Nonlinearity Management in Optics: Experiment, Theory, and Simulation
Martin Centurion$^{1,2}$
Mason A. Porter$^{2,3}$
P. G. Kevrekidis$^4$
and Demetri Psaltis$^{1}$
We conduct an experimental investigation of nonlinearity management in optics using femtosecond pulses and layered Kerr media consisting of glass and air. By examining the propagation properties over several diffraction lengths, we show that wave collapse can be prevented. We corroborate these experimental results with numerical simulations of the (2 + 1)-dimensional focusing cubic nonlinear Schr\"odinger equation with piecewise constant coefficients and a theoretical analysis of this setting using a moment method.
kerr14a.tex
\documentclass[aps,prl,showpacs,twocolumn]{revtex4}

%\documentclass{article}

\usepackage{amsmath,epsfig,amsfonts,epsfig,graphicx,
%amsgen,amsbsy,amsopn,amstext,
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% created 1/29/06 by MAP (kerr00)
% updated 2/12/06 by MAP (kerr00)
% experimental discussion added by MC (kerr00a)
% some EP discussion added by MAP on 4/4/06 (kerr01)
% experimental results added by MC 4/07/06 (kerr01a)
% turned into a coherent paper by PGK 4/08/06 (kerr03)
% updated by MAP 4/09/06 (kerr04)
% updated by MC 4/10/06 (kerr05)
% updated by MC 4/13/06 to include D. Psaltis' comments (kerr06)
% steamlined slightly by MAP for length 4/13/06 (kerr07)
% PGK, 4/13/06 (kerr08)
% MAP, 4/14/06 (kerr09)
% Modified by MC to address reviewer comments, 5/30/06 (kerr10)
%, MAP, 5/31/06 (kerr11); dealing with other referee comments; some cuts to ensure <= 4 pages
% PGK, kerr12
% MC, kerr13
% MAP, kerr14
% MAP, kerr14a (only difference is to correct the typo Martin found)


\begin{document}
\title{Nonlinearity Management in Optics: Experiment, Theory, and Simulation}
\author{Martin Centurion$^{1,2}$, Mason A. Porter$^{2,3}$, P. G. Kevrekidis$^4$, and Demetri
Psaltis$^{1}$ }
%\\
\affiliation{ $^1$Department of Electrical Engineering,
%California Institute of Technology, Pasadena, CA  91125, USA
$^2$Center for the Physics of Information,
%California Institute of Technology, Pasadena, CA 91125, USA
$^3$Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA \\
$^4$Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA }

\begin{abstract}

We conduct an experimental investigation of nonlinearity management in optics using femtosecond pulses and layered Kerr media consisting of glass and air.  By examining the propagation properties over several diffraction lengths, we show that wave collapse can be prevented.  We corroborate these experimental results with numerical simulations of the (2 + 1)-dimensional focusing cubic nonlinear Schr\"odinger equation with piecewise constant coefficients and a theoretical analysis of this setting using a moment method.

\end{abstract}

%PACs: 05.45.Yv, 42.65.Sf, 42.65.-k

\pacs{05.45.Yv, 42.65.Sf, 42.65.Tg, 42.65.-k}

\maketitle


%\section{Introduction}

{\it Introduction}. In the past decade, techniques for managing dispersion \cite{DM} and nonlinearity \cite{borisbook} have attracted considerable
%continuously increasing
attention in diverse branches of physics, including optics, atomic/condensed-matter physics, and other areas. Dispersion management (DM), originally proposed for optical fibers based on periodically alternating the group-velocity dispersion with opposite signs, was found to be a robust method for supporting breathing solitary waves \cite{DMS}.  It was subsequently implemented in atomic matter waves \cite{ober}, where the sign of dispersion was controlled via periodic potentials.
%On the other hand, while n
Nonlinearity management (NM) was originally proposed in the
%optics
context of layered optical media \cite{mezentsev}, but it has garnered special attention in the context of ultracold physics \cite{borisbook}, where it was reformulated as Feshbach Resonance Management \cite{frm}. In this latter setting, the Feshbach resonance was proposed as a means of periodically modifying the scattering length of interatomic interactions (and hence the mean-field nonlinearity)
%. This was suggested as a means
to avoid collapse in higher-dimensional Bose-Einstein condensates \cite{saito} and sustain robust breathing coherent structures of atomic matter waves \cite{montesinos,others}.  The idea of NM has
%, similar to its DM counterpart \cite{DM},
also inspired considerable mathematical research,
%taking a more mathematical perspective
with a focus on its averaged and collapse-preventing properties \cite{pelinovsky}.


%In part motivated by this research effort, recent
%theoretical works have also considered the possibility of spatial
%nonlinearity management. This has been discussed both in the context
%of optical media with spatial microstructure \cite{ieee}, and
%in Bose-Einstein condensates in the presence of spatially dependent
%magnetic fields \cite{djfetal}.


However, to the best of our knowledge, there has not yet been any {\it experiment} that addresses the theoretically proposed framework of NM. The present Letter aims to close this gap by offering an experimental investigation of NM in an optical setting.  In particular, we study the propagation of femtosecond pulses in layered Kerr media.  In a nonlinear Kerr medium, the index of refraction can be written $n = n_0 + n_2I$, where $n_0$ is the linear contribution, $n_2$ is the Kerr coefficient, and $I$ is the beam intensity.  A beam propagating through such a medium with self-focusing (i.e., $n_2>0$)
%, while self-defocusing corresponds to $\eta_2<0$)
in two-dimensions (2D)
%with a positive Kerr coefficient
will
%experience
collapse \cite{sulem} if the beam power is above a critical threshold,
%\begin{align}
% $P_c = \frac{\pi(0.61)^2\lambda^2}{8n_0n_2}$,
$P_c = \pi(0.61)^2\lambda^2/(8n_0n_2)$,
%\end{align}
where $\lambda$ is the light beam's wavelength.
 %of the light beam.
 Our medium has layers of
 %we study propagation in a layered medium consisting of
 glass and air, which are both self-focusing but with very different Kerr coefficients ($n_2^{(1)} = 3.2 \times 10^{-16} \rm{cm}^2/\rm{W}$ for glass \cite{Tzortzakis} and $n_2^{(2)} = 3.2 \times 10^{-19} \rm{cm}^2/\rm{W}$ for air \cite{Couairon}). The
 %particular
 aim of our experiment is to showcase what is arguably NM's most striking feature---preventing the wave collapse in 2D that would
%otherwise
occur in a homogeneous medium.  We illustrate this effect for powers in the interval ($2 P_c, 6P_c$), where $P_c$ refers to the critical power in glass, that would otherwise be on the verge of collapse before the occurrence of more complex processes, such as the formation of a plasma through multiphoton absorption \cite{Braun, Liu}. The latter offers a defocusing, lossy mechanism that prevents this catastrophic phenomenon.  Optical beams that do not diverge due to diffraction (filaments or spatial solitons) have been obtained through a variety of physical mechanisms \cite{Segev}. Self-guiding and filamentation of femtosecond pulses
%in gases and condensed matter
relies on nonlinear losses and negative index changes from plasma formation to stabilize the beam \cite{Tzortzakis,Couairon,Braun,Liu,Dubietis}. Here we demonstrate a new guiding mechanism that does not rely on plasma formation
%(or other nonlinearities)
and can, in principle, be lossless. The variation of the Kerr coefficient in the layered medium prevents collapse, sustaining an oscillatory variation of the beam width for considerable propagation distances (before eventual dispersion due to weak losses arising at the interfaces between air-glass slides). We
 %directly
 compare our experimental results to numerical simulations of the nonlinear Schr{\"o}dinger (NLS) equation,
 %model
 adapted to the detailed experiment setup, and find good qualitative agreement (and quantitative agreement within the appropriate propagation regime) between the two. An additional theoretical understanding of the experimental trends is offered by a moment approach \cite{montesinos}.
 %, which yields an ordinary differential equation governing a diagnostic of the beam width.

%The remainder of this Letter is organized as follows.  We present our experimental setup and then our theoretical/numerical setup. We subsequently give the results of both setups, which we compare directly for different settings.  We subsequently summarize our findings.


\begin{figure}[tbp]
\centering \includegraphics[width=7.5cm]{setup_new.eps}
\caption{Experimental setup. ND = Neutral density filters, NLM =
Nonlinear medium, and L1--L3 = lenses.} \label{setup}
\end{figure}


\begin{figure}[tbp]
%\centering \includegraphics[height=5.75cm,width=7.5cm]{Fig2v3.eps}
\centering \includegraphics[height=5.75cm,width=7.5cm]{Fig2a_new.eps}
%\centering \includegraphics[height=5.75cm,width=7.5cm]{Fig2b1.eps}
% \centering \includegraphics[height=5.75cm,width=7.5cm]{loss.eps}
\begin{center}
\hskip-0.2cm\includegraphics[height=5cm,width=3.75cm]{Loss_Layered_Glass.eps}
%\includegraphics[height=5cm,width=3.75cm]{Fig2b1.eps}
\includegraphics[height=5cm,width=3.75cm]{Fig2b1jpg.eps}
\end{center}
\caption{(Color online) Dynamics for power $P = 5.9 P_c$.  (Top) Beam FWHM (in $\mu$m) as a function of propagation distance (in mm), for air, the layered medium, and glass. The respective experimental results are denoted by diamonds, circles, and x's (the solid curves are visual guides). The FWHM computed from Eq.~(\ref{nls}) is shown by the thin dash-dotted curve, and simulations
incorporating time-propagation are shown by the thick dashed curve.
%for comparison.
(Bottom right) Spatial dependence of the field in Eq.~(\ref{nls}).
 %as $|E|^2=|E(r,z)|^2$ for our radial simulations.
 (Bottom left) The (normalized) experimentally measured beam energy for glass and the layered medium as a function of the number of glass slides.} \label{energy}
\end{figure}


{\it Experimental Setup}.  We show a schematic diagram of the experimental setup in Fig.~\ref{setup}.  A Titanium:Sapphire laser amplifier system generates pulses with a duration of 160 femtoseconds and an energy of 2 mJ at $\lambda=800$ nm. The beam profile is approximately Gaussian with a measured diameter (that is, a full-width half-maximum or FWHM) of 5 mm.  The beam is attenuated using neutral density filters (ND) and then focused with a 300 mm focal length lens (L1). The nonlinear medium (NLM) is composed of 1 mm thick glass slides (Corning MicroSlides part number 2947) separated by 1 mm gaps of air. The input face of the nonlinear medium is placed 2 mm after the beam focus (the beam is diverging when it enters the nonlinear medium). At the input face of the nonlinear medium,
%we have
FWHM $\approx 43 \mu$m. After traversing the NLM, the beam is attenuated and imaged on a CCD camera (Pulnix TM-7EX) using two lenses (L2 and L3) in a 4-F configuration, with a magnification of $M = 8$.
% of the beam.
We perform multiple experiments in which we vary the number of glass slides from one to nine. For each experiment, the beam profile is captured both at the output face of the NLM and after further propagation through 1 mm of air. The 4-F system allows us to image different planes along the propagation direction %simply
by changing the position of the CCD camera with respect to L3. For comparison, we also measure the propagation in glass by placing multiple glass slides together (without air gaps).
%The position of lens L1 is adjusted to maintain a fixed distance between
%the focal position and the input face of the layered medium.
%Finally, the CCD camera is moved 64 mm to capture the beam profile after it propagates in 1 mm of air. We repeat the experiments for layered media with one--nine glass slides. For comparison, we also measure the propagation in glass by placing multiple glass slides together (without air gaps).
%and in air.


{\it Theoretical Setup}. The standard model for beam propagation in optical media, incorporating the dispersive and Kerr effects, is the NLS equation \cite{sulem}, which we adapt
%.   We adapt the NLS
to our experimental setting.
% by using the properties of air and glass in an appropriately rescaled setting.
%version of the model.
We rescale space by the wavenumber, $(\xi,\eta,\zeta)=k^{(1)}\! \ast \!(x,y,z)$ and the electric field envelope with $u=(n_2^{(1)}/n_0^{(1)})^{1/2} E$ \cite{yaron}.  (The superscript $(j)$ denotes the medium, with  $j = 1$ for glass and $j = 2$ for air.)  The physical setting is then described by
%The resulting partial differential equation (PDE) describing the physical setting is then
\begin{align}
    iu_\zeta & = -\frac{1}{2}{\nabla}^2_\perp u - |u|^2u \,, \quad 0 < \zeta < \tilde{l} \mbox{ (glass)}\,, \notag \\
    iu_\zeta & = -\frac{1}{2}\frac{\eta_0^{(1)}}{\eta_0^{(2)}}{\nabla}^2_\perp u - \frac{\eta_2^{(2)}}{\eta_2^{(1)}}|u|^2u \,, \quad \tilde{l} < \zeta < \tilde{L} \mbox{ (air)}\,, \label{nls}
\end{align}
%\begin{align}
%    iu_\zeta & = -\frac{1}{2}{\nabla}^2_\perp u - |u|^2u \,, \quad 0 < \zeta < \tilde{l} \mbox{ (glass)}\,, \notag \\
%    iu_\zeta & = -\frac{1}{2}[\eta_0^{(1)}/\eta_0^{(2)}]{\nabla}^2_\perp u - [\eta_2^{(2)}/\eta_2^{(1)}]|u|^2u \,, \quad 0 < \zeta < \tilde{L} \mbox{ (air)}\,, \label{nls}
%\end{align}
where glass of scaled length $\tilde{l}$ alternates with air of scaled
length $\tilde{L}-\tilde{l}$.  Our numerical simulations consider radially symmetric solutions of (\ref{nls}), an approximation supported by the experimental data.  We compare the simulation results directly to the experiments.
%with $u = u(r,\zeta)$, where $r = \sqrt{\xi^2 + \eta^2}$.

To gain
%also gain some
%theoretical/
analytical insight,
 %on the present setting,
 we use the moment method
 %(see, e.g., Refs.~
 \cite{montesinos} to reduce the radial NLS equation to a set of ordinary differential equations (ODEs) for its moments.
  %as follows.
  Define
%\begin{align}
 $I_1(\zeta)
%&
= \int |u|^2d\xi d\eta$,
%\,, \quad
$I_2(\zeta) = \int |u|^2r^2d\xi d\eta$,
%\,, \notag \\
$I_3(\zeta)
%&
= i\int\left(u\frac{\partial u^*}{\partial r} - u^*\frac{\partial u}{\partial r}\right)rd\xi d\eta$,
%\,, \notag \\
$I_{4}^{(j)}(\zeta)
%&
=\int\left(|\nabla u|^2 - g_j|u|^4\right)d\xi d\eta$ (with $g_1 = 1$ and $g_2 = {\eta_2^{(2)}}/{\eta_2^{(1)}}$),
%\,, \quad
and $I_5(\zeta) = \frac{1}{2}\int|u|^4d\xi d\eta$.
%\,, \label{moments}
%\end{align}
%where $d\xi d\eta = rdrd\theta$.
%The intensity
The quantity $I_1$ is the beam power and is conserved by the dynamics of Eq.~(\ref{nls}).  The Gaussian profile
%$u =  V_0\exp\{-(\xi^2 + \eta^2)/(2\sigma^2)\}$
$u =  V_0\exp\{-r^2/(2\sigma^2)\}$ gives $I_1 = V_0^2\pi\sigma^2$.  The remaining quantities are
%diagnostics
associated with the beam width ($I_2$), momentum ($I_3$), energy ($I_4^{(j)}$), and nonlinearity ($I_5$).
%Approximating the phase of $u$ by a spherical wavefront (i.e., taking
Assuming
%the phase varies quadratically with the radius,
$\mbox{arg}(u) = I_3r^2/(4I_2)$,
%[which is
a good approximation for $\zeta=0$ in our experiments, yields a closed set of coupled ODEs for the $I_i$. With the invariants
%\begin{align}
$Q_1^{(j)} = 2(I_4^{(j)} + g_jI_5)I_2 - I_3^2/4$ and  $Q_2 = 2\sqrt{I_2}I_5$ \cite{montesinos},
%\,,
%\end{align}
we obtain an Ermakov-Pinney (EP) \cite{ep} equation describing the dynamics of the scaled beam width $y(z) = \sqrt{I_2(\zeta)}$:
%in glass ($j = 1$) and air ($j = 2$):
\begin{equation}
    y'' = (Q_1^{(j)} - g_jQ_2)/y^3 \equiv \beta_j/y^3 \,. \label{ep}
\end{equation}
For the initial Gaussian beam, the invariants are $Q_1^{(1)} = \pi^2\alpha^4P^2$, $Q_1^{(2)} = [\eta_0^{(1)}/\eta_0^{(2)}]Q_1^{(1)} = 1.5Q_1^{(1)}$, and $Q_2 = \pi^2\alpha^6P^3$, where $\alpha \approx 1.3556665$ and $P$ is the beam's scaled (by $P_c$) power.
%Because equation (\ref{ep}) is an EP equation, it
%It is interesting to
%Note that Eq.~(\ref{ep}) can be solved {\it explicitly} using the ``nonlinear superposition'' principle of the EP equation \cite{ep}.
In the $k$th segment of the medium, the solution is
%\begin{eqnarray}
$y = \left(A_k + B_k \zeta^2 + C_k\zeta\right)^{1/2}$,
%\label{epresult}
%\end{eqnarray}
with $A_kB_k - C_k^2 = 1$.  The initial value $y(\zeta=0)$ comes from the experimental setup, and the
%solution
coefficients $A_k, B_k$, and $C_k$ can be obtained from continuity conditions at the interfaces between the slides. Results using Eq.~(\ref{ep}) will also be directly compared with the experiments.


\begin{figure}[tbp]
%\centering \includegraphics[width=7.5cm]{loss.eps}
\centering \includegraphics[width=7.5cm]{Fig3_new.eps} \caption{(Color online) Comparison of NM for different powers.  The top left panel shows the experimental results for $P=2.3P_c$ (pluses), $P=3.9P_c$ (stars), $P=4.9P_c$ (triangles), and $P=5.9P_c$ (circles).  We plot the FWHM as a function of the propagation distance $z$.  The individual cases of $2.3 P_c$ (top right), $3.9 P_c$ (bottom left), and $4.9P_c$ (bottom right) are compared with the PDE diagnostic (see
text), shown by the dash-dotted curves, and the ODE results
%for the width
(thick dashed curves).}
%Beam energy as a function of the number of glass slides traversed by the beam. The natural log of the total energy in the beam is calculated for propagation through glass (multiple glass slides without air gaps) and the layered medium. The energy is normalized to the total energy after 1 glass slide in each case. }
\label{period}
\end{figure}


\begin{figure}[tbp]
\centering \includegraphics[width=7.5cm]{Fig4_new.eps} \caption{(Color online) NM for different layered media.  (Top) Numerical results for $P=5.9 P_c$ with 1 mm glass--1 mm air slide layers (thinnest dash-dotted curve), with 1 mm glass--1.5 mm air layers (intermediate thickness dash-dotted curve), and with 1 mm glass--2 mm air layers (thickest dash-dotted curve).  (Bottom) Corresponding solid curves with increasing thicknesses and symbol sizes represent the experimental results for the respective cases.}
\label{widths}
\end{figure}


{\it Results}. We applied our experimental, numerical, and analytical approaches to a variety of settings.
Figure \ref{energy} shows the measured beam diameter (FWHM) as a function of propagation distance in air, the layered medium, and glass. For the layered medium, the beam propagates through glass from 0 to 1 mm, through air from 1 mm to 2 mm, then through another layer of glass followed by air, and so on for nine periods. The pulse power is 5.9 $P_c$.  In air, the effect of the nonlinearity is practically negligible, so the beam diffracts. In glass, the beam focuses to a minimum diameter of 7.5 $\mu$m after a distance of 2 mm,
%then propagates with a diameter of
%approximately
%subsequently
defocuses to 14 $\mu$m between 3 mm and 5 mm (the reason is discussed below), and
%after 5 mm the beam
then starts to diverge. In
%the propagation through
the layered medium, the beam initially focuses through the first layer and then its diameter oscillates over the first three periods with a mean of about 30 $\mu$m. The minimum beam diameter is 20 $\mu$m. The oscillations then die down and the beam starts to diverge, although with a smaller slope than either propagation in air or in glass.

Comparing the propagation through the layered nonlinear medium with the linear propagation (air), it is clear that the beam diameter is sustained over several diffraction lengths.  In both the glass and the layered medium, the beam is self-focused and stabilized. The loss of energy is what ultimately causes the beam to diverge. Their stabilizing mechanisms are different. The bottom panel of Fig.~\ref{energy} shows the total power in the beam as a function of the number of glass slides the beam has traversed for both the layered medium and glass. In the former,
%layered medium,
we measured a total transmission of 37\% for 8 layers (88\% per glass slide),
%which was
independent of the power (linear loss). This arises mainly from reflections at the air-glass interfaces and perhaps also absorption/scattering. In glass, an additional loss appears after 2 mm when the beam reaches the minimum diameter. It has been demonstrated
%shown
for similar propagation
parameters that the stabilizing mechanism for propagation in glass is plasma formation \cite{Tzortzakis}, which occurs through multi-photon absorption that creates a negative index change and
%thereby
balances the Kerr self-focusing.
%The plasma causes a saturation of the index change, and thus a saturation (clamping) of the intensity [REF - intensity clamping paper].
The multi-photon absorption thus accounts for the additional loss measured in glass. The resulting larger beam diameter (lower intensity) prevents multi-photon absorption from becoming significant in the layered medium. The guiding in the layered medium is done purely by Kerr self-focusing and diffraction. In glass, plasma formation starts at about 2 mm, whereas here
%we have shown that
the beam propagates through the layered medium for a much longer distance. We believe the range of stabilization in the layered medium can be greatly improved by reducing the reflection losses due to the refractive index mismatch (e.g., by using anti-reflection coating on the glass surfaces). The experimental results are in good qualitative and quantitative agreement with
%the findings of
the PDE simulations of Eq.~(\ref{nls}).  
%We use definition-based diagnostic of the FWHM
In Fig.~\ref{energy}, we show that the beam's FWHM accurately follows the propagation in both air and glass. In the layered medium, the PDE
%(using a definition-based FWHM diagnostic)
also follows the experiment qualitatively (and even quantitatively at first).  (Our simulations include the losses at each interface.)  We also used a beam propagation code to perform a full 3D simulation that includes dispersion in the temporal domain.  The results are similar to the 2D simulation, but with an improved quantitative agreement with the experiments (Fig.~\ref{energy})  
%include the linear losses, which are incorporated by multiplying the power by 0.94 at each interface).
The effect of temporal dispersion is to increase the duration of the pulse, thereby decreasing the power and the strength of the self-focusing. In the simulation, the pulse broadens by approximately 15\% after 15 mm of propagation, and the shape changes from a Gaussian to a weakly multi-peaked profile. The numerical results show that temporal effects do not play a critical role in the stabilization of the beam, as shown previously for plasma stabilized filaments in glass \cite{Tzortzakis}.  In the rest of the paper, we thus compare the experimental results with the 2D simulation.


%For longer propagation, however, the beam diverges more strongly in the experiment due to temporal processes not included in the model.  (Note, however, that the simulations do include the linear losses, which are incorporated by multiplying the power by 0.94 at each interface in both the PDE and ODE descriptions.) The effect of spectral broadening due to the nonlinearity and temporal dispersion is to increase the duration of the pulse, thereby decreasing the power.


We also studied propagation through the layered medium as a function of pulse power. The top left panel of Fig.~\ref{period} shows the
%respective
evolution of the beam diameter for $P = 5.9, 4.9, 3.9, 2.3 P_c$.
%by circles, triangles, stars, and pluses in the top left panel.
We adjust the
%pulse
power by placing neutral density filters before the focusing lens L1 in Fig.~\ref{setup}. There is a clear trend to stronger self-focusing with increasing power.  In each case, the beam diverges less
  %clearly
than for propagation in air. There are some oscillations in the beam diameter; they increase with power and eventually die down after a few periods as the power decreases. The minimum beam diameter decreases with increasing power, showing that there is no significant intensity clamping for these power values. The measured loss was the same (within experimental errors) for all power levels. The results for $2.3 P_c$ (top right panel), $3.9 P_c$ (bottom left), and $4.9 P_c$ (bottom right) are also compared with PDE and ODE numerical results.
% of both the PDE and the ODE.
%theoretical approximation of the EP equation.
%Two diagnostics are used in the PDE.  One is from the
The PDE diagnostic (shown by dash-dotted curves) is based on the definition of the original beam and
%and a smoother one (dashed curve) is based on a Gaussian beam approximation, according to which $FWHM=2.3548 \sigma$, where the variance $\sigma^2$ is computed by a second moment integral.
%These are shown by dash-dotted and dashed curves, respectively.
%, and the ODE results are shown by the thick dashed curve.
%of the EP formalism ODE is also shown.
correctly follows the experiment's qualitative trends
%of the variation
(and even its quantitative ones for short propagation distances or weaker
powers involving quasi-linear propagation/beam divergence). The ODE
approximation (thick dashed curves) captures the weaker beam divergence for larger $P$ but is less successful with finer features (such as oscillations in the beam width). This may be attributable to the sensitive nature of the
%EP
closure approximations (especially bearing in mind the loss properties of the medium). Nevertheless,
%the EP equation
Eq.~(\ref{ep}) provides a fair, analytically tractable approximation to the observations.

Finally, Fig.~\ref{widths} shows the effect
%on the beam evolution
of changing the periodicity of the layered structure. The initial power was set to 5.9 $P_c$, and the air gaps were increased from 1 mm to 1.5 mm and then 2 mm. The thickness of the glass layers was 1 mm in each case.
%kept constant at 1 mm.
Changing the periodicity effectively changes the divergence of the beam. As the period is increased, the intensity of the beam is lower when it reaches the second and subsequent glass layers, resulting in a weaker self-focusing and a ``faster'' divergence. The PDE results for the FWHM once again accurately capture the relevant trends qualitatively (and also quantitatively at first).

{\it Conclusions}. In sum, we have offered in the present work the first experimental realization of nonlinearity management (NM) in the context of optical physics using femtosecond pulse propagation in layered (glass-air) media.  We have demonstrated stabilization of the beam through NM, which can potentially provide a lossless self-guiding mechanism. We have examined the effects of different beam powers and different layered-media compositions.  We also compared these results with uniform media and (partially) accounted for the relevant loss mechanisms. The experimental results are accurately captured qualitatively (and, when appropriate, also quantitatively) by an NLS model and some of their key features can also be seen using a far more drastic but analytically tractable ODE approximation.  Interesting future experimental directions may involve the reduction of losses or the incorporation of slides of defocusing material that could lead to a complete stabilization (with stable oscillations of the beam) \cite{saito,montesinos}.
%Similar experiments on the atomic physics side are also of interest due to the availability of the 3D setting, where more elaborate structures may be stabilized.



{\it Acknowledgements} We thank Michael Cross for initiating this collaboration and
%the Center for the Physics of Information at CalTech,
%as part of the Caltech Information Science and Technology initiative. We
acknowledge support from the DARPA Center for Optofluidic Integration (DP), the Caltech Information Science and Technology initiative (MC, MAP), and NSF-DMS and CAREER (PGK).
%the Gordon and Betty Moore Foundation (MC, MAP) and NSF-DMS and CAREER (PGK) for funding.


%\bibliographystyle{plain}
\begin{thebibliography}{99}

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%for a
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\bibitem{saito} F.~Kh. Abdullaev {\it et al.}, Phys. Rev. A {\bf 67}, 013605 (2003); H. Saito and M. Ueda, Phys. Rev. Lett. {\bf 90}, 040403 (2003).

\bibitem{montesinos} G.~D. Montesinos {\it et al.}, Physica D {\bf 191}, 193 (2003); J.~J. Garcia-Ripoll, V.~M. Perez-Garcia, patt-sol/9904006.

\bibitem{others} F.~Kh. Abdullaev {\it et al.}, Phys. Rev. Lett. {\bf 90}, 230402 (2003);  M.~A. Porter {\it et al.}, Phys. Rev. E., in press (2006).

\bibitem{pelinovsky} V. Zharnitsky and D. Pelinovsky, Chaos {\bf 15}, 37105 (2005); S. Cuccagna {\it et al.}, J. Diff. Eqs. {\bf 220}, 85 (2006).

%\bibitem{ieee} D.C. Hutchings, IEEE J. Selec. Top. Quantum Electr.
%{\bf 10}, 1124 (2004).

%\bibitem{djfetal} G. Theocharis {\it et al.},
%Phys. Rev. A {\bf 72}, 033614 (2005); cond-mat/0509471;
%F. Kh. Abdullaev {\it et al.}, Phys. Rev. A {\bf 72}, 061605 (2005)
%H. Sakaguchi and B.A. Malomed, Phys. Rev. E {\bf 73}, 26601 (2006);

\bibitem{sulem}  C. Sulem and P.~L. Sulem, \newblock {\it The Nonlinear Schr{\"o}dinger Equation},
Springer-Verlag (New York, 1999).

\bibitem{Tzortzakis} S. Tzortzakis {\it et al.}, Phys. Rev. Lett. {\bf 87}, 213902 (2001).

\bibitem{Couairon} A. Couairon {\it et al.}, J. Opt. Soc. Am. B {\bf 19}, 1117 (2002).
% n2 for air

\bibitem{Braun} A. Braun {\it et al.}, Opt. Lett. {\bf 20}, 73 (1995).

\bibitem{Liu} W. Liu {\it et al.}, Opt. Commun. {\bf 202}, 189 (2002).

\bibitem{Segev} G. Stegeman and M. Segev, Science {\bf 286}, 1518 (1999).

\bibitem{Dubietis} A. Dubietis {\it et al.}, Phys. Rev. Lett. {\bf 92}, 253903 (2004).
%\bibitem{vpg2}

\bibitem{yaron} Y. Silberberg, Opt. Lett. {\bf 15}, 1282 (1990).

\bibitem{ep} V.~P. Ermakov,
%\newblock
Univ. Izv. Kiev {\bf 20}, 1 (1880); E. Pinney,
%\newblock
Proc. Amer. Math. Soc. {\bf 1}, 681 (1950).

%\bibliography{ref,bec}

\end{thebibliography}


\end{document}

%%%%%%%%%%%%%%%%%%% some older notes %%%%%%%%%%%%%


{\it Introduction}  One of the major recent thrusts in the nonlinear dynamics community over the last few years has been {\it nonlinearity management}, in which the coefficients of nonlinear terms in partial differential equations are tuned to simulate the (temporal or spatial) adjustment of appropriate physical parameters in experimental settings \cite{FRM,tsoy,monte,borisbook,mapfesh}. Management techniques have been studied theoretically in diverse physical settings, including the adjustment of two-body interactions via Feshbach resonances in Bose-Einstein condensation, layed Kerr media in nonlinear optics, and ???, but they have not yet been applied experimentally (to our knowledge) despite their exceptional promise.
%{\bf Relevant (?) prior exp:
\cite{ultra}


verifying previously obtained theoretical results \cite{FRM,layer,monte,michi,pacc} for the first time. In our experiments the pulse power is above the critical power in glass but well below the critical power for self-focusing in air. The beam will self-focus in glass when the nonlinearity overcomes the effect of diffraction, and diverge when propagating in air. The beam can be self-guided in a layered medium by the proper balance of self-focusing in glass and diffraction in air, thus preventing the collapse. When a beam with power above the critical value propagates in pure glass (or air), the collapse is prevented by the formation of a plasma through multiphoton absorption. The plasma has a negative contribution to the refractive index that can balance the self-focusing [REFs]. This stabilizing mechanism is intrinsically lossy as it relies on absorption of the light. A layered medium, however, relies only on the Kerr nonlinearity and can in principle conserve the beam energy. In addition, the periodicity of the medium can be adjusted to control the propagation.

We also include theory and numerical computations for our specific experimental setting.


\cite{sulem}
%((map---paragraph on blow-up and collapse in 2 + 1 dim NLS \cite{sulem} and other sources; we need to stress the importance of this finding

%{\bf map: additional background here??? --- do we want to include a paragraph or a couple sentences
%discussing other things that one can get with nonlinearity management?  perhaps we should mention %a couple of these in conclusions and outlooks at the end?  a waveguide to get the 1D breathing
%solutions;switiching the signs of the Kerr nonlinearity to stabilize it rather than "just" delay the
%blow-up/collapse, etc.}


%{\bf map: mention that nonlinearity management was initially motivated by disperson management?}


%{\bf map: here is some text from a Feshbach paper; maybe some small part of this can go in the
%motivation in terms of why nonlinearity management is interesting to study ...  Nonlinearity
%management in the GP equation, with $g = g(t)$, models the use of an applied magnetic field in the
%vicinity of a so-called "Feshbach resonance" to change the BEC's two-body scattering length (which is %proportional to $g$).  The study and exploitation of such resonances, which allow large changes in $g%$ with minute adjustments of a magnetic field, have been among the most active research topics in
%experimental and theoretical atomic physics for several years.  For example, Feshbach resonances
%have provided a key for
%creating molecular BECs, stabilizing and destabilizing condensates, etc.}

in which we compare the propagation of two-dimensional Gaussian waves in layered versus unlayered (uniform) Kerr media and determine which energy levels can successfully stave off blow-up and collapse. We then report analytical and numerical results obtained using the experimental parameters.  Finally, we summarize our results and discussion extensions of this work.

%{\bf map: do we need any references for the exp set-up?  is there are a paper or two that indicates what the standards are ?

%{\bf map: citation of Martin's previous stuff
\cite{martin1,martin2}
%or these citations could go%elsewhere}

%(Alternatively, one can study (\ref{ep}) by constructing and iterating a map from one layer to another \cite{layer} or by integrating the equation "by hand".  The EP approach, however, allows one to generalize the present analysis to other situations, such as [[[???]]].)  The initial condition in the $k = 0$ segment, during which the beam propagates 2.16 mm through air, is determined by the initial conditions in the experiments.  These are $y(0) = \sigma\alpha\sqrt{\pi P}$ and $y'(0) = I_3(0)/(2y(0)) = 0$.  (Alternatively, one can choose not to evolve (\ref{ep}) through the initial 2.16 mm of air and instead obtain $y'(0)$ by curve-fitting the phase function to the experimental value at the beginning of the first layer of glass.  The values of $y'$ obtained at this point are almost exactly the same.)  In simulating the dynamics of $y(\zeta)$, the power loss is implemented at the transition between each layer, and the values of $Q_1^{(j)}$ and $Q_2$ are recomputed.  For visualization purposes, the power loss is scaled out as if the power loss were continuos, $P = (0.94)^\zeta P(0)$.  The obtained values of the FWHM should be compared at the transition points (integer values of $z$).


References to figures... \ref{theory}

%The large jump at 16 mm is probably not a real effect but the result of experimental error.


%\begin{figure}[tbp]
%\centering \includegraphics[width=7.5cm]{layered_power.eps}
\caption{Beam diameter (FWHM) as a function of propagation distance for different values of the pulse power. The power P1 = 5.9 Pcr, P2 = 4.9 Pcr, P3 = 3.9 Pcr and P4 = 2.3 Pcr. } \label{period}
%\end{figure}


%... include discussion of delaying collapse in 2 + 1 dim NLS by a large propagation distance






%\begin{figure}[tbp]
%\centering \includegraphics[width=7.5cm]{theory.eps} \caption{Numerics from the EP result and from
%the direct numerics} \label{theory}
%\end{figure}

%{\bf map: some reiteration of previous stuff and implications (including stressing
%the important of these results via the blow-up/collapse in 2 + 1 dim NLS that we are delaying for long
%propagation distances); we want this to be an impetus for experiments with sing-changing nonlinearity
%management}


...optics version of a BEC experiment comment?; need to talk about doing this in a BEC using Feshbach resonances \cite{inouye,fesh}; also discuss preventing collapse completely by switching between focusing and defocusing...

nlin0604031.extracted_bib
\begin{thebibliography}{99}

\bibitem{DM} C. Kurtzke, IEEE Photon. Technol. Lett. {\bf 5}, 1250 (1993).

\bibitem{borisbook} B.~A. Malomed, {\it Soliton Management in Periodic Systems}, Springer-Verlag (Berlin, 2006).

\bibitem{DMS}  N.~J. Smith {\it et al.}, Electron. Lett. {\bf 32}, 54 (1996); I. Gabitov and S. Turitsyn, Opt. Lett. {\bf 21}, 327 (1996).
%for a
%review, see chapter 5 in: B.A. Malomed, Progr. Opt. {\bf 43}, 71 (2002).


\bibitem{ober} B. Eiermann {\it et al.}, Phys. Rev. Lett. {\bf 91}, 60402 (2003).

\bibitem{mezentsev} L. Berg{\'e} {\it et al.}, Opt. Lett. {\bf 25}, 1037 (2000); I. Towers and B.~A. Malomed, J. Opt. Soc. Am. {\bf 19}, 537 (2002).

\bibitem{frm} P.~G. Kevrekidis {\it et al.}, Phys. Rev. Lett. {\bf 90}, 230401 (2003).

\bibitem{saito} F.~Kh. Abdullaev {\it et al.}, Phys. Rev. A {\bf 67}, 013605 (2003); H. Saito and M. Ueda, Phys. Rev. Lett. {\bf 90}, 040403 (2003).

\bibitem{montesinos} G.~D. Montesinos {\it et al.}, Physica D {\bf 191}, 193 (2003); J.~J. Garcia-Ripoll, V.~M. Perez-Garcia, patt-sol/9904006.

\bibitem{others} F.~Kh. Abdullaev {\it et al.}, Phys. Rev. Lett. {\bf 90}, 230402 (2003);  M.~A. Porter {\it et al.}, Phys. Rev. E., in press (2006).

\bibitem{pelinovsky} V. Zharnitsky and D. Pelinovsky, Chaos {\bf 15}, 37105 (2005); S. Cuccagna {\it et al.}, J. Diff. Eqs. {\bf 220}, 85 (2006).

%\bibitem{ieee} D.C. Hutchings, IEEE J. Selec. Top. Quantum Electr.
%{\bf 10}, 1124 (2004).

%\bibitem{djfetal} G. Theocharis {\it et al.},
%Phys. Rev. A {\bf 72}, 033614 (2005); cond-mat/0509471;
%F. Kh. Abdullaev {\it et al.}, Phys. Rev. A {\bf 72}, 061605 (2005)
%H. Sakaguchi and B.A. Malomed, Phys. Rev. E {\bf 73}, 26601 (2006);

\bibitem{sulem}  C. Sulem and P.~L. Sulem, \newblock {\it The Nonlinear Schr{\"o}dinger Equation},
Springer-Verlag (New York, 1999).

\bibitem{Tzortzakis} S. Tzortzakis {\it et al.}, Phys. Rev. Lett. {\bf 87}, 213902 (2001).

\bibitem{Couairon} A. Couairon {\it et al.}, J. Opt. Soc. Am. B {\bf 19}, 1117 (2002).
% n2 for air

\bibitem{Braun} A. Braun {\it et al.}, Opt. Lett. {\bf 20}, 73 (1995).

\bibitem{Liu} W. Liu {\it et al.}, Opt. Commun. {\bf 202}, 189 (2002).

\bibitem{Segev} G. Stegeman and M. Segev, Science {\bf 286}, 1518 (1999).

\bibitem{Dubietis} A. Dubietis {\it et al.}, Phys. Rev. Lett. {\bf 92}, 253903 (2004).
%\bibitem{vpg2}

\bibitem{yaron} Y. Silberberg, Opt. Lett. {\bf 15}, 1282 (1990).

\bibitem{ep} V.~P. Ermakov,
%\newblock
Univ. Izv. Kiev {\bf 20}, 1 (1880); E. Pinney,
%\newblock
Proc. Amer. Math. Soc. {\bf 1}, 681 (1950).

%\bibliography{ref,bec}

\end{thebibliography}