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
Figure 3 shows the force on a permeable slab(u,=4, E=1)as a function of incident angle Bo, measured from the surface normal. This example requires the calculation of both magnetic currents from Eq (7)and magnetic surface charge from Eq (9), which are used to compute the Lorentz force. As seen in Fig 3, there is excellent agreement between the two force calculation methods at all incident angles The brewster angle for total transmission of a te incident wave is given by [8] and is calculated for this example to be Bb=63. 4. This is evident in Fig 3 by the zero force at this particular angle Stress Tensor Brewster Angle Fig 3. Force density from an oblique incident wave on a quarter-wave slab(d= n1/4 80nm)as a function of incident angle e The free space wavelength is 2o= 640nm and Er=l, ur=4, Ei= l 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. The force is shown as a function of dielectric constant e. for a lossless dielectric slab in Fig. 4. Again, there is excellent agreement between the two force calculation methods. It is seen that the force goes to zero as expected when the slab is impedance matched to free space Er=l. For all positive permittivities, including the region O<Er <l, the force is in the positive z-direction, indicating that the force is pushing the slab 4. Semi-infinite half-space Calculation of the Lorentz force on a half-space is more involved than what might be expected For a lossless half-space, the fields propagate inside the medium without attenuation. This poses problem in finding the radiation pressure by integrating over the Lorentz force from z=0 to z-o0. Previously, this issue has been sidestepped by introducing a small amount of loss in the medium, applying the distributed Lorentz force, and allowing the losses to approach zero after tegration [1, 5]. The problem with this approach is that not all of the force on free charges can be attributed to force on the bulk medium. Some of the wave energy may be lost, such as ohmic losses in a conducting medium which must be considered when calculating the total force on a material body. This problem will be addressed subsequently First, we derive the exact solution to the half-space problem of an incident te plane wave by applying the Maxwell stress tensor to the boundary as described in section 2. 1. The path of #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 9285
Figure 3 shows the force on a permeable slab (µr = 4, εr = 1) as a function of incident angle θ0, measured from the surface normal. This example requires the calculation of both magnetic currents from Eq. (7) and magnetic surface charge from Eq. (9), which are used to compute the Lorentz force. As seen in Fig. 3, there is excellent agreement between the two force calculation methods at all incident angles. The Brewster angle for total transmission of a TE incident wave is given by [8] θb = tan−1 rµ1 µ0 , (17) and is calculated for this example to be θb = 63.4 ◦ . This is evident in Fig. 3 by the zero force at this particular angle. 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 θ0 F z (pN/m 2 ) Lorentz Stress Tensor Brewster Angle Fig. 3. Force density from an oblique incident wave on a quarter-wave slab (d = λ1/4 = 80nm) as a function of incident angle θ0. The free space wavelength is λ0 = 640nm and εr = 1, µr = 4, 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. The force is shown as a function of dielectric constant εr for a lossless dielectric slab in Fig. 4. Again, there is excellent agreement between the two force calculation methods. It is seen that the force goes to zero as expected when the slab is impedance matched to free space εr = 1. For all positive permittivities, including the region 0 ≤ εr ≤ 1, the force is in the positive zˆ-direction, indicating that the force is pushing the slab. 4. Semi-infinite half-space Calculation of the Lorentz force on a half-space is more involved than what might be expected. For a lossless half-space, the fields propagate inside the medium without attenuation. This poses a problem in finding the radiation pressure by integrating over the Lorentz force from z = 0 to z → ∞. Previously, this issue has been sidestepped by introducing a small amount of loss in the medium, applying the distributed Lorentz force, and allowing the losses to approach zero after integration [1, 5]. The problem with this approach is that not all of the force on free charges can be attributed to force on the bulk medium. Some of the wave energy may be lost, such as ohmic losses in a conducting medium, which must be considered when calculating the total force on a material body. This problem will be addressed subsequently. First, we derive the exact solution to the half-space problem of an incident TE plane wave by applying the Maxwell stress tensor to the boundary as described in section 2.1. The path of (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9285 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
Fig. 4. Force density from a normal incident wave on a lossless slab(d= 80nm)as a function of the relative permittivity Er. The free space wavelength is no= 640nm and u, 1, Ei= 1. Shown are the forces calculated from the distributed Lorentz force(circles)and the Maxwell stress tensor (line ). The background media (region 0 and region 2)is fre integration for the Maxwell stress tensor is the same as that shown in Fig. I except that there is only one boundary at z=0. Again we allow the integration to just enclose the boundary such that Sz-0. The force per unit area on the half space medium is given by F=Re{2.1(x=0-)-2·7(=0+)} The contributions on the two sides of the interface are ,个(=0)=2[2E1(=0)P+(H1(=0)-(=0) +[-0H(2=0)H(z=0-)],(19a) 1=0)=2121(2=0+)P+((=0)P-H(=0) +[-1H(x=0)H(z=0+],(19b) and the force tangential to the boundary is seen to be F1=Re-H(z=0-)H(=0-)+1H2(z=0+)H(z=0+) Upon substitution of the fields on both sides of the boundary, the tangential force is simplified Fr=ERe [(1+Rhs)(1-Rhs)-Po1l Thsl- where Rhs and Ths are the half-space reflection and transmission coefficients, respectively and Pol is given by [8] HokI unko #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 9286
0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ε r F z (pN/m 2 ) Lorentz Stress Tensor Fig. 4. Force density from a normal incident wave on a lossless slab (d = 80nm) as a function of the relative permittivity εr. The free space wavelength is λ0 = 640nm and µr = 1, Ei = 1. Shown are the forces calculated from the distributed Lorentz force (circles) and the Maxwell stress tensor (line). The background media (region 0 and region 2) is free space. integration for the Maxwell stress tensor is the same as that shown in Fig. 1 except that there is only one boundary at z = 0. Again we allow the integration to just enclose the boundary such that δz → 0. The force per unit area on the half space medium is given by F¯ = 1 2 Re{zˆ·T ¯¯(z = 0 −)−zˆ·T ¯¯(z = 0 +)}. (18) The contributions on the two sides of the interface are zˆ·T ¯¯(z = 0 −) = zˆ h ε0 2 |Ey(z = 0 −)| 2 + µ0 2 |Hx(z = 0 −)| 2 −|Hz(z = 0 −)| 2 i +xˆ −µ0Hz(z = 0 −)H ∗ x (z = 0 −) , (19a) 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 +) , (19b) and the force tangential to the boundary is seen to be Fx = 1 2 Re − µ0Hz(z = 0 −)H ∗ x (z = 0 −) + µ1Hz(z = 0 +)H ∗ x (z = 0 +) . (20) Upon substitution of the fields on both sides of the boundary, the tangential force is simplified to Fx = 1 2 E 2 i kx ω2 Rek0z µ0 ∗ (1+Rhs) (1−Rhs) ∗ − p ∗ 01|Ths| 2 (21) where Rhs and Ths are the half-space reflection and transmission coefficients, respectively and p01 is given by [8] p01 = µ0k1z µ1k0z . (22) (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9286 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
The simplification to Eq. (21) is a direct result of phase matching, which requires kx to be ontinuous across the boundary. Applying the formulas Ths 1+ Rhs and Por Ths =l-Rp obtained from the boundary conditions, it immediately follows that p1|Th2=(1+Rn)(1-Rn) (23) so that the tangential force is zero(Fr=0). This result is a direct consequence of the boundary conditions and phase matching at a single interface, and it can be generalized to multiple inter faces, including the slab of section 3, the details of which were omitted for brevity. This differs from previous reports of derived nonzero tangential force at a planar boundary [1, 4 The radiation pressure on a lossless half-space medium is found from Eq. (18)to be F=AE? (cos eo(1+Ras)-Er cos, T231 (24) where Er is the relative permittivity of the half-space and B is the transmitted angle found from Snell,s law. Note that this equation also accounts for variations in relative permeability ur, which appears in the reflection and transmission coefficients. Again we can see the contribution from the incident. reflected. and transmitted wave momentums on the force at the medium Equation(24)is applied to evaluate the radiation pressure on the half-space medium due to a normal incident plane wave. By taking the limit Er -oo, it is simple to show that the radiation cache lim F=-ZE0E (25) This limit is seen in Fig. 5 where E;= 1V/ m such that F2 --Eo(IV/ m)=-885 pN/m for very large Er. Thus, the force of a normally incident plane wave on a lossless dielectric half-space is pulling toward the incoming wave, while in the slab case the plane wave pushes the medium in the wave propagating direction. Likewise, the analytical limit of E,+0 gives which is also seen in Fig. 5 Equation(24)differs from the TE incident results F:= 5E?(1+R2s)cos20o and Fr 2E(1-R2s)sin eo cos eo previously reported from the distribution of Lorentz force in a loss- less media [1]. The previously reported results produce different values from Eq. (24)for three ases in particular. First, for an impedance matched interface such as a fictive interface be- tween two free space media, Rhs=0, which would yield a nonzero value of F:=25E. Since uch a fictive boundary could be placed anywhere, it would indicate that forces in free space are present everywhere. Second, a nonzero tangential force Fx+0 at oblique incidence [1, 4 disagrees with Eq. (24). Third, the predicted normal force F is always positive for a lossless dielectric, which contradicts the known theory that the force of a normally incident plane wave on a lossless half-space is pulling tow incident wave[8. The differing results obtained from the distribution of lorentz force and the maxwell stress tensor do not imply that either result is incorrect. Instead, we interpret the force on a sem infinite half-space obtained from the method of [l] to be the total Lorentz force on all bound and free charges and currents with the assumption that the fields attenuate to zero as z- oo due to some finite loss, which holds once the force density is integrated, even if the losses approach cero. To illustrate this point, the force on a lossless half-space due to a normally incident wave can be computed from the distributed Lorentz force by considering the ohmic loss due to the #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 928
The simplification to Eq. (21) is a direct result of phase matching, which requires kx to be continuous across the boundary. Applying the formulas Ths = 1 + Rhs and p01Ths = 1 − Rhs obtained from the boundary conditions, it immediately follows that p ∗ 01|Ths| 2 = (1+Rhs)(1−Rhs) ∗ (23) so that the tangential force is zero (Fx = 0). This result is a direct consequence of the boundary conditions and phase matching at a single interface, and it can be generalized to multiple interfaces, including the slab of section 3, the details of which were omitted for brevity. This differs from previous reports of derived nonzero tangential force at a planar boundary [1, 4]. The radiation pressure on a lossless half-space medium is found from Eq. (18) to be F¯ = zˆ ε0 2 E 2 i cos2 θ0 1+R 2 hs −εr cos2 θ1T 2 hs , (24) where εr is the relative permittivity of the half-space and θ1 is the transmitted angle found from Snell’s law. Note that this equation also accounts for variations in relative permeability µr , which appears in the reflection and transmission coefficients. Again we can see the contribution from the incident, reflected, and transmitted wave momentums on the force at the medium interface. Equation (24) is applied to evaluate the radiation pressure on the half-space medium due to a normal incident plane wave. By taking the limit εr → ∞, it is simple to show that the radiation pressure approaches lim εr→∞ F¯ = −zˆε0E 2 i . (25) This limit is seen in Fig. 5 where Ei = 1V/m such that Fz → −ε0(1V/m) = −8.85 [pN/m 2 ] for very large εr . Thus, the force of a normally incident plane wave on a lossless dielectric half-space is pulling toward the incoming wave, while in the slab case the plane wave pushes the medium in the wave propagating direction. Likewise, the analytical limit of εr → 0 gives lim εr→0 F¯ = +zˆε0E 2 i , (26) which is also seen in Fig. 5. Equation (24) differs from the TE incident results Fz = ε0 2 E 2 i 1+R 2 hs cos2 θ0 and Fx = ε0 2 E 2 i 1−R 2 hs sinθ0 cosθ0 previously reported from the distribution of Lorentz force in a lossless media [1]. The previously reported results produce different values from Eq. (24) for three cases in particular. First, for an impedance matched interface such as a fictive interface between two free space media, Rhs = 0, which would yield a nonzero value of Fz = zˆ ε0 2 E 2 i . Since such a fictive boundary could be placed anywhere, it would indicate that forces in free space are present everywhere. Second, a nonzero tangential force Fx 6= 0 at oblique incidence [1, 4] disagrees with Eq. (24). Third, the predicted normal force Fz is always positive for a lossless dielectric, which contradicts the known theory that the force of a normally incident plane wave on a lossless half-space is pulling toward the incident wave [8]. The differing results obtained from the distribution of Lorentz force and the Maxwell stress tensor do not imply that either result is incorrect. Instead, we interpret the force on a semiinfinite half-space obtained from the method of [1] to be the total Lorentz force on all bound and free charges and currents with the assumption that the fields attenuate to zero as z → ∞ due to some finite loss, which holds once the force density is integrated, even if the losses approach zero. To illustrate this point, the force on a lossless half-space due to a normally incident wave can be computed from the distributed Lorentz force by considering the ohmic loss due to the (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9287 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
Fig. 5. Force density from a normal incident wave on a lossless half-space medium as calculated from the maxwell stress tensor. The 2-directed force is shown as a function of the relative permittivity Er. The free space wavelength is 10= 640nm and ur= l, Ei=1 The incident media(region O)is free space. The dotted lines denote +EoE=8.85 pN/m conductivity o in a slightly lossy medium. Some of the energy transferred from the wave to the conduction current Jc is dissipated as ohmic loss. The time average power dissipated is [ 8] 具={E( (27) which must be subtracted from the Lorentz force calculated on the conduction current/. There- fore, the total radiation pressure on a dielectric half-space due to a normally incident plane wave F= lim =Re JxB-2-UJC.E* where J is the current from Eq. 8),Je=oE is the conduction current, and ve is the energy elocity. After integration, the i-directed force reduces to F2=E2(+R2)-lim3一E22 and attenuation constant ky can be approximated by (8)4k1)).As o -( where kI is the imaginary part of the wavenumber(kI=Im k1). As o-0, the energy velocity Thus, the total force on the dielectric half-space reduces to 0E(1+R)-5E27 (31) where the first term is a result of the calculated Lorentz force [1 and the second term is de- rived from the ohmic losses due to the finite conductivity. This expression is in agreement with Eq(24)at normal incidence 60=0, which is derived from the Maxwell stress tensor without approximation. #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 9288
−3 −2 −1 0 1 2 3 −10 −8 −6 −4 −2 0 2 4 6 8 10 log10(ε r ) F z [pN/m 2 ] Fig. 5. Force density from a normal incident wave on a lossless half-space medium as calculated from the Maxwell stress tensor. The ˆz-directed force is shown as a function of the relative permittivity εr. The free space wavelength is λ0 = 640nm and µr = 1, Ei = 1. The incident media (region 0) is free space. The dotted lines denote ±ε0E 2 i = 8.85 pN/m 2 conductivity σ in a slightly lossy medium. Some of the energy transferred from the wave to the conduction current J¯ c is dissipated as ohmic loss. The time average power dissipated is [8] Pc = 1 2 ReZ V dVJ¯ c(r¯)·E¯ ∗ (r¯) , (27) which must be subtracted from the Lorentz force calculated on the conduction current J¯ c. Therefore, the total radiation pressure on a dielectric half-space due to a normally incident plane wave is F¯ = lim σ→0 1 2 ReZ ∞ 0 dz J¯×B¯ ∗ −zˆ 1 ve (J¯ c ·E¯ ∗ ) , (28) where J¯ is the current from Eq. (8), J¯ c = σE¯ is the conduction current, and ve is the energy velocity. After integration, the ˆz-directed force reduces to Fz = ε0 2 E 2 i (1+R 2 hs)− lim σ→0 1 2 σ ve2kI E 2 i T 2 hs, (29) where kI is the imaginary part of the wavenumber(kI = Im{k1}). As σ → 0, the energy velocity and attenuation constant kI can be approximated by [8] ve ≈ 1 √ µ1ε1 kI ≈ σ 2 rµ1 ε1 . (30) Thus, the total force on the dielectric half-space reduces to Fz = ε0 2 E 2 i (1+R 2 hs)− ε1 2 E 2 i T 2 hs (31) where the first term is a result of the calculated Lorentz force [1] and the second term is derived from the ohmic losses due to the finite conductivity. This expression is in agreement with Eq. (24) at normal incidence θ0 = 0, which is derived from the Maxwell stress tensor without approximation. (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9288 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
5. Cylindrical particle In this section, the force on a dielectric particle due to the superposition of three incident plane waves is calculated by the two methods of section 2. The fields are calculated by the mie theory as in [9] and verified through comparison with results from the commercial package CST Microwave Studio ( The force distribution N/m] inside the particle is then calculated using the Lorentz method. The distributed Lorentz force is numerically integrated throughout the interior surface of the particle to obtain the total force [N/m]. This value is then compared with the force obtained from the Maxwell stress tensor by integrating the tensor around a circle ng the 2D 200-100 Fig. 6. The incident electric field magnitude [V/ m] is due to three plane waves of free space wavelength 7o=532 nm propagating at angles T/2, 7/6, 117/6]. The overlayed (ro, o)=(0, 100)[nm] embedded in water(Eb= 1.69E0 three plane waves of free space wavelength no=532 nm are incident in the(xy) plane with ngles, (t/2, 77/6,11 /6)[rad]. The particle is a polystyrene cylinder(Ep=2.56Eo)of radius a=0. 3no with center position(o, yo)=(0, 100)[nm], and the background medium is water (Eb=1.69E0. The magnitude of the total incident electric field is shown in Fig. 6 with the posi tion of the dielectric particle overlayed. It is seen that the particle is of the same size as a typical radient trap in the Rayleigh regime(seen from the periodicity in the interference pattern), but poses no problem for the method used here. Figure 7 shows the total field magnitudes resulting from the incident fields and the scattered fields. The symmetry of the fields with respect to x produces zero net force in the x-direction, as we shall see The distributed Lorentz force is found from Eq (5)to be f=5Ref-ioP x HoH]. (32) where the electric polarization vector is Pe=2(Ep -Eb)Ez. The distributed Lorentz force com- fy)inside the particle are shown in Fig. 8. It is seen that fr is symmetric abou x=O, as expected. However, the distribution of fy is more complex, and it is not immediately apparent from the distributed force if the total force Fy is positive or negative The total force is obtained from the Lorentz force of Fig. 8 by simple numerical integration and found to be F= y2. 11. 10-8N/m]. Subsequently, the Maxwell stress tensor of Eq. 4 was integrated along a circular path of radius 2a centered at(xo, yo), resulting in a force of y2.12.10-18IN/ m. Note that the exact path of integration for the Maxwell stress tensor is not important as long as it completely encloses the cylinder. The excellent agreement between the two force calculation methods for this example further establishes the method of [1] on #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 9289
5. Cylindrical particle In this section, the force on a dielectric particle due to the superposition of three incident plane waves is calculated by the two methods of section 2. The fields are calculated by the Mie theory as in [9] and verified through comparison with results from the commercial package CST Microwave Studio ®. The force distribution [N/m 3 ] inside the particle is then calculated using the Lorentz method. The distributed Lorentz force is numerically integrated throughout the interior surface of the particle to obtain the total force [N/m]. This value is then compared with the force obtained from the Maxwell stress tensor by integrating the tensor around a circle completely enclosing the 2D particle. Fig. 6. The incident electric field magnitude [V/m] is due to three plane waves of free space wavelength λ0 = 532 nm propagating at angles {π/2,7π/6,11π/6}. The overlayed particle is a 2D polystyrene cylinder (εp = 2.56ε0) of radius a = 0.3λ0 with center position (x0,y0) = (0,100) [nm] embedded in water (εb = 1.69ε0). Three plane waves of free space wavelength λ0 = 532 nm are incident in the (xy) plane with angles, {π/2,7π/6,11π/6} [rad]. The particle is a polystyrene cylinder (εp = 2.56ε0) of radius a = 0.3λ0 with center position (x0,y0) = (0,100) [nm] , and the background medium is water (εb = 1.69ε0). The magnitude of the total incident electric field is shown in Fig. 6 with the position of the dielectric particle overlayed. It is seen that the particle is of the same size as a typical gradient trap in the Rayleigh regime (seen from the periodicity in the interference pattern), but poses no problem for the method used here. Figure 7 shows the total field magnitudes resulting from the incident fields and the scattered fields. The symmetry of the fields with respect to x produces zero net force in the x-direction, as we shall see. The distributed Lorentz force is found from Eq. (5) to be ¯f = 1 2 Re −iωP¯ e × µ0H ∗ , (32) where the electric polarization vector is P¯ e = zˆ(εp −εb)Ez . The distributed Lorentz force components (fx and fy) inside the particle are shown in Fig. 8. It is seen that fx is symmetric about x = 0, as expected. However, the distribution of fy is more complex, and it is not immediately apparent from the distributed force if the total force Fy is positive or negative. The total force is obtained from the Lorentz force of Fig. 8 by simple numerical integration and found to be F¯ = yˆ2.11 · 10−18 [N/m]. Subsequently, the Maxwell stress tensor of Eq. 4 was integrated along a circular path of radius 2a centered at (x0,y0), resulting in a force of F¯ = yˆ2.12·10−18 [N/m]. Note that the exact path of integration for the Maxwell stress tensor is not important as long as it completely encloses the cylinder. The excellent agreement between the two force calculation methods for this example further establishes the method of [1] on (C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9289 #8324 - $15.00 USD Received 29 July 2005; revised 12 October 2005; accepted 1 November 2005
