Babinet’s Principle
This file was generated by AI and may require review.
Babinet’s principle states that the diffraction patterns from complementary screens are identical except for the directly transmitted light.
Statement
Two screens are complementary if their transmission functions sum to unity:
$$ t_1(x, y) + t_2(x, y) = 1 $$For example:
- A circular aperture and a circular opaque disk of the same size
- A single slit and a thin wire of the same width
- Any aperture and its photographic negative
Babinet’s principle states:
$$ U_1(x, y) + U_2(x, y) = U_0(x, y) $$where:
- $U_1$ = diffracted field from screen 1
- $U_2$ = diffracted field from screen 2
- $U_0$ = unobstructed field (no screen present)
In the Angular Spectrum
For the angular spectrum:
$$ A_1(f_x, f_y) + A_2(f_x, f_y) = \delta(f_x, f_y) $$The delta function represents the unobstructed plane wave. Therefore:
$$ A_2(f_x, f_y) = \delta(f_x, f_y) - A_1(f_x, f_y) $$Consequences
Intensity Away from the Axis
For points away from the optical axis (where the delta function doesn’t contribute):
$$ |A_2|^2 = |A_1|^2 $$The diffraction patterns have identical intensity except at the center.
On-Axis Difference
At the center ($f_x = f_y = 0$):
- The aperture transmits the direct beam
- The opaque obstacle blocks it
This is the only difference between the patterns.
Famous Example: Poisson’s Spot
When Fresnel proposed his wave theory of light, Poisson (a skeptic) pointed out it predicted a bright spot at the center of a circular disk’s shadow — seemingly absurd. Arago then demonstrated this spot experimentally, confirming the wave theory.
The spot occurs because diffracted waves from the disk’s edge constructively interfere at the center. Babinet’s principle explains why: the disk’s diffraction pattern matches the aperture’s, which has a central maximum.
Applications
| Application | How Babinet’s Principle Helps |
|---|---|
| Calculating wire diffraction | Use easier slit calculation, add DC |
| Particle sizing | Measure diffraction to determine size |
| Optical testing | Complementary masks give identical fringes |