## Extrema of the magnitude and the phase of a complex function of a real variable: application to attenuated internal reflection

JOSA A, Vol. 5, Issue 8, pp. 1187-1192 (1988)

http://dx.doi.org/10.1364/JOSAA.5.001187

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### Abstract

Given a complex function *F*(*ω*) = |*F*(*ω*)|exp[*j*Δ(*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, *r** _{p}*(

*ϕ*), and the ratio of complex reflection coefficients for the

*p*and

*s*polarizations,

*ρ*(

*ϕ*) =

*r*

*(*

_{p}*ϕ*)/

*r*

*(*

_{s}*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 |

*r*

*| also determines a new angle at which the reflection phase shift*

_{p}*δ*

*= arg*

_{p}*r*

*exhibits a minimum of its own. Likewise, the same (quartic) equation that determines the second Brewster angle of minimum |*

_{p}*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

**History**

Original Manuscript: December 28, 1987

Manuscript Accepted: April 5, 1988

Published: August 1, 1988

**Citation**

R. M. A. Azzam, "Extrema of the magnitude and the phase of a complex function of a real variable: application to attenuated internal reflection," J. Opt. Soc. Am. A **5**, 1187-1192 (1988)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-5-8-1187

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### References

- 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.
- 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]
- 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]
- S. Y. Kim, K. Vedam, “Analytic solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986). [CrossRef]
- R. W. Hunter, “Measurement of optical properties of materials in the vacuum ultraviolet spectral region,” Appl. Opt. 21, 2103–2114 (1982). [CrossRef] [PubMed]
- 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]
- 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.
- 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]
- See, e.g., S. M. Selby, ed., Standard Mathematical Tables, 20th ed. (Chemical Rubber Company, Cleveland, Ohio, 1972), p. 106.
- 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]

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