Abstract

What we believe to be a new principle is introduced for the design and selection of gradient-index antireflection profiles that are effective over a wide range of incident angles as well as wavelengths at a given physical film thickness. It is shown that at oblique incidence the smoothness of the optical path of incident light inside a gradient-index film has a crucial effect on the overall reflection. Thus the smoothness of variations in refractive angle (rather than that of the index profile itself) needs to be maximized for wide-angle operation. As an example, the performance of Gaussian and Quintic profiles at large incident angles are considered in light of this point of view.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Mussett and A. Thelen, "Multilayer antireflection coatings," in Progress in Optics Vol. 8, E. Wolf, ed. (North-Holland, 1970), pp. 203-237.
  2. J. A. Dobrowolski, "Optical properties of films and coatings," in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), pp. 42.19-42.34.
  3. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).
  4. R. Jacobsson, "Light reflection from films of continuously varying refractive index," in Progress in Optics, Vol. 5, E. Wolf, ed. (North-Holland, 1966), pp. 247-286.
    [CrossRef]
  5. W. H. Southwell, "Gradent-index antireflection coatings," Opt. Lett. 8, 584-586 (1983).
    [CrossRef] [PubMed]
  6. E. Spiller, I. Haller, R. Feder, J. E. E. Baglin, and W. N. Hammer, "Graded-index AR surfaces produce by ion implantation on plastic materials," Appl. Opt. 19, 3022-3026 (1980).
    [CrossRef] [PubMed]
  7. P. Yeh and S. Sari, "Optical properties of stratified media with exponentially graded refractive index," Appl. Opt. 22, 4142-4145 (1983).
    [CrossRef] [PubMed]
  8. P. G. Verly, J. A. Dobrowolski, and R. R. Willey, "Fourier-transform method for the design of wideband antirefretion coatings," Appl. Opt. 31, 3836-3846 (1992).
    [CrossRef] [PubMed]
  9. E. B. Grann, M. G. Moharam, and D. A. Pommet, "Optimal design for antireflective tapered two-dimensional subwavelength grating structures," J. Opt. Soc. Am. A 12, 333-339 (1995).
    [CrossRef]
  10. M. J. Minot, "The angular reflectance of single-layer gradient refractive-index films," J. Opt. Soc. Am. 67, 1046-1050 (1977).
    [CrossRef]
  11. A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
    [CrossRef]
  12. Y. Kanamori, M. Ishimori, and K. Hane, "High efficient light-emitting diodes with antireflection subwavelength gratings," IEEE Photon. Technol. Lett. 14, 1064-1066 (2002).
    [CrossRef]
  13. P. B. Clapham and M. C. Hutley, "Reduction of lens reflexion by the 'moth eye' principle," Nature 244, 281-282 (1973).
    [CrossRef]
  14. P. Baumeister, "The transmission and degree of polarization of quarter-wave stacks at nonnormal incidence," Opt. Acta. 8, 105-119 (1961).
    [CrossRef]
  15. J. A. Dobrowolski and S. H. C. Piotrowski, "Refractive index as a variable in the numerical design of optical thin films," Appl. Opt. 21, 1502-1510 (1982).
    [CrossRef] [PubMed]
  16. J. C. Monga, "Anti-reflection coatings for grazing incidence angles," J. Mod. Opt. 36, 381-387 (1989).
    [CrossRef]
  17. J. C. Monga, "Double-layer broadband antireflection coatings for grazing incidence angles," Appl. Opt. 31, 546-553 (1992).
    [CrossRef] [PubMed]
  18. J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
    [CrossRef] [PubMed]
  19. D. Poitras and J. A. Dobrowolski, "Toward perfect antireflection coatings 2: Theory," Appl. Opt. 43, 1286-1295 (2004).
    [CrossRef] [PubMed]
  20. W. H. Southwell, "Coating design using very thin high-and-low-index layers," Appl. Opt. 24, 457-460 (1985).
    [CrossRef] [PubMed]
  21. H. Sankur and W. H. Southwell, "Broadband gradient-index antireflection coating for ZnSe," Appl. Opt. 23, 2770-2773 (1984).
    [CrossRef] [PubMed]
  22. J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T. M. Lu, S. Y. Lin, and J. S. Juneja, "Very low-refractive-index optical thin films consisting of an array of SiO2 nanorods," Opt. Lett. 31, 601-603 (2006).
    [CrossRef] [PubMed]

2006 (1)

2004 (1)

2002 (2)

Y. Kanamori, M. Ishimori, and K. Hane, "High efficient light-emitting diodes with antireflection subwavelength gratings," IEEE Photon. Technol. Lett. 14, 1064-1066 (2002).
[CrossRef]

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

1998 (1)

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

1995 (2)

J. A. Dobrowolski, "Optical properties of films and coatings," in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), pp. 42.19-42.34.

E. B. Grann, M. G. Moharam, and D. A. Pommet, "Optimal design for antireflective tapered two-dimensional subwavelength grating structures," J. Opt. Soc. Am. A 12, 333-339 (1995).
[CrossRef]

1992 (2)

1989 (2)

J. C. Monga, "Anti-reflection coatings for grazing incidence angles," J. Mod. Opt. 36, 381-387 (1989).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

1985 (1)

1984 (1)

1983 (2)

1982 (1)

1980 (1)

1977 (1)

1973 (1)

P. B. Clapham and M. C. Hutley, "Reduction of lens reflexion by the 'moth eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

1970 (1)

A. Mussett and A. Thelen, "Multilayer antireflection coatings," in Progress in Optics Vol. 8, E. Wolf, ed. (North-Holland, 1970), pp. 203-237.

1966 (1)

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

1961 (1)

P. Baumeister, "The transmission and degree of polarization of quarter-wave stacks at nonnormal incidence," Opt. Acta. 8, 105-119 (1961).
[CrossRef]

Acree, M.

Baglin, J. E. E.

Baumeister, P.

P. Baumeister, "The transmission and degree of polarization of quarter-wave stacks at nonnormal incidence," Opt. Acta. 8, 105-119 (1961).
[CrossRef]

Blasi, B.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Clapham, P. B.

P. B. Clapham and M. C. Hutley, "Reduction of lens reflexion by the 'moth eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

Dobrowolski, J. A.

Doll, W.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Dreibholz, J.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Feder, R.

Glaubitt, W.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Gombert, A.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Grann, E. B.

Haller, I.

Hammer, W. N.

Hane, K.

Y. Kanamori, M. Ishimori, and K. Hane, "High efficient light-emitting diodes with antireflection subwavelength gratings," IEEE Photon. Technol. Lett. 14, 1064-1066 (2002).
[CrossRef]

Heinzel, A.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Horbelt, W.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Hutley, M. C.

P. B. Clapham and M. C. Hutley, "Reduction of lens reflexion by the 'moth eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

Ishimori, M.

Y. Kanamori, M. Ishimori, and K. Hane, "High efficient light-emitting diodes with antireflection subwavelength gratings," IEEE Photon. Technol. Lett. 14, 1064-1066 (2002).
[CrossRef]

Jacobsson, R.

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

Juneja, J. S.

Kanamori, Y.

Y. Kanamori, M. Ishimori, and K. Hane, "High efficient light-emitting diodes with antireflection subwavelength gratings," IEEE Photon. Technol. Lett. 14, 1064-1066 (2002).
[CrossRef]

Kim, J. K.

Lin, S. Y.

Lu, T. M.

Luther, J.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Ma, P.

Minot, M. J.

Moharam, M. G.

Monga, J. C.

J. C. Monga, "Double-layer broadband antireflection coatings for grazing incidence angles," Appl. Opt. 31, 546-553 (1992).
[CrossRef] [PubMed]

J. C. Monga, "Anti-reflection coatings for grazing incidence angles," J. Mod. Opt. 36, 381-387 (1989).
[CrossRef]

Mussett, A.

A. Mussett and A. Thelen, "Multilayer antireflection coatings," in Progress in Optics Vol. 8, E. Wolf, ed. (North-Holland, 1970), pp. 203-237.

Piotrowski, S. H. C.

Poitras, D.

Pommet, D. A.

Rose, K.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Sankur, H.

Sari, S.

Schubert, E. F.

Southwell, W. H.

Spiller, E.

Sporn, D.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

A. Mussett and A. Thelen, "Multilayer antireflection coatings," in Progress in Optics Vol. 8, E. Wolf, ed. (North-Holland, 1970), pp. 203-237.

Vakil, H.

Verly, P. G.

Willey, R. R.

Wittwer, V.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Xi, J.-Q.

Ye, D.

Yeh, P.

Zanke, C.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Appl. Opt. (9)

IEEE Photon. Technol. Lett. (1)

Y. Kanamori, M. Ishimori, and K. Hane, "High efficient light-emitting diodes with antireflection subwavelength gratings," IEEE Photon. Technol. Lett. 14, 1064-1066 (2002).
[CrossRef]

J. Mod. Opt. (1)

J. C. Monga, "Anti-reflection coatings for grazing incidence angles," J. Mod. Opt. 36, 381-387 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nature (1)

P. B. Clapham and M. C. Hutley, "Reduction of lens reflexion by the 'moth eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

Opt. Acta. (1)

P. Baumeister, "The transmission and degree of polarization of quarter-wave stacks at nonnormal incidence," Opt. Acta. 8, 105-119 (1961).
[CrossRef]

Opt. Lett. (2)

Sol. Energy (1)

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, C. Zanke, B. Blasi, A. Heinzel, W. Horbelt, D. Sporn, W. Doll, V. Wittwer, and J. Luther, "Grazing with very high solar transmittance," Sol. Energy 62, 177-188 (1998).
[CrossRef]

Other (4)

A. Mussett and A. Thelen, "Multilayer antireflection coatings," in Progress in Optics Vol. 8, E. Wolf, ed. (North-Holland, 1970), pp. 203-237.

J. A. Dobrowolski, "Optical properties of films and coatings," in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), pp. 42.19-42.34.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(Color online) Geometry of a gradient-index ARC.

Fig. 2
Fig. 2

(Color online) Comparison of reflectance under Gaussian and Gaussian-type profiles for a wavelength of λ = 1 μ m with d = 1 μ m physical thickness. Inset is the Gaussian and Gaussian-type profile.

Fig. 3
Fig. 3

Gaussian and Quintic gradient-index coating angular reflectance calculated for a wavelength of 1 μ m with a physical thickness of (a) 1 μ m and (b) 3 μ m .

Fig. 4
Fig. 4

(a) Gaussian and Quintic gradient-index coating angular reflectance calculated with a physical thickness of 1 μ m . The spectral reflectance calculated with a physical thickness of 1 μ m for (b) θ 0 = 70 ° incident angle and (c) θ 0 = 80 ° incident angle.

Fig. 5
Fig. 5

(Color online) (a) Index profile plotted for the Gaussian (solid curve) and Quintic profile (dash curve), respectively. Those two profiles are normalized to optical distance. (b) Refractive angle within the Gaussian and Quintic coating. (c) Reflectance within the Gaussian and Quintic coating at incident angle of θ 0 = 75 ° .

Fig. 6
Fig. 6

(Color online) (a) Five-layer discretized Gaussian profile. (b) Angular reflectance with respect to different numbers of discrete layers for a Gaussian profile.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

z = 0 x d x n ( x ) .
n TE ( x ) = n ( x ) cos ( θ ( x ) ) ,
n TM ( x ) = n ( x ) / cos ( θ ( x ) ) ,
cos ( θ ( x ) ) = [ 1 n min n ( x )   sin ( θ 0 ) ] 1 / 2 .
d r TE ( x ) = [ n TE ( x ) n TE ( x + d x ) n TE ( x ) + n TE ( x + d x ) ] 2 [ d n TE ( x ) 2 n TE ( x ) ] 2 .
n ( x ) = n min   sin   θ 0 sin { θ 0 + ( θ 1 θ 0 ) [ 1 cos ( π x ) / 2 ] } ,
n ( x ) = n min + ( n max n min ) exp [ ( x 1 0.4 ) 2 ] ,
n ( x ) = n min + ( n max n min ) ( 10 x 3 15 x 4 + 6 x 5 ) .

Metrics