Ryo Kawata 1a), Tatsuhiko Watanabe 1,2b), and Yasuo Kokubun 3c) 1 Graduate School of Engineering, Yokohama National University,

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1 LETTER IEICE Elctronics Exprss, Vol.15, No.1, 1 1 Full-st high-spd mod analysis in fw-mod fibrs by polarization-split sgmntd cohrnt dtction mthod: Proposal and simulation of calculation rror Ryo Kawata 1a), Tatsuhiko Watanab 1,b), and Yasuo Kokubun 3c) 1 Graduat School of Enginring, Yokohama National Univrsity, 79 5 Tokiwadai, Hodogaya-ku, Yokohama 4 851, Japan Prsntly with ETH Zurich Institut of Elctromagntic Filds (IEF), ETZ K94, Gloriastrass 35, 89 Zurich, Switzrland 3 Faculty of Enginring, Yokohama National Univrsity a) kawata-ryo-zp@ynu.jp b) watanabt@thz.ch c) kokubun-yasuo-sd@ynu.ac.jp Abstract: W propos a novl mod analysis mthod namd th polarization-split sgmntd cohrnt dtction (PSCD) mthod, which can obtain a full st of amplituds, phass, and polarization stats of guidd mods including dgnrat mods in fw-mod fibrs within th spd limit imposd by th rspons tim of th lctronics. Th dtaild calculation procss of formulas for th complx amplituds of th LP 1,LP vn, and LP odd mods from th masurd intr-frquncy signals of th four lmnts of a quadrant photodtctor is dscribd. In addition, th calculation rrors of th formulas for th LP mods as functions of th rotation of th quadrant photodtctor and th offst of th bam from th local oscillator ar valuatd by numrical simulation. From ths rror analyss, w driv formulas to compnsat for th ffct of th rotation of th quadrant photodtctor and to rduc th rror du to th offst of th bam from th local oscillator. Kywords: fw mod fibr, mod division multiplxing, LP mod, cohrnt dtction, mod analysis Classification: Optical systms Rfrncs IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 [1] H. Kubota, t al.: Intrnal group dlay disprsion in fw-mod fibr, IEICE Elctron. Exprss 7 (1) 155 (DOI: /lx.7.155). 1

2 [] D. Soma, t al.: 1.16 Pta-bit/s dns SDM/WDM transmission ovr low-dmd 6-mod 19-cor fibr across C+L band, ECOC 17 (17) Th.PDP.A.1. [3] K. Iga and Y. Kokubun: An optical fibr mod analyzr using th rfraction from th obliqu sction, IEICE Trans. E6 (1977) 1. [4] I. P. Gils, t al.: Mthod to visualis and masur individual mods in a fw modd fibr, ECOC1 (1) Tu.1.F.5 (DOI: /ECEOC.1. Tu.1.F.5). [5] R. Gabt, t al.: Charactrization of fw mod fibrs by OLCI tchniqu, ECOC 14 (14) Th.1.4. (DOI: 1.9/ECOC ). [6] S. Brdagué and P. Facq: Mod division multiplxing in optical fibrs, Appl. Opt. 1 (198) 195 (DOI: /AO.1.195). [7] J. W. Nicholson, t al.: Spatially and spctrally rsolvd imaging of modal contnt in larg-mod-ara fibrs, Opt. Exprss 16 (8) 733 (DOI: /OE ). [8] D. M. Nguyn, t al.: Modal dcomposition tchniqu for multimod fibrs, Appl. Opt. 51 (1) 45 (DOI: /AO.51.45). [9] D. R. Gray, t al.: Mitigating spctral lakag and sampling rrors in spatial and spctral (S) imaging, OFC 15 (15) W4I.6. [1] N. K. Fontain, Charactrization of spac-division multiplxing fibrs using swpt-wavlngth intrfromtry, OFC 15 (15) W4I.7 (DOI: / OFC.15.W4I.7). [] M. Ohashi, t al.: Longitudinal fibr paramtr masurmnts of multi-cor fibr using OTDR, Opt. Exprss (14) 3137 (DOI: /OE ). [1] M. Nakazawa, t al.: Masurmnt of mod coupling distribution along a fw-mod fibr using a synchronous multi-channl OTDR, Opt. Exprss (14) 3199 (DOI: /OE..3199). [13] Y. Kokubun, t al.: Full-st mod analysis of thr-mod fibrs calculatd from polarization componnts of nar-fild pattrn, Jpn. J. Appl. Phys. 55 (16) 8RB3 (DOI: /JJAP.55.8RB3). [14] T. Watanab, t al.: Instantanous mod analysis of dgnratd orthogonal polarization mods using polarization-splittd cohrnt dtction by quadrant dtctor, IEICE Socity Confrnc (15) B-1-1 (in Japans). [15] Y. Kokubun, t al.: What is a mod in fw mod fibrs?: Proposal of MIMOfr mod division multiplxing using tru ignmods, IEICE Elctron. Exprss 13 (16) (DOI: /lx ). 1 Introduction IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 Mod analysis tchniqu is indispnsabl to undrstand dply th transmission charactristics of fw mod fibrs (FMF s) [1, ]. Thr hav bn svral mod analysis mthods, which can b catgorizd into two groups. On consists of mthods that analyz th mods of output light basd on information on th modal pattrn of th lctromagntic fild, such as prism-coupld mod analysis using an obliqu sction of an optical fibr [3] and a taprd fibr [4], th low-cohrnc intrfromtry mthod [5], th ignmod xpansion tchniqu [6], th spatial and spctral (S) imaging mthod [7, 8, 9], th swpt wavlngth intrfromtry mthod [1], and so on. Th othr consists of tim-of-flight (ToF) mthods such as th OTDR mthod [, 1]. Thr has bn no mod analysis mthod, howvr,

3 that can obtain a full st of th amplituds, phass, and polarization stats of transmittd mods including dgnrat mods. Thus, w proposd and dmonstratd a mod analysis mthod namd th intnsity profils from angld polarizr (IPAP) mthod [13]. This mthod is basd on th calculation of all modal componnts involving th amplituds, phass, and polarization stats from th intnsity profils of an NFP through an angld polarizr, which can b rotatd manually. Thr has bn a problm, howvr, that th masurmnt tim is limitd by th mchanical moving tim of th rotating polarizr. To solv this problm, w prviously proposd a novl high-spd mod analysis mthod namd th polarization-split sgmntd cohrnt dtction (PSCD) mthod [14] and drivd calculation formulas for th complx amplituds of th LP 1,LP vn, and LPodd mods from th masurd intr-frquncy signals of th four lmnts of a quadrant photodtctor. Sinc this mthod dos not involv any mchanically movabl lmnts, this mod rcognition principl can b xtndd to a mod rcivr for mod-division multiplxing transmission without a mod dmultiplxr or mod discrimination optical lmnt if th rspons spd of th quadrant photodtctor is sufficintly high. In this papr, w dscrib th dtaild drivation procss of formulas for calculating th complx amplitud of LP 1,LP vn, and LPodd mods from th dtctd photocurrnts of four lmnts of th quadrant photodtctor, and valuat th calculation rrors of th formulas in th cas of th rotation of th quadrant photodtctor and th offst of th bam from th local oscillator. From ths rror analyss, w driv formulas to compnsat for th ffct of th rotation of th quadrant photodtctor and to rduc th rror du to th offst of th bam from th local oscillator. Outlin of masurmnt systm Fig. 1 shows an outlin of th masurmnt systm. First, w collimat th signal bam from a fw-mod fibr and combin it with a collimatd bam from th local oscillator using a translucnt mirror. Th polarization dirction of th bam from th local oscillator is convrtd to circularly polarizd light using a =4 plat or rotatd by =4 rad using a = plat to involv th x and y polarization componnts with th sam ratio. Nxt, th mixd light is sparatd into th x and y polarization componnts using a polarization bam splittr. Thn ach polarization componnt is dtctd by a quadrant photodtctor, and four intr-frquncy lctric signals IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 Fig. 1. Outlin of mod analysis systm. 3

4 whos frquncy corrsponds to th diffrnc btwn thos of th signal and th local oscillator ar usd to calculat th complx amplituds of th LP 1,LP vn, and LP odd mods by th addition and subtraction of th dtctd signals. 3 Thory of analysis of LP mods 3.1 Dfinition of lctric fild vctor of LP mods and rlationship with htrodyn dtctd signals W driv formulas to calculat th amplitud and phas of th LP mods from htrodyn dtctd signals of individual lmnts of th quadrant photodtctor. Th ara numbrs of th quadrant photodtctor is dfind as shown in Fig.. Fig.. Dfinition of ara numbrs of quadrant photodtctor. First, w dfin th amplitud and rlativ phas of th x and y polarization componnts of th LP 1,LP vn, and LPodd mods as shown in Tabl I. Th rlativ phas is masurd as th phas diffrnc from th x polarization componnt of th LP 1 mod. Tabl I. Dfinitions of amplitud and phas of LP 1,LP vn, and LPodd mods. Lt E total b th lctric fild of th composit wav comprising ths optical filds. Hr, E LP1, E LP, and E LP o ar th transvrs fild amplitud functions of th LP 1,LP vn, and LPodd mods, and x and y ar th unit vctors in th x and y dirctions, rspctivly. E total ¼fA x x E LP1 ðx; yþþa y y E LP1 ðx; yþ j IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 þ B x x E LP ðx; yþj 1 þ B y y E LP ðx; yþj þ B x o x E o LP ðx; yþj 3 þ B y o y E o LP ðx; yþj 4 g j!t : ð1þ 4

5 Hr ω is th angular frquncy of th light sourc. Lt us dfin th lctric fild of th bam from th local oscillator and th polarization vctor of th polarizr by Eqs. () (3), whr D is th amplitud of th light bam from th local oscillator,! is th small diffrnc in th angular frquncy btwn th signal light bam from th FMF and th bam from th local oscillator, and δ is th azimuth angl of th polarizr. E LO ¼ Dð x þ j y Þ jð!!þt ; pol ¼ x cos þ y sin : To simplify th calculation, w assum a plan wav for th bam from th local oscillator, although in th cas of a Gaussian bam, D dpnds on th position in th transvrs cross sction. Also, w assum circular polarization of th light bam from th local oscillator aftr passing through a =4 plat. Th intnsity profil of th lctric fild of th output optical bam from th FMF givn by Eq. (1), which is multiplxd with th bam from th local oscillator givn by Eq. (), can b writtn as rffiffiffiffiffi " Pðx; yþ ¼ jðe total þ E LO Þ pol j : ð4þ This intnsity profil is dtctd by a quadrant dtctor, and th output photocurrnt signals from th four channls of th quadrant photodtctor can b calculatd from Eqs. (1) (3). Sinc th photocurrnt of th combind bam is a sinusoidal signal with angular frquncy!, th signal currnt i AC ðx; y; tþ at position ðx; yþ on th surfac of th photodtctor can b drivd as follows by substituting Eqs. (1), (), and (3) into Eq. (4): i AC ðx; y; tþ /E LP1 ½A x cos cosð!tþþa y sin sinð!t þ Þ þ cos sin fa x sinð!tþþa y cosð!t þ ÞgŠ þ ELP ½B x cos cosð!t þ 1 ÞþB y sin sinð!t þ Þ þ cos sin fb x sinð!t þ 1 ÞþB y cosð!t þ ÞgŠ þ ELP o ½B x o cos cosð!t þ 3 ÞþB o y sin sinð!t þ 4 Þ þ cos sin fb o x sinð!t þ 3 ÞþB o y cosð!t þ 4 ÞgŠ: First, w assum that th lctric fild functions E LP1, E LP, and E LP o can b xprssd in th form of sparat variabls, and thy ar xprssd in trms of Cartsian coordinats by Eqs. (6) (8), rspctivly, ðþ ð3þ ð5þ IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 E LP1 ðx; yþ ¼E ðþ ðxþe ðþ ðyþ; E LP ðx; yþ ¼E ð1þ ðxþe ðþ ðyþ; E LP o ðx; yþ ¼E ðþ ðxþe ð1þ ðyþ; whr th suprscripts () and (1) rprsnt th mod ordr in ithr in th x or y dirction. This assumption is introducd to asily calculat th intgration ovr th dtction ara of th quadrant photodtctor. If an accurat lctric fild profil has bn obtaind for th FMF by masurmnt or thortical calculation, th intgration can b numrically calculatd using th fild profil. Sinc th Gaussian ð6þ ð7þ ð8þ 5

6 w function in th Cartsian coordinat systm can b asily transformd into th polar coordinat systm ðr; Þ using xp x þ y ¼ xp r, th following formulas can also b drivd using th polar coordinat systm. In such a cas, th Lagurr-Gaussian function is usd instad of th Hrmit-Gaussian function. For highr ordr LP 1 and LP mods, th complx amplituds can b masurd using a sgmntd octant photodtctor dividd in th azimuth dirction with dual partitioning in th radial dirction. Th formulas for LP 1 and LP mods can b drivd in a similar way using th polar coordinat systm and Lagurr-Gaussian function. Whn E ðþ ðxþ is approximatd by th Hrmit-Gaussian function, E ðþ ðxþ and E ð1þ ðxþ can b xprssd as E ðþ ðxþ ¼N ð w x Þ ; p E ð1þ ðxþ ¼N 1 ffiffi x w ð x w Þ ; w ð9þ ð1þ whr w is th spot siz and N and N 1 ar th normalization cofficints givn by 1 1 N ¼ p 1= and N 1 ¼ p 1=. E ðþ ðyþ and E ð1þ ðyþ ar obtaind by rplacing x w ffiffi w ffiffi by y in Eqs. (9) and (1). Using Eqs. (9) and (1), Eqs. (6) (8) can b rwrittn as E LP1 ðx; yþ ¼N ð w x Þ ð y w Þ ; p E LP ðx; yþ ¼N N 1 ffiffi x ð w x Þ ð y w Þ ; w p E LP o ðx; yþ ¼N N 1 ffiffi y ð w x Þ ð y w Þ : w ðþ ð1þ ð13þ Using Eqs. () (13), w can calculat th photocurrnts I 1AC I 4AC from th first to fourth channls of th quadrant dtctor. Hr, w assum that th ara of th photodtctor surfac is sufficintly larg to dtct th ntir powr of th light bam. Evn if this is not th cas, although th normalization cofficints ar diffrnt from N and N 1, thr is no ssntial diffrnc in th principl of th calculation. IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, Signal photocurrnt on x and y polarization sids Lt us calculat th intgral of th lctric fild amplitud in ach ara of th quadrant dtctor. Whn th intgration of th lctric fild profil E LP1 ovr th ara μ of th photodtctor, whr μ is th ara numbr of th quadrant photodtctor shown in Fig., is rprsntd by C ;LP1, C 1;LP1 -C 4;LP1 ar obtaind as follows. Z 1 Z 1 Z 1 Z 1 C 1;LP1 ¼ E LP1 ðx; yþdxdy ¼ N ð w x Þ ð y w Þ dx dy ¼ N w ¼ C ;LP1 ¼ C 3;LP1 ¼ C 4;LP1 : ð14þ 4 Nxt, sinc E LP ðx; yþ is an vn function with rspct to th x axis from Eq. (1), w obtain 6

7 C 1;LP ¼ Z 1 Z 1 p N N 1 ffiffi x w ð x w Þ ð y w Þ dx dy ¼ p ffiffiffi Z N N 1 1 w w y ð w Þ dy p ¼ ffiffi N N 1 w w pffiffiffiffiffi pffiffiffi N N 1 w ¼ : ð15þ Similarly, C ;LP C 4;LP and C 1;LP o C 4;LP o ar obtaind. Thn w can driv th photocurrnt signal dtctd in ach dtction ara of th quadrant dtctor using ths cofficints. Th photocurrnt signal of th first channl of th quadrant dtctor is givn by whr I 1AC ¼ ½A x cos cosð!tþþa y sin sinð!t þ Þ þ cos sin fa x sinð!tþþa y cosð!t þ ÞgŠ þ ½B x cos cosð!t þ 1 ÞþB y sin sinð!t þ Þ þ cos sin fb x sinð!t þ 1 ÞþB y cosð!t þ ÞgŠ þ ½B o x cos cosð!t þ 3 ÞþB o y sin sinð!t þ 4 Þ þ cos sin fb o x sinð!t þ 3 ÞþB o y cosð!t þ 4 ÞgŠ; ð16þ ¼ N w ; ð17þ 4 pffiffiffiffiffi N N 1 w ¼ : ð18þ Similarly, w can also calculat I AC (μ: ara numbr) as IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 I AC ¼ ½A x cos cosð!tþþa y sin sinð!t þ Þ þ cos sin fa x sinð!tþþa y cosð!t þ ÞgŠ þ j½ Š int ½B x cos cosð!t þ 1 ÞþB y sin sinð!t þ Þ þ cos sin fb x sinð!t þ 1 ÞþB y cosð!t þ ÞgŠ þ j½ Š int ½B o x cos cosð!t þ 3 ÞþB o y sin sinð!t þ 4 Þ þ cos sin fb o x sinð!t þ 3 ÞþB o y cosð!t þ 4 ÞgŠ: ð19þ For th signal photocurrnt from th μth channl of th quadrant dtctor on th x polarization sid in Fig. 1, substituting ¼ into Eq. (19), w can obtain th following xprssions. I ðxþ 1AC ¼ A x cosð!tþþb x cosð!t þ 1 ÞþB o x cosð!t þ 3 Þ; ðþ I ðxþ AC ¼ A x cosð!tþ B x cosð!t þ 1 ÞþB o x cosð!t þ 3 Þ; ð1þ I ðxþ 3AC ¼ A x cosð!tþ B x cosð!t þ 1 Þ B o x cosð!t þ 3 Þ; ðþ I ðxþ 4AC ¼ A x cosð!tþþb x cosð!t þ 1 Þ B o x cosð!t þ 3 Þ: ð3þ Similarly, w can obtain th formulas for th signal photocurrnt from th μth channl of th quadrant dtctor on th y polarization sid by substituting ¼ = into Eq. (19). 7

8 3.3 Calculation of complx amplitud and phas diffrncs btwn mods From th addition and subtraction of Eqs. () (3), w can driv th six complx amplituds of th x and y polarization componnts of th LP 1,LP vn, and LPodd mods, which consist of th amplitud and phas diffrnc from that of th x polarization of LP 1. Th sum of Eqs. () (3) givs I ðxþ 1AC þ I ðxþ AC þ I ðxþ 3AC þ I ðxþ 4AC ¼ 4A x cosð!tzþ: ð4þ Th sum of th complx amplituds of th x polarization componnts of th LP vn and LP odd mods vanishs, and w can obtain th amplitud of th x polarization componnt of th LP 1 mod. Th subtraction and addition of first and scond channls giv I ðxþ ðxþ 1AC IAC ¼ B x cosð!t þ 1 Þ; I ðxþ 1AC þ I ðxþ AC ¼ B x o cosð!t þ 3 Þ: ð5þ ð6þ Thus, w can obtain th amplitud and phas of th x polarization componnts of th LP vn and odd mods as follows: B x ¼ 1 ðxþ ðxþ Amp½I1AC IAC Š; B o x ¼ 1 ðxþ Amp½I1AC þ I ðxþ AC Š; 1 ¼ 1 ðxþ ðxþ Phas½I1AC IAC Š; 3 ¼ 1 ðxþ Phas½I1AC þ I ðxþ AC Š; ð7þ ð8þ ð9þ ð3þ whr Amp[ ] and Phas[ ] ar th absolut valu and th phas of th argumnt, which is a sinusoidal function, rspctivly. Similarly, w can obtain th amplitud and phas of th y polarization componnts of th LP vn and odd mods from th formulas similar to Eqs. () (3), which giv th photocurrnt on th y polarization sid. In gnral, LP mods ar rprsntd by a linar combination of TE 1,TM 1, and HE 1 mods [15]. Th matrix rlating tru ignmods and LP mods is givn by Eq. (7) in Rf [15], and th invrs matrix is drivd as Eq. (3) in Rf [15]. Thrfor, using th complx amplituds of th LP mods obtaind by th proposd mthod, w can also driv th complx amplituds of th tru ignmods using ths matrix formulas. 4 Evaluation of calculation rror of formulas for LP 1 and LP mods IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 In th drivation of th formulas for calculating th complx amplituds of th LP 1 and LP mods in sctions , w assumd that th axis of th quadrant photodtctor is adjustd to th x and y axs of th Cartsian coordinat systm. In an actual masurmnt stup, howvr, th axs ar considrd to b rotatd and 8

9 shiftd to som xtnt. Thrfor, w valuatd th calculation rrors of th formulas for th LP 1 and LP mods for two cass: i.., th cas whr th quadrant photodtctor is rotatd by σ rlativ to th x axis of th Cartsian coordinats, which is th axis of th polarization splittr, and th cas whr th cntr axis of th bam from th local oscillator is shiftd by ζ. First, spcific valus of th amplitud and phas and ithr of two disturbancs, i.., th rotation of th quadrant photodtctor or th offst btwn th signal bam and th rfrnc bam from th local oscillator, ar assumd. Nxt, th photocurrnts of lmnts of th quadrant photodtctor ar calculatd from Eq. (4). In th calculation of filds of LP 1,LP vn, and LPodd mods, w assumd a stp-indx fibr with V ¼ 3:59 ( ¼ :348%, a ¼ 6:65 µm). Finally, th complx amplituds of th LP mods ar calculatd using Eqs. (7) (3) and similar formulas, and th calculation rror is valuatd by comparing th initially assumd valus of th amplitud and phas with th calculatd valus. 4.1 Calculation rror du to rotation of photodtctor Lt us considr th cas that th quadrant photodtctor is rotatd by σ. Sinc th complx amplitud of th LP 1 mod is obtaind as th sum of all th signals from th four lmnts of th quadrant photodtctor, th calculation rror for th LP 1 mod is zro. Thrfor, w considrd thr typical cass for th valuation of th amplitud rror: i.., only on of th LP vn and LP odd mods is incidnt on th quadrant photodtctor, and both mods ar incidnt on th dtctor with qual amplituds. In ths cass, th phas diffrnc btwn th LP mods and th LP 1 x-polarizd mod is assumd to b zro, i.., 1 and 3 ¼. Th calculation rrors of amplituds B x and Bo x wr valuatd as th diffrnc btwn th calculatd valus of B x and Bo x and th initially givn valus of B x and Bo x, i.., 1.. Th calculation rrors of th phas diffrncs 1 and 3 wr valuatd as th diffrnc btwn th calculatd valus of 1 and 3 and th initially givn valus of 1 and 3. Th simulatd rsults of th calculation rror of th amplitud ar shown in Fig. 3(a). It can b sn from Fig. 3(a) that th rotation of th quadrant photodtctr from th axis of th polarization splittr, i.., th bas axis of th Cartsian coordinats, significantly affcts th calculation rror. On th othr hand, sinc th IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 Fig. 3. (a) Amplitud. (b) Phas diffrnc. Simulatd rsults of calculation rror du to rotation of photodtctor. 9

10 phas diffrncs 1 and 3 ar assumd to b zro, th phas rror was valuatd to b zro. To simulat th calculation rror of th phas, w considrd four typical cass: i.., 1 ¼ and 3 ¼, 1 ¼ and 3 ¼, 1 ¼ 4 and 3 ¼, and 1 ¼ and 3 ¼ 4. In ths cass, th amplituds of LPvn and LP odd mods ar assumd to b unity, i.., B x ¼ Bo x ¼ 1. Th simulatd rsults ar shown in Fig. 3(b). It is sn from this figur that th maximum rror of th phas diffrnc from th phas of LP 1 x-polarizd mod is largr than.3 [rad]. Although th rotation of th quadrant photodtctor affcts th calculation rror, w can compnsat for this ffct by taking into account th rotation of th coordinat axis in th drivation in Sctions , and th corrctd formula for th LP x-polarizd mods is givn by " # B x B o x ¼ 1 " # " # u v I1AC x v u I x AC ð31þ whr u and v ar th corrctd cofficints for th rotation angl of th photodtctor and ar givn by u ¼ cos sin ; v ¼ cos þ sin : ð3þ Using th corrctd formula Eq. (31), th calculation rror can b rducd to almost zro, as shown in Figs. 3(a) and (b). 4. Calculation rror du to offst of light bam Nxt w considr th cas whr th cntr axis of th bam from th local oscillator is shiftd by ζ as shown in Fig. 4. Fig. 4. Dfinition of offst ζ of light bam from local oscillator. IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 In this simulation, th cntr axis of th light bam from th fw-mod fibr is assumd to b adjustd to that of th quadrant photodtctor. In addition, th dirction of th offst is assumd to b along th x axis bcaus th ffct of th offst taks its maximum valu in this cas. Although th light bam from th local oscillator was assumd to b a plan wav in Sctions to simplify th calculation, a Gaussian bam with a spot siz of W was assumd in this simulation bcaus a narly Gaussian bam from a singl-mod fibr is usd in th actual masurmnt stup. Th spot siz of th light bam from a fw-mod fibr was dfind by W f, and W f and ζ wr normalizd by W. Th intgration of th optical powr dnsity givn by Eq. (4) was prformd from r ¼ to 1 assuming that th diamtr of th quadrant photodtctor is sufficintly larg compard with W and 1

11 W f. Th amplituds and th phas diffrncs in Eqs. (7) (3) can b calculatd using (I ðxþ ðxþ ðxþ 1AC IAC ) and (I1AC þ I ðxþ AC ). This is bcaus thr is rdundancy among th photocurrnts from th four lmnts of th quadrant photodtctor, i.., I 1AC ¼ I 3AC and I AC ¼ I 4AC. Thrfor, othr combinations of two photocurrnts may rduc th calculation rror, and w thrfor simulatd th calculation rror of B x using th othr combinations in Eq. (7), i.., (I 1 þ I 4 ), ( I 3 I ), ( I 3 þ I 4 ), and 1 ði 1 þ I 4 I I 3 Þ. Also th rror for B o x, w usd othr combinations in Eq. (8), i.., (I 1 I 4 ), ( I 3 þ I ), ( I 3 I 4 ), and 1 ði 1 þ I I 3 I 4 Þ. In this simulation, both th LP vn and LP odd mods wr assumd to b xcitd qually, i.., B x ¼ Bo x ¼ 1:, and th phas diffrnc btwn th LP mods and th LP 1 x-polarizd mod is assumd to b zro, i.., 1 and 3 ¼. In this cas, th calculation rror for th LP 1 mod was zro, and thrfor th simulatd rsults for th LP mods ar shown in th following figurs. W simulatd th cass of W f =W ¼ :63, 1., and (¼ 1=:63), and th simulatd rsults of th calculation rror of amplituds B x and Bo x ar shown in Figs. 5(a) and (b) for th cas of W f =W ¼ 1: bcaus th rsults for othr cass hav similar tndncy to this cas. It can b sn from ths figurs that th calculation rror can b rducd by using all four photocurrnts I 1AC to I 4AC. This mans that th calculation formula for th LP mods should b xprssd by Eq. (33) instad of Eqs. (7) (3). (a) Simulatd rsults of ΔB x. (b) Simulatd rsults of ΔB o x. Fig. 5. Simulatd rsults of calculation rror of amplituds du to th offst of light bam. " # " # B x / B o x I x 1AC I x AC I x 3AC ð33þ IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 On th othr hand, in th simulation of th calculation rror of th phas diffrnc, thr typical cass wr simulatd: i.., 1 ¼ and 3 ¼ 4, 1 ¼ 4 and 3 ¼, and 1 ¼ 4 and 3 ¼ 4. Hr th amplituds of LPvn and LP odd ar assumd to b unity, i.., B x ¼ Bo x ¼ 1. Sinc th offst most significantly affcts th calculation rror for th cas 1 3 ¼, as shown in Fig. 3(b), w simulatd th cas 1 ¼ 4 and 3 ¼ 4. Th rsults ar shown in Figs. 6(a) and (b). It can b sn from ths figurs that th calculation rror for th phas diffrnc can b rducd significantly to almost zro whn all four photocurrnts I 1AC to I 4AC ar usd, i.., Eq. (33) is usd. I x 4AC

12 (a) Simulatd rsults of Δφ 1. (b) Simulatd rsults of Δφ 3. Fig. 6. Simulatd rsults of calculation rror of phas diffrncs du to th offst of light bam. 5 Conclusion W proposd a nw high-spd mod analysis mthod namd th PSCD mthod that uss a quadrant dtctor and cohrnt dtction. Th complx amplituds of th LP mods can b calculatd from th addition and subtraction of th photocurrnt of th channls of th quadrant photodtctor. Sinc no mchanically movabl lmnt is involvd, this mod rcognition principl can b xtndd to a mod rcivr for mod-division multiplxing transmission without a mod dmultiplxr or mod discrimination optical lmnt if th rspons spd of th sgmntd photodtctor is sufficintly high. W valuatd th calculation rrors of th formulas for th cass of rotation of th quadrant photodtctor and th offst of th light bam from th local oscillator. From ths simulations, it was shown that th calculation rror du to th rotation of th quadrant photodtctor can b rducd to almost zro by taking into account th rotation angl. On th othr hand, it was found that th calculation rror du to th offst of th bam from th local oscillator can b rducd by using all four photocurrnts from th quadrant photodtctor, although thr is rdundancy among ths four photocurrnts. Acknowldgmnts This work was supportd by th National Institut of Information and Communications Tchnology (NICT), Japan, undr th Rsarch on Innovativ Tchnologis of Amplification, Connction, and Transmission for SDM Projct and by JSPS KAKENHI Grant Numbr 16K1463. IEICE 18 DOI: /lx Rcivd Novmbr 15, 17 Accptd Dcmbr 4, 17 Publicizd Dcmbr 15, 17 Copyditd January 1, 18 1