Abstract

We show theoretically as well as experimentally that the Gouy-phase shift, which depends on the exact positioning of a sample in relation to the focus of a probe beam in a pump-probe experiment, may have a pronounced effect on the shape of the pump-probe signal. The effect occurs only when single-cycle probe pulses are used, i.e. when the slowly varying envelope approximation breaks down, while it disappears for multi-cycle pulses. The effect is thus most relevant in THz time-resolved spectroscopy, where such single cycle pulses are most commonly used, but it should not be overlooked also in other spectral regimes when correspondingly short pulses are involved.

© 2014 Optical Society of America

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2013

C.-Y. Chung, J. Hsu, S. Mukamel, E. O. Potma, “Controlling stimulated coherent spectroscopy and microscopy by a position-dependent phase,” Phys. Rev. A 87, 033833, 2013).
[CrossRef]

J. Savolainen, S. Ahmed, P. Hamm, “2D Raman-Thz spectroscopy of water,” Proc. Natl. Acad. Sci. U. S. A. 110, 20402–20407 (2013).
[CrossRef] [PubMed]

2012

P. Hamm, J. Savolainen, “2D-Raman-THz spectroscopy of water: Theory,” J. Chem. Phys. 136, 094516, 2012).
[CrossRef]

T. Kobayashi, J. Liu, K. Okamura, “Applications of parametric processes to high-quality multicolour ultra-short pulses, pulse cleaning and cep stable sub-3fs pulse,” J. Phys. B At. Mol. Opt. Phys. 45, 074005, 2012).
[CrossRef]

K. W. Kim, A. Pashkin, H. Schaefer, M. Beyer, M. Porer, T. Wolf, C. Bernhard, J. Demsar, R. Huber, A. Leitenstorfer, “Ultrafast transient generation of spin-density-wave order in the normal state of bafe2as2 driven by coherent lattice vibrations,” Nat. Mater. 11, 497–501 (2012).
[CrossRef] [PubMed]

L. Zhang, H. Zhong, K. Mu, C. Zhang, Y. Zhao, “Phase characterization in broadband THz wave detection through field-induced second harmonic generation,” Opt. Express 20, 75–80 (2012).
[CrossRef] [PubMed]

M. Cheng, A. Reynolds, H. Widgren, M. Khalil, “Generation of tunable octave-spanning mid-infrared pulses by filamentation in gas media,” Opt. Lett. 37, 1787–1789 (2012).
[CrossRef] [PubMed]

2011

M. Theuer, S. S. Harsha, D. Molter, G. Torosyan, R. Beigang, “Terahertz time-domain spectroscopy of gases, liquids, and solids,” Chemphyschem 12, 2695–2705 (2011).
[CrossRef] [PubMed]

2010

2009

F. Krausz, M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

E. Pickwell, V. P. Wallace., “Biomedical applications of terahertz technology,” J. Phys. D Appl. Phys. 02, R301 (2009).

M. Breusing, C. Ropers, T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. 102, 086809, 2009).
[CrossRef] [PubMed]

2008

P. A. George, J. Strait, J. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, M. G. Spencer, “Ultrafast optical-pump terahertz-probe spectroscopy of the carrier relaxation and recombination dynamics in epitaxial graphene,” Nano Lett. 8, 4248–4251 (2008).
[CrossRef]

2007

K. Reimann, “Table-top sources of ultrafast THz pulses,” Rep. Prog. Phys. 70, 1597–1632 (2007).
[CrossRef]

N. Lastzka, R. Schnabel, “The Gouy phase shift in nonlinear interactions of waves,” Opt. Express 15, 7211–7217 (2007).
[CrossRef] [PubMed]

P. Kužel, H. Němec, F. Kadlec, “Propagation of THz pulses in photoexcited media: Analytical theory for layered systems,” J. Chem. Phys. 127, 024506, 2007).
[CrossRef]

D. Polli, M. Rini, S. Wall, R. W. Schoenlein, Y. Tomioka, Y. Tokura, G. Cerullo, A. Cavalleri, “Coherent orbital waves in the photo-induced insulator-metal dynamics of a magnetoresistive manganite,” Nat. Mater. 6, 643–647 (2007).
[CrossRef] [PubMed]

2005

T. Kampfrath, L. Perfetti, F. Schapper, C. Frischkorn, M. Wolf, “Strongly coupled optical phonons in the ultrafast dynamics of the electronic energy and current relaxation in graphite,” Phys. Rev. Lett. 95, 187403, 2005).
[CrossRef] [PubMed]

T. Tritschler, K. D. Hof, M. W. Klein, M. Wegener, “Variation of the carrier-envelope phase of few-cycle laser pulses owing to the Gouy phase: a solid-state-based measurement,” Opt. Lett. 30, 753–755 (2005).
[CrossRef] [PubMed]

2004

E. Knoesel, M. Bonn, J. Shan, F. Wang, T. F. Heinz, “Conductivity of solvated electrons in hexane investigated with terahertz time-domain spectroscopy,” J. Chem. Phys. 121, 394–404 (2004).
[CrossRef] [PubMed]

J. Faure, J. van Tilborg, R. A. Kaindl, W. P. Leemans, “Modelling laser based table-top THz sources: Optical rectification, propagation and electro-optic sampling,” Opt. Quantum Electron. 36, 681–697 (2004).
[CrossRef]

C. A. Schmuttenmaer, “Exploring dynamics in the far-infrared with terahertz spectroscopy,” Chem. Rev. 104, 1759–1779 (2004).
[CrossRef] [PubMed]

2002

K. J. Siebert, T. Loffler, H. Quast, M. Thomson, T. Bauer, R. Leonhardt, S. Czasch, H. G. Roskos, “Alloptoelectronic continuous wave THz imaging for biomedical applications,” Phys. Med. Biol. 47, 3743–3748 (2002).
[CrossRef] [PubMed]

H. Němec, F. Kadlec, P. Kužel, “Methodology of an optical pump-terahetz probe experiment: An analytical frequency domain approach,” J. Chem. Phys. 117, 8454, 2002).
[CrossRef]

B. Ferguson, X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[CrossRef]

T. Feurer, N. S. Stoyanov, D. W. Ward, K. A. Nelson, “Direct visualization of the Gouy physe by focusing phonon polaritons,” Phys. Rev. Lett. 88, 257402, 2002).
[CrossRef]

2001

P. U. Jepsen, W. Schairer, I. H. Libon, U. Lemmer, N. Hecker, M. Birkholz, K. Lips, M. Schall, “Ultrafast carrier trapping in microcrystalline silicon observed in optical pumpterahertz probe measurements,” Appl. Phys. Lett. 79, 1291, 2001).
[CrossRef]

1999

J. T. Kindt, C. A. Schmuttenmaer, “Theory for determination of the low-frequency time-dependent response function in liquids using time-resolved terahertz pulse spectroscopy,” J. Chem. Phys. 110, 8589, 1999).
[CrossRef]

P. Kužel, M. A. Khazan, J. Kroupa, “Spatiotemporal transformations of ultrashort terahertz pulses,” J. Opt. Soc. Am. B 16, 1795–1800 (1999).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy physe shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

1998

1997

G. Haran, W. D. Sun, K. Wynne, R. M. Hochstrasser, “Femtosecond far-infrared pump-probe spectroscopy: a new tool for studying low-frequency vibrational dynamics in molecular condensed phases,” Chem. Phys. Lett. 274, 365–371 (1997).
[CrossRef]

R. McElroy, K. Wynne, “Ultrafast dipole solvation measured in the far infrared,” Phys. Rev. Lett. 79, 3078–3081 (1997).
[CrossRef]

D. You, P. H. Bucksbaum, “Propagation of half-cycle far infrared pulses,” J. Opt. Soc. Am. B 14, 1651–1655 (1997).
[CrossRef]

1996

J. Zielbauer, M. Wegener, “Ultrafast optical pump THz-probe spectroscopy on silicon,” Appl. Phys. Lett. 68, 1223, 1996).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[CrossRef]

1987

M. C. Nuss, D. H. Auston, F. Capasso, “Direct subpicosecond measurement of carrier mobility of photoex-cited electrons in gallium arsenide,” Phys. Rev. Lett. 58, 2355–2358 (1987).
[CrossRef] [PubMed]

1890

C. R. Gouy, “Sur une propriete nouvelle des ondes lumineuses,” Acad. Sci. Paris 110, 1251, 1890).

Ahmed, S.

J. Savolainen, S. Ahmed, P. Hamm, “2D Raman-Thz spectroscopy of water,” Proc. Natl. Acad. Sci. U. S. A. 110, 20402–20407 (2013).
[CrossRef] [PubMed]

Arnett, D. C.

B. N. Flanders, D. C. Arnett, N. F. Scherer, “Optical pump-terahertz probe spectroscopy utilizing a cavity-dumped oscillator- driven terahertz spectrometer,” IEEE J. Sel. Top. Quantum Electron. 4, 353–359 (1998).
[CrossRef]

Auston, D. H.

M. C. Nuss, D. H. Auston, F. Capasso, “Direct subpicosecond measurement of carrier mobility of photoex-cited electrons in gallium arsenide,” Phys. Rev. Lett. 58, 2355–2358 (1987).
[CrossRef] [PubMed]

Bauer, T.

K. J. Siebert, T. Loffler, H. Quast, M. Thomson, T. Bauer, R. Leonhardt, S. Czasch, H. G. Roskos, “Alloptoelectronic continuous wave THz imaging for biomedical applications,” Phys. Med. Biol. 47, 3743–3748 (2002).
[CrossRef] [PubMed]

Beigang, R.

M. Theuer, S. S. Harsha, D. Molter, G. Torosyan, R. Beigang, “Terahertz time-domain spectroscopy of gases, liquids, and solids,” Chemphyschem 12, 2695–2705 (2011).
[CrossRef] [PubMed]

Bernhard, C.

K. W. Kim, A. Pashkin, H. Schaefer, M. Beyer, M. Porer, T. Wolf, C. Bernhard, J. Demsar, R. Huber, A. Leitenstorfer, “Ultrafast transient generation of spin-density-wave order in the normal state of bafe2as2 driven by coherent lattice vibrations,” Nat. Mater. 11, 497–501 (2012).
[CrossRef] [PubMed]

Beyer, M.

K. W. Kim, A. Pashkin, H. Schaefer, M. Beyer, M. Porer, T. Wolf, C. Bernhard, J. Demsar, R. Huber, A. Leitenstorfer, “Ultrafast transient generation of spin-density-wave order in the normal state of bafe2as2 driven by coherent lattice vibrations,” Nat. Mater. 11, 497–501 (2012).
[CrossRef] [PubMed]

Birkholz, M.

P. U. Jepsen, W. Schairer, I. H. Libon, U. Lemmer, N. Hecker, M. Birkholz, K. Lips, M. Schall, “Ultrafast carrier trapping in microcrystalline silicon observed in optical pumpterahertz probe measurements,” Appl. Phys. Lett. 79, 1291, 2001).
[CrossRef]

Bonn, M.

E. Knoesel, M. Bonn, J. Shan, F. Wang, T. F. Heinz, “Conductivity of solvated electrons in hexane investigated with terahertz time-domain spectroscopy,” J. Chem. Phys. 121, 394–404 (2004).
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Breusing, M.

M. Breusing, C. Ropers, T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. 102, 086809, 2009).
[CrossRef] [PubMed]

Bucksbaum, P. H.

Capasso, F.

M. C. Nuss, D. H. Auston, F. Capasso, “Direct subpicosecond measurement of carrier mobility of photoex-cited electrons in gallium arsenide,” Phys. Rev. Lett. 58, 2355–2358 (1987).
[CrossRef] [PubMed]

Cavalleri, A.

D. Polli, M. Rini, S. Wall, R. W. Schoenlein, Y. Tomioka, Y. Tokura, G. Cerullo, A. Cavalleri, “Coherent orbital waves in the photo-induced insulator-metal dynamics of a magnetoresistive manganite,” Nat. Mater. 6, 643–647 (2007).
[CrossRef] [PubMed]

Cerullo, G.

D. Polli, M. Rini, S. Wall, R. W. Schoenlein, Y. Tomioka, Y. Tokura, G. Cerullo, A. Cavalleri, “Coherent orbital waves in the photo-induced insulator-metal dynamics of a magnetoresistive manganite,” Nat. Mater. 6, 643–647 (2007).
[CrossRef] [PubMed]

G. Cerullo, M. Nisoli, S. Stagira, S. D. Silvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283–1285, 1998).
[CrossRef]

Chandrashekhar, M.

P. A. George, J. Strait, J. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, M. G. Spencer, “Ultrafast optical-pump terahertz-probe spectroscopy of the carrier relaxation and recombination dynamics in epitaxial graphene,” Nano Lett. 8, 4248–4251 (2008).
[CrossRef]

Cheng, M.

Chung, C.-Y.

C.-Y. Chung, J. Hsu, S. Mukamel, E. O. Potma, “Controlling stimulated coherent spectroscopy and microscopy by a position-dependent phase,” Phys. Rev. A 87, 033833, 2013).
[CrossRef]

Czasch, S.

K. J. Siebert, T. Loffler, H. Quast, M. Thomson, T. Bauer, R. Leonhardt, S. Czasch, H. G. Roskos, “Alloptoelectronic continuous wave THz imaging for biomedical applications,” Phys. Med. Biol. 47, 3743–3748 (2002).
[CrossRef] [PubMed]

Dawlaty, J.

P. A. George, J. Strait, J. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, M. G. Spencer, “Ultrafast optical-pump terahertz-probe spectroscopy of the carrier relaxation and recombination dynamics in epitaxial graphene,” Nano Lett. 8, 4248–4251 (2008).
[CrossRef]

Demsar, J.

K. W. Kim, A. Pashkin, H. Schaefer, M. Beyer, M. Porer, T. Wolf, C. Bernhard, J. Demsar, R. Huber, A. Leitenstorfer, “Ultrafast transient generation of spin-density-wave order in the normal state of bafe2as2 driven by coherent lattice vibrations,” Nat. Mater. 11, 497–501 (2012).
[CrossRef] [PubMed]

Elsaesser, T.

M. Breusing, C. Ropers, T. Elsaesser, “Ultrafast carrier dynamics in graphite,” Phys. Rev. Lett. 102, 086809, 2009).
[CrossRef] [PubMed]

Faure, J.

J. Faure, J. van Tilborg, R. A. Kaindl, W. P. Leemans, “Modelling laser based table-top THz sources: Optical rectification, propagation and electro-optic sampling,” Opt. Quantum Electron. 36, 681–697 (2004).
[CrossRef]

Feng, S.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy physe shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and tempral reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

Ferguson, B.

B. Ferguson, X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[CrossRef]

Feurer, T.

T. Feurer, N. S. Stoyanov, D. W. Ward, K. A. Nelson, “Direct visualization of the Gouy physe by focusing phonon polaritons,” Phys. Rev. Lett. 88, 257402, 2002).
[CrossRef]

Flanders, B. N.

B. N. Flanders, D. C. Arnett, N. F. Scherer, “Optical pump-terahertz probe spectroscopy utilizing a cavity-dumped oscillator- driven terahertz spectrometer,” IEEE J. Sel. Top. Quantum Electron. 4, 353–359 (1998).
[CrossRef]

Frischkorn, C.

T. Kampfrath, L. Perfetti, F. Schapper, C. Frischkorn, M. Wolf, “Strongly coupled optical phonons in the ultrafast dynamics of the electronic energy and current relaxation in graphite,” Phys. Rev. Lett. 95, 187403, 2005).
[CrossRef] [PubMed]

Gallmann, L.

L. Gallmann, J. Herrmann, R. Locher, M. Sabbar, A. Ludwig, M. Lucchini, U. Keller, “Resolving intra-atomic electron dynamics with attosecond transient absorption spectroscopy,” Molecular Physics, in press, DOI: (2013).
[CrossRef]

George, P. A.

P. A. George, J. Strait, J. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, M. G. Spencer, “Ultrafast optical-pump terahertz-probe spectroscopy of the carrier relaxation and recombination dynamics in epitaxial graphene,” Nano Lett. 8, 4248–4251 (2008).
[CrossRef]

Gouy, C. R.

C. R. Gouy, “Sur une propriete nouvelle des ondes lumineuses,” Acad. Sci. Paris 110, 1251, 1890).

Hamm, P.

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D. Polli, M. Rini, S. Wall, R. W. Schoenlein, Y. Tomioka, Y. Tokura, G. Cerullo, A. Cavalleri, “Coherent orbital waves in the photo-induced insulator-metal dynamics of a magnetoresistive manganite,” Nat. Mater. 6, 643–647 (2007).
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M. Theuer, S. S. Harsha, D. Molter, G. Torosyan, R. Beigang, “Terahertz time-domain spectroscopy of gases, liquids, and solids,” Chemphyschem 12, 2695–2705 (2011).
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K. J. Siebert, T. Loffler, H. Quast, M. Thomson, T. Bauer, R. Leonhardt, S. Czasch, H. G. Roskos, “Alloptoelectronic continuous wave THz imaging for biomedical applications,” Phys. Med. Biol. 47, 3743–3748 (2002).
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A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy physe shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
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Figures (5)

Fig. 1
Fig. 1

Scheme of a optical-pump-THz-probe experiment. The displacement z is measured relative to the middle focus.

Fig. 2
Fig. 2

(a) Spectrum of the THz pulse THz(ω) (Eq. (14), black) and with negative frequencies zeroed (blue). (b) Fourier transformation of the pump pulse intensity profile Ĩpu(ω) (Eq. (15)). (c) Convolution THz(ω) ⊗ Ĩpu(ω) and (d) convolution with zeroing negative frequencies of the THz pulse THz(ω) before the convolution (blue) and after the convolution (red). The pump-probe delay between the center of both pulses was set to tpp = 0.

Fig. 3
Fig. 3

Outcome of a hypothetical pump probe experiment of a sample with infinitesimally fast response function as a function of THz detection time t, pump-probe delay time tpp between the center of both pulses, and the Gouy-phase error of the sample. Red and blue depict positive and negative fields, respectively. The top panels show the corresponding input THz pulse ETHz(t).

Fig. 4
Fig. 4

Same as in Fig. 2, but for a THz pulse according to Eq. (18). The red line in (d) has been offset a little bit to make it visible.

Fig. 5
Fig. 5

Top panels: THz pulses (red) at various positions relative to the focus from −500 μm to +500 μm. The phases ϕ of these pulses have been estimated by taking the pulse at z=0 μm and adding a phase that has been varied to give the best fit (black dotted lines). Bottom panels: Raman-pump-THz-probe signal from water at various z-positions in dependence of pump-probe delay tpp between the peaks of the Raman-pump and the THz probe pulse. The detection time t was kept fixed at 50 fs relative to the peak of the transmitted THz pulse, where the signal is maximal.

Equations (22)

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P 3 ( t ) 0 0 E T H z ( t t 2 ) I p u ( t t 1 ) R ( t 1 , t 2 ) d t 1 d t 2
e i ϕ P ( 3 ) ( t ) 0 0 e i ϕ E T H z ( t t 2 ) I p u ( t t 1 ) R ( t 1 , t 2 ) d t 1 d t 2
E ( t ) = E 0 ( t ) cos ( ω 0 t )
E ( t ) = E 0 ( t ) cos ( ω 0 t + ϕ )
E ( t ) F T E ( ω ) Θ ( ω ) E ( ω ) FT 1 E ˜ ( t ) ( e i ϕ E ˜ ( t ) )
E T H z ( t ) e i ϕ E ˜ T H z ( t ) Eq . ( 1 ) e i ϕ P ˜ ( 3 ) ( t ) e i ϕ e i ϕ P ˜ ( 3 ) ( t ) ( P ˜ ( 3 ) ( t ) )
E T H z ( t ) Re ( e i ϕ E ˜ T H z ( t ) ) Eq . ( 1 ) P ( 3 ) ( t ) e i ϕ P ˜ ( 3 ) ( t ) ( e i ϕ P ˜ ( 3 ) ( t ) )
P ( 1 ) ( t ) 0 E T H z ( t t 1 ) R ( t 1 ) d t 1
P ˜ ( 1 ) ( ω ) E ˜ T H z ( ω ) R ˜ ( ω )
P ˜ ( 1 ) ( ω ) Θ ( ω ) E ˜ T H z ( ω ) R ˜ ( ω )
R ( t 1 , t 2 ) = δ ( t 1 ) δ ( t 2 )
P 3 ( t ) E T H z ( t ) I p u ( t ) ,
P ˜ 3 ( ω ) E ˜ T H z ( ω ) I ˜ p u ( ω ) .
E T H z ( t ) 0 ω e τ 1 2 ω 2 / 4 e i ω t d ω
I p u ( t ) e t 2 / τ 2 2
Θ ( ω ) ( E ˜ T H z ( ω ) I ˜ p u ( ω ) ) ( Θ ( ω ) E ˜ T H z ( ω ) ) I ˜ p u ( ω )
E ( 3 ) ( t ) d P ( 3 ) ( t ) d t
E T H z ( t ) = cos ( ω 0 t ) e t 2 / τ 1 2
Θ ( ω ) ( E ˜ T H z ( ω ) I ˜ p u ( ω ) ) ( Θ ( ω ) E ˜ T H z ( ω ) ) I ˜ p u ( ω )
P ( t , t ) 0 0 E T H z ( t t 2 ) I p u ( t t 1 ) R ( t 1 , t 2 ) d t 1 d t 2
P ˜ ( ω , ω ) E ˜ T H z ( ω ) I ˜ p u ( ω ) R ˜ ( ω , ω )
P ˜ ( ω ) P ˜ ( ω , ω ω ) d ω

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