Abstract

A simple, rigorous electromagnetic formula is derived for predicting the electromagnetic power provided by sources located in transparent or dissipative planar microcavities. With this simple approach, we compare numerically and experimentally the electromagnetic power that escapes the microcavity when the source is located in a metallodielectric or in an all-dielectric resonant planar structure. Although a strong light-extraction coefficient might be expected for metallodielectric microcavities, we show that these attractive structures suffer from metal absorption even when thin metallic layers are used. Experiments implemented with europium chelates located in metallodielectric or in all-dielectric microcavities confirm this result.

© 1999 Optical Society of America

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References

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  1. E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics: Advances in Atomic, Molecular and Optical Physics, P. R. Berman, ed. (Academic, Boston, 1994), p. 1.
  2. S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics (North Holland, Amsterdam, 1991), p. 767.
  3. H. Rigneault, S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 1: basic concepts and analytical trends,” IEEE J. Quantum. Electron. 34, 1612–1631 (1998); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 2: selected exact simulations and role of photon recycling,” IEEE J. Quantum. Electron. 34, 1632–1643 (1998).
    [CrossRef] [PubMed]
  4. H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
    [CrossRef]
  5. I. Abram, G. Bourdon, “Photonic-well microcavities for spontaneous emission control,” Phys. Rev. A 54, 3476–3479 (1996).
    [CrossRef] [PubMed]
  6. D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
    [CrossRef]
  7. E. R. Brown, O. B. McMahaon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
    [CrossRef]
  8. D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
    [CrossRef] [PubMed]
  9. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11,245–11,251 (1996).
    [CrossRef]
  10. H. A. Macleaod, Thin Film Optical Filters, 2nd ed. (Hilger, Bristol, UK, 1986) pp. 292–313.
  11. P. H. Berning, A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. 47, 230–239 (1957).
    [CrossRef]
  12. C. Amra, S. Maure, “Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns,” J. Opt. Soc. Am. A 14, 3102–3113 (1997).
    [CrossRef]
  13. C. Amra, S. Maure, “Mutual coherence and conical pattern of sources optimally excited within multilayer optics,” J. Opt. Soc. Am. A 14, 3114–3124 (1997).
    [CrossRef]
  14. C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
    [CrossRef]
  15. H. Rigneault, S. Monneret, “Field quantization and spontaneous emission in lossless dielectric multilayer structures,” J. Eur. Opt. Soc. Quantum Semiclass. Opt. 9, 1017–1040 (1997).
    [CrossRef]
  16. E. F. Gudgin Dickson, A. Pollak, E. P. Diamandis, “Time-resolved detection of lanthanide luminescence for ultrasensitive bioanalytical assays,” J. Photochem. Photobiol. B. Biol. 27, 3–19 (1995).
    [CrossRef]
  17. C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).
  18. S. Robert, H. Rigneault, F. Lamarque, “Spontaneous emission of Pr ions located in planar dielectric microcavities,” J. Opt. Soc. Am. B 15, 1773–1779 (1998).
    [CrossRef]

1998 (1)

1997 (4)

H. Rigneault, S. Monneret, “Field quantization and spontaneous emission in lossless dielectric multilayer structures,” J. Eur. Opt. Soc. Quantum Semiclass. Opt. 9, 1017–1040 (1997).
[CrossRef]

H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
[CrossRef]

C. Amra, S. Maure, “Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns,” J. Opt. Soc. Am. A 14, 3102–3113 (1997).
[CrossRef]

C. Amra, S. Maure, “Mutual coherence and conical pattern of sources optimally excited within multilayer optics,” J. Opt. Soc. Am. A 14, 3114–3124 (1997).
[CrossRef]

1996 (5)

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11,245–11,251 (1996).
[CrossRef]

I. Abram, G. Bourdon, “Photonic-well microcavities for spontaneous emission control,” Phys. Rev. A 54, 3476–3479 (1996).
[CrossRef] [PubMed]

H. Rigneault, S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 1: basic concepts and analytical trends,” IEEE J. Quantum. Electron. 34, 1612–1631 (1998); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 2: selected exact simulations and role of photon recycling,” IEEE J. Quantum. Electron. 34, 1632–1643 (1998).
[CrossRef] [PubMed]

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

1995 (2)

E. F. Gudgin Dickson, A. Pollak, E. P. Diamandis, “Time-resolved detection of lanthanide luminescence for ultrasensitive bioanalytical assays,” J. Photochem. Photobiol. B. Biol. 27, 3–19 (1995).
[CrossRef]

E. R. Brown, O. B. McMahaon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

1994 (1)

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

1993 (1)

1957 (1)

Abram, I.

I. Abram, G. Bourdon, “Photonic-well microcavities for spontaneous emission control,” Phys. Rev. A 54, 3476–3479 (1996).
[CrossRef] [PubMed]

Amra, C.

Aspe, D.

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

Autiero, H.

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

Begon, C.

H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
[CrossRef]

Berning, P. H.

Bourdon, G.

I. Abram, G. Bourdon, “Photonic-well microcavities for spontaneous emission control,” Phys. Rev. A 54, 3476–3479 (1996).
[CrossRef] [PubMed]

Brown, E. R.

E. R. Brown, O. B. McMahaon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

Cazaux, L.

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

Diamandis, E. P.

E. F. Gudgin Dickson, A. Pollak, E. P. Diamandis, “Time-resolved detection of lanthanide luminescence for ultrasensitive bioanalytical assays,” J. Photochem. Photobiol. B. Biol. 27, 3–19 (1995).
[CrossRef]

Fan, S.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11,245–11,251 (1996).
[CrossRef]

Galaup, C.

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

Gudgin Dickson, E. F.

E. F. Gudgin Dickson, A. Pollak, E. P. Diamandis, “Time-resolved detection of lanthanide luminescence for ultrasensitive bioanalytical assays,” J. Photochem. Photobiol. B. Biol. 27, 3–19 (1995).
[CrossRef]

Haroche, S.

S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics (North Holland, Amsterdam, 1991), p. 767.

Hinds, E. A.

E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics: Advances in Atomic, Molecular and Optical Physics, P. R. Berman, ed. (Academic, Boston, 1994), p. 1.

Ho, K. M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Jacquier, B.

H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
[CrossRef]

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11,245–11,251 (1996).
[CrossRef]

Kroll, N.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Lamarque, F.

Macleaod, H. A.

H. A. Macleaod, Thin Film Optical Filters, 2nd ed. (Hilger, Bristol, UK, 1986) pp. 292–313.

Maure, S.

McMahaon, O. B.

E. R. Brown, O. B. McMahaon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

Monneret, S.

H. Rigneault, S. Monneret, “Field quantization and spontaneous emission in lossless dielectric multilayer structures,” J. Eur. Opt. Soc. Quantum Semiclass. Opt. 9, 1017–1040 (1997).
[CrossRef]

H. Rigneault, S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 1: basic concepts and analytical trends,” IEEE J. Quantum. Electron. 34, 1612–1631 (1998); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 2: selected exact simulations and role of photon recycling,” IEEE J. Quantum. Electron. 34, 1632–1643 (1998).
[CrossRef] [PubMed]

Moretti, P.

H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
[CrossRef]

Picard, C.

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

Pollak, A.

E. F. Gudgin Dickson, A. Pollak, E. P. Diamandis, “Time-resolved detection of lanthanide luminescence for ultrasensitive bioanalytical assays,” J. Photochem. Photobiol. B. Biol. 27, 3–19 (1995).
[CrossRef]

Rigneault, H.

S. Robert, H. Rigneault, F. Lamarque, “Spontaneous emission of Pr ions located in planar dielectric microcavities,” J. Opt. Soc. Am. B 15, 1773–1779 (1998).
[CrossRef]

H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
[CrossRef]

H. Rigneault, S. Monneret, “Field quantization and spontaneous emission in lossless dielectric multilayer structures,” J. Eur. Opt. Soc. Quantum Semiclass. Opt. 9, 1017–1040 (1997).
[CrossRef]

H. Rigneault, S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 1: basic concepts and analytical trends,” IEEE J. Quantum. Electron. 34, 1612–1631 (1998); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 2: selected exact simulations and role of photon recycling,” IEEE J. Quantum. Electron. 34, 1632–1643 (1998).
[CrossRef] [PubMed]

Robert, S.

S. Robert, H. Rigneault, F. Lamarque, “Spontaneous emission of Pr ions located in planar dielectric microcavities,” J. Opt. Soc. Am. B 15, 1773–1779 (1998).
[CrossRef]

H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Sickmiller, M. E.

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Sievenpiper, D. F.

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Sigalas, M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Smith, D. R.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Tisnes, P.

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

Turner, A. F.

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11,245–11,251 (1996).
[CrossRef]

Yablonovitch, E.

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

E. R. Brown, O. B. McMahaon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

J. Eur. Opt. Soc. Quantum Semiclass. Opt. (1)

H. Rigneault, S. Monneret, “Field quantization and spontaneous emission in lossless dielectric multilayer structures,” J. Eur. Opt. Soc. Quantum Semiclass. Opt. 9, 1017–1040 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

J. Photochem. Photobiol. B. Biol. (1)

E. F. Gudgin Dickson, A. Pollak, E. P. Diamandis, “Time-resolved detection of lanthanide luminescence for ultrasensitive bioanalytical assays,” J. Photochem. Photobiol. B. Biol. 27, 3–19 (1995).
[CrossRef]

New J. Chem. (1)

C. Galaup, C. Picard, L. Cazaux, P. Tisnes, D. Aspe, H. Autiero, “Synthesis and luminescence of Eu3+ complexes derived from novel receptors containing a tetralactam unit,” New J. Chem. 20, 997–999 (1996).

Phys. Rev. A (3)

H. Rigneault, S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 1: basic concepts and analytical trends,” IEEE J. Quantum. Electron. 34, 1612–1631 (1998); H. Benisty, H. De Neve, C. Weisbuch, “Impact of planar microcavity effects on light extraction. Part 2: selected exact simulations and role of photon recycling,” IEEE J. Quantum. Electron. 34, 1632–1643 (1998).
[CrossRef] [PubMed]

H. Rigneault, S. Robert, C. Begon, B. Jacquier, P. Moretti, “Radiative and guided wave emission of Er3+ atoms located in planar multidielectric structures,” Phys. Rev. A 55, 1497–1502 (1997).
[CrossRef]

I. Abram, G. Bourdon, “Photonic-well microcavities for spontaneous emission control,” Phys. Rev. A 54, 3476–3479 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B 54, 11,245–11,251 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Other (3)

H. A. Macleaod, Thin Film Optical Filters, 2nd ed. (Hilger, Bristol, UK, 1986) pp. 292–313.

E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics: Advances in Atomic, Molecular and Optical Physics, P. R. Berman, ed. (Academic, Boston, 1994), p. 1.

S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics (North Holland, Amsterdam, 1991), p. 767.

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Figures (10)

Fig. 1
Fig. 1

Definitions of radiative and trapped light.

Fig. 2
Fig. 2

Coordinate frame.

Fig. 3
Fig. 3

Microcavity structure and spectral-power density emitted by a surface source at λ0 = 550 nm located in the air–HLHLHLH L′ Al L′ HLHLHLH–glass cavity.

Fig. 4
Fig. 4

(a) Sample D design and (b) sample M design: Eu–chelate location.

Fig. 5
Fig. 5

Bieuropium chelate (upper drawing) and the absorption spectrum in a methanol solution (lower graph).

Fig. 6
Fig. 6

Europium chelate’s normalized spectrum and the sample D’s microcavity transmittance. The structure is air–HLH L (Eu–chelate) (3L) HLH–glass. The y axis is in arbitrary units.

Fig. 7
Fig. 7

Europium chelate’s normalized spectrum and the sample M’s microcavity transmittance. The structure is air–HLH 0.979L (Eu–chelate) 0.979L Al 1.958L HLH–glass. The y axis is in arbitrary units.

Fig. 8
Fig. 8

Experimental setup.

Fig. 9
Fig. 9

Predicted radiation patterns for the two samples at λ = 615 nm.

Fig. 10
Fig. 10

Experimental radiation pattern for the two samples at λ = 615 nm.

Equations (12)

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

F=Φ+A,
rot E=jωμH+M,  rot H=-jωE+J,
Fii=14rM¯iHii+Hii-JiE¯ii+E¯iidr,
fiiσ, ϕ=dFiidσdϕ=π2σ[Mˆ¯iHˆii+Hˆii-JˆiEˆ¯ii+Eˆ¯ii,
fiiσ, ϕ=4π22σαii|Mˆi|2+βii|Jˆi|2+γiiziJˆi  Mˆi, αii=YiYiΔYi,  γii=2J ImYiΔYi, βii=-1ΔY¯i,  Yi=HˆiiEˆii,
fiiσ, ϕ=2π2σβii|Jˆi|2.
RedFdϕ=4π2λν*=0maxn0,ns Refν*dν*.
RedFdϕ=jπ pResiduf, σp=4π2λν*=maxn0,nsmaxni fν*, ϕdν*,
RedFdϕ=4π2λν*=0Refν*dν*.
dΦdϕ=ν*=0dΦdϕdν*.
E=Poynting flux powerTotal provided power=02πdϕ ν*=0ν0*minn0,ns RedΦν*, ϕdϕdν*02πdϕ ν*=0 RedFν*, ϕdϕdν*,
01Refν*dν*0Refν*dν*=0.4510.579=0.78,

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