Abstract

Sculpting the subcycle temporal structure of optical waveforms allows one to image and even control the electronic clouds of atoms, molecules, and solids. Here, we show how the transverse spin component arising upon spatial confinement of such optical waveforms enables extremely efficient chiral recognition and control of ultrafast chiral dynamics. When an intense and ultrashort linearly polarized laser pulse is tightly focused into a medium of randomly oriented chiral molecules, the medium generates light that is elliptically polarized, with opposite helicities and opposite rotations of the polarization ellipse in media of opposite handedness. In contrast to conventional optical activity of chiral media, this new nonlinear optical activity is driven by purely electric–dipole interactions. It leads to giant enantio-sensitivity in the near VIS–UV domain, where optical instrumentation is readily available, already in optically thin media. Adding a polarizer turns the rotation of the polarization ellipse into a highly enantio-sensitive intensity of nonlinear optical response. We also show that subcycle optical control of the incident light wave enables full control over the enantio-sensitive response. The proposed all-optical method not only enables chiral discrimination with extreme enantio-efficiency, but also ultrafast imaging and control of chiral dynamics using commercially available optical technology.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Chiral molecules can exist in pairs of mirror-reflected versions, known as the left- and right-handed enantiomers. These mirror twins behave identically unless they interact with another chiral object, such as circularly polarized light, which has long served as a standard tool for chiral recognition [1]. However, circularly polarized light is not an efficient chiral probe: the chiral structure of the light helix emerges only beyond the electric–dipole approximation. The weakness of non-dipole interactions makes the optical response of left- and right-handed media to this field virtually identical, and chiral discrimination hard. One way around this fundamental limitation is to use synthetic chiral light that is locally chiral [2], i.e., within the electric–dipole approximation: the tip of the electric-field vector draws a chiral Lissajous figure in time. We have recently shown [2] that the application of such light to high-harmonic generation spectroscopy can lead to giant enantio-sensitivity at the level of total intensity signals. With the benefit of hindsight, one can recognize that the enantio-sensitive population transfer achieved via microwave spectroscopy [36] and the enantio-sensitive electro–optic effect [7] also rely on the general concept of locally chiral fields.

It has been known, since the time of Louis Pasteur [8], that the chirality of light is not required to detect the chirality of matter. Even linearly polarized light leads to optical activity: when propagating through a chiral medium, its polarization plane rotates, in opposite directions in media of opposite handedness. While this effect relies on weak non-electric-dipole interactions, it benefits from the cumulative addition from all chiral units in optically dense media, eventually leading to large rotation angles. Still, detecting the handedness of dilute samples, such as those routinely produced in chemistry labs, can be challenging.

Several techniques [925] can yield strongly enantio-sensitive signals in thin and low-concentration media by analyzing vectorial observables [26], such as the induced polarization in three-wave mixing experiments with crossed polarizations [912], or the direction of the photoelectron current upon ionization with circularly polarized light [1319], where the chiral response arises due to purely electric–dipole interactions. The application of two-color cross-polarized beams and quantum control strategies may further increase the enantio-sensitivity of the latter method [2022]. Even enantio-sensitive attosecond delays in photo-ionization have recently been reported [23]. However, these signals require angle-resolved detection of photoelectrons, rather than purely optical signals, which restricts their applicability. In the case of three-wave mixing [912], measuring the enantio-sensitive response requires resolving the phase of the emitted light, which can be challenging in the context of electronic excitations and ultrafast processes. Another interesting example of a chiral response to achiral light is the enantio-sensitive orientation of chiral molecules by an optical centrifuge [27,28].

Here, we show how intense linearly polarized laser pulses, when confined both in time and in space, can generate an extremely strong nonlinear analogue of optical activity, enabling efficient chiral discrimination of dilute samples with commercially available optical technology. Intense few-cycle laser pulses have opened a range of new opportunities for ultrafast spectroscopy [29] in atoms [30,31], molecules [32], and solids [33,34]. When the electric field amplitude of the laser pulse $E(t) = {E_0}a(t)\cos (\omega t + {\phi _{{\rm CEP}}})$ is strongly modulated by the rapidly varying envelope $a(t)$, the carrier-envelope phase (CEP) ${\phi _{{\rm CEP}}}$ determines the temporal structure of the electric field oscillations, and thus the nonlinear response of matter. In the frequency domain, the large coherent bandwidth of ultrashort pulses gives rise to interference between different-order multiphoton pathways. This interference is sensitive to the spectral phase in general, and to ${\phi _{{\rm CEP}}}$ in particular, giving rise to ${\phi _{{\rm CEP}}}$-dependent low-order [35] and high-order harmonic generation [36] and photoelectron emission [30,3739].

Here, we show that the interference of different multiphoton pathways also opens new opportunities for highly enantio-sensitive imaging and control of ultrafast chiral dynamics in molecules. Below, we present a simple all-optical setup that encodes the handedness of a chiral medium in the polarization properties of the nonlinear response. We show that the emitted harmonic light exhibits optical activity: opposite ellipticity and opposite rotation angles in media of opposite handedness. Adding a polarizer turns the enantio-sensitive polarization into highly enantio-sensitive intensity.

The proposed all-optical method relies on standard lasers to generate ultrashort pulses, which are then tightly focused into a jet of chiral molecules, as shown in Figs. 1(a) and 1(b). Commercially available technology allows one to generate CEP-stable pulses lasting only a few cycles [confinement in time, see Fig. 1(c)], with broad spectral bandwidth, as shown in Fig. 1(d). By focusing the beam tightly [confinement in space, see Figs. 1(a) and 1(b)], we create a strong longitudinal component [40], and the field acquires forward ellipticity ${\varepsilon _f}$, in the propagation direction. For a Gaussian beam that is linearly polarized along $x$, ${\varepsilon _f} = - \frac{{2x}}{{k{w^2}}}$, where $w$ is the beam waist, $k = \frac{{2\pi}}{\lambda}$ is the wave number, and $\lambda$ is the wavelength, see Fig. 1(b) and Supplement 1. The sign of the forward ellipticity, or transverse spin, is opposite on opposite sides of beam axis, and is locked to the propagation direction. Harmonic generation in a tight focus requires thin media, which can be realized in a liquid microjet [41] judiciously placed before the focus.

 figure: Fig. 1.

Fig. 1. Proposed experimental setup. (a) A few-cycle, $x$-polarized, tightly focused beam acquires forward ellipticity ${\varepsilon _f}$ along the propagation direction $y$, with opposite sign at opposite sides of the beam axis. A microjet carrying chiral molecules, placed before the focus, generates elliptically polarized harmonics. The transverse ellipticity ${\varepsilon _t}$ and rotation angle $\gamma$ of the harmonic polarization ellipses record the molecular handedness, and have opposite signs at opposite sides of the beam. (b) Intensity (black) and forward ellipticity (red) of the driving field. (c–d) Electric field in (c) the time and (d) frequency domains at the beam axis ($x = 0$), for ${\phi _{{\rm CEP}}} = 0,\pi /2$. Laser parameters: intensity $6 \cdot {10^{13}}\;{\rm W} \cdot {{\rm cm}^{- 2}}$, focal diameter $2w = 5\,\,\unicode{x00B5}{\rm m}$, carrier wavelength $\lambda = 780$ nm, and pulse duration 7 fs (FWHM of the field amplitude).

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Consider first the nonlinear response of isotropic achiral media to such light. Because both the field and the medium have mirror symmetry with respect to the xy plane [see Fig. 1(a)], the induced polarization ${\boldsymbol P}(t) = {P_x}(t){\boldsymbol{ \hat x}} + {P_y}(t){\boldsymbol {\hat y}} + {P_z}(t){\boldsymbol {\hat z}}$ is confined to this plane (i.e., ${P_z} = 0$). Since ${P_y}$ is along the propagation direction, it does not contribute to the far-field signal. Thus, the far-field harmonic emission is linearly polarized along $x$. In contrast, in isotropic chiral media, where the mirror symmetry is absent, ${P_z}$ is no longer forbidden, and has an opposite phase in media of opposite handedness; that is, ${\boldsymbol P}(t)$ becomes 3D and enantio-sensitive at the single-molecule response. The chiral polarization component ${P_z}$ can be strong because it is driven by purely electric-dipole interactions. As we show here, the interplay between ${P_x}$ and ${P_z}$ can lead to the generation of harmonic light with enantio-sensitive polarization.

For long pulses, ${P_x}$ and ${P_z}$ have different temporal symmetry; while ${P_x}(t + T/2) \simeq - {P_x}(t)$, where $T$ is the laser period, ${P_z}(t + T/2) \simeq {P_z}(t)$. As a result, the Fourier components of ${P_x}$ and ${P_z}$ do not overlap. However, this does not apply to broadband few-cycle pulses, where the ${\phi _{{\rm CEP}}}$-dependent interference of even- and odd-order multiphoton pathways is the norm, and ${P_x}$ and ${P_z}$ overlap in frequency. As ${P_z}$ is out of phase in opposite enantiomers, the molecular handedness is mapped on opposite ellipticities and opposite rotations of the polarization ellipse of the generated light. As we show here, control of ${\phi _{{\rm CEP}}}$ enables control of this enantio-sensitive response.

We have calculated the nonlinear response of randomly oriented propylene oxide to the field in Fig. 1 using time-dependent density functional theory (see Supplement 1). The emitted harmonic field can be written as ${{\boldsymbol F}_{R,L}} = {F_a}{\boldsymbol x} \pm {F_c}{\boldsymbol z}$, where ${\pm}$ is ${+}$ for right-handed and ${-}$ for left-handed molecules. The intensity, proportional to $|{F_a}{|^2} + |{F_c}{|^2}$, is not enantio-sensitive, as shown in Fig. 2(a), because the incident light is achiral. However, the molecular handedness controls the polarization of the emitted light. Figures 2(b)–2(e) show the ellipticity and rotation angle of the polarization ellipses of the light emitted by left- and right-handed molecules for ${\phi _{{\rm CEP}}} = 0.5\pi$. Both quantities reach high values in wide spatial and spectral regions, allowing for accurate detection of the medium handedness using standard optical instrumentation. Giant enantio-sensitivity is observed near harmonics 4, 6, and 8, corresponding to wavelengths of $\lambda = 195,130$, and 97.5 nm. The modulation of the ellipticity and rotation angle with the harmonic number reflects the anisotropy of the chiral molecular potential.

 figure: Fig. 2.

Fig. 2. Ultrafast and nonlinear optical rotation. (a) Far-field harmonic amplitude emitted by randomly oriented propylene oxide in the setup of Fig. 1 (see caption for laser parameters), for ${\phi _{{\rm CEP}}} = \pi /2$, versus the harmonic number and emission angle. (b)–(c) Ellipticity and (d)–(e) rotation angle of the polarization ellipses of the harmonic light emitted from (b)–(d) left-handed and (c)–(e) right-handed molecules.

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Note that ${\varepsilon _f}$ has opposite sign at opposite sides of the beam axis, as shown in Figs. 1(a) and 1(b). Thus, so does the chiral response of the medium ${P_z}$, which vanishes at $x = 0$ (where the driver is linear), but becomes strong close to the axis in a tight focus [$2w = 5\,\,\unicode{x00B5}\rm m$, see Fig. 1(b)]. The harmonic ellipticity and rotation angle change linearly with $1/w$, and thus the harmonic polarization is still highly enantio-sensitive for larger values of $w$ (see Supplement 1).

This new type of optical activity enables ultrafast and highly enantio-sensitive imaging of chiral matter, and creates new opportunities to determine the enantiomeric excess in mixtures, $ee = \frac{{{C_R} - {C_L}}}{{{C_R} + {C_L}}}$, where ${C_{R/L}}$ is the concentration of right/left molecules. For weak values of $ee$ or $|{F_c}|/|{F_a}|$, both the harmonic ellipticity and rotation angle depend linearly on $ee$ (see Supplement 1).

The polarization of the emitted harmonic light records the relative amplitude and phase of the chiral and achiral components of the light-driven dynamics. Shaping the subcycle temporal structure of the incident wave allows us to control it. Figure 3 shows that the ellipticity and rotation angle of the emitted light can be controlled by the CEP of the driving field. We present values for emission angle $\beta = {3^ \circ}$, although the results for other angles are similar, as shown in Figs. 2(b)–2(e). The mapping between light-driven enantio-sensitive dynamics and polarization of harmonic light opens the way to reconstruct the ultrafast electronic response of a chiral molecule from polarization measurements.

 figure: Fig. 3.

Fig. 3. CEP-control over the enantio-sensitive polarization. (a)–(b) Ellipticity and (c)–(d) rotation angle of the polarization ellipses of the harmonic light emitted from (a)–(c) left-handed and (b)–(d) right-handed propylene oxide at emission angle $\beta = {3^ \circ}$, as functions of the harmonic number and the CEP. See the laser parameters in Fig. 1.

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The highly enantio-sensitive polarization of the emitted harmonic light can be converted into highly enantio-sensitive intensity. Placing a polarizer before the detector, we can project the achiral (${F_a}$) and chiral (${\pm}{F_c}$) components of the harmonic light onto the same polarization axis and make them interfere. The harmonic intensity becomes ${I_{R,L}}(\alpha) \propto |\cos (\alpha){F_a} \pm \sin (\alpha){F_c}{|^2}$, where $\alpha$ is the angle between the selected polarization direction and the $x$ axis. Figures 4(a) and 4(b) show the harmonic intensity emitted from left- and right-handed propylene oxide at the detector for $\alpha = {85^ \circ}$. The controlled interference between ${F_a}$ and ${F_c}$ makes the harmonic intensity strongly enantio-sensitive. The degree of enantio-sensitivity can be quantified using a standard definition of chiral dichroism, $CD = 2\frac{{{I_L} - {I_R}}}{{{I_L} + {I_R}}}$, see Figs. 4(c)–4(f). This quantity is related to ${F_a}$, ${F_c}$ and $\alpha$ via

$${\rm CD} = 2\frac{{\sin (2\alpha)|{F_a}||{F_c}|\cos ({\phi _{\textit{ac}}})}}{{\mathop {\cos}\nolimits^2 \alpha |{F_a}{|^2} + \mathop {\sin}\nolimits^2 \alpha |{F_c}{|^2}}},$$
where ${\phi _{\textit{ac}}} = \arg ({F_c}/{F_a})$. If $\alpha = {45^ \circ}$, then ${\rm CD} = 2{s_2}/{s_0}$, where ${s_0}$ and ${s_2}$ are the Stokes parameters in the $xz$ basis. Figures 4(a) and 4(b) show that adding a polarizer to the optical setup creates left–right asymmetry (along $x$) in the intensity distribution of the emitted light. This is analogous to the forward–backward asymmetry observed in photoelectron circular dichroism [13] or the left–right asymmetry in enantio-sensitive light bending [42]. Here, the vectorial observable that allows enantio-sensitivity in the absence of chiral light [26] is the flux of emitted light along $x$.
 figure: Fig. 4.

Fig. 4. Turning enantio-sensitive polarization into enantio-sensitive intensity. Harmonic intensity emitted from (a) left-handed and (b) right-handed propylene oxide in the setup of Fig. 1 for ${\phi _{{\rm CEP}}} = \pi /2$, using a polarizer with $\alpha = {85^ \circ}$. Chiral dichroism, $CD = 2\frac{{{I_L} - {I_R}}}{{{I_L} + {I_R}}}$, for (c) fixed ${\phi _{{\rm CEP}}} = \pi /2$ and $\alpha = {85^ \circ}$, (d) fixed $\alpha = {85^ \circ}$ and emission angle $\beta = {3^ \circ}$, (e) fixed ${\phi _{{\rm CEP}}} = \pi /2$ and $\beta = {3^ \circ}$, and (f) fixed harmonic number $N = 6$ and $\beta = {3^ \circ}$. See other laser parameters in Fig. 1.

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The relative strength between ${F_a}$ and ${F_c}$ can be adjusted with the polarizer angle $\alpha$, and their relative phase is controlled by the CEP of the driver. Thus, by adjusting $\alpha$ and ${\phi _{{\rm CEP}}}$, we can achieve perfect constructive or destructive interference for any harmonic frequency. Figure 4(d) shows the chiral dichroism as a function of the harmonic number $N$ and ${\phi _{{\rm CEP}}}$, for $\alpha = {85^ \circ}$ and emission angle $\beta = {3^ \circ}$. Note that the sign of the chiral dichroism is not affected by the value of $\alpha$, as shown in Figs. 4(e) and 4(f), as long as ${-}{90^ \circ} \lt \alpha \lt 0$ or $0 \lt \alpha { \lt 90^ \circ}$, and that ${\rm CD}(\alpha) = - {\rm CD}(- \alpha)$ [see Eq. (1)].

The chiral dichroism in the emitted light provides direct access to chiral electron dynamics. In particular, the white lines in Fig. 4(d) indicate the CEP values for which the achiral and chiral components of the induced polarization have a phase delay of ${\pm}\pi /2$; thus, the polarization vector draws an ellipse in the $xz$ plane at frequency $N\omega$ with its major axis along $x$ (or $z$). On the other hand, the CEP values that maximize the chiral dichroism correspond to a phase delay of 0 or $\pi$, so that the polarization vector at frequency $N\omega$ forms a straight line in the $xz$ plane.

We believe that the proposed method opens new opportunities for highly enantio-sensitive imaging and control of ultrafast chiral dynamics using commercially available optical technology. The possibility to reconstruct the electronic response of chiral molecules to light via multidimensional spectroscopy, with the CEP and frequency being the two measurement dimensions, from either polarization or intensity measurements, could be exploited to visualize changes in molecular chirality during photo-induced chemical reactions. Adding more spectroscopic dimensions, e.g., a chirp parameter, may expand these opportunities.

One of the key ingredients of the proposed method is the longitudinal field that naturally arises when light is confined in space. Such fields appear in a number of physical environments, e.g., in optical nanofibers and periodic nanophotonic structures, where light is confined in one or two spatial dimensions [43]. New opportunities may arise from creating controlled waveforms inside these nanostructures. They may allow the design of compact devices for efficient chiral discrimination and imaging that can be used in routine operations in chemistry labs.

Funding

Royal Society (URF\R1\201333); Engineering and Physical Sciences Research Council (MURI-MIR EP/N018680/1); Deutsche Forschungsgemeinschaft (152/6-2, SM 292/5-2, SPP 1840); Horizon 2020 Framework Programme (101029393).

Acknowledgment

The authors thank Felipe Morales for technical support and stimulating discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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References

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  1. N. Berova, P. L. Polavarapu, K. Nakanishi, and R. W. Woody, Comprehensive Chiroptical Spectroscopy (Wiley, 2013).
  2. D. Ayuso, O. Neufeld, A. F. Ordonez, P. Decleva, G. Lerner, O. Cohen, M. Ivanov, and O. Smirnova, Nat. Photonics 13, 866 (2019).
    [Crossref]
  3. S. Eibenberger, J. Doyle, and D. Patterson, Phys. Rev. Lett. 118, 123002 (2017).
    [Crossref]
  4. M. Leibscher, T. F. Giesen, and C. P. Koch, J. Chem. Phys. 151, 014302 (2019).
    [Crossref]
  5. C. Pérez, A. L. Steber, A. Krin, and M. Schnell, J. Phys. Chem. Lett. 9, 4539 (2018).
    [Crossref]
  6. N. V. Vitanov and M. Drewsen, Phys. Rev. Lett. 122, 173202 (2019).
    [Crossref]
  7. P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, Phys. Rev. Lett. 91, 173901 (2003).
    [Crossref]
  8. L. Pasteur, Ann. Chim. Phys. Sér. 24, 442 (1848).
  9. P. Fischer, D. S. Wiersma, R. Righini, B. Champagne, and A. D. Buckingham, Phys. Rev. Lett. 85, 4253 (2000).
    [Crossref]
  10. D. Patterson and J. M. Doyle, Phys. Rev. Lett. 111, 023008 (2013).
    [Crossref]
  11. N. Ji, V. Ostroverkhov, M. Belkin, Y.-J. Shiu, and Y.-R. Shen, J. Am. Chem. Soc. 128, 8845 (2006).
    [Crossref]
  12. M. A. Belkin, T. A. Kulakov, K.-H. Ernst, L. Yan, and Y. R. Shen, Phys. Rev. Lett. 85, 4474 (2000).
    [Crossref]
  13. B. Ritchie, Phys. Rev. A 13, 1411 (1976).
    [Crossref]
  14. I. Powis, J. Chem. Phys. 112, 301 (2000).
    [Crossref]
  15. N. Böwering, T. Lischke, B. Schmidtke, N. Müller, T. Khalil, and U. Heinzmann, Phys. Rev. Lett. 86, 1187 (2001).
    [Crossref]
  16. C. Lux, M. Wollenhaupt, T. Bolze, Q. Liang, J. Köhler, C. Sarpe, and T. Baumert, Angew. Chem. (Int. Ed.) 51, 5001 (2012).
    [Crossref]
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D. Ayuso, A. F. Ordonez, P. Decleva, M. Ivanov, and O. Smirnova, Nat. Commun. 12, 3951 (2021).
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2020 (1)

I. Tutunnikov, J. Floß, E. Gershnabel, P. Brumer, I. S. Averbukh, A. A. Milner, and V. Milner, Phys. Rev. A 101, 021403 (2020).
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2019 (6)

A. A. Milner, J. A. M. Fordyce, I. MacPhail-Bartley, W. Wasserman, V. Milner, I. Tutunnikov, and I. S. Averbukh, Phys. Rev. Lett. 122, 223201 (2019).
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P. V. Demekhin, A. N. Artemyev, A. Kastner, and T. Baumert, Phys. Rev. Lett. 121, 253201 (2018).
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I. Tutunnikov, E. Gershnabel, S. Gold, and I. S. Averbukh, J. Phys. Chem. Lett. 9, 1105 (2018).
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A. F. Ordonez and O. Smirnova, Phys. Rev. A 98, 063428 (2018).
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2017 (4)

S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. Géneaux, F. Légaré, S. Petit, B. Pons, G. Porat, T. Ruchon, R. Taeb, V. Blanchet, and Y. Mairesse, Science 358, 1288 (2017).
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G. Galinis, J. Strucka, J. C. T. Barnard, A. Braun, R. A. Smith, and J. P. Marangos, Rev. Sci. Instrum. 88, 083117 (2017).
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2016 (1)

A. Kastner, C. Lux, T. Ring, S. Züllighoven, C. Sarpe, A. Senftleben, and T. Baumert, ChemPhysChem 17, 1119 (2016).
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2015 (2)

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Nature 521, 498 (2015).
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K. Y. Bliokh and F. Nori, Phys. Rep. 592, 1 (2015).
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2013 (4)

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, Nature 493, 70 (2013).
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D. Patterson, M. Schnell, and J. M. Doyle, Nature 497, 475 (2013).
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2009 (2)

F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
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T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, Nat. Phys. 5, 357 (2009).
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2006 (2)

M. F. Kling, C. Siedschlag, A. J. Verhoef, J. I. Khan, M. Schultze, T. Uphues, Y. Ni, M. Uiberacker, M. Drescher, F. Krausz, and M. J. J. Vrakking, Science 312, 246 (2006).
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N. Ji, V. Ostroverkhov, M. Belkin, Y.-J. Shiu, and Y.-R. Shen, J. Am. Chem. Soc. 128, 8845 (2006).
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2003 (2)

P. Fischer, A. D. Buckingham, K. Beckwitt, D. S. Wiersma, and F. W. Wise, Phys. Rev. Lett. 91, 173901 (2003).
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A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, Nature 421, 611 (2003).
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N. Böwering, T. Lischke, B. Schmidtke, N. Müller, T. Khalil, and U. Heinzmann, Phys. Rev. Lett. 86, 1187 (2001).
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Supplementary Material (1)

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)

Fig. 1.
Fig. 1. Proposed experimental setup. (a) A few-cycle, $x$-polarized, tightly focused beam acquires forward ellipticity ${\varepsilon _f}$ along the propagation direction $y$, with opposite sign at opposite sides of the beam axis. A microjet carrying chiral molecules, placed before the focus, generates elliptically polarized harmonics. The transverse ellipticity ${\varepsilon _t}$ and rotation angle $\gamma$ of the harmonic polarization ellipses record the molecular handedness, and have opposite signs at opposite sides of the beam. (b) Intensity (black) and forward ellipticity (red) of the driving field. (c–d) Electric field in (c) the time and (d) frequency domains at the beam axis ($x = 0$), for ${\phi _{{\rm CEP}}} = 0,\pi /2$. Laser parameters: intensity $6 \cdot {10^{13}}\;{\rm W} \cdot {{\rm cm}^{- 2}}$, focal diameter $2w = 5\,\,\unicode{x00B5}{\rm m}$, carrier wavelength $\lambda = 780$ nm, and pulse duration 7 fs (FWHM of the field amplitude).
Fig. 2.
Fig. 2. Ultrafast and nonlinear optical rotation. (a) Far-field harmonic amplitude emitted by randomly oriented propylene oxide in the setup of Fig. 1 (see caption for laser parameters), for ${\phi _{{\rm CEP}}} = \pi /2$, versus the harmonic number and emission angle. (b)–(c) Ellipticity and (d)–(e) rotation angle of the polarization ellipses of the harmonic light emitted from (b)–(d) left-handed and (c)–(e) right-handed molecules.
Fig. 3.
Fig. 3. CEP-control over the enantio-sensitive polarization. (a)–(b) Ellipticity and (c)–(d) rotation angle of the polarization ellipses of the harmonic light emitted from (a)–(c) left-handed and (b)–(d) right-handed propylene oxide at emission angle $\beta = {3^ \circ}$, as functions of the harmonic number and the CEP. See the laser parameters in Fig. 1.
Fig. 4.
Fig. 4. Turning enantio-sensitive polarization into enantio-sensitive intensity. Harmonic intensity emitted from (a) left-handed and (b) right-handed propylene oxide in the setup of Fig. 1 for ${\phi _{{\rm CEP}}} = \pi /2$, using a polarizer with $\alpha = {85^ \circ}$. Chiral dichroism, $CD = 2\frac{{{I_L} - {I_R}}}{{{I_L} + {I_R}}}$, for (c) fixed ${\phi _{{\rm CEP}}} = \pi /2$ and $\alpha = {85^ \circ}$, (d) fixed $\alpha = {85^ \circ}$ and emission angle $\beta = {3^ \circ}$, (e) fixed ${\phi _{{\rm CEP}}} = \pi /2$ and $\beta = {3^ \circ}$, and (f) fixed harmonic number $N = 6$ and $\beta = {3^ \circ}$. See other laser parameters in Fig. 1.

Equations (1)

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C D = 2 sin ( 2 α ) | F a | | F c | cos ( ϕ ac ) cos 2 α | F a | 2 + sin 2 α | F c | 2 ,

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