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Vodo, Plarenta; Lu, Wentao; Parimi, Patanjali; Sridhar, Srinivas

Negative Refraction in One-Dimensional Photonic Crystal Inproceedings

In: APS Meeting Abstracts, 2006.

@inproceedings{vodo2006negativeb,

title = {Negative Refraction in One-Dimensional Photonic Crystal},

author = {Plarenta Vodo and Wentao Lu and Patanjali Parimi and Srinivas Sridhar},

year = {2006},

date = {2006-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {We demonstrate experimentally negative refraction of plane waves by a one-dimensional Photonic crystal (PhC) prism at microwave frequencies. The prism was fabricated from a microwave alumina PhC acting as a left-handed metamaterial. Focusing of plane waves from a one-dimensional PhC plano-concave lens is predicted theoretically and realized experimentally. The focal point is observed to move with the radius of curvature of the lens. The measured values of refractive index are in complete agreement with those determined from band structure calculations.},

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Vodo, Plarenta; Parimi, PV; Lu, WT; Sridhar, Srinivas

Focusing by planoconcave lens using negative refraction Journal Article

In: Applied Physics Letters, vol. 86, no. 20, pp. 201108, 2005.

@article{vodo2005focusing,

title = {Focusing by planoconcave lens using negative refraction},

author = {Plarenta Vodo and PV Parimi and WT Lu and Srinivas Sridhar},

year = {2005},

date = {2005-01-01},

journal = {Applied Physics Letters},

volume = {86},

number = {20},

pages = {201108},

publisher = {American Institute of Physics},

abstract = {We demonstrate focusing of a plane microwave by a planoconcave lens fabricated from a photonic crystal having a negative refractive index and left-handed electromagnetic properties. An inverse experiment, in which a plane wave is produced from a source placed at the focal point of the lens, is also reported. A frequency-dependent negative refractive index, n(ω) < 0 is obtained for the lens from the experimental data which match well with that determined from band structure calculations.

},

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Lu, Wentao T; Sridhar, Srinivas

Flat lens without optical axis: Theory of imaging Journal Article

In: Optics express, vol. 13, no. 26, pp. 10673–10680, 2005.

@article{lu2005flat,

title = {Flat lens without optical axis: Theory of imaging},

author = {Wentao T Lu and Srinivas Sridhar},

year = {2005},

date = {2005-01-01},

journal = {Optics express},

volume = {13},

number = {26},

pages = {10673--10680},

publisher = {Optical Society of America},

abstract = {We derive a general theory for imaging by a flat lens without optical axis. We show that the condition for imaging requires a material having elliptic dispersion relations with negative group refraction. This medium is characterized by two intrinsic parameters σ and κ. Imaging can be achieved with both negative and positive wave vector refraction if σ is a positive constant. The Veselago-Pendry lens is a special case with σ = 1 and κ = 0. A general law of refraction for anisotropic media is revealed. Realizations of the imaging conditions using anisotropic media and inhomogeneous media, particularly photonic crystals, are discussed. Numerical examples of imaging and requirements for sub-wavelength imaging are also presented.

},

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Gennaro, E Di; Parimi, PV; Lu, WT; Sridhar, S; Derov, JS; Turchinetz, B

Slow microwaves in left-handed materials Journal Article

In: Physical Review B, vol. 72, no. 3, pp. 033110, 2005.

@article{di2005slow,

title = {Slow microwaves in left-handed materials},

author = {E Di Gennaro and PV Parimi and WT Lu and S Sridhar and JS Derov and B Turchinetz},

year = {2005},

date = {2005-01-01},

journal = {Physical Review B},

volume = {72},

number = {3},

pages = {033110},

publisher = {APS},

abstract = {Remarkably slow propagation of microwaves in two different classes of left-handed materials LHM�s is reported from microwave-pulse and continuous-wave transmission measurements. Microwave dispersion in a composite LHM made of split-ring resonators and wire strips reveals group velocity vg ~ c/50, where c is the free-space light velocity. Photonic crystals PhCs made of dielectric Al2O3 rods reveal vg ~ c/10. Group delay dispersion of both the composite LHM and PhCs determined from the experiment is in complete agreement with that obtained from theory. The slow group velocities are quantitatively described by the strong dispersion observed in these materials.

},

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Sridhar, S; Parimi, PV; Lu, WT; Vodo, P; Derov, John S

Imaging and negative refraction in left-handed metamaterials Inproceedings

In: Metamaterials, pp. 595505, International Society for Optics and Photonics 2005.

@inproceedings{sridhar2005imaging,

title = {Imaging and negative refraction in left-handed metamaterials},

author = {S Sridhar and PV Parimi and WT Lu and P Vodo and John S Derov},

year = {2005},

date = {2005-01-01},

booktitle = {Metamaterials},

volume = {5955},

pages = {595505},

organization = {International Society for Optics and Photonics},

abstract = {Negative refraction and left-handed electromagnetism in a photonic crystal are demonstrated in waveguide and free space experiments at microwave frequencies. Precision control to achieve tailor-made refractive indices has been achieved. The negative refraction in these photonic crystals is shown to lead to imaging by a flat lens. We have also developed a generalized theory of flat lens imaging. These results promise potential applications in a variety of optical and microwave systems for communications and imaging.},

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Parimi, PV; Lu, WT; Vodo, P; Sokoloff, J; Derov, JS; Sridhar, S

Negative refraction and left-handed electromagnetism in microwave photonic crystals Journal Article

In: Physical review letters, vol. 92, no. 12, pp. 127401, 2004.

@article{parimi2004negative,

title = {Negative refraction and left-handed electromagnetism in microwave photonic crystals},

author = {PV Parimi and WT Lu and P Vodo and J Sokoloff and JS Derov and S Sridhar},

year = {2004},

date = {2004-01-01},

journal = {Physical review letters},

volume = {92},

number = {12},

pages = {127401},

publisher = {APS},

abstract = {We demonstrate the negative refraction of microwaves in a metallic photonic crystal prism. The spectral response of the photonic crystal prism, which manifests both positive and negative refraction, is in complete agreement with band-structure calculations and numerical simulations. The validity of Snell’s law with a negative refractive index is confirmed experimentally and theoretically. The negative refraction observed corresponds to left-handed electromagnetism that arises due to the dispersion characteristics of waves in a periodic medium. This mechanism for negative refraction is different from that in metamaterials.},

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Vodo, Plarenta; Parimi, Patanjali V; Lu, Wentao T; Gennaro, Emiliano Di; Sridhar, Srinivas

Negative Refraction experiments in Photonic Crystal prisms Journal Article

In: APS, vol. 2004, pp. S19–008, 2004.

@article{vodo2004negative,

title = {Negative Refraction experiments in Photonic Crystal prisms},

author = {Plarenta Vodo and Patanjali V Parimi and Wentao T Lu and Emiliano Di Gennaro and Srinivas Sridhar},

year = {2004},

date = {2004-01-01},

journal = {APS},

volume = {2004},

pages = {S19--008},

abstract = {We have experimentally demonstrated negative refraction in metallic photonic crystal (PC) prisms [1]. The refracted fields in the parallel plate waveguide (PPW) are measured by an automated dipole antenna, which scans the desired area, while the free space (FS) measurements, performed in a anechoic chamber, are measured by a rectangular X-band horn that swings in an arc in far field area. Both TE and TM excitation modes are used in FS experiments. Numerical calculations of the band structure and equi-frequency surface simulations are used to determine frequency regions of negative refraction of the triangular lattice PC. Angle of refraction determined by theoretical simulations and experimental results, are in exceptional good agreement, yielding the negative refraction index. FS and PPW refraction experimental results agree remarkably with simulations. 1. "Negative Refraction and Left-handed electromagnetism in Microwave Photonic Crystals", P.V Parimi, W.T Lu, P.Vodo J. Sokoloff and S.Sridhar, cond-mat/0306109 (2003)},

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Lu, WT; Sokoloff, JB; Sridhar, S

Refraction of electromagnetic energy for wave packets incident on a negative-index medium is always negative Journal Article

In: Physical Review E, vol. 69, no. 2, pp. 026604, 2004.

@article{lu2004refraction,

title = {Refraction of electromagnetic energy for wave packets incident on a negative-index medium is always negative},

author = {WT Lu and JB Sokoloff and S Sridhar},

year = {2004},

date = {2004-01-01},

journal = {Physical Review E},

volume = {69},

number = {2},

pages = {026604},

publisher = {APS},

abstract = {We analyze refraction of electromagnetic wave packets on passing from an isotropic positive to an isotropic negative-refractive-index medium. We definitively show that in all cases the energy is always refracted negatively. For localized wave packets, the group refraction is also always negative},

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Parimi, Patanjali; Vodo, Plarenta; Wentao, Lu; di Gennaro, Emiliano; Sridhar, Srinivas

Image formation by and wave propagation in a photonic crystal Inproceedings

In: APS Meeting Abstracts, 2004.

@inproceedings{parimi2004image,

title = {Image formation by and wave propagation in a photonic crystal},

author = {Patanjali Parimi and Plarenta Vodo and Lu Wentao and Emiliano di Gennaro and Srinivas Sridhar},

year = {2004},

date = {2004-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {Negative refraction and imaging by a flat slab of a material are two of the important consequences of lefthanded electromagnetism. In our recent work we have demonstrated negative refraction and imaging by photonic crystals in the microwave frequency range [1]. The details of image formation are intriguing and urge its investigation. We have carried out microwave measurements in a parallel plate waveguide made of a pair of metallic plates. The Photonic crystal is made of alumina rods arranged on a square lattice such that the electric field is parallel to the axis of the rods. The detector is a dipole antenna which is inserted into the waveguide from outside. HP 8510C network analyzer is used to measure the complex transmission coefficient. The intensity maps of vs. probe position are obtained by scanning the probe using an xy robot, both inside and outside the crystal. The results suggest Bloch wave propagation …},

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Vodo, P; Parimi, PV; Lu, WT; Sridhar, S; Wing, R

Microwave photonic crystal with tailor-made negative refractive index Journal Article

In: Applied physics letters, vol. 85, no. 10, pp. 1858–1860, 2004.

@article{vodo2004microwave,

title = {Microwave photonic crystal with tailor-made negative refractive index},

author = {P Vodo and PV Parimi and WT Lu and S Sridhar and R Wing},

year = {2004},

date = {2004-01-01},

journal = {Applied physics letters},

volume = {85},

number = {10},

pages = {1858--1860},

publisher = {American Institute of Physics},

abstract = {Negative refraction and left-handed electromagnetism in a metallic photonic crystal (PC) wedge are demonstrated in free space for both transverse magnetic and electric mode propagation. The experimental results are in excellent agreement with numerical calculations based on the band structure with no fit parameters used in modeling. The results demonstrate precision control on the design and fabrication of the PC to achieve tailor-made refractive indices between −0.6 and +1.},

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Gennaro, Emiliano Di; Parimi, Patanjali V; Vodo, Plarenta; Lu, Wentao; Sridhar, Srinivas

Microwave pulse propagation measurements in left-handed materials Inproceedings

In: APS Meeting Abstracts, 2004.

@inproceedings{di2004microwave,

title = {Microwave pulse propagation measurements in left-handed materials},

author = {Emiliano Di Gennaro and Patanjali V Parimi and Plarenta Vodo and Wentao Lu and Srinivas Sridhar},

year = {2004},

date = {2004-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {Left handed electromagnetism is well established in media with negative permeability and permittivity and in photonic crystals [1]. In such media the negative refractive index is accompanied by large dispersion dn/dω, and consequently a very low group velocity is predicted for left-handed metamaterial (LHM). It is well known that an artificial material consisting of interleaved arrays of wires and split ring resonators in a certain microwave frequency range shows left handed behavior. We have carried out pulse measurements on LHM using a transition analyzer in order to measure the group velocity. Time delay measurements are performed in an X-band and parallel plate waveguide. Sending a 100ns width pulse with a carrier frequency ranging between 9 and 11 GHz, we analyze the signal delay due to the sample. The results suggest that the group velocity in the LHM is very low. Pulse delay measurements in photonic crystals are also discussed. Work supported by NSF & AFRL [1]. P. V. Parimi et al., submitted (2003).},

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Vodo, P; Parimi, V; Lu, WT; Sridhar, S

Strongly Modulated Photonic Crystal with Tailor Made Negative Refractive Index Technical Report

NORTHEASTERN UNIV BOSTON MA 2004.

@techreport{vodo2004strongly,

title = {Strongly Modulated Photonic Crystal with Tailor Made Negative Refractive Index},

author = {P Vodo and V Parimi and WT Lu and S Sridhar},

year = {2004},

date = {2004-01-01},

institution = {NORTHEASTERN UNIV BOSTON MA},

abstract = {Negative refraction and left-handed electromagnetism in a metallic photonic crystal (PC) wedge are demonstrated in free space for both transverse magnetic and electric mode propagation. The experimental results are in excellent agreement with numerical calculations based on the band structure with no fit parameters used in modeling. The results demonstrate precision control on the design and fabrication of the PC to achieve tailor-made refractive indices between-0.6 to+ 1.},

keywords = {},

pubstate = {published},

tppubtype = {techreport}

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Sridhar, Srinivas

Imaging by Flat Lens Using Negative Refraction in Microwave Photonic Crystals Inproceedings

In: APS March Meeting Abstracts, 2004.

@inproceedings{sridhar2004imaging,

title = {Imaging by Flat Lens Using Negative Refraction in Microwave Photonic Crystals},

author = {Srinivas Sridhar},

year = {2004},

date = {2004-01-01},

booktitle = {APS March Meeting Abstracts},

abstract = {Metamaterials with negative indices of refraction exhibit novel aspects of electromagnetic wave propagation. Here we report the observation of negative refraction (NR) at microwave frequencies in metallic and dielectric photonic crystal (PC) prisms. The NR observed in the PC corresponds to left-handed electromagnetism and arises due to the dispersion characteristics of waves in a periodic medium. This mechanism for NR is different from that in metamaterials where locally both the dielectric permittivity and magnetic permeability are negative. The PC prism exhibits both positive and negative refraction in different spectral windows, and experimental results are in complete agreement with band-structure calculations and numerical simulations. A remarkable consequence of NR noted by Veselago is that of focusing by a flat lens. We have demonstrated imaging of a microwave point source by a flat lens fabricated from a PC. We also demonstrate a unique feature of a flat lens, the absence of an optical axis, by moving the object and showing that the image displaces correspondingly. The PC flat lens has some interesting similarities and differences compared with the Veselago lens. Potential applications of negative refraction and flat lens imaging at microwave and optical frequencies, including sub-wavelength image resolution suggested by Pendry, will be described. Collaborators: P.Parimi, W.T.Lu, P.Vodo, J.S.Derov. Supported by National Science Foundation and the Air Force Research Laboratories, Hanscom. 1. "Imaging by flat lens using negative refraction", P.Parimi, W.T.Lu, P.Vodo and S.Sridhar, Nature, v.426, 404 (2003). 2. "Negative refraction and left-handed electromagnetism in metallic microwave photonic crystals", P. Parimi, W.T. Lu, P. Vodo, J. Sokoloff, J.S.Derov and S. Sridhar, cond-mat/0306109.},

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Parimi, Patanjali V; Lu, Wentao T; Vodo, Plarenta; Sridhar, Srinivas

Imaging by flat lens using negative refraction Journal Article

In: Nature, vol. 426, no. 6965, pp. 404–404, 2003.

@article{parimi2003imaging,

title = {Imaging by flat lens using negative refraction},

author = {Patanjali V Parimi and Wentao T Lu and Plarenta Vodo and Srinivas Sridhar},

year = {2003},

date = {2003-01-01},

journal = {Nature},

volume = {426},

number = {6965},

pages = {404--404},

publisher = {Nature Publishing Group},

abstract = {Metamaterials with negative indices of refraction exhibit novel aspects of electromagnetic wave propagation. Here we report the observation of negative refraction (NR) at microwave frequencies in metallic and dielectric photonic crystal (PC) prisms. The NR observed in the PC corresponds to left-handed electromagnetism and arises due to the dispersion characteristics of waves in a periodic medium. This mechanism for NR is different from that in metamaterials where locally both the dielectric permittivity and magnetic permeability are negative. The PC prism exhibits both positive and negative refraction in different spectral windows, and experimental results are in complete agreement with band-structure calculations and numerical simulations. A remarkable consequence of NR noted by Veselago is that of focusing by a flat lens. We have demonstrated imaging of a microwave point source by a flat lens fabricated from a PC. We also demonstrate a unique feature of a flat lens, the absence of an optical axis, by moving the object and showing that the image displaces correspondingly. The PC flat lens has some interesting similarities and differences compared with the Veselago lens. Potential applications of negative refraction and flat lens imaging at microwave and optical frequencies, including sub-wavelength image resolution suggested by Pendry, will be described. Collaborators: P.Parimi, W.T.Lu, P.Vodo, J.S.Derov. Supported by National Science Foundation and the Air Force Research Laboratories, Hanscom. 1. "Imaging by flat lens using negative refraction", P.Parimi, W.T.Lu, P.Vodo and S.Sridhar, Nature, v.426, 404 (2003). 2. "Negative refraction and left-handed electromagnetism in metallic microwave photonic crystals", P. Parimi, W.T. Lu, P. Vodo, J. Sokoloff, J.S.Derov and S. Sridhar, cond-mat/0306109.},

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Lu, WT; Sokoloff, JB; Sridhar, S

Inproceedings

In: APS Meeting Abstracts, 2003.

@inproceedings{lu2003perfect,

author = {WT Lu and JB Sokoloff and S Sridhar},

year = {2003},

date = {2003-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {We study analytically and numerically imaging by Â" perfect lensÂ" made of negative index medium (NIM)(or left-handed meta-material (LHM)). For far field, the perfect lens is only perfect for n=-1. Away from this value, aberration and caustics will be present. For near field imaging with Re ε=-1, the consequences of departure from dielectric matching and losses are discussed. Work supported by NSF-0098801 and AFRL, Hanscom.},

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Lu, WT; Sridhar, S

Near-field imaging by negative permittivity media Journal Article

In: Microwave and Optical Technology Letters, vol. 39, no. 4, pp. 282–286, 2003.

@article{lu2003near,

title = {Near-field imaging by negative permittivity media},

author = {WT Lu and S Sridhar},

year = {2003},

date = {2003-01-01},

journal = {Microwave and Optical Technology Letters},

volume = {39},

number = {4},

pages = {282--286},

publisher = {Wiley Online Library},

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pubstate = {published},

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Sokoloff, JB; Lu, WT; Sridhar, S

Refraction of Electromagnetic Energy for Wave Packets Incident on a Negative Index Medium: Always Negative Inproceedings

In: APS Meeting Abstracts, 2003.

@inproceedings{sokoloff2003refraction,

title = {Refraction of Electromagnetic Energy for Wave Packets Incident on a Negative Index Medium: Always Negative},

author = {JB Sokoloff and WT Lu and S Sridhar},

year = {2003},

date = {2003-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {Since all physical radiation sources admit their radiation in the form of wave packets, we analyze refraction of electromagnetic wave packets on passing from an isotropic positive (PIM) to an isotropic negative index medium (NIM). We definitively show that in all cases the energy is always refracted negatively. For localized waves, the group refraction is always negative. We accomplish this by treating comprehensively group refraction at a PIM-NIM interface by analytical and numerical treatment of several exhaustive examples: localized wave packets, beams, and also a finite number of plane waves. We focus on physically important quantities such as the energy flow and momentum. In all of the cases we show that the energy and momentum refract negatively. We show that in all cases where the wave group does not extend to infinity in the perpendicular direction of the wave vector, the interference pattern also refracts negatively. Work supported by NSF-0098801 and AFRL, Hanscom.

},

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Parimi, Patanjali; Vodo, Plarenta; Lu, Wentao; Sridhar, Srinivas

Microwave Propagation in Negative Index and Artificial Dielectric Media Inproceedings

In: APS Meeting Abstracts, 2003.

@inproceedings{parimi2003microwave,

title = {Microwave Propagation in Negative Index and Artificial Dielectric Media},

author = {Patanjali Parimi and Plarenta Vodo and Wentao Lu and Srinivas Sridhar},

year = {2003},

date = {2003-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {Negative index media (NIM) are fabricated by interleaving arrays of split ring resonators and wire strips. Microwave (X-band) waveguide measurements on the NIM yield quantitative information on the material parameters, % tilden(ω)=n^' +in^' ', tildevarepsilon% (ω )=varepsilon ^' +ivarepsilon ^' ' and tilde% μ(ω )=μ ^' +iμ ^' ' . Typical NIM features such as passband in the NIM region expected from the theoretical analysis are observed in the measured data. The n^'(ω) determined from the waveguide transmission parameters (tildeS_11,tildeS_21) is found to vary from -4.8 to-0.3 in the passband region 9.6-10.5GHz. The results show that transmission is optimized for n^' (ω ) -1 and low n^' ' . A detailed investigation of several NIM materials suggests that the characteristic properties of the NIM are dependent on the length of the material, choice of the substrate material, and continuity in the wire strips. Artificial dielectric media fabricated with arrays of wire strips exhibit a characteristic microwave plasmon mode in the X-band region, below which varepsilon ^' (ω )<0. Work supported by the National Science Foundation and the Air Force Research Labs, Hanscom.},

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Kaso, A; Parimi, PV; Sridhar, S; Derov, JS

Microwave Transmission in Left-Handed Metamaterials Inproceedings

In: APS Meeting Abstracts, 2002.

@inproceedings{kaso2002microwave,

title = {Microwave Transmission in Left-Handed Metamaterials},

author = {A Kaso and PV Parimi and S Sridhar and JS Derov},

year = {2002},

date = {2002-01-01},

booktitle = {APS Meeting Abstracts},

abstract = {As has been already shown, a material with both ε and mu negative can sustain propagation waves. The permittivity ε and permeability mu should necessarily be frequency dependent for the causality principle to hold. We consider the dispersive expressions obtained by theoretical considerations from JB Pendry. We try to obtain some useful insight for pulse propagation, like group velocity v_g, the spreading of the wave packet in time in the disperssive medium. Transmission and reflectivity are computed for a normal incidence of the pulse on the slab. All the above situations can prove useful in the applications of this new metamaterial, such as in the delay lines where high values of the index of refraction are needed. particular attention is given to the frequency range where the material displays n=-1. Experiments are underway exploring the properties of LHM and the results will be discussed.},

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Lu, WT; Sokoloff, JB; Sridhar, S

Comment on Journal Article

In: arXiv preprint cond-mat/0207689, 2002.

@article{lu2002comment,

title = {Comment on},

author = {WT Lu and JB Sokoloff and S Sridhar},

year = {2002},

date = {2002-01-01},

journal = {arXiv preprint cond-mat/0207689},

abstract = {Valanju, Walser and Valanju (VWV) [1] have shown

that for a group consisting of two plane waves incident

on the interface between a material of positive refractive

index (PIM) and material of negative refractive index

(NIM), the group velocity refracts positively. Here we

show that this is true only for the special two plane wave

case constructed by VWV, but for generic localized wave

packets, the group refraction is generically negative.

The sum of two plane waves of wavevector and frequency (k1, ω1) and (k2, ω2), considered by VWV, can

be written as 2e

i(k0·r−ω0t)

cos[(1/2)(∆k · r − ∆ωt)],

where (k0, ω0) the average wavevector and frequency and

(∆k, ∆ω) denote their differences. Clearly, the argument

of the cosine is constant along planes, which propagate

in time along the direction of their normal, ∆k. We

have carried out numerical simulations of wave packets

incident on the PIM-NIM interface and for the 2-wave

case arrive at conclusions similar to VWV. For arbitrary

number of incident plane waves whose k vectors are all

parallel, the group refraction is again positive. Note that

in all these special cases the packet remains nonzero on

infinite planes.

Here we show that for any wave packet that is spatially

localized, the group refraction is generically negative.

For 3 (or more) waves whose wave vectors not aligned,

the group refraction will be negative. For example, consider three wave vectors in PIM in the x-z-plane, whose

magnitudes are, k − δk, k, k − δk and whose angles with

the z-axis are, θ − δθ, θ, θ + δθ, respectively. The dispersion k

2 = (ω

2 − ω

2

p

)(ω

2 − ω

2

b

)/c2

(ω

2 − ω

2

0

) were used for

NIM. The results are shown in Fig. 1. The wave packet

refracts negatively, in obvious contrast to VWV. As we

have seen, two plane waves result in a wave packet-like

structure which is constant along planes; the addition of

a third wave breaks the planes into localized wave packets

which refract negatively.

A packet constructed from a finite number of plane

waves will always give a collection of propagating wave

pulses, as seen in Fig. 1. A wave packet localized in one

region of space, as occurs in all experimental situations,

can only be constructed from a continuous distribution of

wavevectors. Consider such a wave packet incident from

outside the NIM, E = E0

R

d

2kf(k − k0)e

i(k·r−ω(k)t)

,

where ω(k) = ck. If f(k − k0) drops off rapidly as k

moves away from k0, ω(k) can be expanded in a Taylor

series to first order in k − k0 to a good approximation.

This gives, E = E0e

i(k0·r−ω(k0)t

g(r − ctk0/k0), where

g(R) = R

d

2kf(k−k0)e

i(k−k0)·R. Inside the NIM, k and

k0 in the argument of the exponent get replaced by kr

and kr0 which are related to k and k0 by Snell’s law.

Then the wave packet once it enters the NIM is given by

Er = E

′

0

e

i(kr0·r−ω(kr0)t

gr(r − vgrt), (1)

where gr(R) = R

d

2kf(k − k0)e

iR·[(k−k0)·∇kkr]

. Here

kr0 denotes kr evaluated at k = k0 and vgr = ∇kr ω(kr)

evaluated at kr = kr0. Thus, the refracted wave moves

with the group velocity vgr. Evaluation of Eq. (1) for

a Gaussian wave packet shows that the incident packet

gets distorted but the maximum of the packet moves at

vgr. For the case of an isotropic medium, considered by

VWV [1], the group velocity is anti-parallel to the wave

vector in the medium. Hence, the group velocity will be

refracted the same way as the wavevector is, contrary to

the claim of VWV [1].

Thus VWV’s statement that the “Group Refraction is

always positive” is true only for the very special (and

rare) wave packets constructed by them and is incorrect

for more general wave packets that are spatially localized.

This work was supported by the National Science

Foundation, the Air Force Research Laboratories and the

Department of Energy.

W. T. Lu, J. B. Sokoloff and S. Sridhar

Department of Physics, Northeastern University, 360

Huntington Avenue, Boston, MA 02115.},

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that for a group consisting of two plane waves incident

on the interface between a material of positive refractive

index (PIM) and material of negative refractive index

(NIM), the group velocity refracts positively. Here we

show that this is true only for the special two plane wave

case constructed by VWV, but for generic localized wave

packets, the group refraction is generically negative.

The sum of two plane waves of wavevector and frequency (k1, ω1) and (k2, ω2), considered by VWV, can

be written as 2e

i(k0·r−ω0t)

cos[(1/2)(∆k · r − ∆ωt)],

where (k0, ω0) the average wavevector and frequency and

(∆k, ∆ω) denote their differences. Clearly, the argument

of the cosine is constant along planes, which propagate

in time along the direction of their normal, ∆k. We

have carried out numerical simulations of wave packets

incident on the PIM-NIM interface and for the 2-wave

case arrive at conclusions similar to VWV. For arbitrary

number of incident plane waves whose k vectors are all

parallel, the group refraction is again positive. Note that

in all these special cases the packet remains nonzero on

infinite planes.

Here we show that for any wave packet that is spatially

localized, the group refraction is generically negative.

For 3 (or more) waves whose wave vectors not aligned,

the group refraction will be negative. For example, consider three wave vectors in PIM in the x-z-plane, whose

magnitudes are, k − δk, k, k − δk and whose angles with

the z-axis are, θ − δθ, θ, θ + δθ, respectively. The dispersion k

2 = (ω

2 − ω

2

p

)(ω

2 − ω

2

b

)/c2

(ω

2 − ω

2

0

) were used for

NIM. The results are shown in Fig. 1. The wave packet

refracts negatively, in obvious contrast to VWV. As we

have seen, two plane waves result in a wave packet-like

structure which is constant along planes; the addition of

a third wave breaks the planes into localized wave packets

which refract negatively.

A packet constructed from a finite number of plane

waves will always give a collection of propagating wave

pulses, as seen in Fig. 1. A wave packet localized in one

region of space, as occurs in all experimental situations,

can only be constructed from a continuous distribution of

wavevectors. Consider such a wave packet incident from

outside the NIM, E = E0

R

d

2kf(k − k0)e

i(k·r−ω(k)t)

,

where ω(k) = ck. If f(k − k0) drops off rapidly as k

moves away from k0, ω(k) can be expanded in a Taylor

series to first order in k − k0 to a good approximation.

This gives, E = E0e

i(k0·r−ω(k0)t

g(r − ctk0/k0), where

g(R) = R

d

2kf(k−k0)e

i(k−k0)·R. Inside the NIM, k and

k0 in the argument of the exponent get replaced by kr

and kr0 which are related to k and k0 by Snell’s law.

Then the wave packet once it enters the NIM is given by

Er = E

′

0

e

i(kr0·r−ω(kr0)t

gr(r − vgrt), (1)

where gr(R) = R

d

2kf(k − k0)e

iR·[(k−k0)·∇kkr]

. Here

kr0 denotes kr evaluated at k = k0 and vgr = ∇kr ω(kr)

evaluated at kr = kr0. Thus, the refracted wave moves

with the group velocity vgr. Evaluation of Eq. (1) for

a Gaussian wave packet shows that the incident packet

gets distorted but the maximum of the packet moves at

vgr. For the case of an isotropic medium, considered by

VWV [1], the group velocity is anti-parallel to the wave

vector in the medium. Hence, the group velocity will be

refracted the same way as the wavevector is, contrary to

the claim of VWV [1].

Thus VWV’s statement that the “Group Refraction is

always positive” is true only for the very special (and

rare) wave packets constructed by them and is incorrect

for more general wave packets that are spatially localized.

This work was supported by the National Science

Foundation, the Air Force Research Laboratories and the

Department of Energy.

W. T. Lu, J. B. Sokoloff and S. Sridhar

Department of Physics, Northeastern University, 360

Huntington Avenue, Boston, MA 02115.

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