Abstract

We demonstrate a method to obtain within an arbitrary numerical aperture (NA) the entire scattering matrix of a scatterer by using focused beam coherent Fourier scatterometry. The far-field intensities of all scattered angles within the NA of the optical system are obtained in one shot. The corresponding phases of the field are obtained by an interferometric configuration. This method enables the retrieval of the maximum available information about the scatterer from scattered far-field data contained in the given NA of the system.

© 2016 Optical Society of America

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References

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2016 (1)

2015 (1)

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

2014 (1)

2013 (2)

N. Kumar, O. El Gawhary, S. Roy, S. E. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. 8, 13048 (2013).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15, 075707 (2013).

2012 (1)

V. F. Paz, S. Peterhansel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1, e36 (2012).

2011 (1)

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105, 775–781 (2011).

2010 (1)

B. B. M. Wurm and F. Pilarski, “A new flexible scatterometer for critical dimensional metrology,” Rev. Sci. Instrum. 81, 023701 (2010).
[Crossref]

2009 (1)

2007 (2)

R. M. Silver, B. M. Barnes, R. Attota, J. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
[Crossref]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementation,” Opt. Lasers Eng. 45, 304–317 (2007).

2005 (1)

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

2002 (2)

2001 (1)

H.-T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 3983 (2001).
[Crossref]

1997 (1)

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).

1996 (1)

1981 (2)

1980 (1)

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).

1978 (1)

1907 (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London A 79, 399–416 (1907).

Assafrão, A. C.

Attota, R.

Barnes, B. M.

Boef, A. D.

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

Boher, P.

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

Burton, D. R.

Chandezon, J.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).

Chaton, P.

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

Coene, W. M. J.

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105, 775–781 (2011).

Desieres, Y.

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

El Gawhary, O.

N. Kumar, O. El Gawhary, S. Roy, S. E. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. 8, 13048 (2013).
[Crossref]

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105, 775–781 (2011).

Faridian, A.

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

Foucher, J.

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

Frenner, K.

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

V. F. Paz, S. Peterhansel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1, e36 (2012).

Gaylord, T. K.

Gdeisat, M. A.

Hansen, P.-E.

Hazard, J.

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

Herraez, M. A.

Huang, H.-T.

H.-T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 3983 (2001).
[Crossref]

Jin, M.

D. Kim, M. Jin, H. Lee, S. Kim, and R. Magnusson, “Snapshot conical diffraction phase image measurement in angle-resolved microellipsometry,” in Imaging and Applied Optics (Optical Society of America, 2013), paper. CTu3C.5.

Jun, J.

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementation,” Opt. Lasers Eng. 45, 304–317 (2007).

Kim, D.

D. Kim, M. Jin, H. Lee, S. Kim, and R. Magnusson, “Snapshot conical diffraction phase image measurement in angle-resolved microellipsometry,” in Imaging and Applied Optics (Optical Society of America, 2013), paper. CTu3C.5.

Kim, S.

D. Kim, M. Jin, H. Lee, S. Kim, and R. Magnusson, “Snapshot conical diffraction phase image measurement in angle-resolved microellipsometry,” in Imaging and Applied Optics (Optical Society of America, 2013), paper. CTu3C.5.

Knop, K.

Kong, W.

H.-T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 3983 (2001).
[Crossref]

Kumar, N.

N. Kumar, O. El Gawhary, S. Roy, S. E. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. 8, 13048 (2013).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15, 075707 (2013).

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105, 775–781 (2011).

N. Kumar, “Coherent Fourier scatterometry,” Ph.D. thesis (TU Delft, 2014).

Lalor, M. J.

Lee, H.

D. Kim, M. Jin, H. Lee, S. Kim, and R. Magnusson, “Snapshot conical diffraction phase image measurement in angle-resolved microellipsometry,” in Imaging and Applied Optics (Optical Society of America, 2013), paper. CTu3C.5.

Leger, J.

Li, L.

Madsen, M. H.

Magnusson, R.

D. Kim, M. Jin, H. Lee, S. Kim, and R. Magnusson, “Snapshot conical diffraction phase image measurement in angle-resolved microellipsometry,” in Imaging and Applied Optics (Optical Society of America, 2013), paper. CTu3C.5.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005), Chap. 7.

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005), Chap. 7.

Marx, E.

Maubauch, J.

M. van Kraaij and J. Maubauch, “A more efficient rigorous coupled-wave analysis algorithm,” in Progress in Industrial Mathematics at ECMI 2004, Vol. 8 of Mathematics in Industry, (Springer, 2006), pp. 164–168.

Maystre, D.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).

Moharam, M. G.

Osten, W.

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

V. F. Paz, S. Peterhansel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1, e36 (2012).

Patrick, H. J.

Paz, V. F.

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

V. F. Paz, S. Peterhansel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1, e36 (2012).

Pedrini, G.

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

Pereira, S. E.

N. Kumar, O. El Gawhary, S. Roy, S. E. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. 8, 13048 (2013).
[Crossref]

Pereira, S. F.

S. Roy, A. C. Assafrão, S. F. Pereira, and H. P. Urbach, “Coherent Fourier scatterometry for detection of nanometer-sized particles on a planar substrate surface,” Opt. Express 22, 13250–13262 (2014).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15, 075707 (2013).

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105, 775–781 (2011).

Peterhansel, S.

V. F. Paz, S. Peterhansel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1, e36 (2012).

Petit, J.

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

Pilarski, F.

B. B. M. Wurm and F. Pilarski, “A new flexible scatterometer for critical dimensional metrology,” Rev. Sci. Instrum. 81, 023701 (2010).
[Crossref]

Qian, S.

S. Qian, Introduction to Time-Frequency and Wavelet Transform, 2nd ed. (Prentice-Hall, 2002).

Raoult, G.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).

Rayleigh, L.

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London A 79, 399–416 (1907).

Roy, S.

S. Roy, A. C. Assafrão, S. F. Pereira, and H. P. Urbach, “Coherent Fourier scatterometry for detection of nanometer-sized particles on a planar substrate surface,” Opt. Express 22, 13250–13262 (2014).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15, 075707 (2013).

N. Kumar, O. El Gawhary, S. Roy, S. E. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. 8, 13048 (2013).
[Crossref]

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005), Chap. 7.

Silver, R. M.

Stocker, M.

Teroux, L.

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

Terry, F. L.

H.-T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 3983 (2001).
[Crossref]

Tishchenko, A. V.

Török, P.

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).

Urbach, H. P.

S. Roy, A. C. Assafrão, S. F. Pereira, and H. P. Urbach, “Coherent Fourier scatterometry for detection of nanometer-sized particles on a planar substrate surface,” Opt. Express 22, 13250–13262 (2014).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15, 075707 (2013).

N. Kumar, O. El Gawhary, S. Roy, S. E. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. 8, 13048 (2013).
[Crossref]

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105, 775–781 (2011).

van den Berg, P. M.

van Kraaij, M.

M. van Kraaij and J. Maubauch, “A more efficient rigorous coupled-wave analysis algorithm,” in Progress in Industrial Mathematics at ECMI 2004, Vol. 8 of Mathematics in Industry, (Springer, 2006), pp. 164–168.

Vesperinas, M. N.

M. N. Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Scientific, 2005), Chap. 9.9, pp. 316–319.

Wilson, T.

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).

Wurm, B. B. M.

B. B. M. Wurm and F. Pilarski, “A new flexible scatterometer for critical dimensional metrology,” Rev. Sci. Instrum. 81, 023701 (2010).
[Crossref]

Zhan, Q.

Appl. Opt. (3)

Appl. Phys. B (1)

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105, 775–781 (2011).

Appl. Phys. Lett. (1)

H.-T. Huang, W. Kong, and F. L. Terry, “Normal-incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. 78, 3983 (2001).
[Crossref]

J. Eur. Opt. Soc. (1)

N. Kumar, O. El Gawhary, S. Roy, S. E. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. 8, 13048 (2013).
[Crossref]

J. Micro/Nanolithogr. MEMS MOEMS (1)

A. Faridian, V. F. Paz, K. Frenner, G. Pedrini, A. D. Boef, and W. Osten, “Phase-sensitive structured illumination to detect nanosized asymmetries in silicon trenches,” J. Micro/Nanolithogr. MEMS MOEMS 14, 021104 (2015).

J. Opt. (2)

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15, 075707 (2013).

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).

J. Opt. Soc. Am. (3)

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

Light Sci. Appl. (1)

V. F. Paz, S. Peterhansel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1, e36 (2012).

Opt. Commun. (1)

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).

Opt. Express (3)

Opt. Lasers Eng. (1)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementation,” Opt. Lasers Eng. 45, 304–317 (2007).

Proc. R. Soc. London A (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London A 79, 399–416 (1907).

Proc. SPIE (1)

P. Boher, J. Petit, L. Teroux, J. Foucher, Y. Desieres, J. Hazard, and P. Chaton, “Optical Fourier transform scatterometry for LER and LWR metrology,” Proc. SPIE 5752, 192–203 (2005).
[Crossref]

Rev. Sci. Instrum. (1)

B. B. M. Wurm and F. Pilarski, “A new flexible scatterometer for critical dimensional metrology,” Rev. Sci. Instrum. 81, 023701 (2010).
[Crossref]

Other (7)

D. Kim, M. Jin, H. Lee, S. Kim, and R. Magnusson, “Snapshot conical diffraction phase image measurement in angle-resolved microellipsometry,” in Imaging and Applied Optics (Optical Society of America, 2013), paper. CTu3C.5.

M. van Kraaij and J. Maubauch, “A more efficient rigorous coupled-wave analysis algorithm,” in Progress in Industrial Mathematics at ECMI 2004, Vol. 8 of Mathematics in Industry, (Springer, 2006), pp. 164–168.

M. N. Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Scientific, 2005), Chap. 9.9, pp. 316–319.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005), Chap. 7.

N. Kumar, “Coherent Fourier scatterometry,” Ph.D. thesis (TU Delft, 2014).

M. Gdeisat and F. Lilley, “Two-dimensional phase unwrapping problem,” https://www.ljmu.ac.uk/~/media/files/ljmu/about-us/faculties-and-schools/tae/geri/two_dimensional_phase_unwrapping_finalpdf.pdf?la=en .

S. Qian, Introduction to Time-Frequency and Wavelet Transform, 2nd ed. (Prentice-Hall, 2002).

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Figures (5)

Fig. 1.
Fig. 1. Schematic diagram of our approach to the problem of one shot scattering matrix determination for a large number of incident angles. (A) The x y z coordinate system is attached to the sample with the z = 0 plane chosen at the interface between the grating and the incident medium, which is also the geometric focus plane of the objective. (B) The far-field maps of the complex amplitude of the field is projected on the CCD in the exit pupil of the objective where the coordinate system ( ξ η ) is chosen such that ξ and η are parallel to the x and y direction, respectively. (C) The interface between the half-space z < 0 and the grating and the interface between the grating and the substrate z > d are indicated by the dotted lines.
Fig. 2.
Fig. 2. Schematic overview of the experimental setup. S, He–Ne laser; SMF, single-mode fiber; L1, collimating lens; L2 and L3, telescopic lenses; BS, nonpolarizing beam splitter; POL in and POL out : polarizers; MO, microscope objective; PZT, piezoelectric translation stage; CCD, data acquisition camera.
Fig. 3.
Fig. 3. Correlation function ρ as a function of the piezo translation stage displacement. The maxima and minimums of the correlation curve are examined to monitor the movement of the piezo. Only four experimental images are shown for simplicity.
Fig. 4.
Fig. 4. Measured (left) and simulated (right) intensities of the far field scattered by a grating illuminated by a focused field for different combinations of input and output polarizations. The incident wavelength is 633 nm, and NA = 0.4 . The grating parameters are given in Table 1.
Fig. 5.
Fig. 5. Phase of the scattered far field retrieved from measurements (left) and simulations (right) for different combinations of input and output polarizations. The incident wavelength is 633 nm, and NA = 0.4 . The grating parameters are given in Table 1.

Tables (1)

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Table 1. Physical Dimensions of the Grating Under Investigation

Equations (9)

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E r = m [ E m s r s + E m p r p ] exp [ i ( k m x r x + k m y r y k m z r z ) ] , z < 0 ,
k m x r = k x i + j 2 π Λ , k m y r = k y i .
k m z r = ( k i ) 2 ( k m x r ) 2 ( k m y r ) 2 ,
R m = ( r m s s e i ϕ m s s r m s p e i ϕ m s p r m p s e i ϕ m p s r m p p e i ϕ m p p ) ,
ξ = k x r k i η = k y r k i .
R m = f Ω R m Ω 1 ,
I μ ν = I μ ν ref + I μ ν obj + 2 I μ ν ref I μ ν obj cos ( ϕ r ) ,
ϕ μ , ν = arctan [ 2 ( I 2 I 4 ) 2 I 3 I 5 I 1 ] .
ρ j = ξ , η [ ( I j ( ξ , η ) I j ) ( I 1 ( ξ , η ) I 1 ) ] [ ξ , η ( I j ( ξ , η ) I j ) 2 ] [ ξ , η ( I 1 ( ξ , η ) I 1 ) ] 2 ,

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