Abstract

We investigate the form of the image of a finite sized spherical particle in confocal and conventional microscopes when the illuminating light has an arbitrary polarization. In particular, we take the cases of radial and azimuthal polarizations and use the Mie theory to find the scattered field from differently sized particles for these cases. We present numerical results for the changes in the detected intensity when subresolution and resolvable spherical particles are illuminated with particular wavelengths and polarizations. Further, we find the limiting size of a particle for which it can be considered a point scatterer for a particular wavelength.

© 2008 Optical Society of America

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References

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  1. J. Mertz, T. Pons, L. Moreaux, C. Yang, and S. Chaprak, "Biological imaging with second harmonic generation," in Proceedings of the Conference on Laser and Electro-Optics (CLEO), Vol. 2 of 2004. OSA Technical Digest Series (CD) (Optical Society of America, 2004), paper CThS5.
  2. T. Wilson, R. Juskaitis, and P. Hidgon, "The imaging of dielectric point scatterers in conventional and confocal polarizations microscopes," Opt. Commun. 141, 298-313 (1997).
    [CrossRef]
  3. P. Török, P. Hidgon, R. Juskaitis, and T. Wilson, "Optimizing the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers," Opt. Commmun. 155, 335-341 (1998).
    [CrossRef]
  4. P. Li, S. Kebin, and Z. Liu, "Chromatic confocal microscopy using super continuum light," Opt. Express 13, 9039-9044 (2005).
    [CrossRef] [PubMed]
  5. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1999).
  6. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys 64, 1632-1639 (1988).
    [CrossRef]
  7. T. Wilson, F. Massoumian, and R. Juskaitis, "Generating and focusing of radially polarized electric fields," Opt. Eng. 42, 3088-3089 (2003).
    [CrossRef]
  8. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London 253, 358-379.
  9. D. Cheng, Field and Wave Electromagnetics (Tehran U. Press, 2003).
  10. G. H. Hoster, Analytical, Numerical and Computational Methods for Science and Engineering (Prentice-Hall, 1999).

2005 (1)

2003 (1)

T. Wilson, F. Massoumian, and R. Juskaitis, "Generating and focusing of radially polarized electric fields," Opt. Eng. 42, 3088-3089 (2003).
[CrossRef]

1998 (1)

P. Török, P. Hidgon, R. Juskaitis, and T. Wilson, "Optimizing the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers," Opt. Commmun. 155, 335-341 (1998).
[CrossRef]

1997 (1)

T. Wilson, R. Juskaitis, and P. Hidgon, "The imaging of dielectric point scatterers in conventional and confocal polarizations microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

1988 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys 64, 1632-1639 (1988).
[CrossRef]

J. Appl. Phys (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys 64, 1632-1639 (1988).
[CrossRef]

Opt. Commmun. (1)

P. Török, P. Hidgon, R. Juskaitis, and T. Wilson, "Optimizing the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers," Opt. Commmun. 155, 335-341 (1998).
[CrossRef]

Opt. Commun. (1)

T. Wilson, R. Juskaitis, and P. Hidgon, "The imaging of dielectric point scatterers in conventional and confocal polarizations microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

Opt. Eng. (1)

T. Wilson, F. Massoumian, and R. Juskaitis, "Generating and focusing of radially polarized electric fields," Opt. Eng. 42, 3088-3089 (2003).
[CrossRef]

Opt. Express (1)

Proc. R. Soc. London (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London 253, 358-379.

Other (4)

D. Cheng, Field and Wave Electromagnetics (Tehran U. Press, 2003).

G. H. Hoster, Analytical, Numerical and Computational Methods for Science and Engineering (Prentice-Hall, 1999).

J. Mertz, T. Pons, L. Moreaux, C. Yang, and S. Chaprak, "Biological imaging with second harmonic generation," in Proceedings of the Conference on Laser and Electro-Optics (CLEO), Vol. 2 of 2004. OSA Technical Digest Series (CD) (Optical Society of America, 2004), paper CThS5.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1999).

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Figures (7)

Fig. 1
Fig. 1

(Color online) Plane wave incident on a scatter.

Fig. 2
Fig. 2

Optical system, the stationary coordinates, and the primed temporary coordinates.

Fig. 3
Fig. 3

(Color online) Optical imaging system.

Fig. 4
Fig. 4

(Color online) Normalized image intensities along the plane φ s = 0 , versus the normalized radius of the image plane, ρ s , in the image plane, of a point scatterer—solid curve, a sphere of radius 0.5λ—the curve marked with ( ) , and a sphere of radius 1.25λ—the curve marked with ( * * * ) ; the first column is the confocal microscope result, and the second column is resulted from conventional detection; the first row is for the linear illumination, the second row is for the radial illumination, and the third row is for the azimuthal illumination. (a) In a confocal microscope when illuminated with linearly polarized light; (b) in a confocal microscope when illuminated with radially polarized light; (c) in a confocal microscope when illuminated with azimuthally polarized light; (d) in a conventional microscope when illuminated with linearly polarized light; (e) in a conventional microscope when illuminated with radially polarized light; (f) in a conventional microscope when illuminated with azimuthally polarized light. The NA is 1 in all cases.

Fig. 5
Fig. 5

Detected image intensity in a conventional microscope when the illumination is linearly polarized (x polarized), simulated (a) for a point scatterer and (b) for a scatterer with radius size of 0.5λ. The image size is 12 and 14 normalized units are in the y and x directions, respectively.

Fig. 6
Fig. 6

(Color online) Simulated detected image intensities along the plane φ s = 0 , for different NAs, when the illumination is linearly polarized, in a conventional microscope, (a) for a point scatterer and (b) for a resolvable particle with radius 0.5λ.

Fig. 7
Fig. 7

(Color online) Magnitudes of some of the terms in Mie expansion versus the normalized radius∕π of the spherical particle. (a) Solid curve for the first term, curve marked with a circle for the second term, and the line marked with (*) stands for the third term, which has a zero value in the interval of the normalized radius; (b) solid curve for the seventh term, curve marked with a circle for the eighth term, and curve marked with (*) stands for the nineth term of the Mie expansion, 1.25λ is the radius of the largest particle for which we have simulated the image.

Equations (16)

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E s θ ( r ) = 1 k 1 r l = 1 [ B l ζ l ( 1 ) ( k 1 r ) P l ( 1 ) ( cos θ ) ( E i x cos φ ) × sin θ ] ,
E s φ ( r ) = 1 k 1 r l = 1 [ B l ζ l ( 1 ) ( k 1 r ) P l ( 1 ) ( cos θ ) ( E i x sin φ ) sin θ ] ,
E i = ( α ρ a ρ + β φ a φ ) e i k 1 z ,
α ρ = ( E i · a ρ ) e i k 1 z , β φ = ( E i · a φ ) e i k 1 z .
E i = e i k 1 z { α ρ ( cos φ x ^ + sin φ y ^ ) + β φ ( sin φ x ^ + cos φ y ^ ) } ,
E s θ ( r ) = 1 k 1 r l = 1 { B l ζ l ( 1 ) ( k 1 r ) [ ( α ρ cos 2 φ β φ sin φ cos φ ) P l ( 1 ) ( cos θ ) sin θ + ( α ρ sin 2 φ + β φ cos φ sin φ ) × P l ( 1 ) ( cos θ ) sin θ ] } ,
E s φ = 1 k 1 r l = 1 { B l ζ l ( 1 ) ( k 1 r ) [ ( α ρ cos φ sin φ β φ sin 2 φ ) P l ( 1 ) ( cos θ ) 1 / sin θ ( α ρ sin φ cos φ + β φ cos 2 φ ) × P l ( 1 ) ( cos θ ) × 1 / sin θ ] } .
E 1 = 0 α 0 2 π cos θ i ( α ρ ( cos θ cos φ cos θ sin φ sin θ ) + β φ ( sin φ cos φ 0 ) ) × exp [ i k ρ s cos ( φ i φ s ) sin θ i ] × exp ( i k f cos θ i ) sin θ i d θ i d φ i ,
r ^ = R W 1 ( θ i , φ i ) · r ^ ,
R W ( θ i , φ i ) = [ cos θ i cos 2 φ i + sin 2 φ i sin 2 ( θ i 2 ) sin 2 φ i cos φ i sin θ i sin 2 ( θ i 2 ) sin 2 φ i cos 2 φ i + cos θ i sin 2 φ i sin θ i sin φ i cos φ i sin θ i sin θ i sin φ i cos θ i ] .
E s ( r ) = [ E s θ cos θ cos φ E s φ sin φ E s θ cos θ sin φ + E s φ cos φ E s θ sin θ ] .
E s ( r ) = R W ( θ i , φ i ) · E s ( r ) .
E 1 ( ρ s , φ s ) = 0 2 π 0 α E s ( r ) cos θ i × exp [ i k ρ s cos ( φ i φ s ) sin θ i ] × exp ( i k f cos θ i ) sin θ i d θ i d φ i .
E 2 = [ E 1 ( r ^ × k ^ ) ] ( r ^ × k ^ ) + [ E 1 ( r ^ × ( r ^ × k ^ ) ] ( k ^ × ( r ^ × k ^ ) ) 1 ( k ^ r ^ ) 2 ,
I conv ( ρ s , φ s ) = 0 2 π 0 α | E 2 ( θ , φ ; ρ s , φ s ) | 2 × sin θ cos 1 / 2 θ d θ d φ ,
I conf ( ρ s , φ s ) = | 0 2 π 0 α E 2 ( θ , φ ; ρ s , φ s ) × exp ( i k ρ s sin θ cos ( φ φ s ) ) × sin θ cos 1 / 2 θ d θ d φ | 2 ,

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