Abstract

Imaging interferometric microscopy (IIM) is a synthetic aperture imaging approach providing resolution to the transmission medium (refractive index n) linear systems limit extending to λ4n using only low-numerical-aperture (low-NA) optics. IIM uses off-axis illumination to access high spatial frequencies along with interferometric reintroduction of a zero-order reference beam on the low-NA side of the optical system. For a thin object normal to the optical axis, the frequency space limit is [(1+NA)nλ], while tilting the object plane allows collection of diffraction information up to the material transmission bandpass-limited spatial frequency of 2nλ. Tilting transforms the spatial frequencies; the algorithm to reset to the correct image frequencies is described. IIM involves combining multiple subimages; the image reconstruction procedures are discussed. A mean-square-error metric is introduced. For binary objects, sigmoidal filtering of the image provides significant resolution improvement.

© 2008 Optical Society of America

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References

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2007

2006

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef] [PubMed]

V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia, “Superresolved imaging in digital holography by superposition of tilted wavefronts,” Appl. Opt. 45, 822-826 (2006).
[CrossRef] [PubMed]

V. Mico, Z. Zalevsky, and J. Garcia, “Superresolution optical system by common-path interferometry,” Opt. Express 14, 5168-5177 (2006).
[CrossRef] [PubMed]

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

2005

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102, 13081-13086 (2005).
[CrossRef] [PubMed]

A. Zlotnik, Z. Zalevsky, and E. Marom, “Superresolution with nonorthogonal polarization coding,” Appl. Opt. 44, 3705-3715 (2005).
[CrossRef] [PubMed]

S. A. Alexandrov, T. R. Hillman, and D. D. Sampson, “Spatially resolved Fourier holographic light scattering angular spectroscopy,” Opt. Lett. 30, 3305-3307 (2005).
[CrossRef]

T. M. Tridhavee, B. Santhanam, and S. R. J. Brueck, “Optimal frequency coverages and parsings for imaging interferometric lithography,” J. Microlithogr., Microfabr., Microsyst. 4, 033005 (2005).
[CrossRef]

2004

2003

2002

M. Dyba and S. W. Hell, “Focal spots of size λ/23 open up far-field florescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 016390 (2002).
[CrossRef]

2000

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82-87 (2000).
[CrossRef] [PubMed]

1999

1995

1994

1992

1986

1967

1964

1963

W. Lukosz and M. Marchant, “Optischen Abbildung Unter Ueberschreitung der Beugungsbedingten Aufloesungsgrenze,” Opt. Acta 10, 241-255 (1963).
[CrossRef]

1960

A. I. Kartashev, “Optical system with enhanced resolving power,” Opt. Spectrosc. 9, 204-206 (1960).

1952

M. Françon, “Amélioration de la resolution d'optique,” Nuovo Cimento, Suppl. 9, 283-290 (1952).
[CrossRef]

1873

E. Abbé, “Beiträge zur Theorie des Mikroskops und der Mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413-468 (1873).
[CrossRef]

Appl. Opt.

Arch. Mikrosc. Anat. Entwicklungsmech.

E. Abbé, “Beiträge zur Theorie des Mikroskops und der Mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413-468 (1873).
[CrossRef]

J. Microlithogr., Microfabr., Microsyst.

T. M. Tridhavee, B. Santhanam, and S. R. J. Brueck, “Optimal frequency coverages and parsings for imaging interferometric lithography,” J. Microlithogr., Microfabr., Microsyst. 4, 033005 (2005).
[CrossRef]

J. Microsc.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82-87 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nuovo Cimento, Suppl.

M. Françon, “Amélioration de la resolution d'optique,” Nuovo Cimento, Suppl. 9, 283-290 (1952).
[CrossRef]

Opt. Acta

W. Lukosz and M. Marchant, “Optischen Abbildung Unter Ueberschreitung der Beugungsbedingten Aufloesungsgrenze,” Opt. Acta 10, 241-255 (1963).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

A. I. Kartashev, “Optical system with enhanced resolving power,” Opt. Spectrosc. 9, 204-206 (1960).

Phys. Rev. Lett.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef] [PubMed]

M. Dyba and S. W. Hell, “Focal spots of size λ/23 open up far-field florescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 016390 (2002).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102, 13081-13086 (2005).
[CrossRef] [PubMed]

Other

Z. Zalevsky and D. Mendlovic, Optical Super Resolution (Springer, 2002).

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Optical systems with improved resolving power,” in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 1999), Vol. 15, Chap. 4.

M. V. Klein and T. E. Furtak, Optics (Wiley, 1986).

S. W. Smith, The Scientist and Engineer's Guide to Digital Signal Processing, ISBN 0-7506-7444-X, retrieved February 2007, http://www.dspguide.com/ch11/4.htm.

B. R. Frieden, Probability Statistical Optics and Data Testing (Springer-Verlag, 1983).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Optical arrangement for imaging interferometric microscopy: β = sin 1 ( NA ) , α ill is the incident beam angle of incidence, and α ref is the angle of the reference beam onto the image plane. For convenience, the figure is drawn for a 2 × magnification of the image.

Fig. 2
Fig. 2

(a) Manhattan geometry pattern used for image resolution exploration consisting of five nested “ells” and a large box. (b) Intensity Fourier space components of the pattern with the 240 nm lines and spaces of the “ells” mapped onto the frequency space coverage of the imaging system.

Fig. 3
Fig. 3

Filtering in a Fourier plane. (a) Sharp cutoff in a region where the Fourier intensity is low (example is for a 260 nm C D structure). (b) Schematic representation of apodized filters when the transition is in a region with a strong Fourier intensity (example for a 350 nm C D structure).

Fig. 4
Fig. 4

Optical arrangement using (a) off-axis illumination and (b) off-axis illumination with a tilted object to enhance the frequency space information.

Fig. 5
Fig. 5

Frequency space coverage with a tilted object (a) for a 180 nm C D structure and (b) location of high frequency for 170 nm structure showing the clipping of the strong spectral content associated with the nested lines.

Fig. 6
Fig. 6

Optical arrangement with a tilted object plane.

Fig. 7
Fig. 7

Laboratory-frame (dark background circle) and image-frame (lighter top ellipse) frequency space coverage. The dot at NA x = 1.76 , NA y = 0 corresponds to the spatial frequency calibration of the reference beam, where the two frequencies are the same.

Fig. 8
Fig. 8

MSE versus C D of simulated images without (black) and with (gray) thresholding for NA = 0.4 , coherent illumination, shown with corresponding images. The black vertical dashed line at 870 nm corresponds to the middle of the step and provides a measure of the resolution. The gray dashed line at 740 nm is the comparable resolution for the thresholded images. The dotted lines at 950 nm ( 0.6 λ NA ) and at 790 nm ( λ 2 NA ) correspond to the Rayleigh and Abbé limits, respectively.

Fig. 9
Fig. 9

(a) Filtered low-frequency object, (b) crosscut with threshold value indicated, (c) thresholded image.

Fig. 10
Fig. 10

(a) Low-frequency image; (c) intermediate-frequency image of x-directed structures taken at 53°; (e) intermediate-frequency image of y-directed structures taken at 53°; (f) summation of the low-frequency image with the intermediate-frequency images; (b), (d), (f), (h) corresponding simulations.

Fig. 11
Fig. 11

(a) High-frequency image of vertical structures taken at 80°; (c) high-frequency image of horizontal structures taken at 80°; (e) summation of the low- and high-frequency images; (b), (d), (f) corresponding simulations.

Fig. 12
Fig. 12

(a) Combined image including five subimages: low, intermediate-x, intermediate-y, high-x, and high-y. No filtering of the two offset images to eliminate double coverage was included. (b) Corresponding simulation. (c) High-x filtered image, (d) corresponding simulation, (e) reconstructed image after filtering, (f) corresponding simulation.

Fig. 13
Fig. 13

(a) Row with the minimum of the MSE curves obtained by scanning the high-frequency subimage of the horizontal structures against the corresponding model in the y direction. (b) Column with the minimum of the MSE curves obtained by scanning the high-frequency image of the horizontal structures against the corresponding model in the x direction.

Fig. 14
Fig. 14

Reconstructed image with C D structures: 260 nm , 240 nm (first row); 220 nm , 210 nm (second row). Reference object is in the white circle.

Fig. 15
Fig. 15

Experiment (a), (c) and simulation (b), (d) results showing the impact of the frequency mapping; (e) crosscuts of laboratory frame subimages, (f) crosscuts of image plane subimages.

Fig. 16
Fig. 16

(a) Reconstructed image: 260 nm , 240 nm (first row) structures; 220 nm , 210 nm (second row) structures. (b) Crosscuts of reconstructed images for 260 and 240 nm C D . Black, experimental; gray, simulation (simple bandpass filter at NA = 1.38 ).

Fig. 17
Fig. 17

(a) Reconstructed image: 180 nm , 170 nm (top row) structures; 140 nm , 130 nm (bottom row) structures. (b) Crosscuts of reconstructed images for 180 and 170 nm C D . Black, experimental; gray, simulation (simple bandpass filter at NA = 1.87 ). The average and sigma of the two curves are set equal for easy comparison.

Fig. 18
Fig. 18

Theoretical and experimental MSE curves for NA = 0.4 : (a) on axis illumination, (b) off-axis illumination, and (c) off-axis illumination with tilted mask.

Equations (10)

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NA eff = NA + sin α ill ,
NA eff max , min = [ sin ( θ tilt ± sin 1 ( NA ) ) ] + sin ( α ill ) .
k y , lab = k 0 f y , lab = k y , image = k 0 f y , image .
k x , image = k 0 ( f x , image sin α ill ) ,
k z , image = k 0 2 k x , image 2 k y , image 2 = k 0 2 [ 1 ( f x , image sin ( α ill ) ) 2 ] k y , image 2 .
k x , lab = k x , image cos θ tilt k z , image sin θ tilt .
k 0 f x , lab = k x , lab + k 0 sin ( α ref ) ,
f x , lab = ( f x , image sin α ill ) cos θ tilt ( 1 ( f x , image sin α ill ) 2 f y , image 2 ) sin θ tilt + sin ( α ref ) .
f x , image = ( f x , lab sin ( α ref ) ) cos θ tilt + ( 1 f y , lab 2 ( f x , lab sin ( α ref ) ) 2 ) sin θ tilt + sin ( α ill ) .
MSE = 1 N i ( I i I i ) 2 ,

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