James A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force," Appl. Opt. 43, 2545-2554 (2004)

The efficiency of trapping an on-axis spherical particle by use of laser tweezers for a particle size from the Rayleigh limit to the ray optics limit is calculated from generalized Lorenz-Mie light-scattering theory and the localized version of a Gaussian beam that has been truncated and focused by a high-numerical-aperture lens and that possesses spherical aberration as a result of its transmission through the wall of the sample cell. The results are compared with both the experimental trapping efficiency and the theoretical efficiency obtained from use of the localized version of a freely propagating focused Gaussian beam. The predicted trapping efficiency is found to decrease as a function of the depth of the spherical particle in the sample cell owing to an increasing amount of spherical aberration. The decrease in efficiency is also compared with experiment.

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Contribution to Minimum Radiation Trapping Efficiency Q^{min} of External Reflection (ER), Transmission (T), and Transmission Followed by p - 1 Internal Reflections (IR^{
p-1}) for the Localized Version of a Focused Gaussian Beam with λ = 0.488 μm, n = 1.33, m = 1.2, a = 5.0 μm, w_{
i
} = 0.172 μm, w_{
a
} = 0.200 μm, and z_{0}^{max} = -5.21 μm

Process

Contribution to Q^{min}

Coherent Sum

ER

+0.6589

+0.6589

T

-0.6863

-0.02888

IR^{1}

+0.00211

-0.02657

IR^{2}

+0.00010

-0.02607

IR^{3}

+0.00024

-0.02603

IR^{4}

+0.00033

-0.02605

IR^{5}

-0.00010

-0.02621

IR^{∞}

-0.02626

Table 2

Minimum Value of Radiation Trapping Efficiency Q^{min} As a Function of Particle Radius a for the Localized Version of a Focused Gaussian Beam with λ = 1.06 μm, n = 1.33, m = 1.18, and w_{
a
} = 0.390 μm Incident upon the Particlea

Position z_{0}^{max} is the location of the center of the focused Gaussian beam waist that corresponds to minimum trapping efficiency.
Ref. 2.
Eq. (13).

Table 3

Minimum Value of Radiation Trapping Efficiency Q^{min} As a Function of Particle Radius a for a Gaussian Beam Truncated and Focused by a Lens and Transmitted through a Flat Interface with λ = 1.06 μm, w_{
a
} = 0.390 μm, W/A = 1.5, α = 60°, n_{1} = 1.5, n_{2}= 1.33, and m = 1.18 Incident upon the Particlea

Position z_{0}^{SA} is the location of the spherical aberration’s principal diffraction maximum that corresponds to the minimum trapping efficiency.
Ref. 2.
Eq. (32).

Table 4

Minimum Value of Radiation Trapping Efficiency Q^{min} As a Function of Interface Position d for a Gaussian Beam Truncated and Focused by a Lens and Transmitted through a Flat Interface with λ = 1.06 μm, W/A = 1.5, α = 60°, n_{1} = 1.5, n_{2} = 1.33, m = 1.18, and a = 4.935 μma

Position z_{0}^{SA} is the location of the spherical aberration’s principal diffraction maximum that corresponds to the minimum trapping efficiency. The two columns labeled Ratio are the ratio of the trapping efficiency of the previous column divided by the corresponding trapping efficiency at d = -4.935 μm.
Ref. 2.
Eq. (32).

Tables (4)

Table 1

Contribution to Minimum Radiation Trapping Efficiency Q^{min} of External Reflection (ER), Transmission (T), and Transmission Followed by p - 1 Internal Reflections (IR^{
p-1}) for the Localized Version of a Focused Gaussian Beam with λ = 0.488 μm, n = 1.33, m = 1.2, a = 5.0 μm, w_{
i
} = 0.172 μm, w_{
a
} = 0.200 μm, and z_{0}^{max} = -5.21 μm

Process

Contribution to Q^{min}

Coherent Sum

ER

+0.6589

+0.6589

T

-0.6863

-0.02888

IR^{1}

+0.00211

-0.02657

IR^{2}

+0.00010

-0.02607

IR^{3}

+0.00024

-0.02603

IR^{4}

+0.00033

-0.02605

IR^{5}

-0.00010

-0.02621

IR^{∞}

-0.02626

Table 2

Minimum Value of Radiation Trapping Efficiency Q^{min} As a Function of Particle Radius a for the Localized Version of a Focused Gaussian Beam with λ = 1.06 μm, n = 1.33, m = 1.18, and w_{
a
} = 0.390 μm Incident upon the Particlea

Position z_{0}^{max} is the location of the center of the focused Gaussian beam waist that corresponds to minimum trapping efficiency.
Ref. 2.
Eq. (13).

Table 3

Minimum Value of Radiation Trapping Efficiency Q^{min} As a Function of Particle Radius a for a Gaussian Beam Truncated and Focused by a Lens and Transmitted through a Flat Interface with λ = 1.06 μm, w_{
a
} = 0.390 μm, W/A = 1.5, α = 60°, n_{1} = 1.5, n_{2}= 1.33, and m = 1.18 Incident upon the Particlea

Position z_{0}^{SA} is the location of the spherical aberration’s principal diffraction maximum that corresponds to the minimum trapping efficiency.
Ref. 2.
Eq. (32).

Table 4

Minimum Value of Radiation Trapping Efficiency Q^{min} As a Function of Interface Position d for a Gaussian Beam Truncated and Focused by a Lens and Transmitted through a Flat Interface with λ = 1.06 μm, W/A = 1.5, α = 60°, n_{1} = 1.5, n_{2} = 1.33, m = 1.18, and a = 4.935 μma

Position z_{0}^{SA} is the location of the spherical aberration’s principal diffraction maximum that corresponds to the minimum trapping efficiency. The two columns labeled Ratio are the ratio of the trapping efficiency of the previous column divided by the corresponding trapping efficiency at d = -4.935 μm.
Ref. 2.
Eq. (32).