1. Introduction

When a beam of light propagates freely, diffraction leads to transverse spreading and blurring of spatial details [1,2]. Circumventing this constraint has prompted the development of ‘diffraction-free’ beams, which are monochromatic solutions of the Helmholtz or paraxial wave equations whose transverse spatial profiles evolve self-similarly [3–5]. In one dimension (1D), it can be demonstrated that only (accelerating) Airy Beams [6–8] and cosine waves [5] propagate self-similarly. More options are available in two dimensions (2D), including Bessel [9, 10], Weber [11–13], and Mathieu beams [14], among a hierarchy of possibilities [5,15,16]. Optical nonlinearities can be exploited instead to preserve the beam profile [17], but this typically requires the use of pulsed beams. However, maintaining the spatial profile and pulse shape – so-called ‘light bullets’ – is challenging because such waves require either a specific material dispersion [18–22] or nonlinearity, whereupon they become unstable [23,24]. It is useful to search for free-space diffraction-free solutions in light of their wide range of potential applications. Along these lines, if the spectral and spatial spatial degrees of freedom (DoFs) of the optical field are separable, each can make use of standard solutions; e.g., an Airy pulse waveform (the only known 1D solution) and a Bessel-beam spatial profile [25]. In other configurations, the pulsed beam disperses temporally even when the spatial profile is maintained [26,27].

Here we show that an optical beam having an arbitrary transverse spatial profile in 1D or 2D can in principle become independent of the location along the propagation axis – given the availability of a sufficient spectral bandwidth. The underlying principle is the introduction of correlations between the beam’s spatial and spectral DoFs by assigning each transverse wave vector component to a single wavelength according to a prescribed rule. Spatio-temporal coupling – which is typically undesirable – occurs naturally during the propagation of ultrafast pulsed beams [28–30] or the generation of ultra-high-power laser pulses [31]. Here we exploit intentionally introduced space-time (ST) correlations to eliminate axial variations in a pulsed beam during propagation by compensating for the spatial dispersion that underpins diffraction with spatio-temporal dispersion. We examine several examples, including diffraction-free Gaussian beams, ‘bottle’ beams with a non-diffracting null, and non-accelerating Airy beams. Such ST beams may have applications in laser manufacturing [32], 3D optical lattices, and particle trapping and manipulation [33,34].

2. Diffraction of pulsed beams

A 1D pulsed optical beam evolves according to

 

Fig. 1 Representations of optical beams in (k_x, k_z) space. Each point uniquely identifies a monochromatic plane wave (k_x, k_z; ω), $\omega =c\sqrt{k_x^2+k_z^2}$. A circle in this plane represents all possible iso-frequency plane waves. (a) A monochromatic beam of spatial bandwidth Δk_x is represented by the red arc. This representation captures the plane waves needed in constructing such a beam but not its actual shape, which depends on the amplitudes F(k_x). (b) A pulsed plane wave of spectral bandwidth Δω represented by the vertical red line. (c) A pulsed beam of spatial and spectral bandwidths Δk_x and Δω, respectively, is represented by the red patch. The beam diffracts and the pulse waveform disperses. (d) A diffraction-free space-time beam represented by a horizontal red line corresponding to a constant $k_z=\mathcal{K}$. Both spatial and temporal profiles are z-independent.

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In general, the ST spectrum F(k_x, ω) for a pulsed beam is taken to separate into a product of spatial F_x (k_x) and temporal F_t (ω) contributions [the shaded region in Fig. 1(c)], so that the initial beam in turn separates in space and time f (x, 0; t) = f_x (x) f_t (t). Nevertheless, the beam subsequently loses this separability – a phenomenon known as ‘space-time’ coupling in the propagation and focusing of ultra-short pulses [28–30, 35–37] – as a consequence of the non-separability of k_x and ω in the integral in Eq. (1). We still have $k_x^2+k_z^2={\left(\omega /c\right)}^2$, but the complementarity between k_x and k_z is no longer strict because ω varies over Δω and each k_x can be associated with a continuum of frequencies ω and hence axial wave numbers k_z. Spatial dispersion is thus compounded with ST dispersion: the beam diffracts and the pulse disperses.

3. Diffraction-free space-time beams

We now show that an alternative exists to thwart diffraction. Any monochromatic plane wave (k_x, ω) may be excited when k_x < ω/c and is associated with a particular k_z. The key insight is that there exists a trajectory in (k_x, k_z) space that guarantees all the constituent plane waves have the same k_z: the k_z-iso-surface $k_z(k_x,\omega )=\mathcal{K}$ that fixes a hyperbolic relationship between k_x and ω,

Crucially, in this conception we need not restrict ourselves to any specific 1D beam profile to achieve diffraction-free propagation. The only requirement is to use a sufficient bandwidth Δω of the auxiliary spectral DoF to ‘protect’ the beam’s spatial DoF [Fig. 1(d)],

The generalization to 2D optical beams is straightforward. We assign to each transverse spatial frequency ${\mathbf{k}}_{\mathrm{T}}=(k_x,k_y)$ in the 2D spatial spectrum F_x (k_x, k_y) the frequency ω such that $k_z=\sqrt{{\left|{\mathbf{k}}_{\mathrm{T}}\right|}^2-{\left(\omega /c\right)}^2}=\mathcal{K}$, which guarantees diffraction-free propagation. Therefore, all 2D transverse spatial frequencies such that $k_x^2+k_y^2=\text{constant}$ are encoded on the same frequency ω.

4. Examples of diffraction-free ST beams

To illustrate the concept of a diffraction-free ST beam, we present in Fig. 2 an example of a Gaussian beam in two configurations. The traditional case of a Gaussian beam of FWHM x_o = 10 µm and spectral bandwidth of Δλ = 0.01 nm (corresponding to a FWHM pulse-width of 94 ps) is shown in Fig. 2(a)–2(d). The ST spectrum F(k_x; λ) = F_x (k_x)F_t (λ) is separable [Fig. 2(a)] and so is the initial ST intensity |f(x, 0; t)|² [Fig. 2(b)]. The beam profile diffracts with a Rayleigh range of $z_{\mathrm{R}}=\pi w_o^2/\lambda \approx 283$ µm (w_o is the Gaussian beam radius [2]) [Fig. 2(c)], whether at the pulse center |f(x, z; 0)|² or in the time-averaged profile ∫dt| f (x, z; t)|². The contribution of the beam center to the pulse |f (0, z; t)|² consequently drops [Fig. 2(d)], while the spatially averaged pulse ∫dx| f (x, z; t)|² is axi-symmetric [Fig. 2(d), inset]. Increasing the bandwidth here reduces the pulse-width and increases the ST coupling effects [2].

 

Fig. 2 (a) ST spectrum F(k_x, λ) for a Gaussian beam of initial FWHM x_o = 10 µm and Δλ = 0.01 nm. (b) ST intensity | f (x, 0; t)|² at z = 0. Inset shows |f (x, z; t)|² at z/z_R = 20. (c) Beam profile | f (x, z; 0)|² at z/z_R = 0, 10, 20. The beam profiles correspond to the peaks of the pulses at each z. Inset shows the time-averaged profile ∫dt| f (x, z; t)|² at the same z. (d) Pulse shape | f (0, z; t)|² at the beam center for z/z_R = 0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|², which is independent of z. All color maps are normalized independently. (e)–(h) Correspond to (a)–(d) except that ST correlations are introduced between k_x and λ; δλ =0.01 nm and Δλ =1 nm. (e) The ST spectrum F(k_x, λ). (f) The ST intensity | f (x, 0; t)|² is localized in x and t. The inset is removed since there is no axial variation. (g) Beam profile | f (x, z; 0)|² at z/z_R =0, 10, 20. The beam profile is z-invariant. (h) Pulse shape | f (0, z; t)|² at the beam center for z/z_R =0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|², which is independent of z.

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By introducing hyperbolic ST spectral correlations according to Eq. (2), we obtain the non-separable spectrum in Fig. 2(e) from Fig. 2(a). Each k_x is correlated to a specific λ with an uncertainty of δλ, whereas the bandwidth Δλ is determined by Δk_x (Eq. (4)). The beam size x_o =10 µm requires an associated bandwidth of ≈1 nm. The ST intensity is no longer separable and is instead localized around x = 0 and t = 0 for all z. The transverse sections of the beam |f (x, z; 0)|² and ∫dt| f (x, z; t)|² are invariant upon propagating a distance 20z_R [Fig. 2(g)], whereupon the Gaussian beam of identical spatial bandwidth has increased 20-fold in width [Fig. 2(c)]. Note that the pulsewidth at x = 0 is determined by Δλ [Fig. 2(h)] whereas the spatially averaged pulsewidth is set by δλ [Fig. 2(h), inset]. Note also that a delay in the pulse at x =0 emerges as a result of the reorganization of energy in the time-frame of the pulse after propagating for a long distance [Fig. 2(h)], whereupon different parts of the transverse wave front shift in time with respect to the center of the time-frame (we only plot the pulse at x =0). Once the beam is integrated over the transverse plane, these local time delays vanish [Fig. 2(h), inset]. Here, we also note that the smaller transverse width of the ST beam desired, the bigger bandwidth Δλ is required, resulting in a narrower pulsewidth at x =0.

A unique advantage of our approach is that diffraction-free propagation is ensured by satisfying the constraint in Eq. (2) without reference to the actual ST spectrum. Consequently, by modulating F(k_x, λ) we produce diffraction-free pulsed beams of arbitrary profiles. We present two examples in Fig. 3. In the first [Fig. 3(a)–3(d)], we modulate the spectrum in Fig. 2(a) with a phase term of the form $e^{i\pi u(k_x)}$, where u(·) is the unit-step function, so the spectrum is odd along k_x. The corresponding axially self-similar ST intensity [Fig. 3(b)] now has a null at x =0: | f (0, z; t)|² =0 and ∫dt| f (0, z; t)|² = 0. The result is a ‘bottle-beam’ with a diffraction-free null along z. A second example is provided in Fig. 3(e)–3(h) where the spectrum is modulated by the phase $e^{-i\pi {\left(k_x{x}_{\mathrm{o}}\right)}^3}$, which produces a diffraction-free Airy profile [Fig. 3(g)]. Note, crucially, that the Airy profile here is not accelerating transversely and is thus a truly diffraction-free beam, in contrast to accelerating monochromatic Airy beams.

 

Fig. 3 Diffraction-free beams of arbitrary spatial profile with ST correlations introduced between k_x and λ; the correlation uncertainty is δλ = 0.01 nm and the bandwidth is Δλ =1 nm. (a) ST spectrum F(k_x, λ) for a ’bottle-beam,’ where the phase along the spatio-temporal spectrum is phase-modulated to create an odd-parity beam with a minimum at its center. (b) ST intensity | f (x, 0; t)|² at z = 0. The inset is removed since there is no axial variation. (c) Beam profile | f (x, z; 0)|² at z/z_R =0, 10, 20. The beam profiles correspond to the peaks of the pulses at each z. Inset shows the time-averaged profile ∫dt| f (x, z; t)|² at the same z. (d) Pulse shape | f (0, z; t)|² at the beam center for z/z_R =0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|², which is independent of z. All color maps are normalized independently. (e)–(h) Correspond to (a)–(d) except that the phase modulation is $e^{-i\pi {\left(k_x{x}_{\mathrm{o}}\right)}^3}$, which is designed to produce a non-accelerating Airy beam. (e) Real part of the ST spectrum F(k_x, λ), while the inset shows the imaginary part. (f) ST intensity | f (x, 0; t)|² at z = 0. (g) Beam profile | f (x, z; 0)² at z/z_R =0, 10, 20. Inset shows the time-averaged profile ∫dt| f (x, z; t)|² at the same z. (h) Pulse shape | f (0, z; t)|² at the beam center for z/z_R =0, 10, 20. Inset is the spatially averaged pulse shape ∫dx |f (x, z; t)|², which is independent of z.

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5. Impact of uncertainty on diffraction-free length

The diffraction-free length is determined by the uncertainty in the relationship ω(k_x) = ω_x, which is subject to practical restrictions on ST beam generation. In Fig. 4, we further relax the ST correlations and increase δλ to 0.05 nm and find that the initial ST intensity |f (x, 0; t)|² now diffracts [Fig. 4(c)] over the same axial range considered in Fig. 2 and Fig. 3, where the reduced correlation uncertainty (δλ =0.01 nm) arrests the beam spread.

 

Fig. 4 The impact of the ST correlation uncertainty δλ on the diffraction-free propagation length. (a) ST spectrum F(k_x, λ) for a Gaussian beam with ST correlations introduced between k_x and λ; δλ = 0.05 nm and Δλ = 1 nm. (b) ST intensity |f (x, 0; t)|² at z = 0. Insets show | f (x, z; t)|² at z/z_R =10, 20. (c) Beam profile |f (x, z; 0)|² at z/z = 0, 10, 20. The beam profiles correspond to the peaks of the pulses at each z. Inset shows the time-averaged profile ∫dt| f (x, z; t)|² at the same z. (d) Pulse shape | f (0, z; t)|² at the beam center for z/z_R = 0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|², which is independent of z. The pulse at the beam center |f (0, z; t)|² undergoes z-dependent delay and deformation. All color maps are normalized independently.

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We have examined thus far ST beams propagating across distances over which traditional beams of comparable spatial bandwidth diffract significantly (~20z_R). Because of the finite correlation uncertainty δλ, it is critical to examine the propagation of the beam at larger z. The width of a Gaussian beam increases linearly when z ≫ z_R at a rate inversely proportional to the initial width [2]. Considering a ST beam having the same spatial bandwidth and a correlation uncertainty of δλ = 0.05 nm, we find the diffraction dynamics delineated into two regimes: a slow diffraction regime for z < 3 mm followed by a fast diffraction regime in which the width approaches that of the traditional Gaussian beam [47, 48]. Here, we define critical distance z_c that occurs at the knee separating ‘slow’ and ‘fast’ diffraction regimes (Fig. 5). Decreasing the uncertainty to δλ =0.01 nm increases z_c to ≈16 mm [Fig. 5].

 

Fig. 5 Impact of ST correlation uncertainty on the diffraction-free (DF) range z_c of ST beams compared to monochromatic Gaussian beams (Δλ =0.01 nm). Smaller δλ increases z_c.

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6. Proposed realization for diffraction-free ST beams

The question remains of the feasibility of constructing such diffraction-free ST beams. We propose an arrangement that combines traditional approaches of pulse-shaping [49–51] with those for beam-shaping [52–54]. Starting with a separable pulsed beam f (x, t) = f_x (x) f_t (t), where the beam profile f_x (x) is generic (e.g., the TEM₀₀ mode of a pulsed laser), the pulse spectrum is dispersed along the y-direction via a diffraction grating and then directed to a 2D spatial light modulator (SLM) that imprints a phase distribution in the transverse x–y plane of the form $e^{i{k}_x(y)x}$ (see [55–61]). Since the spectrum is spread along y, each spatial frequency k_x (y) is selected to correlate correctly to the wavelength at position y according to Eq. (2). A second diffraction grating combines the spectrum to reconstitute the pulse, thereby producing the desired ST beam. Preparing a diffraction-free ST beam of width 10 µm at a wavelength of 800 nm requires a 1-nm-wide spectrum and a spatial spectrum Δk_x = 556 rad/nm. These requirements can be fulfilled by spreading the pulse by a diffraction grating with 1800 lines/mm that is illuminated over a 45-mm-wide section, and then directing the spectrum via a cylindrical lens of 200-mm focal length to an SLM with 5-µm-wide pixels. This system can produce the desired ST correlation with an uncertainty of δλ =0.01 nm, corresponding to a diffraction length of 16 mm ≈ 55z_R of a monochromatic beam having the same spatial bandwidth Δk_x.

7. Conclusion

In conclusion, we have described a general approach to preparing diffraction-free ST beams in 1D and 2D with arbitrary cross sections. This is particularly surprising in 1D where it has been conclusively demonstrated that only two spatial profiles (Airy and cosine) propagate with no transverse changes – assuming monochromatic light. Relaxing the monochromaticity constraint enables ST correlations to arrest changes in the transverse beam profile along the propagation axis. Current technology can readily achieve a ST correlation-uncertainty of δλ ≈0.01 nm over a bandwidth of 1 nm, which can maintain a pulsed beam of width 10 µm and pulse-width 94.1 ps over a distance of 16-mm – sufficient for deep-tissue imaging, for instance. The ST-beam concept can be applied to other manifestations of waves, such as acoustic and electron beams, for instance. Finally, the theory outlined here was formulated for coherent light in free space. It remains an open question whether this strategy can be extended to partially coherent light, to propagation in dispersive media, and propagation in the presence of scattering perturbations.

We have recently learned that another group has been studying similar pulsed beams and that their work will appear published concurrenly with ours 62