Application of the Mathieu’s equation for a design of a photonic crystal supporting surface electromagnetic waves

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ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 3, NOVEMBER 2015

# Application of the Mathieu’s equation for a design of a photonic crystal supporting surface electromagnetic waves

Ekaterina Rostova, Giovanni Dietler, and Sergey K. Sekatskii

Laboratoire de Physique de la Matière Vivante, IPSB, BSP,
Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

*corresponding author: `ekaterina.rostova@epfl.ch`

Abstract

Nowadays, unique characteristics of surface
electromagnetic waves, particularly, surface plasmons
supported by a specially designed photonic crystal find
numerous applications. We propose to exploit an evident
analogy between such a photonic crystal and a structure
with a sine-modulated refractive index. The light
propagation inside the latter is described by the famous
Mathieu’s differential equation. This application of the
Mathieu’s equation can be useful for a design of multilayer
structures, and also for fundamental understanding of
electromagnetic phenomena in inhomogeneous media.

1. Introduction

Photonic crystals (PCs) are materials that possess a periodic
modulation of their refractive indices (RIs) on the scale of
the light wavelength [1]. Optical surface modes in onedimensional photonic crystals (multilayer dielectric
mirrors) were extensively studied in the 1970s, both
theoretically [2, 3] and experimentally [4]. Twenty years
later, excitation of optical surface waves (SW) in photonic
crystals in a Kretschmann-like configuration was first
demonstrated [5]. These studies rapidly led to broad use of
surface waves based on photonic crystals in ever-widening
applications in the field of optical sensors [6-14]. This
technique benefits much from its unique peculiarities,
namely, the existence of both _p_ - and _s_ -polarized surface
waves enables to discriminate surface and volume effects.
Moreover, an electromagnetic field penetration depth into
an external medium is significantly increased compared to
surface plasmon polaritons (SPR)-based biosensors.
Surface waves-based biosensors permit to study not only
interactions between relatively thin layers of proteins (and
other biomolecules), but also between such thick objects as
bacteria, cells and cell organelles. A direct experimental
comparison of the sensitivity of biosensors based either on
surface plasmon polaritons or on photonic crystal surface
waves (same as Bloch waves or Tamm states) may be
found elsewhere [15]. The authors showed that a biosensor
based on photonic crystal surface waves can demonstrate
almost two times higher sensitivity that one based on
surface plasmon polaritons.

Quite an important particular application of photonic
crystals is their use to support long-range surface plasmon
polaritons existing in such structures as _photonic crystal –_
_thin metal layer – infinite dielectric_ [16] (of course, such

electromagnetic waves exist only for _p_ -polarized incident
light [17]). These systems have much in common with
well-known symmetric sandwich structures such as a thin
metal layer between two infinite dielectrics, known to
support long-range surface plasmons (from now on, when
this cannot lead to a confusion, we will omit the word
“polariton” and speak simply about plasmons) [18, 19]. For
this case the photonic crystal can be characterized, for a
certain wavelength, by the same refractive index as that of
an infinite dielectric, thus forming a symmetric sandwich
structure. Exploitation of such structures enabled to excite
and use in practice surface plasmons in thin Pd [12, 20] and
Co (to be submitted) layers as well as blue (at 405 nm) [21]
and UV (at 375 nm; to be submitted) plasmons in thin gold
layers. Note that if a photonic crystal is not exploited, for
all these cases there are no reasons to speak about plasmons
at all, because their propagation length is just in the order of
a light wavelength.

All these circumstances well underline the necessity
of deep and physically transparent understanding of the
details of plasmon propagation in the structures involving
photonic crystals. Certainly, rather effective calculation
methods, based essentially on multiple reflections inside the
multilayer structure (Fresnel formulae), do exist and are
successfully used; see _e.g._ Konopsky’s paper [22] which, in
our opinion, is one of the best examples. Still we believe
that these methods are rather formal and do not clearly
reveal the physics of the phenomenon beyond. As a
consequence, they are not indeed “intuition-friendly” and,
correspondingly, their prediction potential without detailed
calculations rests limiting.

In this work, we use an evident analogy between a
one-dimensional photonic crystal and a structure with a
sine-modulated refractive index. The propagation of light
inside the latter is described by the famous Mathieu’s
differential equation, which extensive studies of many years
had a considerable impact on the common knowledge, see
_e.g._ a classic book [23] and numerous references therein.
Strictly speaking, a photonic crystal (Fig. 1), which is a
periodic sequence of dielectric layer pairs of thicknesses _dL_

and _dS_ with respectively large _nL_ = ε _L_ and small

_nS_ = ε _S_ refractive indices, is not exactly a medium with

a sine-modulated refractive index. Similar situation exists,
for example, for electrons in crystals: planes of ions do
create a periodic potential but do not create a sine potential.

ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 3, NOVEMBER 2015

Nevertheless, this circumstance does not make an
application of the Mathieu’s equation for the corresponding
problem meaningless. Rather the contrary, the same
equation appears as the tool to tackle related problems.
However, unfortunately the paradigm of the Mathieu’s
equation and its application are not commonly used to
describe problems related to excitation of optical modes. By
the present study we aim to broad applications of this
equation and introduce it to the area of investigation into
photonic crystals. Briefly, the present article discusses
application of the Mathieu’s differential equation to design
a photonic crystal with a metal film supporting long-range
surface modes.

The article is divided into five chapters: first, we
introduce the Mathieu’s equation and explain how an
electromagnetic field distribution in periodic structures can
be calculated by solving the equation. We emphasize on the
requirements on its solutions which must be fulfilled to
describe the surface modes. Second, we give numerical
examples of photonic crystals with certain parameters. The
given examples illustrate how to design a photonic crystal
to excite surface waves at a desired wavelength. Next,
solutions of the Mathieu’s equation are analyzed to
demonstrate physical limits of the optical properties of the
periodic structures for surface wave excitation. In
conclusion, an analogy was drawn between surface waves
and energy eigenstates for a quantum particle in a periodic
potential.

2. Mathieu’s equation

Below we consider a system _infinite dielectric – finite metal_
_layer – infinite periodic dielectric structure_ with a sine
profile of the refractive index (PC), see Fig. 1. Let the
infinite dielectric has a dielectric constant ε1 and metal –

ε _m_, while a dielectric constant of the periodic structure is

given by ε( _z_ ) =ε _av_ + Δεcos(2π _z_ / _L_ ), where _L_ is the period
of the structure, and for the moment we identify an average
dielectric constant with ε _av_ = (ε _L_ +ε _S_ /) 2,

Δε=ε _av_ −ε _S_ =ε _L_ −ε _av_ (Δε > 0). A finite-thickness layer

characterized by the dielectric constant ε _m_ should not be

necessarily a metal; we will use this notation just to
simplify the terminology.

Based on the properties of surface plasmons, which
are “transverse magnetic (TM) in nature” and on known
properties of TM electromagnetic waves [24], one can write
for the _Hy_ - component of the magnetic field [19]:

Figure 1. A schematic of the studied structure (not to scale):
dielectric (z < 0) − metal (0 < z < h) − photonic crystal
(z > h). The dielectric constants are ε1,ε _m_,ε _S_ and ε _L_,
respectively; the last two correspond to the dielectic
constants of the alternating layers of the photonic crystal
with the period _L_ . The bold red line depicts the electric field
intensity distribution of a surface wave in the structure, the
dashed line images the metal layer boundary. The surface
wave propagates along the x-axis.

The _Hy_ - components of electromagnetic field inside the
homogeneous media are given by solutions of appropriate
wave equations, which in our case reduce to

∂2 _zH_ 2 _y_ + ( _k_ 2ε,1 _m_ - β2 ) _H_ _y_ =

ε,1 _m_ - β2 ) _H_ _y_ =,0 (3)

∂

_H_

_zH_ 2 _y_ + ( _k_ 2ε,1 _m_ - β2 ) _H_ _y_

2 _y_ + ( _k_ 2ε,1 _m_ - β2 ) _H_ _y_ =,0

where we introduce _k_ 2 =ε0ω2 = (2π/λ)2.
The equations (2) and continuity of the tangential _Hy_ field component at the interfaces ?= 0 _,_ ?= ℎ together
with the continuity of the tangential _Ex_ -field component at
the interface ?= 0 yield the following solution of the wave
equation:
## • Inside an infinite dielectric, ?< 0, f 1( z ) = eS 1 z

(this implies _Re_ (S1)>0; exactly this form of a solution
describes the surface wave, _i.e._ a wave localized at the
interface),

- Inside the metal, 0 < ?< ℎ, we have

_f_ _m_ ( _z_ ) = cosh( _Sm_ _z_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Sm_ _z_

_m_ ( _z_ ) = cosh( _Sm_ _z_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Sm_ _z_ ),

( _z_ ) = cosh( _Sm_ _z_ ) + 1ε _m_ sinh( _Sm_ _z_ )

- Inside the medium with a varying refractive index
(PC), ?> ℎ,

_f_ 3( _z_ ) = (cosh( _Smh_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Smh_ )) _P_ ( _z_ - _h_

= (cosh( _Smh_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Smh_ )) _P_ ( _z_

( _z_ ) = (cosh( _Smh_ ) + 1ε _m_ sinh( _Smh_ )) _P_ ( _z_ - _h_ ),

- _H_ = _H_ 0 _e_ _y_ _f_ ( _z_ )exp( _i_ ω _t_ - _i_ β _x_ ), (1)

0 _y_

where _e_ � _y_ is a unit vector along the _y_ -axis (Fig. 1). For TM

modes, if we know the _Hy_ - component of the magnetic
field, the electric field components can be evaluated from
Maxwell’s equations [2, 19]:

_E_ _x_ = ωε _i_ 0ε ∂∂ _Hzy_

β

= - _H_ _y_ . (2)

ωε0ε

_i_ ∂

=
ωε0ε

β

where _P_ ( _z_ ) is an appropriate solution of the Mathieu’s
equation (see eq. (6) below). Here the exponential decay
constant is given by

_S_,12 _m_ = β2 −ε,1 _mk_ 2. (4)

The solution of the Mathieu’s equation is chosen to
exponentially decrease for _z_ → +∞, and _P_ (0) =,1 which
ensures also the continuity of _Hy_ on the interface ?= ℎ.
The continuity of _Ex_ at the interface ?= ℎ will be
established later on when a solution of the Mathieu’s
equation will be discussed.

_z_

β

, _Ez_ = - _H_ _y_
ωε0ε

ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 3, NOVEMBER 2015

Inside a medium with a varying refractive index (PC),
the wave equation reduces to the Mathieu’s equation

ε _m_ ε1

sinh( _Smh_ ) + cosh( _Sm_

_m_ ε1

_Sm_ sinh( _Smh_ ) + _S_ 1 cosh( _Smh_

_m_

sinh( _Smh_ ) + 1 cosh( _Smh_ ) =

_dz_ 2 _H_ 2 _y_ - (β2 −ε _avk_ 2 - Δε _k_ 2 cos(2π _z_ / _L_ )) _H_ _y_ =

_d_ 2 _H_

β2 −ε _avk_ 2 - Δε _k_ 2 cos(2π _z_ / _L_ )) _H_ _y_ = .0 (5)

1

2 _y_ - (β2 −ε _avk_ 2 - Δε _k_ 2 cos(2π _z_ / _L_ )) _H_ _y_ = .0

= ε31(0) (cosh( _Smh_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Smh_ ))π _L_ _y_ ('0), (9)

31(0) (cosh( _Smh_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Smh_ ))π _L_ _y_

1(0) (cosh( _Smh_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Smh_ ))π _L_ _y_ ('0),

_mh_ ) + _S_ 1ε _m_

With the variable change _u_ = π _z_ / _L_, eq. (5) yields a
standard form of the Mathieu’s equation [23] which will be
in the main focus of the present study:

_du_ 2 _H_ 2 _y_ + ( _a_ - 2 _q_ cos(2 _u_ )) _H_ _y_ =

_d_ 2 _H_ _y_ + ( _a_ - 2 _q_ cos(2 _u_ )) _H_ _y_ = .0 (6)

2 _y_ + ( _a_ - 2 _q_ cos(2 _u_ )) _H_ _y_ = .0

Here _a_ and _q_ are parameters given by

_a_ = −β2 _L_ 2π−2 +ε _avk_ 2 _L_ 2π−2, (7)

_q_ = - 1 Δε _k_ 2 _L_ 2π−2. (8)
2

3. Surface electromagnetic waves

To obtain surface waves in the studied structure,
besides a requirement to satisfy the continuity of the _Ex_ field component at ?= ℎ, a solution _y(u)_ of the Mathieu’s
equation should exponentially decrease for _u_ → +∞ . The
exponentially decreasing solutions belong to the instability
regions of _(a, q)_ -chart of the Mathieu’s equation parameters
(hatched areas on Fig. 2, adapted from Ref. _23_ ), which are
associated with photonic band gaps.

Figure 2. The stability chart for the solutions of the
Mathieu’s equation with the parameters _(a, q)_ . The solid
lines correspond to characteristic curves, the hatched areas
depict the instability regions, the empty areas – stability
regions. The solutions marked by the points ‘1’ and ‘2’ are
analyzed in the text.

Let us now analyze the condition of the continuity of
_Ex_ -field component on the metal – PC interface. From now
on we denote the variable as _u_ = π( _z_ - _h_ /) _L_ to place the
interface at _u_ =0. The aforementioned continuity condition
reads:

where ε3(0) is a local _,_ just near the interface, value of the
photonic crystal dielectric constant.

It is natural to start further analysis of propagation
modes in our system using an analogy to long-range surface
plasmons in symmetric sandwich structures, in which a
photonic crystal is replaced by a semi-infinite uniform
dielectric with the dielectric constant ε3 =ε1 . In a thin
metal film with _h_ → 0, the long-range surface plasmons
exist if ?! = ?! = 0 [18, 19], which is necessary to ensure
the continuity of the longitudinal electric field component
at the interface. This means, see (3), that the propagation

constant is given by β = ε1 _k_, so that from (7) we obtain

the photonic crystal period

λ _a_

_L_ = . (10)

λ

.
2 ε _av_ −ε1

2 ε _av_

_av_

Next, the continuity condition (9) results in _y_ ('0) = 0, and

such a solution _y_ ( _u_ ) can be constructed as follows. The
Mathieu’s equation describing a magnetic field inside a
periodic multilayer structure (photonic crystal) is given by
eq. (6) which we rewrite as

_d_ 2 _y_

_du_ 2 + ( _a_ - 2 _q_ cos(2 _u_ )) _y_ =

_d_ 2 _y_

+ ( _a_   - 2 _q_ cos(2 _u_ )) _y_ =,0 (11)

2 + ( _a_ - 2 _q_ cos(2 _u_ )) _y_ =,0

with the following relation between the parameters _a_ and _q_ :

1 _a_

_q_ = - Δε .) (12)

1 _a_
2 Δε(ε _av_ −ε1.)

= - Δε

2 (ε _av_ −ε1

_av_

The exponentially decreasing solution of the equation (11)
has the form of _y_ ( _u_ ) = _e_ −µ _u_ ϕ( _u_ ), where µ> 0 is a
characteristic exponent and ϕ( _u_ ) is a 2π- or π- periodic
function. The continuity condition of the magnetic field _Hy_
at the interface _u=0_ requires that ϕ(0) = ;1 such a solution
exists [23] and can be found. To find it, the easiest
numerical method, called the “shooting method”, supposes
to vary the value of _b_ in the initial condition _y_ ('0) = _b_
while solving numerically the ordinary differential equation
(11), and to analyze the behavior of the solution for large _u_ :
only one certain value of _b_ corresponds to an exponentially
decaying solution. If at certain value of _u0_ the derivative

_y_ (' _u_ |) _u_ = _u_ 0 =,0 then all that we need to construct a solution

which gives rise to surface waves, is to shift the variable _u_
and consider the function _y_ 1( _u_ ) = _Ce_ −µ( _u_ + _u_ 0)ϕ( _u_ + _u_ 0) ;

normalization constant _C_ is so chosen that _y_ 1(0) = 1. The
function

_f_ 3( _z_ ) = (cosh( _Smh_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Smh_ )) _y_ 1(π( _z_ - _h_ /) _L_

= (cosh( _Smh_ ) + _SS_ 1 _m_ εε _m_ 1 sinh( _Smh_ )) _y_ 1(π( _z_ - _h_ /) _L_ ) then

( _z_ ) = (cosh( _Smh_ ) + _S_ 1 _m_ εε _m_ 1 sinh( _Smh_ )) _y_ 1(π( _z_ - _h_ /) _L_ )

describes the _Hy-_ component of the electromagnetic field
inside the photonic crystal.

ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 3, NOVEMBER 2015

This simple consideration quite clearly demonstrates
the physical meaning of a photonic crystal design
supporting long-range surface plasmons. The expression
(10) corresponds to an optimal selection of the thicknesses
of the alternating layers constituting the photonic crystal,
while the aforementioned shift of the initial value of the
variable _u_ is nothing else than the necessity to add an
additional (of large or small refractive index) layer with the
well-defined thickness to ensure such propagation [22].
Moreover, we clearly see also additional possibilities which
appear for photonic crystal-based structures but are lacking
for symmetric sandwich structures. For the case of photonic
crystals, the condition _S1=0_ is not any more necessary: even
without it, the boundary condition (9) which (again in the
limit _h_ → 0 ) reads

ε _S_ 11 = ε31(0) π _L_ _y_ 1('0), (13)

still can be fulfilled. Remember that _S1_ is necessarily
positive, hence we need _y_ 1('0) - 0 at the interface: just
after the metal – PC interface, the absolute value of
electromagnetic field vector should increase (of course, for
larger distances it will eventually decrease according to its
exponential decay when _u_ → ∞ ). This corresponds to
stronger localization and deeper penetration of the
electromagnetic field inside transparent non-absorbing
medium (photonic crystal) and consequently its weaker
localization inside a lossy metal. As a result, a smaller
fraction of the electromagnetic field energy is dissipated in
the metal thus leading to even larger propagation distances
of the surface waves (compare again with Ref. _22_ ). Note,

given by _y_ 1( _u_ ) ≅ _e_ −µ _uce_ 1( _u_, _q_ ) . Here the index µ is much

less than one (see Ref. _23_ Fig. 11) and _ce1(u, q)_ is the
Mathieu function describing a 2π-periodic solution of the
Mathieu’s equation for the pair _(a, q)_ lying exactly on the
corresponding characteristic curve. This curve, when _a_ is
slightly larger than one and _q_ small, is described by the
following expansion (eq. (17), of Ref. _23_ ):

_a_ = 1+ _q_ - _q_ 2 /8 - _q_ 3 / 64 - _q_ 4 /1536 +11 _q_ 5 / 36864 + _O_ ( _q_ 6 ).

(14)
## The above y 1( u ) solution is clearly a complete

analogue of the symmetric sandwich case, in which in the
external medium, the magnetic field amplitude is described
by _f_ 3( _z_ ) = _Ce_ - _S_ 3 _z_ with _S3_ very close to zero: both these
solutions decrease away from the interface very slowly. The
further the chosen _(a, q)_ -pair from the characteristic curves
on the chart is, the larger is the difference of our solution
from one for the symmetric sandwich case: we may call
these solutions “more unstable and less periodic”.

Let us now illustrate the consideration given above by
several numerical calculations corresponding to real
photonic crystal structures. For the case _S1_ =0, the equations
(8) and (10) fix the ratio of the parameters

_q_ 1 Δε

=  - . (15)

_a_ 2 ε _av_ −ε1

1 Δε
.
2 ε _av_ −ε1

1 Δε

=
2 ε _av_

ε _av_ −ε1

however, that if _S_ 1 ≠ 0, also

λ _a_

_L_ ≠ but should

≠

2 ε _av_

2 ε _av_ −ε1

λ

_av_

be recalculated from eq. (7) with the given values of _S_ 1 and
_a_ : this shows certain subtlety of the task to optimize
corresponding structures. This case will be discussed in
details elsewhere.

4. Numerical examples

In the previous sections we showed how the Mathieu’s
equation can be applied to design a multilayer structure
which supports long-range surface plasmons. Such surface
modes can be excited in the structure which design and
physical properties correspond to a pair of _(a, q)_ -parameters
lying in the unstable region on the stability chart (Fig. 2).
These parameters, according to eq. (7), immediately define
a pair of _L, β_ values and also the _S1, Sm_ values using eq. (4).

As a starting point, we will study numerically the
properties of the solutions in the first unstable region of
negative _q_ and positive _a_ . We chose the parameter _a_ close
to 1 and parameter _q_ such that the pair _(a, q)_ is very close to
the characteristic curve separating stability and instability
regions (but, of course, still belongs to the instability
region; see Fig. 2, point ‘1’). For such an _(a, q)_ -pair,
according to the known properties of the Mathieu’s
equation, its vanishing at infinity solution is approximately

Suppose we would like to optimize a structure which
supports the surface waves at the wavelength of 739 nm.
We assume that _nS_ =1.455, _nL_ =2.076 (Ref. _22_, such a
photonic crystal has been quite successfully used to support
long-range surface plasmons in Pd films [12, 20]) so that

ε _av_ = .32134, Δε = .10964 and _q_ / _a_ = −.02477 using eq.
(15). For the parameter _a_ slightly larger than 1, this ratio
defines ?= −0.333 and ?= 1.344 on the characteristic
curve, and the optimal period of the photonic crystal _L_ =288
nm calculated from eq. (10). To approximately describe the
electromagnetic field inside the photonic crystal, we can
use the following _q_ -expansion of the _ce1(u,q)_ function (p.
21 of Ref. _23_ ):

_q_ _q_ _q_ 2
_ce_ 1( _u_, _q_ ) = cos _u_ - 1( + + + ...) cos(3 _u_

( _u_, _q_ ) = cos _u_ - 1( + + + ...) cos(3 _u_ ) +
8 8 192

_q_ 2 _q_ _q_ 2
+ 1( + + + ...) cos(5 _u_ ) + ...(16)

192 6 128

It is immediately clear that _ce_ 1(',0 _q_ ) = 0 hence this
function exactly corresponds to the boundary condition we
are looking for: the photonic crystal refractive index just at
the interface is equal to _nL._ Analogically, for _a_ slightly
smaller than one belonging to the characteristic curve (Fig.
2), the magnetic field inside a photonic crystal will be close
to the function _se_ 1( _u_, _q_ ) (see its _q_ -expansion at p.13 of Ref.

_23_ ). The continuity condition of _se_ 1('π/,2 _q_ ) = 0 is
fulfilled hence the refractive index of the photonic crystal

right at the interface should be equal to ε _av_ .

Now, we will look at those decreasing to zero
solutions in the proximity of _a = 1_ which lie _between_ two

ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 3, NOVEMBER 2015

characteristic curves (Fig. 3). We are interested in the
question either for all values of _a_ one may find such a
solution _y(u)_ which at certain points satisfies the condition

_y_ 1(' _u_ ) = 0, or there are such values of _a_ that always
_y_ 1(' _u_ ) < 0 . The latter would mean that no surface
electromagnetic waves for such parameters _a, q_ exist.
Numerical results attest that in the close proximity of _a = 1_
( _i.e._ for small _q_ ) between two characteristic curves, the
solution and its derivative are almost periodic (Fig. 4,
graphs 1-2). Similar situation occurs when _q_ is approaching
_-6_, so that the pair _(a, q)_ again lies very close to the
characteristic curve. The further the point on the chart from
the characteristic curves is, the faster the solution is
decreasing. This means that the derivative of such a
decreasing solution is mainly negative, however the
numerical results show that the derivative is positive at
certain intervals (and, correspondingly, is equal to zero at
some points, Fig. 4 graphs 3-5). We have checked a number
of pairs _(a, q)_ belonging to the line of ?/?= −0.25 =
????? (Fig. 3, Fig. 4 graph 1), and for all of them the points
## where y 1(' u ) = 0 were found.

Next, we analyzed the points described by the ratio
?/?= −1.5 and lying deeper in the instability region. For
all the chosen points on this line, we succeeded to find such
initial conditions that the derivative was equal to zero at
certain points. Moreover, the chosen points belonging to the
lines ?= 1.1 = ?????, ?= 3 = ?????, and several other
randomly chosen points were still suitable to find the
solutions giving rise to the surface waves, though these
points lie quite far away from the characteristic curves (Fig.
3, Fig. 4 graphs 3-5). Summarizing, despite the analyzed
points were chosen in different locations in the region of
positive parameter _a_ and negative _q_, we could not find any
that would correspond to a decreasing at infinity solution
with a derivative never equals to zero.

Other instability regions of Mathieu equation which
correspond to larger values of _a_, such as
_a=4, 9_, _etc._ [23], also give rise to surface waves which can
be considered quite similarly. These solutions are an
analogue of larger order resonances in Fresnel reflection of
light inside a PC (increase of the relative phase differences
on 2π _n_ -values) and do not seem very interesting.

Now let us note another possibility. The above
analysis suggests that we should still be able to construct a
photonic crystal supporting long-range plasmons even if an
external medium is such that ε1 is larger than ε _av_ . In such
a case, from eq. (7) we see that ?< 0; still having at our
disposal a certain finite value of Δε (and hence of _q_ ) we
are able to find such a pair _(a, q)_ that the derivative of the
corresponding function _y_ (' _u_ ) = 0 for some values of _u_ . For
this, we again should work not too far from the curve
separating unstable region of ?< 0 and small _q_ from a
narrow stable region existing for larger values of _q_ (see Fig.
2 and Ref. _23_ ). For this characteristic curve, under the
condition of a negative _a_ close to zero, we have (p. 15 of
Ref. _23_ ):

_a_ =  - _q_ 2 / 2 + 7 _q_ 4 /128  - 29 _q_ 6 / 2304 + _O_ ( _q_ 8 ). (17)

Figure 3. The points of _(a, q)_ -diagram corresponding to the
analyzed solutions of the Mathieu’s equation in the
instability region (hatched area). Depending on the solution
behavior, it may describe surface modes (filled squares) or
be inappropriate because of the impossibility to fulfill the
continuity condition (empty squares). The black numbered
empty circles denote _(a, q)_ -pairs for which the solution and
its derivative are plotted below (Fig. 4). The dashed black
line corresponds to a = −2q (see explanation in the text).

Let us illustrate this possibility taking the same wavelength
of 739 nm and ?= −1. Then ?/?= 2.184 (Fig. 2, point

ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 3, NOVEMBER 2015

‘2’) which gives _n_ 1 = .186 and, the necessary period

λ − _a_

_L_ =

λ −

λ

λ −

=1092 nm. Under these _(a, q)-_ parameters,
2 ε1 −ε _av_

In this part of the chart, we again tried to find _(a, q)_ pairs that would correspond to such decaying solution for
which always _y_ 1(' _u_ ) < 0, and we succeeded to find many
of them. Indeed, we observed a tendency that for the
majority of |?| > |?|, a solution that can describe surface
modes was not found by varying the initial conditions.

ε1 −ε

_av_

the magnetic field component _Hy_ inside the photonic crystal
is approximately described by π - periodic function

_ce_ 0( _u_, _q_ ) which satisfies the condition _ce_ '0 (,0 _q_ ) = 0 (p. 49
of Ref. _23_ ).

Figure 4. Solutions of the Mathieu’s equation plotted for different _(a, q)_ -pairs depicted on Fig. 3. The solid blue line
corresponds to the magnetic field _Hy_, the dashed black line – to its derivative _H′y_ . A photonic crystal with certain _(a, q)_ parameters can support surface waves if its solution is decreasing on the crystal scale and its derivative goes to zero to fulfill
the continuity requirement. The inset images a zoomed region of the graph to show that the derivative does go to zero at certain
points (graphs 1-4, 8) or never goes to zero (graphs 6, 7).

What is the largest value of the external medium
dielectric constant ε1 achievable if _Δε_ is limited by the
material properties? To answer this question, we need to
use instead of eq. (17) an asymptotic description of the
same characteristic curve for large negative _a_ values. This
question is not completely trivial, but looking _e.g._, into
Langer’s analysis [25] of the Mathieu’s equation solutions
when at least one of the parameters _a, q_ is large, we found

an estimation of the type _a_ = - | 2 _q_ | +2 | _q_ | + _O_ (ln | _q_ |)

(see his eqs. (6.3) and (7.10) and note the difference in
notation, _a_ = Δ, 2 _q_ = Ω, in his paper), which attests that

asymptotically lim _a_ →−∞ (| 2 _q_ /| _a_ ) = −1, and also this is

known that “no characteristic values exist when | _a_ |≥| 2 _q_ | ”
(p. 690). From this and eq. (15), we immediately infer that
surface electromagnetic waves cannot be obtained if

ε1 >ε _L_ . This means that the external medium cannot have
a refractive index higher than the largest refractive index of
the periodic structure. At the same time, for any ε1 <ε _L_
and large negative _a_, an appropriate photonic crystal
supporting long-range plasmon propagation can be
designed. However, for too small differences of refractive
indices this requires large values of _|a|_ and, consequently,
large periods of the structure which might be impossible to
realize in practice.

ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 3, NOVEMBER 2015

5. Conclusions and discussion

Acknowledgements

In the present paper, the application of the Mathieu’s
equation to excitation of surface waves in photonic crystals
was presented. We have discussed a design of a multilayer
structure coated with a thin metal film to support surface
plasmons, and established the necessary requirements for
the solutions of the Mathieu’s equation describing surface
waves. First, such a solution must vanish at the infinity (or
say, on the scale of the multilayer structure) which
immediately defines that one should work in the instability
regions of the Mathieu’s equation stability chart. Further,
given by eq. (9) the continuity condition of the tangential
electric field component across the interface has to be
fulfilled.

A limit of a thin metal film was considered. Here eq.
(9) reduces to a simple condition that the decaying at
infinity solution of the Mathieu’s equation should have a
zero derivative at the interface. We have briefly analyzed
for which pairs _(a, q)_ belonging to the different instability
regions on the Mathieu’s equation stability diagram this
condition can be fulfilled, and for which cannot (Figs. 3, 4).
Several examples of a real multilayer structure were given,
and limits appearing for design of photonic crystal
structures were discussed. These results constitute a base
for further development of a similar approach, for example
for the cases when the layer thicknesses _dL_ and _dS_ of the
photonic crystal are different. Hill’s differential equation

[23], sometimes named also “generalized Mathieu
equation” is appropriate here.

In conclusion, we would like to draw attention to an
analogy of surface waves in a multilayer structure to a
quantum particle in a periodic potential, which is also
described by the Mathieu’s equation [26]. Essentially, the
Mathieu’s equation, if we rewrite it as

- _d_ 2 _u_ / _dy_ 2 + (− _a_ + 2 _q_ cos(2 _y_ )) _u_ = 0, can be seen as a
particular case of the non-relativistic stationary Schrödinger
equation. This equation (with a proper scaling) describes a
quantum particle in a periodic potential
_V_ ( _y_ ) = 2 _q_ cos(2 _y_ ) = _V_ 0 cos(2 _y_ ) with the total energy of

the particle _E_ = _a_ . For a given potential value, there may
be certain values of _a_ (means _En_ ) for which bounded
solutions (eigenstates) can be found. Such bounded
solutions describe surface waves, which we are interested
in, and they certainly have a zero derivative at ?= 0,

_u_ ('0) = 0, due to the potential symmetry. Obviously, there
is no eigenstates with the energies _E_ smaller than the
minimal potential value _–V0_ and therefore, no bounded
solutions. This analogy is an excellent illustration of the
problem of plasmon excitation: none of the Mathieu’s
equations with a negative parameter _a_ such that _a_ < −2 | _q_ |
is suitable for the excitation of surface waves. Moreover,
this reasoning explains well that for small negative _a_,

_a_
surface waves cannot be excited for - - 1, which is
2 _q_

perfectly in line with the asymptotic behavior of the
parameter _a_ mentioned above.

The research has been fulfilled with the partial financial
support of Swiss National Science Foundation, grant N
200021_150161.

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