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Antenna model of the Purcell effect
Alexander E. Krasnok, Alexey P. Slobozhanyuk, Constantin R. Simovski, Sergei A. Tretyakov, Alexander N. Poddubny, Andrey E. Miroshnichenko, Yuri S. Kivshar, Pavel A. Belov
TL;DR
The paper addresses how environment-induced emission-rate changes can be understood classically rather than only through quantum emitters. It develops an equivalent small-antenna approach based on input impedance and applies it across electric and magnetic cases and frequency ranges. The method is experimentally verified for electric and magnetic dipole antennas and applied to dielectric-sphere Mie resonances.
Problem
The paper addresses the need to understand and measure the Purcell effect’s classical counterpart for small antennas as well as quantum emitters.
Method
The approach represents the emitter and resonant object as coupled classical oscillators and expresses the Purcell factor through equivalent-antenna input impedance and radiation resistance.
Results
The technique was experimentally verified for electric and magnetic dipole antennas and successfully applied to Purcell enhancement from dielectric-sphere Mie resonances.
Takeaways & Limitations
The input-impedance formulation provides an alternative route for calculating or measuring electric and magnetic Purcell factors in classical and quantum-emitter settings.
Takeaways & Limitations
The fixed-dipole-moment approximation becomes inadequate for very strong emitter–nanoantenna coupling, requiring a self-consistent eigenmode treatment.
Abstract
from arXiv · showhide
The Purcell effect - the modification of the spontaneous emission rate in presence of resonant cavities or other resonant objects - is a fundamental effect of quantum electrodynamics. However, a change of the emission rate caused by environment different from free space has a classical counterpart. Not only quantum emitters, but any small antenna tuned to the resonance is an oscillator with radiative losses, and the influence of the environment on its radiation can be understood and measured in terms of the antenna radiation resistance. We present a general approach which is applicable to measurements of the Purcell factor for radio antennas and to calculations of these factors for quantum emitters. Our methodology is suitable for calculation and measurement of both electric and magnetic Purcell factors, it is versatile and applies to various frequency ranges. The approach is illustrated by a general equivalent scheme and allows the Purcell factor to be expressed through the continious radiation of a small antenna in presence of the environment.
I. INTRODUCTION
The paper frames the Purcell effect as an environment-induced change in emission and develops a classical antenna perspective for analyzing it. It reviews weak coupling, radiative and non-radiative contributions, and existing measurement approaches.
- The Purcell effect is a modification of a quantum source’s spontaneous emission lifetime caused by interaction with its environment.
- The effect is significant when the environment is a resonator tuned to the emission frequency.
- I. INTRODUCTION: In weak coupling, the resonator changes the decay rate without modifying the spontaneous-emission frequency or transition dipole moment.
- The Purcell factor can describe changes in far-field radiation and, in lossy environments, Joule losses in the environment.
- Quantum-emitter Purcell factors are measured directly through time-resolved photoluminescence or indirectly through techniques such as Raman spectroscopy.
- The paper studies the classical counterpart of the effect and proposes calculating electric and magnetic Purcell factors through the input impedance of an equivalent small antenna.
A. General Method
The general method recasts the Purcell factor for a small dipole in terms of radiation and input resistances. It connects this antenna formulation to scattered fields and the electromagnetic Green function.
- A. General Method: The dipole’s total radiated power splits into free-space radiation and an additional contribution caused by the object’s scattered field.
- The additional mutual resistance R12 equals the real part of the mutual impedance and represents the object’s effect on the emitter.
- The Purcell factor can be written as F = P/P11 = Rrad/R0,rad, providing a radiation-resistance alternative to the conventional formulation.
- For a low-loss short-dipole emitter, the Purcell factor is determined by the modification of the resistive part of its input impedance.
- The input-impedance formulation is linked to the total electric field at the dipole origin through the induced electromotive force method.
- The resulting impedance expression is equivalent to the known Green-function representation and can be used instead of scattered-field or Green-function calculations.
B. Equivalent Circuit for Finding the Purcell Factor
The paper models an optical emitter and nanoantenna as electromagnetically coupled resonant circuits, replacing mutual induced voltages with a mutual impedance. Its real part adds mutual resistance to the emitter and determines the environment-modified emission power.
- Equivalent resonant circuits: Optically small emitters and nanoantennas can both be represented as resonant RLC circuits because their coupling is electromagnetic and governed by Maxwell’s equations.The model treats the emitter and nanoantenna as coupled oscillators while neglecting tunneling effects.
- Mutual impedance: The emitter–nanoantenna coupling is represented by mutual impedance Zm, replacing the mutually induced electromotive force in the equivalent circuit.The mutual impedance is introduced through the equivalent generator theorem and reciprocity.
- Mutual impedance: The final radiating circuit uses a fixed emitter current I1 = Ie and loads it with the emitter impedance plus mutual impedance Zm.The real part of Zm contributes the additional resistance associated with the nanoantenna’s effect on the emitter.
- Equivalent resonant circuits: The nanoantenna is modeled as a Lorentzian scatterer whose impedance contains radiation, inductive, capacitive, and dissipative contributions.Its series impedance is written using R2, L2, and C2, with values obtained from the Lorentzian model.
- Purcell factor: The mutual resistance Rm ≡ R12 combines radiative and dissipative resistance added by the nanoantenna, allowing the Purcell factor to be calculated from the delivered power.In the quasi-static approximation, sufficiently large coupling parameter N can produce very large Purcell factors, while nearby reflectors can instead yield F < 1.
- Purcell factor: The factor depends on the nanoantenna and mutual location rather than on the particular emitter, so it applies to an arbitrary dipole emitter at that point.The formulation also allows reduced enhancement when the antenna resonance is substantially detuned from the emission frequency.
C. Validation of the Equivalent Circuit
The equivalent circuit is validated against analytical and exact electrodynamic descriptions for small spherical particles. Agreement holds within the dipole approximation and across the plasmon-resonance band, while the simple field approximation becomes less accurate away from resonance.
- Analytical validation: The circuit model is tested against the exact solution for a dipole emitter near a small isotropic sphere.For optically small spheres, the analytical series can be reduced to its dipole-polarization term.
- Analytical validation: The strict electrodynamic model and the circuit model coincide within the dipole approximation.
- Resonant validation: In the plasmon-resonance band, the circuit model agrees with the exact calculation for a 40 nm golden sphere 10 nm from the emitter.The sphere is modeled in air using permittivity values from Johnson and Christy experiments.
- Resonant validation: The approximation for Aee works near resonance because the dipole eigenmode produces nearly uniform polarization of the sphere.
- Approximation boundary: Away from resonance, field non-uniformity shifts the effective dipole toward the emitter, so the simple approximation underestimates the Purcell effect at low frequencies.
D. Extension of the Circuit Model
The circuit framework extends from electric to magnetic nanoantenna responses and accommodates multiple resonant modes. It also organizes Purcell-factor behavior for plasmonic and dielectric spherical systems through dipole orientation, wavelength, and impedance-based extraction.
- Magnetic extension: The equivalent circuit remains applicable to artificial magnetic dipoles, such as submicron silicon spheres at magnetic Mie resonances.
- Magnetic extension: A magnetic nanoantenna can be modeled as an optically small loop with effective area S and uniform loop current.
- Multiple modes: Electric and magnetic dipole contributions both enter the total mutual impedance and can combine constructively.
- Multiple modes: Coincident or closely spaced higher-multipole resonances may produce very large Purcell factors.
E. Theoretical Verification of the General Approach
The authors position input-impedance extraction as a practical alternative to field- and Green-function-based Purcell calculations. The approach is presented as applicable to nano-optical systems with or without losses and is tested against an exact spherical-particle solution.
- Motivation: The general approach uses input impedance as an alternative to conventional Green-function and scattered-field calculations.Direct evaluation of microscopic fields and Green functions at the emitter is described as challenging and time-consuming.
- Impedance extraction: A point dipole modeled in commercial software acquires a modified input resistance in the presence of an arbitrary object.
- Scope: The method is described as applicable to nano-optical systems with or without losses.
- Numerical verification: For a dielectric sphere of radius a = 70 nm and relative permittivity 15, simulations compare F = Rin/R0,in with an accurately evaluated exact solution for parallel and orthogonal dipoles.
F. Purcell Factor for Radio Antennas
The paper extends the Purcell-effect concept to resonant radio antennas, interpreting environmental emission changes through radiation resistance and antenna decay. For very poor antennas, radiation gain near an object approximates the object's Purcell factor.
- Classical antenna model: A resonant dipole antenna can model a quantum emitter by radiating stored energy at its resonance after excitation by a short pulse.The antenna's finite emission time is set by its decay rate.
- Classical antenna model: An arbitrary nearby object can increase the antenna's radiation resistance and thereby shorten its emission time without being a resonator tuned to the same frequency.The decay time can decrease from 1/γ0 to 1/γ when the object enhances radiation resistance.
- Radiation resistance and matching: The Purcell factor is most relevant for a short dipole with R0,rad ≪ Rout, because added mutual resistance can substantially improve generator-to-antenna matching.For a resonant half-wavelength antenna, the relative increase in radiation resistance is generally modest.
- Radiation resistance and matching: With a short dipole tuned by series inductance, the environment redistributes pulse energy toward radiation and the decay-rate multiplier is F = Rrad/R0,rad.Without the object, most energy is lost in the 50-ohm feeding resistance rather than radiated.
- Radiation gain and Purcell factor: For a very poor transmitting antenna, the object's Purcell factor can be inferred from radiation gain, but this equivalence does not generally hold for efficient antennas.The measured factor is determined by the object's properties and location, whereas antenna radiation enhancement is a distinct quantity except in the poor-emitter limit.
- Magnetic antennas: The framework also treats magnetic dipole antennas using an optically small loop with effective area S, nearly uniform current I, and moment m = µ0SIn.The magnetic dipole moment is related to the effective magnetic current by Im = ˙m = −jωm.
III. MEASUREMENT OF THE PURCELL FACTOR IN THE MICROWAVE SPECTRAL RANGE
The microwave measurement method retrieves the Purcell factor from antenna input resistance and tests it for electric and magnetic dipoles near a large copper plate. Measurements are compared with analytical expressions for different dipole orientations.
- Measurement method: The method extracts the antenna input resistance from measured S-parameters using the reflection coefficient S11 and waveguide impedance Zw.For a dipole connected to a one-mode waveguide, Rin is related to S11 and Zw.
- Experimental geometry: A flat, optically large copper plate near the antenna emulates a perfectly conducting plane, enabling analytical electric and magnetic Purcell factors for parallel and perpendicular orientations.The setup tests both electric and magnetic dipole configurations.
- Experimental comparison: The experiment compares analytical predictions with F = Rin/R0,in using brass-wire electric dipoles and a 1-cm-diameter wire-ring magnetic dipole.The coaxial cable has characteristic impedance Zw = 50 Ω, and measurements span 5–14 GHz.
- Experimental limitations: At very small antenna-to-metal separations, the analytical formulas become inapplicable because the finite antenna size prevents reliable Purcell-factor measurement.For the magnetic antenna, slight current inhomogeneity around the ring also produces disagreement with the formulas.
- Experimental corrections: The measured input resistance includes radiation from the open cable end, so the separately measured cable contribution δR is subtracted from Rin.Without this correction, disagreement with the predicted results would be more noticeable.
CONCLUSIONS
The paper combines theoretical and experimental analysis of the classical Purcell effect for subwavelength electric and magnetic dipole antennas. It proposes impedance-based measurement and verifies the technique across microwave antennas and dielectric-sphere Mie resonances.
- Conclusions: The study analyzes the classical counterpart of the Purcell effect for subwavelength electric and magnetic dipole antennas.It generalizes a nanophotonics approach to microwave antennas and recovers the Green-function expression for the Purcell factor.
- Conclusions: The proposed method measures the Purcell factor directly through the input impedance of a small antenna.The technique was experimentally verified for both electric and magnetic dipole antennas.
- Conclusions: The technique was also applied to Purcell enhancement associated with Mie resonances of a dielectric sphere.This extends the demonstrated approach beyond the microwave antenna measurements.