Supplementary information for: All-optical signal processing using dynamic Brillouin gratings

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1 Supplementary information for: All-optial signal proessing using dynami Brillouin gratings Maro Santagiustina, Sanghoon Chin 2, Niolay Primerov 2, Leonora Ursini, Lu Thévena 2 Department of Information Engineering University of Padova, Italy 2 Eole Polytehnique Fédérale de Lausanne, Switerland Marh 4, 23 Introdution In this doument, more details are given to explain, to support and to extend the results presented in the main manusript. In partiular, the details refer to the definitions, the methods and the parameters of both the theoretial alulations. Moreover, in this doument higher order differentiation is presented and the effets of self-phase modulation SPM) onsidered. A omment on the role of spontaneous Brillouin noise SpBN)is also provided. 2 Theoretial Model The equations desribing the slowly varying wave envelopes of the optial waves DW, RP, WP, RW respetively A d, t), A w, t), A r, t), A ret, t)) and of the AW, Q, are given by: A d + v s t A d = ηg B 2 QA w + jζ K A d 2 )A d, ) A w + v s t A w = ηg B 2 Q A d + jζ K A w 2 )A w, 2) A ret + v f t A ret = ηg B 2 QA r + jζ K A ret 2 )A ret, 3) A r + v f t A r = ηg B 2 Q A ret + jζ K A r 2 )A r, 4) 2τ B t Q + Q = A d A w + A reta r + ξ. 5) Besides the referene ited in the main manusript [] these equations an be also found in [2]. The symbols, t indiate respetively spae and time derivatives, v s, v f are the group veloities for the slow and fast axis of the PMF, η = nϵ /2 is a normaliation fator is the light speed in vauum, n the refrative index, g B the SBS gain oeffiient) suh that wave intensities A 2 are measured in V 2 /m 2. The equations also take into onsideration the SPM through the term proportional to ζ K = ζγ, where γ is the Kerr oeffiient, ζ = ηa eff, where A eff is the fiber effetive area. The AW is defined by Q = 2vaρ/ 2 ıγ e ϵ Ω B τ B ), where v a is the sound speed, γ e the eletrostrition oeffiient, ϵ the vauum permittivity, τ B the phonon lifetime. The SpBN is also taken into aount through the random variable ξ, as defined in [3]. For the sake of simpliity it will be set v s = v f = = /n.

2 2. Information storage Negleting the SPM and the SpBN, for the writing proess no RP and RW), in a referene frame moving with the AW, i.e. pratially fixed with the fiber) the equations are redued to: t A d + A d = η B A w q exp Γ B t) 6) t A w A w = η B A d q exp Γ B t) 7) t q = Γ B A d A w expγ B t), 8) where η B = ηg B /2. By formally integrate Eq. 8) in time, at a fixed position {, L} the result is: t ) q, t) = Γ B A d t ) A wl + t expγ B t )dt 9) In the ideal ase, the AW is alulated straightforwardly: t ) q, t) = Γ B A d t ) ε w exp ıθ w )δ + t expγ B t )dt ) = Γ B ε w exp ıθ w )A d 2 ) ) exp Γ B where is the point of the ollision between the WP and the DW. 2.2 Information retrieval The stored AW moves at the sound speed v a and deays with the aousti lifetime τ B see Table for pratial values). If the reading proess is arried out within a few ns from the storage time, the AW motion an be ompletely negleted and the deay is expeted to ause a derease of the wave amplitude but no distortion. When the interation between the DW and the WP is over, equations 3)-4)-5) govern the reading proess on the PMF fast axis. Through a formal hange of variables = t referene frame moving with the RW A ret, t)) the governing equations are: t A r 2 A r = η B A ret q exp Γ B t), ) t A ret = η B A r q exp Γ B t), 2) t q v fa q = Γ B A ret A r expγ B t), 3) Let us onsider an ideal Dira funtion RP, injeted from the same fiber side from whih the WP was previously injeted = L) with a delay t ; the RP propagates without distortion till it interats with the stored AW: A r ) ), t) = A rl + 2t = ε r expıθ r ) δ + 2t t, 4) while in the new referene frame the stored AW Eq. )) reads: ) [ q, t) = Γ B ε w exp ıθ w )A d 2 )] + 2t exp Γ B + t. 5) While the RP walks off the AW, a part of it is baksattered into the RW that, similarly to what was done for the writing proess, an be alulated by integrating Eq. 2) also substituting Eqs. 4),5): A ret, t) = η B Γ B ε w ε r exp[ıθ r θ w )] 6) t ) ) [ δ + 2t t A d 2 l + 2t )] exp Γ B + 2t dt ) = A out A d + t, where A out = η B Γ B ε w ε r exp[ Γ B t + ıθ r θ w )] is a onstant. Therefore, in the referene frame fixed with the fiber the RW is: A ret, t) = A out A d t + t ). 7) 2

3 2.3 Arbitrary waveform time differentiation The time differentiation of the DW an be ahieved when the RP, injeted from = L and propagating without distortion till it interats with the stored AW, is the time derivative an ideal Dira distribution: A r ), t) = A rl + 2t = ε ) r expıθ r ) ) δ ) ) + 2t t. 8) The stored AW is still given by Eq. 5). Substituting Eqs. 5) and 8) into Eq. 2) and integrating one gets: A ret, t) = η B Γ B ε w ε ) r exp[ıθ r ) θ w )] 9) t δ ) ) ) [ + 2t t A d 2 + 2t )] exp Γ B + 2t dt = [ ) )] = A ) dad out dt + t Γ B A d + t, where A ) out = η B Γ B ε w ε ) r exp[ Γ B t + ıθ r ) 2.4 True time reversal A ret, t) = A ) out [ dad dt θ w )]/2. In the referene frame fixed with the fiber: + t t) Γ B A d + t t) ]. 2) In order to obtain the DW time reversal, it is neessary to exhange the roles of the waves A r, A ret in Eqs Therefore, a forward-propagating RP is injeted from = and its interation with the stored AW generates a bakward propagating RW at a frequeny ω ret = ω r Ω B. The phase mathing ondition between slow and fast axis beomes ω r = ω d + n/n). Through a hange of variables = + t referene frame is moving with A ret ) the governing equations beome: t A r + 2 A r = η B A ret q exp Γ B t), 2) t A ret = η B A r q exp Γ B t), 22) t q + q = Γ B A reta r expγ B t). 23) Let the RP at the input side = ) be a Dira funtion, propagating without distortion till it interats with the stored AW: A r ) ), t) = A r 2t = ε r expıθ r ) δ 2t t. 24) The stored AW Eq. )) in the new referene frame reads: ) [ q, t) = Γ B ε w exp ıθ w )A d 2 )] 2t exp Γ B t. 25) By formally integrating Eq. 22) and substituting Eqs. 24) and 25): A ret, t) = η B Γ B ε w ε r exp[ıθ r + θ w )] 26) t ) ) [ δ 2t t A d 2 2t )] exp Γ B dt ) [ = A ttr A )] d + t exp Γ B, where A ttr = η B Γ B ε w ε r exp[ıθ r + θ w )]. Finally, in the referene frame fixed with the fiber, one gets: A ret, t) = A ttr A d + t + t ) exp [ Γ B + t )]. 27) 3

4 3 The effet of the Self Phase Modulation The WP and RP powers in experiments are usually large about W peak power) so one might ask what the effets of the SPM ould be. In fat, though the fiber used is short of the order of a few meters), the aumulated nonlinear phase shift an be large at the position where the pulses ollide. As for the ross-phase modulation XPM) its effets are ertainly negligible, beause the overlap time between the WP and the DW is very small due to the ounter-propagation of the waves. Here we show that the SPM ontribution an be analytially predited, with a very good degree of approximation, and its effets are not detrimental for the signal proessing funtions that have been demonstrated. The theory is orroborated by a numerial analysis. In fat, it is legitimate to assume that the SPM affets the propagation only till the point of ollision in Eqs. )-5) and therefore to assume that eah pulse is affeted by a preditable, onstant, phase shift θ = γp d, where d is the distane between the input or output) and the ollision point d = or d = L ). The model is tested in the partiular ase of a seond order time differentiation. 3. Seond order time differentiation In the simulation, we onsider a PMF of length L = 5 m. The other fiber parameters are defined in Table. The DW is a superposition of two real Gaussian pulses, eah of them with a FWHM of 2 ns see Fig. a). The WP is the first derivative of a 5 ps Gaussian pulse see Fig. a). The stored AW profile atually reprodues the first derivative of the DW, spatially ompressed by a fator of 2, as it has been explained in the main manusript. In Fig. b the modulus of the ideal first order derivative of the DW dashed blue urve) is ompared to the numerial red urve) and the theoretial with approximated SPM phase shift - dark green urve) store wave modulus. Similarly, in Fig., the ideal first derivative of the DW dashed blue line) is ompared with the numerial red urves) and the theoretial with approximated SPM phase shift - dark green urves) stored aousti wave. The ontinuous urves refer to the real part, while the dashed urves to the imaginary part. This figure shows that the SPM effet is well desribed by the theory proposed in this setion. In order to obtain the seond order derivative of the DW, let us onsider a RP whih is also the first order derivative of a Gaussian pulse i.e. the same as WP in the simulations), as shown in Fig. 2a. The RW reprodues the seond order time differentiation of the input data pulse, with a very good agreement. In fat, in Fig. 2b the modulus of the ideal seond order derivative of the DW dashed blue urve) is ompared to the numerial blak urve) and the theoretial approximated SPM phase shift - dark red urve) RW modulus. Similarly, in Fig. 2, the ideal seond order derivative of the DW dashed blue urve) is ompared with the numerial blak urves) and the theoretial approximated SPM phase shift - dark red urves) RW. The ontinuous urves represent the real part and the dashed urves the imaginary part. 4 Experimental soures of distortion In the experiments, due to the lak of a suitable modulator, the write and read pulsewidth were 4 ps FWHM pulses, so their duration is omparable to the single pulse of the sequene. Therefore, they are not lose to the ideal delta Dira funtions. This introdues some distortion in the output pulse, as shown in Fig. 3, where a numerial simulation using the experimental input sequene but with shorter 5ps, FWHM) write and read pulses is presented. Note in partiular that both the extintion ratio and the rise and deay times are affeted by the finite pulsewidth. Additional soure of distortion like that appearing in the pulse leading edge) arises from temperature and birefringene gradients, whih auses a shift of the Brilluoin gain and of the peak refletivity wavelength with substantial refleted amplitude hange. 5 The effets of the spontaneous Brillouin noise The ontribution of SpBN is shown by ξ term in eq. 5, whih atually represents a Langevin noise soure desribing the thermal exitation of AW. The statistial properties of ξ, desribed by a white Gaussian random variable, are investigated in [3], in whih the AW is expressed in terms of density ρ in the dynamial equations, so the noise variane is alulated obtaining eq. 6 in [3]. Let us remark that here we defined the AW in terms 4

5 Normalied Amplitude [m] a) Normalied Amplitude [m] b) Normalied Amplitude [m] ) Figure : a): Spatial profile of the DW blue) and the WP magenta). b): Spatial profile of stored AW modulus; omparison between the ideal first order derivative of the DW dashed blue), the numerial obtained red) and the theoretial approximated SPM phase shift - dark green) stored wave. ): Same as b); the ontinuous urves refer to the real part, the dashed urves to the imaginary part. of Q rather than ρ Q = 2v 2 aρ/ ıγ e ϵ Ω B τ B )) so the noise variane an be straightforwardly realulated in terms of Q. 5

6 Normalied Amplitude [m] a) Normalied Amplitude t [ns] b) Normalied Amplitude t [ns] ) Figure 2: a): Spatial profile of the DW blue) and the WP magenta). b): the omparison between the ideal seond order derivative of the DW dashed blue), the numerial blak) and the theoretial approximated SPM phase shift - dark red) RW. ): Same as in b); the ontinuous urves represent the real part, the dashed urves the imaginary part. We onsidered the SpBN in the numerial simulations for ompleteness. However, it has been verified that the SpBN indued flutuations do not affet the SBS signals: noise ontribution is few orders of magnitude 6

7 Normalied Amplitude t [ns] Figure 3: True time reversal: the experimental input data waveform blue urve), the experimental retrieved waveform blak urve), the numerially obtained retrieved waveform green urve) when the data waveform is the experimental but using shorter 5ps, FWHM) write and read pulses. lower than SBS waveforms. This fat has been also orroborated by experiments, whih results are not affeted by SpBN. 6 PM fiber parameters definition In Tab., we give the parameters used in the main manusript and also in this doument. PMF birefringene n = 5 4 Aousti wave veloity v a = m/s Eletrostrition oeffiient γ e =.8 Refrative index n =.5 Light group veloity = 2 8 m/s Permittivity ϵ = /36π 9 A 2 sw m Brillouin frequeny shift Ω B = 2π.93 GH) rad/s Phonon lifetime τ B = 5 ns SBS gain oeffiient g B = 5 m/w Effetive area A eff = 4 µm 2 Kerr s oeffiient γ =.32 W m Table : PM fiber parameters. 7

8 Referenes [] V.P. Kalosha, W. Li, F. Wang, L. Chen, X. Bao, Frequeny-shifted light storage via stimulated Brillouin sattering in optial fibers, Opt. Lett., 33, ). [2] R. W. Boyd, Nonlinear optis, Aademi Press, 2 nd ed. 23, ISBN [3] R.W. Boyd, K. R a`ewski, P. Narum, Noise initiation of stimulated Brillouin Sattering, Phys. Rev. A, 42, , 99. 8