Abstract

A new method for total surface measurement based on reflection ellipsometry is presented. By scanning the surface of the target under test with a focused laser beam, one can measure the surface topography and its material distribution simultaneously with high lateral resolution. Target topography is determined by ellipsometric measurement of local gradient angles γx and γy of the target’s scanned surface elements. To identify the material, one measures the local complex refractive index n, too. The influence of beam focusing on the measurement results is discussed. We describe successful tests with various dielectric and metallic surfaces by use of He-Ne (632-nm) and He-Cd (442-nm) lasers.

© 2002 Optical Society of America

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References

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  1. K. P. Rastogi, ed., Optical Measurement Techniques and Applications (Artech House, Boston, Mass., 1997).
  2. K. J. Gásvik, Optical Metrology (Wiley, Ontario, Canada, 1995).
  3. K. Dickmann, L. Gronewalter, “Dynamischer Autofokus-Sensor zur Erfassung von Mikrostrukturen,” Sensor Magazine 1, 8–10 (1990).
  4. D. J. Whitehouse, “Surface metrology,” Meas. Sci. Technol. 8, 955–972 (1997).
    [CrossRef]
  5. G. Gülker, K. D. Hinsch, “Detection of surface microstructure changes by electronic speckle pattern interferometry,” Opt. Lasers Eng. 26, 165–178 (1997).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987).
  7. K. Leonhard, U. Droste, H. J. Tiziani, “Topometry for locally changing materials,” Opt. Lett. 23, 1772–1774 (1998).
    [CrossRef]
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  9. R. M. A. Azzam, ed., Selected Papers on Ellipsometry, Vol. MS 27 of SPIE Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1991).
  10. “Spectroscopic ellipsometer Woollam M2000,” catalog (Woollam Company, Inc., Lincoln, Nebr., 2001), http://www.jawoollam.com .
  11. “CER ellipsometer SE 500,” catalog (Sentech Instruments GmbH, Berlin, 2000).
  12. D. O. Barsukov, G. M. Gusakov, A. A. Komarskii, “Precision ellipsometry on a focused light beam. 1,” Opt. Spectrosc. (USSR) 64, 782–785 (1988).
  13. U. Neuschaefer-Rube, W. Holzapfel, F. Wirth, “Target analysis by focusing ellipsometry,” in Proceedings of XVI IMEKO World Congress, R. H. Osanna, ed, (Vienna, Austria, 2000), Vol. II, 249–254.
  14. G. Bouwhuis, Principles of Optical Disc Systems (Adam Hilger, Bristol, UK, 1985).
  15. M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. M. Herzinger, “Extension of rotating-analyzer to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
    [CrossRef]
  16. U. Neuschaefer-Rube, W. Holzapfel, J. Doberitzsch, “Measurement of surface topography and material by microellipsometry,” in Proceedings of XV IMEKO World Congress, H. Imai, ed. (Osaka, Japan, 1999), Vol. IX, 63–69.
  17. W. Holzapfel, U. Neuschaefer-Rube, J. Doberitzsch, “Precise structure measurement using laser-based microellipsometry,” Technisches Messen 66, 455–462 (1999).
  18. K. Brammer, G. Siffling, Kalman-Bucy-Filter (Oldenbourg-Verlag, Munich, 1989).
  19. W. Holzapfel, U. Neuschaefer-Rube, M. Sofsky, “Optical surface measurement applying optimal filtering of ellipsometric and autofocus data,” Opt. Eng. to be published.
  20. G. W. Milton, D. J. Eyre, J. V. Mantese, “Finite frequency range Kramers Kronig relations: bounds on the dispersion,” Phys. Rev. Lett. 79, 3062–3064 (1997).
    [CrossRef]
  21. W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1988).
  22. K. H. Hellwege, A. M. Hellwege, eds., Optische Konstanten, Vol. II of Landolt-Börnstein Zahlenwerte und Funktionen (Springer-Verlag, Berlin, 1962), Part 8.
  23. E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985).
  24. T. V. Vorburger, K. C. Ludema, “Ellipsometry of rough surfaces,” Appl. Opt. 19, 561–573 (1979).
    [CrossRef]
  25. M. Czarske, Strukturbreitenmessung auf Photolithographischen Masken und Wafern im Lichtmikroskop, PTB-Bericht Opt-55 (Physikalisch-Technische Bundesanstalt, Braunschweig, Germany, 1997).
  26. D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
    [CrossRef]
  27. C. A. Fenstermaker, F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
    [CrossRef]
  28. D. Rönnow, S. K. Anderson, K. A. Niklasson, “Surface roughness effects in ellipsometry: comparison of truncated sphere and effective medium models,” Opt. Mater. 4, 815–821 (1995).
    [CrossRef]

1999 (1)

W. Holzapfel, U. Neuschaefer-Rube, J. Doberitzsch, “Precise structure measurement using laser-based microellipsometry,” Technisches Messen 66, 455–462 (1999).

1998 (1)

1997 (3)

D. J. Whitehouse, “Surface metrology,” Meas. Sci. Technol. 8, 955–972 (1997).
[CrossRef]

G. Gülker, K. D. Hinsch, “Detection of surface microstructure changes by electronic speckle pattern interferometry,” Opt. Lasers Eng. 26, 165–178 (1997).
[CrossRef]

G. W. Milton, D. J. Eyre, J. V. Mantese, “Finite frequency range Kramers Kronig relations: bounds on the dispersion,” Phys. Rev. Lett. 79, 3062–3064 (1997).
[CrossRef]

1996 (1)

1995 (1)

D. Rönnow, S. K. Anderson, K. A. Niklasson, “Surface roughness effects in ellipsometry: comparison of truncated sphere and effective medium models,” Opt. Mater. 4, 815–821 (1995).
[CrossRef]

1990 (1)

K. Dickmann, L. Gronewalter, “Dynamischer Autofokus-Sensor zur Erfassung von Mikrostrukturen,” Sensor Magazine 1, 8–10 (1990).

1988 (1)

D. O. Barsukov, G. M. Gusakov, A. A. Komarskii, “Precision ellipsometry on a focused light beam. 1,” Opt. Spectrosc. (USSR) 64, 782–785 (1988).

1979 (2)

T. V. Vorburger, K. C. Ludema, “Ellipsometry of rough surfaces,” Appl. Opt. 19, 561–573 (1979).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

1969 (1)

C. A. Fenstermaker, F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

Anderson, S. K.

D. Rönnow, S. K. Anderson, K. A. Niklasson, “Surface roughness effects in ellipsometry: comparison of truncated sphere and effective medium models,” Opt. Mater. 4, 815–821 (1995).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Barsukov, D. O.

D. O. Barsukov, G. M. Gusakov, A. A. Komarskii, “Precision ellipsometry on a focused light beam. 1,” Opt. Spectrosc. (USSR) 64, 782–785 (1988).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987).

Bouwhuis, G.

G. Bouwhuis, Principles of Optical Disc Systems (Adam Hilger, Bristol, UK, 1985).

Brammer, K.

K. Brammer, G. Siffling, Kalman-Bucy-Filter (Oldenbourg-Verlag, Munich, 1989).

Czarske, M.

M. Czarske, Strukturbreitenmessung auf Photolithographischen Masken und Wafern im Lichtmikroskop, PTB-Bericht Opt-55 (Physikalisch-Technische Bundesanstalt, Braunschweig, Germany, 1997).

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1988).

Dickmann, K.

K. Dickmann, L. Gronewalter, “Dynamischer Autofokus-Sensor zur Erfassung von Mikrostrukturen,” Sensor Magazine 1, 8–10 (1990).

Doberitzsch, J.

W. Holzapfel, U. Neuschaefer-Rube, J. Doberitzsch, “Precise structure measurement using laser-based microellipsometry,” Technisches Messen 66, 455–462 (1999).

U. Neuschaefer-Rube, W. Holzapfel, J. Doberitzsch, “Measurement of surface topography and material by microellipsometry,” in Proceedings of XV IMEKO World Congress, H. Imai, ed. (Osaka, Japan, 1999), Vol. IX, 63–69.

Droste, U.

Eyre, D. J.

G. W. Milton, D. J. Eyre, J. V. Mantese, “Finite frequency range Kramers Kronig relations: bounds on the dispersion,” Phys. Rev. Lett. 79, 3062–3064 (1997).
[CrossRef]

Fenstermaker, C. A.

C. A. Fenstermaker, F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

Gásvik, K. J.

K. J. Gásvik, Optical Metrology (Wiley, Ontario, Canada, 1995).

Gronewalter, L.

K. Dickmann, L. Gronewalter, “Dynamischer Autofokus-Sensor zur Erfassung von Mikrostrukturen,” Sensor Magazine 1, 8–10 (1990).

Gülker, G.

G. Gülker, K. D. Hinsch, “Detection of surface microstructure changes by electronic speckle pattern interferometry,” Opt. Lasers Eng. 26, 165–178 (1997).
[CrossRef]

Gusakov, G. M.

D. O. Barsukov, G. M. Gusakov, A. A. Komarskii, “Precision ellipsometry on a focused light beam. 1,” Opt. Spectrosc. (USSR) 64, 782–785 (1988).

Herzinger, C. M.

Hinsch, K. D.

G. Gülker, K. D. Hinsch, “Detection of surface microstructure changes by electronic speckle pattern interferometry,” Opt. Lasers Eng. 26, 165–178 (1997).
[CrossRef]

Holzapfel, W.

W. Holzapfel, U. Neuschaefer-Rube, J. Doberitzsch, “Precise structure measurement using laser-based microellipsometry,” Technisches Messen 66, 455–462 (1999).

U. Neuschaefer-Rube, W. Holzapfel, J. Doberitzsch, “Measurement of surface topography and material by microellipsometry,” in Proceedings of XV IMEKO World Congress, H. Imai, ed. (Osaka, Japan, 1999), Vol. IX, 63–69.

W. Holzapfel, U. Neuschaefer-Rube, M. Sofsky, “Optical surface measurement applying optimal filtering of ellipsometric and autofocus data,” Opt. Eng. to be published.

U. Neuschaefer-Rube, W. Holzapfel, F. Wirth, “Target analysis by focusing ellipsometry,” in Proceedings of XVI IMEKO World Congress, R. H. Osanna, ed, (Vienna, Austria, 2000), Vol. II, 249–254.

Hottier, F.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Johs, B.

Komarskii, A. A.

D. O. Barsukov, G. M. Gusakov, A. A. Komarskii, “Precision ellipsometry on a focused light beam. 1,” Opt. Spectrosc. (USSR) 64, 782–785 (1988).

Leonhard, K.

Ludema, K. C.

Mantese, J. V.

G. W. Milton, D. J. Eyre, J. V. Mantese, “Finite frequency range Kramers Kronig relations: bounds on the dispersion,” Phys. Rev. Lett. 79, 3062–3064 (1997).
[CrossRef]

McCrackin, F. L.

C. A. Fenstermaker, F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

Milton, G. W.

G. W. Milton, D. J. Eyre, J. V. Mantese, “Finite frequency range Kramers Kronig relations: bounds on the dispersion,” Phys. Rev. Lett. 79, 3062–3064 (1997).
[CrossRef]

Neuschaefer-Rube, U.

W. Holzapfel, U. Neuschaefer-Rube, J. Doberitzsch, “Precise structure measurement using laser-based microellipsometry,” Technisches Messen 66, 455–462 (1999).

W. Holzapfel, U. Neuschaefer-Rube, M. Sofsky, “Optical surface measurement applying optimal filtering of ellipsometric and autofocus data,” Opt. Eng. to be published.

U. Neuschaefer-Rube, W. Holzapfel, J. Doberitzsch, “Measurement of surface topography and material by microellipsometry,” in Proceedings of XV IMEKO World Congress, H. Imai, ed. (Osaka, Japan, 1999), Vol. IX, 63–69.

U. Neuschaefer-Rube, W. Holzapfel, F. Wirth, “Target analysis by focusing ellipsometry,” in Proceedings of XVI IMEKO World Congress, R. H. Osanna, ed, (Vienna, Austria, 2000), Vol. II, 249–254.

Niklasson, K. A.

D. Rönnow, S. K. Anderson, K. A. Niklasson, “Surface roughness effects in ellipsometry: comparison of truncated sphere and effective medium models,” Opt. Mater. 4, 815–821 (1995).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985).

Rheinländer, B.

Rönnow, D.

D. Rönnow, S. K. Anderson, K. A. Niklasson, “Surface roughness effects in ellipsometry: comparison of truncated sphere and effective medium models,” Opt. Mater. 4, 815–821 (1995).
[CrossRef]

Schubert, M.

Siffling, G.

K. Brammer, G. Siffling, Kalman-Bucy-Filter (Oldenbourg-Verlag, Munich, 1989).

Sofsky, M.

W. Holzapfel, U. Neuschaefer-Rube, M. Sofsky, “Optical surface measurement applying optimal filtering of ellipsometric and autofocus data,” Opt. Eng. to be published.

Theeten, J. B.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Tiziani, H. J.

Vorburger, T. V.

Whitehouse, D. J.

D. J. Whitehouse, “Surface metrology,” Meas. Sci. Technol. 8, 955–972 (1997).
[CrossRef]

Wirth, F.

U. Neuschaefer-Rube, W. Holzapfel, F. Wirth, “Target analysis by focusing ellipsometry,” in Proceedings of XVI IMEKO World Congress, R. H. Osanna, ed, (Vienna, Austria, 2000), Vol. II, 249–254.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987).

Woollam, J. A.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

D. J. Whitehouse, “Surface metrology,” Meas. Sci. Technol. 8, 955–972 (1997).
[CrossRef]

Opt. Lasers Eng. (1)

G. Gülker, K. D. Hinsch, “Detection of surface microstructure changes by electronic speckle pattern interferometry,” Opt. Lasers Eng. 26, 165–178 (1997).
[CrossRef]

Opt. Lett. (1)

Opt. Mater. (1)

D. Rönnow, S. K. Anderson, K. A. Niklasson, “Surface roughness effects in ellipsometry: comparison of truncated sphere and effective medium models,” Opt. Mater. 4, 815–821 (1995).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

D. O. Barsukov, G. M. Gusakov, A. A. Komarskii, “Precision ellipsometry on a focused light beam. 1,” Opt. Spectrosc. (USSR) 64, 782–785 (1988).

Phys. Rev. B (1)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

G. W. Milton, D. J. Eyre, J. V. Mantese, “Finite frequency range Kramers Kronig relations: bounds on the dispersion,” Phys. Rev. Lett. 79, 3062–3064 (1997).
[CrossRef]

Sensor Magazine (1)

K. Dickmann, L. Gronewalter, “Dynamischer Autofokus-Sensor zur Erfassung von Mikrostrukturen,” Sensor Magazine 1, 8–10 (1990).

Surf. Sci. (1)

C. A. Fenstermaker, F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

Technisches Messen (1)

W. Holzapfel, U. Neuschaefer-Rube, J. Doberitzsch, “Precise structure measurement using laser-based microellipsometry,” Technisches Messen 66, 455–462 (1999).

Other (16)

K. Brammer, G. Siffling, Kalman-Bucy-Filter (Oldenbourg-Verlag, Munich, 1989).

W. Holzapfel, U. Neuschaefer-Rube, M. Sofsky, “Optical surface measurement applying optimal filtering of ellipsometric and autofocus data,” Opt. Eng. to be published.

M. Czarske, Strukturbreitenmessung auf Photolithographischen Masken und Wafern im Lichtmikroskop, PTB-Bericht Opt-55 (Physikalisch-Technische Bundesanstalt, Braunschweig, Germany, 1997).

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 1988).

K. H. Hellwege, A. M. Hellwege, eds., Optische Konstanten, Vol. II of Landolt-Börnstein Zahlenwerte und Funktionen (Springer-Verlag, Berlin, 1962), Part 8.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985).

U. Neuschaefer-Rube, W. Holzapfel, F. Wirth, “Target analysis by focusing ellipsometry,” in Proceedings of XVI IMEKO World Congress, R. H. Osanna, ed, (Vienna, Austria, 2000), Vol. II, 249–254.

G. Bouwhuis, Principles of Optical Disc Systems (Adam Hilger, Bristol, UK, 1985).

U. Neuschaefer-Rube, W. Holzapfel, J. Doberitzsch, “Measurement of surface topography and material by microellipsometry,” in Proceedings of XV IMEKO World Congress, H. Imai, ed. (Osaka, Japan, 1999), Vol. IX, 63–69.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

R. M. A. Azzam, ed., Selected Papers on Ellipsometry, Vol. MS 27 of SPIE Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

“Spectroscopic ellipsometer Woollam M2000,” catalog (Woollam Company, Inc., Lincoln, Nebr., 2001), http://www.jawoollam.com .

“CER ellipsometer SE 500,” catalog (Sentech Instruments GmbH, Berlin, 2000).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987).

K. P. Rastogi, ed., Optical Measurement Techniques and Applications (Artech House, Boston, Mass., 1997).

K. J. Gásvik, Optical Metrology (Wiley, Ontario, Canada, 1995).

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Figures (13)

Fig. 1
Fig. 1

Block diagram of the measurement setup.

Fig. 2
Fig. 2

Trajectory of a beam in the focusing ellipsometer as it hits a tilted surface element.

Fig. 3
Fig. 3

Coordinate rotation at the principal plane of the focusing optics.

Fig. 4
Fig. 4

Procedure for determining surface characteristics.

Fig. 5
Fig. 5

Examples of possible paths of integration to determine surface height h at point P by use of local gradient angles γ x and γ y .

Fig. 6
Fig. 6

Complex refractive index of some materials: literature values22,23 and (circled) our own ellipsometric measurement values (detected with unfocused beams on plane surfaces).

Fig. 7
Fig. 7

Calculated dependence of real part n f and imaginary part k f of the refractive index on beam waist diameter d f [wavelength of light λ = 442 nm; mean irradiation angle εmean = 45°; mean input azimuth ϑin,mean = 30°; device under test: plane glass surface (n = 1.5, k = 0)].

Fig. 8
Fig. 8

Errors e n = n f - n and e k = k f - k caused by the dependence of beam focusing on complex refractive index n = n - jk of the surface material [beam waist diameter d = 2.8 µm; wavelength of light λ = 442 nm; mean irradiation angle εmean = 45°; mean input azimuth ϑin,mean = 30°; horizontal surface].

Fig. 9
Fig. 9

Result of measuring a sinusoidal surface normal made from glass (Mahr PGN 10): (a) ellipsometrically measured topography h(x, y), (b) control signal h AF(x, y) of the AF system.

Fig. 10
Fig. 10

Result of the measurement of a sinusoidal surface standard made from glass (Mahr PGN 10): (a) purely ellipsometrically measured distribution of refractive index n [calculated with the topography shown in Fig. 9(a)], (b) ellipsometrically measured distribution of the refractive index n [calculated with the AF control signal h AF(x, y) shown in Fig. 9(b)].

Fig. 11
Fig. 11

Ellipsometrically measured local distribution of complex refractive index n of a gold stripe of 16-µm width (thickness, 160 µm) upon a glass substrate: (a) real part n, (b) imaginary part k.

Fig. 12
Fig. 12

Measured topography of an arc-shaped chrome surface: (a) ellipsometric measurement with known refractive index (n¯ = 2.2 - j4.35), (b) control signal h AF(x, y) of the AF system.

Fig. 13
Fig. 13

Definitions of the vectors used for calculating slope angles γ x and γ y .

Equations (28)

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

S=RϑoutR-φtan Ψ expjΔ001RφRϑin,
Rξ=cos ξsin ξ-sin ξcos ξ.
a=I0,0-I90,90+I90,0-I0,90, b=I0,0-I90,90+I0,90-I90,0, c=I0,45-I0,-45+I90,45-I90,-45, d=I90,-45-I0,45+I90,45-I0,-45, e=I90,-45+I0,45+I90,45+I0,-45, f=I0,0-I0,90+I90,90-I90,0, g=I0,0+I0,90+I90,90+I90,0,
φ=12arccosba1+c/a21/2ϑin,
ϑout=φ-12arcsinca1+c/a21/2,
Ψ=12arccosde cos2φ+2ϑin,
Δ=arccos×f/g-cos2φ+2ϑincos2φ-2ϑoutsin2Ψsin2φ+2ϑinsin2φ-2ϑout.
tan γx=-nxnz,
tan γy=-nynz.
hxp, yp=hx0, y0+ktan γxdx+tan γydy.
ĥ=DTM-1D-1DTM-1x=Gx.
xT=sx1, 1sxM, 1, sx1, 2sxM, N,sy1, 1syM, 1, sy1, 2syM, N.
D=-DxDx00-DxDxDyDy00DyDy,Dx=12Δx110011,Dy=12Δy-1100-11.
M=Diagσx2σx2  σy2σy2.
ĥT=ĥ1, 1ĥM+1, 1, ĥ1, 2ĥM+1, N+1.
ĥzi, j=¼ĥi, j+ĥi+1, j+ĥi, j+1+ĥi+1, j+1, i=1,, N, j=1,, M.
n2-k2=sin2 α1+tan2 α cos2 2Ψ-sin2 Δ sin2 2Ψ1+sin 2Ψ cos Δ2,
2nk=sin2 α tan2 α sin 4Ψ sin Δ1+sin 2Ψ cos Δ2.
en=nf-n,
ek=kf-k
n=const.tan γxtan γy1.
i=-cos ε cos ϑin-cos ε sin ϑinsin ε.
p=RotzϑinRotyεRotx-φ001.
op=sin ϑout-cos ϑout0.
o=i×p×op
o=signozo|o|,
n=o+i.
n=i1×p1×i2×p2.

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