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\markboth{\hfil second harmonic generation \hfil EJDE--1999/36}
{EJDE--1999/36\hfil Habib Ammari, Gang Bao, \& Kamel Hamdache \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 1999}(1999), No.~36, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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The effect of thin coatings on second harmonic generation
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35C20, 78A45, 78A60.
\hfil\break\indent
{\em Key words and phrases:} Thin coatings, second harmonic generation,
asymptotic analysis.
\hfil\break\indent
\copyright 1999 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted May 30, 1999. Published September 20, 1999.} }
\date{}
%
\author{Habib Ammari, Gang Bao, \& Kamel Hamdache}
\maketitle
\begin{abstract}
The effect of thin coatings of nonlinear material on second harmonic generation
is studied in this paper. Asymptotic expansions of the fields inside a thin
nonlinear coating are performed. The convergence of these formal expansions is
established. Our asymptotic analysis reveals the physical nature of this
nonlinear problem. It also provides an effective method for overcoming the
computational difficulties that arise in thin nonlinear coatings.
\end{abstract}
\newcommand\varep{\varepsilon}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\def\Box{\leavevmode\vbox{\hrule\hbox{\vrule\kern5pt%
\vbox{\kern5pt}\vrule}\hrule}}
\section{Introduction}
Significant technological advances have been made recently in nonlinear optics,
due to rapid developments of laser technology and of nonlinear materials.
Applications of nonlinear optics are everywhere, for example, lasers,
spectroscopy, optical switching, optical disc storages, optical computing, and
communications \cite{nrc}. A particularly remarkable application is to generate
powerful coherent radiation at a frequency that is twice that of available
lasers, so-called second harmonic generation (SHG). The underlying physics is
simple: an intense incident (pump) beam induces a response through a nonlinear
polarization in a given medium, the medium then reacts to modify the field in a
nonlinear fashion. The former and latter processes are governed by the
constitutive equations and a nonlinear system of Maxwell's equations
respectively. In this paper, we shall restrict our attention to second order
nonlinear effects, which are the simplest and representative to other nonlinear
effects. In principle, the computation of the field intensity distributions of
SHG in a nonlinear medium may be done by combining a method of finite elements
and a fixed-point iteration scheme. However, this numerical method fails
completely when the thickness of the coating becomes exceedingly small due to
numerical instabilities.
The present work is devoted to a mathematical study of optical second harmonic
generation in thin nonlinear coatings. We shall adopt a model for the SHG
derived by Bao and Dobson in \cite{bd1}, which is a two-point boundary-value
problem for a nonlinear system of ordinary differential equations (ODEs). Our
method is based on asymptotic expansions of the fields in the
nonlinear coatings. An advantage of working on this simple model is that an
explicit asymptotic analysis becomes available, which provides a complete and
rigorous characterization of the physical nature of thin nonlinear coatings. It
also provides an effective method for the computation and analysis of the nonlinear
model which involves a small scale. This paper is our first attempt to
model and analyze the effects of thin nonlinear coatings. In an upcoming paper,
we shall investigate thin nonlinear coatings on a periodic structure (grating).
In that case, the model problem becomes a system of nonlinear partial
differential equations.
We refer the reader to Engquist and N\'ed\'elec \cite{nedelec} and Bendali and
Lemrabet \cite{bl} for recent results and references on effects of a thin
coating on the scattering by a linear medium. Results on mathematical analysis
and numerical solutions of this and other related SHG models may be found in Bao
and Chen \cite{bc}, Bao and Dobson \cite{bd1}, \cite{bd2}. See Shen \cite{shen}
for a detailed discussion on various physical aspects of SHG. The reader is
also referred to the book \cite{Friedman} for a description of mathematical
problems in nonlinear diffractive optics which arise in industrial applications.
We present the model problem in Section 2. A formal asymptotic expansion of the
fields in the nonlinear coating is performed in Section 3. Section 4 is
concerned with the well-posedness of the model, as well as a convergence
analysis of the asymptotic expansions.
\section{Model problem}
\setcounter{equation}{0}
Assume that
the media are nonmagnetic with constant magnetic permittivity everywhere,
that no external charge or current is present in the field,
and that the electric and magnetic fields are time harmonic, {\em i.e.},
\[
{\bf E}={\bf E}({\bf r}) e^{-i\omega t}\,,\quad {\bf H}={\bf H}({\bf r})
e^{-i\omega t}
\]
where ${\bf r}=(x_1, x_2, x_3)\in {\mathbb R}^3$.
We are interested in studying optical wave interaction in nonlinear media, or
in particular SHG which deals with
the phenomenon of two wave mixing. Suppose that there is a pumping wave
with same frequency $\omega_1=\omega$. Consider the two waves
fields
${\bf E}({\bf r}, \omega_1)$ and ${\bf E}({\bf r}, \omega_2)$, where
$\omega_2=2\omega_1$.
To simplify our notation,
we denote ${\bf E}(\omega_i)={\bf E}({\bf r}, \omega_i)$.
The Maxwell equations yield the following coupled system \cite{shen}:
\begin{eqnarray*}
\left[
\nabla\times(\nabla\times)-{\omega_1^2\varep_1}\cdot\right]
{\bf E}(\omega_1) &=& {4\pi\omega_1^2}\chi^{(2)}(\omega_1=-\omega_2
+\omega_1):
{\bf E}^{*}(\omega_1){\bf E}(\omega_2)\,,\\
\left[\nabla\times(\nabla\times)-{\omega_2^2\varep_2}\cdot\right]
{\bf E}(\omega_2) &=& {4\pi\omega_2^2}\chi^{(2)}(\omega_2=\omega_1+
\omega_1):
{\bf E}(\omega_1){\bf E}(\omega_1)\,, \label{e3}
\end{eqnarray*}
where $k_1^2=\omega_1^2\varep_1$, $k_2^2=\omega_2^2\varep_2$, and
$E^*$ denotes the complex conjugate.
Through the nonlinear coupling, energy can be transfered back and forth
between fields at each frequency. Clearly, the nonlinear nature of the medium
is described by $\chi^{(2)}$, the second order nonlinear
susceptibility tensor.
More discussions on nonlinear susceptibility tensors as well as the
derivation of the above system
from the Maxwell equations may be found in \cite{shen}.
Note that the presence of new frequency components is the most striking
difference between nonlinear and linear optics.
Throughout, we make the following general assumptions: the fields are
transverse; the medium is stratified; the surface is flat; and a normally
incident pump beam. Since the medium is stratified, the fields vary only in one
direction. The transversality assumption further reduces the Maxwell system to
a system of nonlinear Helmholtz equations. The formulation of the model can be
simplified because of the general assumptions made in the Introduction. Since
the medium is stratified, the fields vary only in one direction. The
transversality assumption essentially allows us to reduce the Maxwell system to
a system of Helmholtz equations. For instance, by choosing the polarization
properly, one may assume that ${\bf E}({\bf r}, \omega_1)=E(z, \omega_1)\vec{y}$
and ${\bf E}({\bf r}, \omega_2)=E(z, \omega_2)\vec{x}$.
Let us specify the geometry of the model. Assume that a slab of nonlinear
material of thickness $h$
is placed in between two linear materials. Suppose that the whole space is
filled with material in such a way that the
indexes of refraction $k_1(z)$ and $k_2(z)$ at frequencies $\omega_1$ and
$\omega_2 = 2 \omega_1$,
respectively, satisfy
\[
k_j(z) = \left\{\begin{array}{l}
k_{j1} \quad z \geq h,\\
k_{j0} \quad 0 < z < h,\\
k_{j2} \quad z \leq 0,
\end{array}
\right.
\]
for $j=1, 2$. Here $k_{j1}$ and $k_{j2}$ are fixed constants with the
properties: $k_{j1}>0$, $Re(k_{j2}) >0$, and $ Im(k_{j2})\ge 0$.
However $k_{j0}$ may be a bounded function.
Assume that a plane wave with the electric field
$(0, u^{in} e^{i k_{11} z}, 0)$ is incident on the
slab of nonlinear material from the above. Using the jump conditions,
as derived in \cite{bd1},
we have the following
two-point boundary-value problem:
\begin{equation} \label{0}
\begin{array}{c}
u'' + k_1^2 u = \chi_1\, \overline{u} v \quad \mbox{in } (0, h),\\[3pt]
v'' + k_2^2 v = \chi_2\,u^2 \quad \mbox{in } (0, h),\\[3pt]
u'(0) = i k_{12}u(0),\\[3pt]
u'(h) = - i k_{11}u(h) + 2 i k_{11} e^{i k_{11} h}u^{in},\\[3pt]
v'(0) = i k_{22}v(0),\\[3pt]
v'(h) = - i k_{21}v(h),
\end{array}
\end{equation}
where $u = E(z, \omega_1), v = E(z, \omega_2)$ are the pump and second-harmonic fields,
and $\chi_1(z)$, $\chi_2(z)$ characterize the nonlinearity
of the medium at frequencies $\omega_1, \omega_2$, respectively.
\section{Asymptotic expansions}
\setcounter{equation}{0}
We first scale the model problem.
Let the functions $u_h$ and $v_h$ be defined by
\begin{eqnarray*}
&u_h(y) = u(h y) \quad {\rm for\;} y \in [0, 1],\,&\\
&v_h(y) = v(h y) \quad {\rm for\;} y \in [0, 1]\,.&
\end{eqnarray*}
It is then easily seen from (\ref{0}) by noticing
$\frac{d}{d z} = \frac{1}{h} \frac{d}{d y}$ that $u_h$ and
$v_h$ satisfy the equations
\begin{eqnarray} \label{1}
&\frac{1}{h^2} u_h'' + k_1^2 u_h = \chi_1 \overline{u_h}
v_h \quad \mbox{in } (0, 1), &\\
&\frac{1}{h^2}v_h'' + k_2^2 v_h =
\chi_2 u_h^2 \quad \mbox{in } (0, 1)& \nonumber
\end{eqnarray}
along with the boundary conditions
\begin{eqnarray}
&u'_h(0) = i h k_{12}u_h(0), &\nonumber \\
&u'_h(1) = - i h k_{11}u_h(1) + 2 i h k_{11} e^{i k_{11} h}u^{in}, &\label{2}\\
&v'_h(0) = i h k_{22}v_h(0),&\nonumber \\
&v'_h(1) = - i h k_{21}v_h(1)\,.& \nonumber
\end{eqnarray}
Assume that for $h$ small, $u_h$ and $v_h$ have the
formal expansions
\begin{eqnarray}
\label{3.4}
&&u_h(y) = u_0(y) + hu_1(y) + h^2u _2(y) + 0(h^3),\\ \label{3.5}
&&v_h(y) = v_0(y) + hv_1(y) + h^2v_2(y) + 0(h^3).
\end{eqnarray}
Our goal in this section is to determine the functions $u_j$ and $v_j$ for $j=0,
1, 2$. This may be done by substituting (\ref{3.4}) for $u_h$ and (\ref{3.5})
for $v_h$ into (\ref{1}) and (\ref{2}) and equating the coefficients of each
power of $h$.
From the coefficients of $h^{-2}$ in (\ref{1})
and from the coefficients of $h^0$ in (\ref{2}), we obtain
\begin{eqnarray*}
& u_0'' = 0\; \mbox{ in } (0, 1)\,,&\\
&u'_0(0) = 0,\quad u'_0(1) =0&
\end{eqnarray*}
and
\begin{eqnarray*}
&v_0'' = 0\; \mbox{ in } (0, 1)\,,& \\
&v'_0(0) = 0,\quad v'_0(1) =0\,.&
\end{eqnarray*}
Hence, both $u_0$ and $v_0$ are constants in $[0, 1]$.
Similarly the coefficients of
$h^{-1}$ in (\ref{1}) and $h$ in (\ref{2}) yield the following
problems for $u_1$ and $v_1$:
\begin{eqnarray}
&u_1'' = 0\; \mbox{ in } (0, 1)\,,\nonumber \\
&u'_1(0) = i k_{12} u_0,& \label{4} \\
&u'_1(1) = - i k_{11} u_0 + 2 i k_{11} u^{in}& \nonumber
\end{eqnarray}
and
\begin{eqnarray}
&v_1'' = 0\; \mbox{ in } (0, 1)\,,& \nonumber \\
&v'_1(0) = i k_{22} v_0, &\label{5}\\
&v'_1(1) = - i k_{21} v_0\,. &\nonumber
\end{eqnarray}
We can now determine the constants $u_0$ and $v_0$.
In fact, from (\ref{4}) it follows that
$u_1'$ is a constant in $[0, 1]$. Using the boundary conditions,
it is evident that
\begin{equation} \label{u0}
u_0 = \frac{2 k_{11} u^{in}}{ k_{12} + k_{11}}\,.
\end{equation}
Similarly, from the equations in (\ref{5}), we see that
$v_1'$ is also a constant in $[0, 1]$. In addition,
the boundary conditions in (\ref{5})
yield
\[
v_0 = 0.
\]
Using (\ref{5}) again, we deduce that
$v_1$ is a constant in $[0, 1]$.
\paragraph{Remark 3.1.} In general, it is well known \cite{shen} that
nonlinear optical effects of SHG are very weak. This physical property is
justified by the fact that
$v_0=0$ in the formal expansion for $v_h$. It also indicates that
higher order terms must be used to obtain useful
nonlinear properties of the SHG field.
We next examine the problems
for $u_2$ and $v_2$.
\begin{eqnarray}
&u_2'' + k_1^2 u_0 = 0\; \mbox{ in } (0, 1)\,,&\nonumber\\
&u'_2(0) = i k_{12} u_1(0),&\label{6}\\
&u'_2(1) = - i k_{11} u_1(1) - 2 k^2_{11} u^{in}&\nonumber
\end{eqnarray}
and
\begin{eqnarray}
&v_2'' = \chi_2 u_0^2\; \mbox{ in } (0, 1)\,,&\nonumber\\
&v'_2(0) = i k_{22} v_1(0), &\label{7}\\
&v'_2(1) = - i k_{21} v_1(1) = - i k_{21} v_1(0)\,. &\nonumber
\end{eqnarray}
From (\ref{4}) and (\ref{5}) we know that
\begin{eqnarray} \label{3.11}
&u_1(y) = i k_{12} u_0 y + u_1(0),&\\
&v_1(y) = v_1(0)\,.& \nonumber
\end{eqnarray}
Using the explicit forms of $u'_2$ and $v'_2$,
\begin{eqnarray*}
&u'_2(y) = -u_0 \int_0^{y} k_1^2(x)\, dx + i k_{12} u_1(0),&\\
&v'_2(y) = u^2_0 \int_0^{y} \chi_2(x)\, dx + i k_{22} v_1(0),&
\end{eqnarray*}
we arrive at
\[
u_1(0) = \frac{1}{i (k_{11} + k_{12})}
\Bigr( k_{11} k_{12} u_0 - 2 k^2_{11} u^{in}
+u_0 \int_0^1 k^2_1(x) dx \Bigr)
\]
and
\[
v_1(0) = \frac{i u_0^2
\int_0^1 \chi_2(x)\, dx}{ k_{21} + k_{22}}\,.
\]
The calculation of $u_2$, $v_2$ may be carried out similarly.
Equating the coefficients
of powers $h$ in (\ref{1}) and $h^3$ in (\ref{2}) yields
\begin{eqnarray*}
&u_3'' + k_1^2 u_1 = \chi_1 \overline{u_0} v_1 =
\frac{i \chi_1 | u_0|^2 u_0}{ k_{21} + k_{22}}\int_0^1 \chi_2(x)dx \;
\mbox{ in } (0, 1)\,,&\\
&u'_3(0) = i k_{12} u_2(0),&\\
&u'_3(1) = - i k_{11} u_2(1) - 2 i k^3_{11} u^{in}&
\end{eqnarray*}
and
\begin{eqnarray*}
&v_3'' + k_2^2 v_1 = 2 \chi_2 u_0 u_1\; \mbox{ in } (0, 1)\,,&\\
&v'_3(0) = i k_{22} v_2(0),&\\
&v'_3(1) = - i k_{21} v_2(1)\,.&
\end{eqnarray*}
Thus
\begin{eqnarray}
\label{3.11a}
&u_2(y) = -u_0 \int_0^{y} \int_0^{x}k_1^2(t)\, dt
dx + i k_{12} u_1(0) y + u_2(0),&\\
&v_2(y) = u^2_0 \int_0^{y} \int_0^{x}\chi_2(t)\, dt
dx + i k_{22} v_1(0) y + v_2(0)\,. &\nonumber
\end{eqnarray}
It suffices to determine $u_2(0)$ and $v_2(0)$. From
\[
u'_3(y) = - \int_{0}^{y} k_1^2(x) u_1(x) dx
+ \frac{i|u_0|^2 u_0}{k_{12} + k_{21}} \int_{0}^{y}
\chi_1(x) dx \int_0^1\chi_2(x) dx+ i k_{12}u_2(0),
\]
we deduce, by using the boundary condition of $u_3$
at $y= 1$, that
\begin{eqnarray*}
\lefteqn{ i k_{12} u_2(0) + i k_{11} u_2(1) }\\
&=& - 2 i k_{11}^3 u^{in} +
\int_0^1 k_1^2(x) u_1(x) dx
- \frac{i |u_0|^2 u_0}{k_{21} + k_{22}} \int_0^1\int_{0}^{1}
\chi_1(x) dx \int_0^1 \chi_2(x) \,dx\,.
\end{eqnarray*}
But from (\ref{3.11a})
\[
u_2(1) = u_2(0) - u_0 \int_0^{1}
\int_0^{x}
k_1^2(t)\, dt
dx + i k_{12} u_1(0).
\]
Thus
\begin{eqnarray*}
u_2(0) & = & \frac{1}{i( k_{11} + k_{12})}\Bigr( - 2 i k_{11}^3 u^{in} +
\int_0^1 k_1^2(x) u_1(x) dx \\
&& - \frac{i|u_0|^2 u_0}{k_{21} + k_{22}} \int_0^1 \chi_1(x) dx\int_{0}^{1}
\chi_2(x) dx\\
&& + i k_{11} u_0 \int_0^{1} \int_0^{y}k_1^2(x)\, dx
dy + k_{11} k_{12} u_1(0)\Bigr),
\end{eqnarray*}
which allows the calculation of $u_2$ in $[0, 1]$.
Finally, in order to determine $v_2(0)$, we use
\[
v_3'(y) =- \int_{0}^{y} k_2^2(x) v_1(x) dx
+ 2 u_0 \int_{0}^{y} \chi_2(x) u_1(x) dx+ i k_{22} v_2(0)
\]
together with the boundary condition of $v_3$
at $y= 1$. By a simple computation, we get
\begin{eqnarray*}
v_2(0) & =& \frac{1}{i (k_{21} + k_{22})}
\Bigr(- i k_{21} u_0^2 \int_0^1 \int_0^{x} \chi_2(t) dt
dx + k_{21} k_{22} v_1(0)\\
&& - 2 u_0 \int_{0}^{1} \chi_2(x) u_1(x) dx
+ v_1 \int_0^1 k_2^2(x) dx \Bigr).
\end{eqnarray*}
Therefore, we have completed the formal
asymptotic expansions for $u_h$, $v_h$ up to $O(h^3)$.
\section{Convergence analysis}
\setcounter{equation}{0}
In this section, we validate the formal asymptotic expansions
performed in Section 3 by estimating
$\| u_h - (u_0 + h u_1 + h^2 u_2) \|_{H^1(0,1)}$ and
$\| v_h - (h v_1 + h^2 v_2) \|_{H^1(0,1)}$,
where $u_0$,
$u_1$, $u_2$, $v_1$, $v_2$ are the functions defined in Section 3.
We begin with two stability results for the model problem.
\begin{lemma} \label{l1}
Let $m$ be an integer. If $w_h$ satisfies
\begin{eqnarray}
&w_h'' + k_1^2\,h^2 w_h = 0(h^{m}) \quad \mbox{in } (0, 1),&\nonumber\\
&w_h'(0) - i k_{12} h w_h(0) = 0(h^{m}),&\label{w} \\
&w_h'(1) + i k_{11} h w_h(1) = 0(h^{m}),& \nonumber
\end{eqnarray}
then
\begin{equation}
h^{1/2} \| w_h \|_{L^2(0, 1)} + \| w'_h \|_{L^2(0, 1)}
= 0(h^{m - 1/2}).
\end{equation}
\end{lemma}
\paragraph{Proof.} From
\[
w_h(y) = \int_0^{y} w'_h(x)\, dx + w_h(0),
\]
it is easy to show that
\begin{equation} \label{p1}
\| w_h \|^2_{L^2(0, 1)} \leq \| w'_h \|^2_{L^2(0, 1)}+ 2| w_h(0) |^2.
\end{equation}
Multiplying (\ref{w}) by $\overline{w_h}$ and integrating by
parts over $[0, 1]$,
\begin{eqnarray}
\lefteqn{\int_0^1 | w'_h|^2 - h^2 \int_0^1
k_1^2 | w_h|^2 + i h ( k_{12} | w_h(0)|^2 + k_{11} | w_h(1)|^2) }\nonumber\\
&=& \int_0^1 0(h^m) \overline{w}_h
+ 0(h^m) \overline{w}_h(0)+ 0(h^m) \overline{w}_h(1)\,. \label{f2}
\end{eqnarray}
Taking the imaginary part of (\ref{f2}) yields that
\[
| w_h(0)| +| w_h(1)| \leq C \Bigr( h^{m -1} + h^{\frac{m-1}{2}}
\| w_h \|^{1/2}_{L^2(0, 1)} \Bigr)\,,
\]
where the constant $C$ independent of $h$.
By taking the real part of (\ref{f2}) and using Poincar\'e's inequality
(\ref{p1}), we arrive at
\[
\| w'_h \|_{L^2(0, 1)} \leq C \Bigr( h^{m} \| w_h \|_{L^2(0, 1)}
+ h^{2 m -1} \Bigr).
\]
Combining the above estimates for $\| w'_h \|_{L^2(0, 1)}$
and $| w_h(0)|$ together with (\ref{p1}), we obtain
\[
\| w_h \|_{L^2(0, 1)} \leq C h^{m-1} \mbox { and }
\| w'_h \|_{L^2(0, 1)} \leq C h^{m -\frac{1}{2}},
\]
which completes the proof. \hfill $\Box$ \smallskip
The next result is useful in our well-posedness study of the model.
The proof, which is essentially identical to that of Lemma \ref{l1}, is
omitted here.
\begin{lemma}
\label{ww}
Let $w_h$ be defined as the solution of the boundary-value problem
\begin{eqnarray}
&w_h'' + k_1^2\,h^2 w_h = h^2 \, f \quad \mbox{in } (0, 1)\,, &\nonumber\\
&w_h'(0) - i k_{12} h w_h(0) = 0\,, &\label{w0}\\
&w_h'(1) + i k_{11} h w_h(1) = 0\,,&\nonumber
\end{eqnarray}
where $f \in L^2(0, 1)$.
Then there is a positive constant $C$
independent of $h$, such that
\begin{equation} \label{es23}
\| w_h \|_{H^1(0, 1)} \leq C\,h\, \| f\|_{L^2(0, 1)}.
\end{equation}
\end{lemma}
We now present a well-posedness
result for the boundary-value problem (\ref{1})-(\ref{2}).
\begin{theorem} \label{t1}
There is a constant $h_0 > 0$, such that for any $ 0 < h < h_0$, Problem
(\ref{1})-(\ref{2}) admits a unique solution $(u_h, v_h)$
in $H^1(0, 1) \times H^1(0, 1)$.
\end{theorem}
\paragraph{Remark 4.1.} This existence and uniqueness result was first
established by Bao
and Dobson in \cite{bd1} for an arbitrary $h$ but sufficiently small
nonlinear susceptibilities, {\em i.e.},
\[\|\chi_1 \|_{L^{\infty}(0, 1)} \|\chi_2 \|_{L^{\infty}(0, 1)}
<< 1\;.\]
\paragraph{Proof.} For the sake of completeness, we sketch the proof here.
Following \cite{bd1}, the result is proved by the fixed point theorem.
Let $ w_h $ be defined by
\begin{equation} \label{defw}
w_h = u_h - \frac{2 k_{11} u^{in}}{ k_{12} + k_{11}}
e^{i (k_{11} - k_{12})h} e^{i k_{12} h y}.
\end{equation}
Denote
\[
f_h = \frac{2 k_{11} u^{in}}{ k_{12} + k_{11}}
e^{i (k_{11} - k_{12})h} e^{i k_{12} h y}\,.
\]
It is easy to verify that
$ w_h$ and $v_h$ satisfy respectively the boundary-value problems
\begin{eqnarray*}
&w_h'' + k_1^2\,h^2 w_h = h^2 \chi_1 \overline{w}_h v_h
+ h^2 \chi_1 \overline{f}_h v_h \quad \mbox{in } (0, 1),&\\
&w_h'(0) - i k_{12} h w_h(0) = 0,&\\
&w_h'(1) + i k_{11} h w_h(1) = 0&
\end{eqnarray*}
and
\begin{eqnarray*}
&v_h'' + k_2^2\,h^2 v_h = h^2 \chi_2 w_h^2 + h^2 \chi_2 w_h f_h
+ h^2 \chi_2 f^2_h \quad \mbox{in } (0, 1)\,, &\\
&v_h'(0) - i k_{22} h v_h(0) = 0\,,&\\
&v_h'(1) + i k_{21} h v_h(1) = 0\,.&
\end{eqnarray*}
Let $A^h_j : H^1(0, 1) \mapsto H^{-1}(0, 1)$ be the linear operator defined by
$B^h_j(u_1, u_2) = (A^h_j u_1, u_2),$ where
\[
B^h_j(u_1, u_2) = \int_0^1 u'_1 u_2' -
\int_0^1 k_j^2 u_1 \overline{u}_2
+ i k_{j1} u_1(1) \overline{u}_2(1) + i k_{j2} u_1(0) \overline{u}_2(0).
\]
Lemma \ref{ww} yields that
\begin{equation} \label{na}
\| (A^h_j)^{-1} \| \leq {C_j}{h^{-1}},
\end{equation}
where the constant $C_j$ is independent of $h$.
Thus, the existence and uniqueness of a solution $(u_h, v_h)$ to the
boundary-value problem (\ref{1})-(\ref{2}) is
equivalent to the well-posedness
of the fixed point problem
\begin{eqnarray*}
w_h &=& F(w_h)\\
&=&(A^h_1)^{-1}\Bigr( h^2 \chi_1 \overline{w}_h ((A^h_2)^{-1}(h^2
\chi_2 w_h^2 + h^2 w_h f_h + f^2_h)) \\
&&+ f_h ((A^h_2)^{-1}(h^2 \chi_2 w_h^2 + h^2 w_h f_h + f^2_h)) \Bigr).
\end{eqnarray*}
The rest of Theorem \ref{t1}'s follows immediately from
the contraction mapping principle by using
Estimate (\ref{na}) and the arguments given in \cite{bd1}.\hfill $\Box$ \smallskip
Next we prove strong convergence in $H^1(0, 1)$ of $u_h$ and $v_h$ to
$u_0$ and $0$, respectively.
\begin{theorem} \label{t2} Assume that
$(u_h, v_h)$ is the unique solution of the problem (\ref{1})(\ref{2}).
Then
\begin{eqnarray*}
&u_h \longrightarrow u_0 \mbox{ in } H^1(0, 1),&\\
&v_h \longrightarrow 0 \mbox{ in } H^1(0, 1).&
\end{eqnarray*}
\end{theorem}
\paragraph{Proof.}
By Theorem \ref{t1}, $u_h$ and $v_h$ are uniformly bounded in $H^1$ with
respect to $h$. Multiplying the equation satisfied by $v_h$ on $[0, 1]$
and integrating by parts over $[0, 1]$, we get
\begin{equation} \label{f1}
\int_0^1 | v'_h|^2 - h^2 \int_0^1
k_2^2 | v_h|^2 + i h ( k_{22} | v_h(0)|^2 + k_{21} | v_h(1)|^2)
= h^2\int_0^1 \chi_2 u_h^2 \overline{v_h}.
\end{equation}
Thus
\begin{equation}
\label{4.21}
\int_0^1 | v'_h|^2 \leq C h^2 + C h^2
| \int_0^1 \chi_2 u_h^2 \overline{v_h} |.
\end{equation}
Since $u_h$ is uniformly bounded in $H^1$ with respect to $h$,
the Sobolev imbedding theorem implies that $\| u_h \|_{L^{\infty}(0, 1)}$
is uniformly bounded in $h$. Hence
\[
| \int_0^1 \chi_2 u_h^2 \overline{v_h} | \leq
C \, \| u_h \|_{L^{\infty}(0, 1)} \| u_h \|_{L^2(0, 1)}
\| v_h \|_{L^2(0, 1)}.
\]
Substituting the above estimate into (\ref{4.21}), we obtain
\begin{equation} \label{e1}
\int_0^1 | v'_h|^2 \leq C h^2\,,
\end{equation}
with the constant $C$ independent of $h$.
Therefore, $v'_h$ converges to $0$ strongly in $L^2(0, 1)$.
Hence $v_h$ converges strongly to a constant $v_0$ in $H^1(0, 1)$.
By taking the imaginary part in (\ref{f1}),
we have for a constant $C$ independent of $h$ that
\begin{equation} \label{e2}
| v_h(0)|^2 + | v_h(1)|^2 \leq C h.
\end{equation}
Since the convergence of $v_h$ to $v_0$ is in ${\cal C}^0(0, 1)$, the
above inequality shows that $v_0 = 0$.
Now let $w_h$ be defined by (\ref{defw}).
By arguments similar to the above,
we can show that $w_h$ converges to $0$ strongly in $H^1(0, 1)$.
Hence $u_h$ converges strongly to the constant
$u_0$ given by (\ref{u0}) in $H^1(0, 1)$.\hfill $\Box$ \smallskip
We are now in position to prove our main result of this paper.
\begin{theorem} There exist positive
constants $h_0$, $C_1$, $C_2$, such that the estimates
\begin{eqnarray*}
h^{1/2}\| u_h - (u_0 + h u_1) \|_{L^2(0, 1)}
+ \| u'_h - (u'_0 + h u'_1 + h^2 u'_2) \|_{L^2(0, 1)}
&\leq& C_1 h^{5/2},\\
h^{1/2}\| v_h - h v_1 \|_{L^2(0, 1)}
+ \| v'_h - (h v'_1 + h^2 v'_2) \|_{L^2(0, 1)}
&\leq &C_2 h^{5/2}
\end{eqnarray*}
hold for any $0 < h < h_0$. Here $C_1$, $C_2$ are independent of $h$.
\end{theorem}
\paragraph{Proof.} Let $w_h$ and $t_h$ be defined by
\[
w_h = u_h - (u_0 + h u_1 + h^2 u_2),\quad
t_h = v_h - (h v_1 + h^2 v_2).
\]
It is easy to see that $w_h$ and $t_h$ satisfy respectively
the boundary-value problems
\begin{eqnarray*}
&w_h'' + k_1^2\,h^2 w_h = h^2 \chi_1 \overline{u}_h v_h +
0(h^{3}) \quad \mbox{in } (0, 1),&\\
&w_h'(0) - i k_{12} h w_h(0) = 0(h^{3}),&\\
&w_h'(1) + i k_{11} h w_h(1) = 0(h^{3})&
\end{eqnarray*}
and
\begin{eqnarray*}
&t_h'' + k_2^2\,h^2 t_h = h^2 \chi_2 (u_h^2 - u_0^2) + 0(h^{3})
\quad \mbox{in } (0, 1), &\\
&t_h'(0) - i k_{22} h t_h(0) = 0(h^{3}),&\\
&t_h'(1) + i k_{21} h t_h(1) = 0(h^{3}).&
\end{eqnarray*}
From (\ref{e1}) and (\ref{e2}) we may deduce that
\[
\| v_h \|_{L^{\infty}(0, 1)} \leq \| v_h \|_{H^1(0, 1)} \leq C h,
\]
where $C$ is a constant independent of $h$.
Similarly, we have
\[
\| u_h - u_0 \|_{L^{\infty}(0, 1)} \leq \| u_h - u_0
\|_{H^1(0, 1)} \leq \tilde C h,
\]
with the constant $\tilde C$ independent of $h$.
Using the above estimates, we obtain that
\begin{eqnarray*}
& \|\chi_1 \overline{u}_h v_h \|_{L^2(0, 1)} = 0(h),&\\
&\| \chi_2 (u_h^2 - u_0^2) \|_{L^2(0, 1)} = 0(h).&
\end{eqnarray*}
Hence we have the equations for $w_h$, $t_h$ that
\begin{eqnarray*}
w_h'' + k_1^2\,h^2 w_h &=& 0(h^3)\,,\\
t_h'' + k_2^2\,h^2 t_h &=& 0(h^3)\,.
\end{eqnarray*}
A direct application of Lemma \ref{l1} yields the estimates in the
statement of the theorem. The proof is now complete. \hfill $\Box$ \smallskip
\section{Numerical experiments}
The method presented here is compared to a finite element method stated in
\cite{bd1}. In Figure 1, the $L^2$ norm of the difference of the fundamental
fields by using these two methods is plotted versus the depth of the nonlinear
optical film; while in Figure 2, we plot the difference of the second harmonic
fields versus the depth of the nonlinear medium. Here, the material parameters
and the incident field are chosen exactly the same as in the example of
\cite{bd1} with appropriate units: $u^{in}=100$, $\omega_1=0.3774$, $h=0.1$,
$\varep_{11}=\varep_{21}=1.0$, $\varep_{10}=\varep_{20}= 5.6169$,
$\varep_{12}=-43.5375 +1.98i$, $\varep_{22}=-10.5459+0.8385 i$, and
$\chi_1=\chi_2=3\times 10^{-7}$.
\vskip10pt
\centerline{
\psfig{figure=fig1.ps,height=2.4in}
}
\vskip2pt
\centerline{{\bf Figure 1:} Comparison of asymptotic and
finite element solutions:}
\centerline{The pump (fundamental) fields.}
\vskip10pt
\vskip10pt
\centerline{
\psfig{figure=fig2.ps,height=2.4in}
}
\vskip2pt
\centerline{{\bf Figure 2:} Comparison of asymptotic and
finite element solutions:}
\centerline{The second harmonic fields.}
\vskip10pt
Our results indicate that the numerical solutions by using asymptotic and
finite element methods agree well when $h$ (the depth) is sufficiently small.
However, in such a situation, our asymptotic method is certainly the better
one because of its simple explicit form. In comparison, the method
in \cite{bd1} which is a combination of the finite element method and a
fixed-point approach requires solving systems of linear equations
iteratively. The size of
the systems increases as $h$ decreases. In addition, numerical instabilities
can occur as $h$ becomes exceedingly small.
\paragraph{Acknowledgment}
The research of the second author was partially supported by the NSF grants DMS
98-03604, the NSF University-Industry Cooperative Research Programs grants DMS
97-05139, DMS 98-03809, DMS 99-72292, and the NSF Western Europe Programs grant
INT-9815798. The work began during a visit of the second author with Centre de
Math\'ematiques Appliqu\'ees of Ecole Polytechnique. We also thank John L.
Fleming for his help in the numerical experiments.
\begin{thebibliography}{99}
\bibitem{bc} G. Bao and Y. Chen,
{\em A nonlinear grating problem in diffractive
optics}, SIAM J. Math. Anal. {\bf 28} (1997), 322-337.
\bibitem{bd1} G. Bao and D. C. Dobson,
{\em Second harmonic generation in nonlinear
optical films}, J. Math. Phys. {\bf 35}(1994), 1622-1633.
\bibitem{bd2} G. Bao and D. C. Dobson,
{\em Diffractive optics in nonlinear media with periodic
structure}, Euro. J. Appl. Math. {\bf 6}(1995), 573-590.
\bibitem{BaoLi} G. Bao and G. Li, {\em Optimal design in nonlinear optics},
in ``Encyclopedia of Optimization'', Ed. by
P. M. Pardalos and C. A. Floudas,
Kluwer Academic Pub., to appear.
\bibitem{bl} A. Bendali and K. Lemrabet,
{\em The effect of thin coating on the scattering of a time-harmonic wave for
the Helmholtz equation}, SIAM J. Appl. Math. {\bf 56} (1996),
1664-1693.
\bibitem{nedelec} B. Engquist and J.C. N\'ed\'elec,
{\em Effective boundary conditions for acoustic and electromagnetic
scattering in thin layers}, preprint, 1993.
\bibitem{Friedman}A. Friedman,
{\em Mathematics in Industrial Problems,}
Part 7 (1994), Springer-Verlag,
Heidelberg.
\bibitem{shen} Y. R. Shen, {\em
The Principles of Nonlinear Optics}, John Wiley, New York,
1984.
\bibitem{nrc} {\em Mathematical Opportunities in Nonlinear Optics},
National Research Council, National Academy Press, Washington, D. C., 1992.
\end{thebibliography}
\noindent {\sc Habib Ammari } \\
Centre de Math\'ematiques Appliqu\'ees, UMR CNRS 7641,\\
Ecole Polytechnique\\91128 Palaiseau, France \\
e-mail address: ammari@cmapx.polytechnique.fr \medskip
\noindent {\sc Gang Bao } \\
Department of Mathematics\\ Michigan State University \\
East Lansing, MI 48824-1027 \\
e-mail address: bao@math.msu.edu \medskip
\noindent {\sc Kamel Hamdache }\\
Institut Galil\'ee, UMR CNRS 7539,\\
Universit\'e de Paris XIII, \\
93430 Villetaneuse, France \\
e-mail address: hamdache@cmapx.polytechnique.fr
\end{document}