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Photonic Molecules in Photonic Crystals

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2006 Jpn. J. Appl. Phys. 45 6108

(http://iopscience.iop.org/1347-4065/45/8R/6108)

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Photonic Molecules in Photonic Crystals

Satoru ISHII, Kengo NOZAKI and Toshihiko BABA�

Department of Electrical and Computer Engineering, Yokohama National University,

79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan

(Received February 1, 2006; accepted April 21, 2006; published online August 4, 2006)

We propose a photonic molecule consisting of multiple nanocavities in a photonic crystal and demonstrate its lasing andmodal characteristics by finite-difference time-domain analysis and by experiments. In the analysis, we show that a point-shiftdefect and a point-missing defect have s- and p-orbital cavity modes, respectively, and that two adjacent defects exhibit �- and�-orbital bonding and antibonding modes, which are similar to electronic states in chemical molecules. For such structure, weverify the modal characteristics as photonic molecules, i.e., mode splitting and its dependence on coupling strength, throughtheir fabrication into GaInAsP photonic crystal slabs and the observation of lasing characteristics.[DOI: 10.1143/JJAP.45.6108]

KEYWORDS: photonic molecule, photonic crystal, nanolaser, coupled cavity laser, GaInAsP

1. Introduction

Photonic molecules (PMs) are optical analogs to chemicalmolecules, i.e., coupled cavities whose resonant modes aresimilar to electronic states in chemical molecules. Suchsimilarities have been discussed for PMs consisting ofcoupled two-vertical-cavity surface-emitting lasers1) and forcoupled microspheres.2) In a recent study, we fabricated PMsof two or more microdisk lasers (MDLs) and observed theirphotopumped lasing as well as their anti-crossing modalcharacteristics and mini-band formation arising from modecoupling.3) We also observed a bistability based on saturableabsorption or gain switching in the MDL-PM by separatinggain and absorption regions.4,5) The threshold of the bi-stability is less than 100 mW even under continuous waveoperation at room temperature. Thus, MDLs are effective fordemonstrating the fundamental properties and useful func-tions of PMs. However, one disadvantage of the MDL-PM ishigher order mode coupling, which is less similar to simplechemical molecules such as H2. The MDL has a whisperinggallery mode (WGM) as a cavity mode, which maintains ahigh Q because of total internal reflection at the disk edge.This mode cannot be the fundamental mode or a lower ordermode but a higher order mode beyond the 10th, and itsvolume is typically as large as 10ð�=n1Þ3, where � is theresonant wavelength and n1 is the index of the activelayer.6,7)

In this study, we propose and demonstrate PMs in aphotonic crystal (PC). A point defect introduced into auniform PC can be a nanocavity laser.8–11) It maintains ahigh Q fundamental or lower order mode with a volume ofless than ð�=n1Þ3. Therefore, two or more adjacent pointdefects are a PM exhibiting greater similarities to simplechemical molecules. In the MDL-PM, the coupling is verysensitive to the interdisk spacing, because it takes placebetween exponentially decaying evanescent fields of modes.On the other hand, coupling is easier to control in the PC-PM, because the mode is confined by Bragg reflection,which has a greater penetration depth than the total internalreflection.

In this study, we present finite difference time domain(FDTD) analysis of PC-PMs in §2. Here, we first showthat two kinds of original cavity modes correspond to s- andp-orbital electrons. Then we discuss the bonding andantibonding of the original modes in PC-PMs, which aresimilar to �- and �-orbital electrons in chemical molecules.In §3, we describe the fabrication of the PC-PMs intoGaInAsP PC slabs and the observation of photopumpedlasing. We also evaluate mode coupling from the splitting ofthe resonance and its dependence on the coupling strength.

2. Theoretical Analysis

In the analysis, a PC slab consisting of circular airholesarranged in a triangular lattice is assumed as the fundamentalmodel, and the modal characteristics of defects and PMs arecalculated using a two-dimensional FDTD method with theequivalent index approximation of the slab. The normalizedairhole diameter 2r=a is set to 0.545 for the lattice constanta, the equivalent index of the slab neq ¼ 2:73, the index ofairholes is 1.0, and the Yee cell size is a=31:4. The initialwave is given for the vertical component of the magneticfield Hz by a Gaussian pulse with a center-normalizedfrequency a=� of 0.275. This induces transverse electricpolarization, for which the PC slab has an in-plane photonicbandgap.

Figure 1 shows the distributions of Hz and the in-planeelectric field vector Exy of original cavity modes for twokinds of point defects with the coordinate system andcorresponding Brillouin zone. Figure 1(a) assumes a point-shift defect, in which two adjacent airholes are each shiftedby 0:17a.11) In the following, we refer to this as an H0defect. The figure shows a monopole Hz distribution with nonodes in the cavity. Due to the differential relations Ex /@Hz=@y and Ex / �@Hz=@x, Exy rotates around the Hz pole.Because it looks like an s-orbital electron in an atom, werefer to it as an s mode. Figure 1(b) assumes a point-missingdefect, usually called an H1 defect, in which one airhole isremoved. This structure should maintain three-fold rotation-ally symmetric modes having dipole Hz distributions withone node and figure-of-eight Exy distributions. It is wellknown, however, that the degeneracies in one direction (xdirection in the case of Fig. 1) and in the two other�E-mail address: baba@ynu.ac.jp

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directions are broken, because the symmetry of the actualcalculation model (and fabricated devices) is not perfect. Asa result, two dipole modes in the x and y directions appearwith different resonant frequencies. They act as differentoriginal modes in the PMs, as shown below. Since they looklike p-orbital electrons in an atom, we call them px and pymodes. Thus, the original modes in the PC are very simple,compared with whispering gallery modes in the MDL.

Figure 2 shows the field distributions in a PM consistingof two H0 defects. Here, the defects are aligned on the y axis(�–X PM), and the number of airholes N separating thedefects is set to 1. In general, two modes characterized by in-phase and out-of-phase coupling appear in diatomic PMs.Note here that the coupling phase in Hz is opposite that inExy due to the differential relationships. In most cases, Exy

has a higher intensity than Hz on or near the centerlinebetween the two defects because of the discontinuity in Exy

at the boundary of airholes separating the defects. Thismeans that Exy (particularly the electric field componentparallel to the centerline, which contributes to the coupling)mainly determines the parity of the coupling. Therefore, wedefine the bonding and antibonding modes as those showingin-phase and out-of-phase couplings of Exy, respectively.The distribution in Fig. 2 also shows a similarity to thesituation in which s-orbital electrons form �-orbital bonding

and antibonding states in a chemical molecule. Therefore,we call this �-coupling. Figure 3 shows the field distribu-tions in PMs consisting of the two H1 defects shown inFig. 1(b). As shown in Fig. 3(a), the PM with two defectsaligned on the x axis (�–J PM) exhibits bonding andantibonding modes formed by the coupling of two px or pymodes shown in Fig. 1(b) (four modes in total). In thebonding and antibonding modes, the two px modes exhibit�-coupling, whereas the two py modes exhibit �-coupling,which is similar to the situation in which p-orbital electronsform �-orbital states in a chemical molecule. Four similarlycoupled modes can be seen for the �–X PM with two H1defects, as shown in Fig. 3(b). Thus, PC PMs provide abetter analogy to chemical molecules than MDL PMs.

The formation of bonding and antibonding modes leads toa resonant frequency splitting, the width of which dependson the coupling strength. Figure 4 shows the normalized

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Fig. 1. FDTD model and field distributions of original modes with

coordinate system and Brillouin zone. (a) H0 defect. (b) H1 defect. Colors

show Hz field with relative intensity from �1 (blue), 0 (green) to 1 (red).

Arrows denote Exy vectors.

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Fig. 2. FDTD models of PMs consisting of H0 defects and field

distributions of modes. (a) Bonding mode. (b) Antibonding mode.

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Fig. 3. FDTD model of PMs consisting of H1 defects and field

distributions of modes. (a) �–J PM. (b) �–X PM. Images on the left

shows the bonding mode, and those on the right, the antibonding mode.

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frequency a=� of each mode in the PC-PMs with H1 defectscalculated as a function of N. The splitting becomes wideras N decreases. In addition, it is wider for �-coupling and inthe �–X PM than for �-coupling and in the �–J PM. Inparticular, the splitting is wide in the �–X PM even forN ¼ 5 and a corresponding interdefect spacing of 2

ffiffiffi

3p

aþ2r (�1:8 mm in the experiment shown in the next section).Thus, in PC PMs, the mode coupling and splitting areobtained more easily than in MDL PMs, in which theinterdisk spacing has to be narrower than 0.2 mm.3) Thedifference in the splitting between modes and structuresarises from the different field penetration depths of theoriginal mode; the py mode penetrates more than the pxmode, and the py mode penetrates more in the y directionthan in the x direction. Since the �-coupling is particularlyweak, the splitting cannot be observed for N > 1 in theanalysis. In Fig. 4, circles and triangles denote the bondingand antibonding modes, respectively. In most cases, thebonding mode frequency is lower than the antibonding modefrequency. This supports our definition of bonding andantibonding modes. One exceptional case is the �-couplingof the �–J PM. Frequencies of the bonding and antibondingmodes are close, but the former is slightly higher than thelatter. Such an opposite situation may occur due to the veryweak intensity of Exy in the PM near the centerline betweenthe defects. In this case, Hz mainly determines the parity ofthe coupling, and thus, the frequency relation between thebonding and antibonding modes is inverted (or the definitionof these modes should be changed, although we keep theinitial definition in the following discussion to avoidconfusion).

3. Experiment

In device fabrication, we used GaInAsP/InP epitaxialwafers. The active layer in each wafer consisted of fivecompressively strained quantum wells, and its total thicknessand photoluminescence peak were 0.2 and 1.58 mm, respec-tively. For this wafer, an airbridge PC slab in a triangularlattice was formed using e-beam lithography, HI/Xeinductively coupled plasma etching,12) and HCl wet etchingof InP. Scanning electron microscopy (SEM) images offabricated PC-PMs with H0 defects and H1 defects areshown in Fig. 5. Here, the lattice constant a is 0.46 and0.44 mm for the former and latter, respectively, and the basic

airhole diameter 2r is 0.22 mm (2r=a ¼ 0:50 for the latter).The exceptional airhole diameter 2r0 around the defects is setto be 0.20 mm (2r0=a ¼ 0:45 for the latter) to reduce out-of-plane radiation loss.10) The number of airholes N is changedbetween samples.

In the measurement, the entire area of the PM wasphotopumped by 0.98 mm pulsed laser light at room temper-ature at a duty ratio of 0.075% and a focused spot diameterof 3.5 mm. Output light from the device was coupled to anoptical fiber through the upper lens system after filtering thepump light and analyzed using an optical spectrum analyzer.The lasing characteristics measured for the �–X PM of H0defects with N ¼ 1 and the �–J PM of H1 defects withN ¼ 2 are shown in Fig. 6. Two neighboring mode peakswere observed for both devices. Regarding the PM with H0defects, only this device lased. Therefore, it is difficult todiscuss modal behavior as a PM for this sample. On the otherhand, we observed lasing in many samples of PMs with H1defects. The typical threshold pump power was 1.8–2.0mW. This threshold value is nearly twice that for a single H1defect laser fabricated by the same process.12) This result isreasonable because the optical gain was divided into the twomodes. When 2r was reduced to 0.20 mm (2r=a ¼ 0:46), 2r0

was also reduced due to the proximity effect of e-beamlithography, and the lasing wavelengths became 10–30 nmlonger. This should mainly be caused by the change in thecavity length. For a 10 nm reduction in 2r0, the cavity lengthmeasured from the space between the innermost airholesbecomes �1:5% smaller. It corresponds to a wavelengthshift of �20 nm at a lasing wavelength � � 1:6 mm.

Figure 7 shows the dependence of two lasing wavelengthson N. For the �–J PM, wavelength splitting was clearlyobserved for N < 5, and it was increased by decreasing N, asshown in Fig. 7(a). This result indicates that two-modelasing arises from the bonding and antibonding modes. It is

(a) (b)

N1 3 5

N

0.310

0.300

0.295

1 2 3 4 5

0.305

0.315

{N

N {

Fig. 4. Calculated normalized frequency a=� of coupled modes, when N

is the number of airholes separating defects. (a) �–J PM. (b) �–X PM.

(a)

(b)

Fig. 5. Top views of fabricated devices. (a) PM with H0 defects. Two

white arrows indicate positions of H0 defects. (b) PMs with H1 defects.

The left image shows �–J PM, and the right �–X PM.

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difficult to directly specify whether the lasing occurred by�-coupling or �-coupling. Therefore, we carried out three-dimensional (3D) FDTD simulation by precisely modelingthe experimental device. We found that the observed modesare the bonding and antibonding modes of the �-coupling,because the out-of-plane loss of the �-coupling is very large,and only the �-coupling shows Q factors higher than 1000

for N < 2. The splitting in Fig. 7(a) is much wider than thecalculated one shown in Fig. 4(a). This may be caused bythe size difference between the two defects. As mentionedabove, a 10 nm change in 2r0 shifts the lasing wavelength by�20 nm and changes the mode-splitting characteristics.Although Fig. 7(a) suggests that the splitting arises partlyfrom the coupling as it decreased with larger N. For the �–XPM, two-mode lasing was not confirmed but only single-mode lasing, as shown in Fig. 7(b). The 3D FDTD calcu-lation also indicated that only antibonding modes of the�-coupling and the �-coupling have high Qs and inparticular, the former has a high Q of 3500. Therefore,single-mode lasing is a reasonable result for this PM.

4. Conclusions

We proposed PMs formed by point defect nanocavities inthe PC slab. First, we theoretically showed that the H0 defectand the H1 defect have s- and p-orbital resonant modes,respectively, as original modes, and that PMs formed bythese defects exhibit �- and �-orbital bonding and anti-bonding modes. Such simple original modes and modecouplings analogous to those in chemical molecules andeasy couplings due to the large field penetration from thecavity are unique features of PC PMs which cannot beobtained in MDL PMs. We fabricated PMs with H0 and H1defects in GaInAsP slabs, and observed two-mode lasing byphotopumping. The dependence of the mode splitting on thecoupling strength was suggested for the H1 PM. In PCs,various PMs can be formed by lithography which will be aninteresting area for future study. Another future target is thefabrication of a photonic integrated circuit, in whichfunctional PM devices such as bistable switches aremonolithically integrated with other PC devices.

Acknowledgment

This work was supported in part by a Grant-in-Aid, theIT Program, and the 21st Century COE Program of Ministryof Education, Culture, Sports, Science and Technology, anda Grant-in-Aid from the Japan Society for the Promotionof Science.

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H1 defects. The inset shows the lasing spectrum above the threshold.

Circles and triangles denote longer and shorter wavelengths, respectively.

1.54

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Fig. 7. Lasing wavelengths � (left axis) and corresponding a=� (right

axis) measured as functions of N in PMs with H1 defects. (a) �–J PM.

(b) �–X PM.

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