**2.1. Fundamentals of nonlinear optics**

Radiation-matter interaction processes lead to numerous effects due to light-induced polari‐ zation, which acts as a source of new electromagnetic waves. The polarization induced by the electric field of the electromagnetic wave, *E* → (*r* <sup>→</sup> , *t*), is responsible for several optical processes. If this electric field is relatively weak in comparison to the electrostatic field acting on electrons within an atom or molecule, the polarization has a linear dependence on the electric field:

$$
\vec{P} = \chi \vec{E},
\tag{1}
$$

where χ is the linear susceptibility and *P* <sup>→</sup> is the linear polarization. This is the linear optics regime.

However, if the electric field of the electromagnetic wave is strong, new contributions turn out to be significant. Then, an approximate relation between electric field (cause) and polarization (effect) is a power series in *E* → (*r* <sup>→</sup> , *t*):

$$
\vec{P} = \vec{\mathcal{X}}^{(1)} \cdot \vec{E} + \vec{\mathcal{X}}^{(2)} : \vec{E}\vec{E} + \vec{\mathcal{X}}^{(3)} : \vec{E}\vec{E}\vec{E} + \dots \\
\equiv \vec{P}^{(1)} + \vec{P}^{(2)} + \vec{P}^{(3)} + \dots \tag{2}
$$

$$\begin{aligned} P &= \boldsymbol{\chi}^{(1)} E\_0 \cos \boldsymbol{\alpha} t + \boldsymbol{\chi}^{(2)} \left( E\_0 \cos \boldsymbol{\alpha} t \right)^2 + \boldsymbol{\chi}^{(3)} \left( E\_0 \cos \boldsymbol{\alpha} t \right)^3 + \dots \\ &= \boldsymbol{\chi}^{(1)} E\_0 \cos \boldsymbol{\alpha} t + \boldsymbol{\chi}^{(2)} E\_0 \left( \cos \boldsymbol{\alpha} t \right)^2 + \boldsymbol{\chi}^{(3)} E\_0 \left( \cos \boldsymbol{\alpha} t \right)^3 + \dots \end{aligned} \tag{3}$$

$$P = \mathcal{X}^{(i)} E\_0 cosot + \mathcal{X}^{(2)} \frac{1}{2} E\_0^{\ 2} \left( 1 + cos 2ot \right) + \mathcal{X}^{(3)} \frac{1}{4} E\_0^{\ 3} \left( 3 cosot + cos 3ot \right) + \dots$$

$$\Rightarrow P = \mathcal{X}^{(1)} E\_0 cosot + \mathcal{X}^{(2)} \left[ \frac{1}{2} E\_0^{\ 2} + \frac{1}{2} E\_0^{\ 2} cos 2ot \right] + \dots \tag{4}$$

$$\mathcal{X}^{(3)} \left[ \frac{3}{4} E\_0^{\ 3} cosot + \frac{1}{4} E\_0^{\ 3} cos 3ot \right] + \dots$$

$$P^{(2)} = \mathcal{X}^{(2)} \left( E\_t \cos \alpha\_t t + E\_2 \cos \alpha\_2 t \right)^2 = $$

$$= \mathcal{X}^{(2)} \left[ E\_1^{(2)} + E\_2^{(2)} + E\_1^2 \cos 2\alpha\_1 t + E\_2^{(2)} \cos 2\alpha\_2 t + \frac{1}{2} E\_1 E\_2 \cos \left( \alpha\_1 - \alpha\_2 \right) t + \frac{1}{2} E\_1 E\_2 \cos \left( \alpha\_1 + \alpha\_2 \right) t \right]. \tag{5}$$

$$P\_i\left(\alpha\_3 = \alpha\_1 + \alpha\_2\right) = \sum\_{j,k} \chi\_{jk}^{(2)}\left(\alpha\_3 = \alpha\_1 + \alpha\_2; \alpha\_1, \alpha\_2\right) E\_j\left(\alpha\_1\right) E\_k\left(\alpha\_2\right) \tag{6}$$

$$\mathbf{p}\_l = \alpha\_{lm} E\_m + \beta\_{lmn} E\_m E\_n + \gamma\_{lmn} E\_m E\_n E\_o + \dots \tag{7}$$

In Equation (7), αlm is the (first order) linear polarizability, βlmn and γlmno are the second- and third-order hyperpolarizabilities, respectively.

Through second-order time-dependent perturbation theory [17], we can derive an expression for βlmn that is quite complicated. But a simplification happens when we consider the case of SHG with the fundamental frequency ω away from electronic resonances, but with the secondharmonic 2ω resonant with an electronic transition of the molecule, ωvg [22]:

$$
\beta\_{lmw} = \frac{1}{2\hbar} \frac{Q\_l P\_{mu}}{2\alpha - \alpha\_{\text{vg}} + i\Gamma},
\tag{8}
$$

where Γ is the width of the electronic transition. Terms Pmn and Ql are due to off-resonance and resonant transition moments, respectively. They are given by

$$P\_{mn} = \sum\_{s} \left| \frac{\left\langle \mathbf{v} | \hat{\boldsymbol{\mu}}\_{n} | \mathbf{s} \right\rangle \left\langle \mathbf{s} | \hat{\boldsymbol{\mu}}\_{n} | \mathbf{g} \right\rangle}{\left(\alpha - \alpha\_{\text{v}}\right) \left(\alpha - \alpha\_{\text{g}}\right)} - \frac{\left\langle \mathbf{v} | \hat{\boldsymbol{\mu}}\_{n} | \mathbf{s} \right\rangle \left\langle \mathbf{s} | \hat{\boldsymbol{\mu}}\_{n} | \mathbf{g} \right\rangle}{\left(\alpha + \alpha\_{\text{v}}\right) \left(\alpha + \alpha\_{\text{g}}\right)} \right|,\tag{9}$$

$$\mathcal{Q}\_l = \left\langle \mathbf{g} | \widehat{\boldsymbol{\mu}}\_l | \mathbf{v} \right\rangle. \tag{10}$$

In Equations (9) and (10), *μ* ^ is the electric dipole operator, | *<sup>g</sup>* is the fundamental state, |*ν* is the excited state resonant with the second-harmonic, and |*s* is any other state.

### **2.3. Second-order susceptibility**

The nonlinear second-order susceptibility is a macroscopic average of second-order nonlinear polarizability β. It is defined as the term that describes the interaction between the material medium and the optical electric field [23]. The relation between χ(2) and β can be described by a coordinate transformation from the molecular reference frame to the laboratory frame, as shown in the following equation:

$$\mathcal{Z}\_{\psi k}^{(2)} = N \sum \left\{ R\left(\psi\right)R\left(\theta\right)R\left(\phi\right) \right\} \beta\_{\text{low}}.\tag{11}$$

where N is the number of molecules by volume, and *R*(*ψ*)*R*(*θ*)*R*(*φ*) is the product of three rotation matrices that relate the molecular coordinate system (l, m, n) to the laboratory coordinate system. The symbol represents the orientational average. If we know the β tensor for molecules, measurements of χ(2) elements can give us information about their molecular orientation, as will be described in Section 2.4.

Second-harmonic generation (SHG) arises from a term of Equation (5) proportional to cos(2*ωt*), when just one electric field at frequency ω is applied. In that case, Equation (6) becomes:

In Equation (7), αlm is the (first order) linear polarizability, βlmn and γlmno are the second- and

Through second-order time-dependent perturbation theory [17], we can derive an expression for βlmn that is quite complicated. But a simplification happens when we consider the case of SHG with the fundamental frequency ω away from electronic resonances, but with the second-

<sup>1</sup> , 2 2

where Γ is the width of the electronic transition. Terms Pmn and Ql are due to off-resonance

( )( ) ( )( ) | | || || ˆ ˆ ,

*s vs sg vs sg*

= ||. m

The nonlinear second-order susceptibility is a macroscopic average of second-order nonlinear polarizability β. It is defined as the term that describes the interaction between the material medium and the optical electric field [23]. The relation between χ(2) and β can be described by a coordinate transformation from the molecular reference frame to the laboratory frame, as

where N is the number of molecules by volume, and *R*(*ψ*)*R*(*θ*)*R*(*φ*) is the product of three rotation matrices that relate the molecular coordinate system (l, m, n) to the laboratory coordinate system. The symbol represents the orientational average. If we know the β tensor for molecules, measurements of χ(2) elements can give us information about their molecular

é ù <sup>=</sup> ê ú - -- ++ ë û å *mn nm*

 ww

 mmˆ | ˆ |

> ww

^ is the electric dipole operator, | *<sup>g</sup>* is the fundamental state, |*ν* is

 b

<sup>=</sup> å <sup>j</sup> . *ijk N R RR lmn* (11)

µ *Q gv l l* (10)

*v ss g v ss g <sup>P</sup>* (9)

*vg Q P*

*<sup>i</sup>* (8)

w w <sup>=</sup> <sup>h</sup> - +G *l mn*

harmonic 2ω resonant with an electronic transition of the molecule, ωvg [22]:

b

and resonant transition moments, respectively. They are given by

m

ww

*mn*

In Equations (9) and (10), *μ*

**2.3. Second-order susceptibility**

shown in the following equation:

c

orientation, as will be described in Section 2.4.

 m

the excited state resonant with the second-harmonic, and |*s* is any other state.

( ) ( ) ( ) (2)

 yq

 ww

*lmn*

third-order hyperpolarizabilities, respectively.

34 Advanced Electromagnetic Waves

$$
\vec{P}^{(2)}\left(2\alpha = \alpha + \alpha\right) = \vec{\chi}^{(2)} : \vec{E}\left(\alpha\right)\vec{E}\left(\alpha\right) \tag{12}
$$

SHG is intrinsically sensitive to surfaces and interfaces due to its selection rule. As we will see later [see Equation (13)], the SHG intensity is proportional to the square of the second-order susceptibility χ(2). As a polar third-rank tensor, χ(2) changes sign under the inversion operation (under the electric dipole approximation): *χijk* (2) = −*χ*−*i*<sup>−</sup> *<sup>j</sup>*−*<sup>k</sup>* (2) . However, in centrosymmetric media χ(2) remains unchanged upon inversion of coordinates: *χijk* (2) =*χ*−*i*<sup>−</sup> *<sup>j</sup>*−*<sup>k</sup>* (2) . Therefore, the only possible solution for the two earlier equations is χ(2) = 0. We can conclude that, under the electric dipole approximation, for media with inversion symmetry, no second-order optical process is possible, including SHG. Most bulk molecular materials do have inversion symmetry. This is because the functional groups in the bulk of these systems are, in general, randomly or oppositely oriented [24]. However, because inversion symmetry is usually broken at the surface/interface, SHG is not forbidden in those cases. Figure 2 illustrates SHG at an interface between two centrosymmetric media. The incidence angles obey the momentum conservation along the interface plane: for refraction of incident laser: *k<sup>ω</sup>* (2) sin*αω* (2) =*k<sup>ω</sup>* (1) sin*αω* (1) (Snell's law); for reflective/refractive SHG generation: *k*2*<sup>ω</sup>* (1) sin*α*2*<sup>ω</sup>* (1) =2*k<sup>ω</sup>* (1) sin*αω* (1) ; and *k*2*<sup>ω</sup>* (2) sin*α*2*<sup>ω</sup>* (2) =2*k<sup>ω</sup>* (1) sin*αω* (1) . Index (i), i = 1, 2, refer to media 1 and 2 in Figure 2. For the specific case of thin polymeric films adsorbed on solid substrate such layer-by-layer films, if asymmetric molecules (or functional groups) adsorb with random orientations, the net SHG signal is canceled out. Conversely, if there is a substantial SHG signal, we can conclude that molecules have a net average orientation at the interface. More detailed considerations about the importance of symmetry on the interpretation of SHG (and other second-order processes, such as SFG) can be found elsewhere [13, 17, 18, 22].

For second-harmonic generation at interfaces between two different media, as shown in Figure 2, Y. R. Shen demonstrated [13, 18] that the intensity of the second-harmonic signal is given by

$$I(2o) = \frac{8\pi^3(2o)\sec^2 a}{\text{c}^3 \hbar \left[\varepsilon\_1(2o)\right]^{\frac{1}{2}}\varepsilon\_1(o)} \left|\hat{\mathbf{e}}(2o)\cdot\tilde{\chi}\_s^{(2)}:\hat{\mathbf{e}}(o)\hat{\mathbf{e}}(o)\right|^2 I^2(o),\tag{13}$$

where the SHG signal is expressed in terms of the net second-order susceptibility of the surface χs (2). The term *χeff* (2) = *e* ^(2*ω*) <sup>⋅</sup>*<sup>χ</sup>* ↔ *s* (2) ∶ *e* ^(*<sup>ω</sup>*)*<sup>e</sup>* ^(*ω*) is the effective susceptibility, which also depends on the polarizations of the input and output beams, *є* ^ (*ωi* ) and the Fresnel factors *L nn*(*ω<sup>i</sup>* ), since *e* ^ (*ωi* ) = *є* ^ (*ωi* ) ⋅ *L* <sup>↔</sup>(*ω<sup>i</sup>* ). The Fresnel factors are given by the following expressions:

$$\begin{split} L\_{\text{xx}}\left(o\_{i}\right) &= \frac{2\varepsilon\_{1}\left(o\_{i}\right)k\_{z\_{\text{z}}}\left(o\_{i}\right)}{\varepsilon\_{2}\left(o\_{i}\right)k\_{1z}\left(o\_{i}\right) + \varepsilon\_{1}\left(o\_{i}\right)k\_{2z}\left(o\_{i}\right)}, \\ L\_{\text{yy}}\left(o\_{i}\right) &= \frac{2k\_{1z}\left(o\_{i}\right)}{k\_{1z}\left(o\_{i}\right) + k\_{2z}\left(o\_{i}\right)}, \\ L\_{zz}\left(o\_{i}\right) &= \frac{2\varepsilon\_{1}\left(o\_{i}\right)k\_{2z}\left(o\_{i}\right)\left(\frac{\varepsilon\_{2}}{\varepsilon\_{z}}\right)}{\varepsilon\_{2}\left(o\_{i}\right)k\_{1z}\left(o\_{i}\right) + \varepsilon\_{1}\left(o\_{i}\right)k\_{2z}\left(o\_{i}\right)}. \end{split} \tag{14}$$

**Figure 2.** SHG geometry from the interface between two different media, showing the SHG beams generated in reflec‐ tion and transmission. χ<sup>S</sup> (2) in usually non-vanishing, while in general χ<sup>V</sup> (2) = 0.

From Equation (13), we can see that the SHG signal I(2ω) is proportional to |*χeff* (2) | 2 , the effective second-order susceptibility of surface. From Equations (8) and (11), we can write *χeff* (2) as a complex number:

$$\mathcal{X}\_{\text{eff}}^{(2)} = \frac{A}{2\alpha - \alpha\_{\text{eg}} + i\Gamma} = \left| \mathcal{X}\_{\text{eff}}^{(2)} \right| e^{i\phi},\tag{15}$$

where |*χeff* (2) | is its modulus and *ϕ* is the phase. Frequently, it is necessary to experimentally measure the phase of *χeff* (2) , because it is related to the relative orientation (up or down) of molecules at the interface. This can be accomplished by interference between SHG signals from sample (polymer films) and a nonlinear reference, like crystalline quartz or zinc sulfide, ZnS [25, 26]. In practice, we measure the effective *χeff* (2) that is the sum of the reference and the film signals:

$$\mathcal{X}^{(2)}\_{\circ \circ'} = \left| \mathcal{X}^{(2)}\_{\circ \circ'} \right| e^{i\phi\_{\circ \circ'}} + \left| \mathcal{X}^{(2)}\_{\circ \circ \circ \circ} \right| e^{i\phi\_{\circ \circ \circ \circ}}$$

As the SHG signal is proportional to square of *χeff* (2) , we have

( ) ( ) ( )

<sup>=</sup> <sup>+</sup>

ew

w

*yy i*

*L*

ew

( )

w

*zz i*

*L*

tion and transmission. χ<sup>S</sup>

complex number:

(2)

measure the phase of *χeff*

where |*χeff*

signals:

w

*xx i*

*L*

36 Advanced Electromagnetic Waves

( ) ( )

<sup>=</sup> <sup>+</sup>

( ) ( ) ( ) ( )

*k k k*

*k*

*i zi*

<sup>2</sup> ,

 w

1 2 21 12 1 1 2

ew

 w ew

( ) ( )

*zi zi*

<sup>2</sup> ,

 w

w

*i zi i zi z i*

2

*s*

e

æ ö ç ÷

e

 w

 w .

(14)

( ) ( ) ( ) ( )

*k k*

**Figure 2.** SHG geometry from the interface between two different media, showing the SHG beams generated in reflec‐

second-order susceptibility of surface. From Equations (8) and (11), we can write *χeff*

(2) (2) , <sup>2</sup>

molecules at the interface. This can be accomplished by interference between SHG signals from sample (polymer films) and a nonlinear reference, like crystalline quartz or zinc sulfide, ZnS

(2)

*eff eff vg*

(2) = 0.

f

, because it is related to the relative orientation (up or down) of

*<sup>A</sup> <sup>e</sup> <sup>i</sup>* (15)

that is the sum of the reference and the film

*i*

 c


(2) | 2

, the effective

(2) as a

(2) in usually non-vanishing, while in general χ<sup>V</sup>

From Equation (13), we can see that the SHG signal I(2ω) is proportional to |*χeff*

w w= = - +G

c

(2)

[25, 26]. In practice, we measure the effective *χeff*

*i zi i zi*

 w

( ) ( )

*k*

*i zi*

21 12

1 2

 w ew

è ø <sup>=</sup> <sup>+</sup>

ew

w

*k k*

2

$$\begin{aligned} \left| \mathcal{X}\_{\epsilon\mathcal{Y}}^{(2)} \right|^2 &= \left| \mathcal{X}\_{r\mathcal{Y}}^{(2)} \right| e^{i\phi\_{r\mathcal{Y}}} + \left| \mathcal{X}\_{\mathcal{I}\mathcal{Y}\text{lin}}^{(2)} \right| e^{i\phi\_{\mathcal{I}\text{lin}}} \right|^2 = \\ &= \left| \mathcal{X}\_{r\mathcal{Y}}^{(2)} \right|^2 + \left| \mathcal{X}\_{\mathcal{I}\text{lin}}^{(2)} \right|^2 + 2 \left| \mathcal{X}\_{r\mathcal{Y}}^{(2)} \right| \left| \mathcal{X}\_{\mathcal{I}\text{lin}}^{(2)} \right| \cos(\Delta\phi), \end{aligned} \tag{16}$$

where Δ*<sup>ϕ</sup>* =(*ϕfilm* <sup>−</sup>*ϕref* ) <sup>=</sup> <sup>2</sup>*<sup>π</sup> <sup>λ</sup>* Δl is the phase difference between two signals, and Δl is the difference of optical length due to a compensator inserted in the detection beam path (amor‐ phous quartz window). Figure 3 shows the experimental setup. The angle θ of the compensator determines the additional optical path Δl<sup>=</sup> <sup>Δ</sup>*<sup>n</sup> <sup>d</sup> cosθ* traveled by the SHG and pump beams from the sample to the detector, where Δn is the difference in refractive indices of the compensator for the fundamental and SHG beams. Therefore, the phase difference between the two signals is given by

$$
\Delta \phi = \frac{2\pi}{\lambda} \frac{\Delta n \, d}{\cos \theta}. \tag{17}
$$

10

ሺଶሻ is related to

.

Figure 3 illustrates the (normalized) interference pattern intensity as a function of compensator angle θ.

**Figure 3: (a) Experimental setup for SHG phase measurements. (b) Detected intensity (normalized) as a function of compensator angle θ. Figure 3.** a) Experimental setup for SHG phase measurements. (b) Detected intensity (normalized) as a function of compensator angle θ.

As seen earlier, Equation (11), the macroscopic quantity ߯

dipole moment Ԧof the molecule to local electric field components ܧሬԦ

microscopic quantity ߚ through an orientational average of a coordinate transformation,

where ߚ is a tensor that relates the components of the second-order contribution to the

**Figure 4: Molecular geometry with the azobenzene group (Ph– N = N –Ph) along the axis. The frame (x, y, z) is the sample reference frame, with xz as a mirror plane. The (X, Y, Z) frame is the laboratory coordinate system, with XZ as the incidence plane. The molecule is tilted by the polar angle with respect to the surface normal, and is the azimuthal angle with respect to the sample symmetry** 

hyperpolarizability ߚ will have only one element, that is, ߚకకక, along the molecular axis

ξ, as shown in Figure 4 for an azobenzene group. Then, for the case of a molecular

For molecules with electrons delocalized mainly along a single direction, the

**direction, which in turn is rotated by with respect to the incidence plane (X direction).** 

monolayer adsorbed on the surface, Eq. (11) can be written as

2.4. Molecular orientation from SHG measurements

### **2.4. Molecular orientation from SHG measurements**

As seen earlier, Equation (11), the macroscopic quantity *χijk* (2) is related to microscopic quantity *βlmn* through an orientational average of a coordinate transformation, where *βlmn* is a tensor that relates the components of the second-order contribution to the dipole moment *p* <sup>→</sup> of the molecule to local electric field components *E* → *local* .

**Figure 4.** Molecular geometry with the azobenzene group (Ph– N = N –Ph) along the ξ axis. The frame (x, y, z) is the sample reference frame, with xz as a mirror plane. The (X, Y, Z) frame is the laboratory coordinate system, with XZ as the incidence plane. The molecule is tilted by the polar angle θ with respect to the surface normal, and φ is the azimu‐ thal angle with respect to the sample symmetry direction, which in turn is rotated by Ω with respect to the incidence plane (X direction).

For molecules with electrons delocalized mainly along a single direction, the hyperpolariza‐ bility *βlmn* will have only one element, that is, *βξξξ*, along the molecular axis ξ, as shown in Figure 4 for an azobenzene group. Then, for the case of a molecular monolayer adsorbed on the surface, Equation (11) can be written as

$$\mathcal{Z}\_{\psi k}^{(2)} = N \left( \hat{i} \cdot \hat{\boldsymbol{\xi}} \right) \left( \hat{j} \cdot \hat{\boldsymbol{\xi}} \right) \left( \hat{k} \cdot \hat{\boldsymbol{\xi}} \right) \mathcal{B}\_{\xi \xi \xi}. \tag{18}$$

The transformation of coordinates from the molecular frame *l* ^ , *m* ^, and *<sup>n</sup>* ^ to the sample frame *i* ^ , *j* ^ , and *k* ^ is given by

$$
\hat{i}\cdot\hat{\underline{\varphi}} = \sin\theta\cos\varphi,\tag{19}
$$

Probing the Molecular Ordering in Azopolymer Thin Films by Second-Order Nonlinear Optics http://dx.doi.org/10.5772/61180 39

$$
\hat{j} \cdot \hat{\tilde{\varphi}} = \sin \theta \sin \phi,\tag{20}
$$

$$
\hat{k} \cdot \hat{\xi} = \cos \theta.\tag{21}
$$

Considering a medium with a C1v symmetric distribution of molecules on the plane xy (xz is the sample plane of symmetry), we obtain six independent elements of the tensor *χijk* (2) 27]:

**2.4. Molecular orientation from SHG measurements**

molecule to local electric field components *E*

38 Advanced Electromagnetic Waves

plane (X direction).

*i* ^ , *j* ^ , and *k* ^

the surface, Equation (11) can be written as

is given by

c

The transformation of coordinates from the molecular frame *l*

As seen earlier, Equation (11), the macroscopic quantity *χijk*

*βlmn* through an orientational average of a coordinate transformation, where *βlmn* is a tensor that

**Figure 4.** Molecular geometry with the azobenzene group (Ph– N = N –Ph) along the ξ axis. The frame (x, y, z) is the sample reference frame, with xz as a mirror plane. The (X, Y, Z) frame is the laboratory coordinate system, with XZ as the incidence plane. The molecule is tilted by the polar angle θ with respect to the surface normal, and φ is the azimu‐ thal angle with respect to the sample symmetry direction, which in turn is rotated by Ω with respect to the incidence

For molecules with electrons delocalized mainly along a single direction, the hyperpolariza‐ bility *βlmn* will have only one element, that is, *βξξξ*, along the molecular axis ξ, as shown in Figure 4 for an azobenzene group. Then, for the case of a molecular monolayer adsorbed on

( )( )( ) (2) ˆ ˆ ˆˆˆ <sup>ˆ</sup> .

*ijk* =× × × *Ni j k*

ˆ ˆ *i sin cos* × = x

 qj

 x  xb

xxx

^ , *m* ^, and *<sup>n</sup>*

(18)

, (19)

^ to the sample frame

 x

relates the components of the second-order contribution to the dipole moment *p*

→ *local* . (2) is related to microscopic quantity

<sup>→</sup> of the

$$\chi\_1 = \chi\_{\underline{\dots}\underline{\pi}}^{(2)} = N \cos^3 \theta \beta\_{\xi\xi\xi},\tag{22}$$

$$\chi\_2 = \chi\_{\text{xx}}^{(2)} = N \sin^3 \theta \cos^3 \varphi \mathcal{B}\_{\xi \xi \xi},\tag{23}$$

$$\mathcal{X}^{\sharp} = \mathcal{X}^{\chi\_{\sharp}}\_{(2)} = \mathcal{X}^{(2)}\_{\simeq \mathcal{Y}} = \mathcal{X}^{(2)}\_{\geqslant \mathcal{Y}} = N \left( \cos \theta - \cos^{\flat} \theta \right) \left( 1 - \cos^{2} \theta \right) \mathcal{J}\_{\sharp \mathcal{Y} \xi},\tag{24}$$

$$\mathcal{X}^4 = \mathcal{X}^{\underline{\times}\underline{\times}}\_{(2)} = \mathcal{X}^{\underline{\times}\underline{\times}}\_{\underline{\times}} = \mathcal{X}^{(2)}\_{\underline{\times}\underline{\times}\underline{\times}} = N \left( \cos \theta - \cos^3 \theta \right) \cos^2 \theta \mathcal{Y}\_{\underline{\times}\underline{\times}},\tag{25}$$

$$\mathcal{X}^{\sharp} = \mathcal{X}^{\sharp \equiv \pm}\_{(\sharp)} = \mathcal{X}^{\sharp \equiv \pm}\_{(\sharp)} = \mathcal{X}^{\sharp \equiv \pm}\_{(\sharp)} = N \left( \sin \theta - \sin^{3} \theta \right) \cos \varphi \mathcal{B}\_{\sharp \not\equiv \pm},\tag{26}$$

$$\mathcal{X}^{\rho} = \mathcal{X}^{\alpha \chi}\_{(2)} = \mathcal{X}^{\chi\_{2\chi}}\_{\;\;\;\nu\chi} = \mathcal{X}^{(2)}\_{\;\;\;\nu\chi} = N \left( \cos \rho - \cos^3 \rho \right) \sin^3 \theta \mathcal{\beta}\_{\;\;\;\;\;\mathcal{Y}\zeta},\tag{27}$$

Therefore, measuring these six elements in the preceding equations allows determining up to five parameters of the orientation distribution function of the adsorbed monolayer (since usually the product Nβξξξ is unknown). This can be performed by SHG measurements with several combinations of polarization, such as SinSout, SinPout, PinSout, PinPout, MinSout, MinPout, where the first polarization is for the pump beam at ω and the other is for the generated beam at 2ω. S indicates the polarization with the electric field perpendicular to the incidence plane, and P is with the electric field parallel to the incidence plane. M polarization is that where the electric field has equal components perpendicular and parallel to the incidence plane (mixed polari‐ zation). Figure 2 shows both S and P polarization.

We should note, however, that in general the laboratory coordinate system (XYZ), defined by incidence plane XZ and the sample plane XY, is not coincident with the sample coordinate system (xyz), defined by the sample plane xy and the plane of mirror symmetry xz. We define the angle Ω describing the relation between the two coordinate systems, as shown in Figure 4.

Therefore, to fully determine *χeff* (2) in laboratory frame, we need to do an additional coordinate transformation from the sample frame (xyz) to the laboratory frame (XYZ). Thus, we obtain *χeff* (2) for six possible polarization combinations as a function of independent components χ1 to χ<sup>6</sup> (Equations (22) to (27)), which characterizes the distribution of orientations for the molecules in study:

$$\chi^{(2)}\_{\rm eff, RS} = L\_{\gamma\gamma} \left( \mathfrak{Z}\alpha \right) L^2\_{\gamma\gamma} \left( \alpha \right) \left[ \sin^3 \left( \mathfrak{Q} \right) \chi\_2 + \mathfrak{Z} \sin \left( \mathfrak{Q} \right) \cos^2 \left( \mathfrak{Q} \right) \chi\_6 \right], \tag{28}$$

$$\begin{split} \chi^{(2)}\_{\eta\eta^{\*},\mathcal{B}^{0}} &= \sin(\alpha)L\_{\varpi}\left(2\alpha\right)L^{2}\_{\gamma\eta}\left(\alpha\right)\left[\cos^{2}\left(\Omega\right)\chi\_{3} + \sin^{2}\left(\Omega\right)\chi\_{4}\right] - \\ &\left[\cos\left(\alpha\right)L\_{\varpi}\left(2\alpha\right)L^{2}\_{\gamma\eta}\left(\alpha\right)\right]\cos\left(\Omega\right)\sin^{2}\left(\Omega\right)\chi\_{2} + \left(\cos^{3}\left(\Omega\right) - 2\sin^{2}\left(\Omega\right)\cos\left(\Omega\right)\right)\chi\_{4}\right], \end{split} \tag{29}$$

$$\begin{aligned} \chi\_{\text{eff}\_{\text{IV}},\text{PS}}^{(2)} &= \cos^2\left(\alpha\right) L\_{\text{yy}}\left(2\alpha\right) L\_{\text{zz}}^2\left(\alpha\right) \left[\sin\left(\Omega\right)\cos^2\left(\Omega\right)\chi\_{\text{z}} + \left(\sin^3\left(\Omega\right) - 2\sin\left(\Omega\right)\cos^2\left(\Omega\right)\right)\chi\_{\text{\textell}}\right] \\ &+ 2\cos\left(\alpha\right) \sin\left(\alpha\right) L\_{\text{yy}}\left(2\alpha\right) L\_{\text{zz}}\left(\alpha\right) L\_{\text{zz}}\left(\alpha\right) \left[\sin\left(\Omega\right)\cos\left(\Omega\right)\left(\chi\_{\text{\textell}} - \chi\_{\text{\textell}}\right)\right] \\ &+ \sin^2\left(\alpha\right) L\_{\text{yy}}\left(2\alpha\right) L\_{\text{zz}}^2\left(\alpha\right) \sin\left(\Omega\right)\chi\_{\text{\textell}}, \end{aligned} \tag{30}$$

$$\begin{split} \chi^{(2)}\_{\boldsymbol{\uprho}^{\boldsymbol{\uprho}},\boldsymbol{\uprho}} &= -\cos^3(\boldsymbol{\alpha})L\_{\boldsymbol{\upmu}}(2\boldsymbol{\alpha})L\_{\boldsymbol{\upmu}}^{\boldsymbol{\uprho}}(\boldsymbol{\alpha}) \Big[ \cos^3(\boldsymbol{\varvarOmega})\boldsymbol{\uprho}\_{\boldsymbol{\uprho}} + \left(3\sin^2(\boldsymbol{\varvarOmega})\boldsymbol{\upalpha}(\boldsymbol{\varvarOmega})\right) \boldsymbol{\uprho}\_{\boldsymbol{\uprho}} \Big] \\ &+ \Big[ \sin(\boldsymbol{\alpha})\cos^2(\boldsymbol{\alpha})L\_{\boldsymbol{\upmu}}(2\boldsymbol{\alpha})L\_{\boldsymbol{\upmu}}^{\boldsymbol{\upmu}}(\boldsymbol{\upalpha}) - 2\cos^2(\boldsymbol{\alpha})\sin(\boldsymbol{\varvarOmega})L\_{\boldsymbol{\upmu}}(2\boldsymbol{\varvar}\boldsymbol{\uprho})L\_{\boldsymbol{\upmu}}(\boldsymbol{\upalpha})L\_{\boldsymbol{\upmu}}(\boldsymbol{\upalpha}) \Big] \Big] \\ &\left[ \sin^2(\boldsymbol{\varvarOmega})\boldsymbol{\uprho}\_{\boldsymbol{\uprho}} - \cos^2(\boldsymbol{\varvarOmega})\boldsymbol{\uprho}\_{\boldsymbol{\uprho}} \right] + \left[ 2\sin^2(\boldsymbol{\varvarOmega})\boldsymbol{\upalpha}(\boldsymbol{\upalpha})L\_{\boldsymbol{\upmu}}(2\boldsymbol{\varvar}\boldsymbol{\up$$

$$\begin{split} & \chi\_{\sigma^{0}\_{\sigma},00}^{2} = \frac{1}{2} L\_{\gamma\gamma} (2\alpha) L\_{\gamma\gamma}^{2} (\alpha) \Big[ \sin^{3} \Omega \underline{\chi}\_{2} + \Big( 3 \sin \Omega \cos^{2} \Omega \rfloor \underline{\chi}\_{b} \Big) \\ & + \frac{1}{2} \cos^{2} \alpha L\_{\gamma\gamma} (2\alpha) L\_{\gamma\gamma}^{2} (\alpha) \Big[ \sin \{\Omega\} \cos^{2} \left(\Omega\right) + \Big( \sin^{3} \left(\Omega\right) - 2 \sin \left(\Omega\right) \cos^{2} \Omega \rfloor \underline{\chi}\_{b} \Big] \\ & + \frac{1}{2} \sin^{2} \left(\alpha \right) L\_{\gamma\gamma} (2\alpha) L\_{\gamma\gamma}^{2} (\alpha) \Big[ \sin \{\Omega\} \cos \left(\Omega\right) \Big] \chi\_{\sigma} \\ & + \cos \left(\alpha \right) L\_{\gamma\gamma} (2\alpha) L\_{\gamma\gamma} (\alpha) L\_{\gamma\alpha} \Big( \alpha \Big) \Big[ \cos \Omega \sin^{2} \left(\Omega\right) \chi\_{z} + \Big( \cos^{3} \Omega - 2 \sin^{2} \left(\Omega \right) \cos \left(\Omega \right) \Big) \chi\_{z} \Big] \\ & + \sin \left(\alpha \right) L\_{\gamma\gamma} (2\alpha) L\_{\gamma\gamma} (\alpha) L\_{z} \left( \alpha \right) \Big[ \cos^{2} \Omega \chi\_{z} + \sin^{2} \left(\Omega \right) \chi\_{z} \Big] \\ & + \sin \left(\alpha \right) \cos \left(\alpha \right) L\_{\gamma\gamma} (2\alpha) L\_{z} \left( \alpha \right) L\_{z} \left( \alpha \right) \Big[ \sin \Omega \left$$

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In practice, SHG measurements consist in recording the SH intensity as a function of the sample azimuthal angle Ω for several polarization combinations. From them, we can determine the independent components, from χ1 to χ6, that are related to the orientational distribution of molecules on the sample. For example, for the case of an isotropic sample on its xy plane, only χ1 and χ3 = χ<sup>4</sup> will be nonvanishing, so that all *χeff* (2) are either null or independent of azimuthal angle Ω, as expected. In this case, the ratio *<sup>χ</sup>*<sup>1</sup> *<sup>χ</sup>*<sup>3</sup> <sup>=</sup> cos<sup>3</sup> *θ* cos*θ* − cos<sup>3</sup> *<sup>θ</sup>* depends only on the average molecular tilt with respect to the normal direction (polar angle θ). Therefore, measurements can imme‐ diately be qualitatively interpreted to determine if samples are isotropic or not about the surface, and in that case, if polar orientation changes significantly.

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In order to fully determine the orientation of adsorbed molecules at an interface (film), we assume that this orientation is described by a distribution function such as

$$F\left(\theta,\ \phi\right) = A e^{-\frac{\left(\theta - \theta\_0\right)^2}{2\sigma^2}} \left[d\_0 + d\_1 \cos\left(\phi\right) + d\_2 \cos\left(2\phi\right) + d\_3 \cos\left(3\phi\right)\right].\tag{34}$$

The first term of *F* (*θ*, *φ*) is a Gaussian distribution function of the polar angle θ, where θ0 is the average molecular tilt and σ is the polar distribution width. *A* is a normalization constant, given by

$$A = \frac{1}{(2\pi)^{\frac{1}{2}}\sin(\theta\_0)\sigma\left(1 - \frac{\sigma^2}{2}\right)}.\tag{35}$$

This distribution describes the orientation angle of adsorbed molecules with respect to the zaxis, which is perpendicular to plane of sample (see Figure 4).

The second factor of Equation (34) is a Fourier series on the azimuthal angle φ, truncated at the third term. There, d0 is the normalization constant equal to 1 / 2*π*. This distribution describes the anisotropy of adsorbed molecules along to plane of sample, with respect to the mirror plane xz.

In order to obtain the parameters in the orientational distribution function F(θ, φ), we need to experimentally measure *χeff* (2) in the six polarization combinations as the sample is rotated (varying the sample azimuthal angle Ω). We then adjust the data (simultaneous fitting) to Equations (28) to (33) in order to find the parameters θ0, σ, d1, d2, and d3, as well as the initial sample azimuth Ω<sup>0</sup> (initial angle between sample symmetry direction and laboratory frames, see Figure 4).

In this chapter, we are interested in probing the molecular orientation of layer-by-layer films of azopolymers and their thermal stability using the SHG technique. In the next section, we will briefly describe the most important characteristics of these ultrathin films that are relevant to the physical interpretation of SHG results.
