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Thermodynamics

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Zone Plate

A zone plate is a specially constructed screen such that light is obstructed from every alternate zone. It can be designed so as to cut off light due to the even numbered zones or that due to the odd numbered zones. The correctness of Fresnel’s method in dividing a wavefront into half period zones can be verified with its help.

To construct a zone plate, concentric circles are drawn on white natural numbers, the radii are proportional to the square roots of the natural numbers. The odd numbered zones (i.e. 1^st, 3^rd, 5^th, etc.) are covered with black ink and a reduced photograph is taken. In the developed negative, the odd zones are transparent to incident light and the even zones will cut off light.

If such a plate is held perpendicular to an incident beam of light and a screen is moved on the other side to get the image, it will be observed that maximum brightness is possible at some position at some position of the screen say b cm from the zone plate. XO is the upper half of the incident plane wavefront. P is the point at which the light intensity is to be considered. The distance of the point P from the wavefront is b. OM₁ (=r₁), OM₂ (=r₂) etc. are the radii of the zones,

where λ is the wavelength of light.

If the source is at a large distance from the zone plate, a bright spot will be obtained at P. As the distance of the source is large, the incident wavefront can be taken as a plane one with respect to the small area of the zone plate. The even numbered zones cut off the light and hence the resultant amplitude at P = A = m₁ + m₂+ ….. etc. In this case the focal length of the zone plate ƒ_n is given by

Thus, a zone plate has different foci for different wavelengths. The radius of the nth zone increases with increasing value of λ. It is very interesting to note that as the even numbered zones are opaque the intensity at P is much greater than that when the whole wavefront is exposed to the point P.

In the first case the resultant amplitude is given by

A = m₁ + m₃ + m₅ + …. + m_n …. (n is odd)

When the whole wavefront is unobstructed the amplitude is given by

A = m₁ – m₂+ m₃ ….. + m_n

= m₁/2 (if n is very large and n is odd).

If a parallel beam of white light is incident on the zone plate, different colours come to focus at different points along the line OP. thus, the function of a zone plate is similar to that of a convex (converging) lens and a formula connecting the distance of the object and image points can be obtained for a zone plate also.