## Abstract

Diffraction places a fundamental limitation on the distance an optical beam propagates before its size increases and spatial details blur. We show here that imposing a judicious correlation between spatial and spectral degrees of freedom of a pulsed beam can render its transverse spatial profile independent of location along the propagation axis, thereby arresting the spread of the time-averaged beam. Such correlation introduced into a beam with arbitrary spatial profile enables spatio-temporal dispersion to compensate for purely spatial dispersion that underlies diffraction. As a result, the spatio-temporal profile in the local time-frame of the pulsed beam remains invariant at all positions along the propagation axis. One-dimensional diffraction-free *space-time beams* are described – including non-accelerating Airy beams, despite the well-known fact that cosine waves and accelerating Airy beams are the only one-dimensional diffraction-free solutions to the *monochromatic* Helmholtz equation.

© 2016 Optical Society of America

## 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

*x*and

*z*are the transverse and axial coordinates, respectively,

*k*and

_{x}*k*are the corresponding wave-vector components, and the ST spectrum

_{z}*F*(

*k*,

_{x}*ω*) is the Fourier transform of the initial field

*f*(

*x*, 0;

*t*). For a

*monochromatic beam ω*=

*ω*

_{o},

*F*(

*k*,

_{x}*ω*) =

*F*(

*k*)

_{x}*δ*(

*ω*−

*ω*

_{o}) so that $f(x,z;t)=f(x,z;0){e}^{-i{\omega}_{o}t}$ and diffraction is driven by spatial dispersion ${k}_{z}^{2}={\left({\omega}_{o}/c\right)}^{2}-{k}_{x}^{2}$. Such a beam is represented in (

*k*,

_{x}*k*)-space by an arc of an origin-centered circle whose horizontal extent is the spatial bandwidth Δ

_{z}*k*[in Fig. 1(a)]. Each point in the (

_{x}*k*,

_{x}*k*)-plane identifies a monochromatic plane wave. Alternatively, a

_{z}*pulsed plane wave f*(

*x*,

*z*;

*t*) =

*f*(0,

*z*;

*t*) for which

*F*(

*k*,

_{x}*ω*) =

*F*(

*ω*)

*δ*(

*k*) is represented in (

_{x}*k*,

_{x}*k*)-space by a vertical line whose radial extent is the bandwidth Δ

_{z}*ω*[Fig. 1(b)].

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*) and temporal

_{x}*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*and

_{x}*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*and

_{x}*k*is no longer strict because

_{z}*ω*varies over Δ

*ω*and each

*k*can be associated with a continuum of frequencies

_{x}*ω*and hence axial wave numbers

*k*. Spatial dispersion is thus compounded with ST dispersion: the beam diffracts and the pulse disperses.

_{z}## 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*. The key insight is that there exists a trajectory in (

_{z}*k*,

_{x}*k*) space that guarantees all the constituent plane waves have the

_{z}*same k*: the

_{z}*k*-iso-surface ${k}_{z}({k}_{x},\omega )=\mathcal{K}$ that fixes a hyperbolic relationship between

_{z}*k*and

_{x}*ω*,

*k*,

_{x}*k*)-space by a horizontal line [Fig. 1(d)]. This one-to-one correspondence between |

_{z}*k*| and

_{x}*ω*– by imposing the constraint

*ω*(

*k*) =

_{x}*ω*on the (

_{x}*k*,

_{x}*ω*) pairings – entails that each spatial frequency

*k*is ‘encoded’ on a single frequency

_{x}*ω*to ensure that

*k*is independent of both

_{z}*k*and

_{x}*ω*. Such a beam evolves self-similarly except for an overall phase,

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 Δ*ω* of the auxiliary spectral DoF to ‘protect’ the beam’s spatial DoF [Fig. 1(d)],

*ω*

_{o}= (

*ω*

_{max}+

*ω*

_{min})/2. This bandwidth is necessary to ‘protect’ the beam and arrest its spreading. In general, a

*broader*spectrum is required to protect a beam with larger Δ

*k*(i.e.,

_{x}*finer*details). Therefore, a beam with a highly localized transverse profile becomes associated with a highly localized pulse; that is, there is a correlation between localization in space and time in this conception. We refer to such a beam as a ‘space-time’ (ST) beam since the two DoFs are inextricably intertwined or ‘entangled’. The notion of ‘classical entanglement’ has recently attracted considerable attention [38–46]. Most previous experimental and theoretical studies of classical entanglement have focused on optical beams in which correlations exist between discretized DoFs, such as polarization or a few spatial modes. The diffraction-free ST beams described here rely instead on correlation between two continuous DoFs (

*ω*and

*k*).

_{x}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*) the frequency

_{y}*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 *µ*m and spectral bandwidth of Δ*λ* = 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 ${z}_{\mathrm{R}}=\pi {w}_{o}^{2}/\lambda \approx 283$

*µ*m (

*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].

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 Δ

*λ*is determined by Δ

*k*(Eq. (4)). The beam size

_{x}*x*

_{o}=10

*µ*m 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 20

*z*

_{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 Δ

*λ*[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 Δ

*λ*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*. The corresponding axially self-similar ST intensity [Fig. 3(b)] now has a null at

_{x}*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.

## 5. Impact of uncertainty on diffraction-free length

The diffraction-free length is determined by the uncertainty in the relationship *ω*(*k _{x}*) =

*ω*, which is subject to practical restrictions on ST beam generation. In Fig. 4, we further relax the ST correlations and increase

_{x}*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.

We have examined thus far ST beams propagating across distances over which traditional beams of comparable spatial bandwidth diffract significantly (~20*z*_{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*to ≈16 mm [Fig. 5].

_{c}## 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 TEM

_{00}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

*µ*m at a wavelength of 800 nm requires a 1-nm-wide spectrum and a spatial spectrum Δ

*k*= 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-

_{x}*µ*m-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 ≈ 55

*z*

_{R}of a monochromatic beam having the same spatial bandwidth Δ

*k*.

_{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 *µ*m 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

## Funding

U.S. Office of Naval Research (ONR) contract N00014-14-1-0260.

## Acknowledgments

We thank D. N. Christodoulides and T. M. Yarnall for helpful discussions.

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