Computer-Generated Volume Holograms Optimize Degrees of ...

Computer-Generated Volume Holograms Optimize Degrees of Freedom in 3D Aperiodic Structures Wenjian Cai, Timothy D. Gerke, Theodore J. Reber, Ariel Libertun and Rafael Piestun

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eriodic three-dimensional (3D) structures known as photonic crystals have increasingly caught the attention of the scientific community. Aperiodic 3D structures, however, have remained relatively unexplored. Recent work has demonstrated the potential of such structures to produce novel and improved optical functions in diffractive optical applications.1,2,3 In particular, we have designed and fabricated new structures named computer-generated volume holograms (CGVHs) to control diffraction efficiency, angular selectivity and polarization in unconventional ways. Classical volume holograms are modulated 3D gratings obtained by the interference of light waves in a photosensitive material. However, the physical structure of these holograms is limited by the possible 3D fields that can be obtained by the interference of light. Two-dimensional diffractive optical elements, on the other hand, modulate amplitude or phase in a predetermined way to achieve a given optical function. Unfortunately, current diffractive optical elements are inherently 2D in the sense that their action can be described as a modulation of the wave on a single transverse plane along the direction of propagation, which limits the possible functionalities. In contrast, the new CGVHs offer additional degrees of freedom to encode information in three dimensions in alternative forms. A CGVH is thus a fully 3D refractive index modulation whose previously designed functionality is the result of scattering, diffraction and interference of light. In recent work, we have focused on the diffraction patterns generated by coherent illumination of the structure. As opposed to optically recorded holograms,

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the structure of a CGVH is not limited by optical interference of an object and reference waves in a photosensitive material, thus enabling optimal encoding beyond the classical optical interferometric encoding. Our designs and experiments have shown unique properties. In particular, we have demonstrated that the third dimension can be used to significantly increase the diffraction efficiency,1 to improve the angular selectivity2 and to control the polarization response.3 In order to create the required 3D micro and nano-structures, we scanned focused femtosecond laser pulses to produce permanent refractive index changes inside glass. Furthermore, to create polarization-sensitive devices, we applied the recently discovered effect of femtosecond-laser-induced birefringence in fused silica.4 In this regime, the laser-created plasma gives origin to subwavelength structures that generate anisotropy by the effect of form birefringence. We demonstrated for the first time polarization-selective computer-generated holograms using this effect in three dimensions. These holograms form different reconstructions for different illuminating polarization states (see the figure). Volume holograms are advantageous for applications such as high-density optical data storage, space variant optical processing and space-time pulse shaping. CGVHs not only present the potential to combine some of the advantages of 2D diffractive optical elements and volume holography, but they can also provide new functionalities such as synthetic spacetime coding, engineered space-variant point spread functions and achromatic angular selectivity.2 CGVHs can be localized in the volume of a dielectric material

and among other waveguiding or freespace optical devices—which makes them attractive for photonic integration. t [ The authors are with the department of electrical and computer engineering and the department of physics at the University of Colorado, Boulder, Colo. ] References 1. W. Cai et al. Opt. Lett. 31, 1836-8 (2006). 2. T.D. Gerke and R. Piestun, “Performance of Computer Generated Volume Holograms Directly Written with a Femtosecond Laser,” Frontiers in Optics 2006, Rochester, N.Y. 3. W. Cai et al. Opt. Express 14, 3785-91 (2006). 4. Y. Shimotsuma et al. Phys. Rev. Lett. 91, 247405 (2003). 5. V. R. Bhardwaj et al. Phys. Rev. Lett. 96, 057404 (2006).

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Top: Microscope image of a detail of the fabricated polarization-sensitive computer-generated hologram. Center: Far field reconstruction. Note that the reconstruction is asymmetric. When the polarization is changed between two orthogonal states, the reconstruction switches the +1 and -1 orders. Bottom: Experimental reconstruction at +1 order for different polarizations. From left to right, the polarization is rotated clockwise 0, 60 and 90 degrees.

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Xiaosheng Wang, Igor Makasyuk, Jianke Yang and Zhigang Chen

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ne of the most fascinating features of photonic band-gap (PBG) structures is that they provide a fundamentally different way of waveguiding by defects in otherwise uniformly periodic structures, as opposed to guidance by total internal reflection (TIR).1 Such bandgap guidance has been demonstrated in two-dimensional (2D) arrays of dielectric cylinders with an isolated defect for microwaves, and in holey-core photonic crystal fibers (PCF) or in all-solid PCF with a lower-index core for optical waves, where the PBG refers to time-domain frequency modes. In fact, laser emission based on photonic defect modes has been realized for a wide range of spectra. On the other hand, the PBG of spatial frequency modes (propagation constant vs. transverse wave vector) in waveguide lattices2 represents another possibility for unconventional guidance of light, especially in periodic waveguides with structured defects. For instance, in optically induced lattices containing a

negative defect, a probe beam tends to diffuse away from the defect site where the induced index is lower than that in neighboring sites. However, within the PBG, the probe beam can be localized by defects, forming an evanescent defect mode.3 In a number of recent experiments, we have demonstrated novel spatial confinement of light in optically induced PCF-like structures, including those with a negative defect embedded in 1D stripe waveguide lattices, 2D square lattices, and 2D ring lattices akin to PBG fibers.4,5 In these settings, the guidance of light is distinctively different from traditional guidance by TIR or solitoninduced nonlinear self-guidance. Typical experimental results are summarized in the figure. The refractive index structures (lattices) with a single-site negative defect are optically induced in a nonlinear photorefractive crystal by spatial modulation of an optical beam with an amplitude mask. Such structures (in the x-y plane) remain nearly invariant

after 10 to 20 mm of propagation through the crystal (along the z-direction) under appropriate conditions. The lattice potential or the index change is controlled by parameters such as the lattice beam intensity, polarization, coherence, as well as the bias field across the crystal. The lattice spacing is typically between 20 and 30 mm. To test the “guiding” property of the defect, a focused probe beam of about 16 mm FWHM is aimed into the defect site or the central low-index core, propagating collinearly with the lattice beam. The intensity or the wavelength of the probe beam is so chosen such that it does not experience any nonlinear self-action. Without the lattice, the probe beam diffracts dramatically. However, with the lattice, it exhibits good confinement by the defect channel. Such guidance is not attributed to TIR because the defect forms an anti-guide with self-focusing nonlinearity (top two rows) while the center of the Bessel-like lattice also forms an anti-guide with self-defocusing nonlinearity (bottom row). Instead, this arises from the formation of photonic defect modes, or the anti-resonance effect in the ring PCF-like waveguides.4,5 Our results bring about the possibility to control light in a new type of reconfigurable PBG structure. t This work was supported by NSF, AFOSR, PRF and NSFC. [ X. Wang, I. Makasyuk and Z. Chen are with the department of physics and astronomy, San Francisco State University, San Francisco, Calif. (Chen is also affiliated with the Key Laboratory for Weak-Light Nonlinear Photonics Materials, Ministry of Education and TEDA Applied Physical School at Nankai University in Tianjin, China.) J. Yang is with the department of mathematics and statistics at the University of Vermont and the Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, China. ]

Guiding light in optically induced PCF-like structures. The first two rows are 1D and 2D lattices with a single-site negative defect induced with self-focusing nonlinearity. The bottom row shows the ring lattices with a central low-index core induced with self-defocusing nonlinearity. From left to right, shown are schematic illustration of induced lattice structures (red indicates high index), experimentally created lattice patterns, the probe beam at input, and its 2D and 3D intensity patterns exiting the defect or low-index channel, respectively.

References 1. P. Russell. Science 299, 358 (2003). 2. D.N. Christodoulides et al. Nature 424, 817 (2003). 3. F. Fedele et al. Opt. Lett. 30, 1506 (2005); J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006). 4. I. Makasyuk et al. Phys. Rev. Lett. 96, 223903 (2006). 5. X. Wang et al. Opt. Lett. 31, 1887 (2006); Opt. Express 14, 7362 (2006).

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DIFFRACTIVE OPTICS

Optically Induced PCF-Like Structures

DIFFRACTIVE OPTICS

Switchable Diffractive Lens for Vision Correction

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Guoqiang Li, David Mathine, Pouria Valley, Pekka Äyräs, Joshua Haddock, M. Giridhar, Jim Schwiegerling, Gerald Meredith, Bernard Kippelen, Seppo Honkanen and Nasser Peyghambarian

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ith aging, the eye’s lens loses some of its elasticity and becomes less able to focus incoming light, making it difficult for a person to switch easily between focusing on a near object and a distant one—a condition called presbyopia. Area-divided bifocal or trifocal lenses have limited field of view, requiring the user to gaze down to accomplish near vision tasks, and in some cases causing dizziness and discomfort. An electro-active lens allows a voltagecontrolled change of the focusing power across the entire aperture. Such a lens has stringent requirements, including high light efficiency, relatively large aperture, fast switching time, low driving voltage and power-failure-safe configuration. None of the previous demonstrations1,2,3 satisfies these requirements simultaneously. We report new switchable, flat, thin liquid crystal diffractive lenses4 that meet all of the above requirements. Lenses with eight phase levels, 10 mm diameters and focal lengths of 1 m and 0.5 m (+1.0 diopter and +2 diopter of added power, respectively) have been demonstrated at 555 nm. The operation of these spectacle lenses is based on electrical control of the refractive index of a 5-mm-thick layer of nematic liquid crystal using a circular array of photolithographically defined transparent indium tin oxide (ITO) electrodes. A schematic drawing of the liquid crystal lens is shown in part (a) of the figure. In the patterned substrate (b), a 1 mm gap was required between adjacent electrodes to maintain electrical isolation and ensure a smooth transition of the phase profile. Over the patterned ITO, we sputtered an electrically insulating layer of SiO2, into which small via openings were etched. We subsequently sputtered an electrically conductive layer

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of ITO over the insulating layer to fill the vias and contact the electrodes. We then patterned the layer to form eight independent electrical bus bars. The lens operates with high transmission, low voltage (< 2 VRMS), fast response (about 130 ms), a diffraction efficiency exceeding 90 percent, small aberrations and a power-failure-safe configuration. The focused spot size is close to the diffraction-limit value. Interferometric measurements show excellent imaging capability of the lens. Good spherical profiles were obtained in crosssections, indicating small aberrations. The RMS value of the higher-order aberrations is 0.039l, which is comparable to a high quality reading glass. To test the imaging properties of the lens, we constructed a model human eye and placed a double lens element in front of it to provide near vision correction. As can be seen in (c), the model eye has insufficient power to form a sharp image. However, by switching on the diffractive lens, the image is brought into focus (d). Negative focusing powers can also be achieved with the same lenses by changing the sign of the slope of the applied voltages. These results represent a significant advance in the state-of-theart in liquid crystal diffractive lenses for vision care and other applications. They have the potential of revolutionizing the field of presbyopia correction. t [ G. Li ([email protected]), D. Mathine, P. Valley, P. Äyräs, M. Giridhar, J. Schwiegerling, G. Meredith, S. Honkanen and N. Peyghambarian ([email protected]. edu) are with College of Optical Sciences, University of Arizona, Tucson, Ariz. J. Haddock and B. Kippelen are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Ga. ] References 1. B. Wang et al. Appl. Opt. 43, 3420 (2004). 2. C.W. Fowler et al. Ophthal. Physiol. Opt. 10, 186 (1990). 3. W.W. Chan et al. Appl. Opt. 36, 8958 (1997). 4. G. Li et al. Proc. Natl. Acad. Sci. USA 103, 6100 (2006).

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(a) Structure of the liquid crystal lens, where k is the wave vector and E is the polarization state of the incident light. (b) Layout of the one-layer electrode pattern (two central zones shown). Adjacent zones are distinguished by color. An electrical insulation layer with vias is added (vias shown with white dots). Each bus connects to one electrode (sub-zone) in each zone. Dimensions of the vias, the bus line, and the gap between electrodes are illustrated in the bottom right corner of the figure. (c) and (d) Hybrid imaging using the 1-diopter electroactive diffractive lens with the model eye. The object is placed at a reading distance (~30 cm). (c) The image is severely out of focus in the model eye when the diffractive lens is OFF. (d) When the diffractive lens is activated, the object is imaged clearly.

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Christian R. Rosberg, Dragomir N. Neshev, Yaroslav V. Kartashov, Rodrigo A. Vicencio, Wieslaw Krolikowski, Mario I. Molina, Arnan Mitchell, Victor A. Vysloukh, Lluis Torner and Yuri S. Kivshar

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ack in 1932, the famous Russian scientist Igor E. Tamm predicted that a truncated crystalline lattice could support special types of electronic states that are bound at the very edge of the semi-infinite periodic potential.1 These states, known in many fields as Tamm states, represent a special class of spatially localized surface waves, which, in general, may appear at interfaces between different physical media. An optical analog of linear Tamm states has been described theoretically and demonstrated experimentally for an interface separating periodic and homogeneous dielectric materials.2 Despite the fact that many theoretical concepts have been successfully introduced in the physics of surface waves, nonlinear Tamm states have never been experimentally observed. In a recent paper, we experimentally demonstrated self-action of a narrow laser beam propagating near the edge of a lithium niobate (LiNbO3) waveguide array with defocusing nonlinearity and a semi-infinite periodic refractive index modulation in the transverse direc-

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tion,3 as shown schematically in part (b) of the figure. For the first time to our knowledge, we observed the formation of surface gap solitons or nonlinear Tamm states. While linear surface modes do not exist in this kind of system, light self-trapping is observed in the nonlinear regime above a certain threshold power when the propagation constant is shifted into the gap of the photonic transmission spectrum. The detailed theory of such nonlinear Tamm states was developed a few months earlier by Kartashov et al.4 Encouraged by the recent studies of discrete surface solitons,5 they predicted the existence of surface gap solitons at the interface between a uniform dielectric medium and a photonic lattice with defocusing nonlinearity. In such systems, light localization occurs in the form of nonlinear surface modes exhibiting a staggered phase structure, as shown in (a). The alternating phase of the beam tail inside the periodic medium reflects the fact that the propagation constant of the self-localized mode

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Theoretical prediction (a,b) and experimental observation (c,d) of nonlinear Tamm states in a truncated photonic lattice. (a) Theoretical profile of a nonlinear Tamm state—a surface gap soliton.4 (b) Schematic of the waveguide array geometry. (c) Three-dimensional representation of the nonlinear surface state observed above the localization threshold.3 The horizontal and vertical sample coordinates are x and y, respectively. (d) Experimental plane-wave interferogram demonstrating the staggered phase structure of the nonlinear Tamm state.

lies within the photonic bandgap. This essential feature enables one to draw a direct analogy to the electronic Tamm states and extend this concept to the nonlinear regime, so that the surface gap solitons can be termed nonlinear Tamm states. They possess a unique combination of properties related to both electronic and optical surface waves and gap solitons. Part (c) of the figure depicts a threedimensional representation of the spatial beam intensity distribution of a nonlinear Tamm state observed in experiment.3 The nonlinear mode was excited by injecting a narrow probe beam into the surface waveguide at the edge of the periodic structure. Part (d) shows the corresponding interference pattern created when superimposing an inclined plane reference wave in order to reveal the phase structure of the output beam. A half-period vertical shift of the interference fringes, corresponding to an exact p phase jump in the horizontal beam direction, is clearly observed between each lattice site, as predicted by theory.4 The ability to generate these types of optical surface modes could lead to novel and effective experimental tools for the study of nonlinear effects near surfaces. t [ C.R. Rosberg, D.N. Neshev, W. Krolikowski and Y.S. Kivshar are with CUDOS at the Australian National University, Canberra, Australia. Y.V. Kartashov and L. Torner are with ICFO, Barcelona, Spain. R.A. Vicencio is with the Max-Planck-Institut für Physik komplexer Systeme, Dresden, Germany. M.I. Molina is with the departamento de física, facultad de ciencias, Universidad de Chile, Santiago, Chile. A. Mitchell is with the School of Electrical & Computer Systems Engineering, RMIT University, Melbourne, Australia. V.A. Vysloukh is with the departamento de física y matemáticas, Universidad de las Américas, Puebla, Mexico. ] References 1. I.E. Tamm. Z. Phys. 76, 849 (1932). 2. P. Yeh et al. Appl. Phys. Lett. 32, 102 (1978). 3. C.R. Rosberg et al. Phys. Rev. Lett. 97, 083901 (2006). 4. Y.V. Kartashov et al. Phys. Rev. Lett. 96, 073901 (2006). 5. K.G. Makris et al. Opt. Lett. 30, 2466 (2005).

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DIFFRACTIVE OPTICS

Nonlinear Tamm States in Periodic Photonic Structures