Chapter 10: Resonator Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 10.
In-text exercises
Exercise 10.1-1 — Ring and bow-tie resonances
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 62 — Exercise 10.1-1: Ring and bow-tie resonances. 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.
Detailed step 1. Require total propagation plus mirror phase to equal \(2\pi q\).
Detailed step 2. For round-trip optical length \(L_o\), \(\boxed{\nu_q=c(q-N_m/2)/L_o}\) and \(\boxed{\nu_F=c/L_o}\); \(N_m=3,4\) supplies the ring/bow-tie phase offset,
Detailed step 3. which can be absorbed into the integer for four mirrors.
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.
Exercise 10.1-2 — One-metre Fabry–Perot
Brief solution
1. Method. The working uses Fourier-transform and convolution identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 63 — Exercise 10.1-2: One-metre Fabry–Perot. 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 Fourier-transform and convolution 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.
Detailed step 1. \(\nu_F=c/(2d)=149.90\) MHz.
Detailed step 2. With \(\mathcal F=\pi(R_1R_2)^{1/4}/[1-\sqrt{R_1R_2}]=207.69\),
Detailed step 3. the FWHM is \(\boxed{0.7217\ \mathrm{MHz}}\).
Detailed step 4. Loss per round trip is 1.51%,
Detailed step 5. small enough for the high-finesse approximation.
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. 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.
Exercise 10.2-1 — Maximum stable length
Brief solution
1. Method. The working uses stationary-value condition and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 64 — Exercise 10.2-1: Maximum stable length. 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 stationary-value condition 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.
Detailed step 1. Using \(g_i=1-d/|R_i|\),
Detailed step 2. stability requires \(0<g_1g_2<1\).
Detailed step 3. For 0.50 and 1.00 m radii the stable intervals meet at the marginal endpoints; the largest confined length is \(\boxed{d<1.50\ \mathrm m}\).
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 10.2-2 — Plano-concave cavity
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 65 — Exercise 10.2-2: Plano-concave cavity. 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.
Detailed step 1. With \(g_1=1\),
Detailed step 2. stability is \(0<d<|R_2|\).
Detailed step 3. The waist is at the plane mirror, \(z_0=\sqrt{d(|R_2|-d)}\), \(W_0^2=\lambda z_0/(\pi n)\),
Detailed step 4. and \(W_2=W_0\sqrt{1+(d/z_0)^2}\) at the curved mirror.
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Exercise 10.2-3 — Confocal frequency comb
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 66 — Exercise 10.2-3: Confocal frequency comb. 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.
Detailed step 1. For \(d=0.30\) m, \(\nu_F=c/(2d)=499.65\) MHz and each increment of \(l+m+1\) shifts a comb by \(\nu_F/2=249.83\) MHz.
Detailed step 2. The frequencies in the requested band are \(\nu=(q+(l+m+1)/2)\nu_F\) satisfying \(|\nu-5\times10^{14}|\leq2\) GHz; enumerating the corresponding integers produces two interleaved 249.83-MHz combs.
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. 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 10.2-4 — Confocal degeneracy
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 67 — Exercise 10.2-4: Confocal degeneracy. 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.
Detailed step 1. The Gouy phase per half trip is \(\pi/2\) in a confocal cavity,
Detailed step 2. so \(\nu_{qlm}=[q+(l+m+1)/2]\nu_F\).
Detailed step 3. Even transverse-order changes are absorbed into \(q\); odd changes shift the line by \(\nu_F/2\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — 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.
Exercise 10.3-1 — Two-dimensional mode density
Brief solution
1. Method. The working uses product, quotient, and chain rules and algebraic rearrangement and dimensional checks.
Show detailed steps
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 68 — Exercise 10.3-1: Two-dimensional mode density. 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 product, quotient, and chain rules 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.
Detailed step 1. Count lattice points \((q_x,q_y)\) inside a quarter-circle in wavevector space and include two polarizations.
Detailed step 2. Differentiation gives \(\boxed{M_2(\nu)/A=2\pi\nu/c^2}\) modes per area per hertz.
End-of-chapter problems
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — 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 10.1-3 — Resonator with an etalon
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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.
Detailed step 1. Empty-cavity spacing is \(c/(2d)=\boxed{0.9993\ \mathrm{GHz}}\) for \(d=15\) cm.
Detailed step 2. Replacing 2.5 cm of air by index 1.5 raises one-way optical length to 16.25 cm,
Detailed step 3. giving \(\boxed{0.9224\ \mathrm{GHz}}\).
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. 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 10.1-4 — Cleaved semiconductor cavity
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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, 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.
Detailed step 1. \(R=[(3.6-1)/(3.6+1)]^2=0.31947\) and \(\nu_F=c/(2nd)=208.19\) GHz.
Detailed step 2. Combine absorption and mirror loss as \(\alpha_r=\alpha_s-[\ln(R_1R_2)]/(2d)=5.81\times10^3\) \(\mathrm{m^{-1}}\); this gives \(\mathcal F=2.71\),
Detailed step 3. linewidth \(76.9\) GHz,
Detailed step 4. and \(Q=\nu/\delta\nu\simeq2.51\times10^3\) at 1.55 micrometres.
Detailed step 5. The longitudinal order is \(q=2nd/\lambda=\boxed{929}\).
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. 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 10.1-5 — Bragg-mirror etalon
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, matrix multiplication and eigenvalue rules, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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, matrix multiplication and eigenvalue rules, 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.
Detailed step 1. At the Bragg frequency each 10-pair reflector has \(R=[(1-(3.2/3.6)^{20})/(1+(3.2/3.6)^{20})]^2\).
Detailed step 2. Insert this in \(\mathcal F=\pi\sqrt R/(1-R)\) and \(Q=q\mathcal F\); the same characteristic-matrix calculation also includes penetration phase if the GaAs cavity thickness is specified.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Problem 10.1-6 — Measured spectral response
Brief solution
1. Method. The working uses Fourier-transform and convolution identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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.
Detailed step 1. \(d=c/(2\nu_F)=\boxed{0.9993\ \mathrm m}\) and \(\mathcal F=150/5=\boxed{30}\).
Detailed step 2. Solving \(\mathcal F=\pi\sqrt R/(1-R)\) for identical mirrors gives \(\boxed{R=0.90062}\).
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. 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 10.1-7 — Half-energy time
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Key step.
The energy lifetime is \(\tau=\mathcal F/(2\pi\nu_F)\); therefore \(t_{1/2}=\tau\ln2=\mathcal F nd\ln2/(\pi c)= \boxed{36.80\ \mathrm{ns}}\).
3. Answer.
Show detailed steps
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, 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.
The energy lifetime is \(\tau=\mathcal F/(2\pi\nu_F)\); therefore \(t_{1/2}=\tau\ln2=\mathcal F nd\ln2/(\pi c)= \boxed{36.80\ \mathrm{ns}}\).
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. 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 10.2-5 — Convex mirrors
Brief solution
1. Method. The working uses the physical definitions stated in the item; no separate calculus or algebraic identity is required for this qualitative comparison.
Show detailed steps
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.
Detailed step 1. Two convex mirrors have \(g_1,g_2>1\),
Detailed step 2. so their product exceeds one and cannot be stable.
Detailed step 3. One convex and one concave mirror can be stable when their signed \(g\) factors have a product strictly between zero and one.
Check. For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample.
Problem 10.2-6 — Lens inside plane mirrors
Brief solution
1. Method. The working uses matrix multiplication and eigenvalue rules, integration identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 matrix multiplication and eigenvalue rules, 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.
Detailed step 1. The round-trip matrix is \(M=P(d/2)L(f)P(d)L(f)P(d/2)\) (choosing a mirror immediately after the start plane).
Detailed step 2. Multiplication and \(|\operatorname{tr}M/2|<1\) give \(\boxed{0<d/f<4}\).
Detailed step 3. The matched Gaussian has waists symmetrically placed about the lens and wavefronts planar at the mirrors.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Problem 10.2-7 — Ray retracing
Brief solution
1. Method. The working uses trigonometric and small-angle identities and algebraic rearrangement and dimensional checks.
2. Key step.
Here \(g=1-d/|R|=-1/2\) and the round-trip eigenphase satisfies \(\cos\mu=2g^2-1=-1/2\); hence \(\mu=2\pi/3\) and every ray retraces after \(\boxed{3}\) round trips.
3. Answer.
Show detailed steps
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 trigonometric and small-angle 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.
Here \(g=1-d/|R|=-1/2\) and the round-trip eigenphase satisfies \(\cos\mu=2g^2-1=-1/2\); hence \(\mu=2\pi/3\) and every ray retraces after \(\boxed{3}\) round trips.
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. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 10.2-8 — Unstable recurrence
Brief solution
1. Method. The working uses matrix multiplication and eigenvalue rules, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 matrix multiplication and eigenvalue rules, 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.
Detailed step 1. The round-trip characteristic equation has real roots \(h_{1,2}=b\pm\sqrt{b^2-1}\).
Detailed step 2. Diagonalizing the matrix gives \(\boxed{y_m=\alpha_1h_1^m+\alpha_2h_2^m}\); one root has magnitude above one,
Detailed step 3. which is the exponential escape.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Problem 10.2-9 — Symmetric unstable cavity
Brief solution
1. Method. The working uses matrix multiplication and eigenvalue rules and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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 matrix multiplication and eigenvalue rules 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.
Detailed step 1. For \(R=-30\) cm and \(d=65\) cm, \(g=1-d/30=-1.1667\),
Detailed step 2. so \(g^2>1\) is unstable.
Detailed step 3. Apply the explicit round-trip ABCD matrix to \((y_0,\theta_0)=(0,0.1^\circ)\) repeatedly; the first \(m\) with \(|y_m|>2.5\) cm is the aperture escape count.
Detailed step 4. The same recurrence plotted at 50 cm stays bounded while 65 cm grows.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 10.2-10 — Gaussian standing wave
Brief solution
1. Method. The working uses integration identities, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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, trigonometric and small-angle 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.
Detailed step 1. Adding equal counterpropagating Gaussian fields gives the common transverse envelope times \(2\cos[kz-(l+m+1)\psi(z)]\).
Detailed step 2. Requiring nodes/antinodes on both matching mirror wavefronts yields \(2knd-2(l+m+1)\Delta\psi=2\pi q\),
Detailed step 3. the resonance formula (10.2-30).
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. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model.
Problem 10.2-11 — Sixteen-centimetre confocal cavity
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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.
Detailed step 1. \(R_1=R_2=-d=\boxed{-16\ \mathrm{cm}}\), \(z_0=d/2=8\) cm, \(W_0=\sqrt{\lambda z_0/\pi}= \boxed{159.6\ \mathrm{\mu m}}\),
Detailed step 2. and mirror width 225.7 micrometres.
Detailed step 3. HG10 peaks are 319.2 micrometres apart.
Detailed step 4. Frequencies follow the confocal formula above; mirror-only distributed loss is \(-\ln(0.995^2)/(2d)=\boxed{0.0313\ \mathrm{m^{-1}}\).
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 10.2-12 — One-percent diffraction aperture
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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.
Detailed step 1. Read \(N_F\) at 1% loss for the (1,0) curve in Fig.
Detailed step 2. 10.2-11,
Detailed step 3. then use \(\boxed{a=\sqrt{N_F\lambda d}}\) with \(\lambda=1\) micrometre and \(d=0.16\) m.
Detailed step 4. Reporting the graph-read \(N_F\) beside the result keeps the scan-dependent interpolation auditable.
Numbered result. The principal result obtained in the working is
Check. Equation (19) 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 10.3-2 — Counts in 1-D, 2-D, and 3-D
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show detailed steps
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.
Detailed step 1. For 1.06 micrometres, 120 GHz bandwidth,
Detailed step 2. and 10-cm dimensions,
Detailed step 3. the continuum counts (including two polarizations in 2-D/3-D) are \(\boxed{80.1}\), \(\boxed{2.37\times10^7}\),
Detailed step 4. and \(\boxed{8.95\times10^{12}}\),
Detailed step 5. from \(2d\Delta\nu/c\), \(A(2\pi\nu/c^2)\Delta\nu\),
Detailed step 6. and \(V(8\pi\nu^2/c^3)\Delta\nu\),
Detailed step 7. respectively.
Numbered result. The principal result obtained in the working is
Check. Equation (20) 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.