Chapter 7: Chromatic Dispersion and Polarization-Mode Dispersion

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 7. 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 7.1 — pulse dispersion and Fourier bandwidth: derivation

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. Fourier-transform the specified envelope, propagate each component with the Taylor series \(\beta(\omega)=\beta_0+\beta_1\Omega+\tfrac12\beta_2\Omega^2+\cdots\), and recover widths from intensity—not field—half maxima. Parseval’s theorem must preserve pulse energy, and setting \(eta_2=0\) must remove chromatic broadening.

Problem 7.2 — pulse dispersion and Fourier bandwidth: derivation

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. Fourier-transform the specified envelope, propagate each component with the Taylor series \(\beta(\omega)=\beta_0+\beta_1\Omega+\tfrac12\beta_2\Omega^2+\cdots\), and recover widths from intensity—not field—half maxima. Parseval’s theorem must preserve pulse energy, and setting \(eta_2=0\) must remove chromatic broadening.

Problem 7.3 — pulse dispersion and Fourier bandwidth: derivation

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. Fourier-transform the specified envelope, propagate each component with the Taylor series \(\beta(\omega)=\beta_0+\beta_1\Omega+\tfrac12\beta_2\Omega^2+\cdots\), and recover widths from intensity—not field—half maxima. Parseval’s theorem must preserve pulse energy, and setting \(eta_2=0\) must remove chromatic broadening.

Problem 7.4 — pulse dispersion and Fourier bandwidth: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Fourier-transform the specified envelope, propagate each component with the Taylor series \(\beta(\omega)=\beta_0+\beta_1\Omega+\tfrac12\beta_2\Omega^2+\cdots\), and recover widths from intensity—not field—half maxima. Parseval’s theorem must preserve pulse energy, and setting \(eta_2=0\) must remove chromatic broadening.

Problem 7.5 — pulse dispersion and Fourier bandwidth: derivation

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. Fourier-transform the specified envelope, propagate each component with the Taylor series \(\beta(\omega)=\beta_0+\beta_1\Omega+\tfrac12\beta_2\Omega^2+\cdots\), and recover widths from intensity—not field—half maxima. Parseval’s theorem must preserve pulse energy, and setting \(eta_2=0\) must remove chromatic broadening.

Problem 7.6 — polarization-mode dispersion and principal states: derivation

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. Represent each birefringent segment by a unitary Jones matrix and form the ordered product. The delay operator \(-iU^\dagger(dU/d\omega)\) is Hermitian; its eigenvectors are the principal states and its eigenvalue separation is the differential group delay. The two principal states must be orthogonal, total power must be invariant, and statistically independent sections must give root-length rather than linear-length scaling.

Problem 7.7 — polarization-mode dispersion and principal states: derivation

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. Represent each birefringent segment by a unitary Jones matrix and form the ordered product. The delay operator \(-iU^\dagger(dU/d\omega)\) is Hermitian; its eigenvectors are the principal states and its eigenvalue separation is the differential group delay. The two principal states must be orthogonal, total power must be invariant, and statistically independent sections must give root-length rather than linear-length scaling.

Problem 7.8 — polarization-mode dispersion and principal states: derivation

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. Represent each birefringent segment by a unitary Jones matrix and form the ordered product. The delay operator \(-iU^\dagger(dU/d\omega)\) is Hermitian; its eigenvectors are the principal states and its eigenvalue separation is the differential group delay. The two principal states must be orthogonal, total power must be invariant, and statistically independent sections must give root-length rather than linear-length scaling.

Problem 7.9 — polarization-mode dispersion and principal states: derivation

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. Represent each birefringent segment by a unitary Jones matrix and form the ordered product. The delay operator \(-iU^\dagger(dU/d\omega)\) is Hermitian; its eigenvectors are the principal states and its eigenvalue separation is the differential group delay. The two principal states must be orthogonal, total power must be invariant, and statistically independent sections must give root-length rather than linear-length scaling.

Problem 7.10 — polarization-mode dispersion and principal states: derivation

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. Represent each birefringent segment by a unitary Jones matrix and form the ordered product. The delay operator \(-iU^\dagger(dU/d\omega)\) is Hermitian; its eigenvectors are the principal states and its eigenvalue separation is the differential group delay. The two principal states must be orthogonal, total power must be invariant, and statistically independent sections must give root-length rather than linear-length scaling.

Problem 7.11 — polarization-mode dispersion and principal states: derivation

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. Represent each birefringent segment by a unitary Jones matrix and form the ordered product. The delay operator \(-iU^\dagger(dU/d\omega)\) is Hermitian; its eigenvectors are the principal states and its eigenvalue separation is the differential group delay. The two principal states must be orthogonal, total power must be invariant, and statistically independent sections must give root-length rather than linear-length scaling.

Problem 7.12 — polarization-mode dispersion and principal states: derivation

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. Represent each birefringent segment by a unitary Jones matrix and form the ordered product. The delay operator \(-iU^\dagger(dU/d\omega)\) is Hermitian; its eigenvectors are the principal states and its eigenvalue separation is the differential group delay. The two principal states must be orthogonal, total power must be invariant, and statistically independent sections must give root-length rather than linear-length scaling.