Abstract

Structured illumination microscopy (SIM) was recently adapted to coherent imaging, named structured oblique-illumination microscopy (SOIM), to improve the contrast and resolution of a light-scattering image. Herein, we present high-resolution laterally isotropic SOIM imaging with 2D hexagonal illuminations. The SOIM is implemented in a SIM fluorescence system based on a spatial-light modulator (SLM). We design an SLM pattern to generate diffraction beams at 0° and ± 60.3° simultaneously to form a 2D hexagonal illumination, and undertake calculations to obtain optimal SLM shifts at 19 phases to yield a reconstructed image correctly. Beams of linear and circular polarizations are used to show the effect of polarization on the resolution improvement. We derive the distributions of the electric field of the resultant hexagonal patterns and work out the formulations of the corresponding coherent-scattering imaging for image reconstruction. The reconstructed images of gold nanoparticles (100 nm) confirm the two-fold improvement of resolution and reveal the effect of polarization on resolving adjacent nanoparticles. To demonstrate biological applications, we present the cellular structures of a label-free fixed HeLa cell with improved contrast and resolution. This work enables one to perform high-resolution dual-mode − fluorescence and light-scattering − imaging in a system, and is expected to broaden the applications of SOIM.

© 2017 Optical Society of America

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References

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2016 (2)

F. Jünger, P. V. Olshausen, and A. Rohrbach, “Fast, label-free super-resolution live-cell imaging using rotating coherent scattering (ROCS) microscopy,” Sci. Rep. 6(1), 30393 (2016).
[Crossref] [PubMed]

F. Ströhl and C. F. Kaminski, “Frontiers in structured illumination microscopy,” Optica 3(6), 667–677 (2016).
[Crossref]

2015 (2)

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

M. Saxena, G. Eluru, and S. S. Gorthi, “Structured illumination microscopy,” Adv. Opt. Photonics 7(2), 241–275 (2015).
[Crossref]

2014 (3)

2013 (4)

2012 (3)

2011 (2)

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

B. J. Chang, S. H. Lin, L. J. Chou, and S. Y. Chiang, “Subdiffraction scattered light imaging of gold nanoparticles using structured illumination,” Opt. Lett. 36(24), 4773–4775 (2011).
[Crossref] [PubMed]

2009 (3)

B. J. Chang, L. J. Chou, Y. C. Chang, and S. Y. Chiang, “Isotropic image in structured illumination microscopy patterned with a spatial light modulator,” Opt. Express 17(17), 14710–14721 (2009).
[Crossref] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

L. M. Hirvonen, K. Wicker, O. Mandula, and R. Heintzmann, “Structured illumination microscopy of a living cell,” Eur. Biophys. J. 38(6), 807–812 (2009).
[Crossref] [PubMed]

2008 (3)

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

S. R. P. Pavani, A. R. Libertun, S. V. King, and C. J. Cogswell, “Quantitative structured-illumination phase microscopy,” Appl. Opt. 47(1), 15–24 (2008).
[Crossref] [PubMed]

R. Fiolka, M. Beck, and A. Stemmer, “Structured illumination in total internal reflection fluorescence microscopy using a spatial light modulator,” Opt. Lett. 33(14), 1629–1631 (2008).
[Crossref] [PubMed]

2007 (1)

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

2006 (4)

E. Chung, D. Kim, and P. T. So, “Extended resolution wide-field optical imaging: objective-launched standing-wave total internal reflection fluorescence microscopy,” Opt. Lett. 31(7), 945–947 (2006).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

2000 (3)

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[Crossref] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

1999 (2)

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

T. A. Klar and S. W. Hell, “Subdiffraction resolution in far-field fluorescence microscopy,” Opt. Lett. 24(14), 954–956 (1999).
[Crossref] [PubMed]

Agard, D. A.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Baird, M. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

Beach, J. R.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Beck, M.

Best, G.

Betzig, E.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Carlton, P. M.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Chang, B. J.

Chang, Y. C.

Chen, B.-C.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Chen, J.

Chetty, S.

Chhun, B. B.

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

Chiang, S. Y.

Chou, L. J.

Chowdhury, S.

Chung, E.

Cogswell, C. J.

Cremer, C.

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Davidson, M. W.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

R. Fiolka, L. Shao, E. H. Rego, M. W. Davidson, and M. G. L. Gustafsson, “Time-lapse two-color 3D imaging of live cells with doubled resolution using structured illumination,” Proc. Natl. Acad. Sci. U.S.A. 109(14), 5311–5315 (2012).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Dhalla, A.-H.

Dyba, M.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Egner, A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Eluru, G.

M. Saxena, G. Eluru, and S. S. Gorthi, “Structured illumination microscopy,” Adv. Opt. Photonics 7(2), 241–275 (2015).
[Crossref]

Fiolka, R.

Frohn, J. T.

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[Crossref] [PubMed]

Girirajan, T. P. K.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Golubovskaya, I. N.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Gorthi, S. S.

M. Saxena, G. Eluru, and S. S. Gorthi, “Structured illumination microscopy,” Adv. Opt. Photonics 7(2), 241–275 (2015).
[Crossref]

Griffis, E. R.

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

Gruppetta, S.

Gustafsson, M. G. L.

R. Fiolka, L. Shao, E. H. Rego, M. W. Davidson, and M. G. L. Gustafsson, “Time-lapse two-color 3D imaging of live cells with doubled resolution using structured illumination,” Proc. Natl. Acad. Sci. U.S.A. 109(14), 5311–5315 (2012).
[Crossref] [PubMed]

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

Hammer, J. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Heckenberg, N.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Heintzmann, R.

K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014).
[Crossref]

K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express 21(2), 2032–2049 (2013).
[Crossref] [PubMed]

L. M. Hirvonen, K. Wicker, O. Mandula, and R. Heintzmann, “Structured illumination microscopy of a living cell,” Eur. Biophys. J. 38(6), 807–812 (2009).
[Crossref] [PubMed]

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Hell, S. W.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

T. A. Klar and S. W. Hell, “Subdiffraction resolution in far-field fluorescence microscopy,” Opt. Lett. 24(14), 954–956 (1999).
[Crossref] [PubMed]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Hess, S. T.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Hirvonen, L. M.

L. M. Hirvonen, K. Wicker, O. Mandula, and R. Heintzmann, “Structured illumination microscopy of a living cell,” Eur. Biophys. J. 38(6), 807–812 (2009).
[Crossref] [PubMed]

Huang, H. C.

Izatt, J.

Jakobs, S.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Jünger, F.

F. Jünger, P. V. Olshausen, and A. Rohrbach, “Fast, label-free super-resolution live-cell imaging using rotating coherent scattering (ROCS) microscopy,” Sci. Rep. 6(1), 30393 (2016).
[Crossref] [PubMed]

Kaminski, C. F.

Kim, D.

King, S. V.

Kirchhausen, T.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Klar, T. A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

T. A. Klar and S. W. Hell, “Subdiffraction resolution in far-field fluorescence microscopy,” Opt. Lett. 24(14), 954–956 (1999).
[Crossref] [PubMed]

Knapp, H. F.

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[Crossref] [PubMed]

Kner, P.

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

Lai, K.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Lai, X.

Lee, E. S.

Lee, J. Y.

Li, D.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Libertun, A. R.

Lin, S. H.

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Littleton, B.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Longstaff, D.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Lv, X.

Mandula, O.

K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express 21(2), 2032–2049 (2013).
[Crossref] [PubMed]

L. M. Hirvonen, K. Wicker, O. Mandula, and R. Heintzmann, “Structured illumination microscopy of a living cell,” Eur. Biophys. J. 38(6), 807–812 (2009).
[Crossref] [PubMed]

Mason, M. D.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Milkie, D. E.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Moses, B.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Munroe, P.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Olshausen, P. V.

F. Jünger, P. V. Olshausen, and A. Rohrbach, “Fast, label-free super-resolution live-cell imaging using rotating coherent scattering (ROCS) microscopy,” Sci. Rep. 6(1), 30393 (2016).
[Crossref] [PubMed]

Park, J. H.

Pasham, M.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Pavani, S. R. P.

Rego, E. H.

R. Fiolka, L. Shao, E. H. Rego, M. W. Davidson, and M. G. L. Gustafsson, “Time-lapse two-color 3D imaging of live cells with doubled resolution using structured illumination,” Proc. Natl. Acad. Sci. U.S.A. 109(14), 5311–5315 (2012).
[Crossref] [PubMed]

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

Rohrbach, A.

F. Jünger, P. V. Olshausen, and A. Rohrbach, “Fast, label-free super-resolution live-cell imaging using rotating coherent scattering (ROCS) microscopy,” Sci. Rep. 6(1), 30393 (2016).
[Crossref] [PubMed]

Rubinsztein-Dunlop, H.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

Sarafis, V.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Saxena, M.

M. Saxena, G. Eluru, and S. S. Gorthi, “Structured illumination microscopy,” Adv. Opt. Photonics 7(2), 241–275 (2015).
[Crossref]

Sedat, J. W.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Shao, L.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

R. Fiolka, L. Shao, E. H. Rego, M. W. Davidson, and M. G. L. Gustafsson, “Time-lapse two-color 3D imaging of live cells with doubled resolution using structured illumination,” Proc. Natl. Acad. Sci. U.S.A. 109(14), 5311–5315 (2012).
[Crossref] [PubMed]

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

So, P. T.

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Stemmer, A.

R. Fiolka, M. Beck, and A. Stemmer, “Structured illumination in total internal reflection fluorescence microscopy using a spatial light modulator,” Opt. Lett. 33(14), 1629–1631 (2008).
[Crossref] [PubMed]

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[Crossref] [PubMed]

Ströhl, F.

Wang, C. J. R.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Wicker, K.

K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014).
[Crossref]

K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express 21(2), 2032–2049 (2013).
[Crossref] [PubMed]

L. M. Hirvonen, K. Wicker, O. Mandula, and R. Heintzmann, “Structured illumination microscopy of a living cell,” Eur. Biophys. J. 38(6), 807–812 (2009).
[Crossref] [PubMed]

Winoto, L.

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

Xu, P.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Xu, Y.

Zeng, S.

Zhang, M.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Zhang, X.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

M. Saxena, G. Eluru, and S. S. Gorthi, “Structured illumination microscopy,” Adv. Opt. Photonics 7(2), 241–275 (2015).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (3)

Biophys. J. (2)

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Eur. Biophys. J. (1)

L. M. Hirvonen, K. Wicker, O. Mandula, and R. Heintzmann, “Structured illumination microscopy of a living cell,” Eur. Biophys. J. 38(6), 807–812 (2009).
[Crossref] [PubMed]

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

Micron (1)

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38(2), 150–157 (2007).
[Crossref] [PubMed]

Nat. Methods (3)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

P. Kner, B. B. Chhun, E. R. Griffis, L. Winoto, and M. G. L. Gustafsson, “Super-resolution video microscopy of live cells by structured illumination,” Nat. Methods 6(5), 339–342 (2009).
[Crossref] [PubMed]

L. Shao, P. Kner, E. H. Rego, and M. G. L. Gustafsson, “Super-resolution 3D microscopy of live whole cells using structured illumination,” Nat. Methods 8(12), 1044–1046 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Optica (1)

Proc. Natl. Acad. Sci. U.S.A. (3)

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. U.S.A. 97(13), 7232–7236 (2000).
[Crossref] [PubMed]

R. Fiolka, L. Shao, E. H. Rego, M. W. Davidson, and M. G. L. Gustafsson, “Time-lapse two-color 3D imaging of live cells with doubled resolution using structured illumination,” Proc. Natl. Acad. Sci. U.S.A. 109(14), 5311–5315 (2012).
[Crossref] [PubMed]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Proc. SPIE (1)

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Sci. Rep. (1)

F. Jünger, P. V. Olshausen, and A. Rohrbach, “Fast, label-free super-resolution live-cell imaging using rotating coherent scattering (ROCS) microscopy,” Sci. Rep. 6(1), 30393 (2016).
[Crossref] [PubMed]

Science (2)

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

(a) Bottom (xy plane) and (b) side (xz plane) views of the propagation and polarization vectors of three sets of ± 1-order beams at θ = 0° and ± 60.3° for Uni-s; the beams are s-linearly polarized with their polarization vectors along the y axis at the BFP of the objective; the interaction angle for each set of ± 1-order beams is 2β = 88°.

Fig. 2
Fig. 2

(a) A simulated resolution target, (b) 2D correlation map of Sum_V0, and images reconstructed with (c) optimized V0(1.68π, 0.58π) and (d) non-optimized V0(1π, 1.75π), indicated as red and blue cursors in (b).

Fig. 3
Fig. 3

Schematic of an SLM-based SOIM microscope; SLM spatial light modulator, L lenses, M mirror, HWP/QWP half/quarter wave plate, BS(50/50) beam splitter of 50% transmission and 50% reflection, BFP back focal plane; length in mm. The polarization states of beams are for the Uni-s case; beams at 0° and ± 60° are indicated with red and blue lines, respectively.

Fig. 4
Fig. 4

(a) Portions of the designed SLM pattern with unit cell 8 x 14 pixels and (b) Fourier spectrum of the entire SLM pattern.

Fig. 5
Fig. 5

(a) Comparison of pattern shifts in two sets − SLM-1 and SLM-2, obtained from the calculations of condition number and according to the equal-shift concept, respectively, (b) normalized magnitude maps |A−1| of SLM-1 and SLM-2; the lth row of |A−1| represents the coefficients of spectra D0-18 to retrieve the lth pattern-excitation component Fl (l = 0 – 18), and (c) wide-field image of 100-nm gold nanoparticles and their SOIM images reconstructed with SLM-1 and SLM-2.

Fig. 6
Fig. 6

Polarization vectors of interfering beams at the BFP of the objective and normalized I( r )and | I( k )| of resultant hexagonal patterns at the sample plane calculated for (a) Axi-s, (b) Uni-s, (c) Uni-p, and (d) Uni-c cases, and (e) horizontal and vertical profiles of I( r ) indicated with dash blue lines in (a)-(d). In the calculations, the interaction angle 2β = 88° for each set of diffraction beams at the sample plane was determined experimentally.

Fig. 7
Fig. 7

(a) Light-scattering images of a negative USAF 1951 resolution target at Group 9 and Elements 2 and 3, (b) fluorescence images of hexagonal patterns (Uni-s, -p, and -c), and (c) horizontal and vertical profiles of patterns indicated with dashed yellow lines in (b); the insets in (a) are enlarged structures in the red region.

Fig. 8
Fig. 8

Light-scattering WF and SOIM (Uni-s, -p, and -c) images, enlarged images of a single nanoparticle within a yellow box indicated in the images, and horizontal and vertical profiles of a single nanoparticle indicated with yellow lines for (a) gold nanoparticles and (b) polystyrene beads (both 100 nm). The gold nanoparticles were immersed in water (n = 1.33) but the beads were immersed in index-matching oil (n = 1.52) to enhance the image contrast.

Fig. 9
Fig. 9

(a) WF and SOIM (Uni-s, -p, -c) images of gold nanoparticles (100 nm) in clusters, (b) enlarged images of numbers 1 − 4 in (a), and (c) profiles indicated with dash lines in (b); the arrows indicate the best-resolved nanoparticles among Uni cases and the dotted lines in (c) are the Gaussian fits of two curves.

Fig. 10
Fig. 10

(a) WF and SOIM (Uni-c) images of cellular structures near the edge of a label-free fixed HeLa cell immersed in water, (b) enlarged images in the yellow regions of (a) and profiles indicated as numbers 1 and 2 in the images, and (c) WF and SOIM (Uni-c) images of cellular structures immersed in an index-matching oil (n = 1.52).

Tables (3)

Tables Icon

Table 1 x-Components of Coefficients of Patterned-Excitation Components

Tables Icon

Table 2 y-Components of Coefficients of Patterned-Excitation Components

Tables Icon

Table 3 z-Components of Coefficients of Patterned-Excitation Components

Equations (15)

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

E ( r ,t)= n=1,2,3 P ±n f ±n e i( k ±n r ωt ϕ ±n )
k ±n =k(±cosθsinβ x ^ ,±sinθsinβ y ^ ,cosβ z ^ ).
P ±n = 1 1+ε ( cosθsinθ(cosβ1) x ^ ,( sin 2 θcosβ+ cos 2 θ) y ^ ,sinθsinβ z ^ ) i ε 1+ε ( ( cos 2 θcosβ+ sin 2 θ) x ^ ,cosθsinθ(cosβ1) y ^ ,cosθsinβ z ^ ).
E Unis ( r ,t)= ( cosωt{ 3 (cosβ1) 2 [ cos( k 2 + ϕ 2 )cos( k 3 + ϕ 3 ) ] }, cosωt{ 2cos( k 1 + ϕ 1 )+ 1+3cosβ 2 [ cos( k 2 + ϕ 2 )+cos( k 3 + ϕ 3 ) ] }, sinωt{ 3 sinβ[ sin( k 2 + ϕ 2 )sin( k 3 + ϕ 3 ) ] } ) z=0 .
E Unip ( r ,t)= ( cosωt{ 2cosβcos( k 1 + ϕ 1 )+ 3+cosβ 2 [ cos( k 2 + ϕ 2 )+cos( k 3 + ϕ 3 ) ] }, cosωt{ 3 (cosβ1) 2 [ cos( k 2 + ϕ 2 )cos( k 3 + ϕ 3 ) ] }, sinωt{ sinβ[ 2sin( k 1 + ϕ 1 )+sin( k 2 + ϕ 2 )+sin( k 3 + ϕ 3 ) ] } ) z=0 .
E Unic ( r ,t)= 1 2 [ E Unis ( r ,t)+ E Unip ( r ,t+ π 2ω ) ] z=0 .
E Axis ( r ,t)= ( cosωt{ 3 [ cos( k 2 + ϕ 2 )cos( k 3 + ϕ 3 ) ] }, cosωt[ 2cos( k 1 + ϕ 1 )+cos( k 2 + ϕ 2 )+cos( k 3 + ϕ 3 ) ], 0 ) z=0 .
D( r )= | H ( r )[ E ( r ,t) U g ( r ) ] | 2 .
D( k )= ac{ H ( k )[ E ( k ) U g ( k ) ] } .
D( k )= m=x,y,z ac{ H m ( k )[ E m ( k ) U g ( k ) ] } = l=018 e i Φ l m=x,y,z C l,m F l,m ( k ) . = l=018 e i Φ l F l ( k )
e i Φ 0 F 0 ( k )= m=x,y,z n=13 C 0,mn [ ac( H m ( k ) U g ( k k n ) )+ac( H m ( k ) U g ( k + k n ) ) ] e i Φ 1,2 F 1,2 ( k )= e ±i2 ϕ 1 m=x,y,z C 1,m / C 2,m ( H m ( k ) U g ( k k 1 ) H m ( k ) U g ( k ± k 1 ) ) e i Φ 3,4 F 3,4 ( k )= e ±i2 ϕ 2 m=x,y,z C 3,m / C 4,m ( H m ( k ) U g ( k k 2 ) H m ( k ) U g ( k ± k 2 ) ) e i Φ 5,6 F 5,6 ( k )= e ±i2 ϕ 3 m=x,y,z C 5,m / C 6,m ( H m ( k ) U g ( k k 3 ) H m ( k ) U g ( k ± k 3 ) ) e i Φ 7,8 F 7,8 ( k )= e ±i( ϕ 1 + ϕ 2 ) m=x,y,z C 7,m / C 8,m ( H m ( k ) U g ( k k 1 ) H m ( k ) U g ( k ± k 2 ) + H m ( k ) U g ( k k 2 ) H m ( k ) U g ( k ± k 1 ) ) e i Φ 9,10 F 9,10 ( k )= e ±i( ϕ 1 ϕ 2 ) m=x,y,z C 9,m / C 10,m ( H m ( k ) U g ( k k 1 ) H m ( k ) U g ( k k 2 ) + H m ( k ) U g ( k ± k 2 ) H m ( k ) U g ( k ± k 1 ) ) e i Φ 11,12 F 11,12 ( k )= e ±i( ϕ 1 + ϕ 3 ) m=x,y,z C 11,m / C 12,m ( H m ( k ) U g ( k k 1 ) H m ( k ) U g ( k ± k 3 ) + H m ( k ) U g ( k k 3 ) H m ( k ) U g ( k ± k 1 ) ) e i Φ 13,14 F 13,14 ( k )= e ±i( ϕ 1 ϕ 3 ) m=x,y,z C 13,m / C 14,m ( H m ( k ) U g ( k k 1 ) H m ( k ) U g ( k k 3 ) + H m ( k ) U g ( k ± k 3 ) H m ( k ) U g ( k ± k 1 ) ) e i Φ 15,16 F 15,16 ( k )= e ±i( ϕ 2 + ϕ 3 ) m=x,y,z C 15,m / C 16,m ( H m ( k ) U g ( k k 2 ) H m ( k ) U g ( k ± k 3 ) + H m ( k ) U g ( k k 3 ) H m ( k ) U g ( k ± k 2 ) ) . e i Φ 17,18 F 17,18 ( k )= e ±i( ϕ 2 ϕ 3 ) m=x,y,z C 17,m / C 18,m ( H m ( k ) U g ( k k 2 ) H m ( k ) U g ( k k 3 ) + H m ( k ) U g ( k ± k 3 ) H m ( k ) U g ( k ± k 2 ) )
H ( k )= n=13 C E,n H ( k + k n )+ C E,n H ( k k n ) .
D ( k )=ac{ H ( k ) U g ( k ) }= l=018 m=x,y,z C l,m G l,m ( k )= l=018 G l ( k ) .
[ D 0 ( k ) D 1 ( k ) D q ( k ) D 18 ( k ) ]=[ e i Φ 0,018 e i Φ 1,018 e i Φ q,018 e i Φ 18,018 ][ F 0 ( k ) F 1 ( k ) F l ( k ) F 18 ( k ) ]=AF.
Sum_ V 0 = l=018 | F 0 ( k ) F l ( k + K l ) | .

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