Chromatic and Modal Dispersion in Fibers

A fiber carries information as a train of light pulses, and the whole game is keeping those pulses distinct after hundreds of kilometres of glass. Each pulse spreads in time, and once neighbours overlap the receiver can no longer tell a one from a zero — intersymbol interference sets a hard ceiling on the data rate. This broadening is dispersion, and because it piles up with distance the true figure of merit is not bit rate alone but the bit-rate–distance product, quoted in $\mathrm{Gb\,s^{-1}\cdot km}$ . Two distinct mechanisms do the spreading, and it takes results from earlier in this chapter to separate them. From the guided-mode picture (aq-optics-waveguide-modes-wave-equation) we inherit that a fiber supports a discrete set of modes, each with its own propagation constant $\beta$ . Modal dispersion is the spread in arrival time between those modes; it lives only in multimode fibers and vanishes the instant single-mode operation (aq-optics-single-multimode-operation) leaves just one. Chromatic dispersion survives even then: any real source emits a band of wavelengths, a single mode's group velocity depends on wavelength, and so the colours of one pulse drift apart. It splits again — a material part from the glass index $n(\lambda)$ and a waveguide part from how tightly the mode is confined. Everything is organized by one function: the mode propagation constant $\beta(\omega)$ . Its first derivative is the group delay per length; its second derivative $\beta_2$ — the group-velocity dispersion — is the pulse-broadening engine, repackaged as a coefficient $D$ in $\mathrm{ps\,nm^{-1}\,km^{-1}}$ that you multiply by linewidth and length. In standard silica the material and waveguide parts cancel near $1.31\,\mu\mathrm{m}$ , while loss bottoms out near $1.55\,\mu\mathrm{m}$ — a mismatch that has shaped fiber design for forty years.

Key takeaways

Dispersion broadens optical pulses as they travel, forcing neighbouring bits to overlap and capping the bit-rate–distance product $B\cdot L$ .
Modal dispersion comes from guided modes having different group velocities; it exists only in multimode fiber and is killed by single-mode operation ( $V<2.405$ ).
Chromatic dispersion comes from one mode's group velocity depending on wavelength together with a source of finite linewidth; it splits into material and waveguide parts.
The coefficient $D=-\tfrac{2\pi c}{\lambda^2}\beta_2$ crosses zero near $1.31\,\mu\mathrm{m}$ , but the loss minimum sits at $1.55\,\mu\mathrm{m}$ , where $D\approx+17\,\mathrm{ps\,nm^{-1}km^{-1}}$ ( $\beta_2\approx-21.7\,\mathrm{ps^2\,km^{-1}}$ ).
Chromatic broadening is $\sigma_\tau=|D|\,L\,\sigma_\lambda$ ; step-index modal spread is $\approx n_1\Delta L/c$ (tens of $\mathrm{ns\,km^{-1}}$ ), which graded-index profiles cut by $\sim\Delta/2$ .
Key idea: pulse broadening is governed by how the group delay $\beta_1=\mathrm{d}\beta/\mathrm{d}\omega$ varies — across modes (modal dispersion) or across the source spectrum (chromatic dispersion, set by $\beta_2$ ). Single-mode operation removes the first; wavelength and profile design minimize the second.

The core insight

Pulse broadening is governed by how the group delay $\beta_1=\mathrm{d}\beta/\mathrm{d}\omega$ varies: across modes (modal dispersion) or across the source spectrum (chromatic dispersion, set by $\beta_2$ ). Single-mode operation kills the former; the choice of wavelength and index profile minimizes the latter.

Where this lives today

The zero-dispersion window at $1.31\,\mu\mathrm{m}$ and the loss minimum at $1.55\,\mu\mathrm{m}$ do not coincide, and that mismatch is the axis around which optical telecommunications is engineered. Modern long-haul systems run at $1.55\,\mu\mathrm{m}$ to exploit the $\sim0.2\,\mathrm{dB\,km^{-1}}$ loss and erbium-doped fiber amplifiers, then fight the residual $D\approx17\,\mathrm{ps\,nm^{-1}km^{-1}}$ with dispersion-compensating fiber, chirped fiber Bragg gratings, or — in today's coherent transponders — digital signal processing that inverts the accumulated $\beta_2 L$ phase after detection. Because that phase is deterministic, a DSP chip can undo hundreds of kilometres of chromatic dispersion, which is why single-channel rates have climbed past $400\,\mathrm{Gb\,s^{-1}}$ on fiber whose $D$ has not changed since the 1980s. Elsewhere the same physics is a feature: in the anomalous regime above $\lambda_0$ , $\beta_2<0$ lets dispersion cancel the Kerr self-phase-modulation chirp, forming solitons and — in photonic-crystal fiber and microresonators — the supercontinua and frequency combs that now anchor optical atomic clocks, while data centres run graded-index multimode fiber with cheap VCSELs for tens of gigabits over a few hundred metres.

The physical picture

Start with the ray picture of a fat multimode step-index core. Different rays take different paths: one runs straight down the axis, another zig-zags at the steepest angle total internal reflection allows. Both travel at the same speed $c/n_1$ , but the zig-zag ray covers a longer path, so it arrives late. Because the extra path grows with length, the spread grows linearly with $L$ . That is modal dispersion, set purely by geometry: the fractional index step $\Delta$ between core and cladding.

Now silence every mode but one. The spreading does not stop, because a pulse is a wave packet built from a band of frequencies, and the glass index — plus the way the field spills between core and cladding — changes with wavelength. Each colour has its own group velocity, and the red and blue edges pull apart. This is chromatic dispersion, unavoidable in a real single-mode fiber, and its two parts fight in silica: material dispersion swings from negative to positive through the near-infrared while waveguide dispersion stays negative, so their sum crosses zero near $1.31\,\mu\mathrm{m}$ — moving that crossing is what dispersion-shifted fibers do.

Figure

Chromatic dispersion of standard single-mode silica fiber: material D_M and waveguide D_W add to a total D that passes through zero near λ₀ ≈ 1.31 μm and reaches D ≈ +17 ps nm⁻¹ km⁻¹ at the 1.55 μm loss minimum

Think of $\beta_2$ as a chirp generator: it makes the instantaneous frequency sweep across the stretched pulse — blue crowded to one edge, red to the other. That chirp is deterministic and reversible, which is why a short length of opposite-sign fiber can squeeze the pulse back. Chromatic dispersion is a distortion you can undo — the fact that underwrites dispersion compensation, and the bridge to solitons, where $\beta_2$ is balanced against the Kerr nonlinearity instead of cancelled.

The core equations

The starting object is the mode's propagation constant $\beta(\omega)$ , the axial phase shift per unit length. Because a pulse occupies a narrow band around a carrier $\omega_0$ , we Taylor-expand $\beta$ there; the coefficients are the dispersion parameters that follow.

\beta(\omega)=\beta_0+\beta_1(\omega-\omega_0)+\tfrac{1}{2}\beta_2(\omega-\omega_0)^2+\cdots \qquad (1)

where: $\beta(\omega)$ — axial propagation constant of the guided mode; $\omega_0$ — carrier angular frequency; $\beta_1=\mathrm{d}\beta/\mathrm{d}\omega$ — inverse group velocity $1/v_g$ ; $\beta_2=\mathrm{d}^2\beta/\mathrm{d}\omega^2$ — group-velocity dispersion (GVD). The linear term $\beta_1$ sets when the pulse arrives; the quadratic $\beta_2$ sets how much it spreads. The first-order coefficient is the reciprocal group velocity. It packages the familiar group index — the same $n_g$ used elsewhere for pulse travel time.

v_g=\frac{1}{\beta_1}=\frac{c}{n_g},\qquad n_g=n-\lambda\frac{\mathrm{d}n}{\mathrm{d}\lambda} \qquad (2)

where: $v_g$ — group velocity (speed of the pulse envelope); $n_g$ — group index; $n$ — phase (material) index; $\lambda$ — vacuum wavelength. A pulse's delay per length is $\beta_1=n_g/c$ , and dispersion is just the variation of this delay with wavelength. Engineers rarely quote $\beta_2$ directly; they quote the dispersion coefficient $D$ , the change in group delay per length per unit wavelength. It follows from $\beta_2$ by the chain rule and, for the material part alone, reduces to the curvature of the index.

D=-\frac{2\pi c}{\lambda^2}\,\beta_2=-\frac{\lambda}{c}\frac{\mathrm{d}^2 n}{\mathrm{d}\lambda^2} \qquad (3)

where: $D$ — chromatic dispersion parameter, $\mathrm{ps\,nm^{-1}\,km^{-1}}$ ; $\beta_2$ — GVD parameter, $\mathrm{ps^2\,km^{-1}}$ ; $c$ — vacuum light speed. The two forms are the same physics with opposite signs (positive $D$ , anomalous, means $\beta_2<0$ ); the rightmost shows material dispersion is literally the curvature of the Sellmeier index. With $D$ in hand, pulse broadening is a one-line estimate: multiply the coefficient by length and source linewidth. It is the number a link designer reaches for first.

\sigma_\tau=|D|\,L\,\sigma_\lambda \qquad (4)

where: $\sigma_\tau$ — RMS temporal broadening; $L$ — fiber length; $\sigma_\lambda$ — RMS spectral width of the source. Linear in length and linewidth, so a narrow-line laser and a short span each help in direct proportion. The total chromatic $D$ is the sum of two pieces: a material term fixed by the glass and a waveguide term fixed by the geometry. The waveguide part enters through the normalized parameters $V$ and $b$ .

D=D_M+D_W,\qquad D_W=-\frac{n_2\Delta}{c\lambda}\,V\frac{\mathrm{d}^2(Vb)}{\mathrm{d}V^2} \qquad (5)

where: $D_M$ — material dispersion from $\mathrm{d}^2 n/\mathrm{d}\lambda^2$ ; $D_W$ — waveguide dispersion; $b$ — normalized propagation constant; $\Delta$ — fractional index difference. Since $V\,\mathrm{d}^2(Vb)/\mathrm{d}V^2>0$ for the fundamental mode, $D_W$ is always negative and grows with confinement — the knob that shifts $\lambda_0$ . Modal dispersion has no $\lambda$ in it — it is pure geometry. Comparing the axial ray with the critical-angle ray gives the delay spread directly.

\Delta\tau_{\text{step}}=\frac{L\,n_1}{c}\,\Delta=\frac{L}{c}\,\frac{n_1(n_1-n_2)}{n_2} \qquad (6)

where: $n_1$ — core index; $n_2$ — cladding index; $\Delta=(n_1-n_2)/n_1$ — fractional index difference; $L$ — length. Linear in $L$ and $\Delta$ , typically tens of $\mathrm{ns\,km^{-1}}$ — orders of magnitude above chromatic spread, so multimode links are short. Whatever the mechanism, broadening becomes a ceiling on the bit-rate–distance product. Bits must be spaced by a few times $\sigma_\tau$ , which fixes the maximum $B\cdot L$ .

B\,L\lesssim\frac{1}{4\,|D|\,\sigma_\lambda} \qquad (7)

where: $B$ — bit rate; $L$ — distance; $D$ — dispersion coefficient; $\sigma_\lambda$ — source linewidth. The product $B\cdot L$ , not $B$ alone, is conserved — doubling the reach halves the rate. Finally, modal dispersion is switched off entirely by forcing single-mode operation. The cutoff condition, inherited from the waveguide-modes analysis, caps the $V$ -number.

V=\frac{2\pi a}{\lambda}\sqrt{n_1^2-n_2^2}<2.405 \qquad (8)

where: $V$ — normalized frequency (V-number); $a$ — core radius; $2.405$ — first zero of $J_0$ , the cutoff of the $\mathrm{LP}_{11}$ mode. Below it only the fundamental $\mathrm{LP}_{01}$ propagates, so modal dispersion is exactly zero and only chromatic remains.

Derivation AFrom the propagation constant to the dispersion coefficient

The aim is to connect the three ways people talk about chromatic dispersion — the GVD parameter $\beta_2$ , the engineer's coefficient $D$ , and the material curvature $\mathrm{d}^2 n/\mathrm{d}\lambda^2$ — and show they are one quantity in three costumes. We start from the group delay per length, differentiate with respect to wavelength (sources are specified by linewidth in nm), and carry the plane-wave index through to the curvature term.

Step 1. The group delay per length is $\tau=\beta_1$ , so the coefficient is $D=\mathrm{d}\beta_1/\mathrm{d}\lambda=(\mathrm{d}\omega/\mathrm{d}\lambda)\,\beta_2$ , using $\mathrm{d}\beta_1/\mathrm{d}\omega=\beta_2$ . Why: dispersion is the wavelength-dependence of arrival time, and the chain rule routes it through $\beta_2$ , the quantity the wave equation hands us.

Step 2. From $\omega=2\pi c/\lambda$ , $\mathrm{d}\omega/\mathrm{d}\lambda=-2\pi c/\lambda^2$ , so $D=-(2\pi c/\lambda^2)\,\beta_2$ . Why: this is exactly why $D$ and $\beta_2$ carry opposite signs — the minus comes from $\omega$ and $\lambda$ running opposite.

Step 3. For the material part, use $\beta=n\omega/c$ for a plane wave in the glass, giving $\beta_1=(1/c)(n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda)=n_g/c$ . Why: the group index re-appears exactly as in equation (2), tying the fiber delay to the material group index.

Step 4. Differentiate $\beta_1$ with respect to $\lambda$ : the $n$ and single-derivative terms cancel by the product rule, leaving $D_M=-(\lambda/c)\,\mathrm{d}^2 n/\mathrm{d}\lambda^2$ . Why: only the curvature survives, so material dispersion vanishes where the index has an inflection — near $1.276\,\mu\mathrm{m}$ in silica. *As a check: at $\lambda=1.55\,\mu\mathrm{m}$ , $D=17\,\mathrm{ps\,nm^{-1}km^{-1}}$ gives $\beta_2=-D\lambda^2/(2\pi c)=-21.7\,\mathrm{ps^2\,km^{-1}}$ , the standard SMF value.*

Go deeper · Derivation B · 1 minModal delay spread from the ray picture

Go deeper · Derivation C · 1 minGaussian pulse broadening set by β₂ L

Derivation CGaussian pulse broadening set by β₂ L

Here we prove the headline result that group-velocity dispersion stretches a pulse by an amount controlled by the single combination $\beta_2 L$ , and that the process is lossless and reversible. In the slowly varying envelope approximation, in a frame moving at the group velocity, the envelope obeys a Schrödinger-like equation in which $\beta_2$ plays the role of an inverse mass. Propagating an unchirped Gaussian lets us read the output width off directly and identify the dispersion length.

Step 1. Keeping terms through $\beta_2$ and moving to retarded time $T=t-\beta_1 z$ , the envelope satisfies $\partial A/\partial z=-\tfrac{i}{2}\beta_2\,\partial^2 A/\partial T^2$ . Why: subtracting the group delay $\beta_1 z$ leaves $\beta_2$ as the sole agent of distortion — a diffusion equation with an imaginary coefficient.

Step 2. In frequency this integrates trivially: $\tilde A(z,\omega)=\tilde A(0,\omega)\exp(\tfrac{i}{2}\beta_2\omega^2 z)$ , a pure quadratic phase. Why: the amplitudes are untouched, so dispersion only re-times the spectral components — which is why it is perfectly reversible.

Step 3. Feed in an unchirped Gaussian $A(0,T)=\exp(-T^2/2T_0^2)$ ; the integral back to time gives a complex width $T_0^2-i\beta_2 z$ . Why: the imaginary part is the induced chirp, and its growth with $z$ stretches the pulse.

Step 4. The real half-width is $T_1(z)=T_0\sqrt{1+(\beta_2 z/T_0^2)^2}$ , negligible until $z\sim L_D=T_0^2/|\beta_2|$ and linear beyond. Why: since $L_D\propto T_0^2$ , short pulses disperse fastest, so high bit rates suffer most. *As a check: for $z\gg L_D$ , $T_1\to|\beta_2|z/T_0$ , reproducing $\sigma_\tau=|D|L\sigma_\lambda$ once $\sigma_\lambda$ and $T_0$ are linked at the transform limit.*

Limiting cases

Which mechanism dominates depends on the fiber type, the source, and the wavelength:

Case	Result	Meaning
Multimode step-index	$\Delta\tau_{\text{modal}}\approx\dfrac{n_1\Delta}{c}\,L$	Modal dispersion rules; linear in $L$ and $\Delta$ , tens of $\mathrm{ns\,km^{-1}}$ , limiting reach to short links.
Multimode graded-index	$\Delta\tau_{\text{modal}}\approx\dfrac{n_1\Delta^2}{2c}\,L$	An optimized profile equalizes mode delays, shrinking the spread by $\sim\Delta/2$ versus step-index.
Single-mode fiber	$\sigma_\tau=\|D\|\,L\,\sigma_\lambda$	No modal dispersion; chromatic $D$ and linewidth set the limit, residual is PMD $\propto\sqrt{L}$ .
Zero-dispersion wavelength	$D(\lambda_0)=0\Rightarrow\sigma_\tau\propto\|S\|\,L\,\sigma_\lambda^2$	At $\lambda_0\approx1.31\,\mu\mathrm{m}$ first-order $D$ vanishes; the slope $S=\mathrm{d}D/\mathrm{d}\lambda$ sets the residual.

Worked examples

1 · Modal budget of a step-index multimode link (confidence builder)

A step-index multimode fiber has core index $n_1=1.48$ and cladding $n_2=1.465$ . Find the modal delay spread per kilometre and the modal-limited bit-rate–distance product.

Step 1. Fractional index difference: $\Delta=(n_1-n_2)/n_1=(1.48-1.465)/1.48=\mathbf{0.01014}$ — the single geometric number that sets modal dispersion.

Step 2. Delay spread per length from equation (6): $\Delta\tau/L=n_1\Delta/c=(1.48)(0.01014)/(2.998\times10^8)=\mathbf{50\ ns\,km^{-1}}$ . Over $1\,\mathrm{km}$ the fastest and slowest rays arrive $50\,\mathrm{ns}$ apart — enormous next to chromatic spreads in ps.

Step 3. With bits spaced by a few $\Delta\tau$ , taking $B\lesssim1/(4\,\Delta\tau)$ gives $B\cdot L\lesssim1/(4\times50\times10^{-9}\ \mathrm{s\,km^{-1}})=5\times10^{6}\ \mathrm{(bit/s)\cdot km}$ . So $\Delta=0.0101$ , $\Delta\tau/L\approx50\,\mathrm{ns\,km^{-1}}$ , and $B\cdot L\lesssim5\ \mathrm{(Mb/s)\cdot km}$ — a 1 km link caps near 5 Mb/s, which is why step-index multimode is confined to short, low-rate runs.

2 · Chromatic broadening on a single-mode long-haul span

Standard single-mode fiber at $1.55\,\mu\mathrm{m}$ has $D=17\,\mathrm{ps\,nm^{-1}km^{-1}}$ . A DFB laser has $\sigma_\lambda=0.1\,\mathrm{nm}$ . Find $\beta_2$ , the broadening over $L=100\,\mathrm{km}$ , and the bit-rate ceiling.

Step 1. Convert to GVD: $\beta_2=-D\lambda^2/(2\pi c)=-(17\times10^{-6}\ \mathrm{s\,m^{-2}})(1.55\times10^{-6}\ \mathrm{m})^2/(2\pi\times2.998\times10^8)=\mathbf{-21.7\ ps^2\,km^{-1}}$ ; the negative sign marks the anomalous regime above $\lambda_0$ .

Step 2. Broadening from equation (4): $\sigma_\tau=|D|\,L\,\sigma_\lambda=(17)(100)(0.1)=\mathbf{170\ ps}$ — every kilometre adds $1.7\,\mathrm{ps}$ , accumulating linearly.

Step 3. Bit-rate ceiling: $B\lesssim1/(4\sigma_\tau)=\mathbf{1.47\ Gb\,s^{-1}}$ , equivalently $B\cdot L\lesssim1/(4|D|\sigma_\lambda)\approx147\ \mathrm{(Gb/s)\cdot km}$ . So $\beta_2=-21.7\,\mathrm{ps^2\,km^{-1}}$ , $\sigma_\tau=170\,\mathrm{ps}$ , and $B\lesssim1.5\,\mathrm{Gb\,s^{-1}}$ over 100 km without compensation — thousands of times better than the multimode link, yet far below the rates modern systems reach with compensation.

3 · How far before a pulse doubles?

A $50\,\mathrm{ps}$ (half-width $T_0$ ) transform-limited pulse enters the same fiber, $\beta_2=-21.7\,\mathrm{ps^2\,km^{-1}}$ . Over what distance does it broaden appreciably?

Step 1. The natural scale is the dispersion length $L_D=T_0^2/|\beta_2|=(50)^2/21.7=\mathbf{115\ km}$ ; below it the pulse is essentially unbroadened.

Step 2. At $z=L_D$ the width grows to $T_1=\sqrt{2}\,T_0=\mathbf{71\ ps}$ , a 41% stretch. A 50 ps pulse survives $\sim100\,\mathrm{km}$ before broadening by half; since $L_D\propto T_0^2$ , a 10 ps pulse would broaden as much in only $\sim4.6\,\mathrm{km}$ — the quadratic penalty on high bit rates.

Physics of the portrait

The theory that gives the $b(V)$ curves behind the coefficient $D_W$ came out of Bell Labs at the moment low-loss fiber first became real — the work of Detlef Gloge.

Origins · 1 minDetlef Gloge and how this was discovered

Detlef Gloge

Bell Labs, Crawford Hill · weakly guiding fibers, 1971 · fiber waveguide theory and dispersion

When Corning drew the first low-loss fiber in 1970, the theory to design real systems was still missing: the exact vector modes of a fiber are a thicket of Bessel functions with no transparent picture of dispersion. Detlef Gloge, at Bell Laboratories' Crawford Hill site in Holmdel, cut through it in 1971 with the weakly guiding approximation. Recognizing that a communications fiber has $n_1\approx n_2$ (a fractional step $\Delta$ of order one percent), he showed the modes collapse into nearly-degenerate linearly polarized $\mathrm{LP}$ groups, each described by a single universal normalized propagation constant $b(V)$ . His companion paper that year, "Dispersion in Weakly Guiding Fibers," turned those curves into the working formula for waveguide dispersion, $D_W\propto V\,\mathrm{d}^2(Vb)/\mathrm{d}V^2$ — the very term that lets a designer push the zero-dispersion wavelength around. Two years later, with Enrique Marcatili, he built the multimode theory of graded-core fibers, showing that a near-parabolic profile equalizes the mode delays and slashes modal dispersion by the factor $\Delta/2$ that makes graded-index fiber usable. His normalized-parameter language — $V$ , $b$ , $\mathrm{LP}$ modes — is still the vocabulary every fiber-optics text speaks.

History

1966

— Kao and Hockham argue that purified glass fiber can carry telecommunications, setting the loss target

1971

— Gloge's weakly-guiding analysis yields the LP modes and the b(V) curves behind waveguide dispersion

1973

— Gloge and Marcatili show a near-parabolic graded index equalizes mode delays, cutting modal dispersion

1980

— Mollenauer, Stolen and Gordon observe optical solitons, group-velocity dispersion balancing the Kerr nonlinearity

1982

— dispersion-shifted fiber moves the zero-dispersion wavelength onto the 1.55 μm loss minimum

Going further

Texts. Saleh & Teich, Fundamentals of Photonics, chs. 9–10 · G. P. Agrawal, Fiber-Optic Communication Systems and Nonlinear Fiber Optics (the standard source for $D$ , $\beta_2$ , pulse-broadening) · Hecht, Optics, fiber and dispersion chapters · Loudon, The Quantum Theory of Light (group velocity in media) · Demtröder, Laser Spectroscopy (source spectral width). Key relations and anchor numbers: $D=-(2\pi c/\lambda^2)\beta_2=-(\lambda/c)\,\mathrm{d}^2 n/\mathrm{d}\lambda^2$ ; at $1.55\,\mu\mathrm{m}$ , $D=17\,\mathrm{ps\,nm^{-1}km^{-1}}\Rightarrow\beta_2=-21.7\,\mathrm{ps^2\,km^{-1}}$ ; step-index modal spread $\Delta\tau\approx Ln_1\Delta/c=50\,\mathrm{ns\,km^{-1}}$ for $n_1=1.48$ , $\Delta=0.0101$ ; Gaussian broadening $T_1=T_0\sqrt{1+(\beta_2 L/T_0^2)^2}$ with $L_D=115\,\mathrm{km}$ for $T_0=50\,\mathrm{ps}$ . Standard values: $\lambda_0\approx1.31\,\mu\mathrm{m}$ , loss $\approx0.2\,\mathrm{dB\,km^{-1}}$ , $D\approx+17\,\mathrm{ps\,nm^{-1}km^{-1}}$ at $1.55\,\mu\mathrm{m}$ , cutoff $V=2.405$ .

Builds on: Total Internal Reflection and Optical Guiding · Waveguide Modes and the Wave Equation · Single-Mode and Multimode Operation · Step-Index and Graded-Index Optical Fibers. Leads to: Dispersion Management and Compensation · Polarization-Mode Dispersion and Birefringent Fibers · Nonlinear Optical Fibers and Effects · Fiber Attenuation and Loss Mechanisms.

Ask the tutor: Want to walk through how a coherent receiver's DSP inverts the $\beta_2 L$ phase to undo 100 km of chromatic dispersion in software?

Why is modal dispersion so much bigger than chromatic dispersion?

Because they scale on completely different quantities. Modal dispersion is set by the fractional index step $\Delta\sim0.01$ , and $\Delta\tau/L=n_1\Delta/c$ reaches tens of $\mathrm{ns\,km^{-1}}$ . Chromatic dispersion is the product of a small coefficient $D$ and a small linewidth $\sigma_\lambda$ , landing in the $\mathrm{ps\,km^{-1}}$ range — a thousand times smaller. That is why the industry moved to single-mode fiber for anything long: eliminating modal dispersion alone buys three orders of magnitude of bit-rate–distance product.

If chromatic dispersion just stretches a pulse, why can't I always undo it?

For the linear, first-order part you very nearly can — that is the beauty of $\beta_2$ . It imposes a deterministic quadratic phase, so a length of opposite-sign fiber, a chirped grating, or a DSP filter inverts it exactly, which is how coherent systems clean up hundreds of kilometres of accumulated dispersion. What resists undoing is everything beyond first order: the dispersion slope $S$ that varies across a WDM band, polarization-mode dispersion (random and time-varying), and nonlinear mixing with the Kerr effect — the residuals that ultimately cap a real system.

What actually happens at the zero-dispersion wavelength — is the pulse perfectly preserved?

No. Setting $D(\lambda_0)=0$ cancels only the first-order term. Two things remain. First, the dispersion slope $S=\mathrm{d}D/\mathrm{d}\lambda$ is nonzero, so different parts of the spectrum see slightly different $D$ , giving residual broadening $\propto|S|\,L\,\sigma_\lambda^2$ . Second, polarization-mode dispersion, indifferent to $\lambda_0$ , still spreads the pulse as $\sqrt{L}$ . And with dispersion near zero, nonlinear four-wave mixing between WDM channels phase-matches strongly, so dispersion-shifted fiber can be worse for dense multiplexing.

How does a graded-index profile equalize the mode delays?

In a step-index core all rays travel at $c/n_1$ , so the long-path high-order modes arrive late. A graded index turns $n$ down away from the axis, so the wide-swinging rays spend their detours in lower-index glass where the local speed $c/n(r)$ is higher. With a near-parabolic profile the speed-up almost exactly compensates the extra path, and all modes arrive together. Gloge and Marcatili showed the optimum shrinks the residual spread from $n_1\Delta/c$ to about $n_1\Delta^2/2c$ , a factor of $\Delta/2$ — which is why graded-index multimode fiber carries data-centre traffic.

Why quote a bit-rate–distance product instead of just a bit rate?

Because dispersion broadening grows with length, the maximum bit rate always trades off against how far you push it. From $\sigma_\tau=|D|L\sigma_\lambda$ and $B\lesssim1/(4\sigma_\tau)$ , the length cancels into a constant: $B\cdot L\lesssim1/(4|D|\sigma_\lambda)$ . Doubling the distance forces you to halve the rate, so the product $B\cdot L$ is the invariant that characterizes a fiber-plus-source combination — which is why an engineer's first question about a fiber is not "how fast?" but "how fast times how far?".

Does single-mode fiber literally carry only one mode?

One spatial mode, yes — the fundamental $\mathrm{LP}_{01}$ — provided $V<2.405$ , which the core radius and index step are chosen to guarantee. But "one mode" hides a subtlety: that mode has two orthogonal polarization states, and in a real fiber tiny asymmetries (bends, stress, non-circular cores) make them travel at slightly different speeds. The result is polarization-mode dispersion, a residual spreading that scales as $\sqrt{L}$ rather than $L$ and cannot be designed away like modal dispersion — the whisper of mode-related spreading that survives single-mode operation.