Biomedical Engineering Reference
In-Depth Information
to produce an OAM peak with azimuthal number
m
.However,for
an irrational value of the angle
α
, this condition will never be
exactly met. Nevertheless, using the theory of continued expansion
of rational fractions, we can design structures that approximately
match the integer condition for the
m
α
product, thus producing
well-defined OAM peaks. This is so because an arbitrary divergence
angle
thatadmits
precisely one infinite continued fraction representation (and vice
versa) in the form [32, 33]:
α
isdirectlydeterminedbyanirrationalnumber
ζ
1
ζ
=
[
a
0
;
a
1
,
a
2
,
a
3
,]
=
a
0
+
(11.7)
1
a
1
+
1
a
2
+
1
a
4
+
a
3
+
...
Such infinite continued fraction representation is very useful
because its initial segments provide excellent rational approxima-
tions to the irrational numbers. The rational approximations (i.e.,
fractions) are called the convergents of the continued fraction,
and it can be shown that even-numbered convergents are smaller
than the original number
ζ
while odd-numbered ones are bigger.
Moreover, once the continued fraction expansion of
ζ
has been
obtained, well-defined recursion rules exist to quickly generate the
successive convergents. In fact, each convergent can be expressed
explicitly in terms of the continued fraction as the ratio of certain
multivariate polynomials called continuants. If two convergents are
found,withnumerators
p
1
,
p
2
, . . . anddenominators
q
1
,
q
2
,...then
the successive convergents are givenby theformula:
p
n
q
n
=
a
n
p
n
−
1
+
p
n
−
2
(11.8)
a
n
q
n
−
1
+
q
n
−
2
Thus, to generate new terms into a rational approximation, only the
two previous convergents are necessary. The initial or seed values
required for the evaluation of the first two terms are (0,1) and
(1,0) for (
p
−
2
,
p
−
1
)and(
q
−
2
,
q
−
1
), respectively. It is clear from
the discussion above that for spirals generated from an arbitrary
irrational number
ζ
, azimuthal peaks of order
m
(i.e., Bessel order
m
) will appear in its FHD due to the denominators
q
n
of the
rational approximations (i.e., the convergents) of
ζ
≈
p
n
/
q
n
.In
