October 15, 2003 / Vol. 28, No. 20 / OPTICS LETTERS 1861

Spectrum of three-dimensional photonicquantum-ring microdisk cavities: comparison

between theory and experiment

Joongwoo Bae, Jawoong Lee, and O’Dae Kwon

Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Pohang 790-784, Korea

Vladimir G. Minogin

Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow Region, Russia

Received April 7, 2003

The spectrum of a three-dimensional Rayleigh–Fabry–Perot microdisk cavity of a photonic quantum-ringlaser is analyzed for angle-dependent emission modes. It is shown that joint consideration of the Fabry–Perotresonance condition and the whispering-gallery mode boundary condition explains the emitting angles andspectral distribution of the cavity modes well. © 2003 Optical Society of America

OCIS codes: 140.5960, 300.6260.

Microcavity lasers are of interest for fundamental stud-ies of cavity quantum electrodynamics as well as forreal applications such as photonic integrated circuitsof high-density optical interconnection and optical sig-nal processing.1 The light-emission modes from semi-conductor microdisk lasers based on two-dimensionalwhispering-gallery (WG) resonances were extensivelystudied in the early 1990s,2,3 and the resonant modesof circular and square-shaped microdisks were also in-vestigated further in detail.4,5

The photonic quantum-ring (PQR) laser is a newtype of microdisk laser that is similar in structureto the vertical-cavity surface-emitting laser but witha much lower threshold. Furthermore, its spec-tral characteristics should be classified as a three-dimensional (3D) Rayleigh–Fabry–Perot (RFP) WGmode laser.1,6,7 We previously reported the angle-dependent spectral characteristics and nonequalintermode spacings of the PQR laser in brief andpresented a torus knot model analysis in which thespecific mode distributions remained unknown.8 Inthis Letter we analyze the WG emission modes of the3D RFP microcavity of the PQR laser in detail and findthat the distribution of the discrete angle-dependentemission modes is in excellent agreement with a newangular quantization theory.

Figure 1 shows a schematic diagram of a cylin-drical RFP cavity of a GaAs PQR laser and thegeometry of the angle-dependent spectrum mea-surements. The p-type (top) and n-type (bottom)l�4 AlAs�As�Al0.12Ga0.88As distributed Bragg ref lec-tor (DBR) layers of 19.5 and 30.5 periods, respectively,vertically conf ine photons, in addition to the in-planeannular confinement, to form a toroidal 3D RFPcavity. The active disk sandwiched by the DBRs con-tains three quantum wells of GaAs, each 6 nm thick,separated by 8-nm-thick barriers of Al0.3Ga0.7As.The well thicknesses and compositions are tuned toyield a resonance wavelength of 850 nm in the verticaldirection. Cylindrical mesas that are 5 mm deep arestructured by chemically assisted ion-beam etching

0146-9592/03/201861-03$15.00/0 ©

and are planarized by polyimide for a 5-mm-widestripe electrode extending to the center of the topmesa. Details of the fabrication and structure aredescribed elsewhere.9 A tapered optical f iber probewith a 3-mm tip (core) diameter is placed near thedevice edge, �3 mm above the emission ring of thePQR device, which is mounted on a rotational stageto change the view angle uv. An optical spectrumanalyzer (HP70004A) measures the spectrum ofemitted light coupled into the f iber probe. For thewhole measurement the devices are operated with cwinjection currents at room temperature.

The emission spectra for a PQR laser of diameterf � 20 mm, collected at view angles of uv � 10± and15± with an injection current of 2.5 mA, are shownin Fig. 2(a). Each spectrum has an envelope of an�3-nm span containing several sharp discrete modes.The envelope of the uv � 15± spectrum is blueshifted by�0.9 nm compared with that of the uv � 10± spectrum.The individual modes in the overlapped envelope rangematch each other well. For a given device the wave-lengths of the emission modes are completely definedby the 3D RFP resonance conditions, whereas theirintensity envelopes vary with the observation angle.Figure 2(b) shows a spectrum measured at the view

Fig. 1. Schematic diagram of a cylindrical PQR laser andits angle-dependent spectrum measurement setup.

2003 Optical Society of America

1862 OPTICS LETTERS / Vol. 28, No. 20 / October 15, 2003

Fig. 2. Spectra of a PQR laser of (a) f � 20 mm at viewangles uv � 10± (right) and uv � 15± (left) with an injectioncurrent of 2.5 mA and (b) f � 35 mm at uv � 15± withan injection current of 10 mA. Vertical stems show thecalculated mode wavelengths.

angle uv � 15± with a current of 10 mA injected into aPQR laser of diameter f � 35 mm. The spectral peaksat 849.8 nm for the 20-mm device and at 851.8 nm forthe 35-mm device are the vertical emission modes.

In the PQR microcavity the helical propagation oflaser light is defined by bouncing the light wave be-tween the DBRs and circulating the wave due to totalinternal ref lection (Fig. 3). By combining these twoaspects, one can express the optical modes of the PQRmicrocavity in cylindrical coordinates as10

Cm�r,w, z� ~ Jm�ktr�exp�6imw�exp�6ikzz� , (1)

where Jm is the Bessel function of integer order m�� 0,61,62,63, . . .� and kz and kt are the longitudinaland transverse components of the wave vector. At agiven incident angle uin these components of the wavevector are defined as

kz � k cos uin, kt � k sin uin , (2)

where k � �2p�l�n, with l being the wavelength infree space and n being an effective refractive index at agiven wavelength. Taking into account that the angleuin is related to the emission angle u as sin u � n sin uinand kz � �2p�l0�n0, where l0 is the wavelength of thelight emitting in the vertical direction in free space andn0 is an effective index for the wavelength l0, one canrewrite the f irst equation in set (2) as

l � l0nn0

∑1 2

µsin u

n

∂2∏1�2. (3)

Considering the boundary condition for the WG reso-nance mode, ktR � x1

m, where R is the radius of thedisk and x1

m is the f irst zero of the Bessel function,4 the

quantization rule for the emission angle is extractedfrom the second equation in set (2) as

sin um � x1mlm�2pR . (4)

By combining Eqs. (3) and (4), the quantized emissionwavelengths and their emission angles can be given by

lm � l0nm

n0

∑1 1

µx1m

a

∂2∏21�2, (5)

um � sin21

(nm

x1m

a

∑1 1

µx1m

a

∂2∏21�2), (6)

where nm is the effective index for the mth mode withwavelength lm, n0 is that for the vertical emissionmode with wavelength l0, which is the longest in thespectrum, and a � 2pRn0�l0 is a characteristic dimen-sionless parameter. For the experimentally observedemission modes of the PQR laser the values of nm arenot known except for that of n0, which is given by thecavity medium. In the simplest approximation the de-pendence of the refractive index on the wavelength canbe considered linear,11

nm � n0 1 a�lm 2 l0� , (7)

where a is a phenomenological constant to be deter-mined from comparison of theoretical and experimen-tal spectra. Differentiating both sides of Eq. (5) withrespect to x1

m and simplifying under typical experimen-tal conditions of x1

ml0 ,, 2pRnm, the absolute value ofthe spectral intermode spacing jDlmj (hereafter Dlmfor convenience) can be found as

Dlm � l0Dx1

m

a2 x1m . (8)

Note that we assumed in Eq. (8) that nm � n0, sincethe accuracy in determining Dlm is much less thanthat of determining lm. Equation (8) can be used as astarting point for characterizing the observed modes.

To assign the mode index m to the measured spectralmodes with unknown indices, one can plot quanti-ties Dlm on an x1

m-scaled x axis and properly shiftthe data positions in the x-axis direction until thelinear extrapolation line of data passes the origin�x1

0, 0�. Thereafter all the measured data can be as-signed with the mode index by converting the scale ofthe x axis from x1

m to the mode index m. The resultsof this process are shown in Fig. 4(a) for two micro-cavities with diameter f � 20 and 35 mm, for which

Fig. 3. (a) Side view of a bouncing light wave with anincident angle of uin in a 3D RFP microcavity of thickness d.(b) Top view of a circulating light wave with total internalref lection in a circular cavity of radius R.

October 15, 2003 / Vol. 28, No. 20 / OPTICS LETTERS 1863

Fig. 4. (a) Intermode spacing Dlm measured from thespectra shown in Fig. 2 (squares and circles) and theirlinear extrapolations (solid curves) for two 3D RFP mi-crocavities. (b) Mode wavelengths measured from thespectra (squares and circles) and calculated by Eqs. (5)and (7) (solid curves). (c) Emission angles u evaluatedby the measured wavelengths and Eq. (3) (squares andcircles) and calculated by Eq. (6) (solid curves).

the slopes are found to be 0.020 and 0.007 nm�mode,respectively.

For a chosen value n0 � 3.5 for a GaAs cavity thecomparison of measured wavelengths with the wave-lengths estimated according to Eqs. (5) and (7) allowsone to estimate the value of a. For the devices withdiameter f of 20 and 35 mm the coefficient a is foundto be 0.75 3 1023�nm and 1.27 3 1023�nm, respec-tively. The behavior of a in Eq. (7) is explained bythe resonance-dependent effective-index variation inthe Fabry–Perot cavity, which can be either positiveor negative with the resonance frequency.11 The mea-sured and calculated mode wavelengths for two cavitiesare shown in Fig. 4(b). The calculated wavelengthsare also shown in Figs. 2(a) and 2(b) by vertical stems.

As one can see, the theoretical and experimental re-sults match each other extremely well.

The emission angles of the modes are illustratedin Fig. 4(c). The squares and circles represent theevaluated emission angles by associating the mea-sured wavelengths and Eq. (3). The solid curves arecalculations resulting from Eq. (6). The measuredand calculated angles are also found to be in excellentagreement. They linearly increase with slopes of0.96±�mode and 0.50±�mode for the microcavity ofdiameter f of 20 and 35 mm, respectively.

In summary, we have found that the quantizedspectral and angular properties of the helical emis-sion modes of the 3D RFP toroid of the PQR laserare well characterized by the off-normal Fabry–Perot resonance condition and the WG mode resonancecondition. The linear behavior of the intermodespacing along x1

m led us to assign the specif ic modeindex to the measured modes and to estimate thecoefficient of effective-index change. The positions ofmeasured and calculated emission modes are found tobe in good agreement. The emission angles estimatedwith the measured and calculated mode wavelengthsagree with each other well.

This research was supported by the Samsung Com-pany, the Korea Science and Engineering Foundationand the Brain Korea 21 National Research laboratoryProgram of the Korean government, and the RussianFoundation for Basic Research project 01-02-16337. J.Bae’s e-mail address is joongwoo@postech.ac.kr.

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