# Nonlinear Optical Interactions in Waveguides

This is a continuation from the previous tutorial - Raman and Brillouin devices.

As we have seen in the preceding tutorials, the efficiency of a nonlinear optical interaction generally increases with the intensities of the interacting optical waves and the interaction length.

In a homogeneous bulk medium, the intensity of an optical wave can be increased by tightening the focus of the beam to reduce its cross-sectional spot size, but often at the expense of reducing the effective interaction length due to an increase in the beam divergence as a result of the decrease in the beam spot size.

In an optical waveguide, however, an optical wave is guided and confined to a small cross-sectional area for the entire length of the waveguide. Because of optical confinement, a guided optical wave can maintain a high intensity over a long distance that is practically limited only by the length and the attenuation coefficient of the waveguide.

Therefore, both high intensity and long interaction length desired for efficient nonlinear optical interactions can be simultaneously fulfilled in an optical waveguide. For example, in a low-loss optical fiber, the effective interaction length is on the order of tens of kilometers and the optical intensity can be quite high at a modest power level because of the small core diameter of a typical single-mode fiber.

This unique characteristic makes optical waveguides ideal media for efficient nonlinear optical devices.

Coupled-wave theory is used in the analysis of interactions among waves of different frequencies, including the acousto-optic interactions and the nonlinear optical interactions discussed in the previous tutorials.

In the analysis of the coupling of waveguide modes, however, coupled-mode theory has to be used. In general, both the interaction among different optical frequencies and the characteristics of the waveguide modes have to be considered for a nonlinear optical interaction in an optical waveguide.

Therefore, a combination of coupled-wave and coupled-mode theories has to be employed in the analysis of such an interaction.

First, the total field of the interacting waves is expanded in terms of the fields of individual frequencies:

$\tag{9-215}\mathbf{E}(\mathbf{r},t)=\sum_q\mathbf{E}_q(\mathbf{r})\exp(-\text{i}\omega_qt)$

where $$\mathbf{E}_q(\mathbf{r})$$ is the spatially dependent total field amplitude for the frequency $$\omega_q$$.

Then, instead of taking out a uniquely defined fast-varying spatial variation as done in (5) [refer to the coupled-wave theory tutorial] for the formulation of coupled-wave theory, we expand each field $$\mathbf{E}_q(\mathbf{r})$$ at a given frequency $$\omega_q$$ in terms of the waveguide modes:

$\tag{9-216}\mathbf{E}_q(\mathbf{r})=\sum_\nu{A}_{q,\nu}(z)\hat{\boldsymbol{\mathcal{E}}}_{q,\nu}(x,y)\exp(\text{i}\beta_{q,\nu}z)$

Note that the propagation constant $$\beta_{q,\nu}$$ is a function of both optical frequency $$\omega_q$$ and the waveguide mode $$\nu$$.

Note also that this expansion is valid only when nonlinear polarization $$\mathbf{P}^{(n)}$$ is small compared to linear polarization $$\mathbf{P}^{(1)}$$ so that the waveguide modes defined by the linear optical properties of the medium remain a valid concept.

Because $$\hat{\boldsymbol{\mathcal{E}}}_{q,\nu}$$ is the normalized mode field pattern defined in the "power and orthogonality" section of the wave equations for optical waveguides tutorial, the power contained in waveguide mode $$\nu$$ at frequency $$\omega_q$$ is simply given by

$\tag{9-217}P_{q,\nu}=|A_{q,\nu}|^2=A_{q,\nu}A^*_{q,\nu}$

Following the procedures used in formulating the coupled-mode equations discussed in the coupled-mode theory tutorial and allowing for any possible linear coupling besides nonlinear, we find the following coupled-mode equation that accounts for the $$n$$th-order nonlinear interaction in a waveguide structure:

$\tag{9-218}\pm\frac{\text{d}A_{q,\nu}}{\text{d}z}=\sum_\mu{\text{i}}\kappa_{q,\nu\mu}A_{q,\mu}\text{e}^{\text{i}(\beta_{q,\mu}-\beta_{q,\nu})z}+\text{i}\omega_q\text{e}^{-\text{i}\beta_{q,\nu}z}\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\hat{\boldsymbol{\mathcal{E}}}^*_{q,\nu}\cdot\mathbf{P}_q^{(n)}\text{d}x\text{d}y$

where the plus sign is taken for a forward-propagating mode with $$\beta_{q,\nu}\gt0$$, and the minus sign is for a backward-propagating mode with $$\beta_{q,\nu}\lt0$$.

To be precise, just like that of the linear coupling coefficient $$\kappa_{\nu, \mu}$$, the form of the nonlinear term on the right-hand side of (9-218) also has to be modified when $$\hat{\boldsymbol{\mathcal{E}}}_{q,\nu}$$ represents nonorthogonal modes of different individual waveguides in a structure that consists of multiple waveguides. Such modification is normally not significant and is ignored here. No such approximation is incurred in the use of (9-218), however, if $$\hat{\boldsymbol{\mathcal{E}}}_{q,\nu}$$ represents the modes of the entire structure, which are orthogonal to each other. This is always true in the case of a single waveguide. It is also true if the supermodes of a structure that consists of multiple waveguides are used in the analysis.

In summary, for nonlinear interactions in optical waveguides, the expansion in (9-216) replaces that in (9-53) and (9-218) replaces (9-55) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial].

Besides the nonlinear effect characterized by $$\mathbf{P}^{(n)}$$, linear effects such as a periodic grating or an externally applied voltage also modify the behavior of the waves in a waveguide and lead to coupling between different waveguide modes. Coupling between different waveguides in the presence of optical nonlinearity is also possible. The first term on the right-hand side of (9-218) accounts for the possibility of such linear coupling effects based on the coupled-mode formulation discussed in the coupled-mode theory tutorial].

Therefore, (9-218) can be viewed as an extension of (33) or (39) [refer to the coupled-mode theory tutorial] to include the nonlinear perturbation. In the case of coupling between different waveguides, $$\kappa_{q,\nu\mu}$$ still has to be evaluated using (40) [refer to the coupled-mode theory tutorial] due to the nonorthogonality between modes of different waveguides.

Many guided-wave nonlinear optical devices have direct bulk counterparts. The use of a waveguide for such a device offers the advantages of improved efficiency, phase matching, or miniaturization of the device but is not absolutely necessary for the device function. Guide-wave optical frequency converters typically fall into this category.

Some nonlinear optical devices rely on the waveguide geometry for their functions and thus have no bulk counterparts. All-optical switches and modulators that use waveguide interferometers or waveguide couplers belong to this category.

Sometimes, the use of an optical waveguide is necessary for the practical reason that only a waveguide can provide the long interaction length required for the function of a device though the basic function of the device does not depend on the waveguide geometry. Many nonlinear optical devices that use optical fibers belong to this category.

The next part continues with the Guided-Wave Optical Frequency Converters tutorial.