Temporal model for quasi-phase matching in high-order harmonic generation

Y. Tao; S. J. Goh; H. M. J. Bastiaens; P. J. M. van der Slot; S. G. Biedron; S. V. Milton; K. -J. Boller

doi:10.1364/OE.25.003621

Optics Express
Vol. 25,
Issue 4,
pp. 3621-3638
(2017)
•https://doi.org/10.1364/OE.25.003621

Temporal model for quasi-phase matching in high-order harmonic generation

Y. Tao, S. J. Goh, H. M. J. Bastiaens, P. J. M. van der Slot, S. G. Biedron, S. V. Milton, and K. -J. Boller

Open Access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Y. Tao, S. J. Goh, H. M. J. Bastiaens, P. J. M. van der Slot, S. G. Biedron, S. V. Milton, and K. -J. Boller, "Temporal model for quasi-phase matching in high-order harmonic generation," Opt. Express 25, 3621-3638 (2017)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Related Topics
Table of Contents Category
- Nonlinear Optics
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Multiharmonic generation (190.4160)
- Ultrafast nonlinear optics (190.7110)

About this Article
History
- Original Manuscript: December 13, 2016
- Revised Manuscript: January 28, 2017
- Manuscript Accepted: January 30, 2017
- Published: February 10, 2017

Abstract

We present a model for quasi-phase matching (QPM) in high-order harmonic generation (HHG). Using a one-dimensional description, we analyze the time-dependent, ultrafast wave-vector balance to calculate the on-axis harmonic output versus time, from which we obtain the output pulse energy. Considering, as an example, periodically patterned argon gas, as may be provided with a grid in a cluster jet, we calculate the harmonic output during different time intervals within the drive laser pulse duration. We find that identifying a suitable single spatial period is not straightforward due to the complex and ultrafast plasma dynamics that underlies HHG at increased intensities. The maximum on-axis harmonic pulse energy is obtained when choosing the QPM period to phase match HHG at the leading edge of the drive laser pulse.

Full Article | PDF Article

More Like This

Quasi-phase-matching of high-order harmonics in plasma plumes: theory and experiment

V. V. Strelkov and R. A. Ganeev
Opt. Express 25(18) 21068-21083 (2017)

Quasi-phase matching and characterization of high-order harmonic generation in hollow waveguides using counterpropagating light

Amy L. Lytle, Xiaoshi Zhang, Richard L. Sandberg, Oren Cohen, Henry C. Kapteyn, and Margaret M. Murnane
Opt. Express 16(9) 6544-6566 (2008)

Quasi-phase-matched high-order harmonic generation using tunable pulse trains

Kevin O’Keeffe, David T. Lloyd, and Simon M. Hooker
Opt. Express 22(7) 7722-7732 (2014)

Previous Article Next Article

References

View by:
Article Order
Year
Author
Publication

S. Witte, V. T. Tenner, D. W. E. Noom, and K. S. E. Eikema, “Lensless diffractive imaging with ultra-broadband table-top sources: from infrared to extreme-ultraviolet wavelengths,” Light. Sci. Appl. 3, e163 (2014).
[Crossref]
Y. Mairesse, J. Higuet, N. Dudovich, D. Shafir, B. Fabre, E. Mevel, E. Constant, S. Patchkovskii, Z. Walters, M. Y. Ivanov, and O. Smirnova, “High harmonic spectroscopy of multichannel dynamics in strong-field ionization,” Phys. Rev. Lett. 104, 213601 (2010).
[Crossref] [PubMed]
S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312, 424 (2006).
[Crossref] [PubMed]
T. Popmintchev, M. C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U. S. A. 106, 10516 (2009).
[Crossref] [PubMed]
X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3, 270 (2007).
[Crossref]
A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]
E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft x-ray generation in the water window with quasi-phase matching,” Science 302, 95 (2003).
[Crossref] [PubMed]
M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]
L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A 85, 053823 (2012).
[Crossref]
L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using multi-mode polarization beating,” Phys. Rev. A 87, 023810 (2013).
[Crossref]
A. Hage, B. Landgraf, M. Taylor, M. Wunsche, T. Gangolf, H. Hoppner, M. J. Prandolini, R. Riedel, M. Schulz, F. Tavella, A. Willner, M. Yeung, G. G. Paulus, C. Spielmann, B. Dromey, and M. Zepf, “New design of a multi-jet target for quasi phase matching,” Rev. Sci. Instrum. 85, 103105 (2014).
[Crossref] [PubMed]
A. Willner, F. Tavella, M. Yeung, T. Dzelzainis, C. Kamperidis, M. Bakarezos, D. Adams, M. Schulz, R. Riedel, M. C. Hoffmann, W. Hu, J. Rossbach, M. Drescher, N. A. Papadogiannis, M. Tatarakis, B. Dromey, and M. Zepf, “Coherent control of high harmonic generation via dual-gas multijet arrays,” Phys. Rev. Lett. 107, 175002 (2011).
[Crossref] [PubMed]
A. Willner, F. Tavella, M. Yeung, T. Dzelzainis, C. Kamperidis, M. Bakarezos, D. Adams, R. Riedel, M. Schulz, M. C. Hoffmann, W. Hu, J. Rossbach, M. Drescher, V. S. Yakovlev, N. A. Papadogiannis, M. Tatarakis, B. Dromey, and M. Zepf, “Efficient control of quantum paths via dual-gas high harmonic generation,” New J. Phys. 13, 113001 (2011).
[Crossref]
M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]
D. R. Austin and J. Biegert, “Attosecond pulse shaping using partial phase matching,” New. J. Phys. 16, 113011 (2014).
[Crossref]
E. Constant, D. Garzella, P. Breger, E. Mevel, C. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. 82, 1668 (1999).
[Crossref]
S. Kazamias, D. Douillet, C. Valentin, F. Weihe, F. Auge, T. Lefrou, G. Grillon, S. Sebban, and P. Balcou, “Observation of high-contrast coherence fringes in high-order harmonic generation,” Phys. Rev. A 68, 033819 (2003).
[Crossref]
W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. iii. the photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149 (1992).
[Crossref] [PubMed]
S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]
M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Soviet Physics - JETP 64, 1191 (1986).
H. Dachraoui, T. Auguste, A. Helmstedt, P. Bartz, M. Michelswirth, N. Mueller, W. Pfeiffer, P. Salieres, and U. Heinzmann, “Interplay between absorption, dispersion and refraction in high-order harmonic generation,” J. Phys. B 42, 175402 (2009).
[Crossref]
T. Auguste, B. Carre, and P. Salieres, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Phys. Rev. A 76, 011802(R) (2007).
[Crossref]
C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]
J. F. Hergott, M. Kovacev, H. Merdji, C. Hubert, Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carre, and P. Salieres, “Extreme-ultraviolet high-order harmonic pulses in the microjoule range,” Phys. Rev. A 66, 021801(R) (2002).
[Crossref]
S. Kazamias, D. Douillet, F. Weihe, C. Valentin, A. Rousse, S. Sebban, G. Grillon, F. Auge, D. Hulin, and P. Balcou, “Global optimization of high harmonic generation,” Phys. Rev. Lett. 90, 193901 (2003).
[Crossref] [PubMed]
K. Midorikawa, Y. Nabekawa, and A. Suda, “XUV multiphoton processes with intense high-order harmonics,” Prog. Quant. Electron 32, 43 (2008).
[Crossref]
P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. 84, 073103 (2013).
[Crossref] [PubMed]
Y. Tamaki, J. Itatani, M. Obara, and K. Midorikawa, “Optimization of conversion efficiency and spatial quality of high-order harmonic generation,” Phys. Rev. A 62, 063802 (2000).
[Crossref]
A. S. Sandhu, E. Gagnon, A. Paul, I. Thomann, A. Aytle, T Keep, M. M. Murnane, H. C. Kapteyn, and I. Christov, “Generation of sub-optical-cycle, carrier-envelope-phase-insensitive, extreme-uv pulses via nonlinear stabilization in a waveguide,” Phys. Rev. A 74, 061803 (2006).
[Crossref]
I. Thomann, A. Bahabad, X. Liu, R. Trebino, M. M. Murnane, and H. C. Kapteyn, “Characterizing isolated attosecond pulses from hollow-core waveguides using multi-cycle driving pulses,” Opt. Express 17, 4611 (2009).
[Crossref] [PubMed]
R. A. Ganeev, M. Suzuki, and H. Kuroda, “Quasi-phase-matching of high-order harmonics in multiple plasma jets,” Phys. Rev. A 89, 033821 (2014).
[Crossref]
P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689 (2001).
[Crossref] [PubMed]
L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Optical rotation quasi-phase-matching for circularly polarized high harmonic generation,” Opt. Lett. 37, 2415 (2012).
[Crossref] [PubMed]

2014 (4)

S. Witte, V. T. Tenner, D. W. E. Noom, and K. S. E. Eikema, “Lensless diffractive imaging with ultra-broadband table-top sources: from infrared to extreme-ultraviolet wavelengths,” Light. Sci. Appl. 3, e163 (2014).
[Crossref]

A. Hage, B. Landgraf, M. Taylor, M. Wunsche, T. Gangolf, H. Hoppner, M. J. Prandolini, R. Riedel, M. Schulz, F. Tavella, A. Willner, M. Yeung, G. G. Paulus, C. Spielmann, B. Dromey, and M. Zepf, “New design of a multi-jet target for quasi phase matching,” Rev. Sci. Instrum. 85, 103105 (2014).
[Crossref] [PubMed]

D. R. Austin and J. Biegert, “Attosecond pulse shaping using partial phase matching,” New. J. Phys. 16, 113011 (2014).
[Crossref]

R. A. Ganeev, M. Suzuki, and H. Kuroda, “Quasi-phase-matching of high-order harmonics in multiple plasma jets,” Phys. Rev. A 89, 033821 (2014).
[Crossref]

2013 (2)

P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. 84, 073103 (2013).
[Crossref] [PubMed]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using multi-mode polarization beating,” Phys. Rev. A 87, 023810 (2013).
[Crossref]

2012 (2)

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Optical rotation quasi-phase-matching for circularly polarized high harmonic generation,” Opt. Lett. 37, 2415 (2012).
[Crossref] [PubMed]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A 85, 053823 (2012).
[Crossref]

2011 (3)

A. Willner, F. Tavella, M. Yeung, T. Dzelzainis, C. Kamperidis, M. Bakarezos, D. Adams, M. Schulz, R. Riedel, M. C. Hoffmann, W. Hu, J. Rossbach, M. Drescher, N. A. Papadogiannis, M. Tatarakis, B. Dromey, and M. Zepf, “Coherent control of high harmonic generation via dual-gas multijet arrays,” Phys. Rev. Lett. 107, 175002 (2011).
[Crossref] [PubMed]

A. Willner, F. Tavella, M. Yeung, T. Dzelzainis, C. Kamperidis, M. Bakarezos, D. Adams, R. Riedel, M. Schulz, M. C. Hoffmann, W. Hu, J. Rossbach, M. Drescher, V. S. Yakovlev, N. A. Papadogiannis, M. Tatarakis, B. Dromey, and M. Zepf, “Efficient control of quantum paths via dual-gas high harmonic generation,” New J. Phys. 13, 113001 (2011).
[Crossref]

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

2010 (2)

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]

Y. Mairesse, J. Higuet, N. Dudovich, D. Shafir, B. Fabre, E. Mevel, E. Constant, S. Patchkovskii, Z. Walters, M. Y. Ivanov, and O. Smirnova, “High harmonic spectroscopy of multichannel dynamics in strong-field ionization,” Phys. Rev. Lett. 104, 213601 (2010).
[Crossref] [PubMed]

2009 (3)

T. Popmintchev, M. C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U. S. A. 106, 10516 (2009).
[Crossref] [PubMed]

H. Dachraoui, T. Auguste, A. Helmstedt, P. Bartz, M. Michelswirth, N. Mueller, W. Pfeiffer, P. Salieres, and U. Heinzmann, “Interplay between absorption, dispersion and refraction in high-order harmonic generation,” J. Phys. B 42, 175402 (2009).
[Crossref]

I. Thomann, A. Bahabad, X. Liu, R. Trebino, M. M. Murnane, and H. C. Kapteyn, “Characterizing isolated attosecond pulses from hollow-core waveguides using multi-cycle driving pulses,” Opt. Express 17, 4611 (2009).
[Crossref] [PubMed]

2008 (1)

K. Midorikawa, Y. Nabekawa, and A. Suda, “XUV multiphoton processes with intense high-order harmonics,” Prog. Quant. Electron 32, 43 (2008).
[Crossref]

2007 (3)

T. Auguste, B. Carre, and P. Salieres, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Phys. Rev. A 76, 011802(R) (2007).
[Crossref]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3, 270 (2007).
[Crossref]

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]

2006 (2)

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312, 424 (2006).
[Crossref] [PubMed]

A. S. Sandhu, E. Gagnon, A. Paul, I. Thomann, A. Aytle, T Keep, M. M. Murnane, H. C. Kapteyn, and I. Christov, “Generation of sub-optical-cycle, carrier-envelope-phase-insensitive, extreme-uv pulses via nonlinear stabilization in a waveguide,” Phys. Rev. A 74, 061803 (2006).
[Crossref]

2003 (3)

S. Kazamias, D. Douillet, F. Weihe, C. Valentin, A. Rousse, S. Sebban, G. Grillon, F. Auge, D. Hulin, and P. Balcou, “Global optimization of high harmonic generation,” Phys. Rev. Lett. 90, 193901 (2003).
[Crossref] [PubMed]

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft x-ray generation in the water window with quasi-phase matching,” Science 302, 95 (2003).
[Crossref] [PubMed]

S. Kazamias, D. Douillet, C. Valentin, F. Weihe, F. Auge, T. Lefrou, G. Grillon, S. Sebban, and P. Balcou, “Observation of high-contrast coherence fringes in high-order harmonic generation,” Phys. Rev. A 68, 033819 (2003).
[Crossref]

2002 (1)

J. F. Hergott, M. Kovacev, H. Merdji, C. Hubert, Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carre, and P. Salieres, “Extreme-ultraviolet high-order harmonic pulses in the microjoule range,” Phys. Rev. A 66, 021801(R) (2002).
[Crossref]

2001 (1)

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689 (2001).
[Crossref] [PubMed]

2000 (1)

Y. Tamaki, J. Itatani, M. Obara, and K. Midorikawa, “Optimization of conversion efficiency and spatial quality of high-order harmonic generation,” Phys. Rev. A 62, 063802 (2000).
[Crossref]

1999 (2)

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

E. Constant, D. Garzella, P. Breger, E. Mevel, C. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. 82, 1668 (1999).
[Crossref]

1994 (1)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

1992 (1)

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. iii. the photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149 (1992).
[Crossref] [PubMed]

1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Soviet Physics - JETP 64, 1191 (1986).

Adams, D.

Agostini, P.

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Soviet Physics - JETP 64, 1191 (1986).

Aquila, A.

Attwood, D. T.

Auge, F.

Auguste, T.

T. Auguste, B. Carre, and P. Salieres, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Phys. Rev. A 76, 011802(R) (2007).
[Crossref]

Austin, D. R.

D. R. Austin and J. Biegert, “Attosecond pulse shaping using partial phase matching,” New. J. Phys. 16, 113011 (2014).
[Crossref]

Aytle, A.

Backus, S.

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

Bahabad, A.

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]

Bakarezos, M.

Baker, S.

Balcou, P.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Bartz, P.

Biegert, J.

D. R. Austin and J. Biegert, “Attosecond pulse shaping using partial phase matching,” New. J. Phys. 16, 113011 (2014).
[Crossref]

Breger, P.

Brion, C. E.

Brizuela, F.

Burton, G. R.

Campi, F.

Carre, B.

T. Auguste, B. Carre, and P. Salieres, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Phys. Rev. A 76, 011802(R) (2007).
[Crossref]

Cassou, K.

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

Chan, W. F.

Chen, M. C.

Chirila, C. C.

Christov, I.

Christov, I. P.

Cohen, O.

Constant, E.

Cooper, G.

Corkum, P. B.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Cros, B.

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

Daboussi, S.

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

Dachraoui, H.

Delone, N. B.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Soviet Physics - JETP 64, 1191 (1986).

Dorrer, C.

Douillet, D.

Drescher, M.

Dromey, B.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]

Dudovich, N.

Durfee, C. G.

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

Dzelzainis, T.

Eikema, K. S. E.

Fabre, B.

Foster, P.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]

Gagnon, E.

Ganeev, R. A.

R. A. Ganeev, M. Suzuki, and H. Kuroda, “Quasi-phase-matching of high-order harmonics in multiple plasma jets,” Phys. Rev. A 89, 033821 (2014).
[Crossref]

Gangolf, T.

Garzella, D.

Gaudiosi, D.

Gerrity, M.

Gibson, E. A.

Grillon, G.

Guilbaud, O.

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

Gullikson, E. M.

Guo, X.

Hage, A.

Haworth, C. A.

Heinzmann, U.

Helmstedt, A.

Hergott, J. F.

Herne, C.

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

Heyl, C. M.

Higuet, J.

Hoffmann, M. C.

Hooker, S. M.

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using multi-mode polarization beating,” Phys. Rev. A 87, 023810 (2013).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A 85, 053823 (2012).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Optical rotation quasi-phase-matching for circularly polarized high harmonic generation,” Opt. Lett. 37, 2415 (2012).
[Crossref] [PubMed]

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]

Hoppner, H.

Hu, W.

Hubert, C.

Hulin, D.

Itatani, J.

Y. Tamaki, J. Itatani, M. Obara, and K. Midorikawa, “Optimization of conversion efficiency and spatial quality of high-order harmonic generation,” Phys. Rev. A 62, 063802 (2000).
[Crossref]

Ivanov, M. Y.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Jean, E.

Johnsson, P.

Kamperidis, C.

Kapteyn, H. C.

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

Kazamias, S.

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

Keep, T

Kim, B.

Kovacev, M.

Krainov, V. P.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Soviet Physics - JETP 64, 1191 (1986).

Kuroda, H.

R. A. Ganeev, M. Suzuki, and H. Kuroda, “Quasi-phase-matching of high-order harmonics in multiple plasma jets,” Phys. Rev. A 89, 033821 (2014).
[Crossref]

L’Huillier, A.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Landgraf, B.

Landreman, M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]

Le Blanc, C.

Lefrou, T.

Lein, M.

Lewenstein, M.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Liu, L. Z.

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using multi-mode polarization beating,” Phys. Rev. A 87, 023810 (2013).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A 85, 053823 (2012).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Optical rotation quasi-phase-matching for circularly polarized high harmonic generation,” Opt. Lett. 37, 2415 (2012).
[Crossref] [PubMed]

Liu, X.

Lytle, A. L.

Mairesse, Y.

Mansten, E.

Marangos, J. P.

Maynard, G.

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

Merdji, H.

Mevel, E.

Michelswirth, M.

Midorikawa, K.

K. Midorikawa, Y. Nabekawa, and A. Suda, “XUV multiphoton processes with intense high-order harmonics,” Prog. Quant. Electron 32, 43 (2008).
[Crossref]

Y. Tamaki, J. Itatani, M. Obara, and K. Midorikawa, “Optimization of conversion efficiency and spatial quality of high-order harmonic generation,” Phys. Rev. A 62, 063802 (2000).
[Crossref]

Mueller, N.

Muller, H. G.

Mullot, G.

Murnane, M. M.

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

Nabekawa, Y.

K. Midorikawa, Y. Nabekawa, and A. Suda, “XUV multiphoton processes with intense high-order harmonics,” Prog. Quant. Electron 32, 43 (2008).
[Crossref]

Noom, D. W. E.

O’Keeffe, K.

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using multi-mode polarization beating,” Phys. Rev. A 87, 023810 (2013).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A 85, 053823 (2012).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Optical rotation quasi-phase-matching for circularly polarized high harmonic generation,” Opt. Lett. 37, 2415 (2012).
[Crossref] [PubMed]

Obara, M.

Y. Tamaki, J. Itatani, M. Obara, and K. Midorikawa, “Optimization of conversion efficiency and spatial quality of high-order harmonic generation,” Phys. Rev. A 62, 063802 (2000).
[Crossref]

Papadogiannis, N. A.

Patchkovskii, S.

Paul, A.

Paul, P. M.

Paulus, G. G.

Persson, A.

Pfeiffer, W.

Popmintchev, T.

Prandolini, M. J.

Rading, L.

Rakowski, R.

Riedel, R.

Robinson, J. S.

Ros, D.

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

Rossbach, J.

Rousse, A.

Rudawski, P.

Rundquist, A. R.

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

Salieres, P.

T. Auguste, B. Carre, and P. Salieres, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Phys. Rev. A 76, 011802(R) (2007).
[Crossref]

Salin, F.

Sandhu, A. S.

Schulz, M.

Schwenke, J.

Sebban, S.

Shafir, D.

Sidorenko, P.

Smirnova, O.

Smith, R. A.

Spielmann, C.

Suda, A.

K. Midorikawa, Y. Nabekawa, and A. Suda, “XUV multiphoton processes with intense high-order harmonics,” Prog. Quant. Electron 32, 43 (2008).
[Crossref]

Suzuki, M.

R. A. Ganeev, M. Suzuki, and H. Kuroda, “Quasi-phase-matching of high-order harmonics in multiple plasma jets,” Phys. Rev. A 89, 033821 (2014).
[Crossref]

Tamaki, Y.

Y. Tamaki, J. Itatani, M. Obara, and K. Midorikawa, “Optimization of conversion efficiency and spatial quality of high-order harmonic generation,” Phys. Rev. A 62, 063802 (2000).
[Crossref]

Tatarakis, M.

Tavella, F.

Taylor, M.

Teng, H.

Tenner, V. T.

Thomann, I.

Tisch, J. W. G.

Tobey, R.

Toma, E. S.

Trebino, R.

Valentin, C.

Wagner, N.

Walters, Z.

Weihe, F.

Willner, A.

Witte, S.

Wunsche, M.

Yakovlev, V. S.

Yeung, M.

Zepf, M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]

Zhang, X.

Zhou, X.

J. Phys. B (1)

Light. Sci. Appl. (1)

Nat. Photon. (1)

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]

Nat. Phys. (1)

New J. Phys. (1)

New. J. Phys. (1)

D. R. Austin and J. Biegert, “Attosecond pulse shaping using partial phase matching,” New. J. Phys. 16, 113011 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Optical rotation quasi-phase-matching for circularly polarized high harmonic generation,” Opt. Lett. 37, 2415 (2012).
[Crossref] [PubMed]

Phys. Rev. A (11)

R. A. Ganeev, M. Suzuki, and H. Kuroda, “Quasi-phase-matching of high-order harmonics in multiple plasma jets,” Phys. Rev. A 89, 033821 (2014).
[Crossref]

Y. Tamaki, J. Itatani, M. Obara, and K. Midorikawa, “Optimization of conversion efficiency and spatial quality of high-order harmonic generation,” Phys. Rev. A 62, 063802 (2000).
[Crossref]

T. Auguste, B. Carre, and P. Salieres, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Phys. Rev. A 76, 011802(R) (2007).
[Crossref]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A 83, 063405 (2011).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A 85, 053823 (2012).
[Crossref]

L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using multi-mode polarization beating,” Phys. Rev. A 87, 023810 (2013).
[Crossref]

Phys. Rev. Lett. (6)

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 143901 (2007).
[Crossref] [PubMed]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187 (1999).
[Crossref]

Proc. Natl. Acad. Sci. U. S. A. (1)

Prog. Quant. Electron (1)

K. Midorikawa, Y. Nabekawa, and A. Suda, “XUV multiphoton processes with intense high-order harmonics,” Prog. Quant. Electron 32, 43 (2008).
[Crossref]

Rev. Sci. Instrum. (2)

Science (3)

Soviet Physics - JETP (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Soviet Physics - JETP 64, 1191 (1986).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1 The ionization fraction, η, (red line) and the harmonic dipole amplitude, d₃₉, for the 39^th harmonic (≈ 20 nm, blue line) calculated for argon and a drive laser pulse (Gaussian temporal shape) with a central wavelength at 800 nm, a pulse duration of τ = 40 fs (FWHM) and a peak intensity of I₀ = 2.5 × 10¹⁴ W/cm². The drive laser pulse envelop (black dashed line) is plotted for reference.

Download Full Size | PPT Slide | PDF

Fig. 2 Calculated coherence lengths (black curves) for the 21^st harmonic (a) and the 39^th harmonic (b) as a function of time. Assumed is harmonic generation in 30 mbar of argon in a 5 mm long medium (L_med = 5 mm, dashed blue lines) with a Gaussian-shaped laser pulse of 2.5 × 10¹⁴ W/cm² peak intensity (blue dashed curves). The absorption lengths (dashed pink lines) and the minimum required laser intensity (solid orange lines) are calculated for both harmonics and plotted for comparison. The green areas show within which time interval there is a sufficiently high drive laser intensity for HHG (I > I_thr). The red areas indicate the time interval during which the phase-matching condition is approximately fulfilled, i.e., the harmonic output from the entire medium adds constructively (L_coh > L_med), if the respective harmonic is generated.

Download Full Size | PPT Slide | PDF

Fig. 3 Calculated HH pulse energy for the 21^st (black) and the 39^th harmonic (red) as a function of the medium length. For L_med > 1 mm, the 39^th harmonic output is about four to five orders of magnitude weaker.

Download Full Size | PPT Slide | PDF

Fig. 4 (a) Schematic of a density modulated medium consisting of alternating regions with high and low gas density. (b) The corresponding gas density profile can be described via a maximum gas density, ρ, and a modulation depth, κ. The lengths of the high-density and low-density regions are L_high and L_low, respectively.

Download Full Size | PPT Slide | PDF

Fig. 5 Calculated coherence lengths in the low-density region for the 39^th harmonic (≈ 20 nm) as a function of time for various modulation depths, κ (1 (purple), 0.8 (red), 0.6 (grey), 0.4 (cyan), 0.2 (orange) and 0 (black)). The ionization fraction (dark green curve) and the temporal profile of the drive laser intensity (blue dashed curve) are plotted for reference. The red dashed line is the threshold intensity for the 39^th harmonic generation and the green area indicates the time interval (from t₋₁ to t₁) during which the generation of the 39^th harmonic is possible. The generating medium is 5 mm long and the gas pressure is set at 30 mbar in the high-density region. The drive laser pulse has a peak intensity of 2.5 × 10¹⁴ W/cm².

Download Full Size | PPT Slide | PDF

Fig. 6 Growth of the on-axis HH intensity, I₃₉, throughout the density modulated medium for (a) κ=1, (b) κ=0.8 and (c) κ=0.6 when the modulation period, P_mod, is chosen to induce QPM at the peak of the drive laser pulse (t₀=0 fs). The black curves show the growth of I₃₉ generated at the drive laser peak, while the other curves show the growth of I₃₉ at an earlier and later time in the pulse (t₁=−5.7 fs (red) and t₃=5.7 fs (blue)). The propagation coordinate is expressed relative to the modulation period. The pink dashed line represents I₃₉ generated at t₀ in a homogeneous medium, without density modulation (κ=0).

Download Full Size | PPT Slide | PDF

Fig. 7 (a) calculated coherence lengths enlarged from Fig. 5. The purple horizontal line is the coherence length in the low-density regions, while the black curve is the coherence length in the high-density regions. The modulation depth is set to κ=1. For the calculation of the HH output pulse energy, seven different modulation periods are chosen corresponding to providing QPM at seven selected instances in the drive laser pulse (from t₋₁ to t₁, dotted lines). (b) On-axis HH pulse energy for the 39^th harmonic versus the length of the medium obtained with the seven different modulation periods. The highest output is achieved with P_mod chosen to provide transient QPM in the rising edge of the pulse (t₋₃, thick blue line). The pink dashed line of the lowest output is obtained without density modulation. (c) Variation of HH output behind the medium vs. the modulation period (lower axis) and vs. the time chosen for QPM (upper axis).

Download Full Size | PPT Slide | PDF

Tables (1)

Table 1 List of coherence lengths in the high-density and low-density regions and the modulation periods (P_mod) chosen to provide QPM at seven different instances within the drive laser pulse when the 39^th harmonic is generated.

View Table

Equations (17)

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

E_{q} (t) \propto \int_{0}^{L_{med}} ρ (z) exp (- \frac{L_{med} - z}{2 L_{abs} (z)}) d_{q} (z, t) [1 - η (t)] exp [i ϕ_{q} (z, t)] d z .

d_{q} (z, t) {\begin{array}{l} \propto {(\frac{I (z, t)}{I_{t h r} (q)})}^{4.6} & if & I < I_{t h r} (q) \\ \approx 0 & if & I < I_{thr} (q) \end{array}

ϕ_{q} (L_{med}, t) = \int_{0}^{L_{med}} Δ k (z, t) d z .

Δ k (z, t) = Δ k_{atom} (z, t) + Δ k_{plasma} (z, t) + Δ t_{geo} (z) + Δ k_{dipole} (z, t) .

Δ k_{atom} (z, t) = q [1 - η (z, t)] \frac{2 π P (z)}{λ} Δ n .

Δ k_{plasma} (z, t) = - η (z, t) P (z) N_{a} r_{e} \frac{q^{2} - 1}{q} λ,

Δ k_{geo} (z) = - \frac{1}{z_{R} + \frac{(z - z_{0})}{z_{R}}} (q - 1),

Δ k_{dipole} (z, t) = - α_{q} \frac{\partial I (z, t)}{\partial z} = \frac{2 (z - z_{0})}{z_{R}^{2} {[1 + {(\frac{(z - z_{0})}{z_{R}})}^{2}]}^{2}} α_{q} I_{0},

F_{q} = S \int_{- \infty}^{\infty} I_{q} (t) d t .

ϕ_{q} (z, t) = Δ k (t) \cdot z = \frac{π z}{L_{coh} (t)} .

\begin{array}{l} I_{q} (t) \propto ρ^{2} {[1 - η (t)]}^{2} d_{q} {(t)}^{2} \frac{4 L_{abs}^{2}}{1 + 4 π^{2} [L_{abs}^{2} / L_{coh} {(t)}^{2}]} \\ \times [1 + exp (- \frac{L_{med}}{L_{abs}}) - 2 cos (π \frac{L_{med}}{L_{coh} (t)}) exp (- \frac{L_{med}}{2 L_{abs}})] . \end{array}

E_{q} (t) = \sum_{n = 1}^{N} [E_{q, n}^{high} (t) + E_{q, n}^{low} (t)],

ϕ_{q, n} (z, t) = {\begin{array}{r} (n - 1) & [Δ k_{high} (t) L_{high} + Δ k_{low} (t) L_{low}] \\ + Δ k_{high} (t) \cdot z & in any high - density region, \\ (n - 1) & [Δ k_{high} (t) L_{high} + Δ k_{low} (t) L_{low}] \\ + Δ k_{high} (t) L_{high} + Δ k_{low} (t) \cdot z & in any low - density region . \end{array}

E_{q} (t) = ξ_{q}^{high} (t) + ξ_{q}^{low} (t),

ξ_{q}^{high} (t) \propto \frac{ρ d_{q} (t)}{i π / L_{coh}^{high} (t)} [exp (i \frac{π L_{high}}{L_{coh}^{high} (t)}) - 1] \frac{1 - exp [i N π (\frac{L_{high}}{L_{coh}^{high} (t)} + \frac{L_{low}}{L_{coh}^{low} (t)})]}{1 - exp [i π (\frac{L_{high}}{L_{coh}^{high} (t)} + \frac{L_{low}}{L_{coh}^{low} (t)})]},

\begin{array}{l} ξ_{q}^{low} (t) \propto \frac{(1 - κ) ρ d_{q} (t)}{i π / L_{coh}^{low} (t)} [exp (i \frac{π L_{low}}{L_{coh}^{low} (t)}) - 1] exp [i \frac{π L_{high}}{L_{coh}^{high} (t)}] \\ \times \frac{1 - exp [i N π (\frac{L_{high}}{L_{coh}^{high} (t)} + \frac{L_{low}}{L_{coh}^{low} (t)})]}{1 - exp [i π (\frac{L_{high}}{L_{coh}^{high} (t)} + \frac{L_{low}}{L_{coh}^{low} (t)})]} . \end{array}

I_{q} (t) = \frac{1}{2} c ε_{0} {| ξ_{q}^{high} (t) + ξ_{q}^{low} (t) |}^{2} .

	L_high (mm)	L_low (mm)	P_mod (mm)
t₋₁ =−5.7 fs	0.63	1.50	2.13
t₋₂ =−4.2 fs	0.48	1.50	1.98
t₋₃ =−2.8 fs	0.39	1.50	1.89
t₋₄ =−1.4 fs	0.32	1.50	1.82
t₀ = 0 fs	0.27	1.50	1.77
t₂ = 2.8 fs	0.20	1.50	1.70
t₁ = 5.7 fs	0.17	1.50	1.67