Abstract

Several modern optical coating designs tools are discussed in the frame of a new design paradigm proposing the search not for a formally optimal solution with the lowest possible merit function value but for the most practical solution that takes into account additional feasibility demands. Considered design tools include a stochastic optimization procedure that takes into account upper and lower constraints for layer optical thicknesses. This procedure allows one to obtain multiple solutions to a design problem, which presents additional opportunities for choosing a practically optimal design. Two special design techniques involving integer optimization also take into account additional demands. The first one is aimed at designing multicavity narrow bandpass filters with quarter wave or multiple quarter wave layer optical thicknesses. It enables obtaining bandpass filters with extremely steep transmittance slopes, bandwidths of several tens of nanometers, and very small ripples in transmission zones. The second technique is aimed at covering design problems that have been traditionally solved using the theory of equivalent layers. One more technique considered in this paper is aimed at reducing the influence of noncorrelated thickness errors on design spectral characteristics.

© 2012 Optical Society of America

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    [CrossRef]

2012

2011

2010

2008

2007

2006

2005

2004

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 5250, 312–321 (2004).
[CrossRef]

2003

2002

2001

1997

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, “Phase properties of multilayers,” Appl. Opt. 36, 4382–4392 (1997).
[CrossRef]

J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photon. News 8(6), 24–33 (1997).
[CrossRef]

1996

1995

1993

A. N. Tikhonov, A. V. Tikhonravov, and M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).
[CrossRef]

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

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

1992

1990

1989

1986

W. H. Southwell, W. J. Gunning, and R. L. Hall, “Narrow-bandpass filter using partitioned cavities,” Proc. SPIE 678, 177–184 (1986).

1985

1978

1972

H. A. Macleod, “Turning value monitoring of narrow-band all-dielectric thin film optical filters,” Opt. Acta 19, 1–28 (1972).
[CrossRef]

1965

Amotchkina, T. V.

Angelov, I. B.

Apolonski, A.

Badoil, B.

Baumeister, P.

Baumeister, P. W.

Bertsimas, D. J.

Birge, J. R.

Boos, M.

A. Zoeller, M. Boos, R. Goetzelmann, H. Hagedorn, and W. Klug, “Substantial progress in optical monitoring by intermittent measurement technique,” Proc. SPIE 5963, 105–113 (2005).
[CrossRef]

Bovard, B. G.

Brons, J.

Burton, R.

Cathelinaud, M.

Chen, S.-H.

Cheng, X.

Chun, B.

DeBell, G. W.

Dobrowolski, J. A.

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]

X. Cheng, B. Fan, J. A. Dobrowolski, L. Wang, and Z. Wang, “Gradient-index optical filter synthesis with controllable and predictable refractive index profiles,” Opt. Express 16, 2315–2321 (2008).
[CrossRef]

J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photon. News 8(6), 24–33 (1997).
[CrossRef]

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

L. Li, and J. A. Dobrowolski, “Computation speeds of different optical thin-film synthesis methods,” Appl. Opt. 31, 3790–3799 (1992).
[CrossRef]

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

P. G. Verly, and J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
[CrossRef]

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

P. G. Verly, J. A. Dobrowolski, W. Wild, and R. Burton, “Synthesis of high rejection filters with the Fourier transform method,” Appl. Opt. 28, 2864–2875 (1989).
[CrossRef]

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

J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,” Appl. Opt. 4, 937–946 (1965).
[CrossRef]

J. A. Dobrowolski, Optical Properties of Films and Coatings (McGraw-Hill, 1994), pp. 42.3–42.130.

Ehlers, H.

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C (2008).
[CrossRef]

Fabricius, H.

Fan, B.

Friedrich, K.

Furman, S.

S. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, 1992).

Goetzelmann, R.

A. Zoeller, M. Boos, R. Goetzelmann, H. Hagedorn, and W. Klug, “Substantial progress in optical monitoring by intermittent measurement technique,” Proc. SPIE 5963, 105–113 (2005).
[CrossRef]

Gunning, W. J.

W. H. Southwell, W. J. Gunning, and R. L. Hall, “Narrow-bandpass filter using partitioned cavities,” Proc. SPIE 678, 177–184 (1986).

Hagedorn, H.

A. Zoeller, M. Boos, R. Goetzelmann, H. Hagedorn, and W. Klug, “Substantial progress in optical monitoring by intermittent measurement technique,” Proc. SPIE 5963, 105–113 (2005).
[CrossRef]

Hall, R. L.

W. H. Southwell and R. L. Hall, “Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,” Appl. Opt. 28, 2949–2951 (1989).
[CrossRef]

W. H. Southwell, W. J. Gunning, and R. L. Hall, “Narrow-bandpass filter using partitioned cavities,” Proc. SPIE 678, 177–184 (1986).

Hoffmann, K. H.

Hwangbo, C. K.

Kaiser, N.

Kärtner, F. X.

Kasahara, I.

A. Tikhonravov, M. Trubetskov, and I. Kasahara, “Achievements and challenges in the design and production of high quality optical coatings,” IEICE Trans Electron E91-C, 1622–1629 (2008).
[CrossRef]

Kemp, R. A.

Kim, J. S.

Klug, W.

A. Zoeller, M. Boos, R. Goetzelmann, H. Hagedorn, and W. Klug, “Substantial progress in optical monitoring by intermittent measurement technique,” Proc. SPIE 5963, 105–113 (2005).
[CrossRef]

Klyuev, E. V.

A. V. Tikhonravov, M. K. Trubetskov, I. V. Kozlov, V. G. Zhupanov, and E. V. Klyuev, “Design and production of bandpass filters with steep transmittance slopes,” in Optical Interference Coatings Topical Meeting (Optical Society of America, 2010), paper MA6.

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 5250, 312–321 (2004).
[CrossRef]

Kozlov, I. V.

A. V. Tikhonravov, M. K. Trubetskov, I. V. Kozlov, V. G. Zhupanov, and E. V. Klyuev, “Design and production of bandpass filters with steep transmittance slopes,” in Optical Interference Coatings Topical Meeting (Optical Society of America, 2010), paper MA6.

Krausz, F.

Kruschwitz, J.

J. Kruschwitz, “Software tools speed optical thin-film design,” Laser Focus World 39, 157–166 (2003).

Kuo, C.-C.

Lappschies, M.

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C (2008).
[CrossRef]

Lau, K.

Lee, C.-C.

Lemarchand, F.

Lequime, M.

Li, L.

Lowe, D.

Macleod, H. A.

H. A. Macleod, “Turning value monitoring of narrow-band all-dielectric thin film optical filters,” Opt. Acta 19, 1–28 (1972).
[CrossRef]

H. A. Macleod, Thin Film Optical Filters, 4th ed. (CRC Press, 2010).

Nocedal, J.

J. Nocedal, and S. J. Wright, Numerical Optimization (Springer Verlag, 2006).

Nohadani, O.

Pervak, V.

Pistner, J.

Popov, K. V.

Pronin, O.

Pulker, H. K.

N. Kaiser and H. K. Pulker, Optical Interference Coatings (Springer-Verlag, 2003).

Razskazovskaya, O.

Ristau, D.

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C (2008).
[CrossRef]

Schallenberg, U. B.

Schlichting, S.

D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C (2008).
[CrossRef]

Schulz, U.

Southwell, W. H.

Stenzel, O.

Sullivan, B. T.

Tempea, G.

Thelen, A.

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

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988).

Tikhonov, A. N.

A. N. Tikhonov, A. V. Tikhonravov, and M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).
[CrossRef]

Tikhonravov, A.

A. Tikhonravov, M. Trubetskov, and I. Kasahara, “Achievements and challenges in the design and production of high quality optical coatings,” IEICE Trans Electron E91-C, 1622–1629 (2008).
[CrossRef]

Tikhonravov, A. V.

V. Pervak, O. Pronin, O. Razskazovskaya, J. Brons, I. B. Angelov, M. K. Trubetskov, A. V. Tikhonravov, and F. Krausz, “High-dispersive mirrors for high power applications,” Opt. Express 20, 4503–4508 (2012).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and V. Pervak, “Estimations of production yields for choosing of a practically optimal optical coating design,” Appl. Opt. 50, C141–C147 (2011).
[CrossRef]

V. Pervak, M. K. Trubetskov, and A. V. Tikhonravov, “Robust synthesis of dispersive mirrors,” Opt. Express 19, 2371–2380 (2011).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50, C111–C116 (2011).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Application of constrained optimization to the design of quasi-rugate optical coatings,” Appl. Opt. 47, 5103–5109 (2008).
[CrossRef]

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]

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]

A. V. Tikhonravov and M. K. Trubetskov, “Elimination of cumulative effect of thickness errors in monochromatic monitoring of optical coating production: theory,” Appl. Opt. 46, 2084–2090 (2007).
[CrossRef]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, J. Pistner, F. Krausz, and A. Apolonski, “Band filters: 2-material technology versus rugate,” Appl. Opt. 46, 1190–1193 (2007).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the effect of accumulation of thickness errors in optical coating production using broadband optical monitoring,” Appl. Opt. 45, 7026–7034 (2006).
[CrossRef]

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

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 5250, 312–321 (2004).
[CrossRef]

A. V. Tikhonravov and M. K. Trubetskov, “Automated design and sensitivity analysis of wavelength-division multiplexing filters,” Appl. Opt. 41, 3176–3182 (2002).
[CrossRef]

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, “Phase properties of multilayers,” Appl. Opt. 36, 4382–4392 (1997).
[CrossRef]

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]

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, “Optimal single-band normal incidence antireflection coatings,” Appl. Opt. 35, 644–658 (1996).
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A. V. Tikhonravov, M. K. Trubetskov, I. V. Kozlov, V. G. Zhupanov, and E. V. Klyuev, “Design and production of bandpass filters with steep transmittance slopes,” in Optical Interference Coatings Topical Meeting (Optical Society of America, 2010), paper MA6.

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M. K. Trubetskov and A. V. Tikhonravov, “Robust synthesis of multilayer coatings,” in Optical Interference Coatings Topical Meeting (Optical Society of America, 2010), paper TuA4.

Trubetskov, M.

A. Tikhonravov, M. Trubetskov, and I. Kasahara, “Achievements and challenges in the design and production of high quality optical coatings,” IEICE Trans Electron E91-C, 1622–1629 (2008).
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Trubetskov, M. K.

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[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and V. Pervak, “Estimations of production yields for choosing of a practically optimal optical coating design,” Appl. Opt. 50, C141–C147 (2011).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50, C111–C116 (2011).
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[CrossRef]

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A. V. Tikhonravov and M. K. Trubetskov, “Elimination of cumulative effect of thickness errors in monochromatic monitoring of optical coating production: theory,” Appl. Opt. 46, 2084–2090 (2007).
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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]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, J. Pistner, F. Krausz, and A. Apolonski, “Band filters: 2-material technology versus rugate,” Appl. Opt. 46, 1190–1193 (2007).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the effect of accumulation of thickness errors in optical coating production using broadband optical monitoring,” Appl. Opt. 45, 7026–7034 (2006).
[CrossRef]

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

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 5250, 312–321 (2004).
[CrossRef]

A. V. Tikhonravov and M. K. Trubetskov, “Automated design and sensitivity analysis of wavelength-division multiplexing filters,” Appl. Opt. 41, 3176–3182 (2002).
[CrossRef]

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

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]

A. N. Tikhonov, A. V. Tikhonravov, and M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, I. V. Kozlov, V. G. Zhupanov, and E. V. Klyuev, “Design and production of bandpass filters with steep transmittance slopes,” in Optical Interference Coatings Topical Meeting (Optical Society of America, 2010), paper MA6.

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[CrossRef]

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

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]

P. W. Baumeister, “Design of a wavelength-division multiplexing bandpass with quasi-chebyshev spectral shape,” Appl. Opt. 40, 1132–1137 (2001).
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A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the effect of accumulation of thickness errors in optical coating production using broadband optical monitoring,” Appl. Opt. 45, 7026–7034 (2006).
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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]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, J. Pistner, F. Krausz, and A. Apolonski, “Band filters: 2-material technology versus rugate,” Appl. Opt. 46, 1190–1193 (2007).
[CrossRef]

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A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Application of constrained optimization to the design of quasi-rugate optical coatings,” Appl. Opt. 47, 5103–5109 (2008).
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A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50, C111–C116 (2011).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and V. Pervak, “Estimations of production yields for choosing of a practically optimal optical coating design,” Appl. Opt. 50, C141–C147 (2011).
[CrossRef]

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

Comp. Maths. Math. Phys.

A. N. Tikhonov, A. V. Tikhonravov, and M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).
[CrossRef]

IEICE Trans Electron

A. Tikhonravov, M. Trubetskov, and I. Kasahara, “Achievements and challenges in the design and production of high quality optical coatings,” IEICE Trans Electron E91-C, 1622–1629 (2008).
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[CrossRef]

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[CrossRef]

Other

A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” in Vestnik MGU, Vol. 23 of Physics and Astronomy Series (1982), pp. 91–93.

H. A. Macleod, Thin Film Optical Filters, 4th ed. (CRC Press, 2010).

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M. K. Trubetskov and A. V. Tikhonravov, “Robust synthesis of multilayer coatings,” in Optical Interference Coatings Topical Meeting (Optical Society of America, 2010), paper TuA4.

A. V. Tikhonravov, M. K. Trubetskov, I. V. Kozlov, V. G. Zhupanov, and E. V. Klyuev, “Design and production of bandpass filters with steep transmittance slopes,” in Optical Interference Coatings Topical Meeting (Optical Society of America, 2010), paper MA6.

J. Nocedal, and S. J. Wright, Numerical Optimization (Springer Verlag, 2006).

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Figures (15)

Fig. 1.
Fig. 1.

Main parameters (N, MF, TOT) of the 50 intermediate three-line filter designs obtained by the gradual evolution procedure.

Fig. 2.
Fig. 2.

Spectral transmittance (left) and layer optical thicknesses (right) of the three-line filter design with N=75, MF=0.191, TOT=9687.5nm obtained by the gradual evolution procedure.

Fig. 3.
Fig. 3.

Layer optical thicknesses of the 10 three-line filter designs with N=51, MF and TOT values indicated in Table 1.

Fig. 4.
Fig. 4.

Preproduction estimation of thickness errors for the Designs 2 and 3 from Table 1 in the case of broadband monitoring in the 400–900 nm spectral range with 0.5% random noise in transmittance measurement data.

Fig. 5.
Fig. 5.

Layer optical thicknesses of the two hot mirror designs with N=40 and MF=0.865, TOT=9046.1nm (Design A) and MF=0.913, TOT=9062.3nm (Design B).

Fig. 6.
Fig. 6.

Spectral transmittance of the hot mirror design with N=40, MF=0.865, TOT=9046.1nm (Design A).

Fig. 7.
Fig. 7.

Preproduction estimation of thickness errors for the two hot mirror designs depicted in Fig. 5 in the case of broadband monitoring in the 400–900 nm spectral range with 0.5% random noise in transmittance measurement data.

Fig. 8.
Fig. 8.

Phase shifts on reflection for three quarter wave mirrors with different upper layers: standard HLH mirror (solid curve), mirror with 3H upper layer (dashed curve), mirror with A upper layer (dotted curve).

Fig. 9.
Fig. 9.

Spectral transmittance of the nine-cavity two-material bandpass filter (left) and layer optical thicknesses of this filter (right).

Fig. 10.
Fig. 10.

Spectral transmittance of the nine-cavity three-material bandpass filter (left) and layer optical thicknesses of this filter (right).

Fig. 11.
Fig. 11.

Spectral transmittance of the 68-layer two-material filter obtained by the needle optimization procedure (left) and layer optical thicknesses of this filter (right).

Fig. 12.
Fig. 12.

Spectral transmittance of the 68-layer two-material filter obtained by the design formula optimization (left) and layer optical thicknesses of this filter (right).

Fig. 13.
Fig. 13.

Reflectance of the wide band high reflecting mirror obtained by the design formula optimization (left) and layer optical thicknesses of this mirror (right).

Fig. 14.
Fig. 14.

Refractive index profiles of four AR coating designs: designs AR-6, AR-10, AR-14 are “classical” AR designs, design AR-robust is the design obtained using robust optimization.

Fig. 15.
Fig. 15.

Influence of thickness errors on the reflectances of the designs AR-robust (left) and AR-10 (right).

Tables (3)

Tables Icon

Table 1. Merit Function MF and Total Optical Thickness TOT Values of the 51-Layer Three-Line Filter Designs

Tables Icon

Table 2. Parameters of Five Wide Band Reflector Designs

Tables Icon

Table 3. Parameters of Four AR Designs

Equations (8)

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

MF={1Lj=1L[T(λj)T^(λj)ΔTj]2}1/2,
M1S1M2S2M3SqMq+1,
φ(λ)=φl(λ)+φr(λ)2πsλ0/λ,
c1Hc2Lc3H(a1La2Ha1La1Ha2La1H)ma1La2Ha1Ld1Hd2L.
a2=12a1.
m=10,a1=0.109,c1=0.060,c2=0.186,c3=0.176,d1=0.138,d2=0.399.
c1Hd1L(a1Hb1L)m1c2Hd2L(a2Hb2L)m2c3Hd3L(a3Hb3L)m3c4H.
d˜j=dj+Δj+(δj/100%)dj,

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