Optics of spatiotemporal optical vortices for atto- and nano-photonics

1 Introduction

Spatiotemporal optical vortices (STOVs), spatiotemporal fields with a transverse line phase singularity, were theoretically described about two decades ago [1], [2], and first observed in a nonlinear optics experiment as a side-effect of the arrest of collapse in filamentation [3]. As a fundamentally linear wave object, they were subsequently generated by purely linear means, typically with a 4f pulse shaper with a tilted π-step phase plate, a spiral phase plate [4], or a spatial light modulator [5], placed at the Fourier plane. Other more compact devices have afterwards been proposed or realized [6], [7], [8], see also [9] for a review. Last years, STOVs are once again being generated by nonlinear means from visible and near infrared STOVs that drive perturvative and nonperturbative, strong-field light–matter interactions, and up-convert their structure to second-order harmonic [10], [11], sum-frequency STOVs in ultraviolet [12], and third-order harmonic STOVs [13]. Theoretical studies of high-order harmonic and attosecond STOVs have been reported in [14], [15], and they have been observed recently [16], giving birth to STOVs at the attosecond and nanometer scales.

The precise knowledge of the spatiotemporal structure of STOVs, its change on propagation or focusing systems is crucial for the precise control of the driving field and its interaction with matter. This knowledge is also required for the interpretation of the presumably STOV-like structures observed in the generated harmonics or attosecond pulses. After preliminary numerical studies [17], [18], [19], the free space propagation of STOVs, and of their dual fields, or spatiotemporal tilted Hermite-lobed (THL) pulses, were studied analytically in [20], and focusing of ST THL pulses producing STOVs in [21]. Focusing of STOVs themselves has only been inspected numerically [17], [18], [22].

We develop a theory of the optics of STOVs similar in simplicity to the theory of Gaussian beam propagation through optical systems represented by ABCD matrices. Material dispersion in the elements is assumed to be small, as intended in the experiments. Since STOVs are not continuous beams we develop alternate formulations in spatioemporal (ST) and spatiospectral (SS) domains that would describe the observations with spatial and temporal resolution, and the observations of spatially resolved spectra. The ST description is the most common with visible or near infrared STOVs as they are usually long pulses of several tens of femtoseconds where characterization techniques are standard. Interestingly, second-order harmonic STOVs are still observed in ST domain [4], [10], third-order harmonic STOVs are observed in both ST and SS domains [13], but the high-order harmonics in [16] can only be characterized experimentally in SS domain given the limitations of ST characterization techniques at attosecond and nanometer scales.

A complete picture of the ST propagation of STOVs, as that seen in Figure 1(a) with the temporal abscissa, needs the supplementary ST THL pulse seen in Figure 2(a), also with the temporal abscissa. STOVs and ST THL pulses are the two complementary output pulses from a standard 4f pulse shaper when the tilted π-step phase plate or the spiral phase plate is placed at the common focus, and constitute a spatial Fourier transform pair. The spatiospectrum of STOVs have been repeatedly observed or numerically evaluated [13], [16], but only calculated for unit topological charge (TC) [14]. It turns out that the temporal Fourier transform of a STOV of arbitrary TC is the SS THL pulse, also in Figure 2(a) with the frequency abscissa, and the temporal Fourier transform of the ST THL pulse is the SS optical vortex (SSOV), seen Figure 1(a) with the frequency abscissa. Thus, a complete picture of propagation in SS domain requires the same pair of pulses. In addition, the symmetry of the propagation equations for paraxial and quasi-monochromatic pulses in ST and SS domains implies that the propagation of STOVs or ST THL pulses is identical to the propagation of SSOVs or SS THL pulses. This duality has implications such as, for example, that one may observe the spatially resolved spectrum of a STOV created by a 4f pulse shaper without any spectrometer by simply replacing the tilted π-step phase plate with the spiral phase plate, or vice versa for a ST THL pulse, and the same after propagation though any optical elements. For the extreme ultraviolet harmonic or attosecond STOVs generated at a focus, one would observe their ST structure at that focus with a spatially resolved spectrometer at the far field. It was in fact the numerical observation of this duality that led to the interpretation of the high harmonics in [16] as STOVs.

Figure 1:

Transformation of STOVs and SSOVs through various optical systems. The TC is +2 in this example. (a) STOV or SSOV, and their (b) free space propagation, (c) focusing, (d) Fourier transform, (e) 4f system magnification. First row: Iso-intensity surface. Second row: phase.

Figure 2:

Transformation of ST THL and SS THL pulses through various optical systems. The Hermite polynomial order is 2 in this example. (a) ST THL or SS THL, and their (b) free space propagation, (c) focusing, (d) Fourier transform, (e) 4f system magnification. First row: Iso-intensity surface. Second row: phase.

We illustrate the theory with focusing and propagation up to the far field as the most common ABCD system in experiments with STOVs/ST THL pulses, providing complete, closed-form expressions for STOV and ST THL pulse focusing in ST and SS domains, and unveil unexpected features at the far field that should be taken into account in experiments.

Also, once the debate on the transverse orbital angular momentum (OAM) of STOVs has been closed [23], we find that ABCD systems such as a lens and a Fourier transforming system, or image-forming systems such a single lens or a 4f magnifier, can manipulate the transverse OAM as desired, imparting or removing it, even changing its direction, which will be of interest for the future applications of the transverse OAM.

2 Propagation of STOVs and ST THL pulses through optical systems

Under paraxial and quasi-monochromatic conditions, the ST envelope of an optical field in non-dispersive media is ruled out by the Schrödinger equation ∂ _z ψ = (i/2k ₀)Δ_⊥ ψ, where k ₀ = ω ₀/c is the propagation constant, ω ₀ is the carrier frequency, c the velocity of light, and Δ_⊥ the transverse Laplacian to the propagation direction z. In absence of dispersion, the local time t′ = t − z/c does not appear explicitly in the Schrödinger equation since the second-order derivative in the local time vanishes. Ideally, STOVs and ST THL pulses have separable fields of the form ψ ₀(x, y, t′) = X ₀(x, t′)Y ₀(y). Separability allows us to use the 1D version of Schrödinger equation ∂ z X = ( i / 2 k 0 ) ∂ x 2 X for the propagation of X ₀ and omit from now the independent propagation of Y ₀. The solution of the 1D Schrödinger equation can then be written in terms of the 1D Fresnel diffraction integral as

for propagation a distance z, where t′ appears only as a parameter in the initial pulse. These fields may additionally experience propagation through a series of aligned optical systems that can be represented by an ABCD matrix and that do not introduce significant material dispersion. If these systems do not break the separability in x and y, the propagation of X ₀(x, t′) can be written in terms of the generalized Fresnel diffraction integral as [24]

3 Spectrospatial analysis

From their discovery [3], most of the theoretical analysis and experiments with STOVs/ST THL pulses are performed in the direct ST domain [4], [5], [6], [7], [8], also in the experiments of second harmonic [10], [11] and third harmonic [13] generation. Exceptions are those involving extremely short wave lengths in the extreme ultraviolet and soft X-rays range [14], [15], [16], given the current limitations in the direct ST characterization at these wave lengths, and where a SS characterization is the standard one. STOVs at these wave lengths are in fact being identified from the observation of numerical and experimental spatiospectra. It is then surprising that the spatiospectra of STOVs have only been evaluated analytically for unit TC [14]. A precise knowledge of the SS structure of STOVs/ST THL pulses provided by closed-form, simple expressions is fundamental for understanding and interpreting these and future experiments.

It turns out that the temporal frequency spectrum X ̂ ( x , Ω ) = ( 1 / 2 π ) ∫ − ∞ ∞ d t ′ X ( x , t ′ ) e i Ω t ′ of the STOV in Eq. (3) is the SS THL pulse

where Ω = ω − ω ₀ is the detuning from the carrier frequency ω ₀ and Ω₀ = 2/t ₀ is the (narrow) spectral width. An irrelevant factor ( − i ) n / π Ω 0 is omitted in Eq. (17). Similarly, the temporal frequency spectrum of the ST THL pulse in Eq. (11) is the SSOV

where i n / π Ω 0 is omitted. Actually, the fact that STOVs and ST THL pulses are not only spatial Fourier transform pairs but also temporal Fourier transform pairs is an obvious fact from the symmetric role of x and t′ in their expressions, which however has not been noticed. The two standard outputs pulses from a 4f pulse shaper when a tilted π-step phase plate or a spiral phase plate are placed at the common focus are then doubly related by a spatial Fourier transform and a temporal Fourier transform.

We can then Fourier transform in time the generalized Fresnel diffraction integral in Eq. (2) to write

which is formally the same as its ST counterpart in Eq. (2) On doing this, we are ignoring the slight differences of propagation with frequency, which is justified and the customary approach for the narrow-band, many-cycle pulses considered in this paper, and is ultimately validated by the numerical and experimental observations. The propagated spatiospectrum of the STOV can then be obtained from the propagated ST THL pulse by simply making the replacement t′/t ₀ → Ω/Ω₀ in its expression (12):

Analogously, the propagated spatiospectrum of the ST THL pulse is obtained from the same replacement in the expression (4) of the propagated STOV:

Thus, the spatiospectrum of the STOV is identical and propagates identically to the ST THL pulse, and as such is plotted at the output plane of the various optical systems in the same Figure 2 as the ST THL pulse and indicated by SS THL pulse. Vice versa, the spatiospectrum of the ST THL pulse behaves identically as a STOV, and is plotted at the exit of the optical systems in the same Figure 1 as the STOV with the name SSOV.

Let us illustrate this double duality with focusing the output of a STOV-THL-forming 4f pulse shaper, which is also the standard setup in experiments of high harmonic and attosecond pulse generation, including here the propagation up to the far field as the observation region. Equations (7) and (14) are then modified to their limiting far-field forms as functions of the observation angle θ = x/z:

for the focused STOV far field, and

for the focused ST THL pulse. The decay and curvature factors q / z e i k 0 x 2 / 2 z are omitted, and the divergence angle is given by

which can be approximated by θ ₁ ≃ x ₀/f for strong focusing (meaning negligible focal shift but still paraxial). Equations (7), (14), (22) and (23) provide for the first time the complete picture of propagating STOV-THL pulses in standard experiments as would be observed with field characterization techniques with spatiotemporal resolution. In Figure 3 we have deliberately chosen a soft focusing geometry to enhance the intricate far field structure of STOVs (first row), where the vortices never merge in a perfectly elliptical STOV, as it would contradict the conservation of the transverse OAM in free space propagation. The transverse OAM of the focused elliptical STOV is ±nγ/2ω ₀, whereas an elliptical STOV at the far field would have ∓nγ _far/2ω ₀ → 0 as γ _far = ct ₀/x _far → 0 given the linearly increasing width x _far at the far field. Similarly the far field structure of the focused ST THL pulse (third row) is never another ST THL pulse with n π-steps in the phase, but the vortices remain up to infinite propagation distances even if strongly deformed.

Figure 3:

Focusing and propagation up to the far field of STOVs and ST THL pulses of TC +3, observed spatiotemporally (odd rows) and spatiospectrally (even rows). In this example ω ₀ = 2.8 rad/fs, x ₀ = 0.3 mm, t ₀ = 100 fs, and f = 75 mm.

The second and four rows of Figure 3 illustrate the SS picture as detected by a spectrometer with spatial resolution. The complete picture is obtained in analytical terms by replacing t′/t ₀ with Ω/Ω₀ in Eq. (7) for STOV focusing and in Eq. (14) for ST THL focusing, and interchanging the equations. The same for Eqs. (22) and (23). The result is

for the SS THL pulse of the focused STOV, and

for the SSOV of the focused ST THL pulse, with respective far fields

for the SS THL pulse, and

for the SSOV. As evidenced in Figure 3, focusing of the STOV viewed spatiospectrally is identical to focusing the ST THL pulse in space-time, and focusing of the ST THL viewed spatiospectrally is identical to focusing of the STOV in space-time.

4 Conclusions

Although spatiotemporal optical vortices are not modes of propagation, the spatiotemporal/spatiospectral duality in which vortices appear in the spatial spectrum in the form of spatiospectral vortices when they disappear in space-time, confers them with a sophisticated unity that had not previously been noticed. Once the problem of the transverse orbital angular momentum has been clarified [23], the present theory provides a full understanding from closed-form analytical expressions of the transformation properties of STOVs of arbitrary topological charge through rather general optical systems as ABCD systems. The formulation in spatiospectral domain as spatiospectral tilted Hermite-lobed pulses and spatiospectral optical vortices is particularly relevant in experiments where STOVs are created in the extreme ultraviolet and soft X-ray wave length regimes, where the standard characterization techniques are substantially based on the observation of spatiospectra. Overall, the precise knowledge of the spatiotemporal and spatiospectral structure of STOVs provided here will lead to decisive advances in the design and precision of their applications. Further theoretical and experimental efforts are necessary to describe and generate broadband, few-cycle STOVs propagating non-paraxially, as the electromagnetic STOVs reported in Ref. [29].