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 9285Figure 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