Simple geometric interpretation of signal evolution in phase-sensitive fibre optic parametric amplifier

Transcription

1 Simle geometric interretation of signal evolution in hase-sensitive fibre otic arametric amlifier A.A. REDYUK,,,* A.E. BEDNYAKOVA,, S.B. MEDVEDEV, M.P. FEDORUK,, AND S.K. TURITSYN,3 Novosibirsk State University, Novosibirsk, Pirogova street, 639, Russia Institute of Comutational Technologies SB RAS, Novosibirsk, 6 Acad. Lavrentiev avenue, 639, Russia 3 Aston Institute of Photonic Technologies, Aston University, Aston Triangle, Birmingham B4 7ET, UK * alexey.redyuk@gmail.com Abstract: Visualisation of comlex nonlinear equation solutions is a useful analysis tool for various scientific and engineering alications. We have re-examined geometrical interretation of the classical nonlinear four-wave mixing equations for the secific scheme of a hase sensitive one-um fiber otical arametric amlification, which has recently attracted revived interest in the otical communications due to otential low noise roerties of such amlifiers. Analysis of the hase ortraits of the corresonding dynamical systems rovide valuable additional insight into field dynamics and roerties of the amlifiers. Simle geometric aroach has been roosed to describe evolution of the waves, involved in hase-sensitive fiber otical arametric amlification (PS-FOPA) rocess, using a Hamiltonian structure of the governing equations. We have demonstrated how the roosed aroach can be alied to the otimization roblems arising in the design of the secific PS-FOPA scheme. The method considered here is rather general and can be used in various alications. 6 Otical Society of America OCIS codes: (938) Nonlinear otics, four-wave mixing; (94) Nonlinear otics, arametric rocesses; (63) Fiber otics amlifiers and oscillators. References and links. M. E. Marhic, Fiber Otical Parametric Amlifiers, Oscillators and Related Devices (Cambridge University, 8).. R. H. Stolen, Phase-matched-stimulated four-hoton mixing in silica-fiber waveguides, IEEE J. Quantum Electron., 3 (975). 3. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, & P. O. Hedekvist, Fiber-based otical arametric amlifiers and their alications, IEEE J. Sel. Toics Quantum Electron. 8, 56 5 (). 4. S. Radic & C. J. McKinstrie, Two-um fiber arametric amlifiers, Ot. Fiber Technol. 9, 7 3 (3). 5. C. McKinstrie & S. Radic, Phase-sensitive amlification in a fiber, Ot. Exress, (4). 6. C. J. McKinstrie, S. Radic, & A. H. Gnauck, All-otical signal rocessing by fiber-based arametric devices, Ot. Photonics News 8, 34 4 (7). 7. M. E. Marhic, P. A. Andrekson, P. Petrooulos, S. Radic, C. Peucheret, & M. Jazayerifar, Fiber otical arametric amlifiers in otical communication systems, Laser Photonics Rev. 9, 5 74 (5). 8. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tisuwannakul, B. J. Puttnam, H. Toda, & L. Grüner-Nielsen, Towards ultrasensitive otical links enabled by low-noise hasesensitive amlifiers, Nat. Photonics, 5, (). 9. R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O Gorman, A. Bogris, D. Syvridis, S. Dasguta, P. Petrooulos, & D. J. Richardson, All-otical hase and amlitude regenerator for next-generation telecommunications systems, Nat. Photonics 4, ().. A. A. Redyuk, M. F. C. Stehens, & N. J. Doran, Suression of WDM four-wave mixing crosstalk in fibre otic arametric amlifier using Raman-assisted uming, Ot. Exress 3(), (5).. M. F. C. Stehens, I. D. Phillis, P. Rosa, P. Harer, & N. J. Doran, Imroved WDM erformance of a fibre otical arametric amlifier using Raman-assisted uming, Ot. Exress 3(), 9 9 (5).. A. Bendahmane, A. Mussot, A. Kudlinski, P. Szriftgiser, M. Conforti, S. Wabnitz, & S. Trillo, "Otimal frequency conversion in the nonlinear stage of modulation instability," Ot. Exress 3, (5). 3. J. Kakande, R. Slavik, F. Parmigiani, A. Bogris, D. Syvridis, L. Gruner-Nielsen, R. Phelan, P. Petrooulos, & D. J. Richardson, Multilevel quantization of otical hase in a novel coherent arametric mixer architecture, Nat. Photonics 5, ().

2 4. A. Perentos, S. Fabbri, M. Sorokina, I. D. Phillis, S. K. Turitsyn, A. D. Ellis, & S. Sygletos, QPSK 3R regenerator using a hase sensitive amlifier, Ot. Exress 4, (6). 5. S. Watanabe, F. Futami, R. Okabe, R. Ludwig, C. Schmidt-Langhorst, B. Huettl, C. Schubert, & H. Weber, An Otical Parametric Amlified Fiber Switch for Otical Signal Processing and Regeneration, IEEE J. Sel. Toics Quantum Electron. 4, (8). 6. H. Steffensen, J. R. Ott, K. Rottwitt, & C. J. McKinstrie, Full and semi-analytic analyses of two-um arametric amlification with um deletion, Ot. Exress 9(7), (). 7. J. R. Ott, H. Steffensen, K. Rottwitt, & C. J. McKinstrie, Geometric interretation of four-wave mixing, Phys. Rev. A (3). 8. G. Caellini & S. Trillo, Third-order three-wave mixing in single-mode fibers: exact solutions and satial instability effects, J. Ot. Soc. Am. B 8, (99).. Introduction Otical arametric amlification (OPA) in fibers [, ] is a classical nonlinear henomenon that manifests itself as a rocess of nonlinear energy transfer from owerful uming wave(s) (considered as energy sources) to signal wave(s), leading to effective amlification of the latter. OPA is a articular case of a fundamental nonlinear wave mixing effect when waves of different frequencies are agitated through otical nonlinearity to generate new frequencies. In otical fiber, that is an examle of media without inversion symmetry, the leading third-order nonlinearity (Kerr nonlinearity) involves interaction of four hotons. Therefore, four-wave-mixing (FWM) is the general term covering various hysical effects resulting from the Kerr nonlinearity. Recently, fiber OPA has attracted a great deal of renewed interest as a romising technology for low-noise arametric amlification in high-caacity otical communications [3 ]. This recent revival of rather classical field is strongly suorted by advances in the technology of highlynonlinear fibers (HNLF) with a good figure of merit (high nonlinear coefficient with a reasonably low losses). One of the key reasons for renewed interest in fiber OPA is a otential ossibility to reach a db noise figure for the in-hase comonents (hase sensitive amlification). Desite numerous technical challenges this makes hase-sensitive arametric amlifiers an interesting and timely research direction. Low noise in-line amlifiers would offer significant link erformance imrovement. Moreover, such hase-sensitive amlifiers may oen u new oortunities for all-otical regeneration of coherent otical signal [9, 3 5]. In this work we will aly geometrical aroach develoed in [6 8] for otimisation of secific amlification schemes based on hase sensitive fiber OPA. Geometric resentation of a hase dynamics and visualisation of families of solutions is often more useful than formal exact analytical solutions. We will study energy transfer between um and signal using intuitive hase ortraits (geometric interretation) and find otimum arameters of hase-sensitive fibre otic arametric amlifiers (PS-FOPA) leading to the maximum gain.. Theory.. Basic equations and Hamiltonian formulation General FWM rocess involves interaction of four waves at different frequencies. When alied to fiber OPA this would generally mean the use of two um waves at different frequencies and a signal wave. The interaction of signal with two uming waves create the fourth wave, called the idler. However, in fiber OPA a artially degenerate case (in which the frequencies of two waves/hotons are equal) is also imortant for ractical alications. In this case one uming waves creates a air of Stokes (lower frequency) and anti-stokes (higher frequency) waves, with frequencies symmetrically located with resect to the frequency of the um. In case of one-um otical arametric amlification degenerate three-wave mixing rocess between three continuous waves is described by the well-known three couled differential equations for the owers P i (i =,, 3) and one for the relative hase of the interacting waves

3 = βz : dp dz = dp dz = dp 3 dz = 4P P P 3 sin, () dz = β γ + (P ( ) P P 3 ) + P P P 3 P + P3 4P cos, () where P denote um ower, P and P 3 are signal and idler owers, i is the hase of ith wave, β = β + β 3 β is the roagation constant mismatch, γ is the coefficient of nonlinearity, Z = γz is the normalised distance. The system of Eqs. ()-() admits the following two invariants, reserving at any roagation distance along the fiber: δ = P P 3, (3) P T = P + P + P 3. (4) We consider a one-um hase sensitive fibre otical arametric amlification, where signal and idler wavelengths are symmetrically located around the um wavelength. Assuming signal and idler inut owers to be equal in PS-FOPA, P () = P 3 (), the initial roblem ()-() can be simlified and reduced to: dp dz = dp dz = 4P P sin, (5) dz = β ( ) γ + (P P ) + P P P 4P cos = (6) = β γ + (P P ) + (P 4P ) cos. The system of Eqs. (5)-(6) is a Hamiltonian system that can be resented as follows: dp dz = H = P (P T P ) sin, (7) dz = H P = (P P T ) cos + 3P P T + β γ. (8) Here the Hamiltonian H reads: H = P (P T P ) cos 3 ( P + P T β ) γ P. (9) After introducing new variables (following the rocedure described in [8]) = P /P T, l = Z P T and k = β γp T the Hamiltonian (9) and Eqs. (7)-(8) can be resented in the normalised form: H = ( ) cos 3 + ( k), () d dl = H = ( ) sin, () dl = H = (k ) + 3 ( ) cos, () where is a contribution of the um ower to the total ower lying in the range between and, k can be interreted as the normalised hase mismatch.

4 .. Stationary oints of the dynamical system First, let us find the stationary oints (, ) of the dynamical system ()-() that describe steady state co-roagation of three nonlinearly interacting waves in such systems. This is not the focus of this aer, however, otentially, such steady state describing joint roagation of high ower nonlinear waves at different wavelengths can be interesting for various alications. There are four ossibilities when the right-hand side of the Eq. () becomes zero: A =, B =, C = πn, D = πn + π, n Z. Substituting these values in the right-hand side of the Eq. (), we find equations governing the existence of stationary oints: cos A = k, cos B = k +, C = 3 k 7, D = k +. Now we can determine mathematical conditions for the stationary oints existence. For inner oints C and D: oint C = ( 7 (3 k), πn) lies inside the band [, ] when k [ 4, 3]; oint D = (k +, πn + π) lies inside the band [, ] when k [, ]. For boundary oints A and B: oint A = (, A ) exists when k [, 3]; oint B = (, B ) exists when k [ 4, ]. It should be mentioned that although the boundary stationary oints are not involved in the dynamic rocess, they define the structure of the hase ortrait in their vicinity..3. Phase ortraits A hase ortrait, corresonding to the Hamiltonian (), deends on a set of stationary oints. It turns out that we can obtain imortant information concerning dynamics of the nonlinear system analysing behaviour near critical oints. The rest of the hase ortrait can tyically be deduced from this information. As it was shown in the section., there are three different scenarios of the system dynamics: when k [, 3] only two stationary oints A and C exist, when k [, ] all oints A, B, C and D exist, and finally, when k [ 4, ] two oints B and C exist. Figure demonstrates different tyes of hase ortraits for k =, k =.5 and k =. k= k=.5 B B k= B B C C D C A A A A Fig.. Phase ortraits: k = (left), k =.5 (middle), k = (right). The existence of the a Hamiltonian for differential Eqs. ()-() can be used to reduce the roblem to only one variable. Consider the Hamiltonian () and Eq. (): H = ( ) cos 3 + ( k), (3)

5 d = ( ) sin. (4) dl The relative hase is secifically imortant as it shows the direction of the ower flow. Mathematical solutions governing signal amlification (and corresonding um deletion) corresond to the regime of PS-FOPA oeration that is of major interest for alications in otical communication systems. When the variable takes values between and π (sin is ositive), the ower is transferred from the um to the signal and idler, whereas when is between π and π (sin is negative), the energy is sreading in the oosite direction. At some oint the relative hase can change so much that the ower flow is reversed and the ower begins to flow back to the um. Such behaviour is tyical for nonlinear dynamical regimes described by the eriodic functions. In hysical terms, the system oerates as a nonlinear couler between waves roagating at different wavelengths. The hase can be find from the Hamiltonian as Let us denote using (5): cos = H + 3 ( k). (5) ( ) α (H, ) = sin = ± cos. (6) Thus, inserting the equation for in (4), we obtain the differential equation with searable variables d = dl, (7) ( )α(h, ) where the Hamiltonian H lay a role of a arameter. Due to the fact that the Hamiltonian is indeendent of l, its value defines the articular trajectory of the hase ortrait. The equation (7) can be rewritten in the integral form as (L) () d L ( )α(h, ) = dl L. (8) Exlicit solution (l) of Eq. (8) was found in [8] in terms of Jacobian ellitic functions. As we are going to examine tyical otimization roblems for PS-FOPA, the use of Eq. (8) is more convenient for our further analysis than its analytical solution resented in a formal rather comlicated mathematical form. Equation (8) connects three key imortant arameters: inut ower (), outut ower (L) and length L, and imlicitly, through a Hamiltonian, describes imact of the inut hase (). It is also convenient that Eq. (8) can be easily integrated numerically through e.g. traezoidal method. In the next section, we aly this aroach to several otimization roblems. 3. Otimization of the ractical PS-FOPA Now let us consider ractical imlementation of idler and um ower evolution in PS-FOPA. We study waves evolution in a tyical highly nonlinear fiber with the following arameters: zero disersion wavelength λ zdwl = nm, disersion sloe S =.83 s nm km and nonlinear coefficient γ = 8.7 (W km). For the fixed inut um ower P = 3 W and signal (idler) ower P =. mw, signal wavelength λ s = 55 nm and um wavelength λ = nm the value of arameter k can be calculated directly k =.7. Figure (a) shows the hase ortrait for the case k =.7, corresonding to the system evolution with arameters described above. It is seen that there are three tyes of trajectories, which we call tye "", "" and "3". Trajectories of tye "" and "" are closed trajectories around stationary oints C and D, resectively. Characteristic evolution of the um ower, signal (idler)

6 B ot 3 () () () () () 3 G = 5 db C D (L) A 3 Fig.. (a) Phase ortrait for k =.7. (b) Schematic deiction of the ower transfer rocess in PS-FOPA with a fixed gain. um signal (idler) rel hase 3.5 (rad) Power (W) Length (km) Fig. 3. Evolution of the um ower (solid red), signal (idler) ower (solid green) and relative hase (dashed blue) during two full eriods of the st tye trajectory. ower and relative hase during two full eriods of the st tye trajectory is deicted in Fig. 3. For trajectories of tye "3" hase changes in a whole range of values [, π]. Searatrixes confining trajectories of tyes "", "3" and "","3" are deicted by black dots in Fig. (a). At first glance, one can think that the searatrix enclosing stationary oint C, which gives the maximum signal amlification = () (L), would define an otimal trajectory of signal amlification in the hase lane. In the next section we consider this regime in more details and will see that this is not necessarily truth. 3.. Proerties of searatrixes To analyse searatrixes, first, let us make a simle transformation by adding a constant k + / to the Hamiltonian () leading to its factorization: H (, ) = H(, ) + k + / = ( )((4 cos + 3) + k + ). (9) Equation H (, ) = defines the searatrix enclosing stationary oint C. It consists of two curves: the horizontal line = between the two stationary oints of tye B and the curve given by

7 equation: = k + 4 cos + 3. () Substitution of the exression () in the right-hand side of () yields the following ordinary differential equation: + cos + + k =, () dl or in the equivalent form: = dl. () ( cos + + k) Equation () has a singularity at cos + + k = (coordinate of the stationary oint B), which means that infinite distance (here fiber length) is required to aroach the stationary oint B. It can be shown by analogy, that ower flow roagation along the second searatrix enclosing stationary oint D also requires an infinite fiber san length. Though, formally, a searatrix corresonds to the maximum amlification = m, where m = (k + )/7 (see eq.), this amlification level is achieved for roagation over infinite fiber san. Therefore, in ractical terms, otimisation of the amlification arameters such as gain, should be done at the finite roagation distance L. There are other trajectories on the hase lane that give required high amlification over finite fiber sans. This qualitative observation is suorted by the direct numerical solution of the amlifier otimization roblem resented in the next section. 3.. Otimization of the PS-FOPA using geometrical aroach The first imortant ractical design roblem, that can be easily solved using Eq. (8), is a determination of HNLF length for a given inut signal ower, um ower and required gain. Assuming, for instance, that we are interested in 5 db signal amlification in the shortest ossible PS-FOPA. Note, that the maximum attainable gain (when all um ower is converted to the signal and idler) is limited by the ower conservation law u to G max = P /P + 5 db. To solve the otimization roblem we should find all the trajectories on the hase lane, yielding 5 db signal gain and then choose the trajectory, corresonding to the shortest ossible length of the amlifier roviding for such target gain..35 G = 4 db G = 45 db G = 5 db 5 = = (otimal) = 3 4 Length (km).3 ot = G (db) (rad)..3.5 Length (km) Fig. 4. (a) Deendence of PS-FOPA length on initial relative hase () for G = 4 (green), 45 (blue) and 5 (red) db. (b) Deendence of G on PS-FOPA length for () = (solid red), (dashed red) and 3 (solid black) rad. P = 3 W, P = P 3 =. mw.

8 Fig. 5. Deendence of G on PS-FOPA length and um ower. () =, P = P 3 =. mw. Schematic deiction of the ower transfer rocess in PS-FOPA with the desired gain is resented in Fig. (b). The coordinates () = P ()/P T and (L) = P (L)/P T are known and determined by the initial owers of the um and signal (idler) and target amlification level G = (L) (). The oints of intersection between the trajectories and horizontal lines = () and = (L) determine the start and end oints of the ower flow. Each trajectory of interest is uniquely defined by the relative hase arameter at the fiber inut (). Only one of the trajectories corresonds to the shortest length of the amlifier (shown by red circles in Fig.(b)). Therefore, in order to determine the shortest ossible fiber length we should find the otimum value of (). Equation (8) rovides us with the required deendence of FOPA length on (). In the considered examle (), (L) and () are known and the length of PS-FOPA can be straightforwardly calculated by numerical integration of (8). It can be seen from Fig. 4(a), that an otimum value of the relative hase at L = does exist and it is aroximately equal to for all considered gain levels G = 4, 45, and 5 db. In the other words, we found the otimum trajectory, which rovides us with the shortest fiber length and that does not deend on the gain variation. Next otimization roblem, that can be considered, is to determine signal gain for given inut signal ower, um ower and given length of HNLF. Signal gain can also be found by solving Eq. (8) for known fiber length L, initial normalised um ower () and unknown ower (L) using simle bisection method. Resulting deendence of signal gain on cavity length is shown in Fig. 4(b). Figure 4(b) shows, that gain is almost the same for () = and ot () =, so we can confine the otimization roblem to the simlest case when () =. The maximum signal gain is reached at L.35 km for the initial relative hases () = and ot () =. Finally, another design roblem, that can be solved using roosed aroach, is a multiarametric otimization of PS-FOPA. For examle, to investigate signal gain behaviour as a function of um ower and HNLF length contour lots can be lotted solving Eq. (8) numerically using bisection method as in the revious section. Deendence of the signal gain on the initial um ower and cavity length variation is shown in Fig Conclusion We have erformed theoretical analysis of the classical four-wave mixing equations in the alication to the secific scheme of hase sensitive one-um fibre otical arametric amlification.

9 Simle geometric aroach has been alied to describe the evolution of the waves, involved in the PS-FOPA rocess, using the Hamiltonian structure of governing equations. This method rovides geometrical visualisation of the solutions that might be useful for ractical qualitative and quantitative design analysis. We have demonstrated how the geometrical aroach can be used for solving the otimization roblems directly relevant to the design of the PS-FOPA with secific arameters. Funding Work of AR, SBM and MPF was suorted by Russian Science Foundation (4--). Work of SKT was suorted by the grant of the Ministry of Education and Science of the Russian Federation (4.B5.3.3). Work of AB was suorted by the Ministry of Education and Science of Russian Federation ( ).