Patent Publication Number: US-2023145767-A1

Title: High-gain differential electro-optic modulator

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. Application Serial No. 17/226,730, filed Apr. 9, 2021, now allowed, which is incorporated by reference herein in its entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure generally relates to electro-optic differential modulators. 
     BACKGROUND 
     In optical communication systems, electro-optic 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-optic differential modulators. 
     One general aspect includes an optical modulator including: a Mach-Zehnder interferometer including (i) a first optical waveguide including a first semiconductor junction diode, and (ii) a second optical waveguide including a second semiconductor junction diode. The optical modulator also includes a semiconductor region that connects the first semiconductor junction diode with the second semiconductor junction diode, such that a distance between the first optical waveguide and the second optical waveguide is less than 2.0 µm for at least a portion of a longitudinal direction of the optical modulator. 
     Implementations may include one or more of the following features. The optical modulator where the first semiconductor junction diode includes a first anode and a first cathode, and where the second semiconductor junction diode includes a second anode and a second cathode. The optical modulator where the first anode is connected to the second anode through the semiconductor region that spans the distance between the first optical waveguide and the second optical waveguide. The optical modulator where the semiconductor region between the first anode and the second anode is configured without any external voltage connection that has an impedance less than 100 ohm. The optical modulator further including: a first electrode connected to the first cathode and configured to apply a first electric field to the first optical waveguide. The optical modulator may also include a second electrode connected to the second cathode and configured to apply a second electric field to the second optical waveguide. The optical modulator further including: a radio frequency (RF) transmission line configured to apply (i) a first voltage to the first cathode through a first electrode, and (ii) a second voltage to the second cathode through a second electrode. The optical modulator where the first optical waveguide includes a plurality of first semiconductor junction diodes, and where the second optical waveguide includes a plurality of second semiconductor junction diodes. The optical modulator where the RF transmission line is configured to (i) apply the first voltage to the plurality of first semiconductor junction diodes through a plurality of first electrodes, and (ii) apply the second voltage to the plurality of second semiconductor junction diodes through a plurality of second electrodes. The optical modulator where for a first portion of the optical modulator, the first optical waveguide is wider by at least 0.04 µm than the second optical waveguide, and where for a second portion of the optical modulator, the second optical waveguide is wider by at least 0.04 µm than the first optical waveguide. The optical modulator where for at least a portion of the optical modulator: the first optical waveguide increases in width along the longitudinal direction of the optical modulator, and the second optical waveguide decreases in width along the longitudinal direction of the optical modulator. The optical modulator where the first semiconductor junction diode includes a first p-doped region and a first n-doped region, and where the second semiconductor junction diode includes a second p-doped region and a second n-doped region. The 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 optical modulator where the third p-doped region is configured without any external voltage connection that has an impedance less than 100 ohm. The optical modulator where the first semiconductor junction diode further includes a first oxide layer between the first p-doped region and the first n-doped region, and where the second semiconductor junction diode further includes a second oxide layer between the second p-doped region and the second n-doped region. The optical modulator where the Mach-Zehnder interferometer further includes: an optical splitter configured to receive input light and split the input light into the first optical waveguide and the second optical waveguide. The optical modulator may also include an optical combiner configured to receive first output light from the first optical waveguide and second output light from the second optical waveguide, and combine the first output light with the second output light. The optical modulator where the distance between the first optical waveguide and the second optical waveguide is the distance between an inner sidewall of the first optical waveguide and an inner sidewall of the second optical waveguide. 
     Another general aspect includes an optical modulator including: an optical splitter configured to split an input light into a first optical transmission path and a second optical transmission path. The optical modulator also includes means for modulating a phase difference between light in the first optical transmission path and light in the second optical transmission path without applying a bias voltage through an impedance less than 100 ohm between the first optical transmission path and the second optical transmission path. The optical modulator also includes an optical combiner configured to combine light that is output from the first optical transmission path and light that is output from the second optical transmission path. 
     Implementations may include one or more of the following features. The optical modulator further including: a radio frequency (RF) transmission line configured to apply (i) a first voltage to the first optical transmission path through a first electrode, and (ii) a second voltage to the second optical transmission path through a second electrode. The optical modulator where the first optical transmission path includes a first semiconductor junction diode, the second optical transmission path includes a second semiconductor junction diode, and the first semiconductor junction diode is just below turn-on while the second semiconductor junction diode is at maximum reverse voltage during modulation. The optical modulator where the phase difference between the light in the first optical transmission path and the light in the second optical transmission path is modulated by applying a first electric field to the first optical transmission path and a second electric field to the second optical transmission path in push-pull mode. The optical modulator where the phase difference between the light in the first optical transmission path and the 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. 
     Another general aspect includes a method of modulating an optical signal, the method including: splitting input light into a first optical transmission path and a second optical transmission path. The method of modulating also includes modulating a phase difference between light in the first optical transmission path and light in the second optical transmission path without applying a bias voltage through an impedance less than 100 ohm between the first optical transmission path and the second optical transmission path. The method of modulating also includes combining light that is output from the first optical transmission path and light that is output from the second optical transmission path. 
     Implementations may include one or more of the following features. The method where the phase difference between the light in the first optical transmission path and the 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. 
     Another general aspect includes an optical modulator that includes a Mach-Zehnder interferometer including (i) a first optical waveguide comprising a first semiconductor junction diode, and (ii) a second optical waveguide comprising a second semiconductor junction diode. The optical modulator also includes a semiconductor region that connects a terminal of the first semiconductor junction diode with a terminal of the second semiconductor junction diode. The terminal of the first semiconductor junction diode and the terminal of the second semiconductor junction diode are either both p-doped anodes or both n-doped cathodes. The semiconductor region is not connected to any other circuit element through an impedance less than 100 ohm. 
     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 differential modulator which implements a bias voltage connection between the waveguides; 
         FIG.  2    illustrates an example of a cross section of a modulator which implements a bias voltage connection between the waveguides; 
         FIG.  3    illustrates an example of an equivalent circuit along a cross-section of a modulator which implements a bias voltage connection between the waveguides; 
         FIG.  4    illustrates another example of a cross section of a modulator which implements a bias voltage connection between the waveguides; 
         FIG.  5    illustrates an example of a top view of a modulator, according to implementations of the present disclosure; 
         FIG.  6    illustrates another example of a top view of a modulator, according to implementations of the present disclosure; 
         FIG.  7    illustrates an example of a top view of modulator showing width variations of optical waveguides, 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; 
         FIG.  10    illustrates another example of a cross section of a modulator, according to implementations of the present disclosure; 
         FIG.  11    illustrates an example of performances of different modulators, in terms of conductance per unit length between the V+ and V- terminals as a function of distance between the waveguides; 
         FIG.  12    illustrates an example of performances of different modulators, in terms of voltage across each semiconductor junction diode as a function of applied differential voltage ΔV across the two terminals of the modulator; 
         FIG.  13    illustrates an example of the performances of modulators according to implementations of the present disclosure, in terms of normalized differential refractive index change between the two waveguides of the modulator, as a function of distance between the waveguides, for various waveguide widths; and 
         FIG.  14    is a flowchart illustrating an example of modulating an optical signal, according to implementations of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Systems and techniques are disclosed herein that provide a novel differential electro-optic modulator which can achieve a significantly higher modulation bandwidth and/or efficiency. This is accomplished by novel implementations which enable significant reduction in the physical distance between waveguides of the modulator. In some implementations, the reduced physical distance between the waveguides is achieved by removing a bias voltage connection between semiconductor junction diodes of waveguides of the modulator, while maintaining finite depletion regions in the semiconductor junction diodes. The reduced physical distance between the diodes, in turn, enables significantly reduced electrical resistance between the diodes, which increases the modulation bandwidth and/or efficiency of the modulator. In some implementations, to mitigate detrimental optical coupling that may occur between the closely-spaced waveguides, the waveguides have widths that vary in an alternating manner along the length of the modulator. 
       FIG.  1    illustrates an example of a top view of a differential modulator  100  which implements a bias voltage connection between the waveguides. This example is provided for comparison with modulators according to implementations of the present disclosure which are described further below with reference to  FIGS.  5 - 14   . 
     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. 
     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 “of” 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 of the DC bias connection  118  and the semiconductor junction diodes are provided with reference to  FIG.  2   , below. 
       FIG.  2    illustrates an example cross section of a modulator  200  which implements a bias voltage connection between the waveguides (e.g., the modulator  100  of  FIG.  1   ). This example is provided for comparison with modulators according to implementations of the present disclosure which are described further below with reference to  FIGS.  5 - 14   . 
     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 . The optical waveguides  202  and  204  can be implemented, for example, as silicon ribbed waveguides on top of a slab. 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 ). 
     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. 
     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 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. 
     However, the structure of modulator  200  results in various limitations on modulation performance. In particular, the structure of modulator  200  results in significant electrical series resistance in various regions of the modulator  200 . 
     In particular, the presence of DC bias connection  228  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 . Furthermore, typical techniques to reduce such electrical series resistance, such as increasing the silicon doping of the semiconducting structure, can have other negative consequences such as increasing optical absorption. 
     Furthermore, the semiconducting regions  240  and  242  (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  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.  3    illustrates an example equivalent circuit  300  along a cross-section of a modulator which implements a bias voltage connection between the waveguides (e.g., the cross section of modulator  200  of  FIG.  2   ). This example is provided for comparison with modulators according to implementations of the present disclosure which are described further below with reference to  FIGS.  5 - 14   . 
     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 7.2 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 7.2 mΩ-m. The electrical series resistance  338  between semiconductor junction diodes  312  and  314  (e.g., corresponding to semiconducting region  238  in  FIG.  2   ) is 7.4 mΩ-m (with 3.7 mΩ-m of series resistance between each of semiconductor junction diodes  312  and  314  and DC bias voltage connection  328 ). 
       FIG.  4    illustrates another example of a cross section of a modulator  400  which implements a bias voltage connection between the waveguides (e.g., another example of a cross section of modulator  100  of  FIG.  1   ). This example is provided for comparison with modulators according to implementations of the present disclosure which are described further below with reference to  FIGS.  5 - 14   . 
     The structure of modulator  400  is referred to as a silicon-insulator-silicon capacitor (SISCAP) modulator structure. As compared with modulator  200  of  FIG.  2   , the modulator  400  implements thin oxide layers  444  and  446  in the semiconductor junction diodes  412  and  414  of optical waveguides  402  and  404 . Furthermore, in modulator  400 , the DC bias connection  428  applies a bias voltage at the anodes  434  and  436  (P-doped regions) of semiconductor junction diodes  412  and  414 , while the electrodes  416  and  418  apply varying voltages at the cathodes  430   and  432  (N-doped regions) of semiconductor junction diodes  412  and  414 . The DC bias connection  428  ensures that the semiconductor junction diodes  412  and  414  remain reverse biased. 
       FIGS.  5 - 14    relate to modulators according to implementations of the present disclosure. In contrast with the modulators of  FIGS.  1 - 4   , the modulators of  FIGS.  5 - 14    do not implement any bias voltage connection between the waveguides, resulting in significantly smaller series resistance between electrodes, and thus higher bandwidth of modulation. Furthermore, in  FIGS.  5 - 14   , the modulators implement waveguide structures that vary in width so as to mitigate detrimental optical coupling between the closely-spaced waveguides. 
       FIG.  5    illustrates an example of a top view of a modulator  500  according to implementations of the present disclosure. 
     The modulator  500  is based on an MZI implementation which includes two optical transmission paths  502  and  504 , optical splitter  506 , and optical combiner  508 . The modulator  500  further includes terminals, such as terminal  510  and terminal  512 , through which voltages can be applied. The voltages travel along RF transmission line  514 , which is connected to semiconducting structure  516  via electrodes that apply respective voltages, and resulting electric fields, across one or both of the optical transmission paths  502  and  504 . 
     In contrast to the modulator  100  of  FIG.  1   , the modulator  500  does not implement any DC bias connection between the two optical transmission paths  502  and  504 . This enables the two optical transmission paths  502  and  504  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  502  and  504  is less than 0.5 µm for at least a portion of the longitudinal direction of the optical transmission paths  502  and  504 . 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  502  and  504 . 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  502  and  504 . 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  500  (e.g., at a point  505  in  FIG.  5   ). 
     However, because the two optical transmission paths  502  and  504  are more closely spaced, there is risk of more significant detrimental optical coupling between light in optical transmission path  502  and light in optical transmission path  504 . To mitigate such optical coupling, in some implementations, the waveguide of one of the optical transmission paths ( 502  or  504 ) is designed to have a larger width than the other path, at the same distance along the length of the modulator  500 . This helps ensure that the light traveling in the waveguides of optical transmission paths  502  and  504  are not phase matched, thus mitigating optical coupling between the two waveguides. An alternative way to understand the importance of using different waveguide widths is to look at the two eigenmodes of the coupled waveguides of optical transmission paths  502  and  504 . If the waveguides have equal widths, 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 wider than the other, then the lowest order eigenmode consists of light that is predominantly in the wider waveguide, and the second lowest eigenmode is predominantly in the narrower 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 wider by at least 0.04 µm than the waveguide of the other optical transmission path. In some implementations, the waveguide width difference is within a range of 0.04 µm to 0.4 µm. 
     Furthermore, in such implementations, the width variation of the two waveguides may be exchanged along the modulator  500 , to help ensure that the total length of the wider portions in each waveguide are equal, and also that the total length of the narrower portions in each waveguide are equal. In the example of  FIG.  5   , moving from the left to right, the waveguide of first optical transmission path  502  is wider than the waveguide of the second optical transmission path  504 , and then becomes narrower than the waveguide of the second optical transmission path  504  (alternatively, the first optical transmission path  502  may start narrower and become wider). The example of  FIG.  5    shows one width swap in the middle of modulator  500 , but in some implementations, additional width swaps can be included. Further details of the width variations of the waveguides will be discussed in reference to  FIG.  7   , below. 
     Although the description of  FIG.  5   , above, provided an example of a modulator  500  with variable-width waveguides in the two optical transmission paths  502  and  504 , in other implementations, the waveguides may have constant width along the length of the modulator  500 . 
     Furthermore, although the description of  FIG.  5    provided an example of a modulator  500  without a physical DC bias connection, in some implementations, a DC bias connection may be implemented between the two optical transmission paths  502  and  504 , 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  502  and  504  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.  5 - 10    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  500  implements an example of a continuous traveling-wave structure, in which the RF transmission line  514  is continuously connected to the semiconducting structure  516 . Alternatively, a segmented traveling-wave structure can be implemented, as described with reference to  FIG.  6   , below. 
       FIG.  6    illustrates another example of a top view of a modulator  600  according to implementations of the present disclosure. The modulator  600  is an example of an implementation of a segmented traveling-wave structure. 
     The modulator  600  is also 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 a 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 . The modulator  600  also does not implement any DC bias connection between the two optical transmission paths  602  and  604 , which reduces the distance 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   ). 
     The differences between modulator  500  of  FIG.  5    and modulator  600  of  FIG.  6    arise from the configuration of the semiconducting structure ( 516 ,  616 ) and the manner in which the RF transmission line ( 514 ,  614 ) is connected to the semiconducting structure ( 516 ,  616 ). Modulator  500  of  FIG.  5    implements a continuous traveling wave structure in which RF transmission line  514  is continuously directly connected to the semiconducting structure  516 . By contrast, modulator  600  of  FIG.  6    implements a segmented traveling wave structure in which RF transmission line  614  is intermittently connected to segments of the semiconducting structure  616 , with intermittent regions  620  along the optical transmission paths  602  and  604  in which there is no semiconducting structure. This structure of modulator  600  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  614 , e.g., of the average capacitance per unit length of the RF transmission line  614 . A lumped-element modulator can also benefit from the techniques disclosed herein. 
     Furthermore, in modulator  600 , the waveguides of optical transmission paths  602  and  604  have different widths in different sections of the modulator  600 , similar to the configuration of the waveguides in modulator  500  of  FIG.  5   . Further details of the width variation of the waveguides are provided with reference to  FIG.  7   , below. 
       FIG.  7    illustrates an example of a top view of a width-exchange region of modulator  700  showing width variations of optical waveguides, according to implementations of the present disclosure (e.g., modulator  500  of  FIG.  5    or modulator  600  of  FIG.  6   ). In some implementations, the width-exchange region is implemented between the semiconducting regions  616  of  FIG.  6   . 
     The modulator  700  includes two optical transmission paths  702  and  704 , which can be implemented by silicon ribbed waveguides. Furthermore, as discussed with reference to  FIGS.  5  and  6   , above, modulator  700  does not implement any DC bias connection, thus enabling the two optical transmission paths  702  and  704  to be more closely-spaced together, thus reducing electrical series resistance therebetween. 
     To mitigate detrimental optical coupling between the more closely-spaced waveguides of the optical transmission paths  702  and  704 , one of the optical transmission paths  702  or  704  has a waveguide of a larger width than the waveguide the other optical transmission path. This helps ensure that light traveling in the waveguides of optical transmission paths  702  and  704  are not phase matched , thus mitigating optical coupling between the two waveguides. For example, in some implementations, the waveguide of one of the optical transmission path  702  or  704  is wider by at least 0.04 µm than the waveguide of the other optical transmission path. 
     Furthermore, the width variation of the two waveguides may be exchanged along the modulator  700 . For example, in  FIG.  7   , in portion  722  of modulator  500 , the waveguide of second optical transmission path  704  is wider than the waveguide of the first optical transmission path  702 . Then, in portion  724  of modulator  700 , the waveguide of the first optical transmission path  702  is wider than the waveguide of the second optical transmission path  704 . In some implementations, the difference in waveguide width is at least 0.04 µm. In some implementations, the waveguide width difference is within a range of 0.04 µm to 0.4 µm. 
     The example of  FIG.  7    shows one width exchange in the middle of modulator  700 , but in some implementations, additional width exchanges can be included, e.g., as long as the distance between width 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. 
     A potential complication that arises from varying the widths of waveguides in optical transmission paths  702  and  704  is that wider waveguides have higher effective refractive index than narrower waveguides. As a result, the phase of light in the waveguide is affected differently in wider portions of the waveguide as compared to narrower portions of the waveguide. As such, if the two optical transmission paths  702  and  704  have different lengths of wider portions (e.g., if the length of portion  722  is greater than the length of portion  724 , or vice versa), then this could result in different inherent phase shifts of light in the two waveguides, e.g., due to wavelength or temperature differences, or different speeds of light in the two waveguides. 
     To mitigate such complications, the exchanging of widths of the two waveguides can be implemented to ensure that the total length of the wider portions in each waveguide are equal, and also that the total length of the narrower portions in each waveguide are equal. This helps ensure that the total effective path length of optical transmission path  702  is the same as that of optical transmission path  704 . As a result, this can help ensure non-zero inherent differential phase shift between light propagating along the two optical transmission paths  702  and  704 . 
     In some implementations, the width-exchanging transition can be implemented in a gradual manner. For example, from left to right in  FIG.  7   , the distance between the waveguides of the two optical transmission paths  702  and  704  is gradually increased. This helps ensure that light in the two optical transmission paths  702  and  704  remains largely uncoupled. With this increased separation, the widths of each waveguide is changed, such that the wider waveguide becomes narrower and the narrower waveguide becomes wider. Once the waveguides have exchanged widths, then the two waveguides are gradually brought closer together again. 
     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   ). 
       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  505  of modulator  500  of  FIG.  5    or a cross at point  605  of modulator  600  of  FIG.  6   ). 
     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.  5 - 7   , above, one of the optical waveguides  802  and  804  is wider than the other optical waveguide. For example, in  FIG.  8   , the second optical waveguide  804  is wider by at least 0.04 µm than the first optical waveguide  802 . In some implementations, the waveguide width difference is within a range of 0.04 µm to 0.4 µm. 
     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   ). 
     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. 
       FIG.  10    illustrates another example of a cross section of a modulator  1000  according to implementations of the present disclosure (e.g., another example of a cross section of modulator  500  of  FIG.  5    or modulator  600  of  FIG.  6   ). 
     The structure of modulator  1000  is a silicon-insulator-silicon capacitor (SISCAP) modulator structure. As compared with modulator  800  of  FIG.  8   , the modulator  1000  implements thin oxide layers  1044  and  1046  in the semiconductor junction diodes  1012  and  1014  of optical waveguides  1002  and  1004 . Furthermore, as in modulator  800  of  FIG.  8   , the anodes  1034  and  1036  (P-doped regions) of semiconductor junction diodes  1012  and  1014  are connected together (without a DC bias connection therebetween), and the electrodes  1016  and  1018  apply varying voltages at the cathodes  1030  and  1032  (N-doped regions) of semiconductor junction diodes  1012  and  1014 . 
     These features provide numerous technical advantages for modulator  1000 , as compared to modulator  200  of  FIG.  2   . One advantage is that the absence of a DC bias voltage connection in modulator  1000  enables the two optical waveguides  1002  and  1004  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  1038  connecting semiconductor junction diodes  1012  and  1014 , which significantly reduces the electrical series resistance between semiconductor junction diodes  1012  and  1014 . Another advantage is that higher-resistivity P-doped material is used in the smaller semiconducting region  1038  (between semiconductor junction diodes  1012  and  1014 ), and lower-resistivity N-doped material is used in the larger semiconducting regions  1040  and  1042  (connecting semiconductor junction diodes  1012  and  1014  with electrodes  1016  and  1018 ). As a result, the total series resistance between the electrodes  1016  and  1018  is significantly reduced, thus significantly improving bandwidth and speed of the modulation. For example, in some implementations, the distance (denoted as  1005  in  FIG.  10   ) between the two optical waveguides  1002  and  1004  is less than 0.5 µm. In some implementations, the distance  1005  between the two optical waveguides  1002  and  1004  is less than 2.0 µm. In some implementations, the distance  1005  between the two optical waveguides  1002  and  1004  is within a range of 0.1 µm to 2.0 µm. In some implementations, the distance  1005  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  1000  (e.g., measured at a cross section of the modulator  1000  as shown in  FIG.  10   ). 
     Furthermore, in some implementations, as discussed in regards to  FIGS.  5 - 8   , above, one of the optical waveguides  1002  and  1004  is wider than the other optical waveguide. For example, in  FIG.  10   , the second optical waveguide  1004  is wider by at least 0.04 µm than the first optical waveguide  1002 . In some implementations, the waveguide width difference is within a range of 0.04 µm to 0.4 µm. 
     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 (e.g., implemented as a modulator described with reference to  FIGS.  5 - 10   ). 
       FIG.  11    illustrates an example of the performances of different modulators, in terms of conductance per unit length between the V+ and V– terminals as a function of distance between the waveguides. 
     The plots shown in  FIG.  11    compare the performance of a modulator with a DC bias connection (e.g., modulator  100  of  FIG.  1   ) with the performance of a modulator with a floating anode implementation (e.g., the modulators of  FIGS.  5 - 10   ), and a modulator with a floating cathode implementation, for typical semiconductor doping levels. As shown in  FIG.  11   , if the distance between the waveguides is 0.1 µm, then the floating anode implementation doubles the conductance, as compared to the DC bias connection implementation. The floating cathode implementation also provides an improvement over the DC bias connection implementation, although significantly smaller due to the predominance of the P-doped regions in the semiconducting structure of the modulator. 
       FIG.  12    illustrates an example of the performances of different modulators, in terms of voltage across each semiconductor junction diode as a function of applied differential voltage ΔV across the two terminals of the modulator, where ΔV = V+ – V–, which is the voltage across the two diodes in series in  FIGS.  3  and  9   . 
     The curves with dotted lines represent the voltages across each of the two diodes of a modulator with a DC bias connection (e.g., modulator  100  of  FIG.  1   ), and the curves with solid lines represent the voltages across each of the two diodes of a modulator with a floating anode implementation (e.g., the modulators of  FIGS.  5 - 10   ). 
     For the curves with dotted lines (a modulator with a DC bias connection, e.g., modulator  100  of  FIG.  1   ), the bias voltage is adjusted so that the diodes remain below the diode turn-on voltage. For these curves, the voltage across each diode is approximately linear with a magnitude of slope equal to approximately ½ (i.e., +½ and –½ for the two dotted-line curves). 
     For the curves with solid lines (a modulator with a floating anode implementation, e.g., the modulators of  FIGS.  5 – 10   ), when the differential applied voltage ΔV is equal to zero, the voltage across each diode is also zero, such that curves for the two diodes intersect at point (0, 0) of the graph. If the applied differential voltage ΔV is large, then one diode is just below turn-on, and the other diode has a large reverse voltage. This allows the modulator to operate at the highest possible gain automatically, despite environmental or fabrication process changes. The solid-line curves for both diodes initially transition with slopes of magnitude 1 (i.e., +1 and –1 for the sloped portions of the two solid-line curves). The overall result is a nonlinear behavior of voltage across each diode as a function of applied differential voltage ΔV. The difference of the voltages across the two diodes is strictly proportional to the applied differential voltage ΔV in both cases, but the MZI configuration of the modulator is no longer driven with equal and opposite sign in each optical transmission path. This will introduce a small nonlinear chirp on the resulting optical signal that is output from the optical combiner, which may impact transmission in the presence of chromatic dispersion. However, this effect should be very small. 
       FIG.  13    illustrates an example of the performances of modulators according to implementations of the present disclosure (e.g., the modulators of  FIGS.  5 - 10   ), in terms of normalized differential refractive index change between the two waveguides of the modulator, as a function of distance between the waveguides, for various waveguide widths. 
     In the example of  FIG.  13   , the nominal waveguide width is 0.45 µm, and the wavelength is 1.31 µm. The waveguide thickness is 0.22 µm, and the slab thickness is 0.10 µm. The results of  FIG.  13    were generated by simulating a depletion zone in the center of each waveguide, with the size of the depletion zone having width 0.2 µm and height 0.22 µm. The depletion zone is increased in refractive index by 3×10 -4  in one waveguide and decreased in refractive index by 3×10 -4  in the other waveguide, and the refractive indices of the two waveguide modes are calculated. The sign of the refractive index change is changed, and the refractive indices of the two waveguide modes are calculated again. The mode refractive indices for the two cases are subtracted and averaged for the two waveguides, and the result is normalized to the largest value, to yield the differential index in the plots of  FIG.  13   . 
       FIG.  14    is a flowchart illustrating an example method  1400  of modulating an optical signal, according to implementations of the present disclosure. The method  1400  may be performed by using a modulator as disclosed herein (e.g., a modulator as described with reference to  FIGS.  5 - 10   ). 
     The method  1400  includes splitting input light into a first optical transmission path and a second optical transmission path ( 1402 ). 
     The method  1400  further includes modulating a phase difference between light in the first optical transmission path and light in the second optical transmission path without applying a bias voltage between the first optical transmission path and the second optical transmission path ( 1404 ). In some implementations, the phase difference between the light in the first optical transmission path and the 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 with reference to  FIGS.  5 - 10   . 
     The method  1400  further includes combining light that is output from the first optical transmission path and light that is output from the second optical transmission path ( 1406 ). 
     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.