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
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.
At zero temperature, 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
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.
Replace electron and hole Fermi factors by Boltzmann tails. 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
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.
Differentiation gives \(h\nu_p=E_g+kT/2\). Solve the two half-maximum roots of \(\sqrt{x}e^{-x/kT}\); their difference is proportional to \(kT\), 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
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.
Escape half-angle is \(\sin^{-1}(1/n)\) and isotropic one-face fraction \([1-\cos\theta_c]/2\). Critical angles for GaAs, GaN, polymer are \(16.13^\circ,23.58^\circ,41.81^\circ\); fractions are 1.99%, 4.17%, and 12.73% before Fresnel loss. 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
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 both half-height wavelengths for each of the seven curves, then convert with \(\Delta\nu\simeq c\Delta\lambda/\lambda_0^2\) and \(\Delta E=h\Delta\nu\). Compare to Exercise 17.1-3; the 0.53-micrometre curve’s excess in quadrature/width over the thermal prediction is alloy broadening. 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
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
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.
The internal acceptance satisfies \(n_{LED}\sin\theta_a=\mathrm{NA}\). 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
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.
At each injected density read the two zero-gain frequencies from Fig. 17.2-3, subtract them, and least-squares fit \(B=a\Delta n+b\). 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
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.
Gain is positive between \(E_g\) and \(E_{Fc}-E_{Fv}\) and peaks at the quasi-Fermi separation. 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
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.
For each \(\Delta n\), solve charge neutrality for quasi-Fermi levels; evaluate joint DOS times \(f_c-f_v\) over photon energy at 0 K and 300 K. Extract peak, zero crossings, transparency density, and widths, then compare with Fig. P17.2-3. 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
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.
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. Add that result to nominal \(E_g\); it should coincide, within graph uncertainty, 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
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.
Steady excess density is \(\Delta n=(I/eV)\tau\), with active volume \(dwl\). Use zero-K quasi-Fermi levels to get gain window and peak gain; total gain is \(e^{\gamma_pd}\). 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
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.
Define \(\sigma(\nu)=\gamma(\nu)/(N_2-N_1)\) using the calculated same-k occupation difference. It changes strongly with carrier density and photon energy because both DOS and Fermi levels change; unlike discrete-ion amplifiers, 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
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.
The Airy peak-to-valley ratio for equal facets is \([(1+R)/(1-R)]^2\). 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
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.
In Eq. (17.3-10), index appears in internal optical mode density, photon velocity \(c/n\), Fresnel/escape efficiency, and the conversion between internal and external solid angles. 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
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.
Available gain energy is \(0.96-0.91=0.05\) eV, so \(B=0.05/h\). 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
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.
Facet reflectance is \(R=[(3.5-1)/(3.5+1)]^2=0.3086\). 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
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.
The wavelength spacing is \(\Delta\lambda=\lambda^2/[2d(n-\lambda,dn/d\lambda)]\), involving group index rather than phase index. Insert 0.12 nm to solve \(n_g=\lambda_c^2/(2d\Delta\lambda)\) and then \(a=(n_g-n_0)/\lambda_c\). 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.