Abstract

Optimized chirped mirrors may perform suboptimally, or completely fail to satisfy specifications, when manufacturing errors are encountered. We present a robust optimization method for designing these dispersion-compensating mirror systems that are used in ultrashort pulse lasers. Possible implementation errors in layer thickness are taken into account within an uncertainty set. The algorithm identifies worst-case scenarios with respect to reflectivity as well as group delay. An iterative update improves the robustness and warrants a high manufacturing yield, even when the encountered errors are larger than anticipated.

© 2008 Optical Society of America

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References

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  1. R. Szipöcs, K. Ferencz, C. Spielmann, and F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201-203 (1994).
    [CrossRef] [PubMed]
  2. F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, and T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831-833 (1997).
    [CrossRef] [PubMed]
  3. N. Matuschek, F. X. Kärtner, and U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197-208 (1998).
    [CrossRef]
  4. F. X. Kaertner, U. Morgner, T. R. Schibli, E. P. Ippen, J. G. Fujimoto, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 18, 882-885 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).
  7. V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. 41, 6514-6520 (2002).
    [CrossRef] [PubMed]
  8. A. Ben-Tal and A. Nemirovski, “Robust convex optimization,” Math. Oper. Res. 23, 769-806 (1998).
    [CrossRef]
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    [CrossRef]
  10. D. Bertsimas and M. Sim, “Robust discrete optimization and network flows,” Math. Program. 98, 49-71 (2003).
    [CrossRef]
  11. D. Bertsimas and M. Sim, “Tractable approximations to robust conic optimization problems,” Math. Program. 107, 5-36(2006).
    [CrossRef]
  12. D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
    [CrossRef]
  13. J. A. Kong, Electromagnetic Wave Theory (EMW, 2000).
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    [CrossRef] [PubMed]
  15. J. R. Birge and F. X. Kärtner, “Efficient optimization of multilayer coatings for ultrafast optics using analytic gradients of dispersion,” Appl. Opt. 46, 2656-2662 (2007).
    [CrossRef] [PubMed]
  16. O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, “Self-referenced 200 mHz octave-spanning ti:sapphire laser with 50 attosecond carrier-envelope phase jitter,” Opt. Express 13, 5163-5169 (2005).
    [CrossRef] [PubMed]
  17. D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res. (to be published).
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    [CrossRef] [PubMed]
  19. ILOG CPLEX, “High-performance software dor mathematical programming and optimization,” http://www.ilog.com/products/cplex (2007).
  20. A. Tikhonravov, M. Trubetskov, and G. DeBell, “Applictions of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493-5508 (1996).
    [CrossRef] [PubMed]

2007 (3)

V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[CrossRef]

J. R. Birge and F. X. Kärtner, “Efficient optimization of multilayer coatings for ultrafast optics using analytic gradients of dispersion,” Appl. Opt. 46, 2656-2662 (2007).
[CrossRef] [PubMed]

2006 (2)

J. R. Birge and F. X. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. 45, 1478-1483 (2006).
[CrossRef] [PubMed]

D. Bertsimas and M. Sim, “Tractable approximations to robust conic optimization problems,” Math. Program. 107, 5-36(2006).
[CrossRef]

2005 (1)

2003 (1)

D. Bertsimas and M. Sim, “Robust discrete optimization and network flows,” Math. Program. 98, 49-71 (2003).
[CrossRef]

2002 (2)

V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. 41, 6514-6520 (2002).
[CrossRef] [PubMed]

A. Ben-Tal and A. Nemirovski, “Robust optimization--methodology and applications,” Math. Program. 92, 453-480(2002).
[CrossRef]

2001 (1)

1998 (2)

N. Matuschek, F. X. Kärtner, and U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197-208 (1998).
[CrossRef]

A. Ben-Tal and A. Nemirovski, “Robust convex optimization,” Math. Oper. Res. 23, 769-806 (1998).
[CrossRef]

1997 (1)

1996 (2)

1994 (1)

1992 (1)

Angelow, G.

Apolonski, A.

V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).

Ben-Tal, A.

A. Ben-Tal and A. Nemirovski, “Robust optimization--methodology and applications,” Math. Program. 92, 453-480(2002).
[CrossRef]

A. Ben-Tal and A. Nemirovski, “Robust convex optimization,” Math. Oper. Res. 23, 769-806 (1998).
[CrossRef]

Bertsimas, D.

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[CrossRef]

D. Bertsimas and M. Sim, “Tractable approximations to robust conic optimization problems,” Math. Program. 107, 5-36(2006).
[CrossRef]

D. Bertsimas and M. Sim, “Robust discrete optimization and network flows,” Math. Program. 98, 49-71 (2003).
[CrossRef]

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res. (to be published).

Birge, J. R.

DeBell, G.

Dobrowolski, J.

Ell, R.

Ferencz, K.

Fujimoto, J. G.

Haus, H. A.

Heine, C.

Ippen, E. P.

Kaertner, F. X.

Kärtner, F. X.

Keller, U.

Kim, J.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (EMW, 2000).

Krausz, F.

V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).

R. Szipöcs, K. Ferencz, C. Spielmann, and F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201-203 (1994).
[CrossRef] [PubMed]

Matos, L.

Matuschek, N.

Morf, R.

Morgner, U.

Mücke, O. D.

Naumov, S.

V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).

Nemirovski, A.

A. Ben-Tal and A. Nemirovski, “Robust optimization--methodology and applications,” Math. Program. 92, 453-480(2002).
[CrossRef]

A. Ben-Tal and A. Nemirovski, “Robust convex optimization,” Math. Oper. Res. 23, 769-806 (1998).
[CrossRef]

Nohadani, O.

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[CrossRef]

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res. (to be published).

Pervak, V.

V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).

Scheuer, V.

Schibli, T.

Schibli, T. R.

Sim, M.

D. Bertsimas and M. Sim, “Tractable approximations to robust conic optimization problems,” Math. Program. 107, 5-36(2006).
[CrossRef]

D. Bertsimas and M. Sim, “Robust discrete optimization and network flows,” Math. Program. 98, 49-71 (2003).
[CrossRef]

Spielmann, C.

Sullivan, B.

Szipöcs, R.

Tempea, G.

Teo, K. M.

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[CrossRef]

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res. (to be published).

Tikhonravov, A.

V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).

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

Tilsch, M.

Trubetskov, M.

V. Pervak, A. Tikhonravov, M. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Opt. 87, 5-12 (2007).

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

Tschudi, T.

Verly, P.

Winter, A.

Yakovlev, V.

Appl. Opt. (7)

IEEE J. Sel. Top. Quantum Electron. (1)

N. Matuschek, F. X. Kärtner, and U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197-208 (1998).
[CrossRef]

J. Appl. Phys. (1)

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[CrossRef]

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

Math. Oper. Res. (1)

A. Ben-Tal and A. Nemirovski, “Robust convex optimization,” Math. Oper. Res. 23, 769-806 (1998).
[CrossRef]

Math. Program. (3)

A. Ben-Tal and A. Nemirovski, “Robust optimization--methodology and applications,” Math. Program. 92, 453-480(2002).
[CrossRef]

D. Bertsimas and M. Sim, “Robust discrete optimization and network flows,” Math. Program. 98, 49-71 (2003).
[CrossRef]

D. Bertsimas and M. Sim, “Tractable approximations to robust conic optimization problems,” Math. Program. 107, 5-36(2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Other (3)

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res. (to be published).

J. A. Kong, Electromagnetic Wave Theory (EMW, 2000).

ILOG CPLEX, “High-performance software dor mathematical programming and optimization,” http://www.ilog.com/products/cplex (2007).

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Figures (6)

Fig. 1
Fig. 1

Two-dimensional illustration of the neighborhood. For a design x ^ , all possible implementation errors Δ x U are contained in the shaded circle. The bold arrow d shows a possible descent direction and thin arrows Δ x i * represent worst errors.

Fig. 2
Fig. 2

Robust optimization algorithm improves (left) the worst-case cost in the neighborhood of the current design. Discoveries of new bad neighbors cause the small peaks. (Right) The price of robustness is an increase in the nominal cost.

Fig. 3
Fig. 3

Reflectivity and group delay for each chirped mirror in the pair: (left) nominally optimal design; (right) robustly optimal configuration.

Fig. 4
Fig. 4

Layer thicknesses of nominal optimum and robust optimum of the mirror pair.

Fig. 5
Fig. 5

Comparison of worst-case cost and worst-case GD cost of two designs, the nominal and robust optimum, for increasing size of possible perturbations or errors.

Fig. 6
Fig. 6

Comparison of the nominal and robust design: mean and ninety-fifth percentile of the cost distribution of 10 6 randomly sampled designs for varying perturbation sizes.

Equations (9)

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f ( x ) = k w r ( λ k ) [ R ( λ k ; x ) 1 ] 4 + k w d ( λ k ) [ τ g ( λ k ; x ) τ ^ g ( λ k ) + τ 0 ( x ) ] 2 ,
U { Δ x R n | Δ x 2 Γ } ,
g ( x ) max Δ x U f ( x + Δ x ) .
min x g ( x ) min x max Δ x U f ( x + Δ x ) .
N { x | x x ^ 2 Γ } .
Δ x j = Γ N / 2 k = 1 N / 2 cos ( 2 π k j N + ϕ k ) ,
Δ x 2 2 = k = 1 N | Δ x k | 2 = Γ 2 .
U * ( x ^ ) { Δ x * | Δ x * = arg max Δ x U f ( x ^ + Δ x ) } .
minimize d , β β subject   to d 2 1 d Δ x * β Δ x * U * ( x ^ ) β ϵ ,

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