Abstract:
The invention relates to an electro-optic modulator structure containing an additional set of bias electrodes buried within the device for applying bias to set the operating point. Thus the RF electrodes used to modulate incoming optical signals can be operated with zero DC bias, reducing electrode corrosion by galvanic and other effects that can be present in non-hermetic packages. The bias electrodes are at least partially separated from the substrate with a buffer layer, which in one embodiment has a small amount of conductivity. This conductive buffer layer reduces optical loss from the bias electrodes and also reduces DC drift.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority from U.S. Provisional Patent Application Ser. No. 60/884,653 filed Jan. 12, 2007, which is hereby incorporated by reference for all purposes. 
     
    
     TECHNICAL FIELD 
       [0002]    The present invention relates generally to electro-optic devices and, more specifically, to electro-optic devices used in fiber-optic telecommunications. 
       BACKGROUND OF THE INVENTION 
       [0003]    Electro-optic devices are frequently used in fiber-optic telecommunication systems to manipulate optical signals. In general, these electro-optic devices include at least one optical waveguide formed from and/or in an electro-optic material. When an electric field is generated in the electro-optic material, the refractive index of the optical waveguide(s) change and the optical signal propagating there through can be altered. Some examples of common electro-optic devices used in telecommunication systems include optical modulators, optical switches, optical couplers, etc. 
         [0004]    One example of a particularly successful electro-optic device is the Mach-Zehnder (MZ) optical modulator. Referring to  FIG. 1   a , there is shown an embodiment of a Mach-Zehnder optical modulator having an optical waveguide  20  formed in an electro-optic substrate  10 . The optical waveguide  20  includes a first Y-branch  22 , a first interferometer arm  24 , a second interferometer arm  26 , and a second Y-branch  28 . An electrode structure (not shown in  FIG. 1   a ) is provided near/adjacent the optical waveguide  20  for generating an electric field in one or both interferometer arms  24 / 26 . For example according to one well known configuration, the electrode structure includes a signal electrode (also often referred to as a hot electrode) and two ground electrodes, which are configured to generate oppositely oriented electric fields in the first  24  and second  26  interferometer arms. Conventionally, the electrode structure is formed from a highly conductive metal such as gold (Au). The exact position and design of the electrodes relative to the optical waveguide  20  is generally dependent on the substrate. For example, if the substrate is formed from X-cut lithium niobate (LiNbO 3 ), the signal electrode  40  is typically positioned on top of the substrate substantially between the interferometer arms  24 / 26 , while the ground electrodes  42 / 44  are positioned on top of the substrate outside of the interferometer arms  24 / 26  (e.g., as illustrated in  FIG. 1   b ). In contrast, if the substrate is formed from Z-cut LiNbO 3 , the signal electrode  40  is typically positioned substantially above one interferometer arm  26 , while the ground electrode  42  is positioned substantially above the other interferometer arm  24  (e.g., as illustrated in  FIG. 1   c ). In each case, the ground electrodes  42 , 44  are typically connected to ground, while the signal electrode  40  is connected to a high-frequency power source. 
         [0005]    Referring to  FIG. 1   d  there is shown an embodiment of a Z-cut LiNbO 3  modulator  100 , wherein the signal electrode  140  is connected to a high-frequency power source  145  at one end and to a terminal resistor at the other end, such that it functions as a traveling-wave electrode. In operation, an optical signal is input into the left side of the device  100  where it is transmitted through the optical waveguide  120  until it is split at the first Y-branch  122 , and then propagates equally along the two isolated paths corresponding to the two interferometer arms  124 / 126 . Simultaneously, an RF data signal from the high-frequency power source  145  is transmitted through an RF transmission line  147  (e.g., a co-axial cable) to the signal electrode  140 , which functions as a microwave transmission line. As the modulation voltage is applied between the signal electrode  140  and the ground electrodes  142  and  144 , an electric field is generated in the underlying electro-optic substrate  110 . As illustrated in  FIG. 1   e , the vertical electric field lines in the first  124  and second  126  interferometer arms are oppositely oriented such the light propagating in each of the arms is complementarily phase shifted relative to one another in a push-pull fashion. In accordance with the electro-optic effect, the electric field changes the refractive index within the interferometer arms such that the input optical signal experiences constructive or destructive interference at the second Y-branch  128 . This interference produces an amplitude modulated optical signal that is output the right side of the device, wherein the modulation corresponds to the original RF data signal. 
         [0006]    Notably, since the Z-axis of a LiNbO 3  crystal has the highest electro-optic coefficient, Z-cut LiNbO 3  modulators exhibit a relatively high modulation efficiency. Unfortunately, Z-cut LiNbO 3  modulators are also known to suffer more from charge build up problems, which for example, may lead to temperature induced bias drift and/or DC drift. 
         [0007]    Temperature induced bias drift refers to when the operating (bias) point of the modulator shifts with changes in temperature. In LiNbO 3 , temperature induced bias drift typically arises from the pyroelectric effect, which creates mobile charge when temperature fluctuations occur in the substrate. The mobile charge can generate strong electric fields that can change the operating (bias) point of the electro-optic modulator. In addition, since the electric fields induced by the pyroelectric effect in Z-cut LiNbO 3  are predominantly normal to the substrate, the mobile charge moves toward the surface of the substrate, where the electrodes  140 ,  142 ,  144  are located. Accordingly, a bleed layer  160  is typically required near the surface of Z-cut LiNbO 3  to dissipate accumulated electric charge. Optionally, additional bleed layers (not shown) are used to dissipate charge at the sides or bottom of the substrate. In general, the bleed layer  160  will be formed from a semiconductive material so that the highly conductive electrodes  140 / 142 / 144  are prevented from shorting out. 
         [0008]    DC drift refers to when the operating (bias) point of the modulator shifts as a low frequency or DC voltage is applied to the modulator for extended periods of time. In general, low frequency or DC voltages are required to control the operating (bias) point of the modulator (i.e., the point about which the swing of the modulated RF signal is accomplished). For example in the embodiment described with reference to  FIG. 1   d , the RF data signal corresponds to a modulation signal that includes an RF component superimposed on a DC or low frequency component. 
         [0009]    DC drift, also termed bias drift, is particularly problematic when the modulator includes a buffer layer  150  disposed between the substrate  110  and the signal electrode  140 . If the buffer layer  150  has little conductivity relative to the substrate  110 , mobile charge within the substrate, which may be in the form of electrons, holes, or ions, counteracts the effect of the applied voltage, establishing a positive DC drift. In addition, impurities in the buffer layer  150 , which is typically formed from a dielectric material such as silicon dioxide (SiO 2 ), are believed to form additional mobile charge, which either counteract the effect of the applied voltage, establishing a positive DC drift, or enhance the applied bias voltage, establishing a negative DC drift. The former is more common for undoped SiO 2 . The end result of the mobile charge in the buffer and substrate is that the bias voltage required to operate the electro-optic modulator increases over time. 
         [0010]    The purpose of the buffer layer  150  is two-fold. First, the buffer layer  150  is used to prevent optical absorption of the optical signal by the overlying electrodes  140 / 142 . Notably, this is more important for Z-cut embodiments, wherein the electrodes  140 / 142  lie directly over the interferometer arms  126 / 124 . Secondly, the buffer layer  150  is used to speed up the propagation of the RF modulation signal so that the optical wave and the microwave propagate with equal phase velocities, thus increasing the interaction length, and as a result, increasing modulation bandwidth and/or efficiency at high frequencies. 
         [0011]    Various solutions to prevent dc drift have been proposed. For example, in X-cut LiNbO 3  modulators it has been proposed to provide a separate low-frequency bias electrode structure  270 , optically in series with an RF electrode structure  240 , as illustrated in  FIG. 2 . A buffer layer  250  is provided below the RF electrode structure  240 , to provide velocity matching, but is eliminated below the bias electrode structure  270 , to reduce DC drift. Conveniently, since the bias electrode structure  270  is deposited directly on the substrate, the required drive (bias) voltage is significantly reduced. Unfortunately, in order to accommodate both electrode structures, the length of the modulator is significantly increased. In addition, this design is not ideal for Z-cut LiNbO 3 , wherein the waveguides are located directly below the bias electrodes, because the highly conductive bias electrode material (e.g., Au) may introduce significant optical loss. 
         [0012]    In Z-cut LiNbO 3  modulators, dc drift has been reduced by modifying the buffer layer. For example in U.S. Pat. Nos. 5,404,412 and 5,680,497, the effect of the buffer layer charging in optical modulators is reduced by doping the buffer layer, causing it to be more conductive. The added conductivity in essence shorts out the buffer layer, preventing the buffer layer from charging up and stealing all of the applied voltage from the waveguides. Accordingly, a DC or slowly varying voltage applied to the signal electrode is able to control the bias point of the modulator over time. Unfortunately, it can be difficult to quantitatively control the introduction of the doping elements with a high reproducibility. Furthermore, water may be absorbed by the conductive buffer layer, changing its properties. In addition, the required drive (bias) voltage may be relatively high because the generated electric field must pass through the conductive buffer layer (e.g., which may be quite thick for Z-cut configurations). In US Patent Application Publication No. 2003/0133638, DC drift is reduced by implanting a SiO 2  buffer layer with fluorine ions. The negative fluorine ions (F − ) are believed to react with positive ions, such as lithium (Li + ) from the substrate, to form stable compounds such as LiF. The reduction in the number of mobile Li +  ions then results in a reduction in DC drift. Again, the required drive (bias) voltage may be relatively high because the generated electric field must pass through the ion-implanted buffer layer (e.g., which may be quite thick for Z-cut configurations). 
         [0013]    In US Patent Application Publication No. 2006/0023288 and U.S. Pat. No. 7,127,128, DC drift is reduced by providing a separate low-frequency bias electrode structure substantially aligned with an overlying RF electrode structure. For example, consider the prior art X-cut embodiment illustrated in  FIGS. 3   a  and  3   b , wherein a dielectric buffer layer  350  is provided below the RF electrode structure  340 / 342 / 344 , to provide velocity matching, but is eliminated below the bias electrode structure  370 / 372 / 374 , to reduce DC drift. Advantageously, this configuration provides a relatively short modulator (i.e., since the bias and RF electrode structures are stacked) with a relatively low drive voltage (i.e., since the bias electrode structure is deposited directly on the substrate  310 ). Further advantageously, the bias electrode structure  370 / 372 / 374  is fabricated from a material having a high resistivity, which is conductive at low frequencies and functions as a dielectric at high-frequencies. Accordingly, the bias electrode structure can be deposited on the substrate without introducing significant loss. 
         [0014]    US Patent Application Publication No. 2006/0023288 also describes numerous low bias drift embodiments for Z-cut LiNbO 3  modulators. Referring to  FIGS. 4   a  and  4   b , the Z-cut embodiments typically include two bias signal electrodes, each of which is split into two separate elongated segments. More specifically, each segment of each split bias electrode  470 / 476  is shifted laterally to an opposite side of the corresponding waveguide segment  426 / 424 . Again, a dielectric buffer layer  450  is provided below the bleed layer  460  and RF electrode structure  440 ,  442 ,  444 , to provide velocity matching, but is eliminated below the bias electrode structure  470 ,  472 ,  474 ,  476 , to reduce DC drift. Advantageously, this configuration provides a relatively short modulator (i.e., since the bias and RF electrode structures are stacked) with a relatively low drive voltage (i.e., since the bias electrode structure is deposited directing on the substrate  410 ). Further advantageously, since the bias signal electrodes  470 ,  476  are split, and are not disposed directly over the interferometer arms  426 ,  424 , respectively, optical loss is reduced. 
         [0015]    Yet another advantage of many of the embodiments described in US Patent Application Publication No. 2006/0023288 is improved humidity tolerance. As is well known in the art, the presence of high magnitude electric fields and high humidity often results in corrosion of electro-optic devices. For example, when a metal adhesion layer (e.g., Ti, Ti/W, Cr, etc) is used to promote adhesion between an RF electrode (e.g., Au) and an electro-optic substrate (e.g., LiNbO 3 ), any moisture in direct contact with the multi-layer structure will serve as an electrolyte that induces galvanic corrosion. Galvanic corrosion, which results from the difference in electrochemical potentials of dissimilar metals, can create a conductive deposit between the surface RF electrodes, which causes current leakage, short circuit, or peeling of the RF electrodes. Various schemes have been proposed to obviate galvanic corrosion, and thus reduce the need for a hermetic package. For example, in U.S. Pat. No. 6,867,134 the adhesion layer is eliminated, whereas in US Patent Application Publication No. 2003/0062551 the adhesion layer is encapsulated. Alternatively, the adhesion layer can be made of a thin metal, such as nickel, which has a work function similar to gold. While these methods do suppress galvanic corrosion, electro-migration corrosion can still occur. Electro-migration corrosion occurs when a large DC voltage is applied across closely-spaced electrodes (e.g., gold RF electrodes) in the presence of water or a high humidity level. Similar to galvanic corrosion, electro-migration corrosion negatively impacts the performance and reduces the service life of electro-optic devices. As a result, electro-optic devices are often coated as shown in U.S. Pat. No. 6,560,377 and/or sealed in hermetic packages. However, the coatings negatively impact the RF properties of the RF electrode, and hermetic packaging adds cost to the modulator. 
         [0016]    In US Patent Application Publication No. 2006/0023288 humidity tolerance is increased in various ways. For example in some embodiments, the large DC voltage is applied to bias electrodes that are disposed beneath a buffer layer, whereas in other embodiments the large DC voltage is applied to bias electrodes that are disposed below the substrate. Since these buried bias electrodes are protected from humidity, electro-migration corrosion of the buried bias electrodes is reduced. Moreover, if the buried bias electrodes are DC isolated from the RF electrodes, then electro-migration corrosion of the RF electrodes is also minimized. Furthermore, if the adhesion layer is eliminated, encapsulated, and/or formed of a material with a work function similar to that used to form the RF electrode, then both galvanic and electro-migration corrosion mechanisms are eliminated, enabling low cost non-hermetic packaging of the modulator. 
         [0017]    In addition, improved humidity tolerance is also provided by fabricating the bias electrodes from a high resistivity material (e.g., a material having an electrical resistivity substantially higher than that of the RF electrodes, but substantially lower than the substrate). Notably, these high resistivity bias electrodes have been found to be significantly more robust than prior art high-conductivity bias electrodes (e.g., fabricated from gold). 
       SUMMARY OF THE INVENTION 
       [0018]    The instant inventors have found that electro-optic devices having a separate bias electrode structure, as for example disclosed in US Patent Application Publication No. 2006/0023288, may exhibit improved performance if a thin buffer layer is provided between the bias signal electrode(s) and the substrate. 
         [0019]    According to one aspect of the present invention there is provided an electro-optic device comprising: an electro-optic substrate having an optical waveguide formed therein; an RF electrode structure disposed for generating an RF electric field in the optical waveguide, the RF electrode structure including a first RF electrode; a first buffer layer disposed between the substrate and the first RF electrode; a bias electrode structure disposed for generating a low frequency or DC electric field in the optical waveguide, the bias electrode structure including a first bias electrode at least partially disposed between the first buffer layer and the first RF electrode; and a second buffer layer disposed between the first RF electrode and the first bias electrode. 
         [0020]    According to one aspect of the present invention there is provided a method of fabricating an electro-optic device comprising: providing an electro-optic substrate having an optical waveguide formed therein; forming a first buffer layer on the electro-optic substrate; forming a bias electrode layer on at least one of the first buffer layer and the electro-optic substrate, the bias electrode layer patterned to provide a first bias electrode that at least partially extends over the first buffer layer; forming a second buffer layer on the bias electrode layer; and forming an RF electrode layer on the second buffer layer, the RF electrode layer patterned to include a first RF electrode. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0021]    Further features and advantages of the present invention will become apparent from the following detailed description, taken in combination with the appended drawings, in which: 
           [0022]      FIG. 1   a  is a schematic diagram of a prior art Mach-Zehnder optical modulator, illustrating a common optical waveguide configuration; 
           [0023]      FIG. 1   b  is a schematic diagram of a prior art Mach-Zehnder optical modulator, illustrating a common electrode configuration for X-cut LiNbO 3 ; 
           [0024]      FIG. 1   c  is a schematic diagram of a prior art Mach-Zehnder optical modulator, illustrating a common electrode configuration for Z-cut LiNbO 3 ; 
           [0025]      FIG. 1   d  is a plan view of a prior art Mach-Zehnder optical modulator having a Z-cut LiNbO 3  substrate; 
           [0026]      FIG. 1   e  is a sectional view of the prior art Mach-Zehnder optical modulator illustrated in  FIG. 1   d  taken along line I-I; 
           [0027]      FIG. 2  is a plan view of a prior art low bias drift Mach-Zehnder optical modulator having an X-cut LiNbO 3  substrate; 
           [0028]      FIG. 3   a  is a plan view of another prior art low bias drift Mach-Zehnder optical modulator having an X-cut LiNbO 3  substrate; 
           [0029]      FIG. 3   b  is a sectional view of the prior art Mach-Zehnder optical modulator illustrated in  FIG. 3   a  taken along line II-II; 
           [0030]      FIG. 4   a  is a plan view of a prior art low bias drift Mach-Zehnder optical modulator having a Z-cut LiNbO 3  substrate; 
           [0031]      FIG. 4   b  is a sectional view of the prior art Mach-Zehnder optical modulator illustrated in  FIG. 4   a  taken along line III-III; 
           [0032]      FIG. 5   a  is a sectional view of an electro-optic device in accordance with one embodiment of the instant invention; 
           [0033]      FIG. 5   b  is a plan view of the electro-optic device in  FIG. 5   a  illustrating an embodiment where the optical waveguides are patterned to form a Mach-Zehnder interferometer; 
           [0034]      FIG. 5   c  is a plan view of the electro-optic device in  FIG. 5   a  illustrating an embodiment where the optical waveguides are patterned to form a 2×2 optical switch; 
           [0035]      FIG. 5   d  is a plan view of the electro-optic device in  FIG. 5   a  illustrating an embodiment where the optical waveguides are patterned to form an optical coupler; 
           [0036]      FIG. 6  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0037]      FIG. 7  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0038]      FIG. 8  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0039]      FIG. 9  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0040]      FIG. 10  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0041]      FIG. 11  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0042]      FIG. 12  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0043]      FIG. 13  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a patterned lower buffer layer; 
           [0044]      FIG. 14  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a planarized lower buffer layer; 
           [0045]      FIG. 15  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a planarized lower buffer layer; 
           [0046]      FIG. 16  is a plan view illustrating an embodiment where the bias electrode structure is substantially aligned with the RF electrode structure (not shown); 
           [0047]      FIG. 17   a  is a plan view illustrating an embodiment where the bias electrode structure is segmented; 
           [0048]      FIG. 17   b  is a sectional view of the embodiment illustrated in  FIG. 17   a  taken along line B-B; 
           [0049]      FIG. 17   c  is a sectional view of the embodiment illustrated in  FIG. 17   a  taken along line C-C; 
           [0050]      FIG. 18  is a plan view illustrating another embodiment where the bias electrode structure is segmented; 
           [0051]      FIG. 19  is a plan view illustrating another embodiment where the bias electrode structure is segmented; 
           [0052]      FIG. 20  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a planarized lower buffer layer; 
           [0053]      FIG. 21  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having a planarized lower buffer layer; and 
           [0054]      FIG. 22  is a sectional view of an electro-optic device in accordance with another embodiment of the instant invention having an etched substrate. 
       
    
    
       [0055]    It will be noted that throughout the appended drawings, like features are identified by like reference numerals. 
       DETAILED DESCRIPTION 
       [0056]    Referring to  FIG. 5   a , there is shown a sectional view of an electro-optic device in accordance with one embodiment of the instant invention. The electro-optic device  500  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  570  and second  576  bias signal electrodes, bias ground electrodes  572 ,  574 , and a lower buffer layer  580 ,  586 . 
         [0057]    In this embodiment, the substrate  510  is fabricated from an electro-optic material such as Z-cut lithium niobate (LiNbO 3 ). Alternatively, the substrate is fabricated from another electro-optic material, such as Z-cut lithium tantalite (LiTaO 3 ). The width, length, and thickness of the substrate  510  typically vary with the type of electro-optic device. For example, substrates for conventional Mach-Zehnder modulators are often about 40 mm long, 2 mm wide, and 1 mm thick. Of course, other dimensions are also possible. 
         [0058]    The first  524  and second  526  optical waveguides are embedded in, or otherwise supported by, the substrate  510 . The optical waveguides  524 ,  526  may be fabricated using one of various well-known methods, such as titanium diffusion or annealed proton exchange. For example, in one embodiment the waveguides  524 ,  526  are formed by patterning titanium in or on a Z-cut LiNbO 3  substrate, followed by subjecting the substrate to increased temperatures to allow the titanium to diffuse therein. Conventionally, titanium diffused LiNbO 3  provides waveguide(s) that are about 7 μm wide and about 3 μm deep. The pattern used to form the waveguides  524 ,  526  is dependent on the type of electro-optic device. For example, if the electro-optic device is a Mach-Zehnder modulator, the pattern may be similar to that illustrated in  FIG. 5   b . Alternatively, if the electro-optic device is an optical switch or a tunable directional coupler, the pattern may be similar to that illustrated in  FIG. 5   c  or  5   d , respectively. Further alternatively, the pattern may correspond to another electro-optic device having two substantially parallel waveguides or waveguide segments. In each case, the sectional view along any of lines A-A would correspond to  FIG. 5   a . Notably,  FIGS. 5   b ,  5   c , and  5   d  only show the waveguide patterns (dotted lines) and an exemplary RF electrode structure (thatched structure) to simplify the illustrations (e.g., the buffer layer and/or the bias electrodes are omitted from the figures). 
         [0059]    The RF signal electrode  540  and the RF ground electrodes  544 ,  542  are supported by an substrate  510 . The RF electrodes  540 ,  542 ,  544  are part of an RF electrode structure used to apply a high-frequency RF voltage across the waveguides  524 ,  526 . For example, in one embodiment the RF electrode structure forms a traveling-wave electrode structure used to propagate a microwave signal that generates oppositely oriented electric fields in the first  524  and second  526  optical waveguides. The RF signal electrode  540  and RF ground electrode  542  are positioned substantially over the waveguides  526 ,  524 , respectively. The RF electrodes  540 ,  542 ,  544  are typically formed from a material that exhibits high electrical conductivity such as gold (Au), copper (Cu), silver (Ag), or platinum (Pt). Since these metals do not always readily adhere to conventional bleed layer materials, an adhesion layer is optionally used to promote adhesion. Some examples of suitable adhesion layers include thin film layers of chromium (Cr), titanium (Ti), titanium-tungsten (Ti/W), etc. Alternatively, the adhesion layer is formed from a metal that has a work function similar to the RF electrode material (e.g., nickel (Ni) has a work function similar to that of Au). As discussed above, a matched work function assures that little if any voltage potential arises across the two metals, thus reducing galvanic corrosion. Alternatively, the adhesion layer is encapsulated as described in US Patent Application Publication No. 2003/0062551, or eliminated as described in U.S. Pat. No. 6,867,134, by activating the surface. While the RF electrode structures illustrated in  FIGS. 5   b ,  5   c ,  5   d , show one example of a suitable electrode design (e.g., known as coplanar waveguide), other designs are also possible (e.g., conventional coplanar strip, asymmetric coplanar strip, etc.). The RF electrode structure  540 ,  542 ,  544 , which in one embodiment is about 15-40 μm high, may be fabricated using one of various well-known methods, such as electroplating, sputtering, evaporation, plasma etching, liftoff, etc. 
         [0060]    The upper buffer layer  550  is provided to reduce optical losses due to absorption from the RF electrodes and to provide velocity matching between the optical signal and the RF signal. Accordingly, the upper buffer layer  550  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns) and has a dielectric constant that is lower than the dielectric constant of the substrate  510 . For example, in one embodiment the buffer layer  550  is fabricated with silicon dioxide (SiO 2 ). In another embodiment, the buffer layer  550  is fabricated with benzocyclobutene (BCB). In general, the resistivity of the buffer layer  550  will be in the range from about 10 17 -10 19 Ω-cm @25° C., but could be higher. Using a substantially non-conductive material also allows the buffer layer  550  to provide electrical insulation between the bias signal electrodes and/or between the bias signal electrodes and the RF signal electrode. The buffer layer  550  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). In one embodiment, the buffer layer  550  is planarized throughout the wafer. In another embodiment, the buffer layer  550  is patterned so as to cover only the bias electrode structure. In yet another embodiment, the buffer layer  550  is patterned to cover only the bias signal electrodes. The thickness of the upper buffer layer  550  is typically in the range between about 0.05 and 2 μm, and more commonly between about 0.4 and 1.0 μm. Notably, conventional SiO 2  buffer layers are often subject to an annealing step after deposition. Optionally, this annealing step is eliminated to prevent damage to the bias electrode. 
         [0061]    The bleed layer  560  is used to bleed off electric charge created by the pyroelectric effect. Accordingly, the bleed layer  560 , which is used to dissipate the accumulated electric charge, is typically formed from an electrically conductive film. Preferably, the film is formed from a semiconductor to prevent shorting out the RF electrodes. Some examples of suitable materials for the slightly conductive film  560  include tantalum silicon nitride (TaSiN), titanium silicon nitride (TiSiN), silicon titanium oxynitride (SiTiON), amorphous or polycrystalline silicon (Si), etc. Conveniently, when the bleed layer  560  is formed from materials such as TaSiN, the highly resistive bleed layer also serves as a moisture barrier that prevents impurities from entering the buffer layer and/or prevents voltage induced ion migration near the bias electrodes. Accordingly, the electro-optical device could be described as a humidity tolerant and/or atmospheric tolerant electro-optic device. In addition, when the bleed layer  560  is formed from a material such as TaSiN, the RF electrodes may be deposited directly on the bleed layer (i.e., in the absence of an adhesion layer or an activated surface), thus simplifying the manufacturing process and further improving humidity tolerance. Note that the term TaSiN, as used herein, corresponds to a chemical composition that may be complex, and that is not necessarily represented by the 1:1:1 ratio suggested in the abbreviated chemical name (e.g., the actual formula may be more accurately represented by Ta x Si y N z ). The bleed layer  560  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the bleed layer  560  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0062]    The first bias signal electrode  570 , the second bias signal electrode  576 , the first bias ground electrode  574 , and the second bias ground electrode  572 , are all supported by the substrate  510 . The bias electrodes  570 ,  572 ,  574 ,  576  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  570  and second  576  bias signal electrodes includes a lower split portion disposed directly on the substrate and an upper cap section, which bridges the corresponding lower split portion. 
         [0063]    Each of the bias electrodes  570 ,  572 ,  574 ,  576  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity electrode material allows the bias electrodes  570 ,  572 ,  574 ,  576  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about ˜10 4  to 10 6 Ω-cm @25° C., which is between 2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ωcm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  570 ,  572 ,  574 ,  576  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when the bias electrodes  570 ,  572 ,  574 ,  576  and the bleed layer  560  are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  570 ,  572 ,  574 ,  576  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  570 ,  572 ,  574 ,  576  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0064]    The lower buffer layer  580 ,  586  provides a spacer between the optical waveguides  526 ,  524  and the overlying upper cap section of the bias signal electrodes  570 ,  576 , respectively. Accordingly, the lower buffer layer  580 ,  586  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  580 ,  586  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  580 ,  586  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  580 ,  586  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  580 ,  586  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  580 ,  586  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0065]    According to one embodiment of the instant invention, the electro-optic device  500  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower buffer layer  580 ,  586 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  570 ,  572 ,  574 ,  576 . In this embodiment, the bias signal electrodes  570 ,  576  have been shown having a substantially bracket-shaped (i.e., [-shaped) cross section. In other embodiments, the bias signal electrodes  570 ,  576  may have a substantially U-shaped cross-section such that they provide a conformal coating for each region of the lower buffer layer. In each instance, the lower split portions and upper cap section of each bias signal electrode  570 ,  576  substantially encapsulate the first  580  and second  586  regions of lower buffer layer, respectively. 
         [0066]    Advantageously, the upper cap sections, which extend over the waveguides  526 ,  524 , significantly enhance the bias electrode modulation efficiency. In addition, the lower split portions not only contribute to the applied field, but also help to limit any ionic conduction in the lower buffer layer  580 ,  586 , which would counteract the applied field from each bias signal electrode  570 ,  576 . More specifically, encapsulation of the lower buffer layer by the bias electrode prevents ions from traveling horizontally from a region near one bias electrode to a region near the other bias electrode. However, such an encapsulation does not prevent ion/electron migration from occurring vertically within the lower buffer layer. The lower buffer layer  580 ,  586  helps reduce loss of the optical mode associated with the cap sections of the bias signal electrodes  570 ,  576 . 
         [0067]    The performance of the electro-optic device  500  is generally dependent on the thickness of the lower buffer layer  580 ,  586  and its conductivity. For example, while thinner lower buffer layers are associated with increased bias electrode modulation efficiency, they are also associated with increased optical loss. Similarly, while low resistivity materials are associated with increased bias electrode modulation efficiency and faster response time, they are also generally associated with increased optical loss. With regard to the former, the bias signal electrodes  570 ,  576  may introduce optical loss if the lower buffer layer  580 ,  586  is not thick enough to optically isolate the optical mode in the waveguides  526 ,  524  from the cap sections of the bias signal electrodes  570 ,  576 . 
         [0068]    Referring to  FIG. 6 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  600  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  670  and second  676  bias signal electrodes, bias ground electrodes  672 ,  674 , and a lower buffer layer  680 ,  686 . 
         [0069]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0070]    The first bias signal electrode  670 , the second bias signal electrode  676 , the first bias ground electrode  674 , and the second bias ground electrode  672 , are all supported by the substrate  510 . The bias electrodes  670 ,  672 ,  674 ,  676  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  670  and second  676  bias signal electrodes includes a lower portion disposed directly on the substrate and an upper portion, which covers the lower portion and extends over the first  680  and second  686  regions of the lower buffer layer, respectively. 
         [0071]    Each of the bias electrodes  670 ,  672 ,  674 ,  676  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  670 ,  672 ,  674 ,  676  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  670 ,  672 ,  674 ,  676  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when bias electrodes  670 ,  672 ,  674 ,  676  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  670 ,  672 ,  674 ,  676  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  670 ,  676  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0072]    The lower buffer layer  680 ,  686  provides a spacer between the optical waveguides  526 ,  524  and the overlying upper portions of the bias signal electrodes  670 ,  676 , respectively. Accordingly, the lower buffer layer  680 ,  686  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  680 ,  686  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  680 ,  686  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  680 ,  686  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 7 -10 19 Ωcm @25° C. The lower buffer layer  680 ,  686  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  680 ,  686  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0073]    According to one embodiment of the instant invention, the electro-optic device  600  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower buffer layer  680 ,  686 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  670 ,  672 ,  674 ,  676 . In this embodiment, the bias signal electrodes  670 ,  676  have been shown having a substantially L-shaped cross section. In other embodiments, the top left comers of the bias signal electrodes  670 ,  676  may be rounder. In other embodiments, the thickness of the bias electrodes is constant, conforming to the shape of the lower buffer layer. 
         [0074]    Advantageously, the upper portions, which extend over the waveguides  526 ,  524 , significantly enhance the bias electrode modulation efficiency. The lower portions not only contribute to the applied field, but also help to limit any horizontal ionic conduction in the lower buffer layers  680 ,  686 , which would counteract the applied field from each bias signal electrode  670 ,  676 . The lower buffer layer  680 ,  686  helps reduce loss of the optical mode associated with the upper portions of the bias signal electrodes  670 ,  676 . 
         [0075]    Further advantageously, since the bias signal electrodes  670 ,  676  have a substantially L-shaped cross section rather than the bracket-shaped cross section illustrated in  FIG. 5   a , optical loss is reduced. In particular, optical loss is reduced because optical loading created by one half of the lower split portion is eliminated. Note, that while the substantially L-shaped cross section provides reduced optical loss, it may also provide a small reduction in modulation efficiency. 
         [0076]    In this embodiment, the bias signal electrodes  670 ,  676  are fabricated such that the lower portions of each bias electrode  670 ,  676  are disposed to the same side of the device (e.g. to the left of the waveguides  524 ,  526 , respectively). As a result, symmetry of the optical loading is advantageously maintained even if misalignment errors occur during the manufacturing process. For example, if the narrow lower portions of each bias electrode  670 ,  676  are deposited too far to the left in  FIG. 6 , then the effects will be felt equally in both optical waveguides  524 ,  526 . 
         [0077]    Referring to  FIG. 7 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  700  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  770  and second  776  bias signal electrodes, bias ground electrodes  772 ,  774 , and a lower buffer layer  780 ,  786 . 
         [0078]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0079]    The first bias signal electrode  770 , the second bias signal electrode  776 , the first bias ground electrode  774 , and the second bias ground electrode  772 , are all supported by the substrate  510 . The bias electrodes  770 ,  772 ,  774 ,  776  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  770  and second  776  bias signal electrodes includes a lower portion disposed directly on the substrate and an upper portion, which covers the lower portion and extends over the first  780  and second  786  regions of the lower buffer layer, respectively. More specifically, the first  770  bias signal electrode includes a lower portion  770   a  and an upper portion  770   b  , whereas the second  776  bias signal electrode includes a lower portion  776   a  and an upper portion  776   b.    
         [0080]    Each of the bias electrodes  770 ,  772 ,  774 ,  776  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  770 ,  772 ,  774 ,  776  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C.(LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  770 ,  772 ,  774 ,  776  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when the bias electrodes  770 ,  772 ,  774 ,  776  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  770 ,  772 ,  774 ,  776  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  770 ,  776  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0081]    The lower buffer layer  780 ,  786  provides a spacer between the optical waveguides  526 ,  524  and the overlying upper portions of the bias signal electrodes  770 ,  776 , respectively. Accordingly, the lower buffer layer  780 ,  786  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  780 ,  786  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  780 ,  786  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  780 ,  786  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  780 ,  786  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  780 ,  786  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0082]    According to one embodiment of the instant invention, the electro-optic device  700  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower buffer layer  780 ,  786 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  770 ,  772 ,  774 ,  776 . In this embodiment, the bias signal electrodes  770 ,  776  have been shown having a substantially L-shaped cross section. In other embodiments, the outer comers of the bias signal electrodes  770 ,  776  may be rounder. In other embodiments, the thickness of the bias electrodes is constant, conforming to the shape of the lower buffer layer. 
         [0083]    Advantageously, the upper portions, which extend over the waveguides  526 ,  524 , significantly enhance the bias electrode modulation efficiency. The lower portions not only contribute to the applied field, but also help to limit any horizontal ionic conduction in the lower buffer layers  780 ,  786 , which would counteract the applied field from each bias signal electrode  770 ,  776 . The lower buffer layer  780 ,  786  helps reduce loss of the optical mode associated with the upper portions of the bias signal electrodes  770 ,  776 . 
         [0084]    Further advantageously, since the bias electrodes  770 ,  776  have a substantially L-shaped cross section rather than the bracket-shaped cross section illustrated in  FIG. 5   a , optical loss is reduced. In particular, optical loss is reduced because optical loading created by one half of the lower split portion is eliminated. Note that while the substantially L-shaped cross section does provide reduced optical loss, it may also cause a small reduction in modulation efficiency. 
         [0085]    In this embodiment, the bias electrodes  770 ,  776  are fabricated such that the lower portions  770   a ,  776   a  of each bias electrode  770 ,  776  are disposed on opposite sides of the device (e.g., such that they are shifted laterally to the outside of the waveguides  524 ,  526 ). As a result, the electro-optic device may be easier to fabricate for embodiments where the waveguides  526 ,  524  are close together (e.g., optical couplers and/or optical switches). In addition, alignment between the lower portions  770   a ,  776   a  and the upper portions  770   b ,  776   b , respectively, may be less critical. For example, if the upper portions  770   b ,  776   b  are deposited too far to the right in the embodiment illustrated in  FIG. 7 , then it would not result in a short between the two bias signal electrodes as it would in the analogous scenario for the embodiment illustrated in  FIG. 6 . 
         [0086]    Referring to  FIG. 8 , there is shown a sectional view of an electro-ophic device in accordance with other embodiment of the instant invention. The electro-optic device  800  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  870  and second  876  bias signal electrodes, bias ground electrodes  872 ,  874 , and a lower buffer layer  880 ,  886 . 
         [0087]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0088]    The first bias signal electrode  870 , the second bias signal electrode  876 , the first bias ground electrode  874 , and the second bias ground electrode  872 , are all supported by the substrate  510 . The bias electrodes  870 ,  872 ,  874 ,  876  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  870  and second  876  bias signal electrodes includes a lower portion having three segments disposed directly on the substrate and an upper cap section. The center segment of each lower portion  870 a,  876 a splits the corresponding lower buffer layer into two parts. The cap sections  870 b,  876 b extend over the split buffer layers  880 ,  886 , respectively, to bridge the lower portion segments. 
         [0089]    Each of the bias electrodes  870 ,  872 ,  874 ,  876  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  870 ,  872 ,  874 ,  876  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C.(LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  870 ,  872 ,  874 ,  876  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when bias electrodes  870 ,  872 ,  874 ,  876  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  870 ,  872 ,  874 ,  876  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  870 ,  876  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0090]    The lower buffer layer  880 ,  886  provides a spacer between the optical waveguides  526 ,  524  and the overlying portions of the bias signal electrodes  870 ,  876 , respectively. Accordingly, the lower buffer layer  880 ,  886  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  880 ,  886  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  880 ,  886  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  880 ,  886  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  880 ,  886  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  880 ,  886  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0091]    According to one embodiment of the instant invention, the electro-optic device  800  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower split buffer layers  880 ,  886 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  870 ,  872 ,  874 ,  876 . In this embodiment, the bias signal electrodes  870 ,  876  have been shown having a substantially E-shaped cross section. In other embodiments, the bias signal electrodes  870 ,  876  may provide a more conformal coating of the split lower buffer layers  880 ,  886 , respectively. 
         [0092]    Advantageously, the upper layers  870   b ,  876   b , which extend over waveguides  526 ,  524 , respectively, significantly enhance the bias electrode structure modulation efficiency. The lower portions  870   a ,  876   a  not only contribute to the applied field, but also help to limit any horizontal ionic conduction in the split buffer layers  880 ,  886 , which would counteract the applied field from each bias signal electrode  870 ,  876 . The lower split buffer layers  880 ,  886  help reduce loss of the optical mode due to the presence of the bias signal electrodes  870 ,  876 . 
         [0093]    Further advantageously, since the bias electrodes  870 ,  876  have a substantially E-shaped cross section rather than the bracket-shaped cross section illustrated in  FIG. 5   a , the center segment of the lower portions may significantly improve modulation efficiency. Preferably, the center segment of the lower portions  870   a ,  876   a  is relatively narrow such that it does not introduce excessive optical loss. For example, according to one embodiment the center segments of the lower portions  870   a ,  876   a  are much narrower than the corresponding outer segments. 
         [0094]    Referring to  FIG. 9 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  900  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  970  and second  976  bias signal electrodes, bias ground electrodes  972 ,  974 , and a lower buffer layer  980 ,  986 . 
         [0095]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0096]    The first bias signal electrode  970 , the second bias signal electrode  976 , the first bias ground electrode  974 , and the second bias ground electrode  972 , are all supported by the substrate  510 . The bias electrodes  970 ,  972 ,  974 ,  976  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  970  and second  976  bias signal electrodes includes a lower portion and an upper cap section. Each lower portion  970   a ,  976   a  includes two outer segments disposed directly on the substrate  510  and a center segment that is not in contact with the substrate  510 . The upper cap sections  970   b ,  976   b  extend over the buffer layers  980 ,  986   b , respectively, to bridge the outer segments and to provide contact with the center segment. 
         [0097]    Each of the bias electrodes  970 ,  972 ,  974 ,  976  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  970 ,  972 ,  974 ,  976  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  970 ,  972 ,  974 ,  976  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when bias electrodes  970 ,  972 ,  974 ,  976  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  970 ,  972 ,  974 ,  976  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  970 ,  976  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0098]    The lower buffer layer  980 ,  986  provides a spacer between the optical waveguides  526 ,  524  and the overlying cap sections of the bias signal electrodes  970 ,  976 , respectively. Accordingly, the lower buffer layer  980 ,  986  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  980 ,  986  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  980 ,  986  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  980 ,  986  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  980 ,  986  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  980 ,  986  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0099]    According to one embodiment of the instant invention, the electro-optic device  900  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower buffer layer  980 ,  986 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  970 ,  972 ,  974 ,  976 . Notably, the lower buffer layer is only partially etched in the region directly over each waveguide  526 ,  524  such that the bias signal electrodes  970 ,  976  have substantially E-shaped cross section, wherein the middle arm of the E is shorter than the outer arms. According to other embodiments, the bias signal electrodes  970 ,  976  may provide a more conformal coating of the lower buffer layers  980 ,  986 , respectively. 
         [0100]    Advantageously, the upper layers  970   b ,  976   b , which extend over waveguides  526 ,  524 , respectively, significantly enhance the bias electrode structure modulation efficiency. The outer segments of the lower layers  970   a ,  976   a  not only contribute to the applied field, but also help to limit any horizontal ionic conduction in the buffer layers  980 ,  986 , which would counteract the applied field from each bias signal electrode  970 ,  976 . The center segments  970   c ,  976   c  of the lower portions  970   a ,  976   a , respectively, advantageously help to focus the applied field, while minimizing increases in optical loss. 
         [0101]    Referring to  FIG. 10 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  1000  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  1070  and second  1076  bias signal electrodes, bias ground electrodes  1072 ,  1074 , and a lower buffer layer  1080 ,  1086 . 
         [0102]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0103]    The first bias signal electrode  1070 , the second bias signal electrode  1076 , the first bias ground electrode  1074 , and the second bias ground electrode  1072 , are all supported by the substrate  510 . The bias electrodes  1070 ,  1072 ,  1074 ,  1076  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  1070  and second  1076  bias signal electrodes includes a lower portion and an upper cap section. Each lower portion  1070   a ,  1076   a  includes a center segment disposed directly on the substrate  510  that splits the lower buffer layer into two segments. The upper cap sections  1070   b ,  1076   b  extend over the split buffer layers  1080 ,  1086  and are in contact with the lower portions  1070   a ,  1076   a , respectively. 
         [0104]    Each of the bias electrodes  1070 ,  1072 ,  1074 ,  1076  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  1070 ,  1072 ,  1074 ,  1076  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  1070 ,  1072 ,  1074 ,  1076  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when the bias electrodes  1070 ,  1072 ,  1074 ,  1076  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  1070 ,  1072 ,  1074 ,  1076  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  1070 ,  1076  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0105]    The lower buffer layer  1080 ,  1086  provides a spacer between the optical waveguides  526 ,  524  and the overlying upper cap sections  1070   b ,  1076   b , respectively. Accordingly, the lower buffer layer  1080 ,  1086  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  1080 ,  1086  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  1080 ,  1086  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  1080 ,  1086  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  1080 ,  1086  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  1080 ,  1086  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0106]    According to one embodiment of the instant invention, the electro-optic device  1000  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower split buffer layers  1080 ,  1086 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  1070 ,  1072 ,  1074 ,  1076 . In this embodiment, the bias signal electrodes  1070 ,  1076  have been shown having a substantially T-shaped cross section. In other embodiments, the bias signal electrodes  1070 ,  1076  may provide a more conformal coating of the lower buffer layers  1080 ,  1086 , respectively. 
         [0107]    Advantageously, the upper cap sections  1070   b ,  1076   b , which extend over waveguides  526 ,  524 , respectively, significantly enhance the bias electrode structure modulation efficiency. 
         [0108]    Further advantageously, since the bias electrodes  1070 ,  1076  have a substantially T-shaped cross section rather than the E-shaped cross section illustrated in  FIG. 8 , optical loading from the outer segments is eliminated. 
         [0109]    Referring to  FIG. 11 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  1100  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  1170  and second  1176  bias signal electrodes, bias ground electrodes  1172 ,  1174 , and a lower buffer layer  1180 ,  1186 . 
         [0110]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0111]    The first bias signal electrode  1170 , the second bias signal electrode  1176 , the first bias ground electrode  1174 , and the second bias ground electrode  1172 , are all supported by the substrate  510 . The bias electrodes  1170 ,  1172 ,  1174 ,  1176  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  1170  and second  1176  bias signal electrodes includes a lower split portion  1170   a ,  1176   a  and an upper split portion  1170   b ,  1176   b , respectively. The upper split portions  1170   b ,  1176   b  at least partially extend over the buffer layers  1180 ,  1186 , respectively. 
         [0112]    Each of the bias electrodes  1170 ,  1172 ,  1174 ,  1176  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  1170 ,  1172 ,  1174 ,  1176  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  1170 ,  1172 ,  1174 ,  1176  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when the bias electrodes  1170 ,  1172 ,  1174 ,  1176  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  1170 ,  1172 ,  1174 ,  1176  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  1170 ,  1176  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0113]    The lower buffer layer  1180 ,  1186  provides a spacer between the optical waveguides  526 ,  524  and the overlying upper split portions of the bias signal electrodes  1170 ,  1176 , respectively. Accordingly, the lower buffer layer  1180 ,  1186  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  1180 ,  1186  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  1180 ,  1186  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  1180 ,  1186  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  1180 ,  1186  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  1180 ,  1186  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0114]    According to one embodiment of the instant invention, the electro-optic device  1100  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower buffer layer  1180 ,  1186 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  1170 ,  1172 ,  1174 ,  1176 . 
         [0115]    Advantageously, the upper split portions  1170 b,  1176 b, which extend over waveguides  526 ,  524 , respectively, significantly enhance the bias electrode structure modulation efficiency. The lower split portions  1170   a ,  1176   a  not only contribute to the applied field, but also help to limit any horizontal ionic conduction in the buffer layers  1180 ,  1186 , which would counteract the applied field from each bias signal electrode  1170 ,  1176 . 
         [0116]    Further advantageously, the upper split portions  1170   b ,  1176   b  form a partial cap, thus reducing optical loss. In addition, the inner bottom comers of the split portions  1170   b ,  1176   b  can be used to focus the applied field onto the underlying waveguides  526 ,  524 , respectively. 
         [0117]    Referring to  FIG. 12 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  1200  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  1270  and second  1276  bias signal electrodes, bias ground electrodes  1272 ,  1274 , and a lower buffer layer  1280 ,  1286 . 
         [0118]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0119]    The first bias signal electrode  1270 , the second bias signal electrode  1276 , the first bias ground electrode  1274 , and the second bias ground electrode  1272 , are all supported by the substrate  510 . The bias electrodes  1270 ,  1272 ,  1274 ,  1276  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  1270  and second  1276  bias signal electrodes includes a lower split portion  1270   a ,  1276   a  and a partial cap portion  1270   b ,  1276   b , respectively. The partial cap portions  1270   b ,  1276   b  at least partially extend over the buffer layers  1280 ,  1286 , respectively. 
         [0120]    Each of the bias electrodes  1270 ,  1272 ,  1274 ,  1276  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  1270 ,  1272 ,  1274 ,  1276  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  1270 ,  1272 ,  1274 ,  1276  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when the bias electrodes  1270 ,  1272 ,  1274 ,  1276  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  1270 ,  1272 ,  1274 ,  1276  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  1270 ,  1276  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0121]    The lower buffer layer  1280 ,  1286  provides a spacer between the optical waveguides  526 ,  524  and the overlying partial cap sections of the bias signal electrodes  1270 ,  1276 , respectively. Accordingly, the lower buffer layer  1280 ,  1286  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). In one embodiment, the lower buffer layer  1280 ,  1286  is fabricated with a substantially non-conductive dielectric material, such as silicon dioxide (SiO 2 ) or benzocyclobutene (BCB). In another embodiment, the lower buffer layer  1280 ,  1286  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity. In yet another embodiment, the lower buffer layer  1280 ,  1286  is fabricated with a dielectric material having small amount of conductivity, such as doped or ion implanted SiO 2 . In general, the resistivity of the lower buffer layer will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  1280 ,  1286  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  1280 ,  1286  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0122]    According to one embodiment of the instant invention, the electro-optic device  1200  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower buffer layer  1280 ,  1286 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  1270 ,  1272 ,  1274 ,  1276 . 
         [0123]    Advantageously, the partial cap portions  1270   b ,  1276   b , which extend over waveguides  526 ,  524 , respectively, significantly enhance the bias electrode structure modulation efficiency. The lower split portions  1270   a ,  1276   a  not only contribute to the applied field, but also help to limit any horizontal ionic conduction in the buffer layers  1280 ,  1286 , which would counteract the applied field from each bias signal electrode  1270 ,  1276 . 
         [0124]    Further advantageously, since the partial cap portions  1270   b ,  1276   b  do not extend fully over the waveguides  526 ,  524 , respectively, optical loss is reduced. In addition, the inner bottom corners of the partial caps portions  1270   b ,  1276   b  can be used to focus the applied field onto the underlying waveguides  526 ,  524 , respectively. 
         [0125]    Referring to  FIG. 13 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  1300  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  1370  and second  1376  bias signal electrodes, bias ground electrodes  1372 ,  1374 , and a lower buffer layer  1380 ,  1386 . 
         [0126]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0127]    The first bias signal electrode  1370 , the second bias signal electrode  1376 , the first bias ground electrode  1374 , and the second bias ground electrode  1372 , are all supported by the substrate  510 . The bias electrodes  1370 ,  1372 ,  1374 ,  1376  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. The bias signal electrodes  1370 ,  1376  extend over the first  1380  and second  1386  regions of the buffer layer, respectively. 
         [0128]    Each of the bias electrodes  1370 ,  1372 ,  1374 ,  1376  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  1370 ,  1372 ,  1374 ,  1376  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and 1.3×10 ∫ Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  1370 ,  1372 ,  1374 ,  1376  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when the bias electrodes  1370 ,  1372 ,  1374 ,  1376  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  1370 ,  1372 ,  1374 ,  1376  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  1370 ,  1376  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0129]    The lower buffer layer  1380 ,  1386  provides a spacer between the optical waveguides  526 ,  524  and the overlying bias signal electrodes  1370 ,  1376 , respectively. Accordingly, the lower buffer layer  1380 ,  1386  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). Since the bias signal electrodes  1370 ,  1376  are not in direct contact with the substrate  510 , it is preferred that the lower buffer layer  1380 ,  1386  be fabricated from a material having at least some conductivity. For example, in one embodiment the lower buffer layer  1380 ,  1386  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity, or which has been doped or ion implanted to provide increased conductivity. In general, the resistivity of the lower buffer layer  1380 ,  1386  will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  1380 ,  1386  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  1380 ,  1386  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. The lower buffer layer may require an annealing step after deposition if formed from SiO 2 . 
         [0130]    According to one embodiment of the instant invention, the electro-optic device  1300  is fabricated by first depositing the lower buffer layer material, which is subsequently annealed and etched to form the lower buffer layer  1380 ,  1386 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  1370 ,  1372 ,  1374 ,  1376 . 
         [0131]    Advantageously, the lower buffer layer  1380 ,  1386  significantly reduces optical loss in the waveguides  526 ,  524  resulting from the overlying bias signal electrodes  1370 ,  1376 . 
         [0132]    Further advantageously, the non-split bias signal electrodes illustrated in  FIG. 13  are less complicated to fabricate than the split bias signal electrodes illustrated in  FIGS. 4 ,  11 , and  12 . Further advantageously, the non-split bias electrodes illustrated in  FIG. 13  impart more uniform mechanical stress on the optical waveguides underneath them, and therefore less differential mechanical stress that changes with temperature, resulting in less sensitivity of relative optical phase in the two waveguides to temperature. Changes in relative optical phase result in undesirable shifts in the bias point of an MZ. 
         [0133]    Referring to  FIG. 14 , there is shown a sectional view of an electro-optic device in accordance with other embodiment of the instant invention. The electro-optic device  1400  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  1470  and second  1476  bias signal electrodes, bias ground electrodes  1472 ,  1474 , and a lower buffer layer  1485 . 
         [0134]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0135]    The first bias signal electrode  1470 , the second bias signal electrode  1476 , the first bias ground electrode  1474 , and the second bias ground electrode  1472 , are all supported by the substrate  510 . The bias electrodes  1470 ,  1472 ,  1474 ,  1476  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each bias signal electrode  1470 ,  1476  extends over the buffer layer  1485 . 
         [0136]    Each of the bias electrodes  1470 ,  1472 ,  1474 ,  1476  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  1470 ,  1472 ,  1474 ,  1476  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  1470 ,  1472 ,  1474 ,  1476  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when the bias electrodes  1470 ,  1472 ,  1474 ,  1476  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  1470 ,  1472 ,  1474 ,  1476  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  1470 ,  1476  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0137]    The lower buffer layer  1485  provides a spacer between the optical waveguides  526 ,  524  and the overlying bias signal electrodes  1470 ,  1476 , respectively. Accordingly, the lower buffer layer  1485  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). Since the bias signal electrodes  1470 ,  1476  are not in direct contact with the substrate  510 , it is preferred that the lower buffer layer  1485  be fabricated from a material having at least some conductivity. For example, in one embodiment the lower buffer layer  1485  is fabricated from a dielectric material, such as SiO 2 , which has been sputtered such that it exhibits a small amount of intrinsic conductivity, or which has been doped or ion implanted to provide increased conductivity. In general, the resistivity of the lower buffer layer  1485  will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  1485  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  1485  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. 
         [0138]    According to one embodiment of the instant invention, the electro-optic device  1400  is fabricated by first depositing the planarized lower buffer layer  1485 , and secondly depositing the bias electrode material, which is subsequently etched to form the bias electrodes  1470 ,  1472 ,  1474 ,  1476 . A lift-off process may also be used to pattern the bias electrodes. 
         [0139]    Advantageously, the lower buffer layer  1485  significantly reduces optical loss in the waveguides  526 ,  524  resulting from the overlying bias signal electrodes  1470 ,  1476 , respectively. 
         [0140]    Further advantageously, the non-split bias signal electrodes illustrated in  FIG. 14  are less complicated to fabricate than the split bias signal electrodes illustrated in  FIGS. 4 ,  11 , and  12 . In addition, since the lower buffer layer  1485  does not need to be patterned like the lower buffer layer described with regards to  FIGS. 5   a ,  6 ,  7 ,  8 ,  9 ,  10 ,  11 ,  12 , and  13 , the electro-optic device  1400  is simpler to fabricate. Further advantageously, the non-split bias electrodes illustrated in  FIG. 14  impart more uniform mechanical stress on the optical waveguides underneath them, and therefore less differential mechanical stress that changes with temperature, resulting in less sensitivity of relative optical phase in the two waveguides to temperature. Changes in relative optical phase result in undesirable shifts in the bias point of an MZ. 
         [0141]    Notably, a planarized lower buffer layer (e.g., like  1485 ) could be included in any of the embodiments described with regard to  FIGS. 5   a ,  6 ,  7 ,  8 ,  9 ,  10 ,  11 ,  12 , and  13 . In particular, any of the bias signal electrodes disclosed in these embodiments could be deposited on a planarized buffer layer deposited on the substrate rather than directly on the substrate. For example, consider the embodiment illustrated in  FIG. 15 , which is similar to the embodiment described with regard to  FIG. 5   a.    
         [0142]    The electro-optic device  1500  includes a substrate  510 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  1570  and second  1576  bias signal electrodes, bias ground electrodes  1572 ,  1574 , and a lower buffer layer  1585 . 
         [0143]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0144]    The first bias signal electrode  1570 , the second bias signal electrode  1576 , the first bias ground electrode  1574 , and the second bias ground electrode  1572 , are all supported by the substrate  510 . The bias electrodes  1570 ,  1572 ,  1574 ,  1576  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  1570  and second  1576  bias signal electrodes includes a lower split portion and an upper cap section, which bridges the corresponding lower split portion. 
         [0145]    Each of the bias electrodes  1570 ,  1572 ,  1574 ,  1576  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  1570 ,  1572 ,  1574 ,  1576  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6 Ω-cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to 10 6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  1570 ,  1572 ,  1574 ,  1576  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when bias electrodes  1570 ,  1572 ,  1574 ,  1576  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  1570 ,  1572 ,  1574 ,  1576  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  1570 ,  1576  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0146]    The lower buffer layer  1585  provides a spacer between the optical waveguides  526 ,  524  and the overlying capping sections of the bias signal electrodes  1570 ,  1576 , respectively. Accordingly, the lower buffer layer  1585  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). Since the bias signal electrodes  1570 ,  1576  are not in direct contact with the substrate  510 , it is preferred that the lower buffer layer  1585  be fabricated from a material having at least some conductivity. For example, in one embodiment the lower buffer layer  1585  is fabricated with a dielectric material, such as silicon dioxide (SiO 2 ), which has been sputtered such that it exhibits a small amount of intrinsic conductivity, or that has been doped/ion-implanted such that it exhibits increased conductivity. In general, the resistivity of the lower buffer layer  1585  will be in the range from about 10 17 -10 19 Ω-cm @25° C. The lower buffer layer  1585  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  1585  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. 
         [0147]    According to one embodiment of the instant invention, the electro-optic device  1500  is fabricated by depositing the planarized lower buffer layer  1585  on the substrate  510 , etching the lower buffer layer  1585  (e.g., half way down) to provide slots for accommodating the lower split portions of the bias signal electrodes  1570 ,  1576 , depositing the bias electrode material, and etching the bias electrodes material to form bias electrodes  1570 ,  1572 ,  1574 ,  1576 . The bias electrodes  1570  and  1576  may be of nearly constant thickness, being more conformal to the shape of lower buffer layer  1585 . 
         [0148]    Advantageously, this fabrication method allows the lower portions of the bias electrode to be fabricated relatively consistently. For example, alignment errors in etching the bias electrode material will not result in unequal widths of the lower portions. 
         [0149]    In each of the embodiments described with regard to  FIGS. 5   a ,  6 ,  7 ,  8 ,  9 ,  10 ,  11 ,  12 ,  13 ,  14 , and  15 , the electro-optic device includes a bias electrode disposed substantially over a waveguide, with an intervening lower buffer layer. Notably, this is in contrast to the embodiments described in US Patent Application Publication No. 2006/0023288 and U.S. Pat. No. 7,127,128, wherein bias electrodes are shifted laterally with respect to the waveguides, and wherein the buffer layer below the bias signal electrodes are eliminated to reduce DC drift and improve bias electrode modulation efficiency. 
         [0150]    In each of the above-described embodiments, the lower buffer layer significantly reduces optical loss in the waveguides  526 ,  524  resulting from the overlying bias signal electrodes. For example in the absence of a lower buffer layer, a high resistivity bias electrode material, such as TaSiN, deposited directly over a waveguide in a Z-cut LiNbO 3  optical modulator will result in optical loss of about 1 to 2 dB/mm, which translates to 30 to 60 dB for a 30 mm electrode. With a 0.3 μm thick lower buffer layer, the optical loss drops substantially, becoming nearly negligible compared to other optical loss mechanisms in the device. 
         [0151]    In each of the above-described embodiments, the bias electrode modulation efficiency will be at least partially dependent on the thickness of each lower buffer layer and its conductivity. For example, a thinner lower buffer layer will provide increased modulation efficiency, but at the cost of increased optical loss. 
         [0152]    Improvement in the performance of the electro-optical devices described with regard to  FIGS. 5   a ,  6 ,  7 ,  8 ,  9 ,  10 ,  11 ,  12 ,  13 ,  14 , and  15  may be achieved by appropriate design of the bias electrode structure. 
         [0153]      FIGS. 16 ,  17   a,b,c,    18 , and  19  illustrate some possible bias electrode structure designs. For illustrative purposes the embodiments in  FIGS. 16 and 17   a,b,c  are described using the electro-optic device  500  discussed with reference to  FIG. 5   a , wherein the waveguides  524 ,  526  are patterned to form a Mach-Zehnder interferometer. Of course those skilled in the art, will be able to adapt these bias electrode structure designs to other embodiments of the instant invention. To simplify the drawings, the RF electrode structure  540 ,  542 ,  544  has been omitted from the plan views. 
         [0154]    Referring to  FIG. 16 , there is shown an embodiment wherein the bias electrode structure  570 ,  572 ,  574 ,  576  substantially shadows the RF electrode structure (not shown). For example, each of the bias signal electrodes  570 ,  576  includes an elongated lower split portion, which is capped with an elongated upper cap section, both of which run under the RF electrode structure for the length of the parallel sections of the interferometer arms (i.e., the interaction distance). The ground electrodes  572 ,  574  are provided with ground potential. The bias signal electrode  570  is coupled to a low frequency or DC power source that provides the bias voltage (e.g., +5V). Optionally, the other bias signal electrode  576  is also coupled to a low frequency or DC power source (not shown) that provides another bias voltage (e.g., −5V). In this embodiment, the sectional view along line A-A corresponds to the sectional view provided in  FIG. 5   a.    
         [0155]    Referring now to  FIG. 17   a , there is shown an embodiment wherein the bias ground electrodes  574 ,  572  and bias signal electrodes  570 ,  576  are segmented to suppress propagation of the RF signal along their length and reduce accumulated mechanical stress along their length. More specifically, the bias electrode material is patterned such that the segmented bias signal electrodes  570 ,  576  are coupled to outer bias signal electrodes  571 ,  577  with feed lines that extend between the bias ground electrode segments  574 ,  572 . In general, the outer bias signal electrodes  571 ,  577  are fabricated with a relatively low series resistance to compensate for the high series resistance introduced by the narrow segmented bias electrodes  570 ,  576 , respectively. In particular, the low series resistance provides the means for the bias signal voltage to be applied to the bias signal electrodes  570 ,  576  with minimal voltage drop. In one embodiment, the series resistance is reduced by providing wide outer bias signal electrodes  571 ,  577 . In another embodiment, the series resistance is reduced by providing a thin metal film  573 ,  579 , such as Cr, Ni/Cr (Nickel/Chromium), Ti/W and/or Au, on top of each outer bias signal electrode  571 ,  577 , respectively. 
         [0156]    As discussed above, the segmented bias signal electrodes  570 ,  576  are coupled to the outer bias electrodes  571 ,  577 , respectively, at multiple feed points. For illustrative purposes, 3 feed points per bias signal electrode are shown. In other embodiments, feed points are provided every 0.25 to 5 mm. The feed points may be at regular intervals or may vary over the length of the waveguides. For example, with regard to the latter, the spacing between feed points may vary incrementally (i.e., such that each successive spacing differs) or may vary step-wise (i.e., such that at least some successive spacings are the same). Varying the spacing allows the frequency response of the bias electrodes to be tailored. For example, a design with multiple spacings results in multiple time constants of arbitrary amplitude to define the time domain response of the bias electrode. The time domain response of the bias electrode could be tailored to compensate for other relaxation effects occurring in the bulk or near surface of the substrate. 
         [0157]    As illustrated in  FIG. 17b , which is a sectional view along line B-B, the feed lines coupling the segmented bias signal electrodes  570 ,  576  to the outer bias signal electrodes  571 ,  577 , respectively, are at the same level as the lower split portions. In this embodiment, the fabrication of the electro-optic device is similar to the process described above for fabricating electro-optic device  500 . 
         [0158]    As illustrated in  FIG. 17   c , which is a sectional view along line C-C, each segmented bias ground electrode  574 ,  572  is electrically coupled to the overlying RF ground electrodes  542 ,  544  through a via  575  (e.g., a Au via formed by etching the bleed and buffer layers). 
         [0159]    The RF ground electrodes  542 ,  544 , and thus bias ground electrodes  572 ,  574 , are provided with a ground potential. The outer bias electrode  571  is coupled to a low frequency or DC power source that provides the bias voltage (e.g., +5V). Optionally, the other outer bias signal electrode  577  is also coupled to a low frequency or DC power source (not shown) that provides another bias voltage having the opposite sign (e.g., −5V). 
         [0160]    Referring to  FIG. 18 , there is shown another embodiment having bias electrodes that are segmented to suppress propagation of the RF signal along their length and reduce accumulated mechanical stress along their length. In this embodiment, the bias electrode material is patterned such that each segmented bias signal electrode  1870 ,  1876  is coupled to an outer bias signal electrode  1877 ,  1871 , with feed lines that extend between segments of the other bias signal electrode  1876 ,  1870 , respectively. More specifically, feed lines couple the segmented bias signal electrodes  1870  for the first waveguide  1826  to outer bias electrode  1877 , and couple the segmented bias signal electrodes  1876  for the second waveguide  1824  to outer bias electrode  1871 . As a result, the outer bias electrode  1871  functions as the ground electrode for the segmented bias electrodes  1870 , whereas the outer bias electrode  1877  functions as the ground electrode for the segmented bias electrodes  1876 . The outer bias electrode  1871  is coupled to a low frequency or DC power source that provides a first bias voltage (e.g., −5V), while the other outer bias signal electrode  1877  is coupled to a low frequency or DC power source (not shown) that provides a second bias voltage (e.g., +5V). Advantageously, this push-pull arrangement maximizes the modulation efficiency of the segmented bias electrodes. 
         [0161]    In general, the outer bias signal electrodes  1871 ,  1877  are fabricated with a relatively low series resistance to compensate for the high series resistance introduced by the narrow segmented bias electrodes  1876 ,  1870 , respectively. In particular, the low series resistance provides the means for the bias signal voltages to be applied to the bias signal electrodes  1870 ,  1876  with minimal voltage drop. In one embodiment, the series resistance is reduced by providing wide outer bias signal electrodes  1871 ,  1877 . In another embodiment, the series resistance is reduced by providing a thin metal film  1873 ,  1879 , such as Cr, Ni/Cr, Ti/W and/or Au, on top of each outer bias signal electrode  1871 ,  1877 , respectively. 
         [0162]    As discussed above, the segmented bias signal electrodes  1870 ,  1876  are coupled to the outer bias electrodes  1877 ,  1871 , respectively, at multiple feed points. For illustrative purposes, 2-3 feed points per bias signal electrode are shown. In other embodiments, feed points are provided every 0.25 to 5 mm. The feed points may be at regular intervals or may vary over the length of the waveguides. For example, with regard to the latter, the spacing between feed points may vary incrementally (i.e., such that each successive spacing differs) or may vary step-wise (i.e., such that at least some successive spacings are the same). 
         [0163]    The segmented bias signal electrodes  1870 ,  1876  may have a substantially bracket-shaped, U-shaped, T-shaped, E-shaped, or other-shaped cross section. If the cross-section is substantially bracket-shaped, then the bias electrode material may be patterned such that the feed lines connecting the segmented bias signal electrodes  1870 ,  1876  to the outer bias signal electrodes  1877 ,  1871 , respectively, are at the same level as the lower split portions (e.g., analogous to fabrication method described for the embodiment illustrated in  FIGS. 17   a,b,c ). Alternatively, the bias electrode material may be patterned such that the feed lines connecting the segmented bias signal electrodes  1870 ,  1876  to the outer bias signal electrodes  1877 ,  1871 , respectively, are at the same level as at the cap sections. In this instance, the fabrication of the electro-optic device is similar to the process described above for fabricating electro-optic device  1500 . Further alternatively, the bias electrode material may be patterned such that the feed lines connecting the segmented bias signal electrodes  1870 ,  1876  to the outer bias signal electrodes  1877 ,  1871 , respectively, are part of an intermediate high resistivity layer that is disposed above the cap sections. In this instance, the segmented bias signal electrodes  1870 ,  1876  may be periodically connected to the intermediate high resistivity layer with high resistivity vias, as discussed in  FIGS. 12   a  and  15   a  of US Patent Application Publication No. 2006/0023288. Notably, the intermediate high resistivity layer is too far from the substrate to directly contribute to the electric field generated in the optical waveguides. 
         [0164]    Referring to  FIG. 19 , there is shown another embodiment having bias electrodes that are segmented to suppress propagation of the RF signal along their length and reduce accumulated mechanical stress along their length. In this embodiment, the bias electrode material is also patterned such that each segmented bias signal electrode  1970 ,  1976  is coupled to an outer bias signal electrode  1977 ,  1971 , with feed lines that extend between segments of the other bias signal electrode  1976 ,  1970 , respectively. More specifically, feed lines couple the segmented bias signal electrodes  1970  for the first waveguide  1926  to outer bias electrode  1977 , and couple the segmented bias signal electrodes  1976  for the second waveguide  1924  to outer bias electrode  1971 . As a result, the outer bias electrode  1971  functions as the ground electrode for the segmented bias electrodes  1970 , whereas the outer bias electrode  1977  functions as the ground electrode for the segmented bias electrodes  1976 . The outer bias electrode  1971  is coupled to a low frequency or DC power source that provides a first bias voltage (e.g., −5V), whereas the other outer bias signal electrode  1977  is coupled to a low frequency or DC power source that provides a second bias voltage (e.g., +5V). Advantageously, this push-pull arrangement maximizes the modulation efficiency of the segmented bias electrodes. 
         [0165]    In general, the outer bias signal electrodes  1971 ,  1977  are fabricated with a relatively low series resistance to compensate for the high series resistance introduced by the narrow segmented bias electrodes  1976 ,  1970 , respectively. In particular, the low series resistance provides the means for the bias signal voltages to be applied to the bias signal electrodes  1970 ,  1976  with minimal voltage drop. In one embodiment, the series resistance is reduced by providing wide outer bias signal electrodes  1971 ,  1977 . In another embodiment, the series resistance is reduced by providing a thin metal film  1973 ,  1979 , such as Cr, Ni/Cr, Ti/W and/or Au, on top of each outer bias signal electrode  1971 ,  1977 , respectively. 
         [0166]    As discussed above, the segmented bias signal electrodes  1970 ,  1976  are coupled to the outer bias electrodes  1977 ,  1971 , respectively, at multiple feed points. For illustrative purposes, 2 feed points per bias signal electrode are shown. In other embodiments, feed points are provided every 0.25 to 5 mm. The feed points may be at regular intervals or may vary over the length of the waveguides. For example, with regard to the latter, the spacing between feed points may vary incrementally (i.e., such that each successive spacing differs) or may vary step-wise (i.e., such that at least some successive spacings are the same). 
         [0167]    Each segment of each bias signal electrode  1970 ,  1976  includes a split bias electrode. Advantageously, the design of bias electrodes  1970 ,  1976  is selected to provide a tailored frequency response. For example, in this embodiment the segments on the right hand side of  FIG. 19  are longer and have a wider gap between the split bias electrodes, whereas the segments on the left-hand side of  FIG. 19  are shorter and have a narrower gap between the split bias electrodes. The longer bias signal electrode segments have a larger series resistance and nearly the same capacitance, hence the RC time constant for these bias electrode segments will be much longer. The gap between the split electrodes is wider for the longer electrode segments, reducing modulation efficiency per unit length, but leaving capacitance per unit length about the same. The capacitance changes very little as a function of the gap in the split electrode whereas modulation efficiency as given by V π L changes dramatically. Hence, the modulation efficiency of the segments with a longer time constant can be adjusted independently of the modulation efficiency of the segments with shorter time constant. Optionally, three or more groups of bias signal electrode segments with different RC time constants and V π L, allowing for a wide variety of tailored frequency response. In one embodiment, the bias electrode structure is designed to compensate for other effects that influence bias frequency response, creating a modulation efficiency that changes very little as frequency changes. For example, short term conductivity effects in the substrate or at the substrate surface may influence the bias electrode frequency response. The use of multiple tailored time constants in the bias electrode response may compensate for these effects, leading to a flat frequency response. A flatter bias frequency response improves performance of bias control circuits. A bias electrode with flat low frequency response is also useful for any applications where the bias or other modulation signal must be slowly swept or varied over time in a predictable manner. 
         [0168]    Advantageously, the electro-optic devices described with regard to  FIGS. 5   a ,  6 ,  7 ,  8 ,  9 ,  10 ,  11 ,  12 ,  13 ,  14 ,  15 ,  16 ,  17   a,b,c,    18 , and  19  are humidity tolerant. In particular, humidity tolerance is provided by burying the bias electrodes below the upper buffer layer so that they are protected from humidity, and so that electro-migration corrosion of the buried bias electrodes is reduced. Optionally, humidity tolerance is improved by eliminating the RF electrode adhesion layer, by encapsulating the RF electrode adhesion layer, and/or by using an RF electrode adhesion layer having a work function similar to the material used to form the RF electrodes. Further optionally, humidity tolerance is improved by allowing the bias electrodes to be DC isolated from the RF electrodes  540 ,  542 ,  544 . 
         [0169]    According to one embodiment, DC isolation is provided by passing the signal from an RF generator through a low pass filter onto the bias signal electrode (e.g.,  570 ), and through a high pass filter onto the RF signal electrode (e.g.,  540 ). Advantageously, this arrangement boosts the high end frequency response to the incoming bias signal, accommodating dither signals or other tones in the MHz frequency range, that are often summed in with the slowly varying bias voltage. According to another embodiment, a bias-tee is used to couple the bias signal electrode (e.g.,  570 ) and the RF signal electrode (e.g.,  540 ). Of course, various other bias control circuits are also envisioned. 
         [0170]    In each of the above-described embodiments, the electro-optic device includes an upper buffer layer and a lower buffer layer. While the upper and lower buffer layers may be formed from the same material (e.g., doped SiO 2 ) or different materials, performance may improve when the upper and lower buffer layers have different conductivities. More specifically, performance is expected to improve when the lower buffer layer is more conductive than the upper buffer layer, and particularly when the lower buffer layer has a conductivity that is substantially the same or greater than the conductivity of the substrate. As discussed above, a relatively high conductivity may be provided with a doped SiO 2  buffer layer. Some examples of suitable dopants include Ti, In, Sn, Al, Cr and Zn. Doping of SiO 2  buffer layers to increase conductivity is well known in the art, and is discussed in further detail in U.S. Pat. Nos. 5,404,412 and 5,680,497. 
         [0171]    Notably, an electro-optic device having a stacked design where an upper non-doped buffer layer separates the RF electrodes from the bias electrodes and a lower doped buffer layer at least partially separates the bias electrodes from the substrate, provides a number of advantages compared to an electro-optic device having a single doped buffer layer separating common bias/RF electrodes from the substrate. For example, in the former, the thickness of the doped buffer layer is selected to reduce optical loss from the bias electrodes rather than to provide velocity matching. Accordingly, the doped buffer layer is relatively thin and is likely to affect the required drive (bias) voltage. Also, in the former, the doped buffer layer is protected from the atmosphere by the non-doped buffer layer. Accordingly, impurities (e.g., water) are less likely to be absorbed therein. In addition, since the bias electrodes are protected from the atmosphere by the non-doped buffer layer, humidity tolerance is improved. 
         [0172]    In addition, an electro-optic device having a stacked design where an upper non-doped buffer layer separates the RF electrodes from the bias electrodes and a lower doped buffer layer at least partially separates the bias electrodes from the substrate, provides a number of advantages compared to an electro-optic device having only a non-conductive buffer layer separating the RF electrodes from the bias electrodes. In particular, the lower doped buffer layer substantially reduces optical loss. For example, in the embodiments illustrated in  FIGS. 13 and 14 , optical loss is reduced sufficiently to enable the use of non-split bias signal electrodes with Z-cut LiNbO 3 . 
         [0173]    While the stacked designs discussed above are particularly valuable for electro-optic devices based on Z-cut LiNbO 3  or Z-cut LiTaO 3 , where the bias electrodes are generally disposed above the waveguides, they are also envisaged for use with other substrates and/or where the bias electrodes are not disposed directly above the waveguides. For example, a lower buffer layer may be included in the electro-optic devices described with regard to  FIGS. 3   a  and  4   a  as illustrated in  FIGS. 20 and 21 , respectively. In these embodiments, the lower buffer layers  385 ,  485  are planarized and have a conductivity that is higher than the upper buffer layers  350 ,  450 , respectively. In general, some examples of suitable electro-optic substrates include X-cut LiNbO 3 , X-cut LiTaO 3 , Y-cut LiNbO 3 , Y-cut LiTaO 3 , gallium arsenide (GaAs), and indium phosphide (InP). Alternatively, the substrate may be a polymer substrate having an electro-optic polymer waveguide. 
         [0174]    As shown in  FIG. 22 , the stacked design may also be used in electro-optic devices where the substrate has been etched to form ridges around the waveguides. The electro-optic device  2200  includes a substrate  510 , etched to form slots  2290 ,  2291 , and  2292 , first  524  and second  526  optical waveguides, an RF signal electrode  540 , RF ground electrodes  542 ,  544 , an upper buffer layer  550 , a bleed layer  560 , first  2270  and second  2276  bias signal electrodes, bias ground electrodes  2272 ,  2274 , and a lower buffer layer  2285 . 
         [0175]    The substrate  510 , first  524  and second  526  optical waveguides, RF signal electrode  540 , RF ground electrodes  542 ,  544 , upper buffer layer  550 , and bleed layer  560 , are as described above. 
         [0176]    The first bias signal electrode  2270 , the second bias signal electrode  2276 , the first bias ground electrode  2274 , and the second bias ground electrode  2272 , are all supported by the substrate  510 . The bias electrodes  2270 ,  2272 ,  2274 ,  2276  are part of the bias electrode structure used to apply a low-frequency or DC voltage across the optical waveguides. Each of the first  2270  and second  2276  bias signal electrodes are on top of lower buffer layer  2285 . 
         [0177]    Each of the bias electrodes  2270 ,  2272 ,  2274 ,  2276  is typically formed from a high-resistivity material, such as tantalum silicon nitride (TaSiN), amorphous silicon (Si), indium oxide (In 2 O 3 ), tin oxide (SnO 2 ), zinc oxide (ZnO), etc. Using a high resistivity material allows the bias electrodes  2270 ,  2272 ,  2274 ,  2276  to be conductive at low frequencies and to function as a dielectric at high-frequencies. Accordingly, the bias electrodes are effectively transparent to the electric field generated by the RF electrodes. Suitable resistivity values for the bias electrode material lie between that of the RF electrode material and that of the substrate. For example, TaSiN typically has a resistivity in the range of about 10 4  to 10 6   6 -cm @25° C., which is between ˜2.3×10 −6 Ω-cm @25° C. (Au) and ˜1.3×10 17 Ω-cm @25° C. (LiNbO 3 ). Preferably, the resistivity of the bias electrode material is in the range from about 1 to 10 8  ohm-cm (Ω-cm) @25° C., more preferably from about 10 2  to 10 7  ohm-cm (Ω-cm) @25° C., and most preferably from about 10 4  to  10   6  ohm-cm (Ω-cm) @25° C. The preferred range is determined by the fact that lower resistivity materials typically provide faster response times, whereas higher resistivity materials result in reduced optical loss due to the proximity of the bias electrode to the waveguide(s). The use of a higher resistivity bias electrode material may also reduce coupling with the RF signal, thus decreasing the net RF loss per unit length of the RF signal electrode. Each of the bias electrodes  2270 ,  2272 ,  2274 ,  2276  may be formed from the material used to form the bleed layer  560 , or a different material. Conveniently, when bias electrodes  2270 ,  2272 ,  2274 ,  2276  and the bleed layer are fabricated from the same material (e.g., TaSiN), the fabrication process is relatively simple. The bias electrodes  2270 ,  2272 ,  2274 ,  2276  may be fabricated using one of various well-known methods, including deposition and sputtering. The thickness of the bias electrodes  2270 ,  2276  is typically in the range between about 0.05 and 0.5 μm, and more commonly between about 0.05 and 0.25 μm. 
         [0178]    The lower buffer layer  2285  provides a spacer between the optical waveguides  526 ,  524  and the overlying bias signal electrodes  2270 ,  2276 , respectively. Accordingly, the lower buffer layer  2285  is typically fabricated with a material that is optically transparent in the wavelength of interest (e.g., 1.55 microns). Since the bias signal electrodes  2270 ,  2276  are not in direct contact with the substrate  510 , it is preferred that the lower buffer layer  2285  be fabricated from a material having at least some conductivity. For example, in one embodiment the lower buffer layer  2285  is fabricated with a dielectric material, such as silicon dioxide (SiO 2 ), which has been sputtered such that it exhibits a small amount of intrinsic conductivity, or that has been doped/ion-implanted such that it exhibits increased conductivity. In general, the resistivity of the lower buffer layer  2285  will be in the range from about 10 17 -10 19  Ω-cm @25° C. The lower buffer layer  2285  is typically fabricated using one of various well-known methods, such vacuum deposition, ion-assist vacuum deposition, sputtering, or chemical vapor deposition (CVD). The thickness of the lower buffer layer  2285  is typically in the range between about 0.05 and 1 μm, and more commonly between about 0.1 and 0.5 μm. 
         [0179]    According to one embodiment, the electro-optic device  2200  is fabricated by first etching slots  2290 ,  2291 , and  2292  into a Z-cut lithium niobate substrate, using a process such as Reactive Ion Etching or Ion Milling. The etch depth is typically between 1 to 10 μm. The lower buffer layer  2285  is then deposited on the etched substrate  510 , conformal to the slots. The lower buffer layer is annealed, if necessary. The bias electrode material is then deposited, and etched to form bias electrodes  2270 ,  2272 ,  2274 ,  2276 . Upper buffer layer  550 , bleed layer  560 , and RF electrodes  540 ,  542 ,  544  are fabricated as described above. Upper buffer layer  550  may be fabricated of the same material as the lower buffer layer  2285 , or more likely of a material having higher resistivity, such as undoped silicon dioxide (SiO 2 ). 
         [0180]    Of course, the embodiments of the invention described above have been presented by way of example only. It will be understood by those skilled in the art that various omissions and substitutions may be made without departing from the spirit and scope of the invention as defined by the appended claims.