Abstract

Simple and explicit expressions for the phase shifts that p- and s-polarized light experience in frustrated total internal reflection (FTIR) and optical tunneling by an embedded low-index thin film are obtained. The differential phase shifts in reflection and transmission Δr,Δt are found to be identical, and the associated ellipsometric parameters ψr,ψt are governed by a simple relation, independent of film thickness. When the Fresnel interface reflection phase shifts for the p and s polarizations or their average are quarter-wave, the corresponding overall reflection phase shifts introduced by the embedded layer are also quarter-wave for all values of film thickness. In the limit of zero film thickness (i.e., for an ultrathin embedded layer), the reflection phase shifts are also quarter-wave independent of polarization (p or s) or angle of incidence (except at grazing incidence). Finally, variable-angle FTIR ellipsometry is shown to be a sensitive technique for measuring the thickness of thin uniform air gaps between transparent bulk media.

© 2006 Optical Society of America

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References

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  1. S. Zhu, A. W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: a demonstration and review," Am. J. Phys. 54, 601-607 (1986).
    [CrossRef]
  2. C. K. Carneglia and L. Mandel, "Phase-shift measurement of evanescent electromagnetic waves," J. Opt. Soc. Am. 61, 1035-1043 (1971).
    [CrossRef]
  3. C. K. Carneglia and L. Mandel, "Differential phase shifts of TE and TM evanescent waves," J. Opt. Soc. Am. 61, 1423-1424 (1971).
    [CrossRef]
  4. 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).
    [CrossRef]
  5. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  6. V. V. Efimov, O. V. Ivanov, and D. I. Sementsov, "Phase relation for electromagnetic waves reflected and transmitted by a dielectric slab," J. Opt. A, Pure Appl. Opt. , 3, 514-516 (2001).
    [CrossRef]
  7. R. W. Astheimer, G. Falbel, and S. Minkowitz, "Infrared modulation by means of frustrated total internal reflection," Appl. Opt. 5, 87-91 (1966).
    [CrossRef] [PubMed]
  8. P. W. Baumeister, "Optical tunneling and its application to optical filters," Appl. Opt. 6, 897-905 (1967).
    [CrossRef] [PubMed]
  9. J. J. Tuma, Engineering Mathematics Handbook, 3rd ed. (McGraw-Hill, 1987), p. 75.

2004 (1)

2001 (1)

V. V. Efimov, O. V. Ivanov, and D. I. Sementsov, "Phase relation for electromagnetic waves reflected and transmitted by a dielectric slab," J. Opt. A, Pure Appl. Opt. , 3, 514-516 (2001).
[CrossRef]

1986 (1)

S. Zhu, A. W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: a demonstration and review," Am. J. Phys. 54, 601-607 (1986).
[CrossRef]

1971 (2)

1967 (1)

1966 (1)

Astheimer, R. W.

Azzam, R. M.

Bashara, N. M.

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

Baumeister, P. W.

Carneglia, C. K.

Efimov, V. V.

V. V. Efimov, O. V. Ivanov, and D. I. Sementsov, "Phase relation for electromagnetic waves reflected and transmitted by a dielectric slab," J. Opt. A, Pure Appl. Opt. , 3, 514-516 (2001).
[CrossRef]

Falbel, G.

Hawley, D.

S. Zhu, A. W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: a demonstration and review," Am. J. Phys. 54, 601-607 (1986).
[CrossRef]

Ivanov, O. V.

V. V. Efimov, O. V. Ivanov, and D. I. Sementsov, "Phase relation for electromagnetic waves reflected and transmitted by a dielectric slab," J. Opt. A, Pure Appl. Opt. , 3, 514-516 (2001).
[CrossRef]

Mandel, L.

Minkowitz, S.

Roy, R.

S. Zhu, A. W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: a demonstration and review," Am. J. Phys. 54, 601-607 (1986).
[CrossRef]

Sementsov, D. I.

V. V. Efimov, O. V. Ivanov, and D. I. Sementsov, "Phase relation for electromagnetic waves reflected and transmitted by a dielectric slab," J. Opt. A, Pure Appl. Opt. , 3, 514-516 (2001).
[CrossRef]

Tuma, J. J.

J. J. Tuma, Engineering Mathematics Handbook, 3rd ed. (McGraw-Hill, 1987), p. 75.

Yu, A. W.

S. Zhu, A. W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: a demonstration and review," Am. J. Phys. 54, 601-607 (1986).
[CrossRef]

Zhu, S.

S. Zhu, A. W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: a demonstration and review," Am. J. Phys. 54, 601-607 (1986).
[CrossRef]

Am. J. Phys. (1)

S. Zhu, A. W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: a demonstration and review," Am. J. Phys. 54, 601-607 (1986).
[CrossRef]

Appl. Opt. (2)

J. Opt. A, Pure Appl. Opt. (1)

V. V. Efimov, O. V. Ivanov, and D. I. Sementsov, "Phase relation for electromagnetic waves reflected and transmitted by a dielectric slab," J. Opt. A, Pure Appl. Opt. , 3, 514-516 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Other (2)

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

J. J. Tuma, Engineering Mathematics Handbook, 3rd ed. (McGraw-Hill, 1987), p. 75.

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Figures (7)

Fig. 1
Fig. 1

Reflection and transmission of p- and s-polarized light at an angle of incidence ϕ by a uniform layer (medium 1) of thickness d and refractive index n 1 that is embedded in a bulk medium (0) of refractive index n 0 .

Fig. 2
Fig. 2

Reflection phase shift for the p polarization Δ r p as a function of the angle of incidence ϕ for d λ = 0.005 , 0.02 to 0.20 in equal steps of 0.02, and 10. All curves pass through a common point A at which Δ r p = 90 ° . These results are calculated for an air gap of thickness d between two glass prisms ( N = 1.5 ) .

Fig. 3
Fig. 3

Reflection phase shift for the s polarization Δ r s as a function of the angle of incidence ϕ for d λ = 0.005 , 0.02 to 0.20 in equal steps of 0.02, and 10. All curves pass through a common point B at which Δ r s = 90 ° . These results are calculated for an air gap of thickness d between two glass prisms ( N = 1.5 ) .

Fig. 4
Fig. 4

Average reflection phase shift Δ r a = ( Δ r p + Δ r s ) 2 as a function of the angle of incidence ϕ for d λ = 0.005 , 0.02 to 0.20 in equal steps of 0.02, and 10. As in Figs. 2, 3, all curves pass through a common point C at which Δ r a = 90 ° . These results are calculated for an air gap of thickness d between two glass prisms ( N = 1.5 ) .

Fig. 5
Fig. 5

Differential reflection phase shift Δ r = ( Δ r p Δ r s ) as a function of the angle of incidence ϕ for d λ = 0.005 , 0.02 to 0.20 in equal steps of 0.02, and 10. These results are calculated for an air gap of thickness d between two glass prisms ( N = 1.5 ) .

Fig. 6
Fig. 6

Reflection ellipsometric parameters ψ r as a function of the angle of incidence ϕ for d λ = 0.005 , 0.02 to 0.20 in equal steps of 0.02, and 10. These results are calculated for an air gap of thickness d between two glass prisms ( N = 1.5 ) .

Fig. 7
Fig. 7

Transmission ellipsometric parameters ψ t as a function of the angle of incidence ϕ for d λ = 0.005 , 0.02 to 0.20 in equal steps of 0.02, and 10. These results are calculated for an air gap of thickness d between two glass prisms ( N = 1.5 ) .

Equations (51)

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R ν = R ν exp ( j Δ r ν ) = r 01 ν ( 1 X ) ( 1 r 01 ν 2 X ) ,
T ν = T ν exp ( j Δ t ν ) = ( 1 r 01 ν 2 ) X 1 2 ( 1 r 01 ν 2 X ) ,
ν = p , s ,
r 01 ν = exp ( j δ ν ) ,
δ p = 2 arctan ( N U cos ϕ ) ,
δ s = 2 arctan ( U N cos ϕ ) ,
N = n 0 n 1 > 1 , U = ( N 2 sin 2 ϕ 1 ) 1 2 .
X = exp ( 2 x ) ,
x = 2 π n 1 ( d λ ) U .
Δ r ν = arg ( R ν ) ,
Δ t ν = arg ( T ν ) .
tan Δ r ν = tan δ ν tanh x ,
tan Δ t ν = tanh x tan δ ν .
tan Δ r ν tan Δ t ν = 1 ,
Δ r ν Δ t ν = ± π 2 .
Δ r = Δ r p Δ r s ,
Δ t = Δ t p Δ t s .
tan Δ r = tan Δ t = tanh x ( tan δ p tan δ s ) ( tanh 2 x + tan δ p tan δ s ) ,
Δ r = Δ t .
Δ r a = ( Δ r p + Δ r s ) 2 .
tan ( 2 Δ r a ) = tanh x ( tan δ p + tan δ s ) ( tanh 2 x tan δ p tan δ s ) .
ρ r = R p R s = tan ψ r exp ( j Δ r ) ,
ρ t = T p T s = tan ψ t exp ( j Δ t ) .
ρ r = ( r 01 p r 01 s ) [ ( 1 r 01 s 2 X ) ( 1 r 01 p 2 X ) ] ,
ρ t = [ ( 1 r 01 p 2 ) ( 1 r 01 s 2 ) ] [ ( 1 r 01 s 2 X ) ( 1 r 01 p 2 X ) ] .
ρ r ( X = 1 ) = ( r 01 p r 01 s ) [ ( 1 r 01 s 2 ) ( 1 r 01 p 2 ) ] ,
ρ t ( X = 1 ) = 1 .
ρ r ( X = 0 ) = ( r 01 p r 01 s ) ,
ρ t ( X = 0 ) = ( 1 r 01 p 2 ) ( 1 r 01 s 2 ) .
Γ = ρ r ρ t = ( tan ψ r tan ψ t ) exp [ j ( Δ r Δ t ) ] = ( r 01 s r 01 s 1 ) ( r 01 p r 01 p 1 ) ,
Γ = ρ r ρ t = sin δ s sin δ p .
Δ r Δ t = 0 ,
tan ψ r tan ψ t = sin δ s sin δ p .
tan ψ r tan ψ t = sin δ s sin δ p = ( N 2 + 1 ) sin 2 ϕ 1 .
sin δ p sin δ s = δ p δ s ,
sin 2 ϕ p = ( N 2 + 1 ) ( N 4 + 1 ) ,
sin 2 ϕ s = ( N 2 + 1 ) ( 2 N 2 ) ,
sin 2 ϕ a = 2 ( N 2 + 1 ) ,
ψ t G = arctan ( 1 N 2 ) ,
δ ν = tan δ ν = 0 ,
x = tanh x = 0 .
tan Δ r ν = 0 0 ,
tan Δ t ν = 0 0
Δ r ν ( ϕ c ) = arctan ( δ ν x ) ϕ c ,
Δ r p ( ϕ c ) = arctan [ π 1 ( d λ ) 1 N 2 ( N 2 1 ) 1 2 ] ,
Δ r s ( ϕ c ) = arctan { [ π 1 ( d λ ) 1 ( N 2 1 ) 1 2 ] }
tan ( δ p 2 ) = N 2 tan ( δ s 2 ) .
δ p δ s = sin δ p sin δ s ,
( δ p δ s ) ϕ c = N 2 .
( δ p δ s ) 90 ° = 1 N 2 .
( δ p δ s ) ϕ a = 1 ,

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