Abstract

We demonstrate ultrafast coherent control of multiphoton absorption in a dynamically shifted energy level structure. In a three-level system that models optical interactions with sodium atoms, we control the quantum interference of sequential 2 + 1 photons and direct three-photon transitions. Dynamic change in energy levels predicts an enormous enhancement of |7p〉-state excitation in the strong-field regime by a negatively chirped pulse. In addition, the |4s〉-state excitation is enhanced symmetrically by nonzero linear chirp rates given as a function of laser peak intensity and laser detuning. Experiments performed by ultrafast shaped-pulse excitation of ground-state atomic sodium verifies the various strong-field contributions to |3s〉-|7p〉 and |3s〉-|4s〉 transitions. The result suggests that for systems of molecular level understanding adiabatic control approach with analytically shaped pulses becomes a more direct control than feedback-loop black-box approaches.

© 2011 Optical Society of America

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2010

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

2009

S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic cesium: an analytical control approach,” Opt. Express 17, 7648 (2009).
[CrossRef] [PubMed]

T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, and T. Baumert, “Robust photon locking,” Phys. Rev. Lett. 102, 023004 (2009).
[CrossRef] [PubMed]

2008

H. Suchowski, A. Natan, B. D. Bruner, and Y. Silberberg, “Spatio-temporal coherent control of atomic systems: weak to strong field transition and breaking of symmetry in 2D maps,” J. Phys. At. Mol. Opt. Phys. 41, 074008 (2008).
[CrossRef]

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a threelevel system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[CrossRef] [PubMed]

M. C. Stowe, A. Péer, and J. Ye, “Control of four-level quantum coherence via discrete spectral shaping of an optical frequency comb,” Phys. Rev. Lett. 100, 203001 (2008).
[CrossRef] [PubMed]

Z. Amitay, A. Gandman, L. Chuntonov, and L. Rybak, “Multichannel selective femtosecond coherent control based on symmetry properties,” Phys. Rev. Lett. 100, 193002 (2008).
[CrossRef] [PubMed]

2007

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef] [PubMed]

2006

M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B 82, 183–188 (2006).
[CrossRef]

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef] [PubMed]

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

2005

N. Dudovich, T. Polack, A. Péer, and Y. Silberberg, “Simple route to strong-field coherent control,” Phys. Rev. Lett. 94, 083002 (2005).
[CrossRef] [PubMed]

2003

B. Chatel, J. Degert, S. Stock, and B. Girard, “Competition between sequential and direct paths in a two-photon transition,” Phys. Rev. A 68, 041402 (2003).
[CrossRef]

D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Rep. 374, 385–481 (2003).
[CrossRef]

2002

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[CrossRef] [PubMed]

2001

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[CrossRef] [PubMed]

Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. 267, 161 (2001).
[CrossRef]

2000

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

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

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000).
[CrossRef]

1999

T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

D. Meshulach, and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulse,” Phys. Rev. A 60, 1287 (1999).
[CrossRef]

1998

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[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, 919 (1998).
[CrossRef] [PubMed]

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

1997

V. Blanchet, C. Nicole, M.-A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2717 (1997).
[CrossRef]

1992

R. S. Judson, and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[CrossRef] [PubMed]

1985

D. J. Tanner, and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[CrossRef]

Ahn, J.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic cesium: an analytical control approach,” Opt. Express 17, 7648 (2009).
[CrossRef] [PubMed]

T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Amitay, Z.

Z. Amitay, A. Gandman, L. Chuntonov, and L. Rybak, “Multichannel selective femtosecond coherent control based on symmetry properties,” Phys. Rev. Lett. 100, 193002 (2008).
[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, 919 (1998).
[CrossRef] [PubMed]

Baumert, T.

T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, and T. Baumert, “Robust photon locking,” Phys. Rev. Lett. 102, 023004 (2009).
[CrossRef] [PubMed]

M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B 82, 183–188 (2006).
[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, 919 (1998).
[CrossRef] [PubMed]

Bayer, T.

T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, and T. Baumert, “Robust photon locking,” Phys. Rev. Lett. 102, 023004 (2009).
[CrossRef] [PubMed]

Beaurepaire, E.

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

Bergeman, T.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a threelevel system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[CrossRef] [PubMed]

Bergmann, K.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[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, 919 (1998).
[CrossRef] [PubMed]

Blanchet, V.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

V. Blanchet, C. Nicole, M.-A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2717 (1997).
[CrossRef]

Bouchene, M. A.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

Bouchene, M.-A.

V. Blanchet, C. Nicole, M.-A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2717 (1997).
[CrossRef]

Brixner, 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, 919 (1998).
[CrossRef] [PubMed]

Bruner, B. D.

H. Suchowski, A. Natan, B. D. Bruner, and Y. Silberberg, “Spatio-temporal coherent control of atomic systems: weak to strong field transition and breaking of symmetry in 2D maps,” J. Phys. At. Mol. Opt. Phys. 41, 074008 (2008).
[CrossRef]

Bucksbaum, P. H.

T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Chatel, B.

B. Chatel, J. Degert, S. Stock, and B. Girard, “Competition between sequential and direct paths in a two-photon transition,” Phys. Rev. A 68, 041402 (2003).
[CrossRef]

Cheng, Z.

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000).
[CrossRef]

Chuntonov, L.

Z. Amitay, A. Gandman, L. Chuntonov, and L. Rybak, “Multichannel selective femtosecond coherent control based on symmetry properties,” Phys. Rev. Lett. 100, 193002 (2008).
[CrossRef] [PubMed]

Clow, S. D.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a threelevel system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[CrossRef] [PubMed]

Cohen, J. L.

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef] [PubMed]

Dayan, B.

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[CrossRef] [PubMed]

de Vivie-Riedle, R.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

Débarre, D.

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

Degert, J.

B. Chatel, J. Degert, S. Stock, and B. Girard, “Competition between sequential and direct paths in a two-photon transition,” Phys. Rev. A 68, 041402 (2003).
[CrossRef]

Dudovich, N.

N. Dudovich, T. Polack, A. Péer, and Y. Silberberg, “Simple route to strong-field coherent control,” Phys. Rev. Lett. 94, 083002 (2005).
[CrossRef] [PubMed]

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[CrossRef] [PubMed]

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[CrossRef] [PubMed]

Gallagher Faeder, S. M.

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[CrossRef] [PubMed]

Gandman, A.

Z. Amitay, A. Gandman, L. Chuntonov, and L. Rybak, “Multichannel selective femtosecond coherent control based on symmetry properties,” Phys. Rev. Lett. 100, 193002 (2008).
[CrossRef] [PubMed]

Gerber, G.

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, 919 (1998).
[CrossRef] [PubMed]

Girard, B.

B. Chatel, J. Degert, S. Stock, and B. Girard, “Competition between sequential and direct paths in a two-photon transition,” Phys. Rev. A 68, 041402 (2003).
[CrossRef]

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

V. Blanchet, C. Nicole, M.-A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2717 (1997).
[CrossRef]

Goswami, D.

D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Rep. 374, 385–481 (2003).
[CrossRef]

Guerin, S.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Hakobyan, V.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Hournung, T.

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

Joffre, M.

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

Judson, R. S.

R. S. Judson, and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[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, 919 (1998).
[CrossRef] [PubMed]

Kompa, K.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

Kompa, K.-L.

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

Laude, V.

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000).
[CrossRef]

Lee, S.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic cesium: an analytical control approach,” Opt. Express 17, 7648 (2009).
[CrossRef] [PubMed]

Liese, D.

M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B 82, 183–188 (2006).
[CrossRef]

Lim, J.

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic cesium: an analytical control approach,” Opt. Express 17, 7648 (2009).
[CrossRef] [PubMed]

Martin, J.

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

Meier, R.

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

Melikechi, N.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

Menzel-Jones, C.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef] [PubMed]

Meshulach, D.

D. Meshulach, and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulse,” Phys. Rev. A 60, 1287 (1999).
[CrossRef]

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

Milner, V.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef] [PubMed]

Motzkus, M.

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

Natan, A.

H. Suchowski, A. Natan, B. D. Bruner, and Y. Silberberg, “Spatio-temporal coherent control of atomic systems: weak to strong field transition and breaking of symmetry in 2D maps,” J. Phys. At. Mol. Opt. Phys. 41, 074008 (2008).
[CrossRef]

Nicole, C.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

V. Blanchet, C. Nicole, M.-A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2717 (1997).
[CrossRef]

Ogilvie, J. P.

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

Oron, D.

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[CrossRef] [PubMed]

Péer, A.

M. C. Stowe, A. Péer, and J. Ye, “Control of four-level quantum coherence via discrete spectral shaping of an optical frequency comb,” Phys. Rev. Lett. 100, 203001 (2008).
[CrossRef] [PubMed]

N. Dudovich, T. Polack, A. Péer, and Y. Silberberg, “Simple route to strong-field coherent control,” Phys. Rev. Lett. 94, 083002 (2005).
[CrossRef] [PubMed]

Polack, T.

N. Dudovich, T. Polack, A. Péer, and Y. Silberberg, “Simple route to strong-field coherent control,” Phys. Rev. Lett. 94, 083002 (2005).
[CrossRef] [PubMed]

Präkelt, A.

M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B 82, 183–188 (2006).
[CrossRef]

Proch, D.

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

Rabitz, H.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

R. S. Judson, and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[CrossRef] [PubMed]

Rice, S. A.

D. J. Tanner, and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[CrossRef]

Ruppe, H.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

Rutz, S.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

Rybak, L.

Z. Amitay, A. Gandman, L. Chuntonov, and L. Rybak, “Multichannel selective femtosecond coherent control based on symmetry properties,” Phys. Rev. Lett. 100, 193002 (2008).
[CrossRef] [PubMed]

Sarpe-Tudoran, C.

T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, and T. Baumert, “Robust photon locking,” Phys. Rev. Lett. 102, 023004 (2009).
[CrossRef] [PubMed]

M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B 82, 183–188 (2006).
[CrossRef]

Scheriber, E.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

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, 919 (1998).
[CrossRef] [PubMed]

Shapiro, E. A.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef] [PubMed]

Shapiro, M.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef] [PubMed]

Shore, B. W.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[CrossRef]

Silberberg, Y.

H. Suchowski, A. Natan, B. D. Bruner, and Y. Silberberg, “Spatio-temporal coherent control of atomic systems: weak to strong field transition and breaking of symmetry in 2D maps,” J. Phys. At. Mol. Opt. Phys. 41, 074008 (2008).
[CrossRef]

N. Dudovich, T. Polack, A. Péer, and Y. Silberberg, “Simple route to strong-field coherent control,” Phys. Rev. Lett. 94, 083002 (2005).
[CrossRef] [PubMed]

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[CrossRef] [PubMed]

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[CrossRef] [PubMed]

D. Meshulach, and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulse,” Phys. Rev. A 60, 1287 (1999).
[CrossRef]

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

Solinas, X.

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

Spielmann, Ch.

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000).
[CrossRef]

Stock, S.

B. Chatel, J. Degert, S. Stock, and B. Girard, “Competition between sequential and direct paths in a two-photon transition,” Phys. Rev. A 68, 041402 (2003).
[CrossRef]

Stowe, M. C.

M. C. Stowe, A. Péer, and J. Ye, “Control of four-level quantum coherence via discrete spectral shaping of an optical frequency comb,” Phys. Rev. Lett. 100, 203001 (2008).
[CrossRef] [PubMed]

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, 919 (1998).
[CrossRef] [PubMed]

Suchowski, H.

H. Suchowski, A. Natan, B. D. Bruner, and Y. Silberberg, “Spatio-temporal coherent control of atomic systems: weak to strong field transition and breaking of symmetry in 2D maps,” J. Phys. At. Mol. Opt. Phys. 41, 074008 (2008).
[CrossRef]

Tanner, D. J.

D. J. Tanner, and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[CrossRef]

Theuer, H.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[CrossRef]

Tournois, P.

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000).
[CrossRef]

Trallero-Herrero, C.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a threelevel system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[CrossRef] [PubMed]

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef] [PubMed]

Verluise, F.

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000).
[CrossRef]

Weinacht, T.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a threelevel system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[CrossRef] [PubMed]

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef] [PubMed]

Weinacht, T. C.

T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Weiner, A. M.

Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. 267, 161 (2001).
[CrossRef]

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

Wollenhaupt, M.

T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, and T. Baumert, “Robust photon locking,” Phys. Rev. Lett. 102, 023004 (2009).
[CrossRef] [PubMed]

M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B 82, 183–188 (2006).
[CrossRef]

Wörste, L.

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

Ye, J.

M. C. Stowe, A. Péer, and J. Ye, “Control of four-level quantum coherence via discrete spectral shaping of an optical frequency comb,” Phys. Rev. Lett. 100, 203001 (2008).
[CrossRef] [PubMed]

Zeidler, D.

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

Zheng, Z.

Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. 267, 161 (2001).
[CrossRef]

Appl. Phys. B

T. Hournung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277 (2000).

M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, “Quantum control by selective population of dressed states using intense chirped femtosecond laser pulses,” Appl. Phys. B 82, 183–188 (2006).
[CrossRef]

Chem. Phys.

Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. 267, 161 (2001).
[CrossRef]

Eur. Phys. J. D

M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Scheriber, and L. Wörste, “Temporal coherent control induced by wave packet interferences in one and two photon atomic transitions,” Eur. Phys. J. D 2, 131 (1998).
[CrossRef]

J. Chem. Phys.

D. J. Tanner, and S. A. Rice, “Control of selectivity of chemical reaction via control of wavepacket evolution,” J. Chem. Phys. 83, 5013–5018 (1985).
[CrossRef]

J. Phys. At. Mol. Opt. Phys.

H. Suchowski, A. Natan, B. D. Bruner, and Y. Silberberg, “Spatio-temporal coherent control of atomic systems: weak to strong field transition and breaking of symmetry in 2D maps,” J. Phys. At. Mol. Opt. Phys. 41, 074008 (2008).
[CrossRef]

Nature

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature 418, 512 (2002).
[CrossRef] [PubMed]

Opt. Express

J. P. Ogilvie, D. Débarre, X. Solinas, J. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express 14, 759–766 (2006).
[CrossRef] [PubMed]

S. Lee, J. Lim, and J. Ahn, “Strong-field two-photon absorption in atomic cesium: an analytical control approach,” Opt. Express 17, 7648 (2009).
[CrossRef] [PubMed]

Opt. Lett.

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000).
[CrossRef]

Phys. Rep.

D. Goswami, “Optical pulse shaping approaches to coherent control,” Phys. Rep. 374, 385–481 (2003).
[CrossRef]

Phys. Rev. A

B. Chatel, J. Degert, S. Stock, and B. Girard, “Competition between sequential and direct paths in a two-photon transition,” Phys. Rev. A 68, 041402 (2003).
[CrossRef]

D. Meshulach, and Y. Silberberg, “Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulse,” Phys. Rev. A 60, 1287 (1999).
[CrossRef]

S. Lee, J. Lim, J. Ahn, V. Hakobyan, and S. Guerin, “Strong-field two-photon transition by phase shaping,” Phys. Rev. A 82, 023408 (2010).
[CrossRef]

Phys. Rev. Lett.

N. Dudovich, T. Polack, A. Péer, and Y. Silberberg, “Simple route to strong-field coherent control,” Phys. Rev. Lett. 94, 083002 (2005).
[CrossRef] [PubMed]

C. Trallero-Herrero, J. L. Cohen, and T. Weinacht, “Strong-field atomic phase matching,” Phys. Rev. Lett. 96, 063603 (2006).
[CrossRef] [PubMed]

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a threelevel system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[CrossRef] [PubMed]

T. Bayer, M. Wollenhaupt, C. Sarpe-Tudoran, and T. Baumert, “Robust photon locking,” Phys. Rev. Lett. 102, 023004 (2009).
[CrossRef] [PubMed]

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[CrossRef] [PubMed]

Z. Amitay, A. Gandman, L. Chuntonov, and L. Rybak, “Multichannel selective femtosecond coherent control based on symmetry properties,” Phys. Rev. Lett. 100, 193002 (2008).
[CrossRef] [PubMed]

N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. 86, 47–50 (2001).
[CrossRef] [PubMed]

R. S. Judson, and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68, 1500–1503 (1992).
[CrossRef] [PubMed]

M. C. Stowe, A. Péer, and J. Ye, “Control of four-level quantum coherence via discrete spectral shaping of an optical frequency comb,” Phys. Rev. Lett. 100, 203001 (2008).
[CrossRef] [PubMed]

V. Blanchet, C. Nicole, M.-A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transitions: from optical interferences to quantum interferences,” Phys. Rev. Lett. 78, 2717 (1997).
[CrossRef]

Rev. Mod. Phys.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[CrossRef]

Rev. Sci. Instrum.

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

Science

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

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, 919 (1998).
[CrossRef] [PubMed]

Other

M. Shapiro, and P. Brumer, Principles of the quantum control of molecular processes,(Wiley, New York, 2003).

D. E. Keller, A. E. Kramida, J. R. Fuhr, L. Podobedova, and W. L. WIese, in NIST atomic Spectra database, NIST Standard Reference Database Version 4, http://physics.nist.gov/PhysRefData/ASD/

P. Meystre, and M. SargentIII, Elements of Quantum Optics, 3rd ed. (Springer-Verlag, Berlin, 1991).

D. A. Steck, Alkali D Line Data, http://steck.us/alkalidata/

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Figures (5)

Fig. 1
Fig. 1

Energy levels and transitions of atomic sodium

Fig. 2
Fig. 2

(a) Dressed-state picture for a 2 + 1 photon absorption process in a sodium three-state model system. Three arrows indicate possible transition paths from |g〉 to |r〉 for a negatively chirped pulse interaction. (b) Schematic illustrations of dynamically Stark-shifted resonant frequencies of |3s〉 − |4s〉 and |4s〉 − |7p〉 transitions (dashed lines) of sodium atoms, overlaid with two-photon (blue) and one-photon (red) time-frequency spectrograms for the negatively chirp pulse of a2 = −5000 fs2. (c) Similar illustration for a positively chirped pulse with a2 = 5000 fs2. The spectrograms were vertically shifted to match the corresponding transitions.

Fig. 3
Fig. 3

Three-photon sodium excitation to the|7p〉 state with a chirped optical pulse. The curves show the calculation of total (solid line), sequential (blue dashed line), and direct (green dot-dash line) transitions. The experimental measurements are shown as circles. The inset shows the sequential and direct transition paths from |3s〉 to |7p〉.

Fig. 4
Fig. 4

(a) Theoretical and (b) experimental results of chirped-pulse three-photon excitation probability of sodium atoms as a function of the linear chirp rate and laser peak intensity. For the relative measurement of |7p〉 sodium atoms, the |7s〉 − |3p〉 fluorescence signal was recorded.

Fig. 5
Fig. 5

Chirped-pulse excitation of |4〉-state sodium atoms. (a) The excitation transition probability shown as a function of chirp rate at a laser transform-limited peak intensity of I = 3.0 × 1011 W/cm2. The curves show the calculation of total (solid line), sequential (dashed line), and direct (dot-dash line) transitions. The sequential transition was multiplied by 1000. The experimental results are plotted as circles. The inset shows the sequential and direct transition paths from |3s〉 to |7p〉. (b) Theoretical and (c) experimental results of the chirped-pulse excitation probability of sodium |4s〉 atoms plotted as a function of the linear chirp rate and laser peak intensity.

Equations (19)

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H ^ ( t ) = ( S g ( t ) 1 2 Ω ( t ) e i [ Δ 1 t + 2 ϕ ( t ) ] 0 1 2 Ω ( t ) e i [ Δ 1 t + 2 ϕ ( t ) ] S e ( t ) 1 2 Ω e f ( t ) e i [ Δ 2 t + ϕ ( t ) ] 0 1 2 Ω e f ( t ) e i [ Δ 2 t + ϕ ( t ) ] S f ( t ) ) ,
H ^ ( T ) ( t ) = ( S g ( t ) + Δ 1 + 2 ϕ ˙ ( t ) 1 2 Ω ( t ) 0 1 2 Ω ( t ) S e ( t ) 1 2 Ω e f ( t ) 0 1 2 Ω e f ( t ) S r ( t ) Δ 2 ϕ ˙ ( t ) ) ,
H ^ ( T ) ( t ) = ( 0 1 2 Ω ( t ) e i Q 1 ( t ) 0 1 2 Ω ( t ) e i Q 1 ( t ) 0 1 2 Ω e f ( t ) e i Q 2 ( t ) 0 1 2 Ω e f ( t ) e i Q 2 ( t ) 0 ) ,
a f = d t Ω e f ( t ) 2 e i Q 2 ( t ) t d t Ω ( t ) 2 e i Q 1 ( t ) ,
a e = i d t Ω ( t ) 2 e i Q 1 ( t ) + i d t Ω e f ( t ) 2 e i Q 2 ( t ) t d t Ω e f ( t ) 2 e i Q 2 ( t ) t d t Ω ( t ) 2 e i Q 1 ( t ) ,
a f , res = i π E ( ω r e + δ f e ) d ω E ( ω ) E ( ω e g + δ e g ω ) ,
a f , nonres = d ω E ( ω ) ω ω f e δ f e d ω E ( ω ) E ( ω f g + δ f g ω ω ) ,
E ( ω ) = π E 0 T 0 exp ( 1 4 ( ω ν ) 2 T 0 2 + i a 2 ( ω ν ) 2 ) ,
a f , res = 2 π 3 T o T p E 2 ( ν ) exp [ ( δ 1 ph 2 ( ν ) 4 + δ 2 ph 2 ( ν ) 8 ) ( T o 2 + 2 i a 2 ) ] e i ( θ + π ) / 2 ,
a f , res = 2 π 3 T o T p E 2 ( ν ) exp [ ( δ 3 ph 2 ( ν ) 12 + 3 δ s 2 8 ) ( T o 2 + 2 i a 2 ) ] e i ( θ + π ) / 2 ,
a f = a f , res seq + a f , nonres seq + a f , nonres direct ,
a f , nonres direct = 4 π 3 δ s T o T p E 3 ( ν ) exp [ δ 3 ph 2 12 ( T o 2 + 2 i a 2 ) ] e i θ .
a f , nonres seg = 2 π T o T p E 3 ( ν ) exp [ δ 3 ph 2 12 T o 2 + 2 i a 2 ] e i θ / 2 × d ω ω ω f e δ f e exp [ 3 8 ( ω ω f g + δ r g 3 ) 2 ( T o 2 + 2 i a 2 ) ] .
a f , nonres seg = 2 π 3 T o T p E 2 ( ν ) exp [ ( δ 3 ph 2 ( ν ) 12 + 3 δ s 2 8 ) ( T o 2 + 2 i a 2 ) ] e i ( θ + π ) / 2 sgn ( a 2 δ s ) = a f , res seg sgn ( a 2 δ s ) ,
a e = a e direct + a e seq ,
a e direct = i d t Ω ( t ) 2 e i Q 1 ( t ) ,
a e seq = i d t Ω e r ( t ) 2 e i Q 2 ( t ) t d t Ω e r ( t ) 2 e i Q 2 ( t ) t d t Ω ( t ) 2 e i Q 1 ( t ) .
a e direct = 2 π T o T p exp [ δ 2 ph 2 ( ν ) 8 ( T o 2 + 2 i a 2 ) ] e i ( θ + π ) / 2 ,
P e | a e direct | 2 1 1 + B 2 exp [ A 2 2 ( 1 + B 2 ) ] ,

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