# Traveling-wave modulators

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This is a continuation from the previous tutorial - guided-wave electro-optic modulators.

At a low modulation frequency, the time it takes for the optical wave to travel through an electro-optic modulator is short compared to the modulation period. This situation is characterized by the condition that \(f\tau_\text{tr}\lt1\), where \(f\) is the modulation frequency and \(\tau_\text{tr}\) is the transit time for the optical wave to propagate through the modulator. In this case, the modulator can be considered as a lumped device because its length is small compared to the wavelength of the modulation field. The 3-dB ** modulation bandwidth**, \(f_\text{3dB}\), of a lumped electro-optic modulator is determined by both the transit time \(\tau_\text{tr}\) of the optical wave and the RC time constant \(\tau_\text{RC}\) of the lumped driving circuit including the loading effects of the modulator. Because \(\tau_\text{RC}\gt\tau_\text{tr}\) for most lumped modulators, the modulation bandwidth of a lumped modulator is usually determined by its RC time constant so that

\[\tag{6-92}f_\text{3dB}=\frac{1}{2\pi\tau_\text{RC}}\]

The value of \(\tau_\text{RC}\) for a modulator of a given resistance, such as 50 \(\Omega\), increases with the length \(l\) of its electrode because the capacitance increases with this length. For a lumped LiNbO_{3} waveguide modulator, the product \(f_\text{3dB}l\) typically falls in the range of 1-3 GHz cm. Therefore, the modulation bandwidth of a lumped LiNbO_{3} is limited to a few gigahertz at best.

In many applications, however, the modulation frequency is in the microwave or millimeter-wave range to take advantage of the high-bandwidth capacity of the optical carrier wave. The modulation efficiency of a lumped modulator drops drastically at high modulation frequencies because of its RC-limited frequency response. This problem can be overcome by using a traveling-wave configuration for the modulator and by matching the phase velocity of the microwave modulation field to that of the optical wave in the ** traveling-wave modulator**.

The electrodes of a traveling-wave modulator are made of strip transmission lines, as shown in figure 6-15 below.

The electrodes are specifically designed for traveling-wave interactions. The high-frequency modulation signal is injected at one end, propagates along the same direction as the optical wave, and terminates at the end of the electrode transmission line. The traveling-wave configuration inherently requires the use of the transverse modulation scheme. This, however, is consistent with the configurations of most guided-wave devices. Therefore, traveling-wave modulation can be applied to a large variety of guided-wave devices, including single-waveguide phase modulators, Mach-Zehnder interferometers, and directional coupler switches, to meed the demand for high-frequency modulation and switching very often found in the applications of guided-wave devices.

Two key factors determine the modulation bandwidth of a traveling-wave modulator: (1) phase-velocity mismatch between the optical wave in the waveguide and the microwave in the transmission line and (2) frequency-dependent attenuation of the microwave modulation signal as it propagates along the transmission line. These parameters are determined by the waveguide material, the details of the waveguide structure, and the design of the transmission line. A transmission line has a characteristic impedance, \(Z\), given by

\[\tag{6-93}Z=\sqrt{\frac{L}{C}}\]

where \(L\) and \(C\) are, respectively, the inductance and capacitance per unit length of the transmission line. The phase velocity, \(v_\text{p}^\text{m}\), of a microwave electrical signal propagating in a transmission line is

\[\tag{6-94}v_\text{p}^\text{m}=\frac{1}{\sqrt{LC}}=\frac{c}{n_\text{m}}\]

where \(n_\text{m}\) is the refractive index of the microwave in the transmission line. For a microwave transmission line on a LiNbO_{3} substrate, \(n_\text{m}\approx\) 4.225 with variations around this value caused by variations in the structure and parameters of the transmission line. The microwave traveling in the transmission line suffers a loss characterized by a frequency-dependent ** power attenuation** coefficient of \(\alpha_\text{m}\), thus a

**coefficient of \(\alpha_\text{m}/2\). The modulation signal sent from the input end of the modulator is then characterized by the following space- and time-dependent traveling microwave voltage throughout the electrode:**

*voltage attenuation*\[\tag{6-95}V(z,t)=V_\text{pk}\text{e}^{-\alpha_\text{m}z/2}\cos\left[2\pi{f}\left(\frac{z}{v_\text{p}^\text{m}}-t\right)\right]\]

where \(V_\text{pk}\) is the peak modulation voltage and \(f\) is the modulation frequency of the microwave signal.

The phase velocity, \(v_\text{p}^\text{o}\), of a guided optical wave is determined by its frequency \(\omega\) and propagation constant \(\beta\) as

\[\tag{6-96}v_\text{p}^\text{o}=\frac{\omega}{\beta}=\frac{c}{n_\beta}\]

where \(n_\beta\) is the effective refractive index of the guided optical wave.

Because an optical wavefront entering the input end of the electrode at time \(t\) arrives at location \(z\) at a later time of \(t+z/v_\text{p}^\text{o}\), it sees a space- and time-varying voltage of \(V(z,t+z/v_\text{p}^\text{o})\), rather than \(V(z,t)\), as it travels through the waveguide in the modulator. Therefore, the electro-optically induced change in the propagation constant of the guided optical wave as a function of modulation frequency varies with space and time as follows:

\[\tag{6-97}\Delta\beta(f,z,t)=\Delta\beta_\text{pk}\text{e}^{-\alpha_\text{m}z/2}\cos\left[2\pi{f}\left(\frac{\tau_\text{VM}}{l}z-t\right)\right]\]

where \(\Delta\beta_\text{pk}\) is the change corresponding to a constant peak voltage \(V_\text{pk}\) and

\[\tag{6-98}\tau_\text{VM}=\left|\frac{l}{v_\text{p}^\text{m}}-\frac{l}{v_\text{p}^\text{o}}\right|=\frac{l}{c}|n_\text{m}-n_\beta|\]

is an effective temporal walk-off between the wavefronts of the optical wave and the microwave due to the ** velocity mismatch** between them. The electro-optically induced phase shift for the optical wave at the output end of the modulator as a function of the modulation frequency and time can be found as

\[\tag{6-99}\Delta\varphi(f,t)=\displaystyle\int\limits_0^l\Delta\beta(f,z,t)\text{d}z=\Delta\varphi_\text{pk}(0)\frac{1}{l}\displaystyle\int\limits_0^l\text{e}^{-\alpha_\text{m}z/2}\cos\left[2\pi{f}\left(\frac{\tau_\text{VM}}{l}z-t\right)\right]\text{d}z\]

where \(\Delta\varphi_\text{pk}(0)=\Delta\beta_\text{pk}l\) is the phase shift induced by a constant voltage \(V_\text{pk}\) at \(f=0\). Instead of carrying out the integration in (6-99), we examine two important limiting cases in the following.

1. **Bandwidth limited by velocity mismatch**. If \(\tau_\text{VM}\) is significant but \(\alpha_\text{m}\approx0\), the integral in (6-99) yields

\[\tag{6-100}\Delta\varphi(f,t)=\Delta\varphi_\text{pk}(f)\cos\left[2\pi{f}\left(t-\frac{\tau_\text{VM}}{2}\right)\right]\]

where

\[\tag{6-101}\Delta\varphi_\text{pk}(f)=\Delta\varphi_\text{pk}(0)\frac{\sin\pi{f}\tau_\text{VM}}{\pi{f}\tau_\text{VM}}\]

is the modulation-frequency-dependent peak phase shift. By setting \(\Delta\varphi_\text{pk}(f_\text{3dB})=\Delta\varphi_\text{pk}(0)/2\), the 3-dB modulation bandwidth limited by velocity mismatch is found to be

\[\tag{6-102}f_\text{3dB}^\text{VM}\approx\frac{2}{\pi\tau_\text{VM}}=\frac{2c}{\pi{l}|n_\text{m}-n_\beta|}\]

2. **Bandwidth limited by microwave attenuation**. If the phase velocities of the optical wave and the microwave signal are perfectly matched, we have \(\tau_\text{VM}=0\). The modulation bandwidth is then only limited by the increasing loss of the strip line at high frequencies characterized by \(\alpha_\text{m}(f)\) as a function of the modulation frequency. In this limit, the integral in (6-99) yields

\[\tag{6-103}\Delta\varphi(f,t)=\Delta\varphi_\text{pk}(f)\cos2\pi{ft}\]

where

\[\tag{6-104}\Delta\varphi_\text{pk}(f)=\Delta\varphi_\text{pk}(0)\frac{1-\text{e}^{-\alpha_\text{m}(f)l/2}}{\alpha_\text{m}(f)l/2}\]

is a function of the modulation frequency through the frequency dependence of \(\alpha_\text{m}\). By setting \(\Delta\varphi_\text{pk}(f_\text{3dB})=\Delta\varphi_\text{pk}(0)/2\), the 3-dB modulation bandwidth limited by attenuation in the transmission line is found to be determined by

\[\tag{6-105}\alpha_\text{m}(f_\text{3dB})\approx\frac{3.2}{l}\]

The frequency dependence of the attenuation coefficient of a transmission line is usually characterized by \(\alpha_\text{m}=af^{1/2}\), where \(a\) is a constant. Then the 3-dB modulation bandwidth limited by attenuation can be expressed as

\[\tag{6-106}f_\text{3dB}^\text{att}\approx\left(\frac{3.2}{al}\right)^2\]

A figure of merit for a modulator is its power-bandwidth ratio, \(P/f_\text{3dB}\), which measures the power cost of imposing a unit bandwidth of information on the optical carrier. The required microwave power to drive the modulator depends on the impedance \(Z_\text{s}\) of the microwave source and the impedance \(Z\) of the modulator. The standard impedance for the microwave source is \(Z_\text{s}\) = 50 \(\Omega\). It is desired that the impedance \(Z\) of the modulator matches \(Z_\text{s}\) as closely as possible for the most efficient delivery of the microwave power to the modulator, but perfect match is often not possible because of design constraints. In the general situation when \(Z_\text{s}\ne{Z}\), the power required from the microwave source for a peak modulation voltage \(V_\text{pk}\) on the modulator is

\[\tag{6-107}P=\frac{V_\text{pk}^2}{2Z_\text{s}}\frac{1}{1-[(Z_\text{s}-Z)/(Z_\text{s}+Z)]^2}\]

For a modulator such as a Mach-Zehnder interferometer that requires a total phase variation from 0 to \(\pi\) for the full range of its operation, only a peak voltage of \(V_\text{pk}=V_\pi/2\) is required by biasing the device at \(V_\text{b}=V_\pi/2\) so that the full swing of the microwave voltage from \(-V_\text{pk}\) to \(V_\text{pk}\) provides the voltage variations from 0 to \(V_\pi\).

The \(P/f_\text{3dB}\) ratio for a properly designed traveling-wave modulator is generally much smaller than that for a comparable lumped modulator, reflecting a much improved performance.

**Example 6-9**

Properly designed transmission lines are used for the electrodes of the \(x\)-cut, \(y\)-propagating LiNbO_{3} Mach-Zehnder waveguide interferometric modulator shown in figure 6-8(a) and discussed in Example 6-6 [refer to the guided-wave electro-optic modulators tutorial]. The transmission line electrodes have an impedance of \(Z\) = 30 \(\Omega\); a microwave index of \(n_\text{m}\) = 4.225; and a frequency-dependent microwave power attenuation coefficient of \(\alpha_\text{m}=af^{1/2}\), with \(a\) = 2 \(\text{dB cm}^{-1}\text{GHz}^{-1/2}\). It is driven by a microwave source of impedance \(Z_\text{s}\) = 50 \(\Omega\). Find the 3-dB modulation bandwidth and the \(P/f_\text{3dB}\) ratio of this traveling-wave Mach-Zehnder waveguide interferometric modulator for the TE-like mode at \(\lambda\) = 1.3 μm.

From Example 6-6 [refer to the guided-wave electro-optic modulators tutorial], we find that the length of the electrodes is \(l\) = 12.5 mm, and the half-wave voltage for the TE-like mode is \(V_\pi\) = 6 V. We also find that \(n_\beta\approx{n_e}\) =2.145 for the TE-like mode of this device. We then find that \(\tau_\text{VM}\) = 86.7 ps, and that the bandwidth limited by velocity mismatch is

\[f_\text{3dB}^\text{VM}=\frac{2\times3\times10^8}{\pi\times12.5\times10^{-3}\times|4.225-2.145|}\text{ Hz}=7.3\text{ GHz}\]

using (6-102).

To find the bandwidth limited by attenuation, we first convert the attenuation coefficient measured in decibels per centimeter into that measured per centimeter. Therefore, \(a\) = 2 \(\text{dB cm}^{-1}\text{GHz}^{-1/2}\) = 0.46 \(\text{cm}^{-1}\text{GHz}^{-1/2}\). We then find that the bandwidth limited by attenuation is

\[f_\text{3dB}^\text{att}=\left(\frac{3.2}{0.46\times1.25}\right)^2\text{ GHz}=31\text{ GHz}\]

using (6-106).

Because \(f_\text{3dB}^\text{att}\gt{f}_\text{3dB}^\text{VM}\), the bandwidth of this modulator is limited by velocity mismatch to be \(f_\text{3dB}\) = 7.3 GHz.

By biasing the modulator at \(V_\text{b}=V_\pi/2=3\text{ V}\), the device can be modulated with a peak voltage of \(V_\text{pk}=V_\pi/2=3\text{ V}\). With \(Z\) = 30 \(\Omega\) and \(Z_\text{s}\) = 50 \(\Omega\), it is then found that \(P\) = 96 mW using (6-107). Thus, the modulator has a power-bandwidth ratio of \(P/f_\text{3dB}\) = 13.15 \(\text{mW GHz}^{-1}\).

The next part continues with the magneto-optic effects tutorial.