Abstract

A method for studying photon tunneling in uniaxial crystal films is presented. The complex refractive index and the complex angle of refraction of the evanescent wave in a crystal are calculated for the most general case. The reflectance and transmittance resulting from the tunneling effect in crystal films are discussed, and the relations among these coefficients and the optical parameters of crystal are found. These relations provide a theoretical basis for characterizing crystal films by means of photon tunneling.

© 1998 Optical Society of America

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References

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  1. S. Zhu, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 7, 801–607 (1986).
  2. J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555–3557 (1995).
    [CrossRef]
  3. J. M. Guerra, “Photon tunneling microscopy of polymeric surfaces,” Science 262, 1395–1400 (1993).
    [CrossRef] [PubMed]
  4. J. M. Guerra, “Photon tunneling microscopy,” Appl. Opt. 29, 3741–3752 (1990).
    [CrossRef] [PubMed]
  5. W.-Q. Zhang, “General ray-tracing formulas for crystal,” Appl. Opt. 31, 7328–7331 (1992).
    [CrossRef] [PubMed]
  6. W.-Q. Zhang, Y.-M. Zhuo, “Design and application of a bifocus lens,” Appl. Opt. 32, 4204–4208 (1993).
    [CrossRef] [PubMed]
  7. W.-Q. Zhang, “Reflectance, transmittance and total internal reflection in biaxial crystal,” Optik (Stuttgart) 104, 67–71 (1997).

1997 (1)

W.-Q. Zhang, “Reflectance, transmittance and total internal reflection in biaxial crystal,” Optik (Stuttgart) 104, 67–71 (1997).

1995 (1)

J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555–3557 (1995).
[CrossRef]

1993 (2)

J. M. Guerra, “Photon tunneling microscopy of polymeric surfaces,” Science 262, 1395–1400 (1993).
[CrossRef] [PubMed]

W.-Q. Zhang, Y.-M. Zhuo, “Design and application of a bifocus lens,” Appl. Opt. 32, 4204–4208 (1993).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

1986 (1)

S. Zhu, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 7, 801–607 (1986).

Guerra, J. M.

J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555–3557 (1995).
[CrossRef]

J. M. Guerra, “Photon tunneling microscopy of polymeric surfaces,” Science 262, 1395–1400 (1993).
[CrossRef] [PubMed]

J. M. Guerra, “Photon tunneling microscopy,” Appl. Opt. 29, 3741–3752 (1990).
[CrossRef] [PubMed]

Zhang, W.-Q.

Zhu, S.

S. Zhu, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 7, 801–607 (1986).

Zhuo, Y.-M.

Am. J. Phys. (1)

S. Zhu, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 7, 801–607 (1986).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555–3557 (1995).
[CrossRef]

Optik (Stuttgart) (1)

W.-Q. Zhang, “Reflectance, transmittance and total internal reflection in biaxial crystal,” Optik (Stuttgart) 104, 67–71 (1997).

Science (1)

J. M. Guerra, “Photon tunneling microscopy of polymeric surfaces,” Science 262, 1395–1400 (1993).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Transformation between two coordinates in a uniaxial crystal.

Fig. 2
Fig. 2

Wave vector, wave front, and ray of an e-evanescent wave in total internal reflection.

Fig. 3
Fig. 3

Electric field components of the incident, reflected, and transmitted waves in isotropic media.

Fig. 4
Fig. 4

Relations among the reflectances and transmittances and the thickness d of crystal films (a) for a CeF3–calcite–CeF3 film (solid curve, θ1 = 63°; dashed curve, θ1 = 80°), and (b) for a SiO2–calcite–SiO2 film with θ1 = 80.2°.

Equations (25)

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n 1 sin   θ 1 = n   sin   ϕ ,
1 / n 2 = 1 / n e 2   +   1 / n o 2 - 1 / n e 2 S 2 z ,
Sx = sin   ϕ   sin   α , Sy = - sin   ϕ cos   α   sin   β + cos   β   cot   ϕ , Sz = sin   ϕ cos   α   cos   β   -   sin   β   cot   ϕ ,
Sy = - sin   ϕ cos   α   sin   β + u   cos   β + iv   cos   β = sin   ϕ s y 1 + is y 2 = sin   ϕ   s y , Sz = sin   ϕ cos   α   cos   β - u   sin   β - iv   sin   β = sin   ϕ s z 1 + is z 2 = sin   ϕ   s z ,
u = b   cos   α   cos   β   sin   β / 1 + b   sin 2   β vn 1 sin   θ 1 = n e n 1 2 sin 2   θ 1 cos 2   β   sin 2   α / n e 2 + cos 2   α   cos 2   β   +   sin 2   β / n o 2 - 1 + b   sin 2   β 1 / 2 / 1 + b   sin 2   β ,
ψ = k o n S · r - ω t = k o n Sz 2 Z 2 + Sy 1 Y 1 - ω t = k o n 1 sin   θ 1 Z 2 - k o n 1 sin   θ 1 u + iv Y 1 - ω t ,
E k = n 2 S k E · S / n 2 - n k 2 = sin   ϕ ns k E · S / n 2 - n k 2 ,   k = x ,   y ,   z ,
Ex = n e 2 s x s z A E · S ,   Ey = n e 2 s y s z A E · S , Ez = - n o 2 s x 2 + s y 2 A E · S ,
Ee = Ae Fe 1 + iFe 2 .
H = n 0 Sz - Sy - Sz 0 Sx Sy - Sx 0 E = n   sin   ϕ 0 s z - s y - s z 0 s x s y - s x 0 A e Fe 1 + iFe 2 = n 1 sin   θ 1 Be Fe 1 + iFe 2 = Be Fe 1 + iFe 2 ,
Sx 1 = 0 ,     Sy 1 = - cos   ϕ = - i sin 2   ϕ - 1 1 / 2 , Sz 2 = sin   ϕ .
X Y Z = 1 0 0 0 cos   β sin   β 0 - sin   β cos   β cos   α 0 - sin   α 0 1 0 sin   α 0 cos   α X 1 Y 1 Z 2 .
Eo = - S y + iS y / N S x + iS x / N 0 Eo 1 + iEo 2 = Ao Eo ,
Ho = Bo Eo ,
1 0 0 n 1 cos   θ 1 1 0 0 - n 1 cos   θ 1 0 cos   θ 1 - n 1 0 0 - cos   θ 1 - n 1 0 Eis Ers Eip Erp = Ui ,   r Ei ,   r .  
Ee = L - 1 Ae Fe 1 + iFe 2 = Ne 1 + iNe 2 Fe 1 + iFe 2 , He = L - 1 Be Fe 1 + iFe 2 = Me 1 + iMe 2 Fe 1 + iFe 2 , Eo = No Eo 1 + iEo 2 , Ho = Mo Eo 1 + iEo 2 .
Ere = Are Fre 1 + iFre 2 , Ero = Aro Ero 1 + iEro 2 , Hre = Bre Fre 1 + iFre 2 , Hro = Bro Ero 1 + iEro 2 .
1 0 0 n 1 cos   θ 1 1 0 0 - n 1 cos   θ 1 0 cos   θ 1 - n 1 0 0 - cos   θ 1 - n 1 0 Eis Ers Eip Erp = Nex 1 Nez 2 Mex 1 Mez 2 Nrex 1 Nrez 2 Mrex 1 Mrez 2 Nox 1 Noz 2 Mox 1 Moz 2 Nrox 1 Nroz 2 Mrox 1 Mroz 2 Fe Fre Eo Ero ,
Ui ,   r Eis Ers Eip Erp = U Fe Fre Eo Ero .
ε e = E e   exp i β d - γ d ,
ε o = E o   exp - γ o d ,
ε re = E re exp - i β d + γ d ,     ε ro = E r o exp γ o d .
1 0 0 n 3 cos   θ 3 0 cos   θ 3 - n 3 0 Ets Etp exp i β 3 d = Ut Et exp i β 3 d ,
U Fe   exp i β d - γ d Fre   exp - i β d + γ d Eo   exp - γ o d Ero   exp + γ o d = Ut Ets Etp exp i β 3 d .
W = Re E × H * .

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