Rib waveguides for transverse-magnetic polarization silicon-photonic modulator

Abstract

To provide an electro-optic modulator in silicon photonics which can achieve a higher bandwidth and / or lower loss than other electro-optical modulators.SOLUTION: A silicon-photonic optical modulator includes: an optical input; and an optical waveguide that is connected to the optical input and that is configured to propagate quasi-transverse-magnetic (quasi-TM) polarized light. The optical waveguide is configured as a rib waveguide that includes a rib arranged on a slab. The rib includes at least one dopant. An average concentration of the at least one dopant in a vertical doping profile in a lowermost portion of the rib is larger than an average concentration of the at least one dopant in the vertical doping profile in an uppermost portion of the rib. The uppermost portion of the rib has a height that is between 20% and 80% of a height of the rib waveguide. The lowermost portion of the rib includes a remainder of the rib below the uppermost portion.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present disclosure generally relates to electro-optic modulators in silicon photonics.

Background Art

[0002] In an optical communication system, an electro-optic modulator provides a basic mechanism for modulating an optical waveform to transmit information. Generally, an electro-optic modulator operates by changing one or more characteristics of an optical waveform in response to information such as digital data provided by an electrical signal.

Summary of the Invention

[0003] Embodiments of the present disclosure generally relate to electro-optic modulators in silicon photonics. Some aspects of the present disclosure relate to a silicon photonic optical modulator including an optical input and an optical waveguide connected to the optical input and configured to propagate quasi-transverse magnetic (quasi-TM) polarization. The optical waveguide is configured as a rib waveguide including a rib disposed on a slab. The rib includes at least one dopant. The average concentration of at least one dopant in the bottom vertical doping profile of the rib is greater than the average concentration of at least one dopant in the top vertical doping profile of the rib. The top of the rib has a height between 20% and 80% of the height of the rib waveguide. The bottom of the rib includes the remainder of the rib below the top.

[0004] This modulator and other modulator embodiments described herein can have at least one or more of the following features.

[0005] In some embodiments, the average concentration of at least one dopant at the bottom is at least 1.5 times the average concentration of at least one dopant at the top.

[0006] In some embodiments, the average concentration of at least one dopant at the bottom is at least twice the average concentration of at least one dopant at the top.

[0007] In some embodiments, the height of the top is between 35% and 65% of the height of the rib waveguide.

[0008] In some embodiments, the average concentration of at least one dopant at the bottom is 10 17 cm -3 to 10 18 cm -3 and the average concentration of at least one dopant at the top is less than 5×10 16 cm -3 .

[0009] In some embodiments, at least one dopant includes a first dopant in a first lateral portion of the rib and a second dopant in a second lateral portion of the rib, with the second lateral portion facing the first lateral portion. The first lateral portion and the second lateral portion form a semiconductor junction diode.

[0010] In some embodiments, the concentration of the first dopant in the first vertical doping profile in the first lateral portion of the rib is higher in the lower part of the rib than in the upper part of the rib, and the concentration of the second dopant in the second vertical doping profile in the second lateral portion of the rib is higher in the lower part of the rib than in the upper part of the rib.

[0011] In some embodiments, the silicon photonic optical modulator includes an electrode configured to apply an electric field to the semiconductor junction diode.

[0012] In some embodiments, the silicon photonic optical modulator includes a semiconductor contact region where the electrode contacts. The height of the semiconductor contact region is greater than the height of the slab.

[0013] In some embodiments, the effective refractive index of the TM-polarized two-dimensional (2D) waveguide mode in the rib waveguide is greater than the effective refractive index of the transverse electric field (TE)-polarized one-dimensional (1D) waveguide mode in the slab.

[0014] In some embodiments, the optical waveguide is a first optical waveguide. The silicon photonic optical modulator includes a Mach-Zehnder interferometer including the first optical waveguide and a second optical waveguide. The first optical waveguide includes a first semiconductor junction diode based on at least one dopant, and the second optical waveguide includes a second semiconductor junction diode based on at least one dopant.

[0015] Some aspects of the present disclosure relate to another silicon photonic optical modulator including an optical input; an optical waveguide configured to receive light from the optical input, the optical waveguide being configured as a rib waveguide including a rib disposed on a slab, the rib waveguide having a shape configured to propagate quasi-transverse magnetic (quasi-TM) polarization; and an electrode configured to apply an electric field across the rib waveguide. The width of the rib waveguide is in the range of 250 nm to 400 nm.

[0016] Embodiments of this modulator and other modulators described herein can have at least one or more of the following features.

[0017] In some embodiments, the height of the rib waveguide is greater than the width of the rib waveguide.

[0018] In some embodiments, the height of the rib waveguide is in the range of 300 nm to 400 nm, and the thickness of the slab is in the range of 50 nm to 150 nm.

[0019] In some embodiments, the width of the rib waveguide is in the range of 250 nm to 360 nm.

[0020] In some embodiments, the optical waveguide is a first rib waveguide, and the silicon photonic optical modulator includes a second rib waveguide. The spacing between the first rib waveguide and the second rib waveguide is less than 500 nm.

[0021] In some embodiments, the height of the first rib waveguide is at least 10 nm greater than the height of the second rib waveguide in at least a portion of the silicon photonic optical modulator.

[0022] Some aspects of the present disclosure relate to a method of manufacturing a silicon photonic optical modulator. The method includes forming a rib waveguide on a substrate, the rib waveguide including a rib disposed on a slab; and implanting at least one dopant into the rib. The average concentration of at least one dopant in the vertical doping profile at the bottom of the rib is greater than the average concentration of at least one dopant in the vertical doping profile at the top of the rib. The top of the rib has a height between 20% and 80% of the height of the rib waveguide, and the bottom of the rib includes the remainder of the rib below the top.

[0023] This method and other methods described herein can have at least one or more of the following features.

[0024] In some embodiments, the step of implanting at least one dopant into the rib includes directing a beam of at least one dopant at the rib in a direction from a side of the substrate on which the rib is disposed toward the substrate.

[0025] In some embodiments, the beam has an acceleration energy in the range of 50 keV to 230 keV.

[0026] Some aspects of the present disclosure relate to a method of modulating quasi-transverse magnetic (TM) polarization. The method includes inputting an input quasi-TM polarization into an optical waveguide, the optical waveguide being configured as a rib waveguide including a rib disposed on a slab, the rib including at least one dopant, an average concentration of the at least one dopant in a bottom vertical doping profile of the rib being greater than an average concentration of the at least one dopant in a top vertical doping profile of the rib, a top of the rib having a height between 20% and 80% of a height of the rib waveguide, a bottom of the rib including the remainder of the rib below the top; and applying at least one electric field across the rib waveguide.

[0027] Details of one or more embodiments of the subject matter of the present disclosure are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims.

Brief Description of the Drawings

[0028]

Figure 1

[0029]

Figure 2

[0030]

Figure 3

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Figure 4

[0032] Figure 4B is a diagram showing an example of a vertical doping profile in the waveguide of Figure 4A according to some embodiments of the present disclosure.

[0033] Figures 4D to 4F are diagrams showing an example of a vertical doping profile in the waveguide of Figure 4C according to some embodiments of the present disclosure.

[0034] Figure 4H is a diagram showing an example of a vertical doping profile in the waveguide of Figure 4G according to some embodiments of the present disclosure.

[0035]

Figure 5

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Figure 6

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Figure 7

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Figure 8

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Figure 9

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Figure 10

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Figure 11

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Figure 12

DETAILED DESCRIPTION

[0043] (DETAILED DESCRIPTION) This specification discloses systems and techniques for providing an electro-optic modulator in silicon photonics that can achieve a higher bandwidth and / or lower loss compared to other electro-optic modulators. In some embodiments, this is achieved by a rib waveguide incorporating a non-uniform vertical doping profile. The non-uniform doping can reduce the electrical resistance that occurs when modulating the modulator and can improve the switching speed. In some embodiments, the rib waveguide has a wide rib, significantly reducing the sidewall-related loss of the transverse magnetic (TM) polarization propagating through the waveguide. These and other features described herein can be incorporated into an electro-optic modulator having a structure that can use TM polarization within the modulator instead of transverse electric (TE) polarization. In some embodiments, this is made possible by a rib waveguide structure where the height of the waveguide is greater than the width of the waveguide. As a result, generally within the rib waveguide, TM light having a higher effective refractive index than TE light is obtained based on an appropriate combination of rib dimensions (e.g., rib height and / or width) and slab dimensions (e.g., slab thickness).

[0044] FIG. 1 is a diagram showing an example of a top view of a differential modulator 100 in which embodiments of the present disclosure can be utilized. In this example, the modulator 100 is based on the implementation of a Mach-Zehnder interferometer (MZI), and an optical signal propagates along two optical transmission paths 102 and 104 (e.g., from left to right in FIG. 1) along the length of the modulator 100. At the input of the modulator 100, an optical splitter 106 splits the input optical signal into two optical transmission paths 102 and 104. At the output of the modulator 100, an optical combiner 108 combines the optical signals output from the two optical transmission paths 102 and 104. The optical splitter 106 and the optical combiner 108 can be implemented in various ways using, for example, symmetric, asymmetric, or adjustable optical intensity couplers. The optical transmission paths 102 and 104 can be implemented by waveguides formed in a semiconductor structure 116, as will be described in more detail with reference to FIG. 2 below. In some embodiments, the optical core of the waveguide and / or the optical splitter 106 and / or the optical combiner 108 can include a silicon rib. For example, the transmission paths 102 and 104 can include rib waveguides such as waveguide 420 and / or waveguide 450. In some embodiments, an optical phase rotator that rotates the phase of the input optical signal so that quasi-TM light propagates along the optical transmission paths 102 and 104 may be implemented between the input of the modulator 100 and the optical transmission paths 102 and 104.

[0045] Modulator 100 uses a traveling wave configuration that generates an electrical signal in which the voltages applied to terminals 110 and 112 propagate along radio frequency (RF) transmission line 114 terminated by an RF termination resistor. The electrical signal in RF transmission line 114 propagates at the same speed as the light propagating along the two optical transmission paths 102 and 104, inducing an electro-optic modulation. In particular, RF transmission line 114 is connected to semiconductor structure 116 via electrodes (described in more detail with reference to FIG. 2 below) that apply the respective voltages and the resulting electric fields across one or both of optical transmission paths 102 and 104. The applied voltage induces a phase shift in the light propagating in one or both of optical transmission paths 102 and 104. In some embodiments, the phase shift is differential in that the magnitude of the phase shift is equal between optical transmission paths 102 and 104 and the signs of the phase shifts are opposite.

[0046] Electro-optic modulation is achieved by varying the voltage at one or both of terminals 110 and 112 to modulate the differential phase shift between the phase of the light in the first optical transmission path 102 and the phase of the light in the second optical transmission path 104. For example, if the terminal voltages are controlled such that the differential phase shift causes destructive interference at optical combiner 108, this corresponds to the “off” or logic “0” state of modulator 100. In contrast, if the terminal voltages are controlled such that the differential phase shift between the two optical transmission paths 102 and 104 causes constructive interference at optical combiner 108, this corresponds to the “on” or logic “1” state of modulator 100.

[0047] Also, the differential phase shift between the two optical transmission paths 102 and 104 may be affected by other factors. For example, the physical lengths of the optical transmission paths 102 and 104 may be the same to provide zero intrinsic differential phase shift, or different lengths to provide non-zero intrinsic differential phase shift. Further, in some embodiments, DC (direct current) phase shifters 122 and 124 (such as thermo-optic phase shifters like optical waveguide heaters) may be implemented near the ends of the optical transmission paths 102 and 104 to control the relative phases of the two optical signals before they are combined at the optical combiner 108.

[0048] In some embodiments, phase modulation may be performed by a "push-pull" mechanism in which the phases of the light in both the optical transmission paths 102 and 104 are modulated to control the relative phase shift between the two paths. In the push-pull operation, as the voltage V+ at the terminal 110 increases and the voltage V- at the terminal 112 decreases (or vice versa), corresponding optical phase shifts occur in each of the optical transmission paths 102 and 104. Push-pull modulation can offer various advantages such as lower average power consumption and reduced chirp of the modulation signal compared to non-push-pull modulation.

[0049] In some scenarios, a DC (direct current) bias connection 118 may be connected between the two optical transmission paths 102 and 104. The DC bias connection 118 is implemented such that the semiconductor junction diodes of the optical transmission paths 102 and 104 each maintain a reverse bias even when the data signals applied to the terminals 110 and 112 change between logic 1 and logic 0. Further details will be described with reference to FIG. 2 below.

[0050] FIG. 2 is a diagram showing an example of a cross-section of the modulator 200 (e.g., cross-section 126 of the modulator 100 in FIG. 1).

[0051] The cross-section of the modulator 200 shows the details of the MZI structure. The MZI includes a first optical waveguide 202 and a second optical waveguide 204. In some embodiments, the modulator 200 includes a substrate 206 (e.g., a silicon substrate), an insulating structure / layer 208 (e.g., a dielectric such as an oxide), and a semiconductor structure 210 (e.g., a silicon layer including the optical waveguides 202 and 204).

[0052] The optical waveguides 202 and 204 can be implemented, for example, as rib waveguides on a slab. In the example of FIG. 2, the optical waveguide 202 includes a rib 203 disposed on a slab 205. Similarly, the optical waveguide 204 includes a rib 207 on a slab 209. The ribs 203, 207 and the slabs 205, 209 are all part of the semiconductor structure 210. Further details of the rib waveguide structure will be described with reference to FIGS. 4A-4H below.

[0053] Each of the optical waveguides 202 and 204 includes a semiconductor junction. The semiconductor junction diode can be implemented, for example, as a PIN (p-type / intrinsic / n-type) junction diode, or a p / n junction diode. In the modulator 200, a p / n junction is embedded in each of the optical waveguides 202, 204 to form a diode within each waveguide. These diodes are shown as a first semiconductor junction diode 212 and a second semiconductor junction diode 214. As described with respect to FIGS. 4A-4H, in some embodiments, the vertical doping profiles of the ribs 203, 207 are non-uniform.

[0054] Furthermore, modulator 200 includes electrodes 216 and 218 (e.g., metal electrodes) that are in physical contact with semiconductor structure 210. In some embodiments, electrodes 216 and 218 are in physical contact with p-doped contact regions 220 and 222 of semiconductor structure 210. Electrodes 216 and 218 may be formed, for example, by etching insulating layer 208 and forming metal (e.g., tungsten, copper, and / or aluminum) contacts. In some embodiments, in modulator 200, the p-doped region may alternatively be an n-doped region, and vice versa (e.g., such that contact regions 220 and 222 are n-doped instead of p-doped).

[0055] In some embodiments, one or both of electrodes 216, 218 are in contact with a portion of a semiconductor material (such as silicon) having a thickness greater than the slab thickness (e.g., h slab,1 or h slab,2 ) and, for example, are in contact with a protruding portion of the semiconductor material. For example, the vertical height h con of one or both of contact regions 220, 222 (shown in FIG. 2) can be made greater than the slab thickness. This provides a more robust electrical contact than contacting the slab directly or contacting a portion of the semiconductor material having the same thickness as the slab. In various embodiments, h con can be made greater than, less than, or equal to the height h core of the rib. In some cases, making h con equal to h core can facilitate simplification of manufacturing, for example, because the contact region and the rib can be defined in a common manufacturing process. The foregoing description of contact regions 220, 222 is applicable to other contact regions described herein, such as contact regions 820, 822.

[0056] Furthermore, modulator 200 may include metal layers 224 and 226 over electrodes 216 and 218. In some embodiments, metal layers 224 and 226 may form segments of an RF transmission line (e.g., RF transmission line 114 of FIG. 1).

[0057] In some scenarios, a DC bias connection 228 is implemented between two optical waveguides 202 and 204. The DC bias connection 228 ensures that the semiconductor junction diodes 212 and 214 maintain a reverse bias during modulation. For example, in push-pull mode modulation, a differential voltage (e.g., V+ and V−) is applied to the metal layers 224 and 226 (and thus to the electrodes 216 and 218). When the voltage of the first electrode 216 (e.g., V+) increases and the voltage of the second electrode 218 (e.g., V−) decreases, the width of the depletion region of the first optical waveguide 202 decreases, while the width of the depletion region of the second optical waveguide 204 increases (and vice versa). When the depletion width changes, this causes a change in the effective refractive index experienced by the light traveling along each of the optical waveguides 202 and 204, resulting in a corresponding phase shift of the light. As a result, push-pull modulation can be achieved in the modulator 200.

[0058] In an example of the modulator 200, the DC bias connection 228 is applied to the cathodes 230 and 232 (N-doped regions) of the semiconductor junction diodes 212 and 214, and the varying voltages V+ and V− are applied to the anodes 234 and 236 (P-doped regions) of the semiconductor junction diodes 212 and 214. The DC bias connection 228 ensures that the semiconductor junction diodes 212 and 214 maintain a reverse bias. For example, in an example of the modulator 200, if the bias voltage applied at the DC bias connection 228 is very low (or non-existent), this can lead to the activation of the first semiconductor junction diode 212 (e.g., a forward bias of about 0.6 V or more with respect to silicon) by a significant number of carriers injected into the depletion region of the first semiconductor junction diode 212, resulting in a forward bias and slow operation. By implementing the DC bias connection 228 with a sufficiently large bias voltage, it is ensured that the semiconductor junction diodes 212 and 214 maintain a reverse bias under modulation.

[0059] FIG. 3 is a diagram showing an exemplary equivalent circuit 300 along a cross-section of a modulator (e.g., cross-section 126 of the modulator 100 in FIG. 1).

[0060] In the example of FIG. 3, the electrical series resistance 340 between the first electrode 316 and the first semiconductor junction diode 312 (corresponding to, for example, the semiconductor region 240 in FIG. 2) is denoted as Rp (for example, the unit is mΩ-m). The electrical series resistance 342 between the second electrode 318 and the second semiconductor junction diode 314 (corresponding to, for example, the semiconductor region 242 in FIG. 2) is also denoted as Rp (however, in some embodiments, the actual values of the electrical series resistances 340 and 342 may be different). The electrical series resistance 338 between the semiconductor junction diodes 312 and 314 (corresponding to, for example, the semiconductor region 238 in FIG. 2) is 2*Rn (where Rn is the series resistance between each of the semiconductor junction diodes 312 and 314 and the DC bias voltage connection portion 328).

[0061] In the equivalent circuit of the phase modulator shown in FIG. 3, the resistances Rn and Rp mainly originate from the slab and may limit the bandwidth of the modulator. Increasing the doping of the slab reduces the resistance and increases the bandwidth, but since the doped silicon is absorptive, the optical loss also increases.

[0062] In addition, each semiconductor junction diode 312, 314 has an associated series resistance R (sometimes called rib resistance) corresponding to the charge transport (for example, the lateral charge transport between the n-doped side and the p-doped side of each rib 203, 207 above the slab) through the ribs 203, 207 between the cathodes 230 and 232 and the anodes 234 and 236. r (Sometimes also called rib resistance). For the purposes of the present disclosure, this rib resistance is related to the resistances R n and R pInstead of, or in addition to, it is recognized that it can also be an important or main limiter of the modulator bandwidth. It is further recognized that the upper parts of ribs 203 and 207 contribute relatively little to the phase modulation based on depletion region adjustment compared to the lower parts of ribs 203 and 207. Thus, some embodiments according to the present disclosure feature non-uniform n-doping and / or p-doping in the vertical direction in one or more waveguide ribs (e.g., high doping levels in the lower parts of the ribs and low or no doping levels in the upper parts of the ribs), reducing the rib capacitance C, and as a result, the product R r C 2 and / or R r is reduced, providing a higher bandwidth modulator. Further details regarding the non-uniform doping in the vertical direction will be described with reference to FIGS. 4A - 4H.

[0063] FIG. 4A shows in cross-section an example of a single waveguide of a silicon photonic depletion phase modulator (e.g., a waveguide in one of transmission paths 102 or 104 of modulator 100, or one of waveguides 202 or 204 in FIG. 2). In particular, FIG. 4A shows an example of a waveguide 400 configured for TE polarization and having a uniform vertical doping profile. Waveguide 400 is implemented by a rib waveguide structure having a rib 402 on a slab 404. Light is guided along rib 402 and propagates in the longitudinal direction y of the modulator (the direction perpendicular to the x - z cross-section shown in FIG. 4A as indicated by coordinate system 412) by total internal reflection within rib 402. Rib 402 extends in the vertical direction z from slab 404. For example, the vertical direction z may be orthogonal to the surface of the substrate on which waveguide 400 is formed, such as substrate 206. The lateral direction x is orthogonal to the vertical direction z and the longitudinal direction y, defines a direction that traverses rib 402 and is parallel to the electric field generated at the p / n junction within rib 402.

[0064] The waveguide 400 includes an n-type region 406 on one side of the slab 404 and the rib 402, and includes a p-type region 408 on the opposite lateral side surfaces of the slab 404 and the rib 402. The regions 406 and 408 form a semiconductor junction diode. The intermediate region 410 is a depletion region that is naturally formed between the n-type and p-type regions 406 and 408 (for example, when the n-type and p-type regions 406 and 408 are directly joined laterally to form a p / n diode), and / or may include an intrinsic undoped region when the regions 406, 408, and 410 form a PIN diode.

[0065] The rib structure enables an optical mode confined within the rib 402 while allowing electrical connection to the rib 402 through regions on both sides of the slab 404. As described above with reference to FIG. 2, the phase modulation of the light within the rib 402 is achieved by modulating the voltage difference between the n-doped region 406 and the p-doped region 408 of the waveguide 400. For example, increasing the voltage difference between the n-doped region 406 and the p-doped region 408 widens the width of the depletion region in the intermediate region 410, thereby increasing the effective refractive index of the optical mode and enabling phase modulation of the light within the rib 402.

[0066] The waveguide 400 is different from other waveguides described herein in several respects. First, the waveguide 400 is configured for the propagation of TE-polarization based on the size of the waveguide 400. As shown in FIG. 4A, the thickness of the slab is h slab,1 is. The rib 402 has a rib height h core,1 and a rib width w core,1 and in some embodiments, to configure the waveguide 400 for the propagation of TE polarization, the rib height h core,1 is smaller than the rib width w core,1 For example, the waveguide 400 generally has a wide and short shape. Thereby, the effective refractive index of the TE2D waveguide mode within the rib 402 becomes higher than the effective refractive index of the TM1D slab mode, and it is ensured that the guided TE mode leaks less to the slab 404 than the guided TM mode.

[0067] There are various reasons why silicon photonic modulators and waveguides, such as modulator 400, are often configured for TE - polarization.

[0068] First, in the case of a modulator using a rib waveguide, the TM2 - D rib mode refractive index is usually significantly lower than the TE1 - D slab mode refractive index. The rib waveguide requires special conditions that are usually not met to guide TM light. For example, in some cases, when the effective refractive index of the TM2 - D rib mode is greater than that of the TE1 - D slab mode, TM light is guided. The "slab mode" is a 1 - D mode that is guided when there is no rib 402 and the width of the slab 404 is infinite. When this effective refractive index condition is not met, the TM rib mode will be phase - matched with the TE slab mode propagating at a certain angle with respect to the rib 402. In such a case, due to a small perturbation, the light of the TM mode will leak into the slab 404.

[0069] Second, TE - polarization has a stronger vertical confinement within the rib 402 compared to TM - polarization, reducing losses due to the underlying substrate and the upper layer. For example, in some embodiments, there is a metal wiring layer on silicon, and the metal layer can be brought much closer to the silicon before causing a greater optical loss for TE - polarization than for TM - polarization.

[0070] Third, in most silicon photonic modulators, the height h core,1 of the waveguide is smaller than the width w core,1 of the waveguide. As a result, TE - polarization will have a higher effective refractive index than TM - polarization. This allows the bending radius to be reduced and the size of the silicon photonic device to be reduced.

[0071] Fourth, since most other elements that make up the silicon photonic circuit are designed for TE - polarization, many silicon photonic modulators employ TE - polarization. For example, most grating couplers are configured for TE - polarization.

[0072] Fifth, in many scenarios, for example, since the lithography process is simplified by reducing the etching depth, it is usually easier to fabricate a waveguide structure with a width larger than the height (corresponding to TE transmission).

[0073] However, in some cases, distinct advantages may be obtained by using TM polarization. For example, TM-polarization has the advantage that there is less light in the slab compared to TE-polarization. To understand why there is less light in the slab for TM-polarization compared to TE-polarization, the boundary conditions of the electric field of light given by Maxwell's equations can be considered. In a non-magnetic material such as silicon, the transverse electric field E ∥ is continuous across the boundary, while the longitudinal electric field multiplied by the permittivity (E ⊥ )(ε) is continuous across the boundary. Since the permittivity of silicon is about 5.8 times that of the oxide, when the electric field is perpendicular to a thin silicon sheet surrounded by oxide, the electric field inside the silicon is about 5.8 times lower than that of the surrounding oxide. Therefore, TM polarization has almost no electric field inside the silicon slab.

[0074] In an actual implementation, the guided-wave 2D mode rarely becomes a pure TE or TM mode. Therefore, usually, the guided-wave optical mode is a quasi-TE or quasi-TM mode. In the quasi-TM mode, the main component of the electric field of light is arranged along the z-axis. In the quasi-TE mode, the main component of the electric field of light is arranged along the x-axis. For the sake of brevity in the description, when discussing the polarization of the guided-wave optical mode in this disclosure, the word "quasi" may be omitted.

[0075] Another difference between waveguide 400 and several other waveguides described herein is related to the vertical doping profile. Waveguide 400 has a uniform n-type and p-type doping profile in the vertical direction. As shown in FIG. 4B, for the vertical doping profile (profile along the z-direction) in the n-type region 406, the dopant concentration N D414 is substantially constant over the entire depth of the waveguide 400 (e.g., from a height z = 0 (the bottom of the slab 404) to the height h at the top of the rib 402 core,1 over the entire rib 402 and the underlying slab 404 thereunder). In the p-type region 408, another uniform vertical doping profile (not shown) is applied, and the dopant concentration N A is constant. As will be described in more detail below (e.g., with respect to FIGS. 4C-4H), some waveguides within the scope of the present disclosure can have a non-uniform vertical doping profile.

[0076] FIG. 4C is a cross-sectional view of a waveguide 420 configured to transmit TM polarization. The waveguide 420 has a rib 422 on top of a slab 424, as described with reference to FIG. 4A. Further, as described with reference to FIG. 4A, an n-type region 426 and a p-type region 428 are disposed on opposing lateral sides of the slab 424 and the rib 422. The waveguide 420 has a rib width w core,2 , a rib height h core,2 , and a slab thickness h slab,2 . A depletion region 430 (included in the undoped region of the lateral PIN junction in some embodiments) is formed between the n-type region 426 and the p-type region 428. The elements of the waveguide 420 can have the same characteristics as those described for the elements of the waveguide 400, unless otherwise specified.

[0077] In some embodiments, to configure the waveguide 420 for TM-polarization transmission, the waveguide 420 generally has a narrow and long dimension as compared to the wide and short dimension of the waveguide 400. As one of the other advantages, the narrow and long configuration of the waveguide 420 in FIG. 4C reduces the portion of the optical mode in the slab 424 and allows for a higher doping in the slab 424 for the same optical loss as compared to the waveguide 400 in FIG. 4A. By increasing the doping of the slab 424, in a modulator using the waveguide 420 that uses TM polarization instead of TE polarization, the bandwidth in the modulator using the waveguide 420 can be increased without increasing the optical loss as compared to a modulator using the waveguide 400.

[0078] For example, in some embodiments, the rib height h core,2 is greater than the threshold of 0.85λ / n, and the rib width h core,2 is greater than the thickness h of the slab slab,2 Here, λ is the free-space wavelength of light, and n is the refractive index of silicon. Thereby, in the case of TM polarization, it is ensured that the electric field decreases to a low value at the upper and lower parts of the waveguide, and a large electric field strength is not generated outside the waveguide due to the electric field boundary condition (described above). For example, when the wavelength λ = 1310 nm, the threshold is 0.85λ / n = 320 nm. Therefore, in this example, the rib height h of the waveguide core,2 is greater than 320 nm, and the rib width h of the waveguide core,2 needs to be greater than 90 nm. In some embodiments, TM transmission is efficient when h core,2 > w core,2 For example, the effective refractive index of the TM-polarized two-dimensional (2D) waveguide mode in the rib waveguide can be made greater than at least one, or all, of the effective refractive indices of the transverse electric field (TE)-polarized one-dimensional (1D) waveguide modes in the slab.

[0079] As a comparative example of the dimensions of the waveguide, when the thickness h of the slab slab,1 = h slab,2 = 90 nm, the waveguide 400 can have waveguide rib heights h core,1 and rib widths w core,1 of 220 nm and 420 nm, respectively, while the waveguide 420 can have waveguide rib heights h core,2 and rib widths w core,2 of 400 nm and 220 nm, respectively.

[0080] h core,2 > 0.85λ / n, h core,2 > h slab,2 and / or h core,2 > w core,2The aforementioned conditions may enable preferential transmission of TM-polarization. However, for the purposes of the present disclosure, it has further been recognized that the TM mode is sensitive to the roughness of the rib sidewalls (e.g., sidewall 436). The roughness of the rib sidewalls causes losses to the TM light transmitted through the waveguide, and due to the strong confinement of the TM mode, this loss can be higher than the sidewall loss of TE light in a waveguide configured for TE transmission. However, it has been found that the sidewall loss is unexpectedly sensitive to the rib width. For example, within a certain range of w core,2 even a slight increase in w core,2 results in a significant decrease in the sidewall loss. For example, in some embodiments, a rib width of 250 nm < w core,2 < 400 nm or 300 nm < w core,2 < 400 nm or 250 nm < w core,2 < 360 nm or 300 nm < w core,2 < 360 nm has been found to provide an effective trade-off between low-loss TM transmission and modulation efficiency for light at, for example, 1310 nm (e.g., because the depletion region 430 between the n-type region 426 and the p-type region 428 becomes smaller in proportion to the width w core,2 as the width w core,2 of the rib 422 increases). This rib width can also provide advantages in waveguides that do not incorporate a non-uniform vertical doping profile. Thus, some embodiments according to the present disclosure have a rib width of 250 nm < w core,2 < 400 nm and have a uniform vertical doping profile.

[0081] Furthermore, in some embodiments, it has been found that one or more of the aforementioned conditions can be relaxed while maintaining acceptable device performance (e.g., TM-polarization transmission), and based on the relaxation of the conditions, the design flexibility can be increased, and in some cases, the values of other parameters can be improved. For example, in some embodiments, h core,2 > 0.80λ / n, h core,2 > h slab,2 , and / or h core,2 > 0.9w core,2It has been found that effective transmission and modulation performance can be provided. In some embodiments, one or more of these conditions are provided together with a rib width of 250 nm < w core,2 <400 nm.

[0082] However, other rib widths are also within the scope of the present disclosure. For example, in some embodiments, the rib width is less than 250 nm, or less than 300 nm, for example, in the range of 150 nm to 250 nm. In some embodiments, the rib height is in the range of 300 nm to 400 nm. In some embodiments, the slab thickness is in the range of 50 nm to 150 nm.

[0083] Referring again to FIG. 4C, in some embodiments, the waveguide 420 has a non-uniform vertical doping profile in the rib 422. For example, the waveguide 420 can have a "deep" doping profile in which the lower portion of the rib 422 in the vertical direction (the portion closer to the slab 424 along the z-axis) has a higher dopant concentration (donor of the n-type region 426, acceptor of the p-type region 428, or both) than the higher portion of the rib 422 in the vertical direction. Therefore, the product of the rib resistance (R r ) related to the lateral transmission of the rib 422 and the square of the rib capacitance C can be reduced compared to the case where the rib 422 is uniformly doped.

[0084] The power consumed is I 2 R r (where I is the current), and in a typical state of the waveguide modulator, since I is approximately proportional to C, R r C 2 is a relevant quantity to be considered. Therefore, when the value of R r C 2 decreases, the amount of RF power lost in the rib resistance R rで will decrease. In some embodiments, the product R r C can be reduced based on non-uniform vertical doping, corresponding to a reduction in switching time and faster modulation.

[0085] Product R r C 2 The decrease in R can correspond to a decrease in capacitance per unit length (length in the y - direction) and / or a decrease in loss per unit length. The decrease in capacitance can be based at least on a decrease in the area of the semiconductor junction diode formed by the n - type region 426 and the p - type region 428. The diode can be modeled as a parallel - plate capacitor, and the undoped or lightly doped upper part of the rib 422 contributes little or no contribution to the area (in the y - z plane) of the parallel - plate capacitor. Further, the lower the doping level, the wider the depletion region width (e.g., the width of the depletion region 430 in the higher part of the rib 422 with less doping), and accordingly, the lower the diode capacitance in the rib 422. Thus, based on some embodiments of the non - uniform doping described herein, the increase in R based on the non - uniform doping in the vertical direction rの can be offset by the decrease in capacitance C, and R r C 2 and / or R r C は decreases compared to the case of uniform doping.

[0086] The decrease in loss per unit length is a result of the low doping level, because highly doped silicon has a higher absorption rate than lightly doped silicon. Thus, the light transmitted through the rib 422 may experience less attenuation than when the doping is uniform in the vertical direction, including the light passing through the higher vertical part of the rib 422 with less doping.

[0087] At the higher part of the rib 422, due to the low doping level, the phase modulation by depletion width adjustment may be somewhat less effective compared to a modulator having a uniform high - level doping in the vertical direction. For example, V π L may increase compared to the case of uniform high - concentration doping. However, in some embodiments, the advantages in terms of conductance, capacitance, and optical loss per unit length may outweigh this potential decrease in modulation efficiency.

[0088] Referring again to FIG. 4C, the rib 422 can conceptually be divided into an upper portion 432 and a lower portion 434, and on one or both of the doped side surfaces of the rib 422, the average doping concentration of the lower portion 434 along the vertical doping profile is higher than that of the upper portion 432. The end between the lower portion 434 and the upper portion 432 is the vertical height h with respect to the bottom of the slab 424 doping as defined. This "deep doping" profile can take various forms in various implementations.

[0089] For example, FIGS. 4D-4F show the doping concentration (e.g., in units of cm -3 ) as a function of height along the vertical doping profile of the n-doped side surface of the rib 422, where z = 0 is defined at the bottom of the slab 424. In some embodiments, as shown for the vertical doping profile 440 in FIG. 4D, the lower portion 434 has a substantially constant doping concentration N D,L and the upper portion 432 is substantially undoped. In some embodiments, as shown for the vertical doping profile 442 in FIG. 4E, the lower portion 434 has a substantially constant doping concentration N D,L and the upper portion 432 has a substantially constant doping concentration N D,U , where N D,L > N D,U . For example, in some embodiments, N D,L is at least 1.5 times N D,U , or at least 2 times N D,U , which is a doping difference found to provide the aforementioned advantages related to reducing rib resistance while maintaining an acceptable modulation efficiency.

[0090] FIG. 4F shows, for example, an example of a practically obtained doping profile by injecting dopant ions into the rib 422 from above (downward in the z direction). Instead of the dopant concentration dropping sharply exactly at z = h doping , the value N of the lower portion 434 D,LThe dopant concentration decreases at least somewhat gently from this point and converges to either zero in the upper part 432 (in the case of profile 444) or N D,H (in the case of profile 446). Thus, the condition that N D,L is at least 1.5 times that of N D,U or at least 2 times that of N D,U can instead be expressed as the average concentration in the lower part 434 along the vertical doping profile being at least 1.5 times the average concentration in the upper part 432 along the vertical doping profile, or at least 2 times the average concentration in the upper part 434 along the vertical doping profile.

[0091] The relative height and thickness of the upper part 432 and the lower part 434 can vary in various embodiments. In some embodiments, h doping / h core,2 is 0.2 to 0.8, 0.2 to 0.65, or 0.35 to 0.65. This value has been found to provide a useful trade-off between a reduction in rib resistance and a high modulation efficiency in combination with the above-described average dopant concentration condition. For example, in some embodiments, the lower 0.2 to 0.65 portion of the waveguide 420 has an average dopant concentration that is at least 1.5 times or at least 2 times that of the remaining upper part of the waveguide (e.g., the upper 0.35 to 0.8 portion of the waveguide 420) along the vertical doping profile.

[0092] In some embodiments, as shown in the example of FIG. 4C, the upper part 432 is the uppermost part of the rib 422, and the lower part 434 is the remaining part of the rib 422, e.g., the lowermost part of the rib 422 extending from the height z = h doping to the slab 424.

[0093] "The concentration of at least one dopant in the vertical doping profile within the rib is higher at the lower part of the rib than at the upper part of the rib" does not need to rely on dividing the rib into upper and lower parts of finite height such as in portions 432, 434, or on determining the average dopant level over a finite height. For example, in profile 444, the dopant concentration is greater at z = h1 than at z = h2, satisfying the condition that the concentration of at least one dopant in the vertical doping profile of rib 422 is higher at the lower part of rib 422 than at the upper part of rib 422.

[0094] Figures 4D - 4F show the donor doping profile of the n - type region 426, but a similar description applies to the acceptor doping profile of the p - type region 428. For example, a non - uniform vertical doping profile can exist independently for each of the p - type doping and n - type doping such that the donor concentration is higher at the lower part of the first lateral side of rib 422 than at the upper part of the first lateral side of rib 422, and the acceptor concentration is higher at the lower part of the second opposite lateral side of rib 422 than at the upper part of the second lateral side of rib 422. The donor and acceptor concentrations at the upper and lower parts do not have to be equal (although they can be), and the corresponding parts of rib 422 at the upper and lower parts do not have to be the same for the two doping profiles (although they can be).

[0095] For example, Figure 4G shows an example of a waveguide 450 in which each of the upper and lower parts of the rib is defined to be different for n - type doping and p - type doping. The waveguide 450 has a rib 452 at the top of a slab 454, as described with reference to Figures 4A and 4C. Further, as described with reference to Figures 4A and 4C, an n - type region 456 and a p - type region 458 are disposed on opposite lateral sides of the slab 454 and the rib 452. The waveguide 450 has a rib width w core,2 , a rib height h core,2 , a slab thickness h slab,2It has. The elements of the waveguide 450 can have the same features as those described for the elements of the waveguide 420, unless otherwise specified.

[0096] In this example, the n-type doping is concentrated in the first lower part 434n of the rib 452, and the first upper part 432n of the rib 452 is substantially undoped. Further, the p-type doping is concentrated in the second lower part 434p of the rib 452, and the second upper part 432p of the rib 452 is substantially undoped. The donor and acceptor doping profiles 460n and 460p are shown in FIG. 4H respectively, and the donor and acceptor concentrations N D,L and N A,L in the lower parts 434n, 434p do not have to be equal (although they may be equal). As shown in FIG. 4H, the donor dopant concentration transitions to substantially zero at height h doping,n and the acceptor dopant concentration transitions to substantially zero at height h doping,p , but h doping,n and h doping,p do not have to be equal (although they may be equal). The descriptions provided with respect to FIGS. 4C-4F regarding the vertical doping profile of the n-type region 426 (e.g., relative concentrations at different heights, regions corresponding to upper / lower parts, etc.) can be applied independently to each of the vertical doping profiles 460n, 460p. For example, independently for each of the n-type side and the p-type side, the lower 0.2-0.65 portion of the waveguide 450 along the vertical doping profile can have at least 1.5 times, or at least 2 times, the average dopant concentration of the remaining upper part of the waveguide along the vertical doping profile (e.g., the upper 0.35-0.8 portion of the waveguide 420).

[0097] The examples of FIGS. 4G-4H include a dopant concentration that decreases to zero at the upper part of the rib 452. However, in some embodiments, either or both of the donor dopant concentration or the acceptor dopant concentration do not have to become zero at the upper part of the rib 452, for example, as described with respect to FIGS. 4E-4F. For example, the upper parts 432n and 432p do not have to be undoped.

[0098] As an example of the dopant concentration, in some embodiments, in the p-doped portion of the rib (e.g., boron-doped), the average concentration of the p-type dopant at the bottom of the rib is 10 17 cm -3 to 10 18 cm -3 , and the average concentration of the p-type dopant at the top is less than 10 17 cm -3 , less than 5×10 16 cm -3 , less than 10 16 cm -3 , or less than 5×10 15 cm -3 . For example, the top can have a height between 20% and 80% or between 35% and 65% of the height of the waveguide. In some embodiments, in the n-doped portion (e.g., phosphorus-doped), the average concentration of the n-type dopant at the bottom is 10 17 cm -3 to 10 18 cm -3 , and the average concentration of the n-type dopant at the top is less than 10 17 cm -3 , less than 5×10 16 cm -3 , less than 10 16 cm -3 , or less than 5×10 15 cm -3 . In some embodiments, these value ranges have been found to provide efficient light transmission, high modulation efficiency, and high-speed modulation switching.

[0099] FIG. 12 shows a non-limiting example of the electron and hole concentrations in the non-uniformly doped waveguide rib 1200 described herein. In the p-doped surface 1202 of the rib 1200, as a result of the non-uniform acceptor doping in the vertical direction, the hole concentration ranges from about 3×10 17 cm -3 at the bottom of the rib 1200 to about 1×10 15 cm -3It decreases until. On the n-doped surface 1204 of the rib 1200, as a result of non-uniform donor doping in the vertical direction, the electron concentration is about 3.5×10 at the bottom of the rib 1200 17 cm -3 to about 1×10 at the top of the rib 1200 15 cm -3 and decreases until.

[0100] The foregoing description refers to the doping profile and dopant concentration (e.g., the concentration of dopant atoms such as phosphorus or boron), but in some embodiments, instead of or in addition to, a similar description can be made for the activated dopant concentration, such as after activated high-temperature annealing.

[0101] Figures 5A and 5B respectively show the TE mode and TM mode obtained by calculation in a silicon rib waveguide. Figure 5A shows an example of the calculated mode of a conventional silicon phase modulator using TE-polarization. In particular, Figure 5A shows the magnitude of the x-component of the electric field. Figure 5B shows an example of the calculated mode of a silicon phase modulator using TM-polarization according to an embodiment of the present disclosure. Figures 5A and B show the magnitudes of the x-component and z-component of the electric field respectively defined in Figures 4A and C.

[0102] As shown, there is quite a lot of light in the slab of Figure 5A, but there is little light in the slab of Figure 5B. Therefore, the slab of Figure 5B can have significantly higher doping near the rib, and thus the series resistance is significantly lower.

[0103] As described above, in the examples of FIGS. 5A and 5B, the dimensions of the waveguide ribs can be varied to facilitate TE or TM transmission. In either case, the thickness of the slab is 90 nm. However, in FIG. 5A, the rib height and rib width of the waveguide are 220 nm and 420 nm, respectively, while in FIG. 5B, the rib height and rib width of the waveguide are 400 nm and 220 nm, respectively. The waveguide of FIG. 5A is a typical modulator waveguide configured for TE-polarization. As described above, in such a configuration, since the effective refractive index of the TM2D rib mode is lower than that of the TE1D slab mode, the guided TM mode leaks into the slab (see Table 1 below).

[0104] In contrast, the waveguide of FIG. 5B can guide the TM mode without leaking into the slab because the waveguide rib is higher and narrower. As shown in Table 1, increasing the height of the waveguide rib makes the effective refractive index of the TM2D guided mode higher than that of the TE1D slab mode.

[0105] Table 1 Effective mode refractive index for wavelength λ = 1310 nm

Table 1

[0106] Embodiments of the modulator according to the present disclosure configured for TM-polarization can provide various technical advantages (e.g., compared to a typical modulator configured for TE-polarization). For example, the doping in the slab can be significantly increased and / or a higher doping can be placed closer to the rib. In some embodiments, the doping concentration is in the range of 5×10 3 to 1×10 17 in a first portion of the slab within a range of 50 nm to 500 nm (e.g., 100 nm) from the closest sidewall of the rib, compared to a second portion of the slab more than 100 nm from the closest sidewall of the rib, with a dopant or active dopant having a value in the range of 1×10 19 to 1×10 18 per cm19 can be increased) within the range. Further, as described above, the reduction in rib resistance due to the non-uniform vertical doping profile can reduce the rib resistance instead of or in addition to the reduction in resistance based on the increase in slab doping. Based on the increase in slab doping concentration and / or the non-uniform vertical doping profile within the rib, some embodiments of the present disclosure can provide a series resistance that is approximately 3.8 times lower compared to a typical modulator configured for TE-polarization. The reduction in series resistance can expand the modulation bandwidth (reduce the switching time) and reduce the loss.

[0107] Another advantage associated with embodiments having a configuration for TM transmission is that the phase modulation efficiency can be increased for a given voltage and a given modulator length. This is because the TM-polarization is more confined horizontally within the waveguide rib perpendicular to the depletion region, resulting in a larger change in the effective refractive index for a given voltage change.

[0108] A further advantage associated with embodiments having a non-uniform vertical doping profile within the waveguide rib is that the capacitance per unit length can be reduced (e.g., compared to a waveguide having a uniform vertical doping profile within the waveguide rib), expanding the modulation bandwidth (reducing the switching time).

[0109] Yet another advantage associated with embodiments having a rib width within the above-described range is that sidewall losses can be reduced (e.g., compared to a waveguide having a wider rib) while maintaining a high modulation efficiency (e.g., compared to a waveguide having a narrower rib).

[0110] Some embodiments of modulators according to the present disclosure can be configured to mitigate potential technical challenges. For example, in a modulator configured for TM-polarization, the waveguide rib is taller and thinner (compared to the waveguide rib of a typical modulator designed for TE-polarization), so the series resistance may increase along the vertical edge of the rib connected to the top of the waveguide. To reduce such resistance, some embodiments include a waveguide rib with an effective refractive index of the TM2D rib slightly higher than the threshold 0.85λ / n that is higher than the effective refractive index of the TE1D slab. For example, in some embodiments, for a 90 nm slab configured to transmit light with a wavelength of 1310 nm, the rib height of the waveguide is 350 nm and the rib width of the waveguide is 220 nm.

[0111] Another potential issue is that, as a result of the increased depletion region, the capacitance of the p-n junction of the waveguide (e.g., semiconductor junction diodes 212 and 214 in FIG. 2) may increase. However, in these structures, the fringing field contributes significantly to the capacitance, and as a result, the increase in capacitance is sub-linear with respect to the increase in height. For example, doubling the rib height of the waveguide only results in the capacitance increasing by about 1.5 times. Further, as described above, by doping the top of the rib less than or not at all compared to the bottom of the rib, effectively reducing the height of the depletion region at the top of the rib, and / or increasing the width of the depletion region, the increase in capacitance can be at least partially mitigated.

[0112] Figures 6-9 relate to modulators according to some embodiments of the present disclosure. These modulators can include any of the rib waveguides described above with respect to FIGS. 4C-4H. For example, these modulators can have a non-uniform vertical doping profile, have dimensions configured for TM optical transmission, and / or include one or more rib waveguides having a rib width in the range of 250 nm to 400 nm or 300 nm to 400 nm to reduce sidewall losses. In contrast to the modulator design shown in FIG. 2, the modulators of FIGS. 6-9 do not include a bias voltage connection between waveguides, and as a result, the series resistance between the electrodes is significantly reduced and the modulation bandwidth is further increased. Further, in some embodiments according to FIGS. 6-9, the modulator implements a waveguide structure with varying height to reduce harmful optical coupling between adjacent waveguides.

[0113] The features described with reference to FIGS. 6-9 can help improve the structure of the modulator of FIG. 2 in various aspects. For example, the presence of the DC bias connection 228 in FIG. 2 increases the physical distance of the semiconductor (e.g., silicon) region 238 between the semiconductor junction diodes 212 and 214. For this reason, a large electrical series resistance occurs in the semiconductor region 238 connecting the semiconductor junction diodes 212 and 214.

[0114] Furthermore, the semiconductor regions 240 and 242 in FIG. 2 (connecting the semiconductor junction diodes 212 and 214 to their respective electrodes 216 and 218) are p-doped semiconductor materials having a higher resistance than the n-doped semiconductor material. For this reason, a large electrical series resistance occurs in the semiconductor regions 240 and 242 between the electrodes 216 and 218 and the semiconductor junction diodes 212 and 214.

[0115] As a result, the total electrical series resistance between the electrodes 216 and 218 in FIG. 2 can significantly attenuate the voltage along the modulator 200 by charging and discharging the diode capacitance. Further, this attenuation typically increases as the modulation frequency increases. The RF losses occurring along the modulator 200 can affect the bandwidth of the modulator 200.

[0116] FIG. 6 is a diagram showing an example of a top view of a modulator 600 according to an embodiment of the present disclosure.

[0117] The modulator 600 is based on an MZI implementation including two optical transmission paths 602 and 604, an optical splitter 606, and an optical combiner 608. The modulator 600 further includes terminals to which a voltage can be applied, such as a terminal 610 and a terminal 612. The voltage propagates along an RF transmission line 614 connected to a semiconductor structure 616 via an electrode to which each voltage is applied, and an electric field is generated across one or both of the optical transmission paths 602 and 604. In some embodiments, an optical phase rotator that rotates the phase of the input light so that quasi-TM light propagates through the optical transmission paths 602 and 604 may be implemented between the input of the modulator 600 and the optical transmission paths 602 and 604.

[0118] The optical transmission paths 602 and 604 can include rib waveguides such as waveguide 420 and / or waveguide 450.

[0119] In contrast to the modulator 100 of FIG. 1, the modulator 600 does not implement any DC bias connection between the two optical transmission paths 602 and 604. This makes it possible to arrange the two optical transmission paths 602 and 604 closer to each other to reduce the electrical series resistance therebetween. For example, in some embodiments, the distance between the waveguides of the two optical transmission paths 602 and 604 is less than 0.5 μm for at least a part of the longitudinal direction of the optical transmission paths 602 and 604. In some embodiments, the distance between the waveguides is less than 2.0 μm for at least a part of the longitudinal direction of the optical transmission paths 602 and 604. In some embodiments, the distance between the waveguides is in the range of 0.1 μm to 2.0 μm for at least a part of the longitudinal direction of the optical transmission paths 602 and 604. In some embodiments, the distance between the waveguides is defined as the distance between the inner sidewalls of the two waveguides at a predetermined point (e.g., point 605 in FIG. 6) along the longitudinal direction of the modulator 600.

[0120] However, since the two optical transmission paths 602 and 604 are arranged closer to each other, there is a greater risk of more serious harmful optical coupling between the light in the optical transmission path 602 and the light in the optical transmission path 604. To reduce such optical coupling, in some embodiments, one waveguide of the optical transmission path (602 or 604) is designed to have a greater height (e.g., at least 10 nm greater) than the other waveguide at the same distance along the length of the modulator 600. This prevents the light traveling in the waveguides of the optical transmission paths 602 and 604 from being phase-matched, reducing the optical coupling between the two waveguides.

[0121] Another way to understand the importance of using different waveguide heights is to look at the two eigenmodes of the combined waveguide of the optical transmission paths 602 and 604. When the waveguide heights are equal, the lowest-order eigenmode is an even eigenmode and the second-lowest eigenmode is an odd eigenmode. In such a scenario, differential modulation cannot occur. However, when one waveguide is sufficiently higher than the other, the lowest-order eigenmode consists mainly of the light present in the higher waveguide, and the second-lowest eigenmode is mainly present in the lower waveguide. This enables differential modulation despite the adjacent waveguides. For example, in some embodiments, one waveguide of the optical transmission path 602 or 604 is at least 10 nm or 40 nm higher than the waveguide of the other optical transmission path. In some embodiments, the difference in waveguide height is in the range of 40 nm to 120 nm.

[0122] Furthermore, in such an embodiment, the change in height of the two waveguides may be exchanged along the modulator 600 such that the total length of the high portion in each waveguide is equal and the total length of the low portion in each waveguide is equal. In the example of FIG. 6, moving from left to right, the waveguide of the first optical transmission path 602 becomes higher than the waveguide of the second optical transmission path 604 and then lower than the waveguide of the second optical transmission path 604 (alternatively, the waveguide of the first optical transmission path 602 may start low and become high). There may be one such exchange of relative height at the center of the modulator 600, but in some embodiments, additional height exchanges may be included as long as, for example, the distance between the height exchanges is significantly longer than the beat length (typically 10 μm) between the two eigenmodes in the two waveguides. This can reduce the optical coupling between the two waveguides. In some embodiments, an odd number of exchanges is preferred. This is because it ensures that the start and end transitions cancel each other out.

[0123] In the above description of FIG. 6, an example of the modulator 600 having waveguides with varying heights in the two optical transmission paths 602 and 604 was shown, but in other embodiments, the waveguides may have a constant height along the length of the modulator 600.

[0124] Furthermore, in the description of FIG. 6, an example of a modulator 600 without a physical DC bias connection was shown. However, in some embodiments, a DC bias connection may be implemented between the two optical transmission lines 602 and 604, although via a high impedance. For example, in some embodiments, the high impedance is achieved with an impedance greater than 1 kohm. As another example, in some embodiments, the high impedance is achieved with an impedance greater than 100 ohm. In such a scenario of a DC bias connection via a high impedance, the current would be generated by the voltage difference between (i) an external voltage and (ii) the voltage that would exist between the optical transmission lines 602 and 604 when no external voltage is applied. This generated current is smaller than the sum of the leakage current of the diode and the photo-generated current within the diode, and thus the circuit operates mainly as if no external DC bias voltage were applied (e.g., similar to a true floating voltage). Therefore, it should be understood that the embodiments of the present disclosure as shown in FIGS. 6 - 9 without a physical DC bias connection can also be implemented using a DC bias connection, although via a high impedance.

[0125] The modulator 600 implements an example of a continuous traveling wave structure in which an RF transmission line 614 is continuously connected to a semiconductor structure 616. Alternatively, as will be described below with reference to FIG. 7, a segmented traveling wave structure may be implemented.

[0126] FIG. 7 is a diagram showing another example of a top view of a modulator 700 according to an embodiment of the present disclosure. The modulator 700 is an example of an implementation of a segmented traveling wave structure.

[0127] The modulator 700 is also based on an MZI implementation that includes two optical transmission paths 702 and 704, an optical splitter 706, and an optical combiner 708. The modulator 700 further includes terminals to which a voltage can be applied, such as terminal 710 and terminal 712. The optical transmission paths 702 and 704 can include rib waveguides such as waveguide 420 and / or waveguide 450. The voltage propagates along the RF transmission line 714 connected to the semiconductor structure 716 via electrodes for applying the respective voltages, and an electric field is generated across one or both of the optical transmission paths 702 and 704. The modulator 700 also does not implement any DC bias connection between the two optical transmission paths 702 and 704. Thus, the distance between them is reduced. For example, in some embodiments, the distance between the waveguides of the two optical transmission paths 702 and 704 is less than 0.5 μm for at least a portion of the longitudinal direction of the optical transmission paths 702 and 704. In some embodiments, the distance between the waveguides is less than 2.0 μm for at least a portion of the longitudinal direction of the optical transmission paths 702 and 704. In some embodiments, the distance between the waveguides is in the range of 0.1 μm to 2.0 μm for at least a portion of the longitudinal direction of the optical transmission paths 702 and 704. In some embodiments, the distance between the waveguides is defined as the distance between the inner sidewalls of the two waveguides at a predetermined point (e.g., point 705 in FIG. 7) along the longitudinal direction of the modulator 700.

[0128] The differences between the modulator 600 of FIG. 6 and the modulator 700 of FIG. 7 are due to the configuration of the semiconductor structures (616, 716) and the way the RF transmission lines (614, 714) are connected to the semiconductor structures (616, 716). The modulator 600 of FIG. 6 implements a continuous traveling wave structure in which the RF transmission line 614 is continuously and directly connected to the semiconductor structure 616. In contrast, the modulator 700 of FIG. 7 implements a segmented traveling wave structure in which the RF transmission line 714 is intermittently connected to segments of the semiconductor structure 716 and has intermittent regions 720 where there is no semiconductor structure along the optical transmission paths 702 and 704. This structure of the modulator 700 can be referred to as a capacitively loaded traveling wave structure and has the advantage of providing additional degrees of freedom in the implementation of the RF transmission line 714, such as the average capacitance per unit length of the RF transmission line 714. Lumped element modulators can also benefit from the techniques disclosed herein.

[0129] Furthermore, in some embodiments of the modulator 700, the waveguides of the optical transmission paths 702 and 704 have different heights in different sections of the modulator 700, similar to the configuration of the waveguides in the modulator 600 of FIG. 6. Further details of the height variation of the waveguides will be described below with reference to FIG. 8.

[0130] FIG. 8 shows an example of a cross-section of a modulator 800 according to an embodiment of the present disclosure (e.g., a cross-section at point 605 of the modulator 600 of FIG. 6 or a cross-section at point 705 of the modulator 700 of FIG. 7). In particular, the modulator 800 of FIG. 8 is an example of a differential proximity arrangement design in which one waveguide is higher than the other.

[0131] The cross-section of modulator 800 shows the details of the MZI structure. The MZI includes a first optical waveguide 802 and a second optical waveguide 804. The optical waveguides 802 and 804 can be implemented, for example, as silicon rib waveguides on a slab. In some embodiments, modulator 800 includes a substrate 806 (e.g., a silicon substrate), an insulating structure 808 (e.g., a dielectric such as an oxide), and a semiconductor structure 810 (e.g., a silicon layer including optical waveguides 802 and 804). In some embodiments, one or both of waveguides 802 and 804 have one or more non-uniform vertical doping profiles and / or have a rib width configured to reduce sidewall losses. For example, waveguides 802 and 804 can be waveguide 420 and / or waveguide 450.

[0132] In some embodiments, as described with respect to FIGS. 6 and 7 above, one of optical waveguides 802 and 804 is higher than the other optical waveguide. For example, in FIG. 8, the second optical waveguide 804 is at least 10 nm or 40 nm higher than the first optical waveguide 802. In some embodiments, the difference in height of the waveguides is in the range of 40 nm to 120 nm.

[0133] Each of optical waveguides 802 and 804 includes a semiconductor junction. The semiconductor junction diode can be implemented, for example, as a PIN (p-type / intrinsic / n-type) junction diode or a p / n junction diode. In modulator 800, a P / N junction is embedded in each of optical waveguides 802 and 804 (e.g., at the bottom of the rib of waveguides 802 and 804) to form a diode within each waveguide. These diodes are shown as a first semiconductor junction diode 812 and a second semiconductor junction diode 814.

[0134] In addition, modulator 800 includes electrodes 816 and 818 (e.g., metal electrodes) that are in physical contact with silicon layer 810. In some embodiments, electrodes 816 and 818 are in physical contact with n-doped contact regions 820 and 822 of silicon layer 810. Electrodes 816 and 818 may be formed, for example, by etching insulating layer 808 and forming metal (e.g., tungsten, copper, and / or aluminum) contacts. Further, modulator 800 may include metal layers 824 and 826 on electrodes 816 and 818. In some embodiments, metal layers 824 and 826 may form segments of RF transmission lines (e.g., RF transmission line 114 of FIG. 1). In some embodiments, in modulator 800, the p-doped region may alternatively be an n-doped region, and vice versa (e.g., such that contact regions 820 and 822 are n-doped instead of p-doped).

[0135] There are a number of differences between modulator 800 and modulator 200 of FIG. 2. Most notably, modulator 800 does not implement any DC bias voltage connection between semiconductor junction diodes 812 and 814 (as compared to modulator 200 that implements DC bias connection 228). Instead, semiconductor junction diodes 812 and 814 are connected in series with opposite polarities (anodes 834 and 836 are connected together). This results in no continuous current flowing through semiconductor junction diodes 812 and 814. This configuration ensures that the voltage across the two semiconductor junction diodes 812 and 814 is self-adjusted naturally and the diodes 812 and 814 maintain a reverse bias, despite variations in the modulation voltage (e.g., V+ and V−) that may be applied to electrodes 816 and 818. Implementing a floating voltage between semiconductor junction diodes 812 and 814 biases diodes 812 and 814 automatically at the most efficient point of the modulator in terms of volts per phase shift. This is where diodes 812 and 814 are below turn-on. In some embodiments, this volts per phase shift is the “gain” of the modulator.

[0136] Another difference between modulator 800 and modulator 200 of FIG. 2 is that the polarities of semiconductor junction diodes 812 and 814 are reversed compared to modulator 200. In particular, semiconductor junction diodes 812 and 814 have their respective (p-doped) anodes 834 and 836 near the center of modulator 800 and their respective (n-doped) cathodes 830 and 832 near the ends of modulator 800. Thus, the semiconductor region 838 between semiconductor junction diodes 812 and 814 is p-doped, and semiconductor regions 840 and 842 (connecting semiconductor junction diodes 812 and 814 to their respective electrodes 816 and 818) are n-doped.

[0137] These aforementioned differences can provide a number of technical advantages to modulator 800 compared to modulator 200 of FIG. 2. One advantage is that since modulator 800 has no DC bias voltage connection, two optical waveguides 802 and 804 can be implemented much closer to each other compared to modulator 200 of FIG. 2. This allows the size of semiconductor region 838 connecting semiconductor junction diodes 812 and 814 to be significantly reduced, and the electrical series resistance between semiconductor junction diodes 812 and 814 to be significantly reduced. For example, in some embodiments, the distance between two optical waveguides 802 and 804 (denoted as 805 in FIG. 8) is less than 0.5 μm. In some embodiments, the distance 805 between two optical waveguides 802 and 804 is less than 2.0 μm. In some embodiments, the distance 805 between two optical waveguides 802 and 804 is in the range of 0.1 μm to 2.0 μm. In some embodiments, the distance 805 between waveguides can be defined as the distance between the inner sidewalls of the two waveguides at a predetermined point along the longitudinal direction of modulator 800 (e.g., measured in the cross-section of modulator 800 as shown in FIG. 8).

[0138] Another advantage is that, since p-doped silicon has a higher resistivity than n-doped silicon (for the same light absorption), a p-doped material with a higher resistivity is used for the small semiconductor region 838 (between the semiconductor junction diodes 812 and 814), and an n-doped material with a lower resistance value is used for the large semiconductor regions 840 and 842 (connecting the semiconductor junction diodes 812 and 814 to the electrodes 816 and 818). Alternatively, in some embodiments, an n-doped material can be used for the smaller semiconductor region 838, and a p-doped material can be used for the larger semiconductor regions 840 and 842.

[0139] As a result, the total series resistance between the electrodes 816 and 818 is significantly reduced, and the modulation bandwidth and speed are significantly improved.

[0140] The lack of a DC bias voltage connection in the modulator 800 deprives the freedom in the ability to adjust the amount of reverse bias in the semiconductor junction diodes 812 and 814. However, in some scenarios, for example, the large benefits provided by the configuration of the modulator 8, such as the improvement of the modulation bandwidth and speed, outweigh such limitations.

[0141] The modulator according to the embodiments of the present disclosure can be used in many applications. For example, one application is a high-speed optical intensity modulator that generates an IM-DD (Intensity Modulated Direct Detection) format such as NRZ (Non-Return-to-Zero) or PAM (Pulse Amplitude Modulation). Another application is to use the modulator in combination with a second modulator having a 90-degree relative phase shift as part of a large interferometer to generate more complex modulation formats for coherent detection, such as quadrature phase shift keying (QPSK) modulation or quadrature amplitude modulation (QAM). For example, this can be achieved by an in-phase / quadrature modulator structure including nested modulators (each of the two branches of the modulator (outer modulator) implements another modulator (inner modulator)). In some embodiments, phase shifters can be implemented to set phase differences of 180 degrees and 90 degrees for the inner modulator and the outer modulator, respectively. As described in the present disclosure, each modulator of such a nested modulator structure can be implemented.

[0142] FIG. 9 is a flowchart showing an example of a method 900 for modulating quasi-TM polarization according to an embodiment of the present disclosure. The method 900 can be executed by using a waveguide as disclosed herein, such as waveguide 420 or waveguide 450.

[0143] The method 900 includes inputting quasi-TM polarization into an optical waveguide having a non-uniform vertical doping profile (902). For example, the optical waveguide can be a rib waveguide, and the doping concentration at the lower part of the rib can be made higher than the doping concentration at the upper part of the rib.

[0144] Method 900 further includes applying at least one electric field to the quasi-TM polarization in the optical waveguide (904). For example, a p / n or PIN semiconductor junction can be formed in the waveguide, and the p-dopant, n-dopant, or both have a non-uniform vertical doping profile. The electric field can be applied using one or more electrodes electrically coupled to the semiconductor junction, for example, using the electrode structures described with respect to modulator 100, modulator 200, modulator 600, modulator 700, or modulator 800. The electric field can induce a phase shift of the quasi-TM polarization, for example, for use in optical signal modulation. Any of the control operations described with respect to the use of modulator 100, modulator 200, modulator 600, modulator 700, or modulator 800 can be performed in combination with or using method 900.

[0145] FIG. 10 is a flowchart illustrating an example of a method 1000 for modulating a quasi-TM polarized optical signal according to an embodiment of the present disclosure. Method 1000 can be performed by using a modulator as disclosed herein. For example, method 1000 can be performed using modulator 100, modulator 200, modulator 600, modulator 700, or modulator 800. The modulator used to perform method 1000 can include one or more rib waveguides such as waveguide 420 and / or waveguide 450. For example, the rib waveguide can have at least one non-uniform vertical doping profile and / or can have a rib width in the range of 250 nm to 400 nm or 300 nm to 400 nm.

[0146] Method 1000 includes splitting the quasi-TM polarization into a first optical transmission path and a second optical transmission path (1002). In some embodiments, an optical phase rotator that rotates the phase of the input light such that the quasi-TM light propagates through the optical transmission path may be implemented at the input of the modulator.

[0147] Method 1000 further includes modulating a phase difference between the quasi-TM polarization in the first optical transmission path and the quasi-TM polarization in the second optical transmission path (1004). For example, the phase difference can be modulated by adjusting the voltage applied to the semiconductor junction diode of the rib waveguide in one or both of the transmission paths. In some embodiments, the phase difference is modulated without applying a bias voltage between the first optical transmission path and the second optical transmission path. In some embodiments, the phase difference is modulated without applying a bias voltage via an impedance of less than 1 kΩ or less than 100 Ω between the first optical transmission path and the second optical transmission path. In some embodiments, the phase difference between the quasi-TM polarization in the first optical transmission path and the quasi-TM polarization in the second optical transmission path is modulated while maintaining a finite depletion region in the semiconductor junction diode of each of the first optical transmission path and the second optical transmission path. For example, this modulation can be performed using the floating anode structure of the modulator described above.

[0148] Method 1000 further includes multiplexing the quasi-TM polarization output from the first optical transmission path and the quasi-TM polarization output from the second optical transmission path (10,006).

[0149] FIG. 11 shows an example of a method 1100 for manufacturing a silicon photonic optical modulator. The optical modulator includes one or more waveguides. For example, the optical modulator can be modulator 100, modulator 200, modulator 600, modulator 700, or modulator 800.

[0150] Method 1100 includes forming a rib waveguide including ribs disposed on a slab on a substrate (1102). For example, forming the rib waveguide can include performing one or more etching, lithography, deposition, annealing, oxidation, and / or material growth steps to form the ribs disposed on the slab. For example, the ribs and the slab can be composed of silicon (e.g., crystalline silicon and / or another form of silicon) on a substrate such as a semiconductor substrate, optionally with a dielectric layer interposed therebetween, as shown in FIGS. 2 and 8. The rib waveguide can have any of the waveguide geometries and material properties described with respect to waveguides 420 and 450 and FIGS. 2 and 8.

[0151] Method 1100 further includes implanting at least one dopant into the ribs such that the ribs are doped in an ion implantation process (1104). The at least one dopant can form a p-type region and / or an n-type region within the ribs, resulting in the formation of a semiconductor junction diode within the ribs. The at least one dopant is implanted such that the concentration of the at least one dopant in the vertical doping profile within the ribs is higher at the lower portion of the ribs than at the upper portion of the ribs. Both donor and acceptor implantations can be performed using appropriate masking.

[0152] For example, in some embodiments, the at least one dopant is implanted by directing a beam of the at least one dopant into the ribs from a side of the substrate on which the ribs are disposed, e.g., generally vertically downward. For example, to implant a dopant into rib 422 of FIG. 4C, the beam of dopant can be directed into the rib in the -z direction from above rib 422.

[0153] By appropriately selecting the beam energy, dopants can be implanted such that the concentration of at least one dopant in the vertical doping profile within the rib is higher at the lower part of the rib than at the upper part of the rib. For example, using a high acceleration energy, the implanted dopant ions can be made to penetrate deep into the rib. In some embodiments, the acceleration energy is between 50 keV and 230 keV, an energy at which it has been found that a doping profile suitable for obtaining the advantages described herein is obtained. In some embodiments, an acceleration energy between 100 keV and 200 keV can further provide an advantageous doping profile, such as providing a balance between deep implantation of the dopant and a broad distribution of the dopant for effective modulation.

[0154] Other methods of forming the waveguides described herein are also within the scope of the present disclosure. For example, in some embodiments, the waveguide is doped by dopant diffusion or grown with dopant atoms already included in, for example, the rib and / or slab.

[0155] Although the present disclosure includes details of many specific embodiments, these should not be construed as limitations on the scope of the invention or the claims, but rather as descriptions of features specific to particular embodiments of a particular invention. In the present disclosure, the particular features described in the context of individual embodiments can also be implemented in combination in a single embodiment. Conversely, the various features described in the context of a single embodiment can also be implemented separately in multiple embodiments or in any suitable sub-combination. Furthermore, features are described above as acting in a particular combination and may even initially be claimed as such, but one or more features from the claimed combination may in some cases be deleted from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0156] Similarly, although the operations are depicted in the drawings in a particular order, this should not be construed as requiring that the particular order in which such operations are shown, or that they be performed sequentially, or that all the operations shown be performed, to achieve the desired result.

[0157] The claims are as follows.

Claims

Claim 1 an optical input; and an optical waveguide connected to the optical input and configured to propagate quasi-transverse magnetic (quasi-TM) polarization, the optical waveguide being configured as a rib waveguide including a rib disposed on a slab; comprising the rib including at least one dopant; an average concentration of the at least one dopant in a bottom vertical doping profile of the rib being greater than an average concentration of the at least one dopant in a top vertical doping profile of the rib; a top portion of the rib having a height between 20% and 80% of a height of the rib waveguide; and a bottom portion of the rib including a remainder of the rib below the top portion; a silicon photonic optical modulator. Claim 2 The silicon photonic optical modulator according to claim 1, wherein the average concentration of the at least one dopant at the bottom portion is at least 1.5 times the average concentration of the at least one dopant at the top portion. Claim 3 The silicon photonic optical modulator according to claim 2, wherein the average concentration of the at least one dopant at the bottom portion is at least 2 times the average concentration of the at least one dopant at the top portion. Claim 4 The silicon photonic optical modulator according to claim 1, wherein the height of the top portion is between 35% and 65% of the height of the rib waveguide. Claim 5 The average concentration of the at least one dopant at the lowermost part is 10 17 cm -3 to 10 18 cm -3 and the average concentration of the at least one dopant at the uppermost part is less than 5×10 16 cm -3 The silicon photonic optical modulator according to claim 1. Claim 6 the at least one dopant including a first dopant in a first lateral portion of the rib and a second dopant in a second lateral portion of the rib, the second lateral portion being opposite the first lateral portion; The silicon photonic optical modulator according to claim 1, wherein the first lateral portion and the second lateral portion form a semiconductor junction diode. Claim 7 a concentration of the first dopant in a first vertical doping profile in the first lateral portion of the rib being higher than in an upper portion of the rib in a lower portion of the rib; and a concentration of the second dopant in a second vertical doping profile in the second lateral portion of the rib being higher than in an upper portion of the rib in a lower portion of the rib; Claim 8 The silicon photonic optical modulator according to claim 6, comprising an electrode configured to apply an electric field to the semiconductor junction diode. Claim 9 comprising a semiconductor contact region with which the electrode is in contact; The silicon photonic optical modulator according to claim 8, wherein the height of the semiconductor contact region is greater than the height of the slab. **Claim 10** The silicon photonic optical modulator according to claim 1, wherein the effective refractive index of the TM-polarized two-dimensional (2D) waveguide mode in the rib waveguide is greater than the effective refractive index of the transverse electric field (TE)-polarized one-dimensional (1D) waveguide mode in the slab. **Claim 11** wherein the optical waveguide is a first optical waveguide, the silicon photonic optical modulator comprises a Mach-Zehnder interferometer having the first optical waveguide and a second optical waveguide, the first optical waveguide comprises a first semiconductor junction diode based on the at least one dopant, and the second optical waveguide comprises a second semiconductor junction diode based on the at least one dopant, the silicon photonic optical modulator according to claim 1. **Claim 12** Optical input; an optical waveguide configured to receive light from the optical input, the optical waveguide being configured as a rib waveguide including a rib disposed on a slab, the rib waveguide having a shape configured to propagate quasi-transverse magnetic (quasi-TM) polarization; and an electrode configured to apply an electric field across the rib waveguide, comprising the width of the rib waveguide is in the range of 250 nm to 400 nm, a silicon photonic optical modulator. **Claim 13** The silicon photonic optical modulator according to claim 12, wherein the height of the rib waveguide is greater than the width of the rib waveguide. **Claim 14** the height of the rib waveguide is in the range of 300 nm to 400 nm, and the thickness of the slab is in the range of 50 nm to 150 nm, the silicon photonic optical modulator according to claim 12. **Claim 15** The silicon photonic optical modulator according to claim 12, wherein the height of the rib waveguide is in the range of 250 nm to 360 nm. **Claim 16** wherein the optical waveguide is a first rib waveguide, the silicon photonic optical modulator comprises a second rib waveguide, and the distance between the first rib waveguide and the second rib waveguide is less than 500 nm, the silicon photonic optical modulator according to claim 12. **Claim 17** The silicon photonic optical modulator according to claim 16, wherein the height of the first rib waveguide is at least 10 nm greater than the height of the second rib waveguide in at least a part of the silicon photonic optical modulator.

18. A method of manufacturing a silicon photonic optical modulator, comprising: forming a rib waveguide on a substrate, the rib waveguide comprising a rib disposed on a slab; and injecting at least one dopant into the rib, wherein an average concentration of the at least one dopant in a vertical doping profile at a bottommost portion of the rib is greater than an average concentration of the at least one dopant in a vertical doping profile at a topmost portion of the rib, wherein the topmost portion of the rib has a height between 20% and 80% of a height of the rib waveguide, and wherein the bottommost portion of the rib includes the remainder of the rib below the topmost portion.

19. The method according to claim 18, wherein the step of injecting the at least one dopant into the rib includes directing a beam of the at least one dopant toward the rib in a direction from a side surface of the substrate on which the rib is disposed toward the substrate.

20. The method according to claim 19, wherein the beam has an acceleration energy in a range of 50 keV to 230 keV.

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