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1 Auhor's personal copy Opics Communicaions 84 () 7 76 Conens liss available a ScienceDirec Opics Communicaions ournal homepage: Invied paper Phoonic generaion of microwave arbirary waveforms Jianping Yao Microwave Phoonics Research Laboraory, School of Informaion Technology and Engineering, Universiy of Oawa, Oawa, ON, Canada, KN 6N5 aricle info absrac Aricle hisory: Received 5 December Received in revised form February Acceped 4 February Available online March Keywords: Arbirary waveform generaion Direc space-o-ime pulse shaping Fiber opics Fiber Bragg graing Microwave Phoonics Phoonic microwave delay-line filer Specral-shaping and wavelengh-o-ime mapping Temporal pulse shaping In his paper, echniques o generae microwave arbirary waveforms based on all-fiber soluions are reviewed, wih an emphasis on he sysem archiecures based on direc space-o-ime pulse shaping, specral-shaping and wavelengh-o-ime mapping, emporal pulse shaping, and phoonic microwave delay-line filering. The generaion of phase-coded and frequency-chirped microwave waveforms is discussed. The challenges in he implemenaion of he sysems for pracical applicaions are also discussed. Elsevier B.V. All righs reserved.. Inroducion Microwave arbirary waveforms are widely used in radar, communicaions, medical imaging, and modern insrumenaion sysems. Microwave arbirary waveforms are usually generaed in he elecrical domain using digial elecronics. Due o he limied sampling rae, he generaion of a microwave arbirary waveform in he elecrical domain is limied o a low frequency and small bandwidh. For many applicaions, however, high frequency and large bandwidh waveforms are needed. A soluion is o generae microwave arbirary waveforms in he opical domain, o ake advanage of he high speed and broad bandwidh offered by modern opics. In general, phoonically assised microwave waveform generaion can be classified ino four caegories, ) direc space-o-ime pulse shaping, ) specral-shaping and wavelengh-o-ime mapping, ) emporal pulse shaping, and 4)microwave pulse generaion based on phoonic microwave delay-line filering. These echniques can be implemened in free space where a spaial ligh modulaor (SLM) is usually employed o perform emporal or specral shaping. The key advanage of using an SLM in a microwave arbirary waveform generaion sysem is is flexibiliy. An SLM can be updaed in real ime, making he sysem reconfigurable. However, a pulse shaping sysem based on an SLM is usually implemened in free space, making he sysem bulky and cosly. In addiion, he coupling beween fiber and free space and free space o fiber makes he sysem lossy and sensiive o environmenal changes. Microwave arbirary waveform generaion can also be implemened using pure fiber-opic devices. address: pyao@sie.uoawa.ca. Considering he low loss and small size, a microwave waveform generaion sysem using fiber opic devices is considered a promising alernaive o ha implemened based on free space opics. In his paper, echniques o use fiber opic devices o implemen arbirary microwave waveform generaion are reviewed. All he four differen echniques ha are implemened using fiber opic devices are discussed. The use of he echniques o generae frequency-chirped and phase-coded microwave waveforms is discussed. The challenges in implemenaion he sysems for pracical applicaions are also discussed.. Direc space-o-ime pulse shaping Arbirary waveform generaion can be realized based on direc space-o-ime (DST) mapping [ ], in which an arbirary opical pulse sequence is generaed in he opical domain and hen applied o a high-speed opical-o-elecrical converer o generae a microwave waveform. By his echnique, reprogrammable cycle-by-cycle synhesis of an arbirarily shaped phase-coded or frequency-chirped waveform could be implemened. In he sysem, a bandwidh-limied opical-o-elecrical converer was usually used o conver an opical pulse burs consising of isolaed opical pulses ino a smooh microwave waveform. Fig. shows a DST mapping sysem in which an ulrashor opical pulse from a pulsed laser source is sen o an opical pulse shaper o generae a pulse burs, wih he pulse spacing increasing emporally. The pulse burs is hen applied o an opical-o- -48/$ see fron maer Elsevier B.V. All righs reserved. doi:.6/.opcom...69

2 Auhor's personal copy 74 J. Yao / Opics Communicaions 84 () 7 76 Pulse Burs p T ( ) Bandpass Filering +s +nd Time Frequency Time Fig.. Arbirary waveform generaion based on direc space-o-ime (DST) mapping. Fig.. Generaion of a chirped microwave waveform from a pulse burs wih increasing pulse spacing hrough bandpass filering. elecrical converer or a phoodeecor (PD). Due o he bandwidhlimied naure of he PD, he high-frequency componens are eliminaed and a smooh frequency-chirped microwave waveform is generaed. The key device in he DST mapping sysem is he opical pulse shaper. I could be implemened using free-space opical componens [ 8], bu wih large size, high loss and poor sabiliy. A simpler bu more effecive soluion is o generae a pulse burs using an arrayed waveguide graing (AWG) [9 ], as shown in Fig.. An ulra-shor pulse is launched ino he AWG hrough an ampliude mask. Due o he differen ime delays resuled from he differen physical lenghs of he waveguides in he AWG, a pulse burs wih emporally spaced pulses is generaed. To generae a pulse burs wih he desired emporal paern, he inpu ampliude mask can be configured o block he ligh from going ino some of he opical waveguides. The use of he AWG-based pulse shaper o generae a pulse burs of more han pulses as an ulrafas opical daa packe over approximaely an 8-ps emporal window was demonsraed [9]. The generaion of high-repeiion-rae femo-second WDM pulses was also demonsraed []. The heory behind he generaion of a microwave arbirary waveform using a DST mapping sysem is ha he waveform o be generaed can be obained by filering he pulse burs using a band-limied filer []. We recenly demonsraed ha a emporally spaced pulse burs, such as a pulse burs wih increasing or decreasing emporal spacing, would have a muli-channel specral response, wih one channel having a specrum ha corresponds o he specrum of he waveform o be generaed []. Fig. shows he generaion of a linearly chirped microwave waveform from a pulse burs wih increasing pulse spacing hrough bandpass filering. As canbeseenhe+sorderchannelhasaspecralresponsehaisequalo ha of a linearly chirped microwave waveform. By employing a bandpass filer o selec he +s order channel, a linearly chirped microwave waveform would be generaed. In he following, a brief mahemaical derivaion is provided o show he generaion of an arbirary microwave waveform hrough pulse posiion modulaion [] and bandpass filering. A uniformly-spaced inensiy-modulaed opical pulse burs p T (), in which he kh pulse has a ime delay of τ k =kt, where T is he imedelay difference beween wo adacen pulses, can be expressed as N p T ðþ= k = α k g kt ð Þ; ðþ Arrayed Waveguide Array where g() is a single shor pulse, α k is he coefficien weighed on he kh pulse, and N is he number of he pulses in he pulse burs. The pulse burs can be expressed in anoher form, N p T ðþ= g ðþ k = α k δð ktþ = g ðþa ½ ðþ s ðþš ðþ where a() is he weigh profile which is given as a(kt)=α k for k N, oherwise a()=; s() is an uni impulse rain given by s ðþ= δð ktþ, and denoes he convoluion operaion. k The specrum of he pulse burs, P T (ω), can be calculaed by he Fourier ransform, P T ðωþ = GðωÞ π AðωÞ m π T δω mω ð Þ = m T Gðω ÞA ð ω mω Þ where Ω=π/T, A(ω) is he Fourier ransform of a(), and G(ω) is he specrum of he shor pulse g(). Since he pulse g() is usually ulrashor, is specrum G(ω) changes much slower compared wih A(ω mω) wihin he bandwidh a Ω. Therefore, he change of G(ω) wihin he mh channel could be ignored, and G(ω) can be approximaed as G(mΩ). Thus, he pulse burs has a muli-channel specral response, wih all channels having he same specral profile A(ω) and he mh channel being locaed a mω. If a microwave bandpass filer wih is bandpass locaed a ω=mω is used in he sysem, he specrum of he mh channel of he pulse burs is seleced. As a resul, he signal a he oupu of he microwave bandpass filer is given by PðωÞ T GmΩ ð ÞAðω mωþ: ð4þ ðþ In he ime domain, he oupu microwave signal, p(), is he inverse Fourier ransform of Eq. (4), which is given by p ðþ= T GmΩ ð Þa ðþ exp ð mω Þ ð5þ = T GmΩ ð Þa ðþ exp f ½ φðþ+ mω Šg Mask Beam Expander Inpu Oupu Fig.. Opical pulse shaper based on an arrayed waveguide array. where φ() is he phase response of a(). We can clearly see ha he oupu signal is a microwave signal wih a cenral frequency locaed a mω. The generaed phase modulaion is us he phase of he weigh profile, φ(). Based on Eq. (5), we can see if an arbirary microwave waveform is generaed hen a() should be a funcion having an arbirary phase response. While in a DST mapping sysem, since only he power of he individual pulse is deeced a he PD, he coefficiens, α k,arealways posiive. Therefore, he required phase response φ() canno be inroduced. A soluion o his problem is o inroduce a phase shif hrough varying he spacing of he pulse burs, which is also called pulse posiion modulaion. For a specific frequency, a ime shif corresponds o

3 Auhor's personal copy J. Yao / Opics Communicaions 84 () a phase shif, and he inclusion of he phase shif would make he weigh profile be complex-valued, leading o he generaion of an arbirary microwave waveform. Assume a funcion f() is inroduced o describe he pulse posiion modulaion, ha is, s+ ð fðþ Þ = δð + fðþ kt Þ, a new opical k pulse burs is given by burs wih pulse posiion modulaion, an arbirary microwave waveform can be obained. For example, o generae a phase-coded microwave waveform, if he phase modulaion funcion is φ(), hen he relaionship beween he pulse posiion modulaion funcion f() and he desired phase modulaion can be obained by leing φ()=mωf() or p T ðþ= g ðþfa ðþ s+ ½ fðþ Šg: ð6þ Wih he inroducion of f(), he ime spacing of he pulse burs is no longer uniform. Based on Fourier Series expansion, s() can be expressed by is Fourier series, s ðþ= T exp ð mωþ. By variable m subsiuion, Eq. (6) can be wrien as m p T ðþ= g ðþ T a ðþexp ½ mωf ðþ Š expðmωþ: ð7þ Noe ha he above equaion is obained by variable subsiuion, i is no longer a Fourier series expansion (an aperiodic signal does no have a Fourier series expansion). Since g() is ulra-shor, we may model i as an uni impulse, ha is, g()=δ(), where δ() is he Dirac dela funcion. Thus, Eq. (7) can be approximaed as p T ðþ T a ðþexp ½ mωf ðþ Š expðmωþ: ð8þ m I is clearly seen from Eq. (8) ha he nonuniformly-spaced pulse burs is expressed as he sum of muliple bandpass microwave signals wih differen cenral frequencies. If T is sufficienly small such ha he mh channel is no inerfered by is adacen channels, he specral componen a mω can be filered ou by a microwave bandpass filer wih is cenral frequency a mω. Then he oupu microwave signal is now given by p ðþ= T fa ðþexp ½ mωf ðþ Š g expðmωþ: ð9þ Comparing Eq. (9) wih Eq. (5), we can see ha an addiional phase modulaion is inroduced o he microwave signal due o he pulse posiion modulaion inroduced by f(). Therefore, alhough he weigh coefficiens are all posiive, by using a specially designed pulse fðþ= φðþ mω : ðþ Subsiuing Eq. () ino Eq. (6), we ge he ime delay of each pulse in he pulse burs, τ k + φτ ð kþ = kt: ðþ mω Considering he propery of he Dirac funcion, he coefficiens can also be obained, which are given by aðτ α k = k Þ +φ ðτ k Þ= mω : ðþ Based on he analysis we conclude ha if a pulse burs wih a pulse posiion modulaion described by Eq. () wih he coefficiens given by Eq. (), a microwave signal wih a cenral frequency locaed a mω and a phase modulaion of a()exp[φ()] are hen obained a he oupu of a microwave bandpass filer wih is cenral frequency locaed a mω.. Specral-shaping and wavelengh-o-ime mapping Arbirary waveform generaion can be realized based on specralshaping and wavelengh-o-ime mapping. The fundamenal principle of he echnique is shown in Fig. 4(a). The sysem consiss of a pulsed source, a specral shaper and a dispersive elemen. The specral shaper is used o modify he specrum emied from he pulsed laser source, which can be a passively or acively mode-locked laser source. The shaped specrum hen undergoes wavelengh-o-ime mapping in a dispersive device, which can be a lengh of dispersive fiber or a chirped fiber Bragg graing. A microwave waveform is generaed in he elecrical domain a a high-speed phoodeecor. If he dispersive elemen is a lengh of fiber wih a value of dispersion of Φ, for an inpu Fig. 4. Arbirary waveform generaion based on specral-shaping and wavelengh-o-ime mapping. (a) Schemaic of a microwave waveform generaion sysem based on specralshaping and wavelengh-o-ime mapping. (b) Illusraion of wavelengh-o-ime mapping in a dispersive elemen.

4 Auhor's personal copy 76 J. Yao / Opics Communicaions 84 () 7 76 pulse g() wih a emporal widh of Δ, he signal a he oupu of he dispersive elemen is given y ðþ= g ðþexp = exp Φ exp Φ = exp Φ Φ " # = g ð τ τ Þ exp Φ dτ τ exp Φ gðτþ exp g τ ð Þ exp GðωÞ ω = Φ Φ τ dτ Φ τ dτ ðþ where G(ω) is he Fourier ransform of g(). As can be seen he oupu signal envelope is proporional o he Fourier ransform of he inpu signal envelope. Noe ha Eq. () is obained if he duraion of he inpu ulrashor pulse, Δ, and he second-order dispersion Φ of he dispersive elemen saisfy he following condiion, Δ Φ bb; ð4þ which means he phase erm τ in Eq. () saisfies τ Δ bb, Φ Φ Φ hus we have exp τ [4]. Φ The wavelengh-o-ime mapping is illusraed in Fig. 4(b). If he inpu o he dispersive device is a recangular pulse, hen he oupu emporal waveform should be a sinc funcion. As can be seen, he key device in he arbirary microwave waveform generaor is he specral shaper, which should be designed o have a magniude response ha can make he shaped specrum have he same shape as he microwave waveform o be generaed. Based on his concep, a few approaches o generaing chirped microwave waveforms [5 ] were demonsraed. The main effor in hese approaches is o design an opical specral shaper ha has a magniude response wih is shape idenical o ha of he microwave waveform o be generaed. Fig. 5(a) shows a specral-shaping and wavelengh-o-ime mapping sysem for he generaion of a chirped microwave waveform [7]. An ulra-shor pulse from a mode-locked laser is sen o an opical specral shaper. For chirped microwave waveform generaion, he magniude response of he opical specral shaper should have an increasing or decreasing free specral range (FSR) which is ermed chirped FSR in [7]. The specrum-shaped pulse is hen sen o a dispersive elemen, which is a lengh of single-mode fiber (SMF), as shown in Fig. 5(a). To obain a specral response wih a chirped FSR, he specral shaper is designed by superimposing wo chirped fiber Bragg graings wih differen chirp raes ino a same fiber wih a small longiudinal offse, as shown in Fig. 5(b). Disribued Fabry Pero inerference is hen produced in he fiber due o he reflecions beween he wo chirped fiber Bragg graings. This generaes an in-fiber opical filer wih an FSR inversely proporional o he caviy lengh L. Since he wo chirped fiber Bragg graings have differen linear chirp raes, he equivalen caviy lengh L varies linearly wih respec o opical wavelengh λ. As a resul, he FSR is no consan bu is increasing or decreasing wih respec o opical wavelengh. From Fig. 5(b), we can see he equivalen caviy lengh L is linearly proporional o he wavelengh λ wihin he filer bandwidh, LðλÞ = d + C C ðλ λ C C Þ ð5þ where C and C are he chirp raes of he wo chirped fiber Bragg graings (in nm/mm), d is he longiudinal offse, and λ is he sar wavelengh. The FSR of he disribued Fabry Pero filer is given by FSR λ n eff LðλÞ ð6þ where n eff is he effecive refracive index of he fiber. Afer he dispersion-induced linear frequency-o-ime mapping, he FSR is mapped o he emporal period of he generaed chirped pulse, namely, Δτ, wih a mapping relaionship λ /χ, where χ (ps/nm) is he oal dispersion of he SMF. For simpliciy, he insananeous microwave carrier frequency of he generaed emporal pulse, f RF, can be approximaed by he reciprocal of he emporal period Δτ. f RF ðþ= " # Δτ =n C C eff C C λ + d χ λ χ λ χ ð7þ I can be seen ha he frequency of he microwave waveform is linearly proporional o ime, herefore he generaed microwave waveform is linearly chirped. For he SMF wih a given lengh, he cenral carrier frequency of he generaed chirped pulse a = is dependen only upon he longiudinal offse d. The chirp rae of he generaed pulse is deermined by he chirp raes of he wo chirped fiber Bragg graings. Therefore, by choosing he longiudinal offse and he chirp raes of he chirped fiber Bragg graings, a linearly chirped microwave waveform wih a high cenral frequency and a large chirp rae can be generaed. The maor limiaion of he approach in [7] is ha he opical specral shaper, once fabricaed, is no reconfigurable. For many applicaions, however, i is expeced ha he chirp rae and he cener frequency of he microwave chirped waveform can be unable. To solve Fig. 5. Chirped microwave waveform generaion based on specral-shaping and wavelengh-o-ime mapping. (a) Schemaic of he chirped microwave pulse generaion sysem. (b) Opical specral shaper consising of wo superimposed chirped fiber Bragg graings wih differen chirp raes and a small longiudinal offse. MLL: mode-locked laser; SI-CFBG: superimposed chirped fiber Bragg graing; SMF: single mode fiber; PD: phoodeecor.

5 Auhor's personal copy J. Yao / Opics Communicaions 84 () his problem, we proposed an opical specral shaper ha is a Sagnacloop mirror incorporaing a chirped fiber Bragg graing [8]. Similar o he superimposed chirped fiber Bragg graings in [7], due o he incorporaion of a chirped fiber Bragg graing, he Sagnac-loop mirror would have a disribued Fabry Pero inerference due o he reflecions from he wo opposie direcions of he chirped fiber Bragg graing. Thus, an in-fiber opical filer wih an FSR ha is inversely proporional o he lengh difference would be formed. Since he reflecion poin is wavelengh dependen, he equivalen lengh difference varies linearly wih respec o opical wavelengh λ. As a resul, he FSR is no consan bu is increasing or decreasing wih respec o he opical wavelengh. Fig. 6 shows he opical specral shaper based on an all-fiber Sagnac-loop mirror incorporaing a chirped fiber Bragg graing [8]. The Sagnac-loop mirror is consruced from a fused -db fiber coupler spliced o he erminals of he chirped fiber Bragg graing, which is locaed approximaely a he middle poin of he fiber loop. A unable delay-line (TDL) is locaed in he fiber loop o finely une he imedelay difference beween wo fiber lenghs L and L. A polarizaion conroller (PC) is also placed in he loop o opimize he visibiliy of he inerference paern a he oupu of he loop mirror. A hree-por opical circulaor is used o direc he ulrashor pulse ino he loop mirror and o oupu he specrum-shaped pulse for wavelengh-o-ime mapping. Mahemaically, he Sagnac-loop mirror incorporaing a chirped fiber Bragg graing can be modeled as a wo-ap delay-line filer. The ransfer funcion of he Sagnac-loop mirror is expressed as TðλÞ = WðλÞ + cos πn eff λ ΔL ; λ λ B λ ð8þ where W(λ) is he inensiy reflecion specrum of he chirped fiber Bragg graing wih a bandwidh B λ, and n eff is he effecive refracive index of he fiber core. ΔL=L L is he fiber lengh difference, wih L and L measured from he cener of he chirped fiber Bragg graing o he fiber coupler along he clockwise and counerclockwise pahs as shown in Fig. 6. The fiber lengh difference ΔL comes from wo sources: he wavelengh-independen pah difference, ΔL, and he wavelengh-dependen fiber lengh difference inroduced by he chirp of he chirped fiber Bragg graing, ΔL(λ). ΔL can be conrolled o be eiher a posiive or a negaive value by uning he TDL in he fiber loop. ΔL(λ) is deermined by he bandwidh and he chirp parameer of he chirped fiber Bragg graing, and can be calculaed using ΔL(λ)=δλ/C, where δλ (nm) is he wavelengh deuning from he cener wavelengh λ,andc (nm/cm) is he chirp parameer of he chirped fiber Bragg graing. Then he filer ransfer funcion T(λ) can be rewrien as ( " TðλÞ = WðλÞ + cos 4πn eff λ ΔL λ + δλ # ) : ð9þ C Since he linearly chirped fiber Bragg graing is locaed in he fiber loop, an opical signal wih differen wavelenghs will be refleced from a differen posiion in he chirped fiber Bragg graing. As a resul, an opical specral filer wih a wavelengh-dependen FSR is formed. The FSR of he opical specral filer response is a funcion of he wavelengh and can be expressed as FSR = λ n eff ΔL = λ n eff δλ C + ΔL ðþ According o Eq. (), by properly choosing he parameers of he chirped fiber Bragg graing and by conrolling he TDL in he fiber loop, he FSR of he Sagnac-loop mirror can be conrolled. Afer he specrum-shaped opical pulse propagaes hrough he dispersive elemen and is deeced by he high-speed PD, he shaped specrum is mapped ino a emporal microwave pulse as T(λ) y() hanks o he dispersion-induced linear wavelengh-o-ime mapping. Assume ha he inpu ulrashor opical pulse is a uni impulse, according o he mapping relaionship λ /χ, he ime-domain waveform is given by y ðþ= W ( " + cos 4πn eff χ λ χ ΔL + δ # ) Cχ ðþ where δ is he ime deuning from he cener of he emporal waveform, which is given by he mapping relaionship δλ δ/χ. The ime-domain pulse duraion ΔT of he generaed microwave pulse is deermined by he window funcion W(/χ), and is calculaed by ΔT=B λ χ. Considering ha he pulse widh of he inpu ulrashor opical pulse is no zero, he deeced pulse envelope should be modified by adding an envelope r(), y ðþ= r ðþw ( " + cos 4πn eff χ λ χ ΔL + δ # ) Cχ ðþ where r() is he pulse envelope afer he inpu pulse passing hrough he dispersive elemen. Assuming ha he inpu ulrashor opical pulse has a Gaussian envelope as g() exp( /Δ ), where Δ is he half pulse-widh a /e maximum, hen he envelope of oupu pulse from a dispersive elemen will mainain he Gaussian shape, bu wih a broadened pulse widh of Φ = Δ. Acually, he envelope r()is a scaled version of he specrum envelope of he inpu pulse, which is mapped o he ime domain hanks o he dispersion-induced frequency-o-ime mapping in he chirped fiber Bragg graing. The insananeous microwave carrier frequency of he generaed waveform can be obained from he phase erm of Eq. (), which is expressed as f RF ðδþ = π dψ = n eff d λ χ ΔL þ δ : ðþ Cχ I is shown ha he generaed microwave waveform is linearly chirped. For a given dispersive elemen, he cenral microwave carrier frequency of he generaed chirped microwave pulse is only deermined by he absolue value of wavelengh-independen fiber lengh difference ΔL. Therefore, he cenral frequency of he generaed microwave chirped waveform can be uned by simply uning ΔL. The chirp rae of he generaed microwave waveform, given by CR RF = df RF ðδþ= dδ = þn eff = Cλ χ ; ð4þ Fig. 6. An all-fiber opical specral shaper consising of Sagnac-loop mirror incorporaing a chirped fiber Bragg graing. FC: fiber coupler; TDL: unable delay-line; CFBG: chirped fiber Bragg graing; PC: polarizaion conroller. is only dependen on he dispersion of he chirped fiber Bragg graing. The sign of he chirp rae corresponds o he posiive and negaive values of ΔL. Therefore, by appropriaely conrolling he TDL and choosing he

6 Auhor's personal copy 78 J. Yao / Opics Communicaions 84 () 7 76 dispersion of he chirped fiber Bragg graing, a linearly chirped microwave pulse wih a high cenral frequency and a unable chirp rae can be generaed. Fig. 7 shows he specra of a Gaussian pulse afer specral shaping by he Sagnac-loop mirror for differen ΔL.The corresponding emporal domain waveforms afer wavelengh-o-ime mapping are also shown. In [7] and [8], he specral shaper for specrum shaping and he dispersive elemen for wavelengh-o-ime mapping are wo separae componens. In fac, he wo componens can be a single componen if he magniude response and he group delay response can be individually conrolled o perform simulaneously specrum shaping and wavelengh-o-ime mapping. In [9], a single chirped fiber Bragg graing was employed o perform he wo funcions, as shown in Fig. 8(a). The key componen in he sysem is he linearly chirped fiber Bragg graing, which should be designed o have a magniude as well as a group delay response ha can fulfill he requiremens for boh specral shaping and wavelengh-o-ime mapping. Differen echniques have been proposed o synhesize an FBG, such as he well-known discree layer-peeling (DLP) algorihm [4 6] and he Gelfand Levian Marchenko (GLM) inverse scaering algorihm [7]. Here a simplified approach is employed o synhesize he LCFBG [9]. Fig. 8(b) shows he magniude and he group delay response of a linearly chirped fiber Bragg graing. The magniude response has an increasing FSR and he group delay response is linear. Thanks o he inheren linear group delay response, a linearly chirped fiber Bragg a LCFBG Fig. 8. (a) A microwave arbirary waveform generaor basd on specral-shaping and wavelengh-o-ime mapping using a single linearly chirped fiber Bragg graing. (b) The magniude and phase responses of he linearly chirped fiber Bragg graing. MLL: modelocked laser, LCFBG: linearly chirped fiber Bragg graing, PD: phoodeecor. graing can always ac as a linear wavelengh-o-ime mapper. Therefore, he focus of he work is o synhesize he graing refracive index modulaion profile from he arge graing magniude response. The synhesis is performed based on an accurae mapping of graing reflecion response o he refracive index modulaion [9].We can firs se up he mapping relaionship by applying a linearly increasing index modulaion funcion o a es graing wih he use of a linearly chirped phase mask and hen measuring he graing reflecion specrum. A linear index modulaion funcion is firs consruced, which is expressed as " # z πz πc Δn L ðþ= z Δn max exp exp ð z L= Þ L Λ Λ b ð5þ Normalized ampliude Normalized ampliude Normalized ampliude Wavelengh λ (nm) a c Wavelengh λ (nm) e Wavelengh λ (nm) Normalized ampliude Normalized ampliude Normalized ampliude Time (ns) Time (ns) b d f Time (ns) Fig. 7. The specra of an opical Gaussian pulse afer specral shaping by he Sagnac-loop mirror and he corresponding emporal waveforms. (a) A symmerical FSR (ΔL =), (c) an increasing FSR (ΔL = 9.7 mm), and (e) a decreasing FSR (ΔL =6.9 mm). The generaed ime-domain waveforms wih (b) a symmerical chirp rae and a zero cenral frequency, (d) a negaive chirp rae and a cenral frequency of. GHz, and (f) a posiive chirp rae and a cenral frequency of 6. GHz.

7 Auhor's personal copy J. Yao / Opics Communicaions 84 () where Λ is he graing period a he cener of he linearly chirped fiber Bragg graing, L is he lengh of he linearly chirped fiber Bragg graing, and C is he graing chirp rae. We can imprin he index modulaion funcion in Eq. (5) ino he es graing using a given linearly chirped phase mask. Then he reflecion specrum R es (λ) of he fabricaed es graing is measured. By equally dividing he graing ino N consecuive segmens wih posiions z i ( i N), we can ge he sampled reflecion specrum R es (λ i ) hanks o he unique mapping relaionship beween Δn L (z i ) and R es (λ i ). For a graing wih a arge reflecion specrum R g (λ), we can compare i wih he es graing response R es (λ i ) wavelengh by wavelengh and hen deermine he desired index modulaion funcion Δn D (z i ) by querying he linear index modulaion funcion Δn L (z i ) segmen by segmen. Therefore, by applying he ampliudeonly index modulaion Δn D (z i ) under he same experimenal condiion, a desired linearly chirped fiber Bragg graing wih he arge reflecion specrum can be easily fabricaed wih he curren FBG fabricaion echnology. To improve he reconfigurabiliy, recenly an inegraed arbirary microwave waveform generaor ha incorporaes a fully-programmable specral shaper fabricaed on a silicon phoonic chip was demonsraed []. The specral shaper consiss of a cascade of mulichannel microring resonaors on a silicon phoonics plaform ha is compaible wih elecronic inegraed circui echnology. The sysem reconfigurabiliy is achieved by hermally uning boh he resonan frequencies and he coupling srenghs of he microring resonaors. Two generaions of he microring specral shaper were developed. In he firs generaion sysem, he resonaors have a ring srucure wih he cenral wavelengh of each resonaor independenly uned by a micro-heaer placed above he microring, and have a uning speed a millisecond o microsecond range. Since he heaing of he ring has lile impac on he coupling efficiency, which is crucial in conrolling he magniude profile of he specral shaper, a coupler wih a Mach Zehnder srucure in he inpu por of each microring was added in he second generaion sysem. By hermally uning he phase shif beween he wo arms, he coupling coefficien ino a ring can be adused. I was demonsraed ha for each resonan frequency, full uning from he on sae (no dip) o he off sae can be achieved. By incorporaing he specral shaper ino a phoonic arbirary microwave waveform generaion sysem, a variey of differen waveforms are generaed including hose wih an apodized ampliude profile, muliple π phase shifs, wo-one waveforms and frequency-chirped waveforms []. More recenly, we demonsraed a more flexible approach o he generaion of microwave arbirary waveforms using a single spaially discree chirped fiber Bragg graing (SD-CFBG) []. The SD-CFBG funcions o perform simulaneously specral slicing, frequency-oime mapping, and emporal shifing of he inpu opical pulse, which leads o he generaion of an opical pulse burs wih he individual pulses in he burs emporally spaced by he ime delays deermined by he SD-CFBG. Wih he help of a bandpass filer, a smooh microwave waveform is obained. The SD-CFBG is fabricaed using a linearly chirped phase mask by axially shifing he phoosensiive fiber o inroduce a spaial spacing beween wo adacen sub-graings during he fabricaion process. By properly designing he fiber shifing funcion, a large ime-bandwidh-produc microwave arbirary waveform wih he desired frequency chirping or phase coding can be generaed. The phoonic generaion of large TBWP microwave waveforms wih a linear, nonlinear and sepped frequency chirping was experimenally demonsraed. Insead of using a dispersive elemen wih only he second-order dispersion for linear wavelengh-o-ime mapping, a dispersive elemen wih boh he second- and hird-order dispersion can be used o achieve nonlinear wavelengh-o-ime mapping. For he case of chirped microwave waveform generaion, for example, if a dispersive elemen wih boh he second- and hird-order dispersion is employed, a chirped microwave waveform can be generaed using an opical specral shaper wih a uniform FSR, which would simplify he implemenaion [,]. If he dispersion up o he hird order is considered, hen a new wavelengh-o-ime mapping funcion would be used, which is given by [8] ω = Φ Φ Φ ; ð6þ where Φ is he hird-order dispersion. Assume ha he specral shaper is a Sagnac-loop filer, he ransfer funcion of he wo-ap Sagnac-loop filer is given by HðωÞ = ½ + cos ð ωτ ÞŠ ð7þ where τ is he ime-delay difference beween he wo aps. Based on wavelengh-o-ime mapping, a microwave waveform o be generaed is given " i ðþ= r ðþ # + cos Φ Φ τ Φ ð8þ where r() is again he pulse envelope. If he inpu shor pulse from he mode-locked laser is a Gaussian pulse, he oupu pulse envelope r() can be analyically expressed using he Airy funcion []. The insananeous microwave carrier frequency of he obained waveform can be wrien as ω RF ðþ= d d Φ Φ Φ τ = τ Φ Φ τ : ð9þ Φ As can be seen, he frequency of he microwave carrier is no consan, bu a funcion of ime. The microwave waveform is linearly chirped. 4. Temporal pulse shaping Microwave waveforms can also be generaed based on emporal pulse shaping (TPS) [9 ]. A TPS sysem usually consiss of a mode locked laser source, a pair of complemenary dispersive elemens and a modulaor. The modulaor can be a MZM [,] or a phase modulaor []. Fig. 9 shows a schemaic of a TPS sysem, in which he modulaor is a MZM. Again, if Δ = Φbb, where Δ is he emporal widh of he inpu shor pulse, he elecrical field of he shor pulse g() afer propagaing hrough he firs dispersive elemen wih a dispersion of Φ can be expressed as [4] p ðþ= g ðþexp = exp Φ exp Φ Φ " # = g ð τ τ Þ exp Φ dτ exp τ dτ Φ Φ gðτþ exp = exp Gðω Þ Φ ω = Φ gðτþ exp τ dτ Φ ðþ

8 Auhor's personal copy 7 J. Yao / Opics Communicaions 84 () 7 76 MLL g () where G(ω) is he Fourier ransform of g(). The signal a he oupu of he MZM, q ðþ= p ðþ x ðþ= exp Φ G Φ x ðþ ðþ where x() is he inpu microwave signal o he MZM. Afer propagaing hrough he second dispersive elemen ha has an opposie chromaic dispersion Φ, we obain he oupu emporal signal as he convoluion of q() wih he impulse response of he dispersive elemen exp = Φ, " # y ðþ= q ðþexp = Φ q ð τ τ Þ exp ð Þ dτ Φ = exp Φ qðτþ exp = exp Φ τ exp Φ exp τ dτ Φ Φ τ G τ Φ xðτþ exp exp τ dτ = exp G τ Φ Φ Φ exp τ dτ Φ = exp Φ =π Φexp Φ Sync F G Φ Φ MZM p () q () V bias xðþ x () Paern Generaor Fig. 9. Schemaic a TPS sysem for arbirary waveform generaion. ω = Φ gð ÞX Φ xðτþ τ Φ ðþ where F denoes he Fourier ransform operaion, * denoes he convoluion operaion, and X(ω) is he Fourier ransform of x(). As can be seen from Eq. () he oupu waveform is a convoluion beween he inpu opical pulse and he Fourier ransform of he inpu modulaion signal. If he inpu opical pulse is ulra shor, say, a uni impulse funcion, he convoluion of a funcion wih a uni impulse is he funcion iself, hen he generaed waveform is simply he Fourier ransform of he modulaion signal. Based on he propery of Fourier ransform, a slow waveform would lead o he generaion of a fas waveform wih narrow emporal widh. The maor challenge in implemening a TPS sysem is he complexiy in he modulaion sage since he modulaion signal is usually complex-valued. For example, o generae a non-symmerical waveform, based on Fourier ransform propery, he modulaion signal is complex valued. Therefore, in he modulaion sage an ampliude modulaor and a phase modulaor mus be employed, and he magniude and phase informaion mus be precisely synchronized, which makes he sysem exremely complicaed. Considering he fac ha he Fourier ransform of a real and symmerical Φ y () waveform is sill real and symmerical, i is possible o use an MZM ha is biased a he minimum ransmission poin o perform emporal specrum shaping wih a real signal ha has boh posiive and negaive values []. Wih his concep, waveforms such as square waves, recangular waves, riangular waves or double can be easily generaed. The TPS sysems in [ ] were employed for he generaion of opical waveforms. In fac, he TPS echnique can be exended for he generaion of high-frequency microwave waveforms. Recenly, he generaion of a microwave waveform wih a coninuously unable frequency by use of an unbalanced emporal pulse shaping sysem was proposed [4]. As can be seen from Eq. (), in a convenional TPS sysem he oupu waveform is he Fourier ransformaion of he inpu modulaion signal. If a second Fourier ransformaion is applied o he oupu waveform of a convenional TPS sysem, he finally generaed waveform would be a scaled version of he inpu modulaion signal. The second Fourier ransformaion can be performed by adding a hird dispersive elemen, he enire sysem is hen called an unbalanced TPS sysem. The schemaic of he sysem is illusraed in Fig.. In general, he values of he hird-order dispersion of he wo dispersive elemens are small and negligible, and only he secondorder dispersion or group velociy dispersion (GVD) is considered. The dispersive elemens can hen be characerized by he ransfer funcion given by H i ðωþ = exp Φ i ω = ; ði =; Þ, where Φ and Φ (ps ) are he dispersion of he wo dispersive elemens. In he unbalanced TPS sysem, he dispersion values should saisfy Φ Φ b, and Φ Φ. Therefore, he enire unbalanced TPS sysem can be modeled as a ypical TPS sysem wih a pair of complemenary dispersive elemens, followed by a residual dispersive elemen wih a ransfer funcion of H ðωþ = exp Φ ω =, where Φ = Φ + Φ is defined as he residual dispersion. Mahemaically, when a coninuous-wave x()=exp(ω m ) wih an angular frequency of ω m is applied o a MZM, he modulaed signal e IM () a he oupu of he MZM is given by e IM ()=exp(ω ) {exp [βx()]+exp[ βx()+ϕ ]}. Based on Taylor expansion, we have ½βxðÞ e IM ðþ= expðω Þ n n = Š n + e ϕ ½ βxðþ Š n ; ðþ n = n where β is he phase modulaion index, ϕ is a phase shif inroduced by he dc bias. To make he MZM operae a he minimum ransmission poin for he suppression of he opical carrier, a dc volage is applied o he MZM o inroduce a π phase shif (i.e., ϕ =π) beween he wo arms of he MZM. If he modulaion index β is sufficienly small, e IM () can be approximaed o be e IM () exp(ω ) [βx()]. Therefore, he modulaion funcion of he MZM biased a he minimum ransmission poin is βx(). I is known ha if he firs dispersive elemen has an adequae dispersion, i.e., Δ = Φbb, Δ is he pulse widh of he inpu opical pulse g(), he oupu signal of he ypical TPS sysem, s(), shown in Fig. (b), is he convoluion beween he inpu signal and he Fourier ransform of he modulaion funcion, s ðþ gðþe IM ðω Þ ω = = Φ = J ðβþ½g T ð Þ + g+ ð T ÞŠ ð4þ where E IM (ω) is he Fourier ransform of e IM (), * denoes he convoluion operaion, and T = ω m Φ = π. Therefore, wo imedelayed replicas of he inpu pulse are generaed a he oupu of he ypical TPS sysem, which correspond o he wo opical sidebands a he oupu of he DSB-SC modulaor. The elecrical field a he oupu of he enire unbalanced TPS sysem, y(), is obained by propagaing s() hrough he residual

9 Auhor's personal copy J. Yao / Opics Communicaions 84 () 7 76 a 7 b Φ = Φ + Φ Φ Φ = Φ + Φ Φ Φ s ( ) Fig.. (a) An unbalanced TPS sysem for he generaion of a microwave waveform wih unable carrier frequency. (b) The second linearly chirped fiber Bragg graing can be considered as a cascade of wo linearly chirped fiber Bragg graings wih he firs one having he opposie dispersion o LCFBG and he second one having a residual dispersion. dispersive elemen. If T = Φ bb is saisfied, hen y() can be approximaed by he real-ime FT of s() in he residual dispersive elemen [4], h i yð Þ exp = Φ SðωÞ ω = =Φ h i = exp = Φ J ðβþg = Φ cos T=Φ ð5þ where G(ω) is he Fourier ransform of he inpu pulse g(). The curren a he oupu of he PD is proporional o he inensiy of he inpu elecrical field, which is given by Ið Þ = Ryð Þ = K exp τ " + cos π T Φ # ð6þ where R is he responsiviy ofphe ffiffiffi PD, K = RJ(β)πΔ/ is a ime independen consan, and τ = ΔΦ = Δ is he oupu pulse widh. A frequency-muliplied microwave signal is generaed, which is a pulsed microwave signal wih a Gaussian envelope. The new carrier frequency is ωrf = πt = ΔΦ = ωm Φ = Φ ð7þ From Eq. (7) we can conclude ha he frequency muliplicaion facor M = ωrf = ωm = Φ = Φ is deermined by boh he sreching dispersion Φ and he residual dispersion Φ. Fig. shows he simulaion resuls of an unbalanced TPS sysem for he generaion of a microwave waveform wih unable carrier frequency. Noe ha he MZM in he unbalanced TPS sysem is biased a he minimum ransmission poin o suppress he opical carrier. If he MZM is biased a a he quadraure poin, double-sideband modulaion would be resuled. The use of double-sideband modulaion would cause he dispersion-induced power cancelaion, leading o a reduced fringe visibiliy, as shown in Fig.. If he dispersion up o he hird order is considered, hen a new wavelengh-o-ime mapping funcion would be used, which is given in Eq. (6), hen, he signal q() a he oupu of he MZM is given by qð Þ = eim ð Þ GðωÞ ω= Φ Φ Φ = ½βxð Þ GðωÞ ω= Φ : Φ Φ ð8þ Since he firs and second dispersive elemens have conugae dispersion and he hird-order dispersion of he dispersive elemen is very small, he signal s() a he oupu of he second dispersive elemen, shown in Fig. (b), is given by sð Þ=F ½qð Þ ω = Φ = Φ Φ β Φ g ð Þ X ðωþ ω = Φ Φ Φ ; ð9þ wih is specrum S(ω) given by Φ ω : SðωÞ = βgðωþ x Φ ω + ð4þ The oupu signal in he frequency domain is given by " # Φ ω Φ ω + Þ Y ðωþ = P ðωþ H ðωþ = P ðωþ exp 6 ð4þ where H(ω) is he ransfer funcion of he residual dispersive elemen, Φ and Φ are he second-order and he hird-order dispersion of he residual dispersive elemen. By using he nonlinear wavelengh-o-ime mapping funcion given in Eq. (6), he oupu signal in he ime domain is given by yð Þ = F ½sð Þ ω = Φ Φ Φ Φ ω GðωÞx Φ ω + ω = Φ b Φ Φ : ð4þ If he opical signal a he oupu of he hird dispersive elemen is applied o a PD, we have he oupu curren, given by " c ið Þ=Ryð Þ ½GðωÞ " = Fig.. Simulaion resuls. (a) Signal a he oupu of he ypical TPS sysem. (b) The opical specrum of he signal in (a). (c) The frequency-muliplied microwave signal a he oupu of he enire sysem. πτ " x τ exp 4 # Φ ω x Φ ω + Φ Φ Φ # Φ Φ Φ Φ + Φ Φ ω= Φ Φ Φ 4 # Φ Φ + ; 4 6 4Φ Φ Φ ð4þ

10 Auhor's personal copy 7 J. Yao / Opics Communicaions 84 () 7 76 Ampliude (a.u.) a b c Ampliude (a.u.) d e f Fig.. Simulaed oupu microwave waveforms based on double-sideband modulaion wih a carrier frequency of (a) GHz, (b) GHz, and (c) 6 GHz. and DSB-SC modulaion wih a carrier frequency of (d) GHz, (e) 4 GHz, and (f) GHz. where R is again he responsiviy of he PD. Usually, he hird-order dispersion, Φ, is much smaller han he second-order dispersion Φ, and he condiion T = Φ bb should be saisfied in he sysem where T = ω m Φ = π, we also have NN Φ and NN Φ 4,Eq.(4) Φ Φ 4 Φ 4Φ 6 is hen simplified o " i ðþ exp τ Φ # ( " Φ x + Φ Φ Φ #) Φ : 4 Φ Φ Φ Φ ð44þ To generae a chirped microwave pulse, we assume ha he microwave modulaion signal applied o he MZM is also a sinusoidal waveform wih a frequency of ω m, and he hird-order dispersion of he hree dispersive elemens canno be ignored. I can be seen from Eq. (44) ha he oupu signal is also a produc of a Gaussian-like funcion wih a chirped microwave waveform. Since he angular frequency should be kep posiive, he insananeous angular frequency of he chirped microwave waveform is approximaely given as ω RF ω m Φ + Φ Φ Φ Φ Φ Φ : ð45þ Eq. (45) shows ha a chirped microwave waveform can be generaed if boh he second and he hird dispersive elemens have non-zero hird-order dispersion. The chirp rae is given by Φ Φ CR RF =ω Φ Φ m sgn Φ Φ Φ ð46þ where sgn is a sign funcion. From Eq. (46), i can be seen ha he chirp rae is deermined by he values of he hird-order dispersion Φ and Φ. When =, he frequency is he cenral frequency of he generaed microwave waveform which is also equal o he angular frequency ω RF in Eq. (7). Therefore, a microwave waveform wih a unable cenral frequency and chirp rae can be generaed by uning he second-order dispersion and hird-order dispersion of he second and hird dispersive elemens. By varying he hird-order dispersion of a non-linearly chirped FBG (NLC-FBG) based on a srain-gradien beam uning echnique, he chirp rae can be coninuously unable. On he oher hand, if he frequency chirp in he generaed microwave waveform is no desired, he frequency chirp can be made zero by uning he dispersion o make Φ Φ Φ Φ equal o zero. Fig. shows he experimenal resuls of an unbalanced TPS sysem for he generaion of microwave waveform wihou and wih chirp [5]. In he experimen, he inpu microwave waveform is. GHz and he firs dispersive elemen is a dispersion compensaing fiber (DCF) wih a dispersion value of ps. A chirped fiber Bragg graing glued on a canilever beam was employed in he sysem conneced o he oupu of he MZM. When no force is applied o he canilever beam, he oal second-order dispersion and hird-order dispersion afer he MZM are ps and.4 ps. Since he unbalanced TPS sysem is a linear ime-invarian sysem, he values of he second-order dispersion of he hree dispersive elemens in he enire sysem are Φ =77:6ps, Φ = 77:6ps, and Φ =87:54 ps. Based on M = ω RF = ω m = Φ = ΔΦ, he muliplicaion facor is calculaed o be 5.8. Since he frequency of he inpu microwave waveform is. GHz, he cenral frequency of he generaed microwave waveform should be 6.99 GHz. In addiion, he values of he hird-order dispersion of he hree dispersive elemens in he enire sysem are Φ = 4:44ps, Φ =4:44ps,and Φ = :9 ps.the generaed microwave waveform is shown in Fig. (a). The cenral frequency is measured o be 6.75 GHz, as shown in Fig. (c), which agrees well wih he heoreically calculaed value of 6.99 GHz. Since he hird-order dispersion afer he MZM is.9 ps,accordingo Eq. (46), he chirp rae is calculaed o be 4.75 GHz/ns which is close o he chirp rae of 5.5 GHz/ns esimaed from he generaed waveform shown in Fig. (a). When a force is applied o he canilever beam, he values of he second-order dispersion and he hird-order dispersion afer he MZM are 57. ps and.8 ps. The values of he second-order dispersion of he hree dispersive elemens in he enire sysem are Φ = 77:6ps, Φ = 77:6ps, Φ = 55:6ps. Again, he MF is calculaed o be 6.4. Since he frequency of he inpu microwave waveform is. GHz, a cenral frequency of he generaed microwave waveform should be 7.85 GHz. The cenral frequency of he generaed waveform esimaed from he waveform shown in Fig. (b) is 7.5 GHz, which is close o he heoreical value of 7.85 GHz. The values of he hird-order dispersion of he hree dispersive elemens in he

11 Auhor's personal copy J. Yao / Opics Communicaions 84 () a c b Fig.. Measured microwave waveforms for a unbalanced sysem using a chirped FBG ha is glued on a canilever beam (a) wih no srain, (b) wih srain, and (c) he corresponding frequency chirps wih and wihou mechanical force applied o he free end of he canilever beam. enire sysem are Φ = 4:44ps, Φ =4:44ps, Φ =5:75ps. The chirp rae calculaed based on Eq. (46) is.868 GHz/ns, which is close o he chirp rae of.75 GHz/ns obained from he generaed chirped waveform shown in Fig. (b). In addiion, he chirp rae of he experimenally generaed microwave waveform is also reversed due o he change of he sign of he hird-order dispersion, as shown in Fig. (c). 5. Microwave waveform generaion based on a phoonic microwave delay-line filer The implemenaion of microwave delay-line filers has been a opic of ineres in he las wo decades [6,7]. To avoid opical inerferences which are exremely sensiive o environmenal changes, a phoonic microwave delay-line filer is usually implemened in he incoheren regime based on incoheren deecion. The maor limiaion of an incoheren microwave delay-line filer is ha he ap coefficiens are all posiive, which would limi he filer o operae as a low-pass filer only. Numerous echniques have been proposed o implemen a microwave delay-line filer wih negaive coefficiens, including he use of differenial deecion [8], biasing a pair of MZMs a he opposie slopes [9], cross-gain modulaion in a semiconducor amplifier [4], polarizaion modulaion [4], and phase-modulaion o inensiy-modulaion conversion [4]. A comprehensive overview of hese echniques can be found in [4]. The use of a microwave delay-line filer for he generaion of UWB impulse waveform has been demonsraed [44 47]. However, all he filers implemened based on he echniques in [8 4], including he filers for he generaion UWB waveforms [44 47], have a linear group delay response. For microwave waveform generaion, such as he generaion of a chirped microwave waveform, a group delay response ha is nonlinear is usually required. I is known for a delay-line filer wih nonlinear group delay response, he filer should have ap coefficiens ha are complex, which is exremely difficul o implemen especially for a filer wih many aps [48 5]. Recenly, we demonsraed a new concep o implemen a microwave delay-line filer wih arbirary group delay response based on a delayline srucure wih nonuniformly spaced aps [5]. We heoreically proved ha a filer wih an arbirary frequency response in a specific resonance band can be achieved by inroducing addiional ime delays o he aps, making he aps nonuniformly spaced. I is known ha a regular, uniformly-spaced microwave delay-line filer has an impulse response h () given by N h ðþ= α k δð ktþ ð47þ k = where N is he ap number, α k is he ap coefficien of he kh ap, T=π/Ω is he ime-delay difference beween wo adacen aps, and Ω is he FSR of he filer. The expression of h () can be expressed in anoher form h ðþ= a ðþs ðþ where a() is he coefficien profile which can be complex valued, α k = akt ð Þ and a ðþ=ifbor NT and s() is he sampling funcion given by k ð48þ ð49þ s ðþ= δð ktþ: ð5þ Apply he Fourier Transfer o Eq. (47), we have he frequency response of he filer, H ðω N Þ = k = α k exp k π Ω ω : ð5þ I is known ha H (ω) has a muli-channel frequency response wih adacen channels separaed by an FSR, wih he mh channel locaed a ω=mω. In a regular phoonic microwave delay-line filer based on incoheren deecion, he coefficiens are usually all posiive, or special designs have o be incorporaed o generae negaive or complex coefficiens [4]. However, a phase erm can be inroduced o a specific coefficien by adding an addiional ime delay a he specific ap, which is ermed ime-delay-based phase shif [5]. For example, a ω=mω a ime-delay shif of Δτ will generae a phase shif given by Δφ= Δτ mω. Noe ha such a phase shif is frequency-dependen, which is accurae only for he frequency a mω, bu approximaely accurae for a narrow frequency band a around mω. For mos of he applicaions, he filer is designed o have a very narrow frequency band. Therefore, for he frequency band of ineres, he phase shif can be considered consan over he enire bandwidh. As a resul, if he mh bandpass response, where m, is considered, one can hen achieve he desired phase shif a he kh ap by adusing he imedelay shif by Δτ k. Considering he ime-delay shif of Δτ k, one can ge he frequency response of he nonuniformly-spaced delay-line filer a around ω=mω, N H N ðωþ = ω α k exp k π k = Ω + Δτ k N = α k exp ð ωδτ k Þ exp k π k = Ω ω N ½α k expð mωδτ k ÞŠ exp k π Ω ω : k = ð5þ

12 Auhor's personal copy 74 J. Yao / Opics Communicaions 84 () 7 76 As can be seen from Eq. (5), one can ge an equivalen phase shif for each ap coefficien. Specifically, if he desired phase shif for he kh ap is φ k, he oal ime delay τ k for he kh ap is τ k =kt φ k /mω. As a resul, if he ime delay of each ap is adused, he filer coefficiens would have he required phase shifs o generae he required passband wih he desired bandpass characerisics. For a nonuniformly spaced delay-line filer wih a specral response a he mω ha is idenical o ha of a regular delay-line filer wih a coefficien profile given by a(), he ime delays and he coefficiens can be calculaed by [5] τ k + φτ ð kþ = kt ð5aþ mω aðτ β k = k Þ +φ ðτ k Þ= mω ð5bþ where φ() is he phase erm of a(). As can be seen o realize he desired bandpass response, if a regular delay-line filer is used, he ap coefficiens are deermined by Eq. (49), which may be negaive or complex valued; however, if a nonuniformly spaced delay-line filer is used, he ime delays and he ap coefficiens are deermined by Eqs. (5a) and (5b), and he filer has posiive-only coefficiens. Since he filer can be designed o have an arbirary specral response, he employmen of he filer for he generaion of an arbirary microwave waveform can be realized. Fig. 4 shows a delay-line filer for he generaion of phase-coded microwave waveform [5,5]. To generae he required phase code using a delay-line filer wih all posiive coefficiens, he filer should have nonuniformly spaced aps. The desired phase coding is implemened by adusing he ime-delay differences beween he adacen aps. According o Eq. (5a) we have τ k = T k φ k πm = km φ k T π ; k =; ; ; ; N ð54þ where T is he ime-delay difference beween wo adacen aps for a uniformly spaced filer, and T is he period of he microwave carrier. For example, for a four-ap filer we have N=4.Ifm=4 is seleced, for a code paern of {, π, π, }, he ime delays should be {, 7/8 T, 5/ 8 T, T}. As a design example, a binary phase-coded microwave signal wih a -chip Barker code, [+, +, +, +, +,,, +, +,, +,, and +], is generaed [5]. The Barker codes are usually used in direc-sequence spread-specrum communicaions sysems and pulse compression radar sysems hanks o he excellen correlaion performance. In he design, he carrier frequency is 4 GHz and m=4, i.e., he chip rae is GHz, and T is 5 ps. Based on Eq. (54), he ime delays of all he aps are calculaed which are given [, 4, 8,, 6, 9.5,.5, 8,, 5.5, 4, 4.5, and 48] 5 ps. If he inpu microwave signal is a super-gaussian pulse wih a FWHM of ps, he oupu of he filer is calculaed which is shown in Fig. 5. Fig. 5. Phase-coded microwave pulse wih a -chip Barker code generaed by a nonuniformly spaced delay-line filer. A chirped microwave waveform can also be generaed by using a non-uniformly spaced delay-line filer [5,54]. If a broadband chirpfree microwave pulse is passing hrough a microwave delay line filer wih a quadraic phase response or equivalenly, a linear group-delay response, a pulse burs wih increasing or decreasing pulse spacing is generaed. As i was proved in [] ha a pulse-posiion-modulaed pulse burs would have a muli-channel specral response, wih one channel having a specrum corresponding o he desired phase modulaed microwave signal. By using a microwave bandpass filer o selec he channel of ineres, a frequency-chirped microwave signal would be generaed. Fig. 6 shows a delay-line filer ha can be designed o have nonuniformly spaced aps o produce a quadraic phase response. Noe ha insead of using an MZM, a phase modulaor is employed. I was demonsraed ha he phasemodulaion o inensiy-modulaion conversion in a dispersive elemen will generae a noch a he dc, which is used o eliminae he resonance a he dc. The nonuniform ap spacing can be achieved in he sysem by uning he wavelenghs spacing. For example, if he oal dispersion of he dispersive elemen is χ(ps/nm), hen he wavelenghs of he muliple wavelengh sources can be calculaed by λ k = λ + τ k τ χ ð55þ where λ is he wavelengh for he h ap. The generaion of a chirped microwave pulse using a five-ap nonuniformly spaced microwave FIR filer was experimenally demonsraed. The desired pulse has an FWHM of 55 ps, a mean period of ps, and a chirp rae of. GHz/ns, as shown as he doed line in Fig. 7(a). The inpu pulse is a chirp-free Gaussian pulse generaed in he experimen by a paern generaor wih a FWHM of 65 ps. The five-ap delay line filer is designed based on (5). The ime delays of he aps, calculaed based on Eq. (54), are T [.,.8,.,.8, and.5], where T= ps. The wavelenghs from he muli-wavelengh source are hen calculaed based on (55). In he experimen, λ is 54. nm, and he five wavelenghs are se as [54.7, 54., 54., 54.5, and 54.69] nm. In he experimen, he oupu power of each laser is conrolled such ha Fig. 4. A phoonic microwave delay-line filer for microwave phase coding. Fig. 6. A nonuniformly-spaced delay-line filer for he generaion of a chirped microwave waveform. The baseband resonance is eliminaed by he noch a dc due o he PM-IM conversion. PM: phase modulaor. PD: phoodeecor.