Generalized Nonlinear Wave Equation in Frequency Domain

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1 Downloaded from orbit.dtu.dk on: Dec 03, 208 Generalized Nonlinear Wave Equation in Frequency Domain Guo, Hairun; Zeng, Xianglong; Bache, Morten Publication date: 203 Document Version Early version, also known as pre-print Link back to DTU Orbit Citation (APA): Guo, H., Zeng, X., & Bache, M. (203). Generalized Nonlinear Wave Equation in Frequency Domain. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

2 Generalized Nonlinear Wave Equation in Frequency Domain Hairun Guo, Xianglong Zeng and Morten Bache Group of ultrafast nonlinear optics, Department of photonics engineering (DTU Fotonik), Technical University of Denmark, DK-2800, Kgs. Lyngby, Denmark Corresponding author: arxiv:30.473v physics.optics] 8 Jan 203 Introduction We interpret the forward Maxwell equation with up to third order induced polarizations and get so called nonlinear wave equation in frequency domain (NWEF), which is based on Maxwell wave equation and using slowly varying spectral amplitude approximation. The NWEF is generalized in concept as it directly describes the electric field dynamics rather than the envelope dynamics and because it concludes most current-interested nonlinear processes such as three-wave mixing, four-wave-mixing and material Raman effects. We give two sets of NWEF, one is a +D equation describing the (approximated) planar wave propagation in nonlinear bulk material and the other corresponds to the propagation in a waveguide structure. c 203 Optical Society of America OCIS codes: , There are three parts in the derivation. First, we review the derivation of Maxwell s wave equation in frequency domain and the expression of nonlinear induced polarization. Second, in the approximation of planar wave propagation, +D nonlinear wave equation in frequency domain (NWEF) is derived as an interpretation of the +D forward Maxwell equation with up to third order induced polarizations. Third, considering a waveguide structure, NWEF dependent on spatial mode profile is derived.. Maxwell s wave equation and nonlinear induced polarization We start from Maxwell s equations and material equations shown below 3]: E = B, H = D +J () t t B = 0, D = ρ, J = ρ t D E+P,B = µ 0 H+M,J = σe (3) V is the Laplace operator, E and H indicate the electric field (unit: m ) and magnetic field (unit: A m ) vectors, D and B indicate the electric and magnetic flux densities. J is the current density vector and ρ is the charge density, representing the sources for the electromagnetic field. ε 0 is the vacuum permittivity (unit: F m ) and µ 0 is the vacuum permeability (unit: H m ). P and M indicate the induced electric and magnetic polarization. In the absence of free charges in a nonmagnetic medium, we have M = 0, J = 0 and ρ = 0. Hence the Maxwell s equations is simplified to:

3 E = B D, H = t t After Fourier transforming the above equations ( F(ω) = F(t)e iωt dt, matching the DFT algorithm we have: Ẽ = iω B = iωµ 0 H, H = iω D = iω(ε 0 Ẽ+ ) (5) To derive the wave equation regarding the electric field, we use the relation: in which the left side can be extended as: ( Ẽ) = ( iωµ 0 H) = iωµ 0 ( H) ( Ẽ) = ( Ẽ) 2 Ẽ (6) = iωµ 0 iω(ε 0 Ẽ+ ) = ω 2 µ 0 ε 0 Ẽ+ω 2 µ 0 = k 2 0(ω)Ẽ+ω 2 µ 0 (7) (4) On the right side, we have: ( Ẽ) 2 Ẽ 2 Ẽ (8) which is supported if: ) the high order induced electric polarizations (nonlinear induced polarizations) are considered perturbations to the first order induced polarization (linear induced polarization), i.e. = L + NL L = 2πε 0 χ () Ẽ; 2) the relative permittivity ε r = +2π χ () is independent on the spatial distribution, i.e. Ẽ = D/ε 0 ε r = 0. Therefore, in frequency domain, we have the Maxwell s wave equation for the electric field: 2 Ẽ+k 2 0(ω)Ẽ+ω 2 µ 0 = 0 (9) In time-domain, it is: 2 E 2 E c 2 t 2 µ 2 P 0 t 2 = 0 (0) The induced polarization is further expressed as: P = P () +P +P (3) + +P (m) () wherethegeneralizedexpressionofboth thelinearinduced polarizationp () = P L andthenonlinearinduced polarization P (m) is: P (m) dt where: dt 2 dt dt 2 dω dω 2 dt m χ (m) (t,t 2,,t m ) E(t t )E(t t 2 ) E(t t m ) dt m χ (m) (t t,t t 2,,t t m ) E(t )E(t 2 ) E(t m ) dω m χ (m) (ω,ω 2,,ω m ) Ẽ(ω )Ẽ(ω 2 ) Ẽ(ω m )e it ω i (2) χ (m) (ω,ω 2, ω m ) = (2π) m dt dt 2 dt m χ (m) (t,t 2, t m )e i ω it i (3) χ (m) is the temporal response function of the material, also called susceptibility in frequency domain, which is a (m+)-rank tensor. The calculations among the electric fields are dyadic product which result in an m-rank tensor. indicate the multiple tensor product between two tensors, i.e. a (m+)-rank tensor 2

4 and a m-rank tensor (or dyadic tensor). Therefore, the induced polarization P (m) is a vector. In frequency domain, it has: (m) (ω) = + P (m) e iωt dt + dω dω m χ (m) (ω,,ω m ) Ẽ(ω ) Ẽ(ω m ) e i(ω ω i)t dt = 2πε 0 dω dω m χ (m) (ω,,ω m ) Ẽ(ω ) Ẽ(ω m )δ(ω ω i ) (4) Here, the delta function implies that the induced polarization always corresponds to the frequency which equals to the sum of the frequencies of the contributing electric fields. If setting Ω = ω i, i.e. dω = dω i and + (m) (Ω) = 2πε 0 dω dω 2 dω m χ (m) (ω,ω 2,,Ω m δ(ω Ω)dΩ =, the above equation becomes: ω i ) E(ω )E(ω 2 ) E(Ω m ω i ),m 2 (5) and () (Ω) = 2πε 0 χ () (Ω) Ẽ(Ω). Since P (m) is a vector, it can be written as a sum of its components, each casting to one dimension, i.e. P (m) = ĵp (m). Analogously, The nonlinear response tensor χ (m) has χ (m) = ĵr (m). Now R (m) is an m-rank tensor and has R (m) = α α m ( mπ s= ˆα s ) χ (m) Then, the component of the induced polarization P (m) P (m) (t) dt α α m dt 2 dt In frequency domain, it is: (m) (Ω) = 2πε 0 dω α α m ;α α m ], where,α,,α m is dimension mark. is: dt m R (m) (t t,t t 2,,t t m ) E(t )E(t 2 ) E(t m ) dt m χ (m) ;α α m (t t,,t t m ) E α (t ) E αm (t m ) dω m χ (m) ;α α m (ω,,ω m,ω m ω i ) Ẽα (ω ) Ẽα m (ω m )Ẽα m (Ω m (6) ω i ),m 2 (7) () and (Ω) = 2πε 0 χ () ;α (Ω)Ẽα (Ω). χ (m) ;α α m corresponds to one component of the tensor χ (m), in α which χ () ;α is coming from the matrix χ (). Therefore, each component of the electric field Ẽ has a wave equation which, in frequency domain, is written as: 2 Ẽ +k 2 0 (ω)ẽ +ω 2 µ 0 = 0; = () (m) (8) In particular, in uniaxial and biaxial crystals as well as cubic/isotropic materials, matrix χ () only has () diagonal elements and therefore (Ω) = 2πε 0 χ () ; (Ω)Ẽ(Ω). () we combine with k0ẽ 2 and update the wave equation as: 2 Ẽ +k 2 (ω)ẽ +ω 2 µ 0,NL = 0;,NL = + + (m) (9) where k 2(ω) = k2 0 (+2π χ() ; ) and the refractive index in the dimension is therefore defined as n = +2π χ () ;. 3

5 2. +D NWEF In the approximation of planer wave propagation, we neglect the spacial dynamics in the propagation of the electric field but focus on the temporal and spectral dynamics. the Laplace operator is therefore reduced to the only derivative with respect to propagationaxisz since the spacial dynamics is eliminated, i.e. 2 2 z 2. Therefore, Eq.(9) is reduced to a +D wave equation: 2 z 2Ẽ +k 2 (ω)ẽ +ω 2 µ 0,NL = 0 (20) By factoring out the fast dependence of the propagation coordinate from the electric field for all the frequencies, i.e. Ẽ (z,ω) = Ã(z,ω)e ik(ω)z, we get: 2 z 2à 2ik (ω) +ω 2 µ 0,NL e ikz = 0 (2) zã, In the slowly varying spectral amplitude approximation (SVSAA) 4], i.e. k à we have 2 z 2à z k Ã, which means the second-order derivative can be removed. So, we have: zã = i ω2 µ 0 zã 2k (ω),nl e ikz (22) and the reduced +D Maxwell equation (also called forward Maxwell equation) regarding the electric field 5 7]: Ẽ z +ik (ω)ẽ = i ω2 µ 0 2k (ω),nl (23) Among all the nonlinear induced polarizations, second-order and third order nonlinear induced polarizations are most concerned. For second-order nonlinear induced polarization, the response is always assumed instantaneous, giving rise to three-wave-mixing (TWM). The response function has χ (t,t 2 ) = χ δ(t )δ(t 2 ), where χ is a constant indicating the response intensity. Correspondently, in frequency domain, the susceptibility has χ (ω,ω 2 ) = χ (2π) 2 according to Eq.(3), i.e. constant for all the frequencies ω and ω 2. Hence, the second-order nonlinear induced polarization P has the same form as shown in 8,9]: (ω) P (t) χ E α E α2 χ Ẽ α 2πẼα 2 χ F E α E α2 ] As for third-order nonlinear induced polarization, the response is not fully instantaneous but consists of the instantaneous electronic Kerr response and a fraction of non-instantaneous vibrational Raman response. The response function is therefore written as χ (3) α 3 (t,t 2,t 3 ) = χ (3) α 3 R(t )δ(t 2 t )δ(t 3 ), where R(t) = ( f R )δ(t)+f R h R (t) and h R(t)dt =. f R indicates the amount of Raman fraction. h R (t) is the temporal Raman response function. The remaining instantaneous response is called electronic Kerr response which is the origin of the effects of self-phase modulation (SPM), crossphase modulation (XPM) and four-wave-mixing (FWM). In] frequency domain, the susceptibility is χ (3) α 3 (ω,ω 2,ω 3 ) = χ (3) (2π) 3 α 3 ( f R )+f R hr (ω +ω 2 ). Hence, the third-order nonlinear induced polarization P (3) has: 4 (24) (25)

6 (3) (ω) P (3) (t) α 3 α 3 χ (3) α 3 ( f R )E α E α2 E α3 +f R (h R (E α E α2 ))E α3 ] χ (3) α 3 ( f R )(Ẽα 2πẼα 2 2πẼα 3 )+f R ( h R (Ẽα 2πẼα 2 )) 2πẼα 3 ] α 3 ]]] χ (3) α 3 ( f R )F E α E α2 E α3 ]+f R F E α3 F hr F E α E α2 ] (26) (27) Finally, by substituting the nonlinear induced polarizations in Eq.(23) with their interpretations Eq.(25) and Eq.(27), we get the +D NWEF: Ẽ z +ik ω 2 (ω)ẽ = i 2c 2 k (ω) ω 2 i 2c 2 k (ω) α 3 ( ) χ F E α E α2 ] χ (3) α 3 ( f R )F E α E α2 E α3 ]+f R F ]]] E α3 F hr F E α E α2 ] (28) Equation(28) can numerically be solved by split-step Fourier method together with Runge-Kutta method. Itisnotedthatthefrequencydomainhasarange(,+)andthe contentsofk (ω)innegativefrequencies are required. According to the causality of all the induced polarizations, k (ω) shows a property of complex conugate, i.e. its contents in negative frequencies are linked to what in positive frequencies. 3. NWEF in waveguide structure In a waveguide structure, the spacial dynamics of the electric field is always dominated by the eigen modes in the waveguide, which determine the spacial distribution as well as the propagation constant of the electric field. Since the orthogonality between any of the two eigen modes, we redefine the dimension mark in the wave equation Eq.(9) to be the mode mark. Moreover, the electric field is redefined as 0] Ẽ (x,y,z,ω) = B (x,y,ω)ãϕ (z,ω) = B (x,y,ω)ã(z,ω)e iβ(ω)z, where B is the eigen mode distribution and β indicates the mode propagation constant. Now Eq.(9) can be expanded as: ( ) ( ) à e iβ(ω)z 2 x y 2 +k2 β2 B + B e iβ(ω)z 2 z 2 i2β à +ω 2 µ 0,NL = 0 (29) z Remember all the eigen modes of the waveguide have ( 2 x y +k 2 2 β2 ) = 0 and using SVSAA, we get the reduced Maxwell equation for waveguide structure: Ẽ z +iβ (ω)ẽ = i ω2 µ 0 2β (ω),nl (30) Compared with the +D forward Maxwell equation Eq.(23), the only difference in the above expression is the replacement of the spacial propagation constant k by mode propagation constant β. However, it should be noticed that not only the dispersion characteristics but the nonlinear induced polarizations are all revised by employing the waveguide, which will be revealed later. Instead of focusing on the space-related electric field, now we concentrate on the spectral related amplitude à and get its dynamics by making spacial integral on both sides of Eq.(30), i.e.: dxdy B ( ) z +iβ Ẽ = i ω2 µ 0 dxdy 2β B,NL (3) 5

7 Ãϕ z +iβ(ω)ãϕ = i ω 2 µ 0 dxdy 2 g (ω)β (ω) B 2,NL, dxdy B = g (ω) (32) Analogously, we should interpret the nonlinear induced polarizations in Eq.(32) in which the nonlinear susceptibilities are now space-dependent. For second-order nonlinear induced polarization, we employ Eq.(7) and expand it with the definition of the electric field, getting: dxdy B = 2πε 0 dω à ϕ α (ω )Ãϕ α 2 (Ω ω ) dxdy χ (x,y,ω,ω ω ) B (x,y,ω) B α (x,y,ω ) B α2 (x,y,ω ω ) (33) in which we can define a spacial integral factor as Phillips et al. did in 0]: Θ (ω,ω 2 ) = dxdy χ (x,y,ω,ω 2 ) B (x,y,ω +ω 2 ) B α (x,y,ω ) B α2 (x,y,ω 2 ) = (2π) 2 dxdy χ (x,y) B (x,y,ω +ω 2 ) B α (x,y,ω ) B α2 (x,y,ω 2 ) (34) and simplify the expression of second-order nonlinear induced polarization as: dxdy B Now the spacial integral factor = 2πε 0 dω à ϕ α (ω )Ãϕ α 2 (Ω ω ) Θ (ω,ω ω ) (35) Θ plays the role as the second-order susceptibility, determining the intensity of the nonlinear induced polarization. Moreover, it can be assumed that the variation of the mode distribution B with respect to frequency is slow enough compared with that of Ãϕ, or named slowly varying mode distribution approximation (SVMDA), therefore the spacial integral factor is considered constant Θ (ω,ω 2 ) 2 Θ (2π) and the second-order induced polarization is further simplified to: dxdy B Θ Ã ϕ α 2πÃϕ α 2 Analogously, for third-order nonlinear induced polarization, we have: dxdy B (3) α 3 Θ F A ϕ α ] (36) Θ(3) α 3 ( f R )F A ϕ α A ϕ ] α 3 +fr F A ϕ F hr F A ϕ A ϕ ] ]]] α3 α α2 The spacial integral factor playing a role as the third-order susceptibility is: (37) Θ (3) α 3 (ω,ω 2,ω 3 ) = dxdy χ (3) α 3 (x,y,ω,ω 2,ω 3 ) B (x,y, ω n) B α (x,y,ω ) B α2 (x,y,ω 2 ) B α3 (x,y,ω 3 ) n ] f R +f R hr (ω +ω 2 ) = (2π) 3 dxdy χ (3) α 3 B (x,y, ω n) B α (x,y,ω ) B α2 (x,y,ω 2 ) B α3 (x,y,ω 3 ) n ] (3) 3 Θ ;α (2π) α 2α 3 f R +f R hr (ω +ω 2 ) (38) 6

8 Finally, by substituting the nonlinear induced polarizations in Eq.(32) with their interpretations Eq.(36) and Eq.(37), we get the NWEF in the waveguide structure: Ãϕ z +iβ (ω)ãϕ = i ω 2 2c 2 g (ω)β (ω) ω 2 i 2c 2 g (ω)β (ω) References α 3 Θ F A ϕ α ] Θ(3) α 3 ( f R )F A ϕ α A ϕ α 3 ] +fr F A ϕ α 3 F hr F A ϕ α ] ]]] (39). A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering) (Oxford University Press, Inc., 2006). 2. R. Boyd, Nonlinear Optics, Academic Press (Academic Press, 2008). 3. G. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Elsevier Science, 202). 4. M. Kolesik, P. Townsend, and J. Moloney, Theory and simulation of ultrafast intense pulse propagation in extended media, Selected Topics in Quantum Electronics, IEEE Journal of 8, (202). 5. R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, Solitons in laser physics, Physica Scripta 20, 364 (979). 6. A. Husakou and J. Herrmann, Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers, Physical Review Letters 87, (200). 7. M. Conforti, F. Baronio, and C. De Angelis, Nonlinear envelope equation for broadband optical pulses in quadratic media, Physical Review A 8, (200). 8. M. Conforti, F. Baronio, and C. De Angelis, Ultrabroadband optical phenomena in quadratic nonlinear media, Photonics Journal, IEEE 2, (200). 9. M. Conforti, F. Baronio, and C. De Angelis, Modeling of ultrabroadband and single-cycle phenomena in anisotropic quadratic crystals, Journal of Optical Society of America B 28, (20). 0. C. Phillips, C. Langrock, J. Pelc, M. Feer, I. Hartl, and M. Fermann, Supercontinuum generation in quasi-phasematched waveguides, Optics Express 9, (20). 7