Abstract

A technique based on using optical fiber taper waveguides for probing single emitters embedded in thin dielectric membranes is assessed through numerical simulations. For an appropriate membrane geometry, photoluminescence collection efficiencies in excess of 10 % are predicted, exceeding the efficiency of standard free-space collection by an order of magnitude. Our results indicate that these fiber taper waveguides offer excellent prospects for performing efficient spectroscopy of single emitters embedded in thin films, such as a single self-assembled quantum dot in a semiconductor membrane.

© 2009 Optical Society of America

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2008 (1)

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]

2007 (4)

K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdiskquantum dot system," Nature 450, 862-865 (2007).
[CrossRef] [PubMed]

K. Nayak, P. Melentiev, M. Morinaga, F. Le Kien, V. Balykin, and K. Hakuta, "Optical nanofiber as an efficient tool for manipulating and probing atomic fluorescences," Opt. Express 15, 5431-5438 (2007).
[CrossRef] [PubMed]

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

C. P. Michael, M. Borselli, T. J. Johnson, and O. Painter, "An optical fiber taper probe for wafer-scale microphotonic device characterization," Opt. Express 15, 4745-4752 (2007).
[CrossRef] [PubMed]

2006 (1)

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

2004 (3)

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

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]

1999 (1)

1998 (2)

1996 (1)

H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

1994 (1)

1985 (1)

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

Akiyama, T.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Almeida, V. R.

Aoki, T.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Arakawa, Y.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Atature, M.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

Badolato, A.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

Balykin, V.

Barrios, C. A.

Benisty, H.

Borselli, M.

Bowen, W. P.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Collett, M. J.

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

Dayan, B.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Dreiser, J.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

Ebe, H.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

Gerhardt, I.

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]

Goldberg, B. B.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

Hakuta, K.

Hu, E.

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

Huang, W.-P.

Hwang, J.

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]

Imamoglu, A.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

Jayavel, P.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Johnson, T. J.

Kimble, H. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Kippenberg, T. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Kita, T.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Le Kien, F.

Lipson, M.

Mayer, M.

Melentiev, P.

Michael, C. P.

Monneret, S.

H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

Morinaga, M.

Nakat, Y.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Nayak, K.

Painter, O.

K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdiskquantum dot system," Nature 450, 862-865 (2007).
[CrossRef] [PubMed]

C. P. Michael, M. Borselli, T. J. Johnson, and O. Painter, "An optical fiber taper probe for wafer-scale microphotonic device characterization," Opt. Express 15, 4745-4752 (2007).
[CrossRef] [PubMed]

Parkins, A. S.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Petroff, P. M.

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

Rigneault, H.

H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

Rikken, G. L. J. A.

H. P. Urbach and G. L. J. A. Rikken, "Spontaneous emission from a dielectric slab," Phys. Rev. A 57, 3913-3930 (1998).
[CrossRef]

Sandoghdar, V.

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]

Srinivasan, K.

K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdiskquantum dot system," Nature 450, 862-865 (2007).
[CrossRef] [PubMed]

Stanley, R.

Sugawara, M.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Swan, A. K.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

Tanaka, H.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Tatebayashi, J.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Unlu, M. S.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

Urayama, J.

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

Urbach, H. P.

H. P. Urbach and G. L. J. A. Rikken, "Spontaneous emission from a dielectric slab," Phys. Rev. A 57, 3913-3930 (1998).
[CrossRef]

Vahala, K. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Vamivakas, A. N.

A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

Wada, O.

P. Jayavel, H. Tanaka, T. Kita, O. Wada, H. Ebe, M. Sugawara, J. Tatebayashi, Y. Arakawa, Y. Nakat, and T. Akiyama, "Control of optical polarization anisotropy in edge emitting luminescence of InAs/GaAs selfassemble quantum dots," Appl. Phys. Lett. 84 (2004).
[CrossRef]

Wang, C. F.

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

Wilcut, E.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J. Vahala, "Observation of Strong Coupling between One Atom and a Monolithic Microresonator," Nature 443, 671-674 (2006).
[CrossRef] [PubMed]

Wilson-Rae, I.

C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
[CrossRef]

Wrigge, G.

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Zumofen, G.

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).
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C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, "Optical properties of single InAs quantum dots in close proximity to surfaces," Appl. Phys. Lett. 85, 3423-3425 (2004).
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A. N. Vamivakas, M. Atature, J. Dreiser, S. T. Yilmaz, A. Badolato, A. K. Swan, B. B. Goldberg, A. Imamoglu, and M. S. Unlu, "Strong Extinction of a Far-Field Laser Beam by a Single Quantum Dot," Nano Lett. 7, 2892-2896 (2007).
[CrossRef] [PubMed]

Nat. Phys. (1)

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]

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[CrossRef]

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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, 091 102 (2007).
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A. Muller, E. B. Flagg, P. Bianucci, X. Wang, D. G. Deppe, W. Ma, J. Zhang, G. J. Salamo, M. Xiao, and C. K. Shih, "Resonance Fluorescence from a Coherently Driven Semiconductor Quantum Dot in a Cavity," Phys. Rev. Lett. 99, 187 402 (2007).
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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, 032 509 (2005).

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M. Davanco and K. Srinivasan, "Fiber-coupled semiconductor waveguides as an efficient optical interface to a single quantum dipole," preprint: arxiv.org/abs/0905.2994 (2009).

B. Gerardot, S. Seidl, P. Dalgarno, R. Warburton, M. Kroner, K. Karrai, A. Badolato, and P. Petroff, "Contrast in transmission spectroscopy of a single quantum dot," Appl. Phys. Lett. 90, 221 106 (2007).
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T. Søndergaard and B. Tromborg, "General theory for spontaneous emission in active dielectric microstructures: Example of a fiber amplifier," Phys. Rev. A 64, 033 812 (2001).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

(a) Single emitter probing setup based on a tapered fiber waveguide. (b) Detail of (a), showing the membrane that carries the emitter. (c) Schematic of the substrate cross-section, showing membrane and fiber.

Fig. 2.
Fig. 2.

(a) Envisioned experimental configuration for fiber-based non-resonant photoluminescence (PL) spectroscopy with a tapered fiber waveguide. Emitted light is coupled into both forward and backward channels of the fiber taper waveguide, and can be wavelength resolved with a grating spectrometer or spectral filter. (b) 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 Γ0 and Γν, respectively. Power is transferred with efficiency f0 from the supermode to the fiber mode and vice-versa.

Fig. 3.
Fig. 3.

(a) Total radiated power at λ=1.3µm for a dipole in the composite slab-fiber waveguide, normalized to the radiated power in a homogeneous medium of index n slab. (b) Guided and total radiated powers into TE (TM) waves for a horizontal (vertical and horizontal) dipole at the center of a dielectric slab with n slab=3.406.

Fig. 4.
Fig. 4.

(a) Effective indices of the slab TE0, TE1 and TM0 modes at λ=1.3µm. The HE11 fiber mode effective index is also shown. Inset: field components of TE0, TE1 and TM0 slab modes. (b) Effective lengths, Leff, for the TE0 and TM0 slab modes.

Fig. 5.
Fig. 5.

Fiber-collected PL coupling efficiency η PL for (a) x-, (b) y- and (c) z- oriented dipoles at λ=1.3µm. Collection through both the forward and backward fiber channels is considered (notice changing axis scales). Since η PL oscillates with z, its maximum and minimum along z<5µm are plotted. The inset in (a) shows η PL as function of z. Squares: FDTD results. Circles: Mode-expansion results. Solid lines: Free-space collection efficiency considering a NA=0.7 objective.

Fig. 6.
Fig. 6.

Electric field distribution for (a), (b): hybrid-TE0 types-I (slab-like) and II (fiber-like) supermodes; (d), (e): hybrid-TM0 types-I and II supermodes. Fields are for tslab=100nm and λ=1.3µm. (c) and (f): effective indices for (c) hybrid-TE0 and (f) hybrid-TM0 supermodes of types I and II for varying t slab. Fields have been normalized to the total field amplitude maxima in each case. Color scales are in arbitrary units.

Fig. 7.
Fig. 7.

(a) Effective indices for hybrid-TE supermodes at t slab=190nm and λ=1.3µm. The two supermodes with highest modal emission rates are indicated with filled circles labeled I and II, corresponding to slab-like and fiber-like modes. Open circles indicate leaky modes. (b) Fiber fractions for supermodes I and II. (c) PL collection contributions from the individual supermodes, for an x-oriented dipole moment. The factor of 2 is to account for collection from both fiber ends. Black dots are for leaky modes.(d) Modal emission ratios for supermodes I and II. Filled squares/empty circles are for type I/II supermodes.

Fig. 8.
Fig. 8.

(a) Effective indices for hybrid-TM supermodes at t slab=190nm and λ=1.3µm. The two supermodes with highest modal emission rates are indicated with filled circles labeled I and II. Open circles indicate leaky modes. (b) Fiber fractions for supermodes I and II. (c) and (e): PL collection contributions from individual supermodes, for y- and z-oriented dipole moments respectively (notice changing axis scales). The factor of 2 is to account for collection from both fiber ends. (d) and (f): Modal emission ratios, for y- and z-oriented dipole moments of type-I and type-II supermodes (notice changing axis scales). Filled squares/empty circles are for type I/II supermodes; black dots are for leaky modes.

Fig. 9.
Fig. 9.

(a) Total PL collection efficiencies (maximum and minimum values) for 300 nm and 500 nm radius fibers and an x-oriented dipole moment at λ=1.3µm. (b) PL collection contributions from the individual supermodes of the 300 nm radius fiber. The factor of 2 is to account for collection from both fiber ends. Filled squares/empty circles are for type I/II supermodes; black dots are for leaky modes. (c) Fiber fractions for type-I (filled squares) and type-II (empty circles) supermodes of the 300-nm radius fiber.

Fig. 10.
Fig. 10.

(a) Emitted power and modal spontaneous emission rate at λ=1.56µm, normalized to the corresponding quantities in a homogeneous medium, for a (a) radially and (b) longitudinally oriented dipole at position r in the fiber described in the text. (c) Emitted guided mode power, Pg , and spontaneous emission rate, Γg, normalized to the total emitted power and emission rates, at λ=1.56µm for a radially oriented dipole at position r in the fiber described in the text.

Equations (37)

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ηPL=2 PzPTot. ffiber ,
ffiber=Re{S(ef×h*)·ẑdSS(e×hf*)·ẑdS}Re{S(ef×hf*)·ẑdS}Re{S(e×h*)·ẑdS} .
PHom.=μ04πnslabp2 ω043c .
×(1ε(r)×H)(ωc)2=0 .
Gμ=ωβμ4πε0h̄[peμ(r0)]andGv=ω4πNvh̄[pev(r0)].
SdS(ev×hv*)·ẑ=Nvδ (ββ).
S d S ε (r)eμ2=1
S d S ε (r)(ev·ev*)β=β,n=n=δ(ωω)
Γ=ΣμΓμ+ΣndβΓv.
ΓWGΓHom.=PWGPHom.
ηPL=2Σm=1MfmΓmΓ+2 Σm,n=1mnM ΓmΓ ΓnΓ Re {fmhfneexp[i(ξmξn*)z]} ,
fmh=S(ef×hm*)ẑdSRe{S(em×hm*)ẑdS}Re{S(ef×hf*)ẑdS},
fme=S(em×hf*)ẑdSRe{S(em×hm*)ẑdS}Re{S(ef×hf*)ẑdS},
{EH}=Σj[aj(z)eiβjz+aj(z)eiβjz]{ejhj}+
Σj d β [aj(z,β)eiβz+aj(z,β)eiβz] {ej(β)hj(β)}
a±j(z)={0iω0pej*4Njforz0forz0anda±j(z,β)={0iω0pej*(β)4Nj(β)forz0forz0.
S(ej×hk*).ẑ=Njδj,k,
S[ej(β)×hk*(β)]·ẑ=Nj(β)δ(ββ)δj,k.
Sz=p2ω022Σj{p̂ej2Nj+dβp̂ej(β)2Nj(β)}r=0
βj=ω0μ0SdSε(r)(ej×hj*)·ẑSdSε(r)ej2=SdSε(r)(ej×hj*)·ẑ
vg,j=(dβjdω0)1=cβj(cω0)NjSdSε(r)(ej×hj*)·ẑ
Sz=Σj{6πc3βjnω02p̂ej2+dβ6πcμ0nω02p̂ej(β)2Nj(β)}r=0.
ΓHom.= ω03deg2n3πh̄ε0c3 .
Eguided(+)=iΣf,m0dωh̄ωβμ4πε0aμeμei(ωtfβμz)
Eradiative(+)=iΣn0 d ω d β h̄ω4πNv av ev ei(ωtβz)
SdS(ev×hv*)ẑ=Nvδ(ββ).
H=HA+HR+HIR.
HIR=ih̄{Σf,m0dω[Gμ*aμσei(ωω0)tGμσ+aμei(ωω0)t]}
+i h̄ {Σn0dωdβ[Gv*avσei(ωω0)tGvσ+avei(ωω0)t]} ,
Gμ=ωβμ4πε0h̄(degeμ)andGv=ω4πNvh̄(degev).
aμ,v=aμ,v(t0)+Gμ,v*t0tdtσei(ωω0)t.
dσ˜dt=iω0σ˜Γ2σ˜+σz(Σf,mΓμgμ,in(t)+ΣndβΓvrv,in(t)),
dσzdt=Γ (1+σz) 2σ˜+ (Σf,mΓμgμ,in(t)+ΣnΓvrv,in(t))
2σ˜(Σf,mΓμ*gμ,in(t)+ΣndβΓv*rv,in(t)),
Γ=(Σf,mΓμ+ΣndβΓv),
Γμ=2πGμ(ω0)andΓv=2πGv(ω0).
{gμ,in(t)rv,in(t)}=12+dω{a˜μ(t0)a˜v(t0)}eiω(tt0)

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