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Optical Zitterbewegung effect in arrays of helical waveguides
Open accessAbstract
Owing to its topological properties and band collapse, Floquet helical photonic lattices have gained increasing attention as a purely classical setting to realize the optical analogues of a wide variety of quantum phenomena. We demonstrate both theoretically and numerically that light propagation in an appropriately designed helical superlattice can exhibit spatial photonic Zitterbewegung effect, i.e., a quiver spatial oscillatory motion of the beam center of mass around its mean trajectory, in both one‐ and two‐dimensional cases. The lattice spacing determines the effective coupling strength between adjacent helical waveguides, and further drastically not only affects the oscillation amplitude and frequency, but also invert their direction of drift when the effective coupling strength is tuned from positive to negative. Complete arrest and inversion of the drift direction of Zitterbewegung effect are reported.
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Written from the paper’s full text.
TL;DR
The paper theoretically and numerically shows that light propagating in one- and two-dimensional Floquet photonic lattices of helical waveguides can mimic the relativistic Zitterbewegung (ZB) effect as a spatial quivering motion of the beam center of mass. Tuning the effective coupling via lattice spacing can arrest the ZB oscillation or reverse its drift direction.
Key findings
- In 1D helical waveguide arrays, the Floquet quasi-energy spectrum forms two minibands whose dispersion near the Brillouin-zone edge resembles positive- and negative-energy branches of a massive Dirac particle, enabling optical ZB. For \(d_1/d_2 = 0.8\), the numerical oscillation period is \(T = 0.64\) m, agreeing with the theoretical \(T = 2\pi/\beta\).
- The effective coupling \(c_{\text{eff1}}\) can be tuned from positive to negative by changing \(d_1\). At \(d_{1\text{cr}} = 1.66 d_2\), \(c_{\text{eff1}} = 0\), producing a flat band, complete band collapse, dynamic localization, and arrest of ZB; for larger \(d_1\), the ZB drift direction reverses.
- In 2D helical waveguide arrays based on coupled vertical and horizontal SSH chains, ZB-like lateral motion occurs with oscillation periods \(T = 0.15\) m, \(0.13\) m, and \(0.10\) m for \(d_1/d_2 = 1.5\), \(1.66\), and \(2.2\), respectively. At \(d_1/d_2 = 1.66\), the beam oscillates without lateral shift; for \(d_1/d_2 = 1.5\) and \(2.2\), the lateral shift appears in opposite quadrants.
- The optical ZB is robust against random coupling-strength perturbations up to \(\delta = 0.1\) (10%). Helix spatial period also controls the effect: ZB oscillations occur in regions \(n>0\) and \(n<0\) at \(Z = 0.40\) cm and \(0.23\) cm, respectively, with undamped amplitude at \(Z = 0.31\) cm.
Methods
- The authors derive a Schrödinger-type paraxial equation for light in helical waveguides, transform to a co-moving frame, and reduce the system to tight-binding coupled-mode equations with effective couplings \(c_{\text{eff1}} = c(d_1)J_0(k_0 R\Omega d_1)\) and \(c_{\text{eff2}} = c(d_2)J_0(k_0 R\Omega d_2)\).
- Floquet quasi-energy spectra and band structures are calculated analytically and compared with direct numerical propagation. In 1D, the model uses more than 400 dimers and a broad Gaussian beam with \(w_0 = 10d\); in 2D, a broad circular Gaussian beam with \(w_0 = 6d\) is used.
- Beam dynamics are obtained by directly solving the coupled-mode equations, and the center-of-energy evolution is compared with analytical expressions for the ZB frequency \(\omega_{\text{ZB}} = 2|c_{\text{eff1}} - c_{\text{eff2}}|\) and amplitude \(R_{\text{ZB}} = \pi c_{\text{eff1}}/\omega_{\text{ZB}}\).
Limitations
- The paper does not include an explicit limitations section. It presents theoretical and numerical results only; no experimental realization is reported.
- The text notes that damping of ZB oscillations is visible and attributes it to angular spectral broadening of the launching broad Gaussian beam.
- Boundary effects are said to be reduced by using more than 400 dimers, and robustness is tested only against random coupling-strength perturbations up to 10%; other disorder or fabrication imperfections are not explored in the provided text.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“At c eff1 = 0 (d 1/d 2 = 1.66), corresponding to dynamic localization, the beams oscillation without lateral shift shown in Figure 6(c).”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in two-dimensional helical waveguides
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“further drastically not only affects the oscillation amplitude and frequency”
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
“The lattice spacing affects not only amplitude and frequency of the optical ZB”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The lattice spacing determines the effective coupling strength between adjacent helical waveguides”
- Also stated in: § Abstract · plain
“The lattice spacing determines the effective coupling strength”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
- Also stated in: § Abstract · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“ZB effect with undamped amplitude is also observed at Z = 0.31 cm.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“the motion of the wave packet is accompanied by a transverse drift at a constant velocity v 0”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“At d 1 = 1.66d 2, corresponding to the special case c eff1 = 0, two flat band pinned at E = ±c eff2 are presented.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“By employing an alternation of distances between adjacent waveguides, the quasi-energy bands of the superlattice can be effectively controlled”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Introduction
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“invert their direction of drift when the effective coupling strength is tuned from positive to negative”
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
“Further increasing the distance d 1, corresponding to negative c eff1, subsequent reappearance of ZB effect at d 1 > d 1cr is accompanied by the reversal of the direction of oscillations”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We demonstrate both theoretically and numerically that light propagation in an appropriately designed helical superlattice can exhibit spatial photonic Zitterbewegung effect, i.e., a quiver spatial oscillatory motion of the beam center of mass around its mean trajectory, in both one- and two-dimensional cases.”
- Also stated in: § Introduction · plain
“In this work, we show both theoretically and numerically that the optical ZB effect can be realized in Floquet photonic lattices composed of coupled helical waveguides.”
- Also stated in: § Conclusions · boosted
“propagation of light waves in one-dimensional and two-dimensional Floquet photonic system made of SSH-like helical arrays of evanescently coupled optical waveguides is theoretically and numerically investigated and shown to provide a classic wave optics analog of the ZB effect, where a spatial oscillatory motion of an optical beam is observed.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The array is excited by a broad beam G(x) tilted at the Bragg angle, with modes excited with nearly equal amplitude and a phase difference between adjacent waveguides equal to π/2 across the array.
- The authors state this as an assumption: The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
- Also stated in: § Introduction · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“it is shown that longitudinal rotation of waveguides (helix radius and period) leads to notable variations of Floquet quasi-energy spectrum, that will certainly drastically affect the ZB oscillations.”
- Also stated in: § Conclusions · boosted
“it is shown that longitudinal rotation of waveguides (helix radius and period) leads to notable variations of Floquet quasi-energy spectrum, that can certainly drastically affect the ZB oscillations.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Conclusions · boosted
- It may state two separate claims as one. 16
- The quote only partly states this claim. 3
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Although there contain two oscillation frequencies generated by interferences between the upper and zero modes (E 1 − E 2), and between the upper and low modes (E 1 − E 3). The former plays the leading role in determined the ZB-like motion.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in two-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The frequency of ZB, which can be estimated as ω = 2|ϵ| = 2|c eff2 − c eff1|, grows linearly with increasing a gap β = ω.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“complete collapse and band curvature inversion are obtained.”
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
“the band structure experiences a complete band collapse in the first Brillouin zone”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Introduction
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When d 1/d 2 = 1.5 (for c eff1 > 0), the wave packet expands in both transverse dimensions, optical ZB-like motion occurs with the lateral shift along the bisectrix of the (n, m) plane in the first quadrant(n, m) > 0”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in two-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Further increasing the distance d 1, corresponding to negative c eff1, subsequent reappearance of ZB effect at d 1 > d 1cr is accompanied by the reversal of the direction of oscillations, where ZB effect appears in the region n < 0 (see Figure 2(a3) and (b3)).”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Complete arrest and inversion of the drift direction of Zitterbewegung effect are reported.”
- Also stated in: § Conclusions · plain
“Complete arrest and inversion of the drift direction of ZB effect are reported.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Abstract
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The array is excited by a broad beam G(x) tilted at the Bragg angle, with modes excited with nearly equal amplitude and a phase difference between adjacent waveguides equal to π/2 across the array.
- The authors state this as an assumption: The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
- Also stated in: § Conclusions · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The corresponding oscillation periods are T = 0.15 m, 0.13 m and 0.10 m, respectively, which are agreement with the theoretical predicated.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in two-dimensional helical waveguides
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Even they can be completely arrested or their direction can be reversed.”
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
“it can also arrest this effect and even invert their direction of drift.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Introduction
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
- Also stated in: § Optical ZB effect in one-dimensional helical waveguides · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The ZB oscillations can occur in the regions n > 0 and n < 0 at Z = 0.40 cm and 0.23 cm, respectively.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“With increasing the distance d 1 (d 1 > d 2), the gap is extended, and the oscillation period increases and the amplitude decreases.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“the direction of drift depends on the polarity of c eff1, which can be effectively regulated through properly adjusting the lattice distance d 1.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Figure 3 shows that, the beams exhibit the similar ZB-like lateral shifts with δ up to 10 % (namely, δ = 0.1). Consequently, they show a great robustness against perturbation of the coupling strength.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The ZB oscillations are undamped and they are localized, as shown in Figure 2(a2) and (b2).”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Both completely arrest and inversion of lateral shift are also observed.”
- Also stated in: § Optical ZB effect in two-dimensional helical waveguides · plain
“At c eff1 = 0 (d 1/d 2 = 1.66), corresponding to dynamic localization, the beams oscillation without lateral shift”
- Also stated in: § Optical ZB effect in two-dimensional helical waveguides · boosted
“The inversion of the direction of this optical ZB-like motion is also confirmed with the lateral shifts in the third quadrant”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Introduction
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § Optical ZB effect in two-dimensional helical waveguides · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The corresponding oscillation period along the propagation is T = 0.64 m illustrated in Figure 2(b1), which is in good agreement with the theoretical predictions (T = 2π/β).”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The array is excited by a broad beam G(x) tilted at the Bragg angle, with modes excited with nearly equal amplitude and a phase difference between adjacent waveguides equal to π/2 across the array.
- The authors state this as an assumption: The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This means that excitation in such a lattice will propagate without diffraction, which results in complete localization of the wavepacket in the system. This effect is known as dynamic localization.”
§ Optical Zitterbewegung effect in arrays of helical waveguides > Optical ZB effect in one-dimensional helical waveguides
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The authors state this as an assumption.
The array is excited by a broad beam G(x) tilted at the Bragg angle, with modes excited with nearly equal amplitude and a phase difference between adjacent waveguides equal to π/2 across the array.
“Let us assume that the array is excited by a broad beam G(x) tilted at the Bragg angle, the modes are excited with a nearly equal amplitude, but the phase difference between adjacent waveguides is equal to π/2 across the array.”
The authors state this as an assumption.
The coupling strength c(d1,2) decays exponentially with distance as c = c0 exp(−bd).
“c(d 1,2) is the coupling strength, and assumed to be decay exponentially with the distance c = c 0 exp(−bd) [29], [30]”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Oscillation period is said to increase with the band gap
MajorThe paper states that the ZB frequency equals the gap and that the period is 2π divided by the gap. An increasing gap therefore implies a decreasing period, not an increasing one.
“With increasing the distance d 1 (d 1 > d 2), the gap is extended, and the oscillation period increases and the amplitude decreases.”
“The frequency of ZB, which can be estimated as ω = 2|ϵ| = 2|c eff2 − c eff1|, grows linearly with increasing a gap β = ω.”
Two-dimensional period formula uses a negative eigenvalue
MinorThe paper defines E 3 as negative, but then gives T = 2π/E 3, which would be a negative period. The reported periods are positive.
“The oscillation period is determined by the T = 2π/E 3.”
“At the high symmetry point M, there are three eigenvalues E 1 = −E 3 = 2|c eff2 − c eff1|, and E 2 = 0.”
Dirac equation and stated near-edge dispersion have different quadratic coefficients
MinorThe dispersion of Eq. (6) has a term c eff1 squared times the squared momentum, whereas the stated near-edge dispersion has c eff1 times c eff2. These differ for the unequal couplings under discussion.
“ϵ=±√((ceff1−ceff2)²+ceff1ceff2Q²)”
“i(∂φ)/(∂z)=−iceff1σx(∂φ)/(∂ξ)+(ceff1−ceff2)σyφ”
Citations
Cites
Su, W. P., Schrieffer, J. R., & Heeger, A. J. (1979) · Physical Review Letters · doi:10.1103/physrevlett.42.1698 · Cited by 6,783
Photonic Floquet topological insulators
Rechtsman, M. C., Zeuner, J. M., Plotnik, Y., Lumer, Y., Podolsky, D., Dreisow, F., Nolte, S., Segev, M., & Szameit, A. (2013) · Nature · doi:10.1038/nature12066 · Cited by 3,332
Light propagation and localization in modulated photonic lattices and waveguides
Garanovich, I. L., Longhi, S., Sukhorukov, A. A., & Kivshar, Y. S. (2012) · Physics Reports · doi:10.1016/j.physrep.2012.03.005 · Cited by 507
Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal
Zhang, X. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.113903 · Cited by 315
Zitterbewegung and the internal geometry of the electron
Barut, A. O., & Bracken, A. J. (1981) · Physical Review D · doi:10.1103/physrevd.23.2454 · Cited by 280
Classical Simulation of Relativistic Zitterbewegung in Photonic Lattices
Dreisow, F., Heinrich, M., Keil, R., Tünnermann, A., Nolte, S., Longhi, S., & Szameit, A. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.143902 · Cited by 235
On the Zitterbewegung of the Dirac Electron
Huang, K. (1952) · American Journal of Physics · doi:10.1119/1.1933296 · Cited by 226
A Brief History of Time CrystalsPreprint
Khemani, V., Moessner, R., & Sondhi, S. L. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.10745 · Cited by 211
Observing Zitterbewegung with Ultracold Atoms
Vaishnav, J., & Clark, C. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.153002 · Cited by 210
Unified description of Zitterbewegung for spintronic, graphene, and superconducting systems
Cserti, J., & Dávid, G. (2006) · Physical Review B · doi:10.1103/physrevb.74.172305 · Cited by 198
Flagged first, then by citations
Cited by
Sorted by citations
Landau rainbow based on Floquet helical waveguide systems
Zhou, R., Wang, Z., Ma, W., Zhao, W. K., Liu, Y., & Lu, C. (2025) · Optics Letters · doi:10.1364/ol.563395 · Cited by 4
Symmetric Zitterbewegung effect and symmetric Klein tunneling in optical mesh lattices
Zhan, K., Qin, H., He, H., Yao, G., Jiang, Y., & Zhang, R. (2025) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2025.117146 · Cited by 2
Self-citationZitterbewegung effect in a periodically driven trimer lattice
Jiang, Y., Qin, H., Zhang, R., Chen, W., Liang, Y., Yao, G., He, H., & Zhan, K. (2026) · Physics Letters A · doi:10.1016/j.physleta.2026.131889 · Cited by 1
Self-citation
Metadata
- Authors
- Zhan, K., Chen, Q., Zhang, Q., Zhao, T., Qin, H., He, H., & Yao, G.
- Affiliations
- China University of Petroleum, East China (CN)
- Corresponding author
- Zhan, Kaiyun
- Venue
- Nanophotonics
- Publisher
- Wiley
- Published
- 2024-09-05
- Volume
- 13
- Issue
- 23
- Pages
- 4267-4273
- ISSN
- 2192-8606 (ISSN-L)
- 2192-8614
- PMC
- PMC11635971
- OpenAlex
- W4402228449
- Semantic Scholar
- b97af48824d9b4d40d9a2a4158886465188df23c
- Cited by
- 3
- References
- 31
- Citations by year
- 2026: 1 · 2025: 2
- FWCI
- 0.35
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- Nanophotonics (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- Funders
- National Natural Science Foundation of China (61605251)
- Fundamental Research Funds for the Central Universities (22CX03028A)
- Natural Science Foundation of Shandong Province (ZR2021MA030)
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Nonlinear Photonic Systems
- Keywords
- helical waveguide, photonic superlattices, effective coupling strength, topological photonics
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