Spatiotemporal evolutions of ultrashort vortex pulses generated by spiral multi-pinhole plate

Li Ma; Pan Zhang; Zhenhua Li; Chunxiang Liu; Xing Li; Yuqin Zhang; Ruirui Zhang; Chuanfu Cheng

doi:10.1364/OE.25.029864

1. Introduction

An optical vortex with a phase singularity possesses spiral wavefronts that give rise to an orbital angular momentum (OAM) around the propagation direction [1,2]. The characteristic phase distribution of the optical vortices is represented as exp(ilθ), where θ is the azimuthal angle and the integer l is known as vortex order or topological charge [3]. Owing to their unique properties, optical vortices are being applied in free-space communications [4,5], micromanipulations [6,7], quantum optical information processing [8,9], high-resolution microscopy [10,11], etc. The study of optical vortices, both quantized and classical, is a rich and diverse topic of fundamental and applied research [12–14]. Recently, the ultrashort optical vortex, which introduces a vortex into an ultrashort pulse, has attracted great interest for research. The ultrashort vortex pulses with both broad spectral width and high peak power may serve as a useful tool in the studies of ultrafast nonlinear spectroscopy [15,16] and high-intensity field physics [17].

One of the most fundamental fields of research in the ultrashort optical vortex is its generation process. Compared with that in a monochromatic regime, the challenge in generating an ultrashort optical vortex is to ensure that all the spectrum components are well embedded with their phase singularities. Nevertheless, many modified methods used for generating monochromatic optical vortices are still applicable in the ultrashort domain, such as cylindrical lenses [18,19], spiral phase plates [20,21], or computer-generated holograms (CGH) [22,23]. In such processes, the spiral phase plates or spatial light modulators are often used for imposing phase dislocations on all the spectral components of the ultrashort pulse.-Another alternative to generate optical vortices is the multi-pinhole plate which combines the interference and diffraction effects of the multiple beams from pinholes. Such plate has been proved to be a convenient and feasible way for reading topological charges of vortices [24] and producing optical lattices [25]. As an optical device, pinhole plate can be easily fabricated and has a high damage threshold. It is free of the medium-induced dispersion and can avoid the corresponding temporal broadening of ultrafast laser pulse. Further, the generated vortex field is of satisfying the quality with a low amplitude trough in vortex core. The multi-pinhole plate can optionally be employed to study the spatiotemporal characterizations of the generated ultrashort vortex pulses.

A significant issue in the development of ultrashort science is to characterize the ultrashort vortex in both space and time, simultaneously. In most practical cases, the broadband spectral width of the short pulse leads to spatio-temporal couplings where the temporal and spatial properties cannot be separated, e.g., $E (x, y, t) \neq f (t) g (x, y)$ [26,27]. Some studies have proposed methods to identify the spatiotemporal characteristics of this optical field, such as spatial field reconstruction [28,29] and Poynting vector maps analysis with Shack–Hartmann wavefront sensors [30,31]. Based on these methods, many studies on optical vortices in ultrashort pulses have been reported in the literature [21,32–36]. However, the spatiotemporal evolutions of phase, OAM, and energy current of ultrashort vortices are seldom mentioned.

In this study, a multipoint plate with spirally-arranged pinholes is employed as an optical modulator to generate ultrashort vortex pulses. Based on the field reconstruction, the spatiotemporal evolutions of intensity and phase are studied in a Mach–Zehnder-type interferometer. In addition, we theoretically analyze the characterizations of ultrashort vortices and numerically calculate their variations in the femtosecond regime. As the numerical simulations show high coincidence with the experimental results, we further discuss the distributions and evolutions of both OAM and energy current in the pulse duration.

2. Experiment

Figure 1 depicts our experimental setup where the main part is a Mach–Zehnder-type interferometer. The initial ultrashort pulse is generated by a Ti:Sapphire oscillator (central wavelength of 800 nm, repetition rate of 75 MHz, pulse duration of 12 fs, and average power of 500 mW). Spectral-phase interferometry for direct electric-field reconstruction (SPIDER) is employed to measure the amplitude and phase of the input beam. The laser pulse is split into two pulses of the reference beam and the object beam in separate arms by a beam splitter (BS1). In the object beam, a homemade multi-pinhole plate with spiral arrangement pinholes acts as a vortex beam shaper coaxially embedded. The pinhole plate is fabricated by a focused femtosecond laser (Spitfire, Spectral Physics, wavelength 800 nm, 1 KHz) with punching pinholes with radius of approximately $ρ_{0} = 26 μ m$ on aluminum foils. As the inset shows the multi-pinhole plate in Fig. 1, pinholes with the number of N = 36 and $ρ_{1} = 0.5 c m$ are arranged along a spiral curve around the plate origin, with constant azimuth angle increment and varied radial distances. Such pinhole arrangement ensures the generation of ultrashort vortex pulses on the CCD camera (Cascade 1K, pixel size 8 μm × 8 μm, 1004 × 1002 pixels) at distance of z = 50 cm (see Ref [37]. for details). In the reference beam, a piezoelectric nano-translationstage (PI E-516) with precision of 10 nm is used to scan the time delay between the arms. The beam splitter BS3 recombines the two parts collinearly at the CCD plane where the interferogram is recorded. In the interferometer, the object pulse and the reference pulse pass through the same transmissive elements and the dispersion is carefully balanced. By controlling the stepping of the piezoelectric stage to scan the time delay, a series of interference patterns with equal time intervals are recorded, essentially corresponding to spatially resolved linear cross-correlations [28]. The key idea is to use a known reference pulse that can be measured by SPIDER to standardize the object pulse. Based on the time domain Fourier spectroscopy, the interferograms provide the same information as the spectral interferometry [28,29].

Fig. 1 Schematic diagram of the experimental setup. M1–2 are reflecting mirrors, BS1–3 are beam splitters, and the time delay device is composed of piezoelectric stage and mirrors. The multi-pinhole plate consists of N = 36 pinholes with pinhole radius of approximately $ρ_{0} = 26 μ m$ , radius of first pinhole to the plane origin of $ρ_{1} = 0.5 cm$ , and the designed propagation distance is z = 50 cm. The light waves are measured and recorded by SPIDER and CCD, respectively.

Download Full Size | PDF

Figure 2 shows the experimental patterns. The normalized intensity distribution (red curve) and the phase distribution (blue curve) of the incident ultrashort pulse are measured by the SPIDER shown in Fig. 2(a), and the inset shows the pulse width about 12 fs. The pulse covers a wavelength range from 680 nm to 920 nm, which is not an exact Gaussian pulse and has a variable phase distribution with wavelength. The reference pulse is spatially filtered by an iris and a small portion in the beam center covers the interaction region. Moreover, this region is much smaller than the area of the selected reference beam. For convenience, we assume that the reference wave is homogeneous across the interference area, although this assumption would bring about deviations from the practical measurement. The object wave and its interference pattern on the CCD camera are shown in Figs. 2(b) and 2(c), respectively. Figure 2(b) exhibits a typical optical vortex, where a dark spot is surrounded by a ring-like intensity profile, and the fork pattern in Fig. 2(c) confirms that this vortex is in first-order topological charge. Because the spectral amplitude and phase of the reference beam are obtained with SPIDER, the Fourier transform of the interference patterns can characterize the object pulses in the spectral domain. With the inverse Fourier transform of these spectrally resolved pulses, the corresponding complex field in time domain can be obtained. In this way, both the intensity and phase distributions of the generated vortex pulses are retrieved from the interferograms.

Fig. 2 (a) Information of the reference pulse contains the intensity and phase distributions along wavelength, and the inset shows the pulse width; (b) and (c) are the experimental results of the vortex intensity pattern and the interferogram at the observation plane, respectively.

Download Full Size | PDF

3. Theory

In Fig. 3, the diagram shows the propagation of the ultrashort pulse after the pinhole plate. An ultrashort pulse with central wavelength $λ$ perpendicular to the coordinate plane z = 0 is incident on the interface, which is composed of spiral arrangement pinholes. The ultrashort laser pulse in the time domain can be assumed to be of a Gaussian shape [38, 39]

U_{0} (t)=exp[-(t / τ)^{2}]exp(i ω_{0} t),

where

τ

is the full width at the half-maximum of the pulse and

ω_{0}

denotes the central frequency. The corresponding spectral domain

U_{0} (ω)

of the pulse can be simply obtainedfrom the Fourier transform of

U_{0} (t)

,

U_{0} (ω)= τ \sqrt{π} exp[- τ^{2} {(ω - ω_{0})}^{2} / 4] .

In Eq. (2), we notice that the variable is ω = 2πν, with ν the frequency of the light wave. Behind the multi-pinhole plate, the field can be written as

U_{1} (ω)= τ \sqrt{π} exp[- τ^{2} {(ω - ω_{0})}^{2} / 4] U_{s} (ω, ρ),

where

U_{s} (ω, ρ)

is the transmittance function of vector

ρ

and frequency

ω

. This plate consists of N pinholes arranged along a spiral curve with constant azimuth angle increment

α_{n} = 2 π l / N

and variable radial distances

L_{n} - L_{n - 1} = l λ / N

. When passing through the pinhole plate, the waves will carry a phase variation from the plate to the observation plane origin. We take the central frequency as an example to discuss the phase modulating effect of the spiral multi-pinhole plate. The transmitted wave

U_{s} (ω_{0}, ρ)

immediately behind the scattering plane can be written as

U_{s} (ω_{0}, ρ)= \sum_{n = 1}^{N} δ (ρ - ρ_{n}) A_{n} \exp {i ϕ_{n}},

where

A_{n}

is unit amplitude. Because the pinhole diameters are constant and sufficiently small compared with the diffraction distance, we consider the transmittance function of the plate as a delta function

δ (ρ - ρ_{n})

with an additional phase

ϕ_{n}

, where the subscript n denotes the nth pinhole. The waves transmitted from each pinhole will be of superposition and interference at any point, and an optical vortex with topological charge l will be formed at the observation plane [37]:

U_{1} (ω_{0})= \sum_{m, n = 1}^{N} P_{m n} (r) A_{m} A_{n} \exp [i (ϕ_{m} - ϕ_{n})] = T (ω_{0}) exp[i l α],

where

P_{m n} (r) = \int δ (r - ρ_{m}) \times δ (r - ρ_{n} - ρ) d r

is the correlation of the pinholes,

T (ω_{0})

is the amplitude of the wave field, and α is the azimuth angle. Considering other frequencies of the ultrashort pulse, each frequency component is associated with the modulating effects of the pinhole plate and has a different value of l with the following linear relationship [27]:

l (ω) = l_{0} + \frac{Δ l}{Δ ω} (ω - ω_{0}),

where l₀ is the charge value for the central frequency, ∆ω is the spectral width of the pulse, and ∆l is the variation of l over this width. When all the frequencies are superimposed on the observation plane, light is localized in an azimuthal wave packet around a dark light point, independent of time [27,40]. Their relative phases, varying linearly owing to the relationship of Eq. (4), result in a rotation of the azimuthal wave packet in time. Therefore, the helical ultrashort pulse can form a vortex in space and time.

Fig. 3 Sketch of ultrashort vortex pulse propagation. An ultrashort pulse impinges normally on the multi-pinhole plate $P (ρ)$ . When arriving at the observation plane $P (r)$ at z point, the vortex pulse can be theoretically decomposed as time independent parts.

Download Full Size | PDF

The wave field $U (t, r)$ at an arbitrary point of the observation plane can be given by the integral of all the frequencies’ transmittance functions

\begin{array}{l} U (t, r)= \frac{τ \sqrt{π}}{i λ z} \int_{}^{} \int_{- \infty}^{\infty} exp[- τ^{2} {(ω - ω_{0})}^{2} / 4] U_{s} (ω, ρ) \times exp[i ω (t - L / c)] d ρ d ω \\ \propto T (t, r) \exp [i φ (t, r)], \end{array}

where

L = {[z^{2} + {(r - ρ)}^{2}]}^{1 / 2}

is the length from the point ρ of the scattering plane to the point r of the observation plane. We have used the substitute formula

2 π / λ = ω / c

in Eq. (7), and the length L is approximately z, except in the exponential form. Thus, the spatiotemporal evolutions of the ultrashort vortices can be characterized.

4. Results and discussions

We perform the numerical simulations of the spatiotemporal evolutions of the ultrashort vortices according to the theoretical discussion. Comparing the simulated results with the experimental data, we discuss the evolutions of both intensity and phase distributions in the pulse duration, and the characteristics of OAM and energy current are further analyzed in space and time.

4.1 Spatiotemporal evolutions of the intensity and phase distributions of the ultrashort vortex pulses over time

With the time varying from −24 fs to 24 fs with a step of 6 fs, the evolutions of both the normalized intensity and phase of the ultrashort vortex pulses in simulation and experiment are shown in Fig. 4. The simulated results are arranged above the time axis and the corresponding experimental results are displayed below. From both the simulations and the experimental results, we could see that the bright ring-like intensity profiles are asymmetric, especially in the latter case. This is a reasonable result of the asymmetric spiral arrangement of the pinholes on the multipoint plate. Moreover, the vortices are the most powerful at the time t = 0 fs and decrease gradually when time goes forwards or backwards. The Gaussian distribution of the ultrashort pulse power in the time domain causes such decline, as can be deduced from the inset of Fig. 2(a). However, the power of the vortices at t = ± 24 fs is still considerable in the experiment, while that in the simulation vanishes. Two main reasons lead to such deviation between the two cases. First, the ultrashort pulse does not exactly obey the Gaussian distribution. Second, the assumption that the reference pulse is approximated to homogenous also introduces a deviation in measurement.

Fig. 4 Evolutions of the intensity and phase distributions of the ultrashort vortex pulses over time. Time scale varies from −24 fs to 24 fs with a step of 6 fs. The patterns above the time axis are the simulated results, and the corresponding experimental results are displayed below.

Download Full Size | PDF

The phase evolution in the experiment coincides with that of the simulation. In Fig. 4, the phase contour lines of each phase pattern have as common features that they increase clockwise around their own singularities. Furthermore, the phase distributions rotate clockwise around the propagation axis with time. Although the phase distributions in the experiment are non-uniformly variable over time, their variations still maintain the evolutionary trend of the clockwise rotation. In summary, the spatiotemporal evolution of the intensity in the pulse duration is close to the Gaussian distribution, and the phase distribution rotates clockwise over time.

4.2 Spatiotemporal evolutions of the OAM and energy current of the ultrashort vortex pulses over time

For an intuitive analysis of the OAM, we start from the definition of classical OAM of particles with respect to the vortex core. The z-component of OAM is written as

L_{z} (r)= r_{c} \times p (r) \propto r_{c} \times \nabla φ (r),

where k is the wave vector,

φ (r)

is the phase distribution at point r and r_c is the in-plane position vector of the point with respect to the vortex core. In the above, we have directly used the basic expression

p (r)= i ℏ k \propto \nabla φ (r)

of momentum for a photon in quantum theory. In the classical theory of electromagnetic waves, the energy current can be expressed as

J (r) = I (r) \nabla φ (r),

where

I (r)

is the intensity distribution. Further, the classical momentum density is proportional to energy current

J (r)

, and both

J (r)

and intensity

I (r)

are proportional to the photon number. Then

L_{z} (r)

of a photon defined in Eq. (8) is essentially the average of the classical momentum density of the light waves.

Figure 5 shows the in-plane OAM distributions and the OAM variations over time, including the simulated results in the upper figures and the corresponding experimental results in the lower ones. The backgrounds of Figs. 5(a) and 5(c) are the phase distributions of the ultrashort vortices when t = 0 fs. The vector arrow plots represent the direction, and the length shows the size of the single OAM. Owing to the clockwise increasing of the phases, the arrows in both Figs. 5(a) and 5(c) are distributed in a clockwise rotation around the vortex cores. The longer arrows around the central areas indicate that the OAM carried by single photons here are larger than that of the outside area, except the disordered parts at margin.

Fig. 5 (a) and (c) are the in-plane OAM distributions at t = 0fs in simulation and experiment, respectively. (b) and (d) are the corresponding evolutions of OAM distributions over time at the points (r = 80 μm, θ = 0, π/2, π, 3π/2) marked by the white circles and colored shapes in (a) and (c), respectively.

Download Full Size | PDF

To clearly show the temporal evolution of the OAM, we analyze the OAMs at four observation points along the white circles with radius of 80 μm around the vortex cores at the labeled azimuthal angles of 0, π/2, π, and 3π/2, and show their distributions versus time in Figs. 5(b) and 5(d), respectively. Because the vortex distribution is not strictly symmetric, the OAMs at different angles have different values. For the case of the simulation in Fig. 5(b), the OAMs at each of the observation points keep nearly unchanged along with the flow of time. However, the experimental case in Fig. 5(d) shows a wavy curve with minor fluctuations over time, mainly caused by the inevitable disturbances in the experiment. The tendencies of the OAM distributions in the experiment over time still appear stable, even though there are a few irregular points. Since the Eqs. (8) and (9) are obtained based on the classical wave theory of light except the photon momentum $p (r) = ℏ k$ , then the above discussions are also limited to the scope of the classical electromagnetic theory. However, from the viewpoint of quantum optics, every photon carries the information of the whole beam rather than parts of it, both in space and time. This is similar to the well-known Young's double slit experiment, the interference does not vanish even only one photon existing.

Figure 6 shows the in-plane energy current distributions and their variations over time, including the simulated results in the upper figures and the corresponding experimental results in the lower ones. The backgrounds of Figs. 6(a) and 6(c) are the intensity distributions of the ultrashort vortices at t = 0 fs. The vector arrow plots represent the direction and the length shows the size of the current. In a similar manner to the arrows in Figs. 5(a) and 5(c), those in Figs. 6(a) and 6(c) show a clockwise rotation around the vortex cores, and their values decrease with increasing radius.

Fig. 6 (a) and (c) are the in-plane energy current distributions at t = 0 fs in the simulation and experiment, respectively. (b) and (d) are the corresponding evolutions of energy current distributions over time at the points (r = 80 μm, θ = 0, π/2, π, 3π/2) marked by the black circles and colored shapes in (a) and (c), respectively.

Download Full Size | PDF

We also analyze the temporal evolution of the energy current at the same four observation points of Fig. 5. Because the intensity distribution of the pulse has a dominant effect on the distribution of energy current, all the energy currents at the four observation points in the simulation are the same, while they distribute obeying a Gaussian-like versus time. The curves for the experimental case in Fig. 6(d) show differences between the four observation points, which is due to the asymmetric intensity pattern. However, all the curves in Fig. 6(d) still fit the Gaussian-like distribution in the pulse duration.

It is obvious that the experimental results coincide with the theoretical simulation. The OAMs are maintained almost invariable over time, and the distribution of energy current in the pulse duration presents a Gaussian-like distribution.

5. Conclusion

In conclusion, we study the spatiotemporal characteristics of ultrashort optical vortices, involving the evolutions of intensity, phase, OAM, and energy current in the femtosecond regime. The ultrashort vortex pulses are generated by using a spiral multi-pinhole plate and measured by employing a Mach–Zehnder interferometer system. Based on the field reconstruction, ultrashort vortices in space and time are obtained in the experiment. According to theoretical analyses, their spatiotemporal evolutions are also numerically simulated, and they show a high coincidence with the experimental results. Both the simulation and experimental results demonstrate that the evolution of intensity in the pulse duration presents a Gaussian-like distribution, and the phase contour lines of such vortices rotate around the propagation axis over time. Furthermore, the OAM distribution in the cross-section of the vortex pulse is maintained almost invariable, and the energy current in the vortices follows a Gaussian-like distribution versus time. We hope this work could be helpful for generating vortex pulses in ultrafast optics and understanding their spatiotemporal properties.

Funding

National Natural Science Foundation of China (NSFC) (11574185, 11604183 and 11647015); Project of Shandong Province Higher Educational Science and Technology Program (J16LJ09 and J15LJ05).

References and links

1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).

2. P. Vaity and R. P. Singh, “Topological charge dependent propagation of optical vortices under quadratic phase transformation,” Opt. Lett. 37(8), 1301–1303 (2012). [PubMed]

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [PubMed]

4. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).

5. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).

6. A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).

7. M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).

8. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [PubMed]

9. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003). [PubMed]

10. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [PubMed]

11. J. Keller, A. Schönle, and S. W. Hell, “Efficient fluorescence inhibition patterns for RESOLFT microscopy,” Opt. Express 15(6), 3361–3371 (2007). [PubMed]

12. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics Publishing, 1999).

13. L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Institute of Physics Publishing, 2003).

14. M. J. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).

15. Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express 17(26), 24198–24207 (2009). [PubMed]

16. A. Matos-Abiague and J. Berakdar, “Photoinduced charge currents in mesoscopic rings,” Phys. Rev. Lett. 94(16), 166801 (2005). [PubMed]

17. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).

18. J. Courtial and M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre-Gaussian laser modes,” Opt. Commun. 159(1), 13–18 (1999).

19. Z. Qiao, L. Kong, G. Xie, Z. Qin, P. Yuan, L. Qian, X. Xu, J. Xu, and D. Fan, “Ultraclean femtosecond vortices from a tunable high-order transverse-mode femtosecond laser,” Opt. Lett. 42(13), 2547–2550 (2017). [PubMed]

20. K. J. Moh, X. C. Yuan, D. Y. Tang, W. C. Cheong, L.-S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, “Generation of femtosecond optical vortices using a single refractive optical element,” Appl. Phys. Lett. 88(9), 091103 (2006).

21. M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L’Huilier, and C. L. Arnold, “Spatiotemporal characterization of ultrashort optical vortex pulses,” J. Mod. Opt., https://arxiv.org/abs/1505.06081 (2016).

22. O. Martínez-Matos, J. A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares, “Generation of femtosecond paraxial beams with arbitrary spatial distribution,” Opt. Lett. 35(5), 652–654 (2010). [PubMed]

23. J. Atencia, M.-V. Collados, M. Quintanilla, J. Marín-Sáez, and I. J. Sola, “Holographic optical element to generate achromatic vortices,” Opt. Express 21(18), 21056–21061 (2013). [PubMed]

24. G. C. G. Berkhout and M. W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18(13), 13836–13841 (2010). [PubMed]

25. Z. Li and C. Cheng, “Generation of second-order vortex arrays with six-pinhole interferometers under plane wave illumination,” Appl. Opt. 53(8), 1629–1635 (2014). [PubMed]

26. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12(9), 093001 (2010).

27. G. Pariente and F. Quéré, “Spatio-temporal light springs: extended encoding of orbital angular momentum in ultrashort pulses,” Opt. Lett. 40(9), 2037–2040 (2015). [PubMed]

28. M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L’Huillier, and C. L. Arnold, “Spatiotemporal characterization of ultrashort laser pulses using spatially resolved Fourier transform spectrometry,” Opt. Lett. 39(17), 5142–5145 (2014). [PubMed]

29. G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space–time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).

30. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express 14(25), 11919–11924 (2006). [PubMed]

31. F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express 17(12), 9804–9809 (2009). [PubMed]

32. Y. Toda, K. Nagaoka, K. Shimatake, and R. Morita, “Generation and spatiotemporal evolution of optical vortices in femtosecond laser pulses,” Electr. Eng. Jpn. 167(4), 39–46 (2009).

33. K. Yamane, Z. Yang, Y. Toda, and R. Morita, “Frequency-resolved measurement of the orbital angular momentum spectrum of femtosecond ultra-broadband optical-vortex pulses based on field reconstruction,” New J. Phys. 16(5), 053020 (2014).

34. M. Bock, J. Jahns, and R. Grunwald, “Few-cycle high-contrast vortex pulses,” Opt. Lett. 37(18), 3804–3806 (2012). [PubMed]

35. R. Grunwald, T. Elsaesser, and M. Bock, “Spatio-temporal coherence mapping of few-cycle vortex pulses,” Sci. Rep. 4, 7148 (2014). [PubMed]

36. M. Bock, J. Brunne, A. Treffer, S. König, U. Wallrabe, and R. Grunwald, “Sub-3-cycle vortex pulses of tunable topological charge,” Opt. Lett. 38(18), 3642–3645 (2013). [PubMed]

37. Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21(13), 15755–15764 (2013). [PubMed]

38. M. Lefrançois and S. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express 11(10), 1114–1122 (2003). [PubMed]

39. A. Pack, M. Hietschold, and R. Wannemacher, “Propagation of femtosecond light pulses through near-field optical aperture probes,” Ultramicroscopy 92(3-4), 251–264 (2002). [PubMed]

40. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).

Spatiotemporal evolutions of ultrashort vortex pulses generated by spiral multi-pinhole plate

Abstract

1. Introduction

2. Experiment

3. Theory

4. Results and discussions

4.1 Spatiotemporal evolutions of the intensity and phase distributions of the ultrashort vortex pulses over time

4.2 Spatiotemporal evolutions of the OAM and energy current of the ultrashort vortex pulses over time

5. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (9)

Optics Express