Replay field

The resulting two dimensional sine function has a very strong effect on the structure of the replay field of a hologram as it forms the overall envelope that will contain the desired information as well as having repeating sidelobes that represent the higher order diffraction terms as shown in Fig. 1.3. Now we will look at what... 

The exact structure of the replay field distribution depends on the shape of the fundamental pixel and the number and distribution of these pixels in the hologram. The pattern we generate with this distribution of pixels is repeated in each lobe of the sine function from the fundamental pixel. The lobes can be considered as spatial harmonics of the central lobe, which contains the desired 2D pattern. For example, a line of square pixels with alternate pixels being one or zero (i.e. a square wave) would have the basic replay structure seen in Fig. 1.4. 

If function h(x, y) is real then h(x, y) = h(x, y), hence we cannot differentiate between H(u, v) and H(—u,—v) which means that both must appear in the replay field. Hence any replay field generated by a binary phase or amplitude hologram will always have 180° rotational symmetry. This symmetry restricts the useful area of the replay field to the upper half plane of the sine envelope, as any pattern generated by the hologram will automatically appear as desired as well as rotated about the origin by 180°. 

The next step is to look at 2D patterns such as the chequerboard pattern in Fig. 1.5, of pixels on an equally spaced grid (for simplicity A is restricted to binary values (such as 0 or 1). The chequerboard can be generated by the XOR of a 2D grating with itself rotated by 180°. Hence, the replay field will be made from the convolution of the FT of the two gratings. 

If we take the FT of the target replay field of Fig. 1.7, then take the phase and threshold it about tt/2 to create a binary hologram as shown in the left side of Fig. 1.7. 

Fig. 1.7 CGH generated by thresholding the phase of FT (target). Left, binary phase hologram, right is the replay field generated from it...

Fig. 1.8 Hologram generated by DBS (left) and its replay field (right)...

Define an ideal target replay field, T (desired pattern as in Fig. 1.6). 

The calculation of the diffraction pattern for a periodic system revolves around the construction of the reciprocal lattice and subsequent placement of the first Brillouin zone however, in this case the aperiodicity of the pentagonal array requires a different approach due to the lack of translational symmetry. The reciprocal lattice of such an array is densely filled with reciprocal lattice vectors, with the consequence that the wave vector of a transmitted/reflected light beam encounters many diffraction paths. The resultant replay fields can be accurately calculated by taking the FT of the holograms. To perform the 2D fast Fourier transform (FFT) of the quasi-crystalline nanotube array, a normal scanning electron micrograph was taken, as shown in Fig. 1.13. 

The remarkable richness and intricacy of the optical replay field emanating from the quasi-crystalline MWCNT array is best seen in the spherical diffraction pattern shown in Fig. 1.15. The results clearly show that the quasiperiodic array of nanotube antennas act as holograms (apertures) for the reflected light, producing remarkable and striking diffraction patterns (replay fields). 

To find the replay field of this pattern we need to take its Fourier transform. The square grating can be represented by a sin-... 

When an arbitrary pattern is generated by the FT of a hologram, it is contained within a sine envelope based on the dimensions of the smaller or fundamental pixel. For each lobe in the sine there is an associated replication of the pattern. There is also a replication of the pattern at each zero in the sine, even though the central value of the pattern is suppressed by the zero. With holograms, we are only interested in the central lobe of the sine function. The other orders or lobes merely repeat the desired pattern in the replay field and waste the available intensity which can be placed into that desired pattern. The area of interest in the replay field must be limited to half of the area of the central lobe to prevent overlap of orders. As the pixel pitch decreases, the central lobe of the sine envelope broadens, easing the restrictions that are placed on the replay field pattern. 

With binary phase modulation (T g [+1, -1]), the pixel in the center of the replay field can be defined by the structure of the hologram. A drawback of both these binary modulation schemes is that the hologram will always be a real function, which means that the FT of the hologram is the same as the FT of the hologram rotated about the origin... 

X 128 pixel resolution of the FLC SLM [25]. The replay field was transformed with a 250 mm focal length lens and imaged onto a CCD camera. The remains of the zero order can be seen in the center of the plane, mostly due to the electronic addressing of the SLM. The noise is mostly due to the limited resolution of the SLM. 

Figure 34. Hologram and replay field generated by simulated annealing.

Fourier transform replay field of the CGH and (3ac and are the optical ratios diffracted into the replay field and the unwanted zero order, respect vely, such that a + a 1 ... 

This result tells us that the amount of power diffracted into the replay field r u,v) is constant and independent of the input state of polarization. Hence the light that is normally blocked by the input polarizer and analyzer is just directed to the central zero order. The benefit of this is twofold with the absence of polarizers, there will be no fluctuations due to changes in the input polarization and there is an added bonus of 6 dB of extra power due to the 3 dB from each polarizer removed, due to the physical construction of each polarizer. 

The results of this analysis have been verified by simulating the variation of input polarization states entering the FLC SLM and looking for a change in the intensity of the replay field, with no change observed. This result was also verified by an experimental test, the results of which can be seen in Fig. 36. A 128x128 pixel FLC SLM [25]... 

Figure 36. Replay field of a binary phase CGH (a) with polarizers, (b) without polarizers (reproduced from [47]).

If we use the CGH in Fig. 34 as a binary phase image displayed on an FLC SLM, then it is possible to route light to several fibers in the replay field. We are limited however, by the binary phase modulation of the FLC SLM, which means that a symmetric copy of the desired replay field always ap-... 

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