Abstract

Radiation pressure measurements on mirrors submerged in dielectric liquids have consistently shown an effective Minkowski momentum for the photons within the liquid. Using an exact theoretical calculation based on Maxwell’s equations and the Lorentz law of force, we demonstrate that this result is a consequence of the fact that conventional mirrors impart, upon reflection, a 180° phase shift to the incident beam of light. If the mirror is designed to impart a different phase, then the effective momentum will turn out to be anywhere between the two extremes of the Minkowski and Abraham momenta. Since all values in the range between these two extremes are equally likely to be found in experiments, we argue that the photon momentum inside a dielectric host has the arithmetic mean value of the Abraham and Minkowski momenta.

© 2007 Optical Society of America

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References

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  1. R. V.  Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954).
    [CrossRef]
  2. R. V. Jones and B. Leslie, "The measurement of optical radiation pressure in dispersive media," Proc. R. Soc. London, Series A,  360, 347-363 (1978).
    [CrossRef]
  3. A. Ashkin and J. Dziedzic, "Radiation pressure on a free liquid surface," Phys. Rev. Lett. 30, 139-142 (1973).
    [CrossRef]
  4. J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A 8, 14-21 (1973).
    [CrossRef]
  5. J. D. Jackson, Classical Electrodynamics, 2nd edition, (Wiley, New York, 1975).
  6. L.  Landau, E.  Lifshitz, Electrodynamics of Continuous Media, (Pergamon, New York, 1960).
  7. M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 12, 5375-5401 (2004).
    [CrossRef] [PubMed]
  8. R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
    [CrossRef]
  9. R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004).
    [CrossRef]
  10. S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B 39, S671-S684 (2006).
    [CrossRef]
  11. M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13, 2245-2250 (2005).
    [CrossRef] [PubMed]

2006

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B 39, S671-S684 (2006).
[CrossRef]

2005

2004

2002

R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
[CrossRef]

1978

R. V. Jones and B. Leslie, "The measurement of optical radiation pressure in dispersive media," Proc. R. Soc. London, Series A,  360, 347-363 (1978).
[CrossRef]

1973

A. Ashkin and J. Dziedzic, "Radiation pressure on a free liquid surface," Phys. Rev. Lett. 30, 139-142 (1973).
[CrossRef]

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

1954

R. V.  Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954).
[CrossRef]

Ashkin, A.

A. Ashkin and J. Dziedzic, "Radiation pressure on a free liquid surface," Phys. Rev. Lett. 30, 139-142 (1973).
[CrossRef]

Barnett, S. M.

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B 39, S671-S684 (2006).
[CrossRef]

Dziedzic, J.

A. Ashkin and J. Dziedzic, "Radiation pressure on a free liquid surface," Phys. Rev. Lett. 30, 139-142 (1973).
[CrossRef]

Gordon, J. P.

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

Jones, R. V.

R. V. Jones and B. Leslie, "The measurement of optical radiation pressure in dispersive media," Proc. R. Soc. London, Series A,  360, 347-363 (1978).
[CrossRef]

Jones, R. V.

R. V.  Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954).
[CrossRef]

Leslie, B.

R. V. Jones and B. Leslie, "The measurement of optical radiation pressure in dispersive media," Proc. R. Soc. London, Series A,  360, 347-363 (1978).
[CrossRef]

Loudon, R.

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B 39, S671-S684 (2006).
[CrossRef]

R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004).
[CrossRef]

R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
[CrossRef]

Mansuripur, M.

Richards, J. C. S.

R. V.  Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954).
[CrossRef]

Fortschr. Phys.

R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004).
[CrossRef]

J. Mod. Opt.

R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002).
[CrossRef]

J. Phys. B

S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B 39, S671-S684 (2006).
[CrossRef]

Opt. Express

Phys. Rev. A

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

Phys. Rev. Lett.

A. Ashkin and J. Dziedzic, "Radiation pressure on a free liquid surface," Phys. Rev. Lett. 30, 139-142 (1973).
[CrossRef]

Proc. R. Soc. A

R. V.  Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954).
[CrossRef]

Series A

R. V. Jones and B. Leslie, "The measurement of optical radiation pressure in dispersive media," Proc. R. Soc. London, Series A,  360, 347-363 (1978).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics, 2nd edition, (Wiley, New York, 1975).

L.  Landau, E.  Lifshitz, Electrodynamics of Continuous Media, (Pergamon, New York, 1960).

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Figures (3)

Fig. 1.
Fig. 1.

Reflection of light from a mirror having a purely imaginary refractive index of i n 1. The incidence medium is a transparent dielectric of refractive index n o. The normally incident plane wave has frequency f o, free-space wavelength λ o = c/f o, wave-number kz = n o k o = 2πn o/λ o, and field amplitudes (Ex , Hy ) = (E o, n o E o/Z o). The Fresnel reflection coefficient of the submerged mirror is ρ= exp (iϕ) = exp [-2i arctan (n 1/n o)]. Beneath the mirror’s surface, the transmitted beam is an inhomogeneous plane-wave whose imaginary propagation vector k = i (2πn 1/λ o) z ̂ causes the beam amplitude to drop exponentially along the z-axis.

Fig. 2.
Fig. 2.

Two identical plane-waves propagate in opposite directions (along ± z) to create a standing wave inside a dielectric medium of refractive index n o. The field amplitudes for each plane-wave are (Ex , Hy ) = (E o, n o E o/Z o). The sinusoidal curve having a period of λ o/2n o depicts the intensity of the standing E-field, with the origin of the coordinate system chosen to coincide with one of its nulls; the standing H-field profile (not shown) is similar but shifted along the z-axis by λ o/4n o. Inside the narrow gap (width δ ≪ λo) located at z = z o, the field amplitudes for each of the two counter-propagating plane-waves are (Ex , Hy ) = (Eg , Eg /Z o).

Fig. 3.
Fig. 3.

A linearly polarized plane-wave propagates along the z-axis in a dielectric host of refractive index n o; inside the medium, the field amplitudes are (Ex , Hy ) = (E o, H o). A narrow gap of width δ ≪ λ o is assumed to exist in this medium; the plane of the gap is yz in (a) and xz in (b). The electromagnetic field inside the gap is the superposition of two evanescent plane-waves whose combined fields satisfy the boundary conditions on both walls of the gap.

Equations (20)

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< F z > = 0 1 2 Re [ i ωε o μ o ( ε 1 ) E x ( z ) H y * ( z ) ] dz
= 1 2 Re [ ε o μ o ( n 1 2 + 1 ) ( 1 + ρ ) ( 1 ρ * ) n o E o 2 Z o ] 0 exp ( 2 n 1 k o z ) dz
= [ n o 2 ( 1 + n 1 2 ) ( n o 2 + n 1 2 ) ] ε o E o 2 .
< F z > = [ 1 + ( n o 2 1 ) sin 2 ( ϕ 2 ) ] ε o E o 2 .
E x ( z , t ) = 2 E o sin ( n o k o z ) sin ( ωt ) ,
H y ( z , t ) = 2 ( n o E o Z o ) cos ( n o k o z ) cos ( ωt ) .
E x ( z , t ) = 2 E g sin ( k o z + ψ ) sin ( ωt ) ,
H y ( z , t ) = 2 ( E g Z o ) cos ( k o z + ψ ) cos ( ωt ) ,
E o sin ( n o k o z o ) = E g sin ( k o z o + ψ ) ,
n o E o cos ( n o k o z o ) = E g cos ( k o z o + ψ ) .
tan ( k o z o + ψ ) = ( 1 n o ) tan ( n o k o z o ) ,
E g 2 E o 2 = 1 + ( n o 2 1 ) cos 2 ( n o k o z o ) .
k ± k o = ± i n o 2 1 x ̂ + n o z ̂
E ± = 1 2 ( n o 2 x ̂ i n o n o 2 i z ̂ ) E o
H ± = 1 2 ( n o E o Z o ) y ̂ = 1 2 H o y ̂ .
p = < S z > c 2 = 1 2 Re ( 2 E x × 2 H y * ) c 2 = 1 2 n o 2 E o H o c 2 .
k ± k o = ± i n o 2 1 y ̂ + n o z ̂
E ± = 1 2 E o x ̂
H ± = 1 2 ( n o ŷ i n o 2 1 ) E o Z o = 1 2 ( ŷ i 1 n o 2 ) H o .
p = < S z > c 2 = 1 2 Re ( 2 E x × 2 E y * ) c 2 = 1 2 E o H o c 2 .

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