ELLIPSHEET: Spreadsheet Ellipsometry (Excel Ellipsometer)
EXCEL Worksheets for Basic Ellipsometry Calculation


What is Ellipsometry?

Light reflected from a substrate having thin film(s) on top has different phase and amplitude from the incident light.

Fig. 1
After H.G. Tompkins

When liner polarized light is incident to the film(s), the reflected light becomes ellipsoidally polarized. Ellipsometry determines the phase shift angle, usually abbreviated DELTA or shortly DEL, and the magnetite ratio of total reflection coefficients (Rp and Rs in the above equations). The magnitude ratio is given by angle PSI.

Carefully-designed ellipsometry experiments provide a lot of useful information. The reflected light is so sensitive to the surface condition to study the behavior of surface atoms such as recombination and relaxation.

Fig 2
After H.G. Tompkins

tan(PSI) = |Rp|/|Rs|

tan(PSI)exp(iDEL) = Rp/Rs

Unfortunately, we can not analytically determine the film thickness (d) and refractive index (N) from the measured PSI and DEL values. On the other hand, DEL and PSI can be calculated from d and N. One can therefore determine d and N by using table or chart of DEL and PSI obtained for various d and N combinations. Software bundled with commercial ellipsometers carries out a similar procedure using, for example, the mean-least argorithym.

What good of ELLIPSHEET?

Calculation of PSI and DEL is not straightforward. Hand figuring is very complex, almost impossible when complex refractive indices are used. But thanks to the computer. ELLIPSHEET calculates DEL and PSI from d, N, substrate refractive index (Ns), incident angle (PHI), and wavelength.

  • Easy to form your own PSI-DEL vs d-N tables and charts. Very simple, no VBA or macro, all the calculations are compactly described in one line.
  • Help understanding of the principle of ellipsometry.
  • Help understanding of the meanings of data output of commercial ellipsometers
  • Provides opportunity to cross-check the output of commercial ellipsometers
  • Aid to improve your ellipsometric experiments.

To understand these advantages, let us see weak aspects of ellipsometry.

The main application of ellipsometry is the measurement of d and N of thin films. However, d and N data outputted by a commercial ellipsometer is not always 'correct'. As already stressed, ellipsometry with a fixed incident angle cannot analytically determine the 'true' d and N values. The raw data is only PSI and DEL, and the outputted d and N are a possible---hopefully most likely---combination of them.

In order to limit the possibilities, one must assume a certain optical model of the sample to measure. One example is a dielectric transparent thin film on a sufficiently thick dielectric substrate (Fig. 1). The ellipsometer's computer does iterative calculations to find (d, N) that gives the measured (PSI, DEL) with using this model. Therefore, besides instrumental inaccuracy, there are at least two major origins of measurement error: optical model and computation (convergence error). Is either wrong, the outputted d and N can be very strange and the measurer will sometimes see 'no answer'.

The story stated above is somehow contradictory. Ellipsometry cannot determine d and N but we do 'measure' d and N. Yes, we can determine the d and N values for well-defined and well-understood systems. Thermally-grown silicon dioxide on silicon substrate, such as LOCOS oxide, is an example. In this case, there is no doubt about the optical model from a practical point of view. Moreover, fortunately, a well-trained experimentalist can make a good guess of the optical model. Therefore, in many cases, his/her data is 'correct'. But of course, not always.

It is now clear why we need ELLIPSHEET. Putting off the ellipsometer's black-box software, we need a robust, simple, and explicit way to calculate the basic ellipsometric formulas for any given optical parameters. ELLIPSHEET is a very simple Excel spreadsheet; no special input or output windows; nothing clever VBA programmes or macros. Just fill parameters, copy cells, so as to form a table you want. Thus we understand how DEL and PSI behave!


ELLIPSHEET is designed with Microsoft Excel 97. Analysis add-in tool is necessary to use the spreadsheets. Operations with other Excel versions or similar spreadsheet programs are not guaranteed. The calculation results may involve intrinsic truncation errors. The author, E. Kondoh, holds copyright of ELLPSHEET spreadsheet and any spreadsheets linked to this page and of the mathematical formulas used inside the spreadsheets. To use, distribute, and modify the spreadsheets and formulas are not restricted for any non-commercial purpose without fee. The spreadsheets are originally created for the author's personal use. The author is not liable for any damages resulting from the use of the spreadsheets.

Netscape users, have a download problem? Try zip archives.

Now three-layer calculation sheet is under evaluation! Any suggestions and comments are welcome!

Final revision: 23 May 2001. The older versions may have mathematical error. See the file revised date. Special thanks to T. Yumii and H.G. Tompkins

How to Use

1. ELLIPSHEET, Principal Excel Template ellips.xlt   zip

This worksheet, actually book template, consists of 4 worksheets: "Single Layer", "Substrate Calculation", "Effective Medium", and "Optical Properties". After opening this file, make sure that ANALYSIS TOOL in TOOL/ADD-INs menu is checked.

1) Single Layer

This spreadsheet calculates PSI, DEL, and film phase thickness of a single layer film from the thickness, d, film refractive index (complex) N, substrate refractive index (complex) Ns, incident angle, PHI, and wavelength. Cells A7-H7, and J7 are the parameters to fill, and the results appear in Cell AB7, AC8, I7. Cells K7-AA7 are invisible, as their width is set to zero. In order to make a table, copy the 7th row, or drag A7-AC7 to autofill.

Film and substrate parameters





Real part of the refractive index of environment (vacuum and air = 1)



Imaginary part of the refractive index of environment (vacuum and air = 0)



Real part of the film refractive index



Imaginary part of the film refractive index



Real part of the refractive index of the substrate



Imaginary part of the refractive index of the substrate

Angle (Phi1)


Incident angle (deg)



Light wavelength (angstrom)

Film Thickness














Periodical film thickness (angstrom)

2) Substrate Calculation

This worksheet calculates the complex refractive index of a substrate from its PSI and DEL. Unlike the case of a thin film on substrate, the substrate refractive index can be analytically calculated. If any film present on a substrate, an apparent or effective substrate index is obtained.

Film and substrate parameters





Real part of the refractive index of environment (vacuum and air = 1)



Not used







Angle (Phi1)


Light incident angle (deg)






Real part of the refractive index of the substrate



Imaginary part of the refractive index of the substrate

3) Effective Medium Approximation

This worksheet is to obtain the apparent refractive index of mixed two mediums. For instance porous films and rough surface layers can be modeled as a mixture of the host solid and void. The feature size of the optical body suspended in the host medium should be less than the wavelength.

Film parameters Cell  
n1 A7 Real part of the refractive index of medium 1
k1 B7 Imaginary part of the refractive index of medium 1
Medium 1 vol % C7 (%)
n2 D7 Real part of the refractive index of medium 2
k2 E7 Imaginary part of the refractive index of medium 2
n Q7 Real part of the refractive index of the effective medium
k R6 Imaginary part of the refractive index of the effective medium


4) Optical Properties

List of refractive indices for common materials.


2. Examples

1) Single-layer transparent film on Si. ellips2.xls   zip

DEL-PSI plots of a transparent single-layer film (k2=0) on Si for n2=1.2, 1.46 (SiO2), and 1.8 with varying thickness, d.

In order to determine d and N from the measured DEL and PSI, it is better to make this type of plot first. The number of loops will be 3-5 at first. The thickness d is varied from 0 to ~periodical thickness so as to close the loop. Then put the experimentally obtained (PSI, DEL) point. If this point is far away from the loops you already made, try other N values until the point is located between two loops.

Next, 'zoon in' on the point. The goal is to find the curve that passes through the point. Do not 'magnify' too much. Keep the two neighboring curves in view. And repeat the same procedures, by changing d and N, until a plot point on the curve becomes sufficiently close to the experimental point. If the first decimals are in agreement, for most of measurements, OK. Purists may try the second decimal agreement.

It is important to notice that the N dependence of DEL and PSI is small for PSI ~ 10 deg. This occurs when the film thickness is close to zero or to the film phase thickness, or when the film has no 'colour'. Since the measured DEL and PSI values should involve some unavoidable inaccuracy, the ellipsometer's output may contain a large error in N ifn PSI ~ 10 deg. On the other hand, measurement accuracy improves for large PSI values, as N dependence of PSI and DEL becomes large.

If your film is not transparent, or absorbing, it will take a bit more time (see next section), because there is an additional parameter to vary. You may need additional specimens with different d. Changing wavelength or Phi is also helpful. This sheet helps you to determine appropriate measurement conditions.

Fig. 3

2) Single-layer absorbing film on Si. ellips3.xls   zip

PSI-DEL plot of W film for various thicknesses. The increment is 20A and the wavelength is 6330A.

The trajectory starts from the Si point (10.3, 179.1) and converge to the bulk W point (23.2, 130.1). Absorbing films do not show the periodicity. It is understood that light penetrates into the W film up to about 200A. Thinner films do not show the optical constants of bulk W.

Fig. 4

Next example is an absorbing film having a small extinction coefficient (N=1.46-0.02i). That is, this type of film is almost transparent but not perfectly. Compare the curve with that of transparent film (SiO2, magenta). The transparent film shows a closed loop with a period of 3050A, but the loops the absorbing film make do not close.

Fig. 5

Here is a 'zoom-in' view on silicon point with the curve of a transparent film (N=1.8). The N=1.8 curve goes across the trajectory of the absorbing film many times. This means different (d, N) combinations are possible for the same (PSI, DEL). For instance, the transparent films whose thickness is 285, 2345, 4410, … angstrom gives PSI=16.9 and DEL=105.7, but the absorbing film have a unique thickness of 3420A. Thus is the optical mode important.

Fig. 6

3) Wavelength dependence of DEL and PSI ellips4.xls   zip

PSI and DEL values of a 3000A-thick SiO2 film on Si as a function of wavelength.

This type of measurement is called spectroscopic ellipsometry (SE). SE affords advanced analysis such as multilayer film analysis which is usually very difficult to carry out with a single-wavelength ellipsometry. However, since SE is not more than ellipsometry, we first need an optical model, and calculate PSI and DEL, comparing with the measurement. The tricky part is therefore to have an appropriate model, not calculation itself. This worksheet is applicable to common SE measurements in Si processing.

Fig. 7


H.G. Tompkins, A User's Guide to Ellipsometry, Academic Press, Boston, 1993.