The Luneburg lens is an aberration-free lens that focuses light from all directions equally well. We fabricated and tested a Luneburg lens in silicon photonics. Such fully-integrated lenses may become the building blocks of compact Fourier optics on chips. Furthermore, our fabrication technique is sufficiently versatile for making perfect imaging devices on silicon platforms. (C) 2011 Optical Society of AmericaPublisher PDFPeer reviewe

Andrea Di Falco

Colombini

Ergin

Gabrielli

Huidobro

Kundtz

Leonhardt

Lipson

Minano

Pendry

Reardon

Smolyaninova

Susanne C. Kehr

Ulf Leonhardt

Valentine

Zernike

Optics Express

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Luneburg lens in silicon photonics

The Luneburg lens is an aberration-free lens that focuses light from all directions equally well. We fabricated and tested a Luneburg lens in silicon photonics. Such fully-integrated lenses may become the building blocks of compact Fourier optics on chips. Furthermore, our fabrication technique is sufficiently versatile for making perfect imaging devices on silicon platforms. (C) 2011 Optical Society of America</p

Di Falco, Andrea

Kehr, Susanne C.

Leonhardt, Ulf

University of St Andrews Research Portal

University of St. Andrews - Pure

The Luneburg lens is an aberration-free lens that focuses light from all directions equally well. We fabricated and tested a Luneburg lens in silicon photonics. Our technique is sufficiently versatile for making perfect imaging devices on silicon platforms. Lenses are indispensible optical instruments, but most conventional lenses suffer from aberrations [1]  — the focus depends on the direction of incidence and deteri-orates off axis. The Luneburg lens [2, 3] can focus light from all directions equally well. This lens establishes a classic example of non-Euclidean transformation optics [4, 5, 6, 7] where light in a medium is experiencing a curved spatial geometry [3]. Luneburg lenses are applied in microwave technology [8, 9, 10] but have remained impossible to implement in optics. Here we demonstrate an integrated Luneburg lens on a silicon chip that works for near-infrared light. Such lenses may become the building blocks of compact Fourier optics in silicon photonics [11]. The resolution of the Luneburg lens is limited by the wavelength of light, but our manufactur-ing method is also sufficiently versatile for making future perfect imaging devices [7, 12, 13] on silicon platforms. The Luneburg lens [2, 3] is a rotationally symmetric thick lens with a spatially varying refractive-index profile that focuses light on the rim of the lens (Fig. 1). The focal point lies in the direction of the incident light; the lens thus turns the direction of a light ray into the position of the focus. In terms of light waves [1], it performs a Fourier transformation. Because of its rotational symmetry, the Luneburg lens is an ideal lens free from optical aberrations [1], but it is not a perfect lens [13, 14]: its focus size is limited by the wavelength of light [15] (Fig. 1). Nevertheless, it could become an important optical instrument, in particular in integrated photonics, if it were possible to make Luneburg lenses. The required index profile n of the lens is currently impossible to create by doping optical materials, as the profile is given by the formula [3] n = 2 − (r/a)2 for r ≤ a (1

Susanne C Kehr

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The Luneburg lens is an aberration-free lens that focuses light from all directions equally well. We fabricated and tested a Luneburg lens in silicon photonics. Such fully-integrated lenses may become the building blocks of compact Fourier optics on chips. Furthermore, our fabrication technique is sufficiently versatile for making perfect imaging devices on silicon platforms. (C) 2011 Optical Society of AmericaPeer reviewe

St Andrews Research Repository

Luneburg lens in silicon photonicsAndrea Di Falco, Susanne C. Kehr, and Ulf Leonhardt∗School of Physics and Astronomy, University of St. Andrews,North Haugh, St. Andrews KY16 9SS, UK∗ulf@st-andrews.ac.ukAbstract: The Luneburg lens is an aberration-free lens that focuses lightfrom all directions equally well. We fabricated and tested a Luneburg lensin silicon photonics. Such fully-integrated lenses may become the buildingblocks of compact Fourier optics on chips. Furthermore, our fabricationtechnique is sufficiently versatile for making perfect imaging devices onsilicon platforms.© 2011 Optical Society of AmericaOCIS codes: (220.3630) Lenses; (230.7390) Waveguide Planar; (350.6980) Transforms.References and links1. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).3. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).4. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).5. P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmon-ics,” Nano Lett. 10, 1985–1990 (2010).6. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, ”Transformational plasmon optics,” Nano Lett. 10, 1991–1997(2010).7. V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Maxwell fish-eye and Eaton lensesemulated by microdroplets,” Opt. Lett. 35, 3396–3398 (2010).8. S. Combleet, Microwave Optics: The Optics of Microwave Antenna Design (Academic Press, 1976).9. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, 1981).10. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).11. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Com-mun. 1, 124 (2010).12. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat.Nanotechnol., DOI:10.1038/nnano.2010.282 (2011).13. M. Lipson, “Guiding, modulating and emitting light on silicon—challenges and opportunities,” J. LightwaveTechnol. 23, 4222–4238 (2005).14. J. C. Minano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express 14, 9627–9635 (2006).15. U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).16. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).17. E. Colombini, “Design of thin-film Luneburg lenses for maximum focal length control,” Appl. Opt. 20, 3589–3593 (1981).18. E. Colombini, “Index-profile computation for the generalized Luneburg lens,” J. Opt. Soc. Am. 71, 1403–1405(1981).19. S. K. Yao and D. B. Anderson, “Shadow sputtered diffraction-limited waveguide Luneburg lenses,” Appl. Phys.Lett. 33, 307–309 (1978).20. S. K. Yao, D. B. Anderson, R. R. August, B. R. Youmans, and C. M. Oania, “Guided-wave optical thin-filmLuneburg lenses: fabrication technique and properties,” Appl. Opt. 18, 4067–4079 (1979).21. F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).22. S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructuredLuneburg lens in optical frequencies” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010),Paper Th1-1, pp. 177–178.23. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8,568–571 (2009).#141415 - $15.00 USD Received 19 Jan 2011; revised 21 Feb 2011; accepted 21 Feb 2011; published 3 Mar 2011(C) 2011 OSA 14 March 2011 / Vol. 19,  No. 6 / OPTICS EXPRESS  515624. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at opticalfrequencies,” Nat. Photon. 3, 461–463 (2009).25. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at opticalwavelengths,” Science 328, 337–339 (2010).26. L. H. Gabrielli, U. Leonhardt, and M. Lipson, “Perfect imaging in the optical domain using dielectric materials,”arXiv:1007.2564.27. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test,(SPIE tutorial text, Washington, 2004).28. C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space opticalbeam deflecting,” Opt. Express 17, 3423–3428 (2009).1. IntroductionLenses are indispensible optical instruments, but most conventional lenses suffer from aber-rations [1] — the focus depends on the direction of incidence and deteriorates off axis. TheLuneburg lens [2, 3] can focus light from all directions equally well. This lens establishes aclassic example of non-Euclidean transformation optics [4–7] where light in a medium is ex-periencing a curved spatial geometry [3]. Luneburg lenses are applied in microwave technol-ogy [8–11] and have recently been made for surface plasmons [12] but have remained impossi-ble to implement in integrated photonics. Here we demonstrate an integrated Luneburg lens ona silicon chip that works for near-infrared light. Such lenses may become the building blocksof compact Fourier optics in silicon photonics [13]. The resolution of the Luneburg lens is lim-ited by the wavelength of light, but our manufacturing method is also sufficiently versatile formaking future perfect imaging devices [7, 14, 15] on silicon platforms.Fig. 1. Luneburg lens. The lens (blue disk) focuses all light rays (red) propagating in onedirection at the point on its rim that lies in that direction. The underlying wave pattern showsthe real part of the Fourier component of a plane wave with wavelength 0.5a incident fromthe right (calculated by partial-wave expansion). One sees that the focal spot is about half awavelength wide. As the lens is rotationally symmetric it focuses light from all directionsequally well.The Luneburg lens [2, 3] is a rotationally symmetric thick lens with a spatially varyingrefractive-index profile that focuses light on the rim of the lens (Fig. 1). The focal point liesin the direction of the incident light; the lens thus turns the direction of a light ray into theposition of the focus. In terms of light waves [1], it performs a Fourier transformation. Becauseof its rotational symmetry, the Luneburg lens is an ideal lens free from optical aberrations [1],but it is not a perfect lens [15, 16]: its focus size is limited by the wavelength of light [17](Fig. 1). Nevertheless, it could become an important optical instrument, in particular in inte-#141415 - $15.00 USD Received 19 Jan 2011; revised 21 Feb 2011; accepted 21 Feb 2011; published 3 Mar 2011(C) 2011 OSA 14 March 2011 / Vol. 19,  No. 6 / OPTICS EXPRESS  5157grated photonics, if it were possible to make Luneburg lenses. The required index profile n ofthe lens is currently impossible to create by doping optical materials, as the profile is given bythe formula [3]n =√2− (r/a)2 for r ≤ a, (1)where r denotes the distance from the centre of the lens and a its outer radius. Luneburg’sformula (1) shows that one needs to create an index profile with a contrast of √2≈ 1.4, whichis far from the reach of current doping techniques.Fig. 2. The device. a: the waveguide confining the light also creates the refractive indexprofile (1) of the Luneburg lens for horizontal light propagation (Fig. 1) on a chip. The redcurves show the vertical intensity distributions calculated with a modal solver. b: effectiverefractive index depending on the thickness of the silicon below the polymer in (a). c: meas-ured silicon profile (green) versus the theoretical curve (red) required for implementing theLuneburg lens.2. Lens fabricationWe have made a Luneburg lens in silicon photonics [13]. There light is typically confined inwaveguiding structures on planar silicon chips. Instead of doping, one can exploit the three-dimensional nature of the waveguide to create an effective index profile for two-dimensionalwave propagation on the chip. In our case, the Luneburg lens is a thin graded silicon disk (lessthan 70nm thick) of 98µm radius put between a 2µm silica layer on a silicon substrate andan SU8 polymer layer on top (Fig. 2a). The total thickness of the silicon disk and the polymer#141415 - $15.00 USD Received 19 Jan 2011; revised 21 Feb 2011; accepted 21 Feb 2011; published 3 Mar 2011(C) 2011 OSA 14 March 2011 / Vol. 19,  No. 6 / OPTICS EXPRESS  5158is fixed to be 500nm. We exploit the fact that the effective refractive index in a waveguidedepends on its geometric dimensions. As the thickness of the silicon disk is only graduallychanging compared with the wavelength, we can find the local refractive index n assuming asimple model: a planar sandwich of silicon substrate, silica, silicon layer, polymer and air, witheach layer in the model having the local thickness of the layer in the actual device. Figure 2bshows the effective index n depending on the silicon thickness for light of 1550nm wavelengthand polarized such that the magnetic field points orthogonal to the layer structure (while theelectric field is parallel to the structure). One sees that the index range is sufficient to implementthe Luneburg lens and other, perfectly imaging lenses such as Maxwell’s fish eye [15]. Figure2c shows the silicon profile of our Luneburg lens (measured with a Bruker Dektak 150 stylusprofiler) against the corresponding theoretical curve of formula (1) translated to a thicknessprofile by the relationship illustrated in Fig. 2b. The bandwidth of the lens is given by thefrequency range where formula (1) holds with good accuracy. In our case, we tested the lenswith a source of 110nm bandwidth and found good performance.Tapered waveguides like ours have been applied before for creating effective index profiles.Early Luneburg-type lenses [18] were fabricated by sputtering [19,20] but there the index con-trast was insufficient for creating the proper Luneburg lens with profile (1). A crude Luneburglens was made with a macroscopic, multimode waveguide [21], but such a device is far toolarge to be integrated. Recently, approximate Eaton/Minano [14] and Maxwell fish eye [15]lenses were created by putting liquid droplets on a substrate [7], but they cannot be integratedin a solid-state device and are not fully controllable. Luneburg lenses for surface plasmons [5]were proposed, numerically studied [6] and recently made by grey-scale lithography [12]. Inour case, we use dielectrics that are significantly less absorptive than the metals needed forplasmons and our index range is much larger. One can make Luneburg lenses [22] and similarindex profiles [23–25] with silicon metamaterials, but here scattering losses at the structures ofmetamaterials become a critical limitation [26], whereas the index profiles made by our methodare continuos and sufficiently smooth. With our technique one can controllably fabricate nearlyarbitrary index profiles within the range specified in Fig. 2b on silicon photonics platforms thatcan combine electronics with photonics [13]. Such devices will work for infrared light light inthe telecom band and can form the building blocks of on-chip Fourier optics.For fabricating our Luneburg lens we used grey-scale electron-beam lithography [27, 28].First, the prefabricated silicon on insulator chip (2µm silica, 220nm silicon capping layer)was cleaned in acetone and IPA in an ultrasonic bath and ashed with oxygen plasma. SU8resist of 350nm thickness was then spun and soft-baked for 1min at 60oC followed by 4minat 100oC. For the grey-scale lithography we used a LEO-RAITH modified electron beam witha positional accuracy of 2nm. The resist was exposed at 30kV with a dose ranging from 0to 10µAs/cm2. Particular care had to be exercised in choosing the dose profile that producesthe required silicon profile. We found empirically that a dose D with profile Dmax(1− r/a)1/2results in the Luneburg profile shown in Fig. 2c. After exposure the sample was developed withEC for 45s and hard-baked for 10min at 180oC. We transferred the gray-scale profile to thesilicon via Reactive Ion Etching, using a blend of O2:SF6:CHF3=75:100:100 at 20W for 90s.Our profile follows well the theoretically required curve (Fig. 2c), but has some roughness withRMS 1nm and peak-to-peak value 4nm. Finally the chip was spin coated with a layer of SU8polymer with a maximum thickness of 500nm, soft-baked, exposed for 1min to UV light andhard-baked to cross-link and stabilize the polymer. The sample was then cleaved for end-firecoupling (where we couple light in from the side).#141415 - $15.00 USD Received 19 Jan 2011; revised 21 Feb 2011; accepted 21 Feb 2011; published 3 Mar 2011(C) 2011 OSA 14 March 2011 / Vol. 19,  No. 6 / OPTICS EXPRESS  51593. Experimental resultsAfter fabrication, we tested the optical performance of our Luneburg lens. The experimentalmeasurements were performed via end-fire coupling. The beam from a C+L band amplifiedspontaneous emission source (centre wavelength 1575nm, bandwidth 110nm) was polarized(with the electric field parallel to the top surface of the device) and focused with a 10X objectiveon the facet of the sample. The beam was free to diffract in the planar waveguide until it entersthe Luneburg lens. We observed the Luneburg lens from the top with a 50X Mitutoyo objectivelens, with a numerical aperture of 0.55, and the image was taken by a Vidicon ElectrophysicsMicronViewer Camera (Spectral Response: 0.4µm to 1.9µm). We saw a strong intensity peak atthe focal spot (that originates from the focused infrared light scattered at small imperfections).To make the lens visible in our image we illuminated the device with an independent whiteLED. All the measurements taken with the camera were calibrated using the known radius a ofthe lens, as measured independently with the profiler (Fig. 1c).Fig. 3. Light focusing. a: False-colour image of the observed intensity profile in the Lune-burg lens (Fig. 2). Infrared light incident from the right is guided on the chip and focusedin the lens (red spot). The lens itself is made visible by illuminating it with white lightfrom the top. b: measured intensity profiles of the focused infrared light along the two linesindicated in (a). In our experiment the full width at half maximum (FWHM) of the focus isdominated by the width of the incident Gaussian beam.Figure 3a shows the image acquired with the camera in false color. The light comes fromthe right hand side. The red spot is the focused infrared light, while the outline of the lensappears solely due the independent white LED illumination. From this figure, it is possible tomeasure the spot size in the two marked directions (Fig. 3b). The measured minimum spotsize is 3.77µm. We expect from theory (Fig. 1) that the focal spot is about half a wavelengthin size, but in our experiment it was wider. Several factors could contribute to the spot size— the optical resolution of the microscope used in combination with the propagation in thepolymer layer, the geometrical dispersion of the lens (as we used a broad band of wavelengths),scattering due to roughness and the finite width of the incident beam. The latter aspect turns outto account for the observed broadening. The input beam was not a plane wave but a Gaussianbeam. The input waist was measured to be approximately 12µm. The Luneburg lens was adistance of 1.5mm away from the chip facet, which means that (assuming a Gaussian beamprofile) the beam reached the lens with angles between -1.1 and +1.1 degrees, with a resultingarclenght of 3.8µm, which agrees with the measured focus size in our Luneburg lens.#141415 - $15.00 USD Received 19 Jan 2011; revised 21 Feb 2011; accepted 21 Feb 2011; published 3 Mar 2011(C) 2011 OSA 14 March 2011 / Vol. 19,  No. 6 / OPTICS EXPRESS  5160Fig. 4. Performance tests. The figure shows the observed infrared light focused in theLuneburg lens (yellow-red spots similar to Fig. 3, except that the lens is not made visible)depending on the propagation direction (red arrow with angle θ ) and the offset ∆x of theincident Gaussian beam (profile shown). a-c: the focal spot lies in the propagation direction.d,e: the focal point is independent of the offset. Our device thus behaves as expected froma Luneburg lens.We have performed further experiments to characterize the behavior of our Luneburg lens.In particular we launched light on the lens at different angles, to show that the lens effectivelyimplements a spatial Fourier transform. Figure 4 shows three cases with tilting angle +4, 0 and-4 degrees in panel a), b) and c), respectively. We see that the lens focuses equally well fromall directions. We also displaced the input beam with respect to the central axis of the lens, seepanels d) and e) of Fig. 4. Recall that the beam size was smaller than the lens itself. Remarkably,the focused spot still appeared in the middle of the lens, confirming that the lens implementsa Fourier transform, as it is mostly sensitive to the direction of propagation rather than to thespatial distribution of the beam itself.4. ConclusionWe have demonstrated a fully functional Luneburg lens in silicon photonics that can be used inintegrated imaging devices. For this we exploited the three-dimensional geometric aspects ofwaveguides to implement refractive-index profiles for two-dimensional wave propagation thatare normally impossible to make in optical materials.#141415 - $15.00 USD Received 19 Jan 2011; revised 21 Feb 2011; accepted 21 Feb 2011; published 3 Mar 2011(C) 2011 OSA 14 March 2011 / Vol. 19,  No. 6 / OPTICS EXPRESS  5161AcknowledgmentsADF is supported by an EPSRC Career Acceleration Fellowship (EP/I004602/1), SCK by theUniversity of St Andrews and UL by a Royal Society Wolfson Research Merit Award and aBlue Skies Theo Murphy Award of the Royal Society.#141415 - $15.00 USD Received 19 Jan 2011; revised 21 Feb 2011; accepted 21 Feb 2011; published 3 Mar 2011(C) 2011 OSA 14 March 2011 / Vol. 19,  No. 6 / OPTICS EXPRESS  5162

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