scattering matrix interpolation from the perturbation method: application to calculation of a...

Scattering matrix interpolation from the perturbationmethod: application to calculation of a grating’selectromagnetic and scatterometric signatures

Kofi Edee1,2

1Clermont Université, Université Blaise Pascal, LASMEA, BP 10448, F-63000 Clermont-Ferrand, France2CNRS, UMR 6602, LASMEA, F-63177 Aubrière, France, (kofi.edee@univ‑bpclermont.fr)

Received December 20, 2010; revised March 15, 2011; accepted April 4, 2011;posted April 4, 2011 (Doc. ID 139860); published June 16, 2011

The scatterometric and electromagnetic signatures of a pattern are computed with the perturbation methodcombined with the Fourier modal method in order to reduce computational time. From an electromagnetic pointof view, the grating is characterized by its scattering matrix, which allows the computation of the reflection andtransmission coefficients. A slight variation of profile parameters or electrical ones provides a small fluctuation ofthe scattering matrix; consequently, an analytical expression of the local behavior of its eigenvectors and eigen-values can be obtained by using a perturbation method. © 2011 Optical Society of America

OCIS codes: 050.1950, 050.1755.

1. INTRODUCTIONEvolution of electronic components miniaturization requiresvery precise and preferably nondestructive technical control.Conventional techniques, such as the atomic force micro-scopy or the scanning electron microscopy, commonly usedin the industry are not adapted to a real-time monitoring or aredestructive of the analyzed structure. Techniques based onoptical control, such as scatterometry, provide a very satis-fying alternative in terms of reliability and numerical integra-tion. One of the most popular branches of scatterometry [1–4]is an electromagnetic technical characterization based on theanalysis of the electromagnetic signal diffracted by an objectusing an ellipsometer. This technical characterization can bedivided into two steps.

• The first step, conventionally called the direct problem,consists in managing a database of scatterometric signaturesof periodical structures, similar to the characterized object. Ascatterometric signature is a dispersion curve (representationwith respect to the wavelength) of the ratio of diffracted fieldcomplex amplitudes for both polarizations (TE and TM). Thegeneration of such a database is done through a numericalelectromagnetic simulation. The most commonly usedmethod for the numerical simulation is the Fourier modalmethod (FMM).

• The second step, which seems to be a reverse method,consists in finding the geometrical parameters of the patternsfrom experimental measurements. One of the most commonlyused methods is to establish a library, i.e., a set of electromag-netic signatures obtained through a direct electromagneticsimulation for different values of the pattern’s geometricalparameters: height, middle height (critical dimension [CD]),angle of the slope, curves radii, etc. The establishment of thislibrary can quickly penalize the technique.

The direct electromagnetic simulation therefore appears to beone of the most important step in scatterometry. The electro-magnetic simulation is based on the FMM [5–8] which iscoupled with the staircase approximation. This technique,which consists in regarding the grating as a stack of lamellargratings, was first suggested by Peng et al. [9] and was laterused in the very popular rigorous coupled-wave method [10].In all the methods employing this technique, the electromag-netic field is written as a modal expansion in each layer, andthe tangential components of the field in the different slicesare connected by boundary conditions. Several connecting al-gorithms exist to solve boundary equations, but the most ro-bust and stable is probably the algorithm of the scatteringmatrix S. In this paper, we present an interpolation methodof the S matrix. For a given pattern, the S matrix is a functionof the geometric parameters of the pattern (height, CD, slope,etc) but also in terms of physical parameters of the problemincluding the wavelength. The calculation method is based onevaluation by a perturbation method, local variations of theeigenvalues, and eigenvectors of the scattering matrix [11].The technique is applied to a canonical structure commonlyfound in scatterometry, a trapezoidal grating with roundededges. Section 2 briefly explains the framework of the stair-case approximation with the FMM. In Section 3, the computa-tion of eigenvalues and eigenvectors of the S matrix ispresented. Sections 4 and 5 are devoted to the presentationof numerical results.

2. FRAMEWORK OF THE FMMIn a Cartesian coordinate system ðO; ex; ey; ezÞ, let us considerthe grating depicted in Fig. 1, which is invariant along thez direction. We deal with electromagnetic waves that are zindependent. Therefore, the electromagnetic field may bedecomposed into two fundamental cases of polarization:

1418 J. Opt. Soc. Am. A / Vol. 28, No. 7 / July 2011 K. Edee

1084-7529/11/071418-11$15.00/0 © 2011 Optical Society of America

TE polarization (the only non-null components of the field areEz, Hx, and Hy) and TM polarization (the only non-null com-ponents of the field are Hz, Ex, and Ey). It is well known thatin these cases of polarization, all the field components can beexpressed in terms of Ez in the TE case or in terms ofHz in theTM case. Let us add that throughout this paper, time depen-dence of the field will be held by the term expðiωtÞ, where ωdenotes the circular frequency of the monochromatic incidentradiation. The grating surface is approximated by a staircaseprofile so that the structure is replaced by a stack of layers (nc

denote the number of layers). Each of the layers are char-acterized by thickness and refractive index νj , which onlydepends on the x variable:

νjðxÞ ¼�ν; x ∈ ½0; lj�;1; otherwise:

: ð1Þ

It can be shown in each layer that the z component of theelectromagnetic field, denoted by Fðx; yÞ, satisfy to a secondorder differential equation

LFðx; yÞ ¼ −1

k2∂2yFðx; yÞ: ð2Þ

In Eq. (2) and all following expressions, the subscript j isomitted. L is an operator that is different according to thepolarization

TE∶L ¼ 1k2∂2x þ ν2ðxÞ; ð3aÞ

TM∶L ¼ 1

k2ν2ðxÞ∂x

1

ν2ðxÞ∂x þ ν2ðxÞ: ð3bÞ

k ¼ 2π=λ denotes the vacuum wave number, and λ is thewavelength. Here we use the method of separation of vari-ables to obtain solutions Fðx; yÞ. This method seeks to findsolutions of the form

Fig. 1. Grating configuration: two periods are depicted.

Fig. 2. Staircase approximation: the grating is replaced by a stack oflamellar gratings.

Fig. 3. Comparison between ρ and ρ0 for different values of R1 andλ ¼ 240nm. Numerical parameters: M ¼ 20, Nc ¼ 1500, np ¼ 4,R2 ¼ 20nm, CD ¼ 100nm, α ¼ 88°, θ ¼ 70°, d ¼ 200nm.

K. Edee Vol. 28, No. 7 / July 2011 / J. Opt. Soc. Am. A 1419

Fðx; yÞ ¼ XðxÞYðyÞ: ð4Þ

It can be shown that YðyÞ ¼ e�ikry consequently XðxÞ satisfyto an eigenvalue equation

LXðxÞ ¼ r2XðxÞ; ð5Þ

and to the pseudoperiodic condition, i.e., Xðxþ dÞ ¼e�ik sin θdXðxÞ, where θ is the incidence angle. Hence the spec-trum of the L operator is discrete. Their solutions XmðxÞ forma complete set of vectors [12,13], and this property allows therepresentation of any pseudoperiodic functions. In practice,Fðx; yÞ (i.e., Ezðx; yÞ or Hzðx; yÞ) is approximated by a finitesum of eigenvectors XmðxÞ:

Fðx; yÞ ¼Xm¼2Nþ1

m¼1

ðAmeikrmy þ Bme−ikrmyÞXmðxÞ: ð6Þ

The square root rm of the eigenvalue r2m is chosen in such away that

ImðrmÞ < 0 or rm > 0 if rm is real: ð7Þ

According to Eq. (7), AmeikrmyXmðxÞ is associated with down-going waves and Bme−ikrmyXmðxÞ to up-going waves. The FMMconsists in expanding any eigenvectors Xm in the Fourierseries:

XmðxÞ ¼Xn¼∞

n¼−∞

XmnenðxÞ; ð8Þ



Fig. 6. ζðR1Þ for different values of the wavelength λ ¼ 240nm,λ ¼ 400nm, λ ¼ 600nm. Numerical parameters: M ¼ 20, Nc ¼ 1500,np ¼ 4, R2 ¼ 20 nm, CD ¼ 100nm, α ¼ 88°, θ ¼ 70°, d ¼ 200nm.


where enðxÞ ¼ exp

�−ix

�k sin θ þ 2πn

d

��. The field coeffi-

cients Am and Bm are obtained with the help of the boundaryconditions, i.e., the continuity of the components ðEz;HxÞ inthe TE polarization and ðEx;HzÞ in the TM polarization, at theinterfaces separating the layers. The final equations are per-formed with an S matrix analysis. This matrix relates the re-flected waves to the incident wave. See Fig. 2. For a linearmedium,

B1

A2

� �¼ S11 S12

S21 S22

� �A1

B2

� �: ð9Þ

The coefficients S11, S12, S21, and S22 are known as scatteringcoefficients. For physical interpretation, for the sourceapplied from vector A1, B2 ¼ 0, then

B1 ¼ S11A1; A2 ¼ S21A1: ð10Þ

Thus S11 is just the input reflection coefficients (in magni-tude and phase), and S21 is the transmission coefficients. Thematrix S contains all information on the problem of diffrac-tion. Its construction requires the solution of an nc numbersof the eigenvalues equations and the boundary conditionsequations. These calculations are very expensive in comput-

ing times especially for a high number of nc layers. A smallmodification of the physical or geometrical parameters ofthe structure leading to a new problem of diffraction must re-sult in a light perturbation of the scattering matrix. It is notnecessary to undertake the integral calculation of the matrixS in order to deduce the reflection and transmission coeffi-cients of the new problem. A calculation from a perturbationmethod of the eigenvalues and eigenvectors of the matrix Swould allow to deduce the local behavior of the scatteringmatrix S.

3. PERTURBATIONSThe perturbation theory is a mathematical method used tofind an approximate solution to a problem, from the exactone of a related problem. This method is applicable if the pro-blem at hand can be formulated by adding a “small” term tothe mathematical description of the exact solvable problem.Historically, the perturbation method has its roots in early ce-lestial mechanics and it was first used to solve algebraic equa-tions, before being applied to the operator theory, especiallyin classical quantum mechanics [14].

Let us consider a configuration that consists in a gratingilluminated by a plane wave (Fig. 1). Let ξ denote any geome-trical parameters (curves R1 or R2, width at middle height CD,

Fig. 7. Comparison between ρ and ρ0 for different values of the CDand λ ¼ 240nm. Numerical parameters: M ¼ 20, Nc ¼ 1500, np ¼ 4,R1 ¼ 20nm, R2 ¼ 20nm, α ¼ 88°, θ ¼ 70°, d ¼ 200nm.

Fig. 8. Comparison between ρ and ρ0 for different values of the CDand λ ¼ 400nm. Numerical parameters: M ¼ 20, Nc ¼ 1500, np ¼ 4,R1 ¼ 20nm, R2 ¼ 20 nm, α ¼ 88°, θ ¼ 70°, d ¼ 200nm.


height h, the angle of slope α, etc) or physical parameters(wavelength, angle of incidence, etc) in this configuration.Let Sðξ ¼ aÞ be the scattering operator of this problem. IfψpðaÞ is an eigenvector of SðaÞ associated with the eigenvalueλpðaÞ, we get

SðaÞψpðaÞ ¼ λpðaÞψpðaÞ: ð11Þ

Let us suppose a light modification of this parameter ξ leadingξ ¼ a to ξ ¼ b, thus the operator Sðξ ¼ bÞ may be expressedusing the unperturbated operator SðaÞ plus a correction term

SðbÞ − SðaÞ ¼ P ¼ ðb − aÞ~P; ð12Þ

where ðb − aÞ is presumed to be small. The eigenvalues λpðbÞand eigenfunctions ψpðbÞ of the operator SðbÞ verify

SðbÞψpðbÞ ¼ λpðbÞψpðbÞ; ð13Þ

and may be expressed in terms of the power series of ðb − aÞ:

λpðbÞ ¼Xn

ðb − aÞnλðnÞp ; ψpðbÞ ¼Xn

ðb − aÞnψ ðnÞp : ð14Þ

For the 0th order approximation, λpðbÞ ¼ λð0Þp ¼ λpðaÞ;ψpðbÞ ¼ ψ ð0Þ

p ¼ ψpðaÞ:.To calculate the higher order perturbation correction

terms, we put Eqs. (12) and (14) into Eq. (13):

½SðaÞ þ ðb − aÞ~P�Xn

ðb − aÞnψ ðnÞp ¼

Xm;n

ðb − aÞmþnλðmÞp ψ ðnÞ

p :

ð15Þ

By successively identifying the 0, 1, and 2 rank coefficients ofðb − aÞ we get

SðaÞψ ð0Þp ¼ λð0Þp ψ ð0Þ

p ; ð16aÞ

SðaÞψ ð1Þp þ ~Pψ ð0Þ

p ¼ λð0Þp ψ ð1Þp þ λð1Þp ψ ð0Þ

p ; ð16bÞ

SðaÞψ ð2Þp þ ~Pψ ð1Þ

p ¼ λð0Þp ψ ð2Þp þ λð1Þp ψ ð1Þ

p þ λð2Þp ψ ð0Þp : ð16cÞ

The complex eigenvectors are normalized by setting condi-tions on both the modulus and the phase:

�ψ†p;ψp

�¼ 1;

�ψ†ð0Þp ;ψp

�real; ð17Þ

and ψ†p is the eigenvector of the adjoint of the operator SðbÞ.

By applying this normalization to 0, 1, and 2 orders we get

�ψ†ð0Þp ;ψ ð0Þ

p

�¼ 1; ð18aÞ


p

�¼


p

�¼ 0; ð18bÞ


p

�¼


p

�¼ −

12


p

�; ð18cÞ

where ψ†ð0Þp is the eigenvector of the adjoint of the operator

SðaÞ. In order to arrive at the first order approximation,Eq. (16b) is projected on ψ†ð0Þ

q . By using Eqs. (18a) and (18b),we get the expressions of the eigenvalues

λpðbÞ ¼ λð0Þp þ ðb − aÞ�ψ†ð0Þp ; ~Pψ ð0Þ

p

�þ o

�ðb − aÞ2

�; ð19Þ

and the eigenvectors

ψpðbÞ ¼ ψ ð0Þp þ ðb − aÞ

Xq≠p

�ψ†ð0Þq ; ~Pψ ð0Þ

p

�

λð0Þp − λð0Þq

ψ ð0Þq þ oððb − aÞ2Þ:

ð20Þ

By combining Eqs. (16c) and (18c), we obtain the secondorder approximation of the eigenvalues

Fig. 9. Comparison between ρ and ρ0 for different values of the CDand λ ¼ 600nm. Numerical parameters: M ¼ 20, Nc ¼ 1500, np ¼ 4,R1 ¼ 20nm, R2 ¼ 20nm, α ¼ 88°, θ ¼ 70°, d ¼ 200nm.


λpðbÞ ¼ λð0Þp þ ðb − aÞ�ψ†ð0Þp ; ~Pψ ð0Þ

p

�

þ ðb − aÞ2Xq≠p


p

��ψ†ð0Þp ; ~Pψ ð0Þ

q

�


þ oððb − aÞ3Þ; ð21Þ

and of the eigenvectors

ψpðbÞ ¼ ψ ð0Þp þ ðb − aÞ

Xq≠p


p

�


ψ ð0Þq

þ ðb − aÞ2Xq≠p

Xl≠p

�ψ†ð0Þl ; ~Pψ ð0Þ

p

��ψ†ð0Þq ; ~Pψ ð0Þ

l

�

ðλð0Þp − λð0Þl Þðλð0Þp − λð0Þq Þψ ð0Þq

− ðb − aÞ2Xq≠p

�ψ†ð0Þp ; ~Pψ ð0Þ

p


p

�

ðλð0Þp − λð0Þq Þ2ψ ð0Þq

− ðb − aÞ212

Xq≠p

�ψ†ð0Þp ; ~Pψ ð0Þ

q


p

�


þ oððb − aÞ3Þ: ð22Þ

Let η ¼ b − a for all c ∈ ½a − 0:5η; aþ 0:5η�; we get at a secondorder of approximation:

λpðcÞ ¼ λð0Þp þ c − aη

�ψ†ð0Þp ; Pψ ð0Þ

p

�

þ�c − aη

�2Xq≠p

�ψ†ð0Þq ; Pψ ð0Þ

p

��ψ†ð0Þp ; Pψ ð0Þ

q

�


þ oðη3Þ;

ð23Þ

and

ψpðcÞ ¼ ψ ð0Þp þ c − a

ηXq≠p

�ψ†ð0Þq ; Pψ ð0Þ

p

�


ψ ð0Þq

þ�c − aη

�2Xq≠p

Xl≠p

�ψ†ð0Þl ; Pψ ð0Þ

p

��ψ†ð0Þq ; Pψ ð0Þ

l

�

ðλð0Þp − λð0Þl Þðλð0Þp − λð0Þq Þψ ð0Þq

−

�c� aη

�2Xq≠p


p


p

�


−

�c − aη

�212

Xq≠p


q


p

�


þ oðη3Þ:ð24Þ

The scattering matrix SðcÞ is obtained by the followingexpression:

Fig. 10. ζðCDÞ for different values of the wavelength λ ¼ 240nm,λ ¼ 400nm, λ ¼ 600nm. Numerical parameters: M ¼ 20, Nc ¼ 1500,np ¼ 4, R1 ¼ 20nm, R2 ¼ 20nm, α ¼ 88°, θ ¼ 70°, d ¼ 200nm.

Fig. 11. ζðCDÞ for different values of the wavelength λ ¼ 240nm,λ ¼ 400nm, λ ¼ 600nm. Numerical parameters: M ¼ 20, Nc ¼ 1500,np ¼ 10, R1 ¼ 20nm, R2 ¼ 20nm, α ¼ 88°, θ ¼ 70°, d ¼ 200nm.


SðcÞ ¼ ½ψðcÞ�½λðcÞ�½ψðcÞ�−1; ð25Þ

where ½λðcÞ� is a diagonal matrix resulting from the eigenva-lues λðcÞ and ½ψðcÞ� contains the eigenvectors ψðcÞ.

4. APPLICATION TO CALCULATION OFELECTROMAGNETIC SIGNATURESIn this Section, we explore the possibilities offered by the per-tubation method approach to establish a database from theelectromagnetic signature of the structure described in Fig. 1.The pattern is characterized by its curve radii R1 and R2, itswidth at middle height (CD), its height h, and then its angle ofslope α (for α equal to 90° the slope is vertical). The grating ofthe refractive index ν ¼ 1:51309, period d ¼ 200nm, depthh ¼ 200 nm is illuminated by a plane wave with θ ¼ 70°.For these parameters, there is only one diffracted order. Wewill denote rTE (resp. rTM) the complex amplitudes of the dif-fracted field in polarization TE (resp TM). Ellipsometric mea-surements allow us to obtain Is and Ic:

Is ¼ sin 2ψ sinΔ; Ic ¼ sin 2ψ cosΔ; ð26Þ

where ψ and Δ are related to the ratio ρ by

ρ ¼ rTErTM

¼ tanψeiΔ:

We will denote ρ as the values obtained by the RCWA methodby dividing the pattern into nc ¼ 1500 layers, each one de-scribing a lamellar grating. ρ0 will indicate the value estimatedby the method of perturbation for a small variation of a pa-rameter ξ of profile. This calculation of ρ0 is carried out asfollows.

1. We apply to a given parameter ξ, a small variation η. Thematrix SðaÞ and Sðaþ ηÞ are computed, then the perturbationmatrix P ¼ Sðaþ ηÞ − SðaÞ is deduced.

2. Equations (23) and (24) allow us to express theeigenvectors and eigenvalues for all c belonging to½a − 0:5η; aþ 0:5η�. The matrix SðcÞ is then obtained by therelation (25) and the field amplitude by Eq. (10).

We will present curves showing real and imaginary parts of ρwith respect to the parameters ξ ∈ ½ξmin; ξmax�, for three valuesof λ (λ ¼ 240nm, λ ¼ 400 nm, and λ ¼ 600nm). Our validationcriterion consists in interpolating those curves with the per-turbation technique seen above, from a certain number ofpoints np, chosen distant of η ¼ ðξmax − ξminÞ=ðnp − 1Þ.

Fig. 12. Comparison between ρ and ρ0 for different values of α andλ ¼ 240nm. Numerical parameters: M ¼ 20, Nc ¼ 1500, np ¼ 10,R1 ¼ 20nm, R2 ¼ 20nm, CD ¼ 100nm, θ ¼ 70°, d ¼ 200nm.

Fig. 13. Comparison between ρ and ρ0 for different values of α andλ ¼ 400nm. Numerical parameters: M ¼ 20, Nc ¼ 1500, np ¼ 10,R1 ¼ 20nm, R2 ¼ 20 nm, CD ¼ 100nm, θ ¼ 70°, d ¼ 200nm.


To do so, we define a related error at each point ξp,

ζðξpÞ ¼ −Int�log10

��1 − ρ0ðξpÞρðξpÞ

��; ð27Þ

allowing us to estimate the number of digits common to ρ andρ0. IntðχÞ stands for the integer part of χ.

Figures 3–5 represent ρðR1Þ for R1 ∈ ½10; 30�nm for thethree values of the wavelength λ: λ ¼ 240nm (Fig. 3), λ ¼400 nm (Fig. 4), and λ ¼ 600 nm (Fig. 5). The numerical param-eters for these studies are: α ¼ 88°, CD ¼ 100nm, and R2 ¼20 nm. The curves of Fig. 6 represent the error functionζðR1Þ for the three values of λ and np ¼ 4 (η ¼ 20=3 nm).On these curves, we can observe that:

• the precision is not the same according to the points ofthe curve on an interval ½a − 0:5η; aþ 0:5η�, a being a pointused to make the perturbation calculation. The precision isbetter close to a and decreases by a digit when R1 is approach-ing a − 0:5η and aþ 0:5η.

• for this value of np corresponding to η ¼ 20=3 nm,ζðR1Þ ∈ f2; 3g for those three values of the wavelength λ.

From our point of view, this behavior is predictable sinceR1=λ ≪ 1. Indeed the R1 variations do not cause substantial

modifications of the function ρðR1Þ. See Figs. 3–5. This alsoexplains why a slight number of points np (np ¼ 4, η ¼20=3 nm) allows to get such an accuracy. This may not bethe case in the resonant domain, i.e., ξ is comparable to thewavelength λ.

A study of the variations of the CD illustrates this case withthe following numerical values: R1 ¼ 20nm, R3 ¼ 20nm,α ¼ 88°, CD ∈ ½95; 115�nm. Curves representing ρðCDÞ andρ0ðCDÞ are shown in Fig. 7 for λ ¼ 240nm, Fig. 8 forλ ¼ 400 nm, and Fig. 9 for λ ¼ 600 nm. We can observe onthe curves that, contrary to the previous case, functions pre-sent more significant variations. The curves of Fig. 10 repre-sent the error function ζðCDÞ for the three values of λ andnp ¼ 4 (η ¼ 20=3 nm). For this value of np, results are notvery satisfactory for λ ¼ 240nm and λ ¼ 400nm, i.e.,ζðR1Þ ∈ f1; 2g. For a higher value of λ (λ ¼ 600 nm), the resultis more acceptable, i.e., ζðR1Þ ∈ f2; 3g. The growing numberof points np (np ¼ 10, η ¼ 20=9 nm) improves results. SeeFig. 11. We get a precision between 10−2 and 10−3 for λ ¼240nm and λ ¼ 400nm and a precision between 10−3 and10−4 for λ ¼ 600 nm.

However, it is difficult, not to say impossible, to forecast thevariation way of the ζðξÞwith respect to the wavelength, in thecase of the ξ variation the underlyingly causes are those of themultiple parameters of the profile. This is the case when ξ ¼ α.Indeed, a variation of the slope angle α, with CD being con-stant, causes an opposing variation of upper and lower bor-ders of the trapezoid. A study of a variation of ρðαÞ andρ0ðαÞ is presented in Figs. 12–14. Curves from Fig. 15 representthe error function ζ for the three values of λ and np ¼ 10

Fig. 14. Comparison between ρ and ρ0 for different values of α andλ ¼ 600nm. Numerical parameters: M ¼ 20, Nc ¼ 1500, np ¼ 10,R1 ¼ 20nm, R2 ¼ 20nm, CD ¼ 100nm, θ ¼ 70°, d ¼ 200nm.

Fig. 15. ζðαÞ for different values of the wavelength λ ¼ 240nm,λ ¼ 400nm, λ ¼ 600nm. Numerical parameters: M ¼ 20, Nc ¼ 1500,np ¼ 10, R1 ¼ 20nm, R2 ¼ 20nm, CD ¼ 100nm, θ ¼ 70°, d ¼ 200nm.


(η ¼ 20°=9). We get better results for λ ¼ 240 nm and λ ¼600nm (ζðαÞ ∈ f3; 4g), but the precision decreases in one di-git for a middle value of λ, i.e., ζðαÞ ∈ f2; 3g for λ ¼ 400nm.

Fig. 16. Comparison between ρ and ρ0 for different values of λ,(λ ∈ ½235; 245�nm). Numerical parameters: M ¼ 20, Nc ¼ 1500,α ¼ 88°, np ¼ 6, R1 ¼ 20nm, R2 ¼ 20nm, CD ¼ 100nm, θ ¼ 0°,d ¼ 200nm.


Fig. 18. ζðλÞ, (λ ∈ ½235; 245� nm) for different values of the angle ofincidence θ ¼ 0°, θ ¼ 5°. Numerical parameters: M ¼ 20, Nc ¼ 1500,np ¼ 6, R1 ¼ 20 nm, R2 ¼ 20nm, CD ¼ 100nm, d ¼ 200nm.

Fig. 19. Comparison between ρ and ρ0 for different values of λ,(λ ∈ ½580; 620� nm). Numerical parameters: M ¼ 20, Nc ¼ 1500,α ¼ 88°, np ¼ 6, R1 ¼ 20nm, R2 ¼ 20 nm, CD ¼ 100nm, θ ¼ 0°,d ¼ 200nm.


5. APPLICATION TO CALCULATION OFTHE GRATING’S SCATTEROMETRICSIGNATUREThe scatterometric signature of a grating is a representation ofρ with respect to the wavelength. Its establishment consists incarrying out a direct electromagnetic calculation of ρ for eachwavelength. This important step also is very expensive interms of being time consuming, when one deal with a scattero-metry is the control. We propose in this section to consider-ably reduce the execution time of this step with the helpof the pertubation method. From the calculation of ρðλÞfor λ ¼ a and λ ¼ aþ η, we estimate ρ0ðcÞ, for all c ∈½a − 0:5η; aþ 0:5η�. However a major difficulty may limit thisapproach. Let us consider the grating illuminated by a planewave with a given wavelength λ angle of incidence θ. Let SðλÞdenote the scattering matrix of the grating. This matrix isformed of the eigenvectors of the operator L representedin the basis

enðxÞ ¼ e−i2πx sin θ=λe−i2πnx=d: ð28Þ

For a small variation η of λ, the eigenvectors of the operator Lare naturally represented in the basis

~enðxÞ ¼ e−i2πx sin θ=ðλþηÞe−i2πnx=d; ð29Þ

and a calculation of the perturbation matrix P ¼Sðλþ ηÞ − SðλÞ can seem to be unjustified. This remark re-mains valid for the angle of incidence. Those mathematical

considerations do not take the physical aspect of the issueinto account. Indeed, the determining factor while interpolat-ing ρðξÞ seems to be its local behavior; a slow variation of ρðξÞis the most favorable case. Figures. 16 and 17 show compara-tive results between ρðλÞ and ρ0ðλÞ (λ ∈ ½235; 245�nm) for twovalues of angle of incidence: θ ¼ 0° and θ ¼ 5°. By observingthose curves, we notice that the variation of ρðλÞ is more sig-nificant in normal incidence. In Fig. 18, we present ζðλÞ for θ ¼0° and θ ¼ 5°, with np ¼ 6, (η ¼ 2 nm). Contrary to the math-ematical forecasts, the results are less precise for an incidenceof 0°. Figures. 19 and 20 show comparative results betweenρðλÞ and ρ0ðλÞ (λ ∈ ½580; 620�nm) for θ ¼ 0° and θ ¼ 70°. AsFig. 21 shows, despite the significant gap between thosetwo values of angles of incidence, the same precision, at leastof 10−3, can be reached and those for a higher value of η(η ¼ 8 nm, np ¼ 6). This confirms that it is the local behaviorof the ρ function which strongly influences the validity of theperturbation approach. Indeed, in case of slight local varia-tions, in an indifferent point λ0, the perturbation term Pλ0 ¼Sðλ0 þ ηÞ − Sðλ0Þ remains low for a higher value of η.

6. CONCLUSIONScatterometry is a very precise nondestructive technical con-trol that offers large possibilities for an in situ control in realtime. This method requires a comparison between measure-ments realized from a multiwavelength ellipsometer and a di-rect calculation of electromagnetic signatures by changingprofile and electrical parameters. This step can take a largecalculation time. We have demonstrated that by couplingthe FMM with the pertubation method, to compute theeigenvectors and eigenvalues of the scattering matrix, the


Fig. 21. ζðλÞ, (λ ∈ ½580; 620� nm) for different values of the angle ofincidence θ ¼ 0°, θ ¼ 70°. Numerical parameters:M ¼ 20, Nc ¼ 1500,np ¼ 6, R1 ¼ 20 nm, R2 ¼ 20nm, CD ¼ 100nm, d ¼ 200nm.


establishment of the data library can considerably be acceler-ated. In the case of the profile parameter’s fluctuations, theproposed perturbation method is only limited by the ampli-tude of the fluctuations. Regarding the electrical parametersas wavelength, the calculation of the disrupted matrix canseem significant only if it does not modify the Fourier basis.Nevertheless these mathematical considerations do not takeinto account the physics of the problem. Indeed, the determin-ing factor seems to be the local behavior of the function ρ.

ACKNOWLEDGMENTSThe author wishes to thank Jean-Pierre Plumey for manyfruitful discussions.

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