AcademyDiffraction
Academy
Fresnel and Fraunhofer Regimes
Level 1 - Physics topic page in Diffraction.
Principle
Diffraction is wave spreading caused by a finite aperture or obstacle. The observed pattern depends on wavelength, aperture size, and distance from the aperture.
Fresnel diffraction is near-field diffraction, where wavefront curvature still matters. Fraunhofer diffraction is far-field diffraction, where the pattern depends mainly on angle.
Notation
\(a\)
aperture or slit width
\(\mathrm{m}\)
\(L\)
distance from aperture to screen
\(\mathrm{m}\)
\(\lambda\)
wavelength
\(\mathrm{m}\)
\(N_F\)
Fresnel number
1
\(\theta\)
observation angle
\(\mathrm{rad}\)
Method
Derivation 1: Use Huygens wavelets
Each point across an aperture acts as a source of secondary wavelets. The final pattern is set by the phase relationships between those wavelets.
Wavelet phase difference
\[\Delta\phi=\frac{2\pi\Delta r}{\lambda}\]
Far-field path difference across width
\[\Delta r\approx a\sin\theta\]
Derivation 2: Classify the regime
The Fresnel number compares aperture size with diffraction spreading over distance \(L\).
Fresnel number
\[N_F=\frac{a^2}{\lambda L}\]
Fraunhofer tendency
\[N_F\ll1\]
Fresnel tendency
\[N_F\gtrsim1\]
Derivation 3: Relate angle to screen position
In the far field, positions on a distant screen correspond directly to diffraction angles.
Screen geometry
\[y=L\tan\theta\]
Small-angle form
\[y\approx L\theta\]
Rules
Fresnel number
\[N_F=\frac{a^2}{\lambda L}\]
Far-field path difference
\[\Delta r\approx a\sin\theta\]
Small-angle screen position
\[y\approx L\theta\]
Examples
Question
A slit has
\[a=0.20\,\mathrm{mm}\]
\[L=2.0\,\mathrm{m}\]
and \[\lambda=500\,\mathrm{nm}\]
Estimate \(N_F\).Answer
\[N_F=\frac{(0.20\times10^{-3})^2}{(500\times10^{-9})(2.0)}=0.040\]
This is in the Fraunhofer range.Checks
- Diffraction becomes important when an aperture dimension is comparable with the wavelength.
- Fraunhofer patterns are usually simpler because they are angular patterns.
- Increasing \(L\) lowers \(N_F\), pushing the setup toward the far field.
- The small-angle screen formula needs \(y\\ll L\).