text: embodiments of the present invention are shown in fig2 - 48 and additional equations at fig4 - 52 . the designs are determined by considering the effect of rf loss in the electrode . the voltage along the electrode is given by where v 0 is the voltage amplitude at the beginning of the electrode , a ( f ) is defined in equation 2 , and l is the total length of the mz . for the case where the optical and rf velocities are matched , the modulation imparted on a waveguide in a single section is given by f ⁡ ( f , l s , l f ) = eo v 0 ⁢ l ⁢ ∫ l s l f ⁢ v 0 ⁢ ⅇ - α ⁡ ( f ) ⁢ z ⁢ ⁢ ⅆ z ⁢ ⁢ = eo ⁢ ( ⅇ - α ⁡ ( f ) ⁢ l s - ⅇ - α ⁡ ( f ) ⁢ l f ) α ⁡ ( f ) ⁢ l ( 4 ) where eo is the relative modulation strength , and l s and l f are the length along the mz at the start and end of the section , respectively . f total ⁡ ( f ) = ∑ i = 0 n - 1 ⁢ eo ⁢ ( ⅇ - α ⁡ ( f ) ⁢ l s , i - ⅇ - α ⁡ ( f ) ⁢ l f , i ) α ⁡ ( f ) ⁢ l ( 5 ) where l s , i and l f , i are the length along the mz at the start and end of the ith section , respectively . at dc ( direct current 0 hz ) or low frequency , where f equals zero , equation 5 can be rewritten as hence , at dc the modulation imparted on the waveguide only depends on the relative modulation strength and the length of the different mz sections . chirp ⁡ ( f ) = re ⁡ ( f total , 1 ⁡ ( f ) - f total , 2 ⁡ ( f ) )  f total , 1 ⁡ ( f ) + f total , 2 ⁡ ( f )  ( 7 ) where f total , 1 ( f ) and f total , 2 ( f ) are the net modulation responses for the two waveguides in the mz , as a function of frequency . the frequency dependence arises from the fact that each waveguide samples the rf signal with different strength along the length of the rf electrode . note that equation 7 can be used for the cases where f total , 1 ( f ) and f total , 2 ( f ) are complex numbers . the modulation responses take on complex values for the cases where there is velocity mismatch between the rf and optical waves . for such cases , equations 4 and 5 include an additional multiplicative factor to include the effects of velocity mismatch . if the effects of velocity mismatch are small enough to be neglected , then all the values in equation 7 are real numbers and the equation can be rewritten as f total , 1 ⁡ ( f ) = f total , 2 ⁡ ( f ) ⁡ [ 1 + chirp ⁡ ( f ) 1 - chirp ⁡ ( f ) ] = f total , 2 ⁡ ( f ) ⁢ f c ⁡ ( f ) ( 8 ) equations 5 , 6 , and 8 allows one to create a set of simultaneous equations , where the lengths of the mz sections are the independent parameters to be determined . for the two - section mz , there is only one degree of freedom in the design , corresponding to the normalized length of either the first or second section . this degree of freedom can be used to set chirp equal to zero at one frequency . combining equations 5 and 8 for the two - section mz results in the equations 10 and 11 , 1 − 2 e − α ( f i ) l 1 + e − α ( f 1 )( l 1 + l 2 ) = 0 ( 10 ) equation 10 arises from setting chirp equal to zero at frequency , f 1 . equation 11 simply states that the sum of section lengths must equal the total length , l . the set of two simultaneous equations can be solved numerically using commonly available software . the solved variables are the lengths of first and second sections . the equations 12 , 13 and 14 describe a three - section mz having zero chirp at all frequencies . 1 − 2 e − α ( f 1 ) l 1 + 2 e − α ( f 1 )( l 1 + l 2 )− e − α ( f 1 )( l 1 + l 2 + l 3 ) = 0 ( 12 ) equation 12 sets chirp equal to zero at some frequency greater than zero . equation 13 sets the total length of domain inverted sections equal to the length of uninverted section , thereby setting chirp equal to zero at a frequency of zero ( dc ). equation 14 sets the sum of section lengths to the total length , l . this set of three simultaneous equations can be solved numerically . frequency , f 1 can be selected to be anywhere in the frequency range of interest , but is chosen to be 40 ghz for the examples described later . the solved variables are the lengths of the three different sections . equations 15 , 16 , 17 and 18 describe a four - section mz having zero chirp at all frequencies . 1 − 2 e − α ( 1 ) l 1 + 2 e − α ( f 1 )( l 1 + l 2 ) − 2 e − α ( f 1 )( l 1 + l 2 + l 3 ) + e − α ( f 1 )( l 1 + l 2 + l 3 + l 4 ) = 0 ( 15 ) 1 − 2 e − α ( f 2 ) l 1 + 2 e − α ( f 2 )( l 1 + l 2 ) − 2 e − α ( f 2 )( l 1 + l 2 + l 3 ) + e − α ( f 2 )( l 1 + l 2 + l 3 + l 4 )= 0 ( 16 ) equations 15 and 16 sets chirp equal to zero at two frequencies greater than zero while equation 17 sets the total length of domain inverted sections equal to the total length of uninverted sections , thereby setting chirp equal to zero at a dc . equation 18 sets the sum of section lengths equal to the total length , l . this set of four simultaneous equations can be solved numerically , as before . frequencies f 1 and f 2 can be selected to be anywhere in the frequency range of interest , but are chosen to be 20 ghz and 40 ghz for the examples described later . the frequency range of 0 to 40 ghz is the range of interest , hence f 1 and f 2 are chosen to be at mid - band and maximum , respectively . the solved variables are the lengths of the four different sections . fig4 a - e show the equations 19 - 1 through 19 - n for a zero - chirp mz with n sections , where n is an odd number . equations 19 - 1 through 19 -( n − 2 ) set chirp equal to zero at the n − 2 frequencies , f 1 , f 2 , . . . f n - 2 , which are all greater than zero . the frequencies can be selected to be evenly spaced across the frequency range of interest , or unevenly spaced , to force residual chirp to be extremely low in some critical portion of the frequency band . equation 19 -( n − 1 ) sets the total length of domain inverted sections equal to the total length of uninverted sections , thereby setting chirp equal to zero at a dc . equation 19 - n sets the sum of section lengths equal to the total length , l . the n equations are solved simultaneously . the solved variables are the lengths of the n sections . fig5 a - e shows the equations 20 - 1 through 20 - n for a zero - chirp mz with n sections , where n is an even number . similar to as shown before , equations 20 - 1 through 20 -( n − 2 ) set chirp equal to zero at the n − 2 frequencies , f 1 , f 2 , . . . f n - 2 , which are all greater than zero , while equation 20 -( n − 1 ) sets the total length of domain inverted sections equal to the total length of uninverted sections , thereby setting chirp equal to zero at a dc . equation 20 - n sets the sum of section lengths equal to the total length , l . as before , the solved variables are the lengths of the n sections . fig5 a - h show a more general set of equations for an mz with n sections , where n is an odd number , and the desired chirp is specified at n − 1 frequencies . the specified chirp can be non - zero for this more general set of equations . equations 21 - 1 through 21 -( n − 2 ), fig5 a - c , set chirp equal to n − 2 specified values at the n − 2 frequencies , f 1 , f 2 , . . . f n - 2 , which are all frequencies greater than zero , while equation 21 -( n − 1 ), fig5 d , sets chirp to some specified value at dc . equation 21 - n , fig5 e , sets the sum of section lengths equal to the total length , l . equations 21 -( n + 1 ), 21 -( n + 2 ), and 21 -( n + 3 ), fig5 f - h , define the coefficients in the other equations , which depend on the selected value of chirp at a particular frequency . parameters eo 1 and eo 2 are the relative modulation strength for a waveguide under the signal and ground electrodes , respectively . similar to fig5 a - h , fig5 a - h show a more general set of equations for an mz with n sections , where n is an even number , and the desired chirp is specified at n − 1 frequencies . equations 22 - 1 through 22 -( n − 2 ), fig5 a - c , set chirp equal to n − 2 specified values at the n − 2 frequencies , f 1 , f 2 , . . . f n - 2 , which are all frequencies greater than zero , while equation 22 -( n − 1 ), fig5 d , sets chirp to some specified value at dc . equation 22 - n , fig5 e , sets the sum of section lengths equal to the total length , l . as before , equations 22 -( n + 1 ), 21 ( n + 2 ), and 21 -( n + 3 ) define the coefficients in the other equations , fig5 f - h . parameters eo 1 and eo 2 are defined previously . fig5 a - h show a more general function for the equations shown in fig5 a - h to obtain an arbitrary chirp where n is an odd number of sections . equation 53c expresses the expanded functions of the n − 2 equations 21 - 1 through 21 -( n − 2 ). equations 53 d - h are the same as in fig5 defining the chirp , sum of the lengths and the equation coefficients of the other equations . fig5 a and b show equations that define the coefficients f j , 0 and f j , i shown in the equation in fig5 c . similarly fig5 a - h show a more general function for the equations shown in fig5 a - h to obtain an arbitrary chirp where n is an even number of sections . equation 54c expresses the expanded functions of the n − 2 equations 22 - 1 through 22 -( n − 2 ). equations 54 d - h are the same as in fig5 defining the chirp , sum of the lengths , and the equation coefficients of the other equations . fig5 a and b show equations that define the coefficients f j , 0 and f j , i shown in the equation in fig5 c . fig2 - 48 , describe the invention . two general features of the invention that are different from the prior art : ( 1 ) the lengths of the mz sections are unequal to one another and ( 2 ) the layout of domain inversion sections is not exactly symmetric about the midpoint of the rf electrode . in general , for zero - chirp designs , as was the case for the prior art zero - chirp designs , the total length of domain inverted sections equals the total length of non - inverted sections . the lengths of alternating domain sections are determined from the numerical optimization described earlier . the larger the number of sections , the larger the number of frequencies at which chirp can be set to zero . note that the chirp can also be set to non - zero values at the n − 1 frequencies , or the residual chirp can be made extremely low in a selected critical portion of the frequency band , and even n − 1 different chirp values at n − 1 different frequencies , though , it is typically difficult to find a solution to the equations for a set of chirp values that vary with frequency . in addition , as will be seen later , designs that target non - zero chirp values are more sensitive to variation in the intrinsic chirp ( parameters eo 1 and eo 2 ) and the degree of velocity matching . the choice of chirp set to zero at all n − 1 frequencies is the design of most interest . fig2 shows a three - section mz 80 where the sections 83 , 85 , 87 all have unequal lengths . the normalized lengths l 1 , l 2 , and l 3 are 0 . 22 , 0 . 50 , and 0 . 28 , respectively . the normalized length is the actual length divided by the total length of the active portion of the mz , equal to l 1 + l 2 + l 3 . the portions of the mz where the electrode crosses over are neglected , as there is negligible modulation produced there . note that the domain inverted section 85 is slightly off center compared to the prior art three - section mz 70 , where l 1 , l 2 , and l 3 are 0 . 25 , 0 . 50 , and 0 . 25 , respectively . also , the surrounding pair of sections l 1 and l 3 are arranged so that l 3 is longer than l 1 . sections 83 and 87 in fig2 differ by 6 % of total active mz length . rf signal electrode 86 is disposed over waveguide 82 at rf input 2 in section l 1 , it crosses to waveguide 81 in the domain inverted region 85 at l 2 and returns to waveguide 82 in section l 3 to couple to rf output 4 . the chirp vs . frequency curve is plotted for the nominal choice of design parameters , as well as some cases where certain parameters are slightly different from nominal . for most cases , chirp is flatter vs . frequency for the mz 80 shown in fig2 , compared to the prior art mz 60 shown in fig9 . for example , fig2 shows the chirp vs . frequency for the three - section mz 80 shown in fig2 . note that the 3 % shift in the location of the domain inverted section improves the flatness of chirp vs . frequency as compared to fig1 . the magnitude of the chirp is 0 . 01 or less across the entire frequency range , whereas the prior art three - section mz 60 with a centered domain - inverted section has chirp of 0 . 02 near 40 ghz . fig2 and 25 show chirp vs . frequency for the cases of + and − velocity walk - off . the magnitude of chirp remains near 0 . 02 or less over the entire frequency range , whereas chirp in the prior art mz 60 approaches 0 . 04 at 40 ghz , as seen in fig1 and 13 . however , the prior art mz 60 does have lower chirp at frequencies below 20 ghz . fig2 and 27 show chirp vs . frequency for the cases of 20 % higher and 20 % lower rf loss , respectively . note again that the magnitude of chirp is 0 . 01 or less across the entire frequency range , where as the chirp of the prior art mz 60 approaches 0 . 03 at 40 ghz for the case of 20 % higher rf loss , as seen in fig1 . fig2 and 29 show chirp vs . frequency for the cases of different intrinsic chirp ( 0 . 8 and 0 . 6 , respectively ). the magnitude of chirp is less than 0 . 01 across the frequency range , whereas the chirp of the prior art mz 60 is close to 0 . 02 at 40 ghz , as seen in fig1 and 18 . fig3 a and b shows the simulated crosstalk in the received i and q signals for the three - section mz 80 shown in fig2 . note that the magnitude of crosstalk is about 0 . 03 or less , a level which is comparable to the crosstalk for the prior art three - section mz 60 as shown in fig1 . fig3 shows a four - section mz 90 with unequal lengths . the normalized lengths l 1 , l 2 , l 3 , and l 4 , corresponding to sections 91 , 92 , 93 , 94 of the mz 90 are 0 . 130 , 0 . 335 , 0 . 370 , and 0 . 165 , respectively . the lengths for sections 91 and 94 and the lengths for sections 92 and 93 differ by 3 . 5 % of total length , l 1 + l 2 + l 3 + l 4 . fig3 shows chirp vs . frequency for the four - section mz 90 with unequal lengths . chirp is set to zero at dc , 20 ghz , and 40 ghz . note that the curve is very flat . chirp is 0 . 01 or less over the entire frequency range . fig3 - 38 show the sensitivity of the flatness of the chirp vs . frequency curve to various changes in parameters . fig3 and 34 show chirp vs . frequency with + 0 . 05 and − 0 . 05 velocity walk - off between rf and optical signals ( δn rf ), respectively . the magnitude of chirp is 0 . 02 or less over the entire frequency range . this is an improvement over the prior art three - section mz 60 , see fig1 - 13 , as well as the three - section mz 80 with unequal lengths , see fig2 - 25 . the chirp of the prior art three - section mz 60 is + 0 . 04 at 40 ghz , though , very small for frequencies below 20 ghz . the magnitude of the chirp of the three - section mz 80 with unequal lengths is never much above 0 . 02 , however , chirp is significant at frequencies below 20 ghz . fig3 and 36 show chirp vs . frequency for the cases of 20 % higher and 20 % lower rf loss , respectively , while fig3 and 38 show chirp vs . frequency for the cases of different intrinsic chirp ( 0 . 8 and 0 . 6 , respectively ). note that any of these curves show little change from chirp with the nominal design , shown in fig3 . the magnitude of chirp is 0 . 01 or less across the entire frequency range . this is a significant improvement over the prior art three - section mz 60 , where chirp increases roughly linearly with frequency and equals a value between 0 . 01 and 0 . 03 at 40 ghz , see fig1 - 18 , and a dramatic improvement of the prior art four section design 70 having equal lengths , see fig2 , where chirp increases to 0 . 08 at 34 ghz . fig3 a and b show the simulated crosstalk in the received i and q signals for the four - section mz 90 with unequal lengths and a positive velocity walk - off . note that the magnitude of the crosstalk is 0 . 01 or less , which is less than the magnitude of up to 0 . 03 for either the prior art three - section mz 60 , see fig1 , or the three - section mz 80 with unequal lengths , see fig3 . in general , the chirp vs . frequency curve becomes flatter as the number of sections is increased . the only drawback is that the amount of inactive length , corresponding to locations where the electrode crosses over to the adjacent waveguide , becomes a larger portion of overall mz length . it seems a four - section mz with unequal lengths has adequate performance , but designs with more sections are possible . designs were chirp is set to a non - zero value are also possible , allowing the chirp value to be tailored to the application requirement . fig4 shows chirp vs . frequency for a four - section mz 90 with unequal lengths , were the normalized lengths l 1 , l 2 , l 3 , and l 4 , corresponding to sections 91 , 92 , 93 , 94 of the mz 90 are 0 . 223 , 0 . 149 , 0 . 563 , and 0 . 066 , respectively . the chirp value is in the range 0 . 4 ± 0 . 01 across the entire frequency range for nominal choices of parameters . the intrinsic chirp of individual section is assumed to be 0 . 7 , as before . fig4 and 42 show chirp vs . frequency for the cases of 20 % higher and 20 % lower rf loss , respectively , while fig4 and 44 show chirp vs . frequency for the cases of different intrinsic chirp ( 0 . 8 and 0 . 6 , respectively ). the chirp vs . frequency curve shows little change when the values rf loss is modified , however , the value of chirp changes significantly when the intrinsic chirp is changed , the percentage change being approximately equal to the percentage change in intrinsic chirp . fig4 and 46 show chirp vs . frequency with + 0 . 05 and − 0 . 05 velocity walk - off between rf and optical signals ( δn rf ), respectively . the value of chirp drops to 0 . 3 at about 30 ghz in either case . the severity of this variation in chirp on system performance depends on the application . at any rate , designs where chirp is set to zero at all frequencies seem to be the most robust with regards to variations in such parameters as rf loss , velocity walk - off , and intrinsic chirp . tables 1 and 2 provide calculated examples of optimum normalized lengths for an n section mz , assuming a 01 = 0 . 0311 nepers /( cm -( ghz ) 0 . 5 ), corresponding to 0 . 27 db /( cm -( ghz ) 0 . 5 ) ( table 1 ), or a 01 = 0 . 0518 nepers /( cm -( ghz ) 0 . 5 ), corresponding to 0 . 45 db /( cm -( ghz ) 0 . 5 ) ( table 2 ), and 0 . 70 intrinsic chirp . there are some common patterns to the solutions shown in tables 1 and 2 . the longest section is near the middle of the mz . as one moves from that longest section to either the input or output of the mz , the sections become monotonically shorter and shorter . in addition , if one compares the 1 st section to the nth section , the nth section is longer . for mz &# 39 ; s with four or more sections , the n − 1 section is longer than the 2 nd section . for mz &# 39 ; s with six or more sections , the n − 2 section is longer than the 3 rd section , and so on . even though no two lengths are equal in any given design , the difference between two lengths can be 1 % or less of the total active mz length for the mz &# 39 ; s with larger numbers of sections . fig4 and 48 describe a modulator 100 , 120 that combines the electrode and domain reversal for obtaining zero chirp with the frequency response equalization described in u . s . pat . no . 6 , 580 , 840 . mz parts a and b of length l a and l b correspond to the parts before and after the compensation network 114 having length l 1 and l 2 in fig4 of the &# 39 ; 480 patent . the compensation network 110 is a simple rf delay line in fig4 and 48 , analogous to the rf delay line shown in fig5 of the &# 39 ; 480 patent . the modulators 100 , 120 of fig4 and 48 use a co - planar waveguide ( cpw ) rf electrode 124 with z - cut lithium niobate as the substrate ( not shown ), whereas the embodiments shown in fig4 and 5 of the &# 39 ; 480 patent show a co - planar strip ( cps ) rf electrode with x - cut lithium niobate . as before , only the signal electrode is shown in fig4 and 48 . ground electrodes in the cpw electrodes are not shown . the method by which frequency response flattening is accomplished in the modulators 100 , 120 shown in fig4 and 48 is relatively straightforward . mz part a produces most of the modulation at any frequency . mz part b produces modulation that is opposite to that of the part a at low frequency . at some high enough frequency , the phase shift caused by the delay line 110 adds in an additional 180 degrees of phase shift , canceling out the modulation polarity reversal between the parts a and b . the reduction of modulation at low frequency and increase in modulation at a higher frequency produces a flattening of the frequency response of the modulator 100 , 120 . note that the rf signal electrode 124 in the domain reversed region ( s ) in part b , having length l 6 in modulator 100 , and lengths l 5 , l 7 in modulator 120 , are on the opposite waveguide 102 , relative to the domain reversed regions in part a , l 2 , l 4 in modulator 100 and 120 . likewise , for the electrode locations in the regions without domain reversal , having lengths l 1 , l 3 , l 5 and l 7 in modulator 100 , and lengths l 1 , l 3 , l 6 in modulator 120 , the electrode location is chosen to be on the opposite waveguide , 104 in part b relative to part a in order that the modulation in part b cancels out some portion of the modulation produced in part a at low frequency . the use of domain reversal allows for two different design variants 100 , 120 which are shown in fig4 and 48 . in fig4 , the second section of length l 6 in part b has domain reversal , whereas in fig4 the first and third sections of lengths l 5 and l 7 of part b have domain reversal . one problem for some applications with the equalized design of the &# 39 ; 840 patent is non - zero chirp . the chirp can be made nearly zero by applying the domain and electrode reversal design rules discussed earlier to parts a and b , independently of one another . in particular , the number of mz sections , e . g ., 2 or more , need not be the same for parts a and b . part a may require more sections , as it produces most of the modulation . fig4 and 48 show a four - section mz design for part a , and a three - section mz design for part b . most likely , at least three sections are needed for either part a or b , as a two - section design produces a significant amount of chirp at some frequencies . to clarify the design process for the structures shown in fig4 and 48 , lengths l 1 , l 2 , l 3 , and l 4 normalized to length l a would be the same as for the lengths of the four section mz with unequal lengths , that is l 1 , l 2 , l 3 , and l 4 normalized to length l a are 0 . 130 , 0 . 335 , 0 . 370 , and 0 . 165 , respectively , as defined before . similarly , lengths l 5 , l 6 , and l 7 normalized to length l b correspond would be the same as for the three lengths of the three section mz with unequal lengths , that is lengths l 5 , l 6 , and l 7 normalized to length l b are 0 . 22 , 0 . 50 , and 0 . 28 , respectively , as defined before . hence , the mz sections a and b produce nearly zero chirp at all frequencies , independently of one another . thus the net chirp of the entire structure at all frequencies is also close to zero . note that these designs will have the lowest magnitude of chirp across the frequency range for the case where the rf and optical velocities are matched . the design described in the &# 39 ; 840 patent can accommodate a large amount velocity walk - off to allow for improved modulation efficiency ; however , a large amount of velocity walk - off is likely to cause the magnitude of chirp to be significantly larger at some frequencies than for the case where velocities are matched . hence , for the designs shown in fig4 and 48 , there is likely to be some trade - off between overall modulation efficiency and the degree to which chirp equals zero at all frequencies . the embodiment ( s ) of the invention described above is ( are ) intended to be exemplary only . the scope of the invention is therefore intended to be limited solely by the scope of the appended claims .