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2.5.10 · D5Optics

Question bank — Huygens' principle — wavefront propagation

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True or false — justify

The ray and the wavefront point in the same direction.
False. The ray is the line along which energy travels, and it is perpendicular to the wavefront — never parallel. Confusing them is the single most common Huygens error.
Every point on a wavefront emits wavelets, so a plane wave must smear out into a blur as it advances.
False. Each point does emit a full sphere of wavelets, but only the forward common tangent survives; sideways and backward contributions are out of phase along the envelope and cancel, so the plane stays a sharp plane.
The backward wavelet is ignored because Huygens simply forgot about it.
False. It is ignored for a physical reason: the obliquity factor equals at (backward), killing the amplitude there. Huygens dropped it by hand, but Fresnel–Kirchhoff Diffraction later justified the drop.
When light enters glass and slows down, its frequency drops.
False. Wavefronts join continuously across the boundary, so the number of crests arriving per second cannot change — frequency is conserved. Only speed and wavelength change: .
A slower wave in glass carries less energy than in air.
False. Speed here is the speed of phase propagation; intensity depends on amplitude, not speed. Slowing down is not the same as losing energy.
Huygens' construction can predict the intensity of the diffraction pattern, not just its shape.
False (for the naïve version). Bare Huygens gives you the geometry of new wavefronts, but you need Fresnel's added ingredients — the obliquity factor and correct phase summation — to get intensities. See Fresnel–Kirchhoff Diffraction.
The secondary wavelets from all points of a plane wavefront have the same radius after time .
True. They share the same medium speed and all started in phase at the same instant, so each radius is . Equal radii on collinear centres is exactly what keeps the tangent flat.
For a plane wave hitting a flat mirror, the angle of incidence equals the angle of reflection because "that's just a law."
Misleading. It is a derived result: makes , forcing . Huygens explains why the law holds. See Laws of Reflection.

Spot the error

"Wavefronts near a point source are plane, and far away they become spherical."
Backwards. Near a point source wavefronts are spherical; far away the curvature is negligible and they look plane. A line source gives cylindrical fronts.
"In Snell's law from Huygens, ."
Sign/ratio flipped. Since (incident side) and (refracted side) over a common hypotenuse , you get . The incident speed sits on top.
"Because , a larger refractive index means a faster wave in that medium."
Inverted relationship. means larger ⇒ smaller . Denser (higher-) media slow light and bend it toward the normal. See Snell's Law and Refractive Index.
"The new wavefront is the tangent to the wavelets and the backward tangent, taken together."
The backward tangent must be discarded. Only the forward envelope is a real wavefront; the backward one has zero amplitude from the obliquity factor.
"Since and is fixed, wavelength stays the same going into glass."
The conclusion contradicts the premise. fixed but smaller means is smaller in glass. Wavelength shrinks in the denser medium.
"A ray is what actually bends; the wavefront just tags along."
The wavefront tilt is the cause. Because the end of the wavefront nearest the surface arrives first, the whole front tilts, and the ray (perpendicular to it) turns as a consequence. See Phase and Path Difference.

Why questions

Why do we track surfaces of equal phase rather than surfaces of equal amplitude?
Because phase is what governs interference — where wavelets add or cancel. Tracking equal-phase surfaces lets us predict bright/dark and bending without solving the wave equation everywhere. See Wave Optics — Interference.
Why does a plane wave stay a plane wave in a uniform medium?
All secondary wavelets have equal radius and their centres lie on a straight line, so their common tangent is another straight line parallel to the original — advancing a distance .
Why does Huygens' principle naturally explain diffraction around an edge?
Points on the wavefront at the edge still emit spherical wavelets, and with no neighbours beyond the edge to cancel them sideways, the wave curls into the geometric shadow. This is exactly why Diffraction happens.
Why must the two slits in Young's experiment be treated as sources in phase?
A single wavefront reaches both slits simultaneously, so each slit becomes a secondary source emitting in step. Equal-phase sources are what produce a stable interference pattern in Young's Double Slit Experiment.
Why is frequency, not wavelength, the quantity conserved across a boundary?
The wavefronts cannot break or pile up at the interface — crests pass through continuously — so the count of crests per second (frequency) is forced to match on both sides.
Why does the obliquity factor take the specific value ?
It must be maximal () straight forward () and vanish () straight backward (); is the simplest smooth function meeting both endpoints, killing the unphysical backward wave.

Edge cases

At exactly normal incidence (), does Huygens' reflection/refraction still work?
Yes, degenerately. With the whole wavefront hits the surface at once, so too and becomes — but the ratio of speeds is still recovered as a limit; the wave reflects/refracts straight back or straight through.
What happens to the refracted wavefront when light goes from a slower to a faster medium at a steep enough angle?
Beyond the critical angle the Huygens construction demands , which is impossible — no forward tangent forms in medium 2, so the wave is totally internally reflected. See Snell's Law and Refractive Index.
What does a wavefront look like in the limit of infinite distance from a point source?
The spherical curvature , so the front becomes an ideal plane wave. This is the limiting case that justifies treating starlight or a distant laser as plane waves.
If in the construction, what happens?
Each wavelet radius , so the new wavefront collapses onto the old one — the construction describes the wavefront's instantaneous advance in the limit, i.e. its velocity field.
What if two points on the wavefront are not in the same phase (a wave passing through a distorted medium)?
Their wavelets start at different times, radii differ, and the common tangent tilts or curves — this is precisely how a lens or a graded medium reshapes a wavefront and bends the beam.
Recall One-line takeaways

Ray ⟂ wavefront, never parallel. ::: Perpendicularity is the geometry that makes tilting explain bending. Only the forward envelope is real. ::: The obliquity factor zeroes the backward wave. Frequency is conserved, wavelength and speed change. ::: Continuous wavefronts forbid crests from piling up or vanishing.

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