Zone Plate | Fresnel Diffraction

Diffraction – Fresnel's Class

Topic: Construction, Theory, Multiple Foci and Comparison with a Convex Lens

A zone plate is a diffractive optical device consisting of a series of alternate transparent and opaque concentric circular rings. It works on the principle of Fresnel diffraction and the division of a wavefront into half-period zones.

A zone plate focuses light by diffraction and interference rather than by refraction. It is an important application of Fresnel's half-period zone theory. It demonstrates how the superposition of secondary wavelets from alternate half-period zones can produce constructive interference at certain points on the axis.

A zone plate behaves like a converging lens, although its focusing mechanism is entirely different from that of a conventional convex lens.

Construction of a Zone Plate

A zone plate is constructed by drawing a large number of concentric circles on a plane surface. The radii of these circles are proportional to the square roots of natural numbers.

If r is the radius of the first circle, the radii of successive circles are:

r₁ = r,   r₂ = r√2,   r₃ = r√3,   r₄ = r√4, ...

In general, the radius of the nth circle is:

r_n = r√n

Alternate circular regions are made opaque, while the remaining regions are kept transparent. A photographic reduction of the pattern is made on a glass plate. The resulting plate consists of alternate transparent and opaque concentric zones and is called a zone plate.

Figure 1. Schematic representation of alternate circular zones.

3. Types of Zone Plates

(a) Positive Zone Plate

In a positive zone plate, the odd-numbered zones (1, 3, 5, ...) are transparent, while the even-numbered zones (2, 4, 6, ...) are opaque.

It is produced by preparing a pattern in which the even-numbered zones are black and then making a photographic negative. Thus, the odd-numbered zones become transparent and the even-numbered zones remain opaque.

(b) Negative Zone Plate

In a negative zone plate, the even-numbered zones (2, 4, 6, ...) are transparent, while the odd-numbered zones (1, 3, 5, ...) are opaque.

It is obtained by preparing a pattern in which the odd-numbered zones are black and then making a photographic negative.

Figure 2. Schematic comparison of positive and negative zone plates.

Positive Zone Plate	Negative Zone Plate
Odd-numbered zones are transparent.	Even-numbered zones are transparent.
Even-numbered zones are opaque.	Odd-numbered zones are opaque.
Central zone is transparent.	Central zone is opaque.

Theory of a Zone Plate

Fig. represents the edge-wire section of a zone plate. S is a point source emitting spherical waves of wavelength λ, placed on the axis at a distance a from the centre O of the zone plate. Let P be a point on the screen at a distance b from O, at which the intensity is to be found.

In Fig., the radius of the nth period zone of the M_n plate is given by:

OM_n = r_n = r√n

Where r is the radius of the first circular zone of the plate.

Therefore, the distance SM_n is:

SM_n = √(a² + r²n)

Using the binomial approximation, when r_n ≪ a:

SM_n ≈ a + r²n 2a

Similarly, the distance M_nP is:

M_nP ≈ b + r²n 2b

Adding the above two distances, we get:

SM_n + M_nP = a + b + r²n 2 ( 1 a + 1 b ) (1)

Similarly, for the (n − 1)th half-period zone:

SM_n−1 + M_n−1P

= a + b + r²(n − 1) 2 ( 1 a + 1 b ) (2)

Subtracting equation (2) from equation (1), we get:

(SM_n + M_nP) − (SM_n−1 + M_n−1P)

= r² 2 ( 1 a + 1 b ) (3)

As the path difference between the waves reaching P through two consecutive half-period zones is λ/2, and the path difference is independent of n, therefore:

r² 2 ( 1 a + 1 b ) = λ 2

Hence,

1 a + 1 b = λ r² (4)

Zone Plate as a Concex Lans

Compair equation (4) with Lens formula

1 u + 1 v = 1 f

The focal length of zone plate

f= r²/ λ

Multiple Foci of a Zone Plate in case of Positive zone plate

Case (i)

If E₀₁, E₀₂, E₀₃, ..., E_0n are the amplitudes due to the first, second, third, ... nth zones respectively, then the resultant amplitude at P for a zone plate with alternate transparent zones is:

E₀ = E₀₁ + E₀₃ + E₀₅ + ... (5)

Hence, zone plate equation

1 a + 1 b = λ r²

and The focal length of zone plate

f₁= r²/ λ

Case (ii)

If, the zone plate is designed so that the transparent part allows two zones, while the opaque part blocks next two zones.

E₀ =( E₀₁ - E₀₂)+( E₀₅ - E₀₆)+.....

Hence, zone plate equation

1 a + 1 b = 2λ r²

and The focal length of zone plate

f₂= r²/2λ

Case (ii)

If, the zone plate is designed so that the transparent part allows two zones, while the opaque part blocks next two zones.

E₀ =( E₀₁ - E₀₂)+( E₀₅ - E₀₆)+.....

Hence, zone plate equation

1 a + 1 b = 2λ r²

and The focal length of zone plate

f₂= r²/2λ

Resultant Amplitude

If E₀₁, E₀₂, E₀₃, ..., E_0n are the amplitudes due to the first, second, third, ... nth zones respectively, then the resultant amplitude at P for a zone plate with alternate transparent zones is:

E₀ = E₀₁ + E₀₃ + E₀₅ + ... (5)

Unlike an ordinary convex lens, a zone plate possesses several principal and secondary foci. These are called multiple foci.

The first-order focus is obtained when the path difference between successive half-period zones is λ/2. The odd-numbered transparent zones contribute approximately in phase at this focus.

For higher-order foci, consider the condition in which the path difference between successive zones is:

Δ = pλ/2

where p = 1, 2, 3, 4, ... .

When p = 2, alternate groups of two half-period zones contribute with opposite phase. Their resultant amplitudes approximately cancel each other, so no strong focus is formed.

When p = 3, three successive half-period zones form a group. The contributions from alternate groups add constructively. Thus, a secondary focus is obtained.

Similarly, strong foci are obtained for odd values of p:

p = 1, 3, 5, 7, ...

If the principal focal length is f, the focal lengths of the successive strong foci are:

f₁ = f

f₂ = f/3

f₃ = f/5

f₄ = f/7

In general, the focal length of the mth strong focus is:

f_m = f/(2m−1)

Since f = r₁²/λ:

f_m = r₁²/[(2m−1)λ]

where m = 1, 2, 3, ... .

Intensity at Multiple Foci

Under the usual ideal thin-zone approximation, the amplitude at the mth odd-order focus is proportional to 1/(2m−1). Consequently, the relative intensity is approximately:

I_m/I₁ ≈ 1/(2m−1)²

Focus order	Focal length	Relative intensity
First	f	1
Third	f/3	1/9
Fifth	f/5	1/25
Seventh	f/7	1/49

The even-order foci are strongly suppressed in an ideal zone plate because of destructive interference. In practical zone plates, finite zone width, absorption and manufacturing imperfections may produce residual intensity.

7. Comparison Between a Zone Plate and a Convex Lens

(A) Similarities

1. Both act as converging optical devices and can produce a real image of an object.
2. The focal length of both depends on the wavelength of light. Therefore, both exhibit chromatic effects.
3. Both can focus incident parallel rays to a point on the principal axis.
4. Both obey the paraxial imaging relation:
   1/a + 1/b = 1/f
   for a real object and a real image, using positive distance conventions.

(B) Dissimilarities

S.No.	Convex Lens	Zone Plate
1.	It works on the principle of refraction of light.	It works on the principle of diffraction and interference of light.
2.	An ordinary convex lens has one principal focal length.	A zone plate has multiple foci: f, f/3, f/5, f/7, ...
3.	Its focal length is given by the lens-maker's formula: 1/f = (μ−1)(1/R₁ − 1/R₂) for a thin lens in air, using the usual signed-radius convention.	Its principal focal length is: f = r₁²/λ
4.	For a glass lens under normal dispersion, the refractive index is generally greater for violet light than red light. Therefore, fred > fviolet.	Focal length is inversely proportional to wavelength. Therefore, fred < fviolet.
5.	The focusing action is mainly due to refraction.	The focusing action is due to constructive interference of light diffracted by transparent zones.
6.	The principal image is formed by the refracted light.	The image is formed by diffracted light and its intensity is distributed among different diffraction orders.
7.	A conventional convex lens is generally more efficient for transmitting light.	A zone plate blocks a substantial portion of incident light and distributes transmitted energy among several diffraction orders.

8. Important Formulae

Physical quantity	Formula
Radius of nth zone (finite object and image distances)	rn = √[nλab/(a+b)]
Radius of nth zone (plane wave illumination)	rn = √(nλf)
Principal focal length	f = r1²/λ
Focal length of mth strong focus	fm = r1²/[(2m−1)λ]
Relative intensity at mth strong focus	Im/I1 ≈ 1/(2m−1)²
Focal length dependence	f ∝ 1/λ

9. Applications of Zone Plates

1. Focusing X-rays in X-ray microscopy.
2. High-resolution imaging in microscopy.
3. Micro-optics and diffractive optical systems.
4. Focusing and manipulating light in optical instruments.
5. Research in nanophotonics and diffractive optics.

10. Conclusion

A zone plate is an important optical device based on Fresnel diffraction. It consists of alternate transparent and opaque concentric zones. Its focusing action is caused by constructive interference of diffracted light from the transparent zones.

The principal focal length is given by:

f = r₁²/λ

Unlike a convex lens, a zone plate possesses multiple strong foci at distances f, f/3, f/5, f/7, and so on. Its focal length decreases with increasing wavelength.

Physics Study Material
Topic: Zone Plate | Fresnel Diffraction