Nonlinear envelope equation for broadband optical pulses in quadratic media Matteo Conforti, Fabio Baronio and Costantino De Angelis CNISM and Dipartimento di Ingegneria dell’Informazione, Universit` a di Brescia, Via Branze 38, 25123 Brescia, Italy (Dated: May 22, 2018) We derive a nonlinear envelope equation to describe the propagation of broadband optical pulses in second order nonlinear materials. The equation is first order in the propagation coordinate and is valid for arbitrarily wide pulse bandwidth. Our approach goes beyond the usual coupled wave description of χ (2) phenomena and provides an accurate modelling of the evolution of ultra-broadband pulses also when the separation into different coupled frequency components is not possible or not profitable. PACS numbers: 42.65.-k, 42.65.Ky, 42.65.Re, 42.25.Bs plane waves), we can obtain the 1+1D wave equation for the electric field E ( z, t ): The analysis of optical pulse propagation typically in- volves the definition of a complex envelope whose vari- ation is supposed to be “slow” with respect to the os- cillation of a carrier frequency (“slowly varying envelope approximation”, SVEA [1]). In the frequency domain this assumption is equivalent to require that the band- width of the envelope is narrow with respect to the car- rier frequency. Different works showed that it is possible to extend the validity of a proper generalization of the envelope equation (for example the “Nonlinear Envelope Equatio” (NEE) of Brabec and Krausz) to pulse dura- tion down to the single optical oscillation cycle [2–6] and to the generation of high order harmonics [7]. When second order nonlinearities are considered, the usual ap- proach is to write coupled equations for the separated frequency bands relevant for the process [8, 9]. How- ever when ultra-broadband χ (2) phenomena take place, the different frequency bands might merge, generating a single broad spectrum, as observed in recent experi- ments [10]. Obviously in these cases the coupled NEE description of the propagation fails due to the overlap- ping between different frequency bands. The scope of this Letter is to provide a single wave envelope equation to describe ultra-broadband χ (2) in- teractions. To date, such a model is not available and the only way to numerically describe phenomena as those reported in Ref. [10] is to solve directly Maxwell equa- tions in time domain, with an immense computational burden. Our equation, besides providing a powerful tool for analytical treatment due to its simplicity, can be eas- ily solved with a modest computational effort and can be easily generalized to include other kind of nonlinearities such as Kerr or Raman. As far the linear dispersive terms are concerned, our derivation of the envelope equation builds upon the work of Brabec and Krausz [2], that carry to a sim- ple model that was shown (theoretically and experimen- tally) to be accurate in most situations. Starting from Maxwell equations (written in MKS units), neglecting transverse dimensions (i.e considering the propagation of Particular care must be devoted to the definition of the complex envelope, since we do not want to put any limi- tation to the frequency extent of the signals. This aspect is commonly overlooked in literature, and it is taken for granted that the band of the envelope is “narrow” in some sense. We shall see later that for quadratically nonlin- ear media, a proper definition of the envelope is crucial. As usual in the theory of modulation [11], we define the analytic representation of the electric field: We consider now the electric field E and the nonlinear polarization P$_{NL}$ as the product of a complex envelope and a carrier wave: E ( z, t ) = A ( z, t ) / 2 e iω 0 t − iβ 0 z + c.c. , P$_{NL}$ ( z, t ) = A$_{p}$ ( z, t ) / 2 e iω 0 t − iβ 0 z + c.c [in frequency do- main reads: ˆ E ( z, ω ) = ˆ A ( z, ω − ω$_{0}$ ) / 2 e − iβ 0 $^{z}$+ ˆ A $^{∗}$( z, − ω − ω$_{0}$ ) / 2 e iβ 0 $^{z}$], where ω$_{0}$ is a reference frequency, β$_{0}$ = Re [ k ( ω$_{0}$ )] and k ( ω ) = ( ω/c ) √ ˆ ε ( ω ) is the propagation constant. where c is the vacuum velocity of light, ε$_{0}$ is the vacuum dielectric permittivity, ˆ ε ( ω ) = 1 + ˆ χ ( ω ) and ˆ χ ( ω ) is the linear electric susceptibility. that can be written in frequency domain, by defining the Fourier transform F [ E ]( ω ) = ˆ E ( ω ) = ∫ + ∞ −∞ E ( t ) e − $^{iωt}$dt : ∂ 2 ˆ E ( z, ω ) ∂z 2 + ω 2 c 2 ˆ ε ( ω ) ˆ E ( z, ω ) = − ω 2 ε$_{0}$c 2 ˆ P$_{NL}$ ( z, ω ) , (2) ∂ $^{2}$E ( z, t ) ∂z 2 − 1 c 2 ∂ 2 ∂t 2 ∫ + ∞ −∞ E ( z, t $^{′}$) ε ( t − t $^{′}$) dt ′ = 1 ε$_{0}$c 2 ∂ 2 ∂t $_{2}$P$_{NL}$ ( z, t ) , (1) arXiv:1001.1874v1 [physics.optics] 12 Jan 2010 where ˜ E ( z, t ) = E ( z, t ) + i H [ E ]( z, t ) , (3) H [ E ]( z, t ) = 1 π p.v. ∫ + ∞ −∞ E ( z, t $^{′}$) t − t ′ dt ′ (4)