# Nonlinear Optical Susceptibilities

This is a continuation from the previous tutorial - optical nonlinearity.

The nonlinear optical properties of a material are characterized by its nonlinear optical susceptibilities. In this tutorial, the general properties of nonlinear optical susceptibilities are discussed.

It can be seen from (9-3) and (9-4) [refer to the optical nonlinearity tutorial] that the space- and time-dependent nonlinear susceptibilities \(\boldsymbol{\chi}^{(n)}(\mathbf{r}-\mathbf{r}_1,t-t_1;\mathbf{r}-\mathbf{r}_2,t-t_2;\ldots;\mathbf{r}-\mathbf{r}_n,t-t_n)\) are real tensors because both \(\pmb{P}^{(n)}(\mathbf{r},t)\) and \(\pmb{E}(\mathbf{r},t)\) are real vectors.

Though \(\boldsymbol{\chi}^{(n)}(\mathbf{r}_1,t_1;\mathbf{r}_2,t_2;\ldots;\mathbf{r}_n,t_n)\) is always a real function of space and time, its Fourier transform is generally complex. Therefore, the frequency-dependent nonlinear susceptibilities \(\boldsymbol{\chi}^{(n)}(\omega_q=\omega_1+\omega_2+\cdots+\omega_n)\) defined in the frequency domain are generally complex. This characteristic is common to linear and nonlinear susceptibilities, as can be seen by reviewing the characteristics of the linear susceptibility discussed in the linear optical susceptibility tutorial.

Also common to linear and nonlinear susceptibilities is the fact that ** the imaginary part of a frequency-dependent susceptibility signifies the presence of loss or gain in a medium, meaning that there is a net exchange of energy between the optical field and the medium through the interaction described by this susceptibility**.

The real part of a frequency-dependent susceptibility, irrespective of whether it is linear or nonlinear, does not cause a net energy exchange between the optical field and the medium.

The linear and nonlinear optical properties of a given material are not independent of each other. Indeed, there are close relations, at both microscopic and macroscopic levels, between the linear and nonlinear optical susceptibilities of the same material.

The reason for such connections is simply that both linear and nonlinear optical properties of a material have their roots in the same microscopic material properties, including the atomic compositions, the energy levels, the resonance frequencies, and the relaxation rates, as determine the optical responses of the material.

**Reality condition**

The reality condition discussed in the linear optical susceptibility tutorial and expressed explicitly in (56) [refer to the linear optical susceptibility tutorial] for linear susceptibility can be generalized for nonlinear susceptibilities.

As mentioned above, nonlinear susceptibilities in the real space and time domain are real functions of space and time. This reality condition leads to the following relation for nonlinear susceptibilities in the momentum space and frequency domain:

\[\tag{9-23}{\boldsymbol{\chi}^{(n)}}^*(\mathbf{k}_1,\omega_1;\mathbf{k}_2,\omega_2;\cdots;\mathbf{k}_n,\omega_n)=\boldsymbol{\chi}^{(n)}(-\mathbf{k}_1,-\omega_1;-\mathbf{k}_2,-\omega_2;\cdots;-\mathbf{k}_n,-\omega_n)\]

In the case of spatially local interaction when the relation in (9-7) is valid [refer to the optical nonlinearity tutorial], we can use (9-18) [refer to the optical nonlinearity tutorial] to write the reality condition for nonlinear susceptibilities in the following form:

\[\tag{9-24}{\boldsymbol{\chi}^{(n)}}^*(\omega_q=\omega_1+\omega_2+\cdots+\omega_n)=\boldsymbol{\chi}^{(n)}(-\omega_q=-\omega_1-\omega_2-\cdots-\omega_n)\]

**Elements of susceptibility tensors**

To gain a general perspective of the susceptibility tensor elements, we first review the properties of the linear susceptibility tensor \(\boldsymbol{\chi}^{(1)}\). Because \(\boldsymbol{\chi}^{(1)}=\left[\chi_{ij}^{(1)}\right]\) is a second-rank tensor, it consists of nine tensor elements, as shown explicitly in (105) [refer to the propagation in an anisotropic medium tutorial].

Because the linear susceptibility is a function of a single frequency, only one frequency, \(\omega\), needs to be specified. When both \(\boldsymbol{\chi}^{(1)}(\omega)\) and \(\boldsymbol{\chi}^{(1)}(-\omega)\) are considered, the number of elements doubles. The reality condition, by stating that the elements of \(\boldsymbol{\chi}^{(1)}(-\omega)\) are completely determined by those of \(\boldsymbol{\chi}^{(1)}(\omega)\), reduces the maximum number of independent elements back to nine.

As discussed in the propagation in an anisotropic medium tutorial, the linear susceptibility tensor \(\boldsymbol{\chi}^{(1)}(\omega)\) of a material can always be diagonalized, thus further reducing the nine tensor elements to only three diagonal elements that represent the eigenvalues of the tensor.

Depending on the spatial symmetry of a material, the number of independent linear susceptibility elements needed for characterizing the linear optical properties of the material can be further reduced from three to two or one, as summarized in table 2 [refer to the propagation in an anisotropic medium tutorial] in terms of the relations among the three principal refractive indices.

Similar concepts apply in consideration of the properties of nonlinear susceptibilities. However, the complexity increases dramatically due to the fact that the nonlinear susceptibilities are high-rank tensors and are functions of multiple frequencies.

Being a third-rank tensor, \(\boldsymbol{\chi}^{(2)}=\left[\chi_{ijk}^{(2)}\right]\) has 27 tensor elements. The fourth-rank tensor \(\boldsymbol{\chi}^{(3)}=\left[\chi_{ijkl}^{(3)}\right]\) has 81 tensor elements.

In the most general situation, three different frequencies are involved in a second-order nonlinear process characterized by \(\boldsymbol{\chi}^{(2)}\). The three frequencies, say \(\omega_1\), \(\omega_2\), and \(\omega_3\), are not independent of one another but are subject to the condition: \(\omega_3=\omega_1+\omega_2\), assuming that \(\omega_3\gt\omega_1,\omega_2\).

For each tensor element \(\chi_{ijk}^{(2)}\), there are \(3!\) different permutations of the three frequencies, resulting in the following six different frequency dependencies:

\[\tag{9-25}\begin{align}\chi_{ijk}^{(2)}(\omega_3=\omega_1+\omega_2),\qquad\chi_{ijk}^{(2)}(\omega_2=\omega_3-\omega_1),\qquad\chi_{ijk}^{(2)}(\omega_1=-\omega_2+\omega_3),\\\chi_{ijk}^{(2)}(\omega_3=\omega_2+\omega_1),\qquad\chi_{ijk}^{(2)}(\omega_2=-\omega_1+\omega_3),\qquad\chi_{ijk}^{(2)}(\omega_1=\omega_3-\omega_2).\end{align}\]

The sign of each frequency in every element in (9-25) can be changed simultaneously to have elements such as \(\chi_{ijk}^{(2)}(-\omega_3=-\omega_1-\omega_2)\), and so on. Fortunately, because of the reality condition expressed in (9-24), this sign change does not result in additional susceptibility elements needed for describing a nonlinear process.

Therefore, the total number of frequency-dependent \(\boldsymbol{\chi}^{(2)}\) tensor elements needed to describe a second-order nonlinear interaction among three different optical frequencies completely is \(27\times3!=162\).

For a third-order nonlinear process characterized by \(\boldsymbol{\chi}^{(3)}\), there can in general be four different frequencies involved in the interaction. Therefore, the total number of frequency-dependent \(\boldsymbol{\chi}^{(3)}\) tensor elements is \(81\times4!=1944\).

In most situations of practical interest, the number of independent elements of a nonlinear susceptibility tensor that has to be considered in a particular nonlinear interaction can be greatly reduced by applying the symmetry considerations discussed in the following.

**Permutation symmetry**

There is an ** intrinsic permutation symmetry** that is purely a matter of convention of the notation used for frequency-dependent nonlinear susceptibilities.

As an example, we consider, in the case of \(\omega_3=\omega_1+\omega_2\), a nonlinear polarization \(P_x^{(2)}(\omega_3)\) generated by two orthogonally polarized optical field components \(E_y(\omega_1)\) and \(E_z(\omega_2)\) through a second-order nonlinear process. According to (9-21) [refer to the optical nonlinearity tutorial], we have

\[\tag{9-26}P_x^{(2)}(\omega_3)=\epsilon_0\left[\chi_{xyz}^{(2)}(\omega_3=\omega_1+\omega_2)E_y(\omega_1)E_z(\omega_2)+\chi_{xzy}^{(2)}(\omega_3=\omega_2+\omega_1)E_z(\omega_2)E_y(\omega_1)\right]\]

Both terms on the right-hand side of (9-26) are needed because of the convention used in (9-21) for expanding the product of (9-19) [refer to the optical nonlinearity tutorial].

However, they are equal in magnitude because they represent the same physical process of nonlinear mixing of \(E_y(\omega_1)\) and \(E_z(\omega_2)\) to generate \(P_x^{(2)}(\omega_3)\). Therefore, \(\chi_{xyz}^{(2)}(\omega_3=\omega_1+\omega_2)=\chi_{xzy}^{(2)}(\omega_3=\omega_2+\omega_1)\).

Generalization of this result leads to the following intrinsic permutation symmetry:

\[\tag{9-27}\chi_{ijk}^{(2)}(\omega_3=\omega_1+\omega_2)=\chi_{ikj}^{(2)}(\omega_3=\omega_2+\omega_1)\]

This intrinsic permutation symmetry permits free permutation of ** only the frequencies on the right-hand side** of the equals sign in the argument of a nonlinear susceptibility if the corresponding Cartesian coordinate indices are also permuted simultaneously.

It applies to \(\boldsymbol{\chi}^{(3)}\) as well. It reduces the number of independent \(\boldsymbol{\chi}^{(2)}\) elements from 162 to 81 and that of independent \(\boldsymbol{\chi}^{(3)}\) elements from 1944 to 324 without imposing any qualifying physical conditions.

**Example 9-2**

Simplify the expressions for \(\mathbf{P}^{(2)}(\omega_4)\) and \(P_x^{(2)}(\omega_4)\) in Example 9-1 [refer to the optical nonlinearity tutorial] by using the intrinsic permutation symmetry of \(\boldsymbol{\chi}^{(2)}\). Write out the expressions for \(P_y^{(2)}(\omega_4)\) and \(P_z^{(2)}(\omega_4)\).

The intrinsic permutation symmetry requires that \(\boldsymbol{\chi}^{(2)}(\omega_4=\omega_1+\omega_3):\mathbf{E}(\omega_1)\mathbf{E}(\omega_3)=\boldsymbol{\chi}^{(2)}(\omega_4=\omega_3+\omega_1):\mathbf{E}(\omega_3)\mathbf{E}(\omega_1)\). Therefore, the first two terms in \(\mathbf{P}^{(2)}(\omega_4)\) can be combined to have the following expression:

\[\begin{align}\mathbf{P}^{(2)}(\omega_4)=\epsilon_0&\left[2\boldsymbol{\chi}^{(2)}(\omega_4=\omega_1+\omega_3):\mathbf{E}(\omega_1)\mathbf{E}(\omega_3)\right.\\&\left.+\boldsymbol{\chi}^{(2)}(\omega_4=\omega_2+\omega_2):\mathbf{E}(\omega_2)\mathbf{E}(\omega_2)\right]\end{align}\]

By applying the intrinsic permutation symmetry explicitly to the elements of \(\boldsymbol{\chi}^{(2)}\), we can use the relations \(\chi_{xxz}^{(2)}(\omega_4=\omega_1+\omega_3)=\chi_{xzx}^{(2)}(\omega_4=\omega_3+\omega_1)\) and \(\chi_{xyz}^{(2)}(\omega_4=\omega_2+\omega_2)=\chi_{xzy}^{(2)}(\omega_4=\omega_2+\omega_2)\) to express the \(x\) component of \(\mathbf{P}^{(2)}(\omega_4)\) as follows:

\[\begin{align}P_x^{(2)}(\omega_4)=\epsilon_0&\left[2\chi_{xxz}^{(2)}(\omega_4=\omega_1+\omega_3)E_1E_3+\chi_{xyz}^{(2)}(\omega_4=\omega_2+\omega_2)E_2^2\right.\\&\left.+\chi_{xyy}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}+\chi_{xzz}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}\right]\end{align}\]

The \(y\) and \(z\) components of \(\mathbf{P}^{(2)}(\omega_4)\) are, respectively,

\[\begin{align}P_y^{(2)}(\omega_4)=\epsilon_0&\left[2\chi_{yxz}^{(2)}(\omega_4=\omega_1+\omega_3)E_1E_3+\chi_{yyz}^{(2)}(\omega_4=\omega_2+\omega_2)E_2^2\right.\\&\left.+\chi_{yyy}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}+\chi_{yzz}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}\right]\end{align}\]

and

\[\begin{align}P_z^{(2)}(\omega_4)=\epsilon_0&\left[2\chi_{zxz}^{(2)}(\omega_4=\omega_1+\omega_3)E_1E_3+\chi_{zyz}^{(2)}(\omega_4=\omega_2+\omega_2)E_2^2\right.\\&\left.+\chi_{zyy}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}+\chi_{zzz}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}\right]\end{align}\]

A ** full permutation symmetry** exists when all of the frequencies contained in a susceptibility are far away from any resonance frequencies of a material so that the material causes no loss or gain to the optical field at those frequencies.

Therefore, the full permutation symmetry is valid when the imaginary part of a susceptibility is negligibly small. It breaks down in a nonparametric process, where the imaginary part of the susceptibility is significant.

* The full permutation symmetry allows all of the frequencies in a nonlinear susceptibility to be freely permuted if the Cartesian coordinate indices are also permuted accordingly*.

It permits the interchange of the frequency on the left-hand side of the equals sign in the argument of a nonlinear susceptibility with any one on the right-hand side, which is not permitted by the intrinsic permutation symmetry.

However, the sign of a frequency has to be changed at the time when it is moved across the equals sign in a permutation. For example, \(\chi_{ijk}^{(2)}(\omega_3=\omega_1+\omega_2)=\chi_{jik}^{(2)}(-\omega_1=-\omega_3+\omega_2)\), and so on.

By applying the reality condition given in (9-24) and the fact that the susceptibility is necessarily real when the full permutation symmetry is valid, we then have

\[\tag{9-28}\chi_{ijk}^{(2)}(\omega_3=\omega_1+\omega_2)=\chi_{jik}^{(2)}(\omega_1=\omega_3-\omega_2)=\chi_{kij}^{(2)}(\omega_2=\omega_3-\omega_1)\]

Similar relations can be written for the \(\boldsymbol{\chi}^{(3)}\) elements. This full permutation symmetry further reduces the maximum number of independent \(\boldsymbol{\chi}^{(2)}\) elements from 81 to 27 and that of independent \(\boldsymbol{\chi}^{(3)}\) elements from 324 to 81.

If, in addition to being lossless so that the full permutation symmetry is valid, a medium is also nondispersive in the entire spectral range that covers all of the frequencies contained in a nonlinear susceptibility, the frequencies in the susceptibility can be freely permuted independently of the Cartesian coordinate indices.

Similarly, the Cartesian coordinate indices can also be permuted independently of the frequencies. This permutation symmetry is know as ** Kleiman's symmetry condition**.

Under this condition, we have

\[\tag{9-29}\begin{align}\chi_{ijk}^{(2)}(\omega_3=\omega_1+\omega_2)&=\chi_{ijk}^{(2)}(\omega_1=\omega_3-\omega_2)=\chi_{ijk}^{(2)}(\omega_2=\omega_3-\omega_1)\\&=\chi_{jik}^{(2)}(\omega_3=\omega_1+\omega_2)=\chi_{kij}^{(2)}(\omega_3=\omega_1+\omega_2)\end{align}\]

and so on.

Kleiman's symmetry condition, when applicable, further reduces the number of independent \(\boldsymbol{\chi}^{(2)}\) elements from 27 to a maximum of 10 and that of independent \(\boldsymbol{\chi}^{(3)}\) elements from 81 to a maximum of 15.

**Spatial Symmetry**

As we have seen in the propagation in an anisotropic medium tutorial, the form of the linear susceptibility tensor is determined by the symmetry property of a material.

The forms of the nonlinear susceptibility tensors of a material also reflect the spatial symmetry property of the material structure. As a result, some elements in a nonlinear susceptibility tensor may be zero and others may be related in one way or another, greatly reducing the total number of independent tensor elements.

However, as mentioned in the Pockels effect tutorial, ** the linear susceptibility tensor has its form determined only by the crystal system of a material, whereas the form of a nonlinear susceptibility tensor further depends on the point group of the material**.

Within the 7 crystal systems, there are 32 point groups. Among the 32 point groups, 21 are noncentrosymmetric and 11 are centrosymmetric. The 21 noncentrosymmetric point groupos are those listed in Table 9-1 below.

The 11 centrosymmetric point groups are triclinic \(\bar1\), monoclinic \(3/m\), orthorhombic \(mmm\), tetragonal \(4/m\) and \(4/mmm\), trigonal \(\bar3\) and \(\bar3m\), hexagonal \(6/m\) and \(6/mmm\), and cubic \(m3\) and \(m3m\).

Many materials, including gases, liquids, amorphous solids, and many crystals that belong to the 11 centrosymmetric point groups, possess space-inversion symmetry. In the electric-dipole approximation, nonlinear optical effects of all even orders, but not those of the odd orders, vanish identically in a centrosymmetric material.

Note:

Nonlinear optical effects of even orders that are contributed by magnetic-dipole and electric-quadrupole interactions can still exist in a centrosymmetric material. Nonlinear optical effects of even orders contributed by electric-dipole interaction can also exist on the surfaces or interfaces of centrosymmetric materials where the centrosymmetry is broken.

**Therefore, \(\boldsymbol{\chi}^{(2)}\) contributed by electric-dipole interaction is identically zero in a centrosymmetric material, whereas a nonzero \(\boldsymbol{\chi}^{(3)}\) exists in any material.**

This fact can be easily verified by considering the effect of space inversion on the nonlinear polarizations \(\pmb{P}^{(2)}\) and \(\pmb{P}^{(3)}\) given in (9-3) and (9-4), respectively [refer to the optical nonlinearity tutorial].

The space-inversion transformation can be performed on a centrosymmetric material without changing the properties of the material. Being polar vectors, \(\pmb{P}^{(2)}\), \(\pmb{P}^{(3)}\), and \(\pmb{E}\) all change sign under such a transformation. From (9-3), we then find that \(\pmb{P}^{(2)}=-\pmb{P}^{(2)}\). Therefore, \(\pmb{P}^{(2)}\) cannot exist and \(\boldsymbol{\chi}^{(2)}\) has to vanish identically in a centrosymmetric material.

No such conclusion is drawn for \(\pmb{P}^{(3)}\) and \(\boldsymbol{\chi}^{(3)}\) as we examine (9-4) following the same procedure.

The discussion above about the vanishing electric-dipole \(\boldsymbol{\chi}^{(2)}\) for a centrosymmetric material is valid only for the bulk nonlinear optical property of the material but does not apply to the surface or interface of the material.

Centrosymmetry does not exist on the surface of any material or at an interface between two different materials even when the materials themselves are centrosymmetric. Therefore, \(\boldsymbol{\chi}^{(2)}\) contributed by electric-dipole interaction exists at any material surface or interface.

As a result, second-order nonlinear processes that normally do not occur in the bulk of a certain material, such as silicon, which is centrosymmetric, can take place on its surface or interface. The surface \(\boldsymbol{\chi}^{(2)}\) also depends on the structure of the material surface.

We have seen in the electro-optic effects tutorial that the Pockels effect exists only in noncentrosymmetric materials while the electro-optic Kerr effect exists in all materials. Indeed, Pockels effect can be considered as a special second-order nonlinear optical effect, and the electro-optic Kerr effect is a special third-order nonlinear optical effect.

According to (9-21) [refer to the optical nonlinearity tutorial], a nonlinear polarization \(P_i^{(2)}(\omega)\) induced by the interaction between a static electric field \(E_{0k}=E_k(0)\) polarized along the \(k\) direction and an optical field \(E_j(\omega)\) polarized along the \(j\) direction can be expressed as

\[\tag{9-30}\begin{align}P_i^{(2)}(\omega)&=\epsilon_0\left[\chi_{ijk}^{(2)}(\omega=\omega+0)E_j(\omega)E_k(0)+\chi_{ikj}^{(2)}(\omega=0+\omega)E_k(0)E_j(\omega)\right]\\&=2\epsilon_0\chi_{ijk}^{(2)}(\omega=\omega+0)E_j(\omega)E_{0k}\end{align}\]

Using (6-18) [refer to the electro-optic effects tutorial] and identifying \(\Delta\epsilon_{ij}(\omega)\) with \(P_i^{(2)}(\omega)/E_j(\omega)=2\epsilon_0\chi_{ijk}^{(2)}(\omega=\omega+0)E_{0k}\), we find that the Pockels coefficients are related to the \(\boldsymbol{\chi}^{(2)}\) elements as

\[\tag{9-31}r_{ijk}=-\frac{2}{n_i^2n_j^2}\chi_{ijk}^{(2)}(\omega=\omega+0)=-\frac{2}{n_i^2n_j^2}\chi_{kij}^{(2)}(0=\omega-\omega)\]

where the full permutation symmetry is used in moving the zero frequency to the left-hand side of the equals sign in the argument of \(\boldsymbol{\chi}^{(2)}\).

Similarly, for the electro-optic Kerr coefficients, we have

\[\tag{9-32}s_{ijkl}=-\frac{3}{n_i^2n_j^2}\chi_{ijkl}^{(3)}(\omega=\omega+0+0)\]

It can be seen from the above discussions that though not all noncentrosymmetric crystals are useful, any material that can support a second-order nonlinear process through electric-dipole interaction is necessarily a noncentrosymmetric crystal. The nonvanishing \(\boldsymbol{\chi}^{(2)}\) tensor elements and the relations among them for each of the 21 noncentrosymmetric point groups are listed in Table 9-1 above.

**Example 9-3**

The \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\) directional unit vectors used to define the electric field polarizations in Examples 9-1 and 9-2 are aligned with the principal \(x\), \(y\), and \(z\) axes of a crystal. (a) Use the result obtained in Example 9-2 to find the nonvanishing terms in the three components of \(\mathbf{P}^{(2)}(\omega_4)\) if the nonlinear interaction takes place in a \(\bar43m\) crystal, such as GaAs. (b) Find the nonvanishing terms in the case of a crystal of \(mm2\) point group, such as KTP.

(a) From Table 9-1, the only nonvanishing \(\boldsymbol{\chi}^{(2)}\) elements for the \(\bar43m\) point group are \(\chi_{xyz}^{(2)}=\chi_{yzx}^{(2)}=\chi_{zxy}^{(2)}=\chi_{xzy}^{(2)}=\chi_{yxz}^{(2)}=\chi_{zyx}^{(2)}\). From the expressions for the components of \(\mathbf{P}^{(2)}(\omega_4)\) obtained in Example 9-2, we have, for a \(\bar43m\) crystal,

\[\begin{align}&P_x^{(2)}(\omega_4)=\epsilon_0\chi_{xyz}^{(2)}(\omega_4=\omega_2+\omega_2)E_2^2\\&P_y^{(2)}(\omega_4)=2\epsilon_0\chi_{yxz}^{(2)}(\omega_4=\omega_1+\omega_3)E_1E_3\\&P_z^{(2)}(\omega_4)=0\end{align}\]

(b) For the \(mm2\) point group, the only nonvanishing \(\boldsymbol{\chi}^{(2)}\) elements are \(\chi_{xzx}^{(2)}\), \(\chi_{xxz}^{(2)}\), \(\chi_{yyz}^{(2)}\), \(\chi_{yzy}^{(2)}\), \(\chi_{zxx}^{(2)}\), \(\chi_{zyy}^{(2)}\), and \(\chi_{zzz}^{(2)}\), according to Table 9-1. Then, the expressions for the components of \(\mathbf{P}^{(2)}(\omega_4)\) obtained in Example 9-2 reduce to

\[\begin{align}&P_x^{(2)}(\omega_4)=2\epsilon_0\chi_{xxz}^{(2)}(\omega_4=\omega_1+\omega_3)E_1E_3\\&P_y^{(2)}(\omega_4)=\epsilon_0\chi_{yyz}^{(2)}(\omega_4=\omega_2+\omega_2)E_2^2\\&P_z^{(2)}(\omega_4)=\epsilon_0\left[\chi_{zyy}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}+\chi_{zzz}^{(2)}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}\right]\end{align}\]

Because \(\boldsymbol{\chi}^{(3)}\) exists in all materials, the materials used for the devices that are based on third-order nonlinear optical processes are usually isotropic noncrystalline materials, such as glasses, or cubic crystals, such as the III-V semiconductors. Only occasionally are noncubic crystals used for such devices.

Third-order nonlinear processes are particularly important for isotropic materials because \(\boldsymbol{\chi}^{(2)}\) vanishes identically so that \(\boldsymbol{\chi}^{(3)}\) becomes the leading nonlinear susceptibility of such materials.

Table 9-2 below lists the nonvanishing \(\boldsymbol{\chi}^{(3)}\) tensor elements and the relations among them for the cubic crystal system and for isotropic materials.

It can be seen that for all of the point groups in the cubic system and for isotropic materials, there are only 21 nonvanishing \(\boldsymbol{\chi}^{(3)}\) tensor elements.

For the \(23\) and \(m3\) point groups, there are 7 independent \(\boldsymbol{\chi}^{(3)}\) elements. For the \(432\), \(\bar43m\), and \(m3m\) point groups, there are only 4 independent elements of the types \(\chi_{1111}^{(3)}\), \(\chi_{1122}^{(3)}\), \(\chi_{1212}^{(3)}\), and \(\chi_{1221}^{(3)}\). If Kleiman's symmetry condition is valid, the number of independent elements reduces to 2 of the types \(\chi_{1111}^{(3)}\) and \(\chi_{1122}^{(3)}=\chi_{1212}^{(3)}=\chi_{1221}^{(3)}\) for all point groups in the cubic system.

For an isotropic material, there are only 3 independent \(\boldsymbol{\chi}^{(3)}\) elements among the 4 types of nonvanishing elements because \(\chi_{1111}^{(3)}=\chi_{1122}^{(3)}+\chi_{1212}^{(3)}+\chi_{1221}^{(3)}\). If Kleiman's symmetry condition is valid in an isotropic material, we have

\[\tag{9-33}\chi_{1122}^{(3)}=\chi_{1212}^{(3)}=\chi_{1221}^{(3)}=\frac{1}{3}\chi_{1111}^{(3)}\]

reducing the number of independent \(\boldsymbol{\chi}^{(3)}\) elements to only 1.

**Nonlinear optical \(d\) coefficients**

In the literature, experimentally measured values of second-order nonlinear susceptibilities of a material are commonly quoted in terms of nonlinear coefficients \(d_{ijk}\), or \(d_{i\alpha}\) under index contraction. The relation between the \(d\) coefficients and the \(\boldsymbol{\chi}^{(2)}\) elements is simply

\[\tag{9-34}d_{ijk}=\frac{1}{2}\chi_{ijk}^{(2)}\]

if neither index \(j\) nor index \(k\) is associated with a DC field.

Note:

The nonlinear optical \(d\) coefficients are not to be confused with the piezoelectric \(d\) coefficients though both are second-rank tensors and they have the same matrix form. The piezoelectric \(d\) coefficients define the piezoelectric polarization induced by a strain tensor.

**Index contraction**

In certain situations, the rule of index contraction expressed in (115) [refer to the propagation in an anisotropic medium tutorial] can be applied to \(\chi_{ijk}^{(2)}\) and \(d_{ijk}\) on the last two indices \(j\) and \(k\) by replacing \(jk\) with \(\alpha\). Then the 27 elements of \(\chi_{ijk}^{(2)}\), or \(d_{ijk}\), are reduced to 18 elements of \(\chi_{i\alpha}^{(2)}\), or \(d_{i\alpha}\), for \(i=1,2,3\) and \(\alpha=1,2,\ldots,6\). Clearly, the condition for index contraction to be applicable is when there is no physical significance in interchanging the last two indices \(j\) and \(k\) independently of the frequencies in \(\chi_{ijk}^{(2)}\).

For \(\boldsymbol{\chi}^{(2)}(\omega_3=\omega_1+\omega_2)\) in general, index contraction applies only when Kleiman's symmetry condition is valid so that \(\chi_{ijk}^{(2)}(\omega_3=\omega_1+\omega_2)=\chi_{ikj}^{(2)}(\omega_3=\omega_1+\omega_2)\). However, index contraction applies without the requirement of Kleiman's symmetry condition in the special cases of \(\boldsymbol{\chi}^{(2)}(2\omega=\omega+\omega)\) and \(\boldsymbol{\chi}^{(2)}(0=\omega-\omega)\).

For \(\boldsymbol{\chi}^{(2)}(2\omega=\omega+\omega)\), which characterizes the process of ** second-harmonic generation**, index contraction always applies because \(\chi_{ijk}^{(2)}(2\omega=\omega+\omega)=\chi_{ikj}^{(2)}(2\omega=\omega+\omega)\) by the definition of the intrinsic permutation symmetry.

For \(\boldsymbol{\chi}^{(2)}(0=\omega-\omega)\), which characterizes the process of ** optical rectification** for the generation of a DC electric field by an optical field, index contraction applies only when the medium is lossless at the frequency \(\omega\) so that \(\chi_{ijk}^{(2)}(0=\omega-\omega)=\chi_{ijk}^{(2)}(0=-\omega+\omega)=\chi_{ikj}^{(2)}(0=\omega-omega)\) due to the reality condition and the intrinsic permutation symmetry. From (9-31), we find that

\[\tag{9-35}r_{\alpha k}=-\frac{2}{n_i^2n_j^2}\chi_{k\alpha}^{(2)}=-\frac{4}{n_i^2n_j^2}d_{k\alpha}\]

where \(\chi_{k\alpha}^{(2)}=\chi_{kij}^{(2)}(0=\omega-\omega)\) and the index \(k\) is associated with the DC electric field. Note that Kleiman's symmetry condition is never valid for \(\boldsymbol{\chi}^{(2)}(0=\omega-\omega)\) because no material can be completely nondispersive in the entire spectral range from DC to the optical frequencies.

With index contraction, the second-order nonlinear susceptibilities \(\chi_{i\alpha}^{(2)}\) and, correspondingly, the nonlinear coefficients \(d_{i\alpha}\) can be expressed in the form of a 3 x 6 matrix. From the relation in (9-35), it is clear that the matrix form of \(\chi_{i\alpha}^{(2)}\) and \(d_{i\alpha}\) for each of the noncentrosymmetric point groups is exactly the ** transpose** of the matrix of the Pockels coefficients listed in Table 6-1 [refer to the Pockels effect tutorial].

If Kleiman's symmetry condition is valid, the matrix form of \(\chi_{i\alpha}^{(2)}\) and \(d_{i\alpha}\) is further simplified to result in a maximum of only 10 independent parameters. For example, \(d_{14}=d_{25}=d_{36}\) under Kleiman's symmetry condition.

From Table 6-1 [refer to the Pockels effect tutorial], we find that the only nonlinear coefficients for the \(422\) and \(622\) point groups are \(d_{25}=-d_{14}\), which have to vanish identically under Kleiman's symmetry condition though the Pockels coefficients \(r_{52}=-r_{41}\) do not have to vanish.

We also find that under Kleiman's symmetry condition the 3 independent nonlinear coefficients \(d_{14}\), \(d_{25}\), and \(d_{36}\) for the \(222\) point group reduce to 1 identical parameter, and the 2 independent parameters \(d_{14}=d_{25}\) and \(d_{36}\) for the \(\bar42m\) point group also reduce to a single parameter. The properties of some important nonlinear crystals are listed in Table 9-3 below.

Using (9-21) [refer to the optical nonlinearity tutorial] and (9-34), the second-order nonlinear polarization can be expressed in terms of the \(d_{i\alpha}\) matrix. In the general case of \(\omega_1+\omega_2=\omega_3\) with \(\omega_1\ne\omega_2\), we have

\[\tag{9-36}\begin{bmatrix}P_x^{(2)}(\omega_3)\\P_y^{(2)}(\omega_3)\\P_z^{(2)}(\omega_3)\end{bmatrix}=4\epsilon_0\begin{bmatrix}d_{11}&d_{12}&d_{13}&d_{14}&d_{15}&d_{16}\\d_{21}&d_{22}&d_{23}&d_{24}&d_{25}&d_{26}\\d_{31}&d_{32}&d_{33}&d_{34}&d_{35}&d_{36}\end{bmatrix}\times\begin{bmatrix}E_x(\omega_1)E_x(\omega_2)\\E_y(\omega_1)E_y(\omega_2)\\E_z(\omega_1)E_z(\omega_2)\\E_y(\omega_1)E_z(\omega_2)+E_z(\omega_1)E_y(\omega_2)\\E_z(\omega_1)E_x(\omega_2)+E_x(\omega_1)E_z(\omega_2)\\E_x(\omega_1)E_y(\omega_2)+E_y(\omega_1)E_x(\omega_2)\end{bmatrix}\]

In the case of second-harmonic generation, we have

\[\tag{9-37}\begin{bmatrix}P_x^{(2)}(2\omega)\\P_y^{(2)}(2\omega)\\P_z^{(2)}(2\omega)\end{bmatrix}=2\epsilon_0\begin{bmatrix}d_{11}&d_{12}&d_{13}&d_{14}&d_{15}&d_{16}\\d_{21}&d_{22}&d_{23}&d_{24}&d_{25}&d_{26}\\d_{31}&d_{32}&d_{33}&d_{34}&d_{35}&d_{36}\end{bmatrix}\begin{bmatrix}E_x^2(\omega)\\E_y^2(\omega)\\E_z^2(\omega)\\2E_y(\omega)E_z(\omega)\\2E_z(\omega)E_x(\omega)\\2E_x(\omega)E_y(\omega)\end{bmatrix}\]

Note that each \(d\) coefficient in (9-36) has the same value as the corresponding one in (9-37) if we ignore dispersion due to frequency differences between (9-36) and (9-37). It is true that in the case when \(\omega_1\ne\omega_2\), \(\mathbf{P}^{(2)}(\omega_3)=2\mathbf{P}^{(2)}(2\omega)\) if \(\mathbf{E}(\omega_1)=\mathbf{E}(\omega_2)=\mathbf{E}(\omega)\) as is seen by comparing (9-36) and (9-37).

**Unit conversion**

The SI system, which is essentially MKSA system, is used consistently in our tutorials. Nevertheless, the Gaussian system is also used quite often in the literature. In the Gaussian system, cgs units are used, but the electric field quantities and the susceptibilities are normally given the units of esu, meaning electrostatic units, without explicitly spelling out their true dimensions. In the SI system, the units are explicit. Unit conversion for susceptibilities between the SI and Gaussian systems follows the following relations:

\[\begin{align}\text{SI}\qquad&\qquad \text{Gaussian}\\\chi^{(1)}(\text{dimensionless})&=4\pi\chi^{(1)}(\text{dimensionless})\tag{9-38}\\\chi^{(2)}(\text{m V}^{-1})&=\frac{4\pi}{3\times10^4}\chi^{(2)}(\text{esu})\tag{9-39}\\\chi^{(3)}(\text{m}^2\text{ V}^{-2})&=\frac{4\pi}{9\times10^8}\chi^{(3)}(\text{esu})\tag{9-40}\end{align}\]

Unti conversion for the \(d\) coefficient is the same as that for \(\chi^{(2)}\) given in (9-39) because the relation in (9-34) is independent of the unit system used.

The next part continues with the **nonlinear optical interactions tutorial**.