Two-dimensional spatiotemporal Fourier transform with femtosecond pulses for on-chip devices


 On-chip manipulation of the spatiotemporal characteristics of optical signals is important in the transmission and processing of information. However, the simultaneous modulation of on-chip optical pulses, both spatially at the nano-scale and temporally over ultra-fast intervals, is challenging. Here, we propose a two-dimensional spatiotemporal Fourier transform (FT) method for on-chip control of the propagation of femtosecond optical pulses and verify this method employing surface plasmon polariton (SPP) pulses on metal surface. By varying space- and frequency-dependent parameters, we demonstrate that the traditional SPP focal spot may be bent into a ring shape, and that the direction of propagation of a curved SPP-Airy beam may be reversed at certain moments to create an S-shaped path. Compared with conventional spatial modulation of SPPs, this method offers potentially a variety of extraordinary effects in SPP modulation especially associated with the temporal domain, thereby providing a new platform for on-chip spatiotemporal manipulation of optical pulses with applications including ultrafast on-chip photonic information processing, ultrafast pulse/beam shaping, and optical computing.


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Here, we propose a method based on the two-dimensional (2-D) spatiotemporal Fourier transform (FT) to manipulate the propagation path of a femtosecond optical pulse for on-chip devices, so that during propagation the pulse exhibits different spatial properties at different instances. As a wellknown mathematical tool, the FT method has been applied to an extremely wide range of applications related to optics, including spectroscopic measurements of white light 9 , pulse shaping 10 , optical computing 4 , and holography 11 . Given that the frequency-domain FT is used to reshape the pulse in the time-domain, and similarly the spatial FT is used to reshape the spatial distribution of the light, synchronous spatiotemporal modulations of the on-chip optical pulse can be realized once the two operations are suitably combined.
To demonstrate this 2-D spatiotemporal FT method for on-chip devices, we chose surface plasmon polaritons (SPP) excited on a metal surface as carriers of on-chip optical signals, and studied modulation effects of femtosecond SPP pulses. With their capability of breaking through the optical diffraction limit 12 , SPPs are widely employed in nanophotonic devices used in a variety of applications including optical storage 13 , optical sensing 14 , optical tweezers 15 , and Raman scattering 16 . In addition to nanoscale spatial resolutions, SPP pulses generated by a femtosecond laser beam enable femtosecond-scale temporal resolutions, thereby providing a research platform to investigate the manipulation of light fields and the interaction of light and matter at extremely small spatiotemporal scales. In exploring ultrafast phenomena, researchers are able to characterize the dynamics of physical and chemical events in molecular structures 17 and stimulated Raman scattering (SRS) of molecules 18 . With spatiotemporal FT modulations of femtosecond SPP pulses, we transformed the traditional focal spot of the SPP pulses into a ring shape and changed the path of propagation of an SPP-Airy pulse at any instant, to be described below.
3 / 14 Fig. 1 Schematics of the excitation and spatiotemporal FT modulation of SPP pulses. a Nano-slits in the angular spectrum represented by a reference arc are predesigned in the gold film (200 nm in thickness) on a glass substrate (n =1.515). When illuminated with an angularly dispersed femtosecond pulse, the spatially focused SPP excited by scattering from the slits is temporally modulated through dispersion, while its converging wavefront is spatially modulated by the slits through a displacement ∆ with respect to the reference arc (black dashed curve), thereby realizing both spatial and temporal FT modulation in a SPP focusing structure. b In-plane SPP focusing on metal surface. The field at any point N(x,y) near the focus O(0,0) can be calculated by summing the contributions from all points, e.g., M(, ζ) on a convergent SPP wavefront. γ denotes the angle of inclination of the distance with respect to the normal of the arc ∑;  denotes the angle between MO and the y-axis.
The metallic nano-structure that performs the spatiotemporal FT modulation of SPP pulses [ Fig.   1(a)] is composed of a gold film etched with multiple discrete nano-slits around a circular reference arc (black dashed curve) on a glass substrate. When light is incident from underneath, SPP pulses are excited and scattered from the nano-slits towards the centre of the reference arc to create a SPP focal point. Its wavefront is modulated by the angled displacement of each slit ∆ from the reference arc. In general, the spatial distribution of all the slits is designed to create a specific SPP wavefront and subsequently the desired focal field. For example, an Airy phase modulation along the lateral dimension of the beam 19,20 generates a spatially curved SPP-Airy beam near the focus. Although such displacements of the nano-slits generate various structured focal fields, these fields are nonetheless constant and not dynamically controlled. To achieve dynamic control of the SPP field, we considered applying frequency (dispersive) modulations using a femtosecond pulse of incident light with dispersive attributes, employing the frequency-domain FT method to control the temporal focusing wavefront and corresponding propagation paths of the excited SPP pulses. With frequency modulated SPPs, the wavefront of the excited SPP-Airy pulse is dynamically changed during focusing, resulting in an approximate S-shaped space-time propagation path [see Fig. 1(a)].

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To explain the basic principle of our 2-D spatiotemporal FT method on SPP, we build a simple theoretical model. Consider an SPP pulse excited focused by a simple arc-shaped nano-slit 21 where k SPP (ω) =2π/λ SPP (ω) denotes the wave vector of SPP, with λ SPP (ω) the wavelength of the SPP 12 , and ω the frequency of the incident light. Regardless of the time-dependent term, when the focal length R of the reference arc is much larger than the SPP wavelength λ SPP (ω), then, using the 2-D Helmholtz-Kirchhoff integral theorem 22,23 , the time-averaged spatial distribution of the SPP in the focal region is approximately 4,24 After the SPP focusing structure is illuminated by the incident pulse of light, the electric field is where α denotes the arc measure, Near the focus, γ ≈0 for the field distribution. Hence, we simplify Eq.
which is expressible as a 2-D integral, where П(•) and δ(•) are the rectangular and Dirac-delta functions, respectively. Using approximation, where '' . Ignoring the constant term, Eq. (7) where F 2 denotes the 2-D spatial FT. Consider a SPP pulse with arbitrary frequency ω; its spatiotemporal evolution may be obtained by integrating over all frequency components, where F -1 denotes the 1-D temporal FT. From Eq. (9), there is a 2-D spatial FT and a 1-D temporal To verify that the 2-D spatiotemporal FT equation may be used to control the space-time properties of the SPP pulses, we consider a femtosecond incident plane wave pulse expressed as where ω 0 denotes the centre frequency of the incident pulse, t the pulse transmission time, T 0 the instant when the envelope of the pulse reaches its peak amplitude, and  the full-width at half- and its temporal wavefront [ Fig. 2(b)]. However, when the incident light is changed to an angularly dispersed femtosecond pulse, the focal spot of the SPP pulses is bent into a ring shape at a certain time [ Fig. 2(c)]. To reveal the principle of this extraordinary phenomenon, we calculate the timeaveraged SPP focal field and the corresponding temporal wavefront from the above equations. We spatiotemporal FT modulation of a complex SPP field. Being a non-diffractive wave, the Airy beam has interesting novel features including curved transmission and self-recovery during propagation 28 .
Recently, different nanostructures have been designed to generate the on-chip Airy beam of the SPP 29,30 , referred to as the SPP-Airy beam. Here, based on the spatial FT theory, we generated an SPP-Airy beam on a metal surface by displacing each slit by a fixed distance ∆ from the reference arc [ Fig. 1(a)]. Substituting the Airy function into Eq. (3), the spatiotemporal field distribution of the SPP-Airy beam is obtained, where 32 0 ( φ =) l  −  is the phase of the Airy function with l and 0  constants. Thus, the corresponding offset (∆) of each slit from the reference arc is given by Then, using the same dispersion modulation method [ Fig. 2], the SPP-Airy pulses excited by these slits is further spatiotemporally manipulated, especially in their propagation paths.
When a y-polarized femtosecond light beam is obliquely incident at an angle  =10° to the z-axis but without angular dispersion [ Fig. 3(a) In summary, a novel 2-D spatiotemporal FT modulation method employing femtosecond pulses illuminating on-chip devices was proposed, studied, and then verified using SPP pulses in an optical 12 / 14 setup. The angular dispersion of a femtosecond light pulse is exploited to control the excited SPP pulses in the time domain. By changing the dispersion of the light source, the wavefront of the SPP pulses in the focal plane was changed from a traditional focal point to an inverted arc-shape and even a ring shape. Combining both the frequency-domain modulation with dispersion and the spatial modulation with the nanostructure, we demonstrated that the method can generate a SPP-Airy beam with an S shape, proving the universality of the method in creating complex optical fields. We noted that this method is in theory suitable not only for SPP fields, but also for other on-chip propagating waves such as Bloch surface waves 31 . This work offers a new way for manipulating spatiotemporal properties of ultrafast optical pulses produced in on-chip devices with many potential applications including ultrafast photonic information processing 32 , ultrafast pulse shaping 7,33 , beam steering 8 , ultrafast photo-switching and modulation 34 , optical computing 4 , and super spatiotemporal resolution imaging 35 .