Fabrication error analysis of nonperiodic-multilayer-dielectric gratings

. We analyzed fabrication errors of nonperiodic-multilayer-dielectric gratings that were designed to have a high reflective diffraction efficiency for one wavelength and a high transmittance for another wavelength at the same time. Significant deviations were found between the measured and calculated efficiency values although the groove profile parameters were very closed to the design values. The source of error was attributed to coating errors of the film stack. To explain the deviation well, we estimated the parameters of actual coating stack by reverse calculations and using scanning electron microscope. We recalculated the diffraction efficiencies and the results showed that the actual film stack parameters were inverted well. This provided a foundation for us to reoptimize grating groove parameters on the basis of actual coating stack.


Introduction
Multilayer dielectric gratings are key dispersion elements of many modern optical systems [1], such as spectrometers, chirped-pulse amplification systems and spectral beam combining systems.In most cases, it is sufficient to use periodic multilayer stack as the grating substrate, thus there have been few reports on gratings based on the non-periodic multilayer stack (NPS).In this paper, the grating was expected to achieve simultaneously a high reflected −1st-order diffraction efficiency for wavelength λ1 and a high transmitted 0th-order diffraction efficiency for wavelength λ2.Besides, λ2 − λ1 > 1.6μm, so it was necessary to select NPS as the substrate.While in the fabrication process, resulting from coating errors of the film stack, when the fabricated grating groove profiles were close to design parameters, there is a quite disagreement between the measured and design values of the diffraction efficiencies.We estimated the parameters of the actual film stack and recalculated the diffraction efficiencies according to actual film stack.The results were in good agreement with the measured efficiencies.

Principle
Figure 1 shows the schematic diagram of the grating.The wavelengths of the incident light A1 and A2 are λ1 and λ2, respectively.Both A1 and A2 are incident at the −1st-order Littrow mounting with respect to A1 in TE polarization.The film stack is represented by S|H1L1H2L2…L9H10|C, where S is the silicon substrate, Hi is the ith high-index layer, Li is the ith low-index layer, and C is the cover (air).The total number of the films is 19, and the grating period d is 2 μm.The groove shape is an isosceles trapezoid with a 72° base angle.The parameters h, w, and T represent groove depth, the upper base width of grating ridge, and the thickness of the top film, respectively.We denote the duty cycle of the upper base by ∆ = w/d.

Fig. 1. Schematic diagram of the grating
Eight gratings were used for analyzing the fabrication error.The duty cycle ∆ was in the range of 0.24 to 0.35 and the groove depth h was in the range of 0.3 μm to 0.53 μm.In the fabrication process, the actual thickness and the refractive index of each layer deviated from the nominal values, leading to that the grating groove satisfying the design requirement cannot realize high efficiencies as calculated.To better understand the reasons for the deviation between measured and calculated values, we inverted the parameters of the actual film stack.
In the reverse engineering step, calculation of diffraction efficiencies was made by using the software Kappa [2], which is based on the Fourier modal method.To judge the closeness between the measured and where j is the grating number, N is the total number of gratings used to estimate film stack parameters, and X = {d1, …, d18; T, nH1, nL1, nH2, nL2} represents the parameters describing the actual coating stack.In X, {di} are layer thickness values obtained by measuring the scanning electron microscope (SEM) images of the NPS, which can be considered as the actual thickness with a sufficient accuracy.nH1, nL1, nH2, and nL2 represent the high and low refractive indices at λ1 and λ2, respectively.Pj = {∆j, hj} is the vector of groove parameters of the grating.R(X;Pj) and T(X;Pj) are calculated values of diffraction efficiencies of A1 and A2 under the film stack parameters X and groove parameters Pj.Rm,j and Tm,j are measured values of diffraction efficiencies of A1 and A2.{T, nH1, nL1, nH2, nL2} are parameters that need to be inverted.Because light interacts first with the top layer, the influence of the top layer thickness on diffraction efficiency is significantly higher than that of the other layers.Thus, we chose to estimate T in the range of measurement accuracy of SEM.T was inverted in the range of [T0−10nm, T0+10nm], where T0 is the measured value of the top layer thickness.All the refractive indices were inverted in the range of 5% of the relative error of the nominal values.
According to the measurement results and reverse calculations described above, we obtained the estimated film stack parameters of actual coating stack, denoted by Factual.Correspondingly, the calculated efficiency value under Factual is denoted by Eactual.Besides, we denote the designed film stack parameters by Fdesign and the efficiency value under Fdesign by Edesign.

Experiment and results
We fabricated eight gratings and measured their groove parameters and diffraction efficiencies.The measured and calculated values of diffraction efficiencies are shown in Table 1.We define the residual error as the difference between the measured and calculated values.emean,i is the mean value of residual errors and erms,i is the root mean square of residual errors, i = 1, 2 for A1 and A2, respectively.
As shown in the table, all the measured values are lower than Edesign, and most of the latter are greater than 0.985.Especially for A2, emean,2 under Fdesign is obviously less than zero.In contrast, the absolute value of emean,2 under the parameters Factual is less than 1% of that under Fdesign.erms,2 under Factual is also obviously lower than that under Fdesign.The similar phenomenon can also be observed for A1.
In summary, Factual is better than Fdesign to represent the actual film stack, showing that it is more reasonable to reoptimize the grating groove parameters on the basis of Factual.It can guide us to make gratings with higher efficiencies in the future work.

Conclusion
In this paper, we analyzed fabrication errors of NPSbased gratings that were designed to achieve dualwavelength high diffraction efficiencies (>95%).We first fabricated gratings with the designed groove profiles.Resulted from coating errors, the measured efficiencies were lower than design values.We estimated parameters of the actual film stack according to the SEM measurement results and made reverse calculations based on the measured efficiency values.Finally, diffraction efficiencies were recalculated on the basis of the estimated stack parameters and results showed that there was a good agreement between the measured and calculated efficiency values.

Table 1 .
Measured and calculated diffraction efficiencies of