{"id":"5b5b97e8-f80f-4d0d-916b-0b452291c92b","arxiv_id":"2505.17754","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A reformulation of standard partial coherence propagation as a surface operator followed by propagation, whose equivalence to Van Cittert-Zernike is true by construction and whose claimed computational gains rely on an unfair comparison.","lead":"This paper proposes a two-step operator framework for modeling partially coherent light in wave optics, where coherence is 'encoded' at a surface and then propagated. The claimed equivalence to standard Van Cittert-Zernike theory is true only because the surface kernel is set to the known result, and the method does not actually generate partial coherence from a coherent input.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The deterministic kernel cannot create partial coherence: applying CS to a plane wave with a stationary kernel yields a rank-one constant correlation, so Eq. (32) and Appendix A assume the statistics they claim to produce.","rationale":"I read the paper as claiming that a deterministic surface operator can encode partial coherence and reproduce the Van Cittert-Zernike theorem. That claim requires CS applied to a coherent field to produce a field whose second-order correlation equals a non-rank-one kernel. Since CS is deterministic, its output field is deterministic and its correlation is rank-one; the derivation in Eq. (31) makes the factorization explicit, and Appendix A supplies the missing correlation by hand. This is the same weak point the reader identified, and it is load-bearing because Theorem 2.1, Theorem 2.4, Theorem 4.1, and the claimed complexity advantage all depend on it. I agree with the REJECT verdict and would keep it. I give credit where due: the Fresnel propagation step and the positive-semidefinite kernel lemma (Lemma A.1) are standard, but they do not repair the central construction. The proposed numerical check is cheap and decisive.","tokens_in":24399,"tokens_out":4848,"duration_ms":42288,"concrete_test":"Implement one numerical check: on a 256×256 grid with unit sampling, set Ui=A0=1, R=1, and take KS(r-r')=exp(-|r-r'|^2/(2ρc^2))/(2πρc^2), i.e., Eq. (17) with normalization ∫KS=1. Compute Us=CS(Ui) by FFT convolution, then compute ΓS(r1,r2)=Us*(r1)Us(r2) for all pairs. If ΓS is numerically constant in r1-r2 rather than |A0|^2exp(-|r1-r2|^2/(2ρc^2)), Theorem 2.4 is false. To rule out finite-size edge effects, repeat with ρc much smaller than the grid size and compare interior points; the flatness persists. This single check settles whether CS can encode partial coherence without randomness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim fails inside Theorem 2.4. With Ui=A0 and R=R0, the field entering CS is the constant B0. For any deterministic linear operator with a stationary kernel, Us(r)=B0∫KS(r-r')d2r'; with the normalization used later this equals B0. There is no ensemble, so ΓS(r1,r2)=⟨Us*(r1)Us(r2)⟩=|B0|^2, independent of separation. More generally, any deterministic output field gives a rank-one correlation Us*(r1)Us(r2), which cannot equal a genuine partially coherent kernel such as Eq. (7) or Eq. (17). The double integral in Eq. (31) factorizes into |B0|^2[∫KS(r1-r1')dr1']*[∫KS(r2-r2')dr2'], a constant for stationary kernels, not |B0|^2KS(r1-r2). The proof then 'recognizes' the surface field as Us=B0+ΔUs and assigns ⟨ΔUs*(r1)ΔUs(r2)⟩=|B0|^2[γS(r1,r2)-1] by hand (Appendix A, before Eq. (95)); that is the conclusion, not a derivation. Appendix A's subsequent delta-function evaluation uses an infinite aperture and still only works if the correlation of ΔUs is already taken to be the kernel. Theorem 4.1 repeats the same move by calling grid points 'independent random samples' despite deterministic kernels. Thus Theorem 2.1's equivalence to VCZ is circular, and the framework computes coherent diffraction through a filter, not partial coherence. The only escape is to make CS stochastic, which concedes the paper's deterministic claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a 'Surface-Encoded Coherence Transformation' (SECT) framework for modeling partial coherence in wave optics. The framework consists of a surface operator CS, defined by a kernel KS, and a free-space propagation operator Pz; the central claim is that applying CS to a fully coherent incident field deterministically produces a field with the second-order statistics of a partially coherent field, and that the composition Pz∘CS is mathematically equivalent to the Van Cittert–Zernike (VCZ) theorem when KS is chosen as the Fourier transform of the source intensity. The authors further claim an O(M N^2 log N) complexity for M-surface systems versus O(N^{2M}) for conventional mutual-coherence propagation, and provide convergence, energy-conservation, and operator-property results. The paper contains no numerical experiments, no code, and no comparison with existing partial-coherence simulations.","tokens_in":24821,"tokens_out":3273,"duration_ms":33891,"significance":"If the central claim were correct, the paper would offer a genuinely new viewpoint: encoding partial coherence through a deterministic surface operator rather than through ensembles, together with an exponential complexity reduction for multi-surface systems. The manuscript also correctly reviews standard coherence formalism and includes a correct positive-semidefiniteness argument for Fourier-transform kernels (Lemma A.1). However, the load-bearing assertion—that a deterministic linear operator can turn a coherent field into a field with genuine partially coherent statistics—is not established and is in fact contradicted by the manuscript's own equations. The claimed equivalence to VCZ is circular because the appendix assumes the desired correlation function rather than deriving it. Consequently, the advertised conceptual and computational advantages are not realized by the presented framework. The paper would need a fundamentally different stochastic or ensemble-based construction to support the stated claims, which is outside the current manuscript's scope.","major_comments":[{"comment":"The reduction from the double integral in Eq. (31) to Eq. (32) is unjustified. For a deterministic linear operator with stationary kernel KS applied to the constant field B0, one obtains ΓS(r1,r2)=|B0|^2 [∫ KS(r1−r') d^2r']* [∫ KS(r2−r'') d^2r''], which is a constant in (r1,r2) for any integrable stationary kernel, not |B0|^2 KS(r1−r2). No 'appropriate normalization' can convert a constant into a function of r1−r2 unless KS itself is a delta function, in which case the output field is still B0 and has no fluctuations. Thus Theorem 2.4 does not establish that the SECTS component produces a partially coherent field.","section":"Theorem 2.4, Eqs. (29)–(32)"},{"comment":"The proof of SECT-VCZ equivalence assumes precisely the statistics it is supposed to derive. The field at the surface is written as Us(r)=B0+ΔUs(r) with ⟨ΔUs⟩=0 and ⟨ΔUs*(r1)ΔUs(r2)⟩=|B0|^2[γS(r1,r2)−1], where γS is already taken to be the Fourier-transformed source intensity. This is the desired second-order correlation, inserted by hand. The subsequent Fresnel propagation only propagates this assumed correlation; it does not show that the deterministic operator CS produces such a correlation. Theorem A.2 is therefore circular.","section":"Appendix A, Eq. (94)–(95)"},{"comment":"The same unjustified factorization appears in the combined formulation. Equation (47) gives Γd(r1,r2)=|A0|^2∬ Ktot*(r1,r1')Ktot(r2,r2') d^2r1' d^2r2', which is a rank-one kernel in (r1,r2) for any deterministic Ktot. Equation (48) then claims this reduces to |A0|^2 Ktot(r1−r2). That reduction is valid only if the double integral collapses to a single kernel value, which in general it does not; the step silently assumes that the product of the two integrals factorizes into Ktot(r1−r2), which is false for any non-delta kernel. This invalidates the claimed unified operator equivalence.","section":"Section 2.7, Eqs. (46)–(48)"},{"comment":"The convergence theorem asserts that the SECT framework converges to the ensemble average, but the proof invokes randomness that the framework does not contain. The statement that 'each grid point effectively represents an independent random sample from the ensemble' is unsupported because CS is a deterministic convolution; no probability space or ensemble of realizations is defined. In the ρc→0 limit claimed to produce independent samples, the kernel KS approaches a delta function and the output becomes B0, with no fluctuations at all. The central limit theorem and law of large numbers are therefore inapplicable, and the claimed convergence is not established.","section":"Theorem 4.1, proof surrounding Eqs. (63)–(70)"}],"minor_comments":[{"comment":"Several equations have inconsistent dimensional scalings: Eq. (6) writes KS(r−r')∝F{Is(ρ/λz)}(r−r'), but the Fourier transform of Is(ρ/λz) contains factors of (λz)^2 that are not tracked, and the same issue recurs in Eqs. (9)–(11) and (14)–(16). The notation should be made dimensionally consistent.","section":"Eq. (6) and Appendix A"},{"comment":"The η values listed for each scenario are asserted without any derivation, simulation, or measurement. The claims such as 'η≈0.1 (SECTP dominant)' are unsupported and should be either derived from a concrete model or removed.","section":"Section 3.2, Table 2"},{"comment":"The figure claims coherence length increases from approximately 1–2 units at the surface to 2.5–5.5 units at the detection plane, but no simulation parameters, numerical method, or error bars are given. If this is an illustrative schematic, the caption should say so explicitly.","section":"Figure 4 and caption"},{"comment":"The energy conservation claim ∫|Ctot(U)|^2 = ∫|U|^2 for arbitrary convolution kernels is not generally true; a Gaussian low-pass kernel, for example, reduces total energy. No proof is provided, and the proposition should either be restricted to unitary (phase-only) kernels or corrected.","section":"Proposition 4.2"},{"comment":"The complexity claim O(M N^2 log N) assumes stationary kernels and FFT-based convolution for both CS and Pz. For curved surfaces and non-planar detectors, the manuscript itself notes that FFT shortcuts are unavailable (Section 5.1.3), so the stated complexity reduction does not apply to the generalized framework.","section":"Section 5.3"},{"comment":"There are numerous typos and unfinished expressions, e.g., 'e ffects', 'di fferent', and the dangling 'R' in 'and Appendix A begins by assuming' do not appear in the manuscript, but the text contains repeated spacing errors and inconsistent notation such as σθ used both as an angular width and a radial coordinate in Corollary 2.3. A thorough copyedit is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central contribution is a deterministic operator that is claimed to produce partial coherence. This contradicts the standard definition of partial coherence as an ensemble or time-average property, and the manuscript's own equations show that a deterministic linear operator yields a rank-one correlation. The flaw is not a missing derivation step; it is the core mechanism of the framework. Fixing it would require making CS stochastic, which would abandon the paper's advertised deterministic advantage and change the entire scope. I therefore recommend rejection rather than major revision. Additionally, the paper makes strong claims of 'rigorous mathematical proofs' and 'exponential improvement' without any numerical validation or comparison to existing methods; such claims should be tempered in any future submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know before spending time on this: the central claim—that a deterministic linear operator can turn a coherent field into a partially coherent one—is unsupported, and on its own equations it fails. The paper is otherwise a clean restatement of standard coherence theory (VCZ kernels for circular/Gaussian sources, Fresnel propagation of the mutual coherence, operator notation). If you need a compact summary of those textbook results, Sections 2.1–2.3 and the corollaries are serviceable. The multi-surface complexity point is also sensible in isolation: sequential surface+propagation steps at O(N^2 log N) each is better than a naive 2M-dimensional integral.\n\nThe soft spot is load-bearing, not cosmetic. Theorem 2.4 aims to show that applying CS to a fully coherent plane wave yields a field with mutual coherence Gamma_S = |B0|^2 K_S. But with a constant input and a deterministic, stationary kernel, the output field itself is constant (or at least a fixed function), so Gamma_S(r1,r2) = Us^*(r1)Us(r2) is rank one and independent of separation—it cannot equal K_S(r1-r2). The proof jumps from the double integral in Eq. (31) to the kernel in Eq. (32) without justification; for a stationary kernel that double integral factorizes into a product of two constants times |B0|^2, not K_S. Appendix A then simply asserts the desired correlation <Delta U_s^*(r1)Delta U_s(r2)> = |B0|^2[gamma_S - 1] by hand. That is assuming the conclusion. The only escape would be to make CS stochastic, which concedes the paper's deterministic claim.\n\nThere are also smaller internal inconsistencies. Proposition 4.2 (energy conservation for every input) is incompatible with Lemma 4.3 (all eigenvalues between 0 and 1) unless the operator is unitary, and the surface kernel is a low-pass filter, not unitary. The Gaussian kernel in Eq. (13) and Eq. (56) differ in a way that is never reconciled. The claimed O(M N^2 log N) versus O(N^{2M}) comparison is also apples-to-oranges; standard mutual-coherence propagation is not inherently exponential, and existing fast methods (e.g., FFT-based angular spectrum for the mutual intensity) already achieve similar scaling for many practical cases.\n\nBottom line: this is not a new result. It is a repackaging of VCZ with an operator label, plus a central claim that fails on inspection. The tutorial parts could be useful to a student, but as a research contribution it does not hold up. I would not send it to referees.","headline":"A clean restatement of textbook coherence results wrapped around a deterministic-coherence claim that fails on its own equations; not a new result and not ready for referees.","tokens_in":759,"tokens_out":857,"would_cite":false,"duration_ms":33772,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["42.25.Kb","42.25.Fx"],"model":"deepseek-v4-flash","headline":"This paper argues that a single surface operator can imprint partial coherence on a wavefront, after which standard Fresnel propagation reproduces Van Cittert–Zernike theory.","keywords":["partial coherence","mutual coherence function","Van Cittert-Zernike theorem","surface coherence operator","Fresnel propagation","wave optics simulation","Gaussian-Schell model","spatial filtering"],"falsifier":"For a constant input $U_i=A_0$ and any spatially stationary kernel $K_S(r-r')$, the surface operator output is $U_s(r)=B_0\\int K_S(r-r')\\,d^2r'$, which is constant in $r$; the resulting correlation $\\Gamma_S(r_1,r_2)=|B_0|^2\\,|\\int K_S|^2$ does not depend on $r_1-r_2$, whereas Theorem 2.4 claims it equals $|B_0|^2K_S(r_1-r_2,\\lambda)$. Computing this two-point correlation for a constant input settles the central claim.","tokens_in":24160,"feed_emoji":"🔭","tokens_out":14058,"duration_ms":142269,"temperature":0.7,"pith_summary":"The paper proposes a way to model partially coherent light without generating many random realizations of the field. A single linear operator acting at the reflecting surface—the surface coherence operator—is said to imprint the correct spatial correlations onto an otherwise coherent wavefront, and ordinary Fresnel propagation then carries those correlations to the detector. The central claim is that this two-step map is mathematically equivalent to the Van Cittert–Zernike theorem when the surface kernel is chosen as the Fourier transform of the source intensity distribution. If correct, the approach would make the computational cost of coherence simulations grow with the number of optical surfaces rather than with the full complexity of the correlation function.","feed_headline":"One surface operator imprints partial coherence on light","feed_subtitle":"A two-step model claims to encode partial coherence at the surface, avoiding costly ensemble averaging.","key_machinery":"The central object is the surface coherence operator $C_S$, a linear integral operator with kernel $K_S(r,r',\\lambda)$, composed with the Fresnel propagation operator $P_z$ as $U_d=P_z(C_S(R(U_i)))$. The identity doing the work is the claimed surface correlation $\\Gamma_S(r_1,r_2)=|B_0|^2K_S(r_1-r_2,\\lambda)$, which converts the kernel into a coherence function, and the kernel choice $K_S=\\mathcal{F}\\{I_s(\\rho/\\lambda z)\\}$ that links the construction to the Van Cittert–Zernike theorem. The proof machinery is the coherence-function formalism, the Fresnel propagator, and the convolution theorem applied to the double propagation integral.","core_discovery":"The paper's discovery claim is that the mutual coherence function of a partially coherent field can be built in two deterministic steps rather than tracked as a four-dimensional object through the entire system. At the surface, the coherence operator $C_S$ with kernel $K_S(r,r',\\lambda)$ acts on the reflected field; for a uniform plane-wave input the claimed result is $\\Gamma_S(r_1,r_2)=|B_0|^2 K_S(r_1-r_2,\\lambda)$, so the kernel itself becomes the mutual coherence function. Propagation to the detector is carried by the Fresnel operator $P_z$, and the full composition $U_d=P_z(C_S(R(U_i)))$ is claimed to be equivalent to the Van Cittert–Zernike theorem when $K_S$ is the Fourier transform of the source intensity. The paper derives explicit Airy and Gaussian kernels for circular and Gaussian sources, extends the framework to curved surfaces and polychromatic light, and claims the multi-surface computation scales as $O(M N^2\\log N)$ instead of $O(N^{2M})$.","pith_inferences":["Read constructively, the equivalence proof suggests that $C_S$ is best viewed as a definition of the target coherence function at the surface; a follow-up development would be to formalize $C_S$ as a map on the mutual coherence function itself rather than on field amplitudes.","The convergence theorem is stated as a limit with no rate, so a practical extension is to bound the finite-$N_c$ error and use it to set sampling requirements in simulations.","For segmented-mirror telescopes, the framework suggests a pipeline in which source coherence is imprinted once at the pupil and segment phase errors are applied deterministically; this could be tested against existing stellar interferometry measurements.","Replacing the Fresnel kernel with the exact Euclidean-distance kernel would extend the framework to non-paraxial and curved-surface systems, at the cost of losing the FFT speedup; the paper identifies this trade-off but does not quantify it."],"forward_implications":["For systems with multiple surfaces, the framework claims the cost of a coherence-aware simulation drops from $O(N^{2M})$ to $O(M N^2\\log N)$, because each surface–propagation pair is applied sequentially and stationary kernels can be implemented with FFTs.","The circular and Gaussian source corollaries give direct recipes for modeling stellar and Gaussian-Schell sources: choose the Airy or Gaussian kernel at the surface, then use ordinary Fresnel propagation.","The separation of surface and propagation effects yields a dimensionless parameter $\\eta$ that identifies whether surface roughness or propagation distance dominates the coherence change, guiding where to spend computational effort.","The combined operator $K_{\\text{tot}}$ acts as a spatial filter, so partial coherence effects can be computed and interpreted in the spatial-frequency domain.","Polychromatic and temporal coherence enter by spectral weighting and a tensor-product spatial/temporal operator, so the framework claims to cover broadband sources without changing its structure."],"supporting_citations":[{"why":"Defines the probabilistic basis for the Van Cittert–Zernike theorem, the classical result the SECT map is claimed to reproduce.","marker":"[1]"},{"why":"Introduces the degree of coherence and extends the theorem to optics, giving the target statistics for the surface kernel.","marker":"[2]"},{"why":"Supplies the mutual coherence function formalism and the space-time factorization used in the framework's definitions.","marker":"[3]"},{"why":"Provides the space-frequency coherence theory underlying the spectral generalization and Gaussian-Schell model forms.","marker":"[6]"},{"why":"Gives the statistical optics background and the ensemble-average benchmark that Theorem 4.1 claims to converge to.","marker":"[7]"},{"why":"Supplies the Gaussian-Schell beam propagation parameters used in the Gaussian source equivalence corollary.","marker":"[11]"},{"why":"Provides the standard definition and propagation law for the mutual coherence function used in the SECT propagation component.","marker":"[13]"},{"why":"Provides the Fresnel kernel that implements the propagation operator $P_z$.","marker":"[14]"},{"why":"Supports the stationary-kernel assumption by separating deterministic figure from statistically homogeneous residual roughness.","marker":"[16]"}],"fun_headline_variants":["Surface operator writes coherence, skipping costly averaging","Two-step model: surface kernel becomes mutual coherence","Partial coherence from a single surface integral","Deterministic coherence: no more ensemble stacks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a single deterministic linear operation at the surface can produce the same second-order correlations as a genuinely random partially coherent field; with a constant coherent input, the output field has no fluctuation, so the required correlation statistics are effectively assumed rather than derived.","fun_headline_variants_meta":{"raw":{"variants":["Surface operator writes coherence, skipping costly averaging","Two-step model: surface kernel becomes mutual coherence","Partial coherence from a single surface integral","Deterministic coherence: no more ensemble stacks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000293,"raw_usage":{"total_tokens":1706,"prompt_tokens":941,"completion_tokens":765,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":709}},"tokens_in":557,"tokens_out":765,"duration_ms":6498,"temperature":1.0,"reasoning_tokens":709,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:42:21.694972+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a constant input $U_i=A_0$ and any spatially stationary kernel $K_S(r-r')$, the surface operator output is $U_s(r)=B_0\\int K_S(r-r')\\,d^2r'$, which is constant in $r$; the resulting correlation $\\Gamma_S(r_1,r_2)=|B_0|^2\\,|\\int K_S|^2$ does not depend on $r_1-r_2$, whereas Theorem 2.4 claims it equals $|B_0|^2K_S(r_1-r_2,\\lambda)$. Computing this two-point correlation for a constant input settles the central claim.","supporting_citations":[{"cited_title":"Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,","cited_arxiv_id":null,"evidence_quote":"Defines the probabilistic basis for the Van Cittert–Zernike theorem, the classical result the SECT map is claimed to reproduce."},{"cited_title":"The concept of degree of coherence and its application to optical problems,","cited_arxiv_id":null,"evidence_quote":"Introduces the degree of coherence and extends the theorem to optics, giving the target statistics for the surface kernel."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the statistical optics background and the ensemble-average benchmark that Theorem 4.1 claims to converge to."},{"cited_title":"Propagation parameters of Gaussian Schell- model beams,","cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian-Schell beam propagation parameters used in the Gaussian source equivalence corollary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the standard definition and propagation law for the mutual coherence function used in the SECT propagation component."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Fresnel kernel that implements the propagation operator $P_z$."},{"cited_title":"Bridging Statistical Scattering and Aberration Theory: Ray Deflection Function -- I: Theoretical Framework","cited_arxiv_id":"2505.01019","evidence_quote":"Supports the stationary-kernel assumption by separating deterministic figure from statistically homogeneous residual roughness."}],"review_version":1}