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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 11301–11311
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Self-formed cavity quantum electrodynamics in coupled dipole cylindrical-waveguide systems

S. Afshar V, M. R. Henderson, A. D. Greentree, B. C. Gibson, and T. M. Monro  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 11301-11311 (2014)
http://dx.doi.org/10.1364/OE.22.011301


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Abstract

An ideal optical cavity operates by confining light in all three dimensions. We show that a cylindrical waveguide can provide the longitudinal confinement required to form a two dimensional cavity, described here as a self-formed cavity, by locating a dipole, directed along the waveguide, on the interface of the waveguide. The cavity resonance modes lead to peaks in the radiation of the dipole-waveguide system that have no contribution due to the skew rays that exist in longitudinally invariant waveguides and reduce their Q-factor. Using a theoretical model, we evaluate the Q-factor and modal volume of the cavity formed by a dipole-cylindrical-waveguide system and show that such a cavity allows access to both the strong and weak coupling regimes of cavity quantum electrodynamics.

© 2014 Optical Society of America

1. Introduction

Considering only the first few resonances, Fussell et al. [18

18. D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, “Decay rate and level shift in a circular dielectric waveguide,” Phys. Rev. A 71, 013815 (2005). [CrossRef]

] recognized that the radiation peaks of a dipole at the vicinity of an optiocal fiber have non-Lorentzian and asymmetric profiles and associated them with the presence of skew rays (radiation fields with non-zero longitudinal wave-vector components). For this reason, optical fibers, while providing transverse confinement due to cylindrical symmetry, have to date been believed to lack the confinement along the length of the fiber (due to translational symmetry) required to form a cavity and therefore exhibit only cavity-like behavior [18

18. D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, “Decay rate and level shift in a circular dielectric waveguide,” Phys. Rev. A 71, 013815 (2005). [CrossRef]

]. An approach to cavity formation that has been explored is to perturb the refractive index along the waveguide by some means, for example: truncating the structure, as in the case of microdisk cavities [9

9. A. Kiraz, P. Michler, C. Becher, B. Gayral, A. Imamoglu, L. Zhang, E. Hu, W. V. Schoenfeld, and P. M. Petroff, “Cavity-quantum electrodynamics using a single InAs quantum dot in a microdisk structure,” Appl. Phys. Lett. 78, 3932–3934 (2001). [CrossRef]

, 30

30. E. Peter, I. Sagnes, G. Guirleo, S. Varoutsis, J. Bloch, A. Lematre, and P. Senellart, “High-Q whispering-gallery modes in GaAs/AlOx microdisks,” Appl. Phys. Lett. 86, 021103 (2005). [CrossRef]

, 31

31. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef] [PubMed]

] and optical nanowires [32

32. V. Bordo, “Purcell factor for a cylindrical nanocavity: ab initio analytical approach,” J. Opt. Soc. Am. B 29, 1799–1809 (2012). [CrossRef]

]; producing periodic modulation of the refractive index, as in the case of Bragg gratings in optical fibers [24

24. F. Le Kien and K. Hakuta, “Cavity-enhanced channeling of emission from an atom into a nanofiber,” Phys. Rev. A 80, 053826 (2009). [CrossRef]

, 25

25. K. P. Nayak, F. L. Kien, Y. Kawai, K. Hakuta, K. Nakajima, H. T. Miyazaki, and Y. Sugimoto, “Cavity formation on an optical nanofiber using focused ion beam milling technique,” Opt. Express 19, 14040–14050 (2011). [CrossRef] [PubMed]

] or Bragg-stack mirrors in micropillars [33

33. M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89, 233602 (2002). [CrossRef] [PubMed]

, 34

34. S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43, 033001 (2010). [CrossRef]

]; or by producing a highly prolate shape in optical fibers (bottle microresonators) [35

35. C. Junge, S. Nickel, D. O’Shea, and A. Rauschenbeutel, “Bottle microresonator with actively stabilized evanescent coupling,” Opt. Lett. 36, 3488–3490 (2011). [CrossRef] [PubMed]

].

Here, we consider a dipole-fiber system, consisting of a z-oriented dipole located on the surface of a step index cylindrical glass-air fiber, extended along the z direction and show that such system leads to resonance radiation peaks. We reveal that the dipole-fiber system at resonance itself can provide the longitudinal confinement required to form a cavity, described here as a self-formed cavity. We examine and provide an accurate description of the contribution of different radiation modes, including skew rays, in the formation of resonance peaks of a dipole-fiber system for a wide range of resonance orders. In particular, we also show the relation between the radiation modes and WGM of the underlying 2D microdisk cross-section of the fiber. We discover that, for higher order radiation peaks, the contribution from skew rays shift outside the peak, resulting in resonance peaks that are purely associated with the radiation of the WGM of the underlying 2D microdisk. In this sense, we demonstrate that a dipole-fiber system can form a 2D cavity without any extra requirement for confinement in the longitudinal direction. The formation of this cavity results from locating a single dipole on the interface of the fiber. The physics behind this is similar to the case of a planar dielectric microcavity, in which spontaneous emission enhancement and the formation of localized modes are reported despite a lack of confinement in the lateral direction [20

20. G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A 44, 669–681 (1991). [CrossRef] [PubMed]

22

22. F. Björk, H. Heitmann, and Y. Yamamoto, “Spontaneous-emission coupling factor and mode characteristics of planar dielectric microcavity lasers,” Phys. Rev. A 47, 4451–4463 (1993). [CrossRef] [PubMed]

].

2. Theory

The power captured in each mode is Pj = |aj|2Nj and hence the total emitted power of the dipole-waveguide system is the sum of the power captured into all guided and radiation modes, i.e.
Ptotal=jPj+ν0QmaxPν(Q)dQ.
(3)

The total power calculating using Eq. (3) results in similar values to those obtained through the FDTD method [26

26. M. R. Henderson, S. Afshar. V., A. D. Greentree, and T. M. Monro, “Dipole emitters in fiber: interface effects, collection efficiency and optimization,” Opt. Express 19, 16182–16194 (2011). [CrossRef] [PubMed]

].

We use Eqs. (2) and (3) to find the guided and radiation power of a z–oriented dipole located at the interface of a step-index tellurite-air optical fiber, as shown in Fig. 1. Both guided and radiation powers are normalized to the dipole radiation power in bulk tellurite glass, PTe=μ0p02ω4n/12πc. Duo to the normalization to PTe, the results shown in Figs. 1 and 2 are independent of the dipole strength p0. Note that the ratio of the radiation power to PTe is equal to the Purcell factor. Tellurite is a soft glass with a high refractive index n = 2.025. Incorporating single emitters within tellurite step-index fibers has recently been reported as a platform for hybrid quantum-photonics devices [27

27. M. R. Henderson, B. C. Gibson, H. Ebendorff-Heidepriem, K. Kuan, S. Afshar V., J. O. Orwa, I. Aharonovich, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and T. M. Monro, “Diamond in tellurite glass: a new medium for quantum information,” Adv. Mater. 23, 2806–2810 (2011). [CrossRef] [PubMed]

]. The main results presented here are generally true for any choice of core refractive index. However, it is generally known that higher refractive index contrast leads to smaller mode volume. It has also been shown that higher core-clad refractive index contrast leads to enhanced emission of a dipole at the vicinity of an optical fiber [26

26. M. R. Henderson, S. Afshar. V., A. D. Greentree, and T. M. Monro, “Dipole emitters in fiber: interface effects, collection efficiency and optimization,” Opt. Express 19, 16182–16194 (2011). [CrossRef] [PubMed]

]. We have chosen tellurite glass as it has a wide transparency window from 500nm to 4μm, is amenable to fibre processing and has a high refractive index [27

27. M. R. Henderson, B. C. Gibson, H. Ebendorff-Heidepriem, K. Kuan, S. Afshar V., J. O. Orwa, I. Aharonovich, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and T. M. Monro, “Diamond in tellurite glass: a new medium for quantum information,” Adv. Mater. 23, 2806–2810 (2011). [CrossRef] [PubMed]

].

Fig. 1 Radiation (blue) and guided (green) power of a z–oriented dipole, located at the interface of a step-index tellurite-air fiber. TE-WGMs resonances (m, p) are shown. Insets show transverse distribution of |Ez(x, y)|2, respectively from left, for a peak and off-peak using Eq. (1) and a peak using using the FDTD package Meep [39]. The core-clad interface is shown by white circles and the position of the dipoles is shown by black dots. The position of the dipole in the far right inset is at the top.
Fig. 2 Radiation power of a z–oriented dipole located at the interface of a tellurite-air step-index fiber and due to the integral of all 0 < β < kncl, (blue curves). Other peaks correspond to different ranges of βkncl bins, i.e., 0<βkncl<1 as labeled, and the summation of these peaks is shown (red circles). (a) and (b) correspond to TE-WGM resonances (12, 0) and (29, 0), as labeled by black circles.

In Fig. 1 as the core diameter increases; the guided power (green line) approaches an asymptotic value (≈ 0.3), whereas the radiation power shows strong resonance peaks which become narrower and higher. The guided power includes the contribution from all guided modes as they appear and does not show any correlation to the resonance radiation peaks due to the orthogonality of the guided and the radiation modes.

We investigate the physics behind the radiation peaks by using an independent method that explains the formation of WGMs in a 2D circular disk, which is a representation of a thin cross section of the fibre, [37

37. L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004). [CrossRef]

, 38

38. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2004). [CrossRef]

]. We solve Maxwell’s equations for electric and magnetic fields of the form Fi(r)ei(ωtkϕ), where i = r, ϕ, z. For such a disk, TE (with non-zero components; Ez, Hr, and Hθ) and TM (with non-zero components; Hz, Er, and Eθ) solutions of Maxwell’s equations, for electromagnetic fields propagating around the disk, have the dispersion relations;
Jm(ncox)Hm(nclx)qsHm(nclx)Jm(ncox)=0,
(4)
where x = kD/2, Jm and Hm are the Bessel and Hankel functions, and qs = (ncl/nco) for TE and qs = (nco/ncl) for TM modes. Solutions of Eq. (4) determine the location of WGM resonances (kD/2 values) that can be labeled by (m, p), where m and p are the azimuthal and radial mode numbers, respectively.

Equation (2) implies that the emission of a z-oriented dipole only couples into TE-WGMs, since Ez is zero for TM-WGMs. It is possible to excite the TM-WGMs by using r or θ-oriented dipoles, since both Er and Eθ are non-zero for TM-WGMs. However, the coupling of the emission of a z-oriented dipole into TE-WGM is the strongest, since the electric field has only one non-zero component, Ez, that aligns along the dipole direction. Here, we use a z-oriented dipole, which only excites TE-WGMs.

In addition, we have also constructed |Ez(x, y)|2 of the radiation mode of the dipole-waveguide system for D = 0.8μm (peak (5, 0)) and 0.87μm (off-peak), using Eq. (1), and for D = 1.7μm (peak (12, 0)) using the FDTD package Meep [39

39. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010). [CrossRef]

]. The |Ez(x, y)|2 distributions show 10 and 24 nodes for the peaks at D = 0.8 and 1.7μm, respectively, which matches with the expected 2m nodes of the corresponding m = 5 and m = 12 WGMs resonances. In contrast, the |Ez(x, y)|2 distribution at D = 0.87μm, off-peak, does not show any such node pattern.

Similar behavior is observed when the core diameter is fixed and the radiation peaks are examined as a function of the dipole transition wavelength, indicating that different combination of (D, λ) can be associated with one resonance (m, p). This can be inferred from Eq. (4) if the material dispersion is negligible over the wavelength range of interest. Note that the Q-factor of a radiation peak is calculated by Q = λresλ, where λres and Δλ are the resonance wavelength and linewidth of the radiation peak (for constant D), respectively.

The radiation power has been calculated by integrating the radiation power associated with each β within the range of β = 0 to β = βmax = kncl, see Eq. (3). As β is the propagation constant along the z–direction, the radiation power associated with values of β ≈ 0 are completely contained within the transverse plane, while those associated with 0 < β < kncl consist of non-transverse radiation mode, i.e., skew modes. For these modes within the core region kt2+β2=(knco)2=(2πnco/λ)2, where kt is the transverse component of the propagation constant in the core region. Hence, the angle between the propagating direction of the radiation field and the z axis can be defined as θ = cos−1(β/knco) implying that the condition 0 < β < kncl corresponds to 90° > θ > cos−1(ncl/nco) ≈ 60° (for tellurite glass with n = 2.025).

To explain the relation between the WGM of a 2D disk and the radiation modes, we compare the radiation modes associated with WGM (12, 0) and (29, 0) in Figs. 2(a) and 2(b), with Q-factors of 895 and 1.3×109, respectively. Both modes show asymmetric profiles with steeper leading edge compared to their trailing edges. Such asymmetry has been observed and associated with skew rays; rays with non-zero z-component of the wave vector [18

18. D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, “Decay rate and level shift in a circular dielectric waveguide,” Phys. Rev. A 71, 013815 (2005). [CrossRef]

]. The TE-WGM resonances, shown by vertical black lines and black circles, are located within the leading edge of the radiation peaks and to the left of the peak maxima. The separation between TE-WGM resonance and the radiation peak maximum is much larger for mode (12, 0), 98 pm, than (29, 0), 0.26 fm. For both resonances, we have evaluated the radiation integral in Eq. (3) for different bins of β values, e.g. (0 – 0.02), (0.02 – 0.04),...,(0.1 – 1.0) × kncl for (12, 0) in Fig. 2(a), and presented them as a function of core diameter. The key significant conclusions are; I) overall radiation peaks (12, 0) and (29, 0) are the envelopes of underlying radiation peaks that are associated with different β bins (i.e., radiation modes propagating with different z-component wavevector), II) the (12, 0) and (29, 0) radiation peaks are mainly due to the contribution of radiation power associated with the first five bins of β, III) curves associated with higher β bins are shifted to larger core diameters and they are much broader, IV) the curve associated with the lowest β bin matches the corresponding TE-WGM resonance, shown by black circle in Figs. 2(a) and 2(b). Hence the leading edge of the overall radiation peak is steeper than the trailing edge and curves associated with larger values of β contribute to the trailing edge, determining the linewidth and the asymmetric profile of the overall peak.

Examination of the resonance peaks also reveals the main result of this study; the peak associated with TE-WGM resonance (29, 0) is mainly due to contributions from a very small range of β, i.e. 0 < β < 1 × 10−4 × kncl, compared to that of (12, 0), i.e. 0 < β < 0.1 × kncl. These ranges of βs are equivalent to 90° > θ ⪆ 89.997° and 90° > θ ⪆ 87.169° respectively. This means that the peak associated with the (29, 0) resonance has a very small contribution from non-transverse propagating modes (skew modes) and is almost purely radiating in the transverse plane, which explains the extremely small linewidth (of the order of fm) and ultrahigh quality factor (as shown later) of this peak. This clarifies the concept of self-formed cavity: a 2D cavity forms at high resonance orders, since only radiation modes with β ≈ 0 contribute to the radiation peak.

At such a resonance, the radiation is localized within a transverse plane normal to the fiber axis and at the position of the dipole. This is similar to the case of a dielectric planar microcavity in which a localized mode forms in spite of no lateral confinement. For such a mode, similar to the above discussion, the divergence angle of the emitted radiation narrows as the spatial mode number (or the mode Q-factor) increases [22

22. F. Björk, H. Heitmann, and Y. Yamamoto, “Spontaneous-emission coupling factor and mode characteristics of planar dielectric microcavity lasers,” Phys. Rev. A 47, 4451–4463 (1993). [CrossRef] [PubMed]

].

To qualitatively explain the above conclusion, we consider the radiation within the core region, for which k2=(2πnco/λ)2=kr2+kϕ2+β2, in which k = (kr, kϕ, β) is the wave vector in cylindrical coordinates. We impose a resonance condition similar to one that is used in multilayer optical waveguides [40

40. K. J. Rowland, S. Afshar. V, and T. M. Monro, “Bandgaps and antiresonance integrated-arrows and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008). [CrossRef] [PubMed]

]: the phase accumulation associated with kϕ, over a round trip of the circumference of the fiber cross section, should be 2, i.e. kϕ × πD = 2, where l is an integer. This results in the following relation between resonance diameter D (at fixed wavelength) or resonance wavelength (at fixed core diameter);
Dres=2l/(2πnco/λres)2kr2β2
(5)
In this equation, both β and kr are continuous and bounded by 0 < β (kr) < (2πnco). The above equation leads to an important conclusion; for every mode l, there is a continuum class of WGM resonances that are associated with different values of β, skew rays, and kr, spiral rays. For resonances where β ≈ 0 and kr ≈ 0, the equation reduces to a ideal WGM resonance condition for a circular disk, which represents an electromagnetic (EM) field that is propagating around the circumference of the disk and evanescently decaying outside (due to the imaginary part of kϕ) the disk, but without any power radiating in r or z directions. However, for resonances with non-zero β or kr the EM field propagates around the circumference of the disk, but at the same time has power radiating in r and z directions. For these resonances as β and kr increase, the resonance core diameter Dres shifts to higher values for constant mode order and wavelength, which is consistent with the behavior observed in Figs. 2(a) and 2(b). A striking observation of Eq. (5) is that even for β ≈ 0, i.e., no skew rays, there still exist a class of resonances, 2D spiral rays, for which the EM field is confined within the two-dimensional transverse plane, but radiating in the r direction with different values of kr. These values of kr correspond to different phase velocities along the r direction, which determine how quickly the spiral arm of each wave opens as the wave propagates to infinity. Based on the behavior observed in Fig. 2 and Eq. (5), one can expect that the resonance core diameter associated with kr ≠ 0 shifts to larger values as kr increases, resulting in an asymmetric profile. This indicates that even for β ≈ 0, one can not expect to achieve ideal WG resonances with Lorentzian shape and linewidths of Im(kϕ)/2Re(kϕ) [37

37. L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004). [CrossRef]

, 38

38. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2004). [CrossRef]

].

In addition to radiation modes for which kϕ is quantized (i.e. locked to the fiber circumference), there is another class of radiation modes for which all three wave-vector components (kr, kϕ, β) are continuous. This class of radiation modes are not locked to the structure and are orthogonal to radiation modes with locked kϕ, and hence do not show any resonances and Pur-cell enhancement. They contribute to a uniform background radiation power as it is evident in Fig. 1.

Fig. 3 (a) Purcell factor, (b) Q-factor, (c) effective modal volume, and (d) coupling coefficient and decay rate as functions of fiber core diameter at λ = 700nm. Calculations in (d) are for a quantum dot with a spontaneous emission lifetime of 1ns inside a tellurite glass with n = 2.025. The dashed line represents a decay rate of 1 GHz for the quantum dot used for this calculation [31]

Considering a quantum dot (QD) with a spontaneous emission lifetime of τ = 1 ns, we calculate the coupling rate between the cavity and the QD, g/2π [31

31. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef] [PubMed]

] and the decay rate, κ/2π as ; g/2π=(1/2τsp)(3cλ02τsp)/(2πn3Veff) and κ/2π = c/(λcavQ), respectively, in which λcav is the resonance wavelength of the cavity. Figure 3(d) shows that for fibers with core diameters up to 2.23 μm the decay rate of the cavity is higher than the coupling rate, indicating that the cavity is in the weak coupling regime. However, for larger core diameters the cavity operates in the strong coupling regime. For comparison, we have also considered a decay rate of 1 GHz for the quantum dot used in this study [31

31. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef] [PubMed]

], which has been shown as a dashed line in Fig. 3(d). Such a decay rate corresponds to a linewidth of order of 1.6 pm at the resonance wavelength of 700 nm, which is much higher than the linewidths of radiation resonance peaks (22,0)–(29,0) (last five data points in Fig. 3(d)). This indicates that for these peaks the overall decay rate of the quantum dot, which is a combination of the free space decay rate and the cavity decay rate, is entirely determined by the cavity rate since it is much narrower than that of the free space. This shows the suitability of a dipole-fiber system to study both strong and weak coupling regimes of CQED.

While the above discussion provides a phenomenological evidence that dipole-fiber system can operate in the strong coupling regime of CQED, it is possible to provide a direct evidence of the strong coupling by showing the vacuum Rabi splitting in the emission spectrum. In order to model this, we need to extend the model presented here to include dipoles that can change state or exist in a superposition of the ground and excited states. Such a model will naturally include the dipole decay rate and resonance frequency, which when combined with the decay rates and resonance frequencies of fiber WGMs will lead to Rabi splitting of the emission spectrum.

High Q-factors of the radiation peaks of the dipole-fiber system indicates that this configuration can also be used for refractive index sensing applications. We define the sensitivity of radiation peaks as the amount of shift per unit refractive index (RIU) change of the cladding, normalized to the peak’s linewidths (LW), and plot it as a function of core diameter (resonance modes (12, 0) to (29, 0)) as shown in Fig. 4. The sensitivity increases exponentially for higher order resonance modes. Considering that it is possible to locate the position of a resonance to 1/100 of its linewidth [41

41. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label free detection down to single molecules,” Nat. Methods 5, 591–596 (2008). [CrossRef] [PubMed]

], the minimum shift that can be measured using the resonance peak (29,0) in Fig. 4 (with the linewidth of order of 0.54fm) is of the order of 0.0054fm, which leads to a theoretical minimum detectable δn ≈ 5×10−10 for the dipole-fiber system considered here. These results suggest the potential of the dipole-fiber system as a platform for ultra-sensitive refractive index sensing. The Q-factor of the resonance modes considered here only includes radiation losses. In practice there are other sources of losses such as scattering or material losses that reduce the Q-factor of the resonances and hence the sensitivity of these resonances to refractive index changes.

Fig. 4 Sensitivity; shift per refractive index unit (RIU) per linewidth (LW) as a function of core diameter.

Refractive index sensing based on the WGMs of cylindrical waveguides such as microcapillaries has been studied both experimentally and theoretically [42

42. I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319–1321 (2008). [CrossRef]

46

46. K. J. Rowland, A. Francois, P. Hoffmann, and T. M. Monro, “Fluorescent polymer coated capillaries as optofluidic refractometric sensors,” Opt. Express 21, 11492–11505 (2013). [CrossRef] [PubMed]

]. In these studies the WGMs of the cylindrical waveguide are excited either by extended laser beams of tapered fibers [42

42. I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319–1321 (2008). [CrossRef]

, 43

43. I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” App. Phys. Lett. 89, 191106 (2006). [CrossRef]

] or by fluorescent emission of an ensemble of quantum dots [45

45. C. P. K. Manchee, V. Zamora, J. W. Silverstone, J. G. C. Veinot, and A. Meldrum, “Refractometric sensing with fluorescent-core microcapillaries,” Opt. Express 19, 21540–21551 (2011). [CrossRef] [PubMed]

] or dye-doped polymer, which have been coated on the inner walls of the capillary (see [46

46. K. J. Rowland, A. Francois, P. Hoffmann, and T. M. Monro, “Fluorescent polymer coated capillaries as optofluidic refractometric sensors,” Opt. Express 21, 11492–11505 (2013). [CrossRef] [PubMed]

] and references therein for an overview of different methods). However, the key point that leads to the ultra-sensitivity of the dipole-fiber system and distinguishes this work from others is the selective coupling of a dipole emission to those WGMs that have almost zero skew components (β = 0).

3. Discussion and conclusion

Acknowledgments

The authors would like to thank K. Rowland and A. Francois for helpful discussions. This research was supported under the ARC Discovery Project funding ( DP120100901). A. D. Greentree, and B. C. Gibson acknowledge the support of ARC fundings DP130104381, and FT110100225, respectively. T.M. Monro acknowledges the support of ARC Federation and Laureate Fellowships.

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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006). [CrossRef] [PubMed]

9.

A. Kiraz, P. Michler, C. Becher, B. Gayral, A. Imamoglu, L. Zhang, E. Hu, W. V. Schoenfeld, and P. M. Petroff, “Cavity-quantum electrodynamics using a single InAs quantum dot in a microdisk structure,” Appl. Phys. Lett. 78, 3932–3934 (2001). [CrossRef]

10.

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977). [CrossRef]

11.

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977). [CrossRef]

12.

W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69, 1495–1503 (1979). [CrossRef]

13.

M. Janowicz and W. Żakowicz, “Quantum radiation of a harmonic oscillator near the planar dielectric-vacuum interface,” Phys. Rev. A 50, 4350–4364 (1994). [CrossRef] [PubMed]

14.

W. Żakowicz and M. Janowicz, “Spontaneous emission in the presence of a dielectric cylinder,” Phys. Rev. A 62, 013820 (2000). [CrossRef]

15.

J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981). [CrossRef]

16.

D. Y. Chu and S.-T. Ho, “Spontaneous emission from excitons in cylindrical dielectric waveguides and the spontaneous-emission factor of microcavity ring lasers,” J. Opt. Soc. Am. B 10, 381–390 (1993). [CrossRef]

17.

V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004). [CrossRef]

18.

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, “Decay rate and level shift in a circular dielectric waveguide,” Phys. Rev. A 71, 013815 (2005). [CrossRef]

19.

D. J. Heinzen and M. S. Feld, “Vacuum radiative level shift and spontaneous-emission linewidth of an atom in an optical resonator,” Phys. Rev. Lett. 59, 2623–2626 (1987). [CrossRef] [PubMed]

20.

G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A 44, 669–681 (1991). [CrossRef] [PubMed]

21.

K. Ujihara, “Spontaneous emission and concept of effective area in a very short optical cavity with plane-parallel dielectric mirrors,” Jpn. J. Appl. Phys. 30, 901–903 (1991). [CrossRef]

22.

F. Björk, H. Heitmann, and Y. Yamamoto, “Spontaneous-emission coupling factor and mode characteristics of planar dielectric microcavity lasers,” Phys. Rev. A 47, 4451–4463 (1993). [CrossRef] [PubMed]

23.

F. Le Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, “Spontaneous emission of a Cesium atom near a nanofiber: Efficient coupling of light to guided modes,” Phys. Rev. A 72, 032509 (2005). [CrossRef]

24.

F. Le Kien and K. Hakuta, “Cavity-enhanced channeling of emission from an atom into a nanofiber,” Phys. Rev. A 80, 053826 (2009). [CrossRef]

25.

K. P. Nayak, F. L. Kien, Y. Kawai, K. Hakuta, K. Nakajima, H. T. Miyazaki, and Y. Sugimoto, “Cavity formation on an optical nanofiber using focused ion beam milling technique,” Opt. Express 19, 14040–14050 (2011). [CrossRef] [PubMed]

26.

M. R. Henderson, S. Afshar. V., A. D. Greentree, and T. M. Monro, “Dipole emitters in fiber: interface effects, collection efficiency and optimization,” Opt. Express 19, 16182–16194 (2011). [CrossRef] [PubMed]

27.

M. R. Henderson, B. C. Gibson, H. Ebendorff-Heidepriem, K. Kuan, S. Afshar V., J. O. Orwa, I. Aharonovich, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, and T. M. Monro, “Diamond in tellurite glass: a new medium for quantum information,” Adv. Mater. 23, 2806–2810 (2011). [CrossRef] [PubMed]

28.

T. Schröder, M. Fujiwara, T. Noda, H. Q. Zhao, O. Benson, and S. Takeuchi, “A nanodiamond-tapered fiber system with high single-mode coupling efficiency,” Opt. Express 20, 10490–10497 (2012). [CrossRef] [PubMed]

29.

R. Yalla, F. Le Kien, M. Morinaga, and K. Hakuta, “Efficient channeling of fluorescence photons from single quantum dots into guided modes of optical nanofiber,” Phys. Rev. Lett. 109, 063602 (2012). [CrossRef] [PubMed]

30.

E. Peter, I. Sagnes, G. Guirleo, S. Varoutsis, J. Bloch, A. Lematre, and P. Senellart, “High-Q whispering-gallery modes in GaAs/AlOx microdisks,” Appl. Phys. Lett. 86, 021103 (2005). [CrossRef]

31.

K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef] [PubMed]

32.

V. Bordo, “Purcell factor for a cylindrical nanocavity: ab initio analytical approach,” J. Opt. Soc. Am. B 29, 1799–1809 (2012). [CrossRef]

33.

M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89, 233602 (2002). [CrossRef] [PubMed]

34.

S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43, 033001 (2010). [CrossRef]

35.

C. Junge, S. Nickel, D. O’Shea, and A. Rauschenbeutel, “Bottle microresonator with actively stabilized evanescent coupling,” Opt. Lett. 36, 3488–3490 (2011). [CrossRef] [PubMed]

36.

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

37.

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004). [CrossRef]

38.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2004). [CrossRef]

39.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010). [CrossRef]

40.

K. J. Rowland, S. Afshar. V, and T. M. Monro, “Bandgaps and antiresonance integrated-arrows and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008). [CrossRef] [PubMed]

41.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label free detection down to single molecules,” Nat. Methods 5, 591–596 (2008). [CrossRef] [PubMed]

42.

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319–1321 (2008). [CrossRef]

43.

I. M. White, H. Oveys, X. Fan, T. L. Smith, and J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” App. Phys. Lett. 89, 191106 (2006). [CrossRef]

44.

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photon. 5, 591–597 (2011). [CrossRef]

45.

C. P. K. Manchee, V. Zamora, J. W. Silverstone, J. G. C. Veinot, and A. Meldrum, “Refractometric sensing with fluorescent-core microcapillaries,” Opt. Express 19, 21540–21551 (2011). [CrossRef] [PubMed]

46.

K. J. Rowland, A. Francois, P. Hoffmann, and T. M. Monro, “Fluorescent polymer coated capillaries as optofluidic refractometric sensors,” Opt. Express 21, 11492–11505 (2013). [CrossRef] [PubMed]

OCIS Codes
(230.5750) Optical devices : Resonators
(270.5580) Quantum optics : Quantum electrodynamics
(140.3948) Lasers and laser optics : Microcavity devices

ToC Category:
Quantum Optics

History
Original Manuscript: February 26, 2014
Revised Manuscript: April 17, 2014
Manuscript Accepted: April 25, 2014
Published: May 2, 2014

Citation
S. Afshar V, M. R. Henderson, A. D. Greentree, B. C. Gibson, and T. M. Monro, "Self-formed cavity quantum electrodynamics in coupled dipole cylindrical-waveguide systems," Opt. Express 22, 11301-11311 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-11301


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References

  1. E. M. Purcell, “Proceedings of the American Physical Society,” Phys. Rev. 69, 674 (1946). [CrossRef]
  2. P. Berman, Cavity Quantum Electrodynamics (Academic, 1994).
  3. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003). [CrossRef] [PubMed]
  4. D. Vernooy, A. Furusawa, N. Georgiades, V. Ilchenko, H. Kimble, “Cavity QED with high-Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998). [CrossRef]
  5. H. Schniepp, V. Sandoghdar, “Spontaneous emission of europium ions embedded in dielectric nanospheres,” Phys. Rev. Lett. 89, 257403 (2002). [CrossRef] [PubMed]
  6. S. Spillane, T. Kippenberg, O. Painter, K. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]
  7. D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef] [PubMed]
  8. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006). [CrossRef] [PubMed]
  9. A. Kiraz, P. Michler, C. Becher, B. Gayral, A. Imamoglu, L. Zhang, E. Hu, W. V. Schoenfeld, P. M. Petroff, “Cavity-quantum electrodynamics using a single InAs quantum dot in a microdisk structure,” Appl. Phys. Lett. 78, 3932–3934 (2001). [CrossRef]
  10. W. Lukosz, R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977). [CrossRef]
  11. W. Lukosz, R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977). [CrossRef]
  12. W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69, 1495–1503 (1979). [CrossRef]
  13. M. Janowicz, W. Żakowicz, “Quantum radiation of a harmonic oscillator near the planar dielectric-vacuum interface,” Phys. Rev. A 50, 4350–4364 (1994). [CrossRef] [PubMed]
  14. W. Żakowicz, M. Janowicz, “Spontaneous emission in the presence of a dielectric cylinder,” Phys. Rev. A 62, 013820 (2000). [CrossRef]
  15. J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981). [CrossRef]
  16. D. Y. Chu, S.-T. Ho, “Spontaneous emission from excitons in cylindrical dielectric waveguides and the spontaneous-emission factor of microcavity ring lasers,” J. Opt. Soc. Am. B 10, 381–390 (1993). [CrossRef]
  17. V. V. Klimov, M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004). [CrossRef]
  18. D. P. Fussell, R. C. McPhedran, C. Martijn de Sterke, “Decay rate and level shift in a circular dielectric waveguide,” Phys. Rev. A 71, 013815 (2005). [CrossRef]
  19. D. J. Heinzen, M. S. Feld, “Vacuum radiative level shift and spontaneous-emission linewidth of an atom in an optical resonator,” Phys. Rev. Lett. 59, 2623–2626 (1987). [CrossRef] [PubMed]
  20. G. Björk, S. Machida, Y. Yamamoto, K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A 44, 669–681 (1991). [CrossRef] [PubMed]
  21. K. Ujihara, “Spontaneous emission and concept of effective area in a very short optical cavity with plane-parallel dielectric mirrors,” Jpn. J. Appl. Phys. 30, 901–903 (1991). [CrossRef]
  22. F. Björk, H. Heitmann, Y. Yamamoto, “Spontaneous-emission coupling factor and mode characteristics of planar dielectric microcavity lasers,” Phys. Rev. A 47, 4451–4463 (1993). [CrossRef] [PubMed]
  23. F. Le Kien, S. Dutta Gupta, V. I. Balykin, K. Hakuta, “Spontaneous emission of a Cesium atom near a nanofiber: Efficient coupling of light to guided modes,” Phys. Rev. A 72, 032509 (2005). [CrossRef]
  24. F. Le Kien, K. Hakuta, “Cavity-enhanced channeling of emission from an atom into a nanofiber,” Phys. Rev. A 80, 053826 (2009). [CrossRef]
  25. K. P. Nayak, F. L. Kien, Y. Kawai, K. Hakuta, K. Nakajima, H. T. Miyazaki, Y. Sugimoto, “Cavity formation on an optical nanofiber using focused ion beam milling technique,” Opt. Express 19, 14040–14050 (2011). [CrossRef] [PubMed]
  26. M. R. Henderson, S. Afshar. V., A. D. Greentree, T. M. Monro, “Dipole emitters in fiber: interface effects, collection efficiency and optimization,” Opt. Express 19, 16182–16194 (2011). [CrossRef] [PubMed]
  27. M. R. Henderson, B. C. Gibson, H. Ebendorff-Heidepriem, K. Kuan, S. Afshar V., J. O. Orwa, I. Aharonovich, S. Tomljenovic-Hanic, A. D. Greentree, S. Prawer, T. M. Monro, “Diamond in tellurite glass: a new medium for quantum information,” Adv. Mater. 23, 2806–2810 (2011). [CrossRef] [PubMed]
  28. T. Schröder, M. Fujiwara, T. Noda, H. Q. Zhao, O. Benson, S. Takeuchi, “A nanodiamond-tapered fiber system with high single-mode coupling efficiency,” Opt. Express 20, 10490–10497 (2012). [CrossRef] [PubMed]
  29. R. Yalla, F. Le Kien, M. Morinaga, K. Hakuta, “Efficient channeling of fluorescence photons from single quantum dots into guided modes of optical nanofiber,” Phys. Rev. Lett. 109, 063602 (2012). [CrossRef] [PubMed]
  30. E. Peter, I. Sagnes, G. Guirleo, S. Varoutsis, J. Bloch, A. Lematre, P. Senellart, “High-Q whispering-gallery modes in GaAs/AlOx microdisks,” Appl. Phys. Lett. 86, 021103 (2005). [CrossRef]
  31. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef] [PubMed]
  32. V. Bordo, “Purcell factor for a cylindrical nanocavity: ab initio analytical approach,” J. Opt. Soc. Am. B 29, 1799–1809 (2012). [CrossRef]
  33. M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, Y. Yamamoto, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89, 233602 (2002). [CrossRef] [PubMed]
  34. S. Reitzenstein, A. Forchel, “Quantum dot micropillars,” J. Phys. D Appl. Phys. 43, 033001 (2010). [CrossRef]
  35. C. Junge, S. Nickel, D. O’Shea, A. Rauschenbeutel, “Bottle microresonator with actively stabilized evanescent coupling,” Opt. Lett. 36, 3488–3490 (2011). [CrossRef] [PubMed]
  36. A. Snyder, J. Love, Optical Waveguide Theory (Springer, 1983).
  37. L. Prkna, J. Čtyroký, M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004). [CrossRef]
  38. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2004). [CrossRef]
  39. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010). [CrossRef]
  40. K. J. Rowland, S. Afshar. V, T. M. Monro, “Bandgaps and antiresonance integrated-arrows and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008). [CrossRef] [PubMed]
  41. F. Vollmer, S. Arnold, “Whispering-gallery-mode biosensing: label free detection down to single molecules,” Nat. Methods 5, 591–596 (2008). [CrossRef] [PubMed]
  42. I. M. White, H. Oveys, X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319–1321 (2008). [CrossRef]
  43. I. M. White, H. Oveys, X. Fan, T. L. Smith, J. Zhang, “Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides,” App. Phys. Lett. 89, 191106 (2006). [CrossRef]
  44. X. Fan, I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photon. 5, 591–597 (2011). [CrossRef]
  45. C. P. K. Manchee, V. Zamora, J. W. Silverstone, J. G. C. Veinot, A. Meldrum, “Refractometric sensing with fluorescent-core microcapillaries,” Opt. Express 19, 21540–21551 (2011). [CrossRef] [PubMed]
  46. K. J. Rowland, A. Francois, P. Hoffmann, T. M. Monro, “Fluorescent polymer coated capillaries as optofluidic refractometric sensors,” Opt. Express 21, 11492–11505 (2013). [CrossRef] [PubMed]

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