Abstract

We demonstrate that a highly efficient global optimization of chirped mirrors can be performed with the memetic algorithm. The inherently high sensitivity of chirped-mirror characteristics to manufacturing errors can be reduced significantly by means of the stochastic quasi-gradient algorithm. The applicability of these algorithms is not limited to chirped mirrors.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
  2. A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
    [CrossRef]
  3. A. Stingl, M. Lenzner, Ch. Spielmann, F. Krausz, R. Szipöcs, “Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett. 20, 602–604 (1995).
    [CrossRef] [PubMed]
  4. D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
    [CrossRef]
  5. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  10. Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).
  11. A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
    [CrossRef]
  12. A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
    [CrossRef] [PubMed]
  13. F. X. Kärtner, N. Matuscheck, T. Tschibili, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. G. Tempea, F. Krausz, “Dispersion management over one octave with tilted-front-interface chirped mirrors,” presented at the Twelfth International Conference on Ultrafast Phenomena, Charleston, S.C., 9–13 July 2000.
  18. L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.
  19. N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
    [CrossRef]
  20. G. Tempea, V. Yakovlev, B. Bacovic, F. Krausz, “Tilted-front-interface chirped mirrors,” J. Opt. Soc. Am. B 18, 1747–1750 (2001).
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    [CrossRef]
  27. H. A. Macleod, “Basic theory,” in Thin-Film Optical Filters, W. T. Welford, S. Laurenson, eds. (Institute of Physics Publishing, Bristol, UK, 2001), pp. 40–41.
  28. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Conjugate gradient methods in multidimensions,” in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., W. H. Press, ed. (Cambridge U. Press, Cambridge, 1995), pp. 420–425.
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    [CrossRef]

2001 (2)

2000 (4)

M. A. Vorontsov, G. W. Carhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
[CrossRef]

J. K. Ranka, R. S. Windeler, A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

1999 (2)

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
[CrossRef]

1998 (3)

G. Tempea, F. Krausz, Ch. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

1997 (4)

A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

R. Szipöcs, A. Köhazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135 (1997).
[CrossRef]

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

F. X. Kärtner, N. Matuscheck, T. Tschibili, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

1996 (1)

1995 (1)

1994 (1)

1992 (2)

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithms,” IEE Proc. J 139, 413–420 (1992).

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Automatic Control 37, 332–341 (1992).
[CrossRef]

1990 (1)

Angelow, G.

Apolonski, A.

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

Backus, S.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Bacovic, B.

Baltuška, A.

A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Boiko, A.

Boudet, T.

Brabec, T.

Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).

Carhart, G. W.

Cauwenberghs, G.

Chang, Z.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Chaton, P.

Chen, Y.

Cheng, Z.

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).

Cho, S. H.

Christov, I.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Cohen, M.

de Silvestri, S.

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

Dobrowolski, J. A.

Durfee, C.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Ell, R.

Ferencz, K.

G. Tempea, F. Krausz, Ch. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

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

Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Conjugate gradient methods in multidimensions,” in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., W. H. Press, ed. (Cambridge U. Press, Cambridge, 1995), pp. 420–425.

Fujimoto, J. G.

Gallmann, L.

N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
[CrossRef]

L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.

Gonon, G.

Hänsch, T. W.

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

Haus, H. A.

Heine, C.

Herault, L.

Holzwarth, R.

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

Ippen, E. P.

Jouanet, L.

Jung, I. D.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Kapteyn, H. C.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Kärtner, F. X.

Keller, P.

Keller, U.

N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
[CrossRef]

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

F. X. Kärtner, N. Matuscheck, T. Tschibili, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.

Kemp, R. A.

Kimmel, W. M.

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Köhazi-Kis, A.

R. Szipöcs, A. Köhazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135 (1997).
[CrossRef]

Krausz, F.

G. Tempea, V. Yakovlev, B. Bacovic, F. Krausz, “Tilted-front-interface chirped mirrors,” J. Opt. Soc. Am. B 18, 1747–1750 (2001).
[CrossRef]

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

G. Tempea, F. Krausz, Ch. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

A. Stingl, M. Lenzner, Ch. Spielmann, F. Krausz, R. Szipöcs, “Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett. 20, 602–604 (1995).
[CrossRef] [PubMed]

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

G. Tempea, F. Krausz, “Dispersion management over one octave with tilted-front-interface chirped mirrors,” presented at the Twelfth International Conference on Ultrafast Phenomena, Charleston, S.C., 9–13 July 2000.

Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).

Lederer, M. J.

Lenzner, M.

Luther-Davies, B.

Macleod, H. A.

H. A. Macleod, “Basic theory,” in Thin-Film Optical Filters, W. T. Welford, S. Laurenson, eds. (Institute of Physics Publishing, Bristol, UK, 2001), pp. 40–41.

Matuscheck, N.

Matuschek, N.

N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
[CrossRef]

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.

Michielssen, E.

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithms,” IEE Proc. J 139, 413–420 (1992).

Mittra, R.

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithms,” IEE Proc. J 139, 413–420 (1992).

Morf, R.

Morgner, U.

Morier-Genoud, F.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Murnane, M. M.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Nisoli, M.

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

O’Shea, P.

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Poppe, A.

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Conjugate gradient methods in multidimensions,” in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., W. H. Press, ed. (Cambridge U. Press, Cambridge, 1995), pp. 420–425.

Pshenichnikov, M. S.

A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Radcliffe, N. J.

N. J. Radcliffe, P. D. Surry, “Formal memetic algorithms,” in Evolutionary Computing, T. Fogarty, ed., proceedings of an AISB workshop (Springer-Verlag, Berlin, 1994), pp. 1–14.

Ranjithan, S.

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithms,” IEE Proc. J 139, 413–420 (1992).

Ranka, J. K.

J. K. Ranka, R. S. Windeler, A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Rundquist, A.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Sartania, S.

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

Scheuer, V.

Schreuer, V.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Spall, J. C.

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Automatic Control 37, 332–341 (1992).
[CrossRef]

Spielmann, Ch.

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

G. Tempea, F. Krausz, Ch. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

A. Stingl, M. Lenzner, Ch. Spielmann, F. Krausz, R. Szipöcs, “Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett. 20, 602–604 (1995).
[CrossRef] [PubMed]

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

Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).

Stagira, S.

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

Steinmeyer, G.

N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
[CrossRef]

L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.

Stentz, A. J.

J. K. Ranka, R. S. Windeler, A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Stingl, A.

Stoev, V.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Surry, P. D.

N. J. Radcliffe, P. D. Surry, “Formal memetic algorithms,” in Evolutionary Computing, T. Fogarty, ed., proceedings of an AISB workshop (Springer-Verlag, Berlin, 1994), pp. 1–14.

Sutter, D. H.

N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.

Svelto, O.

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

Szipöcs, R.

A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

R. Szipöcs, A. Köhazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135 (1997).
[CrossRef]

A. Stingl, M. Lenzner, Ch. Spielmann, F. Krausz, R. Szipöcs, “Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett. 20, 602–604 (1995).
[CrossRef] [PubMed]

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

Taft, G.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Tempea, G.

G. Tempea, V. Yakovlev, B. Bacovic, F. Krausz, “Tilted-front-interface chirped mirrors,” J. Opt. Soc. Am. B 18, 1747–1750 (2001).
[CrossRef]

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

G. Tempea, F. Krausz, Ch. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

G. Tempea, F. Krausz, “Dispersion management over one octave with tilted-front-interface chirped mirrors,” presented at the Twelfth International Conference on Ultrafast Phenomena, Charleston, S.C., 9–13 July 2000.

Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Conjugate gradient methods in multidimensions,” in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., W. H. Press, ed. (Cambridge U. Press, Cambridge, 1995), pp. 420–425.

Tilsch, M.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

F. X. Kärtner, N. Matuscheck, T. Tschibili, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

Trebino, R.

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Tschibili, T.

Tschudi, T.

R. Ell, U. Morgner, F. X. Kärtner, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, M. J. Lederer, A. Boiko, B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphire laser,” Opt. Lett. 26, 373–375 (2001).
[CrossRef]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

F. X. Kärtner, N. Matuscheck, T. Tschibili, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.

Udem, Th.

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Conjugate gradient methods in multidimensions,” in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., W. H. Press, ed. (Cambridge U. Press, Cambridge, 1995), pp. 420–425.

Vorontsov, M. A.

Wei, Z.

A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Wiersma, D. A.

A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Windeler, R. S.

J. K. Ranka, R. S. Windeler, A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Xu, L.

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Yakovlev, V.

Zeek, E.

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (4)

N. Matuschek, L. Gallmann, D. H. Sutter, G. Steinmeyer, U. Keller, “Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics,” Appl. Phys. B 71, 509–522 (2000).
[CrossRef]

A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, V. Stoev, “Ultrafast laser and amplifier sources,” Appl. Phys. B 65, 161–174 (1997).
[CrossRef]

A. Baltuška, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, R. Szipöcs, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

R. Szipöcs, A. Köhazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135 (1997).
[CrossRef]

IEE Proc. J (1)

E. Michielssen, S. Ranjithan, R. Mittra, “Optimal multilayer filter design using real coded genetic algorithms,” IEE Proc. J 139, 413–420 (1992).

IEEE J. Quantum Electron. (1)

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (3)

G. Tempea, F. Krausz, Ch. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Schreuer, M. Tilsch, T. Tschudi, U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a SESAM and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

M. Nisoli, S. Stagira, S. de Silvestri, O. Svelto, S. Sartania, Z. Cheng, G. Tempea, Ch. Spielmann, F. Krausz, “Toward a terawatt-scale sub-10-fs laser technology,” IEEE J. Sel. Top. Quantum Electron. 4, 414–420 (1998).
[CrossRef]

IEEE Trans. Automatic Control (1)

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Automatic Control 37, 332–341 (1992).
[CrossRef]

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

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

Opt. Lett. (6)

Phys. Rev. Lett. (1)

A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T. W. Hänsch, F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. 85, 740–743 (2000).
[CrossRef] [PubMed]

Other (9)

G. Tempea, F. Krausz, “Dispersion management over one octave with tilted-front-interface chirped mirrors,” presented at the Twelfth International Conference on Ultrafast Phenomena, Charleston, S.C., 9–13 July 2000.

L. Gallmann, N. Matuschek, D. H. Sutter, V. Scheuer, G. Angelow, T. Tschudi, G. Steinmeyer, U. Keller, “Smooth dispersion compensation: novel chirped mirrors with suppressed dispersion oscillations,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds.,Springer-Verlag, Berlin, 2000), pp. 62–64.

L. Xu, W. M. Kimmel, P. O’Shea, R. Trebino, J. K. Ranka, R. S. Windeler, A. J. Stentz, “Measuring the intensity and phase of an ultrabroadband continuum,” in Ultrafast Phenomena XII, T. Elsässer, S. Mukamel, M. M. Murnane, N. F. Scherer, eds. (Springer-Verlag, Berlin, 2000), pp. 129–131.

Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, Ch. Spielmann, F. Krausz, “Generation of intense diffraction-limited white light and 4-fs pulses,” in Ultrafast Phenomena XI, T. Elsässer, J. G. Fujimoto, D. A. Wiesma, W. Zinth, eds. (Springer-Verlag, Berlin, 1998).

H. A. Macleod, “Basic theory,” in Thin-Film Optical Filters, W. T. Welford, S. Laurenson, eds. (Institute of Physics Publishing, Bristol, UK, 2001), pp. 40–41.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, “Conjugate gradient methods in multidimensions,” in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., W. H. Press, ed. (Cambridge U. Press, Cambridge, 1995), pp. 420–425.

TFCalc, Thin-Film design software for windows, V. 3.0 ( http://www.sspectra.com ) .

N. J. Radcliffe, P. D. Surry, “Formal memetic algorithms,” in Evolutionary Computing, T. Fogarty, ed., proceedings of an AISB workshop (Springer-Verlag, Berlin, 1994), pp. 1–14.

Yu. Ermoliev, R. J.-B. Wets, eds.Numerical Techniques for Stochastic Optimization (Springer-Verlag, Berlin, 1988).

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Figures (4)

Fig. 1
Fig. 1

Comparison of an input-coupler design found by the memetic algorithm (solid curves) with that obtained by local optimization using the conjugate-gradient algorithm (dashed curves). In both cases the starting design was obtained by the double-chirping method. Large transmittance in the wavelength range 527–537 nm as well as high reflectance in the range 650–900 nm together with a target dispersion of -35 fs2 were required. The error bars correspond to perturbation of layer thicknesses with a standard deviation of 1 nm.

Fig. 2
Fig. 2

Example of a broadband TFI chirped mirror designed for the wavelength range 560–1130 nm. The global and the local optimization produced designs of comparable quality in this case, probably because the initial design was close to the global optimum. The mirror consists of 70 layers, the incidence medium is fused silica, the angle of incidence is 5°, and the light is p polarized. Neither the reflectance of the fused-silica wedge to air nor the dispersion of the wedged plate was taken into account in the figure.

Fig. 3
Fig. 3

Comparison of the memetic algorithm with the genetic algorithm formed by exclusion of the partial refinement. The y axis is the value of the merit function of the best member in the population. Each curve is the result of averaging over 11 runs of the optimization code.

Fig. 4
Fig. 4

Result of sensitivity optimization of a modest-bandwidth chirped mirror. The mirror consists of 49 layers and the incidence medium is air; normal incidence. The error bars were obtained for a standard deviation of technological errors equal to 1 nm.

Tables (1)

Tables Icon

Table 1 Comparison of Layer Thicknesses Before (d 1 ) and After (d 2 ) Sensitivity Optimization by the Stochastic Quasi-Gradient Algorithm

Equations (12)

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

Lk=1k=1i=1k-1 Mi1<kn,
Rk=1k=ni=k+1n Mi1k<n.
i=1nMi=M1M2Mk-1MkMk+1Mn=LkMkRk.
J˜x1, x2,  , xn=JT1x1, T2x2,  , Tnxn.
J˜=1mk=1mJ˜k+-J˜k-2ckpk1J˜k+-J˜k-2ckpkn,
J˜k+=JT1x1+ckpk1, T2x2+ckpk2,  ,Tnxn+ckpkn,J˜k-=JT1x1-ckpk1, T2x2-ckpk2,  ,Tnxn-ckpkn,
r=η0-Yη0+Y,
Y=C/B, BC=k=1nMk1ηn+1,
Mk=cos δki sin δk/ηkiηk sin δkcos δk,
δk=2πNkdk cos θkλ,
ηk=Nk cos θkTE waves s wavesNk/cos θkTM waves p waves,
N0 sin θ0=Nk sin θk,

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