Ring modulators with enhanced efficiency based on standing-wave operation on a field-matched, interdigitated p-n junction

Fabio Pavanello; Xiaoge Zeng; Mark T. Wade; Miloš A. Popović

doi:10.1364/OE.24.027433

1. Introduction

Resonant ring modulators are emerging as key components of current high-bandwidth, low latency, energy efficient electro-optical transmitter systems [1, 2]. In silicon platforms they are mostly based on the free carrier-plasma dispersion effect and have been widely employed since early demonstrations [3,4] due to higher efficiencies and smaller footprints compared to broadband geometries such as Mach-Zehnder or slow-light modulators [5,6]. To achieve higher speeds (typically above 1 Gbit/s) avoiding the use of pre-emphasis or amplification [7], ring modulators are generally diodes that are driven in depletion rather than carrier-injection mode. Aside from a faster response, this operation mode is also the preferred one for lower energy per bit consumption (order of fJ/bit) due to near absence of DC current through the junction.

Ring modulators have traditionally been used in traveling wave configurations where the optical field does not have steady intensity maxima or minima at specific locations – the instantaneous intensity nulls circulate the cavity at the speed of light within the cavity. This distribution of field results in sub-optimal modulation of the cavity mode, and unnecessary loss and speed limitations, since at any instant junctions are driving carriers in parts of the cavity where the field is null, yet those nulls move too fast to take advantage of in design. However, fast field nulls are fundamentally present at about half wavelength intervals in at least one direction in all possible optical resonant modes larger than a half wavelength in each direction – hence they present a fundamental inefficiency.

In this paper, we show that a standing-wave operation microring cavity can be designed to increase modulation efficiency, reduce loss and increase speed of a modulator by designing around the inefficiency associated with mode field nulls. Standing-wave modes make all field nulls stationary and hence allow optimal matching of the electrical structure to the mode field. Here we address pure standing wave as well as partial standing wave designs.

As a starting point, we utilize an interdigitated pn junction array along the (azimuthal for a ring geometry) propagation direction [8]. Carrier-depletion modulators commonly use lateral p-n junctions, requiring a ridge waveguide structure [3], or vertical p-n junctions, requiring control of the vertical doping profile [9]. Both of the latter junction geometries have a uniform carrier distribution along the propagation direction, and are thus incapable of addressing the “fast-null inefficiency”. In addition, the aim to maximally deplete the cavity of carriers for modulation efficiency leads to carrier starved conduction paths leading to a trade-off between speed and efficiency, especially for vertical junctions [9] and azimuthal junctions in traveling-wave operation [8]. Finally, lateral and vertical junction geometries are not well-suited for implementation in advanced complementary metal-oxide semiconductor (CMOS) processes due to the lack of partial etch steps and of precise vertical control of the doping distribution in thin sub-100nm silicon transistor device layers, respectively [8].

Recently, a geometry that is well-suited for fabrication in unmodified semiconductor-on-insulator (SOI) CMOS microelectronics processes, especially sub-100nm nodes, was proposed, termed the “spoked-ring” cavity, where the p-n junctions are created in the c-Si layer used for the transistor body and are periodically interdigitated along the propagation direction [8,10–12]. Contacts (spokes) are placed along the inner ring edge of the cavity to avoid scattering/absorption losses because the optical mode is confined mostly along the outer ring edge. Here, the junctions are perpendicular to the light propagation direction [see Fig. 1(a)], so that the depletion region widths are modulated along the propagation direction [8]. Critically, this geometry does not require a partial etch step of the c-Si layer (i.e. ridge waveguides). In previous work, modulators based on photonic crystals and on multimode-width cavities exploiting field minima of the optical field [13] or envelope [14] distribution have been demonstrated. They place high-doping regions and metallic contacts closer to the active modulated region to improve speed or to avoid a partial etch step. However, little attention has been devoted to designs which enhance the modulation efficiency, while reducing also background optical losses (e.g. losses coming from regions which do not experience modulation) for classical traveling-wave geometries such as ring resonators, all of which suffer from the fast-null inefficiency.

Fig. 1 (a) Previous spoked-ring architecture [8] exploiting traveling-wave modes where no doping distribution to optical field matching is present. (b) Example of proposed doping distribution to optical field matching. w indicates the unit-cell width p/n and w_d the depletion width. (c) Architecture for standing-wave excitation based on a splitter and a circulator for input-output separation. (d) Standing-wave architecture equivalent to the one in (c), but no circulator required by using two cavities. (e) Partial standing-wave architecture with excitation of the counter-propagating mode provided by sidewall grating.

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2. Device concept

Here we propose an interdigitated p-n junction modulator which modulates standing rather than traveling optical waves. For such a configuration, the inherent p-n junctions periodicity [see Fig. 1(b)] can be used to improve the overall modulation depth compared to previous interdigitated modulators exploiting traveling waves because the field nulls are stationary. Specifically, the optical field of a standing wave can be matched to the doping distribution in order to have maxima of the optical field corresponding to regions where the modulation takes place, enhancing the modulation effect, and minima of the optical field where the modulation is absent, but carriers are required to provide good electrical conductivity and a conduction path for efficient electrical modulation. While this concept can be applied to all the conventional ring junction geometries, for lateral and vertical junctions the cavity would require to be only periodically, rather than uniformly, doped to efficiently exploit an excited standing-wave pattern in the cavity reducing the device capacitance and optical losses, while preserving a similar modulation efficiency as if the cavity was uniformly doped. Figure 1(b) shows an example of standing-wave mode along the azimuthal direction of the cavity which takes advantage of the doping distribution. The benefits of this approach are two-fold; on the one hand the modulation efficiency improves, while on the other hand the background optical losses associated with unmodulated regions are reduced for a given doping concentration compared to equivalent traveling-wave structures, allowing higher modulation depth or narrower bandwidth and thus even higher modulation efficiency. Moreover, for interdigitated geometries placing the contacts (spokes) at minima of the optical fields enables closer approach to the optical mode by pushing the contacts more radially outward reducing the necessary ring width and consequently reducing capacitance and junction access resistance, thus achieving higher electrical speeds and lower energy per bit consumption. For ridge and vertical junction geometries it is possible to approach the contacts as well reducing the capacitance or to avoid having the entire cavity doped by selectively dope only the regions experiencing modulation thus reducing the cavity losses.

Different architectures can be conceived for exciting a standing-wave mode in the cavity. Figure 1(c) shows the optimal implementation in terms of perfect standing-wave excitation using a 3dB splitter to excite two degenerate contra-directional traveling-wave resonances. However, this architecture requires a circulator in order to separate the CW input from the modulated output signal. Figure 1(d) provides an architecture that does not require a circulator, but at the expense of using two (identical) cavities, to be driven synchronously. This implementation, although equivalent to the one in Fig. 1(c) in terms of perfect standing-wave excitation, has twice the energy consumption and a larger footprint due to the double cavity. Finally, to avoid the need of a circulator or two cavities, another solution is presented in Fig. 1(e) where a sidewall grating can be used to excite the counter-propagating mode leading to a partial standing wave in the cavity. In such a case the sidewall grating, as well as the doping distribution, need to be matched to the optical field distribution. This design has an inherent trade-off between how strongly a standing wave is excited versus how much of the light can be sent to the output port (as opposed to reflection port), but has the simplest by far practical implementation.

3. Theoretical model

We investigate the performance of an interdigitated modulator based on standing or partial standing-wave modes associated with the architectures in Figs. 1(c)–1(e), comparing them to the traveling-wave counterpart [8] in Fig. 1(a). We show that a model based on the coupling of modes in time (CMT) [15] with complex refractive indices for modulation [16], and imaginary coupling [17], can describe all the relevant physics of the system. The main approximations in our model are a cartesian unit-cell with a period weighted by the optical field intensity, instantaneous response of the junction to the driving signal (no RC limitation) and the full-depletion approximation for a p-n junction. The latter is valid for applied voltages far from the built-in voltage so that the current is negligible. The CMT equations in the traveling-wave picture for the architectures in Fig. 1 are given by

\begin{array}{l} \frac{d}{d t} \vec{a} = j \bar{\bar{ω}} \vec{a} - j {\bar{\bar{μ}}}_{g} \vec{a} - j {\bar{\bar{η}}}_{d} \vec{a} - j {\bar{M}}_{i} s_{+} \\ s_{-} = s_{+} - j {\bar{M}}_{o} \vec{a} \end{array}

where

\vec{a}

is the vector of the propagating and counter-propagating mode amplitudes inside the ring,

\bar{\bar{ω}}

is the matrix of the complex resonant angular frequencies when the ring is fully undepleted,

{\bar{\bar{μ}}}_{g}

is the matrix describing the sidewall grating induced contra-directional coupling (for case in Fig. 1(e) otherwise absent),

{\bar{\bar{η}}}_{d}

is the matrix describing the effect of a carrier density change (real part shifts the resonant frequency, imaginary part reduces the losses from the fully undepleted state and both couple a₊ and a₋) for a given doping distribution,

{\bar{M}}_{o} ({\bar{M}}_{i})

are the vectors describing the output (input) coupling and s₋ (s₊) are the output (input) wave amplitudes with the total power transmission coefficient given by their squared magnitudes ratio. The quantities in equations 1 are defined as

\begin{matrix} \vec{a} = (\begin{matrix} a_{+} \\ a_{-} \end{matrix}) \bar{\bar{ω}} = (\begin{matrix} ω_{o} + j r_{t o t} & 0 \\ 0 & ω_{o} + j r_{t o t} \end{matrix}) \\ {\bar{\bar{μ}}}_{g} = (\begin{matrix} 0 & μ \\ μ & 0 \end{matrix}) {\bar{\bar{η}}}_{d} = \frac{1}{2} (\begin{matrix} η_{d, a} + η_{d, s} & - η_{d, a} + η_{d, s} \\ - η_{d, a} + η_{d, s} & η_{d, a} + η_{d, s} \end{matrix}) \\ case (c) {\bar{M}}_{i} = \sqrt{r_{e}} (\begin{matrix} 1 \\ 1 \end{matrix}) \begin{matrix} case (e) {\bar{M}}_{i} = \sqrt{2 r_{e}} (\begin{matrix} 1 \\ 0 \end{matrix}) & {\bar{M}}_{o} = {\bar{M}}_{i}^{†} \end{matrix} \end{matrix}

Here μ is a real positive parameter describing the strength of a sinusoidal sidewall grating. In the definition of

\bar{\bar{ω}}

, r_tot = r_o + r_d + r_e where r_o is the passive loss rate (cavity with no doping), r_d is the doping-induced loss rate of the fully undepleted cavity and r_e is the extrinsic loss/coupling rate at the coupling section. For the case in Fig. 1(c), the two inputs, generated at the 3dB coupler, constructively interfere (same optical path lengths as indicated by

{\bar{M}}_{i}

) in the bus waveguide and in the cavity (for the field coupled into) with maxima where the depletion regions are located. For the case in Fig. 1(e), only the conventional traveling-wave excitation is present as indicated by

{\bar{M}}_{i}

and the grating/doping matrices are responsible for exciting contra-directional resonance to synthesize a partial standing-wave pattern. Equations 1 contain as a limit the traveling-wave case when the grating strength μ is equal to zero and the doping matrix

{\bar{\bar{μ}}}_{d}

has the same on-diagonal terms and null off-diagonal terms i.e. no doping distribution matching with a standing wave. The coupled system in Eqs. (1) can be diagonalized using the following vector basis which represents the symmetric and antisymmetric modes

\begin{array}{l} a_{s} = 1 / \sqrt{2} (a_{+} + a_{-}) \\ a_{a} = j / \sqrt{2} (a_{+} - a_{-}) \end{array}

where a₊ and a₋ are associated with fields with azimuthal dependencies exp(−j(γθ − ωt)) and exp(j(γθ + ωt)), with γ the angular integer propagation constant (mode order). Using the vector basis in Eqs. (2), the quantities in Eqs. (1) are transformed to (sm below stands for supermodes)

\begin{matrix} {\vec{a}}_{s m} = (\begin{matrix} a_{s} \\ a_{a} \end{matrix}) {\bar{\bar{ω}}}_{s m} = (\begin{matrix} ω_{o} + j r_{t o t} & 0 \\ 0 & ω_{o} + j r_{t o t} \end{matrix}) \\ {\bar{\bar{μ}}}_{g, s m} = (\begin{matrix} μ & 0 \\ 0 & - μ \end{matrix}) {\bar{\bar{η}}}_{d, s m} = (\begin{matrix} η_{d, s} & 0 \\ 0 & η_{d, a} \end{matrix}) \\ case (c) {\bar{M}}_{i, s m} = \sqrt{2 r_{e}} (\begin{matrix} 1 \\ 0 \end{matrix}) case (e) {\bar{M}}_{i, s m} = \sqrt{r_{e}} (\begin{matrix} 1 \\ j \end{matrix}) \end{matrix}

Here we are assuming excitation only of the symmetric mode for case (b) which has a maximum of the optical field at the bus-ring minimum distance, opposite to the junction location of the 3dB splitter, where the p/n junction is also located. The grating in this configuration is placed on the middle of the n and p doping regions as shown in Fig. 1(e). The eigenfrequencies of the diagonalized system corresponding to the eigenvectors in Eqs. (2) are

(\begin{matrix} ω_{s} \\ ω_{a} \end{matrix}) = (\begin{matrix} ω_{o} + μ - Re (η_{d, s}) + j (r_{t o t} - Im (η_{d, s})) \\ ω_{o} - μ - Re (η_{d, a}) + j (r_{t o t} - Im (η_{d, a})) \end{matrix})

The complex refractive index change δñ_s/a = δn_s/a + jδk_s/a is evaluated by performing an overlap integral of the index perturbation with the optical mode in the propagation direction over the unit-cell of the cavity. μ_d,s/a can be then found by multiplying δn_s/a and δk_s/a by proper scaling factors (Γ_r and Γ_i), which take also into account the confinement of the optical field in the transverse direction, for conversion to angular frequency shifts δω_s/a and loss rates r_d,s/a i.e. η_d,s/a = Γ_r δn_s/a + jΓ_iδk_s/a. Here, we consider a symmetric (same concentrations) p-n doping distribution [see Fig. 1(b)]. For such a distribution the complex refractive index change with respect to the fully doped cavity which depends on the applied voltage, V, across the depletion region e.g. during modulation (averaged due to field symmetry for the different optical response of electrons and holes) for the supermodes can be found to be

δ {\tilde{n}}_{s / a} (V) = 2 δ \tilde{n} \frac{w_{d} (V)}{w} [1 \pm \sin c (2 π \frac{w_{d} (V)}{w})]

where w is the unit-cell period, δñ is the complex refractive index change for the cavity once fully depleted and w_d (V) the depletion width for a single p-n junction as a function of the voltage. Here we notice that the percentage of unit-cell depleted is given by 2w_d (V)/w as each unit-cell counts 2 p-n junctions. The expression for w_d (V) is given by [18]

w_{d} (V) = \sqrt{\frac{2 ϵ_{r} ϵ_{0}}{q} \frac{N_{a} + N_{d}}{N_{a} N_{d}} (V_{b i} - V)}

where ϵ_r and ϵ₀ are the permittivities of Si (relative) and of the vacuum, q is the elementary charge, N_a and N_d are the acceptor and donor doping concentrations, V and V_bi are the voltage across the junction and the built-in voltage, respectively.

4. Quantitative analysis: static regime

The geometry that has been considered in the analysis is shown in Fig. 2. Modulators based on this geometry have been already experimentally demonstrated achieving good modulation performance in terms of modulation depth and energy per bit consumption at 5 Gbit/s [8]. The ring resonator described here relies on the fundamental TE mode at 1.3 μm wavelength which can be efficiently excited in ring geometries with multi-mode waveguide widths by the use of a distributed coupler [8]. The outer radius of the ring cavity is 5 μm and the ring width is 1.2 μm. For such a radius the intensity of the optical mode, plotted as well in Fig. 2, is negligible on the inner radius of the cavity allowing to place higher doping regions and contacts without suffering from losses as previously mentioned in the Introduction section. The inner spokes in Fig. 2 show how the p-n junction is electrically contacted, but they have not been included in the simulation for the optical field in this example, which uses only the ring part simulated in a circular modesolver.

Fig. 2 Optical intensity at resonance for the cavity used in the quantitative analysis from Ref. [8]. (a) shows the intensity plotted at the middle of the silicon device layer and the p-n junctions arrangement such that the optical intensity has a maximum in correspondence to the interface between p and n regions where the depletion region starts. (b) shows a top-view close-up of a section of the ring cavity. (c) shows the intensity in the cross-section view.

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In Fig. 3(a) we report the absolute and differential frequency shift (the derivative of the absolute shift with respect to the depleted cavity percentage) and, in Fig. 3(b), the absolute unloaded (no bus coupling) quality factors Q (i.e. doping + passive losses) for the standing and traveling-wave cases, and their ratio (standing over traveling), as a function of the percentage of depleted cavity. It can be seen that for a small voltage swing around 50% depleted cavity bias point the differential shift is the same for the two configurations, while approaching zero depletion width the differential shift becomes twice as large for the standing-wave case and null when approaching full depletion. This can be understood by recalling that the modulation effect is stronger when the optical intensity is larger at the location (depletion region edges) of a given incremental depletion region change. In terms of unloaded quality factor Q instead we observe that, approaching full or null depletion percentages, the two configurations are equivalent due to the fact that most of the field is fully present or absent in the depleted region. In our case (passive Q = 250, 000) the maximum Q ratio between the standing and traveling-wave modes is ≈2.2 and is obtained for nearly 67% cavity depleted.

Fig. 3 Relevant optical quantities vs cavity depleted ratio for the architectures in Fig. 1(a) and Figs. 1(c) and 1(d), respectively. (a) Absolute (red) and differential shift (blue), (b) absolute (red) and standing to traveling-wave ratio (blue) of unloaded Q (passive Q = 250, 000). Solid line is associated in all the figures to the standing-wave architecture and dashed line to traveling-wave architectures.

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Figure 4(a) shows a first figure of merit (FOM) where the differential shift (blue curve) of Fig. 3(a) is divided by the linewidth associated to the cavity Q (red curve) of Fig. 3(b). We observe that for a Q of 250,000 the maximum is located around 50% and that with respect to the traveling-wave case the ratio is nearly 2. By increasing the passive Q of the cavity the optimal operation point shifts towards larger carrier depletion percentages. It is worth to notice that for the traveling-wave case the FOM is actually superior to the standing-wave FOM above 80% cavity depletion due to the differential shift which tends to zero for the standing-wave case at depletion percentages approaching 100%. However, this behavior is not taking into account the dependence of the depletion width from the voltage. This is reported in Fig. 4(b) where the FOM is multiplied by the derivative of the unit-cell junction depletion width, normalized to the unit-cell period used in this paper (620 nm), with respect to the voltage for symmetric doping concentrations of 1e18 cm⁻³. Thus, the normalized FOM shows a quite different behavior where the optimum operation in the standing-wave case is located for bias voltages as close as possible to zero depletion width. Here, we consider a maximal driving voltage of 0.5 V (forward voltage) to drive the junction in low injection regime. For larger injections the standard depletion width model used here does not hold anymore. Moreover, for such a regime diffusion capacitance will be dominating slowing down the electrical response of the device thus it is out of our interest for practical implementations. On the other side, we observe that for the traveling-wave case and for passive Qs up to 250,000 the normalized traveling-wave FOM is actually inferior to the standing-wave case at 0.5 V bias on all the depletion width range, while for higher Qs it is still superior to the standing-wave case at 0.5 V bias above 90% cavity depletion. However, associated Qs are at least 4 times higher which practically reduce by the same factor the optical bandwidth in the traveling-wave case. In all the calculations typical values of passive Q = 250, 000 (due to roughness, bend loss, etc.) and confinement factor Γ = 63% at a wavelength λ = 1.3 μm have been assumed. The transmission coefficient T = |s₋/s₊|² for cases (c) and (e) of Fig. 1 can be found from the clock-wise and counter clock-wise amplitudes using Eqs. (2) and then evaluating the output amplitude s₋ in Eqs. (1) leads to

\begin{array}{l} T (V) = {| 1 - 2 r_{e} (\frac{1}{j (ω - ω_{s} (V))}) |}^{2} case (c) \\ T (V) = {| 1 - r_{e} (\frac{1}{j (ω - ω_{s} (V))} + \frac{1}{j (ω - ω_{a} (V))}) |}^{2} case (e) \end{array}

Here we can notice that the transmission for case (c) has an equivalent expression to the one for a traveling-wave modulator where instead of the traveling-wave eigenfrequency, the eigenfrequency for the low-loss standing-wave excited supermode needs to be used. For case (e) the transmission is related to both supermodes due to their simultaneous excitation, rather than only to the low-loss one as for case (c). As a limit, if ω_a = ω_s which is the case associated with fully depleted or undepleted cavity and μ = 0, the expression (e) in Eqs. (6) reduces to the one of the traveling-wave case.

Fig. 4 Figure of merit (FOM) for the different configurations and different passive Q values. (a) FOM defined as differential shift over unloaded linewidth and (b) FOM defined as differential shift with respect to the applied voltage over unloaded linewidth (unit-cell of 620 nm considered). Solid line is associated in all the figures to the standing-wave architecture and dashed line to traveling-wave architectures.

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In Figs. 5(a)–5(c) we report the transmission spectra for the different configurations of Fig. 1. The critical coupling condition has been chosen as an example for half-depleted cavity and the other parameters are those used previously in Fig. 3. It can be seen that the shift for a standing wave in Fig. 5(b) and partial standing wave in Fig. 5(c) is enhanced, while the resonance linewidth is reduced for the same external coupling (r_e) compared to the traveling-wave case of Fig. 5(a). For the grating-assisted approach in Fig. 1(e) it is theoretically possible to achieve critical coupling by requiring the grating splitting to cancel carrier splitting, i.e. be equal to the difference between the supermodes’ eigenfrequencies i.e. μ = 0.5(δω_a − δω_s) and the external loss rate to be equal to the geometric mean of the losses for the supermodes r_e = ((r_d,s + r_o)(r_d,a + r_o))^0.5. Although for such conditions only one resonant dip is present, during modulation (off critical coupling) the spectrum results in resonance splitting for the different supermodes which overlap leading to an effective broader linewidth and reduced effective shift. A solution to this problem consists of using a stronger grating as in Fig. 5(c) which prevents the resonances from overlapping. In this case the same external coupling as in Fig. 5(b) has been used for comparison. The drawback of this choice is that the shift is reduced because of the linewidth splitting and thus the extinction as well. Moreover, the linewidth of the grating assisted design is slightly smaller compared to that of the standing-wave design for same external coupling. The reason is related to the excitation of both supermodes which translates in the fact that the actual coupling rate for the input power (|s₊|²) in the low-loss mode is r_e and not 2r_e as for a pure traveling or perfect standing-wave mode. Although this effect may lead to a larger modulation depth, it also slows down the device and, if the speed is linewidth-limited, then a larger external coupling can be chosen improving the extinction and leading to a larger linewidth.

Fig. 5 Transmission spectra (a)–(c) at different cavity depletion percentages and eye diagrams (d)–(f) for the different configurations of Fig. 1 with symmetric p-n (1e18 cm⁻³) doping. (a) and (d) Prior traveling-wave modulator. (b) and (e) Standing-wave modulator with double excitation. (c) and (f) Grating assisted, partial standing-wave excitation with grating strength μ = 15r_d. Parameters used are: wavelength λ = 1.3 μm, passive Q = 250k, effective index n_ef f = 2.1, group index n_g = 3.1, unit-cell period w = 620 nm and grating strength μ = 15r_d. For (d)–(f) the percentage of cavity depleted was from 20% (critical coupling for (d),(e)) to 11% for −0.5 to 0.5 V swing, respectively. g represents the longitudinal mode order and N the number of unit-cells in the cavity. The x-axis are common to Figs.(a)–(c) and (d)–(f) respectively. The y-axis is common to all figures.

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5. Quantitative analysis: dynamic regime

Finally, we carried out dynamic response simulations by solving numerically the coupled differential system in Eqs. (1) for a pseudo-random bit sequence (PRBS7) pattern to obtain an eye diagram and modulation depth for the different architectures using same optical and electrical properties, i.e. same doping and dielectric distributions and associated dimensions (same RC time constant). We chose a fixed target insertion loss (IL) of 1 dB, a voltage swing from −0.5 to 0.5 V and a 10 Gbit/s speed which are common requirements in integrated photonics-electronic transmitters. Figure 5(d)–5(f) report the traveling, standing and grating-assisted partial standing-wave designs. The modulation depth is less than 2 dB in the traveling-wave case, while for the purely standing-wave case this is calculated higher than 10 dB. Here we consider to have a resonance extinction dip of at least 11 dB in practice (theoretically infinite at zero tranmission) which is very common for modulators. In the grating assisted case the modulation depth is still superior to 4.5 dB and the reduction from the previous case is due to a non-perfect excitation of the standing-wave mode which leads also to a non zero transmission at resonance.

If the RC electrical time constant limits the device performance, this architecture offers another benefit aside the possibility of placing contacts closer reducing already the device capacitance. The symmetric p-n mid-level doping distribution can be modified by adding high doping regions in the cavity where the optical field nodes are present to reduce the access resistance to the junction without strongly affecting the optical losses. For the traveling-wave geometry this choice would not be efficient due to the heavy losses incurred leading to greatly increased linewidth and reduced modulation depth. Further study will consider geometries which employ multiple combinations of doping concentrations to achieve an even more efficient modulation and better electrical configuration.

6. Conclusions

In conclusion we have shown that exploiting standing rather than traveling waves in traveling-wave modulators offers a way to remove a fundamental inefficiency in modulating traveling-wave fields due to their fast-moving nulls. The approach improves the modulation efficiency, speed and energy consumption through lower swing voltages and device capacitance, if high doping regions and contacts are opportunely placed in the optical field nodes. Moreover, modification of the basic symmetric p-n doping distribution by the introduction of high-doping regions directly in the cavity along the radial direction would help to overcome potential RC limitations without severely affecting the modulator performance. The cost is increased device complexity, which may or may not justify the increased performance. The grating-assisted, partial standing-wave design offers an intermediate solution, that provides partial benefit with a simple structure.

Funding

DARPA POEM program (HR0011-11-C-0100); University of Colorado Boulder Libraries Open Access Fund.

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Ring modulators with enhanced efficiency based on standing-wave operation on a field-matched, interdigitated p-n junction

Abstract

1. Introduction

2. Device concept

3. Theoretical model

4. Quantitative analysis: static regime

5. Quantitative analysis: dynamic regime

6. Conclusions

Funding

References and links

Cited By

Figures (5)

Equations (8)

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