Chapter 3: Guided Waves in Dielectric Slabs and Fibers
Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 3. Use each problem number with the book; the original prompts are not reproduced. Each entry supplies the governing model, a decisive solution route, and an independent consistency check.
End-of-chapter problems
Problem 3.1 — slab modes and surface waves: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Solve the transverse Helmholtz equation in each layer, choose oscillatory fields in the guide and decaying fields outside, and match tangential fields. The determinant of the boundary equations supplies the modal dispersion relation.
3. Verification.
Confined solutions must decay away from the guide and carry finite power; a surface plasmon additionally requires opposite dielectric-constant signs.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Solve the transverse Helmholtz equation in each layer, choose oscillatory fields in the guide and decaying fields outside, and match tangential fields. The determinant of the boundary equations supplies the modal dispersion relation. Confined solutions must decay away from the guide and carry finite power; a surface plasmon additionally requires opposite dielectric-constant signs.
Problem 3.2 — slab modes and surface waves: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Solve the transverse Helmholtz equation in each layer, choose oscillatory fields in the guide and decaying fields outside, and match tangential fields. The determinant of the boundary equations supplies the modal dispersion relation.
3. Verification.
Confined solutions must decay away from the guide and carry finite power; a surface plasmon additionally requires opposite dielectric-constant signs.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Solve the transverse Helmholtz equation in each layer, choose oscillatory fields in the guide and decaying fields outside, and match tangential fields. The determinant of the boundary equations supplies the modal dispersion relation. Confined solutions must decay away from the guide and carry finite power; a surface plasmon additionally requires opposite dielectric-constant signs.
Problem 3.3 — slab modes and surface waves: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Solve the transverse Helmholtz equation in each layer, choose oscillatory fields in the guide and decaying fields outside, and match tangential fields. The determinant of the boundary equations supplies the modal dispersion relation.
3. Verification.
Confined solutions must decay away from the guide and carry finite power; a surface plasmon additionally requires opposite dielectric-constant signs.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Solve the transverse Helmholtz equation in each layer, choose oscillatory fields in the guide and decaying fields outside, and match tangential fields. The determinant of the boundary equations supplies the modal dispersion relation. Confined solutions must decay away from the guide and carry finite power; a surface plasmon additionally requires opposite dielectric-constant signs.
Problem 3.4 — fiber V number, numerical aperture, and mode count: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Use \(V=k_0a\sqrt{n_1^2-n_2^2}\) and \(\mathrm{NA}=\sqrt{n_1^2-n_2^2}\). Apply the relevant Bessel zero for cutoff and count the polarization-degenerate states only once per stated convention.
3. Verification.
Recover the single-mode condition near \(V<2.405\), and ensure every estimated mode count is nonnegative and increases with core area or NA.
Show detailed steps
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Use \(V=k_0a\sqrt{n_1^2-n_2^2}\) and \(\mathrm{NA}=\sqrt{n_1^2-n_2^2}\). Apply the relevant Bessel zero for cutoff and count the polarization-degenerate states only once per stated convention. Recover the single-mode condition near \(V<2.405\), and ensure every estimated mode count is nonnegative and increases with core area or NA.
Problem 3.5 — fiber V number, numerical aperture, and mode count: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Use \(V=k_0a\sqrt{n_1^2-n_2^2}\) and \(\mathrm{NA}=\sqrt{n_1^2-n_2^2}\). Apply the relevant Bessel zero for cutoff and count the polarization-degenerate states only once per stated convention.
3. Verification.
Recover the single-mode condition near \(V<2.405\), and ensure every estimated mode count is nonnegative and increases with core area or NA.
Show detailed steps
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Use \(V=k_0a\sqrt{n_1^2-n_2^2}\) and \(\mathrm{NA}=\sqrt{n_1^2-n_2^2}\). Apply the relevant Bessel zero for cutoff and count the polarization-degenerate states only once per stated convention. Recover the single-mode condition near \(V<2.405\), and ensure every estimated mode count is nonnegative and increases with core area or NA.
Problem 3.6 — fiber V number, numerical aperture, and mode count: calculation
Brief solution
1. Method.
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
2. Decisive step.
Use \(V=k_0a\sqrt{n_1^2-n_2^2}\) and \(\mathrm{NA}=\sqrt{n_1^2-n_2^2}\). Apply the relevant Bessel zero for cutoff and count the polarization-degenerate states only once per stated convention.
3. Verification.
Recover the single-mode condition near \(V<2.405\), and ensure every estimated mode count is nonnegative and increases with core area or NA.
Show detailed steps
Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Use \(V=k_0a\sqrt{n_1^2-n_2^2}\) and \(\mathrm{NA}=\sqrt{n_1^2-n_2^2}\). Apply the relevant Bessel zero for cutoff and count the polarization-degenerate states only once per stated convention. Recover the single-mode condition near \(V<2.405\), and ensure every estimated mode count is nonnegative and increases with core area or NA.
Problem 3.7 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.8 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.9 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.10 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.11 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.12 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.13 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.14 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Problem 3.15 — cylindrical-fiber eigenmodes and normalization: derivation
Brief solution
1. Method.
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible.
2. Decisive step.
Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals.
3. Verification.
Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.
Show detailed steps
Begin with the governing equation named in the chapter and carry every algebraic or boundary-condition step explicitly; introduce approximations only after the exact relation is visible. Write core fields with \(J_m(hr)\) and cladding fields with \(K_m(qr)\), where \(h^2=n_1^2k_0^2-\beta^2\) and \(q^2=\beta^2-n_2^2k_0^2\). Match boundary components, then use Bessel recurrences for derivatives and power integrals. Accept only \(n_2k_0<\beta<n_1k_0\); the field and required tangential derivative must be continuous, and the normalized confinement fractions must sum to one.