Abstract

By optimizing the design we show that inhomogeneous electromagnetic resonators with almost uniform field intensity and up to twice the energy density of conventional structures are possible by exploiting the properties of negative refractive index materials. We demonstrate that using negative refractive index materials it is possible to make FWHM of the transmission coefficient independent of cavity length L.

© 2007 Optical Society of America

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References

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
    [CrossRef]
  3. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  5. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
    [CrossRef]
  6. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
    [CrossRef] [PubMed]
  7. While not applicable to the subwavelength structures considered here, we note that an effective negative refractive index may also be achieved using photonic crystal dielectric material as discussed by M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
    [CrossRef]
  8. A. K. Popov and V. M. Shalaev, "Compensating losses in negative-index metamaterials by optical parametric amplification," Opt. Lett. 31, 2169-2171 (2006).
    [CrossRef] [PubMed]
  9. P. Schmidt, S. Haas, and A. F. J. Levi, "Synthesis of electron transmission in nanoscale semiconductor devices," Appl. Phys. Lett. 88, 013502 (2006).
    [CrossRef]
  10. N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Propag. Mag. 1, 10-13 (2002).
  11. K. R. Gegenfurtner, "PRAXIS: Brent's algorithm for function minimization," Behav. Res. Methods Instrum. 24, 560-564 (1993).
  12. R. P. Brent, Algorithms for Minimization Without Derivatives (Dover, 2002).
  13. A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
    [CrossRef] [PubMed]
  14. A. Hakansson, H. T. Miyazaki, and J. Sánchez-Dehesa, "Inverse design for full control of spontaneous emission using light emitting scattering optical elements," Phys. Rev. Lett. 96, 153902 (2006).
    [CrossRef] [PubMed]
  15. V. Dudiy and A. Zunger, "Searching for alloy configurations with target physical properties: impurity design via a genetic algorithm inverse band structure approach," Phys. Rev. Lett. 97, 046401 (2006).
    [CrossRef] [PubMed]

2006 (5)

A. K. Popov and V. M. Shalaev, "Compensating losses in negative-index metamaterials by optical parametric amplification," Opt. Lett. 31, 2169-2171 (2006).
[CrossRef] [PubMed]

P. Schmidt, S. Haas, and A. F. J. Levi, "Synthesis of electron transmission in nanoscale semiconductor devices," Appl. Phys. Lett. 88, 013502 (2006).
[CrossRef]

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

A. Hakansson, H. T. Miyazaki, and J. Sánchez-Dehesa, "Inverse design for full control of spontaneous emission using light emitting scattering optical elements," Phys. Rev. Lett. 96, 153902 (2006).
[CrossRef] [PubMed]

V. Dudiy and A. Zunger, "Searching for alloy configurations with target physical properties: impurity design via a genetic algorithm inverse band structure approach," Phys. Rev. Lett. 97, 046401 (2006).
[CrossRef] [PubMed]

2005 (2)

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
[CrossRef]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

2002 (1)

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Propag. Mag. 1, 10-13 (2002).

2000 (3)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

While not applicable to the subwavelength structures considered here, we note that an effective negative refractive index may also be achieved using photonic crystal dielectric material as discussed by M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

1993 (1)

K. R. Gegenfurtner, "PRAXIS: Brent's algorithm for function minimization," Behav. Res. Methods Instrum. 24, 560-564 (1993).

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Brent, R. P.

R. P. Brent, Algorithms for Minimization Without Derivatives (Dover, 2002).

Cai, W.

Chen, L.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

Chettiar, U. K.

Drachev, V. P.

Dudiy, V.

V. Dudiy and A. Zunger, "Searching for alloy configurations with target physical properties: impurity design via a genetic algorithm inverse band structure approach," Phys. Rev. Lett. 97, 046401 (2006).
[CrossRef] [PubMed]

Engheta, N.

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Propag. Mag. 1, 10-13 (2002).

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Gegenfurtner, K. R.

K. R. Gegenfurtner, "PRAXIS: Brent's algorithm for function minimization," Behav. Res. Methods Instrum. 24, 560-564 (1993).

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Gleeson, H. F.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Gondarenko, A.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Haas, S.

P. Schmidt, S. Haas, and A. F. J. Levi, "Synthesis of electron transmission in nanoscale semiconductor devices," Appl. Phys. Lett. 88, 013502 (2006).
[CrossRef]

Hakansson, A.

A. Hakansson, H. T. Miyazaki, and J. Sánchez-Dehesa, "Inverse design for full control of spontaneous emission using light emitting scattering optical elements," Phys. Rev. Lett. 96, 153902 (2006).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Khrushchev, I. Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Kildishev, A.

Levi, A. F. J.

P. Schmidt, S. Haas, and A. F. J. Levi, "Synthesis of electron transmission in nanoscale semiconductor devices," Appl. Phys. Lett. 88, 013502 (2006).
[CrossRef]

Lipson, H.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

Lipson, M.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

Miyazaki, H. T.

A. Hakansson, H. T. Miyazaki, and J. Sánchez-Dehesa, "Inverse design for full control of spontaneous emission using light emitting scattering optical elements," Phys. Rev. Lett. 96, 153902 (2006).
[CrossRef] [PubMed]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Notomi, M.

While not applicable to the subwavelength structures considered here, we note that an effective negative refractive index may also be achieved using photonic crystal dielectric material as discussed by M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Petrovic, J.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Popov, A. K.

Preble, S.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Robinson, J.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

Sánchez-Dehesa, J.

A. Hakansson, H. T. Miyazaki, and J. Sánchez-Dehesa, "Inverse design for full control of spontaneous emission using light emitting scattering optical elements," Phys. Rev. Lett. 96, 153902 (2006).
[CrossRef] [PubMed]

Sarychev, A. K.

Schmidt, P.

P. Schmidt, S. Haas, and A. F. J. Levi, "Synthesis of electron transmission in nanoscale semiconductor devices," Appl. Phys. Lett. 88, 013502 (2006).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Yuan, H. K.

Zhang, Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Zunger, A.

V. Dudiy and A. Zunger, "Searching for alloy configurations with target physical properties: impurity design via a genetic algorithm inverse band structure approach," Phys. Rev. Lett. 97, 046401 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

P. Schmidt, S. Haas, and A. F. J. Levi, "Synthesis of electron transmission in nanoscale semiconductor devices," Appl. Phys. Lett. 88, 013502 (2006).
[CrossRef]

Behav. Res. Methods Instrum. (1)

K. R. Gegenfurtner, "PRAXIS: Brent's algorithm for function minimization," Behav. Res. Methods Instrum. 24, 560-564 (1993).

IEEE Antennas Propag. Mag. (1)

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Propag. Mag. 1, 10-13 (2002).

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075 (1999).
[CrossRef]

Nature (1)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (1)

While not applicable to the subwavelength structures considered here, we note that an effective negative refractive index may also be achieved using photonic crystal dielectric material as discussed by M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Phys. Rev. Lett. (5)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, "Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures," Phys. Rev. Lett. 96, 143904 (2006).
[CrossRef] [PubMed]

A. Hakansson, H. T. Miyazaki, and J. Sánchez-Dehesa, "Inverse design for full control of spontaneous emission using light emitting scattering optical elements," Phys. Rev. Lett. 96, 153902 (2006).
[CrossRef] [PubMed]

V. Dudiy and A. Zunger, "Searching for alloy configurations with target physical properties: impurity design via a genetic algorithm inverse band structure approach," Phys. Rev. Lett. 97, 046401 (2006).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (1)

R. P. Brent, Algorithms for Minimization Without Derivatives (Dover, 2002).

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Figures (5)

Fig. 1
Fig. 1

(a) Refractive index profile n ( z ) of a conventional dielectric resonator consisting of two Bragg mirrors and a cavity. Resonant wavelength is λ 0 = 980 nm . Each Bragg mirror consists of five layer pairs of n = 1 and n = 2 material, each layer being λ 0 ( 4 n ) thick. The central cavity region has refractive index n = 2 and length L = λ 0 n . (b) Normalized resonant electric field intensity E ( z ) 2 corresponding to the refractive index profile shown in (a). Field intensity is normalized to the intensity of the electromagnetic wave incident on the structure from the left.

Fig. 2
Fig. 2

(a) Refractive index profile n ( z ) of an electromagnetic resonator with a cavity containing dielectric pairs consisting of six alternating layer pairs of n = 2 and n = 2 material, each of thickness d. Resonant wavelength is λ 0 = 980 nm . Inset shows increasing the number of layer pairs that fill the cavity of length L = λ 0 n has the effect of increasing the energy density. With increasing layer pairs the stored energy enhancement factor asymptotically approaches ξ = 2 . (b) Normalized resonant electric field intensity E ( z ) 2 corresponding to the refractive index profile shown in (a).

Fig. 3
Fig. 3

(a) Optimal refractive index profile n ( z ) of a resonator that maximizes electromagnetic field intensity in the cavity. The cavity contains six alternating layer pairs of n = 2 and n = 2 material. Thicknesses of the n = 2 material are d 1 = 28.48 nm , d 2 = 43.42 nm , d 3 = 40.35 nm , d 4 = 41.21 nm , d 5 = 38.58 nm , and d 6 = 52.93 nm . Resonant wavelength is λ 0 = 980 nm and the cavity has length L = λ 0 n . (b) The normalized resonant electric field intensity E ( z ) 2 corresponding to the refractive index profile shown in (a).

Fig. 4
Fig. 4

Transmission coefficient as a function of wavelength λ for resonator structure illustrated in Fig. 1a (solid curve) and Figs. 2a, 3a (dotted curve). The cavity with alternating positive and negative refractive index has a larger line width. The results can be fit to Lorentzian functions with FWHM of 0.203 nm and 0.610 nm .

Fig. 5
Fig. 5

(a) Transmission coefficient for a resonator as a function of wavelength λ and cavity length L normalized to the resonant wavelength in the cavity, λ res . Each DBR mirror consists of 5 alternating layer pairs of n = 2 and n = 1 material with each layer a quarter wavelength thick and a central cavity region of length L with refractive index n = 2 , similar to Fig. 1a. Note the complex structure. The transmission coefficient is plotted using a log 10 scale. (b) Transmission coefficient for the structure in (a) but with the central cavity region consisting of alternating layer pairs of n = 2 and n = 2 material. Note, that unlike in (a), transmission is independent of the cavity length L because of electric field phase cancellation in the cavity.

Equations (3)

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E x z = z 0 δ = E x z = z 0 + δ ,
1 μ 1 ( ω ) E x z z = z 0 δ = 1 μ 2 ( ω ) E x z z = z 0 + δ ,
W = z L z R E ( z ) 2 d z ,

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