PRL97,133902(2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 300 diameter 0.5 um. The choice of an infinite dielectric cyl- inder incident by a TE wave allows for a complete descrip- E tion of the Lorentz force by the distribution inside the particle since the force on bound charges at the boundary is zero. The total electric field intensity is shown in Fig. 2(a) for the lossless dielectric particle. The total force s the force on bound currents F= Fb=84.02X 108N/m, which is computed by integrating the force density (1)over the area of the particle or, equivalently, by FIG. 2 Lorentz force density on bound currents (arrows)over- integrating the stress tensor of(6)along the circular path lain on electric field intensity E2I[(V/m)]resulting from ai shown in Fig. 1(a). The integration is performed by simple polarized plane wave of unit amplitude incident from free space numerical integration as in Ref [11]. Although equivalent with wavelength Ao =1064 nm onto a dielectric cylinder 1.25x 10-6[N/m )(b)The lossy cylinder, described by e= the particle is pulled toward the resulting high intensity ents.(max(f6=3.00×10-9N/m3) being pushed by the transfer of wave momentum. The total electric field intensity and force density on bound currents for a lossy particle is shown in Fig. 2(b). The resulting 2F=E2(1+|R2cos2 (9a) force density Fb=-x205 X 10-8N/m indicates that CEC-IRI)cose, sin;.(9b) which is offset by a positive momentum transfer to free currents represented by F=A5.80 X 108N/m, found which is in agreement with the Lorentz force density equivalently by applying an integration path to the stress integrated over the region z E [O, oo)as in Refs. [7,9]. tensor shown in Fig. 1(b). The total pressure on the particle The result(9b) was originally demonstrated in 1905 by is F=x3.75 X 10-18 N/m, which is less than the total Poynting [20], who observed a tangential force given by force on the transparent particle E sing, cose, for a nearly perfect absorbing medium a physically realistic situation of an electromagnetic (R=0 and Er + O)and zero tangential force for the wave impinging on a spherical particle is studied using reflection from a mirror [RI= 1. Thus, the radiation pres- Mie theory. For ER/Eo=2, Fig 3(a)shows that a maxi- sure is normal to the surface of a perfect reflector and mum optical momentum transfer occurs for a value of e given by the force on free currents at the surface F=F= near maximum absorption(i.e the penetration depth is on ZEolE 2cos20 If the dielectric constant of the background the order of the particle diameter). In contrast, a particle medium is increased by Vn, then the force on the reflector with large value for Er can exhibit reduced momentum increases EolEFcos0;=4;, which has been transfer due to significant wave attenuation in the sphere as observed for mirrors submerged in dielectric liquids shown in Fig 3(b). A further decrease in F for the high [21,22 contrast sphere is observed as e, approaches the limit of a The total fields due to plane-wave incidence on a 2D perfect reflector. The later point is made by comparing the particle are found from Mie theory applied to an absorbing adiabatic momentum transfers to the transparent dielectric infinite cylinder as in Refs. [23, 24]. We consider two sphere and to the reflecting sphere of equal size. However, separate problems of Einc melkor incident from free space the combined effect of Fb and Fc is required to explain the onto a lossless cylinder and onto a lossy cylinder, each of radiation pressure increase of Fig. 3(a) or decrease of f(b log1o(EI/Eo) FIG 3. Forces on a 2 um diameter sphere due to a plane wave of unit amplitude. The wave is incident from free space with 1064 nto a nonmagnetic sphere with(a)∈=2∈o+ iEr and(b)∈=16∈0+i∈p 133902-3z^  F


0 2 E2 i 1 jRj 2cos2i; (9a) x^  F


0 2 E2 i 1  jRj 2 cosi sini; (9b) which is in agreement with the Lorentz force density integrated over the region z 2 0; 1 as in Refs. [7,9]. The result (9b) was originally demonstrated in 1905 by Poynting [20], who observed a tangential force given by
0 2 E2 i sini cosi for a nearly perfect absorbing medium (R  0 and
I
0) and zero tangential force for the reflection from a mirror jRj
1. Thus, the radiation pres￾sure is normal to the surface of a perfect reflector and is given by the force on free currents at the surface F

F
c
z
^ 0jEij 2cos2i. If the dielectric constant of the background medium is increased by  n p , then the force on the reflector increases to  n p
0jEij 2cos2i
n c Scos2i, which has been observed for mirrors submerged in dielectric liquids [21,22]. The total fields due to plane-wave incidence on a 2D particle are found from Mie theory applied to an absorbing infinite cylinder as in Refs. [23,24]. We consider two separate problems of E
inc
ze^ ik0x incident from free space onto a lossless cylinder and onto a lossy cylinder, each of diameter 0:5 m. The choice of an infinite dielectric cyl￾inder incident by a TE wave allows for a complete descrip￾tion of the Lorentz force by the distribution inside the particle since the force on bound charges at the boundary is zero. The total electric field intensity is shown in Fig. 2(a) for the lossless dielectric particle. The total force is the force on bound currents F

F
b
x^4:02 1018 N=m, which is computed by integrating the force density (1) over the area of the particle or, equivalently, by integrating the stress tensor of (6) along the circular path shown in Fig. 1(a). The integration is performed by simple numerical integration as in Ref. [11]. Although equivalent in results, the former approach provides the viewpoint that the particle is pulled toward the resulting high intensity focus, while the latter gives the usual intuition of a particle being pushed by the transfer of wave momentum. The total electric field intensity and force density on bound currents for a lossy particle is shown in Fig. 2(b). The resulting force density F
b
x^2:05 1018 N=m indicates that the bound currents are pulled toward the incident wave, which is offset by a positive momentum transfer to free currents represented by F
c
x^5:80 1018 N=m, found equivalently by applying an integration path to the stress tensor shown in Fig. 1(b). The total pressure on the particle is F

x^3:75 1018 N=m, which is less than the total force on the transparent particle. A physically realistic situation of an electromagnetic wave impinging on a spherical particle is studied using Mie theory. For
R=
0
2, Fig. 3(a) shows that a maxi￾mum optical momentum transfer occurs for a value of
I near maximum absorption (i.e. the penetration depth is on the order of the particle diameter). In contrast, a particle with large value for
R can exhibit reduced momentum transfer due to significant wave attenuation in the sphere as shown in Fig. 3(b). A further decrease in F for the high contrast sphere is observed as
I approaches the limit of a perfect reflector. The later point is made by comparing the adiabatic momentum transfers to the transparent dielectric sphere and to the reflecting sphere of equal size. However, the combined effect of Fb and Fc is required to explain the radiation pressure increase of Fig. 3(a) or decrease of FIG. 2. Lorentz force density on bound currents (arrows) over￾lain on electric field intensity jEzj 2 V=m 2 resulting from a z^ polarized plane wave of unit amplitude incident from free space with wavelength 0
1064 nm onto a dielectric cylinder. (a) The lossless cylinder is defined by

16
0. ( maxjfbj
1:25 106 N=m3 ) (b) The lossy cylinder, described by

16 10i
0, contains an additional force density on free cur￾rents. maxjfbj
3:00 109 N=m3). −3 −2 −1 0 1 2 3 −1 0 1 2 3 x 10−23 log10( I 0) F [N] F Fc Fb (a) −3 −2 −1 0 1 2 3 −1 0 1 2 3 x 10−23 log10( I 0) F [N] F Fc Fb (b) FIG. 3. Forces on a 2 m diameter sphere due to a plane wave of unit amplitude. The wave is incident from free space with wavelength 0
1064 nm onto a nonmagnetic sphere with (a)

2
0 i
I and (b)

16
0 i
I. PRL 97, 133902 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133902-3