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

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References

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  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).
  6. E. E. Hall, “The penetration of totally reflected light into the rarer medium,” Phys. Rev. 15, 73–106 (1902).
  7. P. W. Baumeister, “Optical tunneling and its application to optical filters,” Appl. Opt. 6, 897–905 (1967).
    [CrossRef] [PubMed]
  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.

1981 (1)

1979 (1)

1967 (1)

1966 (1)

1950 (1)

F. Abelès, “Un théoreme relatif à la réflexion métallique,” C. R. Hebd. Seances Acad. Sci. 230, 1942–1943 (1950).

1902 (1)

E. E. Hall, “The penetration of totally reflected light into the rarer medium,” Phys. Rev. 15, 73–106 (1902).

Abelès, F.

F. Abelès, “Un théoreme relatif à la réflexion métallique,” C. R. Hebd. Seances Acad. Sci. 230, 1942–1943 (1950).

Astheimer, R. W.

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Baumeister, P. W.

Bennett, H. E.

J. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw Hill, New York, 1978), Sect. 10.

Bennett, J.

J. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw Hill, New York, 1978), Sect. 10.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Falbel, G.

Fresnel, A.

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).

Hall, E. E.

E. E. Hall, “The penetration of totally reflected light into the rarer medium,” Phys. Rev. 15, 73–106 (1902).

Minkowitz, S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Appl. Opt. (3)

C. R. Hebd. Seances Acad. Sci. (1)

F. Abelès, “Un théoreme relatif à la réflexion métallique,” C. R. Hebd. Seances Acad. Sci. 230, 1942–1943 (1950).

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

E. E. Hall, “The penetration of totally reflected light into the rarer medium,” Phys. Rev. 15, 73–106 (1902).

Other (9)

See Ref. 2, p. 50.

J. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw Hill, New York, 1978), Sect. 10.

For any N, Eq. (20) also has a trivial solution at ϕc,where all phase shifts are zero.

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.

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.

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.

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).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

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Figures (3)

Fig. 1
Fig. 1

TIR phase shifts δp, δs, δa, and Δ at the glass–air interface (N=1.5) plotted as functions of the angle of incidence ϕ between the critical angle ϕc=arcsin(2/3)=41.81° and grazing incidence ϕ=90°. Quarter-wave phase shifts are attained at points A, B, and C where the line δ=90° intersects the curves of δp, δs, and δa at angles of incidence denoted by ϕp, ϕs, and ϕa, respectively.

Fig. 2
Fig. 2

Three angles of incidence ϕp, ϕs, and ϕa, at which the p, s, and average TIR phase shifts [Eqs. (9)–(11)] are quarter-wave are plotted as functions of the refractive index ratio N.

Fig. 3
Fig. 3

As in Fig. 1 except that N=4, which corresponds to the Ge–air interface in the infrared. The significance of the marked points D1, D2, E, and F is discussed in the text.

Equations (53)

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ϕc=arcsin n,
tan(δp/2)=N(N2sin2ϕ-1)1/2/cos ϕ,
tan(δs/2)=(N2sin2ϕ-1)1/2/(Ncosϕ),
tan(Δ/2)=(N2sin2ϕ-1)1/2/(Nsinϕtanϕ).
N=N0/N1
δa=(δp+δs)/2.
tan δa=[(cos ϕ)(N+N-1)(N2sin2ϕ-1)1/2]/[2-(sin2ϕ)(N2+1)].
tan(δp/2)=N2tan(δs/2),
sin2ϕp=(N2+1)/(N4+1),
sin2ϕs=(N2+1)/(2N2),
sin2ϕa=2/(N2+1).
tan(δs/2)=1/N2,δp=π/2,
tan(δp/2)=N2,δs=π/2.
Δ(ϕp)=(π/2)-2 arctan(1/N2),
Δ(ϕs)=2 arctan(N2)-(π/2).
Δ(ϕp)=Δ(ϕs).
tan(Δmax/2)=(N2-1)/(2N).
sin2ϕ1,2=(14)[(n2+1)±(n4-6n2+1)1/2],
(δp/2)+(δs/2)=π/2,
(δp/2)-(δs/2)=Δmax/2,
δp=(π+Δmax)/2,
δs=(π-Δmax)/2.
R=jexp(jΔmax/2)00exp(-jΔmax/2).
δa=Δ,
(δp+δs)/2=δp-δs,
δp=3 δs.
sin2ϕ=(N2+1)/(4N2).
sin2ϕsin2ϕc=1/N2.
N>3=1.732.
12[sec2(Δ/2)](Δ/ϕ)=[(1-N2)tan2 ϕ+2]/[Nsinϕ tan2 ϕ×(N2sin2ϕ-1)1/2].
tan2 ϕm=2/(N2-1),
sin2ϕm=2/(N2+1).
(Δ/ϕ)ϕ=ϕc=,
(δp/ϕ)ϕ=ϕc=(δs/ϕ)ϕ=ϕc=.
(Δ/ϕ)ϕ=90°=-(2/N)(N2-1)1/2=-2(1-n2)1/2=-2cosϕc.
(2Δ/ϕ2)ϕ=90°=0.
θ=90°-ϕ,
Δ2(1-n2)θ.
Δϕ=ϕm-ϕc,
sin Δϕ=sin ϕmcos ϕc-cos ϕmsin ϕc,
sin(Δϕ)=(2-1)(N2-1)1/2/[N(N2+1)1/2].
N=(2+1)1/2=1.55377.
sin(Δϕ)max=(2-1)2,(Δϕ)max=9.879°.
Δ=Δmax=45°.
N=1.49661,
n=sin ϕc.
n2=sin2ϕ[1-tan2 ϕ tan2(Δ/2)].
tan ϕ tan(Δ/2)<1,
Δ<2(90°-ϕ)=2θ.
N=sec(Δmax/2)+tan(Δmax/2),
n=sec(Δmax/2)-tan(Δmax/2).
N2=2 csc2 ϕm-1.
n2=1-0.25[(Δ/ϕ)ϕ=90°]2.

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