We report on first results obtained with two modified hologram optimization algorithms. These algorithms take into account the complex modulation characteristic of the spatial light modulators employed for hologram reconstruction. To this end the Jones matrices of the modulator as well as all other components of the setup are used within a modified direct binary search and an iterative Fourier transform algorithm. Geometrical phase effects are included in the optimization. Elimination of the analyzer behind the spatial light modulator is possible by that approach and for typical setups using twisted-nematic liquid crystal modulators an enhanced overall diffraction efficiency is achieved. Possible applications are the comparative digital holography and optical tweezers. Experimental results for the reconstructions of holograms with a Holoeye LC-R 3000 modulator are presented.
© 2008 Optical Society of America
Liquid Crystal Displays (LCDs) are widely used in optical measurement systems e.g. for fringe projection , optical tweezers , or digital holography [3, 4]. Most of the commercially available devices have been developed for projection applications. Therefore, theses devices are optimized to achieve high contrast amplitude modulation. Most of these devices are constructed as twisted-nematic LCDs and phase modulation of the elements is only a side effect. Nevertheless by proper addressing phase-only or at least phase-mostly modulation is possible if the display is well controlled [5, 6, 7, 8].
One well known technique is to use polarization eigenvectors [5, 7, 9, 10, 11, 12, 13]. If such an “eigenmode” is modulated by the LCD only the complex scalar amplitude of the light is changed. The polarization remains unchanged. Therefore, by proper selection of the input polarization one can avoid problems due to geometric phase. However, one has to consider that the eigenmodes change with the addressing graylevel so one can only use some sort of average eigenvector. Theoretical predictions using typical LCD models [14, 15, 16, 17, 18] as well as measurement  of such eigenvectors is possible.
Nevertheless in most setups optimum results are not achieved [18, 19]. Polarization of the output light for twisted-nematic liquid crystal displays (TN-LCDs) dramatically varies with the addressed gray level. This leads to a change of the so-called geometric or Pancharatnam phase [20, 21, 22]. Consequently, the overall phase change is the sum of the conventional dynamic phase change and the geometric phase. The additional geometric phase results in a quite complicated hologram reconstruction because of a non-transitive overall phase change .
We want to clarify this by a small example:We denote the phase difference between the phase due to the arbitrary gray level GN with the phase due to the gray level G0 = 0 by ϕ (GN,G0). Normally one would expect that ϕ (G1,G0)-ϕ (G2,G0) equals the phase difference of the phases of two pixels addressed by the gray levels G1 and G2. Unfortunately this is not true due this additional polarization dependent effect. Mathematically we have
Such complications normally are avoided by using a linear polarizer behind the modulator because then the polarization is constant for all gray levels . Especially if the modulators are used for the reconstruction of digital or computer-generated holograms a closer look on the setup shows that there is potential for increasing the overall efficiency. Most important, the linear analyzer behind the display will lead to loss of light. Omitting this linear analyzer will not only reduce the loss of light but also possible aberrations, unwanted interferences, and the overall cost of the system.
In the following we suggest two advanced algorithms for hologram optimization based on a combination of the vectorial Jones calculus with traditional methods of hologram optimization. The approach is quite general and it is demonstrated for two well known hologram optimization algorithms but it is likely to work with other optimization methods as well, e.g. simulated annealing  or genetic algorithms . The incorporation of the Jones calculus into the hologram optimization step goes beyond the use of the modulator’s complex transmission as already presented (e.g. in [26, 27]), where a linear polarizer behind the spatial light modulator (SLM) has been used and the effects of the nonlinear geometric phase and complex interference contrast therefore have not been present.
2. Hologram optimization algorithms
The two basic algorithms that we modified are the Direct-Binary-Search method (DBS) [28, 29] and the iterative Fourier transform algorithm (IFTA) . Although DBS originally has been presented for the optimization of binary amplitude holograms it can be easily extended to phase holograms with non-binary quantization , unfortunately accompanied by largely extended computational costs.
The core idea to incorporate the influence of the geometric phase into the optimization is the usage of the modulator’s Jones matrix during optimization. Therefore, as a mandatory step it is necessary to first perform a detailed characterization of the SLM. Different methods were introduced for the measurement of the modulators Jones-Matrix e.g. the model-based approaches presented in [14, 5, 31, 6, 18], as well as other characterizations, e.g presented in [32, 33, 34].
We used polarimetric measurements in order to first obtain the phase-reduced Jones matrix. Then additional phase measurements with a modified double-slit method  lead to the full Jones matrix with eight parameters. A general description of the modulator used, not in including the measurement of the Jones-Matrix, can be found in . For the measurement of the modulators Jones-Matrix we refer to [16, 17]. A detailed description about the setup used at our institute will be presented shortly within a doctoral thesis. Apart from the Jones matrix of the modulator also the Jones matrices of the other elements used in the setup should be known. For the results presented here only the Jones matrix of the modulator was measured. For all the other elements, as an approximation, the theoretical Jones matrices were used.
The basic procedure is the same in first approximation for both algorithms. We start with the illumination represented by the Jones vector of the illuminating wave front. All following elements are considered by multiplication of the illumination Jones vectors with the Jones matrices of the optical elements passed in their physical order. For each pixel of the SLM the Jones matrix corresponding to the gray value written into the SLM at that pixel is used. The simulation of the hologram reconstruction is then done separately for the x- and y-component of the electrical field transmitted by the SLM.
In the examples shown below only Fourier or quasi-Fourier holograms were computed. Because we are interested in the reconstruction with low numerical aperture the reconstructed field can be obtained simply by performing Fourier transforms separately for the x and y components of the electrical field. Of course, it is possible to extend this to any other hologram reconstruction geometry by using Fresnel- or Rayleigh-Sommerfeld propagation  or even rigorous methods .
By employing this simulation of the LCD-based hologram reconstruction it becomes possible to optimize the holograms even if no analyzer behind the LCD is used. The procedures of the two advanced hologram optimization algorithms are shown in Fig. 1 and Fig. 2.
The implementation of the advanced DBS optimization is straight forward because DBS is directly based on the minimization of a chosen error metric based on the simulated reconstruction. Arbitrary modifications within the forward simulation of the hologram reconstruction can be implemented. Provided that the simulation is realistic and that it takes into account geometrical phases we therefore can optimize holograms for setups with arbitrary polarization components.
For the advanced IFTA optimization the procedure is more complicated. In the reconstruction plane the reconstructed amplitudes have to be be replaced by the target amplitudes. This is done, as shown in Fig. 2, by scaling the amplitude of the Jones vector J(x,y) of each pixel in the reconstruction plane. The second and more difficult problem is the transformation of the newly calculated phase and amplitude in the hologram plane into a corresponding gray value of the display. We use a minimization of the magnitude of the complex difference vector between the modulator output and the desired complex light field at a certain pixel |J(x,y)-J(GN)|. In the case of the IFT algorithm the polarization behavior of the modulator is incorporated by the magnitude of the difference vector for the replacement of the gray values written to the newly calculated hologram. This heuristic incorporation of the polarization leads to improved results and we have only minor differences compared to the DBS-based optimization. Since the method of optimization is independent of the modulator’s capabilities a change of the input polarization is not analyzed here. Though, of course, in cases where the chosen input polarization causes less changes in the output polarization the optimization is easier and will give probably better results. The extremal case would be the use of the Jones calculus enhanced hologram optimization with a planar nematic liquid crystal display. In this case there is no polarization modulation if linear input polarization is used and therefore conventional hologram optimization is adequate.
Both methods can be of course used for correcting additional aberrations. One way is to measure the wavefront of the optical system. For the DBS optimization it is sufficient to subtract the measured phase distribution during iteration from the hologram before reconstruction. In case of the IFTA optimization the aberration can be subtracted from the phase during iteration before the computation of the gray values.
3. Results of the optimization
For different setups simulations have been done. Setups with both polarizer and analyzer, with a polarizer, an analyzer and two additional quarter wave plates and with a polarizer only have been considered.
3.1. Simulation results
Simulations were performed to compare the Jones matrix-enhanced optimization with the conventional methods. Holograms reconstructing three single spots, each having a size of one single pixel in the reconstruction plane, were optimized. The simulations were done first for an “ideal linear phase modulator” (0 to 2π) without change of polarization and second for a modulator with the measured graylevel-dependent Jones matrices. In order to achieve a meaningful comparison of the two cases the “ideal linear phase modulator” was implemented using a unity Jones matrix with a different global linear phase factor for each of the gray values. Accordingly, the angles of the polarizer and analyzer within our simulation program should not have any influence on the simulated hologram reconstruction with the ideal phase modulator. Anyway, we used exactly the same simulation steps (including the polarizers) in order to be sure that we do not obtain different results due to numeric problems (inaccuracies caused by different implementations).
The optimization was done for different polarizer and analyzer settings. Table 1 shows the results of the simulation. As expected the linear 2π phase shift (setup #7) yields the highest diffraction efficiency but the hologram optimized for the measured characteristic curve of the modulator without an analyzer leads to almost the same efficiency (setup #6).
Only small differences between the optimizations with the phase-only mode and the phase-mostly mode, in which no quarter wave plates were used, exist. The reconstruction of the hologram optimized for the linear phase shift with the measured Jones matrix shows always a reduced diffraction efficiency. This is as expected, because the characteristic curve used for the reconstruction does not match the one used for the optimization.
The error of the simulation is mainly caused by two aspects: by the error of the Jones Matrices used and by errors caused by effects not included in the model used, e.g. pixel cross talk or surface reflections on the SLM’s cover glass. As the error is depending on the gray level, i.e. the addressing voltage for each pixel it is difficult to give a general measure for the error. We estimate the error to be about 10%.
3.2. Measurement results
Optimization of the holograms using the ITO logo and a blazed grating as objects was carried out with different experimental settings and with both proposed algorithms. A precise measurement of the diffraction efficiency of such extended reconstructions is difficult due to overlapping ghost orders. Therefore, the difference of the intensities of the zeroth diffraction orders has been used for the estimation of the diffraction efficiency. The measured results agree very well with the results of the simulation.
The reconstructions were performed with three different settings: 1. phase mostly mode with a linear polarizer in front of and behind the display, 2. in an almost phase only mode with a linear polarizer and quarter wave plate each in front of and behind the modulator, and 3. with a single linear polarizer in front of the display. Table 2 lists the orientations of the polarizing elements, the amplitude contrasts, and the achieved phase modulations. The employed modulator Holoeye LCR-3000 is a twisted-nematic LCOS display with 1920×1200 pixels and a pixel pitch of 9.5 µm. Additional specifications can be found e.g. in [19, 38] and in the documentation of the manufacturer.
The results of the achieved overall light throughput are listed in table 2. For the settings with polarization elements behind the display the measured light throughput is given for a gray value of zero. This leads to maximum transmission in this configuration. Due to the remaining amplitude modulation in practice the achieved throughput is further reduced and the amount of reduction depends on the gray value distribution of the hologram. Anyway, the loss of light compared to the system without a linear polarizer behind the display is significant, as even the employed linear polarizer (B&W Filter, type KS-MIK) illuminated with the correct linear polarization causes about 10% loss of light.
Table 3 shows the measured diffraction efficiencies of the reconstructed holograms. Although the achieved diffraction efficiency with an analyzer is not maximum the overall light throughput (see #3, table 1) yields a noticeable improvement compared to the simple phase only setup (setup #2) which yields the best diffraction efficiency. Therefore, a trade off has to be made. For applications where a maximum diffraction efficiency is needed the phase only setup is best suited and for applications where a slightly reduced diffraction efficiency is not harmful but where a maximum light transmittance is demanded the setup omitting the analyzer yields the best properties.
For the optimization of the holograms (400 × 400 pixels) the whole area has been used. The holograms have been repeatedly tiled on the SLM to the full resolution of 1920x1200 pixels. The aberrations caused by the modulator were measured with an Twyman-Green interferometer and have been incorporated into the hologram optimization. Figure 3 shows the reconstructions of the holograms for setup # 2 and setup # 3. The holograms were optimized to reconstruct over the full reconstruction field which is only possible with proper phase modulation. The -1. order of the hologram (not the LCD) which is in this case a 180° rotated ITO-Logo appears with only very low intensity. The about 10% lower diffraction efficiency of the hologram reconstructed without analyzer causes a slightly stronger zeroth order of diffraction. This small reduction of diffraction efficiency is considerably overcompensated by the increased transmittance (compare table 1).
Based on the transmission of the different setups given in table 2 and the measured diffraction efficiencies shown in table 3 the overall light efficiencies for a blazed grating written into the SLM can be calculated as follows. For setup # 1 using lossless crystal polarizers we obtain an overall efficiency of 82%·69.2% = 56.7%. For setup # 2 the achieved effiency is slightly lower 69%·75.7% = 52.1%. As expected, the best overall efficiency can be reached with the setup without polarizers and advanced hologram optimization, namely 100%·65% = 65%.
We have shown first results obtained with polarization-based extensions of the direct binary search and the iterative Fourier transform algorithm. By considering polarization in the methods it is possible to optimize holograms for twisted-nematic LCDs taking into account geometric phase effect. Therefore, good results can be obtained even if the conventionally used analyzer behind the LCD is omitted. The experimental results verify the advantages of the proposed procedures.
We thank the Deutsche Forschungsgemeinschaft (DFG) for financial support under the grant number OS-111/23-1.
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