1. Introduction

Freely propagating optical fields diffract: their spatial extent increases and their fine features blur with propagation [1]. Combating diffraction in free space has typically relied on exploiting beams having specific transverse profiles that are propagation-invariant solutions of the wave equation. Examples of monochromatic diffraction-free beams include those whose profiles conform to Bessel [2], Mathieu [3], Weber [4], and Airy [5] functions (see [6] for a review). In the case of pulsed beams (wave packets or optical bullets), different functional forms of the field in space and time have been found to be propagation-invariant (both diffraction-free and dispersion-free). The earliest examples are the Brittingham focus-wave mode [7] and MacKinnon’s wave packets [8], and subsequently X-waves [9–11], Bessel pulses [12], among other examples [13], in addition to more recent proposals [14, 15]. Despite the wide diversity of specific propagation-invariant wave packets identified theoretically [16,17], underlying them all is a single fundamental characteristic: in the spatio-temporal spectral domain, the spatial frequencies (that establish the transverse beam profile in space) are tightly correlated with the temporal frequencies (that determine the pulse profile in time) [14–18]. We thus call all such optical fields ‘space-time’ (ST) wave packets. The spatio-temporal spectra of all such free-space wave packets lie at the intersection of the light-cone with spectral hyperplanes, and the various families of previously identified ST wave packets can be generated by tilting the hyperplane with respect to the light-cone [18, 19]. In other words, instead of the spatio-temporal spectrum of the wave packet occupying a patch on the light-cone as is typical for pulsed beams (where the spatial and temporal degrees of freedom are almost independent), the spectra of ST wave packets are confined to reduced-dimensionality trajectories.

Synthesizing such ST wave packets presents considerable difficulties, especially in ensuring the realization of a tight correlation between spatial and temporal frequencies. We have recently provided an experimental solution to this challenge that combines elements from beam-shaping [20,21] and ultrafast pulse-shaping [22,23]: we use a two-dimensional (2D) spatial light modulator (SLM) to modulate the phase in the orthogonal direction to that of the wavelength spread (similarly to so-called 4− f pulse shaping [24–26]) to imprint the spatial frequency that is to be associated with each particular wavelength. Using this approach we produced diffraction-free ST wave packets of transverse width of 14 μm that propagate for 100 mm, for example, in addition to hollow pulsed beams with a diffraction-free axial null [18]. More recently, we have produced non-accelerating Airy ST wave packets that accelerate instead in space-time; that is, this Airy wave packet travels in a straight line without the usual transverse displacement, and instead accelerates in the local frame of the propagating pulse [19]. Moreover, several very recent theoretical studies have further enriched this newly developing field of ST wave packets [27–31].

As described above, our previous experimental realizations of ST wave packets have made use of a SLM to impress a 2D phase distribution on the impinging spectrally resolved beam in order to introduce the desired spatio-temporal spectral correlations. SLMs have an obvious advantage with respect to their capability for dynamically varying the implemented phase distribution when compared to a static phase plate. Nevertheless, relying on SLMs presents some restrictions that stem from their fundamental limitations. First, because most SLMs make use of liquid-crystal-based technology, their low energy-handling capability precludes them from being utilized to synthesize high-energy ST wave packets. Second, the pixel size sets a lower limit on the width of the uncertainty of the spectral correlations (which dictates the diffraction-free propagation distance), while the finite SLM aperture size sets an upper limit on the utilizable spectral bandwidth as well as the quasi-diffraction-free propagation distance.

Here we synthesize ST wave packets using transmissive phase plates that are lithographically inscribed in a transparent polymer film via a gray-scale lithography process. We thus overcome the major limitations of SLMs and implement phase distributions at a higher spatial resolution (∼ 3 μm) and cover a larger area (∼ 25 × 25 mm²). Consequently, we are now in a position to synthesize ST wave packets with larger spectral bandwidth than previously possible. In Refs. [18, 19] we produced ST wave packets with a bandwidth of < 1 nm. Here, we report on the synthesis of diffraction-free ST wave packets with bandwidths up to ~30 nm. In the experiments, we make use of a typical femtosecond Ti:sapphire laser for a smaller-bandwidth experiment and a Ti:sapphire-based multi-terawatt femtosecond laser for the higher bandwidth cases. Furthermore, the use of a transparent transmissive phase plate opens the path towards exploiting high-energy pulses to synthesize ST wave packets (indeed, phase plates have been used previously with high-energy pulses to produce nondiffracting double helical beams and helical filaments [32, 33]). In all cases, we synthesize the ST wave packets – for simplicity – in the form of light sheets with an approximate Gaussian beam profile at the pulse center, where the diffraction-free behavior is observed along one dimension, while the field is uniform along the second transverse dimension [18, 19].

The paper is organized as follows. We first present a brief overview of the concept of ST wave packets and delineate the role of ST correlations in producing propagation-invariant pulsed beams. Next we describe the design and fabrication of the phase plates before describing the optical setup used for wave-packet synthesis. We then present our measurement results confirming the diffraction-free behavior of the produced ST wave packets before outlining our conclusions.

2. Propagation-invariant ST light sheets

We consider here for simplicity one-dimensional (1D) optical beams or light sheets with electric field E (x, z, t), where x is the transverse coordinate, z is the axial coordinate, and t is time (the field is uniform along the other transverse coordinate y). The initial field at z = 0, E (x, 0, t), has a spatio-temporal spectrum Ẽ(k_x, ω), where k_x is the spatial frequency or the transverse component of the wave vector and ω is the temporal frequency, which are subject to the dispersion relationship $k_x^2+k_z^2={\left(\omega /c\right)}^2$ that represents the surface of a cone known as the light-cone [Fig. 1(a)]; here k_z is the axial component of the wave vector and c is the speed of light in vacuum. Upon free-propagation [1], the field evolves axially according to

 

Fig. 1 Concept of space-time (ST) pulsed beams or wave-packets. (a) The surface of the light-cone $k_x^2+k_z^2={\left(\omega /c\right)}^2$ in the (k_x, k_z, ω) domain is the locus of all propagating plane waves. The intersection of the light-cone with a spectral plane orthogonal to the (k_z, ω)-plane is the spectral locus of a family of diffraction-free and dispersion-free ST wave packets for each value of the tilt angle φ the plane makes with respect to the k_z-axis. (b) The ST pulsed beam is synthesized such that each spatial frequency k_x is associated with one temporal frequency ω. In the specific case depicted here, higher temporal frequencies (smaller wavelengths) are associated with higher magnitude spatial frequencies (depicted here as a colored sinusoid), while k_x = 0 (depicted as a straight line) is associated with the lowest temporal frequency (largest wavelength). (c) Calculated ST intensity profile |E (x, 0, τ)|², where τ corresponds to time in the traveling frame of the pulse. In principle, the profile is invariant along the axial coordinate z. We set φ=90° and make use of a Gaussian spectral profile in the calculation.

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The basic concept underlying the propagation-invariance of a ST wave packet is the tight correlation between the spatial and temporal frequencies k_x and ω, respectively. An optical beam has a finite spatial bandwidth Δk_x, whereas an optical pulse has a finite temporal bandwidth Δω. A pulsed optical beam (or wave packet) combines finite spatial and temporal bandwidths, and thus the associated spatio-temporal spectrum occupies in general a 2D patch Ẽ(k_x, ω) on the side of the light-cone.

If correlations are introduced between the spatial and temporal degrees of freedom such that each spatial frequency k_x is associated with a temporal frequency ω [Fig. 1(b)] and thus Ẽ(k_x, ω) → Ẽ(k_x)δ(ω − f(k_x)), the spatio-temporal spectrum is then confined to a 1D trajectory instead of a 2D patch; here (ω = f(k_x) is the functional form of the correlation between ω and k_x. This relationship is taken here to be one-to-one between |k_x| and ω. If this spatio-temporal spectral trajectory defined by ω = f(k_x) is the result of the intersection of the light-cone with a plane that is parallel to the k_x-axis and orthogonal to the (k_z, ω)-plane as shown in Fig. 1(a), then any wave packet whose spatio-temporal spectrum lies along this trajectory can be shown to be diffraction-free and dispersion-free [18]. This can be understood by noting that the projection of this trajectory onto the (k_z, ω)-plane takes the form of a straight line ω/c = k_o + (k_z − k_o) v_g / c where k_o is a constant wave number and v_g is the group velocity of the wave packet, which is related to the tilt angle φ of the spectral plane with the horizontal (k_x − k_z)-plane, v_g =c tan φ. The resulting wave packet thus takes the form

3. Synthesis of space-time wave packets

3.1. Optical setup

We depict schematically in Fig. 2 the optical setup to synthesize the ST wave packets using a transmissive phase plate that modulates the wave front such as to introduce the desired spatio-temporal correlations. The system can be divided into two major sections that carry out the tasks of field synthesis and analysis, with the latter performed in physical space (x, z, t) and in the spatio-temporal spectral domain (k_x, k_z, ω). As described in the previous section, the synthesis of a ST wave packet requires associating each wavelength λ with a particular spatial frequency k_x. To achieve this, we start from a pulsed plane wave whose spatio-temporal spectrum is thus separable, having approximately the form Ẽ(k_x, λ) ∼ δ(k_x Ẽ(λ). The pulse spectrum Ẽ(λ) having a bandwidth of Δλ is spread spatially along one dimension via a diffraction grating G₁ and directed to the transmissive phase plate (that replaces the SLM in [18, 19]) by a cylindrical lens L₁ in a 2f configuration. The phase plate then imparts a 2D phase distribution to the spectrally resolved field before we reconstitute the pulse via a second cylindrical lens L₂ (identical to L₁) and a second diffraction grating G₂ (identical to G₁). At this stage, the spatio-temporal spectrum has the form Ẽ(k_x, λ) ∼ Ẽ(k_x)δ(λ − f (k_x)). Note that the finite sizes of the apertures and the minimum feature size in the phase plate prevent the realization of an idealized delta-function correlation. Instead, the delta function is replaced by a narrow function that introduces a finite spectral uncertainty δλ, which determines the maximum diffraction-free propagation length [14].

 

Fig. 2 Schematic of the optical setup for synthesizing ST wave packets (pulsed beams) using transmissive phase plates. The same overall setup is utilized in synthesizing two ST wave packets having bandwidths of Δλ =0.25 nm and 30 nm. Top inset: Definitions of the symbols used to identify the optical components in the setup. Middle inset: Values of the distances and focal lengths utilized to synthesize ST wave packets with Δλ =0.25 nm; bottom inset: corresponding values for Δλ =30 nm.

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The resulting ST wave packet is split into two paths by a beam splitter for analysis. First, the diffraction-free behavior is confirmed in physical space by scanning a CCD camera (CCD₁) along the propagation axis after a 4 f telescope. The measured intensity is thus the time-averaged value I(x, z) = ∫dt |E (x, z, t)|². Second, the spatio-temporal spectrum is measured by sampling the field before G₂ (i.e., while the field is still spectrally resolved) and implementing a spatial Fourier transform via a lens in a 2 f configuration to resolve the spatial spectrum along k_x. The spatio-temporal spectrum |Ẽ(k_x, λ)|² is then recorded by a camera (CCD₂).

Note that this system is similar conceptually to that implemented in [18, 19] where the arrangement was folded back on itself by employing a reflective SLM, such that the gratings G₁ and G₂ coincide and the lenses L₁ and L₂ coincide.

3.2. Phase plate design and fabrication

The phase plates were fabricated using gray-scale lithography [37]. First, a positive-tone photoresist (Shipley 1813) [38] was spin-coated on an RCA-cleaned 2-inch-diameter D263 glass wafer at 1000 rpm for 60 seconds, which gives a film thickness of ≈ 2.6 μm. The sample was then baked in an oven at 110 °C for 30 minutes. A calibration was conducted to determine the depth of the photoresist after development for a given exposure dose as described previously [39, 40]. A laser pattern generator (Heidelberg Instruments) [41] was used to write the design. The exposed sample was then developed in AZ 1:1 solution [42] for 90 seconds followed by DI water rinse.

We produced two phase plates with this process. The first phase plate was designed to accommodate a bandwidth of 2 nm, that of a typical femtosecond Ti:Sa laser, with dimensions 15×12 mm². By spatially filtering the spectrum incident on the phase plate, different bandwidths can be utilized. In our experiment, we restrict its use to a bandwidth of ~0.25 nm. The second plate had dimensions of 25 × 25 mm² and was designed to accommodate a larger bandwidth of ∼40 nm. The multi-terawatt femtosecond laser used here has a bandwidth of ∼30 nm. Optical micrographs of the fabricated devices are given in Fig. 4(b) and 6(b).

 

Fig. 3 (a) Measured FROG trace obtained via a GRENOUILLE 8-50 (Swamp Optics) of the pulses from the Ti:Sa laser (Tsunami, Spectra Physics). Although the laser bandwidth is ∼ 8 nm, we spectrally filter a bandwidth Δλ ∼ 0.25 nm from it for our ST wave packet synthesis experiment. (b) The spectral intensity and phase of the pulses obtained from the FROG trace. (c) The corresponding pulse intensity and phase in the time domain obtained from the FROG trace.

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Fig. 4 (a) Measured spatio-temporal spectrum |Ẽ(k_x, λ)|² for synthesized ST wave packets with bandwidth Δλ ~ 0:25 nm. (b) Optical micrograph of the phase plate used. Inset shows magnified (4×) optical micrograph of the section enclosed in the rectangle). (c) Measured time-averaged intensity profile I(x, z) = ∫dt |E(x, z, t)|² obtained with CCD1 in Fig. 2 that is scanned axially along z.

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Fig. 5 (a) Measured FROG trace obtained via a GRENOUILLE 8-50 (Swamp Optics) of the pulses from a Ti:Sa-based multi-terawatt femstosecond laser; Δλ ∼30 nm. (b) The spectral intensity and phase of the pulses obtained from the FROG trace. (c) The corresponding pulse intensity and phase in the time domain obtained from the FROG trace.

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Fig. 6 Measured spatio-temporal spectrum |Ẽ(k_x, λ)|² for synthesized ST wave packets with bandwidth Δλ ∼30 nm. (b) Optical micrograph of the phase plate used. (c) Measured time-averaged intensity profile I(x, z) = ∫dt |E(x, z, t)|² obtained with CCD₁ in Fig. 2 that is scanned axially along.

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4. Space-time wave-packet synthesis and characterization

4.1. Space-time wave packet with 0.25-nm bandwidth

The experimental arrangement for synthesizing and characterizing ST wave packets, which applies for bandwidths of Δλ ∼0.25 nm or 30 nm, is shown in Fig. 2; however, different values for the component parameters are selected for the two bandwidths targeted. The laser used with the first phase plate for a bandwidth of Δλ ∼0.25 nm is filtered from a generic pulsed Ti:Sa laser with a pulse width of ∼100 fs (FWHM) at a wavelength of ~800 nm (Tsunami, Spectra Physics). We first characterize the initial laser pulse in the spectral and temporal domains using the FROG technique (employing a GRENOUILLE 8-50, Swamp Optics); see Fig. 3 for measurement results. Crucially, minimal spectral phase and temporal chirping are observed over the spectral bandwidth exploited in our experiment prior to the phase modulation applied by the phase plate. The beam is directed to a reflective diffraction grating (G₁) with 1200 lines/mm and dimensions 25 × 25 mm² (Newport 10HG1200-800-1) at an incidence angle of 68° to spread the pulse spectrum in space. We select the second diffraction-order at 81° that provides a spatial dispersion of 0.064 nm/mrad. The diffracted beam is normally incident on the transmissive phase plate after a cylindrical lens L_1−y of focal length f = 50 cm in a 2 f configuration (we denote the direction of the spatially dispersed spectrum y).

The phase plate is designed to synthesize ST wave packets whose spatio-temporal spectral loci lie along the intersection of the light-cone with a plane that makes an angle of φ = 90° with the k_z-axis; that is, the izo-k_z plane [Fig. 1(a)]. The phase pattern thus implements the dispersion relationship $k_{\mathrm{o}}^2={\left(2\pi /\lambda \right)}^2-k_{\mathrm{x}}^2$, which produces the required correlation between k_x and λ. Here k_o is a fixed axial wave number that corresponds to the maximum wavelength used and is associated with the spatial frequency k_x =0. For experimental convenience, we implement the spatial frequencies k_x on the phase plate with a reduction in scale by a factor 4× and then subsequently compensate for that reduction at the output via a 4 f telescope system that introduces a demagnification by a factor of 4× along the x direction [18].

After traversing the phase plate, the field is directed through a cylindrical lens L_2−y (that is identical to L_1−y) in a 2 f configuration to a beam splitter BS₁ to split the beam in two paths for characterization. In one path, we confirm that the desired spatio-temporal correlation between k_x and λ has been realized by measuring the spatio-temporal spectrum |Ẽ(k_x, λ)|². In this path, we arrange for a spherical lens L_5−s with a focal length of f =10 cm in a 2f configuration to collimate the spectrum along y and carry out a Fourier transform spatially along x. A CCD₂ camera (The ImagingSource, DMK 33UX178) placed in the focal plane of L_5−s thus directly images the spatio-temporal spectrum |Ẽ(k_x, λ)|², which is presented in Fig. 4(a). The data confirms that we obtain the targeted hyperbolic correlation function between k_x and λ that is imposed by the phase distribution [Fig. 4(b)] provided by the phase plate.

In the other path after BS₁, we characterize the ST wave packet in physical space to confirm its diffraction-free behavior by reconstituting the pulse via a diffraction grating G₂ (that is identical to G₁). After G₂, an imaging 4 f system translates the beam and implements the above-described demagnification factor 4× (the values of all the focal lengths of the lenses are provided in the inset of Fig. 2). Starting from the output plane of this imaging system, we capture the transverse intensity distribution with a camera CCD₁ (The ImagingSource, DMK 33UX178) while scanning its axial position along the z direction. The measurement corresponds to the time-averaged intensity I(x, z )= ∫dt |E (x, z, t)|² (the intensity distribution is uniform along y). The measured intensity distribution is plotted in Fig. 4(c) for a scanned axial distance scanned of 25 mm, confirming the expected diffraction-free behavior. The detailed structure of such a ST wave packet in space and time is presented in [18].

4.2. Space-time wave packet with 30-nm bandwidth

The experimental setup for synthesizing and characterizing the ST wave packets for the case of the larger bandwidth Δλ ∼ 30 nm is also that shown in Fig. 2, except for the change in the values of the focal lengths of the lenses used (provided in the inset of Fig. 2). The laser source of the initial pulses here is the Multi-Terawatt Femtosecond Laser (MTFL) [43, 44] housed at the University of Central Florida. This Ti:Sa-based chirped pulse amplification system consists of 3 stages of amplification and is capable of delivering 500-mJ-energy pulses at a wavelength of 800 nm having a pulse width of ∼50 fs (FWHM) at a repetition rate of 10 Hz. The beam used in this experiment was extracted directly from the regenerative amplifier (first stage of amplification) at an energy of only 1 mJ, which was further attenuated to avoid saturating the CCD camera. We have characterized these pulses in the temporal and spectral domains using the FROG technique as described in the previous section, and the measurements are presented in Fig. 5. We note in Fig. 5(b) the existence of a spectral phase extending across the bandwidth we utilize. Once the spatio-temporal spectral correlations are introduced between k_x and λ by the phase plate, this spectral phase will be shared between the spatial and temporal degrees of freedom. We thus expect a slight tilt in the propagating wave front.

Because of the larger spectral bandwidth involved here, we modify the experimental parameters for spreading the spectrum in space and also use a phase plate of larger dimensions. The input pulsed beam is directed at an incidence angle of 51° to the grating G₁ that has in this case 1800 lines/mm and dimensions 25×25 mm² (Thorlabs GR25-1850), and select the first diffraction-order at an angle of 41°, which is directed to a phase plate having dimensions 25 × 25 mm². The field at the phase plate extends over a width of ≈24 mm. The phase plate is designed to produce a ST wave packet whose spatio-temporal spectrum that lies along the hyperbolic trajectory resulting from the intersection of the light cone with a spectral plane tilted an angle φ=45.3° with respect to the k_z-axis [Fig. 1(a)].

The synthesized wave packet is characterized in physical space I(x, z) and in the spatio-temporal domain |Ẽ(k_x, λ)|² using the same previously described methodology. Measurements of the spatio-temporal spectrum |Ẽ(k_x, λ)|² are shown in Fig. 6(a). The data confirms that the targeted hyperbolic correlation function between k_x and λ has been realized over the full bandwidth after making use of the phase distribution depicted in Fig. 6(b). The measured time-averaged intensity I(x, z) while scanning the camera CCD₁ axially is plotted in Fig. 6(c). The diffraction-free behavior is observed over an axial distance of ∼45 mm. Note that a slight tilt in the wave front is seen in the data as expected from the spectral phase of the initial femtosecond pulses [Fig. 5(b)].

5. Conclusion

In conclusion, we have synthesized broadband one-dimensional diffraction-free ST wave packets (or light sheets) using phase plates produced by a gray-scale lithography process. The phase plates are larger in overall size (25×25 mm²) than a typical SLM, have smaller minimal feature size (∼ 3 μm), and have higher diffraction efficiency. These characteristics have allowed us to synthesize ST wave packets of two types. First, we exploited the fine spatial features of the phase plate to produce a ST wave packet with narrow bandwidth Δλ ~ 0.25. Second, we made use of the additional advantage of the large area of the phase plate to produce a ST wave packet of considerably large bandwidth Δλ 30 nm compared to previous demonstrations that exploited SLMs (Δλ ∼ 1 nm in Refs. [18, 19]). In both cases, we maintain a low spectral uncertainty δλ necessary for the diffraction-free propagation. The possibility of exploiting transparent transmissive phase plates paves the way to implementing high-energy ST wave packets that can be useful in nonlinear optical applications, including laser filamentation studies and laser machining or additive manufacturing. Future work will be directed at increasing the diffraction-free propagation distance, realizing ST wave packets with control over both transverse dimensions (ST needles in lieu of light sheets), and synthesizing high-energy ST wave packets. Finally, we note that the correlations introduced between the spatial and temporal degrees of freedom here are the continuous analog of the correlations between discretized optical degrees of freedom known as ‘classical entanglement’ [45, 46] that has shown significant recent utility in studying optical coherence theory.