Abstract

We present an imaging method with the ability to correct even large optical phase aberrations in a purely numerical way. For this purpose, the complex coherence function in the pupil plane of the microscope objective is measured with the help of an image inverting interferometer. By means of a Fourier transform, it is possible to reconstruct the spatially incoherent object distribution. We demonstrate that aberrations symmetric to the optical axis do not impair the imaging quality of such a coherence imaging system. Furthermore, we show that it is possible to gain an almost complete correction of remaining aberrations with the help of a reference measurement. A mathematical derivation is given and experimentally verified. To demonstrate the ability of our method, randomly generated aberrations with peak-to-valley values of up to 8 λ are corrected.

© 2015 Optical Society of America

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References

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2015 (1)

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

2013 (1)

2011 (2)

X. Tao, O. Azucena, M. Fu, Y. Zuo, D. Chen, and J. Kubby, “Adaptive optics microscopy with direct wavefront sensing using fluorescent protein guide stars,” Opt. Lett. 36, 3389–3391 (2011).
[Crossref] [PubMed]

P. Vermeulen, E. Muro, T. Pons, V. Loriette, and A. Fragola, “Adaptive optics for fluorescence wide-field microscopy using spectrally independent guide star and markers,” J. Biomed. Opt. 16, 076019 (2011).
[Crossref] [PubMed]

2010 (2)

A. A. Wagadarikar, D. L. Marks, K. Choi, R. Horisaki, and D. J. Brady, “Imaging through turbulence using compressive coherence sensing,” Opt. Express 35, 838–840 (2010).

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

2009 (3)

2007 (2)

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274 – 280 (2007).
[Crossref]

2006 (1)

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[Crossref]

2005 (1)

2004 (2)

M. Schwertner, M. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high na adaptive optical microscopy,” Opt. Express 12, 6540–6552 (2004).
[Crossref] [PubMed]

M. Schwertner, M. Booth, M. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

2003 (1)

2002 (1)

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

2001 (3)

2000 (1)

D.-S. Wan, M. Rajadhyaksha, and R. Webb, “Analysis of spherical aberration of a water immersion objective: application to specimens with refractive indices 1.33–1.40,” J. Microsc. 197, 274–284 (2000).
[Crossref] [PubMed]

1999 (2)

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Muson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[Crossref] [PubMed]

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the fresnel domain,” Appl. Opt. 38, 1332–1342 (1999).
[Crossref]

1997 (1)

P. Török, S. Hewlett, and P. Varga, “The role of specimen-induced spherical aberration in confocal microscopy,” J. Microsc. 188, 158–172 (1997).
[Crossref]

1996 (1)

1995 (1)

1994 (1)

1993 (2)

J. Beckers, “Adaptive optics for astronomy-principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[Crossref]

S. Hell, G. Reiner, C. Cremer, and E. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

1988 (2)

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 99–166 (1988).
[Crossref]

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

1986 (1)

K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A. 3, 94–100 (1986).
[Crossref]

1985 (3)

F. Roddier and C. Roddier, “An image reconstruction of alpha orionis,” Astrophys. J. 295, L21–L23 (1985).
[Crossref]

C. Coulman, “Fundamental and applied aspects of astronomical ’seeing’,” Annu. Rev. Astron. Astrophys. 23, 19–57 (1985).
[Crossref]

I. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
[Crossref]

1983 (1)

C. Roddier and F. Roddier, “High angular resolution observations of alpha orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[Crossref]

1981 (1)

1972 (1)

1959 (1)

1934 (1)

v. F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Applegate, R.

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Azucena, O.

Babovsky, H.

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

Beckers, J.

J. Beckers, “Adaptive optics for astronomy-principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[Crossref]

Booker, G.

Booth, M.

M. Schwertner, M. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high na adaptive optical microscopy,” Opt. Express 12, 6540–6552 (2004).
[Crossref] [PubMed]

M. Schwertner, M. Booth, M. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

M. Booth and T. Wilson, “Refractive-index-mismatch induced aberrations in single-photon and two-photon microscopy and the use of aberration correction,” J. Biomed. Opt. 6, 266–272 (2001).
[Crossref] [PubMed]

Booth, M. J.

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

Brady, D. J.

A. A. Wagadarikar, D. L. Marks, K. Choi, R. Horisaki, and D. J. Brady, “Imaging through turbulence using compressive coherence sensing,” Opt. Express 35, 838–840 (2010).

P. Potuluri, M. R. Fetterman, and D. J. Brady, “High depth of field microscopic imaging using an interferometric camera,” Opt. Express 8, 624–630 (2001).
[Crossref] [PubMed]

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the fresnel domain,” Appl. Opt. 38, 1332–1342 (1999).
[Crossref]

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Muson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[Crossref] [PubMed]

Brady, R. B.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Muson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[Crossref] [PubMed]

Breckinridge, J. B.

Cai, L.

Candès, E.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[Crossref]

Chen, D.

Choi, K.

A. A. Wagadarikar, D. L. Marks, K. Choi, R. Horisaki, and D. J. Brady, “Imaging through turbulence using compressive coherence sensing,” Opt. Express 35, 838–840 (2010).

Coulman, C.

C. Coulman, “Fundamental and applied aspects of astronomical ’seeing’,” Annu. Rev. Astron. Astrophys. 23, 19–57 (1985).
[Crossref]

Cremer, C.

S. Hell, G. Reiner, C. Cremer, and E. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Dimotakis, P.

Ezawa, T.

Fetterman, M. R.

Fragola, A.

P. Vermeulen, E. Muro, T. Pons, V. Loriette, and A. Fragola, “Adaptive optics for fluorescence wide-field microscopy using spectrally independent guide star and markers,” J. Biomed. Opt. 16, 076019 (2011).
[Crossref] [PubMed]

Fu, M.

Ghiglia, D.

D. Ghiglia and M. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Goldstein, R.

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

Goodman, J.

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

Han, B.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274 – 280 (2007).
[Crossref]

Heintzmann, R.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

K. Wicker, S. Sindbert, and R. Heintzmann, “Characterisation of a resolution enhancing image inversion interferometer,” Opt. Express 17, 15491–15501 (2009).
[Crossref] [PubMed]

Hell, S.

S. Hell, G. Reiner, C. Cremer, and E. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Hewlett, S.

P. Török, S. Hewlett, and P. Varga, “The role of specimen-induced spherical aberration in confocal microscopy,” J. Microsc. 188, 158–172 (1997).
[Crossref]

Horisaki, R.

A. A. Wagadarikar, D. L. Marks, K. Choi, R. Horisaki, and D. J. Brady, “Imaging through turbulence using compressive coherence sensing,” Opt. Express 35, 838–840 (2010).

Itoh, K.

K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A. 3, 94–100 (1986).
[Crossref]

K. Itoh and Y. Ohtsuka, “Interferometric image reconstruction through the turbulent atmosphere,” Appl. Opt. 20, 4239–4244 (1981).
[Crossref] [PubMed]

Kern, B.

Kiessling, A.

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

Kissel, M.

Kowarschik, R.

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

Kubby, J.

Laczik, Z.

LaHaie, I.

Lang, D.

Leonhardt, H.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

Liu, Q.

Loriette, V.

P. Vermeulen, E. Muro, T. Pons, V. Loriette, and A. Fragola, “Adaptive optics for fluorescence wide-field microscopy using spectrally independent guide star and markers,” J. Biomed. Opt. 16, 076019 (2011).
[Crossref] [PubMed]

Mahajan, V.

Marks, D. L.

A. A. Wagadarikar, D. L. Marks, K. Choi, R. Horisaki, and D. J. Brady, “Imaging through turbulence using compressive coherence sensing,” Opt. Express 35, 838–840 (2010).

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Muson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[Crossref] [PubMed]

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the fresnel domain,” Appl. Opt. 38, 1332–1342 (1999).
[Crossref]

Martin, C.

Miyamoto, Y.

Muro, E.

P. Vermeulen, E. Muro, T. Pons, V. Loriette, and A. Fragola, “Adaptive optics for fluorescence wide-field microscopy using spectrally independent guide star and markers,” J. Biomed. Opt. 16, 076019 (2011).
[Crossref] [PubMed]

Muson, D. C.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Muson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[Crossref] [PubMed]

Naik, D. N.

Neil, M.

M. Schwertner, M. Booth, M. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

Norton, A.

Ohtsuka, Y.

K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A. 3, 94–100 (1986).
[Crossref]

K. Itoh and Y. Ohtsuka, “Interferometric image reconstruction through the turbulent atmosphere,” Appl. Opt. 20, 4239–4244 (1981).
[Crossref] [PubMed]

Pons, T.

P. Vermeulen, E. Muro, T. Pons, V. Loriette, and A. Fragola, “Adaptive optics for fluorescence wide-field microscopy using spectrally independent guide star and markers,” J. Biomed. Opt. 16, 076019 (2011).
[Crossref] [PubMed]

Potuluri, P.

Pritt, M.

D. Ghiglia and M. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Rajadhyaksha, M.

D.-S. Wan, M. Rajadhyaksha, and R. Webb, “Analysis of spherical aberration of a water immersion objective: application to specimens with refractive indices 1.33–1.40,” J. Microsc. 197, 274–284 (2000).
[Crossref] [PubMed]

Reiner, G.

S. Hell, G. Reiner, C. Cremer, and E. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Roddier, C.

F. Roddier and C. Roddier, “An image reconstruction of alpha orionis,” Astrophys. J. 295, L21–L23 (1985).
[Crossref]

C. Roddier and F. Roddier, “High angular resolution observations of alpha orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[Crossref]

Roddier, F.

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 99–166 (1988).
[Crossref]

F. Roddier and C. Roddier, “An image reconstruction of alpha orionis,” Astrophys. J. 295, L21–L23 (1985).
[Crossref]

C. Roddier and F. Roddier, “High angular resolution observations of alpha orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[Crossref]

Romberg, J.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[Crossref]

Rosen, J.

Sasamoto, M.

M. Sasamoto and K. Yoshimori, “First experimental report on fully passive interferometric three-dimensional imaging spectrometry,” Jpn. J. Appl. Phys. 48, 09LB03 (2009).
[Crossref]

Schermelleh, L.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

Schwertner, M.

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

M. Schwertner, M. Booth, M. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

M. Schwertner, M. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high na adaptive optical microscopy,” Opt. Express 12, 6540–6552 (2004).
[Crossref] [PubMed]

Schwiegerling, J.

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Sindbert, S.

Stack, R. A.

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E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
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P. Vermeulen, E. Muro, T. Pons, V. Loriette, and A. Fragola, “Adaptive optics for fluorescence wide-field microscopy using spectrally independent guide star and markers,” J. Biomed. Opt. 16, 076019 (2011).
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M. Booth and T. Wilson, “Refractive-index-mismatch induced aberrations in single-photon and two-photon microscopy and the use of aberration correction,” J. Biomed. Opt. 6, 266–272 (2001).
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M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

D.-S. Wan, M. Rajadhyaksha, and R. Webb, “Analysis of spherical aberration of a water immersion objective: application to specimens with refractive indices 1.33–1.40,” J. Microsc. 197, 274–284 (2000).
[Crossref] [PubMed]

P. Török, S. Hewlett, and P. Varga, “The role of specimen-induced spherical aberration in confocal microscopy,” J. Microsc. 188, 158–172 (1997).
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J. Opt. Soc. Am. A (4)

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L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Jpn. J. Appl. Phys. (1)

M. Sasamoto and K. Yoshimori, “First experimental report on fully passive interferometric three-dimensional imaging spectrometry,” Jpn. J. Appl. Phys. 48, 09LB03 (2009).
[Crossref]

Opt. Commun. (1)

D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Widefield microscopy with infinite depth of field and enhanced lateral resolution based on an image inverting interferometer,” Opt. Commun. 342, 102–108 (2015).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (1)

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274 – 280 (2007).
[Crossref]

Opt. Lett. (3)

Phys. Rep. (1)

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 99–166 (1988).
[Crossref]

Physica (1)

v. F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Radio Science (1)

R. Goldstein, H. Zebker, and C. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Science 23, 713–720 (1988).
[Crossref]

Science (1)

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Muson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[Crossref] [PubMed]

Other (2)

D. Ghiglia and M. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

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

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Figures (8)

Fig. 1
Fig. 1 Schematic illustration of the light propagation from the object point to the lens.
Fig. 2
Fig. 2 Schematic sketch of the proposed coherence imaging microscope.
Fig. 3
Fig. 3 Sketch of the experimental setup. The III is marked by the dashed box.
Fig. 4
Fig. 4 (a) Scheme of the reference measurement (determination of the aberrations) and (b) of the actual measurement of the resolution test chart.
Fig. 5
Fig. 5 From the interferograms at three different phase differences (a)–(c), ΔΦexp (d) can be gained. By unwrapping ΔΦexp and decomposing it into Zernike polynomials up to the 10th radial order, ΔΦfit can be gained (e). The remaining errors ΔΦres are shown in (f). Note that for a better illustration, (d)–(f) represent the modulo of the respective functions.
Fig. 6
Fig. 6 Illustration of the point images. (a) shows the conventional control image before the III. The conventional images on the CCD2, which were observed after passing through arms 1 and 2, respectively, are shown in (b) and (c). In (d) the reconstruction of the point source without the phase correction can be seen, whereas in (e) the phase correction was applied. For comparison, (f) shows a simulation of the diffraction-limited point image.
Fig. 7
Fig. 7 Illustration of the resolution of two-point objects with different distances. This is a comparison of the conventional image without the III (a), to the conventional image behind the III (b) as well as to the reconstruction of the structures by means of Γ without (c) and with correction of the aberrations (d). (e) shows the cross-section through the first column of two-point structures and (f) the cross-section through the second column.
Fig. 8
Fig. 8 Influence of two random wavefront errors on the point images and the associated MTFs and the PTFs, respectively. Besides the conventional images, three reconstructions by means of Γ are shown. The latter represent the uncorrected, the system-corrected, and the fully corrected images. To verify that especially aberrations of odd radial orders influence the imaging process, the Zernike coefficients c n m of the system-corrected wavefront are plotted. Additionally, images of an 2D object under the respective conditions are shown.

Tables (1)

Tables Icon

Table 1 Overview of the Zernike polynomials up to the fourth radial order. Polynomials with even radial orders are marked gray.

Equations (23)

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U L ( r , Δ z , t ) = U 0 ( ρ , t ) exp ( i k 2 d ( r ρ ) 2 ) d ξ d η
= exp ( i k 2 d r 2 ) i λ d { U 0 ( ρ , t ) h ( ρ ) } ( r , Δ z , t ) .
U L ( r , Δ z , t ) = U L ( r , Δ z , t ) exp ( i k 2 f r 2 ) P ( r , R )
Γ ( Δ r ) = U L ( r , Δ z , t ) U L * ( r , Δ z , t ) t
= P ( r , R ) P * ( r , R ) U L ( r , Δ z , t ) U L * ( r , Δ z , t ) t
Γ ( Δ r ) = P ( r , R ) P * ( r , R ) { 1 λ 2 d 2 I ( ρ 2 ) } ( Δ r )
Γ ( ν ) = P ( ν , ν max ) P * ( ν , ν max ) { 1 λ 2 d 2 I ( ρ ) } ( ν )
= OTF IF ( ν , ν max ) I ˜ ( ν ) .
I R ( ρ ) = 1 { OTF IF ( ν , ν max ) I ˜ ( ν ) }
= PSF IF ( ρ ) I ( ρ )
P ( ν , ν max ) = p ( ν , ν max ) exp ( i 2 π λ Φ ( ν ) ) .
p ( ν , ν max ) = { 1 | ν | / ν max 1 0 | ν | / ν max > 1 .
OTF IF ( ν , ν max ) = P ( ν , ν max ) P * ( ν , ν max )
= p ( ν , ν max ) exp ( i 2 π λ Δ Φ ( ν ) ) .
Φ ( ν ) = n = 0 m = n n c n m Z n m ( ν ¯ , ψ ) .
Z n m ( ν ¯ , ψ ) = { N n m R n | m | ( ν ¯ ) cos ( m ψ ) m 0 N n n R n | m | ( ν ¯ ) sin ( m ψ ) m < 0 .
R n | m | ( ν ¯ ) = l = 0 ( n | m | ) / 2 ( 1 ) l ( n l ) ! l ! ( n + | m | 2 l ) ! ( n | m | 2 l ) ! ν ¯ n 2 l
N n m = 2 ( n + 1 ) 1 + δ m 0
Δ Φ ( ν ) = 2 n = 0 , odd m = n n c n m Z n m ( ν ¯ , ψ )
Γ corr ( ν ) = exp ( i 2 π λ Δ Φ exp ( ν , ν max ) ) OTF IF ( ν , ν max ) I ˜ ( ν )
= OTF IF , corr ( ν , ν max ) I ˜ ( ν )
PSF IF ( ρ ) = J 1 ( 2 ρ ) 2 | ( ρ ) |
Δ Φ fit ( ν , ν max ) = 2 n = 0 n max m = n n c n m Z n m ( ν ¯ , ψ ) .

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