Breach of symmetries in rotating arrays and metamaterials observed in their rest frame

Ben Z. Steinberg; Ido Kazma

REVIEW 5 major objections 5 minor 7 references

Breach of symmetries in rotating arrays and metamaterials observed in their rest frame

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper shows that a rotating array of small scatterers reveals its rotation through Sagnac phase factors in the inter-particle Green's function, and that random or spiral geometries make that footprint much larger than periodic arrays…

desk verdict A compact, useful proceedings paper that cleanly shows the single-particle polarizability correction is second-order in rotation and proposes a testable Sagnac-loop heuristic, but its numerical claims rest on an approximate Green's function that the paper never validates against the exact series. read the letter →

arxiv 1908.05172 v1 pith:T4A3W3M5 submitted 2019-08-14 physics.optics

Ido Kazma , Ben Z. Steinberg This is my paper · ORCID

classification physics.optics

keywords rotatingmediumelectrodynamicsrest-frameformulationSagnaceffectdiscretedipoleapproximationpolarizabilitytheorygolden-anglespiralnon-reciprocityarrayrotationsensitivity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes how a rigidly rotating array of small dielectric scatterers reveals its rotation when observed in its own rest frame. The rotation enters the inter-particle interactions through a Sagnac phase factor multiplying the ordinary stationary Green's function, so the array response is the result of many closed scattering loops that collect rotation-dependent phases. The paper argues that this response is strongly geometry-dependent: periodic arrays, which contain collinear triples of scatterers that enclose zero area, are relatively insensitive, whereas random and golden-angle spiral arrays, in which every triple forms a nonzero-area triangle, show markedly larger rotation-induced changes in the polarization currents. If correct, this gives a passive, structure-based route to enhancing rotation sensitivity and non-reciprocal response in optical metamaterials.

What carries the argument

The load-bearing object is the uniform Green's-function approximation $G(\rho,\rho')\approx G_{\mathrm{st}}(\rho,\rho')e^{ik_0(\Omega/c)\hat z\cdot(\rho'\times\rho)}$, where $G_{\mathrm{st}}$ is the ordinary two-dimensional free-space Green's function; the exponential is the Sagnac phase a wave accumulates when its path encloses area relative to the rotation axis. This approximation is inserted into the discrete-dipole equations, turning the rotation problem into a stationary problem with phase-modified couplings. A secondary mechanism is the Sagnac-loop count: the response at each scatterer is the interference of all closed ordered loops that visit that scatterer, and a nonzero enclosed area requires that no three scatterers be collinear. The single-particle polarizability, by contrast, is shown to depend on rotation only at second order in $\Omega$, so all first-order rotation visibility comes from inter-particle loop interference.

What would settle it

Compute the two-scatterer coupling using the exact modal series in Eq. (4) and the approximate phase-factor form in Eq. (5) at the inter-particle distances used in the simulations (minimal spacing $2\lambda$); if the differences are comparable to the reported rotation-induced changes in polarization currents, the predicted geometry sensitivity is not trustworthy. The same comparison can be made experimentally by measuring the rotation-induced change in the scattered field of a random versus a periodic array at fixed minimal spacing.

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Extended reading notes

Core claim

The central discovery is that, in the slow-rotation rest-frame description, the exact rotating-medium Green's function can be replaced by the stationary Green's function times a coordinate-dependent phase $\exp[i k_0(\Omega/c)\hat z\cdot(\rho'\times\rho)]$, and that this single phase factor carries the entire rotation footprint. Feeding this approximation into a discrete-dipole calculation shows that polarization currents in a rotating rectangular array deviate from the stationary values by a few percent at $\Omega/\omega=10^{-7}$, while random and golden-angle spiral arrays with the same minimal spacing deviate substantially more at the same rotation rate. The paper interprets the enhancement through the number of Sagnac loops sharing each scatterer: every ordered triple of non-collinear scatterers forms a loop with nonzero enclosed area, and random or spiral arrays maximize such loops whereas periodic arrays necessarily contain collinear triples. Rotation also breaks reciprocity: reversing the sign of $\Omega$ in asymmetric arrays does not map the response back to an inverted image, as it does for the symmetric periodic array.

Load-bearing premise

The load-bearing premise is that the simple approximate formula for the rotating Green's function—a stationary wave kernel times a Sagnac phase—captures the exact rotation physics even for closely spaced scatterers; the paper states this but does not derive it or bound the error.

Editorial extensions

If this is right

Rotation-induced changes in a periodic array's polarization currents remain small; at $\Omega/\omega=10^{-7}$ the ratio to the stationary response spans about 0.98 to 1.02 in the computed square array.
Random and golden-angle spiral arrays of the same size and minimal spacing show significantly larger rotation footprints, so array geometry itself can be used as a sensitivity knob rather than only rotation speed or material contrast.
The rest-frame pattern is independent of where the rotation axis is placed, matching the Sagnac effect's axis independence and simplifying sensor design.
Reversing the rotation direction in a symmetric periodic array just inverts the response pattern, while in non-symmetric arrays the response changes non-reciprocally, exposing rotation as the symmetry-breaking agent.
Single-scatterer polarizability corrections are second order in $\Omega$, so practical rotation sensing with these arrays relies on inter-scatterer coupling rather than on altered individual particles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the Green's-function approximation survives exact-series checks, the geometry-based enhancement suggests a design rule for rotation sensors: maximize the number of nonzero-area triples at a fixed filling fraction, a criterion that could be optimized beyond the random and spiral examples.
The same rest-frame phase-factor machinery should extend to three-dimensional arrays and to scatterers with magnetic response; the TE/TM decoupling used here would need replacement, but the Sagnac-loop counting should carry over.
A direct experimental test could compare two arrays with identical minimal spacing but different triple-collinearity statistics under controlled rotation; the predicted contrast would isolate the geometric contribution from material or size effects.
Because the first-order rotation effect vanishes inside a single scatterer, the non-reciprocity predicted here is inherently a collective many-body effect, distinguishing it from material-based non-reciprocity and suggesting it will be robust to local fabrication disorder.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a short conference proceedings that does one genuinely useful thing: it shows by a clean symmetry argument that the single-particle polarizability correction in a rotating rest frame is second order in the rotation rate, and it sets up a DDA interaction matrix in that frame. The later numerical section compares periodic, golden-angle spiral, and random arrays and reports that the spiral and random arrays show larger rotation-induced changes in the polarization currents, attributing this to the presence of nonzero-area Sagnac loops. The O(Ω²) polarizability result is nice and the DDA formulation is sensible; the heuristic that array sensitivity scales with the number of Sagnac loops is simple, testable, and worth putting out there.

That said, the load-bearing approximation is Eq. (5), the phase-multiplied static Green's function. The paper asserts it is uniform without derivation or error bound. It is borrowed from the authors' own earlier work [1], so it is not invented here, but the present text gives the reader no way to check whether it holds at the inter-particle separations used (minimum 2λ) and at the rotation rates shown. The stress-test worry about near-field corrections is a real one, though 2λ is not extremely near field; at Ω/ω up to 5×10⁻⁶ the simple phase factor might well dominate. Still, a straightforward numerical comparison of Eq. (5) to the exact series in Eq. (4) would settle this, and its absence is the paper's biggest weakness.

The non-reciprocity claim is also a bit ahead of the evidence. The authors observe asymmetry under Ω→−Ω for the spiral and random arrays, which is consistent with non-reciprocity, but they do not quantify it or rule out other explanations. And there is no code, no convergence analysis, and no specification of the spiral scaling parameters beyond a mention—minor for a proceedings, but it limits reproducibility.

Overall, this is a reasonable contribution to the rotating-medium subfield. If I were refereeing it for a proper journal, I would send it out, not desk-reject it. The referee should ask for a validity check of Eq. (5) against the exact series at the smallest separations used and for a more careful statement of what the non-reciprocity demonstration actually shows. With those additions, the paper would be a solid contribution.

Referee Report

5 major / 5 minor

Summary. The manuscript develops a rest-frame formulation of polarizability theory and discrete dipole approximation (DDA) for arrays of small dielectric scatterers rotating at angular velocity Ω, using a two-dimensional Green's function for a slowly rotating medium. The authors show that the first-order-in-Ω contribution to the single-scatterer polarizability integral I2 vanishes, so the rotation response of the array is dominated by Sagnac phase factors in the inter-particle Green's function. They then compute the normalized polarization currents for three array types—a rectangular periodic array, a golden-angle spiral, and a random array—and report that the rotation-induced changes are significantly larger for the spiral and random arrays than for the periodic array. They interpret this as evidence that arrays in which every triple of scatterers forms a nonzero-area triangle contain more Sagnac loops and are therefore more sensitive to rotation, and they note that the response is independent of the rotation-axis position.

Significance. If the central claim holds, the paper offers a geometry-based route to enhanced rotation sensitivity and non-reciprocal response in passive dielectric arrays, which could be relevant for compact rotation sensors and for understanding non-reciprocity in structured media. The paper's analytic observation that the first-order rotation term in the single-scatterer integral vanishes is clean and useful, and the use of standard DDA equations with a physically motivated Green's function is a reasonable framework. However, the quantitative predictions in Figs. 3–5 rest entirely on the approximate Green's function in Eq. (5), whose uniform validity is asserted rather than demonstrated. Because the approximation is taken from the authors' prior work and is not independently validated here, the enhanced-sensitivity conclusions are conditional. The paper is short conference proceedings, and the central claims need stronger support before they can be accepted as established results.

major comments (5)

[Sec. 2, Eq. (5)] The load-bearing approximation G ≈ G_st exp[i k0(Ω/c) ẑ·(ρ'×ρ)] is asserted to be uniform because it shares the singularity of the exact Green's function and reproduces the Sagnac phase. This is not sufficient: matching the singularity and the phase does not guarantee that the regular part of G is well approximated at the near-field separations used here (inter-particle distances down to 2λ), where the exact series (4) contains angular-momentum-dependent wavenumbers γ_m. The paper provides no derivation, error bound, or numerical comparison of (5) with (4). Since Eq. (5) is used in the interaction matrix (6) to produce Figs. 3–5, the central geometry-dependent sensitivity claims are directly conditional on this unvalidated approximation.
[Sec. 3, TE polarizability] The paper states that the same consequences hold for the TE case but says the derivation is more complicated, without providing the TE polarizability or even specifying which polarization is used in the DDA simulations. If the cylinders are excited in TE polarization, the simulations rest on an unsupported polarizability model. The authors should either derive the TE result, cite a derivation, or state explicitly that all simulations are TM and that the TE claim is only a plausible extrapolation.
[Sec. 4, Figs. 3–5] No convergence or error analysis is reported for the DDA computations. The ratios |I_Ω/I_0| are presented as quantitative evidence for geometry-dependent sensitivity, but there is no check that the results are converged with respect to the number of scatterers, the truncation of the interaction matrix, or the discretization of the Green's function. The authors should provide convergence tests and, ideally, a comparison against a known closed-form limit (e.g., a sparse or infinite periodic array) to rule out numerical artifacts.
[Sec. 4, axis independence] The claim that the rotation pattern is independent of the axis location is stated as an observation, but no figure or quantitative data for the shifted-axis simulation is shown. Since this axis independence is used to support the Sagnac-loop interpretation, it should be documented with actual results, for example by reporting the same extremal ratios for the shifted axis or by overlaying the patterns.
[Sec. 3.1, Eq. (11)] The approximation NSL ≈ e N! − (N^2 + 1) appears algebraically incorrect: the terms omitted when truncating the exponential series at n = N are not N^2 + 1. Also, the quantity NSL1 = N(N−1)SL is not defined. These issues do not affect the qualitative ordering argument, but they should be corrected.

minor comments (5)

[Throughout] The manuscript contains several typographical errors, including 'Therefor', 'rations', 'are are', and 'the the' occurrences. A careful proofread is needed.
[Fig. 5 caption] The caption of Figure 5 says 'Excitation in a rotating GA array', but the text describes the random array. The caption should be corrected to match the content.
[Sec. 2, Eq. (5)] The phrase 'uniform approximation' is used without a precise definition. If the authors intend a particular asymptotic or norm sense of uniformity, it should be stated explicitly; otherwise, the term is misleading.
[Sec. 3, Eq. (8)] The internal-field equation (8) appears to assume a constant field inside the scatterer; this is a standard small-particle approximation but should be stated more explicitly, and the condition for its validity (size versus wavelength and skin depth) should be mentioned.
[References] Reference [1] is the source of the Green's function approximation, but the present manuscript should either reproduce the relevant derivation or at least specify which parts of [1] justify Eq. (5). Currently, the reader must consult the prior paper to evaluate the key assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the array simulations use a self-cited approximate Green's function, but no quantity is fitted to the claimed result and the geometry-dependence claim is not equivalent to the input.

full rationale

The paper's load-bearing input is the approximate rotating-medium Green's function in Eq. (5), imported from the authors' own prior work [1]. This is a self-citation, but it is not circular in the prohibited sense: it is a parameter-free approximation with stated slow-rotation assumptions, it is not fitted to the polarization-current ratios reported in Figs. 3-5, and the target array-level claims are computed consequences of that input rather than restatements of it. The polarizability correction in Sec. 3 is derived by expanding the phase in Eq. (5); the first-order cancellation is argued from parity of the integrand and is not assumed. The Sagnac-loop counting in Sec. 3.1 is a qualitative heuristic, and the simulations support rather than define it. The main weakness, that the 'uniform approximation' claim in Sec. 2 is asserted without error bounds or comparison to the exact series (4), is a correctness/validity gap, not a circularity: Eq. (5) is not defined in terms of the predicted sensitivity differences, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to target data; the numerical examples use chosen rotation rates and geometries. The central derivation imports the slowly rotating medium constitutive relations (Shiozawa [5]) and the 2D rotating medium Green's function (Steinberg et al. [1]), and adds an asserted uniform approximation for the Green's function. The single-particle polarizability result is derived within the paper, but the TE case is asserted rather than derived.

free parameters (1)

Golden-angle spiral scaling parameters = Not specified
Tuned by hand so that the minimal inter-particle distance equals 2λ; the value is not reported, which affects reproducibility but not the theoretical claim.

assumptions (5)

domain assumption Slowly rotating medium constitutive relations: D = εE - c^{-2}(Ω×r)×H and B = μH + c^{-2}(Ω×r)×E (Eq. 1).
Imported from Shiozawa [5]; this is the starting point for all subsequent field equations.
domain assumption The 2D Green's function for a homogeneous rotating medium and the modified Helmholtz equation (Eqs. 2-4) from Steinberg et al. [1].
The paper uses these results without re-deriving them; they are the foundation of the DDA formulation.
ad hoc to paper The uniform approximation G ≈ G_st exp(i k0(Ω/c) ẑ·(ρ'×ρ)) (Eq. 5) encapsulates all essential physics.
Asserted in Section 2; no derivation or numerical validation against the exact series is provided in this paper.
domain assumption Small-scatterer approximation: the internal field in each cylinder is constant, leading to Eq. (8).
Standard in polarizability theory; valid when the scatterer is electrically small (radius λ/50 here).
domain assumption Terms of second order and higher in Ω are negligible in the single-particle polarizability under slow rotation.
The paper shows the first-order term in I_2 vanishes and then drops the O(Ω²) terms, assuming practical rotation rates are small.

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Cite this review

Pith. "Pith review of Breach of symmetries in rotating arrays and metamaterials observed in their rest frame." pith.science (2026). https://pith.science/paper/T4A3W3M5

@misc{pith2026190805172,
  author       = {Pith},
  title        = {Pith review of: Breach of symmetries in rotating arrays and metamaterials observed in their rest frame},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4A3W3M5}},
  note         = {Machine review of arXiv:1908.05172}
}

read the original abstract

Polarizability theory and discrete dipole approximation in a rotating medium rest-frame is developed and discussed. The analysis is based on a rigorous rotating medium Green's function, and is used to study the effect of rotation on various arrays and metamaterials. The non-reciprocal electrodynamics induced by the rotation is exposed and studied. Although it can be interpreted in terms of the multiplicity of Sagnac interference loops inside the structure, the associated rotation footprint exhibits new effects of the structure parameters not previously reported in conventional Sagnac effect. The resulting non-reciprocity will also be demonstrated and discussed.

Figures

Figures reproduced from arXiv: 1908.05172 by the authors.

**Figure 1.** Sagnac loops defined by a set of N pointscatterers. (a) Different Sagnac loops. (b) Different Sagnac loops that share at least one common point-scatterer. (c) Different Sagnac loops that share all their point scatterers [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** A closed loop that consists of four point scatterers, with the ordered interactions 1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 1. (a) All four scatterers reside on a straight line, hence the enclosed area is zero. (b) The four scatterers cannot be aligned along a single straight line, hence the enclosed area does not vanish. (c) Scattering events ordered as 1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 3 ⇒ 2 ⇒ 1, are never counted in our analysis since they have repeate… view at source ↗

**Figure 3.** Excitation in a rotating rectangular array. seen that they span a significantly larger range. Hence, sensitivity to rotation has been increased. This observation is consistent with our qualitative discussion in Sec. 3.1.1. We note that also here the picture is independent of the rotation axis location. However, here Ω 7→ −Ω is not manifested by the inversion (x, y) 7→ (−x,−y), since the GA array possesses no geome… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Excitation in a rotating GA array [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Two- dimensional Greens function theory for the electro- dynamics of rotating medium,

Ben Z. Steinberg, A. Shamir, and A. Boag, “Two- dimensional Greens function theory for the electro- dynamics of rotating medium,” Phys. Rev. E , 74, pp. 016608 1-9, 2006

work page 2006
[2]

Electromagnetic ﬁelds in the Presence of rotating bodies,

J. Van Bladel, “Electromagnetic ﬁelds in the Presence of rotating bodies,” Proc. IEEE 64(3), pp. 301-318, Mar. 1976

work page 1976
[3]

Scattering by a rotating dielectric sphere,

Daniel De Zutter, “Scattering by a rotating dielectric sphere,” IEEE Trans. Ant. Propag. , 28(5), pp. 643- 651, Sept. 1980

work page 1980
[4]

Van Bladel, Electromagnetic Fields, IEEE Press, 2nd Ed., 2007

J. Van Bladel, Electromagnetic Fields, IEEE Press, 2nd Ed., 2007

work page 2007
[5]

Phenomenological and electron- theoretical study of the electrodynamics of rotat- ing systems,

T. Shiozawa, “Phenomenological and electron- theoretical study of the electrodynamics of rotat- ing systems,” Proc. IEEE 61(12), pp. 1694-1702, Dec. 1973

work page 1973
[6]

Sagnac effect,

E. J. Post, “Sagnac effect,” Rev. Mod. Phys. , 39(2) pp. 475-493, 1967

work page 1967
[7]

Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,

Seng Fatt Liew, Heeso Noh, Jacob Trevino, Luca Dal Negro, and Hui Cao, “Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,” Optics Express 19 (24), pp. 23631-23642, Jan. 2011

work page 2011

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Two- dimensional Greens function theory for the electro- dynamics of rotating medium,

Ben Z. Steinberg, A. Shamir, and A. Boag, “Two- dimensional Greens function theory for the electro- dynamics of rotating medium,” Phys. Rev. E , 74, pp. 016608 1-9, 2006

work page 2006

[2] [2]

Electromagnetic ﬁelds in the Presence of rotating bodies,

J. Van Bladel, “Electromagnetic ﬁelds in the Presence of rotating bodies,” Proc. IEEE 64(3), pp. 301-318, Mar. 1976

work page 1976

[3] [3]

Scattering by a rotating dielectric sphere,

Daniel De Zutter, “Scattering by a rotating dielectric sphere,” IEEE Trans. Ant. Propag. , 28(5), pp. 643- 651, Sept. 1980

work page 1980

[4] [4]

Van Bladel, Electromagnetic Fields, IEEE Press, 2nd Ed., 2007

J. Van Bladel, Electromagnetic Fields, IEEE Press, 2nd Ed., 2007

work page 2007

[5] [5]

Phenomenological and electron- theoretical study of the electrodynamics of rotat- ing systems,

T. Shiozawa, “Phenomenological and electron- theoretical study of the electrodynamics of rotat- ing systems,” Proc. IEEE 61(12), pp. 1694-1702, Dec. 1973

work page 1973

[6] [6]

Sagnac effect,

E. J. Post, “Sagnac effect,” Rev. Mod. Phys. , 39(2) pp. 475-493, 1967

work page 1967

[7] [7]

Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,

Seng Fatt Liew, Heeso Noh, Jacob Trevino, Luca Dal Negro, and Hui Cao, “Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,” Optics Express 19 (24), pp. 23631-23642, Jan. 2011

work page 2011