Dispersion tuning with a varifocaldiffractive-refractive hybrid lens

Walter Harm, Clemens Roider, Alexander Jesacher,Stefan Bernet,∗ and Monika Ritsch-Marte

Division for Biomedical Physics, Innsbruck Medical University,A-6020 Innsbruck, Austria∗stefan.bernet@i-med.ac.at

Abstract: We present a hybrid diffractive-refractive optical lens doubletconsisting of a varifocal Moire Fresnel lens and a polymer lens of tunablerefractive power. The wide range of focal tunability of each lens and theopposite dispersive characteristics of the diffractive and the refractiveelement are exploited to obtain an optical system where both the Abbenumber and the refractive power can be changed separately. We investigatethe performance of the proposed hybrid lens at zero overall refractivepower by tuning the Abbe number of a complementary standard lens whilemaintaining a constant overall focal length for the central wavelength. As anapplication example, the hybrid lens is used to tune to an optimal operatingregime for quantitative phase microscopy based on a two-color transport ofintensity (TIE) approach which utilizes chromatic aberrations rather thanintensity recordings at several planes to reconstruct the optical path lengthof a phase object.

© 2014 Optical Society of AmericaOCIS codes: (050.1970) Diffractive optics; (120.3620) Lens system design; (090.1970)Diffractive Optics; (120.5050) Phase measurement.

References and links1. C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B. E. Applegate, and K. C. Maitland, “Chromatic confocal mi-

croscopy for multi-depth imaging of epithelial tissue,” Opt. Express 4, 732–740 (2013).2. S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller,

C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fastmulticolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).

3. A. Jesacher, C. Roider, S. Bernet, and M. Ritsch-Marte, “Enhancing diffractive multi-plane microscopy usingcolored illumination,” Opt. Express 21, 11150–11161 (2013).

4. O. Mendoza-Yero, G. Mınguez-Vega, J. Lancis, and V. Climent, “Diffractive pulse shaper for arbitrary waveformgeneration,” Opt. Lett. 35, 535–537 (2010).

5. P. Cu-Nguyen, A. Grewe, M. Hillenbrand, S. Sinzinger, A. Seifert, and H. Zappe, “Tunable hyperchromatic lenssystem for confocal hyperspectral sensing,” Opt. Express 21, 27611–27621 (2013).

6. L. Waller, S. S. Kou, C. J. R. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express22, 22817–22825 (2010).

7. K. A. Nugent, D. P. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).8. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960-2971 (1988).9. H. Gross, Handbook of Optical Systems (John Wiley, 2005).

10. P. Waibel, D. Mader, P. Liebetraut, H. Zappe, and A. Seifert, “Chromatic aberration control for tunable all-siliconemembrane microlenses,” Opt. Express 19, 18584–18592 (2011).

11. N. Davidson, A. Friesem, and E. Hasman, “Analytic design of hybrid diffractive–refractive achromats,” Appl.Opt. 32, 4770–4774 (1993).

12. E. Tajahuerce, J. Lancis, V. Climent, and P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a noveloptical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151,86–92 (1998).

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5260

13. P. Valley, N. Savidis, J. Schwiegerling, M. R. Dodge, G. Peyman, and N. Peyghambarian, “Adjustable hybriddiffractive/refractive achromatic lens,” Opt. Express 19, 7468–7479 (2013).

14. 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).

15. S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moire elements,” Appl. Opt. 47,3722–3730 (2008).

16. S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moire-lenses,” Opt. Ex-press 21, 6955–6966 (2013).

17. I. M. Barton, S. N. Dixit, L. J. Summers, C. A. Thompson, K. Avicola, and J. Wilhelmsen, “Diffractive Alvarezlens,” Opt. Lett. 25, 1–3 (2000).

18. H. J. Caulfield, “The Alvarez-Lohmann lens as a do-nothing machine,” Opt. Laser Technol. 34, 1–5 (2002).19. M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 23, 1434–1441

(1983).20. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,”

Opt. Commun. 133, 339–9346 (1997).21. G. Della Valle, R. Osellame, and P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,”

J. Opt. A Pure Appl. Opt. 11, 013001 (2009).22. A. Jesacher, P. S. Salter, and M. J. Booth, “Refractive index profiling of direct laser written waveguides: tomo-

graphic phase imaging,” Opt. Mater. Express 3, 1223–1232 (2013).23. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The

effect of noise,” J. Microsc. 214, 51–61 (2004).24. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).25. K. Minoshima, A. M. Kowalevicz, I. Hartl, E. P. Ippen, and J. G. Fujimoto, “Photonic device fabrication in glass

by use of nonlinear materials processing with a femtosecond laser oscillator,” Opt. Lett. 26, 1516–1518 (2010).26. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and

diffraction plane pictures,” Optik 35, 237–246 (1972).27. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).28. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a

support constraint,” J. Opt. Soc. Am. A 4, 118–123(1987).

1. Introduction

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

Chromatic dispersion arises from the wavelength-dependence of the phase velocity of a lightfield when passing through an optical material or, in other words, from the change in refrac-tive index with wavelength. A widespread measure of this phenomenon is the material’s Abbenumber, V = (nd −1)/(nF −nC), which accounts for a varying index of refraction with respectto specific, chosen wavelengths. Here, nd , nF and nC historically denote the refractive indicesat the Fraunhofer lines, λd = 587.6 nm, λF = 486.1 nm and λC = 656.3 nm, respectively [8].In the visible electromagnetic spectrum, commercially available glasses, plastics and crystalsexhibit an increasing refractive index with decreasing wavelength, dn/dλ < 0, resulting inpositive Abbe numbers ranging from 20 to 90, with low values indicating large chromatic dis-persion and vice versa [9]. When focusing, the image focal length decreases with wavelengthand the material-dependent Abbe number provides a linear approximation for the shift in focusfor a red and a blue wavelength with respect to the central green wavelength, hence it is alsoa measure of longitudinal chromatic aberrations or axial color. Reduction of chromatic aberra-tions in optical systems is regularly addressed by combining two or more refractive lenses ofdifferent Abbe numbers and refractive powers. In the simplest case of a lens doublet acting asan achromat φ1/V1 +φ2/V2 = 0 is satisfied, matching the foci of a red and a blue wavelength,

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5261

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 andsuperachromats and was demonstrated with tunable chromatic aberration control with siliconemembrane microlenses [10]. The combination of material disparities can also be employed todesign an optical system with a desired effective overall Abbe number, Veff, but is limited bythe maximal refractive power of the lenses which can be fabricated without introducing furthergeometrical aberrations.

A well known alternative for tailoring chromatic aberrations is to use a combination of adiffractive (holographic) with a refractive lens [11, 12]. A holographic lens exhibits a material-independent fixed-value ”Abbe number” of λd/(λF −λC) =−3.452, which is about one orderof magnitude lower than that of refractive lenses [8] and has a negative sign. Therefore a com-bination of a diffractive and a refractive lens allows one to realize an achromat with a muchlarger refractive power than a combination of two refractive lenses. Very recently such a vari-able focus hybrid lens, consisting of a liquid crystal diffractive lens and a pressure controlledfluidic 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 liquidfilled lens, had already been introduced as a method to reduce chromatic aberrations. But thenthere also exist applications which make use of the enhanced dispersion that can be achievedwith diffractive lenses, which were termed ”hyperchromatic”. In [5] a hybrid lens consistingof a tunable fluidic lens in combination with a fixed focus Fresnel lens was demonstrated toact as an optical monochromator by focusing white light through a pinhole, where - due to thedispersion - only a narrow spectral band with a width of 13 nm was transmitted. In this kindof monochromator the diffractive lens provides for the desired high dispersion, whereas thewavelength tunability is achieved by the adjustable optical power of the liquid lens.

Our approach is similar to this concept, also providing an optical element for dispersioncontrol. However, in our hybrid diffractive-refractive lens doublet (DRD) the focal lengths ofboth, the refractive and the diffractive element, can be tuned continuously from large positive tolarge negative values. This is done by using a recently demonstrated so-called Moire diffractiveoptical element (MDOE) [15, 16], which corresponds to a rotational version of a diffractiveAlvarez lens [17] and acts as a tunable Fresnel lens. Here it is used in combination with acommercially available refractive focus-tunable liquid filled polymer lens. This allows to tuneboth the effective refractive power φeff and the effective Abbe number Veff at the same time in awide range, satisfying φeff/Veff = φ1/V1 +φ2/V2.

A particularly interesting operation modality is to assign complementary optical powers tothe successive refractive and diffractive lenses in a DRD, i.e., the two lens elements have thesame absolute values of their focal lengths, but an opposite sign. At a first glance such an ele-ment acts as a “do-nothing machine” [18], since it has zero optical power and does not seem toaffect the optical setup upon insertion. However, what it does is to introduce dispersion, whichcan be continuously tuned without changing the average effective focal length of an opticalsystem. For example, if such a zero power DRD is inserted close to another refractive lens, itcan be used to adjust the effective Abbe number of this lens in a wide range between positiveand negative values without changing the mean focal length. This can be used to compensatechromatic 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 isadvantageous for certain tasks.

As an example we present an application in phase microscopy, based on the transport ofintensity 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

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5262

set of two (or more) slightly defocused images [22]. In [6] it was demonstrated that such aset of images can be recorded in the different color channels of a RGB camera in a singleexposure by using multi color illumination of the sample and the dispersive characteristics ofan imaging system. But it turns out that the optimal defocus for best imaging depends on thesample itself [23], namely on the magnitude of its phase gradients. In order to obtain an intensitymodification between the two images which is sufficient for numerical post-processing, therequired amount of defocus increases for a decreasing phase gradient of the sample. Here weshow that by inserting a zero optical power DRD in such a setup, the dispersion can be adjustedfor reconstructing images of very weak phase samples (e.g., index variations in laser writtenwaveguides) 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 focustunable diffractive and refractive components, i.e., a diffractive Moire lens [15] and a com-mercially available refractive liquid filled polymer lens (Optotune ML-20-30-VIS-HR). TheMoire lens consists of two identical diffractive elements axially stacked back-to-back with acombined transmission function that correspondes to a Fresnel lens. Mutual rotation about thecentral axis of one element with respect to the other changes the refractive power of the Moirelens in a range determined by the design parameters of the diffractive elements [16]. The proto-type of the Moire lens under investigation is designed for a wavelength λ0 = 632.8 nm and bothrefractive and diffractive elements in the DRD feature a refractive power tuning range from -25dpt to +25 dpt. The refractive power φ of a diffractive lens is proportional to the wavelength λof the incident light and is given by

φ(λ ) =λλ0

φ0, (1)

where φ0 is the refractive power at the design wavelength λ0. Figure 1(a) shows a simulation,according to Eq. (1), of the refractive powers of the Moire lens for the design wavelength (black)and for the wavelengths in the definition of the Abbe number (blue, green and red) versus theexperimental tuning parameter, i.e., the relative rotational angle of the two diffractive elementswhich the Moire lens consists of.

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

φeff

Veff=

φR

VR+

φD

VD, (2)

where φeff and Veff 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 rangeof both the refractive and the diffractive lenses for VR =31 and VD = −3.452. A logarithmiccolorscale is used to avoid the dominance of the singularity of the effective Abbe number alongthe dark red and dark blue lines, which corresponds to the case of an achromat, where the lefthand side of Eq. (2) equals zero.

3. Adjustable effective Abbe number

A special operation mode of a DRD is to adjust opposite refractive powers at a selected centralwavelength λd , i.e. φR(λd) = −φD(λd). If such a DRD is placed close to an additional opticalelement, it will affect the effective dispersion of this element without influencing the total re-fractive power at the selected central wavelength. Here we consider the simple case where the

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5263

Diffractive power [1/m]

Re

fra

ctive

po

we

r[1

/m]

-20 -10 0 10 20

-25

-20

-15

-10

-5

0

5

10

15

20

25

-1010

-108

-106

-104

-102

0

102

104

106

108

(a) (b)

Veff

-200 -150 -100 -50 0 50 100 150 200-25

-20

-15

-10

-5

0

5

10

15

20

25

Re

fra

ctive

po

we

r[1

/m]

Rotational angle [°]

632.8 nm

486.1 nm

587.6 nm

656.3 nm

Fig. 1. (a) Simulation of the refractive power of the Moire lens for the design wavelength(black) and the wavelengths used for the definition of the Abbe number (blue, green andred), see inset. (b) Simulation of the effective Abbe number of the DRD over the entiretuning range of both elements. Note the logarithmic color scale.

additional optical element is a standard refractive lens with a refractive power of φL and derivean expression for the effective Abbe number. The effective refractive power, φeff, of the lenssystem consisting of a standard lens and the DRD is given by

φeff(λ ) = φL +φDRD(λ ), (3)

where, using Eg. (1),

φDRD(λ ) = φD(λd)λ −λd

λd(4)

accounts for φDRD(λ ) �= 0 at wavelengths λ different from the central wavelength λd . Here weassume negligible distances between all lenses and that the dispersion of the compound opticalsystem is solely given by the dispersion of the diffractive lens (which is about an order ofmagnitude larger than that of the refractive lenses). The wavelength-dependence of the effectiverefractive power can then be expressed as

dφeff(λ )dλ

=1

λdφD(λ ). (5)

In contrast, the linear dispersion of a standard refractive lens is given in terms of the Abbenumber as

dφL

dλ=

φL

VL(λF −λC), (6)

where λF = 486.1 nm and λC = 656.3 nm are the Fraunhofer lines used in the definition of theAbbe number VL of the lens.

A comparison between the dispersions of the compound optical system (consisting of theDRD and the additional refractive lens), given by Eq. (5), and the actual dispersion of a singlerefractive lens, given by Eq. (6), shows that the compound system can be described as a singlelens with an effective refractive power of φeff = φL. With λd/(λF −λC) =−3.452 the effectiveAbbe number Veff is given by

Veff =−3.452 · φL

φD. (7)

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5264

(a)Screen

SLDRD

�f

(b)

0 1-1-2 2

[cm]fLd1 d2

Veff

-1.04

-1.29

-1.73

-2.59

-5.18

5.18

2.59

1.73

1.29

1.04

�

Fig. 2. Tuning the dispersive characteristics of a spherical lens with a tunable diffractive-refractive doublet (DRD). (a) Schematic of the DRD, consisting itself of a varifocal Moirediffractive lens and a varifocal polymere membrane refractive lens, used in front of thespherical lens (SL). Illustrated is a negative Abbe number of the spherical lens with fdiff >0, fref < 0 and | fdiff| ∼= | fref|. (b) Photographies of axial focus profiles captured on a screenas illustrated in (a). The corresponding effective Abbe numbers Veff were calculated withEq. (7) and range from negative (top) to postive (bottom) values.

Thus the effective Abbe number of the lens system can be tuned at constant average focal lengthby adjusting the refractive powers of the tunable Moire diffractive lens and the refractive tunablepolymer lens to values for which φR(λd) =−φD(λd) is satisfied. For demonstration we use theDRD for tuning the dispersive characteristics of a spherical lens with a focal length fL =15 cm,corresponding to a refractive power φL = 6.67 dpt, as illustrated in Fig. 2(a). Collimated whitelight from a LED passes through the DRD and the subsequent spherical lens (SL) placed at anaxial distance of d2 = 3 mm. The separation d1 of the diffractive and refractive elements withinthe DRD was 3 mm. The axial spectral widening of the focal spot is qualitatively demonstratedby inserting a screen aligned with the optical axis. Figure 2(b) shows photographies of theaxial profile of the focal spot for positive (top) to negative (bottom) refractive powers of thediffractive Moire lens.

For a more quantitative experimental investigation of the dispersive characteristics of thecompound optical system we collimated incident light from a monochromator (TILL PhotonicsPolychrome IV) and measured the focal lengths for two quasi monochromatic wavelenghts,λF = 486 nm and λC = 656 nm, while stepwise adjusting the refractive powers of the diffractiveMoire lens and the refractive polymere lens at a constant overall focal length of fL = 1/φL =0.15 m at λd = 588 nm.

The blue and red dots in Fig. 3(a) represent the measured shifts in focus Δ f for λF and λC,respectively, as a function of the refractive power of the Moire lens at the design wavelengthλ0 = 632.8 nm. The experimental data is compared to the theoretically expected values for theoverall refractive power φ of the lens triplet, which within the thin lens approximation is givenby

φ(λ ) = φL +φR(λd)+φD(λ ), (8)

with φ(λ )−1 = fL +Δ f . Zero refractive power of the DRD at the central wavelength λd , as setexperimentally for the acquisition of each data point, requires φR(λd) = −φD(λd). The greenline in Fig. 3(a) indicates constant focal length fL for the central wavelength λd . With theexperimentally determined shifts in focus Δ fF and Δ fC for the blue and the red wavelengths,

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5265

(a) (b)

-25 -20 -15 -10 -5 0 5 10 15 20 25

-100

-50

0

50

100

Ab

be

nu

mb

er

Refractive power [1/m]

Measurement

Approximation

-20 -15 -10 -5 0 5 10 15 20

-0.05

0

0.05

0.1

0.15

0.2

0.25

�f

[m]

Refractive power [1/m]

656.3 nm

587.6 nm

486.1 nm

Fig. 3. Experimental results for tuning the Abbe number of a standard spherical lens with afocal length of fL = 0.15 m with the DRD. (a) Measured shift in focus for a red wavelength(656.3 nm) and a blue wavelength (488.1 nm) with respect to the refractive power at thedesign wavelength (632.8 nm) of the Moire lens while maintaining zero effective refractivepower of the DRD for the central wavelength (587.6 nm). The red and blue curves cor-respond to Eq. (8) and the green curve indicates the constant focal length for the centralwavelength. (b) Experimentally determined effective Abbe number of the compound sys-tem compared to its theoretical values (within the thin lens approximation) given by Eq. (7)(black curve).

the corresponding indices of refraction nF and nC of the lens triplet can be calculated using [24]

Δ fF/C =−nF/C −nd

nF/C −1fL. (9)

The corresponding Abbe number is given by

V =nd −1

nF −nC. (10)

Figure 3(b) shows both the experimentally determined and the calculated Abbe numbers as afunction of the refractive power of the Moire lens at its design wavelength λ0 = 632.8 nm.The experimental values of the Abbe number (blue crosses) were calculated from the measuredshifts in focus with Eqs. (9) and (10) and the black curve corresponds to Eq. (7). The deviationsbetween experimental and theoretical data are due to the fact that the distances between alloptical components in the derivation of Eq. (7) have been neglected, i.e. the whole optical sys-tem is just approximated as a single thin lens. On the other hand the deviations are sufficientlymoderate to use Eq. (7) as a ”rule of thumb” to estimate the addressable Abbe number range ofsuch a lens combination.

4. Phase microscopy from optimized chromatic aberrations

Many different optical techniques exist to directly measure refractive index variations of trans-parent specimens, ranging from interferometry with coherent illumination to optical coherencetomography with broadband light [25]. It is, however, also possible to retrieve this informationfrom 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

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5266

attention and has emerged as a robust and easy-to-implement tool to examine refractive indexvariations in phase objects by means of recording intensity images at slightly different axialpositions [7, 19, 20]. The transport of intensity equation, which relates the intensity profiles atadjacent transverse planes to each other, can be derived from the paraxial wave equation andtakes on the form

2πλ

∂ I(x,y)∂ z

=−�∇⊥[I(x,y)�∇⊥φ(x,y)

], (11)

where I(x,y) and φ(x,y) denote the intensity and phase distribution, respectively, �∇⊥ is thetransverse gradient operator and λ the central wavelength of the (partially coherent) illumina-tion. The TIE provides a relation between intensity and phase via the change of intensity that isinduced 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 defo-cused images divided by their axial separation, ∂ I(x,y)/∂ z ≈ [I(x,y,z)− I(x,y,z+ δ z)]/δ z.The amount of defocus, δ z, plays a crucial role for the performance of TIE-microscopy andpresents a trade-off between good signal to noise ratio and significant deviation from the trueintensity 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 thanobjects exhibiting large phase gradients. Recently two-shot TIE-microscopy has been appliedsuccessfully 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 forthese weak phase objects to range between 2 µm and 6 µm. In [6] TIE-microscopy has beensimplified to a single-shot technique exploiting system chromatic aberrations in combinationwith a color camera to simultaneously capture in-focus and out-of-focus images on differentcolor channels.

White lightsource

Waveguides infused silica

Objective

Color cameraL3L2L1

DRD

Fig. 4. Experimental set-up of an inverted microscope for the determination of refractiveindex profiles of direct laser written waveguides in fused silica. The DRD to adjust systemchromatic aberrations to an optimal amount of defocus between the green and the red colorchannel is placed in a plane conjugated to the back focal plane of the objective lens, whichis relayed by lenses L1 and L2. The intermediate image is imaged onto a color camera bylens L3.

As an application of the DRD, we have tuned the system chromatic dispersion of a micro-scope to the optimum focal separation of green and red light for single-shot TIE-microscopy

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5267

from a color image of a sample containing DLW-waveguides located 170 µm below the surfaceof fused silica. Such waveguides are fabricated by translating the focus of a pulsed laser in thebulk of fused silica, resulting in channels of a gaussian shaped refractive index variation. Weinvestigated the performance for different amounts of defocus and compared our results withthe conventional method of taking two intensity images at a single wavelength (green) to re-construct the phase information. The simple geometry of the waveguides allowed us to processthe data with the TIE in one dimension,

dI(x)dz

=− λ2π

ddx

[I(x)

ddx

φ(x)], (12)

where the phase distribution may formally be written as

φ(x) =−2πλ

∫ (1

I(x)

∫dI(x′)

dzdx′

)dx. (13)

This implies that the phase can be retrieved by numerical integration of the difference of theintensities of the green and the red channels in the color image divided by the amount of defocusδ z, which approximates the gradient dI(x)/dz.

(a) (b)

-3 -2 -1 0 1 2 30

1

2

3

4

5

6

7

[µm]

OP

L[n

m]

�z=0.9 �m

�z=2.3 �m

1 1.5 2 2.5 3 3.5 4

4.5

5

5.5

6

6.5

7

7.5

�z [µm]

OP

L[n

m]

System chromatic aberration

Tuned chromatic aberration

Two-shot data acquisition

Fig. 5. Phase microscopy of DLW-waveguides in fused silica. (a) Comparison of peak op-tical path length (OPL) acquired for different amounts of defocus with single-shot TIEfrom system chromatic aberrations (square), single-shot TIE with tuned chromatic aberra-tions (circles) and two-shot TIE (asterisks). Error bars represent standard deviation for 10measurements. (b) Transverse OPL profile with reduced standard deviation at the optimalamount of defocus δ z = 2.3 µm.

The measurements were carried out on an inverted transmission bright-field microscope (seeFig. 4) with a white LED light source and a color camera (Canon EOS 1000D). The DRD totune the dispersive characteristics was placed in the 4f-imaged back-focal plane of the micro-scope objective (Olympus UPlanFL N 60x 1.25 NA). The amount of defocus δ z between thegreen and the red focal planes with and without the DRD was determined by comparing theobjective working distances for a green and a red wavelength. To this end, for each of the twocolors the respective spectral filter was inserted into the optical path and the microscope objec-tive was axially scanned with a piezo stage to bring the waveguide into focus. The differencebetween the working distances represents δ z. Without modification of the dispersive character-istics of the micrcoscope we found δ z = 0.9 µm ±0.05 µm. In Fig. 5(a) we compare the recon-structed peak gaussian phase variations in terms of the optical path length, OPL = φ(x) ·λ/2π ,

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5268

for different amounts of defocus. The square represents the OPL reconstructed with δ z causedby the system chromatic dispersion, while the circles represent the OPLs at larger amounts ofdefocus attainable with the DRD. For comparison, the crosses provide the peak OPLs recon-structed from the green channels of two color images taken at axial positions of the microscopeobjective separated by δ z. 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 re-duced the standard deviation from 8.4% to 2.0% of the mean peak OPL. Figure 5(b) shows theaveraged gaussian phase profiles and transverse standard deviations of a waveguide at a defo-cus of 2.3 µm (green) and 0.9 µm (black), respecively, which again demonstrates that the erroris considerably reduced by tuning the amount of defocus via dispersion control to its optimalvalue of about δ z = 2.3 µm.

5. Summary

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

Conversely, longitudinal chromatic aberrations in a microscope can be exploited to capturefocused and defocused images in the RGB channels of a single color image, which lends itself toapplications in the spirit of two-color TIE imaging. Our approach allows one to always operatein the sample-dependent optimal defocus range, which is dictated by the opposing requirementsof sufficiently large difference in intensities on one hand and sufficiently small deviations fromthe approximated “transport of intensity” condition on the other hand. We explicitly show thisfor a phase sample with laser-written waveguides in fused silica, for which it was possible todeduce 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 controlof broadband laser pulses, or for spectral filtering of polychromatic light. For example it shouldbe 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 beadjusted.

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, andby the Austrian Science Fund (FWF): P19582-N20.

#203450 - $15.00 USD Received 23 Dec 2013; revised 18 Feb 2014; accepted 19 Feb 2014; published 27 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005260 | OPTICS EXPRESS 5269