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Generalized Fibonacci photon sieves

JIE KE

1,2

AND JUNYONG ZHANG

Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of sciences, Shanghai 201800, China

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.

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

 F

n−1



n−2

n>2;n∈ N , its initial two seed elements are F

 1

and F

 1. Obviously, x

1 

5∕2 and x



1 −

5∕2, which are associated with the classical geometrical

problem of the golden section, are the characteristic roots of the

characteristic equation x

− x − 1  0.

When the initial seed elements are given as

 a

;j∈ N



∈ R; (2)

where, if a

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



m1

n−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

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