Abstract

Recently we performed a numerical investigation of antireflection coatings that reduce significantly the reflection over a wide range of wavelengths and angles of incidence, and we proposed some experiments to demonstrate their feasibility. We provide a theoretical description of omnidirectional antireflection coatings that are effective over a wide range of wavelengths.

© 2004 Optical Society of America

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References

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  1. Lord Rayleigh, “On reflection of vibrations at the confines of two media between which the transition is gradual,” Proc. London Math. Soc. 11, 51–56 (1880).
  2. R. N. Gupta, “Reflection of sound waves from transition layers,” J. Acoust. Soc. Am. 39, 255–260 (1965).
    [CrossRef]
  3. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 247–286.
    [CrossRef]
  4. R. W. Wood, “Some experiments on artificial mirages and tornadoes,” Philos. Mag. 47, 349–353 (1899).
  5. R. W. Wood, Physical Optics (Macmillian, New York, 1934), p. 88.
  6. J. A. Dobrowolski, D. Poitras, P. Ma, M. Acree, H. Vakil, “Toward perfect antireflection coatings: numerical investigation,” Appl. Opt. 41, 3075–3083 (2002).
    [CrossRef] [PubMed]
  7. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, Bristol, UK, 2001).
    [CrossRef]
  8. Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontières, Gif-sur-Yvette, France, 1992).
  9. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983).
    [CrossRef] [PubMed]
  10. D. Poitras, “Admittance diagrams of accidental and premeditated optical inhomogeneities in coatings,” Appl. Opt. 41, 4671–4679 (2002).
    [CrossRef] [PubMed]
  11. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993).
    [CrossRef] [PubMed]
  12. D. Poitras, S. Larouche, L. Martinu, “Design and plasma deposition of dispersion-corrected multiband rugate filters,” Appl. Opt. 41, 5249–5255 (2002).
    [CrossRef] [PubMed]
  13. B. G. Bovard, “Graded index rugate filters: power-sine rugate structures,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 109–125 (1993).
    [CrossRef]
  14. P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998).
    [CrossRef]
  15. It is worth noting that when the ratio ηmax/ηmin under the exponent in Eq. (7) is increased, the profile of the ES transition layer becomes similar to the modified profile [Fig. 7(d1)].
  16. It may be preferable to use an uneven distribution of points, since (i) its effect on the performance at short wavelengths and large angles of incidence is not critical and (ii) because many layers with close indices would be hard to achieve in practice.
  17. P. V. Adamson, “Antireflecting surface coatings with continuously varying complex refractive index,” Tech. Phys. Lett. 26, 1003–1006 (2000).
    [CrossRef]

2002 (3)

2000 (1)

P. V. Adamson, “Antireflecting surface coatings with continuously varying complex refractive index,” Tech. Phys. Lett. 26, 1003–1006 (2000).
[CrossRef]

1998 (1)

1993 (1)

1983 (1)

1965 (1)

R. N. Gupta, “Reflection of sound waves from transition layers,” J. Acoust. Soc. Am. 39, 255–260 (1965).
[CrossRef]

1899 (1)

R. W. Wood, “Some experiments on artificial mirages and tornadoes,” Philos. Mag. 47, 349–353 (1899).

1880 (1)

Lord Rayleigh, “On reflection of vibrations at the confines of two media between which the transition is gradual,” Proc. London Math. Soc. 11, 51–56 (1880).

Acree, M.

Adamson, P. V.

P. V. Adamson, “Antireflecting surface coatings with continuously varying complex refractive index,” Tech. Phys. Lett. 26, 1003–1006 (2000).
[CrossRef]

Bovard, B. G.

B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993).
[CrossRef] [PubMed]

B. G. Bovard, “Graded index rugate filters: power-sine rugate structures,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 109–125 (1993).
[CrossRef]

Dobrowolski, J. A.

Furman, Sh. A.

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontières, Gif-sur-Yvette, France, 1992).

Gupta, R. N.

R. N. Gupta, “Reflection of sound waves from transition layers,” J. Acoust. Soc. Am. 39, 255–260 (1965).
[CrossRef]

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 247–286.
[CrossRef]

Larouche, S.

Ma, P.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, Bristol, UK, 2001).
[CrossRef]

Martinu, L.

Poitras, D.

Rayleigh, Lord

Lord Rayleigh, “On reflection of vibrations at the confines of two media between which the transition is gradual,” Proc. London Math. Soc. 11, 51–56 (1880).

Southwell, W. H.

Tikhonravov, A. V.

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontières, Gif-sur-Yvette, France, 1992).

Vakil, H.

Verly, P. G.

Wood, R. W.

R. W. Wood, “Some experiments on artificial mirages and tornadoes,” Philos. Mag. 47, 349–353 (1899).

R. W. Wood, Physical Optics (Macmillian, New York, 1934), p. 88.

Appl. Opt. (5)

J. Acoust. Soc. Am. (1)

R. N. Gupta, “Reflection of sound waves from transition layers,” J. Acoust. Soc. Am. 39, 255–260 (1965).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

R. W. Wood, “Some experiments on artificial mirages and tornadoes,” Philos. Mag. 47, 349–353 (1899).

Proc. London Math. Soc. (1)

Lord Rayleigh, “On reflection of vibrations at the confines of two media between which the transition is gradual,” Proc. London Math. Soc. 11, 51–56 (1880).

Tech. Phys. Lett. (1)

P. V. Adamson, “Antireflecting surface coatings with continuously varying complex refractive index,” Tech. Phys. Lett. 26, 1003–1006 (2000).
[CrossRef]

Other (7)

It is worth noting that when the ratio ηmax/ηmin under the exponent in Eq. (7) is increased, the profile of the ES transition layer becomes similar to the modified profile [Fig. 7(d1)].

It may be preferable to use an uneven distribution of points, since (i) its effect on the performance at short wavelengths and large angles of incidence is not critical and (ii) because many layers with close indices would be hard to achieve in practice.

B. G. Bovard, “Graded index rugate filters: power-sine rugate structures,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 109–125 (1993).
[CrossRef]

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, Bristol, UK, 2001).
[CrossRef]

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontières, Gif-sur-Yvette, France, 1992).

R. W. Wood, Physical Optics (Macmillian, New York, 1934), p. 88.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 247–286.
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Schematic drawings and pictures adapted from Wood’s publication in 18994 illustrating (a) the physical cause leading to a mirage effect, (b) an experimental setup for artificially creating mirage effects, and (c) some pictures showing the effect obtained as the sand was heated (from left to right).

Fig. 2
Fig. 2

Polarization effect at oblique incidence: splitting of admittance for materials with different refractive indices.

Fig. 3
Fig. 3

Polarization effect at oblique incidence: Fresnel coefficients at interfaces with different refractive-index-step sizes.

Fig. 4
Fig. 4

Reduction factor sec(θ) for the phase thickness at oblique angles of incidence, as a function of θ, the angle of propagation in materials of different refractive indices.

Fig. 5
Fig. 5

Schematic representation of a multilayer system.

Fig. 6
Fig. 6

Variation of the critical angle of incidence as a function of the ratio n i /n a .

Fig. 7
Fig. 7

Original and modified quintic and ES refractive-index profiles of inhomogeneous transition layers [column (1)]; the performance of the coatings as a function of angle of incidence, for a wavelength λ = 0.01625d [column (2)], and as a function of wavelength, for various angles of incidence [column (3)].

Fig. 8
Fig. 8

Variation of the reflectance at an angle of incidence of 89°, as a function of the total metric thicknesses of the original and modified quintic profiles divided by the wavelength.

Fig. 9
Fig. 9

Variation with thickness of the admittances for s- and p-polarized light of the modified quintic refractive-index profile of Fig. 7(c1), plotted for different angles of incidence.

Fig. 10
Fig. 10

Variation of the Fresnel reflection coefficient for different departures of the admittance of the inhomogeneous layer from the medium admittance at the interface of the layer and the medium.

Fig. 11
Fig. 11

Comparison of the performances of an inhomogeneous AR coating with a modified quintic profile designed for an interface between media of refractive indices 3.0 and 1.0 and of an AR coating for a substrate of refractive index 1.5, which consists of a fraction of the same quintic profile: (a) refractive-index profiles and (b) average reflectance spectra for 0–80° and 89° angles of incidence.

Fig. 12
Fig. 12

Effect of truncation of the overall thickness of an AR coating with a modified quintic refractive-index profile on the performance at angles of incidence of 0°, 80°, and 89°: (a) refractive-index profile and (b) reflectance at λ = 0.01625d as a function of the amount of truncation.

Fig. 13
Fig. 13

Effect of discretization of the refractive-index profile on the average spectral reflectance: (row a) 1000 sublayers, (row b) 10 layers, and (row c) 5 layers and (column 1) refractive-index profiles, (column 2) reflectance spectra, and (columns 3 and 4) admittance diagrams at wavelengths of 15 and 5 μm, respectively. The thickness of the five-layer solution was reduced so that its reflectance matched more closely those shown in (a2) and (b2).

Fig. 14
Fig. 14

Discrete five-layer profile reoptimized to reduce the reflectance in the 3.0–4.0-μm spectral region, for angles of incidence as great as 89°: (a) refractive index profile and (b) reflectance spectra.

Fig. 15
Fig. 15

Effect of absorption on the performance of graded refractive-index AR coatings. (Column 1) transparent inhomogeneous layer matches the real part of the substrate refractive index, (column 2) layer with a graded refractive index and extinction coefficient, and (column 3) graded transparent coating with a thin correction layer [with n = Re(η s ) and d = 18.3 nm] at the substrate interface. Row (b) shows the refractive-index profiles, and rows (a) and (c) show their respective reflectance spectra and admittance diagrams at the wavelength of 600 nm.

Equations (12)

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ddznzsin θz=0,
ηs=nzcos θz, ηp=nz/cos θz.
nzdzθ=nzdz cos θz,
θc=arcsinni-1/ni.
θCinc=arcsinni-1/na.
nQz=nmax-nmax-nmin10zd3-15zd4+6zd5
nESx=ηmax exp12lnηmax/ηmin×sinπxxtot+π/2-sinπ/2,
Rλ2πd2m,
ρs=12ηsdηsdz=n2n-θ2tan θ, ρp=12ηpdηpdz=n2n+θ2tan θ.
dznew=dz1-nM2nz2sinθ0max21/2.
Rηd-ηaηd+ηa2.
Yj=iηj sin φj+Yj-1 cos φjcos φj+iηjYj-1 sin φj,

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