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Abstract
An information retrieval technique from superimposed holograms representing 2D and 3D objects using complementary fringes is presented. By adding two different computer generated holograms with quasi-complementarity information is possible to retrieve information at a specific depth.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Information retrieval through wavefront and hologram reconstruction techniques such as wavefront at different planes [1], holograms by phase retrieval algorithm [2], Babinet’s principle and complementary [3], information retrieval using wavelet [4], coherent superposition by vector operations [5] are well known. Some techniques use fractional Fourier transform, where the rigor of information encryption has been increased. Applied to digital holography, holographic multiplexing techniques, and holograms with 3D object information [6–9].
In recent years different approaches have been introduced and new holographic encryption techniques were developed. The use of diffuse object in the Fourier plane as 2D encryption element [10,11], multiple object encryption [12,13] or by introducing three-dimensional objects as encryption information [14,15] has been proposed. These works have in common the use of holographic masks in different configurations to produce encryption. Depending on the physical implementation of the technique it uses either an optical setup using a CCD to capture the information or focus only on the digital reconstruction.
It is possible to combine both approaches to improve encryption. This is achieved by positioning a holographic mask element operating as an optical key throughout optical propagation and subsequently processing the information throughout digital propagation [16,17]. In these works the process is obtained by digitally subtracting the key code to retrieve the desired information.
In this paper we present the recovery from the holographic recording of a whole set of objects $O_q$ at different positions $Z_q$ (Fig. 1) and a reference beam $R_q$(x,y), called the Primary Holographic Code (PHC), of a selected object at its depth by optically adding a secondary holographic information code (SHC) which contains the same information than the PHC except the information of the chosen object with a phase changed to obtain the complementary fringe set. In the absence of the SHC, the PHC will reconstruct a mixture of all the objects unfocused.
Fig. 1. Information from different objects $O_q$ (x,y) at different distances $Z_q$ overlapping with a specific reference beam $R_3$(x,y) to form a primary hologram code.
2. Theory
The fundamental process in which we achieve image subtraction is by adding two phase masks, PHC and SHC, where the phase inversion mask SHC corresponds to the complementary interference pattern of PHC eliminating the information of a specific object, to obtain such selected object (see Appendix).
The sum of PHC and SHC in Eq. (1) provides a hologram of the object of interest. It is expected that additional information is canceled giving a clean reconstruction. The wavefront generating PHC is constructed using the Huygens-Fresnel-Kirchhoff equation [18,19], expressed in the Eq. (2). We use this equation in order to be able to operate on 3D objects. Based on the fact that Fourier theory works best for 2D plane object propagation using angular spectrum theory, 3D objects are treated as 2D slices in different planes. If we consider 2D flat objects we obtain [20,21]:The variables $(u, v)$ are the spatial position in the propagation plane, $(x_a, y_b, z_c)$ represent the positions of each point of the object, k is the wavenumber and, finally, $n, m, p$ are the total number of elements in the object for each coordinate. The objects $O_q$ can be either flat, 3D objects, or a mixture of both with different positions for each one. An important point to consider is that the sampling is the same in all wavefronts, in such a way that they can be properly combined without unwanted effects.
2.1 Formation of the PHC
The wavefronts of the objects represented by Eq. (2) can be superimposed with a single spherical reference beam with a radius of curvature equal to the distance of the object to be reconstructed. Thus, each particular wavefront is superimposed to form the PHC using optimized calculation techniques [22–26]. This can be generalized for all objects $Oq$ in the following way (see Appendix):
Equation (3) represents the general shape of the PHC as the wavefront superposition of $M$ objects. In this equation the term $R (x,y) exp(ikr)$ represents the reference beam at distance $z$ to one object. This physical condition is essential to obtain lensless Fourier holograms only for this object, facilitating its location in the hologram reconstruction.2.2 Formation of the SHC
The secondary hologram SHC is constructed similarly to PHC using the same reference beams and wavefronts for each object, except of the wavefront of the object to be reconstructed (see Appendix):
Another difference found in Eq. (4) is the pi phase shift factor. This term causes a shift in the fringes pattern, creating a hologram partially complementary to the PHC. This phase shift factor can be included in either each of the wavefronts prior to the calculation or the sum of these. Furthermore, because we use the interference pattern of the holographic code, the phaseshift can be applied on the reference beam providing the same result.
The primary and secondary holographic encryption codes can also be on axis holograms such as the Gabor hologram [18,27]. In this case the reference beam is at an angle relative to the central axis and has an uniform distribution amplitude with spherical wave phase front. All objects in the data sampling window are centered on the propagation axis as depicted in Fig. 1. By using a reference beams with the same distance that the object to decrypt, the PHC and SHC superposition will produce a lensless Fourier-type hologram for that object [18,28,29].
3. Results
To exemplify the above procedure described different objects were used at specific distances. We used a mixture of figures with simple properties such as binary objects, gray scale images or random structures. The selection was made in order to improve reconstruction visualization based on calculation at different propagation planes with defined information.
Figure 2 shows the objects which were propagated at $z_1$ = 10 mm, $z_2$ = 13 mm, $z_3$ = 15 mm, $z_4$ = 17 mm, $z_5$ = 20 mm and $z_6$ = 25 mm, respectively. The selected objects have 300x300 pixels resolution. This resolution allows good reconstruction with modest computational requirements, which can also be accommodated in a space light modulator with adequate performance.
Fig. 2. Images used in our experiments (a) Letter A, (b) INAOE Logo [30], (c) Image of Augustin Fresnel [31], (d) gray level picture, (e) Letter E, and (f) distribution of random numbers between 0 and 255.
Each object is individually propagated using Eq. (2) to construct its wavefront. Later they were combined with a reference beam to form the PHC, as in the Eq. (3). To recover the objects in the PHC hologram, we used the reference beam which has the same distance as the object to be reconstructed. This procedure facilitates its location during reconstruction but this is neither a requirement nor a limitation of the method.
Fig. 3. (a) PHC constructed by all the objects represented in Fig. 2 and (b) reconstruction at z=15mm, the distance of the reference beam (full resolution PHC in Visualization 1.
Figure 3(a) shows a PHC presented in 256 gray levels, generated with the wavefronts of all the objects in Fig. 2 and Eq. (3). All the values of the calculated information were stored as a double precision data file, then scaled from 0 to 255 to represent the hologram image in gray levels and reproduce it in a spatial light modulator (SLM). Using numerical reconstruction the information from multiple objects can be observed in Fig. 3, making it difficult to isolate each of the original images.
In a similar way a SHC is represented by Fig. 4(a). This SHC corresponds to the superposition of all the individual wavefront of objects Fig. 2, with the exception of the wavefront from Fig. 2(c). Once the calculations are made according to Eq. (4), the process is similar to the PHC with the exception for a scaling from 0 to 180 due to scale corrections.
Fig. 4. (a) SHC secondary holographic code of object desired , (b) reconstruction of SHC using FFT algorithm (full resolution SHC in Visualization 2.
Gray level holograms are reconstructed using Huygens Fresnel Kirchhoff propagation or the Fast Fourier Transform algorithm (FFT). When the reconstruction is done at its particular distance ($z_3$ = 15mm in our case) we have a the reconstruction shown in Figs. 3(b) and 4(b). The reconstructions show a conjugate pair of overlapped images, difficult to interpret and see the details, at the most we can talk of some differences among the reconstructed images.
3.1 Flat image reconstruction at selective distances
The individual reconstructions of the PHC and SHC have too much information in addition to the desired reconstructed image. But, getting a clear reconstruction is possible by adding the PHC and the SHC if the reference beam is the same for both holograms.
The holograms in Fig. 3(a) and Fig. 4(a) are added as shown in Eq. (1) and the sum of them extracts a chosen hologram from one of the objects Fig. 5(a). The hologram is numerically reconstructed with fast Fourier transform, and the result can be seen in Fig. 5(b).
Fig. 5. (a) Hologram resulting from the sum of holograms from Fig. 3(a) and Fig. 4(a), (b) reconstruction of the sum of both holograms (undersampled, full resolution image in Visualization 3).
For a successful recovery of the object and its conjugate in the same plane, it is necessary that the reference beam has the same distance as the object to be reconstructed. The Fig. 6 shows the reconstruction at different distances, causing a blur and magnification effect [32] due to the distance from the focus plane. Taking a distance of z = 15 mm, that was the original distance of the object, the reconstruction is in the correct position and a sharp image is obtained (fourth from left to right in Fig. 6).
Fig. 6. Numerical reconstruction of the hologram in Fig. 5(a) using a FFT algorithm with different distances z=10mm, 13mm, 14mm, 15mm, 16mm, 17mm, 20mm, 25mm, from left to right.
There is no remarkable difference if we compare the reconstruction of the hologram resulting from Eq. (1) with the reconstruction of a hologram designed only with the desired object. Correlating the reconstructed image and the original a maximum correlation value of 0.9993 is obtained, thus confirming the proposed procedure (Fig. 7).
Fig. 7. Correlation plot between the hologram of the object alone and the resulting hologram when the sum of PHC and SHC.
The above results were obtained with the object in Fig. 2(c). Similar reconstructions were obtained for each object by selecting the corresponding reference distance in the PHC to the position of the desired object and with the appropriate SCH.
Figure 8 depicts the reconstruction of the objects in Fig. 2, where we can appreciate size difference between the reconstructed objects with respect to the original ones. This scaling is due to the different recording distances and the use of spherical waves. The device used to obtain the optical reconstructions is a bbs HD Kit: LCD L3C07U-85G13, with a transmission screen of 1920 x1080 (HD) 0.74 "diagonal. The quality of the reconstruction is determined by the dimensions used in the holograms size (600x600 pixels), in conjunction with the restriction imposed by limited representation of the SLM used in the optical reconstruction.
Fig. 8. Numerical (a) (b) (c) (d) (e) (f) and optical(g) (h) (i) (j) (k) (l) reconstructions at the corresponding distances for each objects on Figs. 2(a)–2(f), respectively.
Due to transmission losses in the optical setup a gray level scaling was introduced on each experiment, as previously mentioned. This is done to obtain a correct leveling of the fringes, in such a way that additional information becomes a constant background. The result of this is the cancellation of the common information of both holograms, leaving only the diffractive information of the desired object. In the same way, if desired using the resultant hologram as an image on the SLM is required a new grayscale correction.
3.2 Three-dimensional objects
Using the propagation equation Eq. (2), we can work with three-dimensional objects with relative comfort in different configurations. The concept of Fourier hologram without lenses becomes a little different and the visualization of the object is a bit complicated.
To exemplify the use of three-dimensional objects, we placed images on a three dimensional spiral (Fig. 9). To construct a 3D object three flat objects in different positions along the spiral were used. Figure 10(a) shows the hologram formed with the images on the spiral: Fig. 2(b) at the spiral top, Fig. 2(f) at the middle of the spiral and, Fig. 2(c) at the spiral bottom. The reconstruction at the focal plane (Fig. 10(b)) shows a noisy reconstruction in which we can see a blurry the object in the middle and very few signals of the spiral can be seen. Here the reference beam was aligned to the middle of the spiral. Using this configuration the reconstruction of the objects and their conjugates within the same propagation space are obtained.
Fig. 9. Perspectives of a three-dimensional spiral used to construct a 3D object. (a) front and (b) lateral views
Fig. 10. (a) Detail of the PHC of various objects on a spiral wavefront (complete PHC in Visualization 4), (b) Zero order numerical reconstruction at the focal plane.
Considering that we are working with flat objects, we can prepare an SHC to recover an object when combined with the PHC of Fig. 10. Unlike to the previous section, in Fig. 11(b) there is a numerical reconstruction obtained without changing the reference beam. This causes the object to focus on a different plane as can be observed because the zero order reconstruction and its conjugate image do not focus on the same plane.
Fig. 11. (a) Detail of the SHC necessary for the reconstruction of INAOE logo (Fig. 2(b)) placed on the top of the spiral (complete SHC in Visualization 5). (b) Numerical reconstruction of the PHC and the SHC combination in the object focal plane.
Moving the reconstructing plane, Fig. 12 depicts an image of the reconstruction of the PHC of Fig. 10(a), the SHC used (Fig. 12(a)) and the reconstruction of the sum of holograms (Fig. 12(b)). To obtain a better interpretation, these reconstructions were prepared as a video format, so these images are only a capture of some calculations (see Visualization 6). In these results we can appreciate the difficult location of the spiral in the assembly against the filtered and clean version of it.
Fig. 12. (a) Numerical reconstruction of the PHC. (b) Numerical reconstruction of the sum of the PHC and SHC (see Visualization 6).
It must be clarified again that, due to the nature of each object, it is necessary to perform an appropriate scaling for each of the SHCs that are to be used. For example, the spiral presents a very low intensity to the hologram so Fig. 13(a) practically does not need to be scaled, while, for the reconstruction of Fig. 11 it was necessary to scale the SHC to attenuate it by 20% and 25% for the SHC of Fig. 13(b).
Fig. 13. Pieces of SHC needed to get (a) the spiral. (b) the image of Fig. 2(c) (undersampled, full SHC in Visualization 7).
3.3 Experimental setup description
The experimental setup used is depicted in Fig. 14. We used a 532 nm laser source. The laser beam passes through a microscope objective, with the purpose of expanding the beam as a spherical wave. A lens is added to ensure that the focal length coincides with the radius of curvature of the spherical wave, so that it works as a collimator system.
The collimator produces flat waves illuminating the SLM (see Fig. 14). The modulator is a micro LCD screen transmission (Liquid Crystal Display) with a resolution of 1920x1080 pixels HD. This micro monitor is the link between computer-generated holograms using algorithms and rules of mathematical physics, and experimental optics, where optical physical phenomena of diffraction are optically reconstructed.
Having the PHC and SHC calculated they overlap to use that information in the SLM to obtain its reconstruction with propagation. At the exit of the modulator the light is focused with a lens forming the diffraction patterns of the corresponding code to be reconstructed in the focal plane. This location is known because it was used to calculate the formation of lensless Fourier holograms.
4. Discussion
The technology to manipulate information through holograms is attractive due to the versatility and plasticity with which the information can be modulated for specific purposes. We present an arrangement to retrieve information of overlapping objects at different depth planes. The concept of correct information retrieval is found on complementary fringes, as observed in amplitude division interferometers, such as Michelson, Twyman-Green, Mach Zehnder [33].
We propose a PHC formed with superimposed wavefronts containing all the data from the original objects. The SHC is formed with all the original information except for the object to be reconstructed. Taking care of multiplying the wavefronts in the SHC by a factor of $exp(i\pi )$ with the intended consequence of a partial complementing, and hence erasure, of the interference patterns from the first and second holograms, leaving behind the chosen object to be reconstructed.
The number of images which can be overlaid to form the primary and secondary holograms depends on different factors. If the data from object are developed in a virtual environment, the restrictions are minor and the result is more reliable. In this case the limitations are related to the rounding error and computer accuracy. A virtual environment tolerates an unrestricted range of values to generate a hologram. However, when the holograms are displayed on a SLM there are limitations to the technique in which the mapping of the hologram in discrete values is the most important one. In this case all the information representing the virtual hologram is mapped to match 256 values, which is the standard in SLMs. Nowadays, an extended range of gray levels are being implemented on SLM technology [34].
As the objects overlap one behind the other, the recovery of each individual image is complicated, as it is affected by the blurring of the others. In the reconstruction of Figs. 12, the presence of the spiral is practically nil, until it is cleaned by the appropriate SHC.
Because of these restrictions, when the PHC is overlapped with the SHC, a gray levels correction factor must be included in one of the holograms to obtain a clean reconstruction. This factor depends on the number of images, the distance between them, system losses and the dimensions in number of pixels that make up each image. For example, in images of holograms Fig. 3(a) is mapped to 256 gray levels, while Fig. 4(a) is mapped to 180 gray levels. When overlaid, they generate the hologram in Fig. 5(a), to obtain an optimally reconstructed image of Fresnel.
This gray level compensation is applied for each image with different values for the compensation factor. SLMs are also limited in their spatial resolution, which is the number of pixels that form the display such as HD with 1920x1080 or 4k with 3840x2160. When dealing with an object with a high level of detail, the resolution must be high enough to be able to calculate the propagated field to form the hologram with less constraints, and then reconstruct the object preserving the original details.
At the same time, the resolution and modulation of the SLM is a very important role in the use of three-dimensional objects. Being these more complex, they usually need a higher resolution to perceive the details. On the other hand, the sizes necessary for diffraction produce small reconstructions that limit us in their perception and the use of spherical waves introduces certain magnification effects.
As can be seen in Fig. 12, the visualization of objects in space is complicated due to the projection in planes, as well as their propagation information since it is obstructed by other unfocused parts of it. This is why the spiral was used, avoiding some of these problems and managing to appreciate each focused segment of it.
The complexity of encryption can be deepened by varying and combining parameters such as: propagation wavelength, arbitrary separation between objects, arbitrary dimensions of objects, color palettes of objects, and spatial dimension of each front of the spread wave. All contained in a primary hologram and a secondary hologram, where both form a powerful tool.
In this work, the combination of both PHC and SHC was done digitally to accommodate the use of only one modulator (see Fig. 14). But, conceptually it is possible to construct an experimental arrangement using two SLM systems one aligned immediately after the other, where the PHC is shown on one and the SHC on the other. The propagation trough both devices would produce the described effect and thus obtaining the desired reconstruction. It is important to clarify that this experimental setup should work if both systems, SLM1 and SLM2, are perfectly aligned (active area from pixel to pixel) and are in optical contact.
5. Conclusion
The theory described in this manuscript open new possibilities in information encryption information. Taking into account the difficulties discussed involving the use of three-dimensional objects, they have enough complexity and high encryption capability which reinforce this technique, improving the performance as well as their potential. The information retrieval, achieved by applying the principle of complementarity, makes this theory a useful and robust tool in information recovery limited only by calculation precision in the virtual field, or by the current limitations of the SLM in the experimental field.
6. Appendix
In order to understand the holograms addition in Eq. (1) and its terms in Eq. (3) and Eq. (4) to recover the object $W_q (u,v)$ the following algebra was used. We started with Eq. (1) rewrote in Eq. (5).
The terms contained in Eq. (5) are represented by Eq. (3) and Eq. (4). Rewriting Eq. (3) to homologate with Eq. (4) have.1.- The zero order: $2R(x,y)^2$
2.- Object information through the wavefront and its conjugate, depending on the phase of the reconstruction reference beam: $R(x,y)[W_q (u,v)exp(-ikr)+W_q (u,v)^* exp(ikr)]$
3.- Halo scatter noise and other terms involved:
Disclosures
The authors declare no conflicts of interest
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