A novel method for determining and improving the quality of a quadrupolar fiber gyro coil under temperature variations

Size: px

Start display at page:

Download "A novel method for determining and improving the quality of a quadrupolar fiber gyro coil under temperature variations"

Lucy Horn
5 years ago
Views:

1 A novl mthod for dtrmining and improving th quality of a quadrupolar fibr gyro coil undr tmpratur variations Zhihong Li, 1,2 Zhuo Mng, 1,2 Tign Liu, 1 and X. Stv Yao 1,* 1 Polarization Rsarch Cntr, Collg of Prcision Instrumnt & Opto-lctronics Enginring and Ky Laboratory of Opto-lctronics Information and Tchnical Scinc, Ministry of Education, Tianjin Univrsity, Tianjin , China 2 Suzhou Opto-ring Co. Ltd., Suzhou , China * stvyao888@yahoo.com Abstract: W introduc a paramtr calld pointing rror thrmal snsitivity (PETS) for quantitativly dtrmining th quality of a quadrupolar (QAD) fibr coil undr radial tmpratur variations. W show both analytically and xprimntally that th pointing rror of a fibr gyro incorporating th fibr coil is linarly proportional to th final radial thrmal gradint on th coil, with PETS as th proportional constant. W furthr show that PETS is linarly proportional to anothr paramtr calld ffctiv asymmtric lngth of th coil. By thrmally inducing diffrnt radial thrmal gradints on th fibr coil and masuring th corrsponding pointing rrors in a gyroscopic masurmnt stup, w can confidntly dtrmin th PETS of th fibr coil and its associatd ffctiv asymmtric lngth causd by imprfctions in coil winding. Consquntly, w ar abl to prcisly trim th coil to achiv bst thrmal prformanc Optical Socity of Amrica OCIS cods: ( ) Fibr optics and optical communications; ( ) Fibrs, singlmod; ( ) Gyroscops. Rfrncs and links 1. V. Vali and R. W. Shorthill, Fibr ring intrfromtr, Appl. Opt. 15(5), (1976). 2. H. C. Lfvr, Fundamntal of intrfromtric fibr optic gyroscop, in Fibr Optic Gyros: 20 Annivrsary Conf., Proc. SPIE 2837, (1996). 3. G. A. Sandrs, B. Szafranic, R. Y. Liu, M. S. Bilas, and L. Strandjord, Fibr-optic gyro dvlopmnt for a broad rang of applications, Proc. SPIE, Fibr Optic and Lasr Snsors XIII 2510, 2 11(1995). 4. G. A. Sandrs, B. Szafranic, R. Y. Liu, C. L. Laskoski, L. K. Strandjord, and G. Wd, Fibr optic gyros for spac, marin, and aviation applications, in Fibr Optic Gyros: 20 Annivrsary Conf., Proc. SPIE 2837, (1996). 5. M. Chomát, Efficint supprssion of thrmally inducd nonrciprocity in fibr-optic Sagnac intrfromtrs with novl doubl-layr winding, Appl. Opt. 32(13), (1993). 6. R. Dyott, Rduction of th Shup ffct in fibr optic gyros; th random-wound coil, Elctron. Ltt. 32(23), (1996). 7. R. P. Gottsch and R. A. Brgh, Trimming of fibr optic winding and mthod of achiving sam, U.S.Patnt: , 6 18 (1996). 8. F. Mohr, Thrmooptically inducd bias drift in fibr optical Sagnac intrfromtrs, J. Lightwav Tchnol. 14(1), (1996). 9. D. M. Shup, Thrmally inducd nonrciprocity in th fibr-optic intrfromtr, Appl. Opt. 19(5), (1980). 10. W. Fng, X. Wang, and W. Wang, Effct of turns diffrnc in ach layr on tmpratur prformanc of fibr optic gyroscopo, Journal of Chins Inrtial Tchnology 19, (2011). 11. C. Mao, Z. Gong, X. Mou, and G. Yang, Finit diffrnc mthod for tmpratur fild in fibr optical gyroscop, Pizolctrics and Acoustooptics 25, (2003). 12. M. Li, T. Liu, Y. Zhou, J. Jiang, L. Hou, J. Wang, and X. S. Yao, A 3-D modl for analyzing thrmal transint ffcts in fibr gyro coils, Proc. SPIE, Advancd Snsor Systms and Applications III, (2007). 13. C. M. Lofts, P. B. Ruffin, M. D. Parkr, and C. C. Sung, Invstigation of th ffcts of tmporal thrmal gradints in fibr optic gyroscop snsing coils, Opt. Eng. 34(10), (1995). 14. J. Sawyr, P. B. Ruffin, and C. Sung, Invstigation of th ffcts of tmporal thrmal gradints in fibr optic gyroscop snsing coils, part II, Opt. Eng. 36(1), (1997). # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2521

2 1. Introduction It has bn mor than 35 yars sinc fibr-optic gyroscop (FOG) was first proposd by V. Vali and R. Shorthill [1]. Nowadays, FOGs ar in mass production and ar widly usd in various military and civilian applications [2 4], such as in aircrafts, vssls, and land vhicls for prcision rotation rat and angl dtction. A ky componnt in a FOG is a fibr coil, which must b wound painstakingly with spcial gomtris [5 8], such as quadrupolar winding pattrn, to rduc a nonrciprocal ffct causd by thrmal variations known as Shup ffct [9]. Dspit a larg amount of FOGs hav bn producd for various applications, th making of good fibr coils still rmains a work of art, with larg variations in prformanc for coils mad with vn th sam procss. Tst standards ar availabl for tsting th prformanc of a whol fibr gyro in opration, including combind contributions from th light sourc, Y-couplr, passiv componnts, fibr coil, mchanical structur and nclosur, and lctronics. Howvr, to th bst of authors knowldg, no publishd tst standards or comprhnsiv mthods ar availabl for tsting a fibr coil s prformanc undr tmpratur variations alon, although som attmpts wr mad to masur coils asymmtry with inconclusiv rsults [10]. Such a tst is important for nsuring that good quality fibr coils can b mass producd with good uniformity and fficincy. In ordr to dtrmin th quality of a coil alon, maningful tst paramtrs and associatd tst mthods must b dfind and dvlopd. Such a paramtr can rlat to winding imprfctions for nginrs to find out th root causs of th problm coils, and implmnt corrctiv masurs or procss improvmnts. In this papr, w introduc a paramtr calld pointing rror thrmal snsitivity (PETS) for quantifying th prformanc of a quadrupolar fibr coil subjct to a tim-dpndnt radial tmpratur gradint. W show that th pointing rror (asymptotic angular rror) of a gyro systm incorporating th quadrupolar fibr coil is linarly proportional to th final tmpratur gradint btwn th innr and outr coil surfacs in th radial dirction, with PETS as th proportional constant. W furthr show that PETS is linarly proportional to a paramtr calld ffctiv asymmtric lngth of th fibr coil, causd by imprfctions in fibr winding. In addition, by thrmally inducing diffrnt tim-varying radial tmpratur gradints on th fibr coil (as will b shown in inst b of Fig. 3) and masuring th corrsponding pointing rrors in a gyroscopic masurmnt stup, w can confidntly dtrmin th PETS of th fibr coil and its associatd ffctiv asymmtric lngth. Such paramtrs can b dirctly usd to trim th fibr coil [7, 11] for bttr tmpratur prformanc, fdback to coil production for improving winding procss, or simply b usd to quantitativly dtrmin th quality of fibr gyro coils. Finally, w trim multipl coils according to th obtaind asymmtric lngths to achiv bst thrmal prformanc, and xprimntally vrify th usfulnss and corrctnss of th PETS dfinition and th asymmtry lngth rlations. It should point out that our xprimnts indicat that th PETS masurmnt rsults ar robust against nvironmnt tmpratur variations and snsitiv only to coil s asymmtry, and thrfor is a practical paramtr for charactrizing th thrmal prformanc of fibr coils, dtrmining th coil asymmtry lngth, and guiding th coil trimming. 2. Pointing rror causd by tim-dpndnt radial thrmal gradint In this sction, w first driv an xprssion for th angular rror of a gyro systm incorporating an imprfctly wound fibr coil with a crtain lngth asymmtry. W thn introduc and dfin PETS to dscrib th prformanc of th coil against tmpratur variations, and anothr paramtr calld ffctiv asymmtric lngth to rflct th accumulativ winding imprfctions. With th origin of th coordinat locatd at th fibr coil midpoint s = L/2 (L is th total fibr lngth), th nonrciprocal phas shift ϕ inducd by th Shup ffct in th singlmod fibr can b xprssd as [12 14]: # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2522

3 2 τ n L 2 T (,) s t L 2 T ( s,) t ϕ () t = β0 sds s ds L T 0 t 0 t (1) whr τ is th transit tim of light through th fibr coil, β0 = 2π λ0 is th wav numbr, n is th ffctiv rfractiv indx, n T is its tmpratur cofficint, T(,) st t is th tmpratur variation along th fibr, and s and s ar distancs along th fibr coil. Thy ar both positiv, but ar masurd in opposit dirctions, with s incrasing in th clockwis (CW) dirction and s in th countr clockwis (CCW) dirction. Accordingly, th rotation rat rror inducd by a thrmal transint is: 4 γ L 2 T (,) s t L 2 T ( s,) t Ω () t = sds s ds L 0 t 0 t (2) whr γ is a paramtr rlating to th thrmal cofficint of th fibr and th diamtr D of th coil and is dfind as: By intgrating th rat rror Ω () t ψ inducd by th tmpratur variation is obtaind as: 0 n n γ = (3) 2D T ovr tim, th pointing rror (asymptotic angular rror) t 4γ L 2 L 2 ψ ( t ) = Ω ( t ) dt = T ( s, t ) sds T ( s, t ) s ds L 0 0 (4) whr T ( s, t) = T ( s, t) T ( s,0). For a quadrupol-wound coil with 8 layrs of fibr shown in Fig. 1(a), w us full lins labld A to dnot fibr layrs winding in th CCW dirction and brokn lins labld B to dnot th fibr layrs winding in th CW dirction [7]. As shown in Fig. 1(b), w us A i and B i to numbr th fibr layrs in CCW and CW dirctions rspctivly. Th corrsponding quadrupolar winding pattrn can thrfor b dscribd as ABBA-ABBA. In Tabl 1.w dsignat fibr layr numbrs and list th hight and lngth of ach fibr layr. Fig. 1. Illustration of a quadrupol-wound fibr coil, whr A and B ar usd to dnot fibr layrs wound in CCW (solid lin) and CW (brokn lin) dirctions rspctivly, and d is th fibr coil thicknss. (a) Th rlativ positions of CCW and CW layrs. (b) Th numbring of ach fibr layr in CCW and CW dirctions, whr th hight of ach grid cll indicats th diamtr of th fibr and th lins rprsnt th position of fibr s cntr lin in th vrtical axis. W xamin th cas in which th tmpratur variation is only along th radial dirction of th coil, and is indpndnt of th axial position and azimuth angl. Thrfor, th tmpratur on ach fibr layr is constant and Eq. (4) can b simplifid to th following # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2523

4 xprssion, assuming that th fibr coil is symmtrically wound around coil midpoint, as illustratd in Fig. 2(a): 4 4γ ψ () t = T ( A,) t T ( B,) t s ds il /8 [ ] (5) 0 i i L i = 1 ( i 1) L/8 Tabl 1. Hight and Lngth of Each Fibr Layr Winging pattrn A B B A A B B A Global layr numbr CCW layr numbr CCW layr lngth (m) 0~L/8 L/8~L/4 L/4~3L/8 3L/8~L/2 CCW layr hight (m) d/16 7d/16 9d/16 15d/16 CW layr numbr CW layr lngth (m) 0~L/8 L/8~L/4 L/4~3L/8 3L/8~L/2 CW layr hight (m) 3d/16 5d/16 11d/16 13d/16 Howvr, if th coil is not wound prfctly symmtric, th position of th midpoint whr two countr propagating light wavs mt will chang. As shown in Fig. 2(b), if th CCW wound fibr (dnotd by A) is shortnd by l, th midpoint for th countr-propagating light wavs to mt will mov to th longr fibr sid (dnotd by B) by l/2. In practic, vn if th fibr lngths for CCW and CW fibr sctions ar th sam, th optical path lngths may b diffrnt du to rfractiv indx variations along th fibr causd by imprfction of th fibr and diffrnt strsss on th fibr. Thrfor, w introduc a quantity calld ffctiv asymmtric lngth l ff to dnot th shortnd optical path lngth of CCW fibr sction, which will rsult in a shift of coil midpoint by l ff 2 fibr lngth to th CW sction. Consquntly, th pointing rror rsulting from such an imprfct coil can b xprssd as: il lff il lff [ ] [ ] L l + 1 l ff il (6) ff i= 1 ( i 1) + i= γ ψ ( t) = T ( A, t) T( B, t) s ds + T ( A, t) T( B, t) s ds L i i i i Fig. 2. Illustration of asymmtrical fibr wounding. (a) CCW (dnotd by A) and CW (dnotd by B) wound fibr sctions hav th sam lngth. (b) CWW sction is short by a lngth of l, rsulting in a shift of l/2 of th coil midpoint to th CW sid. Th total accumulatd angular rror ψ (pointing rror), which rflcts th prformanc of th fibr coil undr a tmpratur variation, can b dtrmind at thrmal quilibrium (t ) whn th thrmally inducd nonrciprocity in th fibr coil diminishs. Not that th tmpratur chang T ( x) on ach layr can b dscribd as follows: # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2524

5 x i T ( Ai ) = Tinnr + δt,( x i = d /16,7d 16,9d 16,15 d /16) (7a) d x i T ( Bi ) = Tinnr + δt,( x i = 3d 16, 5d 16,11d 16,13d 16) (7b) d whr δ T = Toutr Tinnr, Toutr = TC1 Troom, Tinnr = TC2 Troom, T room tmpratur, and d is th coil width. Substitution of (7) in (5) and (6) yilds: is th room L ψ 0 = γ 2 δt (8a) 8 L 7 ψ = γ( l ) 2 ff δt (8b) 8 8 whr ψ 0 and ψ ar th pointing rrors for coils without and with winding inducd lngth asymmtry l ff. It can b shown that for a fibr coil with N layrs, th pointing rrors causd by a radial tmpratur variation can b xprssd as: L ψ 0 = γ δt (9a) 2 N N 1 ψ = γt δt = ψ 0 ( γδt ) lff (9b) N γ γ L N 1 T = ( l ) 2 ff = lall N N γ (9c) 2 whr lall = L N ( N 1) N lff is th ovrall asymmtric lngth of th coil to b discussd shortly. γ is an important paramtr and is namd as th pointing rror thrmal snsitivity T (PETS) of th coil. It is dirctly proportional to coil s asymmtric lngth, and as will b shown nxt, it can b obtaind xprimntally by changing δ T whil masuring th pointing rrorψ. Although a similar quation to Eq. (9a) for a prfct symmtric quadrupl-winding coil was obtaind prviously [8], it cannot usd to valuat imprfct fibr coils. As will b shown nxt, Eq. (9b) obtaind in this papr is th most important quation for charactrizing th symmtry proprty of a quadrupl-winding fibr coil. It is intrsting to notic from Eq. (9c) that for a prfct gomtrically symmtric coil ( l ff = 0 ), th thrmally inducd pointing rror is non-zro and dcrass quadratically with N. Th pointing rror can b rducd to zro if an asymmtry lff = L N ( N 1) is purposly introducd. Such an asymmtry is undrstandabl, considring that th A portion fibr has two layrs at th innr and outr most surfacs, as shown in Fig. 1 and Fig. 5(a), and thy ar impactd mor by th tmpratur changs than th B layrs bnath thm. Consquntly A portion should b shortnd to match th tmpratur ffct on th B portion by L N ( N 1). In othr words, L N ( N 1 ) is th intrinsic asymmtric lngth rlating to quardruplar winding pattrn and can b compnsatd by trimming A portion fibr by th qual amount. For a fibr coil with 8 and 24 layrs, th rquird trimming lngths # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2525

ar 1.78% and 0.18% of th coil lngth, rspctivly. Thrfor, w introducd an ovrall coil asymmtric lngth l all in Eq. (9c) to tak into account of both intrinsic and winding asymmtris. 3.

6 ar 1.78% and 0.18% of th coil lngth, rspctivly. Thrfor, w introducd an ovrall coil asymmtric lngth l all in Eq. (9c) to tak into account of both intrinsic and winding asymmtris. 3. Exprimntal rsult and discussions Thr framlss SM fibr coils with conductiv potting adhsivs ar usd in this study and all of thm wr wound with th quadrupolar winding schm. Th coils ar approximatly 1.25cm in hight with an innr diamtr of 5.3cm. Th total lngth of fibr on ach coil is 240m, producing 24 fibr layrs. Each fibr coil is placd in an opn-loop gyroscopic stup shown in Fig. 3 to form a complt fibr gyro and to b valuatd for its thrmal prformanc. As shown in insts a) and b), a tmpratur gradint is applid to a coil in th radial dirction such that th fibr in ach layr xprincs th sam tmpratur. Th tmpraturs both at coil s innr and outr surfacs wr monitord with two sparat thrmocoupls. In th xprimnts, w always kp th lngths of fibr pigtails constant at 2 mtrs ach. Fig. 3. Gyroscopic stup for masuring thrmal inducd rror signals. Inst a) shows a ribbon hatr with a width and a lngth qual to th width and th primtr of th coil is wrappd around coil to apply a radial tmpratur gradint. TC1 and TC2 ar th tmpraturs masurd at th outr and innr surfacs of th fibr coil. Inst b) shows TC1 and TC2 as a function of tim whn a radial tmpratur xcitation is applid, and ar stabilizd at 70.2 C and 54.3 C rspctivly, rsulting a tmpratur gradint δ T = TC1 TC2 = 15.9 C Fig. 4. (a) Masurd rotation rat rror (solid lin) of a gyro systm in Fig. 3 whn th fibr coil s outr layr is subjct to a tmpratur chang TC1 (dottd lin). (b) Angular rror of th gyro systm stadily approachs asymptotic valu (pointing rror), dspit th fast fluctuation of th corrsponding rat rror (solid lin). # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2526

A varying tmpratur gradint on th fibr coil will caus a nonrciprocal phas shift, which will rsult in rotation rat and pointing rrors for a gyro systm incorporating th fibr coil.

7 A varying tmpratur gradint on th fibr coil will caus a nonrciprocal phas shift, which will rsult in rotation rat and pointing rrors for a gyro systm incorporating th fibr coil. Figur 4(a) shows th rotation rat rror of a gyro systm in Fig. 3 as a function of tim whn a tmpratur gradint shown in th insts of Fig. 3 is applid to th fibr coil. By intgrating th rat rror Ω () t ovr tim, th angular rror can b obtaind, as shown in Fig. 4(b). It should b pointd out that th angular rror signal stadily approachs an asymptotic valu ψ around 1.04 dgrs (pointing rror), dspit larg fluctuations of th masurd rat rror. 3.1 Exprimntal vrification of Eq. (9) Equation (9b) shows that th pointing rror is linarly proportional to th ffctiv asymmtry lngth l ff of th coil and can b xprimntally vrifid by fixing δ T whil changing l ff by trimming th lngth of A sid (.g. CCW sid) of th coil. In Eq. (9b), th ffctiv asymmtry lngth l ff can b xprssd as: l = l + l (10a) ff ff 0 whr l ff 0 is th initial ffctiv asymmtry lngth bfor trimming th coil and l is th trimming lngth. Substitution of Eq. (10a) in Eq. (9b) yilds: whr th pointing rror bfor coil trimming (l = 0) is N 1 ψ = ψ 0 ( γδt ) l (10b) N N 1 ψ 0 = ψ 0 ( γδt ) lff 0 (10c) N Whn trimming a coil, w unwrap som turns from th coil outr layr ( A sid or CCW sid), as shown in Fig. 5(a) and cut a lngth of fibr qual to th unwrappd portion to kp th fibr pigtail lngth constant. Th trimming lngth l is dfind positiv whn cutting fibr from th outr layr ( A sid) of a coil. Fig. 5. (a) Illustration of coil trimming with two fibr turns unwrappd from th outr layr of th coil. (b) Masurd pointing rror ψ of fibr coil as a function of th trimming lngths l whn a tmpratur xcitation profil from room tmpratur to 70 C is applid to th outr layr of th coil. W trimmd and tstd thr fibr coils of th sam diamtr and mad with th sam fibr. Th masurd pointing rror ψ of th gyro systm incorporating th coils as th function of th trimming lngth l is shown in Fig. 5(b). Th sam tmpratur profil shown in Fig. 3 was applid to ach coil s outr layr and ach coil is trimmd at six trimming lngths (l = 0m, 0.5m, 1m, 1.5m, 2m, 2.5m). It is vidnt that th pointing rror is linarly # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2527

8 proportional to th trimming lngths l and hnc to th ffctiv asymmtry lngth l ff, agrs wll with Eq. (9b). Bcaus th thr coils hav th sam diamtrs and ar mad with th sam fibrs, γ in Eq. (9) ar xpctd to b th sam according to Eq. (3). In addition, th tmpratur gradints applid to th thr coils ar also th sam so that that th slop ( N 1) N γδt in Eq. (9b) of th thr coils ar also xpctd to th sam. Indd, th xprimntal rsults in Fig. 5(b) agr wll with th prdiction of Eq. (9b). It is important to point out that by curv fitting to Eq. (9b) on can obtain γδt and consquntly γ bcaus δ T can b obtaind indpndntly in th xprimnt. Tabl 2. Masurmnt Rsults of γδt and 0 ψ of Thr Fibr Coils Coil-1 Coil-2 Coil-3 Masurd δ T ( N 1) N γδt γ ψ 0 (Dg) Tabl 2. shows th slop γδ T and th intrcpt ψ 0 of ach fibr coil. Th slops ar slightly diffrnt du to th slightly fluctuation of δt among thr coils, possibly causd by non-idntical winding and potting of th coils. Th avrag γ obtaind for this coil typ is /( C m) (for ach coil typ γ can b dtrmind by prforming th trimming procdur dscribd abov). Th diffrnt intrcpts ψ 0 of th thr curvs in Fig. 5(b) and in Tabl 2 indicat that th thr coils hav diffrnt initial ffctiv asymmtric lngths l ff 0, according to Eq. (10c). 3.2 Dtrmination of coil s pointing rror thrmal snsitivity and asymmtric lngth Aftr vrifying Eq. (9) xprimntally, w now can confidntly us it to dtrmin th pointing rror thrmal snsitivity (PETS) γ T of a fibr coil and its ffctiv asymmtric lngth l ff by masuring ψ at diffrnt δ T. Fig. 6. Th pointing rror takn bfor coil trimming. Th lins indicat th bst linar fit through th data points. Using th stup in Fig. 3, w applid six diffrnt tmpratur xcitation profils to th outr layr of ach coil (TC 2 = 30 C, 40 C, 50 C, 60 C, 70 C, 80 C). Th corrsponding tmpratur diffrncs δt for th thr coils ar shown in Tabl 3. Figur 6 shows th masurd ψ as a function of δ T of th thr coils without trimming. Th pointing rror # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2528

9 incrass linarly with δ T, agring wll with Eq. (9b). By curv-fitting ψ to Eq. (9b), w obtain PETS γ T of thr coils and th rsults ar shown in Tabl 3. Tabl 3. Outr Surfac Tmpratur, Final Tmpratur Gradint, and γt of Thr Coils Undr Tst Outr surfac tmpratur ( C) γ ( dg / C ) T δ T ( C) Coil Coil Coil From Eq. (9c), w obtain th ovrall asymmtry lngths l all of th thr coils as 4.3 m, 4.1 m, and 3.33 m rspctivly, whr γ = is usd in th calculation. Th corrsponding ffctiv asymmtric lngths lff 0 b found as: [ ] of th coils from winding imprfctions can lff 0 = L N ( N 1) N lall ( N 1) (11) Substitution of L = 240m and N = 24 in Eq. (11), w obtain th initial ffctiv asymmtry lngths of th thr coils as 4.16m, 3.98m and 3.12m, rspctivly. Th minors sign indicats that th lngth of A portion fibr is longr than that of B portion fibr in th coils. W furthr masurd th pointing rror as a function of final tmpratur gradint of th thr coils with diffrnt trimming lngths, with th rsult of coil-2 shown in Fig. 7(a). Clarly, pointing rror is almost insnsitiv to variations in thrmal gradint with a trimming lngth about 4.5 m, corrsponding to a nar zro ovrall asymmtric lngth. Pointing rror starts to incras in th ngativ dirction with th thrmal gradint whn th coil is ovr trimmd with a trimming lngth of 5 m. Ths rsults ar consistnt with th prdiction of Eqs. (9b) and (9c). Evidntly, PETS is an xcllnt paramtr to quantify th thrmal prformanc of fibr coils: a largr PETS corrsponds to a largr ovrall asymmtric lngth. Th smallr th PETS of a coil, th smallr th thrmal inducd pointing rror of th gyro incorporating th coil. Figur 7(b) shows PETS obtaind by curv fitting in Fig. 7(a) as a function of trimming lngth. Th solid lin is th plot of Eq. (9c) using xprimntal paramtrs (L = 240m, N = 24 and γ = /( C m) ), and it agrs with th xprimntal data rasonably wll. Fig. 7. (a) Pointing rror of Coil-2 with diffrnt trimming lngths on th A portion fibr. Th slop of ach curv is th PETS of th coil. (b) PETS of Coil-2 as a function of trimming lngth l and th corrsponding thortical curv of Eq. (9c) for L = 240m, N = 24 and γ = # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2529

10 4. Summary In summary, w hav proposd and dmonstratd a practical approach to quantitativly dtrmin th prformanc of a fibr gyro coil subjct to thrmal variations. W show both thortically and xprimntally that th pointing rror of a fibr gyro is linarly proportional to its fibr coil s radial thrmal gradint, and dfin th proportional constant as th pointing rror thrmal snsitivity (PETS) to quantify th thrmal prformanc of fibr coils. In addition, w introduc a paramtr calld ovrall asymmtric lngth of th coil and show that it includs two contributions: on is intrinsic to th coil winding pattrn and th othr from imprfctions mad during coil winding. W furthr show both analytically and xprimntally that th asymmtric lngth is linarly proportional to PETS and can b obtaind onc PETS is masurd by masuring th pointing rror as a function of radial thrmal gradint. Finally, w prov xprimntally that th thrmal snsitivity of a fibr coil can b rducd to nar zro by trimming th fibr coil by th amount qualing th ovrall asymmtric lngth obtaind. Our xprimnts indicat that th proposd PETS masurmnt is robust against nvironmnt tmpratur variations and snsitiv only to coil s asymmtry, and thrfor is a practical paramtr for dtrmining th thrmal prformanc of fibr coils, dtrmining th coil asymmtry lngth, and guiding th coil trimming. Mor xprimnts ar undr th way to quantitativly dtrmin th influnc, if any, of nvironmntal tmpratur on PETS masurmnts. Acknowldgmnts This work was supportd by th National Basic Rsarch Program of China (973 Program) undr grant 2010CB327806, Intrnational Scinc & Tchnology Coopration Program of China undr Grants No. 2009DFB10080 and No. 2010DFB13180, China Postdoctoral Scinc Foundation undr Grant No # $15.00 USD Rcivd 20 Nov 2012; rvisd 14 Jan 2013; accptd 15 Jan 2013; publishd 25 Jan 2013 (C) 2013 OSA 28 January 2013 / Vol. 21, No. 2 / OPTICS EXPRESS 2530

INFLUENCE OF GROUND SUBSIDENCE IN THE DAMAGE TO MEXICO CITY S PRIMARY WATER SYSTEM DUE TO THE 1985 EARTHQUAKE

13 th World Confrnc on Earthquak Enginring Vancouvr, B.C., Canada August 1-6, 2004 Papr No. 2165 INFLUENCE OF GROUND SUBSIDENCE IN THE DAMAGE TO MEXICO CITY S PRIMARY WATER SYSTEM DUE TO THE 1985 EARTHQUAKE