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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 11 — May. 24, 2010
  • pp: 10995–11007
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Hybrid gap modes induced by fiber taper waveguides: Application in spectroscopy of single solid-state emitters deposited on thin films

Marcelo Davanço and Kartik Srinivasan  »View Author Affiliations


Optics Express, Vol. 18, Issue 11, pp. 10995-11007 (2010)
http://dx.doi.org/10.1364/OE.18.010995


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Abstract

We show, via simulations, that an optical fiber taper waveguide can be an efficient tool for photoluminescence and resonant, extinction spectroscopy of single emitters, such as molecules or colloidal quantum dots, deposited on the surface of a thin dielectric membrane. Placed over a high refractive index membrane, a tapered fiber waveguide induces the formation of hybrid mode waves, akin to dielectric slotted waveguide modes, that provide strong field confinement in the low index gap region. The availability of such gap-confined waves yields potentially high spontaneous emission enhancement factors (≈ 20), fluorescence collection efficiencies (≈ 23 %), and transmission extinction (≈ 20 %) levels. A factor of two improvement in fluorescence and extinction levels is predicted if the membrane is instead replaced with a suspended channel waveguide. Two configurations, for operation in the visible (≈ 600 nm) and near-infrared (≈ 1300 nm) spectral ranges are evaluated, presenting similar performances.

© 2010 Optical Society of America

1. Introduction

In this paper, we use electromagnetic simulations to show that an optical fiber taper waveguide (sometimes called a micro- or nanofiber waveguide) can be used as an efficient probe for resonant and non-resonant spectroscopy of individual emitters bound to the surface of thin dielectric membranes. High probing efficiency is possible due to the availability of hybrid guided waves akin to the air-slot waveguide modes introduced in [1

1. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

]. Such hybrid waves, referred to as gap modes, display strong field concentration in the gap between fiber and membrane and at the same time couple efficiently to the access optical fiber waveguide mode. The first feature leads to spontaneous emission rate enhancement, while the second, combined with high percentage coupling of spontaneous emission into gap modes, leads to high fluorescence collection efficiency.

Fig. 1. (a) Tapered fiber waveguide-based probing configuration for emitters deposited on the surface of a dielectric membrane. An individual emitter, embedded in a host thin film, is depicted under the fiber. (b) Cross-section of structure in (a). (c) Schematic of single emitter excitation and PL collection via the tapered fiber probe. A non-resonant pump signal is injected into the input fiber and converted into a guided supermode of the composite waveguide, illuminating the slab-embedded dipole. The dipole radiates into guided and radiative supermodes, with rates Γm and Γrad respectively. Power is transferred with efficiency fm from the supermode to the fiber mode and vice-versa. The interaction length L c is the length in which fiber and slab are in close proximity.

The optical fiber taper waveguide is a single mode optical fiber whose diameter is adiabatically and symmetrically reduced to a wavelength-scale minimum, resulting in a low-loss, double-ended device with standard fiber input and output. The manner in which emitters on the surface of a dielectric membrane may be optically accessed through the fiber is shown in Figs. 1(a)–1(c). The wavelength-scale single mode region of the optical fiber waveguide is brought into proximity with the top surface of the membrane, over a length of several wavelengths. Fiber and membrane together form the composite dielectric waveguide with cross-section shown in Fig. 1(b), which supports a complete set of guided, leaky, and radiation supermodes originating from the hybridization of fiber and slab modes [11

11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).

] (note that gap modes are indeed supermodes of the hybrid fiber-slab waveguide, as discussed in Section 4). As illustrated in Fig. 1(c), for non-resonant photoluminescence (PL) spectroscopy measurements, part of the non-resonant pump power initially carried by the fiber is coupled to supermode waves that reach the emitter. Illuminated by the pump, the emitter radiates (at a red-shifted wavelength) into supermodes of the composite waveguide, and a portion of the radiated power is outcoupled through the two fiber ends. Using the same method, we also envision the possibility of resonantly exciting the dipole, in which case the radiated fields (resonance fluorescence) would be coherent and at the same wavelength as the pump signal. Backward-propagating resonance fluorescence can potentially be collected from the input fiber port by way of a directional coupler. In the forward direction, the power at the output fiber is given by the interference between the excitation and resonant fluorescence signals, and may result in an enhanced or diminished transmission level [10

10. M. Davanço and K. Srinivasan, “Fiber-coupled semiconductor waveguides as an efficient optical interface to a single quantum dipole,” Opt. Lett. 34, 2542–2544 (2009), http://ol.osa.org/abstract.cfm?URI=ol-34-16-2542 [CrossRef] [PubMed]

, 12

12. I. Gerhardt, G. Wrigge, P. Bushev, G. Zumofen, M. Agio, R. Pfab, and V. Sandoghdar, “Strong Extinction of a Laser Beam by a Single Molecule,” Phys. Rev. Lett. 98, 033601 (2007). [CrossRef] [PubMed]

].

In [9

9. M. Davanço and K. Srinivasan, “Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides,” Opt. Express 17, 10542–10563 (2009). [CrossRef] [PubMed]

], we analyzed the collection efficiency of the fiber taper waveguide for light emitted by a dipole embedded in a high refractive index membrane. We showed that the collection efficiency, achieved by tapping into emitted slab-confined waves, is potentially highly superior to that attainable with standard free-space collection using high numerical aperture optics. While the existence of the aforementioned gap modes was mentioned in that article, their potential application in performing efficient spectroscopy of individual surface-bound dipoles was only briefly discussed, and not analyzed at any level of detail. Such an analysis is accomplished in the following sections.

This paper is organized as follows. In Section 2, our simulation model and methods are explained. In Section 3, simulation results of fluorescence collection for two configurations appropriate for visible and infrared wavelengths are given. Section 4 analyzes the collection efficiency results in terms of a hybrid waveguide mode decomposition. Parameters from this analysis are used in Section 5 to show the possibility of resonant field extinction by a single dipole. A discussion of the results follows in Section 6, and Section 7 concludes the paper.

2. Model and methods

We envision a general spectroscopy configuration for probing individual emitters in highly dilute guest-host material systems, examples of which are: colloidal quantum dots (CdSe, CdSe/ZnO, PbS, PbSe, etc) in polymer [13

13. J. Lee, V. C. Sundar, J. R. Heine, M. G. Bawendi, and K. F. Jensen, “Full Color Emission from II-VI Semiconductor Quantum Dot-Polymer Composites,” Adv. Mater. 12, 1102–1105 (2000). [CrossRef]

] or sol-gel [14

14. R. D. Schaller, M. A. Petruska, and V. Klimov, “Tunable Near-Infrared Optical Gain and Amplified Spontaneous Emission Using PbSe Nanocrystals,” J. Phys. Chem. B 107, 13765–13768 (2003). [CrossRef]

] hosts; organic dyes in polymer [15

15. A. Zumbusch, L. Fleury, R. Brown, J. Bernard, and M. Orrit, “Probing individual two-level systems in a polymer by correlation of single molecule fluorescence,” Phys. Rev. Lett. 70, 3584–3587 (1993). [CrossRef] [PubMed]

] or small-molecule crystalline hosts [6

6. W. E. Moerner, “Single-photon sources based on single molecules in solids,” N. J. Phys. 6, 88 (2004). [CrossRef]

, 16

16. M. Orrit and J. Bernard, “Single pentacene molecules detected by fluorescence excitation in a p-terphenyl crystal,” Phys. Rev. Lett. 65, 2716–2719 (1990). [CrossRef] [PubMed]

, 17

17. G. S. Harms, T. Irngartinger, D. Reiss, A. Renn, and U. P. Wild, “Fluorescence lifetimes of terrylene in solid matrices,” Chem. Phys. Lett. 313, 533–538 (1999). [CrossRef]

]; rare-earth ions in transparent solid-state hosts [18

18. T. Bottger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μ in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B: Condens. Matter Mater. Phys. 73, 075101 (2006). [CrossRef]

]. In our configuration, a thin film of the guest-host material system would be produced on top of a dielectric, high refractive index material, chosen appropriately according to the guest emitter transition wavelength ranges. For instance, a Si (n ≈ 3.5) membrane could be used for near-infrared wavelengths above 1 μm, while SiNx (n ≈ 2.0) could be used for visible light emission. We point out that such thin-film structures may be produced in most cases with well-known, standard nanofabrication techniques. As depicted in Figs. 1(a) and 1(b), a tapered optical fiber waveguide brought into contact with the host material provides both the excitation and collection channels to the guest emitters. We analyze this structure with the same method as in [9

9. M. Davanço and K. Srinivasan, “Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides,” Opt. Express 17, 10542–10563 (2009). [CrossRef] [PubMed]

], where single emitter collection efficiency from dipoles embedded in a dielectric membrane was studied. The simulation model and analysis methods are briefly described below.

2.1. Simulation model

We model the problem as in Fig. 1(b). An individual electric point dipole embedded in a thin host film on the surface of a dielectric membrane of thickness t slab and refractive index n slab is probed by an optical fiber taper waveguide of radius R and index n fiber = 1.45. The dipole, oriented normal to the membrane surface, is assumed to be at the center of a dielectric host film of index n host and thickness t host. It is also assumed to be aligned with the center of the probing fiber. As in Fig. 1(c), an excitation signal, resonant or non-resonant with one of the emitter’s transitions, is launched into the fiber input and adiabatically reduced in size as the fiber is tapered, exciting supermodes of the coupler structure. Supermodes with sufficient lateral confinement illuminate the dipole, at a position z = z 0 along the coupler. Under non-resonant excitation, the dipole emits coupler supermodes in the ±z directions, at a red-shifted wavelength. The emitted supermodes are converted into input and output fiber modes through the taper transition regions, after which emission is detected.

2.2. Fluorescence collection simulation

To estimate the PL collection efficiency of our fiber-based probing scheme, we simulated a single classical electric dipole radiating in the composite dielectric waveguide of Fig. 1(a), using the Finite Difference Time Domain (FDTD) method [9

9. M. Davanço and K. Srinivasan, “Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides,” Opt. Express 17, 10542–10563 (2009). [CrossRef] [PubMed]

]. The simulation provided the steady-state fields over the entire computational window, which was cubic, with more than six wavelengths in size. These were used to calculate an upper bound for the percentage of the total emitted power P Tot. coupled to the fundamental optical fiber mode at an arbitrary position z along the guide, with the expression

ηPL=2PzPTot.ffiber.
(1)

Here, Pz is the power flowing normally through the constant-z plane, f fiber is the overlap integral in Eq. (2), taken between the radiated field at position z and the fundamental (isolated) fiber mode. The factor of 2 accounts for collection from both fiber ends. The symmetry of the geometry allowed us to choose symmetric ( × H = 0) boundary conditions on the yz-plane, as only y-polarized dipoles were considered. Perfectly-matched layers (PMLs) were used around the domain limits to simulate an open domain. Simulations ran until no field amplitude could be detected in the domain. As in [9

9. M. Davanço and K. Srinivasan, “Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides,” Opt. Express 17, 10542–10563 (2009). [CrossRef] [PubMed]

], η PL oscillates with z, due to the back-and-forth power exchange between the guide and the slab along the waveguide. The values of η PL reported below correspond to maxima obtained within the computational window.

2.3. Supermode analysis

Supermode field profiles and the respective complex propagation constants βm are obtained with a finite-element based eigenvalue solver, with a vectorial formulation. Supermode m’s individual contribution to the total PL collection efficiency η PL, considering one fiber channel, is η PL,m = fm·Γm/Γ = fm·γm, where Γm is the supermode emission rate, and Γ the total emission rate. The fraction γm is supermode m’s spontaneous emission coupling factor (β-factor), which, since emission in both ±z directions is equally likely, is such that 0 ≤ γm ≤ 0.5. The fiber mode fraction, fm, is the percentage of supermode m’s power that is transferred to the output fiber mode. Assuming that the fiber is abruptly removed from the slab at the end of the probing region, and that reflections at the interface are small, fm may be approximated with an overlap integral between the fundamental fiber mode and supermode m [19

19. W.-P Huang, “Coupled-mode theory for optical waveguides: and overview,” J. Opt. Soc. Am. A 11, 963–983 (1994). [CrossRef]

, 20

20. Following Ref [19], the fiber mode fraction, Eq. (2), would be given by the expression fm = 〈fm〉〈mf〉 (〈ff〉〈mm〉)-1, where 〈fm〉 = ∣ (ef × hm* + hf × em*ẑdS/4. Considering no reflections at the interface between the isolated fiber and the contact region, (i.e., the field just after the interface is identical to the incident, foward propagating, field), Eq. (2) gives the same result.

]:

fm=Re{S(ef×hm*)·ẑdSS(em×hf*)·ẑdS}Re{S(ef×hf*)·ẑdS}Re{S(em×hm*)·ẑdS}.
(2)

In this expression, valid for purely dielectric waveguides, {e m, h m} and {e f, h f} are the supermode and fundamental fiber mode fields, respectively. The supermode emission rates Γm is also obtained from the field profiles, according to the expressions given in [9

9. M. Davanço and K. Srinivasan, “Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides,” Opt. Express 17, 10542–10563 (2009). [CrossRef] [PubMed]

].

2.4. Field Extinction

For coherent, resonant excitation, the power detected at the output fiber port is a result of the interference between the excitation signal and the resonance fluorescence from the emitter (which are at the same wavelength), and may be either higher or lower than the detected power in the absence of the dipole. In order to determine the variation in the transmitted power level due to the presence of a single dipole, we make use of the quantum optics input-output formalism of [21

21. S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813 (2004). [CrossRef]

], with which we obtain operators for the multimode field for z > z 0, i.e., past the dipole location:

E(+)(z,t)=i2πmh̄ω4πSmeme1(ωtβmz)×
×[âinm(tnmz/c)+Γm*σ(tnmz/c)].
(3)

Here, σ is the atomic lowering operator, âinm is (incident) supermode m’s input field annihilation operator, e m is the electric field distribution, βm the propagation constant, nm the phase index, and Sm = Re{∫S dS(e m × h m *}, with S the xy plane. The expression in brackets is a well-known result of the input-output formalism, with explicit input (or ”free”) fields and radiated (”source”) contributions [21

21. S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813 (2004). [CrossRef]

], expanded in terms of supermodes. The field operators are then inserted in the fiber mode power operator [10

10. M. Davanço and K. Srinivasan, “Fiber-coupled semiconductor waveguides as an efficient optical interface to a single quantum dipole,” Opt. Lett. 34, 2542–2544 (2009), http://ol.osa.org/abstract.cfm?URI=ol-34-16-2542 [CrossRef] [PubMed]

],

F̂={SdS(E()×hf)·ẑSdS(ef*×H(+))·ẑ+
SdS(H()×ef)·ẑSdS(hf*×E(+))·ẑ}Sf1,
(4)

where e f and h f are the fiber mode electric and magnetic field distributions, Sf = Re{∫S dS(e f × h f *}. In short, the fiber mode power operator allows us to determine the total photon flux coupled into the output fiber mode, based on multimode field operators that describe the coherent interference between the incident (’free’) and emitted, resonance fluorescence (’source’) supermode waves; the operator is the quantum optics equivalent to the overlap integral in Eq. (2), between the total field at a position z along the waveguide and the optical fiber mode. This corresponds to the power coupled into the output fiber at the end of the coupling region, assuming an abrupt transition and small reflections at the interface [19

19. W.-P Huang, “Coupled-mode theory for optical waveguides: and overview,” J. Opt. Soc. Am. A 11, 963–983 (1994). [CrossRef]

]. Considering coherent, steady-state, multimode field excitation, the photon flux F at the output fiber (normalized to the input field power) is found to be

F=h̄ωRe{m,mfmfmei(βmβm)(zz0)×
×[BmBm*+ΓmΓm*ζΓ2(Bm*Γm*ξ+BmΓmξ*)(Γ2)22ζ]},
(5)

with ξ=mΓmBm,ζ=m,mRe{ΓmΓm*Bm*Bm}. In this expression, Bm is the complex amplitude of the m-th supermode incident on the dipole. The magnitude of Bm is determined by the manner with which the fiber is brought into contact with the slab. For instance, for abrupt contact (e.g., very short transition regions in Fig. 1), it approaches ∣fm1/2, where fm is the fiber-mode fraction. Longer transition regions could lead to a power distribution among the excited supermodes different from that obtained with the fiber-mode fractions. The phase of the Bm coefficients at the dipole position is determined through the supermode propagation constants βm, assuming all modes are in-phase at the start of the coupler region.

To gain some insight into the mechanisms at play at resonant excitation, we consider a situation in which only one supermode of the fiber and slab structure may be accessed: in Eq. (5), we set all fiber-mode fractions fm and incident supermode amplitudes Bm to be null except for those of an arbitrary M-th supermode (i.e., fm≠M = 0, Bm≠M = 0). We furthermore make the assumption that the excitation signal drives the transition far from saturation, so that ζ2 ≪ 1 and the denominator of the second term in brackets becomes unity. In this case, the power detected at the output fiber is proportional to 1 − 4γM(1 − γM). Thus the magnitude of optical field extinction by a single dipole is determined by the supermode spontaneous emission coupling factor, and is complete when γM = 0.5.

3. Fluorescence Collection Efficiency

Two configurations were analyzed, for operation at visible (λ = 600 nm) and near-infrared (λ = 1300 nm) wavelengths. For the visible range, a 130 nm thick SiN membrane (refractive index n SiN = 2.0) and 400 nm diameter single mode fiber taper waveguide were considered. In the near-infrared case, our model consists of a 160 nm thick Si membrane (refractive index n Si = 3.505) and a 1 μm diameter fiber taper, which supports a well-confined and a near-cutoff mode. The first configuration models a system suitable for probing visible wavelength emitters such as single molecules or CdTe/ZnSe nanocrystal quantum dots attached to the SiN membrane, while the second would be suitable for infrared emitters, such as PbS and PbSe nanocrystal quantum dots [22

22. F. Wise, “Lead salt quantum dots: The limit of strong quantum confinement,” Acc. Chem. Res. 33, 773–780 (2000). [CrossRef] [PubMed]

]. In both cases, the emitters are considered to be embedded in a 20 nm thick, purely dielectric host film on top of the SiN or Si membranes, as shown in Fig. 1(b). The host film refractive index nhost is allowed to vary between 1.0 and 1.7, a range that includes typical values for possible organic (e.g., PMMA) or inorganic (e.g., silica) transparent host materials. The emitters are modeled as electric dipoles oriented in the y-direction. While this consideration limits our analysis to a best-case scenario for emitters with random dipole orientation, it may be well suited to model certain organic crystal guest-host systems, where emitting molecules embedded in the host crystal tend to align in specific orientations. For instance, in the guest-host system presented in [23

23. R. J. Pfab, J. Zimmermann, C. Hettich, I. Gerhardt, A. Renn, and V. Sandoghdar, “Aligned terrylene molecules in a spin-coated ultrathin crystalline film of p-terphenyl,” Chem. Phys. Lett. 387, 490–495 (2004). [CrossRef]

], consisting of Terrylene molecules in a crystalline p-terphenyl host -a system fit for molecular quantum optics [6

6. W. E. Moerner, “Single-photon sources based on single molecules in solids,” N. J. Phys. 6, 88 (2004). [CrossRef]

]-, the guest molecules have been shown to display dipole moments perpendicularly oriented to the substrate.

Fig. 2. (a)Maximum total spontaneous emission rate enhacement Γ/Γhom, where Γhom is the spontaneous emission rate of a dipole in a homogeneous dielectric medium of refractive index n host. (b) PL collection efficiencies η PL, including both fiber ends, for y-polarized dipoles in the SiN (λ = 600 nm) and Si(λ = 1300 nm) membrane configurations, as functions of the host layer refractive index, n host. Results calculated with finite difference time domain simulations.

Figure 2(a) shows the FDTD-calculated total spontaneous emission rate Γ, for a y-polarized dipole in the SiN (λ = 600 nm) and Si(λ = 1300 nm) membrane configurations, as functions of the host layer refractive index, n host. All rates are normalized to the rate for a dipole in a homogeneous space of index n hostHom.) and thus correspond to the spontaneous emission enhancement (Purcell) factor. Purcell factors as high as 20 and 9 respectively are observed for n host = 1.0 in the Si and SiN cases, and decrease with increasing host index. Simulations with progressively denser meshes were used to verify that these results converged to within at least 3%.

In Fig. 2(b), the corresponding maximum PL collection efficiencies η PL including collection from both fiber ends are plotted. As mentioned earlier, since η PL oscillates in z due to power exchange between the fiber and slab, the plotted values correspond to the maximum efficiency within the computational window (with dimensions > 6 wavelengths). It is apparent that collection efficiencies are above 18 % for all n host in both Si and SiN cases, with maxima of 22.8 % and 20.5 % at n host =1.0 respectively, and decreasing for higher values.

4. Supermode Analysis

In both, Si and SiN, systems, the strong field concentration originates in the large index steps between the membrane and host layer. Such gap modes are the main contributors to the total collection efficiency, η PL. This is apparent in Figs. 4(a) and 4(e), where the individual contributions, η PL,m, to the total collection efficiency, η PL, are plotted. The contributions of all secondary supermodes [black dots in Fig. 4(a) and 4(e)] are at least an order of magnitude smaller than the main ones. Despite providing small individual contributions, secondary super-modes altogether make up for a large portion of the total collected power. It is important to note that, even though many of the secondary supermodes are indeed gap modes, poor lateral confinement causes these to exhibit large effective areas, and therefore low emission rates Γm. Correspondingly, low emission coupling factors γm (γm = Γm/Γ, with Γ the total spontaneous emission rate) are observed in Figs. 4(b) and 4(f). For the main modes, γm > 10 %, so these gap modes carry a considerable percentage of the total spontaneous emission.

Fig. 3. Amplitude of the major electric field component (Ey) of laterally bound gap modes (normalized to the maximum electric field amplitude, ∣Emax) for the (a), (b) Si slab configuration (λ = 1300 nm) with (a) n host = 1-0 and (b) n host = 1.7; (c), (d) SiN configuration (λ = 600 nm) with (c) n host = 1.0 and (d) n host = 1.7. In all cases, t host = 20 nm. Line plots show ∣Ey∣/∣Emax on the x = 0 plane (dotted line in the contour plots).

The oscillation of the total efficiency η PL as a function of z, mentioned in Section 3, can be traced to the beating of the two main contributing supermodes. The beat length is given by L z = 2π/(∆β), where ∆β is the difference between the propagation constants. For the Si, λ = 1.3 μm configuration, the beat length varies between 4.78 μm, for n host = 1.0 and 3.89 μm, for n host = 1.7. In the SiN, λ = 0.6 μm case, it varies between 2.72 μm, for n host = 1.0 and 2.17 μm, for n host = 1.7. In order to maximize the collected power, therefore, control of the interaction length L c on the scale a few microns is desirable.

At the same time, short interaction lengths L c are desirable because all supermodes exhibit some degree of lateral power leakage due to imperfect field confinement. Figures 4(d) and 4(h) show the effective lengths L eff,δ, defined in Section 2.3, necessary for 10 % supermode amplitude decay, for the Si and SiN case respectively. It is apparent that in both cases the main supermodes have considerably longer (more than two decades) effective lengths, as expected, in view of their lower lateral power leakage rate. For the Si structure at λ = 1300 nm, the necessary length for the main supermode to decay by 10 % varies between 3650 μm and 1400 μm for the host index range considered. In the SiN case at λ = 600 nm, the 10 % decay length ranges between 1840 μm and 225 μm. For interaction lengths L c of only a few microns, the main supermode contributions in Figs. 4(a) and 4(e) may thus be taken as lower bounds for the total achievable collection efficiency.

Fig. 4. Supermode contributions to the total PL collection efficiency (η PL,m), modal spontaneous emission coupling factors (γm), fiber mode fractions fm and effective supermode lengths L eff,δ=0.1 for the (a)-(d) Si slab, λ = 1300 nm and (e)-(h) SiN, λ = 600nm systems. Circles: main supermode; dots: secondary supermodes

5. Resonant Extinction Spectroscopy

Fig. 5. (a) Normalized, off- and on-resonance transmission (F 0 and F) and contrast ∆T = (F - F 0)/F 0 as functions of separation from a single, y-oriented dipole at z 0. The dipole is embedded in a host material with n host = 1.7 on top of a 130 nm thick SiN membrane, and emits at λ = 600 nm. (b) Achievable transmission contrast ∆T as a function of the host film index n host. Squares: results obtained assuming dipole excitation with the main supermode only; circles: assuming multimode excitation (see text for details).

Similar performance may in principle be achieved with the Si system, for an appropriate set of parameters. The situation, however, is less favorable for the parameters considered here: in Fig. 4(c), for n host > 1.2, the main supermode fiber fraction is surpassed by that of the second mode, which is only weakly affected by the dipole [γm < 0.01 in Fig. 4(b)]. Assuming that the second supermode is completely transmitted, that 40 % of the power in the fiber is carried by the main supermode [see Fig. 4(c)], and that the latter is extinguished by 31 % [γm = 0.22, Fig. 4(b)], the maximum achievable overall extinction would be only 12 %. Although lower than in the SiN cases studied above, such an extinction level is still quite reasonable for spectroscopy purposes, and is compatible with experimentally observed levels using an NSOM tip [12

12. I. Gerhardt, G. Wrigge, P. Bushev, G. Zumofen, M. Agio, R. Pfab, and V. Sandoghdar, “Strong Extinction of a Laser Beam by a Single Molecule,” Phys. Rev. Lett. 98, 033601 (2007). [CrossRef] [PubMed]

] and a solid immersion lens [26

26. G. Wrigge, I. Gerhardt, J. Hwang, G. Zumofen, and V. Sandoghdar, “Efficient coupling of photons to a single molecule and the observation of its resonance fluorescence,” Nat. Phys. 4, 60–66 (2008). [CrossRef]

].

6. Discussion

We point out that the cases in which the highest Purcell enhancements are observed, in which n host = 1.0, would be challenging or not achievable in practice. These situations, which otherwise provide upper bounds for the achievable enhancement, imply the absence of a host material supporting the emitter, or, in the best case, the ability to produce a 20 nm thick host layer of extremely low refractive index material, for instance an aerogel. It is also worthwhile noting that, although semiconductor nanocrystal quantum dots are composed of high refractive index materials, sufficiently small nanocrystals may not constitute a significant disturbance to the environment, that could lead to large deviations from our calculated results. Evidence of this can be found in [27

27. M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, “Spectroscopy of 1.55 fim PbS Quantum Dots on Si Photonic Crystal Cavities with a Fiber Taper Waveguide,” arXiv:0912.1365v1 (2009).

], where small (< 5 nm) PbS nanocrystal quantum dots were shown to not considerably affect the modes of a high quality factor microresonator. A similar conclusion may be drawn from the simulations involving diamond nanocrystals in nanocavities reported in [28

28. M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16, 19136–19145 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-23-19136 [CrossRef]

].

We reiterate that the results presented here are best-case estimates, obtained for vertical dipole moments aligned with the gap mode field. Horizontal dipole moments are expected to radiate at lower rates. In the case of z-dipoles, gap waves are generated at lower rates because the dipole moment is aligned with the minor supermode field component, Ez. If located at the x = 0 plane, an x-dipole is completely uncoupled from gap modes, producing, rather, anti-symmetric (ŷ × Ex=0=0) supermode waves with a major x-field component. These do not offer the same strong field concentration as gap modes, due to the continuity of the electric field across the gap.

For the channel waveguide situation just described, γm = 0.215 for the main supermode, which would lead to a single-mode extinction of 67 %.

Finally, we point out that the recently reported plasmonic laser [29

29. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef] [PubMed]

], composed of a cylindrical CdS nanowire placed on top of a Ag substrate, with a thin MgS separator [the cross section closely resembles that in Fig. 1(b)] supports gap modes similar to those studied here. While here the supermodes are hybrids of fiber and slab modes, those in [29

29. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef] [PubMed]

] are hybrids of the cylinder and surface plasmon modes. Although hybrid plasmonic waveguides offer substantially stronger Purcell enhancement for emitters located in the MgS spacer or in the nanowire, the supermodes suffer from high propagation losses associated with the surface plasmon. Indeed, supermode propagation lengths (i.e., the required length for the mode power to drop in half) quoted in [29

29. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef] [PubMed]

] are on the order of 10 wavelengths. In the present case, supermode effective lengths are two to three orders of magnitude longer, since the waveguides are purely dielectric, and lateral power leakage is small.

7. Conclusions

Acknowledgement

This work has been supported in part by the NIST-CNST/UMD-NanoCenter Cooperative Agreement.

References and links

1.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

2.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457, 71–75 (2009). [CrossRef] [PubMed]

3.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009). [CrossRef]

4.

Y. C. Jun, R. M. Briggs, H. A. Atwater, and M. L. Brongersma, “Broadband enhancement of light emission insilicon slot waveguides,” Opt. Express 17, 7479–7490 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-9-7479 [CrossRef] [PubMed]

5.

W. Moerner, “Examining nanoenvironments in solids on the scale of a single, isolated inpurity molecule,” Science 265, 46–53 (1994). [CrossRef] [PubMed]

6.

W. E. Moerner, “Single-photon sources based on single molecules in solids,” N. J. Phys. 6, 88 (2004). [CrossRef]

7.

J. Hwang, M. Pototschnig, R. Lettow, G. Zumofen, A. Renn, S. Gotzinger, and V. Sandoghdar, “A single-molecule optical transistor,” Nature 460, 76–80 (2009). [CrossRef] [PubMed]

8.

K. Srinivasan, O. Painter, A. Stintz, and S. Krishna, “Single quantum dot spectroscopy using a fiber taper waveguide near-field optic,” Appl. Phys. Lett. 91, 091102 (2007). [CrossRef]

9.

M. Davanço and K. Srinivasan, “Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides,” Opt. Express 17, 10542–10563 (2009). [CrossRef] [PubMed]

10.

M. Davanço and K. Srinivasan, “Fiber-coupled semiconductor waveguides as an efficient optical interface to a single quantum dipole,” Opt. Lett. 34, 2542–2544 (2009), http://ol.osa.org/abstract.cfm?URI=ol-34-16-2542 [CrossRef] [PubMed]

11.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).

12.

I. Gerhardt, G. Wrigge, P. Bushev, G. Zumofen, M. Agio, R. Pfab, and V. Sandoghdar, “Strong Extinction of a Laser Beam by a Single Molecule,” Phys. Rev. Lett. 98, 033601 (2007). [CrossRef] [PubMed]

13.

J. Lee, V. C. Sundar, J. R. Heine, M. G. Bawendi, and K. F. Jensen, “Full Color Emission from II-VI Semiconductor Quantum Dot-Polymer Composites,” Adv. Mater. 12, 1102–1105 (2000). [CrossRef]

14.

R. D. Schaller, M. A. Petruska, and V. Klimov, “Tunable Near-Infrared Optical Gain and Amplified Spontaneous Emission Using PbSe Nanocrystals,” J. Phys. Chem. B 107, 13765–13768 (2003). [CrossRef]

15.

A. Zumbusch, L. Fleury, R. Brown, J. Bernard, and M. Orrit, “Probing individual two-level systems in a polymer by correlation of single molecule fluorescence,” Phys. Rev. Lett. 70, 3584–3587 (1993). [CrossRef] [PubMed]

16.

M. Orrit and J. Bernard, “Single pentacene molecules detected by fluorescence excitation in a p-terphenyl crystal,” Phys. Rev. Lett. 65, 2716–2719 (1990). [CrossRef] [PubMed]

17.

G. S. Harms, T. Irngartinger, D. Reiss, A. Renn, and U. P. Wild, “Fluorescence lifetimes of terrylene in solid matrices,” Chem. Phys. Lett. 313, 533–538 (1999). [CrossRef]

18.

T. Bottger, C. W. Thiel, Y. Sun, and R. L. Cone, “Optical decoherence and spectral diffusion at 1.5 μ in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration,” Phys. Rev. B: Condens. Matter Mater. Phys. 73, 075101 (2006). [CrossRef]

19.

W.-P Huang, “Coupled-mode theory for optical waveguides: and overview,” J. Opt. Soc. Am. A 11, 963–983 (1994). [CrossRef]

20.

Following Ref [19], the fiber mode fraction, Eq. (2), would be given by the expression fm = 〈fm〉〈mf〉 (〈ff〉〈mm〉)-1, where 〈fm〉 = ∣ (ef × hm* + hf × em*ẑdS/4. Considering no reflections at the interface between the isolated fiber and the contact region, (i.e., the field just after the interface is identical to the incident, foward propagating, field), Eq. (2) gives the same result.

21.

S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813 (2004). [CrossRef]

22.

F. Wise, “Lead salt quantum dots: The limit of strong quantum confinement,” Acc. Chem. Res. 33, 773–780 (2000). [CrossRef] [PubMed]

23.

R. J. Pfab, J. Zimmermann, C. Hettich, I. Gerhardt, A. Renn, and V. Sandoghdar, “Aligned terrylene molecules in a spin-coated ultrathin crystalline film of p-terphenyl,” Chem. Phys. Lett. 387, 490–495 (2004). [CrossRef]

24.

V. S. C. M. Rao and S. Hughes, “Single quantum-dot Purcell factor and beta factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007). [CrossRef]

25.

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very Large Spontaneous-Emission beta Factors in Photonic-Crystal Waveguides,” Phys. Rev. Lett. 99 (2007). [CrossRef] [PubMed]

26.

G. Wrigge, I. Gerhardt, J. Hwang, G. Zumofen, and V. Sandoghdar, “Efficient coupling of photons to a single molecule and the observation of its resonance fluorescence,” Nat. Phys. 4, 60–66 (2008). [CrossRef]

27.

M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, “Spectroscopy of 1.55 fim PbS Quantum Dots on Si Photonic Crystal Cavities with a Fiber Taper Waveguide,” arXiv:0912.1365v1 (2009).

28.

M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16, 19136–19145 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-23-19136 [CrossRef]

29.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef] [PubMed]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Optical Devices

History
Original Manuscript: February 19, 2010
Revised Manuscript: May 4, 2010
Manuscript Accepted: May 5, 2010
Published: May 11, 2010

Citation
Marcelo Davanco and Kartik Srinivasan, "Hybrid gap modes induced by fiber taper waveguides: Application in spectroscopy of single solid-state emitters deposited on thin films," Opt. Express 18, 10995-11007 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-10995


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References

  1. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and confining light in void nanostructure," Opt. Lett. 29, 1209-1211 (2004). [CrossRef] [PubMed]
  2. A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, "Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides," Nature 457, 71-75 (2009). [CrossRef] [PubMed]
  3. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, "All-optical high-speed signal processing with silicon-organic hybrid slot waveguides," Nat. Photon. 3, 216-219 (2009). [CrossRef]
  4. Y. C. Jun, R. M. Briggs, H. A. Atwater, and M. L. Brongersma, "Broadband enhancement of light emission insilicon slot waveguides," Opt. Express 17, 7479-7490 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-9-7479 [CrossRef] [PubMed]
  5. W. Moerner, "Examining nanoenvironments in solids on the scale of a single, isolated inpurity molecule," Science 265, 46-53 (1994). [CrossRef] [PubMed]
  6. W. E. Moerner, "Single-photon sources based on single molecules in solids," N. J. Phys. 6, 88 (2004). [CrossRef]
  7. J. Hwang, M. Pototschnig, R. Lettow, G. Zumofen, A. Renn, S. Gotzinger, and V. Sandoghdar, "A singlemolecule optical transistor," Nature 460, 76-80 (2009). [CrossRef] [PubMed]
  8. K. Srinivasan, O. Painter, A. Stintz, and S. Krishna, "Single quantum dot spectroscopy using a fiber taper waveguide near-field optic," Appl. Phys. Lett. 91, 091102 (2007). [CrossRef]
  9. M. Davanc¸o and K. Srinivasan, "Efficient spectroscopy of single embedded emitters using optical fiber taper waveguides," Opt. Express 17, 10542-10563 (2009). [CrossRef] [PubMed]
  10. M. Davanc¸o and K. Srinivasan, "Fiber-coupled semiconductor waveguides as an efficient optical interface to a single quantum dipole," Opt. Lett. 34, 2542-2544 (2009), http://ol.osa.org/abstract.cfm?URI=ol-34-16-2542 [CrossRef] [PubMed]
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).
  12. I. Gerhardt, G. Wrigge, P. Bushev, G. Zumofen, M. Agio, R. Pfab, and V. Sandoghdar, "Strong Extinction of a Laser Beam by a Single Molecule," Phys. Rev. Lett. 98, 033601 (2007). [CrossRef] [PubMed]
  13. J. Lee, V. C. Sundar, J. R. Heine, M. G. Bawendi, and K. F. Jensen, "Full Color Emission from II-VI Semiconductor Quantum Dot-Polymer Composites," Adv. Mater. 12, 1102-1105 (2000). [CrossRef]
  14. R. D. Schaller, M. A. Petruska, and V. Klimov, "Tunable Near-Infrared Optical Gain and Amplified Spontaneous Emission Using PbSe Nanocrystals," J. Phys. Chem. B 107, 13765-13768 (2003). [CrossRef]
  15. A. Zumbusch, L. Fleury, R. Brown, J. Bernard, and M. Orrit, "Probing individual two-level systems in a polymer by correlation of single molecule fluorescence," Phys. Rev. Lett. 70, 3584-3587 (1993). [CrossRef] [PubMed]
  16. M. Orrit and J. Bernard, "Single pentacene molecules detected by fluorescence excitation in a p-terphenyl crystal," Phys. Rev. Lett. 65, 2716-2719 (1990). [CrossRef] [PubMed]
  17. G. S. Harms, T. Irngartinger, D. Reiss, A. Renn, and U. P. Wild, "Fluorescence lifetimes of terrylene in solid matrices," Chem. Phys. Lett. 313, 533-538 (1999). [CrossRef]
  18. T. Bottger, C. W. Thiel, Y. Sun, and R. L. Cone, "Optical decoherence and spectral diffusion at 1.5 μ in Er3+:Y2SiO5 versus magnetic field, temperature, and Er3+ concentration," Phys. Rev. B: Condens. Matter Mater. Phys. 73, 075101 (2006). [CrossRef]
  19. W.-P. Huang, "Coupled-mode theory for optical waveguides: and overview," J. Opt. Soc. Am. A 11, 963-983 (1994). [CrossRef]
  20. Following Ref. [19], the fiber mode fraction, Eq. (2), would be given by the expression fm = _ f |m__m| f _ (_ f | f __m|m_)-1, where _ f |m_ =∬ ∫ (ef ×h*m +hf×e* mm_ · ˆzdS/4. Considering no reflections at the interface between the isolated fiber and the contact region, (i.e., the field just after the interface is identical to the incident, foward propagating, field), Eq. (2) gives the same result.
  21. S. J. van Enk, "Atoms, dipole waves, and strongly focused light beams," Phys. Rev. A 69, 043813 (2004). [CrossRef]
  22. F. Wise, "Lead salt quantum dots: The limit of strong quantum confinement," Acc. Chem. Res. 33, 773-780 (2000). [CrossRef] [PubMed]
  23. R. J. Pfab, J. Zimmermann, C. Hettich, I. Gerhardt, A. Renn, and V. Sandoghdar, "Aligned terrylene molecules in a spin-coated ultrathin crystalline film of p-terphenyl," Chem. Phys. Lett. 387, 490-495 (2004). [CrossRef]
  24. V. S. C. M. Rao and S. Hughes, "Single quantum-dot Purcell factor and beta factor in a photonic crystal waveguide," Phys. Rev. B 75, 205437 (2007). [CrossRef]
  25. Q4. G. Lecamp, P. Lalanne, and J. P. Hugonin, "Very Large Spontaneous-Emission beta Factors in Photonic-Crystal Waveguides," Phys. Rev. Lett. 99 (2007). [CrossRef] [PubMed]
  26. G. Wrigge, I. Gerhardt, J. Hwang, G. Zumofen, and V. Sandoghdar, "Efficient coupling of photons to a single molecule and the observation of its resonance fluorescence," Nat. Phys. 4, 60-66 (2008). [CrossRef]
  27. M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, "Spectroscopy of 1.55 m PbS Quantum Dots on Si Photonic Crystal Cavities with a Fiber Taper Waveguide," arXiv:0912.1365v1 (2009).
  28. M.W. McCutcheon and M. Loncar, "Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal," Opt. Express 16, 19136-19145 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-23-19136 [CrossRef]
  29. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, "Plasmon lasers at deep subwavelength scale," Nature 461, 629-632 (2009). [CrossRef] [PubMed]

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