  We here derive the so-called Nonlinear Analytical Envelope Equation (NAEE)
inspired by the work of Conforti et al. [M. Conforti, A. Marini, T. X. Tran, D.
Faccio, and F. Biancalana, "Interaction between optical fields and their
conjugates in nonlinear media," Opt. Express 21, 31239-31252 (2013)], whose
notation we follow. We present a complete model that includes $\chi^{(2)}$
terms [M. Conforti, F. Baronio, and C. De Angelis, "Nonlinear envelope equation
for broadband optical pulses in quadratic media," Phys. Rev. A 81, 053841
(2010)], $\chi^{(3)}$ terms, and then extend the model to delayed Raman effects
in the $\chi^{(3)}$ term. We therefore get a complete model for ultrafast pulse
propagation in quadratic nonlinear crystals similar to the Nonlinear Wave
Equation in Frequency domain [H. Guo, X. Zeng, B. Zhou, and M. Bache,
"Nonlinear wave equation in frequency domain: accurate modeling of ultrafast
interaction in anisotropic nonlinear media," J. Opt. Soc. Am. B 30, 494-504
(2013)], but where the envelope is modelled rather than the electrical field
while still keeping a sub-carrier level resolution. The advantage of the
envelope formation is that the physical origin of the additional terms that are
included to model the physics at the carrier level becomes more clear, in
contrast to the electric field equations that are more "black box" expansions
of the electrical field. We also point out that by comparing our results to a
very similar model and widely used model [G. Genty, P. Kinsler, B. Kibler, and
J. M. Dudley, "Nonlinear envelope equation modeling of sub-cycle dynamics and
harmonic generation in nonlinear waveguides," Opt. Express 15, 5382-5387
(2007)], the Raman terms presented there will most likely lead to an
artificially lower Raman effect.
