Larg phas shift via plarizatin-cupling cascading Juan Hu and Xianfng Chn* Dpartmnt f Physics, th Stat Ky Labratry f Advancd Optical Cmmunicatin Systms and Ntwrks Shanghai Jia Tng Univrsity, Shanghai 200240, China *xfchn@sjtu.du.cn Rfrncs Abstract: Hrin, w prps a phnmnn f plarizatin-cupling (PC) cascading gnratd in MgO dpd pridically pld lithium nibat crystal (PPMgLN). PC cascading cntributs t th ffctiv lctr-ptical (EO) Krr ffct that is svral rdrs f magnitud strngr than th classical ns. Exprimnt f Nwtn s rings dmnstrats th larg phas accumulatin during th PC cascadd prcsss, and th xprimntal data is idntical with th thrtical simulatin. 2012 Optical Scity f Amrica OCIS cds: (160.2100) Elctr-ptical matrials; (160.4330) Nnlinar ptical matrials; (260.5430) Plarizatin; (260.1180) Crystal ptics; (160.3730) Lithium nibat. 1. J. M. R. Thmas and J. P. E. Taran, Puls distrtins in mismatchd scnd harmnic gnratin, Opt. Cmmun. 4(5), 329 334 (1972). 2. G. I. Stgman, D. J. Hagan, and L. Trnr, χ(2) cascading phnmna and thir applicatins t all-ptical signal prcssing, md-lcking, puls cmprssin and slitns, Opt. Quantum Elctrn. 28(12), 1691 1740 (1996). 3. R. DSalv, D. J. Hagan, M. Shik-Baha, G. Stgman, E. W. Van Stryland, and H. Vanhrzl, Slffcusing and slf-dfcusing by cascadd scnd-rdr ffcts in KTP, Opt. Ltt. 17(1), 28 30 (1992). 4. W. E. Trrullas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stgman, L. Trnr, and C. R. Mnyuk, Obsrvatin f tw-dimnsinal spatial slitary wavs in a quadratic mdium, Phys. Rv. Ltt. 74(25), 5036 5039 (1995). 5. L. Misguti, S. Backus, C. G. Durf, R. Bartls, M. M. Murnan, and H. C. Kaptyn, Gnratin f bradband VUV light using third-rdr cascadd prcsss, Phys. Rv. Ltt. 87(1), 013601 013604 (2001). 6. A. Fratalcchi, R. Asquini, and G. Assant, Intgratd lctr-ptic switch in liquid crystals, Opt. Exprss 13(1), 32 37 (2005). 7. J. Zhang, J. S. Nlsn, and Z. Chn, Rmval f a mirrr imag and nhancmnt f th signal-t-nis rati in Furir-dmain ptical chrnc tmgraphy using an lctr-ptic phas mdulatr, Opt. Ltt. 30(2), 147 149 (2005). 8. Y. Q. Lu, Z. L. Wan, Q. Wang, Y. X. Xi, and N. B. Ming, Elctr-ptic ffct f pridically pld ptical suprlattic LiNbO3 and its applicatins, Appl. Phys. Ltt. 77(23), 3719 3721 (2000). 9. A. Yariv and P. Yh, Optical Wavs in Crystals: Prpagatin and Cntrl f Lasr Radiatin (Jhn Wily & Sns, Inc., 1984). 10. X. Chn, J. Shi, Y. Chn, Y. Zhu, Y. Xia, and Y. Chn, Elctr-ptic Slc-typ wavlngth filtr in pridically pld lithium nibat, Opt. Ltt. 28(21), 2115 2117 (2003). 11. J. Hu, K. Liu, and X. Chn, 1 x 2 prcis lctr-ptic switch in pridically pld lithium nibat, Opt. Exprss 18(15), 15603 15608 (2010). 12. K. Liu, J. Shi, and X. Chn, Linar plarizatin-stat gnratr with high prcisin in pridically pld lithium nibat, Appl. Phys. Ltt. 94(10), 101106 (2009). 13. Y. H. Chn and Y. C. Huang, Activly Q-switchd Nd:YVO4 lasr using an lctr-ptic pridically pld lithium nibat crystal as a lasr Q-switch, Opt. Ltt. 28(16), 1460 1462 (2003). 14. S. Zhu, Y. Zhu, and N. Ming, Quasi-phas-matchd third-harmnic gnratin in a quasi-pridic ptical suprlattic, Scinc 278(5339), 843 846 (1997). 15. G. Alxakis, N. Thfanus, A. Arapyianni, M. Aillri, C. Carabats-Ndlc, and M. Fntana, Masurmnt f quadratic lctrptic cfficints in LiNbO3 using a variatin f th FDEOM mthd, Opt. Quantum Elctrn. 26(12), 1043 1059 (1994). 16. Y. L, N. Yu, C. S. K, D. K. K, Y. C. Nh, B. A. Yu, W. Shin, T. J. Em, K. Oh, and J. L, All-ptical wavlngth tuning in Slc filtr basd n Ti: PPLN wavguid, Elctrn. Ltt. 44(1), 30 32 (2008). #167955 - $15.00 USD Rcivd 7 May 2012; accptd 9 May 2012; publishd 31 May 2012 (C) 2012 OSA 4 Jun 2012 / Vl. 20, N. 12 / OPTICS EXPRESS 13419
1. Intrductin Th histry f ptical cascading can b tracd back t th 1970 s, whn J. M. Thmas t al. bsrvd a cascad intractin btwn lights [1]. Th imprtanc f cascading, hwvr, was nt fully apprciatd until 1996, whn G. I. Stgman studid cascading systmatically and cnsidrd it as a prmising dirctin t xplr ptical phnmna [2]. Cnsquntly, in th fllwing yars, rsarchrs hav shwn nrmus intrst in th pssibl xplratin f varius nnlinar ffcts such as nnlinar phas shifts, slw light, md-lcking, puls cmprssin, all-ptical transistr actin, and spatial slitns [3 5]. In this lttr, w prps a nw phnmnn namd plarizatin-cupling (PC) cascading. PC cascading is mdld aftr th scnd harmnic gnratin (SHG) cascading [2] within MgO dpd pridically pld lithium nibat crystal (PPMgLN). Basd n this PC cascading ffct, th ffctiv nnlinarity f PPMgLN is dtrmind by th transvrsly applid lctric fild, which is similar t th lctr-ptical (EO) Krr ffct, thus PC cascading ffct culd b dfind as ffctiv EO Krr ffct. Intrstingly, th Krr cnstant f this ffctiv EO ffct is svral rdrs f magnitud largr than that in th classic cuntrparts, and culd lad t larg nnlinar phas shifts. Th nhancmnt f th ffctiv EO Krr ffct is prvd by thrtical calculatin, and th larg phas shifts ar dmnstratd by xprimntal rsults. It has bn shwn that larg phas shifts hav xtnsiv applicatins [6,7]. Thrfr, th subjct f cntrlling th phas shifts by lctr-ptical ffct is widly studid and vry imprtant frm bth scintific and tchnlgical pints f viw. 2. Thrtical mdl and simulatin As is knwn, scnd harmnic gnratin (SHG) cascading ccurs via up-cnvrsin ( ω + ω 2ω ) fllwd by dwn-cnvrsin ( 2ω ω ω ) [2]. Th physical pictur f th classic SHG cascading is shwn in Fig. 1(a). Du t th phas vlcity f th nw fundamntal frquncy phtn is incnsistnt with th input n, th nnlinar phas shift is yildd via cascadd prcsss. Fig. 1. Th physic picturs f SHG cascading prcsss and PC cascading prcsss. (a), (b) Th physical illustratins f th SHG cascading and PC cascadd prcsss. (c) Th phas shifts insid PPMgLN, th xtrnal lctric fild is 0.245 V / µ m. By fllwing th xampl f SHG cascading, w prps a nw cascadd phnmnn PC cascading in PPMgLN undr nn quasi-phas matching (NQPM) cnditin. Ths cascadd prcsss ar dividd int tw stps: Taking rdinary wav (OW) incidnc fr xampl, th nrgy f OW flws t xtrardinary wav (EW), but ds nt caus cmplt dpltin f OW; thn, th nrgy flws back frm EW int OW, aftr apprximatly n chrnc lngth. Bcaus th rgnratd OW is n lngr in phas with th nn-cnvrtd, a nt OW phas is yildd (shwn in Fig. 1(b)). Th vctrs in Fig. 1(b) shw th nrgy #167955 - $15.00 USD Rcivd 7 May 2012; accptd 9 May 2012; publishd 31 May 2012 (C) 2012 OSA 4 Jun 2012 / Vl. 20, N. 12 / OPTICS EXPRESS 13420
variatin f OW and EW during th cupling prcsss. Frm Fig. 1(c), th incrmnt f th nnlinar phas primarily ccurs during th cycl whrin nrgy strngly xchangs btwn th tw bams. Hr, PC cascading in a PPMgLN is cnsidrd whn a transvrs xtrnal lctric fild is applid alng th PPMgLN. Basd n th lctr-ptical ffct, th ptical axis f ach dmain ar altrnatly alignd at th angls f + θ and θ, with rspct t th plan f plarizatin f th input light [8]. Enrgy cupling btwn OW and EW happns in ths fldd dmains. Th rlativ azimuth angl btwn th dilctric axs f tw adjacnt dmains is vry small s that th pridic altrnatin f th azimuth can b cnsidrd as a pridic small prturbatin. In this cas, th cupld-md quatins f th rdinary and xtrardinary wavs ar [9]: i da1 / dz = iκ A2 da2 / dz = iκ A1 β z i β z, (1) with 2 2 ω n n γ 51Ey i(1 cs mπ ) β = k1 k2 Gm, Gm = 2 π m / Λ and κ =, ( m = 1, 3, 5, 7...), 2c n n mπ whr A1 and A 2 ar th nrmalizd cmplx amplituds f OW and EW, rspctivly. β is th th vctr-mismatch; k1 and k2 ar th crrspnding wav vctrs; G m is th m rciprcal vctr crrspnding t th pridicity f pling; Λ is th prid f th PPMgLN, γ 51 is th lctr-ptical cfficint (its valu is rfrrd t [8]), Ey is th transvrs lctric fild intnsity, n and n ar th rfractiv indics fr th rdinary and xtrardinary wavs rspctivly, and with th initial cnditin (Assuming that th incidnt bam is OW): A1 (0) = 1. A2 (0) = 0 Th slutin f Eq. (1) is givn by i( β /2) z β A1 ( z) = [cs sz i sin sz] 2s, i( β /2) z * sin sz A2 ( z) = ( iκ ) s (2) (3) with s 2 = κκ * + ( β / 2) 2. Fr prfct quasi-phas matching (QPM) ( β = 0), th slutin f Eq. (1) is simplifid t A 1 ( z) = cs( κ z) A 2 ( z) = sin( κ z). This wll-knwn QPM cnditin has pnd th way t a hst f ptical dvics such as narrwband Slc-typ filtrs, lctrptical switchs, prcis plarizatin cntrllrs and lasr-q switchs [10 14]. Hwvr, th cascading phnmnn f phas-mismatch has nt yt bn undrstd. Thus, this study cncntrats n th nn-qpm slutin ( β 0), by studying th phas f OW and th EW, in which a rich varity f cascadd phnmna can ccur. * Frm Eq. (3), w hav A2 = C xp[ i( β / 2) z], with C = ( iκ / s)sin( sz). A2 is a ral numbr. Th phas f EW is asily btaind: β z, C > 0 NL 2 Φ =, β z ± π, C < 0 2 (4) #167955 - $15.00 USD Rcivd 7 May 2012; accptd 9 May 2012; publishd 31 May 2012 (C) 2012 OSA 4 Jun 2012 / Vl. 20, N. 12 / OPTICS EXPRESS 13421
with Φ cnfind t intrval [ π π ] NL,. Eq. (4) suggsts that NL Φ f EW varis linarly with phas-mismatch, and th half-wav lss happns. Th phas f OW is mr cmplicatd. Thus, btaining th apprximat slutin fr larg phas mismatch r lw xtrnal lctric fild was attmptd. Assum A1 1, and hnc A 1 ( z) = xp[ i Φ NL ( z)]. Frm Eq. (3), th nnlinar phas shift f OW satisfis th fllwing quatin: NL β L * 2 Φ ( z = L) = (1 1 + (2 κ κ / β ) ). 2 (5) Fr larg phas-mismatch r lw xtrnal lctric fild, w hav β >> κ. This nnlinar phas shift varis in prprtin t th squar f th lctric fild Ey. Th variatin is similar t th classic lctr-ptical Krr ffct and can b shwn as: κ L NL Φ. (6) β L Fllwing th xampl f changing in rfractiv indx f EO Krr ffct, an ffctiv EO nnlinar rfractiv indx by PC cascading can b intrducd by: 3 2 2 ff ωn n γ 51 2 n = E. 2 y (7) π c β In ralm f nnlinar ptics, th plarizability f a mdium mdifid by ptical filds can ff b givn by P = P0 + χ1e + χ2ee + χ3 EE + χ3 EEE, whr χ ff 3 EE crrspnds t th ffctiv third rdr nnlinarity which is inducd by scnd rdr nnlinarity χ 2 in SHG cascadd prcsss [2]. Similarly, during PC cascading th indx f rfractin mdifid by cascadd lctr-ptical ffct can b givn by: 1 3 1 ff 3 2 1 3 2 n = n0 + γ n0 E + s n0 E + sn0 E +. (8) 2 2 2 ff ff Just lik χ 2 inducing an χ 3 in SHG cascading, γ lads t an s in PC cascading. Hr ff s is rfrrd as th ffctiv EO Krr cnstant which culd b nhancd during th ffctiv lctr-ptical Krr ffct. Sinc in a PPMgLN th classical transvrs lctr-ptical ffct can nly rtat th ptical axis f th lithium nibat, th scnd trm n th right sid f Eq. (8) ds nt cntribut t th nnlinar phas shift, and th actual scnd rdr Krr ffct is t wak t caus bsrvabl phnmnn and is ngligibl. In all, th bsrvd larg phas shift dirctly rlatd t th ffctiv EO Krr ffct, as indicatd by th third trm. Suppsing β = 1 π / m, λ = 600 nm, n = 2.1930, n = 2.2829 and γ 51 = 32.6 pm / V, th ffctiv Krr cnstant ff s 2 14 2 2 will b 1.04 10 m / V, which is thr rdrs f magnitud 17 2 2 largr than th classical EO Krr cnstant f lithium nibat, 3.39 10 m / V [15]. It shuld b ntd that th nhancd EO Krr cnstant is an ffctiv ffct which has nthing t d with th particular matrial but mainly gvrnd by th pridical indx mdulatin. Th prcis phas-variatins with xtrnal lctric fild f EW and OW gvrnd by Eq. (3) ar shwn in Fig. 2 which indicats th ptntial f nnlinar EO phas mdulatrs. Th fur slid lins prsnt th variatin f th nnlinar phas shifts with xtrnal lctric fild at diffrnt vctr-mismatchs, β =121 π / m, 16 π / m, 16 π / m and 121 π / m, whil dash lins prsnt th transmissin with lctric filds, shwing th nrgy cupling btwn OW and EW in PPMgLN. By kping an apprpriat nnzr valu f th phas-mismatch, th transmissin f EW and OW vary frm 0 t 0.99 and 0.01 t 1, rspctivly. Just by adjusting th vctr-mismatch, th magnitud and sign f th ffctiv nnlinarity, i.. nnlinar phas can b varid. As shwn in Fig. 2(a), th larg phas shifts nly ccur whn th EW #167955 - $15.00 USD Rcivd 7 May 2012; accptd 9 May 2012; publishd 31 May 2012 (C) 2012 OSA 4 Jun 2012 / Vl. 20, N. 12 / OPTICS EXPRESS 13422
xprincs a half-wav lss, which mans th phas maintains cnstant xcpt at sm critical lctric filds. Whil in Fig. 2(b), in th vicinity f sm critical lctric filds, tiny changs f th lctric fild can caus larg changs f phas fr OW. It shuld b ntd that th cnvntinal phas mdulatrs basd n Pckls ffct nly raliz linar phas shift and ar unabl t btain sharp phas shift r half-wav lss. Fig. 2. Typical variatin f th nnlinar phas shifts with xtrnal lctric fild fr EW and OW. Th dashd lins prsnt th transmissin as a functin f lctric filds fr T = 21 Th lngth f PPMgLN fr calculatin was st as 50 mm. C. 3. Exprimnt and rsults Th schmatic f th xprimntal stup is shwn in Fig. 3. A schm f March-Zhndr intrfrnc was utilizd t invstigat th phas shifts. Th wavlngth f th H-N lasr is 632.8nm, which almst satisfis th third rdr phas-matching cnditin with pling prid f PPMgLN t b 21.1 µm. Th lasr pwr is 8 milliwatts. First, th hrizntally plarizd incidnt bam was sparatd by a bam splittr (BS) with n bam passing thrugh PPMgLN and th thr in fr spac. Thus, th incidnt bam in PPMgLN is OW. Fig. 3. Exprimntal stup fr dmnstrating th nnlinar phas shifts yildd in PC cascading. A PPMgLN crystal, which is 5ml%MgO: PPLN, with th prid f 21.1µm and th lngth f 50 mm. High-vltag is usd t supply transvrs lctric filds. Th xprimntal rsults ar shwn in Fig. 4(a) and (b). By varying th transvrs lctric fild frm 0 t 0.56V/µm, w fund that th intrfrnc frings light-dark changd at diffrnt tmpraturs (diffrnt vctr-mismatchs). Whn th vctr-mismatch was arund 1000 π / m (23 C, Fig. 4(a)), th Nwtn s rings had hardly any chang in varying lctric fild. It cncluds that th nnlinar phas shift cannt b accumulatd t π at vry larg vctr-mismatch. Whn th vctr-mismatch was changd t apprximat 150 π / m by lwring tmpratur t 21.3 C (Fig. 4(b)), th intrfrnc frings xprincd thr tims light-dark altrnatin at sm crtain lctric filds, 0.28 V/µm, 0.44 V/µm and 0.56 V/µm. Each chang mans a π phas shift. By this tkn, transvrs lctric fild is nt th dtrminant f th larg phas shifts. In a PPMgLN, th classical transvrs lctr-ptical #167955 - $15.00 USD (C) 2012 OSA Rcivd 7 May 2012; accptd 9 May 2012; publishd 31 May 2012 4 Jun 2012 / Vl. 20, N. 12 / OPTICS EXPRESS 13423
ffct is nly abl t rtat th ptical axis f th lithium nibat, which indicats th larg phas shift cannt b th classic EO Pckls ffct. Figur 4(c) shws th cmparisn f th simulatd curv and th xprimntal rsults at th tmpratur f 21.3 C. Th critical lctric filds lading t π phas shifts with th vctr-mismatch f apprximat 150 π / m (in Fig. 4(b)) ar shwn as th clr dts in Fig. 4(c). Thus, w can s that th xprimntal data wll fit th thrtical curv. Bcaus f sm unavidabl rrrs, fr instanc, th rfractiv indics f th PPMgLN sampl w usd ar nt cnsistnt with th simulatd valus; th xprimntal data can t b in full agrmnt with th thrtical ns. It shuld b ntd that th PPLN wavguid has bn succssfully prpsd rcntly [16], whr th gap btwn th lctrds can b as shrt as 10 µm, s that nly svral Vlts is nugh t gnrat th larg phas shifts. 4. Cnclusin Fig. 4. Th cmparisn f xprimntal rsults and thrtical simulatin fr dmnstrating th nhancd phas shift yildd in PC cascading; (a), (b) th cntr xprincs dark-light changs by changing th transvrs lctric fild at diffrnt vctr-mismatchs. (c), th cmparisn f simulatd curv and xprimntal rsults at 21.3 C. In summary, PC cascading was dmnstratd in PPMgLN thrugh xprimnt and thry. Nnlinar phas shifts gnratd frm th cascadd prcsss btwn a pair f rthgnal bams i.. OW and EW hav bn invstigatd. Th rsults prvid a mthd which can b usd t achiv nhancd EO Krr ffct. It shuld b ntd that th PC cascading prpsd hr is diffrnt frm th SHG cascading phnmnn, bcaus th frmr blngs t linar ptics, and th lattr is classifid t nnlinar ptics. With a diffrnt physical undrstanding, PC cascading may triggr intrst in a wid rang f filds. Acknwldgmnts This rsarch was supprtd by th Natinal Natural Scinc Fundatin f China (Grant N. 61125503, 61078009), th Natinal Basic Rsarch Prgram 973 f China (Grant N. 2011CB808101), th Fundatin fr Dvlpmnt f Scinc and Tchnlgy f Shanghai (Grant N. 11XD1402600) and th Opn Fund f th Stat Ky Labratry f High Fild Lasr Physics. #167955 - $15.00 USD Rcivd 7 May 2012; accptd 9 May 2012; publishd 31 May 2012 (C) 2012 OSA 4 Jun 2012 / Vl. 20, N. 12 / OPTICS EXPRESS 13424