1Sector for Biological and Soft Systems, Department of Physics, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK
Lars Boyde, Kevin J. Chalut, and Jochen Guck, "Exact analytical expansion of an off-axis Gaussian laser beam using the translation theorems for the vector spherical harmonics," Appl. Opt. 50, 1023-1033 (2011)
The interaction of a Gaussian laser beam with a particle that is located off axis is a fundamental problem encountered across many scientific fields, including biological physics, chemistry, and medicine. For spherical geometries, generalized Lorenz–Mie theory affords a solution of Maxwell’s equations for the scattering from such a particle. The solution can be obtained by expanding the laser fields in terms of vector spherical harmonics (VSHs). However, the computation of the VSH expansion coefficients for off-axis beams has proven challenging. In the present study, we provide a very viable, theoretical framework to efficiently compute the sought-after expansion coefficients with high numerical accuracy. We use the existing theory for the expansion of an on-axis laser beam and employ Cruzan’s translation theorems [Q. Appl. Math. 20, 33 (1962)QAMAAY0033-569X] for the VSHs to obtain a description for more general off-axis beams. The expansion coefficients for the off-axis laser beam are presented in an analytical form in terms of an infinite series over the underlying translation coefficients. A direct comparison of the electromagnetic fields of such a beam expansion with the original laser fields and with results obtained using numerical quadratures shows excellent agreement (relative errors are on the order of ). In practice, the analytical approach presented in this study has numerous applications, reaching from multiparticle scattering problems in atmospheric physics and climatology to optical trapping, sorting, and sizing techniques.
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Comparison of the Exact () and Expanded Polar Laser Field Components () at Various Points and for Different Beam Waists a
Parameters
Field Components
Relative Errors
θ
ϕ
(%)
5
0
0
8
5
0
8
5
8
—
20
0
0
4
20
0
4
20
4
—
The primed and unprimed coordinate systems, S and , coincide, and the location of the laser focus is at their common coordinate origin, (i.e., ). The magnitude of the relative errors of the absolute values of the fields is shown in the last column.
Table 2
Comparison of the Exact () and Expanded Laser Field Components for Zero () and Nonzero () Offset of the Laser Focusa
Parameters
Field Components
Relative Errors
(%)
(%)
5
16
20
10
20
20
15
26
20
20
31
20
25
36
20
30
41
20
The fields are evaluated at the point (, , ) in . The laser focus is confined to on-axis positions (i.e., and ), while its coordinate is varied either by increasing the axial translation distance, , or by shifting the offset, , in the unprimed coordinate system, S. The different numbers of expansion terms for zero and nonzero offset ( and ) are also quoted. The beam waist of the laser is .
Table 3
Comparison of Different Orders of Expansion Coefficients Computed Using Either Numerical Quadrature (, ) or Analytical Expressions (, )a
Orders
Expansion Coefficients
n
m
5
5
10
10
20
20
The focus of the laser (, ) is located off axis at . For conciseness, only the absolute values of the expansion coefficients are shown.
Table 4
Comparison of the Analytical Expansion Coefficients of a Plane Wave () with the Field Expansion Coefficients of a Gaussian Laser Beam () in the Plane Wave Limit (, )a
Orders
Expansion Coefficients
Relative Errors
n
m
(%)
5
5
10
10
20
20
The focus of the Gaussian laser beam () is off axis ( and ).
Tables (4)
Table 1
Comparison of the Exact () and Expanded Polar Laser Field Components () at Various Points and for Different Beam Waists a
Parameters
Field Components
Relative Errors
θ
ϕ
(%)
5
0
0
8
5
0
8
5
8
—
20
0
0
4
20
0
4
20
4
—
The primed and unprimed coordinate systems, S and , coincide, and the location of the laser focus is at their common coordinate origin, (i.e., ). The magnitude of the relative errors of the absolute values of the fields is shown in the last column.
Table 2
Comparison of the Exact () and Expanded Laser Field Components for Zero () and Nonzero () Offset of the Laser Focusa
Parameters
Field Components
Relative Errors
(%)
(%)
5
16
20
10
20
20
15
26
20
20
31
20
25
36
20
30
41
20
The fields are evaluated at the point (, , ) in . The laser focus is confined to on-axis positions (i.e., and ), while its coordinate is varied either by increasing the axial translation distance, , or by shifting the offset, , in the unprimed coordinate system, S. The different numbers of expansion terms for zero and nonzero offset ( and ) are also quoted. The beam waist of the laser is .
Table 3
Comparison of Different Orders of Expansion Coefficients Computed Using Either Numerical Quadrature (, ) or Analytical Expressions (, )a
Orders
Expansion Coefficients
n
m
5
5
10
10
20
20
The focus of the laser (, ) is located off axis at . For conciseness, only the absolute values of the expansion coefficients are shown.
Table 4
Comparison of the Analytical Expansion Coefficients of a Plane Wave () with the Field Expansion Coefficients of a Gaussian Laser Beam () in the Plane Wave Limit (, )a
Orders
Expansion Coefficients
Relative Errors
n
m
(%)
5
5
10
10
20
20
The focus of the Gaussian laser beam () is off axis ( and ).