Optical Fibers, Modes and Propagation Impairments

Introduction

An optical fiber is simultaneously a dielectric waveguide, a transmission medium and a nonlinear electromagnetic system. Its refractive-index profile determines which modes are guided, while material composition, structural imperfections and operating conditions determine attenuation, dispersion, polarization evolution, bending loss and nonlinear distortion. These effects ultimately constrain transmission distance, data rate, launch power and system architecture.

Ray optics provides useful intuition for numerical aperture and multimode propagation, but postgraduate analysis requires modal solutions of Maxwell’s equations. This chapter therefore connects geometric optics, scalar modal theory, graded-index design and practical link impairments. Unless otherwise stated, fibers are assumed to be circular, isotropic, nonmagnetic and weakly guiding.

Learning Objectives

After studying this chapter, the reader should be able to:

• analyze step-index and graded-index fiber profiles;
• calculate numerical aperture, acceptance angle and normalized frequency;
• formulate and interpret the scalar modal eigenvalue equation;
• distinguish spatial-mode counts from polarization-resolved mode counts;
• establish single-mode conditions and cutoff wavelength;
• quantify attenuation, chromatic dispersion and polarization-mode dispersion;
• explain macro- and microbending loss;
• evaluate Kerr and stimulated nonlinear effects; and
• integrate fiber impairments into practical link-design decisions.

1. Fiber Structure and Refractive-Index Profiles

A conventional fiber contains a core of refractive index \(n_1\), a lower-index cladding of index \(n_2\), and one or more protective polymer coatings. Guidance requires \(n_1>n_2\).

The relative index difference is

\[

\Delta=\frac{n_1^2-n_2^2}{2n_1^2}\approx\frac{n_1-n_2}{n_1},

\]

where the approximation is valid when \(\Delta\ll1\). Typical silica communication fibers are weakly guiding, with \(\Delta\) of order \(10^{-3}\) to \(10^{-2}\).

1.1 Step-index fiber

For core radius \(a\),

\[

n(r)=\begin{cases}

n_1,&r<a,\\

n_2,&r\ge a.

\end{cases}

\]

The abrupt boundary provides simple modal analysis but produces substantial intermodal dispersion when many modes are present.

1.2 Graded-index fiber

A useful power-law model is

\[

n^2(r)=n_1^2\left[1-2\Delta\left(\frac{r}{a}\right)^g\right],\qquad 0\le r<a,

\]

with \(n(r)=n_2\) outside the core and approximately \(n_2^2=n_1^2(1-2\Delta)\). The profile exponent \(g=2\) describes an ideal parabolic profile, while \(g\rightarrow\infty\) approaches a step-index profile.

In a parabolic profile, high-order rays travel longer geometric paths but spend more time in lower-index regions, where their velocity is higher. These effects largely cancel, reducing intermodal delay.

2. Numerical Aperture and Normalized Frequency

2.1 Numerical aperture

For a meridional ray entering from a medium of index \(n_0\), Snell’s law and the total-internal-reflection condition give

\[

n_0\sin\theta_a=\sqrt{n_1^2-n_2^2}\equiv \mathrm{NA},

\]

where \(\theta_a\) is the maximum acceptance half-angle. For launch from air, \(n_0\approx1\), so \(\theta_a=\sin^{-1}(\mathrm{NA})\).

Using weak guidance,

\[

\mathrm{NA}\approx n_1\sqrt{2\Delta}.

\]

This geometrical expression is most directly applicable to meridional rays. Coupling from a real laser also depends on spot size, wavefront curvature, alignment and overlap with individual modes.

2.2 Normalized frequency

The dimensionless normalized frequency, or V-number, is

\[

V=k_0a\sqrt{n_1^2-n_2^2}=\frac{2\pi a}{\lambda}\mathrm{NA},

\]

where \(k_0=2\pi/\lambda\) and \(\lambda\) is the vacuum wavelength. The V-number governs the number of guided modes and transverse confinement.

Worked Example 1: NA, V-number and cutoff

A step-index fiber has \(n_1=1.450\), \(n_2=1.444\), \(a=4.1\,\mu\text{m}\), and operates at \(1.55\,\mu\text{m}\).

\[

\mathrm{NA}=\sqrt{1.450^2-1.444^2}

=\sqrt{2.102500-2.085136}=0.1318.

\]

The air acceptance angle is

\[

\theta_a=\sin^{-1}(0.1318)\approx7.57^\circ.

\]

The normalized frequency is

\[

V=\frac{2\pi(4.1)}{1.55}(0.1318)\approx2.19.

\]

Because \(V<2.405\), the fiber is single-mode at this wavelength. Its approximate LP-mode cutoff wavelength is

\[

\lambda_c=\frac{2\pi a\mathrm{NA}}{2.405}

\approx1.41\,\mu\text{m}.

\]

Thus, the fiber is nominally single-mode for wavelengths longer than approximately \(1.41\,\mu\text{m}\), subject to manufacturing tolerances and bend-induced cutoff changes.

3. Modal Theory of Circular Fibers

3.1 Propagation constant and guided-wave condition

A monochromatic modal field varies as

\[

\mathbf{E}(r,\phi,z,t)=\mathbf{e}(r,\phi)e^{j(\omega t-\beta z)},

\]

where \(\beta\) is the longitudinal propagation constant. A bound mode of a step-index fiber satisfies

\[

n_2k_0<\beta<n_1k_0.

\]

The effective modal index is

\[

n_{\mathrm{eff}}=\frac{\beta}{k_0},

\]

so that \(n_2<n_{\mathrm{eff}}<n_1\).

Exact vector solutions are classified as TE, TM, HE and EH modes. In weakly guiding fibers, nearly degenerate vector modes can be grouped into linearly polarized, or LP, mode families.

3.2 Scalar eigenvalue equation

Under the weak-guidance scalar approximation, a field component \(\psi\) obeys

\[

\nabla_t^2\psi+\left[n^2(r)k_0^2-\beta^2\right]\psi=0.

\]

For azimuthal order \(l\), the bound step-index solutions are

\[

\psi(r,\phi)=

\begin{cases}

A J_l(ur/a)e^{jl\phi},&r<a,\\

C K_l(wr/a)e^{jl\phi},&r>a,

\end{cases}

\]

where \(J_l\) is a Bessel function and \(K_l\) is a modified Bessel function. The normalized transverse parameters are

\[

u=a\sqrt{n_1^2k_0^2-\beta^2},\qquad

w=a\sqrt{\beta^2-n_2^2k_0^2},

\]

and therefore

\[

u^2+w^2=V^2.

\]

Continuity of the scalar field and its radial derivative at \(r=a\) gives the characteristic equation

\[

\boxed{\frac{uJ_l'(u)}{J_l(u)}=\frac{wK_l'(w)}{K_l(w)}}.

\]

Primes denote differentiation with respect to the complete function argument. For \(l=0\), this can equivalently be written

\[

\frac{uJ_1(u)}{J_0(u)}=\frac{wK_1(w)}{K_0(w)}.

\]

The allowed roots determine \(\beta\). The scalar equation is appropriate for weak guidance; exact vector boundary conditions must be used when index contrast is large or polarization splitting is important.

A normalized propagation parameter is often introduced:

\[

b=\frac{n_{\mathrm{eff}}^2-n_2^2}{n_1^2-n_2^2}=\frac{w^2}{V^2},

\]

with \(0<b<1\). Values near one indicate strong core confinement, while values near zero indicate proximity to cutoff.

3.3 Single-mode condition

The fundamental HE\(_{11}\), represented approximately as LP\(_{01}\), has no finite cutoff. The next mode family, LP\(_{11}\), appears when

\[

V=2.405,

\]

which is the first positive zero of \(J_0\). Hence a conventional step-index fiber is single-mode when

\[

V<2.405.

\]

“Single mode” normally means one spatial mode. That mode still supports two approximately orthogonal polarization states.

3.4 Mode count

For a highly multimode step-index fiber,

\[

M_{\mathrm{pol}}\approx\frac{V^2}{2},

\]

where \(M_{\mathrm{pol}}\) counts polarization-resolved modes. If the two polarization states are treated as one spatial mode,

\[

M_{\mathrm{spatial}}\approx\frac{V^2}{4}.

\]

For a power-law graded-index fiber,

\[

M_{\mathrm{pol}}\approx\frac{g}{g+2}\frac{V^2}{2},

\qquad

M_{\mathrm{spatial}}\approx\frac{g}{g+2}\frac{V^2}{4}.

\]

Thus, an ideal parabolic fiber \((g=2)\) supports approximately \(V^2/4\) polarization-resolved modes. These are asymptotic large-\(V\) formulas; exact counting near cutoff requires solving the eigenvalue problem.

4. Graded-Index Modal Theory and Intermodal Dispersion

For a parabolic profile,

\[

n^2(r)=n_1^2\left(1-2\Delta r^2/a^2\right),

\]

and the scalar wave equation inside the core becomes

\[

\left[\nabla_t^2-q^2r^2+n_1^2k_0^2-\beta^2\right]\psi=0,

\]

where

\[

q=\frac{n_1k_0\sqrt{2\Delta}}{a}.

\]

This equation has the mathematical form of a two-dimensional harmonic oscillator. Neglecting finite-boundary corrections, its eigenvalues satisfy

\[

n_1^2k_0^2-\beta^2=2q(N+1),

\]

where

\[

N=2p+|l|,

\]

and \(p=0,1,2,\ldots\) is the radial order. Therefore,

\[

\beta^2\approx n_1^2k_0^2\left[1-\frac{4\Delta(N+1)}{V}\right].

\]

For small \(\Delta\),

\[

\beta\approx n_1k_0\left[1-\frac{2\Delta(N+1)}{V}\right].

\]

Modes with the same value of \(N\) form nearly degenerate mode groups. Because \(V\propto\omega\), the second term is approximately independent of frequency if material dispersion is neglected. Consequently, \(d\beta/d\omega\), and therefore group delay, is nearly identical for all mode groups. This demonstrates from modal theory why an ideal parabolic profile suppresses first-order intermodal dispersion.

For the general power-law profile, a weak-guidance ray or WKB estimate of the maximum first-order intermodal delay spread is

\[

\frac{\Delta\tau_{\mathrm{im}}}{L}\approx

\frac{n_1\Delta}{c}\left|\frac{g-2}{g+2}\right|.

\]

This expression assumes a smooth ideal profile, small \(\Delta\), negligible material dispersion and excitation of the full modal range. It approaches the step-index result \(n_1\Delta/c\) as \(g\rightarrow\infty\), while the first-order term vanishes for \(g=2\). Actual parabolic fibers retain second-order delay of order \(\Delta^2\), together with errors caused by profile imperfections, material dispersion, dopant diffusion, mode coupling and wavelength-dependent optimum profile.

Worked Example 2: Profile error in a graded-index fiber

Consider \(L=1\,\text{km}\), \(n_1=1.47\), \(\Delta=0.010\), and \(g=2.1\). Then

\[

\Delta\tau_{\mathrm{im}}

\approx\frac{(1.47)(0.010)(1000)}{3.00\times10^8}

\left|\frac{0.1}{4.1}\right|.

\]

The step-index delay scale is \(49.0\,\text{ns}\), while the profile factor is \(0.02439\). Therefore,

\[

\Delta\tau_{\mathrm{im}}\approx1.20\,\text{ns}.

\]

The example shows that a small departure from the optimum exponent can produce measurable broadening even though it remains far smaller than the approximately \(49\,\text{ns}\) first-order delay of a comparable step-index fiber.

5. Attenuation

If power obeys

\[

P(z)=P(0)e^{-\alpha_p z},

\]

then \(\alpha_p\) is the power-attenuation coefficient in inverse-length units. In decibels,

\[

A_{\mathrm{dB}}=10\log_{10}\frac{P(0)}{P(L)}=\alpha_{\mathrm{dB}}L.

\]

The conversion is

\[

\alpha_p=\frac{\ln 10}{10}\alpha_{\mathrm{dB}},

\]

provided both coefficients use the same distance unit.

5.1 Physical mechanisms

• Intrinsic absorption: electronic absorption dominates at short wavelengths and vibrational absorption at long wavelengths.
• Extrinsic absorption: impurities, especially hydroxyl ions and transition metals, create absorption bands.
• Rayleigh scattering: microscopic density and composition fluctuations produce loss approximately proportional to \(\lambda^{-4}\).
• Structural scattering: core–cladding irregularities and defects create additional loss.
• Bending and joint losses: curvature, connectors, splices and misalignment remove power from guided modes.

Modern silica fiber can exhibit attenuation near \(0.2\,\text{dB/km}\) around \(1550\,\text{nm}\), although installed-link loss also includes splices, connectors and engineering margin.

6. Dispersion and Pulse Broadening

Expanding the propagation constant around carrier angular frequency \(\omega_0\),

\[

\beta(\omega)=\beta_0+\beta_1(\omega-\omega_0)

+\frac{1}{2}\beta_2(\omega-\omega_0)^2+

\frac{1}{6}\beta_3(\omega-\omega_0)^3+\cdots.

\]

Here, \(\beta_1=d\beta/d\omega\) is group delay per unit length, \(\beta_2\) is group-velocity dispersion, and \(\beta_3\) is third-order dispersion.

6.1 Chromatic dispersion

The standard dispersion parameter is

\[

D=\frac{1}{L}\frac{d\tau_g}{d\lambda}

=-\frac{2\pi c}{\lambda^2}\beta_2,

\]

usually expressed in \(\text{ps}/(\text{nm}\cdot\text{km})\). For a source with sufficiently narrow spectral width \(\Delta\lambda\), the approximate delay spread is

\[

\Delta\tau_D\approx |D|L\Delta\lambda.

\]

Chromatic dispersion contains:

• material dispersion, caused by wavelength dependence of the glass refractive index; and
• waveguide dispersion, caused by wavelength-dependent redistribution of modal power between core and cladding.

Standard single-mode silica fiber has near-zero total dispersion around \(1310\,\text{nm}\) and typically about \(+17\,\text{ps}/(\text{nm}\cdot\text{km})\) around \(1550\,\text{nm}\). Exact values depend on the fiber standard and manufacturer.

Worked Example 3: Chromatic broadening

For \(D=17\,\text{ps}/(\text{nm}\cdot\text{km})\), \(L=80\,\text{km}\), and \(\Delta\lambda=0.10\,\text{nm}\),

\[

\Delta\tau_D=(17)(80)(0.10)=136\,\text{ps}.

\]

The result must be compared with symbol duration and receiver equalization capability rather than interpreted in isolation.

6.2 Polarization-mode dispersion

Real fibers contain random birefringence from asymmetry, stress, bends and temperature changes. Two principal states of polarization acquire a differential group delay (DGD). For long fibers with randomly varying birefringence,

\[

\langle\Delta\tau_{\mathrm{PMD}}\rangle\approx D_{\mathrm{PMD}}\sqrt{L},

\]

where \(D_{\mathrm{PMD}}\) is usually stated in \(\text{ps}/\sqrt{\text{km}}\). This square-root law is statistical and is not appropriate for a uniform deterministic birefringent element, where delay grows linearly with length.

If \(D_{\mathrm{PMD}}=0.08\,\text{ps}/\sqrt{\text{km}}\) and \(L=80\,\text{km}\), the mean-scale DGD is

\[

0.08\sqrt{80}=0.716\,\text{ps}.

\]

Higher-order PMD and time variation become important in high-symbol-rate systems.

7. Bending Loss

7.1 Macrobending

Macrobending arises from smooth curvature over a physically large bend radius. Curvature changes the effective refractive-index landscape, allowing part of the guided field to tunnel into radiation modes. Practical critical bend radii are commonly on millimetre-to-centimetre scales and are therefore far larger than the micrometre-scale mode-field radius. Loss rises rapidly as bend radius decreases and is generally greater at longer wavelengths because confinement is weaker.

7.2 Microbending

Microbending consists of small, often irregular axial perturbations caused by cabling pressure, manufacturing defects, thermal contraction or rough support surfaces. Spatial-frequency components of these perturbations can couple guided power into radiation modes or other guided modes.

Bend-insensitive fibers modify the refractive-index profile, for example through trench-assisted cladding, to improve confinement. Installation must nevertheless comply with manufacturer-specified minimum static and dynamic bend radii.

8. Nonlinear Propagation

At sufficiently high optical intensity, fiber properties depend on the field itself. The effective nonlinear area is

\[

A_{\mathrm{eff}}=\frac{\left(\iint |F|^2dA\right)^2}{\iint |F|^4dA},

\]

where \(F\) is the transverse modal field. The Kerr nonlinear coefficient is

\[

\gamma=\frac{n_2^{(K)}\omega_0}{cA_{\mathrm{eff}}}

=\frac{2\pi n_2^{(K)}}{\lambda A_{\mathrm{eff}}},

\]

where \(n_2^{(K)}\) is the nonlinear refractive-index coefficient, not the cladding index.

For constant launched power \(P_0\) and attenuation coefficient \(\alpha_p\),

\[

L_{\mathrm{eff}}=\frac{1-e^{-\alpha_pL}}{\alpha_p},

\qquad

\phi_{\mathrm{NL}}=\gamma P_0L_{\mathrm{eff}}.

\]

Important nonlinear processes include:

• self-phase modulation: a channel modulates its own phase;
• cross-phase modulation: one wavelength channel changes another channel’s phase;
• four-wave mixing: new frequencies are generated through phase-matched mixing;
• stimulated Brillouin scattering: narrowband backward scattering involving acoustic phonons; and
• stimulated Raman scattering: transfer of energy toward lower optical frequencies through molecular vibrations.

Dispersion, effective area, channel spacing, polarization, amplification and modulation format determine the system impact. Increasing launch power improves optical signal-to-noise ratio only until nonlinear penalties dominate.

9. Illustrative Case: Upgrading an 80 km Campus Link

Background

An illustrative operator plans to upgrade an \(80\,\text{km}\) standard single-mode link to \(10\,\text{Gb/s}\) intensity-modulated transmission. Fiber attenuation is \(0.20\,\text{dB/km}\), connector and splice loss totals \(2\,\text{dB}\), \(D=17\,\text{ps}/(\text{nm}\cdot\text{km})\), source spectral width is \(0.05\,\text{nm}\), and \(D_{\mathrm{PMD}}=0.08\,\text{ps}/\sqrt{\text{km}}\). Launch power is \(+3\,\text{dBm}=2.0\,\text{mW}\), and \(\gamma=1.3\,\text{W}^{-1}\text{km}^{-1}\).

Problem and analysis

The total passive loss is

\[

(0.20)(80)+2=18\,\text{dB},

\]

so received power is approximately \(-15\,\text{dBm}\). The power budget may be adequate for a receiver sensitivity of \(-18\,\text{dBm}\), but only a \(3\,\text{dB}\) margin remains.

Chromatic broadening is

\[

\Delta\tau_D=(17)(80)(0.05)=68\,\text{ps}.

\]

A \(10\,\text{Gb/s}\) bit period is \(100\,\text{ps}\); therefore, this broadening is operationally significant. PMD contributes only

\[

0.08\sqrt{80}=0.716\,\text{ps},

\]

and is not the dominant limitation.

For attenuation in natural units,

\[

\alpha_p=\frac{\ln10}{10}(0.20)=0.04605\,\text{km}^{-1}.

\]

Thus,

\[

L_{\mathrm{eff}}=\frac{1-e^{-0.04605(80)}}{0.04605}=21.17\,\text{km},

\]

and

\[

\phi_{\mathrm{NL}}=(1.3)(0.0020)(21.17)=0.055\,\text{rad}.

\]

The Kerr phase shift is small for this unamplified link.

Approach and outcome

The engineering priority is chromatic-dispersion control rather than reducing launch power. Suitable options include a lower-chirp transmitter, electronic dispersion compensation, a dispersion-compensating module, or migration to coherent transmission. Additional power margin should be created by improving connectors or using an optical amplifier where noise and nonlinear consequences are acceptable.

Learning points

• A satisfactory power budget does not guarantee an acceptable dispersion budget.
• PMD and Kerr nonlinearity may be negligible even when chromatic dispersion is critical.
• Launch-power decisions must account for both receiver noise and nonlinear propagation.
• Installed-link measurements should validate specification-based calculations.

10. Engineering Practice, Standards and Emerging Directions

Good fiber engineering includes verifying cutoff wavelength, mode-field diameter, attenuation, dispersion, PMD, bend performance and connector cleanliness. Optical time-domain reflectometry locates loss events, while optical frequency-domain methods, dispersion analyzers and polarization measurements provide more specialized characterization.

Relevant international recommendations include ITU-T G.652 for standard single-mode fiber and G.657 for bend-insensitive access fiber. Laser systems must be operated under applicable laser-safety requirements, such as IEC 60825. Invisible infrared radiation can injure the eye; technicians must never inspect an active connector directly. Governance also requires calibration records, traceable test data and realistic design margins rather than selective reporting of favorable measurements.

Contemporary developments include multicore and few-mode fibers for spatial-division multiplexing, hollow-core fibers for reduced latency and nonlinearity, and specialty fibers with engineered dispersion or large effective area. Machine-learning tools are increasingly used to detect anomalies and estimate nonlinear or polarization states, but they do not replace physical validation, representative training data or safety controls.

11. Challenges and Design Best Practices

Analytical models rely on ideal geometry, smooth index profiles and known material parameters. Real fibers contain longitudinal variations, random coupling and environmental sensitivity. Modal counts are asymptotic, PMD is statistical, and simple dispersion broadening formulas may be inadequate for broadband or strongly chirped signals.

Recommended practices are to:

1. distinguish spatial modes from polarization-resolved modes;

2. use vector analysis for high-index-contrast waveguides;

3. measure installed-link parameters when operating margins are narrow;

4. include connector, splice, aging and repair allowances;

5. respect wavelength-dependent bend specifications;

6. optimize launch power jointly with amplifier noise and nonlinearity; and

7. validate simplified calculations using numerical mode solvers or propagation models when necessary.

Chapter Summary

Optical-fiber guidance is controlled by refractive-index profile, numerical aperture and normalized frequency. In weakly guiding step-index fibers, scalar modal fields are represented by Bessel functions in the core and modified Bessel functions in the cladding. Correct boundary continuity produces the modal eigenvalue equation, while the condition \(V<2.405\) ensures single-spatial-mode operation.

Large-V step-index fibers support approximately \(V^2/2\) polarization-resolved modes, or \(V^2/4\) spatial modes. Graded-index profiles reduce both mode count and intermodal delay. The parabolic profile produces nearly equally spaced mode groups and cancels first-order intermodal dispersion under ideal assumptions.

Transmission performance is limited by attenuation, chromatic dispersion, PMD, bending and nonlinear effects. Chromatic delay grows approximately as \(|D|L\Delta\lambda\), whereas mean PMD delay grows statistically as \(\sqrt{L}\). Macrobending occurs at bend radii much larger than the mode-field scale. Kerr nonlinearity is governed by effective area, nonlinear coefficient, launch power and effective length. Reliable link design requires simultaneous assessment of power, dispersion, polarization, bending and nonlinear budgets.

Key Terms

Numerical aperture: Measure of a fiber’s light-acceptance capability.

Acceptance angle: Maximum launch half-angle for guided meridional rays.

Normalized frequency: Dimensionless parameter \(V\) governing modal behavior.

Propagation constant: Longitudinal spatial frequency \(\beta\) of a mode.

Effective index: Modal phase index \(n_{\mathrm{eff}}=\beta/k_0\).

LP mode: Approximate linearly polarized mode family in a weakly guiding fiber.

Cutoff wavelength: Wavelength at which a higher-order mode ceases to be guided.

Mode group: Set of nearly degenerate modes with similar propagation constants.

Intermodal dispersion: Delay spread between different spatial modes.

Material dispersion: Dispersion caused by wavelength-dependent material index.

Waveguide dispersion: Dispersion caused by wavelength-dependent field confinement.

Polarization-mode dispersion: Statistical delay between principal polarization states.

Rayleigh scattering: Scattering from microscopic refractive-index fluctuations.

Macrobending: Radiation loss caused by smooth, large-scale fiber curvature.

Microbending: Mode coupling and loss caused by small irregular perturbations.

Effective area: Nonlinear interaction area of a guided mode.

Kerr effect: Intensity-dependent change in refractive index.

Effective length: Loss-adjusted interaction length for nonlinear processes.

Review Questions

Short Answer Questions

1. Define numerical aperture and state the assumptions underlying its common step-index expression.

2. What physical meaning do the normalized parameters \(u\), \(w\) and \(V\) possess?

3. Why does the LP\(_{01}\) mode have no finite cutoff?

4. Distinguish macrobending from microbending.

5. Why does PMD commonly scale with the square root of fiber length?

Long Answer Questions

1. Derive the scalar characteristic equation for a circular step-index fiber from boundary continuity.

2. Explain the distinction between exact vector modes and LP mode families in weakly guiding fibers.

3. Develop the modal explanation for first-order intermodal-dispersion cancellation in an ideal parabolic graded-index fiber.

4. Discuss the physical origins and wavelength dependence of attenuation in silica fibers.

5. Compare material, waveguide, intermodal and polarization-mode dispersion, including their system consequences.

Analytical / Application Questions

1. A fiber has \(a=4.5\,\mu\text{m}\), \(\mathrm{NA}=0.12\), and operates at \(1.31\,\mu\text{m}\). Calculate \(V\), determine whether it is single-mode, and find its approximate cutoff wavelength.

2. A \(60\,\text{km}\) link has \(D=16.5\,\text{ps}/(\text{nm}\cdot\text{km})\), source width \(0.08\,\text{nm}\), and \(D_{\mathrm{PMD}}=0.10\,\text{ps}/\sqrt{\text{km}}\). Calculate chromatic and mean-scale PMD delays and identify the dominant impairment.

3. For \(\gamma=1.5\,\text{W}^{-1}\text{km}^{-1}\), launch power \(5\,\text{mW}\), attenuation \(0.22\,\text{dB/km}\), and length \(50\,\text{km}\), calculate effective length and nonlinear phase shift.

Multiple Choice Questions with Answers

Question 1

Which equation is the scalar step-index characteristic equation obtained by continuity of the field and its radial derivative?

A. uJ_l'(u)/J_l(u) = wK_l'(w)/K_l(w)

B. J_l(u)/J_l'(u) = K_l(w)/K_l'(w)

C. uJ_l(u)/J_l'(u) = -wK_l(w)/K_l'(w)

D. uJ_l'(u) = -wK_l'(w)

Correct Answer: A. uJ_l'(u)/J_l(u) = wK_l'(w)/K_l(w)

Question 2

For a highly multimode step-index fiber, the approximate number of spatial modes excluding polarization degeneracy is

A. V²

B. V²/2

C. V²/4

D. V/2

Correct Answer: C. V²/4

Question 3

A step-index fiber has n₁ = 1.450 and n₂ = 1.444. Its numerical aperture is approximately

A. 0.0060

B. 0.0910

C. 0.1318

D. 0.4500

Correct Answer: C. 0.1318

Question 4

What is the conventional single-mode condition for a weakly guiding step-index fiber?

A. V < 1.000

B. V < 2.405

C. V > 2.405

D. V > π

Correct Answer: B. V < 2.405

Question 5

Which power-law profile exponent ideally cancels first-order intermodal dispersion?

A. g = 1

B. g = 2

C. g = 4

D. g → ∞

Correct Answer: B. g = 2

Question 6

A fiber has attenuation 0.20 dB/km over 50 km, with no joint losses. What fraction of input power remains?

A. 0.01

B. 0.10

C. 0.20

D. 0.50

Correct Answer: B. 0.10

Question 7

For D = 17 ps/(nm·km), L = 40 km and spectral width 0.10 nm, the approximate chromatic delay spread is

A. 17 ps

B. 40 ps

C. 68 ps

D. 680 ps

Correct Answer: C. 68 ps

Question 8

For D_PMD = 0.10 ps/√km and L = 100 km, the mean-scale differential group delay is

A. 0.10 ps

B. 1.0 ps

C. 10 ps

D. 100 ps

Correct Answer: B. 1.0 ps

Question 9

Which statement about macrobending is correct?

A. It occurs only when bend radius equals the optical wavelength.

B. Its practical bend radius is generally far larger than the mode-field radius.

C. It cannot cause radiation from a guided mode.

D. Its loss is independent of wavelength.

Correct Answer: B. Its practical bend radius is generally far larger than the mode-field radius.

Question 10

For γ = 1 W⁻¹km⁻¹, launched power 10 mW and effective length 20 km, the nonlinear phase shift is

A. 0.02 rad

B. 0.20 rad

C. 2.0 rad

D. 200 rad

Correct Answer: B. 0.20 rad