4.1: Introduction

In the previous chapters we have discussed how to guide light and generate light. However, an optical network would not be complete if we were unable to control where the light went, or modulate the light to communicate. Although there are many different ways to do each of these tasks, we will only cover a few solutions that are based on integrated optics, which happen to be most common in optical communications. Integrated optical circuits are the photonic equivalent of integrated circuits for electronics. Instead of wires, waveguides control the flow of light, while electronic and thermal variations are used to switch the light.

Integrated optics makes use of three basic fundamental properties: the refractive index of materials is temperature dependent, and, for some materials, the refractive index also changes with an applied DC (or low frequency) voltage. These two properties allow for switching. The third property is that of directional coupling: when two waveguides are in close proximity to each other, the light from one waveguide will flow into the other waveguide. The consequences of these three properties are outlined below, and are used to explain the functional properties of a number of devices, including:

• Mach-Zhender interferometers
• Add-Drop multiplexers
• Array waveguide gratings
• Modulators

Integrated optics: materials

Integrated optical circuits can be fabricated in a number of materials. Glass (silicon oxide) on silicon wafers is a common choice, as is silicon nitride on silicon wafers. These material choices offer very low loss, and can be controlled using temperature. However, glass devices (in contrast to silicon nitride) have to be quite large. Indium phosphide provides for more compact designs, at the expense of higher losses. Indium phosphide can also provide on-chip sources (lasers and LEDs) and detectors (photodiodes). In addition, Indium phosphide, being a semiconductor can be used to control light via various high-speed mechanisms (electro-optic modulation, for instance). A final option that should be mentioned is lithium niobate. Lithium niobate is closer to glass in terms of losses, but has a strong eletro-optic effect, providing for high-speed modulation. Unfortunately, circuit fabrication in lithium niobate is more difficult compared to the other materials (which use standard lithography process developed for the semiconductor industry), thus lithium niobate is mainly limited to high-speed modulation.

Electro and thermo-optical effect

The electro and thermo-optical describe how the refractive index of a material changes with temperature or applied voltage. The change in refractive index with respect to temperature is often given by a Sellmeier equation, which is an empirical fit to measured data, given by:

\[\begin{equation}n^2 = A_0 + A_1f(T) + \sum_{i=0}^{i=N}\frac{B_i +D_if(T)}{\lambda^2-(C_i - E_i)^2}\label{eqn:sellmeier}\end{equation}\]

where \(f(T)\) is a temperature dependent correction, which may be of the form

\[\begin{equation}f(T) = \left(T-k_0\right)\left(T + k_1\right)\label{eqn:T-correct}\end{equation}\]

It is important to remember that equations \(\eqref{eqn:sellmeier}\) and \(\eqref{eqn:T-correct}\) are general examples. The most basic form of the Sellmeier equation can be derived from a general understanding of the interaction of light with matter. However, the goal is to provide the most accurate function of refractive index with wavelength and temperature, and, therefore, deviations from the general form are very common. Table \(\PageIndex{1}\) shows the measured constants for lithium niobate. Even here, care must be taken, because how lithium niobate is grown (it is a crystal and must be grown), and what dopants are added will change the Sellmeier equation. Hence, it is important to know the details of the material type before a published Sellmeier equation is used.

Table \(\PageIndex{1}\): coefficients for Sellmeier equation for lithium niobate with an additional term that does not fit the general form of equation \(\eqref{eqn:sellmeier}\).

	\(i=0\)	\(i=1\)
\(A\) coefficients	5.35583	4.629\(\times 10^{-7|\)
\(B\) coefficients	0.100473	100
\(C\) coefficients	0.20692	11.34927
\(D\) coefficients	3.862\(\times 10^{-8}\)	2.657(\times 10^{-5}\)
\(E\) coefficients	0.89\(\times 10^{-8}\)	0
\(k\) coefficients	24.5	570.82
Additional term	1.5334\(\times 10^{-2}\lambda^2\)

The electro-optic effect is similar: the refractive index is linearly or quadratically dependent on an applied voltage. The quadratic dependence can be ignored for our purposes, leaving only the linear dependence, called the Pockels effect, which is given by:

\[\begin{equation} n(E) \approx n - \frac{1}{2}\tau n^3E\label{eqn:pockels}\end{equation}\]

where \(n\) is the refractive index when the applied electric field, \(E=0\), and \(\tau\) is the Pockels coefficient, which has typical values of between 1-100 pm/V (p = pico = 10^-12). With such a small coefficient, it initially seems impossible to make use of the Pockels effect in practical high-speed devices. The field is dependent on the applied voltage divided by the distance between the terminals. In a material like lithium niobate (\(\tau = 25\) pm/V), the wafer thickness might be between 100-500 µm, meaning that 5 V results in 10-50 kV/m, and a change in refractive index of \(6.7\times 10^{-6}\). Thus, a clever design is required.

To give an impression of how these effects are used, consider two waves that are combined at a 50/50 beam splitter, as shown in Figure \(\PageIndex{1}\). The phase relationship between the two waves determines the balance between the intensities at ports 1 and 2.

\[\begin{align} E_{p1} &= \frac{E_1 + E_2}{2}\nonumber\\ I_{p1} &= E_{p1}E^*_{p1}\nonumber\\ &= \frac{1}{4}\left(E_1E_1^* + E_2E_2^* + E_1^*E_2 + E_1E_2^*\right)\label{eqn:interference}\end{align}\]

Figure \(\PageIndex{1}\): the combination of two light waves via a 50% reflective mirror.

The first two terms in equation \(\eqref{eqn:interference}\) are constants given by the amplitudes of the individual waves. The second two terms are so-called interference terms and determine at which port the light will exit. To make that clear, let us assume that \(E_1 = E_0\exp(-jk_1l_0)\) and \(E_2 = E_0\exp(-jk_2l_0)\), where \(k_1 = 2\pi n_1/\lambda\) and \(k_2= 2\pi n_2/\lambda\). We can, thus, see that \(E_1E_1^* = E_0^2\) and that \(E_2E_2^* = E_0^2\), but the interference terms are

\[\begin{align} E_1^*E_2 &= E_0^2\exp(j(k_1 - k_2)l_0)\nonumber\\ E_1E_2^* &= E_0^2\exp(-j(k_1-k_2)l_0)\nonumber\\ E_1^*E_2 + E_1E_2^* &= E_0^2\left(\exp(j(k_1 - k_2)l_0) + \exp(-j(k_1 - k_2)l_0)\right)\label{eqn:expVersion}\\ &= E_0^2\left(\cos(k_1 - k_2)l_0)+ j\sin((k_1 - k_2)l_0) + \cos((k_1 - k_2)l_0) - j\sin((k_1 - k_2)l_0)\right)\nonumber\\ &= 2E_0^2\cos((k_1 - k_2)l_0)\label{eqn:cosine}\end{align}\]

Recombining the results from equation \(\eqref{eqn:cosine}\) with equation \(\eqref{eqn:interference}\) and recognizing that the intensity is the square of the amplitude, leads to

\[\begin{equation}I_{p1} = \frac{I_0}{2}\left(1 + \cos((k_1 - k_2)l_0)\right)\label{eqn:fringes}\end{equation}\]

Thus, the intensity at ports one and two depend on the refractive index of the medium through which waves one and two propagated

\[\begin{equation}k_1 - k_2 = \frac{2\pi}{\lambda}\left(n_1-n_2\right) = k_0\delta n\label{eqn:wavevectordiff}\end{equation}\]

to fully redirect the intensity from port 1 to port 2, the cosine term must change from unity to negative one, which requires a change of \(\pi\). Thus, in the balanced case considered above, \(k_0\delta n l_0 = 0.5\) is necessary. If we consider that \(\lambda_0 = \) 1550 nm and \(\delta n = 6.7\times 10^{-6}\), then we see that \(l_0 \approx 13\) m. However, this number is without any cleverness in the design, and typical lengths are 2-3 mm.

Waveguide coupling

The field in a waveguide is partially in the core and partially in the cladding. The natural phase velocity of the wave in the cladding is slightly faster than the phase velocity in the core. However, the phase velocity doesn't instantaneously change as the wave crosses from cladding to core because that would cause a discontinuity in the electric field amplitude. Instead, the wave in the cladding gradually speeds up as it moves further from the core, creating a curved wavefront (a line of constant phase) that focuses the wave's energy into the core (this is another way to think about how guiding works), as illustrated in Figure \(\PageIndex{2}\)a. Now imagine that there is a second core placed close to the first. The wavefront must slow down again if it enters the core, so the wavefront in the cladding focuses energy into both cores, thus any energy from one core that is beyond a certain point in the cladding will be transported to the other core (see Figure \(\PageIndex{2}\)b).

Figure \(\PageIndex{2}\): (a) the wavefronts (pink) in a standard waveguide. Energy always flows at right angle to the wavefront, as indicated by the blue arrows. The outward and inward energy flows balance for a mode of the waveguide. (b) For two waveguides that are close enough to have overlapping fields, the power flow is no longer balanced, and energy will travel from one waveguide to another.

The description above is an over simplification of the true situation. In fact, when two waveguide cores are close enough to couple, then in a distance, \(L_c\), all the energy for a specific wavelength in one core will be transported to the other core. If the distance is doubled, all the energy will return to the original core. By controlling the separation between two cores and the distance at which they remain coupled, it is possible to choose how much energy is coupled from one core to another. And, because the required distance is wavelength dependent, it is also possible to separate different wavelengths from each other as well.

The coupling between two waveguides is given by a pair of coupled differential equations:

\[\begin{align} \frac{\textrm{d}E_1}{\textrm{d}z} &= -jC_{21}\exp(j\Delta kz)E_2(z)\label{eqn:coupling1}\\ \frac{\textrm{d}E_2}{\textrm{d}z} &= -jC_{12}\exp(-j\Delta kz)E_1(z)\label{eqn:coupling2}\end{align}\]

where \(\Delta k = k_1-k_2\) is the mismatch in propagation constants between the two waveguides, and \(C_{12}, C_{21}\) are the coupling coefficients, which are themselves given by a so-called overlap integral

\[\begin{align} C_{21} &= \frac{1}{2}\left(n_2^2 - n_{cl}^2\right)\frac{k_0}{k_1}\int_{0}^{a + d/2}u_1(y)u_2(y)dy\label{eqn:overlap1}\\ C_{12} &= \frac{1}{2}\left(n_1^2 - n_{cl}^2\right)\frac{k_0}{k_2}\int_{-a -d/2}^{0}u_1(y)u_2(y)dy\label{eqn:overlap2}\end{align}\]

where \(a\) is the separation between the waveguides and \(d\) is width of the waveguide. Equations \(\eqref{eqn:overlap1}\) and \(\eqref{eqn:overlap2}\) tell us that the coupling coefficient is proportional to how much overlap there is between the modes (\(u_1(y)\) and \(u_2(y)\) in the two waveguides. Equations \(\eqref{eqn:coupling1}\) and \(\eqref{eqn:coupling2}\) have an analytical solution for parallel waveguides, which for the case of transferring power from waveguide one to waveguide two (and no power in waveguide two initially) is

\[\begin{align} P_1(z) &= P_1(0)\left(\cos^2\gamma z + \left(\frac{\Delta k}{2\gamma}\right)^2\sin^2\gamma z\right)\nonumber\\ P_2(z) &= P_1(0)\frac{|C_{21}|^2}{\gamma}\sin^2\gamma z\label{eqn:sol1}\\ \gamma &= \left(\frac{\Delta k}{2}\right)^2 + C^2\nonumber\\ C&= \sqrt{C_{12}C_{21}}\nonumber\end{align}\]

For the case of two identical waveguides, then \(n_1 = n_2\) and \(\Delta k = 0\), then the transmitted field amplitudes are:

\[\begin{align} E_1(z) &= E_1(0)\cos Cz -jE_2(0)\sin Cz\label{eqn:1to2}\\ E_2(z) &= -jE_1(0)\sin Cz + E_2(0)\cos Cz\label{eqn:2to1}\end{align}\]

An example of the power transfer for perfectly matched waveguides is shown in Figure \(\PageIndex{3}\). A 3 dB directional coupler requires a length of about 900 µm for this particular coupling coefficient, while complete power transfer occurs at about 1.8 mm

Figure \(\PageIndex{3}\): power transfer between two identical waveguides that are in close proximity to each other. By choosing the length of the coupler, the ratio of power transfer can be controlled. The vertical axis is the amount of power transferred relative to the power incident at inport 1.

Exercise \(\PageIndex{1}\)

The code below models the coupling between two waveguides. The phasematch (\(\Delta k\)) and coupling coefficients can be adjusted to control the directional coupler's behaviour. What happens when the phasematch is not zero? What happens with the coupling constants are not symmetric?

Missing code

Answer

As the phase mismatch increases (absolute value), the maximum power transfer reduces. If the coupling coefficients are not identical and \(\Delta k = 0\) we get see a kind of power loss: all the power leaves waveguide 1, but does not appear in waveguide 2. However, it is physically very difficult to achieve different coupling coefficients and \(\Delta k = 0). If both \(\Delta k = 0\) and the coupling coefficients are unequal, the power transfer discrepancy gets smaller, but does not completely disappear.