As described in the previous section, the transition from regular lens arrays to randomly distributed lenses represents a solution that not only keeps the advantages of refractive micro lens systems (e.g., maximum efficiency, large-angle spectrum, and high degree of homogeneity) but also gives an extension of the intensity profiles that are producible (e.g., Gaussian and super-Gaussian instead of only top hats).

In spite of this extended variability, the generation of completely arbitrary, non-regular intensity distributions is not possible with randomized lens arrays because of their rotationally symmetric character [18].

Of course, to get a higher flexibility in the addressed angular spectrum of intensity, more freedom in the profile shape of the optical element is needed. Here, in principle, the traditional diffractive diffusers provide such ‘high freedom’ as arbitrary statistic element profiles and allow the shaping of nearly each arbitrary angular distribution [19]. Unfortunately, compared to the previously discussed refractive micro-lenses, diffractive elements also have significant disadvantages. Due to their diffractive character and also as a result of the usually used Fourier transformation-based design methods, the profiles of diffractive diffusers contain phase dislocations. Phase dislocations are points of discontinuity with a spiral shaped phase distribution. The alteration of the phase φ along a closed curve S encircling such a point is an integer multiple of 2π. Discontinuities as phase dislocations and 2π-jumps act as scattering centers which produce disturbing stray light and limit the achievable efficiency. Another important consequence is that phase profiles with spiral phase dislocations cannot be transferred into continuous profiles by phase unwrapping to overcome their drawbacks.

However, following the idea of the transition from regular micro-lenses to non-regular lens shapes, an interesting further step is leading to more or fully randomized structures which exhibit no phase discontinuity [20, 21]. This approach, on the one hand, will allow an extension of the application potentials and, on the other hand, it will keep all advantages of lens like continuous structures discussed before. For this purpose, a calculation method has been developed, which avoids the above-mentioned occurrence of phase dislocations at the very onset.

The design algorithm developed includes an optimization procedure for a merit function. It implies manufacturing aspects such as lateral resolution limits of the structuring process and the maximum achievable profile depth.

To fulfill all demands, a modified IFTA was developed that is comparable with the algorithm described in [20]. However, the algorithm described in [20] was applied to form only high-order super-Gaussian profiles. In most cases, IFTAs use a 2π-modulo phase representation that enables or encourages the occurrence of phase dislocations. To avoid phase dislocations at all, an IFTA-based design algorithm was developed which uses a non-modulo, biunique phase representation for producing arbitrary intensity profiles.

The fabrication of the continuous randomized design was done by direct laser beam writing. As the lateral resolution of the laser beam writer is in the order of 1 μm and the efficiently producible maximum profile depth is about 10–20 μm, a spatial frequency filter and a profile-height limitation procedure were implemented into the design algorithm corresponding to the manufacturing limitations.

To demonstrate the performance and test the potential of the continuous phase design algorithm, a non-symmetric seven-ray star target intensity distribution with a bright-dark contrast of 100% was chosen. The required NA of the distribution was 0.2 (=23° FWHM).

For comparison of the continuous design approach with the conventional diffractive design, we applied a classical IFTA and the modified IFTA for continuous phase distributions to find the same target distribution. The left side of Figure 13 shows a section of the conventional diffractive design. The expected phase dislocations are visible. The diffractive design was made for a target wavelength of 633 nm. For the simulated far field intensity distribution, which is depicted in Figure 14, an incoming laser beam with 2.5 mrad divergence (FWHM) was assumed. For the classical design approach, an intensity contrast of about 1:30 was reached at the design wavelength. Changing the wavelength leads to the well-known occurrence of the zero-order peak, which is caused by a phase shift mismatch at the positions of the 2π steps. Exemplarily, in the case of an illumination wavelength of 543 nm, the simulation shows a zero-order share of 3.3%. With the 2.5 mrad incoming divergence, a ratio between the zero-order intensity and the target intensity of 5/1 was detected, which has a significant disturbing influence on the final intensity distribution (Figure 14, right).

Figure 13 Section of a conventional diffractive design (left) and of a continuous randomized design (right).

Figure 14 Simulated far field intensity distribution of the conventional diffractive design for two wavelengths (left: 633 nm, right: 543 nm with a zero-order peak).

The ratio of the zero-order intensity to the integrated intensity of the target distribution increases in a quadratic way with decreasing incoming divergence. Specifically, the ratio measures 30/1 for an incoming divergence of 1 mrad. Usually, due to fabrication imperfectness which leads to profile height and duty cycle errors, diffractive diffusers, fabricated by state-of-the-art methods, have a typical zero-order efficiency which is larger than 0.1–0.5%, depending on the structure size and NA. In the case of a large illumination NA, the zero-order contribution is also influenced by rigorous diffraction effects. The minimization of the zero-order contribution becomes even more difficult when the diffuser shows a wide structure size spectrum [8].

The right side of Figure 13 shows a part of the calculated continuous phase design in a 2π-modulo representation. The design was made also for a target wavelength of λ=633 nm. At the design wavelength, no disturbing zero-order peak was detected in the simulated far field intensity distribution. The simulated bright-dark intensity contrast of the produced intensity distribution was 1:17. In comparison to the classical diffractive design, the continuous design shows no phase dislocations. Therefore, in the 2π-modulo representation of the designed phase only closed 2π-steps occur (Figure 13, right). In contrast to diffractive profiles with spiral-shaped dislocations, such phase distributions can be unwrapped.

After unwrapping, the calculated phase profile was transferred into a height profile and fabricated by using direct laser beam writing. The measured surface profile is depicted in Figure 15 (top), and a cross-section of the profile is shown in Figure 15 (bottom).

Figure 15 Measured height profile of the fabricated element with the continuous randomized design (top: 3D representation, bottom: a cross-section).

The maximum profile depth of the manufactured element was about 10 μm. The measured intensity distributions using HeNe lasers with both 633 nm and 543 nm are shown in Figure 16. The measured NA of the star-shaped distribution was 0.2, which is close to the target value. The camera pictures show that the intensity contrast decreases with increasing angle. At the center, the contrast was measured to 1/2.5. At the largest angle, the contrast decreases to a value of 1/1. The measurement result can be explained with the resolution limitation of the laser writing system.

Figure 16 Measured far field intensity distributions of the fabricated element with the continuous randomized design for two wavelengths (top: 633 nm, bottom: 543 nm).

The fabricated profile shape can be regarded as a convolution of the target distribution and the writing resolution function. The convolution smears the small structure details, which are responsible both for the intensities propagating to the large angles and also for the high contrast. To produce large angles, either deep profiles with large profile gradients or small structure sizes are needed.

Despite the discussed profile aberration of the micro-optical element, the realized intensity distribution shows no zero-order contribution at the target wavelength of 633 nm. By changing the illumination wavelength to 543 nm, the measured intensity still shows no zero-order contribution. This measurement result at 543 nm impressively demonstrates the robustness of the design principle to wavelength changes or profile aberrations of the element structure. The main reason of this effect is the absence of 2π-phase steps in the design.

Additionally, comparing the sizes of the angular distributions at different wavelengths leads to the following interesting result. Both the measured intensity distributions at 633 nm and at 543 nm were added.

Figure 17 shows the resulting intensity pattern. Obviously the red color distribution shows a larger divergence in comparison to the green one. Measuring the ratio of angles for 633 nm/543 nm gives a value of 1/0.84. Taking into account the grating equation for a simple diffractive grating that produces a first diffraction order at the same angle, the angle ratio is 1/0.857 for the same wavelength change. Taking into account the measurement error, these two ratios are in the same order of magnitude. That means, the manufactured element acts much more as a diffractive element than as a refractive.

Figure 17 Summation of both measured intensity distributions at 633 nm and 543 nm.

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