Optimal design of photonic crystals 51

Chapter 5
Single-Polarization
Single-Mode Photonic Crystal
Fiber
A waveguide that is single-polarization single-mode (SPSM) is one that guides
only one polarization of the fundamental modes, while the orthogonal polariza-
tion is eliminated. The design objective of the SPSM waveguide is to optimize
the frequency range in which only one mode is guided. Shown in Figure 5.1,
the frequency range, also known as the band width, starts with the frequency
λ
W G
1
(β
L
) when the fundamental mode falls below the light line and becomes
effectively guided , while the second mode still lies inside the light cone, until
the frequency λ
W G
2
(β
R
) at which the second mode eventually becomes guided
when it falls below the light line. This problem is analogous to the band gap
optimization problem of the two-dimensional photonic crystals, but instead of
maximizing the difference between the maximum and minimum frequencies over
the entire irreducible Brillouin zone (a set of k
⊥
), the band width optimiza-
tion attempts to maximize the difference between two frequencies over a certain
range of propagation constants (a set of k
z

). Hence, it is natural to extend the
algorithm developed for the band gap optimization problem, to the design of the
SPSM photonic crystal fiber (PCF) with optimal band width.
In this chapter, we will first state the optimization problem, and then pro-
105
( )
λ β
,1
light line
CL
h
( )
λ β
,1
fundamental
guided mode
WG
h
( )
λ β
,2
high order
guided mode
WG
h
( )
( )
λ β
λ β
=

,2
,1
WG
h R
CL
h R
light cone
propagation constant
β
eigenvalue
λ
β
L
β
max
β
=
min
0
β
R
( )
( )
λ β
λ β
=
,1
,1
WG
h L

CL
h L
( )
λ β
,2CL
h
Figure 5.1: Example of dispersion relation of a three-dimensional photonic crys-
tal fiber. The inset illustrates an example of the cross section of the waveguide.
β
L
and β
R
are the propagation constants at which the first and second waveguide
modes intersect the light line respectively.
pose several formulations,starting with a very intuitive formulation. However,
this first attempt presents some serious limitations, so a second formulation is
proposed in the hope of overcoming the identified issues. A trust region approach
is also incorporated into the optimization formulation to increase the validity of
our model.
5.1 The Optimal Design Problem
The resonance problem of PCF is governed by two eigenvalue problems derived
from the Maxwell equations. The first one models the infinitely periodic photonic
crystal that makes up the cladding of the PCF without defect:
A
CL
h
(ε
CL
, β)u
CL,j

h
= λ
CL,j
h
M
CL
h
u
CL,j
h
, j = 1, . . . , N, β ∈ P
h
, in Ω.
(5.1.1)
The operator A
CL
h
can be assembled according to (3.2.19) over the computation
domain of only one primitive cell Ω with periodic boundary condition, and the
106
dielectric function is determined by ε
CL
(as in Figure 3.17 of subsection 3.2.2).
The second eigenvalue problem models the case when a core is introduced as
the defect to the otherwise periodic photonic crystal cladding:
A
W G
h
([ε
CL

; ε
CO
], β)u
W G,j
h
= λ
W G,j
h
M
W G
h
u
W G,j
h
, j = 1, . . . , N
s
, β ∈ P
h
, in Ω
s
.
(5.1.2)
Similarly, the operator A
W G
h
can be assembled according to (3.2.19) over the
computation domain of the super cell Ω
s
with periodic boundary condition,
and the dielectric function is determined by ε

W G
= [ε
CL
; ε
CO
] (as in Figure
3.17 of subsection 3.2.2). Recall that both operators A
W G
h
and A
CL
h
are affine
with respect to the reciprocal of the dielectric variables: 
−1
i
, i = 1, . . . , n
ε
,(see
(3.2.19)). We introduce a change of variables and define γ as the design variables
of the optimization problem:
γ
i
= 
−1
i
, i = 1, . . . , n
γ
= n
ε

, γ
L
≤ γ
i
≤ γ
H
, γ
L
= 1/
H
, γ
H
= 1/
L
. (5.1.3)
We shall only work with γ for the rest of this chapter. The operators A
CL
h
(γ
CL
, β)
and A
W G
h
([γ
CL
; γ
CO
], β) are now linear in their respective design variables.
5.1.1 Formulation I

The most straightforward strategy to the optimization problem is to directly
maximize the difference of the first two frequencies of the waveguide at the
intersections with the light line (fundamental space filling mode of the periodic
cladding), as illustrated in Figure 5.1. It can be stated mathematically as follows:
max
[γ
CL
;γ
CO
]∈Q
W G
h
λ
W G,2
h
([γ
CL
; γ
CO
], β
R
) − λ
W G,1
h
([γ
CL
; γ
CO
], β
L

) ,
s.t. A
W G
h
([γ
CL
; γ
CO
], β)u
W G,j
h
= λ
W G,j
h
M
W G
h
u
W G,j
h
, β = β
L
, β
R
, j = 1, 2,
A
CL
h
(γ
CL

, β)u
CL,1
h
= λ
CL,1
h
M
W G
h
u
W G,1
h
, β = β
L
, β
R
,
β
L
= arg

λ
W G,1
h

[γ
CL
; γ
CO
], β


= λ
CL,1
h
(γ
CL
, β)

,
β
R
= arg

λ
W G,1
h

[γ
CL
; γ
CO
], β

= λ
CL,1
h
(γ
CL
, β)


.
(5.1.4)
We use arg(f(x) = g(x)) to mean “the argument x
∗
” such that f(x
∗
) = g(x
∗
).
The superscript
W G
on the admissible range Q
h
indicates that the decision vari-
ables being considered are those defining the cross-section of the waveguide (su-
107
per cell Ω
s
), i.e., γ
W G
= [γ
CL
; γ
CO
] ∈ Q
W G
h
. We will also encounter another
admissible range denoted by Q
CL

h
, where the superscript
CL
indicates that the
decision variables being considered are those defining the periodic cladding (only
one primitive unit cell Ω), i.e., γ
CL
∈ Q
CL
h
:= {γ : γ ∈ [γ
L
, γ
H
]
n
γ
CL
}; analo-
gously, γ
CO
∈ Q
CO
h
= {γ : γ ∈ [γ
L
, γ
H
]
n

γ
CO
}.
Besides being a non-convex optimization problem in the design variables, for-
mulation (5.1.4) is also nonlinear due to the design-variable-dependent β
L
and
β
R
. The strategy is therefore to apply the subspace approximation and reduction
algorithm developed in chapter 4 to convert it to a convex semi-definite program,
and to solve the resulting approximate subproblems at the approximately fixed
{

β
L
,

β
R
} range at each linearization. We first rediscretize the propagation con-
stant between {β
L
, β
R
} as
P
h
:=


β | β ∈ {β
1
(= β
L
), . . . , β
n
β
(= β
R
)}

. (5.1.5)
We also prescribe a left-hand limit β
min
= 0, and right-hand limit β
max
for the
propagation constant, whose purpose is related to the computation of β
L
and
β
R
, which will be discussed later. Let us also define two additional variables:
U := max
β∈[β
L
,β
R
]
λ

W G,2
h
([γ
CL
, γ
CO
], β), L := min
β∈[β
1
,β
2
]
λ
W G,1
h
([γ
CL
; γ
CO
], β).
(5.1.6)
108
The original problem (5.1.4) can be rewritten as
P
0
: max
γ
CL
,γ
CO

,u,
U −L ,
s.t. λ
W G,1
h

[γ
CL
; γ
CO
], β

≥ λ
CL,1
h
(γ
CL
, β), ∀β ∈ [β
min
, β
L
],
λ
W G,1
h

[γ
CL
; γ
CO

], β

≤ λ
CL,1
h
(γ
CL
, β), ∀β ∈ [β
L
, β
max
],
λ
W G,2
h

[γ
CL
; γ
CO
], β

≥ λ
CL,1
h
(γ
CL
, β), ∀β ∈ [β
min
, β

R
],
λ
W G,2
h

[γ
CL
; γ
CO
], β

≤ λ
CL,1
h
(γ
CL
, β), ∀β ∈ [β
R
, β
max
],
λ
W G,1
h

[γ
CL
; γ
CO

], β
L

≤ L,
λ
W G,2
h

[γ
CL
; γ
CO
], β
R

≥ U,
γ
L
≤ γ
CL
i
, γ
CO
j
≤ γ
H
, i = 1, . . . , n
γ
CL
, j = 1, . . . , n

γ
CO
,
U, L ≥ 0,
β
L
= arg

λ
W G,1
h

[γ
CL
; γ
CO
], β

= λ
CL,1
h
(γ
CL
, β)

,
β
R
= arg


λ
W G,2
h

[γ
CL
; γ
CO
], β

= λ
CL,1
h
(γ
CL
, β)

.
(5.1.7)
The eigenvalues λ
CL,1
h
and λ
W G,j
h
, j = 1, 2, are to satisfy both (5.1.1) and (5.1.2)
at all times. For succinctness, these eigenvalue-related equality constraints are
not explicitly included in P
0
and all the subsequent formulations. The fifth and

sixth constraints are derived from the monotonicity of λ
W G,j
h
, j = 1, 2.
Depending on various design requirements, we have formulated three different
scenarios: (a) cladding γ
CL
are known, and core γ
CO
are variables; (b) core γ
CO
are known, and cladding γ
CL
are variables; (c) both cladding γ
CL
and core
γ
CO
are variables. Each scenario is described in detail below. For notational
clarity, an overline • is used to denote known quantities, or quantities that are
independent of the design variables; an overhat
ˆ
• is used to denote temporarily
fixed quantities at each linearization.
Scenario a: [γ
CL
; γ
CO
] At the beginning of each linearization, a given


γ
CO
is assumed. Together with the known cladding γ
CL
, the intersections with the
fundamental space-filling mode of the waveguide cladding λ
CL,1
h
can be calculated
109
using various root-finding algorithms (e.g., bisection method
1
):

β
L
= arg

λ
W G,1
h
(γ
CL
, γ
CO
β) = λ
CL,1
h
(γ
CL

, β)

,

β
R
= arg

λ
W G,2
h
(γ
CL
, γ
CO
β) = λ
CL,1
h
(γ
CL
, β)

.
(5.1.8)
We relax (5.1.7) and obtain the following:
max
γ
CO
,U,L
U −L,

s.t. λ
W G,1
h

[γ
CL
; γ
CO
], β

≤ λ
CL,1
h
(γ
CL
, β
j
), j = 1, . . . , n
β
,
λ
W G,2
h

[γ
CL
; γ
CO
], β


≥ λ
CL,1
h
(γ
CL
, β
j
), j = 1, . . . , n
β
,
λ
W G,1
h

[γ
CL
; γ
CO
], β
1

≤ L,
λ
W G,2
h

[γ
CL
; γ
CO

], β
n
β

≥ U,
γ
L
≤ γ
CO
i
≤ γ
H
, i = 1, . . . , n
γ
CO
,
U, L ≥ 0.
(5.1.9)
It is important to note that we eliminated all the constraints associated with
β < β
L
and β > β
R
, because these two impose very tight restriction on the
eigenvalues at the current approximate β
L
and β
R
, and no progress could be
made during the optimization process if they are to be satisfied. It also turns

out the first two constraints are crucial in ensuring the validity of the physical
problem: relaxing these two constraints might lead to various “false optimal”
scenarios. For example, we could get into the trouble of lacking intersections β
L
or β
R
: λ
W G,1
h
and λ
W G,2
h
are well separated but are both below the light line
λ
CL,1
h
for β ∈ [β
1
, β
n
β
]. This is an undesirable physical situation, as the first two
modes of the waveguide become both guided, rather than a single-guided mode.
Next, we introduce the approximate and reduced matrices that eventually
1
The bisection method requires two initial points at which the function evaluations have
opposite signs. In our case, β
min
= 0 and β
max

are used. Although function values are not
guaranteed to be of opposite signs, one of the two limits will be taken as a reasonable “approx-
imate” root in such case.
110
span the subspaces upon which our convex optimization models are built:
Θ
W G
1+a
j
(γ
CL
,

γ
CO
, β
j
) := [Φ
W G
1
(γ
CL
,

γ
CO
, β
j
) | Ψ
W G

a
j
(γ
CL
,

γ
CO
, β
j
)]
:= [u
W G,1
h
(γ
CL
,

γ
CO
, β
j
) | u
W G,2
h
(γ
CL
,

γ

CO
, β
j
), . . . , u
W G,2+a
j
h
(γ
CL
,

γ
CO
, β
j
)],
(5.1.10)
for j = 1, . . . , n
β
. The construction of these matrices has been discussed
in detail in section 4.2.1. Notice that in this problem, the matrix that spans
the lower subspace at any propagation constant is of rank 1, and consists of
precisely only one eigenvector u
W G,1
h
(γ
CL
,

γ

CO
, β
j
); the matrix that spans the
upper subspace at propagation constant β
j
is of dimension a
j
. We obtain the
following equivalent convex formulation with semi-definite inclusions:
P
γ
CO
Ia
: max
γ
CO
,U,L
U − L,
s.t. Φ
W G∗
1
(γ
CL
,

γ
CO
, β
j

)


n
γ
CO
i=1
γ
CO
i
A
W G
h,i
(β
j
) + A
W G
h,0
(γ
CL
, β
j
)
−λ
CL,1
h
(γ
CL
, β
j

)M
W G
h

Φ
W G
1
(γ
CL
,

γ
CO
, β
j
)  0,
j = 1, . . . , n
β
,
Ψ
W G∗
a
j
(γ
CL
,

γ
CO
, β

j
)


n
γ
CO
i=1
γ
CO
i
A
W G
h,i
(β
j
) + A
W G
h,0
(γ
CL
, β
j
)
−λ
CL,1
h
(γ
CL
, β

j
)M
W G
h

Ψ
W G
a
j
(γ
CL
,

γ
CO
, β
j
)  0,
j = 1, . . . , n
β
,
Φ
W G∗
1
(γ
CL
,

γ
CO

, β
1
)


n
γ
CO
i=1
γ
CO
i
A
W G
h,i
(β
1
) + A
W G
h,0
(γ
CL
, β
1
)
−LM
W G
h

Φ

W G
1
(γ
CL
,

γ
CO
, β
1
)  0,
Ψ
W G∗
a
n
β
(
γ
CL
,

γ
CO
, β
n
β
)


n

γ
CO
i=1
γ
CO
i
A
W G
h,i
(β
n
β
) + A
W G
h,0
(γ
CL
, β
n
β
)
−UM
W G
h

Ψ
W G
a
n
β

(γ
CL
,

γ
CO
, β
n
β
)  0,
γ
L
≤ γ
CO
i
≤ γ
H
, i = 1, . . . , n
γ
CO
,
U, L ≥ 0.
(5.1.11)
The discrete operator A
W G
h
([γ
CL
; γ
CO

], β) is decomposed into the γ
CO
−independent
matrix A
W G
h,0
(γ
CL
, β), plus a summation of the γ
CL
− and γ
CO
− independent ma-
trices A
W G
h,i
(β) multiplying each decision variable γ
CO
i
. The mass matrix M
W G
h
is independent of both decision variable and propagation constant. Therefore
P

γ
CO
Ia
is a tractable convex program containing 2n
β

+ 2 semi-definite inclusions,
and 2n
γ
CO
+ 2 linear constraints.
111
Scenario b: [γ
CL
; γ
CO
] When γ
CL
are the design variables and are allowed
to vary, the light line is now design-variable-dependent. We need to introduce
more variables L
j
, U
j
, j = 1, . . . , n
β
such that
λ
W G,1
h
([γ
CL
, γ
CO
], β
j

) ≤ L
j
≤ λ
CL,1
h
(γ
CL
, β
j
), j = 1, . . . , n
β
λ
W G,2
h
([γ
CL
, γ
CO
], β
j
) ≥ U
j
≥ λ
CL,1
h
(γ
CL
, β
j
), j = 1, . . . , n

β
.
(5.1.12)
In addition, more reduced and approximate matrices for the periodic cladding
modes are defined as
Θ
CL
1+b
j
(

γ
CL
, β
j
) := [Φ
CL
1
(

γ
CL
, β
j
) | Ψ
CL
b
j
(


γ
CL
, β
j
)]
:= [u
CL,1
h
(

γ
CL
, β
j
) | u
CL,2
h
(

γ
CL
, β
j
), . . . , u
CL,2+b
j
h
(

γ

CL
, β
j
)],
(5.1.13)
for j = 1, . . . , n
β
. Similar to the subspaces for the waveguide modes, the matrix
that spans the lower subspace of the cladding modes is also of rank 1 at any prop-
agation constant, and consists of precisely only one eigenvector u
CL,1
h
(

γ
CL
, β
j
);
the matrix that spans the upper subspace is of dimension b
j
at propagation
constant β
j
. We obtain the following equivalent convex formulation:
P
γ
CL
Ib
: max

γ
CL
, L
1
, . . . , L
n
β
,
U
1
, . . . , U
n
β
U
n
β
− L
1
s.t. Φ
W G∗
1
(

γ
CL
, γ
CO
, β
j
)



n
γ
CL
i=1
γ
CL
i
A
W G
h,i
(β
j
) + A
W G
h,0
(γ
CO
, β
j
)
−L
j
M
W G
h

Φ
W G

1
(

γ
CL
, γ
CO
, β
j
)  0, j = 1, . . . , n
β
,
Ψ
W G∗
a
j
(

γ
CL
, γ
CO
, β
j
)


n
γ
CL

i=1
γ
CL
i
A
W G
h,i
(β
j
) + A
W G
h,0
(γ
CO
, β
j
)
−U
j
M
W G
h

Ψ
W G
a
j
(

γ

CL
, γ
CO
, β
j
)  0, j = 1, . . . , n
β
,
Φ
CL∗
1
(

γ
CL
, β
j
)


n
γ
CL
i=1
γ
CL
i
A
CL
h,i

(β
j
) + A
CL
h,0
(β
j
)
−L
j
M
CL
h

Φ
CL
1
(

γ
CL
, β
j
)  0, j = 1, . . . , n
β
,
Ψ
CL∗
b
j

(

γ
CL
, β
j
)


n
γ
CL
i=1
γ
CL
i
A
CL
h,i
(β
j
) + A
CL
h,0
(β
j
)
−U
j
M

CL
h

Ψ
CL
b
j
(

γ
CL
, β
j
)  0, j = 1, . . . , n
β
,
γ
L
≤ γ
CL
i
≤ γ
H
, i = 1, . . . , n
γ
CL
,
U
j
, L

j
≥ 0, j = 1, . . . , n
β
.
(5.1.14)
The discrete operator A
CL
h
(γ
CL
, β) can also be decomposed into the γ
CL
112
independent matrix A
CL
h,0
(β), plus a summation of the γ
CL
independent matri-
ces A
CL
h,i
(β) multiplying each decision variable γ
CL
i
. The mass matrix M
CL
h
is
independent of both decision variable and propagation constant too as M

W G
h
.
Compared to P

γ
CO
Ia
, program P

γ
CL
Ib
contains 2n
β
more semi-definite inclusions
to account for the light line mode.
Scenario c: [γ
CL
; γ
CO
] The extension from scenario (b) to (c), where both
γ
CL
and γ
CO
are the design variables, is minor with the inclusion of γ
CO
being
part of the decision variables:

P
γ
CL
,γ
CO
Ic
: max
γ
CL
, L
1
, . . . , L
n
β
,
γ
CO
, U
1
, . . . , U
n
β
U
n
β
− L
1
,
s.t. Φ
W G∗

1
(

γ
CL
,

γ
CO
, β
j
)


n
γ
CL
i=1
γ
CL
i
A
W G
h,i
(β
j
) +

n
γ

CO
i=1
γ
CO
i
A
W G
h,n
CO
γ
+i
(β
j
)
+A
W G
h,0
(β
j
) − L
j
M
W G
h

Φ
W G
1
(


γ
CL
,

γ
CO
, β
j
)  0, j = 1, . . . , n
β
,
Ψ
W G∗
a
j
(

γ
CL
,

γ
CO
, β
j
)


n
γ

CL
i=1
γ
CL
i
A
W G
h,i
(β
j
) +

n
γ
CO
i=1
γ
CO
i
A
W G
h,n
CO
γ
+i
(β
j
)
+A
W G

h,0
(β
j
) − U
j
M
W G
h

Ψ
W G
a
j
(

γ
CL
,

γ
CO
, β
j
)  0, j = 1, . . . , n
β
,
Φ
CL∗
1
(


γ
CL
, β
j
)


n
γ
CO
i=1
γ
CO
i
A
CL
h,i
(β
j
) + A
CL
h,0
(β
j
)
−L
j
M
CL

h

Φ
CL
1
(

γ
CL
, β
j
)  0, j = 1, . . . , n
β
,
Ψ
CL∗
b
j
(

γ
CL
, β
j
)


n
γ
CO

i=1
γ
CO
i
A
CL
h,i
(β
j
) + A
CL
h,0
(β
j
)
−U
j
M
CL
h

Ψ
CL
b
j
(

γ
CL
, β

j
)  0, j = 1, . . . , n
β
,
γ
L
≤ γ
CL
i
, γ
CO
j
≤ γ
H
, i = 1, . . . , n
γ
CL
, j = 1, . . . , n
γ
CO
,
U
j
, L
j
≥ 0, j = 1, . . . , n
β
.
(5.1.15)
Program P


γ
CL
,

γ
CO
Ic
has the most decision variables and constraints. Compared
to P

γ
CL
Ib
, the increase is n
γ
CO
more decision variables, and 2n
γ
CO
more linear
inequality constraints.
Despite being the most direct formulation of the optimization problem, for-
mulation I has some fundamental drawbacks:
• Unlike the previous band gap optimization problem, where a dimensionless
quantity gap-midgap ratio is modeled as the objective function, one must
have noticed that the absolute band width is chosen instead as the objective
function in this formulation. This is caused by the possibility of L (or L
1
in

scenarios (b) and (c)) being zero, in which case the objective value would
have turned out to be unity whatever value U or (U
n
β
) takes. Thus it
defeats the purpose of band width optimization.
113
• Probably the most serious caveat with this formulation lies in the diffi-
culty of accurate computation of the intersections β
L
and β
R
(or the corre-
sponding eigenfrequencies). As explained before, when a waveguide mode
is guided, it decays exponentially away from the core into the cladding.
This modal diameter increases rapidly with wavelength, i.e., when the fre-
quency approaches the light line, the transverse decay rate slows down.
In fact, it is explained in [38] that the modal diameter seems to increase
exponentially with the wavelength. Given that any real structure has a
finite cladding, this makes it difficult, both numerically and experimen-
tally, to study the long-wavelength regime, especially the behavior of the
fundamental guided mode (λ
W G,1
h
) as it approaches the less than rigor-
ously defined intersection β
L
with light line (λ
CL,1
h

). The situation for the
second guided mode (λ
W G,2
h
) becomes more subtle if it has a cut-off with
the light line away from long-frequency. One can loosely view λ
W G,2
h
as
a perturbed mode to the second light line, λ
CL,2
h
(a degenerate light line
if the cladding symmetry had not been broken). By the same token, it is
hard to capture the regime when λ
W G,2
h
is approaching λ
CL,2
h
with finite
cladding, even if more sophisticated boundary condition treatment had
been prescribed, e.g., perfect matching layer. However, β
R
is defined as
the intersection of the seconded guided mode with the first light line. So
as long the degeneracy of two light lines is broken, β
R
can be numerically
computed.

This brings us to the next formulation, where we avoid computing β
L
by
picking a fixed propagation constant β
2
.
5.1.2 Formulation II
In formulation II, we start with a prescribed propagation constant β
2
, and try
to optimize the corresponding frequency difference between the light line and
the fundamental guided mode while requiring the second mode to be above the
light line. The band width where only one mode is guided is defined below as a
114
dimensionless ratio for the band gap optimization problem:
J
β
2
h
(γ
CL
, γ
CO
) =
min{λ
CL,1
h
(γ
CL
, β

2
), λ
W G,2
h

[γ
CL
; γ
CO
], β
2

} − λ
W G,1
h

[γ
CL
; γ
CO
], β
2

min{λ
CL,1
h
(γ
CL
, β
2

), λ
W G,2
h

[γ
CL
; γ
CO
], β
2

} + λ
W G,1
h

[γ
CL
; γ
CO
], β
2

.
(5.1.16)
Note that λ
W G,1
h

[γ
CL

; γ
CO
], β
2

= 0, for β
2
= 0, thus it is a well defined
objective function. This concept is illustrated in Figure 5.2. The goal is to
( )
λ β
,1
fundamental
guided mode
WG
h
( )
λ β
,2
high order
guided mode
WG
h
light cone
propagation constant
β
β
1
β
2

L
1
U
( )
2
or U U
( )
λ β
,1
light line
CL
h
( )
λ β
,2
CL
h
eigenvalue
λ
Figure 5.2: Illustration of formulation II for the optimal design of single-mode
single-polarization photonic crystal fiber. The propagation constant β
2
is pre-
scribed, while β
1
is a computed quantity.
max
γ
CL
,γ

CO
J
β
2
h
(γ
CL
, γ
CO
).
115
This can be described as
P
1
: max
γ
CL
,γ
CO
,u,
U−L
U+L
,
s.t. λ
W G,1
h

[γ
CL
; γ

CO
], β
2

≤ L,
λ
W G,2
h

[γ
CL
; γ
CO
],
β
2

≥ L,
λ
CL,1
h
(γ
CL
, β
2
) ≥ U,
λ
W G,2
h


[γ
CL
; γ
CO
], β
2

≥ λ
CL,1
h
(γ
CL
, β
2
),
λ
W G,1
h
([γ
CL
; γ
CO
], β
1
) ≤ L,
λ
W G,2
h
([γ
CL

; γ
CO
], β
1
) ≥ L,
γ
L
≤ γ
CL
i
, γ
CO
j
≤ γ
H
, i = 1, . . . , n
γ
CL
, j = 1, . . . , n
γ
CO
,
U, L ≥ 0,
β
1
= arg

λ
CL,1
h

(γ
CL
, β) = λ
W G,1
h
([γ
CL
; γ
CO
], β
2
)

.
(5.1.17)
Here U and L are two auxiliary variables indicating the lower bound of the
light line and upper bound of the fundamental guided mode at β
2
. The second
waveguide mode is also required to be unguided at β
2
, and at β
1
as a tighter
constraint. β
1
is defined as the propagation constant at which the light line
takes the upper bound value of λ
W G,1
h

at β
2
, and it should be that β
1
< β
2
. This
translates to the fifth inequality in P
1
.
We again formulate the programs into three different scenarios depending
on the various design requirements. Same notations on prescribed and given
quantities are used as formulation I, scenarios (a), (b), and (c).
Scenario a: [γ
CL
; γ
CO
] Assuming a prescribed cladding γ
CL
throughout, and
a given core

γ
CO
at the beginning of each linearization,

β
1
defined as


β
1
= arg

λ
CL,1
h
(γ
CL
, β) = λ
W G,1
h
(γ
CL
,

γ
CO
, β
2
)

,
116
can be computed using root-finding algorithms (e.g., the bisection method is
chosen in our implementation). P
1
is relaxed to
max
γ

CO
,L
U −L
U + L
,
s.t. λ
W G,1
h
(γ
CL
, γ
CO
,

β
1
) ≤ L ,
λ
W G,2
h
(γ
CL
, γ
CO
,

β
1
) ≥ L ,
λ

W G,1
h
(γ
CL
, γ
CO
, β
2
) ≤ L ,
λ
W G,2
h
(γ
CL
, γ
CO
, β
2
) ≥ U ,
γ
L
≤ γ
CO
j
≤ γ
H
, j = 1, . . . , n
γ
CO
,

L ≥ 0.
(5.1.18)
Here U is a constant, and can be computed with given γ
CL
and β
2
as
U = λ
LC,1
h
(γ
CL
, β
2
).
The approximate and reduced matrices for the waveguide modes are defined as
Θ
W G
1+a
j
(γ
CL
,

γ
CO
, β
j
) := [Φ
W G

1
(γ
CL
,

γ
CO
, β
j
) | Ψ
W G
a
j
(γ
CL
,

γ
CO
, β
j
)]
:= [u
W G,1
h
(γ
CL
,

γ

CO
, β
j
) | u
W G,2
h
(γ
CL
,

γ
CO
, β
j
), . . . , u
W G,2+a
j
h
(γ
CL
,

γ
CO
, β
j
)],
j = 1, 2.
(5.1.19)
117

The equivalent convex formulation with semi-definite inclusions can be written as
P
γ
CO
IIa
max
γ
CO
,L
:
U−L
U+L
s.t. Φ
W G∗
1
(γ
CL
,

γ
CO
,

β
1
)


n
γ

CO
i=1
γ
CO
i
A
W G
h,i
(

β
1
) + A
W G
h,0
(γ
CL
,

β
1
)
−LM
W G
h

Φ
W G
1
(γ

CL
,

γ
CO
,

β
1
)  0,
Ψ
W G∗
a
1
(γ
CL
,

γ
CO
,

β
1
)


n
γ
CO

i=1
γ
CO
i
A
W G
h,i
(

β
1
) + A
W G
h,0
(γ
CL
,

β
1
)
−LM
W G
h

Ψ
W G
a
j
(γ

CL
,

γ
CO
,

β
1
)  0,
Φ
W G∗
1
(γ
CL
,

γ
CO
, β
2
)


n
γ
CO
i=1
γ
CO

i
A
W G
h,i
(β
2
) + A
W G
h,0
(γ
CL
, β
2
)
−LM
W G
h

Φ
W G
1
(γ
CL
,

γ
CO
, β
2
)  0,

Ψ
W G∗
a
2
(γ
CL
,

γ
CO
, β
2
)


n
γ
CO
i=1
γ
CO
i
A
W G
h,i
(β
2
) + A
W G
h,0

(γ
CL
, β
2
)
−UM
W G
h

Ψ
W G
a
2
(γ
CL
,

γ
CO
, β
2
)  0,
γ
L
≤ γ
CO
i
≤ γ
H
, i = 1, . . . , n

γ
CO
,
L ≥ 0.
(5.1.20)
The stiffness matrices can again be expressed as a linear combination of the variable independent
parts, while the decision variables are the coefficients. P
γ
CO
IIa
is a convex program containing
four SDP inclusions, and 2n
γ
CO
+ 2 linear inequalities. This is a drastic reduction in terms of
the number of constraints as compared to those in formulation I.
118
Scenario b: [γ
CL
; γ
CO
] When core γ
CO
is fixed, and cladding γ
CL
takes on a given
value

γ
CL

at the beginning of each linearization, we can compute

β
1
= arg

λ
CL,1
h
(

γ
CL
, β) = λ
W G,1
h
(

γ
CL
, γ
CO
, β
2
)

.
The light line is now variable dependent, and we need to introduce two auxiliary variables as
its upper bound U
1

and lower bound U
2
at β
2
. To make sure the second waveguide mode is
not guided, we require it to be above the upper bound of the light line at β
2
, i.e.,
λ
W G,1
h
≤ L ≤ U
2
≤ λ
CL,1
h
≤ U
1
≤ λ
W G,2
h
.
These translate to the following formulation:
max
γ
CL
,L,U
1
,U
2

U
2
−L
U
2
+L
,
s.t. λ
W G,1
h
(γ
CL
, γ
CO
,

β
1
) ≤ L ,
λ
W G,2
h
(γ
CL
, γ
CO
,

β
1

) ≥ L ,
λ
W G,1
h
(γ
CL
, γ
CO
, β
2
) ≤ L ,
λ
W G,2
h
(γ
CL
, γ
CO
, β
2
) ≥ U
1
,
λ
CL,1
h
(γ
CL
, β
2

) ≤ U
1
,
λ
CL,1
h
(γ
CL
, β
2
) ≥ U
2
,
γ
L
≤ γ
CL
i
≤ γ
U
, i = 1, . . . , n
γ
CL
,
L, U
1
, U
2
≥ 0.
(5.1.21)

The second inequality ensures the unguidedness of λ
W G,2
h
at β
1
by remaining above the light
line. In addition to the matrices defined in II(a), more approximate and reduced matrices are
necessary for the cladding modes:
Θ
CL
1+b
2
(

γ
CL
, β
2
) := [Φ
CL
1
(

γ
CL
, β
2
) | Ψ
CL
b

2
(

γ
CL
, β
2
)]
:= [u
CL,1
h
(

γ
CL
, β
2
| u
CL,1
h
(

γ
CL
, β
2
), . . . , u
W G,1+b
2
h

(

γ
CL
, β
2
)].
(5.1.22)
With these matrices, we can rewrite (5.1.21) into a convex program of the following form:
119
P
γ
CL
IIb
: max
γ
CL
,L,U
1
,U
2
U
2
−L
U
2
+L
s.t. Φ
W G∗
1

(

γ
CL
, γ
CO
,

β
1
)


n
γ
CL
i=1
γ
CL
i
A
W G
h,i
(

β
1
) + A
W G
h,0

(γ
CO
,

β
1
)
−LM
W G
h

Φ
W G
1
(

γ
CL
, γ
CO
,

β
1
)  0,
Ψ
W G∗
a
1
(


γ
CL
, γ
CO
,

β
1
)


n
γ
CL
i=1
γ
CL
i
A
W G
h,i
(

β
1
) + A
W G
h,0
(γ

CO
, β
1
)
−LM
W G
h

Ψ
W G
a
j
(

γ
CL
, γ
CO
,

β
1
)  0,
Φ
W G∗
1
(

γ
CL

, γ
CO
, β
2
)


n
γ
CL
i=1
γ
CL
i
A
W G
h,i
(β
2
) + A
W G
h,0
(γ
CO
, β
2
)
−LM
W G
h


Φ
W G
1
(

γ
CL
, γ
CO
, β
2
)  0,
Ψ
W G∗
a
2
(

γ
CL
, γ
CO
, β
2
)


n
γ

CL
i=1
γ
CL
i
A
W G
h,i
(β
2
) + A
W G
h,0
(γ
CO
, β
2
)
−U
1
M
W G
h

Ψ
W G
a
2
(


γ
CL
, γ
CO
, β
2
)  0,
Φ
CL∗
1
(

γ
CL
, β
2
)


n
γ
CL
i=1
γ
CL
i
A
CL
h,i
(β

2
) + A
CL
h,0
(γ
CL
, β
2
)
−U
1
M
CL
h

Φ
CL
1
(

γ
CL
, β
2
)  0,
Ψ
CL∗
b
2
(


γ
CL
, β
2
)


n
γ
CL
i=1
γ
CL
i
A
CL
h,i
(β
2
) + A
CL
h,0
(γ
CL
, β
2
)
−U
2

M
CL
h

Ψ
CL
a
2
(

γ
CL
, β
2
)  0,
γ
L
≤ γ
CL
i
≤ γ
H
, i = 1, . . . , n
γ
CL
,
L, U
1
, U
2

≥ 0.
(5.1.23)
P
γ
CL
IIb
includes two additional decision variables U
1
and U
2
, as well as two more SDP inclusions
accounting for the light line modes, compared to P
γ
CO
IIa
in which only the cladding region is to
be designed and optimized.
Scenario c: [γ
CL
; γ
CO
] Designing both the core and the cladding does not require
significantly more effort than the design when only the cladding is considered. First,

β
1
can be
computed almost the same way as before:

β

1
= arg

λ
CL,1
h
(

γ
CL
, β) = λ
W G,1
h
(

γ
CL
,

γ
CO
, β
2
)

.
The formulation is almost the same as (5.1.21), except that both γ
CL
, and γ
CO

are decision
120
variables:
max
γ
CL
,γ
CO
,L,U
1
,U
2
U
2
− L
U
2
+ L
,
s.t. λ
W G,1
h
(γ
CL
, γ
CO
,

β
1

) ≤ L ,
λ
W G,2
h
(γ
CL
, γ
CO
,

β
1
) ≥ L ,
λ
W G,1
h
(γ
CL
, γ
CO
, β
2
) ≤ L ,
λ
W G,2
h
(γ
CL
, γ
CO

, β
2
) ≥ U
1
,
λ
CL,1
h
(γ
CL
, β
2
) ≤ U
1
,
λ
CL,1
h
(γ
CL
, β
2
) ≥ U
2
,
γ
L
≤ γ
CL
i

, γ
CO
j
≤ γ
H
, i = 1, . . . , n
ε
CL
, j = 1, . . . , n
ε
CO
,
L, U
1
, U
2
≥ 0.
With all the approximate and reduced matrices defined in both scenarios (a) and (b), we
derive the following convex program for scenario (c):
P
ˆ
γ
CL
,
ˆ
γ
CO
IIc
: max
γ

CL
, γ
CO
,
L, U
1
, U
2
U
2
−L
U
2
+L
,
s.t. Φ
W G∗
1
(

ε
CL
,

ε
CO
, β
1
)



n
ε
CL
i=1
γ
CL
i
A
W G
h,i
(β
1
) +

n
ε
CO
i=1
γ
CO
i
A
W G
h,n
CO
ε
+i
(β
1

)
+A
W G
h,0
(β
1
) − LM
W G
h

Φ
W G
1
(

ε
CL
,

ε
CO
, β
1
)  0,
Φ
W G∗
1
(

ε

CL
,

ε
CO
, β
2
)


n
ε
CL
i=1
γ
CL
i
A
W G
h,i
(β
2
) +

n
ε
CO
i=1
γ
CO

i
A
W G
h,n
CO
ε
+i
(β
2
)
+A
W G
h,0
(β
2
) − LM
W G
h

Φ
W G
1
(

ε
CL
,

ε
CO

, β
2
)  0,
Ψ
W G∗
a
1
(

ε
CL
,

ε
CO
, β
1
)


n
ε
CL
i=1
γ
CL
i
A
W G
h,i

(β
1
) +

n
ε
CO
i=1
γ
CO
i
A
W G
h,n
CO
ε
+i
(β
1
)
+A
W G
h,0
(β
1
) − LM
W G
h

Ψ

W G
a
1
(

ε
CL
,

ε
CO
, β
1
)  0,
Ψ
W G∗
a
2
(

ε
CL
,

ε
CO
, β
2
)



n
ε
CL
i=1
γ
CL
i
A
W G
h,i
(β
2
) +

n
ε
CO
i=1
γ
CO
i
A
W G
h,n
CO
ε
+i
(β
2

)
+A
W G
h,0
(β
2
) − U
1
M
W G
h

Ψ
W G
a
2
(

ε
CL
,

ε
CO
, β
2
)  0,
Φ
CL∗
1

(

ε
CL
, β
2
)


n
ε
CO
i=1
γ
CO
i
A
CL
h,i
(β
2
) + A
CL
h,0
(β
2
)
−U
1
M

CL
h

Φ
CL
1
(

ε
CL
, β
2
)  0,
Ψ
CL∗
b
2
(

ε
CL
, β
2
)


n
ε
CO
i=1

γ
CO
i
A
CL
h,i
(β
2
) + A
CL
h,0
(β
2
)
−U
2
M
CL
h

Ψ
CL
b
2
(

ε
CL
, β
2

)  0,
γ
L
≤ γ
CL
i
, γ
CO
j
≤ γ
H
, i = 1, . . . , n
γ
CL
, j = 1, . . . , n
γ
CO
,
L, U
1
, U
2
≥ 0.
As one can see, the only addition in P
ˆ
γ
CL
,
ˆ
γ

CO
IIc
is the inclusion of the decision variables γ
CL
representing the cladding design, as well as the upper and lower bounds on these variables.
While scenarios (c) in both formulations I and II comprise the most decision variables and con-
121
straints, their extension from the simple scenarios (a) and (b) are natural and simply minimal;
moreover, they provide more flexible design needs.
5.1.3 Trust region
Computing a solution to a nonlinear and non-convex problem is nontrivial. Although the SDP
and linear relaxation introduced via our algorithm have proven to be efficient for the band gap
optimization of the two-dimensional photonic crystal, the linearization and approximation did
not seem to be sufficiently reliable in this band width optimization problem with an underlying
quasi-three-dimensional vectorial PDE.
The idea behind a trust region method is very simple. A bound is levied on the step size
of the solution to an approximate subproblem
B
k
s
k
 ≤ ∆
k
,
where B
k
is a scaling matrix and ∆
k
is a positive scalar representing the trust region size.
The step size between two sequential optimal solutions is denoted by s

k
. In our problem, the
step size can be represented by the difference between the optimal solution and the linearizer,
i.e.,γ −

γ. Given the lower bound γ
L
and the upper bound γ
H
on the γ-related decision
variables, we propose the following two trust region methods.
Method I:
1
n
γ
n
γ

j=1

γ
j
− γ
j
γ
j
− γ
L

2

+
n
γ

j=1

γ
j
− γ
j
γ
H
− γ
j

2
≤ ∆, (5.1.24)
which gives a diagonal scaling matrix satisfying
B
2
j,j
= D
j,j
=
1
n
γ

1
γ

j
− γ
L

2
+

1
γ
H
− γ
j

2
. (5.1.25)
If we examine this method I, it is not hard to realize that it would limit the solution strictly
inside the feasible region, and away from the boundary faces or vertices. This is due to the
implicit non-zero requirement on the denominators. In practical photonic crystal design how-
ever, one wishes to achieve a solution comprising of either low or high dielectric materials. In
other words, the solution should always be on the faces or vertices of the polyhedron defining
the feasible region. This brings us to the following modified method II.
Method II:
1
n
γ
n
γ

j=1


γ
j
− γ
j
γ
H
− γ
L

2
≤ ∆, (5.1.26)
which gives another diagonal scaling matrix satisfying
B
2
j,j
= D
j,j
=
1
n
γ

1
γ
H
− γ
L

2
. (5.1.27)

122
We rewrite it in matrix-vector multiplication form
(γ −

γ)
∗
D(γ −

γ) = (γ −

γ)
∗
B
∗
B(γ −

γ) ≤ ∆, (5.1.28)
and simplify the notation
γ
∗
Dγ + χγ + β ≤ ∆, (5.1.29)
where χ = −2

γ
∗
D, and σ =

γ
∗
D


γ − ∆. Finally, we define a second-order cone K
Q
:= {x =
[x
0
;
¯
x] ∈ R
n
γ
+2
: x
0
≥ 
¯
x}, and introduce a new variable x = [x
0
;
¯
x] ∈ K
Q
such that
























−


x
0
¯
x



Q
0,
¯
x =



Bγ
1
2
(χ
∗
γ + σ + 1)


,
x
0
=
1
2
(1 − χ
∗
γ − σ).
(5.1.30)
These new decision variables as well as the new constraints are to be incorporated to the
previous formulations. Effectively, there will be n
γ
+ 2 additional decision variables, one more
SCOP inclusion, and n
γ
+ 2 extra equalities (or 2(n
γ
+ 2) equivalent inequalities).
5.2 Results and Discussion

5.2.1 Model setup
We consider a three-dimensional photonic crystal with z-invariant cross-section set up schemati-
cally according to Figure 3.17, on either a rectangular lattice with pitch distance Λ
x
a = 1.5a = 3
and Λ
y
a = a = 2, or a rhombic lattice with lattice constant a = 2. The primitive cell domain
Ω is decomposed into a uniform grid of mesh size a/40, and the super cell domain Ω
s
is de-
composed into a uniform grid of the same mesh size, but of more elements. The cladding
consists of two rings of Ω surrounding the core, hence, the total number of elements in Ω is
40 × 60 = 2, 400, and in Ω
s
is 2, 400 × 25 = 60, 000 in a rectangular lattice; the total number of
elements in Ω is 40 × 40 = 1, 600, and in Ω
s
is 1, 600 × 25 = 40, 000 in a rhombic lattice setup.
We use the same dielectric materials as in section 3.2.3 to construct the waveguide, i.e.,
epoxy of 
L
= 2.25, and silicon carbide of 
H
= 7.02. Note that since both materials are solid
at room temperature, connectivity is no longer a concern in our formulations. The periodic
cladding in the rectangular lattice has cylinders of material 
L
of radius 0.485a 
L

in the
background of material 
H
; while the periodic cladding in the rhombic lattice has elliptical
holes of material 
L
of major axis length 0.485a and minor axis 0.194a in the background of
material 
H
. The initial configurations are chosen simply as a periodic photonic crystal with
one defect, i.e., an 
H
−filled circular hole in the rectangular lattice , and a 
H
− filled elliptical
hole in the rhombic lattice.
The trust region, if chosen appropriately, provides us with the confidence of the approx-
123
imations made in each run, and the effectiveness of the optimal solution. If we go back and
examine, for example, the third inequality constraint in (5.1.17), we really wish it is never
active. In other words, the progress in each run should not be too radical to violate this in-
equality, and to avoid the second mode being also guided. As a rule of thumb, the size of the
trust region is chosen to be ∆ = 1%, which is shown to be sufficient in our implementation.
5.2.2 Optimal structures
Formulation I
Despite it being unsuccessful, we would like to briefly report the results obtained with formu-
lation I for completeness, and attempt an intuitive explanation for its misfortune.
In Figure 5.3 we can examine the evolution of the optimization process of a sample problem,
constructed with epoxy cylinders of radius 0.3a (in amber color) in silicon carbide background
(in black color) on a rectangular lattice (Λ

x
a = 1.5a, Λ
y
a = a). The starting configuration
is shown at the top left corner. It has been shown ([38]) that in this case, the first guided
mode is cut-off free, and is asymptotically close to the light line at the long wavelength limit,
or, the left intersection β
L
= 0. The second guided mode has a cut-off with the light line
at β
R
> 0. To achieve wider band width, intuitively, β
R
should be pushed further to the
right thanks to the monotonicity of the dispersion relations. Shown in Figure 5.3, as the
optimization progresses, the optimal structure approaches a homogeneous configuration of the
low dielectric material, which leads to the degeneracy of the light lines. At the same time,
the second guided mode is approaching the (degenerated) light line closer from above, and
becoming a cut-off free mode (β
R
→ 0), while the fundamental guided mode is approaching
from below. Numerically however, the first guided mode remains cut-off free: β
L
= 0 while it
is asymptotically approaching the light line at β
L
; the second mode remains a false cut-off β
R
,
and it is asymptotically approaching the light line at an increasing β

R
. Hence it is the increase
of β
R
that is pushing the band width wider. It is also the failure of computing a physical β
R
that caused the overshoot of the objective value in the last panel of Figure 5.3. Recall that in
the root finding procedure, an artificial maximum limit β
max
is imposed on the propagation
constant. β
max
is set to 6π in our implementation for practical reasons. The last panel of
Figure 5.3 corresponds to the situation when β
R
= β
max
, i.e., the bisection search procedure
failed to compute a root within the given search range [β
min
, β
max
] = [0, 6π].
Formulation II
In this section, we will demonstrate some optimal structures obtained via formulation II, and
validate the solutions by examining the confinement of the field variables. Recall that the design
objective is to find an optimal frequency range, or band width (in terms of a dimensionless
ratio), such that the fundamental mode of the waveguide is guided, or it lies below the light
line, while the second order mode is un-guided, or, above the light line. In other words, we
124

1 2 3 4 5 6 7 8 9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
run
Width = U−L
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
0
2

4
6
8
10
12
14
0 10 20 30 40 50 60
0
50
100
150
200
250
300
350
400
Width = U−L
Width = U−L
Width = U−L
Width = U−L
run
run
run
run
Figure 5.3: Evolution of the optimization process based on P
Ic
(5.1.15) with the
initial configuration shown at the top left corner. The objective value: band
width U −L (vertical axis) v.s. the number of runs (horizontal axis) are plotted.
The insets show the current optimal structure at each run: run = 10, 20, 40, 50,

52.
125
would like to achieve a situation where the field variable (closely related to an eigenfunction)
corresponding to the fundamental eigenvalue is highly confined within the core, and decays
exponentially away into the cladding, while the second eigenfunction is attenuated inside Ω
s
,
at least within the propagation constant range [
ˆ
β
1
,
β
2
]. Keep in mind that in this formulation,
β
2
is prescribed and
ˆ
β
1
is approximated at each linearization.
For the purpose of illustration, the intensity of the field variable in the following plots is
calculated as
u(x, y) = u
x
(x, y)
∗
u
x

(x, y) + u
y
(x, y)
∗
u
y
(x, y) + u
z
(x, y)
∗
u
z
(x, y).
In most of the simulations we performed, it seems that the optimization of the core is
the least interesting case, especially when the core consists of only one primitive cell Ω. It is
almost certain that the optimal scenario is when the core is filled up with the high dielectric
material 
H
. That is if one were to start with an initial configuration of a random distribution
of the dielectric materials 
H
and 
L
, it would eventually converge to a uniform 
H
core at
optimum. Nevertheless, the situation gets interesting when we allow a bigger design region
by enlarging the core to include 3 × 3 primitive cells Ω. Shown in the top left corner of
Figure 5.4, the initial configuration is one with a uniform 
H

core of 3 × 3 Ω, surrounded by
2 rings of Ω as the cladding. The final optimal core (top right corner) has an X−shaped 
H
region. The initial (left) and optimal (right) dispersion relations in the second row of Figure
5.4 clearly illustrate the movement of the bands as a result of the optimization. In particular,
before optimization, the second waveguide mode λ
W G,2
h
was a guided mode below the light
line λ
CL,1
h
between [
ˆ
β
1
, β
2
], which can be clearly observed from the last two intensity plots in
the left column. After optimization, the second waveguide mode λ
h
W G, 2 is now above the
light line λ
CL,1
h
between [
ˆ
β
1
, β

2
] with a clear intersection at β
2
(which indicates that the forth
constraint in (5.1.18) is active). The last two intensity plots in the right column of Figure 5.4
also demonstrate the attenuation of the corresponding field variables. At the same time, the
confinement of the first guided mode has been enhanced thanks to the optimization, as shown
in the intensity plots in the third and forth rows of Figure 5.4.
The situation in the rectangular lattice however, is not quite as positive. The observation
of the core optimization in a rectangular lattice when the core consists of one Ω, is the same as
that of the rhombic lattice: the filling with 
H
is the optimized structure. However, if the core
region is enlarged to include 3×3 Ω, no positive band width is obtained. The results of cladding
and cladding+core optimizations are shown in Figure 5.5 in which β
2
= 1, and in Figure 5.6
in which β
2
= 2. In each case, the initial configuration is shown in the left column, including
the initial crystal structure (top), the dispersion relation (second row), the first eigenfunction
intensity at
ˆ
β
1
and β
2
respectively (third and forth rows), as well as the second eigenfunction
intensity at
ˆ

β
1
and β
2
respectively (fifth and last rows). The middle column illustrates the
optimal configuration when only the cladding is the design region. Note that at the optimum,
the value of
ˆ
β
1
is different and often smaller than the initial value. The last column on the
126
0.085
0.095
0.105
0.115
0.125
β
1
=0.74
β
2
=0.8


λ
h
CL,1
λ
h

WG,1
λ
h
WG,2
^
_
J
h
= 0.0063
0.085
0.095
0.105
0.115
0.125
β
1
=0.75
β
2
=0.8


^
_
J
h
= 0.060
||H
2
(β

1
)||
2
−
||H
2
(β
1
)||
2
^
||H
2
(β
2
)||
2
−
||H
1
(β
1
)||
2
^
Figure 5.4: β
2
= 0.8. The figures in the columns from left to right each represent
the case: (left)initial configuration, with band width J
h

= 0.0063 (
ˆ
β
1
= 0.74);
(right) optimal configuration when only core is optimized, with band width J
h
=
0.060 (
ˆ
β
1
= 0.75). The first row illustrates the PCF cross-sections. The second
row shows the corresponding dispersion relations between [
ˆ
β
1
, β
2
]. From the
third row to the last, the magnetic field intensities at both
ˆ
β
1
and β
2
are shown
in 3D surf plots.
127
right hand side contains the optimal configuration if both the cladding and the core, i.e., the

whole cross-section of the waveguide is optimized. Again, the final value of
ˆ
β
1
is even smaller
than the middle column, which allows one a wider range of operation. In both figures, we can
observe a clear improvement of the band width, and the desired movement of the waveguide
modes: first mode (λ
W G,1
h
) being below the light line, while the second mode (λ
W G,2
h
) being
shifted above the light line. However, it is not hard to notice the contradictions in the intensity
plots below. In Figure 5.5, besides exhibiting weak confinement strength of the fundamental
guided mode after optimization, the second waveguide mode failed to get attenuated. At a
larger propagation constant β
2
= 2 in Figure 5.6 we can observe a slight improvement. In the
last column, the optimized structure provided better confinement for the fundamental mode,
attenuated the second mode at
ˆ
β
1
= 1.7807, and weakened its localization at β
2
= 2.
Finally we return to the rhombic lattice and the optimization of the cladding, and the
cladding+core. Shown in Figures 5.7 and 5.8 are the cases of β

2
= 0.8, and β
2
= 1.5 in a
rhombic lattice respectively, with the same arrangement of the subplots as in the previous
figures. In both Figures 5.7 and 5.8, the band widths have clearly been improved after the
cladding optimization, and drastically improved after the cladding+core optimization. The
confinement of the first mode and the attenuation of the second mode shown in the intensity
subplots can further validate the desired results.
Based on all these results, several crucial observations can be made:
• Formulation I can generate infeasible crystal structures due to the numerical difficulty of
computing accurate intersections β
L
and β
R
. This has been predicted theoretically and
also shown experimentally. Formulation II is proposed as a remedy to address the flaws
of formulation I, and it is shown to be a reasonable model for the purpose of optimal
design of the single-mode single-polarization photonic crystal fiber.
• Rectangular lattices setup as in our simulations did not provide satisfactory physical
confinement and attenuation of the field variables, despite the numerically optimized
dispersion relations. Photonic crystal fibers set on the rhombic lattice on the other
hand, have demonstrated both optimal dispersion relations, as well as desirable fields
confinement and attenuation. They are shown to be favorable candidates for the SPSM
photonic crystal fibers. Interestingly enough, this observation is consistent with the
band gap optimization problems in chapter 4, in which the crystal structures with the
largest band gaps are also obtained in the hexagonal lattice (a special case of the rhombic
lattice).
5.3 Conclusions
In this chapter, we studied the band width optimization problem arising in the photonic crystal

fibers due to the index guiding mechanism. Applying the eigenvalue design concept devel-
oped previously for the band gap optimization problem, we proposed several formal convex
128
λ
h
CL,1
λ
h
WG,1
λ
h
WG,2
0.2
0.25
0.3
0.2
0.25
0.3
0.2
0.25
0.3
||H
2
(β
1
)||
2
−
||H
2

(β
1
)||
2
^
||H
2
(β
2
)||
2
−
||H
1
(β
1
)||
2
^
Figure 5.5: β
2
= 1. The figures in the columns from left to right each represent
the case: (left)initial configuration, with band width J
h
= 0.0111 (
ˆ
β
1
= 0.988);
(middle) optimal configuration when only cladding is optimized, with band width

J
h
= 0.0387 (
ˆ
β
1
= 0.962); (right) optimal configuration when both cladding
and core are optimized, with band width J
h
= 0.0478 (
ˆ
β
1
= 0.953). The first
row illustrates the PCF cross-sections. The second row shows the corresponding
dispersion relations between [
ˆ
β
1
, β
2
]. From the third row to the last, the magnetic
field intensities at both
ˆ
β
1
and β
2
are shown in 3D surf plots.
129