Atelier Bonryu(E)

zone plate photography

 
 

Laboratory: Zone Plate Photography

Taking Zone Plate Photographs

- Remark -

Remark#6: Chromatic Aberration

Chromatic aberration:  Since the diffraction and interference phenomena are utilized for image formation by a zone plate, attainment of focusing directly depends on a wavelength of a light.  In other words there is strong chromatic aberration in a zone plate photography.  The chromatic aberration of a glass lens is caused by a fact that the refraction factor of glass of the lens depends on a wavelength of a light and, therefore,  the focal length of a lens composed of such glass is different for a different wavelength of a light.  In the case of a zone plate the cause of the chromatic aberration is not a feature of a medium to a light but a feature of structure of the medium.  Though a focal length depends on the wavelength of the light in both the cases, mathematical procedures to evaluate the amount of the chromatic aberrations are different.

A sinusoidal wave and a phase

Function of a zone plate: A state of a periodic phenomenon as a light wave propagation is represented by a “phase”.  A phase is an argument of a periodic function such as a sinusoidal function and usually one complete cycle of the argument is normalized to
as shown in the above figure.  Therefore, the phase difference between two states kept away by one period is 
.  If a phase difference between two states is an integer times
, these two states have the same value and are regarded as the same state.  And when two waves are superimposed in the same phase the intensity of the wave becomes  doubled.  Contrarily if the phase difference of two states kept away by
plus an integer times
the intensities of the two states compensate each other and the wave amplitude becomes zero.  The principle of a zone plate is based on this phenomenon for waves with the same phases to build up at the focal point.
 
Limiting wavelength by chromatic aberration:  When a zone plate is designed for a light with the wavelength
, the relation among the radius of the outermost zone
, the number of zones
and the focal length
is given as
, which means
.  What happens when a wavelength
of the incident light to this zone plate is different from the designed value
?  In this case the above equation cannot be satisfied for the incident light with the wavelength
.  If the difference between the two wavelengths is large and the phase difference becomes
, two waves compensate each other and this phase difference (
) is regarded as the limiting phase difference for the permissible range of the wavelength.  Then, as the relation among
is
, the limiting wavelength is given as
.    Therefore, the half width of the range of the permissible wavelength
is as follows,
  

By substituting concrete values to
and
, it is understandable that this condition is rather stringent.  For example, for
, the half width
is found to be about 20 nm.  Effect of the chromatic aberration is described in the main text.  Strictly speaking, therefore, only a light wave within a very narrow range of the wavelength can be focused by a zone plate because of the chromatic aberration.  To cope with the problem concerning the chromatic aberration in scientific or engineering fields, ingenious plans such as (1) to utilize positively the chromatic aberration, (2) to treat only a homogeneous light, and (3) to use an achromatic system such as a fractal zone plate, are being considered.
 

Numerical simulation of chromatic aberration: By the way it seems that the above limitation is not so stringent, at least, in the practical case and a photograph with a rather natural color tone can be taken.  The reason is, as described in the main text on the chromatic aberration, that even when the wavelength range is fur wider than the permissible one, the color of the object is reproduced rather satisfactory if the shape of the image of the object by the design wavelength is clearly projected.  The resolving power of the image by the wave other than the wave with the design wavelength must be still bad.  Consequentially, we carry out a numerical simulation to know how much the size of the image of the point source with various wavelength at infinity by using zone plates designed for the wavelength of 550 nm, where the focal lengths are 100 mm and 300 mm.

Result of simulation of chromatic aberration(1)

Intensity distribution of an image of a point source at infinity.  Wavelengths (lam) of the incident lights are 520, 550, and 580 nm , and a zone plate with the design value of the wavelength of 550 nm, the focal length of 100 mm, and 17 zones is used.

Result of simulation of chromatic aberration(2)

Intensity distribution of an image of a point source at infinity.  Wavelengths (lam) of the incident lights are 540, 550, and 560 nm , and a zone plate with the design value of the wavelength of 550 nm, the focal length of 300 mm, and 65 zones is used.

From the result of the simulation shown in the above figure it is concluded that zone plates with 17 zones and 65 zones can focalize a light with the wavelength between 520 and 580 nm and a light with the wavelength between 545 and 555 nm, respectively, but cannot focalize a light with the wavelength outside of these ranges.  As for the theoretical estimation a light with the wavelength between 518 to 582 nm and a light with the wavelength between 542 to 558 nm are focalized by zone plates with 17 zones and 65 zones, in the respective order, which is approximately in agreement with the result of the simulation.  In an actual situation since a photogenic object emits lights with a wide range of wavelength effects of chromatic aberration are observed in various manner.  In our experience, for example, since the zone plate with the design value of wavelength of 550 nm, the focal length of 300 mm, and 65 zones can focalize only a green light with very narrow wavelength range, it is expected that a photograph with unnatural color tone is taken.  But in an actual situation a photograph with a rather natural color tone is taken.  There are various reasons for this phenomenon, i.e., the color we see is not necessarily expressed by a single wavelength light, the color of an object is decomposed to RGB channels at the sensor of a digital camera and also at the optic nerve, and so on.  Anyway, the relation between the number of zones and the color is very interesting problem and we would like to research this problem in more detail.

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