Abstract

The conditions under which light interference in a transparent quarter-wave layer of refractive index n1 on a transparent substrate of refractive index n2 leads to 50% reflectance for incident unpolarized light at an angle φ are determined. Two distinct solution branches are obtained that correspond to light reflection above and below the polarizing angle, φp, of zero reflection for p polarization. The real p and s amplitude reflection coefficients have the same (negative) sign for the solution branch φ>φp and have opposite signs for the solution branch φ<φp. Operation at φ<φp is the basis of a 50%–50% beam splitter that divides an incident totally polarized light beam (with p and s components of equal intensity) into reflected and refracted beams of orthogonal polarizations [Opt. Lett. 31, 1525 (2006) ] and requires a film refractive index n1(2+1)n2. A monochromatic design that uses a high-index TiO2 thin film on a low-index MgF2 substrate at 488nm wavelength is presented as an example.

© 2007 Optical Society of America

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References

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  1. 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).
    [CrossRef] [PubMed]
  2. J. A. Dobrowolski, "Optical properties of films and coatings," in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 42.
  3. R. M. A. Azzam, "Simultaneous reflection and refraction of light without change of polarization by a single-layer-coated dielectric surface," Opt. Lett. 10, 107-109 (1985).
    [CrossRef] [PubMed]
  4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  5. Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985).
  6. 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.
  7. R. M. A. Azzam and A. De, "Optimal beam splitters for the division-of-amplitude photopolarimeter," J. Opt. Soc. Am. A 20, 955-958 (2003).
    [CrossRef]
  8. R. M. A. Azzam, "Variable-reflectance thin-film polarization-independent beam splitters for 0.6328 and 10.6 μm laser light," Opt. Lett. 10, 110-112 (1985).
    [CrossRef] [PubMed]

2006 (1)

2003 (1)

1985 (2)

Azzam, R. M. A.

Bashara, N. M.

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

De, A.

Dobrowolski, J. A.

J. A. Dobrowolski, "Optical properties of films and coatings," in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 42.

Harris, T. J.

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.

Thomas, M. E.

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.

Tropf, W. J.

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.

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

Opt. Lett. (3)

Other (4)

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

Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985).

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.

J. A. Dobrowolski, "Optical properties of films and coatings," in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 42.

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Figures (14)

Fig. 1
Fig. 1

Angle of incidence φ versus film refractive index n 1 such that 50% of incident unpolarized light is reflected by a quarter-wave pellicle. The continuous and dashed curves represent two independent solution branches. The middle curve gives the Brewster angle as a function of n 1 , φ B = tan 1 n 1 .

Fig. 2
Fig. 2

Amplitude reflection coefficients R p and R s that are associated with the HAB of Fig. 1 plotted as functions of the film refractive index n 1 .

Fig. 3
Fig. 3

Amplitude reflection coefficients R p and R s that are associated with the LAB of Fig. 1 plotted as functions of the film refractive index n 1 .

Fig. 4
Fig. 4

Plot of R p versus R s associated with the high- and low-angle branches (Figs. 2, 3) yields two arcs of the unit circle in the third and fourth quadrants, respectively. The arrows indicate the direction of increasing n 1 , and points A and B correspond to the similarly marked points in Fig. 1.

Fig. 5
Fig. 5

Intensity reflectances R p 2 and R s 2 that are associated with the HAB of Fig. 1 plotted as functions of the film refractive index n 1 . The average reflectance, ( R p 2 + R s 2 ) 2 , is constant at 0.5.

Fig. 6
Fig. 6

Intensity reflectances R p 2 and R s 2 that are associated with the LAB of Fig. 1 plotted as functions of the film refractive index n 1 . The average reflectance, ( R p 2 + R s 2 ) 2 , is constant at 0.5.

Fig. 7
Fig. 7

Angle of incidence φ versus the refractive index n 1 of a transparent quarter-wave coating on a transparent substrate ( n 2 = 1.5 ) that reflects 50% of incident unpolarized light. The continuous and dashed curves represent two distinct solution branches. The middle dash-dot curve gives the polarizing angle φ p as a function of n 1 as obtained from Eqs. (19, 20).

Fig. 8
Fig. 8

Amplitude reflection coefficients R p and R s that are associated with the HAB of Fig. 7 plotted as functions of the film refractive index n 1 . The significance of the point of intersection C is discussed in the text.

Fig. 9
Fig. 9

Amplitude reflection coefficients R p and R s that are associated with the LAB of Fig. 7 plotted as functions of the film refractive index n 1 .

Fig. 10
Fig. 10

Intensity reflectances R p 2 and R s 2 that are associated with the HAB of Fig. 7 plotted as functions of the film refractive index n 1 . The average reflectance, ( R p 2 + R s 2 ) 2 , is constant at 0.5.

Fig. 11
Fig. 11

Intensity reflectances R p 2 and R s 2 that are associated with the LAB of Fig. 7 plotted as functions of the film refractive index n 1 . The average reflectance, ( R p 2 + R s 2 ) 2 , is constant at 0.5.

Fig. 12
Fig. 12

Average reflectance ( R p 2 + R s 2 ) 2 and ellipsometric parameters ψ r , ψ t , Δ r , Δ t as functions of the angle of incidence φ for a beam splitter that consists of a quarter-wave ( 43.51 nm ) thin film of Ti O 2 on a Mg F 2 substrate at 488 nm wavelength. The angle of incidence is varied by ± 5 ° around the design angle φ = 46.04 ° .

Fig. 13
Fig. 13

Average reflectance ( R p 2 + R s 2 ) 2 and ellipsometric parameters ψ r , ψ t , Δ r , Δ t of a beam splitter as functions of the thickness d 1 of a Ti O 2 film on a Mg F 2 substrate at λ = 488 nm and φ = 46.04 ° . The film thickness is varied by ± 5 % around the design value of 43.5 nm .

Fig. 14
Fig. 14

Average reflectance ( R p 2 + R s 2 ) 2 and ellipsometric parameters ψ r , ψ t , Δ r , Δ t as functions of the wavelength λ (in nanometers) for a beam splitter that consists of a 43.51 nm Ti O 2 thin film on a Mg F 2 substrate at an angle of incidence of φ = 46.04 ° . The wavelength λ is changed by ± 5 % around 488 nm .

Equations (33)

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R s = S 0 S 2 S 1 2 S 0 S 2 + S 1 2 ,
R p = n 1 4 S 0 S 2 n 2 2 S 1 2 n 1 4 S 0 S 2 + n 2 2 S 1 2 ,
S i = ( n i 2 u ) 1 2 , i = 0 , 1 , 2 ,
u = sin 2 φ .
P = S 0 S 2 S 1 2 ,
R s = P 1 P + 1 , R p = n 1 4 P n 2 2 n 1 4 P + n 2 2 .
R p 2 + R s 2 = 1 .
a 4 P 4 + a 3 P 3 + a 2 P 2 + a 1 P + a 0 = 0 ,
a 4 = n 1 8 ,
a 3 = 2 n 1 4 ( n 1 4 + n 2 2 ) ,
a 2 = n 1 4 ( n 1 4 12 n 2 2 ) + n 2 4 ,
a 1 = 2 n 2 2 ( n 1 4 + n 2 2 ) ,
a 0 = n 2 4 .
P 2 = ( 1 u ) ( n 2 2 u ) ( n 1 2 u ) 2 .
b 2 u 2 + b 1 u + b 0 = 0 ,
b 2 = P 2 1 ,
b 1 = ( n 2 2 + 1 ) 2 n 1 2 P 2 ,
b 0 = n 1 4 P 2 n 2 2 .
φ = arcsin ( u 1 2 ) .
R s = ( P 1 ) ( P + 1 ) = 1 2 ,
P = ( 2 1 ) 2 .
n 1 = ( n 2 P ) 1 2 .
n 1 = ( 2 + 1 ) n 2 .
P = n 2 2 n 1 4 .
c 2 u 2 + c 1 u + c 0 = 0 ,
c 2 = n 1 8 n 2 8 ,
c 1 = 2 n 1 2 n 2 4 n 1 8 ( n 2 2 + 1 ) ,
c 0 = n 1 4 n 2 2 ( n 1 4 n 2 2 ) .
R p = R s = 1 2 = 0.7071 , n 1 = n 2 = 1.2247 .
D 1 = ( λ 2 ) ( n 1 2 sin 2 φ ) 1 2 ,
d 1 = D 1 2 ,
Δ r = π , Δ t = 0 ,
ψ r + ψ t = π 2 .

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