Abstract

Three-dimensional confocal fluorescence imaging of in vivo tissues is challenging due to sample motion and limited imaging speeds. In this paper a novel method is therefore presented for scanning confocal epi-fluorescence microscopy with instantaneous depth-sensing based on self-interference fluorescence microscopy (SIFM). A tabletop epi-fluorescence SIFM setup was constructed with an annular phase plate in the emission path to create a spectral self-interference signal that is phase-dependent on the axial position of a fluorescent sample. A Mach-Zehnder interferometer based on a 3 × 3 fiber-coupler was developed for a sensitive phase analysis of the SIFM signal with three photon-counter detectors instead of a spectrometer. The Mach-Zehnder interferometer created three intensity signals that alternately oscillated as a function of the SIFM spectral phase and therefore encoded directly for the axial sample position. Controlled axial translation of fluorescent microsphere layers showed a linear dependence of the SIFM spectral phase with sample depth over axial image ranges of 500 µm and 80 µm (3.9 × Rayleigh range) for 4 × and 10 × microscope objectives respectively. In addition, SIFM was in good agreement with optical coherence tomography depth measurements on a sample with indocyanine green dye filled capillaries placed at multiple depths. High-resolution SIFM imaging applications are demonstrated for fluorescence angiography on a dye-filled capillary blood vessel phantom and for autofluorescence imaging on an ex vivo fly eye.

© 2017 Optical Society of America

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2015 (2)

2014 (2)

M. G. Gräfe, A. Hoffmann, and C. Spielmann, “Ultrafast fluorescence spectroscopy for axial resolution of flurorophore distributions,” Appl. Phys. B 117(3), 833–840 (2014).
[Crossref]

M. Smolla, M. Ruchty, M. Nagel, and C. J. Kleineidam, “Clearing pigmented insect cuticle to investigate small insects’ organs in situ using confocal laser-scanning microscopy (CLSM),” Arthropod Struct. Dev. 43(2), 175–181 (2014).
[Crossref] [PubMed]

2013 (2)

W. Loh, S. Yegnanarayanan, R. J. Ram, and P. W. Juodawlkis, “Unified theory of oscillator phase noise I: white noise,” IEEE Trans. Microw. Theory Tech. 61(6), 2371–2381 (2013).
[Crossref]

D. R. Lee, Y. D. Kim, D. G. Gweon, and H. Yoo, “Dual-detection confocal fluorescence microscopy: fluorescence axial imaging without axial scanning,” Opt. Express 21(15), 17839–17848 (2013).
[Crossref] [PubMed]

2012 (1)

2010 (2)

S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
[Crossref] [PubMed]

D. M. de Bruin, R. H. Bremmer, V. M. Kodach, R. de Kinkelder, J. van Marle, T. G. van Leeuwen, and D. J. Faber, “Optical phantoms of varying geometry based on thin building blocks with controlled optical properties,” J. Biomed. Opt. 15(2), 025001 (2010).
[Crossref] [PubMed]

2009 (2)

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
[Crossref] [PubMed]

Q. Xu, K. Shi, S. Yin, and Z. Liu, “Chromatic two-photon excitation fluorescence imaging,” J. Microsc. 235(1), 79–83 (2009).
[Crossref] [PubMed]

2008 (2)

M. Dogan, A. Yalcin, S. Jain, M. B. Goldberg, A. K. Swan, M. S. Unlu, and B. B. Goldberg, “Spectral self-interference fluorescence microscopy for subcellular imaging,” IEEE J. Sel. Top. Quantum Electron. 14(1), 217–225 (2008).
[Crossref]

R. A. Schwarz, W. Gao, D. Daye, M. D. Williams, R. Richards-Kortum, and A. M. Gillenwater, “Autofluorescence and diffuse reflectance spectroscopy of oral epithelial tissue using a depth-sensitive fiber-optic probe,” Appl. Opt. 47(6), 825–834 (2008).
[Crossref] [PubMed]

2007 (2)

L. Thiberville, S. Moreno-Swirc, T. Vercauteren, E. Peltier, C. Cavé, and G. Bourg Heckly, “In vivo imaging of the bronchial wall microstructure using fibered confocal fluorescence microscopy,” Am. J. Respir. Crit. Care Med. 175(1), 22–31 (2007).
[Crossref] [PubMed]

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. 46(10), 1819–1829 (2007).
[Crossref] [PubMed]

2006 (4)

A. Bilenca, A. Ozcan, B. Bouma, and G. Tearney, “Fluorescence coherence tomography,” Opt. Express 14(16), 7134–7143 (2006).
[Crossref] [PubMed]

B. N. Giepmans, S. R. Adams, M. H. Ellisman, and R. Y. Tsien, “The fluorescent toolbox for assessing protein location and function,” Science 312(5771), 217–224 (2006).
[Crossref] [PubMed]

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8(1), 1–33 (2006).
[Crossref] [PubMed]

L. E. Meyer, N. Otberg, W. Sterry, and J. Lademann, “In vivo confocal scanning laser microscopy: comparison of the reflectance and fluorescence mode by imaging human skin,” J. Biomed. Opt. 11(4), 044012 (2006).
[Crossref] [PubMed]

2005 (3)

J. W. Lichtman and J. A. Conchello, “Fluorescence microscopy,” Nat. Methods 2(12), 910–919 (2005).
[Crossref] [PubMed]

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 microm,” Opt. Express 13(11), 3931–3944 (2005).
[Crossref] [PubMed]

2003 (2)

M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3 x 3 fiber-optic couplers,” Opt. Lett. 28(22), 2162–2164 (2003).
[Crossref] [PubMed]

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13(1), 195–208 (2003).
[PubMed]

2002 (2)

R. K. Jain, L. L. Munn, and D. Fukumura, “Dissecting tumour pathophysiology using intravital microscopy,” Nat. Rev. Cancer 2(4), 266–276 (2002).
[Crossref] [PubMed]

P. Lam, K. J. Wynne, and G. E. Wnek, “Surface-tension-confined microfluidics,” Langmuir 18(3), 948–951 (2002).
[Crossref]

2001 (1)

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
[Crossref] [PubMed]

2000 (2)

S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
[Crossref] [PubMed]

R. Birngruber, U. Schmidt-Erfurth, S. Teschner, and J. Noack, “Confocal laser scanning fluorescence topography: a new method for three-dimensional functional imaging of vascular structures,” Graefes Arch. Clin. Exp. Ophthalmol. 238(7), 559–565 (2000).
[Crossref] [PubMed]

1999 (1)

M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
[Crossref] [PubMed]

1998 (2)

G. A. Wagnières, W. M. Star, and B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68(5), 603–632 (1998).
[Crossref] [PubMed]

P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23(13), 986–988 (1998).
[Crossref] [PubMed]

1995 (2)

F. C. Delori, C. K. Dorey, G. Staurenghi, O. Arend, D. G. Goger, and J. J. Weiter, “In vivo fluorescence of the ocular fundus exhibits retinal pigment epithelium lipofuscin characteristics,” Invest. Ophthalmol. Vis. Sci. 36(3), 718–729 (1995).
[PubMed]

D. Stavenga, “Insect retinal pigments: spectral characteristics and physiological functions,” Prog. Retin. Eye Res. 15(1), 231–259 (1995).
[Crossref]

1994 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

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]

1951 (1)

W. C. Duesterhoeft, M. W. Schulz, and E. Clarke, “Determination of instantaneous currents and voltages by means of alpha, beta, and zero components,” Trans. Am. Inst. Electr. Eng. 70(2), 1248–1255 (1951).
[Crossref]

Adams, S. R.

B. N. Giepmans, S. R. Adams, M. H. Ellisman, and R. Y. Tsien, “The fluorescent toolbox for assessing protein location and function,” Science 312(5771), 217–224 (2006).
[Crossref] [PubMed]

Aguirre, A. D.

C. Vinegoni, S. Lee, A. D. Aguirre, and R. Weissleder, “New techniques for motion-artifact-free in vivo cardiac microscopy,” Front. Physiol. 6, 147 (2015).
[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]

Anderson, R. R.

M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
[Crossref] [PubMed]

Arend, O.

F. C. Delori, C. K. Dorey, G. Staurenghi, O. Arend, D. G. Goger, and J. J. Weiter, “In vivo fluorescence of the ocular fundus exhibits retinal pigment epithelium lipofuscin characteristics,” Invest. Ophthalmol. Vis. Sci. 36(3), 718–729 (1995).
[PubMed]

Bilenca, A.

Birngruber, R.

R. Birngruber, U. Schmidt-Erfurth, S. Teschner, and J. Noack, “Confocal laser scanning fluorescence topography: a new method for three-dimensional functional imaging of vascular structures,” Graefes Arch. Clin. Exp. Ophthalmol. 238(7), 559–565 (2000).
[Crossref] [PubMed]

Bouma, B.

Bourg Heckly, G.

L. Thiberville, S. Moreno-Swirc, T. Vercauteren, E. Peltier, C. Cavé, and G. Bourg Heckly, “In vivo imaging of the bronchial wall microstructure using fibered confocal fluorescence microscopy,” Am. J. Respir. Crit. Care Med. 175(1), 22–31 (2007).
[Crossref] [PubMed]

Bremer, C.

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13(1), 195–208 (2003).
[PubMed]

Bremmer, R. H.

D. M. de Bruin, R. H. Bremmer, V. M. Kodach, R. de Kinkelder, J. van Marle, T. G. van Leeuwen, and D. J. Faber, “Optical phantoms of varying geometry based on thin building blocks with controlled optical properties,” J. Biomed. Opt. 15(2), 025001 (2010).
[Crossref] [PubMed]

Cable, A.

S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
[Crossref] [PubMed]

Carney, M.

S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
[Crossref] [PubMed]

Cavé, C.

L. Thiberville, S. Moreno-Swirc, T. Vercauteren, E. Peltier, C. Cavé, and G. Bourg Heckly, “In vivo imaging of the bronchial wall microstructure using fibered confocal fluorescence microscopy,” Am. J. Respir. Crit. Care Med. 175(1), 22–31 (2007).
[Crossref] [PubMed]

Cense, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Chen, C. W.

S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
[Crossref] [PubMed]

Chen, Y.

S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
[Crossref] [PubMed]

Cheung, E. L.

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B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
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F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
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B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
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R. K. Jain, L. L. Munn, and D. Fukumura, “Dissecting tumour pathophysiology using intravital microscopy,” Nat. Rev. Cancer 2(4), 266–276 (2002).
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G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
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Giepmans, B. N.

B. N. Giepmans, S. R. Adams, M. H. Ellisman, and R. Y. Tsien, “The fluorescent toolbox for assessing protein location and function,” Science 312(5771), 217–224 (2006).
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Gillette, J. M.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
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F. C. Delori, C. K. Dorey, G. Staurenghi, O. Arend, D. G. Goger, and J. J. Weiter, “In vivo fluorescence of the ocular fundus exhibits retinal pigment epithelium lipofuscin characteristics,” Invest. Ophthalmol. Vis. Sci. 36(3), 718–729 (1995).
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M. Dogan, A. Yalcin, S. Jain, M. B. Goldberg, A. K. Swan, M. S. Unlu, and B. B. Goldberg, “Spectral self-interference fluorescence microscopy for subcellular imaging,” IEEE J. Sel. Top. Quantum Electron. 14(1), 217–225 (2008).
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M. Dogan, A. Yalcin, S. Jain, M. B. Goldberg, A. K. Swan, M. S. Unlu, and B. B. Goldberg, “Spectral self-interference fluorescence microscopy for subcellular imaging,” IEEE J. Sel. Top. Quantum Electron. 14(1), 217–225 (2008).
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M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
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M. G. Gräfe, A. Hoffmann, and C. Spielmann, “Ultrafast fluorescence spectroscopy for axial resolution of flurorophore distributions,” Appl. Phys. B 117(3), 833–840 (2014).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
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Gweon, D. G.

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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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Helmchen, F.

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
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G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
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M. G. Gräfe, A. Hoffmann, and C. Spielmann, “Ultrafast fluorescence spectroscopy for axial resolution of flurorophore distributions,” Appl. Phys. B 117(3), 833–840 (2014).
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Jain, R. K.

R. K. Jain, L. L. Munn, and D. Fukumura, “Dissecting tumour pathophysiology using intravital microscopy,” Nat. Rev. Cancer 2(4), 266–276 (2002).
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M. Dogan, A. Yalcin, S. Jain, M. B. Goldberg, A. K. Swan, M. S. Unlu, and B. B. Goldberg, “Spectral self-interference fluorescence microscopy for subcellular imaging,” IEEE J. Sel. Top. Quantum Electron. 14(1), 217–225 (2008).
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B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
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W. Loh, S. Yegnanarayanan, R. J. Ram, and P. W. Juodawlkis, “Unified theory of oscillator phase noise I: white noise,” IEEE Trans. Microw. Theory Tech. 61(6), 2371–2381 (2013).
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G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
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Kleineidam, C. J.

M. Smolla, M. Ruchty, M. Nagel, and C. J. Kleineidam, “Clearing pigmented insect cuticle to investigate small insects’ organs in situ using confocal laser-scanning microscopy (CLSM),” Arthropod Struct. Dev. 43(2), 175–181 (2014).
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D. M. de Bruin, R. H. Bremmer, V. M. Kodach, R. de Kinkelder, J. van Marle, T. G. van Leeuwen, and D. J. Faber, “Optical phantoms of varying geometry based on thin building blocks with controlled optical properties,” J. Biomed. Opt. 15(2), 025001 (2010).
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L. E. Meyer, N. Otberg, W. Sterry, and J. Lademann, “In vivo confocal scanning laser microscopy: comparison of the reflectance and fluorescence mode by imaging human skin,” J. Biomed. Opt. 11(4), 044012 (2006).
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J. W. Lichtman and J. A. Conchello, “Fluorescence microscopy,” Nat. Methods 2(12), 910–919 (2005).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
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Q. Xu, K. Shi, S. Yin, and Z. Liu, “Chromatic two-photon excitation fluorescence imaging,” J. Microsc. 235(1), 79–83 (2009).
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W. Loh, S. Yegnanarayanan, R. J. Ram, and P. W. Juodawlkis, “Unified theory of oscillator phase noise I: white noise,” IEEE Trans. Microw. Theory Tech. 61(6), 2371–2381 (2013).
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S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
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G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
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L. E. Meyer, N. Otberg, W. Sterry, and J. Lademann, “In vivo confocal scanning laser microscopy: comparison of the reflectance and fluorescence mode by imaging human skin,” J. Biomed. Opt. 11(4), 044012 (2006).
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Munn, L. L.

R. K. Jain, L. L. Munn, and D. Fukumura, “Dissecting tumour pathophysiology using intravital microscopy,” Nat. Rev. Cancer 2(4), 266–276 (2002).
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M. Smolla, M. Ruchty, M. Nagel, and C. J. Kleineidam, “Clearing pigmented insect cuticle to investigate small insects’ organs in situ using confocal laser-scanning microscopy (CLSM),” Arthropod Struct. Dev. 43(2), 175–181 (2014).
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S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
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S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
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Otberg, N.

L. E. Meyer, N. Otberg, W. Sterry, and J. Lademann, “In vivo confocal scanning laser microscopy: comparison of the reflectance and fluorescence mode by imaging human skin,” J. Biomed. Opt. 11(4), 044012 (2006).
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Ozcan, A.

Park, B.

Peltier, E.

L. Thiberville, S. Moreno-Swirc, T. Vercauteren, E. Peltier, C. Cavé, and G. Bourg Heckly, “In vivo imaging of the bronchial wall microstructure using fibered confocal fluorescence microscopy,” Am. J. Respir. Crit. Care Med. 175(1), 22–31 (2007).
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Pierce, M. C.

Piyawattanametha, W.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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Quimby, L.

S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
[Crossref] [PubMed]

Rajadhyaksha, M.

M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
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Ram, R. J.

W. Loh, S. Yegnanarayanan, R. J. Ram, and P. W. Juodawlkis, “Unified theory of oscillator phase noise I: white noise,” IEEE Trans. Microw. Theory Tech. 61(6), 2371–2381 (2013).
[Crossref]

Richards-Kortum, R.

Roney, C. A.

S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
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Ruchty, M.

M. Smolla, M. Ruchty, M. Nagel, and C. J. Kleineidam, “Clearing pigmented insect cuticle to investigate small insects’ organs in situ using confocal laser-scanning microscopy (CLSM),” Arthropod Struct. Dev. 43(2), 175–181 (2014).
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Schmidt-Erfurth, U.

R. Birngruber, U. Schmidt-Erfurth, S. Teschner, and J. Noack, “Confocal laser scanning fluorescence topography: a new method for three-dimensional functional imaging of vascular structures,” Graefes Arch. Clin. Exp. Ophthalmol. 238(7), 559–565 (2000).
[Crossref] [PubMed]

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B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
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Schulz, M. W.

W. C. Duesterhoeft, M. W. Schulz, and E. Clarke, “Determination of instantaneous currents and voltages by means of alpha, beta, and zero components,” Trans. Am. Inst. Electr. Eng. 70(2), 1248–1255 (1951).
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Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Schwarz, R. A.

Shi, K.

Q. Xu, K. Shi, S. Yin, and Z. Liu, “Chromatic two-photon excitation fluorescence imaging,” J. Microsc. 235(1), 79–83 (2009).
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Shtengel, G.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
[Crossref] [PubMed]

Smolla, M.

M. Smolla, M. Ruchty, M. Nagel, and C. J. Kleineidam, “Clearing pigmented insect cuticle to investigate small insects’ organs in situ using confocal laser-scanning microscopy (CLSM),” Arthropod Struct. Dev. 43(2), 175–181 (2014).
[Crossref] [PubMed]

Sougrat, R.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009).
[Crossref] [PubMed]

Spielmann, C.

M. G. Gräfe, A. Hoffmann, and C. Spielmann, “Ultrafast fluorescence spectroscopy for axial resolution of flurorophore distributions,” Appl. Phys. B 117(3), 833–840 (2014).
[Crossref]

Star, W. M.

G. A. Wagnières, W. M. Star, and B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68(5), 603–632 (1998).
[Crossref] [PubMed]

Staurenghi, G.

F. C. Delori, C. K. Dorey, G. Staurenghi, O. Arend, D. G. Goger, and J. J. Weiter, “In vivo fluorescence of the ocular fundus exhibits retinal pigment epithelium lipofuscin characteristics,” Invest. Ophthalmol. Vis. Sci. 36(3), 718–729 (1995).
[PubMed]

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D. Stavenga, “Insect retinal pigments: spectral characteristics and physiological functions,” Prog. Retin. Eye Res. 15(1), 231–259 (1995).
[Crossref]

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Sterry, W.

L. E. Meyer, N. Otberg, W. Sterry, and J. Lademann, “In vivo confocal scanning laser microscopy: comparison of the reflectance and fluorescence mode by imaging human skin,” J. Biomed. Opt. 11(4), 044012 (2006).
[Crossref] [PubMed]

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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Summers, R. M.

S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
[Crossref] [PubMed]

Swan, A. K.

M. Dogan, A. Yalcin, S. Jain, M. B. Goldberg, A. K. Swan, M. S. Unlu, and B. B. Goldberg, “Spectral self-interference fluorescence microscopy for subcellular imaging,” IEEE J. Sel. Top. Quantum Electron. 14(1), 217–225 (2008).
[Crossref]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
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Teschner, S.

R. Birngruber, U. Schmidt-Erfurth, S. Teschner, and J. Noack, “Confocal laser scanning fluorescence topography: a new method for three-dimensional functional imaging of vascular structures,” Graefes Arch. Clin. Exp. Ophthalmol. 238(7), 559–565 (2000).
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S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
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B. N. Giepmans, S. R. Adams, M. H. Ellisman, and R. Y. Tsien, “The fluorescent toolbox for assessing protein location and function,” Science 312(5771), 217–224 (2006).
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M. Dogan, A. Yalcin, S. Jain, M. B. Goldberg, A. K. Swan, M. S. Unlu, and B. B. Goldberg, “Spectral self-interference fluorescence microscopy for subcellular imaging,” IEEE J. Sel. Top. Quantum Electron. 14(1), 217–225 (2008).
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L. Thiberville, S. Moreno-Swirc, T. Vercauteren, E. Peltier, C. Cavé, and G. Bourg Heckly, “In vivo imaging of the bronchial wall microstructure using fibered confocal fluorescence microscopy,” Am. J. Respir. Crit. Care Med. 175(1), 22–31 (2007).
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[Crossref] [PubMed]

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M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
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[Crossref] [PubMed]

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G. A. Wagnières, W. M. Star, and B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68(5), 603–632 (1998).
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S. Yuan, C. A. Roney, J. Wierwille, C. W. Chen, B. Xu, G. Griffiths, J. Jiang, H. Ma, A. Cable, R. M. Summers, and Y. Chen, “Co-registered optical coherence tomography and fluorescence molecular imaging for simultaneous morphological and molecular imaging,” Phys. Med. Biol. 55(1), 191–206 (2010).
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M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
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S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
[Crossref] [PubMed]

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L. Thiberville, S. Moreno-Swirc, T. Vercauteren, E. Peltier, C. Cavé, and G. Bourg Heckly, “In vivo imaging of the bronchial wall microstructure using fibered confocal fluorescence microscopy,” Am. J. Respir. Crit. Care Med. 175(1), 22–31 (2007).
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M. G. Gräfe, A. Hoffmann, and C. Spielmann, “Ultrafast fluorescence spectroscopy for axial resolution of flurorophore distributions,” Appl. Phys. B 117(3), 833–840 (2014).
[Crossref]

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M. Smolla, M. Ruchty, M. Nagel, and C. J. Kleineidam, “Clearing pigmented insect cuticle to investigate small insects’ organs in situ using confocal laser-scanning microscopy (CLSM),” Arthropod Struct. Dev. 43(2), 175–181 (2014).
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V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13(1), 195–208 (2003).
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Front. Physiol. (1)

C. Vinegoni, S. Lee, A. D. Aguirre, and R. Weissleder, “New techniques for motion-artifact-free in vivo cardiac microscopy,” Front. Physiol. 6, 147 (2015).
[Crossref] [PubMed]

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R. Birngruber, U. Schmidt-Erfurth, S. Teschner, and J. Noack, “Confocal laser scanning fluorescence topography: a new method for three-dimensional functional imaging of vascular structures,” Graefes Arch. Clin. Exp. Ophthalmol. 238(7), 559–565 (2000).
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M. Dogan, A. Yalcin, S. Jain, M. B. Goldberg, A. K. Swan, M. S. Unlu, and B. B. Goldberg, “Spectral self-interference fluorescence microscopy for subcellular imaging,” IEEE J. Sel. Top. Quantum Electron. 14(1), 217–225 (2008).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

W. Loh, S. Yegnanarayanan, R. J. Ram, and P. W. Juodawlkis, “Unified theory of oscillator phase noise I: white noise,” IEEE Trans. Microw. Theory Tech. 61(6), 2371–2381 (2013).
[Crossref]

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F. C. Delori, C. K. Dorey, G. Staurenghi, O. Arend, D. G. Goger, and J. J. Weiter, “In vivo fluorescence of the ocular fundus exhibits retinal pigment epithelium lipofuscin characteristics,” Invest. Ophthalmol. Vis. Sci. 36(3), 718–729 (1995).
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D. M. de Bruin, R. H. Bremmer, V. M. Kodach, R. de Kinkelder, J. van Marle, T. G. van Leeuwen, and D. J. Faber, “Optical phantoms of varying geometry based on thin building blocks with controlled optical properties,” J. Biomed. Opt. 15(2), 025001 (2010).
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L. E. Meyer, N. Otberg, W. Sterry, and J. Lademann, “In vivo confocal scanning laser microscopy: comparison of the reflectance and fluorescence mode by imaging human skin,” J. Biomed. Opt. 11(4), 044012 (2006).
[Crossref] [PubMed]

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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).
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M. Rajadhyaksha, S. González, J. M. Zavislan, R. R. Anderson, and R. H. Webb, “In vivo confocal scanning laser microscopy of human skin II: advances in instrumentation and comparison with histology,” J. Invest. Dermatol. 113(3), 293–303 (1999).
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Q. Xu, K. Shi, S. Yin, and Z. Liu, “Chromatic two-photon excitation fluorescence imaging,” J. Microsc. 235(1), 79–83 (2009).
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S. Zill, S. F. Frazier, D. Neff, L. Quimby, M. Carney, R. DiCaprio, J. Thuma, and M. Norton, “Three-dimensional graphic reconstruction of the insect exoskeleton through confocal imaging of endogenous fluorescence,” Microsc. Res. Tech. 48(6), 367–384 (2000).
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Supplementary Material (1)

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Figures (9)

Fig. 1
Fig. 1

SIFM microscope setup. The microscope uses epi-fluorescence detection mode in which a dichroic mirror (DM) separates the excitation and emission optical paths. Telescopes are used to image the SIFM phase plate (PP) and two galvanometer scanners (HS&VS) onto the back focal plane of the microscope objective. The blue dashed frame denotes the Olympus IX71 inverted microscope platform. Component abbreviations: MO: microscope objective, VS: vertical galvanometer scanner, HS: horizontal galvanometer scanner, DM: dichroic mirror, BPF: band pass filter, LPF: long pass filter, PP: phase plate, SMF: single-mode fiber, Lx: lenses. Lens focal lengths: L1: 25 mm, L2: 200 mm, L3: 50 mm, L4: 2 × 60 mm, L5: 60 mm, L6: 180 mm. Inset: The propagation of the wavefront through the phase plate for the in (upper diagram) and out (lower diagram) of focus case, respectively. The wavefront curvature changes the OPD (black arrows) between the edge and center wavefront parts and induces a phase shift onto the SIFM self-interference signal.

Fig. 2
Fig. 2

Simulation of the SIFM method. (A) SIFM intensity spectra according to Eq. (8) for an in focus sample (blue) and a 100 µm axially displaced out of focus sample (red) measured with a 4 × microscope objective. (B) The SIFM phase ϕ(δ) as a function of depth as simulated by the exact model of Eqs. (14)-(15) in blue and for the linear approximation of Eq. (16) in red. (C) The normalized spectrally integrated intensities of Icenter(k,δ) in blue and Iedge(k,δ) in red that show the signal decay with depth for both wavefront parts.

Fig. 3
Fig. 3

Fiber-based Mach-Zehnder Interferometer (MZI) for SIFM detection with SPADs. The MZI generated three reference signals with a 120° phase-separation to create three DC output signals that alternately oscillated as a function of the SIFM signal spectral phase. SPAD photon counting digitized these output signals. Component abbreviations: G: glass plate, PC: polarization controller, p: optical path length difference, SPAD: single photon avalanche diode.

Fig. 4
Fig. 4

Simulated MZI signals. (A) An unmodulated Gaussian input spectrum (left) and the corresponding three MZI reference spectra as a function of k (right). (B) An input SIFM signal from an out of focus fluorescent sample with ϕ(δ) = -π/2 rad (left) and the corresponding output at the three SPADs (right). (C) The k-integrated SPAD intensity signals as a function of sample depth. (D) The integrated SPAD intensity signals divided by the axial PSF as a function of ϕ(δ) to show the unique combination of the SPAD intensities for every ϕ(δ). Plots (A) - (C) are normalized for the peak of their spectra. Visualization 1 provides a movie that dynamically shows the change in the SPAD signals with depth δ.

Fig. 5
Fig. 5

SIFM signal response as a function of axial position with the 4 × (A)-(C) and the 10 × (D)-(F) microscope objectives. (A)&(D) ISPAD(δ) signals for SPAD1 (blue), SPAD2 (red) and SPAD3 (green). (B)&(E) SIFM fluorescence intensity Iflu(δ). (C)&(F) SIFM phase Φ(δ) with the exact model phase of Eq. (15) in green and the linear phase model of Eq. (16) in red. The calibration allowed determination of the correct Iflu(δ) and Φ(δ) curves in the presence of small setup alignment errors. The calibration indicated relative error factors for the measurements with the 4 × objective as (o2/o1) = 0.88, (o3/o1) = 0.88, (a2/a1) = 0.85, (a3/a1) = 0.93, (φ2-φ1) = 0.31 rad, (φ3-φ1) = 0.55 rad; and for the 10 × objective as (o2/o1) = 1.01, (o3/o1) = 0.85, (a2/a1) = 0.67, (a3/a1) = 0.83, (φ2-φ1) = −0.20 rad, (φ3-φ1) = −0.11 rad.

Fig. 6
Fig. 6

(A) The intensity SNR of Iflu(δ) and (B) the phase noise performance of Φ(δ) as a function of integration time. In both plots the experimental data is shown in black dots, the theoretical shot noise limited performance is shown as the green line, and the theoretical optimal setup as the blue line. In (B) the red line shows the simplified phase noise model of Eq. (39).

Fig. 7
Fig. 7

SIFM imaging validation. (A) Schematic drawing of the sample consisting of a microscope slide, layers of double-sided tape and rectangular capillaries (c1, c2 and c3). (B) OCT cross-section image of the sample showing the capillaries as rectangles on top of the double-sided tape layers. The OCT image size is 0.93 mm × 2.0 mm (height × width) and the scale-bar indicates 100 µm. (C) SIFM intensity Iflu(δ). (D) SIFM phase Φ(δ). (E) SIFM depth δ. The imaged sample area for SIFM was 1.7 mm × 1.7 mm and the scale-bars in (C)-(E) indicate 200 µm.

Fig. 8
Fig. 8

SIFM imaging on a capillary blood vessel phantom with ICG-soaked diaper fibers. (A)&(B) were obtained with the 4 × microscope objective, while for (C)-(F) a 10 × microscope objective was used. (A), (C) and (E) show the SIFM Intensity Iflu(δ). (B), (D) and (F) show the SIFM depth δ. The scale-bars indicate 200 µm in (A)&(B) and 80 µm in (C)-(F). The locations of (C)-(D) and (E)-(F) are indicated in (A) by respectively red and blue dashed frames.

Fig. 9
Fig. 9

SIFM imaging on a crane fly. (A) White light photo of the fly over a 14.7 mm × 19.0 mm field-of-view. Scale-bar indicates 2 mm. (B) Zoomed white light photo over 2.6 mm × 2.6 mm that shows the fly’s head including the compound eyes and the small antennae. Scale-bar indicates 200 µm. (C)&(D) SIFM Iflu(δ) and depth δ measured with the 4 × objective over a 1.7 mm × 1.7 mm field-of-view of the fly’s head. Scale-bar indicates 200 µm. (E)&(F) SIFM Iflu(δ) and depth δ obtained with the 10 × objective over a 0.27 mm × 0.27 mm field-of-view of the left eye. Scale-bar indicates 30 µm. The inset image in (F) magnifies 4 facets for improved visualization of the depth difference between the facets and within the interfacet grooves.

Equations (39)

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R( δ )= m 2 f MO 2 δ
Δ( r,δ )= r 2 2R(δ)
E s (k,r,δ)= E 0 (k) e ikΔ(r,δ) = E 0 (k) e ik r 2 2R(δ)
E mf (k,r)=( kw 2π f L1 ) e ( rkw 2 f L1 ) 2
P(k)= e ikd
E center (k,δ)=2π 0 a E s (k,r,δ) E mf (k,r)rdr
E edge (k,δ)=2π a b E s (k,r,δ) E mf (k,r)P(k)rdr
I SIFM (k,δ)= E center (k,δ) E center * (k,δ)+ E edge (k,δ) E edge * (k,δ)+2Re{ E center (k,δ) E edge * (k,δ)} = I center (k,δ)+ I edge (k,δ)+ I interference (k,δ)
I center (k,δ)= I 0 (k)C(k,δ)( 1+ e k 2 w 2 a 2 2 f L1 2 2 e k 2 w 2 a 2 4 f L1 2 Cos( k a 2 2R(δ) ) )
C(k,δ)= 8π f L1 2 R (δ) 2 w 2 4 f L1 4 + k 2 R (δ) 2 w 4 .
I edge (k,δ)= I 0 (k)C(k,δ)( e k 2 w 2 a 2 2 f L1 2 + e k 2 w 2 b 2 2 f L1 2 2 e k 2 w 2 ( a 2 + b 2 ) 4 f L1 2 Cos( k a 2 k b 2 2R(δ) ) ),
I interference (k,δ)=2 I 0 (k)C(k,δ)( e k 2 w 2 a 2 2 f L1 2 Cos( dk )+ e k 2 w 2 a 2 4 f L1 2 Cos( dk k a 2 2R(δ) ) + e k 2 w 2 ( a 2 + b 2 ) 4 f L1 2 Cos( dk k( a 2 b 2 ) 2R(δ) ) e k 2 w 2 b 2 4 f L1 2 Cos( dk k b 2 2R(δ) ) ).
I SIFM (k,δ)= I center (k,δ)+ I edge (k,δ)+2 I center (k,δ) I edge (k,δ) Cos(Ψ(k,δ))
Ψ(k,δ)=Arccos( I interference (k,δ)/( 2 I center (k,δ) I edge (k,δ) ) ).
Ψ(k,δ)=kd+ϕ(δ).
ϕ(δ) k 0 ( a b Δ(r,δ) E mf ( k 0 )rdr a b E mf ( k 0 )rdr 0 a Δ(r,δ) E mf ( k 0 )rdr 0 a E mf ( k 0 )rdr ) ( k 0 2R(δ) )( a 2 1 e k 0 2 w 2 a 2 4 f L1 2 (ab)(a+b) 1 e k 0 2 w 2 ( b 2 a 2 ) 4 f L1 2 )
I SPADx (k,δ)=(1/3){ 1+Cos( kp(x1)2π/3 ) } I SIFM (k,δ)
I SPADx (k,δ)=(1/3){ 1+Cos( kp(x1)2π/3 ) } { I center (k,δ)+ I edge (k,δ)+2 I center (k,δ) I edge (k,δ) Cos(kd+ϕ(δ)) }.
Ι SPADx (δ)= I SPADx (k,δ)dk =(1/3){ I center ( k 0 ,δ)+ I edge ( k 0 ,δ) + I center ( k 0 ,δ) I edge ( k 0 ,δ) Cos(ϕ(δ)+(x1)2π/3) } I 0
I=MR
I=[ Ι SPAD1 (δ) Ι SPAD2 (δ) Ι SPAD3 (δ) ]
R=[ R 1 R 2 R 3 ]=[ I center ( k 0 ,δ)+ I edge ( k 0 ,δ) I center ( k 0 ,δ) I edge ( k 0 ,δ) cos(ϕ(δ)) I center ( k 0 ,δ) I edge ( k 0 ,δ) sin(ϕ(δ)) ](1/3) I 0 .
M=[ 1 1 0 1 cos(2π/3) sin(2π/3) 1 cos(-2π/3) sin(-2π/3) ].
R= M 1 I
M 1 =[ 1/3 1/3 1/3 2/3 -1/3 -1/3 0 1/ 3 -1/ 3 ].
Ι SPADx (δ)=(1/3){ o x ( I center ( k 0 ,δ)+ I edge ( k 0 ,δ) ) + a x I center ( k 0 ,δ) I edge ( k 0 ,δ) Cos(ϕ(δ)+(x1)2π/3+ φ x ) } I 0
R=[ R 1 R 2 R 3 ]=[ o 1 ( I center ( k 0 ,δ)+ I edge ( k 0 ,δ) ) a 1 I center ( k 0 ,δ) I edge ( k 0 ,δ) cos(ϕ(δ)+ φ 1 ) a 1 I center ( k 0 ,δ) I edge ( k 0 ,δ) sin(ϕ(δ)+ φ 1 ) ](1/3) I 0
M=[ 1 1 0 ( o 2 o 1 ) ( a 2 a 1 )cos(2π/3+ φ 2 φ 1 ) ( a 2 a 1 )sin(2π/3+ φ 2 φ 1 ) ( o 3 o 1 ) ( a 3 a 1 )cos(-2π/3+ φ 3 φ 1 ) ( a 3 a 1 )sin(-2π/3+ φ 3 φ 1 ) ].
I flu (δ)=3 R 1 = o 1 ( I center ( k 0 ,δ)+ I edge ( k 0 ,δ) ) I 0
Φ(δ)=atan2( R 3 , R 2 )=ϕ(δ)+ φ 1 .
D(δ)= R 2 2 + R 3 2 R 1 = a 1 I center ( k 0 ,δ) I edge ( k 0 ,δ) o 1 ( I center ( k 0 ,δ)+ I edge ( k 0 ,δ) ) .
S Ry = M y1 I ¯ SPAD1 + M y2 I ¯ SPAD2 + M y3 I ¯ SPAD3
n Ry = M y1 2 I ¯ SPAD1 + M y2 2 I ¯ SPAD2 + M y3 2 I ¯ SPAD3 .
SN R Ry = S Ry n Ry
SN R I flu ( δ ) =SN R R1 = S R1 n R1 .
n Q = ( sin(Φ(δ)) n R2 ) 2 + ( cos(Φ(δ)) n R3 ) 2 ρ( n R2 , n R3 ) n R2 n R3 sin(2Φ(δ))
ρ( R 2 , R 3 )= Cov( R 2 , R 3 ) n R2 n R3 = n=1 3 m=1 3 M 2n M 3m Cov( I SPADn , I SPADm ) n R2 n R3 .
σ Φ(δ) = n Q S Q .
σ Φ(δ) = 2 1/2 SN R Q 1

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