Picture-generating freeform surfaces are able to generate a picture in a defined plane by incoherent beam shaping comparable to illumination purposes. No classical imaging is performed. Therefore the classical Rayleigh criterion of the diffraction limit cannot be applied. In this paper, we investigate the physical light formation of picture-generating freeform surfaces using Fresnel-Huygens-based simulations. A criterion for the diffraction limit was found. The resolution of such surfaces is significantly inferior to the resolution of classical imaging systems. However, in many cases, such systems are limited by the geometrical resolution. The influence of those two limitations were examined and a maximum of resolution, being limited by diffraction and by geometrical parameters can be found.
© 2012 Optical Society of America
CorrectionsS. Zwick, R. Feßler, J. Jegorov, and G. Notni, "Erratum: Resolution limitations for tailored picture-generating freeform surfaces," Opt. Express 20, 26743-26743 (2012)
Freeform optics are getting more and more important in optical systems, as they offer new degrees of freedom in the design of optical systems. The application of freeforms enables e.g. to improve the energy efficiency of illumination systems (see e.g. ), the image quality in imaging systems [2, 3] or to miniaturize an optical system. Especially in illumination applications, freeform surfaces are already state of the art [4–7]. Classical incoherent beam shaping systems generate usually intensity distributions consisting of low spatial frequencies.
Recent design algorithm developments allow to design (or tailor) freeform surfaces that generate intensity distributions including middle to high spatial frequencies [8–12]. This allows to generate pictures in the so-called design or picture plane of the freeform surface (see Fig. 1). The picture in such a system is generated by a single, tailored (especially designed) freeform lens or mirror. In contrast to conventional imaging, the light modulation is done by a ray-optical redistribution of light comparable to classical beam shaping.
Until now, these novel picture-generating freeform surfaces are mainly used to demonstrate the potential of the developed algorithms. This is due to the main drawback of a picture-generating freeform projector in comparison to a conventional imaging projector: The freeform is designed and produced for one certain picture only, i.e. the projected picture is fixed, no dynamical adaption (as e.g. with digital projectors) is possible. However, for some applications such projectors offer various advantageous. Using freeform mirrors, those systems are wavelength independent. This enables polychromatic applications (without any chromatic aberrations) or applications in UV- or IR-ranges, where no commercial projectors are available (Projectors suffer from technical problems when using light sources in UV- or IR-range. This is mainly due to the limited choice of appropriate materials for optical components and light modulators as well as the limited variety of light sources.). Additionally, they are highly energy efficient, as nearly 100% of the incident light is directed to the picture plane. In comparison, conventional projectors using e.g. transmittive LCDs and an absorption-based modulation direct only around 5–10% of the collected light to the image plane . Furthermore, as no classical imaging is performed, exceptionally high depth of focus can be generated .
One central evaluation criterion for image projectors is the resolution of the projected image. The resolution of an image displayed by a classical projector system is mainly limited by two parameters: First, by the resolution of the input image, i.e. the light modulator or slide, that is imaged. Second, it is limited by the optical transfer function of the optical system, which includes in diffraction limited systems mainly diffraction effects.
The resolution of pictures displayed by a picture-generating freeform surface can be classified accordingly. However, for picture-generating freeform projectors, no classical imaging is performed. Therefore neither the classical Rayleigh criterion of the diffraction limit can be applied nor can the resolution of the input picture be investigated straight forward. In this contribution, we investigate and compare the diffraction limit and the geometrical resolution limitations of picture-generating freeform surfaces designed with the algorithm described in section 2. As a first step, this invesitgation focuses on the resolution limit of one-dimensional sinusodial line patterns.
2. Designing picture-generating freeform surfaces
Assume that light from some light source illuminates a screen S after passing through a lens L where it is refracted. Assume that the light is unidirectional, i.e. at every point on the entrance surface of the lens we have light rays in a single direction only and this direction is continuously differentiable distributed. For example, this condition is fulfilled for light from point light sources or for parallel light. Then, we observe that the light rays passing through the lens define a unique map15].
We introduce the following definitions: A brightness distribution is called admissible if it is nowhere zero or infinite and if it is continuous. A light source is called admissible if its light is unidirectional (explained above) and if its brightness distribution on the entrance surface of the lens is admissible. A lens surface is called C2-continuous if it has continuous curvatures. A geometrical situation is called prescribed if one of the two surfaces of the lens and its spatial position is C2-continuously prescribed and if in addition one point of the other lens surface and the screen plane are given. A focal point is a point where different light rays cross each other. In such a point the light is not unidirectional. Then, the following assertion can be demonstrated mathematically: For any given admissible light source and prescribed geometrical situation there exists at most one (free form) C2-continuous lens which realizes a prescribed admissible brightness distribution on the screen without any focal points between lens and screen. The associated map Ψ is one to one, continuously differentiable and has an everywhere positive Jacobian. The lens really exists if all prescribed data are such that it is a priori assured that there will be no total reflection within the lens. An analogous statement holds for freeform mirrors.
Based on this solution, the Fraunhofer ITWM  has developed a method for computing freeform optics (lenses and mirrors). The investigations in this contribution are based on freeform optics being computed with this method throughout.
The assertion made above can be slightly extended to situations with non-continuous brightness distributions. For instance, if we have sharp light/dark boundaries along some line, then the corresponding lens has a jump of its curvature across some line. However, it is not yet known to the authors what happens if the brightness distributions have regions with zero brightness. Then, in general the map Ψ will be no longer continuous and the corresponding lens, if it exists at all, will have discontinuities in its gradients. However, in many practical cases one can find workarounds in these situations.
3. Diffraction-limited resolution
The freeform surfaces are designed in the geometric-optical approximation. According to geometric optics, the design of a freeform-based projector can be scaled down without changing (apart from the size) the resulting picture. However, at some point, diffraction gets relevant and limits the point resolution Δx.
In classical incoherent imaging systems, there is a fundamental physical limit of resolution due to diffraction. The resolution limit is described by the so-called Rayleigh criterion for points Fig. 2(a)) and the refraction index n of the medium.
In classical incoherent imaging, each point of the image is formed by a converging beam, being brought to focus in the image plane (see Fig. 2(a)). Due to the diffraction at the system’s pupil, the image of a point is not represented by a point but by a diffraction pattern (e.g. the Airy pattern in case of a homogeneously illuminated circular pupil). The size of the Airy-pattern is given by the NA and λ of the converging beam (see Eq. (1)). The whole image is composed as an incoherent superposition of Airy-patterns. The Rayleigh criterion expresses the minimum resolvable distance of two points.
In contrast to this, the picture-generating freeform projectors treated here are based on incoherent beam shaping, and no classical imaging is performed. In this case, the light modulation is achieved on a redistribution of light in such way, that bright respectively dark regions are generated in the picture plane. The distance d between freeform and picture plane is fixed. Each point on the freeform surface corresponds one to one to a point on the picture plane (see Fig. 2(b)). I.e. the picture in the picture plane is not generated by a superposition of Airy-patterns. Therefore Eq. (1) and Eq. (2) cannot be used to determine the resolution of such picture-generating freeform surfaces.
In order to evaluate the diffraction limit of such picture-generating freeform surfaces, the pictures of different freeform surfaces were simulated using a wave-optical propagation method and analyzed. For this investigation, we concentrated on a one-dimensional approach in order to limit the computational effort (see section 3.1). Therefore we used freeform mirrors generating line patterns of different period p in the picture plane. Commonly, there are binary line patterns used to evaluate the resolution of an optical system. However the design of freeform surfaces generating binary intensity distributions is very demanding due to the jumps in the intensity distribution (see section 2). During the time of this investigation, the algorithm for the design of intensity jumps was still under development/improvement. Instead we used sinusoidal line patterns, as they are much easier to design and as well suited to evaluate the resolution.
3.1. Simulation method
Depending on the geometrical parameters of the optical setup, the surface of the freeform can be modulated by up to several millimeters (see e.g. ). Therefore, the thin element approximation  is not valid anymore and wave propagation methods based on this approximation  cannot be used. Additionally, the large height differences of the freeform surface require a very high sampling, which (depending on the propagation algorithm) may lead to problems.
Therefore, the wave-optical computations performed here are based on the Huygens-Fresnel principle, which is valid for (b,d) ≫ λ (see Fig. 3). As it can be derived from Fresnel-Kirchhoff diffraction, the diffracting element does not have to be planar. In case of a coherent point light source, the field in the plane of interest can be expressed mathematically as follows :
Equation (3) can be interpreted as follows: Each point of the wave front can be considered as a source of a new spherical wave. The field in point Pl is formed by a superposition of diverging spherical waves originating from secondary sources located at each point of the diffracting element. The complex amplitudes of these secondary point sources are denoted by U(Fm).
Applied to our problem, each point of the freeform surface is the source of a secondary spherical wave. After propagation they superimpose coherently with each other in the picture plane (see Fig. 3).
In our investigation, the freeform surface is illuminated by a homogeneous collimated light source (which corresponds to a point light source placed at infinity). Therefore the start amplitude at the freeform is assumed to be constant and equal to one. The start phase is given by the propagation of the wave front over the distance rQF = |r⃗QF| between light source and point Fm at the freeform. Therefore U(Fm) is given by:
Constant coefficients (1/iλ) and amplitude changes due to the incident angle θ have not been taken into account in our simulation, as the angles are relatively small. With these approximations, we obtain the intensity distribution
On one hand, using this approach, one can avoid sampling respectively memory problems as discussed in  for FFT-based computational methods. On the other hand, computations in two dimensions are very time-consuming. This is why the following simulations have been performed only in one dimension.
3.2. Results and discussion
In order to evaluate the diffraction limit, various freeform mirrors generating sinusoidal line patterns with different picture distances d and a varying number of displayed periods N have been used. In order to identify the diffraction limit, the period p of the sinusoidal line pattern in the picture plane was scaled down. This was done by scaling the freeform surface in such a way, that d was constant, but the diameters of the freefrom Dff and the picture plane Dpicture were scaled down correspondingly (Mff = Dpicture/Dff = const.). By analyzing if p is resolved, the diffraction limit could be determined.
In contrast to ray-optics, the fringe pattern is superimposed by diffraction oscillations in the wave-optical approximation, when scaling down p. Therefore for a first analysis the generated pictures have been evaluated using Fourier analysis. For this first analysis, the diffraction limit was defined by a threshold for the ratio of the intensity of the main frequency Imain to the whole spectral intensity Imain/Ispectrum = 4%. Even though this is an arbitrary classification, it coincides well with a subjective evaluation of the diffraction limit. Therefore this method is well suited for a first estimation of diffraction limit and its dependencies.
Figure 4 depicts the result of this analysis. It can clearly be seen, that the resolution depends linearly on 1/NA. In this arrangement we define NA ≈ Dff/(2d). As expected, the resolution also depends on the wavelength (compare results for freeform with the number of periods N = 5.5, simulated for λ = 400 nm and λ = 600 nm). For comparison, the resolution according to the Rayleigh criterion for lines is plotted (see Eq. (2)). The resolution of lines generated with these freeform surfaces is significantly inferior to the resolution according to Rayleigh.
The simulation indicates a dependency on the number of periods N, as the resolution for line patterns with different N differs significantly (see Fig. 4(a)). A large number of displayed periods N leads to a low resolution and vice versa. Indeed, when assuming a dependency of a so-called sub-NA NAsub = NA/N, the resolution of the different tested freeforms with λ = 400 nm coincide (see Fig. 4(b)). Hence, the resolution of such freeform surfaces is a function of NAsub.
Figure 4(b) shows for comparison the Rayleigh criterion for lines as a function of NAsub. Even though the resolution of the freeform surfaces approaches in this case, this criterion is not suitable for estimating the diffraction limit of such picture-generating freeform surfaces.
In order to investigate the dependency on NAsub in more detail, further simulations have been performed. Figure 5 shows the propagation of the light field generated by a freeform, which has been designed to generate a sinusoidal line pattern with N = 4 and d =50 mm. At a distance of z≈100 mm, a periodic pattern of focuses can be found. This indicates, that the line pattern is generated by spherical submirrors with a focal length of around fsub=100 mm, leading to a nearly diffraction limited spot in the focal plane (see Fig. 5(b)). The pattern generated in the design plane (z = d =50 mm) is therefore a superposition of strongly defocused spots.
Figure 6(a) shows a cross section of the freeform surface generating the intensity distribution in Fig. 5(a). It can be clearly seen, that the surface consists of four subapertures (SA1–SA4), corresponding to the number of line periods. The diameter of the subaperture Dsub depends on the intensity to be generated in the picture plane and the incident light (in case of an intensity variation of the light source over the freeform mirror). In our case, the freeform is illuminated with homogeneous collimated light parallel to the optical axis. Furthermore, the amplitude of all periods are equal. Therefore the diameter of the subapertures Dsub are equal to each other.
We performed the light propagation for each subaperture SA separately. Figure 6(b) shows the results. The dashed line depicts the intensity in the design plane, when including the whole freeform surface (SA1–SA4). The red lines show the resulting intensity, when illuminating the subapertures separately (for SA2 and SA3 exemplarily). The black line shows the coherent superposition of the separately computed subapertures (SA2+SA3). It can be seen, that the intensity pattern generated by the whole freeform is formed by a coherent superposition of the light fields generated by the single subapertures.
As the picture formation is a superposition of the light distributions generated by the sub-apertures, the diffraction is limited by the numerical aperture of the subapertures Δx(NAsub). In case of equal Dsub, NAsub is given by20, 21] or Shack-Hartmann sensors , even though the freeform mirrors are commonly treated as macro-optical elements.
As each period is generated by one subaperture, the period size is limited by the diameter Dsub of this subaperture. When scaling down Dsub, the influence of the diffraction increases and therefore the size of the resulting diffraction pattern of each subaperture increases.
The Rayleigh line criterion is not applicable in this case, as it is defined for incoherent superposition. However the Sparrow criterion is suitable and leads in the coherent case to the resolvable distance 
This means, the diffraction limit is given, when the period of the line pattern p (corresponding the distance of two points) is equal to the Sparrow distance p = ΔxS. This corresponds very well with first the subjective finding of diffraction limit and second, with the criterion of our first investigation (see Fig. 4).
Figure 7 (dashed line) shows the intensity distribution of the whole freeform (SA1–SA4) close to the diffraction limit according to Sparrow with p > xS (p = 163.6 μm and ΔxS = 162 μm). The sinusoidal line pattern is strongly disturbed due to diffraction. Due to the coherent superposition of the subapertures, beside the four main maxima, there are additional side lobes, which disturb the picture. This corresponds to the superposition of two defocused spots, shown in .
4. Geometric resolution limit
In this section, we address the resolution of the input picture for picture-generating freeform surfaces designed with the algorithm developed by Fraunhofer ITWM (see section 2). In contrast to conventional imaging systems, the resolution of this surfaces is not given by the number of pixels of the bitmap used for the freeform design but by the number of subapertures. The intensity modulation of this picture results in a certain number of subapertures, which corresponds to the number of local maxima, i.e. the number of picture details. That is to say, each picture detail is assigned to one subaperture.
Therefore, the picture detail size Δxgeom in the picture plane results from the geometrical dimensions of the freeform surface respectively its subapertures Dsub. For a surface with a picture-to-freeform-ratio Mff, Δxgeom is given by a simple relation:
In order to improve the geometric resolution in the picture plane, the freeform mirror and therefore Dsub can be downscaled. This is a tolerable approach in the geometric-optical approximation. However, when leaving the geometric optical approximation, diffraction becomes important. So for low NAsub, There are two opposing tendencies: On one hand, when down-sizing Dsub, the resolution Δxgeom is improved. On the other hand, downsizing Dsub leads to a stronger influence of diffraction and therefore to an enlargement of the resulting diffraction pattern.
Figure 8 shows these two opposing tendencies. The hatched parts below ΔxS and Δxgeom depict the areas with limitations due to either diffraction (red hatched area) or geometric reasons (black hatched area). For each geometrical arrangement, there can be found a minimal Δx for a certain NAsub, limited by diffraction as well as geometrical reasons.
E.g. using a setup with a working distance of z = 100 mm and Mff = 1, the minimal possible feature size is Δx = Dsub = 0.25 mm. A higher resolution cannot be achieved using this geometrical arrangement. However, when using a Mff < 1, this geometrical resolution can be improved (compare Fig. 8, black dashed and black solid line). Using the same setup with a Mff = 0.5, a resolution of Δx = 0.17 mm is possible. This corresponds to Dsub = 0.34 mm.
In this contribution, the diffraction limit and the geometric resolution limit of picture-generating freeform surfaces designed with the algorithm developed by Fraunhofer ITWM was investigated and compared.
These picture-generating surfaces are able to generate a picture in a defined plane by incoherent beam shaping. As no classical imaging is performed, the Rayleigh criterion for diffraction limited resolution cannot be applied. In order to analyze the diffraction limit, freeform surfaces generating sinusoidal line patterns with varying period length were investigated using a propagation algorithm and analyzing the resulting pictures. It could be found, that the picture is formed as a coherent superposition of pictures generated by subapertures. Therefore the resolution of the whole picture depends on the numerical aperture NAsub of the subapertures. It could be shown, that the Sparrow criterion for lines in the coherent case is a suitable criterion to determine the diffraction limit of such picture-generating freeform surfaces.
Additionally, we investigated the geometrical resolution limitations, which are given by the number N of subapertures on the freeform. For low NAsub, there are two opposing tendencies: On one hand, when downsizing Dsub, the resolution Δxgeom is improved. On the other hand, it leads to a stronger influence of diffraction and therefore to an enlargement of the resulting diffraction pattern. From this results a minimal limit for Δx for each geometrical freeform arrangement for the geometrical and diffraction limited resolution.
In case of a line pattern with equal amplitudes, the determination of Dsub respectively NAsub is straight forward, as Dsub = Dff/N. In case of more sophisticated pictures with heterogeneous intensity modulation, an analysis of the geometrical resolution and wave-optical limits might be more difficult. However, this investigation gives a good estimations of geometrical limitations of picture-generating freeform surfaces.
In conclusion, picture-generating freeform surfaces designed with the algorithm from Fraunhofer ITWM and picture-generating freeform surfaces following the same design conditions are not capable to realize as high resolutions as conventional imaging systems. However, for low-and middle-resolution applications, they offer various advantageous, especially the high energy efficiency, the exceptionally large depth of focus and (for freeform mirrors) the wavelength independence.
A more general investigation of the resolution and of such freeform surfaces including also two-dimensional and arbitrary intensity distributions have to be performed further in detail.
The authors would like to thank Dr. Herbert Gross (Carl Zeiss AG), Dr. Tobias Haist (ITO, University of Stuttgart) and Dr. Dirk Michaelis (Fraunhofer IOF) for very fruitful discussions. This work was supported by the FhG Internal Programs under Grant No. WISA 821 004.
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