Waveguide coupling in the few-cycle regime
Herv´e Leblond1and Said Terniche1,2
1LUNAM Universit´e, Universit´e d’Angers, Laboratoire de Photonique d’Angers, EA 4464, 2 Boulevard Lavoisier, 49000 Angers, France
2Laboratoire Electronique Quantique, USTHB, BP 32 El-Alia, 16011 Bab Ezzouar Alger, Algeria (Received 21 January 2016; published 22 April 2016)
We consider the coupling of two optical waveguides in the few-cycle regime. The analysis is performed in the frame of a generalized Kadomtsev-Petviashvili model. A set of two coupled modified Korteweg–de Vries equations is derived, and it is shown that three types of coupling can occur, involving the linear index, the dispersion, or the nonlinearity. The linear nondispersive coupling is investigated numerically, showing the formation of vector solitons. Separate pulses may be trapped together if they have not initially the same location, size, or phase, and even if their initial frequencies differ.
DOI:10.1103/PhysRevA.93.043839
I. INTRODUCTION
There are many studies devoted to nonlinear propagation of ultrashort light pulses in the few- and subcycle regimes.
The first theoretical descriptions of the phenomena [1] were based on the slowly varying envelope approximation (SVEA), by means of generalizations of the nonlinear Schr¨odinger equation. This approach has evidenced various original effects, e.g., the limitation of the soliton size by a second-order dispersion of the nonlinearity [2]. However, several models have been proposed to go beyond the SVEA. Some of them are still spectral approaches, such as the so-called unidirectional pulse propagation equation [3,4]. Other works are based on the Drude approximation and the Maxwell equations [5,6]. An alternative approach is to seek for generic approximate models for the electric field itself, such as the modified Korteweg–
de Vries (mKdV) [7], the short-pulse [8], the sine-Gordon (sG) [9], the double sine-Gordon [10], the mKdV-sG [11]
equations, and others [12–14]. It has been shown that the most general of these models is the mKdV-sG one [15]. However, most of these studies are one-dimensional. It has been proved that in bulk medium, the cubic nonlinearity can lead either to catastrophic self-focusing or to enhancement of the diffraction effect [16], depending on the sign of the nonlinear coefficient, while in the presence of quadratic nonlinearity, either stable plane waves or stable spatiotemporal solitons can form [17].
One-dimensional media supporting light propagation are optical waveguides and fibers, based on either index variations or photonic crystal properties. Experiments of supercontinuum generation in fibers, especially hollow-core photonic ones (see, e.g., the review [18]) are closely related to few-cycle pulse propagation [19,20]. The propagation of few-cycle pulses in coupled waveguides has been considered theoretically in the frame of the unidirectional pulse propagation equation [21], and experiments on ultrashort pulse propagation in waveguide arrays have also been reported [22].
However, the generalization of the guiding phenomenon to the few-cycle regime beyond SVEA is most frequently treated as if it should not give rise to any difficulty, which is far from being true. Indeed, guided propagation modes are intrinsically objects of the linear theory and, from the mathematical point of view at least, they in principle lose any meaning in a strongly nonlinear regime. In the frame of a generalized Kadomtsev-Petviashvili (GKP) model, it was
shown that guided modes suffer an important widening in the few-cycle regime, insofar as it may happen that guiding is lost in the subcycle regime, while nonlinear waveguiding of the ultrashort pulses can occur [23].
In the present paper we consider the coupling between two waveguides in the few-cycle regime. We use the same approach as in [23], starting from an approximate generic non-SVEA model of GKP type, in a simple waveguide structure. We show that three types of coupling can exist, one of which is a linear coupling which straightforwardly generalizes the one which is known to occur in the continuous regime and for pulse of moderate shortness. The other type of coupling is also linear, but due to dispersion properties, while the third type is fully nonlinear and specific to the considered situation. The derivation of the coupled model equations is then illustrated by a set of numerical computations assuming linear coupling only.
II. LINEAR COUPLING
We consider a two-dimensional waveguiding structure consisting of two cores labeled by 1 and 2 surrounded by some dielectric cladding; cf. Fig. 1. The propagation of ultrashort optical pulses is governed by the generalized Kadomtsev-Petviashvili (GKP) equation [16,24,25], which reads in dimensionless form as
∂zu=Aα∂t3u+Bα∂tu3+Vα∂tu+wα 2
t
∂x2udt, (1) whereα=gin the cores 1 and 2,α=cin the cladding.
The dimensionless variables are defined asz=ζ /L,t= (τ −ζ /v0)/τw,x=ξ / l, andu=E/E0,ζ,τ,ξ, andEbeing the dimensioned propagation distance, time, space variable, and electric field, respectively. The quantities L and l are longitudinal and transverse reference lengths, andτwandE0 are reference time and reference electric field, respectively.
The velocity v0=cv/n0 (we denote the speed of light in vacuum by cv to avoid confusion) is chosen to be close to both linear velocitiesvgandvcin the guide core and cladding and these velocities are replaced withv0inwα, in front of the antiderivative (integral) term in Eq. (1).
The dimensionless nonlinear coefficient is Bα =
−γαLE20 τw
, (2)
FIG. 1. Schema of the guiding structure.
where its dimensioned counterpart is γα= 1
2nαcv
χα(3), (3) in whichnα (α=g,c) are the linear refractive indices, χα(3) (α=g,c) are the third-order susceptibilities at low frequency limit, and the dispersion parameters are, in dimensionless form,
Aα= (−βαL)
τw3 , (4)
where
βα= −nα
2cv
, (5)
the prime denoting the derivative with respect to the angular frequency.
The dimensionless parameters accounting for velocity mismatch and diffraction are
Vα= L τw
1 v0 − 1
vα
(6) and
wα= vαLτw
l2 v0Lτw
l2 . (7)
The reference time τw is fixed arbitrarily to the order of magnitude of the optical period and pulse length; i.e., we work in the femtosecond range.
In the linearized model (Bα=0), each waveguide con- sidered separately possesses some guided mode profile fj, (j =1, 2), so that the field of a wave propagating in guidej has the form
u=fj(x)ei(ωt−βz)=fj(x)eiϕ. (8) The equation satisfied byfj is
∂x2fj +Kαfj =0, (9) with
Kα= 2ω wα
(Aαω3−β−ωVα), (10) in each medium. We seek for a solution of the model equations in the form of two interacting modes, i.e.,
u=R(z)f1(x)eiϕ+S(z)f2(x)eiϕ, (11) wheref1is a mode of guide 1 alone, andf2a mode of guide 2 alone. Reporting (11) into the GKP equation (1), we get
∂zRf1+∂zSf2 = −iwα 2ω
Kα(Rf1+Sf2)−R∂x2f1−S∂x2f2 , (12) which reduces to
∂zRf1+∂zSf2 =0 (13)
in the cladding,
∂zRf1+∂zSf2= iwg
2ω(Kc−Kg)Sf2 (14) in guide 1, and
∂zRf1+∂zSf2=iwg
2ω(Kc−Kg)Rf1 (15) in guide 2. Multiplying Eqs. (13) to (15) byf1and integrating over allx yields
∂zR+I1∂zS =iwg
2ω(Kc−Kg)I2(R+S), (16) with I1= −∞∞ Rf1f2dx, I2= g1f1f2dx= g2f1f2dx, where “ g
j·dx” designates the integration over the core of the guidej =1 or 2. The mode profilesf1 andf2 are real, see [23], and normalized so that −∞∞ fj2dx =1 forj =1 or 2. In the same way we have
I1∂zR+∂zS =iwg
2ω(Kc−Kg)I2(R+S), (17) with the same integrals I1 and I2, since we assume two identical channels. The set of equations (16) and (17) is straightforwardly solved to yield
∂zR =∂zS= iwg 2ω
(Kc−Kg)I2
1+I1 (R+S). (18) The few-cycle pulse is not monochromatic, but can be con- sidered as a Fourier integral of such modes,u1= Reiϕdω, u2= Seiϕdω. We derive u1 with respect toz, report (18) into the obtained integral, replaceKαusing (10), which yields
∂zu1= −iβR− I2
1+I1[(Ac−Ag)(iω)3 +iω(Vc−Vg)](R+S)
eiϕdω. (19) The factorsiωcorrespond to derivatives in the time domain, and Eq. (19) can be rewritten as
∂zu1=
(−iβ)Reiϕdω− I2 1+I1
(Ac−Ag)∂t3
+(Vc−Vg)∂t
(u1+u2). (20) In the absence of the second waveguide, we getI2=0, and the above equation reduces to
∂zu1=
(−iβ)Reiϕdω, (21) which hence is nothing but the equation satisfied by the amplitude of the guide profile in a single guide. According to [23], we can approximate it by the modified Korteweg–
de Vries (mKdV) equation which is the (1+1)-dimensional reduction of Eq. (1) in the waveguide core, as
∂zu1 =Ac∂t3u1+Bc∂tu31+Vc∂tu1. (22) The system of partial differential equations which governs the evolution of the fields in both channels when linear
coupling is taken into account is thus
∂zu1=A∂t3u1+B∂tu31+V ∂tu1+C∂tu2+D∂t3u2, (23)
∂zu2=A∂t3u2+B∂tu32+V ∂tu2+C∂tu1+D∂t3u1, (24) where we have set
C= − I2
1+I1(Vc−Vg), (25) D= − I2
1+I1(Ac−Ag), (26) A=Ac+D,B=Bc,V =Vc+C. The first coupling terms, with the first derivative, generalize the coupling which occurs in continuous waves or long pulses. The second terms are related to the dispersion, which has a much higher effect in the two-cycle regime than for longer pulses.
III. NONLINEAR COUPLING
In order to derive the nonlinear coupling term, we rewrite Eq. (1) as
∂zu=Lα(u)+Bα∂tu3, (27) Lαbeing the linear differential operator on the right-hand side of (1). We again look for an approximate solution in the form u=R(z)f1(x)eiϕ+S(z)f2(x)eiϕ+c.c., (28) where c.c. stands for complex conjugate. We set u1= R(z)eiϕ+c.c.,u2=S(z)eiϕ+c.c., and then
∂zu=Lα(u)+Bα∂t
u31f13+3u21u2f12f2 +3u1u22f1f22+u32f23
. (29)
Multiplying byf1and integrating over allx yields
∂zu1+I1∂zu2 = ∞
−∞
Lα(u)dx+∂t
u31I3+3u21u2I4,
+3u1u22I5+u32I6
, (30)
where we have setI3= −∞∞ Bαf14dx,I4= −∞∞ Bαf13f2dx, I5= −∞∞ Bαf12f22dx, andI6= −∞∞ Bαf1f23dx.
Let us denote bydthe distance between the two channels.
The integrals decompose as ∞
−∞
Bαf1nf2mdx=
g1
Bαf1nf2mdx+
g2
Bαf1nf2mdx. (31) Since the mode profile f2 evolves as f2∼C2e−κx as x becomes large, the integral
g1
Bαf1nf2mdx∼
g1
Bαf1n(C2e−κd)mdx (32) decreases as e−mκd as d becomes large. In the same way, f1∼C1eκxasx tends to−∞and
g2
Bαf1nf2mdx∼
g2
Bα(C1e−κd)nf2mdx (33) decreases ase−nκdfor larged. Consequently −∞∞ Bαf1nf2mdx decreases ase−min(m,n)κd. It is thus seen thatI2,I4,I6are small
with respect toI3, and thatI1andI5are much smaller. HenceI1 andI5can be neglected. Further, the symmetry of the structure shows thatI6 =I4. Finally,
∂zu1= ∞
−∞
Lα(u)dx+∂t
I3u31+I4
3u21u2+u32 . (34) It is seen that in the absence of coupling, Eq. (34) reduces to
∂zu1= ∞
−∞Lα(u)dx+I3∂tu31; (35) henceI3is nothing but the effective nonlinear coefficient of the model equation for a single mode, and −∞∞ Lα(u)dxcoincides with the linear part of this equation.
Identifying u1 andu2 with the variables of the previous section with the same name (which is an approximation), we get the equations
∂zu1=A∂t3u1+B∂tu31+V ∂tu1 +C∂tu2+D∂t3u2+E∂t
3u21u2+u32 , (36)
∂zu2=A∂t3u2+B∂tu32+V ∂tu2 +C∂tu1+D∂t3u1+E∂t
3u1u2+u31 , (37) where the coefficientsA,B,C, andDare the same as in Sec.II, andE=I4.
Preliminary computations (see the following section for the description of the numerical procedure) show that the nonlinear coupling can be effective even with a low value of the coefficientE. As an example, we assumed two identical initial pulses, one in each channel, slightly shifted one with respect to the other: due to the nonlinear coupling, they evolve into two vector solitons with different velocities, which separate in time. Complete separation of the two pulses was observed for a ratioE/Bas low as 0.002. Owing to its great complexity, the detailed analysis of evolution of few-cycle-pulse solitons in the presence of nonlinear coupling is left for further investigation.
IV. EXAMPLES OF BEHAVIOR OF THE LINEAR NONDISPERSIVE COUPLING
We solve the model derived above in dimensionless form, assuming a purely linear and nondispersive coupling, namely
∂zu= −∂t(u3)−∂t3u−C∂tv, (38)
∂zv = −∂t(v3)−∂t3v−C∂tu, (39) using a standard fourth-order Runge-Kutta scheme in the Fourier domain. The nonlinear terms are computed by means of one inverse and one direct fast Fourier transform at each substep of the scheme. The initial data are
u=Ausin (ωut+ϕu)e−(t−tu)2/τu2, (40) v=Avsin (ωvt+ϕv)e−(t−tv)2/τv. (41) The angular frequencyωj and pulse half-duration at 1/e2 τj
(j =1 for u and 2 for v) are written in terms of some wavelengths λj and full width at half maximum FWHMj
(a)
(b)
(c)
FIG. 2. Formation of a vector soliton. (a) Input, (b) output atz=2 without coupling (C=0), (c) output atz=2 with couplingC= −1.
The solid blue line is u, the dashed red line isv. The parameters areAu=2,Av=0.2,λu=λv=1, FWHMu=FWHMv =3,ϕu= ϕv=0,tu=tv=0.
according to the standard relations ωj =2π c/λj andτj = FWHMj/√
2 ln 2, wherec=0.3 is the speed of light in vac- uum expressed inμm/fs, so that the dimensionless parameters λjand FWHMj can be easily compared to wavelengths inμm and a pulse duration in fs.
A soliton is launched in one channel (sayu), and a smaller input with analogous duration in the other (v). In the absence of coupling, the pulse uforms a soliton, and the pulse v is spread out by dispersion. In the presence of linear coupling, a vector soliton forms, which involves localized intensity on both channels (Fig. 2). Obviously it assumes that some energy is transferred from one channel to the other. This
0 0.2 0.4 0.6 0.8 1
0.8 2.2 3.6 = +1
= 0
= −1
= 0
= +1 = −1
u, c
v, c
v, c
u, c
v, c u, c
z
maxt|u|
t|v|
max
FIG. 3. Evolution of the amplitudes in both channels: plots of maxt|u|, maxt|v|versus the propagation distancez. ForC= +1, u grows andvdecreases first; forC= −1, it is the contrary. The parameters areAu=Av =2,λu=λv =1, FWHMu=FWHMv = 3,ϕu=0,ϕv=π/2,tu=tv=0.
(a)
(b)
(c)
FIG. 4. (a) Input, (b) output atz=2 without coupling (C=0), (c) output atz=2 with couplingC= −1. The solid blue line isu, the dashed red line isv. The parameters areAu=Av=3,λu=λv=1, FWHMu=FWHMv =2,ϕu=ϕv=0,tu=1,tv = −1.
energy exchange can occur periodically as in the case of monochromatic waves; see Fig.3. It depends on the sign of the coupling.
The mutual trapping of the two solitons can occur even if their centers do not coincide exactly at the beginning of the process. An example is shown in Fig. 4: the two input pulses are identical, but shifted alongt; without coupling, they propagate as two identical solitons shifted in position, but with the coupling, they form a single vector soliton, whose intensity mostly propagates in theuchannel, plus some radiation.
The output strongly depends on the relative phase between the pulses, as can be seen from the example shown in Fig.5, in which we start from an input close to the previous one, but assume that the two initial pulses are in phase quadrature.
Without coupling, the pulses propagate as solitons, while with a positive coupling coefficient, they form a single vector soliton [Fig.5(c)]. However, when the coupling coefficient is negative, two vector solitons are formed [Fig.5(d)].
A remarkable feature is that two solitons with different frequencies can lock together to form some vector soliton (Fig. 6). Due to dispersion, the two solitons with different frequencies have different velocities and separate in the absence of coupling [Fig.6(b)]. With the coupling, they merge into a single vector soliton [Fig. 6(c)]. The same type of frequency locking is also shown in Fig.5(c).
V. CONCLUSION
We considered the coupling of two optical waveguides, in which few-cycle pulses are launched. Starting from a sim- plified model of a generalized Kadomtsev-Petviashvili type,
(a)
(b)
(c)
(d)
FIG. 5. The effect of a phase difference on the evolution of the coupled pulses. (a) Input, (b) output atz=2 without coupling (C=0), (c) output atz=2 with couplingC= +1, (d) output with couplingC= −1. The solid blue line isu, the dashed red line is v. The parameters are Au=Av=3, λu=1, λv=0.8, FWHMu
=FWHMv=2,ϕu=0,ϕv=π/2,tu=1,tv= −1.
we were able to derive the system of two coupled modified Korteweg–de Vries equations which describe the nonlinear propagation in the coupled waveguides. We obtained three
(a)
(b)
(c)
FIG. 6. Two solitons with different frequencies can lock together.
(a) Input, (b) output atz=2 without coupling (C=0), (c) output atz=2 with couplingC= −1. The solid blue line isu, the dashed red line isv. The parameters areAu=Av=1.8,λu=1,λv=0.8, FWHMu=FWHMv=3,ϕu=ϕv =0,tu=tv =0.
types of coupling terms, one of which is the generalization of the coupling which occurs in the linear regime, while the others are driven by the dispersion or by the nonlinearity.
Leaving the study of the dispersive linear coupling, and of the nonlinear one, for further consideration, we investigated numerically the evolution of two input few-cycle pulses in the presence of a linear nondispersive coupling. The formation of vector solitons is evidenced. Separate pulses can be mutually trapped, with initial mismatch in location, size, or phase, and even if their initial frequencies differ. The complete analysis of the vector solitons of the system (38) and (39) is left for further investigation, as well as the inclusion of the other coupling terms.
[1] Th. Brabec and F. Krausz, Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,Phys. Rev. Lett.78,3282(1997).
[2] Sh. Amiranashvili, U. Bandelow, and N. Akhmediev, Disper- sion of nonlinear group velocity determines shortest envelope solitons,Phys. Rev. A84,043834(2011).
[3] A. V. Husakou and J. Herrmann, Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers, Phys. Rev. Lett.87,203901(2001).
[4] Sh. Amiranashvili, U. Bandelow, and N. Akhmediev, Few-cycle optical solitary waves in nonlinear dispersive media,Phys. Rev.
A87,013805(2013).
[5] E. V. Kazantseva, A. I. Maimistov, and J.-G. Caputo, Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media,Phys. Rev. E71,056622(2005).
[6] N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, Maxwell-Drude- Bloch dissipative few-cycle optical solitons,Phys. Rev. A81, 043815(2010).
[7] I. V. Mel’nikov, D. Mihalache, F. Moldoveanu, and N.-C.
Panoiu, Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor, Phys. Rev. A 56, 1569 (1997).
[8] T. Sch¨afer and C. E. Wayne, Propagation of ultra-short op- tical pulses in cubic nonlinear media, Physica D 196, 90 (2004).
[9] H. Leblond and F. Sanchez, Models for optical solitons in the two-cycle regime,Phys. Rev. A67,013804(2003).
[10] A. Nazarkin, Nonlinear Optics of Intense Attosecond Light Pulses,Phys. Rev. Lett.97,163904(2006).
[11] S. V. Sazonov, Extremely short and quasi-monochromatic electromagnetic solitons in a two-component medium, JETP 92,361(2001).
[12] S. A. Skobelev, D. V. Kartashov, and A. V. Kim, Few-Optical- Cycle Solitons and Pulse Self-Compression in a Kerr Medium, Phys. Rev. Lett.99,203902(2007).
[13] V. G. Bespalov, S. A. Kozlov, Yu. A. Shpolyanskiy, and I. A.
Walmsley, Simplified field wave equations for the nonlinear propagation of extremely short light pulses,Phys. Rev. A66, 013811(2002).
[14] H. Leblond and D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Phys. Rep.523,61(2013).
[15] H. Leblond and D. Mihalache, Few-optical-cycle solitons:
Modified Korteweg–de Vries sine-Gordon equation ver- sus other non–slowly-varying-envelope-approximation models, Phys. Rev. A79,063835(2009).
[16] H. Leblond, D. Kremer, and D. Mihalache, Collapse of ultra- short spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation, Phys. Rev. A 81, 033824 (2010).
[17] H. Leblond, D. Kremer, and D. Mihalache, Ultrashort spatiotem- poral optical solitons in quadratic nonlinear media: Generation of line and lump solitons from few-cycle input pulses, Phys.
Rev. A80,053812(2009).
[18] J. M. Dudley, G. Genty, and S. Coen, Supercontinuum gen- eration in photonic crystal fiber, Rev. Mod. Phys. 78, 1135 (2006).
[19] A. M. Heidt, J. Rothhardt, A. Hartung, H. Bartelt, E. G. Rohwer, J. Limpert, and A. T¨unnermann, High quality sub-two cycle pulses from compression of supercontinuum generated in all- normal dispersion photonic crystal fiber,Opt. Express19,13873 (2011).
[20] H. Leblond, Ph. Grelu, and D. Mihalache, Models for su- percontinuum generation beyond the slowly-varying-envelope approximation,Phys. Rev. A90,053816(2014).
[21] I. Babushkin, A. Husakou, J. Herrmann, and Y. S. Kivshar, Frequency-selective self-trapping and supercontinuum genera- tion in arrays of coupled nonlinear waveguides,Opt. Express 15,11978(2007).
[22] A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, Spatiotempo- ral nonlinear optics in arrays of subwavelength waveguides, Phys. Rev. A82,041802(R)(2010).
[23] H. Leblond and D. Mihalache, Linear and nonlinear waveguid- ing of few-cycle optical solitons in a planar geometry,Phys. Rev.
A88,023840(2013).
[24] S. A. Kozlov and S. V. Sazonov, Nonlinear propa- gation of optical pulses of a few oscillations dura- tion in dielectric media, Zh. Eksp. Teor. Fiz. 111, 404 (1997) [JETP84,221(1997)].
[25] H. Triki, H. Leblond, and D. Mihalache, Derivation of a mod- ified Korteweg–de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian,Opt. Commun.285, 3179(2012).
