Single-arm 3-wave interferometer for measuring dispersion in short lengths of fiber

Transcription

1 Single-arm 3-wave interfermeter fr measuring dispersin in shrt lengths f fiber By Michael Anthny Galle St# A thesis submitted in cnfrmity with the requirements fr the degree f Master f Applied Science at the Graduate Department f Electrical & Cmputer Engineering, University f Trnt Cpyright 007 by Michael Anthny Galle

2 i Abstract Single-arm three wave interfermeter fr measuring dispersin in shrt lengths f fiber Michael Anthny Galle Master f Applied Science Graduate Department f Electrical & Cmputer Engineering University f Trnt 007 A simple fiber-based single-arm spectral interfermeter t measure the dispersin parameter in shrt lengths (DL) f fiber (< 50 cm) with a measurement precisin f ps/nm is develped. Dispersin is measured by examining the envelpe f the interference pattern prduced by three interfering waves: tw frm the facets f the test fiber and ne frm a mirrr placed behind it. The peratinal cnstraints n system parameters are discussed and a methd fr extending ne f them is intrduced. Experimental verificatin f this technique is carried ut via cmparisn f measurements made n SMF8 TM and DCF with thse made using cnventinal techniques. Mrever, this new technique is used t measure the dispersin f twin-hle fiber fr the first time.

3 ii Acknwledgements I wuld like t express my sincere gratitude t Prfessr Li Qian and Dr. Waleed Mhammed fr their inspiratin, visin, guidance and supprt ver the last year. I wuld never have cmpleted this wrk had it nt been fr their unwavering encuragement and dedicatin t supprting my studies and my research. Fr this I am truly grateful. I wuld like t thank Chris Sapian fr his supprt and friendship thrughut the years in bth Undergraduate and Graduate schl. I wuld like t thank my friends and clleagues in graduate schl fr their insights, supprt and friendship thrughut the last year. Special thanks g t Fei Ye and Jiawen Zhang. I wuld like t thank my parents and family fr their lve, supprt and encuragement thrughut all my years in schl. I wuld never have cme this far withut their help, inspiratin and guidance.

4 iii Cntents Abstract... i Acknwledgements... ii Cntents... iii List f Figures... v List f Tables... vii Chapter 1: Intrductin Mtivatin Objectives Organizatin f Thesis... 6 Chapter : Thery n Chrmatic Dispersin f a Waveguide Dispersin in a Waveguide Material Dispersin Waveguide Dispersin Chapter 3: Cnventinal Measurement Techniques Time f Flight Technique Mdulatin Phase Shift Technique Dispersin Measurements n Shrt Length Fiber Tempral Interfermetry (Furier Transfrm Spectrscpy) Spectral Interfermetry...0 General Case: Unbalanced...0 Special Case: Balanced Cmparisn f Dispersin Measurement Techniques... 9 Chapter 4: Thery f Single Arm Interfermetry A New Cncept Mathematical Descriptin Equal Amplitude Case Unequal Amplitude Cases System Parameters Wavelength Reslutin f the Dispersin Measurement Minimum Required Surce Bandwidth Measurable bandwidth f the dispersin curve B mea Minimum Fiber Length Maximum Fiber Length The Effect f Wavelength Windwing Mdel Develpment Simulatin Results... 63

5 iv Prbability vs. Windw Size Prbability vs. Average Step Size Prbability vs. Fiber Length Prbability vs. Tlerance...71 Chapter 5: Experiments & Analysis Experimental Prcess Experimental Challenges Experimental Instrumentatin & Specific Limits Experiments Single Mde Fiber Dispersin Cmpensating Fiber Twin Hle Fiber Errr Analysis Chapter 6: Cnclusins Expected Significance t Academia Expected Significance t Industry Patent Applicatin Cnclusins Appendix A: Matlab Cde A.1: Generating the Interference Pattern and the Envelpe A. Calculating Neff A.3: Prbability vs. Several ther Parameters A.3.1: Prbability vs. windw size A.3.: Prbability vs. average step size A.3.3: Prbability vs. fiber length...10 A.3.4: Prbability vs. tlerance A.3.5: The Prbability calculating functin A.4: Determining the Precisin f the Measurements A.4.1: Standard deviatin f the SMF8 TM Measurement A.4.: Standard deviatin f the DCF Measurement A.4.3: Standard deviatin f the THF Measurement Appendix B Crning SMF8 TM Data Sheet References and links... 11

6 v List f Figures Fig. 1-1: Intersymbl interference caused by dispersin leads t reductin in system bandwidth. Page Fig. -1: Cntributins f bth waveguide and material dispersin. 13 Fig. 3-1: Time f flight dispersin measurement technique. 16 Fig. 3-: Mdulatin Phase Shift Dispersin Measurement Technique. 17 Fig. 3-3: Experimental setup fr dual arm tempral interfermetry. 18 Fig. 3-4: Sample Tempral Interfergram. 19 Fig. 3.5: Interference pattern prduced by tw time delayed pulses. 1 Fig. 3-6: Filtering ut all but the f(t-τ) terms s that the phase infrmatin can be extracted. Fig. 3-7: Amplitude and phase spectrum f f(ω). 3 Fig. 3-8: Experimental setup fr Spectral Interfermetry. 4 Fig. 3-9: Sample spectral interfergram. 5 Fig. 3-10: Balanced path requirements fr a Michelsn interfermeter. 6 Fig. 3-11: Interference f the cupler arm reflectins. 7 Fig. 3-1: Fringe cancellatin technique fr a Michelsn interfermeter. 8 Fig. 4-1: Single-arm three waves interfermeter. 33 Fig. 4-: Fig. 4-3: Interference when reflectins frm the facets and mirrr have equal amplitudes. Calculated 3 wave interference pattern and envelpe fr a 30 cm piece f SMF8TM Fig. 4-4: Simulated interference pattern prduced by the SAI setup fr a 30- cm-lng SMF8TM test fiber, with =0.9, =1. 41

7 vi Fig.4-5: Simulated interference pattern prduced by the SAI setup fr a 30- cm-lng SMF8TM test fiber, with =0.4, =1. 41 Fig.4-6: Simulated interference pattern prduced by the SAI setup fr a 30- cm-lng SMF8TM test fiber, with =0.1, =1. 4 Fig. 4-7: Simulated interference pattern prduced by the SAI setup fr a 30- cm-lng SMF8TM test fiber, with =1, = Fig. 4-8: Simulated interference pattern prduced by the SAI setup fr a 30- cm-lng SMF8TM test fiber, with =1, = Fig. 4-9: Simulated interference pattern prduced by the SAI setup fr a 30- cm-lng SMF8TM test fiber, with =1, = Fig. 4-10: Dependence f the wavelength reslutin n the dispersin-length prduct. 47 Fig. 4-11: Minimum required surce bandwidth. 48 Fig. 4-1: Minimum bandwidth required as a functin f the dispersin length prduct. Fig. 4-13: The dependence f the measurable bandwidth (Bmea), n the DLf prduct Fig. 4-14: Minimum fiber length vs. surce bandwidth. 54 Fig. 4-15: The maximum measurable fiber length, Lf as a functin f the step size f the tunable laser. 56 Fig. 4-16: Tracing the envelpe f the interfergram by wavelength windwing. 58 Fig. 4-17: Measured Prbability density functin (histgram) and a Gaussian fit fr the step size f the Agilent 8164A tunable laser. Fig. 4-18: Mdel shwing the prbability density functins fr the step size and the carrier fr determining the prbability f hitting a peak in a given wavelength windw Fig. 4-19: Prbability vs. windw size. 64 Fig. 4-0: Prbability vs. Step Size. 65 Fig. 4-1: Prbability that at least ne peak is sampled in a given windw vs. fiber length. 67

8 vii Fig. 4-: Prbability vs. Fiber length fr different step sizes. 69 Fig. 4-3: Prbability vs. Tlerance. 71 Fig. 5-1: Experimental prcess fr the develpment and testing f the Single Arm Interfermeter. 74 Fig. 5-: Experimental Setup f a Single Arm Interfermeter 76 Fig. 5-3: Fig. 5-4: Fig. 5-5: Measured dispersin cmpared t published Dispersin equatin fr a 39.5cm SMF8TM fiber. (a) Measured upper envelpe (experimental) fringe pattern. (b) Simulated interference pattern and upper envelpe. Measured dispersin parameter plt fr DCF using the Agilent 8347A and Single Arm interfermetry Fig. 5-6: Crss sectin f a typical Twin-Hle Fiber. 83 Fig. 5-7: Fig. 5-8: Fig. 5-9: Measured dispersin fr the Twin-Hle Fiber perfrmed using Single Arm Interfermetry. Errr in calculating B due t the errr in lcating the peaks f the interfergram Cnceptual design fr a dispersin measurement mdule fr a tunable laser system Fig. 5-10: Agilent 8164A/B Lightwave measurement system mainframe. 94 Fig. 5-11: Agilent A/B Phtnic Dispersin and Lss Analyzer 94 List f Tables Table 3-1: Summary f the varius dispersin measurement techniques 30 Page Table 4-1: Table 4-: Differences & Similarities between the Michelsn Interfermeter, CP-OCT and the Single Arm Interfermeter The dips where the prbability drps t zer in Fig. 4- ccur m when the carrier perid is a multiple f G n the step size

9 Chapter 1: Intrductin The design f phtnic devices heavily depends n an accurate characterizatin f the cmpnents used. One f the main cmpnents in a phtnic device is an ptical fiber which serves as a lw-lss medium fr light transmissin. An imprtant characteristic f fiber is the dispersin that light experiences as it travels inside the fiber. Dispersin is the phenmenn that causes different frequencies f light t travel at different velcities. The phenmenn f dispersin is cmmnly bserved thrugh the spreading f light by a prism. When white light, which cntains a brad spectrum f frequencies, enters a prism the different wavelengths are bent at different angles since each frequency sees a different index f refractin, a phenmenn first quantified by Newtn in the 17th century [1]. Inside a fiber this variatin in the index f refractin with frequency is what causes the frequency dependence f the velcity. A mre mdern example f the phenmenn f dispersin is the affect it has n the perfrmance f phtnic devices used in cmmunicatin systems. In these systems, dispersin, r mre specifically secnd rder dispersin, leads t a bradening f the pulses used t represent 1 r 0 in a digital cmmunicatin system. Pulse bradening causes adjacent bits t verlap and leads t intersymbl interference []. Intersymbl interference ccurs when a pulse is bradened beynd its allcated bit slt t such an extent that it begins t verlap with adjacent bits and it is n lnger pssible t determine whether r nt a specific bit cntains a 1 r a 0. This effect is illustrated in Fig. 1-1: 1

10 Chapter 1: Intrductin Fig. 1-1: Intersymbl interference caused by dispersin leads t reductin in system bandwidth. As a result f intersymbl interference the allcated bit slts must be widened and this effectively lwers the number f bits that can be transmitted in a given perid f time and reduces the system bandwidth []. As a result mdern cmmunicatin systems have evlved methds t mitigate the effects f dispersin. Current methds f cuntering the effects f dispersin in an ptical fiber use dispersin cmpensating devices such as chirped fiber Bragg gratings and dispersin cmpensating fiber (DCF) []. In rder t effectively use these techniques it is critical t knw the exact magnitude f the dispersin that is being cmpensated fr. As a result knwledge f the dispersin in bth the transmissin system and the dispersin cmpensatin system is critical t the design f the verall cmmunicatin system. Fr example, in rder t determine the length f dispersin cmpensating fiber required t cmpensate fr the dispersin incurred in a span f standard single mde fiber, ne must knw the dispersin in bth types f fiber as well as the exact length f single mde fiber fr which the dispersin is t be cmpensated []. The dispersin in the ptical fiber can then be cmpensated by splicing a length f DCF given by: LFiber DFiber ( ) LDCF Eq. 1-1 [] D ( ) DCF

11 Chapter 1: Intrductin 3 D is knwn as the secnd rder dispersin parameter which is a functin f the secnd rder dispersin f the fiber. Its significance and its effect n an ptical signal will be discussed in detail in chapter. Knwledge f dispersin in a fiber is als critical fr the study f fiber based nnlinear wave interactin phenmena. An ptical slitn is a pulse that maintains a cnstant shape (width) as it prpagates alng a fiber (first rder slitn) r has a shape that is peridic with prpagatin (higher rder slitn) [3, 4]. This is due t the fact that the effects f dispersin and self phase mdulatin (SPM) are in balance [4-5]. SPM is the effect whereby the phase f a given pulse is mdified by its wn intensity prfile [6]. Knwledge f the dispersin in an ptical fiber allws fr the determinatin f the required intensity fr the frmatin f an ptical slitn. This effect has als been used in the area f slitn effect pulse cmpressin [5, 7, 8] where the cmbinatin f the chirping effect f SPM and subsequent distributed cmpressin effect f negative dispersin is used t cmpress an ptical pulse [7]. Knwledge f dispersin is als imprtant fr the study f nnlinear effects such as secnd harmnic generatin, threewave mixing and fur-wave mixing since it determines the interactin lengths between the varius wavelengths. Dispersin is particularly imprtant in techniques that aim t extend this interactin length such as in Quasi Phase Matching (QPM) devices [9-11]. 1.1 Mtivatin The mtivatin fr this thesis is t measure the dispersin parameter in shrt lengths f ptical fiber. Mre accurately the methd is required t measure fiber with small dispersin length prducts (DL). The initial need fr a shrt length characterizatin scheme came frm the need t measure the dispersin f a type f specialty fiber knwn

12 Chapter 1: Intrductin 4 as the twin-hle fiber (THF) (Ch 5.6). This fiber is nt easy t acquire and is expensive t prduce therefre the use f cnventinal dispersin measurement techniques requiring lng lengths f fiber (Ch. 3) are nt pssible. The principle reasn fr measuring the dispersin in THF is t study nnlinear wave interactin phenmena in these fibers. Knwledge f the dispersin was als f practical imprtance since we planned t use QPM t increase the interactin length [9-11]. Shrt length characterizatin was als required because the fiber gemetry f THF is nt unifrm alng its length which results in a variatin f the dispersin alng the fiber length. The dispersin measurement n a lng length f fiber, therefre, is different than the dispersin in a given sectin f that fiber. Typically nly a small sectin f THF is used in QPM experiments and therefre the dispersin f the specific sectin f THF used in the experiment must be measured. Shrt length (small DL) characterizatin is nt nly required fr THF but it is als necessary fr ther types f specialty fiber as well. Phtnic Crystal Fiber (PCF) [1-14], fr example, can be used fr dispersin cmpensatin (DC-PCF) [13]. Fr devices with small dispersin length (DL) prducts, such as fiber laser cavities [3], the length f the dispersin cmpensatin fiber required is very shrt. As a result, it is necessary t measure the dispersin in the exact sectin f DC-PCF that will be used in the system. Recent advances in Micrstructured fiber r PCF allw fr a high degree f cntrl ver the dispersin [14]. This has led t a need fr experimental testing t determine hw clse the dispersin in the fabricated device is t the predicted theretical dispersin. Experimental verificatin f the theretical dispersin is less expensive when nly shrt lengths f this fiber are required and therefre it is bth cnvenient and ecnmical if dispersin can be measured n shrt lengths f fiber.

13 Chapter 1: Intrductin 5 Gain fiber is anther type f specialty fiber fr which it is desirable t have a shrt length dispersin measurement technique. Typically shrt lengths f gain fiber are used t cmpensate fr lsses in a lng haul ptical transmissin line [15]. The dispersin in these shrt lengths f gain fiber must be knwn in rder fr dispersin cmpensatin schemes t accurately cmpensate fr the dispersin prduced in the entire channel. The dispersin is f particular imprtance when these gain fibers are used t make mde lcked fiber lasers [4] since dispersin affects the grup velcity f a pulse within the cavity [] it als affects mde lcking schemes. 1. Objectives The primary bjective f this thesis is the develpment f a technique t measure the dispersin parameter in fiber lengths belw 50 cm (small DL prducts). The first bjective is t develp and test the technique by cmparing its results with published (r cnventinally measured) dispersin parameter curves fr SMF8 TM and Dispersin Cmpensating Fiber (DCF). Secnd, the thery fr the technique will be further investigated and peratinal cnstraints will be utlined. Third, the dispersin parameter f Twin Hle fiber (THF) will be measured. The dispersin parameter fr this fiber has nt yet been reprted in the literature. The furth and final bjective is t shw that this technique is cnducive t cmmercial develpment, since it can measure waveguides and ptical fibers frm several centimeters t a few meters in length, withut the cmplicatins intrduced by cnventinal interfermetric dispersin techniques such as the dual arm techniques.

14 Chapter 1: Intrductin Organizatin f Thesis This thesis is rganized int six chapters. The first chapter intrduced the tpic f dispersin and the mtivatin and bjective behind this wrk. The secnd chapter utlines the basic thery behind light prpagatin in a fiber and intrduces the cncept f chrmatic dispersin. It als utlines hw bth the material and the waveguide dispersin are cmbined t yield the ttal chrmatic dispersin in a waveguide. The third chapter surveys the cnventinal techniques fr measuring chrmatic dispersin in ptical fiber. The furth chapter describes the thery and limitatins f the nvel single arm interfermeter develped in this thesis fr dispersin characterizatin f shrt length ptical elements. It als shws hw sme f these limitatins can be relaxed s that a larger range f fiber length can be characterized using the technique. The fifth chapter describes the experimental results used t verify and implement the new technique. Characterizatin is first perfrmed n Crning SMF8 TM since the dispersin curves are well knwn and can be used t verify the validity f the thery and the technique. As a secnd verificatin the technique is applied t dispersin cmpensating fiber. Once the technique has been verified and tested it is used t characterize specialty fiber knwn as Twin Hle Fiber fr which the dispersin curves have nt yet been reprted. The sixth chapter cncludes the thesis by summarizing the benefits f the single arm interfermeter and by describing the cntributins this new technlgy can make t the field f ptical characterizatin. The thesis is cncluded with an examinatin f the future wrk that is required in rder t develp a cmmercial device frm this technlgy.

15 Chapter : Thery n Chrmatic Dispersin f a Waveguide Dispersin is the phenmenn whereby the index f refractin f a material varies with the frequency r wavelength f the radiatin being transmitted thrugh it [1]. The term Chrmatic Dispersin is ften used t emphasize this wavelength dependence. The ttal dispersin in a waveguide r an ptical fiber is a functin f bth the material cmpsitin (material dispersin) and the gemetry f the waveguide (waveguide dispersin). This chapter utlines the cntributins f bth material and waveguide dispersin, identifies their physical surce and develps the mathematical terminlgy fr their descriptin..1 Dispersin in a Waveguide When light is cnfined in an ptical fiber r waveguide the index is a prperty f bth the material and the gemetry f the waveguide. The waveguide gemetry changes the refractive index via ptical cnfinement by the waveguide structure. The refractive index is therefre a functin f bth the material and waveguide cntributins. Fr this reasn in a fiber r a waveguide the index is knwn as an effective index. The relatinship between the effective index and the first, secnd and higher rder dispersin can be understd mathematically via a Taylr expansin: n eff n eff 3 dneff d n eff d n 3 eff ( ) ( ) ( ) ( ) 3 d d d... Eq. -1 7

16 Chapter : Thery n Chrmatic Dispersin f a Waveguidewww.inmetrix.cm 8 The first term in Eq. -1 represents the linear prtin f the effective index as a functin f wavelength and shws hw dispersin manifests itself in the wavelength dependence f the phase velcity fr a wave inside a medium. The relatinship between the first term and the phase velcity is described in Eq. -: V p c ( ) Eq. - [3] n ( ) eff The secnd term in Eq. -1 is related t the grup velcity f an ptical pulse and represents the first rder dispersin. The grup velcity is the velcity that the envelpe f an ptical pulse prpagates. It depends n a quantity knwn as the grup index, N G, which is a functin f bth the index f refractin and the slpe f the index f refractin at a particular wavelength. The grup velcity relates t the secnd term via Eq. -3 where c is the velcity f light in vacuum: V g c ) N G c n( dn ) d c n( dn ) d ( Eq. -3 [3] The third term in Eq. -1 represents the variatin in the grup velcity as a functin f wavelength. This variatin in the grup velcity is knwn as Grup Velcity Dispersin, GVD, which is related t the third term via Eq. -4, where λ is the particular wavelength fr which the GVD is calculated and c is the speed f light in vacuum: GVD( ) d n eff c c d Eq. -4 [] The term in the brackets in Eq. -4 is knwn as the dispersin parameter, D, which represents secnd rder dispersin since it describes hw the secnd derivative f the effective index varies with respect t wavelength:

17 Chapter : Thery n Chrmatic Dispersin f a Waveguidewww.inmetrix.cm 9 d n eff D( ) Eq. -5 [16] c d The dispersin parameter is imprtant since it is related t pulse bradening which critically limits the bit rate f a cmmunicatin system. Eq. -6 shws hw an increase in the dispersin parameter directly relates t an increase in pulse bradening: T D ) L ( Eq. -6 [] In Eq. -6 Δλ is the range f wavelengths traveling thrugh the medium and L is the length f the medium. The dispersin parameter, D(λ ), which is related t pulse bradening, is the mst significant parameter since it limits the bit rate f an ptical cmmunicatin system. The dispersin parameter f a waveguide such as an ptical fiber is given by the ttal dispersin due t bth the material and waveguide cntributins. The ttal dispersin is the cmbinatin f the material dispersin and the waveguide dispersin and thus the dispersin parameter f a waveguide is given by: c D 1 VG d d D M D W Eq. -7 [] The next tw sectins discuss the cntributins that bth material and waveguide dispersin make individually t the ttal dispersin.

18 Chapter : Thery n Chrmatic Dispersin f a Waveguidewww.inmetrix.cm 10. Material Dispersin Material dispersin riginates frm the frequency r wavelength dependent respnse f the atms/mlecules f a material t electrmagnetic waves. All media are dispersive and the nly nn-dispersive medium is vacuum [1]. The surce f material dispersin can be examined frm an understanding f the atmic nature f matter and the frequency dependent aspect f that nature [1]. Material dispersin ccurs because atms absrb and re-radiate electrmagnetic radiatin mre efficiently as the frequency appraches a certain characteristic frequency fr that particular atm called the resnance frequency [1]. When an applied electric field impinges n an atm it distrts the charge clud surrunding that atm and induces a plarizatin that is inversely prprtinal t the relative difference between the frequency f the field and the resnance frequency f the atm [1]. Thus the clser the frequency f the electrmagnetic radiatin is t the atms resnance frequency the larger the induced plarizatin and the larger the displacement between the negative charge clud and the psitive nucleus. The relative displacement between the electrn clud and the nucleus is given by the Lrentz Oscillatr Mdel [1] as: x qe / me E ( ) Eq. -8 [1] The induced plarizatin is given by: P q e x P ( ) E Eq. -9 [1]

19 Chapter : Thery n Chrmatic Dispersin f a Waveguidewww.inmetrix.cm 11 The index f refractin is given by the relatinship between the induced plarizatin and the incident electric field. It is knwn as the dispersin equatin [1] and is given by Eq. -10: P qe 1 n ( ) 1 1 Eq. -10 [1] E 0me In this equatin n(ω) is knwn as the abslute index f refractin [1] since it is the index f refractin seen by light f frequency ω in bulk media. It illustrates mathematically hw the index f refractin varies fr different frequencies (wavelengths) accrding t hw clse they are t a resnance frequency f the atm. Given this knwledge f n(ω), the grup index f the material can be determined via N G n( ) dn d n( ) dn d. The material dispersin is then determined by taking the derivative f the grup index f the material with respect t wavelength r equivalently the secnd derivative f the abslute index with respect t wavelength: D.3 Waveguide Dispersin 1 dng d n c d c d M Eq. -11 [] Waveguide dispersin ccurs because waveguide gemetry variably affects the velcity f different frequencies f light. Mre technically, waveguide dispersin is caused by the variatin in the index f refractin due t the cnfinement f light an ptical mde [3]. Waveguide dispersin is a functin f the material parameters f the waveguide such as n n the nrmalized cre-cladding index difference, cre cladding cre n, and

20 Chapter : Thery n Chrmatic Dispersin f a Waveguidewww.inmetrix.cm 1 gemetrical parameters such as the cre size, a [, 17]. The index in a waveguide is knwn as an effective index, n eff, because f the prtin f the index change caused by prpagatin in a cnfined medium. Cnfinement is best described by a quantity knwn as the V parameter, which is a functin f bth the material and gemetry f the waveguide. The V parameter is given by Eq. -1: V ( ) a( n cre n cladding) 1/ ancre Eq. -1 [] Prpagatin in a waveguide is described by a quantity knwn as the nrmalized prpagatin cnstant, b, which is als a functin f the material and gemetry f the waveguide. This quantity is given in Eq. -13: b n n eff cre n n cladding Eq. -13 [] cladding The cntributin f the waveguide t the dispersin parameter depends n the cnfinement and prpagatin f the light in a waveguide and hence it is a functin f bth the V parameter and the nrmalized prpagatin cnstant, b. The waveguide dispersin can be calculated via knwledge f V and b via Eq. -14: D W N n G( cladding G( cladding) cladding ) Vd ( Vb) dn dv d d( Vb) dv Eq. -14 [] In mst cases the main effect f the waveguide dispersin in standard single mde fibers is a reductin in dispersin cmpared t dispersin in bulk []. In cmparisn t material dispersin the cntributin f waveguide dispersin is quite small

21 Chapter : Thery n Chrmatic Dispersin f a Waveguidewww.inmetrix.cm 13 and in mst standard single mde fibers it nly shifts the zer dispersin wavelength frm 176nm t 1310nm []. This effect is illustrated in Fig. -1: Fig. -1: Cntributins f bth waveguide and material dispersin [] In summary, the dispersin in a waveguide r an ptical fiber is caused nt nly by the material but als by the effect f cnfinement and prpagatin in the waveguide. Thus accurate knwledge f the dispersin in a waveguide cannt be made by simple knwledge f the material dispersin but must include the effect f the waveguide. As a result either the dimensins f the waveguide must be knwn t a high degree f accuracy s that the waveguide dispersin can be calculated (which is nt easy since fabricatin prcesses are hardly perfect) r the dispersin must be measured empirically. Accurate measurement f the (ttal) dispersin parameter, D, is critical t the design f

22 Chapter : Thery n Chrmatic Dispersin f a Waveguidewww.inmetrix.cm 14 phtnic systems. Measurement techniques fr the determinatin f the dispersin parameter will nw be discussed in the next chapter.

23 Chapter 3: Cnventinal Measurement Techniques There are 3 categries f dispersin measurement techniques: Time f flight (TOF) [18], Mdulatin phase shift (MPS) [17, 19] and Interfermetric [16]. TOF and MPS are the mst widely used cmmercial dispersin measurement techniques. Interfermetric techniques are nt widely used cmmercially but have been used in labratries fr dispersin measurements. Interfermetric techniques cme in tw frms; tempral and spectral. This chapter surveys the existing techniques, their advantages and disadvantages and cncludes with a quantitative cmparisn f the varius dispersin measurement techniques in terms f measurement precisin and fiber length requirements. 3.1 Time f Flight Technique In the TOF technique the secnd rder dispersin parameter, D, hereafter referred t simply as the dispersin parameter, can be determined either by measuring the relative tempral delay between pulses at different wavelengths r by measuring the pulse bradening itself. The relative tempral delay between pulses at different wavelengths is measured t determine the grup velcity which can then be used t determine the dispersin parameter using Eq. 3-1: D( ) t L( ) Eq. 3-1 [16] The abve equatin can als be used t determine the dispersin parameter frm the pulse bradening itself if Δt is the measured pulse bradening and Δλ is the bandwidth f the wavelengths in the puilse. The measurement precisin achievable by the 15

24 Chapter 3: Cnventinal Measurement Techniques 16 TOF technique is n the rder f 1 ps/nm [17]. The setup fr such a system is shwn in the Fig. 3-1: Tunable λ 1 Detectr (t 1 ) Tunable λ Detectr (t ) Fig. 3-1: Time f flight dispersin measurement technique One f the main prblems with the TOF technique is that it generally requires several kilmeters f fiber t accumulate an appreciable difference in time fr different wavelengths. Anther issue with the TOF technique when the pulse bradening is measured directly is that the pulse width is affected by changes in the pulse shape which leads t errrs in the measurement f the dispersin parameter. As a result, in rder t measure the dispersin parameter with a precisin near 1 ps/nm-km several kilmeters f fiber are required [16]. Anther lng fiber measurement technique is nw discussed in the next sectin. 3. Mdulatin Phase Shift Technique The MPS technique is anther dispersin characterizatin technique that requires lng lengths f fiber. In the MPS technique, a cntinuus-wave ptical signal is amplitude mdulated by an RF signal, and the dispersin parameter is determined by measuring the RF phase delay experienced by the ptical carriers at the different wavelengths. A diagram f the experimental implementatin f this technique is shwn in Fig. 3-:

25 Chapter 3: Cnventinal Measurement Techniques 17 Amplitude Mdulated Envelpe Tunable λ 1 Detectr RF analyzer λ 1 λ Fig. 3-: Mdulatin Phase Shift Dispersin Measurement Technique The RF phase delay infrmatin is extracted by this technique, and by taking the secnd derivative f the phase infrmatin, the dispersin parameter can be determined. Measurement precisin achievable by the MPS technique is n the rder f 0.07 ps/nm [0]. Due t its higher precisin, MPS has becme the industry standard fr measuring dispersin in ptical fibers. Hwever, MPS, has several disadvantages. The first is that it is expensive t implement since the cmpnents required such as an RF analyzer and a tunable laser, are cstly. The secnd is that its precisin is limited by bth the stability and jitter f the RF signal [1, ]. MPS has several limitatins n the minimum device length that it is capable f characterizing. In the MPS methd the width f the mdulated signal limits the minimum characterizable device length. This methd als typically requires fiber lengths in excess f tens f meters t btain a precisin t better than 1 ps/nm-km [16]. Therefre it is nt desirable fr the characterizatin f specialty fibers r gain fibers [3], f which lng fiber lengths are expensive t acquire r nt available. Als, when fiber unifrmity changes significantly alng its length, the dispersin f a lng span f fiber cannt be used t accurately represent that f a shrt sectin f fiber. In such cases, dispersin

26 Chapter 3: Cnventinal Measurement Techniques 18 measurement perfrmed directly n shrt fiber samples is desirable. As a result a technique fr measuring the dispersin f shrt lengths f fiber is desired. 3.3 Dispersin Measurements n Shrt Length Fiber Interfermetric techniques are capable f characterizing the dispersin n fiber lengths belw 1m [16] (fiber with small DL prducts). There are tw categries f interfermetric techniques fr making dispersin measurements n fiber f shrt length: tempral and spectral. These tw categries will be discussed in detail in the fllwing sectins Tempral Interfermetry (Furier Transfrm Spectrscpy) B. Band Surce Test Fiber U1 Detectr Cupler Tracking Laser Cupler Lens U Mirrr Fig. 3-3: Experimental setup fr dual arm tempral interfermetry Dual Arm tempral interfermetry emplys a bradband surce and a variable ptical path t prduce a tempral interfergram between a fixed path thrugh the test fiber and variable air path. It invlves mving ne arm f the interfermeter at a cnstant speed and pltting the interference pattern as a functin f delay length (time) [3-3]. The spectral amplitude and phase are then determined frm the Furier transfrm f the tempral interfergram. A sample tempral interference pattern is shwn in Fig. 3-4:

27 Chapter 3: Cnventinal Measurement Techniques 19 Fig. 3-4: Sample Tempral Interfergram [4] A tempral interfergram gives the phase variatin as a functin f time. The spectral phase variatin can be extracted frm the tempral interfergram if a Furier Transfrm is applied t it. The spectral phase cntains the dispersin infrmatin which can be indirectly btained by taking the secnd derivative f the spectral phase. A precisin f ps/nm measured n a m-lng phtnic crystal fiber [9] was recently reprted using tempral interfermetry. The main disadvantage f tempral interfermetry is that it is susceptible t nise resulting frm bth translatin inaccuracy and vibratin f the ptics in the variable path. A tracking laser is typically required t calibrate the delay path length [6, 9]. Anther prblem with this technique is that a secnd derivative f the phase infrmatin must be taken t btain the dispersin parameter which means that it is less accurate than a methd that can btain the dispersin parameter directly. A methd fr btaining the dispersin parameter directly is nw discussed in the next sectin.

28 Chapter 3: Cnventinal Measurement Techniques Spectral Interfermetry Spectral interfermetry, like tempral interfermetry, is capable f characterizing the dispersin in shrt length fiber (< 1m) (r fiber with a small DL prduct). In spectral interfermetry, instead f stepping the length f ne f the arms, a scan f the wavelength dmain perfrmed t prduce a spectral interfergram. Spectral interfermetry is generally mre stable than tempral interfermetry since the arms f the interfermeter are kept statinary. Thus it is simpler than tempral interfermetry since n tracking laser is necessary. There are tw types f spectral interfermetry, ne is general and des nt require balancing, and anther, the special case, is balanced. In the balanced case it is pssible t directly measure the dispersin parameter frm the interfergram. This makes it mre accurate than tempral interfermetry and it is fr this reasn that spectral interfermetry is discussed as a dispersin measurement technique. We first examine the mre general case f spectral interfermetry. General Case: Unbalanced In general spectral interfermetry the dispersin parameter is btained frm the interference spectrum prduced by tw time delayed light pulses/beams in an unbalanced dual arm interfermeter. Tw pulses/beams frm the tw arms f the interfermeter are set up t interfere in a spectrmeter and a spectral interfergram is prduced. The interference pattern prduced fr a given time r phase delay is given by:

29 Chapter 3: Cnventinal Measurement Techniques 1 ) )exp( ( ) )exp( ( ) ( ) ( ) )exp( ( ) ( ) )exp( ( ) ( ) ( ) ( ) )exp( ( ) ( ) ( * * * i f i f E E i E E i E E E E i E E I Eq. 3- [33] The last tw terms in Eq. 3- result in spectral interference pattern via a ) ) ( cs( term. This interference pattern is seen in Fig Fig. 3.5: Interference pattern prduced by tw time delayed pulses [34] There are several ways t extract the phase infrmatin frm the csine term but the mst prevalent way t d s is t take the Inverse Furier transfrm f the spectral interference pattern. Nte that )) ( exp( ) ( ) ( ) (.. ) ( * i E E t f F T f [33] cntains all the phase infrmatin n the spectral phase difference ) (. Therefre, if ) ( f can be extracted frm the interference pattern then the phase difference infrmatin can be knwn. If an Inverse Furier Transfrm f the spectral interference is perfrmed n the interference pattern the fllwing is btained: * * * 1 ) ( ) ( ) ( ) ( ) ( ) ( )) ( (.. t f t f t E t E t E t E I T F Eq. 3-3 [33]

30 Chapter 3: Cnventinal Measurement Techniques If all terms except the f ( t ) term get filtered ut via a band pass filter then the phase infrmatin can be extracted frm a Furier Transfrm n f ( t ). A graphical descriptin f this prcess is given in Fig. 3-6: Fig. 3-6: Filtering ut all but the f(t-τ) terms s that the phase infrmatin can be extracted [34] The phase infrmatin can then be extracted if a Furier Transfrm is applied t the filtered cmpnent f(t-τ) t transfer it back t the spectral dmain. The cmplex amplitude therefre becmes f ( ) E ( ) E( ) exp( i( ) ) [33, 34]. The phase f this cmplex amplitude minus the linear part (ωτ) that is due t the delay, yields the spectral phase difference between the tw pulses as a functin f ω and is independent f the delay between the tw pulses [33, 34]. In this way the phase difference between the tw pulses can be btained. A sample plt f the amplitude and phase infrmatin retrieved using this methd is shwn in Fig. 3-7:

31 Chapter 3: Cnventinal Measurement Techniques 3 Fig. 3-7: Amplitude and phase spectrum f f(ω) [34] If ne f the pulses travels thrugh a nn-dispersive medium such as air and the ther pulse travels thrugh a dispersive medium such as an ptical fiber then the phase difference spectrum will be directly related t the dispersin in the fiber. Thus the dispersin parameter plt can be determined by taking the secnd derivative f the phase difference spectrum with respect t wavelength. The main issue with this frm f spectral interfermetry, hwever, is that the dispersin parameter is nt determined directly but rather via a secnd rder derivative f the phase infrmatin with respect t wavelength. Therefre, like tempral interfermetry, this general unbalanced frm f spectral interfermetry is nt as accurate as the methd capable f measuring the dispersin parameter directly which will be discussed in the next sectin n balanced interfermetry.

32 Chapter 3: Cnventinal Measurement Techniques 4 Special Case: Balanced In balanced spectral interfermetry the arm lengths f an interfermeter are kept cnstant and they are balanced fr a given wavelength called the central wavelength such that the grup delay in bth arms is the same. This allws fr the remval f the effect f the large linear dispersin term in the interfergram. Balanced interfermetry measures the dispersin parameter D at the wavelength at which the grup delay is the same in bth arms. This wavelength is hencefrth referred t as the central wavelength. The dispersin parameter in this case can be directly determined frm the phase infrmatin in the spectral interfergram withut differentiatin f the phase. Fr this reasn it is mre accurate than bth unbalanced general spectral interfermetry and tempral interfermetry. As a result balanced spectral interfermetry is ften used t btain accurate dispersin measurements in shrt length waveguides and fibers. A precisin f ps/nm has been reprted n a 1 m lng SMF using this technique [16]. The experimental setup fr balanced spectral interfermetry is shwn in Fig Bradband surce OSA Cupler Lens Test Fiber U U1 Mirrr Fig. 3-8: Experimental setup fr Spectral Interfermetry A sample spectral interference pattern prduced frm the setup in Fig. 3-8 is shwn in Fig The central wavelength can be seen in this interfergram and is

33 Chapter 3: Cnventinal Measurement Techniques 5 labeled. The dispersin parameter can be determined at the central wavelength,, frm the phase infrmatin given by the wavelength separatin between the peaks/trughs f the interfergram [16]. Fig. 3-9: Sample spectral interfergram [16] Bth frms f spectral interfermetry are cnsidered t be less susceptible t nise since the arms f the interfermeter are kept still and there are n mving parts. It is fr this reasn that spectral interfermetry in general is cnsidered t be mre accurate than tempral interfermetry. Spectral interfermetry is therefre cnsidered t be the technique f chice fr measuring the dispersin f phtnic cmpnents [34-37] and spectral depth reslved ptical imaging [38, 39]. One well knwn and imprtant class f spectral interfermetry is ptical cherence tmgraphy (OCT) [40-45]. The reslutin f balanced spectral interfermetry, in particular, can be imprved by replacing the cmbinatin bradband surce and Optical Spectrum Analyzer shwn in Fig. 3-8 with a tunable laser and detectr system. Current tunable laser technlgy allws

34 Chapter 3: Cnventinal Measurement Techniques 6 fr a bandwidth f 130 nm and a 1 picmeter reslutin. This imprves the range f fiber lengths that can be measured using this technique. Als f nte is that the use f tunable lasers fr dispersin measurement is becming mre widespread [46] as they decrease in cst. Balanced dual arm spectral interfermeters are typically in a Michelsn r a Mach Zehnder cnfiguratin in which the path lengths are equalized at the given wavelength in which the dispersin is t be measured [3, 4, 3]. The mst ften used cnfiguratin, hwever, is the Michelsn and the discussin that fllws cnsiders the Michelsn interfermeter. In a balanced Michelsn interfermeter the dispersin is measured frm the interference between tw waves: ne that passes thrugh the test fiber and anther that passes thrugh an air path. Balancing the air path length with the fiber eliminates the effect f the grup index f the fiber in the interference pattern. This allws fr the measurement f the secnd derivative f the effective index with respect t wavelength directly frm the interference pattern [16]. The main disadvantage f this cnfiguratin is that, fr this t wrk, tw types f path balancing must ccur simultaneusly. The first type f path balancing is cupler arm balancing illustrated in red in Fig. 3-10: B. Band Surce OSA Cupler Lens Test Fiber U U1 Mirrr Fig. 3-10: Balanced path requirements fr a Michelsn interfermeter

35 Chapter 3: Cnventinal Measurement Techniques 7 The path lengths f bth arms cming ut f the cupler (highlighted in red) need t be balanced exactly r an extra set f interference fringes will be created frm the reflectins at the tw end facets f the cupler arms as shwn in Fig B. Band Surce OSA Cupler Lens Test Fiber U U1 Mirrr Fig. 3-11: Interference f the cupler arm reflectins The secnd type f balancing is test fiber-air path balancing t ensure that the grup delay in the air path exactly equals that f the fiber fr a given central wavelength. This ensures that the central wavelength in the interference pattern is within the viewable bandwidth f the OSA. The main prblem in implementing a Michelsn interfermeter is that the arms f the cupler cannt be balanced exactly and as a result the effect f the extra set f reflectins prduced at the cupler facets cannt be remved. One methd f canceling ut the extra set f fringes prduced at the facets f the cupler arms is by having a relatively lng difference between the cupler arms as shwn in Fig. 3-1:

36 Chapter 3: Cnventinal Measurement Techniques 8 B. Band Surce Test Fiber U1 OSA Cupler Lens Mirrr U Fig. 3-1: Fringe cancellatin technique fr a Michelsn interfermeter This fringe cancellatin technique, depicted in Fig. 3-1, dramatically reduces the perid f the fringes prduced by the extra set f reflectins frm the cupler facets t a level in which they are smaller than the reslutin f the OSA. As a result they becme lw-pass filtered by the OSA and d nt shw up in the plt f the interference. This technique, hwever, requires cmpensatin f the added dispersin due t the ptical path difference between the cupler arms. T d this, hwever, requires knwledge f the exact difference in length between the tw arms f the cupler and the exact dispersin parameter curve fr the arms f the cupler. Bth f which are generally nt easy t determine accurately. Als f nte is that this technique requires a much lnger air path which intrduces mre nise int the measurement due air path disturbances. As a result f the difficulties inherent in the fringe cancellatin technique I will intrduce a new methd (which is a subset f balanced spectral interfermetry) fr the measurement f dispersin. This new methd, knwn as Single Arm Interfermetry, will nt require the cancellatin f any extra fringes as was the case fr the Michelsn. In the next sectin I cmpare the perfrmance f Single Arm Interfermetry t the cnventinal techniques in rder t shw hw it is a natural prgressin in the develpment f dispersin measurement technlgy. The perfrmance f Single Arm Interfermetry is

37 Chapter 3: Cnventinal Measurement Techniques 9 intrduced befre the details f the technique are described in rder entice the reader study the technical/theretical discussin in chapter Cmparisn f Dispersin Measurement Techniques There have been several techniques develped fr the measurement f chrmatic dispersin in fiber. Especially imprtant are thse develped fr the measurement f shrt lengths f fiber [16, 47]. One reasn that shrt length characterizatin techniques are imprtant stems frm recent develpments in the design and fabricatin f specialty fiber. Specialty fiber such as Twin Hle Fiber (THF) [48] and Phtnic Crystal Fiber (PCF) [9] have made shrt length fiber characterizatin desirable due t their high cst. Because f this it is nt ecnmical t use TOF and MPS techniques t characterize these types f fiber. Anther impetus fr shrt length characterizatin cmes frm the fact that in many specialty fibers the gemetry is ften nn-unifrm alng its length. As a result f this nn-unifrmity the dispersin in these fibers varies with psitin. Thus measurement f the dispersin in a lng length f this fiber will be different than that measured in a sectin f the same fiber. In the last few sectins several dispersin measurement techniques have been discussed and it has been shwn that it is desirable t seek a shrt length characterizatin scheme. The techniques discussed fr shrt length dispersin characterizatin were tempral and spectral interfermetry. Tempral interfermetry and unbalanced general spectral interfermetry are bth capable f characterizing shrt length fiber, hwever, since they btain the dispersin parameter indirectly via secnd rder differentiatin f the phase term they are nt as accurate as balanced spectral interfermetry which directly

38 Chapter 3: Cnventinal Measurement Techniques 30 measures the dispersin parameter. As a result the technique f chice fr dispersin measurement is balanced spectral interfermetry since it will prvide the mst accurate measurements. As a result the new technique will emply balanced spectral interfermetry. The tw imprtant parameters in cmparing dispersin measurement techniques is the minimum device length that each is capable f characterizing and the precisin t which the characterizatin is achieved. It is generally desirable t characterize as shrt a fiber as pssible with as high a precisin as pssible. It is als desirable t perfrm the measurement in the simplest way pssible. A summary f the length requirements and the precisin f the varius dispersin measurement techniques is summarized in Table 3-1: Table 3-1: Summary f the varius dispersin measurement techniques and their precisin Technique Time f Flight (Film laser pulse) Mdulatin Phase Shift Tempral Interfermetry Dual Arm Spectral Interfermetry (Balanced) Single Arm Interfermetry (Balanced Spectral Interfermetry) Measures Shrt Fiber? Precisin (Shrtest length) Reference Cmments N 1 ps nm -1 (7.8 m) 40 -Need km s f fiber N Yes <1 m Yes <1 m 0.1 ps nm-1 (1. km) [19] 0.07 ps nm -1 (Agilent 86038B ) [0] 0.01 ps nm -1 (1 m) [16], ps nm -1 (0.814m) [49] 19, 0, 16, ps nm -1 (1 m) 16 Yes <0.5 m ps nm -1 (0.395 m) This wrk -Need 10 s f meters f fiber -System is expensive esp. RF cmpnents -Nise due t translatin f mirrr: -Stepping accuracy, drift in psitin, vibratin -Less accurate, Indirect measure f D -N mving parts less nise -Mre accurate, directly measures D -Technique f chice -Subset f Balanced SI but simpler -Details in the next chapter *Nte that in calculating the reslutin f the Single Arm Interfermetry technique the standard deviatin f the measurement fr single mde fiber (0.8 ps/nm-km) was multiplied by the length f SMF used ( km).

39 Chapter 3: Cnventinal Measurement Techniques 31 In the summary given in Table 3-1 it is evident that the rder f magnitude fr the measurement in dual arm spectral interfermetry [16] is the same as the rder f magnitude reprted fr Single arm Interfermetry. The technique used in single arm interfermetry, hwever, is significantly simpler as will be shwn in the next chapter. The next chapter intrduces the thery and implementatin f Single Arm Interfermetry and utlines the parameters affecting perfrmance.

40 Chapter 4: Thery f Single Arm Interfermetry A Single Arm Interfermeter (SAI) can be prduced by flding the tw arms f a Michelsn interfermeter tgether int a cmmn path (as in a cmmn path interfermeter) and placing a mirrr behind the test fiber. This cnfiguratin was designed t eliminate the calibratin step required by dual arm interfermeters in which the cupler arms are made t be disprprtinate in length t eliminate the effect f the extra reflectins frm the cupler-test fiber/cupler-air path facets. Since calibratin is nt required this technique is als mre accurate than a dual arm interfermeter. 4.1 A New Cncept This chapter intrduces a balanced Single-Arm Interfermeter (SAI) fr the direct measurement f dispersin in shrt fibers. A balanced SAI is depicted in Fig This cnfiguratin is nt nly much simpler than a dual arm interfermeter but it als eliminates the need fr system calibratin (assuming the dispersin intrduced by the cllimating lens is negligible and the air path is stable). Its simpler cnstructin als makes it less susceptible t plarizatin and phase instabilities. 3

41 Chapter 4: Thery f Single Arm Interfermetry 33 Balanced Circulatr U0 U1 U Surce Detectr Surce Fiber (APC) Test Sample Mirrr Fig. 4-1: Single-arm interfermeter where three waves interfere; U, U1 and U3. The SAI is a balanced interfermeter since the grup delay in the fiber is the same as the grup delay in the air path. It will be shwn mathematically that this balancing f the grup delay in each path allws the dispersin parameter t be measured directly frm the interference pattern. The cnceptual difference between SAI and Dual Arm interfermeters is that, in SAI, the interference pattern is prduced by three waves: tw frm the reflectins at the facets f the test fiber and ne frm a mirrr placed behind it (as shwn by U, U 1, and U in Fig. 4-1). The beating between the interference fringes prduced by the test fiber and thse by the air path generates an envelpe which is equivalent t the interference pattern prduced by tw waves (U 1 and U in Fig. 4-1) in a dual-arm interfermeter. Frm the phase infrmatin in this envelpe the dispersin parameter can be extracted. Bth dual and single arm balanced interfermeters have in cmmn this ability t directly measure the dispersin parameter frm the interference pattern. The SAI cnfiguratin appears similar t cmmn path interfermeters, ften used fr depth imaging as in Cmmn-Path Optical Cherence Tmgraphy (CP-OCT)

42 Chapter 4: Thery f Single Arm Interfermetry 34 [50, 51]. The SAI, hwever, is fundamentally different frm CP-OCT since it utilizes 3 reflectins, and extracts the dispersin parameter directly frm the envelpe f the interference pattern. The main difference between cmmn path interfermeters and single arm interfermeters is the fact that there is a path balancing f the grup delay in the fiber path and the air path. The differences between the Michelsn Interfermeter, CP-OCT and balanced Single Arm Interfermetry are utlined in Table 4-1: Table 4-1: Differences & Similarities between the Michelsn Interfermeter, CP-OCT and the Single Arm Interfermeter Balanced Michelsn Interfermeter CP-OCT (Cmmn path) Balanced SAI # f interfering 3 waves # f lngitudinally 1 1 separate paths Path balancing yes n yes Dispersin infrmatin entire interfergram n/a envelpe f interfergram Dispersin directly n/a directly parameter measured Measures dispersin parameter ptical path length difference dispersin parameter In the next sectin, we will briefly present the theretical representatin f the interference pattern, the phase between the adjacent peaks/trughs f the envelpe, and its relatinship t the dispersin. 4. Mathematical Descriptin Equal Amplitude Case Dispersin measurements can be made using a single-arm interfermeter by extracting the secnd derivative f the effective index with respect t wavelength frm the envelpe f the interference pattern generated by three waves U, U 1 and U depicted in Fig. 4-:

43 Chapter 4: Thery f Single Arm Interfermetry 35 U0 U1 U Lf Lair Fig. 4-: Interference when reflectins frm the facets and mirrr have equal amplitudes The extra reflectin frm the surce fiber is eliminated using angle plished fiber as shwn in Fig. 4-. Nte that this methd is insensitive t the lss intrduced by the angle plished cnnectr since the dispersin infrmatin is cntained within the phase f the three reflected waves. The ptical path length f the air path is made t cancel ut the strng linear effective grup index term f the test fiber at a central wavelength,. The amplitudes f U and U 1 are assumed t be equal t the magnitude f the reflectin at the fiber end facets. The amplitude f U depends n the amunt f light cupled back t the fiber. This cupling efficiency can be adjusted by varying the alignment f the mirrr such that U has the same amplitude as U and U 1. In this simplified presentatin: U U 1 U U 0 0 e e jl f jl jk L f 0 air Eq. 4-1 In Eq. 4-1, L f and L air are the lengths f the test fiber and the air path, respectively. β and k are the prpagatin cnstant f the fundamental mde in the fiber and the prpagatin cnstant in free space. The interference pattern is prduced by the interference f the three reflectins is given by Eq. 4-:

44 Chapter 4: Thery f Single Arm Interfermetry 36 I U U 0 U U 1 3 cs(l f k L air ) 4cs( L f k L air )cs( L f k L air ) Eq. 4- Eq. 4- cntains tw fast terms, with a phase ( L k L ) and 1 f air ( Lf klair ). Since 1 is slwer than it will amplitude mdulate the faster term. As a result the perid f the carrier will be that f the slwest f the fast terms, carrier 1. This carrier is then itself amplitude mdulated by the slwer term envelpe L k ( f Lair) t prduce the envelpe f the interference pattern. This envelpe is equivalent t the interference pattern prduced by Michelsn interfermeter [16] and it can be written as: U 5 4cs( ) Eq. 4-3 envelpe The calculated interference pattern generated by the setup fr a 39.5 cm SMF8 TM test fiber is shwn in Fig It depicts the envelpe functin (highlighted) which is a gd apprximatin f the envelpe f the actual envelpe f the carrier.

45 Chapter 4: Thery f Single Arm Interfermetry 37 λ λ 4 λ 3 λ 1 λ Fig. 4-3: Calculated 3 wave interference pattern and envelpe fr a 39.5 cm piece f SMF8 TM Applying a Taylr expansin t the phase f the slw envelpe and replacing with n eff, where n eff is the effective index f the fiber, gives the phase relatin in Eq. 4-4: envelpe 1 ( ) n eff dneff ( ) d L f L air L f dn eff d L f ( )! d n eff d L f ( ) 3! 3 d 3 n eff 3 d Eq. 4-4 The first term in Eq. 4-4 (in the square brackets) disappears when L air is adjusted t balance ut the grup delay f the test fiber at λ, the balanced wavelength. Taking the difference between the phases at tw separate wavelengths; 1 and results in [16]:

46 Chapter 4: Thery f Single Arm Interfermetry 38 envelpe envelpe envelpe 1 ( 0 )! m ( 1 0 )! 1 d neff d 0 ( 0 ) 3! 3 ( 1 0 ) 3! d neff 3 d 0 L f Eq. 4-5 Nte that m is the number f fringes between the tw wavelengths. If this phase difference is taken using a different pair f peaks/trughs (i.e. λ 3 & λ 4 ) the result is a system f equatins in which d n eff d and d n eff d 3 3 can be slved directly [16]. Since the trughs in the interference pattern are mre sharply defined it is mre accurate t chse the wavelength lcatins f the trughs f the envelpe as the wavelengths used in Eq Nte that, if we ignre the third-rder dispersin, then nly tw wavelengths (e.g., and ) are required t calculate the secnd-rder dispersin. This, hwever, wuld be less accurate. The dispersin parameter D can then be fund as fllws: d n eff D( ) c d Eq. 4-6 The next sectin presents a mre general analysis f the interference pattern and details the effect f having variable reflectin magnitudes frm each f the facets. It will shw hw the variatin in the magnitude f the reflectins has n effect n the phase infrmatin in the envelpe and as such the simplified analysis presented here is generally applicable.

47 Chapter 4: Thery f Single Arm Interfermetry Unequal Amplitude Cases T prve that this methd is insensitive t the lss intrduced by the angle plished cnnectr since the dispersin infrmatin is cntained within the phases f the three reflected waves we will nw shw the effect that is btained if the reflectins d nt have equal magnitudes. The interference pattern prduced by three reflectins with unequal amplitudes is nt as simple as presented in the previus sectin. Here we shw that despite this fact the previus results still hld since the lcatins f the trughs f the envelpe, which are used t btain the dispersin infrmatin, remain the same even thugh the fringe cntrast varies. In general the reflectins frm the facet and the mirrr, shwn in Fig. 4-, d nt have the same magnitude and we express the magnitudes f the reflectins in terms f the first reflectin in the fllwing way. U U 1 au bu 0 0 e e jl f jl jk L f 0 air Eq. 4-7 In Eq. 4-7 L f and L air are the lengths f the test fiber and the air path, respectively. β and k are the prpagatin cnstant f the fundamental mde in the fiber and the prpagatin cnstant in free space. a is the fractin f the amplitude reflected frm the secnd facet in terms f the first and b is the fractin f the amplitude reflected frm the mirrr in terms f the fractin reflected frm the first facet. The interference pattern f the spectral interfergram can be expressed as: I U U 0 U {1 a 1 U b a( b 1) cs(k L 4a cs( L air ) b cs(( L f k L air f )cs( L k L air f ))} k L air ) Eq. 4-8

48 Chapter 4: Thery f Single Arm Interfermetry 40 The expressin in Eq. 4-8 can be treated as a fast-varying carrier (with respect t frequency r wavelength) mdified by an upper and a lwer slw-varying envelpe, as shwn in Fig. 4-3, which depicts the simulated spectral interfergram generated by the 3- wave SAI with a 39.5-cm SMF8 fiber as the test fiber. Upn clser examinatin (Fig. 4-3, lwer right), the carrier is nt a pure sinusidal functin, because there are three fastvarying phases in Eq. 4-8, (L f + k L air ), (L f + k L air ), and k L air, all f which vary much faster than the phase f the envelpe ( envelpe ), which equals L f k L air. When is large (>0.5), it can be shwn that the upper envelpe is apprximated by U 1 a b a( b 1) b 4a cs( ) Eq. 4-9 envelpe It will nw be shwn that althugh the magnitude f the interference pattern is nt the same as the envelpe fr cases in which b 1, the peak and trugh lcatins f the tw match exactly. As a result the phase infrmatin f the interfergram is preserved and the dispersin infrmatin can be extracted frm the interfergram. Nte that a = b=1 is a special case f this mre general analysis and was presented in the previus sectin. Several cases will be shwn fr the variatin in the magnitudes f the reflectins frm each f the facets. The Matlab cde used t generate these interference patterns is presented in Appendix A.1. The first few cases will be shwn t determine the effect f the variatin f a while keeping b cnstant. Figs. 4-4 t 4-6 shw that the variatin f a des nt change the interference pattern and the envelpe in Eq. 4-3 still matches the upper peaks interference pattern prduced using Eq. 4-. In the figures belw the envelpe functin as determined by Eq. 4-9 is pltted alng with the fringe pattern t shw that it is a gd

49 Chapter 4: Thery f Single Arm Interfermetry 41 apprximatin f the actual upper envelpe f the carrier and that the lcatins f the peaks and trughs are the same. Fig. 4-4: Simulated interference pattern prduced by the setup in Fig. 4-1 fr a 30-cm-lng SMF8 TM test fiber, with a =0.9, b =1. The parameters used fr the SMF8 TM is published in [Appendix B]. The envelpe calculated by Eq. 4-9 is superimpsed n the fringe pattern in a thick line, which is a clse apprximatin f the upper envelpe. Fig. 4-5: Simulated interference pattern prduced by the setup in Fig. 4-1 fr a 30-cm-lng SMF8 TM test fiber, with a =0.4, b =1. The parameters used fr the SMF8 TM is published in Appendix B. The envelpe calculated by Eq. 4-9 is superimpsed n the fringe pattern in a thick line, which is a clse apprximatin f the upper envelpe.

50 Chapter 4: Thery f Single Arm Interfermetry 4 Fig.4-6: Simulated interference pattern prduced by the setup in Fig. 4-1 fr a 30-cm-lng SMF8 TM test fiber, with a =0.1, b =1. The parameters used fr the SMF8 TM is published in [Appendix B]. The envelpe calculated by Eq. 4-9 is superimpsed n the fringe pattern in a thick line, which is a clse apprximatin f the upper envelpe. The next few cases will shw the effect f a variatin f b while keeping a cnstant. Figs. 4-7 t 4-9 shw that the variatin f b des change the magnitude f the interference pattern and the magnitude f the envelpe in Eq. 4-9 des nt match the upper peaks f the interference pattern prduced using Eq. 4-8 but that the phases f bth equatins still match. Since the dispersin infrmatin is cntained within the phase f the interference pattern it can still be used as in sectin t determine the dispersin.

51 Chapter 4: Thery f Single Arm Interfermetry 43 Fig. 4-7: Simulated interference pattern prduced by the setup in Fig. 4-1 fr a 30-cm-lng SMF8 TM test fiber, with a =1, b =0.9. The parameters used fr the SMF8 TM is published in [Appendix B]. The envelpe calculated by Eq. 4-9 is superimpsed n the fringe pattern in a thick line, which is a clse apprximatin f the upper envelpe. Fig. 4-8: Simulated interference pattern prduced by the setup in Fig. 4-1 fr a 30-cm-lng SMF8 TM test fiber, with a =1, b =0.4. The parameters used fr the SMF8 TM is published in [Appendix B]. The envelpe calculated by Eq. 4-9 is superimpsed n the fringe pattern in a thick line, which is a clse apprximatin f the upper envelpe.

52 Chapter 4: Thery f Single Arm Interfermetry 44 Fig. 4-9: Simulated interference pattern prduced by the setup in Fig. 4-1 fr a 30-cm-lng SMF8 TM test fiber, with a =1, b =0.1. The parameters used fr the SMF8 TM is published in [Appendix B]. The envelpe calculated by Eq. 4-9 is superimpsed n the fringe pattern in a thick line, which is a clse apprximatin f the upper envelpe. Since the phase f the upper envelpe, φ envelpe (and therefre the dispersin infrmatin) is unaffected by the magnitude f the reflectins frm the facets and the mirrr, the methd fr determining the dispersin parameter as presented in Eqs. 4-4 t 4-6 is valid even in the general case. The dispersin parameter, therefre, can always be btained frm an SAI. As mentined earlier, the main difference between the fringes prduced in this setup and thse prduced by dual arm interfermeters is the presence f a fast carrier (Eq. 4-8) slwly mdulated by the desired envelpe. The presence f this carrier sets extra peratinal cnstraints that will be discussed in the next sectin.

53 Chapter 4: Thery f Single Arm Interfermetry System Parameters There are fur factrs f interest with regard t the dispersin measurement system. These factrs are imprtant since they will determine the quality and range f the utput f the dispersin measurements. The first factr f interest is the wavelength reslutin f the measurement, the secnd is the minimum required bandwidth f the surce, the third is the measurable bandwidth f the dispersin curve and the furth is the test fiber length. The sectins that fllw discuss hw each f these factrs affect the utput f the dispersin measurement Wavelength Reslutin f the Dispersin Measurement The wavelength reslutin f the pints in the plt f the dispersin parameter is determined by the minimum step size f the translatin stage. With smaller step increments in the translatin stage there are smaller step increments in the plt f the dispersin parameter vs. wavelength. This is because variatin f the air path changes the wavelength where the air path and test fiber are balanced and prduces a new interfergram frm which the dispersin parameter can be determined. Examinatin f Eq. 4-4 shws that the first term can be remved if the grup delay in the air path is equal t that in the fiber path fr the central wavelength, (central wavelength at which the grup delay in fiber and air paths are balanced). The relatinship between the air path length and the fiber length at the wavelength is given by Eq. 4-10: L air n eff dn eff L f d Eq. 4-10

54 Chapter 4: Thery f Single Arm Interfermetry 46 Taking the derivative f L air with respect t and using the definitin given by Eq. 4-6, we btain: dlair d eff L cd L. f f d n d Eq Therefre the change f with respect t the change f L air can be written as d dl air cl 1 f D Eq. 4-1 Thus, the relatinship between a change in the central (balanced) wavelength and the change in the air path length is given by: 1 d dlair Eq cl D The minimum amunt by which we can change the air path sets the minimum increment f the central wavelength in the interfergram. This amunt must be several times smaller than the bandwidth f the surce. Thus the minimum step size f the air path sets the wavelength reslutin f the measured dispersin curve. Nte the wavelength reslutin is als inversely prprtinal t the dispersin-length prduct f the test fiber. I will nw shw the dependence f the wavelength reslutin n the dispersin length prduct. As a numerical example, fr a step size f 0.1m, assuming a 50-cm-lng SMF8 TM test fiber, the wavelength reslutin is 0.1nm, which is sufficient fr mst applicatins. As a graphical example the wavelength reslutin is pltted against the dispersin-length prduct f standard SMF8 TM fiber. f

55 Chapter 4: Thery f Single Arm Interfermetry 47 Fig. 4-10: Dependence f the wavelength reslutin n the dispersin-length prduct. Nte we assume the values λ = 1550nm and L air = 5μm and B surce = 130nm Minimum Required Surce Bandwidth A minimum number f envelpe fringes are required fr accurate measurements f dispersin. As lng as the balanced wavelength, 0, and fur ther wavelengths crrespnding t the peaks (r trughs) f the envelpe fringes are captured within the surce bandwidth, B surce, (Fig. 4-11), it is sufficient t determine dispersin D( 0 ). It is fund in practice that mre accurate measurements require selecting tw peaks (r trughs) n either side f 0, as indicated by B min n Fig

56 Chapter 4: Thery f Single Arm Interfermetry 48 Fig. 4-11: Minimum required surce bandwidth and the lcatins f the trughs Fr a given test fiber, the dispersin-length prduct is fixed. Therefre, the nly factr that limits the number f envelpe fringes is the surce bandwidth, B surce. The lnger the fiber, r the larger the dispersin, the mre clsely spaced the envelpe fringes, and hence the smaller the required bandwidth. In rder t determine B min quantitatively, we need t determine the maximum value fr the wavelength spacing ( ), as shwn in Fig Frm Eq. 4-4, ignring the 3 rd -rder term, we can btain the envelpe phase difference envelpe ( envelpe, which has an upper bund f since the first trugh ccurs at : envelpe 1 envelpe 0 d n 1 0 eff Lf Eq. 4-14! d 1 0

57 Chapter 4: Thery f Single Arm Interfermetry 49 Applying the definitin f dispersin in Eq. 4-6, we can therefre find the upper bund f the wavelength spacing ( ): Eq cdl f Next, we examine the wavelength spacing between and. Frm 4-5, ignring the 3 rd -rder term and applying Eq. 4-6 gives, cdl Eq Cmbining Eqs and 4-16, we get the upper bund fr the wavelength spacing : f cdl Eq The minimum required surce bandwidth B min shuld be nt less than the upper bund f ( ), therefre, B cdl f f 0 min Eq It is clear that the dispersin-length prduct f the test fiber als affects the minimum required bandwidth. Using a similar numerical example, assuming a 50-cmlng SMF test fiber and 1.55μm as the balanced wavelength, the minimum required bandwidth is 85 nm. As a graphical example the minimum bandwidth required is pltted against the dispersin-length prduct fr a standard single mde fiber and the values assumed fr the calculatin are Nte we assume the values λ = 1550nm and dlair = 5μm and Bsurce = 130nm..

58 Chapter 4: Thery f Single Arm Interfermetry 50 Fig. 4-1: Minimum bandwidth required as a functin f the dispersin length prduct. Nte we assume the values λ = 1550nm and L air = 5μm and B surce = 130nm Measurable bandwidth f the dispersin curve B mea Since each spectral interfergram prduces ne dispersin value at the balanced wavelength,, t btain dispersin versus wavelength, a number f interfergrams are recrded at varius balanced wavelengths by setting the apprpriate air path lengths. Since each interfergram shuld be taken ver a bandwidth f at least B min, frm Fig ne can see that the measurable bandwidth f the dispersin curve is the difference between the available surce bandwidth B surce and the minimum required bandwidth B min, that is, B mea 0 Bsurce Bmin Bsurce Eq cdl f

59 Chapter 4: Thery f Single Arm Interfermetry 51 Alternatively, if we d nt require tw f the trughs t be n each side f, then the measurable bandwidth B mea can be larger. In rder t accurately determine, the central fringe (frm 1 t 1 in Fig. 4-11) is required t be entirely visible within the measured spectral range. Therefre, B mea 0 Bsurce 1 0 Bsurce Eq. 4-0 cdl f Equatin 4-19 r 4-0 gives the lwer bund fr the measurable bandwidth, which assumes the widest pssible central fringe. In practice, since envelpe ( ) cannt be cntrlled, the width f the central fringe can be anywhere between zer and twice the limit f Eq Therefre, B mea can be as large as B surce in certain cases. Examinatin f Eq r 4-0 shws that increasing the dispersin-length prduct f the test fiber increases B mea. Nte that fr a given measurement system, B surce is fixed, s the nly parameter that can be used t extend B mea is L f. The dispersin length prduct is, in fact, the main independent variable in determining the system parameters. As a graphical example the minimum measurable bandwidth is pltted against the dispersin-length prduct fr a standard single mde fiber.

60 Chapter 4: Thery f Single Arm Interfermetry 5 Fig. 4-13: The dependence f the measurable bandwidth (B mea ), n the DL f prduct. Nte we assume the values λ = 1550nm and L air = 5μm and B surce = 130nm. The dispersin length-prduct has been shwn t be the main independent variable in determining the measurable bandwidth and the minimum bandwidth. But the range f this parameter is itself affected by the surce used. The bandwidth f the surce determines the minimum fiber length that can be characterized using this technique and the minimum wavelength step f the surce leads t a maximum characterizable fiber length. The next sectin discusses hw the surce bandwidth and minimum wavelength step size affect the range f fiber lengths that can be measured using the SAI technique.

61 Chapter 4: Thery f Single Arm Interfermetry Minimum Fiber Length The bandwidth f the surce determines the minimum fiber length that can be characterized using SAI. A smaller fiber length prduces a wider spectral interfergram as determined by Eq Thus in rder fr a certain fiber length t be characterizable using SAI the interfergram prduced must fit inside the surce bandwidth. Therefre the requirement is that, Using Eq. 4-18, we have: Bmin B surce Eq. 4-1 L f 8 cdb surce Eq. 4- Nte that fr a lnger fiber there will be a greater measurement bandwidth (accrding t Eq r 4-0) and a higher wavelength reslutin (Eq. 4-13). As a numerical example, fr a surce bandwidth f 130nm, the minimum length fr a SMF8 fiber is 0.3m. The maximum fiber length is pltted as a functin f the surce bandwidth in Fig

62 Chapter 4: Thery f Single Arm Interfermetry 54 Fig. 4-14: Minimum fiber length vs. surce bandwidth. Nte λ = 1550 and D = 18 ps/nm-km Maximum Fiber Length The SAI methd uses the slw-varying envelpe functin t btain dispersin. Thugh the carrier fringes are nt f interest, they still need t be reslved during measurement therwise the envelpe shape cannt be preserved. The carrier fringe spacing is directly affected by the length f the fiber under test, L f. A lnger fiber will lead t narrwer carrier fringes. The minimum step size f the tunable laser, hwever, sets a limit n the minimum carrier fringe perid that can be detected due t aliasing. Since a lnger fiber length has a higher frequency carrier this minimum detectable fringe perid results in a limit n the maximum fiber length. The carrier fringe perid is the wavelength difference that causes

63 Chapter 4: Thery f Single Arm Interfermetry 55 the fast varying phase t shift by. The Fast phase term in Eq. 4- fr a balanced air path, L air N g L f, can be written as: L ) Eq. 4-3 ( kneff L f kn g f Using a first rder apprximatin f n eff and N g N g n n Where n is the cre index, the phase term is written as Eq nL f eff Eq. 4-5 The fringe perid,, crrespnds t a phase shift 4nL f Eq. 4-6 Hence, nl Eq. 4-7 f In rder t detect ne fringe accurately, we apply the Nyquist criterin that at least sample pints have t be included in ne fringe. This sets the fllwing limit ver the fiber length: L f 4 n Eq. 4-8 Where Δ is the minimum wavelength step size f the tunable laser. If the fiber length limit is exceeded aliasing ccurs. The maximum fiber length fr aliasing t be avided is pltted as a functin f step size in Fig

64 Chapter 4: Thery f Single Arm Interfermetry 56 Fig. 4-15: The maximum measurable fiber length, L f as a functin f the step size f the tunable laser. The detectr reslutin is 1 picmeter, =1550 nm and n = The preceding analysis assumes that it is necessary t avid aliasing t ensure that all f the peaks f the interfergram are sampled in rder t accurately plt the envelpe f the interfergram. It is this assumptin that leads t the upper limit in the fiber length given in Eq This upper limit hwever can be exceeded by dividing the interfergram int small windw sectins and selecting a single pint in each windw t plt the envelpe. The thery behind this technique, called wavelength windwing, will be explained in detail in the next sectin.

65 Chapter 4: Thery f Single Arm Interfermetry The Effect f Wavelength Windwing The prblem with trying t measure a fiber lnger than Eq. 4-8 allws is that the perid f the carrier gets shrter with increasing fiber length. Accrding t Nyquist thery the sampling perid, determined by the average step size f the tunable laser, must be at least times smaller than the perid f the carrier in rder t avid aliasing. This ensures that all the sampled peaks f the carrier match the true envelpe f the interference pattern. Aliasing is a phenmenn that prevents every peak f the carrier frm being sampled but it des nt mean that sme f the peaks in a given wavelength windw range will nt be sampled. We can therefre divide the interfergram int small windw sectins, as shwn in Fig. 4-16, each cntaining many sampled pints. Thus when aliasing des ccur there will be a certain prbability that at least ne f the sampling pints will land n a peak f the interfergram within each wavelength windw (assuming a slw variatin in the envelpe within that windw). Therefre, the envelpe f the interfergram can be pltted under cnditins where aliasing des ccur by taking the maximum in each wavelength windw and cnnecting them tgether, as shwn in Fig

66 Intensity (a.u.) Chapter 4: Thery f Single Arm Interfermetry 58 Wavelength (a.u.) Fig. 4-16: Tracing the envelpe f the interfergram by wavelength windwing. Detailed statistical analysis (develped in the next sectin) shws hw the prbability that at least ne f the peaks will be sampled within a wavelength windw is determined. This technique shws that the upper limit in Eq. 4-8 can be exceeded by many flds by wavelength windwing. 4.5 Mdel Develpment This technique uses a tunable laser system t sample the peaks f an interfergram. A real wrld tunable laser system, hwever, des nt step the wavelength with equal step sizes but has a certain standard deviatin in its step size. In rder t prduce an accurate mdeling f a real wrld prcess this variatin in the step size f the tunable laser must be taken int accunt by the mdel. The tunable laser system used in the experiments was

67 Chapter 4: Thery f Single Arm Interfermetry 59 the Agilent 8164A which has an average step size f 1 pm and a standard deviatin f 0.17 pm as determined frm the histgram and the Gaussian PDF in Fig. 4-17: σ Fig. 4-17: Measured Prbability density functin (histgram) and a Gaussian fit fr the step size f the Agilent 8164A tunable laser. In rder fr the mdel t accurately determine the prbability f a sampled pint matching at least ne peak f the carrier wave within a certain wavelength windw, certain parameters must be determined. The mdel that will be develped requires knwledge f the fiber length, the width f wavelength windw, the average step size f the tunable laser, the standard deviatin f this step size and the tlerance in detecting the peak as a percentage f the carrier perid.

68 Chapter 4: Thery f Single Arm Interfermetry 60 In this mdel we will designate the fiber length as L f, the wavelength windw within which we wish t detect a peak as W, the average step size f the tunable laser as μ, the standard deviatin f the step size f the tunable laser as σ and the tlerance in detecting the peak as a percentage f the carrier perid as ε. If we call λ the separatin between the first carrier peak and the maximum sampling prbability density f the first step, as shwn in Fig. 4-18, then the wavelength lcatin f the next maximum sampling prbability ccurs at λ + μ and the fllwing ne ccurs at λ + μ and s n. Fig illustrates the prbability density functins alng with the carrier functins. Fig. 4-18: Mdel shwing the prbability density functins fr the step size and the carrier fr determining the prbability f hitting a peak in a given wavelength windw. The prbability density functins fr the step size and the carrier fringes are shwn. Nte that even with aliasing the tunable laser has a chance f hitting the peaks f the carrier at least nce fr a given wavelength windw since the perid f the peaks f the carrier is different than the perid f the wavelength steps f the tunable laser.

69 Chapter 4: Thery f Single Arm Interfermetry 61 Fig als illustrates the fact that even with aliasing, where all the peaks f the interfergram are nt sampled, there is still a chance that at least ne f the peaks f the interfergram will be sampled fr a given wavelength windw since the perid f the peaks f the carrier is different than the perid f the wavelength steps f the tunable laser. Thus, fr any given windw size there will be a number f peaks f the carrier. If we assume the lcatin f the first carrier peak t be at λ 1, as shwn in Fig. 4-18, then the prbability that this first peak is sampled by the first step f the tunable laser is given by: ) ( d e P Eq. 4-9 Therefre the prbability that the first peak is nt sampled by the first step is: ) ( d e P P Eq Here ε, shwn in Fig. 4-18, is a fractin f the width f the carrier perid and this measure translates int a tlerance in the measurement f the peak amplitude. If we let N be the number f steps f the tunable laser in a given windw size then the prbability f nt sampling the first peak with any f the N steps is given by: N n n N N d e P P P P 1 ) ( ( Eq If we let M be the number f peaks f the carrier in a given windw size then the prbability f nt sampling any f the M peaks with any f the N steps is given by:

70 Chapter 4: Thery f Single Arm Interfermetry 6 M m N n M m N n n nm NM erf erf d e P P m m ) ( ( ) ( ) ( Eq. 4-3 Where λ m is the lcatin f the m th peak in the wavelength windw and is given by m λ 1 and Λ + and Λ - are the nrmalized wavelength parameters given by: n m m Eq Since the mdel assumes a fixed relatinship between the first carrier peak and the maximum f the prbability density functin this prbability shuld be averaged fr λ varying ver ne carrier wave perid. This gives the prbability that n carrier peak is sampled in a given windw fr a randm alignment between the carrier peaks and the maximum f the prbability density functin. The result is given as: M m N n NM erf erf P 1 1 ) ( ) ( 1 1 Eq Thus the prbability that at least ne f the peaks is sampled fr a given windw size is determined as: M m N n erf erf P 1 1 ) ( ) ( Eq. 4-35

71 Chapter 4: Thery f Single Arm Interfermetry Simulatin Results The calculated effective index and dispersin f SMF8 TM is used in the fllwing sectins t determine the prbability that at least ne peak is sampled as ne f five parameters is varied. The parameters varied are the windw size, the step size, the fiber length (which determines the peak spacing), and the tlerance (which determines the hw clse the sampled peak is t the actual carrier amplitude). The results are shwn in the fllwing five sectins. The parameters held cnstant in these simulatins are chsen t be the same as the experimental cnditins that will be implemented in sectin 5. in the experiment n SMF8 TM. The Matlab cde used t perfrm these simulatins is given in Appendix A Prbability vs. Windw Size The prbability that at least ne peak is sampled in a given windw size, W, is shwn in Fig 4-19, as a functin f the windw size. The parameters held cnstant fr this simulatin are the fiber length (Lf = 39.5 cm), the average step size (μ = 1 pm) and the tlerance (ε = 0.0 x average carrier perid). The prbability is pltted fr 3 different cases f the standard deviatin in Fig. 4-19: σ = 0.05pm, which is as clse t the σ = 0 case (i.e. cnstant step size case) that we can get using the mdel since σ = 0 leads t a Λ m+ = 1/0 (undefined) in Eq The ther tw cases pltted in Fig are σ = 0.17pm, and σ = 1pm.

72 Chapter 4: Thery f Single Arm Interfermetry 64 Fig. 4-19: Prbability vs. windw size. The parameters held cnstant fr this simulatin are the fiber length (L f = 39.5 cm), the average step size (μ = 1 pm) and the tlerance (ε = 0.0 x average carrier perid). The prbability is pltted fr 3 different cases f the standard deviatin: σ = 0.05pm, σ = 0.17pm, and σ = 1 pm Fig shws that fr the given parameters a unity prbability can be btained fr a windw size f > 0.9nm. The windw size, hwever, is nt the nly parameter that affects the prbability that the tunable laser step will sample the peak f the interfergram in a given windw. The next sectin shws that the average step size f the tunable laser als affects this prbability.

73 Chapter 4: Thery f Single Arm Interfermetry Prbability vs. Average Step Size The prbability that at least ne peak is sampled in a given windw size, W, is shwn in Fig. 4-0 as a functin f the average step size, μ, f the tunable laser. The parameters held cnstant fr this simulatin are the fiber length (L f = 39.5cm), the windw size (W = 0.5 nm) and the tlerance (ε = 0.0 x average carrier perid). The prbability is pltted fr 3 different cases f the standard deviatin in Fig. 4-0: σ = 0.05pm, which is as clse t the σ = 0 case (i.e. cnstant step size case) that we can get using the mdel since σ = 0 leads t a Λ m+ = 1/0 (undefined) in Eq. 4-33, σ = 0.17pm, and σ = 1pm. Fig. 4-0: Prbability vs. Step Size. The parameters held cnstant fr this simulatin are the fiber length (L f = 39.5cm), the windw size (W = 0.5 nm) and the tlerance (ε = 0.0 x average carrier perid). The prbability is pltted fr 3 different cases f the standard deviatin: σ = 0.05pm, σ = 0.17pm and σ = 1 pm

74 Chapter 4: Thery f Single Arm Interfermetry 66 Fig. 4-0 shws that fr the given parameters there is a near unity prbability fr an average step size belw 0.5 pm and that it decreases as the step size increases. The average step size f the tunable laser, hwever, is nt the nly parameter that affects the prbability that the tunable laser step will sample the peak f the interfergram in a given windw. The next sectin shws that the length f the test fiber als affects this prbability Prbability vs. Fiber Length The prbability that at least ne peak is sampled in a given windw size, W, is shwn in Fig 4-1 as a functin f the fiber length, L f. The parameters held cnstant fr this simulatin are the average step size f the tunable laser (μ = 1 pm), the windw size (W = 0.5 nm) and the tlerance (ε = 0.0 x average carrier perid). The prbability is pltted fr 3 different cases f the standard deviatin in Fig. 4-1: σ = 0.05pm, which is as clse t the σ = 0 case (i.e. cnstant step size case) that we can get using the mdel since σ = 0 leads t a Λ m+ = 1/0 (undefined) in Eq. 4-33, σ = 0.17 pm and σ = 1 pm.

75 Chapter 4: Thery f Single Arm Interfermetry 67 Fig. 4-1: Prbability that at least ne peak is sampled in a given windw vs. fiber length. The parameters held cnstant fr this simulatin are the average step size f the tunable laser (μ = 1 pm), the windw size (W = 0.5 nm) and the tlerance (ε = 0.0 x average carrier perid). The prbability is pltted fr 3 different cases f the standard deviatin: σ = 0.05 pm, σ= 0.17 pm, and σ=1 pm. Fig. 4-1 shws sme peculiar dips where the prbability drps t zer fr the cases where the standard deviatin is small (σ = 0.05 pm and σ = 0.17 pm). We can see that when the standard deviatin is high (σ = 1pm) these dips disappear. We als ntice frm Fig. 4-1 that fr higher standard deviatin the prbability curves drp mre quickly t the asympttic value. Thus a lwer standard deviatin in the step size f the tunable laser prduces curves with higher initial prbabilities, but large dips in the prbability curve where the prbability drps t zer. A higher standard deviatin in the step size prduces curves with lwer initial prbabilities but eliminates the dips where the prbability drps t zer. It is therefre beneficial t have sme amunt f variatin in the step size f the tunable laser in rder t eliminate these dips in the prbability.

76 Chapter 4: Thery f Single Arm Interfermetry 68 These dips where the prbability drps t zer can be explained by the fact that certain fiber lengths lead t a carrier spacing that is a multiple f the wavelength step size and as a result nne f the peaks in a windw get sampled. Fig. 4- shws the prbability as a functin f fiber length fr σ = 0.05pm and fr tw different step sizes μ = 1.3pm (pltted in blue and μ = 1pm (pltted in green). Fig. 4- shws that the lcatin f the dips are different fr each case since the dips ccur at different fiber lengths (different carrier spacing). The dips ccur whenever the carrier spacing is a certain multiple f the step size f the tunable laser. This multiple is given in Eq n G m Eq n and m are psitive integers. Whenever the carrier perid is a multiple f G there is a high prbability that nne f the peaks get sampled.

77 Chapter 4: Thery f Single Arm Interfermetry 69 Fig. 4-: Prbability vs. Fiber length fr the σ = 0.05pm case fr a step size f μ = 1 pm and fr the step size f μ = 1.3 pm. The lcatin f the dips in prbability ccur at different fiber lengths (carrier perids) fr different step sizes. They ccur when the carrier perid is a certain multiple f the step size and there is a chance that nne f the peaks within the windw get sampled. The parameters held cnstant fr this simulatin are the windw size (W = 0.5 nm), the tlerance (ε = 0.0 x average carrier perid) and the standard deviatin f the step size σ = 0.05 pm. The average carrier perid is determined by taking the average f all the carrier perid in the bandwidth as described by Eq. 4-37: p Eq n L eff f Bandwidth This is easily calculated using the Matlab prgram written in Appendix A..5. As a numerical example Fig. 4- shws several dips where the prbability drps t zer. In the case where μ = 1.3 pm in Fig. 4- when the fiber length is 0.05m the average carrier perid is determined t be 13 pm which is 10 times the step size. Table 4- shws several ther numerical examples using the dips in Fig. 4-.

78 Chapter 4: Thery f Single Arm Interfermetry 70 Table 4-: The dips where the prbability drps t zer in Fig. 4- ccur when the carrier perid is a multiple f G n m the step size. Fiber length Step Size Carrier Perid m n Multiple 0.04 m 1.3 pm 0.8 pm m 1.3 pm 9.1 pm m 1 pm 6 pm pm pm m 1.3 pm.6 pm m 1.3 pm.85 pm m 1.3 pm 1.95 pm Nte that a dip ccurs whenever the perid f step size appraches G times the carrier perid (fr the cases with lw standard deviatin). This is nt illustrated in Fig. 4- since it is impssible t get a high enugh reslutin s that the simulated pints fall exactly n the fiber length where every dip ccurs. This is als the reasn that the dips in Fig. 4- d nt fall cmpletely t zer. We als ntice that fr the given parameters that we have held cnstant in this simulatin the prbability f sampling a peak asympttically appraches a cnstant value as the length is increased. We ntice that this cnstant is the same, regardless f the standard deviatin f the step size. The cnclusin, therefre, is that this technique can be used t measure the dispersin f lng lengths f fiber (assuming f curse that a lng enugh air path can be prduced by the experimental setup and that the perid f the carrier peaks is still abve the laser linewidth).

79 Chapter 4: Thery f Single Arm Interfermetry Prbability vs. Tlerance The prbability that at least ne peak is sampled in a given windw size is shwn in Fig. 4-3 as a functin f the tlerance. The parameters held cnstant fr this simulatin are the average step size f the tunable laser (μ = 1 pm), the windw size (W = 0.5 nm), and the fiber length L f = 39.5 cm. The prbability is pltted fr 3 different cases f the standard deviatin in Fig. 4-3: σ = 0.05pm, which is as clse t the σ = 0 case (i.e. cnstant step size case) that we can get using the mdel since σ = 0 leads t a Λ m+ = 1/0 (undefined) in Eq. 4-33, σ = 0.17 pm and σ = 1 pm. Fig. 4-3: Prbability vs. Tlerance. The parameters held cnstant fr this simulatin are the average step size f the tunable laser (μ = 1 pm), the windw size (W = 0.5 nm) and the fiber length L f = 39.5 cm. The prbability is pltted fr 3 different cases f the standard deviatin: σ= 0.05pm, σ = 0.17pm and σ = 1pm.

80 Chapter 4: Thery f Single Arm Interfermetry 7 Fig. 4-3 shws that the prbability f hitting a peak increases as the definitin f where the peak actually is becmes relaxed. As the tlerance is increased the degree t which the peaks f the envelpe match the amplitude f the actual interference pattern is reduced. It can be seen frm this figure that the minimum prbability f hitting a peak is zer and that it appraches unity if the tlerance is.6% fr the given parameters that are held cnstant. This chapter has develped the thery f single arm interfermetry, discussed hw it is implemented and hw it can be explained via rigrus mathematical analysis f three wave interference. The technical limits f the SAI have been discussed by shwing the effects n the dispersin measurements f fur factrs f interest. The range f characterizable fiber lengths can be extended via a wavelength windwing technique in which the envelpe is pltted by selecting a few pints in a given bandwidth. The result f this range extensin is that the ultimate limit n the test fiber length is the laser linewidth (which shuld be much smaller than the carrier fringe perid) and the maximum air path length. The next chapter will describe the practical applicatin f the thery that has been develped in this chapter.

81 Chapter 5: Experiments & Analysis In this chapter the results f experiments using Single Arm Interfermetry are presented t substantiate the thery f single arm interfermetry, intrduced in the last chapter. An utline f the steps in the experimental prcess is first prvided t give an verview f the experimental prcess. Then the challenges encuntered during the setup f the Single arm interfermetry experiments are described. Fllwing these challenges is a descriptin f the instruments used in the experiments and their specific limitatins. The last three sectins f this chapter utline the results f the experiments perfrmed t characterize three different types f fiber: Single mde fiber (SMF8 TM ), Dispersin Cmpensating Fiber (DCF) and Twin Hle Fiber (THF). 5.1 Experimental Prcess The experiments in this chapter were carried ut t validate the thery presented in the previus chapter and t characterize the dispersin f a Twin Hle fiber fr which the dispersin has never been published. The first step in the experiment is t set up the Single Arm Interfermeter and t assemble the cntrl and data acquisitin hardware. The secnd step in the experiment is t test the technique by using it t measure the dispersin f fibers fr which the dispersin curves are knwn r that can easily be measured using cnventinal techniques. T d this, the dispersin curves f Single Mde Fiber (SMF8 TM ) and Dispersin Cmpensating Fiber (DCF) were measured. After careful analysis f the results fr the experiments n SMF8 TM and DCF the new technique was then used t measure the dispersin f a fiber that has never befre been 73

82 Chapter 5: Experiments & Analysiswww.inmetrix.cm 74 characterized. The entire experimental prcess fr this prject is utlined in Fig. 5-1 belw. Set up f the Apparatus, Cntrl and Data Acquisitin Test the technique by measuring fiber with knwn dispersin parameter curves - SMF8 TM - DCF Analyze the results using develped cmputer prgrams (Matlab) Characterize fiber with unknwn dispersin parameter curves - Twin-Hle Fiber Fig. 5-1: Experimental prcess fr the develpment and testing f the Single Arm Interfermeter. The first step is t set up the apparatus as well as the cntrl and data acquisitin hardware and sftware. The secnd and third steps test the technique and the furth step uses the verified technique t characterize a fiber with unknwn dispersin. 5. Experimental Challenges In rder t cmpare Single Arm Interfermetry t ther dispersin measurement techniques the challenges f setting up such an interfermeter must als be well understd. There were several challenges assciated with the setup f the system and the implementatin f the experiments. One challenge in the setup included alignment f the APC cnnectr with the test fiber which was especially difficult fr Twin-hle fiber since the fiber was different in size t SMF s cre t cre alignment was nt easy. Using a bare fiber adapter and a fiber

83 Chapter 5: Experiments & Analysiswww.inmetrix.cm 75 t fiber cnnectr helped but cupling was still a difficult task since the cre f THF is slightly ff centre (see Fig. 5-6) whereas the cre f SMF8 TM is at the centre f the fiber. Anther challenge is t prevent the angle plished cnnectr (APC) frm being brken by being pushed t frcefully against the flat plished cnnectr (FPC). One way t eliminate the pssibility f this ccurrence is t prduce the APC with a lcking pin t prevent a standard FPC frm breaking it. Anther challenge in the setup was placing the test fiber at the right lcatin in the bare fiber adapter s that light culd be prperly cllimated by the cllimating lens. Trial and errr using an infrared card and a pinhle t cllimate the beam helped in this regard. Anther challenge was alignment f the mirrr such that the beam culd be reflected back exactly int the cllimating lens and thus back t the detectr with a magnitude n the same rder as the reflectins frm the facets f the test fiber. Trial and errr was used t achieve maximum fringe visibility. Air flw in the air path is an effect that leads t changes in the density and therefre the ptical path length in the air path. T slve this prblem the system was encased in a cntainer t reduce air flw in the air path. Because f its simplicity the challenges presented in the set up f a single arm interfermetry experiment are rather straightfrward and it is fr this reasn that it will be very cmpetitive as a dispersin measurement technique. This simplicity cupled with the advantage f high precisin make the SAI a pwerful methd fr characterizing the dispersin f shrt length fibers. The next sectin utlines the instruments and tls used in the setup f an SAI and their specific limitatins.

84 Chapter 5: Experiments & Analysiswww.inmetrix.cm Experimental Instrumentatin & Specific Limits Tunable Laser Surce Detectr Circulatr Cllimating lens L f Test fiber Lair Mirrr DAQ, cmputer Fig. 5-: Experimental Setup f a Single Arm Interfermeter fr dispersin characterizatin. The tunable laser surce and detectr used are the Agilent 8164A Lightwave Measurement System with a bandwidth f 130 nm centered arund 1550 nm, and a minimum average wavelength step f 1 pm (standard deviatin 0.17 pm). An angle-plished cnnectr is used at the launch fiber t eliminate the reflectin frm this facet. The reflectins frm the cllimatin lens surfaces are suppressed by using an antireflectin cated lens. The mirrr tilt is adjusted t btain maximum fringe visibility. The mirrr translatin is cntrlled manually, and the minimum step is apprximately 5m. The experimental set up is shwn in Fig. 5-. The tunable laser surce and detectr used are plug-in mdules f the Agilent 8164A Lightwave Measurement System. The surce has a bandwidth f 130 nm centered arund 1550 nm, and a minimum average wavelength step f 1 pm (standard deviatin σ = 0.17 pm). The unit recrds the detectr readings and the wavelength readings as the surce wavelength is swept. The spectral interference pattern is then analyzed. The fibers are aligned by a standard cnnectr r using a bare fiber adapter in cases where the fiber is nt cnnectrized. An angleplished cnnectr (APC) is used at the launch fiber as shwn in Fig. 5- in rder t eliminate the reflectin frm this facet. A lcking mechanism can be used t prevent the APC frm being brken by the FPC. The reflectins frm the cllimatin lens surfaces

85 Chapter 5: Experiments & Analysiswww.inmetrix.cm 77 are suppressed by using an antireflectin cated lens. The dispersin f the lens is negligible. The mirrr tilt is adjusted t btain maximum fringe visibility. The mirrr translatin is cntrlled manually, and the minimum step is apprximately 5m. In the fllwing sectins, we will apply the SAI technique t measure the dispersin f three different fibers: a standard SMF8 TM single mde fiber, a Dispersin Cmpensating Fiber (DCF) and a Twin-Hle Fiber (THF). In measuring the envelpe f the spectral interfergram, the ttal scanning regin is divided int 0.5-nm-wide wavelength windws, ver which the envelpe is cnsidered cnstant. The peak value within each band is extracted t prduce the spectral envelpe as described in sectins Experiments Single Mde Fiber The dispersin prperties f SMF8 TM are well knwn and hence it was used t verify the thery f single arm interfermetry. In this experiment we used a cm piece f the SMF8 TM fiber in a SAI in rder t characterize its dispersin. Fig. 5-3 shws a plt f bth the experimental dispersin parameter pints and the simulated dispersin f SMF8 TM. Frm this figure we can see that the slpe f the measured dispersin pints clsely match the simulated dispersin curve. The simulated dispersin curve fr SMF8 TM was calculated using the dispersin equatin given in Appendix B: 4 S D ( ) 3 Eq Where λ 0 = 1313 nm and S = ps/nm-km and D(λ) is measured in ps/nm-km.

86 Dispersin (ps/nm.km) Chapter 5: Experiments & Analysiswww.inmetrix.cm Measured SMF-8 Simulated SMF (nm) Fig. 5-3: Measured dispersin cmpared t published dispersin equatin [Appendix B] fr a cm SMF8 TM fiber. The standard deviatin f the measured dispersin is determined using a linear fit and calculating the standard deviatin f the difference between the measured values and the linear fit. The simulatin is calculated using the Matlab cde in Appendix A.3.1 t be is 0.8 ps/nm-km (crrespnding t a relative errr f 1.6%). When this standard deviatin is multiplied by the length f the fiber, this translates int a standard deviatin f ps/nm. The wavelength reslutin f the measured dispersin curve, as determined by Eq. 4-13, is.4 nm. The measurable bandwidth accrding t Eq. 4-0 is 30nm, which is the bandwidth actually used, as shwn in Fig The standard deviatin f the measured dispersin is calculated by taking the difference between the measured pints and a linear fit and then calculating the standard deviatin frm the difference. The standard deviatin, as calculated using the Matlab cde in Appendix A.3.1, is 0.8 ps/nmkm (this crrespnds t a relative errr f 1.6%). When this standard deviatin is multiplied by the length f the fiber, this translates int a standard deviatin f ps/nm. A cmparisn between the measured and simulated interference patterns fr SMF8 TM is shwn in Figs. 5-4 (a) and (b).

87 Intensity (a.u.) Intensity (a.u.) Chapter 5: Experiments & Analysiswww.inmetrix.cm Wavelength (nm) (a) Wavelength (nm) (b) Fig. 5-4: (a) Measured upper envelpe (experimental) fringe pattern. (b) Simulated interference pattern and upper envelpe. The experimental and simulated cnditins are: fiber length L f = 0.395m effective grup index at central wavelength = , L air = L f.

88 Chapter 5: Experiments & Analysiswww.inmetrix.cm 80 The simulated interference pattern is generated using Eq. 4-8 and the envelpe f the interference pattern is generated using Eq The Matlab cde used in the simulatin is given in Appendix A.1. In the simulatin a fiber length f m is assumed in rder t match the experimental cnditins. The path length f the air path is determined via a calculatin f the effective grup index f the fiber was determined t be at the central wavelength, λ, via Eq. 5-: Where ( ( ) ) J l1 ( ( ) a) ( ( ) a) K l1 ( ( ) a) J ( ( ) a) K ( ( ) a) l Eq. 5- l ( ) ( ) n n cre eff ( ) ( ) n n eff ( ) cladding ( ) Eq. 5-3 Nte that a is the cre size f the fiber and J and K are Bessel functins f the first and secnd kind. The lcatins f equality in Eq. 5- determine the values f κ(λ) and γ(λ) as well as a mde f the fiber. The first f these mdes is called the fundamental mde f the fiber. The values f n cre (λ) and n cladding (λ) are the index f bulk glass with the cmpsitin f the cre and cladding respectively. The effective grup index as a functin f wavelength in SMF8 TM fiber is determined using the simulatin in Appendix A.1.. In Fig 5-4 there are differences between the upper envelpe f the experimental fringe pattern and the upper envelpe f the simulated fringe pattern. These differences are in the cntrast and amplitude f the experimental fringe pattern. The larger cntrast in the experimental data is due t the fact that in the experiment the magnitude f the reflectins frm the facets f the fiber and the mirrr were nt equal. The aim f the

89 Chapter 5: Experiments & Analysiswww.inmetrix.cm 81 experiment was t simply maximize the fringe visibility s that the lcatins f the peaks/trughs f the envelpe culd be determined s that the dispersin culd be calculated. The simulatin has a different cntrast since it assumes equal reflectins frm the fiber facets and the mirrr. The analysis that shws hw the differences in the reflectins frm the facets and the mirrrs lead t variatin in the fringe cntrast was presented in chapter The variable amplitude in the experimental fringe pattern is due t the fact that there is a backgrund amplitude spectrum that has nt been remved frm the measurement Dispersin Cmpensating Fiber As a secnd methd f verificatin, we measured dispersin n a shrt piece f DCF, whse dispersin value is apprximately ne rder f magnitude higher than that f SMF8 TM, and has an ppsite sign. We used a cm piece f DCF fiber, and the measurement results are given in Fig T verify the accuracy f ur measurement, we als measured dispersin n an identical m DCF using a cmmercial dispersin measurement system (Agilent 8347A), which emplys the MPS technique. Again, ur measured dispersin values are in gd agreement with thse measured by the cmmercial device, thugh the fiber length we used is almst 3-rders f magnitude smaller. The standard deviatin f the measured dispersin is calculated by taking the difference between the measured pints and a linear fit and then determining the standard deviatin f the difference. The standard deviatin f the measured data (as calculated using the Matlab cde in Appendix A.3. ) is 0.99 ps/nm-km, which crrespnds t a

90 Dispersin (ps/nm.km) Chapter 5: Experiments & Analysiswww.inmetrix.cm 8 relative errr f 0.9%. When multiplied by the length f the fiber, this translates int a standard deviatin f ps/nm wave interference meas. Mdulatin phase meas (nm) Fig. 5-5: Measured dispersin parameter plt fr DCF using the Agilent 8347A and Single Arm interfermetry. The standard deviatin f the measured data (as calculated using the Matlab cde in Appendix A.3. - with reference t a linear fit) using the SAI is 0.99 ps/nm-km, which crrespnds t a relative errr f 0.9%. When multiplied by the length f the fiber, this translates int a standard deviatin f ps/nm. Since DCF has negative dispersin values a prcedure fr determining the sign f the dispersin was develped. By examinatin f Eq repeated belw fr cnvenience 1 d dlair Eq. 5-4 cl D We can see that if the sign f the dispersin is negative then the lcatin f the central wavelength will decrease as the path length f the air path is increased. This is a quick methd fr determining the sign f the dispersin. f

91 Chapter 5: Experiments & Analysiswww.inmetrix.cm Twin Hle Fiber Twin Hle Fiber (THF) has been used in fiber pling t facilitate parametric generatin in fibers [48, 5] r making fiber-based electr-ptic switching devices [53]. In such nnlinear applicatins, dispersin f the fiber is an imprtant parameter t be determined. The dispersin prperties f THF, hwever, have never been reprted. This is partly due t the lack f unifrmity in the diameter f the THF alng its length. The fiber has a 3- μm-diameter cre and a numerical aperture that is higher than that f SMF8TM. The crss sectin f a typical THF is shwn in Fig. 5-6: Fig. 5-6: Crss sectin f a typical Twin-Hle Fiber The cre is Ge-dped silica, and has an index similar t that f SMF8 TM. Therefre, we expect the dispersin f THF t be slightly lwer than that f SMF8 TM. Since we did nt knw the magnitude f the dispersin fr THF we decided t chse the largest length f THF available t increase the chance that the minimum bandwidth

92 Chapter 5: Experiments & Analysiswww.inmetrix.cm 84 required fr a measurement wuld fit in the available bandwidth f the tunable laser surce. The largest length f THF available was cm. This length f fiber is slightly lnger than the length allwed by Eq. 4-8 but since we used the technique f wavelength windwing described in sectins the measurement f the envelpe was still pssible in this experiment. The measurement results frm the experiment n THF are given in Fig The standard deviatin f the measured dispersin is calculated by taking the difference between the measured pints and a linear fit and then calculating the standard deviatin frm the difference. The standard deviatin f the measured data, as calculated using the Matlab cde in Appendix A.3.3, is ps/nm-km (which crrespnds t a relative errr f.9%). When multiplied by the fiber length, this standard deviatin translates int a precisin f ps/nm. The slightly larger standard deviatin cmpared t thse fr the SMF and DCF measurement is due t the higher lss in fiber cupling between the SMF and the THF, and hence the lwer and mre nisy signal level during the THF measurement.

93 Disperisn (ps/nm.km) Chapter 5: Experiments & Analysiswww.inmetrix.cm Measured THF Linear fit f THF (nm) Fig. 5-7: Measured dispersin fr the cm Twin-Hle Fiber perfrmed using Single Arm Interfermetry. The standard deviatin f the measured data (as calculated using the Matlab cde in Appendix A with reference t the linear fit) is ps/nm-km, which crrespnds t a relative errr f.9%. Multiplied by the fiber length, this translates int a standard deviatin f ps/nm. An imprtant aspect f the previus three sectins is the errr assciated with the measurement f each pint in the dispersin parameter plts. The next sectin utlines the surce and magnitude f the errr assciated with the measurement f the dispersin parameter.

94 Chapter 5: Experiments & Analysiswww.inmetrix.cm Errr Analysis It is imprtant t understand the surce and magnitude f the errr in the measurement f the dispersin parameter in the previus experiments t gain an understanding f the precisin and accuracy that can be attained with an SAI. There are several surces f errr in the measurement f the dispersin parameter. Errrs intrduced by the envirnment in which the experiment takes place are the first types f errrs in the experiment. These errrs are nt quantifiable s they were mitigated by encasing the system in a sealed cntainer in which the temperature and density f the air was stabilized. Encasing the system in a sealed cntainer mitigates the errr that causes a variatin in the ptical path length f the air path due t air currents and the errr that causes a variatin in the length f the fiber due t temperature fluctuatins in the air. There are three ther quantifiable surces f errr in the experiment. Instrument errr in accurately measuring the wavelength f the tunable laser is the first, human errr in measuring the lengths f the fiber used in the experiment is the secnd, and systematic errr due t the wavelength windwing prcess (which puts an uncertainty with a magnitude f + ne half the windw size n the pints in the envelpe) is the third. Instrument errr in the measurement f the wavelength is much smaller than the wavelength windw used t plt the envelpe and as a result, it can be ignred in cmparisn t the systematic errr. Thus the majr quantifiable cntributins t the errr in measuring the dispersin parameter are human errr and systematic errr. Hw these tw quantities cmbine t

95 Chapter 5: Experiments & Analysiswww.inmetrix.cm 87 prduce an verall errr in the measurement f the dispersin parameter is nw discussed. The dispersin parameter is measured (at the central wavelength, λ 0 ) using equatin Eq. -5: d n eff D( ) Eq. 5-5 c d There are tw surces f errr in this calculatin; the errr in the measurement f the lcatin f the central wavelength, λ 0, due t systematic errr caused by the use f wavelength windwing t plt the envelpe and the errr in the measurement f the secnd derivative f the effective index with respect t wavelength. Fr simplicity this quantity is hencefrth referred t as B. When tw measurements are made independently the errrs are added in quadrature. Fr example, given the functin z = f(x, y) the errr in z can be calculated: B df df z ( x) ( y) Eq. 5-6 dx dy B and λ 0 are nt independent since B depends n λ 0, hwever, fr simplicity we assume that the tw are independent and later we will shw that the errr in λ 0 is much smaller than the errr in B and thus the errr in measuring the dispersin parameter, D, really nly depends n the errr in measuring B. At this pint, hwever, we prceed with the analysis assuming that the measurement f λ 0 and B are independent. Under this assumptin the errr in the dispersin parameter can be fund via the additin f the errrs in quadrature:

96 Chapter 5: Experiments & Analysiswww.inmetrix.cm ) ( ) ( ) ( ) ( B c c B B db dd d dd D Eq. 5-7 Where Δλ is the errr assciated with measuring the central wavelength, which is + half the wavelength windw and ΔB is the errr in calculating the secnd derivative f the effective index with respect t wavelength. Since B is calculated using the phase infrmatin in the envelpe f the interference pattern via Eq. 4-5 we use this equatin in rder t determine ΔB. In rder t simplify the calculatin f ΔB, we ignre the third rder dispersin term s that Eq. 4-5 becmes: B eff A f eff f d n d L d n d L m ),, ( Eq. 5-8 S that: ),, ( A ml B f Eq. 5-9 Thus if we assume that all variables in the experiment are independent then their errrs can be added in quadrature: ) ( ) ( ) ( ) ( f f f f f L A L m A A L m L dl db A da db B Eq ΔB is the ttal errr in measuring the secnd derivative f the effective index with respect t wavelength, B, and it is due t bth ΔA and the human errr in measuring the length f the test fiber, L f. ΔA is the errr in calculating the B due t the errr in lcating the peaks f the envelpe as shwn in Fig The magnitude f this errr is again + half the width f

97 Chapter 5: Experiments & Analysiswww.inmetrix.cm 89 the wavelength windw used t plt the envelpe, i.e. it is the systematic errr. ΔA is calculated by adding the errr in measuring the lcatin f the trughs in quadrature: ) ( 1 ) ( 1 ) ( 1 1 ) ( ) ( ) ( d da d da d da A Eq In rder t reduce the systematic errr it is best t chse the wavelength lcatins λ, λ 1 and λ t be the trughs f the envelpe since their lcatins are mre sharply defined. Therefre this is the reasn why the trughs f the envelpe lcatins were used in the experiments instead f the peaks. The systematic errr is illustrated in Fig 5-8. Fig. 5-8: Errr in calculating B due t the errr in lcating the peaks f the interfergram λ 0 = 1580nm Δλ 0 =0.15n m λ = 1553nm Δλ 1 = 0.15nm Δλ = 0.15nm λ 1 = 1540nm

98 Chapter 5: Experiments & Analysiswww.inmetrix.cm 90 A numerical example f the errr assciated with the measurement f the dispersin parameters in the plts f the previus three sectins is nw presented fr ne f the SMF8 TM measurements. The single mde fiber was measured t be m. Thus the human errr in the measurement f the fiber length is estimated t be ΔL f = m. One f the interfergrams frm the measurement is shwn in Fig The wavelength windw size used in the experiments was 0.5nm therefre Δλ 0,1, =0.15nm. Frm Fig. 5-8 we can see that λ 0 = 1580nm, λ 1 = 1540nm and λ = 1553nm. Thus frm Eq. 5-11, ΔA = 0.008nm. Substitutin f ΔA= 0.008nm and ΔL f = m int Eq (assuming m = 1 separatin is used as in Fig 5-8) yields, B 3.913x x x10 / m which shws that the errr in lcating the peaks f the envelpe has a larger effect than the human errr in measuring the length f the fiber. Substitutin f this value int Eq. 5-7 yields: D 1.917x x ps / nm km. Which shws that the errr in measuring B, has a larger effect than the errr in determining the central wavelength. Thus ΔD is mainly determined by the errr in measuring B regardless f whether r nt λ 0 and B are independent. This value fr ΔD is cnsistent with the bserved spread in the dispersin pattern in Fig In cnclusin, the experimental results f Single Arm Interfermetry cnfirm the thery develped in chapter 4. They shw that the dispersin parameter can be calculated frm the envelpe f the fringe pattern prduced by the interference f 3 waves in a balanced SAI. The experiments n Single mde fiber (SMF8 TM ) and Dispersin

99 Chapter 5: Experiments & Analysiswww.inmetrix.cm 91 Cmpensating Fiber (DCF) were used t cnfirm the thery behind the technique and nce the technique was cnfirmed it was used t measure the unknwn dispersin parameter plt fr THF. The length f Twin hle fiber used in the experiment was larger than allwed by Eq. 4-8 s the technique f wavelength windwing, described in sectins , had t be used. This technique was shwn theretically and via simulatin t extend the maximum length f fiber that can be characterized by this technique. Ultimately the largest length f fiber that can be characterized is limited by the largest air path that can be prduced in the experiment and the laser linewidth.

100 Chapter 6: Cnclusins 6.1 Expected Significance t Academia The single arm interfermeter is intrduced as an alternative t the Michelsn r the Mach Zehnder cnfiguratin fr interfermetric measurements f the dispersin parameter. It will be mst useful fr measurements f the dispersin parameter in shrt lengths f fiber. The technlgy will be used t eliminate the need fr the arm balancing required by dual arm interfermeters and by ding s allw fr greater ease in the cmmercializatin f Interfermetric dispersin measurement techniques. The new interfermeter is significant fr Academia since it can be studied and used alngside the earlier types f interfermeters like the Michelsn, the Mach-Zehnder and the Fabry Pert. This new interfermeter prvides academia with anther tl fr studying dispersin in shrt length fibers and waveguides which will be useful in the develpment f specialty fibers. These specialty fibers require simple and accurate shrt length characterizatin since they are generally made in very small quantities and their gemetry tends t vary as a functin f psitin alng the fiber. Anther significant academic achievement f the Single Arm Interfermeter is that a paper has been written fr this technique and it will be submitted shrtly fr review t the Jurnal Optics Express. If it is accepted fr publicatin the new technique will be accessible t anyne interested in measuring dispersin n shrt length fibers. This technique increases the ease f dispersin characterizatin and as a result it will lead t a greater number f dispersin measurements being perfrmed, especially in the area f specialty fiber. 9

101 Chapter 6: Cnclusins Expected Significance t Industry The new interfermeter is significant t Industry since it eliminates the need t cmpensate fr unwanted reflectins by eliminating the need fr a cupler altgether. As a result this interfermeter is a simpler (less expensive) interfermetric dispersin measurement device capable f characterizing the dispersin f shrt length ptical fiber. As a result it is a viable cmmercial cmpetitr t the current Mdulatin Phase Shift (MPS) based devices currently n the market. The new interfermeter, hwever, has an advantage ver MPS based devices since it has the ability t measure shrt length fiber with high accuracy. Als, since it can measure shrt lengths f fiber it has the ability fr anther type f measurement as well. Dispersin is a functin f bth material and dimensinal (waveguide) prperties f a fiber but if the dimensins, particularly the diameter f the fiber, vary then the dispersin will vary. If several small sectins can be cut frm varius pints n a lng length fiber and the dispersin is measured in each f them then the variatin in the dispersin can be pltted as a functin f psitin in the fiber. This can then be directly related t the variatin in the fiber diameter. The main pint here is that a great deal f accuracy in measuring the fiber diameter can be achieved by measuring it indirectly via the dispersin and it wuld be an easy way fr a fiber drawing cmpany t perfrm quality cntrl. Greater cmmercial interest in this device will enable measurement f dispersin in smaller lengths f fiber since larger bandwidth tunable lasers will be develped. Als the advancement in the speed f the tunable laser and scanning prcess will make each measurement faster t btain.

102 Chapter 6: Cnclusins Patent Applicatin One f the mst interesting features f a single arm interfermeter is the ease with which it can be built. This ease f cnstructin lends itself very nicely t ecnmical cmmercial assembly f a dispersin measurement device. An idea which is currently under patent is t prduce a cheap add-n mdule fr a tunable laser system t allw it t make dispersin measurements. A cnceptual design f such a mdule is illustrated in Fig. 5-9: T detectr Input Circulatr Test fiber Cllimating lens Mirrr Fig. 5-9: Cnceptual design fr a dispersin measurement mdule fr a tunable laser system. The cnnectr labeled T detectr is the input t a pwer detectr, the cnnectr labeled input is cnnected t the utput f a tunable laser. The test fiber can then be cnnected as shwn in the diagram in rder t perfrm the dispersin measurement. * [U f T Inventin Disclsures: RIS ID # & RIS ID # Patent applicatins nw underway] This dispersin measurement mdule culd be prduced t wrk with, fr example, the Agilent 8164A r 8164B Lightwave measurement system mainframe depicted in Fig. 5-10:

103 Chapter 6: Cnclusins 95 Fig. 5-10: Agilent 8164A/B Lightwave measurement system mainframe. The Agilent 8164A r 8164B Lightwave measurement mainframe is a mainframe which cntrls mdules such as tunable lasers and measurement devices that are inserted int the slts n the mainframe. The cst f the mainframe and a tunable laser mdule is $0,000. A dispersin characterizatin system sld by Agilent, namely the Agilent 86038A/B Phtnic Dispersin and Lss Analyzer depicted belw in Fig csts $130,000. Fig. 5-11: Agilent A/B Phtnic Dispersin and Lss Analyzer

104 Chapter 6: Cnclusins 96 Since this system includes the mainframe and tunable laser their value must be subtracted. This leaves abut $110,000 fr the dispersin and lss characterizatin devices in the system. Since an SAI has a higher precisin, can characterize bth shrt and lng length fiber and it is less expensive t implement it is very easy t see that this technlgy is disruptive t the industry. As a result the cmmercial ptential f this characterizatin technlgy is quite extrardinary. 6.4 Cnclusins In this paper we presented a nvel fiber-based SAI t measure directly the dispersin cefficient in shrt lengths f fiber (< 50 cm) with a standard deviatin (precisin) as lw as ps/nm. The technique utilizes the spectral interfergram created by three reflectins and extracts the secnd-rder dispersin frm the envelpe f the interfergram. The technique is shwn t be a simpler alternative t the Michelsn r Mach Zehnder interfermeters. By eliminating ne f the interfermeter arms, the technique des nt require calibratin and are less susceptible t plarizatin and phase fluctuatins. The cnstraints n the perating parameters f this technique, such as wavelength reslutin, fiber length, and measurable bandwidth, were discussed in detail. We verified the technique experimentally by perfrming a dispersin measurement n SMF8 TM and DCF. Our measured dispersin results n SMF8 TM shwed gd agreement with the simulated dispersin values based n published fiber gemetry and material prperties. Our measurement results n DCF agreed well with the measurement perfrmed n a much lnger DCF using a cmmercial dispersin measurement system. In additin t SMF8 TM and DCF, single arm interfermetry was used t measure the dispersin parameter f a twin-hle fiber fr the first time.

105 Chapter 6: Cnclusins 97 The perating parameters f this technique were discussed in detail and it was shwn that the range f measurable fiber lengths can be extended using wavelength windwing and a tunable laser with a randm step size. This methd can als be used t measure the dispersin f any waveguide in general and is nt limited t ptical fiber.

106 Appendix A: Matlab Cde A.1: Generating the Interference Pattern and the Envelpe % Envelpe and Interference pattern prgram clear all clse all clc % Parameters step_size = 1*10^-1;% 1 pm step size Lf = 0.395; % Length f fiber in meters Lair = *Lf; % is the grup index U=1; % First Fresnel reflectin gamma=1; % Fractin f first Fresnel reflectin % reflected frm first facet alpha=1; % Fractin f the first Fresnel % reflectin reflected frm the mirrr % Interference pattern lad neff.mat % neff fr single mde fiber neff_fit = plyfit(lambda1, neff, 3); % Interplated lambda = 1510*10^-9:step_size:1640*10^-9; % Interplated neff_sim = plyval(neff_fit, lambda); % Interplated beta = (*pi./lambda).* neff_sim; % Beta values interplated k=*pi./lambda; % Entire interference pattern I=abs(1+alpha*exp(i*beta**Lf)+gamma*exp(i*(beta**Lf+k**Lair))).^; % Envelpe f the interference pattern envelpe_full = U^*(1 + alpha^ + gamma^ + 4*alpha*abs(cs(beta*Lf - k*lair)) + *alpha*(gamma-1) + *gamma); figure plt(lambda,i,lambda,envelpe_full, 'x'); xlabel('lambda (nm)') ylabel('intensity (a.u.)') A. Calculating Neff clc; clear; warning ff; glbal Ks K r0 rj n_j tj m beta w l eps0 mu0 ns n lambda0 V Uj Wj Rs R1 p a % Fiber parameters ================================================ 98

107 Appendix A: Matlab Cde 99 fr lambda_i=0:100 end lambda_i lambda0=1.5e-6+.1e-6*lambda_i/100 lambda1(lambda_i+1)=lambda0; k=*pi/lambda0 % SMF parameters m=1; %NA=.1; NA=0.11 Delta_n=0.0036; n1=silica_index(lambda0*1e6,1); % Taken frm data file n=silica_index(lambda0*1e6,0); % Taken frm data file Dn=n1-n; % Surce fiber Rs=.3e-6; V=k*Rs*sqrt(n1^-n^); ws=rs*( *v^ *v^-6); n=n1; ns=n; U=fzer(@LP,V-.4); % Functin LP defined belw X=U/Rs; W=sqrt(V^-U.^); beta(lambda_i+1) = sqrt(k^*n1^-(u/rs).^); neff(lambda_i+1) = beta(lambda_i+1)/k; save neff lambda1 beta neff %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% functin S1=LP(U) glbal V n ns m W=sqrt(V^-U.^); Jp=(besselj(m-1,U)-besselj(m+1,U))/; Kp=(besselk(m-1,W)+besselk(m+1,W))/; J=besselj(m,U); K=besselk(m,W); %S1=(Jp./(U.*J) + Kp./(W.*K)).*((ns/n)^*Jp./(U.*J)+Kp./(W.*K)) - m^*(1./u.^+1./w.^).*((ns/n)^./u.^+1./w.^); S1=besselj(0,U)./(U.*besselj(1,U)) - besselk(0,w)./(w.*besselk(1,w)); end

108 Appendix A: Matlab Cde A.3: Prbability vs. Several ther Parameters A.3.1: Prbability vs. windw size % Prbability versus WINDOW SIZE clear all clse all warning ff clc % Independent parameters that may be varied %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lf = 0.395; % Fiber length in meters step_size = 1*10^(-1); % Average wavelength step f the tunable laser tlerance = 0.0; % Tlerance in lcating the peak (gives >99.9% % f peak) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.17*10^-1; fr i = 1:30 carse_sampling_bandwidth(i) = i* 0.01*10^-9; % Width f windw Pnne_average1(i) = 1 - Prbability(Lf, carse_sampling_bandwidth(i), step_size, tlerance, sigma) end % Cnvert t nm carse_sampling_bandwidth = carse_sampling_bandwidth * 10^9; % Plt the curve figure plt(carse_sampling_bandwidth,pnne_average1, 'b') xlabel('windw Size in nm') ylabel('prbability') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.5*10^-13; % sigma = 0 % Prbability vs windw size fr i = 1:30 carse_sampling_bandwidth(i) = i* 0.01*10^-9; Pnne_average(i) = 1 - Prbability(Lf, carse_sampling_bandwidth(i), step_size, tlerance, sigma) % Width f windw end % Cnvert t nm carse_sampling_bandwidth = carse_sampling_bandwidth * 10^9; % Plt the curve hld n plt(carse_sampling_bandwidth,pnne_average, 'g')

109 Appendix A: Matlab Cde %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 1*10^-1; % Prbability vs windw size fr i = 1:30 carse_sampling_bandwidth(i) = i* 0.01*10^-9; % Width f windw Pnne_average3(i) = 1 - Prbability(Lf, carse_sampling_bandwidth(i), step_size, tlerance, sigma) end % Cnvert t nm carse_sampling_bandwidth = carse_sampling_bandwidth * 10^9; % Plt the curve hld n plt(carse_sampling_bandwidth,pnne_average3, 'r') A.3.: Prbability vs. average step size % Prbability versus STEP SIZE clear all clse all warning ff clc % Independent parameters that may be varied %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lf = 0.395; % Fiber length in meters tlerance = 0.0; % Tlerance in lcating the peak carse_sampling_bandwidth = 0.5*10^-9; % Width f windw fr finding peak f envelpe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.05*10^-1; % Test prgram fr i = 1:0 step_size(i) = i*0.1*10^-1 Pnne_average(i) = 1 - Prbability(Lf, carse_sampling_bandwidth, step_size(i), tlerance, sigma) end % Cnvert t pm step_size = step_size * 10^1; % Plt the curve figure plt(step_size,pnne_average, 'g') xlabel('step Size in picmeters') ylabel('prbability')

110 Appendix A: Matlab Cde 10 hld n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.17*10^-1; % Test prgram fr i = 1:0 step_size(i) = i*0.1*10^-1 Pnne_average(i) = 1 - Prbability(Lf, carse_sampling_bandwidth, step_size(i), tlerance, sigma) end % Cnvert t pm step_size = step_size * 10^1; % Plt the curve plt(step_size,pnne_average, 'b') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 1*10^-1; % Test prgram fr i = 1:0 step_size(i) = i*0.1*10^-1 Pnne_average(i) = 1 - Prbability(Lf, carse_sampling_bandwidth, step_size(i), tlerance, sigma) end % Cnvert t pm step_size = step_size * 10^1; % Plt the curve plt(step_size,pnne_average, 'r') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A.3.3: Prbability vs. fiber length % Prbability versus FIBER LENGTH clear all clse all warning ff clc % Independent parameters that may be varied %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tlerance = 0.0; % Tlerance in lcating the peak carse_sampling_bandwidth = 0.5*10^-9; % Width f windw step_size = 1*10^(-1); % Average wavelength step f the tunable laser %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.05*10^-1;

111 Appendix A: Matlab Cde %Pnne_average = Prbability(Lf, carse_sampling_bandwidth, step_size, tlerance) % % Prbability vs Fiber length fr i = 1:150 Lf(i) = 0.01*i; Pnne_average(i) = 1 - Prbability(Lf(i), carse_sampling_bandwidth, step_size, tlerance, sigma); i end % Plt the curve figure plt(lf,pnne_average, 'g' ) xlabel('fiber Length in meters') ylabel('prbability') hld n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.17*10^-1; % % Prbability vs Fiber length fr i = 1:150 Lf(i) = 0.01*i; Pnne_average(i) = 1 - Prbability(Lf(i), carse_sampling_bandwidth, step_size, tlerance, sigma); end % Plt the curve plt(lf,pnne_average, 'b') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 1*10^-1; % Prbability vs Fiber length fr i = 1:150 Lf(i) = 0.01*i; Pnne_average(i) = 1 - Prbability(Lf(i), carse_sampling_bandwidth, step_size, tlerance, sigma); end % Plt the curve plt(lf,pnne_average, 'r') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A.3.4: Prbability vs. tlerance % Prbability vs. Tlerance clear all clse all warning ff clc

112 Appendix A: Matlab Cde % Independent parameters that may be varied size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lf = 0.395; % Fiber length in meters step_size = 1*10^(-1); % Average wavelength step tlerance = 0.0; % Tlerance in lcating the peak carse_sampling_bandwidth = 0.5*10^-9; % Width f windw %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.05*10^-1; % Standard deviatin f the surce fr i = 1:50 tlerance(i) = i* 0.001; % Width f windw Pnne_average(i) = 1 - Prbability(Lf, carse_sampling_bandwidth, step_size, tlerance(i), sigma) end % Cnvert t % tlerance = tlerance * 100; % Plt the curve figure plt(tlerance,pnne_average, 'g') xlabel('tlerance: % f the peak spacing f the carrier ') ylabel('prbability') hld n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 0.17*10^-1; % Standard deviatin f the surce fr i = 1:50 tlerance(i) = i* 0.001; % Width f windw Pnne_average(i) = 1 - Prbability(Lf, carse_sampling_bandwidth, step_size, tlerance(i),sigma) end % Cnvert t % tlerance = tlerance * 100; % Plt the curve plt(tlerance,pnne_average, 'b') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma = 1*10^-1; % Standard deviatin f the surce fr i = 1:50 tlerance(i) = i* 0.001; % Width f windw Pnne_average(i) = 1 - Prbability(Lf, carse_sampling_bandwidth, step_size, tlerance(i), sigma) end % Cnvert t % tlerance = tlerance * 100; % Plt the curve plt(tlerance,pnne_average, 'r') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

113 Appendix A: Matlab Cde A.3.5: The Prbability calculating functin functin Pnne_average = Prbability(Lf,carse_sampling_bandwidth,step_size,tlerance,sigma) % Dependent parameters N = carse_sampling_bandwidth/step_size; % n = Step number, N = number f steps f the tunable laser % Dependent n fiber lad neff.mat % neff fr single mde fiber calc frm prg in A..6 neff_fit = plyfit(lambda1, neff, 4); % Interplated lambda = 1510*10^-9:step_size:1640*10^-9; neff_sim = plyval(neff_fit, lambda); beta = (*pi./lambda).* neff_sim; k=*pi./lambda; lambda_p = lambda.^./(*neff_sim*lf); % Fringe perid as a functin f wavelength % Determine M summatin = 0; M = 1; while summatin < carse_sampling_bandwidth summatin = summatin + lambda_p(m); M = M+1; % m = Peak number, M = number f peaks f % carrier in the carse sampling bandwidth end lambda_p = summatin/m; % lambda_p is nw the average carrier perid % Dep n required tlerance epsiln = tlerance*lambda_p; % Prbability calculatin lambda0 = 0:lambda_p/100:lambda_p; % Average f Pnne fr different lambda0's ver the perid % f ne carrier wave using 100 slts Pnne = 1; % Initialize fr m = 1:M fr n = 0:N-1 t_upper = ((m*lambda_p+(epsiln/))- (n*step_size+lambda0))/(()^0.5*sigma); t_lwer = ((m*lambda_p-(epsiln/))- (n*step_size+lambda0))/(()^0.5*sigma); Pmn = 0.5*(erf(t_upper)-erf(t_lwer)); Pnne = Pnne.* (1 - Pmn); end end Pnne_average = (1/100) * sum(pnne); % Equivalent t taking (1/perid) * integral --> Averaging % functin end

114 Appendix A: Matlab Cde A.4: Determining the Precisin f the Measurements A.4.1: Standard deviatin f the SMF8 TM Measurement % Standard deviatin f measured pints fr SMF clear all clse all clc lambda = [ ]; D = [ ]; D_eq = plyfit(lambda, D, 1); D_fit = plyval(d_eq, lambda); figure plt(lambda, D, '.', lambda, D_fit) x = D - D_fit; mu = mean(x) sigma = std(x) A.4.: Standard deviatin f the DCF Measurement % Standard deviatin f measured pints fr DCF clear all clse all clc lambda = [ ]; D = [ ]; D_eq = plyfit(lambda, D, 1); D_fit = plyval(d_eq, lambda); figure plt(lambda, D, '.', lambda, D_fit) x = D - D_fit; mu = mean(x) sigma = std(x)

115 Appendix A: Matlab Cde A.4.3: Standard deviatin f the THF Measurement % Standard deviatin f measured pints fr THF clear all clse all clc lambda = [ ]; D = [ ]; D_eq = plyfit(lambda, D, 1); D_fit = plyval(d_eq, lambda); figure plt(lambda, D, '.', lambda, D_fit) x = D - D_fit; mu = mean(x) sigma = std(x)

116 Appendix B Crning SMF8 TM Data Sheetwww.inmetrix.cm Appendix B Crning SMF8 TM Data Sheet 108