Abstract

Two different versions of an optical theorem for a scattering body embedded inside a lossy background medium are derived in this paper. The corresponding fundamental upper bounds on absorption are then obtained in closed form by elementary optimization techniques. The first version is formulated in terms of polarization currents (or equivalent currents) inside the scatterer and generalizes previous results given for a lossless medium. The corresponding bound is referred to here as a variational bound and is valid for an arbitrary geometry with a given material property. The second version is formulated in terms of the T-matrix parameters of an arbitrary linear scatterer circumscribed by a spherical volume and gives a new fundamental upper bound on the total absorption of an inclusion with an arbitrary material property (including general bianisotropic materials). The two bounds are fundamentally different as they are based on different assumptions regarding the structure and the material property. Numerical examples including homogeneous and layered (core-shell) spheres are given to demonstrate that the two bounds provide complimentary information in a given scattering problem.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (6)

H. Shim, L. Fan, S. G. Johnson, and O. D. Miller, “Fundamental limits to near-field optical response over any bandwidth,” Phys. Rev. X 9(1), 011043 (2019).
[Crossref]

S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
[Crossref]

S. Nordebo, M. Mirmoosa, and S. Tretyakov, “On the quasistatic optimal plasmonic resonances in lossy media,” J. Appl. Phys. 125(10), 103105 (2019).
[Crossref]

M. I. Mishchenko and J. M. Dlugach, “Multiple scattering of polarized light by particles in an absorbing medium,” Appl. Opt. 58(18), 4871–4877 (2019).
[Crossref]

R. Dezert, P. Richetti, and A. Baron, “Complete multipolar description of reflection and transmission across a metasurface for perfect absorption of light,” Opt. Express 27(19), 26317–26330 (2019).
[Crossref]

A. K. Skrivervik, M. Bosiljevac, and Z. Sipus, “Fundamental limits for implanted antennas: Maximum power density reaching free space,” IEEE Trans. Antennas Propag. 67(8), 4978–4988 (2019).
[Crossref]

2017 (4)

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

M. I. Mishchenko, G. Videen, and P. Yang, “Extinction by a homogeneous spherical particle in an absorbing medium,” Opt. Lett. 42(23), 4873–4876 (2017).
[Crossref]

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. H. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, J. J. H. T. Furtenbacher, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

2016 (1)

2015 (1)

W. Al-Taay, S. F. Oboudi, E. Yousif, M. A. Nabi, R. M. Yusop, and D. Derawi, “Fabrication and characterization of nickel chloride doped PMMA films,” Adv. Mater. Sci. Eng. 2015, 1–5 (2015).
[Crossref]

2014 (2)

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

S. Tretyakov, “Maximizing absorption and scattering by dipole particles,” Plasmonics 9(4), 935–944 (2014).
[Crossref]

2012 (6)

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radio-frequency electric field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
[Crossref]

A. E. Krasnok, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “All-dielectric optical nanoantennas,” Opt. Express 20(18), 20599–20604 (2012).
[Crossref]

F. Merli, L. Bolomey, F. Gorostidi, B. Fuchs, J.-F. Zurcher, Y. Barrandon, E. Meurville, J. R. Mosig, and A. K. Skrivervik, “Example of data telemetry for biomedical applications: An in vivo experiment,” IEEE Antennas Wirel. Propag. Lett. 11, 1650–1654 (2012).
[Crossref]

J. Wang, H. Zhang, T. Lv, and T. A. Gulliver, “Capacity of 60 GHz wireless communication systems over fading channels,” J. Networks 7(1), 203–209 (2012).
[Crossref]

A. E. Miroshnichenko, B. Luk’yanchuk, S. A. Maier, and Y. S. Kivshar, “Optically induced interaction of magnetic moments in hybrid metamaterials,” ACS Nano 6(1), 837–842 (2012).
[Crossref]

2011 (2)

F. Merli, B. Fuchs, J. R. Mosig, and A. K. Skrivervik, “The effect of insulating layers on the performance of implanted antennas,” IEEE Trans. Antennas Propag. 59(1), 21–31 (2011).
[Crossref]

M. Gustafsson and D. Sjöberg, “Physical bounds and sum rules for high-impedance surfaces,” IEEE Trans. Antennas Propag. 59(6), 2196–2204 (2011).
[Crossref]

2008 (1)

X. Huang, P. K. Jain, I. H. El-Sayed, and M. A. El-Sayed, “Plasmonic photothermal therapy (PPTT) using gold nanoparticles,” Lasers Med. Sci. 23(3), 217–228 (2008).
[Crossref]

2007 (5)

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A 24(9), 2943–2952 (2007).
[Crossref]

C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on metamaterials: Restrictions on scattering and absorption over a frequency interval,” J. Phys. D: Appl. Phys. 40(22), 7146–7151 (2007).
[Crossref]

M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. London, Ser. A 463(2086), 2589–2607 (2007).
[Crossref]

C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on broadband scattering by heterogeneous obstacles,” J. Phys. A: Math. Theor. 40(36), 11165–11182 (2007).
[Crossref]

C. Park and T. S. Rappaport, “Short-range wireless communications for next-generation networks: UWB, 60 GHz millimeter-wave WPAN, and ZigBee,” IEEE Wirel. Commun. 14(4), 70–78 (2007).
[Crossref]

2006 (2)

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D: Appl. Phys. 39(6), 1226–1229 (2006).
[Crossref]

J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. A 23(11), 2784–2796 (2006).
[Crossref]

2002 (1)

2001 (2)

2000 (1)

K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48(8), 1230–1234 (2000).
[Crossref]

1999 (1)

A. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: Application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

1998 (1)

1985 (1)

N. D. Hawkins, R. Steele, D. C. Rickard, and C. R. Shepherd, “Path loss characteristics of 60 GHz transmissions,” Elect. Lett. 21(22), 1054–1055 (1985).
[Crossref]

1980 (1)

1979 (1)

C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
[Crossref]

1971 (1)

R. Progelhof, J. Franey, and T. W. Haas, “Absorption coefficient of unpigmented poly(methyl methacrylate), polystyrene, polycarbonate and poly(4-methylpentene-1) sheets,” J. Appl. Polym. Sci. 15(7), 1803–1807 (1971).
[Crossref]

1963 (1)

M. L. Meeks and A. E. Lilley, “The microwave spectrum of oxygen in the Earth’s atmosphere,” J. Geophys. Res. 68(6), 1683–1703 (1963).
[Crossref]

1958 (1)

R. Harrington, “On the gain and beamwidth of directional antennas,” IEEE Trans. Antennas Propag. 6(3), 219–225 (1958).
[Crossref]

1947 (1)

J. H. V. Vleck, “The absorption of microwaves by oxygen,” Phys. Rev. 71(7), 413–424 (1947).
[Crossref]

Ackerson, B. J.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Ackerson, C. J.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Adams, R. A.

R. A. Adams, Calculus: a complete course (Addison-Wesley, 1995), 3rd ed.

Al-Taay, W.

W. Al-Taay, S. F. Oboudi, E. Yousif, M. A. Nabi, R. M. Yusop, and D. Derawi, “Fabrication and characterization of nickel chloride doped PMMA films,” Adv. Mater. Sci. Eng. 2015, 1–5 (2015).
[Crossref]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 2001), 5th ed.

Auwera, J. V.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. H. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, J. J. H. T. Furtenbacher, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
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Figures (6)

Fig. 1.
Fig. 1. Problem setup. Here, $\epsilon _{\mathrm {b}}$ and $\mu _{\mathrm {b}}$ denote the relative permittivity and permeability of the passive background medium, respectively, and $\hat {\boldsymbol {n}}$ the outward unit vector.
Fig. 2.
Fig. 2. Optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$ of a sphere in a lossy medium, plotted as a function of the number of included multipoles $L$.
Fig. 3.
Fig. 3. Comparison between the optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, the absorption of a homogeneous sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$, and the corresponding variational bound $Q_{\mathrm {a}}^{\mathrm {var}}$; all plotted as functions of the electrical size $k_0a$. The plots are for various levels of background loss $\epsilon _{\mathrm {b}}^{\prime \prime }$, and the calculations of $Q_{\mathrm {a}}^{\mathrm {var}}$ and $Q_{\mathrm {a}}^{\mathrm {Au}}$ are for two different radii of the sphere $a_1=20$ nm (to the left) and $a_2=89$ nm (to the right).
Fig. 4.
Fig. 4. Comparison of the various upper bounds $Q_{\mathrm {a}}^{\mathrm {opt}}$, $Q_{\mathrm {a},2}^{\mathrm {opt}}$ (the electric multipole contribution), $Q_{\mathrm {a},21}^{\mathrm {opt}}$ (the electric-dipole upper bound), $Q_{\mathrm {a}}^{\mathrm {var}}$ and the absorption of a sphere $Q_{\mathrm {a}}$ tuned to optimal electric (plasmonic) dipole resonance at $k_0a=0.1$ for: a) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-3}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$, and the permittivity of the sphere is $\epsilon =-2.024+\textrm{i} 0.0040$; b) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$, and the permittivity of the sphere is $\epsilon =-2.024 + \textrm{i} 0.0020$.
Fig. 5.
Fig. 5. Comparison between the optimal total normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, the total absorption of a sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$ (ratio $r/d=0$), and the total absorption of a multilayered sphere made of silicon (core of radius $r$) and gold (shell of thickness $d)Q_{\mathrm {a}}^{\mathrm {Si,Au}}$; all are plotted as functions of the electrical size $k_0a$. The plots are made for a fixed level of a background loss $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-1}$ and a fixed total radius of spheres $a=89$ nm.
Fig. 6.
Fig. 6. Comparison between the optimal total normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, variational bound on the normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {var}}$, and total absorption of a multilayered sphere made of germanium (core of radius $r$) and gold (shell of thickness $d)\;Q_{\mathrm {a}}^{\mathrm {Ge,Au}}$. All the results are plotted as functions of the electrical size $k_0a$ for different levels of losses in the background medium with relative permittivity $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$: a) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-1}$; b) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-3}$; c) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$.

Equations (70)

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{ × E { i , s } = i k 0 η 0 μ b H { i , s } , × H { i , s } = i k 0 η 0 1 ϵ b E { i , s } ,
{ D = ϵ 0 ϵ E + 1 c 0 χ e m H , B = 1 c 0 χ m e E + μ 0 μ H ,
{ × E = i k 0 χ m e E + i k 0 η 0 μ H , × H = i k 0 η 0 1 ϵ E i k 0 χ e m H ,
{ × E = i k 0 η 0 μ b H J m , × H = i k 0 η 0 1 ϵ b E + J e ,
{ J e = i k 0 η 0 1 χ e e E i k 0 χ e m H , J m = i k 0 χ m e E i k 0 η 0 χ m m H .
P a = P s + P t + P i ,
P a = 1 2 R e { V E × H n ^ d S } ,
P s = 1 2 R e { V E s × H s n ^ d S } ,
P t = 1 2 R e { V ( E i × H s + E s × H i ) n ^ d S } ,
P i = 1 2 R e { V E i × H i n ^ d S } ,
{ n ^ × ( E i + E s ) = n ^ × E , n ^ × ( H i + H s ) = n ^ × H ,
P a = k 0 2 η 0 I m { V F M a F d v } ,
P t = k 0 2 η 0 I m { V F i M t F d v } 2 P i ,
P i = k 0 2 η 0 I m { V F i M b F i d v } ,
F = ( E η 0 H ) , F i = ( E i η 0 H i ) ,
M a = ( ϵ χ e m χ m e μ ) = χ + M b
χ = ( χ e e χ e m χ m e χ m m ) , M b = ( ϵ b I 0 0 μ b I ) ,
M t = ( ϵ ϵ b I χ e m χ m e μ μ b I ) = χ + i 2 I m { M b } .
m a x i m i z e P a s u b j e c t   t o P s 0 ,
P a o p t = k 0 α 2 8 η 0 V F i M t ( I m { M a } ) 1 M t F i d v ,
α 2 + 2 α = q ,
q = 4 V F i I m { M b } F i d v V F i M t ( I m { M a } ) 1 M t F i d v .
α = 1 1 q .
M a , i = ϵ i I , M b = ϵ b I , M t , i = ( ϵ i ϵ b ) I ,
P a v a r = k 0 α 2 8 η 0 i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } V i | E i ( r ) | 2 d v ,
q = 4 I m { ϵ b } V | E i ( r ) | 2 d v i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } V i | E i ( r ) | 2 d v .
V a | E i ( r ) | 2 d v = | E 0 | 2 2 π τ = 1 2 l = 1 ( 2 l + 1 ) W τ l ( k b , a ) ,
W τ l ( k , a ) = V a | v τ m l ( k r ) | 2 d v ,
V a | E i ( r ) | 2 d v = | E 0 | 2 V a ,
I i = | E 0 | 2 R e { ϵ b } / 2 η 0 ,
σ a v a r = k 0 R e { ϵ b } α 2 4 i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } V i | e i k b k ^ r | 2 d v .
Q a v a r = k 0 a R e { ϵ b } α 2 4 i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } 1 π a 3 V i | e i k b k ^ r | 2 d v ,
1 π a 3 V i | e i k b k ^ r | 2 d v = 2 τ = 1 2 l = 1 ( 2 l + 1 ) a 3 [ W τ l ( k b , a i ) W τ l ( k b , a i 1 ) ] ,
W 1 l ( k b , a i ) = a i 2 I m { k b j l + 1 ( k b a i ) j l ( k b a i ) } I m { k b 2 } ,
W 2 l ( k b , a i ) = ( l + 1 ) W 1 , l 1 ( k b , a i ) + l W 1 , l + 1 ( k b , a i ) ( 2 l + 1 ) ,
P a v a r = k 0 2 η 0 | ϵ ϵ b | 2 I m { ϵ } | E 0 | 2 V a ,
P s = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l A τ l | f τ m l | 2 ,
P t = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l 2 R e { B τ l a τ m l i f τ m l } ,
P i = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l C τ l | a τ m l i | 2 ,
A τ l = 1 R e { k b } { I m { k b ξ l ξ l } τ = 1 , I m { k b ξ l ξ l } τ = 2 ,
B τ l = 1 i 2 R e { k b } { k b ξ l ψ l k b ψ l ξ l τ = 1 , k b ξ l ψ l + k b ψ l ξ l τ = 2 ,
C τ l = 1 R e { k b } { I m { k b ψ l ψ l } τ = 1 , I m { k b ψ l ψ l } τ = 2 ,
P a , τ m l = R e { ϵ b } 2 | k b | 2 η 0 [ A τ l | f τ m l | 2 + 2 R e { B τ l a τ m l i f τ m l } + C τ l | a τ m l i | 2 ] ,
f n = n T n , n a n i ,
A τ l a n i n a n i T n , n = B τ l a n i a n i ,
g = τ , m , l | a τ m l i | 2 ,
T n , n = B τ l A τ l g a n i a n i .
P a , τ m l = R e { ϵ b } 2 | k b | 2 η 0 { A τ l | τ m l ( T τ m l , τ m l B τ l a τ m l i a τ m l i A τ l g ) a τ m l i | 2 + ( | B τ l | 2 A τ l + C τ l ) | a τ m l i | 2 } .
P a o p t = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l ( | B τ l | 2 A τ l + C τ l ) | a τ m l i | 2 .
P a o p t = π R e { ϵ b } | E 0 | 2 | k b | 2 η 0 τ = 1 2 l = 1 ( 2 l + 1 ) ( | B τ l | 2 A τ l + C τ l ) ,
Q a o p t = 2 | k b a | 2 τ = 1 2 l = 1 ( 2 l + 1 ) ( | B τ l | 2 A τ l + C τ l ) .
Q a o p t , L = 2 ( k b a ) 2 τ = 1 2 l = 1 L 2 l + 1 4 = 1 ( k b a ) 2 L ( L + 2 ) ,
ψ l ( z ) = k = 0 α k l z l + 1 + 2 k ,
ξ l ( z ) = i k = 0 l β k l z l + 2 k + α 0 l z l + 1 + O { z l + 2 } ,
A τ l { 2 β 0 l β 1 l sin 2 θ | z | 2 l 1 cos θ τ = 1 , l β 0 l 2 sin 2 θ | z | 2 l + 1 cos θ τ = 2 ,
B τ l { 1 2 e i θ ( 2 l + 1 ) cos θ τ = 1 , 1 2 l e i θ ( 2 l + 3 ) + ( l + 1 ) e i θ ( 2 l 1 ) ( 2 l + 1 ) cos θ τ = 2 ,
C τ l { 2 α 0 l α 1 l | z | 2 l + 3 sin 2 θ cos θ τ = 1 , ( l + 1 ) α 0 l 2 | z | 2 l + 1 sin 2 θ cos θ τ = 2 ,
L ( F , λ ) = ( 1 λ ) I m { V F M a F d v } + λ I m { V F i M t F d v } λ I m { V F i M b F i d v } ,
δ L ( F , λ ) = I m { V δ F [ ( 1 λ ) ( M a M a ) F λ M t F i ] d v } ,
F = α 2 i ( I m { M a } ) 1 M t F i ,
I m { M a } = M a M a 2 i .
P a o p t = k 0 α 2 8 η 0 V F i M t ( I m { M a } ) 1 M t F i d v .
f ( M a ) = V F i M t ( I m { M a } ) 1 M t F i d v ,
δ f ( M a ) = V F i δ M a ( I m { M a } ) 1 ( M a M b ) F i d v V F i ( M a M b ) ( I m { M a } ) 1 I m { δ M a } ( I m { M a } ) 1 ( M a M b ) F i d v + V F i ( M a M b ) ( I m { M a } ) 1 δ M a F i d v ,
δ ( I m { M a } ) 1 = ( I m { M a } ) 1 I m { δ M a } ( I m { M a } ) 1 .
R e { δ M a ( I m { M a } ) 1 ( M a M b ) ( M a M b ) ( I m { M a } ) 1 δ M a 2 i ( I m { M a } ) 1 ( M a M b ) } = 0 ,
R e { [ ( M a M b ) ( I m { M a } ) 1 2 i I ] δ M a ( I m { M a } ) 1 ( M a M b ) } = 0 .
I m { M a } = M a M b 2 i ,
f ( M a ) | M a = M b = V F i ( M b M b ) ( I m { M b } ) 1 ( M b M b ) F i d v = 4 V F i I m { M b } F i d v .
{ E ( r ) = τ , m , l a τ m l v τ m l ( k r ) + f τ m l u τ m l ( k r ) , H ( r ) = 1 i η 0 η τ , m , l a τ m l v τ ¯ m l ( k r ) + f τ m l u τ ¯ m l ( k r ) ,

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