Initial and refined reflectance spectra for a four-layer broadband antireflection coating

The first two tutorials used thicknesses calculated outside TFStudio and then checked the result. This time TFStudio will calculate the thicknesses.

The task is to reduce the front-surface reflectance of BK7 over 450–650 nm. We will start with a reasonable but poor four-layer coating, define what the optimizer should minimize, and run a thickness refinement. The number and order of the layers will stay fixed.

The design conditions

Use the following conditions throughout the example:

  • Incident medium: Air
  • Substrate: BK7 (Schott)
  • Coated surface: Front
  • Evaluation: Ignore other side
  • Angle of incidence: 0°
  • Polarization: Average
  • Reference wavelength: 550 nm
  • Optimization band: 450–650 nm

This is a calculation for one coated surface, not a complete glass plate with two reflecting surfaces.

Use these entries from the built-in material catalog. Their values at 550 nm are:

Catalog entry nn kk
MgF2 1.378506 0
HfO2 2.011238 3.01×10−63.01\times10^{-6}
SiO2 (Fused Silica) 1.459911 0
BK7 (Schott) 1.518522 8.76×10−98.76\times10^{-9}

The material names matter because another catalog entry can use different dispersion or extinction data.

Build the starting coating

The physical order, read from the incoming air toward the substrate, is

Air∣MgF2 100 nm∣HfO2 70 nm∣SiO2 50 nm∣HfO2 35 nm∣BK7.\text{Air}|\mathrm{MgF_2}\ 100\ \text{nm} |\mathrm{HfO_2}\ 70\ \text{nm} |\mathrm{SiO_2}\ 50\ \text{nm} |\mathrm{HfO_2}\ 35\ \text{nm} |\text{BK7}.

The 100 nm MgF₂ and 70 nm HfO₂ layers are close to quarter-wave thicknesses at 550 nm. The two thinner layers give the refiner two more phase adjustments. It is only a starting point; it is not expected to meet the broadband target.

The Design Editor lists the substrate-adjacent layer first. Enter the same stack in this order:

Design Editor layer Material Starting thickness
1 HfO2 35 nm
2 SiO2 (Fused Silica) 50 nm
3 HfO2 70 nm
4 MgF2 100 nm

Leave all four thicknesses unlocked.

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Before refinement, the average reflectance over 450–650 nm is 5.148%. The worst point is 13.66% at 450 nm, so the starting stack is not a broadband AR coating yet.

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To calculate values mentioned above you can add a monitor in the bottom left corner of the program:

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After clicking 'Add' you will see the average value of R over specified 450-650 nm range.

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Define the merit function

A single-wavelength target such as R(550 nm)=0R(550\ \mathrm{nm})=0 would allow the optimizer to make another narrow V-coat. We instead need every sampled wavelength in the band to affect the result.

Open Merit Function Editor

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and use the filter-type wizard:

  1. Choose AR → Broadband AR.
  2. Set the wavelength range to 450–650 nm.
  3. Choose a continuous target.
  4. Set the angle to 0° and polarization to Average.
  5. Generate the operands.

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The wizard creates two equal-weight optical operands:

Operand Band Target Weight
RGT 450–650 nm R=0R=0 1
TGT 450–650 nm T=1T=1 1

RGT is the root-mean-square deviation of reflectance from zero over the band. TGT does the same for transmission relative to one. Using both matters when layer's material contains extinction coefficients: a coating should not satisfy an AR target by absorbing the light instead of transmitting it.

The optical merit function is

OMF=εR2+εT22,\mathrm{OMF} = \sqrt{\frac{\varepsilon_R^2+\varepsilon_T^2}{2}},

where εR\varepsilon_R and εT\varepsilon_T are the two band RMS errors. OMF is a dimensionless error, not the average reflectance.

Add these thickness constraints in the same editor:

Operand Layers Target
MNT 1-4 30 nm minimum
MXT 1-4 250 nm maximum

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The limits keep the local refiner away from vanishingly thin or unnecessarily thick layers. Because the starting stack satisfies both limits, its total merit function (MF) value and optical-only merit function value (OMF) are equal: 0.057881.

You'll notice on Optical Evaluation plot that region that is affected by set optical targets is now highlighted:

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Run the refinement

Open Refinement

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and use:

  • Method: Sequential QP
  • Maximum iterations: 200

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Click Run. In this case it converges after 35 iterations and the resulting MF is 0.001014.

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The resulting Design Editor table is:

Layer Material Start Refined
1 HfO2 35.00 nm 30.97 nm
2 SiO2 (Fused Silica) 50.00 nm 30.79 nm
3 HfO2 70.00 nm 51.54 nm
4 MgF2 100.00 nm 105.99 nm

The total coating thickness is 219.28 nm, and the thinnest layer is 30.79 nm.

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Compare the spectra

The plot uses the same wavelength range and vertical scale for both designs.

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The refined curve sits close to the bottom of the plot, so the table below gives its values more clearly.

Wavelength Starting RR Refined RR Refined TT
450 nm 13.6634% 0.2858% 99.7138%
500 nm 5.7872% 0.0791% 99.9204%
550 nm 3.8629% 0.0995% 99.8999%
600 nm 3.4452% 0.0084% 99.9909%
650 nm 3.0635% 0.2546% 99.7445%

Checked on a 1 nm grid from 450 to 650 nm:

Metric Starting design Refined design
Average reflectance 5.148% 0.08%
Maximum reflectance 13.663% 0.286%
MF / OMF 0.057881 0.001014
Total thickness 255.00 nm 219.28 nm

What refinement did and did not do

Refinement moved four existing interfaces. It did not choose the materials, change their order, or add another layer. A different starting stack can settle into a different local minimum, so this result is not proof of a globally best four-layer design.

The result also applies only to the stated material data, one coated surface, normal incidence, and the 450–650 nm band. It has not yet been checked for oblique incidence, polarization splitting, or thickness errors. Those are
separate analysis and tolerance problems.