ALLOCATING TOLERANCES FOR VEE-GROOVE FIBER ALIGNMENT

Transcription

1 ALLOCATING TOLERANCES FOR VEE-GROOVE FIBER ALIGNMENT Maheu Barraa and R. Ryan Vallance Precson Sysems Laboraory Unversy of Kenucky Lengon KY * S. Kan J. Lehman and Burke Hunsaker Teradyne Connecon Sysems Nashua NH Absrac Ths paper presens a mehod for allocang olerances o dmensons n mcro scale vee-grooves used o algn opcal fbers. The obecve s o reduce he manufacurng cos whou eceedng a lm on he msalgnmen beween wo mang fbers. The allocaon procedure s performed on a D geomerc model of a connecor ha algns an array of mulple fbers. An analycal model of he connecon based upon sascs s used for provdng a relaon beween varaon n manufacured dmensons and varaon n he resulng msalgnmen of he fbers conaned n he array. Opmal olerances are deermned usng a non-lnear consraned opmzaon algorhm ha mnmzes he manufacurng cos whle sasfyng consrans on he varaon of he msalgnmen of any par of fbers n he array. The mehod provdes a useful ool when desgnng mass-produced connecors for mul-fber cables for whch manufacurng cos and accuracy are crcal parameers. Keywords: olerance allocaon vee-grooves opcal fbers algnmen manufacurng cos Inroducon Vee-grooves are wdely used n mcroscale devces especally for posonng cylndrcal obecs. A common applcaon s o algn opcal fbers whch s of prmary neres for communcaons []. When connecng wo fbers a laeral msalgnmen due o he manufacurng errors of he vee-groove generaes a consequenal amoun of sgnal loss. Consderng Marcuse s [] model for sgnal loss due o laeral msalgnmen of sngle-mode fbers he cores of wo fbers should be algned whn abou µm o acheve ~ 0. db sgnal loss. Ths s a challengng proposon for only wo ndvdual fbers bu he challenge s even greaer for mass produced nerconnecs whch on cables conanng egh or more fbers as shown n Fg. Our obecve s o selec olerances ha are suffcen for algnng opcal fbers whou ecessve loss bu smulaneously mnmzng manufacurng coss ha arse from ecessvely gh olerances. Tolerance allocaon s generally formulaed as an opmzaon problem wh an obecve funcon and se of consrans. In hs case he obecve s o mnmze he manufacurng cos whch s a funcon of he olerances. Tolerance relaons for eched slcon vee-grooves are no avalable. However for oher maerals lke zrconum a secondary maeral removal process lke grndng may mprove he olerances. In hs case we employ relaons developed by Chase [3] o relae manufacurng coss o olerances. Boh cos and sgnal loss can be defned as funcons of he olerances allocaed by he desgner. The opmzaon consrans are formulaed as mamum olerable sgnal loss whn an enre connecor. Fbers Ferrules Fg : Mulfber Connecor wh Vee-Grooves In formulang he opmzaon problem he greaes challenge s deermnng a relaonshp * Precson Sysems Laboraory Mechancal Engneerng Unversy of Kenucky 0-A CRMS Buldng Lengon KY hp:// Teradyne Connecon Sysems Nashua NH. hp://

2 beween he olerances and he performance crera. For comple assembles such as a fber-opc connecor Mone Carlo smulaons are effecve means for relang fnal olerance of an assembly o he olerances of he componens [4]. However may be dffcul o mplemen Mone Carlo smulaons whn he opmzaon algorhm due o compuaonal me. We nsead use an alernave approach n whch a few Mone Carlo smulaons proved a mahemacal model relang assembly olerances o componen olerances. Knowng he geomery and he dmensons of he vee-groove s possble o defne he sgnal loss of he fber connecon as a funcon of he olerances of he veegroove. Ths paper presens a process o effcenly allocae he olerances for vee-groove fber algnmen. The frs sep s o consruc a mahemacal model of he dmensons and geomery of vee-grooves. The second sep s o defne hrough a sascal sudy he msalgnmen of he fber as a funcon of he olerances n he vee-grooves. The hrd sep s o esmae a relaon beween he olerances and he manufacurng coss. Fnally olerances are allocaed wh an opmzaon algorhm ha mnmzes he manufacurng cos for a gven mamum lm on sgnal loss. An eample llusraes he mehod. Mahemacal Model of he Dmensons and Geomery of a Vee-Groove Fber-o-fber connecons are a maor source of opcal loss. There es hree ypes of connecon losses [] ha are drecly relaed o he manufacurng errors whn he connecors. The frs one s caused by he laeral msalgnmen due o he offse of he cenerlnes of he mang fbers. The second comes from he end-separaon msalgnmen whch s he gap beween he ends of he conneced fbers. And fnally he hrd loss s generaed by he angular msalgnmen whch occurs when here ess an angle beween he wo aes of he fbers. The laeral msalgnmen s of mos concern for connecon loss snce angular msalgnmen s neglgble and end-separaon s usually resolved by mechancal conac beween he fbers or ndemachng compounds. In consderng only laeral msalgnmen a D model of he vee-grooves s a reasonable appromaon for represenng he veegroove geomery. The modelng plane s he z-plane y beng he drecon along he aes of he fbers. Geomery of a ferrule usng vee-grooves s llusraed n he plane perpendcular o he aes of he fbers n Fg. 0 z Fbers Ferrule Vee-Grooves Fg : Connecor n he z-plane In he D confguraon s possble o esablsh a mahemacal relaon beween he varaon of he dmensons n he ferrules and he laeral msalgnmen beween wo mang fbers. As shown n Fg 3 he h vee-groove of an array can be paramercally represened n wo dmensons wh four parameers and her manufacurng errors:. aperure angle α and angle error δ α. nclnaon angle γ whch s he angle beween he groove s bsecor and a vercal lne (deally γ equals zero) and angle error δ 3. deph o he vrual vere h and error γ δ h and 4. radus of curvaure r a he boom of he groove and error δ r. 0 z r + δ α +δα r Vere δ γ h + δ Fg 3: Manufacurng Errors n a Vee-Groove Merology appled o a ferrule canno drecly measure he value of he nclnaon angle so was decded o represen he aperure angle and he nclnaon angle as a combnaon of wo half-angles: L on he lef sde and R one he rgh sde. The aperure angle s defned as he sum of he wo half- h

3 angles whle he nclnaon angle s calculaed as half her dfference. The geomery of a sngle vee-groove s hen defned by four dmensons ( L R h r ) and her varaons. Furhermore varaon n he pch beween wo successve vee-grooves s also a crcal parameer for algnng fbers. The pch p s hen a ffh dmenson used for modelng he connecor. Varaon Analyss by he Law of Error Propagaon Tolerance allocaon requres a relaon beween dmensonal varaon and connecon loss. Marcuse [] presened a relaon gven n Eq () for he connecon loss T of sngle-mode fbers as a funcon of he laeral msalgnmen d and he wdh parameers w and w of he wo fbers. The dmensons d w and w are epressed n he same lengh un and T has no dmenson. Usually connecon loss s epressed n decbels. w w d T = ep () w + w w + w For mul-mode fbers epermenal daa are used o esablsh he relaon beween laeral msalgnmen and connecon loss as shown n Fg 4. Fg 4: Epermenal Deermnaon of a Relaon beween Loss and Msalgnmen by Curve Fng For boh sngle-mode and mul-mode cases s necessary o esablsh a relaon beween dmensonal varaon and laeral msalgnmen. Ths s done by applyng he law of error propagaon [5] on he mahemacal model of he connecors. Ths mehod s compuaonally effcen when used n a olerance allocaon algorhm. The geomerc model of he vee-groove mus be epressed n erms of sascs. Every dmenson ξ s defned as a randomly dsrbued varable. Is mean µ equals he value of he nomnal dmenson whle s sandard devaon σ s a hrd of he olerance. For a complee represenaon of he connecor he same procedure s appled for he dmensons of he fbers and for he dmensons of he sysem ha algns he wo ferrules. The dsrbuon of errors beween he wo ferrules may be measured epermenally or predced usng anoher geomerc/varaon model. The laeral msalgnmen d for he h par of fbers s modeled as a vecor n he z-plane. I s possble o defne s coordnaes ( d z ) d as a funcon of he dmensons of he vee-grooves he fbers and he algnng sysem as shown n Eqs ()- (3). = f ξ ξ ξ... ξ () z d d (... n ) ( ξ ξ ξ... ξ ) = f z... n (3) n beng he oal number of assgned dmensons whn he connecor. Accordng o he law of error propagaon f he dmensons are ndependen (whch s a reasonable assumpon for mos applcaons) hen he sandard devaon σ of he laeral msalgnmen n he - d drecon s gven by Eq (4). A smlar equaon gves he laeral msalgnmen n he z-drecon. d n = f σ ξ σ (4) For perfec dmensons he msalgnmen equals zero. Hence for random dmensons s varaon s drecly relaed o s sandard devaon. The law of propagaon error hen gves a drec analycal epresson of he varance n laeral msalgnmen as funcon of he varances n he dfferen dmensons of he connecor. Four dmensons L R h and p defne he geomery of a vee-groove. Snce he curren sudy analyzes he sensvy of he laeral msalgnmen o he geomery of he vee-grooves he sandard devaons of he componens n he and z drecons for he laeral msalgnmen are epressed as shown n Eqs (5)-(6):

4 σ d f + h σ zd f + h f σ L z σ f h z σ L h L f + p σ L f + f + p z σ R p f + σ z p σ R + Cons R σ R + Cons z (5) (6) where he consan erms are due o he varaons n he dmensons of he fbers and he algnng sysem. Eqs (5)-(6) reurn he varances of he componens n he and z drecons for he laeral msalgnmen bu Marcuse s model and Eq () requre he magnude of he msalgnmen d. Is value could be epressed wh a on probably dsrbuon for d and z d bu an unknown correlaon coeffcen beween he wo componens compromses he accuracy of he calculaon. Therefore a Mone Carlo smulaon of he connecor s used o deermne an emprcal relaon beween he connecon loss and he sandard devaons of and z by a wo-sep process. d d The frs sep consss n collecng daa from he Mone Carlo smulaon. Is npus are he nomnal values and he olerances of he dfferen dmensons defnng he geomery of a connecor. A large number of connecors are vrually generaed usng he mahemacal model prevously presened. Ther dmensons are normally dsrbued wh a mean equal o her nomnal value and a sandard devaon equal o one hrd of her olerance. The algorhm calculaes he msalgnmen of each randomly generaed sample hen performs a sascal reamen on he colleced resuls. Fnally reurns he sandard devaon of he componens n and z of he laeral msalgnmen as well as a cumulave dsrbuon funcon (cdf) of he connecon loss (n db) for every par of fbers as shown n Fg 5. Fg 5: Oupus of Mone Carlo Smulaon Every cdf s curve fed wh a wo-varable connuous funcon. Snce he olerance analyss focuses on he hghes par of he cdf (beyond 90%) he curve fng s performed eclusvely on hs par of he cdf n order o ge more relable appromaons. I has been found ha for snglemode fbers he cdf of a Webull random varable s a good appromaon whle a Gamma ncomplee funcon fs well he cdf of he mul-mode fbers. The smulaon s run many mes wh dfferen npu olerances. The resulng cdf s are reduced o wo parameers defnng he fed curve. Hence he frs sep of he process reurns a se of values for he wo fng parameers n funcon of he sandard devaons of and z. d d The second sep s a new curve fng procedure. Ths me one of he fng parameers s ploed n funcon of he sandard devaons of d and z d and s curve fed. The resulng relaons are fnally compared o new Mone Carlo smulaons and has appeared ha hey were eremely relable. These funcons are used as emprcal models of he connecon loss. Thereby a varaon analyss based upon he law of propagaon error followed by an emprcal ye accurae model of he connecor performance provdes a relaon beween he connecon loss and he olerances of he vee-grooves. Cos-Tolerance Relaons The cos of a manufacured par depends upon he seleced manufacurng process and dmensonal olerances. The cos of achevng a parcular olerance depends upon boh he dmenson's nomnal value and olerance. The manufacurng cos generally ncreases f he olerance s ghened and

5 s more epensve o make a gven olerance on a large nomnal dmenson. Based on hs Chase [3] recommends epressng olerances as recprocal power funcons for maeral removal processes. Eq (7) epresses he olerance for he h dmenson as a funcon of cos C range R and hree consans a b and c. The values of he hree consans depend upon he range and he manufacurng process. Alhough a consan erm would be necessary for accuracy s praccally mpossble o evaluae and doesn affec he olerance allocaon. a b R = c (7) C Smlar funcons are no avalable for echng processes commonly used wh slcon. Knowng he range and he manufacurng process of every dmenson enables generang he cos-olerance funcons requred o esmae he manufacurng cos of a connecor n funcon of he olerances assgned o s dfferen dmensons. The poron of he oal manufacurng cos ha s arbuable o vee-groove olerancng s hen he sum of he coss for he four dmensons L R h and p as shown n Eq (8). c Cos = c + h h L ah h L al L bh c + b L p c + p a p p R R bp ar R b R (8) can be specfed o preven he opmzaon from drvng he assgned olerances o unreasonably hgh or low values. Snce hs opmzaon only deals wh allocang olerances o he vee-grooves s assumed ha he olerances for he fbers and he sysem ha algns he ferrules s already known emprcally or predced by anoher analyss. The varables of he opmzaon problem are hen he olerances for he four dmensons ( L R h p ) defnng he vee-grooves. The radus of curvaure s no ncluded snce doesn affec he posonng of he fber. Ths mehod was used o allocae olerances o an eemplary connecor. The obecve was o mnmze he manufacurng cos of an 8-fber connecor whle he connecon loss of every par of sngle-mode fbers should be less han 0.5 db. The calculaed connecon losses along he connecors are dsplayed n Fg 6 and he resulng olerances allocaed by he opmzaon procedure are lsed n Table. When olerancng he connecor only one olerance s assgned o a nomnal dmenson even f hs feaure s repeaed several mes n he produc. Tha s he reason why he manufacurng cos depends upon he dmensons of one sngle veegroove nsead of he complee array of grooves. Tolerance Allocaon by Opmzaon Opmal olerances for he dmensons are deermned usng nonlnear consraned opmzaon. The problem s formulaed as a mnmzaon subec o consrans. The funcon o mnmze s he manufacurng cos of he connecor wh respec o s olerances as defned n he prevous secon. Consrans are formulaed by specfyng ha he sandard devaon of he laeral msalgnmen σ d for every par of fbers whn he connecor mus be posve ye below a crcal value. Addonal bounds Fg 6: Compued Losses Table : Compued Tolerances Dmenson Assgned Tolerance Lef Angle radans Rgh Angle radans Deph of he Vere 0.59 mcrons Pch beween Vees mcrons Conclusons Manufacurng cos s a crcal parameer for mass-produced feaures. On he oher hand accurae devces need low varaon n her dmensons. Massproduced mul-fber connecors usng vee-grooves

6 face boh problems. Ther desgn can be opmzed by a olerance allocaon mehod. Ths paper presens a mehod for allocang olerances o he dmensons of he vee-grooves. A mahemacal model of he geomery s consruced and used n a sascal analyss. Applyng he law of error propagaon allows dervaon of a relaon beween dmensonal varance and varance n sgnal loss. Opmal olerances of he vee-grooves are compued by mnmzng he relave manufacurng cos whle respecng consrans on mamum loss accepable for every par of mang fbers. The mehod s demonsraed for an eemplary 8-fber veegroove algnmen sysem. References [] Zanger H. & Zanger C. Fber Opcs Communcaon and Oher Applcaons Macmllan Publshng Company New York 99. [] Marcuse D. Loss Analyss of Sngle-Mode Fber Splces. The Bell Sysem Techncal Journal. May- June 977. pp [3] Chase K. W. Tolerance Allocaon Mehods for Desgners ADCATS Repor No Brgham Young Unversy 999. [4] Rachakonda P. Barraa M. Vallance R.R. Kan S. & Lehman J. D Error Models and Mone Carlo Smulaons for Budgeng Varaon n Opcal-Fber Array Connecors Proceedngs of he ASPE 00 Annual Meeng pp [5] Arenberg J. W. On he Orgns of he Law of Error Propagaon and s Uses Proceedngs of he ASPE 00 Summer Topcal Meeng pp