Synthesis of Nano-Optical Elements for Forming 3D Images at Zero Diffraction Order

Abstract

A method is proposed to compute and synthesize a nano-optical element to produce a new visual effect: a 3D image formed in the vicinity of zero diffraction order. Usual relief rainbow holograms or OVDs can form 3D effect, but at +1 or − 1 diffraction order only and they provide 3D parallax in left/right direction only, and after rotation/inclination of an element, a 3D image changes its color and further disappears completely. The new visual effect provides with full 3D parallax. Moreover, a 3D zero-order image is well visible when an optical element is rotated through 360 degrees; the color of 3D image does not depend on the viewing angle. A synthesis technology is developed incorporating the computation of scattering patterns in elementary areas, computation of the phase function of the entire optical element, and the formation of its microrelief by using e-beam lithography. The microrelief consists of multilevel kinoforms with an accuracy of 10 nm in terms of depth. It was demonstrated by experimental results that the new visual effect is easy for visual perception under white light illumination. A sample of nano-optical element is manufactured, which when illuminated by white light, forms a 3D image in the vicinity of zero-order of diffraction (video available at: https://bit.ly/3QtzxbI).

1. Introduction

The development of methods for synthesizing optical elements to form 3D images began in the 1970s after Denis Gabor was awarded the Nobel Prize for the development of holographic recording principle [1]. Gabor’s follower [2] developed method for recording 3D holograms, which formed 3D images when illuminated by a white point light source. It was the invention of the surface relief holograms, which became known as rainbow holograms. These holograms formed visual 3D parallax in the left/right direction only, and when tilted up/down, the 3D image changes its color, that’s why they were called rainbow holograms. The 3D effect was formed in 1st diffraction order. In 1976, it was suggested the approach for forming 3D image at 1st diffraction order using binary computer-generated holograms [3]. Later, it was invented efficient technology of rainbow holograms microrelief replication (embossing technology) and rainbow holograms became widely used for protection against counterfeit [4]. The first optical security element used on Visa credit cards was a rainbow hologram with the original recorded on the optical table using an analog laser recording technique. At the same time, such optical elements appeared on bank notes and IDs.

The microrelief can be formed using both electron-beam lithography and optical recording methods. Modern approaches for synthesizing optical elements using laser microrelief recording have a resolution of 0.5 microns at best [5, 6], which is insufficient for the recording of asymmetric microrelief of optical elements that produce 3D full-parallax images. In this chapter, we form asymmetric microrelief of nano-optical elements using e-beam lithography with a resolution of 0.1 microns [7, 8]. This makes it possible to synthesize nano-optical elements that cannot be reproduced using standard widespread methods of microrelief recording based on optical origination techniques.

E-beam lithography has already been used to synthesize nano-optical elements that form 3D images [9, 10]. In these studies, a 3D image was formed at the first order of diffraction with the microrelief of the optical element shaped using symmetrical structures. The resulting 3D images can be observed only within a limited range of viewing angles near the first order of diffraction when the element is tilted left/right and up/down; however, the 3D image formed disappears when the element is rotated.

We discuss the possibilities of synthesizing nano-optical elements to form 3D images at the zero-order of diffraction. This is the first time that methods of synthesizing nano-optical elements to form 3D images in the zero-order of diffraction have been developed. For such elements, the 3D effect can be observed near the zero-order of diffraction both by tilting the element and rotating it by 360 degrees.

2. Formulation of the problem of the synthesis of nano-optical elements for the formation of 3D images at zero diffraction order

The standard scheme for observing a 3D image formed by a rainbow hologram [2] is shown in Figure 1a, where the area of observation (the area in which the observer’s eyes can be located) is indicated by the yellow band. The 3D image is formed in the first order of diffraction, and the observation area is a limited narrow band. An observer can see 3D parallax in the left/right direction only. When the optical element is slightly tilted or rotated, observer’s eyes leave the area of observation, and the 3D image disappears for an observer.

Figure 1.

Scheme for observing 3D image: a) formed by rainbow hologram at 1st diffraction order, b) formed by CGH with full 3D parallax at 1st diffraction order, and c) in the vicinity of zero order.

Figure 1b shows the observation scheme of a 3D image with full parallax at the 1st diffraction order [11]. The nano-optical element consists of calculated binary kinoforms recorded by e-beam lithography. Here the area of observation is a thick rectangle. An observer can see 3D parallax in both left/right and up/down directions. However, the viewing angle range is also limited, and after rotation of the optical element observer’s eyes leave the area of observation and the 3D image disappears.

Figure 1c shows the new proposed authors’ observation scheme of a 3D image that is formed in the vicinity of the zero order of diffraction. The observation area is a large square centered on the zero order. As long as observer’s eyes are within this region, an observer sees a full parallax 3D image, and the 3D image is observed over a wide range of tilt angles and even when the optical element is rotated through a full range of 360 degrees.

Figure 2 schematically shows the formation of 3D images by a flat reflecting optical phase element at diffraction angles within plus or minus 30° of the zero order.

Figure 2.

Schematic diagram of the formation of 3D images by a flat optical element.

The optical element is located in the z = 0 plane. Figure 2 shows a fragment of observation points (five points along the horizontal direction and three points along the vertical direction). The centers of observation points are marked with the letter “R”. The number of frames is several hundred for real optical elements that form a zero-order 3D image. The light source “S” is located in the Oxz plane of the Cartesian coordinate system. The source is located at an angle of θ₀ to the Oz axis. The direction to the zero order is denoted as L₀. At different angles φ,θ, the observer sees different 2D frames K_n, n = 1 … N of a 3D image. Here, φ and θ are angles in the spherical coordinate system. The angle θ is counted from the Oz axis, and φ is the azimuthal angle. The ray L in Figure 2 points towards one of the observation points and has angular coordinates φ,θ. Let us assume that the angles (φ_n, θ_n) set the directions towards the observation point of frame K_n, n = 1 … N.

Figure 3 shows the observation scheme in the Oxz plane at small diffraction angles. The diffraction angle is defined as the angle between the zero order of diffraction and the direction towards the observation point. Let us denote the diffraction angle as β. For small diffraction angles, the angle is defined by the formula β = θ-θ₀. A 3D image is observed at diffraction angles within 30° of the zero order of diffraction. The angle θ₀ between the light source S and the normal to the plane of the optical element coinciding with the Oz axis in the diagram determines the zero-order diffraction by beam L₀.

Figure 3.

Scheme of observation at small diffraction angles.

Synthesis of a nano-optical element to form zero-order 3D images is quite a complex and challenging task. If we use a grid with a step of 0.1 x 0.1 micron for an optical element of, for example, 28 × 33 mm² size, then the number of points at which it is necessary to calculate the phase function of the optical element is about 10¹¹. However, the proposed method for calculating the phase function of an optical element efficiently solves this problem.

The method that we propose for the first time in this chapter allows the use of different 3D models to form 3D greyscale images. To demonstrate the method for calculating the phase function of the diffractive optical element (DOE) a 3D object was chosen. Figure 4 shows a computer 3D model of the object.

Figure 4.

3D model of the object.

Figure 5 shows a fragmented set of the 2D frames of the 3D computer model, and Figure 6 presents a scheme of partitioning the flat optical element into the elementary regions (G_ij). The size of the elementary region does not exceed 80 microns, which is beyond the resolution of the human eye.

Figure 5.

Fragment of the 2D-frames of the 3D computer model.

Figure 6.

Schematic diagram of the partitioning of an optical element into elementary regions.

Figure 7 schematically shows the formation of the angular pattern in the elementary area G_ij i = 1… L, j = 1… M. The formation involves all rays from the center of the elementary area to all observation points (R). The ray L_n directed towards the center of the observation point K_n is defined by the angles φ_n, θ_n. The number of rays coincides with the number of 2D frames of the 3D image and is equal to several hundred. The intensity of beam L_n in the direction (φ_n,θ_n) for each n, n = 1… N, is determined as follows. The brightness of point (x_i,y_j) in frame K_n determines the intensity of beam L_n.

Figure 7.

Schematic diagram of the formation of the angular pattern of the area G_ij.

The angular pattern of the light scattered from each elementary region G_ij is formed at all observation angles (φ_n, θ_n) of the 3D image. Here, n = 1… N. The angular pattern of the region G_ij is a set of N rays, and each ray L_n has a given intensity. Figure 8 shows the angular patterns computed for twelve elementary G_ij regions.

Figure 8.

Angular patterns of twelve different areas G_ij.

Thus, in the approximation of geometrical optics, radiation angular patterns are determined for all elementary regions G_ij. In the next step, we use the determined angular patterns to compute the phase function of the optical element for each elementary region G_ij.

3. Method for computing the phase functions in elementary regions

We use the scalar Fresnel wave model to compute the phase functions in the all elementary G_ij regions. In this Fresnel model, the scalar wave field $u\left(x,y,f\right)$ in the z = f plane is related to the scalar wave field $u\left(\xi ,\eta ,0\right)$ by the following formula:

$$\gamma {\iint }_{G_{\mathit{ij}}}u\left(\xi ,\eta ,0\right)exp\left\{\mathit{ik}\frac{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}{2f}\right\}\mathit{\text{dξdη}}=u\left(x,y,f\right).$$E1

Here k = 2π/λ and γ = exp.(ikf)/iλf is a given constant where λ is a wavelength. Figure 9 shows a scheme of the formation of the 2D image formed by the angular pattern of the elementary region G_ij. The plane wave falls onto the reflecting flat phase optical element whose microrelief forms an image in the z = f plane.

Figure 9.

Optical scheme of the formation of the image in elementary region G_ij.

The peculiarity of the inverse problem of forming a 2D image is that the right-hand side of Eq. (1) does not contain the wave function $u\left(x,y,f\right)$ but only its absolute value $F\left(x,y\right)=\left|u\left(x,y,f\right)\right|$.

Let us represent the wave function on the plane z = 0 in the form $u\left(\xi ,\eta ,0\right)=\overline{u}\left(\xi ,\eta \right)exp\left(\mathit{ik\phi }\left(\xi ,\eta \right)\right)$. Here $\overline{u}\left(\xi ,\eta \right)$ is the amplitude and φ(ξ,η) is the phase function of the planar optical element in the elementary region G_ij, We, thus, have the following operator equation:

$$\mathrm{A}\phi \left(\xi ,\eta \right)=F\left(x,y\right).$$E2

In the Fredholm operator equation of the first kind (2)F(x,y) is a given function. The operator A is defined by the following relation:

$$\mathit{A\phi }=\left|\left.\gamma {\iint }_{G_{\mathit{ij}}}\bar{u}\left(\xi ,\eta \right)exp\left(\mathit{ik\phi }\left(\xi ,\eta \right)\right)exp\left\{\mathit{ik}\frac{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}{2f}\right\}\mathit{\text{dξdη}}\right|.\right.$$E3

Eq. (3) is a nonlinear operator equation with respect to the desired function φ(ξ,η) and describes an ill-posed problem. Efficient numerical algorithms were developed to solve ill-posed nonlinear problems [12, 13]. However, one of the most efficient methods for the approximate solution of Eq. (2) is the method proposed by Lesem and his colleagues in 1969 [14]. This method later came to be called the Gerchberg-Saxton algorithm [15]. Many studies were dedicated to investigating this algorithm [16, 17, 18]. All these methods have the same property. The value of the functional decreases monotonically quite rapidly during the first 10 iterations, and then the decrease rate falls off rapidly.

We follow Lesem [14] to use an algorithm for the approximate solution of nonlinear Eq. (2). Let us introduce the following notation:

$$\Phi \left\{\nu \right\}\left(x,y\right)=\gamma {\iint }_{G_{\mathit{ij}}}\nu \left(\xi ,\eta \right)·\mathit{exp}\left(\mathit{ik}\frac{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2}{2f}\right)\mathit{\text{dξdη}}.$$E4

Here $\mathrm{\Phi }\left\{\nu \right\}\left(x,y\right)$ is the Fresnel transform of function v. We construct the iterative process of building the phase function that is an approximate solution of inverse problem (3) as follows. Four steps have to be taken to perform one iteration in the iterative algorithm for solving problem (3). Let v^(m)(x,y) be given at the m-th iteration. We write function v^(m)(x,y) in the form v^(m)(x,y) = A₀exp(ikφ₀^(m)(x,y)) and function w^(m)(x,y) in the form w^(m)(x,y) = A₁exp(ikφ₁^(m)(x,y)). Both A₀ and A₁ are real functions. Let A₀(x,y) be the given intensity distribution of incident light in the z = 0 plane. As we are considering phase only DOE, then the amplitude A₀(x,y) will be equal to one in the elementary region G_ij. Let A₁(x,y) = F(x,y) be the given intensity distribution in the focal plane z = f. The algorithm for solving the inverse problem consists of the following four steps performed in sequence:

$$\begin{array}{c}\left.1\right){\phi }_1^{\left(m\right)}\left(x,y\right)=\mathit{arg}\left(\Phi \left\{v^{\left(\mathrm{m}\right)}\right\}\left(x,y\right)\right)\\ 2)w^{\left(m\right)}\left(x,y\right)=A_1\left(x,y\right)\mathit{exp}\left(\mathit{ik}{\phi }_1^{\left(m\right)}\left(x,y\right)\right)\\ 3){\phi }_0^{\left(m+1\right)}\left(x,y\right)=\mathit{arg}\left({\Phi }^{-1}\left\{w^{\left(m\right)}\right\}\left(x,y\right)\right)\\ 4)v^{\left(m+1\right)}\left(x,y\right)=A_0\left(x,y\right)\mathit{exp}\left(\mathit{ik}{\phi }_0^{\left(m+1\right)}\left(x,y\right)\right)\end{array}$$E5

The function ${\phi }_0^{\left(m+1\right)}$ is an approximate solution of Eq. (2). A phase distribution equal to a constant can be used as an initial approximation. The phase function φ(ξ, η) computed by the iterative process (5) uniquely determines the microrelief in the region G_ij. For example, for a normal wave incident on an optical element, the depth of the microrelief in the region G_ij is equal to 0.5 φ(ξ, η) for any point (ξ, η) in this region.

Figure 10 shows two calculated fragments of the microrelief of the multilevel kinoform in the two elementary G_ij regions. The fragments size is 10 × 10 μm². The depth of the microrelief does not exceed 0.5λ and is equal to approximately 0.3 μm.

Figure 10.

Calculated fragments of the microrelief in the two elementary G_ij regions.

Thus, the solution of the inverse problem for each elementary region, G_ij, i = 1… L, j = 1… M, yields the microrelief on the entire area of the nano-optical element. The above algorithm for computing the phase function can be applied to the 3D model of any 3D object.

4. Example of the synthesis of multilevel nano-optical element for forming 3D images at zero diffraction order

To demonstrate the efficiency of the proposed method, we made a 28 × 33 mm² nano-optical element to form a zero-order 3D image. A 28 × 33 mm² flat optical element was partitioned into 369,600 50 × 50 μm² elementary G_ij regions, i = 1… L, j = 1… M, as shown in Figure 6. The number of frames N was 1440 (60 frames horizontally, 24 frames vertically). We compute the microrelief of the flat optical element at the fixed green wavelength λ = 547 nm for each elementary region G_ij. To compute the phase function in the area G_ij, it was used a 500 × 500 grid to solve the inverse problem (2) of computing the phase functions in the elementary regions, and it takes more than 4 hours to compute the phase function for the entire optical element on a PC (AMD Phenom II X6 3.2 GHz CPU and 16 Gb DDR3 memory).

We used a shaped beam e-beam lithography system with a minimum beam size of 0.1 μm x 0.1 μm to record the microrelief of the nano-optical element and used a positive PMMA electron beam resist to form the nano-structures. The maximum microrelief depth was 0.3 μm, and the depth accuracy of microrelief formation was 10 nanometers in terms of depth. The nickel master shim of the diffractive optical element was produced using a standard electroforming procedure.

Figure 11 shows photographs of the nano-optical element taken from different viewing angles at diffraction angles of plus or minus 30° relative to the zero order of diffraction. A cell phone flash with green color filter was used as the light source.

Figure 11.

3D images formed by the nano-optical element at different angles under green light source (see video V1 and video V2) (video available at: https://bit.ly/3QtzxbI).

We computed the microrelief at the fixed wavelength of λ = 547 nm, which corresponds to green light, and the quality of the images formed are good when the nano-optical element is under green light source illumination. However, of course, the main interest is how the element will be seen when illuminated with a white light source. Figure 12 shows photographs of the same nano-optical element taken from different viewing angles by using a white light source, cell phone flash without any filter.

Figure 12.

3D images formed by the nano-optical element at different angles under white light source (see video W1 and video W2) (video available at: https://bit.ly/3QtzxbI).

As can be seen from the Figure 12, the formed 3D image remains clear and contrasting despite the illumination with a white light source. The resulting 3D image can be observed well when illuminated by white light, and the observer sees the 3D image with full parallax both when tilting the optical element and when rotating it by 360 degrees. In addition, unlike rainbow holograms, the color of the formed 3D image does not depend on the viewing angle. That is, it turns out that the formed 3D image behaves like a real 3D object in a full range of viewing angles.

5. Discussion and conclusion

In this chapter, we develop methods for synthesizing nano-optical elements to form 3D images at the zero-order diffraction for the first time. The synthesis methods include both the computation of the phase function of the nano-optical element and the formation of its microrelief by using of e-beam lithography. From a mathematical point of view, the computation of the phase function is a typical inverse problem, which we solve in two steps. In the first step, we use all the image frames that define a 3D object to generate the angular patterns in each elementary region. In the second stage, we compute the phase functions of the nano-optical element in each elementary region. The latter problem reduces to solving a nonlinear integral equation. Despite the large number of elementary regions (∼370,000), a personal computer is definitely sufficient to compute the phase function of the entire nano-optical element.

We used e-beam lithography for the formation of the microrelief. The accuracy of microrelief formation is 10 nanometers in terms of depth. We produced a sample nano-optical element that forms a 3D image in the zero order of diffraction. The resulting 3D image can be observed when illuminated by white light. A 3D image can also be formed in the first order of diffraction, as we did, for example, in our earlier study [11], using a binary microrelief. In this case, the diffraction efficiency of the optical element does not exceed 40%. The use of multilevel microrelief makes it possible not only to increase the diffraction efficiency but also to significantly widen the viewing angles of the 3D image. The observer sees a 3D image with full parallax both when tilting the optical element and even when rotating it by 360 degrees. The 3D image is stable and behaves like a real 3D object. The authors also believe that it is possible to use 2D frames captured from 3D computer models with some animation, and thus the formation of 3D zero order images with the effect of animation is possible; however, this is the subject of future numerical and real experiments on the formation of 3D animated images in the zero order of diffraction.

The structure of an optical element forming a 3D image in the zero order of diffraction can be modified to make the kinoforms fill the G_ij regions partially rather than completely [19, 20]. The remaining parts of the elementary G_ij regions can be filled with diffraction gratings with periods less than 0.6 μm. These diffraction gratings can form an additional 2D image visible over the entire area of the DOE at diffraction angles greater than 60 degrees.

The nano-optical element can be replicated using standard embossing equipment for the production of surface relief holograms. The synthesis methods developed are designed first of all to protect bank notes, IDs and brands against counterfeiting. The technology of the synthesis of nano-optical elements is expensive, knowledge intensive and not widespread, thereby ensuring high protection level of the developed DOEs against counterfeiting.

Methods for calculating the phase functions of nano-optical elements can be used in advanced 3D displays and 3D projectors. At present, supercomputer technologies are widely used to speed up calculations. The phase function in each elementary region is calculated independently, which makes it easy to parallelize the numerical algorithm. The use of a graphics processing unit (GPU) cluster can speed up the calculation of the phase function of a nano-optical element by hundreds or even thousands of times. At present, processors with hundreds thousand cores have been developed and are available [21]. The use of such technologies can make it possible to compute the phase function of the entire optical element in a fraction of a second, thereby opening up opportunities for the synthesis of animated 3D images in prospective 3D design systems and 3D displays [22, 23].