The use of a negative index planoconcave lens array for wide-viewing angle integral imaging

Wide-viewing angle integral imaging by means of a negative refractive index planoconcave lens array is theoretically investigated. The optical properties of a negative refractive index lens are analyzed from the point of view of integral imaging. The effective focal length of a positive index planoconvex lens and a negative index planoconcave lens with the same surface spherical curvature R are approximated as , 2 P eff f R = and , 0.4 N eff f R = , respectively. This short effective focal length of the negative index lens is advantageous for extending the viewing angle of the integral imaging. In addition, some other optical properties of a negative index lens are analyzed and compared for a positive index lens. Three-dimensional ray-tracing observation simulations of integral imaging systems with a negative index lens array and a positive index lens array are then performed, in a comparative study of the wide-viewing angle mode for integral imaging. A three-dimensional ray-tracing simulator for an integral imaging system is then developed. Some interesting issues that appear in the wide-viewing mode of integral imaging are discussed. The negative refractive index planoconcave lens was found to give a wider viewing angle of -60(deg.) ~ +60(deg.) and reduces aberration with only a single spherical planoconcave lens. ©2008 Optical Society of America OCIS codes: (160.3918) Metamaterials, (080.0080) Geometric optics, (100.6890) Threedimensional image processing. References and links 1. B. Javidi and F. Okano, eds., Three Dimensional Television, Video, and Display Technologies (Springer, 2002). 2. B. Lee, J.-H. Park, and S.-W. Min, “Three-dimensional display and information processing based on integral imaging,” in Digital Holography and Three-Dimensional Display, T.-C. Poon, ed. (Springer, 2006), pp. 333-378. 3. H. Liao, M. Iwahara, H. Nobuhiko, and T. Dohi, “High-quality integral videography using a multiprojector,” Opt. 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Javidi, “Three-dimensional integral imaging with electronically synthesized lenslet arrays,” Opt. Lett. 27, 1767-1769 (2002). #101915 $15.00 USD Received 23 Sep 2008; revised 22 Nov 2008; accepted 11 Dec 2008; published 17 Dec 2008 (C) 2008 OSA 22 December 2008 / Vol. 16, No. 26 / OPTICS EXPRESS 21865 10. J.-H. Park, S. Jung, H. Choi, and B. Lee, “Viewing-angle-enhanced integral imaging by elemental image resizing and elemental lens switching,” Appl. Opt. 41, 6875-6883 (2002). 11. J.-H. Park, Y. Kim, J. Kim, S. -W. Min, and B. Lee, “Three-dimensional display scheme based on integral imaging with three-dimensional information processing,” Opt. Express 12, 6020-6032 (2004). 12. S.-W. Min, J. Kim, and B. Lee, “Wide-viewing projection-type integral imaging system with an embossed screen,” Opt. Lett. 29, 2420-2422 (2004). 13. H. Choi, J.-H. Park, J. Hong, and B. 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Lett. 88, 081101 (2006). 19. C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, “Performance of a negative index of refraction lens,” Appl. Phys. Lett. 84, 3232-3234 (2004). 20. P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, “Focusing by planoconcave lens using negative refraction,” Appl. Phys. Lett. 86, 201108 (2005). 21. K.-Y. Kim, “Photon tunneling in composite layers of negativeand positive-index media,” Phys. Rev. E 70, 047603 (2004). 22. H. M. Ozaktas, Z Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, New York, 2001). 23. J.-H. Park, S.-W. Min, S. Jung, and B. Lee, “Analysis of viewing parameters for two display methods based on integral photography,” Appl. Opt. 40, 5217-5232 (2001). 24. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780nm wavelength,” Opt. Lett. 32, 53-55 (2007). 25. G. Dolling, C. Enkrich, M. 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Abstract: Wide-viewing angle integral imaging by means of a negative refractive index planoconcave lens array is theoretically investigated.The optical properties of a negative refractive index lens are analyzed from the point of view of integral imaging.The effective focal length of a positive index planoconvex lens and a negative index planoconcave lens with the same surface spherical curvature R are approximated as , 2 , respectively.This short effective focal length of the negative index lens is advantageous for extending the viewing angle of the integral imaging.In addition, some other optical properties of a negative index lens are analyzed and compared for a positive index lens.Three-dimensional ray-tracing observation simulations of integral imaging systems with a negative index lens array and a positive index lens array are then performed, in a comparative study of the wide-viewing angle mode for integral imaging.A three-dimensional ray-tracing simulator for an integral imaging system is then developed.Some interesting issues that appear in the wide-viewing mode of integral imaging are discussed.The negative refractive index planoconcave lens was found to give a wider viewing angle of -60(deg.)~ +60(deg.) and reduces aberration with only a single spherical planoconcave lens.

Introduction
Integral imaging (InIm) is considered to be one of the more feasible technologies for displaying three-dimensional (3D) objects [1][2][3].The InIm display uses a lens array to multiplex directional parallax views of a 3D target object into the two-dimensional (2D) image plane in the form of an elemental image and demultiplexes the elemental image into a full parallax 3D image of the target object without the need for an additional viewing apparatus such as polarization glasses.
One of the challenging problems in InIm is extending the viewing angle.In the high viewing angle mode of InIm, problems associated with image distortion and flipping occur.Some theoretical studies on these problems have been reported [4][5][6].Various approaches for enhancing the viewing angle of InIm have been made.The curved lens array [7] widens the viewing angle, but the curve feature is disadvantageous because of its thickness.Viewing angle enhancement approaches using time multiplexing schemes [8][9][10][11] can also extend the viewing angle, but require additional loads as high speed polarization switching or a mechanical lens array moving apparatus.The critical problem for the time multiplex approach is the high-speed driving of liquid crystal display panels.
The conventional setup employs a lens array composed of a simple spherical singlet.To date, a few studies on the lens array itself such as a grouped lens system or a functional material lens have been reported.Studies of the lens array structure have also been reported.An embossed screen [12] and an aspherical inhomogeneous lens array [13,14] were used in these studies, but they fail to provide a fundamental solution to the problem of viewing angle enhancement.Innovations and new ideas regarding lens array itself to enhance the viewing angle continue to be needed.
In this paper, we theoretically investigate wide-viewing angle InIm using a negative refractive index lens array.Advances associated with the negative index medium have led to an explosion of research and has highlighted some interesting potentials for creating unprecedented optical functions.Although the practical realization of negative index media is currently limited to a very narrow frequency bandwidth and limited structures, previous and theoretical studies indicate that its potential applications are considerable [15][16][17][18][19][20][21].The compactness, aberration compensation properties, and imaging characteristics, and evanescent field component enhancement are known to be advantageous optical properties of a negative index lens.Despite of the theoretical and practical complexity associated with the realization of negative index media in the broad wavelength range, the resulting macroscopic physical law of light refraction and total internal reflection on the negative index media can be described by no other than the Snell's law of refraction.Thus, it is possible to analyze the imaging characteristics of InIm with a negative index lens array as well as the optical properties of the negative index lens itself using only the Snell's law.Regarding InIm, the most important feature of the negative index lens is its short effective focal length.Using this unique property of the negative index lens, it becomes possible to construct wide-viewing angle InIm and study the imaging characteristics of the wide-viewing angle InIm by means of a negative index lens array.
In this paper, we expand out the meaning of the negative index lens to InIm and the imaging characteristics of wide-viewing angle InIm with a negative index lens array.In Sec. 2, the focusing properties of the negative index planoconcave lens are accounted for with a comparative analysis of the positive index planoconvex lens.In Sec. 3, imaging simulations of InIm with the negative index lens array, performed by a self-developed ray-tracing simulator, are described.The comparative analysis of an InIm-based negative index lens and a conventional positive index lens array InIm is presented.Concluding remarks are found in Sec. 4.

Focusing property of negative index planoconcave lens
In this section, the focusing properties of a negative index planoconcave lens (NIL) are analyzed and the results compared with those for a conventional positive index planoconvex lens (PIL).The focusing characteristics of a specific lens are prerequisites for a better understanding of the imaging characteristics and performance of InIm, when a specific lens array is used.
In Figs.1(a) and 1(b), the structures and ray traces of NIL and PIL with spherical curvature R are illustrated, respectively.For NIL, Snell's law at the incidence plane and at the spherical surface of the NIL is described, respectively, by sin sin where i θ and n n are the incidence angle of a ray and the absolute value of the negative refractive index (Note that light rays are incident from the right in Fig. 1.).The refractive index of the surroundings is set to 1 (air).The ray trace in the left half-infinite air region is obtained by For PIL, Snell's law at the incidence plane and at the spherical surface of the PIL is described, respectively, by sin sin ( ) where i θ and p n are the incidence angle of a ray and the positive refractive index.The ray trace in the left half-infinite air region is obtained by By default, let the lens curvature radius, R (taken as a positive value), the negative refractive index, n n − , and the positive refractive index, p n , be set to 1 (in the unit of coordinates), -1.5, and 1.5, respectively.
As shown in Figs.1(a), 1(b), Eqs.(1c), and (2c), the crossing angle of the ray to the optic axis (z-axis) are ψ φ + and ψ φ − for NIL and PIL, respectively.With respect to the incidence position ( ) , the ray crossing angle varies.The parameter φ is given by a function of the incidence position ( ) The crossing angle is closely related to the effective focal length of the lens.In geometric optics, the focal length of a lens can be precisely defined for a paraxial region near the optic axis.Based on the lens maker's formula, the effective focal lengths for NIL and PIL in the paraxial region around the optic axis are obtained by , 0.4 , respectively.With the same spherical curvature, the focal length of NIL is five times shorter than that of PIL.
To estimate the focal point and spherical aberration for NIL and PIL, we determine the position of the crossing point position of the rays in the focal plane (x-axis).For an incidence point ( ) z x in the incidence plane, the x-directional focus profiles, ( ) i x x , of the NIL and the PIL at a fixed f z are given, respectively, as, from Eqs. 1(c) and 2(c), where f z is defined by the parameter that satisfies the minimization of the focus constraint defined as ( ) , max min The meaning of the minimization of Eq. ( 5) is that the major portion of the rays that are incident on the incident on the incidence plane, i R x R − ≤ ≤ , are maximally focused around a single focal point ( ) i x x at the optimal f z , which is referred to as the effective focal plane.This definition of the effective focal plane is available for the entire range of incidence including non-paraxial incidence as well as the case of paraxial incidence case.The ray crossing point distribution with respect to the incidence position i x is parameterized by the effective focal length.The focusing at the normal incidence condition is estimated.The minimum f z is the effective focal plane of the lens.According to the numerical calculation, f z for NIL and PIL at a normal incidence is obtained as, respectively, 0.6 f z = and 3 f z = − in the geometries of Fig. 2, which is matched to the lens maker's formula.Figure 2 shows the ray tracing profiles of NIL and PIL at a normal incidence.On the other hand, in InIm, the spatial multiplexing of 3D images is obtained in the form of an elementary image.The elemental image is actually the image of the 3D target object captured by the same lens array.When displaying the elemental image through the lens array, spatial point image information distributed in the image plane is directionally displayed.A specific point information in the elemental image is brought along the corresponding directional ray through the corresponding elementary lens.Observers at different positions can visualize different parallax images.This Fourier transform property of the lens is the core of the InIm based 3D display system.The Fourier transform property of a lens is very important with respect to the applications to InIm systems.In Fig. 3  As can be seen in Fig. 4(a), the aberration level for PIL is no less than that for NIL.Near 1 , The spot width physically defines the width of the area where all rays incident on the interval ( ) x x impinge at the focal plane.The spot position is the center of the area.Inversely, for a fixed spot width ρ , we can define the effective transmission window of a lens by the continuous interval ( ) x x in the incidence plane, such that the interval length, ( ) ,2 ,1 , where the effective transmission window varies with the incidence angle i θ and so ( ) w θ is parameterized by the incidence angle i θ .
In InIm, a point source is placed at a position in the focal plane under the lens.The rays that radiate from this point source are collimated to form a plane wave bundle that passes along a specific direction and the radiation is limited by the effective transmission window.The brightness of the pixel in InIm toward the viewing angle i θ is determined by the effective transmission window.From the point of view of InIm, the parameter ρ is a measure of the degree of collimation.In Figs.In the analysis of Fig. 5, it is possible to extract the effective focal lengths for NIL and PIL, which are the derivative, ( ) ( ) In the paraxial region ( sin 0.5 i θ ≤ ), the linear relationship between the transverse vector of the incidence ray bundle, sin i θ , and the transversal shift of the focal point from the center, ( ) i μ θ is confirmed.In the plot, the effective focal length eff f is defined as ( ) In the cases of NIL and PIL, the effective focal length can be extracted, from the plots shown in Fig. 5, as, respectively, , 0.4 , 2 This is in good agreement with results from the lens maker's formula.However, outside the paraxial region ( sin 0.5

Simulation of integral imaging with negative index lens array
In this section, the effect and meaning of using an NIL array in InIm are discussed with selfdeveloped 3D ray-tracing simulations.In particular, the imaging characteristics of wideviewing InIm with the use of a NIL array are of main concern.For a comparative study, the InIm simulations with a perfect paraxial lens array and a PIL array are performed simultaneously.
A perfect paraxial lens is defined as a mathematical lens with zero-thickness, noaberration, and a focal length of f .The key function of the perfect paraxial lens lies in its Fourier transform property such that an obliquely incident plane wave with an incidence angle of θ is focused at the position shifted by sin f θ from the center in the focal plane.As shown in Fig. 5, both NIL and PIL have approximately linear Fourier transform properties, similar to this, perfect paraxial lens.Thus, the perfect paraxial lens is the ideal realization of spherical lenses as NIL and PIL.A critical difference between a perfect paraxial lens and NIL and PIL is the effective window size.The perfect paraxial lens focuses all of the incidence plane waves into an exact focus with no aberration and, as a result, the effective widow is the entire lens aperture.The mathematical transform for the perfect paraxial lens is represented by the 4 4  InIm is capable of displaying both virtual and real objects when the gap between the lens array and elemental images is equal to the focal length of the lens.In the simulation setup, the letters S and N are the real objects and the letter U is a virtual object [23].The observation camera is denoted by a simple perfect paraxial lens with a focal length of e f .The center of the camera lens is placed at ( ) ( ) that can capture the whole lens array.The reverse projected image of the 3D scene through the lens array onto the 2D flat display panel as shown in Fig. 6(a) is referred to as the elemental image.The elementary image plane is placed on the focal plane of the lens array.The observer can then perceive the pixelate 3D image through the lens array in the display mode, as illustrated in Fig. 6(b).In this setup, the number of elemental lenses is the 3D image resolution.In the case of our simulation, the 3D image resolution is 41 41 × , the number of used elemental lenses used.comparing the images for a real object observation in Fig. 8, it is clear that the synthesized 3D images of both the perfect paraxial lens array and the NIL array correctly express the views of the real object.In the case of the NIL array, the low brightness and pixelate 3D images are due to the finite effective window size, as explained in Fig.

Figures 4 (Fig. 4 .
Figures 4(a) and 4(b) show the crossing point distributions ( ) i x x for the NIL (at

ix
= ± , incident rays are totally internal-reflected and, as a result, do not contribute to the formation of the focus.In Fig. 4(b), the crossing point distribution ( ) i x x for an oblique incidence of 70(deg.) is shown.In this case, the PIL forms a highly aberrated focus, which is indicated by the inclined, non-flat ( ) i x x distribution (red lined).Meanwhile, the NIL forms a well-defined focus at 0.5 x = − , which is indicated by a relatively flat crossing point distribution (blue lined).In this oblique incidence case, the incident rays on 0 i x > are totally internal-reflected for both cases of NIL and PIL.Let us look into, more carefully, the Fourier transform property and the transmission efficiency of NIL and PIL, as they are related to the #101915 -$15.00USD Received 23 Sep 2008; revised 22 Nov 2008; accepted 11 Dec 2008; published 17 Dec 2008 (C) 2008 OSA total internal reflection, more carefully.To accomplish this, let us define the spot width ρ and the spot position μ , for a specific continuous interval, ( ) 5(a) and 5(b), the focus position ( ) i μ θ and effective window ( ) i w θ for various incidence angles for NIL, obtained by ray-tracing analysis, are shown as a function of sin i θ , respectively.For comparison, in Figs.5(c) and 5(d), the focus position ( ) i μ θ and the effective window ( ) i w θ for various incidence angles for PIL are shown, respectively.In this simulation, the spot width ρ is set to 0.1.The spot position and the effective transmission window are defined for a specific incidence angle.

#l and 2 l
101915 -$15.00USD Received 23 Sep 2008; revised 22 Nov 2008; accepted 11 Dec 2008; published 17 Dec 2008 (C) 2008 OSAwhere λ is the wavelength, 1 are the distances between the ( ) , u v plane and the lens plane and the lens plane and ( ) , x y plane, respectively.f is the focal length of the lens.

Fig. 8 .
Fig. 8. Images of the 3D target object observed at various viewing angle position.The synthesized elemental image may look quite different with respect to the lens focal length.In this paper, we examine the InIm systems with NIL with a focal length of

Fig. 11 .
Fig. 11.Images of the InIm with a perfect paraxial lens array of focal length , 1cm P eff f = .

5
. The particularly low brightness of the ( ) 60 deg.± images is mainly due to a slight elongation of the focal length.In spite of the low brightness, the synthesized images are well matched to the observation images for the real object at those viewing angles.#101915 -$15.00USD Received 23 Sep 2008; revised 22 Nov 2008; accepted 11 Dec 2008; published 17 Dec 2008 (C) 2008 OSA

Fig. 12 .
Fig. 12. Images of the InIm with a PIL array of effective focal length , 1cm P eff f = .
set to 4.97cm , 2m , and 5.1cm , respectively.The camera image plane is set to 12mm 12mm × are