Abstract

An efficient first-order algorithm with an exact calculation of the gradients is described and applied to the refinement of inhomogeneous dielectric thin films. Numerical results calculated at normal and oblique angles of incidence are correlated with an important mathematical theorem. It is shown that, when appropriate conditions are satisfied, the synthesis process generates binary type solutions based on two materials, including classical multilayers.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
    [CrossRef]
  2. 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 methods,” Appl. Opt. 27, 2832–2840 (1988).
    [CrossRef] [PubMed]
  3. J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [CrossRef] [PubMed]
  4. A. Premoli, M. L. Rastello, “Minimax refining of wideband antireflection coatings for wide angular incidence,” Appl. Opt. 33, 2018–2024 (1994).
    [CrossRef] [PubMed]
  5. J. A. Dobrowolski, S. Piotrowski, “Refractive index as a variable in the numerical design of optical thin film systems,” Appl. Opt. 21, 1502–1510 (1982).
    [CrossRef] [PubMed]
  6. A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).
    [CrossRef]
  7. A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings,” Appl. Opt. 32, 4265–4275 (1993).
    [CrossRef] [PubMed]
  8. J. A. Dobrowolski, “Substractive method of optical thin film interference filter design,” Appl. Opt. 12, 1885–1893 (1973).
    [CrossRef] [PubMed]
  9. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  10. J. A. Dobrowolski, P. G. Verly, eds., Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, Proc. SPIE2046 (1993).
  11. R. Jacobson, Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications, Vol. 8 of Physics of Thin Films Series (Academic, New York, 1976).
  12. W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
    [CrossRef] [PubMed]
  13. R. W. Bertram, M. F. Ouellette, P. Y. Tse, “Inhomogeneous optical coatings: an experimental study of a new approach,” Appl. Opt. 28, 2935–2939 (1989).
    [CrossRef] [PubMed]
  14. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993).
    [CrossRef] [PubMed]
  15. R. L. Hall, “Experimental aspects of rugate filter fabrication,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 14–16.
  16. J. Allen, B. D. Herrington, P. G. Girow, T. Slim, “Digitized rugate filters for laser rejection,” in Ref. 10, pp. 126–131.
  17. C. G. Snedacker, “New numerical thin film synthesis technique,” J. Opt. Soc. Am. 72, 1732A (1982).
  18. P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous coatings at the NRC,” in Ref. 10, pp. 36–45.
  19. C. Van der Laan, H. J. Frankena, “Fast computation method for derivatives of multilayer stack reflectance,” Appl. Opt. 17, 538–541 (1978).
    [CrossRef]
  20. J. A. Dobrowolski, “Comparison of the Fourier transform and flip-flop thin film synthesis methods,” Appl. Opt. 25, 1966–1972 (1986).
    [CrossRef]
  21. T. C. Chen, “Turning point method of optimized multilayer design,” Proc. Inst. Electr. Eng. 137, 101–107 (1990).
  22. S. Furman, A. V. Tikhonravov, Optics of Multilayer Systems (Editions Frontières, Gif-sur-Yvette, 1992).
  23. A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
    [CrossRef] [PubMed]
  24. P. E. Gill, W. Murray, “Minimization subject to bounds on the variables,” (National Physical Laboratory, Teddington, Middlesex, UK, 1976).
  25. User’s Manual, IMSL Math/Library, FORTRAN Subroutines for Mathematical Applications (IMSL Problem Solving Software Systems, Houston, Tex., 1989), p. 847.
  26. T. R. Cuthbert, Optimization Using Personal Computers (Wiley, New York, 1987), Chap. 5.
  27. W. H. Press, S. A. Teukdlski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992), pp. 418–423.
  28. 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]
  29. J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, P. G. Verly, “Optimal single-band normal-incidence antireflection coatings,” Appl. Opt. 35, 644–658 (1996).
    [CrossRef] [PubMed]
  30. P. Baumeister, “Simulation of a rugate filter via a stepped-index dielectric multilayer,” Appl. Opt. 25, 2644–2645 (1986).
    [CrossRef] [PubMed]
  31. M. Zukic, K. H. Guenther, “Design of nonpolarizing achromatic beamsplitters with dielectric multilayer coatings,” Opt. Eng. 28, 165–171 (1989).
    [CrossRef]
  32. G. W. DeBell, A. V. Tikhonravov, M. K. Trubetskov, “Use of a new synthesis algorithm to design polarisation insensitive optical coatings,” in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. SPIE2262, 187–197 (1994).
  33. P. G. Verly, “Fourier-transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995).
    [CrossRef] [PubMed]
  34. A. V. Tikhonravov, M. K. Trubetskov, I. V. Zuev, P. G. Verly, “Efficient refinement of inhomogeneous optical coatings,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series, pp. 22–24.

1996 (1)

1995 (1)

1994 (1)

1993 (3)

1992 (1)

1990 (2)

J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
[CrossRef] [PubMed]

T. C. Chen, “Turning point method of optimized multilayer design,” Proc. Inst. Electr. Eng. 137, 101–107 (1990).

1989 (2)

M. Zukic, K. H. Guenther, “Design of nonpolarizing achromatic beamsplitters with dielectric multilayer coatings,” Opt. Eng. 28, 165–171 (1989).
[CrossRef]

R. W. Bertram, M. F. Ouellette, P. Y. Tse, “Inhomogeneous optical coatings: an experimental study of a new approach,” Appl. Opt. 28, 2935–2939 (1989).
[CrossRef] [PubMed]

1988 (1)

1986 (2)

1985 (1)

1982 (2)

1978 (2)

1973 (1)

1958 (1)

Aguilera, J.

Aguilera, J. A.

Baumeister, P.

Bertram, R. W.

Bloom, A.

Bovard, B. G.

Chen, T. C.

T. C. Chen, “Turning point method of optimized multilayer design,” Proc. Inst. Electr. Eng. 137, 101–107 (1990).

Coursen, D.

Cuthbert, T. R.

T. R. Cuthbert, Optimization Using Personal Computers (Wiley, New York, 1987), Chap. 5.

DeBell, G. W.

G. W. DeBell, A. V. Tikhonravov, M. K. Trubetskov, “Use of a new synthesis algorithm to design polarisation insensitive optical coatings,” in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. SPIE2262, 187–197 (1994).

Dobrowolski, J. A.

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, P. G. Verly, “Optimal single-band normal-incidence antireflection coatings,” Appl. Opt. 35, 644–658 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings,” Appl. Opt. 32, 4265–4275 (1993).
[CrossRef] [PubMed]

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]

J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
[CrossRef] [PubMed]

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 methods,” Appl. Opt. 27, 2832–2840 (1988).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Comparison of the Fourier transform and flip-flop thin film synthesis methods,” Appl. Opt. 25, 1966–1972 (1986).
[CrossRef]

J. A. Dobrowolski, S. Piotrowski, “Refractive index as a variable in the numerical design of optical thin film systems,” Appl. Opt. 21, 1502–1510 (1982).
[CrossRef] [PubMed]

J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Substractive method of optical thin film interference filter design,” Appl. Opt. 12, 1885–1893 (1973).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukdlski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992), pp. 418–423.

Frankena, H. J.

Furman, S.

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

Gill, P. E.

P. E. Gill, W. Murray, “Minimization subject to bounds on the variables,” (National Physical Laboratory, Teddington, Middlesex, UK, 1976).

Goldstein, F. T.

Guenther, K. H.

M. Zukic, K. H. Guenther, “Design of nonpolarizing achromatic beamsplitters with dielectric multilayer coatings,” Opt. Eng. 28, 165–171 (1989).
[CrossRef]

Gustafson, D. E.

Hall, R. L.

R. L. Hall, “Experimental aspects of rugate filter fabrication,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 14–16.

Jacobson, R.

R. Jacobson, Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications, Vol. 8 of Physics of Thin Films Series (Academic, New York, 1976).

Kemp, R. A.

Lowe, D.

Murray, W.

P. E. Gill, W. Murray, “Minimization subject to bounds on the variables,” (National Physical Laboratory, Teddington, Middlesex, UK, 1976).

Ouellette, M. F.

Piotrowski, S.

Premoli, A.

Press, W. H.

W. H. Press, S. A. Teukdlski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992), pp. 418–423.

Rastello, M. L.

Snedacker, C. G.

C. G. Snedacker, “New numerical thin film synthesis technique,” J. Opt. Soc. Am. 72, 1732A (1982).

Southwell, W. H.

Sullivan, B. T.

Teukdlski, S. A.

W. H. Press, S. A. Teukdlski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992), pp. 418–423.

Tikhonravov, A. V.

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, P. G. Verly, “Optimal single-band normal-incidence antireflection coatings,” Appl. Opt. 35, 644–658 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings,” Appl. Opt. 32, 4265–4275 (1993).
[CrossRef] [PubMed]

A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, I. V. Zuev, P. G. Verly, “Efficient refinement of inhomogeneous optical coatings,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series, pp. 22–24.

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).
[CrossRef]

G. W. DeBell, A. V. Tikhonravov, M. K. Trubetskov, “Use of a new synthesis algorithm to design polarisation insensitive optical coatings,” in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. SPIE2262, 187–197 (1994).

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

Trubetskov, M. K.

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, P. G. Verly, “Optimal single-band normal-incidence antireflection coatings,” Appl. Opt. 35, 644–658 (1996).
[CrossRef] [PubMed]

G. W. DeBell, A. V. Tikhonravov, M. K. Trubetskov, “Use of a new synthesis algorithm to design polarisation insensitive optical coatings,” in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. SPIE2262, 187–197 (1994).

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, I. V. Zuev, P. G. Verly, “Efficient refinement of inhomogeneous optical coatings,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series, pp. 22–24.

Tse, P. Y.

Van der Laan, C.

Verly, P. G.

Vetterlin, W. T.

W. H. Press, S. A. Teukdlski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992), pp. 418–423.

Willey, R. R.

Zuev, I. V.

A. V. Tikhonravov, M. K. Trubetskov, I. V. Zuev, P. G. Verly, “Efficient refinement of inhomogeneous optical coatings,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series, pp. 22–24.

Zukic, M.

M. Zukic, K. H. Guenther, “Design of nonpolarizing achromatic beamsplitters with dielectric multilayer coatings,” Opt. Eng. 28, 165–171 (1989).
[CrossRef]

Appl. Opt. (17)

J. A. Dobrowolski, “Substractive method of optical thin film interference filter design,” Appl. Opt. 12, 1885–1893 (1973).
[CrossRef] [PubMed]

C. Van der Laan, H. J. Frankena, “Fast computation method for derivatives of multilayer stack reflectance,” Appl. Opt. 17, 538–541 (1978).
[CrossRef]

J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
[CrossRef] [PubMed]

J. A. Dobrowolski, S. Piotrowski, “Refractive index as a variable in the numerical design of optical thin film systems,” Appl. Opt. 21, 1502–1510 (1982).
[CrossRef] [PubMed]

W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Comparison of the Fourier transform and flip-flop thin film synthesis methods,” Appl. Opt. 25, 1966–1972 (1986).
[CrossRef]

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 methods,” Appl. Opt. 27, 2832–2840 (1988).
[CrossRef] [PubMed]

R. W. Bertram, M. F. Ouellette, P. Y. Tse, “Inhomogeneous optical coatings: an experimental study of a new approach,” Appl. Opt. 28, 2935–2939 (1989).
[CrossRef] [PubMed]

J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
[CrossRef] [PubMed]

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]

A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings,” Appl. Opt. 32, 4265–4275 (1993).
[CrossRef] [PubMed]

A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

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

A. Premoli, M. L. Rastello, “Minimax refining of wideband antireflection coatings for wide angular incidence,” Appl. Opt. 33, 2018–2024 (1994).
[CrossRef] [PubMed]

P. G. Verly, “Fourier-transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995).
[CrossRef] [PubMed]

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, P. G. Verly, “Optimal single-band normal-incidence antireflection coatings,” Appl. Opt. 35, 644–658 (1996).
[CrossRef] [PubMed]

P. Baumeister, “Simulation of a rugate filter via a stepped-index dielectric multilayer,” Appl. Opt. 25, 2644–2645 (1986).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

C. G. Snedacker, “New numerical thin film synthesis technique,” J. Opt. Soc. Am. 72, 1732A (1982).

P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
[CrossRef]

Opt. Eng. (1)

M. Zukic, K. H. Guenther, “Design of nonpolarizing achromatic beamsplitters with dielectric multilayer coatings,” Opt. Eng. 28, 165–171 (1989).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

T. C. Chen, “Turning point method of optimized multilayer design,” Proc. Inst. Electr. Eng. 137, 101–107 (1990).

Other (13)

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

G. W. DeBell, A. V. Tikhonravov, M. K. Trubetskov, “Use of a new synthesis algorithm to design polarisation insensitive optical coatings,” in Optical Thin Films IV: New Developments, J. D. Rancourt, ed., Proc. SPIE2262, 187–197 (1994).

A. V. Tikhonravov, M. K. Trubetskov, I. V. Zuev, P. G. Verly, “Efficient refinement of inhomogeneous optical coatings,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series, pp. 22–24.

P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous coatings at the NRC,” in Ref. 10, pp. 36–45.

P. E. Gill, W. Murray, “Minimization subject to bounds on the variables,” (National Physical Laboratory, Teddington, Middlesex, UK, 1976).

User’s Manual, IMSL Math/Library, FORTRAN Subroutines for Mathematical Applications (IMSL Problem Solving Software Systems, Houston, Tex., 1989), p. 847.

T. R. Cuthbert, Optimization Using Personal Computers (Wiley, New York, 1987), Chap. 5.

W. H. Press, S. A. Teukdlski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, 1992), pp. 418–423.

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).
[CrossRef]

J. A. Dobrowolski, P. G. Verly, eds., Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, Proc. SPIE2046 (1993).

R. Jacobson, Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications, Vol. 8 of Physics of Thin Films Series (Academic, New York, 1976).

R. L. Hall, “Experimental aspects of rugate filter fabrication,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 14–16.

J. Allen, B. D. Herrington, P. G. Girow, T. Slim, “Digitized rugate filters for laser rejection,” in Ref. 10, pp. 126–131.

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 (8)

Fig. 1
Fig. 1

Series of intermediate solutions obtained in the synthesis of a cavity filter.

Fig. 2
Fig. 2

Convergence of the merit function value in Fig. 1. The filled circles correspond to the intermediate solutions of Figs. 1C1E.

Fig. 3
Fig. 3

Series of intermediate solutions obtained in the synthesis of a wideband AR coating.

Fig. 4
Fig. 4

Convergence of the merit function value in Fig. 3. The filled circles correspond to the intermediate solutions of Figs. 3C3E.

Fig. 5
Fig. 5

Quasirugate filter with suppressed second and third harmonics.

Fig. 6
Fig. 6

P-polarization admittance of a medium of refractive index n when light is incident from glass at angle θ a .

Fig. 7
Fig. 7

Glass-immersed beam splitter for p-polarized light at 50° incidence.

Fig. 8
Fig. 8

Glass-immersed beam splitter for p-polarized light at 40°-55° incidence.

Equations (9)

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

Mj=cosϕjiηjsinϕjiηj sinϕjcosϕj, j=1, 2, , J,
ϕj=2πλnj dj cos θj
ηj=nj cos θj s polarization, ηj=njcos θj p polarization,
F=1Kk=1KRλk-RˆkδRk21/2,
F=1KFk=1KRλk- RˆkδRk2Rλk,
Rj=Rnj=2 Rer*rnj,
rnj=TrMj+1MjnjMj-1ψr,
Mj-1=Πm=1j-1Mj, Mj+1=Πm=j+1JMj,
ψr=t2nana1-r-1+rnans1-r-ns1+r.

Metrics