Propagation of Airy Related Beams Generated From Flat Topped Gaussian Beams Through a Chiral Slab 2014 Optics and Lasers in Engineering

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Propagation of Airy-related beams generated from flat-toppedGaussian beams through a chiral slab

Zhirong Liu a,b, Daomu Zhao a,n

a Department of Physics, Zhejiang University, Hangzhou 310027, Chinab School of Basic, East China Jiaotong University, Nanchang 330013, China

a r t i c l e i n f o

Article history:

Received 30 March 2013Received in revised form

12 July 2013

Accepted 13 July 2013Available online 9 August 2013

Keywords:

ABCD transforms

Chiral media

Laser beam shaping

Propagation

a b s t r a c t

We derived the analytical expression for the propagation of Airy-related beams generated from flat-

topped Gaussian beams through an ABCD optical system, and use it to study the propagation of this typeof beams through a chiral slab. Several influence factors, such as the optical beams order Nand the chiral

parameter of the chiral medium, on the beam propagation properties both in near- and far-zones are

discussed in detail. It is shown that the Airy tails of high order beams decay more quickly than those of

low order beams in the chiral medium; the constructive interference effect between the LCP

(left-circularly polarized) and RCP (right-circularly polarized) beams becomes more significant as the

chiral parameter increases; the LCP and RCP beams are not separated in the near-zone, while the two

beams are obviously separated in the far-zone, and accordingly the interference peaks decrease as the

propagation distance increases in the chiral slab.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Non-spreading or non-diffraction (also named diffraction-free)beams are by definition localized optical wave packets that remain

invariant during propagation. Due to their novel features this

intriguing class of wave packets have received sustained attention

[19]. In 1979, Scholars Berry and Balazs made an important

observation within the context of quantum mechanics: they

theoretically demonstrated that the Schrdinger equation describ-

ing a free particle can exhibit a non-spreading Airy wave packet

solution [1]. Bessel beams, initially predicted theoretically and

demonstrated experimentally by Durnin et al. in 1987, is perhaps

the best known example of diffraction-free wave [2]. In 2007,

a finite-energy Airy beam is first introduced theoretically and

demonstrated experimentally by extending Berry and Balazss

infinite-energy Airy model by Siviloglou and Christodoulides

[3,4]. Owing to their non-diffraction properties, Airy beams haveability to reconstruct themselves during propagation even though

parts of the beams are distorted or obstructed [5]. The Airy beams

have been identified for unique features, such as weak diffraction,

transverse acceleration [3,4], and self-healing [5]. And over very

recent years, Airy beams have attracted a great deal of interest in

applications such as optical clearing micro-particles [6], plasma

physics [7], optical micro-manipulation [8,9] and other fields.

In general, the finite energy Airy beam can be generated from the

fundamental Gaussian beam through a Fourier transformation pro-

vided that a cubic phase is imposed [4]. Meanwhile, many Airy-related beams have been proposed or generated by making some

changes in the methods of generating Airy beams [1016]. For

example, using the partially coherent Gaussian beam as the incident

beam, a broadband white light Airy beam can be generated, and its

decay parameter depends on the spatial coherence of the incident

beam [10]; by adding a special apodization mask in the light path, a

reduced side-lobe Airy beam can be generated that has an effectively

enhanced central lobe, and the side lobe is reduced compared with

the common Airy beam [11].

It is known that chirality can lift the degeneracy of two circular

polarizations [17]. When a linearly polarized beam is incident

normally upon a chiral slab, it will be split into two circularly

polarized beams (left-circularly polarized and right-circularly polar-

ized beams) at the interface with different phase velocity propagat-ing in the chiral slab. This circular birefringence of the chiral medium

is of crucial importance in the fields of biochemistry, chemistry, and

medicine. In this work, we investigate the propagation of Airy-

related beam, which is generated from Airy transform from flat-

topped Gaussian beams [18], through a chiral slab [1925].

2. Propagation characteristics of the Airy-related beam

through a chiral slab

Matrix optics is a powerful and analytical method to deal with

the beam propagation problem. For a beam through a chiral slab,

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http://dx.doi.org/10.1016/j.optlaseng.2013.07.011

n Corresponding author. Tel.: +86 571 888 63887; fax: +86 571 879 51328.

E-mail address: [email protected] (D. Zhao).

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the corresponding transfer matrix of the optical system can be

written as [26]

AL BL

CL DL

" # 1 z=n

L

0 1

" #;

AR BR

CR DR

" # 1 z=n

R

0 1

" #: 1

In Eq. (1), nL n0=1 n0k0; nR n0=1n0k0 denote therefractive indices of the left-circularly polarized (LCP) and right-

circularly polarized (RCP) beams, respectively. n0 and represent

the original refractive index and the chiral parameter of the chiral

slab, respectively. k0 2=0 is the wave number of the opticalwave with 0 being the wavelength of the Airy-related beam in

vacuum. Thus, the wave number of the left-circularly polarized

and right-circularly polarized Airy-related beams in the chiral slab

is kJ nJk0, where J L,R.Here, we investigate the dynamics of one-dimensional (1D)

Airy-related beam through a chiral slab. A symmetric 2D case can

be obtained by replacing the coordinate x with y, and multiplying

the two scalar fields together. The electric field profile of Airy-

related beams generated from flat-topped Gaussian beams can be

expressed as [18]

E1x1;z 0 2ffiffiffi

pA0

N

n 1

1n1 ffiffiffiffiffianpN

N

n

exp

2a3n3

Ai

x1 a2n

exp

anx1

:

2In Eq. (2), Ai indicates the Airy function, x1= represents adimensionless transverse coordinate, denotes an arbitrary trans-

verse scale, an w20=4n2 represents the modulation parameterso as to ensure containment of the infinite Airy tail, A0 is a

constant related with the beam power, N denotes the order of

the Airy-related beams,N

n

is a binomial coefficient, and w0 is

the waist size of fundamental Gaussian beam with N 1. It is clearfrom Eq. (2) that the Airy-related beams generated from flat-

topped Gaussian beams can be considered as a finite sum ofcommon finite energy Airy beams, and Eq. (2) becomes the

common finite energy Airy beam when N 1.To visualize the shape of Airy-related beams characterized by Eq.

(2), a preliminary demonstration is shown in Fig. 1(a) for the flat-

topped Gaussian beams, and (b) for the Airy-related beams gener-

ated from flat-topped Gaussian beams of different orders, respec-

tively. All of the curves in Fig. 1 have been normalized to the peak

intensity value. It is apparent from Fig. 1(a) that the irradiance

profile becomes flat-topped when N41, and the flat zone occupies

a larger fraction of the beam profile when the value of N increases.

And it is also clearly seen from Fig. 1(b) that the Airy-related beams

generated from flat-topped Gaussian beams N41 possess profilessimilar to those of the Airy beam generated from the fundamental

Gaussian beamN

1, and the Airy tails of higher order beams

decay more quickly than that of lower order beams. Furthermore,

the peak of the first lobe is a little shifted in the positive x-direction

when the value of N increases, which can be inferred from the Airy

function in Eq. (2). So the Airy-related beams can be modulated by

varying the beam order N.

Within the framework of paraxial approximation, the propaga-

tion of a beam through the first order ABCD optical system is

described by the generalized HuygensFresnel diffraction integral,

which is known as Collins formula [27,26]

EJx;z ffiffiffiffiffiffiffiffiffiffiffiffi

ik0

2BJ

s Z11

exp ik02BJ

AJx212x1x DJx2

E1x1;z 0dx1

3In Eq. (3) the superscripts J L,R denote LCP and RCP, respectively.To calculate this integral, the Airy function can be written in terms

of the integral representation [28]

Aix 12

Z11

expiu3

3 ixu

du: 4

On substituting Eqs. (2) and (4) into Eq. (3), after tedious integral

calculations, one obtains the expression of Airy-related beamsfield distribution in an arbitrary plane z40:

EJx;z

2A0ffiffiffiffiffiffiffiAJ

p Nn 1

1

n1 ffiffiffiffiffiffiffiffianpN

N

n

exp ik0C

Jx2

2AJ" #

Airy xAJ

BJ2

44k20AJ2

iBJanAJk02

a2n !

exp anxAJ

anBJ2

2AJ2k204 iB

J3

8AJ3k306

"

ia2nB

J

AJk02 iB

Jx

2AJ2k03 iB

J

62k0AJ

2a3n3

#: 5

In Eq. (5) AJ, BJ, CJ and DJ are elements of the optical transfermatrix for the LCP and RCP beams in the chiral slab, respectively.

When the Airy-related beam propagates along the z-direction, the

optical field is split into two components, e.g., the LCP and RCP

beams. And the total field profile in the chiral slab is given by

Ex ELx ERx: 6

Thus, the total intensity of Airy-related beam in the plane z40 can

be expressed as

I jELxj2 jERxj2 Iint; 7

with

Iint ELxERnx ERxELnx; 8

where Iint

denotes the interference term, and n indicates the

complex conjugation. By complex but straightforward deduction,

Fig. 1. The normalized intensity distribution of (a) flat-topped Gaussian beams and (b) Airy-related beams generated from flat-topped Gaussian beams of different orders N.

Z. Liu, D. Zhao / Optics and Lasers in Engineering 52 (2014) 131814


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one can find that the interference intensity satisfies the following

relation

Iintexp 2a0x

x0

exp a0

z21 n20k202k

20x

40n

20

" #: 9

Eq. (9) indicates that Iint is sensitive to the propagating distance z

and the chiral parameter of the medium. Eqs. (5)(9) are the

main analytical results obtained in this paper, which are conve-

nient for studying the propagation characteristics of Airy-relatedbeams in a chiral slab.

3. Numerical simulation and analysis

Without loss of generality, in the following simulations we

choose the parameters of the Airy-related beam as: 0 532 nm, 100 m, w0 50 m; and the original refractive index n0 3.On substituting from Eqs. (1) and (5) into Eq. (7), one can obtain

the analytical expression for the Airy-related beam in the chiral

slab. The propagation of the Airy-related beams of different orders,

which are generated from Gaussian beams and flat-topped Gaus-

sian beams, respectively, in the chiral slab with the chiral para-

meter 0:16=k0 is shown in Fig. 2(a)(d) for N 1, and(e)(h) for N 10, respectively.

The propagation of Airy-related beams of different order in the

chiral slab with 0:16=k0 is shown in Fig. 2(a)(d). From thenormalized intensity distributions of the LCP and RCP beams [see

from Fig. 2(a) and (b)], one can see that the circularly polarized Airy

beams of opposite handedness propagate along two different curved

trajectories in the chiral slab. And one can also see from Fig. 2(c) that

the interference pattern between LCP and RCP beams is quite different

from the situation of independent propagation of each beam. Due to

the strong interference effect, the total field is greatly distorted in the

overlapping areas of the two beams [see from Fig. 2(d)]. For the

convenience of comparison, the case for N 10 is shown in Fig. 2(e)(h), and one can find from Fig. 2(a) and (e) that both the LCP and

RCP beams of high order have a shorter nonspreading distance, in

which distance the beam can maintain its original intensity distribu-

tion with no measurable spreading. Meanwhile, one can also see from

Fig. 2(c) and (g) that the interference fringes of lower order LCP and

RCP beams is clearer than that of higher order beams, which indicates

that the interference effect between lower order LCP and RCP beams is

stronger than that between higher order beams.

The case of the chiral parameter 0:28=k0 is shown in Fig. 3.

Other parameters are the same as those in Fig. 2. Comparing Fig. 2(c) and Fig. 3(c), or Fig. 2(g) and Fig. 3(g), one can find that the

constructive interference effect between the LCP and RCP beams

becomes more significant as the chiralitys parameter increases.

Fig. 4(a)(j) shows the intensity distribution of Airy-related

beams at the near-zone with z 200 mm. The chiral parameter inFig. 4(a)(e) is 0:16=k0, and in Fig. 4(f)(j) 0:28=k0, andother parameters are the same. It is clearly seen from Fig. 4(a) and

(f) that the LCP and RCP beams of the same order are not

separated, and nearly overlapped together at the near-zone, so

the interference between them is fairly strong [see Fig. 4(d) and

(i)], and accordingly the oscillating intensity curves can be seen in

Fig. 4(e) and (j).

Fig. 5(a)(j) shows the intensity distribution of Airy-related

beams in the far-zone with z

800 mm. Other parameters are

same as Fig. 4(a)(j). Obviously, one can see from Fig. 5(a) and

(f) that the LCP and RCP beams of the same order are separated

and not overlapped at the far-zone, for the two beams propagate

along different curved trajectories in the chiral slab. The amplitude

of oscillation decreases as the propagation distance increases, so

interference between them becomes weaker and more compli-

cated [see Fig. 5(d) and (i), and Fig. 5(e) and (j)].

4. Conclusion

In conclusion, we have investigated the propagation of Airy-

related beams generated from flat-topped Gaussian beams through

Fig. 2. The propagation of the Airy-related beams of different order in the chiral slab with 0:16=k0; (a) and (e) the LCP beam; (b) and (f) the RCP beam; (c) and (g) theinterference term Iint; (d) and (h) the total intensity; (a)(d) for N 1, and (e)(h) for N 10; respectively.

Z. Liu, D. Zhao / Optics and Lasers in Engineering 52 (2014) 1318 15


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Fig. 3. The evolution of the Airy-related beams of different orders propagating through the chiral medium with 0:28=k0; (a)(d) N 1, and (e)(h) N 10:

Fig. 4. The normalized intensity distribution of Airy-related beams of different orders propagating through the chiral media when z 200 mm with (a)(e) 0:16=k0and (f)(j) 0:28=k0:



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a chiral slab by using the Matrix Optics. And we have discussed

several influence factors, such as the optical beam order N and thechiral parameter of the chiral slab, on the beam propagation

properties both in near- and far-zones in detail. It is shown that the

Airy tails of high order beams decay more quickly than those of low

order beams in the chiral slab; the constructive interference effect

between the LCP and RCP beams becomes more significant as the

chiral parameter increases; the LCP and RCP beams are not

separated in the near-zone, while the two beams are obviously

separated in the far-zone, and accordingly the interference peaks

decrease as the propagation distance increases.

Acknowledgments

This work was supported by the National Natural Science

Foundation of China (11274273 and 11074219), the Jiangxi Provin-

cial Natural Science Foundation of China (20132BAB212007), the

East China Jiaotong University Startup Outlay for Doctor Scientific

Research (09132007), and the East China Jiaotong University

Foundation (11JC02 and 10JC01).

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Fig. 5. The normalized intensity distribution of Airy-related beams of different orders propagating through the chiral slab when z 800 mm with (a)(e) 0:16=k0 and(f)(j) 0:28=k0 :

Z. Liu, D. Zhao / Optics and Lasers in Engineering 52 (2014) 1318 17
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