1Ovhrgtsalhgiwawpsbhn

AwbswsattoFt

Wang et al. Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. B 719

Long rectangle resonator 1550 nmAlGaInAs/InP lasers

Shi-Jiang Wang, Yong-Zhen Huang,* Yue-De Yang, Jian-Dong Lin, Kai-Jun Che, Jin-Long Xiao, and Yun Du

State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences,P.O. Box 912, Beijing 100083, China

*Corresponding author: yzhuang@semi.ac.cn

Received December 9, 2009; revised January 29, 2010; accepted February 7, 2010;posted February 17, 2010 (Doc. ID 121205); published March 22, 2010

1550 nm AlGaInAs/InP long rectangle resonator lasers with three sides surrounded by SiO2 and p electrodelayers are fabricated by planar technology, and room-temperature continuous-wave lasing is realized for a la-ser with a length of 53 �m and a width of 2 �m. Multiple peaks with wavelength intervals of Fabry–Pérotmode intervals and mode Q factors of about 400 and a lasing mode with a Q factor over 8000 are observed fromthe lasing spectrum at threshold current. The numerical results of the FDTD simulation indicate that the las-ing mode may be a whispering-gallery mode, which is a coupled mode of two high-order transverse modes ofthe waveguide. © 2010 Optical Society of America

OCIS codes: 140.3945, 140.4780, 140.5960.

2TftswsaltstboetltbmstipASrStgTdaw

. INTRODUCTIONptical microcavities with the advantages of ultrasmallolume, high-mode Q factor, and large free-spectral rangeave attracted a great deal of attention. The Fabry–Pérotesonator, photonic crystal resonator, and whispering-allery mode (WGM) resonator are three basic types of op-ical microcavities [1]. Vertical-cavity surface-emitting la-ers and microdisk lasers are examples of Fabry–Pérotnd WGM-type microresonators. In addition to microdiskasers [2–4], equilateral-triangle [5], square [6,7], andexagonal resonator [8] microlasers are also whispering-allery-type microlasers. Recently, lasing in metal–nsulator–metal subwavelength plasmonic waveguidesas realized with the plasmon mode [9]. For rectangularnd square microresonators, mode field characteristicsere analyzed and numerically simulated with an em-hasis on the WGMs of total internal reflection on theides of the resonators [10–14], and high-Q WGMs formedy mode coupling in the rectangle microresonators, whichave very weak radiation loss in the vertices of the reso-ators, were analyzed [15].In this paper, long rectangle resonator 1550 nm

lGaInAs/InP lasers with a cavity length of 53 �m and aidth of 2 �m are fabricated with three sides surroundedy an insulating SiO2 layer, p electrode metals, and oneide of a cleaved mirror. Room-temperature continuous-ave (CW) operation is realized, and the lasing output

pectra are observed with series modes with Q factors ofbout 500 and a lasing mode with a Q factor over 8000 athreshold current. The FDTD simulation results indicatehat the high-order transverse modes, such as the fourth-rder transverse mode, can be assigned as the WGMs.urthermore, the lasing mode may be the coupled mode ofwo high-order transverse modes.

0740-3224/10/040719-6/$15.00 © 2

. DEVICE FABRICATIONhe IQE (Europe) Ltd AlGaInAs/InP laser wafer is used

or fabrication of long rectangle resonator lasers. The ac-ive region of the laser wafer is five compressivelytrained quantum wells, with thicknesses of the quantumells and barrier layers of 6 and 10 nm, respectively,

andwiched between 60 nm undoped graded AlGaInAsnd 60 nm doped AlGaInAs cladding layers. The upperayers are p-InP and InGaAs contacting layers with a to-al thickness of 1920 nm. The long rectangle resonator la-ers with a width of 2 �m are fabricated under the sameechnique process [5]. First, an 800 nm SiO2 is depositedy plasma-enhanced chemical vapor deposition (PECVD)n the AlGaInAs/InP laser wafer as a hard mask for drytching. Then, the rectangle cavity resonator patterns areransferred onto the SiO2 layer by using standard photo-ithography and inductively coupled plasma (ICP) etchingechniques, and the AlGaInAs/InP laser wafer is etchedy another ICP process with patterned SiO2 as hardasks. Figure 1(a) shows the scanning electron micro-

cope image of one side of a rectangle waveguide resona-or after the ICP etching process where the etching depths about 5 �m. After the ICP etching, a chemical etchingrocess is used to improve the smooth side walls of thelGaInAs/InP rectangle waveguide. Then the residualiO2 hard masks on top of the rectangular waveguide areemoved using a diluted HF solution. Finally, a 450 nmiO2 insulating layer is deposited on the laser wafer, andhe SiO2 insulating layer on top of the rectangular wave-uide is etched using the ICP etching process again. A topi–Pt–Au p contact is formed using a standard metaleposition process, and Au–Ge–Ni metallization is useds an n-type contact metal after lapping down the laserafer to a thickness of about 100 nm. Figure 1(b) shows

010 Optical Society of America

trFlem

3AsbCTaaw=chc

aoca0�si

oplpia1sah1ttn5t

Faame

Fcrw

For1

720 J. Opt. Soc. Am. B/Vol. 27, No. 4 /April 2010 Wang et al.

he scanning electron microscope image of a rectangleesonator after the top Ti–Pt–Au p contact is formed, andig. 1(c) is the schematic of the long rectangle resonator

asers encapsulated by the insulating SiO2 layer and plectrode metal on three sides and one side of the cleavedirror.

. LASER OUTPUT CHARACTERISTICSfter cleaving one side of the long rectangle resonator la-ers, the lasers are measured through the cleaved mirrory directly placing them on a heat sink and injecting aW current through a golden probe at room temperature.he output power coupled into a multimode optical fibernd the applied voltage versus the CW injection currentre plotted in Fig. 2 for a long rectangle resonator laserith a cavity length of L=53 �m and a width of a2 �m. The curve of output power versus the injectionurrent shows the threshold current of about 15.6 mA. Weave two lasers in the cleaved wafer, which is too small toleave again. The other laser has a threshold current of

ig. 1. (Color online) Scanning electron microscope images of (a)rectangle resonator after the ICP etching process and (b) a rect-ngle resonator after top Ti–Pt–Au p contact is formed. (c) Sche-atic of the long rectangle resonator lasers encapsulated by p

lectrode metal.

bout 14 mA. Assuming the current is injected uniformlyn the two lasers, we estimate the practical thresholdurrent of the long rectangle laser to be 3.9 mA. Thepplied voltage V versus the injection current I from.5 mA to 30 mA can be fitted by I=2.510−13�exp��V-IR� /1.55kT�−1� �mA� with the series re-

istor R=16 �, where k is the Boltzmann constant and Ts the absolute temperature.

The laser spectra measured at room temperature by anptical spectrum analyzer at a resolution of 0.1 nm arelotted in Fig. 3 at an injection current of 25 mA. Theasing-mode wavelength is 1560 nm, and distinct multipleeaks with near-uniform wavelength intervals appearn the laser spectrum. The mode wavelength intervalsnd mode Q factors of the multiple peaks from480 to 1530 nm are plotted as open circles and solidquares in Fig. 4, where the mode Q factors are obtaineds the ratio of mode wavelength to the full widths of thealf-maximum of each peak by fitting the spectrum from480 to 1530 nm with a multiple-peak Lorentzian func-ion. The solid lines are mode spacing ��=�2 /2ngL withhe group index varied with the photon energy E (eV) asg=3.0715+0.366E [16] with cavity lengths of 53 and6.3 �m. The mode Q factors are around 400 with a fluc-uation of about 100 for the modes from 1480 to 1530 nm.

ig. 2. Output power coupled to a multimode optical fiber (solidurve) and applied voltage (dashed curve) versus injection cur-ent for a rectangular waveguide laser with length of 53 �m andidth of 2 �m at room temperature.

ig. 3. Laser spectra of a rectangle resonator laser with lengthf 53 �m and width of 2 �m at injection current of 25 mA atoom temperature. The inset shows the lasing mode at current of7 mA.

H�ta

dtjdcohti��m�aTts

4F�

tinatt

wLnrntdf

wcrattov

Ig

Ftbt

Adri

Ut

F2

FooF+

Wang et al. Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. B 721

owever, the Q factor of the lasing mode is about 8.86103 obtained from the inset spectrum in Fig. 3 near the

hreshold current of 17 mA, which has a linewidth ofbout 0.176 nm.Figure 5 shows detailed spectra of the lasing mode un-

er injection currents of 20, 25, 30, and 35 mA, respec-ively. The lasing peaks are evident double modes at in-ection currents of 20 and 25 mA, which may beegenerate modes in the rectangle resonator [15]. Be-ause the resonator is surrounded by p electrode metalsn three sides of the resonator, the degenerate modes mayave comparative-mode Q factors instead, if only one ofhem has a high-Q factor. Mode wavelength shift �� var-es with the increase of injection current �I at the rate of� /�I=0.091 nm/mA. Thermal resistance of ��� /�T�−1

�� /�IV�=652 K/W is obtained from variation of theode wavelength with the injection current, where

�� /�T� is taken as a typical value of 0.1 nm/K and thepplied voltage V is 1.4 V from the V-I curve in Fig. 2.he high thermal resistance can be partly explained, ashe tested laser wafer is only put on a Cu heat sink in-tead of being soldered on the heat sink.

. MODE ANALYSISor a 2D rectangle resonator with length L and width a

L�a� surrounded by air, we can assume that the fields of

ig. 5. Detail spectra of a lasing mode at injection currents of0, 25, 30, and 35 mA, respectively.

ig. 4. Mode interval (circle) and mode Q factors (solid square)f multiple peaks from 1480 to 1530 nm are plotted as functionsf mode wavelengths. The solid line is the mode interval ofabry–Pérot resonator with a group index of ng=3.07150.366E.

he confined modes are exponentially decayed when mov-ng away from the rectangle sides. Choosing the coordi-ate origin at the center of the rectangle resonator with xnd y axes of the symmetry axes of the rectangle resona-or, we can express magnetic field Hz for the TE mode inhe rectangle resonator as Fz

p,q�x ,y�=Fp�x�Fq�y� [12,15],

Fp�x� = �cos��xx − �x� �x� L/2

cos��xL/2 − �x�exp�− x�x − L/2�� x � L/2

cos�− �xL/2 − �x�exp�x�x + L/2�� x � − L/2� ,

�1�

here Fq�y� takes the same form in Eq. (1) by replacing p,, and x with q, a, and y. The mode numbers p and q de-ote the number of wave nodes in the x and y directions,espectively, and �x and �y are zero or /2 when the modeumbers p and q are even or odd. �v and v �v=x ,y� arehe propagation constants in the rectangle resonator andecay constants in the external region. They satisfy theollowing relations:

�x2 + �y

2 = N2k02, �2�

�v2 + v

2 = �N2 − 1�k02, v = x,y, �3�

here N is the refractive index of the resonator. For theonfined modes, their incident angles at the sides of theectangle resonator should be larger than the criticalngle of the total internal reflection, which is equivalento v�0 in Eq. (3). Based on the continuous conditions ofhe tangential electric field for the TE modes at the sidesf the rectangle resonator, we can obtain the mode eigen-alue equations as

�x tan��xL/2 − �x� = N2x,

�y tan��ya/2 − �y� = N2y. �4�

n the total confinement approximation, the mode propa-ation constants are

�kxL = �p + 1�

kya = �q + 1� . �5�

or the long rectangle resonator, we define the WGM ashe mode with totally internal reflection (TIR) at theoundary of the resonator. With the approximation solu-ions of Eq. (5), the mode wavelength is

� =2 N

��q + 1�

a �2

+ � �p + 1�

L �2. �6�

ssuming the mode number p is much larger than unityue to the long cavity length and ignoring the dispersiveelation of the refractive index, we can deduce the modenterval for different transverse modes as

�� �3

4 N2L�2N

��2

− � �q + 1�

a �2

. �7�

nder the condition of total confinement approximation,he incident angles of 6.7°, 13.5°, 20.6°, 27.8°, and 35.9°

cfiovtflrtrttocLdarmm

5Fsntwwtrngydr0tsrlrlBwSipic[tccF1dfiiattf

tbw

=cvn=ttcltfm�tTacQtvtp

Fcls

722 J. Opt. Soc. Am. B/Vol. 27, No. 4 /April 2010 Wang et al.

an be obtained from Eqs. (2) and (5) for the fundamental,rst-, second-, third-, and fourth-order transverse modesn the output face as �=1500 nm and N=3.2. The trans-erse mode with mode number q�2 will be total reflec-ivity, because the critical angle of the total internal re-ection is 18.2°. The longitudinal mode interval of theectangle resonator decreases with the increase of theransverse mode number q, as shown in Eq. (7). For theesonator with N=3.2, a=2 �m, and L=53 �m, the longi-udinal mode intervals are 6.6 nm and 5.4 nm, respec-ively, for the fundamental transverse and the fourth-rder transverse modes at �=1500 nm. Theorresponding mode intervals are 34.9 nm and 28.5 nm as=10 �m. Considering that the mode interval should beetermined by the group index instead of the mode indexnd the group index ng=3.55 at 1500 nm from the fittingelation in Fig. 4, we can confirm that the experimentode intervals agree very well with those of the funda-ental modes.

. FDTD NUMERICAL SIMULATIONinally, the mode characteristics of the TE modes areimulated for the long rectangle resonator by FDTD tech-iques [17]. The optical confinement in the vertical direc-ion of the resonator is provided by a multiple-layer slabaveguide of the laser wafer, and the resonator is etchedith the etched sidewalls covering the vertical field dis-

ribution of the slab waveguide. So the long rectangleesonator can be simply simulated by a 2D FDTD tech-ique with the effective refractive index of the slab wave-uide of the laser wafer. The 2D FDTD simulation canield correct mode wavelengths but ignore the vertical ra-iation loss [18]. In the simulation, the three sides of theectangle resonator are assumed to be surrounded by.2 �m insulating SiO2 and 0.2 �m gold with the refrac-ive indices of 1.45 and 0.18+10.2i [19]. The other side isurrounded by air as the output mirror, and the effectiveefractive index of the resonator is taken to be 3.2. So theong rectangle resonator is equivalent to a Fabry–Pérotesonator with one side confined by the insulating SiO2ayer and p electrode and the other side of cleaved mirror.ut the transverse direction of the resonator with theidth of 2 �m is also strongly confined by the insulatingiO2 layer and p electrode. A Gaussian modulated cosine

mpulse covering a wide frequency band is used as the in-ut wave P�x0 ,y0 , t�=exp�−�t− t0�2 / tw

2 �cos�2 f0t�, where t0s the pulse center, tw is the pulse half-width, and f0 is theenter frequency of the pulse and the “bootstrapping”20]. The spatial cell size and the time step �t are takeno be 20 nm and the Courant limit, respectively, and a 50-ell perfectly matched layer (PML) absorbing boundaryondition is used as the boundaries to terminate theDTD computation window with the PML boundaries�m away from the rectangle resonator. The time-

omain outputs of one component of the electromagneticelds are recorded at an arbitrarily selected monitor point

nside the resonator as an FDTD output. Then the Padépproximation with Baker’s algorithm [21] is used toransform a late FDTD output from the time domain tohe frequency domain, and the mode frequencies and Qactors are obtained from the peak frequency and the ra-

io of the peak frequency to the corresponding 3 dBandwidth of the peak by fitting the intensity spectraith a multipeak Lorentzian function.Choosing the exciting sources for the fundamental �q

0� and the fourth-order �q=4� transverse modes of theorresponding 2-�m-wide waveguide, respectively, we in-estigate mode characteristics for the long rectangle reso-ators. Using a wide frequency excitation pulse with tw500�t, ,t0=1000�t, and f0=200 THz, the intensity spec-

ra of the modes with the exciting sources correspondingo q=0 and q=4 are plotted in Fig. 6 as dashed and solidurves, respectively, for the long rectangle resonator withengths of (a) 10 �m and (b) 53 �m. The results show thathe mode spacing is about 6.6 �35.2� nm and 5.5 �28.9� nmor the fundamental and the fourth-order transverseodes in the long rectangle resonator with a length of 53

10� �m, which agrees very well with Eq. (7). The ob-ained mode wavelengths and mode Q factors are listed inable 1 for the modes with q=0 and 4 in the long rect-ngle resonator with a length of 53 �m. The results indi-ate the fourth-order transverse modes have much larger

factors than the fundamental modes. The highest Q fac-or of 1.31�105 is obtained for the fourth-order trans-erse mode at a wavelength of 1523.7 nm, which may behe high-Q whispering-gallery mode formed by mode cou-ling as discussed in [15]. For the long rectangle resona-

ig. 6. Intensity spectra of q=0 (dashed curve) and q=4 (solidurve) modes for a rectangle resonator with width of 2 �m, andengths of (a) 10 �m and (b) 53 �m obtained by FDTDimulation.

t8�1d31

ttrmtqgttptewtfnnoBrfiw F

mwqs+nrim

6Wc

F=1fTa

Ffir

Wang et al. Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. B 723

or with a length of 10 �m, we have mode Q factors of.29�103, 6.75�103, 2.12�103, 1.20�103, and 3.51103 for fourth-order transverse modes at wavelengths of

452.8, 1480.6, 1509.5, 1538.8, and 1568.5 nm. The fun-amental transverse modes have mode Q factors of 216,81, 345, and 330 at mode wavelengths of 1449, 1481.9,517.1, and 1551.5 nm.Using a single-mode exciting source with tw=20000�t,

0=40000�t, and f0 as the mode frequency, we simulatehe mode field pattern for the 53 �m�2 �m rectangleesonator by the FDTD technique. The field pattern of theagnetic field components Hz for the TE modes are plot-

ed in Fig. 7 for (a) the q=0 mode at 1509.7 nm and the=4 mode at (b) 1501.2 nm and (c) 1523.7 nm in the re-ion of 3.5 �m�2 �m near the output facet of the resona-or, where the positions of the p electrode metal, SiO2, andhe resonator are indicated by the solid lines. The fieldatterns of Figs. 7(a) and 7(b) are those of the fundamen-al and the fourth-order transverse modes, clearly. How-ver, Fig. 7(c) shows a more complicated field pattern,hich corresponds to the coupled mode with small radia-

ion loss at the vertices of the resonator and the highest Qactor of 1.31�105, as shown in [15]. To explain this phe-omenon, we calculate the field distributions in the waveumber k space by transforming the field distributionver 2 �m�2 �m into the k space and plot them in Fig. 8.ecause the field distributions in the k space are symmet-

ic relative to the kx=0 and ky=0 axes, we only plot theeld distribution in the space of kx�0 and ky�0 in Fig. 8,here Figs. 8(a)–8(c) correspond to the field patterns in

Table 1. Mode Wavelengths and Q Factors forFundamental and Fourth-order Transverse Modes

in Long Rectangle Resonator with Length of53 �m and Width of 2 �m Obtained by

FDTD Simulation

q=4 q=0

� (nm) Q � (nm) Q

1495.7 2.02�104 1483.5 1.37�103

1501.2 5.24�103 1490.0 1.97�103

1506.8 4.01�103 1496.4 6.28�102

1512.4 2.98�103 1503.0 1.48�103

1518.2 2.57�103 1509.7 2.19�103

1523.7 1.31�105 1516.3 1.57�103

ig. 7. Field pattern of magnetic field component Hz for (a) q0 mode at 1509.7 nm; q=4 modes at (b) 1501.2 nm and (c)523.7 nm in the region of 3.5 �m�2 �m near the output facetor the 53 �m�2 �m rectangle resonator by FDTD simulation.he solid curves indicate positions of the p electrode metal, SiO2,nd the resonator.

igs. 7(a)–7(c), respectively. The peak value of the q=0ode in Fig. 8(a) is at ky=0, which is induced by the twoaves with ky= /a and ky=− /a. The peak value of the=4 mode in Fig. 8(b) is near ky=7.5, which is a littlemaller than the total confined approximation value �q1� /a=7.85. Two distinct peaks are found in Fig. 8(c)ear ky=7.5 and ky=10.5, which correspond to q=4 and 6,espectively. The field distribution in the k space clearlyndicates that the high-Q mode is the coupled mode of two

odes with different amplitudes.

. SUMMARYe have fabricated 1550 nm AlGaInAs/InP lasers using

onventional photolithography and the ICP etching tech-

ig. 8. (Color online) Fourier transform (FT) of cavity-modeeld patterns in Figs. 7(a)–7(c) plotted in (a), (b), and (c),espectively.

ntrromwttgom

e6stT

R

1

1

1

1

1

1

1

1

1

1

2

2

724 J. Opt. Soc. Am. B/Vol. 27, No. 4 /April 2010 Wang et al.

ique. Cw electrically injected operations are realized forhe laser with a length of 53 �m and a width of 2 �m atoom temperature. The laser output spectra indicate a se-ies of low-Q modes and a lasing mode with a Q factor onerder larger than that of the low-Q modes. The lasingode can be assigned as the whispering gallery modes,hich correspond to the coupled mode of higher-order

ransverse modes. FDTD numerical simulation verifieshe experimental results, and a high-Q whispering-allery mode with field distribution of coupled modes isbserved from the field distribution in the k space nu-erically.

This work was supported by the National Nature Sci-nce Foundation of China (NNSFC) under grants0777028, 60723002, and 60838003; the Major State Ba-ic Research Program under grant 2006CB302804; andhe Project of National Lab for Tsinghua Informationechnologies.

EFERENCES1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846

(2003).2. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and

R. A. Logan, “Whispering-gallery mode microdisk lasers,”Appl. Phys. Lett. 60, 289–291 (1992).

3. M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralowthreshold GaInAsP-InP microdisk injection lasers: design,fabrication, lasing characteristics, and spontaneous emis-sion factor,” IEEE J. Sel. Top. Quantum Electron. 5, 673–681 (1999).

4. Q. Song, H. Cao, S. T. Ho, and G. S. Solomon, “Near-IR sub-wavelength microdisk lasers,” Appl. Phys. Lett. 94, 061109(2009).

5. Y. Z. Huang, Y. H. Hu, Q. Chen, S. J. Wang, and Z. C. Fan,“Room-temperature continuous-wave electrically injectedInP-GaInAsP equilateral-triangle-resonator lasers,” IEEEPhotonics Technol. Lett. 19, 963–965 (2007).

6. H. J. Moon, K. An, and J. H. Lee, “Single spatial-mode se-lection in a layered square microcavity laser,” Appl. Phys.Lett. 82, 2963–2965 (2003).

7. Y. Z. Huang, K. J. Che, Y. D. Yang, S. J. Wang, and Z. C.Fan, “Directional emission InP/InGaAsP square-resonatormicrolasers,” Opt. Lett. 33, 2170–2172 (2008).

8. C. F. Zhang, F. Zhang, X. W. Sun, Y. Yang, J. Wang, and J.Xu, “Frequency-upconverted whispering-gallery mode las-

ing in ZnO hexagonal nanodisks,” Opt. Lett. 34, 3349–3351(2009).

9. M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. C.Zhu, M. H. Sun, P. J. Veldhoven, E. J. Geluk, F. Karouta, Y.S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing inmetal–insulator–metal subwavelength plasmonicwaveguides,” Opt. Express 17, 11107–11112 (2009).

0. M. Lohmeyer, “Mode expansion modeling of rectangular in-tegrated optical microresonators,” Opt. Quantum Electron.34, 541–557 (2002).

1. M. Hammer, “Resonant coupling of dielectric opticalwaveguides via rectangular microcavities: the coupledguided mode perspective,” Opt. Commun. 214, 155–170(2002).

2. W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes insquare resonators,” IEEE J. Quantum Electron. 39, 1563–1566 (2003).

3. W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Mode qual-ity factor based on far-field emission for square resonator,”IEEE Photonics Technol. Lett. 16, 479–481 (2004).

4. D. V. Batrak, A. P. Bogatov, A. E. Drakin, and D. R. Mifta-khutdinov, “Modes of a semiconductor rectangular micro-cavity,” Quantum Electron. 38, 16–22 (2008).

5. Y. D. Yang and Y. Z. Huang, “Mode analysis and Q-factorenhancement due to mode coupling in rectangle resona-tors,” IEEE J. Quantum Electron. 43, 497–502 (2007).

6. K. J. Che, Y. D. Yang, and Y. Z. Huang, “Multimode reso-nances of metallic-confined square-resonator microlasers,”Appl. Phys. Lett. 96, 051104 (2010).

7. A. Taflove and S. C. Hagness, in Computational Electrody-namics: The Finite-Difference Time-Domain Method (ArtechHouse, 2005).

8. Y. D. Yang, Y. Z. Huang, and Q. Chen, “Comparison ofQ-factors between TE and TM modes in 3-D Microsquaresby FDTD simulation,” IEEE Photonics Technol. Lett. 19,1831–1833 (2007).

9. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R.W. Alexander, Jr., and C. A. Ward, “Optical properties of themetals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W inthe infrared and far infrared,” Appl. Opt. 22, 1099–1020(1983).

0. S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove,“FDTD microcavity simulations design and experimentalrealization of waveguide-coupled single-mode ring andwhispering-gallery mode disk resonators,” J. LightwaveTechnol. 15, 2154–2165 (1997).

1. W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of reso-nant frequencies and quality factors of cavities by FDTDtechnique and Padé approximation,” IEEE Microw. Wirel.Compon. Lett. 11, 223–225 (2001).