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J. Eur. Opt. Society-Rapid Publ. 21, 33( 2025)
Fig
. 5. Experimental reconstruction for the combination of two triplicator holograms encoded in the x – y directions.( a, b) The same triplicator hologram is encoded in the x and y directions for( a) the flower and( b) the circle.( c) Equivalent result for the circle object when a different realization of the added noise r R ðx 0 Þ is applied to the horizontal and the vertical triplicator hologram.( d) Result when the hologram in( c) is combined with the complex-conjugate version of the phase function / 1 ðxÞ encoded in the horizontal direction.
Therefore, the diffracted field dðx 0 Þ ¼ F e iu h1 e
iu
½ h2
Š is given by dðx 0 Þ ¼ c 2
0 � h �
½ 1h 2 Š x 0 � x 0 0; � y0 � y 0 0 þ ih2 x 0; y 0 � y 0 0 �
� ½ h 1 Hh 2 Š x 0 þ x 0 0; � y0 � y 0 0 þ ih1 x 0 � x 0 0; y0 þ d ð x 0; y 0 Þ � þ ih
1 �x0 � x 0 0; � �y0 � ½ h1 Hh 2 Š �x 0 þ x 0 0; �y0 � y 0 0
� � þih 2 �x0; �y 0 � y 0 0 � ½ h1 h 2 Š �x 0 � x 0 0; �y0 � y 0 0 g: ð12Þ where x 0 0 ¼ y0 0
¼ kf = p. This relation describes the diffracted field as an array of 3 3 terms. The delta function represents a central spot located on axis. The positive first diffraction order diffracted in the horizontal and the vertical directions reproduce h 1( x 0) and h 2( x 0) respectively. The corresponding negative first orders carry their inverted complex-conjugate versions. Finally, the diagonal and anti-diagonal terms reproduce the convolution [ h 1 * h 2 ], the correlation [ h 1 H h 2 ] terms, respectively.
Figure 5 shows some experiments illustrating this situation. In Figure 5a the functions / 1( x) and / 2( x) used to calculate the triplicator holograms are identical, h 1( x 0)= h 2( x 0), the flower pattern filled with the same added noise. Note that the direct flower is retrieved at the positive horizontal and vertical orders, while the inverted flower is retrieved at the negative orders. The anti-diagonal elements show three equally-intense bright spots, the central one corresponding to the delta function in equation( 12)) and the other two corresponding to the autocorrelation terms. Finally, the two diagonal terms give the autoconvolution.
Figure 5b shows equivalent results now for the circular shape. Like in Figure 5a, the same random noise is used to calculate the phase functions / 1( x) and / 2( x). This is not the case in Figure 5c, where a different realization of the added random phase r R( x 0) is selected for the circular pattern encoded in the horizontal triplicator and in the vertical triplicator. As a consequence, although the circular shape is identical, the noise is different, and therefore the anti-diagonal autocorrelation peaks appearing in Figure 5b