The method of multipole fields for 3D vector tomography

Abstract

To investigate the spherical tokamak plasmas the multipole fields method for the inversion of Doppler spectroscopy measurements is proposed. The method is based on the expansion of unknown vector field and ray transform on the special basis vector functions. An analytical expression for the ray transform of basis vector functions and the results of computer simulation are given.

Appendix

1.1 The Sobolev spaces

Let $m$ be integer $\geqslant 1$. The Sobolev space $H^m(\Omega )$ of order $m$ on $\Omega$ is defined by

$$\begin{aligned} H^m(\Omega )&=\{u \ |\ D^\alpha \in L^2(\Omega )\ \forall \alpha , |\alpha | \leqslant m \}, \nonumber \\ D^\alpha&=\frac{\partial ^{\alpha _1 + \cdots \alpha _n}}{\partial x_1^{\alpha _1}\cdots \partial x_n^{\alpha _n}}, \ \ \ \alpha ={\alpha _1, ... \alpha _n}, \ \ \ |\alpha |=\alpha _1+ \cdots + \alpha _n. \nonumber \end{aligned}$$

$L^2(\Omega )$ is the set of all measurable functions $u(x)$ in $\Omega$ such that the norm

$$\begin{aligned} \Vert u\Vert _{L^2(\Omega )}=\Bigr (\int \limits _{\Omega }|u(x)|^2 dx \Bigl )^{1/2} < \infty \nonumber \end{aligned}$$

In the particular case $\Omega = \mathbb R^N$ the definition is given in (Lions and Margenes 1972, p. 5).

In our case the $H^1$ space have a norm given by

$$\begin{aligned} \Vert u\Vert _{H^1}^2 = \Vert u \Vert _{L_2}^2 + \Vert \nabla u \Vert _{L_2}^2, \end{aligned}$$

(33)

where we have abbreviated:

$$\begin{aligned} \Vert \nabla u\Vert _{L^2}^2 \mathop {=}\limits ^\mathrm{def} \sum \limits _{m=1}^n \Vert \partial _m u \Vert _{L^2}^2 , \ \ \partial _m= \frac{\partial }{\partial x_m}, \nonumber \end{aligned}$$

That is the space $H^1({\mathbb R}^n)$ is the Hilbert space consisting of all $u$ in $L^2({\mathbb R}^n)$ that have finite values of the norm (33).

Theorem 1

(Funk-Hecke)

$$\begin{aligned} \int \limits _{S^{N-1}}\!\!\!\!\! F({\varvec{\omega }}\cdot {\varvec{\sigma }}) Y_{lm}({\varvec{\omega }}) d{\varvec{\omega }} \!=\! \frac{l! (4\pi )^{\frac{N}{2}-1} \ \Gamma (\frac{N}{2}-1)}{(l+N-3)!}Y_{lm}({\varvec{\sigma }}) \!\!\int \limits _{-1}^{1}\!\!\! F(t) C_l^{\frac{N}{2}-1}(t)(1-t^2)^{\frac{N-3}{2}} dt \end{aligned}$$

Proof

The proof can be found in Bateman and Erdelyi (1953), Erdelyi et al. (1953), Natterer (1986). $\square$

1.2 Vector spherical harmonics. Some properties

Since vector fields (a special case of tensor fields with spin $S=1$) are important in applied problems, some properties and the multipole expansion of vector fields have been considered in this section. Components of vector spherical harmonics in the rotation specified by the Euler angles $\alpha , \beta , \gamma ,$ are transformed likewise the components of irreducible tensors Varshalovich et al. (1988). For example, scalar spherical functions $Y_{lm}(\hat{\mathbf{r}})$, vector spherical harmonics $\mathbf{Y}_{lm}^{L}(\hat{\mathbf{r}})$ are irreducible rank $l$ tensors. Other examples of irreducible tensor operators are also the functions $f(r)Y_{lm}(\hat{\mathbf{r}}),$ where $f(r)$ is an arbitrary function of $r$, $\hat{\mathbf{r}}=\frac{\mathbf{r}}{r} .$

There exist two types of sets of vector spherical harmonics. The first type of vector spherical harmonics $\mathbf{Y}_{JM}^{L}(\hat{\mathbf{r}})$ writes in the form of a linear combination of scalar spherical harmonics.

$$\begin{aligned} \mathbf{Y}_{lm}^{L}(\hat{\mathbf{r}})= \sum \limits _{M,\sigma }C^{lm}_{LM1\sigma }Y_{LM}(\hat{\mathbf{r}}) \mathbf{e}_{\sigma } . \end{aligned}$$

(34)

Here $C^{lm}_{LM1\sigma }$ are the Clebsh-Gordan coefficients, $\mathbf{e}_{\sigma }$ are cyclic covariant basis vectors, $\sigma = \pm 1,0,$ $\mathbf{e}_{\pm 1}=\mp \frac{1}{\sqrt{2}}(\mathbf{e}_x\pm i\mathbf{e}_y),\ \ \mathbf{e}_{0}=\mathbf{e}_z.$

Vectors $\mathbf{Y}_{lm}^{L}(\hat{\mathbf{r}})$ satisfy the condition of orthogonality and completeness.

$$\begin{aligned} \int \limits _{S^2}\mathbf{Y}_{l^\prime m^\prime }^{L*}(\hat{\mathbf{r}})\cdot \mathbf{Y}_{lm}^{L}(\hat{\mathbf{r}}) ds(\hat{\mathbf{r}})= \delta _{l^\prime l}\delta _{m^\prime m} \end{aligned}$$

(35)

$$\begin{aligned} \sum \limits _{Llm}\mathbf{Y}_{lm}^{L*}(\hat{\mathbf{r}}^\prime ) \mathbf{Y}_{lm}^{L}(\hat{\mathbf{r}})=\sum \limits _{\sigma } \mathbf{e}_{\sigma }\mathbf{e}_{\sigma }^*\delta (\hat{\mathbf{r}} -\hat{\mathbf{r}}^\prime ). \end{aligned}$$

(36)

Under rotation of the frame of reference specified by the Euler angles $\alpha ,\beta ,\gamma ,$ the vector spherical harmonics are transformed as follows Varshalovich et al. (1988).

$$\begin{aligned} \mathbf{Y}_{lm^{\prime }}^{L}(\hat{\mathbf{r}}^{\prime })=\sum \limits _{m=-l}^l \mathbf{Y}_{lm}^{L}(\hat{\mathbf{r}}) D_{mm^{\prime }}^l(\alpha ,\beta ,\gamma ) \end{aligned}$$

(37)

Another set of vector spherical harmonics has been introduced in Hansen (1935), Morse and Feshbach (1953).

$$\begin{aligned} \mathbf{P}_l^m(\hat{\mathbf{r}})&=\hat{\mathbf{r}} Y_{lm}(\hat{\mathbf{r}}) \end{aligned}$$

(38a)

$$\begin{aligned} \mathbf{C}_l^m(\hat{\mathbf{r}})&=\frac{1}{\sqrt{l(l+1)}}(\hat{\mathbf{r}} \times \nabla _{\hat{\mathbf{r}}})Y_{l m}(\hat{\mathbf{r}})= \frac{-1}{\sqrt{l(l+1)}} \Bigr ( \frac{1}{\sin \theta }\frac{\partial }{\partial \varphi }\hat{\varvec{\theta }}- \frac{\partial }{\partial \theta }\hat{\varvec{\varphi }} \Bigl )Y_{lm}(\hat{\mathbf{r}}) \end{aligned}$$

(38b)

$$\begin{aligned} \mathbf{B}_l^m(\hat{\mathbf{r}})&=\frac{1}{\sqrt{l(l+1)}} \nabla _{\hat{\mathbf{r}}}Y_{l m}(\hat{\mathbf{r}})= \frac{1}{\sqrt{l(l+1)}} \Bigr ( \frac{\partial }{\partial \theta }\hat{\varvec{\theta }}+ \frac{1}{\sin \theta }\frac{\partial }{\partial \varphi }\hat{\varvec{\varphi }} \Bigl )Y_{lm}(\hat{\mathbf{r}}), \end{aligned}$$

(38c)

where $Y_{lm}(\hat{\mathbf{r}})$ are scalar spherical harmonics, and operators $\nabla _\mathbf{r}, \nabla _{\hat{\mathbf{r}}}$ have the following form:

$$\begin{aligned} \nabla _\mathbf{r} = \hat{\mathbf{r}}\frac{\partial }{\partial r} + \frac{1}{r}\nabla _{\hat{\mathbf{r}}}, \ \ \nabla _{\hat{\mathbf{r}}}= \hat{\varvec{\theta }}\frac{\partial }{\partial \theta }+\frac{\hat{\varvec{\varphi }}}{\sin \theta }\frac{\partial }{\partial \varphi }. \end{aligned}$$

(39)

Under rotation of the frame of reference, the vector spherical harmonics $\mathbf{P}_l^m, \mathbf{C}_l^m, \mathbf{B}_l^m$ are transformed as irreducible tensor operators (37).

The vector spherical harmonics are pointwise orthogonal,

$$\begin{aligned} \mathbf{P}_l^m \cdot \mathbf{C}_l^m = \mathbf{C}_l^m \cdot \mathbf{B}_l^m = \mathbf{B}_l^m \cdot \mathbf{P}_l^m = 0, \end{aligned}$$

(40)

and satisfy the orthogonality relation

$$\begin{aligned} \int \limits _{S^2}\mathbf{P}_l^m(\hat{\mathbf{r}})\cdot {\mathbf{P}_{l^{\prime }}^{m^{\prime }}}^{*}(\hat{\mathbf{r}}) \ ds(\hat{\mathbf{r}})&= \int \limits _{S^2}\mathbf{C}_l^m(\hat{\mathbf{r}})\cdot {\mathbf{C}_{l^{\prime }}^{m^{\prime }}}^{*}(\hat{\mathbf{r}})\ ds(\hat{\mathbf{r}}) \nonumber \\&= \int \limits _{S^2}\mathbf{B}_l^m(\hat{\mathbf{r}})\cdot {\mathbf{B}_{l^{\prime }}^{m^{\prime }}}^{*}(\hat{\mathbf{r}}) \ ds(\hat{\mathbf{r}}) =\frac{4\pi }{2l+1}\frac{(l+|m|)!}{(l-|m|)!}\delta _{ll^{\prime }}\delta _{mm^{\prime }}.\quad \qquad \end{aligned}$$

(41)

for any $l=0,1,2,\ldots$ $|m|\leqslant l$.

An example of vectors $\mathbf{C}_1^0(\hat{\mathbf{r}})$ and $\mathbf{C}_2^0(\hat{\mathbf{r}})$ is shown in Fig. 2. The set of vector spherical harmonics forms a complete orthogonal set in the space $[L^2(S^2)]^3$; therefore, any function $\mathbf{u} \in [L^2(S^2)]^3$ may be represented in the form

$$\begin{aligned} \mathbf{u}(\hat{\mathbf{r}}) =\sum \limits _{l=0}^{\infty }\sum \limits _{m=-l}^{l}[a_{lm}\mathbf{P}_l^m(\hat{\mathbf{r}})+ b_{lm}\mathbf{B}_l^m(\hat{\mathbf{r}})+ c_{lm}\mathbf{C}_l^m(\hat{\mathbf{r}})]= \mathbf{u}_{p}(\hat{\mathbf{r}})+\mathbf{u}_{s}(\hat{\mathbf{r}}), \end{aligned}$$

(42)

$$\begin{aligned} \mathbf{u}_{p}(\hat{\mathbf{r}})\!=\!\sum \limits _{l=0}^{\infty }\sum \limits _{m=-l}^{l} a_{lm}\mathbf{P}_l^m(\hat{\mathbf{r}}),\ \mathbf{u}_{s}(\hat{\mathbf{r}})\!=\!\sum \limits _{l=0}^{\infty }\sum \limits _{m=-l}^{l} [b_{lm}\mathbf{B}_l^m(\hat{\mathbf{r}})+ c_{lm}\mathbf{C}_l^m(\hat{\mathbf{r}})], \end{aligned}$$

(43)

where $\mathbf{u}_{p},\mathbf{u}_{s}$ are, respectively, the potential and the solenoidal components of the field.

Fig. 2

Vector spherical harmonics. The view from the positive direction of axis $z$.

1.3 Hansen vectors

$$\begin{aligned} \mathbf{M}_l^m(\mathbf{r})&=\nabla _\mathbf{r}\times [\mathbf{r}\, z_l(\varkappa r) Y_{lm}(\hat{\mathbf{r}})]= \sqrt{l(l+1)}\, z_l(\varkappa r)\,\mathbf{C}_l^m(\hat{\mathbf{r}}),\end{aligned}$$

(44a)

$$\begin{aligned} \mathbf{N}_l^m(\mathbf{r})&=\frac{1}{\varkappa }\nabla _\mathbf{r}\times \nabla _\mathbf{r}\times [\mathbf{r}\, z_l(\varkappa r) Y_{lm}(\hat{\mathbf{r}})]= l(l+1)\frac{z_l(\varkappa r)}{\varkappa r }\, \mathbf{P}_l^m(\hat{\mathbf{r}}), \end{aligned}$$

(44b)

$$\begin{aligned} \mathbf{L}_l^m(\mathbf{r})&= \nabla _\mathbf{r}[ z_l(\varkappa r)Y_{lm}(\hat{\mathbf{r}})]\nonumber \\&=\sqrt{l(l+1)}\frac{z_l(\varkappa r)}{\varkappa r }\, \mathbf{B}_l^m(\hat{\mathbf{r}})\!+\!\frac{d z_l(\varkappa r)}{d(\varkappa r)} \,\mathbf{P}_l^m(\hat{\mathbf{r}}), \end{aligned}$$

(44c)

where $z_l(\varkappa r)= j_l(\varkappa r), y_l(\varkappa r), h_l^{(1)}(\varkappa r), h_l^{(2)}(\varkappa r)$ designate any spherical Bessel function of the first, the second and the third kind, respectively.

1.4 $D$– Wigner functions

Matrices $D^J_{M_1\,M_2}(\alpha ,\beta ,\gamma )$ are representations of the group of rotation SO(3) and are usually written as production three functions, each depending only of one Euler angle $(\alpha , \beta , \gamma )$:

$$\begin{aligned} D^J_{M_1\,M_2}(\alpha ,\beta ,\gamma )=e^{-\mathsf{i}M_1\alpha }\, d^J_{M_1\,M_2}(\beta )\,e^{-\mathsf{i}M_2 \gamma }. \end{aligned}$$

Here $\alpha$ $(0 \leqslant \alpha < 2\pi )$ is an angle of rotation with respect to the initial axis $z$, $\beta$ $(0 \leqslant \beta \leqslant \pi )$ is an angle of rotation with respect to a new (turned) axis $y^\prime$, and $\gamma$ $(0 \leqslant \gamma < 2\pi )$ is an angle of rotation with respect to a new (turned) axis $z^\prime$. Any rotation of frame of reference may be executed sequentially by rotation of axis $z$, next, a new axis $y^\prime$ and a new axis $z^\prime$ at angles, respectively, $(\alpha ,\beta ,\gamma )$. Real functions $d^J_{M_1\,M_2}(\beta )$ have the following precise representation

$$\begin{aligned} d^J_{M_1\,M_2}(\beta )&= (-1)^{M_1-M_2}[(J+M_1)!(J-M_1)!(J+M_2)!(J-M_2)!]^{1/2} \nonumber \\&\times \sum _k(-1)^k\frac{(\cos \beta /2)^{2J-2k-M_1+M_2} (\sin \beta /2)^{2k+M_1-M_2}}{k!(J-M_1-k)!(J+M_2-k)!(M_1-M_2+k)!}. \end{aligned}$$

Index $k$ runs through all the integer values for which the factorial’s argument has the positive values

$$\begin{aligned} \max (0,M_2-M_1)\le k\le \min (J-M_1,J+M_2) \end{aligned}$$

[see Varshalovich et al. (1988)]. Useful recurrent formulas for computing $d^J_{M_1\,M_2}(\beta )$ may be found in Biedenharn and Louck (1981). In Edmonds (1957) one can find the method for computing $D^J_{M_1\,M_2}(\alpha ,\beta ,\gamma )$ for arbitrary arguments with the use of the following relation:

$$\begin{aligned} D^J_{M_1\,M_2}(\alpha ,\beta ,\gamma )= \sum _{m}e^{-\mathsf{i}M_1 \alpha }\cdot d^J_{M_1\,m}(\pi /2)\cdot e^{-\mathsf{i}m \beta } \cdot d^J_{m\,M_2}(\pi /2)\cdot e^{-\mathsf{i}M_2 \gamma }, \end{aligned}$$

where $d^J_{M_1\,M_2}(\pi /2)$ may be easily computed.

Functions $D_{M_1\,M_2}^J(\alpha ,\beta ,\gamma )$, when $\gamma =0$, satisfy the conditions of orthogonality and normalisation.

$$\begin{aligned} \int \!\!\! D^{J}_{M_1\,M_2}(\varvec{\nu })D^{J^\prime *}_{M_1^\prime \,M_2^\prime }(\varvec{\nu }) d\Omega&\!=\! \frac{4\pi }{2J+1}\delta _{JJ^\prime }\delta _{M_1M_1^\prime }, \ {\varvec{\nu }}\!=\!(\alpha ,\beta ), \\ \ d\Omega =sin\beta d\beta d\alpha ,&\ 0 \leqslant \alpha < 2\pi , \ \ \ 0\leqslant \beta \leqslant \pi . \nonumber \end{aligned}$$

(45)