## Radiation pressure and the linear momentum of the electromagnetic field

Optics Express, Vol. 12, Issue 22, pp. 5375-5401 (2004)

http://dx.doi.org/10.1364/OPEX.12.005375

Acrobat PDF (322 KB)

### 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

## 1. Introduction

**of the classical electrodynamics, and that, in free-space, the momentum density**

*S**(i.e., momentum per unit volume) is given by*

**p***p*=

*2, where*

**S**/c*c*is the speed of light in vacuum. What has been a matter of controversy for quite some time now is the proper form for the momentum of the electromagnetic waves in dielectric media. The question is whether the momentum density in a material medium has the form

*=*

**p***×*

**D***, due to Minkowski [1,2*

**B**2. H. Minkowski, Math. Annalon68, 472 (1910). [CrossRef]

*=*

**p****×**

*E*

**H**/c^{2}, due to Abraham [3

3. M. Abraham and R. C. Circ. Mat. Palermo28, 1 (1909). [CrossRef]

4. M. Abraham and R. C. Circ. Mat. Palermo30, 33 (1910). [CrossRef]

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

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

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

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

*p*-polarized plane wave. The torque is calculated directly from the Lorentz law applied to the induced (bound) charges at the surfaces of the slab, then shown to be consistent with the change in the angular momentum of the incident light. The case of an anti-reflection coated, semi-infinite dielectric medium is taken up in Section 8, where the increase in the momentum of the incident beam upon transmission into the dielectric medium is shown to result in a net force on the anti-reflection coating layer that tends to peel the layer away from its substrate; another new result that requires experimental verification. In Section 9 we analyze the case of a dielectric slab of finite thickness, and show that optical interference within the slab is responsible for the (longitudinal) stress induced by the electromagnetic radiation.

*E*-field, we show that the lateral pressure on the medium can be compressive or expansive, and that the magnitude and direction of this radiation force are in complete accord with the results of Sections 5 and 6. The generality of this lateral pressure (and the dependence of its direction on the state of polarization) are brought to the fore in Section 11, where the simple fringes produced by the interference between two plane waves are shown to exhibit the same phenomena.

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]

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]

## 2. Notation and basic definitions

*(*

**E̱***x, y, z, t*)=

*(*

**E***x, y, z*) exp(-i

*ω t*), where

*ω*=2π

*f*is the angular frequency. For brevity, we omit the explicit dependence of the fields on

*x, y, z, t*. To specify their magnitude and phase, complex amplitudes such as

*E*are expressed as |

*E*| exp(iϕ

_{E}). Time-averaged products of two fields, say,

*Real*(

*A̱*)=|

*A*| cos(

*ωt*-ϕ

_{A}) and

*Real*(

*Ḇ*)=|

*Ḇ*| cos(

*ω t*-ϕ

_{B}), given by ½|

*AB*| cos(ϕ

_{A}-ϕ

_{B}), may also be written ½

*Real*(

*AB**).

**∇·**=0, the density of bound charges

*D**ρ*

_{b}=-

**∇·**may be expressed as

*P**ρ*

_{b}=

*ε*

_{o}

**∇·**. Inside a homogeneous and isotropic medium,

*E**being proportional to*

**E***and*

**D****∇·**=0 imply that

*D**ρ*

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

*perpendicular to the interface,*

**D**

**D**_{⊥}, must be continuous. The implication is that

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

*σ*=

*ε*

_{o}(

*E*

_{2⊥}-

*E*

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

*E*-field, these charges give rise to an electric Lorentz force

*=½*

**F***Real*(

*σ**), where

**E***is the force per unit area of the interface. Since the tangential*

**F***E*-field,

**E**_{‖}, is generally continuous across the interface, there is no ambiguity as to which field must be used in conjunction with the Lorentz law. As for the perpendicular component, the average

*across the boundary, ½(*

**E**

**E**_{1⊥}+

**E**_{2⊥}), must be used in calculating the interfacial force. (The use of the average

**E**_{⊥}in this context is not a matter of choice; it is the only way to get the calculated force at the boundary to agree with the time rate of change of the momentum that passes through the interface. From a physical standpoint, the interfacial charges produce a local

**E**_{⊥}that has the same magnitude but opposite directions on the two sides of the interface. It is this locally-generated

**E**_{⊥}that is responsible for the

*E*-field’s discontinuity. Averaging

**E**_{⊥}across the interface eliminates the local field, as it should, since the charge cannot exert a force on itself.)

**∇·**=0 and

*B**=*

**B***µ*

_{o}

*, the perpendicular*

**H***H*-field,

**H**_{⊥}, at the interface between adjacent media must remain continuous. The tangential

*H*-field at such interfaces, however, may be discontinuous. This, in accordance with Maxwell’s equation ∇×

*=*

**H̱***+ ∂*

**J̱***/∂*

**Ḏ***t*, gives rise to an interfacial current density

**J**_{s}=

**H**_{2‖}-

**H**_{1‖}. Such currents can exist on the surfaces of good conductors, where

*E*_{‖}is negligible, yet the high conductance of the medium permits the flow of the surface current. Elsewhere, the only source of electrical currents are bound charges, with the bound current density being

*J̱*_{b}=∂

**/∂**

*P̱**t*=

*ε*

_{o}(

*ε*-1)∂

**/∂**

*E̱**t*. Assuming time-harmonic fields with the time-dependence factor exp(-i

*ω t*), we can write

*J*_{b}=-i

*ω ε*

_{o}(

*ε*-1)

*. The*

**E***H*-field of the electromagnetic wave then 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.*

**F****Note**: For time-harmonic fields, the contribution of conduction electrons to current density may be combined with that of bound electrons. Since

*=*

**J**_{c}*σ*, where

_{c}**E***σ*

_{c}is the conductivity of the medium, the net current density

*+*

**J**_{c}*may be attributed to an effective dielectric constant*

**J**_{b}*ε*+i(

*σ*

_{c}

*/ε*

_{o}

*ω*). In general, since

*ε*is complex-valued, there is no need to distinguish conduction electrons from bound electrons, and

*ε*may be treated as an effective dielectric constant that contains both contributions. An exception will be made in Section 3 in the case of perfect conductors, where

*σ*

_{c}→∞. Here both

*E*- and

*H*-fields inside the medium tend to zero, and the contributions of bound charges/currents become negligible. The effect of the radiation in this case will be the creation of a surface current density

*and a surface charge density*

**J**_{s}*σ*, both of which may be attributed in their entirety to the conduction electrons. The conservation of charge then requires that ∇·

*+∂*

**J̱**_{s}*̱σ*/∂

*t*=0.

## 3. Reflection of plane wave from a perfect conductor

_{o}, having

*E*-field amplitude

*E*

_{o}(units=V/m) and

*H*-field amplitude

*H*

_{o}=

*E*

_{o}/

*Z*

_{o}(units=A/m), where

*=½*

**S***Real*(

*×*

**E****), and the momentum density (per unit volume) is*

**H***p*=

**/**

*S**c*

^{2}(vacuum speed of light

*A*=1.0 m

^{2}and height

*c*. The same momentum returns to the source after being reflected from the mirror, so the net rate of change of the field momentum over a unit area d

*/d*

**p***t*=2

*, which is equal to the force per unit area,*

**S**/c*, exerted on the reflector. The force density of the light on a perfect reflector in free-space is thus given by*

**F***=*

**F***q*(

**+**

*E**×*

**V***) in conjunction with the surface current density*

**B***and the magnetic field*

**J**_{s}*at the surface of the conductor. Here there are neither free nor (unbalanced) bound charges, and the motion of the conduction electrons constitutes a surface current density*

**H***=*

**J**_{s}*q*. For time-harmonic fields, the force per unit area may thus be written

**V***J*

_{s}=2

*H*

_{o}(because ∇×

**=**

*H̱***+ ∂**

*J̱***/∂**

*Ḏ**t*; the factor of 2 arises from the interference between the incident and reflected beams where the two

*H*-fields, being in-phase at the mirror surface, add up.) Since

*=*

**B***µ*

_{o}

*, one might conclude that*

**H***F*

_{z}=2

*µ*

_{o}

*H*

_{o}, is assumed to exert a force on the

*entire J*

_{s}. The problem is that the field is 2

*H*

_{o}at the top of the mirror and zero just under the surface, say, below the skin-depth. (Here we are using a limiting argument in which a good conductor, having a finite skin-depth, approaches an ideal conductor in the limit of zero skin-depth.) Therefore, the average

*H*-field through the “skin-depth” must be used in calculating the force, and this average is

*H*

_{o}not 2

*H*

_{o}. The force per unit area thus calculated is

*F*

_{z}=

*µ*

_{o}

*ε*

_{o}

^{2}of incident optical power, for example, the radiation pressure on the mirror will be 6.67 nN/mm

^{2}.

*s*-polarized. Compared to normal incidence, the component of the magnetic field

*H*on the surface is now multiplied by cosθ, which requires the surface current density

*J*

_{s}to be multiplied by the same factor (remember that

*J*

_{s}is equal to the magnetic field at the surface). The component of force density along the

*z*-axis,

*F*

_{z}, is thus seen to have been reduced by a factor of cos

^{2}θ. This result is consistent with the alternative derivation based on the time rate of change of the field’s momentum in the

*z*-direction, d

*p*

_{z}/d

*t*, which is multiplied by cosθ in the case of oblique incidence. Since the beam has a finite diameter, its foot-print on the mirror is greater than that in the case of normal incidence by 1/cosθ. Thus the force density

*F*

_{z}, obtained by normalizing d

*p*

_{z}/d

*t*by the beam’s foot-print, is seen once again to be reduced by a factor of cos

^{2}θ.

*p*-polarized beam at oblique incidence on a mirror. The magnetic field component at the surface is 2

*H*

_{o}, which means that the surface current

*J*

_{s}must also have the same magnitude as in normal incidence. We conclude that the force density on the mirror must be the same as that at normal incidence, namely,

*F*

_{z}=

*ε*

_{o}

*z*-direction, however, is similar to that in Fig. 1(b), which means that the force density of normal incidence must have been multiplied by cos

^{2}θ in the case of oblique incidence. The two methods of calculating

*F*

_{z}for

*p*-light thus disagree by a factor of cos

^{2}θ.

*z*). This additional force pulls on the electric charges induced at the surface by

*E*

_{⊥}. Note that

*E*

_{⊥}=2

*E*

_{o}sinθ just above and

*E*

_{⊥}=0 just below the surface. The discontinuity in

*E*

_{⊥}gives the surface charge density as

*σ*=2

*ε*

_{o}

*E*

_{o}sinθ. The perpendicular

*E*-field acting on these charges is the average of the fields just above and just below the surface, namely,

*E*

_{o}sinθ. The electric force density is thus

*F*

_{z}=½

*Real*(

*ε*

_{o}

^{2}θ. The upward force on the charges thus reduces the downward force on the current, leading to a net

*F*

_{z}=

*ε*

_{o}

^{2}θ), which is the same as that in the case of normal incidence multiplied by cos

^{2}θ.

*σ*=2

*ε*

_{o}

*E*

_{o}sinθ exp(i2π

*x*sinθ/λ

_{o}) in the above example is produced by the spatial variations of the current density

*J*

_{s}=2

*H*

_{o}exp(i2π

*x*sinθ/λ

_{o}). Conservation of charge requires ∇·

**+∂**

*J̱**ρ*/∂

*t*=0, which, for time-harmonic fields, reduces to ∂

*J*

_{s}/∂

*x*-i

*ωσ*=0. Considering that

*H*

_{o}=

*E*

_{o}/

*Z*

_{o}and

*ω*=2π

*c*/

*λ*

_{o}, it is readily seen that the above

*J*

_{s}and

*σ*satisfy the required conservation law.

**Note:**The separate contributions of charge and current to the radiation pressure discussed in this section were originally discussed by Max Planck in his 1914 book,

*The Theory of Heat Radiation*[10]. Our brief reconstruction of his arguments here is intended to facilitate the following discussion of electromagnetic force and momentum in dielectric media.

## 4. Semi-infinite dielectric

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

*E*- and

*H*-fields have magnitudes

*E*

_{o}and

*H*

_{o}=

*E*

_{o}/

*Z*

_{o}. Assuming a beam cross-sectional area of unity (

*A*=1.0m

^{2}), the time rate of flow of momentum onto the surface is ½

*ε*

_{o}

*r*|

^{2}is reflected back. The net rate of change of linear momentum, which must be equal to the force per unit area exerted on the surface, is thus

*F*

_{z}=½

*ε*

_{o}(1+|

*r*|

^{2})

*ε*is not purely real, but has a small imaginary part. The complex refractive index of the material is

*n*+i

*κ*=√ε, and the reflection coefficient is

*r*=(1-√ε)/(1+√ε).

*E*-field is

*E*(

*z*)=

*E*

_{t}exp(i2π√

*ε*

_{z}

*/λ*

_{o}), where

*E*

_{t}=(1+

*r*)

*E*

_{o}, the

*H*-field is

*H*(

*z*)=√

*ε*(

*E*

_{t}

*/Z*

_{o})exp(i2π√

*ε*

_{z}

*/λ*

_{0}), the

*D*-field is

*D*(

*z*)=

*ε*

_{o}

*E*(

*z*)+

*P*(

*z*)=

*ε*

_{o}

*εE*(

*z*), and the dipolar current density is

*J*(

*z*)=-i

*ωP*(

*z*)=-i

*ωε*

_{o}(ε-1)

*E*(

*z*), where

*ω*=2π

*f*=2π

*c/λ*

_{o}is the optical frequency. The force per unit volume is thus given by

*F*

_{z}from

*z*=0 to ∞. The multiplicative coefficient

*κ*disappears after integration, and the force per unit area becomes

*F*

_{z}=¼ (

*n*

^{2}+

*κ*

^{2}+1)

*ε*

_{o}|

*E*

_{t}|

^{2}. Upon substitution for

*E*

_{t}and

*r*, this expression for

*F*

_{z}turns out to be identical to that obtained earlier based on momentum considerations.

*κ*→0 and write the radiation force per unit surface area of the dielectric as

*F*

_{z}=¼ (

*n*

^{2}+1)

*ε*

_{o}|

*E*

_{t}|

^{2}. (A similar trick has been used by R. Loudon in his calculation of the photon momentum inside dielectrics [7

**49**, 821–838 (2002). [CrossRef]

*H*

_{t}=

*nE*

_{t}

*/Z*

_{o}, one may also write

*F*

_{z}=¼

*ε*

_{o}|

*E*

_{t}|

^{2}+¼

*µ*

_{o}|

*H*

_{t}|

^{2}. This must be equal to the rate of the momentum entering the medium at

*z*=0. Since the speed of light in the medium is

*c/n*, the momentum density (per unit volume) within the dielectric may be expressed as follows:

*=¼ (*

**p****×**

*D***)+¼(**

*B**×*

**E****)/**

*H**c*

^{2}. Historically, there has been a dispute as to whether the proper form for the momentum density of light in dielectrics is Minkowski’s ½

**×**

*D**or Abraham’s ½*

**B***×*

**E***/*

**H***c*

^{2}[5

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

*p*. In the limit when

*ε*→1, the two terms in the expression for

*become identical, and the familiar form for the free-space,*

**p****=**

*p***/**

*S**c*

^{2}, emerges.

**with**

*D**ε*

_{o}

*+*

**E****and**

*P**with*

**B***µ*

_{o}

*H*, we obtain

*=¼(*

**p****×**

*P***)+½(**

*B***×**

*E***)/**

*H**c*

^{2}, which shows the separate contributions to a plane-wave’s momentum density by the medium and by the radiation field. The mechanical momentum of the medium, ¼

**×**

*P***, arises from the interaction between the induced polarization density**

*B**and the light’s*

**P***-field. The contribution of the radiation field, ½*

**B***×*

**E**

**H**/c^{2}, has the same form,

**S**/c^{2}, as the momentum density of electromagnetic radiation in free space. Since

**=**

*P**ε*

_{o}(

*ε*-1)

*, the mechanical momentum density may be written as ¼*

**E***×*

**P***=½(*

**B***ε*-1)

**S**/c^{2}. For a dilute medium having refractive index

*n*≈1, the coefficient of

**S**/c^{2}in the above formula reduces to ½(

*ε*-1)≈

*n*-1, which leads to the expression (

*n*-1)

**S**/c^{2}derived in [5

**8**, 14–21 (1973). [CrossRef]

**Note:**In a recent paper [15

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

*×*

**E**

**H**/c^{2}, where

*and*

**E***are evaluated inside the dielectric. In contrast, our approach in the present section, which involves the introduction of a small (but non-zero)*

**H***κ*, followed by an integration of the feeble magnetic Lorentz force over the infinite thickness of the dielectric, ensures that the mechanical momentum of the medium is properly taken into consideration. This yields the term ¼(

**×**

*P**) in our last expression for the momentum density, which is missing from Obukhov and Hehl’s Eq. (27).*

**B**## 5. Oblique incidence with s-polarized light

*s*-polarization here, leaving a discussion of

*p*-polarized light at oblique incidence for the next section. The two cases turn out to be fundamentally different, although both retain the expression for momentum density derived in the case of normal incidence.

*s*-polarized light at the interface between the free-space and a dielectric medium. Again, we assume that the dielectric constant

*ε*is complex, allowing it to approach a real number only after calculating the total force by integrating through the thickness of the medium. Inside the medium, the

*E*- and

*H*-field distributions are

*s*-light. Since there are no free charges inside the medium (nor on its surface), the only relevant force here is the magnetic Lorentz force on the dipolar current density

*J*

_{y}(

*x, z*)=-i

*ω ε*

_{o}(

*ε*-1)

*E*

_{t y}(

*x, z*). Following the same procedure as before, we find the net force components along the

*x*- and

*z*-axes to be

*ε*, whether real or complex. Considering that the incident beam’s cross-sectional area must be

*A*=cosθ to produce a unit area footprint at the interface, the time rate of change of the incident beam’s momentum upon reflection from the surface gives rise to

*F*

_{x}=½

*ε*

_{o}sinθ cosθ (1-|

*r*

_{s}|

^{2})

*F*

_{z}=½

*ε*

_{o}cos

^{2}θ (1+|

*r*

_{s}|

^{2})

*ε*to be real, and set the refractive index

*n*=√

*ε*. Since|

*E*

_{t}|=|1+

*r*

_{s}|

^{2}

*n*sinθ′, and

*A*of the transmitted beam is not unity, but cosθ′, which means that the above expressions for

*F*

_{x}and

*F*

_{z}must be divided by cosθ′ if force per unit area is desired. The direction of the force

*=*

**F***F*+F

_{x}**x**_{z}z is

*not*the same as the propagation direction (i.e., at angle θ′ to the surface normal); the reason for this will become clear shortly.

*=¼*

**F***ε*o(

*ε*+1)|

*E*

_{t}|

^{2}(sinθ′

*+cosθ′*

**x***z*). Multiplying this force by the beam’s cross-sectional area

*A*=cosε′, then subtracting it from the previously calculated force in Eq. (7) yields the following residual force:

*+ cosθ′*

**x***. Moreover, the magnitude of Δ*

**z****is proportional to sinθ′, the cross-sectional area of a segment from the edge of the beam just below the interface; see Fig. 4. The force per unit area of the beam’s edge is thus**

*F**F*

^{(edge)}=¼

*ε*

_{o}(

*ε*-1)|

*E*

_{t}|

^{2}. (For example, if the optical power density inside a glass medium having

*n*=1.5 happens to be 1.0 W/mm

^{2}, the lateral pressure on each edge of the beam will be 1.39 nN/mm

^{2}.) This force, which the light exerts on the dielectric at both edges of the beam, is orthogonal to the edge and compressive in this case of

*s*-polarized light. (We will see in the next section that the force at the edges of a

*p*-polarized beam has exactly the same magnitude but opposite direction, tending to expand the dielectric medium.) Once the component of the force acting on the beam’s edge has been subtracted from Eq. (7), the remaining force turns out to be in the propagation direction and in full agreement with the results of the preceding section.

## 6. Oblique incidence with p-polarized light

*p*-polarized light depicted in Fig. 5 differs somewhat from the case of s-polarized light in that, in addition to the magnetic Lorentz force on its bulk, the medium also experiences an electric Lorentz force on the (bound) charges induced at its interface with the free space. Inside the medium the field distributions are

*p*-light. Since there are no net bound charges inside the medium, the only relevant force in the bulk is the Lorentz force of the

*H*-field on the dipolar current density

**(**

*J**x, z*)=-i

*εε*

_{o}(

*ε*-1)

*(*

**E**_{t}*x, z*). Following the same procedure as before, we find the force components exerted on the bulk along the

*x*- and

*z*-axes as follows:

*z*-component of

*is*

**E***E*

_{z}(

*x, z*=0)=(1-

*r*

_{p}) sinθ

*E*

_{o}exp(i2π

*x*sinθ/λ

_{o}). Immediately below the interface, the continuity of

**D**_{⊥}requires that the above

*E*

_{z}be divided by

*ε*. The surface charge density

*σ*, being equal to the discontinuity in

*ε*

_{o}

*E*

_{z}, is thus given by

*F*

_{x}=½

*Real*[

*σ*(

*x*)

*E*

_{x}*(

*x, z*=0)], where

*E*

_{x}is continuous across the interface, and

*F*

_{z}=½

*Real*[

*σ*(

*x*)

*E*

_{z}*(

*x, z*=0)], where

*E*

_{z}is the average

*E*

_{⊥}across the boundary, namely,

*E*

_{z}=½(1+1/

*ε*)(1-

*r*

_{p})sinθ

*E*

_{o}exp(i2π

*x*sinθ/λ

_{o}). Thus

*ε*, real or complex. Considering that the incident beam’s cross-sectional area must be

*A*=cosθ to yield a unit-area footprint at the interface, the time rate of change of the incident beam’s momentum upon reflection gives rise to

*F*

_{x}=½ sinθ cosθ (1-|

*r*

_{p}|

^{2})

*µ*

_{o}

*F*

_{z}=½cos

^{2}θ (1+|

*r*

_{p}|

^{2})

*µ*

_{o}

*s*- and

*p*-beams on the bulk of the medium given by Eqs. (6) and (10). In Eq. (6) the

*E*-field inside the medium is

*E*

_{t}=(1+

*r*

_{s})

*E*

_{o}, whereas in Eq. (10) the

*H*-field in the medium is

*H*

_{t}=(1-

*r*

_{p})

*H*

_{o}. From Eqs. (5) and (9) we find

*E*

_{t}|is the same in both cases, the force components turn out to be different. This is caused by the direction of the force on the beam’s edges being different in the two cases, as will become clear below when we analyze the case of media with real-valued

*ε*.

*ε*to be real, and set the refractive index

*n*=√

*ε*. Since|

*H*

_{t}|

^{2}=|1-

*r*

_{p}|

^{2}

*H*

_{t}=

*nE*

_{t}

*/Z*

_{o}, sinθ=

*n*sinθ′, and

*A*of the transmitted beam is not unity, but cosθ′. From the earlier discussions we know that, inside the dielectric, the force per unit cross-sectional area of the beam is

*=¼*

**F***ε*o(

*ε*+1)|

*E*

_{t}|

^{2}(sinθ′

*+cosθ′*

**x***), aligned with the propagation direction. Multiplying this force by the beam’s cross-sectional area*

**z***A*=cosθ′, then subtracting it from the previously calculated force in Eq. (15), yields the following residual force:

*+ cosθ′*

**x***. Moreover, its magnitude is proportional to sinθ′, the cross-sectional area of a segment from the edge of the beam just below the interface; see Fig. 4. The force per unit area of the beam’s edge is thus*

**z***F*

^{(edge)}=¼

*ε*

_{o}(

*ε*-1)|

*E*

_{t}|

^{2}. This force, exerted on the dielectric at both edges of the beam, is orthogonal to the edge and expansive in this case of

*p*-polarized light. Once the force component acting on the beam’s edge has been subtracted from Eq. (15), the remaining force turns out to be in the propagation direction and in full agreement with the results of Section 4.

## 7. Dielectric slab illuminated at Brewster’s angle

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

*p*-polarized plane wave is incident on a dielectric slab of thickness

*d*and refractive index

*n*at the Brewster’s angle θ

_{B}(tanθ

_{B}=

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

_{B}=1/

*n*. Since the reflectivity at Brewster’s angle is zero, the only beams in this system are the incident beam, the refracted beam inside the slab, and the transmitted beam; see Fig. 6(a). Inside the slab, 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 slab and just beneath the surface,

*E*

_{x}=

*E*

_{o}cosθ

_{B}, and

*E*

_{z}=(

*E*

_{o}/

*n*

^{2})sinθ

_{B}, which means that the magnitude of

*E*inside the slab is

*E*

_{o}/

*n*.

**∇·**=

*D**ε*

_{o}

*ε*

**∇·**=0. The bound currents have a density of

*E*

**J̱**_{b}=∂

*/∂*

**P̱***t*=

*ε*

_{o}(

*ε*-1)∂

*/∂*

**E**̱*t*. However, the force exerted by the magnetic field of the light on these currents is zero because of the 90° phase between

*and ∂*

**H̱***/∂*

**E**̱*t*. Thus the Lorentz force inside the volume of the slab is zero. The only force is due to the bound charges induced on the two surfaces of the plate, the density of which is obtained from the discontinuity in

*ε*

_{o}

**E**_{⊥}across the interface, namely,

*=*

**P***ε*

_{o}(

*ε*-1)

*, where*

**E***inside the medium has magnitude*

**E***E*

_{o}/

*n*. Consider a (tilted) cylindrical volume, aligned with the internal

*E*-field and stretched between the two facets, as shown in Fig. 6(b). If the base area of this cylinder is denoted by

*a*, its volume will be

*ad*, where

*d*is the thickness of the slab. The electric dipole moment of this cylindrical volume is thus

*ad*, which must be equal to the surface charge

**P***aσ*on either base multiplied by the length of the cylinder,

*d*/sinθ′

_{B}=

*nd*/sinθ

_{B}. The charge density on each surface is thus

*σ*=

*ε*

_{o}(

*ε*-1)

*E*

_{o}sinθ

_{B}/

*n*

^{2}=

*ε*

_{o}

*E*

_{o}(1-1/

*n*

^{2})sinθ

_{B}, consistent with our earlier finding in Eq. (17).

*x*). If the incident beam’s cross-sectional area is denoted by

*A*, its footprint on the slab will have an area

*A*/cosθ

_{B}. The force density

*F*

_{x}, when integrated over the footprint and multiplied by the distance

*d*between the two surfaces, yields the following torque

*T*on the slab:

*F*

_{x}component yields a torque that tends to rotate the slab around the

*y*-axis. Now, the electromagnetic field’s momenta before and after the slab are the same, both in magnitude (

*p*=½

*A*

*ε*

_{o}

*is displaced parallel to itself by Δ=*

**p***d*(1-1/

*n*

^{2})sinθ

_{B}, as shown in Fig. 6(a). The change

*p*Δ in the angular momentum of the beam is thus seen to be identical with the torque

*T*exerted by

*F*

_{x}, as given by Eq. (19). As a numerical example, consider the case of a glass slab having

*A*=1.0 mm

^{2},

*d*=10 µm, and

*n*=1.5, illuminated at θ

_{B}=56.3° with a 1.0 W/mm

^{2}beam of light. Using Eq. (19), the torque on the slab is found to be

*T*=15.4 f N.m.

*F*

_{z}on the top surface of the slab. Since

*E*

_{z}is discontinuous across the boundary, we must average

*E*

_{z}just above and just below the interface. Thus

*F*

_{z}=½

*σE*

_{z}=¼

*ε*

_{o}

*n*

^{2})(1+1/

*n*

^{2}) sin

^{2}θ

_{B}, with the factor ½ introduced again to account for time- and space-averaging. Integrating over the footprint of the beam (area=

*A*/cosθ

_{B}), we obtain

*F*

_{z}=¼

*ε*

_{o}

*n*

^{2}-

*n*

^{-2})

*A*cosθ

_{B}. The forces

*F*

_{z}on the top and bottom facets of the slab, having equal magnitudes and opposite signs, cancel out.

*F*

_{z}, being laterally displaced by

*d*tanθ′

_{B}=

*d/n*between the top and bottom facets, must also exert a torque on the plate. This torque, however, is exactly cancelled out by an equal and opposite torque originating from the force of the beam exerted at its right and left edges. As derived in Section 6, the force density

*F*

^{(edge)}=¼

*ε*

_{o}(

*n*

^{2}-1)|

*E*

_{t}|

^{2}is normal to the edge and expansive in the case of

*p*-light. This force is of equal magnitude and opposite sign on the two edges of the beam inside the slab; while its horizontal components are aligned (and, therefore, do not give rise to a torque), its vertical components

*F*

^{(edge)}sinθ′

_{B}produce a torque. The product of the area of each edge of the beam and the distance between the edges is (

*d*/cosθ′

_{B})(

*A*/cosθ

_{B})=

*Ad*(

*n*

^{2}+1)cosθ

_{B}/cosθ′

_{B}, and the

*E*-field inside the medium is

*E*

_{t}=

*E*

_{o}/

*n*, so the torque produced by

*E*-field.

**Note**: In a 1912 paper, G. Barlow (stimulated by J. H. Poynting) reports on an experiment similar to that described in this section, although incidence is not at Brewster’s angle [16

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

*p*assigned to the light inside the glass medium results in the same overall torque.

## 8. Force experienced by an anti-reflection coating layer

*n*is shown in Fig. 7, coated with a dielectric layer of index √

*n*and thickness

*d*=λ

_{o}/(4√

*n*). This quarter-wave thick layer is a perfect anti-reflection coating that allows the incident beam into the semi-infinite medium with no reflection whatsoever. Conservation of energy requires the

*E*-field to enter the semi-infinite medium with an amplitude|

*E*

_{t}|=

*E*

_{o}/√

*n*, while the quarter-wave thickness of the coating layer shifts the phase of

*E*

_{t}by 90° relative to that of the incident beam. The

*H*-field in the semi-infinite medium is then

*H*

_{t}=

*nE*

_{t}

*/Z*

_{o}=i√

*n*

*E*

_{o}/

*Z*

_{o}. Inside the coating layer, the counter-propagating beams shown in Fig. 7 have the following distributions:

*z*=0 to

*z*=λ

_{o}/(4√

*n*), to yield the total force per unit area exerted on the layer, namely,

*p*

_{i}/d

*t*=½

*ε*

_{o}

*p*

_{t}/d

*t*=¼

*ε*

_{o}(

*ε*+1)|

*E*

_{t}|

^{2}=¼

*ε*

_{o}(

*n*

^{2}+1)

*n*, it is clear that

*F*

_{z}=d(

*p*

_{i}

*-p*

_{t})/d

*t*; in other words, the upward force experienced by the coating layer is exactly equal to the time rate of change of the light’s linear momentum upon crossing the layer. As a numerical example, consider the case of a glass substrate of index

*n*=2.0, coated with a quarter-wave thick layer of index √

*n*=1.414. At the incident power density of 1.0 W/mm

^{2}, the computed net force on the coating layer is

*F*

_{z}=-0.83 nN/mm

^{2}.

## 9. Homogeneous slab in free space

*d*and complex refractive index

*n*+i

*κ*=√

*ε*, surrounded by free-space and illuminated at normal incidence. The slab’s (complex) reflection and transmission coefficients are denoted by

*r*and

*t*, respectively. The counter-propagating beams within the slab have

*E*-field amplitudes

*E*

_{1}and

*E*

_{2}; the total field distribution is given by

*ρ*=[(

*√*ε-1)/(

*√*ε+1)]exp (i4p√

*εd/λ*

_{o}), the various parameters of the system of Fig. 8 may be written as follows:

*H*-field on the dipolar current distribution, then integrating through the thickness of the slab. The final result is

*ε*is real,

*κ*goes to zero and the above formula reduces to

*r*|

^{2}+|

*t*|

^{2}=1 for a non-absorbing slab. According to Eq. (27), the net force is zero when

*d*=λ

_{o}/2

*n*. This is consistent with the fact that a half-wave-thick plate does not reflect at all and, therefore, the incident and transmitted momenta are identical. The force density is a maximum for a quarter-wave-thick slab, and is given by

^{2}, the radiation pressure on a quarter-wave slab of

*n*=1.5 is thus predicted to be 0.98 nN/mm

^{2}.

## 10. Gaussian beam in homogeneous, isotropic medium

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

*s*- and expansive forces for

*p*-polarized light. These differences can be traced to the formula used for calculating the

*E*-field component of the Lorentz force. Alternative formulations of the Lorentz force and the conditions under which each can be considered valid are discussed in Section 15.

*z*-axis in an isotropic, homogeneous medium, is shown in Fig. 9. The

*E*-field in this system is along the

*y*-axis, and has the following Gaussian profile:

*E*

_{o}is the magnitude of the field at the origin (

*x, z*)=(0, 0), λ=λ

_{o}/

*n*is the wavelength inside the homogeneous dielectric of refractive index

*n*, and α(

*z*)=1/

*z*)-iπ/λ

*R*

_{c}(

*z*) is the Gaussian beam’s complex parameter defined in terms of the beam’s 1/e radius

*r*

_{o}and radius of curvature

*R*

_{c}. The parameter α(

*z*) evolves from its initial value a(0) according to the Gaussian beam formula 1/α(

*z*)=1/α(0)+iλ

*z*/π. The beam’s

*H*-field may be found from the Maxwell equation ∇×

**=-∂**

*E̱***/∂**

*Ḇ**t*, or its time-harmonic equivalent ∇×

**=iω**

*E**µ*

_{o}

*. The only components of*

**H***H*turn out to be

*H*

_{x}and

*H*

_{z}, given by

*H*

_{x}diminish for a reasonably large beam size

*r*

_{o}, so only the first term in Eq. (31a) is significant. Also, since the Gaussian beam profile in Eq. (30) is valid only in the paraxial approximation, if the beam’s

*E*-field is re-calculated from the

*H*-field components of Eq. (31) using ∇×

**=-i**

*H**ωε*

_{o}

*ε*, the

**E***E*

_{y}(

*z*) of Eq. (30) will be recovered only after higher-order terms similar to those in Eq. (31a) have been neglected.

*F*

_{x}and, therefore, the only relevant fields are

*E*

_{y}and

*H*

_{z}. The lateral component of force per unit volume along the

*x*-axis is given by

*x*, is positive on one side of the Gaussian beam and negative on the other side. It is also compressive, meaning that it tends to pull the dielectric toward the beam center at

*x*=0. If the force density of Eq. (33) is integrated from

*x*=0 to ∞, the magnitude of the net force on either side of the center turns out to be ¼

*ε*

_{o}(

*n*

^{2}-1)

*s*-polarized beam.

*F*

_{x}, the relevant field components are

*E*

_{z}and

*H*

_{y}. The lateral component of force per unit volume along the

*x*-axis is then given by

*z*)=α(0)=1/

*x*, is positive on one side of the Gaussian beam and negative on the other side. It is also expansive, in the sense that it tends to push the dielectric away from the beam center at

*x*=0. If the force density of Eq. (37) is integrated from

*x*=0 to ∞, the magnitude of the net force on either side of the center turns out to be ¼

*µ*

_{o}(1-1/

*n*

^{2})

*r*

_{o}, Eq. (35a) yields

*E*

_{x}(

*x*=0,

*z*=0)=(

*Z*

_{o}/

*n*)

*H*

_{o}.

*s*-light, expansive for

*p*-light, and has the exact same magnitude of ¼

*ε*

_{o}(

*n*

^{2}-1)

## 11. Interference fringes and lateral radiation pressure

*E*- and

*H*-fields, thus resulting in a non-zero value of the magnetic Lorentz force. We have seen examples of this in Sections 8 and 9, where interference inside a dielectric layer of finite thickness gave rise to local forces. We now examine the case of two plane-waves, both polarized in the

*p*-direction, that propagate at angles ±θ relative to the

*z*-axis; see Fig. 10. Assuming that the field amplitudes inside the medium of refractive index

*n*are

*E*

_{o}and

*H*

_{o}=(

*n/Z*

_{o})

*E*

_{o}, the field distributions are written

*E*

_{x}and

*H*

_{y}are in-phase, they do not contribute to the force component along the

*z*-axis; the only force component is

*s*-polarized plane waves, where the fields are

*F*

_{x}turns out to be

*x*=0 to

*x*=λ

_{o}/(4

*n*sinθ), the result turns out to be

*E*-field at the origin is 2

*E*

_{o}. The force on each side of the fringe is thus seen to be identical to that for a Gaussian beam studied in Section 10, or for plane waves studied in Sections 5 and 6.

## 12. Light pulse and the photon momentum

**49**, 821–838 (2002). [CrossRef]

*n*. For concreteness, we assume a Gaussian temporal profile for the pulse, although any other profile should, in the end, lead to the same conclusions. The

*E*- and

*H*-fields are thus given by

*E*

_{o}is the amplitude of the

*E*-field at the pulse center where

*z*=(

*c/n*)

*t*, and

*w*is a measure of the pulse width along the

*z*-axis. The instantaneous density (per unit volume) of the Lorentz force, which has a component in the

*z*direction only, is given by

*w*>>λ

_{o}, the local force density at each point (

*x, y, z*) can be time-averaged over one oscillation period. The squared cosine term in Eq. (44) thus averages to ½ and the sine term to zero, yielding the following expression for the average local force density:

*z*=(

*c/n*)

*t*to ∞, or from

*z*=-∞ to (

*c/n*)

*t*, to yield the total force (per unit cross-sectional area) at the leading and trailing edges of the pulse as follows:

*F*

_{z}is positive, the pulse tends to push the medium forward, whereas at the trailing edge, where

*F*

_{z}is negative, the medium is being pulled backward.

*is directed along the*

**S***z*-axis and, at the pulse center, has a magnitude

*S*

_{z}=½(

*n/Z*

_{o})

*A*and the effective pulse duration by Δ

*T*, its total energy may be written as

*S*

_{z}

*A*Δ

*T*. [The same result is obtained by adding the

*E*-field and

*H*-field energy densities, ¼

*ε*

_{o}

*ε*

*µ*

_{o}

*c/n*)Δ

*T*and also with the cross-sectional area

*A*. Incidentally, for the Gaussian pulse of Eq. (43),

*hf*, where

*h*is Planck’s constant and

*f*is the optical frequency. The photon intensity is thus found to be ½

*Z*

_{o}

*hν*/(

*nA*Δ

*T*).

*M*–this could include the mass of the Earth, to which the slab is attached–its acquired momentum will be given by the integrated force over the pulse duration, namely,

*MV*=¼

*ε*

_{o}(

*ε*-1)

*A*Δ

*T*. [This reduces to

*MV*=½(

*n*

^{2}-1)

*hf/nc*for a single photon.] So long as the pulse stays within the slab this acquired momentum remains constant. However, as soon as the leading edge of the pulse exits through the slab’s rear facet, the trailing edge begins to exert a braking force to slow down the slab’s motion. By the time the trailing edge leaves the slab, the motion has come to a halt, and all the momentum initially acquired by the slab has returned to the light pulse.

**S**/c^{2}and ½(

*ε*-1)

**S**/c^{2}, respectively. With the pulse fully contained within the slab, its effective length along the

*z*-axis is (

*c/n*)Δ

*T*while its cross-sectional area is

*A*; the total electromagnetic momentum of the pulse should, therefore, be

*S*

_{z}

*A*Δ

*T/nc*. Since

*S*

_{z}

*A*Δ

*T*is the pulse’s energy, which equals

*hf*for a single photon, the photon’s electromagnetic momentum (when inside the slab) turns out to be

*hf/nc*. By the same token, the photon’s mechanical momentum (again, when fully contained within the slab) should be ½(

*n*

^{2}-1)

*hf/nc*. This is precisely the momentum

*MV*that the slab temporarily acquires from a single photon entry, as explained in the preceding paragraph. The physical basis for assigning a mechanical momentum to individual photons should thus be abundantly clear in light of the above considerations.

*n*

^{2}+1)/(2

*n*)]

*hf/c*, turns out to be greater than

*hf/c*, the momentum of the same photon in free space. If the facets of the slab happen to be anti-reflection coated, the difference between the inside and outside momenta will be balanced by the force exerted by the entering and exiting photons on the coating layer (see Section 8 for a discussion of this force). In the absence of anti-reflection layers, the facets reflect a fraction of the photons, in which case the momentum transferred to the slab by the reflected photons can be used in a statistical analysis to account for the difference between the photon momenta inside and outside the slab.

## 13. Mirror immersed in liquid dielectric

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

*n*. [The indices of the liquids used ranged from 1.33 (water) to 1.61 (carbon disulphide)]. In the present section we analyze the case of a perfect metallic conductor immersed in a liquid of arbitrary refractive index

*n*, and show that a direct application of our method yields results that are in full agreement with the reported experiments.

*ρ*=(1-

*n*)/(1+

*n*), and the single-path phase shift through the liquid column as

*ϕ*=2π

*n d/λ*

_{o}. Upon matching the

*E*- and

*H*-fields at the liquid surface, we find

*J*

_{s}=2

*H*

_{t}=2

*nE*

_{t}

*/Z*

_{o}, the force per unit area at the mirror surface is given by

*ε*

_{o}

*d*of the liquid column happens to be an integer-multiple of λ

_{o}/2

*n*, then

*F*

_{z}in Eq. (48) turns out to be equal to the radiation pressure in free-space. However, if

*d*differs from the above value by λ

_{o}/4

*n*, the coefficient of

*ε*

_{o}

*n*

^{2}. We conclude that, for an essentially monochromatic beam of light, the ratio of the radiation pressure on the immersed mirror to that in free-space could be anywhere between unity and

*n*

^{2}. In practice, the light source is never perfectly monochromatic and, in fact, its coherence length in many experiments is substantially smaller than the depth

*d*of the liquid. Under such circumstances, the radiation pressure is estimated by averaging

*F*

_{z}of Eq. (48) over the source’s bandwidth. Assuming equal likelihood for all values of

*ϕ*, the averaged coefficient of

*ε*

_{o}

*n*, which is precisely what has been measured in the experiments.

*E*- and

*H*-fields inside the liquid may be written similarly to those in the dielectric slab of Section 9 (see Eqs. (24)), and the magnetic Lorentz force on the bound currents can be straightforwardly calculated as follows:

*z*=0 to

*d*, it yields the force per unit cross-sectional area of the liquid column as follows:

*ε*

_{o}

*d*of the liquid column, the fringes beneath the liquid surface are washed out, but those closer to the mirror survive. These surviving fringes impart a net negative pressure to the liquid column that accounts for the aforementioned factor of

*n*increase in the radiation pressure experienced by the mirror. It is remarkable that the hydrostatic response of the liquid volume to this (nonuniform) pattern of radiation pressure does not appear to have affected the final result of the experiments.

## 14. Optical tweezers

*s*-polarized beam, by virtue of its compressive pressure, can pull a glass bead (laterally) to the point of highest intensity, a

*p*-polarized beam, because of the expansive nature of its force, tends to push the glass bead away. In order to clarify this misunderstanding (in the case of

*p*-light) let us examine the passage of a localized beam through a glass slab in the vicinity of one of the slab’s sidewalls; see Fig. 13. Unlike the analyses elsewhere in the paper, the present discussion cannot be made rigorous without extensive numerical calculations, so we limit our analysis to a few qualitative remarks.

*x*-component of the

*E*-field in the vicinity of the sidewall in Fig. 13 have magnitude

*E*

_{o}just outside the slab. The continuity of

**D**_{⊥}at this sidewall then requires that

*E*

_{x}just inside the slab have a magnitude equal to

*E*

_{o}/

*n*

^{2}. The p-polarized beam inside the slab thus exerts an expansive force on the bulk in the positive

*x*-direction, with a force per unit area

*F*

^{(edge)}=¼

*ε*

_{o}(

*n*

^{2}-1)(

*E*

_{o}/

*n*

^{2})

^{2}. The (bound) charge density at the sidewall is obtained from the discontinuity in

**E**_{⊥}as

*σ*=-

*ε*

_{o}(1-1/

*n*

^{2})

*E*

_{o}. When the (average)

*E*-field at the sidewall,

*E*

^{(eff)}=½(1+1/

*n*

^{2})

*E*

_{o}, interacts with the above charge density, it gives rise to a force per unit area

*F*

^{(wall)}=½

*Real*(

*σE*

^{(eff)}*)=-¼

*ε*

_{o}(1-1/

*n*

^{4})

*F*

_{x}=

*F*

^{(wall)}+

*F*

^{(edge)}=-¼

*ε*

_{o}(1-1/

*n*

_{2})

## 15. Concluding remarks

*of electromagnetic plane-waves in isotropic and homogeneous dielectric media. The old debate as to whether*

**p***=½*

**p***×*

**D***or*

**B***=½*

**p***×*

**E**

**H**/c^{2}has been settled by showing that the correct expression is obtained by averaging the two forms. We examined several cases involving a single plane-wave in a dielectric medium, and verified that the new expression for

*is valid in each case. The utility of the formula is limited, however, as in most cases of practical interest multiple plane-waves interfere with each other, and the force experienced by the media must then be calculated from a direct consideration of the (conduction and/or bound) charges and currents, and their interactions with the electric and magnetic components of the radiation field. Examples of these calculations were given, and the results in all cases were found to be consistent with the known properties of electromagnetic radiation. The case of a metallic mirror immersed in a dielectric liquid, which yields to exact analysis, was shown to be in agreement with reported experiments as well.*

**p***p*-polarized, finite-diameter beams. We arrived at this result from four different perspectives in Sections 6, 7, 10, and 11, and showed that the magnitude, direction, and polarization-dependence of the lateral pressure are exactly the same in all the cases considered. In light of these analyses, it is plausible to argue that the strength of the lateral pressure is independent of the cross-sectional profile of the beam, in general, and of the detailed structure of the beam’s edge, in particular.

**8**, 14–21 (1973). [CrossRef]

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]

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

*E*- and

*B*-fields on the induced electric dipole density

*, which, in turn, is assumed to be proportional to the applied*

**P***E*-field, that is,

*=*

**P***ε*

_{o}(

*ε*-1)

**. It is apparent that wherever both**

*E**and the*

**P***E*-field gradient happen to have non-zero projections along any coordinate axis, the medium experiences an electric Lorentz force. This, however, is

*not*the same as the force of the

*total E*-field on the induced charges, which is what we have been using throughout the paper. Whereas

*in Eq. (51) is produced by external sources alone, our total E-field has included the contributions of the induced charges and currents within the dielectric medium as well. In the latter case, the Maxwell equation*

**E****∇·**=0 in conjunction with the constitutive relation

*D**=*

**D***ε*

_{o}

*ε*ensures the absence of (unbalanced) bound charges from the interior of the medium; the only charges then are those induced at the interface(s) between adjacent media of differing dielectric constants. The difference between some of the results in this paper and those in the literature (e.g., the existence of an expansive lateral pressure at the edge of a beam) may thus be traced back to the

**E***internal E*-field contribution to the Lorentz force, which is absent from Eq. (51), but has been properly accounted for in our formulation.

*E*-field component of the Lorentz force in Eq. (51) as (

**)**

*P*·∇*may be understood if one considers the*

**E***E*-field’s force on individual atoms, molecules, and even larger particles which, nonetheless, are small compared to the light’s wavelength. In dealing with a dense continuum such as a solid or a liquid, however, one must avoid the use of ad hoc formulas and, instead, embrace the universal form of the Lorentz force,

*ρ*(

*)*

**r**, t*(*

**E***), where*

**r**, t*ρ*is the local charge density. This approach assumes that the constituent atoms of the medium under consideration are vanishingly small, and that these atoms are uniformly and densely distributed throughout the volume. If one opts for the “clumpy” model of the material, in which individual atoms (or groups of atoms) occupy certain isolated locations in space, then the local

*E*-field acting on each such clump will have to be calculated for the “clumpy situation” as well. This, however, will introduce variations in the

*E*-field over and above the smooth distribution calculated from the macroscopic Maxwell equations. We thus believe that, if one uses the macroscopic equations to derive the optical

*E*-field inside the medium, one must also ignore the “lumpiness” of the actual material and accept the smooth distribution of (bound) charges and dipole moments. This is the approach taken throughout this paper, in stark contrast to a majority of the published literature in the field, and is the source of the disagreement between some of our results and those published by others. While the published results obtained on the basis of Eq. (51) should remain applicable to low-density gases and dilute collections of small particles, it is our belief that, for dense, homogeneous solids and liquids, the approach taken in this paper is more suitable.

## Acknowledgments

## References

1. | H. Minkowski, Nachr. Ges. Wiss. Gottingen53 (1908). |

2. | H. Minkowski, Math. Annalon68, 472 (1910). [CrossRef] |

3. | M. Abraham and R. C. Circ. Mat. Palermo28, 1 (1909). [CrossRef] |

4. | M. Abraham and R. C. Circ. Mat. Palermo30, 33 (1910). [CrossRef] |

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

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

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

8. | L. Landau and E. Lifshitz, |

9. | J. D. Jackson, |

10. | M. Planck, |

11. | R. V. Jones and J. C. S. Richards, |

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. |

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

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

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

16. | G. Barlow, Proc. Roy. Soc. Lond. |

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

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 10, 2004

Revised Manuscript: August 10, 2004

Published: November 1, 2004

**Citation**

Masud Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express **12**, 5375-5401 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-22-5375

Sort: Journal | Reset

### References

- H. Minkowski, Nachr. Ges. Wiss. Gottingen 53 (1908).
- H. Minkowski, Math. Annalon 68, 472 (1910). [CrossRef]
- M. Abraham, R. C. Circ. Mat. Palermo 28, 1 (1909). [CrossRef]
- M. Abraham, R. C. Circ. Mat. Palermo 30, 33 (1910). [CrossRef]
- J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8, 14-21 (1973). [CrossRef]
- R. Loudon, “Radiation Pressure and Momentum in Dielectrics,” De Martini lecture, to appear in Fortschritte der Physik (2004).
- R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821-838 (2002). [CrossRef]
- 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).
- R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. A 221, 480 (1954). [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 the presence of spherical aberrations,” Appl. Opt. 41, 2494 (2002). [CrossRef] [PubMed]
- Y. N. Obukhov and F. W.Hehl, “Electromagnetic energy-momentum and forces in matter,” Phys. Lett. A, 311, 277-284 (2003). [CrossRef]
- G. Barlow, Proc. Roy. Soc. Lond. A 87, 1-16 (1912). [CrossRef]
- R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A 68, 013806 (2003). [CrossRef]

## 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.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 7. |
Fig. 8. |
Fig. 9. |

Fig. 10. |
Fig. 11. |
Fig. 12. |

Fig. 13. |
||

« Previous Article | Next Article »

OSA is a member of CrossRef.