Acoustics, optics & radiative materials

Five independent physics-formula modules for sound and light — decibel levels and room acoustics, medical/industrial ultrasound imaging physics, blackbody and photon radiometry, fiber/optical photonic

Five small, stateless modules for the closed-form physics equations behind sound and light measurement: decibel arithmetic and room acoustics (acoustics), medical and industrial ultrasound imaging physics (ultrasonics), blackbody radiation and photon counting (radiometry), fiber-optic and diffraction physics (photonics), and geometric-optics lens, refraction and reflection formulas (optics). None of these are simulators — each function is a single textbook formula (Snell's law, the Stefan-Boltzmann law, the Sabine reverberation equation, the sonar/medical-ultrasound equations) evaluated on scalar Float64 inputs. There is no shared struct or engine state between them; every module reads its own docstring header as its complete public API.

from algenta import MojoRuntime

runtime = MojoRuntime(mode="local")

# Reflection coefficient at a soft-tissue/bone interface (typical values in Rayl)
runtime.execute("ultrasonics", "reflection_coefficient", [1.63e6, 7.8e6])["result"]
# 0.4281016903027161 — roughly 43% of incident intensity reflects back toward the transducer,
# which is why bone shows up bright and casts an acoustic shadow on B-mode images

runtime.execute("optics", "thin_lens_equation", [10.0, 30.0])["result"]
# 15.0 — object at 3x the focal length forms a real image at 1.5x the focal length

The modules

Module	What it does
acoustics	Sound-pressure/intensity/power decibel levels, the inverse-square law, pipe resonance, wavelength, Doppler shift, Sabine reverberation time, and combining two decibel sources.
ultrasonics	Medical/industrial ultrasound physics: acoustic impedance and interface reflection, tissue attenuation, Doppler blood-flow shift, imaging depth/PRF trade-offs, axial/lateral resolution, and a simplified thermal safety index.
radiometry	Blackbody and photon-counting radiometry: Planck spectral radiance, Stefan-Boltzmann exitance, photon energy/flux, solid angle, Lambertian irradiance, luminous efficacy, and detector noise-equivalent power.
photonics	Fiber-optic and wave-optics formulas in radians and meters: Snell's law, critical/Brewster angle, Fresnel reflectance, fiber numerical aperture, photon energy, Beer-Lambert absorption, diffraction-grating and Rayleigh-resolution angles.
optics	Geometric-optics formulas in degrees and nanometers: Snell's law (with a total-internal-reflection guard), critical/Brewster angle, thin-lens imaging, magnification, diffraction-limited resolution in arcseconds, and objective-lens numerical aperture.

photonics and optics are not duplicates — they disagree on units

Both modules implement Snell's law, the critical angle, the Brewster angle, Fresnel reflectance at normal incidence, numerical aperture, and photon energy, and five of those pairs even share a function name. Reading mojo/engine/photonics.mojo and mojo/engine/optics.mojo side by side shows they are two independent implementations with incompatible calling conventions, not one formula called twice:

Function (shared name)	photonics	optics
Angle units	radians in, radians out	degrees in, degrees out
Wavelength units	meters	nanometers
numerical_aperture(a, b)	fiber NA: sqrt(n_core² - n_clad²) from core/cladding index	objective NA: n·sin(half_angle_deg) from medium index and half-angle
critical_angle(n1, n2) beyond validity (n2 ≥ n1)	no guard — passes asin(n2/n1) straight through with an out-of-domain ratio	explicit guard — returns -1.0
snells_law* at angles past total internal reflection	no guard — same out-of-domain asin exposure	explicit guard — returns -1.0

Calling photonics.numerical_aperture(1.5, 0.85) expecting the objective-lens formula computes sqrt(1.5² - 0.85²) = 1.236 — a fiber NA, not the optics.numerical_aperture(1.5, 0.85) result of 1.5·sin(0.85°) ≈ 0.0222. The two modules exist because they were built for different domains (fiber/waveguide photonics vs. classical lens design) and neither refers to the other's constants or helpers — photonics uses PLANCK = 6.626e-34 and SPEED_OF_LIGHT = 3.0e8 as module-level comptime values, optics redeclares the same two constants plus EV_JOULE independently.

photonics.critical_angle and photonics.snells_law_angle have no domain guard. Both compute an asin() of a ratio that can legitimately exceed 1.0 in magnitude — critical_angle(n1, n2) whenever n2 ≥ n1 (there is no total internal reflection going from a less-dense to a more-dense medium), and snells_law_angle whenever the incidence angle is past the critical angle for the given index pair. optics.critical_angle and optics.snells_law both check for exactly this case and return the sentinel -1.0 instead. Code calling the photonics versions with medium indices it does not control should check n1 > n2 (and, for snells_law_angle, that the input angle is below photonics.critical_angle(n1, n2)) itself before calling.

Worked examples

Every number below was computed independently in Python from the real formulas in mojo/engine/acoustics.mojo, ultrasonics.mojo, radiometry.mojo, photonics.mojo, and optics.mojo — see Verification for why, and reproduce any of them with the same formula.

Acoustics — combining two equal noise sources. decibel_sum_2 adds levels in the power domain, not linearly:

runtime.execute("acoustics", "decibel_sum_2", [70.0, 70.0])["result"]
# 73.01029995663981 — two 70 dB sources combine to ~73 dB, not 140 dB;
# L = 10*log10(10^7 + 10^7) = 10*log10(2) + 70

resonant_frequency_pipe_open and _closed give the two ends of the same organ-pipe/HVAC-duct problem — a 0.5 m pipe with c = 343 m/s (speed of sound in air) resonates at 343/(2·0.5) = 343.0 Hz open at both ends, but 343/(4·0.5) = 171.5 Hz — exactly one octave lower — with one end closed, because a closed pipe only fits a quarter-wavelength standing wave instead of a half.

Ultrasonics — a soft-tissue/bone reflection and a diagnostic thermal index. Continuing the opening example: acoustic_impedance(1060.0, 1570.0) (liver density and sound speed) gives 1,664,200 Rayl, close to the ~1.63 MRayl and ~7.8 MRayl textbook values for soft tissue and cortical bone used above. At a 5 MHz transducer with 0.72 W of acoustic output and a tissue attenuation coefficient of 0.3 dB/cm/MHz:

runtime.execute("ultrasonics", "thermal_index", [0.72, 5.0, 0.3])["result"]
# 0.005142857142857143 — TIS = P*f*alpha/210, well under the TI <= 1 diagnostic-safety guideline

axial_resolution(1540.0, 2.0e6) (tissue sound speed, 2 MHz pulse bandwidth) gives 0.000385 m — 0.385 mm, the depth-direction resolution limit set by pulse bandwidth, independent of frequency; lateral_resolution at the same 5 MHz center frequency, a 5 cm focal depth and a 1 cm aperture gives wavelength(0.000308 m) * 0.05 / 0.01 = 0.00154 m (1.54 mm) — coarser than axial resolution, which is the reason B-mode ultrasound images are sharper along the beam axis than across it.

Radiometry — a blackbody at skin temperature and the sun. stefan_boltzmann_radiance(0.98, 310.0) (human skin emissivity, body temperature in kelvin) gives 513.1998808832 W/m² of thermal exitance — the physical basis for a thermal (IR) camera reading. planck_spectral_radiance at 0.5 μm (green light) and the sun's 5778 K photosphere temperature gives 82,855,556.4 (W/sr/m²/μm on the module's simplified scale) — the module's own docstring calls this a "simplified form," so treat it as relative spectral shape, not an SI-calibrated radiance.

runtime.execute("radiometry", "photon_energy", [532.0])["result"]
# 2.330820966273361 — a 532 nm (green laser) photon carries about 2.33 eV

runtime.execute("radiometry", "solid_angle_cone", [0.5235987755982988])["result"]
# 0.8417872144769325 — a 30-degree half-angle cone subtends about 0.84 steradian,
# roughly 1/15 of the full 4*pi sr sphere

Photonics vs. optics — the same physical Snell's-law question, two unit systems. Light going from air (n=1.0) into glass (n=1.5) at 30 degrees of incidence:

import math
runtime.execute("optics", "snells_law", [1.5, 30.0, 1.0])["result"]
# 48.59037789072913 — degrees, going glass (1.5) -> air (1.0) at 30 deg incidence

runtime.execute("photonics", "snells_law_angle", [1.0, 1.5, math.radians(45.0)])["result"]
# 0.4908826782893113 rad (28.13 deg) — radians, air (1.0) -> glass (1.5) at 45 deg incidence

fresnel_reflectance_normal(1.0, 1.5) — identical formula and identical 1.0/1.5 inputs in both modules — returns 0.04 either way: 4% of light reflects at a normal-incidence air/glass interface, the textbook value for an uncoated lens surface.

Guards and edge cases

• Every division-prone function in acoustics (sound_pressure_level, _intensity_level, _power_level, inverse_square_law_pressure, wavelength, resonant_frequency_pipe_open/ _closed, doppler_frequency, reverberation_time_sabine) checks its denominator against 1e-30 and returns 0.0 (or, for inverse_square_law_pressure, the unmodified reference SPL) rather than dividing by a near-zero value.

• radiometry.absorptance_from_reflectance clamps its result at a floor of 0.0 — passing reflectance=0.6, transmittance=0.6 (a physically inconsistent pair that sums past 1.0) gives 0.0 rather than the negative value the raw 1 - R - T energy-balance formula would produce. It does not clamp or warn on the upper end, so R = T = 0.0 returns 1.0 (fully absorbing) without checking that assumption either.

• radiometry.planck_spectral_radiance returns 0.0 for non-positive wavelength or temperature, and again if the Planck exponent c2 / (wavelength_um * temperature) exceeds 700.0 — the point past which exp() would overflow Float64 before the division ever happens.

• optics.thin_lens_equation returns a sentinel 1.0e18 ("effectively infinity") instead of dividing by zero when 1/f - 1/do_ lands on exactly 0.0 — the physical case of an object placed exactly at the focal point, which forms an image at infinity.

• ultrasonics.reflection_coefficient returns 0.0 if z1 + z2 <= 0.0 (impedances cannot be negative in the source physics, but the guard fires on a literal zero sum from two zero inputs too), and pulse_repetition_frequency/axial_resolution/lateral_resolution all return 0.0 for a non-positive depth, bandwidth, or aperture rather than dividing by it.

Verification

The five mojo/tests/libs/test_{acoustics,ultrasonics,radiometry,photonics,optics}.mojo files are auto-generated coverage smoke tests, matching the pattern documented on Classical ML algorithms: every function in each module is called exactly once against a shared, arbitrary 0.5-valued fixture, and the only assertion is that the result is finite (not isnan(x) and not isinf(x)) — none of the five files pin a specific numeric output. Every number in this page's worked examples was instead computed independently against the real formulas in mojo/engine/{acoustics,ultrasonics,radiometry,photonics,optics}.mojo and can be reproduced by re-running the same arithmetic in plain Python.

Next: see the engine catalog for how these five modules are exposed as engines, or Classical ML algorithms for the same "auto-generated smoke test, worked examples verified independently" documentation pattern applied to a different module family.