Trapping metallic Rayleigh particles with radial polarization

doi:10.1364/OPEX.12.003377

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Trapping metallic Rayleigh particles with radial polarization

Qiwen Zhan »View Author Affiliations

Optics Express, Vol. 12, Issue 15, pp. 3377-3382 (2004)
http://dx.doi.org/10.1364/OPEX.12.003377

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▸ Article Outline
– Abstract
1. Introduction
2. Focal field calculation of highly focused radial polarization
3. Optical trapping of metallic Rayleigh particles using radial polarization
4. Discussions
5. Conclusions
– References and links

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Abstract

Metallic particles are generally considered difficult to trap due to strong scattering and absorption forces. In this paper, numerical studies show that optical tweezers using radial polarization can stably trap metallic particles in 3-dimension. The extremely strong axial component of a highly focused radially polarized beam provides a large gradient force. Meanwhile, this strong axial field component does not contribute to the Poynting vector along the optical axis. Consequently, it does not create axial scattering/absorption forces. Owing to the spatial separation of the gradient force and scattering/absorption forces, a stable 3-D optical trap for metallic particles can be formed.

1. Introduction

It is well-known that the linear momentum of light wave can exert radiation pressure on microscopic physical objects and perform mechanical manipulations on them. Since Ashkin demonstrated the first practical laser traps and showed the use of radiation pressure to capture and manipulate micrometer sized particles[1

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159, (1970). [CrossRef]

], optical traps and tweezers have become powerful tools for the trapping and manipulations of atoms, molecules, nano particles, living biological cells and organelles within cells[2

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron 6, 841–856, (2000). [CrossRef]

Two types of radiation forces were identified in the optical tweezers: gradient force and scattering/absorption forces. The gradient force is proportional to the gradient of the square of the electric field (energy density) and is responsible to pull the particles towards the center of the focus. The scattering/absorption forces are due to the net momentum transfer caused by scattering/absorption of photons from the particles. These forces are proportional to the Poynting vector (power density) of the field. If the particle is located at the center of the focus, the scattering/absorption forces tend to push the particles out of the focus and destabilize the optical trap. For dielectric particles, the gradient force can easily surpass the scattering force. Thus, it is relatively easier to trap dielectric particles in 3-dimension (3-D). However, for metallic particle, stable 3-D trapping is generally considered difficult due to strong scattering and absorption. Although trapping of both Rayleigh and Mie metallic particles has been reported previously [3

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932, (1994). [CrossRef] [PubMed]

–5

S. Sato, Y. Harada, and Y. Waseda, “Optical trapping of microscopic metal particles,” Opt. Lett. 19, 1807–1809 (1994) [CrossRef] [PubMed]

], the formation of stable trap have been largely limited to either extremely small particles or need to be optimized at a longer wavelength. As wavelength decreases, the scattering/absorption forces increases much faster than the gradient force, which could potentially destabilize the trap.

Recently, a new method of shaping the focus using highly focused cylindrical vector beams was proposed and its application to optical tweezers was suggested [6

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-324 [CrossRef] [PubMed]

, 7

Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A: pure appl. opt. , 5, 229–232, (2003). [CrossRef]

]. It has been shown that dielectric Rayleigh particles with different refractive index can be stably trapped using either a strongly focused radial polarization or a strongly focused azimuthal polarization. In this paper, we further show that metallic Rayleigh particles can also be stably trapped in 3-D by a strongly focused radial polarization. In section 2, we illustrate the geometry that is used in our optical trapping calculations and briefly review the focal field calculation for radially polarized beam. We then present the numerical simulations of the radiation forces exerted on metallic Rayleigh particles and show that 3-D stable optical trap can be formed due to a lacking of power density along the optical axis for highly focused radially polarized beam. A short discussion is then given in section 4.

2. Focal field calculation of highly focused radial polarization

There is an increasing interest in laser beams with radial polarization. Particular interest has been given to the high numerical aperture (NA) focusing properties of these beams and their applications as high-resolution probes [8

R. Dorn, S Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901, (2003). [CrossRef] [PubMed]

–11

H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B. 15, 1381–1386 (1998). [CrossRef]

]. Due to the polarization symmetry, the electric field at the focus of a radially polarized beam has an extremely strong axial component, and the transverse size of the axial component is much smaller than that of the transversal component. This property may find applications in high-resolution microscopy, microlithography, metrology and nonlinear optics, etc. In this paper, we show that this unique focusing property allows stable 3-D trapping of metallic nano-particles.

2.1 Geometry and focal field calculation for highly focused radial polarization

The field distribution near the focus of highly focused polarized beams is analyzed with the Richards-Wolf vectorial diffraction method [12

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Ser. A 253, 349–357 (1959). [CrossRef]

, 13

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959). [CrossRef]

]. Details of focal field calculation for radial, azimuthal and generalized cylindrical polarization using Richards-Wolf method can be found in [6

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-324 [CrossRef] [PubMed]

, 10

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-2-77. [CrossRef] [PubMed]

]. The geometry of the problem is shown in Fig. 1. The illumination is a radially polarized beam with a planar wavefront over the pupil. An aplanatic lens produces a converging spherical wave towards the focus of the lens. The particle to be trapped is assumed to be immersed in an ambience with refractive index of n1. For simplicity, the refractive index on the object side is assumed to be the same as the ambience. The focal field can be written as

\vec{E} (r, φ, z) = E_{r} {\vec{e}}_{r} + E_{z} {\vec{e}}_{z}

(1)

where e⃗r, e⃗z are the unit vectors along the radial and axial directions. The amplitudes of two orthogonal components E_r, E_z can be expressed as

E_{r} (r, φ, z) = 2 A \int_{0}^{θ_{mx}} \cos^{\frac{1}{2}} (θ) P (θ) \sin θ \cos θ J_{1} (k r \sin θ) e^{i k_{1} z \cos θ} d θ

(2)

E_{z} (r, φ, z) = i 2 A \int_{0}^{θ_{mx}} \cos^{\frac{1}{2}} (θ) P (θ) \sin^{2} θ J_{0} (k r \sin θ) e^{i k_{1} z \cos θ} d θ

(3)

where θ_max is the maximal angle given by the NA of the objective lens, P(θ) is the pupil apodization function, k₁ is the wave number in the ambient medium, J_n(x) is the Bessel function of the first kind with order n, and A=n₁πf/λ with f being the focal length. For many applications, such as high resolution imaging, it is sufficient to simply calculate the relative strength of each field components. However, quantitative estimation of the radiation forces requires the calculation of absolute field strength with given beam parameters. In the calculations throughout this paper, we choose λ=1.047µm, n₁=1.33 and a simple annulus pupil apodization function

P (θ) = {\begin{matrix} P_{0} & if \sin^{- 1} (N A_{1}) \leq θ \leq \sin^{- 1} (N A ⁄ n_{1}) \\ 0 & otherwise \end{matrix}

(4)

where P₀ is a constant amplitude factor and NA is the lens numerical aperture determined by the outer radius of the annulus and n₁. The NA is chosen to be 0.95n₁ in this study. NA₁ corresponds to the inner radius of the annulus which is variable. The laser beam power is assumed to be 100 mW. As NA₁ is varied, the amplitude P₀ is adjusted accordingly to maintain the power level. As one example, field distribution for NA₁=0.6 is shown in Fig. 2.

Fig. 1. Geometry used in the optical trapping calculation. Q(r, φ) is an observation point in the focal plane. The particle to be trapped is at the focus.

Fig. 2. Line scan of calculated focal plane energy density distribution for a highly focused radial polarization. It can be seen that axial component is dominant. For comparison, result for linearly polarized incident is also shown (green line).

2.2 Properties of highly focused radial polarization

From Fig. 2, it can be seen that the overall distribution of field strength is dominated by the axial component, giving a more compact focus. Thus, focused radial polarization provides stronger gradient force to pull the metallic particles towards the center of the focus. Another important feature of such a focal field is that the strong axial component is a non-propagating field and does not contribute to the energy flow along the propagation direction. Consequently, the time averaged Poynting vector <S>z=Re{(E⃗×H⃗*)z}/2=Re{E_rH*_φ}/2 along the optical axis has a null at center. To demonstrate this, the corresponding time averaged Poynting vector is calculated and its axial component is shown in Fig. 3. The magnetic field is calculated in a similar way to that of the electric field calculation; assuming the magnetic field at the pupil plane is aligned along the azimuthal direction. From Fig. 3, it is clear that axial Poynting vector near optical axis is substantially zero. As we show in the next section, this has important meaning for metallic particle trapping in optical tweezers.

Fig. 3. Calculated axial component of time averaged Poynting vector for a highly focused radial polarization. (a) 2-dimensional distribution near focus; (b) line scan of (a) at the focal plane.

3. Optical trapping of metallic Rayleigh particles using radial polarization

3.1 Radiation forces on metallic Rayleigh particles

For Rayleigh metallic particles, one can use the dipole approximation to calculate the radiation force [14

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation , Academic (New York), 1969.

, 15

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996). [CrossRef]

]. Assuming a spherical metallic Rayleigh particle with radius a (a≪λ), the gradient force on this particle can be expressed as F⃗ _grad=Re(α)ε ₀∇|E⃗|²/4, where α is the polarizability of the metallic particles given by α=4πa ³ ε ₁( $\hat{ε}$ -ε ₁)/( $\hat{ε}$ +2ε ₁), with $\hat{ε}$ ,ε ₁ being the dielectric constants of the metallic particle and the ambient, respectively. In this paper, we use gold particle in the simulations ( $\hat{ε}$ =-54+5.9i at 1.047 µm [3

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932, (1994). [CrossRef] [PubMed]

]).The axial components of scattering and absorption forces are F_scat,z =n ₁<S>_z C_scat /c, and F_abs,z =n ₁<S>_z C_abs/c, where <S>_z is the time averaged axial component of the Poynting vector, C_scat and C_abs are the scattering and absorption cross sections. For Rayleigh particles, these cross sections are given by C_scat=

k_{0}^{4}

|α|²/6π and C_abs =k ₁ Im(α)/ε ₁, respectively. It is clear that the forces can be divided into two types. The gradient force F_grad is proportional to the gradient of the square of the electric field (energy density). The absorption/scattering forces F_scat and F_abs are proportional to the Poynting vector (power density).

3.2 Formation of 3D stable trap for metallic Rayleigh particles

To this point, the advantage of using highly focused radial polarization for metallic particle trapping should be clear. The strong confinement of axial component contribute to an enhanced gradient force, while the scattering/absorption forces are zero along the optical axis and remains negligible near the optical axis due to the spatial separation of the energy density and power density. Stable 3-D optical trap (optical tweezers) is thus formed.

Simulation results of radiation forces for NA₁=0.6 are shown in Fig. 4. In comparison, calculation results with 100 mW linearly polarized light illumination are also included. It can be seen that annular illumination with radial polarization provides higher gradient force as well as lower scattering/absorption forces. Along the optical axis, the scattering/absorption force is substantially zero, eliminating the major potential cause of trap destabilization.

Fig. 4. Calculated radiation forces on 38.2 nm (diameter) gold particle. (a) Transverse gradient force along x-axis; (b) axial gradient force along z-axis; (c) sum of axial scattering and absorption forces along z-axis; (d) sum of axial scattering and absorption forces on x-axis. For comparison, result for highly focused linearly polarized incident is also shown (red dotted lines)

3.3 Trapping stabilities

To obtain stable 3-D trap, several stability criterion need to be satisfied. To balance the forward scattering/absorption forces and stably trap metallic particles in 3-D, gradient force should provide a sufficient large component opposite to the propagation direction, i.e. R=F_grad,z /(F_scat,z +F_abs,z )≥1, where R is called the stability criterion. From the previous sections, it is clear that this can be easily satisfied by the optical trap using radial polarization, since the scattering force around optical axis is negligible. For a more conservative estimation of R, we use R=(F_grad,z )_max/[(F_scat,z )_max+(F_abs,z )_max≥1 to estimate the stability. We calculated R for three different NA₁ values of 0.01, 0.6 and 0.8 and the stability parameters are 24.9, 26.2, and 22.8. In comparison, the corresponding stabilities parameter for 100 mW linearly polarized illumination are also calculated to be 10.5 (NA₁=0.01), 7.6 (NA₁=0.6) and 4.5 (NA₁=0.8), respectively. This clearly demonstrates the advantage of using radial polarization over linear polarization.

In addition, in order to form a stable trap, the potential well generated by the gradient forces must be deep enough to overcome its kinetic energy in Brownian motion. A generally accepted criterion is

R_{Thermal} = e^{- U_{max} ⁄ k_{B} T} ≪ 1

, where the potential depth is given by U _max=ε ₀Re(α)|E

|_{max}^{2}

/2 [16

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]

]. Assuming a temperature of 300K, R_Thermal for the situations considered above are calculated to be 3.34 ×10^-12 (NA₁=0.01), 1.21×10^-12 (NA₁=0.6) and 2.57×10^-9 (NA₁=0.8). The corresponding R_Thermal for linear polarization illumination are 3×10^-19 (NA₁=0.01), 2.97×10^-12 (NA₁=0.6) and 1.18×10^-6 (NA₁=0.8). We notice a difference from the thermal stability parameter estimated in [3

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932, (1994). [CrossRef] [PubMed]

]. This may be caused by different optical models being used. In Ref. [3

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932, (1994). [CrossRef] [PubMed]

], a simple Gaussian beam model was used, while we use uniform illumination with rigorous vectorial model. At extremely low NA₁, linear polarization illumination actually gives better thermal stability. For intermediate and high NA1, radial polarization illumination has advantages in terms of thermal stability.

Finally, in some cases, the gradient force needs to be large enough to balance the gravity. Using the density of gold (~19.3 g/cm³) and ignoring the buoyancy force, the gravity of a gold nano-sphere with radius of 18.1 nm is 4.7×10^-18 N, which is much smaller than the maximal gradient force 7.4×10^-14 N (NA₁=0.01), 5.8×10^-14 N (NA₁=0.6) and 2.52×10^-14 N (NA₁=0.8) in the optical trap using radial polarization.

4. Discussions

Although 100mW power is used in the previous calculation, the stabilities calculation indicates that much lower power can be used to form a stable 3-D trap. Its advantages over optical trap with linear polarization allow lower laser power to be used. The lower laser power and the low power density along the optical axis lead to lower absorption and make it less likely to damage the sample. In addition, the trap produced by radial polarization is not limited by wavelength. Because of the skin depth, in conventional trap the gradient force grows at a weaker dependence than a³ as particle size increases, eventually settling on a², while the scattering force grows at a dependence of a⁶ [3

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932, (1994). [CrossRef] [PubMed]

]. Thus destabilization caused by scattering force becomes increasingly severe as particle size increases. This is not the case for optical traps using radial polarization. Metallic particles with larger sizes can be stably trapped, as long as the particles can fit in the dark hollow region shown in Fig. 3. From the simulation, we can see that intermediate NA₁ is preferred to achieve higher gradient forces for trapping and better stability. The axial component is mostly contributed by the higher NA portion of the illumination. For lower NA₁, the axial component is less significant and the field distribution is more spread out than linearly polarized incident. Thus, for lower NA₁, radial polarization does not show advantage in terms of absolute value of gradient force. However, it still provides the advantage of negligible scattering/absorption forces. For extremely large NA₁, high sidelobes lead to weaker radiation forces and lower stability.

The capability to trap metallic particles with shorter wavelength, extremely small sizes at low power has significant meaning for bio-imaging. Metallic nano-particles have important applications in florescence bio-imaging to probe specific targets. Trapping these metallic nano-particles allows various spectroscopic methods to be attached to optical tweezers, opening new possibilities in intracellular imaging at single molecule level.

5. Conclusions

In summary, optical tweezers using highly focused radial polarization is proposed to trap metallic Rayleigh particles. Calculations show that there is a separation of the gradient force and the absorption/scattering force due to a dominant non-propagating axial component of the electric field. The extremely strong axial field only contributes to gradient force. The spatial separation of the forces and the dominance of the non-propagating axial field allow one to trap metallic particles in 3-D stably.

References and links

1.	A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159, (1970). [CrossRef]
2.	A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron 6, 841–856, (2000). [CrossRef]
3.	K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932, (1994). [CrossRef] [PubMed]
4.	M. Gu and D. Morrish, “Three-dimensional trapping of Mie metallic particles by the use of obstructed laser beams,” J. Appl. Phys. , 91, 1606–1612, (2002). [CrossRef]
5.	S. Sato, Y. Harada, and Y. Waseda, “Optical trapping of microscopic metal particles,” Opt. Lett. 19, 1807–1809 (1994) [CrossRef] [PubMed]
6.	Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-324 [CrossRef] [PubMed]
7.	Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A: pure appl. opt. , 5, 229–232, (2003). [CrossRef]
8.	R. Dorn, S Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901, (2003). [CrossRef] [PubMed]
9.	S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
10.	K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-2-77. [CrossRef] [PubMed]
11.	H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B. 15, 1381–1386 (1998). [CrossRef]
12.	E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Ser. A 253, 349–357 (1959). [CrossRef]
13.	B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959). [CrossRef]
14.	M. Kerker, The Scattering of Light and Other Electromagnetic Radiation , Academic (New York), 1969.
15.	Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996). [CrossRef]
16.	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]

OCIS Codes
(020.7010) Atomic and molecular physics : Laser trapping
(170.4520) Medical optics and biotechnology : Optical confinement and manipulation
(260.5430) Physical optics : Polarization

ToC Category:
Research Papers

History
Original Manuscript: April 15, 2004
Revised Manuscript: July 7, 2004
Published: July 26, 2004

Citation
Qiwen Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12, 3377-3382 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3377

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References

A. Ashkin, �??Acceleration and trapping of particles by radiation pressure,�?? Phys. Rev. Lett. 24, 156-159, (1970). [CrossRef]
A. Ashkin, �??History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,�?? IEEE J. Sel. Top. Quantum Electron 6, 841-856, (2000). [CrossRef]
K. Svoboda and S. M. Block, �??Optical trapping of metallic Rayleigh particles,�?? Opt. Lett. 19, 930-932, (1994). [CrossRef] [PubMed]
M. Gu and D. Morrish, �??Three-dimensional trapping of Mie metallic particles by the use of obstructed laser beams,�?? J. Appl. Phys., 91, 1606-1612, (2002). [CrossRef]
S. Sato, Y. Harada and Y. Waseda, �??Optical trapping of microscopic metal particles,�?? Opt. Lett. 19, 1807-1809 (1994) [CrossRef] [PubMed]
Q. Zhan and J. R. Leger, "Focus shaping using cylindrical vector beams," Opt. Express 10, 324-331 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-324">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-324</a> [CrossRef] [PubMed]
Q. Zhan, �??Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,�?? J. Opt. A: pure appl. opt., 5, 229-232, (2003). [CrossRef]
R. Dorn, S Qubis, and G. Leuchs, �??Sharper focus for a radially polarized light beam,�?? Phys. Rev. Lett. 91, 233901, (2003). [CrossRef] [PubMed]
S. Quabis, R. Dorn, M. Eberler, O. Glöckl and G. Leuchs, �??Focusing light into a tighter spot,�?? Opt. Commun. 179, 1-7 (2000). [CrossRef]
K. S. Youngworth and T. G. Brown, �??Focusing of high numerical aperture cylindrical vector beams,�?? Opt. Express 7, 77-87 (2000), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-2-77.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-2-77."</a> [CrossRef] [PubMed]
H. Kano, S. Mizuguchi and S. Kawata, �??Excitation of surface-plasmon polaritons by a focused laser beam,�?? J. Opt. Soc. Am. B. 15, 1381-1386 (1998). [CrossRef]
E. Wolf, �??Electromagnetic diffraction in optical systems I. An integral representation of the image field,�?? Proc. R. Soc. Ser. A 253, 349-357 (1959). [CrossRef]
B. Richards and E. Wolf, �??Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,�?? Proc. R. Soc. London Ser. A 253, 358-379 (1959). [CrossRef]
M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, Academic (New York), 1969.
Y. Harada, T. Asakura, �??Radiation forces on a dielectric sphere in the Rayleigh scattering regime,�?? Opt. Commun. 124, 529-541 (1996). [CrossRef]
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]

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