Abstract

Starting from the causality of the permittivity and permeability of a medium, we investigate the causality of the propagation constant. We show that a reduced dispersion relation, obtained from the frequency dependence of the propagation constant by neglecting a linear frequency dependent term, obeys causality. The propagation constant is identical to the reduced propagation constant under appropriate limiting values of the physical parameters. We illustrate the causality of the reduced propagation constant through examples of (a) a nonmagnetic material where the permittivity is given by the Lorentz model, (b) a material where the permittivity and permeability are both Lorentz-type, and (c) an effective medium comprising a nonmagnetic material with Lorentz-type permittivity in a dispersionless host medium, where the effective permittivity is given by the Maxwell–Garnett rule. Causality of the propagation constant enables the use of simple operator formalisms to derive the underlying partial differential equations for baseband and envelope wave propagation, as demonstrated through an illustrative example of a negative index medium with gain.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  2. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
    [CrossRef]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  4. A. Korpel and P. P. Banerjee, “A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions,” Proc. IEEE 72, 1109–1130 (1984).
    [CrossRef]
  5. 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]
  6. P. P. Banerjee, R. Aylo, and G. Nehmetallah, “Baseband and envelope propagation in media modeled by a class of complex dispersion relations,” J. Opt. Soc. Am. B 25, 990–994 (2008).
    [CrossRef]
  7. R. J. Beerends, H. G. terMorsche, J. C. van denBerg, and E. M. deVries, Fourier and Laplace Transforms (Cambridge Univ. Press, 2003).
  8. V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).
  9. S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
    [CrossRef] [PubMed]
  10. T. Christensen and N. B. Olsen, “Determination of the frequency-dependent bulk modulus of glycerol using a piezoelectric spherical shell,” Phys. Rev. B 49, 15396–15399 (1994).
    [CrossRef]
  11. V. Yannopapas, “Negative refraction in random photonic alloys of polaritonic and plasmonic microspheres,” Phys. Rev. B 75, 035112 (2007).
    [CrossRef]
  12. G. Kristensson, S. Rikte, and A. Sihvola, “Mixing formulas in the time domain,” J. Opt. Soc. Am. A 15, 1411–1422 (1998).
    [CrossRef]
  13. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  14. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [CrossRef]
  15. R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
    [CrossRef]
  16. A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
    [CrossRef]

2009 (1)

S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (2)

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

V. Yannopapas, “Negative refraction in random photonic alloys of polaritonic and plasmonic microspheres,” Phys. Rev. B 75, 035112 (2007).
[CrossRef]

2000 (2)

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]

R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

1998 (2)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

G. Kristensson, S. Rikte, and A. Sihvola, “Mixing formulas in the time domain,” J. Opt. Soc. Am. A 15, 1411–1422 (1998).
[CrossRef]

1994 (1)

T. Christensen and N. B. Olsen, “Determination of the frequency-dependent bulk modulus of glycerol using a piezoelectric spherical shell,” Phys. Rev. B 49, 15396–15399 (1994).
[CrossRef]

1989 (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

1984 (1)

A. Korpel and P. P. Banerjee, “A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions,” Proc. IEEE 72, 1109–1130 (1984).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Aylo, R.

Banerjee, P. P.

P. P. Banerjee, R. Aylo, and G. Nehmetallah, “Baseband and envelope propagation in media modeled by a class of complex dispersion relations,” J. Opt. Soc. Am. B 25, 990–994 (2008).
[CrossRef]

A. Korpel and P. P. Banerjee, “A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions,” Proc. IEEE 72, 1109–1130 (1984).
[CrossRef]

Beerends, R. J.

R. J. Beerends, H. G. terMorsche, J. C. van denBerg, and E. M. deVries, Fourier and Laplace Transforms (Cambridge Univ. Press, 2003).

Christensen, T.

T. Christensen and N. B. Olsen, “Determination of the frequency-dependent bulk modulus of glycerol using a piezoelectric spherical shell,” Phys. Rev. B 49, 15396–15399 (1994).
[CrossRef]

deVries, E. M.

R. J. Beerends, H. G. terMorsche, J. C. van denBerg, and E. M. deVries, Fourier and Laplace Transforms (Cambridge Univ. Press, 2003).

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Govyadinov, A. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kim, C. K.

S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
[CrossRef] [PubMed]

Korpel, A.

A. Korpel and P. P. Banerjee, “A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions,” Proc. IEEE 72, 1109–1130 (1984).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kristensson, G.

Lee, S. H.

S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
[CrossRef] [PubMed]

Lucarini, V.

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Nehmetallah, G.

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]

Noginov, M. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Olsen, N. B.

T. Christensen and N. B. Olsen, “Determination of the frequency-dependent bulk modulus of glycerol using a piezoelectric spherical shell,” Phys. Rev. B 49, 15396–15399 (1994).
[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]

Park, C. M.

S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
[CrossRef] [PubMed]

Peiponen, K. E.

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).

Podolskiy, V. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

Rikte, S.

Ruppin, R.

R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

Saarinen, J. J.

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[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]

Seo, Y. M.

S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
[CrossRef] [PubMed]

Sihvola, A.

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]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

terMorsche, H. G.

R. J. Beerends, H. G. terMorsche, J. C. van denBerg, and E. M. deVries, Fourier and Laplace Transforms (Cambridge Univ. Press, 2003).

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

van denBerg, J. C.

R. J. Beerends, H. G. terMorsche, J. C. van denBerg, and E. M. deVries, Fourier and Laplace Transforms (Cambridge Univ. Press, 2003).

Vartiainen, E. M.

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ε 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]

Wang, Z. G.

S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Yannopapas, V.

V. Yannopapas, “Negative refraction in random photonic alloys of polaritonic and plasmonic microspheres,” Phys. Rev. B 75, 035112 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. Condens. Matter (1)

S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative modulus,” J. Phys. Condens. Matter 21, 175704 (2009).
[CrossRef] [PubMed]

Opt. Commun. (1)

R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

Phys. Rev. B (4)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

T. Christensen and N. B. Olsen, “Determination of the frequency-dependent bulk modulus of glycerol using a piezoelectric spherical shell,” Phys. Rev. B 49, 15396–15399 (1994).
[CrossRef]

V. Yannopapas, “Negative refraction in random photonic alloys of polaritonic and plasmonic microspheres,” Phys. Rev. B 75, 035112 (2007).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Proc. IEEE (1)

A. Korpel and P. P. Banerjee, “A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions,” Proc. IEEE 72, 1109–1130 (1984).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (4)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

R. J. Beerends, H. G. terMorsche, J. C. van denBerg, and E. M. deVries, Fourier and Laplace Transforms (Cambridge Univ. Press, 2003).

V. Lucarini, J. J. Saarinen, K. E. Peiponen, and E. M. Vartiainen, Kramers–Kronig Relations in Optical Materials Research (Springer, 2005).

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Real and imaginary parts of the relative permittivity, found from Eq. (7), with ω T = 2 π × 4.25 × 10 12   rad / s , ω L = 2 π × 7.46 × 10 12   rad / s , γ = 0.94 × 10 12   rad / s , and ε = 13.4 ε 0 .

Fig. 2
Fig. 2

(a) Real and (b) imaginary parts of K ( ω ) from Eq. (6); (c) numerically calculated Hilbert transform of the imaginary part in (b); (d) numerically calculated Hilbert transform of the real part in (a). Note that (a) is identical to (c) and (b) is identical to (d).

Fig. 3
Fig. 3

Real and imaginary parts of the (a) relative permittivity and (b) relative permeability, found from Eqs. (8, 9), respectively; ω p e = 1.1543 × 10 11   rad / s , ω 1 e = ω 1 m = 9.42 × 10 10   rad / s , ω p m = 1.3024 × 10 11   rad / s , γ e = 2 γ m = 3.769 × 10 8   rad / s .

Fig. 4
Fig. 4

(a) Real and (b) imaginary parts of K ( ω ) from Eq. (6); (c) numerically calculated Hilbert transform of the imaginary part in (b); (d) numerically calculated Hilbert transform of the real part in (a). Note that (a) is identical to (c) and (b) is identical to (d).

Fig. 5
Fig. 5

(a) Real and (b) imaginary parts of K eff ( ω ) from Eq. (6); (c) numerically calculated Hilbert transform of the imaginary part in (b); (d) numerically calculated Hilbert transform of the real part in (a). Note that (a) is identical to (c) and (b) is identical to (d).

Fig. 6
Fig. 6

(a) Initial ( Z = 0 ) baseband initial pulse in time domain and (b) after propagation a distance Z = 10 . The initial Gaussian pulse is taken as ψ n ( 0 , T ) = exp ( T 2 / τ 2 ) , with τ = 5 .

Fig. 7
Fig. 7

Gaussian pulse envelope in time domain in a negative index material for ω c n = 4 and initial width of τ = 20 . (a) Initial pulse; (b) after propagation by Z = 10 .

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

n 2 ( ω ) = ε ( ω ) μ ( ω ) / ( ε 0 μ 0 ) ,
I t 1 [ n 2 ( ω ) ] = [ ε ̃ ( t ) * μ ̃ ( t ) ] / ( ε 0 μ 0 ) ,
k r ( ω ) = ± { ω 2 2 c 2 [ ε r + ε r 2 + ε i 2 ] } 1 / 2 ,
k i ( ω ) = ± { ω 2 2 c 2 [ ε r ε r 2 + ε i 2 ] } 1 / 2 .
ω m ( n ( ω ) 1 ) m
K ( ω ) = ω ( n ( ω ) 1 ) c = k ( ω ) ω c .
K ( ω ) = ω ε ( ω ) μ ( ω ) ε μ ,
ε ( ω ) = ε ( 1 + ω L 2 ω T 2 ω T 2 ω 2 + j ω γ ) ,
ε ( ω ) = ( 1 + ω p e 2 ω 1 e 2 ω 2 + j γ e ω ) ε 0 ,
μ ( ω ) = ( 1 + ω p m 2 ω 1 m 2 ω 2 + j γ m ω ) μ 0 ,
K eff ( ω ) = ω ( ε eff ε eff , ) μ 0 .
ε eff = ε h [ ε s ( 1 + 2 f s ) + 2 ε h ( 1 f s ) ε s ( 1 f s ) + ε h ( 2 + f s ) ] ,
ε eff = [ ε h ( 2 3 f s ) + ε s ( 3 f s 1 ) 4 + 1 4 [ ε h ( 2 3 f s ) + ε s ( 3 f s 1 ) ] 2 + 8 ε h ε s ] ,
k ± ( ω ) = ± ( ω 0 v ) [ ( ω / ω 0 ) 1 + ( ω / ω 0 ) 2 + j 1 1 + ( ω / ω 0 ) 2 ] ,
k G ( ω ) = k ( ω ) + 1 j = ( ω 0 v ) ( ( ω / ω 0 ) 1 + ( ω / ω 0 ) 2 j ( ω / ω 0 ) 2 1 + ( ω / ω 0 ) 2 ) ,
ε r ( ω ) = ( c ω ) 2 ( k r 2 k i 2 ) ,     ε i ( ω ) = ( c ω ) 2 ( 2 k r k i ) ,
ε ( ω ) = ( ε r j ε i ) ε 0 = ( c v ) 2 ε 0 { ( ω / ω 0 ) 2 1 [ 1 + ( ω / ω 0 ) 2 ] 2 } j { 2 ( ω / ω 0 ) [ 1 + ( ω / ω 0 ) 2 ] 2 } .
ψ z 1 ω 0 2 3 ψ t 2 z ( ω 0 v ) ( 1 ω 0 ) ψ t + ( ω 0 v ) ( 1 ω 0 2 ) 2 ψ t 2 = 0.
ψ n Z 3 ψ n T 2 Z ψ n T + 2 ψ n T 2 = 0.
Ψ n ( Z , ω n ) = Ψ n ( 0 , ω n ) exp [ j ( ω n j ω n 2 1 + ω n 2 ) Z ] = Ψ n ( 0 , ω n ) exp [ j ( ω n 1 + ω n 2 ) Z ] exp ( ω n 2 1 + ω n 2 Z ) ,
a ψ e Z + b ψ e T + c 2 ψ e T 2 + d 2 ψ e T Z + e 3 ψ e T 2 Z + f ψ e = 0 ,
ψ e n ( Z , T ) = I t 1 { Ψ e n ( Z , ω n ) } = I t 1 { Ψ e n ( 0 , ω n ) exp [ ( f j b ω + c ω 2 a e ω 2 + j d ω ) Z ] } .

Metrics