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Tailoring the axial shape of the point spread function using the Toraldo concept

M. Martínez-Corral, M. T. Caballero, E. H. K. Stelzer, and J. Swoger

Open Access Open Access

Abstract

A novel procedure for shaping the axial component of the point spread function of nonparaxial focusing systems by use of phase-only pupil filters is presented. The procedure is based on the Toraldo technique for tailoring focused fields. The resulting pupil filters consist of a number of concentric annular zones with constant real transmittance. The number of zones and their widths can be adapted according to the shape requirements. Our method is applied to design filters that produce axial superresolution in confocal scanning systems.

©2002 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. (a) Normalized axial intensity PSF corresponding to the circular pupil and to the three-zone Toraldo filter; (b) contour plot of the 3D intensity PSF in the meridian plane corresponding to the circular pupil; (c) as b) but for the three-zone Toraldo filter. The normalized radial coordinate is rN =(n/λ)r sin2 α.

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Fig. 2.
Fig. 2. Three-zones Toraldo filter: (a) mapped transmittance; (b) actual 2D representation for the cases of α = 10° (1) and α = 67.5° (2). Note that the form of the pupil filter strongly depends on the value of α.

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Fig. 3.
Fig. 3. (a) Seven-zone Toraldo filter designed for obtaining axial superresolution in confocal fluorescence microscopy. The filter was calculated for the case of α = 67.5° ; (b) axial intensity PSF.

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Fig. 4.
Fig. 4. (a) Seven-zone Toraldo filter designed to obtain axial superresolution in reflection confo-cal microscopy. The filter was calculated for the case of α = 67.5° ; (b) 3D intensity PSF of a reflection confocal microscope with two circular pupils; (c) as b) but with the seven-zone Toraldo filter in the illumination system.

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Equations (11)

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h ( z ) = ∫ 0 α A ( θ ) exp ( i 2 π n cos θ λ z ) sin θ dθ ,
ζ = cos θ − cos α 1 − cos α − 0.5 ; q ( ζ ) = A ( θ )
h ( z N ) = ( 1 − cos α ) exp ( i π 1 + cos α 1 − cos α z N ) ∫ − 0.5 0.5 q ( ζ ) exp ( i 2 πζ z N ) dζ ,
z N = n λ ( 1 − cos α ) z .
q ( ζ ) = ∑ i = 1 m [ k i a i ( ζ ) − k i − 1 a i − 1 ( ζ ) ] , where a i ( ζ ) = { rect ( ζ / Δ i ) if i = 1 , … , m 0 if i = 0 .
h ( z N ) = ∑ i = 1 m k i a ˜ i ( z N ) , where a ˜ i ( z N ) = Δ i sin c ( Δ i z N ) − Δ i − 1 sin c ( Δ i − 1 z N ) .
h ( z N ) = ( k 1 − k 2 ) Δ sin c ( Δ z N ) + k 2 sin c ( z N ) , with Δ = Δ 1 .
( k 1 − k 2 ) Δ + k 2 = 1 .
( k 1 − k 2 ) Δ sin c ( Δ z 1 ) + k 2 sin c ( z 1 ) = 0 .
k 1 = sin ( πΔ z 1 ) − sin ( π z 1 ) sin ( πΔ z 1 ) − Δ sin ( π z 1 ) and k 2 = sin ( πΔ z 1 ) sin ( πΔ z 1 ) − Δ sin ( π z 1 ) .
Δ = sin c ( z 1 ) 2 sin c ( Δ z 1 ) .
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