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ARTICLE IN PRESS Model

JLEO-54043; No. of Pages 3

Optik xxx (2014) xxx– xxx

Contents lists available at ScienceDirect

Optik

jou rn al homepage: www.elsev ier .de / i j leo

nhanced beaming of light from a photonic crystal waveguide via aelf-collimation photonic crystal

ifeng Shena,∗, Lulu Lia, Hao Zhaoa, Yueqin Caob

Science College, China University of Mining and Technology, Xuzhou 221116, ChinaCollege of Liberal Arts, Jiangsu Normal University, Xuzhou 221116, China

r t i c l e i n f o

rticle history:eceived 14 May 2013ccepted 12 October 2013vailable online xxx

ACS:2.70.Qs

a b s t r a c t

A simple method to control the light transmission from a photonic crystal waveguide (PCW) is proposed.The self-collimation (SC) effect in PC is utilized to depress the diffractive behavior of the light beamemitting out from the PCW. The finite difference in time domain (FDTD) simulation results show thatchoosing appropriate self-collimation frequency and the space between the PCW and the SC-PC canachieve a very good collimating beam in the outgoing side.

© 2013 Elsevier GmbH. All rights reserved.

2.82.−m

eywords:hotonic crystal waveguideelf-collimationDTD

eaming

. Introduction

The control of light transmission of waveguide becomes a dif-cult issue when the width of the waveguide is comparable tohe wavelength due to strong diffractions of light. In traditionalptics a light beam with low divergence can be hardly achievedrom a waveguide with a subwavelength width. Recently it washown that enhanced transmission and beaming of light throughubwavelength structures can be achieved by modulating the exiturfaces [1–7]. The beaming effect arises from the coherent inter-erence of light emitted from the subwavelength structure and thatf the excited leaky surface modes at the exit surface. However,his method does not provide any simple rules for determininghich surfaces support surface modes and any simple conditions

or producing the beaming effect. In PCs, the flow of light can be con-rolled by utilizing the photonic band gap or other specific spatialispersion property. Self-collimation effect is of particular inter-sting that a beam of electromagnetic wave can propagate withlmost no diffraction in a perfectly periodic PC [8,9]. Utilizing the

Please cite this article in press as: S. Yifeng, et al., Enhanced beaming of ligcrystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/

elf-collimation effect Tang et al. achieved a highly efficient beam-ng emission from the PCW [10]. They add another PC to providehe self-collimation effect. And they attributed the beaming effect

∗ Corresponding author.E-mail address: shenyf@cumt.edu.cn (S. Yifeng).

030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.ijleo.2013.10.053

to the interference of the multiple self-collimation beams excitedby the waveguide. However, they only study the PC system con-sisting of dielectric cylinders in air, which is not widely applied inpractice due to larger loss in the direction of cylinders. Moreover,they removed four cylinders at the exit surface of the PCW to obtaina better angle match, which introduced a resonator between thePCW and the PC with the self-collimation effect. This leads to morecomplexity in the system. At last, they choose only three arraysof cylinders to form the PC with the self-collimation effect, whichis hardly to contain a good self-collimation behavior because theself-collimation effect is a bulk property of the crystal.

In this paper we study the PC system consisting of air holes inhigh refractive index dielectric media and propose a simple methodto control the emitting beam from the PCW with dimensions ofcomparable to wavelength. We design a PC, which contains a largenumber of arrays so that it can provide a good self-collimation effectat the working frequency, to serve as a collimating lens to controlthe emitting beam.

2. Two-dimensional PCW and self-collimation PC

We consider a two-dimensional triangular lattice of air holes in a

ht from a photonic crystal waveguide via a self-collimation photonicj.ijleo.2013.10.053

background media with an effective refractive index of n = 4.36. Wechoose the hole radius r = 0.49a, where a is the lattice parameter.For TM polarization (electric field pointing along the axis of holes),a wide frequency gap for reduced frequencies ωa/2�c in the range

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space S between the right edge of the PCW and the left edge of theSC-PC is only S = 9.8a − (9.5 ×

√2a2 + 0.5r2) − (5a + r) ≈ 0.202a.

Fig. 1. The photonic crystal waveguide structure.

0.380, 0.534) can be found by using the plane wave expansionPWE) method, where ω is the angular frequency and c is the speedf light in vacuum.

In our simulations, the PC system has a finite size of 10a × 12√

3asee Fig. 1). A waveguide is obtained by removing one row ofoles in the x direction. This waveguide has a effective width of√

3a − 2r) ≈ 0.75a. And a Gaussian beam source with a width of 4as launched at (−6a, 0) to excite the waveguide modes. The work-ng frequency is arbitrarily selected as ω0 = 0.499(2�c/a), which isear the center of the band gap so that the light filed is well guidedy the PCW. The wavelength corresponding to ω0 in the backgroundedia is � ≈ 0.46a, which is comparable with the waveguide width.

o the radiated light field will show a broad angular distribution dueo the inevitable diffractions. Fig. 2 shows the electric-field for theCW without any modulations. It is easily found that the radiatedight field has a large angular distribution in an azimuthal angleange of −53.1◦ ≤ � ≤ +53.1◦. A power monitor I with a width of 20as launched at (+8a, 0) to detect the transmitted power. The wholeransmission is about 42.6% in this case because of high reflectionst the input side of the PCW. We also place another power monitorI with a same width launched at (+20a, 0) to measure the far-fieldransmission distribution. Thus, for the PCW without any modula-ions, the radiated light field contains a large beam divergence dueo strong diffractions at the exit of the PCW.

To obtain a very low beam divergence we utilize the self-ollimation effect to depress the diffraction effect. As we know that

self-collimating beam can keep almost unchanged shape whenransmitting in the photonic crystal as if it does not meet withny diffraction. The reason is that in PCs the energy flow can onlyass along the directions orthogonal to the flat parts of the equi-requency contour (EFC) [8]. That is, the strong anisotropic disper-

Please cite this article in press as: S. Yifeng, et al., Enhanced beaming of ligcrystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/

ion relation of PCs can be helpful to depress the diffraction effect.Thus we will introduce a collimating lens, which is designed

y a PC with a self-collimation effect, to collimate the outgoingight beam. We name this additional PC as SC-PC. We choose a

Fig. 2. The electric-field for the PCW without any modulations.

Fig. 3. (a) The SC-PC structure and (b) the EFCs of the SC-PC.

two-dimensional square lattice of air holes in the same backgroundmedia to constitute this SC-PC (see in Fig. 3(a)). The radius of theair holes is r2 = 0.4a2, where a2 is the lattice constant of the SC-PC.Fig. 3(b) shows the EFCs of two frequencies in the first band. Onecan find that at the circular frequency ω = 0.1500(2�c/a2) there is aself-collimation effect occurring along the �–M direction. We needthe system working at the same frequency, so ω = 0.1500(2�c/a2)should be equal to ω = 0.499(2�c/a2). Thus we obtain a2 = 0.3006a.To serve as a collimating lens the SC-PC is truncated as a squareshape of 19

√2a2 × 19

√2a2 (see in Fig. 3(a)), and the �–M direction

is aligned along the x direction so that we can obtain the collimatingbeam along this direction.

3. Simulations and discussions

Fig. 4 gives the real-space electric-field amplitude for the PCWwith the SC-PC. Noting that the SC-PC is placed symmetrically alongthe x axis and the center of the SC-PC is located at (X2 = 9.8a, Z2 = 0),where X2 and Z2 are the coordinates. Then we can calculate the

ht from a photonic crystal waveguide via a self-collimation photonicj.ijleo.2013.10.053

We can clearly find that a perfectly collimating beam emits out fromthe SC-PC and the scattering loss between the PCW and the SC-PC is

Fig. 4. The real-space electric-field amplitude for the PCW with the SC-PC.

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Fig. 5. (a) Static beam intensity recorded by monitor I at (8a, 0); (b) static beamintensity recorded by monitor II at (20a, 0). In both figures the blue curves representtti

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R10096–R10100.

he data for the case of PCW without any modulations while the red curves representhe data for the case of PCW with SC-PC. (For interpretation of the references to colorn this text, the reader is referred to the web version of the article.)

ery weak along the z direction. In our system the SC-PC has a mucharger number of arrays than the PC-2 used in Tang’s paper (Refer-nce [10]), so the self-collimation effect is much stronger and theabry–Pérot resonant effect is much weaker in our case. Further-ore, in our case there are no multiple self-collimation beams like

CLB-0 and SCLB-1 in Tang’s paper, so the beaming effect does notome from the interference of the multiple self-collimation beamsut only from the self-collimation effect itself.

We also compare the static field distributions of both the neareld and far field for these two cases. The static beam intensityistributions along the z axis are recorded by the monitor I at8a, 0) and monitor II at (20a, 0). First, we consider the near fieldt x = 8a (see Fig. 5(a)). For the PCW system the emitting beamas a broad static field distribution (from −4a to +4a) along the zirection, which is corresponding to an angle of divergence � from53.1◦ to 53.1◦. But for the PCW with SC-PC system the static fieldistribution is limited to −1.2a to +1.2a along the z direction, which

s just corresponding to a width of the self-collimated beam in theC-PC. Moreover, the peak intensity is improved by 1.77 times com-

Please cite this article in press as: S. Yifeng, et al., Enhanced beaming of ligcrystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/

aring the red curve with the blue curve. Second, we analyze the fareld at x = 20a (see Fig. 5(b)). For the PCW system the far field stillhows a much broad static distribution from −10a to +10a alonghe z direction. But for the PCW with SC-PC system the static field

[

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distribution is still limited to a narrow range from −2a to +2a alongthe z direction. And the peak intensity is improved by 4.84 times.So we can conclude that with the help of the SC-PC a much bet-ter collimating beam can pass through the PCW. Furthermore, thiscollimating beam contains almost 95% of the whole transmittedlight energy from the exit of the PCW. That means the couplingbetween the PCW and the SC-PC is perfect and the insert loss of theSC-PC is very low. It should be noted that this coupling efficiencyis very sensitive to the value of S, the location of the center of theSC-PC. If we choose another value of X2 = 9.7a, the efficiency willbe depressed greatly, and the emitting beam does not keep so goodcollimating shape. However, if we redesign the structure of the SC-PC choosing the parameter a2 = 0.3006a, which is corresponding toω = 0.1474(2�c/a2), we can still obtain a perfect coupling betweenthe PCW and SC-PC and a perfectly collimating beam even whenX2 = 9.7a.

4. Conclusions

We have designed a SC-PC to serve as a collimating lens toreshape the emitting beam from a PCW with a width of wavelength.We have showed that this method can effectively improve thebeaming effect and depress the diffractions. We have pointed outthat the self-collimation frequency and the space between the PCWand the SC-PC play a central role in the achievement of perfectlycollimating beam. This method may be promising for connectionof PCW or semiconductor laser to other optical elements such asfibers or ridge waveguides.

Acknowledgment

The authors gratefully acknowledge support from the Fun-damental Research Funds for the Central Universities (no.2012LWB47).

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