## Radiation pressure on a dielectric wedge

Optics Express, Vol. 13, Issue 6, pp. 2064-2074 (2005)

http://dx.doi.org/10.1364/OPEX.13.002064

Acrobat PDF (1732 KB)

### Abstract

The force of electromagnetic radiation on a dielectric medium may be derived by a direct application of the Lorentz law of classical electrodynamics. While the light’s electric field acts upon the (induced) bound charges in the medium, its magnetic field exerts a force on the bound currents. We use the example of a wedge-shaped solid dielectric, immersed in a transparent liquid and illuminated at Brewster’s angle, to demonstrate that the linear momentum of the electromagnetic field within dielectrics has neither the Minkowski nor the Abraham form; rather, the correct expression for momentum density has equal contributions from both. The time rate of change of the incident momentum thus expressed is equal to the force exerted on the wedge plus that experienced by the surrounding liquid.

© 2005 Optical Society of America

## 1. Introduction

1. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375. [CrossRef] [PubMed]

*p*=½

*×*

**D***) and M. Abraham (*

**B***p*=½

*×*

**E***/*

**H***c*

^{2}). Another conclusion was that the local electromagnetic force density may be obtained by adding the Lorentz force of the

*E*-field on the induced bound charges to the Lorentz force of the

*H*-field on the induced bound currents. We argued that, whereas bound currents are typically distributed throughout the volume of a given medium, bound charges (in isotropic, homogeneous dielectrics) appear only on the surfaces and/or interfaces with adjacent media.

1. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375. [CrossRef] [PubMed]

*p*-polarized plane-wave at Brewster’s angle of incidence. The prism’s apex angle is such that, at the exit facet of the wedge, the beam arrives once again at Brewster’s angle. The choice of this particular geometry simplifies the analysis by suppressing all interfacial reflections. In Section 3 the total force exerted on a dielectric wedge in free-space is shown to be equal to the time rate of change of the incident optical momentum. This is reassuring, in light of the fact that the electromagnetic momentum in free-space is not the subject of any controversies [2]. The case of a dielectric wedge immersed in a transparent liquid is investigated in Section 4, where, once again, the net force is shown to be equal to the time rate of change of the incident optical momentum. This time around, however, the momentum density is assumed to be given by the average of Minkowski and Abraham expressions, both of which have been the subject of extensive debate for nearly a century [3

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

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

## 2. Notation and basic definitions

*E*- and

*H*-fields (both inside and outside the medium). We then apply the Lorentz law

*=0, the bound-charge density*

**D***ρ*

_{b}=-∇·

*may be expressed as*

**P***ρ*

_{b}=

*ε*

_{o}∇·

*. Inside a homogeneous and isotropic medium,*

**E***E*being proportional to

*and ∇·*

**D***=0 imply that*

**D***ρ*

_{b}=0; no bound charges, therefore, exist inside such media. However, at the interface between two adjacent media, the component of

*perpendicular to the interface,*

**D**

**D**_{⊥}, must be continuous, implying that

**E**_{⊥}is discontinuous and, therefore, bound charges exist at such interfaces; the interfacial bound charges thus have areal density

*σ*

_{b}=

*ε*

_{o}(

*E*

_{2⊥}-

*E*

_{1⊥}). Under the influence of the local

*E*-field, these charges give rise to a Lorentz force density

*=½*

**F***Real*(

*σ*), where

_{b}**E****is the force per unit area of the interface. Since the tangential*

**F***E*-field,

**E**_{‖}, is continuous across the interface, there is no ambiguity as to the value of

**E**_{|}| that should be used in computing the force. As for the perpendicular component, the average

**E**_{‖}across the boundary, ½(

*E*

_{1}⊥+

*E*

_{2}⊥), must be used in calculating the interfacial force [1

1. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express **12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375. [CrossRef] [PubMed]

*J̱*

_{b}=

*∂P̱/∂t*=

*ε*

_{o}(ε-1)

*∂E/∂t*. Assuming time-harmonic fields with the time-dependence factor exp(-

*iω t*), we can write

*J*

_{b}=-

*iωε*

_{o}(

*ε*-1)

*E*. The

*B*-field of the electromagnetic wave exerts a force on the bound current according to the Lorentz law, namely,

*=½*

**F***Real*(

*J*

_{b}×

**), where*

**B***is force per unit volume. We have shown in [1] that, among other things, this magnetic Lorentz force is responsible for a lateral pressure exerted on the host medium at the edges of a finite-diameter beam; the force per unit area at each edge (i.e., side-wall) of the beam is given by*

**F***E*| is the magnitude of the

*E*-field of a (finite-diameter) plane-wave in a medium of dielectric constant

*ε*. If the

*E*-field is parallel (perpendicular) to the beam’s edge, the force is compressive (expansive); in other words, the opposite side-walls of the beam tend to push the medium toward (away from) the beam center. The edge force does not appear to be sensitive to the detailed structure of the beam’s edge; in particular, a one-dimensional Gaussian beam exhibits the edge force described by Eq. (2) when its (magnetic) Lorentz force on the host medium is integrated laterally on either side of the beam’s center [1

**12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375. [CrossRef] [PubMed]

*n*(=√

*ε*) was shown in [1

**12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375. [CrossRef] [PubMed]

**×**

*D**and Abraham’s ½*

**B***×*

**E***/*

**H***c*

^{2}, namely,

*c/n*, the rate of flow of optical momentum per unit area per unit time is thus ¼(

*n*

^{2}+1)

*ε*

_{o}|

*E*|

^{2}. These results will be used in the following sections to establish the equivalence of force and the time rate of change of incident optical momentum on a dielectric prism.

## 3. Dielectric prism illuminated at Brewster’s angle

*p*-polarized plane wave is incident on a dielectric prism of apex angle

*ϕ*and refractive index

*n*at Brewster’s angle θ

_{B}(tanθ

_{B}=

*n*). The refracted angle inside the slab is given by tanθ′

_{B}=1/

*n*. The apex angle is

*ϕ*=2θ′

_{B}, so the internal angle of incidence on the exit facet is also equal to Brewster’s angle. Since the reflectivity at Brewster’s angle is zero, the only beams in this system are the incident beam, the refracted beam inside the prism, and the transmitted beam. Inside the prism, the

*H*-field is the same as that outside, as required by the continuity of

**H**_{‖}at the interfaces. Similarly, the continuity of

**E**_{‖}and

**D**_{⊥}at the interfaces require that, inside the prism and just beneath the surface,

*E*

_{‖}=

*E*

_{o}cosθ

_{B}, and

*E*

_{⊥}=(

*E*

_{o}/

*n*

^{2})sinθ

_{B}, which fixes the magnitude of

*E*inside the prism at

*E*

_{o}

*/n*.

*ρ*

_{b}is zero, and the magnetic field of the light everywhere (except at the beam’s edges) is 90° out of phase relative to the bound-current density

*. The electric component of the Lorentz force is due to the bound charges induced on the entrance and exit facets, the density of which is obtained from the*

**J**_{b}*ε*

_{o}**E**_{⊥ }discontinuity, namely,

*E*

_{‖}=

*E*

_{o}cosθ

_{B}is continuous across the boundary, and the Lorentz force on the induced surface charges is

*a*the footprint area of the beam on each facet of the prism, the projection of

*F*

_{‖}on the

*x*-axis (accounting for both facets) is given by

*F*

_{⊥}on the top surface of the prism. Averaging

*E*

_{⊥}just above and just below the surface, we find

*a*), projecting onto the

*x*-axis, and adding the forces on the top and bottom facets, the contribution of

*E*

_{⊥}to the total force is found to be

*F*

_{w}=¼

*ε*

_{o}(

*n*

^{2}-1)|

*E*|

^{2}, is normal to the side-walls and expansive in this case of

*p*-polarized light. Here the left wall area is larger than the right wall area by 2

*a*sinθ′

_{B}, so the net force on the side-walls, directed along the negative

*x*-axis, is

*E*

_{o}and cross-sectional area

*a*cosθ

_{B}momentum arrives at the rate of ½(

*a*cosθ

_{B})

*ε*

_{o}

_{B}with the

*x*-axis, where cos(2θ′

_{B})=(

*n*

^{2}-1)/(

*n*

^{2}+1), the total force

*F*

_{x}in Eq. (10) is seen to be equal to the time rate of change of the incident optical momentum.

## 4. Wedge immersed in a liquid

8. 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]

10. 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]

*n*

_{2}immersed in a liquid of index

*n*

_{1}. The Brewster angle in this case is given by tanθ

_{B}=

*n*

_{2}

*/n*

_{1}and, as before, θ

_{B}+θ′

_{B}=90° and the apex angle of the prism is

*ϕ*=2θ′

_{B}. Keeping the same incident

*E*-field amplitude

*E*

_{o}as before, the previous

*H*-field magnitude

*H*

_{o}will have to be multiplied by

*n*

_{1}, and the

*E*-field magnitude inside the prism becomes

*n*

_{1}

*E*

_{o}/

*n*

_{2}. We continue to assume the same footprint,

*a*, for the beam at the entrance and exit facets of the prism. The density of bound charges at the interface between the prism and the liquid may be written as follows:

*σ*

_{b}was solely due to the bound charges on the exterior facets of the prism, in the present case the interfacial charge is the superposition of two adjacent and oppositely charged layers. One such layer belongs to the solid dielectric, the other to the liquid. A simple way to visualize these layers is to imagine the existence of a narrow gap around the wedge, separating the solid from its liquid host. The continuity of

*D*

_{⊥}at the interface requires the existence of an

*E*

_{⊥}within the gap itself. Thus the

*E*

_{⊥}discontinuity on one side of the gap yields the charge density on the solid surface, while the corresponding discontinuity on the other side of the gap determines the charge density on the liquid surface. Presently we are concerned only with the net force on the interfacial charges and, therefore, ignore the detailed composition of the charged layer at the solid-liquid interface. In the following sections we shall return to computing the force on the solid object alone, at which point due attention will be paid to the composition of the charged layer.

*x*-axis, and its magnitude is given by

*x*-axis is given by

*on the host liquid. As shown in Fig. 2, the uncompensated area of each edge within the liquid is*

**F**_{w}*a*sinθ

_{B}, and the angle between

*and the*

**F**_{w}*z*-axis is 2θ′

_{B}; therefore,

*a*cosθ

_{B}is the cross-sectional area of the incoming and outgoing beams, that the rate of flow of optical momentum per unit area per unit time is ¼ (

*ε*

_{o}

_{B}with the x-axis, it is clear that the total

*F*

_{x}exerted on the media (prism plus liquid) is equal to the time rate of change of the incident optical momentum.

11. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**, 569–582 (1992). [CrossRef] [PubMed]

## 5. Nature of the force exerted on interfacial charged layer

_{o}of the light - at the interface between the liquid of dielectric constant

*ε*

_{1}and solid of dielectric constant

*ε*

_{2}shown in Fig. 3. If the perpendicular

*E*-field in the gap is denoted by

*E*

_{g}, the continuity of

*⊥ yields the amplitude of*

**D****⊥ in the two media as**

*E**E*

_{g}

*/ε*

_{1}and

*E*

_{g}

*/ε*

_{2}, as indicated. The surface charge densities are then given by the

*E*⊥ discontinuity at each interface, namely,

*σ*

_{1}=

*ε*

_{o}[(1/

*ε*

_{1})-1]

*E*

_{g}and

*σ*

_{2}=

*ε*

_{o}[1-(1/

*ε*

_{2})]

*E*

_{g}. In the limit when the gap shrinks to zero, the net charge density becomes

*σ*=

*σ*

_{1}+

*σ*

_{2}=

*ε*

_{o}[(

*E*

_{g}/

*ε*

_{1})-(

*E*

_{g}

*/ε*

_{2})], as expected. The effective

**E**_{⊥}acting on each charge layer (thus exerting a force on the corresponding medium) is the average

*E*-field at the boundary. Therefore, the effective (perpendicular) force per unit area acting on

*σ*

_{1}is

*F*

_{1}=¼

*σ*

_{1}[(1/

*ε*

_{1})+1]

*E*

_{g}, which, upon substitution for

*σ*

_{1}, becomes

*σ*

_{2}is

*σ*turns out to be

*F*=

*F*

_{1}+

*F*

_{2}=¼

*ε*

_{o}[(1/

*ε*

_{1})2- (1/

*ε*

_{2})

^{2}] |

*E*

_{g}|

^{2}, which is identical to the total charge density σ times the net (effective)

*E*-field

*E*

_{eff}=½[(

*E*

_{g}/

*ε*

_{1})+(

*E*

_{g}/

*ε*

_{2})] at the solid-liquid interface. Note, however, that

*F*

_{1}and

*F*

_{2}

*cannot*be obtained by simply multiplying this

*E*

_{eff}into

*σ*

_{1}and

*σ*

_{2}, respectively. The reason is that an additional (attractive) Coulomb force exists between the two charged layers, namely,

*F*

_{1}and

*F*

_{2}emerge when the above

*F*

_{o}is added to σ

_{1}

*E*

_{eff}and σ

_{2}

*E*

_{eff}, respectively.

*ε*

_{1}=

*ε*

_{2}=

*ε*≠1. In the absence of the gap,

*E*

_{⊥}is continuous and there are no net charges, nor forces, at the boundary. When the gap is introduced, however,

*E*

_{g}=

*εE*

_{⊥},

*σ*

_{2}=-

*σ*

_{1}=

*ε*

_{o}(

*ε*-1)

*E*

_{⊥}and

*F*

_{2}=-

*F*

_{1}=¼

*ε*

_{o}(

*ε*

^{2}-1)|

*E*⊥|

^{2}. The net charge and the net force are still zero, of course, but each medium experiences a force arising from two sources: the force of

*E*

_{⊥}on the corresponding interfacial charge density, given by ±½

*ε*

_{o}(ε-1)|

*E*⊥|

^{2}, and the force Fo of Eq. (18) between the two charged layers, given by ±¼

*ε*

_{o}(

*ε*-1)2|

*E*⊥|

^{2}. For each medium, the two contributions to the force are in the same direction and thus reinforce each other. Note that the parallel component

*E*

_{‖}of the

*E*-field, if any, will exert on each medium a corresponding force ±½

*ε*

_{o}(

*ε*-1)

*Real*(

*E*

_{⊥}

*E*

_{‖}*) in the direction parallel to the junction. These are quite strong forces whose existence is masked by the continuity of

*ε*at the boundary, but brought out clearly when a gap is imagined to exist between the two media.

## 6. Force experienced by prism immersed in a liquid

*E*-field at the solid-liquid interface is

*E*

_{‖}=

*E*

_{o}cosθ

_{B}, which, by virtue of its continuity, is the same in the solid, liquid, and the gap. The perpendicular E-field on the liquid side of the interface is

*E*

_{⊥}=

*E*

_{o}sinθ

_{B}, which leads to a gap field of

*E*

_{g}=

*ε*

_{1}

*E*

_{o}sinθ

_{B}. The surface charge density on the solid side of the interface is thus given by

*E*

_{‖}on these charges, when projected onto the

*x*-axis and multiplied by 2

*a*(to account for the illuminated area on both facets of the prism) is

*E*

_{⊥}to the force exerted on the prism, we substitute for

*E*

_{g}in Eq. (17), then multiply by 2

*a*cosθ

_{B}to account for the illuminated areas on both sides of the wedge as well as for projection onto the

*x*-axis. We thus find

^{2}θ

_{B}=

*ε*

_{2}/(

*ε*

_{1}+

*ε*

_{2}), we obtain

*ε*

_{o}(

*ε*

_{1}+1)

*a*cosθ

_{B}, and the angle between the momentum vector and the

*x*-axis is 2θ′

_{B}, where cos2θ′

_{B}=(

*ε*

_{2}-

*ε*

_{1})/(

*ε*

_{2}+

*ε*

_{1}). The time rate of change of the incident momentum must, therefore, be multiplied by

*ε*

_{1}(

*ε*

_{2}-1)/(

*ε*

_{2}-

*ε*

_{1}) to yield the net force on the prism. That this coefficient exceeds unity (when

*ε*

_{1}>1) should not be surprising considering that the force of radiation on the body of the liquid is directed along the negative

*x*-axis, and is given by

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

*F*

_{‖x}of the electric Lorentz force, given by Eq. (20), are active participants in pushing the solid object around. The role of

*F*

_{⊥x}as given by Eq. (21), however, may be questioned on the grounds that the mutual attraction between the adjacent charged layers (denoted by

*σ*

_{1}and

*σ*

_{2}in Fig. 3) cannot possibly contribute to the prism’s motion through its liquid environment. This is especially true if the two charged layers are found to intermix at the (atomically) rough exterior surface of the solid. It thus seems appropriate to remove from

*F*

_{⊥x}the contribution of the attractive force between the two charged layers on each facet of the prism. This is done by using the average

*E*

_{⊥}at the solid-liquid boundary, namely, <

*E*

_{⊥}>=½ [1+(

*ε*

_{1}/

*ε*

_{2})]

*E*

_{o}sinθ

_{B}, when computing the perpendicular force on the charge density

*σ*

_{2}that appears on the solid side of the interface. We then find

*F̂*

_{x}on the prism is greater than the time rate of change of the incident momentum, this time by a factor of 2

*ε*

_{1}(

*ε*

_{2}-1)/[(

*ε*

_{1}+1)(

*ε*

_{2}-

*ε*

_{1})]. That this coefficient exceeds unity (when

*ε*

_{1}>1) may be understood in light of the fact that the force of radiation on the body of the liquid - again with the attractive force

*F*

_{o}between the adjacent charged layers removed - is directed along the negative

*x*-axis, and is given by

*, when combined with the force of*

**F**_{w}

**E**_{‖}on the surface charge density

*σ*

_{1}at the liquid side of the interface, has no component on the

*x*-axis. The net force of the radiation on the body of the liquid, whether the attractive force

**F**_{o}between the adjacent charged layers is included, as in Eq. (23), or excluded, as in Eq. (26), is therefore due solely to the action of

**E**_{⊥}on the surface charge density

*σ*

_{1}.

**E**_{⊥}on the

*net*charge accumulated at the interface (i.e.,

*σ*

_{1}+

*σ*

_{2}), then the

*net*force on the body of the liquid will be zero, and the force on the prism becomes

*ε*

_{1}and

*ε*

_{2}in order to settle the question.

## 7. Computer simulations

_{o}=0.65 µm) illuminating a prism of (relative) index

*n*=1.5 are shown in Fig. 4. The incidence is at Brewster’s angle θ

_{B}=56.31°, and the apex angle of the prism is

*ϕ*=2θ′

_{B}=67.38°.

*H*-field profile at the outset is

*H*

_{y}(

*x, z*=0)=

*H*

_{o}exp[-(

*x/x*

_{o})

^{2}], where

*H*

_{o}=2.6544 A/m and

*x*

_{o}=3.9 µm. A time-snapshot of this

*H*-field is shown in Fig. 4(a), where the beam is seen to propagate along the negative

*z*-axis, enter the prism, then re-emerge into the free-space without any significant reflection losses at the boundaries. In free space, the initial

*E*-field amplitude at the center of the Gaussian beam is

*E*

_{o}=

_{o}

*H*

_{o}=1000 V/m, where

*E*-field as it propagates along the

*z*-axis and passes through the prism is depicted in Fig. 4(b). Inside the prism, the beam is seen to broaden by a factor of

*n*, while its

*E*-field amplitude decreases by the same factor. Ignoring for the moment the complication that the

*E*-field consists of both

*E*

_{x}and

*E*

_{z}, we approximate the

*z*-component of the initial beam’s Poynting vector as follows:

*y*-axis) is then found to be

*n*

_{1}, we multiply the

*H*-field by

*n*

_{1}, thus maintaining the

*E*-field strength at

*E*

_{o}=1000 V/m. The integrated power of the incident beam in its liquid environment will then be 6.49

*n*

_{1}×10

^{-3}W/m.

*n*

_{1}=1.33,

*n*

_{2}=1.995,

*H*

_{o}=3.53 A/m, incident power=8.63×10

^{-3}W/m). Here θ

_{B}and θ′

_{B}remain the same as in the previous case, since the relative index of the prism,

*n*

_{2}/

*n*

_{1}, has not changed. Visible in Fig. 4(d) are interference fringes formed between the incident beam and the residual reflections at both facets of the prism, although the contribution of these fringes to the overall force remains negligible. From Eq. (15) the total force must be ½(

*F*

_{x}=22.95 pN/m. The simulated force on the entire system (prism plus water) is found to be

*F*

_{x}=22.82 pN/m, in excellent agreement with the aforementioned theoretical value.

## 8. Concluding remarks

## Acknowledgments

## References

1. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express |

2. | J. D. Jackson, |

3. | J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A |

4. | R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. |

5. | R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A |

6. | R. Loudon, “Radiation Pressure and Momentum in Dielectrics,” De Martini lecture, Fortschr. Phys. |

7. | Y. N. Obukhov and F. W. Hehl, “Electromagnetic energy-momentum and forces in matter,” Phys. Lett. A |

8. | 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. |

9. | A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science |

10. | A. Rohrbach and E. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. |

11. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 6, 2005

Revised Manuscript: March 8, 2005

Published: March 21, 2005

**Citation**

Masud Mansuripur, Armis Zakharian, and Jerome Moloney, "Radiation pressure on a dielectric wedge," Opt. Express **13**, 2064-2074 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-2064

Sort: Journal | Reset

### References

- M. Mansuripur, �??Radiation pressure and the linear momentum of the electromagnetic field,�?? Opt. Express 12, 5375-5401 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375</a>. [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics, 2nd edition (Wiley, New York, 1975).
- J. P. Gordon, �??Radiation forces and momenta in dielectric media,�?? Phys. Rev. A 8, 14-21 (1973). [CrossRef]
- R. Loudon, "Theory of the radiation pressure on dielectric surfaces,�?? J. Mod. Opt. 49, 821-838 (2002). [CrossRef]
- R. Loudon, �??Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,�?? Phys. Rev. A 68, 013806 (2003). [CrossRef]
- R. Loudon, �??Radiation Pressure and Momentum in Dielectrics,�?? De Martini lecture, Fortschr. Phys. 52, 1134-1140 (2004). [CrossRef]
- Y. N. Obukhov and F. W. Hehl, �??Electromagnetic energy-momentum and forces in matter,�?? Phys. Lett. A 311, 277-284 (2003). [CrossRef]
- 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]
- A. Ashkin and J. M. Dziedzic, �??Optical trapping and manipulation of viruses and bacteria,�?? Science 235, 1517-1520 (1987). [CrossRef] [PubMed]
- A. Rohrbach and E. Stelzer, �??Trapping forces, force constants, and potential depths for dielectric spheres in thepresence of spherical aberrations,�?? Appl. Opt. 41, 2494 (2002). [CrossRef] [PubMed]
- A. Ashkin, �??Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,�?? Biophys. J. 61, 569-582 (1992). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.