# Guided-Wave Optical Frequency Converters

This is a continuation from the previous tutorial - nonlinear optical interactions in waveguides.

All of the optical frequency converters discussed in the optical frequency converters tutorial can be made in waveguide structures.

The basic principles and characteristics of these devices are the same as their bulk counterparts, except that the characteristics of the waveguide modes have to be considered.

Though a guided-wave optical frequency converter generally takes the form of a single waveguide, there is often a possibility that multiple waveguide modes are involved in the frequency conversion process.

Each individual frequency component can consist of multiple waveguide modes, as expressed by (9-216) [refer to the nonlinear optical interactions in waveguides tutorial]. Even when each frequency component is represented by only one waveguide mode, it is still possible for the different interacting frequency components to be in different waveguide modes.

For a parametric second-order process in a waveguide involving three different frequencies with $$\omega_3=\omega_1+\omega_2$$, we have

$\tag{9-219}\hat{\boldsymbol{\mathcal{E}}}^*_{3,\nu}\cdot\mathbf{P}_3^{(2)}=2\epsilon_0\sum_{\mu,\xi}\hat{\boldsymbol{\mathcal{E}}}^*_{3,\nu}\cdot\boldsymbol{\chi}^{(2)}(\omega_3=\omega_1+\omega_2):\hat{\boldsymbol{\mathcal{E}}}_{1,\mu}\hat{\boldsymbol{\mathcal{E}}}_{2,\xi}A_{1,\mu}A_{2,\xi}\text{e}^{\text{i}(\beta_{1,\mu}+\beta_{2,\xi})z}$

to replace (9-56) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial], and similar expressions for $$\hat{\boldsymbol{\mathcal{E}}}^*_{1,\mu}\cdot\mathbf{P}_1^{(2)}$$ and $$\hat{\boldsymbol{\mathcal{E}}}^*_{2,\xi}\cdot\mathbf{P}_2^{(2)}$$ to replace (9-57) and (9-58), respectively.

The interacting waves in an efficient frequency converter normally propagate in the same direction though contradirectional geometry is also possible. Here, we consider only codirectional geometry with all of the interacting waves propagating in the forward direction.

Then, using (9-218) [refer to the nonlinear optical interactions in waveguides tutorial], we can write

$\tag{9-220}\frac{\text{d}A_{3,\nu}}{\text{d}z}=\text{i}\omega_3\sum_{\mu,\xi}C_{\nu\mu\xi}A_{1,\mu}A_{2,\xi}\text{e}^{\text{i}\Delta\beta_{\nu\mu\xi}z}$

$\tag{9-221}\frac{\text{d}A_{1,\mu}}{\text{d}z}=\text{i}\omega_1\sum_{\nu,\xi}C^*_{\nu\mu\xi}A_{3,\nu}A^*_{2,\xi}\text{e}^{-\text{i}\Delta\beta_{\nu\mu\xi}z}$

$\tag{9-222}\frac{\text{d}A_{2,\xi}}{\text{d}z}=\text{i}\omega_2\sum_{\nu,\mu}C^*_{\nu\mu\xi}A_{3,\nu}A^*_{1,\mu}\text{e}^{-\text{i}\Delta\beta_{\nu\mu\xi}z}$

where

\tag{9-223}\begin{align}C_{\nu\mu\xi}&=2\epsilon_0\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\hat{\boldsymbol{\mathcal{E}}}^*_{3,\nu}\cdot\boldsymbol{\chi}^{(2)}(\omega_3=\omega_1+\omega_2):\hat{\boldsymbol{\mathcal{E}}}_{1,\mu}\hat{\boldsymbol{\mathcal{E}}}_{2,\xi}\text{d}x\text{d}y\\&=4\epsilon_0\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\hat{\boldsymbol{\mathcal{E}}}^*_{3,\nu}\cdot\mathbf{d}(\omega_3=\omega_1+\omega_2):\hat{\boldsymbol{\mathcal{E}}}_{1,\mu}\hat{\boldsymbol{\mathcal{E}}}_{2,\xi}\text{d}x\text{d}y\end{align}

is the effective nonlinear coefficient that accounts for the overlapping of the field distribution patterns of different waveguide modes, as well as for any possible spatial variations in $$\boldsymbol{\chi}^{(2)}$$ due to the waveguide structure, and

$\tag{9-224}\Delta\beta_{\nu\mu\xi}=\beta_{1,\mu}+\beta_{2,\xi}-\beta_{3,\nu}$

is the phase mismatch.

In comparison to the effective nonlinear susceptibility, $$\chi_\text{eff}$$, defined in (9-59) [refer to the coupled-wave analysis of nonlinear optical interactions tutorial] for the interaction of plane waves, the effective nonlinear coefficient defined above for the interaction of waveguide modes has the following relation:

$\tag{9-225}|C_{\nu\mu\xi}|^2=\frac{|\chi_\text{eff}|^2}{2c^3\epsilon_0n_{3,\nu}n_{1,\mu}n_{2,\xi}}\frac{\Gamma_{\nu\mu\xi}}{\mathcal{A}}=\frac{2|d_\text{eff}|^2}{c^3\epsilon_0n_{3,\nu}n_{1,\mu}n_{2,\xi}}\frac{\Gamma_{\nu\mu\xi}}{\mathcal{A}}$

where $$n_{q,\nu}=c\beta_{q,\nu}/\omega_q$$ is the effective refractive index of a waveguide mode, $$\Gamma_{\nu\mu\xi}$$ is the overlap factor for the interacting waveguide modes, and $$\mathcal{A}$$ is the cross-sectional area of the waveguide core.

The overlap factor $$\Gamma_{\nu\mu\xi}$$ accounts for the differences in the mode field distributions among the interacting waves and any transverse spatial variations in the nonlinear susceptibility.

An effective area for the interaction can be defined as $$\mathcal{A}_\text{eff}=\mathcal{A}/\Gamma_{\nu\mu\xi}$$.

In the case of second-harmonic generation in a waveguide, we have

$\tag{9-226}\mathbf{P}_{2\omega}^{(2)}=\epsilon_0\sum_{\mu,\xi}\boldsymbol{\chi}^{(2)}(2\omega=\omega+\omega):\hat{\boldsymbol{\mathcal{E}}}_{\omega,\mu}\hat{\boldsymbol{\mathcal{E}}}_{\omega,\xi}A_{\omega,\mu}A_{\omega,\xi}\text{e}^{\text{i}(\beta_{\omega,\mu}+\beta_{\omega,\xi})z}$

$\tag{9-227}\mathbf{P}_{\omega}^{(2)}=2\epsilon_0\sum_{\nu,\xi}\boldsymbol{\chi}^{(2)}(\omega=2\omega-\omega):\hat{\boldsymbol{\mathcal{E}}}_{2\omega,\nu}\hat{\boldsymbol{\mathcal{E}}}^*_{\omega,\xi}A_{2\omega,\nu}A^*_{\omega,\xi}\text{e}^{\text{i}(\beta_{2\omega,\nu}-\beta_{\omega,\xi})z}$

Therefore, the coupled equations for second-harmonic generation in a waveguide are

$\tag{9-228}\frac{\text{d}A_{2\omega,\nu}}{\text{d}z}=\text{i}\omega\sum_{\mu,\xi}C_{\nu\mu\xi}A_{\omega,\mu}A_{\omega,\xi}\text{e}^{\text{i}\Delta\beta_{\nu\mu\xi}z}$

$\tag{9-229}\frac{\text{d}A_{\omega,\mu}}{\text{d}z}=\text{i}\omega\sum_{\nu,\xi}C^*_{\nu\mu\xi}A_{2\omega,\nu}A^*_{\omega,\xi}\text{e}^{-\text{i}\Delta\beta_{\nu\mu\xi}z}$

where

\tag{9-230}\begin{align}C_{\nu\mu\xi}&=2\epsilon_0\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\hat{\boldsymbol{\mathcal{E}}}^*_{2\omega,\nu}\cdot\boldsymbol{\chi}^{(2)}(2\omega=\omega+\omega):\hat{\boldsymbol{\mathcal{E}}}_{\omega,\mu}\hat{\boldsymbol{\mathcal{E}}}_{\omega,\xi}\text{d}x\text{d}y\\&=4\epsilon_0\displaystyle\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\hat{\boldsymbol{\mathcal{E}}}^*_{2\omega,\nu}\cdot\mathbf{d}(2\omega=\omega+\omega):\hat{\boldsymbol{\mathcal{E}}}_{\omega,\mu}\hat{\boldsymbol{\mathcal{E}}}_{\omega,\xi}\text{d}x\text{d}y\end{align}

and

$\tag{9-231}\Delta\beta_{\nu\mu\xi}=\beta_{\omega,\mu}+\beta_{\omega,\xi}-\beta_{2\omega,\nu}$

All of the general concepts discussed in the coupled-wave analysis of nonlinear optical interactions tutorial for parametric second-order interactions are equally valid for such interactions in an optical waveguide, except that the form of each relation must be modified to factor in the characteristics of the waveguide modes.

For instance, Manley-Rowe relations still exist but such relations have to be expressed in terms of optical powers in the waveguide modes. Phase matching is also most important for an efficient interaction, but it is now determined by the propagation constants of the interacting waveguide modes. Therefore, the coherence length for the coupling of the mode fields $$\boldsymbol{\mathcal{E}}_{3,\nu}$$, $$\boldsymbol{\mathcal{E}}_{1,\mu}$$, and $$\boldsymbol{\mathcal{E}}_{2,\xi}$$ is

$\tag{9-232}l_\text{coh}=\frac{\pi}{|\Delta\beta_{\nu\mu\xi}|}$

Because the propagation constant $$\beta_{q,\nu}$$ for a given frequency $$\omega_q$$ is mode dependent due to modal dispersion, the phase mismatch, and, consequently, the efficiency of an interaction are dependent on specific combinations of the modes among the interacting frequency components.

In a multimode waveguide, there can be many different mode combinations, as is indicated by the summation over the mode indices on the right-hand side of the coupled equations in (9-220) - (9-222) and those in (9-228) and (9-229).

However, it is unlikely and undesirable, though not impossible, for multiple mode combinations to be simultaneously phase matched in a particular interaction. In a practical optical frequency converter, normally only one waveguide mode for each frequency component is efficiently coupled to other frequency components in the interaction.

When an interaction involves only one waveguide mode in each frequency component, the coupled equations have the form of those for the corresponding interaction in a bulk medium though the mode amplitudes are used instead of the field amplitudes and the coefficients in the equations look different.

Then, the characteristics of any guided-wave optical frequency converter can be obtained by converting those of its bulk counterpart described in the optical frequency converters tutorial with the following modifications:

(1) the mode power $$P_{q,\nu}$$ is used in place of $$I_q\mathcal{A}_q$$

(2) $$\Delta\beta$$ is used instead of $$\Delta{k}$$

(3) the nonlinear coefficient $$C_{\nu\mu\xi}$$ is used in place of $$\chi_\text{eff}$$ by replacing a compound coefficient of the form on the right-hand side of (9-225) in any relation for a bulk device with $$|C_{\nu\mu\xi}|^2$$ for a guided-wave device.

For example, consider a second-harmonic generator in which the fundamental and second-harmonic waves each contain only one mode. Then, the coupled equations in (9-228) and (9-229) are reduced to

$\tag{9-233}\frac{\text{d}A_{2\omega,\nu}}{\text{d}z}=\text{i}\omega{C}A_{\omega,\mu}^2\text{e}^{\text{i}\Delta\beta{z}}$

$\tag{9-234}\frac{\text{d}A_{\omega,\mu}}{\text{d}z}=\text{i}\omega{C^*}A_{2\omega,\nu}A^*_{\omega,\mu}\text{e}^{-\text{i}\Delta\beta{z}}$

where $$C=C_{\nu\mu\mu}$$ and $$\Delta\beta=2\beta_{\omega,\mu}-\beta_{2\omega,\nu}$$.

In the low-efficiency limit when depletion of power in the fundamental wave is negligible, we can obtain, by integrating (9-233) or by converting (9-110) [refer to the optical frequency converters tutorial], the following relation for a waveguide of length $$l$$:

$\tag{9-235}P_{2\omega,\nu}(l)=\omega^2|C|^2P_{\omega,\mu}^2l^2\frac{\sin^2(\Delta\beta{l}/2)}{(\Delta\beta{l})^2}=\frac{4\pi^2c^2}{\lambda^2}|C|^2P_{\omega,\mu}^2l^2\frac{\sin^2(\Delta\beta{l}/2)}{(\Delta\beta{l}/2)^2}$

In the high-efficiency limit with perfect phase matching, we have

$\tag{9-236}P_{2\omega,\nu}(l)=P_{\omega,\mu}(0)\tanh^2\kappa{l}$

$\tag{9-237}P_{\omega,\mu}(l)=P_{\omega,\mu}(0)\text{sech}^2\kappa{l}$

with the coefficient $$\kappa$$ given by

$\tag{9-238}\kappa=[\omega^2|C|^2P_{\omega,\mu}(0)]^{1/2}=\left[\frac{4\pi^2c^2}{\lambda^2}|C|^2P_{\omega,\mu}(0)\right]^{1/2}$

The techniques for phase matching discussed in the phase matching for nonlinear optical processes tutorial are also applicable to guided-wave devices. Besides, the modal dispersion in a waveguide can also be used for phase matching if modes of different orders are involved in an interaction.

Often, a combination of different techniques is employed. For example, a waveguide is fabricated along a certain direction in a crystal for collinear birefringent phase matching, but temperature is used for fine tuning once the wave propagation direction is fixed by the waveguide structure.

Quasi-phase matching is particularly useful for guided-wave devices because of its advantages discussed in the phase matching for nonlinear optical processes tutorial and because of its compatibility with micro-fabrication technology.

For a guided-wave device that is quasi-phase matched using a periodic structure with a duty factor $$\xi$$, the coupled equations can be transformed in a manner similar to the transformation shown in (9-96) [refer to the phase matching for nonlinear optical processes tutorial], resulting in a phase mismatch of $$\Delta\beta_\text{Q}=\Delta\beta+qK$$ that is minimized with a particular integer $$q$$ and a nonlinear coefficient $$C_\text{Q}$$ given by

$\tag{9-239}C_\text{Q}=2C\frac{\sin\xi{q}\pi}{q\pi}\text{e}^{-\text{i}\xi{q}\pi}$

according to (9-100) [refer to the phase matching for nonlinear optical processes tutorial].

With quasi-phase matching, we have to replace the phase mismatch $$\Delta\beta$$ in (9-235) by $$\Delta\beta_\text{Q}$$, and the parameter $$C$$ in (9-235) and (9-238) by $$C_\text{Q}$$.

For a first-order structure with a 50% duty factor, $$|C_\text{Q}|=2|C|/\pi$$. In a slab waveguide, the fanned structure shown in Figure 9-14(c) [refer to the phase matching for nonlinear optical processes tutorial] can also be used for continuous wavelength tuning through quasi-phase matching.

Example 9-23

A PPLN waveguide is used for second-harmonic generation of a fundamental wave at 1.10 μm wavelength. The waveguide is a diffused channel waveguide formed by Ti diffusion into a PPLN crystal similar to the one described in Example 9-11 [refer to the phase matching for nonlinear optical processes tutorial].

It has a diffusion depth of $$d=2\text{ μm}$$ and a width of $$w=3\text{ μm}$$, for an effective waveguide core are of $$\mathcal{A}=wd=6\text{ μm}^2$$. It is a single-mode waveguide for the fundamental wavelength at 1.1 μm. The overlap factor for second-harmonic generation at this wavelength in this waveguide is $$\Gamma=0.4$$. The grating period of the PPLN is properly chosen as a first-order grating with a 50% duty factor for quasi-phase matching of the interacting waveguide modes.

For easy comparison to second-harmonic generation in the bulk PPLN crystal considered in Example 9-12(d) [refer to the optical frequency converters tutorial], we take the waveguide length to be $$l=1\text{ cm}$$. Find the normalized second-harmonic conversion efficiency for this device.

By replacing $$\Delta\beta$$ with $$\Delta\beta_\text{Q}$$ and $$C$$ with $$C_\text{Q}$$ in (9-235), we have the following normalized efficiency in the low-efficiency limit for the PPLN waveguide:

$\tag{9-240}\hat{\eta}_\text{SH}=\frac{P_{2\omega}(l)}{P_\omega^2}=\frac{4\pi^2c^2}{\lambda^2}|C_\text{Q}|^2l^2\frac{\sin^2(\Delta\beta_\text{Q}l/2)}{(\Delta\beta_\text{Q}l/2)^2}$

For perfect quasi-phase matching with a first-order grating that has a 50% duty factor, we have

$\tag{9-241}\hat{\eta}_\text{SH}=\frac{4\pi^2c^2}{\lambda^2}|C_\text{Q}|^2l^2=\frac{32|d_\text{eff}|^2}{c\epsilon_0n_\omega^2n_{2\omega}\lambda^2}\frac{\Gamma}{\mathcal{A}}l^2$

where $$n_\omega$$ and $$n_{2\omega}$$ are the effective refractive index, $$n_\beta$$, of the waveguide modes at the fundamental and second-harmonic frequencies, respectively.

Because the index change created by Ti diffusion is very small, typically on the order of 0.5%, we can simply take the refractive index of the bulk PPLN as a very good approximation for the effective refractive index of a waveguide mode.

From Example 9-11 [refer to the phase matching for nonlinear optical processes tutorial], we have $$d_\text{eff}=d_{33}=-25.2\text{ pm V}^{-1}$$, $$n_\omega=n_\omega^\text{e}=2.1536$$, and $$n_{2\omega}=n_{2\omega}^\text{e}=2.2260$$. We then find the following normalized conversion efficiency:

\begin{align}\hat{\eta}_\text{SH}&=\frac{32\times(25.2\times10^{-12})^2\times0.4\times(1\times10^{-2})^2}{3\times10^8\times8.85\times10^{-12}\times(2.1536)^2\times2.226\times(1.1\times10^{-6})^2\times6\times10^{-12}}\text{ W}^{-1}\\&=409\%\text{ W}^{-1}\end{align}

This normalized conversion efficiency for the PPLN waveguide is $$159$$ times that obtained in Example 9-12(d) for the bulk PPLN [refer to the optical frequency converters tutorial].

Further increase in the efficiency is possible by increasing the length of the waveguide. In the waveguide device, the efficiency continues to increase quadratically with length $$l$$ because the optical waves remain confined as the waveguide length is increased. In a bulk device, the best efficiency only increases linearly with length $$l$$, as seen in (9-121) [refer to the optical frequency converters tutorial], because of the limitation imposed by diffraction.

Note that (9-240) and (9-241) are valid only in the low-efficiency limit. Clearly, $$\hat{\eta}_\text{SH}=409\%\text{ W}^{-1}$$ obtained in this example does not mean that it is possible to obtain an unphysical efficiency of $$409\%$$ by launching a fundamental beam of $$1\text{ W}$$ power into the waveguide. Nor does it mean that a conversion efficiency of $$100\%$$ is obtained by launching a fundamental beam of $$244\text{ mW}$$ into the waveguide.

It only means that a very low input power of the fundamental wave is needed to obtain a decent conversion efficiency. For example, an input fundamental power of only $$P_\omega=24.4\text{ mW}$$ is required to have an output second-harmonic power of $$P_{2\omega}=2.44\text{ mW}$$ for a conversion efficiency of $$10\%$$.

A conversion efficiency approaching $$100\%$$ is theoretically possible, but with an input fundamental power found by using the relation in (9-236) for the high-efficiency limit.

The next part continues with the guided-wave all-optical modulators and switches tutorial