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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 16370–16380
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Directional emission of dielectric disks with a finite scatterer in the THz regime

S. Preu, S. I. Schmid, F. Sedlmeir, J. Evers, and H. G. L. Schwefel  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 16370-16380 (2013)
http://dx.doi.org/10.1364/OE.21.016370


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Abstract

In the Terahertz (THz) domain, we investigate both numerically and experimentally the directional emission of whispering gallery mode resonators that are perturbed by a small scatterer in the vicinity of the resonators rim. We determine quality factor degradation, the modal structure and the emission direction for various geometries. We find that scatterers do allow for directional emission without destroying the resonator’s quality factor. This finding allows for new geometries and outcoupling scenarios for active whispering gallery mode structures such as quantum cascade lasers and passive resonators such as evanescent sensors. The experimental results agree well with finite difference time domain simulations.

© 2013 OSA

1. Introduction

In this paper, we present another way to achieve directional emission: we study directional emission from a WGM resonator with a finite scatterer within the resonator both experimentally and theoretically in the THz frequency range. Theoretical calculations initiated by Wiersig et al. [11

11. J. Wiersig and M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. A 73, 031802 (2006) [CrossRef] .

14

14. R. F. M. Hales, M. Sieber, and H. Waalkens, “Trace formula for a dielectric microdisk with a point scatterer,” J. Phys. A 44, 155305 (2011) [CrossRef] .

] already described efficient outcoupling of a high Q mode via interaction with a low Q mode without ruining the Q factor. Here, however, we address the mode directly by perturbing the mode with the scatterer. We will show that the perturbation may be weak enough to maintain a considerably high Q factor while the power is coupled out directionally. Besides directional emission at high Q factors, finite scatterers may further assist to improve the threshold behaviour of WGM-based lasers [15

15. S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176–178 (1999) [CrossRef] .

]: they can be used to perturb non-lasing modes which suppresses spontaneous emission. Symmetrically spaced perturbations on the rim or above the disk can act as gratings and efficiently couple light out of the resonator [2

2. L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie, “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009) [CrossRef] .

, 16

16. D. C. Aveline, L. Baumgartel, B. Ahn, and N. Yu, “Focused ion beam engineered whispering gallery mode resonators with open cavity structure,” Opt. Express 20, 18091–18096 (2012) [CrossRef] [PubMed] .

] and also provide a novel way to generate optical angular momentum beams [17

17. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012) [CrossRef] [PubMed] .

]. Scatterers therefore present an excellent alternative to improve both directionality and lasing performance of WGM-based lasers, in particular, quantum cascade lasers.

For our studies, we examine waveguide-coupled, passive resonators at THz frequencies. The THz frequency range offers unique advantages for studying WGM resonators: the wavelength is roughly a factor of 1000 larger than in the visible domain. Much larger resonators and feature sizes have to be used compared to the optical domain, simplifying the experimental setup. Optical and physical effects remain the same due to the scalability of electromagnetism. It is therefore much easier to characterize the near and far field [21

21. S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, “Coupled whispering gallery mode resonators in the terahertz frequency range,” Opt. Express 16, 7336–7343 (2008) [CrossRef] [PubMed] .

]. However, materials are less advanced than in the optical domain, resulting in losses and therefore much lower quality factors. We developed a THz system based on n-i-pn-i-p superlattice photomixers [22

22. S. Preu, F. H. Renner, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, A. C. Gossard, T. L. J. Wilkinson, and E. R. Brown, “Efficient terahertz emission from ballistic transport enhanced n-i-p-n-i-p superlattice photomixers,” Appl. Phys. Lett. 90, 212115 (2007) [CrossRef] .

, 23

23. S. Preu, G. H. Döhler, S. Malzer, L. J. Wang, and A. C. Gossard, “Tunable, continuous-wave terahertz photomixer sources and applications,” J. Appl. Phys. 109, 061301 (2011) [CrossRef] .

] that allows for characterizing WGM resonators between 60 GHz and several hundreds of GHz [21

21. S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, “Coupled whispering gallery mode resonators in the terahertz frequency range,” Opt. Express 16, 7336–7343 (2008) [CrossRef] [PubMed] .

].

2. Experimental setup

We use a n-i-pn-i-p superlattice photomixer [23

23. S. Preu, G. H. Döhler, S. Malzer, L. J. Wang, and A. C. Gossard, “Tunable, continuous-wave terahertz photomixer sources and applications,” J. Appl. Phys. 109, 061301 (2011) [CrossRef] .

] as THz source that has been developed in house. The measurement setup is illustrated in Fig. 1(a) and (b). The source is mounted on a silicon lens for pre-collimation and efficient out-coupling of the THz beam. The beam is then collimated with a parabolic mirror (PM1). A second parabolic mirror (PM2) focuses the THz beam on a feed horn (FH) with a rectangular Teflon waveguide (dimensions 1 mm x 1.5 mm) mounted inside the horn. The feed horn is used to increase the coupling efficiency. The Teflon waveguide delivers the THz power to the WGM resonator. A Golay cell detector (D2) is used to monitor the transmission through the waveguide. The WGM resonator is situated in the center of a rotation stage. The Golay cell can be mounted on the rotation stage to scan the far field emission from the resonator. A horizontal cylindrical (HL) lens collects THz power emitted perpendicular to the detection plane. A second, vertical cylindrical lens (DL) is mounted in front of the Golay cell to collect power within a 10°–15° angle to improve the signal to noise ratio. The scan range was limited by the Teflon waveguide and the size of the Golay cell and the lenses to about 140°. Alternatively, the Golay cell detector can also be attached to a probe waveguide that touches the resonator in order to scan the near field. In order to reduce the necessary integration times of the Golay cell novel field-effect transistors could also be used [24

24. S. Preu, S. Kim, R. Verma, P. G. Burke, M. S. Sherwin, and A. C. Gossard, “An improved model for non-resonant terahertz detection in field-effect transistors,” J. Appl. Phys. 111, 024502 (2012) [CrossRef] .

, 25

25. S. Preu, H. Lu, M. S. Sherwin, and A. C. Gossard, “Detection of nanosecond-scale, high power THz pulses with a field effect transistor,” Rev. Sci. Instrum. 83(2012) [CrossRef] [PubMed] .

].

Fig. 1 Schematic setup. The resonator is mounted at a fixed position in the center of a rotation stage. The detector (Golay cell, D1) scans the angular far field pattern of the disk. Two cylindrical lenses (horizontal lens, HL, and detector lens, DL) improve the signal to noise ratio by focusing the emitted power from the resonator on the detector. For most measurements, the lenses collect radiation within a 10° angle. Alternatively, the lenses can be replaced with a probe waveguide that touches the boundary of the disk to scan the near field. b Photograph of the resonator in the center of the ration mount. c Definition of variables used throughout the paper. Angles are defined with respect to the coupling position, the angle of incidence is defined as positive in the direction of the light. The thickness of the boundary between hole and disk circumference is labelled as b. The hole (radius rh) position relative to the coupling position is defined as αh. Parameters for the numerical calculation, such as the incidence angle of a light ray on the surface, χ, and the far field emission angle θ are also included.

The frequency resolution (120 MHz) of the system is sufficient to measure Q factors up to 1200 in the frequency range from 500 − 350 GHz. Due to the lack of ultra-low-loss materials at THz frequencies, this resolution is sufficient since the ideal loss-limited Q factors are in the range of a few 1000. As an example, the imaginary part of the dielectric constant of polyethylene and Teflon [26] is in the range of ε″ > 1.8 × 10−3, resulting in an absorption-limited Q factor of < 1100, even if no radiative losses are present. For the investigated disks and frequency ranges, however, the measured Q factor was mostly smaller than 1200, limited by radiative losses.

Fig. 2 The evanescent near field of a polyethylene WGM resonator (diameter 25 mm, in the following referenced as “large resonator”) without hole is probed with a waveguide. a) Transmission through the feed waveguide while the near field probe is scanned along the circumference of the resonator. b) Outcoupled power through the near field probe.

Fig. 3 a) Q factors of several disks without and with a hole with a boundary thickness of b (see Fig. 1(c)). The closer the hole is drilled to the resonator circumference, the stronger are the perturbations of the mode and the stronger the reduction of the Q factor. The inset shows the extrapolated loss-limited Q factor in the limit of large frequencies. b) Modal structure of the disk with b = 0.38 mm before and after the hole was drilled. Despite a strong shift in the resonance frequency, the Q factor was only slightly altered.

The Q factors did not degrade drastically due to the perturbation of the hole. This demonstrates that finite scatterers can be used for resonators and laser cavities. In the following, we discuss the theoretical framework for determining the outcoupling performance by the hole.

3. Theoretical modeling

We now turn to our theoretical approach to model the experimental data. Such modeling is desirable mainly for two reasons. On the one hand, the model gives insight into the physical mechanism for the directional emission, not least since it provides access to system properties which may be difficult to obtain in the experiment. On the other hand, in order to design the emission pattern of WGM resonators towards specific applications, an accurate theoretical modeling is desirable in order to perform numerical parameter studies. For stationary resonance problems a number of numerical methods exist, such as scattering methods [27

27. H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” in “Progress in Optics” 47E. Wolf, ed. (Elsevier Science Bv, Amsterdam), pp. 75–137 (2005) [CrossRef] .

], boundary integral methods [28

28. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003) [CrossRef] .

30

30. C.-L. Zou, H. G. L. Schwefel, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Quick root searching method for resonances of dielectric optical microcavities with the boundary element method,” Opt. Express 19, 15669–15678 (2011) [CrossRef] [PubMed] .

] and multipole methods [31

31. H. G. L. Schwefel and C. G. Poulton, “An improved method for calculating resonances of multiple dielectric disks arbitrarily positioned in the plane,” Opt. Express 17, 13178–13186 (2009) [CrossRef] [PubMed] .

]. These methods however do not allow for studying the effects of the coupled waveguide.

The light propagation is thus simulated by the Finite-Difference Time-Domain (FDTD) Method [32

32. S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997) [CrossRef] .

34

34. S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011) [CrossRef] .

]. The geometry of waveguide and resonator as well as the material properties are chosen as in the experiment, without fine-tuning of any of the parameters. We however restrict the simulation to a two dimensional rectangular grid in the resonator plane, with grid constant Δx = Δy = 0.02 mm. This corresponds to about λ/90 or λ/110 for the two wavelengths in the mm range we consider below. In two dimensions, Maxwell’s equations for the six components of E⃗ and H⃗ separate into two disjunct sets of three equations each. As in the experiment, we study the propagation of the EM field in TEz polarization, such that Hz, Ex and Ey need to be considered. We drive the Hz component using a soft source placed at one edge of the waveguide, with a transverse mode profile obtained by solving the corresponding Helmholtz equation. Unwanted reflections from the boundaries of the simulation area are suppressed by Berenger type perfectly matched absorbing boundaries [35

35. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. of Comput. Phys. 114, 185–200 (1994) [CrossRef] .

].

In the second step, we excite the system with a monochromatic continuous-wave field at the resonance frequency of the mode identified in the first step, and evolve the system into its stationary state (modulo the oscillations at the incident field frequency). This evolution can be monitored, e.g., via the time-dependent power flux through the waveguide downstream the resonator. The steady state, where the resonator is fully loaded, is reached on time scales short enough to neglect the initial cavity build-up time.

In the third step, we calculate the emission pattern of the resonator. A principle problem arises from restraining the computation time to a moderate level. The FDTD simulations therefore are constrained to a small region around the resonator (∼ tens of mm) which are small compared to the experimental detector distance (∼ 10 cm). Thus it is impractical to include the detector itself into the simulation. But while the detector exclusively monitors the far field radiation component of the emission, the vicinity of the resonator is strongly influenced by near field contributions. Thus, a transformation of the FDTD results into the far field is required [32

32. S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997) [CrossRef] .

34

34. S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011) [CrossRef] .

]. To this end, we calculate the amplitude and the phase of the electric and magnetic fields on a circle around the resonator center, with radius larger than the resonator radius by a variable offset ρ. At each FDTD grid point r⃗ on the circle, we fit the FDTD time evolution over few cycles of the incident field to the function a sin(ωt + ϕ). Here, ω is the frequency of the incident field; the amplitude a > 0 and the phase ϕ are the fit parameters. From the field configuration, we evaluate the time averaged Poynting vector S⃗(r⃗) at each point r⃗ on the circle. Next, we determine the intersection point R⃗(r⃗) of a ray starting at r⃗ in the direction S⃗ with the detection sphere of radius R around the resonator center. As a result, the point r⃗ contributes with magnitude |S⃗| to the detection signal at detector position R⃗(r⃗). By incoherently summing up the contribution of all points on the circle around the resonator we obtain a prediction for the emission pattern as observed by the detector.

We compared the emission patterns evaluated for different offsets ρ from the resonator edge, and found that for small offsets, the results strongly depend on ρ. However, already with offsets ρ of few wavelengths of the incident light field, the radiation pattern becomes largely independent of the offset, indicating the suppression of near field components. It should be noted, however, that neither the simulation range nor the detector distance in the experiment are large enough to neglect the finite size of the resonator.

4. Comparison of theoretical and experimental results

We investigated two types of resonators for comparison with our theoretical framework. First we consider a polyethylene WGM resonator with R = 5.2 mm, rh = 0.51 mm, b = 0.41 mm and αh = 90°. The relative permittivity is εWGM = 2.56 (nPE = 1.6) for the resonator and εh = 1 (air) inside the hole. This resonator is referred to as small resonator in the following. Its resonance frequencies and Q factors are illustrated in Fig. 3. The far field emission was characterized for angles between 160° and 290° at a resonance frequency of 170 GHz (λ = 1.76 mm).

The dimensions of our second resonator are R = 12.5 mm, rh = 1.25 mm and b = 1 mm. In the following, this resonator is referred to as large resonator. The larger size was chosen to have less radiative losses of the unperturbed resonator and to simplify probing the near field with a probe waveguide. The large resonator was studied at αh = 47° and αh = 315°.

Figure 4(a) shows a FDTD snapshot indicating the considered geometry. Figure 4(b) depicts the result of the Poynting vector analysis for the small resonator. The Poynting vector structure clearly resembles the interference pattern visible in the field configuration. However, the radiation pattern cannot be read off directly from the Poynting vector structure, as the projection of the pattern onto the detection sphere has not yet been applied. The experimental data are compared to the prediction for the far-field pattern from the Poynting analysis in Fig. 4(c). The comparison between experiment and theory in (c) shows excellent agreement. The theoretical curve was obtained for a detector size of 10°, and a detector distance of 10 cm, consistent with the experiment. Apart from an overall scaling, no further free parameter had to be adjusted. The scaling was chosen such that the largest relative emission from the theoretical and the experimental data coincide. The main lobes as well as the intensity minima observed in the experiment are clearly reproduced in the theoretical analysis. The main difference arises at 270°, where the theoretical data exhibits a reduction in the emission intensity. This most likely is due to the waveguide, which is situated at this emission angle in the theoretical calculation, such that an angular range around 270° had to be excluded from the Poynting vector analysis. In the experiment, the waveguide was bent, such that it did not interfere with the emitted radiation at this angle, see Fig 1. The Poynting vector analysis and the comparison between experiment and theory for the large resonator with hole at αh = 315° are shown in Fig. 5. Again, a detector size of 10°, and a detector distance of 10 cm were chosen. Good qualitative matching between experiment and theory is achieved, even though the quantitative agreement is not as good as for the small resonator. The main difference is the width of the second lobe around 130° − 180°, which is wider in the experimental data than in the theoretical prediction. One possible origin for this could be slightly incorrect values for the index of refraction or the distance to the coupling waveguide to the resonator which could not be measured accurately, altering the perturbation of the resonator. Furthermore, the disk center is centered in the rotation stage. There may be some imaging error from radiation emitted at the disk boundary (being 12.5 mm off-axis) to the detector despite the 10° collimation angle (see Fig. 1). This effect is much weaker for the small resonator, where theory and experiment agreed excellently. Finally, the results for the large resonator with hole at αh = 47° are shown in Fig. 5. The first main lobe at around 120° predicted in the theoretical analysis is missing in the experimental data, but experiment and theory agree well for the second lobe around 200°. From the Poynting vector analysis, it can be concluded that the peak around 120° originates from the coupling region of the resonator.

Fig. 4 Small Resonator (R = 5.2 mm) with a hole at 180° ± 5°. a) Snapshot of the FDTD calculation on a logarithmic scale. The small circle indicates the hole. A movie ( Media 1) is provided in the supplementary material. b) Result of the Poynting vector analysis. The background shows a snapshot of the FDTD simulation. The arrows are the Poynting vectors, originating from the point at which they are evaluated, respectively. The Poynting vectors clearly reflect the pattern of the radiated field. The distance between the resonator edge and the circle on which the Poynting vectors are calculated is ρ = 8 mm. c) Experimental data (black dots) as well as far-field radiation pattern predicted from the theoretical Poynting-analysis (red) on a linear scale. The dip in the theoretical emission intensity at 270° is due to the waveguide region which was excluded from the Poynting vector analysis. In the experiment, the waveguide is curved and thus does not lie in this direction.
Fig. 5 Large Resonator, the hole with d = 12.5 mm sits at 315° ±5° (top row) and 47° ±5° (bottom row); a) snapshot of the FDTD calculation. A movie ( Media 2) is provided in the supplementary material. b) Result of the Poynting vector analysis, as in Fig. 4(c). The distance between the resonator edge and the circle on which the Poynting vectors are calculated is ρ = 4 mm. c) Experimental data (black dotted) as well as far-field radiation pattern predicted from the theoretical Poynting-analysis on a linear scale (red). While the second main lobe observed experimentally at 130° − 180° is more narrow in the theoretical prediction, good overall agreement is achieved. d) snapshot of the FDTD calculation, the emission direction can be anticipated visually. A movie ( Media 3) is provided in the supplementary material. e) Result of the Poynting vector analysis, as in Fig. 4. The distance between the resonator edge and the circle on which the Poynting vectors are calculated is ρ = 4 mm. f) Experimental data (black dotted) as well as far-field radiation pattern predicted from the theoretical Poynting-analysis on a linear scale (red). While the first main lobe predicted from the theoretical calculation around 120° is missing in the experimental data, experiment and theory agree well for the second lobe at 200°, most likely due to the slightly curved waveguide in the experiment.

5. Conclusion

Acknowledgments

We thank H. Lu and A. C. Gossard at MRL Santa Barbara, CA, USA, for growing the substrate of the THz devices. We further would like to thank G. Leuchs and his devision at the MPL for the creative atmosphere and the financial support. We acknowledge support by the Deutsche Forschungsgemeinschaft and the Friedrich-Alexander-Universität Erlangen-Nürnberg within the funding program Open Access Publishing.

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22.

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S. Preu, G. H. Döhler, S. Malzer, L. J. Wang, and A. C. Gossard, “Tunable, continuous-wave terahertz photomixer sources and applications,” J. Appl. Phys. 109, 061301 (2011) [CrossRef] .

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S. Preu, S. Kim, R. Verma, P. G. Burke, M. S. Sherwin, and A. C. Gossard, “An improved model for non-resonant terahertz detection in field-effect transistors,” J. Appl. Phys. 111, 024502 (2012) [CrossRef] .

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H. E. Türeci and H. G. L. Schwefel, “An efficient Fredholm method for the calculation of highly excited states of billiards,” J. Phys. A 40, 13869–13882 (2007) [CrossRef] .

30.

C.-L. Zou, H. G. L. Schwefel, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Quick root searching method for resonances of dielectric optical microcavities with the boundary element method,” Opt. Express 19, 15669–15678 (2011) [CrossRef] [PubMed] .

31.

H. G. L. Schwefel and C. G. Poulton, “An improved method for calculating resonances of multiple dielectric disks arbitrarily positioned in the plane,” Opt. Express 17, 13178–13186 (2009) [CrossRef] [PubMed] .

32.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997) [CrossRef] .

33.

A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method,3rd ed. (Artech HouseBoston, 2005).

34.

S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011) [CrossRef] .

35.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. of Comput. Phys. 114, 185–200 (1994) [CrossRef] .

OCIS Codes
(230.5750) Optical devices : Resonators
(140.3945) Lasers and laser optics : Microcavities
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Optical Devices

History
Original Manuscript: March 6, 2013
Revised Manuscript: May 10, 2013
Manuscript Accepted: May 13, 2013
Published: July 2, 2013

Citation
S. Preu, S. I. Schmid, F. Sedlmeir, J. Evers, and H. G. L. Schwefel, "Directional emission of dielectric disks with a finite scatterer in the THz regime," Opt. Express 21, 16370-16380 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-16370


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  35. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. of Comput. Phys.114, 185–200 (1994). [CrossRef]

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