Abstract

Given a complex function F(ω) = |F(ω)|exp[jΔ(ω)] of a real argument ω, the extrema of its magnitude |F(ω)| and its phase Δ(ω), as functions of ω, are determined simultaneously by finding the roots of one common equation, Im[G(ω)] = 0, where G = (F/F)2 and F = ∂F/ω. The extrema of |F| and Δ are associated with Re G < 0 and Re G > 0, respectively. This easy-to-prove theorem has a wide range of applications in physical optics. We consider attenuated internal reflection (AIR) as an example. In AIR the complex reflection coefficient for the p polarization, rp(ϕ), and the ratio of complex reflection coefficients for the p and s polarizations, ρ(ϕ) = rp(ϕ)/rs(ϕ), are considered as functions of the angle of incidence ϕ. It is found that the same (cubic) equation that determines the pseudo-Brewster angle of minimum |rp| also determines a new angle at which the reflection phase shift δp = arg rpexhibits a minimum of its own. Likewise, the same (quartic) equation that determines the second Brewster angle of minimum |ρ| also determines angles of incidence at which the differential reflection phase shift Δ = arg ρ experiences a minimum and a maximum. Angular positions and magnitudes of all extrema are calculated exactly for a specific case that represents light reflection by the vacuum–Al or glass–aqueous-dye-solution interface. As another example, the normal-incidence reflection of light by a birefringent film on an absorbing substrate is examined. The ratio of complex principal reflection coefficients is considered as a function of the film thickness normalized to the wavelength of light. The absolute value and the phase of this function exhibit multiple extrema, the first 13 of which are determined for a specific birefringent film on a Si substrate.

© 1988 Optical Society of America

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References

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  1. I could not find this theorem in standard textbooks of complex function mathematics, physical optics, or electric circuit theory. Considering its simplicity, I would be surprised if it were truly new. Its significance, in all likelihood, has largely been missed.
  2. See, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.
  3. R. M. A. Azzam, “Maximum minimum reflectance of parallel-polarized light at interfaces between transparent and absorbing media,”J. Opt. Soc. Am. 73, 959–962 (1983).
    [CrossRef]
  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. London 77, 949–957 (1961).
    [CrossRef]
  5. S. Y. Kim, K. Vedam, “Analytic solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986).
    [CrossRef]
  6. R. W. Hunter, “Measurement of optical properties of materials in the vacuum ultraviolet spectral region,” Appl. Opt. 21, 2103–2114 (1982).
    [CrossRef] [PubMed]
  7. S. G. Jennings, “Attenuated total reflectance measurements of the complex refractive index of polystyrene latex at CO2laser wavelengths,” J. Opt. Soc. Am. 71, 923–927 (1981).
    [CrossRef]
  8. Cubic equations are solvable exactly and explicitly. See, e.g., S. M. Selby, ed., Standard Mathematical Tables, 20th ed. (Chemical Rubber Company, Cleveland, Ohio, 1972), pp. 103–105.
  9. (rp′/rp) is given by Eq. (14) of Ref. 3.
  10. See, e.g., R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 274.
  11. R. M. A. Azzam, “Explicit equations for the second-Brewster angle of an interface between a transparent and an absorbing medium,”J. Opt. Soc. Am. 73, 1211–1212 (1983); errata, J. Opt. Soc. Am. A 1, 325 (1984).
    [CrossRef]
  12. R. M. A. Azzam, “Contours of constant principal angle and constant principal azimuth in the complex ∊ plane,” J. Opt. Soc. Am. 71, 1523–1528 (1981).
    [CrossRef]
  13. See, e.g., S. M. Selby, ed., Standard Mathematical Tables, 20th ed. (Chemical Rubber Company, Cleveland, Ohio, 1972), p. 106.
  14. R. M. A. Azzam, “PIE: perpendicular-incidence ellipsometry—application to the determination of the optical properties of uniaxial and biaxial absorbing crystals,” Opt. Commun. 19, 122–124 (1976); “NIRSE: normal-incidence rotating-sample ellipsometer,” Opt. Commun. 20, 405–408 (1977).
    [CrossRef]

1986 (1)

1983 (2)

1982 (1)

1981 (2)

1976 (1)

R. M. A. Azzam, “PIE: perpendicular-incidence ellipsometry—application to the determination of the optical properties of uniaxial and biaxial absorbing crystals,” Opt. Commun. 19, 122–124 (1976); “NIRSE: normal-incidence rotating-sample ellipsometer,” Opt. Commun. 20, 405–408 (1977).
[CrossRef]

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. London 77, 949–957 (1961).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Maximum minimum reflectance of parallel-polarized light at interfaces between transparent and absorbing media,”J. Opt. Soc. Am. 73, 959–962 (1983).
[CrossRef]

R. M. A. Azzam, “Explicit equations for the second-Brewster angle of an interface between a transparent and an absorbing medium,”J. Opt. Soc. Am. 73, 1211–1212 (1983); errata, J. Opt. Soc. Am. A 1, 325 (1984).
[CrossRef]

R. M. A. Azzam, “Contours of constant principal angle and constant principal azimuth in the complex ∊ plane,” J. Opt. Soc. Am. 71, 1523–1528 (1981).
[CrossRef]

R. M. A. Azzam, “PIE: perpendicular-incidence ellipsometry—application to the determination of the optical properties of uniaxial and biaxial absorbing crystals,” Opt. Commun. 19, 122–124 (1976); “NIRSE: normal-incidence rotating-sample ellipsometer,” Opt. Commun. 20, 405–408 (1977).
[CrossRef]

See, e.g., R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 274.

Bashara, N. M.

See, e.g., R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 274.

Born, M.

See, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.

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. London 77, 949–957 (1961).
[CrossRef]

Hunter, R. W.

Jennings, S. G.

Kim, S. Y.

Vedam, K.

Wolf, E.

See, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

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

Opt. Commun. (1)

R. M. A. Azzam, “PIE: perpendicular-incidence ellipsometry—application to the determination of the optical properties of uniaxial and biaxial absorbing crystals,” Opt. Commun. 19, 122–124 (1976); “NIRSE: normal-incidence rotating-sample ellipsometer,” Opt. Commun. 20, 405–408 (1977).
[CrossRef]

Proc. Phys. Soc. London (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. London 77, 949–957 (1961).
[CrossRef]

Other (6)

Cubic equations are solvable exactly and explicitly. See, e.g., S. M. Selby, ed., Standard Mathematical Tables, 20th ed. (Chemical Rubber Company, Cleveland, Ohio, 1972), pp. 103–105.

(rp′/rp) is given by Eq. (14) of Ref. 3.

See, e.g., R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 274.

I could not find this theorem in standard textbooks of complex function mathematics, physical optics, or electric circuit theory. Considering its simplicity, I would be surprised if it were truly new. Its significance, in all likelihood, has largely been missed.

See, e.g., M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.

See, e.g., S. M. Selby, ed., Standard Mathematical Tables, 20th ed. (Chemical Rubber Company, Cleveland, Ohio, 1972), p. 106.

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Figures (7)

Fig. 1
Fig. 1

Amplitude reflectance |rp| versus the angle of incidence ϕ for AIR of p-polarized light at an interface with complex relative dielectric constant = 0.64 − j0.024. |rp| is minimum at the pseudo-Brewster angle ϕpB = 38.657°.

Fig. 2
Fig. 2

AIR phase shift for the p polarization δp plotted versus the angle of incidence ϕ for an interface with = 0.64 − j0.024. δp goes through a minimum of its own at an angle, ϕ δ p min = 45.836 °, that is determined by another root of the same cubic equation of the pseudo-Brewster angle ϕpB.

Fig. 3
Fig. 3

Ratio of amplitude reflectances for the p and s polarizations |ρ| = |rp/rs| plotted versus the angle of incidence ϕ for an interface with = 0.64 − j0.024. |ρ| is minimum at the second Brewster angle ϕ2B = 38.666°.

Fig. 4
Fig. 4

Difference of the AIR phase shifts for the p and s polarizations Δ = δpδs plotted versus the angle of incidence ϕ for an interface with = 0.64 − j0.024. Δ goes through a minimum and a maximum at angles of incidence ϕ Δ min = 47.979 ° and ϕ Δ max = 61.945 °, which are determined by two other roots of the same quartic equation that specifies the second Brewster angle ϕ2B.

Fig. 5
Fig. 5

Ratio of normal-incidence amplitude reflectances for the x and y linear polarizations, |ρ| = |Ry/Rx|, parallel to the principal axes of a birefringent film with principal refractive indices N1x = 1.55 and N1y = 1.50, plotted versus the normalized film thickness ζ = d/λ. The birefringent film overlays a Si substrate with a complex refractive index N2 = 3.85 − j0.02 at a wavelength λ = 632.8 nm.

Fig. 6
Fig. 6

The differential reflection phase shift Δ = δyδx for the same birefringent film on Si as in Fig. 5 (also plotted versus ζ).

Fig. 7
Fig. 7

The function Im(UyUx)2 plotted versus ζ. Its roots [Eq. (40)], represented by the points of intersection of the curve with the ζ axis, determine the location of the extrema of |ρ| and Δ that appear in Figs. 5 and 6.

Tables (1)

Tables Icon

Table 1 First 13 Roots of Eq. (40) for a Birefringent Film (N1x = 1.55, N1y = 1.50) on a Si Substrate (N2 = 3.85 − j0.02) at λ = 632.8 nma

Equations (48)

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F ( ω ) = | F ( ω ) | exp [ j Δ ( ω ) ]
( ln F ) / ω = F / F = ( | F | / | F | ) + j Δ ,
G = ( F / F ) 2 ,
Re G = ( | F | / | F | ) 2 ( Δ ) 2 ,
Im G = 2 ( | F | / | F | ) Δ .
| F | = 0 , Δ = 0 ,
Im [ G ( ω ) ] = 0 ,
Re G < 0 if | F | = 0 ,
Re G > 0 if Δ = 0.
r p ( ϕ ) = cos ϕ ( sin 2 ϕ ) 1 / 2 cos ϕ + ( sin 2 ϕ ) 1 / 2 ,
= 1 / 0
= r j i ,
0 < r < 1 , i / r 1.
r p = | r p | exp ( j δ p ) ,
Im ( r p / r p ) 2 = 0 ,
α 3 u 3 + α 2 u 2 + α 1 u + α 0 = 0 ,
u = sin 2 ϕ
α 0 = | | 4 , α 1 = 2 | | 4 , α 2 = | | 4 3 | | 2 , α 3 = 2 r + 2 | | 2 .
= 0.64 j 0.024.
u 1 = 0.3990 , u 2 = 0.3902 , u 3 = 0.5146.
ϕ 2 = 38.657 ° , ϕ 3 = 45.836 ° .
| r p | min = 0.0053.
δ p min = 14.2865 ° .
ϕ 3 a = ( ϕ B nom + ϕ C nom ) / 2 = ( tan 1 r 1 / 2 + sin 1 r 1 / 2 ) / 2 .
lim i 0 ( ϕ 3 ϕ 3 a ) = 0
lim i 0 δ p min = 0.
r p / r s = ρ ( ϕ ) = sin ϕ tan ϕ ( sin 2 ϕ ) 1 / 2 sin ϕ tan ϕ + ( sin 2 ϕ ) 1 / 2 ,
ρ = | ρ | exp ( j Δ ) ,
Im ( ρ / ρ ) 2 = 0.
i = 0 4 a i u i = 0 ,
ϕ 1 = 38.666 ° , ϕ 2 = 47.979 ° , ϕ 3 = 61.945 ° .
| ρ | ( ϕ 1 ) = | ρ | min = 0.024
Δ ( ϕ 2 ) = Δ min = 6.406 ° , Δ ( ϕ 3 ) = Δ max = 25.459 ° .
R ν = ( r 01 ν + r 12 ν X ν ) / ( 1 + r 01 ν r 12 ν X ν ) , ν = x , y ,
X ν = exp ( j 4 π N 1 ν ζ ) , ν = x , y ,
ζ = d / λ .
ρ = R y / R x = | ρ | exp ( j Δ ) ,
( ρ / ρ ) = ( j 4 π ) ( U y U x )
U ν = N 1 ν r 12 ν X ν [ ( r 01 ν + r 12 ν X ν ) 1 r 01 ν ( 1 + r 01 ν r 12 ν X ν ) 1 ] , ν = x , y ,
Im ( U y U x ) 2 = 0
F ( ω ) = F r ( ω ) + j F i ( ω ) .
F ( ω ) = F r ( ω ) + j F i ( ω ) .
H = ( F ) 2 ,
Re H = ( F r ) 2 ( F i ) 2 ,
Im H = 2 F r F i .
Im [ H ( ω ) ] = 0 ,
Re H < 0 if F r = 0 ,
Im H > 0 if F i = 0.

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