# Electromagnetic Boundary Conditions Defined by Reflection Properties of
Eigen Plane Waves

Ismo V. Lindell and Ari Sihvola
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Radio Science and Engineering
Aalto University, School of Electrical Engineering
Espoo, Finland

###### Abstract

It is known that the two eigen plane waves incident to the generalized soft-
and-hard/DB (GSHDB) boundary are reflected as from the PEC or PMC boundary,
i.e., with reflection coefficients $-1$ or $+1$, for any angle of incidence.
The present paper discusses a more general class of boundaries by requiring
that the reflection coefficients $R_{+}$ and $R_{-}$, corresponding to the two
eigen plane waves, have opposite values, $R_{\pm}=\pm R$ with $R$ independent
of the angle of incidence. It turns out that, there are two possibilities,
$R=1$ for the class of GSHDB boundaries, and $R=j$ for another class,
extending that of the perfect electromagnetic conductor (PEMC) boundaries.
Matched waves at, and plane-waves reflected from, boundaries of the latter
class are studied in the paper.

##  1 Introduction

Boundary conditions are fundamental building blocks in problems of
mathematical physics, including electromagnetics. For given sources in a given
space, Maxwell equations require additional constraints on the boundaries of
the domain that force uniqueness and existence of the solution. These
conditions between the electric and magnetic fields and fluxes at the boundary
can in principle be rather general [1]; however, in finite real-world
problems, the boundaries often are characterized by approximate surface
conditions [2, 3].

Examples of such boundary conditions are perfectly electrically conducting
(PEC) surfaces, artificial magnetic conductor and high-impedance surfaces [4,
5], and anisotropic soft-and-hard surfaces [6]. But the possibilities to
engineer the electromagnetic response of complex boundaries have a great
variety. Thin two-dimensional structures which are capable of modifying the
reflection and transmission of the incident wave in multiple manners have been
recently designed and fabricated, and the label metasurface has appeared in
the electromagnetics literature to cover such advanced surfaces [7].
Generalizing the traditional frequency-selective surfaces and transmit/reflect
arrays [8, 9], these metasurfaces can control the amplitude and phase of the
wavefront in various ways [10], the effects can be nonreciprocal [11], non-
local, or even active [12]. In the optical range of the electromagnetic
spectrum, these developments within the metasurface paradigm are known by
labels like planar photonics [13] and flat optics [14].

It is the purpose of the present paper to investigate still uncharted classes
of complex electromagnetic boundaries and surfaces. The treatment is purely
theoretical, concentrating on certain aspects of electromagnetic boundary
conditions beyond those in the book [1] by these authors. Theoretical
predictions of unconventional boundaries, like, as example, the perfect
electromagnetic conductor PEMC [15] have found experimental realizations [16],
and found practical applications in the manipulation of properties of
electromagnetic waves. The novel class of boundary conditions introduced in
the present paper may be of use in the future design of novel metaboundaries.

##  2 General Boundary Conditions

A typical electromagnetic problem involves solving the Maxwell equations in a
region bounded by a surface with imposed boundary conditions. The most general
(linear and local) boundary conditions (GBC) can be shown to take the form [1,
17, 18],

$\displaystyle\alpha_{1}c{\mbox{{n}}}\cdot{\mbox{{B}}}+\frac{\beta_{1}}{\epsilon_{o}}{\mbox{{n}}}\cdot{\mbox{{D}}}+{\mbox{{a}}}_{1t}\cdot{\mbox{{E}}}+\eta_{o}{\mbox{{b}}}_{1t}\cdot{\mbox{{H}}}$
$\displaystyle=$ $\displaystyle 0,$ (1)
$\displaystyle\alpha_{2}c{\mbox{{n}}}\cdot{\mbox{{B}}}+\frac{\beta_{2}}{\epsilon_{o}}{\mbox{{n}}}\cdot{\mbox{{D}}}+{\mbox{{a}}}_{2t}\cdot{\mbox{{E}}}+\eta_{o}{\mbox{{b}}}_{2t}\cdot{\mbox{{H}}}$
$\displaystyle=$ $\displaystyle 0,$ (2)

with

$c=1/\sqrt{\mu_{o}\epsilon_{o}},\ \ \eta_{o}=\sqrt{\eta_{o}/\epsilon_{o}}.$
(3)

The unit vector n is normal, and the vectors
${\mbox{{a}}}_{1t}\cdots{\mbox{{b}}}_{2t}$ are tangential, to the boundary
surface. They are dimensionless and so are the scalars
$\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$. Because one of the scalar
coefficients in each equation can be chosen, the number of free parameters
defining the general boundary conditions is 10. Realization of a boundary
defined by general conditions of the form (1) and (2), in terms of the
interface of a bianisotropic medium, has been discussed in [1], Section 5.14.

In the present paper we assume for simplicity that the boundary surface
${\mbox{{n}}}\cdot{\mbox{{r}}}=0$ is planar, as defined by a constant unit
vector n. Also, we assume that the medium is isotropic with parameters
$\epsilon_{o}$ and $\mu_{o}$, in which case the boundary conditions (1) and
(2) can be rewritten in the simpler equivalent form [1, 18]

$\displaystyle{\mbox{{a}}}_{1}\cdot{\mbox{{E}}}+{\mbox{{b}}}_{1}\cdot\eta_{o}{\mbox{{H}}}$
$\displaystyle=$ $\displaystyle 0,$ (4)
$\displaystyle{\mbox{{a}}}_{2}\cdot{\mbox{{E}}}+{\mbox{{b}}}_{2}\cdot\eta_{o}{\mbox{{H}}}$
$\displaystyle=$ $\displaystyle 0,$ (5)

where the four vectors ${\mbox{{a}}}_{1}\cdots{\mbox{{b}}}_{2}$ may contain
components normal and tangential to the boundary,

${\mbox{{a}}}_{i}=a_{in}{\mbox{{n}}}+{\mbox{{a}}}_{it},\ \ \
{\mbox{{b}}}_{i}=b_{in}{\mbox{{n}}}+{\mbox{{b}}}_{it},\ \ \ i=1,2.$ (6)

It is assumed that (4) and (5) represent two linearly independent conditions,
because, otherwise, they do not define a unique boundary-value problem.

###  2.1 Special Cases

The boundary conditions (4) and (5) contain a lot of familiar special cases
starting from the perfect electric conductor (PEC), defined by

${\mbox{{a}}}_{1t}\cdot{\mbox{{E}}}={\mbox{{a}}}_{2t}\cdot{\mbox{{E}}}=0,\ \ \
\ {\mbox{{a}}}_{1t}\times{\mbox{{a}}}_{2t}\not=0,$ (7)

and the perfect magnetic conductor (PMC), defined by

${\mbox{{b}}}_{1t}\cdot{\mbox{{H}}}={\mbox{{b}}}_{2t}\cdot{\mbox{{H}}}=0,\ \ \
\ {\mbox{{b}}}_{1t}\times{\mbox{{b}}}_{2t}\not=0.$ (8)

A straightforward generalization of these two conditions is that of the
perfect electromagnetic conductor (PEMC) [15, 19, 20],

${\mbox{{b}}}_{1t}\cdot({\mbox{{H}}}+M{\mbox{{E}}})={\mbox{{b}}}_{2t}\cdot({\mbox{{H}}}+M{\mbox{{E}}})=0,$
(9)

or, in vector form,

${\mbox{{n}}}\times({\mbox{{H}}}+M{\mbox{{E}}})=0,$ (10)

where $M$ is the PEMC admittance.

As other special cases of the GBC conditions (1) and (2) we may add

  * •

The DB conditions [21],

${\mbox{{n}}}\cdot{\mbox{{D}}}=\epsilon_{o}{\mbox{{n}}}\cdot{\mbox{{E}}}=0,\ \
\ \ {\mbox{{n}}}\cdot{\mbox{{B}}}=\mu_{o}{\mbox{{n}}}\cdot{\mbox{{H}}}=0.$
(11)

  * •

The soft-and-hard (SH) conditions [22],

${\mbox{{a}}}_{t}\cdot{\mbox{{E}}}=0,\ \ \ \
{\mbox{{a}}}_{t}\cdot{\mbox{{H}}}=0.$ (12)

  * •

The generalized soft-and-hard (GSH) conditions [23],

${\mbox{{a}}}_{t}\cdot{\mbox{{E}}}=0,\ \ \ \
{\mbox{{b}}}_{t}\cdot{\mbox{{H}}}=0.$ (13)

  * •

In the EH boundary conditions [1],

${\mbox{{a}}}\cdot{\mbox{{E}}}=0,\ \ \ \ {\mbox{{b}}}\cdot{\mbox{{H}}}=0,$
(14)

the vectors a and b are not restricted, whence (14) are more general than
(13).

  * •

The soft-and hard/DB (SHDB) conditions [24],

${\mbox{{a}}}_{t}\cdot{\mbox{{E}}}+\alpha{\mbox{{n}}}\cdot\eta_{o}{\mbox{{H}}}=0,\
\ \ \
\alpha{\mbox{{n}}}\cdot{\mbox{{E}}}-{\mbox{{a}}}_{t}\cdot\eta_{o}{\mbox{{H}}}=0,$
(15)

contain SH and DB conditions (12), (11) as special cases.

  * •

The generalized soft-and-hard/DB (GSHDB) conditions [25],

${\mbox{{a}}}_{t}\cdot{\mbox{{E}}}+\alpha{\mbox{{n}}}\cdot\eta_{o}{\mbox{{H}}}=0,\
\ \ \
\beta{\mbox{{n}}}\cdot{\mbox{{E}}}+{\mbox{{b}}}_{t}\cdot\eta_{o}{\mbox{{H}}}=0,$
(16)

represent a further generalization of (15).

  * •

The impedance conditions

${\mbox{{a}}}_{1t}\cdot{\mbox{{E}}}+{\mbox{{b}}}_{1t}\cdot\eta_{o}{\mbox{{H}}}=0,\
\ \ \
{\mbox{{a}}}_{2t}\cdot{\mbox{{E}}}+{\mbox{{b}}}_{2t}\cdot\eta_{o}{\mbox{{H}}}=0.$
(17)

###  2.2 Purpose of this Study

In a previous study [25], it has been shown that a boundary defined by the
GSHDB conditions (16) has the following property: any plane wave incident from
an isotropic half space to the boundary can be decomposed in two eigen plane
waves, one of which is reflected as from the PEC boundary, and, the other one,
as from the PMC boundary. Since the GSHDB conditions (16) contain (11) – (13)
and (15) as special cases, the same property is shared by all the
corresponding boundaries. The converse property was shown in [17, 1]: if the
PEC/PMC decomposition property is required to be valid for the eigenwaves
associated to the GBC conditions (4), (5) (reflection coefficient either $+1$
or $-1$), the boundary must actually satisfy the GSHDB conditions (16). Thus,
the generalized soft-and-hard/DB boundary is the most general boundary with
the PEC/PMC equivalence property.

It is the purpose of the present paper to extend this theory by requiring that
the reflection coefficients $R_{+},R_{-}$ corresponding to the two eigen plane
waves satisfy $R_{\pm}=\pm R$, where $R$ is independent of the angle of
incidence (k vector of the incident wave). For the special case of the GSHDB
boundary we have $R=1$.

##  3 Reflection of Eigen Plane Waves

Let us consider time-harmonic plane waves satisfying the plane-wave equations

$\displaystyle{\mbox{{k}}}\times{\mbox{{E}}}$ $\displaystyle=$ $\displaystyle
k_{o}\eta_{o}{\mbox{{H}}},$ (18)
$\displaystyle{\mbox{{k}}}\times\eta_{o}{\mbox{{H}}}$ $\displaystyle=$
$\displaystyle-k_{o}{\mbox{{E}}}.$ (19)

and denote fields incident to, and reflecting from, the boundary surface,
respectively by

${\mbox{{E}}}^{i}({\mbox{{r}}})={\mbox{{E}}}^{i}\exp(-j{\mbox{{k}}}^{i}\cdot{\mbox{{r}}}),\
\ \ \
{\mbox{{E}}}^{r}({\mbox{{r}}})={\mbox{{E}}}^{r}\exp(-j{\mbox{{k}}}^{r}\cdot{\mbox{{r}}}),$
(20)

with (see Figure 1)

${\mbox{{k}}}^{i}={\mbox{{k}}}_{t}-k_{n}{\mbox{{n}}},\ \ \ \
{\mbox{{k}}}^{r}={\mbox{{k}}}_{t}+k_{n}{\mbox{{n}}},$ (21)
${\mbox{{k}}}^{i}\cdot{\mbox{{k}}}^{i}={\mbox{{k}}}^{r}\cdot{\mbox{{k}}}^{r}=k_{o}^{2}=\omega^{2}\mu_{o}\epsilon_{o}.$
(22)

GBC$\mathbf{k}^{i}$$\mathbf{k}^{r}$$\mathbf{k}_{t}$$\mathbf{k}_{t}$$k_{n}$$k_{n}$$\mathbf{n}$
Figure 1: Plane wave incident on a surface characterized by a general boundary
condition GBC. The incident and reflected wave vectors
$\mathbf{k}^{i},\mathbf{k}^{r}$ are decomposed into their tangential and
normal components.

Eigen plane waves are defined by requiring that the tangential components of
the electric fields at the boundary satisfy a relation of the form

${\mbox{{E}}}_{t}^{r}=R{\mbox{{E}}}_{t}^{i},$ (23)

i.e., that the incident and reflected fields tangential to the boundary have
similar polarization and are related by of a scalar factor $R$, the reflection
coefficient.

Polarizations and reflection coefficients of the eigen plane waves depend on
the vector parameters of the GBC boundary conditions (4), (5). Applying a
previous analysis [1, 17], the reflection coefficient can be shown to satisfy
the condition

${\mbox{{n}}}\cdot((1+R){\mbox{{a}}}_{1t}^{\prime}-(1-R){\mbox{{b}}}_{1t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})\times((1+R){\mbox{{a}}}_{2t}^{\prime}-(1-R){\mbox{{b}}}_{2t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})=0,$
(24)

where we denote

$\displaystyle{\mbox{{a}}}_{1t}^{\prime}$ $\displaystyle=$
$\displaystyle{\mbox{{a}}}_{1t}+(b_{1n}/k_{o}){\mbox{{n}}}\times{\mbox{{k}}}_{t},$
(25) $\displaystyle{\mbox{{a}}}_{2t}^{\prime}$ $\displaystyle=$
$\displaystyle{\mbox{{a}}}_{2t}+(b_{2n}/k_{o}){\mbox{{n}}}\times{\mbox{{k}}}_{t},$
(26) $\displaystyle{\mbox{{b}}}_{1t}^{\prime}$ $\displaystyle=$
$\displaystyle{\mbox{{b}}}_{1t}-(a_{1n}/k_{o}){\mbox{{n}}}\times{\mbox{{k}}}_{t},$
(27) $\displaystyle{\mbox{{b}}}_{2t}^{\prime}$ $\displaystyle=$
$\displaystyle{\mbox{{b}}}_{2t}-(a_{2n}/k_{o}){\mbox{{n}}}\times{\mbox{{k}}}_{t},$
(28)

and

${\overline{\overline{\mathsf{J}}}}_{t}=\frac{1}{k_{o}k_{n}}(({\mbox{{n}}}\times{\mbox{{k}}}){\mbox{{k}}}_{t}+k_{n}^{2}{\mbox{{n}}}\times{\overline{\overline{\mathsf{I}}}}_{t}).$
(29)

Here we assume $k_{n}\not=0$, which excludes the possibility of lateral
eigenwaves, ${\mbox{{k}}}={\mbox{{k}}}_{t}$. The dyadic
${\overline{\overline{\mathsf{J}}}}_{t}$ is known to satisfy the properties
[1]

${\overline{\overline{\mathsf{J}}}}_{t}\cdot{\overline{\overline{\mathsf{J}}}}_{t}=-{\overline{\overline{\mathsf{I}}}}_{t}=-{\overline{\overline{\mathsf{I}}}}+{\mbox{{n}}}{\mbox{{n}}},$
(30)

and

$({\mbox{{c}}}_{t}\cdot{\overline{\overline{\mathsf{J}}}}_{t})\times({\mbox{{d}}}_{t}\cdot{\overline{\overline{\mathsf{J}}}}_{t})={\mbox{{c}}}_{t}\times{\mbox{{d}}}_{t},$
(31)

valid for any two vectors ${\mbox{{c}}}_{t},{\mbox{{d}}}_{t}$ tangential to
the boundary. The rule

$k_{o}k_{n}{\mbox{{b}}}^{\prime}_{it}\cdot{\overline{\overline{\mathsf{J}}}}_{t}=({\mbox{{b}}}_{it}\cdot{\mbox{{n}}}\times{\mbox{{k}}}-k_{o}a_{in}){\mbox{{k}}}_{t}-k_{n}^{2}{\mbox{{n}}}\times{\mbox{{b}}}_{it},$
(32)

for $i=1,2$, appears useful in the subsequent analysis.

In the general case, the solutions $R$ of the equation (24) depend on the
boundary parameters ${\mbox{{a}}}_{1}\cdots{\mbox{{b}}}_{2}$ and the
tangential component of the wave vector, ${\mbox{{k}}}_{t}$. Requiring that
the solutions $R$ be independent of the wave vector sets conditions for the
boundary parameter vectors. For example, for the GSHDB boundary, (24) has the
solutions $R=+1$ and $R=-1$ for any ${\mbox{{k}}}_{t}$ [17].

Equation (24) is of the quadratic form

$C_{2}R^{2}+C_{1}R+C_{0}=0,$ (33)

with

$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle{\mbox{{n}}}\cdot({\mbox{{a}}}_{1t}^{\prime}+{\mbox{{b}}}_{1t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})\times({\mbox{{a}}}_{2t}^{\prime}+{\mbox{{b}}}_{2t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t}),$
(34) $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle{\mbox{{n}}}\cdot({\mbox{{a}}}_{1t}^{\prime}+{\mbox{{b}}}_{1t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})\times({\mbox{{a}}}_{2t}^{\prime}-{\mbox{{b}}}_{2t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})+$
(35) $\displaystyle+$
$\displaystyle{\mbox{{n}}}\cdot({\mbox{{a}}}_{1t}^{\prime}-{\mbox{{b}}}_{1t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})\times({\mbox{{a}}}_{2t}^{\prime}+{\mbox{{b}}}_{2t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t}),$
$\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle{\mbox{{n}}}\cdot({\mbox{{a}}}_{1t}^{\prime}-{\mbox{{b}}}_{1t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})\times({\mbox{{a}}}_{2t}^{\prime}-{\mbox{{b}}}_{2t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t}).$
(36)

For $C_{2}\not=0$ the two solutions are obtained from

$R_{\pm}=-\frac{C_{1}}{2C_{2}}\pm\sqrt{\frac{C_{1}^{2}}{4C_{2}^{2}}-\frac{C_{0}}{C_{2}}}.$
(37)

The coefficients $C_{0}$ and $C_{2}$ are closely related. From (34) and (36)
we can write

$k_{o}k_{n}C_{0}=A-B,\ \ \ \ k_{o}k_{n}C_{2}=A+B,$ (38)

with

$\displaystyle A$ $\displaystyle=$ $\displaystyle
k_{o}k_{n}{\mbox{{n}}}\cdot({\mbox{{a}}}_{1t}^{\prime}\times{\mbox{{a}}}_{2t}^{\prime}+{\mbox{{b}}}_{1t}^{\prime}\times{\mbox{{b}}}_{2t}^{\prime}),$
(39) $\displaystyle B$ $\displaystyle=$ $\displaystyle
k_{o}k_{n}{\mbox{{n}}}\cdot({\mbox{{a}}}_{1t}^{\prime}\times({\mbox{{b}}}_{2t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})-{\mbox{{a}}}_{2t}^{\prime}\times({\mbox{{b}}}_{1t}^{\prime}\cdot{\overline{\overline{\mathsf{J}}}}_{t})).$
(40)

Expanding these expressions as shown in the Appendix, the three coefficients
take the form

$\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\frac{-1}{k_{o}k_{n}}(k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot{\mbox{{k}}}^{i}+({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}-{\mbox{{k}}}^{i}{\mbox{{k}}}^{i})),$
(41) $\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
2{\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}-{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})-\frac{2}{k_{o}}({\mbox{{a}}}_{1}\times{\mbox{{b}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{a}}}_{2})\cdot({\mbox{{n}}}\times{\mbox{{k}}}_{t}),$
(42) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{o}k_{n}}(k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot{\mbox{{k}}}^{r}+({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}-{\mbox{{k}}}^{r}{\mbox{{k}}}^{r})).$
(43)

As a special case, for the GSHDB boundary (16), defined by $a_{1n}=b_{2n}=0$
and ${\mbox{{a}}}_{2t}={\mbox{{b}}}_{1t}=0$, we have $C_{1}=0$ and
$C_{0}=-C_{2}$, whence, from (37), $R_{\pm}=\pm 1$.

Let us now concentrate on finding the boundary conditions corresponding to the
relation $R_{+}=-R_{-}$.

##  4 Boundaries Defined by $R_{+}=-R_{-}$

From (37) we find that $R_{+}=-R_{-}$ is equivalent to the condition

$C_{1}=0,$ (44)

without any restriction on $C_{0}$ and $C_{2}$. From (42) we obtain the
corresponding relation between the vectors
${\mbox{{a}}}_{1}\cdots{\mbox{{b}}}_{2}$,

$k_{o}{\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}-{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})+({\mbox{{n}}}\times({\mbox{{a}}}_{1}\times{\mbox{{b}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{a}}}_{2}))\cdot{\mbox{{k}}}_{t}=0.$
(45)

Requiring this to be valid for any angle of incidence, i.e., for any
tangential vector ${\mbox{{k}}}_{t}$, leads to the two conditions

$\displaystyle{\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}-{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})$
$\displaystyle=$ $\displaystyle 0,$ (46)
$\displaystyle{\mbox{{n}}}\times({\mbox{{a}}}_{1}\times{\mbox{{b}}}_{2}-{\mbox{{a}}}_{2}\times{\mbox{{b}}}_{1})$
$\displaystyle=$ $\displaystyle 0.$ (47)

They are respectively equivalent to

$\displaystyle{\mbox{{a}}}_{1t}\times{\mbox{{a}}}_{2t}-{\mbox{{b}}}_{1t}\times{\mbox{{b}}}_{2t}$
$\displaystyle=$ $\displaystyle 0,$ (48) $\displaystyle
b_{2n}{\mbox{{a}}}_{1t}-b_{1n}{\mbox{{a}}}_{2t}+a_{2n}{\mbox{{b}}}_{1t}-a_{1n}{\mbox{{b}}}_{2t}$
$\displaystyle=$ $\displaystyle 0.$ (49)

For the GSHDB boundary (16), these are identically satisfied.

With (44), the condition (33) becomes

$C_{2}R^{2}+C_{0}=0,$ (50)

whence

$R_{\pm}=\pm\sqrt{-C_{0}/C_{2}}.$ (51)

Requiring that the solution $R$ of (50) be independent of ${\mbox{{k}}}_{t}$,
(50) can be split in the following set of equations relating the boundary
parameters ${\mbox{{a}}}_{1}\cdots{\mbox{{b}}}_{2}$ and the reflection
coefficient $R$:

$\displaystyle(R^{2}-1)({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\times{\mbox{{n}}}$
$\displaystyle=$ $\displaystyle 0,$ (52)
$\displaystyle(R^{2}+1){\mbox{{n}}}\cdot({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}+{\mbox{{a}}}_{2}{\mbox{{b}}}_{1}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}-{\mbox{{a}}}_{1}{\mbox{{b}}}_{2})\times{\mbox{{n}}}$
$\displaystyle=$ $\displaystyle 0,$ (53)
$\displaystyle(R^{2}+1){\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2})$
$\displaystyle=$ $\displaystyle 0,$ (54)
$\displaystyle(R^{2}-1)(b_{1n}a_{2n}-b_{2n}a_{1n})$ $\displaystyle=$
$\displaystyle 0,$ (55)
$\displaystyle(R^{2}-1)({\mbox{{b}}}_{1t}{\mbox{{a}}}_{2t}+{\mbox{{a}}}_{2t}{\mbox{{b}}}_{1t}-{\mbox{{b}}}_{2t}{\mbox{{a}}}_{1t}-{\mbox{{a}}}_{1t}{\mbox{{b}}}_{2t})$
$\displaystyle=$ $\displaystyle 0.$ (56)

Derivation of this set of equations is given in the Appendix.

Now we can separate three possible cases:

  * •

$R^{2}=1$

  * •

$R^{2}=-1$

  * •

$R^{2}\not=1$ and $R^{2}\not=-1$.

In the last case, all expressions in (52) – (56) multiplying $(R^{2}-1)$ or
$(R^{2}+1)$ must vanish. Actually, this corresponds to setting
$C_{0}=C_{1}=C_{2}=0$ in (33), which means that $R$ may have any value.
Ignoring this, we have only two possibilities, either $R^{2}=1$, or
$R^{2}=-1$. Each of these defines a class of boundaries through certain
conditions for the vector parameters ${\mbox{{a}}}_{1}\cdots{\mbox{{b}}}_{2}$
arising from (52) – (56). Since it has been previously shown [1, 17] that the
case $R^{2}=1$ corresponds to the class of GSHDB boundaries defined by (16),
let us concentrate on the case $R^{2}=-1$, which yields a second possibility
for the class of boundaries satisfying $R_{+}=-R_{-}$. In this case, the
reflection coefficients for the two eigenwaves are $+j$ and $-j$.

##  5 Boundaries Defined by $R_{+}=-R_{-}=j$

From (52), (55) and (56) we obtain conditions for the boundary corresponding
to the case $R^{2}=-1$:

$\displaystyle
a_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{a}}}_{1t}+b_{1n}{\mbox{{b}}}_{2t}-b_{2n}{\mbox{{b}}}_{1t}$
$\displaystyle=$ $\displaystyle 0,$ (57) $\displaystyle
b_{1n}a_{2n}-b_{2n}a_{1n}$ $\displaystyle=$ $\displaystyle 0.$ (58)
$\displaystyle{\mbox{{b}}}_{1t}{\mbox{{a}}}_{2t}+{\mbox{{a}}}_{2t}{\mbox{{b}}}_{1t}-{\mbox{{b}}}_{2t}{\mbox{{a}}}_{1t}-{\mbox{{a}}}_{1t}{\mbox{{b}}}_{2t}$
$\displaystyle=$ $\displaystyle 0,$ (59)

which, together with (48) and (49), define the present class of boundaries.

It has been shown that the case $R^{2}=1$ corresponds to the class of GSHDB
boundaries, for which the vector parameters satisfy
${\mbox{{n}}}\cdot{\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}=0$ and
${\mbox{{n}}}\cdot{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2}=0$, as is seen from
(16). Similarly, from (48) and (54) we can conclude that the case $R^{2}=-1$
corresponds to ${\mbox{{n}}}\cdot{\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}\not=0$
and ${\mbox{{n}}}\cdot{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2}\not=0$. Because
of this, we can assume that the tangential vectors
${\mbox{{b}}}_{1t},{\mbox{{b}}}_{2t}$ and
${\mbox{{a}}}_{1t},{\mbox{{a}}}_{2t}$ form two linearly independent pairs,
whence we can expand

$\left(\begin{array}[]{c}{\mbox{{b}}}_{1t}\\\
{\mbox{{b}}}_{2t}\end{array}\right)={\cal
B}\left(\begin{array}[]{c}{\mbox{{a}}}_{1t}\\\
{\mbox{{a}}}_{2t}\end{array}\right),$ (60)

for some matrix

${\cal B}=\left(\begin{array}[]{cc}B_{11}&B_{12}\\\
B_{21}&B_{22}\end{array}\right),$ (61)

defined by four scalars $B_{11}\cdots B_{22}$. The condition (48) requires

${\rm det}{\cal B}=B_{11}B_{22}-B_{12}B_{21}=1.$ (62)

From (49) we obtain a relation between the normal components of the four
vectors as

$\left(\begin{array}[]{c}b_{1n}\\\
b_{2n}\end{array}\right)=-\left(\begin{array}[]{cc}B_{22}&-B_{12}\\\
-B_{21}&B_{11}\end{array}\right)\left(\begin{array}[]{c}a_{1n}\\\
a_{2n}\end{array}\right)=-{\cal B}^{-1}\left(\begin{array}[]{c}a_{1n}\\\
a_{2n}\end{array}\right).$ (63)

Combining these, we can write

$\displaystyle{\mbox{{b}}}_{1}$ $\displaystyle=$ $\displaystyle
B_{11}{\mbox{{a}}}_{1}+B_{12}{\mbox{{a}}}_{2}-{\rm tr}{\cal B}\
a_{1n}{\mbox{{n}}},$ (64) $\displaystyle{\mbox{{b}}}_{2}$ $\displaystyle=$
$\displaystyle B_{21}{\mbox{{a}}}_{1}+B_{22}{\mbox{{a}}}_{2}-{\rm tr}{\cal B}\
a_{2n}{\mbox{{n}}},$ (65)

with

${\rm tr}{\cal B}=B_{11}+B_{22}.$ (66)

Applying (64) and (65), the boundary conditions (4) and (5) take the form

$\displaystyle{\mbox{{a}}}_{1}\cdot{\mbox{{E}}}+(B_{11}{\mbox{{a}}}_{1}+B_{12}{\mbox{{a}}}_{2}-{\rm
tr}{\cal B}\ a_{1n}{\mbox{{n}}})\cdot\eta_{o}{\mbox{{H}}}$ $\displaystyle=$
$\displaystyle 0,$ (67)
$\displaystyle{\mbox{{a}}}_{2}\cdot{\mbox{{E}}}+(B_{21}{\mbox{{a}}}_{1}+B_{22}{\mbox{{a}}}_{2}-{\rm
tr}{\cal B}\ a_{2n}{\mbox{{n}}})\cdot\eta_{o}{\mbox{{H}}}$ $\displaystyle=$
$\displaystyle 0,$ (68)

for some vectors ${\mbox{{a}}}_{1},{\mbox{{a}}}_{2}$ and a matrix ${\cal B}$
restricted by (62).

Applying (60), (59) becomes

$-2B_{21}{\mbox{{a}}}_{1t}{\mbox{{a}}}_{1t}+(B_{11}-B_{22})({\mbox{{a}}}_{1t}{\mbox{{a}}}_{2t}+{\mbox{{a}}}_{2t}{\mbox{{a}}}_{1t})+2B_{12}{\mbox{{a}}}_{2t}{\mbox{{a}}}_{2t}=0,$
(69)

which for ${\mbox{{a}}}_{1t}\times{\mbox{{a}}}_{2t}\not=0$ yields

$B_{12}=B_{21}=0,\ \ \ B_{11}=B_{22}=B,\ \ \ {\rm tr}{\cal B}=2B,$ (70)

for some scalar $B$. From (62) we obtain

${\rm det}{\cal B}=B_{11}B_{22}-B_{12}B_{21}=B^{2}=1.$ (71)

with two possible solutions $B=+1$ and $B=-1$.

From (64) and (65) we obtain the corresponding possibilities,

${\mbox{{b}}}_{1}={\mbox{{a}}}_{1t}-{\mbox{{n}}}a_{1n},\ \ \
{\mbox{{b}}}_{2}={\mbox{{a}}}_{2t}-{\mbox{{n}}}a_{2n},$ (72)

and

${\mbox{{b}}}_{1}=-{\mbox{{a}}}_{1t}+{\mbox{{n}}}a_{1n},\ \ \
{\mbox{{b}}}_{2}=-{\mbox{{a}}}_{2t}+{\mbox{{n}}}a_{2n}.$ (73)

(57) and (58) are now satisfied for any given vectors ${\mbox{{a}}}_{1}$ and
${\mbox{{a}}}_{2}$. Substituting (72) or (73), the conditions (46), (47) and
(52) – (56) are also satisfied.

Thus, the above analysis offers two possibilities for the boundary conditions
corresponding to the case $R^{2}=-1$,

$\displaystyle{\mbox{{a}}}_{1t}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})+a_{1n}{\mbox{{n}}}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})$
$\displaystyle=$ $\displaystyle 0,$ (74)
$\displaystyle{\mbox{{a}}}_{2t}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})+a_{2n}{\mbox{{n}}}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})$
$\displaystyle=$ $\displaystyle 0,$ (75)

and

$\displaystyle{\mbox{{a}}}_{1t}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})+a_{1n}{\mbox{{n}}}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})$
$\displaystyle=$ $\displaystyle 0,$ (76)
$\displaystyle{\mbox{{a}}}_{2t}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})+a_{2n}{\mbox{{n}}}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})$
$\displaystyle=$ $\displaystyle 0.$ (77)

The conditions (74), (75) can be expressed in vector form as

${\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})+{\mbox{{p}}}_{t}{\mbox{{n}}}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})=0,$
(78)

and the conditions (76), (77) as

${\mbox{{n}}}\times({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})+{\mbox{{p}}}_{t}{\mbox{{n}}}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})=0,$
(79)

with

$\displaystyle{\mbox{{p}}}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{A_{n}}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2})\times{\mbox{{n}}},=\frac{1}{A_{n}}(a_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{a}}}_{1t}),$
(80) $\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle{\mbox{{n}}}\cdot{\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}.$ (81)

In the present case $R^{2}=-1$ we have assumed $A_{n}\not=0$.

In the special case of impedance boundaries, the boundary parameters satisfy
$a_{1n}=a_{2n}=0$, which corresponds to ${\mbox{{p}}}_{t}=0$. In this case,
the condition (78) becomes

${\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})=0,$ (82)

and, the condition (79) becomes

${\mbox{{n}}}\times({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})=0.$ (83)

(82) and (83) equal special cases of the PEMC boundary conditions (9), with
admittances $M=1/\eta_{o}$ and $M=-1/\eta_{o}$, respectively. The reflection
coefficients for the two eigenwaves at the general PEMC boundary are known to
have the form [1], Eq. (2.38),

$R_{\pm}=\frac{1\pm jM\eta_{o}}{1\mp jM\eta_{o}},$ (84)

which satisfies $R_{\pm}^{2}=-1$ for both $M\eta_{o}=1$ and $M\eta_{o}=-1$.

One can easily verify that, for both of the boundary conditions (78) and (79),
the two solutions for the reflection coefficient equation (33) satisfy both
$R_{+}=-R_{-}$ and $R_{\pm}^{2}=-1$. The details are given in the Appendix.

In conclusion, requiring that the reflection coefficients $R_{+}$ and $R_{-}$
of the two eigenwaves satisfy the condition $R_{+}=-R_{-}=R$, with $R^{2}=-1$
for any angle of incidence, the boundary is defined either by the conditions
(78) or (79).

##  6 Extended PEMC Boundaries

The conditions (78) and (79) are special cases of the vector condition

${\mbox{{n}}}\times(M{\mbox{{E}}}+{\mbox{{H}}})+{\mbox{{p}}}_{t}{\mbox{{n}}}\cdot(M{\mbox{{E}}}-{\mbox{{H}}})=0,$
(85)

defining what can be called the class of Extended PEMC (EPEMC) boundaries. The
class depends on two parameters, a scalar $M$ and a vector ${\mbox{{p}}}_{t}$.
The boundary vector parameters corresponding to (85) are

${\mbox{{b}}}_{1}=({\mbox{{a}}}_{1t}-{\mbox{{n}}}a_{1n})/M\eta_{o},\ \ \
{\mbox{{b}}}_{2}=({\mbox{{a}}}_{2t}-{\mbox{{n}}}a_{2n})/M\eta_{o}.$ (86)

For ${\mbox{{p}}}_{t}=0$, the condition of the EPEMC boundary (85) is reduced
to that of the PEMC boundary (9), in which case the parameter $M$ equals the
admittance of the PEMC boundary.

Because (78) and (79) are special cases of (85), respectively corresponding to
$M=1/\eta_{o}$ and $M=-1/\eta_{o}$, they can be called conditions of the
EPEMC+ and EPEMC- boundaries, respectively. For ${\mbox{{p}}}_{t}=0$ both of
them are reduced to the corresponding special PEMC boundaries.

It has been previously shown that a plane wave incident normally to the PEMC
boundary is reflected cross polarized, i.e., satisfying
${\mbox{{E}}}_{t}^{r}\cdot{\mbox{{E}}}_{t}^{i}=0$, exactly when the PEMC
admittance satisfies either $M\eta_{o}=+1$ or $=-1$ [1]. It will turn out that
the same property is valid for the EPEMC boundary.

###  6.1 Matched Waves

By definition, a plane wave is matched to a boundary when the boundary
conditions are identically satisfied for the single plane wave, i.e., there is
no reflected wave [1]. Applying the plane-wave equations (18) and (19), the
conditions of the EPEMC+ and EPEMC- boundaries, (78) and (79), can be
respectively expanded as

$\displaystyle k_{o}{\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})$
$\displaystyle=$ $\displaystyle-{\mbox{{p}}}_{t}{\mbox{{n}}}\cdot
k_{o}({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})$ (87) $\displaystyle=$
$\displaystyle-{\mbox{{p}}}_{t}{\mbox{{k}}}_{t}\cdot({\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})),$
$\displaystyle k_{o}{\mbox{{n}}}\times({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})$
$\displaystyle=$ $\displaystyle-{\mbox{{p}}}_{t}{\mbox{{n}}}\cdot
k_{o}({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})$ (88) $\displaystyle=$
$\displaystyle{\mbox{{p}}}_{t}{\mbox{{k}}}_{t}\cdot({\mbox{{n}}}\times({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})).$

Thus, the conditions for the matched waves at the respective EPEMC+ and EPEMC-
boundaries become

$\displaystyle(k_{o}{\overline{\overline{\mathsf{I}}}}_{t}+{\mbox{{p}}}_{t}{\mbox{{k}}}_{t})\cdot({\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}}))$
$\displaystyle=$ $\displaystyle 0,$ (89)
$\displaystyle(k_{o}{\overline{\overline{\mathsf{I}}}}_{t}-{\mbox{{p}}}_{t}{\mbox{{k}}}_{t})\cdot({\mbox{{n}}}\times({\mbox{{E}}}-\eta_{o}{\mbox{{H}}}))$
$\displaystyle=$ $\displaystyle 0.$ (90)

Let us consider these two cases separately.

####  6.1.1 EPEMC+ Boundary

From (89) we have two possibilities: either

${\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})=0,$ (91)

or the two-dimensional dyadic in (89) has no inverse.

In the latter case, the two-dimensional determinant of the dyadic must be zero
[1, 26], whence

$\frac{1}{k_{o}}{\rm
tr}(k_{o}{\overline{\overline{\mathsf{I}}}}_{t}+{\mbox{{p}}}_{t}{\mbox{{k}}}_{t})^{(2)}=k_{o}+{\mbox{{p}}}_{t}\cdot{\mbox{{k}}}_{t}=0,$
(92)

which is the dispersion equation of the matched wave. Because the dyadic can
be expressed as

$k_{o}{\overline{\overline{\mathsf{I}}}}_{t}+{\mbox{{p}}}_{t}{\mbox{{k}}}_{t}={\mbox{{k}}}_{t}\times({\mbox{{p}}}_{t}\times{\overline{\overline{\mathsf{I}}}}_{t})=-({\mbox{{n}}}\times{\mbox{{k}}}_{t})({\mbox{{n}}}\times{\mbox{{p}}}_{t}),$
(93)

from (89) the fields of the matched wave must satisfy

$({\mbox{{n}}}\times{\mbox{{p}}}_{t})\cdot({\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})={\mbox{{p}}}_{t}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})=0.$
(94)

Applying the orthogonality
${\mbox{{k}}}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})=0$, the polarization of
the matched wave corresponding to the dispersion equation (92) is

${\mbox{{E}}}+\eta_{o}{\mbox{{H}}}\sim{\mbox{{k}}}\times{\mbox{{p}}}_{t}.$
(95)

Considering the case (91), from (78) we obtain

${\mbox{{n}}}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})=0.$ (96)

Applying (18) and (19) we can expand

$\displaystyle k_{o}{\mbox{{n}}}\times({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})$
$\displaystyle=$
$\displaystyle{\mbox{{n}}}\times(-{\mbox{{k}}}\times\eta_{o}{\mbox{{H}}}+{\mbox{{k}}}\times{\mbox{{E}}})$
(97) $\displaystyle=$
$\displaystyle{\mbox{{k}}}{\mbox{{n}}}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})-({\mbox{{n}}}\cdot{\mbox{{k}}})({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})$
$\displaystyle=$
$\displaystyle-({\mbox{{n}}}\cdot{\mbox{{k}}})({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})=0.$

Thus, the second possible matched wave is governed by the dispersion equation

${\mbox{{n}}}\cdot{\mbox{{k}}}=0,$ (98)

which corresponds to a lateral wave. From (96) we obtain the polarization
condition

${\mbox{{E}}}-\eta_{o}{\mbox{{H}}}\sim{\mbox{{n}}}\times{\mbox{{k}}}.$ (99)

####  6.1.2 EPEMC- Boundary

This case can be handled similarly. From (90), for
${\mbox{{n}}}\times({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})\not=0$, the dispersion
equation becomes

$k_{o}-{\mbox{{p}}}_{t}\cdot{\mbox{{k}}}_{t}=0.$ (100)

Because the fields must satisfy

${\mbox{{p}}}_{t}\cdot({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})=0,$ (101)

the polarization condition becomes

${\mbox{{E}}}-\eta_{o}{\mbox{{H}}}\sim{\mbox{{k}}}\times{\mbox{{p}}}_{t}.$
(102)

For the other possibility
${\mbox{{n}}}\times({\mbox{{E}}}-\eta_{o}{\mbox{{H}}})=0$, the matched wave is
again a lateral wave with dispersion equation (98). From (79) we now obtain

${\mbox{{n}}}\cdot({\mbox{{E}}}+\eta_{o}{\mbox{{H}}})=0,$ (103)

whence the polarization of the wave is

${\mbox{{E}}}+\eta_{o}{\mbox{{H}}}\sim{\mbox{{n}}}\times{\mbox{{k}}}.$ (104)

###  6.2 Reflection of Plane Waves

Let us finally consider plane-wave reflection from the EPEMC+ and EPEMC-
boundaries, defined by (78) and (79).

Applying (18) and (19), the sum of incident and reflected fields, at the
EPEMC+ boundary can be shown to satisfy

$\displaystyle
k_{o}{\mbox{{n}}}\times({\mbox{{E}}}^{i}+\eta_{o}{\mbox{{H}}}^{i}+{\mbox{{E}}}^{r}+\eta_{o}{\mbox{{H}}}^{r})$
(105) $\displaystyle=$ $\displaystyle-{\mbox{{p}}}_{t}{\mbox{{n}}}\cdot
k_{o}({\mbox{{E}}}^{i}-\eta_{o}{\mbox{{H}}}^{i}+{\mbox{{E}}}^{r}-\eta_{o}{\mbox{{H}}}^{r}))$
$\displaystyle=$
$\displaystyle{\mbox{{p}}}_{t}{\mbox{{n}}}\cdot({\mbox{{k}}}^{i}\times({\mbox{{E}}}^{i}+\eta_{o}{\mbox{{H}}}^{i})+{\mbox{{k}}}^{r}\times({\mbox{{E}}}^{r}+\eta_{o}{\mbox{{H}}}^{r}))$
$\displaystyle=$
$\displaystyle-{\mbox{{p}}}_{t}{\mbox{{k}}}_{t}\cdot{\mbox{{n}}}\times({\mbox{{E}}}^{i}+\eta_{o}{\mbox{{H}}}^{i}+{\mbox{{E}}}^{r}+\eta_{o}{\mbox{{H}}}^{r}).$

The result can be written as

$(k_{o}{\overline{\overline{\mathsf{I}}}}_{t}+{\mbox{{p}}}_{t}{\mbox{{k}}}_{t})\cdot{\mbox{{n}}}\times({\mbox{{E}}}^{i}+\eta_{o}{\mbox{{H}}}^{i}+{\mbox{{E}}}^{r}+\eta_{o}{\mbox{{H}}}^{r})=0.$
(106)

Omitting the case when k satisfies the dispersion equation (92) of the matched
wave, the dyadic in (106) has a two-dimensional inverse. In this case, the
fields at the boundary must satisfy

${\mbox{{n}}}\times({\mbox{{E}}}^{i}+{\mbox{{E}}}^{r}+\eta_{o}({\mbox{{H}}}^{i}+{\mbox{{H}}}^{r}))=0,$
(107)

for any vector ${\mbox{{p}}}_{t}$. Actually, (107) equals the boundary
condition of the PEMC boundary (10), for $M=1/\eta_{o}$.

The EPEMC- boundary can be handled similarly by starting from the condition
(79), leading to

$(k_{o}{\overline{\overline{\mathsf{I}}}}_{t}-{\mbox{{p}}}_{t}{\mbox{{k}}}_{t})\cdot{\mbox{{n}}}\times({\mbox{{E}}}^{i}-\eta_{o}{\mbox{{H}}}^{i}+{\mbox{{E}}}^{r}-\eta_{o}{\mbox{{H}}}^{r})=0,$
(108)

whence the result (107) will be replaced by

${\mbox{{n}}}\times({\mbox{{E}}}^{i}+{\mbox{{E}}}^{r}-\eta_{o}({\mbox{{H}}}^{i}+{\mbox{{H}}}^{r}))=0,$
(109)

for any vector ${\mbox{{p}}}_{t}$. Again, this equals the boundary condition
of the PEMC boundary (10), for $M=-1/\eta_{o}$.

To summarize, the class of boundaries defined by the property $R^{2}=-1$
consists of two subclasses, EPEMC+ and EPEMC-, which are equivalent in
reflection to two subclasses of PEMC boundaries, respectively defined by
$M=1/\eta_{o}$ and $M=-1/\eta_{o}$, for any vector ${\mbox{{p}}}_{t}$.

As a example, let us consider a plane wave with normal incidence,
${\mbox{{k}}}^{r}=-{\mbox{{k}}}^{i}={\mbox{{n}}}k_{o}$, reflecting from an
EPEMC boundary. The fields are tangential to the boundary:
${\mbox{{n}}}\cdot{\mbox{{E}}}^{i}={\mbox{{n}}}\cdot{\mbox{{E}}}^{r}=0$,
${\mbox{{n}}}\cdot{\mbox{{H}}}^{i}={\mbox{{n}}}\cdot{\mbox{{H}}}^{r}=0$. From
(85), the reflected field can be expressed as

${\mbox{{E}}}^{r}=-\frac{1}{M^{2}\eta_{o}^{2}+1}\left((M^{2}\eta_{o}^{2}-1){\overline{\overline{\mathsf{I}}}}_{t}+2M\eta_{o}{\mbox{{n}}}\times{\overline{\overline{\mathsf{I}}}}\right)\cdot{\mbox{{E}}}^{i},$
(110)

whence

${\mbox{{E}}}^{r}\cdot{\mbox{{E}}}^{i}=C({\mbox{{E}}}^{i}\cdot{\mbox{{E}}}^{i}),$
(111)

where $C$, defined by

$C=-\frac{M^{2}\eta_{o}^{2}-1}{M^{2}\eta_{o}^{2}+1},$ (112)

is a measure of copolarization in the reflected field. This function is
depicted in Figure 2.

The reflected field appears cross polarized when
${\mbox{{E}}}^{r}\cdot{\mbox{{E}}}^{i}=0$, or $C=0$. This happens for
$M\eta_{o}=\pm 1$, i.e., when the EPEMC boundary is either EPEMC+ or EPEMC-.
The same property is known to exist for PEMC boundaries with $M=1/\eta_{o}$ or
$M=-1/\eta_{o}$ ([1], Section 2.4). For $M=0$ (PMC, $C=1$) and
$M\rightarrow\pm\infty$ (PEC, $C=-1$), the reflected field is totally
copolarized. Note that, for a circularly polarized incident wave, satisfying
${\mbox{{E}}}^{i}\cdot{\mbox{{E}}}^{i}=0$, from (110) we have
${\mbox{{E}}}^{r}\cdot{\mbox{{E}}}^{i}=0$ for any $M\not=\pm j/\eta_{o}$.

Figure 2:  Visualization of the co-polarization quantity $C$ of (112), as a
function of the parameter $M\eta_{o}$, for a normally incident plane wave
reflecting from an EPEMC boundary defined by (85). The reflected field is co-
polarized for $M\eta_{o}=0$ (PMC) and $M\eta_{o}\rightarrow\pm\infty$ (PEC)
while, for $M\eta_{o}=1$ (EPEMC+) and $M\eta_{o}=-1$ (EPEMC-), the reflected
field is cross polarized, $C=0$.

##  7 Conclusion

The problem considered in this paper has been to generalize the property of
the class of electromagnetic boundaries known as that of generalized soft-and-
hard/DB (GSHDB) boundaries. The class of boundaries is known to possess the
unique property that the two eigen plane-waves associated to the boundary are
reflected with reflection coefficients $R_{+}=1$ and $R_{-}=-1$ for any angle
of incidence. Such a property has been called PEC/PMC equivalence, because the
two eigenwaves are reflected as from a PEC or a PMC boundary. In the present
case, the possibility of a class of boundaries with a more general property
was studied by requesting that the reflection coefficients of the two
eigenwaves satisfy a condition of the form $R_{\pm}=\pm R$, where $R$ is
independent of the angle of incidence.

It was shown that, under this condition, $R$ must be either $1$ or $j$. While
the former case corresponds to the class of GSHDB boundaries, the latter case
was shown to correspond to another class of boundaries defined by two possible
sets of boundary conditions, (78) and (79). Both of these were shown to be
special cases of a more general class of Extended Perfect Electromagnetic
Conductor (EPEMC) boundaries defined by conditions of the form (85), depending
on a scalar $M$ and a vector ${\mbox{{p}}}_{t}$ tangential to the boundary
surface. The two special cases of (85) were respectively called EPEMC+ and
EPEMC- boundaries as they correspond to the special parameter values
$M=1/\eta_{o}$ and $M=-1/\eta_{o}$, respectively. For ${\mbox{{p}}}_{t}=0$,
the EPEMC boundary is reduced to the PEMC boundary with $M$ equal to the PEMC
admittance. Matched waves and plane-wave reflection properties were studied
for both EPEMC+ and EPEMC- classes of boundaries. While the matched waves
depend on the ${\mbox{{p}}}_{t}$ vector, reflection of the eigenwaves does
not. Actually, in reflection, the EPEMC+ and EPEMC- classes of boundaries are
equivalent to the corresponding classes of PEMC boundaries defined by the
respective admittance values $M=1/\eta_{o}$ and $M=-1/\eta_{o}$, for any
vector ${\mbox{{p}}}_{t}$.

##  8 Appendix

###  8.1 Expanding $A$ and $B$

Applying (32), we can expand the expressions (39) and (40) as

$\displaystyle A$ $\displaystyle=$ $\displaystyle
k_{o}k_{n}{\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})+k_{n}{\mbox{{n}}}\cdot({\mbox{{b}}}_{2}{\mbox{{a}}}_{1}-{\mbox{{b}}}_{1}{\mbox{{a}}}_{2}+{\mbox{{a}}}_{1}{\mbox{{b}}}_{2}-{\mbox{{a}}}_{2}{\mbox{{b}}}_{1})\cdot{\mbox{{k}}}_{t}$
(113) $\displaystyle=$ $\displaystyle
k_{o}k_{n}{\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})-({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):k_{n}({\mbox{{n}}}{\mbox{{k}}}_{t}+{\mbox{{k}}}_{t}{\mbox{{n}}}),$
$\displaystyle B$ $\displaystyle=$
$\displaystyle({\mbox{{b}}}_{1t}^{\prime}{\mbox{{a}}}_{2t}^{\prime}-{\mbox{{b}}}_{2t}^{\prime}{\mbox{{a}}}_{1t}^{\prime}):({\mbox{{n}}}\times{\mbox{{k}}})({\mbox{{n}}}\times{\mbox{{k}}})+k_{n}^{2}({\mbox{{a}}}_{2t}^{\prime}\cdot{\mbox{{b}}}_{1t}^{\prime}-{\mbox{{a}}}_{1t}^{\prime}\cdot{\mbox{{b}}}_{2t}^{\prime})$
(114) $\displaystyle=$
$\displaystyle({\mbox{{b}}}_{1t}^{\prime}{\mbox{{a}}}_{2t}^{\prime}-{\mbox{{b}}}_{2t}^{\prime}{\mbox{{a}}}_{1t}^{\prime}):(k_{t}^{2}{\overline{\overline{\mathsf{I}}}}_{t}-{\mbox{{k}}}_{t}{\mbox{{k}}}_{t}+k_{n}^{2}{\overline{\overline{\mathsf{I}}}}_{t})$
$\displaystyle=$
$\displaystyle({\mbox{{b}}}_{1t}^{\prime}{\mbox{{a}}}_{2t}^{\prime}-{\mbox{{b}}}_{2t}^{\prime}{\mbox{{a}}}_{1t}^{\prime}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}_{t}-{\mbox{{k}}}_{t}{\mbox{{k}}}_{t}).$

Substituting

$\displaystyle{\mbox{{b}}}_{1t}^{\prime}{\mbox{{a}}}_{2t}^{\prime}-{\mbox{{b}}}_{2t}^{\prime}{\mbox{{a}}}_{1t}$
$\displaystyle=$
$\displaystyle{\mbox{{b}}}_{1t}{\mbox{{a}}}_{2t}-{\mbox{{b}}}_{2t}{\mbox{{a}}}_{1t}$
(115) $\displaystyle+$
$\displaystyle\frac{1}{k_{o}}\bigg{(}(b_{2n}{\mbox{{b}}}_{1t}-b_{1n}{\mbox{{b}}}_{2t}){\mbox{{n}}}\times{\mbox{{k}}}-{\mbox{{n}}}\times{\mbox{{k}}}(a_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{a}}}_{1t})\bigg{)}$
$\displaystyle-$
$\displaystyle\frac{1}{k_{o}^{2}}(a_{1n}b_{2n}-a_{2n}b_{1n})({\mbox{{n}}}\times{\mbox{{k}}})({\mbox{{n}}}\times{\mbox{{k}}}),$

we obtain

$\displaystyle B$ $\displaystyle=$
$\displaystyle({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}_{t}-{\mbox{{k}}}_{t}{\mbox{{k}}}_{t})$
(116) $\displaystyle+$ $\displaystyle
k_{o}(b_{2n}{\mbox{{b}}}_{1}-b_{1n}{\mbox{{b}}}_{2}-a_{1n}{\mbox{{a}}}_{2}+a_{2n}{\mbox{{a}}}_{1})\cdot({\mbox{{n}}}\times{\mbox{{k}}})-(a_{1n}b_{2n}-a_{2n}b_{1n})k_{t}^{2},$
$\displaystyle=$
$\displaystyle({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}-{\mbox{{k}}}_{t}{\mbox{{k}}}_{t}-k_{n}^{2}{\mbox{{n}}}{\mbox{{n}}})+k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot{\mbox{{k}}}_{t}.$

These can be combined as

$\displaystyle A\pm B$ $\displaystyle=$ $\displaystyle\pm
k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot({\mbox{{k}}}_{t}\pm
k_{n}{\mbox{{n}}})$ (117)
$\displaystyle\pm({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}-{\mbox{{k}}}_{t}{\mbox{{k}}}_{t}-k_{n}^{2}{\mbox{{n}}}{\mbox{{n}}}\mp
k_{n}({\mbox{{n}}}{\mbox{{k}}}_{t}+{\mbox{{k}}}_{t}{\mbox{{n}}}))$
$\displaystyle=$ $\displaystyle\pm
k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot({\mbox{{k}}}_{t}\pm
k_{n}{\mbox{{n}}})$
$\displaystyle\pm({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}-({\mbox{{k}}}_{t}\pm
k_{n}{\mbox{{n}}})({\mbox{{k}}}_{t}\pm k_{n}{\mbox{{n}}}))$

Thus, we can finally write

$k_{o}k_{n}C_{0}=-k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot{\mbox{{k}}}^{i}-({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}-{\mbox{{k}}}^{i}{\mbox{{k}}}^{i}),$
(118)

and

$k_{o}k_{n}C_{2}=k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot{\mbox{{k}}}^{r}+({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}-{\mbox{{k}}}^{r}{\mbox{{k}}}^{r}).$
(119)

###  8.2 Expanding Condition (50)

Applying (41) and (43), the condition (50) can be expressed in compact form as

$(R^{2}-1)(K_{o}+K_{e})+(R^{2}+1)(L_{o}+L_{e})=0,$ (120)

with terms odd in ${\mbox{{k}}}_{t}$,

$\displaystyle K_{o}$ $\displaystyle=$ $\displaystyle
k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\cdot{\mbox{{k}}}_{t}$
(121) $\displaystyle L_{o}$ $\displaystyle=$ $\displaystyle-
k_{n}{\mbox{{n}}}\cdot({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}+{\mbox{{a}}}_{2}{\mbox{{b}}}_{1}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}-{\mbox{{a}}}_{1}{\mbox{{b}}}_{2})\cdot{\mbox{{k}}}_{t},$
(122)

and, terms even in ${\mbox{{k}}}_{t}$

$\displaystyle K_{e}$ $\displaystyle=$
$\displaystyle({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):(k_{o}^{2}{\overline{\overline{\mathsf{I}}}}_{t}-{\mbox{{k}}}_{t}{\mbox{{k}}}_{t}+({\mbox{{k}}}_{t}\cdot{\mbox{{k}}}_{t}){\mbox{{n}}}{\mbox{{n}}})$
(123) $\displaystyle L_{e}$ $\displaystyle=$ $\displaystyle
2k_{n}k_{o}{\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}).$ (124)

For (120) to be satisfied for any ${\mbox{{k}}}_{t}$, parts odd and even in
${\mbox{{k}}}_{t}$ must vanish separately. This leads to the respective
conditions

$\displaystyle(R^{2}-1)K_{o}+(R^{2}+1)L_{o}$ $\displaystyle=$ $\displaystyle
0,$ (125) $\displaystyle(R^{2}-1)K_{e}+(R^{2}+1)L_{e}$ $\displaystyle=$
$\displaystyle 0.$ (126)

As a special case, for the GSHDB boundary (16) with
${\mbox{{a}}}_{1}={\mbox{{a}}}_{t}$, ${\mbox{{b}}}_{1}=\alpha{\mbox{{n}}}$,
${\mbox{{a}}}_{2}=\beta{\mbox{{n}}}$ and ${\mbox{{b}}}_{2}={\mbox{{b}}}_{t}$,
we find $L_{o}=0$ and $L_{e}=0$. Since ${\mbox{{a}}}_{t}$ and
${\mbox{{b}}}_{t}$ may be any tangential vectors and $\alpha,\beta$ any
scalars, $K_{o}$ and $K_{e}$ are nonzero. Thus, both (125) and (126) yield
$R^{2}=1$, or $R_{\pm}=\pm 1$, a result known from [1, 25].

Let us consider the general case. For (125) to be valid for any polarization
of ${\mbox{{k}}}_{t}$, we must have

$\displaystyle(R^{2}-1)k_{o}({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})\times{\mbox{{n}}}$
$\displaystyle+$
$\displaystyle-(R^{2}+1)k_{n}{\mbox{{n}}}\cdot({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}+{\mbox{{a}}}_{2}{\mbox{{b}}}_{1}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}-{\mbox{{a}}}_{1}{\mbox{{b}}}_{2}))\times{\mbox{{n}}}$
$\displaystyle=$ $\displaystyle 0.$ (127)

Since only the second term depends on ${\mbox{{k}}}_{t}$ (through $k_{n}$),
the two terms must vanish independently, whence we can write

$\displaystyle(R^{2}-1)(a_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{a}}}_{1t}+b_{1n}{\mbox{{b}}}_{2t}-b_{2n}{\mbox{{b}}}_{1t})$
$\displaystyle=$ $\displaystyle 0,$ (128)
$\displaystyle(R^{2}+1)(b_{1n}{\mbox{{a}}}_{2t}+a_{2n}{\mbox{{b}}}_{1t}-b_{2n}{\mbox{{a}}}_{1t}-a_{1n}{\mbox{{b}}}_{2t})$
$\displaystyle=$ $\displaystyle 0.$ (129)

These two conditions correspond to the equation (125). Similarly, (126) can be
split in three conditions, based on different orders of magnitude in
${\mbox{{k}}}_{t}$, as

$\displaystyle(R^{2}-1)({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):{\overline{\overline{\mathsf{I}}}}_{t}$
$\displaystyle=$ $\displaystyle 0,$ (130)
$\displaystyle(R^{2}+1){\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2})$
$\displaystyle=$ $\displaystyle 0,$ (131)
$\displaystyle(R^{2}-1)({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):({\mbox{{k}}}_{t}{\mbox{{k}}}_{t}-({\mbox{{k}}}_{t}\cdot{\mbox{{k}}}_{t}){\mbox{{n}}}{\mbox{{n}}})$
$\displaystyle=$ $\displaystyle 0.$ (132)

The condition (132) can be further split in two parts because one of the terms
does not depend on the polarization of ${\mbox{{k}}}_{t}$,

$\displaystyle(R^{2}-1)({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):{\mbox{{k}}}_{t}{\mbox{{k}}}_{t}$
$\displaystyle=$ $\displaystyle 0,$ (133)
$\displaystyle(R^{2}-1)(b_{1n}a_{2n}-b_{2n}a_{1n})$ $\displaystyle=$
$\displaystyle 0.$ (134)

Setting
${\mbox{{k}}}_{t}={\mbox{{k}}}_{t}^{\prime}+{\mbox{{k}}}_{t}^{\prime\prime}$
in (133), we obtain

$(R^{2}-1)({\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}):({\mbox{{k}}}_{t}^{\prime}{\mbox{{k}}}_{t}^{\prime\prime}+{\mbox{{k}}}_{t}^{\prime\prime}{\mbox{{k}}}_{t}^{\prime})=0,$
(135)

which must be valid for any ${\mbox{{k}}}_{t}^{\prime}$ and
${\mbox{{k}}}_{t}^{\prime\prime}$, whence (135) equals the dyadic condition

$(R^{2}-1)({\mbox{{b}}}_{1t}{\mbox{{a}}}_{2t}+{\mbox{{a}}}_{2t}{\mbox{{b}}}_{1t}-{\mbox{{b}}}_{2t}{\mbox{{a}}}_{1t}-{\mbox{{a}}}_{1t}{\mbox{{b}}}_{2t})=0.$
(136)

Since the trace of (136) covers the condition (130), the latter can be
omitted.

In conclusion, the medium parameters ${\mbox{{a}}}_{1}\cdots{\mbox{{b}}}_{2}$
and the reflection coefficient $R$ must satisfy the conditions (128), (129),
(131), (134) and (136)

###  8.3 Expanding $C_{1}=0$

Let us find the conditions for the boundary parameters
${\mbox{{a}}}_{1}\cdots{\mbox{{b}}}_{2}$ in the case when the coefficient
$C_{1}$ vanishes for all wave vectors ${\mbox{{k}}}_{t}$. Because the second
term of (42) changes sign in ${\mbox{{k}}}_{t}\rightarrow{\mbox{{k}}}_{t}$,
from $C_{1}=0$ we obtain the two conditions

$\displaystyle{\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}-{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})$
$\displaystyle=$ $\displaystyle 0,$ (137)
$\displaystyle({\mbox{{a}}}_{1}\times{\mbox{{b}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{a}}}_{2})\times{\mbox{{n}}}$
$\displaystyle=$ $\displaystyle 0.$ (138)

Assuming ${\mbox{{n}}}\cdot{\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}\not=0$, the
tangential vectors ${\mbox{{a}}}_{1t},{\mbox{{a}}}_{2t}$ make a planar basis
and we can expand

$\left(\begin{array}[]{c}{\mbox{{b}}}_{1t}\\\
{\mbox{{b}}}_{2t}\end{array}\right)=\left(\begin{array}[]{cc}B_{11}&B_{12}\\\
B_{21}&B_{22}\end{array}\right)\left(\begin{array}[]{c}{\mbox{{a}}}_{1t}\\\
{\mbox{{a}}}_{2t}\end{array}\right).$ (139)

Inserting in (137), we obtain

$B_{11}B_{22}-B_{12}B_{21}=1.$ (140)

Expanding (138) as

$a_{1n}{\mbox{{b}}}_{2t}-b_{2n}{\mbox{{a}}}_{1t}+b_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{b}}}_{1t}=0,$
(141)

and applying (139), with ${\mbox{{a}}}_{1t}\times{\mbox{{a}}}_{2t}\not=0$,
after some steps we obtain the expansion

$\left(\begin{array}[]{c}b_{1n}\\\
b_{2n}\end{array}\right)=\left(\begin{array}[]{cc}-B_{22}&B_{12}\\\
B_{21}&-B_{11}\end{array}\right)\left(\begin{array}[]{c}a_{1n}\\\
a_{2n}\end{array}\right)=-\left(\begin{array}[]{cc}B_{11}&B_{12}\\\
B_{21}&B_{22}\end{array}\right)^{-1}\left(\begin{array}[]{c}a_{1n}\\\
a_{2n}\end{array}\right).$ (142)

In conclusion, vectors ${\mbox{{b}}}_{1}$ and ${\mbox{{b}}}_{2}$, depending
through (139) and (142) on any vectors ${\mbox{{a}}}_{1}$ and
${\mbox{{a}}}_{2}$ whose tangential components are linearly independent, in
terms of any parameters $B_{11}\cdots B_{22}$ satisfying (140), yield
$C_{1}=0$ for any ${\mbox{{k}}}_{t}$.

###  8.4 Verifying EPEMC+ and EPEMC- boundary conditions

Let us verify that the EPEMC+ boundary conditions (74) and (75), and the
EPEMC- boundary conditions (76) and (77), depending on the two vectors
${\mbox{{a}}}_{1}$ and ${\mbox{{a}}}_{2}$, both correspond to the property
$R_{\pm}^{2}=-1$.

Substituting (72) and (73) and expanding

$\displaystyle{\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}$ $\displaystyle=$
$\displaystyle{\mbox{{a}}}_{1t}\times{\mbox{{a}}}_{2t}+{\mbox{{n}}}\times(a_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{a}}}_{1t}),$
(143) $\displaystyle{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2}$ $\displaystyle=$
$\displaystyle{\mbox{{a}}}_{1t}\times{\mbox{{a}}}_{2t}-{\mbox{{n}}}\times(a_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{a}}}_{1t}),$
(144)

we obtain

${\mbox{{n}}}\cdot({\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}-{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2})=0.$
(145)

Similarly, we can expand

$({\mbox{{a}}}_{1}\times{\mbox{{b}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{a}}}_{2})\times{\mbox{{n}}}=\pm(a_{1n}{\mbox{{a}}}_{2t}+a_{2n}{\mbox{{a}}}_{1t}-a_{1n}{\mbox{{a}}}_{2t}-a_{2n}{\mbox{{a}}}_{1t})=0.$
(146)

Substituting these in (42) we obtain

$C_{1}=0,$ (147)

which, from (37), ensures the validity of the condition $R_{+}=-R_{-}$.

Further, we can expand

$\displaystyle{\mbox{{a}}}_{1}\times{\mbox{{a}}}_{2}+{\mbox{{b}}}_{1}\times{\mbox{{b}}}_{2}$
$\displaystyle=$ $\displaystyle 2A_{n}{\mbox{{n}}},$ (148)
$\displaystyle{\mbox{{b}}}_{1}{\mbox{{a}}}_{2}-{\mbox{{b}}}_{2}{\mbox{{a}}}_{1}$
$\displaystyle=$ $\displaystyle\mp
A_{n}({\mbox{{n}}}\times{\overline{\overline{\mathsf{I}}}}+{\mbox{{n}}}{\mbox{{p}}}_{t}+{\mbox{{p}}}_{t}{\mbox{{n}}}),$
(149)

with the upper sign corresponding to EPEMC+, and the lower sign to EPEMC-,
boundary conditions.

Substituting these in (41) and (43) yields

$\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\frac{-2{\mbox{{n}}}\cdot{\mbox{{k}}}^{i}}{k_{o}k_{n}}A_{n}(k_{o}\pm{\mbox{{p}}}_{t}\cdot{\mbox{{k}}}_{t}),$
(150) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{2{\mbox{{n}}}\cdot{\mbox{{k}}}^{r}}{k_{o}k_{n}}A_{n}(k_{o}\pm{\mbox{{p}}}_{t}\cdot{\mbox{{k}}}_{t})=C_{0},$
(151)

whence, from (37), the reflection coefficients are $j$ and $-j$ for both of
the EPEMC+ and EPEMC- conditions.

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# Part-level Action Parsing via a Pose-guided Coarse-to-Fine Framework

Xiaodong Chen1 Xinchen Liu2 Wu Liu2 Kun Liu2 Dong Wu2 Yongdong Zhang1 Tao Mei2
1University of Science and Technology of China, Hefei, China 2JD.com, Beijing,
China
cxd1230@mail.ustc.edu.cn,{liuxinchen1,liuwu1,liukun167,wudong}@jd.com,zyd73<EMAIL_ADDRESS>
###### Abstract

Action recognition from videos, i.e., classifying a video into one of the pre-
defined action types, has been a popular topic in the communities of
artificial intelligence, multimedia, and signal processing. However, existing
methods usually consider an input video as a whole and learn models, e.g.,
Convolutional Neural Networks (CNNs), with coarse video-level class labels.
These methods can only output an action class for the video, but cannot
provide fine-grained and explainable cues to answer why the video shows a
specific action. Therefore, researchers start to focus on a new task, Part-
level Action Parsing (PAP), which aims to not only predict the video-level
action but also recognize the frame-level fine-grained actions or interactions
of body parts for each person in the video. To this end, we propose a coarse-
to-fine framework for this challenging task. In particular, our framework
first predicts the video-level class of the input video, then localizes the
body parts and predicts the part-level action. Moreover, to balance the
accuracy and computation in part-level action parsing, we propose to recognize
the part-level actions by segment-level features. Furthermore, to overcome the
ambiguity of body parts, we propose a pose-guided positional embedding method
to accurately localize body parts. Through comprehensive experiments on a
large-scale dataset, i.e., Kinetics-TPS, our framework achieves state-of-the-
art performance and outperforms existing methods over 31.10% ROC score.

###### Index Terms:

Part-level Action Parsing, Action Recognition, Video Understanding, Pose-
Guided Positional Embedding

##  I Introduction

Action recognition [1, 2, 3, 4, 5, 6], which can be treated as a high-level
video classification problem, is a hot topic of video understanding in
computer vision. With the development of rich representations based on neural
networks [7, 8, 9], significant progress has been made on this task. Although
these existing action recognition methods [10, 11, 12, 13] can predict the
high-level human action of the whole video, they neglect the detailed and
middle-level understanding of human actions.

To fill this gap, the Part-level Action Parsing (PAP) task, which aims to
recognize frame-level human action of all body parts and the whole body from a
video in the wild, was firstly focused on by researchers recently. The PAP
task is to address the following problem: given a human action video, a system
needs to predict the human location, body part location, part state/action in
each frame, then integrates these results to predict human action in the video
level. For instance, as video of the fitness center shown in Fig. 1, we not
only need to predict its video-level label as “clean_and_jerk”, but also need
to detect each body part such as “right_arm” and “right_hand” in each frame
and predict their part-level action labels like “carry” and “hold”.

Figure 1: An example of the Part-level Action Parsing (PAP) task. The upper
part is the input video, and the lower part is the labels need to predict,
which include video-level action, location of each person, location of each
body part, and frame-level action of each body part.

By decomposing an action into a human part graph, the PAP task advances the
area of human action understanding with a shift from the traditional action
recognition task to deeper understanding tasks of part-level action parsing.
Moreover, it may have many potential applications such as intelligent
manufacturing, sports analysis, fitness instruction, and so on. Therefore,
this paper concentrates on the part-level action parsing task which is
valuable yet overlooked by the community of multimedia and computer vision.

However, part-level action parsing is a non-trivial task that faces several
challenges as shown in Fig. 1. First of all, there are many obstacles when
predicting the spatial position of the human body and body parts accurately.
Different from traditional object detection tasks, body part detection needs
to overcome the ambiguity of body parts and capture the prior of human body
structure. Moreover, even if there are only two or three people in every frame
of the video, the number of part-level actions that need to be predicted is
very large due to the fine-grained division of human body parts. It is more
strenuous to represent the relationship between these parts. Furthermore, the
trade-off between computational power and accuracy in prediction should also
be considered due to the densely predicted frame-level and part-level actions.

Figure 2: The overall architecture of our method, which includes two main
parts: a) person detection and part detection, and b) video-level action
recognition and segment-level action parsing.

Action recognition has been studied for several years. From data-driven
representations learned by deep Convolutional Neural Network (CNN) [9, 14, 15]
to Transform-based Neural Network [16] with large parameters, the accuracy of
action recognition has been significantly improved. However, traditional
action recognition methods often consider the whole video or clip as the
smallest unit. Despite their excellent performance for video-level action
recognition, these methods cannot work well for frame-level part actions. In
addition, although some researchers have studied frame-level human action
location [3], they only focus on the whole body action and ignore the fine-
grained part-level actions. Due to the small size of body parts, the
traditional methods, e.g., RoIAlign [17], that used in frame-level human
action location have little performance improvement on the PAP task.

To this end, we propose a Pose-guided Coarse-to-Fine framework, named PCF, for
part-level action parsing. We first adopt the existing action recognition
methods, e.g., CSN [18], to predict the coarse action of the whole video,
since it is the State-of-The-Art (SoTA) CNN-based model in the action
recognition task. After that, we predict the fine-grained segment-level body
part action instead of the frame-level action based on the persistence of
human actions, which greatly improves the computational efficiency with less
precision reduction. Moreover, due to the ambiguity of body parts, e.g., the
similarity of the appearance of the left leg and the right leg, traditional
existing object detectors are often unable to predict the body part
effectively. To solve this problem, we propose the pose-guided positional
embedding method which guides the detector to predict the part locations with
human pose keypoints. By encoding each human keypoints with different colored
dots on the original images, the feature representations of different parts
are more easily distinguished by the detector, which effectively reduces the
body part ambiguity.

In summary, the contributions of this paper include: 1) we make one of the
first attempts for part-level action parsing which is a valuable yet
unexplored task; 2) we design a PCF framework to exploit the potential
performance of existing object detectors with pose-guided positional embedding
and predict both the coarse video-level action and the fine-grained body part
action; 3) our method achieves SoTA results on the Kinetics-TPS dataset[19],
which shows the effectiveness of our method.

##  II THE PROPOSED FRAMEWORK

###  II-A Overall Framework

Figure 2 shows the overall structure of the Pose-guided Coarse-to-Fine (PCF)
framework for the PAP task. It includes three stages, i.e., instance and part
detection, video-level action recognition, and part action parsing. In the
first stage, as shown in the upper part of Figure 2, we adopt YOLOF [20] as
the backbone of the person detector and part detector to locate each person
and body parts. To overcome the ambiguity of body parts, we insert the pose
estimator and the positional embedding module between the person detector and
part detector to improve the accuracy of the part location. The second stage
is shown in the lower part of Figure 2. Based on the short-term persistence of
human actions, we exploit segment-level action prediction to approximate the
frame-level action state to balance the accuracy and computation cost. In
particular, we divide the original video into multiple segments that last
three seconds or so. Then tag each segment with six segment-level pseudo
action labels based on the original frame-level part action labels which
significantly reduce the computation cost for frame-level action parsing.
After that, we train models for segment-level action and video-level action
respectively. In the final stage, we integrate the output of all previous
stages to get the final output for the PAP task. Next, we will introduce each
stage of our framework in detail.

###  II-B Pose-guided Part Detection

To our knowledge, body part detection is an unprecedented task in traditional
object detection tasks. Different from general object detection, body part
detection needs to overcome the ambiguity of body parts and model the prior of
human body structure. For example, normally people only have one left foot and
one right foot, but the local features of the left one and the right one are
often very similar. Fortunately, this structural prior is very common in human
pose estimation and has been widely explored. To maximize the usage of this
structural prior, we propose a pose-guided part detection method, which is
shown in the top half of Figure 2.

In detail, the person detector $F_{person}(\cdot)$ first extracts the bounding
box of a person $X_{p}$ from the input frame $X_{f}$ by

$X_{p}=F_{person}(X_{f}).$ (1)

Then the pose estimator $F_{pose}(\cdot)$ takes $X_{p}$ as the input and
outputs the keypoints $K_{p}=\\{(x_{i},y_{i})\\}^{N}_{i=1}$ of the person,
which is formulated by

$K_{p}=F_{pose}(X_{p}),$ (2)

where $(x_{i},y_{i})$ is the coordinates of keypoint $i$, $N$ is the number of
keypoints. After that, the keypoints $K_{p}$ are integrated by the positional
embedding module $G$ with dots of different colors and radius on the original
$X_{p}$ to generate an augmented person image $X^{\prime}_{p}$. This process
can be formulated by

$X^{\prime}_{p}=X_{p}+G(K_{p}).$ (3)

By this means, we can increase the appearance difference between different
body parts and facilitate the learning of body parts detector $F_{body}$.
Finally, the part detector $F_{body}$ is implemented to localize each body
part $Y_{part}$ by

$Y_{Part}=F_{body}(X^{\prime}_{p}).$ (4)

In addition, we also fine-tune the person detection box with the results of
the pose estimator. In a nutshell, the pose estimator has the ability to
predict the possible human keypoints outside the person box, and we fine-tune
the detected person box until all possible human keypoints are included.

###  II-C Part State Parsing and Action Recognition

Fine-grained frame-level part state parsing requires more computation and
hardware cost than coarse video-level action recognition. However, we find
this frame-level action parsing problem can be transformed into a simpler
segment-level action recognition task due to the overwhelming “Long Tail
Effect” caused by the short-term persistence of actions in video segments. For
example, in the segment of “hurling sport” that lasts about three seconds, we
just need to predict “None” for the heads in every frame and easily achieve
97.7% frame-level part state accuracy. To take advantage of the significant
“Long Tail Effect”, as shown in the bottom half of Figure 2, we tag each video
segment with six part-level pseudo labels based on the original frame-level
action state label. The fine-grained part-level label consists of three parts:
coarse video-level action, body part, and the most frequent frame-level action
of the body part. As the example shown in Figure 2, “(hurling_sport) head:
none” means the video-level action of this video is “hurling_sport”, and the
most frequently frame-level action of the “head” in this segment is “none”.
Through this transformation, we can directly apply individual action
recognition networks without sharing parameters, such as CSN [18]. By this
means, we can predict each fine-grained segment-level label and coarse video-
level label respectively without any other models related to the computation-
intensive frame-level action prediction.

##  III EXPERIMENTS

###  III-A Experimental Setting

TABLE I: Results of different settings on the Kinetics-TPS testing set. “ROC Score” refers to the final scores of the methods. Methods | Input | backbone | Video Acc (%) | ROC Score (%)
---|---|---|---|---
baseline [21] | RGB |  TSN-Res50 [10] | - | 29.79
PCF (TSN_RGB) | RGB | TSN-Res50 | 74.03 | 49.23 (+19.44)
PCF (TSN_Flow) | Flow | TSN-Res50 | 83.48 | 54.33 (+5.1)
PCF (Ours) | RGB |  ip-CSN-152 [18] | 96.46 | 60.89 (+6.56)
TABLE II: Effect of Pose-guided Positional Embedding.  Model | Pose |  $AP$ (%) |  $AP_{50}$ (%)
---|---|---|---
$YOLOF_{person}$ | ✗ | 74.60 | 93.40
$YOLOF_{person}$ | ✓ |  74.80 (+0.20) |  93.80 (+0.40)
$YOLOF_{part}$ | ✗ | 36.40 | 53.10
$YOLOF_{part}$ | ✓ |  57.10 (+20.7) |  79.70 (+26.6)

Dataset. The experiments are performed on the Kinetics-TPS dataset [19] that
provides 7.9 M annotations of 10 body parts, 7.9 M part state (i.e., how a
body part moves) of 74 body actions, and 0.5 M interactive objects of 75
objects in the video frames of 24 human action classes. Kinetics-TPS contains
3,809 training videos (4.96 GB in size) and 932 test videos (1.26 GB in size).
It’s worth noting the source videos of Kinetics-TPS come from Kinetics 700.
Hence, all the Kinetics-pretrained models are forbidden in the PAP task.

Evaluation Metrics. We adopt the official evaluation metric, i.e., ROC score,
of the Kinetics-TPS dataset [19]. ROC scores are calculated based on the Part
State Correctness (PSC) and the action recognition conditioned on PSC. The PSC
calculates the accuracy of the whole human detection results and body part
action parsing in each frame. The action recognition conditioned on PSC draws
the ROC curve and calculates the ROC score according to the top-1 video-level
action recognition accuracy and PSC accuracy. Please refer to [22] for more
details.

Details of Detector and Pose Estimator. For person detector and part detector
on keyframes, we adopted the YOLOF [20], which is an anchor-free model with a
ResNet-101 [7] backbone. The model is pre-trained on the COCO dataset [23] and
then fine-tuned on Kinetics-TPS. The final models obtain 93.8 AP@50 in the
person category and 79.7 AP@50 in the 10 body part categories on the Kinetics-
TPS validation set. For the pose estimator, we directly adopted the HRNet-w48
[24] pre-trained on COCO [23] to extract the keypoints of each person without
any fine-tuning.

Details of Action Parsing and Action Recognition Network. We use the CSN
networks [18] as the backbone in our action recognition and action parsing
framework. We use the ip-CSN-152 implementation pre-trained on the IG-65M [25]
dataset with input sampling $T\times\tau=32\times 2$. In particular, we freeze
the Batch Normalization (BN) layers in the backbone during fine-tuning on
Kinetics-TPS.

Details of Training. The detection model and action recognition models are
trained separately. Each model is trained in an end-to-end manner. In detail,
we train the YOLOF detector using SGD with a mini-batch size of six on four
V100 GPUs and train it for 24 epochs with a base learning rate of 0.01, which
is decreased by a factor of 10 at epoch 16 and 22. We perform linear warm-up
[26] during the first 1800 iterations. For the CSN model, we train it using
SGD with a mini-batch size of four on four V100 GPUs for 58 epochs with a base
learning rate of 8e-5, which is decreased by a factor of 10 at epoch 32 and
48. We perform linear warm-up [26] during the first 16 iterations. By default,
we use weight decay of 1e-4 and Nesterov momentum of 0.9 for all models.

Details of Inference. Following the official guideline [22], we extract the
top-10 results from the person detector and the top-1 results of each body
part from the part detector during testing. For the action recognition task
and action state parsing task, we set the number of sampling clips as seven
for each video segment at test time and scale the shorter side of input frames
to 256 pixels.

###  III-B Main Results

To demonstrate the effectiveness of our method, We compare our PCF framework
with the official baseline method. Meanwhile, to illustrate the fairness of
comparison, we replace our ip-CSN-152 backbone with the TSN-Res50 used by the
official baseline.

We present our results on Kinetics-TPS in Table I. The ”Input” in the second
column refers to the video input form, and the calculation of optical flow is
based on the tvl1 algorithm [27]. The “video acc” in the fourth column refers
to the top-1 video-level action recognition accuracy, while the “ROC score” in
the fifth column refers to the final ROC score of the methods.

From the results, we can first find that directly applying our PCF framework
with TSN-Res50 backbone and RGB input form, our performance achieves a
significant enhancement of +19.44 ROC score. Out of our expectation simply
changing the input mode from RGB to optical flow gives a total boost of +5.1
ROC score improvement. This may indicate that the body part action encoded by
optical flow carries more effective information than RGB input when using
2D-CNN based network in the PAP task. Furthermore, with the strong CNN-based
model ip-CSN-152 pretrained on IG-65M, our PCF framework achieves the 60.89%
ROC score on the Kinetics-TPS dataset.

TABLE III: Frame-level Prediction v.s. Segment-level Prediction. Method | Duration | The action prediction accuracy | TFLOPs
---|---|---|---
head | arm | hand | hip | leg | foot
Frame | - | 92.46 | 68.11 | 68.50 | 84.19 | 66.27 | 65.47 | 36.12
Segment | 3.00s | 92.46 | 68.07 | 68.43 | 84.14 | 66.14 | 65.36 | 11.50
Segment | 10.0s | 92.44 | 67.69 | 68.10 | 83.99 | 65.59 | 65.02 | 3.482

###  III-C Ablation Experiments

Effect of Adding Pose Estimator. We investigate the effect of the pose
estimator on detection mAP. For person detector and part detector, we train
the lightweight CNN-based model YOLOF with human location and body parts
location respectively. As shown in Table II, adding the pose estimator brings
consistent AP and AP@50 increases for these two models. More specifically,
equipped with the pose estimator, our $YOLOF_{part}$ model achieves a
significant enhancement of $+26.6$ AP@50 on the Kinetics-TPS dataset.

Frame-level Prediction v.s. Segment-level Prediction In this subsection, we
quantitatively compare the action prediction accuracy and the computation cost
between the frame-level action parsing and the segment-level action parsing in
Table III. The action prediction accuracy and the “TFLOPs” are calculated with
the ip-CSN-152 model on the Kinetics-TPS dataset. From the results, we can see
that the computation “TFLOPs” decreases greatly with the increase of segment
duration, while the loss of the accuracy is just 0.4% or less. Especially when
the duration of the segment is less than three seconds, the accuracy of frame-
level prediction and segment-level prediction is almost the same (decrease
less than 0.1%), while the computation decreases about 68.16%.

##  IV CONCLUSION

This paper presents a pose-guided coarse-to-fine framework for the part-level
action parsing task. In our PCF framework, the pose-guided part detector is
one of the first attempts toward body part detection and brings considerable
improvement in the AP@50 (+26.60%). Meanwhile, we convert the frame-level part
state parsing problem into segment-level action recognition based on the
persistence of human actions, which greatly improves the computational
efficiency with less precision reduction. At last, our method achieves SoTA
results at the Kinetics-TPS dataset, which shows the effectiveness of our PCF
framework. With these three contributions, we provide one of the first
attempts for the part-level action parsing task.

## Acknowledgment

This work was supported by the National Key R&D Program of China under Grant
No. 2020AAA0103800.

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