Patent Publication Number: US-2023152662-A1

Title: Transverse-magnetic polarization silicon-photonic modulator

Description:
TECHNICAL FIELD 
     The present disclosure generally relates to electro-optical modulators in silicon photonics. 
     BACKGROUND 
     In optical communication systems, electro-optical modulators provide a fundamental mechanism of modulating optical waveforms to carry information. In general, electro-optical modulators operate by modifying one or more properties of optical waveforms according to information, such as digital data, provided by electrical signals. 
     SUMMARY 
     Implementations of the present disclosure are generally directed to electro-optical modulators in silicon photonics. 
     One general aspect includes a silicon-photonic optical modulator including: at least one optical input and at least one optical waveguide that is connected to the at least one optical input. The at least one optical waveguide is configured to propagate quasi-transverse-magnetic (quasi-TM) polarized light, where each of the at least one optical waveguide is configured as a rib waveguide that includes a rib arranged on a slab. The silicon-photonic optical modulator also includes at least one electrode configured to apply at least one electric field to the quasi-TM polarized light in the at least one optical waveguide. 
     Implementations may include one or more of the following features. The silicon-photonic optical modulator where the silicon-photonic optical modulator is configured as a silicon-photonic depletion modulator in which the at least one optical waveguide includes at least one semiconductor junction diode. The silicon-photonic optical modulator where the at least one electrode is configured to apply the at least one electric field to the quasi-TM polarized light in the at least one semiconductor junction diode. The silicon-photonic optical modulator where an effective refractive index of a TM polarization 2-dimensional (2D) guided mode in the rib waveguide is greater than an effective refractive index of a transverse-electric (TE) polarization 1-dimensional (1-D) guided mode in the slab. The silicon-photonic optical modulator where a doping concentration is increased by more than 10 17  activated dopants per cm 3  in a first portion of the slab that is within 100 nm of a nearest sidewall of the rib, as compared to a second portion of the slab that is farther than 100 nm from the nearest sidewall of the rib. The silicon-photonic optical modulator where the doping concentration is increased by a value within a range of 5×10 17  to 1×10 19  activated dopants per cm 3  in the first portion of the slab that is within a range of 50 nm to 500 nm of the nearest sidewall of the rib, as compared to the second portion of the slab that is farther away from the nearest sidewall of the rib. The silicon-photonic optical modulator, further including a Mach-Zehnder interferometer including the at least one optical waveguide, where the at least one optical waveguide includes: (i) a first optical waveguide including a first semiconductor junction diode, and (ii) a second optical waveguide including a second semiconductor junction diode. The silicon-photonic optical modulator, further including a semiconductor region that connects the first semiconductor junction diode with the second semiconductor junction diode. The silicon-photonic optical modulator where a distance between the first optical waveguide and the second optical waveguide is less than 500 nm for at least a portion of a longitudinal direction of the silicon-photonic optical modulator. The silicon-photonic optical modulator where the first semiconductor junction diode includes a first p-doped region and a first n-doped region. The silicon-photonic optical modulator where the second semiconductor junction diode includes a second p-doped region and a second n-doped region. The silicon-photonic optical modulator where the first p-doped region is connected to the second p-doped region through a third p-doped region in the semiconductor region that connects the first semiconductor junction diode with the second semiconductor junction diode. The silicon-photonic optical modulator where the third p-doped region is configured without any external voltage connection that has an impedance less than 100 ohm. 
     Another general aspect includes a silicon-photonic optical modulator including: at least one optical input and at least one optical waveguide. The at least one optical waveguide is configured to receive light from the at least one optical input, where each of the at least one optical waveguide is configured as a rib waveguide that includes a rib arranged on a slab. The silicon-photonic optical modulator also includes at least one electrode configured to apply at least one electric field to the light in the at least one optical waveguide. The silicon-photonic optical modulator where a height of the rib waveguide is greater than 0.85 λ/n, where λ is a free-space wavelength of light and n is a refractive index of silicon in the silicon-photonic optical modulator. The silicon-photonic optical modulator where a width of the rib waveguide is greater than a thickness of the slab. 
     Implementations may include one or more of the following features. The silicon-photonic optical modulator where the height of the rib waveguide is greater than the width of the rib waveguide. The silicon-photonic optical modulator where the height of the rib waveguide is within a range of 320 nm to 500 nm. The silicon-photonic optical modulator where the width of the rib waveguide is within a range of 150 nm to 270 nm. The silicon-photonic optical modulator where the thickness of the slab is within a range of 50 nm to 140 nm. The silicon-photonic optical modulator where for the free-space wavelength of the light equal to 1310 nm: the height of the rib waveguide is within a range of 330 nm to 370 nm. The silicon-photonic optical modulator where the width of the rib waveguide within a range of 200 nm to 240 nm. The silicon-photonic optical modulator where the thickness of the slab is within a range of 70 nm to 110 nm. The silicon-photonic optical modulator where the at least one optical waveguide includes a first rib waveguide and a second rib waveguide. The silicon-photonic optical modulator where a distance between the first rib waveguide and the second rib waveguide is less than 500 nm. The silicon-photonic optical modulator where a height of the first rib waveguide is greater than a height of the second rib waveguide in at least part of the silicon-photonic optical modulator. The silicon-photonic optical modulator where for a first portion of the silicon-photonic optical modulator, the height of the first rib waveguide is greater than the height of the second rib waveguide by at least 40 nm. The silicon-photonic optical modulator where for a second portion of the silicon-photonic optical modulator, the height of the second rib waveguide is greater than the height of the first rib waveguide by at least 40 nm. The silicon-photonic optical modulator where a doping concentration is increased by more than 10 17  activated dopants per cm 3  in a first portion of the slab that is within 100 nm of a nearest sidewall of the rib, as compared to a second portion of the slab that is farther than 100 nm from the nearest sidewall of the rib. 
     Another general aspect includes a method of modulating quasi-transverse-magnetic (TM) polarized light, the method including: inputting an input quasi-TM polarized light into at least one optical waveguide, and applying at least one electric field to quasi-TM polarized light in the at least one optical waveguide. 
     Implementations may include one or more of the following features. The method further including: splitting the input quasi-TM polarized light into a first optical waveguide and a second optical waveguide. The method may also include modulating a phase difference between quasi-TM polarized light in the first optical waveguide and quasi-TM polarized light in the second optical waveguide, without applying a bias voltage through an impedance that is less than 100 ohm between the first optical waveguide and the second optical waveguide. The method may also include combining quasi-TM polarized light that is output from the first optical waveguide and the quasi-TM polarized light that is output from the second optical waveguide. The method where the phase difference between the quasi-TM polarized light in the first optical waveguide and the quasi-TM polarized light in the second optical waveguide is modulated while maintaining finite depletion regions in semiconductor junction diodes in each of the first optical waveguide and the second optical waveguide. 
     The details of one or more implementations of the subject matter of this disclosure are set forth in the accompanying drawings and the description below. 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 
         FIG.  1    illustrates an example of a top view of a modulator in which implementations of this disclosure may be utilized; 
         FIG.  2    illustrates an example of a cross section of a modulator according to implementations of this disclosure; 
         FIG.  3    illustrates an example of an equivalent circuit along a cross-section of a modulator according to implementations of this disclosure; 
         FIGS.  4 A and  4 B  illustrate examples of a detailed cross section of a single waveguide of a modulator according to implementations of this disclosure; 
         FIGS.  5 A and  5 B  illustrate examples of TE and TM modes in silicon rib waveguides, respectively; 
         FIG.  6    illustrates an example of a top view of a modulator, according to implementations of the present disclosure; 
         FIG.  7    illustrates another example of a top view of a modulator, according to implementations of the present disclosure; 
         FIG.  8    illustrates an example of a cross section of a modulator, according to implementations of the present disclosure; 
         FIG.  9    illustrates an example of an equivalent circuit along a cross-section of a modulator, according to implementations of the present disclosure; and 
         FIG.  10    is a flowchart illustrating an example of modulating a TM polarized optical signal, according to implementations of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Systems and techniques are disclosed herein that provide a novel electro-optic modulator in silicon photonics which can achieve a higher bandwidth and/or a lower drive voltage as compared with conventional electro-optical modulators. This is accomplished by novel implementations which reduce the amount of light that leaks into the slab portion of the optical waveguide of the modulator. This enables a higher doping in the slab for the same optical loss, thereby enabling a higher-bandwidth modulator without an increase in the optical loss. These technical advantages are achieved by a modulator structure that enables use of transverse-magnetic (TM) polarized light in the modulator, instead of transverse-electric (TE) polarized light. In some implementations, this is enabled by a rib waveguide structure in which the waveguide height is greater than the waveguide width. This, in turn, results in TM light having a higher effective index than TE light in the rib waveguide. 
       FIG.  1    illustrates an example of a top view of a differential modulator  100  in which implementations of this disclosure may be utilized. In this example, the modulator  100  is based on a Mach-Zehnder interferometer (MZI) implementation, in which optical signals propagate along the length of the modulator  100  (e.g., from left to right in  FIG.  1   ) along two optical transmission paths  102  and  104 . At the input of modulator  100 , optical splitter  106  splits an input light into the two optical transmission paths  102  and  104 . At the output of the modulator  100 , the optical combiner  108  combines light output from the two optical transmission paths  102  and  104 . The optical splitter  106  and the optical combiner  108  may be implemented in various ways, for example, using symmetric, asymmetric, or tunable optical intensity couplers. The optical transmission paths  102  and  104  can be implemented by waveguides formed in a semiconducting structure  116 , as described in further detail with reference to  FIG.  2   , below. In some implementations, the optical cores of the waveguides, and/or the optical splitter  106 , and/or the optical combiner  108  can include silicon ribs. In some implementations, an optical phase rotator may be implemented between the input of modulator  100  and the optical transmission paths  102  and  104 , which rotates a phase of the input light so that quasi-TM light propagates in the optical transmission paths  102  and  104 . 
     The modulator  100  uses a travelling wave configuration in which voltages applied at terminals  110  and  112  create an electrical signal that propagates along a radio frequency (RF) transmission line  114 , which is terminated at an RF termination resistance. The electrical signal in RF transmission line  114  travels at the same speed as and induces electro-optic modulation in the light that propagates along the two optical transmission paths  102  and  104 . In particular, the RF transmission line  114  is connected to the semiconducting structure  116  via electrodes (described in further detail with reference to  FIG.  2   , below), that apply respective voltages, and resulting electric fields, across one or both of the optical transmission paths  102  and  104 . The applied voltage(s) induce a phase shift in the light that propagates in one or both of the optical transmission paths  102  and  104 . In some implementations, the phase shift is differential in that the phase shift magnitude is equal and the phase shift sign is opposite between the optical transmission paths  102  and  104 . 
     Electro-optic modulation is achieved by varying the voltage at one or both of the terminals  110  and  112  to modulate the differential phase shift between the phase of light in the first optical transmission path  102  and the phase of 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 the optical combiner  108 , then this corresponds to an “off” or logic “ 0 ” state of the modulator  100 . By 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 the optical combiner  108 , then this corresponds to the “on” or logic “ 1 ” state of the modulator  100 . 
     The differential phase shift between the two optical transmission paths  102  and  104  can also be influenced by other factors. For example, the physical lengths of the optical transmission paths  102  and  104  can be the same to provide zero inherent differential phase shift, or can be different lengths to provide non-zero inherent differential phase shift. Furthermore, in some implementations, direct current (DC) phase shifters  122  and  124  (e.g., thermo-optic phase-shifters, such as 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 light signals before being combining in the optical combiner  108 . 
     In some implementations, the phase modulation can be performed by a “push-pull” mechanism, in which the phases of light in both of optical transmission paths  102  and  104  are modulated, to control the relative phase shift between the two paths. In push-pull operation, the voltage V+ at terminal  110  is increased and voltage V− at terminal  112  is decreased (or vice versa), resulting in corresponding phase shifts of light in each of the optical transmission paths  102  and  104 . Push-pull modulation can provide various advantages over non-push-pull modulation, such as achieving smaller average energy consumption and reduced chirp in the modulated signal. 
     In some scenarios, a direct current (DC) bias connection  118  can be connected between the two optical transmission paths  102  and  104 . The DC bias connection  118  is implemented such that semiconductor junction diodes in each of the optical transmission paths  102  and  104  remain reverse biased, even when data signals applied at the terminals  110  and  112  vary between logical  1  and logical  0 . Further details are provided with reference to  FIG.  2   , below. 
       FIG.  2    illustrates an example of a cross section of a modulator  200  (e.g., cross section  126  of the modulator  100  of  FIG.  1   ). 
     The cross-section of modulator  200  shows details of the MZI structure. The MZI includes a first optical waveguide  202  and a second optical waveguide  204 . In some implementations, the modulator  200  includes a substrate  206  (e.g., a silicon substrate) an insulating structure  208  (e.g., a dielectric, such as an oxide), and a semiconducting structure  210  (e.g., a silicon layer which includes optical waveguides  202  and  204 ). 
     The optical waveguides  202  and  204  can be implemented, for example, as silicon ribbed waveguides on top of a slab. In the example of  FIG.  2   , optical waveguide  202  includes a rib  203  which is arranged on top of a slab  205 . Similarly, optical waveguide  204  includes a rib  207  on top of a slab  209 . The ribs  203 ,  207  and the slabs  205 ,  209  are all parts of the semiconducting structure  210 . Further details of the ribbed waveguide structure are discussed with reference to  FIGS.  4 A and  4 B , below. 
     Each of the optical waveguides  202  and  204  includes a semiconductor junction. The semiconductor junction diodes can be implemented, for example, by a PIN (P-type/intrinsic/N-type) junction diode or a P/N junction diode. In modulator  200 , a P/N junction is implanted into each of the optical waveguides  202  and  204 , forming a diode in each waveguide. These diodes are shown as first semiconductor junction diode  212  and second semiconductor junction diode  214 . 
     The modulator  200  also includes electrodes  216  and  218  (e.g., metal electrodes) which are in physical contact with the silicon layer  210 . In some implementations, the electrodes  216  and  218  are in physical contact with P-doped contact regions  220  and  222  of the silicon layer  210 . The electrodes  216  and  218  may be formed, for example, by etching the insulator layer  208  and forming metal (e.g., tungsten, copper, and/or aluminum) contacts. In some implementations, the P-doped regions may instead be N-doped regions, and vice-versa, in modulator  200  (e.g., so that contact regions  220  and  222  are N-doped instead P-doped). 
     The modulator  200  may also include metal layers  224  and  226  on top of the electrodes  216  and  218 . In some implementations, the metal layers  224  and  226  may form segments of an RF transmission line (e.g., RF transmission line  114  in  FIG.  1   ). 
     In some scenarios, a DC bias connection  228  is implemented between the two optical waveguides  202  and  204 . The DC bias connection  228  ensures that the semiconductor junction diodes  212  and  214  remain reverse biased during modulation. For example, in a push-pull mode of modulation, a differential voltage (e.g., V+ and V−) is applied at the metal layers  224  and  226  (and hence at electrodes  216  and  218 ). If the voltage (e.g., V+) at first electrode  216  is increased while the voltage (e.g., V−) at the second electrode  218  is decreased, then a width of the depletion region in the first optical waveguide  202  decreases while a width of the depletion region in the second optical waveguide  204  increases (and vice versa). As the depletion widths change, this changes the effective refractive index experienced by the light traveling along each of the optical waveguides  202  and  204 , resulting in corresponding phase shifts of the light. As a result, push-pull modulation can be achieved in the modulator  200 . 
     In the example of modulator  200 , the DC bias connection  228  is applied at the cathodes  230  and  232  (N-doped regions) of the semiconductor junction diodes  212  and  214 , while the varying voltages V+ and V− are applied at 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  remain reverse biased. For example, in the example of modulator  200 , if the bias voltage applied at the DC bias connection  228  is very low (or non-existent), then this may result in activation of the first semiconductor junction diode  212  (e.g., forward bias above approximately 0.6 V for silicon) with a significant number of carriers injected into the depletion region of the first semiconductor junction diode  212 , resulting in forward bias and slower operation. Implementing the DC bias connection  228  with a sufficiently large bias voltage ensures that the semiconductor junction diodes  212  and  214  remain reverse biased under modulation. 
       FIG.  3    illustrates an example equivalent circuit  300  along a cross-section of a modulator (e.g., the cross section  126  of the modulator  100  of  FIG.  1   ). 
     In the example of  FIG.  3   , the electrical series resistance  340  between first electrode  316  and first semiconductor junction diode  312  (e.g., corresponding to semiconducting region  240  in  FIG.  2   ) is denoted R p  (e.g., in units of mΩ-m). The electrical series resistance  342  between second electrode  318  and second semiconductor junction diode  314  (e.g., corresponding to semiconducting region  242  in  FIG.  2   ) is also denoted R p  (although the actual values of electrical series resistances  340  and  342  may be different, in some implementations). The electrical series resistance  338  between semiconductor junction diodes  312  and  314  (e.g., corresponding to semiconducting region  238  in  FIG.  2   ) is 2*R n  (with R n  series resistance between each of semiconductor junction diodes  312  and  314  and DC bias voltage connection  328 ). 
     In the equivalent circuit for the phase modulator shown in  FIG.  3   , the resistances R n  and R p  originate primarily from the slab and is the main limitation of the modulator bandwidth. Increasing the doping in the slab reduces the resistance, thus increasing the bandwidth, but it also increases the optical loss, because doped silicon is absorptive. 
       FIGS.  4 A and  4 B  illustrate examples of a detailed cross section 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.  4 A  illustrates an example waveguide  400  configured for TE-polarized light, which may be implemented in some systems, and  FIG.  4 B  illustrates an example waveguide  420  which is configured for TM-polarized light, according to implementations of the present disclosure. 
     In both  FIGS.  4 A and  4 B , waveguide  400  (and waveguide  420 ) is implemented by a rib waveguide structure, with a rib  402  (rib  422 ) on top of a slab  404  (slab  424 ). Light is guided along the rib  402  (rib  422 ) and propagates in a longitudinal direction of the modulator (normal to the cross section shown in  FIGS.  4 A and  4 B ) by total internal reflection inside the rib  402  (rib  422 ). The rib structure allows for a confined optical mode in the rib  402  (rib  422 ) while enabling electrical connections to the rib  402  (rib  422 ) through the regions on both sides of the slab  404  (slab  424 ). As discussed with reference to  FIG.  2   , above, phase modulation of light in the rib  402  (rib  422 ) is achieved by modulating the voltage difference between the n-doped and p-doped regions of the waveguide  400  (waveguide  420 ). For example, increasing the voltage difference between the n-doped and p-doped regions widens the depletion width, thereby increasing the effective refractive index of the optical mode, and allowing for phase modulation of the light in the rib  402  (rib  422 ). 
     The waveguides  400  and  420  in  FIGS.  4 A and  4 B  differ in several aspects. Most noticeably, the waveguides  400  and  420  differ in dimension, with waveguide  400  (configured for TE-polarized light) being wider and shorter, and waveguide  420  (configured for TM-polarized light) being narrower and taller. The narrower and taller configuration of waveguide  420  in  FIG.  4 B  enables a reduction in the portion of the optical mode that is in the slab  424 , allowing for a higher doping in the slab  424  for the same optical loss, as compared to waveguide  400  of  FIG.  4 A . The higher doping in the slab  424 , in turn, allows for a higher bandwidth in modulator  420 , as compared with modulator  400 , without having to increase the optical loss. This is done by using transverse-magnetic (TM) polarized light in the modulator  420  instead of transverse-electric (TE) polarized light. In practical implementations, the guided optical mode in modulators  400  and  420  is actually a quasi-TE or quasi-TM mode, because guided 2D modes are almost never purely TE or TM modes. In quasi-TM mode, the dominant polarization component of the light is aligned along the y-axis. In quasi-TE mode, the dominant polarization component of the light is aligned along the x-axis. For the sake of brevity in exposition, the word “quasi” may be omitted when discussing the polarization of a guided optical mode in this disclosure. 
       FIG.  4 A  illustrates a cross section of an example waveguide  400  configured for TE-polarized light. The modulator  400  has a rib  402  on top of a slab  404 . The rib has a height  406  (denoted h core ) and a width  408  (denoted w core ). As shown, in typical implementations of waveguide  400  configured to TE-polarized light, the rib height  406  (h core ) is smaller than the rib width  408  (w core ). This ensures that the effective index of the TE 2D waveguide mode in the rib  402  is higher than the effective index of the TM 1D slab mode, thus ensuring that a guided TE mode will suffer less leakage to the slab  404 , as compared to a guided TM mode. 
     There are various reasons for why silicon-photonic modulators, such as modulator  400 , are configured for TE-polarized light. 
     First, for modulators that employ rib waveguides, the TM 2-D rib mode index is typically significantly lower than the TE 1-D slab mode index. The rib waveguides need special conditions to guide transverse-magnetic (TM) light which are not normally met. This condition is that the effective index of the TM 2-D rib mode must be larger than that of the TE 1-D slab mode. Slab mode means refers to the 1-D mode that would be guided if there is was no rib  402 , and if the slab  404  was infinitely wide. Otherwise, the TM rib mode will be phase matched to the TE slab mode propagating at certain angles with respect to the rib  402 . In such a case, small perturbations will cause the light in the TM mode to leak away into the slab  404 . 
     Second, TE-polarized light has a tighter vertical confinement in the rib  402 , as compared to TM-polarized light, which mitigates losses due to the substrate below and layers on top. For example, in some implementations, there are metal routing layers above the silicon, and the metal layers can be significantly closer to the silicon before causing significant optical losses for TE-polarized light than TM-polarized light. 
     Third, in most silicon photonic modulators, the waveguide height  406  is less than the waveguide width  408 , which results in TE-polarized light having a higher effective index than TM-polarized light. This allows for a smaller bend radius, decreasing the size of the silicon photonic devices. 
     Fourth, most silicon-photonic modulators employ TE-polarized light because most of the other elements in a silicon photonic circuit are designed for TE polarization. For example, most grating couplers are configured for TE polarization. 
     Fifth, in many scenarios, it is typically easier to fabricate a waveguide structure that has a width greater than its height, e.g., because the lithography process is simplified by a shallower depth of etching. 
     However, TM polarized light has distinct advantages. For example, TM-polarized light has the advantage of having less light in the slab  424 , as compared to TE-polarized light. To understand why TM-polarized light has less light in the slab  424  than TE-polarized light, one can consider the boundary conditions on the electric field of light that are given by Maxwell&#39;s equations. In non-magnetic materials, such as silicon, the transverse electric field, E ∥ , is continuous across a boundary; whereas the normal electric field times the permittivity, (E ⊥ ) (ε), is continuous across a boundary. Because the permittivity of silicon is approximately 5.8 times than that of oxide, when the electric field is normal to a thin piece of silicon surrounded by oxide, the electric field inside that silicon is approximately 5.8 times lower than in the surrounding oxide. Thus, TM-polarized light has very little electric field inside the silicon slab  424 . 
     This can be seen visually in  FIGS.  5 A and  5 B , which show TE and TM modes in silicon rib waveguides, respectively.  FIG.  5 A  shows an example of calculated modes of a conventional silicon phase modulator using TE-polarized light. In particular,  FIG.  5 A  shows the magnitudes of the x-component of the electric field.  FIG.  5 B  shows an example of calculated modes of a silicon phase modulator using TM-polarized light, according to implementations of the present disclosure. In particular,  FIG.  5 B  shows the magnitudes of the y-component of the electric field. 
     As shown, there is significant light in the slab in  FIG.  5 A  but very little light in the slab in  FIG.  5 B . Thus the slab in  FIG.  5 B  can have significantly higher doping near the rib and thus significantly lower series resistance. 
     In addition, the waveguide rib dimensions are different in the examples of  FIGS.  5 A and  5 B . In both cases, the slab thickness is 90 nm. However, in  FIG.  5 A , the waveguide rib height and rib width are 220 nm and 420 nm, respectively, whereas in  FIG.  5 B  the waveguide rib height and rib width are 400 nm and 220 nm, respectively. The waveguide of  FIG.  5 A  is a typical modulator waveguide configured for TE-polarized light. As discussed above, in such a configuration, a guided TM mode will leak into the slab, because the effective index of the TM 2D rib mode is lower than the effective index of the TE 1D slab mode (see Table 1, below). 
     By contrast, the waveguide of  FIG.  5 B  is able to guide a TM mode without leakage into the slab, because the waveguide rib is taller and narrower. Having a taller waveguide rib increases the effective index of the TM 2D waveguide mode above that of the TE 1D slab mode, as seen in Table 1. This occurs when the waveguide rib height is greater than a threshold of approximately 0.85 λ/n, and when the waveguide rib width is wider than the slab height, where λ is the free-space wavelength of light and n is the refractive index of silicon. This guarantees that for TM-polarized light, the electric field has fallen to a low value at the top and bottom of the waveguide so that the boundary condition does not cause significant field to fall outside the waveguide. For instance, for a wavelength of λ=1310 nm, the threshold is 0.85 λ/n=320 nm. Thus, in this example, the waveguide rib height should be larger than 320 nm and the waveguide rib width should be larger than 90 nm. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Effective mode indices, for a wavelength of λ = 1310 nm. 
               
            
           
           
               
               
               
            
               
                   
                 Waveguide configured 
                 Waveguide configured 
               
               
                   
                 for TE-polarized light 
                 for TM-polarized light 
               
               
                   
                 (420 × 220 nm 2 ) 
                 (220 × 400 nm 2 ) 
               
               
                   
                   
               
            
           
           
               
               
            
               
                 TE 1D slab (90 nm) 
                 2.25 
               
            
           
           
               
               
               
            
               
                 TE 2D waveguide 
                 2.70 
                 2.47 
               
               
                 TM 2D waveguide 
                 2.13 
                 2.58 
               
               
                   
               
            
           
         
       
     
     Implementations of modulators according to the present disclosure which are configured for TM-polarized light can provide various technical advantages (as compared to typical modulators configured for TE-polarized light). For example, the doping in the slab can be increased significantly and/or higher doping can be placed closer to the rib. In some implementations, a doping concentration can be increased by a value within a range of 5×10 17  to 1×10 19  (e.g., increased within a range of 1×10 18  to 1×10 19 ) activated dopants per cm 3  in a first portion of the slab that is within a range of 50 nm to 500 nm (e.g., 100 nm) of a nearest sidewall of the rib, as compared to a second portion of the slab that is farther than 100 nm from the nearest sidewall of the rib. Some implementations of the present disclosure can provide approximately 3.5 times lower series resistance as compared to a typical modulator that is configured for TE-polarized light. Another advantage is that the phase modulation efficiency can be increased for a given voltage and a given modulator length. This is a consequence of TM-polarized light being more confined horizontally in the waveguide rib, perpendicular to the depletion region, thus resulting in a larger effective index change for a given voltage change. 
     In addition, modulator implementations according to the present disclosure can be configured to mitigate potential technical challenges. For example, in modulators configured for TM-polarized light, because the waveguide rib is configured to be taller and thinner (as compared with waveguide ribs of typical modulators designed for TE-polarized light), there can be increased series resistance along the vertical edges of the rib, connecting to the top of the waveguide. To mitigate such resistance, a preferred embodiment is to configure the waveguide rib to be only a small amount taller than the threshold to make the effective TM 2D rib index higher than the TE 1D slab effective index. For example, in some implementations, the waveguide rib height is 350 nm and the waveguide rib width is 220 nm, with a 90-nm slab at 1310-nm wavelength. 
     Another challenge is that the capacitance of the p-n junction of the waveguide (e.g., semiconductor junction diodes  212  and  214  in  FIG.  2   ) could be increased, as a consequence of the depletion region being taller. However, the fringing fields contribute significantly to the capacitance in these structures, and consequently the capacitance increase is sublinear to the height increase. As such, increasing the waveguide rib height by a factor of 2 results in an increase of the capacitance by only a factor of approximately 1.5. 
       FIGS.  6 - 9    relate to modulators according to other implementations of the present disclosure. In contrast with the modulators of  FIGS.  1 - 3   , the modulators of  FIGS.  6 - 9    do not implement any bias voltage connection between the waveguides, resulting in significantly smaller series resistance between electrodes, and thus even higher bandwidth of modulation. Furthermore, in  FIGS.  6 - 9   , the modulators implement waveguide structures that vary in height so as to mitigate detrimental optical coupling between the closely-spaced waveguides. 
     The features described with reference to  FIGS.  6 - 9    can help improve upon the structure of the modulators in  FIGS.  1 - 3    in various aspects. For example, the presence of DC bias connection  228  in  FIG.  2    increases the physical distance of the semiconducting (e.g., silicon) region  238  between the semiconductor junction diodes  212  and  214 . This results in significant electrical series resistance in the semiconducting region  238  that connects the semiconductor junction diodes  212  and  214 . Typical techniques to reduce such electrical series resistance, such as increasing the silicon doping of the semiconducting structure, can have negative consequences such as increasing optical absorption. 
     Furthermore, the semiconducting regions  240  and  242  in  FIG.  2    (which connect each of semiconductor junction diodes  212  and  214  with their respective electrodes  216  and  218 ) are P-doped semiconducting material, which has higher resistance than N-doped semiconducting material (for the same optical absorption). This results in significant electrical series resistance in the semiconducting regions  240  and  242  between electrodes  216  and  218  and the semiconductor junction diodes  212  and  214 . 
     Consequently, the total electrical series resistance between electrodes  216  and  218  in  FIG.  2    can significantly attenuate the voltage along the modulator  200  due to charging and discharging of the diode capacitance. Furthermore, this attenuation typically increases as modulation frequency increases. The resulting RF loss along the modulator  200  can detrimentally impact the bandwidth of the modulator  200 . 
       FIG.  6    illustrates an example of a top view of a modulator  600  according to implementations of the present disclosure. 
     The modulator  600  is based on an MZI implementation which includes two optical transmission paths  602  and  604 , optical splitter  606 , and optical combiner  608 . The modulator  600  further includes terminals, such as terminal  610  and terminal  612 , through which voltages can be applied. The voltages travel along RF transmission line  614 , which is connected to semiconducting structure  616  via electrodes that apply respective voltages, and resulting electric fields, across one or both of the optical transmission paths  602  and  604 . In some implementations, an optical phase rotator may be implemented between the input of modulator  600  and the optical transmission paths  602  and  604 , which rotates a phase of the input light so that quasi-TM light propagates in the optical transmission paths  602  and  604 . 
     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 enables the two optical transmission paths  602  and  604  to be more closely-spaced together, thus reducing electrical series resistance therebetween. For example, in some implementations, the distance between the waveguides of the two optical transmission paths  602  and  604  is less than 0.5 μm for at least a portion of the longitudinal direction of the optical transmission paths  602  and  604 . In some implementations, 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  602  and  604 . In some implementations, the distance between the waveguides is within a range of 0.1 μm to 2.0 μm for at least a portion of the longitudinal direction of the optical transmission paths  602  and  604 . In some implementations, the distance between the waveguides is defined as the distance between the inner sidewalls of the two waveguides, at a given point along a longitudinal direction of the modulator  600  (e.g., at a point  605  in  FIG.  6   ). 
     However, because the two optical transmission paths  602  and  604  are more closely spaced, there is risk of more significant detrimental optical coupling between light in optical transmission path  602  and light in optical transmission path  604 . To mitigate such optical coupling, in some implementations, the waveguide of one of the optical transmission paths ( 602  or  604 ) is designed to have a greater height than the other path, at the same distance along the length of the modulator  600 . This helps ensure that the light traveling in the waveguides of optical transmission paths  602  and  604  are not phase matched, thus mitigating optical coupling between the two waveguides. An alternative way to understand the importance of using different waveguide heights is to look at the two eigenmodes of the coupled waveguides of optical transmission paths  602  and  604 . If the waveguides have equal heights, then the lowest order eigenmode is the even eigenmode, and the second lowest eigenmode is the odd eigenmode. In such a scenario, no differential modulation can occur. However, if one waveguide is sufficiently taller than the other, then the lowest order eigenmode consists of light that is predominantly in the taller waveguide, and the second lowest eigenmode is predominantly in the shorter waveguide. This enables differential modulation to occur despite the closely-spaced waveguides. For example, in some implementations, the waveguide of the one of the optical transmission paths  702  or  704  is taller by at least 40 nm than the waveguide of the other optical transmission path. In some implementations, the waveguide height difference is within a range of 40 nm to 120 nm. 
     Furthermore, in such implementations, the height variation of the two waveguides may be exchanged along the modulator  600 , to help ensure that the total length of the taller portions in each waveguide are equal, and also that the total length of the shorter portions in each waveguide are equal. In the example of  FIG.  6   , moving from the left to right, the waveguide of first optical transmission path  602  is taller than the waveguide of the second optical transmission path  604 , and then becomes shorter than the waveguide of the second optical transmission path  604  (alternatively, the first optical transmission path  602  may start shorter and become taller). There may be one such exchange in relative heights in the middle of modulator  600 , but in some implementations, additional height exchanges can be included, e.g., as long as the distance between height exchanges is significantly longer than the beat length between the two eigenmodes in the two waveguides, which is typically 10 μm. This helps mitigate optical coupling between the two waveguides. In some implementations, an odd number of exchanges is preferred, since this will help ensure that the beginning and end transitions cancel each other out. 
     Although the description of  FIG.  6   , above, provided an example of a modulator  600  with variable-height waveguides in the two optical transmission paths  602  and  604 , in other implementations, the waveguides may have constant height along the length of the modulator  600 . 
     Furthermore, although the description of  FIG.  6    provided an example of a modulator  600  without a physical DC bias connection, in some implementations, a DC bias connection may be implemented between the two optical transmission paths  602  and  604 , but through a high impedance. For example, in some implementations, the high impedance is achieved with an impedance greater than 1 kohm. As another example, in some implementations, the high impedance is achieved with an impedance greater than 100 ohm. In such scenarios of a DC bias connection through a high impedance, a current would be generated by the voltage difference between (i) the external voltage and (ii) the voltage that would be between the optical transmission paths  602  and  604  if there were no applied external voltage. This generated current would be less than the diode leakage current plus any photo-generated current in the diodes, and thus the circuit would act primarily as if there were no applied external DC bias voltage (e.g., similar to a true floating voltage). Therefore, it should be appreciated that implementations of the present disclosure, such as those shown in  FIGS.  6 - 9    in which there is no physical DC bias connection, can also be implemented with a DC bias connection but through a high impedance. 
     The modulator  600  implements an example of a continuous traveling-wave structure, in which the RF transmission line  614  is continuously connected to the semiconducting structure  616 . Alternatively, a segmented traveling-wave structure can be implemented, as described with reference to  FIG.  7   , below. 
       FIG.  7    illustrates another example of a top view of a modulator  700  according to implementations of the present disclosure. The modulator  700  is an example of an implementation of a segmented traveling-wave structure. 
     The modulator  700  is also based on an MZI implementation which includes two optical transmission paths  702  and  704 , optical splitter  706 , and optical combiner  708 . The modulator  700  further includes terminals, such as terminal  710  and terminal  712 , through which voltages can be applied. The voltages travel along RF transmission line  714 , which is connected to a semiconducting structure  716  via electrodes that apply respective voltages, and resulting electric fields, 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 , which reduces the distance therebetween. For example, in some implementations, 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 implementations, 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 implementations, the distance between the waveguides is within a 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 implementations, the distance between the waveguides is defined as the distance between the inner sidewalls of the two waveguides, at a given point along a longitudinal direction of the modulator  700  (e.g., at a point  705  in  FIG.  7   ). 
     The differences between modulator  600  of  FIG.  6    and modulator  700  of  FIG.  7    arise from the configuration of the semiconducting structure ( 616 ,  716 ) and the manner in which the RF transmission line ( 614 ,  714 ) is connected to the semiconducting structure ( 616 ,  716 ). Modulator  600  of  FIG.  6    implements a continuous traveling wave structure in which RF transmission line  614  is continuously directly connected to the semiconducting structure  616 . By contrast, modulator  700  of  FIG.  7    implements a segmented traveling wave structure in which RF transmission line  714  is intermittently connected to segments of the semiconducting structure  716 , with intermittent regions  720  along the optical transmission paths  702  and  704  in which there is no semiconducting structure. This structure of modulator  700  can also be referred to as a capacitively loaded traveling wave structure, and has an advantage of providing an additional degree of freedom in implementing the RF transmission  714 , e.g., of the average capacitance per unit length of the RF transmission line  714 . A lumped-element modulator can also benefit from the techniques disclosed herein. 
     Furthermore, in modulator  700 , the waveguides of optical transmission paths  702  and  704  have different widths in different sections of the modulator  700 , similar to the configuration of the waveguides in modulator  600  of  FIG.  6   . Further details of the width variation of the waveguides are provided further below. 
       FIG.  8    illustrates an example of a cross section of a modulator  800  according to implementations of the present disclosure (e.g., a cross section at point  605  of modulator  600  of  FIG.  6    or a cross at point  705  of modulator  700  of  FIG.  7   ). In particular, the modulator  800  of  FIG.  8    is an example of a differential, close-spaced design in which one waveguide is taller than the other. 
     The cross-section of modulator  800  shows 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 ribbed waveguides on top of a slab. In some implementations, the 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 semiconducting structure  810  (e.g., a silicon layer which includes optical waveguides  802  and  804 ). 
     In some implementations, as discussed in regards to  FIGS.  6  and  7   , above, one of the optical waveguides  802  and  804  is taller than the other optical waveguide. For example, in  FIG.  8   , the second optical waveguide  804  is taller by at least 40 nm than the first optical waveguide  802 . In some implementations, the waveguide height difference is within a range of 40 nm to 120 nm. 
     Each of the optical waveguides  802  and  804  includes a semiconductor junction. The semiconductor junction diodes can be implemented, for example, by a PIN (P-type/intrinsic/N-type) junction diode or a P/N junction diode. In modulator  800 , a P/N junction is implanted into each of the optical waveguides  802  and  804 , forming a diode in each waveguide. These diodes are shown as first semiconductor junction diode  812  and second semiconductor junction diode  814 . 
     The modulator  800  also includes electrodes  816  and  818  (e.g., metal electrodes) which are in physical contact with the silicon layer  810 . In some implementations, the electrodes  816  and  818  are in physical contact with N-doped contact regions  820  and  822  of the silicon layer  810 . The electrodes  816  and  818  may be formed, for example, by etching the insulator layer  808  and forming metal (e.g., tungsten, copper, and/or aluminum) contacts. The modulator  800  may also include metal layers  824  and  826  on top of the electrodes  816  and  818 . In some implementations, the metal layers  824  and  826  may form segments of an RF transmission line (e.g., RF transmission line  114  in  FIG.  1   ). In some implementations, the P-doped regions may instead be N-doped regions, and vice-versa, in modulator  800  (e.g., so that contact regions  820  and  822  are N-doped instead P-doped). 
     There are numerous 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  which implements DC bias connection  228 ). Instead, the semiconductor junction diodes  812  and  814  are connected in series with opposite polarity (with anodes  834  and  836  connected together). This ensures that a continuous current can never flow through the semiconductor junction diodes  812  and  814 . This configuration enables the voltages across the two semiconductor junction diodes  812  and  814  to naturally self-adjust to ensure that the diodes  812  and  814  remain reverse-biased, despite variations in modulation voltages (e.g., V+ and V−) that may be applied at electrodes  816  and  818 . Implementing a floating voltage between the semiconductor junction diodes  812  and  814  automatically biases the diodes  812  and  814  at the most efficient point of the modulator in terms of phase shift per volt, which is where the diodes  812  and  814  are just below turn-on. In some implementations, this phase shift per volt is the “gain” of the modulator. 
     Another difference between modulator  800  and modulator  200  of  FIG.  2    is that the polarities of semiconductor junction diodes  812  and  814  are flipped, as compared with modulator  200 . In particular, semiconductor junction diodes  812  and  814  have their respective (P-doped) anodes  834  and  836  closer to the center of modulator  800 , and their respective (N-doped) cathodes  830  and  832  closer to the edges of modulator  800 . As such, the semiconducting region  838  between the semiconductor junction diodes  812  and  814  is P-doped, while semiconducting regions  840  and  842  (connecting each of semiconductor junction diodes  812  and  814  with their respective electrodes  816  and  818 ) are N-doped. 
     These aforementioned differences provide numerous technical advantages for modulator  800 , as compared to modulator  200  of  FIG.  2   . One advantage is that the absence of a DC bias voltage connection in modulator  800  enables the two optical waveguides  802  and  804  to be implemented significantly closer to each other, as compared to modulator  200  of  FIG.  2   . This enables significant reduction in the size of semiconducting region  838  connecting semiconductor junction diodes  812  and  814 , which significantly reduces the electrical series resistance between semiconductor junction diodes  812  and  814 . For example, in some implementations, the distance (denoted as  805  in  FIG.  8   ) between the two optical waveguides  802  and  804  is less than 0.5 μm. In some implementations, the distance  805  between the two optical waveguides  802  and  804  is less than 2.0 μm. In some implementations, the distance  805  between the two optical waveguides  802  and  804  is within a range of 0.1 μm to 2.0 μm. In some implementations, the distance  805  between waveguides may be defined as the distance between the inner sidewalls of the two waveguides, at a given point along the longitudinal direction of the modulator  800  (e.g., measured at a cross section of the modulator  800  as shown in  FIG.  8   ). 
     Another advantage is that, since P-doped silicon has a higher resistivity than N-doped silicon (for the same optical absorption), higher-resistivity P-doped material is used in the smaller semiconducting region  838  (between semiconductor junction diodes  812  and  814 ), and lower-resistivity N-doped material is used in the larger semiconducting regions  840  and  842  (connecting semiconductor junction diodes  812  and  814  with electrodes  816  and  818 ). Alternatively, in some implementations, N-doped material can be used in the smaller semiconducting region  838 , and P-doped material can be used in the larger semiconducting regions  840  and  842 . 
     As a result, the total series resistance between the electrodes  816  and  818  is significantly reduced, thus significantly improving bandwidth and speed of the modulation. 
     Although the lack of a DC bias voltage connection in modulator  800  takes away a degree of freedom in the ability to adjust the amount of reverse bias in semiconductor junction diodes  812  and  814 , such limitations are, in some scenarios, outweighed by the significant benefits offered by the configuration of modulator  800 , such as improved bandwidth and speed of modulation. 
       FIG.  9    illustrates an example equivalent circuit  900  along a cross-section of a modulator according to implementations of the present disclosure (e.g., the cross section of modulator  800  of  FIG.  8   ). 
     In the example of  FIG.  9   , the electrical series resistance  940  between first electrode  916  and first semiconductor junction diode  912  (e.g., corresponding to semiconducting region  840  in  FIG.  8   ) is 3.7 mΩ-m. The electrical series resistance  942  between second electrode  918  and second semiconductor junction diode  914  (e.g., corresponding to semiconducting region  842  in  FIG.  8   ) is 3.7 mΩ-m. The electrical series resistance  938  between semiconductor junction diodes  912  and  914  (e.g., corresponding to semiconducting region  838  in  FIG.  8   ) is 4.6 mΩ-m (without any DC bias voltage connection between the diodes). 
     As seen in this example, the total series resistance between electrodes  916  and  918  is reduced by about a factor of two, as compared with the equivalent circuit  300  of  FIG.  3   . This reduction in total series resistance can significantly improve modulator performance. For example, the modulation bandwidth is increased, by reducing the RF loss along the modulator. Alternatively, modulator efficiency can be improved. For example, a thinner slab can be utilized, which increases total series resistance but also increases optical confinement in the optical waveguides  802  and  804 , thus improving modulator efficiency. Alternatively, a thicker waveguide can be utilized, which increases capacitance but also increases optical confinement. 
     The modulators according to implementations of the present disclosure can be used in many applications. For example, one application is a high-speed optical intensity modulator to generate intensity-modulated direct-detection (IM-DD) formats such as non-return-to-zero (NRZ) or pulse amplitude modulation (PAM). Another application is to use the modulator in conjunction with a second modulator with a 90-degree relative phase shift as part of a larger 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 (IQ) modulator structure that includes nested modulators, with each of the two branches of a modulator (the outer modulator) implementing another modulator (the inner modulators). In some implementations, phase shifters can be implemented that set 180-degree and 90-degree phase differences for the inner and outer modulators, respectively. Each modulator in such a nested modulator structure can be implemented as described in the present disclosure. 
       FIG.  10    is a flowchart illustrating an example method  1000  of modulating a quasi-TM polarized optical signal, according to implementations of the present disclosure. The method  1000  may be performed by using a modulator as disclosed herein. 
     The method  1000  includes splitting quasi-TM polarized light into a first optical transmission path and a second optical transmission path ( 1002 ). In some implementations, an optical phase rotator may be implemented at the input of the modulator, which rotates a phase of the input light so that quasi-TM light propagates in the optical transmission paths. 
     The method  1000  further includes modulating a phase difference between quasi-TM polarized light in the first optical transmission path and quasi-TM polarized light in the second optical transmission path without applying a bias voltage between the first optical transmission path and the second optical transmission path ( 1004 ). In some implementations, the phase difference between the quasi-TM polarized light in the first optical transmission path and the quasi-TM polarized light in the second optical transmission path is modulated while maintaining finite depletion regions in semiconductor junction diodes in 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 modulators discussed above. 
     The method  1000  further includes combining quasi-TM polarized light that is output from the first optical transmission path and quasi-TM polarized light that is output from the second optical transmission path ( 1006 ). 
     While this disclosure contains many specific implementation details, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features that may be specific to particular implementations of particular inventions. Certain features that are described in this disclosure in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination. 
     Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results.