FRESNEL ZONE PLATES

Diffraction optics - Diffraction gratings - Fresnel zone plates - Gabor Hologram - Order - Contact us

1. Intreduction and theory.

Let us place the transparent plate with a round hole between the point source of light and the screen of observation S. The intensity of light in the centre of the screen will depend upon the number of Fresnel zones in the hole. Every Fresnel zone is a ring area of the hole. The light from two adjacent zones comes to point of observation in anti-phase, so the corresponding waves vanish each other because of destructive interference. On the contrary, waves from only odd or only even zones will amplify each other because of constructive interference. Number of the Fresnel zones in the hole of diameter D equals m = D²(1/a+1/b)/4λ, where à and b are the distances from the hole to the source and to the image respectively.  The outer radius of m-th Fresnel zone equals R_m = (abmλ/(a+b))^1/2. If the source of light is situated in infinity and the Fresnel zone plate is illuminated by parallel light beam then behind zone plate the light will be focused in a point similar to lens. Moreover, length à from the plate to source and distance b from the plate to the image  are interrelated by the same equation as for lens: 1/a + 1/b = 1/F, where focal length F is defined as F=R_m²/mλ. If the center of the zone plate is transparent, then m is odd and we have take for formula outer radius R of the transparent ring of the plate. If the center of the plate is black, then R is the outer radius of the dark (black) ring. With the aid of Fresnel zone plate we can even take an optical images, but quality of such an image will be rather low (as compared to lens). in contrast to lens Fresnel zone plate has several focal distances: F_n=F/(2n+1), where n is integer.

2. Fresnel zone plates FZP-01.

For investigation of the wave properties of the light we produce and can send to you by mail the set of zone plates  FZP-01. Zone plates are made on a transparent plastic film of size À5 (148 x 210 ìì). Point size is about 6 microns. There are 7 rows of plates as given below:

Row 1 - plates with the number of Fresnel zones equal to N printed above the row. The odd zones from 1 to N are opened (transparent).  So, in the left plate only the first Fresnel zone is opened (N=1), in the 2-nd left plate (N=3) 1-st and 3-rd Fresnel zones are opened, in the 3-rd (N=5) one 1-st, 3-rd and 5-th zones are opened etc. In the very right plate (N=19) all the odd Fresnel zones from 1 to 19 are opened. The intensity of the light in the focus of the plate equals approximately I = 4MI₀, where M = (N+1)/2 - number of the opened Fresnel zones.

Row 2  - plates with the number of Fresnel zones equal to N printed above the row. The even zones from 1 to N are opened (transparent). So, in the left plate (N=1) all Fresnel zones are closed, in the 2-nd left one (N=3)  only 2-nd Fresnel zone is opened, in the 3-rd one (N=5) 2-nd and 4-th zones are opened, etc. In the very right plate (N=19) all even Fresnel zones from 1 to 19 are opened. The intensity of the light in the focus of the plate equals approximately I = 4MI₀, where M = (N-1)/2 is number of the opened Fresnel zones.

Row 3  - plates with the number of Fresnel zones equal to N printed above the row. All the zones from 1 to N are opened (transparent). So, in the left plate (N=1) only the first Fresnel zone is opened, in the 2-nd left plate (N=3) 1-st, 2-nd and 3-rd Fresnel zones are opened, in the 3-rd one (N=5) all the zones from 1-st to 5-th are opened and etc. In the very right plate (N=19) all the Fresnel zones from 1 to 19 are opened. The intensity of the light in the focus of the plate equals approximately I = 4I₀.

Row 4  - plates with the number of Fresnel zones equal to N printed above the row.  Intermediate plates correspond to intermediate numbers of the Fresnel zones N. All the zones from 1 to N are opened (transparent). So, in the very left plate (N=1) only the 1-st Fresnel zone is opened, in the 2-nd one first two zones are opened, in the 3-rd plate first three zones are opened etc.  In the very right plate (N=19) all the Fresnel zones from 1 to 19 are opened. The intensity of the light in the focus of the plates with the odd N is the same and equals approximately I = 4I₀, while in the focus of the plates with the even N the intensity equals approximately 0.

Row 5  - plates with the number of Fresnel zones equal to N printed above the row. "W" means that all the odd Fresnel zones are transparent (center of the plate is transparent), while "B" means that  all the even Fresnel zones are transparent (center of the plate is black).  The intensity of the light in the focus of the plate equals approximately I ~ 4(N/2)I₀=2NI₀.

For all the plates in the rows 1-5 the focus distance F equals 10 cm for the wavelength 632,8 nm. Source of the light is considered to be in infinity.

Row 6 - Fresnel zone plates with the focal length equal in centimeters to F above the row. Wavelength is  632,8 nm (if illuminated be the light with different wavelength, then the focal length must be recalculated). All the odd Fresnel zones from 1 to 200 are opened (center of the plate is transparent).   Source of the light is considered to be in infinity.

Row 7 - Fresnel zone plates with the distance to image equal in centimeters to F above the row.  Source of the light is considered to be located symmetrically the plate at the same distance a=F from it.  Wavelength is  632,8 nm (if illuminated be the light with different wavelength, then the focal length must be recalculated). All the odd Fresnel zones from 1 to 200 are opened (center of the plate is transparent).

3. Scheme of experiment.

	The source of the coherent optical radiation is required for our experiments. For example, we can use semiconductor laser pointer with the wavelength 650 nm or any other laser. We  used He-Ne laser with the wavelength 632,8 nm. If we use the laser with different wavelength, then the focal distances must be recalculated in accordance with the new wavelength. Diameter of the laser beam equals about 1 mm. It must be extended to diameter > 1 cm. For such a purposes two lenses are used:  short focus lens L1 with focal distance F1 < 2mm and long focus lens L2 with focal distance F2 ~ 8 cm. System assembled of these lenses produce a parallel and coherent light beam of diameter 4-5 cm, which is incident on Fesnel zone plate FZP. Changing the distance between FZP and the screen S we can observe the focusing of light in the moment when the distance to screen equals the focal length F of FZP.
	If we remove lens L2, then FZP will be illuminated by coherent radiation of the point source located in focus of lens L1. In this case the image will appear at the distance b = aF/(a-F) > F. The sizes of zone plates in row 7 of set FZP-01 are chosen by such a way, that if we place a point source at the distance "a" given above the row 7, then the image will appear at the same distance "a" behind the zone plate. The distances "a" are calculated for the wavelength 632,8 nm.  If you use the laser with different wavelength, then these distances have to be recalculated accordingly.
	And finally we can use for experiments an ordinary tungsten light bulb. Let us place such a bulb at the distance L = 3-10 m from the zone plate, while the screen S will be placed in the focus of it.  In this case we shall see on the screen the image of the glower. Taking into account that the bulb is situated not in the infinity, but at the distance "a" from a zone plate, we should move the screen by a small distance  F2/a ≤ 1 cm away from the zone plate. To achieve the maximal resolution of the zone plate the size d of the source should be small. So small, that the coherence length Lλ/d near the plate would be larger that the diameter of the bigger ring on the plate.  Calculations with parameters of experimental setup shows that the size of the source should be less than the parts of millimeter. Diameter of the glower of a small bulb suits this requirement. Therefore every part of the glower makes a rather sharp image in the corresponding point of the focal plane of the zone plate. Because the radiation of a tungsten bulb is not monochromatic, the light of different wavelengths will be focused at different distances from the zone plate. To avoid this a color filter can be placed in front of the zone plate.

Moving the screen towards the zone plate we can observe several points in which image is focusing. If  a photodetector with a small hole is situated in the place of screen, then intensity of the radiation behind the zone plate can be measured.