Technical Issues - Double Dispersion

We apply the term Double Dispersion to materials wherein n,k varies with thickness AND wavelength. Experts have long advised that thickness-dependent dispersion be ignored in thin film design. Consider the following statement in a comprehensive review of coatings containing metal-dielectric composite films.

"But the price for introducing these effective optical constants is their thickness dependence. The thickness of a metal island film must not be varied during a design procedure. A metal island film with given thickness and given effective optical constants has to be tackled as a fixed building block which can be introduced into an interference stack, but should not be modified during synthesis and refinement." O. Stenzel and A. Macleod, Adv. Opt. Techn., 1(6), 463-481 (2012).

This makes sense for design software limited to wavelength dispersion. Double Dispersion, based on bilinear interpolation, removes this restriction, offering new possibilities for utilizing thickness-dependent materials. See "Designing with very thin optical films", R. R. Willey, A. Valavičius, F. T. Goldstein, Appl. Opt. 59, A213-A218 (2020). FilmStar DESIGN's Index Formulator gives smooth transition between n,k tables. In cases where the Formulator is insufficient, indices can be computed in Excel, the FilmStar Workbook or FilmStar BASIC.  Click here for our 2021 SVC PowerPoint presentation Specifying n&k in Optical Thin Film Calculations (also available on YouTube).

The success of our approach depends on the ability to characterize n,k for a range of thicknesses. Our model assumes that n,k values at 45 nm are half way between n,k at 40 and 50 nm. Assuming interpolation is valid, "must not be varied" no longer applies. For those curious about the physics of such materials, please contact Ron Willey. According to Willey, Double Dispersion will be included in his Training Course. Click here to learn about importing n,k data and dispersion functions from other sources.

CAUTION/LIMITATION Double Dispersion assumes that films grow homogeneously. Unfortunately, a film with inhomogeneous layers cannot be simply replaced by a homogeneous single layer film. It is also assumed that the interpolated dispersion curve for t=35 nm lies halfway between t=30 nm and t=40 nm curves. That is, nothing 'strange' happens at interpolated wavelengths. Interpolation accuracy requires close t spacing.

Since the Index Formulator includes ANG/IFP/IFS variables, Triple Dispersion is also possible.

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1. Index as a Function of Thickness

This topic discusses advanced features which might
be unfamiliar: Stack Mode and Index Formulator.
Please contact FTG Software for assistance.
 

DLS, L-M and Simplex are supported in Stack Mode. If optimizing with User-Defined Index Functions containing THK variables, please update to DESIGN 2.61.4417 (05 Aug 2019) or newer.

 

Coating engineers generally design with dielectrics, for which it is assumed that dispersion depends only on wavelength. But what about materials where dispersion changes with layer thickness, e.g. metal films thinner than the percolation threshold? A general solution, independent of theoretical approximation, is now available through variable THK in User-Defined Index Functions (Formula tab). Consider Function LTHICK where L=1.3 for layers <75 nm and 1.46 for layers >=75 nm and Function HTHICK where H=2.1 for layers <75 nm and 2.3 for layers >=75 (not realistic materials, just a simple example to understand and verify Formulas).

 

HTHICK Dispersion Function
Why minus signs? The logic operator () returns -1 if True and 0 if False.

This gives us a coating with four layers and two film materials where n,k varies with thickness. We compare and verify with a four layer coating as follows:

4 Layers with 2 THK-Variable Materials: 23.91H 37.67L 119.57H 94.18L 
  where L = -1.3*(THK<75) - 1.46*(THK>=75)   {LTHICK}
    and H = -2.1*(THK<75) - 2.3*(THK>=75)    {HTHICK)

4 Layers with 4 Fixed Materials: 23.91A 37.67B 119.57H 94.18L 
where A = 2.1, B = 1.3, H = 2.3, L = 1.46 

Two Material Stack Mode Design using LTHICK and HTHICK
Stack Mode Required

Equivalent Four Material Design

-(THK<75)*N1-(THK>=75 & THK<150)*N2-(THK>=150)*N3

Examples may be downloaded here. Index file TiO2_Mod.itw was created by manipulating TiO2_Orig in Excel, wherein n was multiplied by 0.8 and k by 1.2.

Thanks to Ron Willey for suggesting these modifications. Please contact Ron for further information about his work in this area. Note also that the Groups Mode editor includes Inhomogeneous Layer Replacement which applies to materials like ZrO2.

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2. The Index Formulator

As three index tables may be insufficient to describe metal films, the number was increased to nine in DESIGN 2.61.4400 and INDEX 2.51.0600 (NOTE below). A simple approach is to utilize multiple tables where each is valid over a given thickness range. Unfortunately that gives discontinuities wreaking havoc in refinement. It seems reasonable to interpolate n,k for intermediate thicknesses. That requires formulas beyond the ability to edit and verify in the single line editor shown below.

 

 

We therefore added the Index Formulator. Our four Cr index tables correspond to 15 nm, 30 nm, 90 nm and (Palik) 600 nm. We assign (N1,K1) for t<=15 nm, mix (N1,K1) and (N2,K2) for t between 15 and 30 nm, mix (N2,K2) and (N3,K3) for t between 30 and 90 nm, etc. Index tables were deduced by Ron Willey from data supplied by Ketan Patel (Luma Optics Mumbai). Unfortunately there were no measurements for films > 90 nm so are making do with Palik's value.

 

 

Users can verify calculations in INDEX. As shown in the plot below, interpolation eliminates jumps (blue trace) resulting from small thickness changes. The success of these algorithms depends on one's ability to determine n,k for a range of film thicknesses. It also assumes, for example, that n,k at 45 nm is halfway between n,k at 30 nm and n,k at 60 nm. Results from ellipsometry (high angle) are useful, but may not accurately predict %R/%T at normal incidence. Users can verify index values in DESIGN as described here.

Click here to download files. Copy *.itw to C:\Winfilm\Index, *.itf to C:\Winfilm\Config, and *.faw to C:\Winfilm\Designs. Note: *.itf files are User-Index Function Collections. File..Open in the User-Index Function dialog opens a Collection. There should be two Index Functions: CHROME4 and CHROME3. CHROME4 interpolates n,k values; CHROME3 does not, illustrating why interpolation is critical.

The plot below indicates interesting possibilities for metal-dielectric designs. In this case there are four Ag files, corresponding to n,k deduced at 2, 5, 10, 20 nm. A tolerancing plot is displayed at the bottom of this page. Attempts to design AR coatings by fixing Ag layers ("must not be varied") and varying only SiO2 layers have so far been fruitless. Of course, the proof is in the pudding, and we look forward to hearing from users who have fabricated such designs. How well do they correspond to theory? Are results repeatable? It seems reasonable that single Ag layers for n,k determination be deposited onto SiO2 substrates coated with an SiO2 film. The first design layer would then be SiO2 so that a metal is never deposited on a bare substrate.

The Index Formulator does not require THK variables or Stack Mode. It is especially useful for mixed materials and alloys, as in this example: Al(x)Ga(1-x)As.

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3. Tolerancing

The following BASIC program generates tolerancing plots in FSPlot.

Option Explicit
DefInt I-N
DefSng A-H, O-Z
Option Base 1

Sub Main
    Dim i, Iter, j, nRows
    PlotClose
    PlotActivate       ' Open FSPlot Module
    Const nIter = 100  ' # of iterations
    nRows = StackRows()
    If nRows = 0 Then
        MsgBox "Stack Mode required", vbCritical, "Tolerancing"
        End ' quit
    End If
    ReDim tLay0(nRows - 2)
    For i = 3 To nRows
        tLay0(i - 2) = Val(StackIndex(i, "Thick"))
    Next i
    AxesDraw
    For Iter = 1 To nIter
        For i = 3 To nRows
            Select Case StackIndex(i)
            Case "SIO2"
                StackIndexSet i, "Thick", tLay0(i-2) * (1+.02*RndNorm)  ' 2% SD
            Case Else
                StackIndexSet i, "Thick", tLay0(i-2) + 0.2*RndNorm  ' 0.2 nm SD
            End Select
        Next i
        Calculate
        PlotNext
    Next Iter
    For i= 3 To nRows ' restore original design
        StackIndexSet i, "Thick", tLay0(i - 2)
    Next i
End Sub

To gain insight we implement Monte-Carlo tolerancing, varying all layers and then only Ag and only SiO2. The following was generated with the 9 layer Ag-SiO2 with SiO2 SD = 2% and Ag SD = 0.2 nm.

In the following we varied only Ag with SD = 0.2 nm (2 Ångstroms). This was accomplished by commenting-out the "SIO2" StackIndexSet command in the above FilmStar BASIC code.

In the following we varied only SiO2 with SD = 2%.

The sensitivity to Ag thickness variation suggests that the ability to optimize metal layers during optimization is crucial for achieving AR design goals. FilmStar users needing help with the above are urged to contact FTG Software to arrange a ZOOM meeting.

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