Abstract

Heterodyne detection with a local oscillator generated by a diffractive optic provides good phase stability, but suffers from interfering single-beam-bleach signals. This interference is a problem in four-beam transient gratings, but becomes more severe in higher order transient gratings. A simple and low noise method to eliminate this interference is demonstrated in both a four-wave-mixing, first-order–transient-grating experiment and in a six-wave-mixing, second-order–transient-grating experiment.

© 2009 Optical Society of America

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  1. A. A. Maznev, K. A. Nelson, and T. A. Rogers, “Optical heterodyne detection of laser-induced gratings,” Opt. Lett. 23, 1319-1321 (1998).
    [Crossref]
  2. G. D. Goodno, G. Dadusc, and R. J. D. Miller, “Ultrafast heterodyne-detected transient-grating spectroscopy using diffractive optics,” J. Opt. Soc. Am. B 15, 1791-1794 (1998).
    [Crossref]
  3. G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. Theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103, 10619-10629 (1999).
    [Crossref]
  4. M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
    [Crossref]
  5. Q.-H. Xu, Y.-Z. Ma, and G. R. Fleming, “Heterodyne detected transient grating spectroscopy in resonant and nonresonant systems using a simplified diffractive optics method,” Chem. Phys. Lett. 338, 254-262 (2001).
    [Crossref]
  6. K. J. Kubarych, C. J. Milne, S. Lin, and R. J. D. Miller, “Diffractive optics implementation of time- and frequency-domain heterodyne-detected six-wave mixing,” Appl. Phys. B 74, S107-S112 (2002).
    [Crossref]
  7. M. J. Ammend and D. A. Blank, “Passive optical interferometer without spatial overlap between the local oscillator and signal generation,” Opt. Lett. 34, 548-550 (2009).
    [Crossref] [PubMed]
  8. E. van Veldhoven, C. Khurmi, X. Zhang, and M. A. Berg, “Time-resolved optical spectroscopy with multiple population dimensions: a general method of resolving dynamic heterogeneity,” ChemPhysChem 8, 1761-1765 (2007).
    [Crossref] [PubMed]
  9. C. Khurmi and M. A. Berg, “Analyzing nonexponential kinetics with multiple population-period transient spectroscopy (MUPPETS),” J. Phys. Chem. A 112, 3364-3375 (2008).
    [Crossref] [PubMed]
  10. C. Khurmi and M. A. Berg, “Parallels between multiple population-period transient spectroscopy (MUPPETS) and multidimensional coherence spectroscopies,” J. Chem. Phys. 129, 064504 (2008).
    [Crossref] [PubMed]
  11. M. A. Berg has prepared a manuscript to be called “Hilbert-space treatment of incoherent, time-resolved spectroscopy. I. Formalism, a tensorial classification of high-order orientational gratings and generalized MUPPETS 'echoes' .”
  12. M. A. Berg has prepared a manuscript to be called “Hilbert-space treatment of incoherent, time-resolved spectroscopy. II. Pathway description of optical MUPPETS.”
  13. J. P. Ogilvie, M. Plazanet, G. Dadusc, and R. J. D. Miller, “Dynamics of ligand escape in myoglobin: Q-band transient absorption and four-wave mixing studies,” J. Phys. Chem. B 106, 10460-10467 (2002).
    [Crossref]
  14. G. Giraud, C. M. Gordon, I. R. Dunkin, and K. Wynne, “The effects of anion and cation substitution on the ultrafast solvent dynamics of ionic liquids: a time-resolved optical Kerr-effect spectroscopic study,” J. Chem. Phys. 119, 464-477 (2003).
    [Crossref]
  15. E. C. Fulmer, P. Mukherjee, A. T. Krummel, and M. T. Zanni, “A pulse sequence for directly measuring the anharmonicities of coupled vibrations: two-quantum two-dimensional infrared spectroscopy,” J. Chem. Phys. 120, 8067-8078 (2004).
    [Crossref] [PubMed]
  16. J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes,” J. Chem. Phys. 125, 194521 (2006).
    [Crossref] [PubMed]

2009 (1)

2008 (2)

C. Khurmi and M. A. Berg, “Analyzing nonexponential kinetics with multiple population-period transient spectroscopy (MUPPETS),” J. Phys. Chem. A 112, 3364-3375 (2008).
[Crossref] [PubMed]

C. Khurmi and M. A. Berg, “Parallels between multiple population-period transient spectroscopy (MUPPETS) and multidimensional coherence spectroscopies,” J. Chem. Phys. 129, 064504 (2008).
[Crossref] [PubMed]

2007 (1)

E. van Veldhoven, C. Khurmi, X. Zhang, and M. A. Berg, “Time-resolved optical spectroscopy with multiple population dimensions: a general method of resolving dynamic heterogeneity,” ChemPhysChem 8, 1761-1765 (2007).
[Crossref] [PubMed]

2006 (1)

J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes,” J. Chem. Phys. 125, 194521 (2006).
[Crossref] [PubMed]

2004 (1)

E. C. Fulmer, P. Mukherjee, A. T. Krummel, and M. T. Zanni, “A pulse sequence for directly measuring the anharmonicities of coupled vibrations: two-quantum two-dimensional infrared spectroscopy,” J. Chem. Phys. 120, 8067-8078 (2004).
[Crossref] [PubMed]

2003 (1)

G. Giraud, C. M. Gordon, I. R. Dunkin, and K. Wynne, “The effects of anion and cation substitution on the ultrafast solvent dynamics of ionic liquids: a time-resolved optical Kerr-effect spectroscopic study,” J. Chem. Phys. 119, 464-477 (2003).
[Crossref]

2002 (2)

J. P. Ogilvie, M. Plazanet, G. Dadusc, and R. J. D. Miller, “Dynamics of ligand escape in myoglobin: Q-band transient absorption and four-wave mixing studies,” J. Phys. Chem. B 106, 10460-10467 (2002).
[Crossref]

K. J. Kubarych, C. J. Milne, S. Lin, and R. J. D. Miller, “Diffractive optics implementation of time- and frequency-domain heterodyne-detected six-wave mixing,” Appl. Phys. B 74, S107-S112 (2002).
[Crossref]

2001 (1)

Q.-H. Xu, Y.-Z. Ma, and G. R. Fleming, “Heterodyne detected transient grating spectroscopy in resonant and nonresonant systems using a simplified diffractive optics method,” Chem. Phys. Lett. 338, 254-262 (2001).
[Crossref]

2000 (1)

M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
[Crossref]

1999 (1)

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. Theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103, 10619-10629 (1999).
[Crossref]

1998 (2)

Ammend, M. J.

Berg, M. A.

C. Khurmi and M. A. Berg, “Analyzing nonexponential kinetics with multiple population-period transient spectroscopy (MUPPETS),” J. Phys. Chem. A 112, 3364-3375 (2008).
[Crossref] [PubMed]

C. Khurmi and M. A. Berg, “Parallels between multiple population-period transient spectroscopy (MUPPETS) and multidimensional coherence spectroscopies,” J. Chem. Phys. 129, 064504 (2008).
[Crossref] [PubMed]

E. van Veldhoven, C. Khurmi, X. Zhang, and M. A. Berg, “Time-resolved optical spectroscopy with multiple population dimensions: a general method of resolving dynamic heterogeneity,” ChemPhysChem 8, 1761-1765 (2007).
[Crossref] [PubMed]

M. A. Berg has prepared a manuscript to be called “Hilbert-space treatment of incoherent, time-resolved spectroscopy. I. Formalism, a tensorial classification of high-order orientational gratings and generalized MUPPETS 'echoes' .”

M. A. Berg has prepared a manuscript to be called “Hilbert-space treatment of incoherent, time-resolved spectroscopy. II. Pathway description of optical MUPPETS.”

Blank, D. A.

Dadusc, G.

J. P. Ogilvie, M. Plazanet, G. Dadusc, and R. J. D. Miller, “Dynamics of ligand escape in myoglobin: Q-band transient absorption and four-wave mixing studies,” J. Phys. Chem. B 106, 10460-10467 (2002).
[Crossref]

G. D. Goodno, G. Dadusc, and R. J. D. Miller, “Ultrafast heterodyne-detected transient-grating spectroscopy using diffractive optics,” J. Opt. Soc. Am. B 15, 1791-1794 (1998).
[Crossref]

Demirdoven, N.

M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
[Crossref]

Dunkin, I. R.

G. Giraud, C. M. Gordon, I. R. Dunkin, and K. Wynne, “The effects of anion and cation substitution on the ultrafast solvent dynamics of ionic liquids: a time-resolved optical Kerr-effect spectroscopic study,” J. Chem. Phys. 119, 464-477 (2003).
[Crossref]

Fecko, C. J.

M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
[Crossref]

Fleming, G. R.

Q.-H. Xu, Y.-Z. Ma, and G. R. Fleming, “Heterodyne detected transient grating spectroscopy in resonant and nonresonant systems using a simplified diffractive optics method,” Chem. Phys. Lett. 338, 254-262 (2001).
[Crossref]

Fulmer, E. C.

E. C. Fulmer, P. Mukherjee, A. T. Krummel, and M. T. Zanni, “A pulse sequence for directly measuring the anharmonicities of coupled vibrations: two-quantum two-dimensional infrared spectroscopy,” J. Chem. Phys. 120, 8067-8078 (2004).
[Crossref] [PubMed]

Giraud, G.

G. Giraud, C. M. Gordon, I. R. Dunkin, and K. Wynne, “The effects of anion and cation substitution on the ultrafast solvent dynamics of ionic liquids: a time-resolved optical Kerr-effect spectroscopic study,” J. Chem. Phys. 119, 464-477 (2003).
[Crossref]

Golonzka, O.

M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
[Crossref]

Goodno, G. D.

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. Theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103, 10619-10629 (1999).
[Crossref]

G. D. Goodno, G. Dadusc, and R. J. D. Miller, “Ultrafast heterodyne-detected transient-grating spectroscopy using diffractive optics,” J. Opt. Soc. Am. B 15, 1791-1794 (1998).
[Crossref]

Gordon, C. M.

G. Giraud, C. M. Gordon, I. R. Dunkin, and K. Wynne, “The effects of anion and cation substitution on the ultrafast solvent dynamics of ionic liquids: a time-resolved optical Kerr-effect spectroscopic study,” J. Chem. Phys. 119, 464-477 (2003).
[Crossref]

Khalil, M.

M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
[Crossref]

Khurmi, C.

C. Khurmi and M. A. Berg, “Analyzing nonexponential kinetics with multiple population-period transient spectroscopy (MUPPETS),” J. Phys. Chem. A 112, 3364-3375 (2008).
[Crossref] [PubMed]

C. Khurmi and M. A. Berg, “Parallels between multiple population-period transient spectroscopy (MUPPETS) and multidimensional coherence spectroscopies,” J. Chem. Phys. 129, 064504 (2008).
[Crossref] [PubMed]

E. van Veldhoven, C. Khurmi, X. Zhang, and M. A. Berg, “Time-resolved optical spectroscopy with multiple population dimensions: a general method of resolving dynamic heterogeneity,” ChemPhysChem 8, 1761-1765 (2007).
[Crossref] [PubMed]

Krummel, A. T.

E. C. Fulmer, P. Mukherjee, A. T. Krummel, and M. T. Zanni, “A pulse sequence for directly measuring the anharmonicities of coupled vibrations: two-quantum two-dimensional infrared spectroscopy,” J. Chem. Phys. 120, 8067-8078 (2004).
[Crossref] [PubMed]

Kubarych, K. J.

K. J. Kubarych, C. J. Milne, S. Lin, and R. J. D. Miller, “Diffractive optics implementation of time- and frequency-domain heterodyne-detected six-wave mixing,” Appl. Phys. B 74, S107-S112 (2002).
[Crossref]

Lin, S.

K. J. Kubarych, C. J. Milne, S. Lin, and R. J. D. Miller, “Diffractive optics implementation of time- and frequency-domain heterodyne-detected six-wave mixing,” Appl. Phys. B 74, S107-S112 (2002).
[Crossref]

Loparo, J. J.

J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes,” J. Chem. Phys. 125, 194521 (2006).
[Crossref] [PubMed]

Ma, Y. -Z.

Q.-H. Xu, Y.-Z. Ma, and G. R. Fleming, “Heterodyne detected transient grating spectroscopy in resonant and nonresonant systems using a simplified diffractive optics method,” Chem. Phys. Lett. 338, 254-262 (2001).
[Crossref]

Maznev, A. A.

Miller, R. J. D.

K. J. Kubarych, C. J. Milne, S. Lin, and R. J. D. Miller, “Diffractive optics implementation of time- and frequency-domain heterodyne-detected six-wave mixing,” Appl. Phys. B 74, S107-S112 (2002).
[Crossref]

J. P. Ogilvie, M. Plazanet, G. Dadusc, and R. J. D. Miller, “Dynamics of ligand escape in myoglobin: Q-band transient absorption and four-wave mixing studies,” J. Phys. Chem. B 106, 10460-10467 (2002).
[Crossref]

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. Theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103, 10619-10629 (1999).
[Crossref]

G. D. Goodno, G. Dadusc, and R. J. D. Miller, “Ultrafast heterodyne-detected transient-grating spectroscopy using diffractive optics,” J. Opt. Soc. Am. B 15, 1791-1794 (1998).
[Crossref]

Milne, C. J.

K. J. Kubarych, C. J. Milne, S. Lin, and R. J. D. Miller, “Diffractive optics implementation of time- and frequency-domain heterodyne-detected six-wave mixing,” Appl. Phys. B 74, S107-S112 (2002).
[Crossref]

Mukherjee, P.

E. C. Fulmer, P. Mukherjee, A. T. Krummel, and M. T. Zanni, “A pulse sequence for directly measuring the anharmonicities of coupled vibrations: two-quantum two-dimensional infrared spectroscopy,” J. Chem. Phys. 120, 8067-8078 (2004).
[Crossref] [PubMed]

Nelson, K. A.

Ogilvie, J. P.

J. P. Ogilvie, M. Plazanet, G. Dadusc, and R. J. D. Miller, “Dynamics of ligand escape in myoglobin: Q-band transient absorption and four-wave mixing studies,” J. Phys. Chem. B 106, 10460-10467 (2002).
[Crossref]

Plazanet, M.

J. P. Ogilvie, M. Plazanet, G. Dadusc, and R. J. D. Miller, “Dynamics of ligand escape in myoglobin: Q-band transient absorption and four-wave mixing studies,” J. Phys. Chem. B 106, 10460-10467 (2002).
[Crossref]

Roberts, S. T.

J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes,” J. Chem. Phys. 125, 194521 (2006).
[Crossref] [PubMed]

Rogers, T. A.

Tokmakoff, A.

J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes,” J. Chem. Phys. 125, 194521 (2006).
[Crossref] [PubMed]

M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
[Crossref]

van Veldhoven, E.

E. van Veldhoven, C. Khurmi, X. Zhang, and M. A. Berg, “Time-resolved optical spectroscopy with multiple population dimensions: a general method of resolving dynamic heterogeneity,” ChemPhysChem 8, 1761-1765 (2007).
[Crossref] [PubMed]

Wynne, K.

G. Giraud, C. M. Gordon, I. R. Dunkin, and K. Wynne, “The effects of anion and cation substitution on the ultrafast solvent dynamics of ionic liquids: a time-resolved optical Kerr-effect spectroscopic study,” J. Chem. Phys. 119, 464-477 (2003).
[Crossref]

Xu, Q. -H.

Q.-H. Xu, Y.-Z. Ma, and G. R. Fleming, “Heterodyne detected transient grating spectroscopy in resonant and nonresonant systems using a simplified diffractive optics method,” Chem. Phys. Lett. 338, 254-262 (2001).
[Crossref]

Zanni, M. T.

E. C. Fulmer, P. Mukherjee, A. T. Krummel, and M. T. Zanni, “A pulse sequence for directly measuring the anharmonicities of coupled vibrations: two-quantum two-dimensional infrared spectroscopy,” J. Chem. Phys. 120, 8067-8078 (2004).
[Crossref] [PubMed]

Zhang, X.

E. van Veldhoven, C. Khurmi, X. Zhang, and M. A. Berg, “Time-resolved optical spectroscopy with multiple population dimensions: a general method of resolving dynamic heterogeneity,” ChemPhysChem 8, 1761-1765 (2007).
[Crossref] [PubMed]

Appl. Phys. B (1)

K. J. Kubarych, C. J. Milne, S. Lin, and R. J. D. Miller, “Diffractive optics implementation of time- and frequency-domain heterodyne-detected six-wave mixing,” Appl. Phys. B 74, S107-S112 (2002).
[Crossref]

Chem. Phys. Lett. (1)

Q.-H. Xu, Y.-Z. Ma, and G. R. Fleming, “Heterodyne detected transient grating spectroscopy in resonant and nonresonant systems using a simplified diffractive optics method,” Chem. Phys. Lett. 338, 254-262 (2001).
[Crossref]

ChemPhysChem (1)

E. van Veldhoven, C. Khurmi, X. Zhang, and M. A. Berg, “Time-resolved optical spectroscopy with multiple population dimensions: a general method of resolving dynamic heterogeneity,” ChemPhysChem 8, 1761-1765 (2007).
[Crossref] [PubMed]

J. Chem. Phys. (4)

C. Khurmi and M. A. Berg, “Parallels between multiple population-period transient spectroscopy (MUPPETS) and multidimensional coherence spectroscopies,” J. Chem. Phys. 129, 064504 (2008).
[Crossref] [PubMed]

G. Giraud, C. M. Gordon, I. R. Dunkin, and K. Wynne, “The effects of anion and cation substitution on the ultrafast solvent dynamics of ionic liquids: a time-resolved optical Kerr-effect spectroscopic study,” J. Chem. Phys. 119, 464-477 (2003).
[Crossref]

E. C. Fulmer, P. Mukherjee, A. T. Krummel, and M. T. Zanni, “A pulse sequence for directly measuring the anharmonicities of coupled vibrations: two-quantum two-dimensional infrared spectroscopy,” J. Chem. Phys. 120, 8067-8078 (2004).
[Crossref] [PubMed]

J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. I. Vibrational dynamics in two-dimensional IR line shapes,” J. Chem. Phys. 125, 194521 (2006).
[Crossref] [PubMed]

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

J. Phys. Chem. A (3)

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. Theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103, 10619-10629 (1999).
[Crossref]

M. Khalil, N. Demirdoven, O. Golonzka, C. J. Fecko, and A. Tokmakoff, “A phase-sensitive detection method using diffractive optics for polarization-selective femtosecond Raman spectroscopy,” J. Phys. Chem. A 104, 5711-5715 (2000).
[Crossref]

C. Khurmi and M. A. Berg, “Analyzing nonexponential kinetics with multiple population-period transient spectroscopy (MUPPETS),” J. Phys. Chem. A 112, 3364-3375 (2008).
[Crossref] [PubMed]

J. Phys. Chem. B (1)

J. P. Ogilvie, M. Plazanet, G. Dadusc, and R. J. D. Miller, “Dynamics of ligand escape in myoglobin: Q-band transient absorption and four-wave mixing studies,” J. Phys. Chem. B 106, 10460-10467 (2002).
[Crossref]

Opt. Lett. (2)

Other (2)

M. A. Berg has prepared a manuscript to be called “Hilbert-space treatment of incoherent, time-resolved spectroscopy. I. Formalism, a tensorial classification of high-order orientational gratings and generalized MUPPETS 'echoes' .”

M. A. Berg has prepared a manuscript to be called “Hilbert-space treatment of incoherent, time-resolved spectroscopy. II. Pathway description of optical MUPPETS.”

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Figures (5)

Fig. 1
Fig. 1

Schematic of the optical setup: L1–L11, lenses; G1 and G2, transmission gratings; P1–P3, reflective prisms; D1–D3, delay lines; C, chopper; ND0 and ND1, neutral density filters; S, sample; M1 and M2, masks; P, pinhole; PD1 and PD2, matched photodiodes; VND, linear variable neutral density filter; A-B, differential inputs of a lock-in amplifier. Different masks are used for the four-beam, first-order grating and for the six-beam, second-order grating as shown.

Fig. 2
Fig. 2

Conventional one-detector heterodyne detection of a first-order (left) and a second-order (right) transient grating. Top row: fractional change in the probe-beam intensity with the phase changed by 180° in (A) a first-order transient grating and (B) a second-order grating. Bottom row: normalized grating signal (black, +) and single-beam bleach (red, −) obtained by addition and subtraction of the signals of opposite phase in (C) a first-order transient grating and (D) a second-order grating. The undesirable single-beam bleach is the same size as the desired grating signal in the first-order grating, but much larger in the second-order grating.

Fig. 3
Fig. 3

Differential heterodyne detection of a first-order transient grating (left) and a second-order grating (right). Top row: fractional change in the probe-beam intensity with the phase changed by 180° in (A) a first-order transient grating and (B) a second-order grating with τ 1 = 0   ps . Bottom row: normalized differential grating signal (black, +) and single-beam bleach (red, −) obtained by addition and subtraction of the signals of opposite phase in (C) a first-order transient grating and (D) a second-order grating.

Fig. 4
Fig. 4

Comparison of MUPPETS signal obtained by using standard detection (blue, noisy curves) and differential heterodyne detection (black, smooth curves) using the same data collection times. Main panel: τ 1 = 0   ps . Inset: τ 1 = 10   ps . The signal-to-noise ratio is improved substantially by using differential detection.

Fig. 5
Fig. 5

Effect of sample position on differential heterodyne detection: the in- and out-of-phase scans of a second-order-grating signal at τ 1 = 10   ps at two sample positions (black, inner curves and blue, outer curves). A small 0.25 mm movement of the 1 mm thick sample affects the degree to which the single-beam bleach (visible before 10 ps) is suppressed.

Equations (14)

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E n j ( t , r ) = E n j δ ( t t n + k n j r ) e i ( k n j r + φ n j + ω n t ) .
δ ρ = ρ n = 1 N [ m = 1 n ( σ m ) ( I m a + E m a E m b + E m a E m b + I m b ) ] C ( n ) ( τ n , , τ 1 ) .
P = χ δ ρ ( E N + 1 , a + E N + 1 , b + c .c . ) .
n = 1 N + 1 k n b k n a = 0 ,
P N + 1 , b = ρ χ e i φ N + 1 , b [ E N + 1 , b n j σ n I n j C ( 1 ) + ( 1 ) N e i Φ E N + 1 , a ( n = 1 N σ n E n a E n b ) C ( N ) ] ,
Φ = n = 1 N + 1 φ n b φ n a .
S b [ 1 a ] = I N + 1 , b ( I 1 a ) I N + 1 , b ( I 1 a = 0 ) 2 I N + 1 , b ( I 1 a = 0 ) 4 π ω L c I N + 1 , b Im   E N + 1 , b e i φ N + 1 , b P N + 1 , b .
S b ( N ) [ 1 a ] = 4 π ω L ρ c [ ( Im   χ ) σ 1 I 1 C ( 1 ) + ( 1 ) N 1   Im ( χ e i Φ ) I N + 1 , a I N + 1 , b ( n = 1 N σ n I n ) C ( N ) ] .
S b [ 1 a , Φ ± ] = 1 2 [ S b [ 1 a ] ( Φ 0 ) ± S b [ 1 a ] ( Φ 0 + π ) ] .
S b ( N ) [ 1 a , Φ ] = ( 1 ) N 1 4 π ω L ρ c ( χ   cos   Φ + χ   sin   Φ ) I N + 1 , a I N + 1 , b ( n = 1 N + 1 σ n I n ) C ( N ) .
S D = S b [ 1 a ] S a [ 1 a ] .
S D = ( 1 ) N 1 4 π ω L ρ c ( χ   sin   Φ I N + 1 , a + I N + 1 , b I N + 1 , a I N + 1 , b + χ   cos   Φ I N + 1 , a I N + 1 , b I N + 1 , a I N + 1 , b ) ( n = 1 N + 1 σ n I n ) C ( N ) .
S D [ I a = I b ] = ( 1 ) N 1 4 π ω L ρ c ( 2 χ   sin   Φ ) ( n = 1 N + 1 σ n I n ) C ( N ) .
S D [ I a I b ] = ( 1 ) N 1 4 π ω L ρ c ( χ   cos   Φ + χ   sin   Φ ) I N + 1 , a I N + 1 , b ( n = 1 N + 1 σ n I n ) C ( N ) .

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