Patent Publication Number: US-2021194586-A1

Title: Electro-optic modulator and microwave photonic link including an electro-optic modulator

Description:
BACKGROUND 
     The invention relates to an electro-optic modulator, and a microwave photonic link including an electro-optic modulator. 
     SUMMARY 
     In one embodiment, the invention provides a highly linear dual ring resonator modulator design. The dual ring resonator modulator provides high spur-free dynamic range in a wide bandwidth. Harmonic and intermodulation distortions are theoretically analyzed in a single ring resonator modulator structure and a strategy is proposed to enhance the linearity of the single ring resonator design for wide bandwidth applications by utilizing dual ring resonator modulators. Third order intermodulation distortion is suppressed with proper splitting ratio of optical and radio frequency power and optimum direct current biasing of the ring resonators. The structure shows the ability to operate at a 40 gigahertz bandwidth with a spur-free dynamic range more than 130 decibels at 1 hertz microwave operating frequency while keeping the spur-free dynamic range greater than 124 decibels. 
     In another embodiment, the invention provides an electro-optic modulator for coupling to an optical source providing an optical power and a radio frequency source providing a radio frequency power. The electro-optic modulator includes a waveguide receiving the optical power, a first ring resonator modulator and a second ring resonator modulator receiving the radio frequency power. The first ring resonator modulator and the second ring resonator modulator are coupled to the waveguide for modulating the optical power with the radio frequency power, thereby providing a resultant power. The first ring resonator modulator includes a first ring waveguide and a first travelling wave electrode in communication with the first ring waveguide. The second ring resonator modulator includes a second ring waveguide and a second travelling wave electrode in communication with the second ring waveguide. 
     In yet another embodiment, the invention provides a microwave photonic link having the just-described electro-optic modulator. The microwave photonic link includes an optical source providing the optical power and a radio frequency source providing a radio frequency power. The electro-optic modulator is coupled to the optical source and the radio frequency source. 
     In a further embodiment the invention provides, another microwave photonic link. The microwave photonic link includes an optical source providing an optical power, a radio frequency source providing a radio frequency power, and an electro-optic modulator coupled to the optical source and the radio frequency source. The electro-optic modulator includes a waveguide receiving the optical power. The waveguide includes an optical splitter, a first waveguide segment, and a second waveguide segment. The optical splitter splits the optical power into a first optical power for traversing the first waveguide segment and a second optical power for traversing the second waveguide segment. The electro-optic modulator further includes a first ring resonator modulator including the first waveguide segment and a second ring resonator modulator including the second waveguide segment. The first ring resonator modulator and the second ring resonator modulator receives the radio frequency power and modulates the first optical power and the second optical power with the radio frequency power. The microwave photonic link further includes a first photo detector coupled to the first waveguide segment, and a second photo detector coupled to the second waveguide segment. 
     Other aspects of the invention will become apparent by consideration of the detailed description and accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic illustrating a microwave photonic link including an electro-optic modulator. 
         FIG. 2  is a graph of normalized output intensities in fundamental, second-harmonic, third-harmonic and fifth-harmonic frequencies versus bias voltages in 1 hertz (lines), and 500 megahertz (dot lines) operating frequencies. Results are for input radio frequency power of −10 decibel-milliwatts and a ring resonator modulator with 9 millimeters perimeter. Output intensity is in logarithmic scale and normalized versus input intensity. 
         FIG. 3  is a graph of spur-free dynamic range versus radio frequency operating frequency for single ring resonator modulator and dual ring resonator modulator in two perimeters (circle: 3 millimeters, cross: 9 millimeters). Dotted line shows the spur-free dynamic range of a Mach-Zehnder Interference with the same link parameters. Spur-free dynamic range is calculated for 1 hertz noise bandwidth. 
         FIG. 4  is a graph of output fundamental and third order intermodulation distortion powers against the radio frequency input power. Lines are the results for a single ring resonator modulator and dots are for the results of a dual ring resonator modulator. Results are for 9 millimeter rings biased at V A . Noise level is at −164 decibel-milliwatts in 1 hertz bandwidth. 
         FIG. 5  is a graph of spur-free dynamic range in dual ring resonator modulators versus bias voltage for three perimeters of the ring (line: 9 millimeters, dash: 6 millimeters and, dot: 3 millimeters). The radio frequency operating frequency is 1 hertz and system noise bandwidth is 1 hertz. 
         FIG. 6  is a schematic illustrating an enlarged portion of a waveguide in  FIG. 1 . 
         FIG. 7  is a schematic illustrating an enlarged portion of a straight waveguide and a ring waveguide in  FIG. 1 . 
         FIG. 8  is a schematic illustrating a microwave photonic link including a second electro-optic modulator. 
         FIG. 9  is a graph showing a modulation index of a single ring resonator versus frequency. 
         FIG. 10  is a graph showing spur free dynamic range versus radio frequency operating frequency for a ring resonator modulator and a dual ring resonator modulator. The spur free dynamic range is calculated for 1 hertz noise bandwidth. 
         FIG. 11  is a schematic illustrating a microwave photonic link including a third electro-optic modulator. 
         FIG. 12  is a cross-section schematic illustrating a travelling wave electrode capable of being used in the microwave photonic link of  FIG. 8 . 
     
    
    
     DETAILED DESCRIPTION 
     Before any embodiments of the invention are explained in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of components set forth in the following description or illustrated in the following drawings. The invention is capable of other constructions and of being practiced or of being carried out in various ways. 
     Introduction 
     Microwave photonic links (MPLs) are critical for applications such as wireless-access networks, antenna remoting, and video distribution networks. See, for example, “Wireless future drives microwave photonics,”  Nat Photon , vol. 5, pp. 724-724, 12//print 2011, M. L. VanBlaricum, “Photonic systems for antenna applications,”  Antennas and Propagation Magazine, IEEE , vol. 36, pp. 30-38, 1994, and M. Nazarathy, J. Berger, A. J. Ley, I. M. Levi, and Y. Kagan, “Progress in externally modulated AM CATV transmission systems,”  Lightwave Technology, Journal of . vol. 11, pp. 82-105, 1993. Nonlinear distortions remain as one of the fundamental limitations for full scale implementation of microwave photonic links. See, for example, V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, et al., “Long-haul analog photonics,”  Journal of Lightwave Technology , vol. 29, pp. 1182-1205, 2011. To evaluate linearity in microwave photonic links, the spur-free dynamic range (SFDR) is a widely accepted performance metric. See, for example, C. H. Cox III and C. H. Cox,  Analog optical links: theory and practice : Cambridge University Press, 2006. The dominating distortion in microwave photonic links is the third order intermodulation distortion (IMD3) occurring at frequencies very close to the signal frequencies. 
     Current microwave photonic links may utilize Mach-Zehnder interferometers (MZI) as electro-optic (E/O) modulators, resulting in nonlinear distortions. Ongoing research efforts have been focused on achieving a highly linear modulator for microwave photonic links with limited success. See, for example, J. Cardenas, P. A. Morton, J. B. Khurgin, A. Griffith, C. B. Poitras, K. Preston, et al., “Linearized silicon modulator based on a ring assisted Mach Zehnder inteferometer,”  Optics express , vol. 21, pp. 22549-22557, 2013; N. Madamopoulos, B. Dingel, and A. Prescod, “Interferometric modulator with phase-modulating and cavity-modulating components (IMPACC) for high linearity microwave applications: technology review,” in  International Conference on Photonics Solutions  2013, 2013, pp. 88830O-88830O-12. While gains have been made, the devices are very complex. Mach-Zehnder interferometers have an inherent non-linear sinusoidal transfer function. To increase the linearity in Mach-Zehnder interferometers various approaches have been proposed including predistortion, postcompensation, and dual parallel Mach-Zehnder interferometers. See, for example, A. Agarwal, T. Banwell, P. Toliver, and T. Woodward, “Predistortion compensation of nonlinearities in channelized RF photonic links using a dual-port optical modulator,”  Photonics Technology Letters, IEEE , vol. 23, pp. 24-26, 2011; G. Zhang, X. Zheng, S. Li, H. Zhang, and B. Zhou, “Postcompensation for nonlinearity of Mach-Zehnder modulator in radio-over-fiber system based on second-order optical sideband processing,”  Optics letters , vol. 37, pp. 806-808, 2012; S.-K. Kim, W. Liu, Q. Pei, L. R. Dalton, and H. R. Fetterman, “Nonlinear intermodulation distortion suppression in coherent analog fiber optic link using electro-optic polymeric dual parallel Mach-Zehnder modulator,”  Optics express , vol. 19, pp. 7865-7871, 2011. To improve the linearity of Mach-Zehnder interferometer modulators two categories have proposed utilizing ring resonators, namely, 1) resonator-assisted Mach-Zehnder interferometer (RAMZI), and 2) interferometric modulator with phase-modulating and cavity-modulating (IMPAAC) components. See, for example, J. Cardenas, P. A. Morton, J. B. Khurgin, A. Griffith, C. B. Poitras, K. Preston, et al., “Linearized silicon modulator based on a ring assisted Mach Zehnder inteferometer,”  Optics express , vol. 21, pp. 22549-22557, 2013; N. Madamopoulos, B. Dingel, and A. Prescod, “Interferometric modulator with phase-modulating and cavity-modulating components (IMPACC) for high linearity microwave applications: technology review,” in  International Conference on Photonics Solutions  2013, 2013, pp. 88830O-88830O-12. In these modulators one ring resonator is coupled to a branch of a Mach-Zehnder interferometer. In a resonator-assisted Mach-Zehnder Interferometer, the phase response of the ring resonator is used to enhance the linearity in the Mach-Zehnder interferometer. Ring resonator phase response is controlled by the coupling coefficient between the ring resonator and the Mach-Zehnder interferometer branch. The exact value of the coupling coefficient is highly dependent upon the fabrication process and design layout making it difficult to predict this value to great precision. In an interferometric modulator with phase-modulating and cavity-modulating component, the amplitude response of the ring resonator on one branch of a Mach-Zehnder interferometer is used along with a phase modulator on another branch. The radio frequency signal is divided between amplitude and phase modulators in a specific ratio. The resonator-assisted Mach-Zehnder interferometer and interferometric modulator with phase-modulating and cavity-modulating component both require a Mach-Zehnder interferometer and thus inherit the sinusoidal transfer function, required size, and power requirements. The resonator-assisted Mach-Zehnder interferometer and interferometric modulator with phase-modulating and cavity-modulating component structures add complexity to the fabrication and integration process and their operation is sensitive to the structure properties including loss factor of the ring, coupling coefficient, and working frequency. See for example, N. Madamopoulos, B. Dingel, and A. Prescod, “Interferometric modulator with phase-modulating and cavity-modulating components (IMPACC) for high linearity microwave applications: technology review,” in  International Conference on Photonics Solutions  2013, 2013, pp. 88830O-88830O-12; H. Tazawa and W. H. Steier, “Bandwidth of linearized ring resonator assisted Mach-Zehnder modulator,”  Photonics Technology Letters, IEEE , vol. 17, pp. 1851-1853, 2005; and A. Prescod, B. B. Dingel, N. Madamopoulos, and R. Madabhushi, “Effect of ring resonator waveguide loss on SFDR performance of highly linear optical modulators under suboctave operation,”  Photonics Technology Letters, IEEE , vol. 22, pp. 1297-1299, 2010. 
     Ring resonator modulators (RRM) attracted interest because of its potential to enhance linearity, reduce the size, and power consumption in microwave photonic links. See, for example, A. Ayazi, T. Baehr-Jones, Y. Liu, A. E.-J. Lim, and M. Hochberg, “Linearity of silicon ring modulators for analog optical links,”  Opt. Express 20 (12), pp. 13115-13122, 2012. The ring resonator modulator is used as intensity modulators with a Lorentzian shaped transfer function versus applied voltage that yields suppressing of various nonlinear distortions at special bias voltages. See, for example, W. Fegadolli, J. E. B. Oliveira, and V. R. Almeida, “Highly linear electro-optic modulator based on ring resonator,” Microwave and Optical Technology Letters , vol. 53, pp. 2375-2378, 2011. Another advantage of the Lorentzian transfer function of the ring resonator modulator is its repetition in frequency because of the periodicity of the ring resonator resonance in frequency domain. The frequency difference between the two subsequent resonances is called Free-Spectral Range (FSR) of the ring resonator. It has been shown that ring resonator modulators with traveling wave electrodes can be operated at multiples of the Free-Spectral Range, as high as 165 gigahertz in polymer. See, for example, B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. Luo, A. K.-Y. Jen, et al., “Electrooptic polymer ring resonator modulation up to 165 GHz,”  Selected Topics in Quantum Electronics, IEEE Journal of . vol. 13, pp. 104-110, 2007. These advantages provide more freedom in designing modulators in comparison with Mach-Zehnder Interferometer modulators. However, the bandwidth of ring resonator modulators is limited because of the resonance characteristics similar to other resonance type of modulators. See, for example, I. L. Gheorma and R. M. Osgood, “Fundamental limitations of optical resonator based high-speed EO modulators,”  Ieee Photonics Technology Letters , vol. 14, pp. 795-797, June 2002. In all of the resonator type modulators, including resonator-assisted Mach-Zehnder interferometer and interferometric modulator with phase-modulating and cavity-modulating components, bandwidth is a limiting factor in dynamic ranges. See, for example, N. Madamopoulos, B. Dingel, and A. Prescod, “Interferometric modulator with phase-modulating and cavity-modulating components (IMPACC) for high linearity microwave applications: technology review,” in  International Conference on Photonics Solutions  2013, 2013, pp. 88830O-88830O-12; H. Tazawa and W. H. Steier, “Bandwidth of linearized ring resonator assisted Mach-Zehnder modulator,”  Photonics Technology Letters, IEEE , vol. 17, pp. 1851-1853, 2005. 
     A microwave photonic link is shown in  FIG. 1 . The microwave photonic link includes an electro-optic modulator  10  having dual ring resonator modulator. The electro-optic modulator  10  provides for wide bandwidth applications utilizing a highly linear dual ring resonator structure resulting in a significant increase of the spur-free dynamic range. The dynamic response of a single ring resonator modulator is analyzed by deriving analytical equations for nonlinear distortions. Obtained equations show the influence of the bandwidth in suppressing the spur-free dynamic range making single ring resonator modulators not suitable for the wide bandwidth applications. It is shown herein how the fundamental signal power and distortions power are related to the driving radio frequency power and feeding optical power. Using an analytical method a dual ring resonator that can suppress third order intermodulation distortion in wide bandwidth is presented. 
     Single Ring Resonator 
     To evaluate the linearity of the transfer function with the chosen properties of the electro-optic material, a dynamic transfer function is determined according to the multiple round-trip approach given by (Eq. 1). See, for example, H. Tazawa and W. H. Steier, “Analysis of ring resonator-based traveling-wave modulators,”  Photonics Technology Letters, IEEE , vol. 18, pp. 211-213, 2006 and H. Zhang, M.-C. Oh, A. Szep, W. H. Steier, C. Zhang, L. R. Dalton, et al., “Push-pull electro-optic polymer modulators with low half-wave voltage and low loss at both 1310 and 1550 nm,”  Applied Physics Letters , vol. 78, pp. 3136-3138, 2001; 
     
       
         
           
             
               
                 
                   
                     
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     In (Eq. 2) n 0  is the refractive index, L is the perimeter of the ring, r is the electro-optic coefficient of the material, λ is the optical wavelength, g is the electrode gap and φ=ω m /FSR where FSR is defined by c/(n 0 L). The microwave signal in (Eq. 2) is assumed to be in the form of V(t)=V m  sin(ω m t). 
     Nonlinearity and third order intermodulation distortion signal power can be analyzed by calculating output power of harmonic frequencies versus the applied voltage. Third order intermodulation distortion signal amplitude can be expanded to the linear mixing of amplitudes in odd harmonics starting from third harmonic utilizing multinomial theorem as shown in (Eq. 3). See, for example, S. Hunziker and W. Baechtold, “Simple model for fundamental intermodulation analysis of RF amplifiers and links,”  Electronics Letters , vol. 32, pp. 1826-1827, 1996. 
     
       
         
           
             
               
                 
                   
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     Using these equations, output powers for fundamental, second, third, and fifth harmonics are calculated in the range of bias voltages and in 1 hertz and 500 megahertz frequencies ( FIG. 2 ). Two critical bias voltages are well noticed (V A  and V B ). At bias point V A , second and fifth harmonics are suppressed while at V B , the third harmonic has minimum power. The Lorentzian-shaped transfer function of ring resonator modulator has a bias point where the output power in third harmonic is minimum, while the fundamental signal has considerable amount of power, resulting in higher spur-free dynamic range compared to Mach-Zehnder Interferometer modulators. Therefore, V B  is mentioned in literatures as the optimum bias for the ring resonator modulators to obtain high spur-free dynamic range. However, by increasing the radio frequency operating frequency from 1 hertz to 500 megahertz the suppressing of the harmonics at V A  and V B  is diminished considerably resulting in spur-free dynamic range reduction. 
     To obtain spur-free dynamic range, a two tone test and numerical Fourier method is utilized with typical link parameters. See, for example, W. B. Bridges and J. H. Schaffner, “Distortion in linearized electrooptic modulators,” Microwave Theory and Techniques, IEEE Transactions on , vol. 43, pp. 2184-2197, 1995.  FIG. 3  presents the calculated spur-free dynamic range for the ring resonator modulator with two perimeter sizes that is biased at V B  in the range of radio frequency operating frequency up to 5 gigahertz. Decreasing the length of the ring results in improving the spur-free dynamic range in wider bandwidth. However the sensitivity of the modulator is decreased for a smaller ring. Biasing the single ring resonator modulator at V B  will provide relatively high spur-free dynamic range at narrow bandwidths. To increase the spur-free dynamic range at wider bandwidths it is necessary to suppress the third harmonic power utilizing other methods. 
     Dual Ring Resonator 
     An approach of using dual ring resonator modulators (DRRM) to cancel or reduce the third order harmonic portion of the third order intermodulation distortion is presented. Producing equal powers of the third order harmonic at dual ring resonator modulators with 180 degrees phase difference the third order harmonic can exactly be cancelled. However, the cancellation process of the third harmonic power will also slightly suppress the fundamental signal. To yield the maximum spur-free dynamic range, the optical and radio frequency powers are divided between the two modulators in a specific ratio to have minimum cancellation of the fundamental signal while keeping the maximum cancellation for the third harmonic distortion. 
     In a single ring resonator modulator by applying two tone signal to (Eq. 1) and using Bessel expansions the following relations are obtained for output radio frequency signal and third order intermodulation distortion powers: 
     
       
         
           
             
               
                 
                   
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     third order intermodulation distortion signal power is related to the product of Bessel functions, J 1 (δ n ) J 2 (δ n ), however fundamental signal is proportional to the J 0 (δ n ) J 1 (δ n ) product. For small radio frequency signals the δ n &lt;&lt;1 is satisfied and the J 0 (δ n ) J 1 (δ n ) and J 1 (δ n ) J 2 (δ n ) products can be expanded in a linear expansion as follows (see, for example, L. M. Johnson and H. Roussell, “Reduction intermodulation distortion in interferometric optical modulators,”  Optics letters , vol. 13, pp. 928-930, 1988): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     These equations show that the output fundamental signal power has a linear modulation index dependence while the third order intermodulation distortion signal power has a cubic modulation index dependence. For instance, if the radio frequency input power increased 2 times, the output third order intermodulation distortion power is expected to increase eight times, while output signal power will be 2 times more. However, by decreasing the optical carrier power eight times the output third order intermodulation distortion power will remain at the same level. Therefore, to cancel the third order harmonic of third order intermodulation distortion, two ring resonator modulators with divided radiofrequency power at ratio F:1 can be utilized, but the optical power ratio, in some embodiments, is the inverse cube of the radio frequency voltage, i.e. 1:F 3 . Other ratios are possible and may be preferred according to the type of the material used, the fabrication process, and design tolerances. 
     Referring again to  FIG. 1 , the figure illustrates an electro-optic modulator  10  having a dual ring resonator modulator. The electro-optic modulator  10  is part of a microwave photonic link  15 . The microwave photonic link  15  includes an optical source  20 , a first optical fiber  25 , the electro-optic modulator  10 , a radio frequency source  30 , a second optical fiber  35 , a photo detector  40 , and a radio frequency output  45 . 
     The optical source  20  provides optical power. The optical source  20  can consist of a laser, and the optical source provides a carrier signal for the electro-optic modulator  10 . The radio frequency source  30  provides radio frequency power. The radio frequency source  30  can be a radio frequency generator, and the radio frequency power can be a communication signal carrying information. 
     The optical signal from the first optical fiber  25  is provided to a waveguide  50  fabricated on a substrate  55 . The waveguide  50  has multiple segments as discussed further below. The waveguide  50  can be made of a polymer type material, or other known materials (e.g. lithium niobate, silicon) for providing a waveguide, and the substrate can be made of a silicon-based material or other known material for supporting the structure. Other waveguides fabricated on the substrate  55 , such as the ring waveguides discussed below, can be made of the polymer type material, or other known materials for providing a waveguide. Moreover, the structure shown for the electro-optic modulator  10  in  FIG. 1  is a top view of the structure. The cross section structure (e.g., rib type, inverted rib type, step index channel) of the waveguides fabricated on the substrate  55  can also effect the modulation properties of the electro-optic modulator  10 . 
     The waveguide  50  includes an input waveguide  60  coupled to the first optical fiber  25 . The input waveguide  60  receives a first optical power. The input waveguide  60  guides the optical power from the first optical fiber  25  to an optical splitter  65 . The optical splitter  65  splits the first optical power into a first straight waveguide  70  and a second straight waveguide  75 . The optical splitter  65  splits the first optical power into a second optical power O for the first straight waveguide  70  and a third optical power  1 -O for the second straight waveguide  75 . The second optical power O and the third optical power  1 -O are related values. For example, if the first straight waveguide  70  receives 60% of the first optical power, then the second straight waveguide  75  receives 40% of the first optical power. 
     The optical splitter  65  includes a tab portion  67  having a length L. Also the tab portion is spaced from the input waveguide  60  by a distance D. The length L and the distance D help specify the ratio of powers being provided to each straight waveguide  70  and  75 . The length L and the distance D are schematically shown in  FIG. 6 , which enlarges a portion of the waveguide  50 . Other techniques for splitting optical power in the waveguide can be used as known in the art. 
     The second optical power O transmits along the first straight waveguide  70  and modulates with a second radio frequency power  1 -F at a first ring resonator modulator  80  (please note that the first radio frequency power will be discussed below). The second optical power O modulates with the second radio frequency power  1 -F to result in a first resultant power. Similarly, the third optical power  1 -O transmits along the second straight waveguide  75  and modulates with a third radio frequency power F at a second ring resonator modulator  85 . The third optical power  1 -O modulates with the third radio frequency power F to result in a second resultant power. 
     The first straight waveguide  70  and the second straight waveguide  75  combine at an optical coupler  90 . The optical coupler  90  couples the first resultant power with the second resultant power to result in a third resultant power. The third resultant power is provided to the second optical fiber  35  at an output waveguide  95 . The second optical fiber  35  communicates the third resultant power to the photo detector  40 . The photo detector  40  converts the optical power into radio frequency power for transmission at the radio frequency output  45 . 
     The radio frequency source  30  provides a first radio frequency power (i.e., electrical power). The radio frequency source  30  can be an antenna receiving data via an electrical signal. The first radio frequency power is provided to an electrical circuit  97 . The electrical circuit  97  includes multiple circuit paths, which one skilled in the art may also refer to as being circuits. The electrical circuit  97  includes a first circuit path  100  for receiving the first radio frequency power from the radio frequency source  30 . 
     The first circuit path  100  provides the first radio frequency power to a radio frequency power divider  105 . The radio frequency power divider  105  splits the first radio frequency power into a second circuit path  110  and a third circuit path  115 . The radio frequency power divider  105  splits the first radio frequency power into a second radio frequency power  1 -F for the second circuit path  110  and a third radio frequency power F for the third circuit path  115 . The second radio frequency power  1 -F and the third radio frequency power F are related values. For example, if the third circuit path  115  receives 60% of the first radio frequency power, then the second circuit path  110  receives 40% of the first optical power. 
     The second radio frequency power  1 -F is communicated along the second circuit path  110  to the first ring resonator modulator  80 . The second radio frequency power  1 -F modulates the second optical power O to result in the first resultant power, as already mentioned. Similarly, the third radio frequency power F is communicated along the third circuit path  115  to the second ring resonator modulator  85 . The third radio frequency power F modulates the third optical power  1 -O to result the second resultant power, as already mentioned 
     The first ring resonator modulator  80  includes an optical portion and an electrical portion. The optical portion includes the first straight waveguide  70  and a first ring waveguide  130 . The first straight waveguide  70  and the first ring waveguide  130  are optically coupled, in part, by a coupling distance D 2  and a coupling length L 2 . The distance D 2  and the length L 2  help specify the optical power being provided from the first straight waveguide  70  to the first ring waveguide  130  and vice-versa. The distance D 2  and the length L 2  are schematically shown in  FIG. 7 , which enlarges a portion of the waveguide  50 . The electrical portion includes a first electrode  125  and a second electrode  135 . The second radio frequency power  1 -F applied to the first electrode  125  changes the effective refractive index of the first ring waveguide  130 , thereby altering the resonance frequency of the first ring resonator modulator  80 . The altering of the resonance frequency modulates the optical carrier signal, i.e., the second optical power, according to the applied RF signal, i.e., the second radio frequency power. 
     The first ring waveguide  130  defines a diameter for the first ring resonator modulator  80 . The distance between the first ring waveguide  130  and the straight waveguide  70  can change according to the design preference, fabrication tolerance, and process and type of the utilized material. For the embodiment shown, the second ring resonator modulator  85  has an analogues design to the first ring resonator modulator  80 . Moreover, other ring resonator modulator designs can be used in place of the shown ring resonator modulators  80  and  85 . For example, the design and location of the electrodes  125  and  135  can change from the structure shown based on the type of the electro-optic material used to maximize the electro-optic coefficient. Also, the number of ring resonator modulators can be changed, such as to cancel out higher order harmonics. 
     In one implementation, the first ring resonator modulator  80  receives higher optical power via the first straight waveguide  70  while being driven by less radio frequency power  1 -F via the second circuit path  110 , and the second ring resonator modulator  85  receives less optical power via the second straight waveguide  75  while being driven by higher radio frequency power F via the third circuit path  115 . The optical power ratio between the ring resonator modulators  80  and  85  can be set by the coupling ratio of the optical splitter  65 . The radio frequency power ratio between the ring resonator modulators  80  and  85  can be set by the coupling ratio of the radio frequency power divider  105 . If the radio frequency power ratio and the optical power ratio are designed properly, then the third harmonic produced from the ring resonator modulators  80  and  85  could be equal and the distortion can theoretically cancel at the output. It is inevitable that with this method there will be some cancellations of the spur-free dynamic range signal, but the spur-free dynamic range will be maximized at wider bandwidths than previously reported. 
     The output power versus input power for the fundamental and the third order intermodulation distortion for ring resonator modulators and dual ring resonator modulators with F=3 are shown in  FIG. 4 . In the dual ring resonator modulator, third order intermodulation distortion power is suppressed more than fundamental signal, thereby yielding an enhancement in spur-free dynamic range as marked in  FIG. 4 . It is worth mentioning that the third order intermodulation distortion versus input power in a single structure has a slope equal to 3 showing third order harmonic contribution in third order intermodulation distortion. Having a linearization process in the dual ring resonator modulator, the slope of third order intermodulation distortion increases to 7, therefore by biasing the modulators at the proper voltage and having dual ring resonator modulators with the proper power ratio of radio frequency and optical powers both third and fifth order harmonics are cancelled. 
     To obtain optimum bias voltage in dual ring resonator modulators, spur-free dynamic range is calculated in a wide range of the bias voltages, as shown in  FIG. 5 . Bias voltage that has higher spur-free dynamic range in dual ring resonator modulator is the V A  bias point in single ring resonator modulator because of the fifth harmonic cancellation at this bias voltage. A spur-free dynamic range of more than 130 decibels is achievable with dual ring resonator modulators indicating the highly linear behavior of the modulator. As shown in  FIG. 5 , by decreasing the perimeter from 9 millimeters to 3 millimeters, the maximum spur-free dynamic range is slightly increased. 
     In  FIG. 3 , variation of the spur-free dynamic range versus operating radio frequency is shown for two ring perimeters. When the operating radio frequency is 1 hertz, the complete suppression of the fifth order harmonic comes with biasing at V A . However, by working at higher frequencies, fifth harmonic contribution in third order intermodulation distortion power increases reducing spur-free dynamic range. Using a dual ring resonator modulation structure provides cancellation of the third harmonic in a wide bandwidth, resulting in a spur-free dynamic range of greater than 125 decibels in 5 gigahertz bandwidth. By decreasing the perimeter of the ring, from 9 millimeters to 3 millimeters, it is observed to have approximately 2 decibel improvement in spur-free dynamic range of more than 127 decibels in 5 gigahertz bandwidth. 
     Travelling Wave Structure 
     A dual ring resonator modulator has a transfer function that repeats periodically in frequency because of the resonance characteristics of the ring resonator structure. For  FIG. 8 , a radio frequency electromagnetic wave travels along the electrodes. For the shown structure, the electrodes are disposed above the optical waveguide ring structures. Whole frequency repetitive transfer function operation can be obtained as shown in  FIG. 9  with a traveling wave design implemented in the dual ring resonator modulator. The matching of velocities through a configuration as disclosed can be highly dependent upon the electro-optic material used in the optical waveguide structure for proper high frequency modulation. Electro-optic material can lose modulation capability at higher frequencies. Therefore, if the structure has appropriate electro-optic material, which can be used in high frequencies the structure itself, i.e., the ring resonator modulator, has no limits in terms of operating frequency. In other words limitation of the operating frequency may be dictated by the type of the material and not the structure. In some constructions, the preferred material candidate for high frequency modulation has been demonstrated to be polymer at an operating frequency as much as 165 gigahertz for just a single ring resonator modulator design. See, for example, B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. Luo, A. K.-Y. Jen, et al., “Electrooptic polymer ring resonator modulation up to 165 GHz,” Selected Topics in Quantum Electronics, IEEE Journal of, vol. 13, pp. 104-110, 2007. The dual ring resonator modulator transfer function provides efficient modulation in up to millimeter frequency ranges with limited bandwidth if the signal frequency matches the free spectral range (FSR) of the ring. High frequency operation within the dual ring resonator modulator is more advantageous when the velocity mismatch and radio frequency loss are taken into account which is supported by previous work using ring resonator modulator that had a higher modulation index than Mach-Zehnder interferometers based modulators. See, for example, H. Tazawa and W. H. Steier, “Analysis of ring resonator-based traveling-wave modulators,” Photonics Technology Letters, IEEE, vol. 18, pp. 211-213, 2006. 
     The inventors studied linearity of the ring resonator modulator that could be applied to the dual ring resonator modulator of  FIG. 8  for high frequency operation utilizing traveling wave electrodes. Theoretical study shows that the same strategy utilized for lumped electrode structure, e.g. as shown in  FIG. 1 , can be used to enhance linearity in a ring resonator modulator with traveling wave electrodes. Suppressing intermodulation distortion in dual ring resonator modulators is not limited by the operating frequency since the proposed method to cancel third harmonic distortion is not frequency dependent. In  FIG. 10 , variation of the spur free dynamic range versus operating radio frequency is shown for a ring resonator modulator and a dual ring resonator modulator. As seen using the dual ring resonator modulator structure provides complete cancellation of the 3rd harmonic in a wide bandwidth resulting in an spur free dynamic range of greater than 124 decibels at frequencies away from exact resonance frequency. 
     Referring again to  FIG. 8 , the figure illustrates an electro-optic modulator  210  having a dual ring resonator modulator. The electro-optic modulator  210  is part of a microwave photonic link  215 . The microwave photonic link  215  includes the optical source  20 , the first optical fiber  25 , the electro-optic modulator  210 , a first radio frequency source  230 , a second radio frequency source  232 , the second optical fiber  35 , the photo detector  40 , and the radio frequency output  45 . The optical source  20 , the first optical fiber  25 , the second optical fiber  35 , the photo detector  40 , and the radio frequency output  45  are similar to the devices discussed for  FIG. 1 . The first radio frequency source  230  and the second radio frequency source  232  provide radio frequency power analogous to the resultant of the radio frequency source  30  and radio frequency power divider  105  of  FIG. 1 . 
     The optical signal from the first optical fiber  25  is provided to the waveguide  50  fabricated on the substrate  55 . The waveguide  50  has multiple segments as discussed earlier in reference to  FIG. 1 . The optical splitter  65  splits the first optical power into the first straight waveguide  70  and the second straight waveguide  75 . The optical splitter  65  splits the first optical power into an optical power O for the first straight waveguide  70  and an optical power  1 -O for the second straight waveguide  75 . The optical power O and the optical power  1 -O are related values as previously discussed. 
     The second optical power O transmits along the first straight waveguide  70  and modulates with a radio frequency power  1 -F at a first ring resonator modulator  280 . The optical power O modulates with the radio frequency power  1 -F to result in a first resultant power. Similarly, the optical power  1 -O transmits along the second straight waveguide  75  and modulates with a radio frequency power F at a second ring resonator modulator  285 . The optical power  1 -O modulates with the radio frequency power F to result in a second resultant power. The first straight waveguide  70  and the second straight waveguide  75  combine at the optical coupler  90 . The optical coupler  90  couples the first resultant power with the second resultant power to result in a third resultant power, similar to what was discussed for  FIG. 1 . 
     The radio frequency power  1 -F is communicated along the circuit path  110  to the first ring resonator modulator  280 . The second radio frequency power  1 -F modulates the second optical power O to result in the first resultant power, as already mentioned. Similarly, the radio frequency power F is communicated along the circuit path  115  to the second ring resonator modulator  285 . The third radio frequency power F modulates the third optical power  1 -O to result the second resultant power, as already mentioned 
     The first ring resonator modulator  280  includes an optical portion and an electrical portion. The optical portion includes the first straight waveguide  70  and a first ring waveguide  130 . The electrical portion includes an electrode  282  providing a travelling wave design, which is basically a waveguide enabling radio frequency propagation. The electrode  282  is shown in  FIGS. 8 and 12  as a strip-line waveguide that propagates a wave between two layers of metal with a dielectric (i.e., the ring waveguide  130 ) in between. As shown in  FIG. 12 , the travelling wave electrode includes top and bottom electrodes, upper and lower claddings, and an optical core. The travelling wave electrode can have other designs and placements from what is shown in  FIGS. 8 and 12 . The electrode  282  has a ring-like shape similar to the first ring waveguide  130 . The electrode  282  is connected to the second radio frequency source  232  and ground. The signal of the second radio frequency source  232  travels around the electrode  282  in one of a clockwise or counterclockwise direction (counterclockwise is shown). The signal of the first radio frequency source  230  also travels around the electrode  287  in the one of a clockwise and counterclockwise direction (counterclockwise is shown). The radio frequency power  1 -F applied to the electrode  282  changes the effective refractive index of the first ring waveguide  130 , thereby altering the resonance frequency of the first ring resonator modulator  280 . The altering of the resonance frequency modulates the optical carrier signal, i.e., the optical power O, according to the applied RF signal, i.e., the radio frequency power  1 -F. The radio frequency power F applied to the electrode  280  would affect the other ring waveguide similarly. 
     Dual Output Waveguide 
     To provide a variant design utilizing a dual ring resonator modulator, the modulator output can be combined electrically instead of optically. In this design, output of each modulator is connected individually through an optical fiber to a detector in the receiver side. The output of the detectors is combined to provide final output which has suppressed third order intermodulation distortion. The theory behind this variation is very similar to the theory and explanation above for the already discussed designs. 
     With reference to  FIG. 11 , the figure illustrates an electro-optic modulator  310  having a dual ring resonator modulator. The electro-optic modulator  310  is part of a microwave photonic link  315 . The microwave photonic link  315  includes the optical source  20 , the first optical fiber  25 , the electro-optic modulator  310 , the radio frequency source  30 , a second optical fiber  335 , a third optical fiber  337 , a first photo detector  340 , a second photo detector  342 , and the radio frequency output  45 . The optical source  20 , the first optical fiber  25 , the radio frequency source  30 , and the radio frequency output  45  are similar to the devices discussed for  FIG. 1 . It should also be understood that the radio frequency source can be split similar to the first radio frequency source  230  and the second radio frequency source  232  of  FIG. 8 . 
     The optical signal from the first optical fiber  25  is provided to the waveguide  350  fabricated on the substrate  55 . The waveguide  350  has multiple segments. The optical splitter  65  splits the first optical power into the first straight waveguide  70  and the second straight waveguide  75 . The optical splitter  65  splits the first optical power into an optical power O for the first straight waveguide  70  and an optical power  1 -O for the second straight waveguide  75 . The optical power O and the optical power  1 -O are related values as previously discussed. 
     The second optical power O transmits along the first straight waveguide  70  and modulates with a radio frequency power  1 -F at a first ring resonator modulator  80 . The optical power O modulates with the radio frequency power  1 -F to result in a first resultant power. Similarly, the optical power  1 -O transmits along the second straight waveguide  75  and modulates with a radio frequency power F at a second ring resonator modulator  85 . The optical power  1 -O modulates with the radio frequency power F to result in a second resultant power. While first and second ring resonator modulators  80  and  85  are shown, other structures can be used in place of the modulators, such as first and second ring resonator modulators  280  and  285  of  FIG. 8 . However, unlike  FIG. 1 , the first straight waveguide  70  and the second straight waveguide  75  do not combine at the modulator portion of the microwave photonic link to make modulator design and fabrication easier for some constructions. Rather, the first resultant power is provided to the second optic fiber  335  and the second resultant power is provided to the third optic fiber  337 . The second optic fiber  335  provides the first resultant power to the first photo detector  340 . The third optic fiber  337  provides the second resultant power to the second photo detector  342 . The first and second photo detectors  340  and  342  convert the optical powers (i.e., the first and second resultant powers) into radio frequency powers. The radio frequency powers are combined by a radio frequency power coupler and are transmitted by the radio frequency output. 
     Before concluding, it should be noted that the waveguides  50  and  350  show distinct angles when transitioning from one optical segment to another optical segment. For example, the transition from the optical splitter  65  to the first straight waveguide  70  includes two distinct angle transitions. In other structures, the transition may be more arcuate (such as via an S-bend transition) than what is shown. 
     CONCLUSION 
     A dual ring resonator is disclosed as an electro-optic modulator to obtain high spur-free dynamic range at wide bandwidths for microwave photonic links. The nonlinear distortion in ring resonator modulators was analyzed to determine that by dividing the radio frequency and optical powers in specific ratios between two ring resonator modulators, the third harmonic distortion of the third order intermodulation distortion can be cancelled. This highly linear modulator with optimum bias points achieves a theoretical spur-free dynamic range of more than 130 decibels for a 1 hertz bandwidth for some structures. For wide bandwidth operations increasing the operating frequency resulted in a diminished suppression of fifth harmonic, however the third order harmonic can still be suppressed using the dual ring resonator modulators yielding a highly linear modulator for wide bandwidth applications. One proposed dual ring resonator modulator structure provides greater than 125 decibel spur-free dynamic range in a 40 gigahertz bandwidth. Additionally this modulator can be easily fabricated, has a smaller size, low power consumption, and less complexity in comparison with Mach-Zehnder Interferometers based linearized modulators. 
     The following references were cited above and are incorporated herein by reference:
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     Thus, the invention provides, among other things, a new and useful electro-optic modulator, a microwave photonic link including an electro-optic modulator, and a method of communicating a signal with an electo-optic modulator. Various features and advantages of the invention are set forth in the following claims.