Four-wave mixing stability in hybrid photonic crystal fibers with two zero-dispersion wavelengths Benoit S´evigny,1,∗ Olivier Vanvincq,1 Constance Valentin,1 Na Chen,1,2 Yves Quiquempois,1 and G´eraud Bouwmans1 1 Laboratoire
de Physique des Lasers, Atomes et Mol´ecules, UMR 8523, IRCICA Research Institute for Advanced Communication, Parc Scientifique de la Haute Borne, CNRS USR 3380, 50, avenue Halley, 59658 Villeneuve d’Ascq cedex, France
2 Key
Lab of Specialty Fiber Optics and Optical Access Networks, School of Communication and Information Engineering, Shanghai University, Shanghai 200072, China ∗ [email protected]
Abstract: The four-wave mixing process in optical fibers is generally sensitive to dispersion uniformity along the fiber length. However, some specific phase matching conditions show increased robustness to longitudinal fluctuations in fiber dimensions, which affect the dispersion, even for signal and idler wavelengths far from the pump. In this paper, we present the method by which this point is found, how the fiber design characteristics impact on the stable point and demonstrate the stability through propagation simulations using the non-linear Schr¨odinger equation. © 2013 Optical Society of America OCIS codes: (060.2280) Fiber design and fabrication; (060.5295) Photonic crystal fibers; (190.4380) Nonlinear optics, four-wave mixing.
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#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30859
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#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30860
1.
Introduction
Four-wave mixing (FWM) in optical fibers has been extensively studied, increasingly so in the last two decades [1–3] as it presents a nuisance in WDM communication systems but also because it provides a leveraging platform for a range of interesting applications [4–6]. For instance, generation of correlated light at different wavelengths through this process, giving rise to entangled photons [7, 8], allows for quantum cryptography, [9–11] and a range of other applications necessitating entangled light sources [12, 13]. Additional applications, also based on FWM, include parametric amplification [14–16], phase-conjugation [17–19], frequency conversion (FOPO) [15, 20] and pulse generation [21] amongst others. It has been reported in various publications that the efficiency of this process is generally impaired by random longitudinal fluctuations of the dispersion along the fiber, which can limit the gain bandwidth in FOPAs and substantially diminish wavelength conversion efficiency [22–24]. Furthermore, this degradation is usually worse as the phase-matching condition occurs farther from the pump [23]. Highly nonlinear fiber designs have been studied which stabilize the dispersion in the presence of scale fluctuations [25, 26]. The idea relies on negating the scale-sensitivity of the dispersion by compensating the contributions of different waveguiding structures in a multi-layered index configuration. The effect of this is the flattening of the waveguide dispersion contribution in a region of interest and, for instance, can result in a fixed zero-dispersion wavelength invariant to scale fluctuations, which can be of use in a number of applications. This method, however, may prove challenging when the wavelength range becomes quite large (well in excess of 100 THz) and higher-order dispersion becomes important. It was also noted elsewhere that some phasematched sets of wavelengths are invariant with small scale perturbations for some PCF designs with two ZDW [27] and could potentially be used as stable degenerate FWM (DFWM) points but a full investigation on the properties of these phase-matching conditions and a gain analysis to validate the usability of these points remained to be performed. In this paper, we investigate, with the help of an elaborate DFWM model and numerical simulations, stable phase-matching points, both in wavelength and gain, that do not require the ZDW (or dispersion) to remain fixed and are particularly useful in terms of fiber manufacturing process, especially for large frequency spacings [28]. The procedure by which the dispersion can be analyzed to find the point of stability and a numerical investigation that shows the stability of the gain under these conditions (in a normal dispersion pumping regime) will be presented. As well, a method by which the design parameters of a hybrid photonic-crystal fiber (HPCF) can be tuned to have this stable point occur at a wide range of different sets of pump, signal and idler wavelengths will be discussed. 2.
Four-wave mixing and longitudinal dispersion fluctuations
To begin this study, let us first consider the general case of quasi-continuous-wave (CW) degenerate FWM where three waves (an intense, undepleted pump “p” and weak signal “s” and idler “i” with wavelengths λs < λ p < λi ) interact together through the third-order susceptibility tensor in an isotropic medium: 2 in2 ω p ∂ Ap = f pp A p A p (1) ∂z c 2 ∂ As in2 ωs = 2 fsp A p As + fsipp A2p A∗i exp(−iΔkz) (2) ∂z c 2 in2 ωi ∂ Ai = 2 fip A p Ai + fispp A2p A∗s exp(−iΔkz) (3) ∂z c where A denotes the envelope of the wave, fi j and fi jkl are the overlap integrals (considered constant for small variations in fiber scale and which can be readily derived, in vectorial form, #199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30861
by keeping the full fields at pump, signal and idler wavelengths in the non-linear polarization term of the wave equation) [6], n2 is the nonlinear refractive index of the material (assumed constant over the cross-section), ω is the angular frequency of the wave and Δk is the linear phase-mismatch term defined as: Δk = βs + βi − 2β p
(4)
where βm is the propagation constant at wavelengths λm . Under these conditions, the pump can 1/2 be trivially integrated to A p = Pp exp(iγ Pp z) with γ = n2 ω p f pp /c. The coupled equations for the signal and idler waves can thus be recast as follows:
∂ As ∂z ∂ A∗i ∂z
in2 ωs [2 fsp Pp As + fsipp Pp A∗i exp(−iθ z)] c in2 ωi ∗ ∗ − 2 fip Pp A∗i + fispp Pp As exp(iθ z) c
= =
(5) (6)
where θ = Δk − 2γ Pp . At this point, it is necessary to consider that the dispersion fluctuates the fiber length. In this case, the phase argument can readily be from θ z to along z generalized z (ζ )dζ ]/2} and simiθ ( ζ )d ζ . Taking this into account and substituting B = A exp{i[ θ s s 0 0 larly for the idler wave, we can reform the system of coupled differential equations as:
∂ Bs ∂z ∂ B∗ i i ∂z i
κsp Bs − γsipp Pp B∗i 2 ∗ κip ∗ = − B∗i + γispp Pp Bs 2 =
(7) (8)
where γmnpp = n2 ωm fmnpp /c. If we consider that the fluctuation of the dispersion can be modeled as a stochastic process around a constant mean value along the fiber length, we can write the global phase-mismatch term, κi j , as the sum of a constant term and a fluctuating (z-dependent) term:
κsp κip
= =
2(γ − 2γsp )Pp − Δkmean − Δkvar (z) 2(γ − 2γip )Pp − Δkmean − Δkvar (z)
(9) (10)
where γmp = n2 ωm fmp /c. Grouping the constant terms together, κmp /2 = Δmp + W (z), where Δmp = −[2(2γmp − γ )Pp + Δkmean ]/2 and W (z) = −Δkvar (z)/2, the whole system of differential equations can be re-written in matrix form:
∂B = [G + HW (z)]B ∂z where the matrices B, G and H are given by −iΔsp Bs B= , G= ∗ P −iγispp B∗i p
iγsipp Pp iΔ∗ip
(11)
,
H=
−i 0 0 i
.
(12)
Since W (z) is a stochastic variable, Eq. (11) is now considered to be a stochastic differential equation, or SDE [29], and the solution must also be a stochastic process. It is to be noted that the analysis proposed here is a generalization of the derivation presented in [23], keeping the loss terms and full overlap integrals as the case where signal and idler far from the pump will be considered, so the constant γ approximation becomes invalid. In order to push the analysis further, more information is needed on W (z). The first thing to mention is that if several components of very different statistical nature were to contribute to the stochastic fluctuation of Δk, #199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30862
several terms Wi (z) should be added to the SDE. However, in this case we will consider that the use of one process is enough and such formulation has been shown to produce accurate predictions [23]. In first approximation, we can assume that Δk is linearly related to physical stochastic processes that are occurring along the fiber. Let us thus consider a number of contributions X to the fluctuations of the phase-mismatch. The fluctuation of those parameters, δ X(z), can be characterized in terms of their correlation length Lc,X , which can be derived from the autocorrelation function RX (ζ ) = δ X(z)δ X(z + ζ ) (where “. . .” denotes the expected value operator with RX (0) = δ X(z)δ X(z) = σX2 , σX being the standard deviation of X) as: Lc,X = σX−2
∞ 0
RX (ζ )dζ
(13)
If we make the assumption that all contributions to the fluctuation in the dispersion are relatively small (i.e. can be expressed as a first order expansion), have zero mean and are symmetric around a nominal value Xi,0 , have the same correlation length Lc and are statistically independent, they can be combined in a single stochastic variable W (z) with standard deviation given by: 2 N ∂ Δk 1 σW2 = ∑ σX2i (14) 2 ∂ Xi i=1
Xi =Xi,0
For instance, lets assume that the change in the phase-mismatch is primarily related to a change in fiber dimensions (for instance, through the pitch Λ), which is a major contributor [28], neglecting other contributions, we can write the standard deviation of the stochastic variable W (z) as: 2 1 ∂ Δk 2 σW = σΛ2 (15) 2 ∂ Λ Λ=Λ0 A representative process that can be applied to optical fiber drawing is the OrnsteinUhlenbeck process (OUP) [30], which is basically used to model noisy relaxation and can be easily implemented [31]. It can be noted that this process has an autocorrelation function given by
|ζ | . (16) R(ζ ) = R(0) exp − Lc The parameter of interest in this study is the power at the signal and idler wavelengths (Ps = |As |2 and Pi = |Ai |2 ) for which a new set of four coupled SDEs for P = [Ps , Pi , 2Re(Bs Bi ), 2Im(Bs Bi )]T can be derived from the amplitude differential equations. This system of equations can be written exactly in the same form as Eq. (11) where G and H are now 4×4 matrices. To simplify the following calculations, one can derive that fispp = fsipp and is real. To further simplify the remaining calculations and illustrate the contribution of the frequencies ωs and ωi , it is convenient to define h = n2 fispp /c = n2 fsipp /c. The complete SDE system for power is thus given by ⎡ ⎤ 0 0 ωs hPp 2Im(Δsp ) ⎢ ⎥ 0 2Im(Δip ) 0 ωi hPp ⎥ G=⎢ (17) ⎣ 0 0 Im(Δsp + Δip ) Re(Δsp + Δip ) ⎦ 2ωs hPp −Re(Δsp + Δip ) Im(Δsp + Δip ) 2ωi hPp with H having only two non-zero elements H34 = −H43 = 2. Careful integration of Eq. (11) can be performed, yielding P(z) = exp (Gz + HB(z)) P(0),
(18)
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30863
with B(z) =
z
0 W (z
)dz .
Taking the average, as shown in [23], we get
H2 f (z) P(0). P(z) = exp Gz + 2
with, assuming the same type of autocorrelation function as Eq. (16),
z z −1+ . f (z) = 2Lc2 σW2 exp − Lc Lc
(19)
(20)
It is to be noted that the complete effect of the perturbations is contained in f (z); the behavior of this factor is thus of critical importance in characterizing the impact of fluctuations. This solution is generally complicated but has a simpler form in some cases. Let us first consider that Δsp and Δip are real in the absence of loss since, in this case, the propagation constants are real. Furthermore, γsp and γip are also real. Thus G11 = G22 = 0. For further simplification, we consider the gain at the nominal phase-matching condition such that Δsp + Δip = 0. For the case of spontaneous amplification of a small noise at the signal wavelength, equivalent to P(0) = [Ps0 , 0, 0, 0]T , the expected gain G = Ps /Ps0 is calculated to be (as shown in [23])
f (z) 1 1 + cosh[g(z)] + sinh[g(z)] exp[− f (z)] (21) G = 2 g(z) 1/2 where, however, g(z) is now defined as g(z) = ωs ωi (2hPp z)2 + f 2 (z) . Equation (21) is always maximum for f (z)=0, which occurs if σΛ = 0 or ∂ Δk/∂ Λ = 0. This implies that maximum gain should be obtained at the phase-matching condition, in the presence of fluctuations (σΛ = 0), when ∂ Δk/∂ Λ = 0. This formally illustrates, in other words, that if the phase matching does not change with small changes in the scale of the structure, then the gain should be stable as well. 3.
Investigation of FWM stability in the case of a hybrid PCF design with two ZDW
Let us consider the case of a fiber with two zero-dispersion wavelengths (ZDW). Such fibers are pertinent in this study as they can show two sets of phase-matched wavelengths and as we will show, under certain conditions, a stable point can be found at one set but not at the other. A particular fiber design will serve for our study, which exhibits two ZDWs; the hybrid PCF presented in [32], and shown on Fig. 1, represents a good example of such a fiber as the fundamental mode of the first all-solid photonic bandgap (ASPBG, at short wavelengths) smoothly transitions to the LP02 mode of a 7-cell defect PCF (at long wavelengths), both of which exhibit a ZDW [32]. More specifically, the design considered here consists of six Ge-doped inclusions, with maximum index difference of n0 −nSiO2 = 0.032 and dGe /Λ = 0.7, embedded in the sevencell defect of a PCF structure of air holes with dAir /Λ = 0.6. The nominal pitch Λ0 is selected to be around 1 μ m. Simulations were performed to compute the dispersion, at the nominal conditions described above, from the point where the mode becomes the fundamental PBG mode (its effective index neff < nSiO2 ) to the point where neff falls below nFSM , the effective index of the fundamental space-filling mode of the 7-cell hole lattice [32, 33]. These simulations were carried out by finite elements method, using commercially available software, and material dispersion was added pertubatively [6]. Although other fibers with similar two-ZDW have been studied [20, 34], the advantage of such a design is that both ZDW can theoretically be tuned by changing the parameters of the germanium inclusions and air-holes separately. More complicated structures giving additional design flexibility could also be envisaged, as suggested in [35]. #199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30864
Fig. 1. Illustration of the hybrid PCF fiber design under study (Fiber A). The gray area represents pure silica, the white areas represent air holes and the blue areas represent graded-index (parabolic) Ge-doped inclusions with a maximum index difference n0 − nSiO2 = 0.032. In this particular design, dAir /Λ = 0.6 and dGe /Λ = 0.7. The pitch is Λ0 = 1.03 μ m.
From Eqs. (15) and (20), one can see that the condition ∂ Δk/∂ Λ = 0 represents a point where the noise on the phase-matching vanishes (σW = 0, thus f (z) → 0) and maximal gain is obtained. If material dispersion is taken into account, this stable point will be a function of several design parameters. The concept of phase-matching stability can be further visualized through the dynamics of how the signal and idler wavelengths vary with a change in pitch. Figure 2 represents the linear phase-matching condition for fiber A (Fig. 1) at the nominal pitch Λ0 as well as Λ0 ± 1%. As it can be seen, if the pitch is changed by a percent, the phasematched wavelengths generally move around except at one point, where the Δk and ∂ Δk/∂ Λ are simultaneously equal to zero. It is to be noted too that the trivial condition λs = λi = λ p also satisfies this condition but is not of interest in this study as our focus is on wavelength pairs far from the pump. For such a fiber design, assuming fixed dGe /Λ, the stable point will be a function of relative hole size dAir /Λ, pitch Λ and pump wavelength λ p . For a given design, the pump, signal and idler wavelengths at the stable point will be identified as λ˜ p , λ˜ s and λ˜ i respectively. The trajectory of the stable point in (Λ,λ˜ p ,λ˜ s )-space can be plotted to reveal trends on its behavior as a function of the design choices, as seen on Fig. 3. From this data, it can be noted that smaller dAir /Λ will generally increase the signal wavelength (λ˜ s ). It can also be noted that the relationship between pitch and pump wavelength at the stable point is roughly proportional to λ˜ p at high values (panel A of Fig. 3). One last trend to be observed is that λ˜ s becomes less sensitive to changes in Λ, as seen on panel B of Fig. 3 (or λ˜ p , as seen on panel C) for higher pitch values. Indeed, at high pitch, λ˜ s (Λ) is, for all intents and purposes, a vertical line on panel B. This is very interesting from a manufacturing point of view as it suggests that, for a selected pump wavelength and dAir /Λ, obtaining a draw at the right diameter to generate the right stable point with a small deviation on the generated wavelength could be practically achieved by, for instance, making a very long taper in the fiber and then retrieving the segment with maximum gain. However, an additional consideration has to be taken into account and it is related to the type of dispersion at the pump wavelength. Indeed, the stable point may lie in the normal or abnormal dispersion regime of the pump. In the case where the pump is in the abnormal regime, supercontinuum generation from the pump fission into solitons [6] is likely to drain the pump
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30865
1.4
1.2
λ s,i (μm)
1 Δk = 0 (Fiber A) ∂Δk/∂Λ = 0 0.8
0.6
0.4
0.58
0.59
0.6
0.61
0.62 λ p (μm)
0.63
0.64
0.65
0.66
Fig. 2. Linear phase matching curve obtained for fiber A (Fig. 1). The thick solid line is the phase matching at the nominal pitch while the two thinner solid lines illustrate the phasematching for variations in the pitch Λ = Λ0 ± 1 % with nominal pitch Λ0 = 1.03 μ m. The dashed line represents the stability condition ∂ Δk/∂ Λ = 0. For this design, the stable point is at pump λ˜ p = 602.6 nm. The outer set of phase-matched wavelengths are λ˜ s =1,218.9 nm and λ˜ i = 400.2 nm. The inner set of phase-matched wavelengths are λs = 906.7 nm and λi = 451.2 nm. The first ZDW for fiber A is located at 617.5 nm
energy and reduce the efficiency of the phase-matched FWM at the stable point, although this remains to be investigated experimentally. An example of how the regime changes with pump wavelength for dAir /Λ = 0.6 in our design is shown on Fig. 4. It is to be noted that the abnormal dispersion regime is located between the two ZDW of any given curve. If the stable point lies in the normal dispersion regime however, it is generally at a point where two sets of phase-matched wavelengths exist, with only one (the outer set in our case) showing stability, as shown on Fig. 2. In order to perform this investigation, let us focus on DFWM in the normal dispersion regime which is achieved by selecting the design of fiber A (with λ˜ p around 600 nm) as, generally, the gain bands of the outer set are very narrow and far from the pump, which suggests they should be more sensitive to fluctuations of the dispersion. Furthermore, comparison with numerical propagation simulations will be much easier to interpret as no supercontinuum will be generated at the pump wavelength, thus greatly helping with visualization and comparison with the simple gain model presented in the previous section. In general, there are three factors that need to be examined to verify the practicality of the stationary phase-matching point: the correlation length of the dimensional noise Lc , pump power Pp and pump wavelength λ p . In order to evaluate the impact of those factors on the expected gain, Eq. (17) will be used, without approximations (i.e. full phase-mismatch, overlap integrals and loss terms included) to compute the expected gain G = Ps /Ps (0) from Eq. (19), seeding with noise of one photon per frequency and random phase at the fiber input. The following subsections contain computational results showing the influence of those three parameters on G. 3.1.
Impact of correlation length
The impact of correlation length on gain degradation for FWM interaction in standard fiber has been reported in several publications [22,23,36] and is known to be worse for longer correlation
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30866
dAir /Λ =0.6
Stable point parameters
dAir /Λ =0.62 dAir /Λ =0.64 dAir /Λ =0.66 dAir /Λ =0.68 1.4
A
1.3 1.2
B
Λ (μm)
1.1 1 0.9 0.8 0.7 0.6
C 0.55 0.5 0.65
0.45 0.6 0.4
0.55
˜ i (μm) λ
˜ p (μm) λ
Fig. 3. Mapping of the stable point as a function in (Λ,λ˜ p ,λ˜ s )-space for different dAir /Λ and Λ, all other parameters being the same as for fiber A. The projection of the 3D trajectories on each couple of parameters is also plotted on panels A, B and C to help illustrate the different trends. 2.2 ˜ p = 0.65 μm, Λ = 1.49 μm λ
2
˜ p = 0.60 μm, Λ = 1.05 μm λ
1.8
˜ p = 0.54 μm, Λ = 0.70 μm λ
λ i,s (μm)
1.6 1.4 1.2 1 0.8 0.6 0.4 0.5
0.6
0.7
0.8 λ p (μm)
0.9
1
1.1
Fig. 4. Example of the phase matching curve with the stable point at different pump wavelengths (and corresponding pitch) for dAir /Λ = 0.6. The black circles denote the position of the stable set of wavelengths λ˜ i,s for each pump wavelength λ˜ p and the black “×” indicate the position of the ZDW. The dashed line represents the trivial solution λi,s = λ p . We notice that, as the wavelength λ˜ p gets shorter, the stable point moves closer to the ZDW as the pump then crosses into the abnormal dispersion regime. Missing points in the phase matching curves lie outside the simulation domain and thus have been discarded. It is to be noted that the pump is in the abnormal dispersion regime when it is located between the two ZDW for a given curve.
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30867
log 10(P /P0 )
10
log 10(P /P0 )
10
log 10(P /P0 )
10
log 10(P /P0 )
lengths. Experiments have also been performed to effectively reduce the correlation length by cutting a fiber into short pieces and randomly splicing them back together to notice an improvement in FWM gain [37]. Close inspection of Eq. (20) also shows that long correlation lengths will have a worse impact on gain as, for fixed z, f (z) is a monotonically increasing function of Lc . One thing to note, however, is that f (z) is always proportional to σW2 , which means that the degradation of gain, at the phase-matched condition, is always related to the amplitude of the underlying perturbation. Computation of the gain spectrum for fiber A is shown on Fig. 5, all plots sharing the same scale, and confirms the degradation of the gain as Lc increases, especially for the inner set of phase-matched wavelengths while the (stable) outer set is virtually unaffected. It is also interesting to note the broadening of the gain bands. In more detail, we note that the Lc = 0 graph shows lower gain at the outer wavelengths, which is caused both by the frequency spacing Ω = ωi − ω p = ω p − ωs and monotonically decreasing overlap integral term for FWM, h. Although this drop in h will diminish gain, its impact is reasonable as fully-vectorial computation of fmp and fmnpp from simulation data yields values that stay within the same order of magnitude across the band, as seen on Fig. 6(a), suggesting that the drop in gain is not very large when going from the inner to the outer set of phase-matched wavelength pairs. Compared with the standard gain term γ Pp = n2 f pp ω p Pp /c, usually considered constant √ for all wavelengths (i.e. ωi ≈ ωs ≈ ω p and fispp = fsipp ≈ f pp ), the effective gain, ωi ωs hPp has to drop also due to the frequency spacing Ω increase when moving away from the pump as ωs ωi = ω p2 − Ω2 ≤ ω p2 . But as the effect of Lc is introduced, the stability of the outer set of wavelengths to perturbations changes the balance and the gain at the inner set of phase-matched wavelengths becomes lower than that of the outer set (to the point of vanishing, whereas the gain on the outer set drops by less than 15 % for Lc = 10 m) and constitutes the key property of the stability point. Indeed, this property indicates that a set of phase-matched wavelengths located very far from the pump and relying on higher-order dispersion (generally meaning narrow gain bandwidth) can be virtually insensitive to perturbations in the fiber scale, which
10
L =0m c
5 0
0.4
0.6
0.8
1
1.2
1.4 Lc = 1e−03 m
0.4
0.6
0.8
1
1.2
1.4 L = 1e−01 m
5 0
c
5 0
0.4
0.6
0.8
1
1.2
1.4 L = 1e+01 m c
5 0
0.4
0.6
0.8 λ (μm)
1
1.2
1.4
Fig. 5. Illustration of the dependance of the expected gain G to spatial noise correlation length Lc . In these calculations, σΛ = 0.5 %, z = 0.2 m and Pp = 1 kW. It is to be noted that all graphs share the same scale in order to better appreciate the change in gain as Lc is increased. One can also notice that the outer set of wavelengths, where the stability condition occurs, suffers very little gain degradation whereas the inner set vanishes.
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30868
11
x 10
10
10 |fmnpp |
3.5
|fmp |
8
10
6
2.5
10
Ideal gain
G
Overlap integrals (m −2 )
3
2
4
10 1.5
2
10 1
0
0.4
0.6
0.8 λ (μm)
(a)
1
1.2
1.4
10
−4
10
−3
10
−2
10
−1
σΛ (μm)
10
0
10
1
10
(b)
Fig. 6. (a) Values of the overlap integrals fmnpp and fmp as a function of wavelength for a fixed λ p = 602.6 nm. The four-wave mixing term (blue) is symmetric (in frequency) and both curves cross at the pump wavelength with the value fmnpp = fmp = f pp = 1/Aeff,p . Black circles have been added on the curves at the set of wavelengths under study. As we can see, both factors vary by less than an order of magnitude over the entire domain. The abscissa represents λm , λn being the conjugate wavelength. (b) Gain at the stable point as a function of perturbation amplitude σΛ for Lc = 10 m and z = 0.2 m.
appears counter-intuitive. Let us take as an example a standard PCF design operated in the LP02 mode, such as described in [28]. With d/Λ = 0.9 and nominal pitch Λ0 = 3.27 μ m, such a design would have the same phase matched wavelengths as the stable set of fiber A for the same pump wavelength. Comparing peak gain for both the hybrid and standard PCF, calculated using Eq. (19), we note an improvement of over four orders of magnitude in scale fluctuation tolerance σΛ , favoring the hybrid PCF for any given Lc . This tolerance criteria is defined where the logarithm of the expected gain, i.e. log10 (G), falls to half of its ideal value [25]. Indeed, if compared numerically, this improvement should be in the ratio of ∂ Δk/∂ Λ for each fiber, which yields 1.075 × 104 at the phase-matched wavelengths. In terms of actual fluctuations amplitude, the hybrid PCF should be able to have a tolerance in excess of 3.5% in σΛ /Λ for Lc = 10 m and z = 0.2 m, as seen on Fig. 6(b), which is easily achievable. This gives hope to increase conversion efficiency in FWM applications involving wavelengths far from the pump, regardless of the presence of small fluctuations in scale, which can otherwise become very demanding from the fiber manufacturing point of view. Another feature to notice is the presence of gain where there is normally no phase-matching, for instance, the bumps around the pump for large values of Lc . There is no analytical way to write P for non-zero phase-mismatch (Δip + Δsp ), as expressed before, however looking at the solution of the SDE, Eqs. (17) and (18), can provide some insight on the cause. For small values of z, there is a probability that the stochastic part of the integrated fluctuation over z, H34,43 B(z) compensates the uniform phase-mismatch term, G34,43 z, thus giving rise to non-zero net gain. The likelihood of this happening is higher for small values of phase-mismatch and the gain generated thus is always going to be relatively small as it relies on the probability of a nonnominal condition to be satisfied, keeping in mind that the highest probability is that B(z) is zero. This very well explains the broadening of the gain and the apparition of bumps around phase-matched conditions but in places where there is always a non-zero phase-mismatch term. This reasoning also implies that this condition will be a function of σW and Lc , i.e. a fluctuating process is necessary for those bumps to appear as, otherwise, the phase-mismatch term could
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30869
never be compensated. The small feature located around the pump and present on all graphs is attributed to numerical inaccuracies coming from the quasi-singular nature of the matrix being exponentiated in Eq. (19), which is related to vanishing values of both Δmp and σW and interpolation errors in the computation of Δk. 3.2.
Impact of pump power
P = 1e+02 W p
0.5 0
log 10(P /P0 )
log 10(P /P0 )
log 10(P /P0 )
log 10(P /P0 )
Pump power has both an impact on the magnitude of the gain as well as the width and position of the generated bands, as seen on Fig. 7 (the noisy feature around the pump is attributed to the same numerical phenomena mentioned in the previous section). One can notice that the position of the inner set of phase-matched wavelengths tends to shift away from the pump as the power increases compared to the outer set of wavelengths, which remains relatively stationary. This can also be easily visualized by plotting the phase matching curve at different power levels, as seen on Fig. 8. Furthermore, as expected, the gain increases with increasing pump power. Taking this into account, it is theoretically possible to compensate for gain degradation by increasing pump intensity, depending on how important the fluctuations are. One can also note that the position of the outer set of wavelengths is much more stable as a function of pump power, since the slope of linear phase- mismatch Δk as a function of detuning is much more important in that area, given that this solution is a consequence of higher-order dispersion terms. Thus, a higher parametric shift (i.e. Pp ) is required to significantly move the phase-matching condition than it is at lower values of Ω. The behavior of gain as a function of pump power also suggests that a differential gain analysis with respect to Pp (for different fiber lengths) might help in quantifying the amplitude of stochastic variable W (z). Such quantification may
0.4
0.6
0.8
1
1.2
1.4 Pp = 5e+02 W
0.4
0.6
0.8
1
1.2
1.4 Pp = 1e+03 W
0.4
0.6
0.8
1
1.2
1.4 P = 5e+03 W
2 0
5
0
p
20 0
0.4
0.6
0.8 λ (μm)
1
1.2
1.4
Fig. 7. Illustration of the dependance of the expected gain G to pump power Pp . In these calculations, σΛ = 0.5 %, z = 0.2 m and Lc = 5 × 10−3 m.
actually prove to be quite useful, in particular when trying to decouple the effects of multiple contributions to the stochastic part of Δk. 3.3.
Impact of pump wavelength
As discussed on Fig. 2, the condition for stability, at a given nominal pitch, occurs for a specific set of pump and phase-matched wavelengths. However, inaccuracies in the fiber fabrication
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30870
1.4
Pp =0e+00 W
1.2
Pp =1e+02 W Pp =5e+02 W λ s,i (μm)
1 Pp =1e+03 W Pp =5e+03 W 0.8
0.6
0.4
0.58
0.59
0.6
0.61
0.62 λ p (μm)
0.63
0.64
0.65
0.66
Fig. 8. Evolution of the mean phase-matching condition (Δsp + Δip )/2 = 0 for different pump powers. The thicker solid line represents the linear phase-matching condition and “×” marks the position of the ZDW. The vertical line illustrates the position of the stable pump λ˜ p .
process could very well throw off the ZDW by a few nanometers, thus compromising both the intended phase-matching and stability. To characterize the impact of pump wavelength on stability, the expected gain was computed for a wide set of pump wavelengths, spanning the range around the stable point. A 3D plot of the gain map is shown on Fig. 9. One can notice that, generally, the gain falls as one gets further away from the pump λi,s ≈ λ p or the stable point (λ˜ p , λ˜ i,s ) in the phase-matching diagram. Around the pump, where the lower dispersion orders dominate, ∂ Δk/∂ Λ vanishes as the phase-matched wavelength pair gets closer to the pump, explaining the increase in gain of the inner set of phase-matched wavelengths as the pump gets closer to the ZDW. As for the outer set of wavelengths, in a small range of λ p around the stable point, the gain remains high with a FWHM (in log-space) of about 6 nm. This can be easily explained by looking at Fig. 2; indeed, as the pump gets closer to the stable point, all three phase-matching curves tend to pinch together, thus indicating that the global σW falls to zero, following the phase-matching curve, as one moves closer to the point of stability. Combining this effect with a careful selection of the phase matched wavelengths and fiber design, one could simultaneously benefit from a very low sensitivity to nominal pitch as well as some amount of leeway given by the low sensitivity to the pump wavelength. However, a change in the generated wavelength would still occur to some extent, but could be used, alongside other fiber characterization procedures, to determine how far one is from the intended design point. 4.
Propagation simulations using noisy longitudinal pitch profiles with different correlation lengths
In this section, we investigate the validation of the theory developed in the first section of this paper through propagation simulations using the Modified Generalized Nonlinear Schr¨odinger Equation (MGNLSE) integrated using a fourth-order Runge-Kutta in the interaction picture method (RK4IP) [38–40] with adaptive step-size and interpolation of the fiber parameters as a function of pitch (Aeff , β2 , etc.) This equation being a deterministic differential equation
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30871
8
log 10(P /P0 )
6 4 2 0 0.61 0.4
0.605 0.6
0.8
0.6 1
1.2
1.4
0.595
λ p (μm)
λ s,i (μm)
Fig. 9. Plots of the expected gain G as a function of pump wavelength λ p and signal/idler wavelength λs,i for fiber A. In these calculations, σΛ = 0.5 %, z = 0.2 m, Pp = 1kW and Lc = 5 mm. The FWHM (in log-space) of the stable point gain in terms of λ p is about 6 nm.
(by opposition to stochastic), several noisy pitch profiles Λ(z) have to be generated, propagated through and the average gain taken in order to compare these results with the solution of Eq. (11). First and foremost, OUP profiles for Λ have to be generated for different correlation lengths. The OUP presents a normal distribution around its mean and has an autocorrelation function given by Eq. (16). A method by which to generate such processes has been detailed in [31] and can be used to generate the noisy longitudinal pitch profiles needed for the simulations. Examples for two different correlation lengths Lc are shown on Fig. 10(a). Given the typical drawing parameters, response times of the control systems and the physics of the process, realistic values for Lc are expected to range from tens of centimeters to several meters or more [23]. Generally speaking, numerical results obtained using this method are in fairly good agreement with the ones described in the previous section. In particular, the fact that the gain at the outer set of phase-matched wavelengths is more robust to longitudinal scale variations, thanks to the stability condition, is corroborated. This confirms the validity of the analytical approach developed in section 2 and that this original robustness property is preserved even when taking into account several effects disregarded in the model such as SPM, XPM, Raman, dispersion (walk-off) and supercontinuum generation [6]. More precisely, the results presented on Fig. 10(b) represent the average gain for several simulations (10 for Lc = 1 mm, 30 for Lc = 10 cm) using different random Λ(z) profiles having the same correlation length Lc for λ p = 603 nm, z = 200 mm, a pulse length of 200 ps and a peak power Pp =1 kW. The choice of the pulse duration and power is intended to be a compromise between representing a CW pump in a broad spectral window with sufficient spectral resolution and keeping the number of points for each simulation reasonable (220 points). A noise of one photon per mode with random phase was used as a seed. As we can see, the general trend confirms that the gain on the inner bands decreases with increasing Lc while the gain of the outer bands remains high in spite of the other effects considered.
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30872
0.015 Lc = 0.001m Lc = 1m
Lc = 1e−03 m
0 −20 P (dB)
0.01
0.005
−40 −60
(Λ − Λ0 )/Λ0
−80 −100
0
0.4
0.5
0.6
0.7
0.8
0
−0.005
0.9
1
1.1
1.2
1.3
1.4
1
1.1
1.2
1.3
1.4
Lc = 1e−01 m
P (dB)
−20
−0.01
−40 −60 −80
−0.015
0
0.05
0.1
0.15
0.2
0.25 z (m)
(a)
0.3
0.35
0.4
0.45
0.5
−100 0.4
0.5
0.6
0.7
0.8
0.9 λ (µm)
(b)
Fig. 10. Numerical simulation data. (a) Example of noise profiles used for pitch perturbations in the MGNLSE simulations (the same step-size resolution has been used to generate both profiles). The standard deviation used for the OUP is 0.5% of the nominal pitch and the mean is zero. Initial conditions are random and normally distributed with the same standard deviation as the process and zero mean. (b) MGNLSE simulations for Lc = 1 mm (10 simulations) and Lc = 10 cm (30 simulations). The results shown are the average of the gain for simulations with different random scale profiles with the same correlation length Lc with z = 200 mm, a pulse duration of 200 ps and a peak power of Pp =1 kW.
5.
Conclusion
In this paper, we demonstrated the existence of a phase-matched FWM condition that is insensitive to scale fluctuations along the fiber length. The method by which to identify this point of stability has also been discussed. We noticed a number of different trends on the behavior of the stable point as a function of the design parameters, which can be of use as a starting point for design optimization. It has also been discussed that some solutions lie in a region where the pump is in abnormal dispersion regime, which may prove less useful as supercontinuum generation would occur. Propagation simulations were performed and agree fairly well with the analytical prediction of expected gain developed in this paper. Furthermore, the general steps for solving the FWM equations could be extended to other models with moderated longitudinally varying dispersion (such as dispersion-oscillating fibers [41] or tapered fibers where autoresonance is observed [42]) to assess the impact of fluctuations on such processes. Combined with dispersion engineering, this work can help on improving designs for nonlinear applications previously difficult to achieve due to dispersion instabilities. Acknowledgments This work was partly supported by French Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and FEDER through the “Contrat de Projets Etat R´egion (CPER) 2007-2013”, the “Campus Intelligence Ambiante” (CIA), and the FLUX Equipex Project (“Programme Investissement d’Avenir”).
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30873
de Physique des Lasers, Atomes et Mol´ecules, UMR 8523, IRCICA Research Institute for Advanced Communication, Parc Scientifique de la Haute Borne, CNRS USR 3380, 50, avenue Halley, 59658 Villeneuve d’Ascq cedex, France
2 Key
Lab of Specialty Fiber Optics and Optical Access Networks, School of Communication and Information Engineering, Shanghai University, Shanghai 200072, China ∗ [email protected]
Abstract: The four-wave mixing process in optical fibers is generally sensitive to dispersion uniformity along the fiber length. However, some specific phase matching conditions show increased robustness to longitudinal fluctuations in fiber dimensions, which affect the dispersion, even for signal and idler wavelengths far from the pump. In this paper, we present the method by which this point is found, how the fiber design characteristics impact on the stable point and demonstrate the stability through propagation simulations using the non-linear Schr¨odinger equation. © 2013 Optical Society of America OCIS codes: (060.2280) Fiber design and fabrication; (060.5295) Photonic crystal fibers; (190.4380) Nonlinear optics, four-wave mixing.
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#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30859
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#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30860
1.
Introduction
Four-wave mixing (FWM) in optical fibers has been extensively studied, increasingly so in the last two decades [1–3] as it presents a nuisance in WDM communication systems but also because it provides a leveraging platform for a range of interesting applications [4–6]. For instance, generation of correlated light at different wavelengths through this process, giving rise to entangled photons [7, 8], allows for quantum cryptography, [9–11] and a range of other applications necessitating entangled light sources [12, 13]. Additional applications, also based on FWM, include parametric amplification [14–16], phase-conjugation [17–19], frequency conversion (FOPO) [15, 20] and pulse generation [21] amongst others. It has been reported in various publications that the efficiency of this process is generally impaired by random longitudinal fluctuations of the dispersion along the fiber, which can limit the gain bandwidth in FOPAs and substantially diminish wavelength conversion efficiency [22–24]. Furthermore, this degradation is usually worse as the phase-matching condition occurs farther from the pump [23]. Highly nonlinear fiber designs have been studied which stabilize the dispersion in the presence of scale fluctuations [25, 26]. The idea relies on negating the scale-sensitivity of the dispersion by compensating the contributions of different waveguiding structures in a multi-layered index configuration. The effect of this is the flattening of the waveguide dispersion contribution in a region of interest and, for instance, can result in a fixed zero-dispersion wavelength invariant to scale fluctuations, which can be of use in a number of applications. This method, however, may prove challenging when the wavelength range becomes quite large (well in excess of 100 THz) and higher-order dispersion becomes important. It was also noted elsewhere that some phasematched sets of wavelengths are invariant with small scale perturbations for some PCF designs with two ZDW [27] and could potentially be used as stable degenerate FWM (DFWM) points but a full investigation on the properties of these phase-matching conditions and a gain analysis to validate the usability of these points remained to be performed. In this paper, we investigate, with the help of an elaborate DFWM model and numerical simulations, stable phase-matching points, both in wavelength and gain, that do not require the ZDW (or dispersion) to remain fixed and are particularly useful in terms of fiber manufacturing process, especially for large frequency spacings [28]. The procedure by which the dispersion can be analyzed to find the point of stability and a numerical investigation that shows the stability of the gain under these conditions (in a normal dispersion pumping regime) will be presented. As well, a method by which the design parameters of a hybrid photonic-crystal fiber (HPCF) can be tuned to have this stable point occur at a wide range of different sets of pump, signal and idler wavelengths will be discussed. 2.
Four-wave mixing and longitudinal dispersion fluctuations
To begin this study, let us first consider the general case of quasi-continuous-wave (CW) degenerate FWM where three waves (an intense, undepleted pump “p” and weak signal “s” and idler “i” with wavelengths λs < λ p < λi ) interact together through the third-order susceptibility tensor in an isotropic medium: 2 in2 ω p ∂ Ap = f pp A p A p (1) ∂z c 2 ∂ As in2 ωs = 2 fsp A p As + fsipp A2p A∗i exp(−iΔkz) (2) ∂z c 2 in2 ωi ∂ Ai = 2 fip A p Ai + fispp A2p A∗s exp(−iΔkz) (3) ∂z c where A denotes the envelope of the wave, fi j and fi jkl are the overlap integrals (considered constant for small variations in fiber scale and which can be readily derived, in vectorial form, #199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30861
by keeping the full fields at pump, signal and idler wavelengths in the non-linear polarization term of the wave equation) [6], n2 is the nonlinear refractive index of the material (assumed constant over the cross-section), ω is the angular frequency of the wave and Δk is the linear phase-mismatch term defined as: Δk = βs + βi − 2β p
(4)
where βm is the propagation constant at wavelengths λm . Under these conditions, the pump can 1/2 be trivially integrated to A p = Pp exp(iγ Pp z) with γ = n2 ω p f pp /c. The coupled equations for the signal and idler waves can thus be recast as follows:
∂ As ∂z ∂ A∗i ∂z
in2 ωs [2 fsp Pp As + fsipp Pp A∗i exp(−iθ z)] c in2 ωi ∗ ∗ − 2 fip Pp A∗i + fispp Pp As exp(iθ z) c
= =
(5) (6)
where θ = Δk − 2γ Pp . At this point, it is necessary to consider that the dispersion fluctuates the fiber length. In this case, the phase argument can readily be from θ z to along z generalized z (ζ )dζ ]/2} and simiθ ( ζ )d ζ . Taking this into account and substituting B = A exp{i[ θ s s 0 0 larly for the idler wave, we can reform the system of coupled differential equations as:
∂ Bs ∂z ∂ B∗ i i ∂z i
κsp Bs − γsipp Pp B∗i 2 ∗ κip ∗ = − B∗i + γispp Pp Bs 2 =
(7) (8)
where γmnpp = n2 ωm fmnpp /c. If we consider that the fluctuation of the dispersion can be modeled as a stochastic process around a constant mean value along the fiber length, we can write the global phase-mismatch term, κi j , as the sum of a constant term and a fluctuating (z-dependent) term:
κsp κip
= =
2(γ − 2γsp )Pp − Δkmean − Δkvar (z) 2(γ − 2γip )Pp − Δkmean − Δkvar (z)
(9) (10)
where γmp = n2 ωm fmp /c. Grouping the constant terms together, κmp /2 = Δmp + W (z), where Δmp = −[2(2γmp − γ )Pp + Δkmean ]/2 and W (z) = −Δkvar (z)/2, the whole system of differential equations can be re-written in matrix form:
∂B = [G + HW (z)]B ∂z where the matrices B, G and H are given by −iΔsp Bs B= , G= ∗ P −iγispp B∗i p
iγsipp Pp iΔ∗ip
(11)
,
H=
−i 0 0 i
.
(12)
Since W (z) is a stochastic variable, Eq. (11) is now considered to be a stochastic differential equation, or SDE [29], and the solution must also be a stochastic process. It is to be noted that the analysis proposed here is a generalization of the derivation presented in [23], keeping the loss terms and full overlap integrals as the case where signal and idler far from the pump will be considered, so the constant γ approximation becomes invalid. In order to push the analysis further, more information is needed on W (z). The first thing to mention is that if several components of very different statistical nature were to contribute to the stochastic fluctuation of Δk, #199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30862
several terms Wi (z) should be added to the SDE. However, in this case we will consider that the use of one process is enough and such formulation has been shown to produce accurate predictions [23]. In first approximation, we can assume that Δk is linearly related to physical stochastic processes that are occurring along the fiber. Let us thus consider a number of contributions X to the fluctuations of the phase-mismatch. The fluctuation of those parameters, δ X(z), can be characterized in terms of their correlation length Lc,X , which can be derived from the autocorrelation function RX (ζ ) = δ X(z)δ X(z + ζ ) (where “. . .” denotes the expected value operator with RX (0) = δ X(z)δ X(z) = σX2 , σX being the standard deviation of X) as: Lc,X = σX−2
∞ 0
RX (ζ )dζ
(13)
If we make the assumption that all contributions to the fluctuation in the dispersion are relatively small (i.e. can be expressed as a first order expansion), have zero mean and are symmetric around a nominal value Xi,0 , have the same correlation length Lc and are statistically independent, they can be combined in a single stochastic variable W (z) with standard deviation given by: 2 N ∂ Δk 1 σW2 = ∑ σX2i (14) 2 ∂ Xi i=1
Xi =Xi,0
For instance, lets assume that the change in the phase-mismatch is primarily related to a change in fiber dimensions (for instance, through the pitch Λ), which is a major contributor [28], neglecting other contributions, we can write the standard deviation of the stochastic variable W (z) as: 2 1 ∂ Δk 2 σW = σΛ2 (15) 2 ∂ Λ Λ=Λ0 A representative process that can be applied to optical fiber drawing is the OrnsteinUhlenbeck process (OUP) [30], which is basically used to model noisy relaxation and can be easily implemented [31]. It can be noted that this process has an autocorrelation function given by
|ζ | . (16) R(ζ ) = R(0) exp − Lc The parameter of interest in this study is the power at the signal and idler wavelengths (Ps = |As |2 and Pi = |Ai |2 ) for which a new set of four coupled SDEs for P = [Ps , Pi , 2Re(Bs Bi ), 2Im(Bs Bi )]T can be derived from the amplitude differential equations. This system of equations can be written exactly in the same form as Eq. (11) where G and H are now 4×4 matrices. To simplify the following calculations, one can derive that fispp = fsipp and is real. To further simplify the remaining calculations and illustrate the contribution of the frequencies ωs and ωi , it is convenient to define h = n2 fispp /c = n2 fsipp /c. The complete SDE system for power is thus given by ⎡ ⎤ 0 0 ωs hPp 2Im(Δsp ) ⎢ ⎥ 0 2Im(Δip ) 0 ωi hPp ⎥ G=⎢ (17) ⎣ 0 0 Im(Δsp + Δip ) Re(Δsp + Δip ) ⎦ 2ωs hPp −Re(Δsp + Δip ) Im(Δsp + Δip ) 2ωi hPp with H having only two non-zero elements H34 = −H43 = 2. Careful integration of Eq. (11) can be performed, yielding P(z) = exp (Gz + HB(z)) P(0),
(18)
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30863
with B(z) =
z
0 W (z
)dz .
Taking the average, as shown in [23], we get
H2 f (z) P(0). P(z) = exp Gz + 2
with, assuming the same type of autocorrelation function as Eq. (16),
z z −1+ . f (z) = 2Lc2 σW2 exp − Lc Lc
(19)
(20)
It is to be noted that the complete effect of the perturbations is contained in f (z); the behavior of this factor is thus of critical importance in characterizing the impact of fluctuations. This solution is generally complicated but has a simpler form in some cases. Let us first consider that Δsp and Δip are real in the absence of loss since, in this case, the propagation constants are real. Furthermore, γsp and γip are also real. Thus G11 = G22 = 0. For further simplification, we consider the gain at the nominal phase-matching condition such that Δsp + Δip = 0. For the case of spontaneous amplification of a small noise at the signal wavelength, equivalent to P(0) = [Ps0 , 0, 0, 0]T , the expected gain G = Ps /Ps0 is calculated to be (as shown in [23])
f (z) 1 1 + cosh[g(z)] + sinh[g(z)] exp[− f (z)] (21) G = 2 g(z) 1/2 where, however, g(z) is now defined as g(z) = ωs ωi (2hPp z)2 + f 2 (z) . Equation (21) is always maximum for f (z)=0, which occurs if σΛ = 0 or ∂ Δk/∂ Λ = 0. This implies that maximum gain should be obtained at the phase-matching condition, in the presence of fluctuations (σΛ = 0), when ∂ Δk/∂ Λ = 0. This formally illustrates, in other words, that if the phase matching does not change with small changes in the scale of the structure, then the gain should be stable as well. 3.
Investigation of FWM stability in the case of a hybrid PCF design with two ZDW
Let us consider the case of a fiber with two zero-dispersion wavelengths (ZDW). Such fibers are pertinent in this study as they can show two sets of phase-matched wavelengths and as we will show, under certain conditions, a stable point can be found at one set but not at the other. A particular fiber design will serve for our study, which exhibits two ZDWs; the hybrid PCF presented in [32], and shown on Fig. 1, represents a good example of such a fiber as the fundamental mode of the first all-solid photonic bandgap (ASPBG, at short wavelengths) smoothly transitions to the LP02 mode of a 7-cell defect PCF (at long wavelengths), both of which exhibit a ZDW [32]. More specifically, the design considered here consists of six Ge-doped inclusions, with maximum index difference of n0 −nSiO2 = 0.032 and dGe /Λ = 0.7, embedded in the sevencell defect of a PCF structure of air holes with dAir /Λ = 0.6. The nominal pitch Λ0 is selected to be around 1 μ m. Simulations were performed to compute the dispersion, at the nominal conditions described above, from the point where the mode becomes the fundamental PBG mode (its effective index neff < nSiO2 ) to the point where neff falls below nFSM , the effective index of the fundamental space-filling mode of the 7-cell hole lattice [32, 33]. These simulations were carried out by finite elements method, using commercially available software, and material dispersion was added pertubatively [6]. Although other fibers with similar two-ZDW have been studied [20, 34], the advantage of such a design is that both ZDW can theoretically be tuned by changing the parameters of the germanium inclusions and air-holes separately. More complicated structures giving additional design flexibility could also be envisaged, as suggested in [35]. #199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30864
Fig. 1. Illustration of the hybrid PCF fiber design under study (Fiber A). The gray area represents pure silica, the white areas represent air holes and the blue areas represent graded-index (parabolic) Ge-doped inclusions with a maximum index difference n0 − nSiO2 = 0.032. In this particular design, dAir /Λ = 0.6 and dGe /Λ = 0.7. The pitch is Λ0 = 1.03 μ m.
From Eqs. (15) and (20), one can see that the condition ∂ Δk/∂ Λ = 0 represents a point where the noise on the phase-matching vanishes (σW = 0, thus f (z) → 0) and maximal gain is obtained. If material dispersion is taken into account, this stable point will be a function of several design parameters. The concept of phase-matching stability can be further visualized through the dynamics of how the signal and idler wavelengths vary with a change in pitch. Figure 2 represents the linear phase-matching condition for fiber A (Fig. 1) at the nominal pitch Λ0 as well as Λ0 ± 1%. As it can be seen, if the pitch is changed by a percent, the phasematched wavelengths generally move around except at one point, where the Δk and ∂ Δk/∂ Λ are simultaneously equal to zero. It is to be noted too that the trivial condition λs = λi = λ p also satisfies this condition but is not of interest in this study as our focus is on wavelength pairs far from the pump. For such a fiber design, assuming fixed dGe /Λ, the stable point will be a function of relative hole size dAir /Λ, pitch Λ and pump wavelength λ p . For a given design, the pump, signal and idler wavelengths at the stable point will be identified as λ˜ p , λ˜ s and λ˜ i respectively. The trajectory of the stable point in (Λ,λ˜ p ,λ˜ s )-space can be plotted to reveal trends on its behavior as a function of the design choices, as seen on Fig. 3. From this data, it can be noted that smaller dAir /Λ will generally increase the signal wavelength (λ˜ s ). It can also be noted that the relationship between pitch and pump wavelength at the stable point is roughly proportional to λ˜ p at high values (panel A of Fig. 3). One last trend to be observed is that λ˜ s becomes less sensitive to changes in Λ, as seen on panel B of Fig. 3 (or λ˜ p , as seen on panel C) for higher pitch values. Indeed, at high pitch, λ˜ s (Λ) is, for all intents and purposes, a vertical line on panel B. This is very interesting from a manufacturing point of view as it suggests that, for a selected pump wavelength and dAir /Λ, obtaining a draw at the right diameter to generate the right stable point with a small deviation on the generated wavelength could be practically achieved by, for instance, making a very long taper in the fiber and then retrieving the segment with maximum gain. However, an additional consideration has to be taken into account and it is related to the type of dispersion at the pump wavelength. Indeed, the stable point may lie in the normal or abnormal dispersion regime of the pump. In the case where the pump is in the abnormal regime, supercontinuum generation from the pump fission into solitons [6] is likely to drain the pump
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30865
1.4
1.2
λ s,i (μm)
1 Δk = 0 (Fiber A) ∂Δk/∂Λ = 0 0.8
0.6
0.4
0.58
0.59
0.6
0.61
0.62 λ p (μm)
0.63
0.64
0.65
0.66
Fig. 2. Linear phase matching curve obtained for fiber A (Fig. 1). The thick solid line is the phase matching at the nominal pitch while the two thinner solid lines illustrate the phasematching for variations in the pitch Λ = Λ0 ± 1 % with nominal pitch Λ0 = 1.03 μ m. The dashed line represents the stability condition ∂ Δk/∂ Λ = 0. For this design, the stable point is at pump λ˜ p = 602.6 nm. The outer set of phase-matched wavelengths are λ˜ s =1,218.9 nm and λ˜ i = 400.2 nm. The inner set of phase-matched wavelengths are λs = 906.7 nm and λi = 451.2 nm. The first ZDW for fiber A is located at 617.5 nm
energy and reduce the efficiency of the phase-matched FWM at the stable point, although this remains to be investigated experimentally. An example of how the regime changes with pump wavelength for dAir /Λ = 0.6 in our design is shown on Fig. 4. It is to be noted that the abnormal dispersion regime is located between the two ZDW of any given curve. If the stable point lies in the normal dispersion regime however, it is generally at a point where two sets of phase-matched wavelengths exist, with only one (the outer set in our case) showing stability, as shown on Fig. 2. In order to perform this investigation, let us focus on DFWM in the normal dispersion regime which is achieved by selecting the design of fiber A (with λ˜ p around 600 nm) as, generally, the gain bands of the outer set are very narrow and far from the pump, which suggests they should be more sensitive to fluctuations of the dispersion. Furthermore, comparison with numerical propagation simulations will be much easier to interpret as no supercontinuum will be generated at the pump wavelength, thus greatly helping with visualization and comparison with the simple gain model presented in the previous section. In general, there are three factors that need to be examined to verify the practicality of the stationary phase-matching point: the correlation length of the dimensional noise Lc , pump power Pp and pump wavelength λ p . In order to evaluate the impact of those factors on the expected gain, Eq. (17) will be used, without approximations (i.e. full phase-mismatch, overlap integrals and loss terms included) to compute the expected gain G = Ps /Ps (0) from Eq. (19), seeding with noise of one photon per frequency and random phase at the fiber input. The following subsections contain computational results showing the influence of those three parameters on G. 3.1.
Impact of correlation length
The impact of correlation length on gain degradation for FWM interaction in standard fiber has been reported in several publications [22,23,36] and is known to be worse for longer correlation
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30866
dAir /Λ =0.6
Stable point parameters
dAir /Λ =0.62 dAir /Λ =0.64 dAir /Λ =0.66 dAir /Λ =0.68 1.4
A
1.3 1.2
B
Λ (μm)
1.1 1 0.9 0.8 0.7 0.6
C 0.55 0.5 0.65
0.45 0.6 0.4
0.55
˜ i (μm) λ
˜ p (μm) λ
Fig. 3. Mapping of the stable point as a function in (Λ,λ˜ p ,λ˜ s )-space for different dAir /Λ and Λ, all other parameters being the same as for fiber A. The projection of the 3D trajectories on each couple of parameters is also plotted on panels A, B and C to help illustrate the different trends. 2.2 ˜ p = 0.65 μm, Λ = 1.49 μm λ
2
˜ p = 0.60 μm, Λ = 1.05 μm λ
1.8
˜ p = 0.54 μm, Λ = 0.70 μm λ
λ i,s (μm)
1.6 1.4 1.2 1 0.8 0.6 0.4 0.5
0.6
0.7
0.8 λ p (μm)
0.9
1
1.1
Fig. 4. Example of the phase matching curve with the stable point at different pump wavelengths (and corresponding pitch) for dAir /Λ = 0.6. The black circles denote the position of the stable set of wavelengths λ˜ i,s for each pump wavelength λ˜ p and the black “×” indicate the position of the ZDW. The dashed line represents the trivial solution λi,s = λ p . We notice that, as the wavelength λ˜ p gets shorter, the stable point moves closer to the ZDW as the pump then crosses into the abnormal dispersion regime. Missing points in the phase matching curves lie outside the simulation domain and thus have been discarded. It is to be noted that the pump is in the abnormal dispersion regime when it is located between the two ZDW for a given curve.
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30867
log 10(P /P0 )
10
log 10(P /P0 )
10
log 10(P /P0 )
10
log 10(P /P0 )
lengths. Experiments have also been performed to effectively reduce the correlation length by cutting a fiber into short pieces and randomly splicing them back together to notice an improvement in FWM gain [37]. Close inspection of Eq. (20) also shows that long correlation lengths will have a worse impact on gain as, for fixed z, f (z) is a monotonically increasing function of Lc . One thing to note, however, is that f (z) is always proportional to σW2 , which means that the degradation of gain, at the phase-matched condition, is always related to the amplitude of the underlying perturbation. Computation of the gain spectrum for fiber A is shown on Fig. 5, all plots sharing the same scale, and confirms the degradation of the gain as Lc increases, especially for the inner set of phase-matched wavelengths while the (stable) outer set is virtually unaffected. It is also interesting to note the broadening of the gain bands. In more detail, we note that the Lc = 0 graph shows lower gain at the outer wavelengths, which is caused both by the frequency spacing Ω = ωi − ω p = ω p − ωs and monotonically decreasing overlap integral term for FWM, h. Although this drop in h will diminish gain, its impact is reasonable as fully-vectorial computation of fmp and fmnpp from simulation data yields values that stay within the same order of magnitude across the band, as seen on Fig. 6(a), suggesting that the drop in gain is not very large when going from the inner to the outer set of phase-matched wavelength pairs. Compared with the standard gain term γ Pp = n2 f pp ω p Pp /c, usually considered constant √ for all wavelengths (i.e. ωi ≈ ωs ≈ ω p and fispp = fsipp ≈ f pp ), the effective gain, ωi ωs hPp has to drop also due to the frequency spacing Ω increase when moving away from the pump as ωs ωi = ω p2 − Ω2 ≤ ω p2 . But as the effect of Lc is introduced, the stability of the outer set of wavelengths to perturbations changes the balance and the gain at the inner set of phase-matched wavelengths becomes lower than that of the outer set (to the point of vanishing, whereas the gain on the outer set drops by less than 15 % for Lc = 10 m) and constitutes the key property of the stability point. Indeed, this property indicates that a set of phase-matched wavelengths located very far from the pump and relying on higher-order dispersion (generally meaning narrow gain bandwidth) can be virtually insensitive to perturbations in the fiber scale, which
10
L =0m c
5 0
0.4
0.6
0.8
1
1.2
1.4 Lc = 1e−03 m
0.4
0.6
0.8
1
1.2
1.4 L = 1e−01 m
5 0
c
5 0
0.4
0.6
0.8
1
1.2
1.4 L = 1e+01 m c
5 0
0.4
0.6
0.8 λ (μm)
1
1.2
1.4
Fig. 5. Illustration of the dependance of the expected gain G to spatial noise correlation length Lc . In these calculations, σΛ = 0.5 %, z = 0.2 m and Pp = 1 kW. It is to be noted that all graphs share the same scale in order to better appreciate the change in gain as Lc is increased. One can also notice that the outer set of wavelengths, where the stability condition occurs, suffers very little gain degradation whereas the inner set vanishes.
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30868
11
x 10
10
10 |fmnpp |
3.5
|fmp |
8
10
6
2.5
10
Ideal gain
G
Overlap integrals (m −2 )
3
2
4
10 1.5
2
10 1
0
0.4
0.6
0.8 λ (μm)
(a)
1
1.2
1.4
10
−4
10
−3
10
−2
10
−1
σΛ (μm)
10
0
10
1
10
(b)
Fig. 6. (a) Values of the overlap integrals fmnpp and fmp as a function of wavelength for a fixed λ p = 602.6 nm. The four-wave mixing term (blue) is symmetric (in frequency) and both curves cross at the pump wavelength with the value fmnpp = fmp = f pp = 1/Aeff,p . Black circles have been added on the curves at the set of wavelengths under study. As we can see, both factors vary by less than an order of magnitude over the entire domain. The abscissa represents λm , λn being the conjugate wavelength. (b) Gain at the stable point as a function of perturbation amplitude σΛ for Lc = 10 m and z = 0.2 m.
appears counter-intuitive. Let us take as an example a standard PCF design operated in the LP02 mode, such as described in [28]. With d/Λ = 0.9 and nominal pitch Λ0 = 3.27 μ m, such a design would have the same phase matched wavelengths as the stable set of fiber A for the same pump wavelength. Comparing peak gain for both the hybrid and standard PCF, calculated using Eq. (19), we note an improvement of over four orders of magnitude in scale fluctuation tolerance σΛ , favoring the hybrid PCF for any given Lc . This tolerance criteria is defined where the logarithm of the expected gain, i.e. log10 (G), falls to half of its ideal value [25]. Indeed, if compared numerically, this improvement should be in the ratio of ∂ Δk/∂ Λ for each fiber, which yields 1.075 × 104 at the phase-matched wavelengths. In terms of actual fluctuations amplitude, the hybrid PCF should be able to have a tolerance in excess of 3.5% in σΛ /Λ for Lc = 10 m and z = 0.2 m, as seen on Fig. 6(b), which is easily achievable. This gives hope to increase conversion efficiency in FWM applications involving wavelengths far from the pump, regardless of the presence of small fluctuations in scale, which can otherwise become very demanding from the fiber manufacturing point of view. Another feature to notice is the presence of gain where there is normally no phase-matching, for instance, the bumps around the pump for large values of Lc . There is no analytical way to write P for non-zero phase-mismatch (Δip + Δsp ), as expressed before, however looking at the solution of the SDE, Eqs. (17) and (18), can provide some insight on the cause. For small values of z, there is a probability that the stochastic part of the integrated fluctuation over z, H34,43 B(z) compensates the uniform phase-mismatch term, G34,43 z, thus giving rise to non-zero net gain. The likelihood of this happening is higher for small values of phase-mismatch and the gain generated thus is always going to be relatively small as it relies on the probability of a nonnominal condition to be satisfied, keeping in mind that the highest probability is that B(z) is zero. This very well explains the broadening of the gain and the apparition of bumps around phase-matched conditions but in places where there is always a non-zero phase-mismatch term. This reasoning also implies that this condition will be a function of σW and Lc , i.e. a fluctuating process is necessary for those bumps to appear as, otherwise, the phase-mismatch term could
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30869
never be compensated. The small feature located around the pump and present on all graphs is attributed to numerical inaccuracies coming from the quasi-singular nature of the matrix being exponentiated in Eq. (19), which is related to vanishing values of both Δmp and σW and interpolation errors in the computation of Δk. 3.2.
Impact of pump power
P = 1e+02 W p
0.5 0
log 10(P /P0 )
log 10(P /P0 )
log 10(P /P0 )
log 10(P /P0 )
Pump power has both an impact on the magnitude of the gain as well as the width and position of the generated bands, as seen on Fig. 7 (the noisy feature around the pump is attributed to the same numerical phenomena mentioned in the previous section). One can notice that the position of the inner set of phase-matched wavelengths tends to shift away from the pump as the power increases compared to the outer set of wavelengths, which remains relatively stationary. This can also be easily visualized by plotting the phase matching curve at different power levels, as seen on Fig. 8. Furthermore, as expected, the gain increases with increasing pump power. Taking this into account, it is theoretically possible to compensate for gain degradation by increasing pump intensity, depending on how important the fluctuations are. One can also note that the position of the outer set of wavelengths is much more stable as a function of pump power, since the slope of linear phase- mismatch Δk as a function of detuning is much more important in that area, given that this solution is a consequence of higher-order dispersion terms. Thus, a higher parametric shift (i.e. Pp ) is required to significantly move the phase-matching condition than it is at lower values of Ω. The behavior of gain as a function of pump power also suggests that a differential gain analysis with respect to Pp (for different fiber lengths) might help in quantifying the amplitude of stochastic variable W (z). Such quantification may
0.4
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1.4 Pp = 5e+02 W
0.4
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1.4 Pp = 1e+03 W
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2 0
5
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p
20 0
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Fig. 7. Illustration of the dependance of the expected gain G to pump power Pp . In these calculations, σΛ = 0.5 %, z = 0.2 m and Lc = 5 × 10−3 m.
actually prove to be quite useful, in particular when trying to decouple the effects of multiple contributions to the stochastic part of Δk. 3.3.
Impact of pump wavelength
As discussed on Fig. 2, the condition for stability, at a given nominal pitch, occurs for a specific set of pump and phase-matched wavelengths. However, inaccuracies in the fiber fabrication
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30870
1.4
Pp =0e+00 W
1.2
Pp =1e+02 W Pp =5e+02 W λ s,i (μm)
1 Pp =1e+03 W Pp =5e+03 W 0.8
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0.62 λ p (μm)
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Fig. 8. Evolution of the mean phase-matching condition (Δsp + Δip )/2 = 0 for different pump powers. The thicker solid line represents the linear phase-matching condition and “×” marks the position of the ZDW. The vertical line illustrates the position of the stable pump λ˜ p .
process could very well throw off the ZDW by a few nanometers, thus compromising both the intended phase-matching and stability. To characterize the impact of pump wavelength on stability, the expected gain was computed for a wide set of pump wavelengths, spanning the range around the stable point. A 3D plot of the gain map is shown on Fig. 9. One can notice that, generally, the gain falls as one gets further away from the pump λi,s ≈ λ p or the stable point (λ˜ p , λ˜ i,s ) in the phase-matching diagram. Around the pump, where the lower dispersion orders dominate, ∂ Δk/∂ Λ vanishes as the phase-matched wavelength pair gets closer to the pump, explaining the increase in gain of the inner set of phase-matched wavelengths as the pump gets closer to the ZDW. As for the outer set of wavelengths, in a small range of λ p around the stable point, the gain remains high with a FWHM (in log-space) of about 6 nm. This can be easily explained by looking at Fig. 2; indeed, as the pump gets closer to the stable point, all three phase-matching curves tend to pinch together, thus indicating that the global σW falls to zero, following the phase-matching curve, as one moves closer to the point of stability. Combining this effect with a careful selection of the phase matched wavelengths and fiber design, one could simultaneously benefit from a very low sensitivity to nominal pitch as well as some amount of leeway given by the low sensitivity to the pump wavelength. However, a change in the generated wavelength would still occur to some extent, but could be used, alongside other fiber characterization procedures, to determine how far one is from the intended design point. 4.
Propagation simulations using noisy longitudinal pitch profiles with different correlation lengths
In this section, we investigate the validation of the theory developed in the first section of this paper through propagation simulations using the Modified Generalized Nonlinear Schr¨odinger Equation (MGNLSE) integrated using a fourth-order Runge-Kutta in the interaction picture method (RK4IP) [38–40] with adaptive step-size and interpolation of the fiber parameters as a function of pitch (Aeff , β2 , etc.) This equation being a deterministic differential equation
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30871
8
log 10(P /P0 )
6 4 2 0 0.61 0.4
0.605 0.6
0.8
0.6 1
1.2
1.4
0.595
λ p (μm)
λ s,i (μm)
Fig. 9. Plots of the expected gain G as a function of pump wavelength λ p and signal/idler wavelength λs,i for fiber A. In these calculations, σΛ = 0.5 %, z = 0.2 m, Pp = 1kW and Lc = 5 mm. The FWHM (in log-space) of the stable point gain in terms of λ p is about 6 nm.
(by opposition to stochastic), several noisy pitch profiles Λ(z) have to be generated, propagated through and the average gain taken in order to compare these results with the solution of Eq. (11). First and foremost, OUP profiles for Λ have to be generated for different correlation lengths. The OUP presents a normal distribution around its mean and has an autocorrelation function given by Eq. (16). A method by which to generate such processes has been detailed in [31] and can be used to generate the noisy longitudinal pitch profiles needed for the simulations. Examples for two different correlation lengths Lc are shown on Fig. 10(a). Given the typical drawing parameters, response times of the control systems and the physics of the process, realistic values for Lc are expected to range from tens of centimeters to several meters or more [23]. Generally speaking, numerical results obtained using this method are in fairly good agreement with the ones described in the previous section. In particular, the fact that the gain at the outer set of phase-matched wavelengths is more robust to longitudinal scale variations, thanks to the stability condition, is corroborated. This confirms the validity of the analytical approach developed in section 2 and that this original robustness property is preserved even when taking into account several effects disregarded in the model such as SPM, XPM, Raman, dispersion (walk-off) and supercontinuum generation [6]. More precisely, the results presented on Fig. 10(b) represent the average gain for several simulations (10 for Lc = 1 mm, 30 for Lc = 10 cm) using different random Λ(z) profiles having the same correlation length Lc for λ p = 603 nm, z = 200 mm, a pulse length of 200 ps and a peak power Pp =1 kW. The choice of the pulse duration and power is intended to be a compromise between representing a CW pump in a broad spectral window with sufficient spectral resolution and keeping the number of points for each simulation reasonable (220 points). A noise of one photon per mode with random phase was used as a seed. As we can see, the general trend confirms that the gain on the inner bands decreases with increasing Lc while the gain of the outer bands remains high in spite of the other effects considered.
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30872
0.015 Lc = 0.001m Lc = 1m
Lc = 1e−03 m
0 −20 P (dB)
0.01
0.005
−40 −60
(Λ − Λ0 )/Λ0
−80 −100
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Lc = 1e−01 m
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0.9 λ (µm)
(b)
Fig. 10. Numerical simulation data. (a) Example of noise profiles used for pitch perturbations in the MGNLSE simulations (the same step-size resolution has been used to generate both profiles). The standard deviation used for the OUP is 0.5% of the nominal pitch and the mean is zero. Initial conditions are random and normally distributed with the same standard deviation as the process and zero mean. (b) MGNLSE simulations for Lc = 1 mm (10 simulations) and Lc = 10 cm (30 simulations). The results shown are the average of the gain for simulations with different random scale profiles with the same correlation length Lc with z = 200 mm, a pulse duration of 200 ps and a peak power of Pp =1 kW.
5.
Conclusion
In this paper, we demonstrated the existence of a phase-matched FWM condition that is insensitive to scale fluctuations along the fiber length. The method by which to identify this point of stability has also been discussed. We noticed a number of different trends on the behavior of the stable point as a function of the design parameters, which can be of use as a starting point for design optimization. It has also been discussed that some solutions lie in a region where the pump is in abnormal dispersion regime, which may prove less useful as supercontinuum generation would occur. Propagation simulations were performed and agree fairly well with the analytical prediction of expected gain developed in this paper. Furthermore, the general steps for solving the FWM equations could be extended to other models with moderated longitudinally varying dispersion (such as dispersion-oscillating fibers [41] or tapered fibers where autoresonance is observed [42]) to assess the impact of fluctuations on such processes. Combined with dispersion engineering, this work can help on improving designs for nonlinear applications previously difficult to achieve due to dispersion instabilities. Acknowledgments This work was partly supported by French Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and FEDER through the “Contrat de Projets Etat R´egion (CPER) 2007-2013”, the “Campus Intelligence Ambiante” (CIA), and the FLUX Equipex Project (“Programme Investissement d’Avenir”).
#199150 - $15.00 USD Received 8 Oct 2013; revised 27 Nov 2013; accepted 28 Nov 2013; published 6 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.030859 | OPTICS EXPRESS 30873
