Abstract

A confocal reflectance theta line-scanning microscope demonstrates imaging of nuclear and cellular morphology in human skin and oral mucosa in vivo. The illumination and detection are through a divided objective lens pupil, resulting in a theta-microscope configuration. A line is directly scanned in the pupil and descanned onto a linear detector array such that the theta line scanner consists of only seven main optical components. The experimentally measured lateral resolution is 1.0  μm and optical section thickness is 1.7   μm under nominal conditions at 830   nm wavelength. Through full-thickness human epidermis (i.e., in the dermis) the measured lateral resolution is 1.7   μm and the optical section thickness is 9.2   μm. The lateral resolution, sectioning, and image quality in epidermal (epithelial) tissue is comparable to that of point scanning confocal microscopy.

© 2007 Optical Society of America

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  1. K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2006 (1)

2005 (1)

2004 (2)

T. D. Wang, M. J. Mandella, N. Y. Chan, and G. S. Kino, "Confocal fluorescence microscope with dual-axis architecture and biaxial postobjective scanning," J. Biomed. Opt. 9, 735-742 (2004).
[CrossRef] [PubMed]

A. R. Rouse, A. Kano, J. A. Udovich, S. M. Kroto, and A. F. Gmitro, "Design and demonstration of a miniature catheter for a confocal microendoscope," Appl. Opt. 43, 5763-5771 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (1)

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (4)

1995 (1)

M. Rajadhyaksha, S. Gonzalez, J. M. Zavislan, R. R. Anderson, and R. H. Webb, "In vivo confocal scanning laser microscopy of human skin II: melanin provides strong contrast," J Investig. Dermatol. 104, 946-952 (1995).
[CrossRef] [PubMed]

1994 (1)

E. H. K. Stelzer and S. Lindek, "Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy," Opt. Commun. 111, 536-547 (1994).
[CrossRef]

1993 (2)

A. F. Gmitro and D. Aziz, "Confocal microscopy through a fiber-optic imaging bundle," Opt. Lett. 18, 565-567 (1993).
[CrossRef] [PubMed]

P. Corcuff, C. Betrand, and J. L. Leveque, "Morphometry of human epidermis in vivo by real-time confocal microscopy," Arch. Dermatol. Res. 285, 475-481 (1993).
[CrossRef] [PubMed]

1992 (1)

G. J. Brakenhoff and K. Visscher, "Confocal imaging with bilateral scanning and array detectors," J. Micros. 165, 139-146 (1992).
[CrossRef]

1991 (1)

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

1990 (1)

C. J. Koester, "High efficiency optical sectioning with confocal slits," Trans. R. Microsc. Soc. 90, 327-332 (1990).

1980 (1)

1975 (1)

R. A. Laing, M. M. Sandstrom, and H. M. Leibowitz, "In vivo photomicrography of the corneal endothelium," Arch. Ophthalmol. 93, 143-145 (1975).
[CrossRef] [PubMed]

1974 (1)

D. M. Maurice, "A scanning slit optical microscope," Ophthalmologie 13, 1033-1037 (1974).

Anderson, R. R.

W. M. White, M. Rajadhyaksha, S. Gonzalez, R. L. Fabian, and R. R. Anderson, "Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy," Laryngoscope 109, 1709-1717 (1999).
[CrossRef] [PubMed]

M. Rajadhyaksha, R. R. Anderson, and R. H. Webb, "Video-rate confocal scanning laser microscope for imaging human tissues in vivo," Appl. Opt. 38, 2105-2115 (1999).
[CrossRef]

M. Rajadhyaksha, S. Gonzalez, J. M. Zavislan, R. R. Anderson, and R. H. Webb, "In vivo confocal scanning laser microscopy of human skin II: melanin provides strong contrast," J Investig. Dermatol. 104, 946-952 (1995).
[CrossRef] [PubMed]

Aziz, D.

Betrand, C.

P. Corcuff, C. Betrand, and J. L. Leveque, "Morphometry of human epidermis in vivo by real-time confocal microscopy," Arch. Dermatol. Res. 285, 475-481 (1993).
[CrossRef] [PubMed]

Boyde, A.

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Brakenhoff, G. J.

G. J. Brakenhoff and K. Visscher, "Confocal imaging with bilateral scanning and array detectors," J. Micros. 165, 139-146 (1992).
[CrossRef]

Cavanagh, H. D.

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Chan, N. Y.

T. D. Wang, M. J. Mandella, N. Y. Chan, and G. S. Kino, "Confocal fluorescence microscope with dual-axis architecture and biaxial postobjective scanning," J. Biomed. Opt. 9, 735-742 (2004).
[CrossRef] [PubMed]

T. D. Wang, C. H. Contag, M. J. Mandella, N. Y. Chan, and G. S. Kino, "Dual-axis confocal microscope with post-objective scanning and low conherence heterodyne detection," Opt. Lett. 28, 1915-1917 (2003).
[CrossRef] [PubMed]

Collier, T.

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

Contag, C. H.

Corcuff, P.

P. Corcuff, C. Betrand, and J. L. Leveque, "Morphometry of human epidermis in vivo by real-time confocal microscopy," Arch. Dermatol. Res. 285, 475-481 (1993).
[CrossRef] [PubMed]

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Descour, M.

K. B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. Descour, "Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo," Opt. Express 11, 3171-3181 (2003).
[CrossRef] [PubMed]

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

DiMarzio, C. A.

Donaldson, L.

Dwyer, P. J.

Fabian, R. L.

W. M. White, M. Rajadhyaksha, S. Gonzalez, R. L. Fabian, and R. R. Anderson, "Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy," Laryngoscope 109, 1709-1717 (1999).
[CrossRef] [PubMed]

Follen, M.

K. B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. Descour, "Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo," Opt. Express 11, 3171-3181 (2003).
[CrossRef] [PubMed]

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

Fox, W. J.

Gmitro, A. F.

Gonzalez, S.

W. M. White, M. Rajadhyaksha, S. Gonzalez, R. L. Fabian, and R. R. Anderson, "Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy," Laryngoscope 109, 1709-1717 (1999).
[CrossRef] [PubMed]

M. Rajadhyaksha, S. Gonzalez, J. M. Zavislan, R. R. Anderson, and R. H. Webb, "In vivo confocal scanning laser microscopy of human skin II: melanin provides strong contrast," J Investig. Dermatol. 104, 946-952 (1995).
[CrossRef] [PubMed]

Han, S.

Hopkins, M. F.

Im, K.

Jester, J. V.

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Kano, A.

Kim, B.

Kim, D.

Kino, G. S.

Koester, C. J.

C. J. Koester, "High efficiency optical sectioning with confocal slits," Trans. R. Microsc. Soc. 90, 327-332 (1990).

C. J. Koester, "Scanning mirror microscope with optical sectioning characteristics: applications to ophthalmology," Appl. Opt. 19, 1749-1757 (1980).
[CrossRef] [PubMed]

Kroto, S. M.

Laing, R. A.

R. A. Laing, M. M. Sandstrom, and H. M. Leibowitz, "In vivo photomicrography of the corneal endothelium," Arch. Ophthalmol. 93, 143-145 (1975).
[CrossRef] [PubMed]

Leibowitz, H. M.

R. A. Laing, M. M. Sandstrom, and H. M. Leibowitz, "In vivo photomicrography of the corneal endothelium," Arch. Ophthalmol. 93, 143-145 (1975).
[CrossRef] [PubMed]

Lemp, M. A.

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Leveque, J. L.

P. Corcuff, C. Betrand, and J. L. Leveque, "Morphometry of human epidermis in vivo by real-time confocal microscopy," Arch. Dermatol. Res. 285, 475-481 (1993).
[CrossRef] [PubMed]

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Liang, C.

K. B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. Descour, "Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo," Opt. Express 11, 3171-3181 (2003).
[CrossRef] [PubMed]

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

Lindek, S.

E. H. K. Stelzer and S. Lindek, "Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy," Opt. Commun. 111, 536-547 (1994).
[CrossRef]

Malpica, A.

K. B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. Descour, "Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo," Opt. Express 11, 3171-3181 (2003).
[CrossRef] [PubMed]

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

Mandella, M. J.

Martin, L.

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Maurice, D. M.

D. M. Maurice, "A scanning slit optical microscope," Ophthalmologie 13, 1033-1037 (1974).

New, K. C.

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Park, H.

Petroll, W. M.

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Rajadhyaksha, M.

P. J. Dwyer, C. A. DiMarzio, J. M. Zavislan, W. J. Fox, and M. Rajadhyaksha, "Confocal reflectance theta line-scanning microscope for imaging human skin in vivo," Opt. Lett. 31, 942-944 (2006).
[CrossRef] [PubMed]

M. Rajadhyaksha, R. R. Anderson, and R. H. Webb, "Video-rate confocal scanning laser microscope for imaging human tissues in vivo," Appl. Opt. 38, 2105-2115 (1999).
[CrossRef]

W. M. White, M. Rajadhyaksha, S. Gonzalez, R. L. Fabian, and R. R. Anderson, "Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy," Laryngoscope 109, 1709-1717 (1999).
[CrossRef] [PubMed]

M. Rajadhyaksha, S. Gonzalez, J. M. Zavislan, R. R. Anderson, and R. H. Webb, "In vivo confocal scanning laser microscopy of human skin II: melanin provides strong contrast," J Investig. Dermatol. 104, 946-952 (1995).
[CrossRef] [PubMed]

Richards-Kortum, R.

K. B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. Descour, "Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo," Opt. Express 11, 3171-3181 (2003).
[CrossRef] [PubMed]

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

Rogomentich, F.

Rouse, A. R.

Sabharwal, Y. S.

Sandstrom, M. M.

R. A. Laing, M. M. Sandstrom, and H. M. Leibowitz, "In vivo photomicrography of the corneal endothelium," Arch. Ophthalmol. 93, 143-145 (1975).
[CrossRef] [PubMed]

Stelzer, E. H. K.

E. H. K. Stelzer and S. Lindek, "Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy," Opt. Commun. 111, 536-547 (1994).
[CrossRef]

Sung, K. B.

K. B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. Descour, "Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo," Opt. Express 11, 3171-3181 (2003).
[CrossRef] [PubMed]

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

Udovich, J. A.

Visscher, K.

G. J. Brakenhoff and K. Visscher, "Confocal imaging with bilateral scanning and array detectors," J. Micros. 165, 139-146 (1992).
[CrossRef]

Wang, T. D.

Webb, R. H.

White, W. M.

W. M. White, M. Rajadhyaksha, S. Gonzalez, R. L. Fabian, and R. R. Anderson, "Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy," Laryngoscope 109, 1709-1717 (1999).
[CrossRef] [PubMed]

Zavislan, J. M.

P. J. Dwyer, C. A. DiMarzio, J. M. Zavislan, W. J. Fox, and M. Rajadhyaksha, "Confocal reflectance theta line-scanning microscope for imaging human skin in vivo," Opt. Lett. 31, 942-944 (2006).
[CrossRef] [PubMed]

M. Rajadhyaksha, S. Gonzalez, J. M. Zavislan, R. R. Anderson, and R. H. Webb, "In vivo confocal scanning laser microscopy of human skin II: melanin provides strong contrast," J Investig. Dermatol. 104, 946-952 (1995).
[CrossRef] [PubMed]

Appl. Opt. (5)

Arch. Dermatol. Res. (1)

P. Corcuff, C. Betrand, and J. L. Leveque, "Morphometry of human epidermis in vivo by real-time confocal microscopy," Arch. Dermatol. Res. 285, 475-481 (1993).
[CrossRef] [PubMed]

Arch. Ophthalmol. (1)

R. A. Laing, M. M. Sandstrom, and H. M. Leibowitz, "In vivo photomicrography of the corneal endothelium," Arch. Ophthalmol. 93, 143-145 (1975).
[CrossRef] [PubMed]

J Investig. Dermatol. (1)

M. Rajadhyaksha, S. Gonzalez, J. M. Zavislan, R. R. Anderson, and R. H. Webb, "In vivo confocal scanning laser microscopy of human skin II: melanin provides strong contrast," J Investig. Dermatol. 104, 946-952 (1995).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

T. D. Wang, M. J. Mandella, N. Y. Chan, and G. S. Kino, "Confocal fluorescence microscope with dual-axis architecture and biaxial postobjective scanning," J. Biomed. Opt. 9, 735-742 (2004).
[CrossRef] [PubMed]

J. Micros. (1)

G. J. Brakenhoff and K. Visscher, "Confocal imaging with bilateral scanning and array detectors," J. Micros. 165, 139-146 (1992).
[CrossRef]

J. Microsc. (1)

K. B. Sung, C. Liang, M. Descour, T. Collier, M. Follen, A. Malpica, and R. Richards-Kortum, "Near real time in vivo fibre optic confocal microscopy: sub-cellular structure resolved," J. Microsc. 207, 137-145 (2002).
[CrossRef] [PubMed]

Laryngoscope (1)

W. M. White, M. Rajadhyaksha, S. Gonzalez, R. L. Fabian, and R. R. Anderson, "Noninvasive imaging of human oral mucosa in vivo by confocal reflectance microscopy," Laryngoscope 109, 1709-1717 (1999).
[CrossRef] [PubMed]

Ophthalmologie (1)

D. M. Maurice, "A scanning slit optical microscope," Ophthalmologie 13, 1033-1037 (1974).

Opt. Commun. (1)

E. H. K. Stelzer and S. Lindek, "Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy," Opt. Commun. 111, 536-547 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Scanning (1)

K. C. New, W. M. Petroll, A. Boyde, L. Martin, P. Corcuff, J. L. Leveque, M. A. Lemp, H. D. Cavanagh, and J. V. Jester, "In Vivo imaging of human teeth and skin using real-time confocal microscopy," Scanning 13, 369-372 (1991).
[CrossRef]

Trans. R. Microsc. Soc. (1)

C. J. Koester, "High efficiency optical sectioning with confocal slits," Trans. R. Microsc. Soc. 90, 327-332 (1990).

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Figures (10)

Fig. 1
Fig. 1

Optical design of the confocal theta line-scanning microscope. The expanded view on the left is the rear view of the divided objective lens pupil.

Fig. 2
Fig. 2

Optical path of the beam through the cylindrical lens ( C L 1 ) and the objective lens ( L 3 ) in two planes illustrating the primary and secondary lines.

Fig. 3
Fig. 3

Video timing and control electronics for the linear CMOS detector, galvanometric scanner, and frame grabber.

Fig. 4
Fig. 4

(Color online) Schematic of the objective lens pupil, divided into an illumination pupil and a detection pupil (4A). The illumination and detection paths intersect at a half-angle θ. (B) The actual D-shaped pupils are modeled as circular pupils of equivalent area. The intersection of the illumination and detection LSFs at the focus is shown as an expanded view in (B).

Fig. 5
Fig. 5

Predicted diffraction-limited lateral resolution and section thickness versus the ratio of the divider strip half-width to radius of the objective lens pupil ( u / r ) .

Fig. 6
Fig. 6

Axial theta LSF plots with a detection slit width of (A) 5   μm and (B) 50   μm under nominal instrumental conditions (—) and scattering and aberrating conditions through full-thickness human epidermis (- - -).

Fig. 7
Fig. 7

Optical section thickness versus detection slit width under nominal instrument conditions (—) and under scattering and aberrating conditions through full-thickness human epidermis (- - -). Each point represents the mean and standard deviation of 30 measurements. The standard deviation indicates the effects of the natural biological variability of human epidermis in terms of its corrugated cell-layered architecture, presence of pigment melanin, protein keratin, and variable thickness.

Fig. 8
Fig. 8

(A) Lateral theta LSF measurement with a 25   μm slit ( 25   μm corresponds to five times the lateral resolution). (B) Lateral resolution versus detection slit width under nominal instrument conditions (—) and under scattering and aberrating conditions through full-thickness human epidermis (- - -). Each point in (B) represents the mean and standard deviation of five measurements.

Fig. 9
Fig. 9

Confocal theta line-scanning images of human skin in vivo, showing (A) stratum corneum; (B) granular layer, illustrating the dark nuclei in the cells (arrows); (C) the spinous layer, showing the smaller dark nuclei (arrows); (D) the basal cells at the dermal-epidermal junction, where the nuclei (arrows) are visible around the dermal papillae. The dermal papillae appear as ring-shaped clusters of basal cells. Scale bar 100   μm .

Fig. 10
Fig. 10

Confocal theta line-scanning images of human oral mucosa (inner lip) in vivo, showing (A) superficial epithelium with bright nuclei (arrows) at 25   μm depth; (B) deeper epithelium at 75   μm ; (C) blood flow within the capillaries (arrows) below the papillae at the epithelial junction at a depth of 200   μm ; (D) superficial lamina propria at a depth of 300   μm .

Tables (1)

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Table 1 Specifications of the Confocal Theta Line-Scanning Microscope

Equations (3)

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d = 2 { r 2 π [ cos 1 ( u r ) ( u r ) 1 ( u r ) 2 ] } 1 / 2 ,
   Δ z θ = [ 0.76 f o b j λ d ] [ r 2 4 ( 1 + u r ) 2 + f o b j 2 ] 1 / 2 r 2 ( 1 + u r ) ,
Δ x θ = 0.76 f o b j λ d cos θ .

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