二维周期结构逆电磁散射：近场成像与数值方法
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更新于2024-07-15
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"这篇论文研究的是二维横向电极化条件下的周期性非均匀介质的逆电磁散射问题。目标是从周期结构上方和下方直线上的散射场数据中恢复介质的折射率，这些问题通过优化方法和正则化技术来解决。作者提出了一种基于拟牛顿法的数值求解策略，并在伯恩近似的基础上使用多波数连续技术来提高成像分辨率。文章还包括数值示例来验证反演算法的效能。"
详细知识点说明：
1. **逆介质问题 (Inverse Medium Problem)**: 这篇文章的核心是逆电磁散射问题，即从散射场的数据中推断出介质的特性，这里是折射率。这种问题在地震成像、医学成像和无损检测等领域都有重要应用。
2. **周期性非均匀介质 (Periodic Inhomogeneous Media)**: 文章探讨的对象是具有周期性结构的非均匀介质，这种介质在空间上呈现出周期性的折射率变化，这在光学、声学和电磁学中常见，如光子晶体和声子晶体等。
3. **二维横向电极化 (Two-Dimensional Transverse Electric (TE) Polarization)**: 描述的是电磁波的极化状态，TE极化是指电磁波的磁场分量垂直于传播方向，而电场分量沿传播方向的垂直平面内变化。
4. **优化问题 (Optimization Problem)**: 反问题被重构为一个优化问题，目的是寻找最佳的折射率分布，使预测的散射场与观测数据最匹配。
5. **拟牛顿法 (Quasi-Newton Method)**: 是一种求解非线性优化问题的数值方法，通过迭代更新来逼近问题的最优解，文中用于解决反问题。
6. **正则化 (Regularization)**: 由于反问题通常高度不适定，容易受到噪声影响，正则化技术用于稳定解决方案，防止过度敏感，并提高成像质量。
7. **伯恩近似 (Born Approximation)**: 伯恩近似是一种在散射理论中常用的简化方法，用于处理弱散射情况，这里作为反演算法的初始猜测。
8. **快速积分方程法 (Fast Integral Equation Method)**: 在每一步迭代中，这个方法用于高效求解直接问题，即计算给定折射率分布下的散射场。
9. **数值示例 (Numerical Examples)**: 文章通过几个数值模拟案例展示了反演算法的性能，证明了其在解决周期性非均匀介质成像问题中的有效性。
这些知识点结合在一起，构建了一个复杂的数学模型，用于解决实际的科学问题，即通过测量数据恢复周期性结构内部的折射率分布，从而提供了一种理解和利用周期性非均匀介质性质的新方法。
Inverse Problems 30 (2014) 045004 R Zhang and B Zhang
with β
n
= 0. From the scattering problem (1.1)–(1.4) it follows that the total field u satisfies
the Lippmann–Schwinger equation:
u(x) = u
i
(x) + k
2
D
G(x, y)q(y)u(y) dy, x ∈ D. (2.1)
If (x, y) is the free-space Green’s function of the Helmholtz equation u + k
2
u = 0inR
2
,
then G − is continuous in D. Thus, by [6, theorem 8.2], the volume potential operator V
defined by
V (φ)(x) :=
D
G(x, y)φ(y) dy, x ∈ D
is a linear bounded operator from L
2
α
(D) to H
2
α
(D), so it is also compact from L
2
α
(D) to L
2
α
(D).
It is then easy to obtain the following equivalence result between the scattering problem
(1.1)–(1.4) and the Lippmann–Schwinger integral equation (2.1).
Theorem 2.1. Let q ∈ L
∞
per
(D) and assume that k = α
n
for all n ∈ N.Ifu∈ H
2
α
(D) is a
solution of the scattering problem (1.1)–(1.4), then u is a solution of the Lippmann–Schwinger
integral equation (2.1). Conversely, if u ∈ L
2
α
(D) is a solution of (2.1), then u ∈ H
2
α
(D) is a
solution of the scattering problem (1.1)–(1.4).
Theorem 2.2. If q ∈ L
∞
per
(D), then the scattering problem (1.1)–(1.4) has a unique solution in
H
2
α
(D) for all but a discrete set of positive wave numbers k.
Proof. For k ∈ R
+
\{α
n
|n ∈ N}, the integral equation (2.1) can be written in the form
(I − k
2
V
q
)u = u
i
, (2.2)
where V
q
is defined by V
q
φ := V (qφ) for φ ∈ L
2
α
(D) and u
i
is an incident plane wave. We first
consider the integral equation in H
1
α
(D). Clearly, −k
2
V
q
is a compact operator on H
1
α
(D), since
q ∈ L
∞
per
(D) and V is compact from L
2
α
(D) to H
1
α
(D). Then, by the analytic Fredholm theory
(see [6]), I − k
2
V
q
has a bounded inverse for all k ∈ R
+
\{α
n
|n ∈ N} except for a discrete
set C. This means that the integral equation (2.1)or(2.2) has a unique solution in H
1
α
(D) for
k ∈ R
+
\({α
n
|n ∈ N}∪C). From the integral equation (2.1), it follows that u is also in the
space H
2
α
(D), since q ∈ L
∞
per
(D) and V is bounded from L
2
α
(D) to H
2
α
(D). The theorem is thus
proved.
Remark 2.3. In the following sections, we always assume that for the fixed wave numbers
k > 0, the scattering problem (1.1)–(1.4) is uniquely solvable.
For a given contrast q and an incident plane wave u
i
, define the scattering map A(q, u
i
) by
A(q, u
i
) = u, where u is the solution of the scattering problem (1.1)–(1.4) or the Lippmann–
Schwinger integral equation (2.1). From theorem 2.2 and remark 2.3, it follows that the
scattering map A(q, u
i
) is well defined. It is easy to see that the scattering map A(q, u
i
) is
linear with respect to u
i
but is nonlinear with respect to q. Therefore, we may denote A(q, u
i
)
by A(q)u
i
.
We now study the Fr
´
echet differentiability of the scattering map A(q), which is needed in
the study of the IP in section 4. To this end, define the Dirichlet-to-Neumann operators T
±
:
T
±
v =
n∈N
iβ
n
ˆv
±
n
e
iα
n
x
1
on
±
,
where ˆv
±
n
are the Fourier coefficients of [v exp(−iαx
1
)]
±
. It is easy to see that the Rayleigh
expansion conditions are equivalent to the conditions:
∂u
s
∂x
2
= T
+
u
s
on
+
(2.3)
5
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