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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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. .. _fop-exercise-10-1-1-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_10_01_01.svg :alt: Illustrated calculation map for Exercise 10.1-1, Ring and bow-tie resonances :align: center :width: 95% **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 :ref:`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. Require total propagation plus mirror phase to equal :math:`2\pi q`. For round-trip optical length :math:`L_o`, :math:`\boxed{\nu_q=c(q-N_m/2)/L_o}` and :math:`\boxed{\nu_F=c/L_o}`; :math:`N_m=3,4` supplies the ring/bow-tie phase offset, which can be absorbed into the integer for four mirrors. **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-10-1-1-result \boxed{\nu_F=c/L_o} **Step 5 — Check.** Equation :eq:`fop-exercise-10-1-1-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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. .. _fop-exercise-10-1-2-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_10_01_02.svg :alt: Illustrated calculation map for Exercise 10.1-2, One-metre Fabry–Perot :align: center :width: 95% **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 :ref:`Fourier-transform and convolution identities ` and :ref:`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. :math:`\nu_F=c/(2d)=149.90` MHz. With :math:`\mathcal F=\pi(R_1R_2)^{1/4}/[1-\sqrt{R_1R_2}]=207.69`, the FWHM is :math:`\boxed{0.7217\ \mathrm{MHz}}`. Loss per round trip is 1.51%, small enough for the high-finesse approximation. **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-10-1-2-result \boxed{0.7217\ \mathrm{MHz}} **Step 5 — Check.** Equation :eq:`fop-exercise-10-1-2-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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. .. _fop-exercise-10-2-1-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_10_02_01.svg :alt: Illustrated calculation map for Exercise 10.2-1, Maximum stable length :align: center :width: 95% **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 :ref:`stationary-value condition ` and :ref:`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. Using :math:`g_i=1-d/|R_i|`, stability requires :math:`0`. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. With :math:`g_1=1`, stability is :math:`0`. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. For :math:`d=0.30` m, :math:`\nu_F=c/(2d)=499.65` MHz and each increment of :math:`l+m+1` shifts a comb by :math:`\nu_F/2=249.83` MHz. The frequencies in the requested band are :math:`\nu=(q+(l+m+1)/2)\nu_F` satisfying :math:`|\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 .. math:: :label: fop-exercise-10-2-3-result \nu=(q+(l+m+1)/2)\nu_F **Step 5 — Check.** Equation :eq:`fop-exercise-10-2-3-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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. .. _fop-exercise-10-2-4-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_10_02_04.svg :alt: Illustrated calculation map for Exercise 10.2-4, Confocal degeneracy :align: center :width: 95% **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 :ref:`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. The Gouy phase per half trip is :math:`\pi/2` in a confocal cavity, so :math:`\nu_{qlm}=[q+(l+m+1)/2]\nu_F`. Even transverse-order changes are absorbed into :math:`q`; odd changes shift the line by :math:`\nu_F/2`. **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-10-2-4-result \nu_{qlm}=[q+(l+m+1)/2]\nu_F **Step 5 — Check.** Equation :eq:`fop-exercise-10-2-4-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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. .. _fop-exercise-10-3-1-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_10_03_01.svg :alt: Illustrated calculation map for Exercise 10.3-1, Two-dimensional mode density :align: center :width: 95% **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 :ref:`product, quotient, and chain rules ` and :ref:`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. Count lattice points :math:`(q_x,q_y)` inside a quarter-circle in wavevector space and include two polarizations. Differentiation gives :math:`\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 .. math:: :label: fop-exercise-10-3-1-result \boxed{M_2(\nu)/A=2\pi\nu/c^2} **Step 5 — Check.** Equation :eq:`fop-exercise-10-3-1-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`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. Empty-cavity spacing is :math:`c/(2d)=\boxed{0.9993\ \mathrm{GHz}}` for :math:`d=15` cm. Replacing 2.5 cm of air by index 1.5 raises one-way optical length to 16.25 cm, giving :math:`\boxed{0.9224\ \mathrm{GHz}}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-1-3-result \boxed{0.9224\ \mathrm{GHz}} **Check.** Equation :eq:`fop-problem-10-1-3-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`Fourier-transform and convolution identities `, :ref:`exponential, logarithmic, and phasor identities `, and :ref:`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. :math:`R=[(3.6-1)/(3.6+1)]^2=0.31947` and :math:`\nu_F=c/(2nd)=208.19` GHz. Combine absorption and mirror loss as :math:`\alpha_r=\alpha_s-[\ln(R_1R_2)]/(2d)=5.81\times10^3` :math:`\mathrm{m^{-1}}`; this gives :math:`\mathcal F=2.71`, linewidth :math:`76.9` GHz, and :math:`Q=\nu/\delta\nu\simeq2.51\times10^3` at 1.55 micrometres. The longitudinal order is :math:`q=2nd/\lambda=\boxed{929}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-1-4-result q=2nd/\lambda=\boxed{929} **Check.** Equation :eq:`fop-problem-10-1-4-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`Fourier-transform and convolution identities `, :ref:`matrix multiplication and eigenvalue rules `, and :ref:`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 the Bragg frequency each 10-pair reflector has :math:`R=[(1-(3.2/3.6)^{20})/(1+(3.2/3.6)^{20})]^2`. Insert this in :math:`\mathcal F=\pi\sqrt R/(1-R)` and :math:`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 .. math:: :label: fop-problem-10-1-5-result Q=q\mathcal F **Check.** Equation :eq:`fop-problem-10-1-5-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`Fourier-transform and convolution identities ` and :ref:`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. :math:`d=c/(2\nu_F)=\boxed{0.9993\ \mathrm m}` and :math:`\mathcal F=150/5=\boxed{30}`. Solving :math:`\mathcal F=\pi\sqrt R/(1-R)` for identical mirrors gives :math:`\boxed{R=0.90062}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-1-6-result \boxed{R=0.90062} **Check.** Equation :eq:`fop-problem-10-1-6-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`Fourier-transform and convolution identities `, :ref:`exponential, logarithmic, and phasor identities `, and :ref:`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 :math:`\tau=\mathcal F/(2\pi\nu_F)`; therefore :math:`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 .. math:: :label: fop-problem-10-1-7-result t_{1/2}=\tau\ln2=\mathcal F nd\ln2/(\pi c)= \boxed{36.80\ \mathrm{ns}} **Check.** Equation :eq:`fop-problem-10-1-7-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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. Two convex mirrors have :math:`g_1,g_2>1`, so their product exceeds one and cannot be stable. One convex and one concave mirror can be stable when their signed :math:`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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`matrix multiplication and eigenvalue rules `, :ref:`integration identities `, and :ref:`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 round-trip matrix is :math:`M=P(d/2)L(f)P(d)L(f)P(d/2)` (choosing a mirror immediately after the start plane). Multiplication and :math:`|\operatorname{tr}M/2|<1` give :math:`\boxed{0` and :ref:`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 :math:`g=1-d/|R|=-1/2` and the round-trip eigenphase satisfies :math:`\cos\mu=2g^2-1=-1/2`; hence :math:`\mu=2\pi/3` and every ray retraces after :math:`\boxed{3}` round trips. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-2-7-result \boxed{3} **Check.** Equation :eq:`fop-problem-10-2-7-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`matrix multiplication and eigenvalue rules `, :ref:`exponential, logarithmic, and phasor identities `, and :ref:`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 round-trip characteristic equation has real roots :math:`h_{1,2}=b\pm\sqrt{b^2-1}`. Diagonalizing the matrix gives :math:`\boxed{y_m=\alpha_1h_1^m+\alpha_2h_2^m}`; one root has magnitude above one, which is the exponential escape. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-2-8-result \boxed{y_m=\alpha_1h_1^m+\alpha_2h_2^m} **Check.** Equation :eq:`fop-problem-10-2-8-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`matrix multiplication and eigenvalue rules ` and :ref:`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 :math:`R=-30` cm and :math:`d=65` cm, :math:`g=1-d/30=-1.1667`, so :math:`g^2>1` is unstable. Apply the explicit round-trip ABCD matrix to :math:`(y_0,\theta_0)=(0,0.1^\circ)` repeatedly; the first :math:`m` with :math:`|y_m|>2.5` cm is the aperture escape count. The same recurrence plotted at 50 cm stays bounded while 65 cm grows. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-2-9-result (y_0,\theta_0)=(0,0.1^\circ) **Check.** Equation :eq:`fop-problem-10-2-9-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`integration identities `, :ref:`trigonometric and small-angle identities `, and :ref:`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. Adding equal counterpropagating Gaussian fields gives the common transverse envelope times :math:`2\cos[kz-(l+m+1)\psi(z)]`. Requiring nodes/antinodes on both matching mirror wavefronts yields :math:`2knd-2(l+m+1)\Delta\psi=2\pi q`, the resonance formula (10.2-30). **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-2-10-result 2knd-2(l+m+1)\Delta\psi=2\pi q **Check.** Equation :eq:`fop-problem-10-2-10-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`exponential, logarithmic, and phasor identities ` and :ref:`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. :math:`R_1=R_2=-d=\boxed{-16\ \mathrm{cm}}`, :math:`z_0=d/2=8` cm, :math:`W_0=\sqrt{\lambda z_0/\pi}= \boxed{159.6\ \mathrm{\mu m}}`, and mirror width 225.7 micrometres. HG10 peaks are 319.2 micrometres apart. Frequencies follow the confocal formula above; mirror-only distributed loss is :math:`-\ln(0.995^2)/(2d)=\boxed{0.0313\ \mathrm{m^{-1}}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-2-11-result -\ln(0.995^2)/(2d)=\boxed{0.0313\ \mathrm{m^{-1}} **Check.** Equation :eq:`fop-problem-10-2-11-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`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 :math:`N_F` at 1% loss for the (1,0) curve in Fig. 10.2-11, then use :math:`\boxed{a=\sqrt{N_F\lambda d}}` with :math:`\lambda=1` micrometre and :math:`d=0.16` m. Reporting the graph-read :math:`N_F` beside the result keeps the scan-dependent interpolation auditable. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-2-12-result \boxed{a=\sqrt{N_F\lambda d}} **Check.** Equation :eq:`fop-problem-10-2-12-result` 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **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 :ref:`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 1.06 micrometres, 120 GHz bandwidth, and 10-cm dimensions, the continuum counts (including two polarizations in 2-D/3-D) are :math:`\boxed{80.1}`, :math:`\boxed{2.37\times10^7}`, and :math:`\boxed{8.95\times10^{12}}`, from :math:`2d\Delta\nu/c`, :math:`A(2\pi\nu/c^2)\Delta\nu`, and :math:`V(8\pi\nu^2/c^3)\Delta\nu`, respectively. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-10-3-2-result \boxed{8.95\times10^{12}} **Check.** Equation :eq:`fop-problem-10-3-2-result` 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.