Cherenkov Spread Functions (CSFs) Overview

• Cherenkov Spread Functions (CSFs) are quantitative descriptors that map the redistribution of Cherenkov light based on experimental geometry and inverse problem requirements.
• Their formulations vary by context—from telescope point-spread functions and air-shower angular distributions to lateral density profiles and depth-dependent imaging kernels.
• CSFs enable precise design and reconstruction calculations, highlighting the need for domain-specific calibration, parameterization, and validation across different Cherenkov applications.

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Cherenkov spread functions (CSFs) denote quantitative descriptions of how Cherenkov light is distributed after production, propagation, and detection, but the term is not used identically across all subfields. In imaging atmospheric Cherenkov telescope design, the CSF is the ensemble of optical point-spread functions across off-axis angle for a reflector-camera system [1301.6556]. In air-shower reconstruction, it denotes the differential angular distribution of Cherenkov photons about the shower axis [2007.13812]. In Tunka-133 studies, it is identified with the lateral distribution function of Cherenkov photon density on the ground [1810.11382]. In Cherenkov luminescence imaging, it is a depth-dependent discrete point-spread kernel used for attenuation and scatter correction in quantitative dosimetry [2509.13533]. This suggests that the shared content of the term is not a unique formula, but a family of spread kernels adapted to distinct geometries and inverse problems.

1. Scope of the term across Cherenkov methodologies

The literature uses “Cherenkov spread function” for several related but non-equivalent observables. The common structure is a mapping from an underlying source distribution to an observable angular, lateral, or image-plane light distribution.

Context	CSF quantity	Primary role
Davies–Cotton telescope optics	Off-axis ensemble of PSFs	Telescope design and pixel matching
Air-shower angular modeling	$\frac{d^2N_\gamma}{d\theta\,dX}$	Shower reconstruction
Tunka-133 array analysis	$Q(E_0,R)$ or $Q(n,R)$	LDF fitting and mass sensitivity
Cherenkov luminescence imaging	$h(z,r)$	Depth-dependent scatter correction

This terminological plurality matters because normalization, coordinates, and validity ranges differ. A telescope CSF is tied to focal ratio, tessellation, and pixel field-of-view; an air-shower CSF is tied to shower age, refractive index, and primary energy; a lateral-distribution CSF is tied to core distance and primary type; and an imaging CSF is tied to source depth and tissue optical properties. A common misconception is that CSF refers to a single standard function. The cited works show instead that it is a domain-specific spread description whose meaning depends on the forward model being solved.

2. Optical CSFs in tessellated Davies–Cotton reflectors

For a Davies–Cotton reflector, the optical CSF is constructed from the point-spread function as a function of off-axis angle. Using the scale convention $D=1$ and focal ratio $f=F/D$, Bretz and Ribordy derive an analytic PSF for the ideal reflector from the surface
[
z = f-\sqrt{f^2-r^2}, \qquad r^2\equiv x^2+y^2,
]
with local normals chosen so that each differential element focuses onto $(0,0,2f)$. The tangential and sagittal root-mean-square spot sizes on the focal plane are written as
[
\Delta\xi^2(\phi,f)=\frac{1}{16}\sum_{i,j} c^{\xi}_{i,j}\,\phi^i\,f^{-j}, \qquad
\Delta\eta^2(\phi,f)=\frac{1}{16}\sum_{i,j} c^{\eta}_{i,j}\,\phi^i\,f^{-j},
]
with $\phi$ in radians. In practice, the expansion is truncated to $i\le 4$, $j\le 10$, keeping terms contributing at the $10^{-3}$ level or above. Finite tessellation is incorporated by rescaling $f\to f/w(N)$ and interpolating between spherical and ideal-Davies–Cotton coefficients; for $N\ge 5$, one finds $w\approx 1\pm 0.02$ and residuals below $5\,\%$ over $0^\circ\le \phi\le 6^\circ$. A compact representation of the combined PSF is
[
P(f,\phi,N)\simeq c_0(\phi,N)\,f^{-c_1(\phi,N)},
]
with accuracy better than a few percent for $0^\circ\le \phi\le 6^\circ$, $1\le f\le 2$, and $N\ge 5$ [1301.6556].

The CSF at field angle $\phi$ is then the root-mean-square spot size convolved with the pixel response. Defining the ratio
[
r\equiv \theta_{\rm pixel}/P,
]
the edge-of-field design condition becomes
[
\theta_{\rm pixel}=r\,P(f,\phi_{\max},N),
]
where the “$4\sigma$-rule” often uses $r\approx 4$. Since $\theta_{\rm pixel}=\delta/F$ and $F=fD$, this yields the analytic linkage
[
f=\left[\frac{\theta_{\rm pixel}}{r\,c_0}\right]^{-1/c_1},
\qquad
D^2=\frac{4g}{k\,\theta_{\rm pixel}^2}.
]
In this framework, the PSF-to-CSF mapping is explicit:
[
\mathrm{CSF}(\phi)=P(f,\phi,N)\propto f^{-c_1(\phi,N)}.
]

The design consequences are stated directly. Camera field-of-view only shifts the allowed $f$-range along the same $D$–$\theta_{\rm pixel}$ curve. Tessellation $N\ge 5$ makes nearly no difference, whereas $N=3$ is measurably worse. The pixel field-of-view scales as $\propto 1/\sqrt{A'}\,(n'/n)^2$, so doubling the cone refractive index doubles $D$ for the same $\theta_{\rm pixel}$. Tighter PSF containment, corresponding to smaller $r$, requires larger $f$ and hence larger $D$ at fixed $\theta_{\rm pixel}$. Solid cones with $n'\approx 1.5$ yield almost twice the effective aperture compared with air-filled cones, at the price of about $2$–$5\,\%$ absorption loss; those losses affect overall optical efficiency but do not change the PSF. Timing dispersion is not covered by the DC-PSF itself, though the paper notes that it can be controlled by slight modification of facet focal lengths. A worked example with $D=4\,\mathrm{m}$, $f=1.5$, $F=6\,\mathrm{m}$, $N=5$, $\theta_{\rm pixel}=0.2^\circ$, and $\phi_{\max}=5^\circ$ gives $P(0^\circ,1.5,5)\approx 0.0057^\circ$ on-axis and $P(5^\circ,1.5,5)\approx 0.076^\circ$ at the field edge, implying $\theta_{\rm pixel}\approx 0.30^\circ$ for $r=4$ containment.

3. Angular CSFs for air-shower Cherenkov emission

Arbeletche and de Souza formulate the CSF as the differential angular distribution of Cherenkov photons per unit slant depth,
[

\frac{d^2N_\gamma}{d\theta\,dX}(\theta,s,h,E_0)

\frac{1}{\pi}\,N_e(s)\,\sin\theta\,I(\theta,h)\,K(\theta,s,h,E_0),
]
where $N_e(s)$ is the total number of electrons at shower age $s=3X/(X+2X_{\max})$. The factor $\sin\theta/\pi$ is the Jacobian from uniform azimuthal emission. The geometry-cone factor is
[

I(\theta,h)

\frac{1}{\sin\langle\theta_p\rangle}
\begin{cases}
\pi-\ln!\left(1-\frac{\theta}{\theta_{\rm em}}\right), & \theta\le \theta_{\rm em},\[1ex]
\pi-\ln!\left(1-\frac{\theta_{\rm em}}{\theta}\right), & \theta>\theta_{\rm em},
\end{cases}
]
with
[
\theta_{\rm em}(h)=\arccos!\bigl[1/n(h)\bigr],
\qquad
\langle\theta_p\rangle(\theta)=
\begin{cases}
\theta_{\rm em}, & \theta\le \theta_{\rm em},\
\theta, & \theta>\theta_{\rm em}.
\end{cases}
]
The cascade-convolution factor is
[

K(\theta,s,h,E_0)

C(s,h,E_0)\,
\langle\theta_p\rangle^{\nu-1}
\exp!\bigl[-\langle\theta_p\rangle/\theta_1\bigr]
\bigl[1+\epsilon\,\exp!\bigl(\langle\theta_p\rangle/\theta_2\bigr)\bigr].
]
The four free parameters are parametrized as
[
\nu(s,n)=p_{0,\nu}(n-1)^{\,p_{1,\nu}+p_{2,\nu}\ln s},
]
[

\theta_1(s,n,E_0)

p_{0,\theta_1}
(n-1)^{\,p_{1,\theta_1}(E_{\rm TeV})^{\,p_{2,\theta_1}+p_{3,\theta_1}\ln s}},
]
[
\theta_2(s,n)=\theta_1(s,n)\,[\,p_{0,\theta_2}+p_{1,\theta_2}s\,],
\qquad
\epsilon(E_0)=p_{0,\epsilon}+p_{1,\epsilon}(E_{\rm TeV})^{\,p_{2,\epsilon}},
]
with coefficients fitted separately for $\gamma$ and proton primaries [2007.13812].

The physical interpretation is explicit. $I(\theta,h)$ captures the finite opening angle of the Cherenkov cone and approximates the elliptic integral through a logarithmic form. $\langle\theta_p\rangle$ is the mean electron-scattering angle over the narrow interval $\theta\pm\theta_{\rm em}$. The factor $K$ summarizes the Cherenkov yield, the electron energy spectrum, the electron angular distribution, and the geometrical track-length correction. The assumptions are also explicit: wavelength-independent refractive index over $300$–$450\,\mathrm{nm}$, $\theta_{\rm em}(E)\approx \theta_{\rm em}(h)$ for $E\gg E_{\rm thr}$, slow variation of $dN_e/d\theta_p$ and $dN_e/dE$ across the narrow integration window, and factorization of the full two-dimensional electron distribution into the one-dimensional functions $I$ and $K$.

The reported precision is one of the main technical outcomes. For $\gamma$ showers from $100\,\mathrm{GeV}$ to $1\,\mathrm{EeV}$, the average relative deviation is below $5\,\%$ for $\theta<25^\circ$ across ages $s=0.8$–$1.2$, and below $10\,\%$ even out to $\theta\approx 30^\circ$ at low energies. For proton showers from $1\,\mathrm{TeV}$ to $1\,\mathrm{EeV}$, the deviation is below $10\,\%$ for $\theta<30^\circ$ at $E_0\ge 10\,\mathrm{TeV}$, improving with energy. The paper contrasts this with earlier parametrizations that typically exceeded $20$–$30\,\%$ error near the Cherenkov peak or at large angles and were valid only in one angular regime. A common misconception addressed by this work is that one parametrization for the small-angle imaging-atmospheric-Cherenkov-telescope regime automatically remains accurate at large angles relevant to fluorescence-detector reconstruction; the paper’s stated motivation is precisely that such a description had been unavailable.

4. Lateral-distribution CSFs in the Tunka-133 array context

In the Tunka-133 study, the CSF is identified with the lateral distribution function of Cherenkov photon density,
[
Q(E_0,R)=\frac{\Delta N_\gamma(E_0,R)}{\Delta S},
]
where $N_\gamma(E_0,R)$ is the total number of Cherenkov photons from a shower of primary energy $E_0$ reaching a ring at core distance $R$, and $\Delta S$ is the collecting area. For the analysis presented, the primary energy is fixed at $E_0=3\times 10^{15}\,\mathrm{eV}$ and the notation is simplified to $Q(n,R)$, with $n$ labeling the primary nucleus. The adopted analytic form is a four-parameter Breit–Wigner-type parameterization,
[

Q(n,R)

\frac{a(n)}
{\beta(n)\,\Bigl[\,(R/B(n))^{2}+(R-k(n))^{2}/\beta(n)^{2}+R_{0}^{2}/B(n)\Bigr]},
]
with $R_0$ taken as a constant length scale and, in practice, chosen to be $100\,\mathrm{m}$ in the cited references. The intermediate quantity $\beta(n,R)$ is introduced by the authors through an implicit relation that must be solved iteratively, and each parameter
[
Y(n)\in{a(n),B(n),\beta(n),k(n)}
]
is modeled by
[
Y(n)=C_0+C_1\log_{10}n+C_2(\log_{10}n)^2+C_3(\log_{10}n)^3.
]
Tabled coefficients are given for the four functions $Y=a,B,\beta,k$ and are reported as valid for approximately $0\,\mathrm{m}\le R\le 600\,\mathrm{m}$, corresponding to the extent of the Tunka-133 optical stations [1810.11382].

The simulation setup is specific: vertical showers, primaries H, He, Li, Be, Ne, Na, Mg, Al, Sc, Ti, V, and Cr, CORSIKA v.6.x, QGSJET for the high-energy hadronic model, GHEISHA for the low-energy model, Tunka-133 array conditions at about $675\,\mathrm{m}$ above sea level, and detector response idealized as counting photons on a flat $1\,\mathrm{m}^2$ area. The fit quality is quantified by minimizing
[

\chi_{\min}

\sum_i \left|
\frac{Q_{\rm par}(n,R_i)}{Q_{\rm CORSIKA}(R_i)}-1
\right|
\longrightarrow \min.
]
The tabulated minima include $1.4699\times 10^{-3}$ for hydrogen, $2.2873\times 10^{-3}$ for helium, $8.5901\times 10^{-3}$ for lithium, $7.1613\times 10^{-2}$ for sodium, and $3.5013\times 10^{-3}$ for chromium. The paper states that the analytic curves overlay the raw CORSIKA points and demonstrate agreement at the sub-percent level over nearly the full radial range.

The principal physical observation is that lighter primaries yield broader, somewhat flatter lateral distributions, whereas heavier nuclei produce a more sharply peaked LDF with higher central density and steeper fall-off. The paper attributes this to earlier fragmentation of heavy nuclei in the atmosphere, which produces a more compact electromagnetic sub-cascade and hence a narrower Cherenkov light pool. Because $B(n)$, $\beta(n)$, and $k(n)$ vary smoothly with $\log_{10}n$, the parametrization is presented as a handle on mass-composition reconstruction from the slope and curvature of the measured LDF.

5. Depth-dependent CSFs in Cherenkov luminescence imaging

In quantitative Cherenkov luminescence imaging of $^{86}$Y-NM600, the CSF is a depth-indexed two-dimensional discrete kernel $h(z,r)$ derived from GEANT4 and normalized so that $\sum_r h(z,r)=1$. The simulation geometry is a $2\,\mathrm{cm}\times 2\,\mathrm{cm}\times 2\,\mathrm{cm}$ cubic “skin” phantom suspended in air, with a $0.1\,\mathrm{cm}$ thick planar scoring volume attached to the top face. Point-source locations of $^{86}$Y positron emitters are placed at depths $z=2,3,\ldots,10\,\mathrm{mm}$ below the phantom surface. Optical properties are specified over $400$–$1000\,\mathrm{nm}$: refractive index $n=1.4$, absorption coefficient
[

\mu_a(\lambda)

B\,\mu_{a,\mathrm{oxy}}(\lambda)
+
(1-B)\,\mu_{a,\mathrm{deoxy}}(\lambda)
+
W\,\mu_{a,\mathrm{water}}(\lambda)
+
F\,\mu_{a,\mathrm{fat}}(\lambda),
]
with oxygenated blood fraction $B=0.34$, water $W=0.214$, and fat $F=0.277$; and reduced scattering split into Rayleigh and Mie terms,
[
\mu'{s,\mathrm{Ray}}(\lambda)=a' f{\mathrm{Ray}}(\lambda/500\,\mathrm{nm})^{-4},
\qquad
\mu'{s,\mathrm{Mie}}(\lambda)=a'(1-f{\mathrm{Ray}})(\lambda/500\,\mathrm{nm})^{-b},
]
with $a'=48\,\mathrm{cm}^{-1}$, $f_{\mathrm{Ray}}=0.409$, and $b=0.702$. Mie anisotropy uses a double Henyey–Greenstein model with $g_F=0.91$, $g_B=0.87$, and forward/backward ratio $\mathrm{MieFB}=0.85$. Full electron transport and optical photon tracking are simulated, and each Cherenkov photon crossing into the scoring volume is recorded and then terminated [2509.13533].

The resulting radial profiles broaden strongly with source depth: the full-width at half-maximum increases linearly with $z$, from about $2.9\,\mathrm{mm}$ at $2\,\mathrm{mm}$ depth to about $11.8\,\mathrm{mm}$ at $10\,\mathrm{mm}$ depth, while the peak amplitude decreases with depth. No closed-form analytic formula is imposed; the kernels can be used directly in discrete form, though Gaussian approximations may be adequate for some depths. Calibration is then performed with a 96-well phantom containing known $^{86}$Y activities and overlaid with tissue-mimicking fluid at depths $1.66$, $3.31$, $4.97$, $6.62$, $8.28$, and $9.93\,\mathrm{mm}$. CLI radiance after background subtraction is linear in plated activity at each depth, and the depth-dependent calibration coefficient $C(d)$ defined by
[
R(d,A)=C(d)\,A
]
is fit across $1.66$–$9.93\,\mathrm{mm}$ to a line with slope about $-613.8$ ((\mathrm{photons}\cdot \mathrm{s}^{-1}\cdot \mathrm{cm}^{-2}\cdot \mathrm{sr}^{-1})/(\mathrm{MBq}\cdot \mathrm{mm})) and intercept about $9734.2$.

These CSFs are integrated into a depth-weighted Richardson–Lucy deconvolution and reconvolution framework. With measured image $g(x,y)$, source estimate $f(x,y)$, depth-specific kernels $h_z(x,y)$, and depth weights $p_z(x,y)$ normalized so that $\sum_z p_z=1$, the forward model is
[
g(x,y)\simeq \sum_z [f(x,y)\otimes h_z(x,y)]\,p_z(x,y).
]
For each depth layer,
[

f_z^{(k+1)}

f_z^{(k)}
\cdot
\left[
h_z^{\mathrm{flip}}
\otimes
\frac{g}{h_z\otimes f_z^{(k)}}
\right],
]
and after $K=10$ iterations the depth-specific images are averaged using weights $w_z$ that depend on voxel counts at depth $z$ in the region of interest and on the integral of $h_z$ over its FWHM:
[
f_{\mathrm{deconv}}(x,y)=\sum_z w_z\,f_z^{(K)}(x,y).
]
After masking and reconvolution,
[
f_{\mathrm{reconv}}(x,y)=\sum_z w_z\,[f_{\mathrm{deconv}}\otimes h_z],
\qquad
\hat{A}(x,y)=\frac{f_{\mathrm{reconv}}(x,y)}{C[d(x,y)]}.
]
The reported quantitative outcome is that CLI–PET activity quantification yields mean errors of $15.4\,\%$ in liver and $10.3\,\%$ in tumor over the first three timepoints, while tumor absorbed doses from CLI-derived synthetic PET images, $3.4\pm 0.3\,\mathrm{Gy/MBq}$, are statistically indistinguishable from PET-based estimates, $3.2\pm 0.2\,\mathrm{Gy/MBq}$ with $p=0.31$. The paper further states that discrepancies increase at late timepoints because of low activity and background auto-luminescence. A common misconception rejected explicitly by the method is that a single optical blur kernel can be applied independent of isotope or depth: the work states that CSFs were generated only for $^{86}$Y and that other isotopes require regeneration because of differing $\beta$ spectra.

6. Comparative interpretation, assumptions, and limitations

Across these usages, CSFs are forward kernels, but their domains of validity are sharply constrained. For the Davies–Cotton telescope case, the analytic PSF is stated to be valid for prime-focus $f/D$ between $1.0$ and $2.0$, field angles up to about $6^\circ$, and tessellation $N\ge 5$, with accuracy better than $5\,\%$; the Taylor series are stated to remain valid to about $10^\circ$, and larger fields up to $12^\circ$ in small-SST designs are handled by increasing $\phi_{\max}$ in the same formalism [1301.6556]. For the air-shower angular CSF, the assumptions include wavelength-independent $n(h)$ over $300$–$450\,\mathrm{nm}$, $\theta_{\rm em}\lesssim 1.5^\circ$, slow variation of electron distributions across the integration window, and factorization into $I(\theta,h)$ and $K(\theta,s,h,E_0)$ [2007.13812]. For the Tunka-133 LDF parameterization, the analysis is fixed to vertical showers at $3\,\mathrm{PeV}$ and specific primary nuclei under Tunka-133 conditions [1810.11382]. For CLI dosimetry, the assumptions include planar imaging approximation, homogeneous optical properties within each ROI, CT-derived depth maps, and accurate segmentation; mouse curvature and out-of-plane diffusion are neglected [2509.13533].

These differences explain why direct transfer of a CSF from one application to another is generally inappropriate. The telescope CSF is a field-angle optical aberration model; the shower CSF is an emission-angle distribution; the Tunka-133 CSF is a ground-level photon-density profile; the CLI CSF is a tissue-scatter kernel. This suggests that the unifying scientific role of CSFs is methodological rather than taxonomic: they encode the propagation-induced redistribution of Cherenkov light in whatever coordinates are natural to the experiment. Their scientific value lies in enabling explicit inversion or design calculations—pixel matching and cost-performance tradeoffs in telescope optics, unified reconstruction over small and large angles in air showers, primary-mass sensitivity in lateral-distribution analyses, and attenuation/scatter-corrected activity recovery in luminescence imaging.

Within that shared role, several clarifications follow directly from the cited works. Light concentrators in the telescope design formalism enter through the system constant $k(n,n',A')$ and affect effective aperture, but do not change the PSF itself [1301.6556]. In the angular air-shower formalism, the normalization factor $C$ is absorbed into $N_e(s)$ in practice, so the model is primarily a shape parametrization rather than a stand-alone absolute-yield prediction [2007.13812]. In the Tunka-133 parameterization, the smooth dependence on $\log_{10}n$ is part of the fitted model and not a general first-principles law of hadronic composition [1810.11382]. In CLI, depth-dependent attenuation calibration and Monte Carlo-derived scatter correction are both required for quantitative recovery; the CSF alone does not correct attenuation [2509.13533].

The resulting picture is a technically coherent but non-uniform concept: a CSF is a spread description of Cherenkov light whose exact definition is set by experimental geometry, detector response, and the inverse problem under study.