Abstract

Developing a theory based on a spectral Green function for light emission from a point–dipole source embedded in a chiral sculptured thin film (CSTF), we found that the intensity and polarization of the emitted light are strongly influenced by the structural handedness of the CSTF as well as the placement and orientation of the source dipole. The emission patterns across both pupils of the dipole–containing CSTF can be explained in terms of the circular Bragg phenomenon exhibited by CSTFs when illuminated by normally as well as obliquely incident plane waves. The emission characteristics augur well for the future of CSTFs as optical biosensors as well as light emitters with controlled circular polarization and bandwidth.

© 2007 Optical Society of America

Full Article  |  PDF Article

Corrections

Tom G. Mackay and Akhlesh Lakhtakia, "Theory of light emission from a dipole source embedded in a chiral sculptured thin film: erratum," Opt. Express 16, 3659-3659 (2008)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-6-3659

References

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  1. A. Lakhtakia, R. Messier, M. J. Brett, and K. Robbie, "Sculptured thin films (STFs) for optical, chemical and biological applications," Innovations Mater. Res. 1, 165176 (1996).
  2. I. Hodgkinson and Q. h. Wu, "Inorganic chiral optical materials," Adv. Mater. 13, 889-897 (2001).
    [CrossRef]
  3. A. Lakhtakia and R. Messier, Sculptured Thin Films: Nanoengineered Morphology and Optics (SPIE Press, Bellingham, WA, USA, 2005).
    [CrossRef]
  4. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, New York, NY, USA, 1993).
  5. Q. Wu, I. J. Hodgkinson, and A. Lakhtakia, "Circular polarization filters made of chiral sculptured thin films: experimental and simulation results," Opt. Eng. 39, 1863-1868 (2000).
    [CrossRef]
  6. I. J. Hodgkinson, Q. H. Wu, A. Lakhtakia, and M. W. McCall, "Spectral-hole filter fabricated using scultured thin-film technology," Opt. Commun. 177, 79-84 (2000).
    [CrossRef]
  7. J. A. PoloJr., "Sculptured thin films," in: Micromanufacturing and Nanotechnology, pp. 357-381, N. P. Mahalik, ed., Springer, Heidelberg, Germany (2005).
  8. A. Lakhtakia, M. C. Demirel, M.W. Horn, and J. Xu, "Six emerging directions in sculptured-thin-film research," Adv. Solid State Phys. 46 (2008); in press.
  9. R. Messier, V. C. Venugopal, and P. D. Sunal, "Origin and evolution of sculptured thin films," J. Vac. Sci. Technol. A 18, 1538-1545 (2000).
    [CrossRef]
  10. A. Lakhtakia, "On bioluminescent emission from chiral sculptured thin films," Opt. Commun. 188, 313-320 (2001).
    [CrossRef]
  11. H. Tan, O. Ezekoye, J. van der Schalie, M.W. Horn, A. Lakhtakia, J. Xu, andW. D. Burgos, "Biological reduction of nanoengineered iron III oxide sculptured thin films," Environ. Sci. Technol. 40, 5490-5495 (2006).
    [CrossRef] [PubMed]
  12. S. Chan, Y. Li, L. J. Rothberg, B. L. Miller, and P. M. Fauchet, "Nanoscale silicon microcavities for biosensing," Mater. Sci. Eng. C 15, 277-282 (2001).
    [CrossRef]
  13. L. De Stefano, I. Rendina, A. M. Rossi, M. Rossi, L. Rotiroti, and S. D’Auria, "Biochips at work: porous silicon microbiosensor for proteomic diagnostic," J. Phys.: Condens. Matter 19, 395007 (2007).
    [CrossRef]
  14. K. Robbie, M. J. Brett, and A. Lakhtakia, "Chiral sculptured thin films," Nature 384, 616 (1996).
    [CrossRef]
  15. P. C. P. Hrudey, K. L. Westra, and M. J. Brett, "Highly ordered organic Alq3 chiral luminescent thin films fabricated by glancing-angle deposition," Adv. Mater. 18, 224-228 (2006).
    [CrossRef]
  16. J. Xu, A. Lakhtakia, J. Liou, A. Chen, and I. J. Hodgkinson, "Circularly polarized fluorescence from light- emitting microcavities with sculptured-thin-film chiral reflectors," Opt. Commun. 264, 235-239 (2006).
    [CrossRef]
  17. S. K. Arya, A. Chaubey, and B. D. Malhotra, "Fundamentals and applications of biosensors," Proc. Ind. Natn. Sci. Acad. 72, 249-266 (2006).
  18. A. Dorfman, N. Kumar, and J.-i. Hahm, "Highly sensitive biomolecular fluorescence detection using nanoscale ZnO platforms," Langmuir 22, 4890-4895 (2006).
    [CrossRef] [PubMed]
  19. F. Zhang, J. Xu, A. Lakhtakia, S. M. Pursel, and M. W. Horn, "Circularly polarized emission from colloidal nanocrystal quantum dots confined in microcavities formed by chiral mirrors," Appl. Phys. Lett. 91, 023102 (2007). [Interchange the labels LCP and RCP in Fig. 2c of this paper.]
    [CrossRef]
  20. B. Valeur, Molecular Fluorescence: Principles and Applications (Wiley-VCH, Weinheim, Germany, 2002).
  21. A. Ishchenko, "Molecular engineering of dye-doped polymers for optoelectronics," Polym. Adv. Technol. 13, 744-752 (2003).
    [CrossRef]
  22. F. Boxberg and J. Tulkki, "Quantum dots: Phenomenology, photonic and electronic properties, modeling and technology," in: Nanometer Structures — Theory, Modeling, and Simulation, pp. 107-143, A. Lakhtakia, ed., SPIE Press, Bellingham, WA, USA (2004).
  23. A. Lakhtakia, "On radiation from canonical source configurations embedded in structurally chiral materials," Microwave Opt. Technol. Lett. 37, 37-40 (2003).
    [CrossRef]
  24. M. P. C. M. Krijn, "Electromagnetic wave propagation in stratified anisotropic media in the presence of sources," Opt. Lett. 17, 163-165 (1992). [Although Eq. (10) of this paper is not rigorously valid unless the matrix ⊗(z) therein is either diagonal or independent of z, it can be useful with the piecewise uniform approximation technique provided a space-ordering operator is implemented on its right side [25].]
    [CrossRef] [PubMed]
  25. K. Eidner, "Light propagation in stratified anisotropic media: orthogonality and symmetry properties of the 4×4 matrix formalisms," J. Opt. Soc. Am. A 6, 1657-1660 (1989).
    [CrossRef]
  26. A. Lakhtakia and W. S. Weiglhofer, "Green function for radiation and propagation in helicoidal bianisotropic mediums," IEE Proc.-Microw. Antennas Propag. 144, 57-59 (1997).
    [CrossRef]
  27. A. Lakhtakia and M. W. McCall, "Response of chiral sculptured thin films to dipolar sources," Int. J. Electron. Commun. (AE ¨ U) 57, 23-32 (2003).
    [CrossRef]
  28. D. W. Berreman, "Optics in stratified and anisotropic media: 4×4-matrix formulation," J. Opt. Soc. Am. 62, 502-510 (1972).
    [CrossRef]
  29. M. Born and E. Wolf, Principles of Optics, Appendix III, 7th ed. (Pergamon, Oxford, UK, 1999).
  30. F. Wang, "Note on the asymptotic approximation of a double integral with an angular-spectrum representation," Int. J. Electron. Commun. (AEU) 59, 258-261 (2005).
    [CrossRef]
  31. F. Wang and A. Lakhtakia, "Response of slanted chiral sculptured thin films to dipolar sources," Opt. Commun. 235, 133-151 (2004).
    [CrossRef]
  32. M. D. Pickett, A. Lakhtakia, and J. A. PoloJr., "Spectral responses of gytrotropic chiral sculptured thin films to obliquely incident plane waves," Optik 9, 393-398 (2004).
    [CrossRef]
  33. X.-H. Xu and A. J. Bard, "Immobilization and hybridization of DNA on an aluminum (III) alkanebisphophonate thin film with electrogenerated chemiluminescent detection," J. Am. Chem. Soc. 117, 2627-2631 (1995).
    [CrossRef]
  34. W. Tabbara, V. Rannou, and S. Salio, "Statistical approaches to scattering," in: Introduction to Complex Mediums for Optics and Electromagnetics, pp. 591-608,W. S.Weiglhofer and A. Lakhtakia, eds., SPIE Press, Bellingham, WA, USA (2003).

2008 (1)

A. Lakhtakia, M. C. Demirel, M.W. Horn, and J. Xu, "Six emerging directions in sculptured-thin-film research," Adv. Solid State Phys. 46 (2008); in press.

2007 (2)

L. De Stefano, I. Rendina, A. M. Rossi, M. Rossi, L. Rotiroti, and S. D’Auria, "Biochips at work: porous silicon microbiosensor for proteomic diagnostic," J. Phys.: Condens. Matter 19, 395007 (2007).
[CrossRef]

F. Zhang, J. Xu, A. Lakhtakia, S. M. Pursel, and M. W. Horn, "Circularly polarized emission from colloidal nanocrystal quantum dots confined in microcavities formed by chiral mirrors," Appl. Phys. Lett. 91, 023102 (2007). [Interchange the labels LCP and RCP in Fig. 2c of this paper.]
[CrossRef]

2006 (4)

P. C. P. Hrudey, K. L. Westra, and M. J. Brett, "Highly ordered organic Alq3 chiral luminescent thin films fabricated by glancing-angle deposition," Adv. Mater. 18, 224-228 (2006).
[CrossRef]

J. Xu, A. Lakhtakia, J. Liou, A. Chen, and I. J. Hodgkinson, "Circularly polarized fluorescence from light- emitting microcavities with sculptured-thin-film chiral reflectors," Opt. Commun. 264, 235-239 (2006).
[CrossRef]

A. Dorfman, N. Kumar, and J.-i. Hahm, "Highly sensitive biomolecular fluorescence detection using nanoscale ZnO platforms," Langmuir 22, 4890-4895 (2006).
[CrossRef] [PubMed]

H. Tan, O. Ezekoye, J. van der Schalie, M.W. Horn, A. Lakhtakia, J. Xu, andW. D. Burgos, "Biological reduction of nanoengineered iron III oxide sculptured thin films," Environ. Sci. Technol. 40, 5490-5495 (2006).
[CrossRef] [PubMed]

2004 (2)

F. Wang and A. Lakhtakia, "Response of slanted chiral sculptured thin films to dipolar sources," Opt. Commun. 235, 133-151 (2004).
[CrossRef]

M. D. Pickett, A. Lakhtakia, and J. A. PoloJr., "Spectral responses of gytrotropic chiral sculptured thin films to obliquely incident plane waves," Optik 9, 393-398 (2004).
[CrossRef]

2003 (2)

A. Ishchenko, "Molecular engineering of dye-doped polymers for optoelectronics," Polym. Adv. Technol. 13, 744-752 (2003).
[CrossRef]

A. Lakhtakia, "On radiation from canonical source configurations embedded in structurally chiral materials," Microwave Opt. Technol. Lett. 37, 37-40 (2003).
[CrossRef]

2001 (3)

S. Chan, Y. Li, L. J. Rothberg, B. L. Miller, and P. M. Fauchet, "Nanoscale silicon microcavities for biosensing," Mater. Sci. Eng. C 15, 277-282 (2001).
[CrossRef]

A. Lakhtakia, "On bioluminescent emission from chiral sculptured thin films," Opt. Commun. 188, 313-320 (2001).
[CrossRef]

I. Hodgkinson and Q. h. Wu, "Inorganic chiral optical materials," Adv. Mater. 13, 889-897 (2001).
[CrossRef]

2000 (3)

Q. Wu, I. J. Hodgkinson, and A. Lakhtakia, "Circular polarization filters made of chiral sculptured thin films: experimental and simulation results," Opt. Eng. 39, 1863-1868 (2000).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, A. Lakhtakia, and M. W. McCall, "Spectral-hole filter fabricated using scultured thin-film technology," Opt. Commun. 177, 79-84 (2000).
[CrossRef]

R. Messier, V. C. Venugopal, and P. D. Sunal, "Origin and evolution of sculptured thin films," J. Vac. Sci. Technol. A 18, 1538-1545 (2000).
[CrossRef]

1997 (1)

A. Lakhtakia and W. S. Weiglhofer, "Green function for radiation and propagation in helicoidal bianisotropic mediums," IEE Proc.-Microw. Antennas Propag. 144, 57-59 (1997).
[CrossRef]

1996 (2)

A. Lakhtakia, R. Messier, M. J. Brett, and K. Robbie, "Sculptured thin films (STFs) for optical, chemical and biological applications," Innovations Mater. Res. 1, 165176 (1996).

K. Robbie, M. J. Brett, and A. Lakhtakia, "Chiral sculptured thin films," Nature 384, 616 (1996).
[CrossRef]

1995 (1)

X.-H. Xu and A. J. Bard, "Immobilization and hybridization of DNA on an aluminum (III) alkanebisphophonate thin film with electrogenerated chemiluminescent detection," J. Am. Chem. Soc. 117, 2627-2631 (1995).
[CrossRef]

1989 (1)

1972 (1)

Adv. Mater. (2)

I. Hodgkinson and Q. h. Wu, "Inorganic chiral optical materials," Adv. Mater. 13, 889-897 (2001).
[CrossRef]

P. C. P. Hrudey, K. L. Westra, and M. J. Brett, "Highly ordered organic Alq3 chiral luminescent thin films fabricated by glancing-angle deposition," Adv. Mater. 18, 224-228 (2006).
[CrossRef]

Adv. Solid State Phys. (1)

A. Lakhtakia, M. C. Demirel, M.W. Horn, and J. Xu, "Six emerging directions in sculptured-thin-film research," Adv. Solid State Phys. 46 (2008); in press.

Appl. Phys. Lett. (1)

F. Zhang, J. Xu, A. Lakhtakia, S. M. Pursel, and M. W. Horn, "Circularly polarized emission from colloidal nanocrystal quantum dots confined in microcavities formed by chiral mirrors," Appl. Phys. Lett. 91, 023102 (2007). [Interchange the labels LCP and RCP in Fig. 2c of this paper.]
[CrossRef]

Environ. Sci. Technol. (1)

H. Tan, O. Ezekoye, J. van der Schalie, M.W. Horn, A. Lakhtakia, J. Xu, andW. D. Burgos, "Biological reduction of nanoengineered iron III oxide sculptured thin films," Environ. Sci. Technol. 40, 5490-5495 (2006).
[CrossRef] [PubMed]

Innovations Mater. Res. (1)

A. Lakhtakia, R. Messier, M. J. Brett, and K. Robbie, "Sculptured thin films (STFs) for optical, chemical and biological applications," Innovations Mater. Res. 1, 165176 (1996).

J. Am. Chem. Soc. (1)

X.-H. Xu and A. J. Bard, "Immobilization and hybridization of DNA on an aluminum (III) alkanebisphophonate thin film with electrogenerated chemiluminescent detection," J. Am. Chem. Soc. 117, 2627-2631 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys.: Condens. Matter (1)

L. De Stefano, I. Rendina, A. M. Rossi, M. Rossi, L. Rotiroti, and S. D’Auria, "Biochips at work: porous silicon microbiosensor for proteomic diagnostic," J. Phys.: Condens. Matter 19, 395007 (2007).
[CrossRef]

J. Vac. Sci. Technol. A (1)

R. Messier, V. C. Venugopal, and P. D. Sunal, "Origin and evolution of sculptured thin films," J. Vac. Sci. Technol. A 18, 1538-1545 (2000).
[CrossRef]

Langmuir (1)

A. Dorfman, N. Kumar, and J.-i. Hahm, "Highly sensitive biomolecular fluorescence detection using nanoscale ZnO platforms," Langmuir 22, 4890-4895 (2006).
[CrossRef] [PubMed]

Mater. Sci. Eng. C (1)

S. Chan, Y. Li, L. J. Rothberg, B. L. Miller, and P. M. Fauchet, "Nanoscale silicon microcavities for biosensing," Mater. Sci. Eng. C 15, 277-282 (2001).
[CrossRef]

Microw. Antennas Propag. (1)

A. Lakhtakia and W. S. Weiglhofer, "Green function for radiation and propagation in helicoidal bianisotropic mediums," IEE Proc.-Microw. Antennas Propag. 144, 57-59 (1997).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

A. Lakhtakia, "On radiation from canonical source configurations embedded in structurally chiral materials," Microwave Opt. Technol. Lett. 37, 37-40 (2003).
[CrossRef]

Nature (1)

K. Robbie, M. J. Brett, and A. Lakhtakia, "Chiral sculptured thin films," Nature 384, 616 (1996).
[CrossRef]

Opt. Commun. (4)

J. Xu, A. Lakhtakia, J. Liou, A. Chen, and I. J. Hodgkinson, "Circularly polarized fluorescence from light- emitting microcavities with sculptured-thin-film chiral reflectors," Opt. Commun. 264, 235-239 (2006).
[CrossRef]

A. Lakhtakia, "On bioluminescent emission from chiral sculptured thin films," Opt. Commun. 188, 313-320 (2001).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, A. Lakhtakia, and M. W. McCall, "Spectral-hole filter fabricated using scultured thin-film technology," Opt. Commun. 177, 79-84 (2000).
[CrossRef]

F. Wang and A. Lakhtakia, "Response of slanted chiral sculptured thin films to dipolar sources," Opt. Commun. 235, 133-151 (2004).
[CrossRef]

Opt. Eng. (1)

Q. Wu, I. J. Hodgkinson, and A. Lakhtakia, "Circular polarization filters made of chiral sculptured thin films: experimental and simulation results," Opt. Eng. 39, 1863-1868 (2000).
[CrossRef]

Optik (1)

M. D. Pickett, A. Lakhtakia, and J. A. PoloJr., "Spectral responses of gytrotropic chiral sculptured thin films to obliquely incident plane waves," Optik 9, 393-398 (2004).
[CrossRef]

Polym. Adv. Technol. (1)

A. Ishchenko, "Molecular engineering of dye-doped polymers for optoelectronics," Polym. Adv. Technol. 13, 744-752 (2003).
[CrossRef]

Other (11)

F. Boxberg and J. Tulkki, "Quantum dots: Phenomenology, photonic and electronic properties, modeling and technology," in: Nanometer Structures — Theory, Modeling, and Simulation, pp. 107-143, A. Lakhtakia, ed., SPIE Press, Bellingham, WA, USA (2004).

M. P. C. M. Krijn, "Electromagnetic wave propagation in stratified anisotropic media in the presence of sources," Opt. Lett. 17, 163-165 (1992). [Although Eq. (10) of this paper is not rigorously valid unless the matrix ⊗(z) therein is either diagonal or independent of z, it can be useful with the piecewise uniform approximation technique provided a space-ordering operator is implemented on its right side [25].]
[CrossRef] [PubMed]

A. Lakhtakia and M. W. McCall, "Response of chiral sculptured thin films to dipolar sources," Int. J. Electron. Commun. (AE ¨ U) 57, 23-32 (2003).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, Appendix III, 7th ed. (Pergamon, Oxford, UK, 1999).

F. Wang, "Note on the asymptotic approximation of a double integral with an angular-spectrum representation," Int. J. Electron. Commun. (AEU) 59, 258-261 (2005).
[CrossRef]

W. Tabbara, V. Rannou, and S. Salio, "Statistical approaches to scattering," in: Introduction to Complex Mediums for Optics and Electromagnetics, pp. 591-608,W. S.Weiglhofer and A. Lakhtakia, eds., SPIE Press, Bellingham, WA, USA (2003).

A. Lakhtakia and R. Messier, Sculptured Thin Films: Nanoengineered Morphology and Optics (SPIE Press, Bellingham, WA, USA, 2005).
[CrossRef]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, New York, NY, USA, 1993).

J. A. PoloJr., "Sculptured thin films," in: Micromanufacturing and Nanotechnology, pp. 357-381, N. P. Mahalik, ed., Springer, Heidelberg, Germany (2005).

S. K. Arya, A. Chaubey, and B. D. Malhotra, "Fundamentals and applications of biosensors," Proc. Ind. Natn. Sci. Acad. 72, 249-266 (2006).

B. Valeur, Molecular Fluorescence: Principles and Applications (Wiley-VCH, Weinheim, Germany, 2002).

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Figures (8)

Fig. 1.
Fig. 1.

Projections of PLCP onto the z = 0 plane for the regions zobs > L (top row) and z < - L (bottom row), when z -1 (d,h)∙u J ∈ {u n ,u τ,u b }, λ0 = 727nm, d = 0, and ε a = ε b = ε c = 1.

Fig. 2.
Fig. 2.

Projections of PLCP and PRCP onto the z = 0 plane for the regions zobs > L (top two rows) and z < - L (bottom two rows), when u J = z (d,h)∙u n 0 ∈ {652,727,802} nm and h = 1. The relative permittivity parameters are prescribed by eqn. (40). The dipole source is located at d = 0.

Fig. 3.
Fig. 3.

As Fig. 2 but with u J = z (d,h)∙u τ.

Fig. 4.
Fig. 4.

As Fig. 2 but with u J = z (d,h)∙u b .

Fig. 5.
Fig. 5.

Projections of P LCP onto the z = 0 plane for the reqions zobs > L (top row) and z < - L (bottom row), when z (d,h)∙u J ∈ {u n ,u τ,u b }, λ0 = 727nm and h = 1. The CSTF has been replaced by an isotropic material with relative permittivity that is the arithmetic mean of εa, ε b , and ε c of eqn. (40). The dipole source is located at d = 0.

Fig. 6.
Fig. 6.

Projections of PLCP and PRCP onto the z = 0 plane for the regions zobs > L (top two rows) and zobs < -L (bottom two rows), when u J = z (d,h)∙u n 0 ∈ {652,727,802} nm and h = 1. The relative permittivity parameters are prescribed by eqn. (40). The dipole source is located at d = L - 40 nm.

Fig. 7.
Fig. 7.

As Fig. 6 but with u J = z (d,h)∙u τ.

Fig. 8.
Fig. 8.

As Fig. 6 but with u J = z (d,h)∙u b .

Equations (40)

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

D ( r , ω ) = ε 0 ε ͇ r CSTF ( z , ω ) E ( r , ω ) , B ( r , ω ) = μ 0 H ( r , ω ) .
ε ͇ r CSTF ( z , ω ) = S ͇ z ( z , h ) S ͇ y ( χ ) ε ͇ ref ( ω ) S ͇ y 1 ( χ ) S ͇ z 1 ( z , h ) ,
ε ͇ ref ( ω ) = ε b ( ω ) u x u x + ε c ( ω ) u y u y + ε a ( ω ) u z u z ,
S ͇ y ( χ ) = ( u x u x + u z u z ) cos χ + ( u z u x u x u z ) sin χ + u y u y ,
S ͇ z ( z , h ) = ( u x u x + u y u y ) cos [ h π ( z + L ) Ω ] + ( u y u x u x u y ) sin [ h π ( z + L ) Ω ] + u z u z ,
J r ω =− iωp 2 u J δ ( z d ) δ ( x ) δ ( y ) ,
J ( r , ω ) = 1 4 π 2 0 d κ κ 0 2 π d ψ j z κ ψ ω exp [ i κ ( x cos ψ + y sin ψ ) ) ,
u τ = u x cos χ + u z sin χ , u n = u x sin χ + u z cos χ , u b = u y .
J r κ ψ ω = j z κ ψ ω exp [ ( x cos ψ + y sin ψ ) ] ,
E r κ ψ ω = e z κ ψ ω exp [ ( x cos ψ + y sin ψ ) ] ,
H r κ ψ ω = h z κ ψ ω exp [ ( x cos ψ + y sin ψ ) ] .
e z z κ ψ ω = 1 ε a cos 2 χ + ε b sin 2 χ ( ( ε a ε b ) sin χ cos χ × { e x z κ ψ ω cos [ h π ( z + L ) Ω ] + e y z κ ψ ω sin [ h π ( z + L ) Ω ] } + 1 ω ε 0 { κ [ h x z κ ψ ω sin ψ h y z κ ψ ω cos ψ ] ij z z κ ψ ω } )
h z z κ ψ ω = κ ω μ 0 [ e x z κ ψ ω sin ψ e y z κ ψ ω cos ψ ] ;
d dz [ f z κ ψ ω ] = i [ P z κ ψ ω ] [ f z κ ψ ω ] iωp 2 δ ( z d ) [ g z κ ψ ω ] , z L L ,
[ f z κ ψ ω ] = [ e x z κ ψ ω e y z κ ψ ω h x z κ ψ ω h y z κ ψ ω ]
[ g z κ ψ ω ] = u J u z ε a cos 2 χ + ε b sin 2 χ [ κ cos ψ ω ε 0 κ sin ψ ω ε 0 ( ε a ε b ) sin χ cos χ sin [ h π ( z + L ) Ω ] ( ε b ε a ) sin χ cos χ cos [ h π ( z + L ) Ω ] ] + [ 0 0 u J u y u J u x ] .
[ f z κ ψ ω ] = [ G z L κ ψ ω ] [ f L κ ψ ω ] + L z [ G z z s κ ψ ω ] [ g z s κ ψ ω ] d z s , z [ L , L ] .
[ G ( z , z s , ψ , ω ) ] = [ M ( z , κ , ψ , ω ) ] [ M ( z s , κ , ψ , ω ) ] 1
d dz [ M ( z , κ , ψ , ω ) ] = i [ P ( z , κ , ψ , ω ) ] [ M ( z , κ , ψ , ω ) ]
[ f ( L , κ , ψ , ω ) ] = [ M ( 2 L , ψ , ω ) ] [ f ( L , ψ , ω ) ] + [ M ( 2 L , ψ , ω ) ] [ M ( d + L , ψ , ω ) ] 1 [ g ( d , κ , ψ , ω ) ]
E ( r , κ , ψ , ω ) = 1 2 [ b L ( κ , ψ , ω ) ( i s p ) + b R ( κ , ψ , ω ) ( i s + p ) ] × exp { i [ κ ( x cos ψ + y sin ψ ) α 0 ( z + L ) ] } ,
H ( r , κ , ψ , ω ) = 1 η 0 2 [ b L ( κ , ψ , ω ) ( i s p ) + b R ( κ , ψ , ω ) ( i s + p ) ] × exp { i [ κ ( x cos ψ + y sin ψ ) α 0 ( z + L ) ] } ,
E ( r , κ , ψ , ω ) = 1 2 [ c L ( κ , ψ , ω ) ( i s p + ) c R ( κ , ψ , ω ) ( i s + p + ) ] × exp { i [ κ ( x cos ψ + y sin ψ ) + α 0 ( z L ) ] } ,
H ( r , κ , ψ , ω ) = 1 η 0 2 [ c L ( κ , ψ , ω ) ( i s p + ) c R ( κ , ψ , ω ) ( i s + p + ) ] × exp { i [ κ ( x cos ψ + y sin ψ ) + α 0 ( z L ) ] } ,
s = u x sin ψ + u y cos ψ , p ± = ( α 0 k 0 ) ( u x cos ψ + u y sin ψ ) + ( κ k 0 ) u z
[ f ( L , κ , ψ , ω ) ] = 1 2 [ K ( κ , ψ , ω ) ] [ 0 0 i [ b L ( κ , ψ , ω ) b R ( κ , ψ , ω ) ] [ b L ( κ , ψ , ω ) + b R ( κ , ψ , ω ) ] ]
[ f ( L , κ , ψ , ω ) ] = 1 2 [ K ( κ , ψ , ω ) ] [ i [ c L ( κ , ψ , ω ) c R ( κ , ψ , ω ) ] [ c L ( κ , ψ , ω ) + c R ( κ , ψ , ω ) ] 0 0 ]
[ K ( κ , ψ , ω ) ] = [ sin ψ ( α 0 k 0 ) cos ψ sin ψ ( α 0 k 0 ) cos ψ cos ψ ( α 0 k 0 ) sin ψ cos ψ ( α 0 k 0 ) sin ψ η 0 1 ( α 0 k 0 ) cos ψ η 0 1 sin ψ η 0 1 ( α 0 k 0 ) sin ψ η 0 1 sin ψ η 0 1 ( α 0 k 0 ) sin ψ η 0 1 cos ψ η 0 1 ( α 0 k 0 ) cos ψ η 0 1 cos ψ ]
E r ω = 1 4 π 2 0 0 2 π κ E ( z , κ , ψ , ω ) exp [ ( x cos ψ + y sin ψ ) ] ,
H r ω = 1 4 π 2 0 0 2 π κH ( z , κ , ψ , ω ) exp [ ( x cos ψ + y sin ψ ) ]
E r obs ω i cos θ obs 2 2 π [ b L obs ( i s obs p obs ) b R obs ( i s obs + p obs ) ] exp ( i k 0 r ˜ obs ) k 0 r ˜ obs
H r obs ω cos θ obs η 0 2 2 π [ b L obs ( i s obs p obs ) b R obs ( i s obs + p obs ) ] exp ( i k 0 r ˜ obs ) k 0 r ˜ obs
E r obs ω i cos θ obs 2 2 π [ c L obs ( i s obs p + obs ) c R obs ( i s obs + p + obs ) ] exp ( i k 0 r ˜ + obs ) k 0 r ˜ + obs ,
H r obs ω cos θ obs η 0 2 2 π [ c L obs ( i s obs p + obs ) c R obs ( i s obs + p + obs ) ] exp ( i k 0 r ˜ + obs ) k 0 r ˜ + obs
P r obs ω { 1 2 η 0 ( b L obs 2 + b R obs 2 ) ( cos θ obs 2 π k 0 r ˜ obs ) 2 r ̂ obs , z obs < L 1 2 η 0 ( c L obs 2 + c R obs 2 ) ( cos θ obs 2 π k 0 r ˜ + obs ) 2 r ̂ obs , z obs > L .
P LCP r obs ω { 1 2 η 0 ( b L obs 2 ) ( cos θ obs 2 π k 0 r ˜ obs ) 2 r ̂ obs , z obs < L 1 2 η 0 ( c L obs 2 ) ( cos θ obs 2 π k 0 r ˜ + obs ) 2 r ̂ obs , z obs > L
P RCP r obs ω { 1 2 η 0 ( b R obs 2 ) ( cos θ obs 2 π k 0 r ˜ + obs ) 2 r ̂ obs , z obs < L 1 2 η 0 ( c R obs 2 ) ( cos θ obs 2 π k 0 r ˜ obs ) 2 r ̂ obs , z obs > L ,
P j r obs ω = P j r obs ω ω 2 p 2 × 10 13 , j { LCP , RCP } .
Γ j = 10 3 ψ obs = ρ 1 ρ 2 obs ψ obs = 0 2 π obs ( r obs ) 2 sin θ obs P j r obs ω , j { LCP , RCP } ,
ε a , b , c = 1 + q a , b , c 1 + ( 1 N a , b , c i λ a , b , c λ 0 ) 2 .

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