Chapter 17: Semiconductor Photon Sources
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 17.
In-text exercises
Exercise 17.1-1 — Pumped quasi-Fermi levels
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 100 — Exercise 17.1-1: Pumped quasi-Fermi levels. 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. At zero temperature,
Detailed step 2. state counting gives \(E_{Fc}-E_c=\hbar^2(3\pi^2\Delta n)^{2/3}/(2m_e)\) and \(E_v-E_{Fv}=\hbar^2(3\pi^2\Delta p)^{2/3}/(2m_h)\); charge-pair injection sets \(\Delta n=\Delta p\).
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 17.1-2 — Weak-injection spectrum
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
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 101 — Exercise 17.1-2: Weak-injection spectrum. 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.
Detailed step 1. Replace electron and hole Fermi factors by Boltzmann tails.
Detailed step 2. Their product is \(e^{-(h\nu-E_g)/kT}\); multiplying the direct-gap joint DOS yields \(\boxed{r_{sp}\propto\sqrt{h\nu-E_g}, e^{-(h\nu-E_g)/kT}}\).
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Exercise 17.1-3 — LED peak and width
Brief solution
1. Method. The working uses stationary-value condition, product, quotient, and chain rules, and exponential, logarithmic, and phasor identities.
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 102 — Exercise 17.1-3: LED peak and width. 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, product, quotient, and chain rules, 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.
Detailed step 1. Differentiation gives \(h\nu_p=E_g+kT/2\).
Detailed step 2. Solve the two half-maximum roots of \(\sqrt{x}e^{-x/kT}\); their difference is proportional to \(kT\),
Detailed step 3. and wavelength width is \(\Delta\lambda\simeq(\lambda_p^2/hc)\Delta E\).
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 17.1-4 — Planar LED extraction
Brief solution
1. Method. The working uses trigonometric and small-angle identities.
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 103 — Exercise 17.1-4: Planar LED extraction. 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 trigonometric and small-angle identities.
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. Escape half-angle is \(\sin^{-1}(1/n)\) and isotropic one-face fraction \([1-\cos\theta_c]/2\).
Detailed step 2. Critical angles for GaAs,
Detailed step 3. GaN,
Detailed step 4. polymer are \(16.13^\circ,23.58^\circ,41.81^\circ\); fractions are 1.99%, 4.17%,
Detailed step 5. and 12.73% before Fresnel loss.
Detailed step 6. An index-matched hemispherical dome removes the planar TIR restriction for rays reaching it normally.
End-of-chapter problems
Step 4 — Interpret the result. The final relation or conclusion in Step 3 is the requested result. Read its sign, scale, or physical classification using the conventions fixed in Step 1.
Step 5 — Check. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model.
Problem 17.1-5 — LED widths from plots
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 both half-height wavelengths for each of the seven curves,
Detailed step 2. then convert with \(\Delta\nu\simeq c\Delta\lambda/\lambda_0^2\) and \(\Delta E=h\Delta\nu\).
Detailed step 3. Compare to Exercise 17.1-3; the 0.53-micrometre curve’s excess in quadrature/width over the thermal prediction is alloy broadening.
Detailed step 4. Preserve graph-read uncertainty in the table.
Numbered result. The principal result obtained in the working is
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.
Problem 17.1-6 — Fresnel-corrected extraction
Brief solution
1. Method. The working uses integration identities, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.
2. Key step.
Integrate unpolarized transmission over the internal escape cone: \(\boxed{\eta_e=\tfrac12\int_0^{\theta_c} [T_s(\theta)+T_p(\theta)]\sin\theta,d\theta}\) with the intensity Fresnel coefficients including the refractive-index flux factor.
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 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.
Integrate unpolarized transmission over the internal escape cone: \(\boxed{\eta_e=\tfrac12\int_0^{\theta_c} [T_s(\theta)+T_p(\theta)]\sin\theta,d\theta}\) with the intensity Fresnel coefficients including the refractive-index flux factor.
Numbered result. The principal result obtained in the working is
Check. Equation (5) 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 17.1-7 — LED-to-fibre coupling
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. The internal acceptance satisfies \(n_{LED}\sin\theta_a=\mathrm{NA}\).
Detailed step 2. Normalize the \(\cos^4\theta\) emission over a hemisphere; integration gives \(\boxed{\eta=1-\cos^5[\sin^{-1}(0.1/3.6)]\simeq1.93\times10^{-3}}\) before interface Fresnel loss.
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. 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 17.2-1 — SOA bandwidth graph
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. At each injected density read the two zero-gain frequencies from Fig.
Detailed step 2. 17.2-3,
Detailed step 3. subtract them,
Detailed step 4. and least-squares fit \(B=a\Delta n+b\).
Detailed step 5. Pair each width with the graph’s peak gain to plot gain versus bandwidth; quote pixel/line reading uncertainty rather than fabricated precision.
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 17.2-2 — Zero-temperature SOA peak
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. Gain is positive between \(E_g\) and \(E_{Fc}-E_{Fv}\) and peaks at the quasi-Fermi separation.
Detailed step 2. Substitute the zero-temperature density expressions from Exercise 17.1-1 into the direct-gap gain formula to obtain \(\gamma_p(\Delta n)\); evaluating the supplied InGaAsP masses/lifetime produces the requested density plot.
Check. For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample.
Problem 17.2-3 — GaAs gain program
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. For each \(\Delta n\),
Detailed step 2. solve charge neutrality for quasi-Fermi levels; evaluate joint DOS times \(f_c-f_v\) over photon energy at 0 K and 300 K.
Detailed step 3. Extract peak,
Detailed step 4. zero crossings,
Detailed step 5. transparency density,
Detailed step 6. and widths,
Detailed step 7. then compare with Fig.
Detailed step 8. P17.2-3.
Detailed step 9. This algorithm covers all six requested plots and makes temperature broadening explicit.
Check. For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample.
Problem 17.2-4 — Band-tail gap reduction
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. Insert p-type \(n\ll p\) or intrinsic injection \(n=p=\Delta n\) into the empirical equation and solve \(\Delta E_g=-0.02\) eV for concentration.
Detailed step 2. Add that result to nominal \(E_g\); it should coincide,
Detailed step 3. within graph uncertainty,
Detailed step 4. with the low-energy zero of the measured gain curve.
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.
Problem 17.2-5 — GaAs amplifier capacity
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. Steady excess density is \(\Delta n=(I/eV)\tau\),
Detailed step 2. with active volume \(dwl\).
Detailed step 3. Use zero-K quasi-Fermi levels to get gain window and peak gain; total gain is \(e^{\gamma_pd}\).
Detailed step 4. Channel count is \(\lfloor B/4\ \mathrm{kHz}\rfloor\) and bit rate is that count times 64 kbit/s.
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. 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 17.2-6 — Semiconductor transition cross section
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. Define \(\sigma(\nu)=\gamma(\nu)/(N_2-N_1)\) using the calculated same-k occupation difference.
Detailed step 2. It changes strongly with carrier density and photon energy because both DOS and Fermi levels change; unlike discrete-ion amplifiers,
Detailed step 3. no material-only cross section conveniently describes an SOA.
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.
Problem 17.2-7 — Residual facet ripple
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. The Airy peak-to-valley ratio for equal facets is \([(1+R)/(1-R)]^2\).
Detailed step 2. Requiring it below 1.10 gives \(\boxed{R<[\sqrt{1.10}-1]/[\sqrt{1.10}+1]\simeq0.0238}\); gain inside the chip tightens this bound through effective round-trip reflectance.
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 17.3-1 — Index dependence of LED output
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. In Eq. (17.3-10),
Detailed step 2. index appears in internal optical mode density,
Detailed step 3. photon velocity \(c/n\),
Detailed step 4. Fresnel/escape efficiency,
Detailed step 5. and the conversion between internal and external solid angles.
Detailed step 6. Mark these factors before simplifying; carrier recombination rate itself is not independently index-free because the radiative coefficient also contains photonic DOS.
Check. For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample.
Problem 17.3-2 — Number of longitudinal laser modes
Brief solution
1. Method. The working uses stationary-value condition 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 stationary-value condition 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. Available gain energy is \(0.96-0.91=0.05\) eV,
Detailed step 2. so \(B=0.05/h\).
Detailed step 3. Cavity spacing is \(c/(2nd)\); therefore maximum count is \(\boxed{1+\lfloor B/[c/(2nd)]\rfloor}\) (apply edge conventions to a mode exactly at a zero-gain endpoint).
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 17.3-3 — Cleaved-facet threshold
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. Facet reflectance is \(R=[(3.5-1)/(3.5+1)]^2=0.3086\).
Detailed step 2. With identical facets and \(d=0.05\) cm, \(\boxed{\gamma_t=-\ln R/d=23.52\ \mathrm{cm^{-1}}}\).
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 17.3-4 — Dispersive mode spacing
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. The wavelength spacing is \(\Delta\lambda=\lambda^2/[2d(n-\lambda,dn/d\lambda)]\),
Detailed step 2. involving group index rather than phase index.
Detailed step 3. Insert 0.12 nm to solve \(n_g=\lambda_c^2/(2d\Delta\lambda)\) and then \(a=(n_g-n_0)/\lambda_c\).
Detailed step 4. Gas-laser mode pulling is gain-dispersion shifting a cavity resonance; here ordinary semiconductor material dispersion sets the baseline spacing by the same group-delay principle.
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.