Abstract

We study the effect of primary aberrations on the 3-D polarization of the electric field in a focused lowest order radially polarized beam. A full vector diffraction treatment of the focused beams is used. Attention is given to the effects of primary spherical, astigmatic, and comatic aberrations on the local polarization, Strehl ratio, and aberration induced degradation of the longitudinal field at focus.

© 2004 Optical Society of America

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References

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  5. D. P. Biss, K. S. Youngworth, and T. G. Brown, �??Longitudinal field imaging,�?? in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing X, J. Conchello, C. J. Cogswell, and T.Wilson, eds., Proc. SPIE 4964, 73-83, (2003)
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Appl. Opt. (6)

J. Mod. Opt. (2)

R. Kant, �??An analytical method of vector diffraction for focusing optical systems with Seidel aberrations. II. Astigmatism and coma,�?? J. Mod. Opt. 42, 299-320, (1995)
[CrossRef]

R. Kant, �??An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberration, curvature of field, and distortion,�?? J. Mod. Opt. 40, 2293-2310, (1993)
[CrossRef]

J. Opt. Soc. Am. (2)

Pamela L. Greene and Dennis G. Hall, �??Diffraction characteristics of the azimuthal Bessel-Gauss beam,�?? J. Opt. Soc. Am. 13, 962-966, (1996)
[CrossRef]

Colin J. R. Sheppard and Peter T¨or¨ok, �??Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,�?? J. Opt. Soc. Am. 20, 2156-2162, (2003)
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (4)

S. Quabis, R. Dorn, M. Eberler, and O. Glockl, and G. Leuchs, �??Focusing light to a tighter spot,�?? Opt. Commun. 179, 1-7, (2000
[CrossRef]

L. E. Helseth, �??Roles of polarization, phase, and amplitude in solid immersion lens systems,�?? Opt. Commun. 191, 161-172, (2001)
[CrossRef]

R. Kant, �??Vector diffraction in paraboloidal mirrors with Seidel aberrations. I. Spherical aberration, curvature of field aberration and distortion,�?? Opt. Commun. 128, 292-306, (1996)
[CrossRef]

C.J.R. Sheppard, �??Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,�?? Opt. Commun. 138, 262-264, (1997)
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, �??Longitudinal field modes probed by single molecules,�?? Phys. Rev. Lett. 86, 5251-5253, (2001)
[CrossRef] [PubMed]

Proc. SPIE (4)

K. S. Youngworth, D. P. Biss, and T. G. Brown, �??Point spread functions for particle imaging using inhomogeneous polarization in scanning optical microscopy,�?? in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VIII, J. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 4261, 14-23, (2001)

D. P. Biss, K. S. Youngworth, and T. G. Brown, �??Longitudinal field imaging,�?? in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing X, J. Conchello, C. J. Cogswell, and T.Wilson, eds., Proc. SPIE 4964, 73-83, (2003)

K. S. Youngworth and T. G. Brown, �??Inhomogeneous polarization in scanning optical microscopy,�?? in Three- Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J. Conchello, C. J. Cogswell, A. G. Tescher, and T. Wilson, eds., Proc. SPIE 3919, 75-85, (2000)

C.J.R. Sheppard and P. Torok, �??Confocal imaging through a dielectric interface: aberration modeling,�?? in Three- Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, and T.Wilson, eds. Proc. SPIE 2655, 280-286, (1996)

Other (2)

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, UK ,1999)

Peter Torok and Fu-Jen Kao, Optical imaging and microscopy : techniques and advanced systems, (Springer Press, New York, NY, USA, 2003)

Supplementary Material (6)

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Figures (7)

Fig. 1.
Fig. 1.

A figure describing the parameters of the focusing system. A field is being focused by a lens with focal length f. The local field positions at focus are described by coordinates x, y, and z. Focused plane wave components can be described by azimuthal angle φ and polar angle θ.

Fig. 2.
Fig. 2.

a(left) (2.7 MB), b(right) (1.4 MB) two animations of focused radially polarized beams with varying amounts of spherical aberration. The amount of aberration varies from -2λ to 2λ. Axis dimensions are given in units of wavelengths (λ). Movie (a) is an animation of the field in a transverse plane (x, y) at the geometrical focus of the beam (z=0). Movie (b) is an animation of the field in the ρ,z plane of the field, centered at y=0.

Fig. 3.
Fig. 3.

a(left) (1.4 MB), b(right) (2.4 MB) two animations of focused radially polarized beams with varying amounts of coma. The amount of aberration varies from -2λ to 2λ. Axis dimensions are given in units of wavelengths (λ). Movie (a) is an animation of the field in a transverse plane (x, y) at the geometrical focus of the beam (z=0). Movie (b) is an animation of the field in the ρ,z plane of the field, centered at y=0.

Fig. 4.
Fig. 4.

a(left) (1.8 MB), b(right) (1.7 MB) two animations of focused radially polarized beams with varying amounts of astigmatism. The amount of aberration varies from -2λ to 2λ. Axis dimensions are given in units of wavelengths (λ). Movie (a) is an animation of the field in a transverse plane (x, y) at the geometrical focus of the beam (z=0). Movie (b) is an animation of the field in the ρ,z plane of the field, centered at y=0.

Fig. 5.
Fig. 5.

Strehl Ratio comparison. The solid lines are the Strehl ratios for a linearly polarized gaussian input beam with varying amounts of spherical aberration. The dotted lines are the Strehl ratios for a linearly polarized annular input beam with varying amounts of spherical aberration. The dashed lines are the Strehl ratios for a radially polarized input beam with varying amounts of spherical aberration. The blue lines correspond to a system with 0λ of spherical aberration. The green lines correspond to a system with 0.5λ of spherical aberration. The red lines correspond to a system with 1λ of spherical aberration.

Fig. 6.
Fig. 6.

Strehl ratio comparison for different amounts of tube length error w 020. In the figure the blue lines represent Strehl ratios of beams with w 020=0λ, the green lines are Strehl ratios of beams with w 020=0.5λ, and the red lines are Strehl ratios of beams with w 020=1λ. The solid lines are Strehl ratios for focused radially polarized beams, the dashed lines are for focused linearly polarized beams.

Fig. 7.
Fig. 7.

Figure showing the ratio of maximum longitudinal intensity to maximum radial intensity for varying amounts of coma, astigmatism, and spherical aberration. The red line is the ratio for varying amounts of spherical aberration. The green line is the ratio for varying amounts of astigmatism. The blue line is the ratio for varying amounts of coma. The x axis is the amount of aberration in waves (λ). The y axis is the ratio of the maximum longitudinal field intensity at focus versus the maximum radial field intensity at focus, which is a unit-less quantity.

Equations (11)

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E f ( x , y , z ) = i k 2 π e i k f k x 2 + k y 2 k m a x l o ( k x , k y ) k z k 1 k z ×
[ k x k ρ k z k k y k ρ k z k k ρ k ] e i ( k x x + k y y + k z z ) d k x d k y ,
l o ( r ) = exp ( r 2 w o 2 ) J 1 ( β r ) ,
l o ( k x , k y ) = exp ( β 2 k ρ 2 k max 2 ) J 1 ( 2 β k ρ k max ) .
exp ( i k W ( ρ , φ ) ) ,
W ( k x , k y ) = w 040 ( k ρ k max ) 4 + w 131 k x k ρ ( k ρ k max ) 3 + w 222 ( k x k ρ ) 2 ( k ρ k max ) 2 ,
E f ( x , y , z ) = i k 2 π e i k f k x 2 + k y 2 k m a x
× exp ( i k ( w 040 ( k ρ k max ) 2 + w 131 k x k ρ ( k ρ k m a x ) 3 + w 22 ( k x k ρ ) 2 ( k ρ k m a x ) 2 ) )
× exp ( β 2 k ρ 2 k m a x 2 ) J 1 ( 2 β k ρ k m a x ) k z k 1 k z [ k x k x 2 + k y 2 k z k k y k x 2 + k y 2 k z k k x 2 + k y 2 k ]
× e i ( k x x + k y y + k z z ) d k x d k y .
S = I a b I o ,

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