## Abstract

We present a hybrid diffractive-refractive optical lens doublet consisting of a varifocal Moiré Fresnel lens and a polymer lens of tunable refractive power. The wide range of focal tunability of each lens and the opposite dispersive characteristics of the diffractive and the refractive element are exploited to obtain an optical system where both the Abbe number and the refractive power can be changed separately. We investigate the performance of the proposed hybrid lens at zero overall refractive power by tuning the Abbe number of a complementary standard lens while maintaining a constant overall focal length for the central wavelength. As an application example, the hybrid lens is used to tune to an optimal operating regime for quantitative phase microscopy based on a two-color transport of intensity (TIE) approach which utilizes chromatic aberrations rather than intensity recordings at several planes to reconstruct the optical path length of a phase object.

© 2014 Optical Society of America

## 1. Introduction

Chromatic dispersion is inherent to materials that are transparent in the visible spectrum, giving rise to chromatic aberrations in optical instruments. Thus dispersion compensation plays a major role in the design of optical systems. Contrariwise, in recent years methods have been developed, which *exploit* the resulting wavelength-dependent focal length, for example for synchronous multilayer imaging [1–3], laser pulse shaping [4], spectral filtering [5], or phase microscopy based on the transport of intensity equation [6, 7]. These developments demand for dispersion-tunable optical units.

Chromatic dispersion arises from the wavelength-dependence of the phase velocity of a light field when passing through an optical material or, in other words, from the change in refractive index with wavelength. A widespread measure of this phenomenon is the material’s Abbe number, *V* = (*n _{d}* − 1)/(

*n*−

_{F}*n*), which accounts for a varying index of refraction with respect to specific, chosen wavelengths. Here,

_{C}*n*,

_{d}*n*and

_{F}*n*historically denote the refractive indices at the Fraunhofer lines,

_{C}*λ*= 587.6 nm,

_{d}*λ*= 486.1 nm and

_{F}*λ*= 656.3 nm, respectively [8]. In the visible electromagnetic spectrum, commercially available glasses, plastics and crystals exhibit an increasing refractive index with decreasing wavelength,

_{C}*dn/dλ*< 0, resulting in positive Abbe numbers ranging from 20 to 90, with low values indicating large chromatic dispersion and vice versa [9]. When focusing, the image focal length decreases with wavelength and the material-dependent Abbe number provides a linear approximation for the shift in focus for a red and a blue wavelength with respect to the central green wavelength, hence it is also a measure of longitudinal chromatic aberrations or axial color. Reduction of chromatic aberrations in optical systems is regularly addressed by combining two or more refractive lenses of different Abbe numbers and refractive powers. In the simplest case of a lens doublet acting as an achromat

*ϕ*

_{1}/

*V*

_{1}+

*ϕ*

_{2}/

*V*

_{2}= 0 is satisfied, matching the foci of a red and a blue wavelength, where

*ϕ*and

*V*are the optical powers and the Abbe numbers of the two lenses, respectively. Reduction of the residual secondary spectrum is achieved with the design of apochromats and superachromats and was demonstrated with tunable chromatic aberration control with silicone membrane microlenses [10]. The combination of material disparities can also be employed to design an optical system with a desired effective overall Abbe number,

*V*

_{eff}, but is limited by the maximal refractive power of the lenses which can be fabricated without introducing further geometrical aberrations.

A well known alternative for tailoring chromatic aberrations is to use a combination of a diffractive (holographic) with a refractive lens [11, 12]. A holographic lens exhibits a material-independent fixed-value ”Abbe number” of *λ _{d}*/(

*λ*−

_{F}*λ*) = −3.452, which is about one order of magnitude lower than that of refractive lenses [8] and has a negative sign. Therefore a combination of a diffractive and a refractive lens allows one to realize an achromat with a much larger refractive power than a combination of two refractive lenses. Very recently such a variable focus hybrid lens, consisting of a liquid crystal diffractive lens and a pressure controlled fluidic refractive lens, was investigated with respect to its achromatic properties [13]. Similarly, in [14] another tunable hybrid lens, consisting of a fixed focus Fresnel lens and a tunable liquid filled lens, had already been introduced as a method to reduce chromatic aberrations. But then there also exist applications which make use of the enhanced dispersion that can be achieved with diffractive lenses, which were termed ”hyperchromatic”. In [5] a hybrid lens consisting of a tunable fluidic lens in combination with a fixed focus Fresnel lens was demonstrated to act as an optical monochromator by focusing white light through a pinhole, where - due to the dispersion - only a narrow spectral band with a width of 13 nm was transmitted. In this kind of monochromator the diffractive lens provides for the desired high dispersion, whereas the wavelength tunability is achieved by the adjustable optical power of the liquid lens.

_{C}Our approach is similar to this concept, also providing an optical element for dispersion control. However, in our hybrid diffractive-refractive lens doublet (DRD) the focal lengths of *both*, the refractive and the diffractive element, can be tuned continuously from large positive to large negative values. This is done by using a recently demonstrated so-called Moiré diffractive optical element (MDOE) [15, 16], which corresponds to a rotational version of a diffractive Alvarez lens [17] and acts as a tunable Fresnel lens. Here it is used in combination with a commercially available refractive focus-tunable liquid filled polymer lens. This allows to tune both the effective refractive power *ϕ*_{eff} and the effective Abbe number *V*_{eff} at the same time in a wide range, satisfying *ϕ*_{eff}/*V*_{eff} = *ϕ*_{1}/*V*_{1} +*ϕ*_{2}/*V*_{2}.

A particularly interesting operation modality is to assign complementary optical powers to the successive refractive and diffractive lenses in a DRD, i.e., the two lens elements have the same absolute values of their focal lengths, but an opposite sign. At a first glance such an element acts as a “do-nothing machine” [18], since it has zero optical power and does not seem to affect the optical setup upon insertion. However, what it does is to introduce dispersion, which can be continuously tuned without changing the average effective focal length of an optical system. For example, if such a zero power DRD is inserted close to another refractive lens, it can be used to adjust the effective Abbe number of this lens in a wide range between positive and negative values without changing the mean focal length. This can be used to compensate chromatic aberrations in complex optical setups without the need to redesign the optical layout, or, on the other hand, to introduce a precisely controlled amount of linear dispersion, which is advantageous for certain tasks.

As an example we present an application in phase microscopy, based on the transport of intensity equation (TIE) [7, 19, 20], to infer the refractive index profiles of direct laser written (DLW) waveguides in fused silica [21] from the chromatic aberrations in a microscope [6]. Using the TIE method, the phase profile of a sample can be numerically calculated from a set of two (or more) slightly defocused images [22]. In [6] it was demonstrated that such a set of images can be recorded in the different color channels of a RGB camera in a single exposure by using multi color illumination of the sample and the dispersive characteristics of an imaging system. But it turns out that the optimal defocus for best imaging depends on the sample itself [23], namely on the magnitude of its phase gradients. In order to obtain an intensity modification between the two images which is sufficient for numerical post-processing, the required amount of defocus increases for a decreasing phase gradient of the sample. Here we show that by inserting a zero optical power DRD in such a setup, the dispersion can be adjusted for reconstructing images of very weak phase samples (e.g., index variations in laser written waveguides) with the TIE method at a maximal signal-to-noise level.

## 2. Hybrid diffractive-refractive optical lens

Our hybrid diffractive-refractive lens doublet (DRD) is a combination of independently focus tunable diffractive and refractive components, i.e., a diffractive Moiré lens [15] and a commercially available refractive liquid filled polymer lens (Optotune ML-20-30-VIS-HR). The Moiré lens consists of two identical diffractive elements axially stacked back-to-back with a combined transmission function that correspondes to a Fresnel lens. Mutual rotation about the central axis of one element with respect to the other changes the refractive power of the Moiré lens in a range determined by the design parameters of the diffractive elements [16]. The prototype of the Moiré lens under investigation is designed for a wavelength *λ*_{0} = 632.8 nm and both refractive and diffractive elements in the DRD feature a refractive power tuning range from −25 dpt to +25 dpt. The refractive power *ϕ* of a diffractive lens is proportional to the wavelength *λ* of the incident light and is given by

*ϕ*

_{0}is the refractive power at the design wavelength

*λ*

_{0}. Figure 1(a) shows a simulation, according to Eq. (1), of the refractive powers of the Moiré lens for the design wavelength (black) and for the wavelengths in the definition of the Abbe number (blue, green and red) versus the experimental tuning parameter, i.e., the relative rotational angle of the two diffractive elements which the Moiré lens consists of.

Assuming negligible axial seperation of the refractive and the diffractive element, the dispersive characteristics of the DRD can be expressed as

*ϕ*

_{eff}and

*V*

_{eff}are the effective refractive power and the effective Abbe number of the DRD, respectively, and the indices

*R*and

*D*denote refractive and diffractive properties, respectively. In Fig. 1(b) the effective Abbe number of the DRD is plotted throughout the entire tuning range of both the refractive and the diffractive lenses for

*V*=31 and

_{R}*V*= −3.452. A logarithmic colorscale is used to avoid the dominance of the singularity of the effective Abbe number along the dark red and dark blue lines, which corresponds to the case of an achromat, where the left hand side of Eq. (2) equals zero.

_{D}## 3. Adjustable effective Abbe number

A special operation mode of a DRD is to adjust opposite refractive powers at a selected central wavelength *λ _{d}*, i.e.

*ϕ*(

_{R}*λ*) = −

_{d}*ϕ*(

_{D}*λ*). If such a DRD is placed close to an additional optical element, it will affect the effective dispersion of this element without influencing the total refractive power at the selected central wavelength. Here we consider the simple case where the additional optical element is a standard refractive lens with a refractive power of

_{d}*ϕ*and derive an expression for the effective Abbe number. The effective refractive power,

_{L}*ϕ*

_{eff}, of the lens system consisting of a standard lens and the DRD is given by

*ϕ*

_{DRD}(

*λ*) ≠ 0 at wavelengths

*λ*different from the central wavelength

*λ*. Here we assume negligible distances between all lenses and that the dispersion of the compound optical system is solely given by the dispersion of the diffractive lens (which is about an order of magnitude larger than that of the refractive lenses). The wavelength-dependence of the effective refractive power can then be expressed as

_{d}*λ*= 486.1 nm and

_{F}*λ*= 656.3 nm are the Fraunhofer lines used in the definition of the Abbe number

_{C}*V*of the lens.

_{L}A comparison between the dispersions of the compound optical system (consisting of the DRD and the additional refractive lens), given by Eq. (5), and the actual dispersion of a single refractive lens, given by Eq. (6), shows that the compound system can be described as a single lens with an effective refractive power of *ϕ*_{eff} = *ϕ _{L}*. With

*λ*/(

_{d}*λ*−

_{F}*λ*) = −3.452 the effective Abbe number

_{C}*V*

_{eff}is given by

*ϕ*(

_{R}*λ*) = −

_{d}*ϕ*(

_{D}*λ*) is satisfied. For demonstration we use the DRD for tuning the dispersive characteristics of a spherical lens with a focal length

_{d}*f*=15 cm, corresponding to a refractive power

_{L}*ϕ*= 6.67 dpt, as illustrated in Fig. 2(a). Collimated white light from a LED passes through the DRD and the subsequent spherical lens (SL) placed at an axial distance of

_{L}*d*

_{2}= 3 mm. The separation

*d*

_{1}of the diffractive and refractive elements within the DRD was 3 mm. The axial spectral widening of the focal spot is qualitatively demonstrated by inserting a screen aligned with the optical axis. Figure 2(b) shows photographies of the axial profile of the focal spot for positive (top) to negative (bottom) refractive powers of the diffractive Moiré lens.

For a more quantitative experimental investigation of the dispersive characteristics of the compound optical system we collimated incident light from a monochromator (TILL Photonics Polychrome IV) and measured the focal lengths for two quasi monochromatic wavelenghts, *λ _{F}* = 486 nm and

*λ*= 656 nm, while stepwise adjusting the refractive powers of the diffractive Moiré lens and the refractive polymere lens at a constant overall focal length of

_{C}*f*= 1/

_{L}*ϕ*= 0.15 m at

_{L}*λ*= 588 nm.

_{d}The blue and red dots in Fig. 3(a) represent the measured shifts in focus Δ*f* for *λ _{F}* and

*λ*, respectively, as a function of the refractive power of the Moiré lens at the design wavelength

_{C}*λ*

_{0}= 632.8 nm. The experimental data is compared to the theoretically expected values for the overall refractive power

*ϕ*of the lens triplet, which within the thin lens approximation is given by

*ϕ*(

*λ*)

^{−1}=

*f*+ Δ

_{L}*f*. Zero refractive power of the DRD at the central wavelength

*λ*, as set experimentally for the acquisition of each data point, requires

_{d}*ϕ*(

_{R}*λ*) = −

_{d}*ϕ*(

_{D}*λ*). The green line in Fig. 3(a) indicates constant focal length

_{d}*f*for the central wavelength

_{L}*λ*. With the experimentally determined shifts in focus Δ

_{d}*f*and Δ

_{F}*f*for the blue and the red wavelengths, the corresponding indices of refraction

_{C}*n*and

_{F}*n*of the lens triplet can be calculated using [24]

_{C}*λ*

_{0}= 632.8 nm. The experimental values of the Abbe number (blue crosses) were calculated from the measured shifts in focus with Eqs. (9) and (10) and the black curve corresponds to Eq. (7). The deviations between experimental and theoretical data are due to the fact that the distances between all optical components in the derivation of Eq. (7) have been neglected, i.e. the whole optical system is just approximated as a single thin lens. On the other hand the deviations are sufficiently moderate to use Eq. (7) as a ”rule of thumb” to estimate the addressable Abbe number range of such a lens combination.

## 4. Phase microscopy from optimized chromatic aberrations

Many different optical techniques exist to directly measure refractive index variations of transparent specimens, ranging from interferometry with coherent illumination to optical coherence tomography with broadband light [25]. It is, however, also possible to retrieve this information from multiple plane intensity measurements with subsequent iterative phase retrieval [26–28], but generally this involves significant experimental and computational efforts. In recent years, phase microscopy based on the transport of intensity equation (TIE) has gained considerable attention and has emerged as a robust and easy-to-implement tool to examine refractive index variations in phase objects by means of recording intensity images at slightly different axial positions [7, 19, 20]. The transport of intensity equation, which relates the intensity profiles at adjacent transverse planes to each other, can be derived from the paraxial wave equation and takes on the form

*I*(

*x*,

*y*) and

*ϕ*(

*x*,

*y*) denote the intensity and phase distribution, respectively, ∇⃗

_{⊥}is the transverse gradient operator and

*λ*the central wavelength of the (partially coherent) illumination. The TIE provides a relation between intensity and phase via the change of intensity that is induced after infinitesimal propagation along the optical axis due to refractive index variations. In practice the derivative

*∂I*(

*x*,

*y*)/

*∂z*is approximated by the difference of two slightly defocused images divided by their axial separation,

*∂I*(

*x*,

*y*)/

*∂z*≈ [

*I*(

*x*,

*y*,

*z*) −

*I*(

*x*,

*y*,

*z*+

*δz*)]/

*δz*. The amount of defocus,

*δz*, plays a crucial role for the performance of TIE-microscopy and presents a trade-off between good signal to noise ratio and significant deviation from the true intensity derivative [23]. Weak phase objects with structures of low spatial frequency content, i.e., a small

*∂I*(

*x*,

*y*)/

*∂z*, require larger defocus for the signal to compete against noise than objects exhibiting large phase gradients. Recently two-shot TIE-microscopy has been applied successfully to accurately deliver the change in refractive index of direct laser written (DLW) waveguides [21] in fused silica [22]. This work identified the optimal amount of defocus for these weak phase objects to range between 2 μm and 6 μm. In [6] TIE-microscopy has been simplified to a single-shot technique exploiting system chromatic aberrations in combination with a color camera to simultaneously capture in-focus and out-of-focus images on different color channels.

As an application of the DRD, we have tuned the system chromatic dispersion of a microscope to the optimum focal separation of green and red light for single-shot TIE-microscopy from a color image of a sample containing DLW-waveguides located 170 μm below the surface of fused silica. Such waveguides are fabricated by translating the focus of a pulsed laser in the bulk of fused silica, resulting in channels of a gaussian shaped refractive index variation. We investigated the performance for different amounts of defocus and compared our results with the conventional method of taking two intensity images at a single wavelength (green) to reconstruct the phase information. The simple geometry of the waveguides allowed us to process the data with the TIE in one dimension,

where the phase distribution may formally be written as*δz*, which approximates the gradient

*dI*(

*x*)/

*dz*.

The measurements were carried out on an inverted transmission bright-field microscope (see Fig. 4) with a white LED light source and a color camera (Canon EOS 1000D). The DRD to tune the dispersive characteristics was placed in the 4f-imaged back-focal plane of the microscope objective (Olympus UPlanFL N 60× 1.25 NA). The amount of defocus *δz* between the green and the red focal planes with and without the DRD was determined by comparing the objective working distances for a green and a red wavelength. To this end, for each of the two colors the respective spectral filter was inserted into the optical path and the microscope objective was axially scanned with a piezo stage to bring the waveguide into focus. The difference between the working distances represents *δz*. Without modification of the dispersive characteristics of the micrcoscope we found *δz* = 0.9 μm ±0.05 μm. In Fig. 5(a) we compare the reconstructed peak gaussian phase variations in terms of the optical path length, OPL = *ϕ*(*x*) · *λ*/2*π*, for different amounts of defocus. The square represents the OPL reconstructed with *δz* caused by the system chromatic dispersion, while the circles represent the OPLs at larger amounts of defocus attainable with the DRD. For comparison, the crosses provide the peak OPLs reconstructed from the green channels of two color images taken at axial positions of the microscope objective separated by *δz*. The error bars show the standard deviations for 10 measurements. Tuning the system chromatic dispersion from a magnitude of defocus of 0.9 μm to 2.3 μm reduced the standard deviation from 8.4% to 2.0% of the mean peak OPL. Figure 5(b) shows the averaged gaussian phase profiles and transverse standard deviations of a waveguide at a defocus of 2.3 μm (green) and 0.9 μm (black), respecively, which again demonstrates that the error is considerably reduced by tuning the amount of defocus via dispersion control to its optimal value of about *δz* = 2.3 μm.

## 5. Summary

Due to their different wavelength dependence, combinations of diffractive and refractive elements are suitable to control the chromatic dispersion of optical systems. Here we have introduced and investigated an optical hybrid lens that provides tunable chromatic dispersion at adjustable overall refractive power. In particular, we have demonstrated the tuning of the dispersive characteristics of a standard refractive lens without affecting its refractive power at the central wavelength of the illumination. An equation for the overall relative dispersion of a refractive lens and this hybrid lens in cascade was derived.

Conversely, longitudinal chromatic aberrations in a microscope can be exploited to capture focused and defocused images in the RGB channels of a single color image, which lends itself to applications in the spirit of two-color TIE imaging. Our approach allows one to always operate in the sample-dependent optimal defocus range, which is dictated by the opposing requirements of sufficiently large difference in intensities on one hand and sufficiently small deviations from the approximated “transport of intensity” condition on the other hand. We explicitly show this for a phase sample with laser-written waveguides in fused silica, for which it was possible to deduce refractive index variations from a single color image with improved signal to noise ratio. Further applications of such a dispersion tunable hybrid lens might arise in dispersion control of broadband laser pulses, or for spectral filtering of polychromatic light. For example it should be possible to use a similar concept as introduced in [5] for realizing a tunable monochromator, where both the central wavelength and the spectral band width of the transmitted light can be adjusted.

## Acknowledgments

The autors gratefully thank Dr. Martin Booth from Oxford University for providing the DLW-waveguide sample. This work was supported by the ERC Advanced Grant 247024 catchIT, and by the Austrian Science Fund (FWF): P19582-N20.

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