Abstract

An empirically derived formula, which can be used to predict the average residual reflection that can be expected from an antireflection (AR) coating design as a function of bandwidth, overall thickness, available indices of the coating materials, number of layers, etc., is presented. This formula can be a useful tool not only for the thin-film designer but also for the nondesigner or system engineer to estimate the performance limits of an AR coating for a given application before the design is accomplished. The general predictions are also found to be consistent with the results of two recent AR design competitions involving many independent investigators. Some insight with respect to the basic underlying principles of AR coatings can also be gleaned from the results and the process by which they are found.

© 1993 Optical Society of America

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References

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  1. R. R. Willey, “Rugate broadband antireflection coating design,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 224–228 (1989).
  2. R. R. Willey, P. G. Verly, J. A. Dobrowolski, “Design of wideband antireflection coating with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).
  3. P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
    [CrossRef] [PubMed]
  4. R. R. Willey, “Realization of a very broad band AR coating,” in 33rd Annual Technical Conference Proceedings, V. H. Maddox, ed. (Society of Vacuum Coaters, New Orleans, La., 1990), pp. 232–236.
  5. A. Thelen, R. Langfeld, “Coating design problem,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).
  6. A. Macleod, “Design of an antireflection coating for glass over the region 400nm to 900nm,” in Thin Films for Optical Systems, K. H. Guenther, Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).
  7. M. L. Rastello, A. Premoli, “Continuation method for synthesizing antireflection coatings,” Appl. Opt. 31, 6741–6746 (1992).
    [CrossRef] [PubMed]
  8. J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, R. A. Kemp, “Antireflection coatings for germanium IR optics: a comparison of numerical design methods,” Appl. Opt. 27, 2832–2840 (1988).
    [CrossRef] [PubMed]

1992 (2)

1988 (1)

Aguilera, J.

Aguilera, J. A.

Baumeister, P.

Bloom, A.

Coursen, D.

Dobrowolski, J. A.

Goldstein, F. T.

Guenther, K. H.

A. Macleod, “Design of an antireflection coating for glass over the region 400nm to 900nm,” in Thin Films for Optical Systems, K. H. Guenther, Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

Gustafson, D. E.

Kemp, R. A.

Langfeld, R.

A. Thelen, R. Langfeld, “Coating design problem,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

Macleod, A.

A. Macleod, “Design of an antireflection coating for glass over the region 400nm to 900nm,” in Thin Films for Optical Systems, K. H. Guenther, Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

Premoli, A.

Rastello, M. L.

Thelen, A.

A. Thelen, R. Langfeld, “Coating design problem,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

Verly, P. G.

P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
[CrossRef] [PubMed]

R. R. Willey, P. G. Verly, J. A. Dobrowolski, “Design of wideband antireflection coating with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

Willey, R. R.

P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
[CrossRef] [PubMed]

R. R. Willey, “Realization of a very broad band AR coating,” in 33rd Annual Technical Conference Proceedings, V. H. Maddox, ed. (Society of Vacuum Coaters, New Orleans, La., 1990), pp. 232–236.

R. R. Willey, P. G. Verly, J. A. Dobrowolski, “Design of wideband antireflection coating with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

R. R. Willey, “Rugate broadband antireflection coating design,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 224–228 (1989).

Appl. Opt. (3)

Other (5)

R. R. Willey, “Realization of a very broad band AR coating,” in 33rd Annual Technical Conference Proceedings, V. H. Maddox, ed. (Society of Vacuum Coaters, New Orleans, La., 1990), pp. 232–236.

A. Thelen, R. Langfeld, “Coating design problem,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

A. Macleod, “Design of an antireflection coating for glass over the region 400nm to 900nm,” in Thin Films for Optical Systems, K. H. Guenther, Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

R. R. Willey, “Rugate broadband antireflection coating design,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 224–228 (1989).

R. R. Willey, P. G. Verly, J. A. Dobrowolski, “Design of wideband antireflection coating with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

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Figures (8)

Fig. 1
Fig. 1

Variation of the percentage of reflectance versus wavelength with bandwidth, while T = 3.0, L = 1.38, and D = 0.89. This illustrates how the minimum average reflectance decreases with bandwidth.

Fig. 2
Fig. 2

Average reflectance in the band versus optical thickness in cycles (or waves at the geometric mean wavelength of the band). The curve is from the formula, while the X's are from empirical data.

Fig. 3
Fig. 3

Average reflectance in the band versus bandwidth as seen in Fig. 1. The curve is from the formula, while the X's are from empirical data.

Fig. 4
Fig. 4

Average reflectance in the band versus index of refraction of the last layer. The curve is from the formula, while the X's are from empirical data.

Fig. 5
Fig. 5

Average reflectance in the band versus the number of layers. This implies that the minimum number of layers for best results equals 6T + 2. The X's are empirical results.

Fig. 6
Fig. 6

Average reflectance in the band versus the index of refraction difference in all layers except the last. The curve is from the formula, while the X's are from empirical data.

Fig. 7
Fig. 7

Thelen–Langfeld5 results of merit function versus physical thickness from the contributions to the Berlin 1992 design problem. The number of layers is shown for selected designs. The upper line connects the best designs that used no tantala while the lower line connects those that did use tantala.

Fig. 8
Fig. 8

Reflectance versus thickness at 900 nm, which illustrates the undulatory nature of the results and that various designs can be described in terms of these cycles of thickness T. Curves A and B are one- and three-cycle designs from the best of the Berlin design problem contributions.5 Curve C is an adaptation and refinement of the best of the contributions.

Equations (1)

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R AVE ( B , L , T , D ) % = ( 4.378 / D ) ( 1 / T ) 0.31 [ exp ( B 1.4 ) 1 ] ( L 1 ) 3.5 .

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