## Abstract

In this contribution, we investigate the use of holographic optical elements (HOEs) as progressive addition lenses (PALs). We design HOEs with high diffraction efficiency (DE) using the Fourier Modal Method (FMM) and optimize an optical system comprising two of these HOEs to fulfill the optical function of a 2 diopter (dpt) PAL. The resulting design is a holographic PAL (hPAL) exhibiting high DE and limited angular color error (CE) with a distribution of spherical power and astigmatism equivalent to its refractive counterpart. To our knowledge, our contribution is the first complete design of an hPAL. While our HOE design method is shown for PALs here, it has the potential to improve other applications of HOEs as well.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Presbyopia is an age-related eye condition reducing the quality of life of a significant share of the human population. It is a stiffening process of the eye lens, which makes near vision increasingly difficult as the eye lens loses the ability to change its curvature with age [1]. As a result, the eye requires additional spherical power for near vision tasks.

The usual treatment for presbyopia are progressive addition lenses (PALs). PALs solve refractive errors of the eye by providing zones for near and far vision with different lens curvatures, i.e. with different spherical powers. The typical distribution of spherical power over the lens surface of a sample PAL is shown in Fig. 1(a). This PAL is designed for a late-stage presbyopic, but emmetropic eye. This means that this eye has no refractive errors aside from presbyopia. For eyes with additional refractive errors, their individual prescription would need to be included in the PAL design. Here, the entire upper half of the lens at 2 mm < = *y* < = 25 mm is designed for far vision with a spherical power of 0 diopters (dpt). In the lower half of the lens, there is a zone at −10 mm < *x* < 10 mm and −25 mm < *y* < −12 mm with a spherical power of about 2 dpt for near vision. The zones for near and far vision are connected by a continuous increase of spherical power known as the progressive corridor, which is found at −3 mm < *x* < 3 mm and −17 mm < *y* < 2 mm. As stated by Minkwitz’s theorem [2], increasing spherical power along one direction at a certain rate causes unwanted astigmatism to increase at twice the rate in the perpendicular direction. In PALs, this generally leads to large areas of the lens with significantly increased astigmatism on the left and on the right of the progressive corridor [3,4]. Consequently, it is a fundamental requirement for PALs that the astigmatism in the viewing zones and within the progressive corridor is minimal. Figure 1(b) shows the distribution of astigmatism over the lens surface for the PAL whose spherical power is shown in Fig. 1(a). Here, astigmatism is expressed as the difference between the foci of rays along two orthogonal axes in dpt. Clearly, astigmatism is very low for the zones for near and far vision as well as the progressive corridor (blue areas). The respective distributions of spherical power and astigmatism shown in Fig. 1 will serve as a benchmark for the holographic PAL (hPAL) design.

Due to the spatially varying spherical power, refractive PALs inherently require freeform surfaces that can only be realized by high precision manufacturing processes. These are usually based on computer-numerically controlled (CNC) machines with complex soft polishing opposed to the easier hard polishing of spherical lenses [5]. Cheap and scalable processes such as compression molding are known for small elements such as camera lenses or larger lenses like condenser lenses for projectors with lower accuracy requirements [6], but not for high end eyeglasses with individualized prescriptions. As such, it is highly desirable to provide a solution where the PAL functionality can be imprinted on a standard single-vision lens substrate. Holographic optical elements (HOEs) are promising candidates with this regard since their optical function can be realized in a thin film and can be tuned to work without curvature [7–9]. Additionally, state of the art holographic materials and processing techniques enable fast and scalable roll-to-roll mass production [10–13].

The use of HOEs for human visual systems in general and hPALs in particular faces four basic challenges. The first challenge lies in dispersion compensation. HOEs exhibit grating dispersion, which needs to be compensated to minimize angular color error (CE) in the perceived image. Secondly, the HOEs need to operate over a broad bandwidth. The HOEs need to operate with high diffraction efficiency (DE) in a single diffraction order over the entire visible spectrum of light (VIS) [14]; otherwise, the eye will see double images corresponding to the 1st and 0th diffraction orders. The third challenge is introduced by the continuous motion of the eye. Transmission HOEs are characterized by a narrow angular bandwidth and therefore the HOE design needs to account for this constraint so a high DE is achieved for all eye rotation angles. Finally yet importantly, the hPAL design should recreate the spherical power distribution of refractive PALs at minimal astigmatism. Challenges 2) and 3) are very prominent, as HOEs are known for their limited wavelength and angular bandwidth [15–17]. Furthermore, they are mutually dependent as varying the grating parameters controlling DE influences the wavelength and angular bandwidth at the same time.

In this contribution, we present an optimized HOE design method for hPALs recreating the optical function of a PAL. We describe how each of the mentioned challenges is addressed and provide a performance evaluation of the resulting hPAL design in terms of spherical power and astigmatism distribution. Furthermore, we benchmark the hPAL performance against the refractive PAL shown in Fig. 1.

## 2. Methods

In the following, methods to overcome the four basic challenges for the use of HOEs as hPALs are presented.

As a starting point for our considerations, we choose an optical system made up of two HOEs and the eye of the observer as illustrated in Fig. 2 for a 2D case without loss of generality. In our calculations, we consider a material stack of a flat sheet of polymer, the holographic foil for HOE A, the holographic foil of HOE B and another flat sheet of polymer. All HOEs are volume holograms. For simplicity, we only show the HOEs in Fig. 2. Since we are interested in a lens application with high transmission, both HOEs are transmission and phase volume holograms. When using eyeglasses, the eye of the observer rotates with angle θ and creates a continuum of viewing directions. For each viewing direction, we can draw a principal ray connecting the rotational center of the eye M with the center of the pupil. The principal ray can be considered as the optical axis of the current viewing direction. The eye pupil, which acts as the aperture of the optical system, moves as the eye rotates between different viewing directions. This movement of the optical axis and system aperture makes the design of eyeglasses significantly different from that of other optical systems and results in an additional requirement for the HOE. Namely, the HOE design has to be optimized for each viewing direction.

To minimize grating dispersion, we use two HOEs in tandem. One HOE operates in the + 1st diffraction order, while the other HOE operates in the – 1st diffraction order. The Ewald spheres depicted in Fig. 2 indicate how, in this configuration, a principal ray travelling through the system is diffracted by grating vectors of equal magnitude and opposite direction indicated as $\left|\stackrel{\u20d7}{{K}_{B}}\right|$v and $\left|\stackrel{\u20d7}{{K}_{A}}\right|$. In this scenario, the grating dispersion induced by HOE B is canceled by the dispersion induced by HOE A [18]. It should be noted here that while the tandem configuration allows to design dispersion compensated systems, its main advantage is that by altering the direction of the ray between the HOEs, the grating vectors $\left|\stackrel{\u20d7}{K}\right|$ can be adjusted without affecting the directions of the incident and outgoing ray. In this way, the deflection angle is decoupled from the grating vector, which we will use as a free parameter at a later stage for the optimization of the DE. Without the tandem configuration enabling this free parameter, high DE could not be achieved at all. Later we will discuss methods to design HOE tandems as hPALs with the corresponding spherical power distribution. Our configuration for dispersion compensation requires the outgoing ray to have the same direction as the incident ray. Spherical power, however, is induced if the direction of an incident ray is changed. Therefore, it follows that a HOE tandem with a certain spherical power will not be perfectly dispersion compensated. We will explore this trade-off in more depth later.

We estimate the required angular bandwidth for our HOE. Typically, the angular bandwidth would be defined on the input side of a HOE. Here, we estimate the angular bandwidth based on geometry on the output side of the HOE for design convenience. However, due to reciprocity in ray optics, both descriptions are identical. We will ensure that the angular bandwidth at each HOE position is centered on the principal ray of the corresponding viewing direction later in this section. When a parallel bundle of rays enters the eye, this means the DE will drop as the ray position in the bundle moves away from that of the principal ray. This happens because the direction of the outer rays of the bundle differs from those of the principal rays, corresponding to different viewing directions, for which the grating orientation was optimized. This situation is illustrated in Fig. 3, where the red rays belong to a typical ray bundle corresponding to a certain viewing direction. The center red ray and the two blue rays are principal rays corresponding to three different viewing directions. As can be observed the outer bundle rays experience an angular difference φ with respect to the principal rays and therefore causing a drop in DE. To calculate the drop in DE across the ray bundle we need to estimate the angular detuning observed by the bundle rays. As the principal ray would be in the center of the parallel ray bundle, it makes sense to use its position as the center of the angular detuning bandwidth. The required angular bandwidth by +/− φ is then given by the difference between the detuning angles of the outermost rays of the bundle. For a pupil diameter of 2 mm, which is reasonable for day light vision [14], and a distance of 17 mm between the pupil and the HOE surface, we find that an angular bandwidth of about 3.4° (+/− 1.7°) around each principal ray is required. We choose the distance of 17 mm between the pupil and the HOE surface, because this is a typical value for the distance between the pupil and the lens in eyeglass design. It should be noted that the required angular bandwidth is determined by the system geometry using geometrical optics. Therefore, the required angular bandwidth is the same at all wavelengths. The derivation of the required angular bandwidth presented here assumes parallel rays, which correspond to an object at infinity. For a close distance object, a cone of rays would need to be considered. However, the maximal focal length of relevance for a 2 dpt PAL would be 500 mm. For a pupil of 2 mm, the corresponding angular range of the cone would be +/− 0.1°. Hence, we omit this negligible effect for the sake of simplicity.

To maximize the wavelength and angular bandwidth of the HOE, we investigate the influence of the HOE’s grating period, thickness and index modulation on DE as a function of wavelength and incidence angle using the Fourier Modal Method (FMM) [19,20]. A more detailed description of HOEs as gratings characterized by a modulation of the refractive index in a volume material can be found in ref [21]. Our goal is to find a parameter space, in which high DE of above 0.7 is achieved for a wavelength bandwidth covering the entire VIS and the angular bandwidth of +/− 1.7° required by a 2 mm pupil. We choose a DE threshold of 0.7, because it was previously found to be reasonable for practical applications [6]. For this purpose, we investigate the influence of grating period, refractive index modulation and grating thickness on DE. FMM simulation results of DE as a function of angle of incidence (AOI) and wavelength for grating periods between 1.0 and 3.0 µm, a grating thickness of 25 µm and a refractive index modulation of 0.02, which is easily achievable with state-of-the-art materials [22,23], are shown in Fig. 4(a)-4(f). In all cases, the area of high DE is a red, elliptical shape. We notice that the inclination angle of the ellipse relative to the wavelength axis is maximal for a grating period of 1.0 µm and decreases for larger grating periods. In our application, it is highly desirable that the elliptical shape of high DE is parallel to the wavelength axis as this translates to a large wavelength bandwidth. We find that this is the case for grating periods between 2.0 and 2.7 µm (500 and 370 lines/mm, respectively) with a preferred value of 2.4 µm (417 lines/mm). For larger grating periods, the elliptical shape remains almost parallel to the wavelength axis, but peak DE starts to drop. Although high DE over the entire VIS is achieved with this approach, the required angular bandwidth of +/− 1.7° is only obtained in a limited wavelength range between 490 nm and 580 nm as indicated by the black rectangle in Fig. 4(d).

Multiplexing is a known way to increase the bandwidth of HOEs [24], involving multiple exposures of a holographic film. To increase the angular bandwidth, we multiplex an additional HOE whose DE distribution is shifted by 3.5° relative to the first HOE into the holographic film shown in Fig. 4(b). Both HOEs are written into a holographic film that has a maximal refractive index modulation of 0.02. One way to realize this experimentally is to bleach the holographic film so that the available refractive index modulation is limited to the required 0.02. Both HOEs are then written into the holographic film so that each HOE alone has a maximal refractive index modulation of 0.02. At the intersection points of the fringes of the two HOEs, this means that we deviate slightly from the optimal condition since the refractive index modulation there is fixed to 0.02. However, since we limit the multiplexing to two, this approach allows increasing the angular bandwidth while maintaining high peak values of DE. The resulting DE distribution is shown in Fig. 4(d). The black rectangle shows that now the required angular and wavelength bandwidth are achieved simultaneously. The median DE of the multiplexed HOE within the required angular and wavelength bandwidth is 0.88. It should be noted that we discuss angular bandwidth limitations in the recording plane i.e. the plane spanned by incident and outgoing ray. In the orthogonal direction, the angular bandwidth is much higher by design.

To account for the rotating eye, we have to modify the HOE configuration for each viewing direction in order to provide maximum efficiencies for all viewing directions. The grating vector of HOEs is defined as $\stackrel{\u20d7}{K}=\text{}\frac{2\text{\pi}}{\Lambda}\left(\begin{array}{c}\mathrm{cos}\text{\phi}\\ \mathrm{sin}\text{\phi}\end{array}\right)$ with the grating period $\Lambda $, slant angle $\text{\phi}$, the *x* coordinate denoting the direction of propagation and the *y* coordinate being orthogonal to the plane of incidence [15] (assuming invariance in the *z* coordinate). We can use the slant angle $\text{\phi}$ to shift the efficiency for each angular spectrum to match the respective viewing direction. This can be visualized as rotating the HOE the HOE within the 3D volume of a holographic film. This allows transforming a HOE configuration suitable for one viewing direction into a HOE configuration suitable for another viewing direction. In practice, this can be implemented by writing the HOE in sequential unit areas and rotating the holographic film or recording setup. The angle difference between the principal rays of the two viewing directions then is the rotation angle for the slant angle. The HOE's DE properties over the angular or wavelength bandwidth are not affected by this procedure. In the case of eyeglasses, the slant angle at each point on the lens ensures that the angular bandwidth of each HOE is centered on the angle of the principal ray of the corresponding viewing direction.

We want to design HOE tandems with custom spherical power distributions over the lens surface and look for a suitable implementation of HOEs in our optical design software. Due to limitations of our optical design software, we can only implement surface gratings and not volume gratings. This is not a problem, as it is known that the ray direction change induced by diffraction in volume gratings such as HOEs can be described by the grating equation known from surface gratings. In other words, the ability of a HOE to change ray directions i.e. induce spherical power depends solely on the projection of the grating period $\Lambda $ onto the grating surface. It can be found by projecting the grating vector onto the grating surface$\stackrel{\u20d7}{K}=\text{}\frac{2\text{\pi}}{\Lambda}\left(\begin{array}{c}0\\ \mathrm{sin}\text{\phi}\end{array}\right)$ and calculating its projected grating period ${\Lambda}_{y}=\raisebox{1ex}{$\Lambda $}\!\left/ \!\raisebox{-1ex}{$\mathrm{sin}\text{\phi}$}\right.$ . As mentioned earlier, the *x* coordinate is the propagation direction and the *y* coordinate is orthogonal to the plane of incidence. Instead of controlling the grating period $\Lambda $ and slant angle $\text{\phi}$ of a volume grating such as a HOE, it is sufficient to control its projected grating period ${\Lambda}_{y}$. This enables us to use the code for surface gratings to implement HOEs in our optical design software. We parametrize surface grating periods of the HOEs using a global polynomial, which describes the variations of the local surface grating period ${\Lambda}_{y}$ over the lens. Varying the coefficients of the polynomial then translates to changing the magnitude of the surface grating period ${\Lambda}_{y}$ and therefore controlling the spherical power induced by the HOE. Of course, we need to reconnect the HOEs implemented as surface gratings with their volume grating description to ensure that the grating period in the volume $\Lambda $ stays within the proposed interval for large angular and wavelength bandwidth and to ensure that the available angular bandwidth is centered on the principal ray in all cases. This can be done by considering the relationship between grating vector, incident and outgoing ray. As can be seen in the Ewald spheres shown in Fig. 2, the orientation of the grating vector is perpendicular to the bisecting line of the angle between incident and outgoing ray. To calculate the slant angles for the surface grating corresponding to HOE A, we trace a principal ray of a green wavelength e.g. 532 nm starting from the eye of the observer. Using the surface grating period and the grating equation, we can calculate the outgoing ray. The slant angle for HOE A is then calculated using the principal ray as the incident ray and the outgoing ray found via the grating equation. Then, we take the outgoing ray, propagate it to the surface grating corresponding to HOE B and calculate the diffracted ray via the grating equation. Again, we have the directions of an incident and an outgoing ray and calculate the slant angle for HOE B. Once the slant angles are known, we can use ${\Lambda}_{y}=\raisebox{1ex}{$\Lambda $}\!\left/ \!\raisebox{-1ex}{$\mathrm{sin}\text{\phi}$}\right.$ to calculate the grating periods in the volume from the projected grating periods. This method ensures that the Bragg condition is met for the principal rays at all HOE positions and therefore the available angular bandwidth is centered on them. This method allows calculating the grating period in the volume, but does not enforce the grating periods to be within a favorable interval for high DE. This is done within the optimization algorithm.

We implement our optimization method using an optical design software. The two main functions of optical design software are optical system simulation via ray tracing and optimization [25,26]. For the optical system simulation, it is necessary to implement all system components, light sources and parameters necessary for the evaluation of the systems performance. In our case, we implement the optical system, following the sketch in Fig. 2, as lens consisting of two HOEs embedded into a 1.4 mm thick polymer substrate with refractive index 1.59, which is a standard eyeglass material. The HOEs are implemented as surface gratings as described in detail in the previous paragraph. Since we want to evaluate the systems performance for all possible viewing directions, we create a discrete sampling of light sources in the object space, each light source corresponding to a principal ray. We adjust the distance between the light sources and the lens so that light sources in the upper half of the lens, where a spherical power of 0 dpt is desired, are placed at infinity. Light sources corresponding to lens positions where nonzero spherical power is desired are placed at the correct focal lengths, which are the inverse of the spherical power values in dpt. The evaluation of our system performance requires calculating spherical power, astigmatism and grating periods. For each viewing direction, we trace a bundle of rays from the corresponding light source. The ray bundle is limited by the pupil of the eye. We propagate the ray bundle from the source into the eye and evaluate the spherical power and astigmatism by Zernike decomposition of the corresponding field. We also evaluate grating periods over the lens surface to ensure that our condition for high DE is met. Since we fix the refractive index modulation and HOE thickness, controlling the grating period will ensure high DE. In the optimization step, we define a merit function, which comprises the differences between the spherical power and the astigmatism of the HOE tandem and the benchmark PAL. Additionally, we put a penalty term on deviations of the grating period from our proposed interval. The optimization algorithm then varies the coefficients of the HOE parametrization and evaluates the merit function by tracing rays through the current version of the system after each step. A least-squares solver is used to iteratively approach a minimal value of the Merit function corresponding to an optimal system design.

## 3. Results and discussion

In this section, we present and discuss the hPAL design results that were obtained using the outlined optimization method while satisfying the four challenges mentioned in the last section. Following the order in which we introduced the design challenges for HOEs as hPALs, we start by evaluating CE as a means of assessing the effect of grating dispersion. We then investigate the distribution of grating periods over the HOE surfaces to ensure that our conditions for high DE are met versus wavelength and angle. Finally, we benchmark the distributions of spherical power and astigmatism of the hPAL against its refractive counterpart.

We evaluate the CE of the hPAL to assess how well the approach followed for grating dispersion compensation works. Due to the visual perception and different tolerances between individuals, it is hard to find an objective function to scale how much CE interferes with human vision when using eyeglasses. For refractive eyeglasses, approximate formulas based on the Abbe number are known. The perception threshold for color fringes calculated with this method is reported as 0.12 cm/m in literature [27].

To benchmark our refractive device, we simulated a refractive unifocal −4 dpt lens made from a standard eyeglass material (polymer with refractive index 1.595 and Abbe number 40) with tolerable CE. This lens would exhibit color fringes of up to 0.3 cm/m and is comparable to commercially available devices. If the CE of our hPAL is in the same order of magnitude as the benchmark device, we can assume that CE is not an issue for the use of the hPAL. We calculate the distance in the pupil between red and blue rays at 620 nm and 430 nm wavelength, respectively. Since CE changes sign depending on the spherical power of the lens, only absolute values are of interest here. We scale the results so that a value of 5 arbitrary units (a.u.) corresponds to the perception threshold and a value of 16 a.u. corresponds to the maximal CE induced by the refractive unifocal lens. We then calculate the CE of our hPAL using the same scaling. In Fig. 5(a), the CE is plotted over the entire lens surface. The magnitude of CE observed ranges between −10 and 25. However, the regions with high magnitude CE coincide with regions of high astigmatism at −20 mm < *x* < −10 mm as well as 10 mm < *x* < 20 mm and −20 mm < *y* < −10 mm. Regions with high astigmatism strongly distort the users vision when looking through them. Therefore, high CE does not further impair user experience in the regions with high astigmatism. As such, they are of minor importance to the use of the hPAL. Outside of these regions and especially in the progressive corridor at −15 mm < *y* < 0 mm shown in Fig. 5(b), the CE is mostly below absolute values of 10 a.u. This means that we have CE above the perception threshold, but not above values found in commercially available eyeglasses. Therefore, we can conclude, that grating dispersion is not a problem for the operation of our hPAL.

A closer investigation of CE reveals that it is linked to the spherical power of the hPAL. In Fig. 2, we illustrated how to configure the HOEs’ grating vectors to achieve dispersion compensation for the principal rays. This setup has negligible CE thanks to dispersion compensation, but it also has a spherical power distribution of about 0 dpt. In practice, changing the grating configuration to increase spherical power coincides with increased CE since the condition for dispersion compensation is no longer met. This means that a maximum tolerable CE translates to a maximum achievable spherical power for a given geometry. For the geometry investigated here, we find that spherical powers above 2 dpt cannot be realized at acceptable CE within our current approach.

To assess the DE of the hPAL, we plot the grating periods of the two HOEs in the tandem in Fig. 6. The variation of the grating periods over the whole hPAL is less than +/− 0.1 µm from the optimum value of 2.4 µm in both cases. This ensures that the DE of the hPAL is above 0.7 over the whole PAL surface for the entire VIS and the required angular bandwidth of 3.4°. The angular bandwidth is automatically centered on the principal rays thanks to our grating period parametrization method detailed in the last section. Since the median DE is at 0.88 for a single HOE, we can approximate the DE of the tandem structure to be 0.77 by multiplying DE of the both HOEs.

To evaluate whether our hPAL design qualitatively replicates the refractive PAL, we compare their spherical power and astigmatism. The distributions of spherical power and astigmatism are shown in Fig. 7(a) and 7(b), respectively, while those corresponding to the refractive PAL are shown in Fig. 1. Clearly, our hPAL design qualitatively replicates the distributions of spherical power and astigmatism of the refractive PAL. Particularly, the spherical power increases from 0 dpt to 2 dpt as shown in Fig. 7(a) with small astigmatism within the progressive corridor as seen in Fig. 7(b).

A quantitative comparison between our hPAL introduced in Fig. 7 and the refractive PAL introduced in Fig. 1 is shown in Fig. 8. For more convenient visualization of the spherical power and astigmatism distributions in the progressive corridor and the zone for near vision, Fig. 8(a) and 8(b) depict cuts along the *x* = 0 mm and *y* = −23 mm lines along the lens, respectively. Figure 8(a) allows for evaluation of the spherical power and astigmatism in the progressive corridor, which is found in the region between *y* = - 15 mm and *y* = 0 mm at *x* = 0. Here, the gradient in spherical power of 0.10 dpt/mm for the hPAL is very close to the benchmark of about 0.13 dpt/mm for the refractive PAL. At the same time, the maximum astigmatism within the progressive corridor is slightly larger with a value of 0.16 dpt for the hPAL as compared to 0.04 dpt in the refractive PAL. In terms of astigmatism in the zone for near vision shown in Fig. 8(b) the hPAL performs better than its classical counterpart does. In the center of the zone between *x* = −3 to *x* = 3 mm astigmatism is considerably lower, with the minimum value being 0.16 dpt at *x* = 0 mm opposed to 0.31 dpt in the refractive PAL. Outside of the center of the zone, the gradient of astigmatism is much lower for the hPAL than for the refractive PAL, which is a further indication that the provided near vision zone of the hPAL is preferable to the one in the refractive device.

These results show that due to the increased set of free parameters in the design optimization, a holographic PAL can outperform a refractive PAL for example by reducing astigmatism. We expect that by utilizing cubic splines instead of a global polynomial to describe the grating period distribution over the surface one could achieve better results, especially astigmatism in the zone for near vision. This is particularly important since the near vision zone is designed for key activities such as reading or computer work, so any reduction of astigmatism strongly benefits the user’s quality of life.

## 4. Conclusion

We have designed a HOE-based PAL, which is, to the best of our knowledge, the first complete hPAL design reported. We evaluated the CE of the hPAL and found it to be below a benchmark taken from refractive eyeglasses. We ensured that the angular and wavelength bandwidth of the hPAL remain within the specified requirements. Furthermore, we were able to replicate the spherical power and astigmatism distribution of refractive PAL with equal or better performance. Our work shows that, in principle, HOEs can be a viable alternative to refractive eyeglasses in some specialized applications such as PALs. The HOE design rules we identify are shown for PALs, but have the potential to improve other applications as well. However, to date the feasibility of experimental realization needs to be investigated in future work.

## Funding

EU’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 675745.

## Acknowledgments

We thank Philipp Jester, Helmut Wietschorke, Gerhard Kelch and Katharina Rifai from Carl Zeiss Vision GmbH for helpful discussions and providing data on refractive progressive addition lenses. We thank Christoph Menke from Carl Zeiss AG for helpful discussions on optical design and optimization.

## References

**1. **A. Glasser and M. C. W. Campbell, “Biometric, optical and physical changes in the isolated human crystalline lens with age in relation to presbyopia,” Vision Res. **39**(11), 1991–2015 (1999). [CrossRef] [PubMed]

**2. **G. Minkwitz, “On the surface astigmatism of a fixed symmetrical aspheric surface,” Opt. Acta (Lond.) **10**(3), 223–227 (1963). [CrossRef] [PubMed]

**3. **D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 1: Design and development of progressive lenses,” Clin. Exp. Optom. **91**(3), 240–250 (2008). [CrossRef] [PubMed]

**4. **J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes, “Progressive powered lenses: the Minkwitz theorem,” Optom. Vis. Sci. **82**(10), 916–922 (2005). [CrossRef] [PubMed]

**5. **D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin. Exp. Optom. **91**(3), 251–264 (2008). [CrossRef] [PubMed]

**6. **F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. Manuf. Technol. **62**(2), 823–846 (2013). [CrossRef]

**7. **W. Osten, A. Faridian, P. Gao, K. Körner, D. Naik, G. Pedrini, A. K. Singh, M. Takeda, and M. Wilke, “Recent advances in digital holography [invited],” Appl. Opt. **53**(27), G44–G63 (2014). [CrossRef] [PubMed]

**8. **M. Rostykus, M. Rossi, and C. Moser, “Compact lensless subpixel resolution large field of view microscope,” Opt. Lett. **43**(8), 1654–1657 (2018). [CrossRef] [PubMed]

**9. **G. Li, D. Lee, Y. Jeong, J. Cho, and B. Lee, “Holographic display for see-through augmented reality using mirror-lens holographic optical element,” Opt. Lett. **41**(11), 2486–2489 (2016). [CrossRef] [PubMed]

**10. **D. Jurbergs, F.-K. Bruder, F. Deuber, T. Fäcke, R. Hagen, D. Hönel, T. Rölle, M.-S. Weiser, and A. Volkov, “New recording materials for the holographic industry,” in *Proc. SPIE 7233, Practical Holography XXIII: Materials and Applications*(2009), pp. 72330K–72330K–72310.

**11. **D. Vather, I. Naydenova, D. Cody, M. Zawadzka, S. Martin, E. Mihaylova, S. Curran, P. Duffy, J. Portillo, D. Connell, S. McDonnell, and V. Toal, “Serialized holography for brand protection and authentication,” Appl. Opt. **57**(22), E131–E137 (2018). [CrossRef] [PubMed]

**12. **J. M. Russo, F. Dimov, J. Padiyar, and S. Coe-Sullivan, “Mass production of holographic transparent components for augmented and virtual reality applications,” in *SPIE Digital Optical Technologies*(SPIE2017), p. 9.

**13. **F.-K. Bruder, T. Fäcke, F. Grote, R. Hagen, D. Hönel, E. Koch, C. Rewitz, G. Walze, and B. Wewer, “Mass production of volume holographic optical elements (vHOEs) using Bayfol(R) HX photopolymer film in a roll-to-roll copy process,” in *SPIE OPTO*(SPIE2017), p. 20.

**14. **H. Gross, F. Blechinger, and B. Achtner, *Handbook of Optical Systems, Survey of Optical Instruments* (Wiley, 2008).

**15. **B.-R. David, “Understanding diffraction in volume gratings and holograms,” in *Holography - Basic Principles and Contemporary Applications*, E. Mihaylova, ed. (InTech, 2013), p. Ch. 01.

**16. **J. Marín-Sáez, J. Atencia, D. Chemisana, and M.-V. Collados, “Characterization of volume holographic optical elements recorded in Bayfol HX photopolymer for solar photovoltaic applications,” Opt. Express **24**(6), A720–A730 (2016). [CrossRef] [PubMed]

**17. **J. Han, J. Liu, X. Yao, and Y. Wang, “Portable waveguide display system with a large field of view by integrating freeform elements and volume holograms,” Opt. Express **23**(3), 3534–3549 (2015). [CrossRef] [PubMed]

**18. **C. A. Palmer, *Diffraction Grating Handbook* (Richardson Grating Laboratory, 2000).

**19. **L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**(10), 2758–2767 (1997). [CrossRef]

**20. **J. Liu, R. T. Chen, B. M. Davies, and L. Li, “Modeling and design of planar slanted volume holographic gratings for wavelength-division-multiplexing applications,” Appl. Opt. **38**(34), 6981–6986 (1999). [CrossRef] [PubMed]

**21. **H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. **48**(9), 2909–2947 (1969). [CrossRef]

**22. **A. Zanutta, E. Orselli, T. Fäcke, and A. Bianco, “Photopolymer materials for volume phase holographic optical elements,” in *International Conference on Space Optics — ICSO**2016*(SPIE2017), p. 9.

**23. **B. J. Chang, “Dichromated gelatin holograms and their applications,” Opt. Eng. **29**(5), 195642 (1980).

**24. **S. Altmeyer, Y. Hu, P. Thiée, J. Matrisch, M. Wallentin, and J. Silbermann, “Multiplexing of transmission holograms in photopolymer,” in *DGaO Proceedings*(2013).

**25. **R. F. Fischer and B. Tadic, *Optical System Design* (Mcgraw-hill, 2000).

**26. **W. J. Smith, *Modern optical engineering* (McGraw-Hill, 1966).

**27. **H. Goersch, *Wörterbuch der Optometrie* (DOZ-Verlag, 2004).