Abstract

Based on truncated inverse filtering, a theory for deconvolution of complex fields is studied. The validity of the theory is verified by comparing with experimental data from digital holographic microscopy (DHM) using a high-NA system (NA=0.95). Comparison with standard intensity deconvolution reveals that only complex deconvolution deals correctly with coherent cross-talk. With improved image resolution, complex deconvolution is demonstrated to exceed the Rayleigh limit. Gain in resolution arises by accessing the objects complex field - containing the information encoded in the phase - and deconvolving it with the reconstructed complex transfer function (CTF). Synthetic (based on Debye theory modeled with experimental parameters of MO) and experimental amplitude point spread functions (APSF) are used for the CTF reconstruction and compared. Thus, the optical system used for microscopy is characterized quantitatively by its APSF. The role of noise is discussed in the context of complex field deconvolution. As further results, we demonstrate that complex deconvolution does not require any additional optics in the DHM setup while extending the limit of resolution with coherent illumination by a factor of at least 1.64.

© 2010 OSA

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2010 (1)

2009 (2)

2008 (1)

2007 (3)

G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46, 993–1000 (2007).
[CrossRef] [PubMed]

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

C. J. Sheppard, “Fundamentals of superresolution,” Micron 38, 165–169 (2007).
[CrossRef]

2006 (5)

2005 (1)

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. 244, 37 – 49 (2005).
[CrossRef]

2002 (1)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

2000 (1)

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 (3)

1997 (4)

D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, and Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36, 2353–2359 (1997).
[CrossRef] [PubMed]

V. Torczon, “On the convergence of pattern search algorithms,” SIAM J. Optim. 7, 125 (1997).
[CrossRef]

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2d diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

C. J. Sheppard and K. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

1993 (1)

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

1987 (1)

1972 (1)

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Aert, S. V.

Aguet, F.

Angell, D.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1987), 6th ed.

Brooker, G.

Charrière, F.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

Chen, J.

Colicchio, B.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. 244, 37 – 49 (2005).
[CrossRef]

Colomb, T.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

C. Depeursinge, P. Jourdain, B. Rappaz, P. Magistretti, T. Colomb, and P. Marquet, “Cell biology explored with digital holographic microscopy,” Biomed. Opt. p. BMD58 (2008).

Conchello, J. A.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Cotte, Y.

Y. Cotte, M. F. Toy, E. Shaffer, N. Pavillon, and C. Depeursinge, “Sub-rayleigh resolution by phase imaging,” Opt. Lett. 35, 2176–2178 (2010).
[CrossRef] [PubMed]

Y. Cotte and C. Depeursinge, “Measurement of the complex amplitude point spread function by a diffracting circular aperture,” in “Focus on Microscopy,” (2009), Advanced linear and non-linear imaging, pp. TU-AF2-PAR-D.

Cuche, E.

Cui, X. Q.

Debailleul, M.

den Dekker, A. J.

Depeursinge, C.

Y. Cotte, M. F. Toy, E. Shaffer, N. Pavillon, and C. Depeursinge, “Sub-rayleigh resolution by phase imaging,” Opt. Lett. 35, 2176–2178 (2010).
[CrossRef] [PubMed]

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
[CrossRef]

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009).
[CrossRef] [PubMed]

C. Depeursinge, P. Jourdain, B. Rappaz, P. Magistretti, T. Colomb, and P. Marquet, “Cell biology explored with digital holographic microscopy,” Biomed. Opt. p. BMD58 (2008).

Y. Cotte and C. Depeursinge, “Measurement of the complex amplitude point spread function by a diffracting circular aperture,” in “Focus on Microscopy,” (2009), Advanced linear and non-linear imaging, pp. TU-AF2-PAR-D.

Dieterlen, A.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. 244, 37 – 49 (2005).
[CrossRef]

Dyck, D. V.

Ferreira, C.

Garcia, J.

García, J.

Geissbühler, S.

Georges, V.

Gerchberg, R.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning (Addison-Wesley, 1989).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gu, M.

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

Guo, H.

Gustafsson, M. G. L.

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]

Haeberl, O.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. 244, 37 – 49 (2005).
[CrossRef]

Haeberle, O.

Heng, X.

Indebetouw, G.

Jourdain, P.

C. Depeursinge, P. Jourdain, B. Rappaz, P. Magistretti, T. Colomb, and P. Marquet, “Cell biology explored with digital holographic microscopy,” Biomed. Opt. p. BMD58 (2008).

Jung, G.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. 244, 37 – 49 (2005).
[CrossRef]

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Kiryuschev, I.

Knapp, D. W.

Konforti, N.

Kuei, C. P.

Kühn, J.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009).
[CrossRef] [PubMed]

Larkin, K.

C. J. Sheppard and K. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

Lasser, T.

Lauer, V.

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

Leitgeb, R. A.

Leith, E. N.

Leutenegger, M.

Liang, Z.

Lohmann, A. W.

Magistretti, P.

C. Depeursinge, P. Jourdain, B. Rappaz, P. Magistretti, T. Colomb, and P. Marquet, “Cell biology explored with digital holographic microscopy,” Biomed. Opt. p. BMD58 (2008).

Marian, A.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

Märki, I.

Marquet, P.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
[CrossRef]

C. Depeursinge, P. Jourdain, B. Rappaz, P. Magistretti, T. Colomb, and P. Marquet, “Cell biology explored with digital holographic microscopy,” Biomed. Opt. p. BMD58 (2008).

Martinez, P. G.

McDowell, E. J.

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Mendlovic, D.

Mico, V.

Montfort, F.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

Morin, R.

Nehorai, A.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-d fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

Pavillon, N.

Y. Cotte, M. F. Toy, E. Shaffer, N. Pavillon, and C. Depeursinge, “Sub-rayleigh resolution by phase imaging,” Opt. Lett. 35, 2176–2178 (2010).
[CrossRef] [PubMed]

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009).
[CrossRef] [PubMed]

Psaltis, D.

Rao, R.

Rappaz, B.

C. Depeursinge, P. Jourdain, B. Rappaz, P. Magistretti, T. Colomb, and P. Marquet, “Cell biology explored with digital holographic microscopy,” Biomed. Opt. p. BMD58 (2008).

Rosen, J.

Sarder, P.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-d fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

Saxton, W.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Schaefer, L. H.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy.” Biotechniques31 (2001).
[PubMed]

Seelamantula, C. S.

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009).
[CrossRef] [PubMed]

Shaffer, E.

Shemer, A.

Sheppard, C. J.

C. J. Sheppard, “Fundamentals of superresolution,” Micron 38, 165–169 (2007).
[CrossRef]

C. J. Sheppard and K. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

Sheppard, C. J. R.

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

Simon, B.

Swedlow, J. R.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy.” Biotechniques31 (2001).
[PubMed]

Tada, Y.

Tiziani, H. J.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2d diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

Torczon, V.

V. Torczon, “On the convergence of pattern search algorithms,” SIAM J. Optim. 7, 125 (1997).
[CrossRef]

Totzeck, M.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2d diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

Toy, M. F.

Unser, M.

F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-d steerable filters,” Opt. Express 17, 6829–6848 (2009).
[CrossRef] [PubMed]

N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009).
[CrossRef] [PubMed]

Vonesch, C.

C. Vonesch, “Fast and automated wavelet-regularized image restoration in fluorescence microscopy,” Ph.D. thesis, EPFL, LIB Laboratoire d’imagerie biomédicale (2009).

Wallace, W.

W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy.” Biotechniques31 (2001).
[PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1987), 6th ed.

Wu, J. G.

Xu, C.

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. 244, 37 – 49 (2005).
[CrossRef]

Yang, C. H.

Yaqoob, Z.

Zalevsky, Z.

Zhuang, S.

Appl. Opt. (4)

IEEE Signal Process. Mag. (1)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-d fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

J. Microsc. (3)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

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]

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Methods (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Micron (1)

C. J. Sheppard, “Fundamentals of superresolution,” Micron 38, 165–169 (2007).
[CrossRef]

Opt. Commun. (2)

B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. 244, 37 – 49 (2005).
[CrossRef]

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2d diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Optik (1)

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Optik (Stuttg.) (1)

C. J. Sheppard and K. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

SIAM J. Optim. (1)

V. Torczon, “On the convergence of pattern search algorithms,” SIAM J. Optim. 7, 125 (1997).
[CrossRef]

Other (10)

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N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009).
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J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

Y. Cotte and C. Depeursinge, “Measurement of the complex amplitude point spread function by a diffracting circular aperture,” in “Focus on Microscopy,” (2009), Advanced linear and non-linear imaging, pp. TU-AF2-PAR-D.

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[PubMed]

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Figures (7)

Fig. 1.
Fig. 1.

SEM image of pair of nano-holes drilled by FIB in aluminum film at 100 000× magnification. The images show nominal center-to-center pitches η of 600nm (a), 500nm (b), 400nm (c) and 300nm (d) with according scale bars.

Fig. 2.
Fig. 2.

Experimental and synthetic transfer functions in focal plane at λ =532nm and NA=0.95. The experimental amplitude CTF ∣cexp ∣ (a) and phase CTF arg[cexp ] (c) are imaged from a single nano-metric aperture. According to Eq. (22), (b) shows the fitted synthetic amplitude CTF ∣csyn ∣ and (d) its phase part arg[csyn ].

Fig. 3.
Fig. 3.

Experimental transfer functions in focal plane at λ =532nm and NA=0.95 of the test target (cf. Fig. 1). The Γ-masked amplitude spectrum ∣∣ (a) and phase spectrum arg[] (b) are illustrated for η=400nm. (c) compares ∣∣ cross-sections in ky for kx = 0 with the experimental CTF of a single nano-metric hole. (d) shows the same comparison of the experimental OTF and ∣∣.

Fig. 4.
Fig. 4.

Comparisons of unresolved and super-resolved profiles of two nano-holes of test target (cf. Fig. 1) with center-to-center distances η=600nm in (a), η=500nm in (b), η=400nm in (c), and η=300nm in (d). The raw data images I are reconstructed in the focal plane at λ =532nm and NA=0.95 (dmin,coh =460nm). The ‘raw’ profile shows the central y cross-section of the resolution limited raw data I (cf. ‘rw’ insert). The ‘deconvolution complex’ profile shows the corresponding section of ∣o2 (cf. ‘cd’ insert) resulting from complex deconvolution by the experimental CTF. Additionally, ‘deconvolution intensity’ compares the profile of oi resulting from intensity deconvolution by the experimental OTF.

Fig. 5.
Fig. 5.

Influence of kmax (dmin ) on complex deconvolution results according to Eq. (4). (a–c) statistics for η=400nm for deconvolution of Uexp with cexp ‘experimental’, for deconvolution of Uexp with csyn ‘experimental-synthetic’, for deconvolution of Usyn with csyn synthetic (no noise)’, and deconvolution of Unoise with cnoise ‘synthetic (SNR=35)’. (d) statistics of p-t-p in dependence of dmin for all targets η. The grey bars indicate error margin of 25nm.

Fig. 6.
Fig. 6.

Experimental transfer functions in focal plane for λ =532nm and NA=0.95 of test target (cf. Fig. 1) after deconvolution. The amplitude spectrum ∣O∣ (a) and phase spectrum arg[O] (b) are illustrated for η=400nm after division by CTF. (c) compares ∣O∣ cross-section in ky for kx = 0 with for the η=400nm case. (d) shows the same comparison for ∣Oi ∣ and ∣∣.

Fig. 7.
Fig. 7.

XY images in focal plane of test target with sub-resolution pitch η=400nm [cf. insert (a) imaged by SEM]. Insert (b) shows the unresolved test target’s raw image I at λ =532nm and NA=0.95. Insert (c) shows oi resulting from intensity deconvolution. Insert (d) shows ∣o2 resulting from complex deconvolution by the synthetic CTF and insert (e) the according result for deconvolution by the experimental CTF.

Tables (2)

Tables Icon

Table 1. Results of fit of experimental data from optical system at λ =532nm and NA=0.95

Tables Icon

Table 2. Results of peak-to-peak (p-t-p) distance measurements of the test target at λ =532nm and NA=0.95. The standard precision is based on the lateral sampling of 56nm, the complex deconvolution is determined in Fig. 5(d)

Equations (29)

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v c , incoh = 2 v c , coh ,
d min = α λ NA ,
k = k = 2 π ν = 2 π d ,
k c , coh = 2 π d min , coh .
I ( x 2 , y 2 ) = h ( x 1 + M x 2 , y 1 + My 2 ) 2 o i ( x 1 , y 1 ) d x 1 d y 1 ,
J ( k x , k y ) = C ( k x , k y ) O i ( k x , k y ) ,
and I ( x 2 , y 2 ) = J ( k x , k y ) exp [ i 2 π ( k x M x 2 + k y M y 2 ) ] d k x d k y ,
C ( k x , k y ) = h ( x 1 , y 1 ) 2 exp [ i 2 π ( k x x 1 + k y y 1 ) ] d x 1 d y 1 .
o i ( x 1 , y 1 ) = O i ( k x , k y ) exp [ i 2 π ( k x x 1 + k y y 1 ) ] d k x d k y = 1 { J ˜ ( k x , k y ) C ( k x , k y ) } .
J ˜ ( k x , k y ) = J ( k x , k y ) Γ k max ( k x , k y ) where Γ k max ( k x , k y ) = { 1 k x 2 + k y 2 k s f s k s k x 2 + k y 2 k max 0 k x 2 + k y 2 > k max
U ( x 2 , y 2 ) = h ( x 1 + M x 2 , y 1 + M y 2 ) o ( x 1 , y 1 ) d x 1 d y 1 ,
G ( k x , k y ) = c ( k x , k y ) O ( k x , k y ) ,
and U ( x 2 , y 2 ) = G ( k x , k y ) exp [ i 2 π ( k x M x 2 + k y M y 2 ) ] d k x d k y ,
c ( k x , k y ) = h ( x 1 , y 1 ) exp [ i 2 π ( k x x 1 + k y y 1 ) ] d x 1 d y 1 ,
o ( x 1 , y 1 ) = O ( k x , k y ) exp [ i 2 π ( k x x 1 + k y y 1 ) ] d k x d k y = 1 { G ˜ ( k x , k y ) c ( k x , k y ) } .
G ˜ ( k x , k y ) = G ( k x , k y ) Γ k max ( k x , k y ) .
U ( x , y ) = a n A ( x , y ) exp [ i Φ ( x , y ) ] ,
NA = n m λ N δ x m px ,
{ x 1 = f sin θ cos ϕ , y 1 = f sin θ sin ϕ , z 1 = f cos θ , which satisfies f 2 = x 1 2 + y 1 2 + z 1 2 ,
{ x 2 = r 2 cos Ψ , y 2 = r 2 sin Ψ , z 2 , which satisfies r 2 2 = x 2 2 + y 2 2 .
U δ ( r 2 , Ψ , z 2 ) = i λ 0 2 π 0 α P ( θ , ϕ ) exp [ i k r 2 sin θ cos ( ϕ Ψ ) i k z 2 cos θ i k Φ ( θ , ϕ ) ] sin θ d θ d ϕ ,
P ( θ , ϕ ) = cos θ .
h ( z 2 ) = Ω U δ ( r 2 , Ψ , z 2 ) d r 2 d Ψ .
f ( A n , m ) = Σ k x , k y arg [ c exp ] arg [ c syn ( A n , m ) ] 2 ,
U syn = h syn ( x , y + η 2 ) + h syn ( x , y η 2 ) .
d min cd = min [ λ 2 ( 1 NA ± Δ ϕ π ) ] .
I ( x , y ) = U ( x , y ) 2 ,
g ( x ) = a 1 exp [ ( x μ 1 ) 2 2 b 1 2 ] + a 2 exp [ ( x μ 2 ) 2 2 b 2 2 ] .
FWHM = 2 ln 2 ( b 1 + b 2 ) .

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