Second harmonic field generated in reflection by an inhomogeneous nonlinear polarization

1 January 1998

Optics Communications 145 Ž1998. 135–140

Second harmonic field generated in reflection by an inhomogeneous nonlinear polarization A. Le Calvez ) , E. Freysz, A. Ducasse Centre de Physique Moleculaire Optique et Hertzienne, UniÕersite´ Bordeaux 1, 351 cours de la Liberation, Talence Cedex 33405, France ´ ´ Received 15 April 1997; revised 16 June 1997; accepted 23 June 1997

Abstract The second harmonic field reflected at an interface by a spatially inhomogeneous nonlinear polarization is calculated. We show that, through the spatial derivative of the nonlinear polarization, this spatial profile adds a new term to the reflected field calculated by Bloembergen and Pershan for a homogeneous nonlinear medium. The amplitude and phase of the additional term strongly alter both the amplitude and phase of the reflected field. The importance of this effect is illustrated in some meaningful physical cases. q 1998 Elsevier Science B.V.

1. Introduction Optical second harmonic generation ŽSHG. has been proven to be a most effective, versatile and non-destructive probe for surface and interface studies w1–8x. Surface SHG was first considered and formulated by Bloembergen and Pershan in 1962 w9x. In their pioneer work, these authors calculated the harmonic electric field generated by the interface separating a linear medium from a bulk nonlinear medium. In this paper, the reflected harmonic signal was deduced from the general equation of propagation in a spatially homogeneous nonlinear medium. However, in a number of theoretical or experimental cases, the nonlinear medium cannot be assumed as homogeneous within the layer of about one wavelength thickness which mainly contributes to the SH reflected wave. For instance, it is well known that the silica surface becomes negatively charged in neutral or basic aqueous solutions. This charged region results in the creation of a large electric field Ž E f 10 6 Vrcm. at the water–quartz interface. For such a large field, an SH signal can be generated at the interface by a third-order process. A similar mechanism may be at

the origin of the SHG in poled glasses. Since, in the two cases, the electric field varies rapidly close to the interface, the induced nonlinear polarization can no longer be considered as homogeneous and the formulation of Bloembergen and Pershan needs to be modified. However, to our knowledge, no theoretical calculation of the generated reflected harmonic field has been carried out in such a case. These calculations are important to deduce accurate information about the surface parameters. Therefore in this paper, we propose a generalization of the Bloembergen and Pershan method w9x to a nonlinear polarization varying along the normal to the interface. In the first part, the reflected harmonic field generated by such an interface is calculated. In the second part, the general expressions are applied to particular configurations that may be met in some experimental cases.

2. General case

2.1. Equation of propagation

)

E-mail: [email protected].

We will consider an interface between a linear medium Ž z - 0. and a second-order nonlinear medium Ž z ) 0. ŽFig.

0030-4018r98r$17.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 3 6 0 - X

A. Le CalÕez et al.r Optics Communications 145 (1998) 135–140

136

Ef Ž2 v . corresponds to the solution forced by the source term P Ž2. Ž2 v ,r,t .. Ef Ž2 v . has therefore an expression similar to P 2 Ž2 v ,r,t .. In particular, according to Eq. Ž1., the wave vector k s associated to Ef Ž2 v . is 2 k 1Ž v . Ž k 1Ž v . s ´ 2 Ž v . vrc .. We will restrict our study to plane and monochromatic waves. The fields associated respectively to the fundamental and to the second harmonic waves propagating in the nonlinear region are:

(

Fig. 1. Geometry of the interface between the linear and the nonlinear semi-infinite media. The wave vectors of the different waves are indicated.

E Ž v . s 12 w Ev exp i Ž k 1 P r y v t . q c.c. x ,

Ž3.

ET Ž 2 v . s El Ž 2 v . q Ef Ž 2 v . s 21 w ET Ž 2 v . q c.c x s 12 w A exp i Ž k 2 P r y 2 v t . q c.c. x q 12 E f q c.c. ,

Ž4. 1.. The dielectric constants of the two media are ´ 1 and ´ 2 , respectively. The second-order nonlinear polarization, ‘ P Ž2. Ž 2 v ,r ,t . s 12 x Ž2. Ž 2 v , v , v . E Ž v ,r ,t . E Ž v ,r ,t . , Ž1. is induced by the fundamental wave E Ž v ,r,t . s E Ž v . of frequency v . The incident wave vector of E Ž v ,r,t . is k 1 in the nonlinear region. The nonlinear polarization at the frequency 2 v radiates a harmonic wave both in transmission in the nonlinear zone ET Ž2 v ,r,t . s ET Ž2 v . and in reflection in the linear region ER Ž2 v ,r,t . s ER Ž2 v .. The electric field ET Ž2 v ,r,t . must satisfy the general equation of propagation in a nonlinear medium: = n w = n ET Ž 2 v ,r ,t . x q

sy

4p E 2 P Ž2. Ž 2 v ,r ,t . c2

Et 2

1 E 2 D L Ž 2 v ,r ,t . c2 ,

where c.c. denotes the complex conjugate quantity, 5 A 5 s const and A H k 2 with k 2 s ´ 2 Ž 2 v . Ž2 vrc . for the free wave. We will express the second-order polarization depending along the coordinate z as

(

P Ž2. Ž 2 v ,r ,t . s 12 P Ž2. Ž z . exp i Ž k S P r y 2 v t . q c.c s 12

1 2

x Ž2. Ž 2 v , v , v , z . Ev Ev q c.c . Ž 5 .

This expression shows that P Ž2. Ž z . can vary with z if either the second-order nonlinear susceptibility x Ž2. Ž2 v , v , v , z . or the amplitude Ev of the fundamental wave depends on z. The forced solution is then given by E f s C Ž z . exp i Ž k S P r y 2 v t . .

Et 2

Ž2.

where c is the light velocity and D LŽ2 v ,r,t . s ´ 2 ET Ž2 v ,r,t . is the linear displacement current. At an interface, crudely one may say that a layer of about one wavelength thickness contributes to the radiation of the reflected ray w9x. Therefore, the conversion efficiency is small enough to neglect the depletion of the fundamental wave. Thus, only the equation of propagation of the harmonic wave ŽEq. Ž2.. will be taken into account. The solution of Eq. Ž2. is here generalized to a nonlinear polarization depending on the coordinate z where zˆ is the axis normal to the surface. The existence of ER Ž2 v ,r,t . results from the boundary conditions at the interface z s 0 w9,10x. Eq. Ž2. is a second-order differential equation with constant coefficients and a non-null right-hand side. The general solution, therefore, consists of the solution of the homogeneous equation plus one particular solution: El Ž2 v . q Ef Ž2 v .. El Ž2 v . corresponds to the free harmonic wave, which is the solution of the linear equation and propagates with the wave vector k 2 . This wave behaves as if the medium was linear. The particular solution

Ž6.

It is important to notice that the introduction of the inhomogeneous nonlinear polarization P Ž2. Ž z . implies that the coefficient C Ž z . of the forced harmonic wave also varies with z. This constitutes the major difference from the work of Bloembergen and Pershan w9x. The forced field E f satisfies the following equation: = n = n E f q 4v 2 16pv 2 sy

c2

´2 c2

Ef

P Ž2. Ž z . exp i Ž k S P r y 2 v t . .

Ž7.

Finally, we can write ET Ž2 v . and ER Ž2 v . as SCET Ž 2 v . s 12  w A exp Ž ik 2 P r . q C Ž z . exp Ž ik S P r . x =exp Ž y2 i v t . q c.c 4 , SCER Ž 2 v . s

1 2

Ž8.

w ER Ž2 v . exp i Ž k R P r y 2 v t . q c.c x . Ž9.

According to Fig. 1, the Snell–Descartes laws impose that the three vectors k S s 2 k 1 , k R and k 2 lie in the plane of incidence Ž y, z ., and that: kR y s k2 y s kS y s k y .

Ž 10.

A. Le CalÕez et al.r Optics Communications 145 (1998) 135–140

137

2.2. Determination of the forced solution

2.3. Boundary conditions

We can deduce from Eq. Ž7. a system of three coupled differential equations:

The boundary conditions of the electromagnetic harmonic field Ž E, B . at the interface z s 0 imply the existence of a second harmonic wave ER Ž2 v . reflected in the linear medium. For a non-magnetic medium, they impose the continuity of the components E 5 and B 5 where 5 denotes the directions parallel to the surface Ži.e., x and y .. The expression of the magnetic field B is obtained from the Maxwell equation. For free waves like ER Ž2 v . and A, it is simply given by c Bsy =nE 2iv

Ž k S2 y k 22 . C x y 2 ik S z

dC x

d2 Cx

16pv 2

PxŽ2. Ž z . , dz d z2 c2 dC y dC z Ž k S2 z y k 22 . C y y 2 ik S z d z y k S z k y C z q ik y d z 16pv 2 s

c2

y

s

PyŽ2. Ž z . ,

yk 22 z C z y k S z k y C y q ik y

dC y

16pv 2 s

c2

dz

PzŽ2. Ž z . .

Ž 11.

c sy

The resolution of Eq. Ž11. leads to Cz Ž z . s

1

ik y

k 22 z

dC y dz

2v

1

y k S z k yC y y

16pv 2 c2

PzŽ2. Ž z . ,

Ž 12. Ci Ž z . s a i Ž z . exp w yi Ž k S z y k 2 z . z x q bi Ž z . exp w yi Ž k S z q k 2 z . z x ,

with is x or y, Ž 13.

k n Esy

i 2 k2 z i

bi Ž z . s

2 k2 z

H F Ž z . exp w i Ž k

Sz y k2 z

H F Ž z . exp w i Ž k

Sz q k2 z

i

i

16pv 2

Fy Ž z . s y

c2

Bf s y

k y C z y k S z C y q idC yrd z

c 2v

c 2 k 22

qik y

Again, it is important to notice that the inhomogeneity of C Ž z . modifies the expression of B f . Here again, the additional term idC x, yrd z is 908 out of phase with respect to the first usual term. The relationships of continuity for the components E 5 and B 5 yield four equations containing six unknown quantities ERi and A i Ž i s x, y, z .. For the free waves ER Ž2 v . and A, two additional relationships result from the Maxwell equation =E s 0:

. z x d z,

k 22 z PyŽ2. Ž z . y k S z k y PyŽ2. Ž z .

d PzŽ2. Ž z . dz

.

exp i Ž k S r y 2 v t . k S z C x y idC xrd z yk y C x

Ž 16.

PxŽ2. Ž z . ,

16pv 2

Ž 15.

. z x d z,

and Fx Ž z . s y

k z Ex yk y E x

2v

On the other side, the expression of the magnetic field B f associated to the forced wave E f is more complex since = n E f s w = n C q ik S n C x exp iŽ k S P r y 2 v t .. Then, we have

where ai Ž z . s y

k y Ez y k z E y

c

Azsy

Ž 14 .

From Eq. Ž14., we can see that the additional term proportional to the gradient d PzŽ2. Ž z .rd z, introduced by the inhomogeneity of the nonlinear polarization is 908 out of phase with respect to the other terms. Once the expression of the forced wave is known, the expressions for the amplitudes of both the free and reflected waves Ž A and ER Ž2 v .. can be determined by writing the boundary conditions.

ky k2 z

Ay,

ER z s

kR z

ER y ,

Ž 17.

where k R z is a geometric quantity so that k R z ) 0. The resolution of this set of six equations leads to the reflected harmonic field ER Ž2 v .: E xR s

1 k2 z q kR z

E yR s

Ž k 2 z y k S z . C x Ž0. q i

kR z k2 z k R2 k 2 z q k 22 k R z qk y C z Ž 0 . q i

1 The calculations are often simplified if one notices that: k 22 z q k 22 y s k 22 , k S2 z q k S2 y s k S2 and k 2 y s k S y so that k 22 y k S2 s k 22 z y k S2 y .

ky

EzR s

ky kR z

E yR ,

ž

k 22 k2 z

dC x dz

Ž0. ,

/

y k S z C y Ž0.

dC y Ž 0 . dz

with k R z ) 0,

,

Ž 18.

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138

where the functions C x Ž z ., C y Ž z ., C z Ž z . are given by Eqs. Ž12. – Ž14. and where C x Ž0., C y Ž0., dC x Ž0.rd z and dC y Ž0.rd z denote the corresponding quantities at z s 0. Eq. Ž18. clearly shows that the spatial variation of the nonlinear polarization adds a new contribution to the harmonic reflected wave. It is interesting to notice that to compute ER Ž2 v ., both the values of the nonlinear polarization and that of its first derivative at z s 0 are necessary. We will now compute these quantities for particular cases.

3. Examples Hereafter, we will examine some particular cases such as a homogeneous nonlinear medium to deduce the already known expression of ER Ž2 v . w9,10x, and then a nonlinear polarization varying either linearly or exponentially along z. 3.1. Homogeneous nonlinear medium If P Ž2. Ž z . s P Ž2. Ž0. and in consequence C Ž z . are constants, Eqs. Ž18. and Ž12. – Ž14. lead to the classical expressions of the reflected harmonic field. We will express here the s- and p-polarized fields ER0 sŽ2 v . and ER0 pŽ2 v . Žwhere s and p denote the polarizations perpendicular and parallel to the plane of incidence, respectively, and the index 0 indicates the homogeneous case.: ER0 s Ž 2 v . s y ER0 p Ž 2 v . s y

16pv 2 c

2

l CS k2 z q kR z k R l CS

c2

k R2 k 2 z q k 22 k R z

= k 2 z PyŽ2. Ž 0 . q k y PzŽ2. Ž 0 . ,

for example, in experiments using total reflection prisms. In this case, the forced wave must also be evanescent. However, the nonlinear polarization may produce travelling waves both in reflection Ž ER Ž2 v .. and transmission Ž A ., unless an unusual dispersion is present. In their paper, Bloembergen and Pershan w9x considered such a case but neglected the decreasing amplitude of the nonlinear polarization. The following equations Ž20. and Ž21. describe the nonlinear polarization components PiŽ2. Ž z . for the linear and exponential profiles, respectively ŽFig. 2a and 2b.. L denotes a characteristic length of the profile. PiŽ2. Ž z . s PiŽ2. Ž 0 .Ž 1 y zrL . , PiŽ2.

Ž z.

s PiŽ2.

i s x , y or z ,

Ž 20.

Ž 0 . exp Ž yzrL . , i s x , y or z.

Ž 21.

The expressions of C x and C y are then obtained from Eqs. Ž12. – Ž14. while Eq. Ž18. gives the harmonic reflected field ER Ž2 v .. The solution for the linear profile is ` 0 S ERlinear s, p Ž 2 v . s ER s, p Ž 2 v . Ž 1 y il C rL . ,

PxŽ2. Ž 0 . ,

16pv 2

Fig. 2. Ža. Nonlinear polarization depending linearly on z. Žb. Nonlinear polarization depending exponentially on z.

Ž 22.

where ` denotes a semi-infinite medium, while the solution for the exponential profile is

ž

ERexponential Ž 2 v . s ER0 s, p Ž 2 v . 1 q s, p with k R z)0,

Ž 19. where l CS s 1rŽ k S z q k 2 z . stands for the length of coherence for a surface second harmonic phenomenon. 3.2. Case of a nonlinear polarization depending linearly or exponentially on z Such situations are of particular interest. For example, the linear case can be met in a sample charged like a capacitor in which the created electrostatic field E0 presents a linear profile in function of depth. This field can Ž2. Ž . induce an effective second-order nonlinearity xeff z s Ž3. x E0 Ž z . in an isotropic medium. This phenomenon may be at the origin of SHG in poled vitreous materials w11,12x. An exponentially decreasing nonlinear polarization can be created either by an exponentially decreasing nonlinear susceptibility or by an evanescent wave exciting a homogeneous nonlinear medium. The latter situation can be met,

1 iLrl CS y 1

/

.

Ž 23 .

The expressions Ž22., Ž23. show that the reflected harmonic field consists of the sum of the harmonic field reflected by a homogeneous nonlinear medium and an additional term introduced by the spatial derivative of the nonlinear polarization. As noticed already, this term has its own phase. The relative phase of ER s, p Ž2 v . with respect to ER0 s, p Ž2 v . depends on the spatial profile of P Ž2. Ž z .. Moreover, it is important to notice that the ratio Lrl CS is a crucial parameter. Indeed, if l CS is much smaller than L, the nonlinear polarization is nearly constant over the distance l CS and the second harmonic reflected field ER Ž2 v . is nearly equal to the field generated by a homogeneous nonlinear medium ER0 Ž2 v .. On the contrary, if l CS is much greater than the length L, then ER Ž2 v . is quite different from ER0 Ž2 v .. For instance, in the exponential case, if l CS 4 L, ER Ž2 v . tends towards 0. For the linear case, notice that when L becomes very small, the nonlinear polarization is very great within a slab of thickness l CS . Therefore, the corresponding reflected harmonic field

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139

rapidly increases. A more physical situation would correspond to a nonlinear slab in which the profile is linear ŽFig. 3b.. The expression of P Ž2. Ž z . can then be written as PiŽ2. Ž z . s PiŽ2. Ž 0 .Ž 1 y zrL . , if 0 - z - L, PiŽ2.

i s x , y or z ,

Ž z . s 0, if z ) L.

Ž 24. Ž2. Ž

The discontinuity of the first derivative of P z . implies the existence of a new interface in z s L and in consequence the appearance of two new free waves resulting from the reflection and the transmission of the free harmonic wave El Ž2 v . ŽFig. 3a.. Writing boundary conditions both at z s L and at z s 0 leads to the expression of the reflected harmonic field ERlinear Ž2 v .. If we consider only the interferences between the harmonic fields reflected by the first and second nonlinear interfaces, we obtain 0 ERlinear s, p Ž 2 v . s ER s, p Ž 2 v . 1 y i

l CS L

ž

L

ž //

1 y exp i

l CS

.

Fig. 4. Normalized intensity IR r IR0 of the reflected harmonic waves as a function of the normalized length Lr l CS . IR0 s < ER0 Ž2 v .< 2 is the second harmonic intensity generated by an homogeneous semi-infinite nonlinear medium. The three curves correspond respectively to a homogeneous nonlinear slab ŽP P P., to an exponentially decreasing nonlinear polarization Ž – – – . and to a slab of linearly decreasing nonlinear polarization Ž — — ..

Ž 25. Expression Ž25. shows that if L is much greater than l CS , ERlinear Ž2 v . is nearly equal to ER0 Ž2 v .. On the contrary, if L < l CS , the harmonic reflected field ER Ž2 v . now tends as expected towards zero. Indeed, this configuration corresponds to a nonlinear slab of negligible thickness, so the associated reflected harmonic field must be negligibly small.

Fig. 3. Ža. Geometry of an interface between a linear medium and a slab of nonlinear medium embedded in a semi-infinite region of dielectric constant ´ 2 . The discontinuity of the nonlinear polarization implies the appearance of both a reflected and a transmitted wave at the second interface at z s L. Žb. Linear profile of the nonlinear polarization inside the slab w z s 0, z s L x.

In Fig. 4, we have quantified the effect induced by the inhomogeneity of the nonlinear polarization. The normalized intensity IRrIR0 of the reflected harmonic waves assoexponential Ž Ž . ciated to ERlinear 2 v . are plotted as a s, p 2 v and ER s, p function of the normalized length Lrl CS . IR0 s < ER0 Ž2 v .< 2 is the second harmonic intensity generated by a homogeneous semi-infinite medium. In comparison, we have also plotted the normalized intensity that would be reflected by a homogeneous nonlinear slab. The observed oscillations of the intensity are related to the interferences between the two harmonic waves reflected by the two nonlinear interfaces. However, for the linear profile, both a phase shift and a decrease in the oscillations are visible. The phase shift is due to the previously discussed relative phase of the additional term ŽEqs. Ž22. and Ž23... The decrease in the contrast is connected to the smaller amplitude of the field reflected by the second nonlinear interface. As can be seen in Fig. 4, the amplitude of the reflected harmonic wave strongly depends on the spatial profile of the nonlinear polarization. In consequence, a quantitative calibration of the nonlinearity of the sample, which needs an accurate calculation of ER Ž2 v ., implies the knowledge of the profile P Ž2. Ž z .. However, this parameter is rarely known in real experimental situations. This shows that a theoretical modelling of the physical process at the origin of the second-order nonlinearity is often necessary. In fact, this conclusion is also valid for SHG experiments performed in transmission.

4. Conclusion We have proposed an extension of Bloembergen and Pershan’s calculation of the harmonic field reflected by a

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A. Le CalÕez et al.r Optics Communications 145 (1998) 135–140

nonlinear medium. By assuming a spatial dependence of the nonlinearity, we have shown that the spatial derivative of the nonlinear polarization adds a new term to the reflected harmonic field. The amplitude and phase of this field strongly depend on both the amplitude and phase of this additional term. This effect has been illustrated for some meaningful physical cases. Our procedure can be easily extended to other experimental situations. Moreover, the method could also be applied to an interface third-order nonlinear process.

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