Rotationally tunable diffractive optical elements (DOEs) consist of two stacked diffractive optical elements which are rotated with respect to each other around their central optical axis. The combined diffractive element acts as a highly efficient diffractive lens, which changes its optical power as a function of the mutual rotation angle. Here we show that the principle can be extended to produce polychromatic tunable lenses, i.e. lenses which have the same optical power, and the same diffraction efficiency within the full tuning range at three or more selectable wavelengths. The basic principle is to use higher order DOEs, which will be polychromatic at harmonics of a fundamental wavelength. The method can be applied to other types of optical elements which are tunable by rotation, like axicons, or generalized lenses with arbitrary radial phase profiles, or to elements tunable by a mutual translation, like diffractive Alvarez lenses.
© 2017 Optical Society of America
Lenses with adjustable optical power are a current topic of research, since they allow one to construct compact and flexible optical systems, such as focussing or zooming optics for imaging applications, or adaptive beam steering which is advantageous for illumination systems, or laser material processing . Various types of tunable lenses have been developed, as. e.g. tunable liquid lenses, where an elastomeric membrane lens is deformed in a controlled way by external forces [2, 3]. Another variant is based on electro-wetting, where by application of an electric voltage the shape of the active lens material can be controlled [4, 5]. Furthermore there are tunable acoustic lenses [6, 7], and liquid crystal based systems [8–10]. An interesting method, named after their inventors Alvarez lenses, or Alvarez-Lohmann lenses [11–14], uses two specially designed optical plates, which are placed on top of each other, and form a lens which is tunable by a mutual translation of the plates. Diffractive versions of these Alvarez-Lohmann lenses [15,16] have the advantage of being more compact, which, for example, allows for faster refocusing.
However, in many optical systems rotational elements are preferred with respect to translating elements since they are easier to implement. The first rotational version of a diffractive Alvarez-Lohmann lens has already been suggested in 1967 , where it was pointed out that its operation principle is based on the Moiré effect. Modified versions later appeared in several publications (e.g. [18–22]), and have been termed Moiré diffractive optical elements (MDOEs)  referring to their underlying principle. The working principle of these lenses is sketched in Fig. 1. Advantages of these elements are that they have a large tuning range of their optical power, they achieve diffraction limited focussing, their optical power is very precisely and reproducibly adjustable, and they can be utilized for high laser power applications. These elements provide a high diffraction efficiency, on the order of 80% to almost 100%, within their tuning range . But a disadvantage of the rotational MDOEs is their strong dispersion, which is common to diffractive elements. The optical power of a MDOE is proportional to the diffracted wavelength. Therefore the use of MDOE lenses in imaging applications with colored light is challenging, although numerical methods have been developed for post-processing colored images recorded with adjustable diffractive lenses, which correct for the color aberration .
Here we show that MDOEs can be also designed to be polychromatic, i.e. both their efficiency and their optical power is corrected at two, three, or even more wavelengths. For that purpose one can use higher order MDOEs, which manipulate the phase of the incident light not only in a range between 0 and 2π, but in a range up to k2π, where k > 1 is the diffraction order of the lens. It has been already shown that higher order diffractive lenses are polychromatic for a number of harmonic wavelengths, i.e. at these wavelengths the dispersion is corrected [23, 24]. Here we show that this principle can be generalized for tunable diffractive lenses, which can be optimized to be polychromatic at a number of harmonic wavelengths.
2. Polychromatic MDOE lenses
In the following the working principle of monochromatic MDOE lenses will be briefly repeated, and then generalized to polychromatic lenses.
In [17, 18] it was shown that a parabolic lens with an optical power adjustable by a mutual rotation between two successive DOEs can be generated, if the two DOEs have corresponding transmission functions T1,218]: Fig. 1(b). The corresponding two phase functions are actually the same, one of them is just flipped upside down.
If T2 is rotated by an angle θ around its center, it transforms intoEq. (4) describes two different parabolic lenses in two sectors, with optical powers f−1 of 18] Figure 2 shows the results of a numerical simulation of the diffraction efficiency of such a sectorless MDOE lens as a function of the mutual rotation angle θ (left: 15°, middle 30°, right 60°) between the two lens elements.
The simulation was performed by defining two transmission functions according to Eq. (7) in a quadratic grid with 1280 × 1280 pixels, assuming a pixel size of 1 μm, a diameter of the circular DOE area of 1.28 mm, and a factor a = 2.6 × 108 m−2. One of the DOEs was numerically rotated with respect to the other, and the common transmission function was obtained by multiplication of the two individual ones. The common transmission function was then propagated (using numerical fast-Fourier-transform based Fresnel propagation) into several distances, and the corresponding diffraction efficiency was evaluated as the intensity in the central focal spot divided by the total intensity. The whole procedure was repeated for a set of wavelengths (since the propagation operation is wavelength dependent), and the corresponding efficiencies are plotted as a function of both wavelength and inverse propagation distance (i.e. optical power). Note that the simulation also incorporates the effect of the numerical aperture (NA) of the lens, i.e. the vertical “thickness” of the efficiency plot becomes narrower with increasing NA, e.g. by increasing the diameter of the simulated lens.
The graphs at the bottom of Fig. 2 indicate cross sections of the corresponding upper graphs along the dashed lines. These dashed lines correspond to the wavelength-dependent diffraction efficiency at the nominal optical powers, which correspond to the respective rotation values according to Eq. (6). Note that with increasing rotation angle, and correspondingly increasing (nominal) optical powers, the dispersion of the optical power (which corresponds to the slope of the efficiency plots in the upper graphs) also increases linearly. Within the visible wavelength range between 400 and 750 nm the MDOE lens has a diffraction efficiency significantly higher than 50%, with a maximum at the design wavelength of 532 nm. The resulting transmission function Tquant,joint is now only a digitized approximation of the ideal parabolic phase function Tparab = exp[iπr2/λf], where the effective number Neff of digitalization levels corresponds to
Note that Neff is not the number of actually used quantization steps during production of the MDOE plates, but the number of effectively realized quantization levels in the combined MDOE. Thus one observes that with increasing rotation angle θ the combined transmission function will correspond to a Fresnel lens in which fewer and fewer quantization levels are expressed, until, at an angle of θ = 180°, it will become a binary lens . The diffraction efficiency of a blazed grating lens depends on the number of quantization levels Neff, on the diffraction order m, and the maximal height of the sawtooth shaped phase profile Φmax (which depends on the maximal etching depth of a DOE) according to 
On the other hand, if we consider a “perfectly” blazed diffractive lens with N → ∞, the corresponding efficiency becomes
Let us now investigate the effect of higher order MDOEs. The phase profiles Φ1,2(r) of two MDOE elements forming a compound lens with included sector according to Eq. (1) are given byEq. (14) the diffraction order m which has maximal diffraction efficiency depends on Φmax as:
The maximal diffraction efficiency (of nominal 100%) in the mth order is reached, if the argument of the round operation equals the integer number m. If we neglect material dispersion (i.e. if we set n(λ) = n0), this happens for a readout wavelength of λm = λ0k/m, where m is any integer number. This, however, means that whenever we reach maximal diffraction efficiency, the optical power is the same, namelyEq. (12), if the argument of the rounding operation in Eq. (20) is not an exact integer number. But as long as the material dispersion is low enough such that
Figure 3 sketches the results of a numerical simulation of a 5th order MDOE lens, with a phase profile described by Eq. (15). Similar to Fig. 2 the efficiency is plotted as a function of wavelength and optical power, for three mutual rotation angles of 15°, 30°, and 60° in three successive columns, respectively. The parameters of the numerical simulation correspond to those used for Fig. 2, assuming a design wavelength of 532 nm, and a diameter of 1.28 mm of the element. However, due to the fact that the MDOE acts as a 5th order lens, the constant a is decreased by the factor 5 (to a = 5.2 × 107m−2), which yields the same dependence between rotation angle and optical power as in Fig. 2. Correspondingly, the pixel size (edge length) is increased from 1μm of the first order element to 5μm of the 5th order element. Thus the ratio between phase amplitude and pixel size is the same in the two cases. The simulation shows that there exist 3 wavelengths, namely 665 nm, 532 nm, and 443 nm, which are the 4th order, 5th order and 6th order harmonics of the fundamental wavelength 2660 nm, where the optical power is the same. This is due to the fact that at each harmonic wavelength the diffractive dispersion is exactly compensated by a jump of the diffraction order.
Figure 4 exemplifies the effect of using an even higher order MDOE lens, namely one of order 10 at a wavelength of 532 nm. The other parameters, like numerical aperture, and the dependence of optical power on the rotation angle have been chosen as in Fig. 3. The simulation shows that within the visible range the diffraction orders 12, 11, 10, 9, and 8 (indicated in the figure) have the same optical powers at their respective wavelengths of 443 nm, 483 nm, 532 nm, 591 nm and 665 nm. At smaller rotation angles, e.g. 15° (see Fig. 4 at the left), the maxima of the efficiency are progressively merging, yielding a more and more uniform optical power in the whole wavelength range. This is due to the fact that with decreasing rotation angle the slope of the efficiency plots (i.e. the dispersion) decreases, and simultaneously also the numerical aperture of the combined lens decreases, which enhances the Rayleigh range (i.e. the axial extension) of the focal spot. Thus, higher order elements might be preferable to cover the whole visible wavelength range. However, as will be discussed later, the fabrication of higher order elements is more challenging, requiring a higher number of phase steps.
Actually Eq. (20) together with Eq. (12) describes the transition between the dispersion behavior of a (first order) diffractive lens up to a bulk refractive lens , which may be seen as a diffractive lens of very high order. The principle how a higher order MDOE lens suppresses dispersion is basically that the (positive) linear dependence of the optical power as a function of the wavelength is compensated by the (negative) linear dependence of the diffraction order in which the beam is reconstructed.
3. Polychromatic MDOE lenses without sector formation
The quantization method described above can be also applied to higher order lenses without sector formation, i.e. to produce a lens with the same optical power within its whole area. A quantized lens with a transmission function according to Eq. (7) may be produced in higher order k by just multiplying the corresponding phase function with the factor k. However, in this case the number of quantization levels within an interval between 0 and 2π is decreased by the factor k, which correspondingly decreases the diffraction efficiency at higher rotation angles, and thus reduces the usable optical power range of the tunable lens. Nevertheless the quantization method can be adapted for higher diffraction orders such that the number of quantization levels (per 2π phase shift) remains constant, namely by setting
For λ = λ0 the difference between the phase functions in the two sectors is still a multiple of 2π, and therefore the corresponding optical powers are equal. The number Neff of quantized phase levels as a function of the rotation angle is now given byEq. (11), setting Neff = 2πk/θ, and thus becomes
This shows that the efficiency increases at lower diffraction orders m (corresponding to larger wavelengths). The corresponding rotation angle θ80% where the efficiency is reduced to 80% is given by approximately ±90°k/m. The tuning range for the optical power within this range of rotational angles is an interval between −k2aλ0/2m and +k2aλ0/2m. But it has to be noted that according to Eq. (25) a phase offset between the two lens sectors in integer multiples of 2π is only exactly obtained at the design wavelength λ0, i.e. only at that wavelength the lens has a uniform optical power without sector formation. At the other harmonic wavelengths the sector will gradually reappear.
Figure 5 shows the results of a numerical simulation of an MDOE lens designed according to Eq. (24). The parameters are the same as those used for Fig. 3, and the only difference is the quantization of the phase profile according to Eq. (24), using a MDOE lens of order k = 5 at the design wavelength 532 nm. As expected, the resulting lens has the same optical power at all harmonic wavelengths (indicated in the figure). However, in this case the efficiency of the optimized 5th diffraction order at the wavelength of 532 nm is significantly enhanced with respect to the non-quantized lens sketched in Fig. 3. This is due to the fact that the 5th order lens (appearing at λ = 532 nm) behaves like a quantized lens whose diffraction efficiency depends on the rotation angle according to Eq. (27), whereas the other diffraction orders correspond to non-quantized lenses (with sector formation), with a diffraction efficiency which is basically given by the area of the suitable sector divided by the total area, i.e. by (2π − |θ|)/2π, as has been shown both theoretically and experimentally in . Due to these different behaviours, the efficiency of the quantized 5th order lens (at λ = 532 nm) exceeds that of the other non-quantized lenses (at the other wavelengths) in a range of rotation angles θ between −125° and +125°. Due to the improved performance of one selectable diffraction order, the total diffraction efficiency (i.e. the sum over the efficiencies of the individual lenses at different wavelengths) of the quantized multi-order lens designed according to Eq. (24) exceeds that of a non-quantized lens (according to Eq. (15)) within the specified range of rotation angles.
We have shown that the concept of MDOE lenses can be extended to higher order lenses of phase shifts of several multiples of 2π. In this case one can choose a set of harmonic wavelengths, at which the optical power of the MDOE lens is the same, independent of the rotation angle, and (up to a certain diffraction order) even independent of material dispersion. The quantization scheme introduced in  for first order MDOE lenses can also be used in a modified form for higher order lenses, thus allowing for a multi-color MDOE lens which avoids sector formation, at least at one selected wavelength.
Comparing multi-order MDOE lenses with first order lenses, one finds that the technical complexities are comparable: In order to cover the same range of optical powers, the etching depth of a kth order lens needs to be increased by a factor of k. However, at the same time the magnification factor a may be decreased by the factor k, since a kth order MDOE lens has also a k-times increased optical power as compared to a single order MDOE lens. Thus the pixel size may be also enlarged by the factor k, without undersampling of the diffractive structures according to Eq. (2). Thus, the ratio between pixel size and etching depth, i.e. the steepness of the etched structures, remains the same. However, an additional complexity of quantized multi-order lenses consists in the fact that the number of etching levels has to be increased, whereas the step size is kept the same. Thus, in order to reach the same efficiency one has to increase the number of phase quantization levels by a factor of k, keeping the same step size. To summarize, multi-order MDOE lenses which cover the same optical power range than a single order MDOE lens need a lower lateral resolution (i.e. a larger pixel size can be used), but a higher number of phase quantization levels.
In our simulations we therefore have investigated MDOEs of the rather low order k = 5, which is the lowest order which yields a constant optical power for at least three wavelengths in the visible range (at 443 nm, 532 nm, and 665 nm). As has been demonstrated in Fig. 4, MDOE lenses with higher diffraction order may be advantageous, since they provide an almost homogeneous optical power distribution in the whole wavelength range. Regarding the results obtained in , where a single order MDOE with high efficiency was demonstrated using 16 depth quantization levels, a kth order DOE will require 16k levels. Since it is currently possible to produce DOEs with 256 quantization levels, one may extrapolate that a polychromatic MDOE of order k = 16 is feasible, which will have an almost homogeneous optical power within the whole visible range, while still keeping the advantageous feature of being a “thin” diffractive element, which has, for example, an etching depth of about 20 μm in fused silica.
The scheme of producing polychromatic MDOE lenses using higher order diffractive optics can be straightforwardly applied to other types of tunable elements, consisting of arbitrary radial phase profiles with rotation symmetry, as for example, to tunable axicons, axicon lenses, higher order lenses, or even oscillating structures (like a “Mexican hat”). Furthermore, the application to tunable devices which are based on a lateral translation between two diffractive elements, like “standard” diffractive Alvarez lenses, is possible. The principle may be even used for tunable elements without rotation symmetry. Such elements, like tunable spiral phase plates , which may also be sectorless , or tunable lens arrays , have already been experimentally demonstrated, and can be designed in an analogous way to work as polychromatic elements in higher diffraction orders.
ERC Advanced Grant 247024 catchIT.
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