Incidence angle (deg)
Modulation rate (gamma) Delta (rad)
Figure 3.3: Graph of the modulation rate and the ellipsometric parameter
∆as a function of the incidence angle
8 Appendix 1: ellipsometric angles
8.1 Definition of the ellipsometric angles Ψ and ∆
Consider a stratified planar system, that is to say composed of parallel layers of media having different indices. When this system is illuminated under oblique incidence, it is possible that the polarization of the reflected wave is modified with respect to the polarization of the incident wave. This modification comes from the difference between Fresnel’s amplitude coef-ficient of reflectivity rT E for the Transverse Electric polarization (E per-pendicular to the incidence plane) andrT M for the Transverse Magnetic polarization (E contained in the incidence plane)3. These two linear po-larizations are theeigenpolarization states of the system if all the layers of the stratified structure are isotropic and show no rotatory activity (cf.
3One can show that the difference between the two reflectivity coefficients reaches a max-imum for incidence angles close to Brewster’s angle for the substrate.
electromagnetism class 1A). The parameters ρ, Ψ and ∆ are defined as follows:
ρ= rT E rT M
= tan(Ψ) exp(i∆)withΨ∈[0˚,90˚]and∆∈]−180˚,+180˚]
The angles Ψ and ∆ are called the ellipsometric angles. They have a straightforward meaning:
• tan Ψ characterizes theattenuation ratio(of complex amplitudes, tan2Ψof “intensities”) between a TE and a TM polarized wave when reflected on the sample.
• ∆characterizes the relative phase shift of a TE polarized wave with respect to a TM one when reflected on the sample.
The so-called ellipsometry consists firstly in determining precisely by the experiment the anglesΨand∆. The measurements are usually performed for several incidence angles and different wavelengths. The second step allows to deduce from these raw data the characteristics of the sample layers.
8.2 Experimental determination of the ellipsometric angles
The ellipsometric angles can be determined from any polarimeter. This part develops the calculations in the case of a polarimeter with a rotating analyzer.
Definition of the axesx,yandz Thezaxis is defined as the direction of propagation of the incident wave. Thexaxis is such that thexz plane is the incidence plane. Theyaxis is perpendicular to the incidence plane.
In that frame, the electric field of a TM polarized wave is along thexaxis while the electric field of a TE polarized wave is along theyaxis (see Figure 1). After reflection, the axes are calledx0,y0andz0.
Fig. 1. Stratified system under study. The electric field of the in-cident wave is in the xy plane.
If the electric field is along thex axis, the wave is TM polarized ; if it is along theyaxis, the wave is TE polarized.
8. APPENDIX 1: ELLIPSOMETRIC ANGLES 41 Calculation of the light flux exiting the rotating analyzer Let us consider an incident wave, linearly polarized along a direction having an angleθwith respect to the xaxis. The complex amplitude of the electric field vector of the incident wave is proportional to:
Ex=E0cosθ Ey=E0sinθ
The complex amplitude of the electric field vector of the wave reflected on the stratified sample is thus written:
Ex0
Fig. 2. How to write the orien-tations of the polarization states before and after reflection. z and z0 are directed toward the reader (convention of the “light coming to you”). P corresponds to the di-rection of the linear polarization of the incident beam. A corre-sponds to the (instantaneous) di-rection of the analyzer.
Fig. 3. Projection of the reflected fields on the direction of the ana-lyzer.
The complex amplitude, written a, of the wave exiting the analyzer A is obtained by projecting the componentsEx0 andEy0 of the reflected field on the direction of the analyzer (figure 3) :
a=Ex0cosϕ+Ey0sinϕ ∝ cosθcosϕ+ tan Ψ exp(i∆) sinθsinϕ The flux received by the detector is proportionnal to the squared module of the complex amplitudeaof the wave exiting the analyzer. Let us writea∗
the complex conjugate ofa: F ∝ aa∗
∝ [cosθcosϕ+ tan Ψ exp(i∆) sinθsinϕ][cosθcosϕ+ tan Ψ exp(−i∆) sinθsinϕ]
The development and the simplification of the previous equation gives:
F(ϕ)∝tan2Ψ sin2θ+cos2θ+cos(2ϕ)[cos2θ−tan2Ψ sin2θ]+sin(2ϕ) sin(2θ) tan Ψ cos ∆ This expression shows that the flux received by the detector is asine wave of the azimuthϕof the analyzer,with a period of 180˚. If the analyzer rotates evenly, the detected flux varies thus in time as a sine wave whose frequency is twice the one of the analyzer rotation speed, notedf.
The division of the previous equation by the part independent of ϕ gives:
αandβ are the Fourier coefficients of the sine wave at the frequency2f. One can also write:
F(ϕ)∝1 +αcos(2ϕ) +βsin(2ϕ) = 1 +γcos(2ϕ+φ) withα=γcosφandβ=−γsinφ.
The valuesγandφ, and thusαandβ can be very easily deduced from the measurements as displayed by the VI, as long as you have understood that ϕ has the direction x (x0) for origin, that is to say the direction of the TM polarization in the previous calculations (cf. Fig.3) and not the arbitrary one given byTOP0.
Practical calculation ofΨand∆ The parameterφdefined above can be easily determined experimentally from the reference phaseφT M for the TM polarization TM and from the phase φθ given by the reflected beam when the incident beam is linearly polarized along a direction making an angleθwith the directionx0(TM), and from the direction of rotation of the analyzer usingφ=±(φθ−φT M)(+ if the analyzer rotated anti clockwise, - if not).
The coefficientsαandβ are then calculated using the measurement of γof the reflected beam when the incident beam is linearly polarized along a direction making an angleθwith the directionx(we still haveα=γcosφ andβ =−γsinφ). One can now find the value ofρusing the fact that:
9. APPENDIX 2: CALCULATION OF THE INDEX OF A METALLIC MIRROR43
∆’s sign, connected to the handedness of the elliptic polarization, cannot be determined by the previous experiments. To disambiguate, we must introduce an additional measurement with a wave plate of slow and fast axes known, usually calledcompensator.