# Guided-Wave Photodetectors

This is a continuation from the previous tutorial - * avalanche photodiodes*.

The photodetectors discussed in the previous tutorials are vertically illuminated. In a vertically illuminated photodetector (VIPD), the optical signal propagates in a direction perpendicular to the junction interfaces of the device.

This situation leads to a trade-off between the carrier transit time and the quantum efficiency, resulting in a limitation on the bandwidth-efficiency product of the device.

Another limitation of a high-speed VIPD arises from the trade-off between its bandwidth and its saturation power. A large bandwidth for a VIPD requires a small absorption volume, which results in a high carrier concentration at a give power level for the optical signal.

The space-charge effect in the active region caused by the high carrier concentration sets a limit on the saturation power of the photodetector. Although the bandwidth-efficiency product of a VIPD can be improved by using a multipass structure as discussed in the junction photodiodes tutorial, the saturation power is not increased by such a strategy but can only be improved by increasing the effective absorption volume.

Guided-wave photodetectors are developed to overcome these limitations. A well-designed guided-wave photodetector can have both a large bandwidth-efficiency product and a high saturation power.

Most of the photodetectors, including the MSM (metal-semiconductor-metal) photodetectors, the p-i-n photodiodes, the Schottky photodiodes, and the APDs, that are discussed in the previous tutorials can be made in guided-wave device form.

In a guided-wave photodetector, the guided optical signal propagates in a direction that is parallel to the junction interfaces and is perpendicular to the drift of the photogenerated carriers.

This geometry decouples the absorption length of the optical signal from the drift length of the photogenerated carriers. The optical signal is absorbed along the length \(l\) of the active region, while the carriers drift across the thickness \(d\) of the active region.

Thus, the quantum efficiency of a guided-wave photodiode is not that given by (14-84) [refer to the junction photodiodes tutorial] but can be expressed as

\[\tag{14-121}\eta_\text{e}=\eta_\text{coll}\eta_\text{t}\eta_\text{i}=\eta_\text{coll}(1-R)\eta_\text{c}(1-\text{e}^{-\alpha_\text{eff}l})\]

where \(\eta_\text{coll}\) is the collection efficiency of the photogenerated carriers, \(\eta_\text{t}=(1-R)\eta_\text{c}\), and \(\eta_\text{i}=1-\text{e}^{-\alpha_\text{eff}l}\).

Here \(R\) is the reflectivity at the incident surface of the waveguide, \(\eta_\text{c}\) is the coupling efficiency of the optical signal into the waveguide, \(\alpha_\text{eff}\) is the effective absorption coefficient of the active region, and \(l\) is the length of the active region measured along the waveguide direction.

The effective absorption coefficient \(\alpha_\text{eff}\) has to be calculated according to the device structure in order to take into account the fact that only a fraction of the guided optical wave overlaps with the absorbing active region of a thickness \(d\).

In the case when the entire core of the waveguide is the active region, \(\alpha_\text{eff}=\Gamma\alpha\), where \(\Gamma\) is the confinement factor of the waveguide and \(\alpha\) is the absorption coefficient of the material in the active region.

In other device structures, \(\alpha_\text{eff}\ne\Gamma\alpha\) because the active region might be located outside the waveguide core or it might occupy only a fraction of the core.

Because the signal absorption length is no longer tied to the thickness of the active region, a guided-wave photodetector can have both a very thin active region for a very short transit time and a long absorption length for a high quantum efficiency and a high saturation power.

**The major advantage of a guided-wave photodetector over a VIPD is that its carrier transit time can be independently reduced without sacrificing its quantum efficiency and saturation power.**

The transit-time-limited bandwidth of a guided-wave photodetector is easily made larger than its RC-time-limited bandwidth by using a sufficiently thin active region. As a result, the primary bandwidth limitation is not the carrier transit time but is the RC time constant of the device.

Besides this major advantage, a guided-wave photodetector with a thin active region has a few additional advantages.

A thin active region allows the device to operate at a low bias voltage, resulting in a small dark current and reduce noise contributed by the dark current.

The waveguide geometry is also compatible with other guided-wave photonic devices and components, making it easy to incorporate photodetectors into an integrated photonic circuit with reduced input and output coupling losses.

From the standpoint of considering the RC-time-limited bandwidth, guided-wave photodetector can be classified into two major categories:

- Lumped-circuit devices, commonly called
*waveguide photodetectors* - Distributed-circuit devices, commonly called
*traveling-wave photodetectors*

## Waveguide Photodetectors

A waveguide photodetector (WGPD) differs from a VIPD mainly in its optical waveguide structure. Being a lumped-circuit device, its electrical structure and equivalent circuit are similar to those of a VIPD.

The major advantage of a WGPD over a VIPD is that it can maintain a high quantum efficiency for a high cutoff frequency, thus a large bandwidth-efficiency product. The saturation power of a WGPD is comparable to that of a VIPD.

A WGPD is formed by integrating the active region of a photodetector with an optical waveguide. There are two basic integration schemes: the ** butt-coupling** configuration and the

**configuration, which are illustrated in Figure 14-31.**

*evanescent-coupling*The optical structure of a WGPD belongs to one or other of these two integration schemes though its details and sophistication may vary from one device to another.

In the butt-coupling scheme, also called the ** end-coupling** or

**scheme, shown in Figure 14-31(a), the active photoabsorption region of the photodetector is located in the waveguide core or is directly aligned with the core of a feeding waveguide.**

*end-firing*In this coupling scheme, the coupling efficiency from the feeding waveguide, if one is used, to the active region of the photodetector can be as high as 100% in principle but the coupling efficiency, \(\eta_\text{c}\) in (14-121), from free space to the waveguide core can be small if the waveguide core is defined by the thickness of a thin active region.

One approach to improving the coupling efficiency \(\eta_\text{c}\), but at the expense of reducing the effective absorption coefficient \(\alpha_\text{eff}\) to a fraction of \(\Gamma\alpha\), is to use a large-core waveguide with the thin active region occupying only a fraction of the waveguide core.

In the evanescent-coupling scheme, shown in Figure 14-31(b), the entire waveguide is nonabsorbing at the optical signal wavelength because the active photoabsorption region of the photodetector is located outside the waveguide core, typically on top of the waveguide.

In this scheme, a large-core waveguide is normally used to maximize the optical coupling efficiency \(\eta_\text{c}\) to the waveguide. The effective absorption coefficient \(\alpha_\text{eff}\) is also only a fraction of \(\Gamma\alpha\) because of a small overlap factor between the optical field and the active region in this evanescent-coupling configuration.

The overall quantum efficiency of a WGPD can be improved by using a large-core waveguide in either scheme to maximize \(\eta_\text{c}\) because the reduction in \(\alpha_\text{eff}\) can be compensated by increasing the length \(l\) of the active region.

Because a WGPD is a lumped-circuit device, its equivalent circuit and RC-time-limited bandwidth have the same form as those of a corresponding VIPD discussed in earlier tutorials.

Although the transit-time-limited bandwidth is independent of the quantum efficiency and can be made large enough that it is not a limiting factor, there is still a trade-off in maximizing both the RC-time-limited bandwidth and the quantum efficiency.

As the active region is made thin to shorten the carrier transit time and is made long to increase the quantum efficiency, the junction capacitance increases. Any parasitic capacitance that exists tends to increase also.

A high-speed photodetector normally has a fixed load resistance of \(R_\text{L}=50\) Ω. Therefore, increasing the transit-time-limited bandwidth and the quantum efficiency by making the active region thin and long results in a reduction in the RC-time-limited bandwidth.

To reduce the capacitance, the area of the active region can be reduced to a minimum by a given quantum efficiency. However, a small area for the active region leads to a large series resistance \(R_\text{s}\). When \(R_\text{s}\) becomes larger than \(R_\text{L}\), there is a trade-off between the capacitance and the resistance in maximizing the RC-time-limited bandwidth.

Consequently, the bandwidth-efficiency product of a WGPD is typically in the range of 20-40 GHz, which is comparable to that of a multipass VIPD, though a bandwidth-efficiency product larger than 50 GHz with a bandwidth larger than 100 GHz is possible for a well-designed WGPD.

The saturation power of a WGPD is similar to that of a VIPD of a comparable absorption volume.

In a WGPD, there is a trade-off between the saturation power and the bandwidth. The saturation power can be increased by increasing the thickness and the length of the active region while reducing \(\alpha_\text{eff}\) so that the absorption of the optical signal is distributed over a large volume, but the bandwidth will be reduced by such an action.

A WGPD has no advantage in the saturation power if its absorption volume is limited by the consideration of reducing both the carrier transit time and the device capacitance for a large bandwidth.

*Example 14-17*

High-speed InP/InGaAsP/InGaAs/InGaAsP/InP p-i-n VIPD and WGPD of the same device parameters are compared in this example. Both are used for the detection of optical signals at \(\lambda=1.55\) μm.

The intrinsic InGaAs active region has a thickness of \(d_\text{i}=0.2\) μm, which is sandwiched between two InGaAsP layers that form a double-core waveguide.

The absorption coefficient for InGaAs at \(\lambda=1.55\) μm is \(\alpha=6.6\times10^5\text{ m}^{-1}\). The electron and hole saturation velocities are \(v_\text{e}^\text{sat}=8\times10^4\text{ m s}^{-1}\) and \(v_\text{h}^\text{sat}=6\times10^4\text{ m s}^{-1}\), respectively.

The device area \(\mathcal{A}=50\text{ μm}^2\), which can take any shape for the VIPD but is formed by a stripe of \(w=2\text{ μm}\) and \(l=25\text{ μm}\) for the WGPD.

With these dimensions, both devices have a capacitance \(C=30\text{ fF}\), a series resistance from the contacts and the materials of \(R_\text{s}=40\) Ω. The load resistance is \(R_\text{L}=50\) Ω.

For the WGPD, the confinement factor of the active region is \(\Gamma=15\%\), and the optical coupling efficiency \(\eta_\text{c}=70\%\). Assume that both devices have \(\eta_\text{coll}=\eta_\text{t}=1\).

(a) Find the 3-dB cutoff frequencies of both devices.

(b) Find the bandwidth-efficiency product of the VIPD in a single-pass configuration for the optical signal through the active region.

(c) Find the bandwidth-efficiency product of the VIPD in a double-pass configuration with 100% back reflection.

(d) Find the bandwidth-efficiency product of the WGPD.

**Solution:**

**(a)**

The VIPD and the WGPD have the same 3-dB cutoff frequency because they have identical dimensions and device parameters. We first find that

\[\begin{align}v_\text{sat}&=\left[\frac{1}{2}\left(\frac{1}{v_\text{e}^\text{sat}}+\frac{1}{v_\text{h}^\text{sat}}\right)\right]^{-1}=\left[\frac{1}{2}\left(\frac{1}{8\times10^4}+\frac{1}{6\times10^4}\right)\right]^{-1}\text{ m s}^{-1}\\&=6.86\times10^4\text{ m s}^{-1}\end{align}\]

With \(d_\text{i}=0.2\) μm, we find that

\[\tau_\text{tr}=\frac{d_\text{i}}{v_\text{sat}}=\frac{0.2\times10^{-6}}{6.86\times10^4}\text{ s}=2.9\text{ ps}\]

With \(C=30\text{ fF}\), \(R_\text{s}=40\) Ω, and \(R_\text{L}=50\) Ω, we have

\[\tau_\text{RC}=(R_\text{s}+R_\text{L})C=90\times30\times10^{-15}\text{ s}=2.7\text{ ps}\]

Therefore, the 3-dB cutoff frequency

\[f_\text{3dB}=\frac{0.443}{[(2.9\times10^{-12})^2+(2.78\times2.7\times10^{-12})^2]^{1/2}}\text{ Hz}=55\text{ GHz}\]

**(b)**

We have \(\alpha{d}_\text{i}=6.6\times10^5\times0.2\times10^{-6}=0.132\). For the single-pass VIPD with \(\eta_\text{coll}=\eta_\text{t}=1\), we have

\[\eta_\text{e}=\eta_\text{i}=1-\text{e}^{-\alpha{d_\text{i}}}=1-\text{e}^{-0.132}=12.4\%\]

Therefore, its bandwidth-efficiency product

\[\eta_\text{e}f_\text{3dB}=0.124\times55\text{ GHz}=6.8\text{ GHz}\]

**(c)**

For the double-pass VIPD with \(\eta_\text{coll}=\eta_\text{t}=1\) and 100% back reflection, we have

\[\eta_\text{e}=\eta_\text{i}=1-\text{e}^{-2\alpha{d_\text{i}}}=1-\text{e}^{-2\times0.132}=23.2\%\]

Therefore, its bandwidth-efficiency product

\[\eta_\text{e}f_\text{3dB}=0.232\times55\text{ GHz}=12.8\text{ GHz}\]

**(d)**

For the WGPD, we find that \(\alpha_\text{eff}=\Gamma\alpha=0.15\times6.6\times10^5\text{ m}^{-1}=9.9\times10^4\text{ m}^{-1}\) and \(\alpha_\text{eff}l=9.9\times10^4\times25\times10^{-6}=2.475\). Thus, with \(\eta_\text{coll}=\eta_\text{t}=1\) and \(\eta_\text{c}=70\%\), we have

\[\eta_\text{e}=\eta_\text{c}\eta_\text{i}=\eta_\text{c}(1-\text{e}^{-\alpha_\text{eff}l})=0.7\times(1-\text{e}^{-2.475})=64.1\%\]

Therefore, the bandwidth-efficiency product of the WGPD is

\[\eta_\text{e}f_\text{3dB}=0.641\times55\text{ GHz}=35.3\text{ GHz}\]

We find that the bandwidth-efficiency product of the WGPD is 2.75 times that of the double-pass VIPD and is more than five times that of the single-pass VIPD though all of them have the same 3-dB cutoff frequency.

## Traveling-Wave Photodetectors

A traveling-wave photodetector (TWPD) differs from a WGPD mainly in its distributed electrical structure. Its optical structure is similar to that of a WGPD. A TWPD can have a larger bandwidth-efficiency product and a higher saturation power than a WGPD.

A TWPD is formed by integrating an electrical transmission line with one or more guided-wave photodetectors. There are two different basic configurations: the distributed configuration and the periodic configuration, both of which are illustrated in Figure 14-32.

A distributed TWPD, shown in Figure 14-32(a), is a traveling-wave version of a WGPD. It consists of a transmission line built on a fully distributed WGPD.

A periodic TWPD, shown in Figure 14-32(b), consists of a set of photodetectors that are periodically located along a transparent optical waveguide and are serially connected by a transmission line.

A distributed TWPD is often simply called a TWPD. A periodic TWPD is also called a ** velocity-matched distributed photodetector** (VMDP) if it is designed so that the optical wave and microwave are velocity matched.

Both the butt-coupling and evanescent-coupling schemes discussed above for WGPDs can be used for distributed and periodic TWPDs.

For a traveling-wave device, the bandwidth limitation due to the RC time constant of a lumped circuit is replaced by a bandwidth limitation due to the velocity mismatch between the optical wave propagating in the optical waveguide and the microwave propagating in the transmission line.

For a traveling-wave electro-optic modulator discussed in the traveling-wave modulators tutorial, the bandwidth is determined by the mismatch between the ** phase** velocities of the optical wave and microwave, \(v_\text{p}^\text{o}\) and \(v_\text{p}^\text{m}\), respectively.

Because electro-optic modulation acts upon the phase of the optical wave, it is necessary to synchronize the wavefronts of the optical wave and microwave by matching their phase velocities.

For a TWPD, however, velocity matching is considered between the ** group** velocity of the optical wave, \(v_\text{g}^\text{o}\), and the

**velocity of the microwave, \(v_\text{p}^\text{m}\), rather than between the phase velocities of the two waves.**

*phase*In a TWPD, the microwave signal is generated by absorption of the optical signal energy that propagates at the optical group velocity, but propagation of the microwave is determined by its phase velocity because the electrical signals generated along the transmission line add coherently.

The group velocity of a guided optical wave of frequency \(\omega\) and propagation constant \(\beta\) is

\[\tag{14-122}v_\text{g}^\text{o}=\frac{\text{d}\omega}{\text{d}\beta}=\frac{c}{N_\beta}\]

where \(N_\beta\) is the effective group index of the guided mode.

The phase velocity of a microwave electrical signal propagating in a transmission line is

\[\tag{14-123}v_\text{p}^\text{m}=\frac{1}{\sqrt{LC}}=\frac{1}{ZC}\]

where \(L\) and \(C\) are, respectively, the inductance and capacitance per unit length of the transmission line and \(Z=\sqrt{L/C}\) is the characteristic impedance of the transmission line.

The velocity-mismatch-limited bandwidth, \(f_\text{3dB}^\text{VM}\), of a TWPD is characterized by a time constant \(\tau_\text{VM}\), which measures the temporal walk-off between the optical wave and the microwave at the output of the TWPD:

\[\tag{14-124}f_\text{3dB}^\text{VM}=\frac{1}{2\pi\tau_\text{VM}}\]

Replacing \(\tau_\text{RC}\) in (14-99) [refer to the junction photodiodes tutorial] with \(\tau_\text{VM}\), the 3-dB cutoff frequency of a TWPD including transit-time and velocity-mismatch limitations can be approximated as

\[\tag{14-125}f_\text{3dB}\approx\frac{0.443}{[\tau_\text{tr}^2+(2.78\tau_\text{VM})^2]^{1/2}}=\frac{1}{2\pi[\tau_\text{VM}^2+(0.36\tau_\text{tr})^2]^{1/2}}\]

As discussed below, the form of the velocity-mismatch time constant \(\tau_\text{VM}\) depends on the structure of a TWPD.

In a TWPD, microwaves that propagate in both forward and backward directions in the transmission line are generated by absorption of the optical signal along the waveguide.

The velocity mismatch for the forward-propagating microwave is \(v_\text{g}^\text{o}-v_\text{p}^\text{m}\), but that for the backward-propagating microwave is \(v_\text{g}^\text{o}+v_\text{p}^\text{m}\). It is clearly not possible to velocity match the optical signal to both forward- and backward-propagating microwaves.

The bandwidth and the efficiency of a TWPD depend on whether or not the backward-propagating microwave is allowed to contribute to the electrical output.

At the optical input end, the termination of the transmission line can be either (1) connected with a matching impedance to eliminate the reflection of the backward-propagating microwave or (2) left open to allow total reflection of the backward-propagating microwave.

The efficiency of a TWPD with an impedance-matched input electrical termination is half that of the same TWPD with an open input electrical termination.

The velocity-mismatch time constant, \(\tau_\text{VM}\), of a TWPD is a function of the velocity mismatch and the effective length, \(l_\text{eff}\), of the device. Because forward- and backward-propagating microwaves have different velocity mismatches, \(\tau_\text{VM}\) has different forms for TWPDs of different input terminations.

For a TWPD with a matched input electrical termination,

\[\tag{14-126}\tau_\text{VM}=\left|\frac{l_\text{eff}}{v_\text{g}^\text{o}}-\frac{l_\text{eff}}{v_\text{p}^\text{m}}\right|=l_\text{eff}\left|\frac{v_\text{g}^\text{o}-v_\text{p}^\text{m}}{v_\text{g}^\text{o}v_\text{p}^\text{m}}\right|\]

For a TWPD with an open input electrical termination,

\[\tag{14-127}\tau_\text{VM}\approx\frac{3}{2}\frac{l_\text{eff}}{v_\text{p}^\text{m}}\]

The velocity-mismatch time constant for a TWPD with an open input electrical termination is independent of the optical group velocity because the mismatches on the forward- and backward-propagating microwaves have opposite effects.

The effective length, \(l_\text{eff}\), of a TWPD depends on the physical length \(l\) of the optical waveguide, the distribution of the photoabsorption region in the waveguide, and the effective absorption coefficient \(\alpha_\text{eff}\) of the device.

It is the lesser of the physical length and the propagation distance of the optical signal.

For a distributed TWPD,

\[\tag{14-128}l_\text{eff}=\begin{cases}l,\qquad\quad\text{if}\qquad{l}\lt\frac{1}{\alpha_\text{eff}},\\\frac{1}{\alpha_\text{eff}},\qquad\text{if}\qquad{l}\gt\frac{1}{\alpha_\text{eff}}\end{cases}\]

For a periodic TWPD, the optical waveguide of a total physical length \(l\) is only periodically loaded with photodetectors. If the length of each period is \(l_\text{p}\) and the length of the photoabsorption region of each photodetector is \(l_\text{d}\), then

\[\tag{14-129}l_\text{eff}=\begin{cases}l,\qquad\quad\text{if}\qquad{l}\lt\frac{l_\text{p}}{\alpha_\text{eff}l_\text{d}},\\\frac{l_\text{p}}{\alpha_\text{eff}l_\text{d}},\qquad\text{if}\qquad{l}\gt\frac{l_\text{p}}{\alpha_\text{eff}l_\text{d}}\end{cases}\]

Because \(l_\text{p}\) can be much larger than \(l_\text{d}\) in a periodic TWPD, both the physical length and the effective length of a periodic TWPD can be much larger than those of a distributed TWPD.

It can be seen from (14-126) and (14-127) that the velocity-mismatch-limited bandwidth of a TWPD with a matched termination can be unlimitedly improved by velocity matching for \(v_\text{p}^\text{m}/v_\text{g}^\text{o}=1\), whereas that of a TWPD with an open termination can be improved by simply increasing the phase velocity \(v_\text{p}^\text{m}\) of the microwave.

In principle, a TWPD with a matched termination can be velocity matched to have an arbitrarily large velocity-mismatch-limited bandwidth so that its bandwidth is purely transit-time limited.

In reality, however, \(v_\text{p}^\text{m}\lt{v}_\text{g}^\text{o}\) for a transmission line on an optical waveguide loaded with a photodetector, and \(v_\text{p}^\text{m}\gt{v}_\text{g}^\text{o}\) for a transmission line on an unloaded optical waveguide.

For a distributed TWPD, the microwave phase velocity is always lower than the optical group velocity with a ratio \(v_\text{p}^\text{m}/v_\text{g}^\text{o}\) typically falling in the range of 30-80%.

Therefore, perfect velocity matching is not possible in a distributed TWPD but is only possible in a properly designed periodic TWPD.

The major advantage of a TWPD over a WGPD is that its physical length can be made larger than its effective length without degrading its bandwidth because \(\tau_\text{VM}\) depends only on \(l_\text{eff}\).

The bandwidth of a TWPD becomes independent of its physical length when it is longer than the absorption length, whereas that of a WGPD continues to decrease as its length increases.

To maximize the efficiency, a distributed TWPD is normally made long enough that \(l\gg{l}_\text{eff}\). This flexibility allows a distributed TWPD to have an efficiency that is about 1.3 times the efficiency of a comparable WGPD designed for the same bandwidth, thus a 30% advantage in the bandwidth-efficiency product for the TWPD.

For the same reason, a distributed TWPD can have a higher saturation power than a comparable WGPD, particularly when they are both designed for high-speed operation.

On the other hand, when a TWPD and a WGPD are designed to have the same efficiency, the TWPD will have a larger bandwidth, thus a larger bandwidth-efficiency product, than the WGPD.

*Example 14-18*

A distributed TWPD based on the WGPD described in Example 14-17 is made by using a properly designed transmission line for its electrodes. The waveguide has an optical group velocity of \(v_\text{g}^\text{o}=8.9\times10^7\text{ m s}^{-1}\) at the 1.55 μm signal wavelength. The microwave phase velocity of the transmission line is \(v_\text{p}^\text{m}=2.9\times10^7\text{ m s}^{-1}\).

Find the cutoff frequency and the bandwidth-efficiency product of the TWPD if

(a) it has a matched termination

(b) it has an open termination.

**Solution:**

From Example 14-17, we have \(\alpha_\text{eff}=9.9\times10^4\text{ m}^{-1}\). Because \(l=25\) μm \(\gt\alpha_\text{eff}^{-1}=10.1\) μm, we have \(l_\text{eff}=10.1\) μm for this distributed TWPD. From Example 14-17, we also have \(\tau_\text{tr}=2.9\) ps.

**(a)**

With a matched termination, we have

\[\tau_\text{VM}=\left|\frac{l_\text{eff}}{v_\text{g}^\text{o}}-\frac{l_\text{eff}}{v_\text{p}^\text{m}}\right|=\left|\frac{10.1\times10^{-6}}{8.9\times10^7}-\frac{10.1\times10^{-6}}{2.9\times10^7}\right|\text{ s}=235\text{ fs}\]

Therefore, the 3-dB cutoff frequency

\[f_\text{3dB}=\frac{0.443}{[(2.9\times10^{-12})^2+(2.78\times235\times10^{-15})^2]^{1/2}}\text{ Hz}=149\text{ GHz}\]

For the TWPD with a matched termination, the efficiency is only half of the 64.1% efficiency found in Example 14-17. Thus, the bandwidth-efficiency product

\[\eta_\text{e}f_\text{3dB}=\frac{1}{2}\times0.641\times149\text{ GHz}=47.8\text{ GHz}\]

**(b)**

With an open termination, we have

\[\tau_\text{VM}=\frac{3l_\text{eff}}{2v_\text{p}^\text{m}}=\frac{3\times10.1\times10^{-6}}{2\times2.9\times10^7}\text{ s}=522\text{ fs}\]

Therefore, the 3-dB cutoff frequency

\[f_\text{3dB}=\frac{0.443}{[(2.9\times10^{-12})^2+(2.78\times522\times10^{-15})^2]^{1/2}}\text{ Hz}=136.6\text{ GHz}\]

For the TWPD with an open termination, the efficiency is just the 64.1% efficiency found in Example 14-17. Thus, the bandwidth-efficiency product

\[\eta_\text{e}f_\text{3dB}=0.641\times136.6\text{ GHz}=87.6\text{ GHz}\]

Because perfect velocity matching is not achieved in the TWPD with a matched termination, the velocity-mismatch-limited bandwidth of the TWPD with a matched termination is only about twice that of the TWPD with an open termination in this example.

However, the cutoff frequencies for both cases are limited by the transit time because \(\tau_\text{tr}\gg\tau_\text{VM}\) in both cases. As a result, the TWPD with a matched termination only has a slightly larger bandwidth than the TWPD with an open termination.

The one with an open termination then has a larger bandwidth-efficiency product because it has twice the efficiency of the one with a matched termination.

Compared to the WGPD, which is further limited by the RC time constant, however, the TWPD with either type of termination has a larger bandwidth and a larger bandwidth-efficiency product.

In a periodic TWPD, the transmission line runs along the optical waveguide with periodically alternating loaded regions, where \(v_\text{p}^\text{m}\lt{v}_\text{g}^\text{o}\), and unloaded regions, where \(v_\text{p}^\text{m}\gt{v}_\text{g}^\text{o}\).

By properly designing the size and spacing of the periodically distributed photodetectors, it is then possible to achieve close velocity matching in a periodic TWPD.

A velocity-matched periodic TWPD is known as a VMDP. The unique advantage of a VMDP is that the transmission line, the optical waveguide, and the individual photodetectors can be independently optimized.

The transmission line is optimized for its impedance and for velocity matching. Because close velocity matching is possible, a VMDP always has a matched input electrical termination to realize the benefit of velocity matching. Therefore, its efficiency is limited to a maximum of 50%.

The optical waveguide is optimized with the characteristics of large-core, low coupling-loss, and single-mode operation for the VMDP to have a high efficiency and a high saturation power.

With close velocity matching, the bandwidth of a VMDP is no longer velocity-mismatch limited but is essentially that of the individual photodetectors. Each individual photodetectors in a VMDP are optimized for a large bandwidth. Each individual photodetector is kept below its saturation current.

Because velocity matching permits a large device length without degrading the bandwidth, a high saturation power and a high efficiency can both be achieved with a long device without sacrificing the bandwidth.

**Example 14-19**

What changes have to be made to the structure and dimensions of the TWPD considered in Example 14-18 in order to make it into a VMDP? What are the bandwidth and bandwidth-efficiency product that can be obtained for this VMDP instead?

**Solution:**

To make a VMDP, it is necessary to break the 25-μm length of the TWPD into individually optimized photodetectors that are then properly spaced to achieve velocity matching.

Thus, if we keep the sum of the lengths of these individual photodetectors to be 25 μm, the entire length of the VMDP will be much longer.

If perfect, or nearly perfect, velocity matching is accomplished, we have \(\tau_\text{VM}\approx0\). Then, the bandwidth of the device is purely transit-time limited:

\[f_\text{3dB}=\frac{0.443}{2.9\times10^{-12}}\text{ Hz}=152.8\text{ GHz}\]

Because a VMDP is required to have a matched termination, its efficiency is only half of the 64.1% efficiency found for the WGPD. Thus, its bandwidth-efficiency product

\[\eta_\text{e}f_\text{3dB}=\frac{1}{2}\times0.641\times152.8\text{ GHz}=49\text{ GHz}\]

We find that \(f_\text{3dB}\) of this VMDP is only slightly larger than that of the TWPD with a matched termination considered in Example 14-18. As a consequence, its bandwidth-efficiency product is also only slightly larger than that of the TWPD with a matched termination but is smaller than that of the TWPD with an open termination.

The reason for this insignificant improvement by velocity matching is that \(\tau_\text{VM}\ll\tau_\text{tr}\) for both cases of TWPD considered in Example 14-18. Thus, the 3-dB cutoff frequencies of the two cases of TWPD are already close to the transit-time limit.

A VMDP can realize a significant bandwidth increase over a TWPD only when the bandwidth of the TWPD is limited by velocity mismatch rather than by the transit time.

The next tutorial talks about ** the history of semiconductor lasers**.