Abstract

We derive the force of the electromagnetic radiation on material objects by a direct application of the Lorentz law of classical electro-dynamics. The derivation is straightforward in the case of solid metals and solid dielectrics, where the mass density and the optical constants of the media are assumed to remain unchanged under internal and external pressures, and where material flow and deformation can be ignored. For metallic mirrors, we separate the contribution to the radiation pressure of the electrical charge density from that of the current density of the conduction electrons. In the case of dielectric media, we examine the forces experienced by bound charges and currents, and determine the contribution of each to the radiation pressure. These analyses reveal the existence of a lateral radiation pressure inside the dielectric media, one that is exerted at and around the edges of a finite-diameter light beam. The lateral pressure turns out to be compressive for s-polarized light and expansive for p-polarized light. Along the way, we derive an expression for the momentum density of the light field inside dielectric media, one that has equal contributions from the traditional Minkowski and Abraham forms. This new expression for the momentum density, which contains both electromagnetic and mechanical terms, is used to explain the behavior of light pulses and individual photons upon entering and exiting a dielectric slab. In all the cases considered, the net forces and torques experienced by material bodies are consistent with the relevant conservation laws. Our method of calculating the radiation pressure can be used in conjunction with numerical simulations to yield the distribution of fields and forces in diverse systems of practical interest.

© 2004 Optical Society of America

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References

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  1. H. Minkowski, Nachr. Ges. Wiss. Gottingen 53 (1908).
  2. H. Minkowski, Math. Annalon 68, 472 (1910).
    [CrossRef]
  3. M. Abraham, R. C. Circ. Mat. Palermo 28, 1 (1909).
    [CrossRef]
  4. M. Abraham, R. C. Circ. Mat. Palermo 30, 33 (1910).
    [CrossRef]
  5. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8, 14-21 (1973).
    [CrossRef]
  6. R. Loudon, “Radiation Pressure and Momentum in Dielectrics,” De Martini lecture, to appear in Fortschritte der Physik (2004).
  7. R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821-838 (2002).
    [CrossRef]
  8. L. Landau, E. Lifshitz, Electrodynamics of Continuous Media, Pergamon, New York, 1960.
  9. J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.
  10. M. Planck, The Theory of Heat Radiation, translated by M. Masius form the German edition of 1914, Dover Publications, New York (1959).
  11. R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. A 221, 480 (1954).
    [CrossRef]
  12. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288-290 (1986).
    [CrossRef] [PubMed]
  13. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517-1520 (1987).
    [CrossRef] [PubMed]
  14. A. Rohrbach and E. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494 (2002).
    [CrossRef] [PubMed]
  15. Y. N. Obukhov and F. W.Hehl, “Electromagnetic energy-momentum and forces in matter,” Phys. Lett. A, 311, 277-284 (2003).
    [CrossRef]
  16. G. Barlow, Proc. Roy. Soc. Lond. A 87, 1-16 (1912).
    [CrossRef]
  17. R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A 68, 013806 (2003).
    [CrossRef]

Appl. Opt.

Fortschritte der Physik

R. Loudon, “Radiation Pressure and Momentum in Dielectrics,” De Martini lecture, to appear in Fortschritte der Physik (2004).

J. Mod. Opt.

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

Math. Annalon

H. Minkowski, Math. Annalon 68, 472 (1910).
[CrossRef]

Nachr. Ges. Wiss. Gottingen

H. Minkowski, Nachr. Ges. Wiss. Gottingen 53 (1908).

Opt. Lett.

Phys. Lett. A

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

Phys. Rev. A

R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A 68, 013806 (2003).
[CrossRef]

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

Proc. Roy. Soc. A

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

Proc. Roy. Soc. Lond. A

G. Barlow, Proc. Roy. Soc. Lond. A 87, 1-16 (1912).
[CrossRef]

R. C. Circ. Mat. Palermo

M. Abraham, R. C. Circ. Mat. Palermo 28, 1 (1909).
[CrossRef]

M. Abraham, R. C. Circ. Mat. Palermo 30, 33 (1910).
[CrossRef]

Science

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

Other

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

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

M. Planck, The Theory of Heat Radiation, translated by M. Masius form the German edition of 1914, Dover Publications, New York (1959).

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Figures (13)

Fig. 1.
Fig. 1.

A linearly-polarized plane wave is reflected from a perfectly conducting mirror. Whereas the parallel component of the E-field at the mirror surface is zero, the parallel component of the H-field is at its maximum. The surface current J s is equal in magnitude and perpendicular in direction to the magnetic field at the surface. (a) Normal incidence. (b) Oblique incidence with s-polarization. (c) Oblique incidence with p-polarization.

Fig. 2.
Fig. 2.

A linearly-polarized plane wave is normally incident on the surface of a semi-infinite medium of complex dielectric constant ε. The Fresnel reflection coefficient at the surface is r. Shown are the E- and H-field magnitudes for the incident, reflected, and transmitted beams.

Fig. 3.
Fig. 3.

Obliquely incident s-polarized plane wave arrives at the surface of a semi-infinite medium of (complex) dielectric constant ε. The Fresnel reflection coefficient is denoted by rs .

Fig. 4.
Fig. 4.

Oblique incidence on a semi-infinite dielectric at angle θ. The beam’s footprint at the surface has unit area, while the incident and transmitted beams’ cross-sectional areas are cosθ and cosθ′, respectively. A segment from the beam’s left edge (area proportional to sinθ′) exerts a force Δ F on the dielectric; this force is not compensated by an equal and opposite force on the right-hand edge of the beam, as is the case elsewhere at the opposite edges of the beam. The compressive Δ F shown here corresponds to an s-polarized beam. (For p-light Δ F retains the same magnitude but reverses direction, so the edge force becomes expansive.)

Fig. 5.
Fig. 5.

A p-polarized plane wave is obliquely incident at the surface of a semi-infinite medium of (complex) dielectric constant ε. The Fresnel reflection coefficient is denoted by rp .

Fig. 6.
Fig. 6.

(a) Dielectric slab of thickness d and index n, illuminated with a p-polarized plane wave at Brewster’s angle θB. The H-field’s magnitude inside the slab is the same as that outside, but the E-field intside is reduced by a factor of n compared to the outside field. The bound charges on the upper and lower surfaces feel the force of the E-field. The transmitted beam is displaced by d/n horizontally and by d(1-1/n 2) vertically. The force Fx x +Fz z on the upper surface is cancelled out by the force on the lower surface, but the slab experiences a net torque from Fx . The torque of Fz is cancelled out by the forces exerted at the beam’s edges within the dielectric. (b) Tilted cylinder of base area a and length d/sinθ′B, aligned with internal E-field.

Fig. 7.
Fig. 7.

Semi-infinite medium of index n, coated by a quarter-wave layer of index √n, a perfect anti-reflection layer. A normally incident plane wave is fully transmitted to the semi-infinite substrate. The standing wave within the coating layer produces an upward force equal to the time rate of change of the field’s momentum in the free-space minus that in the substrate.

Fig. 8.
Fig. 8.

A plate of thickness d and dielectric constant ε is illuminated at normal incidence by a linearly-polarized plane wave. The magnetic field’s Lorentz force on the standing wave within the slab gives rise to a downward force that is precisely equal to the difference between the rates of incoming and outgoing momenta of the light beam.

Fig. 9.
Fig. 9.

One-dimensional Gaussian beam propagating along the z-axis in a dielectric medium of refractive index n. For the s-light shown here, the E-field has one component, Ey , while the H-field has both Hx and Hz . The lateral force Fx is positive on the left- and negative on the right-hand side, producing compressive pressure toward the beam center. For p-light (not shown), the field components are Hy, Ex, Ez , and the lateral force Fx , while having the same magnitude as in the case of s-light, is expansive in nature.

Fig. 10.
Fig. 10.

Two linearly-polarized plane waves in a medium of refractive index n create interference fringes parallel to the z-axis. The lateral pressure on individual bright fringes is expansive in the case of p-polarization, and compressive in the case of s-polarization.

Fig. 11.
Fig. 11.

A pulse of light (free-space wavelength=λo) travels along z in a dielectric of refractive index n. The light amplitude distribution in the xy-plane is uniform, similar to that of a plane wave, but the beam profile along z is Gaussian. The beam is linearly polarized, with its E-field along the x-axis and H-field along the y-axis. The assumed non-dispersive nature of the medium ensures that the pulse’s group velocity is equal to the phase velocity c/n.

Fig. 12.
Fig. 12.

A perfectly conducting mirror is immersed at depth d below the surface in a liquid of refractive index n. The normally incident beam is a linearly-polarized plane wave of wavelength λo and E-field magnitude E o. The incident E-field’s magnitude beneath the liquid surface is denoted by Et . Upon reflection from the mirror the E-field retains its magnitude but undergoes a 180° phase shift. The reflected beam is also phase-shifted by 2ϕ=4πn do in its round trip to the mirror and back. The induced surface current J s at the top of the mirror is parallel to the E-field, but its magnitude is equal to the net H-field at the mirror surface. Interference between the incident and reflected beams creates standing-wave fringes within the liquid that exert a spatially varying, ± z-directed force on the liquid. In the free-space region above the liquid, the reflected E- and H-fields have the same magnitudes as their incident counterparts, namely,|r|=1. The relative phase between the incident and reflected fields (i.e., the phase of r) is determined by the boundary conditions at the liquid surface.

Fig. 13.
Fig. 13.

A one-dimensional Gaussian beam, linearly-polarized along the x-axis, passes through a glass plate near one of the plate’s sidewalls. The (bound) charges induced on the plate’s wall experience a Lorentz force F (wall) from the E-field of the beam. This force is opposite in direction to the internal force F (edge) exerted on the bulk of the medium at the beam’s edge that is inside the plate. Since F (wall) is generally stronger than F (edge), the net force tends to pull the plate to the left.

Equations (82)

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F = ε o E o 2 .
F = ½ Real ( J s × B * ) .
F z = ½ Real ( J × B * ) = ½ ( 2 π / λ o ) Real [ i ε * ( ε 1 ) ] ε o E t 2 exp ( 4 π κ z / λ o )
= ½ ( 2 π / λ o ) ( n 2 + κ 2 + 1 ) κ ε o E t 2 exp ( 4 π κ z / λ o ) .
p z = ¼ ( n 2 + 1 ) n ε o E t 2 / c = ¼ ( ε + 1 ) ε o E t B t .
E t y ( x , z ) = ( 1 + r s ) E o exp [ i 2 π ( x sin θ + z ε sin 2 θ ) / λ o ]
H t x ( x , z ) = ε sin 2 θ E t y ( x , z ) / Z o
H t z ( x , z ) = sin θ E t y ( x , z ) / Z o
F x = ½ ε o sin θ Real ( ε sin 2 θ ) 1 + r s 2 E o 2
F z = ¼ ε o ( cos 2 θ + ε sin 2 θ ) 1 + r s 2 E o 2
F x = ½ ε o ε sin θ cos θ E t 2 ,
F z = ¼ ε o ( 1 ε sin 2 θ + ε cos 2 θ ) E t 2 .
Δ F = ¼ ε o ( ε 1 ) sin θ ( cos θ x sin θ z ) E t 2 .
H t y ( x , z ) = ( 1 r p ) H o exp [ i 2 π ( x sin θ + z ε sin 2 θ ) / λ o ] ,
E t x ( x , z ) = ( ε sin 2 θ / ε ) Z o H t y ( x , z ) ,
E t z ( x , z ) = ( sin θ / ε ) Z o H t y ( x , z ) .
F x ( bulk ) = ½ μ o sin θ Real ( ε sin 2 θ ) ε 2 1 r p 2 H o 2 ,
F z ( bulk ) = ¼ μ o ( ε 2 sin 2 θ + ε sin 2 θ ) ε 2 1 r p 2 H o 2 .
σ = ε o ( 1 1 / ε ) ( 1 r p ) sin θ E o exp ( i 2 π x sin θ / λ o ) .
F x ( surface ) = ½ μ o sin θ Real [ ( ε * 1 ) ε sin 2 θ ] ε 2 1 r p 2 H o 2
F z ( surface ) = ¼ μ o sin 2 θ ( 1 ε 2 ) 1 r p 2 H o 2
F x ( total ) = ½ μ o sin θ Real ( ε * ε sin 2 θ ) ε 2 1 r p 2 H o 2
F z ( total ) = ¼ μ o ( ε 2 cos 2 θ + ε sin 2 θ ) ε 2 1 r p 2 H o 2
ε E t y / Z o = H tx 2 + H t z 2
H t y = ε E tx 2 + ε E t z 2 / Z o .
F x ( bulk ) = ½ ε o sin θ cos θ E t 2 ,
F z ( bulk ) = ¼ ε o ( ε sin 2 θ + cos 2 θ ) E t 2 .
Δ F = ¼ ε o ( ε 1 ) sin θ ( cos θ x sin θ z ) E t 2 .
σ = ε o E o ( 1 1 / n 2 ) sin θ B .
F x = ½ σ E x = ½ ε o E o 2 ( 1 1 / n 2 ) sin θ B cos θ B .
T = ½ ε o E o 2 Ad ( 1 1 / n 2 ) sin θ B .
T = ¼ ε o E o 2 A d ( n n 3 ) cos θ B .
E x ( z ) = ½ ( 1 + 1 / n ) E o exp ( i 2 π n z / λ o ) + ½ ( 1 1 / n ) E o exp ( i 2 π n z / λ o ) ,
H y ( z ) = ½ ( 1 + n ) H o exp ( i 2 π n z / λ o ) + ½ ( 1 n ) H o exp ( i 2 π n z / λ o ) ,
F z = ½ Real ( J x × B y * ) = ½ Real [ i ω ε o ( ε 1 ) E x × μ o H y * ]
= [ π ( n 1 ) μ o ε o / λ o ] Imag ( E x × H y * )
= ½ ε o [ π ( n 1 ) 2 / ( n λ o ) ] sin ( 4 π n z / λ o ) E o 2 .
F z = ¼ [ ( n 1 ) 2 / n ] ε 0 E o 2 .
E x ( z ) = E 1 exp ( i 2 π ε z / λ o ) + E 2 exp ( i 2 π ε z / λ o )
H y ( z ) = ( ε E 1 / Z o ) exp ( i 2 π ε z / λ o ) ( ε E 2 / Z o ) exp ( i 2 π ε z / λ o )
E 1 = 2 E o / [ ( 1 + ρ ) + ε ( 1 ρ ) ]
E 2 = 2 ρ E o / [ ( 1 + ρ ) + ε ( 1 ρ ) ]
r = { ρ [ ( ε 1 ) / ( ε + 1 ) ] } / { 1 ρ [ ( ε 1 ) / ( ε + 1 ) ] }
t = 4 ε / [ ( ε + 1 ) 2 exp ( i 2 π ε d / λ o ) ( ε 1 ) 2 exp ( + i 2 π ε d / λ o ) ]
F z = ¼ ε o ( n 2 + κ 2 + 1 ) [ 1 exp ( 4 π κ d / λ o ) ] E 1 2
+ ¼ ε o ( n 2 + κ 2 + 1 ) [ 1 exp ( + 4 π κ d / λ o ) ] E 2 2
+ ½ ε o ( n 2 + κ 2 1 ) [ cos ( 4 π n d / λ o + ϕ E 1 ϕ E 2 ) cos ( ϕ E 1 ϕ E 2 ) ] E 1 E 2
F z = ε o E o 2 1 + ( 2 n ( n 2 1 ) sin ( 2 π n d / λ o ) ) 2 .
F z = ½ ε o E o 2 ( 1 + r 2 t 2 ) = ε o r 2 E o 2 ,
F z = ε o [ ( n 2 1 ) / ( n 2 + 1 ) ] 2 E o 2 .
E y ( z ) = E o α ( z ) / α ( 0 ) exp ( i 2 π z / λ ) exp [ α ( z ) x 2 ] .
H x = ( n / Z o ) E o α ( z ) / α ( 0 ) { 1 + ( λ / 2 π ) 2 [ 2 α 2 ( z ) x 2 α ( z ) ] } exp ( i 2 π z / λ ) exp [ α ( z ) x 2 ] ,
H z = i ( n / Z o ) E o α ( z ) / α ( 0 ) [ ( λ / π ) α ( z ) x ] exp ( i 2 π z / λ ) exp [ α ( z ) x 2 ] .
F x ( z ) = ½ Real ( J y B z * ) = ε o E o 2 ( n 2 1 ) α ( z ) / α ( 0 ) [ x / r o 2 ( z ) ] exp [ 2 x 2 / r o 2 ( z ) ] .
F x ( z = 0 ) = ε o E o 2 ( n 2 1 ) ( x / r o 2 ) exp ( 2 x 2 / r o 2 ) .
H y ( z ) = H o α ( z ) / α ( 0 ) exp ( i 2 π z / λ ) exp [ α ( z ) x 2 ] .
E x = ( Z o / n ) H o α ( z ) / α ( 0 ) { 1 + ( λ / 2 π ) 2 [ 2 α 2 ( z ) x 2 α ( z ) ] } exp ( i 2 π z / λ ) exp [ α ( z ) x 2 ]
E z = i ( Z o / n ) H o α ( z ) / α ( 0 ) [ ( λ / π ) α ( z ) x ] exp ( i 2 π z / λ ) exp [ α ( z ) x 2 ] .
F x ( z ) = ½ Real ( J z B y * ) = μ o H o 2 ( 1 1 / n 2 ) α ( z ) / α ( 0 ) [ x / r o 2 ( z ) ] exp [ 2 x 2 / r o 2 ( z ) ]
F x ( z = 0 ) = μ o H o 2 ( 1 1 / n 2 ) ( x / r o 2 ) exp ( 2 x 2 / r o 2 ) .
E x ( x , z ) = 2 E o cos θ cos ( 2 π n x sin θ / λ o ) exp ( i 2 π n z cos θ / λ o ) ,
E z ( x , z ) = 2 i E o sin θ sin ( 2 π n x sin θ / λ o ) exp ( i 2 π n z cos θ / λ o ) ,
H y ( x , z ) = 2 H o cos ( 2 π n x sin θ / λ o ) exp ( i 2 π n z cos θ / λ o ) .
F x = ½ Real ( J z B y * ) = ( 2 π n sin θ / λ o ) ( n 2 1 ) ε o E o 2 sin ( 4 π n x sin θ / λ o ) .
E y ( x , z ) = 2 E o cos ( 2 π n x sin θ / λ o ) exp ( i 2 π n z cos θ / λ o ) ,
H x ( x , z ) = 2 H o cos θ cos ( 2 π n x sin θ / λ o ) exp ( i 2 π n z cos θ / λ o ) ,
H z ( x , z ) = 2 i H o sin θ sin ( 2 π n x sin θ / λ o ) exp ( i 2 π n z cos θ / λ o ) .
F x = ½ Real ( J y B z * ) = ( 2 π n sin θ / λ o ) ( n 2 1 ) ε o E o 2 sin ( 4 π n x sin θ / λ o ) .
F x = ± ¼ ε o ( n 2 1 ) ( 2 E o ) 2 .
E x ( z , t ) = E o exp { [ ( n z ct ) / w ] 2 } cos [ 2 π ( n z ct ) / λ o ] ,
H y ( z , t ) = ( n / Z o ) E x ( z , t ) .
F z ( z , t ) = J x ( z , t ) B y ( z , t ) = μ o ε o ( ε 1 ) ( E x / t ) H y ( z , t )
= ε o ( ε 1 ) E o 2 { 2 ( n / w ) [ ( n z ct ) / w ] cos 2 [ 2 π ( n z ct ) / λ o ] + ( π n / λ o ) sin [ 4 π ( n z ct ) / λ o ] }
× exp { 2 [ ( n z ct ) / w ] 2 } .
< F z ( z , t ) > = ε o ( ε 1 ) ( n / w ) [ ( n z c t ) / w ] E o 2 exp { 2 [ ( n z c t ) / w ] 2 } .
F z = ± ½ ε o ( ε 1 ) E o 2 .
r = [ ρ exp ( i 2 ϕ ) ] / [ 1 ρ exp ( i 2 ϕ ) ] ,
E t / E o = ( 1 + ρ ) / [ 1 ρ exp ( i 2 ϕ ) ] .
F z ( Mirror ) = ½ Real ( J s × μ o H t * ) = [ n 2 / ( sin 2 ϕ + n 2 cos 2 ϕ ) ] ε o E o 2 .
F z ( z ) = ( 2 π n / λ o ) [ ( n 2 1 ) / ( sin 2 ϕ + n 2 cos 2 ϕ ) ] ε o E o 2 sin [ 4 π n ( z d ) / λ o ] .
F z ( Liquid ) = [ ( n 2 1 ) sin 2 ϕ / ( sin 2 ϕ + n 2 cos 2 ϕ ) ] ε o E o 2 .
F = ( P ) E + ( d P / dt × B ) .

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