Abstract

Noncontact optical metrology based on the chromatic confocal principle is becoming increasingly important for fast and accurate measurements of surface topography, distance, and layer thickness in engineering and industry. These sensors are based on the wavelength dependence of longitudinal chromatic aberration of optical systems, and the distance or thickness of the measured sample is coded into spectral information. We provide a theoretical analysis of a problem of the thickness measurement of transparent samples (glass plane-parallel plates or lenses) with respect to material dispersion. Our work deals with a description and analysis of induced measurement errors in the cases of measurement of the thickness of a plane-parallel plate and the central thickness of a lens. Relations are derived for a quantitative evaluation of these errors and a method is presented for minimizing the influence of these errors on the accuracy of measurement.

© 2010 Optical Society of America

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    [CrossRef]
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2009 (2)

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80, 073706 (2009).
[CrossRef] [PubMed]

S. Li, T. Thorsen, Z. Xu, Z. P. Fang, J. Zhao, and S. F. Yoon, “Microvalve thickness and topography measurements in microfluidic devices by white-light confocal microscopy,” Appl. Opt. 48, 5088–5094 (2009).
[CrossRef] [PubMed]

2008 (1)

J. Garzón, T. Gharbi, and J. Meneses, “Real time determination of the optical thickness and topography of tissues by chromatic confocal microscopy,” J. Opt. A Pure Appl. Opt. 10104028 (2008).
[CrossRef]

2005 (3)

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

M. Kunkel and J. Schulze, “Noncontact measurement of central lens thickness,” Glass Sci. Technol. 78, 2–4 (2005).

J. Novak and A. Miks, “Hyperchromats with linear dependence of longitudinal chromatic aberration on wavelength,” Optik (Jena) 116, 165–168 (2005).
[CrossRef]

2004 (7)

J. G. Reyes, J. Meneses, A. Plata, G. Tribillon, and T. Gharbi, “Chromatic confocal method for determination of the refractive index and thickness,” Proc. SPIE 5622, 805–810 (2004).
[CrossRef]

R. J. Garzón, J. Menese, G. Tribillon, T. Gharbi, and A. Plata, “Chromatic confocal microscopy by means of continuum light generated through a standard single-mode fibre,” J. Opt. A Pure Appl. Opt. 6, 544–548 (2004).
[CrossRef]

A. K. Ruprecht, K. Korner, T. F. Wiesendanger, H. J. Tiziani, and W. Osten, “Chromatic confocal detection for high speed micro-topography measurements,” Proc. SPIE 5302, 53–60(2004).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

J. W. McBride and C. Maul, “The 3D measurement and analysis of high precision surfaces using con-focal optical methods,” IEICE Trans. Electron. E87-C, 1261–1267 (2004).

K. Shi, P. Li, S. Yin, and Z. Liu, “Chromatic confocal microscopy using supercontinuum light,” Opt. Express 12, 2096–2101 (2004).
[CrossRef] [PubMed]

A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Chromatic confocal microscopy with a finite pinhole size,” Opt. Lett. 29, 2130–2132 (2004).
[CrossRef] [PubMed]

2000 (2)

1998 (1)

1997 (1)

1996 (1)

H. J. Tiziani, R. Achi, and R. N. Krämer, “Chromatic confocal microscopy with microlenses,” J. Mod. Opt. 43, 155–163(1996).
[CrossRef]

1994 (2)

M. Maly and A. Boyde, “Real-time stereoscopic confocal reflection microscopy using objective lens with linear longitudinal chromatic dispersion,” Scanning 16, 187–192 (1994).

H. J. Tiziani and H. M. Uhde, “3-Dimensional image sensing by chromatic confocal microscopy,” Appl. Opt. 33, 1838–1843(1994).
[CrossRef] [PubMed]

1992 (1)

M. A. Browne, O. Akinyemi, and A. Boyde, “Confocal surface profiling using chromatic aberration,” Scanning 14, 145–153(1992).
[CrossRef]

1984 (1)

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49, 229–233 (1984).
[CrossRef]

Achi, R.

H. J. Tiziani, R. Achi, and R. N. Krämer, “Chromatic confocal microscopy with microlenses,” J. Mod. Opt. 43, 155–163(1996).
[CrossRef]

Akinyemi, O.

M. A. Browne, O. Akinyemi, and A. Boyde, “Confocal surface profiling using chromatic aberration,” Scanning 14, 145–153(1992).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2000).

Boyde, A.

M. Maly and A. Boyde, “Real-time stereoscopic confocal reflection microscopy using objective lens with linear longitudinal chromatic dispersion,” Scanning 16, 187–192 (1994).

M. A. Browne, O. Akinyemi, and A. Boyde, “Confocal surface profiling using chromatic aberration,” Scanning 14, 145–153(1992).
[CrossRef]

Browne, M. A.

M. A. Browne, O. Akinyemi, and A. Boyde, “Confocal surface profiling using chromatic aberration,” Scanning 14, 145–153(1992).
[CrossRef]

Cha, S.

Chun, B. S.

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80, 073706 (2009).
[CrossRef] [PubMed]

Dobson, S.

Fainman, Y.

Fang, Z. P.

Garzón, J.

J. Garzón, T. Gharbi, and J. Meneses, “Real time determination of the optical thickness and topography of tissues by chromatic confocal microscopy,” J. Opt. A Pure Appl. Opt. 10104028 (2008).
[CrossRef]

Garzón, R. J.

R. J. Garzón, J. Menese, G. Tribillon, T. Gharbi, and A. Plata, “Chromatic confocal microscopy by means of continuum light generated through a standard single-mode fibre,” J. Opt. A Pure Appl. Opt. 6, 544–548 (2004).
[CrossRef]

Gharbi, T.

J. Garzón, T. Gharbi, and J. Meneses, “Real time determination of the optical thickness and topography of tissues by chromatic confocal microscopy,” J. Opt. A Pure Appl. Opt. 10104028 (2008).
[CrossRef]

R. J. Garzón, J. Menese, G. Tribillon, T. Gharbi, and A. Plata, “Chromatic confocal microscopy by means of continuum light generated through a standard single-mode fibre,” J. Opt. A Pure Appl. Opt. 6, 544–548 (2004).
[CrossRef]

J. G. Reyes, J. Meneses, A. Plata, G. Tribillon, and T. Gharbi, “Chromatic confocal method for determination of the refractive index and thickness,” Proc. SPIE 5622, 805–810 (2004).
[CrossRef]

Gweon, D.

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80, 073706 (2009).
[CrossRef] [PubMed]

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

Kim, K.

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80, 073706 (2009).
[CrossRef] [PubMed]

Korner, K.

A. K. Ruprecht, K. Korner, T. F. Wiesendanger, H. J. Tiziani, and W. Osten, “Chromatic confocal detection for high speed micro-topography measurements,” Proc. SPIE 5302, 53–60(2004).
[CrossRef]

Krämer, R. N.

H. J. Tiziani, R. Achi, and R. N. Krämer, “Chromatic confocal microscopy with microlenses,” J. Mod. Opt. 43, 155–163(1996).
[CrossRef]

Kunkel, M.

M. Kunkel and J. Schulze, “Noncontact measurement of central lens thickness,” Glass Sci. Technol. 78, 2–4 (2005).

Last, A.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

Lehmann, P.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

Li, P.

Li, S.

Lin, P. C.

Liu, Z.

Lucke, P.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

Lücke, P.

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

Maly, M.

M. Maly and A. Boyde, “Real-time stereoscopic confocal reflection microscopy using objective lens with linear longitudinal chromatic dispersion,” Scanning 16, 187–192 (1994).

Maul, C.

J. W. McBride and C. Maul, “The 3D measurement and analysis of high precision surfaces using con-focal optical methods,” IEICE Trans. Electron. E87-C, 1261–1267 (2004).

McBride, J. W.

J. W. McBride and C. Maul, “The 3D measurement and analysis of high precision surfaces using con-focal optical methods,” IEICE Trans. Electron. E87-C, 1261–1267 (2004).

Menese, J.

R. J. Garzón, J. Menese, G. Tribillon, T. Gharbi, and A. Plata, “Chromatic confocal microscopy by means of continuum light generated through a standard single-mode fibre,” J. Opt. A Pure Appl. Opt. 6, 544–548 (2004).
[CrossRef]

Meneses, J.

J. Garzón, T. Gharbi, and J. Meneses, “Real time determination of the optical thickness and topography of tissues by chromatic confocal microscopy,” J. Opt. A Pure Appl. Opt. 10104028 (2008).
[CrossRef]

J. G. Reyes, J. Meneses, A. Plata, G. Tribillon, and T. Gharbi, “Chromatic confocal method for determination of the refractive index and thickness,” Proc. SPIE 5622, 805–810 (2004).
[CrossRef]

Miks, A.

J. Novak and A. Miks, “Hyperchromats with linear dependence of longitudinal chromatic aberration on wavelength,” Optik (Jena) 116, 165–168 (2005).
[CrossRef]

A. Miks, Applied Optics (Czech Technical U. Press, 2009).
[PubMed]

Mohr, J.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

Molesini, G.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49, 229–233 (1984).
[CrossRef]

Novak, J.

J. Novak and A. Miks, “Hyperchromats with linear dependence of longitudinal chromatic aberration on wavelength,” Optik (Jena) 116, 165–168 (2005).
[CrossRef]

Osten, W.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

A. K. Ruprecht, K. Korner, T. F. Wiesendanger, H. J. Tiziani, and W. Osten, “Chromatic confocal detection for high speed micro-topography measurements,” Proc. SPIE 5302, 53–60(2004).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

Pedrini, G.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49, 229–233 (1984).
[CrossRef]

Plata, A.

J. G. Reyes, J. Meneses, A. Plata, G. Tribillon, and T. Gharbi, “Chromatic confocal method for determination of the refractive index and thickness,” Proc. SPIE 5622, 805–810 (2004).
[CrossRef]

R. J. Garzón, J. Menese, G. Tribillon, T. Gharbi, and A. Plata, “Chromatic confocal microscopy by means of continuum light generated through a standard single-mode fibre,” J. Opt. A Pure Appl. Opt. 6, 544–548 (2004).
[CrossRef]

Poggi, P.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49, 229–233 (1984).
[CrossRef]

Pruss, C.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

Quercioli, F.

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49, 229–233 (1984).
[CrossRef]

Reyes, J. G.

J. G. Reyes, J. Meneses, A. Plata, G. Tribillon, and T. Gharbi, “Chromatic confocal method for determination of the refractive index and thickness,” Proc. SPIE 5622, 805–810 (2004).
[CrossRef]

Ruprecht, A. K.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Chromatic confocal microscopy with a finite pinhole size,” Opt. Lett. 29, 2130–2132 (2004).
[CrossRef] [PubMed]

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

A. K. Ruprecht, K. Korner, T. F. Wiesendanger, H. J. Tiziani, and W. Osten, “Chromatic confocal detection for high speed micro-topography measurements,” Proc. SPIE 5302, 53–60(2004).
[CrossRef]

Schulze, J.

M. Kunkel and J. Schulze, “Noncontact measurement of central lens thickness,” Glass Sci. Technol. 78, 2–4 (2005).

Shi, K.

Steudle, D.

D. Steudle, M. Wegner, and H. J. Tiziani, “Confocal principle for macro- and microscopic surface and defect analysis,” Opt. Eng. 39, 32–39 (2000).
[CrossRef]

Sun, P.

Sun, P. C.

Thorsen, T.

Tiziani, H.

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

Tiziani, H. J.

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Chromatic confocal microscopy with a finite pinhole size,” Opt. Lett. 29, 2130–2132 (2004).
[CrossRef] [PubMed]

A. K. Ruprecht, K. Korner, T. F. Wiesendanger, H. J. Tiziani, and W. Osten, “Chromatic confocal detection for high speed micro-topography measurements,” Proc. SPIE 5302, 53–60(2004).
[CrossRef]

D. Steudle, M. Wegner, and H. J. Tiziani, “Confocal principle for macro- and microscopic surface and defect analysis,” Opt. Eng. 39, 32–39 (2000).
[CrossRef]

H. J. Tiziani, R. Achi, and R. N. Krämer, “Chromatic confocal microscopy with microlenses,” J. Mod. Opt. 43, 155–163(1996).
[CrossRef]

H. J. Tiziani and H. M. Uhde, “3-Dimensional image sensing by chromatic confocal microscopy,” Appl. Opt. 33, 1838–1843(1994).
[CrossRef] [PubMed]

Tribillon, G.

J. G. Reyes, J. Meneses, A. Plata, G. Tribillon, and T. Gharbi, “Chromatic confocal method for determination of the refractive index and thickness,” Proc. SPIE 5622, 805–810 (2004).
[CrossRef]

R. J. Garzón, J. Menese, G. Tribillon, T. Gharbi, and A. Plata, “Chromatic confocal microscopy by means of continuum light generated through a standard single-mode fibre,” J. Opt. A Pure Appl. Opt. 6, 544–548 (2004).
[CrossRef]

Uhde, H. M.

Wegner, M.

D. Steudle, M. Wegner, and H. J. Tiziani, “Confocal principle for macro- and microscopic surface and defect analysis,” Opt. Eng. 39, 32–39 (2000).
[CrossRef]

Wiesendanger, T. F.

A. K. Ruprecht, K. Korner, T. F. Wiesendanger, H. J. Tiziani, and W. Osten, “Chromatic confocal detection for high speed micro-topography measurements,” Proc. SPIE 5302, 53–60(2004).
[CrossRef]

A. K. Ruprecht, T. F. Wiesendanger, and H. J. Tiziani, “Chromatic confocal microscopy with a finite pinhole size,” Opt. Lett. 29, 2130–2132 (2004).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2000).

Xu, Z.

Yin, S.

Yoon, S. F.

Zhao, J.

Zhu, L.

Appl. Opt. (5)

Glass Sci. Technol. (1)

M. Kunkel and J. Schulze, “Noncontact measurement of central lens thickness,” Glass Sci. Technol. 78, 2–4 (2005).

IEICE Trans. Electron. (1)

J. W. McBride and C. Maul, “The 3D measurement and analysis of high precision surfaces using con-focal optical methods,” IEICE Trans. Electron. E87-C, 1261–1267 (2004).

J. Mod. Opt. (1)

H. J. Tiziani, R. Achi, and R. N. Krämer, “Chromatic confocal microscopy with microlenses,” J. Mod. Opt. 43, 155–163(1996).
[CrossRef]

J. Opt. A Pure Appl. Opt. (2)

R. J. Garzón, J. Menese, G. Tribillon, T. Gharbi, and A. Plata, “Chromatic confocal microscopy by means of continuum light generated through a standard single-mode fibre,” J. Opt. A Pure Appl. Opt. 6, 544–548 (2004).
[CrossRef]

J. Garzón, T. Gharbi, and J. Meneses, “Real time determination of the optical thickness and topography of tissues by chromatic confocal microscopy,” J. Opt. A Pure Appl. Opt. 10104028 (2008).
[CrossRef]

Opt. Commun. (1)

G. Molesini, G. Pedrini, P. Poggi, and F. Quercioli, “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49, 229–233 (1984).
[CrossRef]

Opt. Eng. (1)

D. Steudle, M. Wegner, and H. J. Tiziani, “Confocal principle for macro- and microscopic surface and defect analysis,” Opt. Eng. 39, 32–39 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (Jena) (1)

J. Novak and A. Miks, “Hyperchromats with linear dependence of longitudinal chromatic aberration on wavelength,” Optik (Jena) 116, 165–168 (2005).
[CrossRef]

Proc. SPIE (4)

A. K. Ruprecht, K. Korner, T. F. Wiesendanger, H. J. Tiziani, and W. Osten, “Chromatic confocal detection for high speed micro-topography measurements,” Proc. SPIE 5302, 53–60(2004).
[CrossRef]

P. Lücke, A. Last, J. Mohr, A. K. Ruprecht, W. Osten, H. Tiziani, and P. Lehmann, “Confocal micro-optical distance sensor for precision metrology,” Proc. SPIE 5459, 180–184 (2004).
[CrossRef]

A. K. Ruprecht, C. Pruss, H. J. Tiziani, W. Osten, P. Lucke, A. Last, J. Mohr, and P. Lehmann, “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135(2005).
[CrossRef]

J. G. Reyes, J. Meneses, A. Plata, G. Tribillon, and T. Gharbi, “Chromatic confocal method for determination of the refractive index and thickness,” Proc. SPIE 5622, 805–810 (2004).
[CrossRef]

Rev. Sci. Instrum. (1)

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80, 073706 (2009).
[CrossRef] [PubMed]

Scanning (2)

M. A. Browne, O. Akinyemi, and A. Boyde, “Confocal surface profiling using chromatic aberration,” Scanning 14, 145–153(1992).
[CrossRef]

M. Maly and A. Boyde, “Real-time stereoscopic confocal reflection microscopy using objective lens with linear longitudinal chromatic dispersion,” Scanning 16, 187–192 (1994).

Other (10)

http://www.frt-gmbh.com/.

http://www.stilsa.com/.

http://www.micro-epsilon.com/.

http://www.nanofocus-us.com/.

http://www.precitec.com/.

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2000).

A. Miks, Applied Optics (Czech Technical U. Press, 2009).
[PubMed]

www.osram.com.

www.philips.com.

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Figures (4)

Fig. 1
Fig. 1

Optical system for topography measurement.

Fig. 2
Fig. 2

Optical system for thickness measurement.

Fig. 3
Fig. 3

Ray propagation through the plane-parallel plate.

Fig. 4
Fig. 4

Ray propagation through the lens.

Equations (22)

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Δ = ( 1 cos ε n 2 sin 2 ε ) d = n 1 n d + ( 1 cos ε 1 ( sin ε / n ) 2 ) d n = Δ 0 + δ s .
Δ = d n 1 n + d n 2 1 2 n 3 ε 2 = Δ 0 + δ s I I I .
Δ = Δ 0 ν [ n 2 cos ε ( n 2 sin 2 ε ) n 2 sin 2 ε ] ,
ν ( λ 0 ) = n 1 n = n ( λ 0 ) 1 n ( λ 1 ) n ( λ 2 ) , λ 0 λ 1 , λ 2 , λ 1 < λ 2 ,
Δ ¯ = 1 ε max 0 ε max Δ 0 ν [ n 2 cos ε ( n 2 sin 2 ε ) n 2 sin 2 ε ] d ε = ( Δ 0 n ν ) g ( n , ε max ) ,
g ( n , ε max ) = 1 1 ( sin ε max / n ) 2 ( sin ε max ε max ) ( sin ε max ε max ) ( 1 + sin 2 ε max 2 n 2 ) .
Δ ¯ Δ 0 n ν .
Δ ¯ Δ 0 n d ν d ,
σ 1 = h 1 / s 1 , n σ 2 σ 1 = h 1 ( n 1 ) / r 1 , h 2 = h 1 d σ 2 , σ 3 + σ 2 = 2 h 2 / r 2 , h 3 = h 2 + d σ 3 , σ 4 + n σ 3 = h 3 ( n 1 ) / r 1 ,
σ 3 = σ 2 , h 3 = h 1 , σ 4 = σ 1 , s 4 = V 1 A 3 ¯ = s 1 = V 1 A 1 ¯ .
s 1 = d n ( n 1 ) d / r 1 d n [ 1 + d ( n 1 ) n r 1 ] , d = n s 1 1 + s 1 ( n 1 ) / r 1 n s 1 [ 1 s 1 ( n 1 ) r 1 ] ,
d n s 1 ,
d n 1 n ν d [ 1 ( 2 1 / n ) d / r 1 ] n 1 n ν d ,
d = ( n d 1 n d ν d ) d .
S I = i = 1 i = 3 h i ( Δ σ i Δ ( 1 / n i ) ) 2 Δ ( σ i n i ) ,
S I = 2 s 1 σ 1 4 ( 1 s 1 r 1 ) 2 ( s 1 n d 1 ) .
δ s = ( S I 2 σ 1 4 ) sin 2 σ 1 = K sin 2 σ 1 ,
K = s 1 ( 1 s 1 r 1 ) 2 ( s 1 n d 1 ) d 1 n 2 n 3 .
δ s ¯ = K 1 σ 1 max 0 σ 1 max sin 2 σ 1 d σ 1 = K 2 ( 1 sin 2 σ 1 max 2 σ 1 max ) .
W max = W 40 = δ s max 4 sin 2 σ 1 max = K 4 sin 4 σ 1 max ,
W 40 λ / N , and d 4 λ n 3 N ( n 2 1 ) sin 4 σ 1 max .
4 45 W 40 2 λ 2 196 .

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