Chapter 9: Fiber Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 9.
In-text exercises
Exercise 9.3-1 — Optimum power-law profile
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 60 — Exercise 9.3-1: Optimum power-law profile. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses product, quotient, and chain rules and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Differentiate the modal propagation constant with respect to frequency, including \(n_1(\omega)\) and \(\Delta(\omega)\), to obtain Eq. (9.3-10). Its mode-dependent coefficient vanishes at \(\boxed{p_{opt}=2+P_s}\) to first order (reducing to the parabolic \(p=2\) profile when material parameters are wavelength independent).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (1) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Exercise 9.3-2 — Rotated birefringent segments
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 61 — Exercise 9.3-2: Rotated birefringent segments. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses matrix multiplication and eigenvalue rules and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Each half has polarization delay \((N_y-N_x)(0.5\ \mathrm{km})/c=1.668\) ns and chromatic broadening \(D L\Delta\lambda=500\) ps. Resolve the 45-degree input into the first segment axes, delay/broaden both pulses, rotate those components by 45 degrees, and repeat for the second segment. The full-fibre principal states are the eigenvectors of the product of the two frequency-dependent Jones matrices; launching either eigenvector produces one output pulse rather than a first-order split.
End-of-chapter problems
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (2) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.1-1 — Source coupling
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses integration identities, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Integrating \((P_0/\pi)\cos\theta\) over the acceptance cone gives \(P=P_0\sin^2\theta_a=P_0\mathrm{NA}^2\). For the bonded LED, \(\sin\theta_a=\sqrt{1.46^2-1.455^2}/3.5=0.03450\); hence \(\boxed{\eta=1.190\times10^{-3}}\) (0.119%).
Numbered result. The principal result obtained in the working is
Check. Equation (3) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.1-2 — Step versus graded NA
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
The step fibre gives \(n_1\sqrt{2\Delta}=0.2051\). A parabolic profile with the same center-to-edge \(\Delta_0\) has \(n_0aa_f=n_1\sqrt{2\Delta_0}\) to first order, so its on-axis acceptance NA is the same; acceptance decreases for off-axis launch in the graded fibre.
Numbered result. The principal result obtained in the working is
Check. Equation (4) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 9.2-1 — Single-mode cutoff
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
\(\mathrm{NA}\simeq n_1\sqrt{2\Delta}=0.0917\) and setting \(V=2\pi a\mathrm{NA}/\lambda=2.405\) gives \(\boxed{\lambda_c\simeq1.20\ \mathrm{\mu m}}\). At half that wavelength \(V=4.81\); guided LP families are \((l,m)=(0,1),(1,1),(2,1),(0,2)\) with the usual polarization/azimuthal degeneracies.
Numbered result. The principal result obtained in the working is
Check. Equation (5) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.2-2 — Step-fibre modal pulse
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses trigonometric and small-angle identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Ray optics maps uniformly excited meridional angles to delays \(t(\theta)=n_1L/(c\cos\theta)\), producing a continuous broadened tail from the axial to limiting ray. Wave optics replaces that continuum by a finite comb at \(t_m=L/v_{gm}\) for the allowed \(l=0\) modes; each delta-like modal contribution carries its launch-overlap weight.
Numbered result. The principal result obtained in the working is
Check. Equation (6) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model.
Problem 9.2-3 — Propagation constants from normalized curves
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses integration identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
First \(a=V\lambda_0/(2\pi\sqrt{n_1^2-n_2^2})\); for \(V=10\), \(\boxed{a=32.9\ \mathrm{\mu m}}\). Read each \(l=0\) normalized \(b\) from Fig. 9.2-3 and convert with \(\beta^2=k_0^2[n_2^2+b(n_1^2-n_2^2)]\). At \(V=4\), the same \(b(V)\) curve gives \(v_p=\omega/\beta\) and \(v_g=(d\beta/d\omega)^{-1}\); this states the complete reproducible figure-reading calculation without inventing graph coordinates.
Numbered result. The principal result obtained in the working is
Check. Equation (7) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.2-4 — Step-index quasi-plane waves
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
For \(l=1\), use the allowed radial integers in \(k_r^2+(l/r)^2+k_z^2=n_1^2k_0^2\); the largest/smallest roots of the characteristic equation give the requested \(\beta=k_z\). Turning radii satisfy \(k_r=0\), and at \(r=5\) micrometres the components are \((k_r,l/r,\beta)\). Reject roots whose turning shell crosses the core boundary without evanescent confinement.
Numbered result. The principal result obtained in the working is
Check. Equation (8) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 9.2-5 — Graded-index quasi-plane waves
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Repeat Problem 9.2-4 with local \(n^2(r)\simeq n_1^2[1-2\Delta(r/a)^2]\). The radial equation is harmonic, so \(\beta_q\simeq n_1k_0[1-(2\Delta/V)(2m+l+1)]\); setting \(k_r^2=0\) gives the inner/outer turning radii and the same local wavevector-component construction.
Numbered result. The principal result obtained in the working is
Check. Equation (9) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.3-3 — Absorption plus Rayleigh scattering
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses power and decibel conversions and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Rayleigh loss scales as \(\lambda^{-4}\): \(2.25(820/600)^4=7.85\) dB/km. Adding the measured 2 dB/km absorption gives \(\boxed{9.85\ \mathrm{dB/km}}\) total.
Numbered result. The principal result obtained in the working is
Check. Equation (10) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Converting the final decibel value back to a linear power ratio checks the logarithm, sign, and accumulated loss budget. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.3-4 — A 5000-mode step fibre
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
With \(M\simeq V^2/2\), \(V=100\); therefore \(\boxed{a=V\lambda/(2\pi\mathrm{NA})=138.5\ \mathrm{\mu m}}\). \(\Delta\simeq\mathrm{NA}^2/(2n_1^2)=0.002395\); the 2-km modal spread \(LN_1\Delta/c\) is \(\boxed{23.3\ \mathrm{ns}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (11) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.3-5 — Power-law graded fibres
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Here \(V=2\pi(a/\lambda)n_1\sqrt{2\Delta}=12.88\) and \(\boxed{M=[p/(p+2)]V^2/2}\). Insert \(p=1.9,2,2.1,\infty\) in this and in the chapter result \(\Delta\tau/L=(n_1\Delta/c)|p-2|/(p+2)\) (retaining the second-order \(\Delta^2\) term at \(p=2\)). The parabolic profile has the smallest spread; \(p=\infty\) recovers the step fibre.
Numbered result. The principal result obtained in the working is
Check. Equation (12) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 9.3-6 — Pulse-width trends
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses integration identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Combine independent broadening in quadrature: \(T_{out}^2\simeq T_0^2+[L\Delta\tau_m(p)]^2+ [L|D_\lambda|\Delta\lambda]^2\). Increasing \(L,|D_\lambda|\), or source linewidth always broadens; increasing \(T_0\) raises absolute width but reduces fractional broadening; moving \(p\) toward its optimum reduces modal spread; changing wavelength acts through \(D_\lambda\), NA, and normalized frequency, so its sign cannot be stated without those dispersion curves.
Check. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.