Impact of Nitrogen-Vacancy Color Centers on the Optical Kerr Effect in Diamond: A Femtosecond Z-Scan Study

Adam M. Wojciechowski; Krzysztof Dzier\.z\k{e}ga; Mariusz Mr\'ozek; Wojciech Talik

REVIEW 3 major objections 4 minor 26 references

Impact of Nitrogen-Vacancy Color Centers on the Optical Kerr Effect in Diamond: A Femtosecond Z-Scan Study

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read NV centers measurably reduce diamond's Kerr nonlinearity at 1032 nm.

desk verdict Solid Z-scan data, four-fold n2 anisotropy in diamond at 1032 nm, but the NV-concentration attribution rests on unmatched commercial samples and a qualitative model; worth reviewing with requests for better controls. read the letter →

arxiv 2501.01379 v1 pith:WGSQFXUR submitted 2025-01-02 physics.optics physics.atom-ph

Wojciech Talik , Mariusz Mrózek , Adam M. Wojciechowski , Krzysztof Dzierżęga This is my paper · ORCID

classification physics.opticsphysics.atom-ph PACS 42.65.-k42.65.An

keywords nitrogen-vacancycentersopticalKerreffectnonlinearrefractiveindexZ-scandiamondtwo-levelsystemthird-ordersusceptibilityanisotropy

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 tries to establish that the optical Kerr effect in diamond is not fixed by the pristine crystal alone: nitrogen-vacancy color centers pull the nonlinear refractive index down, and the size of the pull grows with NV concentration. The claim, based on closed-aperture Z-scan measurements at 1032 nm on three commercial crystals (undoped, 0.3 ppm NV, and 4.5 ppm NV), is that $n_2$ is positive and shows a four-fold anisotropy in all three, decreasing from $3.155\times10^{-20}$ to $2.397\times10^{-20}$ m$^2$/W as NV density rises. The paper attributes this decrease to a negative third-order contribution from the NV ensemble, modeled as an effective two-level system that is red-detuned from its $\sim$520 nm resonance. A sympathetic reader would care because it suggests NV concentration is a practical tuning knob for ultrafast all-optical switching and modulation in diamond, and because it is the first reported observation of the four-fold anisotropy.

What carries the argument

The argument is carried by two coupled objects. The first is the Z-scan signal model of Eq. (2), with the closed-aperture transmittance fit to extract the nonlinear phase shift $\Delta\Phi_0 = k n_2 I_0 L_{\mathrm{eff}}$; this is what converts measured traces into $n_2$ values and their angular dependence. The second is the effective two-level-system description of the NV centers (Boyd's susceptibility formula), expanded to third order to give Eq. (8), $n_{2,\mathrm{NV}} = N|\mu_{ba}|^4 T_1 T_2^2 \Delta T_2 / (n_0^2 \epsilon_0^2 c \hbar^3 (1+\Delta^2 T_2^2)^2)$, whose sign is set by the detuning $\Delta$; since the 1032 nm laser is red-detuned from the $\sim$520 nm NV absorption band, the NV contribution is negative. The four-fold angular dependence is encoded in $\chi^{(3)}_{\mathrm{eff}}(\theta) = \chi^{(3)}_{xxxx}(1 - \frac{\sigma}{2}\sin^2 2\theta)$, so the anisotropy is attributed to the diamond host, while the NV population reduces the overall magnitude.

What would settle it

Measure $n_2$ at 1032 nm in a diamond plate that is identical in orientation, supplier, and fabrication to the Thorlabs NV-doped crystals but contains $<$0.005 ppm nitrogen; if its $n_2$ is not close to $3.155\times10^{-20}$ m$^2$/W, then the EGSC-to-MCNV/HCNV difference cannot be attributed to NV concentration alone. Alternatively, tune the laser across the red-detuned side of the 520-nm band and check that the $n_2$ reduction grows as detuning shrinks, as Eq. (9) predicts.

Watch

Extended reading notes

Core claim

The central discovery claimed is that the third-order nonlinear susceptibility of diamond at 1032 nm is anisotropic, with four-fold rotational symmetry about the [001] axis, and is reduced by the presence of NV centers, with the reduction attributed to the negative $n_2$ contribution of the NV ensemble when it is treated as a two-level system driven by a strong, red-detuned field. From fits of the closed-aperture transmittance, the paper obtains $n_2(0^{\circ})$ = 3.155, 2.702, and $2.397\times10^{-20}$ m$^2$/W for the undoped, 0.3 ppm, and 4.5 ppm crystals, respectively, and an anisotropy coefficient $\sigma$ that ranges from 0.42 to 0.58. The reduction is interpreted through $n_{2,\mathrm{NV}} \propto N|\mu_{ba}|^4 T_1/(T_2 \Delta^3)$ in the non-resonant red-detuned limit, which is negative for $\Delta<0$, and the paper shows that with reasonable values of $N$, $\mu_{ba}$, and $T_1/T_2$ the magnitude of $n_{2,\mathrm{NV}}$ can reach $\sim 6\times10^{-18}$ m$^2$/W, enough to account for the observed drop.

Load-bearing premise

The measured drop in $n_2$ from one crystal to the next is attributed entirely to NV concentration, but the undoped crystal comes from a different supplier, has different edge orientation, and a different total nitrogen content than the NV-doped ones; if any of those uncontrolled differences also changes the third-order response, the NV-centered interpretation would weaken.

Editorial extensions

If this is right

If NV centers contribute a negative $n_2$ that scales with their density, then the Kerr nonlinearity of diamond can be engineered by doping, not just by crystal choice.
The four-fold anisotropy measured here is a property of the diamond host and persists in NV-doped crystals, so polarization control of the Kerr response remains available even at high NV density.
At 4.5 ppm NV, saturable two-photon absorption appears (up to a few percent), meaning nonlinear absorption must be accounted for when using highly doped crystals in switching or mode-locking applications.
The fitted $n_2$ values provide a quantitative benchmark for NV-doped diamond at 1032 nm, a wavelength relevant to ultrafast ytterbium lasers.

Reading between the lines

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

The two-level model predicts $n_{2,\mathrm{NV}}$ should scale linearly with NV density, so testing an intermediate doping level (say 1–2 ppm) would distinguish a genuine NV concentration effect from a crystal-specific offset; the paper does not report such a measurement.
Because the sign of $n_{2,\mathrm{NV}}$ flips for blue detuning, the same NV ensemble should enhance, rather than reduce, the Kerr nonlinearity when probed on the short-wavelength side of the NV band—an effect not stated in the paper but directly implied by Eq. (8).
The measured anisotropy coefficient $\sigma$ changes between crystals (0.454, 0.418, 0.580), suggesting NV centers contribute their own anisotropic susceptibility; isolating that contribution by measuring the difference between doped and undoped samples as a function of $\theta$ could place the two-level model on a firmer footing.

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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

Here's the short version: this is a cleanly executed Z-scan study of three commercial diamond plates, and the headline observation—four-fold anisotropy in n2 at 1032 nm that decreases with vendor-stated NV density—is new and plausible. The fits follow standard practice, the error bars look reasonable, and the authors are explicit that the two-level model is qualitative. I'd trust the measured numbers more than the attribution.

What's new: to my knowledge no prior paper reports polarization-resolved n2 anisotropy in volume-doped NV diamond near 1 μm. Motojima et al. saw near-surface enhancement at 800 nm, a different regime. So the dataset stands on its own.

The soft spot is the assignment of the n2 drop to NV concentration. The three crystals are not matched: EGSC is Element Six with [110] edges and <0.005 ppm total nitrogen; MCNV and HCNV are Thorlabs with [100] edges and much higher total nitrogen. There is no NV-free Thorlabs plate, so the drop could come partly from substitutional nitrogen, other defects, or different fabrication. The paper acknowledges the model is qualitative, but the model's own scaling also undermines it: Eq. (9) says n2,NV ∝ N, so raising NV from 0.3 to 4.5 ppm should increase the negative contribution fifteen-fold. The measured additional drop is only ~0.3 x 10^-20 m^2/W, smaller than the drop already seen at 0.3 ppm. That would require concentration-dependent T1/T2 or extra mechanisms, none measured.

Minor but real: Data Availability is a placeholder ('Ref. [x] after review'). For a measurement paper, that should be fixed before publication; otherwise the fitted ΔΦ0 values in Fig. 3(d) can't be independently checked.

There's also a small orientation subtlety: EGSC has [110] edges, so θ=0 on that sample might not correspond to the same crystallographic axis as on the Thorlabs [100] plates. The anisotropy fit may still capture the four-fold shape, but a 45° offset could affect the comparison of absolute n2 values.

Bottom line: this is a genuine new dataset, and the central observation—positive, four-fold-anisotropic n2 that shrinks with NV doping—is probably right. The NV-attribution, however, is a hypothesis, not a demonstration. I'd send this to review: a good referee should ask for a matched undoped control (or at least a direct NV-density measurement on the same crystals) and for the raw Z-scan data. The paper deserves a shot after revision, but not acceptance as is.

Referee Report

3 major / 4 minor

Summary. The paper reports femtosecond Z-scan measurements at 1032 nm on three commercial diamond plates: an undoped electronic-grade single crystal (EGSC, Element Six) and two Thorlabs crystals with nominal NV concentrations of 0.3 ppm (MCNV) and 4.5 ppm (HCNV). The authors find a positive nonlinear refractive index n2 with four-fold rotational anisotropy in all samples, with n2 decreasing from 3.155 x 10^-20 m^2/W in EGSC to 2.702 in MCNV and 2.397 in HCNV. They attribute this reduction to a negative third-order contribution from NV centers, modeled as a single effective two-level system red-detuned from a resonance near 520 nm. The paper also reports a saturable two-photon absorption in HCNV and compares the diamond response with the two-band model of Sheik-Bahae.

Significance. The measurements provide a useful dataset on the nonlinear refractive index of commercially available NV-doped diamond in the near-infrared, including the first reported four-fold anisotropy of n2 for these crystals and an intensity-dependent phase shift consistent with the optical Kerr effect. The Z-scan fits and reported uncertainties appear standard and internally consistent. However, the significance of the central attribution of the n2 reduction to NV concentration is undermined by the lack of matched control samples and by the explicitly qualitative nature of the two-level model. If confirmed with proper controls, the effect could be relevant for NV-based nonlinear photonics, but the present evidence does not yet secure the central claim.

major comments (3)

[Table 1 and Section 3] The comparison between the three crystals does not isolate the NV concentration as the cause of the observed n2 reduction. The EGSC sample is from Element Six with [110] edges and total nitrogen below 0.005 ppm, while MCNV and HCNV are Thorlabs samples with [100] edges and total nitrogen of 0.8 and 13 ppm. No NV-free Thorlabs control is measured, so the drop from n2 = 3.155 x 10^-20 m^2/W (EGSC) to 2.702 (MCNV) and 2.397 (HCNV) could be partly or wholly caused by the different supplier, the 45-degree crystallographic orientation offset, higher substitutional nitrogen content, or other fabrication-related differences. The attribution to NV centers is therefore not secure.
[Section 3, Eqs. (8)-(9)] The two-level model predicts n2_NV proportional to N at fixed detuning and T1/T2 ratio, but the measured concentration scaling is inconsistent with this. The n2 drop from EGSC to MCNV (0.3 ppm) is about 0.45 x 10^-20 m^2/W, while the additional drop from MCNV to HCNV (4.5 ppm, a 15-fold increase in NV density) is only about 0.31 x 10^-20 m^2/W. Reproducing this behavior would require a concentration-dependent T1/T2 or dipole moment, which the paper neither measures nor includes. Thus the model does not provide a quantitative explanation of the observed trend.
[Section 3, Fig. 5(d) and text after Eq. (9)] The parameters entering the model are assumed rather than determined: N = 5.3 x 10^22 m^-3, T1 = 10 ns, T2 varied over a wide range, and mu_ba derived via the Einstein relation. The paper explicitly states that determining the absolute values of n2_NV is problematic. Because the model is not fitted to the measured n2 values and its key parameters are unconstrained, the conclusion in the Abstract and Summary that the reduction is attributed to the negative contribution of NV centers is stronger than the evidence supports. The authors should either provide an independent measurement of at least some parameters or explicitly restrict the claim to a qualitative possibility.

minor comments (4)

[Data Availability Statement] The Data Availability Statement contains a placeholder 'Ref. [x] (after review)', which prevents independent verification of the fitted DeltaPhi0 values in Fig. 3(d). The statement should be completed before publication.
[References] Reference 1 contains a typo: 'diaomond' should be 'diamond'.
[Section 2, paragraph on reference measurements] The text states that reference measurements below 30 nJ (7 GW/cm^2) showed no nonlinear effects, but does not provide a check of this condition against the expected n2 of diamond; a brief justification or a control scan at lower intensity would strengthen the claim.
[Section 3, Eq. (7)] The susceptibility in Eq. (7) is written in SI units, while Table 2 reports chi^(3) in both SI and esu; for clarity, the conversion convention used should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity beyond a minor, non-load-bearing self-citation for the standard Z-scan fit formula.

full rationale

Walking the derivation chain: Z-scan traces are fit with Eq. (2) to extract DeltaPhi0, and the slope of DeltaPhi0 versus I0 gives n2 via Eq. (1). The NV explanation then uses Eqs. (8)-(9), derived from Boyd's standard two-level susceptibility, evaluated with parameters that are not taken from the Z-scan data: NV density N = 5.3e22 m^-3 from the supplier concentration, T1 = 10 ns, mu_ba from the Einstein relation, resonance energy 2.4 eV from the measured transmittance spectra, and a scanned T1/T2 ratio bounded by the absorption bandwidth. The observed reduction in n2 is therefore not an input renamed as a prediction; the model could in principle disagree (e.g., for small T1/T2 or blue detuning). The central claim is thus post hoc and qualitative, but not circular. The only self-citation is Ref. [20] for the fitting formula in Eq. (2); this formula is standard Z-scan theory and is not load-bearing for the physical attribution. Separate concerns - unmatched crystals (different supplier, edge orientation, total nitrogen), lack of an NV-free Thorlabs reference, inconsistent concentration scaling of Eq. (9) with the 0.3 vs 4.5 ppm data, and the placeholder Data Availability statement - are correctness/verifiability limitations, not circularity. The score reflects only the minor self-citation, not any reduction of the conclusion to its inputs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

All model constants are either from supplier specifications, textbook formulas, or assumed values. No new physical entity is postulated, but the quantitative NV contribution remains illustrative because T1, T2, and the dipole moment are not measured. No data or code is shipped.

free parameters (4)

T1/T2 ratio = 0.5 to 5 x 10^5
In the two-level NV model, T1 is fixed to 10 ns and T2 is varied over six orders of magnitude to show the possible magnitude of n2,NV; no measured value is available.
Effective resonance energy = 2.4 eV (520 nm)
Taken from the broad absorption band in the transmittance spectra of the NV-doped crystals and used to set the red detuning in the model.
Dipole moment mu_ba = 2.23 x 10^-29 C m
Obtained from the Einstein relation using the assumed T1 of 10 ns; not directly measured.
NV number density N = 5.3 x 10^22 m^-3
Assumed equal to the supplier-stated 0.3 ppm NV concentration for the MCNV crystal and used in the model plot in Fig. 5(d).

assumptions (4)

domain assumption Diamond has m3m symmetry, and for degenerate frequencies the third-order susceptibility tensor has four independent components, producing Eq. (3) for the angular dependence.
Invoked in Section 3 to fit the four-fold anisotropy; a standard crystallographic symmetry fact.
domain assumption The Sheik-Bahae Kramers-Kronig dispersion model for bound-electronic Kerr nonlinearity applies to diamond at 1032 nm.
Used in Eq. (5) and Fig. 5(b) to estimate the diamond host contribution; relies on prior semiconductor theory.
ad hoc to paper NV- and NV0 centers can be treated as a single effective two-level system with conserved total population despite charge-state conversion.
Introduced in Section 3 to compute a negative n2 contribution; justified only qualitatively by charge-state cycling arguments.
domain assumption The standard Gaussian-beam Z-scan model in Eq. (2) accurately describes the closed-aperture transmittance and residual nonlinear absorption.
Used to extract the nonlinear phase shift from the experimental curves; standard for the Z-scan method.

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Cite this review

Pith. "Pith review of Impact of Nitrogen-Vacancy Color Centers on the Optical Kerr Effect in Diamond: A Femtosecond Z-Scan Study." pith.science (2026). https://pith.science/paper/WGSQFXUR

@misc{pith2026250101379,
  author       = {Pith},
  title        = {Pith review of: Impact of Nitrogen-Vacancy Color Centers on the Optical Kerr Effect in Diamond: A Femtosecond Z-Scan Study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGSQFXUR}},
  note         = {Machine review of arXiv:2501.01379}
}

read the original abstract

This study explores nonlinear optical effects in diamond crystals doped with nitrogen-vacancy (NV) centers, focusing on the optical Kerr effect in the infrared spectral region. By employing the Z-scan technique with 230 fs laser pulses at a wavelength of 1032~nm, we investigate the nonlinear refractive index of high-purity and NV-containing diamond crystals at NV concentrations of 0.3 ppm and 4.5 ppm. The results demonstrate a pronounced anisotropy in the nonlinear refractive index, which exhibits a four-fold rotational symmetry, with its amplitude reduced by the NV concentration. The reduction in non-linear susceptibility is linked to the negative contribution of NV centers, modeled as two-level systems upon a strong, laser field, red-detuned relative to their resonance frequency. These findings provide new insights into the interplay between NV centers and nonlinear optical properties of diamond, offering potential pathways for tunable nonlinear optics applications.

Figures

Figures reproduced from arXiv: 2501.01379 by the authors.

**Figure 2.** Schematic diagram of the experimental Z-scan setup. RefDet, OADet, and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** (a-c) Closed aperture (CA) Z-scan results for different laser beam intensities and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The anisotropy of the third-order nonlinear susceptibility [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: (a) In diamond, when the photon of energy [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

The nitrogen-vacancy centre in diaomond,

M. W. Doherty, N. B. Manson, P. Delaney,et al., “The nitrogen-vacancy centre in diaomond,” Phys. Reports528, 1–45 (2013)

work page 2013

[2] [2]

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C. Shinei, Y. Masuyama, M. Miyakawa,et al., “Nitrogen related paramagnetic defects: Decoherence source of ensemble of NV- center,” J. Appl. Phys.132, 214402 (2022)

work page 2022

[3] [3]

Diamonds with a high density of nitrogen-vacancy centers for magnetometry applications,

V. M. Acosta, E. Bauch, M. P. Ledbetter,et al., “Diamonds with a high density of nitrogen-vacancy centers for magnetometry applications,” Phys. Rev. B80, 115202 (2009)

work page 2009

[4] [4]

Precise temperature sensing with nanoscale thermal sensors based on diamond nv centers,

S. Choe, J. Yoon, M. Lee,et al., “Precise temperature sensing with nanoscale thermal sensors based on diamond nv centers,” Curr. Appl. Phys.18, 1066–1070 (2018)

work page 2018

[5] [5]

Magnetic measurements on micrometer-sized samples under high pressure using designed nv centers,

M. Lesik, T. Plisson, L. Toraille,et al., “Magnetic measurements on micrometer-sized samples under high pressure using designed nv centers,” Science366, 1359–1362 (2019)

work page 2019

[6] [6]

Nanoscale sensing and imaging in biology using the nitrogen-vacancy center in diamond,

L. Hall, D. Simpson, and L. Hollenberg, “Nanoscale sensing and imaging in biology using the nitrogen-vacancy center in diamond,” MRS Bull.38, 162–167 (2013)

work page 2013

[7] [7]

Quantum sensors in diamonds for magnetic resonance spectroscopy: Current applications and future prospects,

R. Rizzato, N. R. von Grafenstein, and D. B. Bucher, “Quantum sensors in diamonds for magnetic resonance spectroscopy: Current applications and future prospects,” Appl. Phys. Lett.123, 260502 (2023)

work page 2023

[8] [8]

Quantum information processing with closely-spaced diamond color centers in strain and magnetic fields,

Z. Xu, Z. qi Yin, Q. Han, and T. Li, “Quantum information processing with closely-spaced diamond color centers in strain and magnetic fields,” Opt. Mater. Express9, 4654–4668 (2019)

work page 2019