Statistical error analysis

Analysis –> Errors –> Error analysis

The Errors Analysis initiates a computational procedure that provides a statistical evaluation of the effect that errors in the layer thicknesses and refractive indices in a design will have on the spectral response of a designated spectral characteristic.  

As the computations proceed, a sequence of curves corresponding to the statistical tests being performed is plotted.  During the computations, the progress of the statistical procedure is indicated in the percentage bar at the bottom of the window. This tool allows estimating the influence of errors in layer thicknesses and refractive indices on spectral characteristics of the current design.

Fig. 1. Error analysis setup.

Fig. 2.  Example. For a beamsplitter BS_7, absolute errors of 0.5 nm and relative errors of 1% are specified, i.e. \(\delta=0.5\) nm and \(\Delta=1\%\).

In the Error Analysis Setup window, there are two more tabs (Thickness and Refractive index) that allow you to specify different kinds of expected (estimated) deposition errors.

In Thickness tab there are three sub-tabs: Coating, Materials, and Thicknesses (see left pane). Important: Settings in all three tabs are independent.  In the course of calculations, the errors specified in the last opened tab are taken into account. 

In the Coating sub-tab, you can specify absolute (Abs. RMS) or/and relative (Rel. RMS) level of errors in all coating layers. In the course of the calculations, instead of layer thicknesses \(d_1,…,d_m\), disturbed thicknesses are used:

\[ d_1^{(j)}=d_1+\delta_1^{(j)},…,\;d_m^{(j)}=d_m+\delta_m^{(j)}\; \mbox{(absolute errors)} \;\; d_1^{(j)}=d_1+\Delta_1^{(j)}\cdot d_1,…,d_m^{(j)}=d_m+\Delta_m^{(j)}\cdot d_m\; \mbox{(relative errors),} \]

where \(\{\delta_i^{(j)}\}\) [nm] and \(\{\Delta_i^{(j)}\}\) [%] are absolute and relative errors in layer thicknesses, respectively. The errors are normally distributed numbers with zero average and specified rms deviations \(\delta\) and \(\Delta\).

\(j=1,…,N\) where \(N\) is the number of statistical tests.

Fig. 3. Statistical error analysis: Calculation process with settings from Fig. 2.

Fig. 4. Statistical error analysis with settings from Fig. 2: expected and averaged (Exp) spectral characteristics, two curves, Exp-D and Exp+D, indicate the probability corridor for a given error level. The width of probability corridor corresponds to the level of the errors that were set in Error Analysis Setup window (Fig. 1).

In the course of the calculations, instead of layer thicknesses \(d_1,…,d_m\), disturbed thicknesses will be used. In the case of a two-material coating:

\[ d_1^{(j)}=d_1+\delta_{H,1}^{(j)},…,\;d_m^{(j)}=d_m+\delta_{L,m}^{(j)} \; \mbox{(absolute errors)} \]and

\[ d_1^{(j)}=d_1+\Delta_{H,1}^{(j)}\cdot d_1,…,d_m^{(j)}=d_m+\Delta_{L,m}^{(j)}\cdot d_m, \; \mbox{(relative errors)} \]

where \(\{\delta_{i,H}^{(j)}\}\), \(\{\delta_{i,L}^{(j)}\}\) [nm] and \(\{\Delta_{i,H}^{(j)}\}\), \(\{\Delta_{i,L}^{(j)}\}\),  [%] are absolute and relative errors in layer thicknesses, respectively. The errors ar)e normally distributed numbers with zero average and specified rms deviations \(\delta_{H,L}\) and \(\Delta_{H,L}\).

This option is very convenient if you have not two but multiple thin-film materials or if you would like to estimate effect of errors on the spectral characteristics of stacks.

In the Materials sub-tab, you can specify absolute (Abs. RMS) or/and relative (Rel. RMS) level of errors in layers. The errors in layers of different materials can be specified separately.

Fig. 5.  Example. For a beamsplitter BS_7, relative errors of 0.5% in H-layers and absolute errors of 1nm are specified, i.e. \(\Delta_H=0.5\)% and \(\delta_L=1\) nm.

Fig. 6.  Statistical error analysis: Calculation process with setting from Fig. 5.

Fig. 7. Statistical error analysis with settings from Fig. 5: Result.

In the Thicknesses sub-tab, you can specify absolute (Abs. RMS) or/and relative (Rel. RMS) level of errors in all layers separately. 

Fig. 8.  Example. For a beamsplitter BS_7, relative errors of 1% in H-layers and relative errors of 0.5% in L-layers are specified.

In the course of the calculations, instead of layer thicknesses \(d_1,…,d_m\), disturbed thicknesses will be used. In the case of a two-material coating:

\[ d_1^{(j)}=d_1+\delta_{H,1}^{(j)},…,\;d_m^{(j)}=d_m+\delta_{L,m}^{(j)}\]in the case of absolute errors and

\[ d_1^{(j)}=d_1+\Delta_{H,1}^{(j)}\cdot d_1,…,d_m^{(j)}=d_m+\Delta_{L,m}^{(j)}\cdot d_m,\]

where \(\{\delta_{i,H}^{(j)}\}\), \(\{\delta_{i,L}^{(j)}\}\) [nm] and \(\{\Delta_{i,H}^{(j)}\}\), \(\{\Delta_{i,L}^{(j)}\}\),  [%] are absolute and relative errors in layer thicknesses, respectively. The errors are normally distributed numbers with zero average and rms deviations specified in corresponding columns.

Using this sub-tab is recommended for sophisticated error analysis.

Fig. 9.  Statistical error analysis: Calculation process with setting from Fig. 8.

 Fig. 10.  Statistical error analysis with settings from Fig. 8: Result.

Fig. 11.  Example. For a beamsplitter BS_7, a systematic offset of 0.04 in high-index layers and a systematic offset of 0.005 in low-index layers are specified.

In Refractive index tab, you can specify absolute (RMS) or/and relative (Rel. RMS(%)) offsets in optical constants of layers (refractive indices and extinction coefficients), Fig. 11.

Important: if the Per Material Errors box is checked, then equal offsets for all layers of the same material are specified.

In the course of the calculations, instead of nominal refractive indices \(n_H, n_L\), disturbed refractive indices are used:

\[ n_{H,i}^{(j)}=n_H+\Sigma_{H}, \; n_{L,i}^{(j)}=n_L+\Sigma_{L}\]in the case of absolute errors and

\[ n_{H,i}^{(j)}=n_H\cdot(1+\Sigma_{H}), \; n_{L,i}^{(j)}=n_L\cdot (1+\Sigma_{L})\]

where \(\Sigma_H, \Sigma_L\) are systematic offsets refractive indices of high- and low-index materials, respectively. The errors are normally distributed numbers with zero average and specified rms deviations \(\Sigma_H, \Sigma_L\).

The option allows you to specify errors in substrate refractive indices and refractive index of the incident medium.

Fig. 12.  Statistical error analysis: Calculation process with setting from Fig. 11.

Fig. 13. Statistical error analysis with settings from Fig. 11: expected and averaged (Exp) spectral characteristics, two curves, Exp-D and Exp+D, indicate the probability corridor for a given error level. The width of probability corridor corresponds to the level of the errors that were set in Error Analysis Setup window (Fig. 1).

If the Per Material Errors box is unchecked, then different errors for all layers of the same material are specified. From the practical point of view, it can be addressed to instability from layer to layer of optical constants in the course of the deposition.

In the course of the calculations, instead of nominal refractive indices \(n_H, n_L\), disturbed refractive indices are used:

\[ n_{H,i}^{(j)}=n_H+\Sigma_{i,H}^{(j)}, \; n_{L,i}^{(j)}=n_L+\Sigma_{i,L}^{(j)}\]in the case of absolute errors and

\[ n_{H,i}^{(j)}=n_H\cdot(1+\Sigma_{i,H}^{(j)}), \; n_{L,i}^{(j)}=n_L\cdot (1+\Sigma_{i,L}^{(j)})\]

where \(\{\Sigma_{i,H}^{(j)}\}\) and \(\{\Sigma_{i,L}^{(j)}\}\) are random offsets in layer refractive indices, respectively. The errors are normally distributed numbers with zero average and specified rms deviations \(\Sigma_H\) and \(\Sigma_L\).

Fig. 14.  Example. For a beamsplitter BS_7, a random offset of 0.04 in high-index layers and a random offset of 0.0.005 in low-index layers are specified.

Fig. 15.  Statistical error analysis: Calculation process with setting from Fig. 14.

Fig. 16. Statistical error analysis with settings from Fig. 14: expected and averaged (Exp) spectral characteristics, two curves, Exp-D and Exp+D, indicate the probability corridor for a given error level. The width of probability corridor corresponds to the level of the errors that were set in Error Analysis Setup window (Fig. 1).

Fig. 17. In the process of Error Analysis, a corridor is displayed that corresponds to the deviations of the spectral characteristics from their mathematical expectations. The width of the corridor depends on the probability for the selected characteristic value to fall within such a corridor. This probability can be set in the edit box Corridor Probability. By default, a reasonable value corresponding to one standard deviation is assigned to corridor. probability.

Fig. 18. Statistical error analysis with a standard probability corridor of one sigma specified in Fig. 17.

Fig. 19. Statistical error analysis with the probability corridor of two-sigma specified in Fig. 20. Obviously, expected disturbed spectral characteristics will exhibit larger deviations from the theoretical ones.

Fig. 20. The corridor of 95% corresponds to two-sigma.

Fig. 21. The corridor of 99.7% corresponds to three-sigma.

Pause before computing summary and Pause after each plot options allow you to slow down the Error Analysis procedure in order to save intermediate results in graphics form.

Fig. 22. Expected disturbed spectral characteristics with three-sigma settings will exhibit even larger deviations from the theoretical ones.

Fig. 23. Statistical error analysis in EFI (electric field intensity). Using Wavelength slider you can vary the current wavelength.

OptiLayer provides the statistical error analysis even of electric field intensity distribution. For this purpose, a check box EFI Error Analysis is to be checked in Error Analysis Setup window (Fig. 1).

The calculations are performed for the angle of incidence and polarization state specified on the bottom panel of EFI Error Analysis window. By changing the angle of incidence/polarization, calculations are performed on-the-fly.