A simple and fast frequency domain analysis method for calculating the frequency response and linearity of electro-optic microring modulators

A simple and fas t frequency do main a nalysis metho d for cal culating the fr equency response a nd linearity of ele ctro - optic micro - r ing modula tors P AYAM RABIE I Partow Technologi es LL C, 1487 Poi nsett ia Ave, Suite 1 19, Vi sta, C A 92081 *pr@par tow - tech.com Abstract: A fas t and sim ple f requency domain m ethod i s introduc ed for the analysis of micro - ring modula tor resp onse us ing t he Jacobi – Anger expa nsio n met hod . Reso nance freq uenc y modulated micro - ring (FMMR) m odula tors an d coupl ing m odulate d micr o - ring m odu lators (CMMR) are analyzed using this method. The l inearity of these modulators is analyzed . T he t hird ord er intercept point (I P 3 ) is cal culated for C MMR devices and compared to Mach - Zehnder interferometer (MZI) modulator devices . It is shown that CMMR devices can achieve a 12d B highe r IP 3 compared to MZI devices. CMMR devices have high second order nonli near ity , while MZI devices’ seco nd o rde r nonlinea rit y is zero. A novel g eometry based on dual C MMR modul ators is introdu ced to i mprov e the s econd order n on linearity of CMMR modulators. Keywords : Integrated o ptics ; Lithi um niobate; Optic al resonato rs; Modulator s . 1. Introduct ion O ptoelec tronic sign al conv ersion i s the m ain bu ilding bl ock of any an alog ph otonic a pplicat ion. RF sig nal s modulate o ptical c arriers in ord er to utilize photonic signal pr ocessing cap abilities. M odulat ors w ith g ood line arity and modu lati on speeds bey ond 100 GHz are needed f or a variety of ana log p hot onic a pplication s . Among various materials and technolog ies, lithi um niob a te MZI modu lator s are wid e l y us ed in analog photonic ap plications. T his is due to their h igh linearit y , as w ell as the high - speed mod ul at ion performance. The electro - optic e ffect in lithi um niobate is e xtremel y fast , a nd is related to t he displacement of electrons with respect to cr ystal lattice s, which has a time constant of less tha n a femtosecond. T here is n o material - related b andwidth limitatio n for an electro - optic m odula tor made from lithium niobate . Ho wever , the mod ulato rs that are currently ma de using this material ha ve limit ed bandwi dth du e to t he absorption of R F sign als or phas e mismatch between RF si gnals and o ptical sig nals in the long modula tion el ectrodes of MZI device s [1] . Micro - ring modulators are a class of modulators that use the resonance effect to achieve small form factor devices. Various ty pes of micro - ring modulators hav e been dem onstra ted in th e p ast [2 -3 ]. The modulation can be achiev ed by a ch ange of the resonance frequen cy of the resonator (FMMR), a change in the a bsorpti on or Q of the resona tor , or a change i n the co up ling strength to the micr o- ring resonat or (CMMR). F ig. 1 sho ws FMMR and CMMR modul ator s . FMMR modul ator s are ver y co mpact; ho wever , the y have an i nher ent modul atio n spe ed l imit that is inversely prop ortional to resonato r Q , s ince the reson ator need s to be “ charged” or “discharged” to achieve m odulation . Pr evio usly , w e have exp eri m entally demonstrated FMMR mod u lators u sing lithium niobate. [4 -5]. (a) (b) Fig 1. (a ) FMMR modula tor ; (b) C MMR modu lator CMMR modulators , on the o ther hand , are always “ charged ” . Thr ough modulation o f the coupling it is po ssible to achieve a modulator that ca n completel y switch the light on and o ff at the output while the resonator is always charged [6 ]. This device is very compa ct co mpared to Mach - Zehnde r modu lators due to high sensit ivity cause d by the re sonator. A s opposed t o Mach - Zehnder devices , where a complete 180 - degree phase shift is n eeded for switching, in a CMMR modulat or , only a few degrees of shift is suffi cie nt to t urn t he mo dulat or o n and o ff. It has be en sho wn t hat CM MR modula to rs ha ve no in her ent op tical m odul ation ban dwidth limitati on imposed by cavity lifetime [7 -9 ]. A limitation will be imposed on t he b and widt h b y electronics and RF signal losse s similar to MZI modulators. H o we v er , s ince the electrodes for CMMR are shorter by a factor of 20 - 40 com pared to MZI devices, it is expected that much high er modu lation speeds are feasi ble using these dev ices. In thi s paper we introduce a m ethod to analyze th e linearity o f CMMR modu lators and sh ow in addit ion t o superi or bandw i dth th ey also exhibit hi gher linearity. Pr eviousl y , different methods have been introduced to analyze the dynamic performance of micr o - ri ng reso nato r s . Sacher and Poo n [7 -9 ] deri ved the dyna mic o f modulato rs for the fir st time using ti me do main simulation methods. T heir meth od for the first time sho wed that CMMR devices d o not ha ve a modulatio n speed limitatio n , as opposed to FMMR devices. Their calculation m ethod is a time domain method , and it is not easy to calcu late the frequency doma in respons e an d nonline arities as neede d for m any applic ation s. Hong and Enam i [12 ] also calculated t he d ynami c of mic ro - ri ng mod ulato rs us ing t he ti me domain method. Some other smal l signa l ana lysis freq ue ncy do main met hods ha ve b een introduced more recent ly [13 - 14 ]. Here we provide a f requency domain an alysis method that is v ery sim ple and is not limited to small signals. T he calc ulation is ver y fast and requires o nly a ti ny matr ix i nver sion. T his calculatio n meth od is qu it e general and can be applied to any resonance - based m odulator device . It provi des th e optic al res ponse of th e device and can be used to analyze any ph otonic modul ator devi ce i ncludi ng sil icon photonic micro - ri ng modul ator s. Usin g thi s anal ysis method, t he lin earity of these modulat ors for an alog ph otoni c applica tions are then easil y calculated. We will sho w that the critica l IP 3 of t hese m odulators is much better compared to MZI devices . We sho w that CMMR devices have a high second order nonline ari ty com pared to MZI . Thi s is not cr itical f or man y applicati ons since the second harmonic signal can be easily f iltered out. We also introduce a dual CMMR device stru cture that improves the secon d order non linearity . 2. Analysis of micro - ring modu lators for anal og photoni cs 2.1 Anal ysis of FMMR In order to model the FMMR modulator, we use the Jacobi – Anger funct ion e xpa nsio n met hod of th e pha se modul ated opti cal s ignal , similar to what has been done previ ously to an alyze the MZI and similar devices [9]. First we consider a simple FMMR device in which the reso nance frequency is shifted when a voltage is applied to the electrodes . Using the nota tion sho wn i n Fig. 1a. w e have: 0 o i i t E ae ω − = . (1) as the inp ut fi el d to the device. Assuming a multitude of frequen cies are generated in the m icro - ring mo dulat or, we write: 00 () () , RF RF fn i nt i nt n on n d E be E e ωω ωω ∞∞ −+ −+ = −∞ = −∞ = = ∑∑ . (2) After passin g a round t rip in th e electro - optic mic ro - ri ng mod ulato r , the p hase modulated generated signals can be expan ded usi ng the Jacobi – Ange r exp ansi on metho d : 00 ( ) (( )( ) ) () RF RF d DC i n t i n tt mn n n nm r n E e J be c ωω ωω ϕ βα ∞ ∞∞ − + − + −+ − = −∞ = −∞ = −∞ = = ∑ ∑∑ . (3) w here J is the Bessel fu nction ,  is the modulation dep th ,  is the round trip loss , and t d is the round trip d elay time. We introduce the nota tion: 11 1 00 0 0 11 1 0 , ,, 0 bc d ab c d ab c d bc d −− −                         = = = =                               . (4) We can then relate the c j coefficients to b j using t he matri x ide ntit y c Mb = ⋅ . (5) w here M is g iven b y: 2( ) 2( ) 2( ) 0 12 22 2 10 1 2( ) 2( ) 2( ) 21 0 () () () () () () () () () o RF d o RF d RF d DC od od od o RF d o RF d o RF d it it i t i it it it it it it Je J e J e Me Je J e J e J e Je J e πω ω πω ω πω ω ϕ πω πω πω πω ω πω ω πω ω ββ β α ββ β ββ β − −− −− − − ++ +     =                . (6) For the directional coupler of the res onator we can write : b ci a ρτ = ⋅+ ⋅ . (7) w e can the n so lve e quati ons 5 and 7 to o bta in the c oeffic ient   using 1 ( ). bi I M a τρ − = ⋅⋅ − . (8) w here ρ and τ are the coupling coefficients of coupler between the m icro - ring a nd the wave guide . Also , t he outp ut si gnal a mpl itud e co effic ient s are calculated : . d i Mb a τρ = −⋅ ⋅ ⋅ + . (9) Fig. 2 a and Fig. 2 b show the modul ated electric field a mplitude respons e of a FMMR for th e laser freq uenc y a nd t he fir st fo ur side band (i.e. coeffic ient d 0 , d 1 ,d -1 ,d 2 , d -2 ) for 5 GHz modul atio n fre que ncy and (a) 50GHz modula tion fr eque ncy , a nd (b ) as a funct ion o f the DC bias phase of th e resonator (or the e quival ent ly d etuni ng o f the las er fre que ncy fro m the resonator resonance frequen cy). The calculation is performed for α = 0.98 and a coup ling fact or r= 0.97 a nd a ring l ength o f 300 micro ns. T he β value is selected to be 0.0942 by assuming a V π .L of 4V - cm an d an appli ed RF v oltage of 2.1 volts to t he 3 00 - m icron l ong ele ctrodes of a device. As can be seen from these figure s, the amplitude of the generated sideband s i s signific antly lowe r for 50GHz modula tion fr eque ncy compared to 5GH z , whic h is a n ind icat ion o f st rong frequency dependence for the device. Also, th ere are peak s in th e generated sideband response. The bias phase φ DC to ge ner ate the peak i n mod ula tion r esp onse varie s f or the different frequencies. The peaks happen when generated sideband frequency is at resonance. Similar resul ts ha ve be en pr eviou sl y obta ined usin g other methods o f calculation i n the past [6 - 8]. (a ) (b) Fig . 2 T he g ene rate d fir st h arm oni c, s e cond harmo nic , an d f undame nt al ( las er fr eque ncy ) for (a) 5GH z a nd (b) 50GH z m odul atio n fr eq uency f or the FMMR m odul ato r. Once the amplitudes of electr ic - fi el d side - bands are calc ulated, we can calculate the optical inte nsity s ignal freq uenc y resp onse of the modulat or b y convertin g the side - band electric fiel d signa l amp lit ude to an inte nsi ty si gna l usin g the equat ion . n m nm m I dd ∞ ∗ − = −∞ = ∑ . (10) w here n is the n th generate d harmonic in the output modulated o ptical intensity signa l at the detector. Notice that here for n=0 , we get a DC si gnal. Figure 3 shows the calculated m odulated intensity signal frequency response of a FMMR modulator for the first ( n =1) and se cond ( n =2) harmonic s . The response is w ith respect to input laser power. As can be seen here, the f requ ency res ponse drops at h igher f requ ency and ha s a peak at 5GHz . The bias phase was selected such that the laser frequen cy is 6GH z away f rom the re sona nce fr eque nc y. As can also be seen i n the figure, a large second h armonic signal is generated. It is clear that FMMR devices do not addr ess the high - speed performance needed for RF photo nic s and we wil l not furthe r e xplo re the se de vice s in thi s pap er. Fig 3 . Ca lc ulat ed detec tor curr ent fr equ ency res pons e f or F MMR mod ula tor for f irst an d s econ d harmonics 2.2 Anal ysis of CMMR m odulat ors In order to achieve micro - r ing m odula tors w ith high - speed perform ance, one can use CMMR modul ator s as s hown in Fi g 1b . It ha s bee n sho wn that i n t his geo met ry, t here is theo reti cal ly no o ptical related limitation for high - frequency performance [6 - 8]. Here we develop a simple and q uick met hod to obt ain the d evice fr eque ncy re spo nse . We a lso analy ze the linearity of these modulators. Similar to the previous an alysis , we can analyze CMMR modulator response using the Jaco bi – Ange r func tio n exp ansi on me thod . T he MZI sectio n of the micro - ring mo dula to r mixes t he optical signals tha t are received by its two inputs with an RF si gnal applied to its electrodes. Since there is a f eedback loop , the generated side - lobes are fed back into the MZI section and mix aga in to pr oduc e othe r opt ical ha rmonic fre quenc y c ompone nt s. T he sy stem can be mode led as follows (using Fig 1b): 0 o it i E ae ω − = . (11) 00 () () , RF RF fn i nt i nt n on n d E be E e ωω ωω ∞∞ −+ −+ = −∞ = −∞ = = ∑∑ . (12) 00 () () ( ) RF RF d in t in t t nn nn r E e be c ωω ωω α ∞∞ −+ −+ − = −∞ = −∞ = = ∑∑ . (13) S im ila r to th e FM MR a nal ysis ,  is the ro und trip loss in the micro - ring resonat or. Aga in, u si ng the notation 11 1 00 0 0 11 1 0 , ,, 0 bc d ab c d ab c d bc d −− −                         = = = =                               . (14 ) we c a n wr i t e 3 c Mb = ⋅ . (15 ) w here her e  3 is a diagonal matr ix . I ts element are constant in ti me and can be easily writte n usin g ( 13 ) . Assumi ng a mul ti - tone i nput optical signal is fed to the MZI section fro m t he feedback section of th e devic e , each arm o f MZI section produces si de - bands rela ted to its modulation RF frequency that are related to its input optical signal via t he Bessel functio ns. In MZI section there are two - 3dB coupl ers , t wo modulating paths for the optical signal, as well as t wo inp ut s and two out puts . W e shoul d cal culate the contribution of each of the two input arms to each of t he t wo output arms via the two paths. The contr ibut ion fro m E r to E f after passi ng tw o mod ulati ng pat hs can be writ ten as: 00 0 2 () () 2 () () ( ) 2 () ( ) 2 DC RF RF DC RF i i nt i nt r mn f mn n nm i i nt mn mn n r n m nn e E e i J ce ie i J ce b ϕ ωω ωω ϕ ωω β β ∞ ∞∞ −+ −+ − − = −∞ = −∞ = −∞ − ∞∞ −+ − − = −∞ = −∞ = = ⋅ +− ∑ ∑∑ ∑∑ (16 )  is the modulatio n depth of t he phase modulatio n section s of MZI a nd is given b y  = (  /  ),   is the D C bia s phase difference on t he two ar ms of t he MZI sectio n , and J i s the Bessel function. We can write this equation using matrix notation: 1 r b Mc = ⋅ . (17 ) Similarl y, the contribution from E r to E o after passing two paths of the Mach - Zehnde r can be written as: 00 0 () () () () ( ) 2 () ( ) 2 DC RF RF DC RF i i nt i nt r mn o mn n nm i i r nn n nt mn mn nm ie E e i J ce e i J ce d ϕ ωω ωω ϕ ωω β β ∞ ∞∞ −+ −+ − − = −∞ = −∞ = −∞ − ∞∞ −+ − − = −∞ = −∞ − = = +− ∑ ∑∑ ∑∑ (18 ) We c an write this e qua tion using matr ix no tatio n: 2 R d Mc = ⋅ . (19 ) Next, we need t o add th e contr ibuti on f rom input sign al E i to E F and E R . The transfe r func tio n fr o m E i to E F and E R is simila r to equati ons 15 and 17 , an d is not repeated here. The results can be summarized by: 12 ri b b b Mc M a = + = ⋅+ ⋅ . (20 ) 21 . ir d d d Ma M c = += + ⋅ . (21 ) These matrices relate the signal co efficient of E F and E O to E R and E i . The y are similar to te r ms τ and ρ for a simple directional coupler , but ha ve Be ssel f unct ion el eme nts due to an elec tro - optic modulatio n section . B y s o lving e quat ions 15 and 20 , one obt ain s   value s: 1 13 2 ( .) b I MM M a − = −⋅ (22 ) The outp ut signa l co effi cie nts is then ob tained b y using eq uat ion 21 : 2 13 . d Ma M M b = +⋅ . (23 ) Here we us e this analysis method to inv estigate a few di fferent conditions f or the system respo nse. Fig. 4 sho ws the la ser freq uen cy, fir st a nd second har monic s ( i.e. , co efficient d 0 ,d 1 ,d -1 ,d 2 ,d -2 ) , amplitu de coefficients generated for electric field side - band s in a CMMR m odulator for 5G Hz and 50GH z as a f unction of DC bias point   . As oppos ed to the FMMR devices discussed abo ve, in t his ca se , bet ween z ero coup ling a nd critica l coupling bia s points, there are slight chan ges in t he a mpli tude o f the first harm onic generated electric Field signal s for 5GH z and 50 GHz modulati on f requenci es . In this case, for certain bias values , t he generated side - ba nd s at 50GH z are even slightly higher tha n the ge nerat ed si gnal at 5 GHz. There are also second har monic ge nera ted si gna l s a nd hig her har mo nics as s ho wn in F ig ure 4 . Th ese results theoreticall y d emonstrate that CMMR modulator s do not have hi gh - fre que ncy ro ll - off , as opposed t o FMMR modul ators . (a ) (b) Fig . 4 (a) The generated f irst h armon ic, second harmo n ic , a nd f undame ntal (lase r fre quency ) for (a) 5GH z and (b) 50GH z modul atio n fre quency for CM MR. Similar to the p revious ca se , we calculate the intensity signal that is generated after dete ction for CMMR devices using eq uat ion 1 0. Fi g. 5 sho ws the i ntens ity s i gnal frequ ency r esponse of the m odulator for two di ff erent bias valu es. The signal is relative to input laser po wer, As can be seen here, depending on the selected bias value it is po ssible to ob tain d iffere nt fre que ncy respo nses fo r the syste m. As can be seen later, w e wi ll us e a bias poin t between ze ro coupl ing and critical co upling for linear m odulat ors . He nce , t he lo w freque nc y res ponse f or linear modulators d evices will be simil ar to the case where t he bias is 0.15 in Fig. 5 . Si milar results have been publi shed pre viousl y for the frequ e ncy respons e of C MMR modula tors. [ 6-8] Fig . 5. The f re quen cy re spons e o f CMM R fo r two d iffe rent bias val ues 3. Line arity of C M MR m odulators For analog photo nic applicatio ns, the li nearity of a modulator i s very critical . Here w e analyze the lineari ty of these mod ulator s. Fig. 6 shows th e calculated fundamental and the se cond and thi rd harm onic f requency respon se of a CMMR m odulator a s fun ction of modu lation frequen cy. These signal le vels are with respect to input la ser po w er. The bias phase is appropriately selected between zero coupling and critical co upling po int and is equal to 0 .12 rad ian. Similar to MZI m odulators , t his bias phase result s in hi gh fir st har mo nic po wer and lo w higher harmonic signal pow er. The second harm onic is the do minant d isto rtio n harmo nic for a CMMR device. The third harmonic is very low for a CMMR device. As a co mparison, for a standard M ZI device, there is no second harmonic signal at the quadrature bia s poin t a nd the t hird har monic s ignal i s 3 5 dB lower than fir st har moni c si gnal for si milar fundamental si gnal power levels (i.e. ~ -50 dB in Fig . 6). Hence the CMMR has lower third harmonic distortion compared to MZI for h igh - fr equency modulat ion s peeds. F or m any applic ations , the s econd har monic distortion is not i mportant sin ce it can be filtered o ut. Henc e , CMMR de vice s may not o nly allow signi ficantly wideband oper ation s , but can also provide better linearit y. F ig. 6 Th e firs t, sec ond , and third harmon ic for a CMMR for different modulati on frequ encies For specific w ideband applications where the second harm onic might be critical, we propos e the device s truct ure t hat i s sho wn in F ig. 7 (a) . In t his d evice , two CMMR modulators are used. B y careful selec tion o f the RF signal t hat is app lied to the ele ctrodes o f this modulator, it is possible to elim inate the second harmonic si de - band generated in CMMR modulators. In order to achieve this , the modul ated RF si gnal a pplied t o the s econd mi cro - ring devi ce electrodes must be phase shifted by 90 degrees wi th respect to the first device. Also , the bi as poin t for the com binati on of modul ated s ignals before th e out put coupl er sh ould be 90 degrees o ut of p hase with respect to each oth er. Figure 7(b) shows the calculated first, second , and third har monic signal levels with respe ct to input optical po wer after d etectio n for a dual CMMR (DCMMR) device . In thi s devic e t he second harmonic is completely e l imina ted i n the modulated electric field signal. Howev er, t her e is still a second har monic si gnal in the intensit y signal. Ho wever, the amplit ude of the second har monic i ntensi t y signa l is signi fica ntly l ower for D CMMR modulators compared to sin gle - CMMR m odulators. One disadvantage of DCMMR devices is inherent optical los s of the devic e. A n add itiona l inhe re nt - 9d B loss e xists in D CMM R devi ces . (a) (b) Fig . 7 . Prop osed devic e to achie ve high line arity based o n D CMMR modulator s ; (b) Cal cula ted fun dame ntal , seco nd harm onic , and t hird harmo nic sig nal d istortion l evel s for D CMMR (a) (b ) Fig. 8. The SFDR a nd its compar ison with a simple M ach - Zehnder modula tor for double C MMR (solid line) a nd Mach - Ze hnder (das hed line ) modul ato rs Figur e 8 sho ws the fund ame ntal , seco nd ha rmonic , and thir d har monic l eve ls fo r CMMR and DCMMR devices and com pa r es the results with MZI devices. The results are plotted fo r a modul atio n fre que ncy o f 10 0GHz. For single - CMM R devic e s, the third order harmoni c powe r is approx i m ately 25 dB l ow er than MZI devi ces for si milar funda mental signal po wer levels . Thi s translates to a 12 .5 dB impro veme nt in the IP 3 compared to MZ I devices. For l ow er frequencies the im provement is less. For exam ple, at 50GHz the improv e men t is 10dB i n IP 3 and a t 200GHz the i m provement is 17dB for IP 3 . For ve ry low frequencies (less t han 20GHz ) the third har monic distortion IP 3 is similar to MZI. For dua l - CMMR, d ue to additio nal i nherent insertion los s devices, t he IP 3 and thi rd ha rm onic distortion is si milar , slightly wor se , o r slightl y bet ter d epe nding on t he fr equenc y co mpar ed to MZI d evice s. One iss ue with the se devices is the low frequency distortions that are caused by the resonator cavit y d ynamic s , wh ic h al so repeats once the modulation frequency reaches the free spectral range of the device (i.e. , 0 Hz and 500GHz in our device). This pro blem can be easily solved usin g the metho ds p revio usly described initially by Popovi c [15 ], an d wa s r e - iter ated in a mo r e recent publication by Kodanev and Orenste in [10]. Basically , anothe r port is added to compensat e for energy loss from the mi cro - resonator which completely eliminate t he cavity dyna mics i n the modula tio n res ponse . We expect a frequency independent IP 3 im provem e nt of 12 dB to be practical u sing these de sign s. 4. Conclusion We have developed a novel si mp le method to analyze the micro - ring modul ator freq uenc y response. We applied this method to resonance FMMR m odulators as well as CMMR modulators. The frequency response results obtained are similar to the results obtained pre viousl y usi ng the t ime d o ma in simulati on meth od [7 -9 ]. The analysis shows that CMMR s do no t have a modul ation s peed limit imposed by photon life time in the resonato r . Thi s theor y was then applied t o analyze th e linearity of m icro - ring modulators. Single - CM MR device s a nd doubl e - CMMR d evices were analyzed for their linearity and were compared to MZI devices. The resul t s show tha t CMM R s ha ve superior performan ce due to much low er third ord er distortio n compared to MZI devices. However, CMMR devices have a large second order dist ortion that might be problematic for so me applicatio ns. D ouble - CMMR devices have low se cond order di storti on , but the im provemen t in thi rd order dist ortion is com pe nsa ted by addition al inheren t insert ion loss of the device. There i s lo w freq uenc y di stor tion i n th e d evice cause d b y the ca vity d ynami cs that can be fixed by using a dditional po rts in the device to as was sh own in [11] and [ 15 ] . Acknow ledge m ent: This work is suppor ted by the NASA STTR Prog ram un der contract numbe r T8.02 - 9806 (STTR 2016 - 1). References [1] E. Wooten, et. 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