Polarization microscopy is a useful optical techniquefor the study of materials that have anisotropic opti-cal properties, such as birefringent crystals and bio-logical tissues.1 The study is performedtraditionally by introduction of polars, i.e., a polarizerinto the illumination path and an analyzer into thedetection path of an imaging system. The imagingsystem is typically a conventional or a confocal scan-ning microscope. If there is no sample in a trans-mission system or if there is a perfect reflector in areflection system, and if the polarizer–analyzer pairhas infinite extinction and the lenses do not introduceany cross-polarization component, the field of viewthrough crossed polars ideally will be completelydark. That is to say, the extinction coefficient �theratio of the intensity of light transmitted throughparallel polars to that through crossed polars� of thesystem should be infinite. However, Kubota and In-oue2 have shown that the polarization of incidentlight will be distorted by an objective lens, which maylead to a finite extinction coefficient. Later on, Wil-son and Juskaitis3 showed that in the conventionalpolarization microscope the signal intensity is inte-grated over the exit pupil of the objective lens and
The authors are with the Department of Physics, Trinity CollegeDublin, Dublin 2, Ireland. L. Yang’s e-mail address is [email protected].
Received 7 April 2003; revised manuscript received 14 April2003.
results in a finite value of the extinction coefficient.The extinction coefficient in an ideal confocal system,in which the pinhole is infinitely small, however, isinfinite because the confocal system integrates thefield amplitude over the pupil area.3 As a result,when birefringent materials are studied, confocal po-larization microscopy can provide polarization infor-mation more usefully from the sample and can give ahigher polarization image contrast of the samplethan can conventional microscopy.
The sample discussed in Ref. 3 is a perfect reflector.Wilson et al.4 and Torok et al.5,6 also studied theimages of the dielectric point scatterers with the sizesmaller than the wavelength in conventional and con-focal polarization microscopes. In this paper we fo-cus our study on a microsphere sample whose size iscomparably larger than the wavelength and investi-gate the three-dimensional �3D� imaging of micro-spheres by use of confocal and conventional scanningpolarization microscopes.
Micrometer-sized spherical particles, whose size iscomparably larger than a wavelength, have beenwidely used in the field of microlasers,7,8 as has lasertweezer trapping,9 in recent years. To promote anunderstanding of the interaction of a laser beam withthese microspheres, the imaging properties of micro-spheres have been intensely studied.10–12 Ashkinand Dziedzic10 experimentally studied the optical res-onances of a dielectric sphere by continuously tuningthe wavelength of the incident light. By viewing theparticles under a conventional polarization micro-scope with a crossed analyzer they found that, whenthe wavelength of the light was tuned off resonance,little signal was transmitted. On resonance, how-
ever two arc pairs obviously appeared near thesphere edge, showing a maximum intensity at 45° tothe initial polarization of the light. This phenome-non has been explained in terms of diattenuation ofthe incident polarization as a result of absorption ofthe resonant polarization component. Fayngold11
theoretically investigated the backscattered imagingof a dielectric microsphere and found good agreementwith the experimental results of Ashkin and Dzied-zic.
One may notice, however, that the image of thesphere described in Ref. 10 has a weak arc pair struc-ture even off resonance, which may result from thefinite extinction coefficient of the conventional polar-ization imaging system. More recently, Inami andKawata12 theoretically studied the 3D backscatteredimage of a microsphere with conventional and confo-cal scanning polarization microscopes by using ex-tended Mie scattering theory.13 In the calculatedimages of a microsphere from a conventional polar-ization microscope with a crossed analyzer, arc pairsappear at 45° that are similar to those described inRefs. 10 and 11 for an incident wave on resonance.It should be noted that Inami and Kawata did notmention whether or the incident wavelength was onresonance. Also, they simulated corresponding im-ages obtained with a confocal scanning polarizationmicroscope, which showed similar patterns but witheven greater contrast than those from a conventionalpolarization microscope.
In this paper, we experimentally investigate the3D imaging of latex particles, melamine formalde-hyde �MF�, that have an arbitrary size of a few mi-crometers in diameter, which is �10 times largerthan the wavelength, by using confocal and conven-tional polarization microscopes. The images of anisolated microsphere and a pair of touching spheresare studied. We show that the image properties ofmicrospheres in the two imaging systems are quitedifferent. Because the sphere is more than 10 timeslarger than the focal spot of the objective lens,throughout this paper we theoretically consider thefocus a point and include the polarization effect in theimaging theory of the two types of microscope. Wepresent what is to our knowledge a new analysis toshow the physical origins of the images. Some re-sults are compared with those described in Refs. 10–12. However, some new phenomena were observed.From this study we aim to understand further froman imaging point of view the interaction of a laserbeam with the MF microspheres.
2. Experiment and Discussion
A. Experimental Setup and Samples
Figure 1 shows a schematic of our confocal scanningimaging system. A linearly polarized He–Ne lasersource, with wavelength ��� 632.8 nm, was used.The laser was initially polarized in the vertical direc-tion �extinction ratio, �500:1�, which we define as thex direction. The expanded collimated light was fo-cused on the sample through an infinity-corrected
objective lens �Leitz, Germany� with a numerical ap-erture �NA� of 0.9. The backscattered light from thesample was collected by the objective lens and thenfocused onto a 30-�m-diameter pinhole by a tubelens. The signal was detected by a photomultipliertube �PMT� and analyzed with a computer. Thecomputer also controlled a motorized stage moving inthe x and y directions and a piezoelectric actuatorthat drove the objective lens in the z direction.
The pinhole that was employed in the confocal sys-tem was placed conjugate to the focus of the objectivelens such that the signals that arose largely from thefocal plane could pass through and the signals fromthe out-of-focus plane would be blocked. This en-sured that information was obtained mainly from aspecific layer of the sample and gave the imagingsystem an optical sectioning property. By scanninga perfect reflector axially through focus and measur-ing the detected signal strength we obtained thedepth response. The FWHM of the axial response inour confocal system was measured to be �0.8 �m,whereas for an ideal confocal system it would approx-imate 0.9��NA2,14, i.e., �0.7 �m.
When the pinhole was removed, we could considerthe size of the detector and objective lens infinitelylarge; the sectioning property of the system vanished,and the system worked as a conventional scanningmicroscope.
Both the confocal and the conventional scanningimaging systems could be converted into a polariza-tion configuration by use of an analyzer in the reflect-ing path. We recall that the extinction coefficient inthe ideal confocal polarization microscope was infi-nite, whereas it was finite in the conventional sys-tem.3
The sample that we studied was a MF latex spherethat had a high optical transparency, a high value ofrefractive index n �n � 1.68 in the visible region�, andhigh thermal and mechanical stability.15 The diam-eter of the sphere was �5.2 �m. Size parameter �was �26. � is a measure of the range of the inter-action in units of incident wavelength, defined as � �
Fig. 1. Schematic of a confocal scanning or conventional �whenthe pinhole is removed� polarization microscope.
2a��,16 where a is the sphere’s radius. The size ofthe sphere was more than 10 times greater than thatof the focal spot of the objective lens; hence in thediscussion that follows we simply consider the focusan ideal point.
B. Imaging by Confocal Polarization Microscopy
First we utilized the sectioning property of the con-focal scanning microscope to locate the microspheresample in three dimensions. The images are shownin Fig. 2. An isolated sphere and a pair of touchingspheres were observed. We obtained the imageswithout using the analyzer.
The light was focused onto the surface of the sub-strate �z � a� near the back surface of the sphere,and only the shadow of the sphere can be seen in Fig.2�a�. Figure 2�b� shows the image at z � a�2 withan increased detector gain. The appearance of therings inside the sphere in Fig. 2�b� is due to interfer-ence of the light from the front and back surfaces ofthe sphere and the reflected light from the substrate.
When the focal plane was moved toward the equa-torial plane of the sphere �z � 0�, the center of thesphere become brighter, as shown in Fig. 2�c�. Thisresult can be explained in terms of geometrical optics:When light is focused on the center of the sphere, rayspropagate along the radial direction and reflect back.Bright rings appear in the periphery of the spherewith the center of the sphere saturated, as shown inFig. 2�c�. The brightness of the rings is strongerthan that of the substrate. The signal can be onlythe light backscattered from the focal point of the
sphere surface. Because the distance from the planez � 0 to the substrate was �2.6 �m and the FWHMof the system’s axial response, �0.8 �m, was consid-erably less than this, light from the out-of-focus sub-strate was blocked by the pinhole. In addition, otherlight scattering away from the focus was also blockedby the pinhole.
We used a crossed analyzer to check the polariza-tion of the detected signals. We found that therewas almost no transmission through a crossed ana-lyzer, which indicated that the polarization of thebackscattered signals was mainly parallel to the in-cident polarization. To explain this phenomenon weassumed that the incident beam propagating alongthe z direction was of amplitude E0 and polarized inthe x direction. We considered the signal obtainedin the center plane of the sphere �z � 0�. We as-sumed that a uniform beam was focused by the ob-jective and was incident upon the sphere’s edge atpoint P �Fig. 3� and that the backscattered light frompoint P was collected by the same objective. Thefield of the backscattered light from the sphericalsurface can be considered a superposition of thetransverse electric mode, E0 cos�� � �S���, �, ��, andthe transverse magnetic mode, E0 sin�� � �S���,�, ��, where S���, �, �� and S���, �, �� are the scat-tering amplitudes with polarization parallel and per-pendicular, respectively, to the backscatteringplane.16 The backscattering plane here is defined bythe optical axis and backscattered light. � is thebackscattering angle, � is the polar angle measuredfrom the normal n, and is the angle between the
Fig. 2. Normalized confocal scanning images of microspheres, 5.2 �m in diameter, with different axial locations. From �a� to �f � z � a,a�2, 0, a�4, a�2, a, respectively. There are two defects, A and B, inside the spheres.
normal at point P and the x axis. The x componentof the field in the exit pupil of the objective can beexpressed as
Ex � P��, �, �� � Q��, �, ��cos 2�� � �, (1)
where P��, �, �� � S���, �, �� � S���, �, �� and Q��,�, �� � S���, �, �� S���, �, ��. Similarly, we cangive the y component of the backscattered image fieldas
Ey � Q��, �, ��sin 2�� � �. (2)
For an ideal confocal system with an infinitelysmall detector and a circular objective pupil, detect-ing signal intensity Iconf involves performing the in-tegral of the electromagnetic field amplitude over theexit pupil of the objective and can be written as3
Iconf � ��0
2
�0
�
E sin � cos �d�d��2
, (3)
where the field is E � Exi � Eyj and sin � is the NAof the objective lens.
For an analyzer placed at an angle � to the x axisthe field amplitude becomes
Suppose that the electromagnetic field backscat-tered from focal point P of a smooth spherical surfacehas an axial symmetry to the normal axis n; scatter-ing amplitudes S���, �, �� and S���, �, �� will havethe formulas S���� � S��2 �� and S���� �S��2 ��. Substitution of Eq. �4� into Eq. �3� gives
Iconf � ��0
2
�0
�
�P��, �, ��cos �
� Q��, �, ��cos 2� cos�2
� ���sin � cos �d�d��2
. (5)
When the analyzer is parallel as � � 0, the inten-sity can be written as
I�conf � ��0
2
�0
�
�P��, �, ��
� Q��, �, ��cos 2� cos 2 �sin � cos �d�d��2
.
(6)
Considering the periodic property of cos 2� withinthe range �0, 2� and that backscattering amplitudesS���, �, �� and S���, �, �� vary slowly in this range,the integral of Q��, �, ��cos 2� over � is always muchless than that of P��, �, ��. The intensity can besimplified as
I�conf � ��0
�
P��, �, ��sin � cos �d�d��2
, (7)
which is constant for different position angles at thesphere edge. Similarly, for arbitrary analyzer angle�, Eq. �5� can be rewritten as
I�conf � ��
0
�
P��, �, ��sin � cos �d�d��2
cos2 �. (8)
Comparing Eqs. �7� and �8�, one can see that thebackscattered light from the edge of the sphere willbe polarized parallel mainly to the incident polariza-tion and that the signal strength will be controlledonly by analyzer setting �. This agrees with ourexperimental observation. When the signal is de-tected with a crossed analyzer, the intensity ap-proaches zero. We can physically understand thesephenomena as the field amplitude average effect ofthe confocal microscope. In Fig. 2�c� it is shown thatthe intensity distribution about the sphere edge is notuniform as expected. We attribute this resultmainly to the roughness in the surface of the sphere.The difference in intensity distribution between theisolated sphere and the two touching spheres in Fig.2�c� is due to the fact that the centers of the spheresare not at exactly the same focal plane because thesize of the sphere and the distance from the center tothe substrate are slightly different for each micro-sphere.
Figures 2�d�, 2�e�, and 2�f � show the images at z �a�4, a�2, a, respectively, with the same gain of thedetector, which is comparatively less than that in Fig.2�c� to show the interference rings in the focusedregion of the sphere. We attribute the two brightspots, A and B, in Fig. 2�d� inside the sphere to de-fects. They are located near the cross sections be-tween z � 0 and z � a�4 and disappear at z � a�2�Fig. 2�d��. These defects can be seen randomly in afew microspheres. Their appearance does not seemto signal a refocusing issue.
We used a crossed analyzer to study the intensitydistribution of a polarization component that was
Fig. 3. Focused light incident upon a sphere edge at point P.
perpendicular to the incident polarization in the con-focal system. The images are shown in Fig. 4. ForFig. 4�a� the fact that the signal of the substratebackground is nonzero may be due to the perfor-mance limit of the analyzer and to the size of thepinhole, which is not infinitely small.3
Figure 4�b� shows the image when light was fo-cused at the center plane of the sphere as z � 0. Itwas found that the signal obtained through a crossedanalyzer was very weak but not zero. When theanalyzer was crossed at � � 90°, the detected signalcould actually be derived from Eq. �5� as
I�conf � �sin2 2 �
� ��0
2
�0
�
Q��, �, ��cos 2� sin � cos �d�d��2
, (9)
which is modulated by the factor sin2 2 and predictsa zero value at the edge of the sphere for � 0, �2,, 3�2, as shown in Fig. 4�b�. These patterns agreewell with the numerical analysis of Inami and Ka-wata.12 However, in Fig. 4�b� the signal is weak,which reminds us that the integral of Q��, �, ��cos 2�over � is rather small. Because of the existence ofenvironmental noise, the image contrast of thesphere is poor.
The two defects A and B still can be seen near theregion of z � 0 shown in Fig. 4�b�. The signals fromthese pointlike defects are much higher than thatfrom the edge of the smooth sphere and are satu-rated.
By scanning different cross sections of the spherewe observed arc pairs about the sphere’s surface nearthe region z � a, 3a�4, with the image brighter inthe second and fourth quadrants as shown in Figs.4�a� and 4�c�. These results are quite different fromthe image that we obtained at z � 0. We attributethe differences in the images to polarization interfer-ence.4
From Eq. �9� we note that the integral of the fieldamplitude over the exit pupil is proportional to sin 2 ,which changes the sign between adjacent quadrantsand indicates the phase change of the field. Here weintroduce a reference beam of amplitude r��, �, z�.The reference beam can be considered reflected lightfrom the substrate and the surface of the sphere.
Assume that the phase difference between the refer-ence and the signal beams is ��z�, which derives fromthe optical path difference between the referencebeam and the signal beam. The intensity of the in-terference image with a crossed analyzer can be writ-ten as
Iintconf� , z� � ��
0
2
�0
�
�r��, �, z� f ��� � Q cos 2��
� �exp� j���sin � cos �d�d��2
, (10)
where f ��� takes the form of sin 2� only for reflectedlight coming from a planar substrate in an ideal con-focal polarization microscope and the integral of r��,�, z� f ��� gives a zero value, where r is independent ofangle �.3 However, here the reflected light comesfrom the substrate and the sphere’s surface; thus theintegral of r��, �, z� f ��� is generally a nonzero value,written as ��z�. We note that when different crosssections of the sphere are scanned, backscatteringamplitude Q is also a function of axial position z of theilluminating point. Consider the symmetrical prop-erty of the scattering amplitude; Eq. �10� can be re-written as
Iintconf� , z� � �2 � 2�� cos � sin 2 � � 2 sin2 2 ,
(11)
where � is given by
� � �0
2
�0
�
Q cos 2� sin � cos �d�d�. (12)
The third term in Eq. �11� results in the maximumintensity at an angle of 45° in all four quadrants.The second term, which is modulated by the factor sin2 , may reach its maximum at 45° in the first andthird quadrants from constructive interference, andthe minimum in the second and fourth quadrantsfrom destructive interference, or vice versa, depend-ing on the sign of �� cos �. Hence for the polariza-tion interference image we could expect brighterpatterns in the first and third quadrants or in thesecond and fourth quadrants for a certain cross sec-tion of the microsphere. This explains why the im-
Fig. 4. Normalized images of spheres, 5.2 �m in diameter, obtained with the confocal scanning polarization microscope with a crossedanalyzer. From �a� to �c�, z � a, 0, 3a�4, respectively. There are two defects, A and B, inside the spheres.
ages in Figs. 4�a� and 4�c� have brighter patterns inthe second and fourth quadrants. In addition, the�� cos � and �2 can be considered weight factors thatinfluence the interference patterns. When the focalplane is near the front or back surface of the sphere�z � a or z � a�, the sphere can be consideredquasi-planar. In that case, amplitude Q is indepen-dent of polar angle � and gives � �� 1. Hence thesecond term in Eq. �12� dominates the interferencepattern shown in Figs. 4�a� and 4�c�. But, for z � 0,the edge of the sphere destroys the circular symmetryof Q and gives a relatively large value of Q, so we canexpect that the third term will dominate and give abright spot at 45° in all four quadrants, as shown inFig. 4�b�.
C. Imaging by Conventional Polarization Microscopy
For comparison, we also experimentally studiedbackscattered images by using the conventional scan-ning microscope from which the pinhole shown in Fig.1 was removed. First we focused on the center planeof the sphere. Figures 5�a�, 5�b�, and 5�c� are theimages with a parallel analyzer, a crossed analyzer,and no analyzer, respectively. Interference ringscan still be seen within the region of the spheres inFigs. 5�a� and 5�c�.
It is well known that for the conventional scanningmicroscope the detected signal Iconv scattered fromthe edge of the sphere is the integral of the intensityover the exit pupil of the objective lens and can becalculated as3
Iconv � �0
2
�0
�
�E�2 sin � cos �d�d�. (13)
Again, we consider the scattering amplitudes to besymmetrical as S���� � S��2 �� and S���� �S��2 ��. When the analyzer is parallel to theincident polarization, from Eqs. �1� and �13� the de-tected intensity can be expressed by
I�conv � �0
2
�0
�
�P2 � 2PQ cos 2� cos 2
� Q2 cos2 2� cos2 2 �sin � cos �d�d�. (14)
We assume that the scattering amplitudes vary onlyslightly with angle �. The third integral term ofQ2��, �, ��cos2 2� over � will be much larger thanthat of Q��, �, ��cos 2�, and it is always positive.The second integral term, PQ��, �, ��cos 2� over �, iscomparatively small. Hence the intensity is modu-lated mainly by the factor cos2 2 . It reaches a max-imum at angles � 0, �2, , 3�2 and a minimumat � �4, 3�4, 5�4, 7�4 along the edge of thesphere. Because of the effect of the second integralterm, the intensities at � 0 or � are slightlydifferent from those at � �2 or � 3�2, as shownin Fig. 5�a�. This image behavior is totally differentfrom that of data obtained with the confocal micro-scope, as we have shown in Eq. �7�; there the intensitydistribution is constant for different angles .
As backscattering amplitudes S� and S� also de-pend strongly on size parameter � and on the refrac-tive index of the sphere, all the periodic imagesdescribed by Eq. �14� may provide a great variety ofpossible images of the backscattered light. Noticethat the parallel polarization images of the sphere,shown in Fig. 5�a�, are different from the surfacewave resonant images obtained by Ashkin and Dzied-zic17 and by Fayngold,11 for which the center brightstrip could not be seen when the light was tuned onresonance.
The distribution of the intensity for a crossed an-alyzer obtained by substituting Eq. �2� into Eq. �13�gives
I�conv � �sin2 2 � �
0
2
�0
�
Q2 cos2 2� sin � cos �d�d�,
(15)
for which the multiplying factor sin2 2 predicts thatfour arcs will be located at an angle of 45° and a darkcross will appear in the image, as shown in Fig. 6�b�.This pattern is similar to that obtained by the confo-cal polarization microscope, but the integral of Q2��,�, ��cos2 2� over � in Eq. �15� is larger than Q��, �,��cos 2� in Eq. �9� for a confocal polarization imagingsystem. Hence the contrast of the image pattern ofthe sphere is better if the conventional polarizationmicroscope is used than for the confocal microscope.
Fig. 5. Normalized images of microspheres, 5.2 �m in diameter, obtained with a conventional scanning optical microscope with �a� aparallel analyzer, �b� a crossed analyzer, and �c� no analyzer at z � 0.
These analyses are in good agreement with the ex-perimental results shown in Figs. 4�b� and 5�b�.
We emphasize that the better image contrast of themicrosphere for conventional polarization microscopyis not related to the birefringence of the sample or tooptical resonant absorption but is due to the image-formation process. These image patterns will ingeneral overlap the polarization information of thesample and degrade the polarization image contrast.
When the analyzer is removed, the signal from thesphere can be expressed as
Iconv � �0
2
�0
�
�P2 � 2PQ cos 2� cos 2
� Q2 cos2 2��sin � cos �d�d�. (16)
The image of the sphere edge varies slightly withdifferent values of , which is modulated by the sec-ond integral term of PQ��, �, ��cos 2� over � and bythe factor cos 2 , as shown in Fig. 5�c�.
We also experimentally investigated the 3D polar-ization images by using a conventional scanning po-larization microscope with a crossed analyzer. Theimages are shown in Fig. 6. It can be seen that thefarther the focal plane is away from the center of thesphere, the larger the image distance from the centerto the bright spots of the pattern is. This findingagrees well with the calculated results for the 3Dbackscattered image of a sphere obtained by Inamiand Kawata,12 who used the extended Mie scatteringtheory. The two defects inside the microspherecould not be observed because of the poor sectioningproperty of the conventional microscope.
From Fig. 6 we can also see that the imaging pat-
terns of the pair of touching microspheres are differ-ent from the pattern of the isolated microsphere.When the focal plane moves from the center �z � 0� ofthe sphere, two arcs near the noncontact edge of thetwo touching spheres become comparatively brighter.Especially for z � a, the bright spots extend along thenormal of the sphere. These phenomena do not ap-pear in the confocal imaging system; we considerthem to be a symptom of the more-complicated polar-ization imaging character of the light beam with thetouching spheres in the conventional microscope.
3. Conclusions
We have experimentally studied the three-dimensional imaging of microspheres by usingconfocal and conventional scanning polarization mi-croscopes. We found that the backscattered imagesof the sphere in the two microscope systems are dif-ferent. In the confocal scanning microscope the po-larization of the detected signal is mostly parallel tothe incident polarization. This is due to the fieldamplitude averaging effect and to the high extinctioncoefficient of the confocal system. As a result, thesignal intensity from the microsphere in the confocalpolarization microscope with a crossed analyzer wasfound to be weaker than in the conventional systemand therefore to give poorer image contrast. How-ever, these polarization images are artifacts of theimage-formation processes of the two systems ratherthan intrinsic polarization properties of the sample.
Utilizing the vector approach, which takes into ac-count the polarization and the imaging theory for thetwo microscope systems, we presented the expres-sions for and qualitatively analyzed the intensity dis-tributions of light backscattered from a microspherewith an analyzer both parallel and perpendicular tothe incident polarization. Polarization interferencewas also considered. The analysis provides a phys-ical picture of the imaging of the microsphere, and theresults of the analysis are in good agreement withexperimental observations.
Some new phenomena were observed in the exper-iment. We found that the backscattering images ofa pair of touching microspheres are different from anisolated image in the conventional polarization imag-ing system, a result that needs further study.
It is well known that a microsphere can be anoptical cavity with high optical quality ��1010�, whichis defined as the ratio between resonance frequencyand bandwidth of a cavity mode.7,8 Several recentreports have described the development of new kindof optical microcavity based on micrometer-sized la-tex spheres coated with monolayers of asemiconductor.18–20 It was shown that control ofpolarization-sensitive mode distribution in a spheri-cal microcavity is crucial for the development of newnanoscopic light emitters. We recently determinedthat a MF microsphere covered by shell of CdTenanocrystals is a promising material for low-threshold laser applications.21 Further study byconfocal and conventional polarization microscopy of
Fig. 6. Polarization images of spheres, 5.2 �m in diameter, ob-tained with a conventional scanning optical microscope with acrossed analyzer. Focal positions from �a� to �d� are z � a, a�2,a�2, a, respectively.
polarization-sensitive mode distribution in a spheri-cal microcavity is under way.
We thank Andrey Rogach and Nikolai Gaponik,Institute of Physical Chemistry, University of Ham-burg, Hamburg, Germany, for providing us with sam-ples. We thank L. Andrea Dunbar, Institut dePhotonique et d’Electronique Quantique, EPFL, Lau-sanne, Switzerland, for helpful discussions.
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