Abstract

The condition of minimum deviation (MD) by a transparent optically isotropic prism is re-derived, and expressions for the intensity transmittances Tp(θ) and Ts(θ) of an uncoated prism of refractive index n and prism angle α for incident p- and s-polarized light and their derivatives with respect to the internal angle of refraction θ are obtained. When the MD condition (θ=α/2) is satisfied, Ts is maximum and Tp is maximum or minimum. The transmission ellipsometric parameters ψt,Δt of a symmetrically coated prism are also shown to be locally stationary with respect to θ at θ=α/2. The constraint on (n,α) for maximally flat transmittance (MFT) of p-polarized light at and near the MD condition is determined. The transmittance Tp of prisms represented by points that lie below the locus (n,α) of MFT exhibits oscillation as a function of θ. No similar behavior is found for the s polarization. Magnitudes and angular positions of the maxima and minima of the oscillatory Tp-versus-θ curves are also calculated as functions of α for a ZnS prism of refractive index n=2.35 in the visible.

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References

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  1. I. Newton, Opticks (Dover, 1979).
  2. R. A. Sawyer, Experimental Spectroscopy, 3rd ed. (Dover, 1963).
  3. G. J. Zissis, “Dispersive prisms and gratings,” in Handbook of Optics, M.Bass, E.W. V.Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 5.
  4. J. L. Holmes, Encyclopedia of Spectroscopy and Spectrometry (Academic, 2000).
  5. A. L. Bloom, “Observation of new visible gas laser transitions by removal of dominance,” Appl. Phys. Lett. 2, 101–102 (1963).
    [CrossRef]
  6. A. D. White, “Reflecting prisms for dispersive optical maser cavities,” Appl. Opt. 3, 431–432 (1964).
    [CrossRef]
  7. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), p. 588.
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  10. J. H. Burnett, R. Gupta, and U. Griesmann, “Absolute refractive indices and thermal coefficients of CaF2, SrF2, BaF2, and LiF near 157 nm,” Appl. Opt. 41, 2508–2513 (2002).
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  12. M. Daimon and A. Masumura, “Measurements of the refractive index of distilled water from the near-infrared region to the ultraviolet region,” Appl. Opt. 46, 3811–3820 (2007).
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  13. R. M. A. Azzam, F. F. Sudradjat, and M. Nazim Uddin, “Prism spectroscopic ellipsometer,” Thin Solid Films 455–456, 54–60 (2004).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 177–180.
  15. E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998), pp. 189–191.
  16. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 270–283.
  17. M. Debenham, “Refractive indices of zinc sulfide in the 0.403–13-μm wavelength range,” Appl. Opt. 23, 2238–2239 (1984).
    [CrossRef] [PubMed]
  18. M. Cervantes, “Brewster angle prisms: a review,” Opt. Laser Technol. 20, 297–300 (1988).
    [CrossRef]

2007 (1)

2004 (1)

R. M. A. Azzam, F. F. Sudradjat, and M. Nazim Uddin, “Prism spectroscopic ellipsometer,” Thin Solid Films 455–456, 54–60 (2004).
[CrossRef]

2002 (2)

1988 (1)

M. Cervantes, “Brewster angle prisms: a review,” Opt. Laser Technol. 20, 297–300 (1988).
[CrossRef]

1984 (1)

1969 (1)

1965 (1)

1964 (1)

1963 (1)

A. L. Bloom, “Observation of new visible gas laser transitions by removal of dominance,” Appl. Phys. Lett. 2, 101–102 (1963).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, F. F. Sudradjat, and M. Nazim Uddin, “Prism spectroscopic ellipsometer,” Thin Solid Films 455–456, 54–60 (2004).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 270–283.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 270–283.

Bloom, A. L.

A. L. Bloom, “Observation of new visible gas laser transitions by removal of dominance,” Appl. Phys. Lett. 2, 101–102 (1963).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 177–180.

Burnett, J. H.

Cervantes, M.

M. Cervantes, “Brewster angle prisms: a review,” Opt. Laser Technol. 20, 297–300 (1988).
[CrossRef]

Daimon, M.

Debenham, M.

Griesmann, U.

Gupta, R.

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998), pp. 189–191.

Holmes, J. L.

J. L. Holmes, Encyclopedia of Spectroscopy and Spectrometry (Academic, 2000).

Malitson, I. H.

Masumura, A.

Nazim Uddin, M.

R. M. A. Azzam, F. F. Sudradjat, and M. Nazim Uddin, “Prism spectroscopic ellipsometer,” Thin Solid Films 455–456, 54–60 (2004).
[CrossRef]

Neu, J. T.

Newton, I.

I. Newton, Opticks (Dover, 1979).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), p. 588.

Sawyer, R. A.

R. A. Sawyer, Experimental Spectroscopy, 3rd ed. (Dover, 1963).

Sudradjat, F. F.

R. M. A. Azzam, F. F. Sudradjat, and M. Nazim Uddin, “Prism spectroscopic ellipsometer,” Thin Solid Films 455–456, 54–60 (2004).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), p. 588.

White, A. D.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 177–180.

Wray, J. H.

Zissis, G. J.

G. J. Zissis, “Dispersive prisms and gratings,” in Handbook of Optics, M.Bass, E.W. V.Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 5.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

A. L. Bloom, “Observation of new visible gas laser transitions by removal of dominance,” Appl. Phys. Lett. 2, 101–102 (1963).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Laser Technol. (1)

M. Cervantes, “Brewster angle prisms: a review,” Opt. Laser Technol. 20, 297–300 (1988).
[CrossRef]

Thin Solid Films (1)

R. M. A. Azzam, F. F. Sudradjat, and M. Nazim Uddin, “Prism spectroscopic ellipsometer,” Thin Solid Films 455–456, 54–60 (2004).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 177–180.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998), pp. 189–191.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), pp. 270–283.

I. Newton, Opticks (Dover, 1979).

R. A. Sawyer, Experimental Spectroscopy, 3rd ed. (Dover, 1963).

G. J. Zissis, “Dispersive prisms and gratings,” in Handbook of Optics, M.Bass, E.W. V.Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 5.

J. L. Holmes, Encyclopedia of Spectroscopy and Spectrometry (Academic, 2000).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), p. 588.

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Figures (12)

Fig. 1
Fig. 1

Ray tracing of p- or s-polarized light as it is transmitted through a prism of refractive index n and prism angle α. The common plane of incidence coincides with the plane of the page which is also a principal section of the prism.

Fig. 2
Fig. 2

Deflection angle γ t ( θ ) calculated from Eqs. (1, 2) is plotted as a function of the internal angle of refraction θ for glass prism of refractive index n = 1.5 and different prism angles α from 30° to 80° in steps of 10°.

Fig. 3
Fig. 3

Intensity transmittances of p- and s-polarized light T p ( θ ) and T s ( θ ) of an uncoated prism of refractive index n = 1.5 are plotted as functions of the internal angle of refraction θ for prism angles α from 30° to 80° in steps of 10°.

Fig. 4
Fig. 4

Continuous curve represents the solution of Eq. (16) for the prism angle α as a function of prism refractive index n that achieves MFT of p-polarized light at and near the condition of MD. In the shaded area below the MFT curve the transmittance T p of p-polarized light is an oscillatory function of the internal angle of refraction θ. The dashed and dotted curves represent conditions of TT and ZT by prisms that satisfy Eqs. (20, 21), respectively.

Fig. 5
Fig. 5

MFT of p-polarized light T p ( α / 2 ) under the MD condition decreases monotonically as a function of prism refractive index n.

Fig. 6
Fig. 6

Intensity transmittances T p ( θ ) and T s ( θ ) of p- and s-polarized light are plotted as functions of the internal angle of refraction θ for a Brewster-angle prism with n = 1.5 and α = 2 tan 1 ( 2 / 3 ) = 67.38 ° .

Fig. 7
Fig. 7

Intensity transmittance of p-polarized light T p ( θ ) is plotted as a function of θ for ZnS prisms of the same refractive index n = 2.35 but three different prism angles. MFT is achieved when α = 40.9144 ° , an oscillatory response appears for α = 37 ° , and a transmittance curve with a single well defined peak is obtained for α = 45 ° .

Fig. 8
Fig. 8

Intensity transmittance of p-polarized light T p ( θ ) is plotted as a function of θ for a ZnS prism with prism angle of α = 40.9144 ° and different refractive indices at three different wavelengths. For n = 2.35 at 0.633 μ m wavelength, a MFT is obtained; n = 2.2 at 10 μ m leads to an oscillatory response; and for n = 2.5 at 0.467 μ m a curve with a single well defined peak is obtained.

Fig. 9
Fig. 9

Family of T p ( θ ) -versus-θ curves for ZnS prism with refractive index n = 2.35 and prism angles α from 5° to 45° in steps of 4°.

Fig. 10
Fig. 10

Internal angles of refraction of maximum and minimum transmittances of p-polarized light θ max , θ min and their difference θ max θ min in degrees are plotted as functions of prism angle α in the range 0 ° α 50 ° for a ZnS prism with refractive index n = 2.35 .

Fig. 11
Fig. 11

Minimum and maximum transmittances of p-polarized light T p   min , T p   max and their ratio T p   min / T p   max are plotted as functions of prism angle α for ZnS prism with n = 2.35 . The significance of points A, B, C, and D is discussed in the text.

Fig. 12
Fig. 12

Transmission of p-polarized light through half of a Brewster-angle prism with prism angle α = 2 θ B i = 2 tan 1 ( 1 / n ) .

Tables (1)

Tables Icon

Table 1 Prism Angles α in Degrees that Lead to MFT of p-Polarized Light, TT of p-Polarized Light, and ZT of the p and s Polarizations Listed Versus Prism Refractive Index n a

Equations (22)

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γ ( θ ) = sin 1 ( n   sin   θ ) θ .
γ t ( θ ) = γ ( θ ) + γ ( α θ ) .
γ t ( θ ) = d γ t / d θ = γ ( θ ) γ ( α θ ) .
γ t   min = 2 { sin 1 [ n   sin ( α / 2 ) ] ( α / 2 ) } .
n = sin [ ( γ t   min + α ) / 2 ] sin ( α / 2 ) .
τ ν = 1 R ν ,     ν = p , s ,
τ p ( θ ) = 4 n   cos   θ ( 1 n 2 sin 2 θ ) 1 / 2 ( n 2 + 1 ) ( n 4 + 1 ) sin 2 θ + 2 n   cos   θ ( 1 n 2 sin 2 θ ) 1 / 2 ,
τ s ( θ ) = 4 n   cos   θ ( 1 n 2 sin 2 θ ) 1 / 2 1 + n 2   cos   2 θ + 2 n   cos   θ ( 1 n 2 sin 2 θ ) 1 / 2 ,
T ν = τ ν ( θ ) τ ν ( α θ ) ,     ν = p , s .
T ν ( θ ) = τ ν ( θ ) τ ν ( α θ ) τ ν ( θ ) τ ν ( α θ ) ,
T ν ( θ ) = τ ν ( θ ) τ ν ( α θ ) 2 τ ν ( θ ) τ ν ( α θ ) + τ ν ( θ ) τ ν ( α θ ) ,
T ν ( θ ) = τ ν ( θ ) τ ν ( α θ ) 3 τ ν ( θ ) τ ν ( α θ ) + 3 τ ν ( θ ) τ ν ( α θ ) τ ν ( θ ) τ ν ( α θ ) .
T ν ( α / 2 ) = 0 ,     ν = p , s .
T ν ( α / 2 ) = 2 τ ν ( α / 2 ) τ ν ( α / 2 ) 2 [ τ ν ( α / 2 ) ] 2 .
T ν ( α / 2 ) = 0 ,
τ ν ( α / 2 ) τ ν ( α / 2 ) [ τ ν ( α / 2 ) ] 2 = 0.
ψ t = tan 1 ( T p / T s ) 1 / 2 ,
Δ t ( θ ) = Δ 1 t ( θ ) + Δ 1 t ( α θ ) ,
Δ t ( α / 2 ) = 0.
α = 2 θ B i = 2 tan 1 ( 1 / n ) .
α = 2 θ c = sin 1 ( 1 / n ) .
T p   min ( 0 ) = 16 n 2 / ( n + 1 ) 4 .

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