Chapter 6: Theory of Laser Oscillation and Laser Systems

Source: Amnon Yariv and Pochi Yeh, Photonics: Optical Electronics in Modern Communications, sixth edition (2007), Chapter 6. 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 6.1 — laser threshold, output coupling, and regenerative gain: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Write one round-trip amplitude or power balance and set it to unity at threshold. Express gain through inversion and line shape, then optimize output coupling only after separating internal loss from useful transmission. Below threshold the round-trip multiplier must be less than one; the passive limit must reproduce the ordinary Fabry–Perot result.

Problem 6.2 — laser threshold, output coupling, and regenerative gain: 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. Write one round-trip amplitude or power balance and set it to unity at threshold. Express gain through inversion and line shape, then optimize output coupling only after separating internal loss from useful transmission. Below threshold the round-trip multiplier must be less than one; the passive limit must reproduce the ordinary Fabry–Perot result.

Problem 6.3 — laser threshold, output coupling, and regenerative gain: 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. Write one round-trip amplitude or power balance and set it to unity at threshold. Express gain through inversion and line shape, then optimize output coupling only after separating internal loss from useful transmission. Below threshold the round-trip multiplier must be less than one; the passive limit must reproduce the ordinary Fabry–Perot result.

Problem 6.4 — laser threshold, output coupling, and regenerative gain: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Write one round-trip amplitude or power balance and set it to unity at threshold. Express gain through inversion and line shape, then optimize output coupling only after separating internal loss from useful transmission. Below threshold the round-trip multiplier must be less than one; the passive limit must reproduce the ordinary Fabry–Perot result.

Problem 6.5 — laser threshold, output coupling, and regenerative gain: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Write one round-trip amplitude or power balance and set it to unity at threshold. Express gain through inversion and line shape, then optimize output coupling only after separating internal loss from useful transmission. Below threshold the round-trip multiplier must be less than one; the passive limit must reproduce the ordinary Fabry–Perot result.

Problem 6.6 — laser threshold, output coupling, and regenerative gain: 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. Write one round-trip amplitude or power balance and set it to unity at threshold. Express gain through inversion and line shape, then optimize output coupling only after separating internal loss from useful transmission. Below threshold the round-trip multiplier must be less than one; the passive limit must reproduce the ordinary Fabry–Perot result.

Problem 6.7 — laser threshold, output coupling, and regenerative gain: 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. Write one round-trip amplitude or power balance and set it to unity at threshold. Express gain through inversion and line shape, then optimize output coupling only after separating internal loss from useful transmission. Below threshold the round-trip multiplier must be less than one; the passive limit must reproduce the ordinary Fabry–Perot result.

Problem 6.8 — laser modes, stabilization, and transition cross sections: 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. Expand the field in longitudinal modes, carry the beat terms through the detector bandwidth, and use the resonator mode condition \(2nL\nu/c=m\). Connect cross section and lifetime by equating the Einstein-coefficient forms of spontaneous and stimulated rates. Mode spacings must approach \(c/(2nL)\) without pulling, and any stabilization discriminator must change sign across its lock point.

Problem 6.9 — laser modes, stabilization, and transition cross sections: discussion

Identify the governing conservation law and compare the relevant asymptotic regimes before drawing the qualitative conclusion. Expand the field in longitudinal modes, carry the beat terms through the detector bandwidth, and use the resonator mode condition \(2nL\nu/c=m\). Connect cross section and lifetime by equating the Einstein-coefficient forms of spontaneous and stimulated rates. Mode spacings must approach \(c/(2nL)\) without pulling, and any stabilization discriminator must change sign across its lock point.

Problem 6.10 — laser modes, stabilization, and transition cross sections: 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. Expand the field in longitudinal modes, carry the beat terms through the detector bandwidth, and use the resonator mode condition \(2nL\nu/c=m\). Connect cross section and lifetime by equating the Einstein-coefficient forms of spontaneous and stimulated rates. Mode spacings must approach \(c/(2nL)\) without pulling, and any stabilization discriminator must change sign across its lock point.

Problem 6.11 — laser modes, stabilization, and transition cross sections: 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. Expand the field in longitudinal modes, carry the beat terms through the detector bandwidth, and use the resonator mode condition \(2nL\nu/c=m\). Connect cross section and lifetime by equating the Einstein-coefficient forms of spontaneous and stimulated rates. Mode spacings must approach \(c/(2nL)\) without pulling, and any stabilization discriminator must change sign across its lock point.

Problem 6.12 — laser modes, stabilization, and transition cross sections: calculation

Convert the supplied data to one unit system, isolate the requested quantity symbolically, and retain guard digits until the final numerical evaluation. Expand the field in longitudinal modes, carry the beat terms through the detector bandwidth, and use the resonator mode condition \(2nL\nu/c=m\). Connect cross section and lifetime by equating the Einstein-coefficient forms of spontaneous and stimulated rates. Mode spacings must approach \(c/(2nL)\) without pulling, and any stabilization discriminator must change sign across its lock point.

Problem 6.13 — laser modes, stabilization, and transition cross sections: 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. Expand the field in longitudinal modes, carry the beat terms through the detector bandwidth, and use the resonator mode condition \(2nL\nu/c=m\). Connect cross section and lifetime by equating the Einstein-coefficient forms of spontaneous and stimulated rates. Mode spacings must approach \(c/(2nL)\) without pulling, and any stabilization discriminator must change sign across its lock point.

Problem 6.14 — mode locking and pulse shaping: 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 finite complex mode series, or insert the trial envelope into the stated master equation and equate independent powers. Obtain pulse duration from the half-maximum condition and grating dispersion by differentiating optical path with respect to wavelength. Random phases should suppress a repeatable sharp pulse, equal phases should produce periodic peaks, and the time-bandwidth or dispersion sign must agree with the chosen Fourier convention.

Problem 6.15 — mode locking and pulse shaping: 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 finite complex mode series, or insert the trial envelope into the stated master equation and equate independent powers. Obtain pulse duration from the half-maximum condition and grating dispersion by differentiating optical path with respect to wavelength. Random phases should suppress a repeatable sharp pulse, equal phases should produce periodic peaks, and the time-bandwidth or dispersion sign must agree with the chosen Fourier convention.

Problem 6.16 — mode locking and pulse shaping: 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 finite complex mode series, or insert the trial envelope into the stated master equation and equate independent powers. Obtain pulse duration from the half-maximum condition and grating dispersion by differentiating optical path with respect to wavelength. Random phases should suppress a repeatable sharp pulse, equal phases should produce periodic peaks, and the time-bandwidth or dispersion sign must agree with the chosen Fourier convention.

Problem 6.17 — mode locking and pulse shaping: 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 finite complex mode series, or insert the trial envelope into the stated master equation and equate independent powers. Obtain pulse duration from the half-maximum condition and grating dispersion by differentiating optical path with respect to wavelength. Random phases should suppress a repeatable sharp pulse, equal phases should produce periodic peaks, and the time-bandwidth or dispersion sign must agree with the chosen Fourier convention.

Problem 6.18 — mode locking and pulse shaping: 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 finite complex mode series, or insert the trial envelope into the stated master equation and equate independent powers. Obtain pulse duration from the half-maximum condition and grating dispersion by differentiating optical path with respect to wavelength. Random phases should suppress a repeatable sharp pulse, equal phases should produce periodic peaks, and the time-bandwidth or dispersion sign must agree with the chosen Fourier convention.

Problem 6.19 — mode locking and pulse shaping: 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 finite complex mode series, or insert the trial envelope into the stated master equation and equate independent powers. Obtain pulse duration from the half-maximum condition and grating dispersion by differentiating optical path with respect to wavelength. Random phases should suppress a repeatable sharp pulse, equal phases should produce periodic peaks, and the time-bandwidth or dispersion sign must agree with the chosen Fourier convention.