Patent Publication Number: US-7899277-B2

Title: Integrated on-chip inductors and capacitors for improved performance of an optical modulator

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present invention claims priority from U.S. Provisional Patent Application No. 61/056,743 filed May 28, 2008, which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention is directed to modulators, and more particularly, to integrated electro-optical modulators with velocity mismatched traveling wave electrodes. 
     BACKGROUND OF THE INVENTION 
     Optical-fiber communication systems may include a transmitter with, for example, a high power, low noise laser source whose optical output is modulated with a wide bandwidth external modulator. Integrated electro-optical modulators are finding increasing use to reduce component physical size while increasing performance. To fabricate an integrated electro-optical modulator, a common substrate material for both the laser and modulator is generally required. Laser sources have been typically fabricated on compound semiconductor materials, to which the optical modulator needs to be adapted. 
     Electro-optical modulators are based on the principle of modifying the complex optical refractive index in an optical waveguide by the application of an electric field. For this purpose a time-varying electric modulating input signal is transmitted to electrodes located in close proximity to or in contact with an optical waveguide to create an electric field in the optical waveguide. When a constant optical power level is coupled into one end of such a waveguide, the properties of an optical signal output from the other end of the optical waveguide can be modified in response to the modulating input signal applied to the electrodes. 
     In the case of a dual drive Mach-Zehnder modulator, differential modulating signals are applied to the Mach-Zehnder arms to modify the phase of optical signals propagating through the respective waveguides. When the outputs of the two waveguides are coherently combined in an optical coupling region, the resulting interference produces a change in amplitude of the combined optical signal. 
     Electro-absorption (EA) modulators typically have a single optical waveguide, in which an electric field applied by the modulating signal modifies optical absorption to produce a change in the amplitude of the optical output signal. 
     Typically, the optical waveguides of a modulator are fabricated as ridges on a conducting semiconductor substrate. The ridges may comprise a p-n junction with one electrode connection made through the conducting semiconductor substrate, while the other connection is made through an electrode formed on top of the ridge so that the p-n junction can be biased to a desired level and polarity. An electric potential applied between the electrodes controls the electric field strength as well as carrier density within the p-n junction, which in turn produce changes in the refractive index and absorption constant within the optical waveguide. 
     In a conventional optical modulator, the factors that determine the performance of the modulator are ridge or junction capacitance, electrode capacitance, electrode inductance, length of the modulator and wire-bond pad capacitance. The equivalent circuit model of a conventional modulator is shown in  FIG. 1 , where C pad  is the pad capacitance, L e  is the electrode inductance, R a  is the series resistance of the electrode, C j  is the ridge or junction capacitance, and R c  is the sum of diode forward and contact resistance. The substrate losses are represented by the loss conductance G, corresponding to a loss resistance R g =1/G. All modulator parameters are functions of the length of the modulator with the exception of the bond pad capacitance. For the circuit diagram shown in  FIG. 1 , the modulator total capacitance, inductance, resistance and conductance are represented by several of circuit elements. This is usually termed a lumped circuit representation of the modulator. In a distributed circuit model of the modulator, the circuit shown in  FIG. 1 , is repeated multiple times (excluding C pad ), so that the total capacitance, inductance, resistance and conductance are distributed and represented by multiple circuit elements. In other words, the circuit in  FIG. 1  (excluding the bond pad capacitance C pad  at each end of the component chain) can be considered as a unit cell, which is repeated multiple times in order to represent an equivalent circuit of the modulator. 
     There are three figures of merit which depend on the modulator parameters mentioned above, which will now be briefly explained: 
     a) The electro-optic (EO) bandwidth; 
     b) The required radio-frequency (RF) drive level; and 
     c) The return loss, s 11 . 
     The electro-optic (EO) bandwidth increases with lower total shunt capacitance, which is the sum of C j  and C pad . The ridge capacitance depends on the electrode length, with a longer modulator resulting in lower EO bandwidth. The required radio-frequency (RF) drive level varies inversely with the length of the modulator, with a longer length requiring a lower drive voltage. To achieve lower RF drive voltage, a longer modulator is preferred but the increased total capacitance will decrease the EO bandwidth. Thus a trade-off is needed between bandwidth and the RF drive level requirements. The return loss is very frequency dependent, as it depends on the pad capacitance, electrode inductance, series resistance and total ridge capacitance. In general, higher pad and ridge capacitance result in higher return loss while higher modulator electrode inductance and resistance reduce return loss (assuming modulator ridge has lower impedance than the electrical driving source). 
     For modulation rates extending to tens of gigabits/second and beyond, the propagation velocity of an optical wave in the optical waveguide is no longer negligible. As the electro-optic interaction length (approximately the length of the modulator signal electrode) becomes comparable to the electrical wavelength at the higher modulation rates, electrical distributed effects become pronounced, so matching of the optical and modulating electrical signal velocities has to be considered. An increase in velocity mismatch tends to reduce the bandwidth of the optical output signal. The electrodes used for applying an electric field to the optical waveguide are designed as electrical transmission lines or micro strip-lines along which the applied electrical modulating signal can propagate as a traveling wave. This is known as a “traveling wave” electrode structure. 
     In prior art the optical waveguide and the electrical transmission line is typically designed such that the group velocity of the optical wave is matched to the phase velocity of the electrical traveling wave. Velocity matching can be achieved by appropriate optical waveguide dimensioning, micro strip-line dimensioning, addition of reactive electrical components such as inductances and capacitances, or a combination of all three. 
     The microwave group index and optical group index are important concepts in velocity matching. The more general definitions are: 
     
       
         
           
             
               
                 
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     where c is the velocity of light in vacuum and v is the optical group velocity or velocity of microwave signal in a medium. More specifically, 
     
       
         
           
             
               
                 
                   
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     where n microwave  is the microwave index and v microwave  is the velocity of an electrical microwave signal traversing the modulator electrodes. 
     
       
         
           
             
               
                 
                   
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     where n optical     —     group  is the optical group index and v optical     —     group  is the group velocity of an optical wave in the waveguide. 
     For instance, Tanbakuchi (U.S. Pat. No. 7,031,558) discloses an electro-absorption modulator shown as a ridge  50  with a signal electrode  44  in  FIG. 2(   a ). Bond pads  18 ,  22 , are used to connect the EA modulator to a signal source and a termination, respectively. The signal electrode  44  is connected to bond pads  18 ,  22  by microstrip lines  20  and  24 , respectively, which act as inductors, forming matching networks in conjunction with the bond pad capacitances and the bond wires to match the electrical impedance of the modulator ridge  50  to the signal source and a termination, respectively. With reference to  FIG. 2(   b ), the cross-section shows microstrip lines  20  and  24 , disposed on an insulating layer  54  using a ground plane provided by electrically conducting layer  55 . 
     The optical and electrical signal velocities are matched, as can be concluded from the phase of scattering parameter S 21  of the simulated structure (Tanbakuchi&#39;s FIGS. 7C, 6B). 
     Tanbakuchi&#39;s FIG. 7C shows that the electrical phase of an electrical signal propagating from the input to the output of modulator shown in  FIG. 6B  is linear with frequency. With linear phase, the microwave index, n microwave , can easily be estimated by choosing an arbitrary point on  FIG. 7C , and applying Equation 2: 
     
       
         
           
             
               
                 
                   
                     
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     Substituting L=0.850 mm (from Tanbakuchi&#39;s Table 2) and choosing f=48 GHz at which φ=π (or 180°), the value n microwave =3.68 can be derived for the microwave index. 
     The optical group index for group III-V compound semiconductors, in particular InGaAsP, generally lies in the approximate range 3.7-3.9, so the electrical and optical indexes are apparently very well matched. 
     A variation of the embodiment above is shown in  FIG. 2(   c ), in which the signal electrode is segmented into sections  52  joined in series by microstrip lines  55  on the ridge  50 . However, the amount of series inductance, which such microstrip lines can provide, is limited by the space on top of the ridge  50  as well as the impedance per length of the microstrip lines  55 . 
     Skeie (U.S. Pat. No. 5,675,673) proposes a solution that places a series of inductors  531 - 534  on a separate circuit board  501  as can be seen in  FIG. 2(   d ). While this provides more latitude in choosing inductance values, the separate circuit board requires additional space as well as making more electrical connections necessary to connect to the modulator. Generally this requires more bond wires, whose electrical characteristics, in particular inductance values, are difficult to control, as well as additional bond pads, each of which contribute additional shunt capacitance. 
     The microwave index of Tanbakuchi-like designs would tend to be in the range of 2.9-3.8, indicating that there is velocity matching between the optical and electrical waves, hence velocity mismatch would not limit the bandwidth in practice. In other words, other factors such as modulator diode capacitance, electrode metal and substrate losses contribute to the predicted bandwidth. 
     Furthermore, with velocity-matched and similar designs, however, additional filters are often required for increasing the roll-off characteristics of the signal channel bandwidth to avoid interference and cross-talk with neighboring channels. Such additional filters can increase losses, reduce efficiency and contribute to increased size and complexity of the device. It would be advantageous to incorporate the filter function in the integrated optical modulator. 
     In this disclosure, the above problems are addressed by integrating a network of inductors on the same chip as the modulator, thereby avoiding an increase in device size. Performance of the modulator is further enhanced by departure from velocity-matched conditions in a controlled manner in order to take advantage of the inherent filter characteristics under such conditions. 
     SUMMARY OF THE INVENTION 
     An optical modulator is disclosed comprising a substrate for supporting electrical components, a conducting layer on a first region of the substrate for providing an electrical ground plane, an insulating layer on a second region of the substrate for accommodating an electrical circuit, an electro-optic optical waveguide disposed on the conducting layer for receiving an input optical signal and transmitting an output optical signal. A traveling wave microstrip electrode with two or more contact segments is in contact with an upper portion of the optical waveguide for modulating the input optical signal in response to an input modulating signal applied to the microstrip electrode. Source and termination bond pads are placed on the insulating layer for receiving the input modulating signal from a signal source and transmitting a residual modulating signal to a termination circuit, respectively. 
     A network of reactive electrical components on the insulating layer electrically interconnects the source bond pad, the termination bond pad and the two more contact segments for modifying a microwave signal velocity in the microstrip electrode. 
     A method is also disclosed for limiting a spectral bandwidth of an optical signal with an optical modulator chip comprising a segmented microstrip electrode, an optical waveguide, and an electrical network of reactive components interconnecting electrode segments of the segmented microstrip electrode. The method includes providing an optical input from a lightsource, applying a microwave signal to the segmented microstrip electrode, modulating the optical input with the segmented microstrip electrode to generate an optical signal having an optical signal velocity in the optical waveguide, and modifying a microwave signal velocity in the segmented microstrip electrode with the network of reactive components to produce a mismatch with the optical signal velocity and the microwave signal velocity. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention will be described in greater detail with reference to the accompanying drawings which represent preferred embodiments thereof, wherein: 
         FIG. 1  is an equivalent circuit of a typical prior art modulator using a lumped element circuit representation, which is also a unit cell of a distributed circuit representation, where it is repeated multiple times (excluding the two C pad  at each end); 
         FIG. 2(   a ) is a plan view of a prior art modulator design; 
         FIG. 2(   b ) is a cross-section of the prior art modulator in  FIG. 2(   a ); 
         FIG. 2(   c ) is a plan view of a prior art modulator design with segmented electrodes; 
         FIG. 2(   d ) is a plan view schematic of a prior art modulator design with segmented electrodes and inductors placed on a separate circuit board; 
         FIGS. 3(   a ) and  3 ( b ) are plan views of modulators with on-chip inductors placed in two different configurations to improve modulator performance according to this invention; 
         FIGS. 3(   c ) and  3 ( d ) are cross-sectional views of the modulator in  FIG. 3(   b ) along sections A-A′ and B-B′, respectively; 
         FIG. 4  is a graph of return loss S 11  measurements for 600 μm long modulators with no added on-chip inductance. The vertical axis is S 11  [dB] and the horizontal axis is frequency [GHz]. The two traces represent different modulators of identical design; 
         FIG. 5  is a graph of return loss measurements (vertical axis s 11  [dB], horizontal axis frequency [GHz]) of an 800 μm long modulator with three different amounts of on-chip inductance added; 
         FIG. 6  is a graph of electrical forward loss (vertical axis s 21  [dB], horizontal axis frequency [GHz]) of 600 μm long modulators with no added on-chip inductance. Measurements are taken with respect to a 50Ω reference impedance. The two traces represent different modulators of identical design; 
         FIG. 7  is a graph of electrical forward loss (vertical axis s 21  [dB], horizontal axis frequency [GHz]) of 800 μm long modulators with different amount of inductance added. Vertical axis is S 21  in dB and the horizontal scale is frequency in GHz; 
         FIGS. 8(   a ) and  8 ( b ) are graphs of simulated and measured electro-optic responses (vertical axis s 21  [dB], horizontal axis frequency [GHz]), respectively, of an 800 μm long modulator, illustrating the use of on-chip inductors to introduce velocity mismatch between the traveling electrical and optical signals to produce a sharper roll-off filter; 
         FIG. 9  is a graph showing the simulated sensitivity [dB] on the vertical scale as a function of transmission distance [km] on the horizontal scale; 
         FIG. 10  is a graph showing return loss measurements (vertical axis s 11  [dB], horizontal axis frequency [GHz]) for a 700 μm long modulator taken at three different bias voltages; 
         FIG. 11  is a graph of the measured return loss (vertical axis s 11  [dB], horizontal axis frequency [GHz]) of an integrated laser-Mach Zehnder (ILMZ) 750 μm long device attached on identical chip-on-carrier for three different voltages; 
     
    
    
     DETAILED DESCRIPTION 
     In optical modulator design, high electro-optic (EO) bandwidth, good return loss (S 11 ) across the operating frequency range and low radio frequency (RF) drive voltage are figures of merit when comparing optical modulator performance. Simultaneous optimization of the EO bandwidth, S 11  and RF drive usually involves trade-off between these 3 figures of merit. 
     At the same time, designs with smaller physical size and fewer parts are preferred, which motivates toward integrating as many components as possible onto a common substrate. 
     Optical modulators according to the invention described here improve the three figures of merit while maintaining a compact physical size through two main features:
         a) integrated on-chip inductors (or capacitors), placed at strategic locations along the length of the optical modulator, to improve the EO bandwidth, return loss s 11  and required voltage drive level; and   b) the use of on-chip inductors (or capacitors) to create a mismatch between the velocity of the optical and electrical signals propagating along the waveguide and traveling wave electrode, respectively. The velocity mismatch results in a bandwidth filtering of the EO frequency response. The filtering has two components of interest: a 50 dB/decade filter roll-off component, representing a 30 dB improvement when compared to the 20 dB/decade typically achieved in a typical modulator resistance-capacitance (RC) roll-off, and a bandwidth limiting component.       

     The electrode design methodology disclosed herein is universal and can be applied to all modulator designs such as semiconductor based Mach-Zehnder (MZ) modulators and electro-absorption (EA) modulators, as well as lithium niobate (LiNBO 3 ) modulators. 
     A first embodiment comprising on-chip inductors to improve modulator performance is shown in plan view in  FIG. 3(   a ). A conducting layer  301 , typically n-doped, on modulator chip  300  extends to one side from under a ridge waveguide  303  for providing a connection from an n-side of a p-n junction ( 310  in  FIG. 3(   c )) within the ridge waveguide  303  to a metal ground electrode (not shown). An insulating layer  302  extends over the rest of the modulator chip  300  up to the other side of the ridge waveguide  303 . The insulating layer  302  may consist of layers of semi-insulating semiconductor materials, or layers of dielectric material, or a combinations of both. The conducting layer  301  may also extend partly or completely underneath insulating layer  302 . 
     Conducting electrode segments  304   a - d  are disposed on top of the ridge waveguide  303  for providing a signal electrode connection to the p-n junction ( 310  in  FIG. 3(   c )) within the ridge waveguide  303 . The length and number of the signal electrode segments  304   a - d  are chosen to achieve a prescribed electric field distribution longitudinally along the ridge waveguide  303 . Four segments are shown in this embodiment by way of example, but the number can vary from 2 to more than 6. A typical total modulator length lies in the range 700 μm-900 μm. The number of signal electrode segments  304   a - d  depends primarily on the total length of the modulator chip  300 . Typically a segment length should be between 200 μm and 350 μm. 
     Two bond pads  306   a ,  306   b , preferably deposited on the insulating layer  302  are connected to electrode end segments  304   a ,  304   d  to provide external connection to a microwave signal source and a resistive termination, respectively (both not shown). The four electrode segments  304   a - d  are connected in series through inductors  305 ′ a - c , which are also located on the insulating layer  302 . The inductance values of the inductors are chosen according to the required impedance matching with the microwave signal source and the resistive termination, as well as to adjust the microwave signal velocity propagating along the electrode segments  304   a - d  with respect to the optical signal velocity in the ridge waveguide  303 . For structures in which only a single electrode segment is desired, each end of such a single electrode segment would be connected through an inductor to the bond pads  306   a ,  306   b , respectively. 
       FIG. 3(   b ) is a plan view of a second embodiment of modulator chip  300  with a different configuration of inductors  305   a - e  and their interconnection with signal electrode segments  304   a - d . The conducting layer  301 , insulating layer  302 , ridge waveguide  303  and bond pads  306   a,b  are the same as in the first embodiment. Again, the length and number of the signal electrode segments  304   a - d  can be suitably chosen for a particular application. Only four segments are shown in this example. The inductors  305   a  and  305   e  connect to bond pads  306   a,b  while the inductors  305   b - d  are connected between the segments  304   a - d  via taps  312   a - d , respectively, which are preferably located at the center of each segment  304   a - d.    
     Each of the inductors  305   a - e  forming a network, shown in this example as a series connection between the bond pads  306   a,b , comprises meandering metal traces, but can assume other path shapes such as spirals, squares or polygons in general comprising corners or curves having various radii of curvature. The various curves and shapes can be employed for increasing the inductance of the inductors  305   a - e  within a given confined space on the modulator chip  300 . The inductance value of the inductors  305   a - e , as well as the position of the taps  312   a - d  is chosen, as before, according to impedance matching and microwave velocity adjustment requirements. The segments and inductors may be of different values. 
     Each signal electrode segment  304   a - d  still interacts with the additional series inductors  305   a - e  similar to those in the  FIG. 3(   a ), but in this case, there is an advantage of reducing the space required, as the space between the electrode segments  304   a - d  can now be made much smaller. For the same interaction length of the electrode, the modulator length can thus be reduced, resulting in less optical insertion loss. 
       FIG. 3(   c ) is a cross-section of the second embodiment along section A-A′ in  FIG. 3(   b ). As in  FIG. 3(   b ), the modulator chip  300  comprises the substrate  330  with the conducting layer  301 , preferably of semiconductor material, extending over a part or whole of the surface area of the substrate  330 . A modulator diode  310  comprising a p-n junction within an optical ridge waveguide  303  is electrically connected to the conducting layer  301  and a traveling wave electrode  311  on top of the optical ridge waveguide  303 . The modulator diode  310  layer structure may contain an intrinsic layer, single or multiple quantum wells as well as a p-type and n-type semiconductor region designed according to the required electro-optic characteristics for the modulator function. 
     The traveling wave electrode  311  is connected by means of the taps  312   a - d  or other suitable bridge contact structure to the inductors  305   a - e  situated on the insulating layer  302 . 
     The substrate  330  is preferably insulating or semi-insulating in order for the inductors  305   a - e  to have minimal capacitance to the conducting layer  301 , which is typically at ground potential. In effect, the achievable inductance value per substrate area can thus be maximized. On the other hand, if the substrate  330  is of conducting type, it can function together with the conducting layer  301  as a ground reference of the modulator. In that case the thickness of the insulating layer  302  would be increased accordingly to provide the necessary separation between the inductors  305   a - e  and substrate  330 , which would function as a ground plane. 
       FIG. 3(   d ) is a cross-section of the second embodiment along section B-B′ in  FIG. 3(   b ), which is at a point on the optical ridge waveguide  303  where there is a break between the signal electrode segments  304   a - d  shown in  FIG. 3(   b ). All elements have the same reference number as in  FIG. 3(   c ). In this view the inductors  305   a - e  are seen to extend laterally away from the conducting layer  301 , thereby reducing their capacitance. 
     To illustrate the basic working principle of the disclosed modulator structure, consider an InP base optical modulator, as shown in  FIG. 3(   a ), without the bond pads  306   a,b . For simplicity, the on-chip inductors  305 ′ a - c  are connected in series with the optical modulator signal electrode segments  304   a - d  at regular intervals. In other words, for a given length of modulator, an additional inductance L u  nH is added such the total inductance is (L u +L e ). The four sections of the modulator signal electrode segments  304   a - d  can be represented by the equivalent circuit diagram shown in  FIG. 1 , but excluding the pad capacitance C pad . 
     The influence on the performance of the modulator, as a result of the addition of the on-chip inductors  305 ′ a - c , can be understood by considering the circuit model of one section of a modulator with on-chip inductors connected to its ends. At high frequency, the impedance of the modulator alone can be approximated as 
     
       
         
           
             
               
                 
                   
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     where L e  is the ridge metal electrode inductance and C j  is the ridge capacitance. In a modulator ridge structure without additional series inductors, a typical value of input impedance Z in  is in the range of 25-30Ω. Using a 50Ω reference system, the return loss is about −10 dB. Including the additional on-chip inductance increases the input impedance of the modulator to 
     
       
         
           
             
               
                 
                   
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     where C u  is the total parasitic capacitance and L u  is the added inductance of the on-chip integrated inductor connected to the modulator. Using the modulator structure as described in  FIG. 3(   a ) or  FIG. 3(   b ), the modulator input impedance Z in  can be increased to between 34Ω and 40Ω. This improves the return loss S 11  to less than about −17 dB. 
     The EO bandwidth is inversely proportional to the propagation loss, α, of an electrical signal propagating along the modulator traveling wave electrode: 
     
       
         
           
             
               
                 
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     Thus, it can be seen that with the increase in modulator input impedance Z in , the propagation loss α of the electrical signal along the modulator is decreased. With the decrease in the propagation loss, the EO bandwidth of the modulator, which is proportional to the average signal along the electrode length, is increased for a given length of the modulator. In addition to that, with improved return loss S 11 , more voltage is transferred from the signal source to the modulator. These two factors have the effect of increasing the EO bandwidth. 
     The increase in EO bandwidth can then be traded to decrease the modulator RF drive voltage by increasing the length of the modulator. The RF drive voltage required is decreased by the increase in the electro-optic interaction length of the modulator. 
     InP based electro-absorption modulators fabricated according to  FIG. 3(   b ) with different amounts of on-chip inductors  305   a - e  will be used to illustrate another embodiment of this invention. Although InGaAsP on InP devices are described here, the same design principles may be applied to modulators fabricated in other semiconductor material systems such as those based on GaAs. The modulators described below are suitable for 10 Gbit/s Return-to-Zero (RZ), Non-Return-to-Zero (NRZ), or 40 Gbit/s Optical Duo-Binary (ODB) transmitter applications. These applications require electro-optical bandwidth around 10-12 GHz. RF drive voltage limits and phase versus voltage efficiency for the material used in this embodiment require an electro-optic interaction length of about 800 μm. Phase voltage efficiency (for a particular reference modulator length and optical wavelength) is understood here as the phase change (radians or degrees) of the optical signal for a given applied modulator voltage. The modulator signal electrode is divided into four segments  304   a - 304   d , all 200 μm in length for this example, giving a total modulator length of 800 μm (neglecting separations between the signal electrode segments). 
     The range of values for ridge capacitance, C j , for each of the modulator signal electrode segments  304   a - d  can be computed from the ridge capacitance per unit length of the modulator. Measured ridge capacitance per unit length for a preferred design is 0.8 pF/mm to 0.95 pF/mm, although this can vary from about 0.5 pF/mm to 1.4 pF/mm in practice. The corresponding ridge metal electrode inductance lies in the range of 0.55 nH/mm to 0.65 nH/mm, however values from about 0.3 nH/mm to 0.8 nH/mm are possible, with a preferred range around 0.65 nH/mm for the present example. Thus, the ridge capacitance and ridge metal electrode inductance per signal electrode segment lies in the range of 0.16-0.19 pF and 0.11-0.13 nH respectively. Each of the modulator signal electrode segments  304   a - 304   d  is connected to on-chip inductors  305   b - 305   d  of equal amounts of inductance, however some variation in the inductance values among the on-chip inductors  305   b - 305   d  may be possible depending on a specific design. 
     The on-chip inductors  305   b - 305   d  are fabricated such that the added inductance between any two of the modulator signal electrode segments  304   a - d  lies in the range of 0.22 nH and 0.34 nH for the design space pertaining to the application and fabrication process of the present embodiment. For example, in the 800 μm long modulator designs # 1  and # 3  the values of inductance of the on-chip inductors  305   b - 305   d  are 0.34 nH and 0.24 nH, respectively. In other design approaches, the inductance value for the on-chip inductors  305   b - 305   d  may lie in the range from 0.15 nH to 0.4 nH. Inductors  305   a,e  used in this embodiment to connect to bond pads  306   a  and  306   b , respectively, have values typically in the range of 0.055 nH-0.34 nH. 
     Depending on the process, material, and epitaxial structure of the modulator shown in  FIG. 3(   c ), the ridge capacitance can deviate from the range quoted above. A rule-of-thumb guideline on the amount of inductance to be added can still be developed based on designs # 1  and # 3 , provided the parasitic capacitance of the on-chip inductors  305   b - 305   d  is low compared to the ridge capacitance. Here, all capacitance and inductance values are referenced to a unit length along the ridge waveguide  303 . For example, with ridge capacitance values of 0.8 pF/mm to 0.95 pF/mm an on-chip inductance value range of 1.2 nH/mm to 1.9 nH/mm applies. 
     When the modulator is fabricated such that the ridge capacitance falls outside the range quoted earlier, the range of on-chip inductance can be scaled proportionately with respect to the ridge capacitance to arrive at a new range in which the methodology disclosed herein will still work. For example, for a fabrication process that results in a ridge capacitance of 0.7 pF/mm, applying a scaling reference of 0.875 pF/mm (the mean value of the above range for ridge capacitance values, 0.8 pF/mm to 0.95 pF/mm) results in a new range of 0.96 nH/mm to 1.82 nH/mm for the applicable on-chip inductance. With this lower ridge capacitance of 0.7 pF/mm, the modulator bandwidth will be higher than designs # 1  and # 3 . 
     Any excess bandwidth so obtained can be traded for a lower signal voltage required to drive the modulator. For this trade-off the total electro-optic interaction length is scaled inversely with the fabricated ridge capacitance using the reference length of 800 um. For example, assuming the same 0.7 pF/mm for the ridge capacitance as before, the resulting total length of the modulator is 0.875/0.70*800 μm=1000 μm long, which requires approximately 20% less drive voltage. 
     For cases where the parasitic capacitance of the on-chip inductors is high due to limitations of a particular fabrication process, the total capacitance may be more appropriate for the scaling instead of ridge capacitance. Total capacitance values, which include the parasitic capacitance of the on-chip inductors, typically lie in the range 1.0 pF/mm to 1.2 pF/mm with the process used for the examples quoted herein. 
     Standard electro-absorption modulators having a length of 600 μm were also fabricated on the same InP wafer to permit a direct comparison with the 800 μm long embodiments which incorporate the on-chip inductors. All the modulators have identical wirebond pads for ease of performance comparison. The bond pads  306   a,b  can typically have a capacitance in the range of 65 fF to 80 fF. 
     Two-port scattering parameter (S-parameter) tests were conducted for these modulators using a vector network analyzer (VNA). The S-parameter measurement results are presented in  FIGS. 4 and 5 . In all the S-parameter tests, a 50Ω reference impedance was used. 
       FIG. 4  shows a graph of the return loss (S 11 ) measurements for 600 μm long modulators with no added inductors. The vertical axis is S 11  in dB and the horizontal has units of GHz. Traces of different patterns represent different modulator devices with identical design. The S 11  rises above −10 dB for frequencies higher than 5 GHz. 
       FIG. 5  shows a graph of the return loss measurements (s 11  [dB], vertical axis versus frequency [GHz], horizontal axis) of three 800 μm long modulators with different amounts of on-chip inductance added. The amount of inductance added per 200 μm long section of the modulator increases from design # 3  to design # 1 . 
     Return loss, s 11 , of these 800 μm long modulators shows considerable improvement over the frequency range 2 GHz to 20 GHz as compared to their 600 μm counterpart in  FIG. 4 , even though they have a 33% higher junction capacitance. 
     The measured electrical forward loss of standard 600 μm long modulators without added inductors is plotted in  FIG. 6 . The electrical forward loss s 21 [dB] on the vertical axis is plotted against frequency [GHz] on the horizontal axis. The electrical −3 dB bandwidth is seen to have a value of ˜10 GHz. Measurements were made with respect to a 50Ω reference impedance. Each trace represents a different device of identical design. 
     The measured electrical forward loss of 800 μm long modulators with three different amounts of inductance values for the integrated on-chip inductors is plotted in  FIG. 7 . The electrical forward loss s 21 [dB] on vertical axis is plotted against frequency [GHz] on the horizontal axis. The maximum electrical −3 dB bandwidth is seen to have a value of ˜10 GHz, despite the modulators being a 33% longer. The amount of inductance added per 200 μm long section of the modulator increases from design # 3  to design # 1 . 
     The velocity mismatch achievable with the integrated on-chip reactive components (inductors or capacitors) according to this invention is substantial. A microwave index in the range of 10-13 can be obtained by the addition of on-chip inductors, as obtained by designs # 1  to # 3  in  FIGS. 5 and 7 . The optical group index for semiconductors such as InP and InGaAsP typically lies within the range 3.8 to 3.9, not varying significantly as it is largely determined by the optical refractive index of the material used for the waveguide. Thus a velocity mismatch factor of up to approximately 3 to 4 can be achieved in practice. 
     The effect of velocity mismatch between the electrical and optical signal on the characteristics of a modulator with on-chip inductors or capacitors is explained here. If a mismatch between the velocities of the optical and electrical signals limits the optical bandwidth, f 0 , of the modulator, the bandwidth due to velocity mismatch can be estimated from Equation 6: 
     
       
         
           
             
               
                 
                   
                     f 
                     0 
                   
                   ≈ 
                   
                     
                       1.4 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cL 
                     
                     
                       π 
                       ⁢ 
                       
                          
                         
                           
                             n 
                             0 
                           
                           - 
                           
                             n 
                             μ 
                           
                         
                          
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   [ 
                   6 
                   ] 
                 
               
             
           
         
       
     
     where c is the velocity of light, n 0  and n μ  are the optical and microwave index respectively. 
     With the additional on-chip inductors as shown in  FIG. 3(   b ), for instance, the microwave velocity is proportional to the microwave index, n μ , as calculated from Equation 7:
 
 n   μ   ≈c √{square root over (( L   e   +L   μ )( C   j   +C   μ ))}{square root over (( L   e   +L   μ )( C   j   +C   μ ))}  Equation [7]
 
     where the L e  is the inductance of the modulator electrode, L u  is the on-chip inductance, C j  is the modulator junction capacitance, C μ  is the parasitic capacitance of the on-chip inductor (if an on-chip capacitor is used, then C μ  will be the capacitance and L μ  will be the parasitic inductance) and c is the velocity of light. 
     The use of either an inductor or a capacitor have the same effect, as shown in the equations [6] and [7]. Hence, inductor will be used in this case for simplicity of explanation. 
     From equation [7], it can be seen that the addition of on-chip inductors increases the microwave index as a result of its inductance and parasitic capacitance. The increased value of the difference term |n o −n μ | results in bandwidth filtering as a result of velocity mismatch of the optical and electrical signals, shown in equation [6]. 
     Simulation results of the device shown in  FIG. 3(   b ) are graphed in  FIG. 8(   a ), where electro-optic (EO) response [dB] is plotted on the vertical axis against frequency [GHz] on the horizontal axis. They illustrate the difference in roll-off filter characteristics achievable through the use of on-chip inductors to introduce velocity mismatch between the traveling electrical and optical signal. Curve  1  describes the EO response of a conventional modulator, while Curve  2  is the response corresponding to the modulator design with velocity mismatch. The frequency roll-off can be estimated from the slope of the curves between 15 GHz and 20 GHz. The conventional modulator (Curve  1 ) exhibits a 20 dB/decade roll-off, as expected for a typical RC dominated circuit. By incorporating a designed velocity mismatch (Curve  2 ) a roll-off of 50 dB/decade is obtained, which is significantly higher, corresponding to a 5 th  order filter function. 
     The velocity mismatch modulator with the steeper EO roll-off has application in spectral filtering for 40 Gbits/s systems. In Optical Duo-Binary modulation format at 40 Gbits/s, the Non-Return to Zero signal is low-pass filtered at 0.25*40 Gbits/s and experimental results show that a sharp roll-off of the order 40 dB/decade is usually required. 
     The electro-optic response of an 800 μm long modulator graphed in  FIG. 8(   b ), measured with a network analyzer, indicates a bandwidth of 10 GHz. A linear fit to the measured response after the −3 dB point corresponds to a filter roll-off of 50 dB/decade. 
     In  FIG. 9 , receiver sensitivity [dB] is plotted on the vertical axis against transmission distance [km] on the horizontal axis for a signal modulated at 10 Gbits/s employing an optical duo-binary modulation scheme. The receiver sensitivity for two different electrical filters was simulated, each with a 3 dB bandwidth of 0.25*10 GHz, but with filter roll-offs of 3 rd  and 5 th  orders respectively. The simulations show that a transmitter with 5 th  order filter provides better receiver sensitivity versus transmission distance performance compared to a transmitter with a 3 rd  order filter. 
       FIG. 10  shows return loss s 11  [dB] for a 700 μm long modulator without any on-chip inductor plotted on the vertical axis against frequency [GHz] on the horizontal axis with modulator bias voltages of 0, −1.5 and −3.0 volt. The modulator is attached onto a 4 mm×6 mm ceramic carrier that has impedance controlled transmission lines and thin-film matching resistors. The modulator is attached to the transmission lines and termination resistors with wirebonds. 
     The frequency characteristic of return loss for a 750 μm long modulator with on-chip inductors is shown in  FIG. 11 . Return loss s 11  [dB] is plotted on the vertical axis against modulating frequency [GHz] on the horizontal axis with modulator bias voltages of 0, −1.5 and −3.0 volt. The design # 1  modulator having the structure shown in  FIG. 3(   b ) is attached onto a 4 mm×6 mm ceramic carrier that has impedance controlled transmission lines and thin-film termination resistors. The modulator is attached to the transmission lines and termination resistors with wirebonds. The improvement of the return loss compared to the typical lumped element design in  FIG. 10  is clearly evident. 
     Whereas s 11  for the typical lumped element design increases above about −12 dB for frequencies above 9 GHz, for the modulator with on-chip inductors s 11  remains below −12 dB for practically the whole frequency range of 0-20 GHz.