Imaging interferometry of excitons in two-dimensional structures: Can it detect exciton coherence
IImaging interferometry of excitons in two-dimensional structures:Can it detect exciton coherence
Heinrich Stolz, Maria Dietl, Rico Schwartz, and Dirk Semkat
Institut f¨ur Physik, Universit¨at Rostock, D-18051, Rostock, Germany (Dated: October 3, 2018)
Abstract
Using the theory of imaging with partially coherent light, we derive general expressions for dif-ferent kinds of interferometric setups like double slit, shift and mirror interference. We show thatin all cases the interference patterns depend not only on the point spread function of the imagingsetup but also strongly on the spatial emission pattern of the sample. Taking typical experimen-tally observed spatial emission patterns into account, we can reproduce at least qualitatively allthe observed interference structures, which have been interpreted as signatures for spontaneouslong range coherence of excitons, already for incoherent emitters [1, 2]. This requires a criticalreexamination of the previous work.
PACS numbers: 71.35.-y, 42.50.-p, 78.70.-g a r X i v : . [ c ond - m a t . o t h e r] S e p ondensation of excitons is still a fascinating topic of solid state physics. As in everyBose-Einstein condensate, spontaneous coherence of matter waves should emerge in theexciton system. This spatial coherence is transferred to the decay luminescence and thusshould be observable in the light emission from the exciton cloud.The standard way to measure coherence of light is by interferometry. Indeed, in the pastdifferent setups for spatial interferometry have been used to measure the spatial coherenceof the light emitted by two-dimensional semiconductor structures, most often in the form oftwo types of interferometers: • Shift interferometry. Here two images of the same object, shifted by a small vector (cid:126)δ , are superimposed and by varying the phase delay between the two light paths,interference fringes are generated (see e.g. Fig. 5 b of Ref. [5] and Fig. 2 of Ref. [3]).To determine the interference contrast, the simple formula C = ( I − I − I ) / √ I I is used. • Mirror interferometry. Here the second image is either the mirror or the inverted imageof the object. This is obtained in a Michelson type interferometer, where the secondmirror is replaced by e.g. a retro-reflector (see e.g. Fig. 5 c-e of Ref. [5]).In all the systems investigated up to now, namely exciton-polaritons in a quantumwell/microcavities or excitons in double quantum well (DQW) structures, these types ofinterferometry are considered as the “Standard method for detecting long range order inan exciton system” [3]. Quite remarkably, the results of all the experiments reported showa rather uniform behaviour, a pronounced increase of the coherence of the light emittedfrom the excitons either by lowering the temperature or by increasing the power of the laserexciting the excitons (compare Figs. 3 d,e, 4, and 5 of Ref. [3] or Fig. 5 a of Ref. [5]).However, in most of the studies up to now, it has not been taken into consideration thatall interference setups are also imaging systems and that one should apply the well-knowntheory of imaging to describe the interference phenomena properly. Most important, inthe papers one fact has been overlooked, that the coherence property of light is changedby imaging. This is based on the fundamental van Cittert-Zernike theorem of optics andwell-known in the optics community.In this paper we develop a rigorous theory of imaging interferometry for an arbitrarypatterned source. The properties of the interferometric setup can be desribed by an appro-2riate response function which is related to the amplitude point spread function [8, 9]. Theresults show clearly that in addition to the point spread function also the spatial emissionpattern of the sample has profound effects on the resulting interference pattern.The paper is organized as follows. In section 2 we derive a general theory of imaginginterferometry, which we then specialise to the various setups. Explicit results are given forthe case of completely incoherent emitters. In section 3 we show for some different spatialemission patterns, which are typical for the various experiments reported in the literatur,the resulting interference pattern and compare with the experiments. The paper closes witha critical discussion.
I. THEORY OF IMAGING INTERFEROMETRYA. Theory for 2d objects
Since the objects in this study are planar structures with thickness well below the wave-length of light, we can describe both object and image with 2d vectors. We start with asingle point emitter at position (cid:126)ρ o in the object plane at − d , which is imaged by a lens offocal length f in the image plane at d with 1 /f = 1 /d +1 /d and magnification M = d /d .The amplitude of the light field at a point (cid:126)ρ i in the image plain at d is then given by [8] E I ( (cid:126)ρ i ) = Md λ exp [ − ikd (1 + 1 /M )] · exp (cid:34) − ikM d (cid:126)ρ i (cid:35) (1) × E O ( (cid:126)ρ o ) exp (cid:34) − ik d (cid:126)ρ o (cid:35) · P d ( (cid:126)ρ o + M (cid:126)ρ i ) , with E O ( (cid:126)ρ ) denoting the field amplitude of the emitter and P d ( (cid:126)ρ ) the 2d amplitude pointspread function P SF of the lens.In imaging interferometry we superimpose on this image that of an identical object buton which we impose a symmetry operation R . This can be either a shift by a small vector (cid:126)δ or a reflection, e.g. at the yz plane. In addition we impose an additional phase Φ to makethe interference fringes visible. The image field of this object is given by E (cid:48) I ( (cid:126)ρ i , R ) = Md λ exp [ − ikd (1 + 1 /M ) + i Φ] · exp (cid:34) − ikM d (cid:126)ρ i (cid:35) (2) × E O ( (cid:126)ρ o ) exp (cid:34) − ik d ( R ( (cid:126)ρ o )) (cid:35) · P d ( R ( (cid:126)ρ o ) + M (cid:126)ρ i ) . I ( (cid:126)ρ i , R ) = | E I ( (cid:126)ρ i ) + E (cid:48) I ( (cid:126)ρ i , R ) | = I + I + I inter . (3)While I , I are the two images of the object, the interference term is given by I inter = 2Re (cid:104) E I ( (cid:126)ρ i ) E (cid:48)∗ I ( (cid:126)ρ i , (cid:126)δ ) (cid:105) (4) ∝ | E O ( (cid:126)ρ o ) | Re (cid:110) exp (cid:34) − ikd [( (cid:126)ρ o ) − ( R ( (cid:126)ρ o )) ] + i Φ (cid:35) (5) × P d ( (cid:126)ρ o + M (cid:126)ρ i ) P ∗ d ( R ( (cid:126)ρ o ) + M (cid:126)ρ i ) (cid:111) . To obtain the interference pattern in the most general case of many partial coherentemitters, we have to image instead of the fields the first order field correlation function G O ( (cid:126)ρ, (cid:126)ρ (cid:48) ) = (cid:104) E O ( (cid:126)ρ ) E ∗ O ( (cid:126)ρ (cid:48) ) (cid:105) , which is identical to the mutual coherence function of the emitter[9] by applying the van Cittert-Zernike theorem [9]. This gives the following expression forthe interference pattern I inter ( (cid:126)ρ i ) ∝ (cid:90) (cid:90) Re (cid:110) G O ( (cid:126)ρ o , (cid:126)ρ (cid:48) o ) exp (cid:34) − ikd [( (cid:126)ρ o ) − ( R ( (cid:126)ρ (cid:48) o )) ] + i Φ (cid:35) (6) × P d ( (cid:126)ρ o + M (cid:126)ρ i ) P ∗ d ( R ( (cid:126)ρ (cid:48) o ) + M (cid:126)ρ i ) (cid:111) d(cid:126)ρ o d(cid:126)ρ (cid:48) o . Using the property G O ( (cid:126)ρ, (cid:126)ρ (cid:48) ) = I O ( (cid:126)ρ ) δ ( (cid:126)ρ − (cid:126)ρ (cid:48) ) (7)of an incoherent source, we see that Eq. (6) goes over into I inter ( (cid:126)ρ i ) ∝ (cid:90) I O ( (cid:126)ρ o )Re (cid:110) exp (cid:34) − ikd [( (cid:126)ρ o ) − ( R ( (cid:126)ρ o )) ] + i Φ (cid:35) (8) × P d ( (cid:126)ρ o + M (cid:126)ρ i ) P ∗ d ( R ( (cid:126)ρ o ) + M (cid:126)ρ i ) (cid:111) d(cid:126)ρ o . It should be noted that one obtains the same expression for the interference pattern of atotally incoherent source by integrating Eq. (4) directly over the whole emitter, becausethere is no correlation between the emitting excitons.Equations (6) and (8) are the central relations for shift interferometry of any sources.They show that the interference pattern depends not only on the PSF but also on theintensity distribution of the emitting source in a way which is not straightforward but rathercomplicated, a fact which has been overlooked up to now.4
I. INTERFEROMETRIC SETUPSA. Shift interferometer
Inserting in Eq. 8 the shift operation as R ( (cid:126)ρ ) = (cid:126)ρ − (cid:126)δ (9)and introducing the shift interferometer response function P SI ( (cid:126)ρ, (cid:126)δ ) = P d ( (cid:126)ρ ) P ∗ d ( (cid:126)ρ − (cid:126)δ ) , (10)the different terms in Eq. (3) can be written as a two-dimensional convolution integralbetween a phase shifted image and the interferometer response function I ( M (cid:126)ρ i ) = I O ( (cid:126)ρ ) ⊗ P SI ( (cid:126)ρ,
0) (11) I ( M (cid:126)ρ i , M(cid:126)δ ) = I O ( (cid:126)ρ − (cid:126)δ ) ⊗ P SI ( (cid:126)ρ,
0) (12) I inter ( M (cid:126)ρ, M(cid:126)δ ) = 2Re (cid:40) I O ( (cid:126)ρ ) exp (cid:32) − ikd (cid:126)ρ · (cid:126)δ + i Φ (cid:33)(cid:41) ⊗ P SI ( (cid:126)ρ i , (cid:126)δ ) . (13)which allows a very efficient numerical calculation via fast Fourier transform. B. Mirror interferometry
In case of inversion interferometry, the symmetry operation is given as R ( (cid:126)ρ ) = − (cid:126)ρ (14)and Eq. (8) goes over to I inter ( (cid:126)ρ i ) ∝ (cid:90) I O ( (cid:126)ρ o )Re (cid:110) exp [ i Φ] (15) P d ( (cid:126)ρ o + M (cid:126)ρ i ) P ∗ d (( − (cid:126)ρ o ) + M (cid:126)ρ i ) (cid:111) d(cid:126)ρ o . which, however, is not a simple convolution, making the calculation a bit more tedious. III. POINT SPREAD FUNCTIONSA. Standard setup
Since the interference pattern depends on the (amplitude) point spread function of theoptical setup, we first discuss two typical cases. The first one is the standard setup used in5
10 20 3000.20.40.60 10 20 3000.20.40.6 ab po i n t s p r ead f un c t i on r (µm) FIG. 1: Two-dimensional sections through the 3d point spread function of the standard imagingsetup for different defocussing. Panel a) z = 0 and panel b) z = − . µ m. The real part is givenby the dashed line, the imaginary part by the dotted line and the absolute square by the full line(note shift of zero of the y axis). most experiments up to now [5, 6]. Here, the sample is mounted inside an optical cryostatonto a cold finger which cools the sample to liquid Helium temperatures. Optical access isthrough a quartz window with thickness of typical 1 to 2 mm. Imaging is performed witha high numerical aperture microscope objective with e.g. N.A. = 0 . z in the optimal focus, can be approximated quite well by that of a thin lens with spericalaberrations in the paraxial limit [8, 9]: P d ( r, z ) = P (cid:90) ρ exp (cid:32) i (cid:18) ad (cid:19) kz (cid:33) exp ( − ik Φ( ρ )) J ( ka/d rρ ) dρ . (16)with ρ the radial distance from the optical axis in unit of the lens radius a and J ( x ) theBessel function of zeroth order. The wavefront aberration Φ is given byΦ( ρ ) = 1 √ A R ( ρ ) (17)6ith R ( ρ ) denoting Zernike’s circle polynomial of order n = 4 , l = 0 and A the Zernikecoefficient for primary spherical aberration.For a plane parallel plate of thickness D and index of refraction n P this is given by [ ? ] A = −√ D ( n p − n P (arcsin( N.A. )) (18)For typical values D = 1 mm and n P = 1 .
45 we have A = − . µ m, which is of the orderof the wavelength and thus not small. The 3d PSF for this situation obtained by numericalintegration of Eq. 16 is shown in figure 1 for different defocusing. While for the optimumdefocussing conditions the apparent resolution given by the FWHM of the PSF is still quitegood, strong coherent side lobes extend quite far out of the center. As schown below thisresults in such sever distortions of the interference pattern making a simple interpretationof experimental results almost impossible. B. Optimized setup -0.17 1.06 -0.11 0.11-4 6-10-55 y ( µ m ) x (µm)-4 6-10-55 y ( µ m ) x (µm) Re(P
SI)
Im(P
SI) (µm) | P |
2d point-spread function ba c
FIG. 2: Two-dimensional point spread function (Panel a), and real and imaginary parts of thecorresponding shift interferometer response function (panel b and c).
Much less aberrations are introduced, when the imaging lens is inside the cryostate,directly facing the sample, as was the case in Ref. [4]. Here even the PSF of the actualimaging setup can be obtained from Fig. 3a. For our purposes, we also approximate the PSFby that of a thin lens with sperical aberrations. Chosing a/d = 0 .
175 and A = 0 . λ gives a reasonable fit (compare Fig. 2a with Figure 3a of Ref. [4]). In panel b and c of Fig.2, we show an example for the shift interferometer response P SI for δ = 2 µ m. It shows that P SI possesses a substantial imaginary part, which will give rise to highly structured spatialinterference patterns (see Fig. 2 and 3). 7 V. RESULTS
To demonstrate the essential points of our argumentation, we first discuss in the followingtwo examples of intensity patterns which are typical for the observation in Ref. [4]. The firstone is a ring-type structure which is similar to an LBS ring (compare Fig. 1a and Fig. 1S ofthe supplementary information of Ref. [4]) and may be given as I O ( ρ ) ∝ exp[ − ( ρ − ρ R ) /σ R ].The size of the ring was chosen to be 2 ρ R = 4 . µ m diameter and σ R = 2 µ m resulting ina small dip in the middle (see Fig. 2a). For the additional phase we choose Φ( y ) = π/ y to reproduce the experimental interference fringes. The spatial image of the interferencepattern is shown in Fig. 2b, the pattern of interference contrast is given in panel c. Thetwo images show the same pattern as found in the experiment, low contrast in the spotcenter, but a high contrast which reaches almost 1 in a left and right side lobe of the spot.As can be seen in panel b, we even find fork-like interference patterns at the boundary ofthe luminescing spot. It has to be stressed, that all these signatures arise already for a completely incoherent emitting source! Grau -1 1
Farb Farb PL intensity A interf contrast -20 200x (µm)0-2020 y ( µ m ) -20 200x (µm)0-2020 y ( µ m ) -20 200x (µm)0-2020 y ( µ m ) a b c FIG. 3: Luminescence pattern of an incoherently emitting ring (panel a), the interference contrastcalculated by Eq. 8 (panel b) and the interference contrast (panel c).
As second example we consider the interference pattern of a line array of spots, similarto those in the “macroscopically ordered exciton state” of an outer ring (see Fig.2a). Eachspot is represented by a Gaussian as I O ( ρ ) ∝ exp[ − ( ρ/σ O ) ] with σ O = 2 µ m, the distanceof the spots being 4 . µ m. The interference pattern and contrast for such a source is shownin panels b and c. Again, we reproduce almost quantitavely the essential observations ofRef. [4], even the two parallel lines of minimum contrast can be identified.Finally, we show results for the case of mirror interferometry, which e.g. has been used8 rau -1 1 Farb Farb Iinter CON -20 200x (µm)0-2020 y ( µ m ) -20 200x (µm)0-2020 y ( µ m ) -20 200x (µm)0-2020 y ( µ m ) PL intensity A interf contrast a b c FIG. 4: Luminescence pattern of a chain of incoherent emitters (panel a), the interference contrastcalculated by Eq. 8 (panel b) and the interference contrast (panel c). Farb CON -12 120x (µm)0-1212 y ( µ m ) CON -12 120x (µm)0-1212 y ( µ m ) a b FIG. 5: Luminescence pattern of an array of three random incoherent emitters (panel a) and theinterference contrast of a mirror type interferometer setup calculated with the point spread functionof Fig. 1 for a standard optical setup (panel c). in Ref. [5] to claim spontaneous coherence in a dense polariton system. Here the authorsreport that the large emitting spot breaks down to an array of small spots with sizes in therange of 2 − µ m (see Fig. 4 panel f and h of Ref. [5]). We therefore simulated the mirrorinterference pattern by positioning small incoherent emitters of such sizes in the object plane(see Fig. 5 panel a). The resulting contrast image is shown in Fig. 5 panel b. The pattern isvery similar to that observed experimentally, even the maximum amount of contrast (30%)is identical to that found in the experiments.9 . CONCLUSIONS In conclusion, it is obvious from our model calculations that the experimental findingsupon which the claims of detecting spontaneous coherence of excitons and exciton polaritonsin Ref. [3, 4] and [5] are based, can be explained straightforwardly by the properties of partialcoherent light without the assumption of exciton coherence, by taking only into account thespatial emission patterns characteristic of the samples. Indeed, all these patterns havein common that one observes a change in the pattern from a spatially rather large andhomegeneous distribution to one with an array of small spots with sizes approaching thelimit of optical resolution of the imaging setup. The observation of optical coherence inthis case is not surprising, since in the limit of a point source, the emitted light is bydefinition completely coherent. Therefore, the results of the above mentioned papers haveto be reconsidered by taking the effects of imaging properly into account before any claimsto have observed spontaneous coherence in an exciton system can be justified,.Generally, we want to state that using interferometric methods for determining coherenceof exciton systems is highly questionable. Especially, arguments that are not based on arigorous theoretical analysis may turn out to be completely misleading. [1] Butov L. V.; Gossard A. C.; Chemla D. S.
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