WO2020234098A1

WO2020234098A1 - Electro-optic modulator with periodic p-n junction in slow-light waveguide gratings

Info

Publication number: WO2020234098A1
Application number: PCT/EP2020/063425
Authority: WO
Inventors: Lucio Andreani; Dario GERACE; Marco Passoni; John William WHELAN-CURTIN
Original assignee: Universita' Degli Studi Di Pavia; Cork Institute Of Technology
Priority date: 2019-05-20
Filing date: 2020-05-14
Publication date: 2020-11-26
Also published as: IT201900006998A1

Abstract

The invention concerns an electro-optical modulator, wherein a band-edge slow light in silicon waveguide gratings is applied to Mach- Zehnder modulators based on the plasma dispersion effect. An interleaved p-n junction with the same periodicity as the grating is used, in order to achieve optimal matching between the electromagnetic field profile and the depletion regions of the p-n junction. The resulting modulation efficiency is strongly improved as compared to common modulators based on normal rib waveguides, even in a bandwidth of 20-30 nm near the band edge, while the total insertion loss due to free carriers is not increased. The present invention allows production of slow-light modulators for silicon photonics with reduced energy dissipation.

Description

Electro-optic modulator with periodic p-n junction in slow-light waveguide gratings

Applicants: Universita degli Studi di Pavia (75%), Cork Institute of

Technology, Cork, Irlanda (25%)

Inventors: Lucio Andreani, Dario Gerace, Marco Passoni, William

Whelan-Curtin

The present invention concerns an electro-optic modulator with periodic p-n junction in slow-light waveguide gratings.

The invention was supported by the European Union under H2020- ICT27-2015 project no. 688516 COSMICC.

State of the art

Optical modulators are devices for optical communication that encode a bit stream onto a carrier wave. In particular, silicon electro-optical modulators are crucial components of photonic integrated circuits in view of the applications for optical communication.

Reducing the power dissipation of electro-optical modulators is a key step for applications to large traffic volumes in data centers. Common silicon modulators rely on the plasma dispersion effect in a reverse-biased p-n junction, namely the change of the refractive index due to a variation of the free-carrier concentration, often in a Mach-Zehnder (MZ) Interferometer (MZI) Silicon-On-Insulator (SOI) platform configuration that ensures a wide bandwidth. MZ modulators for silicon photonics are optical modulators that exploit the variation of the refractive index in one or both arms of a MZ interferometer.

Standard MZ modulators have two arms made of rib waveguides, which include sections with a lateral p-n junction that acts as on optical phase- shifter. The main limitation of such device is the poor overlap between the depletion region of the p-n junction and the optical mode profile. As the resulting change of the effective refractive index is quite small, the modulator length has to be of the order of millimeters, which in turn determines the capacitance and the power dissipation of the device. Therefore, usual modulators exhibit a power dissipation of the order of a few pj/bit.

Hence, there is a need for a solution able to reduce the level of power dissipation by an order of magnitude.

Electro-optic modulators that combine a periodic p-n junction with a slow-light structure have been realized with two-dimensional (2D) photonic crystal waveguides [1 ,2]. However, 2D photonic crystal waveguides are demanding in terms of fabrication and are subject to higher propagation and insertion losses. So far they have found very limited use in an industrial environment.

Resonator-based modulators with interleaved p-i-n junctions leading to matching with optical mode have been developed [3-6]. Such structures are based on Fabry-Perot resonators and have typically a small bandwidth, which is generally not suited and desirable for many applications such as optic communications, e.g. Wavelength Division Multiplexing, where many wavelength channels must be supported, or where operation of a wide temperature range is required.

In a previous work [7], the Inventors optimized the geometry for slow light in waveguide gratings (also known as corrugated or fishbone waveguides) that are composed alternating wide/narrow sections of a rib waveguide in SOI, as depicted in Fig. 1 . Waveguide gratings are periodic structures along the direction of propagation that lead to photonic stop bands and to slow light close to the band edges. Such waveguide gratings can be incorporated in standard silicon photonic circuits, as they can be connected with silicon rib waveguides by means of low-loss tapers. In Ref.

[7] the Inventors designed optimized parameters of SOI waveguide gratings for a target wavelength in the O-band (A = 1.3 pm) by finding a proper tradeoff between the group index enhancement and the slow-light bandwidth.

Flowever, the waveguide gratings alone are passive components and cannot be used for the realization of MZ modulators, as a mechanism to modify the refractive is required to realise optical modulation.

Further Ref. [12] generically discloses a slow-light waveguide grating as a rib waveguide.

Object and subject-matter of the invention

It is the object of the present invention to solve the problems and to overcome the drawbacks of the prior art.

It is subject-matter of the present invention a system and a method according to the appended claims, that are an integral part of the present description.

Detailed description of invention’s embodiments

List of figures

The invention will be now described by way of illustration but not by way of limitation, with specific reference to the drawing of the enclosed figures, wherein:

- Fig. 1 shows a schematic representation of an exemplary structure of the slow-light waveguide according to the prior art: 3D view in (a) and cross sections (planes A, B, C) with parameters definition in (b)- (d); dashed volumes are Si, while the surrounding medium is Si02 and is dotted;

- Fig. 2 shows a fundamental band dispersion and group index of the two slow-light configurations described as examples in the present specification; - Fig. 3 shows a doping distribution: 3D view in (a) and top view in (b) for the apparatus of the invention, as well as field profile of the optical mode in (c) for the hereafter termed“SLOW50” waveguide at l = 1 .32 pm;

- Fig. 4 shows a doping profile within one single period (A) and cross- section of charge distribution as a function of voltage (B-F) for the SLOW50 waveguide. The snapshots are taken at a height of 150 nm from the bottom of the waveguide;

- Fig. 5 shows the difference in effective (phase) index between the biased and un-biased (no voltage applied) waveguides at different values of applied reverse voltage for hereafter termed“SLOW150”

(A) and SLOW50 (B) configurations;

- Fig. 6 shows an additional loss due to free-carrier absorption for both SLOW150 and SLOW50 configurations;

- Fig. 7 shows a modulation efficiency as a function of wavelength and applied reverse bias for SLOW150 (A) and SLOW50 (B) configurations;

- Fig. 8 shows (A) Structure layout for calculating the cutoff frequency,

(B) resistance times length, (C) capacitance per unit length, (D) 3-dB cutoff frequency /MB; and

- Fig. 9 shows a typical electro-optic modulator with Mach-Zehnder configuration, wherein an embodiment of the phase-shifter of the invention is used.

It is here specified that elements of different embodiments can be combined together to provide further unlimited embodiments respecting the technical concept of the invention, as the person skilled in the art will directly and unambiguously understand from what has been described.

The present description also refers to the prior art for its implementation, with respect to the non-described detailed features, such as for example elements of minor importance usually used in the prior art in solutions of the same type.

When an element is introduced, it always means that it can be "at least one" or "one or more".

When listing a list of elements or features in this description it is meant that the invention according to the invention "includes" or alternatively "is composed of" such elements.

Embodiments

The present invention concerns an electro-optic modulator that uses electro-optic phase shifters in its arms. Although we will refer to the modulator, the phase-shifter is intended as a specific independent element of the invention, that can be used alone in every suited application where a phase-shifter is required or advantageous, such as a phased antenna array or a Michelson interferometer. Moreover, although we will refer to the modulator as a Mach-Zehnder modulator with two arms, any other modulator that is capable of exploiting one or more phase shifters according to the invention is also intended, for example in applications in coherent communications.

The present invention is described with reference to a Mach-Zehnder modulator with two arms (see Fig. 9) that employs at least a slow-light waveguide grating (also called “corrugated waveguide”, or “grating waveguide”, see Fig.1 ) as a phase-shifter. As mentioned in the prior art section, the corrugation in the waveguide produces a band gap for the propagation of light, leading to increased group index and slowing down of light close to the band edge.

The invention introduces a p-n junction into such a grating structure of the phase-shifter, the p-n junction being realized across the waveguide direction and with a doping pattern that is periodic along the waveguide direction, in order to match at least partially the depletion regions of the p-n junction with the intensity profile of the travelling wave, as explained below. As it has been seen by the Inventors, this new configuration maximizes spatial overlap between the electromagnetic field and the regions where the refractive index is changed by the plasma dispersion effect. This, in turn, leads to an improved modulation efficiency and a reduced energy dissipation per bit, while it does not increase the overall insertion losses.

Although both a periodic p-n junction and a slow-light corrugated waveguide are present separately as such in the literature concerning electro-optic modulators, their combination was neither disclosed nor suggested.

In this respect, the Inventors found that, in order to improve electro-optic modulators or phase shifters, a key difficulty to be solved was to design a structure that:

- operates at a given target wavelength also termed “operational wavelength” (e.g. the O-band wavelength l=1 .3 micron, commonly in the range 1200-1600 nm, with values outside this range also being possible);

- employs a slow-light structure with a sufficient optical bandwidth (e.g. 10-30 nm); and

- implements a p-n junction that matches the travelling wave profile of the optical beam.

The efficient combination of the slow light structure and the pn junction described in the invention is non-trivial and has required extensive numerical modelling with both electrical and optical simulators, leading to the design parameters of structures like those of Fig. 3. The final designed and realized (prototype) modulator structure operates with the required specifications and can reach an energy consumption down to 0.3-0.5 pJ/bit, i.e. about one order of magnitude lower than with common modulators employing standard rib (corrugated) waveguides. In the following, we explain the invention in greater detail. In this respect, it is to be understood that the combination of a periodic p-n junction (a plurality of p-n junctions with a periodicity along the waveguide direction) and a slow-light grating is the basic solution leading to the main part of the advantages, but embodiments where specific parameters are specified (such as for example the type of periodicity and other shape parameters of the structure) provide more optimized versions of the basic structure.

Electrical structure design

As mentioned, an exemplary waveguide grating considered by the invention is displayed in Fig. 1 , which was systematically studied in the above previous work of the Inventors [7] Although any periodic grating can be used with the invention, we will refer, for the sake of simplicity, always to this specific grating.

In order to compare these prior art structures with the invention’s configuration, two example configurations optimized in [7] will be analyzed here. As it can be seen from Fig. 1 , such a grating structure WG shows a series of trenches T_w on both sides of the waveguide direction D_w, wherein:

- the trenches T_w are straight trenches substantially perpendicular to the waveguide direction D_w, forming a series of pair of trenches (along a same direction D_T ) with periodicity a along the waveguide direction;

- each pair of trenches has a total end-to-end length Wi across the waveguide direction D_w

- a central length CL_W of the waveguide connects the trenches r_w;

- the central length CL_w has a thickness of Wi perpendicularly to the waveguide direction D_w.

According to an aspect of the invention, the trenches T_w (and in general the rib waveguide used in the invention structure) have a shape that is symmetrical with respect to the waveguide direction. In this way, possible mixing of modes is avoided.

In general, according to the invention, the slow-light waveguide grating WG is a rib waveguide having a rib width Wi, W2 and extending along a waveguide direction D_w. The trenches are only an example of the fact that the rib width is varied with a periodicity a along the waveguide direction D .

The common parameters of both herein examined structures are the waveguide thickness 5 = 310 nm, the fill-factor di/a = 0.5 where a = di + cb is the period, and the widths Wi and W2 that are equal to 100 nm and 800 nm, respectively.

The two exemplary examined configurations differ mainly for the silicon thickness in the etched cladding regions, t: the first structure has t = 150 nm and is named SLOW150, while the second structure has t = 50 nm and is denoted as SLOW50. These values correspond to common technology platforms for optical interconnects [8]. To lock the lowest band edge at l = 1 .3 pm, the period is set as a = 218.1 nm and a = 234.7 nm for SLOW150 and SLOW50 configurations, respectively.

The photonic band dispersion and group index of the lowest TE mode are plotted in Fig. 2 for both SLOW150 and SLOW50 configurations. Both Si and Si02 are treated as dispersionless materials with a dielectric constant e = 12.299 and 2.093, respectively. The increase of the group index in the proximity of the band edge is clearly evident, and it is more prominent in the SLOW50 case, due to the greater strength of the periodic modulation in this configuration. We see, for example, that a group index n_g > 10 is compatible with a bandwidth of ~10 nm. Notice that the photonic dispersion lies entirely below the air light line (dispersion of light in air), thus the lowest TE mode is truly guided with no intrinsic losses, while actual propagation losses will be of extrinsic type and related to disorder-induced scattering at the vertical sidewall roughness. The designs fulfill the requirement of a minimum feature size ~100 nm, thus the structures may be fabricated by deep UV lithography with low losses.

In order to enhance the efficiency of the phase-shifter of the invention based on the plasma dispersion effect, it has been found very important to maximize the overlap between the region in which the refractive index is modified - i.e., the voltage-dependent depletion region of the p-n junction - and the optical mode of the waveguide. In prior art phase shifters based on a conventional rib waveguide, this is usually achieved by using a lateral p-n junction with an optimized junction position. More complex geometries like interleaved or U-shaped p-n junctions have also been considered. In these cases, the geometry of the p-n junction is chosen in order to optimize the overlap with the optical mode in the transverse directions, i.e., perpendicular to the waveguide axis.

The above structure has been modified with the basic design of the invention above to exploit a plasma dispersion effect, and optimization has been further performed. In this respect, note that in the previous work of the Inventors [7] the structure used therein is a passive component, i.e. a non- doped structure that therefore cannot be used in a modulator.

By way of illustration, and making e.g. reference to Fig. 3, the p-n junctions are interleaved. Note that the p-n junctions can also be simply periodic and not interleaved, see e.g. [2] Fig. 4 and that the depletion regions may be in different positions with respect to the transition in width between I/Vi and I/V2. Advantageously, the p-n junctions are interleaved in such a way that, in use, when the slow-light waveguide grating is fed with an optical beam that is at the operational wavelength and has a travelling wave profile along the waveguide direction, as well as powered with an electrical field, the depletion regions of the p-n junctions match the travelling wave profile.

Additionally, the grating WG and each of the interleaved p-n junctions PNJ may have the following areas and dimensions perpendicularly and across the waveguide direction:

- a p-n junction PNJ length Wi;

- a n or p doped region NR, PR connected to respectively the n or p region of the p-n junction PNJ outside the p-n junction length Wi up to an intermediate doped length W_d, and

- a n+ or p+ doped region NR+, PR+ outside of and respectively connected to the n or p doped region NR, PR up to a total doped length W_t.

In order to have a very good optimization of this structure, the condition Wi < W_d < W_t should be respected.

Wi, W_d, W_t are preferably chosen in the range 400 nm - 1 .2 micron, 600 nm - 1 .5 micron, and 2-4 micron, respectively.

The period of the p-n junctions (either interleaved or not) is preferably chosen to be twice the period a of the waveguide grating or a multiple of 2a, with an approximation of ± 0.5_<2. However, only for 4 a and 6 a (with the same approximation) the efficiency is large enough for most applications. Preferably the period can be in the range of 1 .8-2.5 a for the best performance, however the manufacturing of the device proved difficult, so that choosing 4 a can be a good trade-off between efficiency and ease of manufacturing.

A goal of the invention structure is not only to increase the overlap between the depletion region and the optical mode in the transverse direction, but also to exploit the reshaping of the field profile caused by the slow-light effect. Indeed, the electric field - see Fig. 3(c) - has the typical profile of an optical mode near the low energy edge of a band gap, i.e., it is a dielectric mode with an envelope that is stationary and featuring a strong localization of the field in the regions with higher (effective) index, i.e., in the thicker sections of the grating. As a consequence, this particular doping arrangement provides an ideal superposition between the depletion regions of the junction and the dielectric regions in which the electromagnetic field is more intense.

In the following an example of a simulation used to design the new electro-modulator of the invention is given.

In the design process illustrated, we assume doping concentrations N = 1 .4 x 10¹⁸ cm^-3 (in general, in the range 4-20x10¹⁷ cm^-3) and P = 5 c 10¹⁷ cm^-3 (in general, also in the range 4-20x10¹⁷ cm^-3) as in the literature, but this is not to be understood as a limitation to the possibility of the invention structure. Then, we calculate the charge density in the junction as a function of the applied reverse voltage by employing the software “Lumerical DEVICE”. The code solves the drift-diffusion equations, by applying the finite element method over an unstructured grid. Due to the periodic nature of the structure, the analysis of only one period a is needed, and symmetric boundary conditions are applied along the waveguide direction. The obtained distribution is then interpolated on a regular grid, for clearer visualization and further processing.

As we can see from the example in Fig. 4, the depletion region lies in the center of the wider section of the waveguide grating and its width increases with the applied voltage. As a consequence of the increased depletion region width, the refractive index of the waveguide is modified by the plasma dispersion effect. This leads to a phase shift of the propagating optical mode, which is calculated below. While the charge profiles of Fig. 4 are representative of the general behavior of the interleaved p-n junction in the grating waveguide, the detailed features of the depletion region depend on the exact values of the dopings and also on a possible offset OFF (see Fig. 3(b)) between the junction position and the center of the wide sections (in other words, an offset is provided from an end of each p-n junction to the center of the wide section, along the waveguide direction, in the case of an interleaved structure). Using a smaller p-type than n-type doping is justified by the fact that the plasma-dispersion effect (Eq. 1 a below) is stronger for holes than for electrons. A full optimization of doping levels and profiles has to consider the tradeoffs among various figures of merit (modulator efficiency, losses, cutoff frequency, ...) and is left to the adaptation to specific application of the invention by the skilled person. We should also notice that possible deviations or inhomogeneities in doping levels between the unetched and etched regions of the silicon material would not affect the main feature of the present structure, namely the doubled periodicity between electrical and optical profiles.

Optical design

Making reference to Fig. 1 (d), an embodiment of the electro-optic phase-shifter, PSH, of the invention can be based on prior art grating designs, wherein:

- the trenches T_w are straight trenches substantially perpendicular to the waveguide direction, forming a series of pair of trenches with periodicity a along the waveguide direction;

- each pair of trenches has a total end-to-end length Wi across (perpendicularly to) the waveguide direction;

- a central length CL_W of the waveguide connects the trenches;

- the central length CL_W has a length of Wi perpendicularly to the waveguide direction.

In general, trenches can also have shapes other than the rectangular one. Any shape periodically disposed is suitable for the invention.

According to a preferred aspect of the invention, the electro-optic phase-shifter PSH of the invention has Wi < W2.

Preferably, Wi is in the range of 0-400 nm, while Wi is in the range of 400-1200 nm. Wi and Wi are interchangeable without loss of generality.

Once the distribution of the free carriers is known, the effect of the applied voltage on the optical mode propagation can be quantified. For this purpose, we first describe the plasma dispersion effect by converting the charge distribution into a dielectric constant variation using the equations of Soref et al. with textbook coefficients:

An = -3.64 · 10^_10 T² iV - 3.51 · 10 ⁶T² P°·⁸, (l a)

Ak = 2.80 · 1(T⁵T³ JV + 1.91 · 10 ⁵T³ P, (l b) wherein N, P are the electrons and holes densities in cm^-3, l is the wavelength in meters (here l =1.3 pm), An and Ak are the differences in real and imaginary part of the refractive index, respectively. The resulting dielectric constant e + De, where Ae = 2 yT (An + iAk), is entered in the optical simulation, which then calculates the photonic mode dispersion by the aperiodic Fourier-modal method (A-FMM) [7] In the process, the dielectric constant in each unit cell is discretized using e.g. 40 slices, thus with a spatial resolution of about 5nm. The required Fourier transforms within a single slice are handled numerically.

In order to describe the performance of the phase shifter in terms of modulation efficiency and of propagation losses, we need to calculate the photonic mode dispersion in a periodic grating. In practice, two kinds of optical simulations can be performed. When only the change in the dispersion is needed (i.e., for calculating the phase shift), the optical simulation can be done considering only the change in the real part of the dielectric constant. This greatly simplifies the treatment, since the Bloch vector is purely real and the fitting procedure, as described in ref. [7] The more complete simulation, including also the imaginary part of the dielectric constant, is used to calculate the propagation loss due to free-carrier absorption, which can be extracted from the imaginary part of the Bloch vector.

In both cases, we need to calculate the scattering matrix for one period of the structure. The period of the waveguide with interleaved doping is twice the period a of the waveguide grating, see Fig 3(a), (b). Flowever, a simplification can be made by noting that adjacent periods of the waveguide grating, which are different from the electrical point of view, are the mirror image of each other and produce the same phase-shift for the optical mode. Therefore the doped waveguide is still treated as a periodic medium with period a. This assumption greatly simplifies the calculation, as the resulting band dispersion for the structure with period a can be fitted with an analytic formula with few parameters. We checked the validity of this approximation by comparing the resulting photonic dispersion with that calculated for the full doped waveguide with period 2a, finding excellent agreement (deviations on the band dispersion < 10 ⁷, compared to ~10^-3 variation due to the applied voltage).

The effect of the applied voltage on the band dispersion is displayed in Fig. 5 as the difference in the effective (phase) index of the optical modes between the biased and unbiased waveguide. The enhancement due to slow light is evident, as shown by the increase of the effective index difference towards the band edge. This effect is stronger in the SLOW50 than in the SLOW150 structure, since the SLOW50 structure has an increased slowdown factor that enhances the plasma dispersion effect.

In addition, in Fig. 6 we report the excess loss due to free-carrier absorption for both slow-light configurations, extracted from the imaginary part of the effective index of the mode as a = (20 / In 10) w/ceff/c. The plotted results refer to the unbiased waveguides, and represent the worst-case scenario, since reverse-biasing the junction reduces the absorption loss by lowering the free carrier density in the waveguide. It can be seen that both configurations exhibit an increased propagation loss towards the band edge, as is well known to happen for slow light.

Flowever the most relevant quantity is the total insertion loss of the phase shifter, which is evaluated in the next Section.

Results on phase-shifters The main figure of merit for a phase-shifter is usually taken to be VnU, namely the product of the driving voltage and of the length of the shifter that is needed to yield a p phase difference with respect to the unbiased case. VnU is commonly referred to as the modulation efficiency, and it generally increases as a function of the applied reverse voltage V, since the width of the depletion region increases sublinearly with V and so does the phase shift at fixed length. In general, VnU depends also on the wavelength, but in phase-shifters based on conventional rib waveguides this dependence is negligible. This is not the case in the slow-light waveguides that are the subject-matter of the present invention, because the phase modulation is closely related to the group index, which depends strongly on the wavelength.

As for the insertion loss, a common figure of merit is \L(U), obtained as the propagation loss per unit length a times the length U Here a is calculated at V = 0, while depends on the voltage: this should again be viewed as a quantification of losses in a worst-case scenario, and it applies to MZ modulators in a push-pull configuration where one of the two phase shifters is always kept at V = 0. A typical modulator operates with a phase shift ~0.15p in each arm, thus the length has to be L = 0.15 and the actual insertion loss is -0.15- 11.( ).

The obtained VnU as a function of wavelength is reported in Fig. 7 for both SLOW150 and SLOW50 configurations. The effect of slow light is clearly evident, as VnU decreases monotonically when moving towards the band edge, where the group index increases. Taking advantage of this effect, values of VnU around 0.1 Vcm can be reached, although within a small spectral window (1 -2nm wide) in the proximity of the band edge. This value for the modulation efficiency is about one order of magnitude lower than those obtained in conventional ridge waveguides, where VnU is typically above 1 V.cm [15]. Performance of the slow-light structures remains good even when a greater bandwidth is required. In particular, the SLOW50 configuration at low voltage shows a VnU lower than 0.3 Vcm over a spectral window of about 30nm, which is adequate for many telecom and datacom applications. This wide bandwidth follows from the spatial matching of the electromagnetic field profile with the depletion region of the p-n junction, which is maintained in a wide spectral region below the band edge.

To better see the advantage of the present slow-light design, we present in Tab. 1 a comparison between the calculated performance of the SLOW50 structure and those of two prior art representative rib waveguides with lateral and interleaved p-n junctions, respectively [9]. Considering first the modulation efficiency VnU, we notice that the interleaved p-n junction gives an improvement over the lateral p-n junction even in a normal rib waveguide configuration [9]. When the slow-light structure is considered, VnU is further reduced to values as low as 0.1 -0.3 Vcm at 5-10 nm from the band edge l =1 .3 pm. This clearly shows the advantage of combining slow light with the interleaved p-n junction, which strongly and surprisingly improves the performance in a wide bandwidth.

Configuration V_*U <x @0V IL(U)

(V cm) (dB/cm) (dB)

1 V 3V 1 V 3V

Rib Lateral [9] 1 .180 1 .500 9.2 10.9 4.60 Rib Interleaved [9] 0.510 0.700 13.5 6.90 3.10 SLOW50@1330 nm 0.304 0.617 21.7 6.60 4.47 SLOW50@1320 nm 0.257 0.526 25.6 6.57 4.49

SLOW50@1310 nm 0.191 0.395 34.2 6.52 4.50

SLOW50@1305 nm 0.143 0.297 45.0 6.42 4.45

Tab. 1 : Modulation efficiency VnU, loss per unit length at V = 0, and insertion loss \L{U). Both VnU and \L(U) are given for two different values of the applied reverse bias. Rib lateral (interleaved): normal rib waveguide configuration with lateral (interleaved) p-n junction [9] SLOW50: slow-light configuration of this work with t = 50nm, at four different wavelengths.

In terms of losses, the propagation loss in dB/cm increases when going from the normal (non-corrugated waveguide and lateral p-n junction), to the interleaved (non-corrugated but with interleaved p-n junction), to the slow- light structures (invention structure): however, \L{ ) is nearly the same for all configurations. In practice, this means that the optimal length of a slow- light modulator can be a fraction of a mm, i.e. much smaller than in conventional modulators, thus yielding the same total insertion loss but with a strongly improved modulation efficiency.

Reducing the modulator length has also the advantage of improving tolerance against disorder-induced losses due to fabrication processes. A modulator length below 500 pm means that propagation losses up to several 10 dB/cm can be tolerated, which is well within the reach of current fabrication technologies, with margins of improvement thanks to the use of immersion lithography.

It is also interesting to quantify the effect of p-n junction geometry in the slow-light structure. To this purpose, we calculated the figures of merit for the SLOW50 configuration but using a lateral p-n junction with the same doping levels. At l = 1330, 1320, 1310, 1305 nm and V = 1 V we obtain VnU = 0.94, 0.79, 0.59, 0.45 Vcm, respectively, which are nearly three times larger than for the interleaved-only p-n junction. For the same wavelengths, the propagation losses at V = 0 are 28.7, 34.0, 45.8, 60.4 dB/cm, leading to IL(L_TT)~27 dB at 1 V, which is much higher than the values reported in Tab. 1. Thus, we conclude that the use of the interleaved p-n junction greatly improves the efficiency and reduces the insertion loss of the slow-light modulator, thereby demonstrating the unexpected key role of spatial matching between the p-n junction and the optical field profile. To determine the dynamic behavior of the phase shifter, which is a crucial property for applications, we analyse the frequency response in the small-signal approximation with Lumerical DEVICE. The resistance and the capacitance are obtained from the complex impedance Z = R + Mi oiC). The 3dB cutoff frequency is calculated as /3dB = 1 /(27rf?C), and it is independent of the modulator length L, since C oc L and R oc ML. The simulation setup now includes additional highly doped p+ and n+ regions, see Fig. 8(A). We choose Wd =1 .2 pm, a total width W\ = 2.0 pm, and dopings of 1 c 10¹⁹ cm^-3 in the p+ and n+ regions, with metal contacts at the borders. With these values the waveguide propagation losses are not increased over those reported in Fig. 6, as we have verified. The results as a function of bias voltage are reported in Figs. 3(b) and 8. The behavior is similar for the two slow-light configurations, showing an increase of resistance and a decrease of capacitance for increasing voltage, both effects being caused by an expansion of the depletion region. The SLOW50 configuration has higher resistance and lower capacitance compared to the SLOW150 configuration, the first effect dominates and leads to a smaller cutoff frequency. The SLOW150 configuration can reach /MB > 12.5 GFIz, which can sustain a bit rate up to 25 Gbps with non-return to zero format, or even up to 50 Gbps with PAM-4 encoding (as per se shown in the prior art). The corresponding rates for the SLOW50 configuration are about a factor of two smaller. Further improvements of the dynamic behavior can be obtained by the skilled person in specific applications by tailoring the doping levels and the spatial profiles of both low- and highly doped regions.

The advantage of the slow-light modulator design becomes more useful at small driving voltage, which has a beneficial effect on the energy dissipation given by CV²12. To give an example, we consider both SLOW50 and SLOW150 configurations at 1 V reverse bias. The capacitances per unit length (we consider the average values at 0.5V) are C/L = 1.8 pF/mm and 2 pF/mm, respectively. Working at l = 1310 nm, the efficiencies are VnU = 0.191 Vcm and 0.297Vcm, therefore a phase shift of 0.15p is obtained with length L = 0.15 Ln = 290 pm for SLOW50 and L = 450 pm for SLOW150 configuration. The energy dissipation is then 0.26 pJ/bit for SLOW50 and 0.45 pJ/bit for SLOW150. These values are reduced up to one order of magnitude compared to typical modulators based on normal rib waveguides [10, 1 1 ].

The following list reports a range of values for the different parameters of the invention structure, together with preferred single values. These parameters are given as a specific implementation and optimization of the basic technical concept of the invention. Each parameter range may improve the basic invention in a corresponding preferred way.

Tab. 3: Inventions parameters’ values

The operational wavelength l should be understood as a target parameter, obtained by optimization of the others.

The preferred silicon cladding and silicon core thickness are given for two different silicon standards. Another widespread silicon standard is for a core thickness s=220 nm. When using this standard, values of the other parameters can be properly optimized by the skilled person.

According to the invention, a method for encoding a bit stream onto a carrier optical wave, may comprise the following steps:

- Providing the electro-optic modulator MZM as defined above, with one or two electro-optic phase shifters PSH each having a slow-light waveguide grating WG with a waveguide direction D_w and an operational wavelength;

- Feeding the slow-light waveguide grating WG with an optical beam that is a carrier optical wave at the operational wavelength having an bit stream encoded in it;

- Powering the plurality of n-p junctions with an electrical field. Advantages

The invention structure is a novel combination of band-edge slow light in a waveguide grating with an interleaved p-n junction along the propagation direction, which has the same period of the optical waveguide. This leads to optimal matching between the propagating electric field and the depletion regions of the p-n junctions, and results in a strongly improved modulation efficiency over a bandwidth up to 20-30 nm. The use of an interleaved p-n junction has been seen especially useful to increase the bandwidth of the slow-light modulator, as the spatial matching between the depletion region and the field profile in the waveguide grating is wavelength- independent. The modulation efficiency VnU is in the range 0.1 -0.5Vcm, depending on the operating bandwidth and on the driving voltage, the best performances being obtained at low voltage V ~ 1 V.

It is important to notice that the total loss of the invention slow-light design is not increased with respect to modulators based on normal rib waveguides. On the other hand, the use of an interleaved p-n junction significantly reduces the loss of the slow-light structure relative to a lateral p-n junction: e.g., II _~{U)~ 27 dB and ~ 6.5 dB with lateral and interleaved p- n junction, respectively (the corresponding insertion losses for a length L = 0A 5U are 4 and 1 dB). In summary, the use of the interleaved p-n junction in the slow-light modulator improves the efficiency and reduces the free- carrier induced insertion losses. A proper tradeoff between modulation rate, losses, and energy consumption will be determined for specific applications by the skilled person.

Prospects of application

Photonic integrated circuits are fabricated in industry using rib/ridge waveguides, most commonly in the silicon-on-insulator (SOI) platform. A waveguide grating or corrugated waveguide is a modification of a rib waveguide that has wide and narrow sections, and it can be connected with standard rib waveguides by low-loss tapers. Thus, the modulator described in the present invention is compatible with industrial standards on present- day technology platform, as used by leading industries. The use of a periodic (interleaved) p-n junction together with a slow-light waveguide ensures optimal matching between the travelling wave and the depletion region, improving the modulation efficiency and reducing the energy dissipation per bit. The present invention can therefore be implemented in industry with little or no changes to the present-day technology processes.

Bibliography

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[1 1 ] E. Temporiti, A. Ghilioni, G. Minoia, P. Orlandi, M. Repossi, D. Baldi, and F. Svelto. Insights into silicon photonics Mach-Zehnder-based optical transmitter architectures. IEEE J. Sol. State Circuits, 51 (12):3178—3191 , 2016. [12] RAN HAO ET AL: "Increasing the bandwidth of slow light in fishbone like grating waveguides", Photonics Research, vol. 7, no. 2, 1 February 2019 (2019-02-01 ), page 240, XP55660787, ISSN: 2327- 9125, DOI: 10.1364/PRJ.7.000240.

In the foregoing, the preferred embodiments have been described and variations of the present invention have been suggested, but it is to be understood that those skilled in the art will be able to make modifications and changes without thereby departing from the relevant scope of protection, as defined by the attached claims.

Claims

1) An electro-optic phase-shifter (PHS), comprising:

- A slow-light waveguide grating (WG) wherein:

o The slow-light waveguide grating (WG) is a rib waveguide having a rib width (Wi,W2) and extending along a waveguide direction (D_w); o The rib width (Wi,W2) is varied with a periodicity a along the waveguide direction (D_w);

- A plurality of periodic p-n junctions (PNJ);

Wherein the periodic p-n junctions (PNJ) are provided by doped regions made across the waveguide direction (D_w) and having periodicity equal to 2a, or 4 a or 6a, with approximation of ± 0.5a, along the waveguide direction (Dw).

2) An electro-optic phase-shifter (PHS) according to claim 1 , wherein the periodicity of the p-n junctions is in the range of 1.8-2.5 a.

3) An electro-optic phase-shifter (PHS) according to claim 1 or 2, wherein:

- The rib width is varied between values Wi and Wr,

- Wi is smaller than

- Wi is in the range of 0-400 nm; and

- Wi is in the range of 400-1200 nm.

4) An electro-optic phase-shifter (PSH) according to one or more claims 1 to 3, wherein:

- The slow-light waveguide grating (WG) has an operational wavelength; and

- The periodic p-n junctions (PNJ) are interleaved with the slow-light waveguide grating (WG); Wherein the p-n junctions (PNJ) are interleaved in such a way that, in use, when the slow-light waveguide grating (WG) is fed with an optical beam that is at the operational wavelength and has a travelling wave profile along the waveguide direction (D_w), as well as powered with an electrical field, the depletion regions of the p-n junctions (PNJ) match the travelling wave profile.

5) An electro-optic phase-shifter (PSH) according to claim 4, wherein each p-n junction in the plurality of interleaved p-n junctions (PNJ) has the following areas and dimensions perpendicularly and across the waveguide direction:

- a p-n junction (PNJ) length Wi;

- a n or p doped region (NR, PR) connected to respectively the n or p region of the p-n junction outside the p-n junction length Wi up to an intermediate doped length W_d, and

- a n+ or p+ doped region (NR+, PR+) outside of and respectively connected to the n or p doped region (NR, PR) up to a total doped length

Wi.

6) An electro-optic phase-shifter (PSH) according claim 5, wherein:

- Wi is chosen in the range 400 nm - 1 .2 micron;

W_d is chosen in the range 600 nm - 1 .5 micron;

W_t is chosen in the range 2-4 micron; and

Wi < W_d < W_t .

7) An electro-optic phase-shifter (PSH) according to one or more of the previous claims, wherein the slow-light waveguide grating (WG) is made of Silicon on Insulator. 8) An electro-optic phase-shifter (PSH) according to claim 7, wherein the Silicon on Insulator has unetched silicon thickness 5 and etched silicon thickness t, where t < s and 5 is in the range 200-350 nm, while t is in the range 20-200 nm.

9) An electro-optic modulator (MZM) having a Mach-Zehnder configuration with two arms comprising one or two corresponding electro optic phase shifters (PSH), wherein the one or two corresponding electro optic phase shifters are defined as in one or more claims 1 to 8.

10) A method for encoding a bit stream onto a carrier optical wave, comprising the following steps:

- Providing the electro-optic modulator (MZM) as defined in claim 9 with one or two electro-optic phase shifters (PSH) each having a having slow- light waveguide grating (WG) with a waveguide direction (D_w) and an operational wavelength;

- Feeding the slow-light waveguide grating (WG) with an optical beam that is a carrier optical wave at the operational wavelength having a bit stream encoded in it;

- Powering the plurality of n-p junctions with an electrical field.