Abstract

Light incident on a dielectric slab immersed in a different dielectric medium is either reflected, transmitted, or absorbed, and in this process exerts a force on the slab. This force may be evaluated by integrating either the Lorentz force or the Minkowski force over the thickness of the slab. The procedure of cycle averaging for a monochromatic wave is adopted to find the experimentally accessible forces. The two approaches correspond to the use of the (cycle averaged) Maxwell stress tensor or Minkowski stress tensor, respectively, and appear to give different answers. The integrated Lorentz force points towards the incident beam for a non-absorbing slab with a refractive index smaller than the refractive index of the surrounding dielectric, whereas the Minkowski force points away from the incident beam, regardless of the sign of the refractive index difference. The two approaches agree when the slab can be approximated as an ideal mirror. The integrated Minkowski force can be related to a ballistic picture of the momentum flow in which the incident beam is considered as a stream of photons carrying a momentum ℏk, where k is the wavevector in the incidence medium. The integrated Lorentz force cannot, due to an additional force term related to the standing wave arising from the interference between the incident and reflected waves.

© 2006 Optical Society of America

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References

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J. Mod. Opt. (1)

R. Loudon, “Theory of radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821–838 (2002).
[CrossRef]

Opt. Express (2)

Phys. Lett. A (1)

Y. N. Obukhov and F. W. Hehl, “Electromagnetic energy-momentum and forces in matter,” Phys. Lett. A 311, 277–284 (2003).
[CrossRef]

Phys. Rep. (1)

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
[CrossRef]

Phys. Rev. A (3)

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8, 14–21 (1973).
[CrossRef]

D. F. Nelson, “Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy,” Phys. Rev. A 44, 3985–3996 (1991).
[CrossRef] [PubMed]

R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: The photon drag effect,” Phys. Rev. A 71, 063802 (2005).
[CrossRef]

Proc. R. Soc. London A (1)

R. V. Jones and B. Leslie, “Measurement of optical radiation pressure in dispersive media,” Proc. R. Soc. London A 360, 347–363 (1978).
[CrossRef]

Other (1)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of continuous media, 2nd edition (Butterworth-Heinemann, Oxford, 1984).

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Figures (1)

Fig. 1.
Fig. 1.

The normalized maximum force on the slab as a function of the refractive index of the immersing medium. The solid line is the integrated Lorentz-force for n = 1.5, the short-dashed the integrated Minkowski-force for n = 1.5, the medium-dashed line the integrated Lorentz-force for n = 2.0, and the long-dashed the integrated Minkowski-force for n = 2.0.

Equations (71)

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t g α A + β T αβ A = f α A ,
g α A = ε 0 ε αβγ E β E γ ,
T αβ A = ε 0 E α E β 1 μ 0 B α B β + 1 2 ( ε 0 E 2 + B 2 μ 0 ) δ αβ ,
f α A = β P β E α + ε αβγ t P β B γ .
t g α M + β T αβ M = f α M ,
g α M = ε αβγ D β B γ ,
T αβ M = E α E β H α B β + 1 2 ( E · D + H · B ) δ αβ ,
f α M = 1 2 D β α E β 1 2 E β α D β .
f α = β T αβ = β T αβ ,
f α = V dV f α = S d S β T αβ .
E x = { E 0 exp ( i k 0 z ) + r E 0 exp ( i k 0 z ) , z < d 2 , aE 0 exp ( i k 1 z ) + b E 0 exp ( i k 1 z ) , d 2 < z < d 2 t E 0 exp ( i k 0 z ) , d 2 < z ,
H y = { ε 0 cn 0 E 0 exp ( i k 0 z ) ε 0 cn 0 r E 0 exp ( i k 0 z ) , z < d 2 , ε 0 c ( n + ) aE 0 exp ( i k 1 z ) ε 0 c ( n + ) b E 0 exp ( i k 1 z ) , d 2 < z < d 2 ε 0 cn 0 t E 0 exp ( i k 0 z ) , d 2 < z ,
F z = A d 2 d 2 dz f z = A T zz z = d 2 A T zz z = d 2 ,
T zz A = { 1 4 ε 0 E 0 2 ( n 0 2 + 1 ) ( 1 + r 2 ) 1 2 ε 0 E 0 2 ( n 0 2 1 ) r cos ( 2 k 0 z ψ ) , z < d 2 , 1 4 ε 0 E 0 2 ( n 0 2 + 1 ) t 2 , z > d 2 ,
T zz M = { 1 2 ε 0 E 0 2 n 0 2 ( 1 + r 2 ) , z < d 2 , 1 2 ε 0 E 0 2 n 0 2 t 2 , z > d 2 .
F z A = P ( n 0 2 + 1 ) 2 n 0 c ( 1 + r 2 t 2 ) P ( n 0 2 0 ) n 0 c Re [ r exp ( i k 0 d ) ] ,
F z M = P n 0 c ( 1 + r 2 t 2 ) ,
F z A = 2 P n 0 c ( n 2 1 ) ( n 2 n 0 2 ) sin 2 ( nkd ) 4 n 0 2 n 2 + ( n 2 n 0 2 ) 2 sin 2 ( nkd ) ,
F z M = 2 P n 0 c ( n 2 n 0 2 ) 2 sin 2 ( nkd ) 4 n 0 2 n 2 + ( n 2 n 0 2 ) 2 sin 2 ( nkd ) ,
F z A F z M = n 2 1 n 2 n 0 2 ,
F z A = F z M = 2 Pnκkd c = n 0 P abs c ,
F z A = 2 P n 0 c 1 + n 2 + κ 2 ( n 0 + n ) 2 + κ 2 ,
F z M = 2 P n 0 c n 0 2 + n 2 + κ 2 ( n 0 + n ) 2 + κ 2 .
F z A F z M = 1 + n 2 + κ 2 n 0 2 + n 2 + κ 2 ,
F z A = F z M = 2 P n 0 c ,
F z = pc c p c ( 1 + r 2 t 2 ) .
exp ( i k 0 d 2 ) + r exp ( i k 0 d 2 ) = a exp ( i k 1 d 2 ) + b exp ( i k 1 d 2 ) ,
n 0 exp ( i k 0 d 2 ) n 0 r exp ( i k 0 d 2 ) = ( n + ) a exp ( i k 1 d 2 )
( n + ) b exp ( i k 1 d 2 ) ,
t exp ( i k 0 d 2 ) = a exp ( i k 1 d 2 ) + b exp ( i k 1 d 2 ) ,
n 0 t exp ( i k 0 d 2 ) = ( n + ) a exp ( i k 1 d 2 ) ( n + ) b exp ( i k 1 d 2 ) .
r = i ( ( n + i κ ) 2 n 0 2 ) sin ( k 1 d ) 2 n 0 ( n + i κ ) cos ( k 1 d ) i ( ( n + i κ ) 2 + n 0 2 ) sin ( k 1 d ) exp ( i k 0 d ) ,
t = 2 n 0 ( n + i κ ) 2 n 0 ( n + i κ ) cos ( k 1 d ) i ( ( n + i κ ) 2 + n 0 2 ) sin ( k 1 d ) exp ( i k 0 d ) .
cos ( k 1 d ) = cosh ( κ k d ) cos ( n k d ) i sinh ( κ k d ) sin ( n k d ) ,
sin ( k 1 d ) = cosh ( κ k d ) sin ( n k d ) + i sinh ( κ k d ) cos ( n k d ) ,
r = i ( n 2 n 0 2 ) sin ( nkd ) 2 n 0 n cos ( nkd ) i ( n 2 + n 0 2 ) sin ( nkd ) exp ( i k 0 d ) ,
t = 2 n 0 n 2 n 0 n cos ( nkd ) i ( n 2 + n 0 2 ) sin ( nkd ) exp ( i k 0 d ) ,
r = n 0 n i κ n 0 + n + i κ exp ( i k 0 d ) ,
t = 0 .
E x = E ( z ) exp ( i ω t ) ,
H y = ε 0 c ( n + i κ ) E ( z ) exp ( i ω t ) ,
E ( z ) = E 0 exp ( i k z ) ,
g z A = 1 2 ε 0 E ( z ) 2 n c ,
g z M = 1 2 ε 0 E ( z ) 2 ( n 2 + κ 2 ) n c .
ε 0 E x 2 = ε 0 E ( z ) 2 ,
1 μ 0 B y 2 = ε 0 ( n 2 + κ 2 ) E ( z ) 2 ,
E x D x = ε 0 ( n 2 κ 2 ) E ( z ) 2 .
T xx A = 1 4 ε 0 E ( z ) 2 ( n 2 + κ 2 1 ) ,
T yy A = 1 4 ε 0 E ( z ) 2 ( n 2 + κ 2 1 ) ,
T zz A = 1 4 ε 0 E ( z ) 2 ( n 2 + κ 2 + 1 ) ,
T xx M = 1 2 ε 0 E ( z ) 2 κ 2 ,
T yy M = 1 2 ε 0 E ( z ) 2 κ 2 ,
T zz M = 1 2 ε 0 E ( z ) 2 n 2 .
f z A = 1 2 ε 0 E ( z ) 2 ( n 2 + κ 2 + 1 ) κ ω c ,
f z M = ε 0 E ( z ) 2 n 2 κ ω c .
f z A = z T zz A = 2 κ ω c T zz A ,
f z M = z T zz M = 2 κ ω c T zz M ,
E x = E ( z ) exp ( i ω t ) + r E ( z ) exp ( i ω t ) ,
H y = ε 0 c ( n + i κ ) E ( z ) exp ( i ω t ) r ε 0 c ( n + i κ ) E ( z ) exp ( i ω t ) ,
g z A = 1 2 ε 0 E 0 2 n c [ exp ( 2 κ ω c z ) r 2 exp ( 2 κ ω c z ) ]
+ ε 0 E 0 2 κ c r sin ( 2 n ω c z ψ ) ,
g z M = 1 2 ε 0 E 0 2 ( n 2 + κ 2 ) n c [ exp ( 2 κ ω c z ) r 2 exp ( 2 κ ω c z ) ]
ε 0 E 0 2 ( n 2 + κ 2 ) κ c r sin ( 2 n ω c z ψ ) .
T zz A = 1 4 ε 0 E 0 2 ( n 2 + κ 2 + 1 ) [ exp ( 2 κ ω c z ) + r 2 exp ( 2 κ ω c z ) ]
1 2 ε 0 E 0 2 ( n 2 + κ 2 1 ) r cos ( 2 n ω c z ψ ) ,
T zz M = 1 2 ε 0 E 0 2 n 2 [ exp ( 2 κ ω c z ) + r 2 exp ( 2 κ ω c z ) ]
ε 0 E 0 2 κ 2 r cos ( 2 n ω c z ψ ) .
f z A = 1 2 ε 0 E 0 2 ( n 2 + κ 2 + 1 ) κ ω c [ exp ( 2 κ ω c z ) r 2 exp ( 2 κ ω c z ) ]
ε 0 E 0 2 ( n 2 + κ 2 1 ) r n ω c sin ( 2 n ω c z ψ ) ,
f z M = ε 0 E 0 2 n 2 κ ω c [ exp ( 2 κ ω c z ) r 2 exp ( 2 κ ω c z ) ]
2 ε 0 E 0 2 κ 2 n ω c r sin ( 2 n ω c z ψ ) .

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