1. Introduction

Tunable lenses consisting of a pair of adjacent optical elements have been known for more than 50 years. Alvarez-Lohmann lenses named after their two independent inventors [1–4] are translated with respect to each other in order to control the optical power. Later, rotational versions of these lenses (sometimes also called moiré lenses, due to their underlying operation principle), have been investigated first numerically [5], and later experimentally [6–9]. They consist of two adjacent diffractive optical elements (DOEs), which are rotated with respect to each other around their central optical axis in order to change the optical power. Moiré lenses are interesting for laser machining applications, since they are suited for high laser powers, they provide diffraction limited focusing, they introduce no undesired Petzval field curvature and no barrel or pincushion field distortions, and they keep the focus exactly on the optical axis during optical power variation. The same principle is now also employed for designing tunable lenses with nanostructured optical meta-materials [10,11].

The currently demonstrated diffractive tunable lenses are, however, strongly dispersive. On the one hand, this feature can be advantageously used to control the dispersion in optical setups using a hybrid system consisting of a tunable refractive lens and a subsequent tunable diffractive lens [12]. However, on the other hand strong dispersion is usually not desired for imaging applications with broadband light, although chromatic errors may be reduced by computational post-processing [13]. Thus there is a high interest in designing achromatic tunable lenses. One recent approach is based on "metalenses", which correspond to nano-structured meta surfaces. There it was shown that achromatic behavior can be achieved by polarization control of the incident light [14]. Furthermore, tunable meta lenses based on the Alvarez principle have been demonstrated, which showed a considerably reduced chromatic error as compared to static meta lenses [15].

For the case of achromatic diffractive moiré lenses, suggested in the following, numerical studies have already been performed, which demonstrate their feasibility. One approach suggested by [16] shows that full diffraction efficiency over the whole visible light range can be achieved using multi-layer diffractive structures, consisting of combinations of materials with different refraction indices. However, in this case the focal length is still wavelength-dependent. Another approach suggests to employ multi-order diffractive optical elements [17]. This principle has already been demonstrated for static diffractive "harmonic lenses", which were shown to have the same optical power, and the same (in principle unlimited) diffraction efficiency at a set of harmonic wavelengths [18–21]. More recently, methods have been developed to design static diffractive lenses with a deeper surface relief, which can be optimized to have the same focal length at a set of discrete, arbitrarily selectable wavelengths within a broad spectral range [22–25].

In order to design tunable diffractive lenses, we use the principle of harmonic lenses. These have a deeper surface relief than previously used single-order DOEs. For example, a $k$-th order DOE modulates the phase of a transmitted beam within an interval between 0 and $2\pi k$. With increasing thickness of the surface relief, the transmission function of the respective optical element makes a continuous transformation from a diffractive to a refractive structure [26], with correspondingly changing dispersion properties. Numerical investigations show that this principle can be employed for the design of multi-color tunable moiré lenses [17]. In the present work, this theoretically predicted feature is experimentally investigated.

2. Multi-order moiré lenses

The operational principle of multi-order tunable moiré lenses has been described in detail in [17]. Actually two different types of these lenses have been suggested, namely lenses which produce two sectors of different optical power during rotation (type 1), and lenses where the sector formation is avoided by a special quantization of the phase structure (type 2). For a single-order moiré lens of type 2, the sector formation can be completely suppressed. In this case, however, there arises another (mostly) undesired diffraction order, i.e. the lens becomes bifocal. There, the efficiencies of the two superposed sub-lenses depend on the rotation angle. However, numerical simulations show that for multi-order moiré lenses a perfect suppression of the sector is not simultaneously possible for all harmonic wavelengths at the same time. Thus we decided to manufacture a tunable lens of type 1, i.e. a sector-producing lens. These lenses are typically used in a limited range of rotation angles, such that the sector with the desired optical power covers most of the circular lens area. For example, if the rotation range is limited to an interval between $- 90^{\circ }$ and $+ 90^{\circ }$, then the area of the lens with the "correct" optical power covers at least 75% of the total lens area, which also corresponds to its theoretical diffraction efficiency. Imaging experiments with earlier type 1 moiré lenses (sector-producing) have shown that the effect of the undesired sector at the image quality is almost negligible, even if this sector is not covered. This is due to the fact that the "undesired" sector produces an extremely defocused field, such that most of its transmitted intensity is spread over a large solid angle, and only a small fraction of its intensity overlaps with the restricted image area.

Since the operational principle of multi-order moiré lenses has been described earlier in detail [17], the following description will just be a brief summary of the results, adapted for the current case of our experimentally investigated multi-order moiré lens of type 1: The surface relief functions $H_{1,2}(r)$ of two DOE elements forming a compound moiré lens with included sector are given by [17]:

This behavior is sketched in Fig. 1. In the simulation, an example of two complimentary 6-order DOEs is calculated according to Eq. (1). The DOEs are furthermore rotated by +15$^{\circ }$ and -15$^{\circ }$ with respect to the vertical axis. Figure 1(a) and (b) shows the respective surface height profiles (colorbar at the left corresponds to height in µm). In (c), the sum of the profiles of (a) and (b) is plotted. Furthermore, the sum is cropped by a modulo-operation to the surface height, which corresponds to a phase shift of 2$\pi$ for the wavelength $\lambda _0$. Thus, (c) corresponds to the effective phase transmission function for light with a wavelength of $\lambda _0$. The result shows that the surface profile corresponds to that of two Fresnel lenses in two different sectors of the combined moiré lens, as expected by Eq. (2). The optical powers $f_1^{-1}$ and $f_2^{-1}$ in the two sectors are proportional to the relative rotation angle $\theta$, and are given by

 

Fig. 1. (a): Surface relief of a 6-order DOE according to Eq. (1), rotated by 15$^{\circ }$ with respect to the vertical axis. (b): Surface relief of the complimentary DOE rotated by -15$^{\circ }$. (c) Surface relief of the combined DOEs of (a) and (b) according to Eq. (2). It corresponds to the sum of (a) and (b), with a successive modulo operation. The modulo-operation crops the height of the combined surface relief to a height of $\lambda _0/(n-1)$, where light with a wavelength of $\lambda _0$ experiences a maximal phase shift of 2$\pi$. The colorbars show the corresponding heights (in µm) of the surface profiles. The colorbar at the left applies to (a) and (b), whereas the colorbar at the right applies to (c).

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For our experimental demonstration we designed a type 1 (sector-producing) moiré lens of order $k=6$ at a design wavelength of $\lambda _0=532$ nm, according to Eq. (1). Figure 1(a) and (b) show the corresponding phase profiles in the centers of the two DOEs. The structured area of the circular DOEs had a diameter of 8 mm, and a minimal feature size of 0.5 µm. The parameter $a=1.234 \times 10^7$ m$^{-2}$ was chosen such that the optical power within the "desired" sector of the lens was adjustable in a range between -19.7 m$^{-1}$ and +19.7 m$^{-1}$ by a relative rotation of the two DOEs in a respective range between $-90^{\circ }$ and $+90^{\circ }$. The difference of the optical powers within the two sectors is independent of the rotation angle and corresponds to -79 m$^{-1}$, according to Eq. (4). The two DOEs were etched by 8-bit lithography into fused silica wavers ($n=1.4607$), with an intended maximal depth of the surface profile of 6.92 µm, and with a depth resolution of 256 digital, equidistant steps. According to Eq. (5) the corresponding harmonic wavelengths $\lambda _i$ with the same optical powers within the visible color range are $\lambda _5=638.4$ nm, $\lambda _6=532$ nm and $\lambda _7=456$ nm (Eq. (5)). However, after production of the elements interferometric measurements showed that the depth of the surface relief was on average by 7.1% larger than its target value, which resulted in a correspondingly adapted set of actually realized harmonic wavelengths within the visible range of $\lambda _5=682.6$nm (5th order, red), $\lambda _6=569.8$nm (6th order, green/yellow), $\lambda _7=487.5$nm (7th order, blue) and $\lambda _8=427.4$nm (8th order, violet).

A simulation of the diffraction efficiency of the produced moiré lens is shown in Fig. 2 in comparison with that of another available single-order lens with the same aperture diameter of 8 mm, which was produced earlier for a design wavelength of 633 nm [7]. The adjustable range of optical powers of the multi-order lens, and of the single-order lens, are approximately equal. The calculation was done for an assumed optical power of 10.7 dpt at the central harmonic wavelength of 569.8 nm, which corresponds to a rotation angle of $\theta = 45.7^{\circ }$ for the multi-order lens, and of $\theta = 53.5^{\circ }$ for the single-order lens. The corresponding maximal diffraction efficiencies in the ideal case correspond to the ratio between the area of the desired lens sector and the total lens area, which depends on the relative rotation angle of the two sub lenses. For the multi-order lens and the single-order lens, the respective maximal (ideal) efficiencies are thus supposed to be 87.3% and 85.1% at their corresponding design wavelengths, respectively. In the simulation the lenses are treated as thin phase elements, which are illuminated by an incident plane wave of variable wavelength. The transmitted field is calculated by pointwise multiplication of the incident field with the corresponding transmission functions (with a phase relief according to Eq. (6)). The field is then further propagated (using the angular spectrum propagation method) to different planes behind the lens. The distance between the lens and the successive planes corresponds to the inverse of the optical power in the plots of Fig. 2. In each of these planes, the intensity of the central spot was evaluated. The ratio between the intensity of this focal spot, and the total transmitted intensity corresponds to the diffraction efficiency, which is plotted in Fig. 2 as a function of both, readout wavelength, and optical power (i.e. inverse focal length). The simulation does not include more detailed features of the diffraction process, like effects of the anti-reflection lens coating, the absorption of the lens material, possible phase noise due to the fabrication process, or scattering at the edges of the phase profile.

 

Fig. 2. Simulated diffraction efficiency of a single-order lens (left) compared to that of a multi-order lens (right) as a function of the readout wavelength and the optical power of the lenses. The simulation is done for the parameters of the two single-order and multi-order moiré lenses, which later are experimentally investigated. The relative rotation angle of the multi-order moiré lens was chosen to be 45.7$^{\circ }$, which results in an optical power of 10.7 dpt at its central harmonic wavelength ($\lambda _6=569.8$ nm). The correspondingly chosen rotation angle of the available single-order moiré lens, which results in the same optical power, is 53.5$^{\circ }$.

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The direct comparison shows that a single-order lens has a linear dispersion, i.e. at the position of the highest diffraction efficiency (yellow) the optical power is a linear function of the wavelength. Furthermore the diffraction efficiency has a maximum at the design wavelength 633 nm of the lens, and falls off at lower and higher wavelengths. For a multi-order lens, the dispersion has locally the same slope in narrow regions around the specified harmonic wavelengths of $\lambda _5=682.6$ nm, $\lambda _6=569.8$ nm, and $\lambda _7=487.5$ nm. However, a deviation from the harmonic wavelengths leads to a decrease of the diffraction efficiency along the original linear dispersion curve until it jumps to an adjacent dispersion curve, which is parallel shifted with respect to the previous one, and which is centered around the next nearby harmonic wavelength. The diffraction efficiency is equal, and maximal, at the set of harmonic wavelengths, and theoretically corresponds to the fraction of the area covered by the desired lens sector to the total lens area, which is about 87% for the multi-order moiré lens at the specified rotation angle of $45.7^{\circ }$.

The maximal diffraction efficiency of a multi-order DOE is obtained, if it is illuminated by one of its harmonic wavelength $\lambda _m$. A small change of $\Delta \lambda$ results in an efficiency reduction, which can be approximately calculated by the "grating" formula $\eta =\mathrm {sinc}^2 \{\pi [m(1-\Delta \lambda /\lambda _m)] \}$. With this relation it can be estimated that a small deviation of 3% from the $m=6$ harmonic (corresponding to 17 nm in our case) would reduce the efficiency of the 6th-order by about 10%.

In a first experiment we measured the absolute diffraction efficiency of the tunable lens at three wavelengths (633 nm, 532 nm, and 473 nm) as a function of the adjusted optical power of the lens. The results are shown in Fig. 3.

 

Fig. 3. Measured absolute diffraction efficiency (i.e. ratio between light intensity in the focal spot with respect to incident light intensity) of the the multi-order moiré lens as a function of the adjusted optical power for the three wavelengths 633 nm (in red), 532 nm (in green) and 473 nm (in blue). Lower part: Sketch of the experimental setup: A collimated laser beam, which is expanded to fill the aperture of the tunable lens, is focused by the moiré lens at the plane of a pinhole (50 µm diameter). There the light is spatially filtered in order to reject scattered light, and contributions from other diffraction orders. The intensity transmitted through the pinhole is then measured using a calibrated power meter (photo diode).

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For the measurements, we used three laser beams at the respective wavelengths, which were first expanded and collimated in order to fill the aperture of the tunable lens. After adjusting the lens to a certain focal length, a pinhole (with a diameter of 50 µm) was placed in the focal plane as a spatial filter, rejecting diffusely scattered light, and light of undesired diffraction orders. Behind the pinhole, the transmitted light power was measured with a calibrated power meter (photo diode). The absolute diffraction efficiency was then determined as the ratio between the transmitted light power with respect to the incident light power (measured in front of the tunable lens).

The graphs in the figure show the expected linear decrease of the diffraction efficiency as a function of the adjusted optical power. This is due to the fact that the mutual rotation angle between the two DOEs of the moiré lens linearly increases as a function of the optical power, and therefore the area of the used sector with the specified optical power decreases linearly. For adjusting optical powers in a range between 0 and 20 dtp, the rotation angle had to be tuned in a range between 0 and 90$^{\circ }$.

The measured efficiency is, however, significantly reduced with respect to the theoretically expected efficiency of 87%. Although both sides of the DOEs are broadband anti-reflection coated, some loss of transmission is expected, since the anti-reflection coating at the strongly slanted relief of the structured DOE sides supposedly is not very effective. In order to also determine the relative diffraction efficiency, which is defined as the ratio of the correctly focused light intensity and the total transmitted light intensity, we also measured the relative transmission of the MDOE at the respective wavelengths. It turned out that the average transmission of the MDOE at the harmonic wavelengths is on the order of 96%, which means that the relative diffraction efficiency is by a factor of 1.04 higher than the measured absolute diffraction efficiency in Fig. 3.

The main source for the loss of diffraction efficiency is due to the fact that the DOEs are actually "thick" optical structures, which cannot be fully described by the thin element approximations used for our basic simulations. There, diffuse scattering happens at the phase edges within the elements during propagation through the structured bulk material, which is also known as "shadowing effect" [27]. Corresponding numerical simulations based on rigorous electromagnetic theory, or on an step-by-step propagation of the light field through the structured bulk DOE material are computationally extensive, particularly for highly structured large DOE elements, and they will be the topic of future research.

Nevertheless, the data shows that the relation of (correctly) diffracted light between the different wavelengths keeps approximately constant for all adjusted optical powers. Evaluating the data in Fig. 3 shows that the ratio of the efficiencies (normalized to the red beam, which has the highest efficiency) is approximately 1: 0.85 : 0.83 for the red, green and blue components, respectively, at an adjusted optical power of 0 dpt. This ratio changes to 1: 0.84: 0.78 for the maximal adjusted optical power of 20 dpt. Thus the relative efficiency of the blue component changes by approximately 6% with respect to the other two components within the full tuning range of 20 dpt. If such a lens is used in a physiological vision system (e.g. as eyeglasses), the change of the colors within an object will be almost imperceptible. However, for applications in a high quality camera system, these color changes will be more crucial, and they should be compensated by post-processing of the different color channels of the image.

In order to quantify the maximal resolution of our imaging system, the point-spread-function of the lens was measured at various wavelengths. This was done using collimated laser beams (at 633 nm, 532 nm, and 473 nm), which were expanded to overfill the lens aperture. For this measurement we adjusted the optical power of the moiré lens to a value of 18.5 dpt (corresponding focal length: 5.4 cm). The widths of the focused spots in the focal plane were then measured with a camera, which were 8.8 µm, 6.7 µm , and 6.7 µm (FWHM, +/- 10%) for the red, green, and blue wavelengths, respectively. The theoretically expected radius $s$ of the Airy disc, estimated by the formula $s=0.61 \lambda f/r$, where $\lambda$ is the illumination wavelength, $f=0.054$m is the adjusted focal length, $r=4$mm is the radius of the lens, at the respective wavelengths is 5.2 µm, 4.4 µm, and 3.9 µm, respectively. Thus our experimentally determined PSF is by an average factor of approximately 1.7 larger than its ideal diffraction limited values. This difference is mainly due to the appearance of the undesired sector in the sector-producing moiré lens, which slightly corrupts the ideal PSF.

In order to test the imaging performance of the multi-order moiré lens, we used the experimental setup sketched in Fig. 4. For the first experiments we investigated the performance of the system using narrow band illumination provided by a tunable monochromator (Till Photonics, Polychrome IV) with a bandwidth of approximately 15 nm (FWHM). The light emerging from the monochromator illuminates a diffuser (white paper), which produces a spatially incoherent illumination field. At a small distance of 1 cm, a transmissive object (USAF resolution target) is placed. The tunable multi-order moiré lens is centered between the object and an open camera chip (Canon EOS 5D Mark II). A sharp image is obtained by adjusting the rotation angle of the moiré lens. In this case its focal length is a quarter of the distance between the object and the image plane.

 

Fig. 4. Experimental setup, from left to right: A white light LED illuminates a diffuser (white paper). A transmissive object (USAF resolution target) is placed 1 cm behind the diffuser. At a distance of $2 f$ (where $f$ is the adjusted focal length of the moiré lens), the investigated single- or multi-order moiré lens is placed symmetrically between the object and the image plane (right). Imaging is performed with the open CMOS chip of a Canon EOS 5D Mark II color camera. For obtaining images at different optical powers of the moiré lens, both object and camera are shifted symmetrically (with respect to the moiré lens in the center) along the optical axis, and a sharp image is re-adjusted by tuning the focal length of the moiré lens.

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In a first experiment we tested the chromatic performance of the multi-order lens at an adjusted optical power of 10 dpt (corresponding to a focal length of 10 cm) by tuning the wavelength of the illumination light, keeping all other features of the setup constant. Figure 5 shows images recorded at various wavelengths. A corresponding movie (Visualization 1) shows the results of a wavelength scan in the range between 700 nm and 400 nm. Figure 5(a-f) shows a sequence of 6 images from that movie, recorded at the wavelengths 426.6 nm (Fig. 5(a)), 487.5 nm (Fig. 5(b)), 528.7 nm (Fig. 5(c)), 549.8 nm (Fig. 5(d)), 569.8 nm (Fig. 5(e)), and 682.6 nm (Fig. 5(f)). The images shown in (a), (b), (e) and (f) correspond to those taken at the four harmonic wavelengths in the visible range. All of these images are sharply focused, and they appear at exactly the same positions at the camera chip, having also exactly the same sizes. The corresponding image resolutions are determined by finding the smallest resolved elements in the USAF resolution target. They correspond to 39 µm (element 3/5), 31 µm (element 4/1), 44 µm (element 3/4), and 50 µm (element 3/3) for the four respective harmonic wavelengths. Comparing these values with the ideal resolution limit (which is assumed to be the radius of the central focal spot in the corresponding ideal Airy discs) it turns out that the experimental resolution is in all of these cases approximately by a factor of 5 times lower than its ideal value. One reason for the reduced resolution is the fact that the point spread function of the moiré lens is slightly corrupted, due to the sector formation, as discussed in the previous paragraph. Furthermore, the bandwidth of the light source of about 15 nm produces already a decrease of resolution, due to the dispersion of the lens in the vicinity of the ideal harmonic wavelengths.

 

Fig. 5. Images recorded with the multi-order MDOE at different narrow-band wavelengths (bandwidth 15 nm FWHM). Images (a), (b), (e) and (f) show the results at the harmonic wavelengths of 426.6 nm, 487.5 nm, 569.8 nm and 682.6 nm, respectively, where the sharpness is supposed to be optimal. Image (c) shows the result if the wavelength is tuned to the center between two harmonic wavelengths, namely to 528.7 nm. There, the resolution is by a factor of 5 lower than that of the images recorded at the optimal harmonic wavelengths. Image (c) shows the result if the image is detuned by 20 nm with respect to the yellow harmonic wavelength. The resulting blurring of the image reduces the maximal resolution (obtained at the harmonic wavelengths) by a factor of 2. A movie of the images recorded during a linear scan of the illumination wavelength between 700 nm and 400 nm is attached (see Visualization 1).

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In order to further check the effect of the dispersion, images (c) and (d) were recorded at wavelengths detuned from the harmonic frequencies. Image (d) is recorded at a wavelength of 549.8 nm, which is detuned by 20 nm from the next harmonic wavelength (569.8 nm). It has an already blurry appearance, with a resolution of 70 µm (element 2/6). Image (c) is recorded at a wavelength of 528.7 nm, which is maximally detuned (by 40 nm) with respect to its neighboring harmonic wavelengths. In this case the moiré lens becomes actually bifocal, producing images in 7th-order (corresponding to the harmonic at 487.5 nm) and 6th-order (corresponding to the harmonic at 569.8 nm) at the same time, with correspondingly different focal lengths. As a result the image appears maximally blurred, with a resolution of only 200 µm (element 1/3).

In a further experiment we investigated the dependence of the image resolution on the position of the object within our field of view (FOV). For this purpose we used the previous setup, with the optical power of the moiré lens adjusted to 10 dpt. The usable field of view in the experiment covered an angular range of $\pm$5$^{\circ }$. The illumination wavelength was adjusted to the yellow/green harmonic wavelength at 569.8 nm. Figure 6 shows a double exposure of an image of the resolution target, which was laterally shifted between the two exposures by a distance of 17.6 mm, corresponding to a change of the viewing angle (between moiré lens and object) of 5$^{\circ }$. Comparing the resolution of the two shifted images it turns out that the central image has a resolution of 39 µm (corresponding element 3/5 indicated in the figure by a red rectangle), whereas the resolution of the shifted image is 50 µm (element 3/5). This corresponds to a loss of 30% of resolution from the optical axis to the edge of the FOV. The reason for this reduction of the resolution is on the one hand that diffractive lenses have some astigmatism for larger viewing angles (although they have no Petzval field curvature). Furthermore the effect of the bandwidth of the illumination light (which is about 15 nm) becomes more prominent for larger viewing angles, due to the angular dependence of the dispersion of diffractive structures.

 

Fig. 6. Double exposed image of the resolution target. One exposure was recorded with the resolution target centered along the optical axis, whereas for the second exposure the resolution target was laterally shifted by 17.6 mm. The corresponding image resolutions of the centered and the off-axis images are 39 µm and 50 µm, respectively, indicating a 30% loss of resolution at the edges of the field of view (corresponding smallest resolved elements in the resolution chart are indicated by red rectangles).

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A next consistent step is to investigate the white light imaging features of the tunable multi-order moiré lens. This was done by comparing its performance with that of an available single-order moiré lens. We used the same experimental setup as sketched in Fig. 4, but now with a white light LED for illumination. Images at multiple optical powers are recorded by shifting the object and the camera symmetrically to new equidistant positions with respect to the moiré lens in the center, and then readjusting the optical power of the tunable lenses to obtain a sharp image. This procedure assures that an image magnification of 1 is maintained for all images. Note that the diffuse illumination of the resolution chart strongly reduces the depth of sharpness in the camera plane, and thus increases the sensitivity of the setup for chromatic aberrations.

The results of imaging experiments using both a single-order and a multi-order moiré lens in the same setup are compared in Fig. 7. The focal lengths of the lenses had been adjusted in three different experiments to 5 cm, 10 cm, and 20 cm, respectively. Accordingly, the distances between object (resolution target) and image plane (camera chip) have been adjusted symmetrically around the central moiré lenses to be 20 cm, 40 cm, and 80 cm, respectively. Due to the symmetry of the setup, the optical magnification of all images is unity. The sequence of images shows that the single-order lens designed for monochromatic light (left) produces blurred images with strong chromatic aberrations, which are dominant at the edges of the bars within the image. These chromatic aberrations are strongly reduced in the corresponding images taken with the multi-order lens. Furthermore, the noise level in all images recorded by the single-order moiré lens is significantly higher than in the respective images using the multi-order lens. This is partially a result of the fact that the design wavelength of the available single-order lens was 633 nm, which is not optimal for the incident white light spectrum in the range between 450 nm and 680 nm. However, even a single-order lens with a design wavelength in the center of the white-light spectrum is supposed to have a reduced overall efficiency with respect to a multi-order lens, and thus an increased noise level.

 

Fig. 7. White-light imaging with the setup of Fig. 4. The left and the right columns show the results using a single-order moiré lens (designed for monochromatic illumination), and a multi-order moiré lens under white light illumination, respectively. Within the different rows, imaging results for different adjusted focal lengths of the single- and multi-order moiré lenses of 5 cm, 10 cm, and 20 cm are shown. For the experiments, the distances between the resolution target and the camera chip have been changed symmetrically with respect to the central moiré lenses to 20 cm, 40 cm, and 80 cm, respectively.

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A more detailed comparison of the chromatic errors introduced by the single-order and the multi-order moiré lenses is shown in Fig. 8. In this case we used a tunable single-order moiré lens which was designed for a wavelength of 532 nm (provided by Diffratec Optics OG). In the experiment the object was illuminated with thermal white light (halogen bulb). The spectrum of the light source, recorded with a spectrometer (Ocean Optics, USB 4000, wavelength resolution smaller than 1 nm) is plotted at the bottom of the figure. The spectrum shows that the 8th harmonic wavelength (427.4 nm, violet) has an only negligible intensity, whereas the other three harmonics in the visible range are contributing to the white light image. Color images with both the single-order and the multi-order lenses where taken under the same conditions at an adjusted focal length of 10 cm (corresponding to the lens adjustments in the middle row of Fig. 7). The color images then were disassembled into their red, green, and blue (rgb) color components by reading out the different color channels of the original raw camera pictures. According to the data of the camera chip, the corresponding color channels are peaked around (600 $\pm$ 50) nm (red), 530 $\pm$ 50) nm (green), and (480 $\pm$ 40) nm (blue) (the full-width-at half maximum of the spectral sensitivity is indicated by the $\pm$-values) [28].

 

Fig. 8. Comparison of the red, green, and blue components of two white light images (using a halogen bulb for illumination) recorded with a tunable single-order moiré lens (upper row) and a multi-order moiré lens (lower row). In the experiment the two tunable lenses both have been adjusted to an optical power of 10 dpt. The raw white light images have been recorded with a color camera, and been disassembled in their respective red, green, and blue color channels. At the bottom of the image we display a spectrum of the used thermal halogen lamp, recorded by a spectrometer (spectral resolution smaller than 1 nm).

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The upper row shows the resulting red, green, and blue image channels of the image taken with the single-order moiré lens, whereas the lower row displays those recorded with the multi-order moiré lens. Comparing the results, it turns out that the green component is sharply imaged by both of the moiré lenses. This is due to the fact that the focus adjustment of the moiré lenses was optimized for the center of the white wavelength range, which corresponds to the green wavelength. The image resolution estimated from the USAF resolution target corresponds in the two cases to 70 µm (smallest resolved element 2/6), which is by a factor of two lower than that of the corresponding narrow band image in Fig. 5(e), due to the increased effect of dispersion in the white light experiment. However, for the red and the blue spectral components the image resolution and contrast is strongly reduced for the single-order moiré lens with respect to the results of the multi-order lens. The loss of resolution is due to the fact that the optical power of a single-order lens is proportional to the wavelength, as shown in the simulation of Fig. 2, which corresponds to the situation of the investigated imaging experiment. The simulation shows that the optical power changes in the visible range (between 450 nm to 700 nm) in an interval between 8 and 13 dpt, corresponding to a focal length range between 12.5 cm and 7.7 cm. Thus, if the central wavelength is in focus with a focal length of 10 cm, then the focal lengths of the red- and blue image components are supposed to differ by about $\pm 2.5$ cm.

On the other hand, the images taken with the multi-order moiré lens (lower row) show a good efficiency with a high signal to background ratio, and an image resolution of 70 µm for all rgb-components, which also is the resolution within the corresponding white light image. There it has to be considered that according to the color sensitivity of the used camera each color channel covers a spectral bandwidth of approximately 50 nm (FWHM). According to the simulation in Fig. 2 the optical power of the multi-order lens is supposed to vary in a range of $\pm 0.5$ dpt around its target value of 10 dpt within this spectral bandwidth. This produces some image blurring, which is, however, significantly lower than that produced by the tunable single-order lens.

3. Conclusion

We have experimentally investigated the first physical realization of a recently suggested multi-order tunable moiré lens. We tested the achromaticity of the lens for different adjusted focal lengths by comparing its white light imaging properties with those of a similar tunable single-order moiré lens. The experiments show that the chromatic aberrations of single-order lenses are significantly reduced for a multi-order tunable moiré lens. Furthermore, the diffraction efficiency of the multi-order lens is (almost) equal among the red, green and blue color bands of the white light spectrum. These features open a possibility to use diffractive lenses for white light imaging with variable optical power. Multi-order moiré lenses may, for example, offer new opportunities in minimized imaging systems with restricted space limitations. Focusing with such a lens does not require any shift of the lens, or any additional lateral space, as would be necessary using laterally shifting Alvarez lenses. Moreover, the lenses are thin, light weight, and robust with respect to environmental conditions, like accelerations or harsh thermal conditions. However, their imaging performance is still limited by the loss of resolution due to dispersion. One possible method to strongly reduce this issue would be to replace the currently used broadband color filters in front of the individual pixels of a typical color camera chip by narrowband filters, which are adapted to the sharply imaged harmonic wavelengths of the moiré lenses. Furthermore, computational post-processing of the color channels of even broad-band images can drastically improve the resolution [29]. In the future it also may be possible to reduce the residual dispersion of tunable moiré lenses by designing them for an even higher diffraction order, according to the numerical investigations in [17]. This possibility will, however, depend on necessary improvements in the manufacturing process of deep diffractive structures, which may be expected in the near future.