Nanoscale optical positioning of single quantum dots for bright and pure single-photon emission
Luca Sapienza, Marcelo Davanco, Antonio Badolato, Kartik Srinivasan
NNanoscale optical positioning of single quantum dots for bright and pure single-photon emission
Luca Sapienza,
1, 2, 3, ∗ Marcelo Davanc¸o, Antonio Badolato, and Kartik Srinivasan
1, † Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Maryland NanoCenter, University of Maryland, College Park, MD 20742, USA School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA (Dated: August 6, 2015)
Self-assembled, epitaxially-grown InAs/GaAs quantum dots are promising semiconductor quantumemitters that can be integrated on a chip for a variety of photonic quantum information science appli-cations. However, self-assembled growth results in an essentially random in-plane spatial distribution ofquantum dots, presenting a challenge in creating devices that exploit the strong interaction of single quan-tum dots with highly confined optical modes. Here, we present a photoluminescence imaging approachfor locating single quantum dots with respect to alignment features with an average position uncertainty < nm ( < nm when using a solid immersion lens), which represents an enabling technology forthe creation of optimized single quantum dot devices. To that end, we create quantum dot single-photonsources, based on a circular Bragg grating geometry, that simultaneously exhibit high collection efficiency(48 % ± % into a 0.4 numerical aperture lens, close to the theoretically predicted value of 50 % ), lowmultiphoton probability ( g ( ) ( ) < % ), and a significant Purcell enhancement factor ( ≈ . Single InAs/GaAs quantum dots are one of the morepromising solid-state quantum emitters for applications suchas quantum light generation and single-photon level nonlinearoptics . Critical to many such applications is the incorpora-tion of the quantum dot within an engineered photonic envi-ronment so that the quantum dot interacts with only specificoptical modes. A variety of geometries such as photonic crys-tal devices and whispering gallery mode resonators have beenemployed to achieve such behavior for bright single-photonsources and strongly coupled quantum dot-cavity systems .The optical field in many such geometries varies significantlyover distances of ≈
100 nm, setting a scale for how accuratelythe quantum dot position should be controlled within the de-vice for optimal interaction. While site-controlled growth ofquantum dots presents one attractive option , the propertiesof such quantum dots (in terms of homogeneous linewidth,for example) have not yet matched those of quantum dotsgrown by strain-mediated self-assembly (Stranski-Krastanowgrowth) . However, the in-plane location, polarization, andemission wavelength of such self-assembled quantum dotsare not accurately controlled in a deterministic fashion, andthus techniques are required to determine these propertiesprior to device fabrication, in order to create optimally per-forming systems. Several techniques for location of self-assembled InAs/GaAs quantum dots prior to device fabrica-tion have been reported, including atomic force microscopy(AFM) , scanning confocal photoluminescence microscopy (including in-situ, cryogenic photolithography ), photolumi-nescence imaging , and scanning cathodoluminescence . Ofthese approaches, photoluminescence imaging is particularlyattractive given its potential to combine high throughput sub-50 nm positioning accuracy, spectral information, and com-patibility with high-resolution electron-beam lithography thatis typically used to pattern small features such as those used inphotonic crystals. Localization of single molecules to 10 nmscale accuracy by imaging their fluorescence onto a sensitivecamera has proven to be a powerful technique in the biologicalsciences . Here, we present a two-color photoluminescence imagingtechnique to determine the position of single quantum dotswith respect to fiducial alignment marks, with an average po-sition uncertainty <
30 nm obtained for an image acquisitiontime of 120 s (the average position uncertainty is reduced to <
10 nm when using a solid immersion lens). This wide-fieldtechnique is combined with confocal measurements within thesame experimental setup to determine emission wavelengthand polarization. We use this information to fabricate anddemonstrate quantum dot single-photon sources in a circu-lar Bragg grating geometry that simultaneously exhibit highcollection efficiency (48 % ± g ( ) ( ) < ≈ Results
Quantum dot location via photoluminescence imaging
Prior to sample interrogation, an array of metal align-ment marks is fabricated on quantum-dot-containing materialthrough a standard lift-off process (see Methods). The sam-ples are then placed on a stack of piezo-electric stages to allowmotion along three orthogonal axes (x,y,z) within a closed-cycle cryostat that reaches temperatures as low as 6 K. Thesimplest photoluminescence imaging configuration we use isa subset of Fig. 1(a), and starts with excitation by a 630 nmLED, which is sent through a 90/10 (reflection/transmissionpercentage) beamsplitter and through a 20x infinity-correctedobjective (0.4 numerical aperture) to produce an ≈ µ mdiameter spot on the sample. Reflected light and fluorescencefrom the sample goes back through the 90/10 beamsplitter andis imaged onto an Electron Multiplied Charged Couple Device(EMCCD) using a variable zoom system. When imaging thefluorescence from the quantum dots, the 630 nm LED power isset to its maximum power ( ≈
40 mW, corresponding to an in-tensity of ≈
130 W/cm ), and a 900 nm long-pass filter (LPF) a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug EMCCDcameraclosed-cyclecryostat zoom barrelLPF x-y-z samplemotion
HWP
940 nm LED n m L E D
780 nmlaser spectrometerCCD a bf
20 40 60 80 100 120 140468101214161820
Total magnification P o s i ti o n un c e r t a i n t y ( n m ) x axis y axis Quantum dot
400 200 133 100 80 67 57
Field of view ( µ m)
20 40 60 80 100 120 140400 200 133 100 80 67 57
Alignment Mark
Field of view ( µ m)Total magnification Cross
Position ( µ m) P o s i ti o n ( µ m )
34 36 38 40 42 44 46 48
Position ( µ m) x scan E M CC D c o un t s Peak pos. error = 9.1 nm
22 24 26 28 30 32 34 36 380.51.01.52.02.53.03.5x 10 Position ( µ m) E M CC D c o un t s y scanPeak pos. error = 9.7 nm Position ( µ m) P o s i ti o n ( µ m ) -1.5 -1.0 -0.5 0 0.5 1.0 1.5 Position ( µ m) x scanPeak pos. error = 14.0 nm E M CC D c o un t s -2.0 2.0-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 Position ( µ m) y scanPeak pos. error = 21.1 nm E M CC D c o un t s single mode fiber cd e FIG. 1:
Optically locating single quantum dots. (a) Schematic of the photoluminescence imaging setup. An infrared light emitting diode(LED, emission centered at 940 nm) is used for illumination of the sample while either a 630 nm red LED or a 780 nm laser is used forexcitation of the quantum dots, depending on whether excitation over a broad area (LED) or of individual quantum dots (laser) is required.Samples are placed within a cryostat on an x-y-z positioner. Imaging is done by directing the emitted and reflected light into an ElectronMultiplied CCD (EMCCD) camera, while spectroscopy is performed by collecting emission into a single-mode fiber and sending it to agrating spectrometer. (b) Example photoluminescence image from single quantum dots measured under red LED illumination only. A 900 nmlong pass filter (LPF) is inserted into the collection path when measuring the quantum dot emission. (c) Two orthogonal line cuts (horizontal =x-axis, vertical = y-axis) of the photoluminescence image, showing the profiles of the quantum dot emission (symbols) and their Gaussian fits(lines). (d) Example image of the reflected light from the metallic alignment marks under red LED illumination only. (e) Two orthogonal linecuts (horizontal = x-axis, vertical = y-axis) of the image in (d), showing the profiles of the reflected light from the metallic alignment marks(symbols) and their Gaussian fits (lines). (f) Peak position uncertainties measured from the Gaussian fits of linecuts of the EMCCD images,plotted as a function of magnification and field of view for the quantum dot and metallic alignment marks. The uncertainties represent onestandard deviation values determined by a nonlinear least squares fit of the data. is inserted in front of the EMCCD camera to remove reflected630 nm light. Imaging of the alignment marks is done by re-ducing the LED power to 0.8 mW, turning off the EMCCDgain, and removing the 900 nm LPF.Representative images of the quantum dot photolumines-cence and alignment marks are shown in Fig. 1(b) andFig. 1(d). In Fig. 1(b), circular bright spots surrounded byAiry rings - a signature of optimally focused collection - areclearly visible and represent the emission from single quan-tum dots excited within an ≈ µ m x 56 µ m field of view.Orthogonal linescans of the bright spots (Fig. 1(c)) are fitwith Gaussian functions using a nonlinear least squares ap-proach (see Supplementary Note 1), with the extracted peakpositions showing one standard deviation uncertainties as lowas ≈ ≈
15 nm.Figure 1(f) shows how this uncertainty changes as a functionof system magnification (and hence field of view), which isadjusted using the variable zoom system. We see that thequantum dot uncertainty values show a decreasing trend with higher magnification, and values as low as ≈ ac d N u m b e r o f o cc u rr e n c e s Quantum dot onlyAlignment mark onlyQD - Alignment mark xyxyxy mean error = 21.2 nmmean error = 18.8 nmmean error = 28.3 nm P o s i ti o n ( µ m ) Position ( µ m)0 10 20 30 40 50
28 29 30 31 321.41.61.82.0x 10 x scan, quantum dot Position ( µ m) E M CC D c o un t s Peak pos. error = 15.7 nm y scan, quantum dot
Position ( µ m) Peak pos. error = 14.5 nm E M CC D c o un t s -2 -1 0 1 21.41.61.8 x 10 Peak pos. error = 13.6 nmx scan, alignment mark
Position ( µ m)1.41.61.8 x 10 y scan, alignment mark -2 -1 0 1 2 Position ( µ m) Peak pos. error = 15.3 nm P o s i ti o n ( µ m ) Position ( µ m) E M CC D c o un t s Position ( µ m) Peak pos. error = 4.1 nm y scan, quantum dot x 10 y scan, alignment mark Peak pos. error = 6.4 nm -1.0 -0.5 0.0 0.5 1.0
Position ( µ m) E M CC D c o un t s x 10 b e FIG. 2:
Performance of the two-color positioning technique. (a) EMCCD image of the photoluminescence from a single quantum dotand reflected light by the alignment marks (metallic crosses), acquired by illuminating the sample simultaneously with both the red andnear-infrared LEDs. (b) Orthogonal line cuts (horizontal=x axis, vertical=y axis) of the photoluminescence image, showing the profiles ofthe quantum dot emission (solid symbols) and of the image of the alignment marks (open symbols) and their Gaussian fits (solid lines). (c)Histograms of the uncertainties of the quantum dot and alignment mark postions and quantum dot-alignment mark separations, measured fromthe Gaussian fits of linecuts from 45 images. The uncertainties represent one standard deviation values determined by a nonlinear least squaresfit of the data. (d)-(e) Photoluminescence imaging through a solid immersion lens. (d) Image of the photoluminescence from single quantumdots and reflected light from the alignment marks (metallic crosses), collected under the 630 nm/940 nm co-illumination scheme. (e) Y-axisline cuts from the photoluminescence image, showing the profiles of the quantum dot emission (solid symbols) and reflected light from thealignment mark (open symbols). The solid lines are nonlinear least squares fits to Gaussians. setup (Fig. 1(a)) in which a second, infrared LED at 940 nm iscombined with the 630 nm LED when illuminating the sam-ple. Unlike the 630 nm LED, the 940 nm LED does not ex-cite the quantum dots, but instead serves only to illuminatethe alignment marks, with the wavelength chosen to approx-imately match the expected wavelength of the quantum dotemission. By adjusting the 940 nm LED power appropriately,both the quantum dots and alignment marks can be observedin a single image with the 900 nm LPF in place.Figure 2(a) shows an image taken when the sample is co-illuminated by both 630 nm and 940 nm LEDs, with the940 nm power set to be ≈ µ W, about four orders of magni-tude smaller than that of the 630 nm LED power. Orthogonalline scans through the quantum dot and alignment marks un-der this co-illumination scheme are shown in Fig. 2(b). As ex-pected, the uncertainty values determined for quantum dot andalignment mark positions are larger than those obtained whenacquiring two separate images (Fig. 1(c),(e)), for which theLED power can be optimized independently to maximize the image contrast and minimize each uncertainty. However, wehave favoured the co-illumination approach due to its abilityto reduce some potential uncertainties, like sample drift, thatmay occur during schemes requiring multiple images to be ac-quired. Ultimately, one might envision time-multiplexing anddrift compensation techniques being employed to correct forsuch factors.After carrying out a systematic study of the position uncer-tainties as a function of magnification, integration time, andEMCCD gain, we have found optimized settings for image ac-quisition (under 40x magnification), in terms of the combinedquantum dot and alignment mark uncertainty: an integrationtime of 120 s, an EMCCD gain of 200, and the aforemen-tioned LED powers. Under these conditions, we have studiedthe uncertainties in the quantum dot position, alignment markposition, and quantum dot-alignment mark separation for anumber of different quantum dots on our sample. Histogramsof the measured values are reported in Fig. 2(c), and showthat the mean uncertainty in the quantum dot-alignment markseparation is ≈
28 nm. Finally, we note that in the presentsetup, the available 630 nm LED power is below that requiredto saturate the quantum dot emission (a comparison with thesaturation counts obtained under laser excitation shows that itis about half the value required). Higher 630 nm LED powerwould increase the collected photoluminescence and reducethe uncertainty values that we have reported. This pre-eminentrole of collected photon flux is well-established in the singleemitter localization literature . We have confirmed it in ourexperiments by using a solid-immersion lens , which canboth increase the LED intensity at the quantum dot and thefraction of quantum dot emission that is collected by the mi-croscope objective. Placing a hemispherical lens with refrac-tive index n = 2 on the surface of the sample yields individ-ual quantum dot and alignment mark position uncertainties of ≈ <
10 nm (more details provided in Supplementary Note 2).In total, we note that the positioning uncertainties that we ob-tain are 2 × to 5 × smaller than previously reported , andare obtained with a single image, acquired over a 120 s acqui-sition time, and spanning an area of the sample greater than100 µ m × µ m. Realization of circular Bragg grating bullseye cavities
We now use the optical positioning technique to fabricatenanophotonic structures tailored for the properties of a spe-cific quantum dot and engineered to enhance the collectionefficiency of single photons in free space. First, we obtaininformation about the quantum dot emission wavelength byspatially selecting one quantum dot and collecting its emis-sion into a single-mode fiber that is coupled into a gratingspectrometer (a half waveplate and polarizing beamsplitterare used to switch the collection path between the EMCCDcamera and single-mode fiber). Spatial selection is achievedby exciting individual quantum dots with a 780 nm laser,incorporated into the same micro-photoluminescence setup(Fig. 1(a)), and producing a focused spot size of ≈ µ m on thesample surface. The half waveplate and polarizing beamsplit-ter also enable determination of the quantum dot polarization.Having thus obtained emission wavelength to go along withthe spatial position obtained from the imaging setup, a prop-erly calibrated fabrication process can enable the creation ofnanophotonic structures that are tailored to the specific emitterproperties. This allows one to minimise (and potentially avoidaltogether) the need for mutual spectral tuning of the emitterwith respect to the optical resonance of the cavity, which is aclear limitation of the scalability of these sources.The specific nanophotonic structure we focus on is a circu-lar Bragg grating ‘bullseye’ geometry, which has been devel-oped as a planar structure in which quantum dot photons arefunneled into a near-Gaussian far-field pattern over a moder-ate spectral bandwidth (few nm) with high efficiency (theoret-ical efficiency of 50 % into a 0.4 numerical aperture) and withthe potential for Purcell enhancement of the radiative rate .The cavity mode of interest is tightly confined, and optimalperformance requires the quantum dot to be within a couplehundred nanometres of the centre of the bullseye structure.This is illustrated in Fig. 3(a), which plots the normalized
910 920 930 940 950 960 970 980 990 100000.20.40.60.81.0 wavelength (nm) N o r m a li z e d s p e c t r a l i n t e n s i t y P o s i ti o n ( µ m ) Position ( µ m)-160-120-80-400 H e i g h t ( n m ) Position ( µ m) E | /| E | max ab simulated central wavelength (nm) e x p e r i m e n t a l c e n t r a l w a v e l e n g t h ( n m ) FIG. 3:
Circular dielectric gratings tailored to specific quantumdot emitters. (a) Normalized cavity mode electric field intensity | E | superimposed on a scanning electron microscope image of the centerof one of the cavities. Scale bar represents 200 nm. (b) Experimen-tal central wavelength of 50 circular grating cavities with varyingperiod and central radius, plotted as a function of the simulated cen-tral wavelength. When only one peak is observed in the spectrum,black squares are used to denote the peak wavelength. When twopeaks are observed, red circles and blue triangles are used. Suchtwo-peak behavior is also seen in simulations depending on the de-vice parameters, and is due to coupling to a second cavity mode. Topinset: Atomic Force Microscope image of a circular grating cavityand a linecut (along the dashed line) showing the etch depth of thetrenches. Bottom inset: Examples of photoluminescence spectra ofcircular grating cavities, measured from a high-quantum dot densityregion. electric field intensity superimposed on a scanning electronmicroscope image of the center of a fabricated device. Animportant parameter in the fabrication of these devices is theetch depth of the asymmetric grating, as this determines thefraction of emission in the upwards direction (towards our col-lection optics) compared to the downwards direction (towardsthe substrate). Furthermore, given the high refractive indexdifference between GaAs and air, a change in etch depth of1 nm results in a shift of the optical resonances of about 1 nm.We use AFM to determine the GaAs dry etch rate within thegrating grooves (Fig. 3(b), top inset), and based on this cali-bration, we fabricate (see Methods) 50 circular gratings whoseparameters (pitch and central diameter) have been adjusted sothat the cavity resonances cover the 930 nm to 1000 nm rangeof wavelengths. These samples were fabricated in a region ofthe wafer with a high density of quantum dots, so that the re-sulting emission under high power excitation is broad enoughto feed the cavity modes. Example spectra collected from dif-ferent circular grating cavities are shown in the bottom insetof Fig. 3(b). These measurements allow us to calibrate the ex-perimental cavity resonances with respect to simulations, asshown in the main panel of Fig. 3(b), and tailor the design tomatch the specific quantum dot emission wavelength. Optimized quantum dot single-photon source
Using the quantum dot positions with respect to alignmentmarks as determined by photoluminescence imaging, emis-sion wavelengths as determined by grating spectrometer mea-surements, and the aforementioned calibration of the circulargrating geometry to match target wavelengths, we fabricate(see Methods) a series of circular grating cavities containingsingle quantum dots. Photoluminescence imaging of the de-vices after fabrication, as shown in Fig. 4(a) for a representa-tive device excited by the 630 nm LED, qualitatively indicatesthat the quantum dot emission originates from the centre ofthe bullseye structure, as intended. A measurement of the far-field emission from the device on the EMCCD, as shown inFig. 4(b), shows that it is close to a circular Gaussian function,as confirmed by a nonlinear least squares fit. As the overlapwith a perfect circular Gaussian is ≈
70 %, this far-field pattenis expected to mode match well to a single-mode fiber, an im-portant consideration for long-distance transmission of singlephotons for quantum information applications.We now characterise the emission produced by the opti-cally positioned quantum dots within the circular grating cavi-ties, in terms of collection efficiency, single-photon purity, andspontaneous emission rate. For these measurements, a sec-ond cryostat and photoluminescence setup was used, as it pro-vides direct free-space in-coupling to a grating spectrometerthat is also used for spectral isolation of the quantum dot ex-citonic state (Supplementary Fig. 1). First, we determine thecollection efficiency by pumping the devices with a 780 nmwavelength, 50 MHz repetition rate pulsed laser (50 ps pulsewidth), and varying the laser power until the emission fromthe quantum dot saturates (Fig. 4(c)). Assuming a quantumdot radiative efficiency of unity, and taking into account thelosses within the optical setup (see Supplementary Note 3),we measure a collection efficiency as high as 48.5 % ± , where no optical positioning was used, device fab-rication in a material containing a higher density of quantumdots was performed, to ensure that some non-negligible frac-tion (which turned out to be a few percent) of devices wouldhave a quantum dot spectrally and spatially overlapped withthe desired cavity mode (see Supplementary Note 4). In com-parison, the optical positioning used here allows us to workwith a much lower density of quantum dots ( (cid:46) µ m ). One consequence of this is the comparatively cleanemission spectra we observe, even when exciting with pumppowers that completely saturate the quantum dot emission(Fig. 4(d)). Such clean spectra might be expected to corre-spond to clean (low multi-photon probability) single-photonemission, and to test this, the spectrally filtered emission fromthe bright quantum dot exciton line is measured in a stan-dard Hanbury-Brown and Twiss setup. Under non-resonant,780 nm pulsed excitation, we measure g ( ) ( ) = . ± . and limit the device’s single-photon purity.We next consider pumping the device on an excited state ofthe quantum dot, as such excitation (sometimes referred to asquasi-resonant or p-shell pumping) has been shown to reduce g ( ) ( ) . Measurement of the quantum dot emission underpulsed 857 nm excitation shows that, at the saturation pumpintensity (where the collection efficiency is maximized), thespectrum is nearly identical to that under 780 nm excitation(Supplementary Fig. 2(d)). Moreover, increased excitationpower above saturation (achieved using a 857 nm continuouswave laser) yields far less cavity mode feeding than in the cor-responding 780 nm case (Supplementary Fig. 2(c)), suggest-ing that improved single-photon purity should be observed.This is confirmed by intensity autocorrelation measurements,which indicate that on-demand single-photon emission with apurity of 99.1% ( g ( ) ( ) = . ± . g ( ) ( ) levels are deter-mined from raw coincidences, without any background sub-traction, and with an uncertainty value given by the standarddeviation in the area of the peaks away from time zero.We also measure the spontaneous decay rate of the quantumdot emission under 780 nm pulsed excitation (measurementsat 857 nm have also been performed and yield unchanged re-sults). The spontaneous emission decay of a quantum dot inbulk and a quantum dot in a circular grating cavity are shownin Fig. 4(f). The exponential fit of the decay curve allowsus to extract a lifetime of ≈
520 ps for the quantum dot inthe bullseye cavity, corresponding to a Purcell enhancementof the spontaneous emission rate by a factor of ≈
3. A Pur-cell factor as high as 4 is measured in other devices that have asmaller detuning with respect to the cavity mode (the detuningis 1.6 nm for the device we focus on here). Theoretically, Pur- a c P h o t o n fl u x i n t o N A = . l e n s ( s - )
906 907 90802468101214 x 10 wavelength (nm)906.5 907.5 P h o t o n fl u x i n t o N A = . l e n s ( s - ) -100 -50 0 50 1000100200300400500600 Time (ns) C o i n c i d e n c e s g (2) (0) = 0.009 ± -6 -4 -2 0 2 4 60246810 Time (ns) C o i n c i d e n c e s de f η = 48.5 % +/- 5.0 % η = 0.12 % +/- 0.03 % b y-axis x-axis -10 0 10 20 y-axis scan y-axis (pixels)6.4 pixels -20 -10 0 10 2000.20.40.60.81.0 x-axis scan x-axis (pixels) Time (ns)0 1 2 3 4 5 6 7 N o r m a li z e d c o un t s QD in bullseye τ = 0.52 ns +/- 0.05 nsQD in bulk τ = 1.50 ns +/- 0.10 ns y - a x i s ( p i x e l s ) -20-15-10 -5 0 5 10 15 20-20-15-10-505101520 01000200030004000 x-axis (pixels) P h o t o n fl u x a t d e t e c t o r ( s - ) x 10 P h o t o n fl u x a t d e t e c t o r ( s - ) FIG. 4:
Single-photon emission from an optimised device. (a) Image of the photoluminescence from a single quantum dot within the cavity,collected under 630 nm LED illumination. Scale bar represents 5 µ m. (b) Far-field image of the photoluminescence from a quantum dotin a circular grating cavity, along with linecuts from the 2D Gaussian fit to the data along the x- and y-axes, shown as solid white lines.The upper right inset shows a 2D image plot of the interpolated data, while the bottom curves plot the (uninterpolated) experimental data(symbols) and their Gaussian fits (solid lines). (c) Photon flux into the 0.4 numerical aperture collection objective (left y-axis) and at thedetector (right y-axis), plotted as a function of 780 nm excitation power (in saturation units), for a quantum dot (QD) in a circular grating(QD in BE, red symbols) and in unpatterned GaAs (QD in bulk, black symbols). (d) Examples of photoluminescence spectra collected underdifferent excitation power [color coded in panel (c)]. (e) Photon collection coincidence events measured under pulsed 857 nm excitation, usinga Hanbury-Brown and Twiss setup. The disappearance of the central peak (zoomed-in plot in the inset) is the signature of pure single-photonemission. The uncertainty value is given by the standard deviation in the area of the peaks away from time zero. See Supplementary Fig. 2 foradditional relevant data. (f) Time-resolved photoluminescence measurements collected under pulsed 780 nm excitation, showing the excitedstate decays (symbols) fitted by single exponential curves (solid lines). The shaded gray areas correspond to the 95 % confidence intervals inthe fit. cell factors as high as ≈
11 are expected for quantum dotswith perfect spectral and spatial alignment with respect to thecavity mode. Different methods to achieve such precise spec-tral resonance are currently under consideration; preliminarymeasurements indicate that in-situ N deposition is ill-suitedto the circular grating geometry, as the cavity mode degradesbefore a significant wavelength shift is observed.Going forward, it would be relevant to determine the lo-cation of the optically positioned quantum dots within fabri-cated devices, in order to understand sources of error withinour overall fabrication approach (which combines optical po-sitioning with aligned electron-beam lithography). Supple-mentary Note 5 presents a detailed discussion on the results offinite-difference time-domain simulations examining the Pur-cell factor, collection efficiency, and degree of polarization inthe collected far-field as a function of dipole position and ori-entation within the cavity. Our calculations indicate that thePurcell enhancement, in particular, very sensitively dependson the dipole location, while the collection efficiency is not assensitive. For the devices we have focused on in the main text,we find that a simulated offset between 50 nm and 250 nmwith respect to the cavity center produces results that are con- sistent with our measurements. Discussion
There has been much progress in the development of brightquantum dot single-photon sources in recent years, includ-ing micropillar , vertical nanowire waveguide , fiber-coupled microdisk , and photonic crystal cavity geome-tries. Many metrics are needed to characterize these sources,and the choice of which ones are of particular importanceis largely determined by the intended application. Withinthe landscape of these sources, the results presented hereare unique in terms of simultaneously exhibiting high col-lection efficiency, nearly perfect single-photon purity at thehighest measured collection efficiency, and Purcell enhance-ment of the spontaneous emission rate. For example, previousbright, Purcell-enhanced microcavity single-photon sourceshave shown significant non-zero g ( ) ( ) values (e.g (cid:38) . , while brightnanowire sources show g ( ) ( ) ≈ . For some applications, the metrics demon-strated thus far should be combined with a high degree of pho-ton indistinguishability , which is limited in our work by thecoherence time of the quantum dots in this sample ( <
300 ps,as confirmed by measurements with a scanning Fabry Perotinterferometer; other emitters on the same wafer show coher-ence times as long as 500 ps). Future work will focus on res-onant excitation to improve the coherence time and finecontrol of the cavity-quantum dot detuning to achieve shorterradiative lifetimes . Together, these advances may providea route to a source that simultaneously provides bright, pure,and indistinguishable single-photons.In conclusion, we have developed a photoluminescenceimaging technique that enables the location of single quan-tum dots with respect to alignment markers with an averageposition uncertainty <
30 nm and reaching values as low as <
10 nm. We have combined this technique with systematiccalibration of our fabrication process to create single-photonsources based on a circular Bragg grating geometry that si-multaneously exhibit high brightness, purity, and Purcell en-hancement of the spontaneous emission rate. More generally,this technique is an important step forward in the ability tocreate functional single quantum dot nanodevices, includingquantum light sources, strongly-coupled quantum dot - micro-cavity systems for achieving single photon nonlinearities coupled quantum dot - nanomechanical structures , andintegrated systems involving multiple quantum dot nodes.
Methods
Circular Bragg grating cavity fabrication
Devices are fabricated in a wafer grown by molecular beam epi-taxy, consisting of a single layer of InAs quantum dots (QDs) em-bedded in a 190 nm thick layer of GaAs, which in turn is grown on top of a 1 µ m thick layer of Al x Ga − x As with an average x = 0.65.The s-shell peak of the QD ensemble is located near 940 nm, and agradient in the QD density is grown along one axis of the wafer. Low-temperature photoluminescence imaging of portions of the wafer isperformed prior to any device definition to determine the appropriatelocation on the wafer (in terms of QD density) to fabricate devices.Alignment marks are fabricated using positive tone electron-beamlithography and a lift-off process. Polymethyl methacrylate (PMMA)with a molecular weight of 495,000 is spin coated onto the sample,and 2 µ m wide, 50 µ m long crosses are patterned in the resist usinga 100 keV electron-beam lithography tool. After exposure, the resistis developed in a 1:3 (by volume) solution of methyl isobutyl ketone(MIBK) and isopropanol, and 20 nm of Cr and 100 nm of Au are de-posited on the sample using an electron-beam evaporator. Micropositremover 1165 is used for lift-off, with gentle ultra-sonication appliedif necessary.After location of quantum dots with respect to the alignmentmarks through photoluminescence imaging, circular Bragg grating‘bullseye’ microcavities are fabricated as follows. First, the sampleis spin-coated with a positive tone electron-beam resist (ZEP 520A),and aligned electron-beam lithography with a 100 keV tool and fourmark detection is performed. Next, the pattern is transferred into theGaAs layer using an Ar-Cl inductively-coupled plasma reactive ionetch. After removal of the electron beam resist, the sample is under-cut in hydrofluoric acid.Atomic force microscopy (AFM) was used in the calibration ofthe etch rate, with the samples scanned in tapping mode using acommercial, etched silicon probe whose backside is coated with Al.The AFM probe cantilever has a vendor-specified spring constant of42 N/m, frequency of 300 kHz, and probe tip radius and height of8 nm and 10 µ m, respectively. ∗ Electronic address: [email protected] † Electronic address: [email protected] P. Michler,
Single Semiconductor Quantum Dots (Springer Verlag,Berlin, 2009). C. Santori, D. Fattal, and Y. Yamamoto,
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L.S. acknowledges support under the Cooper-ative Research Agreement between the University of Maryland andNIST-CNST, Award 70NANB10H193. The authors thank SerkanAtes and Krishna Coimbatore Balram for useful discussions andearly contributions to this work. They also thank Christopher Longand Santiago Solares for helpful advice regarding atomic force mi-croscopy.
Author Contributions
L.S. and K.S. fabricated the devices, per-formed the measurements and analyzed the experimental data, andwrote the manuscript. A.B. grew the quantum dot material, M.D.and L.S. performed the electromagnetic simulations, and K.S. super-vised the project.The identification of any commercial product or trade name is usedto foster understanding. Such identification does not imply recom-mendation or endorsement or by the National Institute of Standardsand Technology, nor does it imply that the materials or equipmentidentified are necessarily the best available for the purpose.
Additional Information
Correspondence and requests for materialsshould be addressed to L.S. and K.S.
Competing financial interests
The authors declare no competingfinancial interests.
SUPPLEMENTARY INFORMATION
Supplementary Note 1: Quantum dot positioning setup and measurements
In this note, we provide additional details on the quantum dot positioning setup shown schematically in Fig. 1 of the main text.The samples are housed within a cryogen-free cryostat with a base temperature as low as 6 K. Sample motion is achieved using a three-axiscryogenic piezo-positioning stage system. A confocal micro-photoluminescence geometry is utilized, in which a microscope objective (20xmagnification and 0.4 numerical aperture) both focuses excitation light on the sample and collects light emitted and reflected by the sample. Asdescribed in the main text, photoluminescence imaging is done with co-illumination by 630 nm and 940 nm LEDs, where the former is usedto excite the quantum dots and the latter is used to image alignment marks. Excitation of single quantum dots for spectroscopy is performedby focusing a 780 nm laser on the sample.A 90/10 (reflection/transmission percentage) beamsplitter followed by a 900 nm long-pass filter is used to send the light emitted andreflected by the sample towards the imaging and spectroscopic characterisation paths. Selection between the two paths is accomplished witha half waveplate and polarizing beasmplitter. For photoluminescence imaging, the collected light is coupled into a variable zoom system andElectron Multiplied Charged Couple Device (EMCCD), while for spectroscopy, it is coupled to a single mode fiber whose output is sent to agrating spectrometer equipped with a silicon Charged Coupled Device (CCD).In the photoluminescence imaging measurements, the 900 nm longpass filter serves to reject 630 nm excitation light, while allowing boththe quantum dot emission and reflected 940 nm LED light to pass. A total system magnification of 40x (20x from the objective, and 2x from thezoom barrel) is used, corresponding to a field of view of ≈ µ m x 200 µ m. EMCCD images are acquired with an integration time of 120 sand gain of 200. While the 630 nm LED power is always set at its maximum ( ≈
40 mW, corresponding to an intensity of ≈
130 W/cm ) togenerate as much fluorescence from the quantum dot as possible, the 940 nm LED power is set to achieve a reflected signal from the alignmentmarks that is approximately equal to the intensity of the quantum dot emission. This choice of 940 nm LED power is a tradeoff between theimproved alignment mark position uncertainty produced at higher powers, and the degraded quantum dot position uncertainty that results ifthe reflected 940 nm LED signal swamps the quantum dot emission. A typical 940 nm LED power is ≈ µ W.The linecuts of the images taken by the EMCCD camera are analyzed using a commercial software and fitted by Gaussian functions todetermine the location of the quantum dot and centers of the alignment marks. The fit is optimized using a Levenberg Marquardt iterationalgorithm. The central position of the Gaussian function and its error are then translated from a pixel value on the camera to a distance on thesample by using a calibration obtained by imaging, under the same magnification conditions, a microscope calibration target presenting etchedfeatures with known separations.
Supplementary Note 2: Solid immersion lenses for reduced positioning uncertainties
Solid immersion lenses have been used to increase the collection of light emitted by semiconductor quantum dots by increasing the effectivenumerical aperture of the collection optics . Moreover, because the solid immersion lens reduces the focused excitation spot size, it alsoincreases the excitation intensity at the sample. This can lead to an increased photon flux from the quantum dot, because (as was noted inthe main text), without the solid immersion lens, the maximum 630 nm LED excitation intensity at the sample is not enough to saturate thequantum dot emission. Taken together, the increased emission signal from the quantum dot should lead to a lower uncertainty in its position.We also expect that the solid immersion lens can improve the alignment mark uncertainty, since the amount of 940 nm LED power used toimage the mark will be increased (see discussion above) to match the increased quantum dot emission level.We test the above experimentally using a 2 mm diameter, high refractive index ( n ≈
2) half-ball lens placed directly on the sample surface,with a thin layer of cryogenic grease applied between the sample and lens, obtaining the results shown in Fig. 2(d)-(e) from the main text(the x-axis scans are similar). We measure uncertainties in the quantum dot-marker distance as low as 7.6 nm, a reduction of about a factor 4compared to the average error measured without the lens (and a factor of 2 compared to the best error measured without the lens).However, there are some considerations to take into account other than the reduced positioning error possible with a solid immersion lens.First, the solid immersion lenses are generally 1 mm or 2 mm in diameter. Therefore, when using them for imaging the quantum dot emissionand alignment marks, the area of the sample that can be probed within a single measurement session is highly reduced, unless multiple lensesare used. Second, given that the lenses are hemispherical, care must be taken in optimally focusing the imaging and excitation light on theapex of the solid immersion lens, in order to avoid distortions of the image that would affect the inferred distance between the alignment markand the quantum dot.
Supplementary Note 3: Quantum dot single-photon source characterization
A schematic of the experimental setup used to evaluate the collection efficiency and single-photon purity of the quantum dot emission isshown in Supplementary Fig. S1, and is similar to that used in previous work . The sample is mounted on the cold finger of a liquid helium flowcryostat that sits on a two-axis nano-positioning stage. Spectral properties of the quantum dot emission are investigated via low-temperaturemicro-photoluminescence, where a 20x microscope objective (numerical aperture of 0.4) is used for both the illumination of the sample andthe collection of the emission. Four different excitation sources are available for use. The first is a continuous wave 780 nm diode laser forbasic spectroscopy. The second is a continuous wave Ti:sapphire laser, tunable beteween 780 nm and 1000 nm, that can be used to excite thequantum dot on its different excited state transitions. The third is a gain-switched, 780 nm pulsed laser diode (50 ps pulse width; 50 MHzrepetition rate) for photon counting, lifetime, and correlation measurements. The final source is a 820 nm to 950 nm tunable fiber laser ( <
10 pspulse width; 80 MHz repetition rate) used for counting, lifetime, and correlation measurements under excitation of a quantum dot’s excitedstate.The collected signal is directed to a spectrometer either to record an emission spectrum with a Si CCD camera, or to filter a single emissionline for further investigation (Supplementary Fig. S1(a)). The spectrally filtered emission line is coupled into a single mode optical fiberto enable measurements using fiber-coupled single-photon avalanche diodes (SPADs). Single quantum dot fluorescence decay dynamics are optical attenuator white lightsourceSi CCDcameraLHecryostattwo axis sample stage zoom barrel off max LPFSMF
PL (t) cw diode laser
780 nm spectrometerSi CCD high QESPAD g (2) ( τ ) TRIG
Tunable pulsed fiber laser
Laser trigger high QESPADfast SPAD
PL( λ ) a cb flipmirror
50 : 50 splitter
TRIG
Gain-switcheddiode laser
780 nm cw Ti:sa laser
780 nm
FIG. S1: Schematic of the experimental setup: (a) confocal micro-photoluminescence; (b) time-resolved photoluminescence; (c) HanburyBrown and Twiss photon correlation setup. SMF: single-mode fiber, LPF: long-pass filter, CCD: charge-coupled device, PL: photolumines-cence, SPAD: single-photon avalanche diode, QE: quantum efficiency.measured through time-correlated single-photon counting, which relies on measuring the time delay between an excitation pulse and detectionof an emitted photon by a SPAD (Supplementary Fig. S1(b)). We use a thin Si SPAD whose timing jitter is <
50 ps to enable measurement offast quantum dot decay dynamics. For the second-order correlation function g ( ) ( τ ) measurements, the spectrally filtered emission is directedto a Hanbury-Brown and Twiss interferometer that consists of a fiber-coupled, 50/50 non-polarizing beam-splitter and two fiber-coupled single-photon avalanche diodes (SPADs), as shown in Supplementary Fig. S1(c). These SPADs have a timing jitter of ≈
700 ps, and their outputs areconnected to a time-correlated single-photon counting board. A time bin width of 512 ps is chosen for the g ( ) ( τ ) measurements.Calibration of the quantum dot single-photon source collection efficiency into the 20x (0.4 numerical aperture) objective proceeds as follows.First, the transmission of the optical path from the QD source to the detector is determined. The emitted light escapes the cryostat by travelingthrough two fused silica windows (total transmission ≈
87 %), it is then collected by a microscope objective (transmission of ≈
70 %), goesthrough a 90/10 beamsplitter (transmission of ≈
89 %), reflects off four dielectric mirrors and travels through a polarizer (total transmissionof ≈
78 %) before being focused through the slit of the grating spectrometer. The total transmission of the optical path up to the spectrometeris 42 % ± µ W) is sent into the spectrometer after being attenuated by an independently measured attenuation(64.21 dB) using a variable attenuator, and the detected counts on the Si CCD coupled to the spectrometer are recorded. The counts measuredfrom a quantum dot, excited with a 50 MHz repetition rate source, are then recorded and compared to the laser counts (taking into account thetransmission of the optical path) in order to extract the emitter’s single-photon collection efficiency.Figure 4 and the accompanying discussion in the main text present data characterizing single-photon source performance for an opticallypositioned quantum dot within a circular grating ‘bullseye’ cavity. Here, we present supplementary data referred to in the main text discussion.Supplementary Fig. S2(a) shows an intensity autocorrelation measurement ( g ( ) ( τ ) ) under pulsed excitation at 780 nm when the quantum dotemission is saturated. Despite what appears to be a relatively clean emission spectrum (in terms of an absence of spectral features other thanthe quantum dot emission) in Fig. 4(d) of the main text, the relatively significant multi-photon component measured ( g ( ) ( ) = . ± . , are thought to be a potential source of suchmulti-photon emission, particularly in Purcell-enhanced (e.g., microcavity) geometries. This is consistent with our measurements, as moreintense excitation (through a 780 nm continuous wave laser that provides more output power than the pulsed 780 nm laser) yields a spectrumin which the cavity mode emission is clearly visible (Supplementary Fig. S2(b)). Given that the cavity mode linewidths are many nanometerswide, while the quantum dot excitonic states are orders of magnitude narrower, a spectrally broad emission source such as a quasi-continuumstate is needed to reconcile the presence of the cavity mode within the measured spectrum.The contribution of such quasi-continuum states should be limited if the system is pumped on an excited state of the quantum dot, whichwould prevent the generation of high energy carriers that could fill those states. Using a narrow linewidth ( < g ( ) ( τ ) (Fig. 4(e) in the main text) is markedly different, with g ( ) ( ) = . ± . -100 -50 0 50 100020406080 Time (ns) C o i n c i d e n c e s g (2) (0) = 0.15 ± 0.03 a CC D c o un t r a t e ( s - ) c x 10
900 905 910 915wavelength (nm)
857 nm excitation d
906 907 90800.20.40.60.81.0 wavelength (nm)
780 nm857 nm N o r m a li z e d CC D c o un t s
900 905 910 91500.40.81.21.6 wavelength (nm) CC D c o un t r a t e ( s - ) b x 10
780 nm excitation
FIG. S2: Supplementary data for the single photon source characterization of Figure 4 from the main text. (a) g ( ) ( τ ) under 780 nm pulsedexcitation, with the quantum dot emission saturated. (b) Spectrum under intense 780 nm continuous wave excitation (far above quantum dotsaturation), for which the bullseye cavity modes are visible. The quantum dot line of interest is +1.6 nm detuned with respect to the shorterwavelength cavity mode. The red solid line is a fit of the data to the sum of two Gaussians, which is used to determine the center wavelengths ofthe two cavity modes. The two Gaussians making up the sum are shown as black dashed lines. (c) Spectrum under intense 857 nm continuouswave excitation (far above quantum dot saturation), on resonance with a quantum dot excited state. Compared to 780 nm excitation far abovesaturation, significantly reduced cavity mode emission is observed. (d) Comparison of the quantum dot spectra under saturation conditions(the conditions under which the g ( ) ( τ ) data in part (a) and Fig. 4(e) were taken) for both 780 nm and 857 nm pulsed excitation.the quantum dot emission, and had a throughput of ≈
11 %. The output of the monochromator was coupled into single mode fiber and sentinto the Hanbury-Brown and Twiss setup as described above, and the detected count rates on each of the two SPADs was ≈ × counts/s inthe measurements from Fig. 4(e) in the main text. Overall, this detected count rate includes the collection efficiency of quantum dot emissioninto the NA = 0.4 lens ( ≈
48 %), the transmission of the photoluminescence setup ( ≈
42 %), the throughput of the monochromator ( ≈
11 %),coupling from the monochromator output into single mode fiber and throughput of the single-mode-fiber-based Hanbury-Brown and Twisssetup ( ≈
12 %), and the SPAD quantum efficiency ( ≈
20 %).
Supplementary Note 4: Comparison to single-photon sources created without optical positioning
For the purposes of comparison, in this section we present data from quantum dot single-photon sources in which quantum dot positioningwas not employed (so that the position of the quantum dot with respect to bullseye cavity center was uncontrolled). The investigation of thesedevices was described in detail in Ref. 2, where spectroscopy, lifetime, and photon correlation measurements were presented. In SupplementaryFig. S3(a), we show an EMCCD image of a subset of the array of cavities investigated in Ref. 2, where the array has been illuminated by the630 nm red LED. This EMCCD image reveals two new pieces of information. First, only one of twelve displayed devices shows an emissionlobe near the center of the cavity, for which the collection efficiency is expected to be maximized. For this unpositioned sample, the maximumcollection efficiency measured was ≈
10 %, and the fraction of devices producing this efficiency was a couple of percent. Next, the quantumdot density in this sample is significantly higher than that studied in the current manuscript. While the density is still low enough so that only asingle quantum dot can spatially and spectrally interact with a mode of the cavity, it is about two orders of magnitude larger than what we usein the positioned quantum dot devices. The background emission caused by these quantum dots, and in particular, their potential for supportingquasi-continuum states with broad emission bandwidths , may limit the purity of single-photon emission. Given that the yield for this sampleis only a couple of percent, reducing the quantum dot density without locating the quantum dots prior to fabrication is impractical.Spectroscopy and photon counting measurements from Ref. 2 further address these points. A typical photoluminescence spectrum undernon-resonant pulsed excitation is shown in Supplementary Fig. S3(b). In contrast to the clean spectrum shown in the main text in Fig. 4(d), thespectrum of Supplementary Fig. S3(b) shows significant background emission attributed to feeding of the cavity mode by multi-excitonic statesof nearby quantum dots. This emission can be expected to limit the purity of the single-photon source produced by spectrally isolating a singleexcitonic state, and indeed, the g ( ) ( τ ) measurement in Supplementary Fig. S3(c) shows a significant departure from g ( ) ( ) =
0. While thismeasurement is of a device with a particularly high g ( ) ( ) value, in general, unpositioned devices studied in Ref. 2 showed g ( ) ( ) (cid:38)
15 %.That being said, the discussion from the previous section indicates that even the drastically reduced quantum dot density used in the currentwork most likely needs to be supplemented by excited state pumping of the quantum dot in order to achieve g ( ) ( ) ≈ Supplementary Note 5: Electromagnetic simulations
As discussed in the main text and in Ref. 2, the bullseye cavity supports dipole-like resonant modes (shown in Supplementary Figs. S4(b) and(c)) that are well-suited for the creation of bright single-photon sources - a combination of relatively high Purcell-type radiative enhancement,efficient vertical light extraction from the semiconductor, and near-Gassian far-field for efficient collection into an optical fiber. These modesare strongly localized at the center of the cavity (the central intensity peak has a full-width at half-maximum of ≈
100 nm), and a sequenceof satellite peaks along the radial direction. Because the electric dipole coupling to a cavity mode is proportional to the squared electric fieldmagnitude at the dipole location , we expect that the Purcell enhancement factor F p , coupling efficiency η , and emitted polarization state willvary significantly with dipole position. An understanding of these parameters is not just important from a device performance perspective, butalso provides information about the actual quantum dot location. We employ full-wave numerical electromagnetic simulations to investigatethe sensitivity of the emission properties of our single-photon source to the location of the quantum dot within the bullseye cavity.
935 940 94500.20.40.60.8 Wavelength (nm) I n t e n s i t y ( k c n t s / s ) −100 C o i n c i d e n c e C o un t s g (2) (0)0.37 ± X η bca FIG. S3: Representative data from quantum dot - bullseye cavity devices fabricated without using the quantum dot positioning technique, asin Ref. 2. (a) EMCCD image of an array of cavities, illuminated by a 630 nm LED. Only one of the devices (within the dashed box) showsan emission lobe centered with respect to the cavity. Scale bar represents 50 µ m. (b) Photoluminescence spectrum from a device exhibitingcollection efficiency ≈
10 %. (c) g ( ) ( τ ) from the same device. Note that parts (b) and (c) are re-displayed data from Ref. 2.
1. Purcell Factor and Collection Efficiency
Following Ref. 6, we use finite-difference time domain (FDTD) simulations to model the system as an electric dipole radiating inside asuspended bullseye cavity. The dipole is allowed to radiate with a short Gaussian pulse time dependence, and the electromagnetic field isallowed to evolve over a long time span. The steady-state electromagnetic field is recorded at all edges of the computational window, sothat the total dipole radiated power P rad can be determined. The Purcell factor can then be obtained as F p = P rad / P hom , where P hom is thedipole radiated power in a homogeneous medium . We also record the power P z emitted upwards in the + z direction, which in a real settingis partially collected with a microscope objective with numerical aperture NA . The steady-state field recorded at a parallel plane above thebullseye cavity is used to calculate the emitted far-field, which is then integrated within an angular cone corresponding to a numerical aperture NA to yield the collection efficiency η NA . Perfectly matched layers are used to simulate free-space above and below the cavity, so that effectsrelated to the substrate are not taken into account. The dipole is assumed to be on the z = z -components. The latter assumption is appropriatefor epitaxially grown InAs dots, given their few nanometer vertical size, negligible compared to the membrane thickness . d y d x xy ba c |E | d x E y = E z = 0H x = 0 |E | d y H y = H z = 0E x = 0 FIG. S4: (a) Schematic of the simulation geometry, showing a dipole (green arrows) inside a bullseye cavity. Due to the circular symmetry,a dipole located anywhere within the cavity and with any orientation can be represented by a dipole on the x -axis with d x and d y componentsequivalent to the radial and azimuthal ones. (b) Electric field amplitude squared profile for a ’h’-type bullseye cavity mode. The yz -planeboundary conditions satisfied by the mode are given in the figure. This mode is only excited by x -dipoles. (c) Electric field amplitude squaredprofile for a ’v’-type bullseye cavity mode. The yz -plane boundary conditions satisfied by the mode are given in the figure. This mode is onlyexcited by y -dipoles. In the circular geometry of the cavity, an electric dipole with moment d with arbitrary orientation placed anywhere in the cavity is equivalentto a dipole located on the x -axis with components along the x and y directions, corresponding to the radial and azimuthal components. Thisis illustrated in Supplementary Fig. S4(a). The symmetry of the problem allows a description of the electromagnetic fields supported by thecavity in terms of orthogonal, symmetric and anti-symmetric cavity eigenmodes with respect to the y = h ’-modes) and degeneratemodes of the 90-degree rotated geometry (’ v ’-modes), as illustrated in Supplementary Figs. S4(b) and (c). An x -dipole will however only excitesymmetric h -modes and anti-symmetric v -modes, and vice-versa is valid for a y -dipole; in other words, d · E h = d x · E hx and d · E v = d y · E vy .From ref. 6, the total power emitted by the dipole in the cavity is P rad ∝ ∑ n | d · E n | , where d is the dipole moment and E n is the (normalized)electric field for mode n , evaluated at the dipole location. With the symmetry considerations above, we can write P rad ∝ ∑ n (cid:12)(cid:12)(cid:12) d x · E h , nx (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) d y · E v , ny (cid:12)(cid:12) = | d | (cid:26) ∑ n (cid:12)(cid:12)(cid:12) · E h , nx (cid:12)(cid:12)(cid:12) cos φ + (cid:12)(cid:12) · E v , ny (cid:12)(cid:12) sin φ (cid:27) , (S1)where φ is the dipole orientation - the angle the dipole makes with respect to the x-axis - which we assume to be unknown. Equation (S1) thusallows us to determine the dipole emitted power P rad for a dipole positioned anywhere on the x axis, with arbitrary angle given by φ , just basedon the E h and E v modes which are respectively excited by the x and y dipole components.As such, we proceed to calculate P rad and the collection efficiency η . for NA = . x - and y -oriented dipoles located on the x -axis at varying distance x from the cavity center. We then use eq.(S1) to determine the range within which the quantities of interest can varydue to the (unknown) azimuthal dipole orientation φ . To verify that this procedure is valid, we also simulate the case φ = ◦ , and comparethe collection efficiency with that obtained through eq. (S1) and the x − and y − dipole solutions. As shown in Supplementary Fig. S5, thedifference between the two types of calculations is (cid:46) wavelength (nm) d i p o l e d i s p l a c e m e n t ( n m ) collection efficiency, φ = 45 o
940 950 960 97050100150200250300350 0.10.20.30.4wavelength (nm) d i p o l e d i s p l a c e m e n t ( n m )
940 950 960 97050100150200250300350 0.10.20.30.4 wavelength (nm) d i p o l e d i s p l a c e m e n t ( n m ) ∆ (collection efficiency) 940 950 960 97050100150200250300350 246810x 10 −3 collection efficiency, combined x and y dipoles a b c FIG. S5: Collection efficiency η . into a NA = . x -axis inside thebullseye cavity. The dipole is on the xy -plane and is oriented at an azimuthal angle φ = ◦ . (a) Results obtained using eq.(S1) with separatesimulations for x and y dipoles individually. (b) Results obtained by simulating a dipole at φ = ◦ . (c) Absolute value of difference betweenresults shown in panels (a) and (b).In Supplementary Figs. S6(a) and (b), we show the Purcell Factor F p as a function of wavelength for x - and y -dipoles, respectively, located atvarying positions along x . At a wavelength of 948 nm, the x dipole couples to the ’ h ’ resonance shown in Supplementary Fig. S4(b), while the y -dipole couples to the degenerate ’ v ’ mode in Supplementary Fig. S4(c), and the Purcell Factor F p peaks for dipoles at the cavity center. Fory-dipoles displaced from the center, F p shows a sequence of satellite peaks observed for increasing distances, which contrasts with the x -dipolecase. This can be understood based on the variation of of the v and h field profiles along the x -axis (Supplementary Figs. S4(b)-(c)), as F p ∝ | E | .A second resonance centered at 957 nm exists that is also excited by dipoles in both orientations, however displays considerably lower Purcellenhancement and collection efficiency (shown later). Supplementary Figs. S6(c) and (d) show the overall maximum and minimum achievable F p , and the shaded areas in Supplementary Fig. S6(e) correspond to overall allowed values of F p as a function of dipole displacement, forthree wavelengths around the resonance center. Essentially, these ranges correspond to the uncertainty in our knowledge of F p due to lack ofknowledge of the in-plane dipole orientation. Dotted white lines, on the other hand, correspond to the case φ = ◦ , which corresponds to anin-plane isotropic dipole.Supplementary Figs. S7(a) and (b) show the overall maximum and minimum achievable collection efficiency η . , and the shaded areas inSupplementary Fig. S6(c) correspond to overall allowed values of η . as a function of dipole displacement, for three wavelengths around theresonance center. White dotted lines are for the φ = ◦ case. It is evident that the collection efficiency is a much slower function of bothwavelength and dipole displacement than the Purcell factor. As a result, for the QD-cavity wavelength detuning of the device we focus on inthe main text (1.6 nm), there is a ≈ ±
250 nm range of dipole positions consistent with the experimentally observed collection efficiency(48 % ± ≈ φ prevents us from more precisely estimatingthe location of the emitter within the cavity. For example, if φ = ◦ , from Supplementary Figs. S6(e) and Supplementary Fig. S7(c) we canestimate that the dipole is located within 50 nm of the cavity center, in order to display the experimentally observed F p and η . .We note however that the collection efficiency maximum is shifted with respect to the resonance center by approximately -5 nm, as canbe seen in Supplementary Figs. S6(a)-(d) and Supplementary Figs. S7(a)-(b). This is due to far-field collection efficiency, which is actuallyasymmetric with respect to the resonance center, being higher by approximately 0 . wavelength (nm) d i s p l a c e m e n t ( n m ) x−dipole, Purcell Factor 940 9600100200300 246810 wavelength (nm) d i s p l a c e m e n t ( n m ) y−dipole, Purcell Factor 940 9600100200300 246810wavelength (nm) d i s p l a c e m e n t ( n m ) Maximum Purcell Factor 940 9600100200300 246810 wavelength (nm) d i s p l a c e m e n t ( n m ) Minimum Purcell Factor 940 9600100200300 246810 a bc d P u r c e ll F a c t o r e λ = 948.02 nm λ = 949.63 nm λ = 946.41 nm FIG. S6: Purcell Factor F p as a function of wavelength and dipole position along the x -axis, inside the bullseye cavity. (a) Results for an x -oriented dipole; (b) Results for a y -oriented dipole; (c) maximum achievable F p ; (d) minimum achievable F p ; (e) Purcell Factor as a functionof dipole displacement from the bullseye cavity center, at resonance ( λ = .
02 nm) and at ± F p due to lack of knowledge of the dipole azimuthal angle φ . The white dotted line correspondsto the case φ = ◦ . dipole displacement (nm) a b c λ = 948.02 nm λ = 949.63 nm λ = 946.41 nm c o ll e c ti o n e ffi c i e n c y wavelength (nm) d i p o l e d i s p l a c e m e n t ( n m ) Maximum collection efficiency 940 9600100200300 0.10.20.30.40.5 wavelength (nm) d i p o l e d i s p l a c e m e n t ( n m ) Minimum collection efficiency 940 9600100200300 0.10.20.30.4
FIG. S7: Collection efficiency η . into a NA = . x -axis inside the bullseyecavity. (a) Maximum achievable η . ; (b) minimum achievable η . ; (c) Collection efficiency as a function of dipole displacement from thebullseye cavity center, at resonance ( λ = .
02 nm) and ± η . due to lack of knowledge of the dipole azimuthal angle φ . The white dotted line corresponds to the case φ = ◦ .information is still not sufficient to pinpoint the quantum dot location based on our experimental data, it further corroborates our explanationthat the relatively low observed Purcell factors can still exist with high collection efficiencies.
2. Polarization of the light emitted by a dipole embedded within a bullseye cavity
We now study the polarization properties of the light emitted by a dipole in the bullseye cavity. In particular, our goal is to understandthe degree to which polarization-resolved measurements of the far-field intensity can be used to identify the dipole orientation, which in turnwould enable more precise determination of the dipole location from Purcell enhancement and collection efficiency measurements.The ’ h ’ and ’ v ’ bullseye cavity modes overall display major electric field components oriented in the x and y directions, respectively. This can be verified in two ways: plots of | E x | and | E y | for the ’ v ’ mode in Supplementary Figure S8(a) show the former to be overall at least anorder of magnitude larger than the latter; and the ratio R xy = (cid:82) S NA dS | E x | / (cid:82) S NA dS | E y | , where S NA is the spherical surface corresponding to a NA = . R xy = .
47. As such, we expect the far-field produced by a dipole at an arbitrary orientation characterizedby the azimuthal angle φ to display some degree of polarization. This degree of polarization can in principle be resolved by introducing ofa linear polarizer above the cavity and determining the variation of the transmitted power (collected into a 0 . NA optic) with respect to thepolarizer orientation. ° ° ° ° −0. −0. −0. −0. −0. −0. Re{E x } ° ° ° ° ° ° ° ° −0. −0.15−0. −0.0500.050. Im{E y } ° ° ° ° −0. −0.15−0. −0.0500.050. Re{E y } ° ° ° ° −0. −0. −0. −0. −0. −0. −0. Im{E x } ° ° ° ° |E y | |E x | ad be cf FIG. S8: (a) Magnitude-square, (b) Real part, and (c) Imaginary part of the E x far-field for a ’ v ’ mode. d) Magnitude-square, (e) Real part, and(f) Imaginary part of the E y far-field for a ’ v ’ mode.To perform this calculation, we first note that the radial component of the far electric field is much smaller than the azimuthal and polarones ( | E ρ | (cid:28) | E φ | , | E θ | in spherical coordinates). We then assume that the collection cone is narrow enough that the field at the entrance of thecollecting lens can be well represented as E = E x ˆx + E y ˆy , where E x and E y are the x - and y -components of the far-field (in other words, we take E x and E y to be the transverse components of the far-field). This allows us to use Jones matrix formalism to estimate the power transmittedthrough the polarizer. We represent a polarizer oriented at an angle θ p with respect to the x -axis with the Jones matrix M = (cid:20) cos θ p sin θ p − sin θ p cos θ p (cid:21)(cid:20) (cid:21)(cid:20) cos θ p − sin θ p sin θ p cos θ p (cid:21) = (cid:20) cos θ p − sin θ p cos θ p − cos θ p sin θ p sin θ p (cid:21) (S2)The transmitted electric field E out = ME is, then, (cid:20) E x E y (cid:21) out = (cid:20) E x cos θ p − E y sin θ p cos θ p E y sin θ p − E x sin θ p cos θ p (cid:21) . (S3)The transmitted power is proportional to | E | = | E x | + | E y | . If the emitting dipole is at an arbitrary orientation, both ’ h ’ and ’ v ’ modes areproduced in the cavity, so that, in the far-field, E = α h E h + α v E v ( α h , v represent the dipole coupling strength to the h and v modes). In this case,the resulting expression for the transmitted power consists of a sum of terms E ik E j ∗ l , where i , j ∈ { h , v } and k , l ∈ { x , y } . In determining thetransmitted power, all of these terms are integrated over a portion of a spherical surface which represents the acceptance cone of the collectionlens. Because of the cylindrical symmetry of the cavity, the x and y components of the ’ h ’ and ’ v ’ fields obey the following symmetry relations(as seen in Supplementary Figs. S8(b),(c),(d) and (f)): E hx and E vy are even in x and y ; E vx and E hy are odd in x and y . Because the integrationis performed symmetrically in the xy plane, any cross-term E ik E j ∗ l that results odd in x and y has no contribution to the power; these are crossterms with { i (cid:54) = j , k = l } and { i = j , k (cid:54) = l } . We can thus write the integrands | E x | = cos θ p (cid:18)(cid:12)(cid:12)(cid:12) E hx (cid:12)(cid:12)(cid:12) + | E vx | (cid:19) + cos θ p sin θ p (cid:18)(cid:12)(cid:12)(cid:12) E hy (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) E vy (cid:12)(cid:12) (cid:19) − θ p sin θ p ℜ (cid:110) E hx E v ∗ y + E vx E h ∗ y (cid:111) (S4)and (cid:12)(cid:12) E y (cid:12)(cid:12) = sin θ p (cid:18)(cid:12)(cid:12)(cid:12) E hy (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) E vy (cid:12)(cid:12) (cid:19) + cos θ p sin θ p (cid:18)(cid:12)(cid:12)(cid:12) E hx (cid:12)(cid:12)(cid:12) + | E vx | (cid:19) − θ p sin θ p ℜ (cid:110) E h ∗ x E vy + E v ∗ x E hy (cid:111) , (S5)where the substitutions E h , v ← α h , v E h , v were done for simplicity. We use equations (S4) and (S5) to estimate the collected power that istransmitted through the polarizer, at any polarizer orientation angle . The visibility V can be determined from the maximum and minimum intensities with respect to the polarizer angle as V = I max − I min I max + I min , (S6)where I = (cid:82) S NA = . dS | E | , where S NA = . is the spherical surface corresponding to the NA = . E h and E v by a dipole located at an arbitrary position in the cavity depends on the dipole’s orientation; and because the dipoleorientation is not known, we can only determine the possible ranges of V at each dipole location. As such, we can only determine the range ofachievable visibilities V at each dipole location. This is shown as a function of wavelength in Supplementary Fig. S9(s).These plots indicate the non-monotonic dependence of the visibility on dipole location and orientation. As a result, a measurement of thevisibility, taken together with measurements of the Purcell enhancement and collection efficiency, usually does not provide an unambiguousestimate of the dipole location.For example, we have measured V = . − . ≈
100 nm away from the cavity center. While this is consistent with our estimate of dipole location basedon the Purcell enhancement and collection efficiency measurements ( <
250 nm from the bullseye center), it does not provide a significantlyimproved estimate of the dipole location. Additional measurement techniques (for example, spatially-resolved polarization-dependent far-fieldmeasurements) may be required to achieve a better estimate. ab c V i s i b ili t y wavelength (nm) d i p o l e d i s p l a c e m e n t ( n m ) Maximum Visibility 940 9600100200300 0.20.40.60.8wavelength (nm) d i p o l e d i s p l a c e m e n t ( n m ) Minimum Visibility 940 9600100200300 0.20.40.6 λ = 948.02 nm λ = 949.63 nm λ = 946.41 nm FIG. S9: Visibility V as a function of wavelength and dipole position along the x -axis inside the bullseye cavity. (a) Maximum V ; (b) minimum V ; (c) Visibility as a function of dipole displacement from the bullseye cavity center, at resonance ( λ = .
02 nm) and ± V due to lack of knowledge of the dipole azimuthal angle φ . Thewhite dotted line corresponds to the case φ = ◦ . Supplementary References ∗ Electronic address: [email protected] † Electronic address: [email protected] K. A. Serrels, E. Ramsay, P. A. Dalgarno, B. Gerardot, J. O’Connor, R. H. Hadfield, R. Warburton, and D. Reid, “Solid immersion lensapplications for nanophotonic devices,” Journal of Nanophotonics , 021 854 (2008). S. Ates, L. Sapienza, M. Davanco, A. Badolato, and K. Srinivasan, “Bright single-photon emission from a quantum dot in a circular Bragggrating microcavity,” IEEE Journal of Selected Topics in Quantum Electronics , 1711–1721 (2012). M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, and A. Imamoglu,“Explanation of photon correlations in the far-off-resonance optical emission from a quantum - dot cavity system,” Phys. Rev. Lett. ,207 403 (2009). N. Chauvin, C. Zinoni, M. Francardi, A. Gerardino, L. Balet, B. Alloing, L. H. Li, and A. Fiore, “Controlling the charge environment ofsingle quantum dots in a photonic-crystal cavity,” Phys. Rev. B , 241 306 (2009). A. Laucht, M. Kaniber, A. Mohtashami, N. Hauke, M. Michler, and J. Finley, “Temporal monitoring of nonresonant feeding of semiconductornanocavity modes by quantum dot multiexciton transitions,” Phys. Rev. B , 241 302 (2010). Y. Xu, J. S. Vuˇckovi´c, R. K. Lee, O. J. Painter, A. Scherer, and A. Yariv, “Finite-difference time-domain calculation of spontaneous emissionlifetime in a microcavity,” J. Opt. Soc. Am. B , 465–474 (1999). P. Michler,