PHYSICAL REVIEW B

VOLUME 61, NUMBER 7

15 FEBRUARY 2000-I

Diffraction of cylindrical Bragg reflectors surrounding an in-plane semiconductor microcavity D. Ochoa,* R. Houdre´, and M. Ilegems Institut de Micro et OptoElectronique, Ecole Polytechnique Fe´de´rale de Lausanne, 1015 Lausanne, Switzerland

H. Benisty Laboratoire de Physique de la Matie`re Condense´e, Ecole Polytechnique, 91128 Palaiseau, France

T. F. Krauss and C. J. M. Smith Department of Electronics and Electrical Engineering, Glasgow University, Glasgow, United Kingdom 共Received 29 June 1999兲 In-plane microresonators consisting of an Alx Ga1⫺x As heterostructure waveguide and deep etched cylindrical trenches give both out-of-plane and lateral-light confinement. The air trenches, acting as a Bragg reflector also allow diffraction into air, so that the far-field pattern reveals interesting information on the resonant cavity modes. By the use of a two-dimensional cylindrical model and a transfer-matrix method based on Hankel functions, the energy and angular dependences of the diffracted field are calculated and successfully compared to measurements.

I. INTRODUCTION

Spontaneous emission 共SE兲 modification in microcavities1 has been widely studied in semiconductor physics due to its promise for application in optoelectronics,2 especially in the fields of optical interconnects. Microcavity lasers can potentially achieve ultrafast modulation 共⬎10 GHz兲, low threshold current and high efficiency.3 For low-speed, low-power applications, efficient microcavity light emitting diodes4 could be used in printers, displays and light sources.5 There are many ways to realize tridimensional 共3D兲 semiconductor microcavity resonators. Some rely on a single dielectric interface 共micropillars6–8 and whispering gallery modes9–11兲 but do not privilege full use of in-plane light confinement, which seems a requirement for integrated devices. On the other hand, based on the photonic band gap approach,12 cavities of a few micrometers defined in a planar waveguide were realized.13,14 However, their modal structure in this size domain is complex. The structure studied here consists of circular concentric deep etched trenches that act as a Bragg mirror 共‘‘Bragg leek’’ 15兲. Similar structures termed ‘‘fingerprint’’ 16 with shallow trenches and modest light confinement have already produced interesting distributed feedback 共DFB兲 lasers with broad area surface emission 共diameter ␾ ⬃100 ␮ m), low divergence and narrow linewidth.17,18 They were regarded as future candidates for integrated waveband filters, lowloss sharp bends or optical couplers. Compared to such ‘‘fingerprint’’ structures, the Bragg leek has the advantage of confining light in much smaller volumes ( ␾ ⬃3 ␮ m), due to the comparatively large etch depth of its air trenches and the resulting important in-plane effective index modulation. However, for this reason, previous DFB theories16,19,20 based on weakly coupled modes cannot be used here. Therefore, we develop in Sec. II a 2D cylindrical model based on a transfer-matrix method with Hankel function that allows the calculation of all the relevant information: the reflectivity of the cylindrical Bragg mirror,21–23 the energy of the resonant modes, the field inside the structure 0163-1829/2000/61共7兲/4806共7兲/$15.00

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with optical pumping and the angular resolved air diffraction patterns. Far field spectral measurements performed under optical pumping are compared to calculations in Sec. III. The results confirm that this system is a good candidate for inplane light confinement. II. MODEL

Our aim in this section is to find the resonant modes of a microcavity bound by a circular Bragg reflector and the diffraction patterns generated normal to the cavity. We make use of a 2D model restrained to the in-plane aspect of the problem. This model consists of N infinite cylindrical concentric layers of dielectric materials. The layers have outer radii called r 1 ,...,r N⫺1 and refractive indices n 1 ,...,n N alternatively equal to n eff and 1, with n 1 ⫽n N ⫽n eff where n eff is an effective refractive index that will be discussed later. The usual cylindrical coordinates (z,r, ␸ ) are used where ␸ is the azimuthal angle and r the distance to the 共Oz兲 axis. The angular pulsation of the electromagnetic field ␻ corresponds to a wavevector in vacuum k 0 ⫽ ␻ /c. Since the system is invariant along the z coordinate, electric and induction fields can be written: E(r, ␸ )e i ␻ t and H(r, ␸ )e i ␻ t with E⫽Ez ez ⫹Er er ⫹E␸ e␸ and H⫽Hz ez ⫹Hr er ⫹H␸ e␸ . These fields must satisfy the Maxwell equations in each layer j: ⵜ•E⫽ⵜ•H⫽0

共1兲

ⵜ⫻E⫽⫺ ␮ 0 i ␻ H

共2兲

ⵜ⫻H⫽␧ 0 n 2j i ␻ E

共3兲

and their tangential components must be continuous at all layer interfaces. Because of the invariance of the problem with the z→ ⫺z inversion,12 the field solutions can be decomposed into two distinct polarizations: transverse electron 共TE兲 and transverse magnetic 共TM兲 for which the only nonzero components are respectively (Hz ,Er ,E␸ ) and (Ez ,Hr ,H␸ ). In our 4806

©2000 The American Physical Society

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DIFFRACTION OF CYLINDRICAL BRAGG REFLECTORS . . .

case, quantum dots 共QD’s兲 photoluminescence coupled to the guided mode is much larger for TE than for TM polarization,24 and only TE light will be considered hereafter. The present cylindrical geometry suggests the introduction of Bessel functions instead of the traditional plane waves. A function of particular interest is the Hankel function H m ⫽J m ⫹iN m where J m and N m are the Bessel and Neumann functions and m the azimuthal number. For a given wavevector k, the function H m (kr) and its complex conju* (kr) are linearly independent and represent respecgate H m tively outgoing and incoming waves.25 Their correspondence to plane waves would be the propagative waves e ikr and e ⫺ikr . In comparison, the Bessel function J m (kr) * (kr) 兴 /2 would correspond to a standing ⫽ 关 H m (kr)⫹H m wave cos kr. A TE solution of Maxwell Eqs. 共1兲–共3兲 in a medium of refractive index n is Hz (r, ␸ )⫽H m (kr)⌿ m ( ␸ ) if k⫽nk 0 and if ⌿ m ( ␸ ) is of the form e ⫾im ␸ , which gives a ‘‘rotating’’ wave in the ⫾e␸ direction. The two different signs lead to twofold degenerate modes that give symmetric solutions for the diffraction into air. For the sake of clarity, only the ⫹ sign is kept in the following calculations. Note that the mode degeneracy is a consequence of the perfect circularity of the model. It can be lifted by using elliptical cross sections, as was demonstrated on pillar microcavities.26 The general solution of the Maxwell equations can then be written as a linear combination of the independent solutions * (n j k 0 r)e im ␸ ,m⫽0 . . . ⬁ 兴 in each layer 关 H m (n j k 0 r)e im ␸ ,H m j: ⬁

Hzj 共 r, ␸ 兲 ⫽

兺 Hzj,m共 r 兲 e im ␸ m⫽0

共5兲

and is a 1D problem for a given m. By expressing the continuity of the tangential fields E␸ and Hz at each layer’s interface, it is shown in the appendix that the field coefficients in the central cavity are related to those outside the structure by a transfer matrix

冉 冊冉

pm ␣ mN ⫽ ␤ mN * qm

qm

* pm

冊冉 冊

␣ m1 . ␤ m1

A. Bragg transmission

Before going any further, the transmission of the circular Bragg can be derived easily with Eq. 共6兲 and with the assumption

再

t m⫽

兩 p m兩 2⫺ 兩 q m兩 2

* pm

.

共8兲

B. Mode calculation without sources

In the hypothesis 共7兲 there was no continuity relation between outgoing and incoming waves in the central cavity. In order to find resonant modes, such a continuity relation has to be imposed. The equation

␣ m1 ⫽ ␤ m1

共9兲

expresses the fact that the incoming waves instantly become outgoing waves after crossing the 共Oz兲 axis. Without this assumption the field diverges at the origin, due to the presence of the Neumann function in H m . The power in each layer j can be defined by 兩 ␣ mj 兩 2 ⫹ 兩 ␤ mj 兩 2 . Then, using Eqs. 共6兲 and 共9兲, the ratio of the electromagnetic power confined in the cavity to that in the outermost layer 共the external vacuum field兲 is f m⫽

1 2 1 2 兩␣m 兩 ⫹兩␤m 兩

N 2 N 2⫽ 兩␣m 兩 ⫹兩␤m 兩

1 . 兩 p m ⫹q m 兩 2

共10兲

This ‘‘resonance function’’ f m (␭) gives the spectral information on the cavity modes in the absence of any source.

Optical injection and non power conservation are now being investigated. Since only TE polarized light is considered, the pumped QD’s emit like horizontal dipoles having random phases 共incoherence between the QD’s兲 and random dipole orientations in the 共Oxy兲 plane. By generalizing the approach of Lukosz,27 it can be shown that the total time averaged emission of many QD’s in the central cavity is modeled by introducing a source term for outgoing waves. This source term is found to be approximately constant with m and ␭. By normalizing it to unity, the assumptions for the field calculations become:

再

共6兲

The coefficients p m and q m depend on n j ,d j j⫽1 . . . N and on the light wavelength ␭.

␣ m1 ⫽1 ␤ mN ⫽0.

1 the coefficient r m ⫽ ␤ m , transmitted with the coefficient t m N N ⫽ ␣ m and no incoming wave impinges from outside ( ␤ m ⫽0). The field transmission coefficient is found to be:

C. Field calculation with sources

共4兲

with

*共 n j k 0r 兲 Hzj,m 共 r 兲 ⫽ ␣ mj H m 共 n j k 0 r 兲 ⫹ ␤ mj H m

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共7兲

For each m, Eq. 共7兲 represents the standard transmission problem: an outgoing wave of unity amplitude impinges on the Bragg reflector from the central cavity, is reflected with

␣ m1 ⫽ ␤ m1 ⫹1 ␤ mN ⫽0.

共11兲

N in Eq. 共11兲 corresponds again to the abThe choice of ␤ m sence of incoming waves from outside. With Eqs. 共6兲, 共11兲, 共A4兲, and 共4兲, ␣ mj ␤ mj , and Hz (r, ␸ ) are calculated in each layer of the structure, for each wavelength and at each location (r, ␸ ) in the 共Oxy兲 plane.

D. Diffraction

The far-field amplitude H( ␪ ) is simply given by the Fourier decomposition of the near field Hz (r, ␸ ) in the air trenches: H共 ␪ 兲 ⫽

冕冕

trenches

Hz 共 r, ␸ 兲 e ⫺ik"rr dr d ␸ ,

共12兲

D. OCHOA et al.

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FIG. 1. 共a兲 A Bragg leek is based on a fourth-order circular Bragg reflector made of concentric air trenches. Part of the photoluminescence 共PL兲 guided mode becomes resonant in the central microcavity, is diffracted into air by the trenches and collected into a fiber at different angles ␪. 共b兲 Scanning electron micrograph 共SEM兲 of the structure showing 8 air trenches 共black circles兲.

where k⫽k 0 (sin ␪ey⫹cos ␪ez) is the wave vector of the light diffracted into air at an angle ␪. Since the trenches are much thinner than the wavelength in air, the field is almost constant within them, leading to H共 ␪ 兲 ⯝

兺m trenches j

r j Hzj,m 共 r j 兲

冕

2␲

0

e i 共 m ␸ ⫺k 0 r j sin ␪ sin ␸ 兲 d ␸ . 共13兲

Finally, the diffracted power can be defined by I( ␪ ) ⫽ 兩 H( ␪ ) 兩 2 . Because of the finite numerical aperture of the collecting lens in the experiment, I( ␪ ) has to be averaged in a certain angular range that smooths its angular dependence. Physically, the e im ␸ field rotating component in Eq. 共4兲 is responsible for the angular diffraction behavior of the different modes. It can be rewritten e ik ␸ 2 where s⫽ ␸ r is the curvilinear abscissa along the circle of radius r, and k␸ ⫽(m/r)e ␸ the pseudo wavevector in the azimuthal direction. Since the modes are resonant in the fourth order Bragg stopband, the phase difference of the field in the successive trenches is approximately 4␲ and gives a diffraction in the normal direction with a wave vector k⬜ ⫽ 冑k 20 ⫺k ␸2 ez . In each trench j, the field corresponding to a particular resonant mode has a rotating wave vector k␸j ⫽(m/r j )e ␸ that vectorially adds to the corresponding k⬜j , giving a diffraction angle ␪ j ⫽arctan(k␸j /k⬜j ). The observed diffraction angle corresponds to an average of the different ␪ j taking into account the field intensity in each trench. Only m⫽0 modes can diffract at normal incidence. Modes corresponding to larger m numbers appear at larger angles. III. DESCRIPTION OF THE STRUCTURE, EXPERIMENT AND DISCUSSION

The structure used here is that of Ref. 15. On a GaAs substrate, the following layers are deposited by molecular beam epitaxial 共MBE兲 关see Fig. 1共a兲兴: a 300 nm-thick Al0.2Ga0.8As layer, a 240 nm-thick GaAs waveguide in which an InAs QD’s layer is embedded and a 400 nm-thick

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Al0.8Ga0.2As layer. The benefits of QD’s over quantum wells 共QW’s兲 are manyfold: 共i兲 they reduce nonradiative recombination at the etched walls,28 共ii兲 they provide a broadband photoluminescence 共PL兲 spectrum 共between 880 and above 1050 nm兲 in order to easily scan the resonance modes, and 共iii兲 their PL light is allowed to couple to all modal symmetries in the structure. E-beam lithography and reactive ion etching processes29 are used to define ⬇800 nm-deep, 70 nm-wide, concentric trenches in the waveguide. The central cavity 共defined by the inner diameter of the first trench兲 is D cav⫽3 ␮ m, and the period 共defined by the radius difference between two successive trenches兲 is ⌳⫽580, 600, 620, or 640 nm. Along a radius, successive trenches act as a fourth order Bragg mirror, with a central Bragg wavelength between 960 and 1060 nm, well inside the QD PL spectral range. The use of fourthorder Bragg periods and thin trenches minimizes scattering losses, while still giving high reflectivities within a limited number of rows. Finally, the choice of D cav⫽3 ␮ m allows relatively easy optical excitation in the central cavity and good interpretation of the spectral mode features. After proper spatial expansion, a 5 nW He-Ne laser beam with a 1 cm diameter is focused on the Bragg leek with a lens (NA⫽0.17,f ⫽38.1 nm), giving a spot of about 5 ␮m. Photoexcitation of QD’s mainly occurs in the central cavity, giving a guided PL signal that interacts with the trenches, and couples to resonant modes. Part of this guided PL is coherently diffracted into air by the Bragg trenches. By the use of a lens (NA⫽0.25, diameter d⫽12.7 mm) placed at a distance D⫽12 cm from the leek, the diffraction signal is collected from all over the leek at a variable angle ␪ from the normal of the sample surface and injected into a fiber towards a spectrometer. The angular resolution ⌬ ␪ ⫽d/(2D) ⫽3° is sufficient for the present measurements. Measured spectra at different angles 共from 0 to 24°兲 of the PL emitted by a Bragg leek with 8 trenches, ⌳⫽620 and 640 mm are shown at the top of Figs. 2共a兲 and 2共b兲. The link between the 2D model of Sec. II and the real 3D structure is done by regarding 3D guided light propagation in the multistack heterostructure as 2D planar propagation in a medium of effective refractive index n eff .30 Two guided modes can propagate in the heterostructure considered here: a main guided mode, which is confined in the GaAs wave guide with an effective index n eff and a second guided mode which is confined in the Al0.2Ga0.8As top layer with an ef⬘ . This second guided mode is less coupled fective index n eff to the QD’s but may be subject to larger diffraction into air due to its strong interaction with the air-dielectric interface. With standard refractive indices for the Alx Ga1⫺x As materials31 the effective indices are calculated with the matrix method described in Ref. 32: n eff,theory⫽3.4 ⬘ n eff,theory ⫽3.2⫺4.10⫺4 关 ␭(nm) ⫺4.10⫺4 关 ␭(nm)⫺1000兴 , ⫺1000兴 . In this paper we use n eff⫽3.308⫺6.10⫺4 关 ␭(nm) ⬘ ⫽3.15⫺7.10⫺4 关 ␭(nm)⫺1000兴 . The dif⫺1000兴 , and n eff ference with theoretical values is attributed to uncertainties in the actual refractive indices, growth compositions, layer thicknesses and processing. With these two refractive indices, the sum of the resonance functions f m (␭) of Eq. 共10兲 for m⫽0 to 7 is plotted in Fig. 2 below the measured spectra. Solid lines correspond to the main guided mode, dashed lines

DIFFRACTION OF CYLINDRICAL BRAGG REFLECTORS . . .

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FIG. 2. 共a兲 Top: measured spectra corresponding to light diffracted at different angles ␪. Bottom: sum of the resonance functions corresponding to the main guided mode 共solid line兲 and the second guided mode 共dashed line兲. The thick horizontal bar below the curves shows the calculated stopband of the cylindrical Bragg mirror. The structure has 8 trenches with ⌳⫽620 nm. 共b兲: As for 共a兲 except ⌳⫽640 nm. For the sake of clarity, the spectra of 共a兲 have been normalized to the bulk PL spectrum.

to the second guided mode. The calculated mode distribution in each cluster departs from the ever increasing redshift 共in 4m 2 ⫺1) in the perfect disk model of Ref. 15. A clear example appears on the two even clusters m⫽0,2,4 at 932 and

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998 nm for ⌳⫽620 nm. The Bragg stopband calculated with Eq. 共8兲 for m⫽0 is shown as a thick bar at the bottom of the graphs. The measured peaks correspond well with the calculated modes. They are larger due to absorption losses in the wave guide and to diffraction losses that are not included in the model. The particularly large linewidth of the mode measured at (⌳⫽620 nm, ␭⫽957 nm, ␪ ⫽0°) is explained by the stronger diffraction losses of the second guided mode.33 As predicted, the modes diffract at angles increasing with m. In order to further validate the model, the same measurements and calculations were made for the other two periods ⌳⫽600 and 580 nm. Table I shows a comparison between the wavelengths of the measured spectral peaks and the corresponding calculated modes with their azimuthal numbers. ⬘ of the A star indicates that the effective refractive index n eff second guided mode has been used instead of n eff . The reason why only low-order 共mainly m⬍3) modes appear for ⌳⫽600 and 580 nm is that, for these periods, the PL intensity of the QD’s is lower in the Bragg stopband spectral range, and that the high-order modes are not intense enough to be detected. Measurements and calculations are in good agreement for the main guided mode (⌬␭/␭⬍0.3%) proving that our 2D model is correct for describing light propagation in a GaAs waveguide. The agreement is not as good for the second guided mode (⌬␭/␭⬍1%) since the difference between 3D and 2D light propagation is stronger in the Al0.2Ga0.8As layer due to the presence of the close air-dielectric interface. Another consequence of this limitation is that our 2D model cannot explain the reason why diffraction of the second guided mode is so important for short periods of 共⌳兲. 3D calculations are required in this case. Regarding diffraction calculations, the plot in Fig. 3 combines wavelength and angular dependences of the modes, via the product f m (␭)I( ␪ ), for ⌳⫽620 nm. Only the main guided mode has been taken into account in the plot. There is also an additional m⫽0 mode at 958.8 nm, diffracting around ␪ ⫽0° and corresponding to the second guided mode. Moreover, the angular dependence of I( ␪ ) has to be averaged within ⫾⌬ ␪ ⫽3° because of the finite numerical aperture of the collecting lens. Nevertheless, energy and angular behaviors of the calculations are in reasonable agreement with the measurements of Fig. 2共a兲. The same study has been

TABLE I. Measured and calculated positions of the cavity modes for the four periods ⌳ and for different azimuthal numbers m. Calculations corresponding to the second guided mode are indicated by a star. ⌳⫽640 nm Meas. 970 971 973 978 1002 1003 1005 1009 1040

⌳⫽620 nm Calc. 968.5 970.4 974 978.3 1002.7 1003.7 1006 1008 1037.3

m 1 3 5 7 0 2 4 6 1

Meas. 932 957 963 965 968 972 998.8 999.4 1001

Calc. 932.4 958.8* 963.8 965.6 968.7 970.9 998 998.5 999.2

⌳⫽600 nm m 0 0 1 3 5 7 0 2 4

Meas. 898 922.4 929 929 949.5 959.4 960.6 963.3 991.5

Calc. 898 923* 926.9 927.8 949.7* 959.8 961.1 962.4 988.3

⌳⫽580 nm m 1 1 0 2 0 1 3 5 0

Meas. 892.9 896.2 916.9 923.9 939.8 955.4

Calc. 892.8 894.4 915.6* 923.3 930* 953.2

m 1 3 1 0 0 1

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factor兲 is finally found to be less than a factor two.36 This result shows that the confinement of our Bragg leek is not strong enough to produce a large SE lifetime change at the source. A larger Purcell factor would require smaller cavities:7 with the reasonable assumption that the reflectivity does not change with the disk diameter D cav , Q should scale 2 and like D cav whereas the mode volume scales like D cav hence decreases faster than Q. IV. CONCLUSION

FIG. 3. Diffraction spectra calculated at different angles ␪ for the Bragg leek corresponding to Fig. 2共a兲, for the main guided mode only. Angular averaging due to the finite numerical aperture of the light collection has not been taken into account. For the sake of clarity some curves corresponding to ␪ ⫽0°, 4°, and 8° have been multiplied by a factor 0.4.

done for the other three periods and the conclusions are similar. However calculated diffraction intensities are not in complete agreement with the measurements. Again, a complete understanding of the diffraction properties of the structure, in particular of the diffraction intensities, would require extensive 3D calculations. Finally, the linewidth of the measured peaks gives access to the Bragg mirror reflectivity and to the spontaneous emission enhancement factor. 共i兲 The finesse of a cavity mode is given by F⫽⌬␭/ ␦ ␭ where ␦␭ is the full width at half maximum of the mode peak and ⌬␭ is the free spectral range between two modes of the same parity. We have ␦ ␭⫽0.7 nm for the mode at 963 nm as shown in Fig. 2共a兲 and ⌬␭⫽65 nm between the modes m ⫽0 at 932 and 999 nm, thus leading to F⫽93. The approximation F⯝ ␲ /(1⫺R) for a planar Fabry-Pe´rot mode at the limit of R⯝1 is still valid in our case for a large circular mirror of radius R cavⰇ␭/(2 ␲ n), where Hankel functions H m (kr) can be approximated by plane waves25 e ikr . The reflectivity of the Bragg mirror is therefore R⯝1⫺ ␲ /F ⯝97%. 共ii兲 By using the Fermi golden rule, the QD spontaneous emission enhancement in the cavity mode can be calculated34 3Q␭ 30 ␥ cav ⫽ , 3 ␥ bulk 4 ␲ 2 n eff V eff

共14兲

where ␥ cav is the averaged in-plane spontaneous emission rate of QD’s emitting at the resonant wavelength ␭ 0 , ␥ bulk the SE rate in a bulk medium of refractive index n eff , Q the quality factor of the resonant mode and V eff is here an effective mode volume averaged over the QD’s ensemble.35 With a well confined mode, V eff is found to be approximately equal to 2⫻the surface area of the central cavity ( ␲ 1.5 ␮ m 2 )⫻the vertical extent of the guided mode 共300 nm兲. For the mode m⫽1, ␭ 0 ⫽963 nm, Q⫽␭ 0 / ␦ ␭ ⫽963/0.7⫽1375. Taking into account the twofold degeneracy9 due to the e ⫾im ␸ component in Eq. 共4兲 共for m ⫽0), the spontaneous emission enhancement factor 共Purcell

Tridimensional light confinement is reported in an inplane microcavity surrounded by a circular Bragg reflector. The structure consists of deep concentric trenches etched in a GaAs/Alx Ga1⫺x As waveguiding heterostructure. Measurements are performed under optical excitation by collecting the far field pattern diffracted at the successive trenches. Based on a transfer matrix method with Hankel functions, the 2D cylindrical model described in this paper gives a good insight into the mode resonances of the microresonator. Calculated modal features and diffraction behaviors are in good agreement with experimental results. Expectations for future 3D in-plane microcavities based on 2D omnidirectional photonic crystals37,38,14 are high. Such photonic crystals have already achieved high reflectivities39 and are a good candidate for in-plane light confinement mirrors. The physical understanding of the resonance and diffraction properties is however more delicate33 in this case due to the complex symmetry 共generally hexagonal兲 of the resulting cavities. The advantage of the cylindrical Bragg structure is that it can be analyzed almost completely. With quality factors reaching Q⫽1375 and mirror reflectivities approaching 97%, the Bragg leek microcavity proves its ability to effectively confine light. Among the future applications are: multimode fiber coupling, waveband filtering and inplane optical coupling. APPENDIX: TRANSFER-MATRIX METHOD

Using Eqs. 共3兲 and 共4兲 in each layer j, the azimuthal electric field is found to be: ⬁

E␸j 共 r, ␸ 兲 ⫽

i 关␣ j H⬘ 共n k r兲 n j ␧ 0 c m⫽0 m m j 0

兺

⬘ * 共 n j k 0 r 兲兴 e im ␸ , ⫹ ␤ mj H m

共A1兲

⬘ is the derivative of the Hankel function. The conwhere H m tinuity of the tangential fields at the different layers interfaces21,22 is given by: For j⫽1 . . . N⫺1: Hzj 共 r j , ␸ 兲 ⫽Hzj⫹1 共 r j , ␸ 兲

共A2兲

E␸j 共 r j , ␸ 兲 ⫽E␸j⫹1 共 r j , ␸ 兲

共A3兲

or, since the Hankel functions are linearly independent for different m azimuthal numbers

冉 冊冉

a mj ␣ mj⫹1 ⫽ j ␤ mj⫹1 b m*

with

b mj a mj*

冊冉 冊 ␣ mj ␤ mj

共A4兲

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DIFFRACTION OF CYLINDRICAL BRAGG REFLECTORS . . .

a mj ⫽

b mj ⫽

1 D mj

冋

⬘ * 共 n j⫹1 k 0 r j 兲 H m共 n j k 0r j 兲 H m

⫺

n j⫹1 * 共 n j⫹1 k 0 r j 兲 H m⬘ 共 n j k 0 r j 兲 Hm nj

1

冋

D mj ⫺

册

共A5兲

* 共 n j k 0 r j 兲 H m⬘ * 共 n j⫹1 k 0 r j 兲 Hm

n j⫹1 * 共 n j⫹1 k 0 r j 兲 H m⬘ * 共 n j k 0 r j 兲 Hm nj

册

equal to (n j⫹1 /n j ) 2 . For TM-polarized light, the relation between the coefficients of the electric field Ezj (r, ␸ ) is the same as Eq. 共A4兲 if the factor n j /n j⫹1 is replaced n j⫹1 /n j in Eqs. 共A5兲 and 共A6兲. The transfer matrix of the whole Bragg reflector is obtained by putting together the relations 共A4兲 for j⫽1 . . . N ⫺1:

冉 冊 冉 冊

␣ mN ␣ m1 , N ⫽M m ␤m ␤ m1

共A6兲

共A8兲

with

⬘ * 共 n j⫹1 k 0 r j 兲 D mj ⫽H m 共 n j⫹1 k 0 r j 兲 H m * 共 n j⫹1 k 0 r j 兲 H m* 共 n j⫹1 k 0 r j 兲 . ⫺H m

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共A7兲

All these quantities depend on the geometry of the system and on the wavelength of light 共through k 0 ). Note however that the determinant of the transfer matrix is constant and

M m⫽

qm

* qm

* pm

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det M m⫽(nN /n1)2⫽1 since in our case n 1 ⫽n N ⫽n eff . T. Erdogan and D. G. Hall, J. Appl. Phys. 68, 1435 共1990兲. T. Erdogan and D. G. Hall, IEEE J. Quantum Electron. 28, 612 共1991兲. 21 M. Toda, IEEE J. Quantum Electron. 26, 473 共1990兲. 22 Y. Jiang and J. Hacker, Appl. Phys. Lett. 63, 1453 共1993兲. 23 M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, J. Mod. Opt. 46, 875 共1999兲. 24 D. Labilloy, Ph.D. thesis, Ecole Polytechnique, Palaiseau, France, 1998. 25 Handbook of Mathematical Functions, edited by M. Abramowitz and A. Stegun 共Constable, New York, 1972兲. 26 B. Gayral, J. M. Ge´rard, B. Legrand, E. Costard, and V. ThierryMieg, Appl. Phys. Lett. 72, 1421 共1998兲. 27 W. Lukosz, J. Opt. Soc. Am. 71, 744 共1981兲. 28 J. Y. Marzin, J. M. Ge´rard, A. Izrae¨l, and D. Barrier, Phys. Rev. Lett. 73, 716 共1994兲. 29 T. F. Krauss, R. M. De La Rue, and S. Brand, Nature 共London兲 383, 699 共1996兲. 30 Optical Waveguide Theory, edited by A. W. Snyder and J. D. Love 共Chapman and Hall, London, New York, 1983兲. 31 S. Adachi, in Properties of Aluminium Gallium Arsenide, edited by S. Adachi 共Inspec, London, 1993兲, Vol. 7, pp. 126–140. 32 H. Benisty, R. P. Stanley, and M. Mayer, J. Opt. Soc. Am. A 15, 1192 共1998兲. 33 H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Be´raud, and C. Jouanin, Appl. Phys. Lett. 76, 532 共2000兲. 34 E. M. Purcell, Phys. Rev. 69, 681 共1946兲. 35 The effective volume of a dipole emitting into a cavity mode is usually defined as the spatial integral of the vacuum field intensity divided by its value at the location of the dipole. Since our source emitter consists of a multitude of QD’s randomly located in the cavity 共and emitting with random dipolar orientations in the source plane兲, the effective volume has to be averaged on these QD’s locations and orientations according to the procedure of Ref. 8. 36 The SE rate ␥ leak corresponding to the continuum of modes leaking into the substrate and into the air has to be taken into account and is typically 0.6–0.8 times the bulk emission rate 共Ref. 8兲. The total Se rate is the sum of ␥ cav and ␥ leak . Although ␥ cav

*Electronic address: [email protected]

19

1

20

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is found to be close to unity, the total SE enhancement factor of the microcavity is therefore less than a factor of 2. 37 P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 共1996兲. 38 O. Painter, J. Vuckovic, and A. Sherer, J. Opt. Soc. Am. B 16,

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