Computational Progressively Emission Saturated Nanoscopy

• CPSN is a super‐resolution imaging modality that combines nonlinear UCNP photophysics, progressive emission saturation, and PSF engineering to extend spatial-frequency support.
• It leverages differential saturation in two emission channels to generate complementary Gaussian and doughnut PSFs, enabling Fourier-domain and deep learning fusion of image data.
• The method achieves resolutions down to 33–40 nm with high SNR, offering a robust, single‐beam scanning approach for commercial and research applications.

Searching arXiv for the cited CPSN papers and closely related UCNP PSF-engineering work. Computational Progressively Emission Saturated Nanoscopy (CPSN) is a single-beam super-resolution imaging modality that uses progressively saturated emission in upconversion nanoparticles (UCNPs) to generate complementary point spread functions (PSFs), then combines those data computationally into a final image with extended spatial-frequency support. Its physical basis was established in heterochromatic PSF engineering with NaYF$_4$-based UCNPs under a single doughnut-shape scanning beam, where a two-photon 800 nm channel and a four-photon 740 nm channel encoded complementary low- and high-frequency information; a later formulation explicitly described the approach as CPSN and coupled power-dependent PSF evolution to deep learning within single-beam scanning microscopy (Chen et al., 2020, Liu et al., 14 Jul 2025).

1. Conceptual definition and development

CPSN combines nonlinear UCNP photophysics, PSF engineering, and computational reconstruction. In the heterochromatic implementation, diverse PSFs are obtained simultaneously from the same doughnut-shaped excitation by exploiting different saturation thresholds of distinct emission channels. The four-photon state yields a doughnut emission PSF that carries high-frequency information, while the over-saturated two-photon state yields a Gaussian-like emission PSF that carries complementary lower-frequency information. Fourier-domain heterochromatic fusion then synthesizes a PSF covering both bands of spatial information. In the later deep-learning implementation, the PSF evolves smoothly from doughnut-shaped to Gaussian as excitation power increases, and progressively complementary spatial-frequency components are fused by a deep recursive residual network (DRRN) (Chen et al., 2020, Liu et al., 14 Jul 2025).

Implementation	Computational stage	Reported outcome
Heterochromatic PSF engineering	Fourier-domain fusion + Richardson–Lucy	40 nm, $1/24$ of the excitation wavelength
Deep-learning CPSN	DRRN fusion of progressively complementary information	33 nm, corresponding to $1/29$ of the excitation wavelength; SNR up to 55 dB

Within single-beam scanning microscopy, this places CPSN in a class of methods that do not require a separate depletion beam. The 2025 work explicitly situates the approach in SBSM and states that SBSM is one of the most robust strategies for commercial optical systems. A plausible implication is that CPSN is designed to trade optical complexity for calibrated nonlinear response and post-acquisition reconstruction rather than for additional beam paths or structured detection hardware.

2. UCNP photophysics and progressive emission saturation

The canonical CPSN emitter is a NaYF$_4$ host lattice co-doped with Yb$^{3+}$ as sensitizer and Tm$^{3+}$ as activator. Under 980 nm excitation, Yb$^{3+}$ absorbs via the ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$ transition and transfers energy step-wise into Tm$^{3+}$. The relevant Tm$^{3+}$ manifolds are $1/24$0 (ground), $1/24$1, $1/24$2, $1/24$3 (two-photon), $1/24$4 (three-photon), and $1/24$5 (four-photon). The principal emission bands emphasized for CPSN are 800 nm from $1/24$6 and 740 nm from $1/24$7 (Chen et al., 2020).

A simplified steady-state rate model uses fractional populations $1/24$8, $1/24$9, and $1/29$0 in the ground, two-photon, and four-photon levels:

$1/29$1

$1/29$2

where $1/29$3 is the effective $1/29$4-photon absorption cross-section, $1/29$5 is the radiative lifetime, and $1/29$6 is an effective two-step transfer rate to the fourth level. The emitted intensity from level $1/29$7 is described by a Hill-type saturation law,

$1/29$8

Equivalently, the normalized saturated emission response can be written

$1/29$9

The experimentally important feature is the strong separation in saturation thresholds: the two-photon 800 nm band saturates at a much lower $_4$0 than the four-photon 740 nm band. Typical values reported from Chen et al. Fig. 2B are $_4$1 and $_4$2, as order-of-magnitude values. This differential saturation is the basis of progressive emission saturation: one emission channel becomes Gaussian-like while another remains doughnut-like under the same illumination.

3. PSF evolution under doughnut excitation

All scans in the 2020 implementation use the same doughnut-shaped illumination. In the focal plane, with $_4$3, the excitation intensity is approximated as

$_4$4

where $_4$5 is the beam waist and $_4$6 is the peak intensity in the ring; the nonzero dip at $_4$7 is $_4$8 of $_4$9. The 2025 formulation writes the doughnut beam at the field level as

$^{3+}$0

together with a saturable response

$^{3+}$1

so that

$^{3+}$2

For two-photon-like nonlinearity, this is expanded as

$^{3+}$3

The detected emission PSF for channel $^{3+}$4 can be written as

$^{3+}$5

where $^{3+}$6 is the confocal collection PSF. In practice, $^{3+}$7 is measured directly by scanning single UCNPs (Chen et al., 2020, Liu et al., 14 Jul 2025).

The central experimental consequence is a power-tunable PSF transition. For the 800 nm two-photon channel, at moderate $^{3+}$8, the center becomes saturated and $^{3+}$9 is peaked and Gaussian-like. For the 740 nm four-photon channel, $^{3+}$0 even near the ring maximum, so the doughnut PSF persists. In the 2025 CPSN formulation, the point $^{3+}$1 defines the transition between doughnut and Gaussian regimes and controls the cut-off spatial frequency $^{3+}$2. This progressive PSF evolution is the mechanism by which different spatial-frequency components are selectively accessed.

4. Fourier-domain reconstruction and transfer-function fusion

In the heterochromatic scheme, raw images $^{3+}$3 and $^{3+}$4 are acquired simultaneously in the 800 nm and 740 nm channels under identical doughnut scanning. Their Fourier transforms are

$^{3+}$5

The corresponding optical transfer functions are

$^{3+}$6

A crossover frequency $^{3+}$7 is identified where $^{3+}$8, and binary masks are built as

$^{3+}$9

$^{3+}$0

With an optional normalization factor $^{3+}$1,

$^{3+}$2

and similarly

$^{3+}$3

The fused image is

$^{3+}$4

followed by Richardson–Lucy deconvolution,

$^{3+}$5

with $^{3+}$6. In practice, 50–100 iterations suffice.

The 2025 formulation generalizes the fusion step across multiple power levels. If $^{3+}$7 denotes the image at excitation power $^{3+}$8 and $^{3+}$9 its OTF, then

${}^2F_{7/2}\rightarrow{}^2F_{5/2}$0

and

${}^2F_{7/2}\rightarrow{}^2F_{5/2}$1

The super-resolved image is then obtained by inverse Fourier transform and mild deconvolution. The 2025 paper emphasizes the limitations of manual fusion: precise determination of ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$2 points is error-prone, and manual tuning does not generalize across samples or SNR levels (Chen et al., 2020, Liu et al., 14 Jul 2025).

A key transfer-function argument underlies both implementations. In the 2020 exposition, ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$3 is approximately a classical OTF with cutoff ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$4, whereas the four-photon doughnut ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$5 is broadened to ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$6 but contains a low-frequency hole. The fused ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$7 covers the full band ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$8 without gap.

5. Deep-learning CPSN and DRRN fusion

The deep-learning realization of CPSN uses the power dependence curve to infer complementary image states and a DRRN to fuse them. The 2025 report states that, in order to enhance time resolution, the doughnut-shaped beam at low power and the saturated Gaussian-like image were predicted by the doughnut-shaped beam at low saturation threshold based on the power dependence curve. The DRRN then fuses progressively complementary spatial-frequency information into a final super-resolved image that encompasses the full frequency information (Liu et al., 14 Jul 2025).

The network input consists of four real-space images, ${}^2F_{7/2}\rightarrow{}^2F_{5/2}$9, under Gaussian and doughnut PSFs at different powers, including predicted states from the power-dependence curve. The front end is a convolution layer with 64 filters, a $^{3+}$0 kernel, and ReLU, producing feature maps $^{3+}$1. This is followed by 25 recursive residual blocks. Each block contains a first convolution with 64 filters and a $^{3+}$2 kernel with ReLU, a second convolution with 64 filters and a $^{3+}$3 kernel without activation, and a residual skip of the form $^{3+}$4. After recursion, a final $^{3+}$5 convolution merges features, followed by upsampling of $^{3+}$6, to produce the super-resolved output $^{3+}$7.

Training uses mean squared error,

$^{3+}$8

The dataset is synthetically generated from UCNP distributions convolved with measured PSFs at four power levels, with added Poisson noise over an SNR range of 7–55 dB. Data augmentation consists of random flips, $^{3+}$9 rotations, and random $^{3+}$0 pixel crops. Optimization uses Adam with initial learning rate $^{3+}$1, decayed by 0.5 every 50 k iterations; training runs for 200 k iterations with batch size 16. This computational layer distinguishes CPSN from a purely manual spectral-fusion workflow.

6. Experimental implementations, calibration, and performance

The heterochromatic CPSN platform uses NaYF$^{3+}$2:40%Yb$^{3+}$3,4%Tm$^{3+}$4 nanoparticles of size $^{3+}$5 nm, or NaYF$^{3+}$6:40%Yb$^{3+}$7,2%Tm$^{3+}$8 nanoparticles of size $^{3+}$9 nm for lower-power experiments. The particles are oleate-capped and dispersed in cyclohexane on cleaned coverslips or silicon nano-well substrates. Excitation is provided by a 980 nm single-mode diode laser with polarization maintenance. A vortex phase plate (VPP-1a) generates the doughnut, and circular polarization via a $1/24$00 plate minimizes dip asymmetry. Back-aperture powers span 5 mW–150 mW, corresponding to 0.1–5 MW/cm$1/24$01 at focus. Collection uses an oil-immersion $1/24$02, $1/24$03 objective with a confocal pinhole of approximately 1 AU. The 740/60 nm band-pass is directed to SPAD$1/24$04, and the 805/20 nm band-pass to SPAD$1/24$05. Pixel dwell time is 1 ms and pixel size is 10 nm. For the fusion step, $1/24$06 is chosen where $1/24$07, typically $1/24$08, and the amplitude ratio $1/24$09 is taken in $1/24$10 for artifact suppression (Chen et al., 2020, Liu et al., 14 Jul 2025).

The 2020 implementation reports a highest spatial frequency $1/24$11, giving $1/24$12 for $1/24$13. This matches the measured FWHM of single UCNPs after CPSN processing. The 2025 implementation reports spatial resolution down to 33 nm, approximately $1/24$14, under the 800 nm emission channel; this was measured by fitting Gaussian profiles to line scans across two neighboring UCNPs, with FWHM $1/24$15 nm. The same study reports SNR $1/24$16 dB for single-power Gaussian imaging and up to 55 dB for the Deep-Fusion super-resolved output.

Benchmark results in the 2025 study further distinguish learned fusion from conventional Fourier fusion. In numerical simulations of clathrin-coated pits, Fourier-domain fusion yields 49 nm resolution with SNR $1/24$17 dB, whereas Deep-Fusion yields 40 nm resolution with SNR $1/24$18 dB. In experimental comparison, resolution is reported to improve from 91 nm for microtubules to 40 nm, with SNR raised by $1/24$19 dB. A dual-rod spacing of 210 nm is resolved clearly by Deep-Fusion, while Fourier fusion exhibits artifacts. For dense UCNPs under 800 nm, the reported FWHM is 95 nm for Fourier fusion versus 61 nm for Deep-Fusion. The 2025 work also demonstrates multiple emission bands at 455 nm, 550 nm, 650 nm, 740 nm, and 800 nm, and states applicability to UCNPs doped with Tm$1/24$20/Yb$1/24$21 or Er$1/24$22/Yb$1/24$23, and in principle to any nonlinear fluorophore with a measured power dependence.

7. Scope, misconceptions, and open technical constraints

CPSN is not presented in the underlying reports as a purely optical PSF-shrinking scheme. The 2020 implementation depends on heterochromatic Fourier-domain fusion and Richardson–Lucy deconvolution, and the 2025 implementation adds learned fusion through a DRRN. The optical stage creates progressively complementary transfer characteristics; the computational stage constructs an image whose spectrum is assembled from those complementary bands. This suggests that CPSN should be understood as a joint emitter-engineering and reconstruction framework rather than as a stand-alone illumination pattern (Chen et al., 2020, Liu et al., 14 Jul 2025).

Several technical constraints are explicit. Accurate calibration of the power-dependence curve is required for each fluorophore. Training data must cover the full dynamic range of PSF shapes, and experimental ground truth is limited for very dense samples. Manual identification of cut-off frequencies is error-prone and can lead to artifacts and mismatches. At the same time, the reported signal-to-noise behavior is central to the method’s rationale: saturated emission increases contrast by flattening a PSF peak or dip, while fusion assigns mid-frequency content to the typically lower-noise two-photon channel and high-frequency content to the doughnut-like channel, suppressing reconstruction artifacts.

Within the cited literature, CPSN therefore occupies a specific methodological niche: it uses multilevel nonlinear saturation in UCNPs to generate PSFs that change progressively with wavelength channel or excitation power, then exploits those changes computationally to expand spatial-frequency support. The reported implementations differ in reconstruction strategy—binary Fourier fusion with Richardson–Lucy deconvolution in 2020, DRRN-based fusion in 2025—but they share the same underlying premise: progressively saturated emission can be converted into complementary spectral measurements and recombined into a super-resolved image.