Abstract

The condition for obtaining a differential (or ellipsometric) quarter-wave retardation when p- and s-polarized light of wavelength λ experience frustrated total internal reflection (FTIR) and optical tunneling at angles of incidence ϕ the critical angle by a transparent thin film (medium 1) of low refractive index n1 and uniform thickness d, which is embedded in a transparent bulk medium 0 of high refractive index n0 takes the simple form: tanh2x=tanδptanδs, in which x=2πn1(d/λ)(N2sin2ϕ1)1/2, N=n0/n1, and δp, δs are 01 interface Fresnel reflection phase shifts for the p and s polarizations. From this condition, the ranges of the principal angle and normalized film thickness d/λ are obtained explicitly. At a given principal angle, the associated principal azimuths ψr, ψt in reflection and transmission are determined by tan2ψr=sin2δs/sin2δp and tan2ψt=tanδp/tanδs, respectively. At a unique principal angle ϕe given by sin2ϕe=2/(N2+1), ψr=ψt=45° and linear-to-circular polarization conversion is achieved upon FTIR and optical tunneling simultaneously. The intensity transmittances of p- and s-polarized light at any principal angle are given by τp=tanδp/tan(δpδs) and τs=tanδs/tan(δpδs), respectively. The efficiency of linear-to-circular polarization conversion in optical tunneling is maximum at ϕe.

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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2008 (1)

2006 (1)

2005 (1)

2004 (1)

1981 (1)

1977 (1)

1967 (2)

1966 (1)

Alsamman, A.

Astheimer, R. W.

Azzam, R. M. A.

Bashara, N. M.

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

Baumeister, P. W.

Falbel, G.

Holl, H. B.

Minkowitz, S.

Spinu, C. L.

Zaghloul, A.-R. M.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Other (1)

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

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Figures (5)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Graphical construction that illustrates the range of possible solutions of Eq. (8).

Fig. 3
Fig. 3

Normalized thickness d / λ of a uniform air gap between two IR-transparent Ge prisms ( n 1 = 1 , N = 4 ) that produces Δ r = Δ t = π / 2 is plotted as a function of principal angle ϕ over the full range from ϕ 1 = 15.04 ° to ϕ 2 = 42.93 ° .

Fig. 4
Fig. 4

Reflection and transmission principal azimuths ψ r , ψ t [Eqs. (17, 18)] are plotted as functions of the principal angle ϕ for a uniform air gap between two IR-transparent Ge prisms ( n 1 = 1 , N = 4 ) over the full range of principle angles ϕ 1 ϕ ϕ 2 .

Fig. 5
Fig. 5

Normalized thickness d / λ of a low-index embedded layer that produces circular polarization in FTIR and optical tunneling at ϕ e and ψ r = ψ t = 45 ° [Eq. (24)] is plotted as a function of the refractive index ratio N over the range 2.5 N 6.0 . Equal transmittance of p- and s-polarized light τ p ( ϕ e ) = τ s ( ϕ e ) = τ [Eq. (39)] is also shown as a function of N.

Equations (49)

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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 ) .
Δ r = Δ t ,
tan ψ r / tan ψ t = sin δ s / sin δ p = ( N 2 + 1 ) sin 2 ϕ 1 .
ρ r = tanh x cos δ s j sin δ s tanh x cos δ p j sin δ p ,
x = 2 π n 1 ( d / λ ) ( N 2 sin 2 ϕ 1 ) 1 / 2 .
Re ρ r = 0 ,
tanh 2 x cos δ p cos δ s + sin δ p sin δ s = 0 ,
tanh 2 x = tan δ p tan δ s .
tan δ p tan δ s = ( sin 2 ϕ p sin 2 ϕ s ) ( sin 2 ϕ sin 2 ϕ c ) ( 1 sin 2 ϕ ) ( sin 2 ϕ sin 2 ϕ p ) ( sin 2 ϕ sin 2 ϕ s ) .
tan δ p tan δ s = 1 .
sin 2 ϕ 1 , 2 = [ ( N 2 + 1 ) ( N 4 6 N 2 + 1 ) 1 / 2 ] / 4 N 2 .
N 2 + 1 = 2.414 .
X = exp ( 2 x ) = ( 1 tanh x ) / ( 1 + tanh x )
d / λ = ( ln X ) / [ 4 π n 1 ( N 2 sin 2 ϕ 1 ) 1 / 2 ] .
ρ r = j tan ψ r .
Im ρ r = tan ψ r = tanh 2 x sin ( δ p δ s ) tanh 2 x cos 2 δ p + sin 2 δ p .
tan 2 ψ r = sin 2 δ s / sin 2 δ p .
tan 2 ψ t = tan δ p / tan δ s .
sin 2 ϕ e = 2 / ( N 2 + 1 ) .
tan ( δ s / 2 ) = 1 / N , tan ( δ p / 2 ) = N .
( tan δ p tan δ s ) max = 4 N 2 / ( N 2 1 ) 2 .
tanh x = 2 N / ( N 2 1 ) ,
X = ( N 2 2 N 1 ) / ( N 2 + 2 N 1 ) .
d / λ = ( 4 π n 1 ) 1 ( N 2 + 1 N 2 1 ) 1 / 2 ln ( N 2 + 2 N 1 N 2 2 N 1 ) .
d / λ = ( 4 π ) 1 ( 17 / 15 ) 1 / 2 ln ( 23 / 7 ) = 0.100778 .
R ν = | R ν | 2 = cosh 2 x 1 cosh 2 x cos 2 δ ν , ν = p , s ;
τ ν = 1 | R ν | 2 = 1 cos 2 δ ν cosh 2 x cos 2 δ ν , ν = p , s .
cosh 2 x = ( 1 + tanh 2 x ) / ( 1 tanh 2 x ) ,
τ p = tan δ p / tan ( δ p δ s ) , τ s = tan δ s / tan ( δ p δ s ) .
R ν = 1 τ ν ν = p , s .
τ p = 2 N 2 sin 4 ϕ ( N 2 + 1 ) sin 2 ϕ + 1 ( N 4 + 1 ) sin 4 ϕ + ( N 2 + 1 ) sin 2 ϕ ,
τ s = 2 N 2 sin 4 ϕ ( N 2 + 1 ) sin 2 ϕ + 1 2 N 2 sin 4 ϕ ( N 2 + 1 ) sin 2 ϕ .
sin 2 ϕ p max = ( N 2 + 1 ) / ( N 2 1 ) 2 .
sin 2 ϕ s max = ( N 2 + 1 ) / 4 N 2 .
τ p max = τ s max = N 4 6 N 2 + 1 N 4 + 2 N 2 + 1 ,
ψ t ( ϕ s max ) = 90 ° ψ t ( ϕ p max ) ,
ψ t ( ϕ p max ) = arc tan [ N 6 3 N 4 3 N 2 + 1 2 N 2 ( N 2 + 1 ) ] 1 / 2 .
ϕ p max = 15.95 ° , ϕ s max = 31.02 ° ; τ p max = τ s max = 0.5571 ; ψ t ( ϕ p max ) = 67.84 ° , ψ t ( ϕ s max ) = 22.16 ° .
τ p ( ϕ e ) = τ s ( ϕ e ) = 1 1 2 ( N 2 + 1 N 2 1 ) 2 .
tan 2 ψ t = ( u p u s ) ( u u s u u p ) ,
tan 2 ψ r = ( 4 u p u s u e 2 ) ( ( u u s ) ( u 0.5 u e ) 1 / 2 ( u u p ) ) .
u = ( 1 / 4 ) { ( 3 u p + u s ) ( 3 u p + u s ) 2 + 4 u e ( u s u p ) 16 u p u s } .
I t = I i ( τ p cos 2 ψ t + τ s sin 2 ψ t ) .
η LTC = I t / I i = ( τ p cos 2 ψ t + τ s sin 2 ψ t ) .
η LTC = I t / I i = 2 τ p cos 2 ψ t = 2 τ s sin 2 ψ t .
η LTC = [ u p / ( u p u s ) ] [ 2 2 u s u 1 + u c u 2 ] .
d η LTC / d u = 0
u = u c / u s = 2 / ( N 2 + 1 ) .
η LTC max = τ p ( ϕ e ) = τ s ( ϕ e ) 1 1 2 ( N 2 + 1 N 2 1 ) 2 .

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