2562 OPTICS LETTERS / Vol. 33, No. 21 / November 1, 2008

Imaging performance of an isotropic negativedielectric constant slab

Shivanand, Huikan Liu, and Kevin J. Webb*School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Avenue,

West Lafayette, Indiana, 47907-2035, USA*Corresponding author: webb@purdue.edu

Received May 27, 2008; accepted August 10, 2008;posted September 18, 2008 (Doc. ID 96582); published October 31, 2008

The influence of material and thickness on the subwavelength imaging performance of a negative dielectricconstant slab is studied. Resonance in the plane-wave transfer function produces a high spatial frequencyripple that could be useful in fabricating periodic structures. A cost function based on the plane-wave trans-fer function provides a useful metric to evaluate the planar slab lens performance, and using this, the op-timal slab dielectric constant can be determined. © 2008 Optical Society of America

OCIS codes: 110.2990, 160.4670, 260.3910, 310.6860.

Conventional optics suffers from the diffraction limitbecause of a finite aperture and the availability ofonly the propagating components of the plane-wavespectrum. As a consequence, the spot size is limitedto about one wavelength. Information about the sub-wavelength features of the object are carried by theevanescent fields, which decay exponentially in freespace. A negative refractive index slab �n�0� can actas a lens because of negative refraction and, of par-ticular interest, can amplify the evanescent fieldsand produce images with subwavelength features[1–4]. A passive material must have loss, and the eva-nescent fields are very sensitive to this loss (and ma-terial mismatch, in general) [3,4]. The evanescentfield growth is a result of a reflection coefficient mag-nitude at the slab to free-space interface that can begreater than one (or, correspondingly, the existence ofsurface plasmon waves). This is also a feature of ametal slab in a dielectric background, leading to itsproposed use to achieve subwavelength resolution[2], and subsequent demonstrations [5–8]. For ex-ample, such a slab may find applications in photoli-thography.

We explore the near-field imaging properties of asingle slab of material with isotropic �=��+ j��, with���0. This condition is satisfied, for example, by met-als at optical frequencies. It is also possible to synthe-size materials that have these properties, such asmixtures of metals and dielectrics, and fabricationcan be relatively straightforward. Although this sys-tem has received considerable recent attention, weare motivated to understand what resolution range isaccessible, how to measure resolution, and potentialapplications.

We consider a TM field and a free-space back-ground. For an isotropic slab with dielectric constant�, the dispersion relation is kx

2+kz2=�k0

2, where in freespace k0

2=�2 /c2, with c the velocity. Following theexp�j�t� time convention, the transmission coefficientis

T�kx� =�1 − r2�exp�− jkzd�

1 − r2 exp�− j2kzd�, �1�

0146-9592/08/212562-3/$15.00 ©

where r is the reflection coefficient and d is the thick-ness of the slab [9]. The reflection coefficient for Hy isr= �kz /�−kz0� / �kz /�+kz0�, where kz0=�k0

2−kx2, which is

singular when ��k02−kx

2+��k02−kx

2=0. Therefore,r�kx /k0� is singular at kx /k0→� when �=−1, and atkx /k0→1 when �→�. Singularities in Eq. (1) occurwhen r2=exp�j2kzd�. Also, for large kx and real �,T�kx���1−r2�exp�−kxd�, making the slab thickness adominant parameter in imaging performance.

Figure 1 shows a gray scale logarithmic plot of themagnitude of the transmission coefficient for severalslab thicknesses, as a function of real � and kx /k0, as-suming a wavelength of 700 nm. From Fig. 1, itwould appear that ��−1 would provide the best im-age performance. To investigate this neighborhoodmore clearly, Figs. 2(a)–2(c) show the transmissionspectrum obtained for different slab thicknesses with�=−1.01, −1.02, and −1.1, respectively. A perfect lenshas T�kx�=1 and, generally, the resolution can be ex-pected to improve as T�kx� increases deep in the eva-nescent (free-space) plane-wave spectrum. It is clearfrom Fig. 2 that the response improves with decreas-ing slab thickness d, and practical considerationsdrive this standoff distance between the object andimage planes. In our calculations, we have restrictedd to a minimum value of 0.1 �m. The singularities,which dampen with loss, are undesirable in an imag-ing application.

Fig. 1. Transmission spectrum �T�� ,kx /k0��, plotted on alogarithmic scale, for real � and slab thickness (a) d=0.1 �m and (b) d=0.3 �m. The free-space wavelength is

�=700 nm.

2008 Optical Society of America

November 1, 2008 / Vol. 33, No. 21 / OPTICS LETTERS 2563

We consider specific imaging examples using anobject/incident field defined by Hy=1 with a particu-lar spatial support, i.e., the incident field is definedby Hy�x�=p�x ,w�, a unit amplitude pulse centered atx=0 and of width w. A plane-wave expansion of thisexcitation was used to solve for Hy in each region,and E was determined from an inverse Fourier trans-form of the plane-wave spectrum for Hy multiplied bythe impedance. The power dissipated inside a differ-ential volume is proportional to �E�2. Moreover, photo-resists are sensitive to the intensity of the incidentlight. Hence, we choose �E� as the field parameter inour study.

Singularities in T�kx� produce dominant imageplane effects corresponding to those spatial frequen-cies. As an example, Fig. 3(a) shows a particular case

Fig. 2. Transmission spectrum obtained for several slabthicknesses for (a) �=−1.01, (b) �=−1.02, and (c) �=−1.1.The solid, dashed–dotted, dotted, and dashed curves corre-spond to slab thickness d=0.1, 0.2, 0.3, and 0.4 �m, respec-tively. The wavelength is �=700 nm.

with one singularity (�=−2, d=0.4 �m�. Figure 3(b)

shows the the magnitude of the electric field obtainedin the image plane with an incident field defined byHy�x�=p�x ,0.05��. The dominant influence of the sin-gularity causes a ripple at the corresponding spatialfrequency. Through this mechanism, and with controlof the complex dielectric constant, it is in concept pos-sible to generate a periodic spatial ripple with arbi-trarily high frequency. This could be useful in lithog-raphy for very dense (relative to the illuminatingwavelength) periodic systems (photonic crystals).How practical this is rests on how easily the dielec-tric constant can be controlled, in particular, theimaginary part.

The plane-wave spectra of Fig. 2 suggest that��−1.01 may be useful for subwavelength imaging.Passive materials will have some loss, and we ex-plore the influence of �� in Fig. 4. Fig. 4(a) shows themagnitude of the transmission spectrum obtained fora 0.1-�m-thick slab having ��=−1.01. Clearly, theloss suppresses the singularities. With an objectHy�x�=p�x ,0.01�� and a d=0.1 �m thick slab having��=−1.01 and various ��, we obtain the electric fieldimages of Fig. 4(b). The FWHM of the images is0.265� for �=−1.01− j0.01; 0.235� for �=−1.01− j0.02;0.255� for �=−1.01− j0.05; and 0.295� for �=−1.01− j0.1. As a reference, the FWHM after propagationover 0.1 �m in free space is found to be 0.415�. It isclear for this example that the best performance isnot obtained when the loss is minimum. This leads tothe need to optimize the lens parameters for the bestresults.

Fig. 3. (a) Transmission spectrum. (b) �E� (normalized toits maximum value) obtained at the image plane. The slabparameters are �=−2.0 and d=0.4 �m. The object width is

0.05�, and �=700 nm.

2564 OPTICS LETTERS / Vol. 33, No. 21 / November 1, 2008

A well-chosen cost function can be used to optimizethe parameters of the transfer function. We choosethe function

C��,d� =� �F−1�T�kx/k0���

max��F−1�T�kx/k0���dx, �2�

where F−1 is the inverse Fourier transform. Equation(2) takes into consideration both the magnitude aswell as the phase of the transmission coefficient andhas been used for our results. For a perfect lens, C=0. With �=700 nm, we searched the parameterspace for the minimum C�� ,d� at each value of d(with step size 0.05 �m), by evaluating Eq. (2) with astep size of 0.01 for both �� and ��. The results aregiven in Fig. 5(a). The best resolution is when �� isslightly less than −1. Notice that the optimal �� ford /��1 is small but nonzero. Figure 5(b) shows theFWHM of �E� in the image plane for the optimal �,shown in Fig. 5(a). An object of width 0.01� was used.As the slab thickness increases, the lens performancedeteriorates, but it is still substantially better thanthe free-space case, given by the �=1 curve. TheFWHM is subwavelength only for d /��1, makingthis the useful regime. For thicker lenses, a largerloss ���� gives the minimum cost and hence smallerFWHM. This loss produces a propagating component,clearly important with increasing thickness. How-ever, this large loss significantly reduces the ampli-tude of the electric field in the image plane. Our stud-

Fig. 4. (a) Magnitude of the transmission coefficient for a0.1-�m-thick slab having various �=−1.01+ j��. (b) �E� foran object of width 0.01�. The line types correspond to ��=−0.01 (solid curve), ��=−0.02 (dashed curve), ��=−0.05(dashed–dotted curve), ��=−0.1 (dotted curve). The free-space wavelength is �=700 nm.

ies thus far suggest that Eq. (2) is a useful metric.

The optimal isotropic negative dielectric constantlens can provide modest subwavelength resolutionover slab thicknesses up to about one wavelength.The best results can be achieved with a small amountof loss. With no loss, the transmission coefficient hassingularities that dominate the image. This effectmay be useful in applications requiring a high spatialfrequency periodic image. After evaluating a varietyof cost functions to measure the performance, we sug-gest one based on the plane-wave spectrum. Thisshould provide a basis to compare various structuresand results.

This work was supported by the National ScienceFoundation (ECS-0524442), the Department of En-ergy (DE-FG52-06NA27505), and the Army ResearchOffice (W911NF-07-1-0019).

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Fig. 5. (a) Optimal � obtained from the cost function (2) forvarious slab thicknesses d. (b) Optimal FWHM for �E� as afunction of slab thickness, using data from (a). A wave-length �=700 nm was used. The d /� scale should be inter-preted as various d with �=700 nm. The object in (b) haswidth 0.01�.

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