Modal dispersion

Modal dispersionMain

Community hub

Modal dispersion

7 pages, 0 posts

0 subscribers

Recent from talks

Be the first to start a discussion here.

Recent from talks

Be the first to start a discussion here.

Contribute something

About hubMembersContent overviewUpdatesRules

Main reference articles

Modal dispersion

View on Wikipedia

from Wikipedia

Modal dispersion is a distortion mechanism occurring in multimode fibers and other waveguides, in which the signal is spread in time because the propagation velocity of the optical signal is not the same for all modes. Other names for this phenomenon include multimode distortion, multimode dispersion, modal distortion, intermodal distortion, intermodal dispersion, and intermodal delay distortion.^[1]^[2]

In the ray optics analogy, modal dispersion in a step-index optical fiber may be compared to multipath propagation of a radio signal. Rays of light enter the fiber with different angles to the fiber axis, up to the fiber's acceptance angle. Rays that enter with a shallower angle travel by a more direct path, and arrive sooner than rays that enter at a steeper angle (which reflect many more times off the boundaries of the core as they travel the length of the fiber). The arrival of different components of the signal at different times distorts the shape.^[3]

Modal dispersion limits the bandwidth of multimode fibers. For example, a typical step-index fiber with a 50 μm core would be limited to approximately 20 MHz for a one kilometer length, in other words, a bandwidth of 20 MHz·km. Modal dispersion may be considerably reduced, but never completely eliminated, by the use of a core having a graded refractive index profile. However, multimode graded-index fibers having bandwidths exceeding 3.5 GHz·km at 850 nm are now commonly manufactured for use in 10 Gbit/s data links.

Modal dispersion should not be confused with chromatic dispersion, a distortion that results due to the differences in propagation velocity of different wavelengths of light. Modal dispersion occurs even with an ideal, monochromatic light source.

A special case of modal dispersion is polarization mode dispersion (PMD), a fiber dispersion phenomenon usually associated with single-mode fibers. PMD results when two modes that normally travel at the same speed due to fiber core geometric and stress symmetry (for example, two orthogonal polarizations in a waveguide of circular or square cross-section), travel at different speeds due to random imperfections that break the symmetry.

Troubleshooting

[edit]

In multimode optical fiber with many wavelengths propagating, it is sometimes hard to identify the dispersed wavelength out of all the wavelengths that are present, if there is not yet a service degradation issue. One can compare the present optical power of each wavelength to the designed values and look for differences. After that, the optical fiber is tested end to end. If no loss is found, then most probably there is dispersion with that particular wavelength. Normally engineers start testing the fiber section by section until they reach the affected section; all wavelengths are tested and the affected wavelength produces a loss at the far end of the fiber. One can easily calculate how much of the fiber is affected and replace that part of fiber with a new one. Replacement of optical fiber is only required when there is an intense dispersion and service is being affected; otherwise various methods can be used to compensate for the dispersion.

References

[edit]

^ "Multimode distortion". Federal Standard 1037C. August 7, 1996.
^ "Multimode distortion". ATIS Telecom Glossary. Retrieved June 29, 2014.
^ Mitschke, Fedor (2009). Fiber Optics: Physics and Technology. Springer. p. 20. ISBN 978-3-642-03702-3.

External links

[edit]

Interactive webdemo for modal dispersion Institute of Telecommunications, University of Stuttgart

Revisions and contributors Edit on Wikipedia Read on Wikipedia

View on Grokipedia

from Grokipedia

Modal dispersion, also known as intermodal dispersion, is a type of signal distortion that occurs in multimode optical fibers, where light propagates through multiple guided modes with differing group velocities, resulting in the temporal broadening of optical pulses and intersymbol interference.^[1]^[2]^[3] This phenomenon arises primarily from the varying propagation paths and effective refractive indices experienced by different modes in the fiber core, which has a relatively large diameter (typically 50 or 62.5 μm) to support multiple modes.^[4]^[2] In step-index multimode fibers, axial modes travel straight while higher-order modes follow longer, more oblique paths, leading to arrival time differences at the fiber output.^[1]^[2] The pulse spreading time

\Delta T

can be approximated as

\Delta T = \frac{L n_1^2 \Delta}{c n_2}

, where

L

is the fiber length,

n_1

and

n_2

are the core and cladding refractive indices,

\Delta = (n_1 - n_2)/n_1

is the relative index difference, and

c

is the speed of light; for example, this yields about 68 ns/km in a typical step-index fiber with

n_1 = 1.48

and

n_2 = 1.46

.^[2]^[4] The primary effects of modal dispersion include reduced bandwidth-distance product, limiting data transmission rates in multimode links to short distances (e.g., 100–300 m at 10 Gb/s), and degradation of signal integrity due to pulse overlap.^[3]^[4] In legacy multimode fibers, this restricts performance to around 200 MHz·km, though optimized designs can exceed 2 GHz·km.^[4] Unlike single-mode fibers, where modal dispersion is absent due to the support of only one fundamental mode, multimode fibers suffer this limitation despite advantages in coupling efficiency for short-haul applications like local-area networks.^[1]^[4] Mitigation strategies include using graded-index profiles, which quadratically vary the core refractive index to equalize mode velocities and significantly reduce spreading (e.g., to about 0.25 ns/km), as well as advanced techniques like spatial light modulators for adaptive equalization of mode coupling and dispersion.^[1]^[2]^[3]

Fundamentals

Definition and Basic Concept

Modal dispersion, also known as intermodal dispersion, is a phenomenon in guided wave optics where an optical pulse broadens as it propagates through a multimode waveguide due to the differing group velocities of the various spatial modes supported by the structure.^[5] In essence, this dispersion arises because light signals traveling along different mode paths experience varying effective path lengths and refractive indices, resulting in temporal spreading of the pulse over distance.^[5] To understand modal dispersion, it is essential to first grasp the fundamentals of waveguides and propagation modes. A waveguide is a structure designed to confine and direct electromagnetic waves, such as light, along a specific path, typically by means of total internal reflection at boundaries between materials of different refractive indices; optical fibers serve as a primary example, consisting of a core surrounded by a cladding with lower refractive index.^[6] Within such waveguides, light propagates in discrete patterns called modes, which are self-consistent solutions to Maxwell's equations representing stable field distributions. These modes are classified into transverse electric (TE) modes, where the electric field has no component in the direction of propagation, and transverse magnetic (TM) modes, where the magnetic field lacks such a component; higher-order modes include variations in both fields transverse to the propagation direction.^[7] In multimode waveguides like step-index or graded-index optical fibers, numerous such modes can exist simultaneously, each carrying portions of the optical signal but traversing the guide via distinct field patterns and effective paths.^[5] The core concept of modal dispersion centers on how these modes lead to differential delays in multimode fibers, where light injected into the core excites multiple modes that zigzag or propagate helically at different speeds, causing the fastest and slowest modes to arrive at the output with significant time offsets.^[5] This differential group delay primarily affects short-distance, high-bandwidth applications, limiting the achievable data rates without compensation. Modal dispersion was first systematically observed and analyzed in early fiber optic experiments during the 1970s, as researchers investigated pulse broadening in multimode fibers developed for telecommunications; a seminal study by Olshansky and Keck in 1976 quantified this effect in graded-index fibers, highlighting its relation to waveguide mode propagation. In the broader context of optical fiber systems, modal dispersion contributes to total dispersion alongside chromatic dispersion (due to wavelength-dependent refractive index) and polarization-mode dispersion (in single-mode fibers).^[5]

Physical Causes in Waveguides

Modal dispersion in waveguides arises from the propagation of light in multiple transverse modes, each governed by boundary conditions that enforce total internal reflection at the core-cladding interface. The critical angle for total internal reflection, defined as

\theta_c = \sin^{-1}(n_2 / n_1)

where

n_1

is the core refractive index and

n_2

is the cladding refractive index (

n_2 < n_1

), determines the maximum incidence angle for mode confinement, allowing only rays with angles greater than

\theta_c

to be guided without leakage. These boundary conditions, derived from Maxwell's equations, quantize the allowed modes based on the waveguide's geometry and refractive index contrast, with higher-order modes corresponding to steeper ray angles near the critical limit.^[8] In multimode step-index waveguides, where the core has a uniform refractive index abruptly dropping at the cladding boundary, modal dispersion primarily stems from differences in path lengths traversed by various ray trajectories. Axial rays propagate straight along the fiber centerline, covering the shortest distance, while meridional rays zig-zag in planes containing the axis, undergoing total internal reflections at shallower angles. Skew rays, in contrast, follow helical or polygonal helical paths that spiral around the axis without crossing it, resulting in longer effective paths for higher-order modes, leading to arrival time differences at the waveguide output.^[9]^[10] This path length variation causes lower-order modes (axial and low-angle meridional) to travel faster effectively than higher-order modes (helical skew rays), broadening pulses over distance.^[11] Graded-index waveguides mitigate these effects by introducing a radially varying refractive index profile, typically parabolic, where

n(r)

decreases from the core center to the edge, creating velocity differences among modes. In such profiles, higher-order rays near the core edge propagate through regions of lower refractive index, achieving higher phase velocities that partially compensate for their longer helical paths, thus reducing intermodal delay spreads compared to step-index designs.^[11]^[12] However, imperfect grading or deviations from optimal profiles can exacerbate dispersion by failing to equalize velocities fully.^[12] Additionally, mode coupling—energy transfer between modes due to waveguide imperfections like core ellipticity, microbends, or index perturbations—introduces intermodal interference, further contributing to dispersion by randomizing path delays and causing temporal spreading beyond simple geometric differences.^[13] In ideal straight waveguides, modes propagate independently, but real-world coupling leads to differential group delays that accumulate over length.^[14]

Mathematical Description

Mode Propagation Analysis

In optical fibers, the propagation constant

\beta_m

for the

m

-th mode is defined as the axial component of the wave vector,

\beta_m = k_z

, where

k = 2\pi / \lambda

and the effective index

n_{\text{eff},m} = \beta_m / k

lies between the core and cladding refractive indices. For guided modes,

\beta_m

assumes discrete values determined by the transverse confinement, with

\beta_m

decreasing as the mode order increases; in linearly polarized (LP) modes typical of step-index fibers, the fundamental LP

_{01}

mode has the highest

\beta

, approaching the core wavenumber

k n_1

, while higher-order modes have

\beta

closer to the cladding wavenumber

k n_2

. This variation arises from the mode's radial and azimuthal field distributions, which influence the effective path length.^[15]^[16]^[17] The group velocity

v_{g,m}

of the

m

-th mode, defined as

v_{g,m} = \frac{d\omega}{d\beta_m}

where

\omega

is the angular frequency, differs across modes due to the nonlinear dispersion relation

\omega(\beta)

imposed by the waveguide boundaries. In multimode fibers, this relation causes higher-order modes to propagate with lower group velocities compared to lower-order modes, as their effective paths are longer and more curved, leading to slower signal transport for those components. This intermodal variation is the core mechanism of modal dispersion, limiting bandwidth in multimode systems.^[18]^[19]^[20] In weakly guiding fibers, where the core-cladding index contrast

\Delta = (n_1^2 - n_2^2)/(2n_1^2) \ll 1

, modes are classified using the linearly polarized (LP) approximation, which scalarizes the vectorial Maxwell equations for conceptual simplicity and accurate prediction of field patterns. Developed by Gloge, this approach represents exact hybrid modes (HE and EH) as superpositions of two orthogonal LP

_{lm}

modes, with

l

denoting the azimuthal order (integer

\geq 0

) and

m

the radial order (

\geq 1

); for example, LP

_{11}

is the lowest higher-order mode. Each LP

_{lm}

mode has a cutoff normalized frequency

V_{c,lm}

, below which it radiates into the cladding; the fiber's overall

V = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2}

V=λ2πan12−n22

(with $a$ the core radius) governs propagation, supporting only the fundamental mode for $V < 2.405$ and approximately $V^2 / 2$ modes total in the multimode regime for large $V$ .^[17]^[15]^[21] Mode propagation analysis employs solutions to the scalar Helmholtz equation $\nabla^2 \psi + k^2 n^2(r) \psi = 0$ in cylindrical coordinates, separating into radial, azimuthal, and axial components to form an eigenvalue problem for $\beta_{lm}$ . Boundary continuity at the core-cladding interface yields characteristic equations whose roots define the discrete modes, conceptually illustrating mode-dependent phase velocities $v_{p,lm} = \omega / \beta_{lm}$ , which exceed the cladding light speed but decrease with mode order as $\beta_{lm}$ diminishes. This framework highlights how waveguide geometry induces velocity spreads without requiring full vectorial computation in the weak guidance limit.^[22]^[17]

Dispersion Formula Derivation

The group delay experienced by a light pulse propagating in a specific mode of the fiber is given by

\tau = \frac{L}{v_g}

, where

L

is the fiber length and

v_g

is the mode's group velocity. Modal dispersion

D_{\mathrm{modal}}

quantifies the spread in arrival times across modes and is defined as

D_{\mathrm{modal}} = \frac{\Delta \tau}{L}

, where

\Delta \tau

is the differential group delay, typically the difference between the maximum and minimum group delays among the guided modes.^[5] In step-index multimode fibers, the derivation of the dispersion formula relies on the paraxial approximation (small propagation angles) and the geometric optics limit (high mode count, ray-like propagation). The fundamental mode propagates with group velocity

v_{g,0} = c / n_1

, where

c

is the speed of light in vacuum and

n_1

is the core refractive index. Higher-order modes follow longer helical or zigzag paths near the core-cladding interface, effectively experiencing a higher group index

n_{g,\max} \approx n_1 (1 + \Delta)

, where

n_2

is the cladding index and

\Delta = (n_1 - n_2)/n_1

is the relative index contrast. This yields a maximum differential delay

\Delta \tau \approx L n_1 \Delta / c

, so the intermodal dispersion parameter is

\sigma_{\mathrm{modal}} \approx n_1 \Delta / c

.^[5] For graded-index fibers with a power-law refractive index profile

n(r) = n_1 \left[1 - 2\Delta \left(r/a\right)^\alpha \right]^{1/2}

, where

r

is the radial position,

a

is the core radius, and

\alpha

is the profile parameter, the dispersion is significantly reduced. Rays launched at larger angles travel through regions of lower refractive index, which compensates for their longer paths and equalizes group delays. Under the same paraxial and geometric optics assumptions, the optimal profile occurs near

\alpha = 2

(parabolic), yielding

\Delta \tau \approx \frac{n_1 L \Delta^2}{8 c}

. For general

\alpha

, the broadening is minimized near

\alpha = 2

, with intermodal effects scaling favorably compared to the step-index case.^[5] As a numerical example, consider a typical graded-index multimode fiber with 50/125

\mu

m core/cladding dimensions and

\Delta = 0.01

at 850 nm (

n_1 \approx 1.46

). Using the optimal parabolic profile, the modal dispersion is

\sigma_{\mathrm{modal}} \approx 0.06

ns/km (60 ps/km), enabling bandwidth-length products of several hundred MHz

\cdot

km.^[23]

Effects in Optical Systems

Pulse Broadening Mechanisms

Modal dispersion in multimode optical fibers arises primarily from the differential group delays among propagating modes, where each mode travels at a slightly different group velocity due to variations in effective refractive index and path length. This results in the temporal spreading of an input optical pulse into a train of delayed mode arrivals at the fiber output, with the extent of broadening determined by the spread in these group delays. In step-index fibers, the maximum pulse broadening corresponds to the worst-case scenario where all modal power is distributed across the highest and lowest order modes, yielding a delay difference on the order of tens of picoseconds per kilometer. For graded-index fibers, the design reduces this differential, but residual spreading persists, often quantified using root-mean-square (RMS) broadening to account for the statistical distribution of modal power.^[5]^[24] The degree of pulse broadening scales linearly with fiber length L, as the cumulative delay differences accumulate proportionally over distance, making modal dispersion a key limiter for high-bit-rate transmission in longer links. Launch conditions significantly influence the excited modes and thus the effective broadening: a centered launch primarily excites lower-order modes with minimal differential delays, resulting in narrower output pulses, whereas an off-axis or restricted launch excites higher-order modes, increasing the spread and mimicking worst-case conditions for testing. These effects highlight the importance of controlled excitation in system design to optimize bandwidth.^[25] In simulations of pulse propagation, the output temporal profile is conceptually described as the convolution of the input pulse shape with the fiber's modal impulse response, which represents the superposition of delayed delta functions corresponding to each mode's arrival time. After propagating a sufficient distance—typically on the order of the coupling length—the modes reach an equilibrium mode distribution (EMD) through intermodal coupling and scattering, establishing a uniform statistical power allocation across modes independent of the initial launch. This EMD stabilizes the impulse response, transitioning the broadening behavior from launch-dependent to fiber-intrinsic characteristics.^[26] In short multimode fibers, such as those under 1 km, the differential mode delay dominates the pulse broadening since mode mixing remains minimal, preserving the initial mode-specific delays. For longer links, however, progressive mode mixing averages the delays, reducing the effective differential spread and altering the scaling of broadening from linear to potentially square-root dependence in highly coupled regimes.^[27]

Impact on Signal Quality

Modal dispersion significantly degrades signal quality in multimode optical fibers by causing pulse broadening, which limits the overall bandwidth-length product of the system to approximately BL ≈ 1 / (2π σ_modal), where σ_modal is the root-mean-square modal delay spread in seconds per kilometer.^[28] This relationship arises because the temporal spread of optical pulses due to differing mode velocities reduces the effective transmission capacity, often restricting data rates to below 1 Gbps over 1 km in fibers with high modal dispersion, such as early step-index designs.^[28] In practical terms, this degradation manifests as a fundamental constraint on high-speed data transmission, where excessive pulse spreading prevents reliable signal detection without advanced processing. Key metrics of this degradation include inter-symbol interference (ISI), resulting from the overlap of broadened pulses from adjacent symbols, and an elevated bit error rate (BER) due to mode-resolved timing jitter that confuses symbol boundaries at the receiver.^[29] ISI occurs as faster-propagating modes arrive ahead of slower ones, smearing the temporal profile of each bit and reducing eye opening in the received signal, while BER increases exponentially with dispersion-induced jitter, often exceeding acceptable thresholds (e.g., 10^{-12}) beyond certain link lengths.^[30] These effects collectively diminish signal-to-noise margins and necessitate error-correcting codes or equalization to maintain performance. In Gigabit Ethernet systems using multimode fiber, modal dispersion caps transmission distances at 550 m for OM3 and OM4 categories at 1 Gbps, far shorter than the several kilometers possible with single-mode fiber under similar conditions.^[31] For instance, the IEEE 802.3z standard leverages OM3/OM4 graded-index fibers to achieve this reach, but dispersion still imposes a strict limit compared to single-mode alternatives that extend to 5 km or more without significant modal effects.^[31] Historically, modal dispersion posed a major limitation in early fiber optic deployments during the 1970s and 1980s, confining multimode systems to short-haul, low-bit-rate applications until standards like ISO/IEC 11801 introduced graded-index specifications to mitigate delay differences and enable structured cabling for higher performance.^[32] This resolution through standardized graded-index profiles marked a pivotal advancement, transitioning multimode fiber from niche use to widespread local area network infrastructure.^[33]

Mitigation and Applications

Reduction Techniques

Modal dispersion in multimode optical fibers can be significantly reduced through engineering the fiber's refractive index profile to equalize the propagation speeds of different modes. Graded-index (GI) profiling, where the refractive index decreases gradually from the core center to the edge, compensates for the longer path lengths traveled by higher-order modes by slowing them down less than in step-index fibers. The optimal profile follows a power-law form with exponent α ≈ 2, known as the parabolic profile, which theoretically minimizes intermodal dispersion to near-zero by achieving nearly equal group velocities across modes.^[34]^[35] Another practical technique involves selective mode excitation at the fiber launch, where optics are designed to preferentially couple light into lower-order modes that exhibit less differential group delay. For instance, offset launch methods position the input beam away from the fiber core center to excite fewer high-order modes, thereby reducing the overall modal delay spread and improving effective bandwidth without altering the fiber structure itself.^[36] In advanced systems, few-mode fibers (FMFs) supporting only a limited number of modes (typically 2–10) are employed alongside multiple-input multiple-output (MIMO) digital signal processing (DSP) to mitigate residual modal dispersion in spatial division multiplexing (SDM) applications. MIMO DSP algorithms, such as those based on singular value decomposition, digitally compensate for mode coupling and differential group delays, enabling error-free transmission over tens of kilometers despite inherent modal effects.^[37]^[38] Photonic crystal fibers (PCFs), with their microstructured air-hole cladding, can further reduce the mode count by design, often achieving endlessly single-mode operation across broad wavelength ranges, which eliminates intermodal dispersion entirely.^[39]^[40] To quantify and verify the effectiveness of these reduction techniques, modal delay is measured using methods like time-resolved spectroscopy, which identifies individual mode arrival times by spatially resolving the output beam after propagation. Optical time-domain reflectometry (OTDR) adapted for multimode fibers can also assess modal bandwidth indirectly through backscattered signal analysis. These measurements adhere to international standards such as IEC 60793-1-41, which specifies protocols for determining effective modal bandwidth via differential mode delay (DMD) testing under controlled launch conditions.^[41]^[42] While graded-index profiling offers substantial performance gains, it introduces trade-offs in manufacturing, as achieving a precise parabolic index distribution requires advanced vapor deposition or plasma chemical vapor deposition processes, increasing production complexity and cost compared to step-index fibers. However, this enables high-speed applications, such as 10 Gbps transmission over 400 m in OM4 multimode links, far exceeding the capabilities of unoptimized designs.^[43]^[44]

Role in Fiber Optic Design

In fiber optic design, the selection between multimode and single-mode fibers is fundamentally driven by the dispersion budget, which quantifies the allowable signal broadening before unacceptable degradation occurs. Multimode fibers, prone to modal dispersion due to multiple propagation paths, are chosen for applications where the link length is short enough—typically under 500 meters—to keep dispersion within tolerable limits, offering cost-effective solutions with simpler, cheaper light sources like LEDs or VCSELs.^[45]^[46] In contrast, single-mode fibers eliminate modal dispersion entirely by supporting only one propagation mode, making them essential for links exceeding a few kilometers where even minor pulse broadening would limit bandwidth.^[47] This choice directly influences system architecture, balancing factors like transmission distance, data rate, and overall cost against the dispersion-induced intersymbol interference. The evolution of multimode fiber standards reflects ongoing efforts to extend the viable dispersion budget for high-speed short-reach applications. Early OM1 fibers, standardized in 1989 with a 62.5 μm core, supported only up to 1 Gbps over 300 meters but suffered high modal dispersion with LED sources. Subsequent iterations—OM2 (50 μm core, improved to 500 MHz·km bandwidth), OM3 (laser-optimized for 10 Gbps over 300 meters), OM4 (extended to 400 meters at 10 Gbps), and OM5 (wideband, supporting 400 Gbps+ over 100 meters using shortwave wavelength division multiplexing)—have progressively reduced effective modal dispersion through graded-index profiles and optimized launch conditions, enabling denser data center interconnects.^[44]^[48] These standards, developed under TIA-492 and IEC 60793, have shaped 400 Gbps Ethernet deployments in short-reach links by accommodating VCSEL arrays while minimizing fiber count.^[49] Modal dispersion plays a dominant role in data center and local area network (LAN) designs, where multimode fibers prevail for intra-building and rack-to-rack connections under 500 meters, leveraging their ease of alignment and lower cost despite the dispersion penalty.^[50] For long-haul telecommunications spanning tens to hundreds of kilometers, single-mode fibers are universally adopted to avoid modal dispersion, relying instead on chromatic dispersion management techniques.^[51] Emerging trends amplify these considerations, particularly with VCSEL-based transceivers, which operate at 850-940 nm and pair with multimode fibers for high-parallelism in data centers but face modal dispersion limits at speeds beyond 100 Gbps per lane, prompting optimizations like encircled flux launch conditions to enhance effective bandwidth.^[52] Hybrid systems are gaining traction, combining cost-effective multimode segments for access layers with dispersion-compensating modules or single-mode trunks for extended reach, as seen in radio-over-fiber architectures where multimode cores reduce deployment expenses while electronic equalization mitigates residual modal effects.^[53] A key limitation in 2020s parallel optics deployments arises from modal dispersion, which caps multimode scalability in massive parallelism schemes like 800 Gbps Ethernet, where hundreds of fibers per link become impractical due to dispersion-induced bandwidth reduction. This constraint is driving exploration of few-mode fibers as alternatives, which support a controlled number of modes (e.g., 2-6) to increase spatial multiplexing capacity while applying mode-division techniques to manage intermodal crosstalk and dispersion, potentially scaling to terabit-per-second systems with fewer physical fibers.^[54]

Info Pages

Talk Pages

Special Pages

Modal dispersion

Recent from talks

Recent from talks

Contribute something

Contribute something

Media Pages

Timelines

Articles

Notes collections

Notes

Notes

Days in Chronicle

Modal dispersion

Troubleshooting

References

External links