Influence of molecular rotation on the generation of N$_2^+$ air lasing

Hongchuan Du; Jianfeng Guo; Luzhen Yang; PengJi Ding; Ping Li; Shan Xue; Wenli Yang

REVIEW 3 major objections 5 minor 49 references

Influence of molecular rotation on the generation of N$_2^+$ air lasing

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Molecular rotation, not population inversion alone, drives N2+ air lasing.

desk verdict A serious rovibronic density-matrix study of N2+ air lasing with a credible qualitative story, but the 80% inversion enhancement and RC-dominance numbers rest on an asymmetric 30% coherence scaling that the paper doesn't justify. read the letter →

arxiv 2504.18124 v1 pith:HK2WOCOA submitted 2025-04-25 physics.optics

Wenli Yang , Ping Li , Luzhen Yang , Jianfeng Guo , Pengji Ding , Shan Xue , Hongchuan Du This is my paper · ORCID

classification physics.optics

keywords N2+airlasingmolecularrotationrotationalcoherencepopulationinversionwithoutMaxwell-Blochequationsstrong-fieldionizationrovibronicdensitymatrix

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 argues that molecular rotation, and specifically the rotational coherences created when an intense femtosecond pump ionizes nitrogen, is a central driver of 391-nm N$_2^+$ air lasing. By simulating both the pump-driven preparation of the ionic ensemble and the subsequent propagation of a seed pulse, the authors claim that ionization-produced rotational coherences enhance the population inversion between the $X^2\Sigma_g^+(v=0)$ and $B^2\Sigma_u^+(v''=0)$ states by up to about 80%. In the seed stage, they find that rotational coherences amplify the lasing signal more strongly than population inversion does, and can produce net gain even for rotational line pairs that are not inverted. A sympathetic reader would care because the longstanding debate over whether N$_2^+$ lasing requires population inversion is sharpened: the paper offers a concrete mechanism by which coherence alone can supply the gain. If correct, pump-pulse shape, duration, and timing become tunable controls for air-lasing output.

What carries the argument

The central object is the rovibronic density matrix on the basis $|ivJMK\rangle$, evolved by open-system Liouville equations with an instantaneous ionization source term, Eq. (13). That source term places coherences among rotational, vibrational, and electronic levels of the ion, tied to angle-dependent MO-ADK ionization rates and to the geometric alignment of N$_2^+$. A three-state $\Lambda$/V-type interference model is the analytic lens: it shows how a pre-existing coherence between two nearly degenerate rotational levels changes the population of the shared upper state, and how, during seed propagation, that same coherence couples different spectral sidebands into gain. The Maxwell-Bloch propagation equations then convert the ensemble coherences into a growing seed field.

What would settle it

Measure the degree of electronic coherence between the $X$ and $B$ states of N$_2^+$ produced by an 800-nm, 30-fs, roughly $3\times10^{14}$ W/cm$^2$ pump, for example through quantum-beat interference in a delayed probe; if the measured electronic coherence is substantially above 30% of the strong-field upper bound, the predicted ~80% inversion enhancement should shrink or reverse. Conversely, if an R-branch seed at a delay near 4 ps shows no coherence-driven gain for non-inverted low-$J$ lines over a 2.5-mm propagation length, the dominant-role claim for rotational coherence would fail.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that molecular rotation is not a small correction to the vibronic picture of N$_2^+$ 391-nm lasing but a primary mechanism. Strong-field ionization from multiple orbitals places not only populations but also coherences among rotational, vibrational, and electronic levels of the ion. In the pump stage these coherences redistribute rovibronic populations within tens of femtoseconds: rotational coherences act through constructive interference between the $J-1$ and $J+1$ pathways into the $B$ state, raising its population; vibrational coherences add a further increase; electronic coherences, when scaled to the 30% level that preserves agreement with observed R-branch inversion, leave a net enhancement of the $X_0$--$B_0$ inversion of roughly 80%. In the seed stage, Maxwell-Bloch propagation shows that rotational coherences in both the $X$ and $B$ states amplify the seed by coupling P- and R-branch transition pathways, and that this coherence-driven gain dominates the inversion-driven gain; lasing appears even for rotational pairs without population inversion.

Load-bearing premise

The quantitative 80% boost and the survival of population inversion both depend on scaling ionization-produced electronic coherences to 30% of the model's upper-bound value; if the true electronic coherence is closer to 100%, the model itself predicts no inversion, so the reported enhancement and the seed-stage balance would shift.

Editorial extensions

If this is right

Seed amplification can be positive for rotational lines whose populations are not inverted, so observing 391-nm gain at a given $J$ does not by itself prove population inversion.
The delay-dependent oscillations of the lasing signal with sub-picosecond periods can be read as a fingerprint of rotational coherences, with oscillation frequencies set by rotational energy spacings in the $X$ and $B$ states.
B-state rotational coherences chiefly amplify the inversion-region R-branch lines, while X-state rotational coherences chiefly amplify non-inversion-region lines; branch-resolved measurements can separate the two contributions.
Including all ionization-produced coherences raises the modeled $X_0$--$B_0$ R-branch inversion by up to about 80%, so omitting rotation from pump-stage models underestimates the preparation of gain.

Reading between the lines

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

A direct experimental measurement of the $X$--$B$ electronic coherence degree, for example through quantum-beat or transient-absorption signals after ionization, would decide whether the 30% scaling is right; the paper's own 100%-coherence scenario shows how strongly the conclusion swings on that number.
If rotational coherence is the dominant amplifier, pump shaping that maximizes rotational-coherence production, such as pulse trains or polarization schemes, could increase air-lasing output without raising peak intensity.
The same rovibronic-coherence machinery could be applied to other molecular-ion lasing transitions and to longer delays, where rotational revivals should produce periodic re-enhancement of the seed gain; this is a testable extension not reported in the paper.
Because the seed-stage gain couples P- and R-branch pathways through rotational coherence, the measured P/R intensity ratio at a fixed delay contains information about the relative phase and magnitude of the X- and B-state coherences, not just about populations.

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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

Two things to know. This is the first treatment of N2+ air lasing in a full rovibronic basis with ionization-produced coherences, and it gives a clean physical result: rotational coherences, not just population inversion, matter for seed amplification, and they can enhance the X0-B0 inversion by up to 80% when electronic coherences are set to 30%. The extension of the group's earlier vibronic model to rotation is done carefully, and the three-state interference analysis makes the sign structure transparent. The seed-stage decomposition into RC, X RC, B RC, and No RC scenarios is a sensible way to isolate mechanisms, and the claim that B-state RCs dominate in the inversion region while X-state RCs matter in the non-inversion region is concrete enough to test experimentally.

The soft spot is exactly where the stress-test lands. Equation (13) overestimates all ionization-produced coherences because it does not trace over free-electron degrees of freedom; the authors say so themselves in Sec. II B. But then they scale only electronic coherences to 30% and leave rotational and vibrational coherences at 100%. No physical justification is given for that asymmetry, and the same partial-wave physics that suppresses electronic coherence should also suppress coherence between different rotational states of the same electronic state. If RCs are overestimated by a comparable factor, the reported 80% enhancement shrinks, and the "RC dominates" propagation result is weakened because the comparison is full-RC versus zero-RC. A sensitivity scan over RC amplitude would have addressed this cheaply, and its absence is the largest gap in the paper.

The 30% value is itself taken from the authors' own prior work and is effectively chosen to keep the R-branch inversion consistent with experiment. That makes the survival-of-inversion claim less a prediction and more a consistency check. The paper also ships no code or data, so the quantitative results are not independently reproducible yet. On the other hand, the qualitative conclusions are plausible, the literature is engaged honestly, and the self-identified limitations are stated rather than hidden. The reliance on Ref. [37] is legitimate here because it is the same model family, not gratuitous self-citation.

Who should read this: anyone working on air lasing or strong-field molecular dynamics. It deserves a serious referee—not for the exact percentages, but for the mechanism and the rovibronic framework. I would cite it, with a caveat on the coherence scaling.

Referee Report

3 major / 5 minor

Summary. The manuscript presents a theoretical study of N2+ air lasing at 391 nm using an open-system density-matrix description in a rovibronic basis for the pump stage and Maxwell-Bloch propagation for the seed stage. It reports three main results: (i) molecular rotation modifies angle-dependent vibronic populations of N2+ on tens-of-femtosecond timescales; (ii) ionization-produced rotational, vibrational, and electronic coherences enhance the X0-B0 population inversion by up to about 80% when electronic coherences are scaled to 30% of Eq. (13); and (iii) in seed propagation, rotational coherences contribute more than population inversion to lasing amplification. The paper explicitly notes in Sec. II B that Eq. (11) (and by extension Eq. (13)) overestimates ionization-produced coherences because free-electron degrees of freedom are not traced out, and it applies a 30% reduction to electronic coherences only.

Significance. If the quantitative claims hold, the paper is a valuable step beyond vibronic models: it provides a rovibronic treatment that couples gain preparation and seed propagation, and it gives transparent sign analyses in Secs. III B1 and III C2 that explain how rotational coherences constructively or destructively interfere in the transition pathways. The delay-dependent spectra in Figs. 4-7 are falsifiable against pump-probe experiments. However, the headline numbers are conditioned on one estimated scaling parameter (30% for electronic coherences) and on an untested assumption that rotational and vibrational coherences are not subject to the same reduction that Eq. (13) is acknowledged to need. The central claims therefore require additional sensitivity analysis before they can be accepted as quantitative predictions.

major comments (3)

[Sec. II B, Eq. (13); Sec. III B1/B3] The manuscript states that Eq. (11) overestimates ionization-produced coherences because it does not trace out free-electron degrees of freedom, and that the same limitation applies to its rovibronic extension Eq. (13). Nevertheless, only the electronic coherences are reduced to 30% (Sec. III B3), while rotational coherences and vibrational coherences are kept at full amplitude. Since the photoelectron entanglement that reduces electronic coherence can also reduce coherence between rotational states of the same electronic state, the reported ~80% inversion enhancement in Fig. 2(d) and the seed-stage conclusion that RCs dominate amplification (Figs. 6-7) are conditional on full-amplitude RCs. The authors should provide a sensitivity scan over the RC amplitude (e.g., 30%, 50%, 100%) or give a physical argument why the reduction applies only to ECs.
[Sec. III B3] The 30% electronic-coherence scaling is not derived in the present manuscript; the paper states that with 100% ECs no population inversion is achieved, contradicting experiment, and uses 30% to restore agreement. This makes the central 'approximately 80% enhancement' a calibration-dependent statement rather than a prediction of the model. The manuscript should present the 30% value explicitly as a calibration based on Ref. [37] and quantify the sensitivity of the 80% figure to the EC scaling, or provide a fully reproducible derivation of the 30% from the adiabatic SFA coherence model.
[Sec. III C2, Figs. 6-7] The dominance of RCs in seed amplification is established by comparing the full-RC calculation with the 'No RC' calculation. Because the full-RC case uses RCs at 100% of Eq. (13), and because Eq. (13) is acknowledged to overestimate coherences, the comparison does not isolate the physical RC contribution unless the RC amplitude is realistic. An additional calculation with RCs scaled by the same factor used for ECs (30%) would show whether the conclusion that 'RCs play the dominant role' survives. If RCs are overestimated by a similar factor, the reported enhancement and the RC-dominance claim in the seed stage would both be weakened.

minor comments (5)

[Sec. III C] The seed pulse is described as a '400-nm' Gaussian pulse, but the lasing transition is at 391 nm and the theory in Sec. II C centers on the 391-nm transition. Please verify whether this is a typo for 391 nm; if the seed is truly at 400 nm, explain how a 100-fs, 400-nm pulse effectively seeds the 391-nm spectral region shown in Figs. 4-7.
[Fig. 6 caption] The caption refers to '30% RVEC'; based on the text this should be '30% EVRC'.
[Sec. III B1] The sentence 'The ionization-produced RCs originates from the geometric alignment' should be 'originate'.
[Fig. 2(d)] The red dashed line mentioned in Sec. III B1 is not identified in the figure caption; please indicate which curve corresponds to the approximately 80% enhancement.
[Sec. II B] The paper would benefit from specifying the rovibronic basis truncation (maximum J and number of vibrational levels per electronic state) and from a brief convergence check, since the final populations and coherences are obtained by summing over J and M.

Circularity Check

2 steps flagged · score 6.0 of 10

The 30% electronic-coherence scaling is imported from the authors' prior work and is set so that population inversion survives; rotational coherences are left at full strength despite the same admitted overestimation, so the headline 80% enhancement and RC dominance are conditional on fitted inputs.

fitted input called prediction [Sec. III B 3, Eq. (13) and Fig. 2(d), '30% EVRC' scenario]
"In scenario 1, the ECs are directly calculated by Eq. (13) with the degrees of ECs being 100%, labeled as 'EVRC'... In fact, no inversion is achieved, which contradicts experimental observations, as strong R-branch 391-nm lasing signals are clearly detected under the current laser conditions [44]. In scenario 2, the ECs are reduced to 30% of their original values... 30% is an estimated parameter used to mimic the ionization-produced ECs of N2+ predicted by the adiabatic strong-field approximation coherence model [37]."

The paper's central quantitative claim — that including all ionization-produced coherences enhances X0-B0 population inversion by up to about 80% — is computed in the '30% EVRC' scenario. That 30% value is not derived in this paper; it is imported from the authors' own Ref. [37], and the scenario is defined so that the R-branch inversion, which is already known experimentally, remains. The paper itself shows that at 100% ECs no inversion is achieved. Thus the statement that inversion survives when ECs are included is not an independent prediction: the input parameter was set to the value that preserves the experimentally required inversion, so the outcome is enforced by the fitted input rather than derived from the coherence model alone.
other [Sec. II B, Eqs. (11)-(13); Sec. III C, RC scenarios]
"Since Eq. (11) does not incorporate this procedure, the coherence is overestimated. ... In the simulations, we incrementally incorporate the RCs, VCs, and ECs to isolate the effects of these three types of coherences. ... To evaluate the effects of RCs, we perform additional calculations under three artificial RC scenarios: (1) considering only the X-state RCs ... and (3) excluding both the X- and B-state RCs (denoted as 'No RC')."

Equation (13), which generates the rotational coherences, is the rovibronic extension of Eq. (11) and has the same structure; the paper concedes that Eq. (11) overestimates ionization-produced coherence because the free-electron degrees of freedom are not traced out. Yet only the electronic coherences are scaled down to 30%, while the rotational and vibrational coherences are left at 100% with no comparable correction or sensitivity scan. The seed-stage conclusion that RCs dominate lasing amplification is obtained by comparing full-amplitude RCs ('RC') against zero RCs ('No RC'), so the quoted dominance is an artifact of the uncalibrated RC input rather than a robust prediction. A consistent coherence correction could reduce or remove the reported enhancement and dominance.

full rationale

The paper is not circular in its overall structure: the density-matrix and Maxwell-Bloch propagation equations are standard, and many intermediate results (e.g., P/R branch oscillations, three-state interference mechanisms) follow from the stated equations. However, the two headline claims are load-bearing on a self-cited, selectively applied input. First, the 30% EC scaling from Ref. [37] is the value that keeps the R-branch population inversion alive; at 100% ECs the same model destroys inversion. Therefore the claim that coherences enhance inversion by ~80% is not a first-principles prediction but a consequence of choosing a parameter that matches the experimentally known lasing. Second, the same model is admitted to overestimate coherence, but only ECs are corrected; RCs and VCs remain at full amplitude. The 'RC dominates seed amplification' result is then obtained by a 100%-versus-0% RC comparison, so its quantitative support depends on the uncorrected overestimate. These are specific, quotable reductions of the central claims to calibrated inputs rather than to independent derivation. The paper contains substantial independent modeling content, so a score of 6 (partial circularity) is appropriate rather than a higher score.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claims rest on a chain of established approximations (Born-Oppenheimer, dipole, MO-ADK) plus one explicitly adjustable parameter (the 30% EC scaling). The coherence injection model Eq. (13) is a strong assumption, acknowledged by the authors, and it drives the rotational-coherence effects that underpin both the population-inversion and seed-amplification results.

free parameters (3)

Electronic coherence scaling factor (EVRC) = 0.30
The ionization-produced electronic coherences from Eq. (13) are multiplied by 0.30 to mimic the lower degrees of coherence predicted by the adiabatic strong-field approximation (Ref. [37]) and to keep R-branch population inversion consistent with experimentally observed lasing (Sec. III B3). The paper states that with 100% ECs, no inversion is achieved, contradicting experiments.
Pump-stage decoherence time Td = 1 ps
Phenomenological decoherence time used in the Liouville equation during the pump stage, taken from Ref. [38].
Seed-stage decoherence time Td = 5 ps
Collision-induced decoherence time used in the Maxwell-Bloch equations during seed propagation, taken from Ref. [40].

assumptions (7)

standard math The Born-Oppenheimer approximation separates electronic, vibrational, and rotational degrees of freedom in the rovibronic basis (Eq. (1)).
Standard approximation for molecular structure; not tested in this paper.
standard math The molecule-laser interaction is treated in the dipole approximation and length gauge (Eqs. (4)-(6)).
Standard strong-field approximation.
domain assumption Strong-field ionization rates to the X, A, and B ionic states are computed with MO-ADK theory (Sec. II B).
The MO-ADK rate model is an established but approximate method; the paper does not benchmark it against other ionization models for these conditions.
domain assumption The instantaneous ionization-produced coherence is given by Eq. (13), which assumes full coherence (100% DOC) for rotational and vibrational coherences and neglects entanglement with the photoelectron.
The authors explicitly acknowledge in Sec. II B that Eq. (11) overestimates the electronic coherence because the free-electron degrees of freedom are not traced out; the extension Eq. (13) inherits this limitation for rotational coherences.
domain assumption The orientation of the N2+ ion is predominantly set by geometric alignment, and the rotational evolution of the neutral N2 during ionization is neglected (Sec. II B).
The paper cites Ref. [27] for the minor effect of neutral rotation; this assumption underlies Eq. (13).
domain assumption The seed propagation is modeled in a plane-wave, one-dimensional retarded frame with no diffraction or transverse effects (Eq. (17)).
This is a common simplification for short propagation distances in air-lasing models, but it omits spatial beam dynamics.
domain assumption Initial neutral N2 is a thermal Boltzmann ensemble at 300 K over rotational states of the ground vibronic state (Eq. (8)).
Standard room-temperature assumption; the gas is assumed unaligned before the pump.

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Pith. "Pith review of Influence of molecular rotation on the generation of N$_2^+$ air lasing." pith.science (2026). https://pith.science/paper/HK2WOCOA

@misc{pith2026250418124,
  author       = {Pith},
  title        = {Pith review of: Influence of molecular rotation on the generation of N$_2^+$ air lasing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HK2WOCOA}},
  note         = {Machine review of arXiv:2504.18124}
}

abstract

N$_2^+$ air lasing has attracted considerable attention due to its promising applications in remote sensing and the debates surrounding its generation mechanisms. Here, we present a comprehensive theoretical investigation of the role of molecular rotation in N$_2^+$ lasing at 391 nm ($B^2 \Sigma _u^+(v''=0)\rightarrow X^2 \Sigma _g^+ (v=0)$). By solving the open-system density matrix and Maxwell-Bloch equations in a rovibronic-state basis, we examine both the formation of the N$_2^+$ gain medium induced by a femtosecond pump pulse and the subsequent spatial propagation of the seed pulse. During the pump stage, rotational dynamics are found to significantly modify the angle-dependent populations of ionic vibrational-electronic states within tens of femtoseconds. Furthermore, ionization-produced rotational coherences substantially enhance the population inversion between the $X^2 \Sigma _g^+ (v=0)$ and $B^2 \Sigma _u^+(v''=0)$ states. In the seed propagation stage, both population inversion and rotational coherence are found to contribute to the lasing process, with the latter playing a dominant role in amplifying the lasing signals. These findings reveal the crucial role of molecular rotation in N$_2^+$ air lasing and highlight its potential as a tunable parameter for controlling lasing dynamics.

Figures

Figures reproduced from arXiv: 2504.18124 by the authors.

**Figure 2.** (c), the B0J -state populations reach their peaks at higher J values. These differences in the peak positions create population inversions between the X0J+1 (X0J−1) and B0J states at higher J values, which will lead to lasing emission in the P (R) branch [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Rotational coherences between the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (c) shows the delay-averaged difference spectrum over a 10-ps delay range, in which both absorption and emission lines are clearly visible. Since the P-branch signal arises from multiple emission pathways, whereas each order of the R-branch signal is contributed by a …

**Figure 5.** Figure 5: FIG. 5. (a) Lasing spectra as functions of frequency and seed [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Integrated lasing signal intensities as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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Rotational coherence First, we examine the effect of ionization-produced RC. By comparing the “RC” results with the “W/O” results in Figs. 2(a-c), one can seen that the RCs substantially increase the population of the B0J state, while the pop- ulations of the X0J andAv′≥0J states remain largely un- affected. As a result, the population inversion between t...

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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RC” results with the “W/O

Rotational coherence First, we examine the effect of ionization-produced RC. By comparing the “RC” results with the “W/O” results in Figs. 2(a-c), one can seen that the RCs substantially increase the population of the B0J state, while the pop- ulations of the X0J andAv′≥0J states remain largely un- affected. As a result, the population inversion between t...

work page

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VRC” results with the “RC

Vibrational coherence Next, we examine the effect of the ionization-produced VCs. Comparing the “VRC” results with the “RC” re- sults in Figs. 2(a-c), it can be observed that the VCs sig- nificant increase the population of the B0J state, while the populations of the X0J and Av′≥0J states remain 8 nearly unchanged at higherJ. As a result, the population i...

work page

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EVRC” results with the “VRC

Electronic coherence Finally, let us examine the effect of the ionization- produced ECs, which can be evaluated by comparing the “EVRC” results with the “VRC” ones. As stated in Sec. II B, Eq. (11) predict the maximum DOC between electronic states, and so does its extension Eq. (13). In contrast, more accurate coherence models predict signif- icantly lowe...

work page

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30% EVRC

Without considering macroscopic propagation We first examine the response of the seed pulse on the N + 2 ensemble, with macroscopic propagation effects excluded. For this purpose, the propagation distance is set to a very small value of 0.1 mm. Figure 4(b) shows the difference spectrum as a function of the de- lay time td. The spectrum can be divided into...

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30% EVRC

With consideration of macroscopic propagation We now examine the lasing signal with macroscopic propagation effects included. The propagation length is set tozmax = 2.5 mm. Figure 5(a) shows the correspond- ing difference spectrum as a function of td. The oscilla- tory features alongtd are still visible, originating from the RCs within the X0 and B0 state...

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