Universality and non-differentiability: A new perspective on angular dispersion in optics

Ayman F. Abouraddy; Kenneth L. Schepler; Layton A. Hall; Murat Yessenov

REVIEW 3 major objections 5 minor 13 references

Universality and non-differentiability: A new perspective on angular dispersion in optics

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A kink in the angle-versus-frequency curve of a light pulse lets angular dispersion break speed and dispersion rules that had seemed fixed.

desk verdict A well-packaged perspective restating the authors' own STWP work as 'non-differentiable AD'; the 16-class taxonomy is useful, but the abstract overclaims relative to the idealized-limit caveat. read the letter →

arxiv 2506.18858 v1 pith:7ORNC7PB submitted 2025-06-23 physics.optics

Layton A. Hall , Murat Yessenov , Kenneth L. Schepler , Ayman F. Abouraddy This is my paper · ORCID

classification physics.optics

keywords angulardispersionspace-timewavepacketsnon-differentiablegroupvelocitygroup-velocitypropagationinvarianceuniversalADsynthesizerMGFtheorem

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 the strange properties of space-time wave packets (STWPs)—propagation invariance, on-axis group velocities that depart from c, and free-space normal group-velocity dispersion (GVD)—are not exotic accidents but consequences of a new kind of angular dispersion (AD) in which the propagation angle $\phi(\omega)$ is not differentiable at one spectral frequency $\omega_o$. At that kink the angle cannot be Taylor-expanded, so the usual perturbative treatment of AD fails and the standard limits derived from it, including the Martinez–Gordon–Fork theorem that AD produces only anomalous GVD, no longer apply. The authors show that such non-differentiable AD is exactly what is needed to tune the on-axis group velocity, cancel dispersion to all orders, and control the sign and order of GVD in free space, and they describe a universal AD synthesizer that can impose arbitrary AD profiles including non-differentiable ones. If right, this turns a presumed limitation of angular dispersion into an engineering resource.

What carries the argument

The load-bearing object is the spectral cusp: $\phi(\Omega) \propto \sqrt{\Omega}$ near the non-differentiable frequency $\Omega = 0$, the prototypical profile for baseband STWPs. The argument runs through the light-cone representation, where the field's spectral support is a one-dimensional curve whose intersection with the light-cone has a maximum or minimum at $(k_x, k_z, \omega/c) = (0, k_o, k_o)$; at that point the AD is non-differentiable. The companion mechanism is the universal AD synthesizer, a two-step grating–cylindrical-lens–SLM system that deflects each resolved wavelength at an independently chosen angle, providing independent controls for arbitrarily many AD orders and enabling the non-differentiable profile itself.

What would settle it

Measure the pulse-front tilt $\tan \delta_o^{(1)}$ of a baseband STWP as a function of bandwidth $\Delta\omega$ while keeping the spectral endpoint fixed at $\omega_o$: non-differentiable AD predicts $\tan \delta_o^{(1)} \propto 1/\sqrt{\Delta\omega}$ near $\omega_o$, whereas differentiable AD predicts a bandwidth-independent tilt. A crossover to a bandwidth-independent value at accessible $\Delta\omega$ would show the cusp is washed out and the claimed effects do not survive.

Watch

Extended reading notes

Core claim

The central claim is that the behavior of space-time wave packets is governed by 'non-differentiable angular dispersion': an AD profile $\phi(\omega)$ that is finite and continuous but whose derivative $d\phi/d\omega$ is undefined at a single frequency $\omega_o$, the spectral terminus of the packet. Because no Taylor expansion exists at $\omega_o$, the hierarchical relation between AD coefficients and axial dispersion coefficients is bypassed: $k_z$ can be made to satisfy $k_z = k_o + \Omega/\tilde{v} + \tfrac{1}{2} k_2 \Omega^2$ exactly, with all higher orders zero, independent of the sign of $k_2$. This produces, in free space and on-axis, group velocities $\tilde{v} = c/(1-\eta)$ away from $c$, propagation invariance to all orders, and normal as well as anomalous GVD—each of which is impossible for differentiable AD. The paper further claims that a universal AD synthesizer, which assigns each wavelength its own transverse wave number via spectral resolution and a phase modulator, can implement such profiles and thereby reach the nine classes of pulsed fields that conventional AD engineering cannot.

Load-bearing premise

The practical force of the paper rests on the assumption that finite-energy fields whose spectra merely approach a non-differentiable cusp retain the idealized phenomenology—on-axis group velocity different from $c$, exact all-order dispersion cancellation, and normal GVD on-axis—yet no threshold in spectral uncertainty is given below which each effect survives.

Editorial extensions

If this is right

On-axis fields produced this way can travel at controllable group velocities above or below $c$ in the paraxial regime, with the single parameter $\eta$ setting $\tilde{v} = c/(1-\eta)$.
Propagation-invariant STWPs—free of axial dispersion to all orders—follow from a single non-differentiable profile, explaining the previously reported meter-to-kilometer invariant propagation.
GVD cancellation can now be performed in both the normal- and anomalous-GVD regimes of a material, because free-space normal GVD is accessible on-axis.
Any single dispersion order (e.g., $k_2$, $k_3$, $k_4$) can be isolated with magnitude and sign controlled while all others vanish, enabling dispersion landscapes unavailable with gratings, prisms, or metasurfaces.
A taxonomy of pulsed fields with AD falls into 16 classes; 9 require the universal AD synthesizer, and 4 are only realizable through non-differentiable AD, so the synthesizer opens concrete new classes of fields.

Reading between the lines

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

If the cusp is the operative mechanism, the same kink construction should transfer to other wave equations with a cone-like dispersion relation, such as acoustic, water-wave, or electron matter-wave packets, where a non-differentiable angle-frequency relation would likewise decouple group velocity and dispersion from material response.
A quantitative design rule is missing: the paper introduces a Schmidt number $N_S$ to quantify spectral uncertainty $\delta\omega$ but gives no threshold for when the phenomenological effects ($\tilde{v}\ne c$, all-order cancellation) survive; a natural extension would map allowable $\delta\omega$ versus propagation distance for each predicted effect.
The claim implies a sharp experimental discriminator: for non-differentiable AD the pulse-front tilt scales as $1/\sqrt{\Delta\omega}$ near the spectral endpoint, while differentiable AD gives a bandwidth-independent tilt; measuring this scaling across a wide bandwidth range would test the mechanism directly.
Because the universal AD synthesizer assigns each wavelength an independent transverse wave number, it can also implement discontinuous AD profiles that split the spectrum into distinct group-velocity windows; this follows from the same synthesizer design and extends the paper's schema beyond a single cusp.

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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

First thing you should know: this is not a new result paper. It's a Perspective that re-presents the authors' established work on space-time wave packets under a single conceptual umbrella, 'non-differentiable angular dispersion.' If you already follow that literature, the core idea—φ(ω) ∝ √Ω gives a spectral cusp at ω_o and breaks the Taylor expansion that underpins the MGF theorem—is in their earlier papers (refs 54, 63, 70, 73). What's new here is the packaging: a 16-class decision tree for all fields endowed with AD, and a clear statement that 9 of those classes need either non-differentiable AD or independent control over multiple AD orders. That taxonomy is genuinely useful. It explains, in one picture, why STWPs violate the old 'universal' pulse-front-tilt rule and why they can show normal GVD on-axis.

The math in Section VI checks out. The square-root cusp follows from requiring k_z = k_o + Ω/ṽ on the light cone, and the group-velocity derivation with η is correct. I also appreciate the explicit admission in Section X that ideal non-differentiable AD is physically impossible because every realization has some spectral uncertainty δω.

Now the soft spots. The central concept is defined from the STWP spectral support itself, so the 'mechanism' is partly a restatement of the construction. That's not fatal—good classifications often work that way—but it means the paper explains STWPs by name rather than by independent cause. The bigger issue is the abstract. It says non-differentiable AD 'enables circumventing many well-established constraints in optics.' Section X concedes that ideal profiles are impossible, yet no threshold in δω or Schmidt number is given for when the predicted effects (tunable on-axis group velocity, all-order dispersion cancellation, normal GVD) survive. The stress-test note is right: the Schmidt-number discussion is qualitative. A reader in a position to use these fields can't tell from this paper how sharp the cusp must be.

For a Perspective, I think that gap is acceptable if the authors frame the claims as ideal-limit phenomena. Right now the abstract overstates. A serious referee should ask for either a quantitative statement (e.g., 'effects persist when N_S > N_c') or a clear caveat in the abstract. I'd send this to review. The taxonomy and the unified framing are worth having on record, and the authors are the right people to write it.

Referee Report

3 major / 5 minor

Summary. This Perspective proposes that the unusual properties of space-time wave packets (STWPs) can be traced to a new form of angular dispersion (AD) that is non-differentiable at a frequency ω_o. The authors develop the conventional Taylor-expansion framework for differentiable AD, review the Martinez–Gordon–Fork (MGF) theorem, and show that non-differentiable AD, exemplified by φ(Ω) ∝ √Ω, allows on-axis group velocities different from c, propagation-invariant wave packets with all higher-order dispersion eliminated, and normal or anomalous GVD in free space. They classify pulsed fields with AD into 16 classes, assert that 9 of the admissible classes require a universal AD synthesizer, describe such a synthesizer based on grating–lens–SLM spectral shaping, extend the idea to conical AD, and introduce a Schmidt number to quantify the non-differentiable AD resource.

Significance. If the central claims survive scrutiny, this is a valuable unifying perspective: it connects the STWP literature to the classical AD framework, provides a simple derivation of the √Ω cusp from the spectral-support condition, and gives a falsifiable bandwidth-scaling prediction for pulse-front tilt (Eq. 18). The paper is explicit about idealization (Section X) and offers a quantitative resource measure (Schmidt number), which is a strength. It is also candid about the physical impossibility of producing an ideal non-differentiable AD profile. However, the manuscript is a Perspective rather than a new experimental demonstration, and its practical force depends on bridging the ideal cusp to finite-energy fields.

major comments (3)

[Section X, Eqs. (20)–(21)] The manuscript concedes that an ideal AD profile is physically impossible, but it does not provide any quantitative criterion for when finite spectral uncertainty preserves the claimed effects. The central results—on-axis group-velocity tuning (Section VI A), all-order dispersion cancellation (Section VI B), and normal GVD with all higher orders eliminated (Section VI C)—are derived for an exactly singular spectral support. For any finite δω the spatiotemporal spectrum becomes two-dimensional and k_z(ω) is no longer exactly linear; the paper states only that the Schmidt number N_S drops as δω increases, without specifying a threshold in δω or N_S below which, for example, the on-axis group velocity remains within a given tolerance of c/(1−η) or the GVD retains its designed sign. This gap is load-bearing because the claim of circumventing the MGF theorem rests on the cusp being sufficiently sharp. Please add an explicit finite-bandwidth model (e.g., a smoothed or convolved cusp) and report the degradation of group velocity, GVD, and dispersion cancellation as functions of δω/ω_o or N_S.
[Section VI B, Eq. (17) and following text] The derivation of φ(Ω) ∝ √Ω is obtained by imposing the propagation-invariance condition k_z = k_o + Ω/ṽ and then solving k cos φ = k_z on the light cone. The non-differentiability is therefore a consequence of the chosen spectral support, not an independently established cause. Moreover, because ω_o is a spectral boundary with zero amplitude, any physical field with finite bandwidth has an AD profile that is differentiable on its actual spectral support; the singularity itself is never sampled. The paper should either identify an observable that discriminates a true non-differentiable profile from a high-order differentiable approximation on the same support, or reframe the thesis as: STWPs are designed via a non-perturbative spectral-support condition whose endpoint behavior is non-differentiable in the ideal limit. As written, the causal language, such as 'non-differentiable AD can lead to propagation invariance,' is circular.
[Section VII, Fig. 13] The assertion that 9 of the 15 admissible classes 'can be synthesized using a universal AD synthesizer' is not supported by a per-class realizability analysis. The synthesizer described in Section VIII can imprint an arbitrary phase profile in principle, but Section X concedes that an ideal non-differentiable profile cannot be produced; no tolerance or spectral-resolution requirement is given for each of the 9 classes. In particular, the four classes said to require exclusively non-differentiable AD need a statement of how closely the realized profile must approach the cusp for the class-defining property, such as normal GVD on-axis, to be observable. Without such a condition, the classification has only ideal limiting validity.

minor comments (5)

[Abstract and Section XII] The phrase 'circumventing many well-established constraints' should be qualified as applying to idealized fields with an exact spectral cusp; Section X already acknowledges that the ideal is physically unrealizable, so the abstract should carry the same caveat.
[Section VII] The text moves between '16 classes' and '9 classes' without always stating that one class is physically inadmissible; please consistently say '15 admissible classes' or '9 of 15.'
[Section VI D, Eqs. (18)–(19)] The relationship between Eq. (18) and Eq. (19) should be stated more carefully: please clarify that ω_c is the arithmetic mid-spectral frequency and spell out the limit in which Eq. (19) reduces to Eq. (18), including the treatment of the factor (ω_c/ω_o).
[Section X] The Schmidt number is introduced as a quantifier of non-differentiable AD, but no relation is given between N_S and the observable dispersion coefficients; a sentence connecting N_S to, for example, the residual GVD of a realized STWP would make the resource interpretation concrete.
[Section VI B] The statement that 'the truncation of this dispersion relationship at second order is not an approximation' is correct for the ideal 1D support, but it should be accompanied by a reminder that any finite spectral uncertainty reintroduces higher-order terms; this connects to the missing threshold in Section X.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central derivations are parameter-free consequences of the light-cone geometry, and the self-citations are not load-bearing.

full rationale

The paper's central derivations are self-contained and parameter-free. Non-differentiable AD is defined as a cusp in the angle-frequency relation phi(omega), and the propagation-invariant STWP result follows from the light-cone condition k_z = k_o + Omega/v, which yields phi(Omega) proportional to sqrt(Omega) near the non-differentiable frequency (Section VI.B). This is a mathematical consequence of the spectral support, not a fitted parameter disguised as a prediction. Similarly, the group-velocity relation v = c/(1-eta) in Section VI.A is obtained by substituting a specified phi(Omega) into the exact dispersion relation; eta is a free design parameter of the AD profile, and the resulting v is a derived quantity. The pulse-front-tilt scaling in Eq. 18 is initially quoted from the authors' prior work as an ansatz, but Eq. 19 re-derives the same scaling directly from the sqrt(Omega) cusp, so the paper does not rely on the fit as an input. The Schmidt-number discussion in Section X is a resource-quantification tool, and the concession that an ideal non-differentiable AD profile is physically impossible is a limitation or falsifiability concern, not a circular one. The paper contains self-citations to prior STWP demonstrations and to the Schmidt-number work, but these citations are used for context and prior experimental support, not as the load-bearing argument for the new derivations. Overall, no prediction reduces to its own input by construction, and no central conclusion is forced by a self-citation chain.

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

The analysis is built on the free-space scalar light-cone model and an idealized 1D spectral support in which each frequency has a single propagation angle. It introduces no new physical entity; the only hand-chosen design parameter is eta, which sets the position of the cusp. The Schmidt number is a derived metric, not a fitted quantity.

free parameters (1)

eta (non-differentiable cusp slope) = unspecified positive dimensionless constant (design parameter)
Introduced in Section VI A as phi(omega) = sqrt(2 eta Omega / omega_o); sets the on-axis group velocity v = c/(1-eta). It is chosen by hand to illustrate the mechanism, not fitted to data.

assumptions (5)

domain assumption Free-space scalar wave propagation with k = omega/c and no material chromatic dispersion or anisotropy.
Section II A restricts the analysis to this setting; all dispersion coefficients are derived from the light-cone relation.
domain assumption The angular spectrum of an AD-endowed field is a 1D curve on the light cone, with each frequency mapped to a single transverse wave number.
Section II D establishes this; it is the geometric foundation for defining a single-valued phi(omega).
domain assumption Ideal fields with zero spectral uncertainty are physically realizable in the limit; finite-energy fields approximate them.
Section X concedes the ideal non-differentiable profile requires infinite energy; the claims about circumventing constraints use this idealized limit.
ad hoc to paper A pixelated SLM can implement an arbitrary, including non-differentiable, AD profile with sufficient fidelity.
Section VIII C relies on this to claim universal control; no quantitative resolution or phase-range analysis is provided.
domain assumption Only k_z > 0 spectral components are physically permissible.
Section VI B invokes ref 148 to exclude the k_z = -k_o point for sideband STWPs; this is load-bearing for the sideband case.

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Cite this review

Pith. "Pith review of Universality and non-differentiability: A new perspective on angular dispersion in optics." pith.science (2026). https://pith.science/paper/7ORNC7PB

@misc{pith2026250618858,
  author       = {Pith},
  title        = {Pith review of: Universality and non-differentiability: A new perspective on angular dispersion in optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ORNC7PB}},
  note         = {Machine review of arXiv:2506.18858}
}

read the original abstract

Angular dispersion (AD) is a ubiquitous phenomenon in optics after light traverses a diffractive or dispersive device, whereby each wavelength propagates at a different angle. AD is useful in a variety of applications; for example, modifying the group velocity or group-velocity dispersion of pulsed lasers in free space or optical materials, which are essential ingredients in group-velocity matching and dispersion compensation. Conventional optical components introduce `differentiable' AD, so that the propagation angle can be expanded perturbatively around a fixed frequency, in which only a few low AD-orders are typically relevant. However, this model does not encompass newly emerging classes of propagation-invariant pulsed optical fields, such as `space-time wave packets', which incorporate a new form of AD that we call `non-differentiable AD'. This is a surprising feature: there exists a frequency at which the derivative of the propagation angle with respect to frequency is not defined. Consequently, the propagation angle cannot be expanded perturbatively at this frequency, and a large number of independently controllable AD orders are needed to approximate this condition. Synthesizing these new AD-induced field configurations requires constructing a `universal AD synthesizer' capable of accessing the magnitude and sign of any AD order, a capability missing from any single optical component to date. This Perspective article provides a unified schema for studying differentiable and non-differentiable AD, shows that non-differentiable AD enables circumventing many well-established constraints in optics -- thereby giving rise to new applications, and outlines the requirements for a universal AD synthesizer capable of producing both forms of AD.

Figures

Figures reproduced from arXiv: 2506.18858 by the authors.

**Figure 2.** FIG. 2. GDD is introduced into a transform-limited pulse (a) after [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4. The spectral support for optical fields on the surface of the free-space light-cone [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (11 more)

**Figure 5.** Figure 5: FIG. 5. (a) The phase front is orthogonal to [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Conventional AD is introduced into a collimated pulse [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) The first-order AD coefficient [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Algorithmic procedure for synthesizing an arbitrary disper [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: (a) highlights the need for independent control over ϕo and ϕ (1) o to tune ve, However, such control is not always available. For example, a grating does not provide the full span of values for vealong the normal to the grating [ [PITH_FULL_IMAGE:figures/full_fig_p01…

**Figure 10.** Figure 10: FIG. 10. Spectral support for propagation-invariant STWPs on the surface of the free-space light-cone and their associated AD profile [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: FIG. 11. Same as Fig. 10 for dispersive STWPs. The spectral support is the intersection of the light-cone with a planar curved surface that [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: FIG. 12. Impact of the proximity of the STWP-spectrum to the non [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 13.** Figure 13: FIG. 13. Classification of pulsed optical fields incorporating AD. The first tier distinction is on-axis ( [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

**Figure 14.** Figure 14: FIG. 14. (a) A universal AD synthesizer introduces an arbitrary an [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

**Figure 15.** Figure 15: FIG. 15. (a) The spectral surface associated with conical-AD [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

[1]

A large ϕo and modest ωoϕ (1) o , corresponding to the highlighted region to the right in Fig. 9(a). Here, small changes in ωoϕ (1) o at large ϕo can rapidly tuneev

work page
[2]

A large ωoϕ (1) o at modest ϕo, corresponding to the high- 10 FIG. 9. (a) Group velocity ev along the z-axis (Eq. 8) as a function of ϕo and ωoϕ (1) o . We employ three color palettes: green for the subluminal regime 0 <ev <c, red for the superluminal c <ev < ∞, and blue for negative valuesev <0. The black curves are contours of con- stantev. The dashed b...

work page
[3]

Tuning the on-axis group velocity in free space away from c

work page
[4]

Realizing propagation-invariant wave packets endowed with AD; i.e., a pulsed beam that incorporates AD and yet is free of dispersion to all orders. 11 FIG. 10. Spectral support for propagation-invariant STWPs on the surface of the free-space light-cone and their associated AD profile ϕ(ω). (a) Subluminal (θ <45◦) baseband STWP with kz =ko at kx =0; (b) su...

work page
[5]

For example, realizing normal GVD in free space when ϕo = 0

Tuning the magnitude and sign of a single dispersion coefficient while eliminating all others in an on-axis field. For example, realizing normal GVD in free space when ϕo = 0. We emphasize that these three tasks are impossible to achieve when one is restricted to differentiable AD. A. Tuning the on-axis group velocity via non-differentiable AD Can the gro...

work page
[6]

By requiring that ω = ωo propa- gate along z (ϕo =0◦ and ka =ko), then kz =ko + (ω− ωo)/ev, whereupon ϕ(Ω)≈ q 2(1−en) Ω ωo ∝ √ Ω, which is not differ- entiable at Ω = 0

Baseband STWPs. By requiring that ω = ωo propa- gate along z (ϕo =0◦ and ka =ko), then kz =ko + (ω− ωo)/ev, whereupon ϕ(Ω)≈ q 2(1−en) Ω ωo ∝ √ Ω, which is not differ- entiable at Ω = 0. Here ω > ωo whenen < 1 (superluminal), and ω < ωo whenen > 1 (subluminal). This encompasses the family of ‘baseband’ STWPs145,146, whose name refers to the fact that the s...

work page
[7]

Sideband STWPs. A second solution exists when ka =−ko, and kz =−ko + (ω− ωo)/ev, which corresponds to the family of sideband STWPs, including focus-wave modes (FWMs) discovered by Brittingham in 1983 147, for which ev = c and kz =−ko + Ω/c146. The ‘sideband’ moniker refers to the exclusion of spatial frequencies in the vicinity of kx =0 because they are a...

work page 1983
[8]

Finally, maintainingen(ω)=en is possible when ϕ(ω) =ϕo, which corresponds to X-waves that are thus AD- free with en = cos ϕo

X-waves. Finally, maintainingen(ω)=en is possible when ϕ(ω) =ϕo, which corresponds to X-waves that are thus AD- free with en = cos ϕo. Here kz = ω c cos ϕo, so their superlumi- nal phase and group velocities vph =ev = c/cos ϕo stem from a purely geometric origin 112. The spectral support for AD- free X-waves is the pair of straight lines at the intersecti...

work page

Show all 13 references

[9]

STWPs that are propagation invariant over large distances67–69, with controllable group velocity 64–66, and axially encoded propagation characteristics, whether the on-axis wavelength (axial spec- tral encoding) 152,153 or the group velocity (axial acceleration)154–157
[10]

STWPs incorporating controllable magnitude, sign, and order of GVD in free space70
[11]

Discontinuous AD to tune the group velocity in differ- ent spectral windows158
[12]

Discretized spatiotemporal spectra to explore a variety of space-time Talbot effects72,74,159
[13]

Angular dispersion: an en- abling tool in nonlinear and quantum optics,

Hybrid guided spatiotemporal modes coupled to pla- nar waveguides86,87, multimode waveguides88,89,92 and fibers90,94,95, pulses that are omni-resonant with planar cavities81, and that couple light to space-time surface plasmon polaritons96,160–163. It is useful at this point t...

2010 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

A large ϕo and modest ωoϕ (1) o , corresponding to the highlighted region to the right in Fig. 9(a). Here, small changes in ωoϕ (1) o at large ϕo can rapidly tuneev

work page

[2] [2]

A large ωoϕ (1) o at modest ϕo, corresponding to the high- 10 FIG. 9. (a) Group velocity ev along the z-axis (Eq. 8) as a function of ϕo and ωoϕ (1) o . We employ three color palettes: green for the subluminal regime 0 <ev <c, red for the superluminal c <ev < ∞, and blue for negative valuesev <0. The black curves are contours of con- stantev. The dashed b...

work page

[3] [3]

Tuning the on-axis group velocity in free space away from c

work page

[4] [4]

Realizing propagation-invariant wave packets endowed with AD; i.e., a pulsed beam that incorporates AD and yet is free of dispersion to all orders. 11 FIG. 10. Spectral support for propagation-invariant STWPs on the surface of the free-space light-cone and their associated AD profile ϕ(ω). (a) Subluminal (θ <45◦) baseband STWP with kz =ko at kx =0; (b) su...

work page

[5] [5]

For example, realizing normal GVD in free space when ϕo = 0

Tuning the magnitude and sign of a single dispersion coefficient while eliminating all others in an on-axis field. For example, realizing normal GVD in free space when ϕo = 0. We emphasize that these three tasks are impossible to achieve when one is restricted to differentiable AD. A. Tuning the on-axis group velocity via non-differentiable AD Can the gro...

work page

[6] [6]

By requiring that ω = ωo propa- gate along z (ϕo =0◦ and ka =ko), then kz =ko + (ω− ωo)/ev, whereupon ϕ(Ω)≈ q 2(1−en) Ω ωo ∝ √ Ω, which is not differ- entiable at Ω = 0

Baseband STWPs. By requiring that ω = ωo propa- gate along z (ϕo =0◦ and ka =ko), then kz =ko + (ω− ωo)/ev, whereupon ϕ(Ω)≈ q 2(1−en) Ω ωo ∝ √ Ω, which is not differ- entiable at Ω = 0. Here ω > ωo whenen < 1 (superluminal), and ω < ωo whenen > 1 (subluminal). This encompasses the family of ‘baseband’ STWPs145,146, whose name refers to the fact that the s...

work page

[7] [7]

Sideband STWPs. A second solution exists when ka =−ko, and kz =−ko + (ω− ωo)/ev, which corresponds to the family of sideband STWPs, including focus-wave modes (FWMs) discovered by Brittingham in 1983 147, for which ev = c and kz =−ko + Ω/c146. The ‘sideband’ moniker refers to the exclusion of spatial frequencies in the vicinity of kx =0 because they are a...

work page 1983

[8] [8]

Finally, maintainingen(ω)=en is possible when ϕ(ω) =ϕo, which corresponds to X-waves that are thus AD- free with en = cos ϕo

X-waves. Finally, maintainingen(ω)=en is possible when ϕ(ω) =ϕo, which corresponds to X-waves that are thus AD- free with en = cos ϕo. Here kz = ω c cos ϕo, so their superlumi- nal phase and group velocities vph =ev = c/cos ϕo stem from a purely geometric origin 112. The spectral support for AD- free X-waves is the pair of straight lines at the intersecti...

work page