Chapter 4: Optical Resonators
=============================

Source: Amnon Yariv and Pochi Yeh, *Photonics: Optical Electronics in
Modern Communications*, sixth edition (2007), Chapter
4.  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 4.1 — Fabry--Perot transmission and finesse: 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.
Sum the round-trip geometric series.  With phase :math:`\delta=2n\omega d\cos\theta/c`, reduce the intensity to an Airy denominator and obtain width or finesse by expanding about :math:`\delta=2\pi m`.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.2 — Fabry--Perot transmission and finesse: 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.
Sum the round-trip geometric series.  With phase :math:`\delta=2n\omega d\cos\theta/c`, reduce the intensity to an Airy denominator and obtain width or finesse by expanding about :math:`\delta=2\pi m`.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.3 — Fabry--Perot transmission and finesse: 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.
Sum the round-trip geometric series.  With phase :math:`\delta=2n\omega d\cos\theta/c`, reduce the intensity to an Airy denominator and obtain width or finesse by expanding about :math:`\delta=2\pi m`.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.4 — Fabry--Perot transmission and finesse: 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.
Sum the round-trip geometric series.  With phase :math:`\delta=2n\omega d\cos\theta/c`, reduce the intensity to an Airy denominator and obtain width or finesse by expanding about :math:`\delta=2\pi m`.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.5 — Fabry--Perot transmission and finesse: 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.
Sum the round-trip geometric series.  With phase :math:`\delta=2n\omega d\cos\theta/c`, reduce the intensity to an Airy denominator and obtain width or finesse by expanding about :math:`\delta=2\pi m`.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.6 — Fabry--Perot transmission and finesse: 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.
Sum the round-trip geometric series.  With phase :math:`\delta=2n\omega d\cos\theta/c`, reduce the intensity to an Airy denominator and obtain width or finesse by expanding about :math:`\delta=2\pi m`.
At resonance a lossless symmetric etalon must reach unit transmission; in the low-reflectivity limit the fringes must disappear.

Problem 4.7 — Gaussian resonator modes and coupling: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Form the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`.  Use :math:`0<(A+D+2)/4<1` or the equivalent :math:`g_1g_2` test before extracting waist position, spot size, aperture power, or overlap.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.8 — Gaussian resonator modes and coupling: 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.
Form the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`.  Use :math:`0<(A+D+2)/4<1` or the equivalent :math:`g_1g_2` test before extracting waist position, spot size, aperture power, or overlap.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.9 — Gaussian resonator modes and coupling: 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.
Form the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`.  Use :math:`0<(A+D+2)/4<1` or the equivalent :math:`g_1g_2` test before extracting waist position, spot size, aperture power, or overlap.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.10 — Gaussian resonator modes and coupling: calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation.
Form the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`.  Use :math:`0<(A+D+2)/4<1` or the equivalent :math:`g_1g_2` test before extracting waist position, spot size, aperture power, or overlap.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.11 — Gaussian resonator modes and coupling: 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.
Form the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`.  Use :math:`0<(A+D+2)/4<1` or the equivalent :math:`g_1g_2` test before extracting waist position, spot size, aperture power, or overlap.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.12 — Gaussian resonator modes and coupling: 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.
Form the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`.  Use :math:`0<(A+D+2)/4<1` or the equivalent :math:`g_1g_2` test before extracting waist position, spot size, aperture power, or overlap.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.13 — Gaussian resonator modes and coupling: 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.
Form the round-trip ABCD matrix and solve :math:`q=(Aq+B)/(Cq+D)`.  Use :math:`0<(A+D+2)/4<1` or the equivalent :math:`g_1g_2` test before extracting waist position, spot size, aperture power, or overlap.
The selected root must have the physical confinement sign; coupling efficiencies and aperture power fractions must lie between zero and one.

Problem 4.14 — dispersive etalons, reciprocity, and ring filters: 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.
Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay.  Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order.
Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.15 — dispersive etalons, reciprocity, and ring filters: 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.
Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay.  Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order.
Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.16 — dispersive etalons, reciprocity, and ring filters: 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.
Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay.  Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order.
Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.17 — dispersive etalons, reciprocity, and ring filters: 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.
Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay.  Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order.
Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.18 — dispersive etalons, reciprocity, and ring filters: 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.
Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay.  Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order.
Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.

Problem 4.19 — dispersive etalons, reciprocity, and ring filters: 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.
Retain frequency dependence in both propagation phase and coupling coefficients, then differentiate the total complex transmission phase for group delay.  Use a unitary reciprocal scattering matrix for lossless couplers and cascade ring sections in propagation order.
Energy conservation requires each lossless scattering column to have unit norm; setting dispersion or coupling to zero must recover the elementary etalon or uncoupled-ring limit.