**Computational Assessment of Filter Stacks**

**Abstract**

Two important attributes for a filter or stack are (1) how much UV light is transmitted, and (2) how well the filter(s) prevent visible and IR contamination. Figures of merit to describe these attributes were developed and then calculated based on online spectroscopic data from Schott and other manufacturers. Mean transmittance in the UV was chosen for the first attribute, and Safety, defined as the log of the ratio of how much UV is transmitted to how much visible and IR is transmitted, was chosen for a measure of IR blocking ability. Then the results are plotted against each other to give some idea of how the stacks behave in comparison to each other. The conclusion seems to be that S8612 and UG11 produced the best stacks of those computed. The actual behavior of a filter will depend also on the camera sensor, so it is impossible to declare an absolute winner.

**Definitions**

I decided that "UV" meant 310nm to 400nm, and the rest of the spectrum went from 400nm to 1100nm where silicon sensor cameras stop recording. Provided one has high quality data for transmittance across the whole range from 310nm to 1100nm, then it is possible to compute some figures of merit for these attributes. And there's the rub. For most proprietary filters, including the Baader, one can only find rough numbers for the transmittances, usually in the form of linear-scaled graphs which essentially hide any IR leaks. Even when numbers are available, often they don't cover the whole range. Therefore the following discussion will be limited to situations where one can compute the values, and even then, published spectra can be inaccurate, especially for the non-Schott glasses (allegedly).

The two metrics chosen were mean transmittance in the UV, and “Safety”, defined here as the log of the ratio of area under the UV to the area under the rest of the transmittance curve. More Safety means one is less likely to have one’s image contaminated with IR or visible, for a given exposure and camera setup.

A stack that transmits only UV light and has perfect blocking of IR and visible would have infinite Safety, while one that transmits equal amounts of IR and UV would have Safety=0 by this definition, and one that transmits more visible and IR than UV would have negative Safety.

**Example and Results**

The exact calculation is shown in this example figure. Note that although I have plotted the curves on a diabatic graph, the calculation is done with the original transmittances, not their logs or any transformed version.

One can then plot all the stacks on a chart based on the two attributes. The boundary of the pink areas define the median values of the stacks based on each attribute. The median Safety was 3.42, and the median of the average transmittances was 0.27. The best stacks, in the sense of being the best compromise, are in the white area, and the worst are in the dark pink area. The "winner" seems to be something like 1.75mm S8612/ 1mm UG11 of the stacks examined, but clearly other considerations may alter that conclusion. The sensitivity of the camera has not been included in this calculation, so if (for example) one owns a camera with a sensor that is highly IR sensitive relative to UV, then selecting a filter with a higher Safety might lead to better photos in principle. Likewise, a camera with better UV sensitivity would benefit from less Safety and higher mean transmittance.

Finally, here are the actual spectra, again on a diabatic plot so that the IR leaks can be seen.

**Technical Details**

Now to the details of how the transmittances are calculated.

Curves were fitted using cubic splines to the published spectra of each glass, after converting the raw numbers into absorption coefficient form (absorption coefficient is the fraction of the light removed per unit length of filter thickness). Typically Schott publishes these values as the internal transmittance — transmittance without reflection losses at the surface — through a 1mm thickness of glass. Other manufacturers use 2mm as their reference. One can find the absorption coefficients by working backwards from this with the Beer-Lambert law. I also fitted a line to the refractive indexes given by the manufacturer, if two values were available. If only one refractive index was given, that was assumed to apply at all wavelengths. Using this info, it was then possible to get the complex refractive index, N, for each glass as a function of wavelength.

Using the complex refractive indexes, reflection coefficients can be calculated for each wavelength. Then the Transfer Matrix method described by numerous sources, including (C. L. Mitsas and D. I. Siapkas, 1995), (C. Katsidis and D. I. Siapkas, 2002), and (E. Centurioni, 2005), was used to calculate the transmittance for each wavelength. The light was assumed to be incoherent since these stacks don’t have any thin coatings that need a semi-coherent calculation. It was also assumed that the light was orthogonal to the stack. The filters are treated as uncemented with an air gap between, so there will be partial reflections in the gap (which are included in the calculation).

To calculate the metrics for safety and average transmittance, cubic hermite polynomials were used to interpolate the resulting transmittances at each wavelength, and then the integrals were performed analytically on the hermite polynomials.

**References, Data Sources, and the Code**

C. L. Mitsas and D. I. Siapkas. Generalized matrix method for analysis of coherent and incoherent re- flectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates.

*Applied Optics*, 34(10), April 1995.

C. Katsidis and D. Siapkas. General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference.

*Applied optics*, 41(19):3978–3987, 2002.

E. Centurioni. Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers.

*Applied Optics*, 44(35):7532–7539, Dec. 2005.

Schott glass data (all URLs as of Nov 2, 2017)

http://www.us.schott...lass/index.html

QB21 (Newport Glass)

http://www.newportglass.com/opmcat.htm

U360 (Hoya)

http://www.hoyaoptics.com/pdf/U360.pdf

The actual code is here, and can be run by typing "compare_filterstacks" into the command window of MATLAB with this file in your current folder.

Code

Diabatic plotting routine (needed by the code)