Abstract

The purpose of this paper is threefold. The first aim is to discuss the superlensing effect in left-handed material presented originally by Veselago [Sov. Phys. Usp. 10, 509 (1968) ] and Pendry [Phys. Rev. Lett. 85, 3966 (2000) ] for n=1. Our discussion is based on an integral expression for the electromagnetic fields (i.e., the Extinction Theorem), and it allows us to demonstrate that such superlensing effect does not exist. The second aim is to discuss some basic questions about structured metamaterials formed by periodic resonator arrays. We will show that a set of antennas can never emulate a negative refractive index material. The third, concurrent, aim is to discuss the experimental evidence for negative refraction in structured periodic materials. We present another possible explanation of the experimental results concerning negative refraction that is based on the coherent interference of wavelets originated in the subwavelength structures.

© 2008 Optical Society of America

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References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [Crossref] [PubMed]
  2. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
    [Crossref]
  3. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
    [Crossref] [PubMed]
  4. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
    [Crossref] [PubMed]
  5. 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]
  6. N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403(4) (2002).
    [Crossref] [PubMed]
  7. P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401(4) (2002).
    [Crossref] [PubMed]
  8. M. Nieto-Vesperinas, “Problem of image superresolution with a negative-refractive-index slab,” J. Opt. Soc. Am. A 21, 491-498 (2004).
    [Crossref]
  9. D. Maystre and S. Enoch, “Perfect lenses made with left-handed materials: Alice's mirror?” J. Opt. Soc. Am. A 21, 122-131 (2004).
    [Crossref]
  10. D. Maystre, S. Enoch, and R. McPhedran, “Why a harmonic solution for lossless, perfectly homogeneous, left-handed material cannot exist,” J. Opt. Soc. Am. A 25, 1937-1943 (2008).
    [Crossref]
  11. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles, Springer Series in Optical Sciences 124, W.T.Rhodes, ed. (Springer, 2006), Sect. (1.4.1); the Green function is written in dyadic form. By simple mathematical manipulation it is easy to demonstrate that the last equation of Sect. (1.4.1) is equivalent to our Eq. .
    [Crossref]
  12. L. D. Landau and E. M. Lifshitz, Electrodinámica de Medios Continuos (Ed. Reverte, 1975).
  13. J. M. Cabrera, F. J. López, and F. A. Agulló-López, Optica electromagnética (Addison-Wesley Iberoamericana, 1993).
  14. J. D. Jackson, Classical Electrodynamics (Wiley, 1962), Eq. 7.89..
  15. 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, 2075-2084 (1999).
    [Crossref]
  16. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 6.
  17. The problem of the losses along the propagation direction (briefly sketched in Section ) is one of the more relevant issues in the treatment of metamaterials. A deeper investigation of this issue is outside the scope of the present work, and it will be done elsewhere. For the sake of simplicity, the losses will be disregarded in the present discussion.
  18. For simplicity we are dealing here with scalar fields. A vectorial formulation would not change our conclusions substantially; see, e.g., A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
    [Crossref]
  19. G. O. Reynolds, J. B. De Velis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Optical Engineering Press, 1989).
    [Crossref]
  20. G. J. Swanson and E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. A 2, 789-793 (1985).
    [Crossref]

2008 (1)

2004 (2)

2003 (2)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

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]

2002 (3)

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403(4) (2002).
[Crossref] [PubMed]

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401(4) (2002).
[Crossref] [PubMed]

For simplicity we are dealing here with scalar fields. A vectorial formulation would not change our conclusions substantially; see, e.g., A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

2000 (1)

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

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, 2075-2084 (1999).
[Crossref]

1985 (1)

1968 (1)

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

Agulló-López, F. A.

J. M. Cabrera, F. J. López, and F. A. Agulló-López, Optica electromagnética (Addison-Wesley Iberoamericana, 1993).

Brock, J. B.

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]

Cabrera, J. M.

J. M. Cabrera, F. J. López, and F. A. Agulló-López, Optica electromagnética (Addison-Wesley Iberoamericana, 1993).

Chuang, I. L.

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]

Ciattoni, A.

For simplicity we are dealing here with scalar fields. A vectorial formulation would not change our conclusions substantially; see, e.g., A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

Crosignani, B.

For simplicity we are dealing here with scalar fields. A vectorial formulation would not change our conclusions substantially; see, e.g., A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

De Velis, J. B.

G. O. Reynolds, J. B. De Velis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Optical Engineering Press, 1989).
[Crossref]

Di Porto, P.

For simplicity we are dealing here with scalar fields. A vectorial formulation would not change our conclusions substantially; see, e.g., A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

Doicu, A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles, Springer Series in Optical Sciences 124, W.T.Rhodes, ed. (Springer, 2006), Sect. (1.4.1); the Green function is written in dyadic form. By simple mathematical manipulation it is easy to demonstrate that the last equation of Sect. (1.4.1) is equivalent to our Eq. .
[Crossref]

Enoch, S.

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles, Springer Series in Optical Sciences 124, W.T.Rhodes, ed. (Springer, 2006), Sect. (1.4.1); the Green function is written in dyadic form. By simple mathematical manipulation it is easy to demonstrate that the last equation of Sect. (1.4.1) is equivalent to our Eq. .
[Crossref]

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403(4) (2002).
[Crossref] [PubMed]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[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, 2075-2084 (1999).
[Crossref]

Houck, A. A.

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]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962), Eq. 7.89..

Koltenbah, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodinámica de Medios Continuos (Ed. Reverte, 1975).

Leith, E. N.

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodinámica de Medios Continuos (Ed. Reverte, 1975).

López, F. J.

J. M. Cabrera, F. J. López, and F. A. Agulló-López, Optica electromagnética (Addison-Wesley Iberoamericana, 1993).

Maystre, D.

McPhedran, R.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “Problem of image superresolution with a negative-refractive-index slab,” J. Opt. Soc. Am. A 21, 491-498 (2004).
[Crossref]

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403(4) (2002).
[Crossref] [PubMed]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Parrent, G. B.

G. O. Reynolds, J. B. De Velis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Optical Engineering Press, 1989).
[Crossref]

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, 2075-2084 (1999).
[Crossref]

Reynolds, G. O.

G. O. Reynolds, J. B. De Velis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Optical Engineering Press, 1989).
[Crossref]

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, 2075-2084 (1999).
[Crossref]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[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, 2075-2084 (1999).
[Crossref]

Swanson, G. J.

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

Thompson, B. J.

G. O. Reynolds, J. B. De Velis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Optical Engineering Press, 1989).
[Crossref]

Valanju, A. P.

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401(4) (2002).
[Crossref] [PubMed]

Valanju, P. M.

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401(4) (2002).
[Crossref] [PubMed]

Veselago, V. G.

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

Walser, R. M.

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401(4) (2002).
[Crossref] [PubMed]

Wriedt, T.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles, Springer Series in Optical Sciences 124, W.T.Rhodes, ed. (Springer, 2006), Sect. (1.4.1); the Green function is written in dyadic form. By simple mathematical manipulation it is easy to demonstrate that the last equation of Sect. (1.4.1) is equivalent to our Eq. .
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 6.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 6.

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, 2075-2084 (1999).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

For simplicity we are dealing here with scalar fields. A vectorial formulation would not change our conclusions substantially; see, e.g., A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[Crossref]

Phys. Rev. Lett. (5)

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

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[Crossref] [PubMed]

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]

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403(4) (2002).
[Crossref] [PubMed]

P. M. Valanju, R. M. Walser, and A. P. Valanju, “Wave refraction in negative-index media: Always positive and very inhomogeneous,” Phys. Rev. Lett. 88, 187401(4) (2002).
[Crossref] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

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

Other (7)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 6.

The problem of the losses along the propagation direction (briefly sketched in Section ) is one of the more relevant issues in the treatment of metamaterials. A deeper investigation of this issue is outside the scope of the present work, and it will be done elsewhere. For the sake of simplicity, the losses will be disregarded in the present discussion.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles, Springer Series in Optical Sciences 124, W.T.Rhodes, ed. (Springer, 2006), Sect. (1.4.1); the Green function is written in dyadic form. By simple mathematical manipulation it is easy to demonstrate that the last equation of Sect. (1.4.1) is equivalent to our Eq. .
[Crossref]

L. D. Landau and E. M. Lifshitz, Electrodinámica de Medios Continuos (Ed. Reverte, 1975).

J. M. Cabrera, F. J. López, and F. A. Agulló-López, Optica electromagnética (Addison-Wesley Iberoamericana, 1993).

J. D. Jackson, Classical Electrodynamics (Wiley, 1962), Eq. 7.89..

G. O. Reynolds, J. B. De Velis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Optical Engineering Press, 1989).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Unit cell of a periodic structure.

Fig. 2
Fig. 2

Unit cell with simple current rings orthogonal to the x direction.

Fig. 3
Fig. 3

Plane wave traveling inside a prism-shaped structured sample. Inside the material can be assumed the existence of an averaged refractive index n. At the sample surface itself, the periodic nature of the sample is apparent. Positive values of θ are taken counterclockwise.

Fig. 4
Fig. 4

Slab-lensing by two gratings in tandem.

Equations (35)

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

E ( r ) = E 0 ( r ) + 1 4 π [ n 2 ( r ) 1 ] { k 0 2 E ( r ) G ( r , r ) + [ E ( r ) ] G ( r , r ) } d 3 r ,
2 G ( r , r ) + k 0 2 G ( r , r ) = 4 π δ ( r r ) ,
E = ρ ϵ 0 ,
B = 0 ,
E = i w B ,
B = μ 0 J i w ϵ 0 μ 0 E ,
J ( r ) = σ ( r ) E ( r ) ,
σ ( r ) = m , p , q σ m , p , q exp [ i ( 2 π a ) ( m x + p y + q z ) ] ,
B = i w ϵ 0 μ 0 [ 1 + i σ ( r ) w ϵ 0 ] E .
E = k 0 2 [ 1 + i σ ( r ) w ϵ 0 ] E .
n 2 ( r ) = [ 1 + i σ ( r ) w ϵ 0 ] .
k k > 0 .
k k = n 2 k 0 2 ,
k 2 k 2 = k 0 2 ( ε μ ε μ ) ε 0 μ 0 ,
2 k k = k 0 2 ( ε μ + ε μ ) ε 0 μ 0 < 0 ,
B = μ 0 H .
H x a 1 r = ( 0 , 0 , 0 ) r = ( a , 0 , 0 ) H d r ,
B x a 2 S x B d S ,
μ x B x H x ,
B x a 2 S x B d S > 0 ,
H x a 1 r = ( 0 , 0 , 0 ) r = ( a , 0 , 0 ) H d r < 0 ,
μ x B x H x < 0 .
E ( x , 0 ) exp [ i ( 2 π λ 0 ) n sin ( α ) x ] m E 0 m exp [ i 2 π m x a ] ,
n sin ( α ) sin ( θ m ) = m λ 0 a ,
n sin ( α ) = sin ( θ 0 ) ,
n sin ( α m ) sin ( θ i ) = m λ 0 a .
n sin ( α m ) sin ( θ j ) = j λ 0 a .
σ = N 0 e 2 m ( b i w ) = N 0 e 2 ( b + i w ) m ( b 2 + w 2 ) ,
n 2 ( r ) = [ 1 N 0 e 2 m ε 0 ( b 2 + w 2 ) + i N 0 e 2 b m w ε 0 ( b 2 + w 2 ) ] .
E ( x , 0 ) = E 0 exp [ x 2 2 σ x 2 ] exp [ i ( 2 π λ 0 ) x n sin ( α ) ] .
E ( x , 0 + ) = exp [ x 2 2 σ x 2 ] exp [ i ( 2 π λ 0 ) x n sin ( α ) ] m E 0 m exp [ i 2 π m x a ] ,
E ( x , z ) exp [ i 2 π z λ 0 ] exp [ i π x 2 λ 0 z ] i λ 0 z × E ( x , 0 + ) exp [ i π x 2 λ 0 z ] exp [ i 2 π x x λ 0 z ] d x .
E ( x , z ) = m = E m ( x , z ) ,
E m ( x , z ) = E 0 m exp [ i 2 π z λ 0 ] exp [ i π x 2 λ 0 z ] i λ 0 z π [ ( 1 2 σ x 2 ) i ( π λ 0 z ) ] × exp { π 2 ( m λ 0 a n sin ( α ) + x z ) 2 λ 0 2 ( 1 2 σ x 2 ) i ( π λ 0 z ) } .
m λ 0 a n sin ( α ) + x z = 0 .

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