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Optically induced atomic lattice with tunable

near-field and far-field diffraction patterns

FENG WEN,

1,2,†

HUAPENG YE,

2,†

XUN ZHANG,

WEI WANG,

SHUOKE LI,

HONGXING WANG,

ANPENG ZHANG,

AND CHENG-WEI QIU

Key Laboratory for Physical Electronics and Devices of the Ministry of Education & School of Science & Shaanxi Key Laboratory

of Information Photonic Technique & Institute of Wide Bandgap Semiconductors, Xi’an Jiaotong University, Xi’an 710049, China

Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583, Singapore

*Corresponding author: [email protected]

Received 11 August 2017; revised 3 October 2017; accepted 4 October 2017; posted 5 October 2017 (Doc. ID 304051);

published 8 November 2017

Conventional periodic structures usually have nontunable refractive indices and thus lead to immutable photonic

bandgaps. A periodic structure created in an ultracold atoms ensemble by externally controlled light can overcome

this disadvantage and enable lots of promising applications. Here, two novel types of optically induced square

lattices, i.e., the amplitude and phase lattices, are proposed in an ultracold atoms ensemble by interfering four

ordinary plane waves under different parameter conditions. We demonstrate that in the far-field regime, the atomic

amplitude lattice with high transmissivity behaves similarly to an ideal pure sinusoidal amplitude lattice, whereas

the atomic phase lattices capable of producing phase excursion across a weak probe beam along with high trans-

missivity remains equally ideal. Moreover, we identify that the quality of Talbot imaging about a phase lattice is

greatly improved when compared with an amplitude lattice. Such an atomic lattice could find applications in all-

optical switching at the few photons level and paves the way for imaging ultracold atoms or molecules both in the

near-field and in the far-field with a nondestructive and lensless approach.

OCIS codes: (050.0050) Diffraction and gratings; (270.1670) Coherent optical effects; (050.5080) Phase shift; (070.6760) Talbot and

self-imaging effects.

https://doi.org/10.1364/PRJ.5.000676

1. INTRODUCTION

In the past few years, artificial periodic structures, such as pho-

tonic crystals [1–5] and metamaterials [6–9], have attracted

increasing attention due to their unprecedented capacities of

engineering the transmission and reflection properties of waves.

An important property of such applications is the ability to

strongly modify the propagation of light in certain directions

and frequencies. A number of new physical phenomena have

been predicted to occur in these materials, including strong

localization of light [10], inhibited spontaneous emission from

atoms [11], photon– atom bound states [11], all-optical signal

processing, and switching [12].

Conventionally, photolithography and electron beam

lithography are widely used to fabricate the periodic structures

with microsized or nanosized features. However, the refractive

index of the resulting periodic structures is usually nontunable,

thus leading to immutable photonic bandgap (PBG). To fully

explore the potential of photonic crystals, it is crucially impor-

tant to achieve a dynamical tunability of their bandgap [13].

In previou s studies, a distinct approach to generate spatially

periodic structures, based on the electromagnetically induced

grating (EIG) [14], is proposed by Ling et al. [15] and exper-

imentally demonstrated in cold [16] and hot [17] atomic sam-

ples. Very recently, the spatially dependent electromagnetically

induced transparency (EIT) in cold atoms was demonstrated by

using the phase profile as a control parameter for the atomic

opacity [18]. Unlike traditional photonic crystals, here a peri-

odic structure is created by externally controlled light, and a

novel photonic structure with an optically tunable PBG is

achieved [19]. The EIG with tunable first-order diffraction

has attracted considerable interest due to its potential applica-

tions in all-optical switching and routing [17], light storage

[20], probing optical properties of materials [21], optical bista-

bility [22], shaping a biphoton spectrum [23], and beam split-

ting and fanning [24]. However, 2D EIG and its diffraction

pattern in near-field and far-field have not been demonstrated

yet. In this paper, we demonstrate that optical lattices resulting

from amplitude modulation and phase modulation can be real-

ized in an ultracold atoms ensemble by interfering four ordinary

plane waves under different parameter conditions. We analyze

676

Vol. 5, No. 6 / December 2017 / Photonics Research

Research Article

theoretically the families of such optically induced square lat-

tices and then discuss the corresponding diffraction pattern

both in near-field and far-field regions. We identify that the

phase modulation plays a significant role not only in the effi-

ciency of diffracting light into high-order directions but also in

the improvement of the quality of the Talbot carpet pattern

(visibility and the signal-to-noise ratio). This work may offer

a nondestructive and lensless way to image ultracold atoms

or molecules both in the near-field and in the far-field.

It is worth mentioning that our system has the following ad-

vantages. First, the optical lattice written in an atomic assemble is

reconfigurable and can be dynamically tuned, leading the refrac-

tive index change, and the PBG structures of our scheme are

sensitive to the adjusting the frequency detuning, so the trans-

mission and reflection of the propagating light can be dynami-

cally modulated. Further detailed studies of these effects will be

presented elsewhere. Second, using the multibeam interference

method, other complex lattice structures, i.e., quasi-crystals,

kagome lattice, defect mediated lattice, honeycomb lattice,

Bessel lattice, virtual lattice, ring lattice, and 3D photonic lattice,

can all be realized in current systems. Third, formation of the

lattice, as well as the tuning, is all done all optically.

2. THEORETICAL MODEL

The scheme to construct optically induced square lattices relies

on periodically manipulating the refractive index of an ultracold

atoms (or molecules) ensemble. As illustrated in Fig . 1(a), our

model consists of four strong control fields of frequency ω

and

wave number k

, a weak probe field of frequency ω

and wave

number k

, and an ensemble of closed three-level cascade-type

ultracold atoms (or molecules) composing of a ground state jai,

a metastable state jbi, and an excited state jci. The metastable

state jbi is coupled to the excited state jci via the four strong

control fields near resonance on the jbi → jci transition, while

the jai → jbi transition is connected by the probe beam with

Rabi frequency Ω

The four controlling plane waves, which are injected into

the atomic sample and interact with the atomic ensemble

by coupling the atomic transition jbi → jci, are the lattice-

forming lasers. To be specific, two plane waves being symmet-

rically displaced with respect to the z axis are incident upon the

atomic sample at a small angle θ, whose intersection will generate

a standing wave along the x direction within the atomic ensem-

ble. In the same way, another two plane waves generate a stand-

ing wave along the y direction inside the medium. The effective

Rabi frequency of the four strong controlling fields can be writ-

ten as jG

eff

x;yj

jΩ

sinπx∕aj

jΩ

sinπy∕bj

Here, Ω

is the Rabi frequency of one of the four controlling

fields and assumed to be real for simplicity. a (b) is the spatial

period along the X (Y ) direction [see Fig. 1(b)], which can be

made arbitrarily larger than the wavelength of the controlling

fields by varying θ. A 2D optically induced lattice will be pro-

duced within the ultracold atoms (or molecules) ensemble if the

condition Ω

≫ Ω

is satisfied. Two points should be empha-

sized herein. First, the stability of the optically induced lattice

structure is determined by lattice-forming beams. As long as

the laser beams forming the optical lattice are stable, small fluc-

tuations should not affect our main results. Second, by employ-

ing two-photon Doppler-free configurations in the thermal

atomic vapor, our scheme could be realized in the thermal atomic

vapor.

Spanning the Hilbert space with bare states (jai, jbi,

and jci) and applying the rotational wave approximation,

the Hamiltonian in the interaction picture can be represented

by (ℏ  1)

H  G

−iΔ

jaihbjG

eff

−iΔ

jbihcjh:c; (1)

where μ

is the electric dipole matrix element to atomic transi-

tion jii → jji ( i; j  a; b; c ). Δ

 ω

− ω

and Δ

 ω

− ω

are single photon frequency detunings of E

and E

eff

x;y from

Fig. 1. (a) Cascade-type three-level scheme with jai5S

1∕2

F  3, jbi5P

3∕2

F  3, and jci (5D

5∕2

)of

Rb atoms [25], interacting with

three laser beams: probe field E

and two lattice-forming fields E

x and E

y. (b) The geometr y of four laser beams applied upon a cold atoms

ensemble along the z direction, and the corresponding near-field and far-field diffraction patterns of a probe field.

Research Article

Vol. 5, No. 6 / December 2017 / Photonics Research 677

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