## Radiation pressure and the distribution of electromagnetic force in dielectric media

Optics Express, Vol. 13, Issue 7, pp. 2321-2336 (2005)

http://dx.doi.org/10.1364/OPEX.13.002321

Acrobat PDF (11010 KB)

### Abstract

Using the Finite-Difference-Time-Domain (FDTD) method, we compute the electromagnetic field distribution in and around dielectric media of various shapes and optical properties. With the aid of the constitutive relations, we proceed to compute the bound charge and bound current densities, then employ the Lorentz law of force to determine the distribution of force density within the regions of interest. For a few simple cases where analytical solutions exist, these solutions are found to be in complete agreement with our numerical results. We also analyze the distribution of fields and forces in more complex systems, and discuss the relevance of our findings to experimental observations. In particular, we demonstrate the single-beam trapping of a dielectric micro-sphere immersed in a liquid under conditions that are typical of optical tweezers.

© 2005 Optical Society of America

## 1. Introduction

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 [CrossRef] [PubMed]

*ε*— the bound charges appear only at the surface(s) and/or the interface(s) between adjacent dielectrics. The bound currents, however, induced by the local

*E*-field in proportion to the time-rate of change of the polarization

*=*

**P***ε*

_{o}(

*ε*-1)

**, are distributed throughout the medium. The**

*E**E*-field of the light exerts a force on the induced charge density, while the local

*H*-field exerts a force on the induced current density. The net force can then be obtained by integrating these forces over the entire volume of the dielectric.

2. D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Compt. Phys. **159**, 13–37 (2000). [CrossRef]

4. C. Rockstuhl and H. P. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A: Pure Appl. Opt. **6**, 921–31 (2004). [CrossRef]

*E*- and

*B*-fields. We also avoid the use of the two-component approach, where scattering and gradient forces are treated separately, either in the geometric-optical regime of ray-tracing [5

5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**, 569–582 (1992). [CrossRef] [PubMed]

6. A. Rohrbach and E. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. **41**, 2494 (2002). [CrossRef] [PubMed]

*p*-polarized plane-wave at oblique incidence. In Section 4 we show that a one-dimensional Gaussian beam propagating in an isotropic, homogeneous medium, exerts either an expansive or a compressive lateral force on its host medium, depending on the beam’s polarization state. The case of a top-hat-shaped beam entering a semi-infinite dielectric medium at oblique incidence is also covered in Section 4. In Section 5 we study the behavior of a cylindrical glass rod illuminated by a (one-dimensional) Gaussian beam, paying particular attention to the effects of polarization on the force-density distribution. In Section 6 we analyze the single-beam trapping of a small spherical bead, immersed in a liquid, by a sharply focused laser beam; both linear and circular polarization states of the beam are shown to result in strong trapping forces. A dielectric half-slab under a one-dimensional Gaussian beam centered on one edge of the slab is studied in Section 7. General remarks and conclusions are the subject of Section 8.

## 2. Theoretical considerations

*E*- and

*H*-fields (both inside and outside the medium). We then apply the Lorentz law

*=0, the bound-charge density*

**D***ρ*=-∇·

_{b}*may be expressed as*

**P***ρ*

_{b}=ε_{o}∇·

*. Inside a homogeneous, isotropic medium,*

**E***being proportional to*

**E***and ∇·*

**D***=0 imply that*

**D***ρ*=0; no bound charges, therefore, exist inside such media. However, at the interface between two adjacent media, the component of

_{b}*perpendicular to the interface,*

**D**

**D**_{⊥}, must be continuous. The implication is that

**E**_{⊥}is discontinuous and, therefore, bound charges exist at such interfaces; the interfacial bound charges will thus have areal density

*σ*=

_{b}*ε*

_{o}(

*E*

_{2⊥}-

*E*

_{1⊥}). Under the influence of the local

*E*-field, these charges give rise to a Lorentz force density

**=½**

*F**Real*(

*σ**), where

_{b}**E***is the force per unit area of the interface. Since the tangential*

**F***E*-field,

**E**_{‖}, is continuous across the interface, there is no ambiguity as to the value of

**E**_{‖}that should be used in computing the force. As for the perpendicular component, the average

**E**_{⊥}across the boundary, ½(

**E**_{1⊥}+

**E**_{2⊥}), must be used in calculating the interfacial force [1

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 [CrossRef] [PubMed]

8. 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 [CrossRef] [PubMed]

*J̠*in a dispersionless medium being given by

_{b}*ωt*), one can rewrite Eq. (2) as

*J*=-i

_{b}*ωε*(

_{o}*ε*-1)

*. The B-field of the electromagnetic wave exerts a force on the bound current according to the Lorentz law, namely,*

**E***=½*

**F***Real*(

**), where*

**J**_{b}×**B***F*is the (time-averaged) force per unit volume.

*ε*(

*ω*). Maxwell’s equations are integrated in time over a discrete mesh, with the (model-dependent) dispersive behavior of the medium cast onto a time-dependent polarization vector. We assume

*ε*(

*ω*)=

*ε*∞+Δ

*ε*(

*ω*), where the real-valued

*ε*∞ denotes the relative permittivity of the medium in the absence of dispersion; the effects of dispersion enter through the complex-valued function Δ

*ε*(

*ω*). The fraction of the polarization current density that embodies the contribution of Δ

*ε*(

*ω*) is denoted by

*J̠*. As the FDTD simulation progresses, determining the

_{p}*E*- and

*H*-fields as functions of time,

*is computed concurrently by solving the relevant differential equations of the chosen model.*

**J**̠_{p}*ĵ̠*in a generally absorbing and dispersive medium, we begin with the following Maxwell equation:

*ω*; that is, the free carriers’ conductivity is assumed to be dispersionless. Note that setting σ=0 does not necessarily guarantee a transparent medium, as absorption could enter through the imaginary component of the complex permittivity, Δ

*ε*(

*ω*).

*ĵ̠*is given by

*ε*∞ represents the frequency-independent part of the relative permittivity, while

*models dispersion (*

**J**̠_{p}*=0 for non-dispersive dielectrics). When computing*

**J**̠_{p}*, it is convenient to eliminate the time-derivative of*

**ĵ̠***̠ from Eq. (5), as only one time-slice of E̠ is normally stored during FDTD computations. From Eq. (6),*

**E***ε*

_{o}

*∂*̠/

**E***∂t*=(∇×

*̠-*

**H***σ*̠-

**E***)/*

**J**̠_{p}*ε*∞, which, when substituted in Eq. (5), yields

*(i.e., the sum of the conduction and polarization current densities) at a given instant of time thus requires only the contemporary values of*

**ĵ̠***, and*

**E̠, H̠***, which are readily available during the normal progression of an FDTD simulation.*

**J̠**_{p}*and*

**E̠***as functions of spatial coordinates and time, we compute the force density distribution by time-averaging the Lorentz equation, namely,*

**H̠***ρ*(where the total charge density

**E̠***ρ*=∇·

*ε*), and by the magnetic-field,

_{o}**E̠***, can be readily identified in Eq. (8).*

**J**̠×**B**̠## 3. Comparison to exact solutions

*n*and thickness

*d*illuminated by a plane-wave at normal incidence was analyzed in our previous paper [1

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 [CrossRef] [PubMed]

*xy*-plane and the beam propagates in the -

*z*direction. At normal incidence, there are no induced charges, and the electromagnetic pressure in its entirety may be attributed to the magnetic component of the Lorentz force. Inside the slab, a pair of counter-propagating plane-waves (along ±

*z*) interfere with each other and set up a system of fringes. The force is distributed within the fringe pattern, being positive (i.e., in the +

*z*direction) over one-half of each fringe and negative over the other half. The net force is obtained by integrating the local force density through the thickness of the slab.

*F*inside a slab illuminated at normal incidence by a plane-wave of wavelength λ

_{z}_{o}=640nm and

*E*-field amplitude

*E*

_{o}=1.0 V/m. (Computed values of

*F*and

_{x}*F*were zero, as expected.) The slab is suspended in free-space, and has refractive index

_{y}*n*=2.0 and thickness

*d*=110nm. The total force along the

*z*-axis (per unit cross-sectional area) is found to be ∫

*F*(

_{z}*z*) dz=-2.481 pN/m

^{2}(Δ

*z*=5.0 nm in our FDTD simulations) versus the exact value of -2.479 pN/m

^{2}, obtained from theoretical considerations [1

**12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375 [CrossRef] [PubMed]

*d*=80nm, the simulation yields ∫

*F*(

_{z}*z*) dz=-3.192 pN/m

^{2}, where the exact solution is -3.188 pN/m

^{2}.

*p*-polarized plane-wave (λ

_{o}=650 nm) incident at

*θ*=50° on a semi-infinite dielectric of refractive index

*n*=3.4, located in the region

*z*<0. Figure 2 shows computed time-snapshots of the

*E*component of the field. In Fig. 2(a) the transition of the refractive index from

_{z}*n*

_{o}=1.0 to

*n*

_{1}=3.4 is abrupt, and the force per unit area due to the induced surface charges, computed in FDTD with Δ

*y*=Δ

*z*=5.0 nm, is (

*F*)=(1.696, 2.468) pN/m

_{y}, F_{z}^{2}, versus the exact value of (1.699, 2.467) pN/m

^{2}. For

*n*=2.0, FDTD simulations yield (

*F*)=(1.5903, 1.6475) pN/m

_{y}, F_{z}^{2}versus the exact value of (1.5911, 1.6493) pN/m

^{2}. Figure 2(b) corresponds to a semi-infinite dielectric with a rapid (linear) transition of the refractive index from

*n*

_{o}=1.0 to

*n*

_{1}=3.4 over a 40 nm-thick region. In this case, the force per unit surface area computed in FDTD is (

*F*)=(1.744, 2.638) pN/m

_{y}, F_{z}^{2}.

8. 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 [CrossRef] [PubMed]

## 4. Force exerted by beam’s edge on the host medium

**12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375 [CrossRef] [PubMed]

*E*

_{o}| is the magnitude of the

*E*-field of a (finite-diameter) plane-wave in a medium of dielectric constant

*ε*. If the

*E*-field is parallel (perpendicular) to the beam’s edge, the force is compressive (expansive); in other words, the opposite side-walls of the beam tend to push the medium toward (away from) the beam center. The “edge force” does not appear to be sensitive to the detailed structure of the beam’s edge; in particular, a one-dimensional Gaussian beam exhibits the edge force described by Eq. (9) when its (magnetic) Lorentz force on the host medium is integrated laterally on either side of the beam’s center [1

**12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375 [CrossRef] [PubMed]

*x*, Gaussian along

*y*, and propagating in the negative

*z*-direction) in a homogeneous host medium of refractive index

*n*=2.0; the free-space wavelength of the cw beam is λ

_{o}=0.65µm. Figure 3 shows time snapshots of the field profile (first row) and time-averaged force density distributions (second row) in the y z cross-sectional plane. The beam’s full-width at half-maximum-amplitude measured at the waist (located at z=+0.3 µm) is FWHM=1.5 µm.

*p*-polarized on the left- and

*s*-polarized on the right-hand-side. The upper-left frame shows a color-coded plot of the

*H*distribution, on which the

_{x}*E*-field vector (

*E*) is superposed. The frame below it shows the force density plot of

_{y}, E_{z}*F*, together with the (

_{y}*F*) vector field in the case of

_{y}, F_{z}*p*-polarization. The upper-right frame is a time snapshot of the

*E*distribution, with the

_{x}*H*-field vector (

*H*) superposed. The frame below it shows

_{y}, H_{z}*F*for the

_{y}*s*-polarized beam, with the (

*F*) vector field superposed. These force-density fields are computed using time-averages (over one period of the optical wave,

_{y}, F_{z}*T*=λ

_{o}/

*c*) of the time- and space-dependent force density. Note that

*F*is expansive for

_{y}*p*-light and compressive for s-light. The integrated lateral force per unit area on each half of the beam, ∫ F (x, y, z) dy, is 1.6577 pN/m

^{2}for

*p*-light and 1.6581 pN/m

^{2}for

*s*-light. The theoretical value of the force per unit area on each edge of the beam given by Eq. (9) with

*E*

_{o}=0.5 V/m is

*F*

^{(edge)}=±1.66 pN/m

^{2}.

_{o}=0.65µm) having a top-hat cross-sectional profile with smooth edges. The beam is incident from the free-space at

*θ*

_{inc}=50° onto a semi-infinite dielectric of index

*n*=2.0; the dielectric fills the half-space

_{s}*z*<0. The case of

*p*-polarization is shown on the left-, that of s-polarization on the right-hand side. For the p-polarized beam, its angle of incidence being close to Brewster’s angle

*θ*

_{B}=63.43°, there is little reflectivity at the interface and most of the light is transmitted through to the dielectric medium, whereas for the s-polarized beam a good fraction of the incident light is being reflected.

*F*) inside the dielectric medium are displayed in Fig. 5; the interface region has been excluded to avoid, in the case of

_{y}, F_{z}*p*-light, the high force region of the induced surface charges. (For

*s*-light the

*E*-field is continuous at the boundary and, therefore, no surface charges are induced.) In both cases the force fields near the leading edge of the beam show oscillatory behavior, whereas the trailing edge is fairly smooth. Despite oscillations near the edge, the force is seen to be generally expansive for

*p*-light and compressive for

*s*-light, that is, the sign of the integrated lateral force on each side of the center is consistent with the theoretical arguments presented in [1

**12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375 [CrossRef] [PubMed]

^{2}, the integrated force (∫

*F*dy, ∫

_{y}*F*dy) for

_{z}*p*-light is (-2.64, -1.13) for the left-edge and (2.59, 1.096) for the right-edge. These edge forces are nearly identical in strength (to better than ±1.5%), are orthogonal to the propagation direction within the dielectric, and are in fair agreement with the theoretical value of ±(2.51, 1.04) pN/m2 obtained from Eq. (9) with

*E*

_{o}=0.64 V/m. The corresponding edge forces for the

*s*-light depicted in Fig. 5, right-hand column, are (1.957, 0.821) for the left edge and (-1.939, -0.815) for the right edge. Again, this compressive force is orthogonal to the propagation direction, and is in reasonable agreement with the theoretical value of ±(1.92, 0.8) pN/m2 obtained from Eq. (9) with

*E*

_{o}=0.56 V/m.

## 5. Cylindrical rod illuminated by Gaussian beam

*H*components of a

_{x}, E_{y}, E_{z}*p*-polarized, one-dimensional Gaussian beam (λ

_{o}=0.65 µm, amplitude FWHM=0.5 µm), propagating in free-space along the negative

*z*-direction; the beam’s waist is at

*z*=0.5 µm. Given the amplitude profile as

*H*(

_{x}*y, z*=0.5µm)=

*H*

_{o}exp[-(

*y/y*

_{o})

^{2}], the value of

*y*

_{o}at the waist is 0.3µm, which corresponds to a divergence half-angle

*θ=arctan*(

*λ*

_{o}/

*πy*

_{o})≈35°. The small diameter of the beam at the waist thus results in its rapid divergence along the propagation direction. The cone angle of the beam, although large enough to exhibit lateral trapping of small dielectric objects, is not sufficient to produce vertical trapping [5

5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**, 569–582 (1992). [CrossRef] [PubMed]

*H*-field amplitude was

*H*

_{o}=2.6544 A/m, leading to an initial

*E*-field amplitude (at the center of the Gaussian)

*E*=1000 V/m; here

_{o}=Z_{o}H_{o}*E*and

_{y}*E*, the

_{z}*z*-component of the beam’s Poynting vector is approximated as

*x*-axis) is then found to be

*r*=0.2 µm, index

_{c}*n*=2.0) parallel to the

_{c}*x*-axis at (

*y, z*)=(0.2 µm, 0). The cylinder’s index changes linearly at the surface, from

*n*=1.0 to 2.0, within a 20 nm-thick transition layer (grid cell size=5nm). Figure 7, left column, shows computed plots of |

*H*| as well as force-density components Fy and

_{x}*F*for this

_{z}*p*-polarized beam. The integrated force is (∫

*F*(

_{y}*y, z*) d

*y*d

*z*, ∫

*F*(

_{z}*y, z*) d

*y*d

*z*)=(0.17, 0.21) pN/m without including the surface-charge contribution, and (-0.4, -0.58) pN/m when the surface-charge contribution is included. The net force is thus pulling the cylinder to the left while, at the same time, pushing it downward.

*s*-polarized Gaussian beam (same parameters as in Fig. 6) illuminating the same glass cylinder as above. From top to bottom, the figure shows computed plots of |

*E*| and force-density components

_{x}*F*and

_{y}*F*. Here surface charges are absent, and the force is purely due to the magnetic component of the Lorentz force. The integrated force is (∫

_{z}*F*(

_{y}*y, z*) d

*y*d

*z*, ∫

*F*(

_{z}*y, z*) d

*y*d

*z*)=(-0.14, -0.68) pN/m, which, as in the case of

*p*-light, tends to pull the rod to the left while pushing it downward. The pull to the left is consistent with what has been observed in optical trapping experiments, but the downward push is contrary to the known behavior of single-beam traps (i.e., optical tweezers) [9

9. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

10. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science **235**, 1517–1520 (1987). [CrossRef] [PubMed]

5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**, 569–582 (1992). [CrossRef] [PubMed]

## 6. Spherical bead immersed in liquid and trapped by a focused beam

*r*=1.0µm and index

_{s}*n*=1.6 immersed in a liquid of index

_{s}*n*

_{o}=1.3. The particle is displaced along the x-axis, its center being at (

*x, y, z*)=(Δ

*x*, 0, 0). The incident beam, obtained by focusing a λ

_{o}=0.65µm plane-wave through a 1.25NA immersion objective, is sourced at (

*x, y, z*)=(0, 0, 1.2µm) and propagates in the negative

*z*-direction. The beam entering the objective’s pupil may be circularly or linearly polarized. The actual point of focus can be placed at various locations along the

*z*-axis, (

*x, y, z*)=(0, 0, Δ

*z*), inside or outside the spherical particle. The lens is designed for diffraction-limited focusing within the immersion liquid and, in the absence of the spherical particle, the focused spot is free from aberrations and has a total power (i.e., integrated Sz within the focal plane) of

*P*=0.5 mW.

8. 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 [CrossRef] [PubMed]

*E*-field at the particle’s surface — used in computing the

*E*-field contribution to the Lorentz force — may be obtained in several different ways. In the present simulations we used the

*average E*-field across the boundary (i.e., at the sphere surface) in computing the Lorentz force on the interfacial charges accumulated on the sphere.

*x, y, z*)=(0, 0, 0.7µm) onto a particle centered at (0.25µm, 0, 0), Fig. 8 shows color-coded force-component distributions in various cross-sectional planes through the center of the sphere. The superposed arrows show the projection of the force-density vector in the corresponding plane, e.g., plots on the

*yz*-plane show the (

*F*) vector field. Contributions to force-density come from both (bound) surface charges and (bound) electric currents within the volume of the sphere. When the force density exceeds the range of color coding, the corresponding region is depicted as white. This happens in the area where the beam enters the sphere, seen in the

_{y}, F_{z}*xz*and

*yz*cross-sectional planes of Fig. 8 near the top of each frame. Reflection, refraction, and diffraction create complex interference patterns within the spherical particle, as evidenced by the richly textured force density profiles throughout the volume of the sphere. The surrounding liquid also experiences the effects of radiation pressure, as the bound currents within the liquid body feel the Lorentz force of the

*B*-field, while the bound charges on the liquid side of the interface sense the Lorentz force of the

*E*-field, in the same way that the particle itself experiences such forces. The force imparted to the liquid is strongest in the vicinity of the region where the beam enters the particle, but weak forces are felt (by the liquid) also in those regions where the light leaves the solid particle and re-emerges into the liquid environment.

*(*

**F**_{z}*x, y, z*) dz in the central

*xy*-plane appears on the left, that of ∫

*(*

**F**y*x, y, z*) d

*y*in the central

*xz*-plane in the middle, and the profile of ∫

*(*

**F**x*x, y, z*) dx in the central

*yz*-plane on the right-hand side. The range of integration includes the (bound) charges on the surface of the particle, but excludes any forces that might be exerted by the entering/exiting light on the surrounding liquid. The presence of a strong force on the surface charges is indicated by the white regions (i.e., exceeding the color scale) near the top of the frame in the

*xz*and

*yz*cross-sections depicted in Fig. 9.

*x, y, z*)=(0, 0, 0.7µm) onto a glass bead centered at (0.25µm, 0, 0), the net force on the bead, obtained by integrating the force-density through the spherical volume, is ∭

*(*

**F***x, y, z*) d

*x*d

*y*d

*z*=(-0.40, +0.018, +0.34) pN. The direction of the total force clearly indicates trapping of the particle by the focused beam, as

*F*tends to pull the particle to the left while

_{x}*F*tries to lift it up, toward the center of the focused spot. (Similar calculations were carried out for different positions of the bead relative to the focused spot, and trapping was demonstrated in all cases; see Fig. 10.) It must be pointed out that the

_{z}*y*-component of the net force, although small, is not negligible, thus indicating the transfer of a certain amount of angular momentum from the circularly polarized beam to the illuminated particle (this point will be discussed in more detail below).

*F*) versus vertical displacement Δ

_{x}, F_{z}*z*of the focus center from the sphere center are shown in Fig.10 (Δ

*z*>0 when the focused spot is above the sphere center). Different colored curves correspond to different lateral displacements Δx of the particle from the optical axis of the lens (red: Δ

*x*=0.25µm, green: Δ

*x*=0.5µm, blue: Δ

*x*=0.75µm). The top row in Fig. 10 corresponds to a circularly polarized plane-wave entering the pupil of the lens. The middle row corresponds to linear polarization along

*x*, and the bottom row represents the case of linear polarization along the

*y*-axis. Note that for Δ

*z*>0,

*F*is positive only when Δ

_{z}*x*is small; this indicates that the particle is first trapped laterally by the (fairly strong)

*F*, then lifted upward until its center nearly coincides with the center of the focused beam.

_{x}*x*, or perpendicular to that direction,

*y*),

*F*was found to be zero, as expected from symmetry considerations. The non-zero values of

_{y}*F*obtained for circularly polarized light indicate the transfer of a certain amount of angular momentum from the beam to the particle. We found

_{y}*F*to be generally an order of magnitude weaker than

_{y}*F*, fairly independent of Δz, and an increasing function of Δ

_{x}*x*. When the sense of polarization was reversed (say, from right- to left-circular),

*F*switched sign, while its magnitude — as well as the signs and magnitudes of

_{y}*F*and

_{x}*F*— remained unchanged. We thus believe the weak but non-zero values of

_{z}*F*, predicted to exist under circularly-polarized light, are real and should be subjected to experimental verification.

_{y}## 7. Dielectric half-slab illuminated near its side-wall

**12**, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375 [CrossRef] [PubMed]

*p*-polarized beam inside the half-slab tends to exert an expansive force on its host medium (and hence drive it away from the beam’s center), the net force experienced by the half-slab should be dominated by the surface charges induced on the slab’s side-wall, which force tends to draw the slab toward the center of the beam. Attraction of the half-slab to the beam’s center is thus expected to occur irrespective of whether the incident beam is

*p*- or

*s*-polarized.

_{o}=0.65µm, FWHM=1.5µm), normally incident from free-space onto a dielectric half-slab having

*n*=2.0 and thickness

_{s}*d*=110nm. The beam center and the slab-edge are assumed to coincide at

*y*=0. Figure 11 shows the computed field as well as force-density plots when the beam is

*p*-polarized (left column) and

*s*-polarized (right column). In the

*p*-polarized case the color scale has been adjusted to exclude high force-density values at the edges/corners of the half-slab. The horizontal force component

*F*(second row) is mostly positive inside the slab but has very strong negative values on the slab’s vertical edge, where there is a substantial charge accumulation. In the

_{y}*s*-polarized case, no surface charges are induced (because the only

*E*-field component,

*E*, is everywhere parallel to the glass surfaces). The only forces in the case of

_{x}*s*-polarization, therefore, are due to the bound currents within the medium, acted upon by the

*H*-field of the light beam. Clearly, the force distributions are quite complex, and it is not immediately obvious if the total force in the horizontal direction is pulling the half-slab to the left or pushing it to the right; net (integrated) values of the force are thus needed to settle the question.

*z*-integrated force components, ∫

*F*(

_{y}*y, z*) dz and ∫

*F*(

_{z}*y, z*) dz, plotted versus the y-coordinate, for the half-slab depicted in Fig. 11. For

*p*-light the net

*F*is negative, thus tending to pull the slab to the left. Also

_{y}*F*is seen to be negative everywhere, indicating that radiation pressure pushes the slab downward. For

_{z}*s*-light the integrated

*F*and

_{y}*F*show oscillatory behavior between positive and negative values. Overall, however, the negative values dominate, tending to pull the slab to the left and push it downward. Setting the incident

_{z}*E*-field amplitude to

*E*

_{o}=1000 V/m, the total integrated force (∬

*F*d

_{y}*y*d

*z*, ∬

*F*d

_{z}*y*d

*z*) on the half-slab is found to be (-0.407, -1.439) pN/m for

*p*-light and (-0.313, -1.745) pN/m for

*s*-light.

## 8. Concluding remarks

*E*- and

*H*-fields are known, it is straightforward to compute the bound charge density

*ρ*and the bound current density

_{b}*, then apply the Lorentz law to these charges and currents in order to determine the local distribution of force throughout the volume of interest.*

**J**_{b}*E*-field across the solid-liquid interface to compute the electric component of the Lorentz force on the (bound) charges induced on the solid side of the interface. The merits and demerits of this particular approach to force computation were discussed in [8

**13**, 2064–2074 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2064 [CrossRef] [PubMed]

## Acknowledgments

## References

1. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express |

2. | D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Compt. Phys. |

3. | A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. Roy. Soc. Lond. A |

4. | C. Rockstuhl and H. P. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A: Pure Appl. Opt. |

5. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

6. | A. Rohrbach and E. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. |

7. | J. D. Jackson, |

8. | M. Mansuripur, A. R. Zakharian, and J. V. Moloney, “Radiation pressure on a dielectric wedge,” Opt. Express |

9. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

10. | A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 14, 2005

Revised Manuscript: March 15, 2005

Published: April 4, 2005

**Citation**

Armis Zakharian, Masud Mansuripur, and Jerome Moloney, "Radiation pressure and the distribution of electromagnetic force in dielectric media," Opt. Express **13**, 2321-2336 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2321

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### References

- M. Mansuripur, �??Radiation pressure and the linear momentum of the electromagnetic field,�?? Opt. Express 12,5375-5401 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375</a> [CrossRef] [PubMed]
- D. A. White, �??Numerical modeling of optical gradient traps using the vector finite element method,�?? J. Compt. Phys. 159, 13-37 (2000). [CrossRef]
- A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, �??Theory of trapping forces in optical tweezers,�?? Proc. Roy. Soc. Lond. A 459, 3021-3041 (2003). [CrossRef]
- C. Rockstuhl and H. P. Herzig, �??Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,�?? J. Opt. A: Pure Appl. Opt. 6, 921-931 (2004). [CrossRef]
- A. Ashkin, �??Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,�?? Biophys. J. 61, 569-582 (1992). [CrossRef] [PubMed]
- A. Rohrbach and E. Stelzer, �??Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,�?? Appl. Opt. 41, 2494 (2002). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics, 2nd edition (Wiley, New York, 1975).
- M. Mansuripur, A. R. Zakharian, and J. V. Moloney, �??Radiation pressure on a dielectric wedge,�?? Opt. Express 13, 2064-2074 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2064">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2064</a> [CrossRef] [PubMed]
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu, �??Observation of a single-beam gradient force optical trap for dielectric particles,�?? Opt. Lett. 11, 288-290 (1986). [CrossRef] [PubMed]
- A. Ashkin and J. M. Dziedzic, �??Optical trapping and manipulation of viruses and bacteria,�?? Science 235, 1517-1520 (1987). [CrossRef] [PubMed]

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