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Abstract
Recently, a spatiotemporal optical vortex (STOV) with a transverse orbital angular momentum (OAM) has been generated from coherent ultrafast pulses using mode-locked lasers. In contrast, we demonstrate theoretically and experimentally that a STOV can be generated from a light source with partial temporal coherence with fluctuating temporal structures. By eliminating the need of mode-locked laser sources, the partially coherent STOV will serve as a convenient and cost-effective transverse OAM source.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical vortex beams with phase singularities are well-established phenomena in modern-day singular optics. As shown in 1992 by Allen, et al. [1], optical vortex beams, which can be generated by a variety of techniques, have an orbital angular momentum (OAM) [2–5]. A typical vortex beam has a spatial spiral phase of the form ${e^{il\emptyset }}$ where ∅ is the azimuthal angle and l is the topological charge. These optical vortices carry an OAM of $l{\hbar}$ per photon and have attracted significant interest leading to many interesting applications such as optical tweezers [6], Quantum information processing [7,8], super-resolution microscopy [9,10] and free space optical communication [11].
Partially coherent vortex (PCV) beams have also emerged in the last decade drawing significant interest due to their unique properties in beam shaping, beam rotation and self-reconstruction [12]. A key feature of the PCV beams is that the phase singularity is diminished due to random phase fluctuations. PCV beams have very important applications such as optical trapping where focused PCV beams can be utilized for beam shaping. In addition, the self-reconstruction ability of PCVs on propagation has been used for information encryption and decryption [13].
Recently, a new class of optical vortex with a transverse OAM has been generated successfully [14,15]. Here the spiral phase and phase singularity exist in the spatiotemporal (ST) domain as opposed to the regular optical vortex beam. This new form of vortex is called the ST optical vortex (STOV) and it carries the potential for novel applications in many fields such as in optical manipulation, second harmonic generation, ST spin-orbit angular momentum coupling, etc. [14].
Similar to PCVs in the spatial domain, STOVs with partial temporal coherence has also been studied theoretically by M. Hyde [16]. His work was focused on the theoretical descriptions of the formation and propagation of partially coherent vortex beams in the space-frequency and ST domains. Thus, there is interest in further studying the properties of these vortices that possess partial temporal coherence and experimentally observing them in the lab-this would make partially temporally coherent STOVs an addition to the list of optical vortices that can be readily generated and studied.
Here, we present a novel method to generate a STOV using a light source with partial temporal coherence. The experimental results are well verified by numerical simulations. Since cheaper broadband sources such as light-emitting diodes can be used to generate STOVs in the place of expensive mode-locked lasers, this work provides cheap STOV sources for further applications.
2. Theory
Due to the conservation of the OAM under the Fourier transform, a simple and reliable method can generate the STOV without involving nonlinear processes [14]. A simplified numerical model to simulate the STOV is shown by Eq. (1).
where $\rho = \sqrt {{x^2} + {t^2}} $, $\varphi = \textrm{arctan}\left( {\frac{\textrm{x}}{\textrm{t}}} \right)$, and g($\rho $) is a radial basis function that represents the amplitude of a STOV.The same concept is applied to partial temporal coherent STOV generation, with partially coherent sources such as amplified spontaneous emission (ASE), where the degree of temporal coherence is reduced by the randomly emitted photons [17,18,19]
ASE can be numerically simulated by applying a randomly distributed spectral phase [20,21]. To model the ASE, a Gaussian spectrum with a randomly distributed spectral phase is simulated. The corresponding pulse $A(t )$ can be calculated by the Fourier transform, which is denoted by Eq. (2).
Fig. 1. Numerical generation of pulses with different noise spectral phase (the amplitude of each pulse is adjusted to show fine structures).
Now we can apply the spiral phase ${\textrm{e}^{il\phi }}$ on the spatial-frequency and frequency domain of the partial temporal coherent pulse. By the two-dimensional Fourier transform, the STOV with partial temporal coherence is numerically calculated. To be consistent with the experimental condition, a Gaussian spectrum is used with the center wavelength of 1030 nm and full width at half maximum (FWHM) bandwidth ∼8 nm. Figure 2 shows the simulated partial temporal coherent STOVs with a topological charge l = 1. It shows the cross-section of the field amplitude in a fixed transverse plane as a function of time. Each transverse profile will evolve differently on propagation but the presented data are simulated without propagation effects.
Fig. 2. Amplitude of partial temporal coherent STOV with topological charge l = 1. Variance of each normal distribution is: (a) ${\sigma ^2} = 0.2{\pi ^2}$, (b) ${\sigma ^2} = 0.5{\pi ^2}$, (c) ${\sigma ^2} = {\pi ^2}$.
As the phase randomness increases, the shape of the STOV is severely distorted from the ring-shaped profile with multiple singularities occurring at various temporal locations. Since the location of the phase singularity is fixed in the spatial-frequency domain, vortices after the Fourier transform appear around the same height which is the Fourier conjugate spatial location. The small deviations in height are due to the random phase fluctuation. The presented simulation results are only a small portion of the entire random pulse to reveal the characteristics of the partially coherent STOV. Amplitude fluctuations randomly spread throughout the whole temporal domain with random multiple amplitude peaks.
To visualize the phase and the corresponding Poynting vector, the time averaged Poynting vector < S > is calculated by Eq. (3).
Fig. 3. Partial temporal coherent STOV with topological charge l = 1, ${\sigma ^2} = {\pi ^2}$, (a) electric field amplitude with Poynting vectors; (b) phase and Poynting vectors.
Previous research has demonstrated that a STOV with a higher topological charge $l$ > 1 is not stable under propagation due to the ST astigmatism [14]. For the partially temporally coherent STOV, the random phase itself acts as the ST astigmatism, thus, higher-order partially coherent vortices will break up into fundamental singularities even without propagation. The instability of higher order partial temporal coherent STOVs is analogous to the spatial vortex splitting due to the atmospheric turbulence [22].
Figure 4(a) shows the partial temporal coherent STOV with $l$ = 2 without propagation to highlight sole random phase effect on higher-order vortices, which in Figs. 4(b) and 4(c) where we can clearly observe split of one of phase singularity. The splitting of each high-order vortex happens spontaneously because of the intrinsic random phase fluctuation.
Fig. 4. Partial temporal coherent STOV with topological charge l = 2, ${\sigma ^2} = {\pi ^2}$, (a) the train of multiple vortices, (b) amplitude and Poynting vectors and (c) phase and Poynting vectors to show the vortex split.
The existence of the phase singularities of the partial temporal coherent STOV can be experimentally verified by the interference fringes which is the same method used to detect the coherent STOV [14]. Figure 5 shows the principle of the measurement, where we use the temporal domain energy density plot to represent the reference and the object. The reference and object beams are branched from the same partially coherent source, where the object beam is converted to a STOV. We can clearly see that the reference beam has multiple random peaks. In contrast, the corresponding peaks in the object beam tends to have amplitude dips due to the phase singularities. When the reference scans through the object, each peak of the reference will scan through the corresponding phase singularity in the object beam. Since the object and the reference have a high degree of spatial coherence, if they are overlapped with a temporal delay within the coherent time, a clear time averaged object-reference interference pattern can be formed and captured by a CCD camera. Interference pattern at various temporal delays is recorded. This detection method can not only prove the existence of the phase singularities, but also reveal the order of the topological charge. Mathematically, such a process can be expressed by Eq. (4):
Fig. 5. Principle of the measurement: fringe patterns are detected by the CCD camera when the object and the reference beams with different temporal delays merge with a small angle. X-represent phase singularities in the object beam.
Figure 6 shows the numerically simulated fringe patterns for the partial temporal coherent STOV with topological charge $l$ = 1. As the reference scans the object, we can observe the phase transition of the up and down fringes. At the center position, where the time delay is zero, we can see the π phase difference clearly in Fig. 6(b). Since the spectral bandwidth is around 8 nm, the coherent time is around 450 fs. From Fig. 6(a) to 6(c), the total time delay is around 85 fs, which is in the range of the coherent time.
Fig. 6. The theoretical interference fringe pattern of the partial temporal coherent STOV with topological charge l=1, ${\sigma ^2} = {\pi ^2}$ (a) time delay=-43fs; (b) time delay=0fs; (c) time delay=43fs.
The phase transition patterns also reveal the topological charge of the partial temporal coherent STOV, where we can see the π phase transition twice for topological charge l=2. By the argument of the fixed singularity location in the spatial-frequency domain, vortices again appear around same heights for l=2 case as well. The interference pattern indicates averaged vortex locations. From Fig. 7(a) to 7(c), we can see one transition occur in the upper position, and from Fig. 7(d) to 7(f) is another transition that appears in the lower position.
Fig. 7. The theoretical phase transition of the partial temporal coherent STOV with topological charge l=2, ${\sigma ^2} = {\pi ^2}$ (a) time delay=-119fs; (b) time delay=-93fs (c) time delay=-68fs; (d) time delay=59fs; (e) time delay=85fs; (f) time delay=110fs.
3. Experimental results
The initial experiment was performed using amplified spontaneous emission (ASE) of a fiber laser. Figure 8(a) shows the experimental setup which includes the pulse shaper with a 2D spatial light modulator for generating the partial temporal coherent STOV. An ytterbium-doped single-mode fiber laser was used for the experiment with a center wavelength of 1030 nm and was driven below the lasing threshold to achieve ASE. It is noteworthy that the ASE from the single-mode fiber has a high degree of spatial coherence since only the fundamental transverse mode is available. Hence, it is reasonable to assume that the partial coherence exists only in the temporal domain. Also, autocorrelation measurements were performed for this ASE state, but no signal was observed. This actually verifies that the ASE state is temporally incoherent. The spiral phase patterns of the form ${\textrm{e}^{i\phi }},{\textrm{e}^{ - i\phi }}and\; {\textrm{e}^{i2\phi }}$ corresponding to topological charges $l ={+} 1,\; - 1,\; \textrm{and} + 2$ were projected onto the SLM to produce partial temporal coherent STOVs, where $\; \phi $ is the azimuthal angle on the SLM. Figure 8(b) shows the output spectrum of ASE with a FWHM bandwidth of 7.8 nm.
Fig. 8. (a) Schematic of the setup for partial temporal coherent STOV generation. BS-beam splitter (b) Spectrum of the ASE source.
Figure 9 shows the phase transition patterns observed by scanning the reference beam, for topological charges $l ={+} 1,\; - 1,\; \textrm{and} + 2$. Two, phase transitions were observed for topological charge l=+2, which is in agreement with the theoretical prediction.
Fig. 9. Phase transitions for topological charge (a) l=+1, (b) l=−1, and l=+2. Time delays of the phase patterns from left to right are(a) −68 fs,0 fs and 32 fs (b) −113 fs, 0 fs and 50 fs (c) −216 fs,−131 fs,−103 fs,78 fs,131 fs and 180 fs respectively. In (a)-(c), color bars represent the intensities in arbitrary units.
Besides the ASE, a noise-like pulse (NLP) state from the fiber laser was also investigated since the NLP state provides a broad-spectrum partial temporal coherent pulses where the phase relation between longitudinal modes is not strictly fixed [18]. The NLP state was achieved by adjusting the nonlinear polarization evolution (NPE) based saturable absorber of the same fiber laser that was used for generating the ASE state. An autocorrelation trace with a thin spike on top of a broad pedestal and a noisy RF spectrum are typical signatures of the NLP state [23]. Output spectrum of the NLP state which has a FWHM bandwidth of 52 nm, its autocorrelation signal and the RF spectrum are shown in Fig. 10(a), (b) and (c) respectively. A distinct, thin spike is observed in the autocorrelation function and the FWHM of the pulse is measured to be about 2.7 ps. The RF spectrum of the NLP state of the laser has a repetition rate of 46.58 MHz with a SNR of 54 dB. When the laser was mode-locked, SNR of the RF spectrum was observed to be 72 dB and this loss in the SNR, combined with the characteristic autocorrelation trace, verifies the NLP state achieved in the experiment.
NLP states usually have higher peak power than ASE, hence, the STOV from the NPL state can be useful for high-power applications. Similar to the previous experiment, clear phase transitions were observed for topological charge l=+1 for the NLP state which are shown in Fig. 11.
Fig. 11. Phase transition for topological charge l = 1 of the NLP state. Time delays from left to right are −46 fs, 0 fs and 13 fs respectively.
4. Conclusion
In conclusion, we have successfully generated a partial temporal coherent STOV. Simulation results reveal the existence of multiple singularities of the partial coherent STOV due to random phase, while its topological charge can be characterized by phase transitions of the averaged interference pattern, which can be detected experimentally. The method implemented in our experiment will motivate the use of cheap broadband partially coherent sources for generating STOV. We believe that this partially coherent STOV will find new applications soon because it can be generated conveniently at a low cost.
Acknowledgments
The authors thank Monish Chatterjee and Julie Motz for providing the RF spectrum analyzer for the experiments.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.
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