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Abstract
Previous work has shown that a set of two diffractive optical elements arranged in series can form a diffractive lens with variable optical power that can be tuned by relative rotation of its two sub-elements about their common central axis. However, previous designs of these diffractive optical elements did not take advantage of the full spatial resolution required for the fabrication process because the corresponding sub-elements consisted of both, regions with very high phase gradients (requiring full resolution) and other extended regions with nearly vanishing phase gradients, where the available resolution is "wasted". Here, an advanced design is proposed that exploits the full spatial bandwidth of the production system. This is done by increasing the polar (angular) phase gradient of each sub-element such that it approaches the usually much larger radial phase gradient. A pair of these new sub-elements then composes a diffractive lens that has the same tuning range of its optical power than a standard tunable diffractive lens, but advantageously achieved within a much smaller relative rotation range. This has advantages in systems where high speed tuning of optical power is required, and in systems where the rotation angle is limited.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tunable diffractive optical elements based on a pair of two individual diffractive elements in series are based on independent ideas of Alvarez [1] and Lohmann [2]. There it was suggested that a tandem pair of diffrative optical elements (DOEs) can form a diffractive lens (Fresnel lens) with adjustable optical power by translating one element with respect to the other. It was also proposed that a tunable lens can be obtained by a relative rotation of two sub-elements [3]. Later, diffractive rotationally tunable systems have been investigated in more detail both theoretically [4] and experimentally [5]. A sketch of the corresponding optical arrangement is shown in Fig. 1.
Fig. 1. Left side: Principle of a rotationally tunable diffractive optical lens: Two identical, specially designed DOEs, are arranged "face-to-face" (with the structured sides pointing to each other) directly behind each other. One or both of them can be rotated around the center of their common optical axis. The combined "MDOE" acts as a lens with an optical power which can be adjusted in a wide range as a linear function of the relative rotation angle. Right side: Phase map of one of the two identical sub-DOEs. The phase values in a range between 0 and $2\pi$ are indicated by the colorbar.
It was already noted in the earlier work that the underlying principle of these systems is related to the moiré effect. Thus the corresponding elements have also been called moiré diffractive optical elements (MDOEs). The MDOE principle has later been used in the field of diffractive optics [6–10], and more recently by employing optical metamaterials [11–20]. First commercially available MDOE lenses have been used, for example, as an enabling technology for atom trapping experiments [21]. More background information, and an overview of these systems can be found in a recent tutorial [22].
The phase map of one of the two individual DOEs (a "sub-DOE") of an MDOE lens is plotted in Fig. 1. It consists of regions with both low and high spatial frequencies at the upper and lower parts of the element, respectively. The necessary resolution of the manufacturing process of such an element (i.e. the "feature size", which often corresponds to the pixel size in digital fabrication system) is determined by the highest spatial frequency region of the DOE. In the low spatial frequency regions, the available resolution is barely used, i.e. there is a "waste" of the available spatial bandwidth. In the following it will be shown that the corresponding DOEs can be designed with a modified phase structure, which makes optimal use of the bandwidth. Using the same spatial resolution as that required for the standard sub-DOEs, it is possible to create a new "multiplexed" MDOE lens, which can be tuned in the same optical power range, and with the same efficiency as a standard MDOE lens, but now the full optical tuning range can be accessed by rotating the new multiplexed MDOE lens in a considerably lower angular range.
2. Standard MDOE lenses
In the following some basic results for the construction of a standard MDOE lens are briefly summarized. More detailed information is available in [5,22].
In polar coordinates $(r,\varphi )$, the two sub-DOEs of a standard MDOE lens have transmission functions of
where $a$ is a user-defined parameter, and $\mathrm {round}\{\cdots \}$ is a rounding operation to the nearest integer value.An exemplary phase map of one of the sub-DOEs is sketched in Fig. 1 at the right side. Note that the second sub-DOE of a MDOE lens is identical to the first one, since it is flipped in order to arrange the two DOEs face-to-face in the corresponding MDOE lens. This corresponds to a mirroring operation, which switches the sign of the phase factor, due to the symmetry of the elements.
The transmission function $T_\mathrm {mdoe}$ of the resulting MDOE lens is obtained by multiplying the two transmission functions $T_{+1}$ and $T_{-1}$. If $T_{-1}$ is additionally rotated with respect to $T_{+1}$ by an angle $\theta$, the corresponding transmission function becomes
This approximately corresponds to the transmission function of a parabolic lens ($T_\mathrm {parab}=\exp \left [i \, \pi P r^2 /\lambda \right ]$, where $P$ is the optical power of the lens, and $\lambda$ is the readout wavelength) with an optical power (inverse of the focal length) of
Note that $\theta$ can be both positive and negative, i.e. depending on the rotation direction the corresponding MDOE lens can have a positive or a negative optical power.The rounding operation $\mathrm {round}\{\cdots \}$ in Eq. (1) is necessary in order to obtain a MDOE lens with a homogeneous parabolic lens profile, i.e. it suppresses the formation of a sector with another optical power $P_2=(\theta -2\pi ) a \lambda /\pi$, which otherwise would be formed in the region overlapped by the mutual rotation angle $\theta$ [4].
However, due to the rounding operation the transmission function of the resulting MDOE lens (Eq. (2)) is digitized in radial direction, and it is therefore only an approximation to a perfectly smooth parabolic lens. As a result the MDOE lens becomes bifocal, i.e. it simultaneously has a positive and a negative optical power (with a constant spacing of $\Delta P=2 a \lambda$). The two lens terms of the bifocal lens have different diffraction efficiencies, which change as a function of the mutual rotation angle. Within a relative rotation range between +90$^{\circ }$ and −90$^{\circ }$, the efficiency of one of the bifocal lenses theoretically exceeds 81%, whereas that of the other lens remains below 9%. The remaining intensity is distributed between higher diffraction orders. Thus it is usually recommended to limit the usable rotation range to an interval between +90$^{\circ }$ and −90$^{\circ }$.
Equation (3) shows that the user-defined parameter $a$ determines the change in optical power (corresponding to the inverse of the focal length) $P$ of the MDOE lens as a function of the mutual rotation angle $\theta$. However, the parameter $a$ cannot be set to an arbitrarily high value. In order to avoid undersampling of the phase profile, the phase gradient of the two sub-DOEs (Eq. (1)) should not exceed the resolution of the manufacturing system. For example, the resolution of a digital production system is limited by the size $p$ of its smallest pixel. Then a minimal requirement to avoid undersampling would be that any local phase variation with a height of $2 \pi$ should be approximated by at least two pixels, - better by $N$ pixels, where $N \ge 2$. In previously produced MDOE lenses a factor $N=4$ has demonstrated good results.
In mathematical notation the corresponding condition reads
where $\Phi (r,\varphi )$ is the phase profile of a sub-DOE. This condition may be separated into two conditions concerning the radial and the polar phase gradients of the DOEs in Eq. (1). In the following estimation the effect of the rounding operation is ignored by setting $\mathrm {round}\{a r^2\} \approx a r^2$, which is a good approximation at the outer boundary of a DOE, where $r$ approaches its maximal value $r_\mathrm {max}$. Obviously, the two radial and polar phase gradients have their maximal values at the highest values of $r$ (which is $r_\mathrm {max}$) and of $\varphi$ (which is $\pi$, since $\varphi$ is defined in the interval between $-\pi$ and $\pi$). Then the sampling condition for the radial phase gradient becomes whereas that for the polar component becomes These constraints show that the maximal value of the factor $a$ is limited by the maximal radius of the MDOE lens $r_\mathrm {max}$, by the smallest digital pixel size $p$ (limited by the production system), and by the sampling factor $N\ge 2$ (desired number of resolved pixels per $2\pi$ phase interval).Furthermore, a comparison of the two equations shows that the condition imposed by the radial phase gradient (Eq. (5)) is more stringent due to the additional factor of $2\pi$. This observation suggests that it might be possible to increase the bandwidth utilization of a MDOE lens if one could increase the phase gradient of the polar component until it approaches that of the radial component. In the following this idea will be expanded to become the principle of a "multiplexed MDOE lens".
3. Multiplexed MDOE lenses
In order to increase the polar phase gradient, a naive approach would be to increase the polar coordinate $\varphi$ by a factor $s$, i.e. to substitute $\varphi$ in Eq. (6) by $s \varphi$. However, this will not work, since the factor $s$ would equally increase both, the radial and the polar phase gradients. Thus, an increase of the polar phase gradient has to be done such that the angular range of $\varphi$ remains limited to a $2\pi$-interval. This can achieved by the substitution $\varphi \rightarrow \mathrm {mod}_{ 2 \pi }\{s \varphi \}$, where $s$ is an integer number.
The corresponding transmission functions for the two new sub-DOEs of a "multiplexed" MDOE lens then becomes
Due to the modulo operation, the polar angle term is always limited to a $2 \pi$ interval. Thus, there will be only $2 \pi$ phase jumps within the phase profile, which do not disrupt the transmission function, since its phase is only defined modulo-$2 \pi$. Integer multiplexing factors $s$ are required for a periodic behavior of the angular sawtooth phase modulation function $\mathrm {mod}_{2 \pi }\{s\varphi \}$ within the polar angle range between 0 and $2\pi$. Non-integer factors $s$ would lead to a phase discontinuity at the transition line between the polar angles 0 and $2 \pi$, which distorts the symmetry of the sub-elements and leads to the appearance of two different sectors (with different optical power) in the combined MDOE lens, similar to the sector forming standard MDOE lenses described in [4].
The term "multiplexed" MDOE is suggested by the fact that the transmission functions $T_\mathrm {mult, \pm 1}$ can also be obtained by a subsequent multiplication of the transmission functions of $s$ standard sub-DOEs (according to Eq. (1)), which are mutually rotated by 360$^{\circ }/s$.
The corresponding two sub-elements still have the mirror symmetry of the standard sub-DOEs, namely by flipping one of them upside down one obtains the transmission function of the other. Therefore, a multiplexed MDOE lens still consists of two identical sub-DOEs, which are arranged face-to-face.
If one of the two sub-DOEs is rotated by an angle $\theta$, the argument $\varphi$ in Eq. (7) has to be replaced by $\varphi -\theta$ for one of the sub-elements. A multiplication of the two sub-DOEs for obtaining the transmission function of the combined MDOE lens, $T_{\mathrm {MDOE},\theta }$, then yields:
In analogy with the results obtained for quantized MDOE lenses in [22], Eq. (35), where $\theta$ is replaced by $\mathrm {mod}_{2\pi }\{s \theta \}$), the corresponding optical power of the two lens terms within a bifocal $s$-multiplexed MDOE lens is given by:
For demonstration, a gallery of the phase profiles of sub-DOEs with different factors $s=1,2,3$ and 6 is plotted in Fig. 2. The left column shows one of the two sub-elements, whereas the second column shows the corresponding conjugate sub-elements, which are rotated by 60$^{\circ }$, 30$^{\circ }$, 20$^{\circ }$ and 10$^{\circ }$, respectively. The right column shows the phase profile after multiplication of the transmission functions of the two sub-elements, i.e. the phase profiles of the corresponding MDOE lenses. As expected, all resulting MDOE lenses have the same phase profiles, although they are obtained by different rotation angles of the corresponding sub-elements.
Fig. 2. A gallery of sub-DOEs for "multiplexed" MDOE lenses with different multiplexing factors $s=1,2,3$ and 6 within the four rows. The first column shows the phase profile of one of the two sub-DOEs. The second row shows the corresponding conjugate sub-DOEs, which are additionally rotated by angles of 60$^{\circ }$, 30 $^{\circ }$, 20$^{\circ }$ and 10$^{\circ }$, respectively. The last column shows the phase profile of the corresponding multiplexed MDOE lenses, obtained by multiplication of the transmission functions of the two mutually rotated sub-DOEs. Obviously, the resulting MDOE lenses are identical, although obtained by different relative rotation angles of the corresponding sub-DOEs.
Obviously, an increasing value of $s$ leads to an increase of the spatial frequency (i.e. the bandwidth) in all areas of the sub-elements. Whereas the $s=1$ element (corresponding to a sub-element of a "standard" MDOE lens) shows regions with very low phase modulation (upper part of the element), these low modulation regions are progressively reduced with increasing $s$ factors. Actually, the maximal modulation frequency in radial direction is still the same in all $s$-multiplexed sub-elements, but the phase gradient (or spatial frequency) in polar direction is now increased by the factor $s$.
In order to obtain the (almost) same maximal phase gradients in radial and in polar directions, a factor of $s=6$ is optimal, since it increases the polar gradient by about the factor $2 \pi$, which is the difference between the radial and polar phase gradients in a sub-elements of a standard MDOE lens (Eqs. (5) and (6)). This behavior is investigated in Fig. 3 in more detail. The left column shows the phase profile of a sub-element of a standard MDOE lens. The region with the highest modulation at the bottom of the element (indicated by a black box) is magnified and shown below. An analogous plot for a ($s=6$)-multiplexed sub-DOE is shown at the right side. Comparing the magnified regions of the highest spatial frequencies, it can be observed that the radial phase variations (i.e. the distance between the radial phase jumps) have the same spacing. However, for the case of the standard sub-DOE (left), the phase gradient in polar direction is by a factor of $2\pi$ lower than that in the polar direction, as expected by Eqs. (5) and (6). On the other hand, the polar phase gradient of the $s=6$ sub-DOE is on the order of the corresponding radial gradient, since it is increased by the factor $s=6$. Thus, the $s=6$ sub-element has actually the same maximal phase gradient as the standard sub-DOE, and can be produced without the need for a better resolution of the manufacturing system.
Fig. 3. Comparison of the highest spatial frequency regions of a sub-DOE of a standard MDOE lens (left column) and that of a ($s=6$)-multiplexed MDOE lens (right column).
A numerical simulation of the diffraction efficiencies and the corresponding optical power as a function of the relative rotation angle is shown in Fig. 4 for a standard MDOE lens ($s=1$) and a multiplexed MDOE lens ($s=6$). The optical power is plotted according to Eq. (10) in units of $2 a \lambda$ (which has the dimension of diopters), where $a$ is the user defined production parameter (limited by Eq. (5)), and $\lambda$ is the design wavelength. The data of the standard lens (left column) shows that at zero rotation angle (center of the plot) one of the two lens terms within the bifocal MDOE lens has its maximal efficiency (blue) at an optical power of zero. With increasing rotation angle, the efficiency decreases harmonically, whereas the optical power increases linearly. At the same time the efficiency of the other lens term (green) increases harmonically from zero, whereas the absolute value of its (negative) optical power also increases linearly, however, after having started from its maximal (negative) value. At a rotation angle of 360$^{\circ }$ the diffraction efficiency of the first lens term (blue) is zero, whereas that of the second term is maximal. Regarding the corresponding optical powers of the two lens terms, one sees that the two lenses have switched their roles, i.e. the diffraction properties of a standard MDOE lens have a 360$^{\circ }$-periodicity.
Fig. 4. Diffraction efficiency (upper row) and optical power (lower row) of a standard MDOE lens ($s=1$, left column) and a $s=6$ multiplexed MDOE lens (right column) as a function of the relative rotation angle $\theta$ between the respective sub-DOEs. The data for the two main diffraction orders is indicated in blue and green color, respectively.
At the right column the same calculations are performed for a ($s=6$)-multiplexed MDOE lens. As expected, it has the same diffraction characteristics as the standard MDOE lens (i.e. the same diffraction efficiency at the same optical power adjustments), but now with a periodicity of 360$^{\circ }$/$s$ (where $s=6$). For example, the $s=6$ multiplexed MDOE lens has to be rotated only in a range between −15$^{\circ }$ and +15$^{\circ }$ in order to achieve the same variation of the optical power as compared to a standard MDOE lens ($s=1$), which is rotated in the recommended range between −90$^{\circ }$ and +90$^{\circ }$.
The simulations of the diffraction efficiency were done for the MDOE lenses composed by the sub-elements displayed in Fig. 3. The transmission function of each sub-element $T_{\mathrm {mult}, \pm 1}(x,y)$ was calculated according to Eq. (7) as a complex two-dimensional array in cartesian coordinates $(x,y)$ by using the substitutions $r^2 \mapsto (x+y)^2$ , and $\varphi \mapsto \mathrm {arg}(x+iy)$ (complex argument). The size of the array was $400 \times 400$ pixels, with a pixel size of $p=$ 1 $\mathrm {\mu }$m, corresponding to a DOE diameter of 400 $\mathrm {\mu }$m. In order to obtain a circular shape of each DOE, the amplitude of the transmission function was set to zero in regions outside a disc-shaped area with the desired radius. Other parameters were $a=9.0 \times 10^8 \mathrm {m}^{-2}$, and $\lambda =532$ nm (design wavelength). In the next step, one of the two sub-DOEs was numerically rotated by an angle $\theta$, and the transmission function of the corresponding MDOE lens, $T_{\mathrm {MDOE}, \theta }(x,y)$, was calculated by a pixel-wise multiplication of the two sub-DOEs. Furthermore, the expected optical powers $P_{\pm 1}(\theta )$ of the two lens terms within the resulting bifocal MDOE lens were calculated according to Eq. (10). Finally the diffraction efficiencies of the two lens terms were numerically calculated according to:
This method, where only the zero Fourier component of a final transmission function is evaluated, is typically more accurate than the frequently used method of a numerical, fast-Fourier-transform (FFT) based propagation of the electric field into a selected focal plane, followed by a summation of the intensity in a user-defined region around the focal spot, since the respective numerical propagation operators (e.g. angular spectrum of plane wave propagator) may suffer from numerical errors.
It should be noted, however, that this method does not incorporate details, like for example diffraction effects at the edges of the finite lens elements. Thus it almost exactly reproduces the theoretical function of the diffraction efficiency described by Eq. (11).
A practical example for a realizable ($s=6$)-multiplexed MDOE lens might have the following parameters: Radius $r_\mathrm {max}=$ 5 mm, minimal pixel size (resolution) $p=$ 0.5 $\mathrm {\mu }$m, design wavelength $\lambda =$ 532 nm, required sampling factor $N=$ 4 (minimal number of resolved pixels per $2\pi$ phase interval). According to Eq. (5) the user defined factor $a$ is then calculated to be $a=10^8$ m$^{-2}$. If the mutual rotation angle of the two sub-DOEs is limited to an interval between −15$^{\circ }$ and +15$^{\circ }$, then, according to Eq. (10), the corresponding tuning range of the optical power would cover an interval between −26.6 dpt and +26.6 dpt (corresponding to focal lengths of approximately $\pm 4$ cm to infinity), whereas the ideally obtainable diffraction efficiency of the combined ($s=6$)-multiplexed MDOE lens would vary between 81% and 100%.
It should be noted that the current principle of bandwidth increase can be adapted to earlier demonstrated tunable MDOE systems, which do not avoid the formation of different sectors with different optical properties (see e.g. [5]). Such a sector-forming MDOE lens is obtained if the round-operation in Eq. (7) is omitted. In this case the resulting MDOE lens would consist of $2s$ sectors with alternating optical power. For example, if two sub-DOEs of a $s$-multiplexed MDOE lens (without rounding operation) would be mutually rotated by an angle of $\theta$, $s$ equidistantly spaced (360$^{\circ }/s-\theta$)-sectors would be formed corresponding to lenses with an optical power of $P_1=\mathrm {mod}_{2\pi }\{s \theta \} a \lambda /\pi$ (Eq. (10)), whereas the other $s$ sectors (each covering an angular range of $\theta$) would correspond to lenses with an optical power of $P_2= \mathrm {mod}_{2\pi }\{s (\theta -2\pi )\} a \lambda /\pi$. If the corresponding diffraction patterns of standard ($s=1$) sector-forming MDOE lenses are compared with those of multiplexed sector-forming MDOE lenses, it turns out that the focusing properties of the multiplexed lenses are more homogeneous, due to their increased rotational symmetry.
4. Discussion
Compared to standard tunable MDOE lenses, multiplexed MDOE lenses provide the same tuning range of the optical power within a reduced relative rotation angle range. A multiplexing factor $s=6$ makes optimal use of the available bandwidth, since in this case the distance between radial and polar (angular) phase jumps is approximately equal, such that the maximal spatial resolution of the DOEs does not change.
The reduction of the rotation range can be advantageously used to increase the tuning speed. This is highly desired for applications in laser based material processing systems ("scan-heads"), or in scanning microscopy applications. Furthermore, the reduced rotation range allows to employ rotation systems which are mechanically limited to a reduced angular range. For example, the maximal rotation range of a galvo scanner is typically limited to an interval between −15$^{\circ }$ and +15$^{\circ }$, however at a speed in the 10-50 kHz range. Thus, a suitably modified galvo mount for a multiplexed MDOE lens would allow to tune the focal length within (some tens of) microseconds within a large optical power range (for example, within an accommodation range of 50 dpt for a 1 cm diameter lens, as outlined in the previous example).
The principle of decreasing the rotation range by increasing the polar phase gradient of the sub-elements using the modulo-$2\pi$ operation (as in Eq. (7)) can be straightforwardly applied to other tunable diffractive elements, which have been proposed in previous work. This includes tunable axicon lenses, or tunable aberration generators, which are used to compensate for aberrations induced by other optical components. For example, a tunable MDOE lens which simultaneously produces variable spherical aberrations has been employed in [23] in order to shift the axial focus position of a high numerical aperture microscope objective lens without loss in axial resolution, which usually would be unavoidable.
Furthermore, the same principle can be straightforwardly adapted to produce higher order multiplexed MDOE lenses, which would have phase modulation profiles covering a phase range of multiples of $2\pi$. These higher-order MDOE lenses have been shown to simultaneously operate at various wavelengths within the visible range, having the same optical power at these wavelengths [24]. Using the multiplexing principle, also these higher order lenses would operate within a reduced rotation angle range without any reduction of their efficiency.
Disclosures
The author declares no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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