Inverse Source Problem for Acoustically- Modulated Electromagnetic Waves

We will use the following standard spaces to discuss the wellposedness of the Maxwell’s equations [11]. Let $\Omega\subset\mathbb{R}^{3}$ be an open bounded set with a $C^{1,1}$ boundary, and

$$H(\mathrm{curl},\Omega)=\left\{\mathbf{u}\in(L^{2}(\Omega))^{3}:\nabla\times\mathbf{u}\in(L^{2}(\Omega))^{3}\right\}.$$

The norm on $H(\mathrm{curl},\Omega)$ is given by

$$\|u\|_{H(\mathrm{curl},\Omega)}=\big{(}\|\mathbf{u}\|^{2}_{(L^{2}(\Omega))^{3}}+\|\nabla\times\mathbf{u}\|^{2}_{(L^{2}(\Omega))^{3}}\big{)}^{1/2}.$$

The two tangential trace maps $\Gamma_{\tau}$ and $\Pi_{\tau}$ have the following definitions

	$\displaystyle\Gamma_{\tau}:H(\mathrm{curl},\Omega)$	$\displaystyle\rightarrow H^{-1/2}(\text{div},\partial\Omega),$
	$\displaystyle\mathbf{u}$	$\displaystyle\mapsto\hat{\mathbf{n}}\times\mathbf{u}|_{\partial\Omega}$

and

	$\displaystyle\Pi_{\tau}:H(\mathrm{curl},\Omega)$	$\displaystyle\rightarrow H^{-1/2}(\text{curl},\partial\Omega),$
	$\displaystyle\mathbf{u}$	$\displaystyle\mapsto(\hat{\mathbf{n}}\times\mathbf{u})\times\hat{\mathbf{n}}|_{\partial\Omega},$

where

$$H^{-1/2}(\text{div},\partial\Omega)=\left\{\mathbf{u}\in(H^{-1/2}(\partial\Omega))^{3}:\text{div}_{\partial\Omega}\mathbf{u}\in H^{-1/2}(\partial\Omega)\right\},$$

and

$$H^{-1/2}(\text{curl},\partial\Omega)=\left\{\mathbf{u}\in(H^{-1/2}(\partial\Omega))^{3}:\text{curl}_{\partial\Omega}\mathbf{u}\in H^{-1/2}(\partial\Omega)\right\}.$$

Here $\text{div}_{\partial\Omega}$ is the surface divergence and $\text{curl}_{\partial\Omega}$ is the surface curl. The two spaces $H^{-1/2}(\text{div},\partial\Omega)$ and $H^{-1/2}(\text{curl},\partial\Omega)$ are dual to each other. To handle the impedance boundary condition, we define the tangential trace of a vector field

$$\bm{\phi}_{T}:=\Pi_{\tau}(\bm{\phi})=(\hat{\mathbf{n}}\times\bm{\phi})\times\hat{\mathbf{n}}|_{\partial\Omega},$$

(8)

and the space

$$X=\left\{\mathbf{u}\in H(\mathrm{curl},\Omega):u_{T}\in(L^{2}(\partial\Omega))^{3}\right\}.$$

The norm on $X$ is

$$\|\mathbf{u}\|_{X}^{2}=\|\mathbf{u}\|_{H(\mathrm{curl},\Omega)}^{2}+\|\mathbf{u}_{T}\|_{(L^{2}(\partial\Omega))^{3}}^{2}.$$

We denote the $(L^{2}(\Omega))^{3}$-inner product by

$$(\mathbf{u},\mathbf{v})_{(L^{2}(\Omega))^{3}}:=\int_{\Omega}\mathbf{u}\cdot\mathbf{v}^{*}d{\bf x},\quad\mathbf{u},\mathbf{v}\in(L^{2}(\Omega))^{3},$$

and the $(L^{2}(\partial\Omega))^{3}$ inner product by

$$\langle\mathbf{u},\mathbf{v}\rangle_{(L^{2}(\partial\Omega))^{3}}:=\int_{\partial\Omega}\mathbf{u}\cdot\mathbf{v}^{*}d{\bf x},\quad\mathbf{u},\mathbf{v}\in(L^{2}(\partial\Omega))^{3},$$

where $*$ denotes the complex conjugate. We denote the dual paring of $\mathbf{u}\in H^{-1/2}(\text{div},\partial\Omega)$ and $\mathbf{v}\in H^{-1/2}(\text{curl},\partial\Omega)$ by $\langle\mathbf{u},\mathbf{v}\rangle$.

We will make the following assumptions throughout this paper.

1. A-1.

   The domain $\Omega$ is an open bounded connected domain in $\mathbb{R}^{3}$ with $C^{1,1}$ boundary.

2. A-2.

   The medium is nonmagnetic with $\mu=\mu_{0}$ in $\Omega$, where $\mu_{0}$ is the magnetic permeability in vacuum. The coefficients $\varepsilon$ and $\sigma$ are real piecewise $H^{3}(\Omega)$ functions.

3. A-3.

   There exists positive constants $K_{1}$ and $K_{2}$, such that

   $$K_{1}>\varepsilon,\lambda>K_{2}>0,\quad K_{1}>\sigma\geq 0,$$

   (9)

   and the conductivity $\sigma$ is nonzero.

4. A-4.

   The source $\mathbf{J}$ is an $(L^{2}(\Omega))^{3}$ vector field and is compactly supported in $\Omega$.

Remark: We conclude from A-2 that $\varepsilon$ and $\sigma$ are piecewise $C^{1}$ by the Sobolev embedding theorem. We conclude from A-3 that $K_{1}>\varepsilon_{\delta}>K_{2}>0$ and $\sigma_{\delta}\geq 0$, so long as $\delta$ is sufficiently small.

The modulated Maxwell equations (5) and the impedance boundary condition (6) can be written in terms of only the electric field:

$\displaystyle\nabla\times\nabla\times\mathbf{E}_{\delta}-\mu(\omega^{2}\varepsilon_{\delta}+i\omega\sigma_{\delta})\mathbf{E}_{\delta}=i\omega\mu\mathbf{J}_{\delta}\quad\text{in }\quad\Omega,$

(10)

which is subject to the impedance boundary condition

$$\left(\frac{1}{\mu}\nabla\times\mathbf{E}_{\delta}\right)\times\hat{\mathbf{n}}-i\omega\lambda(\hat{\mathbf{n}}\times\mathbf{E}_{\delta})\times\hat{\mathbf{n}}=0\quad\text{on }\quad\partial\Omega.$$

(11)

We say $\mathbf{E}_{\delta}\in X$ is a weak solution of (10) obeying the impedance boundary condition (11) if for all $\bm{\phi}\in X$,

		$\displaystyle\left(\frac{1}{\mu}\nabla\times\mathbf{E}_{\delta},\nabla\times\bm{\phi}\right)_{(L^{2}(\Omega))^{3}}-\left((\omega^{2}\varepsilon+i\omega\sigma)\mathbf{E}_{\delta},\bm{\phi}\right)_{(L^{2}(\Omega))^{3}}-i\omega\langle\lambda\mathbf{E}_{\delta T},\bm{\phi}_{T}\rangle$
		$\displaystyle=i\omega\left(\mathbf{J},\bm{\phi}\right)_{(L^{2}(\Omega))^{3}}.$		(12)

It follows from Assumptions A1-A4, that the weak solution $\mathbf{E}_{\delta}\in X$ exists and is unique [27].

We now derive the internal functional for both classical and weak solutions. To proceed, we consider the fields $\mathbf{F}$ and $\mathbf{G}$ which obey the Maxwell equations without sources:

	$\displaystyle i\omega\varepsilon\,\mathbf{F}^{*}+\nabla\times\mathbf{G}^{*}$	$\displaystyle=\sigma\mathbf{F}^{*}\quad\text{in }\quad\Omega,$		(13)
	$\displaystyle-i\omega\mu\mathbf{G}^{*}+\nabla\times\mathbf{F}^{*}$	$\displaystyle=0\quad\text{in }\quad\Omega,$

along with the impedance boundary condition

$$\mathbf{G}^{*}\times\hat{\mathbf{n}}-\lambda(\hat{\mathbf{n}}\times\mathbf{F}^{*})\times\hat{\mathbf{n}}=\mathfrak{g}\quad\text{on }\quad\partial\Omega,$$

where $\mathfrak{g}\in(L^{2}(\partial\Omega))^{3}$. Equivalently,

		$\displaystyle\nabla\times\nabla\times\mathbf{F}^{*}-\mu(\omega^{2}\varepsilon+i\omega\sigma)\mathbf{F}^{*}=0\quad\text{in }\quad\Omega$		(14)
		$\displaystyle\left(\frac{1}{i{\omega}\mu}\nabla\times\mathbf{F}^{*}\right)\times\hat{\mathbf{n}}-\lambda(\hat{\mathbf{n}}\times\mathbf{F}^{*})\times\hat{\mathbf{n}}=\mathfrak{g}\quad\text{on }\quad\partial\Omega.$

Note that (13) are explicitly solvable, since the required coefficients are known. Next, we take the inner product of (5) with $\mathbf{F}^{*}$, the inner product of (14) with $\mathbf{E}_{\delta}$, and then subtract to obtain

	$\displaystyle\nabla\times\nabla\times\mathbf{E}_{\delta}\cdot\mathbf{F}^{*}-\nabla\times\nabla\times\mathbf{F}^{*}\cdot\mathbf{E}_{\delta}$
	$\displaystyle=\mu[\omega^{2}(\varepsilon_{\delta}-\varepsilon)+i\omega(\sigma_{\delta}-\sigma)]\mathbf{F}^{*}\cdot\mathbf{E}_{\delta}+i\mu\omega\mathbf{J}_{\delta}\cdot\mathbf{F}^{*}.$

Integrating the above result over $\Omega$ and using the vector identity $(\nabla\times\mathbf{A})\cdot\mathbf{B}=\nabla\cdot\left(\mathbf{A}\times\mathbf{B}\right)+(\nabla\times\mathbf{B})\cdot\mathbf{A}$, we find that

		$\displaystyle\int_{\Omega}\left[\nabla\cdot\left(\frac{1}{\mu}\nabla\times\mathbf{E}_{\delta}\times\mathbf{F}^{*}\right)+\left(\nabla\times\mathbf{F}^{*}\right)\cdot\left(\frac{1}{\mu}\nabla\times\mathbf{E}_{\delta}\right)\right]d{\bf x}$
		$\displaystyle-\int_{\Omega}\left[\nabla\cdot\left(\frac{1}{\mu}\nabla\times\mathbf{F}^{*}\times\mathbf{E}_{\delta}\right)+\left(\nabla\times\mathbf{E}_{\delta}\right)\cdot\left(\frac{1}{\mu}\nabla\times\mathbf{F}^{*}\right)\right]d{\bf x}$
	$\displaystyle=$	$\displaystyle\int_{\Omega}[\omega^{2}(\varepsilon_{\delta}-\varepsilon)+i\omega(\sigma_{\delta}-\sigma)]\mathbf{F}^{*}\cdot\mathbf{E}_{\delta}d{\bf x}+i\omega\mathbf{J}_{\delta}\cdot\mathbf{F}^{*}$

We now integrate by parts the divergence terms, which using the relations $\frac{1}{\mu}\nabla\times\mathbf{E}_{\delta}=i\omega\mathbf{H}_{\delta}$ and $\frac{1}{\mu}\nabla\times\mathbf{F}^{*}=i\omega\mathbf{G}^{*}$ yields

	$\displaystyle i\omega\int_{\partial\Omega}\hat{\mathbf{n}}\cdot(\mathbf{H}_{\delta}\times\mathbf{F}^{*})d{\bf x}-i\omega\int_{\partial\Omega}\hat{\mathbf{n}}\cdot(\mathbf{G}^{*}\times\mathbf{E}_{\delta})d{\bf x}$
	$\displaystyle=\int_{\Omega}[\omega^{2}(\varepsilon_{\delta}-\varepsilon)+i\omega(\sigma_{\delta}-\sigma)]\mathbf{F}^{*}\cdot\mathbf{E}_{\delta}d{\bf x}+i\omega\mathbf{J}_{\delta}\cdot\mathbf{F}^{*}.$		(15)

Note that the boundary integral only depends on the tangential components of the fields $\mathbf{H}_{\delta}$, $\mathbf{E}_{\delta}$, $\mathbf{F}$ and $\mathbf{G}$, which are known from the boundary measurements (7). Therefore, the left-hand side can be determined from experiment. For the right-hand side, we consider the asymptotic expansion in the small quantity $\delta$. The $O(1)$ term is

$$i\omega\int_{\Omega}\mathbf{J}_{\delta}\cdot\mathbf{F}^{*}.$$

The $O(\delta$) term is of the form

$$\int_{\Omega}\left[\left(\omega^{2}\varepsilon\gamma_{\varepsilon}+i\omega\sigma\gamma_{\sigma}\right)\mathbf{F}^{*}\cdot\mathbf{E}+i\omega\gamma_{J}\mathbf{J}\cdot\mathbf{F}^{*}\right]\cos(\mathbf{k}\cdot\mathbf{x}+\varphi))d{\bf x}.$$

(16)

Varying $\mathbf{k}$ and $\varphi$ in (16), and performing the inverse Fourier transform, we obtain the internal functional

$$Q:=\left(\omega^{2}\varepsilon\gamma_{\varepsilon}+i\omega\sigma\gamma_{\sigma}\right)\mathbf{F}^{*}\cdot\mathbf{E}+i\omega\gamma_{J}\mathbf{J}\cdot\mathbf{F}^{*},$$

(17)

which is known at every point in $\Omega$.

We make the following hypothesis to extract more information from the internal function (17):

The hypothesis means that, in each $\Omega_{\alpha}$, we can form the non-singular matrix $[\mathbf{F}^{*}_{1\alpha},\mathbf{F}^{*}_{2\alpha},\mathbf{F}^{*}_{3\alpha}]$, where $\mathbf{F}^{*}_{j\alpha}$ is the $j$th column, $j=1,2,3$. Let $Q_{j\alpha}$ be the internal functional defined as in (17), with $\mathbf{F}^{*}$ replaced by $\mathbf{F}_{j\alpha}^{*}$. Given the row vector $[Q_{1\alpha},Q_{2\alpha},Q_{3\alpha}]$, we have

$$[Q_{1\alpha},Q_{2\alpha},Q_{3\alpha}]^{T}=[\mathbf{F}_{1\alpha}^{*},\mathbf{F}_{2\alpha}^{*},\mathbf{F}_{3\alpha}^{*}]^{T}\left[\left(\omega^{2}\varepsilon\gamma_{\varepsilon}+i\omega\sigma\gamma_{\sigma}\right)\mathbf{E}+i\omega\gamma_{J}\mathbf{J}\right],\quad\text{ in }\Omega_{\alpha}$$

where we view $\mathbf{E}$ and $\mathbf{J}$ as column vectors, and $T$ denotes the transpose. Therefore, if we define $\mathbf{Q}\in(L^{2}(\Omega))^{3}$ by specifying its restrictions according to

$$\mathbf{Q}|_{\Omega_{\alpha}}:=[\mathbf{F}_{1\alpha}^{*},\mathbf{F}_{2\alpha}^{*},\mathbf{F}_{3\alpha}^{*}]^{-T}[Q_{1\alpha},Q_{2\alpha},Q_{3\alpha}]^{T},$$

then $\mathbf{Q}$ is well-defined since both $\mathbf{E}$ and $\mathbf{J}$ are global vector fields over $\Omega$, and we have

$$\mathbf{Q}=i\omega\gamma_{J}\mathbf{J}+\left(\omega^{2}\varepsilon\gamma_{\varepsilon}+i\omega\sigma\gamma_{\sigma}\right)\mathbf{E}.$$

(18)

Note that we view $\mathbf{Q}$ as a vector-valued internal functional.

In this situation, the equality (18) does not involve $\mathbf{J}$ directly.

Here (18) implies that

$$\mathbf{J}=\frac{\mathbf{Q}-\left(\omega^{2}\varepsilon\gamma_{\varepsilon}+i\omega\sigma\gamma_{\sigma}\right)\mathbf{E}}{i\omega\gamma_{J}}.$$

(21)

Inserting the above into the Maxwell equations (1), we obtain an equation of the form

$$\nabla\times\frac{1}{\mu}\nabla\times\mathbf{E}-(a+ib)\mathbf{E}=\frac{\mathbf{Q}}{\gamma_{J}},$$

(22)

where

$$a=\omega^{2}\varepsilon\left(1-\frac{\gamma_{\varepsilon}}{\gamma_{J}}\right),\quad\quad b=\omega\sigma\left(1-\frac{\gamma_{\sigma}}{\gamma_{J}}\right).$$

Note that there are boundary constraints for $\mathbf{E}$, including the impedance boundary condition (2) and the measurement (3).

To analyze the stability of the inverse problem, suppose that there is another source $\tilde{\mathbf{J}}$ with corresponding electric field $\tilde{\mathbf{E}}$ and vector internal functional $\tilde{\mathbf{Q}}$, defined by (18). Let $\mathbf{u}:=\mathbf{E}-\tilde{\mathbf{E}}\in X$ be a weak solution of the equation

$$\nabla\times\frac{1}{\mu}\nabla\times\mathbf{u}-(a+ib)\mathbf{u}=\frac{\mathbf{Q}-\tilde{\mathbf{Q}}}{\gamma_{J}}\quad\quad\text{ in }\quad\Omega;$$

(23)

and obey the impedance boundary condition

$$\left(\frac{1}{\mu}\nabla\times\mathbf{u}\right)\times\hat{\mathbf{n}}-i\omega\lambda(\hat{\mathbf{n}}\times\mathbf{u})\times\hat{\mathbf{n}}=0\quad\text{on }\quad\partial\Omega,$$

(24)

where

$$(\hat{\mathbf{n}}\times\mathbf{u})\times\hat{\mathbf{n}}|_{\partial\Omega}=\mathbf{g}-\tilde{\mathbf{g}}.$$

(25)

It remains to establish the solvability of (22) with boundary condition (2) or the solvability of (23) with boundary condition (24). Now (22) and (23) are similar in form to (10). The difference is that in (10), the term $(\omega^{2}\varepsilon_{\delta}+i\omega\sigma_{\delta})$ has a strictly positive real part and a non-negative imaginary part, which ensures the existence and uniqueness of the weak solution by standard methods [27]. These sign conditions no longer hold for the term $a+ib$ in (22) and (23), due to the presence of the elasto-electric constants $\gamma_{\varepsilon},\gamma_{\sigma},\gamma_{J}$. Therefore, we divide the discussion into several sub-cases. By Assumption A-3, we see that $a$ is either identically zero or bounded away from zero, $b$ is either non-positive or non-negative. This observation accounts for the following classification of sub-cases.

The proof is presented in the next few subsections.

The results in Proposition 4.1 and Theorem 4.3 can be summarized by the following table: