Abstract

Fourier modal method based quantitative analysis method of optical power flow and energy loss in general multi-block photonic structures with an internal dipole emitter is described. The analytic expressions of modal power flow and loss are derived for accurate and efficient quantitative analysis. It is revealed that a few dominating excited photonic modes substantially govern the internal energy flow and energy loss. The optical characteristics of the dominant modes are investigated.

© 2014 Optical Society of America

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  1. R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
    [CrossRef]
  2. S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
    [CrossRef]
  3. M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
    [CrossRef]
  4. M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
    [CrossRef]
  5. J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
    [CrossRef] [PubMed]
  6. S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
    [CrossRef] [PubMed]
  7. S. Demtsu and J. Sites, “Quantification of losses in thin-film CdS/CdTe solar cells,” Proc. IEEE Photovolatic Specialists Conf.31, 347–350 (2005).
    [CrossRef]
  8. W. Lee, S.-Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherent length,” Opt. Express 20(S6), A941–A953 (2012).
    [CrossRef]
  9. S. Lee, S. In, D. R. Mason, and N. Park, “Incorporation of nanovoids into metallic gratings for broadband plasmonic organic solar cells,” Opt. Express 21(4), 4055–4060 (2013).
    [CrossRef] [PubMed]
  10. Y. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2010).
    [CrossRef]
  11. K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2004).
  12. H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, Boca Raton, FL, 2012).
  13. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2006).
  14. K. A. Neyts, “Simulation of light emission from thin-film microcavities,” J. Opt. Soc. Am. A 15(4), 962–971 (1998).
    [CrossRef]
  15. D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: Self-interference of Bloch eigenwaves,” Phys. Rev. A 79(3), 033829 (2009).
    [CrossRef]
  16. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18(11), 2865–2875 (2001).
    [CrossRef] [PubMed]

2013 (3)

J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
[CrossRef] [PubMed]

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

S. Lee, S. In, D. R. Mason, and N. Park, “Incorporation of nanovoids into metallic gratings for broadband plasmonic organic solar cells,” Opt. Express 21(4), 4055–4060 (2013).
[CrossRef] [PubMed]

2012 (3)

W. Lee, S.-Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherent length,” Opt. Express 20(S6), A941–A953 (2012).
[CrossRef]

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
[CrossRef]

M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
[CrossRef]

2010 (2)

R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[CrossRef]

Y. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2010).
[CrossRef]

2009 (1)

D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: Self-interference of Bloch eigenwaves,” Phys. Rev. A 79(3), 033829 (2009).
[CrossRef]

2008 (1)

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
[CrossRef]

2001 (1)

1998 (1)

Brütting, W.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
[CrossRef]

Cao, Q.

Char, K.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Chembo, Y.

Y. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2010).
[CrossRef]

Chigrin, D. N.

D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: Self-interference of Bloch eigenwaves,” Phys. Rev. A 79(3), 033829 (2009).
[CrossRef]

Demtsu, S.

S. Demtsu and J. Sites, “Quantification of losses in thin-film CdS/CdTe solar cells,” Proc. IEEE Photovolatic Specialists Conf.31, 347–350 (2005).
[CrossRef]

Frischeisen, J.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
[CrossRef]

Furno, M.

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
[CrossRef]

M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
[CrossRef]

R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[CrossRef]

Gather, M.

M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
[CrossRef]

Hofmann, S.

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
[CrossRef]

R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[CrossRef]

Hugonin, J.-P.

In, S.

Jung, J.-H.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Kang, Y. S.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Kim, J.

Kim, J.-B.

J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
[CrossRef] [PubMed]

Kim, J.-J.

J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
[CrossRef] [PubMed]

Kim, S. C.

Kim, S.-Y.

J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
[CrossRef] [PubMed]

Koh, J. H.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Krummacher, B. C.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
[CrossRef]

Lalanne, P.

Lee, B.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

W. Lee, S.-Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherent length,” Opt. Express 20(S6), A941–A953 (2012).
[CrossRef]

Lee, J.-H.

J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
[CrossRef] [PubMed]

Lee, S.

Lee, S.-Y.

Lee, W.

Lee, Y.-G.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Leo, K.

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
[CrossRef]

M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
[CrossRef]

R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[CrossRef]

Lússem, B.

R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[CrossRef]

Lüssem, B.

M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
[CrossRef]

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
[CrossRef]

Mason, D. R.

Meerheim, R.

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
[CrossRef]

R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[CrossRef]

Moon, C. K.

J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
[CrossRef] [PubMed]

Neyts, K. A.

Nowy, S.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
[CrossRef]

Park, N.

Reinke, N. A.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
[CrossRef]

Rosenow, T.

M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
[CrossRef]

Silberstein, E.

Sites, J.

S. Demtsu and J. Sites, “Quantification of losses in thin-film CdS/CdTe solar cells,” Proc. IEEE Photovolatic Specialists Conf.31, 347–350 (2005).
[CrossRef]

Wooh, S.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Yoon, H.

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Yu, N.

Y. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2010).
[CrossRef]

Adv. Mater. (2)

J.-B. Kim, J.-H. Lee, C. K. Moon, S.-Y. Kim, and J.-J. Kim, “Highly enhanced light extraction from surface plasmonic loss minimized organic light-emitting diodes,” Adv. Mater. 25(26), 3571–3577 (2013).
[CrossRef] [PubMed]

S. Wooh, H. Yoon, J.-H. Jung, Y.-G. Lee, J. H. Koh, B. Lee, Y. S. Kang, and K. Char, “Efficient light harvesting with micropatterned 3D pyramidal photoanodes in dye-sensitized solar cells,” Adv. Mater. 25(22), 3111–3116 (2013).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

R. Meerheim, M. Furno, S. Hofmann, B. Lússem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[CrossRef]

M. Furno, T. Rosenow, M. Gather, B. Lüssem, and K. Leo, “Analysis of the external and internal quantum efficiency of multi-emitter, white organic light emitting diodes,” Appl. Phys. Lett. 101(14), 143304 (2012).
[CrossRef]

J. Appl. Phys. (1)

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brütting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104(12), 123109 (2008).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (2)

Phys. Rev. A (2)

D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: Self-interference of Bloch eigenwaves,” Phys. Rev. A 79(3), 033829 (2009).
[CrossRef]

Y. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2010).
[CrossRef]

Phys. Rev. B (1)

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B 85(11), 115205 (2012).
[CrossRef]

Other (4)

S. Demtsu and J. Sites, “Quantification of losses in thin-film CdS/CdTe solar cells,” Proc. IEEE Photovolatic Specialists Conf.31, 347–350 (2005).
[CrossRef]

K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2004).

H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, Boca Raton, FL, 2012).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2006).

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Figures (6)

Fig. 1
Fig. 1

(a) Dipole emission in free space and (b) vector optical field distributions generated by the dipole emitters with wavelength λ = 532 nm and polarizations of P T E = ( 1 , 0 , 0 ) and P T M = ( 0 , 0 , 1 ) . (c) Dipole emission in a finite-size photonic structure and (d) vector optical field distributions generated by the dipole emitters with respective polarizations.

Fig. 2
Fig. 2

(a) Modal power spectrum analysis scheme and (b) the classification of optical modes associated with the photonic structure: radiative, leaky, bound, and free-space modes

Fig. 3
Fig. 3

Modal power spectra of the (a) (positive) and (b) (negative) radiative modes, and (c) (positive) and (d) (negative) leaky modes for PTE = (1,0,0). The red circles indicate the modes with the highest power selectively and the respective insets present the vector field distributions of the selected Bloch eigenmodes.

Fig. 4
Fig. 4

Modal power spectra of (a) (positive) and (b) (negative) radiative modes, and (c) (positive) and (d) (negative) leaky modes for PTM = (0,0,1). The red circles indicate the modes with the highest power selectively and the respective insets present the vectorial field distributions of the selected Bloch eigenmodes.

Fig. 5
Fig. 5

Modal power spectra of the net radiative modes and net leaky modes for (a) PTE and for (b) PTM

Fig. 6
Fig. 6

Total internal radiation powers of the dipole are varied with changing h. (a) h is the distance between the 8nm thick metal block and the dipole line source. (b) Total internal radiation powers of the dipole source with changing h.

Tables (4)

Tables Icon

Table 1 Modal Power Spectrum of the Dominant Radiative and Leaky Modes

Tables Icon

Table 2 Analysis Result of Total Power and Energy Loss in Total Optical Field

Tables Icon

Table 3 Quantitative Analysis of Total Power Flow and Energy Loss

Tables Icon

Table 4 Contribution Ratio of Dominant Photonic Modes to Total External Radiation

Equations (21)

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

× E = j ω μ 0 H ,
× H = j ω ε 0 ε r E j ω P δ ( r ) ,
G ± ( r ) = j ε r 2 π k x k y 1 k z ( 1 p 2 p q p m q p 1 q 2 q m m p m q 1 m 2 ) e j ( k x x + k y y + k z | z | ) d k y d k x for ± z > 0 ,
( p , q , m ) = ( k x / ( k 0 ε r ) , k y / ( k 0 ε r ) , 1 p 2 q 2 ) ,
E = μ 0 ω 2 4 π ε r G ± ( r ) P for ± z 0 ,
H = ( j ω μ 0 ) 1 × E .
( E H )= g=1 M + C g + ( E (g) + H (g) + )+ g=1 M C g ( E (g) H (g) ) ,
( E ( g ) H ( g ) )= e j( k x,0 x+ k y,0 y+ k z,0 ( g ) z ) m=M M n=N N ( E x,m,n ( g ) E y,m,n ( g ) E z,m,n ( g ) H x,m,n ( g ) H y,m,n ( g ) H z,m,n ( g ) ) e j( 2πm T x x+ 2πn T y y ) ,
S= 1 2 Re{ E× H * }= 1 2 Re{ g=1 G | C g | 2 E ( g ) × H ( g )* }+ 1 2 Re{ g g G C g C g * E ( g ) × H ( g )* } = g=1 G | C g | 2 S g + g g G M g, g .
S ( g ) = 1 2 Re{ E ( g ) × H ( g )* } ,
E ( g ) × H ( g )* = e 2 β ( g ) z m,n m , n ( E y,m,n ( g ) H z,m',n' ( g )* E z,m,n ( g ) H y,m',n' ( g )* E z,m,n ( g ) H x,m',n' ( g )* E x,m,n ( g ) H z,m',n' ( g )* E x,m,n ( g ) H y,m',n' ( g )* E y,m,n ( g ) H x,m',n' ( g )* ) e j( 2π( m m ) T x x+ 2π( n n ) T y y ) .
P z ( g ) ( z= z ± )= 1 W x W y W y /2 W y /2 W x /2 W x /2 S ( g ) z ^ dxdy = e 2 β ( g ) z ± m=M M n=N N m =M M n =N N 1 2 Re[ E x,m,n ( g ) H y, m , n ( g )* E y,m,n ( g ) H x, m , n ( g )* ]sinc( ( m m ) W x T x )sinc( ( n n ) W y T y ) .
Q z ( g ) = | C g | 2 P z ( g ) W x W y .
P x ( g ) | x= ± W x 2 = 1 ( z + z ) W y W y /2 W y /2 z z + S ( g ) x ^ dzdy = [ e 2 β ( g ) z + e 2 β ( g ) z ] 2( z + z )( 2 β ( g ) ) Re( m,n m , n [ E y,m,n ( g ) H z,m',n' ( g )* E z,m,n ( g ) H y,m',n' ( g )* ] e j( 2π( m m ) T x ( ± W x 2 ) ) )sinc( ( n n ) W y T y ),
P y ( g ) | y= ± W y 2 = 1 ( z + z ) W x W x /2 W x /2 z z + S ( g ) y ^ dzdy = [ e 2 β ( g ) z + e 2 β ( g ) z ] 2( z + z )( 2 β ( g ) ) Re( m,n m , n [ E z,m,n ( g ) H x,m',n' ( g )* E x,m,n ( g ) H z,m',n' ( g )* ] e j( 2π( n n ) T y ( ± W y 2 ) ) )sinc( ( m m ) W x T x ).
Q x ( g ) = | C g | 2 P x ( g ) ( z + z ) W y ,
Q y ( g ) = | C g | 2 P y ( g ) ( z + z ) W x .
( E H )=( E x E y E z H x H y H z )= e j( k x,0 x+ k y,0 y ) m,n ( g C g e j k z ( g ) z E x,m,n ( g ) g C g e j k z ( g ) z E y,m,n ( g ) g C g e j k z ( g ) z E z,m,n ( g ) g C g e j k z ( g ) z H x,m,n ( g ) g C g e j k z ( g ) z H y,m,n ( g ) g C g e j k z ( g ) z H z,m,n ( g ) ) e j( 2πm T x x+ 2πn T y y )
P total ( z )= 1 W x W y W y /2 W y /2 W x /2 W x /2 1 2 Re( E× H * ) z ^ dxdy = g g C g C g * e j k z ( g ) z ( e j k z ( g ) z ) * 1 2 Re{ m,n, m , n [ E x,m,n ( g ) H y, m , n ( g )* E y,m,n ( g ) H x, m , n ( g )* ]sinc( ( m m ) W x T x )sinc( ( n n ) W y T y ) } .
P x | x= ± W x 2 = 1 ( z + z ) W y W y /2 W y /2 z z + 1 2 Re( E× H * ) x ^ dzdy = 1 ( z + z ) g g C g C g * z z + e j k z ( g ) z ( e j k z ( g ) z ) * dz 1 2 Re{ m,n, m , n [ E y,m,n ( g ) H z, m , n ( g )* E z,m,n ( g ) H y, m , n ( g )* ] e j( 2π( m m ) T x ( ± W x 2 ) ) sinc( ( n n ) W y T y ) } ,
P y | y= ± W y 2 = 1 ( z + z ) W x W x /2 W x /2 z z + 1 2 Re( E× H * ) x ^ dzdx = 1 ( z + z ) g g C g C g * z z + e j k z ( g ) z ( e j k z ( g ) z ) * dz 1 2 Re{ m,n, m , n [ E z,m,n ( g ) H x, m , n ( g )* E x,m,n ( g ) H z, m , n ( g )* ] e j( 2π( n n ) T y ( ± W y 2 ) ) sinc( ( m m ) W x T x ) } .

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