Lattice Modules
219
The sum over the system is
⎛
⎜
⎜
⎝
x f
a f
y f
b f
⎞
⎟
⎟
⎠ =
n−1
m=0
R
−2mπ/n − φ x
−2mπ/n − φ y
◦
⎛
⎜
⎜
⎜
⎝
T x,aδ [(−α x x + a)/β x ] δ
β
2
x T a,xδ − α
2
x T x,aδ
/β x
xδ + (α x /β x )T x,aδ aδ +
√
β x T a,δ 2 δ
2
T y,bδ [(−α y y + b)/β y ] δ
β
2
y T b,yδ − α
2
y T y,bδ
/β y
yδ + (α y /β y ) T y,bδ bδ
⎞
⎟
⎟
⎟
⎠
◦ R
2mπ/n + φ x
2mπ/n + φ y
.
Since the x and y planes are decoupled, we can separate them. The x plane
is
x f
a f
=
n−1
m=0
cos (2mπ/n + φ x ) − sin (2mπ/n + φ x )
sin (2mπ/n + φ x ) cos(2mπ/n + φ x )
·
− (α x /β x ) T x,aδ δ
(1/β x ) T x,aδ δ
β
2
x T a,xδ − α
2
x T x,aδ
/β x
δ (α x /β x ) T x,aδ δ
·
cos (2mπ/n + φ x ) sin(2mπ/n + φ x )
− sin (2mπ/n + φ x ) cos(2mπ/n + φ x )
x i
a i
+
n−1
m=0
cos (2mπ/n + φ x ) − sin (2mπ/n + φ x )
sin (2mπ/n + φ x ) cos(2mπ/n + φ x )
0
√
β x T a,δ 2 δ
2
,
and the y plane is
y f
b f
=
n−1
m=0
cos (2mπ/n + φ y ) − sin (2mπ/n + φ y )
sin (2mπ/n + φ y ) cos(2mπ/n + φ y )
·
− (α y /β y ) T y,bδ δ
(1/β y ) T y,bδ δ
β
2
y T b,yδ − α
2
y T y,bδ
/β y
δ (α y /β y ) T y,bδ δ
·
cos (2mπ/n + φ y ) sin(2mπ/n + φ y )
− sin (2mπ/n + φ y ) cos(2mπ/n + φ y )
y i
b i
.
The terms
x|δ
2
and
a|δ
2
are
n−1
m=0
sin
2mπ
n
+ φ x
=
n−1
m=0
e
i(2mπ/n+φx)
− e
−i(2mπ/n+φx)
2i
=
1
2i
e
iφx 1 − e
in2π/n
1 − e i2π/n − e
−iφx 1 − e
−in2π/n
1 − e −i2π/n
= 0,