Calculation of spectral characteristics

• Reflectance R, transmittance T, absorptance A for s- and p-polarized light
• Back side reflectance BR
• Absorption in separate layers
• Phase shifts on reflection and transmission for s- and p-polarizations
• Differential phase shifts on reflection and transmission
• Group delay and group delay dispersion

The spectral characteristics are calculated for the design currently loaded into the memory.

Spectral characteristics can be calculated and plotted in spectral mode as well as in angular mode.

The design cross section is represented by the design bar showing layers in different colors, the number of layers N and the overall thickness Th of the coating.

If a target is loaded, the corresponding merit function value MF is presented on the bottom of the plot.

Fig.1. Transmittance of a 43-layer bandpass filter. Total thickness of 2926.7 nm is shown on the bottom right corner.

The setting are available through:

Evaluation Plot –> right mouse click –> Plots

If you compose several spectral characteristics measured in different units in one plot (for example, reflectance and phase), the second right axis appears automatically.

Fig. 3. Settings of axis for spectral characteristics of an edge filter. In Type field, it is possible to choose between linear, logarithmic and diabatic scales. If Reset on Target Load check box is checked, axis will be recalculated automatically according to loaded target.

Fig. 4. Settings of axis for spectral characteristics of an edge filter using standard Chart Editor.

Fig. 5. Settings corresponding to a coating on a semi-infinite substrate (the structure is shown in Fig. 6).

If the angle of incidence (AOI) is \(\alpha\), then the light exits at the angle \(\beta\) connected with  \(\alpha\) through Snell’s law:

\[n_a \sin\alpha=n_s \sin\beta \]

Important: Back-side reflectance is calculated so that the light exits the coating at the same AOI \(\alpha\)  (Fig. 6).

Example: \(n_a=1, n_s=1.44, \alpha=45^{\circ}\). The light enters the back side at AOI \(\beta=29.4^{\circ}\).

If you need to calculate the back-side reflectance so that the light enters the back side of the substrate at AOI=\(\alpha\), you need to recalculate the corresponding AOI \(\alpha\) using Snell’s law.

Example: \(n_a=1, n_s=1.44, \beta=40^{\circ}\). Then AOI \(\alpha=67.7^{\circ}\).

Fig. 6. Schematic of a multilayer coating on a semi-infinite substrate.

Fig. 7. Schematic of a multilayer coating on a substrate.

If exit medium is the same as incident medium, the light exits the substrate at the same AOI \(\alpha\). The light enters the back-side of the substrate at the same AOI \(\alpha\).

Fig. 8. Settings corresponding to a coating on a substrate (the structure is shown in Fig. 7). Exit medium coincides with incident medium (Air).

Fig. 9. Settings corresponding to a coating on a substrate (the structure is shown in Fig. 10). Exit medium (CaF₂) differs from the incident medium (Air).

If the exit medium differs from the incident medium, the light exits the substrate at  AOI \(\beta\). The light enters the back-side of the substrate at the AOI \(\beta\).

Fig. 10. Schematic of a multilayer coating on a substrate. Exit medium and incident medium are different.

If the angle of incidence (AOI) is \(\alpha\), then the light exits at the angle \(\beta\) connected with  \(\alpha\) through Snell’s law:

\[n_a \sin\alpha=n_{e} \sin\beta \]

Important: Back-side reflectance is calculated so that the light exits the coating at the same AOI \(\alpha\)  (Fig. 10).

Example: \(n_a=1, n_s=1.44, n_{e}=1.65, \alpha=45^{\circ}\). The light enters the back side at AOI \(\beta=25.4^{\circ}\).

If you need to calculate the back-side reflectance so that the light enters the back side of the substrate at AOI=\(\alpha\), you need to recalculate the corresponding AOI \(\alpha\) using Snell’s law.

Example: \(n_a=1, n_s=1.44,  n_{e}=1.65, \beta=30^{\circ}\). Then AOI \(\alpha=55.6^{\circ}\).

Fig. 11. Reflectance and Transmittance of a 28-layer edge filter at AOI 45 degrees. The substrate back side is not taken into account (see Figs. 5 and 6).

Fig. 12. Reflectance and Transmittance of a 28-layer edge filter at AOI 45 degrees. The substrate back side is taken into account (see Figs. 7 and 8).

(i) different spectral characteristics of a design;

(ii) in different spectral ranges;

(iii) with different types of axis;

Fig. 12. Multiple evaluation windows corresponding a 28-layer edge filter: R/T at AOI 45 degrees in the entire spectral range of interest (left panel); only transmittance in the high-transmittance range at AOI 44,45, and 46 degrees (middle panel); transmittance in low transmission range in diabatic scale.

Configuration –> Settings –> Units –> Phase limits.

In the case of degrees, you can choose between \([-180^{\circ};+180^{\circ}]\) and \([0^{\circ};360^{\circ}]\) limits.

In the case of radians, you can choose between \([-\pi;\pi]\) and \([0;2\pi]\) limits.

Also, you can calculate phase as continuous plots (Figs. 13, 14).

Fig. 13. Phase in degrees of a 26-layer narrow bandpass filter.

Fig. 14. Phase in degrees of a 26-layer narrow bandpass filter presented continuously.

Fig. 15. Differential phase shift on reflection of a two-layer Ag-Al₂O₃ coating.

\[d\varphi=\varphi^{s}-\varphi^{p} \]

OptiLayer allows you to calculate the phase difference on transmission \(\Delta\varphi t\). This value is calculated as a difference of phases at the substrate boundary between the wave propagated through a coating and the wave propagating the same distance in the incident medium replacing the coating (see schematic in Fig. 17):

\[ \Delta\varphi t=\varphi_A^{(1)}-\varphi_A^{(2)} \]

Fig. 16. Phase and phase difference on transmittance calculated for a 26-layer narrow band pass filter. Exit medium is air.

Fig. 17. Schematic of the phase difference calculation.

Optilayer performs calculations using analytical formulas!

If multi-coating mode is activated, OptiLayer evaluates GD and GDD of separate coatings (in a complementary pair or in a double-angle configuration) as well as of the corresponding coatings combination.

Fig. 18. Group delay dispersion of a 68-layer dispersive mirror.

Fig. 19. UDT spectral characteristic, that is difference between s- and p-polarized reflectance \(R^{(s)}-R^{(p)}\) of a two-layers Ag-Al₂O₃ coating.

Analysis –> More… –> UDT

The evaluation setting and the plot are available through:

Analysis –> More… –> Layer Absorptance

Fig. 20. Absorptance in the third layer (L3), that is an Ag-layer, of a coating reflecting different colors from its front and back sides.

In this case, spectral characteristics are calculated in an angular range for separate wavelengths (Fig. 21).

Fig. 21. Transmittance of a cold mirrors calculated in an angular range from 0 to 70 degrees for transit wavelengths 695 nm, 700 nm and 795 nm.

Fig. 22. Transfer to the angular mode in General configuration window.

Limits and units for \(\Delta\) angle can be specified in the same way as for phase, i.e. through General Configuration window.

Fig. 23. Ellipsometric angles \(\Psi\) and \(\Delta\) of a seven-layer metal-dielectric coating.