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Capabilities of diffractive optical elements for real-time holographic displays Stephan Reichelt, Hagen Sahm, Norbert Leister and Armin Schwerdtner SeeReal Technologies GmbH, Blasewitzer Str. 43, 01307 Dresden, Germany ABSTRACT This paper illustrates one of the various capabilities of static diffractive optical elements (DOE) beneficial to realtime holographic displays. Custom kinoform-type DOE can be used as elements for illumination of the spatial light modulator, i.e. the display where the video hologram is encoded. For an RGB application of diffractive optical elements, particular issues concerning the inherent wavelength-dependence have to be addressed. Multiorder DOE offer a way to compensate for chromatic as well as monochromatic aberrations. We will discuss concepts and performance of multi-order DOE, show their application in holographic displays, describe issues of fabrication and replication, and give experimental results of the multi-order DOE performance. Keywords: Electro-holography, 3D Display, Multi-order diffractive optical elements
1. INTRODUCTION Real-time holography is the ne plus ultra art and science of visualizing fast temporally changing three-dimensional scenes. The integration of the real-time or electro-holographic principle into display technology is one of the most promising and challenging developments for the future consumer display and TV market. Only holography allows the reconstruction of natural-looking 3D scenes, and therefore provides observers with a completely comfortable viewing experience. Recently, we have developed a novel approach to real-time display holography overcoming the challenges of classic holography by combining an overlapping sub-hologram technique with a tracked viewingwindow technology.1, 2 For the first time this enables solutions for large screen interactive holographic displays. Diffractive optical elements (DOE) are versatile elements on their way to becoming state-of-the-art components in a variety of optical systems. DOE offer a flat, space-saving integration, high-accuracy, nearly arbitrary wavefront shaping, and high diffraction efficiency. However, when blazed for an operation at first order, a DOE exhibits a significant dispersion due to the wavelength-dependent diffraction. To overcome this limitation, socalled multi-order or harmonic diffractive lenses were introduced independently by Sweeney and Sommargren3 and Morris and Faklis4 in 1995. They showed that a diffractive lens with multi-wavelength path-length steps rather than a single-wavelength step has the same optical power for a set of discrete wavelength whereby broadband or multi-spectral applications become feasible. It has also been shown that such higher-blazed diffractive elements have both diffractive and refractive properties.5–7 An important design aspect of a 3D holographic display is the flatness of the screen which should be in the same dimension as today’s standard LCD. But the requirements for a backlight used in holographic displays completely differ from that for standard displays. In holography a well-defined, coherent wavefront is needed to reconstruct the 3D scene. When using standard refractive optics for reference wave collimation, the price we have to pay is a relatively large depth of the backlight to avoid aberrations. Here, multi-order DOE can be advantageously used to provide a well-defined reference wave for the hologram reconstruction at the given RGB wavelengths.
E-mail:
[email protected], Phone: +49 (0)351 450 3240, Web: http://www.seereal.com Proceedings of SPIE, Vol. 6912, Practical Holography XXII: Materials and Applications, Editor(s): Hans I. Bjelkhagen; Raymond K. Kostuk, p. 69120P, (2008). http://dx.doi.org/10.1117/12.762887
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2. SEEREAL’S REAL-TIME HOLOGRAPHIC DISPLAY SOLUTION For the sake of a better understanding we briefly recapitulate the basic principles of our novel approach to real-time display holography in a phenomenological way. For a more comprehensive description we refer to the references.8, 9 The fundamental idea in our concept of an electro-holographic display is to give highest priority to reconstruct the wave field at the observer’s eyes and not the three-dimensional object itself. Figure 1 illustrates the principle.
Tracked viewing window
Reconstruction of a single scene point for fixed observer position
Wasted information
SubHologram
Essential information Reconstructed scene point
Hologram
Virtual viewing window
Figure 1. Schematic drawing of SeeReal’s approach to real-time holography, the so-called tracked viewing technology.
In classic holography, the 3D scene information is encoded in the entire hologram, i.e. every pixel of the hologram contributes to each object point. When illuminated by the reference wave, the combination of all of its cells reproduces the complete scene by means of multiple interferences. A classic hologram has a large angular spectrum, which means that a large space-bandwidth-product is required. If a classic hologram is broken into a number of pieces, each piece will reconstruct the original scene, even though with less resolution and in smaller size. All past attempts of transferring classic holography to display and TV applications have failed due to two inherent challenges: (1) display resolution and (2) data volume and processing. To give an example, extreme-resolution displays with a pixel size of roughly 0.5 microns would be required, which translates to the huge demand for calculation of millions to billions of complex values for each of the 2 Mio scene points (1920 x 1080) to determine an HDTV scene in 3D. When considering the human vision system regarding to where the image of a natural environment is received by a viewer, it becomes obvious that only the information aimed at the pupils of the viewer is required to create the complete scene. All other information is wasted. This basic fact allows to realize tracked real-time video holography with the following key features. Viewing window. In our approach a wavefront originating from a 3D scene is reconstructed only in a small region called viewing window - its size being in the range of an eye pupil size. For an observer different viewing windows for left and right eye are generated. Holography is based on diffraction. The pixel pitch of the holographic display determines the diffraction angle that can be used. For a viewing window type hologram only a small diffraction angle from the display to the border of the viewing window is needed whereas in classical holography the diffraction angle is related to the size of the 3D scene. So the constraint of small pixel pitch is distinctly released in our approach. A display with a pitch in the range of 60 µm can be used.
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Sub-hologram. A direct consequence of the viewing window setup is the following: For each reconstructed object point there is only a limited region in the hologram where data from this object point is coded. This limited region is called sub-hologram. In contrast to a classical hologram, if a viewing window-type hologram is broken into a number of pieces, each piece will reconstruct only part of the original scene, but with full resolution (apart from object points close to the border of the reconstructed scene fragment). The refractive analogue of a sub-hologram is a small lens focusing light from the hologram to the object point. Real time calculation. Hologram calculation is done by setting the sub-hologram for each object point as simple lens function and then summing up different sub-holograms. In contrast to classical computer-generated holograms no Fourier transform is required. Larger pixel pitch and sub-hologram approach reduce computation requirements by a factor of 10,000 as compared to the classic approach. This allows for a real time calculation of holograms. Higher orders. Higher diffraction orders of each object point of the 3D scene are also generated. But light from these points is located in higher diffraction orders outside the viewing window. They are not seen by the eye located inside its viewing window. So higher orders do not deteriorate the visible reconstruction of the scene. Tracking. In a classical film hologram with fine resolution and large diffraction angle an observer can move around the hologram and see the reconstruction from different positions. In contrast to this the restriction of limiting the reconstruction to a viewing window would lead to a 3D scene only visible from a fixed observer position unless an important additional feature is added to the setup, namely tracking. User tracking has been known before from autostereoscopic systems where some kind of sweet spots are moved in accordance with the observer’s eye position. Such concepts have been adapted to holography, taking care of the fact, that coherence of the light his to be maintained during tracking. A camera system in combination with fast image processing algorithms is used to detect the actual eye position of the observer. Then the viewing window is shifted in real-time to the new eye position, by optical or electronical means. Simultaneously hologram content is updated by real-time calculation. Each object point shift of the viewing window also means a shift of its sub-hologram in opposite direction. This principle is illustrated in Figure 1 on the right side.
3. BASICS OF MULTI-ORDER DOE Design procedure. The design of a kinoform-type10 diffractive optical element starts with the calculation of its phase function. The continuous phase function φ of a thin diffractive phase element refers to the phase difference between the desired output wave and the incident wave, i.e. that difference before and behind the diffractive element φ(x, y) = φout (x, y) − φin (x, y), (1) where φ is related to the recording eikonal by φ(x, y) = k0 G(x, y), with the free-space wave number k0 = 2π/λ0 . That is, φ is defined by the optical path difference introduced by the diffractive element and by its design wavelength λ0 . The continuous phase function usually refers to the first diffraction order, which is designated as the design order. In a second design step, the blazing procedure transfers the continuous phase function φ to a modulo m2π phase profile Ψ(x, y) = φ(x, y) mod (m2π), (2) where the integer-valued m is the so-called blaze order. The modulo operation – which represents the foundation of diffractive optics – is based on the fact, that a monochromatic wave is insensitive to phase jumps of multiples of 2π since the constructive interference condition is satisfied in any case. It is important to note that different phase profiles Ψ can be created from the same underlying phase function φ, when higher orders than the usual first blaze order are chosen. Blazing the diffractive element for higher orders will alter the diffractive surface profile toward to more moderate feature sizes although with enlarged blaze depth. In general, the blaze order m represents an additional design parameter of diffractive optics which has already been utilized in the past for adapting the minimum feature sizes to the manufacturing feasibility or for realizing multi-order harmonic DOE,
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which are of special interest here. The latter type is associated with a harmonic multi-wavelength operation, i.e. the diffractive element is designed for and illuminated by a number of discrete wavelengths which fulfill the condition of mλ0 = qλblaze . (3)
q=3
q=7
q=5
Blaze-matched wavelength λ
blaze
[nm]
700 650 600 550 500 450 400 1000
2000 3000 4000 5000 6000 7000 Synthetic design wavelength λsyn [nm]
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Figure 2. Wavelength selection for multi-order DOE. For a given synthetic design wavelength λsyn exists N discrete wavelengths λblaze within the VIS spectral range for which the multi-order condition λblaze = λsyn /q is fullfilled. Exemplary, a q = 5, 6, 7 RGB-design for λsyn = 3150 nm is marked. The larger the synthetic wavelength, the higher the number of suited discrete wavelengths.
Here, m is again the blaze order for the design wavelength λ0 , where q is the blaze order for another, socalled blaze-matched wavelength λblaze . To determine suited orders m, q for a given set of discrete operation wavelengths, we search for the lowest common multiple of these wavelengths. This wavelength now becomes the synthetic design wavelength by which the design procedure is traced back to the standard case with a maximum phase modulation of 2π at a blaze order of one. Furthermore, that means λsyn can be substituted for mλ0 in Eq. (3). Figure 2 visualizes the set of discrete blaze-matched wavelengths λblaze within the visible spectral range over a synthetic design wavelength λsyn -range. The diagram may be helpful if one prefers to go the other way around, that is to say defining first a synthetic wavelength and then searching for suitable operation wavelengths at different blaze orders. The final step of the design procedure is the transfer of the modulo m2π phase profile to a surface relief profile t which is afterwards structured in any substrate material. Here, one has to distinguish between reflective mirror-type (M) or transmissive lens-type (L) diffractive elements tM (x, y)|blaze
=
tL (x, y)|blaze
=
mλ0 Ψ(x, y) , 2π 2 mλ0 Ψ(x, y) . 2π [n(λ0 ) − 1]
(4) (5)
From the equations above it becomes obvious that a reflective multi-order DOE is completely chromatically corrected for the set of harmonic wavelengths, whereas for a transmissive multi-order DOE a minor refractive dispersion of the substrate material is still present. Nevertheless, also multi-order diffractive lenses offer an elegant way to correct the diffractive dispersion at distinct wavelengths.
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Figure 3. Diffraction efficiency over the wavelength for a multi-order DOE with m = 6 for λ0 = 525 nm. The marginal shift of the sinc-functions for blue and red wavelengths in (b) is due to material dispersion n(λ). The blaze depth is assumed to be ideal (µ = 1).
Optical performance. The optical performance of diffractive optical elements strongly depends on how accurate the operation corresponds to the ideal situation which the DOE is designed for. This applies to both variations in the optical setup (operation wavelength, alignment), as well as minor fabrication inaccuracies (profile depth, pattern distortion) of the DOE itself. Apart from alignment errors, which can be avoided by a careful adjustment, a wavelength change will have an impact on the diffraction efficiency and the phase reconstruction. To address these influences a parameter α was established11 λ0 n(λ) − 1 α= . (6) λ n(λ0 ) − 1 Profile depth scaling errors will also influence the diffraction efficiency. In case of a constant profile depth error this can be addressed by introducing a parameter6 µ=
t0 , t
(7)
with t as the nominal and t0 as the actual profile depth. With the parameters α and µ the scalar diffraction efficiency of a multi-order DOE for the qth diffraction order is then given by ηq = sinc2 (αµm − q) .
(8)
If the argument of the sinc-function equals to zero, the diffraction efficiency will have a maximum. Fig. 3 shows the diffraction efficiency vs. wavelength of a ’5/6/7’ multi-order DOE designed for λ0 = 525 nm and m = 6. As it can be seen in Fig. 3(a) besides the green efficiency peak there are further maxima at the blaze-matched wavelengths of 450 nm (q = 7) and 630 nm (q = 5). To illustrate the minor influence of material dispersion we have distinguished between (a) reflective and (b) transmissive multi-order DOE. Figure 4 indicates the efficiency dependence on blaze depth scaling errors for different blaze orders. In contrast to a standard first-order blazed DOE, the multi-order DOE’s efficiency is much more sensitive to potential fabrication inaccuracies. For the above mentioned ’5/6/7’-design for example, a variation in the blaze depth of only 5% will result in an efficiency drop to 65% (blue) or 81% (red), respectively. On the other hand, the aberrations of multi-order DOE are highly wavelength-sensitive too. Unaberrated wavefronts for all wavelengths will be generated only if the operation wavelengths exactly match the design 69124
wavelengths. In addition to the amount of ∆λ = λ0 − λ the accompanied phase error depends on the shape and power of the recording eikonal G(x, y), since the design eikonal is a function of the DOE coordinates. In other words, the phase error will be different for a high or low power diffractive lens or for a spherical or aspheric DOE of the same power. The phase error introduced by a wavelength mismatch is given by the difference between the actual reconstructed phase and the nominal design phase φ(λ) − φ(λ0 ), i.e. ∆φ =
2π G (α − 1). λ0
(9)
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Figure 4. Diffraction efficiency in dependence of blaze depth variations µ = t0 /t for a multi-order DOE. The higher the blaze order the more critical are deviations in the blaze depth from its nominal value. For comparison, the dashed line corresponds to the first blaze order of standard diffractive elements.
4. RGB-DOE IN HOLOGRAPHIC DISPLAYS General issues. As the analysis in the previous section shows, multi-order DOE can designed such that they are chromatically as well as monochromatically corrected for a set of discrete wavelengths. These special properties make multi-order DOE very attractive to all optical systems and applications which operate at more than one distinct wavelength. Especially all laser display systems – for projection or direct view – are well-suited for the application of multi-order DOE since by it all color channels can pass through a common optical path, and only three wavelengths are needed. It is well-known that a colored image can be generated by superimposing (simultaneously or time-sequentially) the three basic colors red (R), green (G), and blue (B) simply by adjusting the brightness of each basic color. Therefore, for display applications a multi-order DOE has to be designed for three wavelengths only, which reduces the required maximum order q for each wavelength, and thus the overall profile depth of the diffractive structure. To beneficially apply multi-order DOE as a passive component to displays, the selection of the RGB wavelengths requires the careful consideration of the following three aspects: Light sources. Main criteria is the availability of high-quality light sources with sufficient coherence, beam quality and power that match as best as possible the blaze-constraint of multi-order DOE given in Eq. (3). Due to the phase error and diffraction efficiency sensitivity this will be the key point. Furthermore, the size, cost and system integration are issues that have to be considered as well. Smallest common multiple of the three wavelengths. A larger synthetic wavelength means that there are more blaze-matched wavelengths which can be selected (cf. Fig. 2). Admittedly, this simplifies the light source selection issue, but it comes with an increased profile depth, which is quite unfavorable in terms of the efficiency sensitivity to profile depth errors. Therefore, the smallest common multiple of three RGB-wavelengths which corresponds to the smallest possible synthetic wavelength is the best choice. 69125
Color gamut and color sensitivity of the eye. Nowadays, in standard non-laser display technology there is a demand for large color gamut. By using laser light at RGB a high color gamut is always provided since the spectrally clear colors lay at the edge of the color tongue. However, also the sensitivity of the human eye to different colors must be taken into consideration to find best-suited wavelengths. In fact, there will be always a trade-off between these aspects. Illumination issues for holographic displays. We continue our discussion with an exemplary design of the focusing element of a backlight unit for holographic displays. In contrast to standard liquid-crystal-displays, where the backlight provides an incoherent illumination of the LC-pixel and a large angular spread in its radiation to ensure large viewing angle, holographic displays have completely different demands to its backlight. The backlight of a holographic display have to be coherent and must have a defined or well-known phase and amplitude distribution. To put it into holographic terms, this wave corresponds to the reference wave which is required to reconstruct the object encoded in the hologram. SeeReal’s approach to holography requires coherence in the illumination only over a hologram area which equals approximately the maximum sub-hologram size. This simplifies the demand to the used optical components, since not a single light source must serve for the entire 20-inch or even larger sized hologram. Instead, a light source matrix can be used to illuminate the hologram. By doing so, the depth of the backlight can be dramatically decreased, since the closest distance between the point source and the focusing element is limited by the maximum f -number which is achievable by classic optics. The proposed backlight unit for holographic displays is shown schematically in Fig. 5. It consists basically of a point source array (PSA), at which the three RGB-colors are emitting in a time-sequential manner. The PSA is followed by a multi-order DOE-lens array which is placed at focal distance behind the sources. Thereby, the light coming from one point source is collimated to a size corresponding to the clear aperture of an individual DOE. The entire hologram panel is then illuminated by a ‘quasi-planar’ wave, which is actually composed of an entire set of planar wavefronts. To achieve a homogeneous intensity behind the DOE-lens array, a 100% fill factor of the array is essential.
RGB pointsource array
Multi-order DOElens array
Video hologram (display panel)
Figure 5. Schematic explosion view of an illumination unit (backlight) for direct-view holographic displays.
At this point one may ask what are the specific advantages for using multi-order DOE in holographic display backlights compared to standard spherical or even aspherical refractive singlets? To answer this question we
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have compared the optical performance of these three types of focusing elements, all with an focal length of f 0 = 40 mm and a diameter of 20 mm (f /# = 2). The selected wavelengths to characterize the chromatic behavior are λB = 450 nm (blue), λG = 525 nm (green) and λR = 630 nm (red). These wavelengths correspond to a ‘5/6/7’ multi-order design, which was already taken as example above. The results of the ray-tracing simulation are graphically summarized in Fig. 6. Consider first the plano-convex lens in Fig. 6(a). It is well-known that plano-convex lenses suffer from chromatic aberration and especially spherical aberration: for the exemplary lens made from BK7 the chromatic focal shift is 840 µm and the peak-to-valley (PV) spherical aberration is ∼ 20λG . Such aberrations are not acceptable in holographic applications. A proper means to compensate for spherical aberration is the use of aspherical lenses. By the exemplary asphere in Fig. 6(b) the spherical aberration is well corrected for all wavelengths but there is still chromatic focal shift of 814 µm. Only the multi-order DOE lens is capable of correcting monochromatic (spherical aberration) and chromatic aberration (focal shift) at the same time, cf. Fig. 6(c). This characteristic makes multi-order DOE to an ideal lens for point imaging at a given set of discrete wavelengths.
(a)
(b)
(c)
Figure 6. Chromatic and monochromatic performance comparison between (a) a spherical planoconvex lens, (b) an aspherical planoconvex lens and (c) a multi-order RGB-DOE. The f -number of all elements is 2. Please note, that (c) is here purely diffractive modeled and that in (c) the optical path difference (OPD, middle row) and longitudinal aberration (LA, lower row) diagrams show the correct behavior of the green wavelength only. By varying the blaze order (Zemax diffraction order) to q = 5 for λR = 630 nm and q = 7 for λB = 450 nm respectively, it can be shown that the wavefront is perfect (i.e. with zero aberration) for all design wavelengths.
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5. FABRICATION Basically, any method for generating diffractive optics can be used to fabricate multi-order diffractive elements. In current art of technology, either lithographic or diamond turning followed by etching or molding processes are well established fabrication methods. As mentioned above, the hardest challenge regarding the fabrication of multi-order DOE is the high accuracy demand in the profile depth and the optical quality of the blazed surface. Although the ratio of profile depth to period still indicates a ‘thin’ diffractive interface12 it reaches an unfavorable value with regard to the manufacturing. Greyscale lithography is best choice doing diffractive optics with binary steps or low step height. Regarding to the bit depth of the lithography process the multi-order blazed continuous profile is approximated by a binary profile of N steps. One drawback of this technique is the lateral sampling in pixel, where the achievable resolution depends on the laser beam focus. In high-quality writing machines, the beam size of the focused laser beam is in the range of 1 µm. If the feature size reaches the some tens micron range, the surface will exhibit a strong pixel structure. The main advantage of the lithography is the independence from the shape of the optical function and the lateral geometry. That is why arrays with 100% fill factor and aspheric designs are easily possible. With lithography, both tooling and direct mastering of DOE are available. On the other hand, diamond turning became a favored method for DOE tooling. It overcomes the disadvantages of lithography being a continuously working method, that is problems involved with step-like approximation are avoided. However, the obtained surface quality is directly related to the tool (monolithic diamond). The price for those advantages is the limited depth accuracy according to the axis stability and the technical radius at the diamond tool, which results in a minimum achievable feature size. Aspherical profiles can be generated by means of a synchronized tilting tool during the drilling. The tooling for the multi-order DOE discussed here has been carried out at the Laboratory for Micromachining at the University of Bremen. The used machine was a ultrahigh precision tooling machine ‘Precitech Freeform 3000’ with hydrostatic linear axes and an air bearing spindle. The obtained mold was a 20 mm cylinder which was used as a master in the subsequent molding step. For molding we have applied a UV curable acrylate resin, whose properties have been tailored to our demands. Figure 7 shows photographs of the diamond turned master and the molded multi-order DOE lens.
5 mm
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(b)
Figure 7. Photographs of the diamond turned master (a) and the molded multi-order DOE lens (b).
6. CHARACTERIZATION Profilometry. The surface profile of the molded multi-order DOE-lens was measured by a stylus profiler (KLATencor P15). For the inner zones a profile depth of 5.9 µm was determined. However, it should be noted that the measured surface profile was not compensated for the trajectory of the stylus. Especially at small feature 69128
sizes the geometry of the stylus itself will distort the measurement. The total height is estimated to be approx. 6.0 - 6.1 µm.
Profile [µm]
0 -2 -4 -6 0
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1
Figure 8. Measured surface profile of the fabricated multi-order lens (central zones).
Wavefront quality. To evaluate the wavefront quality a simple interferometric collimation test was performed. The multi-order DOE was placed at its focal distance of f 0 = 40 mm behind a pinhole to image a point source to infinity. A wedge shear plate was used to laterally shear the collimated wavefront with itself. Fig. 9 shows lateral shearing interferograms obtained for red, green and blue laser light illumination.
(a) λ = 633 nm
(b) λ = 532 nm
(c) λ = 475 nm
Figure 9. Lateral shearing interferograms.
For red and green we observed a wavefront quality of about λ/2, which is quite well considering the large numerical aperture of the DOE. As expected from the theory, the strong wavelength dependence to both efficiency and aberrations became apparent during the experiment. In our experiments we used a red HeNe-laser and green and blue solid state laser operating at 633 nm, 532 nm and 475 nm, respectively. That is, the operation wavelengths differ from that of the design by ∆λR = 3 nm, ∆λG = 7 nm and ∆λB = 25 nm. For a small wavelength mismatch as for red and green, this basically results in a focal shift. Therefore, we have had to readjust the best collimation state between the tests at different colors. First, the experimental setup was best adjusted by using the red HeNe-laser. Due to the ∆λR of 3 nm this best adjusted focus is already 195 µm shifted from its nominal value. Second, the test was performed with green light, where the multi-order lens has to shifted about 355 µm, which corresponds well to the ZEMAXr raytracing simulation (348 µm). Third, the collimation with 532 nm was tested. Besides a large focal shift of we observed a primary spherical aberration of ∼ 1λ.
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7. CONCLUSIONS In conclusion, capabilities of multi-order diffractive optical elements for real-time holographic displays have been discussed. A backlight design incorporating an array of multi-order DOE having a large numerical aperture has been proposed. To evaluate its performance experimentally, we have fabricated a single multi-order DOE lens with an f -number of 2 by diamond turning and molding in acrylate. We have characterized the surface quality and profile and analyzed the optical performance by interferometric tests. From both theoretical analysis and experimental studies it became apparent that the most crucial parameters of multi-order DOE are best-matched operation wavelengths and an accurate blaze depth. In spite of their needs in tight tolerances, multi-order diffractive elements possess significant potential for holographic displays, since they can be designed very flexibly and operate multi-spectrally.
ACKNOWLEDGEMENTS The authors wish to gratefully acknowledge Steffen Buschbeck for his help in performing some of the experiments.
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