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Optical Coatings

 TOC

Introduction

Analysis

Description of the Plots

Results Optimised for 450 to 650 nM

Results Optimised for 400 to 800 nM

Comparing Standard and Wideband Optimisations

Conclusions

Appendix – Design Information

References

Introduction

Optical coatings are used both on camera lenses and good quality filters to reduce reflection at the air-glass interface that can cause ghosting and flare.

Anti-Reflection (AR) coatings as they are known, operate using wave cancellation and refractive index transition reduction techniques in combination. There is a simple explanation of operation, manufacturing and fabrication of AR coatings in [3]. 

A major obstruction to practical implementation of AR filters is the limited values of refractive index available to designers [1]. Multicoated designs for high end commercial optical equipment use alternating high and low refractive index materials partly for this reason.

The purpose of this short note is to investigate the constraints on performance of this technology to aid understanding and pure curiosity.

Analysis

Smith [1] provides an analytical method which is fairly straight forward just requiring the use of complex arithmetic. For simplicity, loss and dispersion are neglected in this note, although the method can deal with them. NB that reflectance behaves differently for linearly polarised light, the procedure for unpolarised light is to analyse both polarisations and average. All the results here are for unpolarised light. Further the analysis is for a single surface, there will of course be two surfaces in a filter for example.

However, Smith only provides design equations for the trivial single coating case. Fortunately Sumita [2] provides design equations and examples for 3 and 4 layer solutions; the author has adapted this method with moderate success to an 8 layer design.

A standard part of the design method includes an optimisation process. Sumita [2] specifies minimising the root mean square (RMS) reflectance in the wavelength band 450 to 650 nM (the full optical band is normally considered to be 400 to 700 nM (violet to red). Sumita does not specify the algorithm to use but here a simple random optimisation was used to refine the design from the initial design. I also look at widening the optimisation bandwidth to 400 to 800 nM to try to improve performance when the incidence angle is high or acute (zero degree incidence is 90 degrees to the glass surface).

Nontechnical readers may find the plots of estimated reflection colour of interest. This is calculated for the coating being illuminated with a colour temperature of 6500K (daylight in shade), the human perceptual response is calculated followed by mapping to the sRGB colour space and applying a gamma of 2.2 typical for the sRGB colour space. Out-of-Gamult only occurred in a few places, the strategy for this was to set the largest channel to the maximum value and clip any channels that remained negative to preserve as much as possible the luminosity scaling. This only occurred at large incidence angles for the wide band 8 coat design. [4] [5] [6].

Description of the Plots

Reflection Spectrum at Zero Incidence

The reflection wavelength response is plotted over the band 300 to 800 nM for the light source arriving at 90 degree to the coated surface. This is where the coating performance is normally at its best. Filter manufactures will quote performance at zero incidence. Several coatings are presented on each plot

Reflection Spectrum Over Incidence

Each plot shows the reflection wavelength response over the band 300 to 800 nM for the light source arriving at 90 degree to the coated surface (zero degrees incidence).  A single coating is displayed per plot, with each trace representing the reflection at increasing angles of incidence. Each higher trace represents in increase of the angle of incidence.

Reflection Spectrum Statistic Over Incidence

Additional plots also represent reflection against incidence information plotted against the incidence angle in air (the medium) expressed at the a lens focal length on an APS-C format required for the corner angle of view from the optical axis to give the incidence angle. Data is shown for maximum and RMS within the wavelength band 400-700 nM. Detail plots are provided just covering the ultra-wide to normal lens area where filter performance is most compromised.

Reflection Colour Plots

For each coating the maximum intensity colour and a relative reflection intensity colour (the illuminating light would be pure white) is plotted in the lower and upper strip of the plots respectively. They are plotted against incidence angle in degrees in the air (the medium) on the horizontal axis. So the lower strip provides the colour at maximum luminosity and the upper strip the luminosity correctly scaled for the reflectance magnitude.

Results Optimised for 450 to 650 nM

Reflectance at Zero Incidence

Reflectance at Various Incidences

Reflection Spectrum Statistic Over Incidence

Reflection Colour Plots

Results Optimised for 400 to 800 nM

Reflectance at Zero Incidence

Reflectance at Various Incidences

The uncoated and single coat do not change of course.

Reflection Spectrum Statistic Over Incidence

Reflection Colour Plots

Comparing Standard and Wideband Optimisations

Conclusions  

General

The standard bandwidth 8 layer design is a bit of a disappointment, it seems the optimizer got stuck on a local minima here and the performance is worse than the 4 layer design. 

However the wideband 8 layer design is better than the 4 layer and offers better bandwidth at the red end of the spectrum.

We can see that a 3 or more layer design gives a large improvement over a single layer solution which is only a little better that uncoated glass.

As the angle of incidence increase away from a ray arriving normal to the plane of the glass the reflection increases for all cases. For the coated cases the increase in concentrated at the red end of the spectrum with only a small change at the blue end. The response zeros (spectral minima) also move towards the blue end of the spectrum with increasing incidence. 

This phenomena is the motivation for the wideband solutions where the optomisation bandwidth has been increased to 800nM at the red end of the spectrum and bought in line with the 400nM visible band limit at the blue end.

Standard vs Wideband

Comparing now the standard and wideband optomisations, we can conclude the following:

a) 3 Coat: No Real Effect, this just moved the characteristic upwards so it was centred over the new band.

b) 4 Coat: A small improvement at the red end for a small degradation in the rest of the spectrum, not much use.

c) 8 Coat: A very significant improvement in bandwidth, however the standard design was probably sub-optimal.

This does show however that to improve bandwidth significantly more than four layers is needed.

Also that additional layers do not necessarily improve RMS reflectivity but may permit a wider band solution. This may explain why the 8 layer Heliopan filter coatings have a similar reflectance to the 5 layer B+W and Hoya designs.

Performace

Does the improved bandwidth 8 layer design permit better reflectance performance at more acute angles of incidence?

As the standard band 8 layer design is sub-optimal it is best to compare the statistics over the visible band for the 3, 4 and 8 layer wide band cases. Here the angle of incidence is expressed as the half angle of view in the corner of a given focal length lens on the APS-C format.

We can see that from 100mm upwards the curves are essentially constant with the following RMS reflectance, single air/glass interface don't forget, also keep in mind there are no manufacturing tolerance which can be expected to degrade things considerably as controlling the film thickness in mass production is not simple.

So there is a small difference of about 1 stop between 3 layer and 8 layer in this table which is in broad agreement with commercially available products.

However below 100mm focal length the 3, 4 and 8 layer designs converge rapidly meeting at a corner angle for a 24mm lens on APS-C (~38mm on 35mm or full frame). So we can see that the performance of the more expensive coatings degrades to a basic coating for the corners of a lens slightly shorter than a standard focal length.

What is worse by 14mm on APS-C the corner reflection is no better than a single coat design although all the coated solutions are better than uncoated glass even for an ultra-wide angle lens.

This the conclusion is that even lots of layers will not help that much in the corner of wide angle lenses. This applies just as much to lenses as to the filters that may be put on them. However it will improve performance closer to the optical axis.

UV Rejection

A few words about UV rejecting coatings. No attempt was made to optomise these designs for rejection of UV below 400nM. However it should be noted that the internal transmittance of most glasses drops rapidly to around zero by about 300nM [1] so rejection due to the coating is only useful between 300-400nM.

Reflection Colour

Looking now at the reflected colour plots we can see that the light level reflected is so low in most cases the intensity scaled bar is essentially black. The exception is the obviously gray uncoated and possibly the single coat which might not be dead black.

In practice, looking at the coatings under normal illumination we would probably only be able to discern the reflected light colour at acute incidence angles above 45 degrees where the reflectance has degraded significantly. At lower incidence angles the colour may give us an idea of what ghosting colours may be seen for very bright white light sources via such a lens or filter. However such a ghosting image is liekly to involve reflections from more than one surface which may have different coatings, so in practice the ghosted colour is probably a combination of both reflectance characteristics.

We can see that the reflection colour of a single coat filter is fairly consistently yellow allow can be a little violet at small incidence angles if noticed. 

The 3 coat is fairly consistently blue-green.

The 4 coat tends to a magenta colour turning yellow at high incidence.

For the 8 layer the colour is more dependent on the design bandwidth as it was possible to influence this significantly. The standard design being blue and magenta at acute incidence angles. The wideband design conversely is magenta to yellow but almost pure white at acute incidence angles.

This suggests that reflectance colour is not a good guide to the number of layers or quality of a optical coating, contrary to popular belief.

What do Canon do?

Canon say a little about their coating method in their Lens Work book [7]. The also provide this reflectance plot of their super-spectra coating:

If you plot the 3 coat designs in this note on a linear reflectance scale we can see this looks fairly similar.

Appendix – Design Information

Uncoated

Single

Optimised 3 layers

Optimised WB 3 layer

Own Optimised 4 layer A

Own Optimised WB 4 layer A

Own Optimised 8 layer D

Own Optimised WB 8 layer D

References

[1] Smith, W J, "Modern Optical Engineering", 3rd ed 2000, McGraw-Hill

[2] US Patent 3781090 “Four Layer Anti-Reflection Coating”, Sumita, Haruki (Minolta Camera) Nov 6 1972

[3] Anti-reflective coating, Wikipedia, http://en.wikipedia.org/wiki/Anti-reflective_coating

[4] CIE colour matching function ref CIE 1931 http://cvrl.ioo.ucl.ac.uk/cmfs.htm

[5] Colour Rendering of Spectra http://www.fourmilab.ch/documents/specrend/

[6] RGB Working Space and Conversion Information http://www.brucelindbloom.com/

[7] EF Lens Work 8th ed

Last Updated 27/06/2008

All Content © 2005-09 Lester Wareham All Rights Reserved     

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