Curved Holographic Optical Elements from a Geometric View Point

The two principal problems addressed in this paper are as follows:

The paper is organized as follows:

• •

  In Section 2 we briefly introduce the core system architecture of a retina scanner display (RSD) and discuss some important aspects of industrial replication of HOEs and the embedding of HOEs in ophtalmic lenses.

• •

  In Section 3 we give a cursory introduction to volume holography, starting with the example of a volume grating obtained by interference of two plane waves during the recording process. Based on the invesitgation of the volume grating we introduce the notion of a grating vector. With this background on volume gratings, we turn our attention to HOE obtained from the interference of spherical waves and define the HOE model which is used throughout this paper.

• •

  In Section 4 we define a space of admissible HOE deformations and investigate some of their properties to formalize Problem 1 and provide a geometric framework for a solution. We then turn our attention to Problem 2 using the previously developed methods.

• •

  In Section 5, we show results of numerical experiments using a basic Matlab implementation of the core principles of the methods developed in the previous sections. We also return to the retina scanner display as the motivating application and discuss the effects of mechanical deformations of an HOE converting a divergent spherical wave into a convergent spherical wave.

• •

  We conclude our presentation with a summary of our results and an outlook of on-going and future research in Section 6.

We briefly recall the RSD architecture and the use of HOEs as combiners in such a system[2]. The conept of (virtual) retina scanner displays goes back at least to the 1990s[3, 11]. However, only recent advances in MEMS technology, laser integration, holographic materials, holography, and lens casting have put the concept within reach of industrialization. Figure 3 shows a schematic illustration of an RSD system. A projection unit, typically containing a laser module and a MEMS mirror module, scans the projected light across the lenses. HOEs attached to or embedded in the lenses diffract the light of the projection unit towards the user’s eyes without perturbing the incident light from the surrounding environment due to wavelength selectivity of the HOEs[18].

While it is possible to record an HOE on curved substrate, it may be advantageous or even necessary to record HOEs in a planar setting, even in cases where the application profits from curved HOE. This may for example be due to the manufacturing process such as a roll-to-roll replication process targeting mass production of HOE at industrial scales [9]. The aforementioned process is based on the replication of a master HOE located on a planar carrier substrate using single beam illumination of a photopolymer foil laminated onto the master. The photolpolymer foil is supplied from a roll and is laminated automatically to the master before replication and delaminated automatically and rolled up again afterwards,

One motivation to investigate the deformation of curved HOE after recording is provided by the need to introduce prescription lenses for vision correction into a RSD. The compatibility with vision correction is expected to be an important aspect for smartglasses targeted at consumer applications. Moreover, there already exist industrial processes to embed HOE into polymer lenses[10] compatible with current standards in the ophtalmic industry. However, the process requires the HOEs to be shaped into sphere segments prior to embedding. Moreover, the casting material is cured with UV light and thus requires that the HOEs are already recorded and fixed in the photopolymer prior to embedding.

Aside from the workflow of the process presented in [10], the thickness (and thus the weight) of the lens is another design aspect which favors curved HOE over planar ones. For a prescribed in advance lens diameter (in particular the relatively large diameters of the lens blanks compared to the final lens diameter after edging and fitting in the spectacles frame), the thickness of the lens is determined by the desired optical power and outer radii according to the lens maker equation[19]

where $R_{1}$ and $R_{2}$ are the radii of curvature, $n$ the index of refraction of the lens material, and $d$ the center thickness of the lens. Note that we assume the radii in (1) to be positive since we are primarily interested in convex-concave (or meniscus) lenses for vision correction. Moreover, we assume the lens to be embedded in air and thus use a refractive index of $n_{0}=1.0$ for the surrounding medium. Figure 4 shows in the left column two convex-concave lenses for vision correction, the top lens corresponding to a corrective lens for hyperopia, and the bottom lens is a negative lens to correct myopia. The central column illustrates the embedding of a planar HOE in the same type of lenses. We observe, that for a prescribed lens radius, the HOE plane introduces a lower bound for the lens thickness. Moreover, the embedding material thickness between HOE and the environment on the eye-side at the lens center is also limited by the overall lens thickness. In the right column, we see an HOE which was curved into a sphere segment prior to embedding. This allows a thinner lens compared to the planar HOE or an increased minimal thickness of the lens material for improved mechanical protection.

We begin our disucssion with the following holographic recording setup: the output of a coherent light source (typically a laser) is collimated and the resulting beam is divided into two light paths. The two beams are considered here for simplicity to represent ideal plane waves. A planar substrate carrying a photopolymer foil is placed in the region where these two plane waves interfere. The intensity of the interference pattern is translated by chemical reactions into a refractive index modulation in the photopolymer and can be fixed using UV-light. If the resulting HOE is illuminated with one of the recording waves serving as a probe wave, the light diffracted at the HOE strucuture will propagate in the same direction as the second recording wave. The angles of incidence of the waves at the photopolymer with respect to the surface normal of the substrate and a reference point (e.g. the center of the future HOE) are typically used to describe the recording geometry of such plane wave converting HOEs.

Figure 5 illustrates the above situation for a planar photopolymer. The photopolymer is illuminated by the two interfering waves and a refractive index modulation in the material occurs. After this recording step – and typically an additional fixation step – the HOE can be reconstructed (or replayed). For an ideal HOE, if the probe wave used to reconstruct the hologram corresponds to one of the recording waves or their complex conjugates, the HOE is said to be reconstructed in on-Bragg condition. This is typically where diffraction efficiency, i.e. the ratio of the power in the diffracted wave and the incident wave, is at its maximum. Any deviation from the on-Bragg condition, e.g. changes in the angle of incidence or the wavelength, will result in a different diffraction angle and a reduced diffraction efficiency[18].

We consider two plane waves $E_{l}=A_{l}(\mathbf{r})\exp(\mathrm{i}\mathbf{k}_{l}\cdot\mathbf{r}),l=1,2$ with real amplitudes $A_{l}$.

The interference pattern of these two waves is given by

We assume thath the cosine modulated intensity pattern in (2) is tranferred linearly into a refractive index modulation. Note that $\cos((\mathbf{k}_{2}-\mathbf{k}_{1})\cdot\mathbf{r})$ is constant, if $(\mathbf{k}_{2}-\mathbf{k}_{1})\cdot\mathbf{r}$ is constant. Moreover, the condition

is the representation of a plane (or familiy of parallel planes for varying constants). The normal vector is colinear to the vector

Since the cosine is $2\pi$–periodic, the next plane in direction of $\mathbf{k}_{g}$, where the cosine value is repeated, is located at a distance

The familiy of planes corresponding to fixed constant values $c_{0}+l\Lambda,c_{0}\in\mathbb{R},l\in\mathbb{Z}$ in (3), are often refered to as Bragg planes. They are all parallel and spaced equidistantly with the volume grating period $\Lambda$ as defined in (5).

Thus, the vector $\mathbf{k}_{g}$ encodes the orientation and the spacing of the Bragg planes and is called the grating vector.

Note that the grating vector is constant in this example and the resulting Bragg strucuture is periodic. Therefore, we will also refer to this type of HOE as a volume grating.

In the following, we will use the above observations to briefly introduce methods to determine the direction of propagation of a diffracted wave using vector formalisms.

We briefly review some modelling aspects regarding the diffraction of light at a volume hologram, in particular the direction of propagation of the diffracted light (also refered to as the diffraction order), and the diffraction efficiency. At this point it is also important to point out that the geometric framework, which we are presenting in the following, is quite flexible and can be combined with different modelling approaches with varying complexity, depending on the physical accuracy required.

A common mathematical model used in optics[6] to compute the wavevector $\mathbf{k}_{d}$ of the diffrated wave from the wave vector $\mathbf{k}_{p}$ of the probe wave and the grating vector $\mathbf{k}_{g}$ is based on simple vector addition motivated by (4), i. e.

This method is referred to as k-vector closure by some authors. However, in the off-Bragg condition, the method leads to the physically invalid phenomenon that the length of the diffracted wave vector is different from the wave number of the probe wave. Therefore, the method can be improved[20] by enforcing that the following holds.

Experimental validations, also of alternative theories, exist and have been discussed elsewhere [21]. In our discussion, we will typically assume that the HOE can be approximated locally by a volume grating and use a Cartesian coordinate system spanned by an orthonormal basis $\{\mathbf{t}_{1},\mathbf{t}_{2},\mathbf{n}\}$ with the unit surface normal $\mathbf{n}$ of the HOE substrate as one coordinate axis. Then we can enforce (7) by defining a new grating vector $\mathbf{k^{\prime}}_{d}$ setting

provided that (10) has a real solution corresponding physically to a propagating diffraction order.

Another alternative – which is also helpful in visualizing volume diffraction in some instances – suggests that the Bragg planes serve as partial mirrors or scatterers (due to the Fresenel conditions at boundary interfaces). Thus, in the on-Bragg condition, volume diffraction can be interpreted as reflection at the Bragg planes[22].

To fully predict the optical characteristics of an HOE, we need to know the direction in which the diffracted light propagates, as well as how much of the incident light is actually diffracted into this new direction. In this paper, we will limit the discussion to the simplifying assumption that there are only two propagating waves present in a volume grating. This is a common assumption in optics known as coupled wave theory[6, 20]. An alternative approach is described by the parallel stacked mirror (PSM) methods[22]. For the interested reader, there exists a vast body of work ranging from introduction into diffraction theory for holography [23] to surveys of different diffraction theories and experimental validations, e.g. with the focus on the far off-Bragg regime [21].

For our discussion, we assume that an incident plane wave (the probe wave) represented by a wave vector $\mathbf{k}_{p}$ is diffracted at an HOE with the diffracted light propagating along the wavevector $\mathbf{k}_{d}$ also referred to as the first diffraction order. If the diffraction efficiency $\eta$ of the HOE is less than $1$, i.e. not all the light is diffracted into the first diffraction order, then a so-called zero diffraction order continuing along the direction of $\mathbf{k}_{p}$ will be present. Since for a retina scanner display we are primarily interested in phase holograms anyway, we will neglect possible absorption effects in the holographic material itself. Thus the contribution to the zero diffraction order is assumed to be $1-\eta$.

We recall the complex representation of a spherical wave, i.e. an electromagnetic field where the surfaces of constant phase, i.e. the wavefronts, are contrentic spheres centered at a fixed point $\mathbf{r}_{0}$, to be

where the amplitude is given by

and

is called the wave number.

We consider two spherical waves, one diverging from the point $\mathbf{r}_{1}$, and the second one converging towards the point $\mathbf{r}_{2}$. Note that the fact that one wave is diverging and the other is converging is accounted for by changing the sign in the phase term arising from the scalar product of the wave vector and the spatial coordinate. Thus we have the recording waves

and

where the amplitudes and wavenumber are again defined as in (12) and (13), respectively.

The situation is illustrated in Figure 8 for the recording and reconstruction configuration of such an HOE.

The absolute value squared of interference pattern of these two spherical waves is given by

and in an ideal holographic material is proportional to the refractive index modulation.

We observe that the oscillating (i.e. the cosine) term is constant on sets where

Note that an ellipsoid of revolution $E_{\mathbf{r}_{1},\mathbf{r}_{2}}$ with focii $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ is characterized by the fact that for any point on the boundary $\mathbf{r}\in\partial E_{\mathbf{r}_{1},\mathbf{r}_{2}}$ the sum of the distances from $\mathbf{r}$ to the two foci is constant, i.e. there exists a real number $C>0$ such that

Thus, we make the following observation.

Note that we will also refer to these loci of constant index of refraction as Bragg isosurfaces.

From the above observations, we obtain immediately the following result.

We point out here, that we will use planar volume gratings as local approximations for the curved Bragg isosurfaces in the remainder of this paper. This approximation is expected to be justified if the distances of construction points of the HOE to the HOE substrate are suffciently large, such that the local curvature of the curved Bragg isosurfaces are large enough to allow a local approximation by the tangent plane. It is noteworthy that the use of curved geometric entitites such as ellipsoids may prove useful for a system analysis, when the HOE properties have to be modelled not only pointwise, but for example in an open disc around a reference point. Interestingly, similar techniques, where curved geometric surfaces have been used to replace tangent planes as local approximations, have been employed already in the analysis of freeform reflector and lens design problems[25, 26]. The associated partial differential equations are still attracting researchers’ attention both from an analytical[27, 28] as well as from a numerical[29, 30, 31, 32] point of view. We plan to return to the topic of supporting quadric methods for HOE in a separate investigation.

Although volume holograms are also referred to as thick holograms in the optics literature, their thickness is to be judged with respect to the wavelength of light. Thus, a phototpolymer of 10 to 20 micron can already contain an optically thick hologram. Since the lateral dimensions are typically in the range of tens of millimeters, e.g. for lenses for smartglasses, the thickness of the photopolymer is assumed to be negligible from a macroscopic geometry point of view. This leads to the following geometric model for an HOE.

For the sake of clearity we make the following assumptions:

1. 1.

   HOEs are modelled as embedded two-dimensional surfaces in $\mathbf{R}^{3}$, typically as the graph of a sufficiently smooth function over a fixed reference plane, such as the $(x,y)$-plane. Note that convex functions are in any case twice differentable almost everywhere[33]. Thus, we consider primarily functions which are at least twice differentiable, i.e. of class $\mathcal{C}^{2}$.

2. 2.

   The physical thickness of the photopolymer is included in the local model used to compute the diffraction efficiencies and neglected in the macroscopic geometry of the HOE surface otherwise.

3. 3.

   The planar HOE is contained in the $(x,y)$-plane of a Cartesian coordiante system $(x,y,z)$ centered at the origin $\mathcal{O}=(0,0,0)$.

4. 4.

   A parametrization of the curved HOE as an embedded surface in $\mathbb{R}^{3}$ is given as the graph of a convex function over the $(x,y)$-plane.

5. 5.

   The curved HOE is a subset of a surface that is rotationally symmetric about the $z$-axis.

In the following, we develop a framework to formulate Problems 1 and 2 more rigorously and establish the tools for an analytic and numerical analysis of curved HOE as well as the basis for a design method to precompensate deterministic deformations of a planar HOE during the recording process.

The core idea of our method is as follows:

1. 1.

   We describe the inner structure of the HOE, i.e. the Bragg-surfaces using a grating vector model to encode the local orientation and spacing of the Bragg surfaces, cf. Definition 3.3. We will also refer to this inner structure as the microscopic geometry of the HOE,

2. 2.

   We relate the inner structure of the HOE to the macroscopic geometry of the surface associated with the HOE using local coordinate systems which we derive from moving frames. Each frame is formed by an orthorgonal set of vectors, one being the local surface normal, the other two being tangent vectors to be chosen in a suitable manner.

3. 3.

   At first order, deformation of the HOE surface $\mathcal{H}$ into the HOE surface $\tilde{\mathcal{H}}$ corresponds to a change of the unit tangent vectors and the unit surface normal vector $\{\mathbf{t}_{1},\mathbf{t}_{2},\mathbf{n}\}$ into a new local frame $\{\tilde{\mathbf{t}}_{1},\tilde{\mathbf{t}}_{2},\tilde{\mathbf{n}}\}$.

4. 4.

   We assume a one-to-one coresspondence between points $\mathbf{r}\in\mathcal{H}$ and $\tilde{\mathbf{r}}\in\tilde{\mathcal{H}}$. If $\tilde{\mathcal{H}}$ is a graph of a function over the $(x,y)$-plane this condition is for example fulfilled by the orthorgonal projection along the $z$-axis, other possible projections will be discussed below.

5. 5.

   Thus, the grating vectors of the deformed HOE are obtained from the original grating vector

   $$\mathbf{k}_{g}(\mathbf{r})=g_{1}(\mathbf{r})\mathbf{t}_{1}(\mathbf{r})+g_{2}(\mathbf{r})\mathbf{t}_{2}(\mathbf{r})+g_{3}(\mathbf{r})\mathbf{n}(\mathbf{r})$$

   (21)

   as the vector in the frame $\{\tilde{\mathbf{t}}_{1},\tilde{\mathbf{t}}_{2},\tilde{\mathbf{n}}\}$ with the same coordinate values as $\mathbf{k}_{g}$.

   $$\tilde{\mathbf{k}}_{g}(\tilde{\mathbf{r}}(\mathbf{r}))=g_{1}(\mathbf{r})\tilde{\mathbf{t}}_{1}(\tilde{\mathbf{r}}(\mathbf{r}))+g_{2}(\mathbf{r})\tilde{\mathbf{t}}_{2}(\tilde{\mathbf{r}}(\mathbf{r}))+g_{3}(\mathbf{r})\tilde{\mathbf{n}}(\tilde{\mathbf{r}}(\mathbf{r}))$$

   (22)

In essence, we use a point-wise correspondence between local frames to transform one vector field $\mathbf{k}_{g}$ on a surface $\mathcal{H}$ into another vector field $\tilde{\mathbf{k}}_{g}$ on a surface $\tilde{\mathcal{H}}$. The vector fields $\mathbf{k}_{g}$ and $\tilde{\mathbf{k}}_{g}$ serve to model locally the optical properties of the HOEs $\mathcal{H}$ and $\tilde{\mathcal{H}}$, respectively. The general idea of the method is illustrated graphically in Figure 9.

Next, we will introduce the notion of moving frames to define local coordinate systems associated with the macroscopic HOE geometry.

Using the moving frames from Definition 4.7, we are now in the position to establish a formalism to account for changes in the microscpic geometry of an HOE due to a macroscopic deformation.

With the formalism developed above, we can provide an answer to Problem 1 as follows:

In the following section, we will use the above framework to address Problem 2 in an analogous manner.

We turn our attention to the inverse problem stated in Problem 2. In our newly developed framework, Problem 2 consists of finding for a given recording geometry $\mathcal{H}$, transformation $T_{\mathbf{C}}\in\mathcal{T}_{a}$, and target HOE $(\tilde{\mathcal{H}},\tilde{\mathbf{k}}_{g})$ a grating vector field $\mathbf{k}_{g}$ such that the vector field induced by $(\mathcal{H},\mathbf{k}_{g})$ and $T_{\mathbf{C}}$ on $\tilde{\mathcal{H}}$ coincides with $\tilde{\mathbf{k}}_{g}$.

Based on the formalisms developed in Section 4.2, we can also express an answer to Problem 2 in terms of admissible transformations and induced grating vector fields.

Figure 10 shows a visualization of the simulation results of the diffraction of a plane wave at a planar and two curved HOEs. The planar HOE was recorded using two plane waves. One recording wave was indicent at $0\deg$ with respect to the optical axis perpendicular to the HOE at its center point, the second recording wave was incident at an angle of about $65\deg$ with respect to the optical axis. The left column of figures shows the propagation directions along the diffracted wave vectors (red) for plane wave illumination at the recording wavelength with a probe wave (green) incident at $65\deg$ to the $z$-axis (the optical axis). The top graphic shows a close up of the HOE surface, part of the grating vector field (blue, rescaled for visualization) is also visible. The center column shows the results for an HOE recorded in a curved phototpolymer. Aside from the macrsocopic geometry, there is no difference to the planar HOE. The right column shows the result for an HOE that was deformed after recording. In particular the view in the bottom graphics – where the propagation directions of the diffracted wavevectors have been scaled up for visualization purposes – shows a significant change of the optical characteristics. Instead of a plane wave, the light is diffracted into a converging wave which also shows an astigmatic focal region between 6 and 8 cm along the $z$-axis.

For a first qualitative validation, the simulation results were compared with a simple measurement of a comparable volume grating that had been curved and embedded in a lens previously. Due to the available measurement setup, the HOE was replayed with a plane wave at $0\deg$ incidence with respect to the optical axis, i. e. in reverse compared to the simulation in Figure 10. The simulation results and the camera measurements of the cross section of the diffracted beam are shown in Figure 11 and show both the same qualitative behavior with an astigmatic focal region in the diffracted beams.

We end this section with abrief outlook on how the developed methods can be applied to the optical design of holographic combiners for RSDs.

Figure 12 shows in the top sequence of graphics the simulation results for an HOE combiner recorded with a divergent and a convergent spherical wave, where the convergent wave converged towards a point source located on the z-axis through the HOE center. The left graphic shows the corresponding grating vector field (rescaled for visualization) in blue. the center graphic shows the directions of propagation of the local wave vectors of the diffracted wave when probed on-Bragg with the divergent recording wave. The right graphic shows the intersection points of the propagation directions of the diffracted wave vectors with different parallel detector planes which are distinguished by colour.

The bottom sequence in Figure 12 shows the analogous simulation results for an HOE curved after recording. While the deformation of the grating vector field may not appear obvious at first glance, the propagation directions after diffraction already show an impact on the focal region which is closer to the HOE vertex than before. Moreover, the visualization suggests again an astigmatic focal region. This impression is also confirmed by the interception points in the graphics on the right-hand side.

From the observations on Figure 12 we conclude, that the geometric framework developed in this paper is expected to prove helpful in the simulation and design of holographic combiners for RSDs which need to be curved after recording. Our method is expected to provide a systematic framework for design and optimization tools, e. g. to precompensate in the design process changes in the diffraction properties of an HOE due to deformations of the HOE geometry after recording.

As an outlook for future applications of this approach, we mention here that based on the above results, the Matlab simulation was extended to implement a parametric optimization for curved HOE created from spherical wave interference which were used successfully in a demonstrator setup of a RSD. Thus, we are confident that the framework presented in this paper offers a promising modelling and simulation approach for the optical design of curved HOE for RSD and will be investigated further.