Abstract

The complex-amplitude reflection coefficients of p- and s-polarized light by a transparent freestanding, embedded, or deposited quarter-wave layer (QWL) are derived as explicit functions of the angle of incidence and layer refractive index. This provides the basis for the design of 50%–50% beam splitters for incident s-polarized or unpolarized light that use a high-index (e.g., TiO2 or Ge) QWL embedded in a glass cube in the visible and near infrared spectral range. These simple devices have good angular and spectral response and are insensitive to small film thickness errors to the first order.

© 2008 Optical Society of America

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References

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    [CrossRef]
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  8. R. M. A. Azzam, and C. L. Spinu, “Achromatic angle-insensitive infrared quarter-wave retarder based on total internal reflection at the Si-SiO2 interface,” J. Opt. Soc. Am. A 21, 2019-2022 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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  14. P. Bhattacharya, Semiconductor Optoelectronic Devices (Prentice Hall, 1997).
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  16. R. M. A. Azzam, “Dividing a light beam into two beams of orthogonal polarizations by reflection and refraction at a dielectric surface,” Opt. Lett. 31, 1525-1527 (2006).
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2007 (1)

2006 (2)

2004 (1)

1988 (1)

1984 (1)

1983 (1)

1982 (1)

1969 (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14-33 (1969).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Surf. Sci. (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14-33 (1969).
[CrossRef]

Other (9)

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Optical properties of crystals and glasses,” in Handbook of Optics, M.Bass, E.W.Van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 33.

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

M. Born and E. Wolf, Principles of Optics (Cambridge, 1999).

M. W. Ribarsky, “Titanium oxide (TiO2) (Rutile),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).

http://www.us.schott.com/optics_devices/english/products/flash/abbediagramm_flash.html.

R. F. Potter, “Germanium (Ge),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).

P. Bhattacharya, Semiconductor Optoelectronic Devices (Prentice Hall, 1997).

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001).
[CrossRef]

G. F. Miner, Lines and Electromagnetic Fields for Engineers (Oxford, 1996).

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Figures (8)

Fig. 1
Fig. 1

Reflection of p- and s-polarized light at an angle ϕ by a film–substrate system. Media 0, 1, and 2 are the ambient, film, and substrate, respectively, and d is the film thickness.

Fig. 2
Fig. 2

Family of curves of the reflection coefficient R p ( ϕ , ε ) of a QWL for the p polarization, Eq. (13), as a function of the angle of incidence ϕ for constant values of layer-to-ambient relative refractive index n = ε from 1.5 to 6.0 in steps of 0.5. Note that R p = 0 (dashed line) at the Brewster angle ϕ B = tan 1 n of the 01 interface.

Fig. 3
Fig. 3

Family of curves of the reflection coefficient R s ( ϕ , ε ) of a QWL for the s polarization, Eq. (14), as a function of the angle of incidence ϕ for constant values of layer-to-ambient relative refractive index n = ε from 1.5 to 6.0 in steps of 0.5. Note that Eq. (18) is satisfied at point P on the n = 2 curve at ϕ = 45 ° .

Fig. 4
Fig. 4

Cross section of an s-polarization BS that uses a high- index QWL embedded in a cube.

Fig. 5
Fig. 5

Relative dielectric function ε = n 2 ( Ti O 2 ) / n 2 (glass) is plotted versus wavelength λ in the spectral range 0.4 λ 1.5 μm for a Ti O 2 layer embedded in fused silica ( Si O 2 ) and N-FK5 Schott glass. The dashed lines show that ε = 2 + 2 , n = 1.84776 at λ = 704.6 nm and λ = 605.4 nm for the Si O 2 and N-FK5 Schott glass substrates, respectively.

Fig. 6
Fig. 6

Intensity reflectances ν = R ν 2 ( ν = p , s, a) for incident p- and s-polarized light and their average as functions of the internal angle of incidence ϕ from 40 ° to 50 ° of an s-polarization BS that uses a QWL of Ti O 2 embedded in a fused-silica cube. The wavelength of light and the metric thickness of the Ti O 2 thin film are kept constant at λ = 704.6 nm and d = 70.9 nm , respectively.

Fig. 7
Fig. 7

Intensity reflectances ν = R ν 2 ( ν = p , s, a) for incident p- and s-polarized light and their average as functions of wavelength λ from 650 to 750 nm of an s-polarization BS that uses a QWL of Ti O 2 embedded in a fused-silica cube. The angle of incidence and the metric thickness of the Ti O 2 thin film are fixed at ϕ = 45 ° and d = 70.9 nm , respectively.

Fig. 8
Fig. 8

Intensity reflectances ν = R ν 2 ( ν = p , s, a) for incident p- and s-polarized light and their average as functions of the thickness d of a Ti O 2 layer, which is embedded in a fused-silica cube. The thickness d is varied by ± 5 nm around 71 nm , while the angle of incidence ϕ = 45 ° and wavelength λ = 704.6 nm are fixed. The device performance as a 50%–50% s-polarization BS is essentially independent of small film thickness changes to first order.

Equations (39)

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R ν = r 01 ν + r 12 ν X 1 + r 01 ν r 12 ν X , ν = p , s ,
X = exp ( j 2 π d / D ϕ ) .
D ϕ = ( λ / 2 ) ( ε 1 ε 0 sin 2 ϕ ) 1 / 2
ε 2 = ε 0 ,
d = D ϕ / 2 = ( λ / 4 ) ( ε 1 ε 0 sin 2 ϕ ) 1 / 2 ,
ε = ε 1 / ε 0 .
r 12 ν = r 10 ν = - r 01 ν , ν = p , s .
X = 1 .
R ν = 2 r 01 ν 1 + r 01 ν 2 , ν = p , s .
r 01 ν = tan α ν , ν = p , s , 45 ° α p 45 ° , 45 ° α s 0.
R ν = sin ( 2 α ν ) , ν = p , s .
r 01 p = ε cos ϕ ( ε sin 2 ϕ ) 1 / 2 ε cos ϕ + ( ε sin 2 ϕ ) 1 / 2 ,
r 01 s = cos ϕ ( ε sin 2 ϕ ) 1 / 2 cos ϕ + ( ε sin 2 ϕ ) 1 / 2 ,
R p = ε 2 cos 2 ϕ ε + sin 2 ϕ ε 2 cos 2 ϕ + ε sin 2 ϕ ,
R s = 1 ε cos 2 ϕ + ε .
R p ( 0 ) = R s ( 0 ) = ε 1 ε + 1 .
ε = ( 2 + 1 ) 2 , n = ( 2 + 1 ) = 2.4142 .
R p ( 45 ° ) = ε 2 2 ε + 1 ε 2 + 2 ε 1 ,
R s ( 45 ° ) = 1 + ( 1 / ε ) .
ε = 2 + 2 , n = 1.84776 .
n i 2 = A i + B i λ 2 C i , i = o , e .
( A o , B o , C o ) = ( 5.913 , 0.2441 , 0.0803 ) , ( A e , B e , C e ) = ( 7.197 , 0.3322 , 0.0843 ) ,
( A a , B a , C a ) = ( 6.5390 , 0.2866 , 0.08252 ) ,
( A , B , C ) = [ 1 , 1.09877 , ( 0.0924317 ) 2 ] .
R s 2 ( 45 ° ) + R p 2 ( 45 ° ) = 1 .
ζ 6 16 ζ 4 48 ζ 3 52 ζ 2 24 ζ 4 = 0 , ζ = ε 1 ,
ζ = 5.2223 , ε = 6.2223 .
R 012 ν = r 01 ν r 12 ν 1 r 01 ν r 12 ν , ν = p , s .
R 010 ν = 2 r 01 ν 1 + r 01 ν 2 , ν = p , s .
R 012 ν = r 02 ν = r 01 ν + r 12 ν 1 + r 01 ν r 12 ν , ν = p , s .
r 12 ν = r 02 ν r 01 ν 1 r 01 ν r 02 ν , ν = p , s .
R 012 ν = R 010 ν r 02 ν 1 r 02 ν R 010 ν , ν = p , s .
R 012 p = R 010 p .
r 02 s = cos ( 2 ϕ B 02 ) ,
R 010 s = 1 ε 1 cos 2 ϕ B 02 + ε 1 .
R 012 s = 1 cos ( 2 ϕ B 02 ) ε 1 ε 1 .
cos ( 2 ϕ B 02 ) = ( ε 2 1 ) / ( ε 2 + 1 )
R 012 s ( ϕ B 02 ) = 2 ε 2 ε 1 ε 2 ε 1 ε 1 ( ε 2 + 1 ) .
ε 1 = 2 ε 2 / ( ε 2 + 1 ) , n 1 = 2 n 2 / ( n 2 2 + 1 ) 1 / 2 .

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