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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 19 — Sep. 10, 2012
  • pp: 20908–20919
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Modeling and correction of distorted two-dimensional Fourier transform spectra from pixelated pulse shaping devices

Erik M. Grumstrup and Niels H. Damrauer  »View Author Affiliations


Optics Express, Vol. 20, Issue 19, pp. 20908-20919 (2012)
http://dx.doi.org/10.1364/OE.20.020908


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Abstract

Two-dimensional Fourier transform spectra of a three level model system are simulated using a non-perturbative density matrix formalism. The electric field distortions resultant from using pixelated pulse shaping devices to produce phase-locked pulse pairs are modeled and the effects on the recovered spectra are examined. To minimize spectral distortions, a temporal filtering scheme is employed which eliminates contributions from spurious sample polarizations.

© 2012 OSA

1. Introduction

Two-dimensional Fourier transform (2DFT) spectroscopy is a powerful technique for studying a wide range of ultrafast photophysics by correlating transition dipoles at two frequencies with sensitivity to phase and coherence information. 2DFT spectroscopy can provide insight into chromophores and their environments such as bath-induced relaxation [1

1. J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297(3-4), 307–313 (1998). [CrossRef]

, 2

2. I. Stiopkin, T. Brixner, M. Yang, and G. R. Fleming, “Heterogeneous exciton dynamics revealed by two-dimensional optical spectroscopy,” J. Phys. Chem. B 110(40), 20032–20037 (2006). [CrossRef] [PubMed]

] and energy transfer phenomena [3

3. G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007). [CrossRef] [PubMed]

, 4

4. D. Zigmantas, E. L. Read, T. Mančal, T. Brixner, A. T. Gardiner, R. J. Cogdell, and G. R. Fleming, “Two-dimensional electronic spectroscopy of the B800-B820 light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 103(34), 12672–12677 (2006). [CrossRef] [PubMed]

] that are inaccessible to conventional methods. After Warren’s early work in a fully collinear geometry [5

5. P. Tian, D. Keusters, Y. Suzaki, and W. S. Warren, “Femtosecond phase-coherent two-dimensional spectroscopy,” Science 300(5625), 1553–1555 (2003). [CrossRef] [PubMed]

, 6

6. W. Wagner, C. Li, J. Semmlow, and W. Warren, “Rapid phase-cycled two-dimensional optical spectroscopy in fluorescence and transmission mode,” Opt. Express 13(10), 3697–3706 (2005). [CrossRef] [PubMed]

], several groups have demonstrated the implementation of 2DFTS in the IR [7

7. S. H. Shim, D. B. Strasfeld, E. C. Fulmer, and M. T. Zanni, “Femtosecond pulse shaping directly in the mid-IR using acousto-optic modulation,” Opt. Lett. 31(6), 838–840 (2006). [CrossRef] [PubMed]

, 8

8. S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007). [CrossRef] [PubMed]

], visible [9

9. E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express 15(25), 16681–16689 (2007). [CrossRef] [PubMed]

14

14. K. L. M. Lewis and J. P. Ogilvie, “Probing photosynthetic energy and charge transfer with two-dimensional electronic spectroscopy,” J. Phys. Chem. Lett. 3(4), 503–510 (2012). [CrossRef]

] and UV [15

15. C. H. Tseng, S. Matsika, and T. C. Weinacht, “Two-dimensional ultrafast Fourier transform spectroscopy in the deep ultraviolet,” Opt. Express 17(21), 18788–18793 (2009). [CrossRef] [PubMed]

, 16

16. C. H. Tseng, P. Sándor, M. Kotur, T. C. Weinacht, and S. Matsika, “Two-dimensional Fourier transform spectroscopy of adenine and uracil using shaped ultrafast laser pulses in the deep UV,” J. Phys. Chem. A 116(11), 2654–2661 (2012). [CrossRef] [PubMed]

] spectral regions using pulse shaping devices in a partially collinear geometry. This method has many advantages over conventional techniques including intrinsically high phase stability, the ability to easily collect highly nondegenerate 2DFT spectra, and perhaps most importantly, a dramatic simplification of the experimental apparatus. By employing the methods described in the above papers, 2DFT spectroscopy can be readily performed in a pump-probe apparatus, allowing the ready adoption of this powerful technique by non-specialized groups.

The key component to performing 2DFT spectroscopy in such a straightforward manner is the use of a computer-controlled pulse-shaping device parameterized by an analytical transfer function (described in more detail below). However, it has long been understood that pixelated pulse shaping devices based on liquid-crystal spatial light modulators (LC-SLM) do not ideally replicate the transfer function and as a consequence, the shaped time-domain field is distorted [17

17. A. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]

]. These distortions often manifest in the form of spurious pulses, which cause unwanted contributions to the collected 2DFT spectrum.

In this paper, we employ non-perturbative density matrix theory, which allows an explicit inclusion of the non-ideal electric field arising from pixilation artifacts, to study these distortions on a model three level system. With knowledge of what specific interactions lead to distorted spectra in hand, we present a temporal filtering scheme, which allows the collection of high quality 2DFT spectra even with relatively poor pulse shaper resolution. The remainder of this paper is organized as follows: In Methods, we present the details of the calculation and model system. In Results, we present simulated spectra showing the nature of the distortions before and after application of the temporal filter. In the Discussion section, we examine in detail the spurious contributions that cause spectral distortion before finally concluding.

2. Methods

Collection of 2DFT spectra in the partially non-collinear geometry requires a phase-locked, collinear pulse pair (“pump” beam) which serves as the first two excitation fields and a non-collinear “probe” pulse which serves as both the third excitation field and means to self-heterodyne the detected signal. A computer-controlled pulse-shaping device, parameterized by Eqs. (1) and (2) to respectively modulate the amplitude and phase of the pump beam, trivializes the creation of a phase-locked pulse pair [9

9. E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express 15(25), 16681–16689 (2007). [CrossRef] [PubMed]

, 18

18. M. A. Montgomery, E. M. Grumstrup, and N. H. Damrauer, “Fourier transform spectroscopies derived from amplitude or phase shaping of broadband laser pulses with applications to adaptive control,” J. Opt. Soc. Am. B 27(12), 2518–2533 (2010). [CrossRef]

].
a(ω;τ)=|cos(ωτ2)|
(1)
p(ω;τ)=ωτ2+(π2π2sgn[cos{ωτ2}])
(2)
Here, sgn refers to the sign operator and ω is the frequency of the electric field. Inverse Fourier transform of the spectral electric field, modulated by Eqs. (1) and (2), results in the real electric field comprised of a phase-locked pulse pair separated by τ, E1(tτ)+E2(t)=14(E˜1[tτ]+E˜2[t])+c.c., where E˜1[tτ] and E˜2[t] are the complex amplitudes of the first and second time domain pulses.

In the partially non-collinear implementation of 2DFT spectroscopy, the pulse pair, together with the third non-collinear pulse, induce the third-order polarization of the sample that reflects the system’s response to the three interactions with the electric field [19

19. D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. 54(1), 425–463 (2003). [CrossRef] [PubMed]

, 20

20. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

]. In the case of ideal pulse shaping, the undistorted 2DFT spectrum can be recovered [21

21. S. M. Gallagher Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: theory and model calculations,” J. Phys. Chem. A 103(49), 10489–10505 (1999). [CrossRef]

]. However, because pulse shaping devices utilizing pixilated LC-SLMs only approximately reproduce a desired transfer function, the shaped time-domain electric field is distorted, often manifest in the form of spurious pulses shown in red (and dotted) in Fig. 1
Fig. 1 Ideal pulse shaping produces a pulse pair comprised of E1 and E2 which are separated by . The pulse pair contained in the “pump” beam together with the third pulse, E3, contained in the “probe” beam produce the third order polarization which radiates the signal field, Esig. The parameter T is the interval between the second (E2) and third (E3, probe) pulse. Non-ideal pulse shaping results in spurious pulses (Esp1 and Esp2 are shown above with red, dotted lines). In this work, t = 0 is defined as the arrival time of the probe pulse (E3) in the sample.
. These spurious pulses are located at times roughly corresponding to the inverse difference frequency between pixels [9

9. E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express 15(25), 16681–16689 (2007). [CrossRef] [PubMed]

, 17

17. A. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]

, 22

22. J. Vaughan, T. Feurer, K. Stone, and K. Nelson, “Analysis of replica pulses in femtosecond pulse shaping with pixelated devices,” Opt. Express 14(3), 1314–1328 (2006). [CrossRef] [PubMed]

]. As a consequence, additional contributions (not shown) to the third-order polarization contribute to the acquired signal, causing distortions in the 2DFT spectrum.

To simulate the effects of pixelated pulse shaping on 2DFT spectra, we chose to use the non-perturbative density matrix formalism. In such calculations, numerical propagation of the density matrix explicitly includes the time-dependent electric field, and consequently, distortions arising from pixilation effects are straightforward to include. In this section, we first outline the form of the electric field used to model the distortions that arise from pulse shaping with a pixilated LC-SLM. Following that, the model system is described and details regarding the calculation of the 2DFT spectra are presented.

Pixelation effects in LC-SLMs arise because each pixel of finite physical width imparts a constant modulation to the amplitudes and phases of all field components passing through it. Numerically, the total spectral electric field can be modeled by Eq. (3).
ϵ˜total(ω)=12ndωn|ϵ˜in(ω)|Π[(ωωn)Δn]×{exp[i(ωnτ2+ϕ1)]+exp[i(ωnτ2+ϕ2)]}exp[i(ωnτ2+ωT)]+|ϵ˜3(ω)|exp(iϕ3)
(3)
Here, |ϵ˜in(ω)|is the spectral amplitude of the unshaped pulse before it is modulated by the LC-SLM and |ϵ˜3(ω)| is the spectral amplitude of the third excitation pulse (whose center after inverse Fourier transform defines t = 0 in this work). The ωn are the central frequencies of the portion of the field passing through pixel n, and Π[(ωωn)/Δn] is a rectangle function centered at ωn with a width Δn that corresponds to the resolution in frequency space imparted by the pixelated device that is modeled. Finally, ϕ1, ϕ2, and ϕ3 define the spectral phases of the three component pulses of the total electric field. In the ideal case with no pixelation effects (Δn=0), and Eq. (3) becomes,

ϵ˜total(ω)=12|ϵ˜in(ω)|{exp[i(ωτ2+ϕ1)]+exp[i(ωτ2+ϕ2)]}×exp[i(ωτ2+ωT)]+|ϵ˜3(ω)|exp(iϕ3)
(4)

The spectrum of the electric field used in the manuscript was Gaussian shaped and centered at 2354.56 THz (~800 nm). For the time domain simulations, the field was defined in the spectral domain and numerically inverse Fourier transformed. To determine the set ofωn, the pixel resolution in nm, Δλ, of the LC-SLM is specified and a numerical algorithm was used to determine the width in frequency space (Δn) that corresponded to a given Δλ at each portion of the spectrum. These points were then binned in a “pixel.” The central wavelength of each pixel defines the parameterωn, which was used for all spectral components of the field passing through pixel n regardless of optical frequency. As an illustration, consider Fig. 2
Fig. 2 Spectral amplitude, |ϵ˜out(ω;τ=500fs)|, for ideal pulse shaping (red) and 0.5 nm pixel resolution (black).
which shows a portion of the spectral amplitude |ϵ˜out(ω;τ=500fs)| for ideal pulse shaping with no pixels (red) and for a pixel resolution of 0.5 nm (black). For both cases, the amplitude of the probe pulse, |ϵ˜3(ω)| (see Eqs. (3) and (4), has been set to 0 for clarity.

We chose to simulate a three level system, as it is the simplest model that can exhibit cross peaks in the 2DFT spectrum. The Hamiltonian of the system is given by Eq. (5) which is composed of the field free Hamiltonian, H^0, and the field interaction operatorV^. Here, E(t)is the total (pump and probe) real time-dependent electric field and μ is the transition dipole moment (set to 1 in this work). The ground state frequency, E00/, was set to 0, and E11/=2366.40 THz and E22/=2414.94 THz give the frequencies of the two excited states.
H^(t)=H^0+V^(t)=[E00μ01E(t)μ02E(t)μ10E(t)E110μ20E(t)0E22]
(5)
The density matrix of the system, ρ^(t), was evolved in time under the Liouville-Von Neumann equation complete with a simple Bloch-type relaxation operator,
tρ^(t)=i[H^(t),ρ^(t)]R^(t)ρ^(t)
(6)
subject to the rotating wave and the slowly varying envelope approximations [20

20. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

, 23

23. K. Blum, Density Matrix Theory and Applications, Physics of Atoms and Molecules, 2nd ed., (Plenum Press, 1996).

,24

24. R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier/Academic Press, 2008).

]. The electric field simulated in this work was 24.3 fs FWHM by amplitude and was kept weak enough to excite less than 1% of the population so as to eliminate higher order polarization terms from contributing to the 2DFT spectra. The spectrum of the electric field was centered at 2390.67 THz and had sufficient bandwidth to span transitions to both excited states. Rate constants for the relaxation operator, γ10=γ20=γ12=0.01 fs−1, were chosen to be at the low end of typical electronic dephasing rates in condensed phase systems [25

25. A. M. Weiner, S. DeSilvestri, and E. P. Ippen, “Three-Pulse scattering for femtosecond dephasing studies - theory and experiment,” J. Opt. Soc. Am. B 2(4), 654–662 (1985). [CrossRef]

], although there are a few exceptions [26

26. H. Lee, Y.-C. Cheng, and G. R. Fleming, “Coherence dynamics in photosynthesis: Protein protection of excitonic coherence,” Science 316(5830), 1462–1465 (2007). [CrossRef] [PubMed]

] which are not represented here. Neither inhomogeneous broadening nor population relaxation effects were included in the simulation.

2DFT spectra in the partially collinear geometry are obtained by double Fourier transform (with respect to t andτ) of the modulus of the third-order polarization, P2D(3)(t,T,τ). In the time domain, this polarization (Eq. (7) is given by the triple convolution of the system response function with the electric field.

P2D(3)(t,T,τ)=0dt30dt20dt1E(t+τ+Tt3t2t1)×E(t+Tt3t2)E(tt3)R(t1,t2,t3)
(7)

The response function, R(t1,t2,t3), is comprised of Liouville pathways describing specific light matter interactions such as stimulated emission, ground state bleach, and excited state absorption [19

19. D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. 54(1), 425–463 (2003). [CrossRef] [PubMed]

].

When the density matrix is perturbatively expanded, the separation of various orders of the polarization naturally arises so that calculation of 2DFT spectra is straightforward. However, when the density matrix is treated non-perturbatively as it is in this work, the expectation value of the dipole operator, Eq. (8), gives the overall time-dependent polarization, P(t), which contains all orders of the light matter interaction.
P(t)=nP(n)(t)=Tr[μ^ρ^(t)]
(8)
It is thus necessary to separate those polarization terms that are desired (in this case, P2D(3)(t,T,τ)) from those that do not contribute to the 2DFT spectrum.

To do this, we first write the total induced polarization as a sum of components,
P(t)=l,m,nP˜l,m,n(t)exp[i(lk1+mk2+nk3)r]+c.c.
(9)
where the kn are the wave vectors of each portion of the electric field that interacts with the system, E(t)=E˜1(t)exp(ik1r)+E˜2(t)exp(ik2r)+E˜3(t)exp(ik3r)+c.c.Experimentally, each of the contributions to the overall polarization of the sample can be readily distinguished by the direction in which the signal is emitted, k=lk1+mk2+nk3. For example, rephasing pathways are emitted in the direction corresponding to l = −1, m = + 1, n = + 1. In a non-perturbative calculation, there is no equivalent directional separation. However, for calculation of the 2DFT spectrum the individual contributions can be separated in a similar way by first defining a phaseϕn=knr. The polarization can then be written as,
P˜(t;ϕ1,ϕ2,ϕ3)=l,m,nP˜l,m,n(t)exp[i(lϕ1+mϕ2+nϕ3)]
(10)
where the parametric dependence of the polarization on individual phases of the electric field components is made explicit. Thus “directional” separation of the time dependent polarization can be realized by solving a set of linear equations [27

27. L. Seidner, G. Stock, and W. Domcke, “Nonperturbative approach to femtosecond spectroscopy: General theory and application to multidimensional nonadiabatic photoisomerization processes,” J. Chem. Phys. 103(10), 3998–4011 (1995). [CrossRef]

, 28

28. B. Wolfseder, L. Seidner, G. Stock, and W. Domcke, “Femtosecond pump-probe spectroscopy of electron-transfer systems: a nonperturbative approach,” Chem. Phys. 217(2-3), 275–287 (1997). [CrossRef]

], or equivalently, by inverse Fourier transform with respect to the phase parameters, ϕn [29

29. H. Wang and M. Thoss, “Nonperturbative simulation of pump–probe spectra for electron transfer reactions in the condensed phase,” Chem. Phys. Lett. 389(1-3), 43–50 (2004). [CrossRef]

, 30

30. T. S. Mančal, A. V. Pisliakov, and G. R. Fleming, “Two-dimensional optical three-pulse photon echo spectroscopy. I. Nonperturbative approach to the calculation of spectra,” J. Chem. Phys. 124(23), 234504 (2006). [CrossRef] [PubMed]

]:

P˜l,m,n(t)02πdϕ102πdϕ202πdϕ3P˜(t;ϕ1,ϕ2,ϕ3)exp[i(lϕ1+mϕ2+nϕ3)].
(11)

Since the induced polarization is dependent on the phase of the electric field (as shown in Eq. (7), we are able to recover P2D(3)(t,T,τ)by calculating the overall polarization with different realizations of the parameters, ϕ1, ϕ2, and ϕ3 in the excitation field. In this work, 16 different realizations of the electric field (ϕ1and ϕ2; see Eqs. (3) and (4)) were used in the propagation of the density matrix resulting in 16 total polarizationsP˜(t;ϕ1,ϕ2,ϕ3). The rephasing (l = −1, m = + 1, n = + 1 and non-rephasing (l = + 1, m = −1, n = + 1) components were then isolated by performing the discrete inverse Fourier transform (Eq. (11) on the set of overall polarizations, .

3. Results

In the partially collinear geometry, the fields corresponding to rephasing and nonrephasing pathways are emitted along the same wave vector and as such, are simultaneously detected. Two-dimensional spectra were obtained by Fourier transform along t of the sum of rephasing and nonrephasing pathways and a cosine transfer along positive-only τ which was scanned from 0 to 400 fs [31

31. A. Albrecht, J. Hybl, S. Faeder, and D. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999). [CrossRef]

]. Figure 3
Fig. 3 (top) Real and imaginary spectra for T = 0 fs for the three level model with = 0.01 fs−1. (bottom) Real and imaginary spectra for T = 100 fs. Warm colors indicate positive amplitude and cold colors indicate negative amplitude. The contours demarcate 10% intervals of the total amplitude. The spectra in the top panel show a phase twist due to pulse overlap effects which are not present in the lower panel. This is highlighted by the dashed lines within the two imaginary spectra.
shows the real and imaginary parts of the 2D spectrum for T = 0 fs (top) and T = 100 fs (bottom). These spectra are generated assuming ideal shaping with no pixelation effects and recover peak shapes that are well known in the literature [32

32. J. Hybl, Y. Christophe, and D. Jonas, “Peak shapes in femtosecond 2D correlation spectroscopy,” Chem. Phys. 266(2-3), 295–309 (2001). [CrossRef]

]. The T = 0 fs 2D spectrum exhibits “phase twist” due to the temporal overlap of E3 with E1 or E2 [21

21. S. M. Gallagher Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: theory and model calculations,” J. Phys. Chem. A 103(49), 10489–10505 (1999). [CrossRef]

]. On the other hand, for all spectra generated with T greater than the pulse width, the expected peak shapes (resembling 4-point stars in the real spectra) are observed.

As is discussed in detail below, the distortions arise from a series of interactions of the three-level system with the field that depend specifically on the spurious pulses. Shown in black in Fig. 5
Fig. 5 black: Calculated |P˜2D(3)(t)| for Δλ = 1 nm, T = 100 fs, and τ = 350 fs. green: Hyperbolic tangent filter used in this work.
is the modulus of the calculated 3rd order polarization, |P˜2D(3)(t)|, for the case of Δλ = 1 nm, T = 100 fs, and τ = 350 fs. The total 3rd order polarization is made up of the desired portion, which begins at t = 0, and the unwanted polarization resultant from the spurious pulses at t ~1.7 ps and t ~3.8 ps. It is notable that the unwanted polarization magnitude is highly significant even when the spurious pulses appear weak (this is shown later, for example, in Fig. 7). This is primarily a consequence of two effects. First, the desired polarization is itself weak, having decayed because the electronic dephasing time (100 fs) is significantly shorter than the value of τ (350 fs). Second, the contributions from the spurious polarization arise from pathways with a comparatively short interval (T = 100 fs) which does not change throughout the course of a measurement (i.e., as τ is varied).

Unfortunately, because the spurious pulses are “encoded” with the same phase information as the pulse pair, phase cycling schemes will be ineffective at removing the unwanted polarization from the collected signal. However, because the inverse electronic dephasing rate is typically much shorter thanΔωc, it is often the case that the unwanted induced polarization occurs significantly after the desired polarization has radiated its signal (such as that seen in Fig. 5). This suggests it should be possible to minimize distortion to the spectrum by performing a single point convolution of a temporal filter with the detected signal.

We have chosen to use a hyperbolic tangent filter, given by Eq. (12) and shown in green in Fig. 5, whose parameters can be adjusted to eliminate the spurious signal (which occurs later in time) while maintaining the desired signal which begins at t = 0 fs [33

33. D. Jonas, (personal communication, August 10, 2011).

].

F(t)=12{(tanh(t+t+a)+1)(tanh(t+ta)+1)}
(12)

Here t+ and t- give the cutoff times in positive and negative t, respectively, and determines how quickly the filter maps from 0 to 1 (in this case shown in Fig. 5, = 12 fs, so that the filter transitions from 0 to 1 in ~100 fs). Fourier transform of the product of the filter and the time domain third order polarization recovers nearly undistorted 2DFT spectra. This can be seen in Fig. 6
Fig. 6 Difference spectra (real; subtracting ideal from the temporally filtered data) for cases of a) Δλ= 0.5 nm, b) 1.0 nm, and c) 1.5 nm. Contour lines correspond to 0.2% deviation from the ideal spectrum. In d) is shown the real spectrum for Δλ = 1.5 nm (compare Fig. 4 (bottom left panel)), showing that even with poor resolution, well-shaped peaks can be recovered after temporal filtering.
Fig. 7 Electric field amplitude generated by inverse Fourier transform of Eq. (3). Parameters are τ = 350 fs and Δλ = 1.0 nm, and T = 100 fs. Experimentally, pulse E3 has wavevector k3, while pulses E1 and E2 have wave vectors k1. Note that because Esp1 and Esp2 are a consequence of the pixelated pulse-shaping device, they are collinear with E1 and E2 and also propagate along k1. The spurious pulses are located at t ~1.7 ps and ~3.8 ps, corresponding to~2π/Δωc, 4π/Δωn, etc. Inset: detail of pulses E1, E2, and E3.
. Panels a–c show the differences between the pixelated spectra that have been temporally filtered and the ideal 2DFT spectrum with contours plotted at 0.2% intervals. Panel d shows the real 2DFT spectrum that is achievable using the temporal filter even for the worst of the three pixel resolutions (Δλ= 1.5 nm). Note the significant improvement relative to the real spectrum seen in Fig. 4 (bottom left panel). At the 10% contour interval, the difference between this filtered case (Fig. 6d) and the ideal case (see Fig. 3) is almost unobservable. In fact, the maximum deviation from the ideal spectrum after the filtering procedure has been applied is only 1.1%.

Experimentally, the signal is detected spectrally by frequency resolution of the probe pulse. Thus, in order to implement this technique, one would first perform an inverse Fourier transform to retrieve the time domain signal. After this the filter can be applied to remove any contributions to the signal that are from spurious pulses. Our results suggest that even with relatively poor pixel resolution, dramatically improved 2DFT spectra can be collected.

4. Discussion

The cause of spectral distortion lies in contributions to the polarization created by Liouville pathways that are insidiously dependent on the spurious pulses created by the pixelation of the LC-SLM. To illustrate this, consider the electric field generated by inverse Fourier transform of Eq. (3) with Δλ = 1.0 nm, T = 100 fs, and τ = 350 fs shown in Fig. 7. The majority of the electric field amplitude is taken up by the desired three pulse sequence composed of the pulse pair (E1, E2) separated by , and the third excitation pulse E3, centered at t = 0 fs. This pulse sequence is detailed in the inset of Fig. 7 where it is emphasized that each pulse in the pair created by the pulse shaper has the same wavevector k1, and the third pulse propagates along a different wavevector, k3. In addition to the desired field, pixelation effects from pulse shaping cause formation of spurious pulses (Esp1, Esp2) at ~1.7 ps and ~3.8 ps (at this particular ). The spurious pulses are located in time at integer multiples of 2π/Δωc relative to E1 and are strongly chirped because of nonlinear dispersion of the spectral field in the pulse shaping apparatus. Because the spurious pulses are produced collinearly with the desired pulse pair E1, E2, they also have a wavevector given by k1.

The three pulse sequence, E1, E2, E3 induces the desired polarization, P2D(3)(t,T,τ) (see Eq. (7), which, under the rotating frame approximation and assuming T is greater than the pulse width, arises from 16 different Liouville pathways within the context of the three level Hamiltonian. Under a relatively broad range of conditions, however, eight additional contributions that are dependent on the spurious pulses, contribute to the third order polarization. These pathways, which are represented by the double-sided Feynman diagrams in Fig. 8
Fig. 8 Double sided Feynman diagrams showing the Liouville pathways contributing to the spurious polarization The indices 0, 1, and 2 refer to the states in the model and therefore to elements of the density matrix (so that 00 refers toρ00, etc). Exchange of excited state 2 for excited state 1 in the above diagrams results in the four pathways not explicitly shown.
, have the pulse interaction order of E2, E3, Esp (see the right side of Fig. 8 for the pulse order from bottom to top where Esp can be any of the spurious pulses, Esp1, Esp2, etc.) and create a polarization with exactly the same phase matching conditions as the desired signal.

The unwanted polarization arises from a different time ordering of field interactions than the desired polarization. In the first three pathways shown in Fig. 8, E2 (at t = -T) followed by E3 (at t = 0 fs) produces a population. In the first diagram, an excited-state population (ρ11 (shown) or (not shown)) is created, whereas in the second and third diagrams, a ground-state population,, is produced. In the fourth pathway of Fig. 6, these two interactions produce a coherence between the two excited states (ρ21 (shown) or ρ12 (not shown)). For all of the pathways, the third field interaction due to Esp occurs at t~(n2π/Δωc)τ (n = 1,2,3,...). Because excited state coherences decay rapidly for most condensed phase systems, the fourth pathway in Fig. 8 is not expected to substantially contribute to the distortion of the 2DFT spectrum. On the other hand, populations decay slowly and the first three pathways in Fig. 8 will contribute strongly to distortions.

The extent to which these components of the polarization contribute to the detected signal is dependent on the parameters of the experiment, however in nearly all cases, temporal filtering will be necessary. For example, since T always separates the first two interactions (E2 from E3), spectra collected for T much greater than the inverse electronic dephasing rate between ground and excited states will contain minimal distortions. Unfortunately, it is impractical to restrict T to such values in many experiments, and without a temporal filtering approach, the distorting pathways described will then contribute to the detected signal. A further factor which necessitates a temporal filter involves the spectral resolution of the LC-SLM, where collection of distortion-free 2DFT spectra using visible excitation becomes increasingly difficult because of the higher energy density per unit wavelength. Nevertheless, assuming coherence times are not so long that the signals from desired and spurious polarizations overlap, a LC-SLM of reasonable pixel resolution combined with temporal filtering makes a robust and facile apparatus for collection of 2DFT spectra.

5. Conclusion

Simulations using non-perturbative density matrix theory were performed on a three-level model system to determine how pulse shaper pixelation effects lead to distortions of 2DFT spectra. It was determined that additional light matter interaction pathways, which depend on the spurious pulses generated by pixelated devices, distort the detected spectrum. A temporal filtering scheme is presented which is highly successful at eliminating these spurious contributions. Care must still be taken when long coherence times or poor pixel resolution result in cases where the radiated signal from the unwanted polarization temporally overlaps with that from the desired polarization. However, in nearly all condensed phase samples, applying a simple time-domain filter will easily eliminate problems associated with pixelated pulse shaping devices, making collection of high quality 2DFT spectra by non-specialized groups readily accessible.

Acknowledgments

We gratefully acknowledge support for this work from the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Science, U.S. Department of Energy Grant DE-FG02-07ER15890. The authors would like to thank Prof. David Jonas for useful and informative conversations.

References and Links

1.

J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297(3-4), 307–313 (1998). [CrossRef]

2.

I. Stiopkin, T. Brixner, M. Yang, and G. R. Fleming, “Heterogeneous exciton dynamics revealed by two-dimensional optical spectroscopy,” J. Phys. Chem. B 110(40), 20032–20037 (2006). [CrossRef] [PubMed]

3.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007). [CrossRef] [PubMed]

4.

D. Zigmantas, E. L. Read, T. Mančal, T. Brixner, A. T. Gardiner, R. J. Cogdell, and G. R. Fleming, “Two-dimensional electronic spectroscopy of the B800-B820 light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 103(34), 12672–12677 (2006). [CrossRef] [PubMed]

5.

P. Tian, D. Keusters, Y. Suzaki, and W. S. Warren, “Femtosecond phase-coherent two-dimensional spectroscopy,” Science 300(5625), 1553–1555 (2003). [CrossRef] [PubMed]

6.

W. Wagner, C. Li, J. Semmlow, and W. Warren, “Rapid phase-cycled two-dimensional optical spectroscopy in fluorescence and transmission mode,” Opt. Express 13(10), 3697–3706 (2005). [CrossRef] [PubMed]

7.

S. H. Shim, D. B. Strasfeld, E. C. Fulmer, and M. T. Zanni, “Femtosecond pulse shaping directly in the mid-IR using acousto-optic modulation,” Opt. Lett. 31(6), 838–840 (2006). [CrossRef] [PubMed]

8.

S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007). [CrossRef] [PubMed]

9.

E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express 15(25), 16681–16689 (2007). [CrossRef] [PubMed]

10.

J. A. Myers, K. L. M. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional Fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express 16(22), 17420–17428 (2008). [CrossRef] [PubMed]

11.

P. F. Tekavec, J. A. Myers, K. L. M. Lewis, and J. P. Ogilvie, “Two-dimensional electronic spectroscopy with a continuum probe,” Opt. Lett. 34(9), 1390–1392 (2009). [CrossRef] [PubMed]

12.

P. F. Tekavec, J. A. Myers, K. L. M. Lewis, F. D. Fuller, and J. P. Ogilvie, “Effects of chirp on two-dimensional Fourier transform electronic spectra,” Opt. Express 18(11), 11015–11024 (2010). [CrossRef] [PubMed]

13.

M. Kullmann, S. Ruetzel, J. Buback, P. Nuernberger, and T. Brixner, “Reaction dynamics of a molecular switch unveiled by coherent two-dimensional electronic spectroscopy,” J. Am. Chem. Soc. 133(33), 13074–13080 (2011). [CrossRef] [PubMed]

14.

K. L. M. Lewis and J. P. Ogilvie, “Probing photosynthetic energy and charge transfer with two-dimensional electronic spectroscopy,” J. Phys. Chem. Lett. 3(4), 503–510 (2012). [CrossRef]

15.

C. H. Tseng, S. Matsika, and T. C. Weinacht, “Two-dimensional ultrafast Fourier transform spectroscopy in the deep ultraviolet,” Opt. Express 17(21), 18788–18793 (2009). [CrossRef] [PubMed]

16.

C. H. Tseng, P. Sándor, M. Kotur, T. C. Weinacht, and S. Matsika, “Two-dimensional Fourier transform spectroscopy of adenine and uracil using shaped ultrafast laser pulses in the deep UV,” J. Phys. Chem. A 116(11), 2654–2661 (2012). [CrossRef] [PubMed]

17.

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

18.

M. A. Montgomery, E. M. Grumstrup, and N. H. Damrauer, “Fourier transform spectroscopies derived from amplitude or phase shaping of broadband laser pulses with applications to adaptive control,” J. Opt. Soc. Am. B 27(12), 2518–2533 (2010). [CrossRef]

19.

D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. 54(1), 425–463 (2003). [CrossRef] [PubMed]

20.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

21.

S. M. Gallagher Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: theory and model calculations,” J. Phys. Chem. A 103(49), 10489–10505 (1999). [CrossRef]

22.

J. Vaughan, T. Feurer, K. Stone, and K. Nelson, “Analysis of replica pulses in femtosecond pulse shaping with pixelated devices,” Opt. Express 14(3), 1314–1328 (2006). [CrossRef] [PubMed]

23.

K. Blum, Density Matrix Theory and Applications, Physics of Atoms and Molecules, 2nd ed., (Plenum Press, 1996).

24.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier/Academic Press, 2008).

25.

A. M. Weiner, S. DeSilvestri, and E. P. Ippen, “Three-Pulse scattering for femtosecond dephasing studies - theory and experiment,” J. Opt. Soc. Am. B 2(4), 654–662 (1985). [CrossRef]

26.

H. Lee, Y.-C. Cheng, and G. R. Fleming, “Coherence dynamics in photosynthesis: Protein protection of excitonic coherence,” Science 316(5830), 1462–1465 (2007). [CrossRef] [PubMed]

27.

L. Seidner, G. Stock, and W. Domcke, “Nonperturbative approach to femtosecond spectroscopy: General theory and application to multidimensional nonadiabatic photoisomerization processes,” J. Chem. Phys. 103(10), 3998–4011 (1995). [CrossRef]

28.

B. Wolfseder, L. Seidner, G. Stock, and W. Domcke, “Femtosecond pump-probe spectroscopy of electron-transfer systems: a nonperturbative approach,” Chem. Phys. 217(2-3), 275–287 (1997). [CrossRef]

29.

H. Wang and M. Thoss, “Nonperturbative simulation of pump–probe spectra for electron transfer reactions in the condensed phase,” Chem. Phys. Lett. 389(1-3), 43–50 (2004). [CrossRef]

30.

T. S. Mančal, A. V. Pisliakov, and G. R. Fleming, “Two-dimensional optical three-pulse photon echo spectroscopy. I. Nonperturbative approach to the calculation of spectra,” J. Chem. Phys. 124(23), 234504 (2006). [CrossRef] [PubMed]

31.

A. Albrecht, J. Hybl, S. Faeder, and D. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111(24), 10934–10956 (1999). [CrossRef]

32.

J. Hybl, Y. Christophe, and D. Jonas, “Peak shapes in femtosecond 2D correlation spectroscopy,” Chem. Phys. 266(2-3), 295–309 (2001). [CrossRef]

33.

D. Jonas, (personal communication, August 10, 2011).

OCIS Codes
(020.0020) Atomic and molecular physics : Atomic and molecular physics
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(300.0300) Spectroscopy : Spectroscopy

ToC Category:
Spectroscopy

History
Original Manuscript: June 7, 2012
Revised Manuscript: August 7, 2012
Manuscript Accepted: August 21, 2012
Published: August 28, 2012

Citation
Erik M. Grumstrup and Niels H. Damrauer, "Modeling and correction of distorted two-dimensional Fourier transform spectra from pixelated pulse shaping devices," Opt. Express 20, 20908-20919 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-20908


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References

  1. J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett.297(3-4), 307–313 (1998). [CrossRef]
  2. I. Stiopkin, T. Brixner, M. Yang, and G. R. Fleming, “Heterogeneous exciton dynamics revealed by two-dimensional optical spectroscopy,” J. Phys. Chem. B110(40), 20032–20037 (2006). [CrossRef] [PubMed]
  3. G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature446(7137), 782–786 (2007). [CrossRef] [PubMed]
  4. D. Zigmantas, E. L. Read, T. Mančal, T. Brixner, A. T. Gardiner, R. J. Cogdell, and G. R. Fleming, “Two-dimensional electronic spectroscopy of the B800-B820 light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A.103(34), 12672–12677 (2006). [CrossRef] [PubMed]
  5. P. Tian, D. Keusters, Y. Suzaki, and W. S. Warren, “Femtosecond phase-coherent two-dimensional spectroscopy,” Science300(5625), 1553–1555 (2003). [CrossRef] [PubMed]
  6. W. Wagner, C. Li, J. Semmlow, and W. Warren, “Rapid phase-cycled two-dimensional optical spectroscopy in fluorescence and transmission mode,” Opt. Express13(10), 3697–3706 (2005). [CrossRef] [PubMed]
  7. S. H. Shim, D. B. Strasfeld, E. C. Fulmer, and M. T. Zanni, “Femtosecond pulse shaping directly in the mid-IR using acousto-optic modulation,” Opt. Lett.31(6), 838–840 (2006). [CrossRef] [PubMed]
  8. S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A.104(36), 14197–14202 (2007). [CrossRef] [PubMed]
  9. E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express15(25), 16681–16689 (2007). [CrossRef] [PubMed]
  10. J. A. Myers, K. L. M. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional Fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express16(22), 17420–17428 (2008). [CrossRef] [PubMed]
  11. P. F. Tekavec, J. A. Myers, K. L. M. Lewis, and J. P. Ogilvie, “Two-dimensional electronic spectroscopy with a continuum probe,” Opt. Lett.34(9), 1390–1392 (2009). [CrossRef] [PubMed]
  12. P. F. Tekavec, J. A. Myers, K. L. M. Lewis, F. D. Fuller, and J. P. Ogilvie, “Effects of chirp on two-dimensional Fourier transform electronic spectra,” Opt. Express18(11), 11015–11024 (2010). [CrossRef] [PubMed]
  13. M. Kullmann, S. Ruetzel, J. Buback, P. Nuernberger, and T. Brixner, “Reaction dynamics of a molecular switch unveiled by coherent two-dimensional electronic spectroscopy,” J. Am. Chem. Soc.133(33), 13074–13080 (2011). [CrossRef] [PubMed]
  14. K. L. M. Lewis and J. P. Ogilvie, “Probing photosynthetic energy and charge transfer with two-dimensional electronic spectroscopy,” J. Phys. Chem. Lett.3(4), 503–510 (2012). [CrossRef]
  15. C. H. Tseng, S. Matsika, and T. C. Weinacht, “Two-dimensional ultrafast Fourier transform spectroscopy in the deep ultraviolet,” Opt. Express17(21), 18788–18793 (2009). [CrossRef] [PubMed]
  16. C. H. Tseng, P. Sándor, M. Kotur, T. C. Weinacht, and S. Matsika, “Two-dimensional Fourier transform spectroscopy of adenine and uracil using shaped ultrafast laser pulses in the deep UV,” J. Phys. Chem. A116(11), 2654–2661 (2012). [CrossRef] [PubMed]
  17. A. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum.71(5), 1929–1960 (2000). [CrossRef]
  18. M. A. Montgomery, E. M. Grumstrup, and N. H. Damrauer, “Fourier transform spectroscopies derived from amplitude or phase shaping of broadband laser pulses with applications to adaptive control,” J. Opt. Soc. Am. B27(12), 2518–2533 (2010). [CrossRef]
  19. D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem.54(1), 425–463 (2003). [CrossRef] [PubMed]
  20. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).
  21. S. M. Gallagher Faeder and D. M. Jonas, “Two-dimensional electronic correlation and relaxation spectra: theory and model calculations,” J. Phys. Chem. A103(49), 10489–10505 (1999). [CrossRef]
  22. J. Vaughan, T. Feurer, K. Stone, and K. Nelson, “Analysis of replica pulses in femtosecond pulse shaping with pixelated devices,” Opt. Express14(3), 1314–1328 (2006). [CrossRef] [PubMed]
  23. K. Blum, Density Matrix Theory and Applications, Physics of Atoms and Molecules, 2nd ed., (Plenum Press, 1996).
  24. R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier/Academic Press, 2008).
  25. A. M. Weiner, S. DeSilvestri, and E. P. Ippen, “Three-Pulse scattering for femtosecond dephasing studies - theory and experiment,” J. Opt. Soc. Am. B2(4), 654–662 (1985). [CrossRef]
  26. H. Lee, Y.-C. Cheng, and G. R. Fleming, “Coherence dynamics in photosynthesis: Protein protection of excitonic coherence,” Science316(5830), 1462–1465 (2007). [CrossRef] [PubMed]
  27. L. Seidner, G. Stock, and W. Domcke, “Nonperturbative approach to femtosecond spectroscopy: General theory and application to multidimensional nonadiabatic photoisomerization processes,” J. Chem. Phys.103(10), 3998–4011 (1995). [CrossRef]
  28. B. Wolfseder, L. Seidner, G. Stock, and W. Domcke, “Femtosecond pump-probe spectroscopy of electron-transfer systems: a nonperturbative approach,” Chem. Phys.217(2-3), 275–287 (1997). [CrossRef]
  29. H. Wang and M. Thoss, “Nonperturbative simulation of pump–probe spectra for electron transfer reactions in the condensed phase,” Chem. Phys. Lett.389(1-3), 43–50 (2004). [CrossRef]
  30. T. S. Mančal, A. V. Pisliakov, and G. R. Fleming, “Two-dimensional optical three-pulse photon echo spectroscopy. I. Nonperturbative approach to the calculation of spectra,” J. Chem. Phys.124(23), 234504 (2006). [CrossRef] [PubMed]
  31. A. Albrecht, J. Hybl, S. Faeder, and D. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys.111(24), 10934–10956 (1999). [CrossRef]
  32. J. Hybl, Y. Christophe, and D. Jonas, “Peak shapes in femtosecond 2D correlation spectroscopy,” Chem. Phys.266(2-3), 295–309 (2001). [CrossRef]
  33. D. Jonas, (personal communication, August 10, 2011).

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