Abstract

In this study, we demonstrated a full-range space-division multiplexing optical coherence tomography (FR-SDM-OCT) system. Utilizing the galvanometer-based phase modulation full-range technique, the total imaging range of FR-SDM-OCT can be extended to >20 mm in tissue, with a digitizer sampling rate of 500 MS/s and a laser sweeping rate of 100 kHz. Complex conjugate terms were suppressed in FR-SDM-OCT images with a measured rejection ratio of up to ∼46 dB at ∼1.4 mm depth and ∼30 dB at ∼19.4 mm depth. The feasibility of FR-SDM-OCT was validated by imaging Scotch tapes and human fingernails. Furthermore, we demonstrated the feasibility of FR-SDM-OCT angiography (FR-SDM-OCTA) to perform simultaneous acquisition of human fingernail angiograms from four positions, with a total field-of-view of ∼1.7 mm × ∼7.5 mm. Employing the full-range technique in SDM-OCT can effectively alleviate hardware requirements to achieve the long depth measurement range, which is required by SDM-OCT to separate multiple images at different sample locations. FR-SDM-OCTA creates new opportunities to apply SDM-OCT to obtain wide-field angiography of in vivo tissue samples free of labeling.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

J. Jerwick, Y. Huang, Z. Dong, A. Slaudades, A. J. Brucker, and C. Zhou, “Wide-field Ophthalmic Space-Division Multiplexing Optical Coherence Tomography,” Photonics Res. 8(4), 539–547 (2020).
[Crossref]

2018 (2)

S. Moon and Z. Chen, “Phase-stability optimization of swept-source optical coherence tomography,” Biomed. Opt. Express 9(11), 5280–5295 (2018).
[Crossref]

S. Song, J. Xu, and R. K. Wang, “Flexible wide-field optical micro-angiography based on Fourier-domain multiplexed dual-beam swept source optical coherence tomography,” J. Biophotonics 11(3), e201700203 (2018).
[Crossref]

2017 (3)

2016 (2)

Z. Wang, B. Potsaid, L. Chen, C. Doerr, H.-C. Lee, T. Nielson, V. Jayaraman, A. E. Cable, E. Swanson, and J. G. Fujimoto, “Cubic meter volume optical coherence tomography,” Optica 3(12), 1496–1503 (2016).
[Crossref]

H. Kawagoe, M. Yamanaka, S. Makita, Y. Yasuno, and N. Nishizawa, “Full-range ultrahigh-resolution spectral-domain optical coherence tomography in 1.7 µm wavelength region for deep-penetration and high-resolution imaging of turbid tissues,” Appl. Phys. Express 9(12), 127002 (2016).
[Crossref]

2015 (3)

2013 (3)

2012 (1)

2011 (2)

2010 (2)

2009 (1)

2008 (2)

2007 (3)

2006 (2)

2005 (2)

2004 (3)

2002 (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]

Alex, A.

An, L.

Aoki, G.

Badar, M.

Baran, U.

U. Baran, L. Shi, and R. K. Wang, “Capillary blood flow imaging within human finger cuticle using optical microangiography,” J. Biophotonics 8(1-2), 46–51 (2015).
[Crossref]

Baumann, B.

Bini, A.

S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
[Crossref]

Bonesi, M.

Bouma, B. E.

Brucker, A. J.

J. Jerwick, Y. Huang, Z. Dong, A. Slaudades, A. J. Brucker, and C. Zhou, “Wide-field Ophthalmic Space-Division Multiplexing Optical Coherence Tomography,” Photonics Res. 8(4), 539–547 (2020).
[Crossref]

Cable, A. E.

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]

Chen, L.

Chen, Z.

Choi, W.

Choma, M. A.

Congiu, T.

S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
[Crossref]

de Boer, J. F.

Dhalla, A.-H.

Doerr, C.

Dong, Z.

J. Jerwick, Y. Huang, Z. Dong, A. Slaudades, A. J. Brucker, and C. Zhou, “Wide-field Ophthalmic Space-Division Multiplexing Optical Coherence Tomography,” Photonics Res. 8(4), 539–547 (2020).
[Crossref]

Drexler, W.

Duker, J. S.

Endo, T.

Fabritius, T.

Fercher, A. F.

Fienup, J. R.

Flotte, T.

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]

Fraser, S.

Fujimoto, J.

Fujimoto, J. G.

Ghaemi, A.

Goetzinger, E.

Götzinger, E.

Gregory, K.

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]

Grulkowski, I.

Guizar-Sicairos, M.

Hee, M. R.

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]

Hendargo, H. C.

Hermann, B.

Hitzenberger, C. K.

Hofer, B.

Huang, D.

Huang, Y.

J. Jerwick, Y. Huang, Z. Dong, A. Slaudades, A. J. Brucker, and C. Zhou, “Wide-field Ophthalmic Space-Division Multiplexing Optical Coherence Tomography,” Photonics Res. 8(4), 539–547 (2020).
[Crossref]

Y. Huang, M. Badar, A. Nitkowski, A. Weinroth, N. Tansu, and C. Zhou, “Wide-field high-speed space-division multiplexing optical coherence tomography using an integrated photonic device,” Biomed. Opt. Express 8(8), 3856–3867 (2017).
[Crossref]

Huber, R.

Huo, L.

Iftimia, N.

N. Iftimia, B. E. Bouma, and G. J. Tearney, Speckle reduction in optical coherence tomography by “path length encoded” angular compounding (SPIE, 2003), Vol. 8, pp. 260–263, 264.

Itoh, M.

Izatt, J. A.

Jayaraman, V.

Jerwick, J.

J. Jerwick, Y. Huang, Z. Dong, A. Slaudades, A. J. Brucker, and C. Zhou, “Wide-field Ophthalmic Space-Division Multiplexing Optical Coherence Tomography,” Photonics Res. 8(4), 539–547 (2020).
[Crossref]

Karnowski, K.

Q. Li, K. Karnowski, M. Villiger, and D. D. Sampson, Local birefringence of the anterior segment of the human eye in a single capture with a full range polarisation-sensitive optical coherence tomography, International Conference on Biophotonics V (SPIE, 2017), Vol. 10340, p. 1.

Kawagoe, H.

H. Kawagoe, M. Yamanaka, S. Makita, Y. Yasuno, and N. Nishizawa, “Full-range ultrahigh-resolution spectral-domain optical coherence tomography in 1.7 µm wavelength region for deep-penetration and high-resolution imaging of turbid tissues,” Appl. Phys. Express 9(12), 127002 (2016).
[Crossref]

Klein, T.

Kowalczyk, A.

Lasser, T.

Lee, H.-C.

Leitgeb, R.

Leitgeb, R. A.

Li, J.

Li, Q.

Q. Li, K. Karnowski, M. Villiger, and D. D. Sampson, Local birefringence of the anterior segment of the human eye in a single capture with a full range polarisation-sensitive optical coherence tomography, International Conference on Biophotonics V (SPIE, 2017), Vol. 10340, p. 1.

Li, X. D.

Lim, Y.

Lin, C. P.

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]

Liu, J. J.

Lu, C. D.

Ma, Y.

Makita, S.

Manelli, A.

S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
[Crossref]

Matz, G.

McNabb, R. P.

Michaely, R.

Moon, S.

Motaghiannezam, R.

Nelson, J. S.

Nielsen, T.

Nielson, T.

Nishizawa, N.

H. Kawagoe, M. Yamanaka, S. Makita, Y. Yasuno, and N. Nishizawa, “Full-range ultrahigh-resolution spectral-domain optical coherence tomography in 1.7 µm wavelength region for deep-penetration and high-resolution imaging of turbid tissues,” Appl. Phys. Express 9(12), 127002 (2016).
[Crossref]

Nitkowski, A.

Park, S. Y.

Pilato, G.

S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
[Crossref]

Pircher, M.

Potsaid, B.

Považay, B.

Puliafito, C. 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).
[Crossref]

Rasakanthan, J.

Raspanti, M.

S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
[Crossref]

Reguzzoni, M.

S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
[Crossref]

Sampson, D. D.

Q. Li, K. Karnowski, M. Villiger, and D. D. Sampson, Local birefringence of the anterior segment of the human eye in a single capture with a full range polarisation-sensitive optical coherence tomography, International Conference on Biophotonics V (SPIE, 2017), Vol. 10340, p. 1.

Sangiorgi, S.

S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
[Crossref]

Sarunic, M. V.

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]

Sekhar, S. C.

Shepherd, N.

Shi, L.

U. Baran, L. Shi, and R. K. Wang, “Capillary blood flow imaging within human finger cuticle using optical microangiography,” J. Biophotonics 8(1-2), 46–51 (2015).
[Crossref]

Siddiqui, M.

Slaudades, A.

J. Jerwick, Y. Huang, Z. Dong, A. Slaudades, A. J. Brucker, and C. Zhou, “Wide-field Ophthalmic Space-Division Multiplexing Optical Coherence Tomography,” Photonics Res. 8(4), 539–547 (2020).
[Crossref]

Song, S.

S. Song, J. Xu, and R. K. Wang, “Flexible wide-field optical micro-angiography based on Fourier-domain multiplexed dual-beam swept source optical coherence tomography,” J. Biophotonics 11(3), e201700203 (2018).
[Crossref]

Stinson, W. G.

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]

Sumimura, H.

Swanson, E.

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).
[Crossref]

Tansu, N.

Tearney, G. J.

Thurman, S. T.

Torzicky, T.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt. 18(11), 116010 (2013).
[Crossref]

S. Zotter, M. Pircher, T. Torzicky, M. Bonesi, E. Goetzinger, R. A. Leitgeb, and C. K. Hitzenberger, “Visualization of microvasculature by dual-beam phase-resolved Doppler optical coherence tomography,” Opt. Express 19(2), 1217–1227 (2011).
[Crossref]

Tozburun, S.

Trasischker, W.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt. 18(11), 116010 (2013).
[Crossref]

Unterhuber, A.

Vakoc, B. J.

Vermeulen, D.

Villiger, M.

Q. Li, K. Karnowski, M. Villiger, and D. D. Sampson, Local birefringence of the anterior segment of the human eye in a single capture with a full range polarisation-sensitive optical coherence tomography, International Conference on Biophotonics V (SPIE, 2017), Vol. 10340, p. 1.

Wang, R. K.

S. Song, J. Xu, and R. K. Wang, “Flexible wide-field optical micro-angiography based on Fourier-domain multiplexed dual-beam swept source optical coherence tomography,” J. Biophotonics 11(3), e201700203 (2018).
[Crossref]

U. Baran, L. Shi, and R. K. Wang, “Capillary blood flow imaging within human finger cuticle using optical microangiography,” J. Biophotonics 8(1-2), 46–51 (2015).
[Crossref]

L. An and R. K. Wang, “Use of a scanner to modulate spatial interferograms for in vivo full-range Fourier-domain optical coherence tomography,” Opt. Lett. 32(23), 3423–3425 (2007).
[Crossref]

Wang, Z.

Weinroth, A.

Werkmeister, R. M.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt. 18(11), 116010 (2013).
[Crossref]

Wojtkowski, M.

Xi, J.

Xu, J.

S. Song, J. Xu, and R. K. Wang, “Flexible wide-field optical micro-angiography based on Fourier-domain multiplexed dual-beam swept source optical coherence tomography,” J. Biophotonics 11(3), e201700203 (2018).
[Crossref]

Yamanaka, M.

H. Kawagoe, M. Yamanaka, S. Makita, Y. Yasuno, and N. Nishizawa, “Full-range ultrahigh-resolution spectral-domain optical coherence tomography in 1.7 µm wavelength region for deep-penetration and high-resolution imaging of turbid tissues,” Appl. Phys. Express 9(12), 127002 (2016).
[Crossref]

Yamanari, M.

Yang, C.

Yasuno, Y.

Yatagai, T.

Yun, S. H.

Zhang, E. Z.

Zhang, J. G.

Zhou, C.

Zotter, S.

W. Trasischker, R. M. Werkmeister, S. Zotter, B. Baumann, T. Torzicky, M. Pircher, and C. K. Hitzenberger, “In vitro and in vivo three-dimensional velocity vector measurement by three-beam spectral-domain Doppler optical coherence tomography,” J. Biomed. Opt. 18(11), 116010 (2013).
[Crossref]

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Appl. Opt. (1)

Appl. Phys. Express (1)

H. Kawagoe, M. Yamanaka, S. Makita, Y. Yasuno, and N. Nishizawa, “Full-range ultrahigh-resolution spectral-domain optical coherence tomography in 1.7 µm wavelength region for deep-penetration and high-resolution imaging of turbid tissues,” Appl. Phys. Express 9(12), 127002 (2016).
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R. Motaghiannezam and S. Fraser, “Logarithmic intensity and speckle-based motion contrast methods for human retinal vasculature visualization using swept source optical coherence tomography,” Biomed. Opt. Express 3(3), 503–521 (2012).
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T. Klein and R. Huber, “High-speed OCT light sources and systems [Invited],” Biomed. Opt. Express 8(2), 828–859 (2017).
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J. F. de Boer, R. Leitgeb, and M. Wojtkowski, “Twenty-five years of optical coherence tomography: the paradigm shift in sensitivity and speed provided by Fourier domain OCT [Invited],” Biomed. Opt. Express 8(7), 3248–3280 (2017).
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Y. Huang, M. Badar, A. Nitkowski, A. Weinroth, N. Tansu, and C. Zhou, “Wide-field high-speed space-division multiplexing optical coherence tomography using an integrated photonic device,” Biomed. Opt. Express 8(8), 3856–3867 (2017).
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S. Sangiorgi, A. Manelli, T. Congiu, A. Bini, G. Pilato, M. Reguzzoni, and M. Raspanti, “Microvascularization of the human digit as studied by corrosion casting,” J. Anat. 204(2), 123–131 (2004).
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S. Song, J. Xu, and R. K. Wang, “Flexible wide-field optical micro-angiography based on Fourier-domain multiplexed dual-beam swept source optical coherence tomography,” J. Biophotonics 11(3), e201700203 (2018).
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B. Baumann, M. Pircher, E. Götzinger, and C. K. Hitzenberger, “Full range complex spectral domain optical coherence tomography without additional phase shifters,” Opt. Express 15(20), 13375–13387 (2007).
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C. Zhou, A. Alex, J. Rasakanthan, and Y. Ma, “Space-division multiplexing optical coherence tomography,” Opt. Express 21(16), 19219–19227 (2013).
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M. Siddiqui, S. Tozburun, E. Z. Zhang, and B. J. Vakoc, “Compensation of spectral and RF errors in swept-source OCT for high extinction complex demodulation,” Opt. Express 23(5), 5508–5520 (2015).
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S. Makita, T. Fabritius, and Y. Yasuno, “Full-range, high-speed, high-resolution 1-µm spectral-domain optical coherence tomography using BM-scan for volumetric imaging of the human posterior eye,” Opt. Express 16(12), 8406–8420 (2008).
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Figures (8)

Fig. 1.
Fig. 1. (A) Schematics of Full-range Space-division Multiplexing OCT (FR-SDM-OCT). (B) Schematic illustration of galvo-based full-range technique. Only one incident beam is shown. Red: original path. Magenta: Offset path. (C) Realization of beam offsetting (s) in FR-SDM-OCT system. BOA: Booster optical amplifier; C: Collimator; DBD: Dual balanced detector; FBG: Fiber Bragg grating; Galvo: Galvanometer; L1, L2: Lens; LS: light source; MZI: Mach-Zehnder interferometer; Obj.: Objective; Ref.: Reference arm; Samp.: Sample arm; SDM: Space-division multiplexing component.
Fig. 2.
Fig. 2. Flow chart of the post-processing procedures for standard SDM-OCT (A), additional step for full-range SDM-OCT (B), 3D rendering (C), and OCT angiography (D).
Fig. 3.
Fig. 3. (A-B) Illustration of trigger jitter in MZI signals from different laser sweeps (A-scans). (B) is the zoomed view of the red rectangle region in (A). (C) A screenshot of representative OCT (pink) and MZI (green) fringes, using two FBG peaks in the MZI channel for a correct phase calibration.
Fig. 4.
Fig. 4. Mirror images from standard and full-range SDM-OCT. Sampling rate was 1.0 GS/s. (A) and (C) were four-channel mirror images taken from a standard or a full-range SDM-OCT system. (B) and (D) were intensity profiles across the center rows of (A) and (C). Complex conjugate rejection ratio (CCRR, in dB) of the 1st and the 4th peaks were shown in (B) and (D). Scale bar: 1 mm.
Fig. 5.
Fig. 5. Experimental observation of phase modulation as function of various parameters, including beam offsets (s), transverse scan range xM and number of effective A-scans M. (A - B) Phase modulation at various s. xM = ∼1.5 mm. M = 750. (A) Center rows of FFTx spectra at various s. (B) Linear fitting between normalized modulation frequency u and s. (C - D) Phase modulation at various xM. s = ∼3 mm. M = 750. (C) Center rows of FFTx spectra at various xM. (D) Linear fitting between u and xM. (E - F) Phase modulation at various M. s = ∼3 mm. xM = ∼1.5 mm. (E) Center rows of FFTx spectra as a function of M. (F) Linear fitting between u and 1/M.
Fig. 6.
Fig. 6. A comparison between standard and full-range SDM-OCT images of a Scotch tape. Sampling rate was 0.5 GS/s (half sampling rate). (A) 1×4 standard SDM-OCT image, showing overlapping of sample and complex-conjugated images. (B) 1×4 FR-SDM-OCT image, showing clear tape structure with significant suppression of complex conjugate images. (C, D) 3× zoomed views of tape images from the 3rd beam in (A) and (B). Red and magenta labels on the right side of each sub-figure indicated the channels of the sample images and their complex conjugates. White arrows: residual complex conjugate signals. Scale bar: 1 mm for (A - B) and 0.5 mm for (C, D).
Fig. 7.
Fig. 7. FR-SDM-OCT images of the human fingernail in vivo. Sampling rate was 0.5 GS/s. The full FR-SDM-OCT image were shown in (A). Four fingernail images were displayed separately in a total depth range of ∼21 mm. (B -E) 2× zoomed images of each beam from (A). Proximal nail fold (PNF), nail junction (NJ), dermal/epidermal junction (DEJ), cuticle, separation of nail plate (NP) and nail bed (NF) were clearly visible. Yellow and orange arrows in (A) and (B): artifacts. (F) 3D rendering of scanned region of the fingernail. 0 in red: Zero delay. Scale bar: 2 mm for (A) and 1 mm for (B-E).
Fig. 8.
Fig. 8. FR-SDM-OCT angiograms of the human fingernail junction. (A) A photograph of volunteer’s index finger. Four parallel channels were shown in the white rectangular box, covering a scan area of ∼1.7 mm × ∼7.5 mm. (B) Stitched, cross-sectional FR-SDM-OCT image of the fingernail junction, with the corresponding angiogram overlaid on top of the structural image in the red channel. (C) Depth-encoded FR-SDM-OCT angiograms of the fingernail, merged from a depth range of 200-1200 µm below the surface. (D–H) Five representative FR-SDM-OCT angiograms showing the microvascular networks at different layers. (I-M) Zoomed microvascular layers in the proximal nail fold and nail bed. PL: papillary layer; SPL: sub-papillary layer; RL: reticular layer; PPL: pseudo-papillary layer. Scale bars: (B - H) 500 µm; (I - M) 200 µm

Equations (9)

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

β = 2 Δ α
I ( k , x ) S ( k , x ) n ( α r α n cos ( 2 z n k ) )
Δ z 2 s sin α Δ α 1 sin α = s β sin 2 α
Δ z 2 β s
I ( k , x ) S ( k , x ) n ( α r α n cos ( 2 z n k + Φ ( x ) ) )
Φ ( M ) = 2 k Δ z = 4 π s 2 β λ 4 π s x M λ F
Φ ( M ) = u M
u = Φ ( M ) M 4 π s x M λ F M
cos ( 2 z n k + Φ ( x ) ) = 1 2 [ exp ( i 2 z n k ) exp ( i Φ ( x ) ) + exp ( i 2 z n k ) exp ( i Φ ( x ) ) ]