Abstract

Dispersion relations and sum rules for the dichroic reflectivity and phase shifts of circularly polarized modes are developed for the magneto-optical case. The reduction in crossing-relation symmetry arising from the presence of a magnetic field and the consequent non-Kramers-Kronig form of the dichroism dispersion relations are discussed in terms of the analyticity of the amplitude reflectivity. Sum rules are derived from the low- and high-frequency limits of the dichroism dispersion relations. These rules include the general results that 0ω-1ln[r+(ω)/r-(ω)]dω=0 and 0[θ+(ω)-θ-(ω)]dω=πωc, where r±(ω) and θ±(ω) are the amplitude and phase of the amplitude reflectivity for the circular modes and ωc is the cyclotron frequency. Approximate finite-energy dispersion relations and sum rules are developed and their range of validity examined.

© 1976 Optical Society of America

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  1. M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
    [Crossref]
  2. A. Villani and A. H. Zimmerman, Phys. Lett. A 44, 295 (1973); and Phys. Rev. B 8, 3914 (1973).
    [Crossref]
  3. M. Altarelli and D. Y. Smith, Phys. Rev. B 9, 1290 (1974).
    [Crossref]
  4. F. C. Jahoda, thesis (Cornell University, 1957) (unpublished);and Phys. Rev. 107, 1261 (1957).
  5. B. Velický, Czech. J. Phys. B 11, 541 (1961).
    [Crossref]
  6. M. Gottlieb, Ph. D. thesis (University of Pennsylvania, 1959) (unpublished).
  7. See, J. S. Toll, thesis (Princeton University, 1952) (unpublished);and Phys. Rev. 104, 1760 (1956).
  8. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).
  9. I. M. Boswarva, R. E. Howard, and A. B. Lidiard, Proc. R. Soc. London Ser. A 269, 125 (1962).
    [Crossref]
  10. D. Y. Smith, Proceedings of the International Symposium on Color Centers in Alkali Halides, 1968, Rome (unpublished), p. 252; Bull. Am. Phys. Soc. 19, 93 (1974); and Bull. Am. Phys. Soc. 19, 259 (1974).
  11. D. Y. Smith, J. Opt. Soc. Am. 66, 454 (1976).
    [Crossref]
  12. In this expression η must be restricted to nonzero values in the case of conductors. In these materials the dielectric function has an ω−1 singularity at ω= 0 which leads to an ω1/2 behavior in lnr˜ (ω) near ω= 0. To ensure that V(ω) is square integrable, it is therefore necessary to require η≠ 0 in Eq. (2) and in its analogue for circular modes, In insulators limω→0lnr˜ (ω) is a nonzero constant and no such restriction applies.
  13. The restriction ω≠ 0 in Eq. (12) also occurs for the derivation from the quotient function Q. It arises from having to avoid improper rearrangement under the integral sign when going from the integral from −∞ to ∞ to one from 0 to ∞.
  14. S. E. Schnatterly, Phys. Rev. 183, 664 (1969).
    [Crossref]
  15. D. Y. Smith, Phys. Rev. B13, (1976) (to be published).
  16. F. Stern, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1963), Vol. 15.
    [Crossref]
  17. A second low-frequency sum rule may be obtained from the expression for ln[r˜+(ω)/r˜-(ω)], Eq. (12), if a specific model is assumed. For example, if an insulator consists of a single magnetically active Lorentzian absorption with plasma frequency ωp centered at ω0, the low-frequency limit of ln[r˜+(ω)/r˜-(ω)] islnr˜+(ω)r˜-(ω)=2ωp2 ωcn0(n02-1) ω04ω=2ωcn0 ω02ω,         ω→0where n0 is the static index (1+ωp2/ω02)1/2. Combining this with Eq. (12) yields∫0∞θ+(ω)-θ-(ω)ω2dω=πωcn0 ω02.For more complicated models of the absorption spectrum similar forms hold but with ω0, ωp, etc., replaced with average values.
  18. H. Becquerel, C. R. Acad. Sci. (Paris) 125, 679 (1897).
  19. C. G. Darwin and W. H. Watson, Proc. R. Soc. London 114, 474 (1927).
    [Crossref]
  20. The dielectric function for this system is∊±(ω)=∊b-ωp2/(ω2-ω02±ωωc+iωγ),with the background dielectric function ∊b equal unity for vacuum.
  21. See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun,  12, 1113 (1973).
    [Crossref]
  22. An alternative approximation is to employ the inverse-second-moment phase sum rule discussed in Ref. 17 rather than the zeroth moment phase sum rule to evaluating the left-hand side of Eq. (31). This leads to a result similar to Eq. (32), but with ω¯2 replaced n0 ω¯02, where n0 is the contribution to the static index from transitions at energies ω> β.
  23. As in the vacuum background case a second sum rule derived from the low-frequency limit holds for insulators, but its form depends on the model assumed. For the single Lorentzian absorption treated in Ref. 17 the low-frequency limit of the quotent yieldsln(r˜+(ω)r˜-(ω))=2ωp2 ωcn0′(n′02-1)ω04ω,         ω→0where now n0′=(∊∞+ωp2/ω02)1/2. This leads to∫0∞1ω2[θ+(ω)-θ-(ω)] dω=πωp2 ωcn0′(n′02-1) ω04.

1976 (1)

1974 (1)

M. Altarelli and D. Y. Smith, Phys. Rev. B 9, 1290 (1974).
[Crossref]

1973 (2)

A. Villani and A. H. Zimmerman, Phys. Lett. A 44, 295 (1973); and Phys. Rev. B 8, 3914 (1973).
[Crossref]

See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun,  12, 1113 (1973).
[Crossref]

1972 (1)

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
[Crossref]

1969 (1)

S. E. Schnatterly, Phys. Rev. 183, 664 (1969).
[Crossref]

1962 (1)

I. M. Boswarva, R. E. Howard, and A. B. Lidiard, Proc. R. Soc. London Ser. A 269, 125 (1962).
[Crossref]

1961 (1)

B. Velický, Czech. J. Phys. B 11, 541 (1961).
[Crossref]

1927 (1)

C. G. Darwin and W. H. Watson, Proc. R. Soc. London 114, 474 (1927).
[Crossref]

1897 (1)

H. Becquerel, C. R. Acad. Sci. (Paris) 125, 679 (1897).

Ahrenkiel, R. K.

See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun,  12, 1113 (1973).
[Crossref]

Altarelli, M.

M. Altarelli and D. Y. Smith, Phys. Rev. B 9, 1290 (1974).
[Crossref]

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
[Crossref]

Becquerel, H.

H. Becquerel, C. R. Acad. Sci. (Paris) 125, 679 (1897).

Boswarva, I. M.

I. M. Boswarva, R. E. Howard, and A. B. Lidiard, Proc. R. Soc. London Ser. A 269, 125 (1962).
[Crossref]

Darwin, C. G.

C. G. Darwin and W. H. Watson, Proc. R. Soc. London 114, 474 (1927).
[Crossref]

Dexter, D. L.

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
[Crossref]

Gottlieb, M.

M. Gottlieb, Ph. D. thesis (University of Pennsylvania, 1959) (unpublished).

Howard, R. E.

I. M. Boswarva, R. E. Howard, and A. B. Lidiard, Proc. R. Soc. London Ser. A 269, 125 (1962).
[Crossref]

Jahoda, F. C.

F. C. Jahoda, thesis (Cornell University, 1957) (unpublished);and Phys. Rev. 107, 1261 (1957).

Lee, T. H.

See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun,  12, 1113 (1973).
[Crossref]

Lidiard, A. B.

I. M. Boswarva, R. E. Howard, and A. B. Lidiard, Proc. R. Soc. London Ser. A 269, 125 (1962).
[Crossref]

Lyu, S. L.

See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun,  12, 1113 (1973).
[Crossref]

Moser, F.

See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun,  12, 1113 (1973).
[Crossref]

Nussenzveig, H. M.

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
[Crossref]

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

Schnatterly, S. E.

S. E. Schnatterly, Phys. Rev. 183, 664 (1969).
[Crossref]

Smith, D. Y.

D. Y. Smith, J. Opt. Soc. Am. 66, 454 (1976).
[Crossref]

M. Altarelli and D. Y. Smith, Phys. Rev. B 9, 1290 (1974).
[Crossref]

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
[Crossref]

D. Y. Smith, Phys. Rev. B13, (1976) (to be published).

D. Y. Smith, Proceedings of the International Symposium on Color Centers in Alkali Halides, 1968, Rome (unpublished), p. 252; Bull. Am. Phys. Soc. 19, 93 (1974); and Bull. Am. Phys. Soc. 19, 259 (1974).

Stern, F.

F. Stern, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1963), Vol. 15.
[Crossref]

Toll, J. S.

See, J. S. Toll, thesis (Princeton University, 1952) (unpublished);and Phys. Rev. 104, 1760 (1956).

Velický, B.

B. Velický, Czech. J. Phys. B 11, 541 (1961).
[Crossref]

Villani, A.

A. Villani and A. H. Zimmerman, Phys. Lett. A 44, 295 (1973); and Phys. Rev. B 8, 3914 (1973).
[Crossref]

Watson, W. H.

C. G. Darwin and W. H. Watson, Proc. R. Soc. London 114, 474 (1927).
[Crossref]

Zimmerman, A. H.

A. Villani and A. H. Zimmerman, Phys. Lett. A 44, 295 (1973); and Phys. Rev. B 8, 3914 (1973).
[Crossref]

C. R. Acad. Sci. (Paris) (1)

H. Becquerel, C. R. Acad. Sci. (Paris) 125, 679 (1897).

Czech. J. Phys. B (1)

B. Velický, Czech. J. Phys. B 11, 541 (1961).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Lett. A (1)

A. Villani and A. H. Zimmerman, Phys. Lett. A 44, 295 (1973); and Phys. Rev. B 8, 3914 (1973).
[Crossref]

Phys. Rev. (1)

S. E. Schnatterly, Phys. Rev. 183, 664 (1969).
[Crossref]

Phys. Rev. B (2)

M. Altarelli and D. Y. Smith, Phys. Rev. B 9, 1290 (1974).
[Crossref]

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
[Crossref]

Proc. R. Soc. London (1)

C. G. Darwin and W. H. Watson, Proc. R. Soc. London 114, 474 (1927).
[Crossref]

Proc. R. Soc. London Ser. A (1)

I. M. Boswarva, R. E. Howard, and A. B. Lidiard, Proc. R. Soc. London Ser. A 269, 125 (1962).
[Crossref]

Solid State Commun (1)

See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun,  12, 1113 (1973).
[Crossref]

Other (13)

An alternative approximation is to employ the inverse-second-moment phase sum rule discussed in Ref. 17 rather than the zeroth moment phase sum rule to evaluating the left-hand side of Eq. (31). This leads to a result similar to Eq. (32), but with ω¯2 replaced n0 ω¯02, where n0 is the contribution to the static index from transitions at energies ω> β.

As in the vacuum background case a second sum rule derived from the low-frequency limit holds for insulators, but its form depends on the model assumed. For the single Lorentzian absorption treated in Ref. 17 the low-frequency limit of the quotent yieldsln(r˜+(ω)r˜-(ω))=2ωp2 ωcn0′(n′02-1)ω04ω,         ω→0where now n0′=(∊∞+ωp2/ω02)1/2. This leads to∫0∞1ω2[θ+(ω)-θ-(ω)] dω=πωp2 ωcn0′(n′02-1) ω04.

D. Y. Smith, Proceedings of the International Symposium on Color Centers in Alkali Halides, 1968, Rome (unpublished), p. 252; Bull. Am. Phys. Soc. 19, 93 (1974); and Bull. Am. Phys. Soc. 19, 259 (1974).

F. C. Jahoda, thesis (Cornell University, 1957) (unpublished);and Phys. Rev. 107, 1261 (1957).

M. Gottlieb, Ph. D. thesis (University of Pennsylvania, 1959) (unpublished).

See, J. S. Toll, thesis (Princeton University, 1952) (unpublished);and Phys. Rev. 104, 1760 (1956).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

The dielectric function for this system is∊±(ω)=∊b-ωp2/(ω2-ω02±ωωc+iωγ),with the background dielectric function ∊b equal unity for vacuum.

In this expression η must be restricted to nonzero values in the case of conductors. In these materials the dielectric function has an ω−1 singularity at ω= 0 which leads to an ω1/2 behavior in lnr˜ (ω) near ω= 0. To ensure that V(ω) is square integrable, it is therefore necessary to require η≠ 0 in Eq. (2) and in its analogue for circular modes, In insulators limω→0lnr˜ (ω) is a nonzero constant and no such restriction applies.

The restriction ω≠ 0 in Eq. (12) also occurs for the derivation from the quotient function Q. It arises from having to avoid improper rearrangement under the integral sign when going from the integral from −∞ to ∞ to one from 0 to ∞.

D. Y. Smith, Phys. Rev. B13, (1976) (to be published).

F. Stern, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1963), Vol. 15.
[Crossref]

A second low-frequency sum rule may be obtained from the expression for ln[r˜+(ω)/r˜-(ω)], Eq. (12), if a specific model is assumed. For example, if an insulator consists of a single magnetically active Lorentzian absorption with plasma frequency ωp centered at ω0, the low-frequency limit of ln[r˜+(ω)/r˜-(ω)] islnr˜+(ω)r˜-(ω)=2ωp2 ωcn0(n02-1) ω04ω=2ωcn0 ω02ω,         ω→0where n0 is the static index (1+ωp2/ω02)1/2. Combining this with Eq. (12) yields∫0∞θ+(ω)-θ-(ω)ω2dω=πωcn0 ω02.For more complicated models of the absorption spectrum similar forms hold but with ω0, ωp, etc., replaced with average values.

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Figures (3)

FIG. 1
FIG. 1

Magnetoreflectivity dichroism and associated sum rule integrals for a single isotropic three-dimensional Lorentz oscillator in free space. The oscillator has natural frequency ω0 = 10, damping constant γ = 1 10 ω 0, and plasma frequency ωp = 10. The cyclotron frequency has been taken as ω c = 1 5 ω 0. The reflectivity given is for light of normal incidence propagating parallel to the magnetic field. The two propagating modes are circularly polarized and are denoted by subscripts +/− for left/right handedness. Part a illustrates ln[r+(ω)/r(ω)] and the convergence of the inverse first-moment reflectivity amplitude sum rule to zero. In part b the phase difference θ+(ω) − θ(ω) and the convergence of the zeroth moment phase dichroism sum rule to ω, are shown.

FIG. 2
FIG. 2

Magnetoreflectivity dichroism and associated sum rule integrals for a Lorentz oscillator in a dielectric medium with constant dielectric function ∊ = 2. The oscillator parameters are the same as those used in Fig. 1 where it was assumed that ∊ = 1, i.e., the oscillator was in vacuum. Note that the sum rule integrals for ω−1 ln[r+(ω)/r(ω)] and θ+(ω) − θ(ω) converge far more rapidly in the present case than for the vacuum background case of Fig. 1.

FIG. 3
FIG. 3

Magnetoreflectivity dichroism for a Lorentz oscillator in a dispersive magnetically inactive dielectric medium. The oscillator parameters are the same as those used in Figs. 1 and 2. The dispersive background ∊b(ω) was modeled with a Lorentz oscillator having natural frequency ω0,b = 30, plasma frequency ωp,b = 30 and damping constant γb = 1. The real part of the refractive index for this background is shown in the insert. For comparison the magnetoreflectivity dichroism for the case of ∊ = ∊b(0) = 2 is shown by the dashed curve.

Tables (1)

Tables Icon

TABLE I Numerical test of magneto-optical sum rules for an optically active Lorentz oscillator osc = - ω p 2 / ( ω 2 - ω 0 2 ± ω ω c + i ω γ ) superimposed on a background b ( ω ) = 1 - ω p , b 2 / ( ω 2 - ω 0 , b 2 + i ω γ b ) with negligible optical activity. Parameters for this example are ω0 = ωp = 10, ωc = 2, and γ = 1, with γb = 1 and ωp,b chosen equal to ω0,b to guarantee ∊b(0) = 2.

Equations (64)

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r ˜ ( ω ) = r ( ω ) e i θ ( ω )
r ˜ * ( ω ) = r ˜ ( - ω * ) .
V ( ω ) = [ ln r ˜ ( ω ) - ln r ˜ ( η ) ] / ( ω - η ) ,
θ ( ω ) = - 2 ω π P 0 ln r ( ω ) ω 2 - ω 2 d ω .
ln r ( ω ) - ln r ( η ) = ln r ( ω ) r ( η ) = 2 π P 0 ω θ ( ω ) ( 1 ω 2 - ω 2 - 1 ω 2 - η 2 ) d ω
N ± ( ω ) = [ x x ( ω ) ± i x y ( ω ) ] 1 / 2 ,
r ˜ ± * ( ω ) = r ˜ ( - ω * ) .
r ˜ r ( ω ) = r ˜ + ( ω ) , ω < 0 = r - ( ω ) , ω > 0 ;
r ˜ l ( ω ) = r ˜ - ( ω ) , ω < 0 = r ˜ + ( ω ) , ω > 0.
ln r ± ( ω ) - ln r ± ( η ) ω - η = 1 π P - θ ± ( ω ) - θ ± ( η ) ( ω - ω ) ( ω - η ) d ω
θ ± ( ω ) - θ ± ( η ) ω - η = - 1 π P - ln r ± ( ω ) - ln r ± ( η ) ( ω - ω ) ( ω - η ) d ω ,
ln ( r + ( ω ) r + ( η ) ) + ln ( r - ( ω ) r - ( ω ) ) = 2 π P 0 ω [ θ + ( ω ) + θ - ( ω ) ] × ( 1 ω 2 - ω 2 - 1 ω 2 - η 2 ) d ω ,
θ + ( ω ) + θ - ( ω ) = - 2 π ω P 0 ln r + ( ω ) + ln r - ( ω ) ω 2 - ω 2 d ω ,
ln r + ( ω ) - ln r - ( ω ) = ln r + ( ω ) r - ( ω ) = 2 π ω P 0 θ + ( ω ) - θ - ( ω ) ω 2 - ω 2 d ω ,
[ θ + ( ω ) - θ - ( ω ) ] - [ θ + ( η ) - θ - ( η ) ] = - 2 π P 0 ω ln [ r + ( ω ) - ln r - ( ω ) ] × ( 1 ω 2 - ω 2 - 1 ω 2 - η 2 ) d ω .
Q ( ω ) = ln [ r ˜ + ( ω ) / r ˜ - ( ω ) ] .
θ + ( ω ) - θ - ( ω ) = - 2 π P 0 ω ln [ r + ( ω ) / r - ( ω ) ] ω 2 - ω 2 d ω .
lim ω N ± ( ω ) = 1 - 1 2 ω p 2 / ω 2
lim ω [ N + ( ω ) - N - ( ω ) ] = ω p 2 ω c / ω 3 .
r ˜ ( ω ) = [ N ( ω ) - 1 ] / [ N ( ω ) + 1 ] .
lim ω r ˜ + ( ω ) / r ˜ - ( ω ) = 1 - 2 ω c / ω .
lim ω ln [ r + ( ω ) / r - ( ω ) ] = - 2 ω c / ω ,
[ θ + ( ω ) - θ - ( ω ) ] ( ω - 1 ln - α ω ) ,             α > 1 ,             ω .
ln [ r + ( 0 ) / r - ( 0 ) ] = ln r ˜ + ( 0 ) / r ˜ - ( 0 ) = 0
θ + ( 0 ) - θ - ( 0 ) = 0.
0 ω - 1 ln [ r + ( ω ) / r - ( ω ) ] d ω = 0.
1 π 0 [ θ + ( ω ) - θ - ( ω ) ] d ω = ω c .
θ + ( ω ) - θ - ( ω ) θ ( ω ) ω ω c ,
1 π 0 [ θ + ( ω ) - θ - ( ω ) ] d ω ω c .
g ( ω ) = 1 ω c [ θ + ( ω ) - θ - ( ω ) ] / θ ( ω ) ω .
r + ( ω ) - r - ( ω ) d r ( ω ) ω ω c
ω c - 1 r ( 0 ) 0 [ r + ( ω ) - r - ( ω ) ] d ω ,
ω c - 1 R ( 0 ) 0 [ R + ( ω ) - R - ( ω ) ] d ω ,
0 ω ω - 1 ln ( r + ( ω ) r - ( ω ) ) d ω / 0 ω ω - 1 | ln ( r + ( ω ) r - ( ω ) ) | d ω
θ + ( ω ) - θ - ( ω ) - 2 π ( 0 α ω ln [ r + ( ω ) / r - ( ω ) ] d ω ω 2 - ω 2 + β ln [ r + ( ω ) / r - ( ω ) ] d ω ω ) ,
θ + ( ω ) - θ - ( ω ) - 2 π ω 2 P 0 α ln [ r + ( ω ) / r - ( ω ) ] ω ( ω 2 - ω 2 ) d ω
β θ + ( ω ) - θ - ( ω ) ω 2 - ω 2 d ω 1 ω ¯ 2 0 [ θ b + ( ω ) - θ b - ( ω ) ] d ω = π ω c ω ¯ 2 ,
ln ( r + ( ω ) r - ( ω ) ) 2 ω π 0 α θ + ( ω ) - θ - ( ω ) ω 2 - ω 2 d ω + 2 ω ω c ω ¯ 2 .
lim ω ln [ r ˜ + ( ω ) / r ˜ - ( ω ) ] = [ 2 ω c ω p 2 / n ( n 2 - 1 ) ] 1 / ω 3 ,
θ + ( ω ) - θ - ( ω ) ω - 3 ln - α ω ,             α > 1 ,             ω .
lim ω 0 ln [ r ˜ + ( ω ) / r ˜ - ( ω ) ] = 0.
0 ω ln ( r + ( ω ) r - ( ω ) ) d ω = 0 ,             n 1.
0 ω - 1 ln ( r + ( ω ) r - ( ω ) ) d ω = 0 ,             n 1.
0 ω ω - 1 ln ( r + ( ω ) r - ( ω ) ) d ω / 0 ω ω - 1 | ln ( r + ( ω ) r - ( ω ) ) | d ω
ω / ( ω 2 - ω 2 ) = - 1 / ω + ω 2 / ω ( ω 2 - ω 2 ) .
0 [ θ + ( ω ) - θ - ( ω ) ] d ω = 0
0 ω 2 [ θ + ( ω ) - θ - ( ω ) ] d ω = - π ω c ω p 2 n ( n 2 - 1 ) ,             n 1.
1 π 0 [ θ + ( ω ) - θ - ( ω ) ] d ω = [ f m / ( f m + f b ) ] ω c ,
ln ( r + ( ω ) r - ( ω ) ) = 2 ω π P 0 θ + ( ω ) - θ - ( ω ) ω 2 - ω 2 d ω
θ + ( ω ) - θ - ( ω ) = - 2 π P 0 ω ln [ r + ( ω ) / r - ( ω ) ] ω 2 - ω 2 d ω .
0 ω - 1 ln ( r + ( ω ) r - ( ω ) ) d ω = 0
1 π 0 [ θ + ( ω ) - θ - ( ω ) ] d ω = ω c .
θ + ( ω ) - θ - ( ω ) - 2 π ω 2 P 0 α ln [ r + ( ω ) / r - ( ω ) ] ω ( ω 2 - ω 2 ) d ω
ln ( r + ( ω ) r - ( ω ) ) 2 ω π 0 α θ + ( ω ) - θ - ( ω ) ω 2 - ω 2 d ω + 2 ω ω c ω ¯ 2
0 ω ln ( r + ( ω ) r - ( ω ) ) d ω = 0 ,             n 1
0 ω - 1 ln ( r + ( ω ) r - ( ω ) ) d ω = 0 ,             n 1
0 [ θ + ( ω ) - θ - ( ω ) ] d ω = 0 ,             n 1
0 ω 2 [ θ + ( ω ) - θ - ( ω ) ] d ω = - π ω c ω p 2 n ( n 2 - 1 ) ,             n 1.
ln [ r + ( ω ) / r - ( ω ) ] [ r + ( ω ) - r - ( ω ) ] / r ( ω ) ,
lnr˜+(ω)r˜-(ω)=2ωp2ωcn0(n02-1)ω04ω=2ωcn0ω02ω,         ω0
0θ+(ω)-θ-(ω)ω2dω=πωcn0ω02.
±(ω)=b-ωp2/(ω2-ω02±ωωc+iωγ),
ln(r˜+(ω)r˜-(ω))=2ωp2ωcn0(n02-1)ω04ω,         ω0
01ω2[θ+(ω)-θ-(ω)]dω=πωp2ωcn0(n02-1)ω04.