Abstract

Modern design approaches enable one to construct a series of theoretical designs with excellent spectral properties for almost all optical coating design problems. Selection of a practical optimal design among a variety of possible theoretical designs becomes a key issue. We demonstrate how preproduction estimations of expected production yields can be used for selection of a practical optimal design. The question of reliability of such estimations is also addressed.

© 2011 Optical Society of America

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References

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  1. P. W. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
    [CrossRef]
  2. A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestnik MGU Ser. Fiz.-Astr. 23, 91–93(1982).
  3. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
    [CrossRef] [PubMed]
  4. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46, 704–710 (2007).
    [CrossRef] [PubMed]
  5. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C124–C130 (2008).
    [CrossRef] [PubMed]
  6. A. Thelen, “Design of a hot mirror: contest results,” Appl. Opt. 35, 4966–4977 (1996).
    [CrossRef] [PubMed]
  7. P. W. Baumeister, “Evaluation of the solutions for the 1998 design problems,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 216–222.
  8. A. Thelen, A. V. Tikhonravov, M. K. Trubetskov, M. Tilsch, and U. Brauneck, “Topical meeting on optical interference coatings (OIC ’2001): design contest results,” Appl. Opt. 41, 3022–3038 (2002).
    [CrossRef] [PubMed]
  9. M. Tilsch, K. Hendrix, and P. Verly, “Optical interference coating design contest 2004,” Appl. Opt. 45, 1544–1554 (2006).
    [CrossRef] [PubMed]
  10. M. Tilsch and K. Hendrix, “Optical interference coatings design contest 2007: triple bandpass filter and nonpolarizing beam splitter,” Appl. Opt. 47, C55–C69 (2008).
    [CrossRef] [PubMed]
  11. K. D. Hendrix and J. Oliver, “2010 OSA topical meeting on optical interference coatings: Design problem,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuB1.
  12. J. A. Dobrowolski, “Comprison of the Fourier transform and flip-flop thin-film synthesis methods,” Appl. Opt. 25, 1966–1972 (1986).
    [CrossRef] [PubMed]
  13. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, “Key role of the coating total optical thickness in solving design problems,” Proc. SPIE Int. Soc. Opt. Eng. 5250, 312–321 (2004).
  14. B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. I. Theoretical description,” Appl. Opt. 31, 3821–3835 (1992).
    [CrossRef] [PubMed]
  15. B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360(1993).
    [CrossRef] [PubMed]
  16. C. Clark and H. A. Macleod, “Errors and tolerances in optical coatings,” in 40th Annual Technical Conference Proceedings (Society of Vacuum Coaters, Washington DC, 1997), pp. 274–279.
  17. A. V. Tikhonravov and M. K. Trubetskov, “Computational manufacturing as a bridge between design and production,” Appl. Opt. 44, 6877–6884 (2005).
    [CrossRef] [PubMed]
  18. H. Ehlers, S. Schlichting, C. Schmitz, and D. Ristau, “Hybrid process control for precision optics enhanced by computational manufacturing,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuC6.
  19. K. Friedrich, S. Wilbrandt, O. Stenzel, N. Kaiser, and K. H. Hoffmann, “Computational manufacturing of optical interference coatings: method, simulation results, and comparison with experiment,” Appl. Opt. 49, 3150–3162 (2010).
    [CrossRef] [PubMed]
  20. A. V. Tikhonravov, M. K. Trubetskov, and V. Pervak, “Estimations of production yields for choosing the best practical design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA6.
  21. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Computational manufacturing experiments for choosing optimal design and monitoring strategy,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA5.
  22. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “On the reliability of computational estimations used for choosing the most manufacturable design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA3.
  23. H. Jeffreys, Theory of Probability3rd ed. (Clarendon Press, 1961).
  24. H. A. Macleod, Thin Film Optical Filters3rd ed. (Institute of Physics, 2001).
    [CrossRef]
  25. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Statistical approach to choosing a strategy of monochromatic monitoring of optical coating production,” Appl. Opt. 45, 7863–7870 (2006).
    [CrossRef] [PubMed]
  26. S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47, C49–C54(2008).
    [CrossRef] [PubMed]
  27. H. Macleod, “Monitoring of optical coatings,” Appl. Opt. 20, 82–89 (1981).
    [CrossRef] [PubMed]

2010 (1)

2008 (3)

2007 (1)

2006 (2)

2005 (1)

2004 (1)

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, “Key role of the coating total optical thickness in solving design problems,” Proc. SPIE Int. Soc. Opt. Eng. 5250, 312–321 (2004).

2002 (1)

1996 (2)

1993 (1)

1992 (1)

1986 (1)

1982 (1)

A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestnik MGU Ser. Fiz.-Astr. 23, 91–93(1982).

1981 (1)

1958 (1)

Amotchkina, T. V.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C124–C130 (2008).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Statistical approach to choosing a strategy of monochromatic monitoring of optical coating production,” Appl. Opt. 45, 7863–7870 (2006).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, “Key role of the coating total optical thickness in solving design problems,” Proc. SPIE Int. Soc. Opt. Eng. 5250, 312–321 (2004).

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Computational manufacturing experiments for choosing optimal design and monitoring strategy,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA5.

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “On the reliability of computational estimations used for choosing the most manufacturable design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA3.

Baumeister, P. W.

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

P. W. Baumeister, “Evaluation of the solutions for the 1998 design problems,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 216–222.

Brauneck, U.

Clark, C.

C. Clark and H. A. Macleod, “Errors and tolerances in optical coatings,” in 40th Annual Technical Conference Proceedings (Society of Vacuum Coaters, Washington DC, 1997), pp. 274–279.

DeBell, G. W.

Dobrowolski, J. A.

Ehlers, H.

H. Ehlers, S. Schlichting, C. Schmitz, and D. Ristau, “Hybrid process control for precision optics enhanced by computational manufacturing,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuC6.

Friedrich, K.

Hendrix, K.

Hendrix, K. D.

K. D. Hendrix and J. Oliver, “2010 OSA topical meeting on optical interference coatings: Design problem,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuB1.

Hoffmann, K. H.

Jeffreys, H.

H. Jeffreys, Theory of Probability3rd ed. (Clarendon Press, 1961).

Kaiser, N.

Kokarev, M. A.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, “Key role of the coating total optical thickness in solving design problems,” Proc. SPIE Int. Soc. Opt. Eng. 5250, 312–321 (2004).

Macleod, H.

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters3rd ed. (Institute of Physics, 2001).
[CrossRef]

C. Clark and H. A. Macleod, “Errors and tolerances in optical coatings,” in 40th Annual Technical Conference Proceedings (Society of Vacuum Coaters, Washington DC, 1997), pp. 274–279.

Oliver, J.

K. D. Hendrix and J. Oliver, “2010 OSA topical meeting on optical interference coatings: Design problem,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuB1.

Pervak, V.

A. V. Tikhonravov, M. K. Trubetskov, and V. Pervak, “Estimations of production yields for choosing the best practical design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA6.

Ristau, D.

H. Ehlers, S. Schlichting, C. Schmitz, and D. Ristau, “Hybrid process control for precision optics enhanced by computational manufacturing,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuC6.

Schlichting, S.

H. Ehlers, S. Schlichting, C. Schmitz, and D. Ristau, “Hybrid process control for precision optics enhanced by computational manufacturing,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuC6.

Schmitz, C.

H. Ehlers, S. Schlichting, C. Schmitz, and D. Ristau, “Hybrid process control for precision optics enhanced by computational manufacturing,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuC6.

Stenzel, O.

Sullivan, B. T.

Thelen, A.

Tikhonravov, A. V.

S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47, C49–C54(2008).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C124–C130 (2008).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46, 704–710 (2007).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Statistical approach to choosing a strategy of monochromatic monitoring of optical coating production,” Appl. Opt. 45, 7863–7870 (2006).
[CrossRef] [PubMed]

A. V. Tikhonravov and M. K. Trubetskov, “Computational manufacturing as a bridge between design and production,” Appl. Opt. 44, 6877–6884 (2005).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, “Key role of the coating total optical thickness in solving design problems,” Proc. SPIE Int. Soc. Opt. Eng. 5250, 312–321 (2004).

A. Thelen, A. V. Tikhonravov, M. K. Trubetskov, M. Tilsch, and U. Brauneck, “Topical meeting on optical interference coatings (OIC ’2001): design contest results,” Appl. Opt. 41, 3022–3038 (2002).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestnik MGU Ser. Fiz.-Astr. 23, 91–93(1982).

A. V. Tikhonravov, M. K. Trubetskov, and V. Pervak, “Estimations of production yields for choosing the best practical design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA6.

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Computational manufacturing experiments for choosing optimal design and monitoring strategy,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA5.

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “On the reliability of computational estimations used for choosing the most manufacturable design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA3.

Tilsch, M.

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C124–C130 (2008).
[CrossRef] [PubMed]

S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47, C49–C54(2008).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46, 704–710 (2007).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Statistical approach to choosing a strategy of monochromatic monitoring of optical coating production,” Appl. Opt. 45, 7863–7870 (2006).
[CrossRef] [PubMed]

A. V. Tikhonravov and M. K. Trubetskov, “Computational manufacturing as a bridge between design and production,” Appl. Opt. 44, 6877–6884 (2005).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, “Key role of the coating total optical thickness in solving design problems,” Proc. SPIE Int. Soc. Opt. Eng. 5250, 312–321 (2004).

A. Thelen, A. V. Tikhonravov, M. K. Trubetskov, M. Tilsch, and U. Brauneck, “Topical meeting on optical interference coatings (OIC ’2001): design contest results,” Appl. Opt. 41, 3022–3038 (2002).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “On the reliability of computational estimations used for choosing the most manufacturable design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA3.

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Computational manufacturing experiments for choosing optimal design and monitoring strategy,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA5.

A. V. Tikhonravov, M. K. Trubetskov, and V. Pervak, “Estimations of production yields for choosing the best practical design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA6.

Verly, P.

Wilbrandt, S.

Appl. Opt. (15)

H. Macleod, “Monitoring of optical coatings,” Appl. Opt. 20, 82–89 (1981).
[CrossRef] [PubMed]

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

B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. I. Theoretical description,” Appl. Opt. 31, 3821–3835 (1992).
[CrossRef] [PubMed]

B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360(1993).
[CrossRef] [PubMed]

A. Thelen, “Design of a hot mirror: contest results,” Appl. Opt. 35, 4966–4977 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
[CrossRef] [PubMed]

A. Thelen, A. V. Tikhonravov, M. K. Trubetskov, M. Tilsch, and U. Brauneck, “Topical meeting on optical interference coatings (OIC ’2001): design contest results,” Appl. Opt. 41, 3022–3038 (2002).
[CrossRef] [PubMed]

A. V. Tikhonravov and M. K. Trubetskov, “Computational manufacturing as a bridge between design and production,” Appl. Opt. 44, 6877–6884 (2005).
[CrossRef] [PubMed]

M. Tilsch, K. Hendrix, and P. Verly, “Optical interference coating design contest 2004,” Appl. Opt. 45, 1544–1554 (2006).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Statistical approach to choosing a strategy of monochromatic monitoring of optical coating production,” Appl. Opt. 45, 7863–7870 (2006).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Optical coating design approaches based on the needle optimization technique,” Appl. Opt. 46, 704–710 (2007).
[CrossRef] [PubMed]

S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47, C49–C54(2008).
[CrossRef] [PubMed]

M. Tilsch and K. Hendrix, “Optical interference coatings design contest 2007: triple bandpass filter and nonpolarizing beam splitter,” Appl. Opt. 47, C55–C69 (2008).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C124–C130 (2008).
[CrossRef] [PubMed]

K. Friedrich, S. Wilbrandt, O. Stenzel, N. Kaiser, and K. H. Hoffmann, “Computational manufacturing of optical interference coatings: method, simulation results, and comparison with experiment,” Appl. Opt. 49, 3150–3162 (2010).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Proc. SPIE Int. Soc. Opt. Eng. (1)

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, “Key role of the coating total optical thickness in solving design problems,” Proc. SPIE Int. Soc. Opt. Eng. 5250, 312–321 (2004).

Vestnik MGU Ser. Fiz.-Astr. (1)

A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestnik MGU Ser. Fiz.-Astr. 23, 91–93(1982).

Other (9)

P. W. Baumeister, “Evaluation of the solutions for the 1998 design problems,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 216–222.

K. D. Hendrix and J. Oliver, “2010 OSA topical meeting on optical interference coatings: Design problem,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuB1.

C. Clark and H. A. Macleod, “Errors and tolerances in optical coatings,” in 40th Annual Technical Conference Proceedings (Society of Vacuum Coaters, Washington DC, 1997), pp. 274–279.

A. V. Tikhonravov, M. K. Trubetskov, and V. Pervak, “Estimations of production yields for choosing the best practical design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA6.

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Computational manufacturing experiments for choosing optimal design and monitoring strategy,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA5.

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “On the reliability of computational estimations used for choosing the most manufacturable design,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuA3.

H. Jeffreys, Theory of Probability3rd ed. (Clarendon Press, 1961).

H. A. Macleod, Thin Film Optical Filters3rd ed. (Institute of Physics, 2001).
[CrossRef]

H. Ehlers, S. Schlichting, C. Schmitz, and D. Ristau, “Hybrid process control for precision optics enhanced by computational manufacturing,” in Optical Interference Coatings, 2010 OSA Technical Digest Series (Optical Society of America, 2010), paper TuC6.

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Figures (4)

Fig. 1
Fig. 1

Dependencies of confidence interval width on Y : N = 1000 (solid curve), N = 100 (dotted curve), and N = 10 (dashed curve).

Fig. 2
Fig. 2

Refractive index profiles of four designs with ramp transmittance in the spectral region from 400 to 800 nm ; reference wavelength for refractive index values is 600 nm .

Fig. 3
Fig. 3

Transmittance of the Ramp 21 design and the corridor of ± 2 % deviations from the target transmittance.

Fig. 4
Fig. 4

Measured transmittance of the 21 layer coating (dashed curve), theoretical transmittance of the Ramp 21 design (solid curve), and the corridor of ± 2 % deviations from the theoretical transmittance (gray area).

Tables (3)

Tables Icon

Table 1 Estimated Production Yields and Respective Confidence Intervals for Different Levels of Allowed Transmittance Deviations from the Target Ramp Transmittance: Experiments with Broadband Optical Monitoring, Numbers of Experiments are Equal to 100 if Y < 85 % and to 1000 if Y > 85 %

Tables Icon

Table 2 Estimated Production Yields and Respective Confidence Intervals for Different Levels of Allowed Transmittance Deviations from the Target Ramp Transmittance: Experiments with Monochromatic Optical Monitoring Strategy MSW, Numbers of Experiments are Equal to 1000

Tables Icon

Table 3 Estimated Production Yields and Respective Confidence Intervals for Different Levels of Allowed Transmittance Deviations from the Target Ramp Transmittance: Experiments with Noncorrelated Random Errors in Layer Thicknesses, Numbers of Experiments are Equal to 5000

Equations (3)

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

Y = M / N .
p 2 ( 1 + z α 2 N ) p ( 2 Y + z α 2 N ) + Y 2 = 0 ,
2 erf ( z α ) = α ,

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