Theory of the radiation pressure on dielectric slabs, prisms and single surfaces

Rodney Loudon; Stephen M. Barnett

doi:10.1364/OE.14.011855

Introduction

Much theoretical work on the radiation pressure on media has focused on the magnitude of the momentum carried by a single photon in a dielectric (see Brevik Ref. [1] for a review). A great deal of controversy has centered on the rival claims of the Abraham expression ħω/cη and the Minkowski expression ħωη/c, where η is the refractive index of the medium. The few available experiments seem to favor the latter, while many theories favor the former.

The relevant experiments measure the forces on objects, rather than the photon momentum itself. Gordon Ref. [2] initiated the idea of calculating the Lorentz forces involved in various experiments, which can then be used to deduce values for the effective photon momentum. He used the form of Lorentz force density

f^{d} = (P . \nabla) E + \frac{\partial P}{\partial t} \times B

appropriate to a dipolar medium, and much subsequent work has used this expression. Here E and B are respectively the electric field strength and the magnetic induction associated with the radiation, while P is the electric polarization in the medium. Thus, the linear forces Ref. [3] and torques Refs. [4,5] exerted by single-photon pulses on dielectric interfaces and on bulk dielectrics have been calculated, and the transfer of momentum to charge carriers in the photon drag effect has been evaluated Ref. [6]. In terms of the photon momentum, the results of these calculations, summarized in Ref. [7], assign the Abraham value of ħω/cη to the total momentum in the bulk dielectric but the Minkowski value of ħωη/c to the momentum transferred to objects immersed in the dielectric, for example mirrors or charge carriers. The (negative) difference between these two momenta is taken up by the host dielectric.

The form of Lorentz force density in Eq. (1) has recently been questioned by Mansuripur Ref. [8], with the introduction of an alternative form

f^{c} = - (\nabla . P) E + \frac{\partial P}{\partial t} \times B

appropriate to a medium formed from individual charges. Amongst other examples, calculations in this formalism have covered the torque on a dielectric slab illuminated at the Brewster angle Ref. [8], likewise the force on a dielectric prism Ref. [9], the Minkowski momentum transfer in the photon drag effect Ref. [10], and the angular momentum of light in a dielectric Ref. [11]. While there is a measure of agreement between the results of calculations based on Eq. (1) and Eq. (2), there are also some discrepancies. In addition, it has been suggested Ref. [12] that the electric component of the form Eq. (1) of the Lorentz force produces inconsistencies when applied to problems such as the dielectric slab and prism.

More recently still, it has been shown Ref. [13] that the two formalisms for the Lorentz force density lead to identical results for the total forces and torques on dielectric samples, and this conclusion has been confirmed Ref. [14]. Nevertheless, although the totals are the same, the component parts of the forces have quite different forms. The purpose of the present paper is to explore their similarities and differences, beginning with the slab and prism, where our totals, calculated with the form Eq. (1) of the Lorentz force, agree with those in Ref. [8] and Ref. [9] respectively. The different components of these total forces are discussed and additional calculations are given for the linear displacement of the slab on passage of a singlephoton wavepacket. A third example, of the passage of light through the isolated surface of a dielectric, gives results that differ in the two formalisms, and the reasons for this discrepancy are discussed.

1. Radiation torque and force on transparent dielectric slab

Consider the system shown in Fig. 1, with a light beam incident from free space at the Brewster angle i on the surface of a dielectric slab of thickness D and refractive index η. The Brewster incidence ensures that no reflections occur at either the entrance or exit surfaces of the slab. The lateral shift of the beam on its passage through the slab results in a torque that can be calculated in two ways, (i) by evaluation of the effective torque on the beam and (ii) by evaluation of the Lorentz forces applied to the slab by the light beam. The two quantities should be equal in magnitude but opposite in sign.

Fig. 1. Representation of a dielectric slab with light beam incident at the Brewster angle, showing the orientations of the three forces at the entrance surface. The corresponding forces at the exit surface are identified by the same coloring as for the entrance surface. The red colored sections of beam edge indicate the regions of unbalanced force contributions of type 3.

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2.1 Effective torque on the light beam

Snell’s law and the Brewster angle condition relate i and the angle of refraction r by

\sin i = η \sin r and η \cos i = \cos r

respectively, so that i+r=π2 and

\sin i = \cos r = \frac{η}{\sqrt{η^{2} + 1}} and \cos i = \sin r = \frac{1}{\sqrt{η^{2} + 1}} .

The transmitted light beam is parallel to the incident beam but with lateral displacement

Δ = \frac{D}{\cos r} \sin (i - r) = D \frac{η^{2} - 1}{η \sqrt{η^{2} + 1}} .

For an incident beam with flux f photons of energy ħω per second, the effective torque on the light is anticlockwise for the arrangement shown in Fig. 1, with value

T_{L} = - \frac{ℏ ω f}{c} Δ = - \frac{ℏ ω f D}{c} \frac{η^{2} - 1}{η \sqrt{η^{2} + 1}}

in agreement with a similar calculation in §7 of Ref. [8].

1.2 Torque on the slab

The magnitude of the clockwise torque TS on the slab is expected to equal that of T _L given by Eq. (6). There are three Lorentz force contributions at the entrance surface of the slab:

The magnetic term of the Lorentz force Eq. (1) in the bulk dielectric, which contributes a force parallel to the refracted beam as the optical pulse passes through the surface Ref. [3]. This is the only force contribution in conditions of normal incidence.
A surface Lorentz force from the electric term in Eq. (1) directed perpendicular to the slab into the free space above it.
A further electric force directed normal to the refracted beam edge within the dielectric. The three contributions, illustrated in Fig. 1, all occur with the same magnitudes but opposite signs at the exit surface of the slab.

The field E of the incident beam is converted to E/η in the dielectric, while the B fields of the incident and refracted beams are equal. The E field is related to the incident photon flux by

\frac{1}{2} A ε_{0} c {∣ E ∣}^{2} = ℏ ω f,

where the cross-sectional area A of the incident beam becomes Aη in the dielectric, so that the Poynting vector is conserved. The illuminated surface area is

\frac{A}{\cos i} = \frac{A η}{\cos r} = A \sqrt{η^{2} + 1} .

The entrance force F ₁ for contribution 1 is evaluated in subsection 2.3 as

F_{1} = \frac{ℏ ω f}{c} \frac{η^{2} - 1}{2 η} .

As shown in Fig. 1, this force is exactly aligned with the corresponding exit force, and the associated torque vanishes. It does, however, produce a displacement of the slab, given in Eq. (27) below. The exit forces 2 and 3 have lateral offsets from the corresponding entrance forces. They contribute to the torque on the slab but not to its displacement.

The force contribution 2 is evaluated at the entrance surface, defined as z=0 with the z axis pointing away from the slab. The tangential components of the E field are continuous across the surface but the normal field changes discontinuously from E sin i at z>0 to E sin r/η at z<0. It follows with the use of Eq. (3) that

\frac{\partial E_{z}}{\partial z} = E \sin i \frac{η^{2} - 1}{η^{2}} δ (z) .

The electrical polarization has a step discontinuity at the surface, with

P_{z} = ε_{0} (η^{2} - 1) (\frac{E \sin r}{η}) ϑ (- z) .

The surface force that results from the first term of Eq. (1), with the surface area from Eq. (8), is

F_{2} = A \sqrt{η^{2} + 1} \int d z P_{z} \frac{\partial E_{z}}{\partial z} = \frac{1}{4} A ε_{0} {∣ E ∣}^{2} \frac{{(η^{2} - 1)}^{2}}{η^{2} \sqrt{η^{2} + 1}} = \frac{ℏ ω f}{c} \frac{{(η^{2} - 1)}^{2}}{2 η^{2} \sqrt{η^{2} + 1}},

where one factor of 1/2 in the intermediate expression results from the integration of the product of delta and step functions and the other from a cycle average of the electric field. The lateral offset between the outwards-pointing forces at the entrance and exit surfaces is D tan r=D/η, and the clockwise torque on the slab is therefore

T_{2} = \frac{ℏ ω f D}{c} \frac{{(η^{2} - 1)}^{2}}{2 η^{3} \sqrt{η^{2} + 1}} .

Consider finally the force contribution 3, which occurs at the left edge of the refracted beam adjacent to the entrance surface, as indicated in red in Fig. 1. It results from the lack of balance between forces on the left and right edges of this initial section of the beam. These have the same nature as the inwardly-directed radial forces that occur for light beams of Gaussian intensity profile (see for example Refs. [4, 7]). We assume here a top-hat intensity profile with sharp edges, as in Ref. [8], but a Gaussian beam is treated in section 3. The force again results from the first term of Eq. (1) and, if ζ is an inwards-pointing coordinate normal to the beam propagation direction, the analogues of Eq. (10) and Eq. (11) are

\frac{\partial E_{ζ}}{\partial ζ} = \frac{E}{η} δ (ζ) and P_{ζ} = ε_{0} (η^{2} - 1) (\frac{E}{η}) ϑ (ζ) .

The area of the unbalanced section of the beam is simply A, and the resulting edge force is

F_{3} = \frac{ℏ ω f}{c} \frac{η^{2} - 1}{2 η^{2}} .

The average lateral offset between the edge forces pointing towards the centre of the beam adjacent to the entrance and exit surfaces is D/cos r. Their contribution to the torque is again clockwise, with value

T_{3} = \frac{ℏ ω f D}{c} \frac{η^{2} - 1}{2 η^{3}} \sqrt{η^{2} + 1} .

The total torque is obtained from the sum of the contributions from Eq. (13) and Eq. (16) as

T_{s} = T_{2} + T_{3} = \frac{ℏ ω f D}{c} \frac{η^{2} - 1}{η \sqrt{η^{2} + 1}} = - T_{L}

with T _L given by Eq. (6), as expected. Alternatively, the forces can first be added, with components parallel and perpendicular to the incident beam direction given by

{(F_{2} + F_{3})}_{∥} = - F_{2} \cos i + F_{3} \sin (i - r) = 0

and

{(F_{2} + F_{3})}_{⊥} = F_{2} \sin i + F_{3} \cos (i - r) = \frac{ℏ ω f}{c} \frac{η^{2} - 1}{2 η} .

The resultant of the two forces is thus perpendicular to the incident beam. The lateral displacement between the sum of forces at the entrance and exit surfaces is

\frac{D}{\cos r} \cos (i - r) = \frac{2 D}{\sqrt{η^{2} + 1}} .

The total torque thus agrees with Eq. (17).

The total torque also agrees with the result Eq. (19) of Ref. [8], obtained by use of the form Eq. (2) of Lorentz force, but the makeup of the total is quite different from that found here. Thus, the surface force in Ref. [8] is associated with a surface charge density, and it has nonzero components both perpendicular and parallel to the surface. The former provides a torque identical to that in Eq. (17) while the latter is cancelled by the unbalanced beam-edge forces, which are directed away from the beam center, in contrast to those derived above.

An experiment of the kind envisaged here was performed almost a century ago by Guy Barlow Ref. [15]. He did not use Brewster angle incidence and complicated corrections for the internal and external reflections were needed but, remarkably, he got agreement with a suitably modified theory to within 1 or 2%.

1.3 Slab displacement

Consider now an incident single-photon pulse, which traverses the slab in a time

τ = \frac{D η}{c \cos r} = \frac{D}{c} \sqrt{η^{2} + 1} .

Similarly to the torque, the slab displacement can be calculated by two different methods, using respectively the effects of transmission on the light pulse itself and on the dielectric slab. An important feature of the absence of any reflections for Brewster angle incidence is the conservation of linear momentum ħω/c parallel to the incident light. This applies to the three stages of incident pulse in free space, refracted pulse within the dielectric, and transmitted pulse in free space. The conservation is demonstrated below, with particular emphasis on the second of the three stages.

The light pulse is delayed by propagation through the slab, with the difference between the distances travelled in the absence and presence of the slab in time τ given by

\frac{D}{\cos r} {η - \cos (i - r)} = D \frac{η^{2} - 1}{\sqrt{η^{2} + 1}},

where the distance within the slab is projected on to the incident direction. The displacement ΔZ of the slab is now easily determined by the principle of the Einstein box Refs. [5,16], by which conservation of the center of mass-energy of the system of light pulse and slab is ensured by the condition

M c^{2} Δ Z = ℏ ω D \frac{η^{2} - 1}{\sqrt{η^{2} + 1}},

where M is the mass of the slab. Note that the displacement always occurs in the propagation direction of the incident and transmitted light. The assumption of Brewster-angle incidence in an Einstein-box theory of slab displacement was previously made by Balazs Ref. [17], and Eq. (23) agrees with his calculation of the displacement parallel to the incident direction.

The slab displacement can also be calculated by consideration of the Lorenz forces applied to it and the consequent momentum transfers. The forces F ₂ and F ₃ exerted by a photon flux are converted to the momentum transfers from a single-photon pulse by the simple removal of f. The sum force given by Eq. (18) and Eq. (19) thus corresponds to a slab momentum transfer

p_{2} + p_{3} = \frac{ℏ ω}{c} \frac{η^{2} - 1}{2 η}

in a direction perpendicular to the incident light. The apparent conflict between this transfer and the requirements of momentum conservation is resolved by the presence of an additional contribution p ₁ parallel to the refracted beam. It results from the magnetic term in the Lorentz force Eq. (1) within the dielectric and its magnitude is

p_{1} = \frac{ℏ ω}{c} {\frac{η^{2} - 1}{\underset{surface}{\underset{︸}{2 η}}} + \underset{bulk}{\underset{︸}{\frac{1}{η}}}} = \underset{total}{\underset{︸}{\frac{η^{2} + 1}{2 η} \frac{ℏ ω}{c}}} .

The surface contribution has the explicit form of the magnetic term in Eq. (1). It is caused by a lack of balance between the magnetic contributions to the Lorentz force from the leading and trailing parts of the pulse during passage through the surface Ref. [3]. For an incident photon flux f, it provides the force component F ₁ quoted in Eq. (9). The bulk contribution has the Abraham form of photon momentum in the dielectric. The total momentum has components

p_{1} \cos (i - r) = \frac{ℏ ω}{c} and - p_{1} \sin (i - r) = - \frac{ℏ ω}{c} \frac{η^{2} - 1}{2 η}

parallel and perpendicular to the incident light respectively. The perpendicular component thus cancels the momentum transfer from forces F ₂ and F ₃ given in Eq. (24) and the parallel component ensures momentum conservation with the incident light. The surface part of Eq. (25) represents a momentum transfer to the slab, which is cancelled after a time τ as the pulse passes through the exit surface to leave the slab at rest again, with a displacement

Δ Z = \frac{ℏ ω}{M c} \frac{η^{2} - 1}{2 η} τ \cos (i - r) = \frac{ℏ ω D}{M c^{2}} \frac{η^{2} - 1}{\sqrt{η^{2} + 1}},

in agreement with Eq. (23).

In summary, the combined force F ₂+F ₃, given by Eq. (18) and Eq. (19), acts perpendicular to the incident light beam and it determines the torque on the slab given by Eq. (17). For a single-photon incident pulse, it provides the momentum transfer p ₂+p ₃ to the slab given by Eq. (24). The force F ₁, with magnitude given by Eq. (9), acts parallel to the refracted light beam and it makes no contribution to the torque on the slab. For a single-photon incident pulse, it provides the slab displacement Eq. (27) in agreement with Einstein box theory. In combination with the photon momentum ħω/cη, also in the direction of the refracted beam, it cancels the momentum p ₂+p ₃ perpendicular to the incident beam. In accordance with Eq. (26), it ensures that the linear momentum ħω/c is conserved throughout the transmission process.

2. Radiation force on transparent dielectric prism

Figure 2 shows a light beam incident at the Brewster angle i on the surface of a dielectric prism or wedge with semi-angle r such that i+r=π/2. The refracted beam in these conditions is directed perpendicular to the prism axis and the light also exits the prism in Brewster configuration, with no reflected components. The forces F ₂ and F ₃ at the entrance and exit surfaces considered individually are identical to those for the slab shown in Fig. 1. However, the reversal of the exit angle of the transmitted light from the prism results in quite different effects from those in the slab. Thus there is clearly no torque on the prism but there is now a net horizontal force.

Fig. 2. Representation of a dielectric prism with light beam incident at the Brewster angle, showing the coordinate system used in the calculations. The natures of the three forces at both entrance and exit surfaces are identified by the same color convention as in Fig. 1.

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The force can again be calculated in two ways. For free-space incident and transmitted photon fluxes f, the effective horizontal force on the light is immediately obtained by reference to Fig. 2 as

F_{L} = - 2 \frac{ℏ ω f}{c} \cos 2 r = - 2 \frac{ℏ ω f}{c} \frac{η^{2} - 1}{η^{2} + 1} .

The force on the prism is also immediately obtained, with the use of Eq. (12) and Eq. (15), as

F_{P} = 2 F_{2} \cos i + 2 F_{3} = \frac{ℏ ω f}{c} \frac{{(η^{2} - 1)}^{2}}{η^{2} (η^{2} + 1)} + \frac{ℏ ω f}{c} \frac{η^{2} - 1}{η^{2}} = - F_{L} .

The final result agrees with Eq. (10) of Ref. [9], where its makeup in terms of individual contributions again differs from that found here, despite the agreement of the total force, similar to the discussion that follows Eq. (20) above.

The calculations so far, for both the slab and the prism, assume an incident light beam of constant intensity across an area A, with zero intensity outside this area. We now generalize the prism calculations to a more physical light beam of Gaussian intensity profile and show how the above results are substantiated. The position dependence of the incident field is now given by

E^{incid} (r) = E \exp {- \frac{({x'}^{2} + y^{2})}{2 w^{2}}} \exp [- i (\frac{ω}{c}) (z' + p + c t)] \hat{x}',

where w is the beam waist, p is the x coordinate of the intersection of the incident beam axis with the prism axis, and x̂′ is a unit polarization vector. The coordinates z′ and x′ point parallel and perpendicular to the incident beam propagation in the plane of Fig. 2, with y perpendicular to the plane. For the coordinate system defined in the figure,

x' = (x - p) \cos (i - r) + z \sin (i - r) and z' = - (x - p) \sin (i - r) + z \cos (i - r) .

The field of the Gaussian beam in the interior of the prism is likewise

E^{prism} (r) = (\frac{E}{η}) \exp {- \frac{[{(\frac{x}{η})}^{2} + y^{2}]}{2 w^{2}}} \exp [- i (\frac{ω}{c}) (η z + c t)] \hat{x},

and the field of the transmitted beam is given by a suitably modified form of Eq. (30). The fields in the three regions satisfy the boundary conditions at the prism surfaces. They are valid for a beam that satisfies the paraxial condition c/ω≪w. It is also assumed that essentially all of the light passes through the prism, expressed in the condition

η p \cot (i - r) ≫ w,

where the quantity on the left is the x coordinate of the prism apex.

The total flow of energy, or Poynting vector integrated over a plane perpendicular to the beam propagation direction, is the same in the three regions. Its common value is related to the photon flux by

\frac{1}{2} π w^{2} ε_{0} c {∣ E ∣}^{2} = ℏ ω f .

Thus πw ² replaces A in Eq. (7) as the effective cross sectional area of the Gaussian beam.

The rate at which momentum enters and leaves the prism can be obtained in terms of the energy-momentum tensor T_ij, with a cycle-averaged momentum flux density given by Ref. [18]

T_{i j} = \frac{1}{2} ℜ {\frac{1}{2} δ_{i j} (ε_{0} {∣ E ∣}^{2} + μ_{0}^{- 1} {∣ B ∣}^{2}) - ε_{0} E_{i}^{*} E_{j} - μ_{0}^{- 1} B_{i}^{*} B_{j}} .

This quantity represents the rate of flow of momentum component i per unit area in direction j. The rate of change of the momentum of the incident light is obtained by integration over a plane perpendicular to its propagation direction, with use of the electric field in Eq. (30) and the corresponding magnetic field. This quantity is equivalent to an effective force exerted on the incident beam at the entrance surface of the prism, and the result is

F_{in} = \frac{1}{2} π w^{2} ε_{0} {∣ E ∣}^{2} {- \frac{η^{2} - 1}{η^{2} + 1} \hat{x} + \frac{2 η}{η^{2} + 1} \hat{z}} .

The corresponding force F _out associated with the transmitted beam leaving the exit surface of the prism is given by the same expression but with the opposite sign of z component. The total effective force on the beam thus agrees with Eq. (28) when expressed in terms of the photon flux by means of Eq. (34).

It remains to calculate the Lorentz forces exerted on the prism by the light beams. It is shown in Ref. [13] that the force density in the form Eq. (1) can be re-expressed as

f_{i}^{d} = \frac{1}{2} ℜ (P_{j}^{*} \nabla_{j} E_{i}) = \frac{1}{4} ε_{0} (η^{2} - 1) \nabla_{i} {∣ E (r) ∣}^{2},

where the first form is needed for calculation of the surface force F ₂ but the second form is adequate for calculation of the edge force F ₃. The force F ₂ at the entrance surface is calculated by the same method as used for the slab surface in Eq. (10) to Eq. (11). A coordinate n normal to the surface with its origin at the point where the axis of the incident beam intersects the entrance surface is defined by

n = x \cos i + [z - p \cot (i - r)] \sin i

and the unit vector normal to the surface is

\hat{n} = \frac{(\hat{x} + η \hat{z})}{\sqrt{η^{2} + 1}} .

The field and polarization analogous to Eq. (10) and Eq. (11) or Eq. (14) are

\frac{\partial E_{n}}{\partial n} = (E_{n}^{incid} - E_{n}^{prism}) δ (n) and P_{n} = ε_{0} (η^{2} - 1) E_{n}^{prism} ϑ (- n) .

The expression to be evaluated is

F_{2} = \frac{1}{2} \int d V ℜ (P_{n}^{*} \frac{\partial E_{n}}{\partial n}) \hat{n},

where the integration volume includes the entrance surface. The calculation proceeds with use of the field expressions Eq. (30) and Eq. (32). The z integration is performed first with use of the delta function, the y integration is straightforward, and the range for the remaining x integral is

- \infty < x < η p \cot (i - r) .

Use of the condition Eq. (33) and the conversion in Eq. (34) then leads to the surface force

F_{2} = \frac{1}{4} π w^{2} ε_{0} {∣ E ∣}^{2} \frac{{(η^{2} - 1)}^{2}}{η^{2} \sqrt{η^{2} + 1}} \hat{n} .

The force at the exit surface is given by the same expression except that the sign of the z component in the unit vector n̂ in Eq. (39) is reversed. The total surface force on the prism is thus parallel to the x axis and its magnitude agrees with the F ₂ contribution in Eq. (29).

The edge force is identical to the inwardly-directed radial force that is often calculated for light beams of Gaussian intensity profile. The resultant force is directed parallel to the x axis and it is given by the second form of force density in Eq. (37) as

F_{3} = \frac{1}{4} ε_{0} (η^{2} - 1) \int d V \frac{\partial {∣ E (r) ∣}^{2}}{\partial x} \hat{x} .

The integration runs over the volume of the upper part of the prism with z>0 and the field expression is taken from Eq. (32). The x integration, with limits

- \infty < x < η p \cot (i - r) - η z,

is easily performed and the y integration is again straightforward, with the result

F_{3} = \frac{1}{4} ε_{0} \frac{η^{2} - 1}{η^{2}} \sqrt{π} w {∣ E ∣}^{2} \int_{0}^{\infty} d z \exp {- \frac{{[p \cot (i - r) - z]}^{2}}{w^{2}}} \hat{x}

The error function is equal to unity for the condition Eq. (33) and the conversion in Eq. (34) then leads to an edge force whose magnitude agrees with the F ₃ contribution in Eq. (29).

The surface and edge forces thus combine to produce the same form of total force on the prism as given in Eq. (29). The results for the Gaussian beam therefore confirm those obtained for the unphysical top-hat intensity profile, with the field and photon flux related by Eq. (34). These calculations are performed with the dipole form of Lorentz force density given by Eq. (1) or Eq. (37) and, according to the general proof in Ref. [13], a calculation based on the charge form given by Eq. (2) must produce the same results as here for a Gaussian input beam. Such a calculation, analogous to that for a beam of top-hat profile in Ref. [9], relies on the longitudinal or axial fields, which are smaller than the transverse fields by factors of order c/ωw. However, the spatial derivatives of the fields produce forces of the same magnitudes as those derived here.

It can also be shown that the momentum ħω/c of an incident photon pulse is conserved throughout the transmission process into and through the prism, when account is taken of a contribution analogous to that shown in Eq. (25) for the slab system, but we do not consider this aspect here.

3. Radiation pressure on a free liquid surface

The formation of bulges on the surface of water when a laser beam is propagated through at normal incidence in either direction was demonstrated in a paper with the above title Ref. [19]. The estimated radiation pressure force at the peak laser power of 4 kW used in the experiments is 4 µN. The effect was interpreted as caused by a longitudinal force associated with the change of photon momentum as the light passes from air into the liquid dielectric.

Gordon Ref. [2] proposed an alternative explanation in which the bulges are caused by inward radial forces associated with the Gaussian intensity profile of the laser beam within the dielectric. This is essentially a tube-of-toothpaste effect, where a transverse squeeze produces a longitudinal flow. The radial force is analogous to the edge forces F ₃ calculated in Eq. (15) for a top-hat profile and in Eq. (46) for a Gaussian profile. In conditions of normal incidence, the force is readily calculated for a Gaussian beam profile. The effective outward force on the liquid surface is found to have a magnitude Refs. [2, 7]

F = \frac{2}{c} \frac{η - 1}{η + 1} Q \approx 10^{- 9} N for η = 1.33 and Q = 1 W,

where Q is the beam power. In contrast to the Brewster-angle incidence assumed in sections 2 and 3, there is now significant surface reflection, whose effects are included. The calculated value Eq. (47) of the force thus agrees with the experimental estimate quoted above. The radial force has significant contributions from both the electric and magnetic components of the Lorentz force given by Eq. (1). Although the physical origin of the radial force is quite distinct, its qualitative effect on the liquid surface is the same as that of a hypothetical longitudinal force.

In addition, there exist two kinds of genuine longitudinal force at the surface but both have negligible effects. The first force arises from the lack of balance between the Lorentz forces in the leading and trailing parts of an optical pulse during passage through the surface. This effect is the cause of the force F ₁ given by Eq. (9), and its associated momentum p ₁ given by Eq. (25) is important in determining the displacement of a dielectric slab. However, it has been shown Ref. [7] that this contribution is entirely negligible for the conditions of the experiment Ref. [19].

The second longitudinal force, analogous to the surface forces F ₂ given by Eq. (12) and Eq. (43), arises from the electric field term in Eq. (1) as a result of the discontinuities in P_z and E_z at z=0. We calculate its contribution for a light beam incident from free space at z>0 with a field profile described by the normalized mode function

u (x, y) = \frac{\exp [\frac{- (x^{2} + y^{2})}{2 w^{2}}]}{\sqrt{π} w},

similar to the field in Eq. (30). Linear polarization is assumed, with E mainly parallel to x, but there is also a small z component associated with the finite cross section of the beam. The analogous small component parallel to ẑ′ is ignored in Eq. (30). The E fields can be calculated from the classical versions of Eq. (6.4) and Eq. (6.7) of Ref. [4], and the required field components at the liquid surface are

E_{z} = \sqrt{\frac{2 c Q}{ε_{0} ω^{2}}} \frac{2 η}{η + 1} \frac{\partial u}{\partial x} for z > 0, and E_{z} = \sqrt{\frac{2 c Q}{ε_{0} ω^{2}}} \frac{2}{η (η + 1)} \frac{\partial u}{\partial x} for z < 0 .

Thus

P_{z} = 2 ε_{0} \sqrt{\frac{2 c Q}{ε_{0} ω^{2}}} \frac{η - 1}{η} \frac{\partial u}{\partial x} ϑ (- z) and \frac{\partial E_{z}}{\partial z} = 2 \sqrt{\frac{2 c Q}{ε_{0} ω^{2}}} \frac{η - 1}{η} \frac{\partial u}{\partial x} δ (z),

analogous to Eq. (10) and Eq. (11), and the surface force is

F_{z} = \int d r P_{z} \frac{\partial E_{z}}{\partial z} = \frac{Q}{c w^{2}} {(\frac{c}{ω})}^{2} {(\frac{η - 1}{η})}^{2} .

This force is small in the paraxial region, with a typical value

F_{z} \approx 10^{- 12} N for η = 1.33, Q = 1 W, ω = 3 \times 10^{15} s^{- 1} and w = 1.5 μ m,

three orders of magnitude smaller than the force calculated in Eq. (47).

A surface force that results from interaction between the small E_z component of the incident light and a surface charge density is calculated in §3 of Ref. [12]. The calculation is based on the charge form of Lorentz force given in Eq. (2). A focused Gaussian laser beam of elliptical intensity profile is assumed and the magnitude of the force depends on the optical polarization state. With conversion to the notation used here and simplification for an incident beam of circular intensity profile and linear polarization, the force is

F_{z}^{'} = - \int d r \frac{\partial P_{z}}{\partial z} E_{z} = \frac{2 Q}{c w^{2}} {(\frac{c}{ω})}^{2} \frac{(η - 1) (η^{2} + 1)}{η^{2} (η + 1)},

with a similar magnitude to the force in Eq. (51). It is argued in Ref. [12] that the surface contribution Eq. (53) is the sole force acting on the liquid in the conditions of the relevant experiment Ref. [19] and that there is no bulk contribution similar to that used here in the derivation of Eq. (47). Thus the electric component in Eq. (2) is taken to vanish because of the zero bulk macroscopic charge density ∇.P. The magnetic component also vanishes for a current P^Ý that is π/2 out of phase with B, leading to a zero cycle average of their product for the monochromatic light assumed in the derivation. The nonzero magnetic force is, however, verified Ref. [8] for the optical pulse assumed in previous work Refs. [2,7].

In contrast to the slab and prism problems treated in sections 2 and 3, different results are thus derived for the Lorentz force on a free liquid surface in the dipole formulation Eq. (1) with incident optical pulses and in the charge formulation Eq. (2) with monochromatic incident light. The measurements in Ref. [19] were in fact made with optical pulses but the two theories differ even when the pulse contribution Eq. (47) of the former is neglected. Thus the remaining forces from Eq. (51) and Eq. (53) are not the same and their difference can be expressed as

F_{z} - F_{z}^{'} = \int d r \frac{\partial}{\partial z} (P_{z} E_{z}),

where P_z and E_z are the fields in the liquid adjacent to its surface. This relation is similar to Eq. (15) of Ref. [13], which leads to the equality of the two forms of total force in Eq. (16) when integrated over a surface that includes the whole dielectric sample. The relation Eq. (54) here applies only to a single surface of the liquid dielectric and the integral does not vanish. However, analogous to the passage of light through a dielectric slab treated in section 2, the dipole and charge forms of the Lorentz force do give identical results when the passage of light through both entrance and exit boundaries of the dielectric is included.

4. Discussion

The main aim of the above calculations is an improved understanding of the relation between calculations of radiation pressure based on the Lorentz force expressions in Eq. (1) and Eq. (2). One immediate difference between the calculations reported in Refs. [3–7] and Refs. [8–12] is the use of quantized fields in the former and classical fields in the latter. We believe that there should be no fundamental differences between the two methods of calculation for the specific results derived in these two sets of papers. Discrepancies between quantum and classical optics are restricted to effects that involve beams of the so-called nonclassical light, for example the anti-bunched or squeezed light that cannot be represented in any classical formalism. No such effects are treated here and, in contrast to our previous work, the present paper uses the classical theory for ease of comparison with alternative calculations of radiation pressure.

Our previous use of quantized fields was motivated partly with a view to possible future applications in radiation pressure effects that may occur when the incident light has a nonclassical nature and partly for the convenience of results that are automatically normalized for single-photon states. Thus the Abraham and Minkowski photon momenta refer to single quanta ħω and much of the discussion of radiation pressure phenomena is couched in terms of these momenta. However, with the relations given in Eq. (7) and Eq. (34), a classical E field is readily converted to a photon flux f in expressions for the radiation pressure forces, and removal of f from these same expressions provides the corresponding single-photon momentum transfers to a dielectric sample.

The theories developed in Refs. [2–7] differ from those in Refs. [8–12] by their bases on the dipole and charge forms of Lorentz force given in Eq. (1) and Eq. (2) respectively. These differ particularly in the form of the electric field term in the force. Nevertheless, in accordance with a general equivalence proved in Ref. [13] and Ref. [14], we have shown that the total torque on a dielectric slab and the total force on a dielectric prism during propagation of a light beam are given by the same expressions in the two theories. We emphasize that it is only the totals that are identical and their constituent parts are quite different. Thus, for example, the edge forces of type F ₃ point into the beam for the theory based on Eq. (1) but out of the beam for the theory based on Eq. (2). The final example treated here, of light incident normally on a liquid surface, treats only part of the total system and finds the difference between the results of the two theories shown in Eq. (54). This discrepancy between the predictions of the theories can be resolved only by a decision as to the appropriate choice of dipolar or charge-based model for the dielectric material. In the context of the experiment reported in Ref. [19], both of the contributions on the left of Eq. (54) are in fact negligible, while the presence of the much larger force in Eq. (47) is consistent with the use of a pulsed light source in the measurements.

The magnetic part of the Lorentz force is the same for the forms Eq. (1) and Eq. (2), but its contributions to various processes nevertheless differ for the theories developed in Refs. [2–7] and in Refs. [8–12]. The discrepancies are caused by the assumptions of incident light with the character of optical pulses in the former but as a monochromatic wave in the latter (note, however, the treatment of a normally-incident optical pulse in §12 of Ref. [8]). Thus, the calculations reported here apply to single-photon wave packets or to a flux f of such wave packets. In contrast to the zero cycle average of the magnetic term in Eq. (1) or Eq. (2) for a monochromatic wave, this does not vanish when the fields have a distribution of frequencies ω, as in the Gaussian distribution defined in Eq. (33) of Ref. [3]. There, the magnetic component of the Lorentz force is evaluated in Eq. (37), which is valid when the pulse duration l/c is much longer than the optical period at the central frequency. The spatial and temporal integrals of the force in a bulk dielectric both vanish Refs. [2, 3] but the force itself is nonzero except at the center of the pulse. The spatial integral of the force does not vanish at times when the pulse is in the process of transmission through a dielectric surface, as is illustrated in Fig. 2 of Ref. [3]. In the monochromatic limit l→∞, while the Poynting vector in Eq. (35) of Ref. [3] retains its single-photon property, the force in Eq. (37) tends to zero as expected. Many of the relevant experiments on radiation pressure use pulsed light, as in Ref. [19], and all include a spread of frequencies.

These features of the optical pulse also play important roles in the calculations of the slab displacement and conservation of momentum for the transmission process in subsection 2.3, which relies on the force F ₁ produced by the magnetic term in Eq. (1). Further, the dominant effective force Eq. (47) on a free liquid surface receives significant contributions from both the electric and magnetic terms of the Lorentz force Eq. (1). These contributions are absent from the theory for a monochromatic wave based on the form Eq. (2) of Lorentz force.

It is hoped that the results reported here will help to clarify the relation between the two forms of Lorentz force, as applied to radiation pressure problems. A full understanding of the ways in which light beams transfer momentum to dielectric systems continues to represent a considerable challenge. Calculations based on the Lorentz force appear to offer a clear way of relating theory and experiment. It is thus all the more important to agree on the appropriate form of the Lorentz force itself.

Acknowledgments

We are grateful to Masud Mansuripur for sharing his results with us in advance of publication and for illuminating discussions on the relation between his calculations and ours. The present work was supported by the UK Engineering and Physical Sciences Research Council.

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Theory of the radiation pressure on dielectric slabs, prisms and single surfaces

Abstract