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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 21, Iss. 8 — Aug. 30, 2004
  • pp: 1559–1563

Phase shifts that accompany total internal reflection at a dielectric–dielectric interface

R. M. A. Azzam  »View Author Affiliations


JOSA A, Vol. 21, Issue 8, pp. 1559-1563 (2004)
http://dx.doi.org/10.1364/JOSAA.21.001559


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Abstract

The absolute, average, and differential phase shifts that p- and s-polarized light experience in total internal reflection (TIR) at the planar interface between two transparent media are considered as functions of the angle of incidence ϕ. Special angles at which quarter-wave phase shifts are achieved are determined as functions of the relative refractive index N. When the average phase shift equals π/2, the differential reflection phase shift Δ is maximum, and the reflection Jones matrix assumes a simple form. For N>3, the average and differential phase shifts are equal (hence δp=3 δs) at a certain angle ϕ that is determined as a function of N. All phase shifts rise with infinite slope at the critical angle. The limiting slope of the Δ-versus-ϕ curve at grazing incidence (Δ/ϕ)ϕ=90°=-(2/N)(N2-1)1/2=-2cosϕc, where ϕc is the critical angle and (2Δ/ϕ2)ϕ=90°=0. Therefore Δ is proportional to the grazing incidence angle θ=90°-ϕ (for small θ) with a slope that depends on N. The largest separation between the angle of maximum Δ and the critical angle is 9.88° and occurs when N=1.55377. Finally, several techniques are presented for determining the relative refractive index N by using TIR ellipsometry.

© 2004 Optical Society of America

OCIS Codes
(240.0240) Optics at surfaces : Optics at surfaces
(260.0260) Physical optics : Physical optics
(260.2130) Physical optics : Ellipsometry and polarimetry
(260.5430) Physical optics : Polarization
(260.6970) Physical optics : Total internal reflection

History
Original Manuscript: January 13, 2004
Revised Manuscript: March 18, 2004
Manuscript Accepted: March 18, 2004
Published: August 1, 2004

Citation
R. M. A. Azzam, "Phase shifts that accompany total internal reflection at a dielectric–dielectric interface," J. Opt. Soc. Am. A 21, 1559-1563 (2004)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-8-1559


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References

  1. A. Fresnel, “Mémoire sur la loi des modifications que la réflexion imprime à la lumière polarisée,” in Ouvres Complète de Fresnel, Vol. 1, H. Senarmont, E. Verdet, A. Fresnel, 1866, pp. 767–775 (Johnson Reprint Corporation, New York, 1965).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  3. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  4. R. M. A. Azzam, “Relations between amplitude reflectances and phase shifts of the p and s polarizations when electromagnetic radiation strikes interfaces between transparent media,” Appl. Opt. 18, 1884–1886 (1979). [CrossRef] [PubMed]
  5. F. Abelès, “Un théoreme relatif à la réflexion métallique,” C. R. Hebd. Seances Acad. Sci. 230, 1942–1943 (1950).
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  8. R. W. Astheimer, G. Falbel, S. Minkowitz, “Infrared modulation by means of frustrated total internal reflection,” Appl. Opt. 5, 87–91 (1966). [CrossRef] [PubMed]
  9. See Ref. 2, p. 50.
  10. 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]
  11. J. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw Hill, New York, 1978), Sect. 10.
  12. For any N, Eq. (20) also has a trivial solution at ϕc,where all phase shifts are zero.
  13. From Eq. (26) we also obtain cos2 ϕm=(N2-1)/(N2+1)= -cos(2ϕB),which provides a direct relation between the incidence angle of maximum TIR differential phase shift ϕmand the Brewster angle of external reflection ϕB.
  14. From Eqs. (35) and (26) it is readily verified that sin2(Δϕ)< sin2(90°-ϕm)=cos2 ϕm,so that Δϕ=ϕm- ϕc< 90°-ϕm.This proves that ϕmis always closer to the critical angle than it is to grazing incidence (90°), consistent with Figs. 1and 3.
  15. Equation (40) can also be obtained by substituting (1- ρ)2/(1+ρ)2=-tan2(Δ/2),when ρ=exp(jΔ),in Eq. (4.20a) of Ref. 3.

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