Chapter 9: Electro-optic Modulation of Laser Beams

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 9. 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 9.1 — electro-optic retardation and modulation response: 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. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.2 — electro-optic retardation and modulation response: discussion

Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.3 — electro-optic retardation and modulation response: plot

Derive a dimensionless plotting expression first, evaluate the limiting values and resonance or cutoff points, and then sample densely enough to resolve the narrowest feature. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.4 — electro-optic retardation and modulation response: 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. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.5 — electro-optic retardation and modulation response: design

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.6 — electro-optic retardation and modulation response: 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. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.7 — electro-optic retardation and modulation response: 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. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.8 — electro-optic retardation and modulation response: 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. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.9 — electro-optic retardation and modulation response: 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. Convert the field-induced impermeability change into principal indices, form the retardation \(\Gamma=(\omega L/c)\Delta n\), and pass the Jones vector through the analyzer. Use the Jacobi–Anger expansion when harmonic content is requested. Zero drive must recover the static bias transmission, and pure phase modulation followed directly by an ideal square-law detector must not create intensity modulation.

Problem 9.10 — index ellipsoid, tensor rotation, and waveguide modulation: 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. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.11 — index ellipsoid, tensor rotation, and waveguide modulation: 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. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.12 — index ellipsoid, tensor rotation, and waveguide modulation: 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. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.13 — index ellipsoid, tensor rotation, and waveguide modulation: 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. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.14 — index ellipsoid, tensor rotation, and waveguide modulation: 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. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.15 — index ellipsoid, tensor rotation, and waveguide modulation: 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. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.16 — index ellipsoid, tensor rotation, and waveguide modulation: design

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.17 — index ellipsoid, tensor rotation, and waveguide modulation: design

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits. Diagonalize the symmetric impermeability tensor after the electro-optic perturbation. Transform it with \(\eta'=R\eta R^T\), use its eigenvectors as the new principal axes, and integrate the modal index shift for phase or coupling. The rotated axes must remain mutually orthogonal, tensor trace invariants must be unchanged, and the zero-field limit must restore the original axes.

Problem 9.18 — acousto-optic Bragg diffraction and deflection: 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. Impose photon–phonon frequency conservation and \(\mathbf k_d=\mathbf k_i\pm\mathbf K\). Use the Bragg geometry to obtain angle or shift, then size bandwidth, acoustic power, and scan range from the interaction length and material figure of merit. The diffracted frequency must have the sign of the absorbed or emitted phonon, and the vector triangle must close at the stated Bragg angle.

Problem 9.19 — acousto-optic Bragg diffraction and deflection: design

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits. Impose photon–phonon frequency conservation and \(\mathbf k_d=\mathbf k_i\pm\mathbf K\). Use the Bragg geometry to obtain angle or shift, then size bandwidth, acoustic power, and scan range from the interaction length and material figure of merit. The diffracted frequency must have the sign of the absorbed or emitted phonon, and the vector triangle must close at the stated Bragg angle.

Problem 9.20 — acousto-optic Bragg diffraction and deflection: discussion

Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Impose photon–phonon frequency conservation and \(\mathbf k_d=\mathbf k_i\pm\mathbf K\). Use the Bragg geometry to obtain angle or shift, then size bandwidth, acoustic power, and scan range from the interaction length and material figure of merit. The diffracted frequency must have the sign of the absorbed or emitted phonon, and the vector triangle must close at the stated Bragg angle.

Problem 9.21 — acousto-optic Bragg diffraction and deflection: 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. Impose photon–phonon frequency conservation and \(\mathbf k_d=\mathbf k_i\pm\mathbf K\). Use the Bragg geometry to obtain angle or shift, then size bandwidth, acoustic power, and scan range from the interaction length and material figure of merit. The diffracted frequency must have the sign of the absorbed or emitted phonon, and the vector triangle must close at the stated Bragg angle.

Problem 9.22 — acousto-optic Bragg diffraction and deflection: discussion

Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Impose photon–phonon frequency conservation and \(\mathbf k_d=\mathbf k_i\pm\mathbf K\). Use the Bragg geometry to obtain angle or shift, then size bandwidth, acoustic power, and scan range from the interaction length and material figure of merit. The diffracted frequency must have the sign of the absorbed or emitted phonon, and the vector triangle must close at the stated Bragg angle.

Problem 9.23 — acousto-optic Bragg diffraction and deflection: 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. Impose photon–phonon frequency conservation and \(\mathbf k_d=\mathbf k_i\pm\mathbf K\). Use the Bragg geometry to obtain angle or shift, then size bandwidth, acoustic power, and scan range from the interaction length and material figure of merit. The diffracted frequency must have the sign of the absorbed or emitted phonon, and the vector triangle must close at the stated Bragg angle.

Problem 9.24 — acousto-optic Bragg diffraction and deflection: design

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and reject any component value that violates power, bandwidth, or material limits. Impose photon–phonon frequency conservation and \(\mathbf k_d=\mathbf k_i\pm\mathbf K\). Use the Bragg geometry to obtain angle or shift, then size bandwidth, acoustic power, and scan range from the interaction length and material figure of merit. The diffracted frequency must have the sign of the absorbed or emitted phonon, and the vector triangle must close at the stated Bragg angle.