Abstract

The effects of generalized component imperfections, azimuth-angle errors, and errors of the normalized Fourier coefficients of the detected photoelectric current on the measured ratio of reflection coefficients ρ in rotating-analyzer ellipsometers (RAE) are determined. The problem is formulated in such a way that much of the earlier work done on error analysis for null ellipsometers (NE) can be adapted to RAE. The results are conveniently expressed in terms of coupling coefficients that determine the extent to which a given source of error couples to an error of the measured value of ρ. The optical properties of the compensator (if used) and of the surface can be simultaneously obtained from a set of two measurements using RAE, in a manner similar to two-zone measurements in NE. In addition, novel methods of obtaining and combining the results from two measurements are examined, with the objective of cancelling the effect of many of the systematic sources of errors. One such method employs two incident polarizations of the same ellipticity but with orthogonal azimuths, in which case the measured value of ρ is almost independent of the input optics. If the two incident polarizations are chosen, instead, to have equal but opposite ellipticity and azimuth, the effects of the polarizer imperfection, off-diagonal elements in the compensator, entrance-window, surface, and exit-window imperfection matrices, as well as polarizer and compensator azimuth-angle errors, all disappear upon such two-measurement averaging; the effects of analyzer imperfection or azimuth-angle error and errors of the normalized Fourier coefficients are only partially cancelled. Finally, use of RAE in generalized ellipsometry and its attendant problems are examined.

© 1974 Optical Society of America

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References

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  1. C. V. Kent and J. Lawson, J. Opt. Soc. Am. 27, 117 (1937).
    [CrossRef]
  2. W. Budde, Appl. Opt. 1, 201 (1962).
    [CrossRef]
  3. S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 297 (1964).
  4. S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 379 (1964).
  5. D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).
  6. J. I. Bohnert, Proc. IRE 39, 549 (1951).
    [CrossRef]
  7. B. D. Cahan and R. F. Spainer, Surf. Sci. 16, 166 (1969); also, in Proceedings of the Symposium on Recent Developments in Ellipsometry edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North–Holland, Amsterdam, 1969).
    [CrossRef]
  8. R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
    [CrossRef]
  9. J. C. Suits, Rev. Sci. Instrum. 42, 19 (1971).
    [CrossRef]
  10. D. J. Scholtens, J. F. Kleibeuker, and J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
    [CrossRef]
  11. D. E. Aspnes, Opt. Commun. 8, 222 (1973).
    [CrossRef]
  12. P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
    [CrossRef]
  13. D. E. Aspnes, J. Opt. Soc. Am. 64, 639 (1974).
    [CrossRef]
  14. D. E. Aspnes, J. Opt. Soc. Am. 64, 812 (1974).
    [CrossRef]
  15. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
    [CrossRef]
  16. D. A. Holmes and D. E. Feucht, J. Opt. Soc. Am. 57, 466 (1967).
    [CrossRef] [PubMed]
  17. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
    [CrossRef]
  18. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 61, 600 (1971).
    [CrossRef]
  19. W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355 (1970).
    [CrossRef]
  20. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 700 (1972). For the compensator, TC0 is diagonal; the ratio of the 2,2 to the 1,1 matrix elements is equal to ρC[Eq. (11)]. For the entrance (and exit) window, TW0 is the product of a complex constant times the 2 × 2 identity matrix.
    [CrossRef]
  21. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 61, 773 (1971); J. Opt. Soc. Am. 61, 1236 (1971).
    [CrossRef]
  22. Notice that, because of the inequality (28), the inequality (31) restricts χr to lie inside, on, or not too far outside the unit circle |χr| = 1 in the complex plane of polarization. In other words, the reflected polarization should be more p-like than s-like.
  23. R. M. A. Azzam and N. M. Bashara, Appl. Phys. 1, 203 (1973); Appl. Phys. 2, 59 (1973).
    [CrossRef]
  24. χr is controlled by the optical elements P and C of the polarizing arm of the ellipsometer, whereas χA is adjusted by rotating the analyzer A around the beam axis, Eq. (41).
  25. See Eq. (5), Ref. 15.
  26. Equations (46a) and (46b) can be simplified, without loss of generality, if we choose the reference position of the analyzer so that the major axis of the transmitted elliptical vibration χAO is parallel to the plane of incidence. Thus, in this case, χAO is pure imaginary, Re(χAO) = 0, and the second terms in the numerators of the right-hand sides of Eqs. (46a) and (46b) become zero.
  27. See Fig. 4 and Table II of Ref. 15.
  28. If an elliptical, instead of linear, analyzer is used, the two polarizations χr and χr* that differ only in handedness lead to different normalized Fourier coefficients α and β, hence can be distinguished. This can be seen from Eqs. (46a) and (46b) by noting that, for an elliptical analyzer, the second terms in the denominators of the right-hand sides of these equations are nonzero and switch sign as χr* is substituted instead of χr.
  29. R. J. Archer and C. V. Shank, J. Opt. Soc. Am. 57, 191 (1967).
    [CrossRef]
  30. T. Yolken, R. Waxler, and J. Kruger, J. Opt. Soc. Am. 57, 283 (1967).
    [CrossRef]
  31. W. G. Oldham, J. Opt. Soc. Am. 57, 617 (1967).
    [CrossRef]
  32. F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970).
    [CrossRef]
  33. J. A. Johnson and N. M. Bashara, J. Opt. Soc. Am. 60, 221 (1970).
    [CrossRef]
  34. D. E. Aspnes, J. Opt. Soc. Am. 61, 1077 (1971).
    [CrossRef]
  35. Equation (62) is the same as Eq. (52) except that, in the latter, χr is explicitly expressed in terms of the normalized Fourier coefficients α and β using Eq. (51).
  36. When C= 0, Eqs. (62a) and (62b) become identical and equivalent to one equation, ρ/ρC= tanP/χr, and the subsequent discussion does not apply.
  37. Alternatively, starting with Eq. (62b) and repeating steps similar to those that led to Eq. (63a), we can obtain a quadratic in ρ only.
  38. Ways to achieve this are mentioned in Refs. 12 and 14.
  39. This assumes that |ρC| = TC is very close to unity, which is true for most compensators. From Eq. (51), note that χr is real; hence it represents a linear vibration, if α2+ β2= 1. This leads to an amplitude of the ac component of the photoelectric current equal to its dc component, as may be seen from Eq. (44) after the cosine and sine terms are combined into a single sine or cosine term. Thus, a linear state can be detected by a rotating analyzer from the condition of maximum (unity) modulation depth in the photoelectric current. See Ref. 14.
  40. This may be chosen on the basis of optimum-precision considerations, as discussed in Refs. 12 and 13.
  41. The azimuth of the compensator can be changed 90° electronically, e.g., by use of a KDP crystal mounted to rotate with the (quarter-wave) compensator as one unit, to which a half-wave voltage that can be regulated by a corrective electro-optic feedback loop is applied along the fast axis of the compensator.
  42. The method of Kent and Lawson (Ref. 1) relies on producing a circular reflected state (detected by a rotating analyzer) by varying the azimuth of a linear polarizer in the incident beam and the angle of incidence. In the present discussion, we assume that the angle of incidence is fixed, but that a compensator is used in the incident beam, which, together with the polarizer, can be adjusted to make the reflected polarization circular.
  43. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1375A (1972); J. Opt. Soc. Am. 62, 1521 (1972); J. Opt. Soc. Am. 64, 128 (1974).

1974 (2)

1973 (4)

D. J. Scholtens, J. F. Kleibeuker, and J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[CrossRef]

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Appl. Phys. 1, 203 (1973); Appl. Phys. 2, 59 (1973).
[CrossRef]

1972 (4)

1971 (4)

1970 (4)

F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970).
[CrossRef]

J. A. Johnson and N. M. Bashara, J. Opt. Soc. Am. 60, 221 (1970).
[CrossRef]

W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
[CrossRef]

1969 (1)

B. D. Cahan and R. F. Spainer, Surf. Sci. 16, 166 (1969); also, in Proceedings of the Symposium on Recent Developments in Ellipsometry edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North–Holland, Amsterdam, 1969).
[CrossRef]

1967 (4)

1964 (2)

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 297 (1964).

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 379 (1964).

1962 (1)

1951 (1)

J. I. Bohnert, Proc. IRE 39, 549 (1951).
[CrossRef]

1937 (1)

Archer, R. J.

Aspnes, D. E.

Azzam, R. M. A.

Bashara, N. M.

Bohnert, J. I.

J. I. Bohnert, Proc. IRE 39, 549 (1951).
[CrossRef]

Budde, W.

Cahan, B. D.

B. D. Cahan and R. F. Spainer, Surf. Sci. 16, 166 (1969); also, in Proceedings of the Symposium on Recent Developments in Ellipsometry edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North–Holland, Amsterdam, 1969).
[CrossRef]

Clarke, D.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Dill, F. H.

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[CrossRef]

Eaton, D. H.

W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

Feucht, D. E.

Grainger, J. F.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Greef, R.

R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
[CrossRef]

Hauge, P. S.

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[CrossRef]

Holmes, D. A.

Hunter, W. R.

W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

Johnson, J. A.

Kent, C. V.

Kleibeuker, J. F.

D. J. Scholtens, J. F. Kleibeuker, and J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

Kommandeur, J.

D. J. Scholtens, J. F. Kleibeuker, and J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

Kruger, J.

Lawson, J.

McCrackin, F. L.

Oldham, W. G.

Rajagopalan, S. R.

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 297 (1964).

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 379 (1964).

Ramaseshan, S.

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 379 (1964).

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 297 (1964).

Sah, C. T.

W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

Scholtens, D. J.

D. J. Scholtens, J. F. Kleibeuker, and J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

Shank, C. V.

Spainer, R. F.

B. D. Cahan and R. F. Spainer, Surf. Sci. 16, 166 (1969); also, in Proceedings of the Symposium on Recent Developments in Ellipsometry edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North–Holland, Amsterdam, 1969).
[CrossRef]

Suits, J. C.

J. C. Suits, Rev. Sci. Instrum. 42, 19 (1971).
[CrossRef]

Waxler, R.

Yolken, T.

Appl. Opt. (1)

Appl. Phys. (1)

R. M. A. Azzam and N. M. Bashara, Appl. Phys. 1, 203 (1973); Appl. Phys. 2, 59 (1973).
[CrossRef]

IBM J. Res. Dev. (1)

P. S. Hauge and F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[CrossRef]

J. Opt. Soc. Am. (16)

Opt. Commun. (1)

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[CrossRef]

Proc. Indian Acad. Sci. A (2)

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 297 (1964).

S. R. Rajagopalan and S. Ramaseshan, Proc. Indian Acad. Sci. A 60, 379 (1964).

Proc. IRE (1)

J. I. Bohnert, Proc. IRE 39, 549 (1951).
[CrossRef]

Rev. Sci. Instrum. (3)

R. Greef, Rev. Sci. Instrum. 41, 532 (1970).
[CrossRef]

J. C. Suits, Rev. Sci. Instrum. 42, 19 (1971).
[CrossRef]

D. J. Scholtens, J. F. Kleibeuker, and J. Kommandeur, Rev. Sci. Instrum. 44, 153 (1973).
[CrossRef]

Surf. Sci. (2)

B. D. Cahan and R. F. Spainer, Surf. Sci. 16, 166 (1969); also, in Proceedings of the Symposium on Recent Developments in Ellipsometry edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North–Holland, Amsterdam, 1969).
[CrossRef]

W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355 (1970).
[CrossRef]

Other (15)

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Notice that, because of the inequality (28), the inequality (31) restricts χr to lie inside, on, or not too far outside the unit circle |χr| = 1 in the complex plane of polarization. In other words, the reflected polarization should be more p-like than s-like.

χr is controlled by the optical elements P and C of the polarizing arm of the ellipsometer, whereas χA is adjusted by rotating the analyzer A around the beam axis, Eq. (41).

See Eq. (5), Ref. 15.

Equations (46a) and (46b) can be simplified, without loss of generality, if we choose the reference position of the analyzer so that the major axis of the transmitted elliptical vibration χAO is parallel to the plane of incidence. Thus, in this case, χAO is pure imaginary, Re(χAO) = 0, and the second terms in the numerators of the right-hand sides of Eqs. (46a) and (46b) become zero.

See Fig. 4 and Table II of Ref. 15.

If an elliptical, instead of linear, analyzer is used, the two polarizations χr and χr* that differ only in handedness lead to different normalized Fourier coefficients α and β, hence can be distinguished. This can be seen from Eqs. (46a) and (46b) by noting that, for an elliptical analyzer, the second terms in the denominators of the right-hand sides of these equations are nonzero and switch sign as χr* is substituted instead of χr.

Equation (62) is the same as Eq. (52) except that, in the latter, χr is explicitly expressed in terms of the normalized Fourier coefficients α and β using Eq. (51).

When C= 0, Eqs. (62a) and (62b) become identical and equivalent to one equation, ρ/ρC= tanP/χr, and the subsequent discussion does not apply.

Alternatively, starting with Eq. (62b) and repeating steps similar to those that led to Eq. (63a), we can obtain a quadratic in ρ only.

Ways to achieve this are mentioned in Refs. 12 and 14.

This assumes that |ρC| = TC is very close to unity, which is true for most compensators. From Eq. (51), note that χr is real; hence it represents a linear vibration, if α2+ β2= 1. This leads to an amplitude of the ac component of the photoelectric current equal to its dc component, as may be seen from Eq. (44) after the cosine and sine terms are combined into a single sine or cosine term. Thus, a linear state can be detected by a rotating analyzer from the condition of maximum (unity) modulation depth in the photoelectric current. See Ref. 14.

This may be chosen on the basis of optimum-precision considerations, as discussed in Refs. 12 and 13.

The azimuth of the compensator can be changed 90° electronically, e.g., by use of a KDP crystal mounted to rotate with the (quarter-wave) compensator as one unit, to which a half-wave voltage that can be regulated by a corrective electro-optic feedback loop is applied along the fast axis of the compensator.

The method of Kent and Lawson (Ref. 1) relies on producing a circular reflected state (detected by a rotating analyzer) by varying the azimuth of a linear polarizer in the incident beam and the angle of incidence. In the present discussion, we assume that the angle of incidence is fixed, but that a compensator is used in the incident beam, which, together with the polarizer, can be adjusted to make the reflected polarization circular.

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Figures (1)

Fig. 1
Fig. 1

Arrangement of the optical components in the rotating-analyzer ellipsometer (RAE). P, C, W, S, W′, A, and D represent the polarizer, compensator, entrance window, surface, exit window, rotating analyzer, and detector, respectively. P, C, and A also represent the azimuthal positions of the polarizer, compensator, and analyzer measured from the plane of incidence, positive in a counterclockwise sense looking into the beam. χi and χr are the incident and reflected polarizations, respectively, whereas χr′ is the state of polarization at the output of the exit window W′. The p and s directions are taken parallel and perpendicular to the plane of incidence, respectively. The light source, electronic signal-processing components, and feedback control paths are not shown here.

Tables (2)

Tables Icon

Table I Coupling coefficients for component imperfections in RAE.a

Tables Icon

Table II Coupling coefficients for azimuth-angle and normalized-Fourier-coefficient errors in RAE.a

Equations (98)

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χ i = E i s / E i p ,             χ r = E r s / E r p ,
E r p = R p p E i p ,             E r s = R s s E i s ,
ρ = χ i / χ r ,
ρ = R p p / R s s ,
ρ = tan ψ e j Δ ,
δ ρ δ χ i = ( 1 / χ r ) δ χ i ,
δ ρ δ χ r = - ( χ i / χ r 2 ) δ χ r ,
δ ρ = ( 1 / χ r ) δ χ i - ( χ i / χ r 2 ) δ χ r .
δ ρ ρ = δ χ i χ i - δ χ r χ r ,
δ ψ = 1 2 sin 2 ψ Re ( δ ρ / ρ ) ,             δ Δ = Im ( δ ρ / ρ ) ,
χ i = tan C + ρ C tan ( P - C ) 1 - ρ C tan C tan ( P - C ) ,
ρ C = T C e - j δ C
δ χ i = a P δ ρ P
δ ρ δ ρ P = ( a P / χ r ) δ ρ P
δ ρ δ ρ P = γ P δ ρ P ,
γ P = a P / χ r .
T = T 0 + δ T ,
δ χ i = i j a i j δ T i j ,
δ ρ δ T = ( 1 / χ r ) i j a i j δ T i j
δ ρ δ T = i j γ i j δ T i j ,
γ i j = a i j / χ r .
χ r = - cot A .
γ P ( NE ) = ( - tan A ) a P             for             P ,
γ i j ( NE ) = ( - tan A ) a i j             for             C and W .
a P = ( - cot A ) γ P ( NE )             for             P ,
a i j = ( - cot A ) γ i j ( NE )             for             C and W .
γ P ( RAE ) = ( - cot A / χ r ) γ P ( NE )             for             P ,
γ i j ( RAE ) = ( - cot A / χ r ) γ i j ( NE )             for             C and W .
γ Z ( RAE ) = ( - cot A / χ r ) γ Z ( NE ) ,             Z = P or C .
δ ρ δ Z = γ Z δ Z ,             Z = P or C .
| R p s R s s | ,             | R s p R s s | 1.
χ r = R s s χ i + R s p R p s χ i + R p p ,
χ i = R p p χ r - R s p - R p s χ r + R s s ,
χ i = ( 1 - R p s R s s χ r ) - 1 ( R p p R s s χ r - R s p R s s ) .
| R p s R s s χ r | 1
χ i = ( 1 + R p s R s s χ r ) ( R p p R s s χ r - R s p R s s )
χ i = ( R p p R s s ) χ r + ( R p p R s s ) χ r 2 ( R p s R s s ) - ( R s p R s s ) ,
ρ = χ i χ r - χ i ( R p s R s s ) + 1 χ r ( R s p R s s ) ,
δ ρ R p s = - χ i ( R p s R s s ) ,             δ ρ R s p = 1 χ r ( R s p R s s ) .
δ ρ R p s = γ p s ( R p s R s s ) ,             δ ρ R s p = γ s p ( R s p R s s ) ,
γ p s = - χ i ,             γ s p = 1 / χ r ,
χ r = ( T W p p T W s s ) χ r + ( T W p p T W s s ) ( χ r ) 2 ( T W p s T W s s ) - ( T W s p T W s s ) ,
χ r = χ r + χ r ( δ T W p p - δ T W s s ) + χ r 2 δ T W p s - δ T W s p ,
δ χ r = χ r ( δ T W p p - δ T W s s ) + χ r 2 δ T W p s - δ T W s p ,
δ ρ δ T W = - ( χ i / χ r ) ( δ T W p p - δ T W s s ) - ( χ i ) δ T W p s + ( χ i / χ r 2 ) δ T W s p
δ ρ δ T W = - ρ ( δ T W p p - δ T W s s ) - χ i δ T W p s + ( ρ / χ r ) δ T W s p ,
χ A = ( cos A ) χ A O + ( sin A ) ( - sin A ) χ A O + ( cos A )
χ A = tan A + χ A O 1 - tan A χ A O ,
I D = k I r χ r χ r * χ A χ A * + χ r χ A * + χ r * χ A + 1 χ r χ r * χ A χ A * + χ r χ r * + χ A χ A * + 1 ,
χ r = - 1 / χ A * ,
I D = I ˜ [ 1 + α cos 2 A + β sin 2 A ] ,
Ī = 1 2 k I r [ ( 1 + χ r 2 ) ( 1 + χ A O 2 ) + 4 Im ( χ r ) Im ( χ A O ) ( 1 + χ r 2 ) ( 1 + χ A O 2 ) ] ,
α = ( 1 - χ r 2 ) ( 1 - χ A O 2 ) + 4 Re ( χ r ) Re ( χ A O ) ( 1 + χ r 2 ) ( 1 + χ A O 2 ) + 4 Im ( χ r ) Im ( χ A O ) ,
β = 2 Re ( χ r ) ( 1 - χ A O 2 ) - 2 Re ( χ A O ) ( 1 - χ r 2 ) ( 1 + χ r 2 ) ( 1 + χ A O 2 ) + 4 Im ( χ r ) Im ( χ A O ) .
χ A O = 0.
α = ( 1 - χ r 2 ) ( 1 + χ r 2 ) ,
β = 2 Re ( χ r ) ( 1 + χ r 2 ) .
χ r = ( 1 - α 1 + α ) 1 2 ,
Re ( χ r ) = β ( 1 + α ) .
χ r = Re ( χ r ) ± j { χ r 2 - [ Re ( χ r ) ] 2 } 1 2
χ r = 1 ( 1 + α ) [ β ± j ( 1 - α 2 - β 2 ) 1 2 ] ,
ρ RAE = ( 1 + α ) [ β ± j ( 1 - α 2 - β 2 ) 1 2 ] × tan C + ρ C tan ( P - C ) 1 - ρ C tan C tan ( P - C ) .
δ χ r = ( χ r α ) δ α ,
δ χ r = ( χ r β ) δ β .
χ r α = - 1 ( 1 + α ) 2 { β ± j [ α ( α + 1 ) ( 1 - α 2 - β 2 ) - 1 2 + ( 1 - α 2 - β 2 ) 1 2 ] } ,
χ r β = 1 ( 1 + α ) [ 1 j β ( 1 - α 2 - β 2 ) - 1 2 ] .
δ ρ δ α = γ α δ α ,
δ ρ δ β = γ β δ β ,
γ α = - ( χ i χ r 2 ) ( χ r α ) ,
γ β = - ( χ i χ r 2 ) ( χ r β )
χ A O = δ ρ A ,             δ ρ A 1.
δ α = ± [ j 2 α ( 1 - α 2 - β 2 ) 1 2 ] ( δ ρ A ) ,
δ β = ± [ j 2 β ( 1 - α 2 - β 2 ) 1 2 ] ( δ ρ A )
γ A = 2 j ( 1 - α 2 - β 2 ) 1 2 ( χ i / χ r 2 ) [ α ( χ r α ) + β ( χ r β ) ] ,
δ α = ( 2 β ) δ A ,
δ β = ( - 2 α ) δ A .
γ A = 2 ( χ i χ r 2 ) [ α ( χ r β ) - β ( χ r α ) ] ,
ρ = 1 χ r tan C + ρ C tan ( P - C ) 1 - ρ C tan C tan ( P - C ) ,
ρ C = 1 tan ( P - C ) ρ χ r - tan C 1 + ρ χ r tan C .
ρ C 2 + m 1 ρ C + m 2 = 0 ,
ρ C = - 1 2 m 1 ± 1 2 ( m 1 2 - 4 m 2 ) 1 2 ,
m 1 = 2 ( χ r 1 - χ r 2 ) - 1 csc ( 2 C ) [ cot ( P 2 - C ) - cot ( P 1 - C ) ] ,
m 2 = - cot ( P 1 - C ) cot ( P 2 - C ) .
m 1 = 2 ( χ r 1 - χ r 2 ) - 1 sec 2 P 1 ,
m 2 = 1.
ρ 1 = χ i 1 / χ r 1 , ρ 2 = χ i 2 / χ r 2 ,
ρ = ( χ i 1 χ i 2 / χ r 1 χ r 2 ) 1 2 .
χ i 1 χ i 2 = - 1 ,
ρ = ( - 1 / χ r 1 χ r 2 ) 1 2 .
ρ = ( χ i / χ r ) + k γ k ( χ i , χ r ) δ f k ,
ρ = ρ 0 ( 1 + ( 1 / ρ 0 ) k γ k ( χ i , χ r ) δ f k ) ,
ρ 1 = ρ 1 0 ( 1 + ( 1 / ρ 1 0 ) k γ k ( χ i 1 , χ r 1 ) δ f k ) , ρ 2 = ρ 2 0 ( 1 + ( 1 / ρ 2 0 ) k γ k ( χ i 2 , χ r 2 ) δ f k ) ,
ρ = ( ρ 1 0 ρ 2 0 ) 1 2 ( 1 + 1 2 ( ρ 1 0 ρ 2 0 ) - 1 2 k [ γ k ( χ i 1 , χ r 1 ) + γ k ( χ i 2 , χ r 2 ) ] δ f k ) ,
γ k ( χ i 1 , χ r 1 ) + γ k ( χ i 2 , χ r 2 ) = 0
γ k ( χ i 2 , χ r 2 ) = - γ k ( χ i 1 , χ r 1 ) .
χ i 2 = - χ i 1 , χ r 2 = - χ r 1 .
P 2 = - P 1 , C 2 = - C 1 ,
α 2 = α 1 , β 2 = - β 1 ,