Cross-phase modulation due to a cascade of quadratic interactions in a PPLN waveguide

Cross-phase modulation due to a cascade of quadratic interactions in a PPLN waveguide
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  IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 12, NO. 3, MAY/JUNE 2006 405 Cross-Phase Modulation Due to a Cascade of Quadratic Interactions in a PPLN Waveguide Carlo Liberale, Ilaria Cristiani,  Member, IEEE , Vittorio Degiorgio, Marco Marangoni, Gianluca Galzerano,and Roberta Ramponi  Abstract  —Thecross-phasemodulationofaweaksignalhasbeenobtained in a periodically poled lithium–niobate waveguide as aresult of the quadratic interaction of the signal with the secondharmonic of an intense pump field. The nonlinear phase acquiredby the signal has been measured as a function of the pump fieldintensity and of the phase mismatch with the second harmonicfield. The measurements have been performed with a properly de-signedexperimentalsetupbasedonaMach–Zehnderinterferomet-ricscheme,whichprovedtobequiteinsensitivetoslowmechanicaland thermal drifts. Numerical simulations are in good agreementwith experimental results.  Index Terms —Frequency conversion, lithium niobate, Mach–Zehnder interferometers, nonlinear optics, optical waveguides,phase measurement. I. I NTRODUCTION I N the last few years, several studies have reported about thegeneration of new optical frequencies through a cascade of two second-order interactions in a nonlinear crystal [1]–[3]. Animportant application of such a process is found in wavelengthconversion devices using periodically poled lithium–niobate(PPLN) waveguides [4]–[7]. The cascade here considered in-volves second-harmonic generation (SHG) of a pump field  p at frequency  ω p  followed by difference-frequency generation(DFG) between the second harmonic of   p  and an additionalinput field  s  (signal) at frequency  ω s . The overall result is theamplification of   s  and, at the same time, the generation of anew field  c  (converted signal) at frequency  ω c  = 2  ω p − ω s . Toperform cascading in a quadratic medium is more convenientthan carrying out four-wave mixing (FWM) in a third-order material because the effective nonlinearity can be considerablylarger, and, in addition, it permits phase-matched waveguidedinteractions at the desired wavelength. Since cascading pro-duces effects equivalent to that of a third-order process, oneshould expect not only signal amplification and generation of a frequency-converted signal, but also a nonlinear phase shift(equivalent Kerr effect) coming from the real part of the ef-fective third-order susceptibility. Indeed, many authors [8]–[12]have presented theoretical and experimental results concerningthe cascade-induced Kerr effect, considering the case of the Manuscript received July 26, 2005; revised January 16, 2006. This workwas supported in part by the FIRB-MIUR Project “Miniaturized Systems for Electronics and Photonics” and in part by the ISTC project A-1033.C. Liberale, I. Cristiani, and V. Degiorgio are with the Dipartimento di Elet-tronica, Universit`a di Pavia, 27100 Pavia, Italy (e-mail:; Marangoni, G. Galzerano, and R. Ramponi are with the Dipartimento diFisica, Politecnico di Milano, and IFN-CNR, 20133 Milan, Italy.Digital Object Identifier 10.1109/JSTQE.2006.871928 “degenerate” cascade process in which  ω p  =  ω s . In such a sit-uation, the strong pump beam  p  is autoinducing the nonlinear phase shift and so the process can be described as a case of self-phase-modulation exploiting quadratic nonlinearities. Re-cently numerical solutions of the propagation equations for thenondegenerate cascade have been presented, indicating that asignificant nonlinear phase shift  φ NL  can be induced on the sig-nal field by the presence of a strong pump, provided that theSHG process is phase matched and the DFG process is appro-priately mismatched [13]. An important aspect of the proposedscheme is that both pump and signal frequencies are within theoptical-communication bandwidth. It should be mentioned thatacross-phasemodulationduetothecascadeofsum—frequencygeneration plus DFG was predicted in [14], and that the nonlin-ear phase shift induced by a DFG process was observed in theexperiment described in [15] by using a pump frequency welloutside the optical-communication bandwidth.In this paper, we describe in detail a cascading experiment,performed in a PPLN waveguide, in which  s  is a CW signaland  p  is a train of picosecond pulses. The nonlinear phase shift φ NL  induced on  s  by the pulsed pump  p  is measured by us-ing a Mach–Zehnder interferometer (MZI). The experiment isespecially designed to be insensitive to slow mechanical andthermal drifts. We find that  φ NL  can be as large as  π/ 6  by usingan average pump power of 15 mW and a duty cycle of the order of 0.001. The observed dependence of   φ NL  on the pump power and on the frequency mismatch between pump and signal isin good agreement with the numerical solutions of the propa-gation equation. A short preliminary version of this work hasrecently appeared [16]. This paper is organized as follows. InSection II, we present numerical solutions of the propagationequations by using the experimental parameters. Sections IIIand Sections IV contain a description of the principle of theinterferometric measurement and of the experimental setup, re-spectively. Section V presents and discusses the experimentalresults, while Section VI is devoted to conclusion.II. C ASCADED  I NTERACTIONS IN THE  PPLN W AVEGUIDE The interaction we are considering is the cascade of a SHGprocess of a pump field  p  at frequency  ω p  followed by DFGbetweenthesecondharmonicof   p andaninputfield s (signal)atfrequency  ω s . We assume that the overall process is performedin a LN waveguide. By taking  z  as the waveguide axis, thegeneric field involved in the process can be written as E   j ( x,y,z,t ) = 12 A  j ( z,t )  e i ( ω  j  t − k  j  z ) + c . c .  γ   j ( x,y )  (1) 1077-260X/$20.00 © 2006 IEEE  406 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 12, NO. 3, MAY/JUNE 2006 where  A  j  is the electric field amplitude and  γ   j ( x,y )  is thetransverse profile of the waveguide mode with frequency  ω  j and wavenumber   k  j .The cascade of the two processes can be modelled by thefollowing coupled equations: ∂A P ∂z  +  β  1P ∂A P ∂t  +  i 2 β  2P ∂  2 A P ∂t 2 =  − α P 2  A P  +  iC  0P A ∗ P A SH e ( − i ∆ kz ) ∂A SH ∂z  +  β  1SH ∂A SH ∂t  +  i 2 β  2SH ∂  2 A SH ∂t 2 =  − α SH 2  A SH  +  iC  0P A 2P e ( i ∆ kz ) +  iC  0SH A S A C e ( i ∆ hz ) ∂A S ∂z  +  β  1S ∂A S ∂t  +  i 2 β  2S ∂  2 A S ∂t 2 =  − α S 2  A S  +  iC  0S A SH A ∗ C e ( − i ∆ hz ) ∂A C ∂z  +  β  1C ∂A C ∂t  +  i 2 β  2C ∂  2 A C ∂t 2 =  − α C 2  A C  +  iC  0C A SH A ∗ S e ( − i ∆ hz ) (2)where  β  1  j  and  β  2  j  are, respectively, the first and the secondderivative of   k  j  with respect to  ω , both evaluated at  ω  j ; theparameters  α  j  are the linear losses; and the parameter   C  0  j  isdefined as  C  0  j  = ( π/ 2)( cε 0 /λ  j ) χ (2)eff   I,χ (2)eff   being the effectivesecond-order susceptibility and  I   the overlap integral betweenthe interacting fields [13]. The quantities  ∆ k  and  ∆ h  represent,respectively, the mismatch for the SHG and the DFG processesand are defined as ∆ k  = 2 k P  −  k SH  + 2 π Λ∆ h  =  k S  +  k C  −  k SH  + 2 π Λ where  Λ  is the poling period of the crystal.A coordinate transformation is applied by introducing a newvariable  T   =  t  −  zβ  1P , where  1 /β  1P  represents the group ve-locity of the pump pulse. Through this transformation we im-pose that the reference system travels at the same velocity as thepump pulse.The cascaded interaction described by (2) converts two pho-tons from the pump field into one photon at wavelength  λ s  andone photon at  λ c . As such, it generates the new field  c , and, atthe same time, it amplifies the input field  s . While the previousanalysis discusses the conversion efficiency and the amplifica-tion of   s  without paying attention to the phase of the outputfields [17], the simulation presented in [13], and also reportedhere,calculatesexplicitlythenonlinearphaseshift φ NL  inducedon the signal field by the cascaded interaction.Equations (2) have been numerically solved by the split-stepFourier method by assuming  γ  P  and  γ  S  to be identical due tothe closeness of   ω s  and  ω p . At the waveguide input  ( z  = 0) the signal field is stationary, while the pump field is pulsed and Fig. 1. Calculated time behavior of (a) the optical gain, defined as the ratiobetween the output and the input signal field amplitude and (b) the nonlinear phase shift experienced by the signal for three different values of the signalpump detuning:  ∆ λ  =  λ p  −  λ s  = 58  nm, 35 nm, and 2 nm. described by function A P (0 ,t ) =  A 0 sech   tT  o  exp  − iCt 2 2 T  o   (3)where  C   is the chirp parameter and  T  o  is related to theFWHM pulsewidth  τ  p  by  τ  p  = 1 . 763  T  o . Numerical simula-tions have been performed using the parameters correspond-ing to the experimental conditions described in the followingsections( ∆ k  = 0 ,C   = 0 . 45 ,τ  p  = 7 . 65 ps,waveguidelength  =1 . 9  cm, normalized SHG efficiency  = 16 % W − 1 · cm − 2 , phase-matching wavelength  λ  = 1570  nm, waveguide attenuation α  = 0 . 3  dB/cm at  ω p ,ω s ,ω c  and 0.6 dB/cm at  ω SH ).Sincethepumpispulsed,thenonlinearphaseshiftinducedonthe signal  s  by  p  is time dependent. Fig. 1(a) and (b) shows thetime behavior of, respectively, the gain and nonlinear—phaseshift experienced by the signal for three different values of thesignal pump detuning:  ∆ λ  =  λ p  −  λ s  = 58  nm (corresponds tothe maximum phase shift),  ∆ λ  = 35  nm, and  ∆ λ  = 2  nm. Thedistortion in the phase that is more clearly visible at the smallest ∆ λ  can be ascribed to the chirp of the pump pulses.The behavior of the  φ NL  peak value as a function of thedetuning between pump and signal for different pulse durationsisshowninFig.2,atafixedinput-pumppeakpower  P  p  = 10 W.The figure shows that the phase shift can be rather large andattains the maximum value when the detuning is 58 nm. Onecan also notice that the result obtained by using 7-ps pumppulses is very close to that obtained with longer pulses, thusattesting that, at the chosen pulse duration, the efficiency of thenonlinear interaction is not significantly decreased.  LIBERALE  et al. : CROSS-PHASE MODULATION DUE TO A CASCADE OF QUADRATIC INTERACTIONS 407 Fig. 2. Calculated peak value of the nonlinear phase shift plotted versus thewavelength difference between pump and signal for various pulse durations.Fig. 3. Schematic of the Mach–Zehnder configuration used for the determina-tion of the optical gain and of the nonlinear phase. III. P RINCIPLE OF THE  I NTERFEROMETRIC  M EASUREMENT Inordertomeasurethenonlinearphaseacquiredbythesignalfield, an interferometric scheme, which is naturally phase sensi-tive, was employed. The Mach–Zehnder configuration sketchedinFig.3waschosensinceitallowsastraightforwardcouplingof bothpumpandsignalbeamsintothewaveguide.TheMZIoutputis given by the superposition of the field coming from the refer-ence arm  ˆ E  r  =  E  r  exp[ i ( ω s t − ϕ r )]  with the field at the waveg-uide output  ˆ E  s  =  gE  s  exp[ i ( ω s t − ϕ s − φ NL )] , whose phaseand amplitude are affected by the nonlinear interaction withthe pump field. In these conditions, the power at the MZI outputis given by P  out  = 12  P  s  + P  r  + P  s  ( g 2 − 1) + 2 gF   cos(Φ + φ NL )  (4)where P  s  and P  r  are, respectively, the signal power at the outputofthewaveguideandofthereferencearmoftheMZI, Φ =  ϕ s − ϕ r  isthephaseimbalance, F   =   P  r /P  s k isaparameterrelatedto the fringe visibility, and k ( ≤ 1)  takes into account the spatialoverlapbetweenthetwofields.Thequantities P  s ,P  r ,(and,thus, F  ) are time independent, whereas the third term on the right-hand side of (4) is a train of pulses at the pump repetition rate,containing the two unknown time-dependent quantities,  g  and φ NL .The maximum sensitivity of   P  out  with respect to  φ NL  isobtained for the conditions  Φ = ± π/ 2 , which are the pointsselectedforthemeasurement.Inthesepoints,theinterferometer  Fig. 4. SHG efficiency versus wavelength. output reads P  out  Φ = ± π 2  = 12  P  s  + P  r  + P  s  ( g 2 − 1) ∓ 2 gF   sin( φ NL )  .  (5)From the analysis of (5), one can notice that the interferometricscheme permits to extract information on both the amplitudeand the sign of the nonlinear phase shift.The determination of   g  and  φ NL  is performed after   P  s ,P  r ,and  F   are preliminarily measured in the absence of   p . In par-ticular,  g  is derived by taking the sum of   P  out (Φ = + π/ 2)  and P  out (Φ = − π/ 2) , while  φ NL  is obtained, once  g  is known, bytaking the difference between the same two signals. Since theoperating point of the interferometer is affected by thermal andmechanical drifts and fluctuations, a stabilization system has tobe implemented in order to obtain reliable measurements.IV. E XPERIMENTAL  S ETUP Thewaveguideusedintheexperimentsis7- µ mwideand1.9-cm long, and it was fabricated by annealed proton exchange inPPLNwithaQPMperiodof19.2 µ m.Theprotonexchangewasrealized in a 1% diluted benzoic-acid melt at 247 ◦ C for 49 minand annealing in air at 350 ◦ C for 5 h. With these parametersthe waveguide results in full  α  crystallographic phase [18], it issingle mode in the telecom spectral window with losses below0.5 dB/cm, and it allows noncritical phase matching for the in-teraction  TM ω 00  → TM 2 ω 10  [19]. The nonlinear characterizationof the waveguide in CW regime at a temperature of 100 ◦ C, i.e.,that used in the experiments, leads to the tuning curve shown inFig. 4: phase matching occurs at the wavelength  λ  = 1570  nm,the normalized conversion efficiency is 16% W − 1 cm − 2 , andthe interaction length equals the sample length, i.e., 1.9 cm.The waveguide is positioned in the upper arm of the MZI asshown in Fig. 5. A pair of 20X microscope objectives are usedforcoupling/outcouplinginto/fromthewaveguidethepumpandthe signal radiation. The pump is provided by a parametric os-cillator driven by a mode-locked Nd:YAG laser, delivering atrain of 7-ps-wide (at FWHM) pulses with a repetition rate  408 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 12, NO. 3, MAY/JUNE 2006 Fig. 5. Scheme of the experimental setup. PC: polarization controller; BS:beam splitter; PBS: polarizing beam splitter; M: mirror; O: objective; IF: inter-ference filter; Pd: photodiode; HWP: half-wavelength retardation plate; PZT:piezoelectric transducer; PI: proportional and integral servo electronics; HVA:high-voltage amplifier; Σ : adder. f  o  = 187 MHz.Thepumpbeamisverticallypolarizedalongthewhole beam path. The wavelength  λ p  is maintained at 1570 nm,which is the phase-matching wavelength of the SHG process.The signal beam is generated by a CW extended-cavity diodelaser (SANTEC), whose wavelength can be tuned in the range λ s  = 1500  –  1550  nm. By acting on the polarization controller PC, the signal power can be distributed in arbitrary proportionsbetween the two arms of the MZI. The horizontal polarizationcomponent of the signal (double-ended arrows) is, in fact, de-viated by a polarizing beam splitter (PBS) to the lower arm of the MZI, while the vertical component (circle and dot) is trans-mitted to the upper arm containing the waveguide. A half-waveplate (HWP) is then used to rotate by 90 ◦  the polarization inthe lower arm so as to obtain recombination between equallypolarized signals at the output of the MZI.The beam-splitter BS2 that closes the MZI provides two sep-arated signal outputs whose phases differ by 180 ◦ . One outputis sent to a high-speed photodiode  Pd HF  (1-GHz bandwidth,ThorlabsD400FC)connectedtoanelectricalspectrumanalyzer (HP 8563A) through a low-noise amplifier (Miteq AU1291).The spectrum analyzer detects the first Fourier component (187MHz) of the output signal, which carries the information on  g and  φ NL . The second MZI output is sent to a low-bandwidthphotodiode  Pd LF , allowing the stabilization of the interferom-eter working point. Both photodiodes are protected by tunableinterference filters (IFs) that select the signal at  λ s , while reject-ing the residual pump and the converted beam at wavelength λ c .The stabilization setup makes use of the well-known fringe-side stabilization method [20] to keep the interferometer outputpower to the desired working point, corresponding to the meanvalue between the minimum and maximum output power. Thelow-frequency signal provided by  Pd LF , which is generallyaffected by thermal and mechanical drifts, is compared in a dif-ferential amplifier with the reference voltage corresponding tothe selected working point. The output of the differential ampli-fier represents the error signal of a suitable feedback circuit thatmodifies, through a piezoelectric actuator (PZT), the position of the M mirror of the MZI lower arm. In this way, the optical pathdifference between the arms of the interferometer is controlledand stabilized. The feedback circuit is composed of a propor-tional and integral stage (PI), an inverter amplifier with unitarygain  ( G  = − 1) , and a high-voltage amplifier (HVA). The PIstage has a 60-dB low frequency gain and a 0-dB crossing pointatafrequency( ∼ 2 kHz)sufficientlylowerthanthefirstmechan-ical resonance of the PZT-mirror assembly (located at ∼ 4  kHz).The unitary gain inverter amplifier can be added or excludeddepending on the phase of the selected locking point, either  Φ =  π/ 2  or  − π/ 2 . To characterize the whole time-independentinterferometer parameters, namely  P  s ,P  r , and  F  , the interfer-ometerfringesarerecordedbyapplyingalinearscantothePZTin open loop conditions.V. R ESULTS AND  D ISCUSSION As pointed out above, by measuring  P  out (Φ = ± π/ 2)  in thepresence of   p , the quantities  φ NL  and  g  both can be determined.In practice, the direct measurement of their peak values is notfeasiblesincetheduration τ  p  oftheoutputpulsesismuchshorter than the response time of the detection system. By sending theoutput of   Pd HF  to an electrical spectrum analyzer, we obtainthe amplitude of the fundamental component  f  o  = 187  MHz.As shown below, such amplitude corresponds with an excellentaccuracy to the time average of the third term in (5), that is,  ∆ P  out  =  P  s  g 2 ( t ) − 1 ± 2 Fg ( t )sin[ φ NL ( t )]  .By considering, for sake of simplicity, a Gaussian envelopefor the output pulses, we can write the time-dependent compo-nent of (5) as P  ( t ) =  S  p  exp  − t 2 T  2o  ⊗ + ∞  n = −∞ δ   t −  nf  o   (6)where ⊗ is the convolution operator,  S  p  is the peak power of theoutput pulses, and  T  o  can be linked with good approximation to τ  p  through the relation  τ  p  = 1 . 665 T  o .The electrical spectrum analyzer performs the Fourier trans-form of (6), which can be written as P  ( f  ) =  S  p T  o f  o √  π  exp  − ( πT  o f  ) 2  · + ∞  n = −∞ δ  ( f   − nf  o ) . (7)Takingintoaccountthatthepulsebandwidthismuchlargerthan f  o , that is,  f  o T  o   1 , one can easily see that the  f  o  componentof   P  ( f  )  has almost the same amplitude as the zero-frequencycomponent. Indeed in our case we find a difference of the order of   10 − 5 between the amplitude of the two spectral components.Hence, from the measurement of the spectral amplitude at187 MHz one can reliably obtain the time average of the thirdterm in (5). It is important to notice that the photodetector band-width does not affect the measurement as it is much larger than  f  o .  LIBERALE  et al. : CROSS-PHASE MODULATION DUE TO A CASCADE OF QUADRATIC INTERACTIONS 409 Fig. 6. Measured average values of the time-dependent output of the MZI,plotted as a function of   λ p  − λ s , for the three cases:  Φ =  π/ 2  (full circles), Φ =  − π/ 2  (triangles), blocked reference arm (crosses). Open circles representthe mean value between the  Φ =  ± π/ 2  measurements. The average pumppower at the waveguide input is  P  inp  = 10 . 3  mW. Fig. 6 shows the behavior of    ∆ P  out  /P  s  measured as afunction of the wavelength difference  λ p  − λ s  for   Φ =  π/ 2 (full circles) and  Φ =  − π/ 2  (triangles) by keeping fixed theaverage pump power at the waveguide input  P  inp  = 10 . 3  mW.Since the direct value of   P  inp  is not experimentally accessible,the measured quantity is the average SH power at the output of the waveguide  P  outSH  . Once  P  outSH  is known, the value of   P  inp  isderived  a posteriori  by running the propagation equations withthe assigned value for   P  outSH  .It can be seen that   ∆ P  out (Φ = + π/ 2)  /P  s  is consider-ably different from   ∆ P  out (Φ =  − π/ 2)  /P  s . This fact, by it-self, constitutes clear evidence of the existence of a signifi-cant nonlinear phase shift. The figure also shows the quan-tity   g 2 ( t )  −  1  , derived in our experiment in two differentways: the open circles represent the average of the two terms,  ∆ P  out (Φ =  ± π/ 2)  /P  s ,whereasthecrossesareobtainedbyadirect measurement performed with the reference arm blocked.The good agreement between the two approaches is a proof of the reliability of the interferometric measurements, resultingin insensitivity to slow mechanical and thermal drifts. The dif-ference between the two terms   ∆ P  out (Φ =  ± π/ 2)  /P  s  givesthe quantity   2 Fg ( t )sin[ φ NL ( t )]  , from which we derive thenonlinear phase.Inordertoretrievethepeakvaluesof  g  and φ NL  startingfromthe average quantities, we assume, consistently with the numer-ical solutions of the propagation equations, that the temporalprofile of the output pulses reproduces with good approxima-tion that of the pump input pulses. Fig. 7 presents the peakvalues of the gain (full circles) and of the nonlinear phase shift(squares)obtainedbyusinganaveragepumppowerof10.3mW,as derived from the data of Fig. 6, plotted versus  λ p  − λ s , to-gether with the solutions of the propagation equations (lines).The agreement between experiment and simulation is quite sat-isfactory for all the data.Themeasuredgainisadecreasingfunctionofthewavelengthdifference λ p  − λ s , with an HWHM bandwidth of about 50 nm.The nonlinear phase presents a maximum at  λ p  − λ s  ∼  55  nm,corresponding to  ∆ βL  ∼ =  π , where  ∆ β   is the mismatch in thepropagation constants for the DFG process. Fig. 7. Signal peak gain  ( • )  and peak nonlinear phase  (  )  shift plotted asfunctions of  λ p  − λ s , for  P  inp  = 10 . 3  mW. Lines represent the solutions of thepropagation equations.Fig. 8. Maximum nonlinear phase shift as a function of the input pump power.Comparison between the numerical simulation (line) and experimental mea-surements (circle). Fig. 8 shows the maximum nonlinear phase measured for three different input pump powers, together with the results of numericalsimulations.Thenonlinearphasereachesa π/ 6 valuefor   P  p  = 15  mW and increases with the input pump power invery good agreement with numerical simulations. Taking intoaccount that the optimization of waveguide parameters can leadto an efficiency much larger than that available in our experi-ment, the presented results are quite promising for the realiza-tion of a nonlinear integrated MZI for an all-optical switchingpurpose.VI. C ONCLUSION In this paper, we have provided numerical and experimentalevidence of the possibility of controlling the nonlinear phaseand the amplitude of a weak signal s injected in a PPLN waveg-uide by acting either on the intensity of a pump field or on thephasemismatchbetweenthepumpandthesignal.Thenonlinear phaseandtheopticalgainhavebeenmeasuredbyaproperlyde-signed MZI configuration providing high sensitivity and strongimmunity to slow mechanical and thermal drifts. With an av-erage pump power of 15 mW, nonlinear phase shift up to  π/ 6 havebeenobtained,butsignificantlyhigherphasevaluesareex-pected for optimized waveguiding structures. These results areencouraging for the realization of a nonlinear integrated MZI tobe used for all-optical signal modulation.
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