Abstract

Based on the reciprocity theorem, we present a formalism to calculate the power emitted by a dipole source into a particular propagating mode leaving an open optical system. The open system is completely arbitrary and the approach can be used in analytical calculations but may also be combined with numerical electromagnetic solvers to describe the emission of light sources into complex systems. We exemplify the use of the formalism in numerical simulations by analyzing the emission of a dipole that is placed inside a cavity with connected single mode exit waveguide. Additionally, we show at the example of a practical ring resonator system how the approach can be applied to systems that offer multiple electromagnetic energy decay channels. As a consequence of its inherent simplicity and broad applicability, the approach may serve as a powerful and practical tool for engineering light-matter-interaction in a variety of active optical systems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]

2017 (2)

R. S. Daveau, K. C. Balram, T. Pregnolato, J. Liu, E. H. Lee, J. D. Song, V. Verma, R. Mirin, S. W. Nam, L. Midolo, S. Stobbe, K. Srinivasan, and P. Lodahl, “Efficient fiber-coupled single-photon source based on quantum dots in a photonic-crystal waveguide,” Optica 4(2), 178–184 (2017).
[Crossref] [PubMed]

M. Davanco, J. Liu, L. Sapienza, C.-Z. Zhang, J. V. De Miranda Cardoso, V. Verma, R. Mirin, S. W. Nam, L. Liu, and K. Srinivasan, “Heterogeneous integration for on-chip quantum photonic circuits with single quantum dot devices,” Nat. Commun. 8(1), 889 (2017).
[Crossref] [PubMed]

2016 (7)

K. Liu, N. Li, D. K. Sadana, and V. J. Sorger, “Integrated Nanocavity Plasmon Light Sources for On-Chip Optical Interconnects,” ACS Photonics 3(2), 233–242 (2016).
[Crossref]

P. Lunnemann and A. F. Koenderink, “The local density of optical states of a metasurface,” Sci. Rep. 6(1), 20655 (2016).
[Crossref] [PubMed]

E. A. Muljarov and W. Langbein, “Exact mode volume and Purcell factor of open optical systems,” Phys. Rev. B 94(23), 235438 (2016).
[Crossref]

K. M. Schulz, H. Vu, S. Schwaiger, A. Rottler, T. Korn, D. Sonnenberg, T. Kipp, and S. Mendach, “Controlling the Spontaneous Emission Rate of Quantum Wells in Rolled-Up Hyperbolic Metamaterials,” Phys. Rev. Lett. 117(8), 085503 (2016).
[Crossref] [PubMed]

H. M. Doeleman, E. Verhagen, and A. F. Koenderink, “Antenna–Cavity Hybrids: Matching Polar Opposites for Purcell Enhancements at Any Linewidth,” ACS Photonics 3(10), 1943–1951 (2016).
[Crossref]

S. Mahmoodian, P. Lodahl, and A. S. Sørensen, “Quantum Networks with Chiral-Light-Matter Interaction in Waveguides,” Phys. Rev. Lett. 117(24), 240501 (2016).
[Crossref] [PubMed]

T. Malhotra, R.-C. Ge, M. Kamandar Dezfouli, A. Badolato, N. Vamivakas, and S. Hughes, “Quasinormal mode theory and design of on-chip single photon emitters in photonic crystal coupled-cavity waveguides,” Opt. Express 24(12), 13574–13583 (2016).
[Crossref] [PubMed]

2015 (3)

P. Kolchin, N. Pholchai, M. H. Mikkelsen, J. Oh, S. Ota, M. S. Islam, X. Yin, and X. Zhang, “High Purcell Factor Due to Coupling of a Single Emitter to a Dielectric Slot Waveguide,” Nano Lett. 15(1), 464–468 (2015).
[Crossref] [PubMed]

P. T. Kristensen, R.-C. Ge, and S. Hughes, “Normalization of quasinormal modes in leaky optical cavities and plasmonic resonators,” Phys. Rev. A 92(5), 053810 (2015).
[Crossref]

A. E. Krasnok, A. P. Slobozhanyuk, C. R. Simovski, S. A. Tretyakov, A. N. Poddubny, A. E. Miroshnichenko, Y. S. Kivshar, and P. A. Belov, “An antenna model for the Purcell effect,” Sci. Rep. 5(1), 12956 (2015).
[Crossref] [PubMed]

2014 (7)

L. Ferrari, D. Lu, D. Lepage, and Z. Liu, “Enhanced spontaneous emission inside hyperbolic metamaterials,” Opt. Express 22(4), 4301–4306 (2014).
[Crossref] [PubMed]

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

P. Then, G. Razinskas, T. Feichtner, P. Haas, A. Wild, N. Bellini, R. Osellame, G. Cerullo, and B. Hecht, “Remote detection of single emitters via optical waveguides,” Phys. Rev. A 89(5), 053801 (2014).
[Crossref]

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-Unity Coupling Efficiency of a Quantum Emitter to a Photonic Crystal Waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

A. Rose, T. B. Hoang, F. McGuire, J. J. Mock, C. Ciracì, D. R. Smith, and M. H. Mikkelsen, “Control of Radiative Processes Using Tunable Plasmonic Nanopatch Antennas,” Nano Lett. 14(8), 4797–4802 (2014).
[Crossref] [PubMed]

P. T. Kristensen, J. R. de Lasson, and N. Gregersen, “Calculation, normalization, and perturbation of quasinormal modes in coupled cavity-waveguide systems,” Opt. Lett. 39(22), 6359–6362 (2014).
[Crossref] [PubMed]

2013 (2)

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the Spontaneous Optical Emission of Nanosize Photonic and Plasmon Resonators,” Phys. Rev. Lett. 110(23), 237401 (2013).
[Crossref] [PubMed]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is — and what is not — an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

2012 (1)

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

2011 (1)

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2(1), 539 (2011).
[Crossref] [PubMed]

2010 (2)

A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett. 35(24), 4208–4210 (2010).
[Crossref] [PubMed]

J.-J. Greffet, M. Laroche, and F. Marquier, “Impedance of a Nanoantenna and a Single Quantum Emitter,” Phys. Rev. Lett. 105(11), 117701 (2010).
[Crossref] [PubMed]

2009 (1)

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(7264), 629–632 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (3)

A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1(4), 215–223 (2007).
[Crossref]

O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Strong Enhancement of the Radiative Decay Rate of Emitters by Single Plasmonic Nanoantennas,” Nano Lett. 7(9), 2871–2875 (2007).
[Crossref] [PubMed]

V. S. Rao and S. Hughes, “Single Quantum Dot Spontaneous Emission in a Finite-Size Photonic Crystal Waveguide: Proposal for an Efficient “On Chip” Single Photon Gun,” Phys. Rev. Lett. 99(19), 193901 (2007).
[Crossref] [PubMed]

2006 (1)

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and Quenching of Single-Molecule Fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[Crossref] [PubMed]

2004 (3)

S. Hughes, “Enhanced single-photon emission from quantum dots in photonic crystal waveguides and nanocavities,” Opt. Lett. 29(22), 2659–2661 (2004).
[Crossref] [PubMed]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432(7014), 197–200 (2004).
[Crossref] [PubMed]

2003 (1)

J. R. Lakowicz, J. Malicka, I. Gryczynski, Z. Gryczynski, and C. D. Geddes, “Radiative decay engineering: the role of photonic mode density in biotechnology,” J. Phys. D Appl. Phys. 36(14), R240–R249 (2003).
[Crossref] [PubMed]

1998 (1)

W. L. Barnes, “Fluorescence near interfaces: The role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

1932 (1)

E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. 4(1), 87–132 (1932).
[Crossref]

Akselrod, G. M.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and Quenching of Single-Molecule Fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[Crossref] [PubMed]

Arcari, M.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-Unity Coupling Efficiency of a Quantum Emitter to a Photonic Crystal Waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

Argyropoulos, C.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Badolato, A.

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is — and what is not — an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Balram, K. C.

Barnes, W. L.

W. L. Barnes, “Fluorescence near interfaces: The role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).
[Crossref]

Bartal, G.

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(7264), 629–632 (2009).
[Crossref] [PubMed]

Bellini, N.

P. Then, G. Razinskas, T. Feichtner, P. Haas, A. Wild, N. Bellini, R. Osellame, G. Cerullo, and B. Hecht, “Remote detection of single emitters via optical waveguides,” Phys. Rev. A 89(5), 053801 (2014).
[Crossref]

Belov, P. A.

A. E. Krasnok, A. P. Slobozhanyuk, C. R. Simovski, S. A. Tretyakov, A. N. Poddubny, A. E. Miroshnichenko, Y. S. Kivshar, and P. A. Belov, “An antenna model for the Purcell effect,” Sci. Rep. 5(1), 12956 (2015).
[Crossref] [PubMed]

Bermel, P.

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and Quenching of Single-Molecule Fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[Crossref] [PubMed]

Bienstman, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Blais, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Bogaerts, W.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

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K. Liu, N. Li, D. K. Sadana, and V. J. Sorger, “Integrated Nanocavity Plasmon Light Sources for On-Chip Optical Interconnects,” ACS Photonics 3(2), 233–242 (2016).
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Verhagen, E.

H. M. Doeleman, E. Verhagen, and A. F. Koenderink, “Antenna–Cavity Hybrids: Matching Polar Opposites for Purcell Enhancements at Any Linewidth,” ACS Photonics 3(10), 1943–1951 (2016).
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Figures (3)

Fig. 1
Fig. 1 (a) An oscillating current density distribution J 1 emits radiation into an arbitrary optical system resulting in a field distribution given by E 1 , H 1 . In the shown case, the excitation energy can leave the system through the exit port1 that has a cross section z 1 and other ports summarized by one port denoted as portn with a cross section z n . At the cross section z 1 the fields E 1 and H 1 are expanded into an orthogonal set of modes. The power leaving the system through the cross section z 1 that is carried by a particular mode M k shall be calculated. (b) back excitation of the mode of interest M k into the system with arbitrary amplitude B k . The reciprocal mode has an opposite propagation direction and is denoted as M k . Excitation of the mode results in a field distribution in the system given by E 2 , H 2 . The reciprocity approach enables us to calculate the power of a source emitted into mode M k from the reciprocally excited field distribution E 2 , H 2 .
Fig. 2
Fig. 2 (a) Investigated example system. A narrow single- mode waveguide is attached to a cubic cavity. Both, waveguide and cavity have walls consisting of perfect electric conductor (i.e. loss less metal). (b) Spectrum of the cavity coupled to the waveguide. The cavity Q-factor is 1400 and the resonance frequency is 707 THz. (c) Electric field distribution E z 2 (r) after excitation of the system with the waveguide mode. The waveguide mode is launched into the system from the left hand side as indicated by the arrow. (d) Purcell factor for a dipole placed at different positions inside the cavity along an exemplary cutting line indicated by the dashed purple line in (c). Solid line: Purcell factor as obtained from the reciprocity approach. Red dots: Purcell factor as obtained from emission of a modelled dipole placed at discrete positions along the cutting line.
Fig. 3
Fig. 3 (a) Integrated dielectric ring resonator coupled to an adjacent straight dielectric waveguide with geometric parameters defined in the text. The shown field distribution in the ring and the waveguide originates from excitation of the TE mode in the straight waveguide. (b) Electric field spectrum and Q-factor of the resonator. (c) TE mode Purcell factor evaluated for the cross section of the ring indicated in (a). The TE mode Purcell factor shows the power of an emitter at any position of the system that is radiated into the propagating TE mode of the straight waveguide. It is normalized to the power of the same emitter radiated into vacuum free space. The calculation is based on the reciprocal mode excitation shown in (a) and Eq. (17). (d) Field distribution from excitation of the TE-mode for dielectric ring resonator with adjacent plasmonic nanoantenna. (e) Electric field spectrum and Q factor of the resonator with adjacent plasmonic nanoantenna. (f) TE mode Purcell factor evaluated at the cross section indicated in (d). (g) TE mode Purcell factor along the line cut at the hot spot between the waveguide and the nanoparticle indicated in (f) (green dashed line).

Equations (18)

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F p = P system P 0 ,
( E 1 ( x,y, z 1 ) H 1 ( x,y, z 1 ) )= i A i,z1 ( e i,z1 ( x,y ) e j β i z 1 h i,z1 ( x,y ) e j β i z 1 )
( E 1 ( x,y, z n ) H 1 ( x,y, z n ) )= i A i,zn ( e i,zn ( x,y ) e +j β i z n h i,zn ( x,y ) e +j β i z n )
( E 2 ( x,y, z 1 ) H 2 ( x,y, z 1 ) )= B k,z1 ( e k,z1 ( x,y ) e +j β k z 1 h k,z1 ( x,y ) e +j β k z 1 )+ i B i,z1 ( e i,z1 ( x,y ) e j β i z 1 h i,z1 ( x,y ) e j β i z 1 )
( E 2 ( x,y, z n ) H 2 ( x,y, z n ) )= i B i,zn ( e i,zn ( x,y ) e +j β i z n h i,zn ( x,y ) e +j β i z n )
E 1 × H 2 E 2 × H 1 dA= V E 2 J 1 dV .
z= z 1 ( E 1 × H 2 E 2 × H 1 ) n z dxdy z= z n ( E 1 × H 2 E 2 × H 1 ) n z dxdy= V E 2 J 1 dV
e i = e i * and h i = h i *
2 A k,z1 B k,z1 z= z 1 ( e (x,y) k,z1 ×h (x,y) k,z1 * ) n z dxdy= V E 2 J 1 dV,
4 A k,z1 B k,z1 = V E 2 J 1 dV,
E 2 (r)= α k ( r ) B k,z1 ,
4 A k,z1 B k,z1 = V B k,z1 α k ( r ) J 1 dV.
P k,z1 = 1 16 | V α k ( r ) J 1 dV | 2 .
P k,z1 = 1 16 ω 2 | α k ( r 0 )p | 2 .
P k,z1 = 1 16 ω 2 p 2 | α k ( r 0 ) | 2 .
F k ( r 0 )= P ( r 0 ) k,z1 P 0 = 3πc 4 μ 0 ω 2 | α k ( r 0 ) | 2 ,
P 0 = μ 0 12πc ω 4 p 2 ,
P k,z1 = 1 16 | V α( r )ϱ(r) dV | 2 ,

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