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Focusing of high numerical aperture cylindrical-vector beams

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Abstract

Cylindrical-vector beams are of increasing recent interest for their role in novel laser resonators and their applications to electron acceleration and scanning microscopy. In this paper, we calculate cylindrical-vector fields, near the focal region of an aplanatic lens, and briefly discuss some applications. We show that, in the particular case of a tightly focused, radially polarized beam, the polarization shows large inhomogeneities in the focal region, while the azimuthally polarized beam is purely transverse even at very high numerical apertures.

©2000 Optical Society of America

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Supplementary Material (1)

Media 1: MOV (2183 KB)     

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Figures (10)

Fig. 1.
Fig. 1. Geometry of the problem. The input apodization is defined in plane 0, conjugate to focal sphere 1.
Fig. 2.
Fig. 2. Normalized intensity of the transverse (radial) component of a high-NA (1.32) radially polarized beam at focus and through focus. Intensities of 0 and 1 correspond to black and white, respectively. The units of x, y, ρ, and z are in wavelengths.
Fig. 3.
Fig. 3. Normalized intensity of the longitudinal (z-) component of a high-NA (1.32) radially polarized beam at focus and through focus. Intensities of 0 and 1 correspond to black and white, respectively. The units of x, y, ρ, and z are in wavelengths.
Fig. 4.
Fig. 4. The ratio of the maximum intensities of the longitudinal (max I z) and radial transverse (max Iρ) fields versus the focusing angle (α). The dotted line shows where max Iz becomes larger than max Iρ. For the example, we used an objective with NA=1.32 in oil, or α≈1.05 radians; max Iz is 1.4 times larger than max Iρ.
Fig. 5.
Fig. 5. Normalized intensity of the transverse (azimuthal) component of a high-NA (1.32) azimuthally polarized beam at focus and through focus. Intensities of 0 and 1 correspond to black and white, respectively. The units of x, y, ρ, and z are in wavelengths.
Fig. 6.
Fig. 6. Movie (2.1 MB) of an azimuthally polarized beam that was created by using a modified Mach-Zehnder interferometer to convert linear polarization to azimuthal polarization.
Fig. 7.
Fig. 7. A small scatterer on a metalized region of a silicon integrated circuit taken in dark-field confocal (IPC) mode with azimuthal polarization with a 40x (NA=0.65) objective.
Fig. 8.
Fig. 8. Normalized intensity of the transverse (y-) component of a high-NA (1.32) linearly polarized beam. Intensities of 0 and 1 correspond to black and white, respectively. The units of x and y are in wavelengths.
Fig. 9.
Fig. 9. Normalized intensity of the longitudinal (z-) component of a high-NA (1.32) linearly polarized beam. Note that the intensity is normalized from 0 to 0.1 instead of 0 to 1 as in previous figures. The units of x and y are in wavelengths.
Fig. 10.
Fig. 10. Normalized intensity of the transverse (x-) component of a high-NA (1.32) linearly polarized beam. Note that the intensity is normalized from 0 to 0.01. The units of x and y are in wavelengths.

Equations (14)

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e 0 = l 0 [ e r ( 0 ) g 0 + e ϕ ( 0 ) ( g 0 × k ̂ ) ] ,
e ( s ) = ik 2 π Ω a 1 ( θ , ϕ ) e ik ( s ̂ 1 r ) d Ω
a 1 = f 1 cos 1 2 ( θ ) l 0 ( θ ) [ e r ( 0 ) g 1 + e ϕ ( 0 ) ( g 1 × s 1 ) ] .
g 1 = cos θ ( cos ϕ i ̂ + sin ϕ j ̂ ) + sin θ k ̂ .
e ( s ) = [ e x ( s ) e y ( s ) e z ( s ) ] = iA π 0 α 0 2 π sin θ cos 1 2 θ l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) [ cos θ cos ϕ cos θ sin ϕ sin θ ] d ϕ d θ
e ϕ ( s ) = e y ( s ) cos ϕ S e x ( s ) sin ϕ S
e ρ ( s ) = e x ( s ) cos ϕ S e y ( s ) sin ϕ S
e ρ ( s ) = iA π 0 α 0 2 π cos 1 2 θ sin θ cos θ cos ( ϕ ϕ S ) l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) d ϕ d θ
e z ( s ) = iA π 0 α 0 2 π cos 1 2 θ sin 2 θ l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) d ϕ d θ
e ρ ( s ) ( ρ s , z s ) = A 0 α cos 1 2 θ sin ( 2 θ ) l 0 ( θ ) J 1 ( k ρ s sin θ ) e ikz s cos θ d θ
e z ( s ) ( ρ s , z s ) = 2 i A 0 α cos 1 2 θ sin 2 θ l 0 ( θ ) J 0 ( k ρ s sin θ ) e ikz s cos θ d θ
e ( s ) = [ e x ( s ) e y ( s ) e z ( s ) ] = iA π 0 α 0 2 π sin θ cos 1 2 θ l 0 ( θ ) e ik ( z S cos θ + ρ S sin θ cos ( ϕ ϕ S ) ) [ sin ϕ cos ϕ 0 ] d ϕ d θ
e ϕ ( s ) ( ρ S , z S ) = 2 A 0 α cos 1 2 θ sin θ l 0 ( θ ) J 1 ( k ρ S sin θ ) e ikz S cos θ d θ
l 0 ( θ ) = exp [ β 0 2 ( sin θ sin α ) 2 ] J 1 ( 2 β 0 sin θ sin α )
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