Environment-induced modification of spontaneous emission: Single-molecule near-field probe Adel Rahmani Atomic Physics Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8423 Patrick C. Chaumet Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientificas, Campus de Cantoblanco, Madrid 28049 Spain Fre ´ de ´ rique de Fornel Groupe d’Optique de Champ Proche, LPUB, CNRS UMR 5027, Faculte ´ des Sciences Mirande, Boı ˆte Postale 47870, F-21078 Dijon Cedex, France ~Received 20 June 2000; published 18 January 2001! The modification of lifetime experienced by a fluorescent molecule placed in an arbitrary environment is investigated theoretically within the framework of linear response theory. We present a complete description of the interaction of the particle with arbitrary structures on a plane substrate or inside a cavity. The theory is based on a self-consistent scattering procedure in which retardation effects and contributions from both ho- mogeneous and evanescent modes of the electromagnetic field are included. The decay rate variations are computed and the concept of single-molecule near-field probe is discussed. DOI: 10.1103/PhysRevA.63.023819 PACS number~s!: 42.50.Ct, 32.70.Cs, 32.70.Jz, 07.79.Fc I. INTRODUCTION Since the pioneering work of Drexhage @1#, the study of fluorescence emission in finite geometries has emphasized the influence of the environment on the dynamics of the fluorescent particle @2–4#. In finite geometries, the fluores- cence lifetime, or the spontaneous emission rate, differs from the free-space value because the presence of matter near the decaying particle modifies the boundary conditions imposed on the electromagnetic field @5#. If, for a fluorescent mol- ecule, we adopt the picture of a dipole interacting with its surroundings through its field, it is the reflected field which conveys back to the molecule information concerning its en- vironment. While this interpretation has a classical flavor, it nevertheless remains consistent with the quantum- mechanical aspects of the source-field interactions as both vacuum and radiation modes conform to the same laws and, hence, are modified in a same way when particular boundary conditions are imposed @6,7#. While exact treatments have been proposed for simple geometries @3,8#, the influence of complex structures on the molecular lifetime is usually dealt with by resorting to a perturbative approach for the electro- magnetic field, and/or by neglecting retardation effects, as it is the case, for instance, for a substrate with shallow rough- ness @9–15#. In the two-dimensional case, Bian et al. pre- sented a nonperturbative treatment using the method of finite difference in time domain @16#. They computed the lifetime and the ~classical! frequency shift for a dipole on a substrate as a function of its position relative to the tip of a scanning near-field optical microscope. Their calculation showed a great sensitivity of the dynamics of the dipole with respect to its position beneath the tip consistent with experimental ob- servations @17–19#. Another numerical study of lifetime modification was presented by Girard et al. @20#, however, these authors based their work on a misconception of the coupling of a two level atom with radiation, as they ignored the fundamental relation between the free-space decay rate of spontaneous emission and the atom polarizability, leading to an unsound model. The three-dimensional problem was ad- dressed by Novotny who studied the influence of a scanning near-field optical microscope tip, represented by an alumi- num disk-shaped object, on the fluorescence lifetime of a dipole lying on a substrate @21#. His calculation, using a semianalytical method derived from the multiple multipole method @22#, showed that the orientation of the dipole was critical, especially when the dipole was located close to the rim of the object. The great sensitivity of a fluorescent molecule to its envi- ronment, makes it an interesting candidate for an elementary near-field probe @23–27#. In order to assess the potential of such a single-molecule probe, it is crucial to describe prop- erly the coupling of the molecule to its environment. The purpose of this work is thus to study, in three dimensions, the modification of the fluorescence decay rate of a molecule by arbitrary structures, including ones too large to be used in a Born-type perturbative approach. The formalism presented here is based on a knowledge of the dynamical electromag- netic response of a plane surface or cavity to both homoge- neous and evanescent modes of the field. The dressed, re- tarded field susceptibility pertaining to the environment is derived through a self-consistent procedure. The decay rate of the molecule is then computed according to linear re- sponse theory. Our paper is organized as follows. Using lin- ear response theory, we relate in Sec. II the spontaneous emission rate to the field susceptibility. Using the theory of Agarwal @28,29#, we then derive the exact retarded field sus- ceptibility tensor associated with a surface ~interface be- tween two media!. The relevant quantity in the problem of spontaneous emission is the field correlation function. Since this correlation function can be related to the field linear susceptibility through the fluctuation-dissipation theorem @28#, this approach, while rigorously quantal, avoids an ex- plicit quantization of the field. In Sec. III, we insert this tensor into the self-consistent procedure of the coupled di- PHYSICAL REVIEW A, VOLUME 63, 023819 1050-2947/2001/63~2!/023819~11!/$15.00 ©2001 The American Physical Society 63 023819-1