Abstract

We describe a high speed 3D Acousto-Optic Lens Microscope (AOLM) for femtosecond 2-photon imaging. By optimizing the design of the 4 AO Deflectors (AODs) and by deriving new control algorithms, we have developed a compact spherical AOL with a low temporal dispersion that enables 2-photon imaging at 10-fold lower power than previously reported. We show that the AOLM can perform high speed 2D raster-scan imaging (>150 Hz) without scan rate dependent astigmatism. It can deflect and focus a laser beam in a 3D random access sequence at 30 kHz and has an extended focusing range (>137 μm; 40X 0.8NA objective). These features are likely to make the AOLM a useful tool for studying fast physiological processes distributed in 3D space

© 2010 OSA

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  1. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
    [CrossRef] [PubMed]
  2. K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50(6), 823–839 (2006).
    [CrossRef] [PubMed]
  3. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
    [CrossRef] [PubMed]
  4. F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
    [CrossRef] [PubMed]
  5. J. B. Pawley, Handbook of Biological Confocal Microscopy (Plenum Press, New York, 1995).
  6. J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83(4), 2292–2299 (2002).
    [CrossRef] [PubMed]
  7. R. D. Roorda, T. M. Hohl, R. Toledo-Crow, and G. Miesenböck, “Video-rate nonlinear microscopy of neuronal membrane dynamics with genetically encoded probes,” J. Neurophysiol. 92(1), 609–621 (2004).
    [CrossRef] [PubMed]
  8. V. Iyer, B. E. Losavio, and P. Saggau, “Compensation of spatial and temporal dispersion for acousto-optic multiphoton laser-scanning microscopy,” J. Biomed. Opt. 8(3), 460–471 (2003).
    [CrossRef] [PubMed]
  9. X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
    [CrossRef]
  10. V. Iyer, T. M. Hoogland, and P. Saggau, “Fast functional imaging of single neurons using random-access multiphoton (RAMP) microscopy,” J. Neurophysiol. 95(1), 535–545 (2005).
    [CrossRef] [PubMed]
  11. R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
    [CrossRef] [PubMed]
  12. Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
    [CrossRef] [PubMed]
  13. Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008).
    [CrossRef] [PubMed]
  14. S. Shoham, D. H. O’Connor, D. V. Sarkisov, and S. S. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nat. Methods 2(11), 837–843 (2005).
    [CrossRef] [PubMed]
  15. B. Losavio, V. Iyer, and P. Saggau, “Two photon microscope for multisite microphotolysis of caged neurotransmitters in acute brain slices,” J. Biomed. Opt . 14, 064033 064031–064013 (2009).
    [CrossRef]
  16. J. Xu, and R. Stroud, Acousto-Optic Devices, Principles, Design and Applications (John Wiley and Sons Inc., 1992).
  17. N. Friedman, A. Kaplan, and N. Davidson, “Acousto-optic scanning system with very fast nonlinear scans,” Opt. Lett. 25(24), 1762–1764 (2000).
    [CrossRef]
  18. W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods 4(1), 73–79 (2007).
    [CrossRef]
  19. E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “Aberration-free optical refocusing in high numerical aperture microscopy,” Opt. Lett. 32(14), 2007–2009 (2007).
    [CrossRef] [PubMed]
  20. A. Kaplan, N. Friedman, and N. Davidson, “Acousto-optic lens with very fast focus scanning,” Opt. Lett. 26(14), 1078–1080 (2001).
    [CrossRef]
  21. G. Duemani Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11(6), 713–720 (2008).
    [CrossRef] [PubMed]
  22. G. D. Reddy and P. Saggau, “Fast three-dimensional laser scanning scheme using acousto-optic deflectors,” J. Biomed. Opt. 10(6), 064038 (2005).
    [CrossRef]
  23. G. Reddy, “A multiphoton microscope for three dimensional functional recording of fast neuronal activity,” (Rice University, Houston Texas USA, 2007).
  24. P. A. Kirkby, R. A. Silver, and K. M. N. S. Nadella, “IMAGING APPARATUS AND METHODS,” (20.03.2008).
  25. D. Vucinić, T. J. Sejnowski, and B. Lu, “A compact multiphoton 3D imaging system for recording fast neuronal activity,” PLoS ONE 2(8), e699 (2007).
    [CrossRef] [PubMed]
  26. Z. Kam, D. A. Agard, and J. W. Sedat, “Three-dimensional microscopy in thick biological samples: A fresh approach for adjusting focus and correcting spherical aberration,” Bioimaging 5(1), 40–49 (1997).
    [CrossRef]
  27. A. P. Goutzoulis, D. R. Pape, and S. V. Kulakov, Design and Fabrication of Acousto Optic Devices (Marcel Dekker, 1994).
  28. E. H. Young, H. C. Ho, and L. J. Harrison, “Optically rotated long time aperture TeO2 Bragg cell,” Proc. SPIE 1296, 304–315 (1990).
    [CrossRef]
  29. B. K. A. Ngoi, K. Venkatakrishnan, L. E. Lim, B. Tan, and L. E. N. Lim, “Angular dispersion compensation for acousto-optic devices used for ultrashort-pulsed laser micromachining,” Opt. Express 9(4), 200–206 (2001).
    [CrossRef] [PubMed]
  30. D. Reddy, and P. Saggau, “Fast Three-Dimensional Random Access Multi-Photon Microscopy for Functional Recording of Neuronal Activity,” Proc. SPIE 6630, 66301A 66301–66308 (2007).
  31. L. Zhu, P. C. Sun, and Y. Fainman, “Aberration-free dynamic focusing with a multichannel micromachined membrane deformable mirror,” Appl. Opt. 38(25), 5350–5354 (1999).
    [CrossRef]

2009

B. Losavio, V. Iyer, and P. Saggau, “Two photon microscope for multisite microphotolysis of caged neurotransmitters in acute brain slices,” J. Biomed. Opt . 14, 064033 064031–064013 (2009).
[CrossRef]

2008

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008).
[CrossRef] [PubMed]

G. Duemani Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11(6), 713–720 (2008).
[CrossRef] [PubMed]

2007

D. Vucinić, T. J. Sejnowski, and B. Lu, “A compact multiphoton 3D imaging system for recording fast neuronal activity,” PLoS ONE 2(8), e699 (2007).
[CrossRef] [PubMed]

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods 4(1), 73–79 (2007).
[CrossRef]

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “Aberration-free optical refocusing in high numerical aperture microscopy,” Opt. Lett. 32(14), 2007–2009 (2007).
[CrossRef] [PubMed]

2006

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50(6), 823–839 (2006).
[CrossRef] [PubMed]

X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
[CrossRef]

2005

V. Iyer, T. M. Hoogland, and P. Saggau, “Fast functional imaging of single neurons using random-access multiphoton (RAMP) microscopy,” J. Neurophysiol. 95(1), 535–545 (2005).
[CrossRef] [PubMed]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[CrossRef] [PubMed]

S. Shoham, D. H. O’Connor, D. V. Sarkisov, and S. S. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nat. Methods 2(11), 837–843 (2005).
[CrossRef] [PubMed]

G. D. Reddy and P. Saggau, “Fast three-dimensional laser scanning scheme using acousto-optic deflectors,” J. Biomed. Opt. 10(6), 064038 (2005).
[CrossRef]

2004

R. D. Roorda, T. M. Hohl, R. Toledo-Crow, and G. Miesenböck, “Video-rate nonlinear microscopy of neuronal membrane dynamics with genetically encoded probes,” J. Neurophysiol. 92(1), 609–621 (2004).
[CrossRef] [PubMed]

2003

V. Iyer, B. E. Losavio, and P. Saggau, “Compensation of spatial and temporal dispersion for acousto-optic multiphoton laser-scanning microscopy,” J. Biomed. Opt. 8(3), 460–471 (2003).
[CrossRef] [PubMed]

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

2002

J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83(4), 2292–2299 (2002).
[CrossRef] [PubMed]

2001

2000

1999

1997

Z. Kam, D. A. Agard, and J. W. Sedat, “Three-dimensional microscopy in thick biological samples: A fresh approach for adjusting focus and correcting spherical aberration,” Bioimaging 5(1), 40–49 (1997).
[CrossRef]

1990

E. H. Young, H. C. Ho, and L. J. Harrison, “Optically rotated long time aperture TeO2 Bragg cell,” Proc. SPIE 1296, 304–315 (1990).
[CrossRef]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Agard, D. A.

Z. Kam, D. A. Agard, and J. W. Sedat, “Three-dimensional microscopy in thick biological samples: A fresh approach for adjusting focus and correcting spherical aberration,” Bioimaging 5(1), 40–49 (1997).
[CrossRef]

Booth, M. J.

Bormuth, V.

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Botcherby, E. J.

Bourdieu, L.

Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008).
[CrossRef] [PubMed]

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Candela, Y.

Chatenay, D.

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Chen, W. R.

X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
[CrossRef]

Davidson, N.

Denk, W.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Dieudonné, S.

Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008).
[CrossRef] [PubMed]

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Duemani Reddy, G.

G. Duemani Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11(6), 713–720 (2008).
[CrossRef] [PubMed]

Dugué, G. P.

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Fainman, Y.

Feltz, A.

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Fink, R.

G. Duemani Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11(6), 713–720 (2008).
[CrossRef] [PubMed]

Friedman, N.

Göbel, W.

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods 4(1), 73–79 (2007).
[CrossRef]

Harrison, L. J.

E. H. Young, H. C. Ho, and L. J. Harrison, “Optically rotated long time aperture TeO2 Bragg cell,” Proc. SPIE 1296, 304–315 (1990).
[CrossRef]

Helmchen, F.

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods 4(1), 73–79 (2007).
[CrossRef]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[CrossRef] [PubMed]

Ho, H. C.

E. H. Young, H. C. Ho, and L. J. Harrison, “Optically rotated long time aperture TeO2 Bragg cell,” Proc. SPIE 1296, 304–315 (1990).
[CrossRef]

Hohl, T. M.

R. D. Roorda, T. M. Hohl, R. Toledo-Crow, and G. Miesenböck, “Video-rate nonlinear microscopy of neuronal membrane dynamics with genetically encoded probes,” J. Neurophysiol. 92(1), 609–621 (2004).
[CrossRef] [PubMed]

Honnorat, N.

Hoogland, T. M.

V. Iyer, T. M. Hoogland, and P. Saggau, “Fast functional imaging of single neurons using random-access multiphoton (RAMP) microscopy,” J. Neurophysiol. 95(1), 535–545 (2005).
[CrossRef] [PubMed]

Iyer, V.

B. Losavio, V. Iyer, and P. Saggau, “Two photon microscope for multisite microphotolysis of caged neurotransmitters in acute brain slices,” J. Biomed. Opt . 14, 064033 064031–064013 (2009).
[CrossRef]

V. Iyer, T. M. Hoogland, and P. Saggau, “Fast functional imaging of single neurons using random-access multiphoton (RAMP) microscopy,” J. Neurophysiol. 95(1), 535–545 (2005).
[CrossRef] [PubMed]

V. Iyer, B. E. Losavio, and P. Saggau, “Compensation of spatial and temporal dispersion for acousto-optic multiphoton laser-scanning microscopy,” J. Biomed. Opt. 8(3), 460–471 (2003).
[CrossRef] [PubMed]

Juskaitis, R.

Kam, Z.

Z. Kam, D. A. Agard, and J. W. Sedat, “Three-dimensional microscopy in thick biological samples: A fresh approach for adjusting focus and correcting spherical aberration,” Bioimaging 5(1), 40–49 (1997).
[CrossRef]

Kampa, B. M.

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods 4(1), 73–79 (2007).
[CrossRef]

Kaplan, A.

Kelleher, K.

G. Duemani Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11(6), 713–720 (2008).
[CrossRef] [PubMed]

Kremer, Y.

Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008).
[CrossRef] [PubMed]

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Krichevsky, O.

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Lapole, R.

Lechleiter, J. D.

J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83(4), 2292–2299 (2002).
[CrossRef] [PubMed]

Léger, J. F.

Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008).
[CrossRef] [PubMed]

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Lim, L. E.

Lim, L. E. N.

Lin, D. T.

J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83(4), 2292–2299 (2002).
[CrossRef] [PubMed]

Losavio, B.

B. Losavio, V. Iyer, and P. Saggau, “Two photon microscope for multisite microphotolysis of caged neurotransmitters in acute brain slices,” J. Biomed. Opt . 14, 064033 064031–064013 (2009).
[CrossRef]

Losavio, B. E.

V. Iyer, B. E. Losavio, and P. Saggau, “Compensation of spatial and temporal dispersion for acousto-optic multiphoton laser-scanning microscopy,” J. Biomed. Opt. 8(3), 460–471 (2003).
[CrossRef] [PubMed]

Lu, B.

D. Vucinić, T. J. Sejnowski, and B. Lu, “A compact multiphoton 3D imaging system for recording fast neuronal activity,” PLoS ONE 2(8), e699 (2007).
[CrossRef] [PubMed]

Luo, Q.

X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
[CrossRef]

Lv, X.

X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
[CrossRef]

Mathieu, B.

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Miesenböck, G.

R. D. Roorda, T. M. Hohl, R. Toledo-Crow, and G. Miesenböck, “Video-rate nonlinear microscopy of neuronal membrane dynamics with genetically encoded probes,” J. Neurophysiol. 92(1), 609–621 (2004).
[CrossRef] [PubMed]

Ngoi, B. K. A.

O’Connor, D. H.

S. Shoham, D. H. O’Connor, D. V. Sarkisov, and S. S. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nat. Methods 2(11), 837–843 (2005).
[CrossRef] [PubMed]

Otsu, Y.

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Reddy, G. D.

G. D. Reddy and P. Saggau, “Fast three-dimensional laser scanning scheme using acousto-optic deflectors,” J. Biomed. Opt. 10(6), 064038 (2005).
[CrossRef]

Roorda, R. D.

R. D. Roorda, T. M. Hohl, R. Toledo-Crow, and G. Miesenböck, “Video-rate nonlinear microscopy of neuronal membrane dynamics with genetically encoded probes,” J. Neurophysiol. 92(1), 609–621 (2004).
[CrossRef] [PubMed]

Saggau, P.

B. Losavio, V. Iyer, and P. Saggau, “Two photon microscope for multisite microphotolysis of caged neurotransmitters in acute brain slices,” J. Biomed. Opt . 14, 064033 064031–064013 (2009).
[CrossRef]

G. Duemani Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11(6), 713–720 (2008).
[CrossRef] [PubMed]

G. D. Reddy and P. Saggau, “Fast three-dimensional laser scanning scheme using acousto-optic deflectors,” J. Biomed. Opt. 10(6), 064038 (2005).
[CrossRef]

V. Iyer, T. M. Hoogland, and P. Saggau, “Fast functional imaging of single neurons using random-access multiphoton (RAMP) microscopy,” J. Neurophysiol. 95(1), 535–545 (2005).
[CrossRef] [PubMed]

V. Iyer, B. E. Losavio, and P. Saggau, “Compensation of spatial and temporal dispersion for acousto-optic multiphoton laser-scanning microscopy,” J. Biomed. Opt. 8(3), 460–471 (2003).
[CrossRef] [PubMed]

Salomé, R.

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Sarkisov, D. V.

S. Shoham, D. H. O’Connor, D. V. Sarkisov, and S. S. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nat. Methods 2(11), 837–843 (2005).
[CrossRef] [PubMed]

Sedat, J. W.

Z. Kam, D. A. Agard, and J. W. Sedat, “Three-dimensional microscopy in thick biological samples: A fresh approach for adjusting focus and correcting spherical aberration,” Bioimaging 5(1), 40–49 (1997).
[CrossRef]

Sejnowski, T. J.

D. Vucinić, T. J. Sejnowski, and B. Lu, “A compact multiphoton 3D imaging system for recording fast neuronal activity,” PLoS ONE 2(8), e699 (2007).
[CrossRef] [PubMed]

Shoham, S.

S. Shoham, D. H. O’Connor, D. V. Sarkisov, and S. S. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nat. Methods 2(11), 837–843 (2005).
[CrossRef] [PubMed]

Sieneart, I.

J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83(4), 2292–2299 (2002).
[CrossRef] [PubMed]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Sun, P. C.

Svoboda, K.

K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50(6), 823–839 (2006).
[CrossRef] [PubMed]

Tan, B.

Toledo-Crow, R.

R. D. Roorda, T. M. Hohl, R. Toledo-Crow, and G. Miesenböck, “Video-rate nonlinear microscopy of neuronal membrane dynamics with genetically encoded probes,” J. Neurophysiol. 92(1), 609–621 (2004).
[CrossRef] [PubMed]

Venkatakrishnan, K.

Vucinic, D.

D. Vucinić, T. J. Sejnowski, and B. Lu, “A compact multiphoton 3D imaging system for recording fast neuronal activity,” PLoS ONE 2(8), e699 (2007).
[CrossRef] [PubMed]

Wang, S. S.

S. Shoham, D. H. O’Connor, D. V. Sarkisov, and S. S. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nat. Methods 2(11), 837–843 (2005).
[CrossRef] [PubMed]

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

Wilson, T.

Wong, J.

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Wyart, C.

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Yasuda, R.

K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50(6), 823–839 (2006).
[CrossRef] [PubMed]

Young, E. H.

E. H. Young, H. C. Ho, and L. J. Harrison, “Optically rotated long time aperture TeO2 Bragg cell,” Proc. SPIE 1296, 304–315 (1990).
[CrossRef]

Zeng, S.

X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
[CrossRef]

Zhan, C.

X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
[CrossRef]

Zhu, L.

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

Appl. Opt.

Bioimaging

Z. Kam, D. A. Agard, and J. W. Sedat, “Three-dimensional microscopy in thick biological samples: A fresh approach for adjusting focus and correcting spherical aberration,” Bioimaging 5(1), 40–49 (1997).
[CrossRef]

Biophys. J.

J. D. Lechleiter, D. T. Lin, and I. Sieneart, “Multi-photon laser scanning microscopy using an acoustic optical deflector,” Biophys. J. 83(4), 2292–2299 (2002).
[CrossRef] [PubMed]

J. Biomed. Opt

B. Losavio, V. Iyer, and P. Saggau, “Two photon microscope for multisite microphotolysis of caged neurotransmitters in acute brain slices,” J. Biomed. Opt . 14, 064033 064031–064013 (2009).
[CrossRef]

J. Biomed. Opt.

V. Iyer, B. E. Losavio, and P. Saggau, “Compensation of spatial and temporal dispersion for acousto-optic multiphoton laser-scanning microscopy,” J. Biomed. Opt. 8(3), 460–471 (2003).
[CrossRef] [PubMed]

G. D. Reddy and P. Saggau, “Fast three-dimensional laser scanning scheme using acousto-optic deflectors,” J. Biomed. Opt. 10(6), 064038 (2005).
[CrossRef]

J. Neurophysiol.

R. D. Roorda, T. M. Hohl, R. Toledo-Crow, and G. Miesenböck, “Video-rate nonlinear microscopy of neuronal membrane dynamics with genetically encoded probes,” J. Neurophysiol. 92(1), 609–621 (2004).
[CrossRef] [PubMed]

V. Iyer, T. M. Hoogland, and P. Saggau, “Fast functional imaging of single neurons using random-access multiphoton (RAMP) microscopy,” J. Neurophysiol. 95(1), 535–545 (2005).
[CrossRef] [PubMed]

J. Neurosci. Methods

R. Salomé, Y. Kremer, S. Dieudonné, J. F. Léger, O. Krichevsky, C. Wyart, D. Chatenay, and L. Bourdieu, “Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors,” J. Neurosci. Methods 154(1-2), 161–174 (2006).
[CrossRef] [PubMed]

Y. Otsu, V. Bormuth, J. Wong, B. Mathieu, G. P. Dugué, A. Feltz, and S. Dieudonné, “Optical monitoring of neuronal activity at high frame rate with a digital random-access multiphoton (RAMP) microscope,” J. Neurosci. Methods 173(2), 259–270 (2008).
[CrossRef] [PubMed]

Nat. Biotechnol.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

Nat. Methods

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[CrossRef] [PubMed]

S. Shoham, D. H. O’Connor, D. V. Sarkisov, and S. S. Wang, “Rapid neurotransmitter uncaging in spatially defined patterns,” Nat. Methods 2(11), 837–843 (2005).
[CrossRef] [PubMed]

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods 4(1), 73–79 (2007).
[CrossRef]

Nat. Neurosci.

G. Duemani Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11(6), 713–720 (2008).
[CrossRef] [PubMed]

Neuron

K. Svoboda and R. Yasuda, “Principles of two-photon excitation microscopy and its applications to neuroscience,” Neuron 50(6), 823–839 (2006).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

PLoS ONE

D. Vucinić, T. J. Sejnowski, and B. Lu, “A compact multiphoton 3D imaging system for recording fast neuronal activity,” PLoS ONE 2(8), e699 (2007).
[CrossRef] [PubMed]

Proc. SPIE

E. H. Young, H. C. Ho, and L. J. Harrison, “Optically rotated long time aperture TeO2 Bragg cell,” Proc. SPIE 1296, 304–315 (1990).
[CrossRef]

Rev. Sci. Instrum.

X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101–046103 (2006).
[CrossRef]

Science

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Other

J. B. Pawley, Handbook of Biological Confocal Microscopy (Plenum Press, New York, 1995).

J. Xu, and R. Stroud, Acousto-Optic Devices, Principles, Design and Applications (John Wiley and Sons Inc., 1992).

D. Reddy, and P. Saggau, “Fast Three-Dimensional Random Access Multi-Photon Microscopy for Functional Recording of Neuronal Activity,” Proc. SPIE 6630, 66301A 66301–66308 (2007).

A. P. Goutzoulis, D. R. Pape, and S. V. Kulakov, Design and Fabrication of Acousto Optic Devices (Marcel Dekker, 1994).

G. Reddy, “A multiphoton microscope for three dimensional functional recording of fast neuronal activity,” (Rice University, Houston Texas USA, 2007).

P. A. Kirkby, R. A. Silver, and K. M. N. S. Nadella, “IMAGING APPARATUS AND METHODS,” (20.03.2008).

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Figures (17)

Fig. 1
Fig. 1

The principle of operation for a cylindrical AOL. (a) pair of AODs in which the sound waves are counter propagating. A collimated laser beam is incident on AOD1 and is diffracted into the + 1 mode (away from the transducer) as illustrated. (b) drive frequency versus time. A positive chirp rate is applied to both AODs producing a chirped wave propagating across the crystals, (c) drive frequency versus distance across the crystals for the ramps shown in (b). The dashed lines in (b) and (c) show the limits of AOD drive frequency where diffraction efficiency drops to 80% of its peak value.

Fig. 2
Fig. 2

Acousto-Optic Deflector Designs. (a) A standard high efficiency AOD with 55° walk off angle and ± 20 mrad deflection angle. Note this has only approximately ± 1 mrad input acceptance angle. (b) Wave vector and refractive index sphere for standard AOD. The axes represent refractive index for the gray surfaces and wave number for the wave vector arrows. (The number of waves/cm in the material is proportional to refractive index). The acoustic wave vector (red arrow) of the standard AOD is rotated 5° off the <110> direction. The polished crystal is thus described as having 5° of acoustic rotation (blue arrow). The inset shows that the incident light wave vector is an extraordinary wave with its polarisation shown as the red ellipse above. The output wave vector is an ordinary wave (green ellipse). The design of our custom AODs is shown in (c). There is only 2° of acoustic rotation so that the walk off angle is reduced to 22°. This makes the crystal much thinner. However the 3D wave vector diagrams (d) show that in order to keep the incident and diffracted wave vectors more than 3° off axis (inner pair of blue latitude circles) the crystal must be optically rotated by 3° as well as acoustically rotated by 2°. The red and green ellipses show the polarization of the incident (extraordinary) wave and diffracted (ordinary) output wave.

Fig. 3
Fig. 3

(a) Schematic of custom AOD with wide transducer, (b) schematic of custom AOD with narrow transducer, (c) plot of diffraction efficiency for an AOD model with approximately 3W of drive power showing how efficiency drops as the transducer is narrowed and the input acceptance angle increases [27].

Fig. 4
Fig. 4

(a) Top view of an AOL with telecentric relays. The drive on all four AODs is at the centre frequency. (b) Top or side view of the compact configuration AOL.

Fig. 5
Fig. 5

Geometric arrangement of optical wavefronts and the drive parameters required for use with one axis of the compact configuration of AOL. (a) definition of distances and focal lengths of converging waves from AODs, (b) illustrates how as the sound wave in the first AOD progresses to the left it causes the deflection angle Δθ1 at X = 0 to increase. However the angle of deflection Δθ2 of the dashed and dotted ray at X = 0 on AOD2 which passes through the same focus is greater according to the geometrically derived equations in the text. (c) shows how the ramp rates of AOD1 must be less than the ramp rate of AOD2 if the resulting focused spot is to be stationary.

Fig. 6
Fig. 6

Sequence and orientation of four AODs forming an AOL. Distances d are defined for the equations of the compact configuration. Note the interleaved numbering of AODs, enabling the same equations to be used for both the XZ and YZ pairs of axes.

Fig. 7
Fig. 7

(a) A graphical way to analyse drive frequencies applied to the XZ pair of AODs. The common mode ratio shown in blue is the gradient in frequency space that produces no movement of the focal spot. The orthogonal differential mode gradient shown in red produces only X deflection of the spot with no common mode deflection. (b) Shows how any pair of coordinates (e.g. point P, red star) or difference in pair of frequencies (green arrow) can be analysed into common mode (blue) and differential mode components (red). fmin and fmax are the drive frequency limits of the AODs, and A and B are appropriate scalar multipliers of their respective unit vectors.

Fig. 8
Fig. 8

Diagram to illustrate AOD X1 and AOD X2 drive frequencies vs. time for part of a sequence of miniscans deflecting in X and focusing a spot at a constant Z plane across one line of a raster at a fixed Y value. The pair of green lines corresponds to a single vector in the frequency space diagram (Fig. 9). The red line shows that there is no X deflection in the flyback period between the end of one data gathering sequence and the beginning of the next.

Fig. 9
Fig. 9

Diagram illustrating the full sequence of miniscans making up one full X scan of one line of a raster scan for an AOD pair. The green arrows correspond to a green pair of drive frequencies as shown vs. time in Fig. 8. As the green vector has a small differential mode component it is scanning in X as well as focusing at a particular Z plane. This is shown by the red differential mode lines. The frequency limits for data collection have to be inset with respect to the frequency limits because data can only be gathered from full AODs. The common mode (blue dashed) reset lines ensure that when data gathering starts again in the next miniscan it does so from the same position.

Fig. 10
Fig. 10

Compact configuration of AOL showing mounts used for precise positioning of each AOD. The AOL has a total length of only 150 mm from the centre of the first AOD to the centre of the last. Inset is a close up of the first AOD showing the gold transducer pad along the left hand (front) edge of the AOD. It is followed by a polarizer and half wave plate. The AOD is 15 mm thick and approximately 20 mm square external dimensions.

Fig. 11
Fig. 11

Focusing of a laser beam to arbitrarily chosen focal points at 30 kHz. (a) Simplified diagram of 4 AOD AOL, 200 mm focal length lens, 2 partial reflectors and 3 silicon detectors with 20 μm pinholes. Detectors were placed at arbitrary XY positions and axial distances from the lens of 175 mm (red), 200 mm (green) and 235 mm (blue). (b) The signal from each detector, plotted with the appropriate colour code for its detector, during random access pointing where the laser beam is focused sequentially into each of the pinholes.

Fig. 12
Fig. 12

Schematic diagram of complete acousto-optic lens (AOL) 2-photon microscope. Inset shows details of AOL. The four AODs are placed as close to one another as possible (40-60 mm centre to centre). Before each AOD is a half wave plate (H). After each AOD is a polarizer (P) to absorb the residual undiffracted zero order light. By fixing the galvanometers, AOL deflection and focusing can be used to focus anywhere within the octahedral shaped scan volume below the objective. Fluorescence was either collected through the objective onto a photomultiplier tube (PMT), or additionally through the condenser (not shown).

Fig. 13
Fig. 13

Experimental measurements of the laser pulse width at various positions through the system. The pulse width at the Ti-Sapphire laser (red, 100 fs at 800 nm, MaiTai, Newport Spectra Physics) for the full acousto-optic lens microscope, measured after the 40X objective lens, without pulse compensation (blue, 1.52 ps) and with −50,000 fs2 GVD introduced by the prism based pre-chirper (green, 115 fs).

Fig. 14
Fig. 14

Pointing mode images of a pollen grain optically sectioned through its equator. Each image is 125 × 125 voxels with a dwell time per voxel of 4 μs. Images were acquired at the same settings, with no subsequent image processing so that the images represent the true range of brightness over the Z focusing range. The height of the AOL focal plane with respect to the natural focal plane of the objective was measured by mechanically refocusing by moving the objective axially. The rows of images correspond to a single frame and 16 frames average, respectively. The scale bars are 10 μm.

Fig. 15
Fig. 15

(a) Raster scanned images showing resolution and effect of chromatic aberrations. Expected distortion of 2-photon PSF across the Z = 0 focal plane predicted from Eq. (4). (b) a montage of bead images measured at the centre, edges and corners of a 100 μm square field of view at Z = 0 using a Ti-Sapphire laser with a 100 fs pulse length and 10.6 nm FWHM spectral width. c) Same as b, but imaged with a laser with a 140 fs pulse length and a 7 nm FWHM spectral width. Fluorescent bead diameter b = 200 nm, λ = 800 nm, theoretical 2-photon PSF w = 370 nm [1].

Fig. 16
Fig. 16

Measurement of the 2-photon PSF along Z axis at X = Y = 0 acquired with pointing mode structural imaging of 200 nm beads with a semiscan angle s = 4.3 mrad using the algorithms described in section 2.5 (a) XY FWHM dimensions of beads imaged with a 40X NA = 0.8 objective lens at 800 nm. Red points show long axis of 2D Gaussian fit and blue points show short axis plotted against focus depth (mean, n = 3). (b) Relationship between Z dimension of bead image and axial focus depth (mean, n = 3). The FWHM was calculated at each Z plane from the intensity from 80x80 nm regions in 31 planes spaced 0.5 μm apart in Z (centered on the bead at the mid plane). A single weak astigmatic lens (± 0.05 m−1) was introduced before the AOL for these measurements, to correct for a small fixed astigmatic component introduced in the optical train. This brought the PSF closer to the diffraction limit (dashed lines for 200 nm beads) than the data in Fig. 15, which was acquired without this lens.

Fig. 17
Fig. 17

(a) Raster scan images of pollen grains sectioned through their equator with the AOL set for Z = 0 so that the focal plane is the natural focal plane of the objective. The top row is for conventional 2 AOD XY scanning and the bottom row using the 4 AOD AOL as a scanner. The columns are in order of increasing frame rate 12, 80 and 155 Hz. Scale bars are 10 μm. The image contrast of all the images is equally enhanced with ImageJ to show spines more clearly. (b) Comparison of astigmatism between theoretical (solid red and blue lines based on Eqs. (6) and (7)) and experimental (red and blue circles) for the two-AOD scanning at frame rates 10-155 Hz. The experimental astigmatic data points are measured by eye and by imageJ Gaussian curve fitting from a Z stack of images of 1μm beads. Their accuracy is approximately ± 1 μm. The inset diagram illustrates four rays (black arrows) and the 2-photon astigmatic beam waists (green ellipses) at the focus and shows the astigmatism parameters AstigZ and AstigXY in the Eq. (6) and (7) (red and blue arrows).

Equations (19)

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

T = W V
N r = 2 T B = 4 W s λ 0
N r t p = 2.12 T B = 4.24 W s λ 0
l X Y = ( w 2 + ( r Δ λ 2 λ ) 2 + b 2 )
l Z = ( d 2 + ( z Δ λ 2 λ ) 2 + ( SA ) 2 + ( AstigZ ) 2 + b 2 )
AstigZ = W 2 n 4 V ( 1 ( NA ) 2 1 n ) 2 s Nvox × Dwell × zoom
Astig XY = NA × AstigZ n 2 NA 2
FrameRate = 1 Nvox ( T + Nvox × Dwell )
f 1 = f c + a 1 t
f 2 = f c + a 2 t
Δ θ 2 Δ θ 1 d 1 ' d 1 ' d 1
a 2 a 1 d 1 ' d 1 ' d 1
d 2 ' = d 1 ' d 1 2 ,     a 1 = V 2 λ ( 2 d 2 ' + d 1 ) ,     a 2 = V 2 2 λ d 2 '
R c o m m = Δ f a v 1 Δ f a v 2 = 2 d 2 ' 2 d 2 ' + d 1
R diff = Δ f a v 1 Δ f a v 2 = 2 d 2 ' 2 d 2 ' + d 1
[ f a v 1 , f a v 2 ] = [ f c , f c ] + A × R c o m m + B × R diff
δ θ δ t = 2 s N v o x × D w e l l
a 1 = V 2 2 λ d 2 ' V δ θ 2 λ δ t ,     a 2 = V 2 2 λ d 2 ' + V δ θ 2 λ δ t
a 3 = V λ ( V d 4 ' δ φ δ t ) ( 2 + d 3 d 4 ' d 3 δ φ V δ t ) ,     a 4 = V 2 2 λ d 4 ' + V δ φ 2 λ δ t

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