Abstract

We describe a numerical vector diffraction model based on Mie theory that describes the imaging of spherical particles by bright-field, confocal, and interferometric microscopes. The model correctly scales the amplitude-scattered field relative to the incident field so that the forward-scattered and incident light can be interfered to correctly model imaging with copolarization transmission microscopes for the first time to our knowledge. The model is used to demonstrate that amplitude and phase imaging with an interferometric microscope allows subwavelength particle sizing. Furthermore, we show that the phase channel allows much smaller particles to be sized than amplitude-only measurements. The model is validated by experimental measurements.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  2. T. Wilson, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  3. M. G. Somekh, “Depth discrimination in scanned heterodyne microscope systems,” J. Microsc. (Oxford) 168, 131–151 (1992).
    [CrossRef]
  4. D. M. Gale, M. I. Pether, J. C. Dainty, “Linnik microscope imaging of integrated circuit structures,” Appl. Opt. 35, 131–148 (1996).
    [CrossRef] [PubMed]
  5. S. P. Morgan, E. Choi, M. G. Somekh, C. W. See, “Interferometric optical microscopy of subwavelength grooves,” Opt. Commun. 187, 29–38 (2001).
    [CrossRef]
  6. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  7. T. Wilson, R. Jus̆kaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
    [CrossRef]
  8. P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
    [CrossRef]
  9. P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
    [CrossRef]
  10. W. Inami, Y. Kawata, “Three-dimensional imaging analysis of confocal and conventional polarization microscopes by use of Mie scattering theory,” Appl. Opt. 39, 6369–6373 (2000).
    [CrossRef]
  11. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  12. Mie Scattering Toolbox Department of Electrical and Electronic Engineering, Loughborough University, Leicestershire, UK, http://www.lboro.ac.uk/departments/el/research/optics/matmie/mfiles.html .
  13. W. C. Tsai, R. J. Pogorzelski, “Eigenfunction solution of the scattering of beam radiation fields by spherical objects,” J. Opt. Soc. Am. 65, 1457–1463 (1975).
    [CrossRef]
  14. E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
    [CrossRef]
  15. N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
    [CrossRef]

2002 (1)

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

2001 (2)

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

S. P. Morgan, E. Choi, M. G. Somekh, C. W. See, “Interferometric optical microscopy of subwavelength grooves,” Opt. Commun. 187, 29–38 (2001).
[CrossRef]

2000 (2)

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

W. Inami, Y. Kawata, “Three-dimensional imaging analysis of confocal and conventional polarization microscopes by use of Mie scattering theory,” Appl. Opt. 39, 6369–6373 (2000).
[CrossRef]

1998 (1)

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

1997 (1)

T. Wilson, R. Jus̆kaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

1996 (1)

1992 (1)

M. G. Somekh, “Depth discrimination in scanned heterodyne microscope systems,” J. Microsc. (Oxford) 168, 131–151 (1992).
[CrossRef]

1975 (1)

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Astrakharchik, E.

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

Astrakharchik-Farrimond, E.

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Cao, X. F.

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

Choi, E.

S. P. Morgan, E. Choi, M. G. Somekh, C. W. See, “Interferometric optical microscopy of subwavelength grooves,” Opt. Commun. 187, 29–38 (2001).
[CrossRef]

Dainty, J. C.

Gale, D. M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Higdon, P.

T. Wilson, R. Jus̆kaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Higdon, P. D.

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Inami, W.

Jus?kaitis, R.

T. Wilson, R. Jus̆kaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Kawata, Y.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Morgan, S. P.

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

S. P. Morgan, E. Choi, M. G. Somekh, C. W. See, “Interferometric optical microscopy of subwavelength grooves,” Opt. Commun. 187, 29–38 (2001).
[CrossRef]

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

Pether, M. I.

Pogorzelski, R. J.

Sawyer, N. B. E.

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

See, C. W.

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

S. P. Morgan, E. Choi, M. G. Somekh, C. W. See, “Interferometric optical microscopy of subwavelength grooves,” Opt. Commun. 187, 29–38 (2001).
[CrossRef]

Shekunov, B. Y.

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

Somekh, M. G.

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

S. P. Morgan, E. Choi, M. G. Somekh, C. W. See, “Interferometric optical microscopy of subwavelength grooves,” Opt. Commun. 187, 29–38 (2001).
[CrossRef]

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

M. G. Somekh, “Depth discrimination in scanned heterodyne microscope systems,” J. Microsc. (Oxford) 168, 131–151 (1992).
[CrossRef]

Török, P.

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

Tsai, W. C.

Wilson, T.

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

T. Wilson, R. Jus̆kaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

T. Wilson, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

York, P.

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

Ann. Phys. (Leipzig) (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Appl. Opt. (2)

Exp. Fluids (1)

E. Astrakharchik-Farrimond, B. Y. Shekunov, P. York, N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Dynamic measurements in supercritical flow using instantaneous phase-shift interferometry,” Exp. Fluids 33, 307–314 (2002).
[CrossRef]

J. Microsc. (Oxford) (1)

M. G. Somekh, “Depth discrimination in scanned heterodyne microscope systems,” J. Microsc. (Oxford) 168, 131–151 (1992).
[CrossRef]

J. Mod. Opt. (1)

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (3)

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

T. Wilson, R. Jus̆kaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

S. P. Morgan, E. Choi, M. G. Somekh, C. W. See, “Interferometric optical microscopy of subwavelength grooves,” Opt. Commun. 187, 29–38 (2001).
[CrossRef]

Rev. Sci. Instrum. (1)

N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, X. F. Cao, B. Y. Shekunov, E. Astrakharchik, “Wide field amplitude and phase confocal microscope with parallel phase stepping,” Rev. Sci. Instrum. 72, 3793–3801 (2001).
[CrossRef]

Other (4)

Mie Scattering Toolbox Department of Electrical and Electronic Engineering, Loughborough University, Leicestershire, UK, http://www.lboro.ac.uk/departments/el/research/optics/matmie/mfiles.html .

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

T. Wilson, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Point of illumination in the back focal plane of an aplanatic objective maps to a plane wave in the sample plane.

Fig. 2
Fig. 2

Optical system. Identical speckle patterns are projected into sample and reference arms of the interferometer. These are imaged through the phase-shifting optics to form four interferograms at the CCD cameras.

Fig. 3
Fig. 3

Intensity transmission images of the scattered light for a 1-nm-diameter particle with a refractive index of 1.33 in air and a microscope objective of 0.9 NA. (a) Bright-field copolarized, intensity 0 (black) to 1 (white); (b) bright-field cross-polarized, intensity 0 (black) to 0.05 (white); (c) confocal copolarized, intensity 0 (black) to 1 (white); (d) confocal cross-polarized, intensity 0 (black) to 0.015 (white).

Fig. 4
Fig. 4

Modeled amplitude and phase images of a 10-μm-diameter polystyrene particle (n = 1.59) immersed in oil (n = 1.52) and a microscope objective of 0.2 NA. (a) Copolarized amplitude, 0 (black) to 1 (white); (b) cross-polarized amplitude, 0 (black) to 0.007 (white); (c) copolarized phase, -π (black) to π (white); (d) cross-polarized phase, -π (black) to π (white).

Fig. 5
Fig. 5

Modeled on-axis phase as a function of particle diameter for particles whose refractive index is adjusted to correspond to a geometric phase of 5°. The refractive index of the background medium is 1.52.

Fig. 6
Fig. 6

Modeled (a) amplitude profiles and (b) apparent optical thickness profiles of weakly scattering particles with diameters from of 2 (narrowest profiles), 4, 6, 8, 10, 12, 12.2, and 14 μm (broadest profiles). The refractive index of the particles and the surrounding medium are 1.59 and 1.52, respectively.

Fig. 7
Fig. 7

(a) Modeled and (b) experimental optical thickness profiles of particles with diameters from 0.5 μm (solid curve), 0.6 μm (dotted curve), 1 μm (short dashed curve), 2 μm (long dashed curve) to 3 μm (dotted-dashed curve). Refractive indices of the particles and background media are 1.59 and 1.33, respectively. The experimental profiles were scaled by a common factor to compensate for aberration effects.

Fig. 8
Fig. 8

Modeled (a) amplitude profiles and (b) optical thickness profiles of particles with diameters close to that at which the step in the phase flips. The diameters are 1.5 μm (solid curves), 1.588 μm (dotted curves), 1.589 μm (short dashed curves), and 1.7 μm (long dashed curves). Refractive indices of the particles and background media are 1.59 and 1.33, respectively. Note the nulls in the amplitude profile coinciding with the phase flip.

Fig. 9
Fig. 9

Apparent optical thickness as a function of particle diameter for weak (solid curve) and strong (dashed curve) scatterers calculated from (a) the full width at half-maximum of the phase profile and (b) the peak phase.

Fig. 10
Fig. 10

Size of the dip (solid curve) in the amplitude profiles of small weak scatterers as a function of particle diameter. The dashed curve shows a dip proportional to the sixth power of the particle diameter.

Fig. 11
Fig. 11

Phasor diagram showing the scattering by a phase particle.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

SP=sin ϕi-cos ϕicos ϕisin ϕiEXEY
Ef=Eia2fλsincπaxfλsincπayfλ,
Ef0=Eia2fλ.
a<0.07fλrsphere,
a=2fNAN-1,
Ef0=4EifNA2λN-12.
SiPi=4fNA2λN-12sin ϕi-cos ϕicos ϕisin ϕiEXEY.
ki=ki,xki,yki,z=ksin θi cos ϕisin θi sin ϕicos θi.
ki,x2+ki,y21/2kNA.
ki,m,n=mΔknΔkk2-m2+n2Δk21/2
m=-N-12,, -1, 0, 1,, N-12,n=-N-12,, -1, 0, 1,, N-12,m2+n21/2N-12,
Δk=2πλ2NAN-1,
ko=ko,xko,yko,z=ksin θo cos ϕosin θo sin ϕocos θo.
ko,p,q=pΔkqΔkk2-p2+q2Δk21/2
p=-N-12,, -1, 0, 1,, N-12,q=-N-12,, -1, 0, 1,, N-12p2+q21/2N-12.
ϕs=arctansin θo sinϕo-ϕicos θi sin θo cosϕo-ϕi-sin θi cos θo, θs=arccossin θi sin θo cosϕo-ϕi+cos θi cos θo.
PsSs=1krs2θs00s1θscos ϕs-sin ϕssin ϕscos ϕsPiSi,
So=-Ps sin θi sinϕo-ϕi+Sscos θi sin θo-sin θi cos θo cosϕo-ϕisin θs, Po=Pscos θi sin θo-sin θi cos θo cosϕo-ϕi+Ss sin θi sinϕo-ϕisin θs.
dθodϕo=θo, ϕoko,x, ko,ydko,xdko,y,
sin θodθodϕo=1-ko,x2-ko,y2-1/2dko,xdko,y,
Ω= ko,x-Δkko,x+Δkko,y-Δkko,y+Δk1-ko,x2-ko,y2-1/2dko,xdko,y.
Eo,XEo,Y=r2ΩN-12fNAcos ϕo-sin ϕosin ϕocos ϕoPoSo.
Eo,XEo,Y=EXEY+r2ΩN-12fNAcos ϕo-sin ϕosin ϕocos ϕoPoSo.
Eo,p,q=m=-N-12N-12n=-N-12N-12Eo,m,n,p,q×expiki,x,m,n-ko,x,p,qxscan+ki,y,m,n-ko,y,p,qyscan+ki,z,m,n-ko,z,p,qzscan.
αg=2πdnp-nmλ,

Metrics