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Abstract
Computer generated holograms (CGHs) are powerful optical elements used in many fields, such as wavefront shaping, quality testing of complex optics, and anti-counterfeiting devices. The Lee algorithm is the most used to generate binary amplitude Fourier holograms. Grayscale CGHs are known to give a higher reconstruction quality than binary holograms, but they usually require a cumbersome production process. A very simple and straightforward method of manufacturing rewritable grayscale CGHs is proposed here by taking advantage of two key components: a digital micro-mirror device (DMDs) and a photochromic plate. An innovative algorithm, named Island algorithm, able to generate grayscale amplitude Fourier CGHs, is reported and compared with the standard Lee approach, based on 9 levels. A crucial advantage lies on the fact that the increase or decrease of the quantification does not affect the spatial resolution. In other words, the new coding leads to a higher spatial resolution (for a given CGH size) and a reconstructed image with an order of magnitude higher contrast with respect to the classical Lee-coded hologram. In order to show the huge potential of our approach, a 201 level Island hologram is designed, produced and reconstructed, pushing the contrast to values higher than$\; {10^4}$. These results reveal the potential of our process as well as our algorithm for generating programmable grayscale CGHs.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Computer Generated Holograms (CGHs) are attractive optical elements that are finding applications for beam shaping, particles manipulation, interferometric optical testing and anti-counterfeiting [1–4]. They are classified in different families and types, depending on how they are calculated and in which way the information is stored. The mathematical transformation used for the hologram calculation determine the family: Fresnel holograms are calculated from the light propagation equations and are reconstructed directly upon illumination, while Fourier holograms are the encoding of the inverse Fourier transform of the object, and require a lens for the reconstruction [5]. The complex function representing the hologram may be recorded either in phase or in amplitude: in the former case, holograms are obtained by recording a phase variation in a material with a refractive index or thickness modulation, and in the latter case, holograms are obtained by controlling the local transparency of the sensitive material [6]. They both provide a similar performance in terms of image reconstruction quality, but phase holograms are usually preferred for the higher diffraction efficiency. In fact, binary phase holograms show a maximum diffraction efficiency of 40% in the first order, whereas binary amplitude holograms are limited to 10% [7].
Concerning the discretization, binary holograms are usually preferred even if grayscale holograms (both amplitude and phase) are known to give a higher reconstruction quality [7], with the strong drawback of a more complex and challenging realization process. Phase grayscale CGHs have been obtained by micro-lithography [8], with a process requiring to consecutively and perfectly align a set of masks, and, additionally, to develop the pattern after each exposure step [9]. As for amplitude CGHs, they are produced by means of both lithographic and direct writing techniques, which allow a binary writing only. To our best knowledge, we have been among the first ones to record an amplitude grayscale CGH [10].
Our recording process is based on the use of two key components: a Digital Micro-mirror Device (DMD) and a non-threshold photosensitive material, namely the photochromic material. DMDs are programmable devices, composed of millions of micro-mirrors controllable individually and in real time, for displaying images or spectroscopic applications [11]. They reproduce any binary pattern and are used to generate dynamic binary or grayscale CGHs [12,13] by exploiting the mirrors fast switching at frequencies higher than human vision frame rate, and suitable for real-time holographic display. But in these cases, the grayscale emerges from a dynamic effect and is then not formally recorded. Moreover the CGH resolution is limited by the actual size of the DMD and cannot be adapted to a larger scale. In addition, a background noise due to the uncontrolled reflection on the mirrors edges makes such holograms useless for interferometry and metrology.
DMDs can be also used as programmable masks to project a given pattern on photochromic plates with incoherent light, allowing recording amplitude CGHs on it. Indeed, the photochromic film becomes progressively transparent when illuminated with light of suitable wavelength [14]. Binary amplitude CGHs are obtained when a single mask is projected until the material becomes transparent, but, with a smart control of the DMD, we demonstrated grayscale CGHs obtained in a one-exposure process of the same duration, without any developing step [10]. This ready to use hologram can also be erased and rewritten, thanks to the reversibility of the transformation in P-type photochromic materials. We also demonstrated the manufacturing of photochromic binary CGHs for optical testing obtained by direct laser writing [15].
In the framework of photochromic materials, a few papers are available, showing the possibility to produce (binary) programmable holograms in diarylethenes and fulgides doped films [16,17] both as amplitude and phase hologram [18]. Indeed, it has been shown that diarylethenes exhibit a large modulation of the refractive index in the NIR that makes them suitable for making pure phase holograms [14]. Other interesting works focus on the real-time holography based on imidazole dimers [19,20] and on photorefractive materials [21]. In the latter case, it has been demonstrated the possibility to reach high efficiency in phase holograms, but an electric field is required to enhance the refractive index modulation and this makes the system more complex than the pure photochromic one.
Concerning Fourier CGHs, up to now, only binary Fourier holograms where produced by developing smart coding algorithms, which were necessary to overcome the binary limitation, but with the drawback of a reduced reconstruction quality and diffraction efficiency. Between Lohmann, Lee and Burkhardt algorithms, which all generate binary CGHs with different compactness and quantification of information, the Lee algorithm is the most widely used and taken as a reference for the current work. The grayscale approach opens to new possibilities to increase the hologram information density and hence the reconstruction fidelity against binary holograms of the same size and resolution.
In this paper, we develop an innovative coding, named Island algorithm, based on grayscale discretization to be applied to photochromic active substrate to produce high fidelity grayscale Fourier holograms where the spatial resolution is decoupled from the coding accuracy. Lee-coded holograms will be taken as reference for judging our approach in order to compare the performances of the algorithm we developed to generate grayscale amplitude Fourier CGHs. Very large improvements in terms of Signal to Noise ratio and fidelity are obtained by keeping the same efficiency.
2. Amplitude Fourier CGH
Fourier holograms exploit the property of lenses to apply a Fourier transformation of the pupil plane, and are essentially the encoding of the Fourier transformation of the image. Hence, the complex wavefront represented by the hologram can be calculated using the inverse Fourier transformation of the image to be reconstructed. Inversely, the reconstructed image can be computed using the Fourier transformation of the hologram. Once calculated, the complex function representing the hologram will be coded in a sensitive substrate, if possible with the ability to modulate both the amplitude and the phase of the wavefront. Although some attempts where successfully done to build multi-material CGHs, they require multi-step processes and complex procedures [6]. Accordingly, the traditional approach requires coding the complex wavefront in the form of a phase only or amplitude only map. In the case of amplitude holograms, the complex wavefront is approximated into a map of transparency levels.
2.1. Binary CGH generation algorithms
The first algorithms developed by Lohmann and Brown proposed a binary amplitude modulation detour-phase hologram to partially implement full complex-amplitude modulation using a photographic film. It consists in a map of opaque cells, each one encoding the full complex-amplitude distribution depending on the position and shape of a transparent subcell [22].
Another coding that produces more compact CGHs with the same reconstruction quality is the one proposed by Lee, encoding the real and imaginary parts of the complex Fourier function instead of the phase and amplitude [22]. The binary version of the algorithm, which is the only one developed up to now, foresees a 4 × 4 elements cell for each pixel and encodes the real and the imaginary part of the complex number in 9 quantification levels as in the Fig. 1(b) (8 discrete levels plus the zero position).
Fig. 1. (a) Complex number to be encoded; correspondent cell of the Lee algorithm (b), Lee compact algorithm (c) and Island coding (d); example of the Lee hologram (e), Lee compact hologram (f) and Island hologram (g).
Each column corresponds to one of the semi-vector of the complex plan. The positive or the negative value of the real part is stored in one of the two first columns and the same logic is followed to store the imaginary part in the two last columns. To increase the number of quantification levels, the cell must be increased to 8 × 8 elements cell for 33 levels, and 16 × 16 elements cell for 129 levels, increasing the hologram size.
2.2. Grayscale CGH generation algorithms
Thanks to the properties of the photochromic plate and to the DMD-based recording setup, we are able to control the local transparency of the plate to generate grayscale maps in a straightforward fashion [10]. Our first algorithm uses the transparency to pop-up the columns of Lee cells, squeezing the information of a quadrant of the complex plane into a single element (Lee-Compact).
As shown in Fig. 1(c), the cell is composed by 2 × 2 elements, representing the positive and negative components of the real and imaginary parts of the complex function. This algorithm gives better results than the binary Lee algorithm in terms of compactness and quantification levels, but shows small diffraction efficiency since two pixels are intrinsically black.
To overcome this limitation, we introduce a new coding scheme (named Island), where the cell is composed by 2 × 1 elements corresponding to the real and the imaginary parts of the complex number. An offset is added to obtain positive values only, that can be transformed into transparency levels (Fig. 1(d)). We keep coding the information in a 4 × 4 elements cell to avoid anamorphism in the reconstructed image. Since the transparency levels quantify the information in these two algorithms, once set the cell size, the quantification of the information is only limited by the capability of the recording process to set the required transparency level, in opposition to Lee algorithm quantification, which is directly limited by the size of the cell.
For example, let’s consider a CGH with 129 quantification levels; a 256-elements pixel is required with the Lee-code, while the size of the pixel always relies on a 2 × 2 element cell with the Island code; this is therefore 64 times more compact than the Lee-code in the case of 129 quantification levels.
Examples of the Lee hologram, Lee compact hologram and Island hologram are shown in Figs. 1(e)–1(g).
3. Simulations
Simulations were performed to evaluate the performances of the two different coding strategies. We considered: i) the Lee coding, with 4 by 4 elements cell, corresponding to 9 quantification levels for the real part of the complex Fourier function and 9 for the imaginary part; ii) the Island coding with the same number of gray tones. In both cases, we kept constant the size of the hologram at 512 by 512 pixels. To be consistent, the size of the image to be reconstructed was set to 128 by 128 pixels for the Lee coding, and 256 by 256 pixels for the Island coding. We chose as object the logo of the astronomical instrument BATMAN [11]; the logo is not centered in order to locate the reconstructed image as far as possible from the 0th order signal, increasing the SNR (Fig. 2(a)).
Fig. 2. (a) BATMAN logo original image; regions where parameters are calculated, in red: neighborhood (b), and background (c).
Moreover, we set the size of the object to be reconstructed smaller than the size of the image to be encoded. For a better balance between the frequency and the spatial domain, we set a ratio of one third between the object size and the image size. We applied a random phase between 0 and 2π to the image before the hologram calculation to reconstruct all the spatial frequencies of the image. The complex hologram is calculated by the Inverse Fast Fourier Transformation (IFFT) of the complex image. In order to optimize the dynamical range of the hologram, increasing its mean transparency, we multiplied by a factor f both the real and imaginary parts of the complex hologram. Practically, f is a gain factor applied to the complex hologram before coding. Then the real and imaginary parts are quantized and the coding is applied. The flowchart of the overall process is reported in Fig. 3.
For a comparison of the reconstructed image quality, we considered the following parameters:
- - the correlation between the original and the reconstructed images; the parameter is calculated in the quadrant where the most intense image lays;
- - the local contrast, defined as the signal to noise ratio (S/N) between the object and its neighborhood (Fig. 2(b)), calculated as the ratio between the average intensity of the object and the background in a circle with a diameter 1.5 times that of the object;
- - the global contrast, defined as the signal to noise ratio (S/N) between the object and the background (Fig. 2(c)), calculated as the ratio between the average intensity of the object and of the background;
- - the diffraction efficiency of the most intense object as respect to the total incident light, calculated as where ${T_h}$is the transmission of the hologram, ${I_1}$ is the sum of the intensity of the object in the desired quadrant, and ${I_{tot}}$ the sum of the intensity of the global reconstructed image.
Parameters were calculated for all the quadrants of the reconstructed images. The results obtained in the quadrant with the most efficient image and with the f value of 4 for Lee and 2 for Island are reported in Fig. 4. Original images for the Lee 4 by 4 (128 by 128 pixels) and the Island 2 by 2 (256 by 256 pixels) are shown in Fig. 4(a) and 4(b). The holograms encoded with the Lee and Island algorithms at their best gain factors are reported in Fig. 4(c) and 4(d), respectively, together with the simulated reconstructed images in Fig. 4(e) and 4(f).
Fig. 4. Original images, holograms, and reconstructed images for the Lee 4 by 4 with f = 4 (a, c, e) and the Island 2 by 2 with f = 2 (b, d, f), both coded with 9 quantification levels.
In order to understand the effect of the gain factor f and to fully explore the potential of the Island coding, we produced the plots (Fig. 5) of all the parameters as respect to the factor f and we performed a simulation with 201 quantification levels.
Fig. 5. Behavior of the reconstruction parameters as function of the gain factor for the three different codings (Lee 4 by 4, Island 2 by 2, with 9 quantification levels, and Island 2 by 2, with 201 quantification levels): a) correlation, b) diffraction efficiency, c) local contrast and d) global contrast.
In both codings, the correlation and the S/N ratios show a maximum, while the efficiency grows monotonically with the gain factor, as the hologram transmittance increases. Therefore, we chose the parameter that maximizes the overall reconstruction performances and possibly the efficiency ($f = 4$ for the Lee and $f = 2$ for the Island coding).
The Island coding behaves much better than the Lee coding under all aspects: while the correlation is similar (the image is reconstructed fairly well in any case), the S/N is larger than two orders of magnitude in the Island coding, meaning that the image is better defined and the contrast between light and dark regions is greater. The diffraction efficiency is low (as for any amplitude hologram), especially for the Lee coding since the light is split in 16 different images.
The summary of the results is reported and compared with the former results in Table 1. With the increase of the quantification levels, the correlation does not change, while the S/N ratios dramatically rise.
Table 1. Summary of the performances of the two different codings.
The lower diffraction efficiency increasing the number of levels is due to the lower gain factor chosen to maximize the image quality. A rise of the gain factor would lead to a higher efficiency, but with a deterioration of the S/N ratios to values comparable to the Island, 9 levels.
4. Hologram production
4.1 Photosensitive plate
The plates used to record the CGH consisted in a photochromic thin film deposited on 3 mm thick glass substrates. The active component is a diarylethene molecule [23], namely the 1,2-bis(2-methyl-5-dimethylaminophenyl)perfluorocyclopentene dispersed in CAB (cellulose acetate butyrate) polymer matrix, with a concentration of 16.6% wt. Both the photochromic molecule and the CAB were dissolved in the solvent consisting in chloroform (20% vol.) and butyl acetate (80% vol.), in a concentration of 100 mg (total weight) in 1 ml.
After the complete dissolution, the solution was sonicated for 30 minutes and filtered with a 0.45 µm filter. The films were obtained by spin coating (700 to 800 rpm, 60s) and the thickness was in the range 3.4-3.7 µm. The overall optical quality and homogeneity of the films were very good.
The films were converted to the colored form by an exposure with an UV lamp at 366 nm for 10 minutes each side. The UV-Vis spectra were recorded in both the initial (uncolored) form and the colored one. A maximum contrast of 800 was achieved at 633 nm, the wavelength employed for the hologram reconstruction.
4.2 Recording set-up
The recording setup is shown in Fig. 6, and described in details in one of our previous papers [10]. It projects successively a serial of binary masks reproduced by the DMD on the photochromic plate. The DMD, composed by 2048 × 1080 micro-mirrors with a pitch of 13.68 µm [24], is illuminated by a collimated beam coming from a white light source and filtered by a high bandpass filter (cutoff at 515 nm).
Fig. 6. Set-up for recording CGHs; it is based on an illumination unit towards the DMD, an imaging optical system based on a 1:1 magnification Offner relay from the DMD plane to the CGH plane, and a post-CGH imaging system. Arrowed red lines represent the path of the optical beam.
The setup is also equipped with a CGH imaging system, to monitor the writing process, recording the transparency of the plate while it is illuminated. In order to maximize the contrast between opaque and transparent lines, each pixel of the CGH corresponds to 2 by 2 DMD micro-mirrors.
4.3 Recording process
To write binary CGHs, it is enough to generate a single mask on the DMD and to project it onto the photochromic plate until the complete transparency in the desired region is obtained. To produce grayscale CGHs, a set of masks must be generated by the DMD and sequentially projected onto the plate for the required amount of time, until the desired level of transparency is reached. Clearly, the total exposure time is equal in the two cases and it is determined by the most transparent areas where the photochromic material is fully converted. Because of the nonlinear response of the photochromic material, each mask has to be projected for a different time in order to get a linear grayscale. Accordingly, the calibration curve was measured on the photosensitive plate in a region right next to the area where the hologram was then recorded. This curve corresponds to the response of the plate for a given illumination intensity and tells us how much time the plate must be exposed to reach a given level of transparency.
When the illumination of the DMD is perfectly uniform, the response results uniform too, and this curve can be applied to each pixel. As shown in Fig. 7(a), in our case the illumination was not uniform, therefore we had to refine the procedure in order to take into account this fact. The exposure time is controlled at the pixel level with respect to its position in the field of view, by a homothetic transformation of the experimental flux calibration curve mentioned above.
Fig. 7. (a) Map of the normalized illumination intensity on the DMD area; (b) Response curves of the plate under illumination in visible wavelengths for the two pixels in panel (a); (c) exposure time as function of the musk number in the case of 201 gray level hologram.
As an example, Fig. 7(b) shows the experimental response curve obtained for point 1 in Fig. 7(a) and the extrapolated response curve for point 2 in Fig. 7(b) according to the different illumination level.
In the same way, whatever the non-uniformities are in the writing process, they can be taken into account by following the same procedure to realize a CGH as close as possible to the calculated one.
Figure 7(c) shows the exposure time for each mask applied to the DMD in the case of 201 levels for coding the target image: except the firsts and last masks, the duration of each mask is about 1 min, resembling the slope of the exposure curve shown in Fig. 7(b). We also verified that a precision of 1 min in the exposure of the single mask is enough to ensure a mismatch below 1% between the recorded and the requested transparencies.
5. Experimental results
5.1 Recorded hologram
Starting by the 256 by 256 pixels image of the logo, the hologram based on 201 levels Island coding was calculated and transferred to the DMD. The CGH size is 512 by 512 pixels, obtained with 1024 by 1024 DMD micromirrors, since each hologram cell is constituted by 2 × 2 micromirrors. The CGH was written following the procedure described in 4.3. Figure 8(b) shows a magnification of 50 by 50 pixels of the calculated grayscale CGH (Fig. 4(d)) and Fig. 8(c) reports the image of the real CGH. The blurring in this last image is due only to the quality of the CGH post imaging setup.
Fig. 8. (a) The calculated hologram; (b) a portion of the calculated hologram (50 by 50 pixels); (c) a CCD image of the recorder hologram in a similar region. The blurring in this last image is due only to the quality of the CGH post imaging setup.
5.2 Reconstructed image
The image was reconstructed by means of a collimated beam at 632.8 nm and it is shown in Fig. 9. The two first diffraction orders are the reconstructed images, faithfully reproduced, and are visible around the zero order.
Fig. 9. Reconstruction of the recorded hologram based on the 201 gray levels using the Island coding. The two first diffraction orders, faithfully reproduced, are visible around the zero order.
The continuous, horizontal and vertical spikes, slightly broadened, are probably due to a small distortion of the DMD pattern projected on the photochromic plate. The dashed lines mainly tilted by 45° are additional diffraction patterns of different orders due to the DMD structure where the micro-mirrors are tilted along their diagonal.
The contrast and the diffraction efficiency of the reconstructed image have not been calculated according to the fact that our reconstruction set-up is not properly suited due to the low dynamic range of the 8-bit camera and to the presence of spikes affecting strongly the neighborhood and the background noise. A reliable reconstruction set-up and analysis procedure is under way.
In order to check the fidelity of the image, Fig. 10 reports the comparison of the simulated and the experimental reconstructed images.
Fig. 10. (a) Simulated and (b) experimental reconstructed images. Single pixels and one pixel wide lines appear clearly on the reconstructed image.
Structures are remarkably faithful to the originals, reproduced pixel by pixel. Even one pixel coming out of a bigger structure is visible on the reproduction, as well as bright and dark lines with one-pixel width are clearly reproduced. Speckles are inevitable since the source is coherent, but the brightness is consistent between the two images.
6. Conclusion and perspectives
In this paper, we report the definition of a new grayscale based algorithm to encode Fourier holograms. Such island algorithm generates grayscale amplitude Fourier CGHs in opposition with the common Lee algorithm that gives binary CGHs. Such new code is more compact than the Lee code, generating more resolved image for the same CGH size, with a dramatic increase of the S/N. Moreover, the gray scale quantification and the spatial resolution are decoupled.
The use of photochromic plates, showing a straightforward tuning of the transparency by modulating the illumination duration combined with the DMD set-up, makes possible the production of these new grayscale CGHs. Moreover, no chemical post-processing is required on these materials, thus the plates are ready-to-use just after the pattern transferring. The CGH produced in such a way can quantify the information in tens/hundreds levels of transparency, which is almost equivalent to an analogical recording. As for spatial resolution, our results, based on a 201 gray level, show that a single pixel in the original object is perfectly reproduced in the reconstructed image. Moreover, since the employed photochromic films show a large modulation of the refractive index, the approach here reported could be applied to pure phase CGHs working in the NIR.
The next step would be to encode a non-binary (i.e. grayscale) image, which could help to correct the imperfect homogeneity of the reconstruction. This method will be used also to generate larger holograms and/or high resolution holograms; indeed, the set-up supporting the CGH plate is equipped with motorized stages, allowing mosaicking of several zones for generating large CGHs and the DMD could be demagnified in order to improve the spatial resolution of the hologram.
Funding
OPTICON project, Seventh Framework Programme (FP7) (312430).
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