Abstract

The superposition of two coaxial Gaussian beams with offset foci and orthogonal linear polarizations can be used to produce a right- or left- circular polarization component with a focal spot of volume smaller than that of the Gaussian beam. This polarization-assisted axial and transverse superresolution effect is attributed to the differential Gouy phase shift within the focal region or to the non-Gaussian annular distribution of the circularly-polarized components in the far field.

© 2003 Optical Society of America

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References

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  1. T. R. M. Sales, �??Smallest focal spot�?? Phys. Rev. Lett. 81, 3844-3847 (1998).
    [CrossRef]
  2. W. Lukosz, �??Optical systems with resolving powers exceeding the classical limit I�?? J. Opt. Soc. Am. 56, 1463-1472 (1966).
    [CrossRef]
  3. W. Lukosz, �??Optical systems with resolving powers exceeding the classical limit II�?? J. Opt. Soc. Am. 57, 932-941 (1967).
    [CrossRef]
  4. D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, �??Two- dimensional superresolution optical system for temporally restricted objects�?? Appl. Optics 36, 6687-6691 (1997).
    [CrossRef]
  5. D. Mendlovic, D. Farkas, Z. Zalevsky, and A. W. Lohmann, �??High-frequency enhancement by an optical system for superresolution of temporally restricted objects�?? Opt. Lett. 23, 801-803 (1998).
    [CrossRef]
  6. A. I. Kartashev, �??Optical systems with enhanced resolving power�?? Opt. Spectrosc. (USSR) 9, 204-206 (1960).
  7. W. Gartner and A. W. Lohmann, �??An experiment going beyond Abbe�??s limit of Diffraction�?? Z. Phys. 174, 18-21 (1963).
  8. D. Mendlovic and A. W. Lohmann, �??Space-bandwidth product adaptation and its application to superresolution: Fundamentals�?? J. Opt. Soc. Am. A 14, 558-562 (1997).
    [CrossRef]
  9. D. Mendlovic, A. W. Lohmann, and Z. Zalevsky, �??Space-bandwidth product adaptation and its application to superresolution: Examples�?? J. Opt. Soc. Am. A 14, 563-567 (1997).
    [CrossRef]
  10. Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, �??Understanding superresolution in Wigner space�?? J. Opt. Soc. Am. A 17, 2422-2430 (2000).
    [CrossRef]
  11. W. Denk, J. H. Strickler, andW.Webb, �??Two-photon laser scanning fluorescence microscopy�?? Science 248, 73-76 (1990).
    [CrossRef] [PubMed]
  12. S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, �??Finer features for functional microdevices�?? Nature 412, 697-698 (2001).
    [CrossRef] [PubMed]
  13. S. W. Hell and J. Wichmann, �??Breaking the diffraction resolution limit by stimulated emission: stimulatedemission- depletion fluorescence microscopy�?? Opt. Lett. 19, 780-782 (1994).
    [CrossRef] [PubMed]
  14. M. Dyba and S. W. Hell, �??Focal spots of size ë/23 open up far-field fluorescence microscopy at 33 nm axial resolution�?? Phys. Rev. Lett. 88, 163901 (2002).
    [CrossRef] [PubMed]
  15. T. Brixner and G. Gerber �??Femtosecond polarization pulse shaping�?? Opt. Lett. 26, 557-559 (2001).
    [CrossRef]
  16. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  17. M. Gouy, �??Sur la propagation anomale des ondes,�?? Comp. Rend. Acad. Sci. 111, 33-40 (1890).
    [CrossRef]
  18. J. J. Stamnes, Waves in focal regions (Adam Hilger, Bristol and Boston, 1986).
  19. C. J. Sheppard and Z. S. Hegesdus, �??Axial behavior of pupil-plane filters�?? J. Opt. Soc. Am. A 5, 643-647 (1988).
    [CrossRef]
  20. M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, and G. Saavedra, �??Tunable axial superresolution by annular binary filters - application to confocal microscopy�?? Opt. Commun. 119, 491-498 (1995).
    [CrossRef]
  21. T. R. M. Sales and G. M. Morris, �??Axial superresolution with phase-only pupil filters�?? Opt. Commun. 156, 227-230 (1998).
    [CrossRef]
  22. Y. Li, �??Focal shift and focal switch in dual-focus systems�?? J. Opt. Soc. Am. A 14, 1297-1304 (1997).
    [CrossRef]
  23. B. Bailey, D. Farkas, D. Taylor, and F. Lanni, �??Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,�?? Nature 366, 44-48 (1993)
    [CrossRef] [PubMed]
  24. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, UK, 7th edition, 1999).
  25. S. Feng and H. Winful, �??Physical origin of the Gouy phase shift�?? Opt. Lett. 26, 485-487 (2001).

Appl. Optics (1)

D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann, and D. Farkas, �??Two- dimensional superresolution optical system for temporally restricted objects�?? Appl. Optics 36, 6687-6691 (1997).
[CrossRef]

Comp. Rend. Acad. Sci. (1)

M. Gouy, �??Sur la propagation anomale des ondes,�?? Comp. Rend. Acad. Sci. 111, 33-40 (1890).
[CrossRef]

Femtosecond polarization pulse shaping (1)

T. Brixner and G. Gerber �??Femtosecond polarization pulse shaping�?? Opt. Lett. 26, 557-559 (2001).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Nature (2)

B. Bailey, D. Farkas, D. Taylor, and F. Lanni, �??Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,�?? Nature 366, 44-48 (1993)
[CrossRef] [PubMed]

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, �??Finer features for functional microdevices�?? Nature 412, 697-698 (2001).
[CrossRef] [PubMed]

Opt. Commun. (2)

M. Martinez-Corral, P. Andres, J. Ojeda-Castaneda, and G. Saavedra, �??Tunable axial superresolution by annular binary filters - application to confocal microscopy�?? Opt. Commun. 119, 491-498 (1995).
[CrossRef]

T. R. M. Sales and G. M. Morris, �??Axial superresolution with phase-only pupil filters�?? Opt. Commun. 156, 227-230 (1998).
[CrossRef]

Opt. Lett. (3)

Opt. Spectrosc. (1)

A. I. Kartashev, �??Optical systems with enhanced resolving power�?? Opt. Spectrosc. (USSR) 9, 204-206 (1960).

Phys. Rev Lett (1)

T. R. M. Sales, �??Smallest focal spot�?? Phys. Rev. Lett. 81, 3844-3847 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

M. Dyba and S. W. Hell, �??Focal spots of size ë/23 open up far-field fluorescence microscopy at 33 nm axial resolution�?? Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

Science (1)

W. Denk, J. H. Strickler, andW.Webb, �??Two-photon laser scanning fluorescence microscopy�?? Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Z. Phys. (1)

W. Gartner and A. W. Lohmann, �??An experiment going beyond Abbe�??s limit of Diffraction�?? Z. Phys. 174, 18-21 (1963).

Other (3)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

J. J. Stamnes, Waves in focal regions (Adam Hilger, Bristol and Boston, 1986).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, UK, 7th edition, 1999).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Superposition of x- and y-polarized coaxial Gaussian beams with centers offset by a distance 2Δ. (b) The Gouy phase shifts associated with the two beams. (c) Difference of the Gouy phases.

Fig. 2.
Fig. 2.

(a) Axial distributions of the LCP component (solid curve) and the constituent Gaussian linearly polarized beam (dashed). (b) Radial distributions of the LCP component (solid curve) and the constituent Gaussian linearly polarized beam (dashed).

Fig. 3.
Fig. 3.

Dependence of the axial and radial widths (FWHM) and the peak intensity of the LCP (minority) component on the offset parameter r=Δ/zo . All values are normalized to the corresponding values for the constituent Gaussian beam.

Fig. 4.
Fig. 4.

Intensity distribution of the LCP beam (top) and a reference linearly polarized Gaussian beam (bottom). The distributions in the meridional plane x=0 (right) and the transverse plane z/zo =-5 (left) are shown. The axial position z is in units of zo and the transverse dimensions are in units of Wo . The offset ratio r=Δ/zo =0.25. In each figure, the intensity is normalized such that the maximum value is unity.

Fig. 5.
Fig. 5.

Proposed scheme for creating polarization-assisted superresolution. Two Gaussian beams of the same width and with a slight difference in curvature are focused by a lens. The x- and y-polarized components have a 90° phase shift at the beam axis. The left-circularly polarized component (minority) is highly confined in the focal region.

Fig. 6.
Fig. 6.

Intensity distributions (above) and contours of constant intensity (below) for the minority component (left) and the reference Gaussian beam (right). The contours are at fractions of the peak intensity of 2/3 (solid red line), 1/2 (dashed green line), and 1/3 (dotted blue line).

Equations (20)

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U ( ρ , z ) = f x ( ρ , z ) x ̂ + f y ( ρ , z ) y ̂ ,
U ( ρ , z ) = f R ( ρ , z ) R ̂ + f L ( ρ , z ) L ̂ ,
f R , L ( ρ , z ) = 1 2 [ f x ( ρ , z ) j f y ( ρ , z ) ]
I R , L ( ρ , z ) = 1 2 [ I x ( ρ , z ) + I y ( ρ , z ) ] [ I x ( ρ , z ) I y ( ρ , z ) ] 1 2 sin [ ϕ x ( ρ , z ) ϕ y ( ρ , z ) ] ,
I x ( ρ , z ) = I ( ρ , z Δ ) , I y ( ρ , z ) = I ( ρ , z + Δ ) ,
ϕ x ( ρ , z ) = ϕ ( ρ , z Δ ) , ϕ y ( ρ , z ) = ϕ ( ρ , z + Δ ) + ξ ,
I ( ρ , z ) = [ W o W ( z ) ] 2 exp [ 2 ρ 2 W 2 ( z ) ] , ϕ ( ρ , z ) = k z k ρ 2 2 R ( z ) + η ( z ) .
I ( 0 , z ) = [ 1 + ( z z o ) 2 ] 1 , ϕ ( 0 , z ) = kz + η ( z ) .
I R , L ( ρ , z ) = I o ( ρ , z ) ± I 1 ( ρ , z ) cos [ ζ ( ρ , z ) ] ,
I o ( ρ , z ) = 1 2 [ I ( ρ , z Δ ) + I ( ρ , z + Δ ) ] ,
I 1 ( ρ , z ) = [ I ( ρ , z Δ ) I ( ρ , z + Δ ) ] 1 2 ,
ζ ( ρ , z ) = η ( z Δ ) η ( z + Δ ) k ρ 2 2 R 1 ( z ) ,
1 R 1 ( z ) = 1 R ( z Δ ) 1 R ( z + Δ ) .
z FWHM ( L ) = 2 z o { [ 2 ( 1 + r 4 ) ] 1 2 ( 1 r 2 ) } 1 2 ,
I L ( 0 , 0 ) = 2 r 2 [ 1 + r 2 ] 2
I L ( ρ , 0 ) = 2 I ( ρ , Δ ) sin 2 [ ζ ( ρ , z ) 2 ] ,
= 2 1 + r 2 exp ( 2 1 + r 2 ρ 2 W o 2 ) sin 2 [ arctan ( r ) r 1 + r 2 ρ 2 W o 2 ]
f R , L ( x , y , z ) = dx dy G ( z ) ( x , y ; x , y ) f R , L ( x , y , 0 ) .
dx dy G ( z ) ( x , y ; x , y ) G ( z ) * ( x , y ; x , y ) = δ ( x x ; y y ) .
P R ( z ) = dx dy f R ( x , y , z ) 2 = dx dy f R ( x , y , 0 ) 2 = P R ( 0 ) .

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