J. Eur. Opt. Society-Rapid Publ. 21, 33( 2025) 35
Fig. 4. Experimental hologram reconstruction when the triplicator hologram u h ðxÞ is summed with the original phase function / ðxÞ:( a, b) Result for the flower pattern with added random noise for( a) u h ðxÞþ / ðxÞ and( b) u h ðxÞ� / ðxÞ.( c, d) Result for the circle pattern and u h ðxÞþ / ðxÞ when( c) no random noise is added and( d) a random noise is added to the circle before calculating the hologram.
The first term in equation( 8) is the positive first diffraction order and gives the convolution [ q * h ] centeredatx 0 ¼ x 0. The second term is the convolution of q( x 0) withthed( x 0) term in equation( 6), thus reproducing q( x 0) on axis. Finally, the last term in equation( 8) is the negative first diffraction order and provides the convolution between q( x 0) andh *( �x 0), i. e., the cross-correlation [ q H h ] centered at x 0 = �x 0 0.
A particular case occurs when Q( x)= e i /( x) and consequently q( x 0)= h( x 0). In this case, the phase-only hologram loaded on the SLM is given by
e i ½ u hðxÞþ / ðxÞ
Š ¼ c 0 ie i2 / ðxÞ e icx þ e i / ðxÞ þ ie �icx; ð9Þ
and the corresponding FT diffracted field d( x 0) now gives the autoconvolution of h( x 0) at the positive first order, the replica of h( x 0) at the zero order, and the autocorrelation of h( x 0) at the negative first order. Note that, for this case, because of the perfect phase match in equation( 9) between the input phase and the phase encoded on the negative first order, the autocorrelation term results in a delta function centered at x 0 = �x 0 0.
Figure 4 shows experimental results considering Q( x) = e ± i /( x). In Figure 4a the phase-only hologram encoded on the SLM is the summation u h( x) + /( x) modulo 2p. This is namely the triplicator hologram combined with the original blazed hologram, where the flower pattern has the additional random phase. The central zero order reproduces the flower. Note that it has the same intensity as the two replicas observed in Figure 3d, thus showing the equal efficiency of the three terms, as expected from the triplicator design. Figure 4a shows a narrow autocorrelation peak localized on the negative first order, while the positive first order located on the right is a noisy cloud of light corresponding to the autoconvolution of the flower pattern. Figure 4b shows a similar situation, where now the displayed phase is u h( x) � /( x). Now the central order reproduces the inverted flower and therefore the correlation and the convolution appear on the opposite sides compared to Figure 4a.
Finally, Figures 4c and 4d illustrate two other interesting cases where again the u h( x)+/( x) pattern is displayed, now for a circular shape. In Figure 4c no random phase r R( x 0) was added to the original circle before calculating /( x). Therefore, when the original phase /( x) is added to the triplicator hologram, now the zero-order reproduces the edge enhanced circle. Because of its circular symmetry, its autoconvolution is equal to its autocorrelation, and two equivalent narrow bright spots are obtained at the ± 1st diffraction orders. Figure 4d shows the result when the random phase r R( x 0) was added to the circular object before calculating the hologram. Now, the hologram u h( x)+/( x) reproduces the full circle in the zero order and the correlation obtained in the negative first order is still a bright spot but the convolution obtained in the positive order is a broad noisy distribution.
5 Two-dimensional triplicators
In this final section, we combine two triplicator holograms in two orthogonal directions. Let us consider two different blazed holograms / 1( x) and / 2( x) that respectively
generate h 1 ðx 0
and h2 ðx 0 Þ ¼ F e i / 2ðxÞ in the
Þ ¼ F e i / 1ðxÞ
Fourier domain. The combined hologram derived from their corresponding triplicator versions u h1( x) and u h2( x) encoded in the x and y direction, respectively, is given by
e iu h1 e iu h2
¼ c 2
0 iei / 1 e icx þ 1 þ ie �i / 1 e �icx
ie i / 2 e icy þ 1 þ ie �i / 2 e �icy; ð10Þ
where the( x) dependence in the phase functions has been omitted for simplicity. The same linear phase slope c = 2p / p is considered along the x and y directions. The former equation leads to the following nine terms:
e iu h1 e iu h2
¼ c 2
0 �ei ð / 1þ / 2 Þ ic xþy
e ð Þ þ ie i / 2 e icy
�e i ð � / 1þ / 2 Þ e ic ð �xþy
�e i ð / 1� / 2 Þ ic x�y
e
Þ þ ie i / 1 e icx þ 1 þ ie �i / 1 e �icx ð Þ þ ie �i / 2 e �icy � e �i ð / 1þ / 2
Þ �ic xþy e ð Þ g: ð11Þ