Abstract

Rotationally tunable diffractive optical elements (DOEs) consist of two stacked diffractive optical elements which are rotated with respect to each other around their central optical axis. The combined diffractive element acts as a highly efficient diffractive lens, which changes its optical power as a function of the mutual rotation angle. Here we show that the principle can be extended to produce polychromatic tunable lenses, i.e. lenses which have the same optical power, and the same diffraction efficiency within the full tuning range at three or more selectable wavelengths. The basic principle is to use higher order DOEs, which will be polychromatic at harmonics of a fundamental wavelength. The method can be applied to other types of optical elements which are tunable by rotation, like axicons, or generalized lenses with arbitrary radial phase profiles, or to elements tunable by a mutual translation, like diffractive Alvarez lenses.

© 2017 Optical Society of America

1. Introduction

Lenses with adjustable optical power are a current topic of research, since they allow one to construct compact and flexible optical systems, such as focussing or zooming optics for imaging applications, or adaptive beam steering which is advantageous for illumination systems, or laser material processing [1]. Various types of tunable lenses have been developed, as. e.g. tunable liquid lenses, where an elastomeric membrane lens is deformed in a controlled way by external forces [2, 3]. Another variant is based on electro-wetting, where by application of an electric voltage the shape of the active lens material can be controlled [4, 5]. Furthermore there are tunable acoustic lenses [6, 7], and liquid crystal based systems [8–10]. An interesting method, named after their inventors Alvarez lenses, or Alvarez-Lohmann lenses [11–14], uses two specially designed optical plates, which are placed on top of each other, and form a lens which is tunable by a mutual translation of the plates. Diffractive versions of these Alvarez-Lohmann lenses [15,16] have the advantage of being more compact, which, for example, allows for faster refocusing.

However, in many optical systems rotational elements are preferred with respect to translating elements since they are easier to implement. The first rotational version of a diffractive Alvarez-Lohmann lens has already been suggested in 1967 [17], where it was pointed out that its operation principle is based on the Moiré effect. Modified versions later appeared in several publications (e.g. [18–22]), and have been termed Moiré diffractive optical elements (MDOEs) [18] referring to their underlying principle. The working principle of these lenses is sketched in Fig. 1. Advantages of these elements are that they have a large tuning range of their optical power, they achieve diffraction limited focussing, their optical power is very precisely and reproducibly adjustable, and they can be utilized for high laser power applications. These elements provide a high diffraction efficiency, on the order of 80% to almost 100%, within their tuning range [20]. But a disadvantage of the rotational MDOEs is their strong dispersion, which is common to diffractive elements. The optical power of a MDOE is proportional to the diffracted wavelength. Therefore the use of MDOE lenses in imaging applications with colored light is challenging, although numerical methods have been developed for post-processing colored images recorded with adjustable diffractive lenses, which correct for the color aberration [22].

 

Fig. 1 Principle of MDOE lenses: (a): Two structured DOEs are placed on top of each other to form a MDOE lens. Rotation of one of the elements with respect to the other changes the optical power of the combined lens. (b) One of the two identical MDOE elements which are placed on top of each other (with one of them flipped upside down), with a phase function according to Eq. (1). Gray levels correspond to phase values between 0 and 2π. (c) One of the two quantized MDOE elements according to Eq. (7), which will form, after combination with a second, flipped element, a lens without sectors of different optical powers.

Download Full Size | PPT Slide | PDF

Here we show that MDOEs can be also designed to be polychromatic, i.e. both their efficiency and their optical power is corrected at two, three, or even more wavelengths. For that purpose one can use higher order MDOEs, which manipulate the phase of the incident light not only in a range between 0 and 2π, but in a range up to k2π, where k > 1 is the diffraction order of the lens. It has been already shown that higher order diffractive lenses are polychromatic for a number of harmonic wavelengths, i.e. at these wavelengths the dispersion is corrected [23, 24]. Here we show that this principle can be generalized for tunable diffractive lenses, which can be optimized to be polychromatic at a number of harmonic wavelengths.

2. Polychromatic MDOE lenses

In the following the working principle of monochromatic MDOE lenses will be briefly repeated, and then generalized to polychromatic lenses.

In [17, 18] it was shown that a parabolic lens with an optical power adjustable by a mutual rotation between two successive DOEs can be generated, if the two DOEs have corresponding transmission functions T1,2

T1=exp[iar2φ]T2=exp[iar2φ],
where r and φ are polar coordinates measured from the center of the optical element, and a is a constant which will be seen to be proportional to the tuning range of the assembled MDOE lens. If the MDOE is digitized as a quadratic array with pixels of size p (edge length), then, in order to avoid undersampling of the MDOE structure, the constant a is limited to [18]:
a<(2pR)1,
where R is the maximal radius of the MDOE lens. The corresponding phase profile is shown in Fig. 1(b). The corresponding two phase functions are actually the same, one of them is just flipped upside down.

If T2 is rotated by an angle θ around its center, it transforms into

T2;rot={exp[iar2(φθ)]forθ<φ<2πexp[iar2(φθ+2π)]for0φθ.
When placing the rotated second element on top of the first stationary DOE, the joint transmission function of the combination becomes Tjoint = T1T2;rot
Tjoint={exp[iar2θ]forθφ<2πexp[iar2(θ2π)]for0φ<θ.
Comparing this with the transmission function Tparab of a parabolic lens, namely with
Tparab=exp[iπr2λf],
where λ is the wavelength of the transmitted light, and f the focal length of the lens, one finds that Eq. (4) describes two different parabolic lenses in two sectors, with optical powers f−1 of
f11=θaλ/πforθφ<2πandf21=(θ2π)aλ/πfor0φ<θ,
i.e. the optical power changes as a linear function of the mutual rotation angle θ. For many applications it is desired that the entire area of the lens forms a uniform Fresnel zone lens without sector formation. This can be achieved by a quantization of the phase profile of the two MDOE components according to [18]
Tquant,1,2=exp[±iround{ar2}φ],
where round{...} corresponds to the rounding operation of its argument to the nearest integer. In this case the joint transmission function Tquant,joint = Tquant,1Tquant,2 of the two DOEs which are mutually rotated by an angle of θ becomes
Tquant,joint={exp[iround{ar2}θ]forθφ<2πexp[iround{ar2}(θ2π)]for0φ<θ.
Note that the difference between the phase of the two functions is 2π round{ar2}, i.e. an integer multiple of 2π, which does not affect the diffraction properties of a DOE. Thus the whole area of the MDOE lens now consists of a uniform lens. Figure 2 shows the results of a numerical simulation of the diffraction efficiency of such a sectorless MDOE lens as a function of the mutual rotation angle θ (left: 15°, middle 30°, right 60°) between the two lens elements.

 

Fig. 2 Diffraction efficiency of a first order sectorless (quantized) MDOE according to Eq. (7) as a function of wavelength, rotation angle, and optical power. The color code in the upper graphs refers to the diffraction efficiency. The simulation assumed a design wavelength of 532 nm, a pixel size of 1μm×1μm, a lens diameter of 1.28 mm, and a factor a = 2.6 × 108m−2. The left row shows the situation for a rotation angle of 15°. The upper graph shows the efficiency (corresponding color table at the right) as a function of readout wavelength, and optical power. Below, the efficiency is plotted again as a function of the wavelength, at the optical power level of 11.5 m−1, which is indicated above, and which corresponds to the nominal optical power expected according to Eq. (20) for the design wavelength. Middle: Corresponding graphs for a mutual rotation angle of 30° between the two elements. Right: Results for a mutual rotation angle of 60°.

Download Full Size | PPT Slide | PDF

The simulation was performed by defining two transmission functions according to Eq. (7) in a quadratic grid with 1280 × 1280 pixels, assuming a pixel size of 1 μm, a diameter of the circular DOE area of 1.28 mm, and a factor a = 2.6 × 108 m−2. One of the DOEs was numerically rotated with respect to the other, and the common transmission function was obtained by multiplication of the two individual ones. The common transmission function was then propagated (using numerical fast-Fourier-transform based Fresnel propagation) into several distances, and the corresponding diffraction efficiency was evaluated as the intensity in the central focal spot divided by the total intensity. The whole procedure was repeated for a set of wavelengths (since the propagation operation is wavelength dependent), and the corresponding efficiencies are plotted as a function of both wavelength and inverse propagation distance (i.e. optical power). Note that the simulation also incorporates the effect of the numerical aperture (NA) of the lens, i.e. the vertical “thickness” of the efficiency plot becomes narrower with increasing NA, e.g. by increasing the diameter of the simulated lens.

The graphs at the bottom of Fig. 2 indicate cross sections of the corresponding upper graphs along the dashed lines. These dashed lines correspond to the wavelength-dependent diffraction efficiency at the nominal optical powers, which correspond to the respective rotation values according to Eq. (6). Note that with increasing rotation angle, and correspondingly increasing (nominal) optical powers, the dispersion of the optical power (which corresponds to the slope of the efficiency plots in the upper graphs) also increases linearly. Within the visible wavelength range between 400 and 750 nm the MDOE lens has a diffraction efficiency significantly higher than 50%, with a maximum at the design wavelength of 532 nm. The resulting transmission function Tquant,joint is now only a digitized approximation of the ideal parabolic phase function Tparab = exp[iπr2/λf], where the effective number Neff of digitalization levels corresponds to

Neff=2πθ.

Note that Neff is not the number of actually used quantization steps during production of the MDOE plates, but the number of effectively realized quantization levels in the combined MDOE. Thus one observes that with increasing rotation angle θ the combined transmission function will correspond to a Fresnel lens in which fewer and fewer quantization levels are expressed, until, at an angle of θ = 180°, it will become a binary lens [20]. The diffraction efficiency of a blazed grating lens depends on the number of quantization levels Neff, on the diffraction order m, and the maximal height of the sawtooth shaped phase profile Φmax (which depends on the maximal etching depth of a DOE) according to [25]

η={sin[π(mΦmax/2π)]sin[π(mΦmax/2π)Neff]sin[πmNeff]πm}2.
Although this Equation refers to a blazed grating, to good approximation it also holds for other (sufficiently smooth) diffractive structures, like blazed diffractive lenses. Assuming an ideally blazed lens of order m, with Φmax = 2πm, the digitalization of the phase levels Neff = 2π/θ yields for the efficiency of the mth diffraction order:
ηm=(sin(mπ/Neff)mπ/Neff)2=(sin(mθ/2)mθ/2)2.
This means that with increasing rotation angle θ the diffraction efficiency decreases. For m = 1 a rotation from 0 to 90° results in a decrease of the efficiency from 100% to approximately 80%. But for higher diffraction orders, the loss of efficiency with rotation angle is correspondingly more pronounced.

On the other hand, if we consider a “perfectly” blazed diffractive lens with N → ∞, the corresponding efficiency ηm becomes

ηm={sin[π(mΦmax/2π)]π(mΦmax/2π)}2.
This function has a strongly peaked maximum (with an efficiency of 1) for the mth diffraction order, if
m=Φmax2π,
i.e. a DOE with a height of its blazed (sawtooth) phase pattern of 2πm will diffract with an efficiency of 1 into the mth diffraction order, and will have no efficiency in other orders. In general, the condition that Φmax/2π is an integer number will not be perfectly fulfilled, and in this case the diffraction order with the highest efficiency will be given by:
m=round{Φmax2π}.

Let us now investigate the effect of higher order MDOEs. The phase profiles Φ1,2(r) of two MDOE elements forming a compound lens with included sector according to Eq. (1) are given by

Φ1,2=kmod2π{±ar2φ},
where mod2π{...} is the modulo 2π operator, and k is the diffraction order (e.g. for the previous MDOE elements m was chosen to be 1). The diffraction order k is defined for the design wavelength λ0, and will change if the reconstruction wavelength is changed. The phase profile Φjoint = Φ1 + Φ2 of the compound MDOE lens, where one of the elements is rotated by an angle θ with respect to the other, is then given (except for a constant offset) by
Φjoint={kmod2π{ar2θ}forθφ<2πkmod2π{ar2(θ2π)}for0φ<θ.
If this kth order phase profile at the design wavelength λ0 (and corresponding index of refraction n0) is reconstructed at another wavelength λ, with a different index of refraction n(λ), the corresponding phase profile is
Φjoint={kλ0(n(λ)1)λ(n01)mod2π{ar2θ}forθφ<2πkλ0(n(λ1))λ(n01)mod2π{ar2(θ2π)}for0φ<θ.
The maximal phase amplitude Φmax is thus given by
Φmax=2πkλ0(n(λ)1)λ(n01).
According to Eq. (14) the diffraction order m which has maximal diffraction efficiency depends on Φmax as:
m=round{Φmax2π}=round{kn(λ)1n01λ0λ}.
The corresponding optical power in the two sectors now depends on the mutual rotation angle θ, on the reconstruction wavelength λ, and on the reconstructed diffraction order m according to
f11=mθaλm=round{kn(λ)1n01λ0λ}θaλπforθφ<2πf21=m(θ2π)aλπ=round{kn(λ)1n01λ0λ}(θ2π)aλπfor0φ<θ.

The maximal diffraction efficiency (of nominal 100%) in the mth order is reached, if the argument of the round operation equals the integer number m. If we neglect material dispersion (i.e. if we set n(λ) = n0), this happens for a readout wavelength of λm = λ0k/m, where m is any integer number. This, however, means that whenever we reach maximal diffraction efficiency, the optical power is the same, namely

f11=mθaλπ=mθaλ0kmπ=kθaλ0πforθφ<2πf21=m(θ2π)aλπ=m(θ2π)aλ0kmπ=k(θ2π)aλ0πfor0φ<θ,
which corresponds to the optical power of the lens if its used at its original design wavelength λ0. The next readout wavelengths with the same optical power closest to the design wavelength λ0 are reached for m = k ±1, k ±2, k ±3.... All of these wavelengths are members of a harmonic series with a fundamental wavelength of λf = 0. Note that the optical powers at the harmonic wavelengths are equal for the entire tuning range of the MDOE lens, even if material dispersion is considered. Material dispersion can slightly reduce the corresponding diffraction efficiency, according to Eq. (12), if the argument of the rounding operation in Eq. (20) is not an exact integer number. But as long as the material dispersion is low enough such that
round{k(n(λ)1)λ0(n01)λ}=round{kλ0λ},
the MDOE will be reconstructed in the same diffraction order, and the optical power at the harmonic wavelengths will be the same. This condition can also be reformulated as
|2Δnn01|<1m,
where Δn = n(λ) − n0 is the wavelength dependent change in refractive index due to material dispersion, and m = 0/λ is the diffraction order in which the harmonic wavelength λ would be reconstructed if no material dispersion were present. Inserting the dispersion data of fused silica within a wavelength range between 440 nm and 650 nm, the left hand side of the equation corresponds to a value of approximately 0.006, i.e. the condition is fulfilled up to a diffraction order of about mmax = 160.

Figure 3 sketches the results of a numerical simulation of a 5th order MDOE lens, with a phase profile described by Eq. (15). Similar to Fig. 2 the efficiency is plotted as a function of wavelength and optical power, for three mutual rotation angles of 15°, 30°, and 60° in three successive columns, respectively. The parameters of the numerical simulation correspond to those used for Fig. 2, assuming a design wavelength of 532 nm, and a diameter of 1.28 mm of the element. However, due to the fact that the MDOE acts as a 5th order lens, the constant a is decreased by the factor 5 (to a = 5.2 × 107m−2), which yields the same dependence between rotation angle and optical power as in Fig. 2. Correspondingly, the pixel size (edge length) is increased from 1μm of the first order element to 5μm of the 5th order element. Thus the ratio between phase amplitude and pixel size is the same in the two cases. The simulation shows that there exist 3 wavelengths, namely 665 nm, 532 nm, and 443 nm, which are the 4th order, 5th order and 6th order harmonics of the fundamental wavelength 2660 nm, where the optical power is the same. This is due to the fact that at each harmonic wavelength the diffractive dispersion is exactly compensated by a jump of the diffraction order.

 

Fig. 3 Diffraction efficiency of a 5th order (at 532 nm) non-quantized MDOE lens as a function of wavelength, rotation angle, and optical power. Upper row: Efficiencies at three different rotation angles (left: 15°, middle: 30°, right: 60°) as a function of wavelength and optical powers. Bottom row: Efficiencies at the optical power levels indicated above as a function of the wavelength. Within the visible range there exist 3 wavelengths, namely 665 nm, 532 nm, and 443 nm, which are 4th order, 5th order and 6th order (indicated in the figure) harmonics of the fundamental wavelength 2660 nm, and which have the same optical power for all rotation angles.

Download Full Size | PPT Slide | PDF

Figure 4 exemplifies the effect of using an even higher order MDOE lens, namely one of order 10 at a wavelength of 532 nm. The other parameters, like numerical aperture, and the dependence of optical power on the rotation angle have been chosen as in Fig. 3. The simulation shows that within the visible range the diffraction orders 12, 11, 10, 9, and 8 (indicated in the figure) have the same optical powers at their respective wavelengths of 443 nm, 483 nm, 532 nm, 591 nm and 665 nm. At smaller rotation angles, e.g. 15° (see Fig. 4 at the left), the maxima of the efficiency are progressively merging, yielding a more and more uniform optical power in the whole wavelength range. This is due to the fact that with decreasing rotation angle the slope of the efficiency plots (i.e. the dispersion) decreases, and simultaneously also the numerical aperture of the combined lens decreases, which enhances the Rayleigh range (i.e. the axial extension) of the focal spot. Thus, higher order elements might be preferable to cover the whole visible wavelength range. However, as will be discussed later, the fabrication of higher order elements is more challenging, requiring a higher number of phase steps.

 

Fig. 4 Efficiency of a 10th order MDOE lens with a phase profile according to Eq. (15) as a function of wavelength, rotation angle, and optical power. The parameters are the same as those used for Fig. 3.

Download Full Size | PPT Slide | PDF

Actually Eq. (20) together with Eq. (12) describes the transition between the dispersion behavior of a (first order) diffractive lens up to a bulk refractive lens [26], which may be seen as a diffractive lens of very high order. The principle how a higher order MDOE lens suppresses dispersion is basically that the (positive) linear dependence of the optical power as a function of the wavelength is compensated by the (negative) linear dependence of the diffraction order in which the beam is reconstructed.

3. Polychromatic MDOE lenses without sector formation

The quantization method described above can be also applied to higher order lenses without sector formation, i.e. to produce a lens with the same optical power within its whole area. A quantized lens with a transmission function according to Eq. (7) may be produced in higher order k by just multiplying the corresponding phase function with the factor k. However, in this case the number of quantization levels within an interval between 0 and 2π is decreased by the factor k, which correspondingly decreases the diffraction efficiency at higher rotation angles, and thus reduces the usable optical power range of the tunable lens. Nevertheless the quantization method can be adapted for higher diffraction orders such that the number of quantization levels (per 2π phase shift) remains constant, namely by setting

Φ1,2=kmod2π{round[±kar2]φk}.
In this case the combined MDOE phase becomes
Φjoint={kλ0λmod2π{round[kar2]θk}forθφ<2πkλ0λmod2π{round[kar2](θ2π)k}for0φ<θ.

For λ = λ0 the difference between the phase functions in the two sectors is still a multiple of 2π, and therefore the corresponding optical powers are equal. The number Neff of quantized phase levels as a function of the rotation angle is now given by

Neff=k2πθ.
If the MDOE is reconstructed in the mth diffraction order, its efficiency is given by Eq. (11), setting Neff = 2πk/θ, and thus becomes
ηm=(sin(mθ/2k)mθ/2k)2.

This shows that the efficiency increases at lower diffraction orders m (corresponding to larger wavelengths). The corresponding rotation angle θ80% where the efficiency is reduced to 80% is given by approximately ±90°k/m. The tuning range for the optical power within this range of rotational angles is an interval between −k20/2m and +k20/2m. But it has to be noted that according to Eq. (25) a phase offset between the two lens sectors in integer multiples of 2π is only exactly obtained at the design wavelength λ0, i.e. only at that wavelength the lens has a uniform optical power without sector formation. At the other harmonic wavelengths the sector will gradually reappear.

Figure 5 shows the results of a numerical simulation of an MDOE lens designed according to Eq. (24). The parameters are the same as those used for Fig. 3, and the only difference is the quantization of the phase profile according to Eq. (24), using a MDOE lens of order k = 5 at the design wavelength 532 nm. As expected, the resulting lens has the same optical power at all harmonic wavelengths (indicated in the figure). However, in this case the efficiency of the optimized 5th diffraction order at the wavelength of 532 nm is significantly enhanced with respect to the non-quantized lens sketched in Fig. 3. This is due to the fact that the 5th order lens (appearing at λ = 532 nm) behaves like a quantized lens whose diffraction efficiency depends on the rotation angle according to Eq. (27), whereas the other diffraction orders correspond to non-quantized lenses (with sector formation), with a diffraction efficiency which is basically given by the area of the suitable sector divided by the total area, i.e. by (2π − |θ|)/2π, as has been shown both theoretically and experimentally in [20]. Due to these different behaviours, the efficiency of the quantized 5th order lens (at λ = 532 nm) exceeds that of the other non-quantized lenses (at the other wavelengths) in a range of rotation angles θ between −125° and +125°. Due to the improved performance of one selectable diffraction order, the total diffraction efficiency (i.e. the sum over the efficiencies of the individual lenses at different wavelengths) of the quantized multi-order lens designed according to Eq. (24) exceeds that of a non-quantized lens (according to Eq. (15)) within the specified range of rotation angles.

 

Fig. 5 Diffraction efficiency of a 5th order quantized MDOE (without sector formation) according to Eq. (24) as a function of wavelength, rotation angle, and optical power. The parameters are the same as those used for Fig. 3. The lens is optimized for maximal efficiency (without sector formation) in the 5th diffraction order at a wavelength of 532 nm.

Download Full Size | PPT Slide | PDF

4. Discussion

We have shown that the concept of MDOE lenses can be extended to higher order lenses of phase shifts of several multiples of 2π. In this case one can choose a set of harmonic wavelengths, at which the optical power of the MDOE lens is the same, independent of the rotation angle, and (up to a certain diffraction order) even independent of material dispersion. The quantization scheme introduced in [18] for first order MDOE lenses can also be used in a modified form for higher order lenses, thus allowing for a multi-color MDOE lens which avoids sector formation, at least at one selected wavelength.

Comparing multi-order MDOE lenses with first order lenses, one finds that the technical complexities are comparable: In order to cover the same range of optical powers, the etching depth of a kth order lens needs to be increased by a factor of k. However, at the same time the magnification factor a may be decreased by the factor k, since a kth order MDOE lens has also a k-times increased optical power as compared to a single order MDOE lens. Thus the pixel size may be also enlarged by the factor k, without undersampling of the diffractive structures according to Eq. (2). Thus, the ratio between pixel size and etching depth, i.e. the steepness of the etched structures, remains the same. However, an additional complexity of quantized multi-order lenses consists in the fact that the number of etching levels has to be increased, whereas the step size is kept the same. Thus, in order to reach the same efficiency one has to increase the number of phase quantization levels by a factor of k, keeping the same step size. To summarize, multi-order MDOE lenses which cover the same optical power range than a single order MDOE lens need a lower lateral resolution (i.e. a larger pixel size can be used), but a higher number of phase quantization levels.

In our simulations we therefore have investigated MDOEs of the rather low order k = 5, which is the lowest order which yields a constant optical power for at least three wavelengths in the visible range (at 443 nm, 532 nm, and 665 nm). As has been demonstrated in Fig. 4, MDOE lenses with higher diffraction order may be advantageous, since they provide an almost homogeneous optical power distribution in the whole wavelength range. Regarding the results obtained in [20], where a single order MDOE with high efficiency was demonstrated using 16 depth quantization levels, a kth order DOE will require 16k levels. Since it is currently possible to produce DOEs with 256 quantization levels, one may extrapolate that a polychromatic MDOE of order k = 16 is feasible, which will have an almost homogeneous optical power within the whole visible range, while still keeping the advantageous feature of being a “thin” diffractive element, which has, for example, an etching depth of about 20 μm in fused silica.

The scheme of producing polychromatic MDOE lenses using higher order diffractive optics can be straightforwardly applied to other types of tunable elements, consisting of arbitrary radial phase profiles with rotation symmetry, as for example, to tunable axicons, axicon lenses, higher order lenses, or even oscillating structures (like a “Mexican hat”). Furthermore, the application to tunable devices which are based on a lateral translation between two diffractive elements, like “standard” diffractive Alvarez lenses, is possible. The principle may be even used for tunable elements without rotation symmetry. Such elements, like tunable spiral phase plates [27], which may also be sectorless [28], or tunable lens arrays [21], have already been experimentally demonstrated, and can be designed in an analogous way to work as polychromatic elements in higher diffraction orders.

Funding

ERC Advanced Grant 247024 catchIT.

References and links

1. L. Guoqiang, “Adaptive Lens,” Prog. in Opt. 55, 199–283 (2010). [CrossRef]  

2. H. Ren, D. Fox, P. A. Anderson, B. Wu, and S.-T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14, 8031 (2006). [CrossRef]   [PubMed]  

3. G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009). [CrossRef]   [PubMed]  

4. B. Berge, “Liquid lens technology: Principle of electrowetting based lenses and applications to imaging,” Proc. IEEE MEMS, 227–230 (2004).

5. S. Xu, Y.-J. Lin, and S.-T. Wu, “Dielectric liquid microlens with well-shaped electrode,” Opt. Express 17, 10499 (2009). [CrossRef]   [PubMed]  

6. A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High-speed varifocal imaging with a tunable acoustic gradient index of refraction lens,” Opt. Lett. 33, 2146–2148 (2008). [CrossRef]   [PubMed]  

7. M. Duocastella, G. Vicidomini, and A. Diaspro, “Simultaneous multiplane confocal microscopy using acoustic tunable lenses,” Opt. Express 22, 19293–19301 (2014). [CrossRef]   [PubMed]  

8. T. Nose and S. Sato, “A liquid crystal microlens obtained with a non-uniform electric field,” Liq. Cryst. 5, 1425–1433 (1989). [CrossRef]  

9. H. Ren, D. Fox, B. Wu, and S. T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express 15, 11328–11335 (2007). [CrossRef]   [PubMed]  

10. P. Valley, D. L. Mathine, M. R. Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Tunable-focus flat liquid-crystal diffractive lens,” Opt. Lett. 35, 336–338 (2010). [CrossRef]   [PubMed]  

11. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970). [CrossRef]   [PubMed]  

12. L. W. Alvarez, “Two-element variable-power spherical lens” U.S. patent 3,305,294 (3 December 1964).

13. S. Barbero, “The Alvarez and Lohmann refractive lenses revisited,” Opt. Express 17, 9376–9390 (2009). [CrossRef]   [PubMed]  

14. S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13, 125705 (2011). [CrossRef]  

15. A. Kolodziejczyk and Z. Jaroszewicz, “Diffractive elements of variable optical power and high diffraction efficiency,” Appl. Opt. 32, 4317–4322 (1993). [CrossRef]   [PubMed]  

16. I. M. Barton, S. N. Dixit, L. J. Summers, C. A. Thompson, K. Avicola, and J. Wilhelmsen, “Diffractive Alvarez lens,” Opt. Lett. 25, 1–3 (2000). [CrossRef]  

17. A. W. Lohmann and D. P. Paris, “Variable Fresnel Zone Pattern,” Appl. Opt. 6, 1567–1570 (1967). [CrossRef]   [PubMed]  

18. S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moiré elements,” Appl. Opt. 47, 3722–3730 (2008). [CrossRef]   [PubMed]  

19. J. Ojeda-Castaneda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photon. Lett. Pol. 5, 20–22 (2013).

20. S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Express 21, 6955–6966 (2013). [CrossRef]   [PubMed]  

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

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

23. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34, 2462–2468 (1995). [CrossRef]   [PubMed]  

24. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34, 2469–2475 (1995). [CrossRef]   [PubMed]  

25. B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).

26. S. Sinzinger and M. Testorf, “Transition between diffractive and refractive micro-optical components,” Appl. Opt. 34, 5970–5976 (1995). [CrossRef]   [PubMed]  

27. W. Harm, S. Bernet, M. Ritsch-Marte, I. Harder, and N. Lindlein, “Adjustable diffractive spiral phase plates,” Opt. Express 23, 413–421 (2015). [CrossRef]   [PubMed]  

28. A. Grewe and S. Sinzinger, “Efficient quantization of tunable helix phase plates,” Opt. Lett. 41, 4755–4758 (2016). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. L. Guoqiang, “Adaptive Lens,” Prog. in Opt. 55, 199–283 (2010).
    [Crossref]
  2. H. Ren, D. Fox, P. A. Anderson, B. Wu, and S.-T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14, 8031 (2006).
    [Crossref] [PubMed]
  3. G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009).
    [Crossref] [PubMed]
  4. B. Berge, “Liquid lens technology: Principle of electrowetting based lenses and applications to imaging,” Proc. IEEE MEMS, 227–230 (2004).
  5. S. Xu, Y.-J. Lin, and S.-T. Wu, “Dielectric liquid microlens with well-shaped electrode,” Opt. Express 17, 10499 (2009).
    [Crossref] [PubMed]
  6. A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High-speed varifocal imaging with a tunable acoustic gradient index of refraction lens,” Opt. Lett. 33, 2146–2148 (2008).
    [Crossref] [PubMed]
  7. M. Duocastella, G. Vicidomini, and A. Diaspro, “Simultaneous multiplane confocal microscopy using acoustic tunable lenses,” Opt. Express 22, 19293–19301 (2014).
    [Crossref] [PubMed]
  8. T. Nose and S. Sato, “A liquid crystal microlens obtained with a non-uniform electric field,” Liq. Cryst. 5, 1425–1433 (1989).
    [Crossref]
  9. H. Ren, D. Fox, B. Wu, and S. T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express 15, 11328–11335 (2007).
    [Crossref] [PubMed]
  10. P. Valley, D. L. Mathine, M. R. Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Tunable-focus flat liquid-crystal diffractive lens,” Opt. Lett. 35, 336–338 (2010).
    [Crossref] [PubMed]
  11. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970).
    [Crossref] [PubMed]
  12. L. W. Alvarez, “Two-element variable-power spherical lens” U.S. patent 3,305,294 (3December1964).
  13. S. Barbero, “The Alvarez and Lohmann refractive lenses revisited,” Opt. Express 17, 9376–9390 (2009).
    [Crossref] [PubMed]
  14. S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13, 125705 (2011).
    [Crossref]
  15. A. Kolodziejczyk and Z. Jaroszewicz, “Diffractive elements of variable optical power and high diffraction efficiency,” Appl. Opt. 32, 4317–4322 (1993).
    [Crossref] [PubMed]
  16. I. M. Barton, S. N. Dixit, L. J. Summers, C. A. Thompson, K. Avicola, and J. Wilhelmsen, “Diffractive Alvarez lens,” Opt. Lett. 25, 1–3 (2000).
    [Crossref]
  17. A. W. Lohmann and D. P. Paris, “Variable Fresnel Zone Pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [Crossref] [PubMed]
  18. S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moiré elements,” Appl. Opt. 47, 3722–3730 (2008).
    [Crossref] [PubMed]
  19. J. Ojeda-Castaneda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photon. Lett. Pol. 5, 20–22 (2013).
  20. S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Express 21, 6955–6966 (2013).
    [Crossref] [PubMed]
  21. A. Grewe, P. Fesser, and S. Sinzinger, “Diffractive array optics tuned by rotation,” Appl. Opt. 56, A89–A96 (2016).
    [Crossref]
  22. F. Heide, Q. Fu, Y. Peng, and W. Heidrich, “Encoded diffractive optics for full-spectrum computational imaging,” Sci. Rep. 6, 33543 (2016).
    [Crossref] [PubMed]
  23. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34, 2462–2468 (1995).
    [Crossref] [PubMed]
  24. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34, 2469–2475 (1995).
    [Crossref] [PubMed]
  25. B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).
  26. S. Sinzinger and M. Testorf, “Transition between diffractive and refractive micro-optical components,” Appl. Opt. 34, 5970–5976 (1995).
    [Crossref] [PubMed]
  27. W. Harm, S. Bernet, M. Ritsch-Marte, I. Harder, and N. Lindlein, “Adjustable diffractive spiral phase plates,” Opt. Express 23, 413–421 (2015).
    [Crossref] [PubMed]
  28. A. Grewe and S. Sinzinger, “Efficient quantization of tunable helix phase plates,” Opt. Lett. 41, 4755–4758 (2016).
    [Crossref] [PubMed]

2016 (3)

2015 (1)

2014 (1)

2013 (2)

J. Ojeda-Castaneda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photon. Lett. Pol. 5, 20–22 (2013).

S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Express 21, 6955–6966 (2013).
[Crossref] [PubMed]

2011 (1)

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13, 125705 (2011).
[Crossref]

2010 (2)

2009 (3)

2008 (2)

2007 (1)

2006 (1)

2000 (1)

1995 (3)

1993 (1)

1989 (1)

T. Nose and S. Sato, “A liquid crystal microlens obtained with a non-uniform electric field,” Liq. Cryst. 5, 1425–1433 (1989).
[Crossref]

1970 (1)

1967 (1)

Alvarez, L. W.

L. W. Alvarez, “Two-element variable-power spherical lens” U.S. patent 3,305,294 (3December1964).

Anderson, P. A.

Arnold, C. B.

Avicola, K.

Barbero, S.

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13, 125705 (2011).
[Crossref]

S. Barbero, “The Alvarez and Lohmann refractive lenses revisited,” Opt. Express 17, 9376–9390 (2009).
[Crossref] [PubMed]

Barton, I. M.

Berge, B.

B. Berge, “Liquid lens technology: Principle of electrowetting based lenses and applications to imaging,” Proc. IEEE MEMS, 227–230 (2004).

Bernet, S.

Chau, F. S.

G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009).
[Crossref] [PubMed]

Diaspro, A.

Dixit, S. N.

Dodge, M. R.

Duocastella, M.

Faklis, D.

Fesser, P.

Fox, D.

Fu, Q.

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

Gómez-Sarabia, C. M.

J. Ojeda-Castaneda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photon. Lett. Pol. 5, 20–22 (2013).

Grewe, A.

Guoqiang, L.

L. Guoqiang, “Adaptive Lens,” Prog. in Opt. 55, 199–283 (2010).
[Crossref]

Harder, I.

Harm, W.

Heide, F.

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

Heidrich, W.

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

Jaroszewicz, Z.

Kolodziejczyk, A.

Kress, B.

B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).

Kumar, A. S.

G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009).
[Crossref] [PubMed]

Ledesma, S.

J. Ojeda-Castaneda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photon. Lett. Pol. 5, 20–22 (2013).

Leung, H. M.

G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009).
[Crossref] [PubMed]

Lin, Y.-J.

Lindlein, N.

Lohmann, A. W.

Mathine, D. L.

McLeod, E.

Mermillod-Blondin, A.

Meyrueis, P.

B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).

Morris, G. M.

Nose, T.

T. Nose and S. Sato, “A liquid crystal microlens obtained with a non-uniform electric field,” Liq. Cryst. 5, 1425–1433 (1989).
[Crossref]

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photon. Lett. Pol. 5, 20–22 (2013).

Paris, D. P.

Peng, Y.

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

Peyghambarian, N.

Peyman, G.

Ren, H.

Ritsch-Marte, M.

Rubinstein, J.

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13, 125705 (2011).
[Crossref]

Sato, S.

T. Nose and S. Sato, “A liquid crystal microlens obtained with a non-uniform electric field,” Liq. Cryst. 5, 1425–1433 (1989).
[Crossref]

Schwiegerling, J.

Sinzinger, S.

Sommargren, G. E.

Summers, L. J.

Sweeney, D. W.

Testorf, M.

Thompson, C. A.

Valley, P.

Vicidomini, G.

Wilhelmsen, J.

Wu, B.

Wu, S. T.

Wu, S.-T.

Xu, S.

Yu, H.

G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009).
[Crossref] [PubMed]

Zhou, G.

G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009).
[Crossref] [PubMed]

Appl. Opt. (8)

J. Opt. (1)

S. Barbero and J. Rubinstein, “Adjustable-focus lenses based on the Alvarez principle,” J. Opt. 13, 125705 (2011).
[Crossref]

Liq. Cryst. (1)

T. Nose and S. Sato, “A liquid crystal microlens obtained with a non-uniform electric field,” Liq. Cryst. 5, 1425–1433 (1989).
[Crossref]

Opt. Express (7)

Opt. Lett (1)

G. Zhou, H. M. Leung, H. Yu, A. S. Kumar, and F. S. Chau, “Liquid tunable diffractive-refractive hybrid lens,” Opt. Lett 34, 2793–2795 (2009).
[Crossref] [PubMed]

Opt. Lett. (4)

Photon. Lett. Pol. (1)

J. Ojeda-Castaneda, S. Ledesma, and C. M. Gómez-Sarabia, “Tunable apodizers and tunable focalizers using helical pairs,” Photon. Lett. Pol. 5, 20–22 (2013).

Prog. in Opt. (1)

L. Guoqiang, “Adaptive Lens,” Prog. in Opt. 55, 199–283 (2010).
[Crossref]

Sci. Rep. (1)

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

Other (3)

B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).

B. Berge, “Liquid lens technology: Principle of electrowetting based lenses and applications to imaging,” Proc. IEEE MEMS, 227–230 (2004).

L. W. Alvarez, “Two-element variable-power spherical lens” U.S. patent 3,305,294 (3December1964).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Principle of MDOE lenses: (a): Two structured DOEs are placed on top of each other to form a MDOE lens. Rotation of one of the elements with respect to the other changes the optical power of the combined lens. (b) One of the two identical MDOE elements which are placed on top of each other (with one of them flipped upside down), with a phase function according to Eq. (1). Gray levels correspond to phase values between 0 and 2π. (c) One of the two quantized MDOE elements according to Eq. (7), which will form, after combination with a second, flipped element, a lens without sectors of different optical powers.
Fig. 2
Fig. 2 Diffraction efficiency of a first order sectorless (quantized) MDOE according to Eq. (7) as a function of wavelength, rotation angle, and optical power. The color code in the upper graphs refers to the diffraction efficiency. The simulation assumed a design wavelength of 532 nm, a pixel size of 1μm×1μm, a lens diameter of 1.28 mm, and a factor a = 2.6 × 108m−2. The left row shows the situation for a rotation angle of 15°. The upper graph shows the efficiency (corresponding color table at the right) as a function of readout wavelength, and optical power. Below, the efficiency is plotted again as a function of the wavelength, at the optical power level of 11.5 m−1, which is indicated above, and which corresponds to the nominal optical power expected according to Eq. (20) for the design wavelength. Middle: Corresponding graphs for a mutual rotation angle of 30° between the two elements. Right: Results for a mutual rotation angle of 60°.
Fig. 3
Fig. 3 Diffraction efficiency of a 5th order (at 532 nm) non-quantized MDOE lens as a function of wavelength, rotation angle, and optical power. Upper row: Efficiencies at three different rotation angles (left: 15°, middle: 30°, right: 60°) as a function of wavelength and optical powers. Bottom row: Efficiencies at the optical power levels indicated above as a function of the wavelength. Within the visible range there exist 3 wavelengths, namely 665 nm, 532 nm, and 443 nm, which are 4th order, 5th order and 6th order (indicated in the figure) harmonics of the fundamental wavelength 2660 nm, and which have the same optical power for all rotation angles.
Fig. 4
Fig. 4 Efficiency of a 10th order MDOE lens with a phase profile according to Eq. (15) as a function of wavelength, rotation angle, and optical power. The parameters are the same as those used for Fig. 3.
Fig. 5
Fig. 5 Diffraction efficiency of a 5th order quantized MDOE (without sector formation) according to Eq. (24) as a function of wavelength, rotation angle, and optical power. The parameters are the same as those used for Fig. 3. The lens is optimized for maximal efficiency (without sector formation) in the 5th diffraction order at a wavelength of 532 nm.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

T 1 = exp [ i a r 2 φ ] T 2 = exp [ i a r 2 φ ] ,
a < ( 2 p R ) 1 ,
T 2 ; rot = { exp [ i a r 2 ( φ θ ) ] for θ < φ < 2 π exp [ i a r 2 ( φ θ + 2 π ) ] for 0 φ θ .
T joint = { exp [ i a r 2 θ ] for θ φ < 2 π exp [ i a r 2 ( θ 2 π ) ] for 0 φ < θ .
T parab = exp [ i π r 2 λ f ] ,
f 1 1 = θ a λ / π for θ φ < 2 π and f 2 1 = ( θ 2 π ) a λ / π for 0 φ < θ ,
T quant , 1 , 2 = exp [ ± i round { a r 2 } φ ] ,
T quant , joint = { exp [ i round { a r 2 } θ ] for θ φ < 2 π exp [ i round { a r 2 } ( θ 2 π ) ] for 0 φ < θ .
N eff = 2 π θ .
η = { sin [ π ( m Φ max / 2 π ) ] sin [ π ( m Φ max / 2 π ) N eff ] sin [ π m N eff ] π m } 2 .
η m = ( sin ( m π / N eff ) m π / N eff ) 2 = ( sin ( m θ / 2 ) m θ / 2 ) 2 .
η m = { sin [ π ( m Φ max / 2 π ) ] π ( m Φ max / 2 π ) } 2 .
m = Φ max 2 π ,
m = round { Φ max 2 π } .
Φ 1 , 2 = k mod 2 π { ± a r 2 φ } ,
Φ joint = { k mod 2 π { a r 2 θ } for θ φ < 2 π k mod 2 π { a r 2 ( θ 2 π ) } for 0 φ < θ .
Φ joint = { k λ 0 ( n ( λ ) 1 ) λ ( n 0 1 ) mod 2 π { a r 2 θ } for θ φ < 2 π k λ 0 ( n ( λ 1 ) ) λ ( n 0 1 ) mod 2 π { a r 2 ( θ 2 π ) } for 0 φ < θ .
Φ max = 2 π k λ 0 ( n ( λ ) 1 ) λ ( n 0 1 ) .
m = round { Φ max 2 π } = round { k n ( λ ) 1 n 0 1 λ 0 λ } .
f 1 1 = m θ a λ m = round { k n ( λ ) 1 n 0 1 λ 0 λ } θ a λ π for θ φ < 2 π f 2 1 = m ( θ 2 π ) a λ π = round { k n ( λ ) 1 n 0 1 λ 0 λ } ( θ 2 π ) a λ π for 0 φ < θ .
f 1 1 = m θ a λ π = m θ a λ 0 k m π = k θ a λ 0 π for θ φ < 2 π f 2 1 = m ( θ 2 π ) a λ π = m ( θ 2 π ) a λ 0 k m π = k ( θ 2 π ) a λ 0 π for 0 φ < θ ,
round { k ( n ( λ ) 1 ) λ 0 ( n 0 1 ) λ } = round { k λ 0 λ } ,
| 2 Δ n n 0 1 | < 1 m ,
Φ 1 , 2 = k mod 2 π { round [ ± k a r 2 ] φ k } .
Φ joint = { k λ 0 λ mod 2 π { round [ k a r 2 ] θ k } for θ φ < 2 π k λ 0 λ mod 2 π { round [ k a r 2 ] ( θ 2 π ) k } for 0 φ < θ .
N eff = k 2 π θ .
η m = ( sin ( m θ / 2 k ) m θ / 2 k ) 2 .

Metrics