J. Eur. Opt. Society-Rapid Publ. 21, 33( 2025) 33
Fig
. 1. Scheme of the optical setup. NDF: Variable neutral density filter; L1, L2: converging lenses; P: linear polarizer; LCOS-SLM: Liquid-crystal on silicon spatial light modulator.
Fig
. 2.( a) Phase profile of the optimum triplicator grating.( b) Target pattern used in this work.( c) Corresponding phase-only hologram with a linear lateral phase shift.( d) Triplicator phaseonly hologram.( e) The final phase function to be displayed on the SLM when combining the triplicator hologram and the lens function.
g( x 0) to be ideally reconstructed by the hologram in the Fourier domain. Figure 2c shows the phase-only distribution /( x) obtained by computing the inverse FT of g( x 0), where a carrier linear phase along the horizontal direction was added, i. e. Figure 2c shows the function [/( x)+ cx ] mod2p. The resulting phase-only hologram looks like a deformed blazed diffraction grating, with phase values distributed in the [ �p,+ p ] range. Figure 2d shows the equivalent triplicator hologram u h( x) obtained by equation( 4). Note that the resulting phase values are limited to the [ �0.39p,+ 0.39p ] range, like in the profile in Figure 2a. Thus, this hologram can be displayed in an SLM with a limited phase modulation depth of only 0.78p radians. In our case, since the SLM reaches 2p radians, the triplicator hologram can be combined with a converging lens function to make the finally displayed phase: u h( x) � pr 2 / kf, where p r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x 2 þ y 2 is the radial coordinate at the SLM plane
and f = 160 mm is the encoded focal length. Figure 2e shows this phase distribution modulo 2p; the gray level version of this phase pattern is loaded on the SLM.
Figure 3 illustrates the experimental results. In Figure 3a, the phase /( x) is given directly as the phase of the inverse FT of the flower pattern g( x 0) inFigure 2b. The Fourier transform hðx 0 Þ¼F e i / ðxÞ is obtained at the focal plane, centered at x 0 = x 0 0( actually the image shows the intensity distribution | h( x 0 � x ' 0, y 0)| 2), where x 0 0 stands for the lateral displacement caused by the linear phase term cx in equation( 4). The hologram reconstruction resembles the original pattern g( x 0) but shows the characteristic edge enhancement because the magnitude( amplitude) information of the FT was discarded, therefore reinforcing the high frequency content [ 16 ]. To avoid this effect, one simple classical solution [ 17 ] consists in calculating the inverse FT of a new pattern g( x 0) exp [ ir R( x ')], where r R( x 0) is a random phase that takes random values at each pixel with a uniform probability distribution in the range [ �p,+ p ]. The added random phase r R( x 0) addshighspatial frequency content to the original pattern g( x 0) whichdistributes the energy all over the Fourier domain. Thus, discarding the magnitude does not impact the hologram reconstruction so severely. The corresponding experimental result presented in Figure 3b now shows the flower pattern completely filled. The price to pay, though, is the arising of speckle noise within the pattern. As will be discussed in the next section, this noise plays a relevant role when using these holograms for correlation operations.
The results in Figures 3a and 3b exhibit a single term shifted laterally, and this is why we name them as blazed holograms, in analogy with the blazed grating which generates a single 100 % efficient first diffraction order. Figures 3c and 3d show results equivalent to those in Figures 3a and 3b where now the SLM displays the triplicator hologram u h( x) defined in equation( 4). Now the three terms described in equation( 6) are clearly observed. The zeroorder provides a bright spot, corresponding to the delta function in equation( 6). Theflower in the right part corresponds to the first diffraction order and again generates the function h( x 0 � x 0 0, y 0)( filled or edge enhanced depending on whether the random phase r R( x 0) was added or not before calculating the inverse FT), while the negative first order generates h *( �x 0 � x 0 0, �y 0), the inverted and complex-conjugate version of h( x 0, y 0) centeredatx 0 = �x 0 0. Note that the two flowers appear with the same intensity. The zero-order has the same energy but is focused on a single spot, thus appearing much more intense.