Abstract

A method is described in which approximate correction for spectral slit broadening is performed at the same time that reflectance or transmittance data are being fitted with analytic dispersion curves. The method is valid for spectra whose structure is not too sharp on a scale comparable with the spectrometer resolution. It is not to be used for sharp line spectra. The key problem that is solved is to find the fastest way to compute the slit broadening effect on the spectra obtained from the trial dispersion curves. The method is applied, as an example, to analyze some reststrahlen data taken with a grating spectrometer in the far ir from a sample of PrCl3. The same approximate method could also be used with data from other types of spectrometers having different slit broadening functions, such as from two-crystal x-ray spectrometers, in regions where very sharp spikes do not appear in the spectra.

© 1968 Optical Society of America

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References

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  1. See, for example, P. McWilliams, W. S. Hall, H. E. Wegner, Rev. Sci. Instrum. 33, 70(1962).
    [CrossRef]
  2. A. S. Eddington, Mon. Not. Roy. Astron. Soc. 73, 359 (1913).
  3. R. J. Trumpler, H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, 1953), p. 101.
  4. R. N. Bracewell, J. Opt. Soc. Amer. 45, 873 (1955).
    [CrossRef]
  5. H. W. Verleur, “Determination of Optical Constants from Reflectivity or Transmission Measurements on Bulk Crystals or Thin Films” (to be published in J. Opt. Soc. Amer.).
  6. M. Lax, J. Phys. Chem. Solids 25, 487 (1964).
    [CrossRef]
  7. A. S. Barker, Ferroelectricity, E. F. Weller, Ed. (Elsevier Publishing Company, Amsterdam, 1967), p. 213ff (Eq. 41). For some extreme values of parameters, Barker’s model is not causal.
  8. K. de L. Kronig, J. Opt. Soc. Amer. 12, 547 (1926).
    [CrossRef]
  9. H. A. Kramers, Atti Congr. Intern. Fis. Como 2, 545 (1927).
  10. H. W. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Company, New York, 1945), Chap. 14.
  11. T. S. Robinson, Proc. Phys. Soc. London, Series B, 65, 910 (1952).
    [CrossRef]
  12. C. Kittel, Elementary Statistical Physics (John Wiley & Sons, New York, 1958), p. 206 ff.
  13. D. W. Berreman, Appl. Opt. 6, 1519 (1967).
    [CrossRef] [PubMed]
  14. R. F. Wallis, A. A. Maradudin, Phys. Rev. 125, 1277 (1962).
    [CrossRef]
  15. J. A. Schilling (née Moraller), Bell Telephone Labs., Holmdel, N. J., private communication.
  16. M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1954), Eq. 10.6.
  17. P. Drude, Theory of Optics (Longmans, Green and Co., New York, 1902), Sect. II, Chap. 5, Eq. 18.
  18. See, for example, Ref. 17, Chap. 2, Eq. 23.
  19. M. Born, E. Wolf, Principles of Optics (Macmillan Company, New York, 1964), 2nd ed., p. 48.

1967 (1)

1964 (1)

M. Lax, J. Phys. Chem. Solids 25, 487 (1964).
[CrossRef]

1962 (2)

See, for example, P. McWilliams, W. S. Hall, H. E. Wegner, Rev. Sci. Instrum. 33, 70(1962).
[CrossRef]

R. F. Wallis, A. A. Maradudin, Phys. Rev. 125, 1277 (1962).
[CrossRef]

1955 (1)

R. N. Bracewell, J. Opt. Soc. Amer. 45, 873 (1955).
[CrossRef]

1952 (1)

T. S. Robinson, Proc. Phys. Soc. London, Series B, 65, 910 (1952).
[CrossRef]

1927 (1)

H. A. Kramers, Atti Congr. Intern. Fis. Como 2, 545 (1927).

1926 (1)

K. de L. Kronig, J. Opt. Soc. Amer. 12, 547 (1926).
[CrossRef]

1913 (1)

A. S. Eddington, Mon. Not. Roy. Astron. Soc. 73, 359 (1913).

Barker, A. S.

A. S. Barker, Ferroelectricity, E. F. Weller, Ed. (Elsevier Publishing Company, Amsterdam, 1967), p. 213ff (Eq. 41). For some extreme values of parameters, Barker’s model is not causal.

Berreman, D. W.

Bode, H. W.

H. W. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Company, New York, 1945), Chap. 14.

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan Company, New York, 1964), 2nd ed., p. 48.

M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1954), Eq. 10.6.

Bracewell, R. N.

R. N. Bracewell, J. Opt. Soc. Amer. 45, 873 (1955).
[CrossRef]

Drude, P.

P. Drude, Theory of Optics (Longmans, Green and Co., New York, 1902), Sect. II, Chap. 5, Eq. 18.

Eddington, A. S.

A. S. Eddington, Mon. Not. Roy. Astron. Soc. 73, 359 (1913).

Hall, W. S.

See, for example, P. McWilliams, W. S. Hall, H. E. Wegner, Rev. Sci. Instrum. 33, 70(1962).
[CrossRef]

Huang, K.

M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1954), Eq. 10.6.

Kittel, C.

C. Kittel, Elementary Statistical Physics (John Wiley & Sons, New York, 1958), p. 206 ff.

Kramers, H. A.

H. A. Kramers, Atti Congr. Intern. Fis. Como 2, 545 (1927).

Kronig, K. de L.

K. de L. Kronig, J. Opt. Soc. Amer. 12, 547 (1926).
[CrossRef]

Lax, M.

M. Lax, J. Phys. Chem. Solids 25, 487 (1964).
[CrossRef]

Maradudin, A. A.

R. F. Wallis, A. A. Maradudin, Phys. Rev. 125, 1277 (1962).
[CrossRef]

McWilliams, P.

See, for example, P. McWilliams, W. S. Hall, H. E. Wegner, Rev. Sci. Instrum. 33, 70(1962).
[CrossRef]

Robinson, T. S.

T. S. Robinson, Proc. Phys. Soc. London, Series B, 65, 910 (1952).
[CrossRef]

Schilling (née Moraller), J. A.

J. A. Schilling (née Moraller), Bell Telephone Labs., Holmdel, N. J., private communication.

Trumpler, R. J.

R. J. Trumpler, H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, 1953), p. 101.

Verleur, H. W.

H. W. Verleur, “Determination of Optical Constants from Reflectivity or Transmission Measurements on Bulk Crystals or Thin Films” (to be published in J. Opt. Soc. Amer.).

Wallis, R. F.

R. F. Wallis, A. A. Maradudin, Phys. Rev. 125, 1277 (1962).
[CrossRef]

Weaver, H. F.

R. J. Trumpler, H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, 1953), p. 101.

Wegner, H. E.

See, for example, P. McWilliams, W. S. Hall, H. E. Wegner, Rev. Sci. Instrum. 33, 70(1962).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan Company, New York, 1964), 2nd ed., p. 48.

Appl. Opt. (1)

Atti Congr. Intern. Fis. Como (1)

H. A. Kramers, Atti Congr. Intern. Fis. Como 2, 545 (1927).

J. Opt. Soc. Amer. (2)

R. N. Bracewell, J. Opt. Soc. Amer. 45, 873 (1955).
[CrossRef]

K. de L. Kronig, J. Opt. Soc. Amer. 12, 547 (1926).
[CrossRef]

J. Phys. Chem. Solids (1)

M. Lax, J. Phys. Chem. Solids 25, 487 (1964).
[CrossRef]

Mon. Not. Roy. Astron. Soc. (1)

A. S. Eddington, Mon. Not. Roy. Astron. Soc. 73, 359 (1913).

Phys. Rev. (1)

R. F. Wallis, A. A. Maradudin, Phys. Rev. 125, 1277 (1962).
[CrossRef]

Proc. Phys. Soc. London (1)

T. S. Robinson, Proc. Phys. Soc. London, Series B, 65, 910 (1952).
[CrossRef]

Rev. Sci. Instrum. (1)

See, for example, P. McWilliams, W. S. Hall, H. E. Wegner, Rev. Sci. Instrum. 33, 70(1962).
[CrossRef]

Other (10)

A. S. Barker, Ferroelectricity, E. F. Weller, Ed. (Elsevier Publishing Company, Amsterdam, 1967), p. 213ff (Eq. 41). For some extreme values of parameters, Barker’s model is not causal.

R. J. Trumpler, H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, 1953), p. 101.

H. W. Verleur, “Determination of Optical Constants from Reflectivity or Transmission Measurements on Bulk Crystals or Thin Films” (to be published in J. Opt. Soc. Amer.).

C. Kittel, Elementary Statistical Physics (John Wiley & Sons, New York, 1958), p. 206 ff.

H. W. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Company, New York, 1945), Chap. 14.

J. A. Schilling (née Moraller), Bell Telephone Labs., Holmdel, N. J., private communication.

M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1954), Eq. 10.6.

P. Drude, Theory of Optics (Longmans, Green and Co., New York, 1902), Sect. II, Chap. 5, Eq. 18.

See, for example, Ref. 17, Chap. 2, Eq. 23.

M. Born, E. Wolf, Principles of Optics (Macmillan Company, New York, 1964), 2nd ed., p. 48.

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Figures (5)

Fig. 1
Fig. 1

Slit functions T for triangular and gaussian slits as functions of ν, showing two- and three-delta function approximations to each [see Eqs. (14) and (A-3)].

Fig. 2
Fig. 2

Diagram of spectrometer showing symbols used in equations.

Fig. 3
Fig. 3

The dots are measured values of reflectance of PrCl3 for radiation incident at 45°. from normal and polarized perpendicular to the threefold axis of crystal symmetry. The solid curve is a best fit derived from a three-oscillator model with correction for slit broadening.

Fig. 4
Fig. 4

Real (solid line) and imaginary (dotted line) parts of for PrCl3 plotted on an arctangent distorted scale vs ν in cm−1, as computed from parameters obtained with slit correction. Dashed curves were obtained from parameters obtained without slit correction.

Fig. 5
Fig. 5

Monochromatic reflectance R0 for PrCl3 computed from Eq. (30) as a function of wavenumber ν. Spectral slit functions at ends of range of each grating are shown with widths exaggerated fivefold. Arrowheads indicate computed values of ρ0 at data wavenumbers where ρ0 differs from R0 by more than 0.4%.

Tables (1)

Tables Icon

Table I Harmonic Oscillator Model Parameters and Optic Mode Frequencies Obtained for PrCl3 with and without Slit Broadening Correction

Equations (45)

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P R ( x ) = 0 S ( ν ) T ( ν , x ) d ν ,
P ( x ) = 0 S ( ν ) R ( ν ) T ( ν , x ) d ν ;
P 0 0 [ R 0 S 0 + ( ν - ν 0 ) ( R 0 S 0 + R 0 S 0 ) ( ν - ν 0 ) 2 2 × ( R 0 S 0 + 2 R 0 S 0 + R 0 S 0 ) ] T ( ν , x 0 ) d ν ,
P R 0 0 [ S 0 + ( ν - ν 0 ) S 0 + ( ν - ν 0 ) 2 2 S 0 ] T ( ν , x 0 ) d ν .
0 ( ν - ν 0 ) T ( ν , x 0 ) d ν = 0.
ρ 0 R 0 S 0 T d ν + 1 2 ( R 0 S 0 + 2 R 0 S 0 + R 0 S 0 ) ( ν - ν 0 ) 2 T d ν S 0 T d ν + 1 2 S 0 ( ν - ν 0 ) 2 T d ν .
ρ 0 R 0 + 1 2 ( R 0 + R 0 S 0 S 0 ) 0 ( ν - ν 0 ) 2 T ( ν , x 0 ) d ν 0 T ( ν , x 0 ) d ν .
ρ 0 R 0 + 1 2 R 0 0 ( ν - ν 0 ) 2 T ( ν , x 0 ) d ν 0 T ( ν , x 0 ) d ν .
u 0 = 0 ( ν - ν 0 ) 2 T ( ν , x 0 ) d ν 0 T ( ν , x 0 ) d ν ,
ρ 0 R 0 + u 0 R 0 / 2.
R 0 R ( ν 0 + Δ ν ) + R ( ν 0 - Δ ν ) - 2 R 0 ( Δ ν ) 2 .
ρ R 0 [ 1 - u 0 ( Δ ν ) 2 ] + u 0 2 ( Δ ν ) 2 [ R ( ν 0 + Δ ν ) + R ( ν 0 - Δ ν ) ] .
α 0 = ( u 0 ) 1 2 .
ρ 0 = 1 2 [ R ( ν 0 + α 0 ) + R ( ν 0 - α 0 ) ]
T ( ν , x 0 ) = 0 for ν - ν 0 β 0 = 1 - ( ν - ν 0 ) / β 0 for ν - ν 0 < β 0 ,
Δ x = W / f ;
x 0 = arcsin ( N / 2 ν 0 )
ν 0 = N / ( 2 sin x 0 ) .
β 0 = 1 2 ( d ν / d x ) Δ x .
β 0 = ( W / f ) ν 0 [ ( ν 0 / N ) 2 - 1 4 ] 1 2 .
u 0 = - β 0 β 0 ν 2 ( 1 - | ν β 0 | ) d v / - β 0 β 0 ( 1 - | ν β 0 | ) d ν = β 0 2 / 6.
ρ = R 0 + 1 2 R 0 β 0 2 / 6.
α 0 = β 0 / ( 6 ) 1 2
T ( ν , x 0 ) = exp { - [ ( x - x 0 ) / Δ x ] 2 } ,
x 0 = arccos ( 2 ν 0 d )
ν 0 = ( 2 d cos x 0 ) - 1 ,
x - x 0 ( ν - ν 0 ) d x d ν | ν 0 2 ( ν - ν 0 ) d [ 1 - ( 2 ν 0 d ) 2 ] 1 2 ,
T ( ν , x 0 ) exp { - [ ( ν - ν 0 ) / β 0 ] 2 } .
β 0 = ( Δ x / d ) [ 1 4 - ( ν 0 d ) 2 ] 1 2
u 0 = β 0 2 / 2 ,
α 0 = β 0 / ( 2 ) 1 2 .
= opt + j = 1 N Δ j 1 - ( ν - ν j ) 2 + i γ j ν / ν j ,
Z = ± ( - sin 2 θ ) 1 2 / ( cos θ ) ,
R 0 = ( 1 - Z ) / ( 1 + Z ) 2 ;
R 0 = d 2 R 0 d Z 2 ( d Z d d d ν ) 2 + d R 0 d Z d 2 Z d 2 ( d d ν ) 2 + d R 0 d Z d Z d d 2 d ν 2 .
v 0 = 0 ( ν - ν 0 ) 4 T ( ν , x 0 ) d ν 0 T ( ν , x 0 ) d ν .
ρ 0 R 0 + u 0 R 0 / 2 + v 0 R 0 IV / 24 ,
ρ 0 A R 0 + [ ( 1 - A ) / 2 ] [ R ( ν 0 + ξ 0 ) + R ( ν 0 - ξ 0 ) ] ,
ξ 0 2 = v 0 / u 0
A = 1 - ( u 0 2 / v 0 ) .
v 0 = β 0 4 / 15.
ρ 0 7 1 2 R 0 + 5 2 4 { R [ ν 0 + β 0 ( 2 / 5 ) 1 2 ] + R [ ν 0 - β 0 ( 2 / 5 ) 1 2 ] } .
v 0 = 3 β 0 4 / 4.
ρ 0 2 3 R 0 + 1 6 { R [ ν 0 + β 0 ( 3 / 2 ) 1 2 ] + R [ ν 0 - β 0 ( 3 / 2 ) 1 2 ] } .
T = { β 0 sin [ ( ν - ν 0 ) / β 0 ] / ( ν - ν 0 ) } 2 .

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