Abstract

The cumulative retardance Δt introduced between the p and the s orthogonal linear polarizations after two successive total internal reflections (TIRs) inside a right-angle prism at complementary angles ϕ and 90°ϕ is calculated as a function of ϕ and prism refractive index n. Quarter-wave retardation (QWR) is obtained on retroreflection with minimum angular sensitivity when n=(2+1)1/2=1.55377 and ϕ=45°. A QWR prism made of N-BAK4 Schott glass (n=1.55377 at λ=1303.5  nm) has good spectral response (<5° retardance error) over the 0.5–2μm visible and near-IR spectral range. A ZnS-coated right-angle Si prism achieves QWR with an error of <±2.5° in the 9–11μm (CO2 laser) IR spectral range. This device functions as a linear-to-circular polarization transformer and can be tuned to exact QWR at any desired wavelength (within a given range) by tilting the prism by a small angle around ϕ=45°. A PbTe right-angle prism introduces near-half-wave retardation (near-HWR) with a 2% error over a broad (4λ12.5μm) IR spectral range. This device also has a wide field of view and its interesting polarization properties are discussed. A compact (aspect ratio of 2), in-line, HWR is described that uses a chevron dual Fresnel rhomb with four TIRs at the same angle ϕ=45°. Finally, a useful algorithm is presented that transforms a three-term Sellmeier dispersion relation of a transparent optical material to an equivalent cubic equation that can be solved for the wavelengths at which the refractive index assumes any desired value.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. R. M. A. Azzam and M. M. K. Howlader, "Silicon-based polarization optics for the 1.30 and 1.55 μm communication wavelengths," J. Lightwave Technol. 14, 873-878 (1996).
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    [CrossRef]
  14. Data available at http://www.us.schott.com/optics_devices/english/products/flash/abbediagramm_flash.html.
  15. W. J. Tropf, M. E. Thomas, and T. J. Harris, "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.
  16. R. M. A. Azzam and M. M. Howlader, "Fourth- and sixth-order polarization aberrations of antireflection-coated optical surfaces," Opt. Lett. 26, 1607-1608 (2001).
    [CrossRef]
  17. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), Chap. 2.
  18. See, for example, J. J. Tuma, Engineering Mathematics Handbook (McGraw-Hill, 1987), p. 7.

2004

2001

1996

R. M. A. Azzam and M. M. K. Howlader, "Silicon-based polarization optics for the 1.30 and 1.55 μm communication wavelengths," J. Lightwave Technol. 14, 873-878 (1996).
[CrossRef]

1994

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, and G. W. Day, "Standard polarization components: progress toward an optical retardance standard," Proc. SPIE 2265, 2-8 (1994).
[CrossRef]

A. M. Kan'an and R. M. A. Azzam, "In-line quarter-wave retarders for the IR using total refraction and total internal reflection in a prism," Opt. Eng. 33, 2029-2033 (1994).
[CrossRef]

N. N. Nagib and M. S. El-Bahrawy, "Phase retarders with variable angles of total internal reflection," Appl. Opt. 33, 1218-1222 (1994).
[CrossRef] [PubMed]

1984

1981

R. M. A. Azzam, "Measurement of the Jones matrix of an optical system by return-path null ellipsometry," Opt. Acta 28, 795-800 (1981).
[CrossRef]

1970

1966

R. J. King, "Quarter-wave retardation systems based on the Fresnel rhomb principle," J. Sci. Instrum. 43, 617-622 (1966).
[CrossRef]

Appl. Opt.

J. Lightwave Technol.

R. M. A. Azzam and M. M. K. Howlader, "Silicon-based polarization optics for the 1.30 and 1.55 μm communication wavelengths," J. Lightwave Technol. 14, 873-878 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. Sci. Instrum.

R. J. King, "Quarter-wave retardation systems based on the Fresnel rhomb principle," J. Sci. Instrum. 43, 617-622 (1966).
[CrossRef]

Opt. Acta

R. M. A. Azzam, "Measurement of the Jones matrix of an optical system by return-path null ellipsometry," Opt. Acta 28, 795-800 (1981).
[CrossRef]

Opt. Eng.

A. M. Kan'an and R. M. A. Azzam, "In-line quarter-wave retarders for the IR using total refraction and total internal reflection in a prism," Opt. Eng. 33, 2029-2033 (1994).
[CrossRef]

Opt. Lett.

Proc. SPIE

K. B. Rochford, P. A. Williams, A. H. Rose, I. G. Clarke, P. D. Hale, and G. W. Day, "Standard polarization components: progress toward an optical retardance standard," Proc. SPIE 2265, 2-8 (1994).
[CrossRef]

Other

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), p. 253.

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

W. J. Tropf, M. E. Thomas, and T. J. Harris, "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.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), Chap. 2.

See, for example, J. J. Tuma, Engineering Mathematics Handbook (McGraw-Hill, 1987), p. 7.

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

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Figures (11)

Fig. 1
Fig. 1

Retroreflection of a monochromatic light beam via two successive TIRs at angles of incidence ϕ and 90 ° ϕ inside a prism of refractive index n. Light enters and leaves the prism normal to its entrance face, which is ARC. p and s are the linear polarizations parallel and perpendicular to the common plane of incidence, respectively (s is parallel to the edge of prism E). When the cumulative retardance is quarter-wave, incident linearly polarized light becomes circularly polarized upon retroreflection.

Fig. 2
Fig. 2

(Color online) Plot of each of the three terms that appear in Eq. (1) as a function of ϕ in the range sin 1 ( 1 / n ) = ϕ c ϕ 90 ° ϕ c , when n = ( 2 + 1 ) 1 / 2 = 1.55377 .

Fig. 3
Fig. 3

(Color online) Cumulative retardance Δ t ( ϕ ) versus ϕ in the range ϕ c ϕ 90 ° ϕ c for values of n from 1.5 to 6.0 in equal steps of 0.5 (solid curves) and from 1.6 to 1.9 in steps of 0.1 (dashed curves). The dashed–dotted curve corresponds to n = ( 2 + 1 ) 1 / 2 .

Fig. 4
Fig. 4

(Color online) Locus of all possible combinations ( n , ϕ ) that produce QWR, Δ t ( n , ϕ ) = 90 ° , on retroreflection by a right-angle prism.

Fig. 5
Fig. 5

(Color online) Refractive index n ( λ ) plotted versus wavelength λ for four different Schott glasses with dispersion formulas given by Eq. (11) and Table 1. All the curves intersect the line n = ( 2 + 1 ) 1 / 2 = 1.55377 at wavelengths that are listed in the last row of Table 1.

Fig. 6
Fig. 6

(Color online) Refractive index n ( λ ) and cumulative retardance Δ t ( λ ) plotted versus wavelength λ for an uncoated N-BAK4 Schott glass right-angle prism in the symmetric ϕ = 45 ° orientation over the 0.5 λ 2 μ m spectral range.

Fig. 7
Fig. 7

ZnS-coated Si right-angle prism (in the symmetric orientation) transforms the state of polarization of an IR (e.g., CO 2 laser) beam from linear to circular. The entrance face of the prism is ARC.

Fig. 8
Fig. 8

(Color online) Relative refractive index n ( λ ) [Eq. (12)] and cumulative retardance Δ t ( λ ) of a ZnS-coated Si right-angle prism as functions of λ in the 9 λ 11 μ m spectral range for the symmetric ( ϕ = 45 ° ) orientation of the prism, shown in Fig. 7. The relative refractive index n ( Si ) / n ( ZnS ) = 1.55377 at wavelength λ = 10.039 μ m .

Fig. 9
Fig. 9

(Color online) Wavelength λ in micrometers is plotted versus angle of incidence ϕ in degrees such that the cumulative retardance Δ t ( λ , ϕ ) of a ZnS-coated Si right-angle prism assumes constant values from 88° to 92° in steps of 0.5°.

Fig. 10
Fig. 10

Transformation of incident (a) linearly and (b) circularly polarized light upon retroreflection by a high-index (PbTe) right-angle prism.

Fig. 11
Fig. 11

In-line half-wave retarder using a chevron, dual-Fresnel-rhomb, prism with four TIR at the same angle of incidence ϕ = 45 ° .

Tables (1)

Tables Icon

Table 1 Constants [( B i , C i ), i = 1, 2, 3] of the Dispersion Formulas [Eq. (11)] of Four Schott Optical Glasses and the Wavelengths λ at Which the Refractive Index n = 1.55377

Equations (156)

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Δ t
90 ° ϕ
n = ( 2 + 1 ) 1 / 2 = 1.55377
ϕ = 45 °
n = 1.55377
λ = 1303.5   nm
< 5 °
2 μ m
< ± 2.5 °
11 μ m
CO 2
ϕ = 45 °
2 %
( 4 λ 12.5 μ m )
ϕ = 45 °
Δ t
90 ° ϕ
n = ( 2 + 1 ) 1 / 2 = 1.55377
2 μ m
11 μ m
CO 2
12.5 μ m
( ϕ = 45 ° )
90 ° ϕ
Δ t
Δ t ( ϕ ) = Δ ( ϕ ) + Δ ( 90 ° ϕ ) ,
Δ ( ϕ ) = 2 tan 1 [ ( n 2 sin 2 ϕ 1 ) 1 / 2 / ( n   sin   ϕ   tan   ϕ ) ] ,
sin 1 ( 1 / n ) = ϕ c ϕ 90 ° ϕ c ,
ϕ c
ϕ = 90 ° ϕ = 45 °
Δ t ( 45 ° ) = 4 tan 1 [ ( n 2 2 ) 1 / 2 / n ] .
Δ t ( 45 ° ) = π / 2
[ ( n 2 2 ) 1 / 2 / n ] = tan ( π / 8 ) = 2 1 ,
n = ( 2 + 1 ) 1 / 2 = 1.55377.
n = ( 2 + 1 ) 1 / 2 = 1.55377
ϕ = 45 °
Δ t = 90 ° , Δ t / ϕ = 0.
g ( x ) = f ( x ) + f ( a x ) ,
g ( x ) = f ( x ) f ( a x ) .
x = a / 2
g ( a / 2 ) = 0.
g ( x )
x = a / 2
n = ( 2 + 1 ) 1 / 2 = 1.55377
ϕ m
Δ ( ϕ )
ϕ c
n = ( 2 + 1 ) 1 / 2
ϕ m = 90 ° ϕ c
ϕ = ( ϕ m + ϕ c ) / 2 = 45 °
n = ( 2 + 1 ) = 2.41421
Δ t ( ϕ )
ϕ c ϕ 90 ° ϕ c
n = ( 2 + 1 ) 1 / 2
Δ t ( ϕ ) = 90 °
ϕ = 45 °
[ Δ t ( ϕ ) = 90 ° ]
( ϕ , 90 ° ϕ )
n > 1.55377
( n , ϕ )
Δ t ( n , ϕ ) = 90 °
n ( λ )
n 2 1 = B 1 λ 2 λ 2 C 1 + B 2 λ 2 λ 2 C 2 + B 3 λ 2 λ 2 C 3 .
n = ( 2 + 1 ) 1 / 2 = 1.55377
( B j , C j )
j = 1 , 2 , 3
n = 1.55377
λ 2
n ( λ )
Δ t ( λ )
ϕ = 45 °
0.5 λ 2 μ m
< 5 °
1.3 λ 1.6 μ m
< 2 °
CO 2
9 λ 11 μ m
n = n ( Si ) / n ( ZnS ) .
n ( λ )
Δ t ( λ )
9 λ 11 μ m
ϕ = 45 °
n ( Si ) / n ( ZnS ) = 1.55377
Δ t ( λ ) = 90 °
λ = 10.039 μ m
< ± 2.5 °
2 μ m
Δ t / ϕ = 0
[ Δ t ( λ , ϕ ) = 90 ° ]
Δ t ( λ , ϕ )
Δ t ( ϕ )
ϕ = 45 °
Δ t   max
n
n = 6
Δ t   max = 176.726 °
n 1
R E π ( 2 / n 2 ) ,
n = 6
B 1 = 30.046
C 1 = ( 1.563 ) 2
B 2 = B 3 = 0
4 λ 12.5 μ m
R E = 3.755 °
λ = 12.5 μ m
n = 5.6146
< 2.1 %
θ i
θ r = θ i
e = b / a
e = ( R E / 2 ) sin   2 θ i .
e 2 = b 2 / a 2 = 8.17 × 10 4
n = 6
θ i = 45 °
θ r
ε r
θ r = 45 ° , ε r = 45 ° ( R E / 2 ) ° .
Δ t ( ϕ )
n = 6
ϕ = 45 °
90 ° ϕ
Δ t ( ϕ )
n = ( 2 + 1 ) 1 / 2
CO 2
2 μ m
( n = 6 )
η = n 2 1 , x = λ 2 ,
a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0 ,
a 3 = S B η , a 2 = ( η S B ) S C + S B C ,
a 1 = P C ( S B / C η S 1 / C ) , a 0 = η P C .
S B = i = 1 3 B i , S C = i = 1 3 C i ,
S 1 / C = i = 1 3 ( 1 / C i ) , S B C = i = 1 3 ( B i C i ) ,
S B / C = i = 1 3 ( B i / C i ) , P C = C 1 C 2 C 3 .
90 ° ϕ
sin 1 ( 1 / n ) = ϕ c ϕ 90 ° ϕ c
n = ( 2 + 1 ) 1 / 2 = 1.55377
Δ t ( ϕ )
ϕ c ϕ 90 ° ϕ c
n = ( 2 + 1 ) 1 / 2
( n , ϕ )
Δ t ( n , ϕ ) = 90 °
n ( λ )
n = ( 2 + 1 ) 1 / 2 = 1.55377
n ( λ )
Δ t ( λ )
ϕ = 45 °
0.5 λ 2 μ m
CO 2
n ( λ )
Δ t ( λ )
9 λ 11 μ m
( ϕ = 45 ° )
n ( Si ) / n ( ZnS ) = 1.55377
λ = 10.039 μ m
Δ t ( λ , ϕ )
ϕ = 45 °

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