Abstract

The dependence of macroscopic radiation pressure on the velocity of the object being pushed is commonly attributed to the Doppler effect. This need not be the case, and here we highlight velocity-dependent radiation pressure terms that have their origins in the mixing of s and p polarizations brought about by the Lorentz transformation between the lab and the material rest frame, rather than in the corresponding transformation of frequency and wavevector. The theory we develop may be relevant to the nano-optomechanics of moving bodies.

© 2012 Optical Society of America

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  1. V. B. Braginski and A. B. Makunin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 25, 653–655 (1967).
  2. A. B. Matsko, E. A. Zubova, and S. P. Vyatchanin, “The value of the force of radiative friction,” Opt. Commun. 131, 107–113 (1996).
    [CrossRef]
  3. S. A. R. Horsley, M. Artoni, and G. C. La Rocca,“Radiation damping in atomic photonic crystals,” Phys. Rev. Lett. 107, 043602 (2011).
    [CrossRef]
  4. K. Karrai, I. Favero, and C. Metzger, “Dopper optomechanics of a photonic crystal,” Phys. Rev. Lett. 100, 240801 (2008).
    [CrossRef]
  5. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, The Electrodynamics of Continuous Media (Butterworth–Heinemann, 2004).
  6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
  7. C. Yeh, “Reflection and transmission of electromagnetic waves from a moving plasma medium,” J. Appl. Phys. 37, 3079–3082 (1966).
    [CrossRef]
  8. J. A. Kong and D. K. Chen, “Reflection and refraction of electromagnetic waves by a moving uniaxially anisotropic slab,” J. Appl. Phys. 40, 2206–2212 (1969).
    [CrossRef]
  9. Y.-X. Huang, “Reflection and transmission of electromagnetic waves by a dielectric medium moving in an arbitrary direction,” J. Appl. Phys. 76, 2575–2581 (1994).
    [CrossRef]
  10. M. Artoni, I. Carusotto, G. C. La Rocca, and F. Bassani, “Fresnel light drag in a coherently driven moving medium,” Phys. Rev. Lett. 86, 2549–2552 (2001).
    [CrossRef]
  11. J. B. Pendry, “Shearing the vacuum—quantum friction,” J. Phys. Condens. Matter 9, 10301–10320 (1997).
    [CrossRef]
  12. T. G. Philbin and U. Leonhardt, “No quantum friction between uniformly moving plates,” New J. Phys. 11, 033035 (2009).
    [CrossRef]
  13. L. Novotny and B. Hecht, Principles of Nano–Optics(Cambridge University, 2008).
  14. C. Genet, A. Lambrecht, and S. Reynaud, “Casimir force and the quantum theory of lossy optical cavities,” Phys. Rev. A 67, 043811 (2003).
    [CrossRef]
  15. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express13, 9280–9291 (2005).
  16. M. Born and E. Wolf, Principles of Optics (Cambridge University, 2009).
  17. A. Alú, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
    [CrossRef]
  18. L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
    [CrossRef]
  19. J. Schilling, “Fundamental optical physics: the quest for zero refractive index,” Nat. Photonics 5, 449–451 (2011).
    [CrossRef]
  20. F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009).
    [CrossRef]

2011 (2)

S. A. R. Horsley, M. Artoni, and G. C. La Rocca,“Radiation damping in atomic photonic crystals,” Phys. Rev. Lett. 107, 043602 (2011).
[CrossRef]

J. Schilling, “Fundamental optical physics: the quest for zero refractive index,” Nat. Photonics 5, 449–451 (2011).
[CrossRef]

2010 (1)

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

2009 (2)

T. G. Philbin and U. Leonhardt, “No quantum friction between uniformly moving plates,” New J. Phys. 11, 033035 (2009).
[CrossRef]

F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009).
[CrossRef]

2008 (1)

K. Karrai, I. Favero, and C. Metzger, “Dopper optomechanics of a photonic crystal,” Phys. Rev. Lett. 100, 240801 (2008).
[CrossRef]

2007 (1)

A. Alú, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

2003 (1)

C. Genet, A. Lambrecht, and S. Reynaud, “Casimir force and the quantum theory of lossy optical cavities,” Phys. Rev. A 67, 043811 (2003).
[CrossRef]

2001 (1)

M. Artoni, I. Carusotto, G. C. La Rocca, and F. Bassani, “Fresnel light drag in a coherently driven moving medium,” Phys. Rev. Lett. 86, 2549–2552 (2001).
[CrossRef]

1997 (1)

J. B. Pendry, “Shearing the vacuum—quantum friction,” J. Phys. Condens. Matter 9, 10301–10320 (1997).
[CrossRef]

1996 (1)

A. B. Matsko, E. A. Zubova, and S. P. Vyatchanin, “The value of the force of radiative friction,” Opt. Commun. 131, 107–113 (1996).
[CrossRef]

1994 (1)

Y.-X. Huang, “Reflection and transmission of electromagnetic waves by a dielectric medium moving in an arbitrary direction,” J. Appl. Phys. 76, 2575–2581 (1994).
[CrossRef]

1969 (1)

J. A. Kong and D. K. Chen, “Reflection and refraction of electromagnetic waves by a moving uniaxially anisotropic slab,” J. Appl. Phys. 40, 2206–2212 (1969).
[CrossRef]

1967 (1)

V. B. Braginski and A. B. Makunin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 25, 653–655 (1967).

1966 (1)

C. Yeh, “Reflection and transmission of electromagnetic waves from a moving plasma medium,” J. Appl. Phys. 37, 3079–3082 (1966).
[CrossRef]

Alekseyev, L. V.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Alú, A.

A. Alú, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Artoni, M.

S. A. R. Horsley, M. Artoni, and G. C. La Rocca,“Radiation damping in atomic photonic crystals,” Phys. Rev. Lett. 107, 043602 (2011).
[CrossRef]

M. Artoni, I. Carusotto, G. C. La Rocca, and F. Bassani, “Fresnel light drag in a coherently driven moving medium,” Phys. Rev. Lett. 86, 2549–2552 (2001).
[CrossRef]

Barnakov, Yu. A.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Bassani, F.

M. Artoni, I. Carusotto, G. C. La Rocca, and F. Bassani, “Fresnel light drag in a coherently driven moving medium,” Phys. Rev. Lett. 86, 2549–2552 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2009).

Braginski, V. B.

V. B. Braginski and A. B. Makunin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 25, 653–655 (1967).

Carusotto, I.

M. Artoni, I. Carusotto, G. C. La Rocca, and F. Bassani, “Fresnel light drag in a coherently driven moving medium,” Phys. Rev. Lett. 86, 2549–2552 (2001).
[CrossRef]

Chen, D. K.

J. A. Kong and D. K. Chen, “Reflection and refraction of electromagnetic waves by a moving uniaxially anisotropic slab,” J. Appl. Phys. 40, 2206–2212 (1969).
[CrossRef]

Engheta, N.

A. Alú, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Favero, I.

K. Karrai, I. Favero, and C. Metzger, “Dopper optomechanics of a photonic crystal,” Phys. Rev. Lett. 100, 240801 (2008).
[CrossRef]

Genet, C.

C. Genet, A. Lambrecht, and S. Reynaud, “Casimir force and the quantum theory of lossy optical cavities,” Phys. Rev. A 67, 043811 (2003).
[CrossRef]

Girvin, S. M.

F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009).
[CrossRef]

Grzegorczyk, T. M.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express13, 9280–9291 (2005).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano–Optics(Cambridge University, 2008).

Horsley, S. A. R.

S. A. R. Horsley, M. Artoni, and G. C. La Rocca,“Radiation damping in atomic photonic crystals,” Phys. Rev. Lett. 107, 043602 (2011).
[CrossRef]

Huang, Y.-X.

Y.-X. Huang, “Reflection and transmission of electromagnetic waves by a dielectric medium moving in an arbitrary direction,” J. Appl. Phys. 76, 2575–2581 (1994).
[CrossRef]

Karrai, K.

K. Karrai, I. Favero, and C. Metzger, “Dopper optomechanics of a photonic crystal,” Phys. Rev. Lett. 100, 240801 (2008).
[CrossRef]

Kemp, B. A.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express13, 9280–9291 (2005).

Kong, J. A.

J. A. Kong and D. K. Chen, “Reflection and refraction of electromagnetic waves by a moving uniaxially anisotropic slab,” J. Appl. Phys. 40, 2206–2212 (1969).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express13, 9280–9291 (2005).

Lambrecht, A.

C. Genet, A. Lambrecht, and S. Reynaud, “Casimir force and the quantum theory of lossy optical cavities,” Phys. Rev. A 67, 043811 (2003).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, The Electrodynamics of Continuous Media (Butterworth–Heinemann, 2004).

Leonhardt, U.

T. G. Philbin and U. Leonhardt, “No quantum friction between uniformly moving plates,” New J. Phys. 11, 033035 (2009).
[CrossRef]

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, The Electrodynamics of Continuous Media (Butterworth–Heinemann, 2004).

Makunin, A. B.

V. B. Braginski and A. B. Makunin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 25, 653–655 (1967).

Marquardt, F.

F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009).
[CrossRef]

Matsko, A. B.

A. B. Matsko, E. A. Zubova, and S. P. Vyatchanin, “The value of the force of radiative friction,” Opt. Commun. 131, 107–113 (1996).
[CrossRef]

Metzger, C.

K. Karrai, I. Favero, and C. Metzger, “Dopper optomechanics of a photonic crystal,” Phys. Rev. Lett. 100, 240801 (2008).
[CrossRef]

Narimanov, E. E.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Noginov, M. A.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano–Optics(Cambridge University, 2008).

Pendry, J. B.

J. B. Pendry, “Shearing the vacuum—quantum friction,” J. Phys. Condens. Matter 9, 10301–10320 (1997).
[CrossRef]

Philbin, T. G.

T. G. Philbin and U. Leonhardt, “No quantum friction between uniformly moving plates,” New J. Phys. 11, 033035 (2009).
[CrossRef]

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, The Electrodynamics of Continuous Media (Butterworth–Heinemann, 2004).

Reynaud, S.

C. Genet, A. Lambrecht, and S. Reynaud, “Casimir force and the quantum theory of lossy optical cavities,” Phys. Rev. A 67, 043811 (2003).
[CrossRef]

Rocca, G. C. La

S. A. R. Horsley, M. Artoni, and G. C. La Rocca,“Radiation damping in atomic photonic crystals,” Phys. Rev. Lett. 107, 043602 (2011).
[CrossRef]

M. Artoni, I. Carusotto, G. C. La Rocca, and F. Bassani, “Fresnel light drag in a coherently driven moving medium,” Phys. Rev. Lett. 86, 2549–2552 (2001).
[CrossRef]

Salandrino, A.

A. Alú, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Schilling, J.

J. Schilling, “Fundamental optical physics: the quest for zero refractive index,” Nat. Photonics 5, 449–451 (2011).
[CrossRef]

Silveirinha, M. G.

A. Alú, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Tumkur, T.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Vyatchanin, S. P.

A. B. Matsko, E. A. Zubova, and S. P. Vyatchanin, “The value of the force of radiative friction,” Opt. Commun. 131, 107–113 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2009).

Yeh, C.

C. Yeh, “Reflection and transmission of electromagnetic waves from a moving plasma medium,” J. Appl. Phys. 37, 3079–3082 (1966).
[CrossRef]

Zubova, E. A.

A. B. Matsko, E. A. Zubova, and S. P. Vyatchanin, “The value of the force of radiative friction,” Opt. Commun. 131, 107–113 (1996).
[CrossRef]

Appl. Phys. Lett. (1)

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, Yu. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

J. Appl. Phys. (3)

C. Yeh, “Reflection and transmission of electromagnetic waves from a moving plasma medium,” J. Appl. Phys. 37, 3079–3082 (1966).
[CrossRef]

J. A. Kong and D. K. Chen, “Reflection and refraction of electromagnetic waves by a moving uniaxially anisotropic slab,” J. Appl. Phys. 40, 2206–2212 (1969).
[CrossRef]

Y.-X. Huang, “Reflection and transmission of electromagnetic waves by a dielectric medium moving in an arbitrary direction,” J. Appl. Phys. 76, 2575–2581 (1994).
[CrossRef]

J. Phys. Condens. Matter (1)

J. B. Pendry, “Shearing the vacuum—quantum friction,” J. Phys. Condens. Matter 9, 10301–10320 (1997).
[CrossRef]

Nat. Photonics (1)

J. Schilling, “Fundamental optical physics: the quest for zero refractive index,” Nat. Photonics 5, 449–451 (2011).
[CrossRef]

New J. Phys. (1)

T. G. Philbin and U. Leonhardt, “No quantum friction between uniformly moving plates,” New J. Phys. 11, 033035 (2009).
[CrossRef]

Opt. Commun. (1)

A. B. Matsko, E. A. Zubova, and S. P. Vyatchanin, “The value of the force of radiative friction,” Opt. Commun. 131, 107–113 (1996).
[CrossRef]

Phys. Rev. A (1)

C. Genet, A. Lambrecht, and S. Reynaud, “Casimir force and the quantum theory of lossy optical cavities,” Phys. Rev. A 67, 043811 (2003).
[CrossRef]

Phys. Rev. B (1)

A. Alú, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

S. A. R. Horsley, M. Artoni, and G. C. La Rocca,“Radiation damping in atomic photonic crystals,” Phys. Rev. Lett. 107, 043602 (2011).
[CrossRef]

K. Karrai, I. Favero, and C. Metzger, “Dopper optomechanics of a photonic crystal,” Phys. Rev. Lett. 100, 240801 (2008).
[CrossRef]

M. Artoni, I. Carusotto, G. C. La Rocca, and F. Bassani, “Fresnel light drag in a coherently driven moving medium,” Phys. Rev. Lett. 86, 2549–2552 (2001).
[CrossRef]

Physics (1)

F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009).
[CrossRef]

Sov. Phys. JETP (1)

V. B. Braginski and A. B. Makunin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 25, 653–655 (1967).

Other (5)

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express13, 9280–9291 (2005).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2009).

L. Novotny and B. Hecht, Principles of Nano–Optics(Cambridge University, 2008).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, The Electrodynamics of Continuous Media (Butterworth–Heinemann, 2004).

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

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Figures (2)

Fig. 1.
Fig. 1.

Scattering matrix S ˜ in Eq. (11) connects the laboratory frame input ( α q L ( + ) , α q R ( ) ) and output ( α q L ( ) , α q R ( + ) ) amplitudes of fields scattering from a dielectric slab of thickness d and area A = L y L z ( L y , L z d ). The slab is centred at x = 0 , and moves laterally with velocity V and with surface normal parallel to x ^ . The s and p polarization unit vectors, respectively, e ^ 1 and e ^ 2 and unit wavevector k ^ ( + ) are here displayed for the case of a plane wave propagating in the ( x - y ) plane. More generally, the unit vectors e ^ 1 and e ^ 2 ( + ) lie in the ( z ^ - y ^ ) plane and in the incidence plane ( x ^ - k ^ ) plane, respectively. Likewise for the incident wavevector that is written as k = k x x ^ + k | | k ^ | | = ( ω / c ) [ cos ( θ ) x ^ + sin ( θ ) ( sin ( χ ) y ^ + cos ( χ ) z ^ ) ] where θ is referred to as the angle of incidence.

Fig. 2.
Fig. 2.

(a) Normal force exerted by a light beam of frequency ω on a laterally moving dielectric slab of thickness d = 10 c / ω . The force, plotted as a function of the angle of incidence θ , is computed from Eq. (16), where the optical response parameters r i and t i are evaluated using (a) ϵ = 5.0 + 0.001 i and (b)  ϵ = 0.001 + 0.001 i and μ = 1 . For clarity we do not include the dispersion of the medium in this example, even though Eq. (16) can be applied to such a situation. The incident wavevector is defined as in Fig. 1. Inset shows the s (dashed) and p (solid) reflection coefficients for such a slab. This calculation was performed using the reflection and transmission coefficients for a slab of arbitrary thickness [16].

Equations (16)

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

E ( x , t ) = ± q = 1 , 2 e ^ q ( ± ) α q ( ± ) e i ( k ( ± ) · x ω t ) ,
B ( x , t ) = 1 c ± q = 1 , 2 ( 1 ) q ¯ e ^ q ¯ ( ± ) α q ( ± ) e i ( k ( ± ) · x ω t ) ,
± α 2 ( ± ) ( c k ω ( ± ) ) e i ( k ( ± ) · x ω t ) = ± α 2 ( ± ) ( c k ω ) e i ( k ( ± ) · x ω t ) ,
α l ( ± ) = ( ω ( ± ) / ω ) α l ( ± ) .
± α 2 ( ± ) e ^ 2 x ( ± ) e i ( k ( ± ) · x ω t ) = ± γ [ α 2 ( ± ) ( e ^ 2 x ( ± ) V y e ^ 1 z ( ± ) / c ) + ( V y / c ) α 1 ( ± ) e ^ 2 z ( ± ) ] e i ( k ( ± ) · x ω t ) ,
α 2 ( ± ) = ( ω ω ) α 2 ( ± ) ( V y η / c ) α 1 ( ± ) 1 + V y 2 η 2 / c 2
α = ( ω ω ) M · α
M = 1 1 + η 2 V y 2 c 2 ( 1 2 V y η c σ z V y η c σ z 1 2 ) ,
d P M μ d t = ϵ 0 A c 2 | k x | 2 ω 2 α IN ( ( 1 4 S S ) ω / c ( R S R S ) | k x | ( 1 4 S S ) k y ( 1 4 S S ) k z ) α IN ,
S q q = ( t q r q r q t q ) .
d P M μ d t = ϵ 0 A c 2 | k x | 2 ω 2 α IN ( ( 1 4 S ˜ S ˜ ) ω / c ( R S ˜ R S ˜ ) | k x | ( 1 4 S ˜ S ˜ ) k y ( 1 4 S ˜ S ˜ ) k z ) α IN ,
S ˜ = M ( S 11 S 22 ) M = 1 1 + η 2 V y 2 c 2 ( S 11 + ( η V y c ) 2 σ z S 22 σ z η V y c ( S 11 σ z σ z S 22 ) η V y c ( σ z S 11 S 22 σ z ) S 22 + ( η V y c ) 2 σ z S 11 σ z ) .
d P ˜ M 1 d t = c 2 k x 2 ω 2 ( 1 + η 2 V y 2 / c 2 ) [ 1 + ( | r 1 | 2 | t 1 | 2 ) | α 1 + η V y c α 2 | 2 + ( | r 2 | 2 | t 2 | 2 ) | α 2 η V y c α 1 | 2 ] ,
d P ˜ M 1 d t c 2 k x 2 ω 2 [ 1 + ( | r 1 | 2 | t 1 | 2 ) | α 1 | 2 + ( | r 2 | 2 | t 2 | 2 ) | α 2 | 2 + η V y c ( | r 1 | 2 | t 1 | 2 | r 2 | 2 + | t 2 | 2 ) ( α 1 α 2 + α 1 α 2 ) ] ,
d P ˜ M 2 d t c 2 | k x | k y ω 2 [ 1 ( | r 1 | 2 + | t 1 | 2 ) | α 1 | 2 ( | r 2 | 2 + | t 2 | 2 ) | α 2 | 2 + V y η c ( | r 2 | 2 + | t 2 | 2 | r 1 | 2 | t 1 | 2 ) ( α 1 α 2 + α 1 α 2 ) ] .
d P ˜ M 1 d t 2 c | k x | 2 η V y ω 2 sin ( 2 φ ) ( | r 1 | 2 | t 1 | 2 | r 2 | 2 + | t 2 | 2 ) .

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