1. Introduction

The nature of electromagnetic momentum in dielectric matter has not been clarified despite research spanning decades (see [1] for a discussion). The debate is usually presented in terms of what is the right expression for the momentum density in a dielectric, the most favoured candidates being the Abraham momentum ε ₀ E × B and the Minkowski momentum D × B. This paper deals with the related issue of the force on a dielectric when it interacts with light. Two approaches to evaluate this force are considered, one related to the Abraham momentum density, the other to the Minkowski momentum density. The first approach involves the Lorentz force density and the Maxwell stress tensor, the second involves the Minkowski force density and the Minkowski stress tensor. The relation between momentum density, stress tensor, and force density is rather loose as terms may be shifted and redefined in the momentum balance equation relating these quantities. The implication of time averaged force measurements for the actual momentum density is therefore not unambiguous. Both views on radiation forces seem plausible and it is therefore of interest to see if they give rise to different predictions. In this paper the relatively simple geometry of a slab of dielectric material immersed in a dielectric with different dielectric constant is considered. It turns out that the forces in both pictures are different, so that at least the question of the evaluation of radiation forces can be settled experimentally.

The paper is organized as follows. Section 2 discusses the relations between momentum density, flow and force density, and the role of cycle averaging for stationary monochromatic radiation. The force on a dielectric slab immersed in a different dielectric is calculated in both pictures in section 3, and the paper is concluded with a discussion in section 4. Inessential but necessary math is presented in the appendices.

2. Momentum balance equations

The starting point of calculations of the forces of light on matter is often the balance equation of electromagnetic momentum [1]. This balance equation connects the density and flow of electromagnetic momentum to the density of the electromagnetic force on the medium. There are many conceivable balance equations but we will focus on two of them. Both can be derived directly from Maxwell’s equations, meaning that the verity of both equations is not questioned. The two balance equations can both be interpreted as being ‘the’ momentum balance equation for the electromagnetic field, but no conclusive evidence has been obtained so far as to what permittivity exactly is the physical content of these equations.

The first of the equations connects the Abraham momentum density, the Maxwell stress tensor, and the Lorentz force density:

where:

The second connects the Minkowski momentum density, the Minkowski stress tensor, and the Minkowski force density:

Here E is the electric field, B the magnetic induction, P is the dielectric polarization, D = ε ₀ E + P is the dielectric displacement, and H = B/μ ₀ is the magnetic field. In these equations ∂_t denotes the partial derivative with respect to time, ∂_α with α = x, y, z denotes the partial derivative with respect to the spatial coordinates, the tensor δ_αβ, is the Kronecker tensor (δ_αβ = 1 if α = β and 0 otherwise), the tensor ε_αβγ is the Levi-Civita tensor (ε_αβγ = 1 for αβγ even permutations of xyz, -1 for odd permutations, and 0 otherwise), and the Einstein summation convention is used. We use the nomenclature in which the Maxwell stress tensor is the name of the quantity defined in equation (3). According to a different convention, the Maxwell stress tensor is the electromagnetic part of the stress tensor in general, be it in vacuum or in a dielectric medium. In that convention the quantity defined in equation(7) is also often called the Maxwell stress tensor.

By adding and subtracting terms many more balance equations of momentum-like quantities can be derived. For example, Abraham himself proposed a balance equation different from equation (1). Abraham’s balance equation combines the Abraham momentum density and the (symmetrized) Minkowski stress tensor. The force density is then equal to the Minkowski force density plus an additional term -∂_t (P × B), the Abraham force density [1]. At this point it is emphasized that the superscipt ‘A’ for the Maxwell stress tensor and the Lorentz force density is used for reasons of notational simplicity, it does not reflect Abraham’s original proposal. Einstein and Laub proposed a slightly modified version of the first momentum balance equation (1)where a term ∂_β (P_βE_α) is added to the left and right hand side giving a force density P_β∂_βE_α + ε_αβγ∂_tP_βB_γ, and a stress tensor with the same off-diagonal components as the Minkowski stress tensor [1]. Recently, Mansuripur proposed the average of the Abraham and Minkowski forms as the electromagnetic momentum density [2, 3]. Obukhov and Hehl adhere to the Abraham-Maxwell-Lorentz equation and regard the Minkowski-equation as describing the part of the electromagnetic momentum interacting with free charges [4].

In this paper we will investigate only the forces on a dielectric body when it interacts with electromagnetic radiation at optical frequencies. Furthermore, we will deal with (nearly) monochromatic light only. This implies that we may replace all quantities by their cycle averages (indicated by brackets in the following). In particular we find that 〈∂_tg_α〉 = 0, where g_α is either the Abraham or Minkowski momentum, so that the balance equations become:

where in the second step it is used that the order of cycle averaging and spatial differentiation may be exchanged as the stress tensor is bilinear in the fields, so that its cycle average will not involve terms varying over a wavelength. The total cycle averaged force on a dielectric body is:

As a consequence, the total force may be evaluated by integrating the force density over the volume, or by integrating the stress tensor over the surface. Furthermore, no conclusive evidence regarding the form of the momentum density may be found as a force like the Abraham force will not contribute to the cycle average [1]. It follows that the issue dealt with in this paper is not so much the issue of Abraham vs. Minkowski momentum, but rather the issue of Lorentz vs. Minkowski force density (or equivalently Maxwell vs. Minkowski stress tensor). Another consequence is that for a body “immersed” in vacuum the total force must be the same for all cases, since in vacuum any suitable stress tensor must be equal to the Maxwell stress tensor. Different predictions regarding radiation pressure are only found in the case where the body is immersed in another dielectric medium.

In a more complete physical description the dielectric media under consideration are considered to be deformable media. This means that electrostriction (change in density due to the presence of an electric field), and hydrostatic forces (for fluids) and elastic forces (for solids) must be taken into account. This entails that such a description must also be based on thermodynamic arguments. Landau and Lifshitz consider this case in the electrostatic limit [5] and find that the stress tensor in equilibrium is equal to the Minkowski stress tensor plus a constant diagonal tensor. This constant depends on the medium and ensures the continuity of forces at interfaces. According to Gordon the same will hold at optical frequencies for the cycle averaged quantities and this may be related to the concept of pseudo-momentum [6]. In the terminology of Nelson the Abraham-Maxwell-Lorentz equation relates to the electromagnetic momentum, whereas the Minkowski equation relates to the wave momentum, the wave momentum being the sum of the momentum and the pseudo-momentum [7]. The wave momentum is claimed to be the right quantity for evaluating forces on bodies surrounded by a dielectric. According to a different view it is the Lorentz force integrated over the volume of the body that should be identified as the radiation force on the body [2, 3, 8, 9]. For the case considered here this boils down to using the Maxwell stress tensor. In the next section the radiation force on a dielectric slab immersed in a different dielectric medium is calculated following the two recipes. It turns out that different answers are obtained for this relatively simple configuration.

3. Slab geometry at normal incidence

Consider a slab of thickness d with dielectric constant ε = (n + iκ)², where n is the refractive index and κ is the absorption coefficient, such that the interfaces are at z = -d/2 and z = +d/2. The slab is immersed in a dielectric with dielectric constant ε = n ₀ ², where n ₀ is the refractive index of the medium. We take the z-axis perpendicular to the slab and consider a monochromatic linearly polarized beam of light at normal incidence with angular frequency ω and polarization directed along the x-axis. Effects of dispersion are neglected so that differences between phase and group velocity are not discernable. The relevant electric and magnetic field components are (with an implicit overall exp (-iωt) time dependency):

where k ₀ = n ₀ ω/c and k ₁ = (n + iκ)ω/c are the wavenumbers in the immersing medium and the slab, respectively. The coefficients r, t, a and b may be solved from the boundary conditions at z = -d/2 and z = +d/2 (see appendix A). It is assumed that the coherence length of the beam is much larger than the slab thickness so that multiple reflections must be added coherently.

The total force on the body is directed along the z-axis and is for an incidence area A:

where the stress tensor must be evaluated in the immersing medium. This is necessary in order to take the forces across the interfaces properly into account. Using the expressions given in appendix B it follows that:

This gives the total force on the slab in both pictures as:

where P is the power of the incident beam. Here it has been used that the the energy flow in the z-direction of the incident beam is S_z = ε ₀ ∣E ₀∣² n ₀ c/2 = P/A. These expressions are the main result of this paper.

In the limit of zero absorption κ= 0 the force for the two cases is given by:

with k = ω/c the vacuum wavenumber. It follows that F^M_z > 0 in all cases, whereas F^A_z > 0 if n > n ₀ and F^A_z < 0 if n < n ₀. So, the Minkowski force always points away from the incident beam, whereas the Lorentz force points away or towards the incident beam depending on the relative magnitude of the refractive indices. If the refractive index of the slab is smaller than the refractive index of the surrounding medium the Lorentz force points towards the incident beam. The ratio of the forces is:

so that F^M_z < F^A_z if n > n ₀ and F^M_z > F^A_z if n < n ₀. The forces are only equal in the vacuum incidence case n ₀ = 1. The forces are maximum (in absolute value) if sin(nkd) = ±1, i.e. if nd = (2j + 1)λ/4 with j an integer. This corresponds to the case where the reflection from the slab reaches its maximum value. The normalized maximum force F _max c/2Pn ₀ is plotted as a function of the two refractive indices plotted in Fig. 1 for the two cases. These predictions may be tested experimentally with a torque balance setup similar to the one used by Jones and Leslie to measure the radiation force on a perfect reflector [10].

In the photon drag effect the momentum transfer to charge carriers in a semiconductor that absorb a photon is observed to be proportional to the refractive index of the absorbing medium and the absorbed power [9]. Mansuripur [3] relates this to the Lorentz force on a thin absorbing slab in a non-absorbing medium, i.e. he considers the limit d ≪ λ and n = n ₀. The present formalism gives for the limiting case in which the thickness of the slab is much smaller than the wavelength:

where P_abs is the absorbed power in the thin slab. This result slightly generalizes Mansuripur’s, as we have not taken n ₀ equal to n. Apparently, the Lorentz and Minkowski force expressions give the same result here, so this special case cannot be used to discriminate between the two.

In the limiting case in which the thickness of the slab is much larger than the absorption length, the transmission coefficient t = 0 and the expression for r derived in appendix A may be used to find:

These expressions are similar but not quite the same. The ratio of the two is now:

which is always smaller than unity, i.e. the Minkowski force is now always larger than the Lorentz force. The forces are only equal in the vacuum incidence case n ₀ = 1 or in the case of a perfect mirror κ = ∞. In the latter case:

This dependence on the refractive index of the incidence medium n ₀ has been confirmed experimentally by Jones and Leslie [10].

4. Discussion

The derived force expressions may be compared with a ballistic picture of the radiation pressure effect in which the incident beam is regarded as a stream of photons with energy E = ℏω and momentum p that is incident on the slab with a flux of N photons per unit time and area, where N = S_z/E = P/EA. The Minkowski momentum density corresponds to a momentum p = n ₀ E/c, whereas the Abraham momentum density corresponds to a momentum p = E/n ₀ c per photon. We limit the attention to the zero-absorption case, so that each photon is either reflected or transmitted by the slab with amplitudes r and t, respectively. The momentum transfer then follows as:

The ratio of momentum and energy of an incident photon can only depend on the properties of the incidence medium. Such a form is consistent with the Minkowski force expression (17) and p = n ₀ E/c. The first term of the Lorentz force expression (16) is consistent with the ballistic picture if the Mansuripur momentum p = (n ₀ ² + 1)E/2n ₀ c is adopted, the second term, due to the interference of the incident and reflected waves is not. The Lorentz force expression (16) is entirely inconsistent with the Abraham momentum p = E/n ₀ c. Application of this Abraham momentum to equation (26) leads to a force 2P/n ₀ c for the ideal mirror case, in disagreement with the classical result derived in the previous section and with experiment.

The question arises if this is a mere paradox or a real inconsistency. I take it to be the former, and the conclusion should rather be that the picture of a photon as a classical particle with a momentum that is n ₀ E/c or E/n ₀ c or some other factor times E/c is a too naieve interpretation of wave-particle duality. Interference effects, such as the force term appearing in the Lorentz force approach, should be described by a proper quantum theory of momentum and radiation forces. A simplified quantum approach, as represented by equation (26) for the radiation force, must therefore be applied with caution. The problem of radiation forces is essentially a classical problem which can be discussed without resorting to quantum arguments.

A. Reflection and transmission coefficients in the slab geometry

The boundary conditions for the components of E and H parallel to the interface at z = -d/2 are:

(27)$$\exp \left(\frac{-i{k}_{0}d}{2}\right)+r\exp \left(\frac{i{k}_{0}d}{2}\right)=a\exp \left(\frac{-i{k}_{1}d}{2}\right)+b\exp \left(\frac{i{k}_{1}d}{2}\right),$$

(28)$$n_0\exp \left(\frac{-i{k}_{0}d}{2}\right)-n_{0}r\exp \left(\frac{i{k}_{0}d}{2}\right)=\left(n+\mathrm{i\kappa }\right)a\exp \left(\frac{-i{k}_{1}d}{2}\right)$$
$$-\left(n+\mathrm{i\kappa }\right)b\exp \left(\frac{i{k}_{1}d}{2}\right),$$

resepectively, and for the components of E and H parallel to the interface at z = +d/2:

(29)$$t\exp \left(\frac{i{k}_{0}d}{2}\right)=a\exp \left(\frac{i{k}_{1}d}{2}\right)+b\exp \left(\frac{-i{k}_{1}d}{2}\right),$$

(30)$$n_{0}t\exp \left(\frac{i{k}_{0}d}{2}\right)=\left(n+\mathrm{i\kappa }\right)a\exp \left(\frac{i{k}_{1}d}{2}\right)-\left(n+\mathrm{i\kappa }\right)b\exp \left(\frac{-i{k}_{1}d}{2}\right).$$

The reflection and transmission coefficients may be solved from these equations as:

(31)$$r=\frac{i\left({\left(n+i\kappa \right)}^2-n_0^2\right)\sin \left(k_{1}d\right)}{2n_0\left(n+i\kappa \right)\cos \left(k_{1}d\right)-i\left({\left(n+i\kappa \right)}^2+n_0^2\right)\sin \left(k_{1}d\right)}\exp \left(-i{k}_{0}d\right),$$

(32)$$t=\frac{2n_0\left(n+i\kappa \right)}{2n_0\left(n+i\kappa \right)\cos \left(k_{1}d\right)-i\left({\left(n+i\kappa \right)}^2+n_0^2\right)\sin \left(k_{1}d\right)}\exp \left(-i{k}_{0}d\right).$$

As k ₁ is generally complex we may write:

(33)$$\cos \left(k_{1}d\right)=\cosh \left(\kappa kd\right)\cos \left(nkd\right)-i\sinh \left(\kappa kd\right)\sin \left(nkd\right),$$

(34)$$\sin \left(k_{1}d\right)=\cosh \left(\kappa kd\right)\sin \left(nkd\right)+i\sinh \left(\kappa kd\right)\cos \left(nkd\right),$$

with k = ω/c the vacuum wavenumber.

In the zero absorption case κ = 0 the reflection and transmission coefficients are:

(35)$$r=\frac{i\left(n^2-n_0^2\right)\sin \left(\mathrm{nkd}\right)}{2n_{0}n\cos \left(\mathrm{nkd}\right)-i\left(n^2+n_0^2\right)\sin \left(\mathrm{nkd}\right)}\exp \left(-i{k}_{0}d\right),$$

(36)$$t=\frac{2n_{0}n}{2n_{0}n\cos \left(\mathrm{nkd}\right)-i\left(n^2+n_0^2\right)\sin \left(\mathrm{nkd}\right)}\exp \left(-i{k}_{0}d\right),$$

so that ∣r∣² +∣t∣² = 1, in agreement with conservation of energy.

In the limit κkd ≫ 1 the reflection and transmission coefficients are:

(37)$$r=\frac{n_0-n-i\kappa }{n_0+n+i\kappa }\exp \left(-i{k}_{0}d\right),$$

(38)$$t=0.$$

B. Cycle averages for plane waves

Consider a plane wave propagating through a medium with dielectric constant ε = (n + iκ)², where n is the refractive index and κ is the absorption coefficient. The wave propagates along the z-axis and is monochromatic with angular frequency ω and linearly polarized along the x-axis. The relevant electric and magnetic field components are:

(39)$$E_x=E\left(z\right)\exp \left(-i\omega t\right),$$

(40)$$H_y={\epsilon }_{0}c\left(n+i\kappa \right)E\left(z\right)\exp \left(-i\omega t\right),$$

where:

(41)$$E\left(z\right)=E_0\exp \left(ikz\right),$$

with k = (n + iκ)ω/c. The cycle averaged Abraham and Minkowski momentum densities are:

(42)$$〈g_z^A〉=\frac{1}{2}{\epsilon }_0{\mid E\left(z\right)\mid }^2\frac{n}{c},$$

(43)$$〈g_z^M〉=\frac{1}{2}{\epsilon }_0{\mid E\left(z\right)\mid }^2\left(n^2+{\kappa }^2\right)\frac{n}{c}.$$

For the calculation of the stress tensor components we need the cycle averages:

(44)$$〈{\epsilon }_0{E}_x^2〉={\epsilon }_0{\mid E\left(z\right)\mid }^2,$$

(45)$$〈\frac{1}{{\mu }_0}{B}_y^2〉={\epsilon }_0\left(n^2+{\kappa }^2\right){\mid E\left(z\right)\mid }^2,$$

(46)$$〈E_x{D}_x〉={\epsilon }_0\left(n^2-{\kappa }^2\right){\mid E\left(z\right)\mid }^2.$$

The non-zero components of the Maxwell stress tensor are:

(47)$$〈T_{\mathrm{xx}}^A〉=\frac{1}{4}{\epsilon }_0{\mid E\left(z\right)\mid }^2\left(n^2+{\kappa }^2-1\right),$$

(48)$$〈T_{\mathrm{yy}}^A〉=-\frac{1}{4}{\epsilon }_0{\mid E\left(z\right)\mid }^2\left(n^2+{\kappa }^2-1\right),$$

(49)$$〈T_{\mathrm{zz}}^A〉=\frac{1}{4}{\epsilon }_0{\mid E\left(z\right)\mid }^2\left(n^2+{\kappa }^2+1\right),$$

and the non-zero components of the Minkowski stress tensor are:

(50)$$〈T_{\mathrm{xx}}^M〉=\frac{1}{2}{\epsilon }_0{\mid E\left(z\right)\mid }^2{\kappa }^2,$$

(51)$$〈T_{\mathrm{yy}}^M〉=-\frac{1}{2}{\epsilon }_0{\mid E\left(z\right)\mid }^2{\kappa }^2,$$

(52)$$〈T_{\mathrm{zz}}^M〉=\frac{1}{2}{\epsilon }_0{\mid E\left(z\right)\mid }^2{n}^2.$$

The components T_zz correspond with the stored electromagnetic energy density. The cycle averaged Lorentz and Minkowski force density are:

(53)$$〈f_z^A〉=\frac{1}{2}{\epsilon }_0{\mid E\left(z\right)\mid }^2\left(n^2+{\kappa }^2+1\right)\kappa \frac{\omega }{c},$$

(54)$$〈f_z^M〉={\epsilon }_0{\mid E\left(z\right)\mid }^2{n}^2\kappa \frac{\omega }{c}.$$

These forces also satisfy:

(55)$$〈f_z^A〉=-{\partial }_z〈T_{\mathit{zz}}^A〉=2\kappa \frac{\omega }{c}〈T_{\mathit{zz}}^A〉,$$

(56)$$〈f_z^M〉=-{\partial }_z〈T_{\mathit{zz}}^M〉=2\kappa \frac{\omega }{c}〈T_{\mathit{zz}}^M〉,$$

i.e. the force density is the ratio of the stored energy and the decay length. The integrated force density can thus be evaluated by direct integration or by taking the stress tensor difference over the integration range.

Consider now two counterpropagating plane waves:

(57)$$E_x=E\left(z\right)\exp \left(-i\omega t\right)+rE\left(-z\right)\exp \left(-i\omega t\right),$$

(58)$$H_y={\epsilon }_{0}c\left(n+i\kappa \right)E\left(z\right)\exp \left(-i\omega t\right)-r{\epsilon }_{0}c\left(n+i\kappa \right)E\left(-z\right)\exp \left(-i\omega t\right),$$

where r = ∣r∣exp(iψ) is an arbitrary complex coefficient. The momentum densities are now:

(59)$$〈g_z^A〉=\frac{1}{2}{\epsilon }_0{\mid E_0\mid }^2\frac{n}{c}\left[\exp \left(-2\kappa \frac{\omega }{c}z\right)-{\mid r\mid }^2\exp \left(2\kappa \frac{\omega }{c}z\right)\right]$$
$$+{\epsilon }_0{\mid E_0\mid }^2\frac{\kappa }{c}\mid r\mid \sin \left(2n\frac{\omega }{c}z-\psi \right),$$

(60)$$〈g_z^M〉=\frac{1}{2}{\epsilon }_0{\mid E_0\mid }^2\left(n^2+{\kappa }^2\right)\frac{n}{c}\left[\exp \left(-2\kappa \frac{\omega }{c}z\right)-{\mid r\mid }^2\exp \left(2\kappa \frac{\omega }{c}z\right)\right]$$
$$-{\epsilon }_0{\mid E_0\mid }^2\left(n^2+{\kappa }^2\right)\frac{\kappa }{c}\mid r\mid \sin \left(2n\frac{\omega }{c}z-\psi \right).$$

The relevant zz-component of the stress tensors are:

(61)$$〈T_{\mathrm{zz}}^A〉=\frac{1}{4}{\epsilon }_0{\mid E_0\mid }^2\left(n^2+{\kappa }^2+1\right)\left[\exp \left(-2\kappa \frac{\omega }{c}z\right)+{\mid r\mid }^2\exp \left(2\kappa \frac{\omega }{c}z\right)\right]$$
$$-\frac{1}{2}{\epsilon }_0{\mid E_0\mid }^2\left(n^2+{\kappa }^2-1\right)\mid r\mid \cos \left(2n\frac{\omega }{c}z-\psi \right),$$

(62)$$〈T_{\mathrm{zz}}^M〉=\frac{1}{2}{\epsilon }_0{\mid E_0\mid }^2{n}^2\left[\exp \left(-2\kappa \frac{\omega }{c}z\right)+{\mid r\mid }^2\exp \left(2\kappa \frac{\omega }{c}z\right)\right]$$
$$-{\epsilon }_0{\mid E_0\mid }^2{\kappa }^2\mid r\mid \cos \left(2n\frac{\omega }{c}z-\psi \right).$$

The force densities are now given by:

(63)$$〈f_z^A〉=\frac{1}{2}{\epsilon }_0{\mid E_0\mid }^2\left(n^2+{\kappa }^2+1\right)\kappa \frac{\omega }{c}\left[\exp \left(-2\kappa \frac{\omega }{c}z\right)-{\mid r\mid }^2\exp \left(2\kappa \frac{\omega }{c}z\right)\right]$$
$$-{\epsilon }_0{\mid E_0\mid }^2\left(n^2+{\kappa }^2-1\right)\mid r\mid n\frac{\omega }{c}\sin \left(2n\frac{\omega }{c}z-\psi \right),$$

(64)$$〈f_z^M〉={\epsilon }_0{\mid E_0\mid }^2{n}^2\kappa \frac{\omega }{c}\left[\exp \left(-2\kappa \frac{\omega }{c}z\right)-{\mid r\mid }^2\exp \left(2\kappa \frac{\omega }{c}z\right)\right]$$
$$-2{\epsilon }_0{\mid E_0\mid }^2{\kappa }^{2}n\frac{\omega }{c}\mid r\mid \sin \left(2n\frac{\omega }{c}z-\psi \right).$$

In the limit of non-absorbing media (κ = 0) it follows that the Minkowski force density is zero, but the Lorentz force density is not. The force is modulated by the standing wave and points to the region of maximum electric field.

 

Fig. 1. The normalized maximum force on the slab as a function of the refractive index of the immersing medium. The solid line is the integrated Lorentz-force for n = 1.5, the short-dashed the integrated Minkowski-force for n = 1.5, the medium-dashed line the integrated Lorentz-force for n = 2.0, and the long-dashed the integrated Minkowski-force for n = 2.0.

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Acknowledgments

Martin van der Mark, Mischa Megens, Gert ’t Hooft and Masud Mansuripur are thanked for their valuable suggestions.

References and links

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