Distinguishing ED and MD emission

ED and MD emission differ in symmetry and polarization. Although both emit into the sine-squared radiation pattern associated with point dipoles, their electric and magnetic field vectors are interchanged. In an inhomogeneous environment, the emission rates and patterns are modified by the interference of directly emitted and scattered fields13,14,15,16,17,18. Differences in these self-interference effects can be exploited to distinguish EDs and MDs based on the different modes into which they emit, even for isotropic emitters like the ions studied here.

As an example, Fig. 1a illustrates the different contributions to x-polarized emission into a dielectric substrate (k x =k y =0) for dipoles embedded within a thin film. Of the three Cartesian dipole orientations that describe an isotropic emitter, only the x-oriented ED and y-oriented MD emit into this mode. The observed emission depends on the interference of the directly emitted light and the wave reflected at the air interface. For normal incidence, the electric field acquires no phase upon reflection at the lower-index air interface. As the phase acquired upon propagation is negligible in this thin film (D≪λ), the result depends solely on the electric field symmetry of the emitting dipole. For an ED, the electric field is symmetric with respect to the dipole axis, and thus constructive interference will enhance emission into this mode; whereas, the opposite is true for a MD and emission is suppressed. Such interference differences persist for other emission directions and polarizations and in the presence of multiple reflections; in general, the electric and the magnetic LDOS differ in an inhomogeneous environment29. Consequently, in a thin film, EDs and MDs can be distinguished by their momentum- and polarization-dependent emission patterns.

Figure 1: Momentum-resolved emission patterns for ED and MD transitions within a thin dielectric film. (a) Schematic illustrating differing self-interference effects for ED and MD emission along the z axis. The dashed lines indicate the light reflected by the air interface. Owing to opposite field symmetries, ED emission into this mode is enhanced by constructive interference, whereas MD emission is suppressed by destructive interference. (b) Experimental BFP images of Eu3+:Y 2 O 3 (D=21 nm, n r =1.77) measured for x-polarized emission with bandpass filters for 605–635 nm and 585–595 nm, respectively. The NA of the objective is 1.3. (c) Theoretical BFP images for a 620-nm isotropic ED emitter and a 590-nm isotropic MD emitter. Dashed lines indicate s- and p-polarized cross-sections, and the solid rectangle shows the region near k x =0 that is projected onto the entrance slit for energy-momentum spectroscopy. The refractive index of the substrate is taken as n r =1.5. (d) Comparison of the experimental and theoretical cross-sections for s-polarization (blue) and p-polarization (red). Left plot compares experimental data acquired with the 620-nm bandpass filter (solid lines) with the theory for an isotropic ED emitter (dashed lines); right plot compares experimental data acquired with the 590-nm bandpass filter (solid lines) with the theory for an isotropic MD emitter (dashed lines). Full size image

To demonstrate this principle, we measured the momentum-distributed luminescence from a Eu3+-doped yttrium oxide (Eu3+: Y 2 O 3 ) thin film in the Fourier domain by back focal plane (BFP) imaging30. Figure 1b shows polarized BFP images obtained using bandpass filters to collect Eu3+ emission from the 5D 0 →7F 1 and 5D 0 →7F 2 transitions centred at 590 and 620 nm, respectively. These transitions exhibit very different momentum distributions. A comparison with theoretical calculations (Supplementary Methods) in Fig. 1c shows that the image for the 620 nm transition resembles an isotropic ED, whereas the image for the 590 nm transition resembles an isotropic MD. The differences are particularly striking for the p-polarized cross-sections (red lines in Fig. 1d), which are concave down for EDs but concave up for MDs. Such differences allow us to unambiguously differentiate ED and MD emission.

Quantifying transitions using energy-momentum spectroscopy

As the sample is rotationally symmetric, the s- and p-polarized cross-sections contain all information encoded in the two-dimensional BFP images. Therefore, we can simultaneously characterize the emission in terms of momentum, polarization and wavelength by projecting the BFP image onto the entrance slit of an imaging spectrograph. As illustrated in Fig. 1c, the entrance slit selects emission about k x =0, such that k y =k || . This light is then dispersed by a diffraction grating to obtain an energy spectrum along the x-dimension, and reimaged onto the CCD (charge-coupled device) camera to retain momentum information along the y-dimension. S- or p-polarization states were selected by rotating a polarizer in front of the spectrograph.

The resulting BFP spectra, shown in Fig. 2a,b, contain a wealth of information about the emission process. These combined energy-momentum spectra are experimental dispersion diagrams, which show how emission is characteristically distributed into the available optical modes. Integrating over the momentum axis (in the vertical direction) produces an energy spectrum (Fig. 2c) of the 5D 0 →7F J transitions in Eu3+:Y 2 O 3 (ref. 31; see Supplementary Fig. S1 for an energy level diagram of Eu3+). However, by measuring and examining the momentum distribution at different wavelengths (Fig. 2g), we can now resolve distinct ED and MD transitions.

Figure 2: Quantifying the ED and MD transitions in Eu3+:Y 2 O 3 by energy-momentum spectroscopy. (a,b) Experimental energy- and momentum-resolved BFP spectra for s- and p-polarization, respectively. (c) Integrated energy spectrum showing the total observed emission (N, black line) decomposed into the contributions from ED (N ED , blue line) and MD (N MD , red line) transitions. (d,e) Theoretical BFP spectra for s- and p-polarization, respectively, produced using fits to equation (1). (f) Spectrally resolved intrinsic emission rates, A ED (solid blue line) and A MD (solid red line), deduced from this fitting analysis together with their 95% confidence intervals (dashed lines). (g) Momentum cross-sections for three representative wavelengths showing strong agreement between experimental data (solid lines) and theoretical fits (dashed lines) for both s-polarization (blue) and p-polarization (red). Emission at 592 nm shows a concave up p-polarized cross-section that is consistent with MD emission, and fitting confirms that this transition is predominantly magnetic (MD=90.8±2.0%). In contrast, 612-nm emission shows a concave down p-polarized cross-section with a local maxima at k || =0, and the fit shows that emission at this wavelength is almost purely electric (ED=99.0±1.0%). Emission at 624 nm shows a less-pronounced maxima, and fitting reveals that this is a more mixed line (ED=88.4±0.7%). Full size image

To quantify the nature of emission at each wavelength, we decompose the momentum-distributed emission into parts originating from ED and MD transitions. For this analysis, we make two approximations that have accurately described previous experiments28: first, that each ion is an isotropic emitter; second, that the ensemble of ions is effectively homogeneous, that is, the excitation of each emitter results in the same emission spectrum. As all observed emission originates from the same excited level (5D 0 ), we can thus fit the measured counts Ns,p(ω, k || ) to a sum of ED and MD emission (see Supplementary Methods):

where and are the normalized density of optical states into which ED and MD transitions may emit. A ED and A MD are the ED and MD spontaneous emission rates (that is, the spectrally resolved Einstein A coefficients) that would be observed in a homogeneous medium. C is a proportionality constant that depends on sample- and setup-specific experimental parameters, but which only affects the total number of counts and not their energy-momentum distribution. For example, increased non-radiative decay can reduce the total number of observed counts by depleting the single excited level from which all emission originates, but this would not change the radiation pattern or underlying emission rates of ED and MD transitions. In the simplest form, the constant C is given by the product of the time-averaged number of excited ions, the measurement time and a constant related to the spectral calibration of the setup's detection efficiency.

Equation (1) highlights how the observed BFP spectra depend on the intrinsic properties of the emitter (A ED and A MD ) and its local optical environment ( and ). For this known thin-film sample, and can be calculated from analytical expressions (Supplementary Methods and Supplementary Fig. S2). Therefore, fitting experimental data determine the values of A ED and A MD to within the constant C. The fits were performed at each wavelength on the p-polarized data, which contains the most contrast, and over the momentum range of −1.1k 0 ≤ k || ≤ 1.1k 0 . (Although the nominal numerical aperture (NA) of the oil immersion objective is 1.3, the effective NA of the experimental setup is lower than this value, and a sharp decrease in the collection efficiency was observed for |k || |>1.1k 0 .) The resulting energy-momentum spectra (Fig. 2d,e) accurately recover the experimental BFP spectra (Fig. 2a,b) and momentum curves (Fig. 2g) for both polarizations.

This procedure decomposes the measured emission spectrum (Fig. 2c) into counts originating from ED transitions (N ED ) and MD transitions (N MD ). As this decomposition is based exclusively on the momentum dependence of the emission, it is valid even if the ensemble of emitters is not homogeneous (that is, if C was a function of ω); the origin of the light emitted at each frequency is always encoded in its momentum distribution. The observed percentage of MD counts (N MD /N) does, however, depend strongly on the local optical environment and the modes detected (for example, see Supplementary Fig. S3 for measurements highlighting the role of NA on observed MD counts), in addition to the intrinsic properties of the emitter.

The intrinsic rates A ED and A MD (Fig. 2f) provide a more fundamental and quantitative comparison of competing ED and MD emission processes. These rates are directly related to the transition dipole moments μ ED and μ MD by the following (Supplementary Methods):

and

Even though the contribution of the 5D 0 →7F 1 MD transition appears weaker than the 5D 0 →7F 2 ED transition in Fig. 2c, their intrinsic rates are actually comparable. MD emission is suppressed in this thin-film sample owing to its lowered magnetic density of optical states (as follows from the qualitative argument in Fig. 1a). We define the intrinsic branching ratios for MD and ED transitions at each frequency: MD(ω)=A MD (ω)/(A MD (ω)+A ED (ω)) and ED(ω)=1−MD(ω). For the curves in Fig. 2g, we find that the 592 nm transition is predominantly magnetic (MD=90.8±2.0%) whereas the 612 nm transition is electric (ED=99.0±1.0%).

Although Eu3+ emission is mainly composed of spectrally distinct dipole transitions, the same technique can also be used to quantify degenerate and higher-order transitions. For example, energy-momentum spectroscopy of Tb3+:Y 2 O 3 shows that ED and MD contributions can be distinguished even for the strongly mixed 5D 4 →7F 5 Tb3+ transition (Supplementary Fig. S4). Higher-order transitions, such as electric quadrupoles32, also have unique signatures in momentum space14,16. However, we did not observe evidence for such transitions in the emission spectra of Eu3+ and Tb3+, nor are they expected on the grounds of theory1.

Using ED and MD transitions to probe the optical environment

Having quantified ED and MD emission by energy-momentum spectroscopy, we can use these transitions as local probes of an unknown electromagnetic environment. As a demonstration, we place a several micron thick sheet of mica on top of the Eu3+ sample with immersion oil (Fig. 3a). The measured BFP spectrum (Fig. 3b) exhibits rapidly varying interference fringes owing to reflections at the mica–air interface. Of particular interest are the two transitions near 581 nm (Fig. 3c). From the analysis in Fig. 2, these lines have been identified as the 5D 0 →7F 0 ED transition centred at 580.8 nm and the 5D 0 →7F 1 MD transition centred at 582.2 nm. As these lines are spectrally close, their emission probes the same wavelength to a good approximation. Close inspection (Fig. 3d) reveals that the momentum-resolved emission patterns for these two transitions are completely inverted. The 180° phase shift observed in these interference fringes is a direct visualization of the symmetry difference between ED and MD emission, as schematically illustrated in Fig. 3a.