Wavelength Converter

Every particle has an associated quantum wavelength λ = h/p = h/(mv) for non-relativistic particles. For an electron at v = 10⁶ m/s: λ = 6.626 × 10⁻³⁴ / (9.109 × 10⁻³¹ × 10⁶) = 0.727 nm = 7.27 Å — comparable to atomic spacings, explaining electron diffraction in crystals. For a 1 kg ball at 10 m/s: λ = 6.626 × 10⁻³⁵ m — utterly undetectable, explaining why large objects don't exhibit wave behavior.

Bragg's law: nλ = 2d sin(θ), where d is crystal plane spacing, θ is the diffraction angle, λ is the X-ray wavelength, and n is the order. For CuKα (λ = 1.54 Å) diffracted from calcite (d = 3.036 Å): first-order peak at θ = arcsin(λ/2d) = arcsin(1.54/6.07) = 14.7°. X-ray crystallography maps atomic positions by solving the inverse problem: from measured diffraction pattern angles and intensities, determine the 3D arrangement of atoms.

Fiber optic communications use three low-loss windows of silica glass: O-band (1260-1360 nm), E-band (1360-1460 nm), S-band (1460-1530 nm), C-band (1530-1565 nm, lowest attenuation ~0.2 dB/km), and L-band (1565-1625 nm). The C-band is dominant for long-haul transmission. Each DWDM channel is ~0.8 nm wide, with up to 80 channels per fiber. WDM allows ~100 Tbps data rates per fiber.

The CMB is a nearly perfect blackbody spectrum at T = 2.725 K. The peak wavelength by Wien's law: λ_max = b/T = 2.898 × 10⁻³ / 2.725 = 1.063 mm (microwave). The photon energy at the peak is E = hc/λ = 1.166 meV. The CMB fills space isotropically with a photon number density of about 411 photons/cm³ and energy density 4.17 × 10⁻¹⁴ J/m³.

Shorter wavelengths (harder X-rays) penetrate more deeply. The attenuation coefficient μ ∝ λ³ × Z³ (photoelectric absorption, dominant below ~0.1 MeV), where Z is atomic number. Soft X-rays (1-10 nm): absorbed by air in millimeters, used for soft tissue contrast. Hard X-rays (0.01-0.1 nm, 12-120 keV): penetrate centimetres of soft tissue, used in medical imaging and industrial inspection. Mammography uses 17-35 keV (0.04-0.07 nm).

A photonic crystal is a periodic dielectric structure that creates a photonic bandgap — a range of wavelengths that cannot propagate through the structure (analogous to electronic bandgaps in semiconductors). The bandgap center wavelength is roughly twice the lattice period. Structural coloration in butterfly wings, opals, and peacock feathers arises from photonic crystal effects, creating iridescent colors that depend on viewing angle unlike pigment-based colors.

Sound at 440 Hz (concert pitch A) in air at 20°C (speed of sound = 343 m/s): λ = v/f = 343/440 = 0.780 m = 78.0 cm. Note that this is a sound wavelength (pressure wave in air), not electromagnetic. For comparison, radio waves at 440 Hz would have λ = c/f = 3×10⁸/440 = 682 km — far too large for conventional antennas. Extremely low frequency (ELF) radio at 76 Hz (used for submarine communication) has λ = 3947 km.

Laser wavelength is determined by the transition energy of the gain medium. For solid-state lasers: Nd:YAG at 1064 nm (near infrared), frequency-doubled to 532 nm (green); Ti:sapphire tunable 690-1080 nm. Diode lasers: InGaAs 780-1550 nm (telecom, CD, DVD). Gas lasers: HeNe at 632.8 nm (red), ArF at 193 nm (UV, semiconductor lithography), CO₂ at 10.6 μm (IR, cutting). Free electron lasers can tune from microwave to X-ray.

Resolving power R = λ/Δλ = f/Δf. Grating spectrometers: R = mN (m = diffraction order, N = number of grooves illuminated); a 1200 groove/mm grating at first order with 50 mm illuminated width gives R = 1200 × 50 = 60,000, resolving 550/60,000 = 0.009 nm = 9 pm at 550 nm. Fabry-Perot etalons: R up to 10⁶-10⁷. Fourier transform spectrometers (FTIR): R up to 10⁶. Atomic transition natural linewidth: R up to 10¹³.