I'm curious about how does Quasi Steady-State all-optical Kerr effect should be treated numerically (or analytically)?
(I allow myself to use an abbreviation QSSAOKE below, if no one will complain)
Particularly:
1) Why I can not (or can) do evaluation of nonlinear waveguide modes directly from Maxwell's equations, if I put there induced 2nd order susceptibilities as a tensor for refractive index?
2) Can I neglect transient effects if I have pulse width much larger than media response time, or if I dealing with continuous interactions?
3) Does FWM appears to be most general description of all 3rd order processes?
I'm interested in electric E field dynamics mostly.
FIRST METHOD
I have a suggestion, that QSSAOKE needs to be deduced from the general form of a wave equation \nabla^2 E - {1 \over {c^2}} {{\partial^2}\over{\partial t^2}}(n_0 + \partial n) E = 0 by taking into account the 2nd-order nonlinearity (so-called NLSE):
i { {\partial u} \over {\partial \xi} } + {1 \over 2} {{\partial^2 u} \over {\partial \tau^2}} + f({|u|}^2) u=0 (1),
where u = \sqrt{\gamma L_D} A is a field, \xi = {{z} \over {L_D}} and \tau = {{T} \over {T_0}} – are dimensionless variables, \gamma = {{n_2 \omega_0} \over {c A_{\text{eff}}}} – is a nonlinear parameter, A_{\text{eff}} ={{{\left({\int\limits_{-\infty}^{\infty} {{|F(x,y)|}^2 dx dy}}\right)}^2} \over {\int\limits_{-\infty}^{\infty} {{|F(x,y)|}^4 dx dy}}} – effective core area, L_D = {{T_0^2} \over {|\beta_2|}} – is a dispersion length, T = t - {z \over {v_g}} – normalized time, \beta_2 = {1 \over c}{\left ( {2 {dn \over {d\omega}} + \omega {{d^2 n} \over {d\omega^2}}} \right )} – second term from Taylor's series for mode propagation constant, \omega_0 – central angular wave frequency, \omega – wave frequency, n – refractive index, v_g – group velocity, and F - modes I'm looking for.
I learned this way from here [1] and here [2]. Though in second work they use different type of NLSE, that includes susceptibility tensor terms, but still putting weak guiding waveguide modes into it.
SECOND METHOD
And that give me another idea – what if we just put those terms in Maxwell's equations directly? Especially knowing that the wave equation is a simple form of this equations if you write them down uniformly, e.g. {{\partial^2} \over {\partial t^2}}U = f(U), where U – is the E-field or H-field. Also this equation for \beta and n with second order components not just only some mathematical model – it has also a physical meaning and links with order of susceptibility tensor, e.g. we can not write down indefinite number of terms for any particular material type.
So I've picked another way to treat this process - an example of calculating effective refractive index for simple pump and probe beams from here [3]:
\begin{array}{lll} {P_i^{(3)}(\omega') = {}} & {\sum\limits_{j} {\chi_{1122}^{(3)}( \omega' = \omega' + \omega - \omega) E_i(\omega') E_j(\omega) E^{*}_j(\omega) }} \\ {} & {{} + \chi_{1212}^{(3)}( \omega' = \omega' + \omega - \omega) E_j(\omega') E_i(\omega) E^{*}_j(\omega) } \\ {} & {{} + \chi_{1221}^{(3)}( \omega' = \omega' + \omega - \omega) E_j(\omega') E_j(\omega) E^{*}_i(\omega)} \end{array} (2),
where P^{(3)} – polarization of 3rd order, \chi^{(3)} – susceptibility tensor. If we use this equation for P we can yield induced 2nd order susceptibilities if we divide P vector by probe filed E vector component-wise.
This one doesn't involve NLSE due to Quasi Steady-State nature of the effect.
So all that leave me some ambiguous picture. It's looks like there is a way to analytically describe a behavior of complex interacting fields, like LGnm or other TEMnm modes, and skip NLSE completely from the analysis by using some simplified equations, like equation (2), and than just use Green propagators for Maxwell's equations or plane wave analysis, assuming that we have an anisotropic media with 2nd order tensor refractive index.
Should I be proved wrong on this?
THIRD METHOD
Or maybe there are better way to use Maxwell's-Bloch systems of equations and use response times with Rabi frequencies, totally skipping tensor calculus?
But as I learned – there is a complicated solution or sometimes there are no solutions at all for such systems with more than 2 states. And I think that I will need 4 states for QSSAOKE with two fields if I go this way in general picture.
MY OTHER SUGGESTIONS
As I can understand from here [4] quasi steady-state case is the general case for long pulses or continuous beams. It said that if there is a large pulse width and small medium response time, than it's a quasi steady-state limit.
I look for steady-state case because I'm interested in processes taking place in liquids like anethole or chloronaphtallene, and I assume that they have a very small response time with respect to pulse width or field intensity changes due to interference. Field sources are continuous in this case. Process taking place in a waveguide or in free space.
Links
[1] Agrawal, G. P. (2001). Nonlinear fiber optics. 3rd Edition. Academic press.
Equations: (5.3.8), (5.2.4), (5.2.1), (3.1.5), (3.1.2), (2.3.28), (1.2.10).
[2] Mecozzi, A., Antonelli, C., & Shtaif, M. (2012). Nonlinear propagation in multi-mode fibers in the strong coupling regime. Optics express, 20(11), 11673-11678.
Equations: (1).
[3] Shen, Y. R. (1984). Principles of nonlinear optics. NY, Wiley.
Equations: (16.23).
[4] Hanson, E. G., Shen, Y. R., & Wong, G. K. L. (1977). Experimental study of self-focusing in a liquid crystalline medium. Applied physics, 14(1), 65-77.
Equations: (1), (4-7).
POSTS
I've also questioned this
here http://www.thescienceforum.com/physics/48601-kerr-effect-optical-kerr-effect.html#post624827
P.S.: I'm sorry for long and complicated question, but I'm kind of desperate of searching for the best solution.
P.P.S.: I decided to leave my question here, because I don't want to start another topic with the same title. Or should I move this to separate topic?