Generalized Fibonacci photon sieves
JIE KE
1,2
AND JUNYONG ZHANG
1,
*
1
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of sciences, Shanghai 201800, China
2
University of Chinese Academy of Sciences, Beijing 100049, China
*Corresponding author: zhangjy829@siom.ac.cn
Received 13 April 2015; revised 29 June 2015; accepted 14 July 2015; posted 15 July 2015 (Doc. ID 238012); published 12 August 2015
We successfully extend the standard Fibonacci zone plates with two on-axis foci to the generalized Fibonacci
photon sieves (GFiPS) with multiple on-axis foci. We also propose the direct and inverse design methods based
on the characteristic roots of the recursion relation of the generalized Fibonacci sequences. By switching the
transparent and opaque zones, according to the generalized Fibonacci sequences, we not only realize adjustable
multifocal distances but also fulfill the adjustable compression ratio of focal spots in different directions.
© 2015
Optical Society of America
OCIS codes: (050.1965) Diffractive lenses; (080.2720) Mathematical methods (general); (100.3190) Inverse problems; (340.7480)
X-rays, soft x-rays, extreme ultraviolet (EUV).
http://dx.doi.org/10.1364/AO.54.007278
1. INTRODUCTION
Focusing of x-ray and extreme ultraviolet radiation has many
applications in physics and life sciences, such as high-resolution
microscopy, spectroscopy, and lithography. Traditional Fresnel
zone plates (FZP) can be used for this kind of focusing [1,2],
but its resolution is limited by the width of the outmost zone
[3,4]. In 2001, Kipp et al. proposed photon sieves [5], which
are essentially Fresnel zone plates with the transparent zones
replaced by a great number of completely separate pinholes
to overcome the disadvantages of FZP. Until now, several kinds
of theoretical models [6–8] and experimental work s [9–12]
with respect to photon sieves have been performed and applied
for designing different kinds of photon sieves such as fractal
[13,14], compound [9], Zernike apodized [15], phase zone
[16], spiral [17], square [18], and reflection photon sieves [19].
The Fibonacci sequence, named after the inventor and
Italian mathematician Leonardo de Pisa, has been employed
in the development of different photonic devices. The focusing
and imaging properties of Fibonacci diffractive elements, e.g.,
gratings [20–22], lenses [23–25], zone plates [26], etc., have
been studied in detail. But this aperiodic sequence has not been
used for the design of photon sieves, although this kind of pho-
ton sieves may allow for new applications in ophthalmology,
nanometer lithography, and weapons vision [27].
In this paper, we successfully extend the standard Fibonacci
zone plates to the generalized Fibonacci photon sieves (GFiPS)
and find that the physical foci, which are defined as the ratios
between different focal distances, can overlap with the math-
ematical foci, which are equal to the characteristic roots of
the recursion relation. At the same time, by using the gener-
alized Fibonacci sequences, we successfully design the GFiPS
with the expected ratio of focal distances and realize the physi-
cal foci overlapping with the mathematical foci. Then, the
inverse design can be proposed, and the structure parameters
of GFiPS and the corresponding mathematical sequences
are given. Besides, two significant extended applications are
discussed.
2. PRINCIPLE
A. Generalized Fibonacci sequence
For the standard Fibonacci sequence,
1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; …; (1)
defined by the linear recursion relation F
n
F
n−1
F
n−2
n>2;n∈ N , its initial two seed elements are F
1
1
and F
2
1. Obviously, x
1
1
p
5∕2 and x
2
1 −
p
5∕2, which are associated with the classical geometrical
problem of the golden section, are the characteristic roots of the
characteristic equation x
2
− x − 1 0.
When the initial seed elements are given as
F
j
a
j
;j∈ N
;a
j
∈ R; (2)
where, if a
j
is not a non-negative integer, it can be mapped to a
positive integer field. The generalized Fibonacci sequences can
be defined by the corresponding linear recursion relation as
F
n
X
j
m1
C
m
F
n−m
;C
m
∈ R; n; j ∈ N; n > j ≥ 2:
(3)
Then, the related characteristic equation of Eq. (3) could be
given by
7278
Vol. 54, No. 24 / August 20 2015 / Applied Optics
Research Artic le
1559-128X/15/247278-06$15/0$15.00 © 2015 Optical Society of America
资源评论
