Abstract

We report an optical design of a line-focused forward-viewing optical coherence tomography (OCT) scanner for high-speed endoscopic imaging. To avoid a complex lens system, an off-axis cylindrical mirror is used for focusing the line illumination onto the sample surface. Because of its insensitivity to the broadband spectrum, the mirror-focused scanner improves the image quality compared to a lens-focused scanner. In this work, a feasibility study is carried out on the use of a reflective-optics-focused line scanner in OCT imaging, instead of the traditional refractive optics scanner. The Strehl ratio, chromatic focal shift, and field analysis were carried out for a plano-convex cylindrical lens, an achromatic cylindrical lens, and a cylindrical-mirror-focused scanner. ZEMAX optical modeling analysis showed that mirror-focused scanning provides better Strehl ratio in comparison to plano-convex cylindrical-lens-focused scanning, and that the Strehl ratio is comparable to achromatic cylindrical-lens-focused scanning. However, field anal ysis on the edges of the scanning elements within the scan range shows that mirror-focused scanning is more robust when compared to a cylindrical achromatic lens. Overall, a mirror-focused scanner shows better performance compared to lenses.

© 2010 Optical Society of America

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2009 (3)

M. Chen, Y. Chen, and W. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47, 84–89 (2009).
[CrossRef]

M. Mujat, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Swept-source parallel OCT,” Proc. SPIE 7168, 71681E, 2009).
[CrossRef]

M. Kamal, N. Sivakumar, and M. Packirisamy, “Optical modeling of a line-scan optical coherence tomography system for high-speed three-dimensional endoscopic imaging,” Proc. SPIE 7386, 738607 (2009).
[CrossRef]

2008 (3)

P. Meemon, K. Lee, S. Murali, and J. Rolland, “Optical design of a dynamic focus catheter for high-resolution endoscopic optical coherence tomography,” Appl. Opt. 47, 2452–2457 (2008).
[CrossRef] [PubMed]

R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical System Design (McGraw-Hill, 2008).

S.-W. Lee and B.-M. Kim, “Line-field optical coherence tomography using frequency-sweeping source,” IEEE J. Sel. Top. Quantum Electron. 14, 50–55 (2008).
[CrossRef]

2007 (1)

2006 (1)

Z. Yaqoob, J. Wu, E. J. McDowell, X. Heng, and C. Yang, “Methods and application areas of endoscopic optical coherence tomography,” J. Biomed. Opt. 11, 063001 (2006).
[CrossRef]

2005 (2)

2004 (4)

A. Dubois, G. Moneron, K. Grieve, and A. C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227–1234 (2004).
[CrossRef] [PubMed]

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

C. E. Webb and J. D. C. Jones, eds., Handbook of Laser Technology and Applications (Institute of Physics, 2004).
[CrossRef]

W. Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. 9, 47–74 (2004).
[CrossRef] [PubMed]

2002 (2)

J. M. Geary, Introduction to Lens Design with Practical ZEMAX Examples (Willmann-Bell, 2002).

W. R. Hendee and E. R. Ritenour, Medical Imaging Physics (Wiley-Liss, 2002).
[CrossRef]

2001 (1)

2000 (1)

A. Franks, K. Jackson, and A. Yacoot, “A parabolic mirror x-ray collimator,” Meas. Sci. Technol. 11, 484–488 (2000).
[CrossRef]

1999 (3)

D. J. Schroeder, Astronomical Optics (Academic, 1999).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

1995 (1)

1991 (1)

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

1978 (1)

Abe, F.

Boccara, A. C.

A. Dubois, G. Moneron, K. Grieve, and A. C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227–1234 (2004).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Chang, W.

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

Chen, M.

M. Chen, Y. Chen, and W. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47, 84–89 (2009).
[CrossRef]

Chen, Y.

M. Chen, Y. Chen, and W. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47, 84–89 (2009).
[CrossRef]

Dickensheets, L. D.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

Drexler, W.

W. Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. 9, 47–74 (2004).
[CrossRef] [PubMed]

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Dubois, A.

A. Dubois, G. Moneron, K. Grieve, and A. C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227–1234 (2004).
[CrossRef] [PubMed]

Endo, T.

Fercher, A.

Fercher, A. F.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Ferguson, R. D.

M. Mujat, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Swept-source parallel OCT,” Proc. SPIE 7168, 71681E, 2009).
[CrossRef]

Fischer, R. E.

R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical System Design (McGraw-Hill, 2008).

Flotte, T.

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

Franks, A.

A. Franks, K. Jackson, and A. Yacoot, “A parabolic mirror x-ray collimator,” Meas. Sci. Technol. 11, 484–488 (2000).
[CrossRef]

Fujimoto, J.

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

Geary, J. M.

J. M. Geary, Introduction to Lens Design with Practical ZEMAX Examples (Willmann-Bell, 2002).

Gordon, L. M.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

Grajciar, B.

Gregory, K.

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

Grieve, K.

A. Dubois, G. Moneron, K. Grieve, and A. C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227–1234 (2004).
[CrossRef] [PubMed]

Haig, N. D.

Hammer, D. X.

M. Mujat, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Swept-source parallel OCT,” Proc. SPIE 7168, 71681E, 2009).
[CrossRef]

Hee, M.

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

Hendee, W. R.

W. R. Hendee and E. R. Ritenour, Medical Imaging Physics (Wiley-Liss, 2002).
[CrossRef]

Heng, X.

Z. Yaqoob, J. Wu, E. J. McDowell, X. Heng, and C. Yang, “Methods and application areas of endoscopic optical coherence tomography,” J. Biomed. Opt. 11, 063001 (2006).
[CrossRef]

Himmer, A. P.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

Hitzenberger, C. K.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Hsiao, W.

M. Chen, Y. Chen, and W. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47, 84–89 (2009).
[CrossRef]

Huang, D.

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

Iftimia, N. V.

M. Mujat, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Swept-source parallel OCT,” Proc. SPIE 7168, 71681E, 2009).
[CrossRef]

Itoh, M.

Jackson, K.

A. Franks, K. Jackson, and A. Yacoot, “A parabolic mirror x-ray collimator,” Meas. Sci. Technol. 11, 484–488 (2000).
[CrossRef]

Jones, J. D. C.

C. E. Webb and J. D. C. Jones, eds., Handbook of Laser Technology and Applications (Institute of Physics, 2004).
[CrossRef]

Kamal, M.

M. Kamal, N. Sivakumar, and M. Packirisamy, “Optical modeling of a line-scan optical coherence tomography system for high-speed three-dimensional endoscopic imaging,” Proc. SPIE 7386, 738607 (2009).
[CrossRef]

Kim, B.-M.

S.-W. Lee and B.-M. Kim, “Line-field optical coherence tomography using frequency-sweeping source,” IEEE J. Sel. Top. Quantum Electron. 14, 50–55 (2008).
[CrossRef]

Lee, K.

Lee, S.-W.

S.-W. Lee and B.-M. Kim, “Line-field optical coherence tomography using frequency-sweeping source,” IEEE J. Sel. Top. Quantum Electron. 14, 50–55 (2008).
[CrossRef]

Leitgeb, R.

Lexer, F.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Lieb, M.

Lin, C.

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

Makita, S.

Matsuda, T.

McDowell, E. J.

Z. Yaqoob, J. Wu, E. J. McDowell, X. Heng, and C. Yang, “Methods and application areas of endoscopic optical coherence tomography,” J. Biomed. Opt. 11, 063001 (2006).
[CrossRef]

Meemon, P.

Meixner, A.

Molebny, S.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Moneron, G.

A. Dubois, G. Moneron, K. Grieve, and A. C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227–1234 (2004).
[CrossRef] [PubMed]

Mujat, M.

M. Mujat, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Swept-source parallel OCT,” Proc. SPIE 7168, 71681E, 2009).
[CrossRef]

Murali, S.

Nakamura, Y.

Packirisamy, M.

M. Kamal, N. Sivakumar, and M. Packirisamy, “Optical modeling of a line-scan optical coherence tomography system for high-speed three-dimensional endoscopic imaging,” Proc. SPIE 7386, 738607 (2009).
[CrossRef]

Pircher, M.

Puliafito, C.

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

Qi, B.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

Ritenour, E. R.

W. R. Hendee and E. R. Ritenour, Medical Imaging Physics (Wiley-Liss, 2002).
[CrossRef]

Rolland, J.

Sattmann, H.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, 1999).

Schuman, J.

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

Sivakumar, N.

M. Kamal, N. Sivakumar, and M. Packirisamy, “Optical modeling of a line-scan optical coherence tomography system for high-speed three-dimensional endoscopic imaging,” Proc. SPIE 7386, 738607 (2009).
[CrossRef]

Sticker, M.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Stinson, W.

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

Swanson, E.

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

Tadic-Galeb, B.

R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical System Design (McGraw-Hill, 2008).

Takahashi, H.

Vitkin, I. A.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

Webb, C. E.

C. E. Webb and J. D. C. Jones, eds., Handbook of Laser Technology and Applications (Institute of Physics, 2004).
[CrossRef]

Williams, T. L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Wu, J.

Z. Yaqoob, J. Wu, E. J. McDowell, X. Heng, and C. Yang, “Methods and application areas of endoscopic optical coherence tomography,” J. Biomed. Opt. 11, 063001 (2006).
[CrossRef]

Yacoot, A.

A. Franks, K. Jackson, and A. Yacoot, “A parabolic mirror x-ray collimator,” Meas. Sci. Technol. 11, 484–488 (2000).
[CrossRef]

Yamanari, M.

Yang, C.

Z. Yaqoob, J. Wu, E. J. McDowell, X. Heng, and C. Yang, “Methods and application areas of endoscopic optical coherence tomography,” J. Biomed. Opt. 11, 063001 (2006).
[CrossRef]

Yang, X. D.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

Yaqoob, Z.

Z. Yaqoob, J. Wu, E. J. McDowell, X. Heng, and C. Yang, “Methods and application areas of endoscopic optical coherence tomography,” J. Biomed. Opt. 11, 063001 (2006).
[CrossRef]

Yasuno, Y.

Yatagai, T.

Yoder, P. R.

R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical System Design (McGraw-Hill, 2008).

Appl. Opt. (3)

IEEE J. Sel. Top. Quantum Electron. (1)

S.-W. Lee and B.-M. Kim, “Line-field optical coherence tomography using frequency-sweeping source,” IEEE J. Sel. Top. Quantum Electron. 14, 50–55 (2008).
[CrossRef]

J. Biomed. Opt. (2)

Z. Yaqoob, J. Wu, E. J. McDowell, X. Heng, and C. Yang, “Methods and application areas of endoscopic optical coherence tomography,” J. Biomed. Opt. 11, 063001 (2006).
[CrossRef]

W. Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. 9, 47–74 (2004).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Meas. Sci. Technol. (1)

A. Franks, K. Jackson, and A. Yacoot, “A parabolic mirror x-ray collimator,” Meas. Sci. Technol. 11, 484–488 (2000).
[CrossRef]

Opt. Commun. (1)

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128(2004).
[CrossRef]

Opt. Express (4)

Opt. Lasers Eng. (1)

M. Chen, Y. Chen, and W. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47, 84–89 (2009).
[CrossRef]

Phys. Med. Biol. (1)

A. Dubois, G. Moneron, K. Grieve, and A. C. Boccara, “Three-dimensional cellular-level imaging using full-field optical coherence tomography,” Phys. Med. Biol. 49, 1227–1234 (2004).
[CrossRef] [PubMed]

Proc. SPIE (2)

M. Mujat, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Swept-source parallel OCT,” Proc. SPIE 7168, 71681E, 2009).
[CrossRef]

M. Kamal, N. Sivakumar, and M. Packirisamy, “Optical modeling of a line-scan optical coherence tomography system for high-speed three-dimensional endoscopic imaging,” Proc. SPIE 7386, 738607 (2009).
[CrossRef]

Science (1)

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

Other (6)

C. E. Webb and J. D. C. Jones, eds., Handbook of Laser Technology and Applications (Institute of Physics, 2004).
[CrossRef]

R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical System Design (McGraw-Hill, 2008).

J. M. Geary, Introduction to Lens Design with Practical ZEMAX Examples (Willmann-Bell, 2002).

W. R. Hendee and E. R. Ritenour, Medical Imaging Physics (Wiley-Liss, 2002).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

D. J. Schroeder, Astronomical Optics (Academic, 1999).

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Figures (12)

Fig. 1
Fig. 1

(a) Pre-objective scanning and (b) post-objective scanning.

Fig. 2
Fig. 2

Cylindrical-lens-focused forward-viewing probe (BFL, back focal length; EFL, effective focal length).

Fig. 3
Fig. 3

Scanner positions with different EFLs for a 2 mm scan range.

Fig. 4
Fig. 4

Illumination distribution at imaging plane (a) along the line and (b) in the focusing direction.

Fig. 5
Fig. 5

Strehl ratio of plano-convex cylindrical-lens-focused scanner.

Fig. 6
Fig. 6

Achromatic cylindrical-lens scanner.

Fig. 7
Fig. 7

Strehl ratio of achromatic cylindrical-lens-focused scanner.

Fig. 8
Fig. 8

Mirror-focused forward scanner configuration.

Fig. 9
Fig. 9

Cylindrical-mirror-focused forward-viewing probe layout.

Fig. 10
Fig. 10

Strehl ratio of cylindrical-mirror-focused scanner.

Fig. 11
Fig. 11

Strehl ratio falls off at different fields.

Fig. 12
Fig. 12

Chromatic focal shift for 12.7 mm EFL.

Tables (3)

Tables Icon

Table 1 Plano-Convex Cylindrical-Lens Specifications

Tables Icon

Table 2 Achromatic Cylindrical-Lens Specifications (in Millimeters)

Tables Icon

Table 3 Cylindrical-Mirror Specifications

Equations (1)

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L s = Y + ( f Z ) tan ( θ 2 α ) .

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