Introduction to Yasuura’s Method of Modal Expansion with Application to Grating Problems

Abstract

In this chapter we introduce the theory of the Yasuura’s method based on modal expansion and explain the methods of numerical computation in detail for several grating problems. After a sample problem we discuss the methods for solving two types of problems that require additional knowledge and steps, that is, scattering by a dielectric cylinder and diffraction by a grating. Some numerical results are shown to give an evidence of an experimental rule for the number of linear equations in formulating the least-squares problem that determines the modal coefficients. After confirming the rule we show a couple of examples of practical interest, i.e., scattering by a relatively deep metal grating, plasmon surface waves on a metal grating placed in conical mounting, scattering by a metal surface modulated in two directions, and scattering by periodically located dielectric spheres. To provide supplementary explanations of particular problems, four appendices are given; H-wave scattering from a cylinder, the normal equation and related topics, conical diffraction by a dielectric grating, and comparison of modal functions and the algorithm of the smoothing procedures.

Appendices

Appendix 1: H-Wave Scattering by a PC Cylinder

Let us consider a problem where an H-wave (TM-wave) is incident to the obstacle shown in Fig. 8.1. That is, the incident wave is polarized in the xy-plane so that the incident magnetic field has only a z-component

$$\begin{aligned} \mathbf{H}^\mathrm{i}(\mathbf{r}) = \mathbf{u}_z F(\mathbf{r}) = \mathbf{u}_z \exp [-\mathrm{i}kr \cos (\theta -\iota )]. \end{aligned}$$

(8.92)

The scattered magnetic field has only a z-component

$$\begin{aligned} \mathbf{H}^\mathrm{s}(\mathbf{r}) = \mathbf{u}_z \varPsi (\mathbf{r}) \end{aligned}$$

(8.93)

which is a leading field of the problem. Thus, we have

Problem 1’: H-wave, PC. Find $\varPsi (\mathbf{r})$ that satisfies:  

(N1):

The 2-D Helmholtz equation in $\mathrm{S}_\mathrm{e}$;

(N2):

The 2-D radiation condition at infinity;

(N3):

The boundary condition

$$\begin{aligned} \partial _\nu \varPsi (s) = g(s) \equiv -\partial _\nu F(s) \quad (s \in \text{ C }). \end{aligned}$$

(8.94)

Here, $\partial _\nu$ denotes a normal derivative at s. Equation (8.94) is called Neumann’s or the second-kind boundary condition.

 

Employing the Green’s (or Neumann’s) function of this boundary-value problem satisfying a homogeneous boundary condition

$$\begin{aligned} \partial _\nu N(\mathbf{r}, s) = 0\quad (\mathbf{r}\in \mathrm{S}; s \in \mathrm{C}), \end{aligned}$$

(8.95)

we get a formal representation similar to (8.13)

$$\begin{aligned} \varPsi _N(\mathbf{r}) - \varPsi (\mathbf{r}) = -\int \limits _{s=0}^C N(\mathbf{r},s) \left[ \partial _\nu \varPsi _N(s) - g(s)\right] \,ds\quad (\mathbf{r}\in \mathrm{S}). \end{aligned}$$

(8.96)

Here, $\varPsi _N$ denotes an approximate solution defined by

$$\begin{aligned} \varPsi _N(\mathbf{r}) = \sum _{m=-N}^N B_m(M)\varphi _m(\mathbf{r}). \end{aligned}$$

(8.97)

After a discussion similar to that in Sect. 8.2.2.3, we have a least-squares problem for the H-wave problem:

LSM 1’: H-wave, PC. Find the coefficients $B_m(M)$ $(m=0, \pm 1, \ldots , \pm N)$ that minimize the mean-squares boundary residual

$$\begin{aligned} E_N = \frac{\left\| \partial _\nu \varPsi _N - g \right\| ^2}{\Vert g \Vert ^2} = \frac{1}{\Vert g \Vert ^2} \left\| \sum _{m=-N}^N B_m(M)\partial _\nu \varphi _m - g \right\| ^2. \end{aligned}$$

(8.98)

We can solve this problem on a computer following the procedure in Sect. 8.2.3. Approximations to other nonzero components can be found by

$$\begin{aligned} \mathbf{E}_N^\mathrm{s}(\mathbf{r}) = \frac{1}{\mathrm{i}\omega \varepsilon _0}\nabla \varPsi _N(\mathbf{r}) \times \mathbf{u}_z. \end{aligned}$$

(8.99)

It is worth noting that in an H-wave scattering from a dielectric obstacle, the boundary condition (8.42) should be altered slightly. Let $\mathbf{H}^\mathrm{s}(\mathbf{r})=\mathbf{u}_z\varPsi _\mathrm{e}(\mathbf{r})$, and $\mathbf{H}^\mathrm{t}(\mathbf{r})=\mathbf{u}_z\varPsi _{\,\mathrm i}(\mathbf{r})$, then we have

$$\begin{aligned} \left\{ \begin{array}{l} \varPsi _\mathrm{e}(s)-\varPsi _{\,\mathrm i}(s)=f(s)\equiv -F(s)\\ \partial _\nu \varPsi _\mathrm{e} - n^{-2} \partial _\nu \varPsi _{\,\mathrm i}(s) = g(s) \equiv -\partial _\nu F(s), \end{array}\right. \end{aligned}$$

(8.100)

where the second line means the electric-field continuity and $n^2 = \varepsilon /\varepsilon _0$.

Appendix 2: Solution of LSP 1 by a Normal Equation and Related Topics

Although we do not use a normal equation in numerical analysis, we look over the solution method by the equation because it is an important theoretical tool in working with a least-squares problem. Let us define an inner product between two functions in $\mathbf{H}= L^2(0, C)$ by

$$\begin{aligned} (f,g)=\int \limits _{s=0}^C \overline{f(s)}g(s)\,ds, \end{aligned}$$

(8.101)

then we find that $\Vert f \Vert = \sqrt{(f,f)}$. Employing these relations, we modify (8.22) to obtain

$$\begin{aligned} E_N = \!\! \sum _{m=-N}^N \sum _{n=-N}^N \overline{A_m}(\varphi _m,\varphi _n) A_n - \!\! \sum _{m=-N}^N \overline{A_m} (\varphi _m,f) - \!\! \sum _{n=-N}^N (f,\varphi _n)A_n + \Vert f \Vert ^2. \end{aligned}$$

(8.102)

The predictable M is not shown.

Now we define a subspace of $\mathbf{H}$, $\varPhi _N$, spanned by the boundary values of a finite number of modal functions $\{\varphi _0(s),\varphi _{\pm 1}, \ldots ,$ $\varphi _{\pm N}\}$. An element of $\varPhi _N$ can be represented as

$$\begin{aligned} \varPsi _N(s) = \sum _{n=-N}^N A_n\,\varphi _n(s). \end{aligned}$$

(8.103)

Apparently, there is a minimum value of $E_N$, which is a squared distance between f(s) and a point in $\varPhi _N$.³² The minimum is achieved when (8.103) agrees with the foot of a perpendicular line from f(s) to the surface of $\varPhi _N$. The necessary and sufficient condition for this is that: The $A_m$ coefficients are the solutions of the set of linear equations

$$\begin{aligned} \sum _{n=-N}^N (\varphi _m,\varphi _n)\,A_n = (\varphi _m, f)\quad (m=0,\pm 1,\ldots ,\pm N). \end{aligned}$$

(8.104)

This is referred to as the normal equation (NE) of LSP 1 and is a formal solution to the problem.³³

Next, let us consider the minimization from a computational point of view. That is, we try to find the $A_m$ coefficients using the sampled values of boundary functions; and the functions are represented by J-dimensional complex-valued vectors $\mathbf{f}$, $\varvec{\varphi }_m$, and $\varvec{\varPsi }_N$ as in Sect. 8.2.3. This leads us to DLSP 1. We know the orthogonal decompositions are useful tools for solving the problem. However, setting them aside, we here consider a NE based on DLSP 1. Because the Jacobian matrix $\varPhi$ is $J \times M$ $(J > M)$, the set of linear equations

$$\begin{aligned} \varPhi \mathbf{A} = \mathbf{f} \end{aligned}$$

(8.105)

is over-determined and does not have a usual solution. However, if we multiply (8.105) by $\varPhi ^\dag$ from the left, we have

$$\begin{aligned} \text{ H } \mathbf{A} = \mathbf{b}, \end{aligned}$$

(8.106)

where

$$\begin{aligned} \text{ H } = \varPhi ^\dag \varPhi \end{aligned}$$

(8.107)

is an $M \times M$ positive-definite Hermitian matrix provided $\varPhi$ is full rank. And,

$$\begin{aligned} \mathbf{b} = \varPhi ^\dag \mathbf{f} \end{aligned}$$

(8.108)

is an M-dimensional right-hand side. Usually, (8.106) is referred to as the NE of DLSP 1 and has been employed as a standard method of solution for a long time.

Obviously, (8.106) is an approximation of (8.104). For example, an (m, n)th element of the coefficient matrix of (8.104) can be represented as

$$\begin{aligned} (\varphi _m,\varphi _n) = \int \limits _{s=0}^C\overline{\varphi _m(s)}\varphi _n(s)\, ds \simeq \frac{C}{J}\sum _{j=1}^J\overline{\varphi _m(s_j)}\varphi _n(s_j) = \frac{C}{J}\,\varvec{\varphi }_m^\dag \varvec{\varphi }_n. \end{aligned}$$

(8.109)

The right-hand side of (8.109) is the (m, n)th entry of H multiplied by the line element C / J. Hence, (8.104) and (8.106) are essentially the same thing, and they have common weak points in numerical computations. Widely-accepted key observations are:

• The NE is rigorous, in principle, and can be employed in theoretical considerations;

• The NE combined with Gaussian elimination (diagonal pivoting assumed) is equivalent to the (modified) Schmidt QRD except for the next two items;

• The NE may lose information in constructing $\mathrm{H} = \varPhi ^\dag \varPhi$, and this process is time consuming usually;

• The NE is dominated by the condition number of H that is square of the original condition number: $\text{ cond }(\text{ H }) = [\text{ cond }(\varPhi )]^2$.

The last item means (8.104) and (8.106) are more sensitive to computational errors than LSP 1 and DLSP 1. Therefore, the NE’s are more difficult to solve on a computer than the original least-squares problems. We, hence, do not recommend the use of (8.104) or (8.106). Even if we are working in the case where the inner products in (8.104) can be calculated analytically, we should not employ (8.104) because of the last item.

Before closing this Appendix, we would like to state a couple of comments on (8.105). Apparently, J cannot be less than M because (8.105) is indeterminate for $J < M$. If we set $J = M$, we have a point-matching method (PMM) or a collocation method. The method is known to be effective if the contour C coincides with a part of a coordinate curve of a system of coordinates in which Helmholtz’s equation is separable; and that the modal functions are the separated solutions in that system. Convergence of the PMM solution is related to the validity of the Rayleigh hypothesis [2, 3, 18].

In Yasuura’s method we usually set $J = 2M$ as we see in Sect. 8.3.1. That is, we employ 2M linear equations to determine M unknown coefficients. This may be understood as a small device or improvement of the PMM. However, this produces good results such as proof of convergence, wide range of application, and so on with little increase of computational complexity as a reasonable cost.

Appendix 3: Conical Diffraction by a Grating

In Sect. 8.2.5 we dealt with diffraction by a grating, where all the field components were functions of two variables (X and Y) and two independent cases of polarization [E-wave (TE, s) and H-wave (TM, p)] existed. In addition, the directions of propagating diffraction-orders were parallel to the plane of incidence. These were possible because: (1) the grating surface was uniform in Z; and (2) the plane of incidence was in parallel to the direction of periodicity $\mathbf{u}_X$. Here, we concisely examine the problem of a lossless dielectric grating in which the second condition is not satisfied, i.e., the plane of incidence makes a nonzero angle $\phi$ with the positive X-direction as shown in Fig. 8.14a. We will see that

• The field components are functions of X, Y, and Z, but the dependence on Y—the direction of uniformity—is limited;

• The two cases of polarization are not independent, i.e., both TE and TM diffracted waves exist for TE (or TM) incidence³⁴;

• The direction of propagating orders lie on the surface of a cone whose vertex agrees with the coordinate origin O; the direction of the zeroth mode is on the plane of incidence at the same time.

Because of the third characteristic, this arrangement ($\phi \not = 0$) is called conical mounting and the term conical diffraction is used. In this connection, the arrangement in Sect. 8.2.5 is termed planar mounting.

Let the incident wave be

$$\begin{aligned} \left[ \begin{array}{c} \mathbf{E}^\mathrm{i} \\ \mathbf{H}^\mathrm{i} \end{array} \right] (\mathbf{r}) = \left[ \begin{array}{c}{} \mathbf{e}^\mathrm{i}\\ \mathbf{h}^\mathrm{i} \end{array}\right] \exp (-\mathrm{i}\mathbf{k}^\mathrm{i} \cdot \mathbf{r}). \end{aligned}$$

(8.110)

Here, $\mathbf{e}^\mathrm{i}$ and $\mathbf{h}^\mathrm{i}$ are electric- and magnetic-field amplitude, which are related by

$$\begin{aligned} \mathbf{h}^\mathrm{i} = \frac{1}{\omega \mu _0}\, \mathbf{k}^\mathrm{i} \times \mathbf{e}^\mathrm{i} \end{aligned}$$

(8.111)

and $\mathbf{k}^\mathrm{i}$ is the incident wavevector defined by

$$\begin{aligned} \mathbf{k}^\mathrm{i} = (n_1 k\sin \theta \cos \phi , n_1 k\sin \theta \sin \phi , -n_1 k\cos \theta ) \equiv (\alpha , \beta , -\gamma ) \end{aligned}$$

(8.112)

with $\theta$ being the polar angle between the wavevector $\mathbf{k}^\mathrm{i}$ and the grating normal $\mathbf{u}_Z$. We decompose the incident wave into a TE(s)- and a TM(p)-component, where TE (or TM) means that the electric (or magnetic) field of the relevant incident wave is perpendicular to the plane of incidence. To do this, we define two unit vectors that span a plane orthogonal to the incident wavevector

$$\begin{aligned} \mathbf{e}^\mathrm{TE}=(\sin \phi ,-\cos \phi ,0), \quad \mathbf{e}^\mathrm{TM}=(\cos \theta \cos \phi ,\cos \theta \sin \phi ,\sin \theta ). \end{aligned}$$

(8.113)

They give the directions of the incident electric fields that are in the TE- and TM-polarization.³⁵ Thus the decomposition is

$$\begin{aligned} \mathbf{e}^\mathrm{i} = \mathbf{e}^\mathrm{TE}\cos \delta + \mathbf{e}^\mathrm{TM}\sin \delta , \end{aligned}$$

(8.114)

where $\delta$ is a polarization angle shown in Fig. 8.14b. $\delta = 0$ and $\pi /2$ mean TE- and TM-incidence. Hence, an incident wave has three angular parameters: $\phi$, $\theta$, and $\delta$.

We consider the problem to seek the diffracted electric and magnetic field in the semi-infinite regions $\text{ V }_1$ and $\text{ V }_2$ over and below the grating surface $\text{ S }_\mathrm{G}$.

Problem 4 conical, dielectric grating. Find the solutions that satisfy the following requirements:  

(CD1):

The Helmholtz equation in $\text{ V }_1$ and $\text{ V }_2$;

(CD2):

Radiation conditions in the positive and negative Z-direction;

(CD3):

A periodicity condition that: the relation $f(X+d,Y,Z) = e^{\mathrm{i}\alpha d}f(X,Y,Z)$ holds for any component of the diffracted wave, and the phase constant in Y is $\beta$;

(CD4):

The total tangential component of electric and magnetic field must be continuous across the grating surface $\text{ S }_\mathrm{G}$.

  Dealing with a problem of conical diffraction, we should keep in mind the unique nature of the problem. First, because every field component has a common phase constant $\beta$ in Y, it is sufficient to match the boundary condition on a cross section between the grating surface and a plane $Y = \text{ const }$. The conically-mounted gratings, hence, belong to the class of quasi-3-D structures. Second, because the TE- and TM-wave are not independent, we always need both TE and TM vector modal functions in constructing approximate solutions.

We define the modal functions satisfying (CD1)–(CD3) by

$$\begin{aligned} \left\{ \begin{array}{l} \varvec{\varphi }_{\ell m}^\mathrm{TE}(\mathbf{r}) = \mathbf{e}_{\ell m}^\mathrm{TE} \exp (-\mathrm{i}\mathbf{k}_{\ell m} \cdot \mathbf{r}), \quad \varvec{\varphi }_{\ell m}^\mathrm{TM}(\mathbf{r}) = \mathbf{e}_{\ell m}^\mathrm{TM} \exp (-\mathrm{i}\mathbf{k}_{\ell m} \cdot \mathbf{r}) \\ \quad \quad \quad (\ell = 1, 2;\; m = 0, \pm 1, \pm 2, \ldots ). \end{array} \right. \end{aligned}$$

(8.115)

Here,

$$\begin{aligned} \mathbf{e}_{\ell m}^\mathrm{TE} = \frac{ \mathbf{k}_{\ell m} \times \mathbf{u}_Z}{|\mathbf{k}_{\ell m} \times \mathbf{u}_Z|}, \quad \mathbf{e}_{\mathrm{p}m}^\mathrm{TM} = \frac{ \mathbf{e}_{\ell m}^\mathrm{TE} \times \mathbf{k}_{\ell m}}{|\mathbf{e}_{\ell m}^\mathrm{TE} \times \mathbf{k}_{\ell m}|} \quad (\ell = 1, 2), \end{aligned}$$

(8.116)

$$\begin{aligned} \mathbf{k}_{1m} = (\alpha _m, \beta , \gamma _{1m}),\quad \mathbf{k}_{2m} = (\alpha _m, \beta , -\gamma _{2m}), \end{aligned}$$

(8.117)

and

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \alpha _m = \alpha + \frac{2m\pi }{d}, \quad \gamma _{\ell m} = \sqrt{(n_\ell k)^2 - (\alpha _m^2 + \beta ^2)} \\ \quad \quad \quad (\text{ Re }\,\gamma _{\ell m} \ge 0,\; \text{ Im }\,\gamma _{\ell m} \le 0). \end{array} \right. \end{aligned}$$

(8.118)

Note that the functions in (8.115) are for constructing electric fields. For the magnetic fields we get

$$\begin{aligned} \varvec{\psi }_{\ell m}^\mathrm{q}(\mathbf{r}) = \frac{1}{\omega \mu _0}\, \mathbf{k}_{\ell m} \times \varvec{\varphi }_{\ell m}^\mathrm{q}(\mathbf{r}) \quad (\ell = 1, 2;\; \mathrm{q}= \mathrm{TE, TM}) \end{aligned}$$

(8.119)

through Maxwell’s equations. Finite linear combinations of the modal functions define approximate solutions:

$$\begin{aligned} \left[ \begin{array}{c} \mathbf{E}_{\ell N} \\ \mathbf{H}_{\ell N} \end{array} \right] (\mathbf{r}) =\sum _{m=-N}^N A_{\ell m}^\mathrm{TE} \left[ \begin{array}{c} \varvec{\varphi }_{\ell m}^\mathrm{TE}\\ \varvec{\psi }_{\ell m}^\mathrm{TE}\end{array}\right] (\mathbf{r}) + \sum _{m=-N}^N A_{\ell m}^\mathrm{TM} \left[ \begin{array}{c} \varvec{\varphi }_{\ell m}^\mathrm{TM}\\ \varvec{\psi }_{\ell m}^\mathrm{TM} \end{array}\right] (\mathbf{r}) \quad (\ell = 1, 2) \end{aligned}$$

(8.120)

Here, the number of modal functions M is neglected.

The unknown coefficients in (8.120) should be determined in order that the solutions satisfy the boundary condition (CD4) approximately in the mean-squares sense. For this purpose we first consider the cross section C between the grating surface $\mathrm{S}_\mathrm{G}$ and a plane $Y = 0$. This is the same thing as the periodic curve C in Sect. 8.2.5. In a similar way to one in Sect. 8.2.5, we define the primary period $\mathrm{S}_1$, whose boundary $\mathrm{C}_1$ ($\subset \mathrm{C}$), the function space $\mathbf{H}$ consisting of all the square integrable functions on $\mathrm{C}_1$, and the norm $\Vert f \Vert$ of a function f(s). Then, we can state the least-squares problem that determines the unknown coefficients:

LSP 4: conical, dielectric grating. Find the coefficients $A_{\ell m}^\mathrm{TM}$ and $A_{\ell m}^\mathrm{TM}$ $(\ell = 1, 2;\; m=0, \pm 1, \ldots , \pm N)$ that minimize the mean-square error

$$\begin{aligned} E_N = \left\| \varvec{\nu }\times \left( \tilde{\mathbf{E}}_{1N} + \tilde{\mathbf{E}}^\mathrm{i} - \tilde{\mathbf{E}}_{2N} \right) \right\| ^2 + Z_0^2 \left\| \varvec{\nu }\times \left( \tilde{\mathbf{H}}_{1N} + \tilde{\mathbf{H}}^\mathrm{i} - \tilde{\mathbf{H}}_{2N} \right) \right\| ^2. \end{aligned}$$

(8.121)

Here, $Z_0$ denotes the intrinsic impedance of vacuum and $\tilde{\mathbf{E}}^\mathrm{i}$ etc. mean periodic functions with respect to x defined in the same way as one in (8.69)–(8.71). The method of discretization and the solution method are found in Sect. 8.2.4.

Appendix 4: Comparison of Modal Functions and Algorithm of the SP

Here we show some results of effectiveness comparison between three kinds of modal functions in solving a sample problem³⁶: E-wave scattering from a PC cylinder whose cross section is given by³⁷

$$\begin{aligned} \text{ C }: r_s = a(1+0.2 \cos 3\theta _s). \end{aligned}$$

(8.122)

Let us normalize every quantity having dimension of length by the total length of C. And, we assume the incident wave comes along the x-axis from the negative x-direction (i.e., $\iota =0$).

The modal functions considered here are: (a) the separated solutions, which we defined by (8.7) in Sect. 8.2.2; (b) monopole fields defined by (8.9); (c) monopole fields whose poles are located densely near the convex part of C. Because the separated solutions are known widely, we explain the monopole fields below:  

(b) Equally spaced poles.:

Let L be a similar curve to C with the ratio of similitude $d\ (0< d < 1)$.³⁸ We arrange M poles on L at regular intervals. Then, the distance between two poles is L / M where L is the length of L.

(c) Concentration of poles near the convex parts of L.:

(i) First, we draw the similar curve L. (ii) Next,we calculate the curvature $\kappa (t)$ of L as a function of $t\;(\in \text{ L })$, and add some positive bias c in order that the biased curvature (BC) be no less than 0: $\tilde{\kappa }(t) = \kappa (t) + c\ (\ge 0)$. (iii) Thirdly, we define a probability density function by normalizing the BC.³⁹

$$\begin{aligned} f(t) = \frac{\displaystyle \int \limits _0^t \tilde{\kappa }(t')\, dt'}{\displaystyle \int \limits _0^L \tilde{\kappa }(t')\, dt'}. \end{aligned}$$

(8.123)

Thus we get the number of poles between $t_1$ and $t_2$ by

$$\begin{aligned} n(t_1, t_2) = M\int \limits _{t_1}^{t_2} f(t)\, dt. \end{aligned}$$

(8.124)

 

We have solved the problem using the method explained in Sects. 8.2.2 and 8.2.3. We used three kinds of modal functions (a), (b), and (c); and tried at two frequencies: $ka = 10$ and 30. The parameter d was set to be 0.87. To see the accuracy of a solution we calculated two kinds of errors: the normalized mean-square error $E_M(\text{ m })$ and the error on energy balance (or on the optical theorem) $e_M(\text{ m })$. The former is the same thing as one defined in (8.22) and (8.30)⁴⁰ except that the subscript shows the total number of modal functions. The latter shows the deviation from a proportional relation between the forward scattering amplitude and total cross section.⁴¹ The argument m shows the type of modal functions: m $=$ sep, esp, and pcc, which mean (a) separated solutions, (b) equally-spaced poles, and (c) poles concentrated near the convex parts.  

Results at $ka=10$.:

Because the obstacle size is handy, the $E_M$ errors fall off rapidly: $E_{45}(\mathrm{sep})$, $E_{35}(\mathrm{esp})$, and $E_{31}(\mathrm{pcc})$ are below 1%. As for the $e_M$ errors of the solutions, the situation is different. The solutions with esp or pcc modal functions converge rapidly as $e_M(\mathrm{esp})$ and $e_M(\mathrm{pcc})$ are below 1% at $M \simeq 30$. On the other hand, $e_{31}(\mathrm{sep})$ is about 10%. Increasing M to 70, we have: $e_{70}(\mathrm{esp})=9\times 10^{-5}$%; $e_{70}(\mathrm{pcc})=1\times 10^{-5}$%; and $e_{71}(\mathrm{sep})=4$%.

Results at $ka=30$.:

The advantage of the monopole fields is clear in this range of frequency. Setting $M\simeq 100$, we have $E_{101}(\mathrm{sep})=4$%, $E_{100}(\mathrm{esp})=2\times 10^{-1}$%, $E_{100}(\mathrm{pcc})=2\times 10^{-3}$%, $e_{101}(\mathrm{sep})=7$%, $e_{100}(\mathrm{esp})=2\times 10^{-1}$%, and $e_{100}(\mathrm{pcc})=5\times 10^{-3}$%. The pcc modal functions seems to be the best choice in solving the problem. In fact, we can find an accurate solution with a $10^{-5}$% $e_M$ error by setting $\text{ m } = \text{ pcc }$ and $M=120$.

 

These results mean that the potential of a combination of separated solutions is not so strong in describing scattered fields from obstacles deformed strongly from a circle. We have two ways to cope with this issue: (i) employment of a set of modal functions other than the separated solutions⁴²; and (ii) employment of the SP.

The Algorithm of the SP

Here we include a guidance how to apply the SP in the boundary-matching process based on Yasuura’s method of modal expansion for convenience. We start from DLSP 1, i.e., minimization of the numerator of (8.30), $\Vert \varPhi \mathbf{A}- \mathbf{f} \Vert ^2$. Instead of minimizing it directly, we force a constraint

$$\begin{aligned} (\mathbf{1}, \varPhi \mathbf{A}- \mathbf{f}) = 0 \end{aligned}$$

(8.125)

on the M-dimensional solution vector A, where the parentheses mean an inner product and $\mathbf{1}=[1\ 1\ \cdots \ 1]^\mathrm{T}$ is a J-dimensional constant vector.

An operator of the smoothing procedure (in a discretized form) is a $J\times J$ matrix given by

$$\begin{aligned} \mathrm{K}^{(p)} = \left[ K^{(p)}_{j\ell } \right] , \end{aligned}$$

(8.126)

where p means the order of the SP. The explicit forms of the matrix elements for $p=1, 2$, and 3 are⁴³:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle K^{(1)}_{j\ell } = u(j-\ell ) - \frac{j-\ell }{J} - \frac{1}{2}, \\ \displaystyle K^{(2)}_{j\ell } = -\frac{1}{2} \left[ \frac{(j-\ell )^2}{J^2} - \frac{|j-\ell |}{J} + \frac{1}{6} \right] , \\ \displaystyle K^{(3)}_{j\ell } = \frac{1}{6} \left[ \frac{|j-\ell |^3}{J^3} - \frac{3(j-\ell )^2}{2J^2} + \frac{|j-\ell |}{2J} + \frac{1}{6} \right] . \end{array} \right. \end{aligned}$$

(8.127)

Thus we can state a method of solution with the SP as follows:

DLSP 3: E-wave, PC, SP. Find the solution vector A that minimizes the discretized mean-square error

$$\begin{aligned} E_{MJ}= \frac{\Vert \mathrm{K}^{(p)}(\varPhi \mathbf{A}- \mathbf{f}) \Vert ^2}{\Vert \mathrm{K}^{(p)}{} \mathbf{f\,} \Vert ^2} \end{aligned}$$

(8.128)

under the constraint (8.125).

Two ways are possible to solve this conditioned least-squares problem: (i) employment of Lagrange’s multiplier; and (ii) elimination of a modal coefficient by using the constraint. Although (i) is a standard way in handling a constraint, we take (ii) because our constraint is simple and can eliminate one of the M unknowns to deduce a least-squares problem with $M-1$ unknowns.