## Abstract

We present a modified configuration of a tunable Alvarez lens with a refocusing frequency of 1 kHz or more. In contrast to the classic Alvarez lens, the approach does not utilize a translational motion of two sub-lenses with respect to each other, but uses a 4f-setup to image two diffractive sub-lenses onto each other. Hereby focus tuning is achieved by rotating a galvo-mirror which affects the overlap of the two sub-lenses which together form an effective lens of refractive power which depends on the rotation angle of the galvo-mirror. We have demonstrated tuning of the optical power in a system where the diffractive Alvarez lens is realized by an LCOS-SLM. We consider our Alvarez setup especially suitable for applications where high refocusing rates are important, as for example in 3D life cell monitoring or tracking.

© 2017 Optical Society of America

## 1. Introduction

In traditional optics lenses are moved back and forth to change focus, which is adverse to high tuning speed. Yet the demand for fast tunable lenses has risen recently, especially in the field of laser processing, machine vision, and microscopy. Several types of tunable lens systems have been developed to meet this demand. One of them changes optical power by deformation of liquid-filled polymer membranes [1], another works according to the principle of electro-wetting [2], while the Moiré lens [3–5] is based on diffractive optics. Tunable lenses can also be realized with Micro-Electro-Mechanical System mirrors (MEMS-mirrors) [6,7]. All these lens types have their individual advantages, although none of them offers a tuning frequency of more than several 100 Hz yet. There is only one type of tunable lenses which enables refocusing rates beyond 100 kHz. It relies on acousto-optic laser beam shaping and has the disadvantage that its optical power is always oscillating [8,9].

In this manuscript, we suggest and demonstrate a modified Alvarez lens system which allows for exact, continuous, and fast adjustability (1 kHz) of optical power. The Alvarez lens, invented by Luis Alvarez in the late sixties [10], in its original form is a varifocal lens where the optical power change is brought about by a lateral displacement of two complementary sub-elements of identical cubic surface profiles, with one sub-element flipped and placed back-to-back with the other (see Fig. 1). The most widely used implementation of the lens is the so-called Humphrey analyzer [11, 12], but it has also been applied in scanning microscopes [13], accommodative intraocular lenses [14,15], or in spectacle lenses for selfadjusting focus [16,17].

The limited prevalence of the Alvarez lens technology is presumably owed to the difficulty, at least in the past, of manufacturing high-quality freeform refractive lenses. It can be assumed that the Alvarez lens in its *diffractive* form is likely to gain popularity, since the production of diffractive structures has seen a significant upturn in recent years. Nevertheless, the required lateral displacement of its diffractive sub-elements for rapid focal power changes remains technically unfavorable.

## 2. Modified Alvarez lens

We will show that with a modification of the Alvarez lens configuration sub-element translations are not required for changing the optical power. Instead of arranging the sub-elements in close vicinity, they can also be imaged onto each other via a 4f-lens configuration. In such an Alvarez lens, the change in optical power is generated by a beam deflection rather than a displacement (see Fig. 2(a)). In case the beam deflector is replaced by a rotatable mirror (e.g. a galvo mirror), only *one* sub-element which is imaged onto itself via the galvo is sufficient (see Fig. 2(b)). Since galvo mirrors can be operated at high frequency (up to e.g. 1 kHz), this modification enables an extremely fast Alvarez varifocal lens. But the explained concept (in this form) has the disadvantage that, depending on the adjusted optical power, the resulting focus moves away from the optical axis. A method which we have identified as suitable for compensating this beam “walk-off”, and which we therefore adopted, has been demonstrated by Rainer Heintzmann’s group, in the context of Optical Photon Reassignment Microscopy (OPRA) [18].

Like acousto-optically generated lenses, our Alvarez element produces parabolic phase profiles which are unfavourable for high-NA imaging. Therefore our technique predominantly aims at low-NA applications as for instance used in machine vision. And even in microscopy, many high-speed applications such as the tracking of activities in neuronal networks do not require imaging at the diffraction limit. For these applications our approach should thus likewise be feasible.

## 3. Experimental setup

For the proof-of-principle-experiments of the modified Alvarez lens which are presented here, we employed a spatial light modulator (SLM) to modulate the optical phase. This is, however, not a prerequisite, and for specific scientific or industrial implementations a corresponding laser-lithographically produced diffractive optical element (DOE) or a “standard” refractive Alvarez sub-element could be more appropriate.

The experimental setup of our modified Alvarez lens with correction of the walk-off is shown schematically in Fig. 3. A collimated laser beam with a wavelength of 532 nm passes a polarizer which aligns the polarization of the laser beam with the orientation of the SLM’s liquid crystal molecules in order to maximize the efficiency of the SLM (Hamamatsu X10468-07), which is a parallel aligned LCOS-SLM for pure phase modulation in the visible range (with a display resolution of 600 × 800 and quadratic pixels with an edge length of 20 μm). The laser beam reaches the SLM via a non-polarizing 50/50 beam splitter, where the diffractive structure of an Alvarez sub-element is imprinted on the phase. After being reflected from the SLM, the light is sent through lens A (placed at a distance of *f _{A}* = 300 mm) and subsequently to a galvo mirror (again at a distance

*f*from lens A) which reflects the light back into its original direction, such that the SLM surface is effectively imaged onto itself (conjugated and laterally reversed). Accordingly, the phase structure of the resulting lens is flat and the optical power of the Alvarez lens is zero.

_{A}When the galvo mirror (Cambridge Technology 6220H with broadband antireflection-coating) is rotated by an angle *α*, the position of the SLM image is shifted by a distance *x*_{0} = *f _{A}* tan(2

*α*). As the position of the SLM-panel is unaffected by this displacement, the superposition of the original and the imaged phase profiles results in a parabolic lens whose center is laterally shifted by

*x*

_{0}/2 away from the optical axis. Without correction of this “walk-off”, we would obtain a tunable lens whose center position depends on the optical power. This undesired effect can be counteracted by an additional beam deflection at the galvo mirror (as described in [18]). In this procedure, the modulated beam is guided again to the non-polarizing beam splitter (to separate it from the unmodulated beam) and subsequently to lens B (at distance

*f*=

_{B}*f*/2 from the SLM) which focuses the beam again onto the galvo mirror (at distance

_{A}*f*from lens B). Together, lens A and B form a telescope which images the galvo onto itself with a magnification of −0.5, thus inverting and increasing the galvo-induced phase slope of 2 ·

_{B}*α*by a factor of 2. The resulting phase slope of −2 ·

*α*is finally compensated by the second reflection from the galvo and the “walk-off” is effectively cancelled. Accordingly, we obtain a tunable lens whose focus position remains on the optical axis for any chosen focal length.

The emerging beam is then collimated by lens C with focal length *f _{C}* =

*f*in a distance of

_{B}*f*from the galvo. This lens forms a sharp image of the SLM at a distance of

_{C}*f*, where we place a circular aperture which is adapted to the minimal overlap area of the sub-elements that occurs during the optical power modulation of the Alvarez lens. The aperture is necessary to block light originating from non-overlapping regions of the sub-elements, which would otherwise negatively affect the point spread function (PSF) of the lens. The aperture is in close vicinity to a short-focal-length lens (

_{C}*L*), the optical power of which adds up to that of the Alvarez lens. The lens

_{s}*L*provides the default refractive power of the entire optical system, which can be varied by the preceding Alvarez module. The PSF of the system is investigated using a 10× microscope objective (not shown in the figure), which forms a magnified image of the focus on a camera (BlueFox 120aG, resolution 640 × 480 quadratic pixels with an edge length of 7.4 μm).

_{s}## 4. Lens characterization

#### Effect of the lens and its optical power:

The phase function of a *refractive* Alvarez sub-element is given by

*A*is a design parameter of the lens which basically determines the lens curvature, as will be specified in the next subsection. The cartesian coordinate

*x*points into the direction where the phase profile

*ϕ*is anti-symmetric (see Fig. 1(b)). When we consider the interaction of one shifted sub-element with its conjugate, we get

*x*

_{0}the lateral shift. The last term

*ϕ*

_{0}generates only an uninteresting global phase shift which will be omitted.

The remaining phase modulation is that of a parabolic lens, with its magnitude growing linearly with the displacement *x*_{0} of the sub-elements as well as with the design parameter *A*. If we want to realize this lens as a thin (first-order) diffractive structure, we have to interpret this as a parabolic phase profile which is then taken modulo 2*π* to create the DOE, i.e.

*λ*denoting the imaging wavelength and

*f*the focal length of the lens. From this we infer the optical power

*D*= 1/

*f*of the corresponding Alvarez lens to be given by

*L*in the x-dimension). The wavelength dependence in this equation leads to chromatic aberrations, which (apart from the wavelength dependence of the refractive index) are absent for a refractive Alvarez lens (see discussion in [5]).

In case a diffractive optical element is used (in air), the relationship between height structure *S* of the DOE and its generated phase modulation is expressed by the following formula:

*λ*

_{0}represents the design wavelength (the diffractive structure shows highest diffraction efficiency at this wavelength) and

*n*

_{0}the optical index of the lens material at

*λ*

_{0}.

#### Determination of the maximal possible optical power range:

The largest possible parameter *A* and thus the maximum optical power of a diffractive Alvarez lens depends on the minimum achievable surface processing unit (pixel size *p*) in the fabrication of the diffractive surface pattern. The theoretically smallest pixel size of a surface structure which is able to influence the phase of light is in the range of the optical wavelength. With conventional laser lithography methods, a pixel size of p = 0.5 μm can be realized and is also reasonable for a laser wavelength of e.g. 532 nm. In case of our SLM, the pixel size is 20 μm.

In order to avoid undersampling of a phase structure, the following conditions must hold

*A*

_{max}is given by where

*r*

_{max}denotes the maximal radius of the diffractive element. In case the phase is modulated with our SLM (

*H*·

*L*= 12 · 16 mm

^{2},

*r*

_{max}= 10 mm),

*A*

_{max}is 1.57 · 10

^{9}m

^{−3}. For an equivalent rectangular DOE (with a pixel size of e.g. 0.5 μm),

*A*

_{max}is equal to 6.28 · 10

^{10}m

^{−3}. It should, however, be kept in mind that the choice of such a large coefficient

*A*leads to a binary grating in the periphery of the diffractive structure. The efficiency of the DOE in these regions is then below 50%. In order to keep the efficiency on a high level over the entire DOE surface,

*A*should be chosen considerably smaller, for example, half of the indicated values for

*A*

_{max}. The maximum possible optical power range can now be determined by inserting [Eq. (9)] into [Eq. (5)]:

*x*

_{0}and the twist angle

*α*of the galvo mirror was used.

Recalling that the pixel size for DOE manufacturing is in the order of the design wavelength, we arrive at the following relation for the maximum optical power range:

*λ*= 532 nm, a diffractive structure dimension of

*H*·

*L*= 12 · 16 mm

^{2}and a sub-element displacement of e.g.

*x*

_{0}= 5.33 mm (the optimal value, as will be shown in the next subsection), the following values for optical power ranges are obtained for an SLM- as well as a DOE-based Alvarez lens (

*A*=

*A*

_{max}/2):

*D*

_{max}), since non-resonant galvo mirrors possess a response time of typically 100 μs for a step of 0.1°.

#### Determination of the optimal pupil size:

In order to avoid aberrations of the Alvarez lens, in a practical application one will have to define a fixed pupil size *P*, which is equal to the minimum overlap *OL* of the Alvarez sub-elements during modulation of its optical power (*OL* = *L* − *x*_{0} = *P* and *P* ≤ *H*, see Fig. 4).

The following trade-off is effective: A large pupil provides a high NA-change per unit displacement *dx*_{0}, but the shift range *x*_{0}, and thus the optical power range, is limited. On the other hand, defining a small pupil enables a large shift range, but only a small NA-change per unit displacement. We can find an optimum value for the pupil size *P* by considering another pupil *P*_{0} of constant size onto which *P* needs to be imaged. In practice, *P*_{0} could be the entrance aperture of a subsequent optical system such as a microscope objective. However, as we shall see, the actual size of *P*_{0} is irrelevant, as long as we can map *P* onto *P*_{0}.

If we image *P* onto *P*_{0} using a two-lens telescope of magnification *M* = *P*_{0}/*P*, all focus position variations induced by the Alvarez module and thus also the accomodation range Δ*D* change according to the axial magnification *M*^{2}, i.e.

*P*in view of a maximized accomodation range using the condition $\frac{d}{dP}\mathrm{\Delta}{D}_{0}=0$, from which the condition $\frac{d}{dP}\left[{P}^{2}(L-P)\right]=0$ can be derived using [Eq. (12)] and Δ

*D*∝

*x*

_{0max}, where

*x*

_{0max}= (

*L*−

*P*) is the maximal allowed sub-element displacement. This equation leads to

*P*

_{opt}= 2/3 ·

*L*. An aperture with a diameter of two-thirds of the respective lens dimension delivers the relatively largest possible optical power range for an Alvarez lens. Since

*P*=

*L*−

*x*

_{0}, the maximal displacement

*x*

_{0max}is therefore limted to

*L*/3 (5.33 mm in our case).

## 5. Experimental results

In this section we present and discuss the performance analysis of the Alvarez lens, where we study the lateral- and axial beam projections around the focal spot of lens *L _{s}*, where the focal powers of

*L*(40 dpts) and of the Alvarez lens add up. The individual images of Fig. 5 illustrate the corresponding measurements recorded with the BlueFox camera.

_{s}The projections were recorded for various sub-element shifts (*x*_{0} = 2.2/0/−2.2 ±0.5 mm whereby *x*_{0max} = 5.33 mm) and thus for different optical power states of the Alvarez lens. This can be seen by the change of the absolute focus position in the images. Taking into account the formula for the optical power of the Alvarez lens [Eq. (10)] and that the galvo performs an angular twist of 0.25° for a control voltage change of 0.1 V, we are able to compare the theoretical with the experimental values for the resulting focal length *f*_{res}:

The axial recording range of the measurements depicted in Fig. 5 extends ±900 μm around the focal spot of the zero optical power Alvarez lens. The individual lateral beam sections were stacked with ImageJ and interpolated (Subfigures (a, c and e)). The intensity projections were acquired starting from the individual focuses and stepping away from them in discrete steps at a distance of 100 μm (with micrometer precision of the micrometer-screw). Although the exposure time of the camera was adapted for each position, the axial intensity distribution could not be recorded homogenously. In order to enhance its visibility and comparability, the intensities were magnified computationally by a certain factor (equal for each image) and thus some regions in the center of the beam are brought into saturation, which leads to an artificial homogenization. As visible in the lateral beam profiles of the focal spot (Subfigures (b, d and f)), a diffraction limited point spread function (PSF; lateral extent: *d*_{exp}) can approximately be achieved:

*L*, which is optimized for multicolor light at the price of a poorer performance for monochromatic light. In all these measurements the pupil was adjusted to the optimal extension of

_{s}*P*= 2

*L*/3 = 10.67 mm.

## 6. Conclusions

We have suggested and experimentally demonstrated the feasibility of a modified diffractive Alvarez lens configuration which does not require any physical translational movement of two sub-lenses against each other, but instead *images* two diffractive sub-structures onto each other. Focus tuning is achieved by rotating a galvo-mirror, which allows for a fast tuning frequency of 1 kHz. Diffractive structures with a lateral resolution e.g. on the order of 0.5 μm offer a wide tuning range on the order of ±50 dpts. For low numerical apertures diffraction limited focusing can be achieved. We believe that our Alvarez lens configuration is ideally suited for applications demanding fast refocusing in low-NA microscopy, as for example for large field-of-view 3D life cell imaging and monitoring, for particle tracking, or for profilometry. The technology might also be interesting for fast laser processing or barcode scanning.

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