Abstract

3-photon excitation enables in vivo fluorescence microscopy deep in densely labeled and highly scattering samples. To date, 3-photon excitation has been restricted to scanning a single focus, limiting the speed of volume acquisition. Here, for the first time to our knowledge, we implemented and characterized dual-plane 3-photon microscopy with temporal multiplexing and remote focusing, and performed simultaneous in vivo calcium imaging of two planes beyond 600 µm deep in the cortex of a pan-excitatory GCaMP6s transgenic mouse with a per-plane framerate of 7 Hz and an effective 2 MHz laser repetition rate. This method is a straightforward and generalizable modification to single-focus 3PE systems, doubling the rate of volume (column) imaging with off-the-shelf components and minimal technical constraints.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (4)

M. Yildirim, H. Sugihara, P. T. C. So, and M. Sur, “Functional imaging of visual cortical layers and subplate in awake mice with optimized three-photon microscopy,” Nat. Commun. 10(1), 177 (2019).
[Crossref]

S. Weisenburger, F. Tejera, J. Demas, B. Chen, J. Manley, F. T. Sparks, F. Martínez Traub, T. Daigle, H. Zeng, A. Losonczy, and A. Vaziri, “Volumetric Ca2+ Imaging in the Mouse Brain Using Hybrid Multiplexed Sculpted Light Microscopy,” Cell 177(4), 1050–1066.e14 (2019).
[Crossref]

S. Han, W. Yang, and R. Yuste, “Two-Color Volumetric Imaging of Neuronal Activity of Cortical Columns,” Cell Rep. 27(7), 2229–2240.e4 (2019).
[Crossref]

H. Dana, Y. Sun, B. Mohar, B. K. Hulse, A. M. Kerlin, J. P. Hasseman, G. Tsegaye, A. Tsang, A. Wong, R. Patel, J. J. Macklin, Y. Chen, A. Konnerth, V. Jayaraman, L. L. Looger, E. R. Schreiter, K. Svoboda, and D. S. Kim, “High-performance calcium sensors for imaging activity in neuronal populations and microcompartments,” Nat. Methods 16(7), 649–657 (2019).
[Crossref]

2018 (5)

K. Guesmi, L. Abdeladim, S. Tozer, P. Mahou, T. Kumamoto, K. Jurkus, P. Rigaud, K. Loulier, N. Dray, P. Georges, M. Hanna, J. Livet, W. Supatto, E. Beaurepaire, and F. Druon, “Dual-color deep-tissue three-photon microscopy with a multiband infrared laser,” Light: Sci. Appl. 7(1), 12 (2018).
[Crossref]

K. Charan, B. Li, M. Wang, C. P. Lin, and C. Xu, “Fiber-based tunable repetition rate source for deep tissue two-photon fluorescence microscopy,” Biomed. Opt. Express 9(5), 2304 (2018).
[Crossref]

B. Chen, X. Huang, D. Gou, J. Zeng, G. Chen, M. Pang, Y. Hu, Z. Zhao, Y. Zhang, Z. Zhou, H. Wu, H. Cheng, Z. Zhang, C. Xu, Y. Li, L. Chen, and A. Wang, “Rapid volumetric imaging with Bessel-Beam three-photon microscopy,” Biomed. Opt. Express 9(4), 1992 (2018).
[Crossref]

C. Rodríguez, Y. Liang, R. Lu, and N. Ji, “Three-photon fluorescence microscopy with an axially elongated Bessel focus,” Opt. Lett. 43(8), 1914–1917 (2018).
[Crossref]

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[Crossref]

2017 (1)

D. G. Ouzounov, T. Wang, M. Wang, D. D. Feng, N. G. Horton, J. C. Cruz-Hernández, Y. Cheng, J. Reimer, A. S. Tolias, N. Nishimura, and C. Xu, “In vivo three-photon imaging of activity of GCaMP6-labeled neurons deep in intact mouse brain,” Nat. Methods 14(4), 388–390 (2017).
[Crossref]

2016 (3)

W. Yang, J. K. Miller, L. Carrillo-Reid, E. Pnevmatikakis, L. Paninski, R. Yuste, and D. S. Peterka, “Simultaneous Multi-plane Imaging of Neural Circuits,” Neuron 89(2), 269–284 (2016).
[Crossref]

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, “A large field of view two-photon mesoscope with subcellular resolution for in vivo imaging,” eLife 5, e14472 (2016).
[Crossref]

J. N. Stirman, I. T. Smith, M. W. Kudenov, and S. L. Smith, “Wide field-of-view, multi-region, two-photon imaging of neuronal activity in the mammalian brain,” Nat. Biotechnol. 34(8), 857–862 (2016).
[Crossref]

2015 (2)

J. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112(30), 9236–9241 (2015).
[Crossref]

D. Sinefeld, H. P. Paudel, D. G. Ouzounov, T. G. Bifano, and C. Xu, “Adaptive optics in multiphoton microscopy: comparison of two, three and four photon fluorescence,” Opt. Express 23(24), 31472–31483 (2015).
[Crossref]

2013 (2)

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, and C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[Crossref]

T. Chen, T. J. Wardill, Y. Sun, S. R. Pulver, S. L. Renninger, A. Baohan, E. R. Schreiter, R. A. Kerr, M. B. Orger, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Ultrasensitive fluorescent proteins for imaging neuronal activity,” Nature 499(7458), 295–300 (2013).
[Crossref]

2012 (1)

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juskaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” Proc. Natl. Acad. Sci. U. S. A. 109(8), 2919–2924 (2012).
[Crossref]

2011 (2)

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U. S. A. 108(49), 19504–19509 (2011).
[Crossref]

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous 2-photon calcium imaging at different cortical depths in vivo with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[Crossref]

2010 (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[Crossref]

2008 (1)

E. J. Botcherby, R. Juškaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008).
[Crossref]

2007 (1)

2006 (2)

2005 (1)

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

1999 (1)

1990 (1)

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

1985 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. Royal Soc. Lond. A 253(1274), 358–379 (1959).
[Crossref]

Abbasi-Asl, R.

K. Takasaki, R. Abbasi-Asl, and J. Waters, “Superficial bound of the depth limit of 2-photon imaging in mouse brain,” bioRxiv618454 (2019).

Abdeladim, L.

K. Guesmi, L. Abdeladim, S. Tozer, P. Mahou, T. Kumamoto, K. Jurkus, P. Rigaud, K. Loulier, N. Dray, P. Georges, M. Hanna, J. Livet, W. Supatto, E. Beaurepaire, and F. Druon, “Dual-color deep-tissue three-photon microscopy with a multiband infrared laser,” Light: Sci. Appl. 7(1), 12 (2018).
[Crossref]

Akturk, S.

Anselmi, F.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U. S. A. 108(49), 19504–19509 (2011).
[Crossref]

Arisaka, K.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous 2-photon calcium imaging at different cortical depths in vivo with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[Crossref]

Baohan, A.

T. Chen, T. J. Wardill, Y. Sun, S. R. Pulver, S. L. Renninger, A. Baohan, E. R. Schreiter, R. A. Kerr, M. B. Orger, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Ultrasensitive fluorescent proteins for imaging neuronal activity,” Nature 499(7458), 295–300 (2013).
[Crossref]

Beaurepaire, E.

K. Guesmi, L. Abdeladim, S. Tozer, P. Mahou, T. Kumamoto, K. Jurkus, P. Rigaud, K. Loulier, N. Dray, P. Georges, M. Hanna, J. Livet, W. Supatto, E. Beaurepaire, and F. Druon, “Dual-color deep-tissue three-photon microscopy with a multiband infrared laser,” Light: Sci. Appl. 7(1), 12 (2018).
[Crossref]

Bègue, A.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U. S. A. 108(49), 19504–19509 (2011).
[Crossref]

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[Crossref]

Bifano, T. G.

Booth, M. J.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juskaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” Proc. Natl. Acad. Sci. U. S. A. 109(8), 2919–2924 (2012).
[Crossref]

E. J. Botcherby, R. Juškaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008).
[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]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999).

Botcherby, E. J.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juskaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” Proc. Natl. Acad. Sci. U. S. A. 109(8), 2919–2924 (2012).
[Crossref]

E. J. Botcherby, R. Juškaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008).
[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]

Carandini, M.

M. Pachitariu, C. Stringer, M. Dipoppa, S. Schröder, L. F. Rossi, H. Dalgleish, M. Carandini, and K. D. Harris, “Suite2p: beyond 10,000 neurons with standard two-photon microscopy,” bioRxiv061507 (2017).

Carrillo-Reid, L.

W. Yang, J. K. Miller, L. Carrillo-Reid, E. Pnevmatikakis, L. Paninski, R. Yuste, and D. S. Peterka, “Simultaneous Multi-plane Imaging of Neural Circuits,” Neuron 89(2), 269–284 (2016).
[Crossref]

Charan, K.

K. Charan, B. Li, M. Wang, C. P. Lin, and C. Xu, “Fiber-based tunable repetition rate source for deep tissue two-photon fluorescence microscopy,” Biomed. Opt. Express 9(5), 2304 (2018).
[Crossref]

B. Li, M. Wang, C. Wu, K. Charan, and C. Xu, “An adaptive excitation source for multiphoton imaging,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2018), paper JTh5C.5.

Chen, B.

S. Weisenburger, F. Tejera, J. Demas, B. Chen, J. Manley, F. T. Sparks, F. Martínez Traub, T. Daigle, H. Zeng, A. Losonczy, and A. Vaziri, “Volumetric Ca2+ Imaging in the Mouse Brain Using Hybrid Multiplexed Sculpted Light Microscopy,” Cell 177(4), 1050–1066.e14 (2019).
[Crossref]

B. Chen, X. Huang, D. Gou, J. Zeng, G. Chen, M. Pang, Y. Hu, Z. Zhao, Y. Zhang, Z. Zhou, H. Wu, H. Cheng, Z. Zhang, C. Xu, Y. Li, L. Chen, and A. Wang, “Rapid volumetric imaging with Bessel-Beam three-photon microscopy,” Biomed. Opt. Express 9(4), 1992 (2018).
[Crossref]

Chen, G.

Chen, L.

Chen, T.

T. Chen, T. J. Wardill, Y. Sun, S. R. Pulver, S. L. Renninger, A. Baohan, E. R. Schreiter, R. A. Kerr, M. B. Orger, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Ultrasensitive fluorescent proteins for imaging neuronal activity,” Nature 499(7458), 295–300 (2013).
[Crossref]

Chen, Y.

H. Dana, Y. Sun, B. Mohar, B. K. Hulse, A. M. Kerlin, J. P. Hasseman, G. Tsegaye, A. Tsang, A. Wong, R. Patel, J. J. Macklin, Y. Chen, A. Konnerth, V. Jayaraman, L. L. Looger, E. R. Schreiter, K. Svoboda, and D. S. Kim, “High-performance calcium sensors for imaging activity in neuronal populations and microcompartments,” Nat. Methods 16(7), 649–657 (2019).
[Crossref]

Cheng, A.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous 2-photon calcium imaging at different cortical depths in vivo with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[Crossref]

Cheng, H.

Cheng, Y.

D. G. Ouzounov, T. Wang, M. Wang, D. D. Feng, N. G. Horton, J. C. Cruz-Hernández, Y. Cheng, J. Reimer, A. S. Tolias, N. Nishimura, and C. Xu, “In vivo three-photon imaging of activity of GCaMP6-labeled neurons deep in intact mouse brain,” Nat. Methods 14(4), 388–390 (2017).
[Crossref]

Clark, C. G.

N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, and C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013).
[Crossref]

Cruz-Hernández, J. C.

D. G. Ouzounov, T. Wang, M. Wang, D. D. Feng, N. G. Horton, J. C. Cruz-Hernández, Y. Cheng, J. Reimer, A. S. Tolias, N. Nishimura, and C. Xu, “In vivo three-photon imaging of activity of GCaMP6-labeled neurons deep in intact mouse brain,” Nat. Methods 14(4), 388–390 (2017).
[Crossref]

Cui, M.

J. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112(30), 9236–9241 (2015).
[Crossref]

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Zeng, J.

Zhang, B.

T. Wang, D. G. Ouzounov, C. Wu, N. G. Horton, B. Zhang, C. Wu, Y. Zhang, M. J. Schnitzer, and C. Xu, “Three-photon imaging of mouse brain structure and function through the intact skull,” Nat. Methods 15(10), 789–792 (2018).
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Appl. Opt. (1)

Biomed. Opt. Express (2)

Cell (1)

S. Weisenburger, F. Tejera, J. Demas, B. Chen, J. Manley, F. T. Sparks, F. Martínez Traub, T. Daigle, H. Zeng, A. Losonczy, and A. Vaziri, “Volumetric Ca2+ Imaging in the Mouse Brain Using Hybrid Multiplexed Sculpted Light Microscopy,” Cell 177(4), 1050–1066.e14 (2019).
[Crossref]

Cell Rep. (1)

S. Han, W. Yang, and R. Yuste, “Two-Color Volumetric Imaging of Neuronal Activity of Cortical Columns,” Cell Rep. 27(7), 2229–2240.e4 (2019).
[Crossref]

eLife (1)

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, “A large field of view two-photon mesoscope with subcellular resolution for in vivo imaging,” eLife 5, e14472 (2016).
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J. Opt. Soc. Am. A (1)

Light: Sci. Appl. (1)

K. Guesmi, L. Abdeladim, S. Tozer, P. Mahou, T. Kumamoto, K. Jurkus, P. Rigaud, K. Loulier, N. Dray, P. Georges, M. Hanna, J. Livet, W. Supatto, E. Beaurepaire, and F. Druon, “Dual-color deep-tissue three-photon microscopy with a multiband infrared laser,” Light: Sci. Appl. 7(1), 12 (2018).
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Figures (8)

Fig. 1.
Fig. 1. System schematic of dual plane 3PE microscope with analog demultiplexing electronics. Dashed red line – idler beam path exiting the laser and sent through pulse compressor. Solid red line – idler beam path exiting the compressor at lowered height and sent through remote focusing. Orange line – idler beam path split from the remote focusing path and sent through temporal delay line. Yellow line – signal beam synchronous with idler pulses. Abbreviations: OPA – optical parametric amplifier, HWP – half-wave plate, PBS – polarizing beam splitter, RA – right-angle mirror, RA – vertical right-angle mirror, d/2 – distance between prism and right-angle mirror (see main text), D – “D”-shaped pick-off mirror, BE – beam expander (1.5x), BR – beam reducer (0.5x), QWP – quarter-wave plate, RObj – remote focus objective, ZM – z-translation mirror, L1 – doublet lens (f = 150 mm), L2 – doublet lens (f = 100 mm), L3 – Plössl lens (f = 50 mm), SL – scan lens (f = 50 mm), TL – Plössl tube lens (f = 200 mm), Dcx – dichroic beam splitter, PObj – primary objective, Pz – piezo scanner, PMT – photomultiplier tube, PreAmp – pre-amplifer, MxSig – multiplexed signal, Amp – amplifier, Comp – comparator circuit, PD – photodiode. Inset: Oscilloscope traces of photodiode output (left), gate control pulses (center), and demultiplexed channel detection windows exhibited by switching a 0.5 VDC signal input (right).
Fig. 2.
Fig. 2. Pulse dispersion characterization and compensation. (a), illustration of interferometric autocorrelator setup inserted into the beam path. A 50:50 beam splitter splits incoming pulses between two arms of nearly equal length which are reflected back by mirrors, one of which is mounted on a piezo-actuated translation stage. The pulses are recombined leading to interference and amplitude modulation of the excitation focus. (b), log-log plot of fluorescence vs. excitation power in 50 µM fluorescein with a linear fit of slope n = 3.04. (c), plot of peak-normalized fluorescence detected in fluorescein vs. inter-pulse delay calculated from position of interferometer mirror for minimal optics (d = 44 cm; red circles) and path A (d = 92 cm; blue circles) compared with the minimal best-fit from simulation (FWHM = 48 fs; black dots). (d), plot of peak-normalized fluorescence for path B without the ZnSe disc (d = 92 cm; purple circles) and with the ZnSe disc (5 mm ZnSe, d = 92 cm; blue circles) compared with the best-fit from simulation with negative chirp (GDD = -2400 fs2; black dots). (e), images of 500 nm fluorescent beads in water imaged with (left panel) and without (right panel) the ZnSe disc. Color scale: mean photon rate of 0–4 counts/pixel (left) and 0–0.4 counts/pixel (right, 10x gain).
Fig. 3.
Fig. 3. Characterization of PSF with remote focusing. (a), images of 500 nm fluorescent beads taken in the remote focusing plane for axial displacements of -50 µm (left), 0 µm (center), and + 50 µm (right) with equal average power. Color scale: Mean photon rate 0–4 counts/pixel. (b), xz-projection of bead images. Scale bar: 1 µm. (c), graph of measured axial PSF FWHM for different remote focused plane positions. Dashed line denotes theoretical FWHM for NA = 0.9. (d), graph of measured lateral PSF FWHM for plane positions as in (c). Dashed line denotes theoretical FWHM for NA = 0.9 convolved with a 0.5 µm spherical shell representing the bead sample (see Appendix for details).
Fig. 4.
Fig. 4. Dual plane calcium imaging of neurons deep in cortex. (a), maximum intensity projections of motion corrected movies from simultaneously acquired planes located at z = 600 µm (left) and z = 650 µm (right) below the cortical surface imaged with the remote focus (RF, path A) and temporally delayed conventional focus (TD, path B), respectively. (b), maximum intensity projections of motion corrected, simultaneously acquired movies from planes in A with the remote focus and conventional focus exchanged. (c), dF/F normalized fluorescence traces extracted from neurons highlighted in A from movies acquired 50 µm above (red) and at (orange) the native imaging plane. Top panel: dF/F trace over the entire movie. Bottom panel: underlined section of full trace. (d), dF/F normalized fluorescence traces extracted from neurons highlighted in B from movies acquired at (orange) and 50 µm below (red) the native imaging plane. Top panel: dF/F trace over the entire movie. Bottom panel: underlined section of full trace.
Fig. 5.
Fig. 5. Multi-plane calcium imaging with dual-plane acquisition and objective scanning. (a), illustration of acquisition scheme in which the objective is position-cycled by 50 µm with a piezo scanner between successive frames. (b), maximum intensity projections of motion corrected movies of odd frames from simultaneously acquired planes imaged at 600 µm (left, remote focus) and 650 µm (right, native focus) deep. (c), maximum intensity projections of motion corrected movies from simultaneously acquired planes of even frames imaged at 650 µm (left, remote focus) and 700 µm (right, native focus) deep. (d), dF/F normalized fluorescence traces extracted from identical neurons highlighted in (a) and (b) imaged by the native focus (orange) and the remote focus (red) from alternating frames. Top panel: dF/F trace over the entire movie. Bottom panel: underlined section of full trace.
Fig. 6.
Fig. 6. Theoretical range of diffraction-limited refocusing based on Eq. (6).
Fig. 7.
Fig. 7. Estimation of diffraction-limited refocusing range from pupil wavefront measurements. (a), Theoretical (black markers) and measured (red markers) values for the rms coefficient of defocus (j = 4, Noll ordering) plotted against axial position of the primary focus. (b), rms coefficient of primary spherical aberration (j = 11, Noll ordering). (c), rms coefficient of secondary spherical aberration (j = 22, Noll ordering). (d), Strehl ratio numerically calculated from simulating the focus generated by the composite wavefront from mode amplitudes in (a)-(c).
Fig. 8.
Fig. 8. Theoretical performance of defocus-only refocusing for excitation NA of 0.9. (a), Strehl ratio, S, (solid line) and 3PE efficiency (dashed line) calculated as S3 plotted against the axial position of the focus. (b), axial 3PE PSF for 50 µm defocus showing significant aberration.

Equations (6)

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O P D = f [ 1 2 f z [ 1 ρ 2 sin 2 ( α ) ] 1 / 2 + z 2 f 2 ] 1 / 2 f ,
Ψ ( ρ , z ) = n k f ( [ 1 2 f z [ 1 ρ 2 sin 2 ( α ) ] 1 / 2 + z 2 f 2 ] 1 / 2 1 ) ,
Ψ ( ρ , z ) n k ( z [ 1 ρ 2 sin 2 α ] 1 / 2 z 2 2 f ρ 2 sin 2 α ) .
Ψ + ( ρ 1 , z 1 ; ρ 2 , z 2 ) n 1 k ( z 1 [ 1 ρ 1 2 sin 2 α 1 ] 1 / 2 z 1 2 2 f 1 ρ 1 2 sin 2 α 1 ) + n 2 k ( z 2 [ 1 ρ 2 2 sin 2 α 2 ] 1 / 2 z 2 2 2 f 2 ρ 2 2 sin 2 α 2 ) .
Ψ + ( ρ 2 , z 2 ) = z 2 2 ( k n 2 2 ρ 2 2 sin 2 α 2 f 2 ) ( n 2 f 2 2 n 1 f 1 + 1 2 ) .
S = 1 ( 4 n 2 k 2 z 4 ( 3 + 16 cos α + cos 2 α ) sin 8 ( α / α 2 2 ) 75 f 2 ( 3 + 8 cos α + cos 2 α ) ) ( n f 2 f 1 + 1 2 ) ,