Huygens's principle and the laws of reflection and refraction
Use Huygens's wavelet construction for plane wavefronts at stationary boundaries between uniform isotropic materials. Equal-time geometry gives reflection and Snell's law for a propagating transmitted ray, using the wave speed in each material.
What a learner can do afterwards
- Constructs a refracted wavefront from wavelets and derives Snell's law
- Relates refractive index to the speed of light in the material
- Explains why the frequency is unchanged on crossing a boundary but the wavelength is not
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You have used rays, normals and n = c/v. A wavefront joins points at the same wave phase. In a uniform isotropic material, rays are perpendicular to it. In Huygens's construction, each front point produces a secondary wavelet. After time t, its radius is v*t. The forward tangent envelope of these wavelets gives the next front.
For oblique incidence, take a plane front AB approaching a flat, stationary boundary. A reaches it first. After time t, B reaches boundary point C. From A, draw a transmitted wavelet of radius v2*t. Draw its tangent from C, touching at D. DC is the refracted front. BC = v1*t is perpendicular to AB; AD = v2*t is perpendicular to DC. This construction requires a propagating transmitted wave.
Let L = AC, the shared boundary segment. Measure ray angles θ1 and θ2 from the normal. Because rays are perpendicular to fronts, these equal the corresponding front-boundary angles. The right triangles give sin(θ1) = BC/L = v1*t/L and sin(θ2) = AD/L = v2*t/L. Here sin(θ) is the opposite side divided by the hypotenuse in a right triangle.
Both triangles use the same time t and boundary length L. Dividing each sine relation by its own speed gives sin(θ1)/v1 = sin(θ2)/v2 = t/L. Since n1 = c/v1 and n2 = c/v2, multiplying by c gives Snell's law: n1*sin(θ1) = n2*sin(θ2). For reflection, both wave speeds are equal, so the equal-time construction gives equal incidence and reflection angles.
The law of reflection uses the local normal: the reflected angle equals the incident angle. Both rays and that normal lie in one plane. On a smooth plane mirror, local normals align, so parallel incoming rays reflect in one direction. Rough reflecting surfaces have varying local normals, so reflected rays leave in different directions.
Refractive index is n = c/v, where c is vacuum light speed and v is light's phase speed in the material. Rearrange to v = c/n. For c = 300 million metres per second and n = 1.25, v = 300/1.25 = 240 million metres per second. Use matching speed units in the ratio.
For monochromatic light transmitted across a stationary linear boundary, cycles match in time at the boundary, so frequency f stays fixed. From v = f*λ, wavelength λ = v/f. Entering ordinary glass from air lowers speed and wavelength. At normal incidence, direction stays unchanged even though speed and wavelength change.
The normal is perpendicular to the boundary. A ray's acute surface angle α and normal angle θ therefore satisfy α + θ = 90 degrees. A surface angle of 20 degrees gives θ = 70 degrees. Use the normal angle in Snell's law. The two readings coincide at 45 degrees, but their reference lines still differ.
Equal-time wavelet geometry gives reflection and Snell's law. Measure angles from the normal. At a stationary linear boundary, frequency stays fixed and wavelength follows speed.
Lesson, questions and flashcards written for this dataset.
Readings and flashcards written for this dataset. Includes links to Khan Academy's "Refraction and Snell's law" and "Specular and diffuse reflection". Video licenses are not specified.
More to read
Constructing a transmitted wavefront
Work in a uniform isotropic material on each side of a flat, stationary boundary. For an oblique plane wavefront, mark A where the front first touches the boundary. Let another point B on that front reach boundary point C after time t. The incident travel distance BC is v1*t.
During that same time, a wavelet from A expands into the second material to radius v2*t. Draw a tangent from C to this wavelet, touching at D. The line DC locates the transmitted front. AD is perpendicular to DC, just as BC is perpendicular to the incident front AB.
Set L = AC. The two right triangles give sin(θ1) = v1*t/L and sin(θ2) = v2*t/L. The angles use the boundary normal. Dividing each sine by its corresponding speed gives the shared value t/L. Replacing c/v1 and c/v2 with refractive indices yields Snell's law.
This construction applies when a propagating transmitted wave exists. It gives front positions and directions, not the reflected or transmitted intensity.
Reflection uses the normal at each patch
At one reflection point, draw the perpendicular to the surface. This is the local normal. An incoming ray making 35 degrees with that line reflects at 35 degrees from it. The incident ray, reflected ray and normal lie in one plane.
Consider several parallel incoming rays. A smooth plane mirror presents the same normal direction at their reflection points, so the outgoing rays remain parallel. On a rough reflecting surface, the local normals vary. Each ray can obey the same reflection law while leaving in a different direction from the others.
For a plane mirror, extend the outgoing ray paths backward to locate a virtual image. Its perpendicular distance behind the mirror equals the object's perpendicular distance in front. Actual reflected light travels in front of the mirror rather than through that apparent image position.
Separate frequency, wavelength and direction
Frequency counts cycles per second. Wavelength measures the spacing between neighboring points at the same wave phase. The relation v = f*λ connects both quantities with wave speed.
For monochromatic light transmitted across a stationary linear boundary, the incident and transmitted cycles match in time at the boundary. Their frequencies are equal. If the transmitted speed is three quarters of the incident speed, its wavelength is also three quarters as large.
Direction needs a separate check. A ray along the boundary normal keeps its direction even if its speed changes. At oblique incidence, use n1*sin(θ1) = n2*sin(θ2), with angles from the normal. A propagating ray entering a higher-index material moves closer to the normal; one entering a lower-index material moves away.
Measuring from the surface instead gives the complementary angle. Subtract that reading from ninety degrees before using it as a normal angle.
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