Lithographically Fabricated Magnifying Maxwell Fisheye Lenses

hv photonics

Article Lithographically Fabricated Magnifying Maxwell Fisheye Lenses

Vera Smolyaninova 1,*, Christopher Jensen 1, William Zimmerman 1, Anthony Johnson 1, David Shaefer 1 and Igor Smolyaninov 2

1 Department of Physics, Astronomy and Geosciences, Towson University, 8000 York Rd., Towson, MD 21252, USA; [email protected] (C.J.); [email protected] (W.Z.); [email protected] (A.J.); [email protected] (D.S.) 2 Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-410-704-2608

Academic Editors: Nelson Tansu Received: 11 February 2016; Accepted: 3 March 2016; Published: 8 March 2016

Abstract: Recently suggested magnifying Maxwell ﬁsheye lenses, which are made of two half-lenses of different radii, have been fabricated and characterized. The lens action is based on control of polarization-dependent effective refractive index in a lithographically formed tapered waveguide. We have studied wavelength and polarization dependent performance of the lenses, and their potential applications in waveguide mode sorting.

Keywords: metamaterials; transformation optics; microscopy; waveguide; mode sorting

1. Introduction Transformation optics (TO) has recently become a useful methodology for the design of unusual optical devices, such as novel metamaterial lenses and invisibility cloaks. Unfortunately, typical TO designs require metamaterials with low-loss, broadband performance, which appear difficult to develop. These difficulties are especially severe in the visible frequency range where good magnetic performance is limited. On the other hand, very recently we have demonstrated that many transformation optics and metamaterial-based devices requiring anisotropic dielectric permittivity and magnetic permeability may be emulated by specially designed tapered waveguides [1]. This approach leads to low-loss broadband performance in the visible frequency range, which is difficult to achieve by other means. We have applied this technique to broadband electromagnetic cloaking in the visible range [1] and successfully extended it to birefrigent TO devices, which perform useful and different functions for mutually orthogonal polarization states of light [2]. Therefore, such use of tapered waveguides belongs to the broadly-defined transformation optics approach. In this paper we have applied this approach to lithographically fabricated magnifying Maxwell fisheye lenses, which were originally introduced in a microdroplet form [3].

2. Materials and Methods Unlike the earlier microdroplet design, which is difﬁcult to fabricate and control, our current design is based on lithographically deﬁned metal/dielectric waveguides. Adiabatic variations of the waveguide shape enable control of the effective refractive indices experienced by the transverse electric (TE) and transverse magnetic (TM) modes propagating inside the waveguides, as shown in Figure1a.

Photonics 2016, 3, 8; doi:10.3390/photonics3010008 www.mdpi.com/journal/photonics Photonics 2016, 3, 8 2 of 7

They are defined as neff = kω/c for each polarization, where the k vector at a given frequency ω is calculated via the boundary conditions at two interfaces as follows: Photonics 2016, 3, 8 2 of 7 k ik ik k ik ik 1 ´ 2 k ´ 2 eík2d “ 1 ` 2 k ` 2 eik2d (1) for the TM, and ε ε 3 ε ε ε 3 ε ˆ m ˙ ˆ ˙ ˆ m ˙ ˆ ˙ ik2d ik2d k1  ik 2 k3  ik 2 e  k1  ik 2 k3  ik 2 e (2) for the TM, and ík2d ik2d for the TE polarized guidedpk1 ´ modes,ik2q pk3 where´ ik2q thee vertical“ pk1 components` ik2q pk3 ` ik of2q thee wavevector ki are defined(2) as: for the TE polarized guided modes, where the vertical components of the wavevector ki are defined as: 2 1/ 2  2   k  k  2  1{2 1  m ω2  (3) k “ k2 ´ ε c  (3) 1 m c2 ˆ ˙1/ 2   2  k     k 2 1{2 2 ω22  (4) k “  c ε ´ k2  (4) 2 c2 ˆ ˙ 2 1/ 2   1{2 k  k 2 ω2  (5) k 3“ k2 ´ 2  (5) 3  cc2  ˆ ˙ inin metal,metal, dielectric,dielectric, andand air,air, respectively.respectively. AsAs illustratedillustrated inin FigureFigure1 a,1a, effective effective birefringence birefringence for for the the lowestlowest guidedguided TMTM andand TETE modesmodes appearsappears toto bebe veryvery strongstrong atat waveguidewaveguide thicknessthickness dd< < 0.40.4 µμm,m, andand bothboth polarizationspolarizations demonstratedemonstrate strongstrong indexindex dependencedependence onon thethe waveguidewaveguide thickness.thickness. ThisThis behaviorbehavior maymay bebe usedused inin buildingbuilding non-trivialnon-trivial birefringentbirefringent TOTO devicesdevices ifif aa waveguidewaveguide thicknessthickness asas aa functionfunction ofof spatialspatial coordinatescoordinatesd d((rr)) maymay bebe controlledcontrolled lithographically lithographically with with enough enough precision. precision.

FigureFigure 1.1.( a(a)) Effective Effective refractive refractive index index plotted plotted as as a functiona function of thicknessof thickness of a of tapered a tapered waveguide waveguide for TM for andTM TE and polarizations. TE polarizations. The waveguide The waveguide geometry geometry is shown is in shown the inset. in ( theb) Atomic inset. Force(b) Atomic Microscope Force (AFM)Microscope image (AFM of a lithographically) image of a lithographically defined individual defined magnifying individual fisheye magnifying lens made fisheye of two lens half-lenses made of oftwo different half-lenses radii. of (c ) different Corresponding radii. ( spatialc) Corresponding distribution spatial of the effective distribution refractive of the index. effective (d) COMSOL refractive Multiphysicsindex. (d) COMSOL simulation Multiphysics of the fisheye simulation lens image of magnification. the fisheye lens The image insets illustratemagnification. ray propagation The insets inillustrate the original ray propagation and the magnifying in the origin fisheyeal and lenses. the magnifying fisheye lenses.

We have developed a lithography technique which enables such d(r) shape control of the dielectric photoresist on gold film substrate, as illustrated in Figure 1b. Shieply S1811 photoresist having refractive index n ~ 1.5 was used for device fabrication. In traditional lithographic applications for the best results of pattern transfer the profile of resist should be rectangular or even with overhang. Our purpose is different. We want to create a more gradual edge profile. This can be done by disregarding typical precautions employed to make the edges sharp. Instead of contact

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We have developed a lithography technique which enables such d(r) shape control of the dielectric photoresist on gold film substrate, as illustrated in Figure1b. Shieply S1811 photoresist having refractive index n ~ 1.5 was used for device fabrication. In traditional lithographic applications for the best results of pattern transfer the profile of resist should be rectangular or even with overhang. Our purpose is different. We want to create a more gradual edge profile. This can be done by disregarding typical precautions employed to make the edges sharp. Instead of contact printing (when mask is touching the substrate), we used soft contact mode (with the gap between the mask and the substrate). This allows for the gradient of exposure due to the diffraction at the edges, which leads to a gradual change of thickness of the developed photoresist. Different degrees of separation between the mask and the substrate produced progressively softer photoresist profile. Further variations of the profile were achieved by use of underexposure and underdevelopment, which produce much softer photoresist edges. This technique has been used previously to fabricate such TO-based devices as a modified Luneburg lens [2]. However, no image magnification has been demonstrated in these experiments. On the other hand, as was noted in [3], it is relatively straightforward to incorporate image magnification into such TO lens designs as Eaton and Maxwell fisheye lenses. The refractive index distribution in a Maxwell fisheye lens is defined as

´1 r2 n “ 2n 1 ` (6) 1 R2 ˆ ˙ at r < R, where 2n1 is the refractive index at the center of the lens, and R is the scale factor. A reflective surface is assumed to be placed at r = R, so that n < 1 values of the refractive index do not need to be used. In our experiments the role of such reflective surface is played by the lens edge. On the other hand, an inverted Eaton lens [4] is defined as n = n1 for r < R, and

2R 1{2 n “ n ´ 1 (7) 1 r ˆ ˙ for r > R. Since the refractive index distribution in the fisheye lens is obtained via the stereographic projection of a sphere onto a plane [5,6], points near the lens edge correspond to points located near the equator of the sphere. Therefore, these points are imaged into points located near the opposite lens edge, as shown in the inset in Figure1d. The inverted Eaton lens has similar imaging properties. As demonstrated in [3], both refractive index distributions may be emulated with a suitable d(r) profile of a thin dielectric placed on top of a metal film, which is also evident from Figure1a. As shown in Figure1c,d, two halves of either Maxwell fisheye or inverted Eaton lens having different values of parameter R may be brought together to achieve image magnification. The image magnification in this case is M = R1/R2. Our numerical simulations in the case of M = 2 are presented. Since the sides of the lens play no role in imaging, the overall shape of the imaging device can be altered to smooth the sharp corners, resulting in the magnifying fisheye lens shape shown in Figure1b, which was fabricated using the lithographic technique described above.

3. Results Experimental images in Figure2 demonstrate measured performance of the designed magnifying fisheye lenses. In these experiments a near-field scanning optical microscope (NSOM) fiber tip was brought in close proximity to the arrays of lithographically formed TO devices and used as an illumination source. As expected from the numerical simulations, an image of the NSOM tip was easy to observe at the opposite edge of the lens. Angular and polarization testing of individual lenses in the array agrees well with theoretical modelling presented in Figure1c,d. As illustrated by Figure1a, the same d(r) profile produces a different refractive index distribution for TM and TE polarized light. Photonics 2016, 3, 8 4 of 7 Photonics 2016, 3, 8 4 of 7

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Figure 2. Experimental testing of angular ( a,b) and polarization (c) performanceperformance of the magnifyingmagnifying ﬁsheyefisheye lenses at λ = 488 nm. The The scale scale bar length is 5 μµm in all images.

DueImage to resolution near zero effectiveand magnification refractive indexof thenear fabricated the device magnifying edge, a fisheyeMaxwell lens fisheye for TM lenses light may will operatebe evaluated as a spatial based (directional) on images of filter lens for testing TE light presented [2]. We should in Figure note 3. that Experimental while the NSOM testing fiber of two tip emitsfisheye unpolarized lenses having light, different polarization M = responseR1/R2 ratio of theis shown image inproduced these ima byges. a TO The device image can resolution be clearly separatedappears toFigure intobe close the2. Experimental TM to diffraction and TE testing contributions-limited of angular (~ 0.6 ( witha,bλ) atand respect 488 polarization nm), to thewhile (c plane) per imageformance of incidencemagnification of the magnifying of the is sourceclose to light. the Polarizationdesign valuesfisheye behavior M lenses = 2 and at of λ = lensesM 488 = 3,nm. inrespectively. The Figure scale2 bar demonstrates length We should is 5 μm in excellentalso all images.note agreementa very broad with (almost theory. 180°) angular rangeImage of the resolution magnifying and fisheye magnification lens operation. of the fabricated magnifying Maxwell fisheye lenses may be Image resolution and magnification of the fabricated magnifying Maxwell fisheye lenses may evaluated based on images of lens testing presented in Figure3. Experimental testing of two fisheye be evaluated based on images of lens testing presented in Figure 3. Experimental testing of two lenses having different M = R /R ratio is shown in these images. The image resolution appears to be fisheye lenses having different1 2 M = R1/R2 ratio is shown in these images. The image resolution closeappears to diffraction-limited to be close to diffraction (~ 0.6λ at-limited 488 nm), (~ 0.6 whileλ at 488 image nm), magnificationwhile image magnification is close to theis close design to the values ˝ M = 2design and M values= 3, respectively.M = 2 and M = We3, respectively. should also We note should a very also broadnote a (almostvery broad 180 (almost) angular 180°) rangeangular of the magnifyingrange of fisheye the magnifying lens operation. fisheye lens operation.

Figure 3. Experimental testing of image magnification at λ = 488 nm of two fisheye lenses with

different M = R1/R2 ratio: (a,b) Original magnified images obtained at different source positions. The location of image and source are indicated by the arrows. (c) Digital overlap of the images in (a) and (b) indicates that image magnification is close to the design value M = 2. (d,e) Similar original images FigureFigure 3. Experimental 3. Experimental testing testing of image of image magnification magnification at λ at= 488λ = nm 488 of nm two of fisheye two fisheye lenses lenses with with different and (f) the digital overlap image obtained with a different magnifying lens designed for M = 3. M = Rdifferent1/R2 ratio: M = (Ra1,b/R)2 Originalratio: (a,b magnified) Original magnified images obtained images obtained at different at different source positions.source positions. The location The of location of image and source are indicated by the arrows. (c) Digital overlap of the images in (a) and imageThe projected and source broadband are indicated performance by the arrows. of (c) theDigital magnifying overlap of the Maxwell images in fisheye (a) and lenses (b) indicates has been that image(b) indicates magnification that image is magnification close to the is design close to value the designM = 2.value (d,e M) Similar = 2. (d,e) original Similar original images images and (f ) the verified inand the (f )λ the = digital488–633 overlap nm imagerange. obtained Examples with aof different such testing magnifying at 515 lens nmdesig andned for633 M nm = 3. are presented digital overlap image obtained with a different magnifying lens designed for M = 3. in Figure 4 (compare these images with Figure 2a,b obtained at 488 nm). This experimental result may be understoodThe projected similar broadband to the performancebroadband cloaking of the magnifying performance Maxwell demonstrated fisheye lenses in Ref haserence been [1]. In all verifiedcases the in effectivethe λ = 488 refractive–633 nm range. index Examples of the tapered of such waveguide testing at 515 scales nm and as d 633/λ atnm small are presented d. Thus, light in Figure 4 (compare these images with Figure 2a,b obtained at 488 nm). This experimental result

may be understood similar to the broadband cloaking performance demonstrated in Reference [1]. In all cases the effective refractive index of the tapered waveguide scales as d/λ at small d. Thus, light

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The projected broadband performance of the magnifying Maxwell fisheye lenses has been verified in the λ = 488–633 nm range. Examples of such testing at 515 nm and 633 nm are presented in Figure4 (compare these images with Figure2a,b obtained at 488 nm). This experimental result may bePhotonics understood 2016, 3, 8 similar to the broadband cloaking performance demonstrated in Reference [1]. In5 of all 7 casesPhotonics the 201 effective6, 3, 8 refractive index of the tapered waveguide scales as d/λ at small d. Thus, light5 of at 7 theseat these wavelengths wavelengths (488 (488 nm, nm, 515 515 nm nm and and 633 633 nm) nm) perceives perceives the the waveguide waveguide edgeedge asas havinghaving similar at these wavelengths (488 nm, 515 nm and 633 nm) perceives the waveguide edge as having similar distribution of effective refractive index at all wavelengths. We have also verifiedverified (see Figure5 5)) thatthat distribution of effective refractive index at all wavelengths. We have also verified (see Figure 5) that the same lens used in reversereverse directiondirection maymay bebe utilizedutilized toto achieveachieve imageimage reduction.reduction. This fact is not the same lens used in reverse direction may be utilized to achieve image reduction. This fact is not trivial since the lens geometrygeometry is obtainedobtained by “gluing together” two halves of the Maxwell fisheyefisheye trivial since the lens geometry is obtained by “gluing together” two halves of the Maxwell fisheye lenses having considerably different radii, which from from the ray optics point of view may lead to ray lenses having considerably different radii, which from the ray optics point of view may lead to ray scattering byby thethe lenslens edges. edges. Potentially, Potentially, such such an an arrangement arrangement of theof the magnifying magnifying fisheye fisheye lens lens may may find scattering by the lens edges. Potentially, such an arrangement of the magnifying fisheye lens may lithographicfind lithographic applications. applications. find lithographic applications.

Figure 4. Experimental verification of broadband performance of the magnifying Maxwell fisheye Figure 4.4. ExperimentalExperimental veriﬁcation verification of of broadband broadband performance performance of theof the magnifying magnifying Maxwell Maxwell ﬁsheye fisheye lens. lens. (a,b) Images taken at λ = 515 nm. (c,d) Images taken at λ = 633 nm. The scale bar length is 5 μm (lens.a,b) Images(a,b) Images taken taken at λ =at 515 λ = nm.515 nm. (c,d) (c Images,d) Images taken taken at λ at= λ 633 = 633 nm. nm. The The scale scale bar bar length length is 5isµ 5m μ inm in all images. allin all images. images.

Figure 5. Operation of the magnifying Maxwell fisheye lens in reverse direction, which leads to Figure 5. Operation of the magnifying Maxwell fisheye lens in reverse direction, which leads to Figureimage reduction. 5. Operation (a) COMSOL of the magnifying Multiphysics Maxwell simulation fisheye of the lens fisheye in reverse lens direction, image reduction which using leads image reduction. (a) COMSOL Multiphysics simulation of the fisheye lens image reduction using torefractive image index reduction. distribution (a) COMSOL corresponding Multiphysics to the experimental simulation ofvariation the fisheye of the lens waveguide image reductionthickness. refractive index distribution corresponding to the experimental variation of the waveguide thickness. using(b,c) Experimental refractive index testing distribution of angular corresponding performance of to the fisheye experimental lens used variation in reverse of the direction. waveguide λ = (b,c) Experimental testing of angular performance of the fisheye lens used in reverse direction. λ = thickness.488 nm. Image(b,c reduction) Experimental factortesting M = 1/2 of is angularobserved. performance The scale bar of length the fisheye is 5 μm lens in all used images. in reverse direction.488 nm. Imageλ = 488 reduction nm. Image factor reduction M = 1/2 is factor observed.M = 1/2The isscale observed. bar length The is scale 5 μm bar in lengthall images. is 5 µm in 4. Discussionall images. 4. Discussion Since the developed lithographic Maxwell fisheye lenses are based on tapered waveguide 4. DiscussionSince the developed lithographic Maxwell fisheye lenses are based on tapered waveguide geometry, their high magnification and very compact design are highly suitable in waveguide mode geometry, their high magnification and very compact design are highly suitable in waveguide mode sortingSince applications. the developed Compact lithographic and efficient Maxwell mode fisheye sorters lenses are required are based in on on- taperedchip mode waveguide-division sorting applications. Compact and efficient mode sorters are required in on-chip mode-division geometry,multiplexing their [7] high and magnification sensing [8] applications. and very compact COMSOL design Multiphysics are highly suitable simulations in waveguide of a Maxwell mode multiplexing [7] and sensing [8] applications. COMSOL Multiphysics simulations of a Maxwell sortingfisheye-based applications. mode sorter Compact are presented and efficient in Figure mode 6, where sorters the are signa requiredl is sent in in on-chip through mode-division a multimode fisheye-based mode sorter are presented in Figure 6, where the signal is sent in through a multimode waveguide from the left and out-coupled through three single mode output waveguides to the right. waveguide from the left and out-coupled through three single mode output waveguides to the right. These simulations demonstrate excellent mode-sorting performance of a magnifying Maxwell These simulations demonstrate excellent mode-sorting performance of a magnifying Maxwell fisheye lens, having spatial dimensions of only a few micrometers (the spatial dimensions are chosen fisheye lens, having spatial dimensions of only a few micrometers (the spatial dimensions are chosen to match experimentally fabricated lens shown in Figure 1). As demonstrated by Figure 6a, to match experimentally fabricated lens shown in Figure 1). As demonstrated by Figure 6a, symmetric excitation of the multimode input waveguide leads to the output power being channeled symmetric excitation of the multimode input waveguide leads to the output power being channeled primarily into the central single mode output waveguide. On the other hand, asymmetric excitation primarily into the central single mode output waveguide. On the other hand, asymmetric excitation

Photonics 2016, 3, 8 6 of 7 multiplexing [7] and sensing [8] applications. COMSOL Multiphysics simulations of a Maxwell ﬁsheye-based mode sorter are presented in Figure6, where the signal is sent in through a multimode waveguide from the left and out-coupled through three single mode output waveguides to the right. These simulations demonstrate excellent mode-sorting performance of a magnifying Maxwell ﬁsheye lens, having spatial dimensions of only a few micrometers (the spatial dimensions are chosen to match experimentally fabricated lens shown in Figure1). As demonstrated by Figure6a, symmetric excitation of the multimode input waveguide leads to the output power being channeled primarily into the Photonics 2016, 3, 8 6 of 7 central single mode output waveguide. On the other hand, asymmetric excitation of the multimode waveguideof the multimode leads to propagation waveguide leads of higher to pr spatialopagation modes, of higher which spatial are channeled modes, which preferentially are channeled into the sidepreferentially single mode into waveguides, the side single as shown mode waveguides, in Figure6b. as shown in Figure 6b.

Figure 6. COMSOL Multiphysics simulations of a Maxwell fisheye-based mode sorter. (a) Symmetric Figure 6. COMSOL Multiphysics simulations of a Maxwell fisheye-based mode sorter. (a) Symmetric excitation of the multimode input waveguide leads to the output power being channeled into the excitation of the multimode input waveguide leads to the output power being channeled into the central single mode output waveguide. (b) Asymmetric excitation of the multimode waveguide central single mode output waveguide. (b) Asymmetric excitation of the multimode waveguide leads leads to propagation of higher spatial modes, which are channeled preferentially into the side single to propagation of higher spatial modes, which are channeled preferentially into the side single mode mode waveguides. The spatial dimensions of the magnifying Maxwell fisheye lens in these waveguides. The spatial dimensions of the magnifying Maxwell fisheye lens in these simulations are simulations are the same as in Figure 1. the same as in Figure1. Additional applications of the magnifying Maxwell fisheye lenses may include microscopy and spatiallyAdditional resolved applications fluorescen ofcethe spectroscopy, magnifying as Maxwell well as various fisheye nonlinear lenses may spectroscopy include microscopy techniques, and spatiallywhich require resolved considerable fluorescence local spectroscopy, field enhancement. as well as various nonlinear spectroscopy techniques, which require considerable local field enhancement. 5. Conclusions 5. Conclusions In conclusion, we have reported the first experimental realization of TO-based birefringent lithographicallyIn conclusion, fabricate we haved magnifying reported the Maxwell first experimental fisheye lenses, realization which operate of TO-based over a very birefringent broad lithographicallyangular range. fabricated The lens action magnifying is based Maxwell on control fisheye of polarization lenses, which-dependent operate effective over a refractive very broad angularindex range. in a tapered The lens waveguide. action is based We have on control studied of polarization-dependentwavelength and polarization effective dependent refractive indexperfo inrmance a tapered of the waveguide. lenses. The We fabricated have studied TO designs wavelength are broadband, and polarization which was dependent verified in performance the 488– of6 the33 nm lenses. wavelength The fabricated range. Our TO technique designs are together broadband, with other which related was results verified [9– in11] the opens 488–633 up an nm wavelengthadditional range.degree Ourof freedom technique in optical together design with and other considerably related results impro [ves9–11 our] opens ability up to anmanipulate additional light on submicrometer scale. In particular, it enables extremely compact and efficient waveguide degree of freedom in optical design and considerably improves our ability to manipulate light on mode sorters, which are required in on-chip mode-division multiplexing and sensing applications. submicrometer scale. In particular, it enables extremely compact and efficient waveguide mode Another potentially interesting application of the magnifying Maxwell fisheye lenses would be sorters, which are required in on-chip mode-division multiplexing and sensing applications. Another verification of the recently proposed perfect imaging without negative refraction [6], which is potentially interesting application of the magnifying Maxwell fisheye lenses would be verification of currently under debate. Our experiments already demonstrated that the image resolution of the thefabricated recently proposed lenses appears perfect to imaging be close without to the diffraction negative refraction limit. Further [6], which studies is currently of image under resolution debate. Ourwould experiments be highly already interesting. demonstrated that the image resolution of the fabricated lenses appears to be close to the diffraction limit. Further studies of image resolution would be highly interesting. Acknowledgments: This research was supported by the National Science Foundation (NSF) grant Acknowledgments:DMR-1104676. We Thisare grateful research to wasJeff Klupt supported for experimental by the National help. Science Foundation (NSF) grant DMR-1104676. We are grateful to Jeff Klupt for experimental help. Author Contributions: V.S., I.S. and D.S. conceived and designed the experiments; C.J., W.Z. and A.J. performed the experiments; V.S. and I.S. analyzed the data and wrote the paper.

Conflicts of Interest: The authors declare no conflict of interest.

References

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Author Contributions: V.S., I.S. and D.S. conceived and designed the experiments; C.J., W.Z. and A.J. performed the experiments; V.S. and I.S. analyzed the data and wrote the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

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