Light-carrying orbital angular momentum (OAM) has been an active research topic for decades, traditionally as vortex beams that have longitudinal OAM with spatial phase singularities. Such vortex beams have proven to be useful for a wide variety of applications, such as free-space communications, particle manipulation, quantum entanglement, imaging, and more^[1,2]. Recently, transverse OAM states with spatiotemporal vortices (STOVs) have been demonstrated as having tilted spatiotemporal vortices^[3,4]. Having full spatiotemporal control of the OAM states may open up new possibilities for the applications listed above.

In certain applications, partial coherence is beneficial, especially when utilizing light with OAM^[5,6]. For example, partial spatial coherence improves the signal-to-noise ratio with longitudinal OAM communication and reduces speckle in imaging^[5]. On top of improved performance, incoherent sources are cheaper and have a simpler setup. Partial temporal coherence sources may offer such benefits for STOVs. STOVs with partial temporal coherence have been recently investigated and demonstrated^[7,8]. However, there has not been a report of tilted STOVs with partial temporal coherence.

Here, we investigate the tilted STOV with partial temporal coherence. We theoretically study the time-varying nature of the OAM direction of a partially coherent STOV. We also present experimental evidence of a partially coherent STOV with tilted OAM. Tilted vortices with partial temporal coherence may be beneficial for future applications such as communication and particle manipulation, especially with low-cost and simple-to-use optical sources.

We first simulate a STOV with partial temporal coherence. To simulate partial coherence, we apply a random phase on each temporal frequency of a coherent wave packet^[8,9]. The probability of the phase follows a normal random distribution where we control the amount of randomness with the standard deviation, σω. On top of the partial temporal coherence, we add a spatiotemporal spiral phase to generate a STOV with a transverse OAM^[8]. As the STOV passes through a pair of rotated cylindrical lenses, the OAM direction is tilted following the method in Ref. [4]. To observe a typical behavior of the tilted incoherent STOV, we use parameters near our experiment values discussed later, a Gaussian beam with 1 mm diameter and a spectrum that is also assumed to be Gaussian with 2.35 THz FWHM. Figures 1(a) and 1(b) show a typical iso-intensity profile of the simulation result.

From observation, the time-varying vortex angle clearly takes on a wide range of values. The partially coherent field itself has pulse-like fluctuations of intensity with temporal width near the coherence time^[10], and we see that each fluctuation tends to have an isolated vortex in a certain direction. However, as shown in Fig. 1(c), some areas can have tilt angles that smoothly vary with time over one fully connected vortex. The random temporal phase causes fluctuations in the amount of transverse OAM and the spatial location of the singularity^[8]. The combination of such time-varying transverse OAM components with the longitudinal OAM by the cylindrical lens pair results in the time-varying random vortex direction. The next step is to measure the time-varying OAM direction. However, because the STOVs are in random positions relative to the center axis, the best choice of sampling location and size is not clear due to the addition of extrinsic OAM. While measuring the intrinsic OAM vector would give a reliable angle, adding extrinsic OAMs from sampling off-center could potentially give an incorrect direction measurement. Sampling at the center of each vortex would certainly be a good choice to eliminate extrinsic contributions. However, for our case, we wish to understand how a stationary observer or particle would experience the partially coherent STOV. We calculate the OAM vector within a small volume at the wave-packet center (200μm×200μm×33fs) while the volume is scanned in time. The volume is temporally narrower than the coherence time (186 fs) to measure the fast-varying OAM angle, while the spatial region is small to capture only the OAM vector near the vortices crossing the center. As the spatial area is reduced in size, the values converge to what is expected and further shrinking beyond our chosen integration region, gives no change in average values. The OAM vector is^[3]L→(t0)=∭Vr→×Imag(ψ*∇ψ)dV,(1)where ψ is the field, and r→ and V are the distance vector and integration volume, both centered on the propagation axis at sampling location t=t0. Furthermore, we define the normalized time-average mean OAM vector as the mean over the entire temporally scanned region, T, normalized to the vector magnitude of the fully coherent case (σω=0), ⟨L→⟩=1T∫L→(t0)dt01T|∫L→(t0,σω=0)dt0|,(2)and the vector components are ⟨L→⟩·ei=⟨Li⟩, where i=x,y,z. Figure 2(a) shows the time-average OAM vector components as well as their magnitude as a function of the coherence. Although the angle of the STOV fluctuates in time, it is noteworthy that the time-average vector components do not change much by the source randomness. Figure 2(b) gives a histogram of the x-component for σω=π8 and σω=π. As coherence is reduced, the OAM distribution broadens. However, the mean OAM components are more or less the same as shown in Fig. 2(a).

Next, we calculate the direction of the OAM from the time-varying OAM vector components using the spherical polar system aligned to the propagation axis, ϕ(t0)=arctan(Ly(t0)Lx(t0))(3)as the azimuthal angle and θ(t0)=arctan((Lx(t0))2+(Ly(t0))2Lz(t0))(4)as the polar angle. This can also be related to the tilt from the original STOV direction before the cylindrical lens system, which is purely transverse in the x direction. The tilt angle, γ from the original transverse OAM direction is given by γ=arccos(cos(ϕ)sin(θ)). However, the tilt angle does not tell the OAM direction, but it only tells how much the OAM is off from the original direction. Therefore, to designate the OAM directions, we will use ϕ and θ. The time-average and standard deviation of angles are shown in Figs. 2(c) and 2(d), respectively. The mean and standard deviation of the angles remain mostly the same, regardless of the degree of coherence. The mean polar angle is between 74 deg and 77 deg, while the azimuthal angle is between 39 deg and 45 deg. Because we are sampling with a moving time window with a limited number of time samples, even the fully coherent tilted vortex has a large variance (i.e., each time slice has a different OAM value).

However, the angles clearly become more random as the source coherence decreases. To illustrate this, we plot the magnitude and angles together as points in a polar cross-sectional plot in Fig. 3, with each point representing a time sample. The polar angle in the plot is the ϕ(t0) in Eq. (3), and the radial value is r(t0)=|L(t0)|sin(θ(t0))|⟨L(σω=0)⟩|, the sine of the polar angle multiplied by the OAM magnitude, again normalized to the fully coherent average OAM magnitude. At high coherence, the values congregate near r=0 because most of the STOV is localized in time. In addition, the magnitude smoothly increases and decreases in time. As coherence drops, the OAM magnitude and direction become more dispersed, shown as spreading on the polar plot around r=0 [Fig. 3(b)]. At the higher randomness of σω=π, the plot randomly spreads further. Therefore, though the perceived standard deviation in angles to an observer is not changed much, the angles are more random for incoherent sources. Since the time-averaged OAM magnitude and direction are stable, it is believed that the incoherent tilted OAM can be used for applications with long response times.

Finally, as we cannot directly measure time-varying OAM or OAM tilt direction, we generate a delayed reference interference pattern. The delayed reference interference patterns can be used to reveal the average tilt-angle by tracking the singularity position^[8] and are represented by the equation, I(x,y,τ)=∫|Eo(x,y,t)+Er(x,y,t+τc)|2dt,(5)where Eo and Er are the object and reference, τ is the delay in spatial units, and c is the speed of light. The reference is the incoherent source itself without OAM.

Each interference pattern reveals the time-average spatial phase information at a specific delay, with the singularities of the vortices identifiable by a fork interference fringe (annotated with arrows). Figures 4(a) and 4(b) show the interference patterns at a separation delay of 100 fs or 30 µm, about half the coherence length, with the fork-shaped fringes location annotated with arrows. The direction of the OAM can be revealed by the fork fringe movement, ΔX and ΔY, over the delay difference Δτ. Figure 4(c) plots the fork location over delay positions separated by 5 µm, revealing the time average tilt direction of the OAM. At greater than 30 µm delay, the interference fringe visibility of the forks starts to decrease significantly. We define angles observed from the interference pattern in the spherical coordinates as earlier so that ϕI=arctan(ΔYΔX) is the azimuthal angle and θI=arctan((ΔX)2+(ΔY)2Δτ) is the polar angle of the interference pattern. With ΔX=121μm, ΔY=65.3μm, and Δτ=30μm, the azimuthal and polar angles are ϕI=28.3deg and θI=77.7deg. Unfortunately, since the reference also varies in time, the measured angles from the interference pattern are not exactly the time-average OAM directional angles. They only tell the average direction of the OAM roughly. As a consequence, θI matches well with the value directly calculated in simulation, while ϕI is about 10 deg off.

The experimental setup is shown in Fig. 5(a) and follows the method described in Ref. [4]. The light source is amplified spontaneous emission (ASE) from an ytterbium fiber ring cavity with spectrum shown in Fig. 5(b). A pulse shaper with a spiral phase is used to add transverse OAM to the source [phase map in Fig. 5(c)]. As an example of the full spatiotemporal phase, the spiral phase combined with the expected random phase from ASE is shown in Fig. 5(d). Rotated cylindrical lenses are used to spatiotemporally tilt the vortices. The cylindrical lens pair is rotated at 45 deg, which gives the maximum tilt angle^[4]. A reference, which is the original ASE with variable delay, is used to record the time-average interference onto a CCD.

Figure 6(a) and 6(b) show interference patterns at delays separated by 5 µm. Figure 6(c) plots the fork fringe location for each delay. In the experimental case, the fork fringe visibility begins degrading with distances over 20 µm. With ΔX=145μm, ΔY=195μm, and Δτ=20μm, the angles are ϕI=53.4deg and θI=85.3deg. The angles are off from the simulated result, since the measurement is done with a time-varying reference beam. However, the measurement clearly shows that the OAM is tilted.

In conclusion, we have demonstrated tilted STOVs with partial temporal coherence. We first explored the time-varying OAM and direction in simulation, which were then demonstrated experimentally. We believe cheap and simple light sources such as ASE will encourage further research and applications of tilted STOVs such as communications and particle manipulation. Since the average is largely unaffected, applications utilizing only average OAM would greatly benefit.