Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

where A0 is a constant and ψ contains the coordinates describing the direction of incidence and propagation with respect to the photographic film.

The intensity pattern of the interference of A and a is

The interference pattern is recorded on the photographic film and after development the phase information is contained in the profile of the density of blackening. The relation between the intensity of the light incident on a spot of the film and the resulting blackening is logarithmic. A detailed discussion of the resulting transmission t of the film, which is called a hologram, is presented in Goodmans (1988). Under certain circumstances one can describe the transmission curve in a linear way and have

where c and β are constants. The third and fourth terms are each complex. However, together they are real and therefore tfilm remains real.

10.6.3Recovery of Image with Same Plane Wave Used for Recording

10.6.3.1 Virtual Image

We illuminate the hologram with a plane wave equal to the one used for the recording of the hologram. We use in Eq. (10.53) A0e−iψ and in Eq. (10.54) the conjugate A0ei and have

A0e−iψ tfilm A0e−iψ (cA20 +β a02 +β A0a0e−iφ eiψ +β A0a0e+iφ e−iψ ) (10.53) A0eiψ tfilm A0eiψ (cA20 +β a02 +β A0a0e−iφ eiψ +β A0a0e+iφ e−iψ ). (10.54)

The first term in Eqs. (10.53) and (10.54) is a constant term; the second may be neglected if we assume that a0 is small compared to A. The third term in (10.53) is

This is the important term for the virtual image. It is the doublet of the wavefront of the original and diverges. In Figure 10.6a. we show the recovery of the image using the beam A. As we know from geometrical optics, the diverging light is traced back to a virtual image of the object.

FIGURE 10.6 Recovery of the image: (a) reference beam A illuminates the hologram; wavefront of image diverges. It is traced back to the virtual image (reversed operation of production of hologram); (b) reference beam A illuminates the hologram; wavefront of image converges to the real image.

and is the duplication of the conjugate of the wavefront of the original. In Figure 10.6b we show the illumination by the beam A and the convergence to the real image, which is in the opposite direction to the virtual image we discussed in Section 10.6.3.1.

10.6.4 Recovery Using a Different Plane Wave

If we produce the hologram with a plane wave of amplitude A and illuminate the hologram with a plane wave of amplitude B, all in the same (horizontal) direction, we get

The real and virtual image now both appear in the horizontal direction (see Figure 10.7).

10.6.5 Production of Real and Virtual Image Under an Angle

To see the virtual image separately from the real image, one has to use a reference

wave under an angle with respect to the normal of the object, around which the scattered light emerges. To do this, we use for A the wave A0e−i2π sin θ(x/λ) and

have for the transparency of the film

with the diffraction pattern of the object. It contains the spatial frequency spectrum including phase information. We may then substitute for the hologram the Fourier transform of the object and obtain the image is by a second Fourier transformation.

Using this simple model we can easily demonstrate that the object can be reconstructed using different sizes of the hologram. Smaller physical sizes will give us more deteriorated images of the object. By cutting a large hologram into small ones, we lose spatial frequency information. For simplicity we consider an object with the first Fourier transformation in such a way that the smallest frequencies are in the center and the largest frequencies at increased distances from the center. To demonstrate the effect of the removal of frequencies we consider the example of a grid as an object. Removing certain sections of the frequency pattern, which is accomplished with the first Fourier transformation, will change the image. In FileFig 10.15 we use a blocking function for a low frequency portion, in FileFig 10.16 a blocking function for an intermediate section, and in FileFig 10.17 a blocking function of a grid. In FileFig 10.18 we simulate cutting down the hologram by a blocking function, which symmetrically cuts down the lowest frequencies.

FileFig 10.15 (W15HOGRBLHIS)

The object is a grid. The transfer function removes certain higher frequencies of the first Fourier transformation. The extent of the blocking function depends on a. The modified image is compared with the original object.

W15HOGRBLHIS is only on the CD.

Application 10.15. Observe the changes of the final image by modification of the blocking function, that is, changing a.

FileFig 10.16 (W16HOGRBLLOS)

The object is a grid. The transfer function is a blocking function passing only one portion of the frequencies of the first Fourier transformation. The extent of the blocking function depends on n and a. The modified image is compared to the original object.

W16HOGRBLLOS is only on the CD.

Application 10.16. Observe the changes of the final image by modification of the blocking function, that is, changing n and a.

FileFig 10.17 (W17HOGRPERS)

The object is a grid. The transfer function is a grid-type blocking function blocking periodic parts of the first Fourier transformation. The width of the peaks and the extent of the blocking function depend on q and a. The modified image is compared to the original object.

W17HOGRPERS

Object Is a Periodic Structure

The FT of the object is multiplied by a blocking function. A blocking function has been chosen blocking certain frequencies such that there are twice as many peaks in the image. The Ft (inverse) of (Ft of object) · (blocking function) is the “new” image. The “new” image is compared to the original, that is, the FT of (FT of object). The blocking function removes certain high frequencies of the FT.

FT of object

ω : cff t(y) N : last(ω)

N 127.

w

Application 10.17. Observe the changes of the final image by modification of the blocking function, that is, changing q and a.

FileFig 10.18 (W18HOSTEPS)

The object is a step function composition. The transfer function passes the lowest frequencies on both parts of the diffraction pattern, that is, the first Fourier transformation. The number of passing lowest frequencies is monitored by a and b. The hologram is larger when more frequencies are passing. The modified image is compared to the original object.

W18HOSTEPS

Object y

The object y has a complicated shape. Its FT is the hologram c. It may be produced in the focal plane of a lens, using parallel light. The illumination of the hologram with parallel light will reproduce the object, that is, the FT (inverse) of the FT, called here cc. We study the reproduced object when the information in the hologram is only partly used; that is, we multiply cc with a filder f . We show separately f and the FT of the product of f and cc. The width of the filter f may be changed by using various values for a and b, corresponding to changing the size of the hologram.

The object i : 0, 1 . . . 255.

The hologram

c : cff t(y)N : last(c) N 255

k : 0 . . . 255 j ≡ 0 . . . 255

The inverse FT of the FT (hologram)

cck : ck yy : icff t(cc)

N : last(cc) N 255 j : 0 . . . 255

The filter

fj : (a − j ) + (j − 255 + b) a ≡ 60

b ≡ 60.