A new method for the determination of the surfacesecond-order nonlinear susceptibility

X. Qu�eelina,*, A. Bourdonb

a Laboratoire de Magn�eetisme et d’Optique de Versailles (UMR 8634), Universit�ee de Versailles St Quentin en Yvelines,

45 Avenue des Etats-Unis, Versailles Cedex 78035, Franceb Laboratoire des Milieux D�eesordonn�ees et H�eet�eerog�eenes (UMR 7603), Universit�ee Paris 6 (case 86), 4 Place Jussieu,

75252 Paris Cedex 05, France

Received 9 October 2001; received in revised form 24 January 2002; accepted 1 May 2002

Abstract

A method for determining the complex tensor elements completely, except for only one arbitrary phase, of the

surface second-order nonlinear susceptibility of a nonlinear medium is given from second harmonic intensity mea-

surements. These measurements are performed in a reflection geometry by taking three different angular variables (two

polarization and one incidence angles). Experimental data are processed by a new numerical method involving no a

priori symmetry for the medium under study. It consists in: (i) determining intermediate field-type data from intensity

results at constant incidence angle and (ii) a two-stage iterative procedure that provides complex values for vð2Þ. � 2002

Elsevier Science B.V. All rights reserved.

1. Introduction

In the previous works on the determination ofthe values of surface nonlinear susceptibility tensorvð2Þ elements, studied samples were chosen with acrystalline surface exhibiting a definite nontrivialsymmetry [1–3]. Methods used were based on thefacts that, due to the surface symmetry, most ofthe vð2Þ elements were zero whereas some otherswere not independent. In previous papers, anazimutal analysis was used in s and p geometries[1–4]. The aim of the present work is to provide a

general method for the determination of the sur-face nonlinear susceptibility tensor vð2Þ of a samplefrom second harmonic-intensity measurementswithout making any assumption on its symmetry.To obtain the most complete possible determina-tion of the 18 complex elements of vð2Þ, we analyzefirst the second harmonic intensity as a function ofthe direction of the input polarization and the di-rection of the output second harmonic analyzer, atconstant incidence angle, which allows, as it will beshown hereafter, the use of a new two-stage de-termination. To achieve the 3D study of vð2Þ, theincident angle of the input beam is varied, which isused for the second stage of the evaluation of vð2Þ.It also avoids the experimental difficulty of anazimutal variation.

1 July 2002

Optics Communications 208 (2002) 197–204

www.elsevier.com/locate/optcom

*Corresponding author. Tel.: +33-1-39-25-46-59; fax: +33-1-

39-25-46-52.

E-mail address: quelin@physique.uvsq.fr (X. Quelin).

0030-4018/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0030-4018 (02 )01566-3

The second harmonic intensity being propor-tional to the squared modulus of the output elec-tromagnetic field, there is an inherent uncertaintyon the output E.M. field phase which is reflected inthe determination of the vð2Þ elements. This facthas not been forgotten neither in the previousworks nor in the present one where this phaseuncertainty is minimized.

The experimental technique considered below isSHG in a reflection geometry which is the easiestway of studying surface properties. However ourmethod can be easily transposed in a transmissiongeometry.

2. Theoretical analysis

In this section, we give the expression of thepolarization in a thin layer of a nonlinear mediumsandwiched between two semi-infinite isotropiclinear media (Fig. 1) and then that of the macro-scopic electromagnetic fields it generates. Thefundamental and second harmonic electromag-netic fields will be noted EðxÞ, E andH, respectively

(omitting 2x for the sake of simplicity). Boundaryconditions require that the electrical field compo-nents along the interface ðOx;OyÞ and the dis-placement current component along the surfacenormal ðOzÞ are continuous across the interfacelayer. If we assume no particular symmetry for thesample and in the case of a linear input polariza-tion, the nonlinear polarization P in the nonlinearthin layer can be expressed, as shown by B€oohmeret al. [5], as follows:

PxðhÞPyðhÞPzðhÞ

0B@1CA ¼

vxxx vxyy vxzz vxyz vxxz vxxyvyxx vyyy vyzz vyyz vyxz vyxyvzxx vzyy vzzz vzyz vzxz vzxy

0B@1CA

A cos2 u1

B sin2 u1

C cos2 u1

2D cosu1 sinu1

2E cos2 u1

2F cosu1 sinu1

0BBBBBBBB@

1CCCCCCCCAjEðxÞ

0 j2 ð1Þ

with

A ¼ t2pf2c ; B ¼ t2s ; C ¼ enn2enn

� 4

t2pf2s ;

D ¼ enn2enn� 2

tstpfs; E ¼ enn2enn� 2

t2pfsfc; ð2Þ

F ¼ tstpfc;

where u1 denotes the angle between the linearpolarization direction and the incidence plane(Fig. 1). EðxÞ

0 is the incident electric field amplitudebefore the input polarizer. fc;s are the Fresnel co-efficients and ts;p are the linear transmission coef-ficients for the fundamental beam. The complexindices of refraction at the fundamental frequencyx are denoted enn ¼ nþ ik for the nonlinear thinlayer, ennj ¼ nj þ ikj with j ¼ 1 for the infiniteincoming medium and j ¼ 2 for the infinitebulk medium. The Fresnel coefficients are ex-pressed as

fs ¼enn1enn2 sin h and fc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� f 2

s

q; ð3Þ

where h denotes the incidence angle of the in-coming beam. The linear transmission coefficientsare given by the following expressions:

Fig. 1. Experiment geometry and polarization directions of the

incoming ðxÞ and reflected ð2xÞ beams. h denotes the incidence

angle while angles u1 and u2 give the direction of the linear

polarization of the incident beam and that of the polarizer used

to analyze the reflected second-harmonic beam, respectively

(u1 ¼ 0� corresponds to a p type incident polarization and

u1 ¼ �90� to a s one). The ðOzÞ direction is defined by the

normal to the surface while the ðOy;OzÞ plane is defined by

the incident plane; the ðOx;OyÞ plane is therefore parallel to the

interface.

198 X. Qu�eelin, A. Bourdon / Optics Communications 208 (2002) 197–204

ts ¼2 cos h

cos h þenn2enn1 fc and tp ¼2 cos henn2enn1 cos h þ fc

: ð4Þ

Let us denote E1 and E2 as the reflected and thetransmitted harmonic fields, respectively. By tak-ing the fact that the electric field component Ezalong z varies abruptly across the layer into ac-count, the boundary conditions for a thin nonlin-ear interface, determined by Heinz [6], areexpressed as:

E2x � E1x ¼ � 1

e 2xð ÞoPz

ox;

E2y � E1y ¼ � 1

e 2xð ÞoPz

oy;

ð5Þ

H2x � H1x ¼oPy

ot;

H2y � H1y ¼oPx

ot

ð6Þ

and

e2E2z � e1E1z ¼ � oPx

ox

�þ oPy

oy

; ð7Þ

where

Pl ¼ Pl exp ix2enn1 sin h

cx

��� 2t

ðl ¼ x; y; zÞ;

ej is the dielectric constant at frequency 2x ofmedium j and H is related to the magnetic field Bby the usual linear isotropic response H ¼ l�1B.The complex index of refraction emmj is related to ejby ej ¼ emm2

j . Using Eqs. (5)–(7), the in-plane of in-cidence component E1p and the out-of-planecomponent E1s of the reflected second harmonicfield can be expressed as:

E1p ¼i

e0

� 2xc

� 1emm2Fc1 þ emm1Fc2

� jEðxÞ

0 j2

� fFc2½vxxxA cos2 u1 þ vxyyB sin2 u1

þ vxzzC cos2 u1 þ vxyzD sin 2u1

þ 2vxxzE cos2 u1 þ vxxyF sin 2u1�þ G½vzxxA cos2 u1 þ vzyyB sin

2 u1

þ vzzzC cos2 u1 þ vzyzD sin 2u1

þ 2vzxzE cos2 u1 þ vzxyF sin 2u1�g ð8Þ

and

E1s ¼i

e0

� 2xc

� 1emm1Fc1 þ emm2Fc2

� jEðxÞ

0 j2

� ½vyxxA cos2 u1 þ vyyyB sin2 u1

þ vyzzC cos2 u1 þ vyyzD sin 2u1

þ 2vyxzE cos2 u1 þ vyxyF sin 2u1�; ð9Þ

where e0 is the vacuum permittivity,

Fc1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� enn1emm1

sin h

� 2s

;

Fc2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� enn1emm2

sin h

� 2s

and

G ¼ enn1 emm2emm2

� sin h:

3. Two-stage determination of v

In this section, it is shown how the 18 inde-pendent complex elements of the surface second-order nonlinear tensor v can be completely deter-mined, except for only one arbitrary phase. Forthis purpose, we consider an experimental setup inwhich the incident beam is linearly polarized andthe reflection-output beam is linearly analyzed. Wepropose the following two-stage determination ofv from second harmonic-intensity measurementsmade by varying three different parameters: (i) u1

the incident linear polarization direction, (ii) u2

the output analyzed linear polarization direction,and (iii) h the incidence angle. The first stageconsists in the determination of intermediate data(only six complex numbers), that connect funda-mental and second harmonic fields to intensityvalues; this calculation requires a nonlinear de-termination technique. The last stage provides vijkvalues by a linear iterative procedure from theseintermediate data.

3.1. Phenomenological SHG study at constantincidence angle

In any ‘‘weak’’ SHG phenomenon in a reflec-tion geometry at constant incidence angle, theoutput field can be connected to the input field byan expression of the following type:

X. Qu�eelin, A. Bourdon / Optics Communications 208 (2002) 197–204 199

Ei ¼ Mijk EðxÞj EðxÞ

k ; ð10Þ

where i, j, and k are taken among the values 1 and2 representing the input s and p polarization states,respectively. Eqs. (8) and (9) are both consistentwith Eq. (10) formulation. As EðxÞ

j and EðxÞk are

indistinguishable ðMijk ¼ MikjÞ, we take the usual2D compact notation Mil where l ¼ 1; 2 and 3stands for ðj; kÞ ¼ ð1; 1Þ; ð2; 2Þ and ð1; 2Þ, respec-tively. In this first stage, we will show how thematrix M can be determined by a set of intensitymeasurements under constant incidence. One cannotice that M depends indeed on the incidenceangle h. In this case, Eq. (10) can be written as

Ei ¼ ½EðxÞ0 �2Milal; ð11Þ

where a is a column vector defined by

a ¼cos2 u1

sin2 u1

sin 2u1

0@ 1A: ð12Þ

Similarly, the second harmonic output field is lin-early analyzed, which gives the following ampli-tude:

E ¼ Ep cosu2 þ Es sinu2: ð13ÞIn a more compact way, Eq. (13) reduces to

E ¼ ½EðxÞ0 �2bi Mil al ð14Þ

with b given by

b ¼ cosu2

sinu2

� : ð15Þ

The output second harmonic intensity is givenby I ð2xÞ ¼ K EE where K is the conventional fac-tor proportional to the laser-beam sectionr ðK ¼ 1

2re0cÞ. If cn is defined as cn ¼ albi [Eqs.

(12) and (15)], where n ¼ lþ 3ði� 1Þ, with16 n6 6, I ð2xÞ is expressed as:

I ð2xÞ ¼I ðxÞ0

h i2K

Mncnj j2; ð16Þ

where Mn is a vector component defined byMn ¼ Mil with the same index notation as that ofcn. As cn are real, it gives for I ð2xÞ the followingquadratic form:

I ð2xÞ ¼ ½I ðxÞ0 �2

K½ðReðMnÞcnÞ

2 þ ðImðMnÞcnÞ2�: ð17Þ

As the fundamental and second harmonic fieldsare observed in the same nondispersive medium 1,no incidence-angle correction has to be done on K.If I ð2xÞ is measured from enough distinct values ofu1 and u2, a M exp complex vector can easily bedetermined, using Eq. (17), by a nonlinear best-fitprocedure (Levenberg–Marquardt method for in-stance [7]). Actually in the determination of M exp,there still remains one arbitrary phase per inci-dence angle that is studied below.

3.2. Connection between the phenomenological vec-tor M and dipolar nonlinear characteristics

The two theoretical expressions (8) and (9) ofthe electric dipolar radiation can be unified in thefollowing formula:

Ei ¼ ½EðxÞ0 �2Qiklakvl; ð18Þ

where vl is a nonlinear surface electric suscepti-bility vector defined in two steps: first by taking theusual 3D symmetric compact notation transform-ing vijk into vin½16 n6 6�, secondly by letting l ¼nþ 6ði� 1Þ ½16 l6 18�. Identifying Eq. (11) toEq. (18) 8ak, givesMjk ¼ Qjklvl: ð19ÞSimilarly, by gathering the two indices j and k intoa one-dimensional index m ½m ¼ k þ 2ðj� 1Þ�,Eq. (19) becomes

Mm ¼ Qmlvl: ð20ÞBy writing explicitly the six Eqs. (20) with ½16m6 6�, we find six orthogonal subspaces of the v18D space, each of them corresponding to onevalue of m. The expressions defining these sub-spaces are given in Appendix A; their dimensionsare 6, 2, 4, 3, 1, and 2 for m ranging from 1 to 6,respectively, which gives back indeed a 18D spacefor v. Quantities Qml defined by Eqs. (A.1) and(A.2) can be evaluated from linear optics charac-teristics of the sample and geometrical parametersof the experiment.

3.3. Iterative determination of v from a set of M exp

vectors at various incidence angles

We have now experimental values for M expm and

a theoretical expression [Eq. (20)] for Mm which is

200 X. Qu�eelin, A. Bourdon / Optics Communications 208 (2002) 197–204

linear with respect to vl. It could seem easy todetermine vl from the set of M exp

m ðhqÞ and QmlðhqÞwhere q indexes the incidence angle, by minimizingthe following quantity

Pq jM exp

m ðhqÞ � QmlðhqÞvlj2.

However M expðhqÞ matrices are determined with anarbitrary phase, at each incidence angle. We pro-pose to minimize the quantity ST defined as thefollowing sum over all the subspaces of the vspace:

ST ¼Xm

Sm ð21Þ

with

Sm ¼Xq

jM expm ðhqÞ expði/qÞ � QmlðhqÞvlj

2 ð22Þ

with respect to both vl and /q. As there are manyfitting parameters of different types, linear or ex-ponential, we choose an iterative procedure byfitting alternatively vl at fixed /q, then /q at fixedvl, and then vl at fixed /q again and so on. At thefirst iteration step, we simply take /q ¼ 0 8q.

3.3.1. Linear best-fit determination of vl at fixed /q

If the evaluations of M exp are made at enoughdifferent hq incidence angles, vl can be evaluated byminimizing Sm with respect to vl and v

l (assumedto be independent variables). Minimizing ST isuseless because each vl appears only in one Sm

term. It gives the following linear equations for vlfor a given m subspace

Pl0lvl ¼ Wl0 ; ð23Þwhere

Pl0l ¼Xq

Q ml0 ðhqÞQmlðhqÞ ð24Þ

and

Wl0 ¼Xq

M expm hq

� �exp i/q

� �Q ml0 hq� �

; ð25Þ

without any summation on m. If the matrix P isnot singular, we find v ¼ P�1W . As the highestdimension of the v subspaces is 6, I ð2xÞ must bemeasured under at least six different incidenceangles h to obtain a set of nonsingular P matrices.Otherwise if the P matrix is singular, it is howeverstill Hermitian; v and W can then be expanded ascombinations of eigenvectors V j of P (V j corre-

sponding to the eigenvalues ej of P): v ¼ xjV j andW ¼ wjV j. It gives then for v: vl ¼ V j

l xj wherexj ¼ wj=ej if ej 6¼ 0, and xj is left undetermined forej ¼ 0 (any arbitrary choice for xj leaves v un-changed in this case).

3.3.2. Phase optimization of M expðhqÞ at fixed vlIn the second side of the iteration process, ST

has to be minimized with respect to /q at constantvl. Differentiating the sum ST , at constant vl, withrespect to /q gives

oST

o/q¼ i e�i/qAq

h� ei/qA

q

ið26Þ

and

o2ST

o/2q

¼ e�i/qAqh

þ ei/qA q

i; ð27Þ

where

Aq ¼Xm

M exp m hq

� �Qml hq

� �vl: ð28Þ

If Aq is denoted by jAqjeiwq , the extremum condi-tion ðoST=o/qÞ ¼ 0 gives /q ¼ wq þ kp (k isinteger) while the minimization conditiono2ST=o/

2q > 0 requires k to be even. Therefore, we

obtain

/q ¼ wq: ð29Þ

Values of /q determined according to Eq. (29)should be sent back into Eqs. (23) and (25) which,in turn, would give back a determination of vl andso on. But the very simple relation Eq. (29) couldseem, at a first glance, to be in contradiction withthe inherent uncertainty on the output E.M. fieldphase mentioned above. Actually it is not the case:if an arbitrary phase shift d/ is applied to all theM expðhqÞ vectors (/q changed in /q þ d/ in Eq.(25)), the same phase shift appears in all the vlcomponents determined afterward through Eq.(23) and further on in all the Aq terms (Eq. (28)).Therefore the iteration process described above, isnot perturbed by an arbitrary overall phase shiftd/ on /q. To observe the convergence of the it-erative process easily, we have taken advantage ofthis property by fixing, at each iteration step, thephase shift d/ that makes real an arbitrary chosencomponent vl0 of v. In the hope of minimizing the

X. Qu�eelin, A. Bourdon / Optics Communications 208 (2002) 197–204 201

observed variations of vl during the iterationprocess, we have chosen for l0 the component thathas the largest modulus. During the iterationprocess, the convergence of the arguments of the velements is seen to be slower than that of theirmodules.

3.4. Remarks about the present method

The experimental noise can be very much re-duced by bringing a large redundancy (morecurves than necessary and much more points thannecessary in each curve) in the nonlinear best-fitdetermination of M exp. Gathering all the redun-dant intermediate data M exp for the iterative linearbest-fit determination of v reduces the uncertaintyon v too.

The sensitivity of the method to external pa-rameters, such as the linear optical properties ofthe two media on both sides of the interface, de-pends on the characteristics of the whole sample. Itis very easy to be estimated: as it takes less than 5min to determine v from M exp with our Mapleprogram, an estimation of the sensitivity to all theparameters would be made in less than 1 h.

4. Experimental probe

This method has been probed on a 2D materialexhibiting no symmetry. Our sample, deposited byreactive radio-frequency sputtering on a float glassplate, is a metal–dielectric composite with Aunanosized grains embedded in a TiO2 host. Thethickness of the studied sample measured by pro-filometry technique and X-ray diffraction is 139nm. The experimental setup is shown in Fig. 2: theNd:YAG laser used is Q-switched and mode-locked (pulse-duration 80 ps). Typical intensityresults are given in Fig. 3 for an angle of incidenceof 30�. Experimental measurements have beenperformed according to the following procedure:19 values of linear incident polarization angle u1

and 4 values of u2 the output analyzed linear po-larization direction for each angle of incidence andthree different values of incident angle h (30�, 50�and 75�). For each angle of incidence, a M vectorhas been determined according to the nonlinear

Fig. 2. Experimental setup for the SH measurements in reflec-

tion geometry. (A) Denotes a Glan–Thomson polarizer used,

together with the first rotating polarizer, for controlling the

incident power; (B) denotes successive alignment pinholes and

(C) corresponds to a half-wave plate for the incident wave-

length (k ¼ 1:064 lm) used for rotating the linear polarization

of the incident beam.

Fig. 3. Typical SH experimental intensity curves at a fixed

angle of incidence (h ¼ 30�), as functions of the incident po-

larization angle u1, for four different output analyzer angles u2:

(d) for u2 ¼ 0� (p polarization); (N) for u2 ¼ �45�; (j) for

u2 ¼ þ45� and (r) for u2 ¼ 0� (s polarization). It is clearly

seen that the s output intensity is much weaker than the p one.

At u2 ¼ �45�, the two curves are significantly different which

indicates a lack of C2 symmetry around the normal to the

surface. It can be noticed that the following relation between

the four curves I2xðu1;u2 ¼ �45�Þ þ I2xðu1;u2 ¼ þ45�Þ ¼I2xðu1;u2 ¼ 0�Þ þ I2xðu1;u2 ¼ 90�Þ deduced from Eq. (17) is

verified. To each experimental plot corresponds a best-fit curve

obtained by our method.

202 X. Qu�eelin, A. Bourdon / Optics Communications 208 (2002) 197–204

best-fit method described in Section 3.1 using aFortran program. Afterwards the components ofvl have been obtained with the set of theM vectorsusing the iterative process described above withthe help of a Maple program. The iteration processconverges well without oscillating; after about1000 steps, the best-fit ST value has relative vari-ations of 10�10. This procedure gives back thenumerical intensity curves which are very similarto the experimental points.

5. Conclusion

We have presented above a general numericaldetermination, in the case of electric dipolar ap-proximation, of the surface vð2Þ tensor from secondharmonic-intensity measurements. It is based onan experimental and theoretical system in which anonlinear very thin layer lies at the interface be-tween two linear semi-infinite media. Through aset of I ð2xÞ measurements (at least six different in-cidence angles h, four different polarization direc-tions u1, and three different angles u2, whichimplies at least 72 I ð2xÞ different measurements) ourmethod provides an almost complete determina-tion of vð2Þ.

The 3D angular scanning is complete enough toleave only one unavoidable undetermined phase,due to the character of intensity measurements,over the phases of the 18 vð2Þ elements. This un-determined phase cannot be defined in an absoluteway but can be evaluated relatively to the phase ofa vð2Þ element of a well-known crystal, KDP orquartz for instance, as it is done in the wholeprevious literature. Nevertheless it is possible togive an absolute phase by using a nonlinear ref-erence medium whose vð2Þ elements have knownphases. It is the case of media transparent at x and2x whose vð2Þ elements are real as shown byBloembergen [8].

Despite its apparent heaviness, we believe thatour method could be useful in many cases, inparticular, when the crystallographic structure ofthe interface is not known. Some benign lacks ofgenerality (isotropic surrounding linear media,nondispersive incident medium) can be noticed inthis presentation. However the completely general

case can be treated using the same method: (i) thesame previous nonlinear determination of the M exp

data followed by (ii) an iterative linear determi-nation of v similar to the one given here but withmore complicated formulas.

We think this two-stage procedure could be alsoused in the case of a quadrupolar radiation. Thesame determination of M exp could be used,whereas Section 3.2 (Q terms) should be very dif-ferent.

Acknowledgements

The authors wish to thank J. Sakars for fruitfuldiscussions, E. Verduijn for his collaboration onFortran program and Prof. P. Gadenne for havinginitialized our collaboration.

Appendix A. Relations between v and the six Mvector elements

Writing explicitly Eq. (20), the linear equationsconnecting the 18 elements vl to the 6 matrix ele-ments Mm are:

M1 ¼ RTfFc2½Av1 þ Cv3 þ 2Ev5� þ G½Av13 þ Cv15

þ 2Ev17�g;M2 ¼ RT ½Fc2Bv2 þ GBv14�;M3 ¼ RTfFc2½Dv4 þ F v6� þ G½Dv16 þ F v18�g;M4 ¼ RU ½Av7 þ Cv9 þ 2Ev11�;M5 ¼ RUBv8;

M6 ¼ RU ½Dv10 þ F v12�;ðA:1Þ

where

R ¼ i

e0

� 2xc

� ;

T ¼ 1emm1Fc2 þ emm2Fc1;

U ¼ 1emm1Fc1 þ emm2Fc2:

ðA:2Þ

Factors A, B, C, D, E, and F are defined in Eq. (2)and G is defined after Eq. (9). The six subspaces ofthe v space defined by each value of m (1 to 6), are

X. Qu�eelin, A. Bourdon / Optics Communications 208 (2002) 197–204 203

obviously orthogonal. It enables the determinationof vl belonging to the same subspace separatelyfrom that of vl belonging to other ones.

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