34
J. Eur. Opt. Society-Rapid Publ. 21, 33( 2025)
Fig. 3. Experimental hologram reconstruction for:( a) Blazed hologram of the original flower pattern.( b) Blazed hologram of the original flower pattern multiplied with a random phase.( c, d) Triplicator hologram version of( a) and( b).( e) Triplicator hologram of a circle pattern.( f) Triplicator hologram of a circle pattern multiplied with a random phase.
Finally, Figures 3e and 3f show equivalent experimental results where, instead of the flower, a binary amplitude circular shape is considered. Of course, in this case, the inversion in the negative order bears no difference to the positive one. As a result, the two diffracted orders – the one containing the object reconstruction and the other its complex conjugate and inverted version – produce identical intensity patterns in the focal plane. This symmetry ensures that both diffraction orders can be used equivalently in applications relying on intensity-based analysis.
4 Convolver and correlator triplicator grating
Correlation optics was a very popular topic at the end of the 20th century due to its application in information optics and pattern recognition [ 18, 19 ]. Nowadays it continues to find applications, for instance, in the detection of modes in structured light beams [ 20, 21 ]. Optical convolution is a basic operation in Fourier optics, and it finds modern applications for instance in the design of metasurfaces [ 22 ]. Moreover, recent advances in diffractive neural networks have revived the interest in optical implementations of convolution, offering high-speed and energy-efficient alternatives to electronic processing [ 23, 24 ]. The ability to perform these operations optically – often in a single step and with inherent parallelism – makes them highly valuable for emerging applications that require low latency and high throughput [ 25 ].
Let us now consider again the field hðx 0 Þ ¼ F e i / ðxÞ reconstructed by the hologram /( x), where we add another
complex function such that qðx 0 Þ¼F ½ QðxÞŠ. Theirconvolution( q * h) and cross-correlation( q H h) are given respectively by [ 26 ]
½ q hŠðx 0 Þ ¼
½ q H hŠðx 0 Þ ¼
ZZ þ1
�1
ZZ þ1
�1
qðÞh s ðx 0 � sÞds ¼ F QðxÞe i / ðxÞ; ð7aÞ
qðÞh s ðs � x 0 Þds ¼ F QðxÞe �i / ðxÞ;
ð7bÞ
where s =( s x, s y) denote the 2D integration variable. Note that the cross-correlation between q( x 0) and h( x 0) is equivalent to the convolution between q( x 0) and h *( �x 0).
Therefore, the generation of the functions h( x 0) and h *( �x 0) in the Fourier plane of the triplicator hologram can enable to implement the hologram as an optical convolver and correlator. Let us consider the product
QðxÞe iu hðx
Þ, where the triplicator hologram is multiplied
by Q( x). Taking into account equation( 5), thisproduct contains three equally energetic terms: 1) the positive first order carries the product of Q( x) with the phase-only term e i / ðxÞ; 2) the negative first order carries the product of Q( x) with the complex-conjugate phase e �i / ðxÞ; and 3) the zero order carries Q( x) multiplied simply by a constant factor. Therefore �, the corresponding Fourier transform d x 0 ¼ F QðxÞe
iu h ðxÞ yields the convolution of q( x 0) with the three terms in equation( 6), and results in three new terms given by
dðx 0
Þ ¼ ic 0 ½ q hŠðx 0 � x 0 0; y0 Þþc 0 qðx 0 Þ þ ic 0 ½ qHhŠðx 0 þ x 0 0; y0 Þ: ð8Þ