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Function

특수한 코팅 조건에 따라 Parameter를 입력 하여 설계 및 분석이 가능한

일종의 Macro(Script) 기능 입니다.

그림과 같이 여러 가지 Macro 기능(Script)을 만들어

놓고 필요에 따라 불러다 이용하면 됩니다.

※ 보여지는 내용은 Function 구매 시 기본으로 들어가

있는 Macro(Script)이며 추가 지원도 가능 합니다.

Macleod 64 Optical Coating Design

Figure 17-21. The final extracted results are placed in a new material document and a substrate document. These can be saved as required.

17.5 Optical Constants of a Metal The Envelope Method is exceptionally stable, even in the presence of measurement errors, but

the validity of the envelope approach depends on the absence of significant absorption. When the absorption is high, the fringes no longer remain within fixed envelopes as the film thickness varies. To handle absorbing films a different technique is required. A script in the Function Enhancement is intended to assist in the extraction of constants for a metal or similary highly absorbing film. The necessary measurements are normal incidence reflectance and transmittance, and, like the substrate tool, the transmittance must be finite.

The envelope method succeeds in decoupling the thickness of the film from the estimate of refractive index. Once the index is established, the other parameters follow and there is no need for a physical model of the film. With absorbing films, the fringes tend to disappear and then we have a minimum of three parameters, thickness, refractive index and extinction coefficient to fit at each wavelength to only two measurements. Thickness, of course, is invariant with wavelength and so it might be thought that an independent measure of thickness would remove any ambiguity and permit extraction of n and k over the entire wavelength range. Unfortunately, although T and R are unambiguously and precisely defined by the values of n, k and d, in the reverse calculation, n and k are frequently exceptionally sensitive to the particular value of d. Because the dimensions of a stylus used in mechanical measurements are orders of magnitude different from optical wavelengths, the d derived by mechanical measurements is not always a reliable indicator of the d that is appropriate for the optical properties. Then there are the problems of multiple solutions and inaccuracies in the measurements. Even with an exactly

Macleod Page 265 Optical Coating Design

correct value of d, the extraction process for n and k tends so to magnify both measurement noise and inaccuracy that the results of the extraction are useless.

It has been found that the only really useful way forward is to make use of a suitable model of the variation of n and k with wavelength. This reduces the number of adjustable parameters and acts to smooth the results reducing the effects of measurement noise. In the extraction process, the physical thickness, also, can and should be one of the adjustable parameters. Multiple solutions are not entirely eliminated and there is no easy way of verifying that the derived solution is correct. User vigilance and the use of measurements of different samples of the same material represent the principal ways of dealing with this. Each new sample presents a puzzle involving a considerable degree of trial and error, and of experience, on the part of the user.

The model used in the script is a combination of terms representing both metallic and dielectric properties. For the metallic properties we use the Drude model and for the dielectric properties, the Lorentzian. The formulae are taken from Dobrowolski, J A, F C Ho, and A Waldorf[7], and, as in the examples in the paper, we limit the number of Lorentz oscillators to two.

The expressions for n and k are as follows:

( )( )

( )( )

( )2 2 2 2 2 22

1 1 22 222 2 2 2 2 2 2 2 2 2

1 1 2 21

A B A BBn k AC B C B C

λ λ λ λλαλ λ λ λ λ

− −− = = + + +

+ − + − +

22 (17.2)

( ) ( ) ( )3 33

1 1 2 222 2 2 2 2 2 2 2 2

1 1 2 2

21

A C A CBCnkC B C B C

λλβλ λ λ λ λ

= = + ++ − + − +

2 2

λ (17.3)

from which

( )2 2

2

2n

α β α+ −= (17.4)

( )2 2

2

2k

α β α+ += (17.5)

This form ensures that we have positive real n and positive real k. As in the paper we do not assign any physical meaning to the parameters but rather use them simply as convenient variables that define a suitable model expression. Note that the script evaluates and displays the parameters A, B C, etc. for wavelengths in microns even though the input data for extraction must be in nanometres.

To these parameters we add the physical thickness, d, and an inhomogeneity factor.

17.5.1 Script Input Data The film should be deposited on one surface of the substrate. The second surface should be

left uncoated. Measurements of normal incidence transmittance and reflectance in percent as functions of wavelength in nanometres should be made and stored in separate tables. The reflectance measurement may include the second surface, when the results will be identified as parallel or not, when the results will be labeled as wedged. To these measurements an optional


reflectance in percent at a given angle of incidence and polarization may be added. The material of the substrate should exist in the current material database together with a substrate file should the substrate be absorbing. An estimate of film physical thickness in nanometres should be available. the more accurate this measurement, the better.

17.5.2 Script Operation The script is launched in the normal way with the appearance of an opening dialog. Cancel

stops operation of the script. OK continues operation.

Figure 17-22. The initial dialog with information about the script.

A file chooser appears next inviting selection of a project file. These files are simple text files with details of the necessary parameters for an extraction. The parameters include the necessary input data files, the materials database that was last used in the project and contains the necessary substrate properties, and the parameters of the refinement process that will ultimately produce the optical constants of the film. The file consists of a number of headers of the form: <Header text> followed in most case with one or two parameter values. The headers are largely self explanatory and they can be in any order. The file will be recognized as a project file by the existence of the header <Metal n and k>. The data should terminate with the header <End>. Anything after that is ignored by the script. A header <Debug> inserted on a separate line before any other header (not in between a header and its data otherwise an error will result) will trigger the debug operation. More details on that later. If any header is absent then the corresponding data will be set as the default.

Figure 17-23. Dialog warning that materials folder has changed from that in the project file.


Should the materials folder in the project file differ from the current database then a warning will be triggered. This warning is advisory only. As long as suitable substrate data is available the process will continue.

The dialog for parameter entry is shown next. The two uppermost fields are for entry of the table files containing the normal incidence transmittance and reflectance data. These two fields are required. Selection of the files will automatically complete the path information. The path field is not editable and the path must be the same for both files. The third field is for an optional table file containing reflectance data taken at a given angle of incidence. The polarization can be either p or s or mean. The path of this file must be the same as the other two table files. The angle of incidence is entered in the field below along with the appropriate polarization. The angle field is editable only if there is a chosen reflectance at angle table file. The Remove button clears this field.

Figure 17-24. The dialog for entry of parameters.

The substrate details are shown on the left and must be entered. If there is no absorption in the substrate then the substrate file can be entered as Lossless as is normal. The thickness should be in millimetres, again as is normal. The substrate can be set as Parallel or Wedged. These have their usual meanings. If the rear surface is not included in the reflectance measurement then Wedged should be used.

Various parameters are shown on the right hand side. The iterative technique used is simplex. Max Iterations is the total number of iterations. The process will terminate automatically when


this number is reached. This number should be set fairly high. 25000 as shown is a useful value and is the default. The process can be terminated manually at any time. Skip Interval is the interval between the data points that will actually take part in the fitting process. Note that at the end of the process the calculated results will be compared with the entire data set. Recycle Interval has exactly the same meaning as in the simplex refinement method for designs and stacks.

Approx Film Thickness is an estimate of the physical thickness, in nanometres, of the film in question. This estimate should be as good as possible. A poor estimate will increase the risk either of a false result, or an extraction process that terminates with an appreciable error value. Thickness Increment is the increment that will be used in setting up the simplex for refinement. The value 0 to 1.0 is advisory only. Inhomogeneity Factor is the same factor that appears in a design document or is used in Reverse Engineer. It is intended only for small values of inhomogeneity and becomes progressively less accurate in its effect if the value moves outside the range ±0.2. Inhomogeneity Increment is the increment that will be used in setting up the simplex. It should be kept small. The default value is 0.0001.

The optical constant parameters are set up in a separate dialog that is initiated by the Edit Constants button and shown next. The various parameters correspond to those shown in equations (17.2) and (17.3). The increments are those that will be used in setting up the simplex. They can be adjusted individually and it is difficult to give typical values that should be used because experience shows that the best values depend on the nature of the film. If any increment is set to zero then that parameter will remain constant during the extraction. This applies also to physical thickness and inhomogeneity factor. If all parameters are set to zero then an overflow condition will exist in the calculations. The script does not check for such conditions. The OK button accepts the entered values. The Cancel button restores the original values. Both buttons return to the main parameter dialog.

Figure 17-25. Starting values for the constants in the film model are set up here. the increment is the value that will be sued in the Simplex refinement.

A great deal of trial and error will usually be required before suitable starting parameters are found. The subsequent rate of convergence is a good guide.

The Save, Cancel and OK buttons all close the parameters dialog. Cancel stops operation of the script. Save saves the parameters in a project file. A file chooser appears where an existing file can be chosen or a new one created. OK simply starts the extraction process without saving the parameters.


Figure 17-26. Self explanatory warning dialogs.

The process involves the creation of a stack that is the vehicle for the calculation of film properties. This stack uses a single-layer design, the single layer being the film in question. The film material is given the name nkDummy and is created in the current materials database. The design is named nkDesign and likewise is created and saved in the current data folder. Should such a material or such a design exist already then a warning is issued that permits selection of Cancel to terminate operation of the script so that the existing material or design can be renamed if necessary. OK permits the relevant material or design to be overwritten. In normal termination script, the design is deleted, but the material not, so, normally, the warning can be safely ignored.

Figure 17-27. Progress of the refinement process will be displayed in this table.

The refinement phase begins with the presentation of a table where the progress will be displayed. A dialog carries an OK button that launches the refinement.


Figure 17-28. This dialog is shown at the same time as the initial table and the process is launched from here.

The refinement process should show a continually falling merit figure. A good fit will have final merit values less than unity. If the process seems to stall at rather larger figures it is best to interrupt the process and return to the parameter settings to try different values. Satisfactory convergence is quite fast.

Figure 17-29. The final values of the refinement process in the case used as an illustration.

Figure 17-30. After <Esc> is pressed the dialog on the left appears. No will permit the refinement to continue. Yes will terminate and will be followed by the dialog on the right. Keep current best asks if the results should be retained. No will discard all results of the refinement. Yes will retain them so that they can be examined.


Figure 17-31. A summary of the results of the iteration displayed when responding Yes to the Keep current best query.

Figure 17-32. The comparison of measured and results calculated using the extracted parameters. Usually it will become clear to the user when the process should be terminated, either because a satisfactory conclusion has been reached, or because the process is stalling at a too large merit value. Termination can be achieved at any time by pressing the <Esc> key. The process used as an illustration was terminated in this way with the results shown in Figure 17-29. After <Esc> is pressed the dialogs in Figure 17-30 appear in turn. Terminate script should be answered with Yes if termination is what was intended. If refinement should be continued and the <Esc> action canceled, then No should be selected. Keep current best follows, asking if the results should be retained. No will discard all results of the refinement. Yes will retain them so that they can be examined. After a Yes answer, a summary of the current best set of parameters is displayed, Figure 17-31, followed by a plot, Figure 17-32, comparing the measurements (the complete set, not simply those used in the refinement) with calculations using the best set of parameters, and, simultaneously a dialog, Save Parameters, Figure 17-33, asking if the best parameters should be saved in the project file. This permits the project to be restarted at the current termination point and gives the opportunity of some parameter adjustment.


Figure 17-33. The Save Parameters dialog. Yes will save the values in the project file.

Figure 17-34. The new material document containing the generated n and k values.

Finally the n and k values are calculated from the best parameters and are placed in a new material document, Figure 17-34. This can be saved and also plotted in the usual way, Figure 17-35.

Note especially that a good correspondence between the calculated and measured results does not necessarily guarantee that the extracted parameters are valid. Multiple solutions are a problem and also there is often considerable sensitivity to errors of measurement. It is always a good practice to prepare and measure several samples of different thicknesses and to compare the extracted results.


Figure 17-35. The plotted n and k values that correspond to the best parameters.

Should the heading <Debug> be added to the project file then the script will operate largely as usual but the table will now display, and retain, details of every tenth iteration, Figure 17-36. This interval of ten can be user adjusted in the script. Line 105 contains the statement: DebugInterval = 10 The interval can be increased or decreased. Also in the Debug condition an information message appears each time the simplex is recycled.

The columns in the table can be plotted in the normal way, using the Plot Column command in the File menu and, once the plot is established, by dragging the columns onto the plot. This particular table resulted in the plot of best and worst merit against iteration number shown in Figure 17-37. The vertical scale has been set to logarithmic otherwise it would be very difficult to distinguish the later values of merit. To set the axis to logarithmic, double click on it, and then, in the dialog that appears, check the box labeled Axis is Logarithmic. The best and worst results are usually very close so that the simplex must be quite small in volume. The vertical spikes in the worst merit curve are where the simplex is recycled. It is clear that it shrinks very rapidly till the best and worst values almost coincide again.


Figure 17-36. The appearance of the table if <Debug> has been set in the project file is shown on the left and the information dialog that appears each time the simplex recycles on the right.

Figure 17-37. The plot of best and worst merit as a function of iteration number. Note the spikes on the worst merit curve. These correspond to the recycling of the simplex. The vertical scale has been set to logarithmic


(double click on the scale and check the Axis is Logarithmic check box). Note also a feature of the extraction process, the best and worst merit values are usually very close together.

17.5.3 Incorrect Models The extraction operation is essentially a fitting of a model to the measurements and it the

model is incorrect then the extracted parameters are unlikely to be correct. In the use of the n and k extraction tool, incorrect models are more likely to exist in the interpretation of transmittance measurements than reflectance.

We can look at a simple case. We construct a film with optical thickness 2.0 at λ0 = 1000nm and index varying linearly as a function of optical thickness from 2.05 at the outer surface to 1.95 at the surface of the substrate. This gives a physical thickness of 1000.21nm. We use a dispersionless double-sided substrate of index 1.52. We now attempt to interpret the transmittance of this film using a homogeneous absorbing film model.

We use the envelope approach to give maximum stability.

Figure 17-38. The refractive index and extinction coefficient derived from fitting the transmittance of an inhomogeneous and nonabsorbing layer with a model of a homogeneous absorbing layer. The estimated layer thickness is 1010.7nm.

The results give a film of index 1.979, and a thickness of 1010.7nm and an extinction coefficient that increases linearly with wavelength, Figure 17-38. The fit between the transmittance calculated from these derived results and the original inhomogeneous performance is so good that no difference can effectively be determined, Figure 17-39. Yet the results are incorrect. This simple demonstration illustrates the danger of assuming that accurate recalculation of the input results necessarily confirms the validity of the extracted parameters.


Figure 17-39. The fit between the original inhomogeneous results (red curve) and those calculated from the derived index and extinction coefficient (black curve). The agreement is so good that no difference can be detected.

Figure 17-40. Comparison between the original inhomogeneous reflectance (red curve) and that calculated from the extracted homogeneous constants (black curve).

Calculation of the reflectance now shows a serious difference, Figure 17-40. Reflectance is very sensitive to inhomogeneity but not to absorption.


Now let us look at a more complex example. Figure 17-41 shows a refractive index profile through a film. The variation consists of a gradually increasing index, on which is superimposed a variation that is close to cyclic. The total thickness of the film is 1000.23nm. There is no dispersion and no absorption. The transmittance of this film on a dispersionless substrate of index 1.52 corresponding roughly to glass is in Figure 17-42. A similar variation of index through the film, although usually less regular, often occurs in practice when the control of a process parameter is deficient, or when manual adjustments to deposition rate are frequently made. The appearance of the fringes is usually a good guide. If there is some irregularity in terms of wavelength then a complicated inhomogeneity should be suspected.

Again, envelope methods were used to extract n and k for the film. The models consisted of an inhomogeneous nonabsorbing film with a linear variation of index and a homogeneous absorbing film.

Figure 17-41. Index variation through an inhomogeneous film. The variation consists of a linear increase with optical thickness modulated by a cyclic variation. Total thickness is 1000.23nm.

The results derived using the inhomogeneous model are shown in Figure 17-43. The total thickness was estimated at 999.42nm, a satisfactory figure. Also the degree of inhomogeneity corresponds well with that of the original film. However, around the region where the fringe pattern is irregular there is a pronounced disturbance in index. Sometimes with this type of problem the apparent inhomogeneity can actually be reversed.

In spite of the incorrect model, the model results show good correspondence with the input transmittance, Figure 17-44.

The results using the homogeneous but absorbing model are slightly poorer. The results are shown in Figure 17-45 and Figure 17-46. The thickness of the film was determined as 1009.92nm.


The message in all this is that a correct model is of major importance. If the model does not fit the film then, even though the measured results my be predicted accurately by the extracted parameters, they can be unstable and so predictions using them can be unsafe.

Figure 17-42. Transmittance of the film of Figure 17-41. Dispersion and absorption were both assumed zero. The rear surface of the substrate is assumed uncoated. The fringes around 800nm are misshapen.

Figure 17-43. The refractive indices extracted from the results in Figure 17-42 using a model of a linearly inhomogeneous film. The curves, from top to bottom at the left-hand side are outer index, mean index and inner index. The value of thickness derived for the film was 999.42nm, which compares well with the 1000.23nm of the original film.


Figure 17-44. The fit between the input and the results calculated from the linearly inhomogeneous model are good. The model, however, is incorrect.

Figure 17-45. Refractive index and extinction coefficient derived from the transmittance of Figure 17-42 using a model of a homogeneous and absorbing thin film. The thickness was determined as 1009.92nm while that of the original film was 1000.23nm.


Figure 17-46. Recalculation of the fringes using the homogeneous and absorbing results. Except for the fringe minima at 730nm and 900nm, the fit is exceptionally good.

17.6 References 1. Aspnes, D E and H G Craighead, Multiple determination of the optical constants of thin-

film coating materials: a Rh sequel. Applied Optics, 1986. 25: p. 1299-1310. 2. Hwangbo, C K, L J Lingg, J P Lehan, H A Macleod, J L Makous, and S Y Kim, Ion-

assisted deposition of thermally evaporated Ag and Al films. Applied Optics, 1989. 28(14): p. 2769-2778.

3. Borgogno, J P, B Lazarides, and E Pelletier, Automatic determination of the optical constants of inhomogeneous thin films. Applied Optics, 1982. 21: p. 4020-4029.

4. Manifacier, J C, J Gasiot, and J P Fillard, A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film. Journal of Physics E, 1976. 9: p. 1002-1004.

5. Swanepoel, R, Determination of the thickness and optical constants of amorphous silicon. Journal of Physics E, 1983. 16(12): p. 1214-1222.

6. Arndt, D P, R M A Azzam, J M Bennett, J P Borgogno, C K Carniglia, W E Case, J A Dobrowolski, D P Arndt, U J Gibson, T Tuttle Hart, F C Ho, V A Hodgkin, W P Klapp, H A Macleod, E Pelletier, M K Purvis, D M Quinn, D H Strome, R Swenson, P A Temple, et al., Multiple determination of the optical constants of thin-film coating materials. Applied Optics, 1984. 23: p. 3571-3596.

7. Dobrowolski, J A, F C Ho, and A Waldorf, Determination of optical constants of thin film coating materials based on inverse synthesis. Applied Optics, 1983. 22(20): p. 3191-3200.

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