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J. Eur. Opt. Society-Rapid Publ. 21, 33( 2025)
correlator. Experimental results are provided using a phaseonly liquid-crystal on silicon( LCOS) SLM.
2 Triplicator hologram design
The analytical profile leading to the optimum phase triplicator demonstrated by Gori et al.[ 1 ] is given by the phase function: uðxÞ ¼ arctan ½ a cos ðcxÞŠ: ð1Þ
Here x =( x, y) denotes the two-dimensional spatial coordinates in the plane of the grating and c = 2p / p where p is the grating’ s period. The numerical constant a = 2.65 leads to three equally-intense diffraction orders( the 0th and ± 1st orders) with a total efficiency g 0 ± 1 = I 0 + I 1 + I �1 = 92.6 %. Thus, ignoring the weak higher harmonic orders, the triplicator grating in equation( 1) can be approximated as the following Fourier expansion:
e iuðxÞ ffi c þ1 e icx þ c 0 þ c �1 e �icx;
where the Fourier coefficients in the three target orders take the values c 0 = 0.556 and c ± 1 = ic 0 [ 12 ]. Therefore, the diffraction pattern d( x 0) generated in the Fourier transform domain is written as
� dðx 0 Þ ¼ c 0 idðx 0 � x 0 0; y 0 Þþdðx 0; y 0 Þþid x 0 þ x 0 0; y 0:
ð3Þ
Here x 0 =( x 0, y 0) are the spatial coordinates at the Fourier plane. If the grating is illuminated with light with wavelength k and the optical FT is achieved in the back focal plane of a lens with focal length f, x 0 is related to the grating’ s spatial frequencies u =( u, v) asx 0 = kfu. The separation between orders is given by x 0 0 ¼ kf = p.
Let us now assume a phase-only Fourier hologram / ðxÞ,
whose reconstruction function is hðx 0 Þ¼F e i / ðxÞ, where
F ½Š indicates the FT. The proposed triplicator hologram is then calculated by inserting /( x) intoequation( 1); this transformation results in a new phase triplicator function u h( x) givenby ð2Þ
u h ðxÞ ¼ arctan ½ a cos ð/ ðxÞþcxÞŠ: ð4Þ
where the linear term cx is kept to spatially separate the different orders in the FT domain.
A well-known result from digital holography is that if the phase values of a phase-only function with a uniform probability density function in the range [ �p, p ] are modified, the resulting new phase-only function can be described with a Fourier series expansion due to the phase 2p periodicity [ 15 ]. Thus, the modified hologram given by equation( 4) can be expressed as
e iu hðx
Þ ¼ Xþ1 m¼�1 c m e im ½ / ðxÞþcxŠ ffi c 0 ie i / ðxÞ e icx þ 1 þ ie �i / ðxÞ e �icx; ð5Þ
where the Fourier series is again approximated by the three main terms of the optimum triplicator phase profile used in this study. This relation shows that these three main terms of the triplicator hologram carry, respectively, the original phase hologram, its complex-conjugate version, and a constant DC term. Note that the linear phase cx appears with the opposite sign in the 1storders.
The corresponding FT diffracted field dðx 0 Þ ¼F e iu hðxÞ is therefore given by dðx 0 Þ ffi c 0 ihðx 0 � x 0 0; y0 Þþdðx 0; y 0 Þþih ð�x 0 � x 0 0; �y0 Þ:
ð6Þ
This shows that the triplicator hologram reconstruction contains the original function h( x 0) shifted laterally a distance x 0 0
, the inverted and complex-conjugate version h *( �x 0) shifted laterally a distance – x 0 0 and, finally, the DC order delta function centered on axis.
Therefore, this phase hologram yields two spatially separated dual replicas( original and inverted-complex conjugate) of the original function, centered at x 0 =± x 0 0. A third contribution appears at x 0 = 0, corresponding to a constant( DC) term, which does not contain objectrelated content. This ability to generate multiple spatially separated instances of the same information enables flexible optical processing strategies, such as simultaneous filtering, correlation, within a compact and passive setup as we will show in the next sections.
3 Experimental system and triplicator hologram reconstruction
The experimental setup is shown in Figure 1. A He – Ne laser with wavelength k = 632.8 nm is first spatially filtered and collimated by lenses L1 and L2. A variable neutral density filter( NDF) placed at the laser output allows to adjust the intensity level. The collimated beam illuminates a liquid-crystal on silicon( LCOS) SLM( Thorlabs, model Exulus-HD1), featuring a full resolution( 1920 1080) parallel-aligned display with square pixels of D = 6.4 lm pixel pitch and fill factor F = 93 %. The SLM works as a phase-only display by aligning the input linear polarizer( P) parallel to the liquid-crystal director axis. For the 632.8 nm wavelength, the SLM exhibits a 2p linear phase modulation depth as a function of the addressed gray level. A Fourier phase-only hologram is loaded together with a lens phase function. Thus, the beam reflected on the SLM is focused on a camera detector located at a distance equal to the encoded focal length f, where the hologram reconstruction is obtained. Note that encoding the lens on the SLM together with the hologram, instead of using an external lens, avoids focusing light reflected at the external surface of the SLM, which remains collimated and only contributes to a small background noise.
Figure 2 illustrates the triplicator hologram design and its implementation in the SLM. Figure 2a shows the phase profile of the optimum triplicator grating given by equation( 1). The phase profile is a continuous function with smooth variations between ± 0.39p radians. Figure 2b shows the binary amplitude pattern of a lotus flower that we consider to design the CGHs. This is the original object function