1. Introduction

Field dependent aberrations are a major concern in imaging and projection applications. Complete correction using a single element, e.g. a correction hologram or an asphere, is in most cases impossible for conventional imaging. Multiple corrections have to be applied as it is e.g. the case in multiconjugate adaptive optics [1].

For the projection of information, this limitation can be circumvented if holographic projection is employed. In this case the hologram leads to a reconstruction of individual spots having different aberration cancellation terms.

If we want to realize N diffraction limited projected points P_j(x, y, z) on a given threedimensional projection surface, the hologram h in conventional holography is just the superposition of the reference (illumination) light r(x, y, z) with the light coming from these points:

S[P_j] is a propagator that gives the optical path length from the point P_j and through the aberrated optical system to the hologram plane located at z_h. The wavenumber is denoted by k = 2π/λ.

Unfortunately, such an amplitude hologram is rather inefficient and leads to additional unwanted reconstructions. Therefore, for holographic projection, typically pure phase holograms are employed and the phase hologram ideally just reconstructs the complex field due to the points P_j. The most simple approach to obtain such a phase only hologram is to just use the phase of the desired light field in the hologram plane [2]:

This is basically also the approach that Ren et al. use for aberration correction where the summation is implemented using a three-dimensional Fourier transform of the reconstruction volume [3].

Such non-iterative straight forward approaches work quite well if only a few points are to be projected. The more points are to be controlled in the reconstruction, the more severe are the problems with this simple approximation, especially concerning the appearance of additional unwanted ghost points and wrong intensities of the reconstructed points.

A lot of hologram optimization techniques, therefore, have been developed in the past to achieve high quality and efficient reconstructions using phase-only holograms in typical reconstruction geometries. Most often Fourier holograms are preferred because of the reduced space-bandwidth product for the hologram medium compared to Fresnel geometries. This is especially important for applications using spatial light modulators. In most cases, aberrations have been neglected.

Aberration correction for such holograms is trivial if only global (field independent) aberrations are to be corrected [4, 5]. If we assume a Fourier hologram which will be reconstructed by an aberrated optical system with aberration exp(iϕ(x_p, y_p)) (defined in the exit pupil of the optical system) then the aberration can be easily corrected by using the simple hologram modification h′ = hexp(−iϕ(x_p, y_p)), respectively φ′ = φ − ϕ. Here, we assumed that the hologram itself is located in the exit pupil. Of course, if the location is in a different plane, simple scaling of the coordinates has to be applied [6].

Therefore, the standard approach is to first optimize the hologram for the perfect optical system and then — in a second step — to include the aberration using the simple phase correction.

This approach is not feasible for field-dependent aberrations because there is no single aberration that one could use for correction. Of course, correction using a time-sequential approach is possible if the hologram is implemented with a spatial light modulator and the application allows for time-sequential operation [7].

Several authors proposed methods for optimizing phase holograms under the assumption of field dependent aberration. The most simple approach limits itself to just depth dependent aberrations, which leads to defocus and often spherical aberration. Shamir et al. and Haist et al. have proposed modifications of the well known iterative Fourier transform algorithm (IFTA) [8], also called Gerchberg-Saxton algorithm (GS) for points in 3D [9, 10].

Most work has been done in the direction of the reconstruction of a small number of reconstruction spots for applications like optical micromanipulation, microassembly and structuring [11, 12]. Typically, the so-called weighted Gerchberg-Saxton algorithms has been employed [13–15]. The weighting is used to improve the uniformity of the reconstructed spots.

If one extends this algorithm to all (discrete) points on a fine raster in the reconstruction plane the summation that is done in this algorithms just corresponds again to the conventional Fourier transform. However, the more points one wants to control, the more necessary is it to give the algorithm more variables for optimization. In the aforementioned work only the phases of the reconstruction spots serve as real free parameters because the amplitudes in the reconstruction plane are always set to zero at all other positions during iteration. This leads to a small effective number of degrees of freedom which increases the reconstruction error.

It is therefore advantageous to incorporate support areas in the reconstruction plane where the optimization can create some light energy [8]. This additional degree of freedom (amplitude and phase at these positions) strongly improves reconstruction quality.

In section two we propose an algorithm which extends the conventional IFTA algorithm to the reconstruction of extended scenes with field dependent aberrations given by field dependent Zernike modes. We use support areas, global weighting and a simple speckle reduction to achieve high quality reconstruction of phase holograms. The main contribution of the work is therefore an extension and combination of the IFTA-based algorithms to arbitrary aberrations and extended reconstruction scenes. Experimental results using a spatial light modulator in combination with a strongly aberrated system are shown in section three.

2. Algorithm

The reconstruction of aberrated Fourier holograms h(x_p, y_p) = exp(iφ(x_p, y_p)) is given by the Fourier transform of the aberrated light field:

Of course, to obtain a realistic reconstruction of the whole reconstruction field, Eq. (3) has to be evaluated for all isoplanatic patches. The size of the isoplanatic patches has to be chosen such, that the variation of aberration for the points lying in one isoplanatic patch is tolerable. For our method we take the center of each isoplanatic patch to be the reference for the whole patch. Therefore, the Zernike coefficients for this position with the patch are used during optimization. The computational cost for optimization grows linearly with the number of patches.

In order to optimize for a desired reconstruction g(u, v) we use a modification of the iterative Fourier transform algorithm (Fig. 1). The reconstruction is computed for each isoplanatic patch using the appropriate aberrations (right hand side of Fig. 1). In each patch we set the amplitude to the desired amplitude. Then, the hologram for this patch is computed using the inverse Fourier transform and the conjugate of the appropriate aberrations. Finally, we build the complex sum of all partial holograms and set the amplitude of the resulting hologram to one. This completes one iteration cycle. For the initialization, random object phases are assumed. The aberrations are represented using Zernike polynomials. The Zernike coefficients a_i are determined by raytracing using Zemax.

 

Fig. 1 Modification of the iterative Fourier transform algorithm to optimize phase only holograms in the presence of field-dependent aberrations. The reconstructions of the hologram for different isoplanatic patches is shown on the right hand side. During iteration the hologram is given by the phase of the complex sum of the individual holograms due to the isoplanatic patches.

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For the optimization it is important that enough degrees of freedom (variables) are available in order to achieve the reconstruction of a complicated reconstruction pattern. To this end not only the phases at the desired reconstruction points but additionally amplitude and phase of light in support areas can be employed. Typically, we use 75 per cent of the complete available reconstruction area as support in order to improve the reconstruction.

The constant b shown in Fig. 1 is used to modify the optimization of the algorithm. By using a large b more energy is directed from the support area to the desired reconstruction. This way, the diffraction efficiency is increased at the cost of the final reconstruction quality, defined to be the root mean squared difference between desired and achieved reconstruction intensity. Suitable values for our experiments with extended patterns (e.g. USAF target) turned out to lie between b = 3 and b = 4. The optimum value strongly depends on the target to be reconstructed. Automatic optimization for a specific reconstruction target would be possible.

IFTA-based holograms of complex patterns typically show a speckle-structure due to the phase singularities that are present in the optimized holograms [16, 17]. To reduce these effects one typically tiles the hologram [10, 16]. The repetition in the hologram plane leads to a discretization in the reconstruction which leads to a strongly improved visual appearance. However, simple tiling will not work for these aberration-corrected holograms. Therefore, we enforce the discretization already during optimization by using a discreet pattern for the desired reconstruction. Every second pixel is chosen to be zero. This enlarges the reconstruction as well as the hologram plane and therefore increases the computational speed but strongly reduces the speckles (see Fig. 2).

 

Fig. 2 Speckle reduction using artifical discretization duing optimization. Optical reconstructions of optimized Fourier holograms using the described algorithm.

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3. Experimental setup and results

We use the setup depicted in Fig. 3 to verify the method. The laser (Helium-Neon, λ = 633 nm, 3 mW) is collimated and illuminates a transmissive twisted-nematic liquid crystal light modulator (Holoeye LC 2002, 800×600 pixel, pixel pitch 32 μm, 512 × 512 pixels used). The reconstruction of the holograms written into the modulator is performed using an achromatic lens with a focal length of 400 mm. Strong aberrations are introduced by tilting the achromatic lens by 30 degrees. The intensity of the reconstruction is recorded using a CMOS image sensor (Ximea iQ MQ013MG-E2 USB 3.0, 1280 × 1024 pixels, 5.3 μm pixel pitch). The aberration in the exit pupil of the optical system is dominated by astigmatism (Zernike standard coefficients in Zemax for astigmatism: −16.7, defocus: −1.65 and coma: −0.1).

 

Fig. 3 Experimental setup. The holograms are reconstructed using a strongly aberrated Fourier system (achromatic lens tilted by 30°).

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The polarizers in front of and behind the modulator are used to achieve suitable phase shifting with the LCD. Perfect 2π phase-only modulation cannot be achieved this way. Further improvement would be possible by using elliptic average eigenpolarizations [18] but not necessary for verification of our technique. We limit our reconstruction to one quadrant of the complete reconstruction plane. This corresponds exactly to the 75% support area.

Figure 4 shows results for corrections achieved with 4 × 4 isoplanatic patches compared to global aberration correction. We only show the optimized quadrant. Therefore, the optical axis is always on the top left corner. Without any aberration correction, the reconstructions are hardly visible at all (not shown).

 

Fig. 4 Optical reconstruction under strong field dependent aberrations. Corrected with global aberration correction (conventional method) and the proposed method.

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4. Conclusions

We have shown a modification of the iterative Fourier transform algorithm that allows one to optimize reconstructions through strongly aberrated optical systems where the aberrations are dependent on the field position. To this end the field is divided into isoplanatic patches. The typical trade-off between reconstruction error and diffraction efficiency can be specified by one single parameter. Speckles are reduced using a simple discretization approach during optimization and the reconstruction quality is generally improved using support areas. We have shown experimental results for a strongly aberrated Fourier reconstruction geometry. Of course, the results can be employed also for other holographic projection geometries.