## Electromagnetic force and torque in ponderable media

Optics Express, Vol. 16, Issue 19, pp. 14821-14835 (2008)

http://dx.doi.org/10.1364/OE.16.014821

Acrobat PDF (226 KB)

### Abstract

Maxwell’s macroscopic equations combined with a generalized form of the Lorentz law of force are a complete and consistent set of equations. Not only are these five equations fully compatible with special relativity, they also conform with conservation laws of energy, momentum, and angular momentum. We demonstrate consistency with the conservation laws by showing that, when a beam of light enters a magnetic dielectric, a fraction of the incident linear (or angular) momentum pours into the medium at a rate determined by the Abraham momentum density, **E** × **H**/*c*^{2}, and the group velocity *V**
_{g}
* of the electromagnetic field. The balance of the incident, reflected, and transmitted momenta is subsequently transferred to the medium as force (or torque) at the leading edge of the beam, which propagates through the medium with velocity

*V*

*. Our analysis does not require “hidden” momenta to comply with the conservation laws, nor does it dissolve into ambiguities with regard to the nature of electromagnetic momentum in ponderable media. The linear and angular momenta of the electromagnetic field are clearly associated with the Abraham momentum, and the phase and group refractive indices (*

_{g}*n*

*and*

_{p}*n*

*) play distinct yet definitive roles in the expressions of force, torque, and momentum densities.*

_{g}© 2008 Optical Society of America

## 1. Introduction

**and**

*P***are defined as averages over small volumes that must nevertheless contain a large number of atomic dipoles. Consequently, the macroscopic**

*M***,**

*E***,**

*D***and**

*H***fields are regarded as spatial averages of the “actual” fields; without averaging, these fields would be wildly fluctuating on the scale of atomic dimensions. (The actual fields, of course, are presumed to be well-defined at all points in space and time.) There is also a tendency to elevate**

*B***and**

*E***to the status of “fundamental,” while treating**

*B***and**

*D***as secondary or “derived” fields.**

*H***and**

*P***with the properties of real materials. Stated differently, if material media consisted of dense collections of point dipoles, then any volume of the material, no matter how small, would contain an infinite number of such dipoles, eliminating thereby the need for the introduction of macroscopic averages into Maxwell’s equations. Also, since in their simplest form, the macroscopic equations contain all four of the**

*M***,**

*E***,**

*D***,**

*H***fields, one should perhaps resist the temptation to designate some of these as more fundamental than others. Tellegen [3**

*B*3. B. D. H. Tellegen, “Magnetic-Dipole Models,” Am. J. Phys. **30**, 650 (1962). [CrossRef]

**,**

*E***,**

*D***and**

*H***should be regarded as equally important physical entities, a point of view with which we agree.**

*B**with*

**P****–**

*D**ε*

_{o}

**and**

*E***with**

*M***–**

*B**µ*

_{o}

**H**, thus allowing

**and**

*P***to be designated as secondary fields. Electric and magnetic energy densities and the Poynting vector may now be written as**

*M**ε*

*=*

_{e}**·**

_{E}**,**

*D**ε*

*=*

_{m}**·**

*H**, and*

**B****=**

*S**×*

**E****, respectively, without the need to explain away the appearance of “derived” fields**

*H***and**

*D***in the expressions pertaining to a most fundamental physical entity. [We note in passing that, in deriving Poynting’s theorem, the assumed rate of change of energy density**

*H**only*postulate of the classical theory concerning electromagnetic energy.]

*ε*(

*ω*) and permeability

*µ*(

*ω*), although a case involving a birefringent medium is discussed in section 5 as well. We derive expressions for the total force and torque exerted on magnetic dielectrics, thus clarifying the reasons behind the traditional division of linear and angular momenta into electromagnetic and mechanical parts. Our methods should be applicable not only to transparent media, whose

*ε*(

*ω*) and

*µ*(

*ω*) are real-valued, but also to absorbing media, where at least one of these parameters is complex.

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

14. R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express **14**, 11855–11869 (2006). [CrossRef] [PubMed]

## 2. Force and torque exerted on electric and magnetic dipoles by the electromagnetic field

10. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express **15**, 13502–13518 (2007). [CrossRef] [PubMed]

*µ*and

*ε*parameters:

8. T. B. Hansen and A. D. Yaghjian, *Plane-Wave Theory of Time-Domain Fields*, IEEE Press, New York (1999). [CrossRef]

*ρ*

_{free}= 0,

**J**_{free}= 0). Our homogeneous, linear, isotropic media will be assumed to be fully specified by their permittivity

*ε*=

*ε*′+i

*ε*″ and permeability

*µ*=

*µ*′+i

*µ*″. Any loss of energy in such media will be associated with

*ε*″ and

*µ*″, which, by convention, are ≥ 0. The real parts of

*ε*and

*µ*, however, may be positive or negative; in particular, in negative-index media,

*ε*′ < 0 and

*µ*′ < 0. Using simple examples that are amenable to exact analysis, we have shown in a previous publication [10

10. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express **15**, 13502–13518 (2007). [CrossRef] [PubMed]

10. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express **15**, 13502–13518 (2007). [CrossRef] [PubMed]

*=-∇·*

_{e}*and ρ*

**P***=-∇·*

_{m}**directly experience the force of the**

*M***and**

*E***fields. The alternative formula is**

*H*13. S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B: At. Mol. Opt. Phys. **39**, S671–S684 (2006). [CrossRef]

14. R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express **14**, 11855–11869 (2006). [CrossRef] [PubMed]

13. S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B: At. Mol. Opt. Phys. **39**, S671–S684 (2006). [CrossRef]

14. R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express **14**, 11855–11869 (2006). [CrossRef] [PubMed]

15. A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, “Single-beam trapping of micro-beads in polarized light: Numerical simulations,” Opt. Express **14**, 3660–3676 (2006). [CrossRef] [PubMed]

**×**

*P***(and, by analogy,**

*E***×**

*M***) will be zero in isotropic media and, therefore, the additional terms in Eq. (5) need not be considered. This statement, while valid in some cases, is generally incorrect. In other words, for the total torques in the two formulations to be identical, Eq. (5) must definitely contain the**

*H***×**

*P***and**

*E***×**

*M***contributions.**

*H**(*

**P***r*,

*t*) is not always parallel to

**(**

*E**r*,

*t*), nor is

**(**

*M**r*,

*t*) parallel to

**(**

*H**r*,

*t*), thus making it obvious that the

**×**

*P***and**

*E***×**

*M***terms in Eq. (5) are indeed necessary. Even in the case of isotropic media, these terms are needed in many circumstances, because**

*H**and*

**P***(or*

**E****and**

*M***) could assume differing orientations. For instance, in a time-harmonic (i.e., single-frequency) circularly polarized electromagnetic field,**

*H***lags behind the rotating**

*P***-field when**

*E**ε*is complex; similarly,

**lags behind the rotating Hfield when**

*M**µ*is complex. In general, therefore, when computing the torque in accordance with Eq. (5), one must beware of the possibility that absorption, dispersion, or birefringence could all create conditions under which

*and*

**P****(or**

*E***and**

*M***) will have differing orientations. One such situation will be encountered in section 5 below.**

*H*## 3. Relation between the Lorentz force and the forces obtained from energy gradients

*p*immersed in an electromagnetic field is

*ε*

*=*

_{p}*p*·

**, while that of a single magnetic dipole**

*E**m*is

*ε*

*=*

_{m}*m*·

**. The force experienced by these dipoles is thus expected to be [1,2]:**

*H**p*and

*m*in these expressions represent individual dipoles and, therefore, the gradient operator acts only on the fields

*and*

**E***in which the dipoles are immersed. The force in Eq. (2), however, is not exactly the same as that predicted by these energy gradients. One can rewrite Eq. (2) in terms of energy gradients as follows:*

**H**## 4. Pulse of light entering a transparent, homogeneous, isotropic medium

17. M. Mansuripur, “Angular momentum of circularly polarized light in dielectric media,” Opt. Express **13**, 5315–5324 (2005). [CrossRef] [PubMed]

*z*-axis, exerting a force and a torque on the medium, which account, respectively, for the

*mechanical*linear and angular momenta of the light inside the medium. The medium is transparent, isotropic, and dispersionless, so that, at all points along the beam’s path,

**is parallel to**

*P**and*

**E****is parallel to**

*M***. Also, at normal incidence, there will be no forces or torques exerted at the entrance facet (when the beam diameter is sufficiently large). The entire force and torque will thus arise from the action of the leading edge of the light pulse on electric and magnetic dipoles in accordance with Eqs. (2) and (5).**

*H***(**

*E***r**,

*t*) and

**(**

*H**r*,

*t*). This is possible because

*ε*and

*µ*are real-valued and frequency-independent. We set

*(*

**P***r*,

*t*)=

*ε*

_{o}(

*ε*-1)

*(*

**E***r*,

*t*) and

**(**

*M**r*,

*t*)=

*µ*

_{o}(

*µ*-1)

*(*

**H***r*,

*t*), use Maxwell’s curl equations to replace some of the space-derivatives with time-derivatives, combine various terms, and find the following equivalent of Eq. (2):

**F**_{1}(

*,*

**r***t*) over the volume of the slab, we find that the gradient terms along

*̂ and*

**x***ŷ vanish, leaving only the gradient term along*

**y***̂. The third term in Eq. (9) contains a time derivative, but since the beam travels with a speed of*

**z***c*/

*n*along the

*z*-axis, ∂/∂

*t*can be replaced with -(

*c*/

*n*)∂/∂

*z*, where

*z*= 0

^{+}stabilize and assume sinusoidal behavior. The

*z*-component of the force can then be time-averaged over one period of oscillation. The beam diameter is large enough that the Fresnel reflection coefficient at normal incidence,

*E*- and

*H*-field amplitudes. At

*z*= 0

^{+}, immediately beneath the surface, the amplitudes of

*E*

*,*

_{x}*E*

*,*

_{y}*B*

*are (1+*

_{z}*r*) times the corresponding incident amplitudes, while those of

*H*

_{x},

*H*

_{y},

*D*

*are (1-*

_{z}*r*) times the corresponding incident amplitudes. This is a consequence of the symmetry of reflection as well as the continuity of

*E*

^{‖},

*H*

^{‖},

*B*

^{⊥}and

*D*

^{⊥}field components. Recognizing that the beam’s cross-sectional area is large, we ignore

*E*

_{z}and

*H*

_{z}at

*z*= 0

^{+}, then treat the remaining field amplitudes as uniform in the

*xy*-plane at

*z*= 0

^{+}. Normalizing the integrated force by the cross-sectional area of the beam, the time-averaged force per unit area is found to be

**p**_{EM}=

*×*

**E****/**

*H**c*

^{2}. Since the leading edge moves a distance of

*c*/

*n*in one second, the electromagnetic momentum (per unit area per second) delivered to the medium is (

*cn*)

^{-1}<

*×*

**E***> =¼*

**H***ε*

_{o}

*n*

^{-1}(1-

*r*

^{2})(

*E*

^{2}

*+*

_{xo}*E*

_{yo}^{2})

**z**̂. Adding this to <

*F*

*> of Eq. (12) yields the total rate of flow of linear momentum into the slab as ½*

_{z}*ε*

_{o}(1+

*r*

^{2})(

*E*

^{2}

*+*

_{xo}*E*

^{2}

*). The final result is exactly equal to the rate of flow of incident plus reflected momenta in the free space, thus establishing the conservation of linear momentum.*

_{yo}*V*

*=*

_{g}*c*/

*n*

*. The electromagnetic momentum density will still be given by the Abraham formula,*

_{g}*p*

_{EM}=

*×*

**E***/*

**H***c*

^{2}, and the rate of flow of energy into the medium will still be determined by the Poynting vector

*=*

**S****×**

*E**. If we then proceed to assume that the energy contained in a given volume of the material is*

**H***Nħω*, with

*ħ*being the reduced Planck constant,

*ω*the angular frequency of the light, and

*N*the number of photons, the Abraham momentum per photon turns out to be

*ħω*/(

*n*

_{g}*c*). This is the general formula for the photon’s

*electromagnetic*momentum in a transparent medium having group refractive index

*n*

*. (This result applies to all transparent media, irrespective of the phase refractive index*

_{g}*n*

*being positive or negative.)*

_{p}

**F**_{1}(

*,*

**r***t*) of Eq.(9) on the semi-infinite slab of Fig. 1. (In transparent dispersionless media

*‖*

**P***and*

**E***‖*

**M***; therefore,*

**H***×*

**P***and*

**E***×*

**M***of Eq. (5) do not contribute to the torque.) The total torque is obtained by integrating*

**H***×*

**r**

**F**_{1}(

*,*

**r***t*) over the volume of the material, as follows:

*x*or ∂/∂

*y*terms takes the integrand beyond the beam’s finite diameter, where

*E*- and

*H*-field intensities are zero. Other integrals end up being zero because of the symmetry of the incident beam around the

*z*-axis. (We are assuming circular symmetry around

*z*, because the emphasis of the present analysis is on “spin” angular momentum, which arises from the polarization state of the beam, as opposed to “orbital” angular momentum, which is rooted in the circulation of the phase profile around the

*z*-axis.) The only part of Eq. (13) that survives integration is the very last line, whose ∂/∂

*z*terms, when integrated over

*z*, yield an expression in terms of the field components at

*z*= 0

^{+}. These components may then be time-averaged to yield

**S**_{o}> =½

**E**_{o}×

**H**_{o}is the time-averaged Poynting vector associated with the incident beam. The integral in the final line of Eq.(14) represents the total incident angular momentum per unit time. The integral is multiplied by (1-

*r*

^{2}), which has the effect of subtracting the angular momentum carried away by the reflected beam. (Note that, unlike linear momentum whose contribution due to reflection adds to the exerted force, angular momentum reverses sign upon reflection and, therefore, its contribution to exerted torque must be subtracted.) The torque <

*T*

*> exerted on the medium by the leading edge of the pulse is thus equal to the net angular momentum influx multiplied by (1-*

_{z}*n*

^{-2}). This implies that the electromagnetic (i.e., Abraham) angular momentum flux into the medium is 1/

*n*

^{2}times the flux of “incident minus reflected” angular momenta. In other words, to conserve the total angular momentum of ħ per photon, each photon’s

*electromagnetic*angular momentum inside the slab must reduce to

*ħ*/

*n*

^{2}.

*n*

_{2}becomes

*n*

_{p}*n*

*, the product of phase and group refractive indices. Thus, upon entering a homogeneous, linear, isotropic, and transparent medium, the electromagnetic angular momentum per photon shrinks by a factor of*

_{g}*n*

_{p}*n*

*, while, according to Eq. (14), the balance of the incident, reflected and transmitted angular momenta is transferred to the medium as a torque (via the leading edge of the beam). This result applies to negativeindex media as well, where the sign of the electromagnetic angular momentum is reversed relative to that of the incident beam (because*

_{g}*n*

*< 0).*

_{p}## 5. Torque on a birefringent slab

*ε*

*,*

_{x}*µ*

*) along the*

_{x}*x*-axis and (

*ε*

*,*

_{y}*µ*

*) along the y-axis is depicted in Fig. 2. The semi-infinite slab is illuminated at normal incidence with an elliptically-polarized plane-wave. The incident field amplitudes are (*

_{y}*E*

_{x}_{o},

*E*

_{y}_{o}) and (

*H*

_{x}_{o},

*H*

_{y}_{o})=

*Z*

_{o}

^{-1}(

*E*

_{x}_{o},

*E*

_{y}_{o}). The incident beam may be written as the sum of left- and right-circularly-polarized plane-waves (LCP and RCP) as follows:

*L*

_{z}^{1}=¼(

*ε*

_{o}/

*k*

_{o})|

*E*

*x*

_{o}-i

*E*

_{y}_{o}|

^{2}and

*L*

_{z}^{2}=-¼(

*ε*

_{o}/

*k*

_{o})|

*E*

_{x}*o*+i

*E*

*y*

_{o}|

^{2}, respectively. Therefore, the total angular momentum influx is

*L*

*=*

_{z}*L*

_{z}_{1}+

*L*

_{z}_{2}=(

*ε*

_{o}/

*k*

_{o})Im(

*E**

_{x}_{o}

*E*

_{y}_{o}). The reflected beam has

*E*-field amplitudes

*r*

_{1}

*E*

_{x}_{o}and

*r*

_{2}

*E*

_{y}_{o}, where

*L*

*=(*

_{z}*ε*

_{o}/

*k*

_{o})Im(

*r**

_{1}

*E**

_{x}_{o}

*r*

_{2}

*E*

_{y}_{o}), the angular momentum per unit area per unit time delivered to the semi-infinite slab will be

**×**

*P***+**

*E***×**

*M**part of Eq. (5); the other contributions, embodied in the*

**H***×*

**r**

*F*_{1}(

*,*

**r***t*) term, vanish when the latter is expressed in the form of Eq. (13) and its various integrals evaluated, then time-averaged. (Unlike the problem studied in section 4, here we are dealing with the steady-state situation where the leading edge of the beam has already passed through the medium and an exponentially-decaying field along the

*z*-axis has been established.) The time-averaged torque per unit surface area experienced by the semi-infinite slab of Fig. 2 is thus given by

*µ*

_{x},

*µ*

_{y},

*ε*

_{x},

*ε*

*is complex, so that absorption of light within the semi-infinite slab would cause the exponential function in the integrand to approach zero when*

_{y}*z*→∞. However, since the term (

## 6. Concluding remarks

**,**

*E***,**

*D***,**

*H***should be treated as fundamental, while polarization and magnetization densities**

*B**and*

**P***can be considered as secondary fields derived from the constitutive relations, Eqs.(3b). Two formulations of the generalized Lorentz law are given in Eq.(2) and Eq. (4). These formulas are equally acceptable in the sense that they are both consistent with the conservation laws; moreover, they predict precisely the same*

**M***total*force on a given body of material, although the predicted force

*distributions*at the surfaces and throughout the volume of the material could be drastically different in the two formulations. If Eq. (2) is used as the generalized law of force, then torque density will be given by Eq.(5). On the other hand, if force is given by Eq. (4), then the torque formula will be Eq. (6).

*ħω*, then the electromagnetic (i.e., Abraham) momentum corresponding to each such bundle of energy (i.e., photon) will be

*ħω*/(

*n*

_{g}*c*), where

*n*

*is the group refractive index of the medium. For circularly-polarized light, the spin angular momentum of each photon is*

_{g}*ħ*/(

*n*

_{p}*n*

*), where*

_{g}*n*

*is the phase refractive index, in agreement with the predictions of Ref. [12]. The balance of linear and angular momenta among the incident, reflected, and transmitted beams is always transferred to the medium in the form of force and torque, which are sometimes identified as “mechanical” momenta of the light beam. When a beam of light enters a material medium from the free space, fractions of its linear and angular momenta remain electromagnetic, while the rest are exerted on the medium in the form of mechanical force and torque. Similarly, when a beam of light emerges from a host medium into the free space, its electromagnetic momenta are augmented by additional momenta that are taken away from the medium, resulting in mechanical backlash (i.e., oppositely directed force and torque) on the medium.*

_{p}18. M. Kristensen and J. P. Woerdman, “Is photon angular momentum conserved in a dielectric medium?” Phys. Rev. Lett. **72**, 2171–2174 (1994). [CrossRef] [PubMed]

*ħ*/(

*n*

_{p}*n*

*), but it is also “dressed” with a certain amount of mechanical angular momentum. What is transferred to the antenna is, in general, a combination of the photon’s electromagnetic and mechanical momenta; moreover, the backlash (i.e., momentum transfer to the liquid upon absorption of the photon by the antenna) must also be taken into account.*

_{g}## Appendix A: Electromagnetic momentum in dispersive media

*ω*

_{1}and

*ω*

_{2}. The electric and magnetic field amplitudes of this superposition are expressed in terms of a plane-wave spectrum of spatial frequencies (

*k*

*,*

_{x}*k*

*) as follows:*

_{y}*ω*

_{1}and ±

*ω*

_{2}, where

*ω*

_{1}and

*ω*

_{2}are two distinct but closely-spaced frequencies. In general,

*k*

*and*

_{x}*k*

*are confined to a small region in the vicinity of the origin of the*

_{y}*k*

_{x}*k*

*-plane, we can safely set*

_{y}*xy*-plane is required to be symmetric, say, with respect to the origin, that is, if it is demanded that the field amplitudes remain intact upon switching (

*x*,

*y*) to (-

*x*, -

*y*), we must have

*t*= 0 will be located at integer-multiples of

*T*= 2

*π*/(

*ω*

_{2}-

*ω*

_{1}). The beat’s group velocity is thus

**(**

*S**,*

**r***t*) over the beam’s cross-sectional area

*A*in the

*xy*-plane, normalizing by

*A*, and using the identity

*δ*(

*k*) is Dirac’s delta function, the beam’s Abraham momentum density turns out to be

*ω*and

*ω*′ can each assume four different values, namely, ±

*ω*

_{1}and ±

*ω*

_{2}, for a total of 16 terms in the above summation. When time-averaging is performed over Eq. (A8), the only terms for which (1/

*T*)∫

^{T}_{0}exp[-i(

*ω*+

*ω*′)

*t*]d

*t*≠0 will be those with

*ω*+

*ω*′ = 0, namely, (

*ω*,

*ω*′)=(-

*ω*

_{1},

*ω*

_{1}), (-

*ω*

_{2},

*ω*

_{2}), (

*ω*

_{1}, -

*ω*

_{1}), and (

*ω*

_{2}, -

*ω*

_{2}). Therefore,

## Appendix B: Electromagnetic angular momentum in dispersive media

*z*-component of angular momentum,

*L*

*(*

_{z}*z*= 0,

*t*), is given by

*δ*′(

*k*) is the derivative of

*δ*(

*k*) with respect to

*k*, we find, upon time-averaging,

*ε*

_{x},

*ε*

_{y},

*E*

*using Maxwell’s 3*

_{z}^{rd}equation, namely,

^{st}equation,

*k*

_{x}*ε*

*+*

_{x}*k*

_{y}*ε*

*+*

_{y}*k*

_{z}*ε*

*= 0, to substitute for*

_{z}*ε*

*in terms of*

_{z}*ε*

*and*

_{x}*E*

*. [Note that*

_{y}*k*

*and*

_{x}*k*

*; therefore, ∂*

_{y}*k*

*/∂*

_{z}*k*

*and ∂*

_{x}*k*

*/∂*

_{z}*k*

*must be included when evaluating Eq.(B3).] Upon algebraic manipulations we find*

_{y}*ε*(

*ω*) and

*µ*(

*ω*). Presently we are interested only in spin angular momentum, so we confine our attention to the case of circularly-polarized light where

*ε*

*(*

_{y}*k*

*,*

_{x}*k*

*,*

_{y}*ω*

_{1},

_{2})=i

*ε*

*(*

_{x}*k*

*,*

_{x}*k*

*,*

_{y}*ω*

_{1},

_{2}), with

*ε*

*being a real-valued function of (*

_{x}*k*

*,*

_{x}*k*

*) for both*

_{y}*ω*

_{1}and

*ω*

_{2}. Under these circumstances, those terms of Eq. (B5) that contain derivatives with respect to

*k*

*or*

_{x}*k*

*either vanish (because real-valued functions have no imaginary parts) or cancel each other out. Moreover,*

_{y}*E*

*and*

_{x}*E*

*are smooth functions of (*

_{y}*x*,

*y*), and that the beam’s cross-sectional area

*A*is large compared to a wavelength. Equation (B5) thus simplifies as follows:

*L*

*> independently of each other. According to Parseval’s theorem, the integrated |*

_{z}*E*

*|*

_{x}^{2}in the

*k*

_{x}*k*

*-plane is equal to the integrated |*

_{y}*E*

*|*

_{x}^{2}over the beam’s cross-sectional area in the

*xy*-plane. The light beam’s energy density in a dispersive medium is

*V*

*=*

_{g}*c*/

*n*

*is the beam’s group velocity. The spin angular momentum in a given volume is thus equal to the energy content of the volume divided by*

_{g}*n*

_{g}*n*

_{p}*ω*, where

*n*

*is negative and, therefore, the spin angular momentum must change sign upon entering from the free space.)*

_{p}*E*

*x*

_{o}(

**̂+i**

*x**ŷ*) arrives at normal incidence at the surface of a transparent medium specified by

*ε*(

*ω*) and

*µ*(

*ω*). Using Fresnel’s reflection coefficient given by Eq. (11), we find the

*E*-field amplitude immediately inside the medium to be

*r*^{2}), of course, accounts for the loss of angular momentum upon reflection at the surface. Whereas in the incidence space the angular momentum flows at the vacuum speed of light

*c*, inside the medium the propagation speed is

*c*/

*n*

*, causing the spin angular momentum contained in a given length of the beam to drop by a factor of*

_{g}## Appendix C: Force exerted on a dispersive slab

*ω*′=-

*ω*from those in which

*ω*′≠-

*ω*, we obtain

*ω*′=-

*ω*) vanishes because, according to Maxwell’s 1

^{st}and 4

^{th}equations both

**k**·

**ε**and

**k**·

**ℋ**are zero; moreover, the remaining terms are all real-valued. As for the second sum, we integrate this force along the

*z*-axis from

*z*=0 to

*V*

_{g}*t*, i.e., over the length of the beat waveform that enters the medium during the time interval (0,

*t*). Here

*V*

*=(*

_{g}*ω*

_{1}-

*ω*

_{2})/(

*k*

_{z}_{1}-

*k*

_{z}_{2}) is the group velocity, which, in the limit of small

*k*

*and*

_{x}*k*

*, approaches the expression in Eq. (A5). Next we integrate over a single beat period, from*

_{y}*t*=0 to

*T*=2

*π*/(

*ω*

_{2}-

*ω*

_{1}), noting that (1/

*T*)∫

^{T}_{0}exp[-i(

*ω*+

*ω*′)

*t*]d

*t*= 0 for all allowed combinations of

*ω*and

*ω*′ (since

*ω*′=-

*ω*is already excluded from the second sum). The second sum in Eq.(C2) contains 12 different combinations of ±

*ω*

_{1}and ±

*ω*

_{2}, but only four (

*ω*,

*ω*′) pairs which have opposite signs, namely, (-

*ω*

_{1},

*ω*

_{2}), (-

*ω*

_{2},

*ω*

_{1}), (

*ω*

_{1}, -

*ω*

_{2}), (

*ω*

_{2}, -

*ω*

_{1}), contribute to <

*F*

_{z}> . For these pairs (

*k*

*+*

_{z}*k*′

*)*

_{z}*V*

*≈ (*

_{g}*ω*+

*ω*′) and, therefore,

*ω*

_{1},

*ω*

_{1}) or (-

*ω*

_{1}, -

*ω*

_{2}) which have identical signs, we note that (

*k*

*+*

_{z}*k*′

*)*

_{z}*V*

*-(*

_{g}*ω*+

*ω*′)≈2

*ω*[(

*n*

*/*

_{p}*n*

*)-1]. Now, if the phase index*

_{g}*n*

*happens to differ substantially from the group index*

_{p}*n*

*, the corresponding exponential function will vary rapidly with time, rendering the time-averaged contributions of the remaining eight terms negligible. If, on the other hand,*

_{g}*n*

*and*

_{p}*n*

*happen to be so close as to cause the exponential function to vary only slowly during the beat period, the eight terms split into two groups of four positive and four negative terms. [According to Eqs.(A4), the terms arising from either (*

_{g}*ω*

_{1},

*ω*

_{2}) or (

*ω*

_{2},

*ω*

_{1}) are equal in magnitude but opposite in sign compared to those arising from (

*ω*

_{1},

*ω*

_{1}) and (

*ω*

_{2},

*ω*

_{2})]. The two groups of terms thus cancel each other out. Therefore, under all circumstances, only four terms contribute to the time-averaged force <

**F***> , yielding*

_{z}*ω*

_{1}and

*ω*

_{2}are equal but opposite in sign, Eq. (C4) reduces to

*A*→∞, the integrals of |

*ε*

*|*

_{z}^{2}and |

*ℋ*

*|*

_{z}^{2}approach zero, while

*ω*,

*ω*′)=(

*ω*

_{2}, -

*ω*

_{1}). Replacing for

*ε*

*,*

_{x}*ε*

*,*

_{y}*ℋ*

_{x},

*ℋ*

*in terms of the incident field amplitudes*

_{y}*ε*

_{x}_{o},

*ε*

_{y}_{o},

*ℋ*

_{x}_{o},

*ℋ*

_{y}_{o}and the Fresnel reflection coefficients

*r*

_{1},

*r*

_{2}(given by Eq.(11)), we find the time-averaged force per unit surface area of the slab to be

## Appendix D: Torque exerted on a dispersive slab

*ω*′=-

*ω*from those in which

*ω*′≠-

*ω*, we obtain

*ε*and

*ε*′ stand for

*ε*(

*k*

*,*

_{x}*k*

*,*

_{y}*ω*) and

*ε*(

*k*

*,*

_{x}*k*

*, -*

_{y}*ω*′), respectively, while (

*µ*,

*ε*) and (

*µ*′,

*ε*′) are the material parameters at

*ω*and

*ω*′. The finite diameter of the

*E*- and

*H*-field profiles in the

*k*

_{x}*k*

*-plane makes the first sum in Eq.(D2) exactly equal to zero. We integrate the second sum over*

_{y}*z*(from 0 to

*V*

_{g}*t*) and then again over

*t*(from 0 to

*T*) to obtain:

*orbital*angular momentum, the symmetry of the beam in the cross-sectional

*xy*-plane ensures that the derivatives in Eq. (D3) drop out. We are then left with

*ω*,

*ω*′) pairs, namely, (

*ω*

_{1}, -

*ω*

_{2}) and (

*ω*

_{2}, -

*ω*

_{1}). In the limit when the beam has a large cross-sectional area

*A*, terms in Eq. (D4) that contain

*k*

*and/or*

_{x}*k*

*make negligible contributions to the integral and can safely be ignored. We set*

_{y}*E*

*=i*

_{y}*E*

*. Accounting for the fact that*

_{x}*ε*

*and*

_{x}*ε*

*′ have equal magnitudes and opposite signs, in the limit when*

_{x}*ω*

_{1}→

*ω*

_{2}, Eq.(D4) may be written

*ε*/d

*ω*=[

*ε*(

*ω*

_{1})-

*ε*(

*ω*

_{2})]/(

*ω*

_{1}-

*ω*

_{2}) and d

*µ*/d

*ω*=[

*µ*(

*ω*

_{1})-

*µ*(

*ω*

_{2})]/(

*ω*

_{1}-

*ω*

_{2}). The expression for time-averaged torque per unit area can finally be written in compact form as follows:

*r*

^{2}) times the rate of flow of incident angular momentum in the free space, while the second term accounts for the electromagnetic (i.e., Abraham) angular momentum that pours into the medium with the group velocity

*V*

*. The equation therefore confirms the conservation of angular momentum.*

_{g}## Acknowledgement

## References and links

1. | J. D. Jackson, |

2. | R.P. Feynman, R.B. Leighton, and M. Sands, |

3. | B. D. H. Tellegen, “Magnetic-Dipole Models,” Am. J. Phys. |

4. | W. Shockley and R. P. James, “Try simplest cases discovery of hidden momentum forces on magnetic currents,” Phys. Rev. Lett. |

5. | W. Shockley, “Hidden linear momentum related to the |

6. | P. Penfield and H. A. Haus, |

7. | L. Vaidman, “Torque and force on a magnetic dipole,” Am. J. Phys. |

8. | T. B. Hansen and A. D. Yaghjian, |

9. | A. D. Yaghjian, “Electromagnetic forces on point dipoles,” IEEE Anten. Prop. Soc. Symp. |

10. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express |

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

12. | M. Padgett, S. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. |

13. | S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B: At. Mol. Opt. Phys. |

14. | R. Loudon and S. M. Barnett, “Theory of the radiation pressure on dielectric slabs, prisms and single surfaces,” Opt. Express |

15. | A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, “Single-beam trapping of micro-beads in polarized light: Numerical simulations,” Opt. Express |

16. | R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, “Momentum of an electro-magnetic wave in dielectric media,“ Rev. Mod. Phys. |

17. | M. Mansuripur, “Angular momentum of circularly polarized light in dielectric media,” Opt. Express |

18. | M. Kristensen and J. P. Woerdman, “Is photon angular momentum conserved in a dielectric medium?” Phys. Rev. Lett. |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: August 6, 2008

Revised Manuscript: August 27, 2008

Manuscript Accepted: August 28, 2008

Published: September 5, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Masud Mansuripur, "Electromagnetic force and torque in ponderable media," Opt. Express **16**, 14821-14835 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-19-14821

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### References

- J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.
- R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading (1964).
- B. D. H. Tellegen, "Magnetic-Dipole Models," Am. J. Phys. 30, 650 (1962). [CrossRef]
- W. Shockley and R. P. James, "Try simplest cases discovery of hidden momentum forces on magnetic currents," Phys. Rev. Lett. 18, 876-879 (1967). [CrossRef]
- W. Shockley, "Hidden linear momentum related to the ?·E term for a Dirac-electron wave packet in an electric field," Phys. Rev. Lett. 20, 343-346 (1968). [CrossRef]
- P. Penfield and H. A. Haus, Electrodynamics of Moving Media, MIT Press, Cambridge (1967).
- L. Vaidman, "Torque and force on a magnetic dipole," Am. J. Phys. 58, 978-983 (1990). [CrossRef]
- T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields, IEEE Press, New York (1999). [CrossRef]
- A. D. Yaghjian, "Electromagnetic forces on point dipoles," IEEE Anten. Prop. Soc. Symp. 4, 2868-2871 (1999).
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field in magnetic media," Opt. Express 15, 13502-13518 (2007). [CrossRef] [PubMed]
- R. Loudon, "Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics," Phys. Rev. A 68, 013806 (2003). [CrossRef]
- M. Padgett, S. Barnett, and R. Loudon, "The angular momentum of light inside a dielectric," J. Mod. Opt. 50, 1555-1562 (2003).
- S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B: At. Mol. Opt. Phys. 39, S671-S684 (2006). [CrossRef]
- R. Loudon and S. M. Barnett, "Theory of the radiation pressure on dielectric slabs, prisms and single surfaces," Opt. Express 14, 11855-11869 (2006). [CrossRef] [PubMed]
- A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, "Single-beam trapping of micro-beads in polarized light: Numerical simulations," Opt. Express 14, 3660-3676 (2006). [CrossRef] [PubMed]
- R. N. C. Pfeifer, T. A. Nieminen, N. R Heckenberg, and H. Rubinsztein-Dunlop, "Momentum of an electro-magnetic wave in dielectric media," Rev. Mod. Phys. 79, 1197-1216 (2007). [CrossRef]
- M. Mansuripur, "Angular momentum of circularly polarized light in dielectric media," Opt. Express 13, 5315-5324 (2005). [CrossRef] [PubMed]
- M. Kristensen and J. P. Woerdman, "Is photon angular momentum conserved in a dielectric medium?" Phys. Rev. Lett. 72, 2171-2174 (1994). [CrossRef] [PubMed]

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