ab initio study of the radiation pressure on dielectric and magnetic media Brandon A Kemp, Tomasz M. Grzegorczyk, and Jin Au Kong Research Laboratory of Electronics, Massachusetts Institute of Technology Cambridge, Massachusetts 02139, USA kemp@mit.edu Abstract: The maxwell stress tensor and the distributed lorentz force are applied to calculate forces on lossless media and are shown to be in excellent greement. From the Maxwell stress tensor, we derive analytical formulae for the forces on both a half-space and a slab under plane wave incidence It is shown that a normally incident plane wave pushes the slab in the wave propagation direction, while it pulls the half-space toward the incoming wave. Zero tangential force is derived at a boundary between two lossless media, regardless of incident angle. The distributed Lorentz force is applied to the slab in a direct way, while the half-space is dealt with by introducing a finite conductivity. In this regard, we show that the ohmic losses have to be properly accounted for, otherwise differing results are obtained. This contribution, together with a generalization of the formulation to magnetic materials, establishes the method on solid theoretical grounds. agreemen between the two methods is also demonstrated for the case of a 2-D circular dielectric par o 2005 Optical Society of America OCIS codes: (2602110) Electromagnetic Theory; (140.7010) Trapping, (2905850) Particle References and links L. M. Mansy Radiation pressure and the linear momentum of the electromagnetic field, " Opt. Express 12. 5375-5401(2004),http://www.opticsexpress.org/abstract.cfm?uri=opex-12-22-5375 2. M. Mansuripur, A. R. Zakharian, and J pressure on a 13,2064-2074(2005),http://www.opticsexpress.org/abstract.cfm?uri-opex-13-6-2064 3. M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media, "Opt. Express13.2245-2250(2005),http://www.op cfm?URI=OPEX-13-6-2245 4. A. R. Zakharian, M. Mansuripur, and Moloney,"Radiation pressure and the distribu- tion of electromagnetic force in a dielectric media, Opt. Express 13, 2321-2336 (2005). http://www.opticsexpress.org/abstract.cfm?uri-opex-13-7-2321 5. R. Loudon, "Theory of radiation pressure on dielectric surfaces, J. Mod. Opt. 49, 821-838(2002 6. R. Loudon, S. M. Barnett, and C. Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect. " Phys. Rev. A 71. 063802(2005) 7. J.A. Stratton, Electromagnetic Theory(McGraw-Hill, 1941). ISBN 0-07-062150-0. 8. J. A Kong, Electromag me Theory(EMW, 2005) ISBN 0-9668143-9-8 9. L. Tsang, J.A. Kong, and K Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000,ISBN0471-38799-7 Recently, the radiation pressure exerted by an electromagnetic field impinging on a medium was derived by the direct application of the Lorentz law [1]. This method applies the Lorentz force #8324·$15.00USD Received 29 July 2005, revised 12 October 2005; accepted 1 November 2005 (C)2005OSA 14 November 2005/ Vol 13. No 23/ OPTICS EXPRESS 9280
Ab initio study of the radiation pressure on dielectric and magnetic media Brandon A. Kemp, Tomasz M. Grzegorczyk, and Jin Au Kong Research Laboratory of Electronics, Massachusetts Institute of Technology Cambridge, Massachusetts 02139, USA bkemp@mit.edu Abstract: The Maxwell stress tensor and the distributed Lorentz force are applied to calculate forces on lossless media and are shown to be in excellent agreement. From the Maxwell stress tensor, we derive analytical formulae for the forces on both a half-space and a slab under plane wave incidence. It is shown that a normally incident plane wave pushes the slab in the wave propagation direction, while it pulls the half-space toward the incoming wave. Zero tangential force is derived at a boundary between two lossless media, regardless of incident angle. The distributed Lorentz force is applied to the slab in a direct way, while the half-space is dealt with by introducing a finite conductivity. In this regard, we show that the ohmic losses have to be properly accounted for, otherwise differing results are obtained. This contribution, together with a generalization of the formulation to magnetic materials, establishes the method on solid theoretical grounds. Agreement between the two methods is also demonstrated for the case of a 2-D circular dielectric particle. © 2005 Optical Society of America OCIS codes: (260.2110) Electromagnetic Theory; (140.7010) Trapping, (290.5850) Particle Scattering References and links 1. M. Mansuripur, ”Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375-5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375. 2. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, ”Radiation pressure on a dielectric wedge,” Opt. Express 13, 2064-2074 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2064. 3. M. Mansuripur, ”Radiation pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express 13, 2245-2250 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2245. 4. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, ”Radiation pressure and the distribution of electromagnetic force in a dielectric media,” Opt. Express 13, 2321-2336 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2321. 5. R. Loudon, “Theory of radiation pressure on dielectric surfaces,” J. Mod. Opt. 49, 821-838 (2002). 6. R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063802 (2005). 7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), ISBN 0-07-062150-0. 8. J. A. Kong, Electromagnetic Wave Theory (EMW, 2005), ISBN 0-9668143-9-8. 9. L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000), ISBN 0-471-38799-7. 1. Introduction Recently, the radiation pressure exerted by an electromagnetic field impinging on a medium was derived by the direct application of the Lorentz law [1]. This method applies the Lorentz force (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9280 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
to bound currents distributed throughout the medium and bound charges at the surface of the edium. The approach allows for the computation of force at any point inside a dielectric [1 2, 3, 4] and has been shown applicable to numerical methods, such as the finite-difference- time-domain(FDTD)[4]. A similar approach was taken to study the radiation pressure on a dielectric surface [5] and a semiconductor exhibiting the photon drag effect [6]. Comparison th the established stress tensor approach has not been done previously. The purpose of this aper is to compare this approach with the application of the Maxwell stress tensor [7, 8]to In this paper, two important generalizations of the method proposed in [1] are introduced that allow for the calculation of forces on lossless media. First, the force on a magnetic polarization vector and a bound magnetic charge density are introduced to the distributed Lorentz force to model magnetic materials. The second generalization allows for the discrimination of the force on free carriers in a slightly conducting medium, which may not contribute to the total force on the bulk medium. In addition to generalizing the Lorentz method, we also derive closed-form expressions from the Maxwell stress tensor for the force on a lossless slab At normal incidence, the force on a slab is in the wave propagation direction, while the force on a half-space pulls it toward the incoming wave. At any incident angle, the tangential force at a boundary is shown to be zero. The force on a 2D dielectric cylinder is calculated by both methods and shown to be in agreement. Furthermore, we contrast the two methods in their relative advantages and disadvantages 2. Force calculation methods magnetic waves incident on dielectric and magnetic bodies are calculated from the total complex field vectors due to both the incident waves and the scattered waves from the bodies. The Maxwell stress tensor approach to compute these forces is considered first. Second, the direct application of the Lorentz force is generalized from [1, 4 2. The maxwell stress tenso The momentum conservation theorem is derived from the maxwell equations and the lorentz force and is given by [8] aG(, t) v·T(F;1), where F andt refer to position and time, respectively, G(, t)=D(, t)xB(, r)is the momentum in [N/m]. The momentum density vector is fundamentally defined in terms of the electric flux density D(, t)and the magnetic flux density B(, 1). By integrating over a volume V enclosed by a surface S and using the divergence theorem, the total force can be written as F(r) /aG)-ds小 We are generally interested in the time average force, which F=-需R中ds|h.() since the time average of the first term on the right-hand side of equation(2)is zero. The complex Maxwell stress tensor for lossless media is T(=:(DE*+BA)I-DE-B'A #8324·$15.00USD Received 29 July 2005, revised 12 October 2005; accepted 1 November 2005 (C)2005OSA 14 November 2005/ Vol 13. No 23/OPTICS EXPRESS 9281
to bound currents distributed throughout the medium and bound charges at the surface of the medium. The approach allows for the computation of force at any point inside a dielectric [1, 2, 3, 4] and has been shown applicable to numerical methods, such as the finite-differencetime-domain (FDTD) [4]. A similar approach was taken to study the radiation pressure on a dielectric surface [5] and a semiconductor exhibiting the photon drag effect [6]. Comparison with the established stress tensor approach has not been done previously. The purpose of this paper is to compare this approach with the application of the Maxwell stress tensor [7, 8] to lossless media. In this paper, two important generalizations of the method proposed in [1] are introduced that allow for the calculation of forces on lossless media. First, the force on a magnetic polarization vector and a bound magnetic charge density are introduced to the distributed Lorentz force to model magnetic materials. The second generalization allows for the discrimination of the force on free carriers in a slightly conducting medium, which may not contribute to the total force on the bulk medium. In addition to generalizing the Lorentz method, we also derive closed-form expressions from the Maxwell stress tensor for the force on a lossless slab and a half-space. At normal incidence, the force on a slab is in the wave propagation direction, while the force on a half-space pulls it toward the incoming wave. At any incident angle, the tangential force at a boundary is shown to be zero. The force on a 2D dielectric cylinder is calculated by both methods and shown to be in agreement. Furthermore, we contrast the two methods in their relative advantages and disadvantages. 2. Force calculation methods The time-average forces of electromagnetic waves incident on dielectric and magnetic bodies are calculated from the total complex field vectors due to both the incident waves and the scattered waves from the bodies. The Maxwell stress tensor approach to compute these forces is considered first. Second, the direct application of the Lorentz force is generalized from [1, 4]. 2.1. The Maxwell stress tensor The momentum conservation theorem is derived from the Maxwell equations and the Lorentz force and is given by [8] ¯f(r¯,t) = − ∂G¯(r¯,t) ∂t −∇·T ¯¯(r¯,t), (1) where ¯r and t refer to position and time, respectively, G¯(r¯,t) = D¯(r¯,t)×B¯(r¯,t) is the momentum density vector, T ¯¯(r¯,t) is the time-domain Maxwell stress tensor, and f(r¯,t) is the force density in [N/m 3 ]. The momentum density vector is fundamentally defined in terms of the electric flux density D¯(r¯,t) and the magnetic flux density B¯(r¯,t). By integrating over a volume V enclosed by a surface S and using the divergence theorem, the total force can be written as F¯(t) = − ∂ ∂t Z V dVG¯(r¯,t)− I S dSh nˆ ·T ¯¯(r¯,t) i . (2) We are generally interested in the time average force, which is F¯ = − 1 2 Re(I S dSh nˆ ·T ¯¯(r¯) i ) , (3) since the time average of the first term on the right-hand side of equation (2) is zero. The complex Maxwell stress tensor for lossless media is T ¯¯(r¯) = 1 2 (D¯ ·E¯ ∗ +B¯ ∗ ·H¯) ¯¯I −D¯E¯ ∗ −B¯ ∗H¯, (4) (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9281 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
where I is the 3 x 3 identity matrix and()denotes the complex conjugate Egs. (3)and(4)are applied later to calculate the force on media. 2. 2. Lorentz force on a host medium The Lorentz force is applied to both bound currents due to the polarization of a dielectric and bound charges at the boundaries resulting from discontinuity of A. EoE [1]. The time-average Lorentz force in (N/m is found to be f=sRepee'+JxB*+PmA"+MXD", where()denotes a complex conjugate In Eq. (5), bound magnetic current M and bound mag- The magnetic flux B is modeled by a magnetic polarization vector Pm such that B=urHoH uoH-Pm and it follows that n=-0(μ-1) By application of Gauss's law, the bound magnetic charges are given by pm =V Pm=V HoH Invoking charge continuity leads to an expression for the bound magnetic current density, which can be used directly in equation (5 Pm=i0(-1)H. By duality a similar expression for the bound electric current density is obtained as in [1] J: where Er can be complex to account for material losses [4]. The charge distributions in equa tion(5)are found at a medium boundary by considering the discontinuity in the normal fields For example, at a boundary between two media referenced by the subscripts O and 1 Pe=n(Er-EoEo Pm=h(H1-Ho)Ho where h is a unit vector normal to the surface pointing from region O to region 1. When applyil Eg.(9), the average of the normal field vectors across the boundary should be used [1]. To get the total force on a material body from Eq.(5), the contribution from distributed current densities J and M are integrated over the volume of the medium and the effect of bound charge densities Pe and Pm are included at the medium boundaries First, the problem of TE incidence on a lossless slab is considered, as shown in Fig. 1. The forces from a TM polarized wave are directly obtained from the duality principle. The forces are calculated by using the two methods which are referred to as stress tensor and Lorentz, from section 2.1 and section 2.2, respectively The force on an isotropic slab characterized by u1=urHo and E1=ErEo is evaluated space(u2=Ho= 4T 10-H/m, E2=E0=8.85.10- F/m)due to an incident TE plane E where Ei is the incident field magnitude(see Fig. 1). The total fields in the three regions are found by application of the boundary conditions [8] #8324·$15.00USD Received 29 July 2005, revised 12 October 2005; accepted 1 November 2005 (C)2005OSA 14 November 2005/ Vol 13. No 23/OPTICS EXPRESS 9282
where ¯¯I is the 3×3 identity matrix and ( ∗ ) denotes the complex conjugate. Eqs. (3) and (4) are applied later to calculate the force on media. 2.2. Lorentz force on a host medium The Lorentz force is applied to both bound currents due to the polarization of a dielectric and bound charges at the boundaries resulting from discontinuity of ˆn · ε0E¯ [1]. The time-average Lorentz force in [N/m 3 ] is found to be ¯f = 1 2 Re{ρeE¯ ∗ +J¯×B¯ ∗ +ρmH¯ ∗ +M¯ ×D¯ ∗ }, (5) where ( ∗ ) denotes a complex conjugate. In Eq. (5), bound magnetic current M¯ and bound magnetic charge ρm have been added to the formulation of [1] to account for permeable media. The magnetic flux B¯ is modeled by a magnetic polarization vector P¯m such that B¯ = µrµ0H¯ = µ0H¯ −P¯m and it follows that P¯m = −µ0(µr −1)H¯. (6) By application of Gauss’s law, the bound magnetic charges are given by ρm = ∇·P¯m = ∇·µ0H¯. Invoking charge continuity leads to an expression for the bound magnetic current density, which can be used directly in equation (5) M¯ = −iωP¯m = iωµ0(µr −1)H¯. (7) By duality a similar expression for the bound electric current density is obtained as in [1] J¯= −iωP¯ e = −iωε0(εr −1)E¯, (8) where εr can be complex to account for material losses [4]. The charge distributions in equation (5) are found at a medium boundary by considering the discontinuity in the normal fields. For example, at a boundary between two media referenced by the subscripts 0 and 1, ρe = nˆ ·(E¯ 1 −E¯ 0)ε0 ρm = nˆ ·(H¯ 1 −H¯ 0)µ0, (9) where ˆn is a unit vector normal to the surface pointing from region 0 to region 1. When applying Eq. (9), the average of the normal field vectors across the boundary should be used [1]. To get the total force on a material body from Eq. (5), the contribution from distributed current densities J¯ and M¯ are integrated over the volume of the medium and the effect of bound charge densities ρe and ρm are included at the medium boundaries. 3. Lossless slab First, the problem of TE incidence on a lossless slab is considered, as shown in Fig. 1. The forces from a TM polarized wave are directly obtained from the duality principle. The forces are calculated by using the two methods which are referred to as stress tensor and Lorentz, from section 2.1 and section 2.2, respectively. The force on an isotropic slab characterized by µ1 = µrµ0 and ε1 = εrε0 is evaluated in free space (µ2 = µ0 = 4π · 10−7 H/m, ε2 = ε0 = 8.85 · 10−12 F/m) due to an incident TE plane wave E¯ i = yEˆ ie ik0z z e ikxx , (10) where Ei is the incident field magnitude (see Fig. 1). The total fields in the three regions are found by application of the boundary conditions [8]. (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9282 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
Region 1 1,E1 12,E2 E z=0 Fig. 1. a plane wave is incident onto a slab of thickness d and incident angle eo. The integration path for the application of the Maxwell stress tensor to calculate the force on a slab is shown by the dotted lines. The path is shrunk so that 8z-0. The integration is performed along the surface on both sides of a boundarie The Maxwell stress tensor is applied through Eq. (3)by selecting the integration path shown in Fig. 1. This integration path is chosen such that the fields are evaluated at the boundary as 8z-0. The force per unit area on the slab of thickness d is, therefore, given by F )-27(z=0)+27(z )} where T(z=zo)is the Maxwell stress tensor of equation(4)evaluated at the point z=zo.The contributions from the fields inside the slab are restricted to the terms e,=0)=221(2=0+)P+2(m(=0P-1=0) +[-1H2(2=0+)H(z=0+),(12a) 21E(=d)+(H(=d)P-(= Hi(z=d-)H(z=d-) By substitution of the fields in the slab, it can be shown that and the radiation pressure on the slab reduces to F=Re{2.7(=0-)-27(z=d+)} Therefore, the force on a lossless slab can be computed solely from the knowledge of the fields outside the slab. This is to be expected and can be generalized to media of arbitrary geometry since the divergence of the stress tensor applied to continuous, lossless media is #8324·$15.00USD Received 29 July 2005, revised 12 October 2005; accepted 1 November 2005 (C)2005OSA 14 November 2005/ Vol 13. No 23/OPTICS EXPRESS 9283
Region 0 Region 1 Region 2 0 0 P ,H 1 1 P ,H 2 2 P ,H y ˆ z ˆ x ˆ z 0 z d Ei H i ki T 0 nˆ nˆ Gz Gz nˆ nˆ Fig. 1. A plane wave is incident onto a slab of thickness d and incident angle θ0. The integration path for the application of the Maxwell stress tensor to calculate the force on a slab is shown by the dotted lines. The path is shrunk so that δz → 0. The integration is performed along the surface on both sides of a boundaries. The Maxwell stress tensor is applied through Eq. (3) by selecting the integration path shown in Fig. 1. This integration path is chosen such that the fields are evaluated at the boundary as δz → 0. The force per unit area on the slab of thickness d is, therefore, given by F¯ = 1 2 Re{zˆ·T ¯¯(z = 0 −)−zˆ·T ¯¯(z = 0 +) +zˆ·T ¯¯(z = d −)−zˆ·T ¯¯(z = d +)}, (11) where T ¯¯(z = z0) is the Maxwell stress tensor of equation (4) evaluated at the point z = z0. The contributions from the fields inside the slab are restricted to the terms zˆ·T ¯¯(z = 0 +) = zˆ h ε1 2 |Ey(z = 0 +)| 2 + µ1 2 |Hx(z = 0 +)| 2 −|Hz(z = 0 +)| 2 i +xˆ −µ1Hz(z = 0 +)H ∗ x (z = 0 +) , (12a) zˆ·T ¯¯(z = d −) = zˆ h ε1 2 |Ey(z = d −)| 2 + µ1 2 |Hx(z = d −)| 2 −|Hz(z = d −)| 2 i +xˆ −µ1Hz(z = d −)H ∗ x (z = d −) . (12b) By substitution of the fields in the slab, it can be shown that zˆ·T ¯¯(z = 0 +) = zˆ·T ¯¯(z = d −), (13) and the radiation pressure on the slab reduces to F¯ = 1 2 Re{zˆ·T ¯¯(z = 0 −)−zˆ·T ¯¯(z = d +)}. (14) Therefore, the force on a lossless slab can be computed solely from the knowledge of the fields outside the slab. This is to be expected and can be generalized to media of arbitrary geometry since the divergence of the stress tensor applied to continuous, lossless media is (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9283 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
zero. Simplification of Eq. (14)with knowledge of the reflected and transmitted fields gives the closed-form force per unit area on the slab as F E2 cos2 (15) where bo is the incident angle and rslab and Tlab are the slab reflection and transmission co- efficients, respectively [8]. This analytic expression shows that the i-component of the force is zero(Fx=0), regardless of incident angle. At normal incidence, equation(15)can be written simply as the summation of force components due to momentum conservation, F=20E2, =:E (16b) Fr=-i-EITslabl2 where Fi, Fr, and F are the forces due to the incident, reflected, and transmitted wave momen- ums, respectively Equation(5)is used to calculate the Lorentz force on the slab, and the results are compared with Eq (15). Figure 2 shows excellent agreement between the two methods applied to compute the force on a lossless dielectric slab(a=1, E,=4). The maxima and minima in the force are due to the periodic dependence of the reflection coefficient Rslab and the transmission coefficient Tslab on the slab thickness Minimum force is observed for slab thicknesses equal to multiples of half wavelength n 2 and maximum force is observed for slab thicknesses equal to odd multiples of quarter wavelength(2n+1)4, where Al is the wavelength of the electromagnetic wave inside the slab and n E[0, 1, 2,.. For comparison, note that for d=21/4, f=3.182 pN/m is calculated by the Lorentz method and f=3. 184 pN/m is calculated by the application of the Maxwell stress tensor, which is in reasonable agreement with f=3.188 pN/m2l previously Fig. 2. Force density from a normal incident wave onto a lossless slab as a function of slab thickness d The free space wavelength is 10=640nm and Er=4, r=1, E=1.Shown are the forces calculated from the distributed Lorentz force(circles) and the maxwell stress tensor (line). The background medium(region 0 and region 2)is free space. #8324·$15.00USD Received 29 July 2005, revised 12 October 2005; accepted 1 November 2005 (C)2005OSA 14 November 2005/ Vol 13. No 23/OPTICS EXPRESS 9284
zero. Simplification of Eq. (14) with knowledge of the reflected and transmitted fields gives the closed-form force per unit area on the slab as F¯ = zˆ ε0 2 E 2 i cos2 θ0 1+|Rslab| 2 −|Tslab| 2 , (15) where θ0 is the incident angle and Rslab and Tslab are the slab reflection and transmission coefficients, respectively [8]. This analytic expression shows that the ˆx-component of the force is zero (Fx = 0), regardless of incident angle. At normal incidence, equation (15) can be written simply as the summation of force components due to momentum conservation, F¯ i = zˆ ε0 2 E 2 i , (16a) F¯ r = zˆ ε0 2 E 2 i |Rslab| 2 , (16b) F¯ t = −zˆ ε0 2 E 2 i |Tslab| 2 , (16c) where F¯ i , F¯ r , and F¯ t are the forces due to the incident, reflected, and transmitted wave momentums, respectively. Equation (5) is used to calculate the Lorentz force on the slab, and the results are compared with Eq. (15). Figure 2 shows excellent agreement between the two methods applied to compute the force on a lossless dielectric slab (µr = 1, εr = 4). The maxima and minima in the force are due to the periodic dependence of the reflection coefficient Rslab and the transmission coefficient Tslab on the slab thickness. Minimum force is observed for slab thicknesses equal to multiples of half wavelength n λ1 2 and maximum force is observed for slab thicknesses equal to odd multiples of quarter wavelength (2n + 1) λ1 4 , where λ1 is the wavelength of the electromagnetic wave inside the slab and n ∈ [0,1,2,...]. For comparison, note that for d = λ1/4, f = 3.182 [pN/m 2 ] is calculated by the Lorentz method and f = 3.184 [pN/m 2 ] is calculated by the application of the Maxwell stress tensor, which is in reasonable agreement with f = 3.188 [pN/m 2 ] previously reported [4]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 2.5 3 3.5 d/λ1 F z (pN/m 2 ) Lorentz Stress Tensor Fig. 2. Force density from a normal incident wave onto a lossless slab as a function of slab thickness d. The free space wavelength is λ0 = 640nm and εr = 4, µr = 1, Ei = 1. Shown are the forces calculated from the distributed Lorentz force (circles) and the Maxwell stress tensor (line). The background medium (region 0 and region 2) is free space. (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9284 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005