1

INTRODUCTION

The diffractive Optical Elements (DOE’s) have now reached a state of maturity. The expression DOE which is more general, is now often preferred to Computer Generated Hologram (CGH). Indeed, they can now be computed and encoded with powerful iterative algorithms, such as the Iterative Fourier Transform Algorithm (IFTA)^1,2 or Direct Binary Search (DBS)³ which have been improved and optimized in order to combine high quality reconstruction with high diffraction efficiency. The fabrication process of DOE’s is also better mastered, the association of e-beam lithography techniques with reactive ion beam etching allows the fabrication of multiple phase level elements with a feature size of about 1 µm⁴. Cheaper fabrication techniques such as embossing also exist and allow the fabrication of good DOE’s at low cost. The field of DOE’s applications⁵ has expanded and includes diffractive lenses, beam shaping, structured light generation, optical interconnects…. Moreover, in order to add flexibility to the optical processors, the DOE’s should be dynamic. A dynamic DOE must be displayed onto a Spatial Light Modulator (SLM) and it is only in the past few years that SLM’s suitable for this application became available at a moderate price. Nevertheless, the SLM’s are not perfect and many difficulties remain. In particular, their coding domain is limited and, depending on the technology used, several problems appear: the amplitude is coupled with the phase, the phase shift does not reach 2π, the spatial resolution is limited, the minimum feature size is still much larger than for fixed DOE’s. Therefore, it is very important that the CGH encoding takes account of the SLM characteristics. Several methods have been proposed by different authors^6-8; we propose in this paper a unified approach based on the use of Optimal Characteristics Curves (OCC). Applications of dynamic DOE’s include programmable lenses, reconfigurable optical interconnects, dynamic display, structured light projection, reconfigurable filters….

In this paper, we will emphasize the use of commercially available, low cost SLM’s because even if there are other types of SLM which are better adapted to dynamic DOE’s, some are still in the state of laboratory research or are very expensive.

Section 2 presents an overview of the characteristics of the different SLM’s available. A review of the implementations of CGH’s onto SLM’s is given in Section 3. Section 4 shows the use of OCC for mapping the CGH on the SLM coding domain. Section 5 provides an optical implementation that validates the simulations of Section 4. The last section includes a summary and conclusions.

2

SLM OVERVIEW

SLM’s can be classified into two categories, the ones addressed optically (OASLM) and the ones addressed electrically (EASLM). An EASLM consists of an electrical-optical interface and the optical output signal is pixelated. In the case of an OASLM, there is an optical write beam that controls the modulation of the read beam. The write beam can be coherent or incoherent. As far as DOE’s applications are concerned, EASLM’s are mostly used.

2.1

Coding domains

The coding domain of a SLM represents the range of complex values that can be reproduced. Ideally, we would like to code values in the full complex plane. But in fact, the values are limited. Since SLM’s are passive devices that can only attenuate the beam that is passing through, the values will already be limited to the unity circle. Unfortunately, no existing SLM can address the whole unity circle; some are amplitude only modulators (Fig. 1 b), some phase only modulators (Fig. 1 c) and in many cases the amplitude and the phase are coupled (Fig. 1i).

Fig. 1

Examples of coding domains: unity disk (a), pure amplitude (b), pure phase (c), amplitude only (d), pure binary phase (e), ternary phase (f), quantified amplitude only (g), quantified phase only (h), spiral (i)

2.3

Comparison

2.3.1.

Coding domains

Table 1 presents a summary of the coding domains of each SLM described above. It can be noted that the TN-LC offers the largest range of coding domains even if the pure amplitude modulation is only partially reached.

Table 1:

Summary of the coding domains of the different SLM’s. The cells in shaded boxes mean that the implementation considered consists of an approximation of the desired coding domain.

	TN-LC	PAL-LC	FLC/ELC	MO	DMD	MQW
amplitude	●					●
quantized amplitude	●		●			●
binary amplitude	●		●	●	●	●
phase	●	●	●		●	●
quantized phase	●	●	●		•	●
binary phase	●	●	●	●	●	●
ternary	●		●	●
coupled amplitude and phase	●		●
complex	●				●

2.3.2.

Operating Rate

Fig. 2 shows that the SLM’s can be classified into three categories according to their operating rate. TN-LC, PAL-LC and MO SLM’s operate at video rate. FLC and DMD SLM’s are faster (tens of kiloHertz) and MQW SLM’s reach much higher rates. The figures given here include the addressing time in the case of an electrically addressed SLM.

Fig. 2

Operating rate of various SLM’s.

Fig. 3

Resolution of various SLM’s.

2.3.3.

Resolution

The resolution of the various SLM’s is approximately the same, but continuously increases with technological progress.

Optically addressed SLM’s have a higher resolution compared with electrically addressed SLM’s.

3

IMPLEMENTATION OVERVIEW

The SLM has always been the weakest part among the components of an optical processor. The poor performance and high cost of SLM’s have delayed the fabrication of a practical optical processor. As shown in section 2, the situation is now different, because the SLM’s used in video projectors are now available at low cost and their characteristics are compatible with the constraints of optical information processing. In this section, we will present some uses of SLM’s for DOE’s display. This review is not exhaustive but should show the evolution of the use of SLM’s for dynamic DOE’s with the progress of SLM technology. There are many different SLM’s as shown in section 2, but all of them are not adapted for DOE’s. An SLM displaying a DOE must transform an incoming wavefront into another wavefront with the desired amplitude and phase. Therefore, it is very important to precisely know the SLM transfer function both in its amplitude and in its phase.

4

OCC SIMULATIONS

Many implementations of DOE’s onto SLM’s have already been considered. In this paper, our purpose is to give a general framework rather than to provide a turn-key method. Instead of asking the question: “how can we implement DOE onto SLM’s, particularly on TN-LCD’s ?”, we should ask ourselves “what is the point of implementing DOE onto SLM’s, especially TN-LCD’s ?”.

First, the answer is purely technological: It allows us to implement dynamic reconfigurable DOE’s. But another point should been considered: TN-LCD’s may also alleviate some of the limitations of classic non reconfigurable devices from the point of view of modulating capabilities, for instance.

To highlight the advantages of implementing DOE’s onto TN-LCD’s, we use the practical, rigorous framework of optimal trade-off (OT), already applied to this domain⁴³. We consider two criteria: the optical efficiency η:

and the normalized amplitude error ERRA:

with

where f(q,r) is the desired pattern and g(q,r) is the pattern reconstructed by the hologram. The error is only determined in the signal-reconstruction area ℑ and N.M is the total number of pixels of the DOE. The coding domain (set of possible values for each pixel) is D. Optimal trade-off design is not limited in terms of criteria and if necessary, other criteria can be included in the design operation. In the following, SNR, related to 1/ERRA, may be considered instead of ERRA. Of course, the point will be to maximize SNR or η, whereas ERRA is to be minimized.

In general, the optimal holograms for the normalized amplitude error ERRA and for the diffraction efficiency η are different: in practice, these two criteria are antagonistic and the filter optimizing ERRA shows a very low η (and vice-versa). In this context, it has been shown⁴³ that computing an optimal trade-off DOE can significantly improve this situation. By definition, a DOE d₀ is said to be an optimal trade-off for ERRA and 1/η in D^N,M, if there is no other DOE d that verifies:

with at least one strict inequality. In practice, it comes down to fix one criterion and optimize the other one.

In Fig. 4, a typical locus of points (η, ERRA), named optimal characteristics curve (OCC), is reported in the case of two classic coding domains: binary amplitude and 4 levels of phase. We see that the point of optimal trade-off design is to show that a slight degradation in ERRA widely increases η (and vice-versa).

Fig. 4

Example of optimal characteristics curve (OCC) for the binary coding domain (from ref. ⁴³). The view on the right consists of a close-up of the view on the left.

The method used in the present paper to compute the DOE’s, independently of the optimal-trade-off design, is a multilevel version of the Direct Binary Search (DBS)³, a Monte Carlo technique known to provide excellent results but computer intensive. The reason for such a choice is twofold: no method (except simulated annealing algorithm applied to DBS⁴⁴) is known to be better than DBS^43,45 and the adaptation of DBS to the kind of coding domains we consider in this paper is very simple. Nevertheless, the use of other methods such as IFTA¹, POCS⁴⁶ or HOLOMED⁷ must not be rejected since these methods are known to require much fewer computations than DBS while giving interesting results, sometimes rather close to the ones given by DBS. The classic Euclidean projection, while straightforward and computer inexpensive, is also reported to give results close to the ones provided by DBS⁸. In any case, as stated earlier, providing another method for implementing DOE’s onto SLM’s with an arbitrary coding domain is not within the scope of the present paper: the point is rather to decide if it is worth implementing DOE’s on SLM’s such as TN-LCD’s.

The figures given here concern a large number of simulations, using a set of 10 images with different characteristics. The resolution or space-bandwidth product (SBP) of the images is fixed (32×32 pixels), corresponding to DOE’s with 128×128 pixels. Each point of our OCC’s consists of a mean of the criteria obtained for the 10 corresponding DOE’s.

In Fig. 4, we briefly recall results already obtained with classic coding domains⁴³. Binary amplitude holograms can be displayed on non-reconfigurable devices (photographic emulsion) or SLM’s, such as TN-LCD’s, FLC-LCD’s or MO-LCD’s. They usually provide poor diffraction efficiency and an acceptable SNR, while the 4-level phase domain allows high diffraction efficiency and poor SNR.

In Fig. 5, we report further simulations concerning eight-level pure amplitude DOE’s (it should be reminded that no commercial SLM is able to produce the corresponding coding domain (a similar one is depicted Fig. 1g), and that fabricating such an element still proves expensive. Only one company, Canyon Materials (from San Diego, Ca.), is reported to fabricate multilevel amplitude-only masks. So these simulations just give us a starting point for the discussion. In general, such DOE’s show large SNR and poor diffraction efficiency. A classic alternative consists in using phase-only DOE’s coded onto eight levels (Fig. 1h). Such elements prove more attractive since they can be fabricated on fixed masks (glass substrates, photopolymers,…) or coded onto TN-LCD’s in phase-only (or rather phase-mostly) mode⁸. In this case, since no energy is absorbed by the element and since the conjugated order and the zero-order of the reconstruction can be considerably reduced, great diffraction efficiency is obtained, but on the other hand SNR proves poor (further simulations show that increasing the number of levels can increase the SNR, but cannot compensate for the lack of non-unity amplitude levels).

Fig. 5

optimal characteristics curve (OCC) for various coding domains quantized on 8 levels (include amplitude-only, phase-only, amplitude displayed as is, i.e. with a ‘parasitic’ phase, and the ideal 360° spiral coding domain described in Fig. 1i)(a). The graph on the right (b) consists of a zoom in a region of particular interest of the OCC. In this close-up, the curve related to the ‘dephased amplitude modulation’ does not appear since the lowest error value for this modulation is above 0.025.

Another set of figures reported Fig. 5 concerns the classic (non optimized) implementation of amplitude-only DOE’s on an SLM showing an amplitude-phase coupling. This kind of implementation is not theoretically suitable since it does not take account of the dephasing introduced by the SLM (however, some opticians refuse to explicitly take this dephasing into account: so, in this case, we will call it parasitic). The simulations finally prove that this implementation should be avoided as much as possible, from a pragmatic point of view: it does not give good results. Efficiency remains low while SNR is strongly degraded.

The simulations concerning improved implementations of DOE’s onto a coupled phaseamplitude domain (we chose an ideal spiral with a 360° maximum dephasing, see Fig. 1i) are also reported Fig. 5. They give three reasons for implementing DOE’s on such a domain: (i) this technique allows the use of a dynamic device to write DOE’s (ii) it obtains interesting performance (compared with the classic one described above) (iii) it allows us to investigate a region of the locii of trade-offs that was not attainable with other implementations. In particular, the figures provided show that coupled amplitude and phase DOE’s behave in an intermediate way between amplitude-only and phase-only elements. They show good SNR and good diffraction efficiency.

5

EXPERIMENTAL VALIDATION

We tried to implement DOE’s onto a recently released high SBP (SVGA, 800×600 pixels) TN-LCD from CRL. Unfortunately, these devices show a limited dephasing capability (less than 3π/2 for a red He-Ne ray). It seems that the more recent, the thinner they are. It can be noticed that, earlier this year, CRL released an XGA (1024×768) device⁴⁷ whose phase modulating capabilities have not yet been reported. Nevertheless, we will show in the following that using SLM’s with such low dephasing ability is possible for displaying coupled phase and amplitude DOE’s.

Fig. 6 reports the coding domains of the SLM obtained for various configurations of the polarizer and analyzer denoted (ψ_p,ψ_a). As expected, each of them shows, a phase modulation coupled to the amplitude modulation. The same experiments on similar devices have shown that a phase-only (or at least, phase-mostly) mode is also possible, but we are interested in coupled phase and amplitude modes in order to implement DOE’s with better SNR, as planned in Section 4.

Fig. 6

Various coding domains available for the CRL SVGA TN-LCD

The configuration we chose is the best compromise between contrast and phase ranging (in this case 83°), corresponding to (ψ_p=45°,ψ_a=-45°). The coding domains presented Fig. 6 show low dephasing capabilities. This is only due to the settings of brightness and contrast of the SLM electronics. Larger dephasings can be produced: possible figures are similar to the ones of another device from CRL, VGA2, used by many opticians. Anyway, such a small dephasing will help us to demonstrate that SLM’s with a low dephasing capability can be used as coupled phase and amplitude modulators.

Before performing the implementation, we decided to simulate the optical reconstructions (Fig. 7). The first view (Fig. 7a) corresponds to the classic binary implementation. The second view (Fig. 7b) corresponds to the amplitude-only domain quantized on eight levels. The results improve (the resemblance, at least), but the reader should remember that this implementation is not possible at the moment with currently available SLM’s. Two implementations of the pure 8-level amplitude DOE ‘as is’, when the ‘parasitic’ phase produced by the TN-SLM is not taken into account during the DOE design, are then simulated. Fig. 7c reports the simulation for an ideal 2π spiral, Fig. 7d the simulation for the coding domain we experimentally characterized earlier with our SLM (ψ_p=45°,ψ_a=-45°). As already reported in Section 4, the implementation onto the ideal 2π spiral provides poor results and should be avoided. But surprisingly, the implementation onto our experimental coding domain is not so bad, even if it has not been, in this case, taken into account in the DOE design. The design and the implementation onto an ideal 2π spiral is simulated Fig. 7e. We clearly see that such an implementation, as explained in Section 4, provides good resemblance (high SNR), and the conjugated order has disappeared. Unfortunately, this implementation is not possible onto most SLM’s because their possible maximum dephasing is less than 2π. Finally, we simulated the implementation onto the coding domain we can address on our SVGA TN-SLM in the configuration (ψ_p=45°,ψ_a=-45°), described Fig. 6. The resemblance seems as good as in the previous case. The only difference is in the conjugated order that can be observed, contrary to the previous case, although it is much weaker than in the case of a binary or amplitude-only modulation. The values of the two criteria are reported in Table 2.

Table 2:

results obtained in simulation and experiments for various coding domains. The reader must remember that when a pure amplitude hologram is displayed on a TN-SLM, a coupled phase modulation, inherent to the TN-SLM, is added, and the results are degraded

Fig. 7

Simulation of reconstructions provided by various implementations of DOE’s: binary amplitude (a), 8-level amplitude only that is, at the moment, only possible in simulations (b), 8-level amplitude only displayed with a parasitic phase ranging from 0° to 360° (c),), 8-level amplitude only displayed with a parasitic phase ranging from 0° to 83° (d), 8-level ideal 360° spiral (e), 8-level experimental spiral with (45°,-45°) configuration (f).

A simple validation has been done, with three experiments corresponding to all the possible implementations among the implementations simulated. The first implementation (Fig. 8a) corresponds to a binary DOE, the second one (Fig. 8b) to an 8-level amplitude DOE displayed as it is, with the coupled phase modulation not taken into account during the design, and the third one (Fig. 8c) to a phase-amplitude hologram for which the coupled phase has been taken into account in the DOE design. The values of both criteria have been reported in Table 2. All three implementations succeed in reconstructing the desired pattern. The binary implementation (Fig. 8a) gives good reconstruction, with a high contrast in the region of interest. The zero order is quite strong. The direct implementation of the 8-level amplitude-only DOE onto the TN-SLM is less good than the previous one (contrast is bad), whereas simulations were close (Fie. 7a and 7d and Table 2, SNR).

Fig. 8

Reconstruction obtained from a binary DOE (a), an eight-level amplitude-only DOE displayed on a TN-SLM (b) and an eight-level optimized phase-amplitude DOE (c)

Finally, the implementation of the 8-level coupled phase and amplitude DOE specially designed for the implementation onto the TN-SLM is interesting. Contrast is not so good as in the binary case, but the region of interest is uniform and the conjugated order vanishes a little. Nevertheless, as reported in Table 2, the diffraction efficiency figures are disappointing since in this case, they should be twice as large as in the binary case, and actually they are almost twice as small. Three explanations can be considered: the characterization of the spiral coding domain was imperfect (while, it is much better because it is much simpler in the binary case), the electronics may not reproduce 8 gray levels (the data sheet from CRL is not clear) and we considered a global behavior of the SLM whereas such TN-LCD’s are reported to show large variations between their pixels⁴⁸

In Table 2, we summarize the various results obtained in simulations and experiments. The reader should remember that the purpose of the present paper is not to provide an efficient technique from the computational point of view: the use of DBS implies huge computing time, which may be alleviated with an adapted version of IFTA, as reported in ref.⁴⁹.

6

CONCLUSIONS

In this paper, a review of the implementation of DOE’s onto SLM’s was presented. We particularly focused on SLM’s with modulating capabilities other than binary, e.g. TN-SLM’s which provide phase-mostly or coupled phase-amplitude modulation. We showed the point in using this latter mode. Considering the general framework of OT design, we showed that it allowed us to investigate a region of trade-offs that are very different from the ones provided by classic coding domains: while the phase-only mode or the amplitude-only mode provide a good value for only one criterion (resp. diffraction efficiency and SNR) and a poor value for the other, this phase-amplitude modulation has an intermediate behavior which gives acceptable values for both criteria. An experimental validation was also provided. It confirmed that displaying a pure amplitude hologram on a TN-SLM, without taking account of the device characteristics during the design, was the worse solution and that the use of the coupling characteristic phaseamplitude of the TN-SLM was a valuable alternative for displaying multilevel DOE’s with a high SNR. Nevertheless, from the experimental point of view, the classic easy binary implementation still challenges this optimized multilevel implementation. This latter would probably prove better with an SLM with an improved electronic drive board and with pixels that have the same characteristics and behave exactly the same way.