# Optical Nonlinearity

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This is a continuation from the previous tutorial - guided-wave acousto-optic devices.

The functioning of electro-optic, magneto-optic, and acousto-optic devices discussed in the earlier tutorials is based on the fact that the optical properties of a material depend on the strength of an electric, magnetic, or acoustic field that is present in an optical medium.

At a sufficiently high optical intensity, the optical properties of a material also become a function of the optical field. Such nonlinear response to the strength of the optical field results in various nonlinear optical effects.

Nonlinear optics is an established broad field with applications covering a very wide range. The most important nonlinear optical devices are optical frequency converters. The frequency-converting function of such devices is uniquely nonlinear and is difficult, if not impossible, to accomplish by other means in the absence of optical nonlinearity.

Other unique nonlinear optical devices include all-optical switches and modulators. Many interesting nonlinear optical phenomena, such as optical solitons, stimulated Raman scattering, and optical phase conjugation, also find useful applications.

The origin of optical nonlinearity is the nonlinear response of electrons in a material to an optical field as the strength of the field is increased.

Macroscopically, the nonlinear optical response of a material is described by a polarization that is a nonlinear function of the optical field. In general, such nonlinear dependence on the optical field can take a variety of forms. In particular, it can be very complicated when the optical field becomes extremely strong.

In the situations of most nonlinear optical devices of interest, with the exception of saturable absorbers, the perturbation method can be applied to expand the total optical polarization in terms of a series of linear and nonlinear polarizations:

\[\tag{9-1}\pmb{P}(\mathbf{r},t)=\pmb{P}^{(1)}(\mathbf{r},t)+\pmb{P}^{(2)}(\mathbf{r},t)+\pmb{P}^{(3)}(\mathbf{r},t)+\cdots\]

where \(\pmb{P}^{(1)}\) is the linear polarization and \(\pmb{P}^{(2)}\) and \(\pmb{P}^{(3)}\) are the second- and third-order nonlinear polarizations, respectively. Except in some special cases, nonlinear polarizations of fourth order and beyond are usually not important and thus can be ignored.

Linear polarization, \(\pmb{P}^{(1)}\) is a linear function of the optical field, whereas nonlinear polarizations \(\pmb{P}^{(2)}\) and \(\pmb{P}^{(3)}\) are, respectively, quadratic and cubic functions of the optical field:

\[\tag{9-2}\pmb{P}^{(1)}(\mathbf{r},t)=\epsilon_0\displaystyle\int\limits_{-\infty}^\infty\text{d}\mathbf{r}'\int\limits_{-\infty}^t\text{d}t'\boldsymbol{\chi}^{(1)}(\mathbf{r}-\mathbf{r}',t-t')\cdot\pmb{E}(\mathbf{r}',t')\]

\[\tag{9-3}\pmb{P}^{(2)}(\mathbf{r},t)=\epsilon_0\displaystyle\int\limits_{-\infty}^\infty\text{d}\mathbf{r_1}\int\limits_{-\infty}^\infty\text{d}\mathbf{r_2}\int\limits_{-\infty}^t\text{d}t_1\int\limits_{-\infty}^t\text{d}t_2\boldsymbol{\chi}^{(2)}(\mathbf{r}-\mathbf{r}_1,t-t_1;\mathbf{r}-\mathbf{r}_2,t-t_2):\pmb{E}(\mathbf{r}_1,t_1)\pmb{E}(\mathbf{r}_2,t_2)\]

\[\tag{9-4}\pmb{P}^{(3)}(\mathbf{r},t)=\epsilon_0\displaystyle\int\limits_{-\infty}^\infty\text{d}\mathbf{r_1}\int\limits_{-\infty}^\infty\text{d}\mathbf{r_2}\int\limits_{-\infty}^\infty\text{d}\mathbf{r_3}\int\limits_{-\infty}^t\text{d}t_1\int\limits_{-\infty}^t\text{d}t_2\int\limits_{-\infty}^t\text{d}t_3\boldsymbol{\chi}^{(3)}(\mathbf{r}-\mathbf{r}_1,t-t_1;\mathbf{r}-\mathbf{r}_2,t-t_2;\mathbf{r}-\mathbf{r}_3,t-t_3)\vdots\pmb{E}(\mathbf{r}_1,t_1)\pmb{E}(\mathbf{r}_2,t_2)\pmb{E}(\mathbf{r}_3,t_3)\]

where \(\boldsymbol{\chi}^{(1)}\) is the linear susceptibility and \(\boldsymbol{\chi}^{(2)}\) and \(\boldsymbol{\chi}^{(3)}\) are the second- and third-order nonlinear susceptibilities, respectively.

**Note:**

**In defining polarizations and susceptibilities in the form of (9-2) - (9-4), we consider only the electric-dipole contribution to the material response under the electric-dipole approximation by neglecting the contributions from magnetic dipoles, electric quadrupoles, and other multipoles to the material response. When the electric-dipole contribution vanishes, other contributions can be important.**

The linear susceptibility is that of linear optics discussed in the linear optical susceptibility tutorial. In general, \(\boldsymbol{\chi}^{(1)}\) is a second-rank tensor, \(\boldsymbol{\chi}^{(1)}=\left[\chi^{(1)}_{ij}\right]\), as expressed in (105) [refer to the propagation in an anisotropic medium tutorial], whereas \(\boldsymbol{\chi}^{(2)}\) and \(\boldsymbol{\chi}^{(3)}\) are, respectively, third- and fourth-rank tensors:

\[\tag{9-5}\boldsymbol{\chi}^{(2)}=\left[\chi^{(2)}_{ijk}\right]\]

and

\[\tag{9-6}\boldsymbol{\chi}^{(3)}=\left[\chi^{(3)}_{ijkl}\right]\]

The nonlinear susceptibilities, \(\boldsymbol{\chi}^{(2)}\) and \(\boldsymbol{\chi}^{(3)}\), characterize the nonlinear optical properties of a material. Thus, the relations in (9-3) and (9-4) define the nonlinear polarizations that describe the nonlinear responses of a material to an optical field.

Because of the generally anisotropic nature of nonlinear susceptibility tensors, the nonlinear polarizations \(\pmb{P}^{(2)}\) and \(\pmb{P}^{(3)}\) are expressed in the form of high-order products between the nonlinear susceptibilities and the optical field.

As discussed in the optical fields and Maxwell's equations tutorial, the response of a material to an optical field can be nonlocal in space and noninstantaneous in time. This statement is true for both linear and nonlinear responses.

In consideration of this fact, the linear polarization \(\pmb{P}^{(1)}\) defined in (9-2) and the nonlinear polarizations \(\pmb{P}^{(2)}\) and \(\pmb{P}^{(3)}\) defined in (9-3) and (9-4) are generally expressed in the form of convolution integrals over both space and time.

For material responses that are local in space but not necessarily instantaneous in time, the susceptibilities can be expressed as

\[\tag{9-7}\boldsymbol{\chi}^{(n)}(\mathbf{r}-\mathbf{r}_1,t-t_1;\mathbf{r}-\mathbf{r}_2,t-t_2;\ldots)=\boldsymbol{\chi}^{(n)}(t-t_1,t-t_2,\ldots)\delta(\mathbf{r}-\mathbf{r}_1)\delta(\mathbf{r}-\mathbf{r}_2)\ldots\]

Then the linear and nonlinear polarizations can be expressed as

\[\tag{9-8}\pmb{P}^{(1)}(\mathbf{r},t)=\epsilon_0\displaystyle\int\limits_{-\infty}^t\text{d}t'\boldsymbol{\chi}^{(1)}(t-t')\cdot\pmb{E}(\mathbf{r},t')\]

\[\tag{9-9}\pmb{P}^{(2)}(\mathbf{r},t)=\epsilon_0\displaystyle\int\limits_{-\infty}^t\text{d}t_1\int\limits_{-\infty}^t\text{d}t_2\boldsymbol{\chi}^{(2)}(t-t_1,t-t_2):\pmb{E}(\mathbf{r},t_1)\pmb{E}(\mathbf{r},t_2)\]

\[\tag{9-10}\pmb{P}^{(3)}(\mathbf{r},t)=\epsilon_0\displaystyle\int\limits_{-\infty}^t\text{d}t_1\int\limits_{-\infty}^t\text{d}t_2\int\limits_{-\infty}^t\text{d}t_3\boldsymbol{\chi}^{(3)}(t-t_1,t-t_2,t-t_3)\vdots\pmb{E}(\mathbf{r},t_1)\pmb{E}(\mathbf{r},t_2)\pmb{E}(\mathbf{r},t_3)\]

In the momentum space and frequency domain, spatially local but temporally noninstantaneous responses imply that the linear and nonlinear susceptibilities are functions of optical frequencies but are independent of optical wavevectors:

\[\tag{9-11}\boldsymbol{\chi}^{(n)}(\omega_1,\omega_2,\ldots,\omega_n)=\displaystyle\int\limits_0^\infty\text{d}t_1\int\limits_0^\infty\text{d}t_2\ldots\int\limits_0^\infty\text{d}t_n\boldsymbol{\chi}^{(n)}(t_1,t_2,\ldots,t_n)\text{e}^{\text{i}\omega_1t_1+\text{i}\omega_2t_2+\ldots+\text{i}\omega_nt_n}\]

This situation applies to the interactions discussed in the following tutorials. Therefore, the discussions in the following tutorials are restricted to spatially local interactions where the linear and nonlinear polarizations can be expressed in the form of (9-8) - (9-10) and the linear and nonlinear susceptibilities in the momentum space and frequency domain are functions of optical frequencies only.

The polarizations defined in (9-8) - (9-10) above are expressed in terms of real field quantities, just as any basic definitions of electromagnetic field quantities are.

However, as we have seen throughout the preceding tutorials, it is generally convenient to deal with optical fields in terms of complex field quantities because optical fields are harmonic fields.

As seen in the harmonic fields tutorial, conversion to expressions in terms of complex field quantities is quite straightforward. Maxwell's equations and the wave equation all retain their general form after the conversion. The complex field \(\mathbf{E}(\mathbf{r},t)\) is defined in (39) [refer to the harmonic fields tutorial] through its relation to the real field \(\pmb{E}(\mathbf{r},t)\) as

\[\tag{9-12}\pmb{E}(\mathbf{r},t)=\mathbf{E}(\mathbf{r},t)+\mathbf{E}^*(\mathbf{r},t)=\mathbf{E}(\mathbf{r},t)+\text{c.c.}\]

In line with this definition, a complex nonlinear polarization, \(\mathbf{P}^{(n)}(\mathbf{r},t)\) in the real space and time domain can be defined through its relation to the real nonlinear polarization, \(\pmb{P}^{(n)}(\mathbf{r},t)\), as

\[\tag{9-13}\pmb{P}^{(n)}(\mathbf{r},t)=\mathbf{P}^{(n)}(\mathbf{r},t)+{\mathbf{P}^{(n)}}^*(\mathbf{r},t)=\mathbf{P}^{(n)}(\mathbf{r},t)+\text{c.c.}\]

Note that in our convention, \(\mathbf{E}(\mathbf{r},t)\) and \(\mathbf{P}^{(n)}(\mathbf{r},t)\) contain components that vary with time as \(\exp(-\text{i}\omega{t})\) with positive values of \(\omega\), while \(\mathbf{E}^*(\mathbf{r},t)\) and \({\mathbf{P}^{(n)}}^*(\mathbf{r},t)\) contain those varying with time as \(\exp(\text{i}\omega{t})\) with positive values of \(\omega\) or, equivalently, \(\exp(-\text{i}\omega{t})\) with \(\omega\) assuming negative values.

By substituting (9-12) and (9-13) in (9-8) - (9-10), expressions of complex polarizations in terms of complex fields can be obtained.

The relation between the complex linear polarization \(\mathbf{P}^{(1)}(\mathbf{r},t)\) and the complex field \(\mathbf{E}(\mathbf{r},t)\) has the same form as (9-8), as shown in (45) [refer to the harmonic fields tutorial].

However, the complex nonlinear polarizations \(\mathbf{P}^{(2)}(\mathbf{r},t)\) and \(\mathbf{P}^{(3)}(\mathbf{r},t)\) contain products of \(\mathbf{E}(\mathbf{r},t)\) and \(\mathbf{E}^*(\mathbf{r},t)\) in addition to those of \(\mathbf{E}(\mathbf{r},t)\) alone. Consequently, they have more complicated expressions than those of the real polarizations in (9-9) and (9-10).

The optical field involved in a nonlinear interaction usually contains multiple, distinct frequency components. Such a field can be expanded in terms of its frequency components as is done in (5) [refer to the coupled-wave theory tutorial]:

\[\tag{9-14}\mathbf{E}(\mathbf{r},t)=\sum_q\mathbf{E}_q(\mathbf{r})\exp(-\text{i}\omega_qt)=\sum_q\boldsymbol{\mathcal{E}}_q(\mathbf{r})\exp(\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega_qt)\]

where \(\boldsymbol{\mathcal{E}}_q(\mathbf{r})\) is the slowly varying amplitude and \(\mathbf{k}_q\) is the wavevector of the frequency component \(\omega_q\).

The nonlinear polarizations also contain multiple frequency components and can be expanded as

\[\tag{9-15}\mathbf{P}^{(n)}(\mathbf{r},t)=\sum_q\mathbf{P}^{(n)}_q(\mathbf{r})\exp(-\text{i}\omega_qt)\]

Note that we do not attempt to express \(\mathbf{P}^{(n)}_q(\mathbf{r})\) further in terms of a slowly varying polarization amplitude multiplied by a fast varying spatial phase term as is done for \(\mathbf{E}_q(\mathbf{r})\). The reason is that, as we shall see later, ** the wavevector that characterizes the fast-varying spatial phase of a nonlinear polarization \(\mathbf{P}^{(n)}_q(\mathbf{r})\) is not simply determined by the frequency \(\omega_q\) but is determined by the fields involved in a particular nonlinear interaction of interest**.

In the discussions of nonlinear polarizations, we also use the notations \(\mathbf{E}(\omega_q)\) and \(\mathbf{P}^{(n)}(\omega_q)\) defined respectively as

\[\tag{9-16}\mathbf{E}(\omega_q)=\mathbf{E}_q(\mathbf{r})\qquad\text{and}\qquad\mathbf{P}^{(n)}(\omega_q)=\mathbf{P}^{(n)}_q(\mathbf{r})\]

Field and polarization components with negative frequencies are interpreted as

\[\tag{9-17}\mathbf{E}(-\omega_q)=\mathbf{E}^*(\omega_q)\qquad\text{and}\qquad\mathbf{P}^{(n)}(-\omega_q)={\mathbf{P}^{(n)}}^*(\omega_q)\]

The following notation for nonlinear susceptibilities is also used:

\[\tag{9-18}\boldsymbol{\chi}^{(n)}(\omega_q=\omega_1+\omega_2+\cdots+\omega_n)=\boldsymbol{\chi}^{(n)}(\omega_1,\omega_2,\ldots,\omega_n)\]

for \(\omega_1+\omega_2+\cdots+\omega_n=\omega_q\), where \(\boldsymbol{\chi}^{(n)}(\omega_1,\omega_2,\ldots,\omega_n)\) are the frequency-domain susceptibilities defined in (9-11).

Using the definitions of the complex fields and polarizations in (9-12) and (9-13) as well as their expansions in (9-14) and (9-15), we can obtain, by taking the Fourier transform on (9-9) and (9-10), the following relations:

\[\tag{9-19}\mathbf{P}^{(2)}(\omega_q)=\epsilon_0\sum_{m,n}\boldsymbol{\chi}^{(2)}(\omega_q=\omega_m+\omega_n):\mathbf{E}(\omega_m)\mathbf{E}(\omega_n)\]

and

\[\tag{9-20}\mathbf{P}^{(3)}(\omega_q)=\epsilon_0\sum_{m,n,p}\boldsymbol{\chi}^{(3)}(\omega_q=\omega_m+\omega_n+\omega_p)\vdots\mathbf{E}(\omega_m)\mathbf{E}(\omega_n)\mathbf{E}(\omega_p)\]

The summation is performed over all ** positive and negative** values of frequencies that, for a given \(\omega_q\), satisfy the constraint of \(\omega_m+\omega_n=\omega_q\) in the case of (9-19) and the constraint of \(\omega_m+\omega_n+\omega_p=\omega_q\) in the case of (9-20). More explicitly, by performing the summation over

**and by expanding the product, we have**

*positive frequencies only*\[\tag{9-21}\begin{align}P_i^{(2)}(\omega_q)=\epsilon_0\sum_{j,k}\sum_{\omega_m,\omega_n\gt0}&\left[\chi_{ijk}^{(2)}(\omega_q=\omega_m+\omega_n)E_j(\omega_m)E_k(\omega_n)\right.\\&+\chi_{ijk}^{(2)}(\omega_q=\omega_m-\omega_n)E_j(\omega_m)E_k^*(\omega_n)\\&+\left.\chi_{ijk}^{(2)}(\omega_q=-\omega_m+\omega_n)E_j^*(\omega_m)E_k(\omega_n)\right]\end{align}\]

and

\[\tag{9-22}\begin{align}P_i^{(3)}(\omega_q)=\epsilon_0\sum_{j,k,l}\sum_{\omega_m,\omega_n,\omega_p\gt0}&\left[\chi_{ijkl}^{(3)}(\omega_q=\omega_m+\omega_n+\omega_p)E_j(\omega_m)E_k(\omega_n)E_l(\omega_p)\right.\\&+\chi_{ijkl}^{(3)}(\omega_q=\omega_m+\omega_n-\omega_p)E_j(\omega_m)E_k(\omega_n)E_l^*(\omega_p)\\&+\chi_{ijkl}^{(3)}(\omega_q=\omega_m-\omega_n+\omega_p)E_j(\omega_m)E_k^*(\omega_n)E_l(\omega_p)\\&+\chi_{ijkl}^{(3)}(\omega_q=-\omega_m+\omega_n+\omega_p)E_j^*(\omega_m)E_k(\omega_n)E_l(\omega_p)\\&+\chi_{ijkl}^{(3)}(\omega_q=\omega_m-\omega_n-\omega_p)E_j(\omega_m)E_k^*(\omega_n)E_l^*(\omega_p)\\&+\chi_{ijkl}^{(3)}(\omega_q=-\omega_m+\omega_n-\omega_p)E_j^*(\omega_m)E_k(\omega_n)E_l^*(\omega_p)\\&+\left.\chi_{ijkl}^{(3)}(\omega_q=-\omega_m-\omega_n+\omega_p)E_j^*(\omega_m)E_k^*(\omega_n)E_l(\omega_p)\right]\end{align}\]

Usually only a limited number of frequencies participate in a given nonlinear optical interaction. Consequently, only one or a few terms among those listed in (9-21) or (9-22) contribute to a nonlinear polarization of interest.

**Example 9-1**

Three optical fields at the wavelengths of \(\lambda_1=750\text{ nm}\), \(\lambda_2=600\text{ nm}\), and \(\lambda_3=500\text{ nm}\), corresponding to the frequencies of \(\omega_1=2\pi{c}/\lambda_1\), \(\omega_2=2\pi{c}/\lambda_2\), and \(\omega_3=2\pi{c}/\lambda_3\), respectively, are involved in second-order nonlinear optical interactions. Find the nonlinear polarization \(\mathbf{P}^{(2)}\) at the frequency of \(\omega_4=2\pi{c}/\lambda_4\), where \(\lambda_4=300\text{ nm}\). If \(\mathbf{E}(\omega_1)=E_1\hat{x}\), \(\mathbf{E}(\omega_2)=E_2(\hat{y}+\hat{z})/\sqrt{2}\), and \(\mathbf{E}(\omega_3)=E_3\hat{z}\), what is the \(x\) component of \(\mathbf{P}^{(2)}(\omega_4)\)?

Because \(\lambda_1^{-1}+\lambda_3^{-1}=2\lambda_2^{-1}=\lambda_4^{-1}\), we find that \(\omega_4=\omega_1+\omega_3=\omega_2+\omega_2\). Therefore, using (9-19), we find the following second-order nonlinear polarization at the frequency \(\omega_4\):

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

Note that there are two terms from the mixing of \(\omega_1\) and \(\omega_3\) because of permutation, but there is only one term from \(\omega_2\) mixing with itself. Using the given fields at the three frequencies, we can express the \(x\) component of \(\mathbf{P}^{(2)}(\omega_4)\) as

\[\begin{align}P^{(2)}_x(\omega_4)&=\epsilon_0\left[\chi^{(2)}_{xxz}(\omega_4=\omega_1+\omega_3)E_1E_3+\chi^{(2)}_{xzx}(\omega_4=\omega_3+\omega_1)E_3E_1\right.\\&\qquad+\chi^{(2)}_{xyz}(\omega_4=\omega_2+\omega_2)\frac{E_2}{\sqrt{2}}\frac{E_2}{\sqrt{2}}+\chi^{(2)}_{xzy}(\omega_4=\omega_2+\omega_2)\frac{E_2}{\sqrt{2}}\frac{E_2}{\sqrt{2}}\\&\qquad\left.+\chi^{(2)}_{xyy}(\omega_4=\omega_2+\omega_2)\frac{E_2}{\sqrt{2}}\frac{E_2}{\sqrt{2}}+\chi^{(2)}_{xzz}(\omega_4=\omega_2+\omega_2)\frac{E_2}{\sqrt{2}}\frac{E_2}{\sqrt{2}}\right]\\&\\&=\epsilon_0\left[\chi^{(2)}_{xxz}(\omega_4=\omega_1+\omega_3)E_1E_3+\chi^{(2)}_{xzx}(\omega_4=\omega_3+\omega_1)E_3E_1\right.\\&\qquad+\chi^{(2)}_{xyz}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}+\chi^{(2)}_{xzy}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}\\&\left.\qquad+\chi^{(2)}_{xyy}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}+\chi^{(2)}_{xzz}(\omega_4=\omega_2+\omega_2)\frac{E_2^2}{2}\right]\end{align}\]

The other two components, \(P_y^{(2)}(\omega_4)\) and \(P_z^{(2)}(\omega_4)\), of \(\mathbf{P}^{(2)}(\omega_4)\) can be explicitly spelled out by following a similar procedure.

The next part continues with the nonlinear optical susceptibilities tutorial.