Chapter 7: Photonic-Crystal Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 7. The characteristic-matrix convention is the book’s.
In-text exercise
Exercise 7.1-1 — Quarter-wave antireflection film
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 55 — Exercise 7.1-1: Quarter-wave antireflection film. 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 matrix multiplication and eigenvalue rules, trigonometric and small-angle 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.
Multiplying boundary/propagation matrices makes \(B\propto n_1n_3\sin^2\delta-n_2^2\sin^2\delta\) at \(\delta=\pi/2\). Thus \(B=0\) and \(r=0\) when \(\boxed{d=\lambda_0/(4n_2),\ n_2=\sqrt{n_1n_3}}\).
End-of-chapter problems
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Problem 7.1-2 — Slab beamsplitter
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.
Use the Airy result \(T=[1+4R_{s,p}\sin^2\delta/(1-R_{s,p})^2]^{-1}\), \(R=1-T\), with the TE/TM Fresnel \(R_{s,p}\) at 45 degrees and \(\delta=(2\pi/\lambda_0)nd\cos\theta_t\). This directly supplies the periodic spectral curves; TM contrast vanishes when the internal/external angle is Brewster.
Numbered result. The principal result obtained in the working is
Check. Equation (2) 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 7.1-3 — Air-gap tunnelling
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, 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.
At normal incidence insert \(n_g,1,n_g\) in the slab matrix and \(d=\lambda_0/2\); the round-trip phase gives unity transmission. Above critical angle the gap normal wavevector is \(j\kappa\); replacing \(\sin\delta,j\cos\delta\) by hyperbolic functions gives finite \(T\propto\operatorname{sech}^2(\kappa d)\): frustrated TIR tunnels through a sufficiently thin gap.
Numbered result. The principal result obtained in the working is
Check. Equation (3) 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 7.1-4 — Unmatched incident medium
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.
Composition of the new boundary and old device gives \(\boxed{r=(r_b+r_m)/(1+r_br_m)}\). It reduces respectively to \(r_m,1,r_b,1\) for \(r_b=0,1\) and \(r_m=0,1\).
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 7.1-5 — Oblique quarter-wave coating
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 trigonometric and small-angle identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Replace admittance by \(n\cos\theta\) (TE) or \(n/\cos\theta\) (TM), and phase by \(\delta=2\pi n_2d\cos\theta_2/\lambda_0\); multiplying the one-film matrix gives \(r=(r_{12}+r_{23}e^{-j2\delta})/ (1+r_{12}r_{23}e^{-j2\delta})\). Squaring this expression is the requested angular reflectance.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Problem 7.1-6 — Quarter/half-wave stacks
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.
At the design wavelength a quarter-wave pair has diagonal matrix \(-\operatorname{diag}(n_2/n_1,n_1/n_2)\); after \(N\) pairs the ratio is raised to \(N\), yielding \(r=[n_a(n_2/n_1)^{2N}-n_s]/[n_a(n_2/n_1)^{2N}+n_s]\) up to layer order. A half-wave layer is \(-I\); every pair is transparent at the design wavelength apart from phase, so only the unmatched outer boundary remains.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Problem 7.1-7 — GaAs/AlAs reflector
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
For a GaAs-matched exterior, evaluate the preceding quarter-wave expression with \(n_1=3.57\), \(n_2=2.94\): \(\boxed{R_N=[(1-(n_2/n_1)^{2N})/(1+(n_2/n_1)^{2N})]^2}\) and \(T_N=1-R_N\). Evaluating \(N=1,\ldots,10\) gives the requested monotonic plot.
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 7.1-8 — Matrix-program verification
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, 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.
Initialize \(M=I\); for each layer multiply \(M_i=\begin{bmatrix}\cos\delta_i&j\sin\delta_i/Y_i\\ jY_i\sin\delta_i&\cos\delta_i\end{bmatrix}\). Convert total input admittance to \(r\) and plot \(|r|^2\) versus wavelength or angle. Using the figure’s layer data reproduces its stopband and TE/TM angular splitting.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Problem 7.2-1 — Gap/midgap estimate
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.
Equal optical thickness gives Bragg frequency \(\boxed{\nu_B=c/[2(n_1d_1+n_2d_2)]}\) with \(d_1+d_2=2\ \mathrm{\mu m}\). The first-order relative gap is \(\Delta\nu/\nu_B\simeq(4/\pi)\sin^{-1}|(n_2-n_1)/(n_2+n_1)|\). It is large for 1.5/3.5 and small for 3.4/3.6, demonstrating that index contrast, not mean index, controls the fractional gap.
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. 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 7.2-2 — Off-axis Bloch 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 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.
Keep conserved \(k_x\); in every layer replace \(n_i\omega/c\) by \(k_{zi}=[(n_i\omega/c)^2-k_x^2]^{1/2}\) and admittance by its TE/TM oblique value. The unit-cell trace then gives \(\boxed{\cos(K\Lambda)=\tfrac12\operatorname{tr}M(k_x,\omega)}\); \(|\operatorname{tr}M/2|>1\) is a bandgap.
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. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model.
Problem 7.2-3 — Propagation normal to periodicity
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
For \(K=0\) (wavevector along the layers), tangential phase matching makes the field sample a translationally uniform direction. Substitution in the off-axis dispersion relation leaves real \(k_x\) for every allowed frequency; the Bragg coupling term vanishes, so no axial-period bandgap opens.
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 7.2-4 — Omnidirectional reflector
Definitions and setup. Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution.
Mathematical formulas used. The working uses algebraic rearrangement and dimensional checks.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
For each conserved air \(k_x\leq\omega/c\), evaluate the cell trace with \(n_2=2n_1\) and equal optical thickness. Shade frequencies for which \(|\operatorname{tr}M/2|>1\) for every point inside the air light cone; the intersection of all angular TE/TM stopbands is the omnidirectional range. This construction, rather than a single normal-incidence gap, is the required projected dispersion plot.
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.