Abstract

Line imaging of fluorescent and absorptive objects with a single-pixel imaging technique that acquires one-dimensional cross-sections through a sample by imposing a spatially-varying amplitude modulation on the probing beam is demonstrated. The fluorophore concentration or absorber distribution of the sample is directly mapped to modulation frequency components of the spatially-integrated temporal signal. Time-domain signals are obtained from a single photodiode, with object spatial frequency correlation encoded in time-domain bursts in the electronic signal from the photodiode.

© 2011 Optical Society of America

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2010

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010).
[CrossRef] [PubMed]

A. Vaziri, and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18, 19645–19655 (2010).
[CrossRef] [PubMed]

2009

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090–1092 (2009).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
[CrossRef] [PubMed]

M. Gora, K. Karnowski, M. Szkulmowski, B. J. Kaluzny, R. Huber, A. Kowalczyk, and M. Wojtkowski, “Ultra high-speed swept source oct imaging of the anterior segment of human eye at 200 khz with adjustable imaging range,” Opt. Express 17, 14880–14894 (2009).
[CrossRef] [PubMed]

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef] [PubMed]

S. Gekle, J. M. Gordillo, D. van der Meer, and D. Lohse, “High-speed jet formation after solid object impact,” Phys. Rev. Lett. 102, 034502 (2009).
[CrossRef] [PubMed]

J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009).
[CrossRef] [PubMed]

M. El-Desouki, M. J. Deen, Q. Y. Fang, L. Liu, F. Tse, and D. Armstrong, “Cmos image sensors for high speed applications,” Sensors 9, 430–444 (2009).
[CrossRef]

2008

S. T. Thoroddsen, T. G. Etoh, and K. Takehara, “High-speed imaging of drops and bubbles,” Annu. Rev. Fluid Mech. 40, 257–285 (2008).
[CrossRef]

N. Ji, H. Shroff, H. N. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605–616 (2008).
[CrossRef]

R. A. Niesner, V. Andresen, and M. Gunzer, “Intravital two-photon microscopy: focus on speed and time resolved imaging modalities,” Immunol. Rev. 221, 7–25 (2008).
[CrossRef] [PubMed]

C. Nitschke, A. Garin, M. Kosco-Vilbois, and M. Gunzer, “3D and 4D imaging of immune cells in vitro and in vivo,” Histochem. Cell Biol. 130, 1053–1062 (2008).
[CrossRef] [PubMed]

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

P. J. Scherz, J. Huisken, P. Sahai-Hernandez, and D. Y. R. Stainier, “High-speed imaging of developing heart valves reveals interplay of morphogenesis and function,” Development 135, 1179–1187 (2008).
[CrossRef] [PubMed]

G. Stutzmann, “Seeing the brain in action: how multiphoton imaging has advanced our understanding of neuronal function,” Microsc. Microanal. 14, 482–491 (2008).
[CrossRef] [PubMed]

R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008).
[CrossRef] [PubMed]

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 14–20 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[CrossRef]

2007

K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
[CrossRef] [PubMed]

S. Blonski and T. A. Kowalewski, “Piv analysis of turbulent flow in a micro-channel,” Theor. Appl. Mech. 45, 489–503 (2007).

D. Hessman, M. Lexholm, K. A. Dick, S. Ghatnekar-Nilsson, and L. Samuelson, “High-speed nanometer-scale imaging for studies of nanowire mechanics,” Small 3, 1699–1702 (2007).
[CrossRef] [PubMed]

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

2006

R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

R. Heintzmann, and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45, 5037–5045 (2006).
[CrossRef] [PubMed]

2005

J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. 30, 2760–2762 (2005).
[CrossRef] [PubMed]

K. B. Im, S. M. Han, H. Park, D. Kim, and B. M. Kim, “Simple high-speed confocal line-scanning microscope,” Opt. Express 13, 5151–5156 (2005).
[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, 2292–2299 (2002).
[CrossRef] [PubMed]

2000

D. N. Fittinghoff, P. W. Wiseman, and J. A. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7, 273–279 (2000).
[CrossRef] [PubMed]

1999

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-stokes raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

1998

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998).
[CrossRef] [PubMed]

J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998).
[CrossRef]

G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23, 1152–1154 (1998).
[CrossRef]

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, “Imaging system using frequency modulation time division multiplexing hybrid reticle,” Opt. Eng. 37, 2119–2123 (1998).
[CrossRef]

1997

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

1996

R. Juskaitis, T. Wilson, M. A. A. Neil, and M. Kozubek, “Efficient real-time confocal microscopy with white light sources,” Nature 383, 804–806 (1996).
[CrossRef] [PubMed]

1991

R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991).
[CrossRef] [PubMed]

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991).
[CrossRef]

1990

W. Denk, J. H. Strickler, and W. W. Webb, “2-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

1988

G. Q. Xiao, T. R. Corle, and G. S. Kino, “Real-time confocal scanning optical microscope,” Appl. Phys. Lett. 53, 716–718 (1988).
[CrossRef]

Andresen, V.

R. A. Niesner, V. Andresen, and M. Gunzer, “Intravital two-photon microscopy: focus on speed and time resolved imaging modalities,” Immunol. Rev. 221, 7–25 (2008).
[CrossRef] [PubMed]

Armstrong, D.

M. El-Desouki, M. J. Deen, Q. Y. Fang, L. Liu, F. Tse, and D. Armstrong, “Cmos image sensors for high speed applications,” Sensors 9, 430–444 (2009).
[CrossRef]

Bae, J. K.

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, “Imaging system using frequency modulation time division multiplexing hybrid reticle,” Opt. Eng. 37, 2119–2123 (1998).
[CrossRef]

Bahlmann, K.

K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
[CrossRef] [PubMed]

Barad, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Baraniuk, R. G.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Bartels, R. A.

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090–1092 (2009).
[CrossRef] [PubMed]

Barzda, V.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef] [PubMed]

R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008).
[CrossRef] [PubMed]

Bellve, K.

K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
[CrossRef] [PubMed]

Benedetti, P. A.

R. Heintzmann, and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45, 5037–5045 (2006).
[CrossRef] [PubMed]

Betzig, E.

N. Ji, H. Shroff, H. N. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605–616 (2008).
[CrossRef]

Bewersdorf, J.

J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998).
[CrossRef]

Blonski, S.

S. Blonski and T. A. Kowalewski, “Piv analysis of turbulent flow in a micro-channel,” Theor. Appl. Mech. 45, 489–503 (2007).

Boreman, G. D.

R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991).
[CrossRef] [PubMed]

Bouma, B. E.

J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. 30, 2760–2762 (2005).
[CrossRef] [PubMed]

G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23, 1152–1154 (1998).
[CrossRef]

Brakenhoff, G. J.

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998).
[CrossRef] [PubMed]

Campagnola, P. J.

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

Carriles, R.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef] [PubMed]

R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008).
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J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, “Imaging system using frequency modulation time division multiplexing hybrid reticle,” Opt. Eng. 37, 2119–2123 (1998).
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R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991).
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D. Hessman, M. Lexholm, K. A. Dick, S. Ghatnekar-Nilsson, and L. Samuelson, “High-speed nanometer-scale imaging for studies of nanowire mechanics,” Small 3, 1699–1702 (2007).
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J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009).
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J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991).
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K. B. Im, S. M. Han, H. Park, D. Kim, and B. M. Kim, “Simple high-speed confocal line-scanning microscope,” Opt. Express 13, 5151–5156 (2005).
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P. J. Scherz, J. Huisken, P. Sahai-Hernandez, and D. Y. R. Stainier, “High-speed imaging of developing heart valves reveals interplay of morphogenesis and function,” Development 135, 1179–1187 (2008).
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W. Gobel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3d laser scanning,” Nat. Methods 4, 73–79 (2007).
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G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11, 713–720 (2008).
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W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
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K. B. Im, S. M. Han, H. Park, D. Kim, and B. M. Kim, “Simple high-speed confocal line-scanning microscope,” Opt. Express 13, 5151–5156 (2005).
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J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, “Imaging system using frequency modulation time division multiplexing hybrid reticle,” Opt. Eng. 37, 2119–2123 (1998).
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C. Nitschke, A. Garin, M. Kosco-Vilbois, and M. Gunzer, “3D and 4D imaging of immune cells in vitro and in vivo,” Histochem. Cell Biol. 130, 1053–1062 (2008).
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K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
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M. Gora, K. Karnowski, M. Szkulmowski, B. J. Kaluzny, R. Huber, A. Kowalczyk, and M. Wojtkowski, “Ultra high-speed swept source oct imaging of the anterior segment of human eye at 200 khz with adjustable imaging range,” Opt. Express 17, 14880–14894 (2009).
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R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991).
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M. El-Desouki, M. J. Deen, Q. Y. Fang, L. Liu, F. Tse, and D. Armstrong, “Cmos image sensors for high speed applications,” Sensors 9, 430–444 (2009).
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P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
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S. Gekle, J. M. Gordillo, D. van der Meer, and D. Lohse, “High-speed jet formation after solid object impact,” Phys. Rev. Lett. 102, 034502 (2009).
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O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010).
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O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
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O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090–1092 (2009).
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K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
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W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
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J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009).
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R. Juskaitis, T. Wilson, M. A. A. Neil, and M. Kozubek, “Efficient real-time confocal microscopy with white light sources,” Nature 383, 804–806 (1996).
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R. A. Niesner, V. Andresen, and M. Gunzer, “Intravital two-photon microscopy: focus on speed and time resolved imaging modalities,” Immunol. Rev. 221, 7–25 (2008).
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C. Nitschke, A. Garin, M. Kosco-Vilbois, and M. Gunzer, “3D and 4D imaging of immune cells in vitro and in vivo,” Histochem. Cell Biol. 130, 1053–1062 (2008).
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J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, “Imaging system using frequency modulation time division multiplexing hybrid reticle,” Opt. Eng. 37, 2119–2123 (1998).
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J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009).
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K. B. Im, S. M. Han, H. Park, D. Kim, and B. M. Kim, “Simple high-speed confocal line-scanning microscope,” Opt. Express 13, 5151–5156 (2005).
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J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998).
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G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11, 713–720 (2008).
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K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
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G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci. 11, 713–720 (2008).
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Sahai-Hernandez, P.

P. J. Scherz, J. Huisken, P. Sahai-Hernandez, and D. Y. R. Stainier, “High-speed imaging of developing heart valves reveals interplay of morphogenesis and function,” Development 135, 1179–1187 (2008).
[CrossRef] [PubMed]

Samuelson, L.

D. Hessman, M. Lexholm, K. A. Dick, S. Ghatnekar-Nilsson, and L. Samuelson, “High-speed nanometer-scale imaging for studies of nanowire mechanics,” Small 3, 1699–1702 (2007).
[CrossRef] [PubMed]

Sanders, J. S.

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991).
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R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
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P. J. Scherz, J. Huisken, P. Sahai-Hernandez, and D. Y. R. Stainier, “High-speed imaging of developing heart valves reveals interplay of morphogenesis and function,” Development 135, 1179–1187 (2008).
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Schlup, P.

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090–1092 (2009).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
[CrossRef] [PubMed]

Shank, C. V.

A. Vaziri, and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18, 19645–19655 (2010).
[CrossRef] [PubMed]

Shapiro, J. H.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[CrossRef]

Sheetz, K. E.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef] [PubMed]

R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008).
[CrossRef] [PubMed]

Shroff, H.

N. Ji, H. Shroff, H. N. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605–616 (2008).
[CrossRef]

Sieneart, I.

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

Silberberg, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

So, P. T. C.

K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
[CrossRef] [PubMed]

Squier, J. A.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef] [PubMed]

R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008).
[CrossRef] [PubMed]

D. N. Fittinghoff, P. W. Wiseman, and J. A. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7, 273–279 (2000).
[CrossRef] [PubMed]

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998).
[CrossRef] [PubMed]

Stainier, D. Y. R.

P. J. Scherz, J. Huisken, P. Sahai-Hernandez, and D. Y. R. Stainier, “High-speed imaging of developing heart valves reveals interplay of morphogenesis and function,” Development 135, 1179–1187 (2008).
[CrossRef] [PubMed]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “2-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Stutzmann, G.

G. Stutzmann, “Seeing the brain in action: how multiphoton imaging has advanced our understanding of neuronal function,” Microsc. Microanal. 14, 482–491 (2008).
[CrossRef] [PubMed]

Sylvester, A. W.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef] [PubMed]

Szkulmowski, M.

M. Gora, K. Karnowski, M. Szkulmowski, B. J. Kaluzny, R. Huber, A. Kowalczyk, and M. Wojtkowski, “Ultra high-speed swept source oct imaging of the anterior segment of human eye at 200 khz with adjustable imaging range,” Opt. Express 17, 14880–14894 (2009).
[CrossRef] [PubMed]

Takehara, K.

S. T. Thoroddsen, T. G. Etoh, and K. Takehara, “High-speed imaging of drops and bubbles,” Annu. Rev. Fluid Mech. 40, 257–285 (2008).
[CrossRef]

Takhar, D.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Tearney, G. J.

J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. 30, 2760–2762 (2005).
[CrossRef] [PubMed]

G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23, 1152–1154 (1998).
[CrossRef]

Thoroddsen, S. T.

S. T. Thoroddsen, T. G. Etoh, and K. Takehara, “High-speed imaging of drops and bubbles,” Annu. Rev. Fluid Mech. 40, 257–285 (2008).
[CrossRef]

Tse, F.

M. El-Desouki, M. J. Deen, Q. Y. Fang, L. Liu, F. Tse, and D. Armstrong, “Cmos image sensors for high speed applications,” Sensors 9, 430–444 (2009).
[CrossRef]

Vakoc, B. J.

J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. 30, 2760–2762 (2005).
[CrossRef] [PubMed]

van der Meer, D.

S. Gekle, J. M. Gordillo, D. van der Meer, and D. Lohse, “High-speed jet formation after solid object impact,” Phys. Rev. Lett. 102, 034502 (2009).
[CrossRef] [PubMed]

Vaziri, A.

A. Vaziri, and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18, 19645–19655 (2010).
[CrossRef] [PubMed]

Waitukaitis, S. R.

J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009).
[CrossRef] [PubMed]

Webb, R. H.

G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23, 1152–1154 (1998).
[CrossRef]

Webb, W. W.

W. Denk, J. H. Strickler, and W. W. Webb, “2-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Wei, M. D.

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

Williams, K. F.

R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991).
[CrossRef] [PubMed]

Wilson, K. R.

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998).
[CrossRef] [PubMed]

Wilson, T.

R. Juskaitis, T. Wilson, M. A. A. Neil, and M. Kozubek, “Efficient real-time confocal microscopy with white light sources,” Nature 383, 804–806 (1996).
[CrossRef] [PubMed]

Wiseman, P. W.

D. N. Fittinghoff, P. W. Wiseman, and J. A. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7, 273–279 (2000).
[CrossRef] [PubMed]

Wojtkowski, M.

M. Gora, K. Karnowski, M. Szkulmowski, B. J. Kaluzny, R. Huber, A. Kowalczyk, and M. Wojtkowski, “Ultra high-speed swept source oct imaging of the anterior segment of human eye at 200 khz with adjustable imaging range,” Opt. Express 17, 14880–14894 (2009).
[CrossRef] [PubMed]

Wolleschensky, R.

R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

Xiao, G. Q.

G. Q. Xiao, T. R. Corle, and G. S. Kino, “Real-time confocal scanning optical microscope,” Appl. Phys. Lett. 53, 716–718 (1988).
[CrossRef]

Xie, X. S.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-stokes raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Yelin, D.

J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. 30, 2760–2762 (2005).
[CrossRef] [PubMed]

Zhong, H. N.

N. Ji, H. Shroff, H. N. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605–616 (2008).
[CrossRef]

Zimmermann, B.

R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

Zumbusch, A.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-stokes raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Annu. Rev. Fluid Mech.

S. T. Thoroddsen, T. G. Etoh, and K. Takehara, “High-speed imaging of drops and bubbles,” Annu. Rev. Fluid Mech. 40, 257–285 (2008).
[CrossRef]

Appl. Opt.

R. Heintzmann, and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45, 5037–5045 (2006).
[CrossRef] [PubMed]

R. G. Driggers, C. E. Halford, G. D. Boreman, D. Lattman, and K. F. Williams, “Parameters of spinning fm reticles,” Appl. Opt. 30, 887–895 (1991).
[CrossRef] [PubMed]

Appl. Phys. Lett.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

G. Q. Xiao, T. R. Corle, and G. S. Kino, “Real-time confocal scanning optical microscope,” Appl. Phys. Lett. 53, 716–718 (1988).
[CrossRef]

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Biophys. J.

P. J. Campagnola, M. D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

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

Curr. Opin. Neurobiol.

N. Ji, H. Shroff, H. N. Zhong, and E. Betzig, “Advances in the speed and resolution of light microscopy,” Curr. Opin. Neurobiol. 18, 605–616 (2008).
[CrossRef]

Development

P. J. Scherz, J. Huisken, P. Sahai-Hernandez, and D. Y. R. Stainier, “High-speed imaging of developing heart valves reveals interplay of morphogenesis and function,” Development 135, 1179–1187 (2008).
[CrossRef] [PubMed]

Histochem. Cell Biol.

C. Nitschke, A. Garin, M. Kosco-Vilbois, and M. Gunzer, “3D and 4D imaging of immune cells in vitro and in vivo,” Histochem. Cell Biol. 130, 1053–1062 (2008).
[CrossRef] [PubMed]

IEEE Signal Process. Mag.

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 14–20 (2008).
[CrossRef]

Immunol. Rev.

R. A. Niesner, V. Andresen, and M. Gunzer, “Intravital two-photon microscopy: focus on speed and time resolved imaging modalities,” Immunol. Rev. 221, 7–25 (2008).
[CrossRef] [PubMed]

J. Biomed. Opt.

R. Wolleschensky, B. Zimmermann, and M. Kempe, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

Microsc. Microanal.

G. Stutzmann, “Seeing the brain in action: how multiphoton imaging has advanced our understanding of neuronal function,” Microsc. Microanal. 14, 482–491 (2008).
[CrossRef] [PubMed]

Nat. Methods

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

Nat. Neurosci.

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

Nature

J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit, M. E. Mobius, S. R. Waitukaitis, and H. M. Jaeger, “High-speed tracking of rupture and clustering in freely falling granular streams,” Nature 459, 1110–1113 (2009).
[CrossRef] [PubMed]

R. Juskaitis, T. Wilson, M. A. A. Neil, and M. Kozubek, “Efficient real-time confocal microscopy with white light sources,” Nature 383, 804–806 (1996).
[CrossRef] [PubMed]

Opt. Eng.

J. S. Sanders, R. G. Driggers, C. E. Halford, and S. T. Griffin, “Imaging with frequency-modulated reticles,” Opt. Eng. 30, 1720–1724 (1991).
[CrossRef]

J. K. Bae, Y. H. Doh, D. S. Noh, and S. J. Kim, “Imaging system using frequency modulation time division multiplexing hybrid reticle,” Opt. Eng. 37, 2119–2123 (1998).
[CrossRef]

Opt. Express

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Label-free second harmonic generation holographic microscopy of biological specimens,” Opt. Express 18, 9840–9851 (2010).
[CrossRef] [PubMed]

K. Bahlmann, P. T. C. So, M. Kirber, R. Reich, B. Kosicki, W. McGonagle, and K. Bellve, “Multifocal multiphoton microscopy (mmm) at a frame rate beyond 600 hz,” Opt. Express 15, 10991–10998 (2007).
[CrossRef] [PubMed]

R. Carriles, K. E. Sheetz, E. E. Hoover, J. A. Squier, and V. Barzda, “Simultaneous multifocal, multiphoton, photon counting microscopy,” Opt. Express 16, 10364–10371 (2008).
[CrossRef] [PubMed]

D. N. Fittinghoff, P. W. Wiseman, and J. A. Squier, “Widefield multiphoton and temporally decorrelated multifocal multiphoton microscopy,” Opt. Express 7, 273–279 (2000).
[CrossRef] [PubMed]

A. Vaziri, and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18, 19645–19655 (2010).
[CrossRef] [PubMed]

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998).
[CrossRef] [PubMed]

K. B. Im, S. M. Han, H. Park, D. Kim, and B. M. Kim, “Simple high-speed confocal line-scanning microscope,” Opt. Express 13, 5151–5156 (2005).
[CrossRef] [PubMed]

M. Gora, K. Karnowski, M. Szkulmowski, B. J. Kaluzny, R. Huber, A. Kowalczyk, and M. Wojtkowski, “Ultra high-speed swept source oct imaging of the anterior segment of human eye at 200 khz with adjustable imaging range,” Opt. Express 17, 14880–14894 (2009).
[CrossRef] [PubMed]

Opt. Lett.

G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23, 1152–1154 (1998).
[CrossRef]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090–1092 (2009).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240–1242 (2009).
[CrossRef] [PubMed]

J. Bewersdorf, R. Pick, and S. W. Hell, “Multifocal multiphoton microscopy,” Opt. Lett. 23, 655–657 (1998).
[CrossRef]

J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. 30, 2760–2762 (2005).
[CrossRef] [PubMed]

Phys. Rev. A

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[CrossRef]

Phys. Rev. Lett.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-stokes raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

S. Gekle, J. M. Gordillo, D. van der Meer, and D. Lohse, “High-speed jet formation after solid object impact,” Phys. Rev. Lett. 102, 034502 (2009).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80, 081101 (2009).
[CrossRef] [PubMed]

Science

W. Denk, J. H. Strickler, and W. W. Webb, “2-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Sensors

M. El-Desouki, M. J. Deen, Q. Y. Fang, L. Liu, F. Tse, and D. Armstrong, “Cmos image sensors for high speed applications,” Sensors 9, 430–444 (2009).
[CrossRef]

Small

D. Hessman, M. Lexholm, K. A. Dick, S. Ghatnekar-Nilsson, and L. Samuelson, “High-speed nanometer-scale imaging for studies of nanowire mechanics,” Small 3, 1699–1702 (2007).
[CrossRef] [PubMed]

Theor. Appl. Mech.

S. Blonski and T. A. Kowalewski, “Piv analysis of turbulent flow in a micro-channel,” Theor. Appl. Mech. 45, 489–503 (2007).

Other

J. J. Art and M. B. Goodman, “Rapid-scanning confocal microscopy,” in Methods in Cell Biology, B. Matsumoto, ed. (Academic Press, 1993), Vol 38, pp. 47–77.
[PubMed]

L. M. Biberman, Reticles in electro-optical devices, International series of monographs in infrared science and technology (Pergamon Press, 1966), Vol. 1.

D. Lovell, “Electro-optic position indicator system,” U.S. Patent 2,997,699 (22 Aug. 1961).

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2004).
[PubMed]

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Figures (7)

Fig. 1
Fig. 1

Schematic of imaging method.

Fig. 2
Fig. 2

Example mask pattern of Eq. (11) with parameters k0 = 0, Δk = 3 mm−1, and Δθ = 6 rad.

Fig. 3
Fig. 3

(a) Digitized time-domain signal collected for an unperturbed excitation beam, with the inset showing the coherence burst near t = 0. (b) Fourier transformed frequency domain signal of the shaded area of the time trace.

Fig. 4
Fig. 4

Space-to-frequency calibration, obtained by scanning a 2-μm slit across the modulated beam at the object plane. The beam intensity distribution at the object plane (black line, bottom axis) is found by summing the temporal trace at each slit position. The frequency bandwidth at a single position (left axis) reveals the system’s PSF. When the slit is removed, the resulting frequency distribution, mapped to the spatial coordinate (blue dashed line) shows excellent agreement with the beam profile.

Fig. 5
Fig. 5

(a) The signal obtained from a 150-μm pinhole contains additional structure, arising from variations in the modulation frequency as the disc revolves. (b) Imperfections in the modulation frequency are revealed by a spectrogram. (c) Frequency wobble correction removes structure on the slit transmission measurement.

Fig. 6
Fig. 6

(a) Image of 1951 USAF test pattern, assembled from a series of scanned SPIFI acquisitions. (b)–(d) Slices across bars of varying width showing image resolution performance.

Fig. 7
Fig. 7

Images obtained using (a) absorptive and (b) fluorescent SPIFI of a fluorescent sample image.

Equations (11)

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

I obj ( x , t ) = I 0 | u ( x ) m ( x , t ) g ( x ) | 2
m ( x , t ) = w ( t ) 2 [ 1 + cos ( 2 π κ x t ) ] .
E obj ( x , t ) = E o 2 u ( x ) g ( x ) w ( t ) [ 1 + cos ( 2 π κ x t ) ] e i ω 0 t .
I obj ( x , t ) = I 0 1 4 | w ( t ) u ( x ) g ( x ) | 2 [ 3 2 + 2 cos ( 2 π κ x t ) + 1 2 cos ( 4 π κ x t ) ]
s 1 ( t ) = | w ( t ) | 2 | u ( x ) g ( x ) | 2 e j 2 π κ t x d x + c . c .
s 2 ( t ) = 1 4 | w ( t ) | 2 | u ( x ) g ( x ) | 2 e j 4 π κ t x d x + c . c .
s 1 ( t ) = | w ( t ) | 2 e j 2 π κ x c t | u ( x ) g ( x ) | 2 e j 2 π κ t x d x + c . c .
𝒢 ( f x ) = | u ( x ) g ( x ) | 2 e i 2 π f x x d x 𝔉 { | u ( x ) g ( x ) | 2 } ,
s 1 ( t ) = 2 | w ( t ) | 2 | 𝒢 ( κ t ) | cos ( 2 π f c t + 𝒢 ( κ t ) )
S ^ 1 + ( x = f κ 1 ) = 𝒲 ( κ x ) | u ( x ) g ( x ) | 2
m ( R , θ ) = 1 2 + 1 2 cos [ ( k 0 + Δ k R ) θ ]

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