Abstract

A tilted bilayer membrane, which consists of two thin films of transparent optically isotropic materials of different refractive indices, can function as a transmission quarter-wave retarder (QWR) at a high angle of incidence. A specific design using a cryolite-Si membrane in the infrared is presented, and its tolerances to small shifts of wavelength, incidence angle, and film thickness errors are discussed. Some designs provide a dual QWR in transmission and reflection. Such devices provide simple linear-to-circular (and circular-to-linear) polarization transformers. Bilayer eighth-wave retarders without diattenuation are also introduced.

© 2001 Optical Society of America

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References

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  1. See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, UK, 1971).
  2. D. A. Holmes, “Wave optics theory of rotatory compensators,” J. Opt. Soc. Am. 54, 1340–1347 (1964).
    [CrossRef]
  3. R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
    [CrossRef] [PubMed]
  4. W. H. Southwell, “Multilayer coating design achieving a broadband 90° phase shift,” Appl. Opt. 19, 2688–2692 (1980).
    [CrossRef] [PubMed]
  5. J. H. Apfel, “Phase retardance of periodic multilayer mirrors,” Appl. Opt. 21, 733–738 (1982).
    [CrossRef] [PubMed]
  6. M. M. K. Howlader, R. M. A. Azzam, “Periodic and quasiperiodic nonquarterwave multilayer coatings for 90-deg reflection phase retardance at 45-deg angle of incidence,” Opt. Eng. 34, 869–875 (1995).
    [CrossRef]
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  8. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  9. A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972).
  10. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  11. J. A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, editor-in-chief (McGraw-Hill, New York, 1995), Vol. I, Chap. 42.

1995 (1)

M. M. K. Howlader, R. M. A. Azzam, “Periodic and quasiperiodic nonquarterwave multilayer coatings for 90-deg reflection phase retardance at 45-deg angle of incidence,” Opt. Eng. 34, 869–875 (1995).
[CrossRef]

1991 (1)

1989 (1)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

1982 (1)

1980 (1)

1964 (1)

Apfel, J. H.

Azzam, R. M. A.

M. M. K. Howlader, R. M. A. Azzam, “Periodic and quasiperiodic nonquarterwave multilayer coatings for 90-deg reflection phase retardance at 45-deg angle of incidence,” Opt. Eng. 34, 869–875 (1995).
[CrossRef]

R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
[CrossRef] [PubMed]

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

Bashara, N. M.

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

Chipman, R. A.

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

Clarke, D.

See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, UK, 1971).

Dobrowolski, J. A.

J. A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, editor-in-chief (McGraw-Hill, New York, 1995), Vol. I, Chap. 42.

Grainger, J. F.

See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, UK, 1971).

Holmes, D. A.

Howlader, M. M. K.

M. M. K. Howlader, R. M. A. Azzam, “Periodic and quasiperiodic nonquarterwave multilayer coatings for 90-deg reflection phase retardance at 45-deg angle of incidence,” Opt. Eng. 34, 869–875 (1995).
[CrossRef]

Kyrala, A.

A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972).

Southwell, W. H.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

M. M. K. Howlader, R. M. A. Azzam, “Periodic and quasiperiodic nonquarterwave multilayer coatings for 90-deg reflection phase retardance at 45-deg angle of incidence,” Opt. Eng. 34, 869–875 (1995).
[CrossRef]

Other (5)

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

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

A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972).

J. A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, editor-in-chief (McGraw-Hill, New York, 1995), Vol. I, Chap. 42.

See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, UK, 1971).

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Figures (10)

Fig. 1
Fig. 1

Reflection and transmission of light by a tilted transparent thin slab of an optically isotropic material (medium 1 of thickness d1) in a transparent ambient (medium 0). p and s are the linear polarizations parallel and perpendicular to the plane of incidence, respectively, and ϕ is the angle of incidence.

Fig. 2
Fig. 2

Reflection and transmission of light by a tilted bilayer membrane of two optically isotropic thin films (media 1 and 2 of thicknesses d1 and d2, respectively) in a transparent ambient (medium 0). p and s are the linear polarizations parallel and perpendicular to the plane of incidence, respectively, and ϕ is the angle of incidence.

Fig. 3
Fig. 3

Maximum differential phase shift in light transmission through a bilayer, |Δtmax|, versus angle of incidence φ for five material systems S0, S1, S2, S3, and S4, which correspond to (N1, N2)=(4, 4), (1.38, 2.35), (1.35, 4), (1.25, 4), and (1.35, 3.42), respectively.

Fig. 4
Fig. 4

Family of contours of the ratio of complex-amplitude p and s transmission coefficients ρt in the complex plane for a cryolite-Si bilayer with refractive indices (1.35, 3.42), as calculated from Eq. (3), at ϕ=82° angle of incidence. The normalized thickness of the first layer, ζ1, assumes constant values marked by each curve while the normalized thickness of the second film, ζ2, is scanned.

Fig. 5
Fig. 5

Multiple solutions (ζ1, ζ2) for bilayer transmission quarter-wave retarders (QWR’s), for both Δt=+90° and Δt=-90°, are presented by the closed contours. Superimposed are the corresponding solution loci for reflection phase shifts Δr=+90° and Δr=-90° for the cryolite-Si bilayer at ϕ=82° angle of incidence. The intersection points x, y, u, and v represent bilayers that function as a dual QWR in transmission and reflection.

Fig. 6
Fig. 6

Δt as a function of wavelength around λ=10.6 μm for the cryolite-Si pellicle QWR.

Fig. 7
Fig. 7

Effect on Δt of errors of ±5% of the cryolite film thickness d1 in the cryolite-Si pellicle QWR.

Fig. 8
Fig. 8

Effect on Δt of errors of ±5% of the Si film thickness d2 in the cryolite-Si pellicle QWR.

Fig. 9
Fig. 9

Effect on Δt of shifting the angle of incidence ϕ by ±1 around ϕ=82° for the cryolite-Si pellicle QWR.

Fig. 10
Fig. 10

Two oppositely tilted pellicle eighth-wave retarders in series function as a QWR.

Equations (10)

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Δt max=arctan{[N-(1/N)]/2},
ρt=Tp/Ts=(tan ψt)exp(jΔt).
ρt=a(1+bX1+cX2+dX1X2)/(1+eX1+fX2+gX1X2),
a=(t01p/t01s)(t12p/t12s)(t20p/t20s),
b=r01sr12s,e=r01pr12p,
c=r12sr20s,f=r12pr20p,
d=r01sr20s,g=r01pr20p.
Xi=exp(-j2πζi),i=1, 2,
ζi=di/Di,i=1, 2,
Di=(λ/2)(Ni2-sin2 ϕ)-1/2,i=1, 2,

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