Plasmonic, photonic, or hybrid? Reviewing waveguide geometries for electro-optic modulators

Messner, Andreas; Moor, David; Chelladurai, Daniel; Svoboda, Roman; Smajic, Jasmin; Leuthold, Juerg

doi:10.1063/5.0159166

Electro-optic modulators are key elements in high-speed optical telecommunication links and preferably rely on materials with a linear electro-optic effect. Choosing adequate waveguide geometries is a key challenge in the design of electro-optic modulators. While all-dielectric geometries promise high-speed modulation with low propagation loss, their modulation efficiency suffers from low confinement and weak electrical fields, resulting in lengthy devices. Plasmonic geometries, on the other hand, allow for most compact devices featuring highest electro-optical bandwidths, but at the cost of higher losses. Alternatively, hybrid photonic–plasmonic solutions open a sweet spot for high-speed modulators with moderate loss. In this review, we discuss the three waveguide types by analyzing and comparing their performance and their sensitivity to variations in geometry with respect to a choice of the electro-optical Pockels-effect material.

I. INTRODUCTION

On-chip integrated electro-optic (EO) modulators are essential in communications and many other applications to encode electrical information onto an optical signal. They enable higher-quality and higher-speed encoding than would be possible by direct laser modulation.^1–3 To form these external modulators, electro-refractive and electro-absorptive EO materials are used to activate integrated optical waveguides and to manipulate either the real or the imaginary part of the waveguide’s effective index.⁴ A variety of physical EO effects can be exploited in these devices.⁵ Popular EO effects in semiconductors are based on the free carrier dispersion effect,⁶ the quantum-confined Stark effect,⁷ or the Franz–Keldysh Effect.⁸ On the other hand, the linear electro-optic effect, also called Pockels effect, stands out as an effect that exclusively modulates the phase but not the amplitude of an optical field while offering a large optical bandwidth, a quasi-instantaneous EO response on the femtosecond timescale (for details on the frequency characteristics of the Pockels effect, refer to Wemple et al.⁹ and Dalton et al.¹⁰), and a linear response. In a proper configuration, it hence offers chirp-free operation.^5,10–13 These benefits make modulators relying on the Pockels effect the first choice for high-quality and high-capacity optical links, as bulk lithium niobate modulators exemplify, which have been the workhorse in long-haul links for the past decades.

Given that the silicon photonic platform has become the mainstay in the industry, the integration of Pockels effect materials with different silicon-integrated optical waveguiding platforms currently receives strong attention. For instance, the silicon photonics platform has been functionalized with organic polymers,^14–16 lithium niobate,^17–19 and barium titanate.^20–24 Alternative approaches have long relied on III-V materials,²⁵ whereas currently, the lithium niobate on insulator (LNOI) platform has received strong attention.^26–33 This is mainly because it relies on the well-accepted and reliable LiNbO₃ material, and offers low loss and a relatively compact footprint.

The plasmonic platform extends the photonic platform with plasmonic building blocks, which overcome the classical refraction limit. Plasmonic devices are substrate and platform agnostic,^34–37 and also compatible with silicon photonics,^15,34,38,39 which makes them very flexible. Furthermore, the plasmonic approach is compatible with a wide variety of nonlinear materials such as organic Pockels-effect materials,^15,39 BaTiO₃,⁴⁰ and LiNbO₃.⁴¹ Even more spectacular, its substrate-agnostic nature means that devices can be fabricated by metal elements and nonlinear material alone.^37,42 This allows a monolithic integration of plasmonics with electronic CMOS chips.^43–45

Hybrid photonic–plasmonic modulators combine plasmonic and photonic waveguiding principles in a hybridized optical mode,^44,46,47 and therefore, their properties form a compromise between those of purely photonic or purely plasmonic devices. The question then is, what can one expect, and how do these different approaches compare against each other?

In this work, we focus on an insightful presentation and comparison between the photonic, the plasmonic, and the photonic–plasmonic hybrid modulator concepts. We first introduce figures of merit for electro-refractive EO modulators. After introducing the most common photonic, plasmonic, and hybrid waveguide geometries, their fundamental performance metrics, and impact on the system level will be discussed and compared. This review has been written to complement and complete recent device-focused review articles by Kim et al.,⁴⁸ Rahim et al.,⁴⁹ Sinatkas et al.,⁴ and Tuniz.⁵⁰ Note that the fundamental properties of waveguides and the design trade-offs that come with them and are described here, are not only valid for electro-optic modulators, but can be transferred to a wider scope, such as for second-harmonic generation⁵¹ or for Kerr nonlinear waveguides.⁵²

A note on nomenclature: Seen from a higher level, the field of plasmonics is part of the field of photonics or nanophotonics. In the scope of this work, though, we focus on waveguiding mechanisms and use the word photonic for all-dielectric structures relying exclusively on photons, whereas we use plasmonic for structures relying on surface plasmon polaritons (SPPs).

A. Performance metrics

Various performance metrics can be used to compare electro-optic modulators to each other.^{4,25,49,53–57} A first class of metrics describes modulator characteristics in the scope of data transmission systems, whereas a second class of metrics is focused more on the underlying device and waveguide physics. We mostly follow the definitions by Witzens,⁵⁷ refer to Miller⁵⁶ for the energy consumption per bit, and to Alferness²⁵ and Brosi et al.⁵⁸ for the field interaction factor. Table I lists an overview. Effects that arise from scaling to more complex systems, potentially comprising several electro-optic modulators, such as crosstalk, are beyond the scope of this paper. The work of Winzer and Neilson⁵⁵ and the references therein give a good introduction.

TABLE I.

Overview on the performance metrics used to characterize electro-optical modulators. The first block comprises the metrics that are of interest on the system level, whereas the second block lists the metrics commonly used to describe modulator physics. Refer to Ref. 56 (E_bit), Refs. 25 and 58 (Γ), and Ref. 57 (rest).

Metric	Symbol	Unit
System-level performance metrics
Insertion loss	IL	dB
EO bandwidth	f_3dB	GHz
Half-wave voltage	V_π	V
On-off voltage	V_on/off	V
Energy per bit	E_bit	fJ bit⁻¹
Device-level performance metrics
Propagation losses	α	dB cm⁻¹, …, dB μm⁻¹
Voltage–length product	V_πL	V cm, …, V μm
Voltage–length–loss product	V_πLα	V dB
Field interaction factor	Γ	1; dimensionless

Metric	Symbol	Unit
System-level performance metrics
Insertion loss	IL	dB
EO bandwidth	f_3dB	GHz
Half-wave voltage	V_π	V
On-off voltage	V_on/off	V
Energy per bit	E_bit	fJ bit⁻¹
Device-level performance metrics
Propagation losses	α	dB cm⁻¹, …, dB μm⁻¹
Voltage–length product	V_πL	V cm, …, V μm
Voltage–length–loss product	V_πLα	V dB
Field interaction factor	Γ	1; dimensionless

The system-level metrics can be differentiated into optical, electro-optical, and electrical metrics:

The spectral window or optical bandwidth describes the spectral range in which the modulator can be operated. This can range from tens of nanometers for Mach–Zehnder modulators (MZMs) down to a few picometers for resonant devices. Note that the latter typically have multiple spectral windows, separated by their free spectral range.
The frequency response or electro-optical bandwidth is a measure of how fast the modulator can be driven. Usually, this characteristic is given in terms of the 3-dB bandwidth, f_3dB: The 3-dB bandwidth refers to the frequency where the power of the modulation sideband would drop by 3 dB against its reference at low frequencies. This corresponds to a 6-dB drop of the power of the corresponding electrical frequency component after photodetection (⁠ $P_{electric} \propto I_{photo} \propto P_{optical}^{2}$ ⁠). The electro-optic bandwidth can be limited by several factors including limited carrier velocity in semiconductors, velocity mismatch between the photonic and the RF wave in longer travelling wave modulators, or RC time constants determined by the capacitance of the modulators and the resistance of the feeding electrical lines.
Next, the half-wave voltage V_π describes the required voltage a phase modulator needs to shift the phase of the transmitted light by Δφ = π. Depending on how phase modulators are integrated in intensity modulators, the on–off voltage V_on/off of the intensity modulator might not be equal to the V_π of the constituent phase modulators. This often leads to some ambiguity, most often with Mach–Zehnder modulators driven in the push–pull mode. There, opposite voltages are applied over each arm, resulting in V_on/off = 1/2 · V_π. When using plasmonic modulators, which can be treated as lumped, open RF elements, and feeding them through a common 50 Ω line, another factor 2 is gained through reflection of the voltage signal. Also, driving the lumped elements in a differential way adds another factor of 2,⁵⁹ resulting in a final V_on/off = 1/8 · V_π of a plasmonic Mach–Zehnder modulator.
Finally, the energy consumption per bit, E_bit, gives the average electrical energy that the modulator requires to encode one bit. This quantity is typically given in units of fJ bit⁻¹ and depends both on the device physics and the employed modulation scheme.⁵⁵

The device-level metrics are of practical relevance and mostly linked to the waveguide cross section.

The propagation losses α give the optical attenuation per unit length. Depending on the modulator technology, this can range from dB⋅cm⁻¹ to dB ⋅ μm⁻¹.
The voltage–length product V_πL is a length-independent measure of the electro-optic performance of linear electro-optic modulators. As the acquired phase shift of light propagating through a modulator is approximately proportional to the modulator’s length L, the necessary V_π drops accordingly, so that their product V_πL remains constant. As the voltage–length product is a property of the waveguide cross section, it can be used to compare different geometries.
Further, the combination of α and V_πL can be an interesting figure of merit for the device designers: The so-called voltage–length–loss product V_πLα in the somewhat eccentric unit of VdB gives the resulting losses that are incurred if a certain phase modulation Δφ is to be achieved for a given voltage swing. The V_πLα accounts for the fact that the half-wave voltage V_π and the propagation losses α are often subject to design trade-offs. Increasing the length of the modulator will reduce V_π but increase αL, so that their product remains constant. Therefore, V_πLα allows us to compare the performance of waveguide geometries on different technology platforms.
Finally, the field interaction factor Γ quantifies how strongly the externally applied electric field interacts with the optical mode in the modulator waveguide. In modulators relying on the Pockels effect, the field interaction factor is a measure of the sensitivity of the waveguide’s effective index change Δn_eff to a change of the active material’s refractive index Δn_mat, Γ = Δn_eff/Δn_mat.

II. THE POCKELS EFFECT IN ELECTRO-OPTIC MODULATORS

Mach–Zehnder modulators (MZMs) based on the Pockels effect are unique as they give a perfect control over the phase and amplitude of a signal because the Pockels effect changes the real part of the refractive index n_mat of a material linearly and predictably upon applying an external electric field while it does not change the absorption of that material. Put in a simple relation (the more general approach is presented below), we can write

Δ n_{mat} = - \frac{1}{2} n_{mat}^{3} r E^{Ω},

(1)

where the refractive index change Δn_mat of the Pockels material depends on its Pockels coefficient r, the applied external electrical field E^Ω and a constant factor

- n_{mat}^{3} / 2

⁠.

Also, Pockels materials offer virtually instantaneous electro-optic response (the EO bandwidth of the Pockels effect spans several hundreds of GHz up to hundreds of THz,^{12,27,60–63} corresponding to a femtosecond timescale). Consequently, Pockels modulators allow to encode information at highest quality and up to highest frequency with perfect control over chirp.^11,13

A large variety of different Pockels materials is available, which we can categorize into two classes:

The first class comprises organic electro-optic (OEO) materials. They can be deposited by spin coating and have been used to activate silicon photonic or plasmonic slot waveguides,¹⁵ in the latter case even monolithically integrated on an electronic driver chip.⁴³ In OEO materials, electrons oscillating in an asymmetric potential are responsible for the femtosecond-scaled EO response, leading to a flat frequency response of the Pockels effect from DC to hundreds of THz.¹⁰ However, organic materials have encountered reservations regarding their long-term thermal and photochemical stability.^10,64 More recently, though, new OEO materials, which can be stabilized via a crosslinking process, have been introduced, which can withstand elevated temperatures for extended times.^65–70

The second class comprises solid-state materials, such as lithium niobate, barium titanate, or lead zirconate titanate (PZT).^71,72 These materials are more stable toward temperature influences,^40,71 and forming electro-optic devices relies on epitaxial growth, wafer bonding, or a combination thereof.^24,72,73 Thin films of lithium niobate (LiNbO₃)—in the bulk form a long-time work horse for long-haul communications—have been used to activate the silicon and silicon nitride platforms with a Pockels material.^74,75 More prominently, though, LiNbO₃ is used on silicon oxide to form the lithium niobate on insulator (LNOI) platform.^26,28,73,76 Modulators with barium titanate (BaTiO₃) instead of LiNbO₃ potentially offer a much stronger Pockels effect²⁴ and have also yielded first results,^21,40,77 while the performance of OEO- and LiNbO₃-based modulators has not yet been matched.

In the following, the Pockels effect is introduced in a general form before we consider waveguide-based phase modulators.

A. The Pockels effect

Generally, the simple Eq. above does not capture the reality, particularly when the Pockels material is anisotropic, the light’s polarization is arbitrary and the external electric field has a vectorial form

{\underset{̲}{E}}^{Ω} = {(\begin{matrix} E_{1}^{Ω} & E_{2}^{Ω} & E_{3}^{Ω} \end{matrix})}^{⊤}

⁠. In this case, the Pockels coefficients are expressed by a tensor r with tensor components r_ijk, which act on the elements

{(1 / n^{2})}_{i j}

of the optical impermeability tensor,

Δ {(\frac{1}{n^{2}})}_{i j} = \sum_{k = 1}^{3} r_{ijk} E_{k}^{Ω} .

(2)

In practice, the refractive index of a material is of greater practical interest than its impermeability. As derived in the supplementary material, the linearization of Eq. leads to

Δ n_{i j} \approx - \sum_{k = 1}^{3} \frac{n_{i}^{2} n_{j}^{2}}{n_{i} + n_{j}} r_{ijk} E_{k}^{Ω} .

(3)

In many devices, the terms of interest are Δn_ii, such as in organic electro-optic materials, where r₃₃₃ is the dominant Pockels tensor element. In this case, and when n_i ≅ n_j, Eq. becomes equivalent to Eq. .

An “effective” Pockels coefficient r_eff is often used to express Eq. by the scalar expression

Δ n_{mat} \approx - \frac{1}{2} n_{mat}^{3} r_{eff} E^{Ω} .

(4)

Thereby r_eff ≈ r₃₃₃ for organic electro-optic materials⁷⁸ and for common lithium niobate modulators.⁷¹ However, not all materials have a dominant r₃₃₃ Pockels tensor element. Then, r_eff depends on the active materials’ crystalline orientation, the modulator’s waveguide’s relative orientation, and the polarization directions of both E^Ω and E^ω. For example, in some electro-optic modulators based on the active material barium titanate, it is

r_{eff} = \frac{1}{2 \sqrt{2}} (r_{113} + r_{333} + 2 r_{232})

⁠.^40,79

B. Waveguide-based modulators with Pockels materials

Figure 1 shows a schematic representation of an electro-optic phase modulator. It consists of two parallel electrodes with a distance d and an electro-optical material at their center. An electrical voltage V_ext drops across d. A light field propagates within the slab. It is described by

{\underset{̲}{E}}^{ω} (r, t) = {\underset{̲}{\hat{E}}}_{0} (x, y) e^{j (ω t - β z + Δ φ)},

(5)

where the index ω indicates the fast oscillation frequency of the optical field, characterized by an amplitude

{\underset{̲}{\hat{E}}}_{0}

and a modal propagation constant β = k₀ · n_eff with the vacuum wavenumber k₀ and the effective refractive index n_eff. In a simplified, quasi-static picture, the external voltage applied to the electrodes translates into an electrical field E^Ω ≈ V_ext/d, which induces a refractive index change within the waveguide (a more holistic view is given in the section on photonic modulators). Hence, the light passing through the modulator experiences a phase change Δφ. The total phase change picked up in a modulator of length L is given by

Δ φ = Δ β \cdot L = k_{0} Δ n_{eff} \cdot L .

(6)

To gain more physical insights it is useful to rewrite Δn_eff in terms of the refractive index change Δn_mat obtained in the active section and the field interaction factor Γ,

Δ n_{eff} = Δ n_{mat} \frac{Δ n_{eff}}{Δ n_{mat}} = Δ n_{mat} \cdot Γ .

(7)

The factor Γ states to what extent a refractive index change within the active EO material modulates the effective index n_eff of the optical mode. As Γ is not referring to the optical power in the active zone but the interacting fields, it is called the field interaction factor. With Eqs. , , and , one can write

Δ φ \approx - \frac{1}{2} n_{mat}^{3} r_{eff} \frac{V_{ext}}{d} \cdot Γ \cdot k_{0} L,

(8)

where r_eff is the effective Pockels coefficient and

V_{ext} / d \approx E_{k}^{Ω}

is the electric field applied via the electrodes of distance d. An infinite plate capacitor often adequately models the external electric field. Note that we imply a homogeneous permittivity and isotropic Pockels effect here, which is valid for most geometries. However, in case of dominating off-diagonal Pockels tensor elements as can be found in some ferroelectric materials such as barium titanate, r_eff depends strongly on the crystalline orientation, polarization of the optical mode and direction of the external electrical field,^40,79,81 and the description of r_eff becomes more complex, as will be detailed below. The performance metric V_πL for a phase-shifter can now be derived from by setting

Δ φ \approx {- \frac{π}{λ} n_{mat}^{3} r_{eff} \frac{V_{ext}}{d} L \cdot Γ|}_{V_{ext} = V_{π}} \overset{!}{=} π .

(9)

This way the applied voltage becomes V_ext = V_π and we can solve it for

V_{π} L \approx \frac{λ d}{n_{mat}^{3} r_{eff}} \cdot \frac{1}{Γ},

(10)

which relates the device modulation efficiency to the nonlinearity of the material used. As discussed in Sec. I, the smaller this figure of merit, the stronger the modulation of a specific waveguide.

FIG. 1.

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Geometrical depiction of an electro-optic phase modulator. An electrical signal V_ext is applied to two electrodes (distance $d$ ⁠), which encompass a waveguide structure (length L) with an electro-optic material. The resulting electric field (E^Ω ≈ V_ext/d) induces a phase shift Δφ onto the light passing through the device. Figure adapted from Ref. 80.

Equation (10) tells us that there are three clear routes to increasing the modulation efficiency of a waveguide modulator: First, by selecting a material with a strong Pockels effect, i.e., large $n_{0}^{3} r_{eff}$ ⁠, second, by reducing the electrode distance d, and third, by designing the waveguide geometry such that the efficiency Γ = Δn_eff/Δn_mat becomes large, i.e., that optical and RF fields interact strongly. The second and third options are of geometrical nature and influenced by the waveguide refractive index profile and electrode design, as will be discussed below. However, design compromises are often unavoidable. For example, increasing the field interaction factor Γ = Δn_eff/Δn_mat might require a wider waveguide, which in turn necessitates an increase of the electrode spacing d and hence reduces the electric field strength E^Ω.⁵⁶

C. Field interaction factor and sensitivity

The previously introduced field interaction factor, Γ = Δn_eff/Δn_mat, may become complex to determine, particularly when considering the vectorial nature of both optical and external electrical fields together with the full Pockels tensor: Among the three components $E_{k}^{Ω}$ of the external field ${\vec{E}}^{Ω}$ and the three components of the optical field $E_{j}^{ω}$ all components that are connected by a nonvanishing r_ijk will contribute toward Γ. In the general case, the reduction of dimensionality from a $(6 \times 3)$ tensor to the scalars r_eff and Γ is not trivial and requires simplifying assumptions.

In literature, Γ is often approximated by closed-form expressions,^58,82,83 which mostly rely on the calculation of the modal overlap between external and optical fields in the region of the active electro-optic material. In a simple and intuitive form, the field interaction factor may be approximated by the energy confinement factor relating the modal energy within the active region to the total energy,

Γ \approx \frac{\iint_{active} ⟨S_{z} (x, y)⟩ d x d y}{\iint_{tot} ⟨S_{z} (x, y)⟩ d x d y},

(11)

with

⟨S_{z} (x, y)⟩

being the time-averaged z-component of the Poynting vector. A more elaborate approach would not consider the overall energy, but the field components with the strongest interactions, typically being the transverse field components

E_{x}^{ω}

and

E_{x}^{Ω}

⁠, connected by r₃₃,

\begin{align} Γ \approx Γ_{x} & \approx \frac{\iint_{active} E_{x}^{Ω} E_{x}^{ω} H_{y}^{* ω} {\vec{e}}_{z} d x d y}{V_{ext} / d \cdot \iint Re ({\vec{E}}^{ω} \times {\vec{H^{*}}}^{ω}) {\vec{e}}_{z} d x d y} \\ \approx \frac{1 / Z^{ω} \cdot \iint_{active} E_{x}^{Ω} {|E_{x}^{ω}|}^{2} d x d y}{V_{ext} / d \cdot \iint Re ({\vec{E}}^{ω} \times {\vec{H^{*}}}^{ω}) \cdot {\vec{e}}_{z} d x d y}, \end{align}

(12)

where

Z^{ω} = Z_{0} / n_{mat} = {(c_{0} ϵ_{0} \cdot n_{mat})}^{- 1}

with Z₀ the impedance of free space, c₀ the speed of light, ϵ₀ the vacuum permittivity, and n_mat the refractive index of the electro-optic material. This expression is similar to the one from Ref. 58 [a derivation of Eq. from Eq. is given in the supplementary material]. However, these closed-form solutions are approximations only because they consider merely optical field components along the x-direction transverse to the waveguide. This neglects both the tensorial nature of the Pockels effect and optical field components in the y- and z-directions, as they occur in real, physical waveguides.

A more general route of calculating the field interaction factor could start from Eq. (7) and its formulation Γ = Δn_eff/Δn_mat. Using a numerical mode simulation tool, one could simulate Δn_eff for both Δn_mat = 0 and an arbitrary choice of Δn_mat ≠ 0 and directly calculate Γ. However, this neglects the tensorial nature of the Pockels effect, the inhomogeneity and anisotropy of $Δ n_{mat} (x, y)$ as well as the field distributions of ${\vec{E}}^{Ω}$ and ${\vec{E}}^{ω}$ ⁠. This matters particularly for plasmonic modes, where spatial changes in the field components of ${\vec{E}}^{Ω}$ and ${\vec{E}}^{ω}$ occur over very short distances. When anisotropic EO materials such as BaTiO₃ are employed, a full tensorial description in simulation is necessary.

When Heni compared values of Γ obtained by a closed-form expression based on Eq. (12) with those obtained by simulation as described below, it was found that the approximation underestimates Γ consistently for a sweep of plasmonic slot widths and metal thicknesses⁸⁴ (cf. p. 32).

Therefore, we propose a new numerical method to determine Γ. It relies on simulating the perturbation Δβ of the propagation constant and considers the full field distributions

{\vec{E}}^{ω}

and

{\vec{E}}^{Ω}

as well as the anisotropic change of

Δ n_{eff} (x, y) = Δ n_{eff} ({\vec{E}}^{Ω} (x, y))

⁠. Using Eq. , we get

Δ β \approx - \frac{π}{λ} n_{mat}^{3} \frac{V_{ext}}{d} r_{eff} \cdot Γ,

(13)

which can be solved for Γ,

Γ \approx - \frac{Δ β λ}{π \cdot n_{mat}^{3} r_{eff}} \cdot \frac{d}{V_{ext}},

(14)

and obtain Δβ by simulating both

{\vec{E}}^{Ω}

and

{\vec{E}}^{ω}

and calculating the propagation constant β twice, once in the absence of an external field, and once in the presence of the applied external voltage V_ext,

Δ β = β (V = V_{ext}) - β (V = 0) .

(15)

r_eff can be calculated in a closed form, and for examples, see Refs. 40 and 79. The electrode distance d is typically well defined by the geometry at hand. In the following sections, this method will be applied to discuss various waveguide cross sections.

D. Comparing waveguide geometries and materials

Ultimately, we are interested in minimizing the metrics V_πL and V_πLα. As discussed above, we can maximize the Pockels effect $n_{mat}^{3} r_{eff}$ and the field confinement factor Γ and minimize the propagation losses α over the length L. The latter two parameters Γ and α depend on the waveguide geometry. An important question then is: Which waveguide geometry gives the highest modulation efficiency (with respect to Γ) and lowest losses α? The answer should be given for important nonlinear materials with refractive indices ranging from n_mat = 1.7 to n_mat = 2.3. The challenge now is that both the field interaction factor Γ and the Pockels effect $n_{mat}^{3} r_{eff}$ strongly depend on the refractive index n_mat. To clearly distinguish the effect of the waveguide geometry with the field interaction factor and the nonlinearity, we will maintain a constant nonlinearity of the materials by considering $n_{mat}^{3} r_{eff} = 1000 pm/V$ throughout our discussion of waveguides, corresponding to an organic electro-optic material with n_mat ≈ 1.8 and r_eff ≈ 170 pm/V, as in Ref. 67. This results in the parameter combinations listed in Table II.

TABLE II.

Overview of the active materials used for simulation purposes. The choice of the values results in a constant nonlinear effect $n_{mat}^{3} r_{eff} = 1000 pm/V$ ⁠, consistent with organic electro-optic material with n_mat ≈ 1.8 and r_eff ≈ 170 pm/V.⁶⁷ This allows us to compare geometries independent of the material. Readers can deduce the figures of merit for a certain material with a given n_mat by a linear scaling. Example: To deduce V_πL of a LiNbO₃ modulator (n_mat ≈ 2.2), scale the simulation value of V_πL for n_mat = 2.2 given throughout the paper by a factor of 3.

n_mat	$r_{eff} (pm/V)$	Remarks
1.7	204	Approximately organic EO material, e.g., Ref. 85
1.8	171	Approximately organic EO material, e.g., Ref. 67
1.9	146	Approximately organic EO material, e.g., Ref. 67
2.0	125	Included for completeness
2.1	108	Included for completeness
2.2	94	∼3× stronger than LiNbO₃’s r₃₃ (⁠ $r_{33}^{LNO} \approx 30 pm/V$ ⁠, Ref. 86)
2.3	82	∼4× weaker than BaTiO₃’s r₃₃ (⁠ $r_{33}^{BTO} \approx 342 pm/V$ ⁠, Ref. 24)

n_mat	$r_{eff} (pm/V)$	Remarks
1.7	204	Approximately organic EO material, e.g., Ref. 85
1.8	171	Approximately organic EO material, e.g., Ref. 67
1.9	146	Approximately organic EO material, e.g., Ref. 67
2.0	125	Included for completeness
2.1	108	Included for completeness
2.2	94	∼3× stronger than LiNbO₃’s r₃₃ (⁠ $r_{33}^{LNO} \approx 30 pm/V$ ⁠, Ref. 86)
2.3	82	∼4× weaker than BaTiO₃’s r₃₃ (⁠ $r_{33}^{BTO} \approx 342 pm/V$ ⁠, Ref. 24)

It should also be noted, that in this work, we assume the Pockels coefficient r_eff to be constant for any modulation frequency $r_{eff} (ω) ≅ c o n s t$ ⁠, although this is not the case for all materials: Ferroelectrics, for instance, have an additional strain-optic contribution linked to their piezoelectricity at lower frequencies.¹² However, taking all these effects into account would go beyond the scope of this work, and the frequency response can be considered by inverse-linearly scaling V_πL and V_πLα. Be aware that the scaling holds only if all tensorial contributions to r_eff scale alike with frequency, which might not be the case for low-frequency contributions in mechanically clamped materials with a piezo-electric response.

In the figures illustrating the simulation results per waveguide geometry, we always plot the field interaction factor Γ, the propagation losses α, the voltage–length product V_πL and the voltage–length–loss product V_πL · α. In many geometries, we will find a trade-off between the modulator’s V_πL or Γ and its propagation loss α, and we hope that the consistent depiction helps to illustrate this effect. The method used to perform the simulations for the result diagrams is described in the supplementary material.

III. PLASMONIC, PHOTONIC, AND HYBRID MODULATORS

Subsequently, we compare three fundamental waveguiding concepts that are currently of high interest in electro-optic waveguide modulators.

First, photonic waveguide modulators, where the refractive index profile of the waveguide is formed such that light is guided either in the material with the highest refractive index, or in a narrow, low-index gap between higher-index materials. Photonic waveguides can have very low propagation losses, but the diffraction limit sets a lower boundary to the mode size and confinement.
Second, and on the other extreme, plasmonic waveguide modulators utilizing metal–dielectric interfaces to tightly confine the optical energy. Metal–insulator–metal (MIM) gap waveguides exhibit no cutoff, so that modes can be confined to nanometer-sized gaps.^87,88 The big challenge of plasmonic geometries is their fundamental optical attenuation, which comes from Ohmic losses of electrons in the metal, which oscillate as part of the plasmonic electromagnetic wave.
Third, hybridizing a photonic and a plasmonic mode, e.g., by introducing a thin lower-index layer between a high-index material and a metal electrode, potentially offers a compromise between highest-speed plasmonic and low-loss photonic concepts.^89–91

Figure 2 exemplifies some waveguide geometries, with photonic concepts in the first row, plasmonic in the second, and hybrid photonic–plasmonic in the third. We have tried to list typical geometries, although literature knows much more variety and the usage of other materials, such as transparent conductive oxides could even blur the boundaries between the categories. For photonics, these are silicon–organic hybrid (SOH) waveguides,¹⁵ lithium niobate on insulator,^29,73 and barium titanate on insulator.^21,77 For the plasmonic geometries, we have chosen the established plasmonic–organic hybrid waveguide geometry,¹⁵ its ferroelectric-filled variant with similarities to a rib waveguide,⁴⁰ and a vertically stacked metal–insulator–metal waveguide.^37,92 For the hybrid case, we have chosen the most basic geometry, one width additional lateral contact electrodes as needed for electro-optic modulation.^44,47,93 Based on this second geometry, we propose a variant comprising a partially etched section improving the modal confinement. More references to literature on photonic, plasmonic, and their hybrid waveguide geometries are given in the respective sections below.

FIG. 2.

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Cross sections of exemplary [(a)–(c)] photonic, [(d)–(f)] plasmonic, and [(g)–(i)] hybrid photonic–plasmonic waveguides for electro-optic modulator applications. The marks for the electrode distance d indicate the distance over which the external voltage V_ext drops. (a) Silicon slot waveguide filled with an organic EO material. The silicon rails are doped to induce electrical conductivity. (b) A LiNbO₃ slab waveguide with lateral metal electrodes. (c) A BaTiO₃ modulator, where the optical confinement is provided by a deposited Si strip. (d) A plasmonic slot waveguide filled with an organic EO material. (e) Plasmonic slot waveguide formed with a BaTiO₃ slab. (f) A vertically arranged plasmonic slot waveguide. (g) Archetype of a hybrid photonic–plasmonic waveguide where the organic EO material is sandwiched between a doped Si strip and a metal top electrode. (h) For operation, the bottom Si electrode must be electrically connected. (i) A partial etch, or sub-wavelength grating can improve modal confinement.

A. Photonic

Photonic waveguide-based electro-optic modulators are widely available. Metal-diffused waveguides in lithium niobate crystals⁹⁴ have paved the way for generations of low-loss, high-performance electro-optic modulators.⁷⁶ In parallel, optical platforms based on semiconductors have been investigated for their strong electro-optic effects and the semiconductor industry’s fabrication expertise. Indium phosphide (InP) and gallium arsenide (GaAs) have been explored first for their linear electro-optic effect²⁵ and later incorporated in multi-quantum well heterostructures to form electro-refractive⁹⁵ and electro-absorptive^96–98 modulators based on the quantum-confined Stark effect.^99,100 InGaAs/InP now forms some of the fastest and most compact modulator technologies, benefiting from the availability of light sources on the same platform.¹⁰¹ Silicon (Si) photonics benefits from established CMOS fabrication processes and techniques,⁶ now driving the successful and commercial development of compact, fast, and cost-efficient modulators.⁵⁷ Although Si benefits from the high-yield and high-volume capabilities of the CMOS industry, its carrier-based modulation is limited to bandwidths up to ∼60 GHz in millimeter-long devices.^53,102 Functionalization of Si waveguides with other EO materials is investigated to form potentially faster and more compact modulators. Prominent examples are silicon slot waveguides filled with electro-optic organic material, which provide a strong Pockels effect.^14,15 Ferroelectric materials might alleviate some of the thermal and chemical concerns faced by organics, and lithium niobate and barium titanate have already been successfully integrated with silicon photonics.^{21,23,24,81,103} The lithium niobate on insulator (LNOI) technology questions the necessity of a silicon photonic platform, as waveguides and modulators are directly formed out of the electro-optic material, yielding promising high-speed modulators with low optical insertion losses.²⁸

One of the greatest advantages of photonic modulator designs is their potentially low optical insertion loss. State-of-the-art LNOI modulators can reach fiber-to-fiber insertion losses below 5 dB,^28,104 followed by InP modulators with ∼9 dB losses.^105,106

One of the greatest challenges is the modulation efficiency and the required length of photonic modulators: To avoid metal-induced losses, the metallic electrodes must be kept relatively far from the waveguide. This leads to an increased electrode gap d, which increases the voltage-length product V_πL [see Eq. (10)]. An exception are electrodes formed from semi-transparent materials, such as doped Si,^14,15 or capacitively coupled electrodes.⁸³

Figure 3 shows numerical simulations of the photonic waveguide geometries from Fig. 2, where the optical electric field is color coded and field lines illustrate the external RF field. In the following, the silicon–organic hybrid (SOH) from Figs. 2(a) and 3(a), as well as the LNOI device [see Figs. 2(b) and 3(b)], are discussed in detail.

FIG. 3.

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Optical field distributions (⁠ $|E^{ω}|$ ⁠, colored) and electrical RF fields (⁠ $E^{Ω}$ ⁠, black field lines) for three examples of the photonic waveguide modulator structures plotted in Fig. 2. (a) Silicon–organic hybrid waveguide. The slot between two doped silicon rails is filled with an organic EO material. The RF field is applied over the silicon rails. (b) Lithium niobate on insulator. The active material itself forms the waveguide; the RF field is applied between two metal electrodes. (c) Silicon on barium titanate. The silicon strip provides lateral confinement to the optical mode. The RF field is applied between two metal electrodes.

1. Example: Silicon–organic hybrid

Silicon–organic hybrid (SOH) modulators rely on a silicon slot waveguide filled with a nonlinear organic material. Strongly doped, and hence electrically conductive, silicon slabs connect the slot waveguide to metallic electrodes further away from the optical mode. The slot widths d in such configurations ranges from 100 to 200 nm,¹⁰⁷ and the organic electro-optic materials used can exhibit nonlinearities up to $n_{0}^{3} r_{33} \approx 2300 pm/V$ (n₀ ≈ 1.81, in-device r₃₃ = 390 pm/V¹⁰⁷), exceeding lithium niobate’s properties by a factor of 7. However, the field interaction factor Γ is modest because E^ω and E^Ω have only little overlap within the EO material. Figure 4(a) illustrates the dependence of Γ on the slot width d. Note that the effect of doping and carriers in the silicon rails make the modeling not trivial.⁸³ Here, we assume a doping to ρ = 5.5 · 10⁻⁴ Ω m, which corresponds to a silicon absorption coefficient of 0.1 mm⁻¹,¹⁰⁸ or an imaginary part of the refractive index of $n_{Si}^{″} = 1.25 \cdot 1 0^{- 6}$ ⁠. Note that a higher doping concentration typically improves the EO bandwidth at the cost of higher propagation losses. A study on the effect is given in the discussion of hybrid photonic–plasmonic waveguides below. The electrical field is modeled to drop entirely within the slot.⁸³ The Si thickness is 220 nm, and the slab thickness is 70 nm.¹⁶

FIG. 4.

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Simulation results for the SOH geometry. n_mat is color coded, from n_mat = 1.7 (light) to n_mat = 2.3 (dark). It is always $n_{mat}^{3} r_{eff} = 1000 pm/V$ ⁠. Narrow slots and low-index materials yield the best field interaction (a), with best voltage–length product (b) for narrow slots, too. Higher losses in narrow slots (c) are out-weighed by the stronger field, so that the voltage–length–loss product becomes best for narrow slots, too.

Populating Eq. (10) with the simulation d of Γ = 0.2, geometrical (d = 150 nm) and material (⁠ $n_{0}^{3} r_{33} = 2300 pm/V$ ⁠) parameters yields V_πL = 840 V µm. This is in good agreement with the measurement value V_πL = 640 V µm in Ref. 107.

Although the modulation efficiency V_πL and the voltage–length–loss product V_πLα are best for small slot widths d, one could think that, rationally, only the narrowest slots should be fabricated. However, at least two findings contradict this conclusion: First, wider slots make the electric field poling of the OEO more efficient,¹⁵ yielding higher $n_{0}^{3} r_{33}$ (the record of r₃₃ = 390 pm/V has been measured in a 190-nm-wide slot¹⁰⁷). Second, narrower slots are more difficult to fabricate, and their optical propagation losses tend to be higher than those of wider slots due to higher sensitivity to scattering effects.⁸³

Doping levels do not influence the losses of practically available Si slot waveguides, which typically exceed 0.5 dB/mm¹⁰⁸ and can reach up to 2.5 dB/mm in recent realizations and are likely to be dominated by fabrication imperfections.¹⁶ To mitigate optical losses induced by doped silicon, the metal electrodes can be capacitively coupled to the Si rail via a high-k material.⁸³

2. Example: Lithium niobate on insulator

Figure 5 visualizes the influence of the electrode separation on a state-of-the-art LNOI modulator geometry, see Fig. 3(b), with an 800-nm-wide, 600-nm-high lithium niobate slab waveguide formed by thinning the surrounding lithium niobate to 300 nm and adding 800 nm SiO₂ cladding. The gold electrodes are 1.1 µm thick and their distance d is swept from 2 to 5 µm. The field interaction factor Γ [Fig. 5(a)] stays nearly unchanged by the sweep, with high-index materials yielding the best confinement, Γ ≈ 0.6 for n_LNO = 2.2. At the same time, the modulation efficiency increases linearly with decreasing electrode distance, a consequence of the increase of the external RF field (E^Ω = V_ext/d). On the other hand, the optical propagation losses sharply increase from 0.05 to ∼200 dB/cm when the electrode distance is decreased from 5 to 2 µm. At the same time, the RF propagation losses (not shown) increase with the smaller the electrode distance, amounting already to 7 dB cm⁻¹ for a 5 µm gap and being the main limitation to state-of-the-art LNOI modulators.³¹ Although the modulation efficiency is best for the smallest electrode distance, this regime is not accessible, in practice, because the RF and optical losses forbid modulators that are sufficiently long to provide a low V_π. State-of-the-art LNOI modulators as used in high-data-rate experiments are 10 mm (5 mm) long and have a 6-dB (3-dB) bandwidth roll-off of up to 100 GHz.^28,29,109 A recent breakthrough using segmented electrodes is discussed below, potentially offering 250 GHz 3-dB bandwidth in a 20-mm-long MZM with V_πL = 1.7 V cm and a DC π-voltage of V_π,DC = 1 V.³¹

FIG. 5.

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Design trade-offs in LNOI modulators. (a) Decreasing the electrode distance does not increase the field interaction factor, but (b) brings an increase of the electrical field strengths and improves the modulation efficiency, resulting in smaller V_πL. (c) On the other hand, electrodes closer to the waveguide contribute to more photonic propagation loss, resulting in larger V_πLα (d). However, shorter devices can have a higher electro-optic bandwidth. Normalized to $n_{mat}^{3} r_{eff} = 1000 p m / V$ ⁠. The design trade-offs in BTO-based photonic modulators are similar.

So far, typical LNOI devices with n_mat = 2.2, r₃₃ = 30 pm/V (⁠ $n_{mat}^{3} r_{33} \approx 320 p m / V$ ⁠), and an electrode distance d = 5 µm have measured voltage–length products of V_πL = 4.2 V cm.³¹ This is in very good agreement with the results of Eq. (10), which yields V_πL = 4 V cm with a simulated Γ ≈ 0.6.

3. Frequency response and the challenge of traveling wave electrodes

Electro-optic modulators with traveling wave electrodes show impressive performance and electro-optical bandwidths exceeding 100 GHz.^{17,18,27,29,31,109} However, designing such electrodes is difficult, as their characteristic impedance needs to be matched to the driving electronics to avoid reflections. Furthermore, three factors largely determine the traveling wave modulators’ frequency response: First, the RF attenuation, second, the RF-optical velocity mismatch, and third, impedance matching between driver electronics and on-chip electrodes.^31,57

The 3-dB bandwidth induced by RF field and limited due to attenuation and walk-off depends on the accumulated phase shift

Δ ϕ = \int_{0}^{L} k Δ n_{eff} (z) d z .

(16)

Here, L is the length of the modulator and

Δ n_{eff} (z) = a V (z),

(17)

with a comprising the electro-optical coefficients, and geometry parameters discussed above. The RF voltage propagating along the stripline is given by

\begin{align} V (z, t) & = V_{0} \cos (Ω t - k_{Ω} z) \exp (- \frac{α_{Ω} (ω)}{2} z) \\ = V_{0} \cos (Ω t - \frac{Ω}{c_{0}} n_{Ω} z) \exp (- \frac{α_{Ω} (ω)}{2} z), \end{align}

(18)

with the RF frequency Ω, the RF effective index n_RF, and the RF attenuation constant α_Ω. Now while the RF field propagates with the velocity

c_{Ω} = \frac{c_{0}}{n_{Ω}},

(19)

the partially modulated optical wave propagates with the group velocity

c_{ω} = \frac{c}{n_{g, ω}} .

(20)

Thus, the optical signal at position z is delayed by Δt = z/c_ω = zn_g,ω/c₀ compared to when it has entered the waveguide. The accumulated phase shift on the optical field exposed to the RF field thus becomes

Δ φ (Ω) = \int_{0}^{L} k a V_{0} \cos (\frac{Ω}{c_{0}} (n_{g, ω} - n_{Ω}) z) \exp (- \frac{α_{Ω} (ω)}{2} z) d z .

(21)

The frequency response is then usually plotted relative to the DC shift

Δ φ (0) = k Δ n_{eff} L

⁠. In the attenuation-limited case, with perfect velocity match between the optical and the RF mode, n_g,ω = n_Ω and we find

\frac{Δ φ (Ω)}{Δ φ (Ω = 0)} = \frac{1}{L} \int_{0}^{L} \exp (- \frac{α_{Ω} (Ω)}{2} z) d z .

(22)

The attenuation-limited 3-dB frequency f_3dB,α = Ω_3dB,α/(2π) is found by solving

{|Δ φ (Ω) / Δ φ (0)|}^{2} \overset{!}{=} 1 / 2

⁠. Numerically, one finds for the 3-dB frequency from Eq. that Lα(Ω), the product of length and RF attenuation, is reached when

L α (Ω_{3 dB, α}) \approx 1.48 .

(23)

The limitation from the RF-optical walk-off can be deduced from Eq. . For a negligible α_RF, it is

\frac{Δ φ (Ω)}{Δ φ (Ω = 0)} = \frac{1}{L} \int_{- L / 2}^{L / 2} \cos (j \frac{Ω}{c_{0}} (n_{g, ω} - n_{RF}) z) d z .

(24)

Note that we have balanced off the walk-off by setting the beginning and end of the modulator to ±L/2. The 3-dB bandwidth is then found by

{|Δ φ (Ω_{3 dB, τ,}) / Δ φ (0)|}^{2} \overset{!}{=} 1 / 2

⁠. Solving for the 3-dB frequency response f_3dB,τ = Ω_3dB,τ,/(2π), one finds that the value is determined by the transit times of the respective waves through the modulator. Numerically, one finds

f_{3 dB, τ} = \frac{0.44}{|τ_{g} - τ_{g, Ω}|},

(25)

with τ_g,Ω = L · n_Ω/c₀ and τ_g = L · n_g,ω/c₀.

In a real modulator the frequency response is lower as it may be limited by any or all of the above-mentioned limitations, i.e., by attenuation f_3dB,α, or by a walk-off f_3dB,τ.

Impedance matching of the traveling wave electrode to the drive electronics is crucial to minimize the RF reflections and make use of the full RF signal. The transferred power fraction to the device is $1 - {|r|}^{2}$ ⁠, with the reflection factor $r = (Z_{L} - Z_{0}) / (Z_{L} + Z_{0})$ ⁠, where Z₀ is the source impedance and Z_L is the load impedance of the travelling wave electrode. The source impedance is often 50 Ω, which is particularly challenging to reach for small electrode separations as desired for efficient modulation: Slowly tapering down the electrode gap from a matched, wider, section, to the active, narrower section can add hundreds of micrometers to the electrode length, with the according RF attenuation.²⁷ Furthermore, Z_L is frequency dependent, making the impedance matching even more challenging.

Additionally, the electrodes must be terminated with a matching load to avoid reflections. Reflections may result in standing waves within the modulator, which would lead to resonant enhancement and suppression of certain frequency ranges, resulting in inter-symbol interference in data modulation setups. In laboratory conditions, the termination can be realized with a second microwave probe,^16,29,74,111 whereas integrated solutions use on-chip doped silicon¹¹² or NiCr resistors.¹¹³ Notably, if the termination can be omitted one might benefit from a voltage doubling as a result of the reflection.⁵⁹

4. Advantages, challenges, and recent advances

An advantage of photonic waveguides and modulators is the low optical propagation loss. This is particularly true for geometries, such as the LNOI waveguides, which feature on-chip device losses of <0.5 dB.^114,115

Challenges faced by photonic modulators are the large size of the photonic modes and/or the large distances between modes and RF electrodes. The large dimensions make large voltages necessary or result in low modulation efficiencies. To compensate for the low efficiency, one may increase the length of the modulator. Yet, this leads to a new design challenge: The trade-off between efficiency and bandwidth: Efficient modulators tend to be long—but come with a reduced bandwidth. High bandwidth modulators are small but require high drive voltages. Depending on the geometry, photonic modulator lengths range from 280 µm¹⁶ for high-performance SOH devices to 1–5 mm for LNOI modulators.^28,109 Devices larger than 1/10th of the guided RF wavelength (free space wavelength divided by the RF index n_RF) cannot be considered lumped anymore.⁵⁷ For Ω = 2π · 100 GHz and n_RF ≈ 2,³¹^, λ_RF = 1.5 mm, and devices longer than $\sim 150 μ m$ therefore require traveling wave electrodes. In traveling wave modulators, not only the walk-off between optical and electrical signals poses a limit to the modulator’s electro-optic bandwidth but also the attenuation of the traveling RF wave. To reduce the required length and keep up a high bandwidth one could design RF waveguides with a smaller gap. Yet, this comes at a price of higher RF losses (and for very small electrode distance d with high optical losses, too).³¹

Hence, research focuses on advances in the current photonic modulator designs. In LNOI devices, where RF propagation losses have limited the EO bandwidth, a new segmented electrode design has recently been introduced, which lowers the RF propagation losses at 50 GHz from 7 to 2 dB/cm.³¹ When a quartz substrate is used instead of the Si substrate, 3-dB bandwidths of 250 GHz have been achieved with 2 cm-long devices, offering a π-voltage at DC of V_π,DC = 1 V.³¹

In SOH devices, replacing the resistive coupling between metal electrodes and Si slot by a capacitive coupling via a high-k dielectric has recently been studied.⁸³ However, this approach reduces the field interaction factor as compared to standard SOH modulators and attempts to increase Γ by weakly doping the Si slot waveguide have resulted in higher RF propagation losses.⁸³ For a 50 Ω-matched traveling wave electrode, numerical simulations yield Γ ≈ 0.14 for undoped and Γ ≈ 0.32 for doped silicon, with RF propagation losses increasing from 1.7 to 2.3 dB/mm.⁸³ Measurements show RF propagation losses exceeding 5 dB/mm, yielding similar results as modeled for more traditional, resistively coupled devices.¹⁰⁸

Furthermore, efforts are on the way to develop photonic modulators that use new electro-optical materials with a higher nonlinearity. Notably, improved organic linear-electro-optical materials promise electro-optical coefficients in the order of r₃₃ = 1000 pm/V.¹¹⁶ Thin-film barium titanate (BaTiO₃) has already been shown to exhibit extremely large Pockels coefficients with r₄₂ = 923 pm/V at n_mat ≈ 2.3. According to Eq. (10) and Ref. 40, this should lead to r_eff = 750 pm/V and $n_{mat}^{3} r_{eff} \approx 9100 pm/V$ ⁠. The waveguides of these BaTiO₃ modulators are formed by loading the film with silicon nitride¹⁰³ or silicon.^20,21,24 Electro-optic bandwidths of these devices are still below 30 GHz²⁴ or 5 GHz.^21,103 However, the use of BaTiO₃ is accompanied by at least two major challenges. First, its microwave permittivity is very high, with estimates and measurements ranging around ϵ_r ≈ 1000.^24,117–119 This property is detrimental to the design of travelling wave electrodes, as the microwave signal is slowed down significantly.¹²⁰ Second, BaTiO₃ has a relatively low Curie temperature of ∼120 °C,^121,122 leading to intrinsic instability of the spontaneous polarization and requiring the material to be DC-biased during operation. In photonic devices with an electrode gap of 4.75 µm, a bias of ∼10 V was necessary to ensure strong EO modulation.²⁴ Such high DC voltages complicate RF biasing circuits and are not compatible with highest-speed driving electronics. In later experiments, the bias was reduced to ∼2 V for a gap of 2.6 µm, however, with no indication if the bias voltage ensured full RF modulation.¹²⁰

Apart from BaTiO₃, lead zirconate titanate (PZT) has been co-integrated with a SiN waveguide and a Pockels coefficient of r_eff = 70 pm/V has been measured, which yields $n_{mat}^{3} r_{eff} = 850 p m / V$ ⁠.^49,123

B. Plasmonic metal–insulator–metal

Plasmonic metal–insulator–metal (MIM) modulators offer a way to overcome the challenges of size and speed, which photonic modulators are bearing. Figures 2(d)–2(f) show typical examples of plasmonic MIM slot waveguides, where the two electrodes can be horizontally separated in-plane [Figs. 2(d) and 2(e)] or vertically separated in an out-of-plane configuration [Fig. 2(f)]. The optical energy is guided in a thin dielectric layer, d ≈ 100 nm thick, consisting of an electro-optic Pockels material and separating the RF electrodes.

Plasmonic modulators are very compact and efficient. They are small because the active dielectric layer and the supported plasmonic modes have sub-wavelength dimensions. Therefore, E^Ω = V/d is very large, and the field interaction factor Γ can even become larger than unity due to the slow-down that plasmonic structures bring with them.¹²⁴ This results in most compact modulators with lengths in the order of ∼10 µm. The compact size brings two more major advantages: First, the parasitic resistance and capacitance are very small, so that the electro-optic bandwidth of these modulators exceeds 500 GHz.⁶² Furthermore, the devices can be truly treated as lumped, capacitive loads. Hence, the driving voltage supplied from a 50 Ω line is reflected, leading to its effective doubling. All these factors together result in modulators with voltage–length products as small as ∼60 V μm.¹²⁴ A second benefit arising from the lumped character of the devices is that they can be driven differentially,⁵⁹ effectively reducing the voltage–length product by another factor of 2. Conversely, the strong confinement and modulation efficiency come at the cost of propagation losses, which are about ∼0.5 dBμm⁻¹.¹²⁵ Furthermore, the high mode intensity in plasmonic waveguides may lead to the photodegradation of organic electro-optic material¹⁰⁷ and hence limit the maximal power that the modulator can withstand. As this effect is linked to the presence of oxygen in the material,^126,127 hermetic packaging can mitigate this risk.

Particularly the horizontal configurations of MIM waveguides, Figs. 2(d) and 2(e) have been exploited in experiments to form the most compact, high-speed plasmonic modulators.^{39,40,128–132}

The coupling of a photonic mode to a plasmonic mode needs special attention. Photonic–plasmonic conversion has been successfully solved by, e.g., deploying silicon bus waveguides,^39,133,134 tapered silicon fiber-to-chip grating couplers,³⁵ or via mode conversion from a dielectrically loaded long-range surface plasmon polariton (SPP).³⁴

MIM waveguides [see Figs. 6(a) and 6(b)] rely on very small lithographic features (slot widths d ∼ 100 nm), which can be difficult to realize. This results in maximal aspect ratios of ∼5 (e.g., a 30-nm-wide gap in a 150-nm-high gold film). However, increasing the aspect ratio would be desirable. A large aspect ratio helps to increase the field interaction factor Γ and leakage to the substrate may be better suppressed.¹³⁵ Furthermore, the modal intensity is reduced, potentially reducing the rate of photodegradation.

FIG. 6.

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Three examples of plasmonic metal–insulator–metal waveguides. The plasmonic mode $|E^{ω}|$ is color-coded, the external RF field E^Ω is displayed by the field lines. The metals are 150 nm thick, and the plasmonic slots are d = 100 nm.¹⁵ (a) Horizontal metal slot on glass as used in plasmonic–organic hybrid devices. (b) Ferroelectric plasmonic waveguide with a barium titanate slab.⁴⁰ (c) Vertical MIM slot waveguide with width w = 1000 nm.³⁷

A way to fabricate MIM waveguides with higher aspect ratios of the gap can be realized by transitioning to a vertical layer stack,^{37,92,136–140} see Fig. 6(c). This allows plasmonic slots with a very large aspect ratio, which can help to mitigate detrimental boundary effects at the metal–insulator interfaces.^24,139 Additionally, the top electrode might protect the EO material from detrimental environmental influences, such as oxygen. Furthermore, with horizontal layers, it is easier to obtain very smooth metal surfaces that form the plasmonic waveguide¹⁴¹ and offer lower propagation losses¹⁴² as these surfaces are not defined by etching. Finally, in a vertical stack, the slot thickness d can be defined by well-controllable layered deposition techniques, which increases fabrication tolerance and lowers cost in contrast to highest-resolution lithography methods.³⁷

Figure 6 shows modal simulations of the different MIM modulator configurations from Fig. 2. The slot dimension is chosen to be d = 100 nm, the metal is 150 nm thick, and the upper electrode of the vertical configuration is w = 1000 nm wide. The optical mode is always strongly confined to the gap region and is well aligned with the RF field. From the simulations, one can also see that in the horizontal configurations [Figs. 6(a) and 6(b)] more of the optical field leaks into the cladding and substrate, leading to a slightly reduced field interaction factor Γ and slightly higher propagation losses α. This can also be seen from the plots in Fig. 7, which displays the dependence of the modulator properties on the electrode separation d.

FIG. 7.

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Influence of the plasmonic gap width d on the performance metrics for horizontal slot (dashed) [see Fig. 6(a)] and vertical slot (solid) waveguides [see Fig. 6(c)]. The higher aspect ratio of the vertical slot reduces leakage into non-EO material, so that modulation efficiency and losses are superior.

Increasing the plasmonic gap width d has multiple implications. The field interaction factor and the voltage–length product deteriorate for wider gaps. Conversely, the losses are reduced. Whereas simulations reveal that a minimal gap width d leads to an optimal voltage–length–loss product V_πLα, modulators with a ∼70 nm-wide-gap perform best in experiment.¹⁴³ When comparing horizontal (dashed) and vertical (solid), the vertical configuration outperforms the horizontal in all simulated parameters: Its greater aspect ratio reduces leakage into the surrounding non-EO materials. Therefore, the stronger confinement Γ leads to a smaller voltage–length product, and to less energy guided in plasmonic edge modes, which tend to be more lossy, and more prone to scattering. Simulations show that a larger aspect ratio (determined by a large waveguide width in the vertical structure) improves the modulators’ performance characteristics (see Fig. 8). Note that the larger aspect ratio increases the capacitance of the vertical device, resulting in an EO bandwidth lower than that that of devices with a horizontal slot, but numerical 3D simulations suggest that the frequency response still is larger than 300 GHz.

FIG. 8.

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Influence of the waveguide width w on the performance of the vertical slot waveguide with slot width d = 100 nm. As the confinement increases for wider waveguides and the propagation losses decrease, it is advantageous to select wider waveguides. Note that the device capacitance increases, and the waveguide is not single moded anymore.

The simulations shown above are in good agreement with experiment. For instance, working with a device having a plasmonic slot of width d = 130 nm, a gold thickness t_Au = 150 nm, and $n_{0}^{3} r_{33} = 1.8 5^{3} \cdot 100 pm V^{- 1}$ ⁠, as in Ref. 125, simulations and Eq. (14) lead to Γ = 0.82, so that V_πL ≈ 400 V µm with Eq. (10). Within a Mach–Zehnder modulator and if driven by RF frequencies, the simulations would suggest a V_πL ≈ 100 V µm for the structure from the experiment. This is only slightly overestimating the measured value of 130 V µm.¹²⁵

1. Frequency response

Plasmonic devices outperform the efficiency-bandwidth products of photonic devices. Due to their compactness, they show only low RF parasitics and offer small device capacitances. As a result, plasmonic modulators can be treated as a capacitive termination of their drive line, resulting in the full reflection of the impeding RF wave and an effective voltage doubling. This regime of operation holds to at least 500 GHz.⁶² Furthermore, the metal electrodes are in direct contact with the active section such that there is hardly any resistance. In consequence, the RC-constant is small and accordingly the frequency response of plasmonic organic hybrid modulators is high and flat from DC to at least 500 GHz.⁶² Furthermore, electro-optic modulation in plasmonic gaps at 1.25⁶³ and 2.4 THz¹⁴⁴ has been demonstrated. Based on measurements of R and C, the EO bandwidth of these modulators might be as high as 80 THz.¹²⁸ The extremely large EO bandwidth also makes plasmonic MIM modulators an excellent choice for microwave photonics and radio over fiber applications.^62,145

2. Advantages, challenges, and recent advances

Plasmonic waveguide modulators’ most unique feature is their electro-optic bandwidth exceeding 500 GHz,⁶² with operation up to 2.4 THz¹⁴⁴—unmatched by any other modulator technology. On the other hand, plasmonic propagation losses are fundamental and pose an important challenge to the technology. Nevertheless, state-of-the-art plasmonic devices combined with foundry-quality silicon photonics can result in plasmonic MZMs with less than 12 dB fiber-to-fiber losses and 5 dB on-chip losses.¹⁴⁶ More recently, plasmonic race-track modulators with fiber-to-fiber losses of 6.5 dB and on-chip plasmonic losses of 1.2 dB have been shown.¹⁴⁷ Furthermore, a change from horizontal to vertical plasmonic waveguides can help to further minimize the impact of losses (see Fig. 7).^37,136,137 The vertical geometry also facilitates the use of better plasmonic metals¹⁴¹ or performance-boosting interfacial layers.¹³⁹

Generally, materials with a stronger electro-optic effect can help to keep devices short and losses low. The use of barium titanate has resulted in high-speed ferroelectric plasmonic modulators.⁴⁰ Although the voltage–length product of first-generation phase shifter plasmonic ferroelectric modulators is ∼400 Vμm, it is estimated that this number can decrease by a factor of 10 to ∼40 Vμm,⁴⁰ an efficiency where devices shorter than 5 µm could be deployed.

C. Hybrid photonic–plasmonic waveguides

Hybrid photonic–plasmonic waveguide modulators are a compromise between pure photonic and pure plasmonic modulators. As such, they have the potential to address the confinement-loss trade-off of plasmonic MIM modulators. Typically, these hybrid photonic–plasmonic structures comprise a thin dielectric layer incorporated between a transparent conductor with high refractive index, such as silicon and a metal [see Figs. 2(g)–2(i)]. While the propagation loss of these hybrid waveguides is lower than those of plasmonic geometries, their field confinement Γ (see field simulations in Fig. 9) is higher than those of purely photonic solutions. This makes hybrid photonic–plasmonic waveguides interesting for technological applications, such as lasers,¹⁴⁸ frequency converters,¹⁴⁹ or electro-optic switches¹⁵⁰ and modulators.^90,93,132 Indeed, the hybrid waveguide structure offers a route to easily functionalize well-established photonic platforms, such as silicon or silicon nitride, with the advantages of sub-wavelength confinement in plasmonics.

FIG. 9.

View large Download slide

Cross section and modal simulation of a hybrid photonic–plasmonic modulator. (a) Simplified geometry of a plain hybrid photonic–plasmonic waveguide. (b) Lateral ground contacts have been added to provide a low-resistive path between ground and signal. (c) An additional partial etch helps to improve the confinement and field interaction factor Γ.

Figure 10 shows the field interaction factor Γ, the voltage–length–loss product V_πLα, and the voltage–length product V_πL of a hybrid photonic–plasmonic modulator as depicted in Fig. 9(b). The field interaction factor Γ in Fig. 10(a) is smaller than that of MIM modulators, because energy is transported not only in the slot but also in the silicon waveguide (see Fig. 9). Still, Γ can reach values close to unity, although values between ∼0.4 and ∼0.7 are more realistic. Figure 10(b) shows how the voltage–length product increases with the dielectric thickness (gap dimension d) —this is because the external field E_ext drops with d⁻¹. The same holds true for the voltage–length–loss product in Fig. 10(d), which is also optimal at the smallest gap dimension d even though losses are highest there [Fig. 10(c)]. As for plasmonic MIM devices, the experimental best case is likely to be determined by a trade-off between the theoretical optimum (very small d), fabrication constraints (requiring slightly larger d) , and potential surface effects at the interfaces of the EO active dielectric with the metal electrode and the silicon waveguide.^143,151,152

FIG. 10.

View large Download slide

Performance parameter of a hybrid geometry modulator shown for changing slot dimensions d. Solid lines: 340 nm-thick silicon; dashed lines: 220 nm silicon. (a) Change of the field confinement Γ. The thinner silicon layer consistently provides better confinement. However, this comes at the price of higher losses, resulting in a comparable V_πLα for both silicon thickness. (a) For thicker silicon (solid lines), the field interaction Γ has an optimum when the dielectric is thick enough to confine the energy strongly, and thin enough so that the energy is not yet transported in the silicon waveguide. (b) The voltage–length product increases steadily, as the strength of the external modulating field E_ext = V_extd⁻¹ drops. (c) Losses decrease for larger d. (d) The voltage–length–loss product is best for small d (low V_πL, but large α), or very large d (large V_πL, but low α).

When designing hybrid structures for practical use, additional points require consideration: First, ground electrodes need to be added laterally to the simplified geometry of Figs. 2(g) and 9(a), resulting in a structure as depicted in Fig. 9(b). This requires the conductive dielectric to be wider than the plasmonic metal strip, which comes at the expense of modal confinement. The confinement loss can be mitigated by introducing a partially etched or sub-wavelength-grated section between the waveguide core region and the lateral contacts [see Figs. 9(c) and 2(i), and particularly Fig. 11].

FIG. 11.

View large Download slide

Detailed schematic of a hybrid modulator with lateral contacts and a partial etch between ground contacts and hybrid photonic-plasmonic section. The partial etch contributes to the lateral confinement of the plasmonic mode and hence to a high confinement factor Γ. The equivalent circuit diagram takes the contact resistance R_contact and channel resistance R_channel as well as modulator capacitance C_modulator into account and assumes the active material to be a perfect insulator.

The partial etch can also help to avoid optical absorption losses from contact dopings or silicidation, which could be employed to reduce the metal–silicon contact resistance. On the other hand, the partial etch leads to a higher device resistance and hence reduces its electro-optic bandwidth. Further relations, which are subject to design trade-offs, are detailed in Table III. They can be optimized iteratively to fulfill the system requirements, which usually are given by insertion losses, bandwidth, and V_π.

TABLE III.

Design trade-offs inherent to hybrid plasmonic modulators.

Increasing parameter	Acts positively on	Acts negatively on
Depth of partial etch h_PE, or filling factor of sub-wavelength-grated contact	Resistance, field interaction	Bandwidth V_πL
Doping concentration	Bandwidth, α, V_πLα	Resistance
Waveguide width w_wg	Capacitance	V_πL, α, bandwidth
Electrode separation w_PE	Resistance	α, bandwidth
Gap d	Bandwidth	Capacitance, field interaction, E_ext

Increasing parameter	Acts positively on	Acts negatively on
Depth of partial etch h_PE, or filling factor of sub-wavelength-grated contact	Resistance, field interaction	Bandwidth V_πL
Doping concentration	Bandwidth, α, V_πLα	Resistance
Waveguide width w_wg	Capacitance	V_πL, α, bandwidth
Electrode separation w_PE	Resistance	α, bandwidth
Gap d	Bandwidth	Capacitance, field interaction, E_ext

To help understand the design complexity and the intertwined performance trade-offs, we analyze an exemplary structure based on a recently published geometry.⁴⁴

The device is based on a 30-nm-thick BaTiO₃ film on 220 nm silicon on insulator.⁴⁴ In the waveguide area, the silicon is n-doped to a concentration of 1 · 10⁻¹⁸ cm⁻³. In this study, we consider an additional partial etch between the waveguide and the electrodes, which can increase Γ and V_πL. The schematic is shown in Fig. 11. The waveguide is 200 nm wide; the gap dimension is d = 30 nm, the partial etch is 100 nm deep, and the distance between signal and ground electrodes is 200 nm. To be consistent with the simulations above, we have assumed BaTiO₃’s refractive index to be n_mat = 2.3, with $r_{eff} n_{mat}^{3} = 1000 p m / V$ ⁠.

Figure 12 displays the impact of waveguide width, doping concentration, depth of the partial etch, and distance between ground electrodes on the EO bandwidth (details on its calculation in Sec. III C 1) and the modulation efficiency in terms of V_πLα (the influence of the gap dimension d was already elaborated in Fig. 10). Only n-type doping was considered here because p-type doping results in higher optical loss for the same achieved conductivity.¹⁵³

FIG. 12.

View large Download slide

Modulation efficiency V_πLα (dark blue, left y-axis) and EO bandwidth f_RC (bright blue, right y-axis) when changing (a) doping concentration, (b) partial etch depth, (c) waveguide width, and (d) electrode spacing. A doping concentration up to 10¹⁹ cm⁻³ increases the bandwidth without causing additional optical losses. In all other dimensions, an improvement of V_πLα to small values goes together with a degradation of the cut-off frequency.

1. Frequency response

The plasmonic nature of the device usually allows it to be sufficiently short to allow modeling as lumped elements. In Fig. 11, one can also see an equivalent circuit model, similar to typical models of plasmonic modulators.³⁹ The model consists of a capacitor C_mod for the active region, resistors R_channel for the conductive path through silicon, and contact resistors R_contact between the ground contacts and silicon.

The capacitance of the modulator is approximated with a plate capacitor model since the gap d is small in comparison to the waveguide width w_wg and the modulator length L,

C_{mod} = ϵ_{0} ϵ_{r} \frac{L \cdot w_{wg}}{d} .

(26)

ϵ₀ is the permittivity of vacuum, and ϵ_r the relative permittivity of the used EO material. Note that ϵ_r is typically frequency dependent.^12,154

If silicon is used as the transparent conductor, the channel resistance R_channel can be modeled with a drift model,

R_{channel} = ρ (C_{n}) \cdot (\frac{s}{(h_{Si} - h_{PE}) \cdot L} + \frac{w_{wg} / 4}{h \cdot L}),

(27)

where s is the separation between waveguide and lateral contacts, C_n the doping concentration, h_Si the thickness of the silicon, and h_PE the deepness of the partial etch. The first term models the silicon path between ground electrode and active section, while the second term adds a small correction for the distance that the charges travel within the active section. Bulucea report the doping-dependent silicon resistivity,¹⁵⁵ which is approximated for the case of phosphorous doping,

ρ (C_{n}) \approx 8 \cdot 1 0^{18} Ω m \cdot {(C_{n} \cdot {cm}^{3})}^{- 0.7} .

(28)

Modeling the contact resistance is a more complex problem that goes beyond the scope of this review.

The lumped element model results in an RC cut-off frequency,

f_{RC} = \frac{1}{2 π R C} = \frac{2}{π (R_{channel} + R_{contact}) C_{mod}} .

(29)

With good electrical contact to the transparent conductor, the contact resistance can be neglected because R_contact ≪ R_channel. The resulting cut-off frequency for the exemplary device modeled above then becomes

f_{RC} \approx \frac{1}{2 π} \cdot \frac{2 d}{ρ (C_{n}) \cdot (\frac{s}{(h - h_{PE})} + \frac{w}{4 h}) \cdot ϵ_{0} ϵ_{r} \cdot w} .

(30)

Large gap size d, small waveguide width w, short electrode separation s, and low silicon resistivity ρ(C_n) contribute to high cut-off frequencies. Finally, all of these degrade the voltage–length–loss product V_πLα, measuring the modulation efficiency.

2. Advantages, challenges and recent advances

Hybrid plasmonic waveguide modulators can surpass photonic modulators in bandwidth (see Fig. 12) while maintaining low insertion loss compared to pure plasmonic modulators (compare Figs. 8 and 10). Using a vertical stack to define the active region brings larger fabrication tolerances³⁷ and allows integration of epitaxial materials.⁴⁴ Further investigation is needed to optimize the combination of conductive dielectric, nonlinear material and their electrical contact simultaneously^{140,143,151,156} and interface effects need to be considered, as bulk behavior is usually dissimilar to materials in contact with the metal.^143,151

Recent exciting experiments demonstrate the potential of hybrid photonic–plasmonic devices. Phase modulation in a BaTiO₃-based hybrid modulator has recently been demonstrated,⁴⁴ as well as an organic EO material-based device relying on graphene instead of silicon for the electric path.^93,157 A recent implementation of a plasmonically enhanced graphene organic hybrid electro-optic phase modulator of 10 µm length with low plasmonic losses of 2.5 dB has shown a bandwidth of 270 GHz. It has already been verified for high-speed on-off-keying data modulation at a line rate of 140 Gbit/s. More complex hybrid photonic–plasmonic layer stacks relying on ultra-thin metal layers promise even lower optical losses.⁴⁷

IV. COMPARISON OF PHOTONIC, PLASMONIC, AND HYBRID GEOMETRIES

A qualitative comparison of photonic, plasmonic and hybrid plasmonic–photonic waveguide geometries in their key performance metrics is illustrated in Fig. 13. Photonic modulators are unmatched in their low losses, but lose modal confinement and strength of the modulating, external electric field. This results in an increased voltage–length product that requires long devices. These, at the same time, require travelling wave electrodes, which typically limit the available electro-optic bandwidth to <100 GHz.^{16,27,29,108,109,158} Recent advances on the structuring of the electrodes promise to bring that number up to 250 GHz for a 2 cm-long device.³¹

FIG. 13.

View large Download slide

Illustrative comparison between photonic, plasmonic, and photonic–plasmonic hybrid modulators based on experimental key performance metrics. A larger radius stands for better performance. BW: bandwidth; V_πL: voltage–length product; loss: device loss; α: loss per unit length; and Γ: field confinement.

Plasmonic electro-optic modulators span the other extreme: Their voltage–length product is extremely small, and so is their size. This allows them to be driven as a lumped element, with electro-optic bandwidths exceeding 500 GHz and demonstrated operation up to 2.4 THz. The plasmonic mode confinement that enables these devices comes at the cost of fundamental propagation losses, currently at ∼0.5 dB/μm.³⁵ However, as devices benefit from stronger electro-optic materials,^67,159 they can become shorter and it is expected that fiber-to-fiber losses can drop: Once fiber-to-plasmonic coupling challenges are overcome, we will likely see losses of plasmonic devices below 5 dB. Most recently, plasmonic ring modulators featuring on-chip-losses with as little as 1.2 dB with a bandwidth in excess of 176 GHz have been demonstrated.¹⁴⁷

Hybrid photonic–plasmonic modulator geometries represent a less explored field of ongoing research and their basic idea to give up some of the plasmonic confinement to reduce the device loss must still be proven in experiment. If successful, these hybrid devices could boast relatively low propagation losses together with a low voltage–length product and fill in the gap between plasmonic and photonic devices. Latest results with plasmonically enhanced graphene–organic hybrid phase modulator show that plasmonic hybrid devices with losses as low as 2.5 dB, bandwidths in excess of 270 GHz, and a length of 10 µm can be realized.⁹³

Table IV lists numerical results for the geometries discussed in this work. For the sake of a fair comparison, all geometries are simulated with a material with n_mat = 2.0 and r_eff = 125 pm/V, so that again $n_{mat}^{3} r_{eff} = 1000 p m / V$ ⁠. The results corroborate the qualitative findings.

TABLE IV.

Numerical comparison between photonic, plasmonic, and photonic–plasmonic hybrid waveguiding concepts. Simulations are carried out with n_mat = 2 and $n_{mat}^{3} r_{33} = 1000 p m / V$ (r₃₃ = 125 pm/V). Dimensions for SOH: 150 nm slot width d, 220 nm Si thickness, 70 nm slab thickness, electrode distance. LNOI: 5 µm electrode distance. Plasmonic: 100 nm gap d, 150 nm gold thickness, in the vertical configuration waveguide width w_wg: 1000 nm. Hybrid: 30 nm gap, 340 nm Si thickness, and 500 nm width. The EO bandwidth is estimated from literature or simulation result.

Geometry	V_πL	V_πLα (V dB)	α	Γ	Bandwidth
Photonic (SOH)	0.07 V cm	0.014	0.21 dB/cm	0.23	<100 GHz^16,108,109
Photonic (LNOI)	1.37 V cm	0.3	0.2 dB/cm	0.57	50–100 GHz²⁹
Photonic (LNOI)	1.37 V cm	0.3	0.2 dB/cm	0.57	250 GHz (sim.)³¹
Plasmonic (POH), horizontal	170 V μm	85	0.49 dB/μm	0.9	>500 GHz⁶²
Plasmonic (POH), horizontal	170 V μm	85	0.49 dB/μm	0.9	>1 THz⁶³
Plasmonic (POH), vertical	127 V μm	55	0.43 dB/μm	1.2	>70 GHz³⁷
Plasmonic (POH), vertical	127 V μm	55	0.43 dB/μm	1.2	300 GHz (sim.)a
Hybrid	87 V μm	25	0.13 dB/μm	1.0	∼50–100 GHzb^,⁹³

Geometry	V_πL	V_πLα (V dB)	α	Γ	Bandwidth
Photonic (SOH)	0.07 V cm	0.014	0.21 dB/cm	0.23	<100 GHz^16,108,109
Photonic (LNOI)	1.37 V cm	0.3	0.2 dB/cm	0.57	50–100 GHz²⁹
Photonic (LNOI)	1.37 V cm	0.3	0.2 dB/cm	0.57	250 GHz (sim.)³¹
Plasmonic (POH), horizontal	170 V μm	85	0.49 dB/μm	0.9	>500 GHz⁶²
Plasmonic (POH), horizontal	170 V μm	85	0.49 dB/μm	0.9	>1 THz⁶³
Plasmonic (POH), vertical	127 V μm	55	0.43 dB/μm	1.2	>70 GHz³⁷
Plasmonic (POH), vertical	127 V μm	55	0.43 dB/μm	1.2	300 GHz (sim.)a
Hybrid	87 V μm	25	0.13 dB/μm	1.0	∼50–100 GHzb^,⁹³

^a

When taking RF lines into consideration, based on simulations.

^b

Based on equivalent circuit calculations, similar as presented in Ref. 80.

The experimental situation is more intricate, with results listed in Table V. Whereas the experimental performance of LNOI devices is close to the expectations from simulation, deviations exist for SOH devices, where the losses in experiment exceed those expected from simulation. Part of this discrepancy might be explained by the importance of fabrication process and its effect on the device performance.

TABLE V.

Comparison between photonic, plasmonic, and photonic–plasmonic hybrid modulators. Values have been experimentally extracted from Mach–Zehnder modulators in push–pull configuration, if not otherwise labeled.

Geometry	V_πL	V_πLα (V dB)	L	α	Bandwidth
Photonic (SOH)	0.4 V mm¹⁶	1.0¹⁶	280 µm¹⁶	2.5 dB/cm¹⁶	∼40 GHz¹⁶
Photonic (SOH)	0.4 V mm¹⁶	1.0¹⁶	280 µm¹⁶	2.5 dB/cm¹⁶	∼100 GHz¹⁰⁹
Photonic (LNOI)	∼2 V cm³¹	<1.5	∼2 cm	<0.75 dB/cm³¹	100 GHz²⁹
Photonic (LNOI)	∼2 V cm³¹	<1.5	∼2 cm	<0.75 dB/cm³¹	250 GHza^,³¹
Photonic (BaTiO₃)	0.23 V cmb	1.3	2 mm	5.8 dB/cm	∼2 GHz (MZM)c
Photonic (BaTiO₃)	0.23 V cmb	1.3	2 mm	5.8 dB/cm	∼20 GHz (racetrack)d
Plasmonic (POH), horizontal	130 V μm	75	15 µm	0.6 dB/μm	>500 GHz⁶²
Plasmonic (POH), horizontal	130 V μm	75	15 µm	0.6 dB/μm	>THz⁶³
Plasmonic (POH), verticale	100 V μm (1550 nm), 30 V μm (1260 nm)	50	11 µm	0.5 dB/μm	>70 GHz
Plasmonic (POH), verticale	100 V μm (1550 nm), 30 V μm (1260 nm)	15	11 µm	0.5 dB/μm	∼300 GHzf
Hybrid	350 V μm	87	10 µm	0.25 dB/μm	∼50–100 GHzg^,⁹³
Hybrid	350 V μm	87	10 µm	0.25 dB/μm	>270 GHzh^,⁹³

Geometry	V_πL	V_πLα (V dB)	L	α	Bandwidth
Photonic (SOH)	0.4 V mm¹⁶	1.0¹⁶	280 µm¹⁶	2.5 dB/cm¹⁶	∼40 GHz¹⁶
Photonic (SOH)	0.4 V mm¹⁶	1.0¹⁶	280 µm¹⁶	2.5 dB/cm¹⁶	∼100 GHz¹⁰⁹
Photonic (LNOI)	∼2 V cm³¹	<1.5	∼2 cm	<0.75 dB/cm³¹	100 GHz²⁹
Photonic (LNOI)	∼2 V cm³¹	<1.5	∼2 cm	<0.75 dB/cm³¹	250 GHza^,³¹
Photonic (BaTiO₃)	0.23 V cmb	1.3	2 mm	5.8 dB/cm	∼2 GHz (MZM)c
Photonic (BaTiO₃)	0.23 V cmb	1.3	2 mm	5.8 dB/cm	∼20 GHz (racetrack)d
Plasmonic (POH), horizontal	130 V μm	75	15 µm	0.6 dB/μm	>500 GHz⁶²
Plasmonic (POH), horizontal	130 V μm	75	15 µm	0.6 dB/μm	>THz⁶³
Plasmonic (POH), verticale	100 V μm (1550 nm), 30 V μm (1260 nm)	50	11 µm	0.5 dB/μm	>70 GHz
Plasmonic (POH), verticale	100 V μm (1550 nm), 30 V μm (1260 nm)	15	11 µm	0.5 dB/μm	∼300 GHzf
Hybrid	350 V μm	87	10 µm	0.25 dB/μm	∼50–100 GHzg^,⁹³
Hybrid	350 V μm	87	10 µm	0.25 dB/μm	>270 GHzh^,⁹³

^a

With segmented electrode on quartz substrate, simulation only.

^b

DC value, might overestimate the actual performance.^12,40,60,117

^c

Limited by velocity mismatch between RF and optical wave.

^d

Limited by photon lifetime in the resonator.

^e

Values for V_πL from Ref. 37 have been measured for phase modulators, and are divided by a factor of 2 in this table, to match the measurements reported from Mach–Zehnder modulators in push–pull configuration.

^f

When taking RF lines into consideration, based on simulations.

^g

Based on a Si-OEO-Au stack and equivalent circuit calculations, similar as presented in Ref. 80.

^h

Based on a Si-graphene-OEO-Au stack.

To conclude, integrated electro-optic modulators relying on the Pockels have seen great advances in technological readiness in recent years and can overcome the bandwidth limitations and nonlinearities of silicon–photonic plasma-dispersion modulators. Photonic, plasmonic, and photonic–plasmonic hybrid waveguide structures can all be employed to form modulators, each technology with its own drawbacks and benefits: Photonic silicon–organic hybrid thin-film lithium niobate modulators promise to be low loss and fast, but have a considerable footprint. They might be well-suited for classical optical communication tasks.^28,160 Plasmonic metal–insulator–metal waveguides are extremely compact and exhibit unmatched electro-optical bandwidths but need clever engineering to reduce losses. Their unique properties may make them a good fit for microwave photonics,^62,146,161 THz spectroscopy,^144,162 or close integration with electronics for (short-haul) optical communications.^43,147,163 Photonic–plasmonic hybrid structures allow to address a middle ground between the photonic and plasmonic extremes, but they require more research attention to find their ideal application.

The presented technologies and waveguide structures cover an extremely wide field of possibilities, and researchers as well as engineers will have great flexibility for choosing the best fit for the application at their hands.

SUPPLEMENTARY MATERIAL

The supplementary material comprises a derivation of the anisotropic refractive index change Δnij induced by the Pockels effect, a derivation of the field interaction factor Γ, and an explanation of the method used for the numerical studies of Γ, α, VπL, and VπL · α.

ACKNOWLEDGMENTS

The authors gratefully acknowledge partial support by ETH, aCryComm FETOPEN (Grant No. 899558), NEBULA EU-ICT (Grant No. 871658), PlasmoniAC EU-ICT (Grant No. 871391), plaCMOS EU-ICT (Project No. 980997), PLASILOR ERC (Grant No. 670478), FLEX-SCALE EU JU-SNS (Grant No. 101096909), and ALLEGRO EU CL4 (Grant No. 101092766). ETH would like to acknowledge support by the Binning Rohrer Nano Center (BRNC) cleanroom operated by IBM Rüschlikon and ETH Zurich, Switzerland.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Andreas Messner: Conceptualization (equal); Data curation (lead); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). David Moor: Conceptualization (supporting); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal). Daniel Chelladurai: Conceptualization (supporting); Investigation (equal); Methodology (equal); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal). Roman Svoboda: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Writing – original draft (supporting). Jasmin Smajic: Funding acquisition (supporting); Project administration (supporting); Supervision (supporting); Writing – review & editing (supporting). Juerg Leuthold: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Investigation (equal); Methodology (equal); Supervision (lead); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

REFERENCES

1.

N. H.

Zhu

,

Z.

Shi

,

Z. K.

Zhang

,

Y. M.

Zhang

,

C. W.

Zou

,

Z. P.

Zhao

,

Y.

Liu

,

W.

Li

, and

M.

Li

,

IEEE J. Sel. Top. Quantum Electron.

24

,

1

(

2018

).

https://doi.org/10.1109/jstqe.2017.2720959

Crossref

2.

S.

Yamaoka

,

N.-P.

Diamantopoulos

,

H.

Nishi

,

R.

Nakao

,

T.

Fujii

,

K.

Takeda

,

T.

Hiraki

,

T.

Tsurugaya

,

S.

Kanazawa

,

H.

Tanobe

,

T.

Kakitsuka

,

T.

Tsuchizawa

,

F.

Koyama

, and

S.

Matsuo

,

Nat. Photonics

15

,

28

(

2021

).

https://doi.org/10.1038/s41566-020-00700-y

Crossref

3.

N.-P.

Diamantopoulos

,

H.

Yamazaki

,

S.

Yamaoka

,

M.

Nagatani

,

H.

Nishi

,

H.

Tanobe

,

R.

Nakao

,

T.

Fujii

,

K.

Takeda

,

T.

Kakitsuka

,

H.

Wakita

,

M.

Ida

,

H.

Nosaka

,

F.

Koyama

,

Y.

Miyamoto

, and

S.

Matsuo

,

J. Lightwave Technol.

39

,

771

(

2021

).

https://doi.org/10.1109/jlt.2020.3021727

Crossref

4.

G.

Sinatkas

,

T.

Christopoulos

,

O.

Tsilipakos

, and

E. E.

Kriezis

,

J. Appl. Phys.

130

,

010901

(

2021

).

https://doi.org/10.1063/5.0048712

Crossref

5.

G. L.

Li

and

P. K. L.

Yu

,

J. Lightwave Technol.

21

,

2010

(

2003

).

https://doi.org/10.1109/jlt.2003.815654

Crossref

6.

R. A.

Soref

and

J. P.

Lorenzo

,

Electron. Lett.

21

,

953

(

1985

).

https://doi.org/10.1049/el:19850673

Crossref

7.

M.

Chacinski

,

U.

Westergren

,

B.

Stoltz

,

L.

Thylen

,

R.

Schatz

, and

S.

Hammerfeldt

,

J. Lightwave Technol.

27

,

3410

(

2009

).

https://doi.org/10.1109/jlt.2009.2015773

Crossref

8.

J.

Fujikata

,

M.

Noguchi

,

K.

Kawashita

,

R.

Katamawari

,

S.

Takahashi

,

M.

Nishimura

,

H.

Ono

,

D.

Shimura

,

H.

Takahashi

,

H.

Yaegashi

,

T.

Nakamura

, and

Y.

Ishikawa

,

Opt. Express

28

,

33123

(

2020

).

https://doi.org/10.1364/oe.405447

Crossref

PubMed

9.

S. H.

Wemple

,

M.

Didomenico

, and

I.

Camlibel

,

J. Phys. Chem. Solids

29

,

1797

(

1968

).

https://doi.org/10.1016/0022-3697(68)90164-9

Crossref

10.

L. R.

Dalton

,

P.

Günter

,

M.

Jazbinsek

,

O.-P.

Kwon

, and

P. A.

Sullivan

,

Organic Electro-Optics and Photonics: Molecules, Polymers, and Crystals

(

Cambridge University Press

,

2015

).

Google Scholar

Crossref

11.

W.

Steier

,

IEEE J. Quantum Electron.

3

,

664

(

1967

).

https://doi.org/10.1109/jqe.1967.1074434

Crossref

12.

S. H.

Wemple

and

M.

DiDomenico

,

Appl. Solid State Sci.

3

,

263

(

1972

).

https://doi.org/10.1016/B978-0-12-002903-7.50011-4

Crossref

13.

S.

Walklin

and

J.

Conradi

,

IEEE Photonics Technol. Lett.

9

,

1400

(

1997

).

https://doi.org/10.1109/68.623276

Crossref

14.

J.

Leuthold

,

C.

Koos

,

W.

Freude

,

L.

Alloatti

,

R.

Palmer

,

D.

Korn

,

J.

Pfeifle

,

M.

Lauermann

,

R.

Dinu

,

S.

Wehrli

,

M.

Jazbinsek

,

P.

Günter

,

M.

Waldow

,

T.

Wahlbrink

,

J.

Bolten

,

H.

Kurz

,

M.

Fournier

,

J. M.

Fedeli

,

H.

Yu

, and

W.

Bogaerts

,

IEEE J. Sel. Top. Quantum Electron.

19

,

114

(

2013

).

https://doi.org/10.1109/jstqe.2013.2271846

Crossref

15.

W.

Heni

,

Y.

Kutuvantavida

,

C.

Haffner

,

H.

Zwickel

,

C.

Kieninger

,

S.

Wolf

,

M.

Lauermann

,

Y. M.

Fedoryshyn

,

A.

Tillack

,

L. E.

Johnson

,

D. L.

Elder

,

B. H.

Robinson

,

W.

Freude

,

C.

Koos

,

J.

Leuthold

, and

L. R.

Dalton

,

ACS Photonics

4

,

1576

(

2017

).

https://doi.org/10.1021/acsphotonics.7b00224

Crossref

16.

C.

Kieninger

,

C.

Füllner

,

H.

Zwickel

,

Y.

Kutuvantavida

,

J. N.

Kemal

,

C.

Eschenbaum

,

D. L.

Elder

,

L. R.

Dalton

,

W.

Freude

,

S.

Randel

, and

C.

Koos

,

Opt. Express

28

,

24693

(

2020

).

https://doi.org/10.1364/oe.390315

Crossref

PubMed

17.

P. O.

Weigel

,

J.

Zhao

,

K.

Fang

,

H.

Al-Rubaye

,

D.

Trotter

,

D.

Hood

,

J.

Mudrick

,

C.

Dallo

,

A. T.

Pomerene

,

A. L.

Starbuck

,

C. T.

DeRose

,

A. L.

Lentine

,

G.

Rebeiz

, and

S.

Mookherjea

,

Opt. Express

26

,

23728

(

2018

).

https://doi.org/10.1364/oe.26.023728

Crossref

PubMed

18.

X.

Wang

,

P. O.

Weigel

,

J.

Zhao

,

M.

Ruesing

, and

S.

Mookherjea

,

APL Photonics

4

,

096101

(

2019

).

https://doi.org/10.1063/1.5115243

Crossref

19.

N.

Boynton

,

H.

Cai

,

M.

Gehl

,

S.

Arterburn

,

C.

Dallo

,

A.

Pomerene

,

A.

Starbuck

,

D.

Hood

,

D. C.

Trotter

,

T.

Friedmann

,

C. T.

DeRose

, and

A.

Lentine

,

Opt. Express

28

,

1868

(

2020

).

https://doi.org/10.1364/oe.28.001868

Crossref

PubMed

20.

W. H. P.

Pernice

,

C.

Xiong

,

F. J.

Walker

, and

H. X.

Tang

,

IEEE Photonics Technol. Lett.

26

,

1344

(

2014

).

https://doi.org/10.1109/lpt.2014.2322501

Crossref

21.

C.

Xiong

,

W. H. P.

Pernice

,

J. H.

Ngai

,

J. W.

Reiner

,

D.

Kumah

,

F. J.

Walker

,

C. H.

Ahn

, and

H. X.

Tang

,

Nano Lett.

14

,

1419

(

2014

).

https://doi.org/10.1021/nl404513p

Crossref

PubMed

22.

F.

Eltes

,

D.

Caimi

,

F.

Fallegger

,

M.

Sousa

,

E.

O’Connor

,

M. D.

Rossell

,

B.

Offrein

,

J.

Fompeyrine

, and

S.

Abel

,

ACS Photonics

3

,

1698

(

2016

).

https://doi.org/10.1021/acsphotonics.6b00350

Crossref

23.

F.

Eltes

,

M.

Kroh

,

D.

Caimi

,

C.

Mai

,

Y.

Popoff

,

G.

Winzer

,

D.

Petousi

,

S.

Lischke

,

J. E.

Ortmann

,

L.

Czornomaz

,

L.

Zimmermann

,

J.

Fompeyrine

, and

S.

Abel

, paper presented at the

2017 IEEE International Electron Devices Meeting (IEDM)

,

2017

.

24.

S.

Abel

,

F.

Eltes

,

J. E.

Ortmann

,

A.

Messner

,

P.

Castera

,

T.

Wagner

,

D.

Urbonas

,

A.

Rosa

,

A. M.

Gutierrez

,

D.

Tulli

,

P.

Ma

,

B.

Baeuerle

,

A.

Josten

,

W.

Heni

,

D.

Caimi

,

L.

Czornomaz

,

A. A.

Demkov

,

J.

Leuthold

,

P.

Sanchis

, and

J.

Fompeyrine

,

Nat. Mater.

18

,

42

(

2018

).

https://doi.org/10.1038/s41563-018-0208-0

Crossref

PubMed

25.

R. C.

Alferness

,

IEEE Trans. Microwave Theory Tech.

30

,

1121

(

1982

).

https://doi.org/10.1109/tmtt.1982.1131213

Crossref

26.

H.

Hu

,

L.

Gui

,

R.

Ricken

, and

W.

Sohler

,

Proc. SPIE

7604

,

76040R

(

2010

).

https://doi.org/10.1117/12.842674

Crossref

27.

A. J.

Mercante

,

S.

Shi

,

P.

Yao

,

L.

Xie

,

R. M.

Weikle

, and

D. W.

Prather

,

Opt. Express

26

,

14810

(

2018

).

https://doi.org/10.1364/oe.26.014810

Crossref

PubMed

28.

M.

Zhang

,

C.

Wang

,

X.

Chen

,

M.

Bertrand

,

A.

Shams-Ansari

,

S.

Chandrasekhar

,

P.

Winzer

, and

M.

Lončar

, paper presented at the

Optical Fiber Communication Conference Postdeadline Papers

,

San Diego, CA

,

2018

.

29.

C.

Wang

,

M.

Zhang

,

X.

Chen

,

M.

Bertrand

,

A.

Shams-Ansari

,

S.

Chandrasekhar

,

P.

Winzer

, and

M.

Lončar

,

Nature

562

,

101

(

2018

).

https://doi.org/10.1038/s41586-018-0551-y

Crossref

PubMed

30.

M.

Bahadori

,

Y.

Yang

,

L. L.

Goddard

, and

S.

Gong

,

Opt. Express

27

,

22025

(

2019

).

https://doi.org/10.1364/oe.27.022025

Crossref

PubMed

31.

P.

Kharel

,

C.

Reimer

,

K.

Luke

,

L.

He

, and

M.

Zhang

,

Optica

8

,

357

(

2021

).

https://doi.org/10.1364/optica.416155

Crossref

32.

D.

Pohl

,

A.

Messner

,

F.

Kaufmann

,

M. R.

Escalé

,

J.

Holzer

,

J.

Leuthold

, and

R.

Grange

,

IEEE Photonics Technol. Lett.

33

,

85

(

2021

).

https://doi.org/10.1109/lpt.2020.3044648

Crossref

33.

Y.

Jia

,

L.

Wang

, and

F.

Chen

,

Appl. Phys. Rev.

8

,

011307

(

2021

).

https://doi.org/10.1063/5.0037771

Crossref

34.

M.

Ayata

,

Y.

Fedoryshyn

,

W.

Heni

,

B.

Baeuerle

,

A.

Josten

,

M.

Zahner

,

U.

Koch

,

Y.

Salamin

,

C.

Hoessbacher

,

C.

Haffner

,

D. L.

Elder

,

L. R.

Dalton

, and

J.

Leuthold

,

Science

358

,

630

(

2017

).

https://doi.org/10.1126/science.aan5953

Crossref

PubMed

35.

U.

Koch

,

A.

Messner

,

C.

Hoessbacher

,

W.

Heni

,

A.

Josten

,

B.

Baeuerle

,

M.

Ayata

,

Y.

Fedoryshyn

,

D. L.

Elder

,

L. R.

Dalton

, and

J.

Leuthold

,

J. Lightwave Technol.

37

,

1484

(

2019

).

https://doi.org/10.1109/jlt.2019.2899372

Crossref

36.

U.

Koch

,

C.

Uhl

,

H.

Hettrich

,

Y.

Fedoryshyn

,

C.

Hoessbacher

,

W.

Heni

,

B.

Baeuerle

,

B.

Bitachon

,

A.

Josten

,

M.

Ayata

,

H.

Xu

,

D. L.

Elder

,

L. R.

Dalton

,

E.

Mentovich

,

D.

Tsiokos

,

N.

Pleros

,

M.

Möller

, and

J.

Leuthold

, paper presented at the

2019 European Conference on Optical Communication (ECOC)

,

Dublin

,

2019

.

37.

A.

Messner

,

P. A.

Jud

,

J.

Winiger

,

M.

Eppenberger

,

D.

Chelladurai

,

W.

Heni

,

B.

Baeuerle

,

U.

Koch

,

P.

Ma

,

C.

Haffner

,

H.

Xu

,

D. L.

Elder

,

L. R.

Dalton

,

J.

Smajic

, and

J.

Leuthold

,

Nano Lett.

21

,

4539

(

2021

).

https://doi.org/10.1021/acs.nanolett.0c05069

Crossref

PubMed

38.

J.

Leuthold

,

C.

Hoessbacher

,

S.

Muehlbrandt

,

A.

Melikyan

,

M.

Kohl

,

C.

Koos

,

W.

Freude

,

V.

Dolores-Calzadilla

,

M.

Smit

,

I.

Suarez

,

J.

Martínez-Pastor

,

E. P.

Fitrakis

, and

I.

Tomkos

,

Opt. Photonics News

24

,

28

(

2013

).

https://doi.org/10.1364/OPN.24.5.000028

Crossref

39.

A.

Melikyan

,

L.

Alloatti

,

A.

Muslija

,

D.

Hillerkuss

,

P. C.

Schindler

,

J.

Li

,

R.

Palmer

,

D.

Korn

,

S.

Muehlbrandt

,

D.

Van Thourhout

,

B.

Chen

,

R.

Dinu

,

M.

Sommer

,

C.

Koos

,

M.

Kohl

,

W.

Freude

, and

J.

Leuthold

,

Nat. Photonics

8

,

229

(

2014

).

https://doi.org/10.1038/nphoton.2014.9

Crossref

40.

A.

Messner

,

F.

Eltes

,

P.

Ma

,

S.

Abel

,

B.

Baeuerle

,

A.

Josten

,

W.

Heni

,

D.

Caimi

,

J.

Fompeyrine

, and

J.

Leuthold

,

J. Lightwave Technol.

37

,

281

(

2019

).

https://doi.org/10.1109/jlt.2018.2881332

Crossref

41.

M.

Thomaschewski

,

V. A.

Zenin

,

C.

Wolff

, and

S. I.

Bozhevolnyi

,

Nat. Commun.

11

,

748

(

2020

).

https://doi.org/10.1038/s41467-020-14539-y

Crossref

PubMed

42.

M.

Ayata

,

Y.

Fedoryshyn

,

W.

Heni

,

A.

Josten

,

B.

Baeuerle

,

C.

Haffner

,

C.

Hoessbacher

,

U.

Koch

,

Y.

Salamin

,

D. L.

Elder

,

L. R.

Dalton

, and

J.

Leuthold

,

J. Lightwave Technol.

37

,

1492

(

2019

).

https://doi.org/10.1109/jlt.2019.2900797

Crossref

43.

U.

Koch

,

C.

Uhl

,

H.

Hettrich

,

Y.

Fedoryshyn

,

C.

Hoessbacher

,

W.

Heni

,

B.

Baeuerle

,

B. I.

Bitachon

,

A.

Josten

,

M.

Ayata

,

H.

Xu

,

D. L.

Elder

,

L. R.

Dalton

,

E.

Mentovich

,

P.

Bakopoulos

,

S.

Lischke

,

A.

Krüger

,

L.

Zimmermann

,

D.

Tsiokos

,

N.

Pleros

,

M.

Möller

, and

J.

Leuthold

,

Nat. Electron.

3

,

338

(

2020

).

https://doi.org/10.1038/s41928-020-0417-9

Crossref

44.

D.

Moor

,

J.

Winiger

,

P.

Ma

,

A.

Messner

,

B.

Bitachon

,

S.

Abel

,

F.

Eltes

,

J.

Fompeyrine

, and

J.

Leuthold

, in

Frontiers in Optics 2021

,

2021

.

45.

D.

Moor

,

J.

Leuthold

,

Y.

Fedoryshyn

,

M.

Möller

, and

U.

Koch

,

IEEE J. Sel. Top. Quantum Electron.

(published online 2023).

https://doi.org/10.1109/JSTQE.2023.3298375

46.

M. Y.

Abdelatty

,

A. O.

Zaki

, and

M. A.

Swillam

,

Appl. Phys. A

123

,

11

(

2017

).

https://doi.org/10.1007/s00339-016-0656-9

Crossref

47.

C. C.-C.

Lin

,

P.-H.

Chang

,

Y.

Su

, and

A. S.

Helmy

,

Nano Lett.

20

,

2950

(

2020

).

https://doi.org/10.1021/acs.nanolett.9b04612

Crossref

PubMed

48.

Y.

Kim

,

J.-H.

Han

,

D.

Ahn

, and

S.

Kim

,

Micromachines

12

,

625

(

2021

).

https://doi.org/10.3390/mi12060625

Crossref

PubMed

49.

A.

Rahim

,

A.

Hermans

,

B.

Wohlfeil

,

D.

Petousi

,

B.

Kuyken

,

D.

Van Thourhout

, and

R.

Baets

,

Adv. Photonics

3

,

024003

(

2021

).

https://doi.org/10.1117/1.AP.3.2.024003

Crossref

50.

A.

Tuniz

,

Riv.Nuovo Cimento

44

,

193

(

2021

).

https://doi.org/10.1007/s40766-021-00018-7

Crossref

51.

A.

Rao

and

S.

Fathpour

,

Phys. Status Solidi A

215

,

1700684

(

2018

).

https://doi.org/10.1002/pssa.201700684

Crossref

52.

G.

Li

,

C. M.

de Sterke

, and

S.

Palomba

,

Laser Photonics Rev.

10

,

639

(

2016

).

https://doi.org/10.1002/lpor.201600020

Crossref

53.

G.

Reed

,

G.

Mashanovich

,

F.

Gardes

, and

D.

Thomson

, “

Silicon optical modulators

,”

Nat. Photonics

4

,

518

(

2010

).

https://doi.org/10.1038/nphoton.2010.179

Google Scholar

Crossref

54.

K.

Liu

,

C. R.

Ye

,

S.

Khan

, and

V. J.

Sorger

,

Laser Photonics Rev.

9

,

172

(

2015

).

https://doi.org/10.1002/lpor.201400219

Crossref

55.

P. J.

Winzer

and

D. T.

Neilson

,

J. Lightwave Technol.

35

,

1099

(

2017

).

https://doi.org/10.1109/jlt.2017.2662082

Crossref

56.

D. A. B.

Miller

,

J. Lightwave Technol.

35

,

346

(

2017

).

https://doi.org/10.1109/jlt.2017.2647779

Crossref

57.

J.

Witzens

,

Proc. IEEE

106

,

2158

(

2018

).

https://doi.org/10.1109/jproc.2018.2877636

Crossref

58.

J.-M.

Brosi

,

C.

Koos

,

L. C.

Andreani

,

M.

Waldow

,

J.

Leuthold

, and

W.

Freude

,

Opt. Express

16

,

4177

(

2008

).

https://doi.org/10.1364/oe.16.004177

Crossref

PubMed

59.

B.

Baeuerle

,

W.

Heni

,

C.

Hoessbacher

,

Y.

Fedoryshyn

,

U.

Koch

,

A.

Josten

,

T.

Watanabe

,

C.

Uhl

,

H.

Hettrich

,

D. L.

Elder

,

L. R.

Dalton

,

M.

Möller

, and

J.

Leuthold

,

Opt. Express

27

,

16823

(

2019

).

https://doi.org/10.1364/oe.27.016823

Crossref

PubMed

60.

K.

Laabidi

,

M. D.

Fontana

,

M.

Maglione

,

B.

Jannot

, and

K. A.

Müller

,

Europhys. Lett.

26

,

309

(

1994

).

https://doi.org/10.1209/0295-5075/26/4/012

Crossref

61.

C.

Cochard

,

T.

Spielmann

,

N.

Bahlawane

,

A.

Halpin

, and

T.

Granzow

,

J. Phys. D: Appl. Phys.

50

,

36LT01

(

2017

).

https://doi.org/10.1088/1361-6463/aa80b8

Crossref

62.

M.

Burla

,

C.

Hoessbacher

,

W.

Heni

,

C.

Haffner

,

Y.

Fedoryshyn

,

D.

Werner

,

T.

Watanabe

,

H.

Massler

,

D. L.

Elder

,

L. R.

Dalton

, and

J.

Leuthold

,

APL Photonics

4

,

056106

(

2019

).

https://doi.org/10.1063/1.5086868

Crossref

63.

I.-C.

Benea-Chelmus

,

T.

Zhu

,

F. F.

Settembrini

,

C.

Bonzon

,

E.

Mavrona

,

D. L.

Elder

,

W.

Heni

,

J.

Leuthold

,

L. R.

Dalton

, and

J.

Faist

,

ACS Photonics

5

,

1398

(

2018

).

https://doi.org/10.1021/acsphotonics.7b01407

Crossref

64.

C.

Kieninger

,

Y.

Kutuvantavida

,

H.

Miura

,

J. N.

Kemal

,

H.

Zwickel

,

F.

Qiu

,

M.

Lauermann

,

W.

Freude

,

S.

Randel

,

S.

Yokoyama

, and

C.

Koos

,

Opt. Express

26

,

27955

(

2018

).

https://doi.org/10.1364/oe.26.027955

Crossref

PubMed

65.

G.-W.

Lu

,

J.

Hong

,

F.

Qiu

,

A. M.

Spring

,

T.

Kashino

,

J.

Oshima

,

M.-A.

Ozawa

,

H.

Nawata

, and

S.

Yokoyama

,

Nat. Commun.

11

,

4224

(

2020

).

https://doi.org/10.1038/s41467-020-18005-7

Crossref

PubMed

66.

S. R.

Hammond

,

K. M.

O’Malley

,

H.

Xu

,

D. L.

Elder

, and

L. E.

Johnson

,

Proc. SPIE

11998

,

119980C

(

2022

).

https://doi.org/10.1117/12.2622099

Crossref

67.

H.

Xu

,

F.

Liu

,

D. L.

Elder

,

L. E.

Johnson

,

Y.

de Coene

,

K.

Clays

,

B. H.

Robinson

, and

L. R.

Dalton

,

Chem. Mater.

32

,

1408

(

2020

).

https://doi.org/10.1021/acs.chemmater.9b03725

Crossref

68.

C.

Eschenbaum

,

A.

Mertens

,

C.

Füllner

,

A.

Kuzmin

,

A.

Schwarzenberger

,

A.

Kotz

,

G.

Ramann

,

M.

Chen

,

J.

Drisko

,

B.

Johnson

,

J.

Zyskind

,

J.

Marcelli

,

M.

Lebby

,

W.

Freude

,

S.

Randel

, and

C.

Koos

, paper presented at the

2022 European Conference on Optical Communication (ECOC)

,

2022

.

69.

L. R.

Dalton

,

J.

Leuthold

,

B. H.

Robinson

,

C.

Haffner

,

D. L.

Elder

,

L. E.

Johnson

,

S. R.

Hammond

,

W.

Heni

,

C.

Hosessbacher

,

B.

Baeuerle

,

E.

De Leo

,

U.

Koch

,

P.

Habegger

,

Y.

Fedoryshyn

,

D.

Moor

, and

P.

Ma

,

APL Mater.

11

,

050901

(

2023

).

https://doi.org/10.1063/5.0145212

Crossref

70.

L.

Johnson

,

D.

Elder

,

H.

Xu

,

S.

Hammond

,

S.

Benight

,

K.

O’Malley

,

B.

Robinson

, and

L.

Dalton

,

Proc. SPIE

11812

,

1181202

(

2021

).

https://doi.org/10.1117/12.2595638

Crossref

71.

E. L.

Wooten

,

K. M.

Kissa

,

A.

Yi-Yan

,

E. J.

Murphy

,

D. A.

Lafaw

,

P. F.

Hallemeier

,

D.

Maack

,

D. V.

Attanasio

,

D. J.

Fritz

,

G. J.

McBrien

, and

D. E.

Bossi

,

IEEE J. Sel. Top. Quantum Electron.

6

,

69

(

2000

).

https://doi.org/10.1109/2944.826874

Crossref

72.

B. W.

Wessels

,

Annu. Rev. Mater. Res.

37

,

659

(

2007

).

https://doi.org/10.1146/annurev.matsci.37.052506.084226

Crossref

73.

A.

Boes

,

B.

Corcoran

,

L.

Chang

,

J.

Bowers

, and

A.

Mitchell

,

Laser Photonics Rev.

12

,

1700256

(

2018

).

https://doi.org/10.1002/lpor.201700256

Crossref

74.

M.

He

,

M.

Xu

,

Y.

Ren

,

J.

Jian

,

Z.

Ruan

,

Y.

Xu

,

S.

Gao

,

S.

Sun

,

X.

Wen

,

L.

Zhou

,

L.

Liu

,

C.

Guo

,

H.

Chen

,

S.

Yu

,

L.

Liu

, and

X.

Cai

,

Nat. Photonics

13

,

359

(

2019

).

https://doi.org/10.1038/s41566-019-0378-6

Crossref

75.

L.

Chang

,

M. H.

Pfeiffer

,

N.

Volet

,

M.

Zervas

,

J. D.

Peters

,

C. L.

Manganelli

,

E. J.

Stanton

,

Y.

Li

,

T. J.

Kippenberg

, and

J. E.

Bowers

,

Opt. Lett.

42

,

803

(

2017

).

https://doi.org/10.1364/ol.42.000803

Crossref

PubMed

76.

A.

Rao

and

S.

Fathpour

,

IEEE J. Sel. Top. Quantum Electron.

24

,

3400114

(

2018

).

https://doi.org/10.1109/JSTQE.2017.2779869

Crossref

77.

F.

Eltes

,

W.

Li

,

E.

Berikaa

,

M. S.

Alam

,

S.

Bernal

,

C.

Minkenberg

,

D. V.

Plant

, and

S.

Abel

, “

Thin-film BTO-based modulators enabling 200 Gb/s data rates with sub 1 Vpp drive signal

,” in

Optical Fiber Communication Conference (OFC) 2023, Technical Digest Series

(Optica Publishing Group, 2023), paper Th4A.2.

78.

L. R.

Dalton

,

P. A.

Sullivan

, and

D. H.

Bale

,

Chem. Rev.

110

,

25

(

2010

).

https://doi.org/10.1021/cr9000429

Crossref

PubMed

79.

P.

Castera

,

D.

Tulli

,

A. M.

Gutierrez

, and

P.

Sanchis

,

Opt. Express

23

,

15332

(

2015

).

https://doi.org/10.1364/oe.23.015332

Crossref

PubMed

80.

C.

Hoessbacher

,

Plasmonic Switches and Modulators for Optical Communications

(

ETH Zurich

,

2017

).

Google Scholar

81.

A.

Messner

,

C.

Haffner

,

W.

Heni

,

U.

Koch

, and

J.

Leuthold

, in

Advanced Photonics 2018 (BGPP, IPR, NP, NOMA, Sensors, Networks, SPPCom, SOF)

(

Optical Society of America

,

Zurich

,

2018

), p.

NoM2D.5

.

82.

R. C.

Alferness

,

Science

234

,

825

(

1986

).

https://doi.org/10.1126/science.234.4778.825

Crossref

PubMed

83.

S.

Ummethala

,

J. N.

Kemal

,

A. S.

Alam

,

M.

Lauermann

,

A.

Kuzmin

,

Y.

Kutuvantavida

,

S. H.

Nandam

,

L.

Hahn

,

D. L.

Elder

,

L. R.

Dalton

,

T.

Zwick

,

S.

Randel

,

W.

Freude

, and

C.

Koos

,

Optica

8

,

511

(

2021

).

https://doi.org/10.1364/optica.411161

Crossref

84.

W.

Heni

,

Plasmonic-Organic Hybrid Modulators

(

ETH Zurich

,

Zurich

,

2019

).

Google Scholar

85.

Y. V.

Pereverzev

,

K. N.

Gunnerson

,

O. V.

Prezhdo

,

P. A.

Sullivan

,

Y.

Liao

,

B. C.

Olbricht

,

A. J. P.

Akelaitis

,

A. K. Y.

Jen

, and

L. R.

Dalton

,

J. Phys. Chem. C

112

,

4355

(

2008

).

https://doi.org/10.1021/jp077194q

Crossref

86.

M.

Jazbinšek

and

M.

Zgonik

,

Appl. Phys. B: Lasers Opt.

74

,

407

(

2002

).

https://doi.org/10.1007/s003400200818

Crossref

87.

D. F. P.

Pile

,

T.

Ogawa

,

D. K.

Gramotnev

,

Y.

Matsuzaki

,

K. C.

Vernon

,

K.

Yamaguchi

,

T.

Okamoto

,

M.

Haraguchi

, and

M.

Fukui

,

Appl. Phys. Lett.

87

,

261114

(

2005

).

https://doi.org/10.1063/1.2149971

Crossref

88.

M.

Fujii

,

J.

Leuthold

, and

W.

Freude

,

IEEE Photonics Technol. Lett.

21

,

362

(

2009

).

https://doi.org/10.1109/lpt.2008.2011995

Crossref

89.

R. F.

Oulton

,

V. J.

Sorger

,

D. A.

Genov

,

D. F. P.

Pile

, and

X.

Zhang

,

Nat. Photonics

2

,

496

(

2008

).

https://doi.org/10.1038/nphoton.2008.131

Crossref

90.

F.

Lou

,

D.

Dai

,

L.

Thylen

, and

L.

Wosinski

,

Opt. Express

21

,

20041

(

2013

).

https://doi.org/10.1364/oe.21.020041

Crossref

PubMed

91.

C.

Lin

and

A. S.

Helmy

,

Sci. Rep.

5

,

12313

(

2015

).

https://doi.org/10.1038/srep12313

Crossref

PubMed

92.

S.

Ummethala

,

V. A.

Pammi

,

A. H. M.

Uddin

,

L.

Hahn

,

W.

Freude

, and

C.

Koos

, in

Conference on Lasers and Electro-Optics

(

Optical Society of America

,

2020

).

93.

P.

Ma

,

X. Z.

Zhang

,

B. I.

Bitachon

,

P.

Habegger

,

D.

Chelladurai

,

W.

Heni

,

A.

Messner

,

M.

Eppenberger

,

D.

Moor

,

D. L.

Elder

,

L. R.

Dalton

,

T.

Greber

, and

J.

Leuthold

, paper presented at the

European Conference on Optical Communication (ECOC) 2022

,

Basel

,

2022

.

94.

R. V.

Schmidt

and

I. P.

Kaminow

,

Appl. Phys. Lett.

25

,

458

(

1974

).

https://doi.org/10.1063/1.1655547

Crossref

95.

U.

Koren

,

T. L.

Koch

,

H.

Presting

, and

B. I.

Miller

,

Appl. Phys. Lett.

50

,

368

(

1987

).

https://doi.org/10.1063/1.98201

Crossref

96.

K.

Wakita

,

Y.

Kawamura

,

M.

Nakao

, and

H.

Asahi

,

IEEE J. Quantum Electron.

23

,

2210

(

1987

).

https://doi.org/10.1109/jqe.1987.1073291

Crossref

97.

K.

Wakita

,

S.

Nojima

,

K.

Nakashima

, and

Y.

Kawaguchi

,

Electron. Lett.

23

,

1067

(

1987

).

https://doi.org/10.1049/el:19870746

Crossref

98.

U.

Koren

,

B. I.

Miller

,

T. L.

Koch

,

G.

Eisenstein

,

R. S.

Tucker

,

I.

Bar-Joseph

, and

D. S.

Chemla

,

Appl. Phys. Lett.

51

,

1132

(

1987

).

https://doi.org/10.1063/1.98761

Crossref

99.

D. A. B.

Miller

,

D. S.

Chemla

,

T. C.

Damen

,

A. C.

Gossard

,

W.

Wiegmann

,

T. H.

Wood

, and

C. A.

Burrus

,

Phys. Rev. Lett.

53

,

2173

(

1984

).

https://doi.org/10.1103/physrevlett.53.2173

Crossref

100.

D. A. B.

Miller

,

D. S.

Chemla

,

T. C.

Damen

,

A. C.

Gossard

,

W.

Wiegmann

,

T. H.

Wood

, and

C. A.

Burrus

,

Phys. Rev. B

32

,

1043

(

1985

).

https://doi.org/10.1103/physrevb.32.1043

Crossref

101.

M.

Smit

,

X.

Leijtens

,

H.

Ambrosius

,

E.

Bente

,

J.

van der Tol

,

B.

Smalbrugge

,

T.

de Vries

,

E.-J.

Geluk

,

J.

Bolk

,

R.

van Veldhoven

,

L.

Augustin

,

P.

Thijs

,

D.

D’Agostino

,

H.

Rabbani

,

K.

Lawniczuk

,

S.

Stopinski

,

S.

Tahvili

,

A.

Corradi

,

E.

Kleijn

,

D.

Dzibrou

,

M.

Felicetti

,

E.

Bitincka

,

V.

Moskalenko

,

J.

Zhao

,

R.

Santos

,

G.

Gilardi

,

W.

Yao

,

K.

Williams

,

P.

Stabile

,

P.

Kuindersma

,

J.

Pello

,

S.

Bhat

,

Y.

Jiao

,

D.

Heiss

,

G.

Roelkens

,

M.

Wale

,

P.

Firth

,

F.

Soares

,

N.

Grote

,

M.

Schell

,

H.

Debregeas

,

M.

Achouche

,

J.-L.

Gentner

,

A.

Bakker

,

T.

Korthorst

,

D.

Gallagher

,

A.

Dabbs

,

A.

Melloni

,

F.

Morichetti

,

D.

Melati

,

A.

Wonfor

,

R.

Penty

,

R.

Broeke

,

B.

Musk

, and

D.

Robbins

,

Semicond. Sci. Technol.

29

,

083001

(

2014

).

https://doi.org/10.1088/0268-1242/29/8/083001

Crossref

102.

M.

Li

,

L.

Wang

,

X.

Li

,

X.

Xiao

, and

S.

Yu

,

Photonics Res.

6

,

109

(

2018

).

https://doi.org/10.1364/prj.6.000109

Crossref

103.

P.

Girouard

,

P.

Chen

,

Y. K.

Jeong

,

Z.

Liu

,

S.-T.

Ho

, and

B. W.

Wessels

,

IEEE J. Quantum Electron.

53

,

5200110

(

2017

).

https://doi.org/10.1109/jqe.2017.2718222

Crossref

104.

L.

He

,

M.

Zhang

,

A.

Shams-Ansari

,

R.

Zhu

,

C.

Wang

, and

L.

Marko

,

Opt. Lett.

44

,

2314

(

2019

).

https://doi.org/10.1364/ol.44.002314

Crossref

PubMed

105.

Y.

Ogiso

,

H.

Wakita

,

M.

Nagatani

,

H.

Yamazaki

,

M.

Nakamura

,

T.

Kobayashi

,

J.

Ozaki

,

Y.

Ueda

,

S.

Nakano

,

S.

Kanazawa

,

T.

Fujii

,

Y.

Hashizume

,

H.

Tanobe

,

N.

Nunoya

,

M.

Ida

,

Y.

Miyamoto

, and

N.

Kikuchi

, paper presented at the

Optical Fiber Communication Conference Postdeadline Papers

,

San Diego, CA

,

2018

.

106.

Y.

Ogiso

,

J.

Ozaki

,

Y.

Ueda

,

N.

Kashio

,

N.

Kikuchi

,

E.

Yamada

,

H.

Tanobe

,

S.

Kanazawa

,

H.

Yamazaki

,

Y.

Ohiso

,

T.

Fujii

, and

M.

Kohtoku

,

J. Lightwave Technol.

35

,

1450

(

2017

).

https://doi.org/10.1109/jlt.2016.2639542

Crossref

107.

C.

Kieninger

,

Y.

Kutuvantavida

,

D. L.

Elder

,

S.

Wolf

,

H.

Zwickel

,

M.

Blaicher

,

J. N.

Kemal

,

M.

Lauermann

,

S.

Randel

,

W.

Freude

,

L. R.

Dalton

, and

C.

Koos

,

Optica

5

,

739

(

2018

).

https://doi.org/10.1364/optica.5.000739

Crossref

108.

H.

Zwickel

,

S.

Singer

,

C.

Kieninger

,

Y.

Kutuvantavida

,

N.

Muradyan

,

T.

Wahlbrink

,

S.

Yokoyama

,

S.

Randel

,

W.

Freude

, and

C.

Koos

,

Opt. Express

28

,

12951

(

2020

).

https://doi.org/10.1364/oe.383120

Crossref

PubMed

109.

X.

Chen

,

G.

Raybon

,

D.

Che

,

J.

Cho

, and

K. W.

Kim

, “

Transmission of 200-GBaud PDM probabilistically shaped 64-QAM signals modulated via a 100-GHz Thin-film LiNbO3 I/Q modulator

,”in

Optical Fiber Communication Conference (OFC) 2021

, edited by

P.

Dong

,

J.

Kani

,

C.

Xie

,

R.

Casellas

,

C.

Cole

, and

M.

Li

,

OSA Technical Digest

(Optica Publishing Group, 2021), paper F3C.5.

Crossref

110.

L.

Alloatti

,

R.

Palmer

,

S.

Diebold

,

K. P.

Pahl

,

B.

Chen

,

R.

Dinu

,

M.

Fournier

,

J.-M.

Fedeli

,

T.

Zwick

,

W.

Freude

,

C.

Koos

, and

J.

Leuthold

,

Light: Sci. Appl.

3

,

e173

(

2014

).

https://doi.org/10.1038/lsa.2014.54

Crossref

111.

R.

Ding

,

Y.

Liu

,

Y.

Ma

,

Y.

Yang

,

Q.

Li

,

A. E.

Lim

,

G.

Lo

,

K.

Bergman

,

T.

Baehr-Jones

, and

M.

Hochberg

,

J. Lightwave Technol.

32

,

2240

(

2014

).

https://doi.org/10.1109/jlt.2014.2323954

Crossref

112.

D.

Patel

,

S.

Ghosh

,

M.

Chagnon

,

A.

Samani

,

V.

Veerasubramanian

,

M.

Osman

, and

D. V.

Plant

,

Opt. Express

23

,

14263

(

2015

).

https://doi.org/10.1364/oe.23.014263

Crossref

PubMed

113.

S.

Sun

,

M.

He

,

M.

Xu

,

X.

Zhang

,

Z.

Ruan

,

L.

Zhou

,

L.

Liu

,

L.

Liu

,

S.

Yu

, and

X.

Cai

,

J. Lightwave Technol.

39

,

1108

(

2021

).

https://doi.org/10.1109/jlt.2020.3032786

Crossref

114.

M.

Zhang

,

C.

Wang

,

R.

Cheng

,

A.

Shams-Ansari

, and

M.

Lončar

,

Optica

4

,

1536

(

2017

).

https://doi.org/10.1364/optica.4.001536

Crossref

115.

C.

Wang

,

M.

Zhang

,

B.

Stern

,

M.

Lipson

, and

M.

Lončar

,

Opt. Express

26

,

1547

(

2018

).

https://doi.org/10.1364/oe.26.001547

Crossref

PubMed

116.

B. H.

Robinson

,

L. E.

Johnson

,

D. L.

Elder

,

A. A.

Kocherzhenko

,

C. M.

Isborn

,

C.

Haffner

,

W.

Heni

,

C.

Hoessbacher

,

Y.

Fedoryshyn

,

Y.

Salamin

,

B.

Baeuerle

,

A.

Josten

,

M.

Ayata

,

U.

Koch

,

J.

Leuthold

, and

L. R.

Dalton

,

J. Lightwave Technol.

36

,

5036

(

2018

).

https://doi.org/10.1109/jlt.2018.2865882

Crossref

117.

M. D.

Fontana

,

K.

Laabidi

,

B.

Jannot

,

M.

Maglione

, and

P.

Jullien

,

Solid State Commun.

92

,

827

(

1994

).

https://doi.org/10.1016/0038-1098(94)90322-0

Crossref

118.

B. H.

Hoerman

,

G. M.

Ford

,

L. D.

Kaufmann

, and

B. W.

Wessels

,

Appl. Phys. Lett.

73

,

2248

(

1998

).

https://doi.org/10.1063/1.121691

Crossref

119.

T.

Hamano

,

D. J.

Towner

, and

B. W.

Wessels

,

Appl. Phys. Lett.

83

,

5274

(

2003

).

https://doi.org/10.1063/1.1635967

Crossref

120.

F.

Eltes

,

C.

Mai

,

D.

Caimi

,

M.

Kroh

,

Y.

Popoff

,

G.

Winzer

,

D.

Petousi

,

S.

Lischke

,

J. E.

Ortmann

,

L.

Czornomaz

,

L.

Zimmermann

,

J.

Fompeyrine

, and

S.

Abel

,

J. Lightwave Technol.

37

,

1456

(

2019

).

https://doi.org/10.1109/jlt.2019.2893500

Crossref

121.

W. J.

Merz

,

Phys. Rev.

76

,

1221

(

1949

).

https://doi.org/10.1103/physrev.76.1221

Crossref

122.

K.-i.

Sakayori

,

Y.

Matsui

,

H.

Abe

,

E.

Nakamura

,

M.

Kenmoku

,

T.

Hara

,

D.

Ishikawa

,

A.

Kokubu

,

K.-i.

Hirota

, and

T. I.

Takuro Ikeda

,

Jpn. J. Appl. Phys.

34

,

5443

(

1995

).

https://doi.org/10.1143/jjap.34.5443

Crossref

123.

K.

Alexander

,

J. P.

George

,

J.

Verbist

,

K.

Neyts

,

B.

Kuyken

,

D.

Van Thourhout

, and

J.

Beeckman

,

Nat. Commun.

9

,

3444

(

2018

).

https://doi.org/10.1038/s41467-018-05846-6

Crossref

PubMed

124.

C.

Haffner

,

W.

Heni

,

Y.

Fedoryshyn

,

J.

Niegemann

,

A.

Melikyan

,

D. L.

Elder

,

B.

Baeuerle

,

Y.

Salamin

,

A.

Josten

,

U.

Koch

,

C.

Hoessbacher

,

F.

Ducry

,

L.

Juchli

,

A.

Emboras

,

D.

Hillerkuss

,

M.

Kohl

,

L. R.

Dalton

,

C.

Hafner

, and

J.

Leuthold

,

Nat. Photonics

9

,

525

(

2015

).

https://doi.org/10.1038/nphoton.2015.127

Crossref

125.

W.

Heni

,

Y.

Fedoryshyn

,

B.

Baeuerle

,

A.

Josten

,

C. B.

Hoessbacher

,

A.

Messner

,

C.

Haffner

,

T.

Watanabe

,

Y.

Salamin

,

U.

Koch

,

D. L.

Elder

,

L. R.

Dalton

, and

J.

Leuthold

,

Nat. Commun.

10

,

1694

(

2019

).

https://doi.org/10.1038/s41467-019-09724-7

Crossref

PubMed

126.

M.-C.

Oh

,

H.

Zhang

,

C.

Zhang

,

H.

Erlig

,

Y.

Chang

,

B.

Tsap

,

D.

Chang

,

A.

Szep

,

W. H.

Steier

,

H. R.

Fetterman

, and

L. R.

Dalton

,

IEEE J. Sel. Top. Quantum Electron.

7

,

826

(

2001

).

https://doi.org/10.1109/2944.979344

Crossref

127.

R.

Dinu

,

D.

Jin

,

G.

Yu

,

B.

Chen

,

D.

Huang

,

H.

Chen

,

A.

Barklund

,

E.

Miller

,

C.

Wei

, and

J.

Vemagiri

,

J. Lightwave Technol.

27

,

1527

(

2009

).

https://doi.org/10.1109/jlt.2009.2014178

Crossref

128.

C.

Haffner

,

W.

Heni

,

Y.

Fedoryshyn

,

A.

Josten

,

B.

Baeuerle

,

C.

Hoessbacher

,

Y.

Salamin

,

U.

Koch

,

N.

Dordevic

,

P.

Mousel

,

R.

Bonjour

,

A.

Emboras

,

D.

Hillerkuss

,

P.

Leuchtmann

,

D. L.

Elder

,

L. R.

Dalton

,

C.

Hafner

, and

J.

Leuthold

,

Proc. IEEE

104

,

2362

(

2016

).

https://doi.org/10.1109/jproc.2016.2547990

Crossref

129.

M. P.

Nielsen

,

L.

Lafone

,

A.

Rakovich

,

T. P. H.

Sidiropoulos

,

M.

Rahmani

,

S. A.

Maier

, and

R. F.

Oulton

,

Nano Lett.

16

,

1410

(

2016

).

https://doi.org/10.1021/acs.nanolett.5b04931

Crossref

PubMed

130.

M. P.

Nielsen

,

X.

Shi

,

P.

Dichtl

,

S. A.

Maier

, and

R. F.

Oulton

,

Science

358

,

1179

(

2017

).

https://doi.org/10.1126/science.aao1467

Crossref

PubMed

131.

Y.

Salamin

,

P.

Ma

,

B.

Baeuerle

,

A.

Emboras

,

Y.

Fedoryshyn

,

W.

Heni

,

B.

Cheng

,

A.

Josten

, and

J.

Leuthold

,

ACS Photonics

5

,

3291

(

2018

).

https://doi.org/10.1021/acsphotonics.8b00525

Crossref

132.

C.

Haffner

,

D.

Chelladurai

,

Y.

Fedoryshyn

,

A.

Josten

,

B.

Baeuerle

,

W.

Heni

,

T.

Watanabe

,

T.

Cui

,

B.

Cheng

,

S.

Saha

,

D. L.

Elder

,

L. R.

Dalton

,

A.

Boltasseva

,

V. M.

Shalaev

,

N.

Kinsey

, and

J.

Leuthold

,

Nature

556

,

483

(

2018

).

https://doi.org/10.1038/s41586-018-0031-4

Crossref

PubMed

133.

C.

Delacour

,

S.

Blaize

,

P.

Grosse

,

J. M.

Fedeli

,

A.

Bruyant

,

R.

Salas-Montiel

,

G.

Lerondel

, and

A.

Chelnokov

,

Nano Lett.

10

,

2922

(

2010

).

https://doi.org/10.1021/nl101065q

Crossref

PubMed

134.

M.

Ono

,

H.

Taniyama

,

H.

Xu

,

M.

Tsunekawa

,

E.

Kuramochi

,

K.

Nozaki

, and

M.

Notomi

,

Optica

3

,

999

(

2016

).

https://doi.org/10.1364/optica.3.000999

Crossref

135.

M.

Pramassing

,

M.

Liebtrau

,

H. J.

Schill

,

S.

Irsen

, and

S.

Linden

,

Opt. Express

28

,

12998

(

2020

).

https://doi.org/10.1364/oe.384629

Crossref

PubMed

136.

D. K.

Gramotnev

,

M. G.

Nielsen

,

S. J.

Tan

,

M. L.

Kurth

, and

S. I.

Bozhevolnyi

,

Nano Lett.

12

,

359

(

2012

).

https://doi.org/10.1021/nl203629m

Crossref

PubMed

137.

Y.

Chen

,

J.

Wang

,

X.

Chen

,

M.

Yan

, and

M.

Qiu

,

Opt. Express

21

,

1639

(

2013

).

https://doi.org/10.1364/oe.21.001639

Crossref

PubMed

138.

M. G.

Nielsen

and

S. I.

Bozhevolnyi

,

J. Opt. Soc. Am. B

32

,

462

(

2015

).

https://doi.org/10.1364/josab.32.000462

Crossref

139.

A.

Messner

,

P. A.

Jud

,

J.

Winiger

,

W.

Heni

,

B.

Baeuerle

,

M.

Eppenberger

,

U.

Koch

,

C.

Haffner

,

H.

Xu

,

D. L.

Elder

,

L. R.

Dalton

,

P.

Ma

, and

J.

Leuthold

, “

High-speed plasmonic modulator for simultaneous C- and O-band modulation with simplified fabrication

,” in

Optical Fiber Communication Conference (OFC) 2020, OSA Technical Digest

(Optica Publishing Group, 2020), paper M1D.3.

Crossref

140.

W.

Heni

,

C.

Haffner

,

R.

Cottier

,

Y.

Fedoryshyn

,

D. L.

Elder

,

L. R.

Dalton

, and

J.

Leuthold

, paper presented at the

Advanced Photonics 2018 (BGPP, IPR, NP, NOMA, Sensors, Networks, SPPCom, SOF)

,

Zurich

,

2018

.

141.

K. M.

McPeak

,

S. V.

Jayanti

,

S. J. P.

Kress

,

S.

Meyer

,

S.

Iotti

,

A.

Rossinelli

, and

D. J.

Norris

,

ACS Photonics

2

,

326

(

2015

).

https://doi.org/10.1021/ph5004237

Crossref

PubMed

142.

S.

Coudert

,

G.

Duchateau

,

S.

Dilhaire

, and

P.

Lalanne

, arXiv:1810.07389 (

2018

).

143.

W.

Heni

,

C.

Haffner

,

D. L.

Elder

,

A. F.

Tillack

,

Y.

Fedoryshyn

,

R.

Cottier

,

Y.

Salamin

,

C.

Hoessbacher

,

U.

Koch

,

B.

Cheng

,

B.

Robinson

,

L. R.

Dalton

, and

J.

Leuthold

,

Opt. Express

25

,

2627

(

2017

).

https://doi.org/10.1364/oe.25.002627

Crossref

PubMed

144.

Y.

Salamin

,

I.-C.

Benea-Chelmus

,

Y.

Fedoryshyn

,

W.

Heni

,

D. L.

Elder

,

L. R.

Dalton

,

J.

Faist

, and

J.

Leuthold

,

Nat. Commun.

10

,

5550

(

2019

).

https://doi.org/10.1038/s41467-019-13490-x

Crossref

PubMed

145.

J.

Leuthold

,

R.

Bonjour

,

Y.

Salamin

,

C.

Hoessbacher

,

W.

Heni

,

C.

Haffner

,

A.

Josten

,

B.

Baeuerle

,

M.

Ayata

,

A.

Messner

,

U.

Koch

,

T.

Watanabe

,

Y.

Fedoryshyn

,

P.

Ma

, and

M.

Burla

, in

Optical Fiber Communication Conference

(

Optical Society of America

,

San Diego, CA

,

2018

), p.

Th3G.1

.

146.

Y.

Horst

,

T.

Blatter

,

L.

Kulmer

,

B. I.

Bitachon

,

B.

Baeuerle

,

M.

Destraz

,

W.

Heni

,

S.

Koepfli

,

P.

Habegger

,

M.

Eppenberger

,

E.

De Leo

,

C.

Hoessbacher

,

D. L.

Elder

,

S. R.

Hammond

,

L. E.

Johnson

,

L. R.

Dalton

,

Y.

Fedoryshyn

,

Y.

Salamin

,

M.

Burla

, and

J.

Leuthold

,

J. Lightwave Technol.

40

,

1690

(

2022

).

https://doi.org/10.1109/jlt.2022.3148534

Crossref

147.

M.

Eppenberger

,

A.

Messner

,

B. I.

Bitachon

,

W.

Heni

,

T.

Blatter

,

P.

Habegger

,

M.

Destraz

,

E.

De Leo

,

N.

Meier

,

N.

Del Medico

,

C.

Hoessbacher

,

B.

Baeuerle

, and

J.

Leuthold

,

Nat. Photonics

17

,

360

(

2023

).

https://doi.org/10.1038/s41566-023-01161-9

Crossref

148.

R. F.

Oulton

,

V. J.

Sorger

,

T.

Zentgraf

,

R.-M.

Ma

,

C.

Gladden

,

L.

Dai

,

G.

Bartal

, and

X.

Zhang

,

Nature

461

,

629

(

2009

).

https://doi.org/10.1038/nature08364

Crossref

PubMed

149.

W.

Heni

,

R.

Huber

,

Y.

Fedoryshyn

,

F.

Leyvraz

,

R.

Gao

,

M.

Lewerenz

,

A.

Messner

,

U.

Koch

,

C.

Haffner

, and

J.

Leuthold

, in

Advanced Photonics 2017 (IPR, NOMA, Sensors, Networks, SPPCom, PS)

(

Optical Society of America

,

New Orleans, LA

,

2017

).

150.

C.

Hoessbacher

,

Y.

Fedoryshyn

,

A.

Emboras

,

A.

Melikyan

,

M.

Kohl

,

D.

Hillerkuss

,

C.

Hafner

, and

J.

Leuthold

,

Optica

1

,

198

(

2014

).

https://doi.org/10.1364/optica.1.000198

Crossref

151.

J. F.

Ihlefeld

,

D. T.

Harris

,

R.

Keech

,

J. L.

Jones

,

J.-P.

Maria

, and

S.

Trolier-McKinstry

,

J. Am. Ceram. Soc.

99

,

2537

(

2016

).

https://doi.org/10.1111/jace.14387

Crossref

152.

A.

Messner

,

Materials and Waveguide Platforms for High-Speed Plasmonic Modulators

(

ETH Zurich

,

Zurich

,

2021

).

Google Scholar

153.

R.

Soref

and

B.

Bennett

,

IEEE J. Quantum Electron.

23

,

123

(

1987

).

https://doi.org/10.1109/jqe.1987.1073206

Crossref

154.

S.

Koeber

,

R.

Palmer

,

M.

Lauermann

,

W.

Heni

,

D. L.

Elder

,

D.

Korn

,

M.

Woessner

,

L.

Alloatti

,

S.

Koenig

,

P. C.

Schindler

,

H.

Yu

,

W.

Bogaerts

,

L. R.

Dalton

,

W.

Freude

,

J.

Leuthold

, and

C.

Koos

,

Light: Sci. Appl.

4

,

e255

(

2015

).

https://doi.org/10.1038/lsa.2015.28

Crossref

155.

C.

Bulucea

,

Solid-State Electron.

36

,

489

(

1993

).

https://doi.org/10.1016/0038-1101(93)90257-q

Crossref

156.

M.

Stengel

and

N. A.

Spaldin

,

Nature

443

,

679

(

2006

).

https://doi.org/10.1038/nature05148

Crossref

PubMed

157.

P.

Ma

,

X. Z.

Zhang

,

B. I.

Bitachon

,

W.

Heni

,

P.

Habegger

,

A.

Messner

,

D.

Moor

,

A.

Emboras

,

D. L.

Elder

,

L. R.

Dalton

,

T.

Greber

,

G.

Indiveri

, and

J.

Leuthold

, paper presented at the

2021 European Conference on Optical Communication (ECOC)

,

2021

.

158.

A. J.

Mercante

,

P.

Yao

,

S.

Shi

,

G.

Schneider

,

J.

Murakowski

, and

D. W.

Prather

,

Opt. Express

24

,

15590

(

2016

).

https://doi.org/10.1364/oe.24.015590

Crossref

PubMed

159.

H.

Xu

,

D. L.

Elder

,

L. E.

Johnson

,

Y.

de Coene

,

S. R.

Hammond

,

W.

Vander Ghinst

,

K.

Clays

,

L. R.

Dalton

, and

B. H.

Robinson

,

Adv. Mater.

33

,

2104174

(

2021

).

https://doi.org/10.1002/adma.202104174

Crossref

160.

M.

Xu

,

Y.

Zhu

,

F.

Pittalà

,

J.

Tang

,

M.

He

,

W. C.

Ng

,

J.

Wang

,

Z.

Ruan

,

X.

Tang

,

M.

Kuschnerov

,

L.

Liu

,

S.

Yu

,

B.

Zheng

, and

X.

Cai

,

Optica

9

,

61

(

2022

).

https://doi.org/10.1364/optica.449691

Crossref

161.

S.

Ummethala

,

T.

Harter

,

K.

Koehnle

,

Z.

Li

,

S.

Muehlbrandt

,

Y.

Kutuvantavida

,

J.

Kemal

,

P.

Marin-Palomo

,

J.

Schaefer

,

A.

Tessmann

,

S. K.

Garlapati

,

A.

Bacher

,

L.

Hahn

,

M.

Walther

,

T.

Zwick

,

S.

Randel

,

W.

Freude

, and

C.

Koos

,

Nat. Photonics

13

,

519

(

2019

).

https://doi.org/10.1038/s41566-019-0475-6

Crossref

162.

I.-C.

Benea-Chelmus

,

Y.

Salamin

,

F. F.

Settembrini

,

Y.

Fedoryshyn

,

W.

Heni

,

D. L.

Elder

,

L. R.

Dalton

,

J.

Leuthold

, and

J.

Faist

,

Optica

7

,

498

(

2020

).

https://doi.org/10.1364/optica.384160

Crossref

163.

W.

Heni

,

B.

Baeuerle

,

H.

Mardoyan

,

F.

Jorge

,

J. M.

Estaran

,

A.

Konczykowska

,

M.

Riet

,

B.

Duval

,

V.

Nodjiadjim

,

M.

Goix

,

J.-Y.

Dupuy

,

M.

Destraz

,

C.

Hoessbacher

,

Y.

Fedoryshyn

,

H.

Xu

,

D. L.

Elder

,

L.

Dalton

,

J.

Renaudier

, and

J.

Leuthold

,

J. Lightwave Technol.

38

,

2734

(

2020

).

https://doi.org/10.1109/jlt.2020.2972637

Crossref

Plasmonic, photonic, or hybrid? Reviewing waveguide geometries for electro-optic modulators

I. INTRODUCTION

A. Performance metrics

II. THE POCKELS EFFECT IN ELECTRO-OPTIC MODULATORS

A. The Pockels effect

B. Waveguide-based modulators with Pockels materials

C. Field interaction factor and sensitivity

D. Comparing waveguide geometries and materials

III. PLASMONIC, PHOTONIC, AND HYBRID MODULATORS

A. Photonic

1. Example: Silicon–organic hybrid

2. Example: Lithium niobate on insulator

3. Frequency response and the challenge of traveling wave electrodes

4. Advantages, challenges, and recent advances

B. Plasmonic metal–insulator–metal

1. Frequency response

2. Advantages, challenges, and recent advances

C. Hybrid photonic–plasmonic waveguides

1. Frequency response

2. Advantages, challenges and recent advances

IV. COMPARISON OF PHOTONIC, PLASMONIC, AND HYBRID GEOMETRIES

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Supplementary Material

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Plasmonic, photonic, or hybrid? Reviewing waveguide geometries for electro-optic modulators

I. INTRODUCTION

A. Performance metrics

II. THE POCKELS EFFECT IN ELECTRO-OPTIC MODULATORS

A. The Pockels effect

B. Waveguide-based modulators with Pockels materials

C. Field interaction factor and sensitivity

D. Comparing waveguide geometries and materials

III. PLASMONIC, PHOTONIC, AND HYBRID MODULATORS

A. Photonic

1. Example: Silicon–organic hybrid

2. Example: Lithium niobate on insulator

3. Frequency response and the challenge of traveling wave electrodes

4. Advantages, challenges, and recent advances

B. Plasmonic metal–insulator–metal

1. Frequency response

2. Advantages, challenges, and recent advances

C. Hybrid photonic–plasmonic waveguides

1. Frequency response

2. Advantages, challenges and recent advances

IV. COMPARISON OF PHOTONIC, PLASMONIC, AND HYBRID GEOMETRIES

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Supplementary Material

Citing articles via

Resources

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