Abstract

Volume imaging based on a fast focus-tunable lens (FTL) allows three-dimensional (3D) observation within milliseconds by extending the depth-of-field (DOF) with sub-micrometer transverse resolution on optical sectioning microscopes. However, the previously published DOF extensions were neither axially uniform nor fit with theoretical prediction. In this work, complete theoretical treatments of focus extension with confocal and various multiphoton microscopes are established to correctly explain the previous results. Moreover, by correctly placing the FTL and properly adjusting incident beam diameter, a uniform DOF is achieved in which the actual extension nicely agrees with the theory. Our work not only provides a theoretical platform for volumetric imaging with FTL but also demonstrates the optimized imaging condition.

© 2017 Optical Society of America

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References

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2015 (2)

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

2014 (4)

2013 (1)

2012 (1)

M. Duocastella, B. Sun, and C. B. Arnold, “Simultaneous imaging of multiple focal planes for three-dimensional microscopy using ultra-high-speed adaptive optics,” J. Biomed. Opt. 17(5), 050505 (2012).
[Crossref] [PubMed]

2011 (2)

2009 (1)

2008 (2)

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]

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High-speed varifocal imaging with a tunable acoustic gradient index of refraction lens,” Opt. Lett. 33(18), 2146–2148 (2008).
[Crossref] [PubMed]

2007 (2)

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] [PubMed]

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102(3), 033104 (2007).
[Crossref]

1990 (1)

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

1988 (1)

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10(4), 128–138 (1988).
[Crossref]

1987 (1)

J. G. White, W. B. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105(1), 41–48 (1987).
[Crossref] [PubMed]

Amos, W. B.

J. G. White, W. B. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105(1), 41–48 (1987).
[Crossref] [PubMed]

Arnold, C. B.

M. Duocastella, B. Sun, and C. B. Arnold, “Simultaneous imaging of multiple focal planes for three-dimensional microscopy using ultra-high-speed adaptive optics,” J. Biomed. Opt. 17(5), 050505 (2012).
[Crossref] [PubMed]

N. Olivier, A. Mermillod-Blondin, C. B. Arnold, and E. Beaurepaire, “Two-photon microscopy with simultaneous standard and extended depth of field using a tunable acoustic gradient-index lens,” Opt. Lett. 34(11), 1684–1686 (2009).
[Crossref] [PubMed]

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High-speed varifocal imaging with a tunable acoustic gradient index of refraction lens,” Opt. Lett. 33(18), 2146–2148 (2008).
[Crossref] [PubMed]

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102(3), 033104 (2007).
[Crossref]

Beaurepaire, E.

Benfenati, F.

Carreel, B.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Chen, L.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Chen, X.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Cheng, H.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Cisotto, L.

Cui, M.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Dal Maschio, M.

De Stasi, A. M.

Dean, K. M.

Denk, W.

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

Diaspro, A.

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]

Duocastella, M.

M. Duocastella, G. Vicidomini, and A. Diaspro, “Simultaneous multiplane confocal microscopy using acoustic tunable lenses,” Opt. Express 22(16), 19293–19301 (2014).
[Crossref] [PubMed]

M. Duocastella, B. Sun, and C. B. Arnold, “Simultaneous imaging of multiple focal planes for three-dimensional microscopy using ultra-high-speed adaptive optics,” J. Biomed. Opt. 17(5), 050505 (2012).
[Crossref] [PubMed]

Fan, M.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Fellin, T.

Filho, L. C. C. P.

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]

Fiolka, R.

Fordham, M.

J. G. White, W. B. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105(1), 41–48 (1987).
[Crossref] [PubMed]

Germain, R. N.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

Hua, H.

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] [PubMed]

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]

Kong, L.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Konijnenberg, A. P.

Kumar, N.

Lämmermann, T.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Lin, C. P.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Lin, Y.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Little, J. P.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Liu, S.

Manukyan, G.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

McLeod, E.

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High-speed varifocal imaging with a tunable acoustic gradient index of refraction lens,” Opt. Lett. 33(18), 2146–2148 (2008).
[Crossref] [PubMed]

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102(3), 033104 (2007).
[Crossref]

Mermillod-Blondin, A.

Minsky, M.

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10(4), 128–138 (1988).
[Crossref]

Mishra, K.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Mugele, F.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Murade, C.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Oh, J. M.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Olivier, N.

Pereira, S. F.

Peterka, D. S.

Quirin, S.

Ren, H.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Roghair, I.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Saggau, P.

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]

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, B.

M. Duocastella, B. Sun, and C. B. Arnold, “Simultaneous imaging of multiple focal planes for three-dimensional microscopy using ultra-high-speed adaptive optics,” J. Biomed. Opt. 17(5), 050505 (2012).
[Crossref] [PubMed]

Sun, Y.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Tang, J.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Urbach, H. P.

van den Ende, D.

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Vicidomini, G.

Webb, W. W.

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

Wei, L.

White, J. G.

J. G. White, W. B. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105(1), 41–48 (1987).
[Crossref] [PubMed]

Yu, Y.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Yuste, R.

Zhang, Y.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Zhao, J.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Zhou, Z.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Zong, W.

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

Cell Res. (1)

W. Zong, J. Zhao, X. Chen, Y. Lin, H. Ren, Y. Zhang, M. Fan, Z. Zhou, H. Cheng, Y. Sun, and L. Chen, “Large-field high-resolution two-photon digital scanned light-sheet microscopy,” Cell Res. 25(2), 254–257 (2015).
[Crossref] [PubMed]

J. Appl. Phys. (1)

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102(3), 033104 (2007).
[Crossref]

J. Biomed. Opt. (1)

M. Duocastella, B. Sun, and C. B. Arnold, “Simultaneous imaging of multiple focal planes for three-dimensional microscopy using ultra-high-speed adaptive optics,” J. Biomed. Opt. 17(5), 050505 (2012).
[Crossref] [PubMed]

J. Cell Biol. (1)

J. G. White, W. B. Amos, and M. Fordham, “An evaluation of confocal versus conventional imaging of biological structures by fluorescence light microscopy,” J. Cell Biol. 105(1), 41–48 (1987).
[Crossref] [PubMed]

Nat. Methods (2)

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] [PubMed]

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Nat. Neurosci. (1)

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]

Opt. Express (5)

Opt. Lett. (3)

Scanning (1)

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10(4), 128–138 (1988).
[Crossref]

Sci. Rep. (1)

K. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4(1), 6378 (2014).
[Crossref] [PubMed]

Science (1)

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

Other (3)

J. B. Pawley, Handbook of Biological Confocal Microscopy, 2nd ed. (Plenum Press, 1995).

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

C. B. Arnold, E. McLeod, and A. Mermillod-Blondin, “Tunable acoustic gradient index of refraction lens and system,” US 8576478 B2 (2013).

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Figures (10)

Fig. 1
Fig. 1 Schematic figure of the simulation concept. Inset red curve of FTL is the RI profile. The diameter of the input Gaussian beam fits with the EA. Beam convergence (orange dot-dashed line) and divergence (green dotted line) after FTL and objective are exaggerated, to show the focus extension as Δz.
Fig. 2
Fig. 2 (a) The axial excitation intensity distributions of three points at d = 3 (green), 29 (orange) and 160 cm (blue). Δz and ΔzFWHM are compared with the blue curve. (b) Comparison of Δz and ΔzFWHM as functions of d. Large ΔzFWHM and uniform intensity distribution are obtained when d is as small as possible.
Fig. 3
Fig. 3 Collection efficiency versus focus tuning. (a) CLSM setups, (b) MPM descanned setups. The two simulation setups with different FTL positions marked by green and blue colors are shown below the collection efficiency figures. The left-bottom (Pass) and the right-bottom (Not pass) represents the case of signal passing and not passing through FTL, respectively. The color of FTL corresponds to the color of the individual collection efficiency curve. Green: cases of signal pass pass through FTL when d = 3 cm; Blue: d = 160 cm with signal not passing through FTL. PD: photo-detector; XY: scanner; P/NP: signal pass/not pass through FTL.
Fig. 4
Fig. 4 Collection efficiency versus focus tuning in MPM NDD schemes with different collection area photo-detectors. (a) small-area detector. (b) large-area detector. Different color curves represent different cases with the FTL position and detailed setups shown below. Green/red: cases of signal pass/not pass through FTL when d = 3 cm; Blue: d = 160 cm with signal not passing through FTL. S/L: small/large-area detector.
Fig. 5
Fig. 5 Extended DOF effective axial intensity combined excitation with detection. (a) and (b) are CLSM and MPM descanned cases, respectively, while (c) and (d) are small- and large-area detector used in MPM NDD cases, respectively. d = 3 cm or 160 cm is marked in parenthesis.
Fig. 6
Fig. 6 Simulated axial profile of focus extension when the beam size is (a) 0.8, (b) 2, and (c) 1.4 times of the EA, respectively. All these simulations follow the refractive index in Eq. (1).
Fig. 7
Fig. 7 Experimental setup. BS: beamsplitter; IM: inverted microscope. PD: photodiode, XY: scanning unit, FTL: focus tunable lens.
Fig. 8
Fig. 8 DOF extension characterized by an 80-nm gold nanoparticle. (a)-(c) are axial intensity distributions with corresponding xz images. The incident beam sizes in front of the FTL are (a) ~4.5, (b) ~8, and (c) ~11 mm, respectively, and the most uniform distribution is found in (b). The arrowheads in (a)-(c) mark one endpoint that is influenced by reflection from glass/water interface on the coverslip. The arrow in (b) points the center of extended DOF with the xy image in (d) while the xy image of the endpoint in (e) is indicated as the text in (b). Scale bar in (a)-(c): 10 μm, (d) and (e): 1 μm.
Fig. 9
Fig. 9 (a) 5 CLSM images of 3 fluorescent beads at different depths. (b) Top: average projection from 100 CLSM image of (a). Bottom: single scan image by uniformly extended DOF.
Fig. 10
Fig. 10 (a) ΔzFWHM vs FTL driving frequency. Individual resonant frequencies on the spec of FTL are square-dots marked. (b) EA vs driven frequency.

Tables (1)

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Table 1 Comparisons of references using FTL for DOF extension.

Equations (2)

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n(r,t)= n 0 + n α J 0 ( ωr v )sin(ωt)
Δz= 2 f tube 2 δ FTL M 2 δ FTL 2 (Md- f tube ) 2

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