Optical Path
A centimetre of glass counts as one and a half centimetres of vacuum.
Lesson at a glanceShow the key points
- In a medium E = E₀ sin ω(t − μx/c): a distance Δx changes the phase by ωμΔx/c, the same as μΔx in vacuum.
- Optical path = μ × geometrical path. Interference depends on the optical path difference.
- Frequency is set by the source and doesn't change between media; the wavelength becomes λ = λ₀/μ.
- Rays from a point through a lens to its image have different geometrical paths but equal optical paths: the longer geometrical path goes through less glass.
For interference, what matters is not how far two waves travel but how many wavelengths they fit in along the way. Inside glass a wavelength is shorter, so a centimetre of glass counts for more.
Phase change in a medium
In a medium of refractive index μ, light travels at v = c/μ, so a light wave there can be written
Travelling a distance Δx changes the phase by . In vacuum, the same phase change would need a distance μΔx. So a path Δx in the medium is equivalent, for phase, to μΔx in vacuum.
In air (μ ≈ 1) the optical and geometrical paths are practically equal.
The same idea through wavelength
A wave’s frequency is fixed by its source and doesn’t change when it enters a new medium. Its speed drops by μ, so its wavelength drops by μ too:
Points one wavelength apart have the same phase. In the medium that is λ₀/μ; in vacuum, λ₀. So a length λ₀/μ of medium is “worth” λ₀ of vacuum, which is again the factor μ.
A helium–neon laser emits 633 nm light. What is its wavelength inside glass of refractive index 1.5?
Light takes the same optical path through 3.00 cm of a liquid as through 2.70 cm of glass of refractive index 1.48. Find the liquid’s refractive index.
Why all rays through a lens arrive together
At a boundary, the incident wavefront and the refracted wavefront each have a single phase along their length. The phase change from one wavefront to the next must therefore be the same along every ray. Using Snell’s law one can show that a path in vacuum and the matching shorter path in glass are related by exactly the factor μ.
The same holds for a lens forming an image. Rays from a point to its image travel different geometrical distances, but they have equal optical paths: a ray along the edge has a longer route through air but crosses only a thin part of the glass, while the central ray goes straight but through the thickest glass. All arrive in phase, which is why they reinforce at the image.
- 1A distance Δx in a medium causes the same phase change as μΔx in vacuum: the optical path.
- 2Frequency stays fixed between media; the wavelength becomes λ₀/μ.
- 3Interference depends on optical path differences.
- 4Rays from an object point to its image through a lens all have the same optical path.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.