---
title: Microsphere-Enabled K-Space Folding
url: https://www.emergentmind.com/topics/microsphere-enabled-k-space-folding
type: topic
---

# Microsphere-Enabled K-Space Folding

Microsphere-enabled k-space folding is an optical technique designed to directly probe high in-plane wavevector ($k_\parallel$) excitonic states—in particular, the spin-forbidden “dark” excitonic reservoir in two-dimensional (2D) transition metal dichalcogenides (TMDs) such as monolayer WSe$_2$. This method leverages a dielectric microsphere to overcome the limited collection numerical aperture (NA) of conventional microscopy, enabling efficient mapping of evanescent or highly localized emission into the far field. When combined with Green-tensor quasinormal-mode (QNM) calibration, microsphere-enabled k-space folding yields quantitative, thermodynamically meaningful metrics for the dark-to-bright exciton population, otherwise inaccessible via standard photoluminescence (PL) detection [2602.01324].

## 1. Experimental Geometry and Optical Setup

The microsphere-enabled k-space folding protocol uses a vertical stacking of a 2D material and plasmonic substrate for optimal emission channeling and detection. The key components are:
- **Sample stack**: Monolayer WSe$_2$ is transferred onto a flat Au substrate, supporting both propagating emission and surface-plasmon-mediated decay pathways.
- **Microsphere probe**: A silica microsphere of diameter $D\approx6.5\,\mu\mathrm{m}$ (refractive index $n_\mathrm{sph}=1.45$) is placed in direct contact with the WSe$_2$. The diameter is chosen such that the whispering–gallery mode (WGM) free-spectral range (~43 meV) coincides with both bright and dark excitonic transition energies ($\sim1.669$ eV and $\sim1.625$ eV).
- **Excitation and collection**: A 100×, NA = 0.8 objective focuses a continuous-wave (CW) laser onto the monolayer–sphere junction, simultaneously collecting PL emissions through the same optical path. The back focal plane (BFP) of the objective is imaged onto a camera, yielding angle-resolved PL measurements.

A side-view schematic (verbal “Fig. 1(c)”) depicts the excitation path through air, focusing onto the silica sphere atop the WSe$_2$/Au stack, with emitted radiation—including high-$k_\parallel$ components—coupling into the sphere and being re-directed into the objective [2602.01324].

## 2. Principle and Mechanism of k-Space Folding

Emissions from dark excitons predominantly populate large in-plane momenta ($k_\parallel$) not accessible to standard far-field optics due to the NA limit ($k_\parallel > k_0\cdot\mathrm{NA}$, with $k_0 = \omega/c$). The microsphere disrupts in-plane translational invariance and supports WGMs that can hybridize with plasmonic fields, thereby enabling the coupling of evanescent emission into propagating channels:
- **In free space**, the relation $k_\parallel = k_0 \sin\theta$ ($|\sin\theta| \le \mathrm{NA}$) governs collection.
- **With the sphere**, evanescent fields of $k_{\parallel,\mathrm{evan}} > k_0$ excite WGMs at internal sphere angles $\theta_\mathrm{sph}$, with $k_{\parallel,\mathrm{evan}} = n_\mathrm{sph}\,k_0 \sin\theta_\mathrm{sph}$.
- Subsequently, emission is radiated into air by Snell’s law: $n_\mathrm{sph}\sin\theta_\mathrm{sph} = \sin\theta_\mathrm{out}$, resulting in $k_{\parallel,\mathrm{evan}} = k_0\sin\theta_\mathrm{out}$.

Because $\sin\theta_\mathrm{sph}$ can exceed unity inside the sphere, large $k_\parallel$ components are “folded” down such that their far-field $\sin\theta_\mathrm{out}$ lies within the observable NA. This increases the **effective collection NA** to $\mathrm{NA}_\mathrm{eff} \approx n_\mathrm{sph} \cdot \mathrm{NA}$.

## 3. Imaging and Quantitative Mapping in the Back Focal Plane

The BFP of the objective serves as a momentum-space (k-space) map, with:
\[
k_x = \frac{k_0}{f} x_\mathrm{BFP}, \quad k_y = \frac{k_0}{f} y_\mathrm{BFP}
\]
($f$ denotes objective focal length). For conventional microscopy, only $k_\parallel \leq k_0\,\mathrm{NA}$ is detected. After sphere-enabled k-space folding, however, the mapping brings emission originally spanning up to $k_\parallel = n_\mathrm{sph}k_0\mathrm{NA}$ into the objective’s circular aperture ($k_x^2 + k_y^2 \leq (k_0\mathrm{NA})^2$). This mapping is sphere-radius-independent to leading order but the efficient folding window is governed by the WGM free-spectral range, itself determined by $D$ as $\lambda^2 / [\pi D n_\mathrm{sph}]$.

Evidence for such folding is directly manifested in back-focal-plane PL images (“Fig. 2(c–e)”), which reveal the redistribution of high-$k_\parallel$ dark-exciton emission into the NA-limited detection window when the microsphere is present [2602.01324].

## 4. Performance Metrics and Empirical Results

Performance benchmarks for the k-space folding protocol include:
- **Maximum $k_\parallel$ captured**: $k_\parallel \approx n_\mathrm{sph} k_0 \mathrm{NA} \approx 1.16 k_0$; a $\sim16\%$ extension beyond the air limit.
- **Dark (out-of-plane) collection efficiency**: $n_\perp$ increases from $8.7\%$ (without sphere) to $43.4\%$ (with sphere), a $\sim5\times$ boost.
- **Bright (in-plane) collection efficiency**: Modestly increases from $46.9\%$ to $54.0\%$ with the sphere.
- **Spatial resolution**: Effective lateral scan resolution ($\mathrm{FWHM}$ of sphere-lens focus) $\lesssim 300\ \mathrm{nm}$.

Polar-plot and BFP images (described as “Fig. 1(a,b)” and “Fig. 2(c–e)”) illustrate the clear redistribution of emission patterns, quantitatively demonstrating enhancement in dark-channel collection efficiency [2602.01324].

## 5. Green-Tensor Quasinormal-Mode Calibration

Intensity detected from excitonic transitions is jointly determined by the environment-modified radiative rate $\gamma_\mathrm{rad,\,i}$ and collection efficiency $n_i$. Thus, population quantification requires a separation of these contributions:

### 5.1. Radiative Rate Modification

For a unit dipole along $\hat{e}_i$ at position $r_0$, the radiative enhancement is governed by the projected LDOS:
\[
F_{r,i}(r_0) = \frac{6\pi c}{\omega} \operatorname{Im}\left[ \hat{e}_i^T G(r_0, r_0; \omega) \hat{e}_i \right]
\]
where $G(r, r'; \omega)$ is the Green tensor. The environment-modified radiative rate is
\[
\gamma_\mathrm{rad,i} = F_{r,i} \gamma_{0,i}
\]
with $\gamma_{0,i}$ the free-space dipole radiative rate.

### 5.2. Collection Efficiency

The detected photon fraction $n_i$ is calculated as
\[
n_i = \frac{\int_{|k_\parallel| \leq k_0 \mathrm{NA}} d\Omega\, |E_i(k_\parallel)|^2}{\int_{\text{all}} d\Omega\, |E_i(k_\parallel)|^2}
\]
where $E_i(k_\parallel)$ is the far-field amplitude, obtained either from the Green tensor or the QNM expansion.

### 5.3. Detected Signal and Population Extraction

The relation between the detected PL signal and the exciton populations is:
\[
I_\mathrm{det}(x) = a[N_B n_B(x) \gamma_\mathrm{rad,B}(x) + N_D n_D(x) \gamma_\mathrm{rad,D}(x)]
\]
where $x$ is the lateral offset of the excitation spot, $a$ is a global instrument factor, $N_{B,D}$ are the bright/dark exciton populations, and $R_i(x)=n_i(x)\gamma_\mathrm{rad,i}(x)$. Notably, $R_B(x)$ and $R_D(x)$ have distinct $x$-dependence, enabling two-parameter fits to spectrally resolved $I_\mathrm{det}(x)$ and thereby providing the absolute population ratio $N_D/N_B$.

## 6. Direct Quantification of the Dark-Exciton Reservoir

Applying this protocol to monolayer WSe$_2$, a room-temperature dark-to-bright population ratio $N_D/N_B=4.3$ was extracted. This finding is consistent with a near-thermalized excitonic manifold under continuous-wave excitation and provides a rigorous thermodynamic benchmark for interaction-driven 2D exciton phases. The methodology thus enables the “hidden” dark exciton manifold to be investigated as a direct thermodynamic observable under ambient conditions [2602.01324].

## 7. Context, Implications, and Figures

Microsphere-enabled k-space folding, calibrated via Green-tensor QNM modeling, transforms the detection of dark excitonic states from a qualitative spectroscopic challenge into a quantifiable population measurement. Key figures described in the primary publication include:
- Polar plots distinguishing bright and dark k-space radiation lobes.
- Schematic diagrams of the folding mechanism.
- BFP images demonstrating efficient folding and spatial redistribution.
- LDOS spatial maps with $\geq 40\times$ selectivity in the dark channel.
- Lateral scan profiles used for population fitting.

*A plausible implication is the extension of this measurement protocol to other van der Waals heterostructures and quantum emitters where buried or forbidden transitions dominate the photophysics. This approach may provide new avenues for probing non-equilibrium states and many-body phenomena in low-dimensional materials* [2602.01324].

Source: https://www.emergentmind.com/topics/microsphere-enabled-k-space-folding