Abstract

Composite films of Ta–Si oxide with refractive indices that varied from 1.48 to 2.15 were realized by using rf ion-beam sputtering. All the composite films were amorphous and had a surface roughness of less than 0.3nm. The inhomogeneity of the composite was discussed, and a rugate filter was designed and fabricated by automatic computer control.

© 2006 Optical Society of America

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  1. E. P. Donovan, D. Van Vechten, A. D. F. Kahn, C. A. Carosella, and G. K. Hubler, 'Near infrared rugate filter fabrication by ion beam assisted deposition of Si(1-x)Nx films,' Appl. Opt. 28, 2940-2944 (1989).
    [CrossRef] [PubMed]
  2. W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, and N. S. Gluck, 'Codeposition of continuous composition rugate filters,' Appl. Opt. 28, 2945-2948 (1989).
    [CrossRef] [PubMed]
  3. H. J. Cho, I. G. Yu, and C. K. Hwangbo, 'Inhomogeneous optical SiOxNy thin films prepared by ion assisted deposition,' Nucl. Instrum. Methods Phys. Res. B 121, 137-140 (1997).
    [CrossRef]
  4. H. Bartzsch, S. Lange, P. Frach, and K. Goedicke, 'Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering,' Surf. Coat. Technol. 180-181, 616-620 (2004).
    [CrossRef]
  5. A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
    [CrossRef]
  6. H. Sankur, W. J. Gunning, and J. F. DeNatale, 'Intrinsic stress and structural properties of mixed composition thin film,' Appl. Opt. 27, 1564-1567 (1988).
    [CrossRef] [PubMed]
  7. J. S. Chen, S. Chao, J. S. Kao, H. Niu, and C. H. Chen, 'Mixed films of TiO2-SiO2 deposited by double electron-beam coevaporation,' Appl. Opt. 37, 90-96 (1996).
    [CrossRef]
  8. N. K. Sahoo and A. P. Shapiro, 'Process-parameter-dependent optical and structural properties of ZrO2MgO mixed-composite films evaporated from the solid solution,' Appl. Opt. 37, 698-718 (1998).
    [CrossRef]
  9. T. U. Ryu, S. H. Hahn, S. W. Kim, and E. J. Kim, 'Optical, mechanical and thermal properties of MgF2-ZnS and MgF2-Ta2O5 composite thin films deposited by co-evaporation,' Opt. Eng. 39, 3207-3213 (2000).
    [CrossRef]
  10. B. J. Pond, J. I. DeBar, C. K. Carniglia, and T. Raj, 'Stress reduction in ion beam sputtered mixed oxide films,' Appl. Opt. 28, 2800-2805 (1989).
    [CrossRef] [PubMed]
  11. S. Chao, C. K. Chang, and J. S. Chen, 'TiO2-SiO2 mixed films prepared by the fast alternating sputter method,' Appl. Opt. 30, 3233-3237 (1991).
    [CrossRef] [PubMed]
  12. M. Veszelei, L. Kullman, C. G. Granqvist, N. von Rottkay, and M. Rubin, 'Optical constants of sputter-deposited Ti-Ce oxide and Zr-Ce oxide films,' Appl. Opt. 37, 5993-6001 (1998).
    [CrossRef]
  13. X. Wang, H. Masumoto, Y. Someno, and T. Hirai, 'Microstructure and optical properties of amorphous TiO2-SiO2 composite films synthesized by helicon plasma sputtering,' Thin Solid Films 338, 105-109 (1999).
    [CrossRef]
  14. A. Ritz, 'TaTiOx layers prepared by magnetron sputtering from separate metal targets,' Surf. Coat. Technol. 174-175, 651-654 (2003).
    [CrossRef]
  15. M. Cevro, 'Ion-beam sputtering of (Ta2O5)x-(SiO2)1-x composite thin films,' Thin Solid Films 258, 91-103 (1995).
    [CrossRef]
  16. S. Chao, W. H. Wang, and C. C. Lee, 'Low-loss dielectric mirror with ion-beam-sputtered TiO2-SiO2 mixed films,' Appl. Opt. 40, 2177-2182 (2001).
    [CrossRef]
  17. Y. Katagiri and H. Ukita, 'Ion beam sputtered (SiO2)x(Si3N4)1-x antireflection coatings on laser facets produced using O2-N2 discharges,' Appl. Opt. 29, 5074-5079 (1990).
    [CrossRef] [PubMed]
  18. C. C. Lee, H. L. Chen, J. C. Hsu, and C. L. Tien, 'Interference coating base on synthesized silicon nitride,' Appl. Opt. 38, 2078-2082 (1999).
    [CrossRef]
  19. J. C. Manifacier, J. Gasiot, and J. P. Fillard, 'A simple method for the determination of the optical constant n, k and the thickness of the weakly absorbing thin film,' J. Phys. E 9, 1002-1004 (1976).
    [CrossRef]
  20. W. H. Southwell, 'Spectral response calculations of rugate filters using coupled-wave theory,' J. Opt. Soc. Am. A 5, 1558-1564 (1988).
    [CrossRef]
  21. W. H. Southwell, 'Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,' Appl. Opt. 28, 2949-2951 (1989).
    [CrossRef] [PubMed]
  22. W. H. Southwell, 'Using apodization functions to reduce sidelobes in rugate filters,' Appl. Opt. 28, 5091-5094 (1989).
    [CrossRef] [PubMed]

2004

H. Bartzsch, S. Lange, P. Frach, and K. Goedicke, 'Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering,' Surf. Coat. Technol. 180-181, 616-620 (2004).
[CrossRef]

2003

A. Ritz, 'TaTiOx layers prepared by magnetron sputtering from separate metal targets,' Surf. Coat. Technol. 174-175, 651-654 (2003).
[CrossRef]

2001

2000

T. U. Ryu, S. H. Hahn, S. W. Kim, and E. J. Kim, 'Optical, mechanical and thermal properties of MgF2-ZnS and MgF2-Ta2O5 composite thin films deposited by co-evaporation,' Opt. Eng. 39, 3207-3213 (2000).
[CrossRef]

1999

C. C. Lee, H. L. Chen, J. C. Hsu, and C. L. Tien, 'Interference coating base on synthesized silicon nitride,' Appl. Opt. 38, 2078-2082 (1999).
[CrossRef]

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, 'Microstructure and optical properties of amorphous TiO2-SiO2 composite films synthesized by helicon plasma sputtering,' Thin Solid Films 338, 105-109 (1999).
[CrossRef]

1998

1997

H. J. Cho, I. G. Yu, and C. K. Hwangbo, 'Inhomogeneous optical SiOxNy thin films prepared by ion assisted deposition,' Nucl. Instrum. Methods Phys. Res. B 121, 137-140 (1997).
[CrossRef]

1996

J. S. Chen, S. Chao, J. S. Kao, H. Niu, and C. H. Chen, 'Mixed films of TiO2-SiO2 deposited by double electron-beam coevaporation,' Appl. Opt. 37, 90-96 (1996).
[CrossRef]

1995

M. Cevro, 'Ion-beam sputtering of (Ta2O5)x-(SiO2)1-x composite thin films,' Thin Solid Films 258, 91-103 (1995).
[CrossRef]

1991

1990

1989

1988

1986

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
[CrossRef]

1976

J. C. Manifacier, J. Gasiot, and J. P. Fillard, 'A simple method for the determination of the optical constant n, k and the thickness of the weakly absorbing thin film,' J. Phys. E 9, 1002-1004 (1976).
[CrossRef]

Bartzsch, H.

H. Bartzsch, S. Lange, P. Frach, and K. Goedicke, 'Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering,' Surf. Coat. Technol. 180-181, 616-620 (2004).
[CrossRef]

Carniglia, C. K.

Carosella, C. A.

Cevro, M.

M. Cevro, 'Ion-beam sputtering of (Ta2O5)x-(SiO2)1-x composite thin films,' Thin Solid Films 258, 91-103 (1995).
[CrossRef]

Chang, C. K.

Chao, S.

Chen, C. H.

J. S. Chen, S. Chao, J. S. Kao, H. Niu, and C. H. Chen, 'Mixed films of TiO2-SiO2 deposited by double electron-beam coevaporation,' Appl. Opt. 37, 90-96 (1996).
[CrossRef]

Chen, H. L.

Chen, J. S.

J. S. Chen, S. Chao, J. S. Kao, H. Niu, and C. H. Chen, 'Mixed films of TiO2-SiO2 deposited by double electron-beam coevaporation,' Appl. Opt. 37, 90-96 (1996).
[CrossRef]

S. Chao, C. K. Chang, and J. S. Chen, 'TiO2-SiO2 mixed films prepared by the fast alternating sputter method,' Appl. Opt. 30, 3233-3237 (1991).
[CrossRef] [PubMed]

Cho, H. J.

H. J. Cho, I. G. Yu, and C. K. Hwangbo, 'Inhomogeneous optical SiOxNy thin films prepared by ion assisted deposition,' Nucl. Instrum. Methods Phys. Res. B 121, 137-140 (1997).
[CrossRef]

DeBar, J. I.

DeNatale, J. F.

Donovan, E. P.

Farabaugh, E. N.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
[CrossRef]

Feldman, A.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
[CrossRef]

Fillard, J. P.

J. C. Manifacier, J. Gasiot, and J. P. Fillard, 'A simple method for the determination of the optical constant n, k and the thickness of the weakly absorbing thin film,' J. Phys. E 9, 1002-1004 (1976).
[CrossRef]

Frach, P.

H. Bartzsch, S. Lange, P. Frach, and K. Goedicke, 'Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering,' Surf. Coat. Technol. 180-181, 616-620 (2004).
[CrossRef]

Gasiot, J.

J. C. Manifacier, J. Gasiot, and J. P. Fillard, 'A simple method for the determination of the optical constant n, k and the thickness of the weakly absorbing thin film,' J. Phys. E 9, 1002-1004 (1976).
[CrossRef]

Gluck, N. S.

Goedicke, K.

H. Bartzsch, S. Lange, P. Frach, and K. Goedicke, 'Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering,' Surf. Coat. Technol. 180-181, 616-620 (2004).
[CrossRef]

Granqvist, C. G.

Gunning, W. J.

Hahn, S. H.

T. U. Ryu, S. H. Hahn, S. W. Kim, and E. J. Kim, 'Optical, mechanical and thermal properties of MgF2-ZnS and MgF2-Ta2O5 composite thin films deposited by co-evaporation,' Opt. Eng. 39, 3207-3213 (2000).
[CrossRef]

Hall, R. L.

Haller, W. K.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
[CrossRef]

Hirai, T.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, 'Microstructure and optical properties of amorphous TiO2-SiO2 composite films synthesized by helicon plasma sputtering,' Thin Solid Films 338, 105-109 (1999).
[CrossRef]

Hsu, J. C.

Hubler, G. K.

Hwangbo, C. K.

H. J. Cho, I. G. Yu, and C. K. Hwangbo, 'Inhomogeneous optical SiOxNy thin films prepared by ion assisted deposition,' Nucl. Instrum. Methods Phys. Res. B 121, 137-140 (1997).
[CrossRef]

Kahn, A. D. F.

Kao, J. S.

J. S. Chen, S. Chao, J. S. Kao, H. Niu, and C. H. Chen, 'Mixed films of TiO2-SiO2 deposited by double electron-beam coevaporation,' Appl. Opt. 37, 90-96 (1996).
[CrossRef]

Katagiri, Y.

Kim, E. J.

T. U. Ryu, S. H. Hahn, S. W. Kim, and E. J. Kim, 'Optical, mechanical and thermal properties of MgF2-ZnS and MgF2-Ta2O5 composite thin films deposited by co-evaporation,' Opt. Eng. 39, 3207-3213 (2000).
[CrossRef]

Kim, S. W.

T. U. Ryu, S. H. Hahn, S. W. Kim, and E. J. Kim, 'Optical, mechanical and thermal properties of MgF2-ZnS and MgF2-Ta2O5 composite thin films deposited by co-evaporation,' Opt. Eng. 39, 3207-3213 (2000).
[CrossRef]

Kullman, L.

Lange, S.

H. Bartzsch, S. Lange, P. Frach, and K. Goedicke, 'Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering,' Surf. Coat. Technol. 180-181, 616-620 (2004).
[CrossRef]

Lee, C. C.

Manifacier, J. C.

J. C. Manifacier, J. Gasiot, and J. P. Fillard, 'A simple method for the determination of the optical constant n, k and the thickness of the weakly absorbing thin film,' J. Phys. E 9, 1002-1004 (1976).
[CrossRef]

Masumoto, H.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, 'Microstructure and optical properties of amorphous TiO2-SiO2 composite films synthesized by helicon plasma sputtering,' Thin Solid Films 338, 105-109 (1999).
[CrossRef]

Niu, H.

J. S. Chen, S. Chao, J. S. Kao, H. Niu, and C. H. Chen, 'Mixed films of TiO2-SiO2 deposited by double electron-beam coevaporation,' Appl. Opt. 37, 90-96 (1996).
[CrossRef]

Pond, B. J.

Raj, T.

Ritz, A.

A. Ritz, 'TaTiOx layers prepared by magnetron sputtering from separate metal targets,' Surf. Coat. Technol. 174-175, 651-654 (2003).
[CrossRef]

Rubin, M.

Ryu, T. U.

T. U. Ryu, S. H. Hahn, S. W. Kim, and E. J. Kim, 'Optical, mechanical and thermal properties of MgF2-ZnS and MgF2-Ta2O5 composite thin films deposited by co-evaporation,' Opt. Eng. 39, 3207-3213 (2000).
[CrossRef]

Sahoo, N. K.

Sanders, D. M.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
[CrossRef]

Sankur, H.

Shapiro, A. P.

Someno, Y.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, 'Microstructure and optical properties of amorphous TiO2-SiO2 composite films synthesized by helicon plasma sputtering,' Thin Solid Films 338, 105-109 (1999).
[CrossRef]

Southwell, W. H.

Stempniak, R. A.

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
[CrossRef]

Tien, C. L.

Ukita, H.

Van Vechten, D.

Veszelei, M.

von Rottkay, N.

Wang, W. H.

Wang, X.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, 'Microstructure and optical properties of amorphous TiO2-SiO2 composite films synthesized by helicon plasma sputtering,' Thin Solid Films 338, 105-109 (1999).
[CrossRef]

Woodberry, F. J.

Yu, I. G.

H. J. Cho, I. G. Yu, and C. K. Hwangbo, 'Inhomogeneous optical SiOxNy thin films prepared by ion assisted deposition,' Nucl. Instrum. Methods Phys. Res. B 121, 137-140 (1997).
[CrossRef]

Appl. Opt.

E. P. Donovan, D. Van Vechten, A. D. F. Kahn, C. A. Carosella, and G. K. Hubler, 'Near infrared rugate filter fabrication by ion beam assisted deposition of Si(1-x)Nx films,' Appl. Opt. 28, 2940-2944 (1989).
[CrossRef] [PubMed]

W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, and N. S. Gluck, 'Codeposition of continuous composition rugate filters,' Appl. Opt. 28, 2945-2948 (1989).
[CrossRef] [PubMed]

H. Sankur, W. J. Gunning, and J. F. DeNatale, 'Intrinsic stress and structural properties of mixed composition thin film,' Appl. Opt. 27, 1564-1567 (1988).
[CrossRef] [PubMed]

J. S. Chen, S. Chao, J. S. Kao, H. Niu, and C. H. Chen, 'Mixed films of TiO2-SiO2 deposited by double electron-beam coevaporation,' Appl. Opt. 37, 90-96 (1996).
[CrossRef]

N. K. Sahoo and A. P. Shapiro, 'Process-parameter-dependent optical and structural properties of ZrO2MgO mixed-composite films evaporated from the solid solution,' Appl. Opt. 37, 698-718 (1998).
[CrossRef]

B. J. Pond, J. I. DeBar, C. K. Carniglia, and T. Raj, 'Stress reduction in ion beam sputtered mixed oxide films,' Appl. Opt. 28, 2800-2805 (1989).
[CrossRef] [PubMed]

S. Chao, C. K. Chang, and J. S. Chen, 'TiO2-SiO2 mixed films prepared by the fast alternating sputter method,' Appl. Opt. 30, 3233-3237 (1991).
[CrossRef] [PubMed]

M. Veszelei, L. Kullman, C. G. Granqvist, N. von Rottkay, and M. Rubin, 'Optical constants of sputter-deposited Ti-Ce oxide and Zr-Ce oxide films,' Appl. Opt. 37, 5993-6001 (1998).
[CrossRef]

S. Chao, W. H. Wang, and C. C. Lee, 'Low-loss dielectric mirror with ion-beam-sputtered TiO2-SiO2 mixed films,' Appl. Opt. 40, 2177-2182 (2001).
[CrossRef]

Y. Katagiri and H. Ukita, 'Ion beam sputtered (SiO2)x(Si3N4)1-x antireflection coatings on laser facets produced using O2-N2 discharges,' Appl. Opt. 29, 5074-5079 (1990).
[CrossRef] [PubMed]

C. C. Lee, H. L. Chen, J. C. Hsu, and C. L. Tien, 'Interference coating base on synthesized silicon nitride,' Appl. Opt. 38, 2078-2082 (1999).
[CrossRef]

W. H. Southwell, 'Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,' Appl. Opt. 28, 2949-2951 (1989).
[CrossRef] [PubMed]

W. H. Southwell, 'Using apodization functions to reduce sidelobes in rugate filters,' Appl. Opt. 28, 5091-5094 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Phys. E

J. C. Manifacier, J. Gasiot, and J. P. Fillard, 'A simple method for the determination of the optical constant n, k and the thickness of the weakly absorbing thin film,' J. Phys. E 9, 1002-1004 (1976).
[CrossRef]

J. Vac. Sci. Technol. A

A. Feldman, E. N. Farabaugh, W. K. Haller, D. M. Sanders, and R. A. Stempniak, 'Modifying structure and properties of optical films by coevaporation,' J. Vac. Sci. Technol. A 6, 2969-2974 (1986).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B

H. J. Cho, I. G. Yu, and C. K. Hwangbo, 'Inhomogeneous optical SiOxNy thin films prepared by ion assisted deposition,' Nucl. Instrum. Methods Phys. Res. B 121, 137-140 (1997).
[CrossRef]

Opt. Eng.

T. U. Ryu, S. H. Hahn, S. W. Kim, and E. J. Kim, 'Optical, mechanical and thermal properties of MgF2-ZnS and MgF2-Ta2O5 composite thin films deposited by co-evaporation,' Opt. Eng. 39, 3207-3213 (2000).
[CrossRef]

Surf. Coat. Technol.

H. Bartzsch, S. Lange, P. Frach, and K. Goedicke, 'Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering,' Surf. Coat. Technol. 180-181, 616-620 (2004).
[CrossRef]

A. Ritz, 'TaTiOx layers prepared by magnetron sputtering from separate metal targets,' Surf. Coat. Technol. 174-175, 651-654 (2003).
[CrossRef]

Thin Solid Films

M. Cevro, 'Ion-beam sputtering of (Ta2O5)x-(SiO2)1-x composite thin films,' Thin Solid Films 258, 91-103 (1995).
[CrossRef]

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, 'Microstructure and optical properties of amorphous TiO2-SiO2 composite films synthesized by helicon plasma sputtering,' Thin Solid Films 338, 105-109 (1999).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Schematic of the rf ion-beam-sputtering deposition system.

Fig. 2
Fig. 2

Schematic of the Ta–SiO 2 target. The gray area is the sputtering area.

Fig. 3
Fig. 3

(a) IAD, heated to 130 °C, and with a beam current of 130 mA ; (b) no IAD, heated to 130 °C, and with a beam current of 130 mA ; (c) no IAD, no heating, and with a beam current 110 mA ; (d) no IAD, no heating, and with a beam current of 130 mA .

Fig. 4
Fig. 4

Substrate temperature versus target position during deposition.

Fig. 5
Fig. 5

Spectrum of the composite films with different concentrations of SiO 2 : (a) n = 1.558 ( Ta , 13 % ; SiO 2 , 87 % ) , (b) n = 1.925 ( Ta , 63 % ; SiO 2 , 37 % ) , (c) n = 2.206 ( Ta , 100 % ; SiO 2 , 0 % ) , (d) substrate.

Fig. 6
Fig. 6

Spectrum of the composite films with different SiO 2 targets: (a) hot pressed target, (b) fused-silica target, (c) substrate.

Fig. 7
Fig. 7

Refractive index versus wavelength at different target positions.

Fig. 8
Fig. 8

Extinction coefficient versus wavelength at different target positions.

Fig. 9
Fig. 9

Surface roughness with different Ta 2 O 5 percentages.

Fig. 10
Fig. 10

Deposition rate versus target position.

Fig. 11
Fig. 11

Experimental data and curve fitting of position versus refractive index.

Fig. 12
Fig. 12

Design and measurement of a rugate filter with a hot pressed Si O 2 target.

Fig. 13
Fig. 13

Refractive-index profile of a rugate filter.

Fig. 14
Fig. 14

Design and measurement of a rugate filter with a fused-silica target.

Equations (5)

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

Rate ( y ) = - 0.6696 + 1.5882 y - 0.3927 y 2 + 0.0434 y 3 - 0.0017 y 4 ,
y ( n ) = - 11035 + 30426.34 n - 33496.72 n 2 + 18411.43 n 3 - 5051.59 n 4 + 553.48 n 5 .
n ( z ) = n a + n p 2 f ( z ) sin ( 4 π λ 0 z + ϕ ) ,
f ( z ) = 10 z 3 - 15 z 4 + 6 z 5 ,
n ( z ) = n s + [ n ( z ) - n s ] ( 10 z 3 - 15 z 4 + 6 z 5 ) .

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