Linear rotary optical delay lines

Transcription

1 Lnear rtary ptal delay lnes Maksm Skrgaty * Department f Engneerng Physs, Éle Plytehnque de Mntréal, Quée, Canada * maksmskrgaty@plymtla Astrat: I present several lasses f analytal and sem-analytal slutns fr the desgn f hgh-speed rtary ptal delay lnes that use a mnatn f statnary and rtatng urvlnear refletrs Detaled analyss f fur dstnt lasses f ptal delay lnes s presented Partularly, I nsder delay lnes ased n a sngle rtatng refletr, a sngle rtatng refletr and a sngle statnary refletr, tw rtatng refletrs, and tw rtatng refletrs and a sngle statnary refletr I demnstrate that n eah f these ases t s pssle t desgn an nfnte varety f the ptal delay lnes featurng lnear dependene f the ptal delay n the rtatn angle Ths s aheved va shape ptmzatn f the rtatng and statnary refletr surfaes Mrever, n the ase f tw rtatng refletrs a nvenent spatal separatn f the nmng and utgng eams s pssle Fr the sake f example, all the lades presented n ths paper are hsen t ft nt a rle f 1m dameter and these delay lnes feature n exess f 6ps f ptal delay Fnally, tw prttypes f rtary delay lnes were farated usng CNC mahnng, and ther ptal prpertes are haraterzed 14 Optal Sety f Amera OCIS des: (874 Gemetr ptal desgn; (848 Nnspheral mrrr surfaes; (36495 Spetrspy, terahertz Referenes and lnks 1 RE Beselt, Large Ampltude Hgh Frequeny Optal Delay, Hneywell ASCa, US 7,899,81 (11 D C Edelsten, R B Rmney, and M Sheuermann, Rapd prgrammale 3 ps ptal delay sanner and sgnal averagng system fr ultrafast measurements, Rev S Instrum 6(3, 579 ( J Ballf, R Gantt, Ph Chavanne, R Wält, and R P Salathé, Rapd and salale sans at 1 m/s n ptal lw-herene refletmetry, Opt Lett (11, ( J Szydl, N Delahenal, R Gantt, R Walt, H Bleuler, and P R Salathe, Ar-turne drven ptal lw herene refletmetry at 86-kHz san repettn rate, Opt Cmmun 154(1 3, 1 4 ( G G Lamuhe, M Dufur, B Gauther, V Bartulv, M Hewk, and J-P Mnhaln, Optal delay lne usng rtatng rhm prsms, Pr SPIE 649, 649G (7 6 A L Oldenurg, J J Reynlds, D L Marks, and S A Bppart, Fast-Furer-dman delay lne fr n vv ptal herene tmgraphy wth a plygnal sanner, Appl Opt 4(, (3 7 P-L Hsung, X L, C Chuda, I Hartl, T H K, and J G Fujmt, Hgh-speed path-length sannng wth a multple-pass avty delay lne, Appl Opt 4(4, (3 8 L Lu and N G Chen, Dule-pass rtary mrrr array fr fast sannng ptal delay lne, Appl Opt 45(1, (6 9 X Lu, M J C, and X L, Rapd sannng all-refletve ptal delay lne fr real-tme ptal herene tmgraphy, Opt Lett 9(1, 8 8 (4 1 K Lharenrat and I Jen Hsu, Optal delay lne fr rapd sannng lw-herene refletmeter, Intern J Inf Eletrn Eng, (1 11 C-L Wnag, C-LPan, Sannng Optal Delay Deve Havng a Held Refletng Mrrr, Natnal Sene Cunl, Tawan, US 5,97,43 ( TD Drney, Sannng Optal Delay Lne Usng a Refletve Element Arranged t Rtate, US 7,46,41 (6 13 G J Km, Y S Jn, S G Jen, and J I Km, Rtary Optal Delay Lne, Krea Eletrtehnlgy Researh Insttute, US 7,453,619 (8 14 J Xu and X-C Zhang, Crular nvlute stage, Opt Lett 9(17, 8 84 (4 15 G-J Km, S-G Jen, J-I Km, and Y-S Jn, Hgh speed sannng f terahertz pulse y a rtary ptal delay lne, Rev S Instrum 79(1, 161 (8 #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1181

2 1 Intrdutn Due t relatvely lw lss f mst dry materals n the THz spetral range, and due t strng senstvty f the materal lsses t the humdty ntent, there has een a large nterest n usng THz waves n nn-destrutve magng and press ntrl In partular, varus ndustral applatns f THz spetrspy are urrently under nvestgatn wth ptental applatns n detetn f nrgan nlusns n fd prduts, detetn f dangerus hemals dsgused y pakagng, detetn f defets n pakaged eletrn hps, mntrng f water ntent n paper and textle faratn lnes, t name just a few In ur researh grup we pursue desgn f THz spetrspy and magng systems that are ased n phtndutve antenna sures and detetrs Ideally, suh systems shuld feature lw nse and hgh aqustn rates Phtndutve antenna-ased systems have ganed nsderale attentn due t wde mmeral avalalty f the radand phtndutve antennas, as well as due t avalalty f relatvely heap fer-ased femtsend lasers that are used t pump phtndutve antennas Very hgh sgnal-t-nse rats f up t 5-6dB an e aheved n suh systems when usng lk-n amplfers and hgh ptal hppng rates Mrever, lw effenes f a sngle dple THz antenna an e mpensated y usng hgh numeral aperture antennas r the nterdgtated antenna arrays that an prdue ver ~1μW f THz pwer wth ~1W femtsend pump laser Fr rapd terahertz magng and real-tme press mntrng, a hgh sannng rate ptal delay lne s requred fr effent samplng f the terahertz pulses In addtn t hgh sannng rates, ne als desres hgh reslutn, large ampltude f sannng, hgh repeatalty, and easy endng/dedng f the ptal delay value Currently, majrty f the THz systems use delay lnes ased n lnear mrpstnng stages wth munted retrrefletrs The ppularty f ths slutn s related t the maturty f the mrpstnng tehnlgy and ts relatvely lw pre Mrever, ptal delay prdued y suh delay lnes depends lnearly n the stage dsplaement, whh greatly smplfes endng and readut f the ptal delay value Spetral reslutn Δ ν f a THz spetrspy setup s related t the maxmal usale ptal delay t max and s gven y Δν 1 t max Whle even a small L = 15m-lng lnear mrpstnng stage an prvde as muh as t max = L = 1 ps f ptal delay, n prate, the maxmal usale delay s typally lmted t t max ps 5 ps Ths s due t appearane f varus parast ehes and perturatns n the pulse shape nurred eause f multple refletns f the THz eam n/n the strutural elements f a THz setup Ths lmts reslutn f a standard THz setup t aut GHz Mrever, the andwdth f a THz setup s lmted y the Nyqust frequeny ν max 1 ( Δ t, where Δ t s a reslutn f the ptal delay lne Typally, ne requres that Nyqust frequeny s several tmes larger (say tmes than the THz andwdth f nterest (~5THz, frm whh t fllws the estmate fr the delay stage reslutn f Δ t < 1 fs Ths, n turn, sets a lmt n the auray f the lnear mrpstnng stage that shuld e superr t Δ L<Δt 15μm One f the hallenges assated wth the use f lnear mrpstnng stages as ptal delay lnes s ther lmted sannng speed The maxmal dsplaement speed f a lnear stage wth dsplaement auray superr t 15μ m and dsplaement range f 15m typally des nt exeed 1m s That means that even n the mst ptmal ase the repettn rate f suh a delay lne (and, hene, aqustn rate f a THz system wll nt exeed 5Hz Lw aqustn rates set y the use f lnear mrpstnng stages lead t nvestgatn f varus ther delay lne arhtetures that an prvde hgher sannng rates One f the enhanements n the lnear pstnng stage was desred n [1], where the authrs used a rtary t lnear mehanal uplng deve that allwed them t dsplae the retrrefletr alng a lnear path, whle usng fast rtary atuatn A system apale f 1s f ps f ptal delay and ~5Hz sannng rate was suggested Unfrtunately, resultant ptal delay #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11813

3 depends n a nn-lnear manner n the mtr rtatn angle, thus, neesstatng the use f an ptal ender fr prese readut f the nstantaneus ptal delay A galvanmeter-ased sllatng ptal delay lne [] was prpsed a deade ag fr hgh speed sannng f THz pulses In ths delay lne, ne f the tw mrrrs n the ptal avty sllates wth a fxed ampltude and frequeny Ths system was apale f ahevng 3ps ptal delay wth 3Hz sannng rate Hwever, the pstn and velty f the mrrr n ths setup hanges snusdally, s the ptal delay dd nt hange lnearly n tme Then, a rtatng ue [3,4] (r a prsm [5] was suggested In suh systems the lght s frst passed thrugh the rtatng ue, then refleted frm the statnary mrrr and, fnally, passed ak thrugh the same ue In ths arrangement, the refleted eam fllws the same path as the ndent eam, therefre requrng separatn f the tw at the utput f the delay lne Sannng rates as hgh as 38Hz were demnstrated wth ptal delay n 1s f ps Plang the ue n the ar turne an further mprve the sannng speed y almst an rder f magntude Unfrtunately, the resultant ptal delay s hghly nnlnear wth respet t the rtatn angle f a ue, therefre, requrng an ptal ender fr prese readut f the ptal delay Mrever, when usng larger ues, grup velty dspersn f the ue materal an eme mprtant due t lng prpagatn length nsde f the ue Ths may lead t a femtsend pulse radenng and, hene, lwer effenes f THz generatn and detetn In [6] a Furer-dman delay lne wth a plygnal mrrr array was presented and hgh sannng rates f 4kHz were demnstrated The desgn prfted frm mmeral plygnal mrrr arrays apale f 15kHz sannng rates, the man dsadvantage f ths setup was a relatvely lw ptal delay f 1ps, whh s adequate fr ptal herene tmgraphy applatns, ut nt fr THz spetrspy Addtnally, ptal delay depends n a nn-lnear manner n the rtatn angle f the mrrr array Other ntale ptal delay lnes mstly nstruted fr OCT applatns, and, hene haraterzed y relatvely small ptal delays (<ps, and hgh sannng rates (several khz are multple-pass avty delay lne [7,8] as well as varus mnatns f urved r straght mrrrs and a sannng mrrr [9,1] Reently, several lnear rtary ptal delay lnes ased n the mnatn f rtatng and statnary urvlnear refletrs have een demnstrated [11 15] The man advantages f these delay lnes nlude: lnear dependene f the ptal delay n the refletr rtatn angle, large values f the ptal delay (1s-1s f ps, hgh sannng rates (1s f Hz, and relatve ease f ther faratn usng CNC mahnng When usng rtary delay lnes that feature lnear dependene f the ptal delay n the refletr rtatn angle, ne an, n prnple, frg mplex ptal enders f the rtatn rate s presely ntrlled Mrever, ptal delay prvded y suh systems s prprtnal t the refletr sze As we wll see later n the paper, when usng relatvely small systems that ft nt a 1m-dameter rle, ptal delay n exess f 6ps an e aheved, whh s mparale t the ptal delay generated y a standard 1m-lng lnear mrpstnng stage wth a retrrefletr Fnally, the as sannng rate f a rtary delay lne s typally n the 5Hz range when standard mtrs are used at 3RPM By plang several refletrs n the same rtatng ase, ne an nrease the aqustn speed y an rder f magntude Thus, n [15] the sannng rates f 4Hz and the ttal delay f 14ps have een demnstrated when usng sx refletrs, all fttng nt an 8m dameter rle At ths pnt, t s mprtant t nte that althugh the hgh sannng rates are typally desred, an nrease n speed als requres a mmensurate nrease n the ptal pwer t mantan the same sgnal t nse rat Rtary delay lnes requre areful desgn f the shapes f ther refletr surfaes The man nstrant n the desgn f an ptal delay lne s the demand that nmng and utgng lght paths stay nvarant durng refletr rtatn Mrever, t s als desrale that the resultant ptal delay s lnear wth respet t the refletr rtatn angle S far, desgns avalale n the lterature nly present several partular mplementatns f the refletr shapes, whle general thery ehnd rtary delay lne desgn s stll mssng The man gal f ths wrk s t develp a mprehensve thery fr the desgn f rtary delay lnes Partularly, I nsder theretal frmulatns as well as analytal and sem- #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11814

4 analytal slutns fr fur general lasses f the rtary ptal delay lnes Delay lnes under nsderatn are mprsed f the fllwng mpnents: a sngle rtatng refletr, a sngle rtatng refletr and a sngle statnary refletr, tw rtatng refletrs, and, fnally, tw rtatng refletrs and a sngle statnary refletr Remarkaly, n mst ases t s pssle t desgn an nfnte varety f the lnear ptal delay lnes va ptmzatn f shapes f the rtatng and statnary refletr surfaes Mrever, n the ase f tw rtatng refletrs a nvenent spatal separatn f the nmng and utgng eams s pssle Fr the sake f example, all the lades presented n ths paper are hsen t ft nt a rle f 1m dameter and these delay lnes feature n exess f 6ps f ptal delay Ths s mparale t the ptal delay generated y standard 1m-lng lnear mrpstnng stage wth a retrrefletr Optal delay lne usng a sngle rtatng lade Example f a rtary ptal delay lne wth ne rtatng refletr s presented n Fg 1 There, a urvlnear refletr (whh I all a lade n the rest f the paper s rtatng arund a fxed axs pstned at the rgn The axs f rtatn s perpendular t the lade surfae Lght eam s arrvng parallel t the OX axs and s refleted ak y the lade edge alng the same path The lght eam s dsplaed alng the OY axs y R In ths smplest mplementatn that was reprted n [1], the rtatng lade stays perpendular t the nmng eam regardless f the lade rtatn angle Y R dl R x ( Inmng and utgng eams R R dl X Fg 1 Shemat f a rtary ptal delay lne featurng a sngle rtatng lade By desgn, the lade edge s always perpendular t the nmng lght eam at all rtatn angles Lght eam s arrvng parallel t the OX axs and s refleted ak y the lade alng the same path I wll nw demnstrate that an analytal slutn exsts fr the shape f a rtatng lade, whh results n the lade eng always perpendular t the nmng lght eam at all rtatn angles Mrever, a rrespndng delay lne wll have a lnear dependene f the ptal delay n the lade rtatn angle In what fllws, I defne t e the lade rtatn angle assumng a lkwse rtatn f the lade frm the OY axs When the lade rtates, #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11815

5 the pnt f ts ntat wth the lght eam wll nly hange ts pstn alng the OX axs ardng t x (, whle stayng nstant at R alng the OY axs Frst I shw that f the funtnal frm f the pstn f the lght refletn pnt n the lade rtatn angle s spefed, then, the shape f the lade s mstly x spefed In what fllw I suppse that the lade rtatn angle s defned n the nterval, I assume that at = the shape f the lade n ylndral rdnate system s [ ] gven y: max x y ( ϕ = r( ϕ s( ϕ ( ϕ = r( ϕ ( ϕ sn As the lade rtates lkwse y angle, ts shape an e expressed as: x( ϕ = r( ϕ s( ϕ ( y( ϕ = r( ϕ sn ( ϕ Fr a gven angle f rtatn, the shape f the lade desred y ( shuld ntan a lght refletn pnt havng Cartesan rdnates ( x(, R Frm ( I an then fnd the parametr representatn f the lade urve n terms f the angle f lade rtatn as: y R x( ϕ x tan ( ϕ = = = x x (, (3 y( ϕ = R ( r ϕ = x + y = x + R frm whh t fllws that the lade shape at = s gven y the fllwng parametr expressn: R ϕ( t = t+ ar tan + πn x ( t ( ϕ ( ( [, ], n max r t = x t + R t where I have delerately ntrdued a new parameter t that shuld nt e nfused wth a, as the lade rtatn angle Nte that parameter t s defned n the same nterval [ ] max lade rtatn angle Fnally, n Cartesan rdnates the lade shape s defned y susttutn f (4 nt (1 1 A lade perpendular t the nmng lght As derved n the appendx, the nlnatn angle f the lade wth respet t the dretn f an nmng lght (OX axs s: x ( tan ( α = x R (5 Frm ths t fllws that the he f:,, (1 (4 x = x + R (6 #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11816

6 results n the lade that s always perpendular t the nmng lght ( α = π ndependently f the rtatn angle Mrever, ptal delay prvded y suh a lade has a lnear dependene n the lade rtatn angle, wth the value f the ptal delay gven y: R Δ T = ( x x( = ; [, max ] (7 Thrughut the paper I suppse that the value f ptal delay s zer fr a zer rtatn angle Maxmal ptal delay and multple lade desgns Nw, I derve the value f the maxmal ptal delay ahevale y the delay lne that features ne lade perpendular t the ndent lght As a nstrant I demand that suh a lade shuld e mpletely nsred nt a rle f radus R dl [see Fg (a] The maxmal ptal delay s aheved when the lght refletn pnt spans all the pssle values x(, Rdl R fr the rrespndng rtatn angles [, max ] Addtnally I requre that < max < π n rder t avd lkng the nmng lght wth a part f the same lade Frm (6 I fnd: x = x = ( max dl x = R R R = R + 1 dl max Therefre, frm (7 t fllws that the maxmal ttal delay s gven y: (8 R Δ T dl max = max max + 1 (9 Frm (9 t fllws that there s a fundamental lmt t the maxmal ahevale ptal delay prdued y a rtary delay lne f radus R dl whh s Δ Tmax = 1975 Rdl, when max = π Fr example, n Fgs (a and ( I present the ase f a lade nsred nt a rle f radus Rdl = 5m and havng max 156 π, whh results n a smewhat suptmal delay ΔT 36 ps < ΔTmax 39 ps It s als nterestng t nte frm (8 that multple lades ( p lades an e ammdated n the same rtary delay lne f max s a smple fratn f π, that s, max = π p [see Fgs ( and (d] Ths sets R = Rdl 1+ ( π p and the maxmal ahevale ptal delay emes: R dl π Δ Tmax = p + (1 ( π Usng multple lades ( p lades allws t nrease the sannng rate y a fatr f p Mrever, as fllws frm (1, f the numer f lades s less than 6, the nrease n the sannng rate mes nly wth a slght redutn n the maxmal value f the ptal delay #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11817

7 m (a m Optal delay [ps] Blade rtatn angle [deg] ( m ( m 1 4 Optal delay [ps] Blade rtatn angle [deg] Fg (a Shemat f a rtary ptal delay lne featurng a sngle rtatng lade Blade pstns at varus rtatn angles are als presented as dtted lnes ( Crrespndng ptal delay ( Shemat f a rtary delay lne wth 3 lades and ( ts rrespndng ptal delay 3 Optal delay lne usng a sngle rtatng lade and a sngle statnary refletr In ths setn I nsder several example f rtary ptal delay lnes that mprse ne nlned lade and ne statnary refletr [see Fgs 3(a and 4(a] As efre, I nsder a urvlnear refletr (a lade rtatng arund a fxed axs The lght eam ntats the lade at x ( alng the OX axs The lght eam s arrvng parallel t the OX axs and t s dsplaed y R alng the OY dretn Ths tme, hwever, the lade surfae s nt perpendular t the nmng eam In fat, I wll dstngush tw ases In the frst ase [see Fg 3(a], the lade edge makes a nstant angle α wth respet t the dretn f the nmng eam (OX axs Therefre, I nly need t use a smple planar statnary refletr t send the eam ak alng ts arrval path I wll demnstrate that n ths smple ase I an nly realse a nnlnear ptal delay that hanges expnentally wth the rtatn angle In the send ase, I allw the nlnatn angle f the lade wth respet t the eam dretn t vary wth the rtatn angle [see Fg 4(a] In ths mre general ase I need t use a urvlnear statnary refletr t send the lght ak alng ts arrval path At the same tme, I shw that n ths mre mplated ase there exst an nfnte numer f analytal slutns fr the lade shape that results n lnear dependene f the ptal delay wth respet t the rtatn angle In ne ntale ase, the lade shape s a smple straght lne, whle the shape f (d #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11818

8 a statnary refletr s urvlnear [see Fg 4(a]; ths ase s f partular nterest fr pratal realsatn f the lnear rtary ptal delay lne due t ts smplty Later n the paper, we als present faratn and ptal haraterzatn f ths delay lne 31 A lade havng nstant nlnatn angle wth respet t the nmng lght I nw nsder the ase when the lade frms a fxed angle α π wth the dretn f an nmng lght In ths ase ne an realse the ptal delay lne y usng an addtnal statnary planar refletr t dret the lght ak alng ts rgnal path f arrval [see Fgs 3(a and 3(] Fr that, the statnary refletr shuld e perpendular t the lght refleted y the lade Frm (5 I an nw fnd the dsplaement f the lght refletn pnt as a funtn f the rtatn angle: x ( tan ( α = x R (11 Slutn f Eq (5 satsfes the fllwng dfferental equatn: x ( t ( α x R =, whh allws analytal slutn f the fllwng frm: ( ( α ( α (1 x = x exp t R tan (13 In rder t alulate ptal delay prvded y ths delay lne, I have t frst fnd the rdnates f a lght refletn pnt at the surfae f a statnary refletr If the lade frms a fxed angle α wth the OX axs, then, the refleted lght frms an angle αr = α π wth the OX axs Therefre, lght refleted y the lade travels alng the lne: ( y = R + x x tan α π (14 In the fllwng, I dente L( t e the dstane travelled y lght etween the lade and the statnary refletr Frm (14 t then fllws that rdnates f the lght refletn pnt lated n the surfae f a statnary refletr are gven y: xsr = x L s( α (15 y = R L sn ( α sr At the same tme, the statnary refletr shuld e perpendular t the lght refleted y the lade y makng a α π angle wth the OX axs In ther wrds, the tangent t the statnary refletr surfae shuld take a nstant value as fllws: ysr L x = tan ( α π = s( α (16 x sr The dfferental equatn wth respet t L( n (16 was derved usng defntns (15, whh has a partularly smple slutn: = + ( α (17 L L x s, #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11819

9 where L s sme nstant Fnally, I nlude that the ptal delay prvded y suh a delay lne has a nn-lnear dependene as a funtn f the rtatn angle, wth the aslute value f the ptal delay gven y: Δ T = x L x + L 4 sn = ( α ( x x( ( By mparng the ptal delays prdued y the rtary delay lnes that use perpendular lade (7 and nlned lade (18 [see Fg 3(], I nte that the letter an gve twe as large ptal delay than the frmer, whh s due t addtnal path travelled y the lght etween the rtatng lade and the statnary refletr Due t nn-lnear ptal delay resultng frm ths rtary delay lne we d nt study ths ase further R dl Y (18 Refletr α ( L R x ( α r Inmng and utgng eams X (a m m Optal delay [ps] ( ( Blade rtatn angle [deg] Fg 3 (a Shemat f a rtary ptal delay lne featurng a sngle rtatng lade that frms a fxed nlnatn angle wth the dretn f nmng lght and a lnear statnary refletr ( Blade pstns at varus rtatn angles (dtted urves ( Crrespndng ptal delay 3 A lade havng hangng nlnatn angle wth respet t the nmng lght As I have demnstrated n the prevus setn, usng nlned lades allws expansn f the desgn spae fr the rtary ptal delay lnes A lmtng feature f the desgn presented n the prevus setn s a hghly nnlnear dependene f the ptal delay n the lade rtatn angle In ths setn I nsder a mre general ase when the lade nlnatn α wth respet t the dretn f nmng lght an hange wth the rtatn angle #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 118

10 angle In ths ase, ne has t use a urvlnear statnary refletr n rder t reflet the lght ak alng ts rgnal path f arrval Addtnally, I demnstrate that there exst an nfnte numer f the delay lne desgns that feature lnear dependene f the ptal delay n the rtatn angle Ntaly, there exsts a partularly smple desgn fr the lnear delay lne wth the rtatng lade n the frm f a smple planar surfae [see Fg 4(a] R dl Y R x ( α ( Inmng and utgng eams X (a L( Refletr ( α π m ( m ( Blade rtatn angle [deg] Refletr Fg 4 (a Shemat f a lnear rtary ptal delay lne featurng a straght rtatng lade and a statnary urvlnear refletr ( Rtary stage desgn that an ammdate p = 7 straght lades Blade pstns at varus rtatn angles are presented as dtted lnes ( Crrespndng ptal delay As efre, I nsder a rtatng lade wth the shape that s defned y the ( x(, R rdnates f the lght refletn pnt n the lade surfae Fr the lade nlnatn angle α wth respet t the dretn f nmng lght I use (5: ( α Optal delay [ps] x ( tan = x R (19 In rder t alulate ptal delay prvded y the rtary delay lne I have t frst fnd the rdnates f the lght refletn pnt n the surfae f a statnary refletr If the lade α wth the OX axs, then the refleted lght frms an angle frms an angle α α π r = wth the OX axs Therefre, lght refleted frm the lade travels alng the lne: #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1181

11 ( ( y = R + x x tan α π ( In the fllwng, I dente L( t e the dstane travelled y lght etween the lade and the statnary refletr Frm ( t als fllws that rdnates f the send lght refletn pnt lated n the surfae f a statnary refletr are gven y: = s( α = sn ( α x x L sr y R L sr At the same tme, the statnary refletr shuld e perpendular t the lght refleted y the α π angle wth the OX axs In ther wrds, the tangent t the lade y makng the statnary refletr surfae shuld take a spef value as fllws: ysr = tan ( α π x sr By susttutng (19 and (1 nt ( I fnd that the dstane etween the lade and the statnary refletr satsfes the fllwng dfferental equatn: L x = s( α (3 If x dentty s spefed, then, usng expressn (19 fr the lade nlnatn angle, as well as s α = 1 tan α 1+ tan α, I an slve (3 y ntegratn: ( 1 max x x x L = L + d x R x R (4 31 Rtary delay lne wth lnear dependene f the ptal delay n rtatn angle I nw nvestgate desgnng f rtary delay lne featurng lnear dependene f the ptal delay n the lade rtatn angle In ths ase, the lade shape defned y x ( shuld have a very spef frm, whh I fnd y requrng that the ttal length f the lght path depends lnearly n the lade rtatn angle: β, (1 ( x L = L (5 where susttutn nt (3 I fnd the fllwng dfferental equatn: L s sme nstant Frm (5 I an then express L( n terms f x x β 1 = 1 + tan ( α, and after Fnally, after susttutng expressn (19 fr the lade nlnatn angle nt (6 I get the fllwng dfferental equatn fr the lade shape: x ( β ( x R = + x ( 1 (6 (7 #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 118

12 Remarkaly, a hghly nnlnear dfferental Eq (7 an e slved sem-analytally In partular, (7 an e frst rewrtten as a prlem f fndng the tw rts f the send rder x : plynmal wth respet t the dervatve f whh has the fllwng slutns: x x x R + + ( x ( + R =, β x x x R = R + ± + x β β β ( 1 (8 (9 Fnally, (9 an e slved y ntegratn wth respet t x The resultng expressn gves dependene f the lade rtatn angle n the value f the lght refletn pnt dsplaement: 1 x x x R 1, (3 mn β β β x x = C + dx R + ± + x where C s sme nstant Nte that slutn (9 desres a ntnuum f all pssle desgns fr the lnear rtary delay lnes parametrzed y the tw parameters R and β 3 Lnear delay lne wth a straght lade There exsts a partular smple slutn f (7 that desres a lnear lade and a urvlnear refletr that tgether mprse a lnear rtary delay lne [see Fg 4(a] Partularly, I nsder β = R n the ntegral slutn (3 Whle, n prnple, expressn (3 defnes tw slutns dependng n the he f a sgn, I fnd that ne f the slutns s n the frm f x = x + R that defnes a lade whh s always perpendular t the nmng lght (see Setn 1 I am, therefre, nterested n the send slutn f (3 that has the fllwng frm: x mn x 1 x x R R x = C + dx R + = + ar tan (31 Frm (31 I, fnally fnd the lade parameters: x = R tan ( L = L + R tan R (3 Frm (3 I nfrm that the lade shape s a straght lne that frms an angle π + wth the OX axs The value f the ptal delay prdued y suh a delay lne s then: 4R Δ T = ( x L x + L( = (33 Fnally, I nte that the ptal delay prdued y the straght lade and a statnary refletr (33 s exatly twe as large as the ne prdued y the lade that s always perpendular t the nmng lght (7 Ths s smply due t addtnal path travelled y the lght etween the lade and the statnary refletr #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1183

13 33 Maxmal ptal delay and multple lade desgns Cnsderatns f ths setn are vrtually dental t thse f Setn, therefre I nly present the fnal results Frst, wthut the lss f generalty I assume that = n (3 whh rrespnds t a straght lade that s perpendular t the dretn f the nmng lght fr a zer rtatn angle By demandng that the rtatng lade s mpletely nsred nt a rle f radus R I then nlude that the maxmal ptal delay s aheved when the dl lght refletn pnt spans all the pssle values rrespndng rtatn angles [, ], y ths stage s therefre gven y: max x, Rdl R wth the max = ar s R Rdl The ttal delay aheved 4R R Δ T = ar s, Rdl (34 whle the maxmal delay s aheved fr an ptmal he f R : 44 R Δ Tmax = max ( ΔT R R 65 R dl dl (35 Fr example, fr a lade radus Rdl = 5m ths results n Δ Tmax = 374 ps Nte that a rtary delay lne wth the maxmal ptal delay gven y (35 an ammdate up t 7 nn verlappng lades When dstruted unfrmly (angle wse there wll e small gaps etween the lades In general, p > 4 lades wthut gaps etween them an e ammdated nt the rtary delay lne f the ndvdual lade maxmal rtatn angle max s a smple fratn f π Ths sets the relatn etween the tw rad as R = Rdl s( π p and the maxmal ahevale ptal delay fr a delay lne wth p lades emes: p 4Rdl π π Δ T = s p p The maxmal delay s aheved when p = 7 lades are used and t s gven y: (36 p p 39 Rdl Δ Tmax = max ( ΔT (37 p= 7 Usng multple lades ( p lades allws t nrease the sannng rate y a fatr f p Mrever, as fllws frm (36, f the numer f lades s less than 15, ths nrease n the sannng rate mes nly wth a mnmal redutn n the value f maxmal ptal delay In Fgs 4( and 4( I present the ase f a lnear delay lne desgned t ammdate p = 7 straght lades T smplfy the presentatn, nly ne lade s shwn n the fgure The parameters f ths delay lne are hsen as fllws R = 5m, ( π R = R s 7 31m, thus resultng n a maxmal delay f Δ T = 374 ps dl 4 Lnear rtary ptal delay lne usng tw rtatng lades Smetmes, t s f nterest t separate n spae the nmng and utgng lght eams Fr ptal delay lne applatns t s, hwever, mprtant that the lne f path f the utgng lght eam s nvarale n tme One f suh systems s presented n Fg 5 There, I nsder dl #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1184

14 a mpste lade made f tw urvlnear refletrs (whh I all su-lades n the rest f the paper rtatng arund a fxed axs The nmng lght eam s arrvng parallel t the OX axs and t s dsplaed y R alng the OY dretn The eam s refleted y the rtatng su-lade numer ne nt the rtatng su-lade numer tw, whh, fnally, sends the eam ut f the delay lne Independently f the rtatn angle, the utgng lght eam fllws the fxed lne f path parallel t the OX axs; addtnally, I suppse that the utgng eam s dsplaed y R alng the OY axs In what fllws I demnstrate that an nfnte numer f slutns exst fr suh a mpste lade, all f them parametrsed y the he f the rdnates f the tw lght refletn pnts at a gven rtatn angle Mrever, the resultng ptal delay f a mpste lade system s always lnear wth the rtatn angle R dl Y R x ( α ( L Outgng eam R ( x ( α ( Inmng eam X m (a ( m ( Blade rtatn angle [deg] Optal delay [ps] Fg 5 (a Shemat f a lnear rtary ptal delay lne featurng a mpste rtatng lade wth tw su-lades ( An example f a rtary delay lne wth R = 5m, R = 1m, R = 3m Blade pstns at varus rtatn angles are presented as dtted lnes ( Crrespndng ptal delay I nw nsder a rtatng lade that has tw su-lades featurng lght refletn pnts defned y the rdnates ( x (, R and ( x(, R I use ndex t ndate the sulade that nterepts the nmng eam, whle I use ndex t ndate the su-lade that sends the eam ut f the delay lne The rrespndng lade nlnatn angles wth respet t the OX axs are α ( and α ( As fllws frm Fg 5(a, n rder fr the nmng and utmng eams t fllw tw parallel paths I have t demand that fr any rtatn angle, the nlnatn angles f the tw lades are related as α α = π Usng defntn (5 I then get: dl #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1185

15 tan ( α R x ( 1 = + tan ( R tan ( ( R x tan ( α x x x = x = + α = x = x R α π α α (38 Mrever, as fllws frm Fg 5(a, rdnates f the tw lght refletn pnts are related y the fllwng gemetral relatn: π R R x x = ( R R tan α = (39 tan α By usng the trgnmetr dentty tan ( α tan ( α 1 tan ( α (39 wth respet t the tangent f the angle α ( : x x = I an nw slve Eq ( α ( α 1 tan tan R R = x x x x tan ( α = ± + 1 R R R R Fnally, Eqs (38 and (4 nsttute a system f upled dfferental equatns that an e readly slved numerally Ntng that Eqs (38 and (4 are the frst rder dfferental equatns, therefre, slutn fr the lade shapes wll e unque when spefyng the pstns f the tw lght refletn pnts at a ertan rtatn angle, say x ( = x, ( x x = = In Fgs 5( and 5(, I present example f an ptal delay lne fund y numeral slutn f Eqs (38 and (4 wth ntal ndtns x = Rdl R, x =, and Rdl = 5m, R = 1m, R = 3m In fat, y keepng Rdl = 5m, and varyng the nput and utput rad n an shw that the he f R = 97m, R = 38m results n the maxmal value f the ptal delay ~134ps 41 Lnear respnse f the rtary delay lnes wth tw su-lades Remarkaly, rtary delay lnes that feature tw su-lades and ffer llnear paths f the nmng and utgng lght eams shw a lnear ptal delay that has a very smple frm: (4 1 R + R Δ T = ( x + x L ( x + x L( =, (41 where L( s the dstane etween the tw lght refletn pnts gven y: ( ( = L = x x + R R (4 Lnear ptal delay stated y (41 s relatvely straghtfrward t nfrm y takng a dervatve f the ptal delay wth respet t the rtatn angle and shwng that ths dervatve takes a nstant value #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1186

16 5 Optal delay lne usng tw rtatng lades and a sngle statnary refletr In ths setn I nsder anther type f the rtary ptal delay lne that an separate n spae the nmng and utgng lght eams It mprses a mpste lade and a statnary refletr [see Fg 6(a] A mpste lade s made f tw urvlnear refletrs (su-lades rtatng arund a fxed axs The nmng lght eam s arrvng parallel t the OX axs and t s dsplaed y R alng the OY axs The eam s then refleted y the rtatng su-lade numer ne n the dretn f a statnary refletr Then, the lght eam s redreted y the statnary refletr nt a rtatng su-lade numer tw, whh, fnally, sends the eam ak ut f the ptal delay lne Independently f the rtatn angle, the utgng lght eam fllws the same lne f path parallel t the OX axs whh ntersets the OY axs at R R dl Y (a R R x ( α x ( ( ( α Inmng eam Outgng eam L ( L X αr Refletr ( m Optal delay [ps] ( m ( Blade rtatn angle [deg] Fg 6 (a Shemat f a rtary ptal delay lne featurng a mpste rtatng lade wth tw su-lades and a statnary refletr ( An example f a rtary delay lne Blade pstns at varus rtatn angles are presented as dtted lnes ( Crrespndng ptal delay In the fllwng analyss I dstngush tw ases Frstly, I d nt mpse lnear respnse (wth respet t the rtatn angle n the rtary delay lne In ths ase there ext an nfnte numer f slutns fr the lade shapes In fat, y artrarly hsng the shape f ne f the su-lades, and y spefyng the pstn and nlnatn f the send su-lade fr the ntal value f the rtatn angle, then, the shape f the send su-lade and the shape f the statnary refletr an e fund y slvng numerally a sngle nnlnear dfferental equatn Sendly, I mpse lnear respnse f the rtary delay lne n the rtatn angle In ths ase, the lade shapes an e, n prnpal, fund y slvng a system f tw nnlnear dfferental equatns y spefyng the pstns and nlnatns f the tw lades fr the ntal value f the rtatn angle #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1187

17 I nw nsder a rtatng lade that has tw su-lades featurng lght refletn pnts x, R x, R As efre, I use ndex fr the lade defned y the rdnates ( ( and ( that nterepts the nmng eam, whle I use ndex fr the lade that sends the lght eam ut f the delay lne The rrespndng lade nlnatn angles wth respet t the OX axs α t e the nlnatn f the statnary refletr are α and α ( Fnally, I defne r wth respet t the OX axs at the refletn pnt that rrespnds t the rtatn angle f the delay lne As fllws frm Fg 6(a, n rder fr the nmng and utmng eams t fllw tw parallel paths I have t demand that fr any rtatn angle the nlnatn angles f the tw lades are related t the refletr nlnatn angle as α = α + α π Usng defntn (5 I then get: r tan ( αr t ( α α ( = + = x 1 t( α t( α t ( α + t ( α ( x x R x R t α = ; t α = I nw dente L ( and L t e the dstanes frm the lght refletn pnts n the lade surfaes t the lght refletn pnt n the surfae f a statnary refletr Usng the x, R, and Cartesan rdnates f the lght refletn pnts n the lade surfaes ( ( x(, R, I nw fnd the rdnates ( xr, yr f the lght refletn pnt n the surfae f a statnary refletr Frm gemetral nsderatns [see Fg 6(a] I wrte: = + s( π α = + s( α π = sn ( π α = + sn ( α π, x x L x L r y R L R L r frm whh t fllws that: L L x y r r ( ( ( ( = = x = ( x x tan ( α ( R R tan ( α tan ( α s( α ( x x tan ( α ( R R tan ( α tan ( α s( α ( tan ( α x tan( α + ( R R tan ( α tan ( α t ( α t ( α ( t ( α t ( α R R x x = Fnally, I need t demand that the tangent t the refletr shape gven y (45 equals t (43: yr = tan ( αr, (46 x r (43 (44 (45 #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1188

18 whh results n the dfferental equatn that relates tw lade shapes x ( and Gven the shape f ne su-lade, say x shape f the send su-lade x ( f the startng pnt x, then Eq (46 an e slved numerally fr the x and x = f the send su-lade are knwn at a gven value f the rtatn angle Nte als that all the expressns n Eqs (45 and (46 an e readly evaluated after presentng the trgnmetr funtns f a dule argument n terms f the tangents f a sngle argument that an e alulated frm e used: x, x usng (43 Namely, the fllwng trgnmetr denttes an ( ( α = ( α ( α tan tan 1 tan s α = 1 tan α 1+ tan α 51 Iteratve methd fr fndng the lade shapes Nw, I present example f an teratve algrthm that an e used t fnd numerally the x assumng that the shape f the ther su-lade (say shape f ne su-lade (say x ( s knwn Ths amunts t numeral slutn f Eq (46 Frst I assume that at a gven value f the rtatn angle x,,, = x are knwn Then, Eq (46 an e slved numerally = Partularly I nte that Eq (46 the values f x, and wth respet t the send rder dervatve x = an e wrtten n terms f the partal dervatves wth respet t the funtns x dervatves: yr yr yr yr x + x + x + x yr x x x x = = tan ( αr, xr xr xr xr xr = x + x + x + x x x x x where t smplfy the ntatn I have used x = x (, x x α α and,,,,, (47 and ther (48 =, = r = r Als I nte that all the partal dervatves n (48 an e easly evaluated as x r y r have explt dependene n x, and x, as fllws frm (43 and (45 By ntng that nne f the partal dervatves wth respet t x, r x, depend n x, I an then reslve (48 as: x y x y y x tan x + tan x + tan x ( α ( α ( α r r r r r r r r r x ( x x x x x x = yr x = r tan ( αr x x (49 Fnally, fr the values f x ( + d and I wrte: x = + d #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1189

19 d x x x + d = x + d + = = x x x = + d = + d = = (5 Iteratns (49, (5 an then e repeated multple tme t fnd the funtn x ( In Fgs 6( and 6( I present example f a rtary ptal delay lne fund y slvng Eq (46 usng teratve algrthm presented ave In ths ase I have used x = ( r, x r x = 95 r, r = 1 m, r = 3 m The =, = parameters were hsen t have a statnary refletr pstned utsde f the rtatng part f a delay lne As n all the prevus examples nsdered n ths wrk, the lades are nsred nt a 1 m -dameter rle Fr ths desgn, the angle dependene f the ptal delay s vrtually lnear and t prvdes up t 13 ps ptal delay The usale range f rtatn angles s 8, therefre, up t 13 refletrs an e ntegrated nt a sngle delay lne, thus resultng n an rder f magntude enhanement f the aqustn rate mpared t a delay lne wth a sngle refletr 5 Lnear delay lne In rder t desgn ptal delay lnes wth lnear respnse, n addtn t ndtn (46 I have t demand that: 1 Δ T = ( x + x L L ( x + x L L( = β, (51 where β s a nstant By takng a dervatve f (51 wth respet t the rtatn angle, I fnally arrve t a system f tw upled dfferental equatns: ( yr tan ( αr = xr, Δ T β = whh an e, n prnple, reslved numerally wth respet t the unknwn lade shapes x In rder t fnd a numeral slutn f (5 ne has t spefy the startng x ( and pnts x, x, and dervatves x = gven value f the rtatn angle, x = (5 fr the tw lades at a 6 Examples f pratal realsatn f the urvlnear refletrs usng CNC mahnng In ths fnal setns I dsuss tw prttypes f the rtary delay lnes that were farated usng CNC mahnng The gal f ths setn s nly t shw that desgns presented n ths papers are expermentally feasle nsderng urrent advanes n the mputer-ntrlled mr-mahnng As ths paper s mstly theretal, I defer the detaled dsussn f the ptal haratersatn f the presented prttypes t ur future pulatns The tw prttypes were farated frm the 5mm-thk arylntrle utadene styrene (ABS sheets usng a Charly U CNC mahne that ffers presn f lnear translatn f 5 μm A self-adhesve refletve alumnum-fl f ~5 μm thkness was attahed t the mahned lade surfae t frm urvlnear refletrs #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1183

20 The frst prttype [see Fg 7(a] mprses a sngle rtatng planar refletr and a sngle statnary urvlnear refletr as shwn n Fg 4(a In ths arrangement, the nmng and the utgng lght passes nde n spae A statnary refletr s mahned tgether wth a small rular gude fr the rtary refletr n rder t smplfy algnment f the tw parts Rtatng refletr was munted n the mult-axs ptmehan stage n rder t ntrl the lade spatal pstn, as well as the lade rtatn angle The rtatng refletr was nsred nt the ds f radus Rdl = 5m Shape f the statnary urvlnear refletr was alulated as desred n setn 3 and (3 wth R = Rdl s ( π 7 31m, β = R, L = Rdl, = By varyng the lade rtatn angle etween and 35 deg, an ptal delay f ~5ps was measured expermentally dretly frm the hgh-reslutn mages f the lght path As seen n Fg 7(, ptal delay s lnearly dependent n the rtatn angle, whh s n ardane wth the theretal predtn Nte that presn f ths measurement s lmted t ~6ps, whh s due t lw presn (~mm assated wth the dret measurement f the ptal lght path Fg 7 (a Expermental realsatn f the rtary delay lne featurng ne rtatng and ne statnary refletr ( Optal delay measured dretly frm the hgh-reslutn mages f the lght path ( Optal delay The send prttype [see Fg 8(a] mprses tw rtatng urvlnear refletrs desgned n suh a way that the utgng and nmng lght path are separated n spae The rtatng refletr was nsred nt the ds f radus Rdl = 5m Shape f the statnary urvlnear refletr was alulated as detaled n setn 4 and usng x ( = Rdl R, x ( =, where Rdl = 5m, R = 97m, R = 38m By varyng the lade rtatn angle etween and 35 deg, an ptal delay f ~1ps was expermentally measured frm the mages f the ptal lght path As seen n Fg 8(, ptal delay s lnearly dependent n the rtatn angle, whh s n ardane wth the theretal predtn #714 - $15 USD (C 14 OSA Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11831

21 Fg 8 (a Expermental realsatn f the rtary delay lne featurng ne rtatng and ne statnary refletr ( Optal delay measured dretly frm the hgh-reslutn mages f the lght path ( Optal delay 7 Dsussn Here, we refly dsuss the auray f the lnear rtary ptal delay lnes detaled n ths paper Ths questn wll e fully answered n the upmng expermental paper, whle here we nly present sme ak f the envelpe estmates In the ase f an nfntely thn laser eam (ptal ray, the ptal delay prdued y the rtary ptal delay lnes presented n ths paper s strtly lnear wth the lade rtatn angle The auray f the rtary ptal delay lne s then dretly related t the presn wth whh the lade rtatn frequeny an e mantaned A standard presn mtr wth a feedak ntrl lp an keep the rtatn frequeny f 3 RPM wth auray etter than 1 RPM Therefre, the auray f a delay lne wth the ttal ptal delay f 1ps s expeted t e etter than 3ps Anther sure f mpresn n the nstantaneus value f the ptal delay s due t fnte auray f the CNC mahnng press used n lade faratn Standard lw-end CNC mahnes an easly aheve presn n the urve defntn f less than 3 μm, whle expensve hgh-end CNC mahnes an aheve mrn presn Therefre, even when usng lw-end CNC mahnes tgether wth surfae plshng allws ahevng 1ps auray n the nstantaneus value f the ptal delay Yet anther fatr that nfluenes perfrmane f the rtary delay lnes s the wavefrnt dstrtn aused y the urvlnear refletrs In fat, the urvlnear refletrs presented n ths wrk were desgned fr peratn wth nfntely thn laser eams (ptal rays A fnte dameter parallel eam wll, therefre, e nverted nt a smewhat dvergent eam wth #714 - $15 USD (C 14 OSA Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 1183

22 dvergene angle eng larger fr larger-dameter eams One way f mpensatng fr the dstrtn f a fnte-sze laser eam wuld e t redue the sze f an nmng eam ether y usng eam mpressn r fusng In ths ase, are has t e taken when usng hgh pwer laser eams n rder t avd alatn f the lade surfae r ar reakdwn Fnally we nte that whle theretal ptal delays f the rtary delay lnes shwn n Fgs 7 and 8 are ~374 ps and ~134 ps rrespndently, t was, hwever, dffult t perate these delay lnes near the extremtes f ther urvlnear refletrs, thus resultng n the redued ttal ptal delay Ths prlem was auses y the suptmal he f the CNC tlng (large dameter drll ts and pr qualty refletr materals (self-adhesve refletve tapes used n the faratn f ths frst prttype In prnple, almst mplete theretal ptal delay an e aheved y usng etter CNC tlng, lade materals and plshng 8 Cnlusns In ths wrk I have develped a mprehensve thery fr the desgn f rtary delay lnes that feature a mnatn f rtatng and statnary refletrs Partularly, I have presented theretal frmulatns as well as analytal and sem-analytal slutns fr fur general lasses f the rtary ptal delay lnes Delay lnes under nsderatn are mprsed f the fllwng mpnents: a sngle rtatng refletr, a sngle rtatng refletr and a sngle statnary refletr, tw rtatng refletrs, and, fnally, tw rtatng refletrs and a sngle statnary refletr I have demnstrated that n all these ases t s pssle t desgn a varety f lnear ptal delay lnes va ptmzatn f shapes f the rtatng and statnary refletr surfaes Mrever, n the ase f tw rtatng refletrs a nvenent spatal separatn f the nmng and utgng eams s pssle Fr the sake f example, all the lades presented n ths paper were hsen t ft nt a rle f 1m dameter and these delay lnes feature n exess f 6ps f ptal delay Ths s mparale t the ptal delay generated y a standard 1m-lng lnear mrpstnng stage wth a retrrefletr, whle n addtn, rtary delay lnes ffer hgh aqustn rates f at least several hundred Hz Fnally, tw examples f the CNC mahned lade prttypes are presented and dsussed Appendx Here I derve expressn fr the nlnatn angle f the lade surfae wth respet t the dretn f nmng lght (OX axs at the pnt f lght refletn In rder t d that I nsder the lade pstn at tw values f the rtatn angle and + δ Partularly, fr the lade rtatn angle the pstn f the lght refletn pnt s gven y ( x(, R After lade rtatn y δ, ths pnt wll have the new rdnates gven y: sn s ( x1 s δ δ x x + Rδ = + O ( δ (53 y1 sn δ δ R R x δ At the same tme, rdnates f the new lght refletn pnt wll e ( x, y = ( x( + δ, R, whh allws us t fnd the nlnatn f the lade wth respet t the OX axs at the lght refletn pnt as: y y x 1 ( tan ( α = lm = (54 δ x x x R 1 Aknwledgments Ths wrk was supprted y the NSERC Strateg Grant STPGP enttled A dynamally renfgurale THz-TDS magng system fr ndustral measurement applatns I wuld lke t thank Dr Hang Qu fr hs ntrutn n the prelmnary expermental haraterzatn f the ptal delays presented n setn 6 f ths paper #714 - $15 USD Reeved 5 Fe 14; revsed 18 Apr 14; aepted 3 Apr 14; pulshed 7 May 14 (C 14 OSA 19 May 14 Vl, N 1 DOI:11364/OE1181 OPTICS EXPRESS 11833