Abstract

The effects of polarizer leakage on reflectivity measurements have been determined analytically and experimentally. It is shown that for values of the extinction ratio larger than 10−3 the inferred refractive index of dielectric materials from reflectivity measurements maybe in error by 27% or more. On the other hand, the real and imaginary parts of the index of carbonaceous materials may be in error by more than 30 and 60%, respectively. Furthermore, when the specularity index of smooth surfaces is to be determined, significant differences (68% or more) may result for the same range of values of the extinction ratio. A method is presented for correcting the reflectivity measurements for polarizer leakage based on the extinction ratio of the polarizer. In addition, analytical expressions are provided for the determination of the actual value of the extinction ratio of a given polarizing film at any wavelength. The effects of the typical and actual values of the extinction ratio on the inferred optical constants and specularity index of dielectric and absorbing materials are assessed.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. I. Simon, “Spectroscopy in Infrared by Reflection and Its Use for Highly Absorbing Substances,” J. Opt. Soc. Am. 41, 336–345 (1951).
    [CrossRef]
  2. D. G. Avery, “An Improved Method for Measurements of Optical Constants by Reflection,” Phys. Soc. B 65, 425–428 (1952).
    [CrossRef]
  3. J. R. Beattie, G. K. T. Conn, “Optical Constants of Metals in the Infrared—Principles of Measurement,” Philos. Mag. 46, 222–234 (1955).
  4. S. P. F. Humphreys-Owen, “Comparison of Reflection Methods for Measuring Optical Constants Without Polarimetric Analysis, and Proposal for New Methods Based on the Brewster Angle,” Proc. Phys. Soc. 77, 949–957 (1961).
    [CrossRef]
  5. W. R. Hunter, “Errors in Using the Reflectance vs Angle of Incidence Method for Measuring Optical Constants,” J. Opt. Soc. Am. 55, 1197–1204 (1965).
    [CrossRef]
  6. R. F. Miller, A. J. Taylor, L. S. Julien, “The Optimum Angle of Incidence for Determining Optical Constants from Reflectance Measurements,” J. Phys. D 3, 1957–1961 (1970).
    [CrossRef]
  7. T. S. Robinson, W. C. Price, “The Determination of the Infrared Absorption Spectra from Reflection Measurements,” Proc. Phys. Soc. 66, II-B, 969–974 (1953).
    [CrossRef]
  8. I. Simon, H. O. McMahon, “Study of the Structure of Quartz, Cristobalite, and Vitreous Silica by Reflection in Infrared,” J. Chem. Phys. 21, 23–30 (1953).
    [CrossRef]
  9. J. R. Beattie, “Optical Constants of Metals in the Infrared—Experimental Methods,” Philos. Mag. 46, 235–245 (1955).
  10. J. T. McCartney, S. Ergun, “Optical Properties of Graphite and Coal,” Fuel 37, 272–282 (1958).
  11. L. A. Gilbert, “Refractive Indices and Absorption Coefficients of Coal in Bulk Measured in the Range 6000 to 2400 Å by a Polarized Light Technique,” Fuel 41, 351–358 (1962).
  12. P. J. Foster, C. R. Howarth, “Optical Constants of Carbons and Coals in the Infrared,” Carbon 6, 719–729 (1963).
    [CrossRef]
  13. W. H. Dalzell, A. F. Sarofim, “Optical Constant of Soot and Their Application to Heat-Flux Calculations,” Trans. ASME, 91, 100–104 (1969).
    [CrossRef]
  14. E. A. Taft, H. R. Philipp, “Optical Properties of Graphite,” Phys. Rev. 138, A197–A202 (1965).
    [CrossRef]
  15. M. W. Williams, E. T. Arakawa, “Optical Properties of Glassy Carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
    [CrossRef]
  16. V. P. Tomaselli, R. Rivera, D. C. Edewaard, K. D. Moller, “Infrared Optical Constants of Black Powders Determined from Reflection Measurements,” Appl. Opt. 20, 3961–3967 (1981).
    [CrossRef] [PubMed]
  17. J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the Refractive Indices of Soot Particles from the Reflectivities of Compressed Soot Pellets,” Combust. Sci. Tech. 37, 263–284 (1984).
    [CrossRef]
  18. D. G. Goodwin, M. Mitchner, “Measurements of the Near Infrared Optical Properties of Coal Ash,” in ASME Heat Transfer Conference (1984), 84-HT-41.
  19. C. E. Batten, “Spectral Optical Constants of Soot from Polarized Angular Reflectance Measurements,” Appl. Opt. 24, 1193–1199 (1985).
    [CrossRef] [PubMed]
  20. J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978), pp. 10–13.
  21. Optics and Filters, Vol. 3 (Oriel, CT, USA, 1989), p. 48.
  22. G. K. T. Conn, G. K. Eaton, “On a Systematic Error in the Measurement of Optical Constants,” J. Opt. Soc. Am. 44, 477–483 (1954).
    [CrossRef]
  23. A. N. Rusk, D. Williams, M. R. Querry, “Optical Constants of Water in the Infrared,” J. Opt. Soc. Am. 61, 895–903 (1971).
    [CrossRef]
  24. R. W. Stobie, M. J. Dignam, “Transmission Properties of Grid Polarizers,” Appl. Opt. 12, 1390–1391 (1973).
    [CrossRef] [PubMed]
  25. H. C. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967), p. 33.
  26. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981), p. 100.
  27. Optics Guide 3 (Melles Griot, CA, USA, 1985), p. 59.

1985 (1)

1984 (1)

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the Refractive Indices of Soot Particles from the Reflectivities of Compressed Soot Pellets,” Combust. Sci. Tech. 37, 263–284 (1984).
[CrossRef]

1981 (1)

1973 (1)

1972 (1)

M. W. Williams, E. T. Arakawa, “Optical Properties of Glassy Carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[CrossRef]

1971 (1)

1970 (1)

R. F. Miller, A. J. Taylor, L. S. Julien, “The Optimum Angle of Incidence for Determining Optical Constants from Reflectance Measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

1969 (1)

W. H. Dalzell, A. F. Sarofim, “Optical Constant of Soot and Their Application to Heat-Flux Calculations,” Trans. ASME, 91, 100–104 (1969).
[CrossRef]

1965 (2)

1963 (1)

P. J. Foster, C. R. Howarth, “Optical Constants of Carbons and Coals in the Infrared,” Carbon 6, 719–729 (1963).
[CrossRef]

1962 (1)

L. A. Gilbert, “Refractive Indices and Absorption Coefficients of Coal in Bulk Measured in the Range 6000 to 2400 Å by a Polarized Light Technique,” Fuel 41, 351–358 (1962).

1961 (1)

S. P. F. Humphreys-Owen, “Comparison of Reflection Methods for Measuring Optical Constants Without Polarimetric Analysis, and Proposal for New Methods Based on the Brewster Angle,” Proc. Phys. Soc. 77, 949–957 (1961).
[CrossRef]

1958 (1)

J. T. McCartney, S. Ergun, “Optical Properties of Graphite and Coal,” Fuel 37, 272–282 (1958).

1955 (2)

J. R. Beattie, “Optical Constants of Metals in the Infrared—Experimental Methods,” Philos. Mag. 46, 235–245 (1955).

J. R. Beattie, G. K. T. Conn, “Optical Constants of Metals in the Infrared—Principles of Measurement,” Philos. Mag. 46, 222–234 (1955).

1954 (1)

1953 (2)

T. S. Robinson, W. C. Price, “The Determination of the Infrared Absorption Spectra from Reflection Measurements,” Proc. Phys. Soc. 66, II-B, 969–974 (1953).
[CrossRef]

I. Simon, H. O. McMahon, “Study of the Structure of Quartz, Cristobalite, and Vitreous Silica by Reflection in Infrared,” J. Chem. Phys. 21, 23–30 (1953).
[CrossRef]

1952 (1)

D. G. Avery, “An Improved Method for Measurements of Optical Constants by Reflection,” Phys. Soc. B 65, 425–428 (1952).
[CrossRef]

1951 (1)

Arakawa, E. T.

M. W. Williams, E. T. Arakawa, “Optical Properties of Glassy Carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[CrossRef]

Avery, D. G.

D. G. Avery, “An Improved Method for Measurements of Optical Constants by Reflection,” Phys. Soc. B 65, 425–428 (1952).
[CrossRef]

Batten, C. E.

Beattie, J. R.

J. R. Beattie, “Optical Constants of Metals in the Infrared—Experimental Methods,” Philos. Mag. 46, 235–245 (1955).

J. R. Beattie, G. K. T. Conn, “Optical Constants of Metals in the Infrared—Principles of Measurement,” Philos. Mag. 46, 222–234 (1955).

Bennet, H. E.

J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978), pp. 10–13.

Bennet, J. M.

J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978), pp. 10–13.

Charalampopoulos, T. T.

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the Refractive Indices of Soot Particles from the Reflectivities of Compressed Soot Pellets,” Combust. Sci. Tech. 37, 263–284 (1984).
[CrossRef]

Conn, G. K. T.

J. R. Beattie, G. K. T. Conn, “Optical Constants of Metals in the Infrared—Principles of Measurement,” Philos. Mag. 46, 222–234 (1955).

G. K. T. Conn, G. K. Eaton, “On a Systematic Error in the Measurement of Optical Constants,” J. Opt. Soc. Am. 44, 477–483 (1954).
[CrossRef]

Dalzell, W. H.

W. H. Dalzell, A. F. Sarofim, “Optical Constant of Soot and Their Application to Heat-Flux Calculations,” Trans. ASME, 91, 100–104 (1969).
[CrossRef]

Dignam, M. J.

Eaton, G. K.

Edewaard, D. C.

Ergun, S.

J. T. McCartney, S. Ergun, “Optical Properties of Graphite and Coal,” Fuel 37, 272–282 (1958).

Felske, J. D.

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the Refractive Indices of Soot Particles from the Reflectivities of Compressed Soot Pellets,” Combust. Sci. Tech. 37, 263–284 (1984).
[CrossRef]

Foster, P. J.

P. J. Foster, C. R. Howarth, “Optical Constants of Carbons and Coals in the Infrared,” Carbon 6, 719–729 (1963).
[CrossRef]

Gilbert, L. A.

L. A. Gilbert, “Refractive Indices and Absorption Coefficients of Coal in Bulk Measured in the Range 6000 to 2400 Å by a Polarized Light Technique,” Fuel 41, 351–358 (1962).

Goodwin, D. G.

D. G. Goodwin, M. Mitchner, “Measurements of the Near Infrared Optical Properties of Coal Ash,” in ASME Heat Transfer Conference (1984), 84-HT-41.

Hottel, H. C.

H. C. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967), p. 33.

Howarth, C. R.

P. J. Foster, C. R. Howarth, “Optical Constants of Carbons and Coals in the Infrared,” Carbon 6, 719–729 (1963).
[CrossRef]

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981), p. 100.

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, “Comparison of Reflection Methods for Measuring Optical Constants Without Polarimetric Analysis, and Proposal for New Methods Based on the Brewster Angle,” Proc. Phys. Soc. 77, 949–957 (1961).
[CrossRef]

Hunter, W. R.

Hura, H.

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the Refractive Indices of Soot Particles from the Reflectivities of Compressed Soot Pellets,” Combust. Sci. Tech. 37, 263–284 (1984).
[CrossRef]

Julien, L. S.

R. F. Miller, A. J. Taylor, L. S. Julien, “The Optimum Angle of Incidence for Determining Optical Constants from Reflectance Measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

McCartney, J. T.

J. T. McCartney, S. Ergun, “Optical Properties of Graphite and Coal,” Fuel 37, 272–282 (1958).

McMahon, H. O.

I. Simon, H. O. McMahon, “Study of the Structure of Quartz, Cristobalite, and Vitreous Silica by Reflection in Infrared,” J. Chem. Phys. 21, 23–30 (1953).
[CrossRef]

Miller, R. F.

R. F. Miller, A. J. Taylor, L. S. Julien, “The Optimum Angle of Incidence for Determining Optical Constants from Reflectance Measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

Mitchner, M.

D. G. Goodwin, M. Mitchner, “Measurements of the Near Infrared Optical Properties of Coal Ash,” in ASME Heat Transfer Conference (1984), 84-HT-41.

Moller, K. D.

Philipp, H. R.

E. A. Taft, H. R. Philipp, “Optical Properties of Graphite,” Phys. Rev. 138, A197–A202 (1965).
[CrossRef]

Price, W. C.

T. S. Robinson, W. C. Price, “The Determination of the Infrared Absorption Spectra from Reflection Measurements,” Proc. Phys. Soc. 66, II-B, 969–974 (1953).
[CrossRef]

Querry, M. R.

Rivera, R.

Robinson, T. S.

T. S. Robinson, W. C. Price, “The Determination of the Infrared Absorption Spectra from Reflection Measurements,” Proc. Phys. Soc. 66, II-B, 969–974 (1953).
[CrossRef]

Rusk, A. N.

Sarofim, A. F.

W. H. Dalzell, A. F. Sarofim, “Optical Constant of Soot and Their Application to Heat-Flux Calculations,” Trans. ASME, 91, 100–104 (1969).
[CrossRef]

H. C. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967), p. 33.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981), p. 100.

Simon, I.

I. Simon, H. O. McMahon, “Study of the Structure of Quartz, Cristobalite, and Vitreous Silica by Reflection in Infrared,” J. Chem. Phys. 21, 23–30 (1953).
[CrossRef]

I. Simon, “Spectroscopy in Infrared by Reflection and Its Use for Highly Absorbing Substances,” J. Opt. Soc. Am. 41, 336–345 (1951).
[CrossRef]

Stobie, R. W.

Taft, E. A.

E. A. Taft, H. R. Philipp, “Optical Properties of Graphite,” Phys. Rev. 138, A197–A202 (1965).
[CrossRef]

Taylor, A. J.

R. F. Miller, A. J. Taylor, L. S. Julien, “The Optimum Angle of Incidence for Determining Optical Constants from Reflectance Measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

Tomaselli, V. P.

Williams, D.

Williams, M. W.

M. W. Williams, E. T. Arakawa, “Optical Properties of Glassy Carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[CrossRef]

Appl. Opt. (3)

Carbon (1)

P. J. Foster, C. R. Howarth, “Optical Constants of Carbons and Coals in the Infrared,” Carbon 6, 719–729 (1963).
[CrossRef]

Combust. Sci. Tech. (1)

J. D. Felske, T. T. Charalampopoulos, H. Hura, “Determination of the Refractive Indices of Soot Particles from the Reflectivities of Compressed Soot Pellets,” Combust. Sci. Tech. 37, 263–284 (1984).
[CrossRef]

Fuel (2)

J. T. McCartney, S. Ergun, “Optical Properties of Graphite and Coal,” Fuel 37, 272–282 (1958).

L. A. Gilbert, “Refractive Indices and Absorption Coefficients of Coal in Bulk Measured in the Range 6000 to 2400 Å by a Polarized Light Technique,” Fuel 41, 351–358 (1962).

J. Appl. Phys. (1)

M. W. Williams, E. T. Arakawa, “Optical Properties of Glassy Carbon from 0 to 82 eV,” J. Appl. Phys. 43, 3460–3463 (1972).
[CrossRef]

J. Chem. Phys. (1)

I. Simon, H. O. McMahon, “Study of the Structure of Quartz, Cristobalite, and Vitreous Silica by Reflection in Infrared,” J. Chem. Phys. 21, 23–30 (1953).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. D (1)

R. F. Miller, A. J. Taylor, L. S. Julien, “The Optimum Angle of Incidence for Determining Optical Constants from Reflectance Measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

Philos. Mag. (2)

J. R. Beattie, “Optical Constants of Metals in the Infrared—Experimental Methods,” Philos. Mag. 46, 235–245 (1955).

J. R. Beattie, G. K. T. Conn, “Optical Constants of Metals in the Infrared—Principles of Measurement,” Philos. Mag. 46, 222–234 (1955).

Phys. Rev. (1)

E. A. Taft, H. R. Philipp, “Optical Properties of Graphite,” Phys. Rev. 138, A197–A202 (1965).
[CrossRef]

Phys. Soc. B (1)

D. G. Avery, “An Improved Method for Measurements of Optical Constants by Reflection,” Phys. Soc. B 65, 425–428 (1952).
[CrossRef]

Proc. Phys. Soc. (2)

S. P. F. Humphreys-Owen, “Comparison of Reflection Methods for Measuring Optical Constants Without Polarimetric Analysis, and Proposal for New Methods Based on the Brewster Angle,” Proc. Phys. Soc. 77, 949–957 (1961).
[CrossRef]

T. S. Robinson, W. C. Price, “The Determination of the Infrared Absorption Spectra from Reflection Measurements,” Proc. Phys. Soc. 66, II-B, 969–974 (1953).
[CrossRef]

Trans. ASME (1)

W. H. Dalzell, A. F. Sarofim, “Optical Constant of Soot and Their Application to Heat-Flux Calculations,” Trans. ASME, 91, 100–104 (1969).
[CrossRef]

Other (6)

J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978), pp. 10–13.

Optics and Filters, Vol. 3 (Oriel, CT, USA, 1989), p. 48.

D. G. Goodwin, M. Mitchner, “Measurements of the Near Infrared Optical Properties of Coal Ash,” in ASME Heat Transfer Conference (1984), 84-HT-41.

H. C. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967), p. 33.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981), p. 100.

Optics Guide 3 (Melles Griot, CA, USA, 1985), p. 59.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Calculated state of polarization (RR/R + R) and reflectivity ratio (R/R) as a function of the angle of incidence θ with and without the correction for the extinction ratio .

Fig. 2
Fig. 2

Reflectometer/monochromator/detection system: LS, light source; RCM, rotatable concave mirror, f.l. = 63.5 mm; I = iris; LC, light chopper; P1, plane mirror, diameter = 50 mm; C1, spherical mirror, f.l. = 150 mm, diameter = 50 mm; SH, sample holder; P2, plane mirror, diameter = 50 mm; C2, spherical mirror, f.l. = 200 mm, diameter = 50 mm; RB, rotatable base; P, polarizer; MS, monochromator slit; M, grating monochromator; F, cut-on wavelength filter; D, detector.

Fig. 3
Fig. 3

Angular variations of the polarized components of the specular reflectivities at 320 nm. Points are the data; solid lines are the Fresnel equations with m ¯ = 1.52–i0.20.

Fig. 4
Fig. 4

Angular variations of the polarized components of the specular reflectivities at 320 nm. Points are the data corrected for polarizer leakage effects with = 0.00794; solid lines are the Fresnel equations with m ¯ = 1.54–i0.03.

Tables (7)

Tables Icon

Table I Inferred Specularity and Refractive Indices of a Dielectric( m ¯ = 1.6) and Absorbing Material ( m ¯ = 2.0–i1.0) as a Function of the Polarizer Extinction Ratio

Tables Icon

Table II Characteristics of the Plane and Concave Mirrors Used In the Experiments

Tables Icon

Table III Cut-on Wavelength Filter Characteristics

Tables Icon

Table IV Photodetector Characteristics

Tables Icon

Table V Real Part of the Refractive Index n for Borosillcate Glass BK-7 as Determined from Angular Reflection Measurements With and Without Correction for the Extinction Ratio

Tables Icon

Table VI Real n and Imaginary k Part of the Index of Amorphous Carbon Rod With and Without Correction for Various Values of the Extinction Ratio

Tables Icon

Table VII Specularity (S.I.) and Refractive Index n of BK-7 Glass With and Without Correction for the Extinction Ratio of the Polarizer at 900 nm

Equations (34)

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

R = I , r I , i ,
R = I , r I , i ,
R = a 2 + b 2 - 2 a cos θ + cos 2 θ a 2 + b 2 + 2 a cos θ + cos 2 θ ,
R = ( a 2 + b 2 - 2 a sin θ tan θ + sin 2 θ tan 2 θ a 2 + b 2 + 2 a sin θ tan θ + sin 2 θ tan 2 θ ) R ,
2 a 2 = [ ( n 2 - k 2 - sin 2 θ ) 2 + 4 n 2 k 2 ] 1 / 2 + ( n 2 - k 2 - sin 2 θ ) , 2 b 2 = [ ( n 2 - k 2 - sin 2 θ ) 2 + 4 n 2 k 2 ] 1 / 2 + ( n 2 - k 2 - sin 2 θ ) .
R 2 R | θ = 45 ° = 1.0 ,
L τ p = L τ p .
I d , = r τ p , I i , + r I i , L ,
I d , = r τ p , I i , + r I i , L ,
I d r , = r R τ p I i , + r R L I i , ,
I d r , = r R τ p I i , + r R L I i , .
R m , = r R τ p , I i , + r R L I i , r τ p , I i , + r I i , L ,
R m , = r R τ p , I i , + r R L I i , r τ p , I i , + r L I i , .
R m , = R + ( r r ) ( 1 τ p ) ( I i , I i , ) L R 1 + ( r r ) ( 1 τ p ) ( I i , I i , ) L ,
R m , = R + ( r r ) ( 1 τ p ) ( I i , I i , ) L R 1 + ( r r ) ( 1 τ p ) ( I i , I i , ) L .
α = K + L τ p 1 + K ( L τ p ) = K + 1 + K ,
R m , = R + K R 1 + K ,
R m , = R + K R 1 + K .
R = R m , ( 1 + K ) - K R m , ( 1 + K ) 1 - 2 ,
R = R m , ( 1 + K ) - ( K ) R m , ( 1 + K ) 1 - 2 .
I d r , k = r R k τ p I i , + r R k L I i , ,
I d r , k = r R k τ p I i , + r R k L I i , ,
α = I d r , I d r , k = R + K R R k + K R k ,
α = I d r , I d r , k = R + K R R k + K R k .
K = ( - b ) R k ( b - 1 ) R k ,
R = α ( R k + K R k ) - α ( K R k + 2 R k ) 1 - 2 ,
R = α ( R k + K R k ) - α ( K R k + 2 R k ) 1 - 2 ,
F = i = 1 N ( G th , i - G exp , i ) 2 ,
R 2 R | θ = 45 ° = 1.0 ,
( 1 - α ) - 2 ( R m , - α R m , ) 2 ( α - ) - 1 ( α R m , - R m , ) | θ = 45 ° = 1.0.
3 β + 2 γ + δ + ζ = 0 ,
β = α 2 ( R m , - R m , 2 ) , γ = α ( R m , - 1 ) ( α 2 R m , + 2 R m , ) , δ = ( 1 - R m , ) ( 2 α 2 R m , + R m , ) , ζ = α ( R m , 2 - R m , ) ,
3 ϕ + 2 x + y + ω = 0.
ϕ = b 2 ( α 2 η - α ) , x = b ( b 2 α + 2 α - 2 α α η - b 2 α 2 η ) , y = 2 b 2 α α η + α 2 η - 2 b 2 α - α , ω = b ( α - η α 2 ) ,

Metrics