Chapter 22: Nanophotonics
Source: Hunsperger, sixth edition (2009), Problems 22.1–22.7, printed page 497.
Problem 22.1 — Photonic-crystal guiding
Working. A periodic dielectric distribution changes the allowed electromagnetic Bloch modes. In a photonic bandgap, certain frequencies cannot propagate in the surrounding periodic medium. A line defect can create allowed states inside that gap, confining light to the defect.
Answer. Defect-guided propagation can use the contrast between allowed defect states and forbidden bulk states.
Check. Some photonic-crystal structures instead combine index guiding and directional stop bands. Not every periodic structure has a complete bandgap for both polarizations.
Problem 22.2 — VCSEL advantages
Working. A short vertical cavity favors a large longitudinal-mode spacing. A small active volume can lower threshold current, while the surface-normal output permits wafer-level testing and two-dimensional arrays. A near-circular aperture can produce a less asymmetric beam than an edge-emitting stripe.
Answer. Low threshold, array compatibility, accessible testing, and useful beam geometry are principal advantages.
Check. Large-aperture VCSELs can support several transverse modes. Surface emission alone does not guarantee single-mode operation or a universally lower divergence.
Problem 22.3 — Rayleigh resolution and lithographic feature size
Working. With \(D=0.05\) m, \(f=1\) m, and \(\lambda_0=1.55\,\mu{\rm m}\),
The given approximation yields \({\rm NA}=D/(2f)=0.025\). With resist factor 0.80, \(W_{\min}=0.80\lambda_0/{\rm NA}=49.6\,\mu{\rm m}\).
Answer. (a) Angular Rayleigh separation 37.82 microradians, or 37.82 micrometres in a focal-plane interpretation. (b) Lithographic feature size 49.6 micrometres.
Check. The Airy diameter is 75.64 micrometres, not the Rayleigh point separation. A linear object size at an arbitrary distance would additionally require that distance or magnification.
Problem 22.4 — Electron de Broglie wavelength
Working. At \(v=2\times10^7\) m/s the nonrelativistic estimate is
Answer. Approximately 0.0364 nm.
Check. Speed is about 0.0667 times the speed of light. Using relativistic momentum reduces the wavelength by only about 0.22 percent, much less than the supplied velocity’s precision.
Problem 22.5 — Two anisotropic dry-etch routes
Working. In reactive-ion etching, ions accelerated through a plasma sheath give directional bombardment while reactive species remove exposed material chemically. Appropriate sidewall passivation suppresses lateral etching. In ion-beam or chemically assisted ion-beam etching, a collimated ion beam provides the directionality; reactive gas can increase removal rate or selectivity.
Answer. RIE and ion-beam/chemically assisted ion-beam etching are two routes to high-aspect-ratio patterned structures.
Check. Mask selectivity, ion damage, redeposition, and sidewall chemistry still constrain achievable aspect ratio. Directionality alone is not enough for an arbitrarily deep trench.
Problem 22.6 — What a lattice constant does and does not determine
Working. For a square lattice with \(a=660\) nm, plane spacing is \(d_{hk}=a/\sqrt{h^2+k^2}\). A weak-scattering Bragg estimate for incidence normal to those planes is
Thus the axial first Bragg estimate is \(1320n_{\rm eff}\) nm; the diagonal estimate is \(933.38n_{\rm eff}\) nm.
Answer. There is no unique blocked wavelength or complete bandgap interval from lattice constant alone. The dielectric indices, fill factor, lattice geometry, propagation direction, and polarization are needed for a Bloch-mode calculation. For the explicitly assumed \(n_{\rm eff}=1\) axial estimate, the Bragg centre would be 1320 nm.
Check. A uniform material with an arbitrarily drawn 660 nm grid has no photonic bandgap. A periodic optical contrast, not just a spacing, is required.
Problem 22.7 — Nanolayer epitaxy
Working. MBE delivers controlled atomic/molecular beams to a heated crystal under ultrahigh vacuum. MOVPE/MOCVD reacts controlled vapor-phase precursors at the surface. For nanolayers, reduce and calibrate growth rate, control substrate temperature, switch sources/precursors rapidly, and account for shutter transients, gas residence time, segregation, and interface roughness.
Answer. MBE and MOVPE are the two techniques discussed. Use layer-resolved process monitoring and calibrated growth interruptions or pulse times to set nanometre thickness.
Check. Measuring average thickness after a long growth does not establish atomically abrupt interfaces. Strain and composition control remain necessary alongside timing control.