Babinet's principle

In physics, Babinet's principle^[1] is a theorem concerning diffraction that states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam intensity.

Explanation

Assume B is the original diffracting body, and B' is its complement, i.e., a body that is transparent where B is opaque, and opaque where B is transparent. The sum of the radiation patterns caused by B and B' must be the same as the radiation pattern of the unobstructed beam. In places where the undisturbed beam would not have reached, this means that the radiation patterns caused by B and B' must be opposite in phase, but equal in amplitude.

Diffraction patterns from apertures or bodies of known size and shape are compared with the pattern from the object to be measured. For instance, the size of red blood cells can be found by comparing their diffraction pattern with an array of small holes. One consequence of Babinet's principle is a paradox that in the diffraction limit, the radiation removed from the beam due to a particle is equal to twice the particle's cross section times the flux. This is because the amount of radiation absorbed or reflected is the same as the amount diffracted.

The principle is most often used in optics but it is also true for other forms of electromagnetic radiation and is, in fact, a general theorem of diffraction and holds true for all waves. Babinet's principle finds most use in its ability to detect equivalence in size and shape.

Demonstration experiment

The effect can be simply observed by using a laser. First place a thin (approx. 0.1 mm) wire into the laser beam and observe the diffraction pattern. Then observe the diffraction pattern when the laser is shone through a narrow slit. The slit can be made either by using a laser printer or photocopier to print onto clear plastic film or by using a pin to draw a line on a piece of glass that has been smoked over a candle flame.

Babinet's Principle in Radiofrequency Structures

Babinet's Principle can be used to find complementary impedances. Babinet's Principle states (in optics) that when a field behind a screen with an opening is added to the field of a complementary structure (that is a shape covering the screen hole), then the sum is equal to the field where there is no screen. Demonstration can be found in any good optic or antenna book (such as Balanis, Krauss, Stutzman). The end result (corollary as a matter of fact) of practical interest for antenna engineers is the following formula:

$Z_\text{metal} \, Z_\text{slot} = \frac{\eta^2}{4},$

where Z_metal and Z_slot are input impedances of the metal and slot radiating pieces, and $\eta$ is the intrinsic impedance of the media in which the structure is immersed. In addition, Z_slot is not only the impedance of the slot, but can be viewed as the complementary structure impedance (a dipole or loop in many cases). In addition, Z_metal is often referred to as Z_screen were the screen comes from the optical definition. It is noteworthy that the thin sheet or screen does not have to be metal, but rather any material that supports a $\vec{J}$ (current density vector) leading to a magnetic potential $\vec{A}$. One issue with this equation, is that the screen must be relatively thin to the given wavelength (or range thereof). If it is not, modes can begin to form or fringing fields may no longer be negligible.

For a more general definition of Eta or intrinsic impedance, $\eta = \sqrt{\frac{\mu}{\epsilon}}$. Note that Babinet's principle does not account for polarization. In 1946, H.G. Booker published Slot Aerials and Their Relation to Complementary Wire Aerials to extend Babinet's principle to account for polarization (otherwise known as Booker's Extension). This information is drawn from, as stated above, Balanis's third edition Antenna Theory textbook.

See also

• Bistatic radar

References

1. M. Born and E. Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge.

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