Abstract

We present general analytic solutions for optical coherent control of electromagnetic energy propagation in plasmonic nanostructures. Propagating modes are excited with tightly focused ultrashort laser pulses that are shaped in amplitude, phase, and polarization (ellipticity and orientation angle). We decouple the interplay between two main mechanisms which are essential for the control of local near-fields. First, the amplitudes and the phase difference of two laser pulse polarization components are used to guide linear flux to a desired spatial position. Second, temporal compression of the near-field at the target location is achieved using the remaining free laser pulse parameter to flatten the local spectral phase. The resulting enhancement of nonlinear signals from this intuitive analytic two-step process is compared to and confirmed by the results of an iterative adaptive learning loop in which an evolutionary algorithm performs a global optimization. Thus, we gain detailed insight into why a certain complex laser pulse shape leads to a particular control target. This analytic approach may also be useful in a number of other coherent control scenarios.

© 2009 Optical Society of America

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2009

J. Huang, T. Feichtner, P. Biagioni, and B. Hecht, "Impedance Matching and Emission Properties of Nanoantennas in an Optical Nanocircuit," Nano Lett. 9(5), 1897-1902 (2009).
[CrossRef] [PubMed]

2008

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

R. Selle, P. Nuernberger, F. Langhojer, F. Dimler, S. Fechner, G. Gerber, and T. Brixner, "Generation of polarization-shaped ultraviolet femtosecond pulses," Opt. Lett. 33(8), 803-805 (2008).
[CrossRef] [PubMed]

2007

S. M. Weber, F. Weise, M. Plewicki, and A. Lindinger, "Interferometric generation of parametrically shaped polarization pulses," Appl. Opt. 46(23), 5987-90 (2007).
[CrossRef] [PubMed]

M. Ninck, A. Galler, T. Feurer, and T. Brixner, "Programmable common-path vector field synthesizer for femtosecond pulses," Opt. Lett. 32(23), 3379-3381 (2007).
[CrossRef] [PubMed]

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, "Femtosecond quantum control of molecular dynamics in the condensed phase," Phys. Chem. Chem. Phys. 9(20), 2470-97 (2007). PMID: 17508081,
[CrossRef] [PubMed]

A. Kuzyk, M. Pettersson, J. J. Toppari, T. K. Hakala, H. Tikkanen, H. Kunttu, and P. T¨orm¨a, "Molecular coupling of light with plasmonic waveguides," Opt. Express 15(16), 9908-9917 (2007).
[CrossRef] [PubMed]

M. Durach, A. Rusina, M. I. Stockman, and K. Nelson, "Toward Full Spatiotemporal Control on the Nanoscale," Nano Lett. 7(10), 3145-3149 (2007).
[CrossRef] [PubMed]

M. Sukharev and T. Seideman, "Coherent control of light propagation via nanoparticle arrays," J. Phys. B 40(11), S283-S298 (2007).
[CrossRef]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

2006

M. Sukharev and T. Seideman, "Phase and polarization control as a route to plasmonic nanodevices," Nano Lett. 6(4), 715-719 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440(7083), 508-511 (2006).
[CrossRef] [PubMed]

T. Brixner, F. Garc’ıa de Abajo, C. Spindler, and W. Pfeiffer, "Adaptive ultrafast nano-optics in a tight focus," Appl. Phys. B 84(1), 89-95 (2006).
[CrossRef]

L. Polachek, D. Oron, and Y. Silberberg, "Full control of the spectral polarization of ultrashort pulses," Opt. Lett. 31(5), 631-633 (2006).
[CrossRef] [PubMed]

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, "Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer," Appl Opt 45(32), 8354-9 (2006).
[CrossRef] [PubMed]

2005

P. M¨uhlschlegel, H. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant Optical Antennas," Science 308(5728), 1607-1609 (2005).
[CrossRef] [PubMed]

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

2004

P. Andrew and W. L. Barnes, "Energy Transfer Across a Metal Film Mediated by Surface Plasmon Polaritons," Science 306(5698), 1002-1005 (2004).
[CrossRef] [PubMed]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424(6950), 824- 830 (2003).
[CrossRef] [PubMed]

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

2002

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, "Generation and characterization of polarization-shaped femtosecond laser pulses," Appl. Phys. B 74(0), s133-s144 (2002).
[CrossRef]

2001

T. Brixner and G. Gerber, "Femtosecond polarization pulse shaping," Opt. Lett. 26(8), 557-559 (2001).
[CrossRef]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

2000

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71(5), 1929-1960 (2000).
[CrossRef]

1999

F. Garc’ıa de Abajo, "Interaction of Radiation and Fast Electrons with Clusters of Dielectrics: A Multiple Scattering Approach," Phys. Rev. Lett. 82(13), 2776 (1999).
[CrossRef]

1998

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

1997

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65(6), 779-782 (1997).
[CrossRef]

D. Yelin, D. Meshulach, and Y. Silberberg, "Adaptive femtosecond pulse compression," Opt. Lett. 22(23), 1793- 1795 (1997).
[CrossRef]

1996

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local Excitation, Scattering, and Interference of Surface Plasmons," Phys. Rev. Lett. 77(9), 1889 (1996).
[CrossRef] [PubMed]

1992

R. S. Judson and H. Rabitz, "Teaching lasers to control molecules," Phys. Rev. Lett. 68(10), 1500 (1992).
[CrossRef] [PubMed]

1986

M. Shapiro and P. Brumer, "Laser control of product quantum state populations in unimolecular reactions," J. Chem. Phys. 84(7), 4103-4104 (1986).
[CrossRef]

1985

D. J. Tannor and S. A. Rice, "Control of selectivity of chemical reaction via control of wave packet evolution," J. Chem. Phys. 83(10), 5013-5018 (1985).
[CrossRef]

Aeschlimann, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Andrew, P.

P. Andrew and W. L. Barnes, "Energy Transfer Across a Metal Film Mediated by Surface Plasmon Polaritons," Science 306(5698), 1002-1005 (2004).
[CrossRef] [PubMed]

Assion, A.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

Attema, I.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Atwater, H.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

Atwater, H. A.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

Aussenegg, F. R.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Barnes, W. L.

P. Andrew and W. L. Barnes, "Energy Transfer Across a Metal Film Mediated by Surface Plasmon Polaritons," Science 306(5698), 1002-1005 (2004).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424(6950), 824- 830 (2003).
[CrossRef] [PubMed]

Bauer, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Baumert, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65(6), 779-782 (1997).
[CrossRef]

Bayer, D.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Bergman, D. J.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, "Coherent Control of Femtosecond Energy Localization in Nanosystems," Phys. Rev. Lett. 88(6), 067,402 (2002).
[CrossRef]

Bergt, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

Biagioni, P.

J. Huang, T. Feichtner, P. Biagioni, and B. Hecht, "Impedance Matching and Emission Properties of Nanoantennas in an Optical Nanocircuit," Nano Lett. 9(5), 1897-1902 (2009).
[CrossRef] [PubMed]

Bielefeldt, H.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local Excitation, Scattering, and Interference of Surface Plasmons," Phys. Rev. Lett. 77(9), 1889 (1996).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440(7083), 508-511 (2006).
[CrossRef] [PubMed]

Brixner, T.

R. Selle, P. Nuernberger, F. Langhojer, F. Dimler, S. Fechner, G. Gerber, and T. Brixner, "Generation of polarization-shaped ultraviolet femtosecond pulses," Opt. Lett. 33(8), 803-805 (2008).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

M. Ninck, A. Galler, T. Feurer, and T. Brixner, "Programmable common-path vector field synthesizer for femtosecond pulses," Opt. Lett. 32(23), 3379-3381 (2007).
[CrossRef] [PubMed]

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, "Femtosecond quantum control of molecular dynamics in the condensed phase," Phys. Chem. Chem. Phys. 9(20), 2470-97 (2007). PMID: 17508081,
[CrossRef] [PubMed]

T. Brixner, F. Garc’ıa de Abajo, C. Spindler, and W. Pfeiffer, "Adaptive ultrafast nano-optics in a tight focus," Appl. Phys. B 84(1), 89-95 (2006).
[CrossRef]

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, "Generation and characterization of polarization-shaped femtosecond laser pulses," Appl. Phys. B 74(0), s133-s144 (2002).
[CrossRef]

T. Brixner and G. Gerber, "Femtosecond polarization pulse shaping," Opt. Lett. 26(8), 557-559 (2001).
[CrossRef]

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65(6), 779-782 (1997).
[CrossRef]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, "Ultrafast adaptive optical near-field control," Phys. Rev. B 73(12), 125,437-11 (2006).
[CrossRef]

J. S. Huang, D. V. Voronine, P. Tuchscherer, T. Brixner, and B. Hecht, "Deterministic spatio-temporal control of nano-optical fields in optical antennas and nano transmission lines," Phys. Rev. B 79(19), 195,441-5 (2009).

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, and W. Pfeiffer, "Nanoscopic Ultrafast Space-Time-Resolved Spectroscopy," Phys. Rev. Lett. 95(9), 093,901-4 (2005).
[CrossRef] [PubMed]

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

Brumer, P.

M. Shapiro and P. Brumer, "Laser control of product quantum state populations in unimolecular reactions," J. Chem. Phys. 84(7), 4103-4104 (1986).
[CrossRef]

Byeon, C. C.

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Cerjak, I.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Choi, S.

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Cunovic, S.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Dekker, H.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424(6950), 824- 830 (2003).
[CrossRef] [PubMed]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440(7083), 508-511 (2006).
[CrossRef] [PubMed]

Dimler, F.

R. Selle, P. Nuernberger, F. Langhojer, F. Dimler, S. Fechner, G. Gerber, and T. Brixner, "Generation of polarization-shaped ultraviolet femtosecond pulses," Opt. Lett. 33(8), 803-805 (2008).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Ditlbacher, H.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Durach, M.

M. Durach, A. Rusina, M. I. Stockman, and K. Nelson, "Toward Full Spatiotemporal Control on the Nanoscale," Nano Lett. 7(10), 3145-3149 (2007).
[CrossRef] [PubMed]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440(7083), 508-511 (2006).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424(6950), 824- 830 (2003).
[CrossRef] [PubMed]

Eisler, H.

P. M¨uhlschlegel, H. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant Optical Antennas," Science 308(5728), 1607-1609 (2005).
[CrossRef] [PubMed]

Engelen, R. J. P.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Faleev, S. V.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, "Coherent Control of Femtosecond Energy Localization in Nanosystems," Phys. Rev. Lett. 88(6), 067,402 (2002).
[CrossRef]

Fechner, S.

Feichtner, T.

J. Huang, T. Feichtner, P. Biagioni, and B. Hecht, "Impedance Matching and Emission Properties of Nanoantennas in an Optical Nanocircuit," Nano Lett. 9(5), 1897-1902 (2009).
[CrossRef] [PubMed]

Felidj, N.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Feurer, T.

Fischer, A.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Galler, A.

Garc´ia de Abajo, F.

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, and W. Pfeiffer, "Nanoscopic Ultrafast Space-Time-Resolved Spectroscopy," Phys. Rev. Lett. 95(9), 093,901-4 (2005).
[CrossRef] [PubMed]

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, "Ultrafast adaptive optical near-field control," Phys. Rev. B 73(12), 125,437-11 (2006).
[CrossRef]

Garc’ia de Abajo, F.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

T. Brixner, F. Garc’ıa de Abajo, C. Spindler, and W. Pfeiffer, "Adaptive ultrafast nano-optics in a tight focus," Appl. Phys. B 84(1), 89-95 (2006).
[CrossRef]

F. Garc’ıa de Abajo, "Interaction of Radiation and Fast Electrons with Clusters of Dielectrics: A Multiple Scattering Approach," Phys. Rev. Lett. 82(13), 2776 (1999).
[CrossRef]

Gerber, G.

R. Selle, P. Nuernberger, F. Langhojer, F. Dimler, S. Fechner, G. Gerber, and T. Brixner, "Generation of polarization-shaped ultraviolet femtosecond pulses," Opt. Lett. 33(8), 803-805 (2008).
[CrossRef] [PubMed]

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, "Femtosecond quantum control of molecular dynamics in the condensed phase," Phys. Chem. Chem. Phys. 9(20), 2470-97 (2007). PMID: 17508081,
[CrossRef] [PubMed]

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, "Generation and characterization of polarization-shaped femtosecond laser pulses," Appl. Phys. B 74(0), s133-s144 (2002).
[CrossRef]

T. Brixner and G. Gerber, "Femtosecond polarization pulse shaping," Opt. Lett. 26(8), 557-559 (2001).
[CrossRef]

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65(6), 779-782 (1997).
[CrossRef]

Hakala, T. K.

Harel, E.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

Hecht, B.

J. Huang, T. Feichtner, P. Biagioni, and B. Hecht, "Impedance Matching and Emission Properties of Nanoantennas in an Optical Nanocircuit," Nano Lett. 9(5), 1897-1902 (2009).
[CrossRef] [PubMed]

P. M¨uhlschlegel, H. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant Optical Antennas," Science 308(5728), 1607-1609 (2005).
[CrossRef] [PubMed]

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local Excitation, Scattering, and Interference of Surface Plasmons," Phys. Rev. Lett. 77(9), 1889 (1996).
[CrossRef] [PubMed]

J. S. Huang, D. V. Voronine, P. Tuchscherer, T. Brixner, and B. Hecht, "Deterministic spatio-temporal control of nano-optical fields in optical antennas and nano transmission lines," Phys. Rev. B 79(19), 195,441-5 (2009).

Huang, J.

J. Huang, T. Feichtner, P. Biagioni, and B. Hecht, "Impedance Matching and Emission Properties of Nanoantennas in an Optical Nanocircuit," Nano Lett. 9(5), 1897-1902 (2009).
[CrossRef] [PubMed]

Huang, J. S.

J. S. Huang, D. V. Voronine, P. Tuchscherer, T. Brixner, and B. Hecht, "Deterministic spatio-temporal control of nano-optical fields in optical antennas and nano transmission lines," Phys. Rev. B 79(19), 195,441-5 (2009).

Inouye, Y.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local Excitation, Scattering, and Interference of Surface Plasmons," Phys. Rev. Lett. 77(9), 1889 (1996).
[CrossRef] [PubMed]

Jeong, M. S.

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Judson, R. S.

R. S. Judson and H. Rabitz, "Teaching lasers to control molecules," Phys. Rev. Lett. 68(10), 1500 (1992).
[CrossRef] [PubMed]

Jung, Y. S.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

Kiefer, B.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

Kik, P.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

Kik, P. G.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

Kim, D. S.

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Kim, H. K.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

Ko, D.

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Koel, B.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

Korterik, J. P.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Krampert, G.

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, "Generation and characterization of polarization-shaped femtosecond laser pulses," Appl. Phys. B 74(0), s133-s144 (2002).
[CrossRef]

Krenn, J. R.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

J. R. Krenn and J. Weeber, "Surface plasmon polaritons in metal stripes and wires," Phil. Trans. R. Soc. Lond. A 362(1817), 739-756 (2004).
[CrossRef]

Kubo, A.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

Kuipers, L.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Kunttu, H.

Kuzyk, A.

Laluet, J.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440(7083), 508-511 (2006).
[CrossRef] [PubMed]

Lamprecht, B.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Langhojer, F.

Leitner, A.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Li, X.

X. Li and M. I. Stockman, "Highly efficient spatiotemporal coherent control in nanoplasmonics on a nanometerfemtosecond scale by time reversal," Phys. Rev. B 77(19), 195,109-10 (2008).
[CrossRef]

Lienau, C.

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Lindinger, A.

S. M. Weber, F. Weise, M. Plewicki, and A. Lindinger, "Interferometric generation of parametrically shaped polarization pulses," Appl. Opt. 46(23), 5987-90 (2007).
[CrossRef] [PubMed]

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, "Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer," Appl Opt 45(32), 8354-9 (2006).
[CrossRef] [PubMed]

M¨uhlschlegel, P.

P. M¨uhlschlegel, H. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant Optical Antennas," Science 308(5728), 1607-1609 (2005).
[CrossRef] [PubMed]

Maier, S.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

Maier, S. A.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

Martin, O. J. F.

P. M¨uhlschlegel, H. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant Optical Antennas," Science 308(5728), 1607-1609 (2005).
[CrossRef] [PubMed]

Meltzer, S.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

Meshulach, D.

Nelson, K.

M. Durach, A. Rusina, M. I. Stockman, and K. Nelson, "Toward Full Spatiotemporal Control on the Nanoscale," Nano Lett. 7(10), 3145-3149 (2007).
[CrossRef] [PubMed]

Niklaus, P.

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, "Generation and characterization of polarization-shaped femtosecond laser pulses," Appl. Phys. B 74(0), s133-s144 (2002).
[CrossRef]

Ninck, M.

Novotny, L.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local Excitation, Scattering, and Interference of Surface Plasmons," Phys. Rev. Lett. 77(9), 1889 (1996).
[CrossRef] [PubMed]

L. Novotny, "Effective Wavelength Scaling for Optical Antennas," Phys. Rev. Lett. 98(26), 266,802-4 (2007).
[CrossRef]

Nuernberger, P.

R. Selle, P. Nuernberger, F. Langhojer, F. Dimler, S. Fechner, G. Gerber, and T. Brixner, "Generation of polarization-shaped ultraviolet femtosecond pulses," Opt. Lett. 33(8), 803-805 (2008).
[CrossRef] [PubMed]

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, "Femtosecond quantum control of molecular dynamics in the condensed phase," Phys. Chem. Chem. Phys. 9(20), 2470-97 (2007). PMID: 17508081,
[CrossRef] [PubMed]

Onda, K.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

Oron, D.

Park, D.

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Petek, H.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

Pettersson, M.

Pfeiffer, W.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

T. Brixner, F. Garc’ıa de Abajo, C. Spindler, and W. Pfeiffer, "Adaptive ultrafast nano-optics in a tight focus," Appl. Phys. B 84(1), 89-95 (2006).
[CrossRef]

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, "Ultrafast adaptive optical near-field control," Phys. Rev. B 73(12), 125,437-11 (2006).
[CrossRef]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, and W. Pfeiffer, "Nanoscopic Ultrafast Space-Time-Resolved Spectroscopy," Phys. Rev. Lett. 95(9), 093,901-4 (2005).
[CrossRef] [PubMed]

Plewicki, M.

S. M. Weber, F. Weise, M. Plewicki, and A. Lindinger, "Interferometric generation of parametrically shaped polarization pulses," Appl. Opt. 46(23), 5987-90 (2007).
[CrossRef] [PubMed]

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, "Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer," Appl Opt 45(32), 8354-9 (2006).
[CrossRef] [PubMed]

Pohl, D. W.

P. M¨uhlschlegel, H. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant Optical Antennas," Science 308(5728), 1607-1609 (2005).
[CrossRef] [PubMed]

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local Excitation, Scattering, and Interference of Surface Plasmons," Phys. Rev. Lett. 77(9), 1889 (1996).
[CrossRef] [PubMed]

Polachek, L.

Rabitz, H.

R. S. Judson and H. Rabitz, "Teaching lasers to control molecules," Phys. Rev. Lett. 68(10), 1500 (1992).
[CrossRef] [PubMed]

Requicha, A.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

Requicha, A. A. G.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

Rice, S. A.

D. J. Tannor and S. A. Rice, "Control of selectivity of chemical reaction via control of wave packet evolution," J. Chem. Phys. 83(10), 5013-5018 (1985).
[CrossRef]

Rohmer, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Rusina, A.

M. Durach, A. Rusina, M. I. Stockman, and K. Nelson, "Toward Full Spatiotemporal Control on the Nanoscale," Nano Lett. 7(10), 3145-3149 (2007).
[CrossRef] [PubMed]

Salerno, M.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Sandtke, M.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Schider, G.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Schneider, C.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Schneider, J.

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, and W. Pfeiffer, "Nanoscopic Ultrafast Space-Time-Resolved Spectroscopy," Phys. Rev. Lett. 95(9), 093,901-4 (2005).
[CrossRef] [PubMed]

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, "Ultrafast adaptive optical near-field control," Phys. Rev. B 73(12), 125,437-11 (2006).
[CrossRef]

Schoenmaker, H.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Segerink, B.

M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, "Novel instrument for surface plasmon polariton tracking in space and time," Rev. Sci. Instrum. 79(1), 013,704 (2008).
[CrossRef]

Seideman, T.

M. Sukharev and T. Seideman, "Coherent control of light propagation via nanoparticle arrays," J. Phys. B 40(11), S283-S298 (2007).
[CrossRef]

M. Sukharev and T. Seideman, "Phase and polarization control as a route to plasmonic nanodevices," Nano Lett. 6(4), 715-719 (2006).
[CrossRef] [PubMed]

Selle, R.

Seyfried, V.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65(6), 779-782 (1997).
[CrossRef]

Shapiro, M.

M. Shapiro and P. Brumer, "Laser control of product quantum state populations in unimolecular reactions," J. Chem. Phys. 84(7), 4103-4104 (1986).
[CrossRef]

Silberberg, Y.

Spindler, C.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

T. Brixner, F. Garc’ıa de Abajo, C. Spindler, and W. Pfeiffer, "Adaptive ultrafast nano-optics in a tight focus," Appl. Phys. B 84(1), 89-95 (2006).
[CrossRef]

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, "Ultrafast adaptive optical near-field control," Phys. Rev. B 73(12), 125,437-11 (2006).
[CrossRef]

Steeb, F.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Stockman, M. I.

M. Durach, A. Rusina, M. I. Stockman, and K. Nelson, "Toward Full Spatiotemporal Control on the Nanoscale," Nano Lett. 7(10), 3145-3149 (2007).
[CrossRef] [PubMed]

X. Li and M. I. Stockman, "Highly efficient spatiotemporal coherent control in nanoplasmonics on a nanometerfemtosecond scale by time reversal," Phys. Rev. B 77(19), 195,109-10 (2008).
[CrossRef]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, "Coherent Control of Femtosecond Energy Localization in Nanosystems," Phys. Rev. Lett. 88(6), 067,402 (2002).
[CrossRef]

Str¨uber, C.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Strehle, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65(6), 779-782 (1997).
[CrossRef]

Sukharev, M.

M. Sukharev and T. Seideman, "Coherent control of light propagation via nanoparticle arrays," J. Phys. B 40(11), S283-S298 (2007).
[CrossRef]

M. Sukharev and T. Seideman, "Phase and polarization control as a route to plasmonic nanodevices," Nano Lett. 6(4), 715-719 (2006).
[CrossRef] [PubMed]

Sun, Z.

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

T¨orm¨a, P.

Tannor, D. J.

D. J. Tannor and S. A. Rice, "Control of selectivity of chemical reaction via control of wave packet evolution," J. Chem. Phys. 83(10), 5013-5018 (1985).
[CrossRef]

Tikkanen, H.

Toppari, J. J.

Tuchscherer, P.

J. S. Huang, D. V. Voronine, P. Tuchscherer, T. Brixner, and B. Hecht, "Deterministic spatio-temporal control of nano-optical fields in optical antennas and nano transmission lines," Phys. Rev. B 79(19), 195,441-5 (2009).

Vogt, G.

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, "Femtosecond quantum control of molecular dynamics in the condensed phase," Phys. Chem. Chem. Phys. 9(20), 2470-97 (2007). PMID: 17508081,
[CrossRef] [PubMed]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440(7083), 508-511 (2006).
[CrossRef] [PubMed]

Voronine, D. V.

J. S. Huang, D. V. Voronine, P. Tuchscherer, T. Brixner, and B. Hecht, "Deterministic spatio-temporal control of nano-optical fields in optical antennas and nano transmission lines," Phys. Rev. B 79(19), 195,441-5 (2009).

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Str¨uber, and D. V. Voronine, "Simultaneous Spatial and Temporal Control of Nano- Optical Excitations," (submitted).

Weber, S. M.

S. M. Weber, F. Weise, M. Plewicki, and A. Lindinger, "Interferometric generation of parametrically shaped polarization pulses," Appl. Opt. 46(23), 5987-90 (2007).
[CrossRef] [PubMed]

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, "Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer," Appl Opt 45(32), 8354-9 (2006).
[CrossRef] [PubMed]

Weeber, J.

J. R. Krenn and J. Weeber, "Surface plasmon polaritons in metal stripes and wires," Phil. Trans. R. Soc. Lond. A 362(1817), 739-756 (2004).
[CrossRef]

Weeber, J. C.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

Weiner, A. M.

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71(5), 1929-1960 (2000).
[CrossRef]

Weise, F.

S. M. Weber, F. Weise, M. Plewicki, and A. Lindinger, "Interferometric generation of parametrically shaped polarization pulses," Appl. Opt. 46(23), 5987-90 (2007).
[CrossRef] [PubMed]

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, "Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer," Appl Opt 45(32), 8354-9 (2006).
[CrossRef] [PubMed]

Yelin, D.

Adv. Mater.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001).
[CrossRef]

Appl Opt

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, "Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer," Appl Opt 45(32), 8354-9 (2006).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. B

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, "Generation and characterization of polarization-shaped femtosecond laser pulses," Appl. Phys. B 74(0), s133-s144 (2002).
[CrossRef]

T. Brixner, F. Garc’ıa de Abajo, C. Spindler, and W. Pfeiffer, "Adaptive ultrafast nano-optics in a tight focus," Appl. Phys. B 84(1), 89-95 (2006).
[CrossRef]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65(6), 779-782 (1997).
[CrossRef]

Appl. Phys. Lett.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett. 79(1), 51-53 (2001).
[CrossRef]

J. Chem. Phys.

D. J. Tannor and S. A. Rice, "Control of selectivity of chemical reaction via control of wave packet evolution," J. Chem. Phys. 83(10), 5013-5018 (1985).
[CrossRef]

M. Shapiro and P. Brumer, "Laser control of product quantum state populations in unimolecular reactions," J. Chem. Phys. 84(7), 4103-4104 (1986).
[CrossRef]

J. Phys. B

M. Sukharev and T. Seideman, "Coherent control of light propagation via nanoparticle arrays," J. Phys. B 40(11), S283-S298 (2007).
[CrossRef]

Nano Lett.

M. Durach, A. Rusina, M. I. Stockman, and K. Nelson, "Toward Full Spatiotemporal Control on the Nanoscale," Nano Lett. 7(10), 3145-3149 (2007).
[CrossRef] [PubMed]

A. Kubo, K. Onda, H. Petek, Z. Sun, Y. S. Jung, and H. K. Kim, "Femtosecond Imaging of Surface Plasmon Dynamics in a Nanostructured Silver Film," Nano Lett. 5(6), 1123-1127 (2005).
[CrossRef] [PubMed]

J. Huang, T. Feichtner, P. Biagioni, and B. Hecht, "Impedance Matching and Emission Properties of Nanoantennas in an Optical Nanocircuit," Nano Lett. 9(5), 1897-1902 (2009).
[CrossRef] [PubMed]

M. Sukharev and T. Seideman, "Phase and polarization control as a route to plasmonic nanodevices," Nano Lett. 6(4), 715-719 (2006).
[CrossRef] [PubMed]

Nature

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440(7083), 508-511 (2006).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424(6950), 824- 830 (2003).
[CrossRef] [PubMed]

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. Garc’ıa de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446(7133), 301-304 (2007).
[CrossRef] [PubMed]

Nature Mater.

S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003).
[CrossRef]

Opt. Express

A. Kuzyk, M. Pettersson, J. J. Toppari, T. K. Hakala, H. Tikkanen, H. Kunttu, and P. T¨orm¨a, "Molecular coupling of light with plasmonic waveguides," Opt. Express 15(16), 9908-9917 (2007).
[CrossRef] [PubMed]

S. Choi, D. Park, C. Lienau, M. S. Jeong, C. C. Byeon, D. Ko, and D. S. Kim, "Femtosecond phase control of spatial localization of the optical near-field in a metal nanoslit array," Opt. Express 16(16), 12,075-12,083 (2008).
[CrossRef]

Opt. Lett.

Phys. Chem. Chem. Phys.

P. Nuernberger, G. Vogt, T. Brixner, and G. Gerber, "Femtosecond quantum control of molecular dynamics in the condensed phase," Phys. Chem. Chem. Phys. 9(20), 2470-97 (2007). PMID: 17508081,
[CrossRef] [PubMed]

Phys. Rev. Lett.

F. Garc’ıa de Abajo, "Interaction of Radiation and Fast Electrons with Clusters of Dielectrics: A Multiple Scattering Approach," Phys. Rev. Lett. 82(13), 2776 (1999).
[CrossRef]

R. S. Judson and H. Rabitz, "Teaching lasers to control molecules," Phys. Rev. Lett. 68(10), 1500 (1992).
[CrossRef] [PubMed]

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local Excitation, Scattering, and Interference of Surface Plasmons," Phys. Rev. Lett. 77(9), 1889 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71(5), 1929-1960 (2000).
[CrossRef]

Science

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, "Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses," Science 282(5390), 919-922 (1998).
[CrossRef] [PubMed]

P. M¨uhlschlegel, H. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant Optical Antennas," Science 308(5728), 1607-1609 (2005).
[CrossRef] [PubMed]

P. Andrew and W. L. Barnes, "Energy Transfer Across a Metal Film Mediated by Surface Plasmon Polaritons," Science 306(5698), 1002-1005 (2004).
[CrossRef] [PubMed]

Other

J. R. Krenn and J. Weeber, "Surface plasmon polaritons in metal stripes and wires," Phil. Trans. R. Soc. Lond. A 362(1817), 739-756 (2004).
[CrossRef]

J. S. Huang, D. V. Voronine, P. Tuchscherer, T. Brixner, and B. Hecht, "Deterministic spatio-temporal control of nano-optical fields in optical antennas and nano transmission lines," Phys. Rev. B 79(19), 195,441-5 (2009).

M. I. Stockman, S. V. Faleev, and D. J. Bergman, "Coherent Control of Femtosecond Energy Localization in Nanosystems," Phys. Rev. Lett. 88(6), 067,402 (2002).
[CrossRef]

T. Brixner, F. Garc´ıa de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, "Ultrafast adaptive optical near-field control," Phys. Rev. B 73(12), 125,437-11 (2006).
[CrossRef]

L. Novotny, "Effective Wavelength Scaling for Optical Antennas," Phys. Rev. Lett. 98(26), 266,802-4 (2007).
[CrossRef]

P. Vasa, C. Ropers, R. Pomraenke, and C. Lienau, "Ultra-fast nano-optics," Laser & Photon. Rev. (2009).
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L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

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Supplementary Material (2)

» Media 1: MOV (704 KB)     
» Media 2: MOV (759 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Nanoplasmonic branching waveguide consisting of 50 nm diameter spheres. The target points for coherent control [r 1 = (930,0,10) nm and r 2 = (570,380,10) nm] are chosen between the last two spheres of each arm and 10 nm above the z = 0 symmetry plane. The structure is excited with a tightly-focused Gaussian beam (indicated in red) at the beginning of the chain using the two polarizations 1 and 2 along the x and y direction, respectively (indicated with 1 and 2 in the focus). Inset: total scattering cross sections obtained for plane wave illumination with polarization 1 (black solid) and 2 (red dashed).

Fig. 2.
Fig. 2.

The phase differences for analytic maximization (solid blue line) and minimization (dashed red line) of the linear flux F lin(r) at positions r 1 (a) and r 2 (b) compared to the results obtained by an adaptive optimization for maximization (blue circles) and minimization (red squares). The laser spectrum is indicated by a black dotted line.

Fig. 3.
Fig. 3.

The local response intensities R(r,ω) plotted logarithmically for the position r 1 for the maximum (blue dashed) and minimum (blue solid) local linear flux, and for the position r 2 (red dash-dotted and red dotted lines for the maximum and the minimum local linear flux, respectively). The optimal phase differences for the linear flux as obtained from Eq. (12) and (13) are shown in Fig. 2.

Fig. 4.
Fig. 4.

Phase difference for analytic maximization (blue solid line) and minimization (red dashed line) of the linear flux difference F lin(r 1)-F lin(r 2) and its comparison to the adap-tively optimized phases (blue circles and red squares, respectively). The laser spectrum is indicated by a black dotted line.

Fig. 5.
Fig. 5.

Amplitude weighting coefficients for polarization components 1 (green) and 2 (purple) for controlling the linear flux difference F lin(r 1) - F lin(r 2). The analytic results (solid and dashed lines) are compared with those from adaptive optimization (diamond and triangle symbols, respectively). The maximization or minimization of the flux difference correspond to energy guidance to positions r 1 (a) or r 2 (b), respectively. Pulse amplitudes are obtained by multiplying these weighting coefficients with Gaussian profiles using Eq. (21). The laser spectral intensity is indicated by a black dotted line.

Fig. 6.
Fig. 6.

(a) The local spectrum [Eq. (6)] for the optimization where the “red” and “blue” halves of the spectrum are guided to positions r 1 (dashed blue) and r 2 (solid red), respectively. (b) The local spectrum normalized to the sum of the local spectra at positions r 1 and r 2.

Fig. 7.
Fig. 7.

Analytic (lines) phases (a,b) φ 1 (blue) and φ 2 (red) and amplitudes (c,d) γ 1 (ω) (green) and γ 2(ω) (purple) for the nonlinear guidance of the x component of the near-field by the decoupled process of first maximizing or minimizing the linear flux difference F lin(r 1) - F lin(r 2) and then compressing the signal at the positions r 1 (a,c) and r 2 (b,d), compared to the adaptively optimized (symbols) phases and amplitudes using the difference of the nonlinear signal F nl(r 1) -F nl(r 2) [Eq. (26)] as the fitness function. The laser spectrum is indicated by a black dotted line. The adaptively optimized phases were adjusted with a linear phase and a phase offset to fit to the analytically calculated data since the evolutionary algorithm is not sensitive to these parameters.

Fig. 8.
Fig. 8.

(a) Temporal near-field intensity ∣E(r,t)∣2 at two locations r 1 (solid) and r 2 (dashed dotted). The near-field intensity for the unshaped pulse (black) is compared to the near-field intensities excited by optimal pulses, to guide the nonlinear flux to r 1 (red) or r 2 (blue). (b,c) To show near-field compression the near-field intensities are plotted for the case of guiding the linear flux and choosing φ 1(ω) ≡ 0 (green) and are compared with the near-field intensities for optimal φ 1(ω) for r 1 [(b), red] and r 2 [(c), blue]. The curves in (a) have been shifted in time and normalized for r 1 (b) and r 2 (c).

Fig. 9.
Fig. 9.

Snapshots of the movies of plasmon propagation at t = 0 for the nonlinear flux guidance of the x component to r 1 (a) (Media 1) and r 2 (b) (Media 2). The amplitude of the x component of the near-field in the z = 10 nm plane is plotted logarithmically, where the excitation pulses are obtained by setting bx = 1 and by = bz = 0, and using the analytic approach of Sections 3.2 and 4.1. The insets show the optimal laser pulses. The arrows indicate the optimized locations.

Fig. 10.
Fig. 10.

(a) Spectral amplitudes (blue) and phases (green) of the largest local component of the shaped near-fields at two positions r 1 (solid, x component) and r 2 (dashed, y component) with the linear spectral phase corresponding to a 90 fs delay between the corresponding temporal intensities, which include all three near-field components, shown in (b).

Tables (1)

Tables Icon

Table 1. Analytic and adaptive linear flux control with phase-only shaping of the two polarization components. Flux values are given in the different columns for unshaped pulses corresponding to linear polarization at 45° orientation with respect to the x-y coordinates and for analytic as well as adaptive flux optimization. The different rows indicate maximization and minimization of flux at the locations r 1, r 2, and of the flux difference. The first and the second row correspond to control at one position from Section 3.1.1 and the third row describes the contrast control from Section 3.2.1. In all cases, the Gaussian spectrum was employed without amplitude shaping. All values are normalized to the sum of the linear flux F lin(r 1)+F lin(r 2) excited with an unshaped pulse, i.e. the first two values in the first column sum up to unity.

Equations (34)

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A(i)(r,ω)=(Ax(i)(r,ω)Ay(i)(r,ω)Az(i)(r,ω)).
Eiin(ω)=Ii(ω) exp[iφi(ω)] .
E(r,ω)=(Ax(1)(r,ω)Ay(1)(r,ω)Az(1)(r,ω))I1(ω)exp[iφ1(ω)]+(Ax(2)(r,ω)Ay(2)(r,ω)Az(2)(r,ω))I2(ω)exp[iφ2(ω)].
E(r,ω)={(Ax(1)(r,ω)Ay(1)(r,ω)Az(1)(r,ω))I1(ω)+(Ax(2)(r,ω)Ay(2)(r,ω)Az(2)(r,ω))I2(ω)exp[iΦ(ω)]}×exp[iφ1(ω)],
Φ(ω)=φ1(ω)φ2(ω).
S(r,ω)=α=x,y,zbαEα(r,ω)2=α=x,y,zbαFT[Eα(r,t)]2,
Flin(r)=α=x,y,zbαEα2(r,t)dt=12πωminωmaxS(r,ω)
Flin(r)=δω2πω=ωminωmaxα=x,y,zbαEα(r,ω)Eα*(r,ω),
S(r,ω)=I1(ω)α=x,y,zbαAα(1)(r,ω)2+I2(ω)α=x,y,zbαAα(2)(r,ω)2
+2I1(ω)I2(ω) Re {Amix(r,ω)exp[iΦ(ω)]} ,
Amix(r,ω)=α=x,y,zbαAα(1)(r,ω)Aα(2)*(r,ω)=Amix(r,ω)exp[iθmix(r,ω)],
Fnl(r)=[α=x,y,zbαEα2(r,t)]2dt.
Φmax(ω)=θmix(ω)and
Φmin(ω)=θmix(ω) π ,
R(r,ω)=S(r,ω)IG(ω).
flin[φ1(ω),φ2(ω),I1(ω),I2(ω)]=Flin(r1)Flin(r2),
flin=ω=ωminωmaxα=x,y,zbαEα(r1,ω)2ω=ωminωmaxα=x,y,zbαEα(r2,ω)2
=ω=ωminωmax(I1(ω)C1(ω)+I2(ω)C2(ω)+
2 I1(ω)I2(ω) {Amix(r1,ω)cos[θmix(r1,ω)+Φ(ω)]
Amix(r2,ω)cos[θmix(r2,ω)+Φ(ω)]}),
Ci(ω)=α=x,y,zbα[Aα(i)(r1,ω)2Aα(i)(r2,ω)2],i=1,2,
δδΦ(ω)flin=ω=ωminωmaxglin(ω),
glin(ω)=2I1(ω)I2(ω){Amix(r1,ω)sin[θmix(r1,ω)+Φ(ω)]
+Amix(r2,ω)sin[θmix(r2,ω)+Φ(ω)]}.
Φ(ω)=arctan{Amix(r2,ω)sin[θmix(r2,ω)]Amix(r1,ω)sin[θmix(r1,ω)]Amix(r1,ω)cos[θmix(r1,ω)]Amix(r2,ω)cos[θmix(r2,ω)]}+kπ,
Ii(ω)=γi(ω) IG(ω) .
flin[γ1(ω),γ2(ω)]=IG(ω)[C1(ω)γ12(ω)+C2(ω)γ22(ω)+2Cmix(ω)γ1(ω)γ2(ω)],
Cmix(ω)=Amax(r1,ω)cos [θmix(r1,ω)+Φ(ω)] Amax(r2,ω)cos[θmix(r2,ω)+Φ(ω)].
[γ1(ω),γ2(ω)] {[0,0],[1,Cmix(ω)/C2(ω)],[Cmix(ω)/C1(ω),1],[1,1]} .
φ1α(ω)=arg{Aα(1)(r,ω)I1(ω)+Aα(2)(r,ω)I2(ω)exp[iΦ(ω)]}
fnl[φ1(ω),φ2(ω),I1(ω),I2(ω)]=Fnl(r1)Fnl(r2).
bα ω=ωminωmaxEα(r,ω)2bβ,γω=ωminωmaxbβ,γ(r,ω)2
φ1(ω)=arg{α=x,y,zAα(1)(r1,ω)I1(ω)+α=x,y,zAα(2)(r1,ω)I2(ω)exp[iΦ(ω)]}.
Eα(r,ω)={Aα(1)(r,ω)I1(ω)+Aα(2)(r,ω)I2(ω)exp[iΦ(ω)]} exp [iφ1(ω)],

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