Oct 1, 2003 - Received April 29, 2003. We report on theoretical studies of the inhibition of the spontaneous emission process in subwavelength di-.
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OPTICS LETTERS / Vol. 28, No. 19 / October 1, 2003
Spontaneous emission in nanoscopic dielectric particles Lavinia Rogobete, Hannes Schniepp, and Vahid Sandoghdar Laboratorium für Physikalische Chemie, Eidgenössische Technische Hochschule (ETH), 8093 Zürich, Switzerland
Carsten Henkel Institut für Physik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany Received April 29, 2003 We report on theoretical studies of the inhibition of the spontaneous emission process in subwavelength dielectric media. We discuss the modification of the spontaneous emission rate as a function of the size and shape of the medium as well as the position of the emitter in it. © 2003 Optical Society of America OCIS codes: 270.5580, 290.4020.
Many experimental and theoretical efforts have addressed the modification of the spontaneous emission process since Purcell’s seminal proposal.1 The pioneering experimental work dates back to 1970 when Drexhage2 showed that the f luorescence lifetimes of emitters placed close to a f lat mirror were different from those in free space. In the 1980s and 1990s several other groups demonstrated the possibility of controlling radiative decay rates by putting emitters in conf ined geometries such as the spaces between two f lat substrates, between the mirrors of high-f inesse cavities, and in whispering-gallery mode resonators.3,4 In all these cases the dimensions of the geometry surrounding the emitters were superior to the transition wavelength of interest. However, several theoretical reports also predicted that the emission rate of an atom is changed inside and outside nanoscopic spheroids,5 – 8 in the near field of sharp tips,9 or close to substrates with lateral nanostructures.10 – 12 In fact, it was recently possible to show the change of the radiative decay rate in dielectric spheres as one crossed the border from the superwavelength regime of Mie resonances to the subwavelength realm of Rayleigh scattering.13 In this Letter we present two-dimensional calculations that extend such studies to examining the role of the shape of the dielectric medium. In 1988 Chew7 reported analytical calculations based on the Mie theory for the emission rate g of a dipole placed at a given location in a dielectric sphere of an arbitrary size. For superwavelength spheres he obtained oscillations in g that were caused by the existence of Mie resonances. However, he also showed that, as the size of the sphere becomes smaller than the emission wavelength, the spontaneous emission rate drops steadily until it levels off at a value given by 关9兾共n2 1 2兲2 兴 共g bulk 兾n兲, where n denotes the index of refraction and g bulk is the decay rate in a bulk dielectric medium. In addition, he found that the emission rate becomes independent of the position and orientation of the emitter dipole moment in the nanosphere. The motivation for our present work is to investigate these issues for other geometries and to determine whether these results are universal. Since analytical calculations for particles other than spheroids have not been possible, we tackle 0146-9592/03/191736-03$15.00/0
the problem numerically. Our starting point is to consider a classical oscillating dipole and to calculate the power emitted by it in particles of different shapes with a surface integral approach. The change in the radiated power can be shown to be equivalent to the modification of the spontaneous emission rate of a quantum-mechanical atom in the same particle.3 We describe the nanoparticle surrounding the emitter by a homogeneous, isotropic, nonmagnetic, and linear dielectric. Throughout this Letter our calculations consider a refractive index n 苷 1.59 and l 苷 628 nm. We solve the macroscopic Maxwell equations numerically with the rigorous method of boundary integral equations described by NietoVesperinas.14 In this method the unknowns are the field and its normal derivative at the boundaries between homogeneous dielectrics. The Sommerfeld radiation condition at infinity is taken into account with the proper causal Green tensor. For the sake of numerical simplicity we focused on a two-dimensional model and a molecular dipole in the computational xy plane.15 For this orientation the wave equation reduces to a scalar equation for the z component of the magnetic field. We discretize the boundary into f inite elements much smaller than other characteristic length scales, such as the wavelength and the dipole– boundary distance, and solve the integral equation with a moment method.16 The total emitted power, and hence the radiative emission rate g, is obtained by integration of the Poynting vector over all directions in the far field. The numerical results are checked for convergence by comparison with the analytically soluble case of a circular particle and by an increase in the number of discrete boundary elements. Throughout this Letter we normalize g to g bulk , thereby eliminating the need for local f ield corrections.17 – 19 We note that with this method we also obtain the angular pattern of the radiation, which can also be modif ied in a significant manner.20 Figure 1 shows our findings for a dipole placed at the center of particles with various shape and size. To be able to compare the size dependence of g for different shapes, we take the square root of the surface area as a length measure. For the case of a circular particle we can verify the analytical results obtained by Chew7 © 2003 Optical Society of America
October 1, 2003 / Vol. 28, No. 19 / OPTICS LETTERS
Fig. 1. Normalized radiative decay rate versus host particle size. Symbols, numerical calculation for triangles, square, pentagons, and circles. Solid curve, analytical result for a circular object.
(see the solid curve). For particles of other shapes the decay rates also follow the same trend as the particle size is varied. In particular, we note that the decay rates all converge if the particle becomes much smaller than a wavelength. In other words, if the atom is placed in the middle of a subwavelength medium, g depends more on the size than on the geometrical curvature of the particle. This conclusion might seem nonintuitive at first, because sharp edges are expected to scatter electric fields more strongly in the electrostatic limit.21 To investigate this, in Fig. 2 we plot g for off-centered emitters. Here the dipole is moved along a line that connects the center to a corner of different polygonal particles of the same surface area. As was pointed out by Chew,7 g does not show any significant dependence on the position of the dipole in a circular geometry; only retardation leads to a slight position dependence. In contrast, however, the emitter position matters a great deal if the nanoparticle has sharp edges. In fact, we note that the largest effect is observed when the dipole is placed close to a corner of the sharpest geometry, namely, a triangle. The systems studied in Figs. 1 and 2 possessed quasi-spherical geometries. We now turn our attention to the effect of the particle’s aspect ratio on g. As a model system we have considered a rectangle with side lengths a and b and a dipole placed at its center, oriented parallel to the b edge. Again we take the square root of the surface area as a measure for the particle size, and for each value we examine rectangles of different aspect ratios a:b with an emphasis on situations in which at least one side is much smaller than a wavelength. As displayed in Fig. 3, the decay rate is reduced if the particle is elongated perpendicular to the dipolar orientation, whereas it approaches the bulk value if the elongation is along the dipolar axis. These trends agree with what is known from the dipole radiation in thin f ilms.22 As the size of the particle approaches zero, on the other hand, the decay rate reaches a limiting value, which in contrast to the situation in Fig. 1 is not the same for the different shapes. To gain insight into these observations, we turn to an alternative picture of the spontaneous emission process in which the decay of the atomic excited state is induced by the f luctuations of the vacuum f ield E. Here the emission rate can be calculated by considering Fermi’s
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golden rule g 苷 2p兾 h¯ 2 j具ejE.Djg典j2 r共v兲, where D is the transition dipole operator and r is the density of photon states.23 In a three-dimensional bulk dielectric of refractive index n, r and E are modified compared with their vacuum values such that the spontaneous emission rate is n times faster.24 Therefore g becomes weaker by a factor of 1兾n if the dielectric medium is shrunk much below a wavelength.25 Moreover, within this limit we approach an electrostatic regime in which the field E inside is smaller than the vacuum field outside by a depolarization factor f .21 In the case of a sphere these two effects combine to yield the limiting value obtained by Chew.7 To explain the limiting values obtained in Fig. 3, we can solve Laplace’s equation for a rectangular object to obtain the corresponding depolarization factors. However, since this problem cannot be solved in an analytical manner, we instead consider two-dimensional elliptical particles.20,26 We can then arrive at the limiting value of g共2D兲 ! g bulk j f j2 苷 g bulk
ja 1 bj2 , jn2 a 1 bj2
(1)
for a field polarized along the b axis.25 This simple picture also clarifies that f ! 1 for b ¿ a, where the
Fig. 2. Normalized radiative decay rate versus source position as it approaches a corner of the particle. The same particle geometries were considered as in Fig. 1, but the particle size was chosen to have a surface area of 共56 nm兲2 .
Fig. 3. Normalized decay rate for rectangular particles with different aspect ratios a:b 苷 1:20, 1:5, 1:1, 5:1, 20:1, as indicated by the boxes at the left. The molecular dipole points upward along the b edge. The horizontal lines signify the limiting values 1 and 1兾n4 obtained from Eq. (1) for elliptical particles. Insets, relative orientations and strengths of the relevant vacuum electric f ields and their depolarization f ields inside the particles.
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boundary conditions on the tangential f ields allow the vacuum f ields parallel to the dipole to penetrate the matrix without change (see the inset in Fig. 3). For a ¿ b, however, the relevant vacuum electric fields are perpendicular to the long side and therefore undergo a discontinuity. One can also understand the limit f ! 1兾n2 in this case by considering the continuity of the displacement field eE across the long side. These electrostatic arguments are in excellent agreement with our numerical results for subwavelength particles. As a final issue, we point out that for particle dimensions approaching the molecular scale the concept of a macroscopic permittivity e becomes invalid27 and our model breaks down. In conclusion, we have investigated the inf luence of the shape and size of a dielectric nanoparticle on the spontaneous emission rate of an atom placed in it at an arbitrary position and with an arbitrary orientation. We found that for a quasi-spherical particle the spontaneous emission rate drops steadily and levels off at a limiting value as the particle size diminishes below a fraction of a wavelength. We have also shown that sharp edges and elongated geometries inf luence the emission rate dramatically. These f indings are important for quantitative understanding and control of the radiation from quantum dots, molecules, or other emitters placed in microstructures and nanostructures. We thank R. Carminati for helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft within the focus program on photonic crystals and by the German Academic Exchange Off ice within the French – German program Procope. C. Henkel’s e-mail address is carsten.henkel@ quantum.physik.uni-potsdam.de. References 1. E. M. Purcell, Phys. Rev. 69, 681 (1946). 2. K. H. Drexhage, J. Lumin. 1–2, 693 (1970). 3. P. R. Berman, ed., Cavity Quantum Electrodynamics (Academic, San Diego, Calif., 1994). 4. R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996). 5. J. Gersten and A. Nitzan, J. Chem. Phys. 75, 1139 (1981).
6. C. Girard, O. Martin, and A. Dereux, Phys. Rev. Lett. 75, 3098 (1995). 7. H. Chew, Phys. Rev. A 38, 3410 (1988). 8. V. V. Klimov, M. Ducloy, and V. S. Letokhov, J. Mod. Opt. 43, 549 (1996). 9. L. Novotny, Appl. Phys. Lett. 69, 3806 (1996). 10. C. Henkel and V. Sandoghdar, Opt. Commun. 158, 250 (1998). 11. G. Parent, D. van Labeke, and D. Barchiesi, J. Opt. Soc. Am. A 16, 896 (1999). 12. A. Rahmani, P. C. Chaumet, and F. de Fornel, Phys. Rev. A 63, 023819 (2001). 13. H. Schniepp and V. Sandoghdar, Phys. Rev. Lett. 89, 257403 (2002). 14. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991). 15. A perpendicular dipole in two dimensions is in fact equivalent to an oscillating point charge. At short scales its electric f ield is qualitatively different from a dipole field. Since this orientation is not a good representation of the physical (three-dimensional) situation in the laboratory, we do not consider it in this work. 16. R. F. Harrington, Field Computation by Moment Methods (Institute for Electrical and Electronics Engineers, Piscataway, N.J., 1993). 17. A. Lagendijk, B. Nienhuis, B. A. van Tiggelen, and P. de Vries, Phys. Rev. Lett. 79, 657 (1997). 18. F. J. P. Schuurmans, P. de Vries, and A. Lagendijk, Phys. Lett. A 264, 472 (2000). 19. A. Rahmani, Opt. Lett. 27, 430 (2002). 20. L. Rogobete, V. Sandoghdar, and C. Henkel, “Modif ication of spontaneous emission in nanoscopic environments,” submitted to J. Opt. B. 21. W. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). 22. M. Kreiter, M. Prummer, B. Hecht, and U. P. Wild, J. Chem. Phys. 117, 9430 (2002). 23. R. Loudon, The Quantum Theory of Light (Oxford U. Press, London, 1983). 24. G. Nienhuis and C. Th. J. Alkemade, Physica (Amsterdam) 81C, 181 (1976). 25. In a two-dimensional problem, as a result of the functional form of the mode density r, the factor 1兾n becomes 1. 26. V. V. Batygin and I. N. Toptygin, Problems in Electrodynamics (Academic, London, 1978), problems 193– 200. 27. J. van Kranendonk and J. Sipe, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 15, p. 245.
