Chapter 15: Lasers
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 15.
In-text exercises
Exercise 15.1-1 — Ruby threshold
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 88 — Exercise 15.1-1: Ruby threshold. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Thermal absorption gives \(\sigma_0=0.2/(1.58\times10^{19})= \boxed{1.27\times10^{-20}\ \mathrm{cm^2}}\). Mirror loss is \(\alpha_r=-\ln(R_1R_2)/(2d)\); threshold inversion is \(\boxed{N_t=\alpha_r/\sigma_0}\) and threshold excited population follows from \(N=N_2-N_1=2N_2-N_a\) for this three-level transition.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (1) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 15.2-1 — Gas-laser oscillation band
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 89 — Exercise 15.2-1: Gas-laser oscillation band. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses integration identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Set Gaussian gain equal to loss at both band edges: \(\boxed{B=\Delta\nu_D \sqrt{\ln(\gamma_0/\alpha_r)/\ln2}}\). Divide by cavity FSR \(c/(2nd)\) and count the integer modes whose frequencies lie inside this band; the given He–Ne parameters supply the requested count.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (2) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Exercise 15.4-1 — Four-level population equation
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 90 — Exercise 15.4-1: Four-level population equation. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Fast emptying makes \(N_1\simeq0\), so one stimulated transition changes \(N=N_2-N_1\simeq N_2\) by one atom, not two. Thus \(\dot N=(N_0-N)/t_{sp}-\sigma c nN\), without the two-level factor 2.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (3) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Exercise 15.4-2 — Q-switched ruby pulse
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 91 — Exercise 15.4-2: Q-switched ruby pulse. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses integration identities and algebraic rearrangement and dimensional checks.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
For \(N_i/N_t=6\), read the normalized peak, duration, and extracted population from Fig. 15.4-8; dimensionalize time by \(t_p\), photon density by \(N_t\), power by \(h\nu V/t_p\), and energy by \(h\nu V(N_i-N_f)/2\). Recording the graph-read coordinates avoids false precision from the scanned plot.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (4) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 15.4-3 — Mode-locking computation
Step 1 — Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Figure 92 — Exercise 15.4-3: Mode-locking computation. The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself.
Step 2 — Mathematical formulas used. The working uses expectation, variance, and probability identities, integration identities, and exponential, logarithmic, and phasor identities.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Evaluate \(A(t)=\sum_{q=-5}^{5}A_qe^{j2\pi q\nu_Ft}\) on one period. Equal phases give the squared Dirichlet kernel; Gaussian magnitudes give a Gaussian-like pulse; random phases give irregular low-contrast fluctuations. Normalize each case by \(\sum|A_q|^2\) for a fair power comparison.
End-of-chapter problems
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (5) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. For a probability result, verify the zero-to-one bounds; when a full distribution is present, also verify normalization and nonnegative variance.
Problem 15.2-2 — Argon longitudinal modes
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
\(\nu_F=c/(2d)=149.9\) MHz. Half-peak loss permits the Doppler FWHM \(B=3.5\) GHz, about \(\boxed{23}\) longitudinal spacings. Single mode requires \(c/(2d)>B\), so \(d<4.28\) cm for Ar+ and \(d<2.50\) m for the 60-MHz CO2 line.
Numbered result. The principal result obtained in the working is
Check. Equation (6) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.2-3 — Length range for one/two modes
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Compute \(\alpha_r=-\ln(0.97)/(2d)\), then \(B(d)=\Delta\nu_D\sqrt{\ln(\gamma_0/\alpha_r)/\ln2}\). The required lengths satisfy \(1\leq B/[c/(2d)]<3\); solve the two equality boundaries numerically and exclude lengths for which \(\gamma_0\leq\alpha_r\).
Numbered result. The principal result obtained in the working is
Check. Equation (7) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 15.2-4 — Etalon selector
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses Fourier-transform and convolution identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Choose etalon FSR \(c/(2d_e)>1.5\) GHz (for example \(d_e<10\) cm) and linewidth \(\nu_{F,e}/\mathcal F\) narrower than the laser’s 300-MHz longitudinal spacing; \(d_e=5\) cm and \(\mathcal F>10\) is a workable pair. Intracavity placement suppresses unwanted modes before gain saturation and is therefore superior.
Numbered result. The principal result obtained in the working is
Check. Equation (8) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.2-5 — Multimode He–Ne power
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses integration identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
At gain/loss ratio two, the permitted Gaussian band is one FWHM; FSR is \(c/(0.6)=499.7\) MHz, giving roughly three modes. Centering one mode makes its saturated gain competition strongest; a first equal-sharing estimate is \(\boxed{50/3\simeq16.7\ \mathrm{mW}}\), refined by weighting each mode with its Gaussian excess gain.
Numbered result. The principal result obtained in the working is
Check. Equation (9) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.2-6 — Single-mode output
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
\(\alpha_{m1}=-\ln0.99/(2d)\), \(\alpha_{m2}=0\), and \(\alpha_r=\alpha_{m1}\). Then \(t_p=n/(c\alpha_r)\) and intracavity steady flux is \(\phi_s(\gamma_0/\alpha_r-1)\); multiply by output transmission, photon energy, and \(1\ \mathrm{mm^2}\) area for \(P_o\).
Numbered result. The principal result obtained in the working is
Check. Equation (10) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.2-7 — Reading a passive cavity trace
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Read peak spacing \(\nu_F\) and FWHM \(\delta\nu\) from the supplied plot. Then \(d=c/(2\nu_F)\), \(t_p=1/(2\pi\delta\nu)\), and \(\gamma_t=1/(ct_p)\). Subthreshold pumping narrows and raises peaks symmetrically around \(5\times10^{14}\) Hz but cannot make their width zero.
Numbered result. The principal result obtained in the working is
Check. Equation (11) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 15.2-8 — Four-level rate equations
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Write level equations with pump relaxation and stimulated term \(\sigma cn(N_2-N_1)\); subtraction gives \(\dot N=(N_0-N)/T_s-\sigma cnN\) and \(\dot n=\sigma cnN-n/t_p+N_2/t_{sp}\). Above threshold, \(N\simeq N_t=1/(\sigma ct_p)\) and the excess pump fixes steady \(n\).
Numbered result. The principal result obtained in the working is
Check. Equation (12) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 15.3-1 — Yb:YAG design
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Convert the pump energy by \(\lambda=hc/E\) and its endpoint energies to obtain band width in nanometres. Thermal populations give \(\alpha=-\sigma_0N\); the 6-cm cavity has \(\alpha_r=-\ln0.8/(12\ \mathrm{cm})\), \(t_p=n/(c\alpha_r)\), and \(N_t=\alpha_r/\sigma_0\). A small pump- laser energy defect improves quantum efficiency; YVO4 changes host index, cross section, lifetime, thermal conductivity, and hence threshold/output.
Numbered result. The principal result obtained in the working is
Check. Equation (13) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.3-2 — Ar+ threshold
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
\(t_p=[c(-\ln0.98/2d)]^{-1}\). Convert 0.003-nm linewidth to frequency, use lifetime/lineshape to find \(\sigma_0\), then \(\boxed{N_t=1/(\sigma_0ct_p)}\); the mode diameter affects total excited ion number, not density threshold.
Numbered result. The principal result obtained in the working is
Check. Equation (14) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.3-3 — EUV spontaneous lifetime
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
For equal line strength, Einstein \(A\propto\nu^3\), so \(t_{sp}\propto\lambda^3\). Thus \(\boxed{t_{EUV}=10\ \mathrm{ns}(18.2/500)^3=0.482\ \mathrm{ps}}\), of the same scale as the tabulated value.
Numbered result. The principal result obtained in the working is
Check. Equation (15) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.4-4 — Gain-switch transients
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses integration identities and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
The stated substitutions directly produce \(X'=-X+XY\), \(Y'=a(Y_0-Y)-2XY\). Integrate with an adaptive ODE solver for the three \(a\) values and define switching time by 10–90% photon density. The \(10^{-5}\) seed represents spontaneous emission; small \(a\) gives relaxation spiking, large \(a\) follows pump rapidly.
Numbered result. The principal result obtained in the working is
Check. Equation (16) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 15.4-5 — Q-switched ruby energetics
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses integration identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Maintenance pump is \(N_2V(hc/450\mathrm{nm})/t_{sp}\); spontaneous power is \(N_2V(hc/694.3\mathrm{nm})/t_{sp}\). Compute \(N_t=-\ln(R_1R_2)/(2d_r\sigma)\) and use Q-switch invariants to solve \(N_f-N_t\ln N_f=N_i-N_t\ln N_i\); extracted pulse energy is \(h\nu V(N_i-N_f)/2\), with peak/duration from the normalized rate solution.
Numbered result. The principal result obtained in the working is
Check. Equation (17) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 15.4-6 — Cavity dumping timeline
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses the physical definitions stated in the item; no separate calculus or algebraic identity is required for this qualitative comparison.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
During high-Q storage, threshold is low, \(N\) clamps, internal photons rise, and external output is small. Opening the dump raises threshold abruptly, empties internal photons as one large external pulse, and lets \(N\) recover under pumping; repeat this four-trace sequence for cycle two.
Check. For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample.
Problem 15.4-7 — Lorentzian mode locking
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses stationary-value condition, Fourier-transform and convolution identities, and exponential, logarithmic, and phasor identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
The Fourier series of Lorentzian modal amplitudes is a periodic exponential pulse. Parseval gives mean power \(\sum|A_q|^2\); coherent summation gives peak \(|\sum A_q|^2\); solving the exponential intensity at half maximum gives FWHM proportional to \(1/\Delta\nu\). Evaluating the standard Lorentzian sums supplies the book’s closed constants.
Check. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors.
Problem 15.4-8 — Intracavity second harmonic
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Two fundamental photons make one harmonic: \(\dot n=\epsilon n-n/t_p-2\zeta n^2\) and \(\dot n_2=\zeta n^2-n_2/t_{p2}\). The nonzero steady solution is \(\boxed{n=(\epsilon-1/t_p)/(2\zeta)}\) and \(\boxed{n_2=\zeta t_{p2}n^2}\) above threshold.
Numbered result. The principal result obtained in the working is
Check. Equation (18) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.