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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 9 — Apr. 26, 2010
  • pp: 9026–9033
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More studies on Metamaterials Mimicking de Sitter space

Miao Li, Rong-Xin Miao, and Yi Pang  »View Author Affiliations


Optics Express, Vol. 18, Issue 9, pp. 9026-9033 (2010)
http://dx.doi.org/10.1364/OE.18.009026


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Abstract

We estimate the dominating frequencies contributing to the Casimir energy in a cavity of meta- materials mimicking de Sitter space, by solving the eigenvalue problem of Maxwell equations. It turns out the dominating frequencies are the inverse of the size of the cavity, and the degeneracy of these frequencies also explains our previous result on the unusually large Casimir energy. Our result suggests that carrying out the experiment in laboratory is possible theoretically.

© 2010 Optical Society of America

1. Introduction

The Casimir energy is one of the important predictions in quantum field theory and continues to be source of inspiration for theoretical as well as experimental work [1

1. M. Bordag, The Casimir Effects 50 years later, (World Scientific Press, 1999).

]. It is the regularized difference between two energies: one is the zero point energy of the electromagnetic field in a finite cavity and the other is that in an infinite background. Generically, the expression of the Casimir energy depends on the details of the cavity including properties of its bulk and its boundary. The earliest work [2

2. H. B. G. Casimir, “On the Attraction Between Two Perfectly Conducting Plates,” Indag. Math. 10, 261 (1948).

] on the Casimir energy and the following up studies [3–11

3. T. H. Boyer, ”Quantum electromagnetic zero point energy of a conducting spherical shell and the Casimir model for a charged particle,” Phys. Rev. 174, 1764–1776 (1968). [CrossRef]

] all reported a result that the energy density of the Casimir energy is inversely proportional to the forth power of the typical size of the cavity. Applying this result to the universe, this kind of the Casimir energy can not be taken as a possible origin of dark energy, since 1/L 4 is too small compared with the observed dark energy density [12

12. A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiattia, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry, ”Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant,” Astron. J. 116, 1009–1038 (1998). [CrossRef]

, 13

13. S. Perlmuttter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, and W. J. Couch, ”Measurements of Omega and Lambda from 42 High-Redshift Supernovae,” Astrophys.J. 517, 565–586 (1999).

] if L is chosen to be the a typical size of universe.

Apparently, it seems difficult to carry out such an experiment, because the Casimir energy is expressed as a sum of all the frequencies, meaning that each frequency contributes a part to this energy, while metamaterials have frequency dispersion, in other words, the designed permittivity and permeability is effective only to frequencies in certain brand. However, this difficulty can be circumvented as we will uncover a fact that there is a typical frequency whose contribution to Casimir energy is dominating. Thus a cavity of metamaterials effective at this typical frequency is sufficient to mimic de Sitter space and induces the Casimir energy predicted by theoretical calculation.

The rest of this paper is organized as follows. In Sec. 2, we uncover the fact that the Casimir energy has a typical contributing frequency.We estimate the typical frequency in metamaterials mimicking de Sitter in Sec. 3. We conclude in Sec. 4.

2. General discussion on the typical frequency of Casimir energy

Physically, it is conceivable that there is a typical contributing frequency to the Casimir energy. According to its definition, the Casimir energy measures the difference between vacuum energy in a finite cavity and that in infinite background. The former is usually contributed by a discrete spectrum and the latter comes from a continuous one. Thus the Casimir measures the difference between discrete and continues ones. When frequency is large, the discrete spectrum approaches a continues one, and their contributions cancel with each other; for small frequencies, their contribution is also negligible. Since the contribution from very large and very small frequencies is tiny, there should be some intermediate scale at which the difference between discrete spectrum and continuous spectrum is maximum. Then the frequency at this scale is the typical frequency.

As a heuristic example, we read off this typical frequency from the process of computing Casimir energy in the static Einstein’s universe. In this case, the Casimir energy is given by [31

31. L. H. Ford, “Quantum Vacuum Energy In General Relativity,” Phys. Rev. D 11, 3370–3377 (1975). [CrossRef]

]

Ec=12a0n=0n3a0320ω3dω,
(1)

where the discrete spectrum consists of ω = n/a 0 with degeneracy n 2, and a 0 is the radius of Einstein universe. Reparameterizing ω by t/a 0, then using Abel-Plana formula

n=0F(n)0dtF(t)=12F(0)+i0dtF(it)F(it)e2πt1,
(2)

Eq.(1) is equal to

EC=1a00dtt3e2πt1.
(3)

3. Typical frequency of Casimir energy in metamaterial mimicking de Sitter

The metamaterials mimicking de Sitter space is designed with the following permittivity and permeability [14

14. M. Li, R. X. Miao, and Y. Pang, “Casimir Energy, Holographic Dark Energy and Electromagnetic Metamaterial Mimicking de Sitter,” arXiv:0910.3375 [hep-th].

]

εr˜r˜=μr˜r˜=L2sin2(r˜/L)cos(r˜/L)sinθ,εθθ=μθθ=sinθcos(r˜/L),εφφ=μφφ=1cos(r˜/L)sinθ.
(4)

where (, θ, φ) denote the spherical coordinates. In terms of the Cartesian coordinates

εij=μij=1cos(r˜/L)(δij(L2r˜2sin2(r˜/L)1)xixjr˜2).
(5)

The event horizon at = π L/2 now becomes the boundary of a cavity of metamaterials.

The Maxwell equations in inhomogeneous medium are

iEi=0,iHi=0,
(6)
tEiεijkγjHk=0,tHi+εijkγjEk=0,
(7)

Eθ|r=Ld=Eφr=Ld=0.
(8)

These boundary conditions are acceptable physically, since the the photons emitted from the center of de Sitter space will travel an infinite amount of time to arrive at the horizon or they can never reach there as seen by any static observer.

To solve Maxwell equations we adopt Newman-Penrose formalism [32

32. E. Newman and R. Penrose, “An Approach to gravitational radiation by a method of spin coefficients,” J. Math. Phys. 3, 566–578 (1962). [CrossRef]

]. That is to use four null vectors reexpress the Maxwell tensor F μν as

Fμν=2[ϕ1(n[μlν]+m[μmv]*)+ϕ2l[μmν]+ϕ0m[μ*nν]]+c.c,
(9)

where “[]” denotes the antisymmetrization, and “c.c” means the complex conjugate. The convention about ϕs and the null vectors is given by

ϕ0=Fμνlμmν,ϕ1=12Fμν(lμnν+m*μmν),ϕ2=Fμνm*μnν,
(10)

with

lμ=(11r2/L2,1,0,0),mμ=12r(0,0,1,isinθ)
nμ=(12,1r2/L22,0,0),m*μ=12r(0,0,1,isinθ).
(11)

To solve Maxwell equations conveniently, we have adopted a coordinate system different from that appearing in Eq. (4), but we will transform back to the old coordinates when finding the typical frequency. Then after some standard steps [33

33. S. A. Teukolsky, “Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations,” Astrophys. J. 185, 635–648 (1973). [CrossRef]

], we obtain

ϕ1=eiwtYlm(θ,φ)R(r),
(12)

where Yml (θ ,φ) is the spherical harmonic function satisfies the following equation

[1sinθθ(sinθθ)+φ2sin2θ+l(l+1)]Ylm(θ,φ)=0,
(13)

and R(r) satisfies

[r2(1r2L2)r2+4r(13r22L2)r+r2w2(1r2L2)6r2L2+2l(l+1)]R(r)=0.
(14)

This equation possesses two independent solutions and the one of physical interest is

R(r)=r2(1+Lr)l1(1+Lr)iwL/2(1Lr)iwL/2F(l+1,l+1+iwL,2l+2,2rL+r),
(15)

it is regular at r = 0 and goes back to flat space result jl(wr)/r when L → ∞. The solution of ϕ0 and ϕ2 can also be found, but to find out ω solving ϕ1 is enough.

Recall that in an empty spherical cavity the two independent electromagnetic modes are TE and TM corresponding to Er = 0 and Hr=0 respectively. Here the situation is similar. In the TE modes, Er = 0 the electrowave is transverse. From the definition of ϕs Eq. (10), we read ϕ 1 = -½(Er + iHr) = -i2 Hr. Combining the following Maxwell equations and boundary conditions Eq. (8)

iwHr=1r2sinθ(θEφφEθ),Eθ|r=Ld=Eφ|r=Ld=0,
(16)

We deduce that

ϕ1|r=Ld=0.
(17)

Since dL, L-d ~ L, the boundary conditions can be imposed on the rL behavior of ϕ 1. When rL, the radial part of ϕ 1 has the following asymptotic form

R(r)~Γ(iwL)Γ(l+1iwL)(1r/L1+r/L)iwL/2+c.c.
(18)

Then Eq. (17) requires that

Re(Γ(iwL)Γ(l+1iwL)(1r/L1+r/L)iwL/2)|r=Ld=0,
(19)

In the coordinate system Eq. (5), the cut-off d is imposed on the physical radial coordinate (4), the cut-off on r is therefore d 2/(2L), the above condition amounts to

Re(Γ(iwL)Γ(l+1iwL)(1r/L1+r/L)iwL/2)|r=Ld2/2L=0
(20)

which determines value of ω. For small l, we can solve Eq. (20) to pick out the lowest ω when ln(4L 2/d 2)≫1 (This is guaranteed by the fact that in usual metamaterials d is nanometer, and L is 1cm) and the other corresponds to very large n. The results are exhibited below.

  1. l = 0. We find that (20) leads to

    sin[ωL2In(4L2/d2)]=0ω=2nπLIn(4L2/d2),n=1,2.
    (21)

  2. l = 1. Then (20) implies that

    ωL=tan(ωL2In(4L2/d2)).

Since ln(4L 2/d 2)≫1, the lowest ω ≈ 2π/Lln(4L 2/d 2).

Based on the above results, we infer that the Casimir energy from the frequencies corresponding to small l is of the order 1/Lln(4L 2/d 2) which is much smaller than L/d 2 predicted in [14

14. M. Li, R. X. Miao, and Y. Pang, “Casimir Energy, Holographic Dark Energy and Electromagnetic Metamaterial Mimicking de Sitter,” arXiv:0910.3375 [hep-th].

]. So the typical frequency cannot be around 1/Lln(4L 2/d 2). To estimate the typical frequency, we observe that for l ≫ 1 the Stirling formula can be used to reexpress Eq. (20) as

Re[Γ(iωL)(dl/2L)iωL]=0.
(23)

We see that a critical l denoted by lc emerges at lc = 2L/d (Indeed lc ≫ 1, so our estimation is reasonable). For llc, the term (dl/2L)iωL becomes highly oscillating, and the ω satisfying Eq. (23) approaches a continuous distribution whose effect is canceled by that from the infinite background. For llc, this case is just what we discussed before, their contribution is subleading. Therefore the dominating contribution to the Casimir energy can only come from l ~ lc. When l ~ lc, the corresponding frequency is around 1/L since in this case the constant appearing Eq. (23) is of order 1. Go one step further, we estimate the contribution to the Casimir energy from frequencies corresponding to l ~ lc. For lc/2<l < 3lc/2, for each l the degeneracy is 2l+1,

Ec~lc/23lc/2(2l+1)1/L~lc2/L=L/d2.
(24)

This result is the same order as we obtained in [14

14. M. Li, R. X. Miao, and Y. Pang, “Casimir Energy, Holographic Dark Energy and Electromagnetic Metamaterial Mimicking de Sitter,” arXiv:0910.3375 [hep-th].

] by a different method. It is a strong support to our estimation about the typical frequency.

In the TM modes, the magnetic wave is transverse

Hr=0,ϕ1=12Er.
(25)

Combined with the Gauss law and the boundary conditions Eq. (8)

[r(r2sinθEr)+sinθ1r2L2θEθ+1(1r2L2)sinθφEφ]|r=Ld2/2L=0
(26)

We deduce that

After some simplification, this condition is transformed to

r(r2ϕ1)|r=Ld2/2L=r(r2Er)|r=Ld2/2L=0.
(27)
Im(Γ(iwL)Γ(l+1iwL)(1r/L1+r/L)iwL/2)|r=Ld2/2L=0.
(28)

Then the process of finding frequency ω satisfying above condition is the same as before and we list the results as follows

  1. l = 0 the frequency is given by

    cos[ωL2In(4L2/d2)]=0ω=(2n+1)πLIn(4L2/d2),n=1,2.
    (29)

  2. l = 1.

    ωL=cot(ωL2ln(4L2/d2)).
    (30)

  3. For ln(2L/d)≫1, the lowest frequency is

    ωπ/LIn(4L2/d2).
    (31)

  4. The typical frequency can be read from

    Im[Γ(iωL)(dl/2L)iωL]=0.
    (32)

Since the structure of Eq. (32) and Eq. (23) is similar, the typical frequency is still of the order 1/L with the critical lc given by 2L/d.

We emphasize that to estimate the typical frequency we have assumed that the contribution from ω with large radial quantum number n is suppressed exponentially. As a check of this assumption, we present the expression of ω in large n limit. Both TE and TM modes have the following frequencies

ωnπLIn(4L2/d2).
(33)

Thus for n is large, ω grow with n linearly and their effects will be suppressed by the black body factor.

To end this section, we propose that the measurable quantity for such experiments would be the Casimir force. From Eq. (24), we get

Fc~1d2.
(34)

ε=μ~Ld.
(35)

where d is some microcosmic cutoff to keep the permittivity and the permeability finite but relatively large.

4. Conclusion

We estimate the typical frequency of the vacuum fluctuations in metamaterials mimicking de Sitter and find it proportional to 1/L, the size of the cavity. Assuming d be 1 nanometer, and L be 1cm, then the typical wavelength is about 1cm.

With our estimation of the typical frequency and the typical angular quantum number, we also have an intuitive understanding of our Casimir energy formula.

We hope that one day the metamaterials suggested by us can be made with appropriate size and effective for the corresponding typical frequency, then the predicted brand new Casimir force can be measured, this experiment is important for study cosmology in laboratory.

Acknowledgments

We would like to thank Prof. Mo Lin Ge for informing us of the exciting developments in the field of electromagnetic cloaking and metamaterials. This work was supported by the NSFC grant No.10535060/A050207, a NSFC group grant No.10821504 and Ministry of Science and Technology 973 program under grant No.2007CB815401.

References and links

1.

M. Bordag, The Casimir Effects 50 years later, (World Scientific Press, 1999).

2.

H. B. G. Casimir, “On the Attraction Between Two Perfectly Conducting Plates,” Indag. Math. 10, 261 (1948).

3.

T. H. Boyer, ”Quantum electromagnetic zero point energy of a conducting spherical shell and the Casimir model for a charged particle,” Phys. Rev. 174, 1764–1776 (1968). [CrossRef]

4.

R. Balian and B. Duplantier, “ElectromagneticWaves Near Perfect Conductors. 2. Casimir Effect,” Annals Phys. 112, 165–208 (1978). [CrossRef]

5.

K. A. Milton, L. L. DeRaad, and J. S. Schwinger, “Casimir Selfstress On A Perfectly Conducting Spherical Shell,” Annals Phys. 115, 388–403 (1978). [CrossRef]

6.

G. Plunien, B. Muller, and W. Greiner, “The Casimir Effect,” Phys. Rept. 134, 87–193 (1986). [CrossRef]

7.

C. M. Bender and P. Hays, “Zero Point Energy Of Fields In A Finite Volume,” Phys. Rev. D 14, 2622–2632 (1976). [CrossRef]

8.

K. A. Milton, “Semiclassical Electron Models: Casimir Selfstress In Dielectric And Conducting Balls,” Ann. Phys. 127, 49–61 (1980). [CrossRef]

9.

K. A. Milton, “Fermionic Casimir Stress On A Spherical Bag,” Annals Phys. 150, 432–438 (1983). [CrossRef]

10.

M. Bordag, E. Elizalde, K. Kirsten, and S. Leseduarte, “Casimir energies for massive fields in the bag,” Phys. Rev. D 56, 4896–4904 (1997). [CrossRef]

11.

S. D. Odintsov, “Vilkovisky effective action in quantum gravity with matter,” Theor. Math. Phys. 82, 45–51 (1990). [CrossRef]

12.

A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiattia, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry, ”Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant,” Astron. J. 116, 1009–1038 (1998). [CrossRef]

13.

S. Perlmuttter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, and W. J. Couch, ”Measurements of Omega and Lambda from 42 High-Redshift Supernovae,” Astrophys.J. 517, 565–586 (1999).

14.

M. Li, R. X. Miao, and Y. Pang, “Casimir Energy, Holographic Dark Energy and Electromagnetic Metamaterial Mimicking de Sitter,” arXiv:0910.3375 [hep-th].

15.

Miao Li, “A Model of holographic dark energy,” Phys. Lett. B 603, 1–5 (2004). [CrossRef]

16.

Qing-Guo Huang and Miao Li, “The Holographic Dark Energy in a Non-flat Universe,” J. Cosmol. Astropart. Phys. 0408, 013 (2004). [CrossRef]

17.

J. Plebanski, “Electromagnetic Waves in Gravitational Fields,” Phys. Rev. 118, 1396–1408 (1959). [CrossRef]

18.

U. Leonhardt and T. Philbin, “General Relativity in Electrical Engineering”, New J.Phys. 8, 247 (2006). [CrossRef]

19.

J. B. Pendry, D. Schurig, and D. R. Smith, ”Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

20.

K. Niu, C. Song, and M. L. Ge, ”The geodesic form of light-ray trace in the inhomogeneous media,” Opt. Express 17(14), 11753–11767 (2009). [CrossRef] [PubMed]

21.

R. A. Shelby, D. R. Smith, and S. Shultz, ”Experimental Verification of a Negative Index of Refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

22.

A. A. Houck, J. B. Brock, and I. L. Chuang, ”Experimental Observations of a Left-Handed Material That Obeys Snell’s Law,” Phys. Rev. Lett. 90, 137401 (2003). [CrossRef] [PubMed]

23.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, ”Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

24.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, ”Negative refraction by photonic crystals,” Nature 423, 604–605 (2003). [CrossRef] [PubMed]

25.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, ”Terahertz Magnetic Response from Artificial Materials,” Science 303, 1494–1496 (2004). [CrossRef] [PubMed]

26.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, ”Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004). [CrossRef] [PubMed]

27.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, ”Dielectric Optical Cloak,” [arXiv:0904.3602][physicsoptics].

28.

T. G. Mackay, S. Setiawan, and A. Lakhtakia, “Negative phase velocity of electromagnetic waves and the cosmological constant,” Eur. Phys. J. C 41S1, 1–4 (2005). [CrossRef]

29.

Q. Cheng and T. J. Cui, “An electromagnetic black hole made of metamaterials,” arXiv:0910.2159 [physics.optics].

30.

T. G. Mackay and A. Lakhtakia, “Towards a metamaterial simulation of a spinning cosmic string,” arXiv:0911.4163 [physics.optics].

31.

L. H. Ford, “Quantum Vacuum Energy In General Relativity,” Phys. Rev. D 11, 3370–3377 (1975). [CrossRef]

32.

E. Newman and R. Penrose, “An Approach to gravitational radiation by a method of spin coefficients,” J. Math. Phys. 3, 566–578 (1962). [CrossRef]

33.

S. A. Teukolsky, “Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations,” Astrophys. J. 185, 635–648 (1973). [CrossRef]

OCIS Codes
(000.2780) General : Gravity
(000.6800) General : Theoretical physics

ToC Category:
Physical Optics

History
Original Manuscript: January 5, 2010
Revised Manuscript: March 30, 2010
Manuscript Accepted: April 1, 2010
Published: April 14, 2010

Citation
Miao Li, Rong-Xin Miao, and Yi Pang, "More studies on metamaterials mimicking de Sitter space," Opt. Express 18, 9026-9033 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9026


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References

  1. M. Bordag, The Casimir Effects 50 years later, (World Scientific Press, 1999).
  2. H. B. G. Casimir, "On the Attraction Between Two Perfectly Conducting Plates," Indag. Math. 10, 261 (1948).
  3. T. H. Boyer, "Quantum electromagnetic zero point energy of a conducting spherical shell and the Casimir model for a charged particle," Phys. Rev. 174, 1764-1776 (1968). [CrossRef]
  4. R. Balian and B. Duplantier, "Electromagnetic Waves Near Perfect Conductors. 2. Casimir Effect," Annals Phys. 112, 165-208 (1978). [CrossRef]
  5. K. A. Milton, L. L. DeRaad and J. S. Schwinger, "Casimir Selfstress On A Perfectly Conducting Spherical Shell," Annals Phys. 115, 388-403 (1978). [CrossRef]
  6. G. Plunien, B. Muller and W. Greiner, "The Casimir Effect," Phys. Rept. 134, 87-193 (1986). [CrossRef]
  7. C. M. Bender and P. Hays, "Zero Point Energy Of Fields In A Finite Volume," Phys. Rev. D 14, 2622-2632 (1976). [CrossRef]
  8. K. A. Milton, "Semiclassical Electron Models: Casimir Selfstress In Dielectric And Conducting Balls," Ann. Phys. 127, 49-61 (1980). [CrossRef]
  9. K. A. Milton, "Fermionic Casimir Stress On A Spherical Bag," Annals Phys. 150, 432-438 (1983). [CrossRef]
  10. M. Bordag, E. Elizalde, K. Kirsten and S. Leseduarte, "Casimir energies for massive fields in the bag," Phys. Rev. D 56, 4896-4904 (1997). [CrossRef]
  11. S. D. Odintsov, "Vilkovisky effective action in quantum gravity with matter," Theor. Math. Phys. 82, 45-51 (1990). [CrossRef]
  12. A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiattia, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry, "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant," Astron. J. 116, 1009-1038 (1998). [CrossRef]
  13. S. Perlmuttter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, and W. J. Couch, "Measurements of Omega and Lambda from 42 High-Redshift Supernovae," Astrophys.J. 517, 565-586 (1999).
  14. M. Li, R. X. Miao and Y. Pang, "Casimir Energy, Holographic Dark Energy and Electromagnetic Metamaterial Mimicking de Sitter," arXiv:0910.3375 [hep-th].
  15. Miao Li, "A Model of holographic dark energy," Phys. Lett. B 603, 1-5 (2004). [CrossRef]
  16. Qing-Guo Huang and Miao Li, "The Holographic Dark Energy in a Non-flat Universe," J. Cosmol. Astropart. Phys. 0408, 013 (2004). [CrossRef]
  17. J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396-1408 (1959). [CrossRef]
  18. U. Leonhardt and T. Philbin, "General Relativity in Electrical Engineering", New J. Phys. 8, 247 (2006). [CrossRef]
  19. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
  20. K. Niu, C. Song, and M. L. Ge, "The geodesic form of light-ray trace in the inhomogeneous media," Opt. Express 17(14), 11753-11767 (2009). [CrossRef] [PubMed]
  21. R. A. Shelby, D. R. Smith, and S. Shultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  22. A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental Observations of a Left-Handed Material That Obeys Snell’s Law," Phys. Rev. Lett. 90, 137401 (2003). [CrossRef] [PubMed]
  23. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
  24. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003). [CrossRef] [PubMed]
  25. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004). [CrossRef] [PubMed]
  26. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic Response of Metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004). [CrossRef] [PubMed]
  27. J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, "Dielectric Optical Cloak," arXiv:0904.3602 [physics.optics].
  28. T. G. Mackay, S. Setiawan and A. Lakhtakia, "Negative phase velocity of electromagnetic waves and the cosmological constant," Eur. Phys. J. C 41S1, 1-4 (2005). [CrossRef]
  29. Q. Cheng and T. J. Cui, "An electromagnetic black hole made of metamaterials," arXiv:0910.2159 [physics.optics].
  30. T. G. Mackay and A. Lakhtakia, "Towards a metamaterial simulation of a spinning cosmic string," arXiv:0911.4163 [physics.optics].
  31. L. H. Ford, "Quantum Vacuum Energy In General Relativity," Phys. Rev. D 11, 3370-3377 (1975). [CrossRef]
  32. E. Newman and R. Penrose, "An Approach to gravitational radiation by a method of spin coefficients," J. Math. Phys. 3, 566-578 (1962). [CrossRef]
  33. S. A. Teukolsky, "Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations," Astrophys. J. 185, 635-648 (1973). [CrossRef]

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