ADDA, a uniform average refraction index of $m=1.5$ was assumed for the grating material. The absorption was neglected. Simulations for several laser wavelengths $\lambda$ with $\lambda /\mathrm{\Lambda }$ values between 1.5 and 2 were conducted under varying incidence angles. For comparability, the scattered intensities calculated for every scattering target were normalized by the scattering cross-section corresponding to that sample. This procedure is imposed by the limited size of the target relative to the incident beam and is equivalent to using the same incident intensity for all the scatterers.

 

Fig. 2. Fragment of the dipole ensemble approximating a sinusoidal grating with the grating period $\mathrm{\Lambda }$ and the grating height $H$. For better illustration, the dipoles are depicted as spheres and not as cubes. Here, the case of 20 nm sized dipoles, $\mathrm{\Lambda }=204\mathrm{nm}$ and $H=220\mathrm{nm}$, is shown.

Download Full Size | PDF

The intensities integrated over the area of the first-order diffraction maxima, i.e., the diffracted powers on both the reflection and the transmission side of the sample plane (side $R$ and $T$ in Fig. 1), were extracted from the simulated intensity distributions. Note that the calculated powers $S$ are proportional to the measured diffracted power $P$ by a factor that has to be determined experimentally. As a result, Fig. 3 shows the added simulated powers $S=S_R+S_T$ of the reflected and transmitted diffraction maxima calculated for an incident angle $\alpha =86°$ and a grating period $\mathrm{\Lambda }=204\mathrm{nm}$ as a function of the grating height $H$. The powers $S$ are normalized to the incident power, which in the no-absorption case equals the total scattered light power $S_{\text{tot}}$. The gray area encloses the interval of interest of $H$ for the investigated gratings and ranges from 100 nm to 300 nm. Hence, $\lambda$ has to be larger than 380 nm in order to provide an unambiguous measurement of the height over the whole desired measurement range.

 

Fig. 3. Added powers $S$ of the diffraction maxima normalized to the incident power $S_{\text{tot}}$ for gratings with increasing height $H$ at different laser wavelengths. The gratings have the period $\mathrm{\Lambda }=204\mathrm{nm}$, the incidence angle $\alpha =86°$, and in-plane polarization is considered. The gray area marks the interval of interest for the heights.

Download Full Size | PDF

In order to evaluate the influence of an incidence variation on the $S(H)$ curves, the scattering was simulated for different incidence angles around $\alpha =86°$. Small variations of the incidence angle up to $\mathrm{\Delta }\alpha =\pm 1°$ allow the consequent evaluation of $\mathrm{\Delta }H/\mathrm{\Delta }\alpha$ for the reflected and for the transmitted diffraction maximum, respectively. It was discriminated between the polarization parallel to the sample plane (in-plane) and the perpendicular polarization of the incident light. The lowest sensitivity regarding deviations from the configured incidence angle $\alpha =86°$ occurs in the interval of interest at $\lambda =405\mathrm{nm}$, for the summed $S=S_R+S_T$ and in-plane polarization. The corresponding curve is depicted in bold in Fig. 3. At this laser wavelength, the simulated scattering patterns lead to well-defined broad diffraction maxima separated by 20°–30° on the reflection and the transmission side of the surface plane (Fig. 4).

 

Fig. 4. Plane projection of the two diffraction maxima calculated for a grating with $\mathrm{\Lambda }=204\mathrm{nm}$ and $H=200\mathrm{nm}$ at $\lambda =405\mathrm{nm}$ and $\alpha =86°$, where $\theta$ are the scattering and $\varphi$ the azimuthal angles. For the scattering plane, the azimuthal angles are ${\varphi }_R=90°$ and ${\varphi }_T=270°$. Note that in the spherical depiction, the meridians (${\varphi }_R+{\varphi }_T=360°$) are intersecting at the incident beam position.

Download Full Size | PDF

The calibration function $H=f(P)$ is derived from the corresponding $S(H)$ curve considering the proportionality $S=KP$. For the laboratory setup, with a 10-bit CCD sensor and the exposure time $T=256\mathrm{\mu s}$, the proportionality factor $K=3.3\times {10}^{11}{\mathrm{W}}^{-1}$ between the simulated $S$ and the measured light power $P$ was determined.

3. GRATING HEIGHT MEASUREMENT UNCERTAINTY

For the instrumental measurement uncertainty evaluation, the factors that influence the detected position and intensity of the diffraction maxima must be analyzed. Besides the possible sources of errors in the measurement system, also uncertainties caused by the modeling/evaluation algorithms have to be considered.

A. Limits of Measurability

A physical limit for the measurement uncertainty of laser-based methods is set by the photon shot noise of laser light. For that, a perfectly coherent light source is assumed, only ideally approximated by a single-mode laser operating well above threshold. The standard deviation ${\sigma }_{H,\text{random}}$ of the grating height $H$ due to the photon shot noise is obtained with an error propagation calculation through the calibration function $H=f(P)$:

In order to investigate the in-line capability of the method, the dependence of the measurement uncertainty on the measurement parameters has to be clarified. Apparently, ${\sigma }_{H,\text{random}}$ depends on the particular measuring time through $P$ as well as on the experimentally determined scaling factor of the calibration function. Using the proportionality between the simulated quantity $S$ and the measured diffracted power $P$, $S=KP$, Eq. (9) can be written as

 

Fig. 5. Measurement uncertainty of $H$ due to photon shot noise of the laser light for different incident light powers. The legend gives the energies of the photons diffracted by a grating with height $H=200\mathrm{nm}$ ($\lambda =405\mathrm{nm}$, $E_{\text{photon}}=4.905\times {10}^{-19}\mathrm{J}$).

Download Full Size | PDF

From an information theoretical point of view, the relation (9) is identical to the square root of the Cramér–Rao bound of $H$ for the photon shot noise of laser light, i.e., for white Poissonian noise [26–29]. The result (9) has also a physical meaning, because it follows from Heisenberg’s uncertainty principle. Considering coherent light, the number of photons follows a Poissonian distribution and Heisenberg’s inequality is minimized [30,31]. As a result, the error propagation calculation for photon shot noise, the information theoretical approach for white Poissonian noise, and the Heisenberg uncertainty principle for coherent light are three different approaches to identify the quantum limit of the achievable measurement uncertainty of $H$ given in Eq. (9).

B. Random Error

The variance of the measured light power $P$ due to the photon detection for a CCD camera is modeled as white Gaussian noise with a constant variance [24]. This assumption is valid for CCD cameras with high pixel readout rates, above the $1/f$ corner frequency, which limits the influence of the frequency-dependent $1/f$ noise on the noise floor. The standard deviation of the number of electrons summed over $N_{\text{bin}}$ pixels is ${\sigma }_{Ne,D}={(N_{\text{bin}})}^{1/2}{\sigma }_{Ne}$, where ${\sigma }_{Ne}$ includes all the contributions to the white noise floor, the dark current, readout, and analog-to-digital conversion errors. Starting from expression (6) for the light power $P$, with ${\overline{N}}_e=\eta {\overline{N}}_{\text{photon}}$ the number of electrons generated on average at the detector with the quantum efficiency $\eta$, the power variance reads

Figure 6 shows the Poissonian and the Gaussian noise contributions to the measurement uncertainty of $H$ for a camera with a quantum efficiency $\eta =0.45$ at $\lambda =405\mathrm{nm}$, ${\sigma }_{Ne}=50$ electrons, and binning over $N_{\text{bin}}=4\times {10}^4$ pixel at a low laser power for which the energy of the photons diffracted by a grating with height $H=200\mathrm{nm}$ is ${10}^{-10}\mathrm{J}$. This corresponds to an average of 5100 photons/pixel or 2300 electrons/pixel, respectively. At lower electron numbers, the detector noise contribution increases drastically, and the shot noise becomes negligible. Otherwise, above a total energy of the collected photons of $2.42\times {10}^{-10}\mathrm{J}$, the shot noise contribution dominates ${\sigma }_{H,\text{random}}$, and the measurement uncertainty is then physically limited by the available laser power.

 

Fig. 6. Measurement uncertainty of $H$ due to shot noise and detector noise, respectively, at a low laser light power for which the energy of the photons diffracted by a grating with height $H=200\mathrm{nm}$ is ${10}^{-10}\mathrm{J}$. The intersection of the curves at a photon energy of $2.42\times {10}^{-10}\mathrm{J}$ here corresponds to $H=355\mathrm{nm}$. ($E_{\text{photon}}=4.905\times {10}^{-19}\mathrm{J}$, ${\sigma }_{Ne}=50$ electrons, $N_{\text{bin}}=4\times {10}^4$ pixel).

Download Full Size | PDF

C. Systematic Errors

The dipole dimension of 20 nm chosen for modeling the samples acts as surface roughness [32], but also limits the resolution of the calibration function and impedes on reaching the theoretical measurement uncertainty. Moreover, the uncertainties of the input parameters for the simulations due to unknown systematical errors propagate through the calibration function. Equation (5) has to be extended by

For this work, the uncertainty contributions of the incidence angle $\alpha$, the laser wavelength $\lambda$, and the grating period $\mathrm{\Lambda }$ are considered for the calculation of ${\sigma }_{H,\text{systematic}}$. Their corresponding partial derivatives can be numerically approximated, and the sensitivities $\mathrm{\Delta }H/\mathrm{\Delta }{x}_i$ are calculated from simulations on gratings with $\mathrm{\Lambda }=204\mathrm{nm}$ illuminated with $\lambda =405\mathrm{nm}$ laser light at an incidence angle $\alpha =86°$ by varying the corresponding parameter $x_i\pm 1$ unit (nm or °) while holding the other parameters constant. For the considered interval, the variation of $H$ with the different parameters could be approximated by low-degree polynomials, and therefore the higher terms of the Taylor series for the propagation of errors in Eq. (15) are negligible.

Figure 7 shows the resulting uncertainty contributions of the different parameters. The applied standard deviations ${\sigma }_{\mathrm{\Lambda }}=0.27\mathrm{nm}$ and ${\sigma }_{\alpha }=0.7°$ are measured for the laboratory setup and dominate the measurement uncertainty ${\sigma }_{H,\text{systematic}}$. A laser with an optical bandwidth of less than 0.02 nm full width at half maximum (FWHM) of the wavelength distribution, which means a wavelength standard deviation of ${\sigma }_{\lambda }=\mathrm{FWHM}/2{(2\ln 2)}^{1/2}=0.009\mathrm{nm}$, was used. Therefore, the wavelength stability of the laser leads to a negligible contribution of the wavelength uncertainty to ${\sigma }_{H,\text{systematic}}$, as depicted in Fig. 7.

 

Fig. 7. Uncertainty contributions due to the sensitivity of the calibration to variations of the incidence angle $\alpha =85.7°\pm 0.7°$ for a grating with $\mathrm{\Lambda }=203.5\pm 0.27\mathrm{nm}$ and laser wavelength $\lambda =405\pm 0.009\mathrm{nm}$.

Download Full Size | PDF

For completion, the systematic contribution of the laser power fluctuations was estimated as ($\partial H/\partial P){\sigma }_P=K(\partial H/\partial S){\sigma }_P$ for the laboratory setup. Due to the power noise of less than 0.5% of the laser, value indicated by the manufacturer and confirmed by measurements, this contribution is lower than that of the wavelength uncertainty and is not shown in Fig. 7.

4. DIFFRACTION MEASUREMENTS

A. Measuring Setup

The laboratory measuring setup reproducing the configuration determined by simulations is shown in Fig. 8. The transparent foil imprinted with a 40 cm broad sinusoidal grating was mounted with vertically oriented grooves. Collimated 405 nm laser light illuminated the surface slightly out of the in-plane diffraction configuration, allowing for both diffraction maxima to be simultaneously observed (Fig. 8, insert) on a screen placed on the scanning stage of a two-axis scanning setup. The deviation angle $ϵ$ between the scattering plane and the plane perpendicular to the grating grooves was chosen such that the laser direct beam does not impede on detecting the reflection side diffraction maximum.

 

Fig. 8. Diffraction measuring setup with the schematics of the diffraction geometry, where $ϵ$ is the deviation angle between the scattering plane, which includes the incident beam, and the plane perpendicular to the grating grooves; $2ϵ$ is the angle between the scattering plane and the plane defined by the two diffracted beams. $d$ is the distance between the diffraction maxima on a screen. The camera is solidly connected to the scanning stage of the $xy$ linear axes system, which holds the collimation optics and the screen. Insert: an image of the first-order diffraction maxima on the screen, overexposed for better illustration; the laser beam passing through the optics can be seen above the reflection diffraction maximum.

Download Full Size | PDF

The incident beam was led through a 2 m single-mode, polarization-maintaining optical fiber to the collimation optics. The optics and a cooled SVS-VISTEK11000 CCD camera were solidly connected to the stage in order to conserve the diffraction geometry throughout the measurement. The camera directed on the screen recorded the position and intensity of the two maxima during a $25\mathrm{cm}\times 23\mathrm{cm}$ scan with 2 frames/s. The laser power of 4 μW, at a beam diameter of about 0.5 mm and an exposure time of 256 μs, ensures a reliable detection of the total diffracted light. The cooling and a high pixel readout rate (max. 40 MHz) of the camera preserved a constantly low-noise floor during the measurement. The scan step was 1 cm in both the horizontal ($x$) and vertical ($y$) directions. The upper margin of the grating was located at $y=23\mathrm{cm}$.

B. Diffraction Measurement Results

For the scan area, the incidence angle $\alpha$ was locally measured by detecting the positions of the specularly reflected and directly transmitted beams in a separate scan, resulting in an average of $\alpha =85.7°\pm 0.7°$. The local variations of the incidence angle are due to foil undulations. The in-plane polarized laser beam intersects the foil on an area strongly elongated in the $x$ direction due to the grazing angle, with a ratio of $x/y=14$, leading to an averaging over every step in this direction.

From the image of the two diffraction maxima, the position and the added intensity for every scan point were extracted. Further, the grating period $\mathrm{\Lambda }$ was calculated using the resulting local diffraction angle and the average incidence angle leading to the average grating period $\overline{\mathrm{\Lambda }}=203.43\mathrm{nm}$ with a standard deviation ${\sigma }_{\mathrm{\Lambda }}=0.27\mathrm{nm}$.

The summed power of the diffraction maxima directly leads to the local grating height, through the calibration function. Several $H=f(S)$ curves were computed for $\mathrm{\Lambda }$ between 203 nm and 204 nm, in steps of 0.25 nm. The calibration curve for 203.5 nm, nearest to the measured mean $\overline{\mathrm{\Lambda }}$, was used for the height evaluation illustrated in Fig. 9. Note that the mapping is interpolated in both $x$ and $y$ directions for better illustration. A broad height distribution results, with a mean $\overline{H}=204.54\mathrm{nm}$ and a standard deviation ${\sigma }_H=20.16\mathrm{nm}$. The extended local fluctuations of $H$ are neither fully explained by the choice of an average grating period instead of the locally measured $\mathrm{\Lambda }$, nor by the foil undulations. The estimated measurement uncertainty ${\sigma }_{H,\text{systematic}}=12\mathrm{nm}$ (Fig. 7) introduced by these parameters at $\overline{H}$ is significantly lower than the measured ${\sigma }_H$, leading to the presumption of large grating height variations induced on the foil by the production process. Comparison with an established method with a low measurement uncertainty is needed to verify this assumption.

 

Fig. 9. Grating height $H$ calculated from the added powers of the diffraction maxima measured at a large area scan of the grating foil with in-plane polarized incident light.

Download Full Size | PDF

5. REFERENCE MEASUREMENTS AND DISCUSSION

In order to investigate the large height fluctuations indicated by the diffraction measurements, AFM measurements were performed at several positions in the previously scanned area. The used microscope was a NanoScope Dimension 3100 from Digital Instruments with supersharp NANOSENSORS SSS-NCH tapping mode probes. The tip radius of less than 2 nm and a half cone angle $<10°$ at 200 nm from the apex ensure that the surface modulation of the investigated gratings is fully accessible. Height calibration measurements on a standard grating of inverted pyramids with a depth of 70 nm and sidewall angle of 35.3° to the normal showed a standard uncertainty of 1.2 nm. The low standard uncertainty of the AFM measurements allows the use of the AFM results as reference for the diffraction measurements.

Due to the different scales of spatial resolution of the two measuring methods, a local comparison of the height values is not possible, and a statistical evaluation of the height distributions is needed. Regularly located 5 mm diameter samples were cut from the foil, covering the regions with height fluctuations. For each of the eight samples, an average grating height was derived from measurements on three $1\mathrm{\mu m}\times 0.25\mathrm{\mu m}$ areas. The resulting height distribution of these eight samples, with values between 180 nm and 230 nm, has a mean of ${\overline{H}}_{\mathrm{AFM}}=206.7\mathrm{nm}$ and a standard deviation of 14.2 nm, which includes the negligible uncertainty contribution of the AFM’s calibration procedure.

Table 1 summarizes the uncertainty evaluation for the grating height measurement with the proposed scatterometry approach. For the camera with a 10-bit CCD sensor and the exposure time $T=256\mathrm{\mu s}$, the calculated standard deviation of the diffracted light power amounts to ${\sigma }_{P,\text{random}}=0.61\mathrm{nW}$. With the current measurement setup, the measured standard deviation of the laser power was found to be three times larger than the estimated value due to photon shot noise. Hence, a power stabilization of the laser is expected to reduce ${\sigma }_P$. However, the resulting uncertainty ${\sigma }_{H,\text{random}}<0.1\mathrm{nm}$ is still negligible compared to the estimated uncertainty of the height ${\sigma }_{H,\text{systematic}}=12\mathrm{nm}$ due to systematic effects. Therefore, a total height uncertainty of 18.6 nm is estimated from the sum of squares between the calculated systematic error and the height fluctuations as determined by AFM measurements. Possible cause for the difference between the estimated and the measured height uncertainties is a slight underestimation of the height fluctuations due to the relatively small number of AFM measurements. However, not all potential error contributions were considered in Section 3.C, such as the polarization state of the incident laser light. A small deviation from the in-plane polarization cannot be excluded and was not yet quantified. Note that Table 1 contains the standard uncertainties. For the expanded uncertainty, coverage factors corresponding to the calculated degrees of freedom [33] have to be multiplied for the desired level of confidence.

Table 1. Evaluation of the Height Uncertainties

View Table | View all tables in this article

Due to computational restrictions, a realistic thickness of the substrate could not be simulated. The influence of the substrate thickness on the scattering pattern is currently under investigation. Broadening and relative intensity changes of the two diffraction maxima could result due to backside scattering. However, by detecting the total diffracted power in the first order, it is expected that this effect is negligible for the height measurements.

The undulation induced complex shape of the foil has a direct influence on the diffraction geometry described by the parameters $\alpha$, $ϵ$, and $d$, as shown in Fig. 8. Table 2 summarizes the parameter variations induced by rotation and translation of the local foil surface for the $x$ and $y$ axes defining the grating plane and for the orthogonal $z$ axis, respectively. Translation equals a change of measuring spot position on the foil, which is desired in the scan ($x$) direction. Translation along the $z$ axis also means an error in calculating the diffraction angle due to a distance change between measuring spot and screen, leading to a variation of $d$. Rotation around the $x$ axis represents an overall rotation of the geometry, and $\alpha$ and $ϵ$ remain constant. Rotation around the $z$ axis equals a rotation of the plane perpendicular to the grating grooves, but not of the scattering plane, and $ϵ$ varies and $\alpha$ remains constant. However, a $z$ rotation of the foil is not probable in the roll-to-roll process. A variation of $\alpha$ is induced only by a rotation around the $y$ axis. Due to the elongated measuring spot, a $y$ rotation implies also a $z$ translation, even when the rotation center is located inside the measuring spot. The variation of $ϵ$ has no significant influence on the total diffracted intensity, as verified by simulations. Every movement of the foil other than the $x$ and $y$ translation has an effect on the $\mathrm{\Lambda }$ measurement uncertainty by unknown variations of $ϵ$ and $d$, but only the $y$ rotation influences the height measurement.

Table 2. Influence of the Sample Movement on the Diffraction Geometry: Variations of the Parameters $\mathsf{\alpha }$, $\mathsf{ϵ}$, and $\mathsf{d}$

View Table | View all tables in this article

At present, the fluctuations of $\alpha$ and $\mathrm{\Lambda }$ dominate the height measurement uncertainty. Nevertheless, the foil undulation can be minimized at the roll-to-roll procedure, which reduces ${\sigma }_{\alpha }$. Changes of the grating parameter are not expected during the production, and the grating period step size for computing the calibration function dominates ${\sigma }_{\mathrm{\Lambda }}$. Therefore, ${\sigma }_{\alpha }$ and ${\sigma }_{\mathrm{\Lambda }}$ can be drastically reduced in the planned in-process application, resulting in a much lower ${\sigma }_H$. Figure 10 shows an example calculated for ${\sigma }_{\alpha }=0.1°$ and ${\sigma }_{\mathrm{\Lambda }}=0.1\mathrm{nm}$, which results in a height measurement uncertainty of less than 4 nm for the interval of interest.

 

Fig. 10. Uncertainty contributions due to the sensitivity of the calibration to variations of the incidence angle $\alpha =85.7°\pm 0.1°$ for a grating with $\mathrm{\Lambda }=203.5\pm 0.1\mathrm{nm}$ and laser wavelength $\lambda =405\pm 0.009\mathrm{nm}$.

Download Full Size | PDF

Another aspect that was not yet considered is the effect of form deviations from the desired sinusoidal modulation on the diffracted power. A deterioration of the forming tool could involve not only a height decrease but also local form modifications. Correlations between these parameters are expected and are subject of future work. However, significant form variations were not observed for the investigated foil. Figure 11 shows an AFM image for a sample with $\overline{H}=226\mathrm{nm}$ and mean standard deviation of 2 nm. A sine function fits the average height profile with an adjusted coefficient of determination of $R^2=0.98$.

 

Fig. 11. AFM image of a sample with $\overline{H}=226\mathrm{nm}$ and mean standard deviation of 2 nm. Dotted line: an average height profile of the grating.

Download Full Size | PDF

Due to the wavelength dependence of the refractive index, the results presented in this paper cannot be directly transferred to other grating height measurements with the same ratio $\lambda /\mathrm{\Lambda }$. However, for relatively small differences in $\mathrm{\Lambda }$, for which the change in wavelength does not significantly affect the refractive index, the setup configuration is expected to remain the same. Only the calibration function has to be calculated for the new parameters.

Even though only the influence on diffraction of grating height changes was considered for this work, it has to be noted that the ADDA calculations deliver the full far-field scattering information. That means that also for gratings with the grating parameter smaller than half the laser wavelength, for which the diffraction maxima are no longer accessible, it is possible to estimate the effect of the height on the scattering distribution. Current research [34] for semiconductor applications, where the surface features of interest are in the deep sub-wavelength range, showed promising results for one-dimensional gratings with an aspect ratio of 1 and $\lambda /\mathrm{\Lambda }=5.32$.

6. CONCLUSIONS

Simulations of laser light scattered from sub-wavelength sinusoidal gratings were used to develop an accurate evaluation algorithm for the grating height, avoiding the expensive and time-consuming process of reference sample manufacturing and measurement. The optimal setup parameters were derived from simulations for the in-line measurement of the height of high aspect sinusoidal gratings formed on a transparent foil in a roll-to-roll procedure. For a grating parameter of about 204 nm, measurements of summed light power diffracted in the first-order maxima at an incidence angle of 86° of in-plane polarized 405 nm laser light led to the lowest measurement uncertainties.

Measurement uncertainty calculations, considering the statistical errors, showed that, depending on the chosen detection system, sampling rates up to the MHz range can be achieved in principle with sub-nanometer precision. However, the measurements and the simulative investigations indicated that the foil undulation, leading to local variations of the incidence angle, is an important factor influencing the measurement uncertainty for the grating height. Moreover, the grating period resolution for computing the calibration function bears a significant contribution to the height measurement uncertainty. As a result, the current measurement uncertainty of the grating height is estimated to 12 nm, but with a potential for $<4\mathrm{nm}$.

Large-area scanning measurements were performed offline in order to characterize the variations of the grating parameters across the foil. A good statistical agreement with reference AFM measurements showed that the method can detect mean height deviations of less than 10% to the nominal height.