Diffraction
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Diffraction is the bending and spreading of waves when they encounter obstacles or pass through apertures. It is a fundamental wave phenomenon that explains why light does not travel in perfectly straight lines.
Physical Origin
Diffraction arises from the Huygens-Fresnel principle: every point on a wavefront acts as a source of secondary spherical wavelets. The superposition of these wavelets determines the resulting wave pattern.
Regimes
Fresnel Diffraction (Near-Field)
When the observation distance is comparable to the aperture size:
- The wavefront curvature matters
- Characterized by Fresnel number: $N_F = \frac{a^2}{\lambda z}$
- $N_F \gtrsim 1$ indicates Fresnel regime
See: Fresnel diffraction
Fraunhofer Diffraction (Far-Field)
When the observation distance is much larger than the aperture:
- Wavefronts are approximately planar
- The diffraction pattern is the Fourier transform of the aperture function
- $N_F \ll 1$ indicates Fraunhofer regime
Key Patterns
| Aperture | Diffraction Pattern |
|---|---|
| Single slit | sinc² intensity distribution |
| Circular aperture | Airy disk (Bessel function) |
| Double slit | Interference fringes modulated by sinc |
| Grating | Sharp diffraction orders |
Resolution Limit
Diffraction fundamentally limits optical resolution. The Rayleigh criterion states that two point sources are just resolved when the center of one Airy disk falls on the first minimum of the other:
$$ \theta_{min} = 1.22 \frac{\lambda}{D} $$where $D$ is the aperture diameter.
Mathematical Description
In the Fraunhofer regime, the electric field $U(x', y')$ in the observation plane is:
$$ U(x', y') \propto \iint A(x, y) \, e^{-j\frac{2\pi}{\lambda z}(x x' + y y')} \, dx \, dy $$This is a 2D Fourier transform evaluated at spatial frequencies $f_x = x'/\lambda z$, $f_y = y'/\lambda z$.