1. Introduction

The problem of the synthesis of diffractive optical elements (DOE) on non-flat surfaces (cylindrical, spherical, or conical) was addressed in many publications. The authors of [1-3] discuss analog recording of holograms on a cylindrical surface. Computer-generated holography, which has a number of advantages over analog recording, opens up ample opportunities for the synthesis of diffractive optical elements on non-flat surfaces. This work is dedicated to the development of computer-generated hologram (CGH) on a cylindrical surface with the aim to display a 3D image.

Many publications have been dedicated to the development of methods for computing and producing CGH on non-flat surfaces with the inverse problem of the computation of the phase function. Mathematically, the computation of the phase function reduces to solving a linear Fredholm integral equation of the first kind, where the unknown function depends on two variables [4]. Moreover, under quite reasonable conditions the linear operator equation is an equation for the difference of two variables and can be solved using fast Fourier transform. Typical examples include publications [5-7], whose authors analyze methods of computer synthesis of holograms on a cylindrical surface. They limit their analysis to simulations involving an algorithm with Fast Fourier Transform. They performed simulations with wavelengths exceeding 100 microns.

Problems arising in such a formulation are due to the fact that the mesh step at the surface of the optical element should be smaller than the wavelength, resulting in a huge number of unknowns in the inverse problem, which can amount to 10¹⁰ even for relatively small optical elements. The inverse problem is further complicated by the fact that usually only the amplitude of the 2D and 3D images is known.

Unlike the above studies, here we develop methods for computing DOE on a cylindrical surface for optical wavelength range. We solve the inverse problem of computing the microrelief for displaying the required 3D image in a finite-parametric representation. The DOE is subdivided into hogels with the size no greater than 50 microns. Each hogel consists of diffraction gratings with the size of no greater than 10 microns. Thus each hogel is specified by a finite number of parameters that determine the period and orientation of diffraction gratings. Parametric representation allows the number of unknowns in the inverse problem to be reduced by a factor of 1000. However, the inverse problem of synthesis in parametric representation is nonlinear. In this paper we propose an efficient method for computing the phase function of the reflective DOEs in terms of the parametric model mentioned above. We solve the inverse problem in two stages. First, we use the 2D frames obtained from the virtual 3D model of the object to compute the intensity of diffraction rays from the hogel in the first order of diffraction in the direction to the observing points. At the second stage, we use standard representation adopted from the theory of diffraction gratings to compute the periods and orientations of the diffraction-grating fragments of each hogel. Strictly speaking, the DOE considered here should be better called a computer-generated holographic stereogram on a cylindrical surface. For the sake of brevity, we use the terms CGH and DOE throughout this paper.

Displaying a 3D image of high quality requires a technology of the synthesis of the microrelief of a high-resolution DOE. To this end, we use electron-beam technology, which offers a resolution of 0.1 micron [8]. Electron-beam technology allows synthesizing unique CGHs for producing 2D images [9]. Electron-beam technology also makes it possible to produce a flat DOE for synthesizing 3D images [10,11]. A characteristic feature of electron-beam technology is that it allows producing the microrelief only on a flat base. In this paper we propose methods for computing and synthesizing a flat optical element, which, when placed on a cylindrical surface of given radius and illuminated with white light, produces a 3D image seen by the observer. The DOE original made using electron-beam technology can be replicated in millions of copies using standard technology of the production of reflective security diagrams on a flexible plastic base.

We made sample DOEs on a cylindrical surface. When illuminated with white light, the DOE produces a 3D image with an all-round 360° view. The video presented here demonstrates the high quality of the 3D image obtained.

2. Formulation of the inverse problem and methods for computing the microrelief of the diffraction optical element

Our task is to compute the DOE on a cylindrical surface. The observer should see a 3D image when the DOE is illuminated with white light. The 3D image is synthesized using a set of 2D frames obtained from different viewing points. Figure 1 shows the layout of observations.

 

Fig. 1 The arrangement of the light source and observer. Source of computer 3D model: cadnav.com, http://www.cadnav.com/3d-models/model-37178.html

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The Oz axis of the Cartesian Oxyz coordinate system coincides with the cylinder axis. Source S and the observer are located in the same plane Q. Plane Q passes through the cylinder axis. The observing layout can be most easily implemented in the case of fixed observer and source of light by turning the DOE on the cylinder about the Oz axis. Light from source S falls onto the DOE on the cylinder. The image displayed by the DOE is seen by the observer in the first order of diffraction. The 3D image is displayed as a result of the formation of two different images synthesized separately for the left and right eye of the observer.

We address the problem of the synthesis of the CGH on the cylindrical surface as the problem of computing and manufacturing a flat DOE, which, when placed onto a cylindrical surface, displays the specified 3D image. To create 3D images, we use a set of 2D frames of the 3D object taken from different viewing points. Let us denote frames as K_n, where n = 1...N. Figure 2 shows the arrangement of the observing points for the frames of the 3D image.

 

Fig. 2 Arrangement of the observing points for the frames of the 3D image.

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In Fig. 2 the cylindrical surface of the DOE is subdivided into elementary hogels G_ij. The viewing points for frames K_n are located on the cylindrical observation surface coaxial with DOE surface. The observing angles of frames K_n and K_{n + 1} differ by 2π/N. The size of the hogel does not exceed 50 microns. The observer sees each hogel as a bright or dark point.

Figure 3(а) shows the DOE subdivided into hogels and unfolded onto a plane. Each hogel is subdivided into subareas as shown in Fig. 3(b). Subareas are filled with diffraction gratings of different periods and orientations. The empty place, which is not filled gratings, consists of areas with flat microrelief, and can be used to adjust the intensity of diffracted rays.

 

Fig. 3 Structure of the DOE: (a) partition of the DOE into hogels unfolded onto a plane, (b) structure of a hogel.

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Figure 4 shows the scheme of the formation of diffracted rays coming from the hogel in the direction of the observer. K₁, …, K_N denotes the 2D frames of the 3D image unfolded onto a plane. Let us draw the perpendicular from the center of the selected hogel G_ij. The perpendicular crosses the image corresponding to n-th frame at point (x_i, y_j). The brightness I_n(x_i, y_j) of the image at this point determines the brightness of the hogel as seen by the observer from the observing point of n-th frame. One subarea in the diffraction structure of the hogel is sufficient for synthesizing in the first order of diffraction the ray with given intensity I_n(x_i, y_j).

 

Fig. 4 Computation of the intensity of the diffracted rays of the hogels.

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Let us now consider arbitrary hogel G of the optical element. We treat the hogel as flat because its size does not exceed 50 microns. Figure 5 shows the arrangement of the radiation falling onto the hogel (wave vector ${\overrightarrow{k}}_1$) and the ray diffracted from the hogel toward the observer (wave vector ${\overrightarrow{\mathrm{k}}}_2$). The absolute values of vectors ${\overrightarrow{k}}_1$ and ${\overrightarrow{\mathrm{k}}}_2$ are equal to $2\pi /\lambda$, where $\lambda$is the wavelength.

 

Fig. 5 Arrangement of vectors ${\overrightarrow{k}}_1$ and ${\overrightarrow{\mathrm{k}}}_2$.

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The coordinates of each hogel on the cylindrical surface are known. The coordinates of the source and the observing point are known, making the computation of vectors ${\overrightarrow{k}}_1$ and ${\overrightarrow{\mathrm{k}}}_2$ an easy task. Let us introduce vector $\overrightarrow{\mathrm{D}}=\frac{2\text{π}}{d}{\overrightarrow{e}}_d$, where d is the period of the grating, and ${\overrightarrow{e}}_d$ is the unit vector in the plane of the optical element perpendicular to the lines of the diffraction grating. It is well known [12] that

Here ${\overrightarrow{\mathrm{k}}}_{G,1}$ is the projection of vector ${\overrightarrow{k}}_1$ onto the surface of hogel G, and ${\overrightarrow{\mathrm{k}}}_{G,2}$ is the projection of vector ${\overrightarrow{\mathrm{k}}}_2$ onto the same plane. Relationship (1) determines both the direction of the grating lines in the subarea and the grating period for the given wavelength. We thus synthesized the diffraction structure of one hogel subarea. The parameters of the diffraction gratings in other subareas are computed in the same way. The empty place in the hogel is used to adjust the intensity of diffracted rays from the subareas. Theoretically, every hogel of the DOE may participate in the synthesis of N/2 frames K_n. The number of frames synthesized by each hogel in the sample CGH produced did not exceed 20.

A distinctive feature of the proposed optical scheme of observation is that the position of each source of light with respect to the observer does not change when passing from one viewing angle to another. It is therefore sufficient to compute the parameters of the diffraction gratings that synthesize the image for only one viewing angle. When passing to the next viewing angle the parameters of the gratings that produce the new image remain the same, and only their locations at the DOE surface change – they shift horizontally by $2\text{π}R/\text{N}$.

3. Making of a CGH on a cylindrical surface for the formation of a 3D image

We used the method described in section 2 to compute the microrelief of the DOE on a cylindrical surface to display the 3D image. We used standard 3D modeling software to obtain a set of 2D frames taken from different viewing points. The number of frames used to synthesize the 3D image is equal to 240. The cylinder has a radius and height of r = 15 mm and h = 31 mm, respectively. The angular separation between the neighboring observing points is equal to 1.5 degree. We computed the DOE for the use with daylight illumination, and therefore set the wavelength equal to 0.545 micron.

To produce the microrelief of the flat DOE we used electron-beam technology with a resolution of 0.1 micron. A characteristic feature of electron-beam technology is that it allows producing the microrelief only on a flat surface. The depth of the microrelief of the optical element is of about 0.2 micron. We then made an electrotype copy of the microrelief – a flat master hologram – with a thickness of about 200 microns. Such a master hologram can be used for mass replication on a flexible base [13]. To demonstrate the efficiency of the proposed technology, we made a thin nickel copy of master hologram with a thickness of about 40 microns. The thin copy of the DOE can be placed both on a flat and on a cylindrical surface. The image that forms when the DOE on a flat surface is illuminated with white light bears very little resemblance to a 3D object as shown in Fig. 6.

 

Fig. 6 Image produced by the DOE located on a flat base.

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The required 3D image is displayed when the DOE placed on a cylindrical surface is illuminated with white light. Figure 7 shows image frames obtained by photographing a real optical element located on a cylindrical surface.

 

Fig. 7 Examples of 2D frames obtained by photographing a real optical element located on a cylindrical surface (see video Visualization 1).

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The presented photographs and video of a real optical element placed on a cylindrical surface demonstrates the high quality of the 3D image seen by the observer over 360° viewing angles.

4. Discussion

A DOE on a cylindrical surface for the formation of a 3D image can be made using various observing schemes of the 3D effect [14]. For example, Yamaguchi et al. [15] proposed methods for computing transmission DOE. The optical element is transparent and illuminated from inside. In this study we consider a special scheme: a reflective DOE in a layout where both the light source and the observer are located on the outer side of the cylinder.

The time for computing the whole DOE using ready-made 2D frames amounted to approximately 30 seconds. The computations were carried out on a PC with AMD Phenom II X6 3.2 GHz CPU and 16 Gb DDR3 memory, without a GPU. Since the proposed method is well parallelized, the computation time can be reduced by a factor of ~100.

Optimizing the calculation time of 2D frames from a computer-generated 3D scene was not an objective of this study. Using standard 3D modeling software, the frames were rendered in 7 minutes on a PC (CPU only). However, it is known that on modern graphics cards such rendering could be performed in a fraction of a second. Thus, the proposed method can be potentially useful for real-time 3D display applications.

For comparison, in [16] one of the fastest calculation methods was proposed and a sample cylindrical hologram was produced. The computation time amounted to more than two hours. For our method, the computation time is substantially shorter due to several factors: the stereographic representation of the volume; the finite-parametric representation of the phase function; positioning of the observation points in a single plane.

5. Conclusions

In this paper we propose an efficient method for computing the microrelief of a DOE on a cylindrical base used for displaying a 3D image with a 360° viewing angle. The optical element is subdivided into hogels no greater than 50 microns. Hogels are filled with diffraction gratings with the size no greater than 10 microns. The minimum period of the diffraction grating was of about 0.58 micron. The computation of the microrelief over the entire DOE area from already available 2D frames takes about 30 seconds on a PC.

The size of the 3D object being the same, the smaller the radius of the cylinder the smaller should be periods of diffraction gratings. To produce the microrelief of the DOE original, we used electron-beam technology with a resolution of 0.1 micron. The technology of microrelief creation ensures high quality of the image on a highly curved surface.

We have fabricated sample DOEs on a cylindrical surface of radius 15 mm and height 31 mm. When illuminated with white light the DOE displays a 3D image with an all-round 360° viewing angle. The presented photos and video demonstrate high quality of the 3D image.

In this paper we limited viewing our hologram to horizontal displacement of the observer’s eye exclusively. Such a constraint allows the effect to be observed when the hologram is illuminated with white light from an ordinary lamp or a flashlight of a smartphone. The proposed computation method can also be used to compute a full-parallax DOE on a cylindrical surface.

The DOE on a cylindrical surface for displaying a 3D image that we developed in our study can be used as optical security feature. Reflective DOE can be replicated using standard technologies used to produce relief holograms on flexible bases.