Characterization of dielectric diffraction gratings on multilayer structures by spectroscopic ellipsometry

Guided photonic modes in transparent structures have long inherent propagation lengths due to their evanescent nature outside of the allowed regions, i.e., total internal reflection at the interfaces or photonic bandgaps, which can be exploited for low-loss signal transport¹ and integrated optics.² In planar geometries, their dispersion lies beyond the light line, $E p h=ℏc k ∥$⁠, thus they cannot couple to the ambient field. Therefore, many approaches for coupling light in photonic waveguides have been developed.³ One approach that allows to excite and detect such modes is the use of (one-dimensional) diffraction gratings.⁴ In fact, by a suitable choice of grating dimensions, this mechanism effectively enlarges the in-plane component of the wavevector such that the guided modes can be coupled to vacuum modes, which is otherwise known as a type of Rayleigh–Woods anomaly.⁵ In the case of shallow gratings (height $≪λ$⁠) the wavevector shift is approximately linear and depends on the periodicity $L$ of the grooves.

Efficient coupling to guided modes using diffraction gratings typically requires a groove spacing similar to the wavelength, which can make an optical characterization using microscopy rather challenging. An alternative for grating characterization is the measurement and description of its optical response and an analysis of the corresponding optical modes. Toward this end, a precise knowledge of the optical properties of the involved materials and the geometrical properties of the layer is necessary. Spectroscopic ellipsometry (SE) is a well-established technique for the determination of optical properties and the layer structure of samples,⁶ though it is typically used for planar stratified media.

Recently, ellipsometry was used in order to investigate gratings on high quality GaAs,⁷ Si,^8–10 and other substrates.¹¹ Furthermore, the spectral bandwidth of the measurement instrument is often neglected while only few studies take this into account^12,13 or found it to have only a small effect.¹⁴ Compared to thin film samples and substrates, multilayer samples with shallow gratings exhibit a high number of very narrow modes that are often absent in experimental data.¹⁵ 

Here, we demonstrate the investigation of the optical response of shallow dielectric gratings on a multilayer structure, which consists of a distributed Bragg reflector (DBR) with a top layer that supports the formation of Bloch surface waves (BSWs), a photonic guided mode confined to the sample surface.¹⁶ These modes, which are associated with, e.g., long-range lateral propagation¹⁷ and exceptional sensitivity,^18–20 can be excited using gratings. We show that even for such a complex system spectroscopic ellipsometry allows us to accurately determine the geometrical properties of the grating, which agree very well with those obtained from atomic force microscopy (AFM).

The sample was produced using pulsed laser deposition (PLD). The laser (Lambda Physik LPX Pro, excitation wavelength of 248 nm, pulse energy of 600 mJ) is focused on a (rotating) target material, creating a plasma plume that expands toward a c-sapphire substrate (Crystec, $10×10 mm 2$⁠) for layer deposition. The multilayer was grown at room temperature for a smooth surface and consists of a 150 nm yttria-stabilized zirconia (YSZ) top layer followed by a DBR made of 7 layer pairs of 90 nm $Al 2 O 3$ and 72 nm YSZ. The oxygen partial pressure and pulse repetition rate were 0.02 Pa and 15 Hz for the $Al 2 O 3$ layers and 0.2 Pa and 15 Hz for the YSZ layers.

The fabrication of nanostructures in dielectric or insulating materials, like YSZ, is a challenging task and was done here by producing a gold hard mask using focused ion beam (FIB) milling and subsequent dry etching. The Au layer, which also reduced FIB related charging effects,²¹ had a thickness of $∼$150 nm and was produced by sputter deposition (30 W, 90 s, 100 SCCM Ar). For FIB milling (FEI Nova 200 NanoLab), we used a $Ga +$-beam current of 100 pA at 30 kV, dwell time $1μs$⁠, dose per volume 200 pC/ $μ m 2$⁠, and magnification $3000×$⁠. Next, the sample was dry-etched at $15 °$C using an inductively coupled plasma (ICP) with 30 SCCM $CF 4$ at 1.3 Pa and 150 W and 300 W HF and ICP power, respectively. This resulted in an etch rate of approximately 0.6 nm/s. After this process, residual gold was removed in a potassium-iodide (KI) solution bath (40 g I  $+$ 80 g KI in 600 ml $H 2$O). Three gratings of different nominal geometries were fabricated, i.e., $L∈{400,400,450}$ nm and $h∈{30,40,40}$ nm. In all cases, the grating region spanned $∼50×50μ m 2$⁠.

The gratings were pre-characterized using atomic force microscopy in a contact mode (XE 150) with scan areas (⁠ $1.5×1.5μ m 2$⁠, $512×512$ pixels) and a 3XC-NA cantilever with an aluminum-coated tip and spring constant of $9 N m − 1$⁠. As the sample is highly insulating, significant drift was observed during the scans, which was corrected using a b-spline interpolation implemented in Gwyddion software.²² 

Reflectivity maps $R(E,θ)$ were obtained by illuminating the sample with collimated light from a Xe-lamp. An image of the reflected light in the back focal plane of a Mitutoyo objective (⁠ $50×$⁠, $NA=0.4$⁠) was dispersed in a spectrometer (Horiba Jobin Yvon iHR320, $300$ grooves/mm) and recorded with a CCD camera (Horiba Jobin Yvon Symphony Open STE). The reflectivity data were calibrated using a Si wafer with a 8 nm oxide layer.

For sensitive measurements that allow reliable quantitative analysis of sample geometries, it is preferable to measure spectroscopic ellipsometry as it is based on measuring the ratio of reflection coefficients, i.e., $ρ$⁠, instead of reflection intensities, which are prone to large experimental uncertainties and do not take into account phase information. Thus, we measured spectroscopic ellipsometry data for the grating region with a commercially available imaging ellipsometer (Accurion EP4) averaging the spectra over the patterned area in the spectral range of 2–4 eV (310–625 nm) with an energy spacing of 10 meV. We used an UV objective (⁠ $7×$ magnification, $NA=0.2$⁠). This results in a spatial resolution of $∼1μm$⁠, which is larger than $L$⁠, and thus the averaged optical response of the grating was determined. Furthermore, the angular spread of $± 1 °$ is negligible compared to the influence of the bandwidth and a focus scan was performed over the grating areas. A knife edge was placed in the beam path to suppress the signal from reflections at the backside of the sample. Measurements were taken at $45 °$ and $55 °$ angle of incidence (AOI) and the measured region of interest was adjusted to match the grating area for each angle, individually. Furthermore, the finite bandwidth of the instrument was taken into account, which is described in more detail below.

In the formalism presented above, the response at a given wavelength is not affected by the neighbored one. However, experimental setups exhibit a finite spectral bandwidth (⁠ $δE$⁠) so that the measured response is also affected by the response of the neighbored energies. This plays an important role if the feature response is smaller than the spectral bandwidth. In this case, it will cause an additional broadening in the optical response spectrum. In the following, we expand the numerical method presented above to include the spectral bandwidth, $δE$⁠, mainly caused by the monochromator, arising from the finite width of its entrance slit. For the instrument used here, $δE$ as a function of energy is shown in Fig. 2(a).

Furthermore, the number of points (⁠ $n w$⁠) should be large enough for the convolution to obtain a smooth curve, i.e., the spectral point spacing should be smaller than the mode broadening, cf. Fig. 2(b). However, too many points will strongly increase the computation time and will not change the calculated spectrum. Here, $n w=25$ points were sufficient for a good description of the spectra. Note that computation is not increased by a factor $n w$ as there is a considerable overlap of the distribution of wavelengths for neighboring sampling energies, and observables for these wavelengths have to be computed only once.

For the modeling of the samples patterned with a grating, all film thicknesses were taken from the analysis obtained before the patterning process, except for the layer containing the grating (⁠ $d S$⁠) as initial parameters. For this layer, we allow a thickness variation of about $±5nm$⁠, due to local inhomogeneities. Consequently, free parameters of the model were $L$⁠, $η$⁠, $h$⁠, $d S$⁠, and $φ$⁠, an azimuthal rotation of the sample orientation (cf. Fig. 1).

The AFM topography for the different gratings with nominal pitch $L=400$ nm and two different heights and $L=450$ nm is shown in Fig. 3. The relevant geometric parameters were extracted as follows: $L$ is given as the distance between periodic peaks of the first derivative, $h$ as the distance between peaks of the height histogram with bin sizes of 1 nm and $η$ as the ratio of the number values below half the peak-to-valley maximum and the number of pixels per row. The obtained values are summarized in Table II. These results reveal the successful fabrication of homogeneous gratings with nearly rectangular cross section, though slightly slanted sidewalls, as well as a smooth ridges and grooves in the sample surface. The absolute median deviation is less than $5%$⁠, except for the first groove of the grating with $L=450$ nm [Fig. 3(c)]. This we attribute to droplets in the mask or the grating. Note, we suspect that the sharp features at the edge of the ridges are measurement artifacts as they arise on the other side of the ridges for the opposite scan direction.

As mentioned above, the electromagnetic field of the BSW mode decays on both sides of the surface interface at $z=0$ nm, which is exemplarily shown in Fig. 4(a). Using diffraction gratings allows these modes to couple to the vacuum states since the dispersion is shifted inside the vacuum light cone, bounded by $E=ℏc k ∥$⁠. Hence, BSW can be observed as peaks in reflection spectra for finite $θ$⁠. In Figs. 4(b) and 4(c), the measured and simulated TE-polarized reflectivity maps are shown for the structure patterned with a grating with $L=400$ nm and very good qualitative agreement is found. The dispersion of the BSW, which is symmetric around $θ= 0 °$ corresponding to $k ∥<0$ and $k ∥>0$⁠, can be seen for diffraction orders $m=1$ and $m=2$⁠. At the crossing points of the dispersions, an anti-crossing behavior can be found, most prominently for $m=1$ at $θ= 0 °$⁠, which agrees with findings in the literature, e.g., Ref. 23. The width of the resulting spectral gap depends on the geometry of the grating, especially on $η$ and $h$ and its position is determined mostly by $L − 1$⁠. A comparison of the reflectivity maps for different grating geometries can be seen in the supplementary material.

In order to gain a deeper insight into the optical properties, e.g., layer thicknesses and change of the polarization, we performed spectroscopic ellipsometry. In Fig. 5, the $Ψ$ and $Δ$ spectra for the unpatterned surface as well as for those patterned with different gratings for an angle of incidence of $55 °$ are shown, respectively. The data for angle of incidence of $45 °$ can be found in the supplementary material. For the analysis, the spectra measured at an angle of incidence of $45 °$ and $55 °$ were analyzed simultaneously.

In Figs. 5(a) and 5(e), the model data (green line) are displayed along with the experimental data (black circles whose size grows proportionally to the bandwidth $δE$⁠) for the unpatterned sample, which follows the well-known behavior of a DBR. The stop band with a central photon energy around 2 eV has a width of around $∼$0.4 eV. Outside the stop band, typical Fabry–Pérot oscillations can be seen, which are due to multiple reflections at layer interfaces and damped with increasing absorption in the system for increasing photon energies.

In the spatial region patterned with a grating, additional narrow peaks are observable in the spectrum. These peaks arise due to a coupling of the evanescent modes to the vacuum states. They correspond to guided modes, analogous to the Fabry–Pérot oscillations mentioned above, and surface modes, both of which lie outside the light line. For the three different gratings, $Ψ,Δ$ spectra are shown in Figs. 5(b)–5(d) and Figs. 5(f)–5(h), respectively. Note, due to the broken rotational symmetry of our sample in presence of the gratings, mode conversion takes place in general. However, we arranged the sample such that the grating lines are almost perpendicular to the plane of incidence, i.e., $ϕ∼ 0 °$⁠. Therefore, this effect can be neglected, which was also verified in a limited spectral range by measuring the $(3×4)$ Mueller matrix. The green solid line corresponds to the model fit, taking into account the experimental bandwidth, which yields excellent agreement. For the resulting best-fit parameters, the calculated optical spectrum, assuming a vanishing bandwidth (⁠ $δE=0$⁠), is shown as the red solid line. Good agreement of the energetic position of the peaks is obtained, their shape, i.e., linewidth (⁠ $γ$⁠), and height, is poorly reproduced. In general, the calculated peaks exhibit a much smaller linewidth and amplitude than experimentally obtained. This effect can be attributed to the effect that the linewidth is much smaller than the bandwidth of the experimental setup [cf. Fig. 2(a)], i.e., $γ<δE$⁠, as described above.

Though the spectral information of the modes resulting from the grating agrees for both cases, i.e., $δE=0$ and $δE≠0$⁠, which was exploited in Ref. 15, it is clear that the bandwidth must be considered in order to accurately describe the measured data and explain the absence of the sharp features resulting from the grating and cannot be neglected as it is typically done when $γ>δE$⁠.^7,9,14 In Ref. 12, for example, depolarization effects resulting from finite bandwidth were taken into account to characterize asymmetry in nanoimprinted gratings.

Note the remaining deviations between the model and measurement data we attribute to irregularities of the gratings and deviations from a perfect rectangular profile. Assuming the grating layer to be made up of $M$ thin layers with linearly (or parabolically) growing $η$ has a similar broadening effect). Furthermore, deviations are slightly larger starting at around 3.8 eV with the onset of absorption in the YSZ layers.

The features at high energies around 3.8 eV correspond to the excited BSW modes for transverse electric and transverse magnetic polarization, which are absent in the spectrum from the unpatterned region, because BSW are not able to couple to the incident light, as expected. The peak in $Ψ$ gives the position of the $TM$-polarized BSW, which depends on the shift of the wavevector $Δ k ∥$ introduced by the grating, mainly determined by $L$⁠. As increasing $L$ corresponds to decreasing $Δ k ∥$⁠, the BSW modes of the gratings with $L=450$ nm appear red-shifted with respect to the gratings with $L=400$ nm. The peaks that correspond to the BSW exhibit a larger broadening in the experimental spectrum compared to the calculated one, which was also reported with Ref. 15. These results show that gratings can be used for spatially controlled BSW excitation, which is of interest for BSW-based optical circuits^24–27 or strong light–matter interactions.^28–30 

A comparison between AFM results and our model fit parameters yields overall very good agreement and a comparison is shown in Table II. The lattice constant $L$ seems to be slightly but systematically underestimated in SE analysis by $∼$1%–2% compared to the value obtained by AFM. The model is also sensitive with regard to the grating height $h$ though relative differences are larger at $∼$10%. These differences may be attributed in part to the difference in the AFM scan range (⁠ $1.5×1.5μ m 2$⁠) compared an average over the whole grating (⁠ $50×50μ m 2$⁠) in SE measurements. Furthermore, the assumption of a perfectly rectangular grating profile which was used in the model analysis is not strictly fulfilled by the actual grating geometries here. Nonetheless, the groove ratio $η$ shows excellent agreement between the two measurement techniques and the surface thickness (including the grating layer) is constant for all geometries, which indicates very homogeneous films. The azimuthal angle offset is well within the tolerance of centering the sample by eye on the ellipsometer.

We would like to note that if the experimental spectra are modeled excluding the bandwidth, similar grating parameters can be obtained, whose differences to the AFM values are systematically larger along with their uncertainties, cf. Table II. Though this may suffice for some investigations, all the information regarding the spectral broadening of the modes, i.e., the imperfection and inhomogeneity of the grating itself, are lost. Furthermore, we want to point out that neglecting the bandwidth leads to a more unstable minimizing routine that is very sensitive on the initial parameter values as opposed to a more robust minimization if the bandwidth is considered.

Within this work, we have demonstrated an ellipsometric modeling approach for one-dimensional gratings on top of a transparent multilayer structure supporting Bloch surface waves. The grating allows incident light to couple to these modes which can be seen as peaks in the specular reflected light. Furthermore, it was shown that the ellipsometer’s spectral bandwidth must be included in the model for an accurate description of the $Ψ,Δ$ spectra, which was accounted for here by using a discrete set of Gaussian weights. As a consequence of the grating coupler, it was possible to observe BSW peaks in the spectra, which are otherwise absent. We compared the model parameters to AFM measurements and found very good agreement for the lattice constant and groove ratio, while the grating height showed relative differences of <10%. The remaining deviations between the model and experimental values and the differences between AFM and SE measurements were attributed to a grating profile that was not perfectly rectangular and the different size of the measurement areas, respectively. These results provide an efficient framework for the optical determination of grating dimensions and characterization of BSW launching regions, while stressing the importance to consider instrument limitations, especially the spectral bandwidth.

See the supplementary material for depolarization data, ellipsometric spectra at AOI = 45 °, and angle dependent reflectivity maps for different grating geometries.

The authors would like to thank M. Hahn for PLD target preparation, Dr. Daniel Splith and Dr. Lukas Trefflich for technical support (all at Leipzig University). An Accurion EP4 imaging ellipsometer was used for the measurements, which was purchased as part of the UltraSpec project [Bundesministerium für Bildung und Forschung (BMBF) German Ministry for Education and Research], Association of German Engineers (VDI)/Association of Electrical, Electronic, and Information Technology (VDE), Project No. 03VP08180). Sebastian Henn acknowledges the graduate school BuildMoNa and funding by the Deutsche Forschungsgemeinschaft (DFG, Project No. STU 647/2-1).

The authors have no conflicts to disclose.

Sebastian Henn: Conceptualization (equal); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (equal); Visualization (lead); Writing – original draft (lead). Marius Grundmann: Formal analysis (supporting); Funding acquisition (supporting); Resources (lead); Validation (supporting); Writing – review & editing (equal). Chris Sturm: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (lead); Methodology (supporting); Project administration (lead); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.