Abstract

A pair of adjacent transmissive diffractive optical elements (DOEs) forms a combined DOE with tunable optical properties, as, for example, a diffractive lens with an adjustable focal length. The optical properties are controlled by a relative movement of the two DOEs, such as a translation or a rotation around the optical axis. Here we discuss various implementations of this principle, such as tunable diffractive lenses, axicons, vortex plates, and aberration correction devices. We discuss the limits of the tuning range and of diffraction efficiency. Furthermore, it is demonstrated how chromatic aberrations can be suppressed by using multi-order DOEs.

© 2021 Optical Society of America

1. INTRODUCTION

The history of diffractive optical elements (DOEs) started in the 18th century, when the first studies of optical diffraction gratings were performed by David Rittenhouse (1732–1796) (an overview is found in [1]). In 1922, these diffraction gratings were used for the first time by V. Ronchi for interferometric tests of optical systems [2]. A further precurser of modern diffractive optics was the invention of lighthouse lenses with the structure of a sliced and re-wrapped bulk lens by Augustin-Jean Fresnel, which was first installed in the Cardovan Tower lighthouse (France) in 1822. Although at that time such a Fresnel lens was based on purely refractive effects, its structure was a forerunner of modern diffractive lenses, which have similar structures on a micrometer scale. Consequently, the diffractive versions of these lenses are also now frequently called Fresnel lenses. Actually, it can be shown that a continuous transition between diffractive and refractive optical elements happens, if the structure size (height and period) of a diffractive element is continuously increased. In particular, this also accounts for the dispersion properties of the corresponding element, changing from the negative dispersion of a diffractive element (which is independent of its refractive index) to the positive dispersion of refractive elements (determined by the wavelength dependence of the refractive index) [3].

Nevertheless, the actual beginning of the history of DOEs is usually considered to be in the late 1960s, shortly after the invention of the laser, and during the booming years of holography. At that time, diffractive optical lenses were called “kinoform lenses,” or just “kinoforms” [4,5]. The basic principle for construction of a kinoform is sketched in Fig. 1.

 figure: Fig. 1.

Fig. 1. Construction principle of a kinoform lens. (a) Surface profile of a refractive lens. (b) Surface profile wrapped by a modulo-operation into an almost flat structure with a sawtooth profile. (c) Corresponding phase transmission image. Gray levels correspond to phase shifts impressed on a transmitted wave.

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One starts with the surface profile of a well-known refractive optical component, such as a lens [Fig. 1(a)], which is then wrapped into an almost flat structure by applying a modulo operation [5] [Fig. 1(b)]. This yields blazed grating structures with a sawtooth profile, which diffract, in principle, with a diffraction efficiency of almost 100%. The corresponding phase profile is sketched as a two-dimensional image in Fig. 1(c), where gray levels correspond to the phase shift, which is acquired by a transmitted wave. These sawtooth phase structures are not obtained in classical holography, where symmetric harmonic structures are recorded, which diffract the light symmetrically into conjugate diffraction orders (“Friedel’s law” [6]). A disadvantage of a DOE with respect to a corresponding refractive element is its strong dispersion. For example, the optical power (i.e., the inverse of the focal length) of a DOE lens depends linearly on the readout wavelength. Furthermore, its diffraction efficiency decreases if the readout wavelength does not correspond to the intended design wavelength.

Another achievement during that time was the development of focus-tunable lenses by Luis Walter Alvarez [7], and later independently by Adolf Wilhelm Lohmann [811]. A sketch of such a lens (taken from the original patent of Alvarez [7]) is shown in Fig. 2. An Alvarez lens is composed of two specially designed adjacent refractive optical elements, which are complementary with respect to each other, i.e., one of them has, up to a constant offset, the negative surface profile of the other (upper part of Fig. 2). The combined optical element is obtained by placing one of the elements on top of the other (lower part of Fig. 2). By laterally shifting one of the elements with respect to the other, the combined element acts as a spherical lens with an optical power that depends linearly on the displacement. This type of focus-tunable lens is called an Alvarez lens, or Alvarez–Lohmann lens. Currently, the principle is used, for example, to produce adaptive eyeglasses with a manually adjustable optical power (e.g., by the company Adlens).

 figure: Fig. 2.

Fig. 2. Principle of an Alvarez lens (sketch taken from the original patent of Alvarez [7]).

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Even earlier, it was recognized by A. W. Lohmann and D. P. Paris that diffractive versions of focus-tunable lenses can be obtained by utilizing the moiré effect [9]. However, at that time the manufacturing of high quality structured phase plates was not possible, and thus the principle was demonstrated by means of superposed binary absorptive Fresnel zone plates, which have a low diffraction efficiency. It was also suggested that rotational versions of these tunable lenses are feasible using an arrangement as sketched in Fig. 3.

 figure: Fig. 3.

Fig. 3. Combined DOEs (not to scale), whose optical properties (such as an adjustable focal length) are tunable by a relative rotation of the two sub-elements. (Systems discussed in detail in the tutorial.)

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By rotating one of the elements with respect to the other around the central optical axis, they produce the transmission function of a spherical lens with a variable optical power, which depends linearly on the mutual rotation angle.

In the more recent past, the fabrication of DOEs became more advanced, using diamond turning techniques, lithographic methods originating from semiconductor processing, direct laser ablation, or laser induced polymerization of photosensitive resins. At the same time, replication of DOEs for mass production became feasible, using similar molding techniques as those developed for compact disc fabrication. During that time, new companies evolved, producing DOEs for various purposes, like diffractive lenses, specialized light diffusers, helical phase plates (producing Laguerre-Gaussian beams, or Bessel beams), or as pattern projectors for alignment and entertainment purposes.

 figure: Fig. 4.

Fig. 4. Diffractive moiré lens with a focal length tunable by rotation. (a) Phase profile of one of the two sub-elements (not to scale). Gray levels correspond to phase values in an interval between zero and $2\pi$. (b) Photograph of one of the two sub-elements. (c) Magnifying lens effect of the combined moiré lens placed some millimeters above a text.

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In 2013 the original idea of Lohmann and Paris [9] of producing rotationally tunable diffractive moiré lenses was demonstrated by means of diffractive phase elements [12]. Figure 4(a) sketches the phase profile of one of the two sub-elements of such a moiré lens (not to scale, due to a clearer representation), whereas Fig. 4(b) displays a photographic picture of the lithographically produced element. In Fig. 4(c), two of these elements have been combined into an adjustable moiré lens (rotated with respect to each other), and placed a few millimeters above a text. The corresponding picture shows that the combined element acts as a lens, which magnifies the underlying part of the text. The focal length of the lens can be adjusted by changing the rotation angle between the two sub-elements. It was shown that the same principle can also be employed to produce various other DOEs with adjustable optical properties, like axicons, phase shifters, or helical phase plates [13,14]. Due to their common principle of operation, which is based on the moiré effect [9], these tunable DOE combinations were called moiré lenses, or more generally, moiré optical elements.

Here it has to be mentioned that the field of tunable optical elements, particularly of tunable lenses, has been widely extended in the recent years. There are various approaches to obtain tunable lenses, like mechanically actuated shape-changing lenses based on a combination of optical fluids and a polymer membrane, which are already commercially available (companies Optotune, Nextlens). Other lens types are based on the electro-wetting principle, changing the meniscus between two non-mixing fluids (water/oil) by the application of an electric field (company Varioptic). A recent review article on electrically tunable lenses may be found in [15]. Quite recently, new developments were reported, based on the electric-induced deformation of electroactive polymers, which changes the meniscus of an embedded soft lens by stretching [16,17], or even by temperature control [17]. Furthermore, there are approaches that directly make use of the electroactive adhesive forces applied at special transparent polymer gels, which changes their shape in an external electric field generated by appropriately arranged electrodes [18,19]. Finally, electrically tunable lenses have also been realized with versatile spatial light modulators [20], or with specialized liquid crystal devices. A recent overview over liquid crystal based systems may be found in [21].

A detailed comparison of the advantages and disadvantages of the different realizations of focus-tunable lenses is out of the scope of this tutorial, due to the large number of reported different approaches, and since a comparison comprises a large number of different properties, whose importance depends on the intended applications. For example, important parameters of tunable lens systems include the achievable tuning range, tuning speed, optical quality, light intensity threshold, realizable numerical aperture (NA), size and clear aperture of the elements, or the stability against environmental conditions (inertial forces, temperature, pressure, etc.).

There are several features of diffractive moiré lenses that are advantageous in practical applications. Tunable moiré lenses are lightweight thin elements suited for high laser powers, they provide (almost) diffraction limited focusing within a large tuning range, they introduce no undesired Petzval field curvature and no barrel or pincushion field distortions, they are stable with respect to environmental conditions (temperature, pressure, vibrations), and they keep the focus exactly on the optical axis during refocusing. Therefore, the basic operation principle of moiré lenses is increasingly finding its way into other fields of optics, such as the design of tunable optical elements using nano-structured meta-materials [2230].

The present tutorial will primarily focus on the properties of moiré elements whose optical properties are tunable by a mutual rotation of their sub-elements [3137]. These have some advantages with respect to optical elements tuned by a mutual translation, such as Alvarez–Lohmann lenses. Tuning of a moiré lens by mutual rotation does not require additional (transverse) space, which improves the compactness of a corresponding optical system. Furthermore, it does not reduce the free aperture of the combined moiré lens, whereas the aperture of an Alvarez–Lohmann lens is limited to the overlapping part of its two sub-elements. A further advantage is that the focus of a rotationally tunable moiré lens exactly stays on the optical axis during the tuning operation. This is hard to achieve with Alvarez–Lohmann lenses, since in this case, the two-sub elements would have to be moved exactly symmetrically in opposite directions.

Thus in the following, the general principle of rotationally tunable moiré elements will be presented, and some special realizations, such as tunable lenses, modified lenses for aberration correction, axicons, and helical phase plates, will be discussed.

An issue, which was not reported in the first suggestions of rotationally tunable moiré lenses, is the formation of two different sectors with different optical properties within the lens aperture. It will be shown how this issue can be solved by a quantization of the phase profiles of DOEs. Another problem is the strong dispersion of moiré elements, i.e., the dependence of the optical power and diffraction efficiency on the light wavelength. This problem will be tackled by using higher order DOEs, which have a surface profile with a larger modulation amplitude.

2. BASIC PRINCIPLE OF TUNABLE REFRACTIVE LENSES

After the invention of tunable optical elements by a mutual displacement of two adjacent sub-elements by Alvarez and Lohmann, the operational principle of these elements was generalized and further analyzed in a series of publications, as, for example, in [10,11,3844]. It was recognized that any two-dimensional surface profile $H(x,y)$ can be obtained as a “subtractive moiré pattern” [39] by superposing two sub-profiles ${H_1}(x,y)$ and ${H_2}(x,y)$, which are conjugate with respect to each other, i.e., ${H_1} = - {H_2}$ (up to a constant offset). The amplitude of the resulting surface profile is then adjustable by a mutual movement of the two sub-elements. The height profiles of these sub-elements are calculated by integrating the desired surface profile $H(x,y)$ along the path that corresponds to the intended direction of the mutual movement. For example, in a cartesian coordinate system $(x,y)$, a plano–convex (positive) parabolic lens has a surface height profile $H(x,y)$ of

$$H(x,y) = {H_0} - \frac{1}{{2f(n - 1)}}\left({{x^2} + {y^2}} \right),$$
where $f$ is the focal length of the lens, and $n$ is the refractive index of the lens material. ${H_0}$ is a constant offset chosen such that the surface height is always positive within the lens area. Since ${H_0}$ is constant, it can be omitted in the following considerations without loss of generality. The negative sign of the modulated part is due to the fact that a positive lens (with positive focal length) has its largest thickness in its center, and falls off parabolically with increasing distance from the optical axis.

Note that the profile of the parabolic lens in Eq. (1) differs from that of an ideal (hyperbolic) lens, which would provide an optimal, diffraction limited focus. The profile of a hyperbolic lens would be ${H_{{\rm hyp}}} = - {(n - 1)^{- 1}}{[(f/n{)^2} + {r^2}]^{1/2}}$ (up to a constant offset). However, as will be shown later, the working principle of a tunable Alvarez lens just scales the phase profile by multiplying it with a variable factor. Such a multiplication does not preserve the hyperbolic profile, i.e., to actually change the focal length, the $f$-dependent term within the square root would have to be multiplied by such a scaling factor. On the other hand, the parabolic lens profile of Eq. (1) is preserved by the scaling operation, which always yields a parabolic lens whose focal length changes as an inverse function of the scaling factor. Since the achievable NA (without loss of diffraction efficiency) of diffractive lenses is rather moderate (${\rm NA} \lt 0.2$), the parabolic lens profile is still a good approximation of the optimal hyperbolic profile, and maintains an almost diffration limited point spread function. Thus, in the following, a parabolic lens profile will be investigated.

To obtain the surface profiles of two sub-lenses, which together form such a tunable parabolic lens by a mutual translation in $x$ direction, the desired surface profile $H(x,y)$ has to be integrated along the $x$ direction. One then obtains the height profile ${H_1}(x,y)$ of one of the sub-elements, whereas the other element ${H_2}(x,y)$ will be the negative of ${H_1}(x,y)$. The integration yields

$$\begin{split}{H_1}(x,y) &= - a\int_0^x \frac{1}{{2f(n - 1)}}\left({{x^2} + {y^2}} \right){\rm d}x\\& = - a\frac{1}{{2f(n - 1)}}\left({\frac{1}{3}{x^3} + x{y^2}} \right),\end{split}$$
where $a$ is a selectable constant (with the dimension of $m^{- 1}$), which is proportional to the tuning range of the combined element. ${H_1} \propto {x^3}/3 + x{y^2}$ is the well-known surface profile of the sub-elements of an Alvarez lens. The surface profile of the independently discovered Lohmann sub-lens (${H_1} \propto ({x^3} + {y^3})$) is actually just a 45°-rotated version of the Alvarez lens. Correspondingly, the mutual shift of two Lohmann sub-elements has to be performed in the diagonal direction.

The working principle of such an Alvarez lens is verified by analyzing the total surface profile ${H_{{\rm tot}}}$ of the two sub-elements, assuming that each element is shifted by an amount $\Delta x/2$ into the positive and into the negative $x$ directions. This yields

$$\begin{split}{H_{{\rm tot}}} & = - a\left[{{H_1}(x + \Delta x/2,y) - {H_1}(x - \Delta x/2,y)} \right]\\ & = - a\frac{1}{{2f(n - 1)}}\left[{\frac{1}{3}{{(x + \Delta x/2)}^3} + (x + \Delta x/2){y^2}) - {{(x - \Delta x/2)}^3} - (x - \Delta x/2){y^2})} \right]\\ & = - \frac{1}{{12}}\Delta {x^3} - \Delta xa\frac{1}{{2f(n - 1)}}\left({{x^2} + {y^2}} \right).\end{split}$$
The result is the sum of a constant offset (${-} \Delta {x^3}/12$), and of the surface profile of a scaled parabolic lens [according to Eq. (1)], with a new focal length ${f_{{\rm new}}} = f/(\Delta xa)$, which is proportional to the mutual lens shift $\Delta x$. If the shift of the sub-elements is not performed symmetrically in opposite directions, then the result will correspond to a parabolic lens that is laterally displaced from the optical axis, which will therefore produce an additional tilt of the transmitted beam.

This principle can be simplified in numerical calculations. In this case, one can define the desired lens profile $H(x,y)$ as a two-dimensional numeric array of surface heights (for example, such an array can be handled in MATLAB as an “image”), and then numerically calculate the cumulative sum of the array along the desired shifting direction (for example, this corresponds to the “cumsum” operation in MATLAB). Thus one gets the surface height array of one of the appropriate sub-lenses ${H_1}$, and the other is just its negative, i.e., ${H_2} = - {H_1}$.

The same principle can be applied to produce diffractive tunable elements. This is done by calculating the surface profiles ${H_1}(x,y)$ and ${H_2}(x,y)$ according to the described method, and then applying the kinoform principle, by applying a modulo operation at the two sub-elements. If the modulo operation is performed such that the maximal remaining surface height (i.e., the height of the corresponding sawtooth structure) corresponds to a phase shift of $2\pi l$ (where $l$ is an integer value) of a transmitted wave with a certain wavelength $\lambda$, then one obtains an $l$th order diffractive element, which, in principle, has a 100% diffraction efficiency for the design wavelength $\lambda$ in the $l$th diffraction order. In this case, the combined diffractive element will have the (almost) same optical properties as the underlying refractive tunable lens, however only at the designated wavelength $\lambda$.

A further interesting property of this principle is that it is not limited to a cartesian coordinate system, but can be applied in all orthogonal coordinate systems, as in two-dimensional polar coordinates $(r,\varphi)$. This is of particular interest for calculating rotationally symmetric moiré elements, which are tuned by a mutual rotation of the two sub-elements.

3. DESCRIPTION OF DIFFRACTIVE OPTICAL ELEMENTS BY TRANSMISSION FUNCTIONS

The optical effect of a thin optical element (such as a DOE) can be advantageously described by its transmission function. The basic concept of transmission functions is that (for sufficiently thin optical elements) the transmitted electric field ${E_{{\rm out}}}(x,y)$ can be calculated by just a (pixelwise) multiplication of the incident field ${E_{{\rm in}}}(x,y)$ with the corresponding transmission function $T(x,y)$, i.e.,

$${E_{{\rm out}}}(x,y) = T(x,y) \cdot {E_{{\rm in}}}(x,y).$$

Generally, a transmission function $T(x,y)$ is a complex function that can be expressed as

$$T(x,y) = A(x,y)\exp [i\Phi (x,y)],$$
where $A(x,y)$ is the amplitude, and $\Phi (x,y)$ is the corresponding phase transmission profile. For transparent objects, such as lenses, the amplitude transmission function equals one (no absorption or amplification), i.e., $A(x,y) \equiv 1$. The corresponding phase transmission function $\Phi (x,y,\lambda)$ can be calculated from the surface profile $H(x,y)$ of a given optical element according to
$$\Phi (x,y,\lambda) = - \frac{{2\pi (n - 1)}}{\lambda}H(x,y).$$
The negative sign in front of the term is due to the fact that the lens material produces a phase delay of the transmitted wave, such that a smaller material thickness corresponds to a larger phase. Note that the phase transmission function depends inversely on the transmitted readout wavelength $\lambda$.

The concept of transmission functions also justifies the “kinoform” principle discussed before. According to the kinoform principle, the surface height function ${H_{{\rm DOE}}}(x,y)$ of a $l$th order DOE (where $l$ is an integer number) can be obtained by wrapping the original smooth surface profile ${H_{{\rm ref}}}(x,y)$ of the corresponding refractive optical element with a cutoff value of $l\Lambda$. There, $\Lambda$ is the path length, where a wave with wavelength ${\lambda _d}$ traveling through a medium with a (wavelength-dependent) refractive index ${n_d}$ acquires a phase shift of $2\pi$, with respect to the same path traveled in the vacuum. The wrapping is done with a modulo-$l\Lambda$ operation, i.e.,

$${H_{{\rm DOE}}}(x,y) = {{\rm mod}_{l \Lambda}}\{{H_{{\rm ref}}}(x,y)\} \quad {\rm where}\quad \Lambda = \frac{{{\lambda _d}}}{{{n_d} - 1}}.$$
Note that the final structure of ${H_{{\rm DOE}}}$ depends on the wavelength ${\lambda _d}$ used for the modulo operation. Thus, a DOE is specified for a certain designated wavelength ${\lambda _d}$, which is called the “design wavelength”.

According to Eq. (6), one can also define the phase transmission function of the corresponding refractive optical element, and of the corresponding DOE, by substituting the readout wavelength in Eq. (6) by the design wavelength ${\lambda _d}$ of the DOE, which yields

$$\begin{split}{\Phi _{{\rm ref}}}\left({x,y,{\lambda _d}} \right)& = - \frac{{2\pi\! \left({{n_d} - 1} \right)}}{{{\lambda _d}}}{H_{{\rm ref}}}(x,y)\quad {\rm and}\\{\Phi _{{\rm DOE}}}\left({x,y,{\lambda _d}} \right) &= - \frac{{2\pi \!\left({{n_d} - 1} \right)}}{{{\lambda _d}}}{H_{{\rm DOE}}}(x,y).\end{split}$$

Further evaluation of this equation by inserting Eq. (7) yields

$$\begin{split}{\Phi _{{\rm DOE}}} &= - \frac{{2\pi\! \left({{n_d} - 1} \right)}}{{{\lambda _d}}}{{\rm mod}_{{l\Lambda}}}\{{H_{{\rm ref}}}\}\\& = - \frac{{2\pi\! \left({{n_d} - 1} \right)}}{{{\lambda _d}}}{{\rm mod}_{{l\Lambda}}}\left\{{- \frac{{{\lambda _d}}}{{2\pi\! \left({{n_d} - 1} \right)}}{\Phi _{{\rm ref}}}} \right\} \\&= {{\rm mod}_{2\pi l}}\left\{{{\Phi _{{\rm ref}}}} \right\}.\end{split}$$
There, the property of the modulo operation ${{\rm mod}_y}\{x\} = {{\rm mod}_{\textit{ay}}}\{ax\} /a$ has been used. Equation (9) shows that the phase transmission function of an $l$th order DOE is obtained by wrapping the phase transmission function of the corresponding refractive element with a cutoff phase of $2\pi l$. Note, however, that both the phase transmisson function of the DOE and that of the refractive element are wavelength dependent, i.e., they are defined only for the design wavelength. If the elements are read out with another wavelength, the phase transmission function is changed. Thus the actually invariant characteristic of a DOE is its surface height profile.

For an arbitrary wavelength $\lambda$, the phase ${\Phi _{{\rm trans}}}$ of the transmitted light field becomes [Eq. (6)]

$$\begin{split}{\Phi _{{\rm trans}}} &= - \frac{{2\pi\! \left({{n_\lambda} - 1} \right)}}{\lambda}{{\rm mod}_{{l\Lambda}}}\{{H_{{\rm ref}}}\}\\& = \frac{{{\lambda _d}}}{\lambda}\frac{{{n_\lambda} - 1}}{{{n_d} - 1}}{{\rm mod}_{2\pi l}}\left\{{{\Phi _{{\rm ref}}}} \right\},\end{split}$$
where ${n_\lambda}$ is the refractive index at the readout wavelength $\lambda$. The corresponding complex transmission function ${T_{{\rm DOE}}}(x,y)$ then becomes
$$\begin{split}{T_{{\rm DOE}}}(x,y) &= \exp \left[{i\frac{{{\lambda _d}}}{\lambda}\frac{{{n_\lambda} - 1}}{{{n_d} - 1}}{{{\rm mod}}_{2\pi l}}\left\{{{\Phi _{{\rm ref}}}(x,y)} \right\}} \right]\\& = \exp \left[{i \frac{{{\lambda _d}}}{\lambda}\frac{{{n_\lambda} - 1}}{{{n_d} - 1}}{\Phi _{{\rm DOE}}}(x,y)} \right].\end{split}$$

Only if the readout wavelength $\lambda$ corresponds to the design wavelength ${\lambda _d}$ does the transmission function become

$$\begin{split}{T_{{\rm DOE}}}(x,y,\lambda &= {\lambda _d}) = \exp \left[{i {{{\rm mod}}_{2\pi l}}\left\{{{\Phi _{{\rm ref}}}(x,y)} \right\}} \right] \\&\equiv \exp \left[{i{\Phi _{{\rm ref}}}(x,y)} \right] = {T_{{\rm ref}}}(x,y).\end{split}$$

The equivalence holds for any integer number $l$ due to the $2\pi$ periodicity of the complex transmission function with respect to its phase term. This means that the phase of a wave behind a DOE is the same than that behind a corresponding refractive element, but only if the wavelength of the transmitted light corresponds to the design wavelength of the DOE.

The diffraction efficiency of the $m$th diffraction order of a digital phase grating is given by [45]

$${\eta _m} = {{\rm sinc}^2}\left({\frac{m}{N}} \right){{\rm sinc}^2}\left({m - \frac{{{\Phi _{{\rm max}}}}}{{2\pi}}} \right).$$
There, it is assumed that the grating is composed of periodic sawtooth structures, which cover a phase interval between zero and ${\Phi _{{\rm max}}}$, and which are composed of $N$ equidistant digital phase steps. The normalized sinc-function is defined as ${\rm sinc}(x) = \sin (\pi x)/(\pi x)$. Approximately, this equation also applies to general DOEs, which are composed of sawtooth structures with different periodicities, if their average “structure size” (corresponding to a sawtooth period of the DOE) is large enough to contain $N$ digital pixels (with a pixel size of $p$).

If the phase transmission function of an $l$th order DOE with a design wavelength ${\lambda _d}$ is inserted [Eq. (10)], and it is considered that the maximal phase height of the ${\rm mod}_{2\pi l}$-operation in Eq. (10) is $2\pi l$, this transforms to

$${\eta _m} = {{\rm sinc}^2}\left({\frac{m}{N}} \right){{\rm sinc}^2}\left({m - l\frac{{{\lambda _d}}}{\lambda}\frac{{{n_\lambda} - 1}}{{{n_d} - 1}}} \right).$$

Since the magnitude of the sinc-function is one for an argument of zero, this shows that the theoretical diffraction efficiency of a DOE can reach 100%, if it is composed of an “infinite” number of phase levels $N \mapsto \infty$, which corresponds to a continuous phase profile, and if the readout wavelength and the readout diffraction order $m$ are chosen such that the argument of the second sinc-function becomes zero. Particularly, this is the case for the $l$th diffraction order, if the readout wavelength corresponds to the design wavelength. However, as will be shown later, there is a sequence of other “harmonic” readout wavelengths, and other appropriate diffraction orders $m$, where the diffraction efficiency also approaches 100%.

The dispersion of the refractive index of the DOE material in Eq. (14) leads to a small reduction in diffraction efficiency at the harmonic wavelengths [46]. For many applications, this effect can be neglected, and the diffraction efficiency becomes

$${\eta _m} = {{\rm sinc}^2}\left({\frac{m}{N}} \right){{\rm sinc}^2}\left({m - l\frac{{{\lambda _d}}}{\lambda}} \right).$$

4. MOIRÉ ELEMENTS TUNABLE BY A MUTUAL ROTATION

The Alvarez–Lohmann principle discussed above can also be applied to combined optical elements whose properties are tuned by a mutual rotation. These elements are advantageously described in two-dimensional polar coordinates $(r,\varphi)$. There, $r = \sqrt ({x^2} + {y^2})$ is the radial distance from the center of the coordinate system (which is typically defined as the center of the desired optical element), and $\varphi$ is its polar angle. A sketch of such a setup including the corresponding coordinates is shown in Fig. 5. The two adjacent diffractive elements are supposed to be mutually rotated around their common optical axis. The mutual rotation angle is $\theta$, and $d$ is the axial distance between the adjacent elements. Typically, the diffractive elements are fabricated by lithographic methods, which have a limited resolution. Thus, the minimal pixel size $p$ (also called “feature size”) is an important parameter that, as will be shown later, affects the practically achievable tuning range of the combined element.

 figure: Fig. 5.

Fig. 5. Combined DOEs description in polar coordinated $(r,\varphi)$: ${r_{{\rm max}}}$: maximal radius; $d$: distance between DOEs; $p$: feature size (pixel size); $\theta$: mutual rotation angle.

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For the following discussion, the two sub-elements of a combined optical element are described by their corresponding transmission functions ${T_1}(r,\varphi)$ and ${T_2}(r,\varphi)$. Furthermore, it will be assumed that the two sub-elements are mutually conjugate pairs, in accordance with the original Alvarez principle. This means that the phase transmission functions have opposite signs, i.e., ${\Phi _1}(r,\varphi) = - {\Phi _2}(r,\varphi)$.

The original principle of obtaining the sub-elements of a combined tunable element by integrating the desired phase function along the variable coordinate can also be applied in polar coordinates. In the following, it is assumed that the required optical element is rotationally symmetric. In this case, its corresponding transmission function $T(r,\varphi)$ is just a function of the radial coordinate $r$, but not of the angular coordinate $\varphi$, i.e.,

$$T(r,\varphi) = \exp [i\Phi (r)].$$

Calculating the transmission function for a corresponding tunable element tuned by a mutual rotation between the two sub-elements, one has to integrate the exponent of this function along the angular coordinate $\varphi$, which yields for the transmission functions ${T_1}$ and ${T_2}$ of the two sub-elements:

$${T_1} = \exp [i\Phi (r)\varphi],\\{T_2} = \exp [- i\Phi (r)\varphi].$$

Note that the two elements ${T_1}$ and ${T_2}$ are actually upside-down flipped versions of each other, since the coordinate transform for an upside-down flip means that $r \mapsto r^\prime = r$, and $\varphi \mapsto \varphi ^\prime = - \varphi$. Therefore, the combined optical element can be obtained by producing two identical sub-elements mounted face to face.

If ${T_2}$ is rotated by an angle $\theta$ around a central axis (i.e., around the point $r = 0$), it transforms into

$${T_{{\rm 2,rot}}} = \left\{{\begin{array}{*{20}{l}}{{\rm exp}\left[{- i\Phi (r)(\varphi - \theta)} \right]}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\{{\rm exp}\left[{- i\Phi (r)(\varphi - \theta + 2\pi)} \right]}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}.} \right.$$
The two cases have to be distinguished because of the $2\pi$-periodicity of rotations.

The joint transmission function ${T_{{\rm joint}}}$ of the two combined elements ${T_1}$ and ${T_{2,{\rm rot}}}$ is obtained by a multiplication of the two respective transmission functions of the sub-elements ${T_{{\rm joint}}} = {T_1}{T_{{2,{\rm rot}}}}$, which yields

$${T_{{\rm joint}}} = \left\{{\begin{array}{*{20}{l}}{{\rm exp}\left[{i\Phi (r)\theta} \right]}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\ {{\rm exp}\left[{i\Phi (r)(\theta - 2\pi)} \right]}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}.} \right.$$
This shows that the phase term in the exponent increases linearly with the rotation angle $\theta$, and its sign can be inverted by reversing the rotation direction. Furthermore, this shows that two different sectors are formed, namely, a “main” sector in the angular range between $\theta$ and $2\pi$, and a residual sector in the interval between zero and $\theta$. These two sectors generally contain two different complex transmission functions that have different optical properties. In the following, this will be discussed by means of a tunable parabolic moiré lens.

A. Tunable Parabolic Moiré Lens

The method described above will now be applied to calculate the two sub-elements that form a combined tunable parabolic moiré lens.

The transmission function of the parabolic lens in cylindrical coordinates is

$${T_{{\rm parab}}}(r,\varphi) = \exp \left[{i\frac{\pi}{{\lambda f}}{r^2}} \right],$$
where $f$ is the focal length, and $\lambda$ is the readout wavelength. Applying Eq. (17), the complex transmission functions ${T_{\pm 1}}(r,\varphi)$ of the two corresponding sub-elements are obtained as
$${T_{\pm 1}}(r,\varphi) = \exp (\pm a{r^2}\varphi),$$
where $a$ is a constant that determines the final tuning range.

To obtain the phase transmission functions of the two corresponding diffractive sub-elements of order $l$, the phase term of the complex transmission function has to be wrapped according to

$${\Phi _{\pm 1}}\left({r,\varphi ,{\lambda _d}} \right) = \pm {{\rm mod}_{2\pi l}}\left\{{a{r^2}\varphi} \right\}.$$

Figure 6 shows the phase transmission functions of these two sub-elements with a mutual rotation angle of $\theta ={ 90^ \circ}$, if the order of the DOEs is choosen to be $l = 1$. The phase profiles of the two sub-elements (DOE 1 and DOE2) are sketched as images, where the gray levels correspond to phase values within an interval between zero and $2\pi$. The main axis of each element is indicated as a red bar. In the simulation, the main axis of DOE 2 is rotated by an angle $\theta ={ 90^ \circ}$ (indicated in the figure) with respect to the axis of DOE 1.

 figure: Fig. 6.

Fig. 6. Adding the phase functions of two mutually rotated sub-elements DOE 1 and DOE 2, which are designed to produce the tunable phase profile of a parabolic lens (at the right). Gray levels in the images correspond to phase values between zero and $2\pi$. The resulting phase function shows two sectors that include parabolic lens profiles with different focal lengths.

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According to Eq. (19), the joint transmission function of the two combined elements is

$${T_{{\rm joint}}} = \left\{{\begin{array}{*{20}{l}}{\exp [i\theta a{r^2}]}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\{\exp [i(\theta - 2\pi)a{r^2}]}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}.} \right.$$

The phase of the resulting joint transmission function is sketched in Fig. 6 at the right side. It shows that two sectors are formed, as expected. The main sector is situated in the angular range between $\theta$ and $2\pi$, whereas the residual sector takes up the remaining area of the lens. The transmission functions within the two sectors correspond to those of two parabolic lenses with different focal lengths.

Comparing the phase profiles in the two sectors [according to Eq. (23)] with the phase profile ${T_{{\rm parab}}}$ of the desired parabolic lens [Eq. (20)], optical powers ${P_1}$ and ${P_2}$ (corresponding to the inverse of the respective focal lengths) within the two sectors are obtained as

$$\begin{split}{P_1} & = \theta a\lambda /\pi \quad\quad\quad\quad {\rm for}\;\; \theta \le \varphi \lt 2\pi \quad {\rm and}\\ {P_2} & = (\theta - 2\pi)a\lambda /\pi \quad {\rm for}\;\; 0 \le \varphi \lt \theta .\end{split}$$

This means that the optical power depends linearly on the rotation angle $\theta$, however with a constant difference of $\Delta P = {P_2} - {P_1} = - 2a\lambda$ between the two sectors. An experimental demonstration of this behavior is shown in Fig. 7. The sign of the optical power within the two sectors depends on the sign of the rotation angle $\theta$, i.e., the resulting transmission function corresponds to a positive (convex) or a negative (concave) lens, depending on the rotation direction.

 figure: Fig. 7.

Fig. 7. Sector forming moiré lens. Left: simulation of the phase transmission function of a moiré lens, where the two sub-elements are mutually rotated by an angle of $\theta ={ 90^ \circ}$. Right: experimental demonstration of the formed sector by means of a tunable moiré lens, consisting of two mutually rotated lithographically fabricated sub-elements. The lens is positioned at some distance above a sample text, thus acting as a magnifying glass. Within the two sectors (indicated by red bars) one observes different magnification factors of the text.

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The same principle can also be applied to produce moiré elements that realize any other kind of rotationally symmetric optical components. However, for the case of tunable lenses, it should be noted that this method works only for parabolic moiré lenses in an optimal way, whose transmission functions are defined by Eq. (20). In this case, an increase in the exponent leads to a linear increase in optical power while preserving the structure of the parabolic lens. This feature does not apply to the case of a hyperbolic lens, which has a phase profile of ${\Phi _{{\rm hyperb}}} = 2\pi {({f^2} + {r^2})^{1/2}}$. Actually, this kind of lens would be desirable, since it provides the best possible focus achievable with a flat lens. To change the optical power of such a lens, the focal length within the square root term would have to be changed. This, however, cannot be done by a multiplication of the whole phase term with a $\theta$-dependent constant. The phase profile resulting from such a multiplication would not correspond to that of a hyperbolic lens any more. The corresponding aberrations would degrade the focusing performance of such a tunable hyperbolic moiré lens below that of a tunable parabolic lens.

A second example, where the linear change of the phase profile preserves the structure of the moiré element, is a tunable axicon, which is described in the following.

B. Tunable Axicon Moiré Element

Axicons are refractive optical elements with a (positive or negative) conical surface profile. For an incident Gaussian beam, the transmitted beam has a Bessel beam profile. Such a beam has an elongated focal region that extends along the optical axis. This is an interesting feature for different practical applications, such as laser marking/cutting/drilling, optical sensors (e.g., bar code readers), fiber coupling, or optical trapping [47].

The transmission function of an axicon is given by

$${T_{{\rm axicon}}}(r,\varphi) = \exp \left[{i\frac{{2\pi \sin (\beta)}}{\lambda}r} \right],$$
where $\lambda$ is the wavelength of the transmitted light, and $\beta$ is the opening angle of the light cone behind the axicon for an incident paraxial beam.

Applying Eq. (17), the phase transmission functions ${\Phi _{\pm 1}}(r,\varphi)$ of the corresponding two diffractive sub-elements are obtained as

$${\Phi _{\pm 1}}(r,\varphi ,{\lambda _d}) = {{\rm mod}_{2\pi l}}\{br\varphi \} ,$$
where $b$ is a constant that determines the final tuning range, and $l$ is the order of the DOEs.

According to Eq. (19), the joint transmission function of the two combined elements becomes

$${T_{{\rm joint}}} = \left\{{\begin{array}{*{20}{l}}{\exp [i\theta br]}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\{\exp [i(\theta - 2\pi)br]}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}.} \right.$$

This corresponds to the transmission functions of two different axicons situated within the two corresponding sectors, which have cone angles ${\beta _1}$ and ${\beta _2}$ of

$$\begin{array}{*{20}{l}}{\beta _1} &= {\rm arcsin} \left[{\theta b\lambda /2\pi} \right] & {\rm for}\quad \theta \le \varphi \lt 2\pi \quad {\rm and}\\{\beta _2} &= {\rm arcsin} \left[{(\theta - 2\pi)b\lambda /2\pi} \right]& {\rm for}\quad 0 \le \varphi \lt \theta .\end{array}$$
Thus the cone angle of the transmitted beam can be adjusted by the rotation angle $\theta$, however with an offset within the two sectors.

This behavior is sketched in the simulation in Fig. 8. There, two first-order phase transmission functions (DOE 1 and DOE 2) are calculated according to Eq. (26). DOE 2 is rotated with respect to DOE 1 by an angle $\theta ={ 90^ \circ}$. Multiplying the two respective transmission functions results in the transmission function of the combined moiré element, whose phase is displayed at the right. As expected, it shows again two sectors (indicated red and blue) that correspond to diffractive axicons with different cone angles.

 figure: Fig. 8.

Fig. 8. Phase transmission functions of two sub-elements (DOE 1 and DOE 2), which produce the transmission function of an axicon (right). The situation is sketched for a mutual rotation angle of $\theta ={ 90^ \circ}$ (indicated in the figure). Two sectors (red and blue) are formed, which correspond to different axicon cone angles. Gray values in the images correspond to phase shifts in an intervall between zero and $2\pi$.

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5. TUNABLE MOIRÉ ELEMENTS WITHOUT SECTOR FORMATION

In most cases, it is desired that a tunable moiré element has a homogeneous transmission function, without the formation of two different sectors. Equation (19) shows that the phase transmission function within the two sectors is proportional to the mutual rotation angle $\theta$, and to $\theta - 2\pi$, where $\theta$ replaces the polar angle $\varphi$ in the complex transmission function of the sub-elements. Thus the transmission functions within the two sectors become identical, if the transmission function of the sub-elements is a periodic function of the polar angle $\varphi$. One phase transmission function that fulfills this condition is ${\Phi _{{\rm grat}}} = {\rm arsin} (\varphi)$. If one replaces the polar coordinates with the respective cartesian coordinates ($x,y$), it turns out that this phase transmission function corresponds to ${\Phi _{{\rm grat}}} = ay$, which is just the phase transmission function of a blazed grating with grating constant $2\pi /a$. Therefore, the combination of two identical blazed gratings into a combined moiré element yields the transmission function of another blazed grating, without the formation of two different sectors, which therefore can achieve a theoretical diffracton efficiency of 100%. Its grating constant is tunable by a mutual rotation of the two sub-gratings. Actually, this is a well-known device used, for example, as a so-called grating scanner [48].

Another interesting example is the phase transmission function ${\Phi _{{\rm saddle}}} \propto {r^2}{\sin}^2 (\theta)$ of a so-called “saddle-lens,” or “quadrupole lens.” Such a phase transmission function corresponds to a superposition of two orthogonal cylindrical lenses with opposite optical powers. According to the previous considerations, the combination of two identical saddle lenses into a combined moiré element produces again a saddle lens whose optical power is tunable by a mutual rotation of the sub-elements, and which does not produce two different sectors, such that a theoretical diffraction efficiency of 100% can be achieved. This property was used, for example, for designing a novel zoom system consisting of two tunable moiré saddle elements, whose zoom factor could be continuously adjusted by a controlled rotation of the individual saddle lens elements [49].

Unfortunately, the general complex transmission functions of the sub-elements according to Eq. (17) are not periodic in $\varphi$, which normally results in the formation of two different sectors. However, the elements can be forced to become periodic in $\varphi$ by using a certain quantization “trick.”

Consider the general joint transmission function in Eq. (19). The difference of the phase terms in the corresponding two sectors is independent of the mutual rotation angle $\theta$, and given by $2\pi \Phi (r)$. This suggests to approximate the original continuous phase profile $\Phi (r)$ by a quantized version, namely, by $\Phi (r) \mapsto {\rm round}\{\Phi (r)\}$, where the “round $\{\ldots \}$”-operator corresponds to rounding of the continuous function $\Phi (r)$ to its nearest integer values at each position $r$. Note that the values of $\Phi (r)$ are, for typical moiré elements such as a lens, continuously distributed in an interval between zero up to ${10^4}$, such that the rounding operation is a very close approximation of the original continuous phase function.

Using the new, quantized phase profile, the phase transmission functions of the new sub-elements ${T_{{\rm q,1}}}$ and ${T_{{\rm q,2}}}$ are given as

$${T_{{\rm q,1}}} = \exp [i{\rm round}\{\Phi (r)\} \varphi],\\{T_{{\rm q,2}}} = \exp [- i {\rm round}\{\Phi (r)\} \varphi].$$

The corresponding joint transmission function of the combined element (assuming a mutual rotation angle of $\theta$ between the two elements) then becomes

$${T_{{\rm q,joint}}} = \left\{{\begin{array}{*{20}{l}}{{\rm exp}\left[{i {\rm round}\{\Phi (r)\} \theta} \right]}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\{{\rm exp}\left[{i {\rm round}\{\Phi (r)\} (\theta - 2\pi)} \right]}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}} \right..$$

Due to the $2\pi$-indeterminacy of the phase arguments in Eq. (30), the two sectors have actually the same phase transmission functions, i.e., a transmitted wave will acquire a homogeneous phase profile.

 figure: Fig. 9.

Fig. 9. Adding the phase functions of two mutually rotated DOEs (DOE 1 and DOE 2) with quantized phase levels. The simulated result corresponds to the phase transmission function of a parabolic lens, which is not disturbed by the formation of different sectors. An experimental demonstration of the homogeneous lensing effect of two correspondingly fabricated, mutually rotated sub-elements, which are placed above a sample text, is shown at the right.

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If the phase profiles ${\Phi _{{q,} \pm {1}}}$ of the corresponding two sub-elements are constructed as first order DOEs, the ${{\rm mod}_{2\pi}}$ -operation has to be applied to the phase terms of Eq. (29), resulting in

$${\Phi _{{ q,} \pm {1}}}(r,\varphi) = \pm {{\rm mod}_{2\pi}}\{{\rm round}\{\Phi (r)\} \varphi \} .$$

As an example, consider the transmission function ${\Phi _{{\rm parab}}}(r)$ of a parabolic lens in Eq. (20). The corresponding phase transmission functions of the two sub-elements are then obtained according to Eq. (29) as

$${\Phi _{{\rm parab,} \pm {1}}}(r,\varphi) = \pm {{\rm mod}_{2\pi}}\{{\rm round}\{a{r^2}\} \varphi \} .$$
The phase transmission function of these two elements (DOE 1 and DOE 2) are sketched in Fig. 9 at the left. The second one (DOE 2) is rotated by an angle $\theta ={ 45^ \circ}$ with respect to the first one.

Combining these two sub-elements into a joint tunable moiré element, one obtains a close approximation of a parabolic lens, which has a refractive power of $P = \theta a\lambda /\pi$. The corresponding phase transmission function is plotted in Fig. 9 as the sum of DOE 1 and DOE 2, followed by a ${\rm mod}_{2\pi}$-operation. In contrast to the previous results (see Fig. 6), the lens has a homogeneous phase profile, without the formation of different sectors. An experimental demonstration of this behavior is shown at the right side of Fig. 9. There, a set of two mutually rotated sub-elements is placed at a short distance above a sample text, acting as a magnifying glass. The picture shows that the resulting lens is homogeneous, i.e., the whole area of the lens magnifies the underlying text in the same way.

Other approaches to avoid the sector formation were investigated in [32,34], where the sub-elements were designed by numerical optimization methods to avoid the sector at a discrete set of rotation angles.

The principle of avoiding the formation of different sectors within the combined moiré elements by such a quantization operation can be analogously applied to any kind of rotationally symmetric moiré element. However, there are two drawbacks of this method. First, the suppression of the sector formation works only for the specified design wavelength of the moiré elements. With increasing deviation of the readout wavelength from the design wavelength, the two sectors will progressively reappear. Second, the diffraction efficiency of such a quantized moiré element decreases with increasing mutual rotation angles. This is a second order effect, which is smaller than 2.5% within a rotation range between ${-}{30^ \circ}$ and ${+}{30^ \circ}$, and it is still below 20% in a range between ${-}{90^ \circ}$ and ${+}{90^ \circ}$, but it becomes more and more dominant for larger rotation angles. This effect will be described in the following section in more detail.

A. Diffraction Efficiency of Moiré Elements

According to Eq. (13), the diffraction efficiency $\eta$ of a digitized, first order DOE composed of $N$ phase levels in a phase interval between zero and $2\pi$ and read out at its design wavelength ${\lambda _d}$ is given by

$$\eta = {{\rm sinc}^2}\left({\frac{1}{N}} \right).$$

For the case of a non-quantized (sector-forming) combined moiré element according to Eq. (17), where both sub-elements are composed of an “infinite” number of phase levels, the diffraction efficiency of each sub-element can theoretically reach 100%. However, the combined moiré element produces two sectors, which consist of different phase transmission functions. In this case, the absolute diffraction efficiencies of the two sectors correspond just to their respective normalized areas with respect to the whole area of the moiré element. This is, however, just an approximation, since the appearance of the different sectors also changes the point spread functions of the resulting optical elements, and thus optical properties (such as the focal length of a lens) are not properly defined any more. Nevertheless, the efficiency of the modified point spread functions will be proportional to the corresponding area of the respective sectors. Normally, only one of the two sectors will have the desired phase transmission function, as, for example, a lens with a certain focal length. Thus, for non-quantized (sector-producing) moiré elements, the effective diffraction efficiency (of the desired transmission function) is maximal at zero mutual rotation angle of its two sub-elements, and it decreases linearly with increasing rotation angle.

The situation changes for the case of quantized moiré elements according to Eq. (29), which avoid the formation of different sectors. The transmission function of a general, quantized, rotationally symmetric moiré element [Eq. (30)] can be expressed as

$${T_{{\rm joint}}} = \exp [i {\rm round}\{\Phi (r)\} (\theta - 2\pi k)],$$
where $k = 0, \pm 1, \pm 2\ldots $ is an arbitrary integer number. Note that the effective transmission function will actually be the same for all integer values of $k$, since this adds just integer multiples of $2\pi$ to the phase. Nevertheless, each value of $k$ corresponds to a different pre-factor of the phase transmission function, and therefore to a different optical effect of the corresponding element. For example, for the case of the parabolic lens described by Eq. (32), each value of $k$ corresponds to an optical power ${P_k}$ of the combined moiré lens of
$${P_k} = (\theta - 2\pi k)a\lambda /\pi .$$
Due to the quantization of the phase function, the actually obtained transmission function is composed of an infinite series of “$k$-components” with different optical properties. Note that this is not a specific feature of combined moiré elements, but it applies to all digitized DOEs.
 figure: Fig. 10.

Fig. 10. Diffraction efficiency for the different $k$-components of a quantized moiré element as a function of the mutual rotation angle, according to Eq. (37). As an example, a rotation angle of $\theta = \pi /2$ is assumed. The $k = 0$ component (indicated as a red vertical line at $\theta = \pi /2$) has an efficiency of 81%. The other $k$-components appear in the graph at integer distances of $2\pi$.

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However, the diffraction efficiency of the individual $k$-components is different. The factor ${\rm round}\{\Phi (r)\}$ in the phase term of Eq. (34) can be regarded as an array of subsequent integer numbers between zero and ${N_{{\max}}}$, where ${N_{{\max}}}$ is the largest integer number for which the absolute value of the phase in the exponent of Eq. (34) is smaller than $2\pi$, i.e., ${N_{{\rm max}}}(\theta - 2\pi k) \lt 2\pi$. Thus, each $2\pi$-phase interval in Eq. (34) is sampled by maximally ${N_{{\max}}}$ equidistant phase steps, given by

$${N_{{\max}}} = \frac{{2\pi}}{{\theta - 2\pi k}}.$$
According to Eq. (33), the corresponding diffraction efficiency ${\eta _k}$ of the $k$-component of the phase transmission function in Eq. (34) is given by
$${\eta _k} = {{\rm sinc}^2}\left({\frac{\theta}{{2\pi}} - k} \right).$$
This behavior is sketched in Fig. 10. There, the efficiency of a combined moiré element is plotted as a function of the mutual rotation angle $\theta$ between its sub-elements for the $k = 0$ component. As an example, the efficiency of the $k = 0$ component at a rotation angle of $\theta ={ 90^ \circ}$ (or $\pi /2$) is indicated as the longest red vertical line at the angular position $\theta = \pi /2$. The efficiencies of the other $k$-components are found at integer distances of $2\pi$, as indicated in the figure. In our example of a mutual rotation angle of $\theta = \pi /2$, the $k = 0$ component has the highest efficiency of 81%, whereas the next lower efficiencies of 9.0%, 3.2%, 1.7%, and 1.0 % are obtained for the $k = 1, - 1, 2\;{\rm and}\; - 2$ components, respectively. Note that the corresponding diffraction efficiencies are ideal values, which are obtained if the sub-elements are fabricated as continuous DOEs, i.e., if each sub-element consists of an “infinite” number of phase levels. Nevertheless, due to the quantization procedure, the combined element consists of a limited number of effective phase levels, which depends on the rotation angle, and which reduces the diffraction efficiency.

The previous considerations show that the diffraction properties of a DOE are considerably changed by the quantization of their phase transmission functions. Whereas a continuous (non-quantized) DOE with a phase function of ${{\rm mod}_{2\pi}}\{\Phi (r)\}$ exactly reproduces the optical effect of the original refractive phase element $\Phi (r)$, the quantized element will reproduce this element only in the $k = 0$ component of its diffraction pattern. Theoretically, there will be an infinite sequence of higher $k$-components, however with low efficiencies. For example, for the case of a parabolic moiré lens, these other $k$-components correspond to superposed lenses with different optical powers given by Eq. (35). Accordingly, such a quantized parabolic moiré lens will have multiple focal points corresponding to different optical powers separated by a constant optical power offset of $2a\lambda$ [where $a$ is defined according to Eq. (22)]. However, for a sufficiently small mutual rotation angle $\theta$, the intensity of the focus of the $k = 0$ component will clearly dominate the “parasitic” foci related to the other $k$-components. Sometimes, these additional $k$-components are called “higher diffraction orders.” However, this is not correct, since the DOE actually diffracts into its first diffraction order, but the first order diffraction pattern is changed due to quantization. This effect appears in all quantized DOEs, such as digitized Fresnel lenses.

For the case of rotationally tunable moiré elements, the number ${N_{{\rm max}}}$ of effective phase levels in each $2\pi -$ interval depends reciprocally on the mutual rotation angle $\theta$. Therefore, the diffraction efficiency of the desired $k = 0$ component of the combined element has a ${\rm sinc}^2$-function dependence on the rotation angle (Eq. 37), i.e.,

$$\eta = {{\rm sinc}^2}\left({\frac{\theta}{{2\pi}}} \right).$$

This dependence is the same for any kind of rotationally symmetric combined moiré element. For example, the diffraction efficiency of a tunable axicon moiré element is the same than that of a tunable moiré lens, if the mutual rotation angles of the corresponding sub-elements are equal. Since the tunable optical effect of a combined moiré element (such as the optical power of a tunable moiré lens) is a linear function of the mutual rotation angle, there is also a fixed relation between the diffraction efficiency of a moiré element and its optical effect.

From a practical point of view, the ${\rm sinc}^2$ dependence of the diffraction efficiency as a function of the rotation angle is advantageous, since it has a maximum (of theoretically 100%) at an angle of zero. Therefore, the efficiency change as a function of the mutual rotation angle is a second order effect, which is small for moderate rotation angles. For example, adjusting the rotation angle within a range between ${-}{30^ \circ}$ and ${+}{30^ \circ}$ reduces the maximal diffraction efficiency by only 2.5%. On the other hand, the optical properties of the combined moiré element (such as the optical power of a parabolic lens) change as a linear function of the rotation angle. Thus, it is possible to realize quantized moiré elements whose optical effects are tunable in a large range, whereas the related reduction of their diffraction efficiencies is limited to only a few percent.

6. PRACTICAL CONSIDERATIONS

DOEs are typically produced by technologies such as photolithography, electron beam lithography, direct laser writing using material ablation, or by photoinduced polymerization. A DOE typically consists of an ensemble of “structures” that have smoothly varying shapes in phase intervals between zero and $2\pi$ and that are confined by abrupt phase jumps at their boundaries. The minimal length of the smooth structures is called the “structure size” $s$ of the DOE. For example, the structures of a Fresnel lens consist of concentric sawtooth rings whose thickness decreases with increasing distance from the center of the lens. Thus the minimal structure size $s$ corresponds to the thickness of the outmost sawtooth ring.

For DOE production, one typically has to specify both the transverse resolution and the depth resolution. The transverse resolution corresponds to the minimal “feature size” $p$ of the DOE, which coincides with the “pixel size” in digital fabrication techniques. The minimal reasonable feature size of a DOE is of the order of one half of the design wavelength, since smaller structures would correspond to diffraction angles larger than 90°.

The depth resolution of a DOE is specified as the number of (typically equidistantly spaced) resolved phase levels within a phase interval between zero and $2\pi$. If $N$ phase levels are resolved in a first order DOE (which has a maximal phase amplitude of $2\pi$), the step size is $2\pi /N$, and the corresponding maximal diffraction efficiency $\eta$ is given according to Eq. (33) as $\eta = {{\rm sinc}^2}(1/N)$. However, this maximal diffraction efficiency is achieved only if the minimal structure size $s$ is sufficiently large, such that the number of resolved pixels $s/p$ within this structure corresponds at least to the number of available phase levels $N$. Otherwise, $N$ in Eq. (33) has to be substituted by an effective number of resolved levels, namely, by ${N_{{\rm eff}}} = s/p$. In practice, a DOE will consist of an ensemble of structures with different sizes, and thus its total diffraction efficiency is an area-weighted average over the diffraction efficiencies of its different structures.

An example of such a situation is sketched in Fig. 11. The blue curve indicates a one-dimensional first order DOE (with a maximal phase amplitude of $2\pi$), which consists of two sawtooth structures with different lengths ${s_1}$ and ${s_2}$. The horizontal and vertical grid lines indicate the feature size $p$ (i.e., the minimally achievable pixel size) and the phase step size $2\pi /N$, respectively. The smooth DOE structures are sampled by a digital curve (red) that has the horizontal and vertical resolution of the indicated grid.

 figure: Fig. 11.

Fig. 11. Illustration of feature size, structure size, and phase step height of a digital DOE.

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In the example, the first structure ${s_1}$ is large enough to be sampled by the maximally available number of $N = 10$ phase levels (level indices displayed in the figure). Thus, it corresponds to an $N = 10$ phase structure, which has a diffraction efficiency of 96.8%, according to Eq. (33). However, the size ${s_2}$ of the adjacent structure extends over only four feature sizes $p$, and therefore it only contains ${N_{{\rm eff}}} = {s_2}/p = 4$ resolved phase levels, which yields a diffraction efficiency of 81.1%. If ${s_2} = 4p$ is the minimal structure size of the entire DOE, then the total theoretical diffraction efficiency of the entire DOE will be at least 81.1%. It will be higher, however, if a significant part of the structures is larger.

Due to the sampling theorem, a DOE structure can be resolved only if each phase structure with a height of $2\pi$ is sampled by at least two pixels, i.e., ${N_{{\rm eff}}}$ has to be at least two for the smallest appearing structure size $s$. However, this yields a diffraction efficiency of only 40.5%, according to Eq. (33). Thus it is typically required to resolve a larger effective number of phase levels ${N_{{\rm eff}}} \gt 2$. In this case, the (absolute value of the) gradient of the phase transmission function $\Phi (r,\varphi)$ has to be smaller than $2\pi /{N_{{\rm eff}}}p$, i.e.,

$$||\nabla \Phi (r,\varphi)|| = \sqrt {{{\left[{\frac{{\rm d}}{{{\rm d}r}}\Phi (r,\varphi)} \right]}^2} + {{\left[{\frac{1}{r}\frac{{\rm d}}{{{\rm d}\varphi}}\Phi (r,\varphi)} \right]}^2}} \lt \frac{{2\pi}}{{{N_{{\rm eff}}}p}},$$
where $p$ is the smallest resolved pixel size, and ${N_{{\rm eff}}} \ge 2$ can be selected according to the efficiency requirement of the DOE.

In our case of moiré elements, this condition applies to each sub-element of a combined DOE. The main consequence of this condition is that it limits the maximally achievable tuning range of the combined moiré elements.

As an example, we consider the case of a tunable parabolic moiré lens whose sub-elements have phase transmission functions [see Eq. (22)] of

$${\Phi _{{1,2}}}(r,\varphi) = \pm a{r^2}\varphi ,$$
where the polar angle coordinate $\varphi$ covers a range between ${-}\pi$ and ${+}\pi$. There, the ${\rm mod}_{{2}\pi}$-operation has been omitted without loss of generality. The user defined “production constant” $a$ determines the change of the optical power as a function of the mutual rotation angle $\theta$ (in the dominant sector of the lens), according to [Eq. (41)]
$${P_1} = a\theta \lambda /\pi .$$

Evaluating the gradient in Eq. (39) for the phase transmission function of Eq. (40), one obtains the condition

$$||\nabla {\Phi _1}(r,\varphi)|| = \sqrt {{{\left[{2ar\varphi} \right]}^2} + {{\left[{ar} \right]}^2}} \lt \frac{{2\pi}}{{{N_{{\rm eff}}}p}}.$$

Obviously, the operand of the square root function has its maximal values for $\varphi = {\varphi _{{\max}}} = \pi$ and $r = {r_{{\max}}}$, which is the radius of the DOE. This means that the phase modulation of the DOE is maximal at its boundaries. Using these values, one sees that the first summand below the square root function exceeds the second one by a factor of ${(2\pi)^2}$ (approximately 40), such that the second term can be neglected in practical applications. The sampling condition then becomes

$$2\pi a{r_{{\rm max}}} \lt \frac{{2\pi}}{{{N_{{\rm eff}}}p}}\quad {\rm or} \quad a \lt \frac{1}{{{N_{{\rm eff}}}p{r_{{\rm max}}}}}.$$
As already mentioned above, a value of ${N_{{\rm eff}}} = 2$ yields a local diffraction efficiency of only 40.5% (at the boundaries of the moiré lens), whereas a value of ${N_{{\rm eff}}} = 4$ already yields an efficiency of 81%. Thus in many practical applications, a value of ${N_{{\rm eff}}} = 4$ is a good compromise between achieving a large tuning range of the moiré element and getting a good diffraction efficiency (which has to be averaged over the whole element, and will therefore actually be larger than 81%).

As a practical example, we consider the design of a moiré lens with a diameter of 1 cm (${r_{{\rm max}}} = 5\;{\rm mm}$), which is supposed to be used at a wavelength of 532 nm. Assuming that the minimal feature size $p$ of the fabrication process is 0.5 µm (which is larger than the minimal “reasonable” feature size of one half of the wavelength), and choosing the minimal number of resolved phase levels to be ${N_{{\rm eff}}} = 4$, such that the theoretically achievable diffraction efficiency will be larger than 81%, the maximal value of the fabrication factor $a$ is $10^8\;{\rm m}^{- 2}$. Assuming that the mutual rotation angle of the moiré lens will be limited to an interval between $-90^ \circ$ and $+90^ \circ$ (to limit the size of the “parasitic” sector to maximally one quarter of the total lens area), the tuning range of the optical power of the lens (Eq. 41) will extend from ${+}26.6$ to ${-}26.6\,\,{\rm dpt}$, corresponding to a focal length range from ${\pm}3.76\;{\rm cm}$ to infinity.

7. SPECIAL MOIRÉ ELEMENTS

In the previous sections, we discussed moiré elements that have well-known refractive counterparts, such as lenses and axicons. In the following, two more special moiré elements will be introduced, namely, a tunable vortex plate and a tunable lens element that compensates for spherical aberrations during shifting the focus of an objective lens with high NA.

A. Tunable Vortex Plates

“Vortex plates,” or “spiral phase plates,” are optical elements that change the orbital angular momentum of a transmitted beam. If the incident beam is a Gaussian beam, the transmitted beam is transformed into a Laguerre–Gaussian beam, with an orbital angular momentum that depends on the so-called helical index $l$ of the vortex plate. These beams have a doughnut shaped intensity distribution with an absolutely dark spot in their centers (i.e., where the intensity equals exactly zero). This kind of beam is used, for example, in laser machining applications, in optical trapping applications (optical tweezers or atom traps), in (fiber-)optical communication systems as a means to increase the information capacity, in stimulated-emission-depletion (STED) microscopy for achieving extreme resolution enhancement, and in phase contrast microscopy as a Fourier filter, which generates strong edge contrast enhancement.

 figure: Fig. 12.

Fig. 12. Combination of two sub-DOEs (DOE 1 and DOE 2) into a combined moiré element generating a tunable vortex plate. The figure shows the phase transmission functions (phase values indicated as gray values) of a first DOE 1, and a second complimentary DOE 2, which is rotated by an angle of 14.4° with respect to the first one. The resulting transmission function corresponds within its main sector to a vortex plate with a helicity of $l = 5$.

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The phase transmission function of a vortex plate with a helical index of $l$ (where $l$ is an integer number) is given by

$${\Phi _{{\rm vortex}}}(r,\varphi) = l\varphi ,$$
which is independent of the radial coordinate $r$. Applying the basic method to produce a rotationally tunable moiré element, which generates a variable vortex plate, one has to integrate the phase function along the variable coordinate, which is the angular coordinate $\varphi$. The two resulting sub-elements ${\Phi _{{\rm vort,1}}}$ and ${\Phi _{{\rm vort,2}}}$ thus have the phase transmission functions [50]
$${\Phi _{{\rm vort,1}}}(r,\varphi) = {l_0}{\varphi ^2},\\{\Phi _{{\rm vort,2}}}(r,\varphi) = - {l_0}{\varphi ^2},$$
where ${l_0}$ is a user selectable constant that determines the tuning range of the combined moiré element. There, a constant offset has been omitted without loss of generality. In this form, the two elements act as refractive elements, which, however, can be transformed into diffractive elements by applying a modulo-$2\pi$ operation. If one of the sub-elements is rotated by an angle $\theta$ with respect to the other, the combined transmission function becomes
$${T_{{\rm vort,joint}}} = \left\{{\begin{array}{*{20}{l}}{{\rm exp}\left\{{i{l_0}(2\theta \varphi - {\theta ^2})} \right\}}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\{{\rm exp}\left\{{i{l_0}[(2\theta - 4\pi)\varphi - {\theta ^2} - 4{\pi ^2} + 4\pi \theta]} \right\}}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}.} \right.$$
Again, the combined transmission function consists of two sectors with different optical properties. Within the transmission function, all terms that do not include the polar coordinate $\varphi$ are spatially constant phase offsets, i.e., they produce a uniform phase shift of just the transmitted wave. Therefore they will be omitted in the following considerations. The remaining phase terms within the two sectors correspond to vortex plates, which have [by comparison with Eq. (44)] two different helical indices $l$ of
$$l = \left\{{\begin{array}{*{20}{l}}{2{l_0}\theta}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\ {2{l_0}\theta - 4\pi {l_0}}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}.} \right.$$
Thus the helical indices in the two sectors can be adjusted as a linear function of the mutual rotation angle $\theta$. An example is shown in Fig. 12. There, the phase transmission functions of the two complimentary sub-DOEs [according to Eq. (45), with the parameter ${l_0} = 10$] are sketched (with phase values within an interval between zero and $2\pi$ corresponding to gray values). DOE 2 is rotated with respect to DOE 1 by an angle of 14.4°. The combined phase transmission function is shown at the right. It corresponds in its main sector to a vortex plate with a helical index of $l = 5$, whereas the helical index in its smaller sector corresponds to $l = - 120$.

An experimental verification of the working principle was demonstrated in [50], and an example of the corresponding results is shown in Fig. 13. There, the light field transmitted through a tunable, vortex-producing moiré element is measured by an interferometric setup (using a plane reference wave). Thus, intensity variations (gray levels) in the corresponding interferograms correspond to phase shifts between zero and $2\pi$. The results show that by adjusting the mutual rotation angle between the two moiré sub-elements to 2°, 4°, 6°, 8°, 16°, 20°, 30°, and 40°, the helicity of the resulting vortex modes is adjusted to $l =1$, 2, 3, 4, 8, 10, 15, and 20, respectively. The appearance of the smaller sector (visible in the “northwestern” region of the interferograms) slightly disturbs the point spread function of the moiré element; however, its influence decreases if only small mutual rotation angles $\theta$ are required. This can be achieved if the user specified constant ${l_0}$ in Eq. (45) is chosen to be sufficiently large, which yields a large tuning range of the helical index within a small angular tuning range. A similar method has been suggested for DOEs in [51], and for metamaterial elements in [52]. Furthermore, in [53], a method has been demonstrated that avoids the sector formation by a special phase quantization method. In this case, the helical index cannot be continuously changed, but it is limited to a set of discrete values at fixed angular positions.

 figure: Fig. 13.

Fig. 13. Phase shift of an incident plane wave behind a tunable moiré vortex plate, measured with an interferometric setup. The mutual rotation angles and corresponding helical indices $l$ of the combined element are indicated in the different sub-plots.

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B. Axial Refocusing in High Numerical Aperture Microscopy

An interesting application of tunable moiré lenses is to dynamically control the focus position in optical microscopy. For example, in confocal scanning microscopy a focused laser spot has to be swept through a three-dimensional sample volume for high resolution imaging. There are fast methods to produce a two-dimensional transverse scan [in the $(x,y)$ plane], using galvo scanners or acousto-optic deflectors. However, a fast scanning method for the axial direction ($z$ direction) is still highly desired. In principle, such an axial scan can be performed using a tunable moiré lens situated in the back Fourier plane of the microscope objective, or at a corresponding conjugate position. In this case, fast axial changes of the focus position may be achieved by a motorized rotation of one of the moiré sub-elements. However, this method introduces strong spherical aberrations if a standard parabolic tunable moiré lens [according to Eq. (32)] is used in combination with a high NA microscope. The aberrations result in a significant longitudinal extension of the beam focus (i.e., of the Rayleigh range), which strongly deteriorates the axial resolution of the microscope. In [54,55], it is shown that this issue can be resolved by designing a modified phase profile of a tunable moiré lens. The modified lens is able to produce a shift of the focus position of a high NA microscope objective, and, at the same time, compensate for spherical aberrations that normally would be introduced by such a focus shift.

 figure: Fig. 14.

Fig. 14. Principle of refocussing with a high NA objective. An incident plane wave (at the left) is transformed by an objective into a spherical wavefront (red, right side), which produces a diffraction limited focus in its center, at a distance ${f_n} = n{f_0}$, where $n$ is the refractive index of the medium, and ${f_0}$ is the focal length of the microscope objective in vacuum. Shifting the focus by a distance $\Delta z$ to the right (blue arrow) leads to a new distance $x$ from each point of the spherical wavefront to the new focal position. Thus the phase at the radial position (“height”) $r$ of the wavefront has to be increased by an amount of $\Delta \Phi (r) = 2\pi n(x(r) - {f_n})/\lambda$.

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Figure 14 sketches the principle of how a high NA microscope achieves a diffraction limited focus, and how this focus can be shifted to a new axial position. A high performance microscope objective consists of a considerable number of stacked lenses that shape the wavefront of the outgoing wave (for an incident plane wave, coming from the back focal plane of the microscope objective) into the shape of a perfect sphere. The focus of such a wave corresponds to the center of the sphere, i.e., it has a distance ${f_n}$ from each point of the spherical wavefront. There, $n$ is the refractive index of the immersion medium, and ${f_n} = n{f_0}$ is the focal length in the immersion medium (with ${f_0}$ being the focal length in vacuum). An axial shift of the focus poition by a distance $\Delta z$ (small blue arrow) leads to a new distance $x$ of each point on the spherical wavefront to the new focal position. Defining $r$ as the radial coordinate (i.e., the “heigth” of the wavefront), the corresponding phase shift $\Delta \Phi$ at position $r$ can be expressed as

$$\Delta \Phi (r) = \frac{{2\pi n}}{\lambda}(x(r) - {f_n}),$$
where
$$x(r) = \sqrt {({r^2} + {{(a + \Delta z)}^2}} \approx \sqrt {{r^2} + {a^2} + 2a\Delta z} .$$
There, a term of order $\Delta {z^2}$ under the square root has been neglected, since only small focal shifts $\Delta z$ are considered. Setting ${a^2} = f_n^2 - {r^2}$ and expanding the square root into its first two Taylor series terms leads to
$$x(r) \approx {f_n} + \frac{{a\Delta z}}{{{f_n}}} = {f_n} + \sqrt {1 - \frac{{{r^2}}}{{f_n^2}}} \Delta z.$$
An investigation of the applied approximations shows that its practicality depends on the $f$-number of the microscope objective, i.e., on the ratio ${f_n}/D$, where $D = 2{r_{{\rm max}}}$ is the diameter of the microscope aperture. The $f$-number is inversely related to the NA of the microscope objective. For example, for a geometric NA ${\rm NA}/n$ (where NA is the standard NA and $n$ the refractive index of the immersion medium), the approximation is well satisfied for focal shifts $\Delta z \lt 0.1{f_n}$. For larger $f$-numbers (corresponding to smaller NAs), larger focal shifts are possible.

According to Eq. (48), the corresponding phase shift $\Delta \Phi$ at a radial position $r$ is given by

$$\Delta \Phi (r) = \frac{{2\pi n}}{\lambda}\sqrt {1 - \frac{{{r^2}}}{{f_n^2}}} \Delta z.$$
Actually, this is the wavefront of an elliptic wave, which, if used alone, produces a focus with strong spherical aberrations. However, in combination with a high NA objective, it shifts the focal spot, and simultaneously corrects the spherical aberrations that would normally occur by shifting the focus with a “standard” lens.

Ideally, the phase shift $\Delta \Phi (r)$ has to be applied at the position of the spherical wavefront indicated in Fig. 14. Fortunately, this condition is automatically fulfilled by a high NA objective that satisfies the so-called sine condition, if a planar DOE is imaged by a $4f$-relay system (i.e., a Kepler telescope setup) at the back focal plane of the objective. A corresponding setup is shown in Fig. 15. The reason is that a microscope objective that satisfies the sine condition automatically transforms an original plane image of the DOE at the back focal plane of the microscope objective into a spherical image in the front pupil plane, which adapts to the spherical wavefront indicated in Fig. 14.

The required phase function to shift the focal position by $\Delta z$ given by Eq. (51) is rotationally symmetric, and its magnitude depends linearly on the desired focal shift $\Delta z$. Therefore, this phase transmission function can be straightforwardly implemented as a tunable moiré element using the phase transmission functions of the sub-lenses described in Eq. (17). However, in high resolution microscopic applications, it is desired to keep a rotationally symmetric point spread function, which is achieved with the quantized version of the respective moiré elements according to Eq. (31). The phase transmission functions of the two respective sub-elements for a high NA focus shifting moiré element are thus

$$\begin{array}{l}{\Phi _{{\rm sub,1}}}(r) = {{\rm mod}_{2\pi}}\left\{{{\rm round}\left[{{a_0}\frac{{2\pi n}}{\lambda}\sqrt {1 - \frac{{{r^2}}}{{f_n^2}}}} \right]\varphi} \right\},\\{\Phi _{{\rm sub,2}}}(r) = - {\Phi _{{\rm sub,1}}}(r),\end{array}$$
where ${a_0}$ is a user selectable constant proportional to the magnitude of the focal shift $\Delta z$ as a function of the mutual rotation angle $\theta$ between the two sub-elements, i.e.,
$$\Delta z = {a_0}\theta .$$
The focal shift $\Delta z$ can be performed in both positive and negative directions, depending on the sign of the mutual rotation angle $\theta$. The absolute magnitude of the constant ${a_0}$ is again limited by the sampling condition quoted in Eq. (39). Furthermore, the diffraction efficiency of the quantized moiré element depends again on the mutual rotation angle $\theta$ according to Eq. (38).
 figure: Fig. 15.

Fig. 15. Setup for shifting the focus position of a high NA microscope objective using a rotationally tunable moiré lens (MDOE).

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The phase transmission functions of the two sub-elements contain the focal length ${f_n} = n{f_0}$ within the immersion medium as a parameter, which determines the detailed shape of the rotationally symmetric transmission function. Therefore, the respective moiré element is in principle designed only for a microscope objective with a fixed focal length of ${f_n}$. However, since the curvature of the wavefront projected at the front pupil of the microscope objective depends reciprocally on the magnification of the ${4}f$ setup sketched in Fig. 15, the same moiré element can be used for focal shifting in combination with different microscope objectives. If ${f_n}$ is increased by a certain factor, then also the magnification of the ${4f}$ setup has to be increased by the same factor, which can be achieved by selecting appropriate lens combinations in the ${4}f$ setup.

An experimental investigation of this method in [55] confirms that refocusing is possible while maintaining the diffraction limited resolution of the microscope objective. The same method was applied in [54] for two-photon confocal microscopy.

8. MOIRÉ ELEMENTS FOR POLYCHROMATIC APPLICATIONS

The moiré elements described in the preceding sections are based on the superposition of two sub-elements designed as first order DOEs. Therefore, each sub-element primarily diffracts into the first diffraction order. As a consequence, the moiré elements are strongly dispersive. This means that the optical properties of the combined moiré element depend on the readout wavelength. For example, the focal length of a moiré lens changes as an inverse function of the readout wavelength. Furthermore, the diffraction efficiency of a moiré element decreases with increasing deviation of the readout wavelength from the design wavelength.

It is, however, possible to manufacture DOEs in a way that strongly reduces the dispersion issues, namely, by using higher order DOEs [3,46,5663], or numerically optimized “spectral diffractive lenses” (SDLs) [64,65]. Although such a multi-order DOE is still a thin optical element, with a surface amplitude typically less than 10 µm, it has very different dispersion properties as compared to a corresponding single order element.

For a single order DOE, designed for a wavelength ${\lambda _1}$, the surface phase profile is modulated within a range between zero and $2\pi$, which corresponds to a variation of the optical path length of the transmitted beam between zero and ${\lambda _1}$. If one neglects the dispersion of the refractive index of the DOE material, then a beam with a smaller wavelength of ${\lambda _m} = {\lambda _1}/m$ (where $m =1,2,\ldots$ is an integer number), transmitted through the same DOE, experiences a phase modulation in the range between zero and $2\pi m$. Such a phase modulation range of $2\pi m$ leads to diffraction of the transmitted wave into the $m$th diffraction order. This means that the sinus of the diffraction angle will be increased by a factor of $m$. On the other hand, the reduction in wavelength ${\lambda _m}$ by a factor of $1/m$ (with respect to the design wavelength ${\lambda _1}$) leads to a corresponding reduction of the sinus of the diffraction angle by $1/m$. Altogether, these two effects exactly cancel each other, such that the beam with the new wavelength ${\lambda _m}$ will be diffracted at the same angle, and with the same efficiency, as the original one. Therefore, any first order DOE with design wavelength ${\lambda _1}$ has the same diffraction properties at a set of so-called harmonic wavelengths ${\lambda _m} = {\lambda _1}/m$. Particularly, this applies to diffractive lenses that thus have identical focal lengths at a corresponding set of harmonic wavelengths.

For the case of combined moiré elements, the situation is similar [66,67]. In principle, each of the already discussed sub-elements can be designed as a multi-order DOE to become polychromatic. This method works perfectly well for non-quantized (sector producing) moiré elements. However, for quantized elements, there are some additional issues that will be discussed in the following.

Consider a continuous $l$th order DOE (with $N \mapsto \infty$) with a design wavelength ${\lambda _d}$, according to Eq. (9). Its diffraction efficiency for the $m$th diffraction order for an arbitrary readout wavelength $\lambda$ is given by Eq. (15). The diffraction efficiency has a strongly peaked maximum of magnitude one, if the arguments of the two sinc-functions are zero. For a continuous DOE with an “infinite” number of phase leveles ($N \mapsto \infty$), this requires that the argument of the second sinc-function vanishes. This happens for a series of “harmonic” wavelengths ${\lambda _m}$, and corresponding diffraction orders $m$, if

$${\lambda _m} = \frac{{l{\lambda _d}}}{m}.$$

There, ${\lambda _m}$ is called the $m$th harmonic of the fundamental wavelength $l{\lambda _d}$ (which corresponds to the first harmonic).

For better comprehensibility, the following considerations will be performed by means of a parabolic moiré lens; however, the basic principle of multi-order moiré elements can be analogously applied to all transmission functions of the combined element, which are rotationally symmetric. Thus, in the following, it is assumed that the phase transmission functions of the two sub-elements are designed according to Eq. (22). The optical power of the corresponding moiré lens in its main sector as a function of the readout wavelength $\lambda$ and the mutual rotation angle $\theta$ was shown to be $P = a\theta /\pi$ [Eq. (24)]. However, this equation was derived for a readout diffraction order $l$, which corresponds to the design diffraction order. For another readout diffraction order $m$, this formula has to be adapted by a pre-factor $m/l$. This corresponds to the well-known fact that doubling the readout diffraction order also doubles the optical power of diffractive lenses. Thus the optical power of an $l$th order moiré lens, which is read out in $m$th order, is

$$P(\lambda) = \frac{m}{l}\frac{{a\lambda \theta}}{\pi}.$$

If the moiré lens is read out with one of its harmonic wavelengths ${\lambda _m}$ [Eq. (54)], where it has maximal diffraction efficiency, this leads to

$$P({\lambda _m}) = \frac{{a{\lambda _d}\theta}}{\pi}.$$
Thus, the focal length as a function of the mutual rotation angle $\theta$ is identical for all harmonic wavelengths ${\lambda _m}$. There, the moiré lens also achieves its maximal diffraction efficiency [66,67].
 figure: Fig. 16.

Fig. 16. Phase profiles of three different types of sixth order moiré lenses at a design wavelength of ${\lambda _6} = 532\;{\rm nm} $. Column (a) refers to a non-quantized (sector producing) moiré lens, column (b) to a “weakly” quantized moiré lens type (which suppresses sector formation only at its design wavelength ${\lambda _6}$), and column (c) to a fully quantized moiré lens, which suppresses sector formation at all harmonic wavelengths. For all three cases, it is assumed that the two sub-elements of the combined moiré lens are mutually rotated by an angle of 15°. The rotated sub-elements of each lens type are sketched in the upper row. The next three rows show the phase profiles of the transmitted beams for wavelengths of 456 nm, 532 nm, and 638 nm, which correspond to the seventh, sixth, and fifth diffraction orders, respectively. The phase profiles are drawn modulo-$2\pi$ with respect to the corresponding readout wavelengths.

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This behavior is illustrated in Fig. 16(a) for a $l =6$th order non-quantized parabolic moiré lens with a design wavelength of ${\lambda _d} = 532\;{\rm nm}$. The upper image shows the phase profile of one of the sub-elements, which is rotated by 15° with respect to its conjugate counterpart (not shown). The lower three images show the phase transmission functions (modulo $2\pi$) of three waves of different wavelengths, transmitted through the corresponding combined moiré lens. The three wavelengths of 456 nm, 532 nm, and 638 nm, are the seventh, sixth, and fifth harmonics of the fundamental wavelength $l{\lambda _d} = 3192\;{\rm nm}$. The images demonstrate that three fully modulated “perfect” lenses are formed within the main sector of the combined moiré lens. In principle, the diffraction efficiency of these main sectors can achieve 100%. Note that the spatial frequency of the transmission modulation is largest at the seventh order, and decreases for the lower orders. Nevertheless, all three lenses have the same optical power at their respective wavelengths, since the decrease in optical power by a lower spatial modulation frequency is compensated for by the fact that the readout wavelength increases by the same relative amount, which leads to a larger optical power. However, a sector within an angular range corresponding to the mutual rotation angle of 15° appears in all of the three transmission functions.

As a next step, it will be checked whether the same procedure of increasing the cutoff phase within the modulo function can also be applied to get two sub-elements, which together form an $l$th order quantized moiré lens, which is supposed to produce a sector-free combined transmission function. The corresponding two surface functions ${H_{{q}, \pm 1}}$ (according to Section 5) are

$${H_{q}, {\pm 1}} = \pm \frac{{{\lambda _d}}}{{2\pi ({n_d} - 1)}}{{\rm mod}_{2\pi l}}\{{\rm round}[a{r^2}]\varphi \} .$$
If the two sub-elements are mutually rotated by an angle $\theta$ and combined into a moiré lens, the corresponding phase transmission function for readout with one of the harmonic wavelengths ${\lambda _m}$ becomes
$${\Phi _{{q,{\rm joint}}}} = \left\{{\begin{array}{*{20}{l}}{\frac{m}{l}{{{\rm mod}}_{2\pi l}}\left\{{{\rm round}[a{r^2}]\theta} \right\}}&{{\rm for}\quad \theta \le \varphi \lt 2\pi}\\ {\frac{m}{l}{{{\rm mod}}_{2\pi l}}\left\{{{\rm round}[a{r^2}](\theta - 2\pi)} \right\}}&{{\rm for}\quad 0 \le \varphi \lt \theta}\end{array}.} \right.$$
There, Eqs. (6) and (54) have been applied under the assumption of a negligible dispersion of the refractive index. In contrast to the quantized single order moiré lenses in Section 5, there is a factor $m/l$ in front of the two phase transmission functions of the two sectors. Therefore, the phase transmission function in the second sector does not differ just by integer multiples of $2\pi$ from that in the first sector, as in the case of single order quantized moiré lenses, but by integer multiples of $2\pi m/l$. Therefore, the phase transmission functions within the two sectors are not equal any more, with the exception of the cases where $m/l$ is an integer number, particularly if $m$ equals $l$, which means that the readout wavelength corresponds to the design wavelength. As a consequence, sector formation is suppressed only if the moiré lens is read out at its design wavelength, whereas sectors are reappearing at the adjacent harmonic wavelengths. However, the optical power at all harmonic wavelengths within the main sector is again identical, and corresponds to that calculated according to Eq. (56).

This behavior is demonstrated in Fig. 16(b). There, the sixth order quantized phase function for one of the sub-elements of a quantized moiré lens with a design wavelength of ${\lambda _d} = 532\;{\rm nm}$ according to Eq. (57) is sketched in the upper position. Combining the corresponding two conjugate sub-elements with a mutual rotation angle of  $\theta ={ 15^ \circ}$ yields the three lower phase transmission functions at the harmonic wavelength 456 nm (seventh order), 532 nm (sixth order), and 638 nm (fifth order). As expected from the previous considerations, sector formation is suppressed at the design wavelength of 532 nm, but it reappears at the other diffraction orders. A detailed investigation shows that the diffraction efficiency at the design wavelength ${\lambda _d}$ corresponds to that of a single order quantized moiré lens according to Eq. (38), whereas the diffraction efficiency at the other harmonic wavelengths is lower, due to the re-appearance of the sector.

The joint transmission function of the quantized multi-order lens in Eq. (58) suggests an extended method of quantization, which is supposed to suppress sector formation at all diffraction orders simultaneously. If the design order $l$ would be inserted as an additional factor in front of the “round” operation, then it would be canceled by multiplication with the pre-factor $m/l$. In this case, the phase transmission functions of the two sectors would be different by integer multiples of $2\pi m$ (where $m$ is an integer number), and thus the two phase terms were indistinguishable again. To insert such an additional factor $l$ in front of the rounding operation without violating the undersampling condition discussed in Section 6, this multiplication has to be accompanied by a division with the factor $l$, which is performed within the rounding operation. The corresponding surface profile function then becomes

$${H_{{\rm fq}, \pm 1}} = \pm \frac{{{\lambda _d}}}{{2\pi ({n_d} - 1)}}{{\rm mod}_{2\pi l}}\left\{l {\rm round}\left[{\frac{{a{r^2}}}{l}} \right]\varphi \right\} .$$
In the following, this surface profile function will be called “fully quantized”.

Due to division by the factor $l$ within the rounding operator, the surface profile becomes more “coarse,” i.e., there are fewer quantization levels within a sawtooth period. According to Eq. (33), the decrease in number of quantization levels by the factor $1/l$ leads to a decrease in diffraction efficiency according to

$$\eta = {{\rm sinc}^2}\left({\frac{{l\theta}}{{2\pi}}} \right).$$
Due to the factor $l$ in the sinc-function, the decrease in efficiency with increased mutual tuning angle is larger than that of the “weakly” quantized lens (Eq. 58). Thus, the usable rotation range, and accordingly the tuning range of the optical power, is reduced for the case of a fully quantized moiré lens by the factor $1/l$ with respect to the other lens types.
 figure: Fig. 17.

Fig. 17. Comparison of the dispersion properties of different types of tunable moiré lenses. (a) First order quantized lens according to Eq. (32). (b) Sixth order non-quantized lens (sector producing) according to Eq. (22). (c) Sixth order quantized lens according to Eq. 57. (d) Sixth order fully quantized lens according to Eq. (59). All lenses are simulated for a design wavelength of 532 nm, which corresponds to the first diffraction order in (a), and to the sixth diffraction order in (b)–(d). The mutual rotation angles of 90° in (a), and of 15° in (b)–(d) are chosen to produce the same optical power of 26 dpt at the design wavelength for all types of lenses. The upper plots show the diffraction efficiency (color coded) as a function of the readout wavelength, and the optical power of the lens. The lower graphs show the diffraction efficiency at the nominal optical power of 26 dpt as a function of the wavelength, which corresponds to a horizontal cut through the data of the upper plot along the line indicated in red.

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This is demonstrated in Fig. 16(c). There, the sixth order quantized phase function for one of the sub-elements of a fully quantized moiré lens with a design wavelength of ${\lambda _d} = 532$ nm [Eq. (59)] is sketched in the upper position. As compared to the phase transmission functions of the other two lens types in (a) and (b), the spatial structure is more coarse. Combining the corresponding two conjugate sub-elements with a mutual rotation angle of $\theta ={ 15^ \circ}$ yields the three lower phase transmission functions at the harmonic wavelength 456 nm (seventh order), 532 nm (sixth order), and 638 nm (fifth order). As expected, the sector formation is suppressed at all of these diffraction orders, i.e., the phase transmission functions are rotationally symmetric. However, a closer view of these phase transmission functions reveals that their quantization is coarser that that of the other lens types. For example, the sixth order phase transmission function consists of only four resolved phase levels (corresponding to four distinct gray values in the plot). The structure becomes even coarser for higher diffraction orders (e.g., the seventh order at 456 nm). Therefore, the diffraction efficiency of fully quantized lenses is lower than that of the other lens types, and it decreases with increasing diffraction order.

The advantage of a fully quantized moiré lens is that the phase transmission function for all harmonic wavelengths is rotationally symmetric, which means that its point spread function is almost diffraction limited at these wavelengths.

If the readout wavelength is not one of the harmonic wavelengths, then the diffraction efficiency will mainly be distributed between the two adjacent diffraction orders. Thus, if one scans the wavelength between ${\lambda _m}$ and ${\lambda _{m + 1}}$, then the diffraction efficiency changes continuously from full efficiency in the $m$th order, to full efficiency in the ($m + 1$)th order. The focal length of the $m$th and $(m + 1)$th orders will be the same. In the range between the two diffraction orders, dispersion of the optical power will occur according to Eq. (24). However, due to the small wavelength interval between two harmonic wavelengths, the optical power power changes in a range that is by a factor $1/m$ smaller than that of a single order moiré lens.

Such a comparison of the dispersion characteristics of different types of tunable moiré lenses is shown in Fig. 17. The figure compares the diffraction efficiencies (color coded) as a function of the readout wavelength, and of the corresponding optical power of the lenses. In Fig. 17(a), the performance of a first order quantized lens [with a phase transmission function according to Eq. (32)] with a design wavelength of 532 nm (and the design parameter $a = 9.8 \cdot {10^7} \;{{\rm m}^{- 2}}$) is shown. The mutual rotation angle between its two sub-elements was selected to be 90°, which corresponds to an optical power of 26 dpt at the design wavelength. The upper plot in the figure shows the corresponding dispersion properties, whereas the lower plot shows the diffraction efficiency at the nominal optical power of 26 dpt as a function of the readout wavelength (which corresponds to a horizontal cut through the upper graph at that optical power, indicated by a red horizontal line). As expected, the optical power of the lens is a linear function of the readout wavelength, with a maximal efficiency of 81% at the design wavelength 532 nm, which is expected for the rotation angle $\theta ={ 90^ \circ}$ according to Eq. (13).

Figure 17(b) shows analogous plots for the situation of a sixth order non-quantized lens (which produces a sector) with a design wavelength of 532 nm (corresponding to the sixth diffraction order) according to Eq. (32). Since the spatial frequency of the higher order element is lower by a factor corresponding to the design diffraction order, the parameter $a$ has been increased by the same factor to $a = 5.9 \cdot {10^8} \;{{\rm m}^{- 2}}$ without violating the undersampling condition. Therefore, the corresponding moiré lens has the same optical power of 26 dpt at a mutual rotation angle of just 15°, for which the simulation was performed. The upper graph shows that the diffraction efficiency is now peaked at a sequence of diffraction orders (numbers indicated in the plot). All indicated maxima of the efficiency appear at the same optical power of 26 dpt, as expected. The plot below shows the efficiency as a function of the wavelength at that optical power value. It shows that the maximal efficiency of all diffraction orders is the same, and corresponds to about 96%, which is expected for the mutual rotation angle of 15°, since the efficiency of a sector lens corresponds to the ratio of the main sector area to the whole lens area, which is ${345^ \circ}{/360^ \circ} = 96\%$. An experimental demonstration and investigation of such a lens are reported in [67].

The next plot in Fig. 17(c) shows the situation for a quantized sixth order lens [Eq. (32)]. The upper graph shows that the dispersion properties are similar to those of the non-quantized lens in (b). However, now the diffraction efficiency is maximal (with a value of 99.4%) at the design wavelength of 532 nm. This is expected, because the quantization scheme favors this wavelength by suppressing sector formation. There, the diffraction efficiency is given by Eq. (38). However, the efficiency of the other diffraction orders is lower, since in these cases, the sector is not suppressed.

Finally, Fig. 17(d) shows the diffraction properties of the fully quantized lens at the same mutual rotation angle of 15°. Again, the upper plot is similar to the plots in (b) and (c), i.e., the efficiency is peaked at the harmonic wavelengths indicated in the plot, and the optical power at these peaks is 26 dpt for all diffraction orders. A more detailed view of the diffraction efficiency in the lower graph shows that it is on average lower than that of the lens types in (b) and (c), and that the efficiency of the diffraction orders is not constant, which is expected from Eq. (60). Further simulations, which are not depicted here, show that the decrease in diffraction efficiency as a function of the mutual rotation angle is by a factor $l$ faster than that of the other lens types. For example, to obtain an average diffraction efficiency of at least 80%, the mutual rotation angle of the quantized lens in (c) has to be limited to a range between ${-}{90}^\circ$ and ${+}{90}^\circ$, whereas the corresponding rotation range of the fully quantized lens in (d) is reduced by the factor $1/l = 1/6$ to a range between ${-}{15}^\circ$ and ${+}{15}^\circ$. This also reduces the tuning range of the optical power, since the dependence of the optical power on the rotation angle is the same for the “weakly” and fully quantized moiré lenses. Nevertheless, a fully quantized moiré lens may be the best choice for applications where a smaller tuning range of the optical power is sufficient, since in this case, the point spread function is rotationally symmetric and almost diffraction limited for all harmonic readout wavelengths.

Figures 17(b)–17(d) also show that the dispersion behavior of multi-order lenses in the vicinity of the harmonic wavelengths is rather moderate, i.e., the deviation of optical power from the nominal optical power of 26 dpt at the harmonic wavelengths is only on the order of 10%. The residual error even decreases if moiré lenses of increasing order are used [3,66]. This makes the concept also suitable for medium performance imaging applications with white light, if a slight reduction in sharpness within the color spectrum can be accepted.

9. DISCUSSION AND CONCLUSION

In the preceding sections, different types and implementations of tunable combined diffractive elements have been discussed, with the main focus on rotationally tunable moiré elements. Advantageous features of these elements are their wide tuning range, their exact and potentially fast tuning characteristics, their high light intensity tolerance, the almost diffraction limited focus quality, and the robustness against environmental conditions such as accelerations and vibrations. These are interesting properties for laser beam steering applications such as laser cutting/welding or marking, or for 3D laser printing. For imaging applications, a tunable moiré lens can be straightforwardly used for focusing at objects in different distances. There, the dispersive properties of first order diffractive moiré lenses may be a disadvantage, which, however, can be partially eliminated by computationally post processing of the recorded images [68,69]. This task is facilitated by the fact that the dispersion behavior of a certain moiré lens is exactly known. It is, however, also possible to design polychromatic tunable moiré elements that have the same optical properties at a set of preselected harmonic wavelengths. In this case, the concept of multi-order DOEs can be applied, which turns out to be compatible with an adapted quantization method, such that the formation of different sectors is suppressed at all harmonic wavelengths simultaneously.

On the other hand, the strong dispersion of diffractive moiré elements may also be an advantage that can be utilized to control the overall dispersion of combined diffractive and refractive optical systems [56,70,71], or to shape the temporal profile of ultrashort laser pulses [72,73].

An obvious application of tunable lenses is to use a set of them to construct more sophisticated optical systems, such as a zoom objective [74]. There, it might be assumed that the limited diffraction efficiency of moiré lenses is a significant problem, since it is supposed to produce a background of non-focused light. However, in practice, it turns out that this effect is rather low, since the undesired diffraction orders of moiré lenses typically have a very different optical power than the main order. Therefore, most of the light diffracted into parasitic orders does not pass through the optical system, and does not reach the detector. Thus, in practical applications, an increase in the background level due to undesired diffraction orders is often negligible, even if non-quantized (sector producing) moiré lenses are employed.

Currently, there are efforts to increase the speed of focus tuning in optical systems such as microscopes or laser beam manipulation. There, an increase in the speed of optical manipulation has been reported by a different implementation of the Alvarez concept, where one of the sub-elements is optically imaged onto a second one, or imaged onto itself in a reflective setup [55,75]. Applying this idea to the case of a reflective moiré sub-element, a potentially faster rotation mount could be constructed, since the reflective element can be mounted on its rear side on the axis of a fast galvo motor (note that this is not possible in a transmissive setup, since in this case, the motor and the axis would be located within the beam path). It is supposed that this will enable rotation frequencies in the kilohertz range, which can be used for high speed dynamic refocusing.

Recently, an interesting application was reported where a tunable moiré lens was successfully used to transport ultracold atoms trapped in the adjustable focal spot of a laser beam [76].

Quite recently, a company (Diffratec) has been founded that produces tunable moiré elements for scientific and industrial applications.

The concept of tunable moiré elements is not restricted to the visible range, but it may also have interesting applications in the ultraviolet [77], or, on the other side, in the infrared [78,79], terahertz, or microwave regime. There, the corresponding elements might be fabricated by 3D plotting [80,81]. The concept may also be useful for manipulation of acoustic waves [82], for example, in medical ultrasound devices. Currently, there is also a considerable effort to transfer the same concept into the field of optics with meta-materials, which has interesting potential advantages, but also poses new challenges [8386].

Funding

Österreichische Forschungsförderungsgesellschaft (864729).

Acknowledgment

The author thanks Monika Ritsch-Marte for her continuous advancement and contribution to the development of the described tunable moiré elements. The concept of refocusing with aberration control in high NA microscopes (Section 7.B) has been devised and realized by Alexander Jesacher. The company Diffratec was involved in designing and investigating various types of tunable moiré elements, which are discussed in this tutorial.

Disclosures

The author declares no conflicts of interest.

Data Availability

No data were generated or analyzed in the presented research.

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References

  • View by:

  1. I. D. Bagbaya, “On the history of the diffraction grating,” Sov. Phys. Usp. 15, 660–661 (1972).
    [Crossref]
  2. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
    [Crossref]
  3. S. Sinzinger and M. Testorf, “Transition between diffractive and refractive micro-optical components,” Appl. Opt. 34, 5970–5976 (1995).
    [Crossref]
  4. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13, 150–155 (1969).
    [Crossref]
  5. J. A. Jordan, P. M. Hirsch, L. B. Lesem, and D. L. Van Rooy, “Kinoform lenses,” Appl. Opt. 9, 1883–1887 (1970).
    [Crossref]
  6. G. Friedel, “Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen,” Comptes Rendus 157, 1533–1536 (1913).
  7. L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent3,305,294 (December3, 1964).
  8. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9,1669–1671 (1970).
    [Crossref]
  9. A. W. Lohmann and D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [Crossref]
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2021 (4)

L. Chen, M. Ghilardi, J. J. C. Busfield, and F. Carpi, “Electrically tunable lenses: a review,” Front. Robot. AI 8, 678046 (2021).
[Crossref]

F. Hartmann, L. Penkner, D. Danninger, N. Arnold, and M. Kaltenbrunner, “Soft tunable lenses based on zipping electroactive polymer actuators,” Adv. Sci. 8, 2003104 (2021).
[Crossref]

M. K. Chen, Y. Wu, L. Feng, Q. Fan, M. Lu, T. Xu, and D. P. Tsai, “Principles, functions, and applications of optical meta-lens,” Adv. Opt. Mater. 9, 2001414 (2021).
[Crossref]

Y. Luo, C. H. Chu, S. Vyas, H. Y. Kuo, Y. H. Chia, M. K. Chen, X. Shi, T. Tanaka, H. Misawa, Y.-Y. Huang, and D. P. Tsai, “Varifocal metalens for optical sectioning fluorescence microscopy,” Nano Lett. 21, 5133–5143 (2021).
[Crossref]

2020 (15)

J. Engelberg and U. Levy, “The advantages of metalenses over diffractive lenses,” Nat. Commun. 11, 1991 (2020).
[Crossref]

K. Iwami, C. Ogawa, T. Nagase, and S. Ikezawa, “Demonstration of focal length tuning by rotational varifocal moiré metalens in an IR-A wavelength,” Opt. Express 28, 35602–35614 (2020).
[Crossref]

E. Castro-Camus, M. Koch, and A. I. Hernandez-Serrano, “Additive manufacture of photonic components for the terahertz band,” J. Appl. Phys. 127, 210901 (2020).
[Crossref]

Y. Wei, Y. Wang, X. Feng, S. Xiao, Z. Wang, T. Hu, M. Hu, J. Song, M. Wegener, M. Zhao, J. Xia, and Z. Yang, “Compact optical polarization-insensitive zoom metalens doublet,” Adv. Opt. Mater. 8, 2000142 (2020).
[Crossref]

I. Sieber, R. Thelen, and U. Gengenbach, “Assessment of high-resolution 3D printed optics for the use case of rotation optics,” Opt. Express 28, 13423–13431 (2020).
[Crossref]

S. Banerji, J. Cooke, and B. Sensale-Rodriguez, “Impact of fabrication errors and refractive index on multilevel diffractive lens performance,” Sci. Rep. 10, 14608 (2020).
[Crossref]

N. Bregenzer, M. Bawart, and S. Bernet, “Zoom system by rotation of toroidal lenses,” Opt. Express 28, 3258–3269 (2020).
[Crossref]

C. M. Gómez-Sarabia, L. M. Ledesma-Carrillo, and J. Ojeda-Castaneda, “Helical phase masks for controlling optical vortices: necessary and sufficient conditions,” Opt. Commun. 470, 126047 (2020).
[Crossref]

F. Balli, M. A. Sultan, and J. T. Hastings, “Rotationally tunable polarization-insensitive metasurfaces for generating vortex beams,” Proc. SPIE 11460, 114602F (2020).
[Crossref]

M. A. May, M. Bawart, M. Langeslag, S. Bernet, M. Kress, M. Ritsch-Marte, and A. Jesacher, “High-NA two-photon single cell imaging with remote focusing using a diffractive tunable lens,” Biomed. Opt. Express 11, 7183–7191 (2020).
[Crossref]

M. Bawart, M. A. May, T. Öttl, C. Roider, S. Bernet, M. Schmidt, M. Ritsch-Marte, and A. Jesacher, “Diffractive tunable lens for remote focusing in high-NA optical systems,” Opt. Express 28, 26336–26347 (2020).
[Crossref]

D. Werdehausen, S. Burger, I. Staude, T. Pertsch, and M. Decker, “General design formalism for highly efficient flat optics for broadband applications,” Opt. Express 28, 6452–6468 (2020).
[Crossref]

L. L. Doskolovich, R. V. Skidanov, E. A. Bezus, S. V. Ganchevskaya, D. A. Bykov, and N. L. Kazanskiy, “Design of diffractive lenses operating at several wavelengths,” Opt. Express 28, 11705–11720 (2020).
[Crossref]

N. Bregenzer, T. Öttl, M. Zobernig, M. Bawart, S. Bernet, and M. Ritsch-Marte, “Demonstration of a multi-color diffractive lens with adjustable focal length,” Opt. Express 28, 30150–30163 (2020).
[Crossref]

C. M. Gómez-Sarabia and J. Ojeda-Castaneda, “Hopkins procedure for tunable magnification: surgical spectacles,” Appl. Opt. 59,D59–D63 (2020).
[Crossref]

2019 (9)

Y. Cui, G. Zheng, M. Chen, Y. Zhang, Y. Yang, J. Tao, T. He, and Z. Li, “Reconfigurable continuous-zoom metalens in visible band,” Chin. Opt. Lett. 17, 111603 (2019).
[Crossref]

Z. Liu, Z. Du, B. Hu, W. Liu, J. Liu, and Y. Wang, “Wide-angle moiré metalens with continuous zooming,” J. Opt. Soc. Am. B 36, 2810–2816 (2019).
[Crossref]

N. Yilmaz, A. Ozdemir, A. Ozer, and H. Kurt, “Rotationally tunable polarization-insensitive single and multifocal metasurface,” J. Opt. 21, 045105 (2019).
[Crossref]

Y. Guo, M. Pu, X. Ma, X. Li, R. Shi, and X. Luo, “Experimental demonstration of a continuous varifocal metalens with large zoom range and high imaging resolution,” Appl. Phys. Lett. 115, 163103 (2019).
[Crossref]

C. Lan, Z. Zhou, H. Ren, S. Park, and S. H. Lee, “Fast-response microlens array fabricated using polyvinyl chloride gel,” J. Mol. Liq. 283, 155–159 (2019).
[Crossref]

D. Tarrazó-Serrano, S. Pérez-López, P. Candelas, A. Uris, and C. Rubio, “Acoustic focusing enhancement in Fresnel zone plate lenses,” Sci. Rep. 9, 7067 (2019).
[Crossref]

H. Liang, A. Martins, B.-H. V. Borges, J. Zhou, E. R. Martins, J. Li, and T. F. Krauss, “High performance metalenses: numerical aperture, aberrations, chromaticity, and trade-offs,” Optica 6, 1461–1470 (2019).
[Crossref]

S. Banerji, M. Meem, A. Majumder, F. G. Vasquez, B. Sensale-Rodriguez, and R. Menon, “Imaging with flat optics: metalenses or diffractive lenses?” Optica 6, 805–810 (2019).
[Crossref]

M. Meem, S. Banerji, A. Majumder, F. G. Vasquez, B. Sensale-Rodriguez, and R. Menon, “Broadband lightweight flat lenses for longwave-infrared imaging,” Proc. Natl. Acad. Sci. USA 116, 21375–21378 (2019).
[Crossref]

2018 (6)

S. Colburn, A. Zhan, and A. Majumdar, “Varifocal zoom imaging with large area focal length adjustable metalenses,” Optica 5, 825–831 (2018).
[Crossref]

S. Wang, P. C. Wu, V. C. Su, Y. C. Lai, M. K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T. T. Huang, J. H. Wang, R. M. Lin, C. H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic metalens in the visible,” Nat. Nanotechnol. 13, 227–232 (2018).
[Crossref]

M. Bawart, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Remote focusing in confocal microscopy by means of a modified Alvarez lens,” J. Microsc. 271, 337–344 (2018).
[Crossref]

L. L. Doskolovich, E. A. Bezus, A. A. Morozov, V. Osipov, J. S. Wolffsohn, and B. Chichkov, “Multifocal diffractive lens generating several fixed foci at different design wavelengths,” Opt. Express 26, 4698–4709 (2018).
[Crossref]

J. Yang, P. Twardowski, P. Gérard, W. Yu, and J. Fontaine, “Chromatic analysis of harmonic Fresnel lenses by FDTD and angular spectrum methods,” Appl. Opt. 57, 5281–5287 (2018).
[Crossref]

I. Sieber, P. Stiller, and U. Gengenbach, “Design studies of varifocal rotation optics,” Opt. Eng. 57, 125102 (2018).
[Crossref]

2017 (5)

S. F. Busch, J. C. Balzer, G. Bastian, and G. E. Town, “Extending the Alvarez-lens concept to arbitrary optical devices: tunable gratings, lenses, and spiral phase plates,” IEEE Trans. Terahertz Sci. Technol. 7, 320–325 (2017).
[Crossref]

S. Bernet and M. Ritsch-Marte, “Multi-color operation of tunable diffractive lenses,” Opt. Express 25, 2469–2480 (2017).
[Crossref]

D.-S. Choi, J. Jeong, E.-J. Shin, and S.-Y. Kim, “Focus-tunable double convex lens based on non-ionic electroactive gel,” Opt. Express 25, 20133–20141 (2017).
[Crossref]

Y.-H. Lin, Y.-J. Wang, and V. Reshetnyak, “Liquid crystal lenses with tunable focal length,” Liq. Cryst. Rev. 5, 111–143 (2017).
[Crossref]

I. Sieber, T. Martin, and P. Stiller, “Tunable refraction power by mutual rotation of helical lens parts,” Proc. SPIE 10375, 103750L (2017).
[Crossref]

2016 (8)

A. Grewe, P. Fesser, and S. Sinzinger, “Diffractive array optics tuned by rotation,” Appl. Opt. 56, A89–A96 (2016).
[Crossref]

F. Heide, Q. Fu, Y. Peng, and W. Heidrich, “Encoded diffractive optics for full-spectrum computational imaging,” Sci. Rep. 6, 33543 (2016).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

S. Petsch, S. Schuhladen, L. Dreesen, and H. Zappe, “The engineered eyeball, a tunable imaging system using soft-matter micro-optics,” Light Sci. Appl. 5, e16068 (2016).
[Crossref]

W. Peng, M. Nabil, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6, 21545 (2016).
[Crossref]

Y. Peng, Q. Fu, F. Heide, and W. Heidrich, “The diffractive achromat full spectrum computational imaging with diffractive optics,” ACM Trans. Graph. 35, 31 (2016).
[Crossref]

A. Grewe and S. Sinzinger, “Efficient quantization of tunable helix phase plates,” Opt. Lett. 41, 4755–4758 (2016).
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Data Availability

No data were generated or analyzed in the presented research.

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Figures (17)

Fig. 1.
Fig. 1. Construction principle of a kinoform lens. (a) Surface profile of a refractive lens. (b) Surface profile wrapped by a modulo-operation into an almost flat structure with a sawtooth profile. (c) Corresponding phase transmission image. Gray levels correspond to phase shifts impressed on a transmitted wave.
Fig. 2.
Fig. 2. Principle of an Alvarez lens (sketch taken from the original patent of Alvarez [7]).
Fig. 3.
Fig. 3. Combined DOEs (not to scale), whose optical properties (such as an adjustable focal length) are tunable by a relative rotation of the two sub-elements. (Systems discussed in detail in the tutorial.)
Fig. 4.
Fig. 4. Diffractive moiré lens with a focal length tunable by rotation. (a) Phase profile of one of the two sub-elements (not to scale). Gray levels correspond to phase values in an interval between zero and $2\pi$. (b) Photograph of one of the two sub-elements. (c) Magnifying lens effect of the combined moiré lens placed some millimeters above a text.
Fig. 5.
Fig. 5. Combined DOEs description in polar coordinated $(r,\varphi)$: ${r_{{\rm max}}}$: maximal radius; $d$: distance between DOEs; $p$: feature size (pixel size); $\theta$: mutual rotation angle.
Fig. 6.
Fig. 6. Adding the phase functions of two mutually rotated sub-elements DOE 1 and DOE 2, which are designed to produce the tunable phase profile of a parabolic lens (at the right). Gray levels in the images correspond to phase values between zero and $2\pi$. The resulting phase function shows two sectors that include parabolic lens profiles with different focal lengths.
Fig. 7.
Fig. 7. Sector forming moiré lens. Left: simulation of the phase transmission function of a moiré lens, where the two sub-elements are mutually rotated by an angle of $\theta ={ 90^ \circ}$. Right: experimental demonstration of the formed sector by means of a tunable moiré lens, consisting of two mutually rotated lithographically fabricated sub-elements. The lens is positioned at some distance above a sample text, thus acting as a magnifying glass. Within the two sectors (indicated by red bars) one observes different magnification factors of the text.
Fig. 8.
Fig. 8. Phase transmission functions of two sub-elements (DOE 1 and DOE 2), which produce the transmission function of an axicon (right). The situation is sketched for a mutual rotation angle of $\theta ={ 90^ \circ}$ (indicated in the figure). Two sectors (red and blue) are formed, which correspond to different axicon cone angles. Gray values in the images correspond to phase shifts in an intervall between zero and $2\pi$.
Fig. 9.
Fig. 9. Adding the phase functions of two mutually rotated DOEs (DOE 1 and DOE 2) with quantized phase levels. The simulated result corresponds to the phase transmission function of a parabolic lens, which is not disturbed by the formation of different sectors. An experimental demonstration of the homogeneous lensing effect of two correspondingly fabricated, mutually rotated sub-elements, which are placed above a sample text, is shown at the right.
Fig. 10.
Fig. 10. Diffraction efficiency for the different $k$-components of a quantized moiré element as a function of the mutual rotation angle, according to Eq. (37). As an example, a rotation angle of $\theta = \pi /2$ is assumed. The $k = 0$ component (indicated as a red vertical line at $\theta = \pi /2$) has an efficiency of 81%. The other $k$-components appear in the graph at integer distances of $2\pi$.
Fig. 11.
Fig. 11. Illustration of feature size, structure size, and phase step height of a digital DOE.
Fig. 12.
Fig. 12. Combination of two sub-DOEs (DOE 1 and DOE 2) into a combined moiré element generating a tunable vortex plate. The figure shows the phase transmission functions (phase values indicated as gray values) of a first DOE 1, and a second complimentary DOE 2, which is rotated by an angle of 14.4° with respect to the first one. The resulting transmission function corresponds within its main sector to a vortex plate with a helicity of $l = 5$.
Fig. 13.
Fig. 13. Phase shift of an incident plane wave behind a tunable moiré vortex plate, measured with an interferometric setup. The mutual rotation angles and corresponding helical indices $l$ of the combined element are indicated in the different sub-plots.
Fig. 14.
Fig. 14. Principle of refocussing with a high NA objective. An incident plane wave (at the left) is transformed by an objective into a spherical wavefront (red, right side), which produces a diffraction limited focus in its center, at a distance ${f_n} = n{f_0}$, where $n$ is the refractive index of the medium, and ${f_0}$ is the focal length of the microscope objective in vacuum. Shifting the focus by a distance $\Delta z$ to the right (blue arrow) leads to a new distance $x$ from each point of the spherical wavefront to the new focal position. Thus the phase at the radial position (“height”) $r$ of the wavefront has to be increased by an amount of $\Delta \Phi (r) = 2\pi n(x(r) - {f_n})/\lambda$.
Fig. 15.
Fig. 15. Setup for shifting the focus position of a high NA microscope objective using a rotationally tunable moiré lens (MDOE).
Fig. 16.
Fig. 16. Phase profiles of three different types of sixth order moiré lenses at a design wavelength of ${\lambda _6} = 532\;{\rm nm} $. Column (a) refers to a non-quantized (sector producing) moiré lens, column (b) to a “weakly” quantized moiré lens type (which suppresses sector formation only at its design wavelength ${\lambda _6}$), and column (c) to a fully quantized moiré lens, which suppresses sector formation at all harmonic wavelengths. For all three cases, it is assumed that the two sub-elements of the combined moiré lens are mutually rotated by an angle of 15°. The rotated sub-elements of each lens type are sketched in the upper row. The next three rows show the phase profiles of the transmitted beams for wavelengths of 456 nm, 532 nm, and 638 nm, which correspond to the seventh, sixth, and fifth diffraction orders, respectively. The phase profiles are drawn modulo-$2\pi$ with respect to the corresponding readout wavelengths.
Fig. 17.
Fig. 17. Comparison of the dispersion properties of different types of tunable moiré lenses. (a) First order quantized lens according to Eq. (32). (b) Sixth order non-quantized lens (sector producing) according to Eq. (22). (c) Sixth order quantized lens according to Eq. 57. (d) Sixth order fully quantized lens according to Eq. (59). All lenses are simulated for a design wavelength of 532 nm, which corresponds to the first diffraction order in (a), and to the sixth diffraction order in (b)–(d). The mutual rotation angles of 90° in (a), and of 15° in (b)–(d) are chosen to produce the same optical power of 26 dpt at the design wavelength for all types of lenses. The upper plots show the diffraction efficiency (color coded) as a function of the readout wavelength, and the optical power of the lens. The lower graphs show the diffraction efficiency at the nominal optical power of 26 dpt as a function of the wavelength, which corresponds to a horizontal cut through the data of the upper plot along the line indicated in red.

Equations (60)

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H ( x , y ) = H 0 1 2 f ( n 1 ) ( x 2 + y 2 ) ,
H 1 ( x , y ) = a 0 x 1 2 f ( n 1 ) ( x 2 + y 2 ) d x = a 1 2 f ( n 1 ) ( 1 3 x 3 + x y 2 ) ,
H t o t = a [ H 1 ( x + Δ x / 2 , y ) H 1 ( x Δ x / 2 , y ) ] = a 1 2 f ( n 1 ) [ 1 3 ( x + Δ x / 2 ) 3 + ( x + Δ x / 2 ) y 2 ) ( x Δ x / 2 ) 3 ( x Δ x / 2 ) y 2 ) ] = 1 12 Δ x 3 Δ x a 1 2 f ( n 1 ) ( x 2 + y 2 ) .
E o u t ( x , y ) = T ( x , y ) E i n ( x , y ) .
T ( x , y ) = A ( x , y ) exp [ i Φ ( x , y ) ] ,
Φ ( x , y , λ ) = 2 π ( n 1 ) λ H ( x , y ) .
H D O E ( x , y ) = m o d l Λ { H r e f ( x , y ) } w h e r e Λ = λ d n d 1 .
Φ r e f ( x , y , λ d ) = 2 π ( n d 1 ) λ d H r e f ( x , y ) a n d Φ D O E ( x , y , λ d ) = 2 π ( n d 1 ) λ d H D O E ( x , y ) .
Φ D O E = 2 π ( n d 1 ) λ d m o d l Λ { H r e f } = 2 π ( n d 1 ) λ d m o d l Λ { λ d 2 π ( n d 1 ) Φ r e f } = m o d 2 π l { Φ r e f } .
Φ t r a n s = 2 π ( n λ 1 ) λ m o d l Λ { H r e f } = λ d λ n λ 1 n d 1 m o d 2 π l { Φ r e f } ,
T D O E ( x , y ) = exp [ i λ d λ n λ 1 n d 1 m o d 2 π l { Φ r e f ( x , y ) } ] = exp [ i λ d λ n λ 1 n d 1 Φ D O E ( x , y ) ] .
T D O E ( x , y , λ = λ d ) = exp [ i m o d 2 π l { Φ r e f ( x , y ) } ] exp [ i Φ r e f ( x , y ) ] = T r e f ( x , y ) .
η m = s i n c 2 ( m N ) s i n c 2 ( m Φ m a x 2 π ) .
η m = s i n c 2 ( m N ) s i n c 2 ( m l λ d λ n λ 1 n d 1 ) .
η m = s i n c 2 ( m N ) s i n c 2 ( m l λ d λ ) .
T ( r , φ ) = exp [ i Φ ( r ) ] .
T 1 = exp [ i Φ ( r ) φ ] , T 2 = exp [ i Φ ( r ) φ ] .
T 2 , r o t = { e x p [ i Φ ( r ) ( φ θ ) ] f o r θ φ < 2 π e x p [ i Φ ( r ) ( φ θ + 2 π ) ] f o r 0 φ < θ .
T j o i n t = { e x p [ i Φ ( r ) θ ] f o r θ φ < 2 π e x p [ i Φ ( r ) ( θ 2 π ) ] f o r 0 φ < θ .
T p a r a b ( r , φ ) = exp [ i π λ f r 2 ] ,
T ± 1 ( r , φ ) = exp ( ± a r 2 φ ) ,
Φ ± 1 ( r , φ , λ d ) = ± m o d 2 π l { a r 2 φ } .
T j o i n t = { exp [ i θ a r 2 ] f o r θ φ < 2 π exp [ i ( θ 2 π ) a r 2 ] f o r 0 φ < θ .
P 1 = θ a λ / π f o r θ φ < 2 π a n d P 2 = ( θ 2 π ) a λ / π f o r 0 φ < θ .
T a x i c o n ( r , φ ) = exp [ i 2 π sin ( β ) λ r ] ,
Φ ± 1 ( r , φ , λ d ) = m o d 2 π l { b r φ } ,
T j o i n t = { exp [ i θ b r ] f o r θ φ < 2 π exp [ i ( θ 2 π ) b r ] f o r 0 φ < θ .
β 1 = a r c s i n [ θ b λ / 2 π ] f o r θ φ < 2 π a n d β 2 = a r c s i n [ ( θ 2 π ) b λ / 2 π ] f o r 0 φ < θ .
T q , 1 = exp [ i r o u n d { Φ ( r ) } φ ] , T q , 2 = exp [ i r o u n d { Φ ( r ) } φ ] .
T q , j o i n t = { e x p [ i r o u n d { Φ ( r ) } θ ] f o r θ φ < 2 π e x p [ i r o u n d { Φ ( r ) } ( θ 2 π ) ] f o r 0 φ < θ .
Φ q , ± 1 ( r , φ ) = ± m o d 2 π { r o u n d { Φ ( r ) } φ } .
Φ p a r a b , ± 1 ( r , φ ) = ± m o d 2 π { r o u n d { a r 2 } φ } .
η = s i n c 2 ( 1 N ) .
T j o i n t = exp [ i r o u n d { Φ ( r ) } ( θ 2 π k ) ] ,
P k = ( θ 2 π k ) a λ / π .
N max = 2 π θ 2 π k .
η k = s i n c 2 ( θ 2 π k ) .
η = s i n c 2 ( θ 2 π ) .
| | Φ ( r , φ ) | | = [ d d r Φ ( r , φ ) ] 2 + [ 1 r d d φ Φ ( r , φ ) ] 2 < 2 π N e f f p ,
Φ 1 , 2 ( r , φ ) = ± a r 2 φ ,
P 1 = a θ λ / π .
| | Φ 1 ( r , φ ) | | = [ 2 a r φ ] 2 + [ a r ] 2 < 2 π N e f f p .
2 π a r m a x < 2 π N e f f p o r a < 1 N e f f p r m a x .
Φ v o r t e x ( r , φ ) = l φ ,
Φ v o r t , 1 ( r , φ ) = l 0 φ 2 , Φ v o r t , 2 ( r , φ ) = l 0 φ 2 ,
T v o r t , j o i n t = { e x p { i l 0 ( 2 θ φ θ 2 ) } f o r θ φ < 2 π e x p { i l 0 [ ( 2 θ 4 π ) φ θ 2 4 π 2 + 4 π θ ] } f o r 0 φ < θ .
l = { 2 l 0 θ f o r θ φ < 2 π 2 l 0 θ 4 π l 0 f o r 0 φ < θ .
Δ Φ ( r ) = 2 π n λ ( x ( r ) f n ) ,
x ( r ) = ( r 2 + ( a + Δ z ) 2 r 2 + a 2 + 2 a Δ z .
x ( r ) f n + a Δ z f n = f n + 1 r 2 f n 2 Δ z .
Δ Φ ( r ) = 2 π n λ 1 r 2 f n 2 Δ z .
Φ s u b , 1 ( r ) = m o d 2 π { r o u n d [ a 0 2 π n λ 1 r 2 f n 2 ] φ } , Φ s u b , 2 ( r ) = Φ s u b , 1 ( r ) ,
Δ z = a 0 θ .
λ m = l λ d m .
P ( λ ) = m l a λ θ π .
P ( λ m ) = a λ d θ π .
H q , ± 1 = ± λ d 2 π ( n d 1 ) m o d 2 π l { r o u n d [ a r 2 ] φ } .
Φ q , j o i n t = { m l m o d 2 π l { r o u n d [ a r 2 ] θ } f o r θ φ < 2 π m l m o d 2 π l { r o u n d [ a r 2 ] ( θ 2 π ) } f o r 0 φ < θ .
H f q , ± 1 = ± λ d 2 π ( n d 1 ) m o d 2 π l { l r o u n d [ a r 2 l ] φ } .
η = s i n c 2 ( l θ 2 π ) .

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