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Abstract
Novel forms of light beams carrying orbital angular momentum (OAM) have recently gained interest, especially due to some of their intriguing propagation features. Here, we experimentally demonstrate the generation of near-diffraction-free two-dimensional (2D) space-time (ST) OAM wave packets (ℓ = +1, +2, or +3) with variable group velocities in free space by coherently combining multiple frequency comb lines, each carrying a unique Bessel mode. Introducing a controllable specific correlation between temporal frequencies and spatial frequencies of these Bessel modes, we experimentally generate and detect near-diffraction-free OAM wave packets with high mode purities (>86%). Moreover, the group velocity can be controlled from 0.9933c to 1.0069c (c is the speed of light in vacuum). These ST OAM wave packets might find applications in imaging, nonlinear optics, and optical communications. In addition, our approach might also provide some insights for generating other interesting ST beams.
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1. Introduction
Structured light has gained growing interest, partially due to the unique spatial distribution of its amplitude and phase [1–4]. One interesting form of structured light are pulses of light that carry orbital-angular-momentum (OAM), which might be valuable to applications that make use of high-speed dynamics [5,6].
In general, an OAM pulse has a two-dimensional (2D) transverse spatial profile with a ring-like intensity distribution. In addition, its phasefront “twists” in a helical fashion as it propagates. The amount of OAM (ℓ) is the number of 2π phase shifts in the azimuthal direction [7,8]. Moreover, other properties that may benefit such pulses are: (a) the near-diffraction-free performance if power density is desired [9,10]; (b) the near-dispersion-free performance to maintain the pulse-shape integrity [11,12]; (c) the control of the pulse’s group velocity (e.g., time delay) to make it tunable in the superluminal and subluminal domains [13].
Previously, OAM pulses have been experimentally demonstrated by shining a pulsed beam on a spatial phase modulator (SLM) [14]. A specific phase pattern on the SLM is designed to reshape the incoming pulsed beam into a hypergeometric-Gaussian (HyGGℓ) mode whose complex amplitude is proportional to the degenerate (confluent) hypergeometric function [15]. Moreover, a subluminal group velocity was achieved and analyzed for different ℓ values. However, due to the divergence of LG modes, such OAM pulses are not diffraction-free or dispersion-free.
One potential approach to generate near-diffraction-free pulses relies on establishing a correlation between the spatial and temporal degree of freedoms of wave packets [16–20]. Moreover, by varying such a space-time correlation in this approach, wave packets with different group velocities might also be potentially achieved [21,22]. Based on this approach, there have been some experimental demonstrations of different optical wave packets including: (a) diffraction-free light sheets with one-dimensional (1D) transverse spatial profiles and controllable group velocities in free space [21,22]; (b) diffraction-free 2D non-OAM Bessel-based wave packets with superluminal or subluminal group velocities in free space or plasma [23–29].
Recently, a simulation paper [30] showed that: (a) based on space-time correlation, a near-diffraction-free and near-dispersion-free 2D OAM wave packet can be generated by a coherent combination of multiple OAM Bessel beams, each beam having the same ℓ value but a different temporal frequency; (b) by varying the relation between the spatial frequency, ${k_r}$ and the temporal frequency, ${f_n}$, both superluminal and subluminal group velocities could be achieved for such OAM wave packets; (c) the discrete frequency lines utilized in the simulation can be potentially achieved by an optical frequency comb, which contains a contiguous group of equally spaced optical frequency lines. Therefore, a laudable goal would be to develop a technique to experimentally generate such 2D diffraction-free OAM wave packets with controllable group velocities in free space using frequency comb [31].
In this article, we experimentally demonstrate this recent concept by generating near-diffraction-free OAM wave packets (ℓ = +1, +2, or +3) with 2D transverse spatial profiles having a controllable group velocity from 0.9933c to 1.0069c (c is the speed of light in vacuum). A diffraction grating and SLMs are employed to establish a controlled relationship between the temporal frequencies and spatial frequencies of the Bessel beams. By uploading different sets of phase patterns on the SLMs, this controlled relationship can be tuned, allowing for the control of the group velocity (${v_g}$) of the generated OAM wave packet from 0.9933c to 1.0069c. In addition, the high modal purity of pulses is maintained (91.31%, 89.59%, and 86.58% for OAM charges of +1, +2, and +3, respectively at ${v_g}$=1.0069c. Investigation of the diffraction performance of these spatiotemporal (ST) OAM wave packets verifies their near-diffraction-free properties.
2. Concept for the generation of near-diffraction-free OAM wave packets with a controllable group velocity
The basic concept for the generation of near-diffraction-free OAM wave packets with a controllable group velocity is shown in Fig. 1. In general, a monochromatic Bessel beam in free space with a specific $({f,\; {k_r}} )$ pair can be represented by a single point on the surface of the light-cone $k_z^2 + k_r^2 = {({2\pi f/c} )^2}$, as shown in Fig. 1. In order to generate an OAM wave packet, multiple frequency comb lines might be potentially utilized for a coherent combination, each carrying the same topological charge ℓ, and its associate spatial frequency, ${k_r}$. Such a coherent combination can be represented by
Fig. 1. Concept for the generation of near-diffraction-free OAM wave packets with a controllable group velocity (a) Multiple near-diffraction-free frequency lines are coherently combined, with each carrying a specific Bessel mode with the same ℓ but a unique ${k_r}$, deduced from a linear relation between f and ${k_z}$ based on ${k_z} = {k_0} + ({2\pi f/c - {\; }{k_0}} )/\textrm{tan}\theta $ (together with $\sqrt {k_{z,n}^2 + k_{r,n}^2} = 2\pi {f_n}/c$). As a result, a near-diffraction-free OAM wave packet is generated. In addition, its group velocity can be tuned by changing θ. (b) The ST spectrum of the OAM wave packet with a near-diffraction-free OAM wave packet with a subluminal group velocity. The simulated (c) transverse profiles at the pulse intensity peak and (d, e) intensity profiles ${|{E({x,y = 0,t} )} |^2}$ at two different z positions and for such an OAM wave packet. (f) The ST spectrum of the OAM wave packet with a near-diffraction-free OAM wave packet with a superluminal group velocity. (g, h) The simulated intensity profiles ${|{E({x,y = 0,t} )} |^2}$ at two different z positions and for such an OAM wave packet. In (b-h), six discrete frequency lines with a spacing of 192 GHz are utilized and coherently combined in the simulation model. The moving time frame (τ) is defined as τ = t - z/c.
Figure 1(b-e) and (f-h) present the results of two simulations of such near-diffraction-free OAM wave packets with superluminal and subluminal group velocities, respectively. In this simulation, six discrete frequency lines with a spacing of 192 GHz are utilized and coherently combined, each carrying a Bessel mode with the same charge (ℓ = 2) and but different $\{ {f_n},{\; }{k_{r,n}},{\; }n = 1, \ldots ,6\} $ pair chosen using the space-time relationship of Eq. (3). An example of a superluminal OAM wave packet is shown Fig. 1(b) for ${k_{z,{\; }0}} = 4.0140 \times {10^6}$ rad/m and θ = 45.2°. Figures 1(c, d) presents the case of $z\; $= 0. Since the initial relative phase delay Δφ between neighboring Bessel modes is selected as 0 at t = 0 and z = 0 in the simulation, the intensity peak of the generated OAM wave packets occurs at t = 0, when all 6 beam components interfere constructively. In the transverse phase profile, the resulting intensity distribution is doughnut-shaped, while the number of 2π phase shifts in the azimuthal direction around the phase profile equals the common OAM charge of 2, as shown in Fig. 1(c). Figures 1(e) present the case of $z\; = \; {z_1}\; > \; 0$. Based on the periodicity of the wave, the intensity and phase profiles at ${z_1}$ are the same as the ones in Fig. 1(c). According to Eq. (3), the envelop of the wave packet resumes its peak whenever $(\Delta {k_z}z - 2\pi \Delta ft) = 0$, which means $t = z/({c \cdot \textrm{tan}\theta } )$. Consequently, when $z = {z_1} > 0$ and θ = 45.2°, the intensity peak of the generated OAM wave packet occurs at $\boldsymbol{\tau } = t - ({{z_1} / c} )= {z_1}/({c \cdot \textrm{tan}\theta \; } )- {z_1} / c < 0$, which indicates its superluminal group velocity, as shown in Fig. (e). In contrast, for the case of the subluminal OAM wave packet, the value of θ is selected as 44.8° and for ${k_{z,{\; }0}}$ is $4.0422 \times {10^6}$ rad/m. When $z = {z_1} > 0$, the intensity peak of the generated OAM wave packet is at $\boldsymbol{\tau } > 0$, which indicates a subluminal group velocity, as shown in Fig. 1(h).
3. Experiment
3.1 Experimental setup of generating and detecting near-diffraction-free OAM wave packets with a controllable group velocity
A single-soliton Kerr frequency comb with a frequency spacing of ∼192 GHz is used, as shown in Fig. 2(b) [33]. Six frequency lines ranging 1555.95 nm to 1563.74 nm are first selected by a waveshaper and subsequently amplified by an erbium-doped fiber amplifier (EDFA). Figure 2(b) shows the optical spectra of the six frequency lines after the EDFA. After a collimator, the output Gaussian pulse (beam waist is ∼ 3.5 mm) is split into two branches by a beam splitter (BS). One is the reference pulse, which is sent to an autocorrelator. An initial relative phase delay of 0 between neighbouring frequency lines is realized by tuning their phases using the waveshaper to achieve a smallest pulse width on the autocorrelator [34]. Here, the smallest pulse duration we achieved is ∼ 1.06 ps. The other branch is sent to a diffraction grating with a ruling of 600 lines/mm and a 4f configuration for beam separation and beam demagnification. Such a grating has a pattern with parallel straight lines of grooves. Consequently, these six frequencies are spatially separated and locate in one row. By using one mirror, three of the six frequencies can be selected and directed to the top area of SLM 1. In addition, the horizontal and vertical positions of the other three frequencies can be controlled by another two mirrors and then directed to the bottom area of SLM 1. As a result, these six frequencies shine on different positions on SLM 1 with an array (2 rows × 3 columns) of phase patterns. The beam waist is ∼1.5 mm. Using the same approach in previous work [35], different phase patterns are designed and loaded for different incoming frequencies to generate the corresponding desired Bessel beams with specific ℓ and ${k_r}$ values. Therefore, each frequency line is assigned to a specific ${k_r}$ value based on the desired space-time correlation. Moreover, by tuning the phase patterns on the SLM 1, a programmable and controllable space-time correlation can be achieved for the generation of OAM wave packets with different group velocities. It should be noted that different grating patterns are also added on SLM 1 for these six incoming beams. By tuning each grating period and the direction, these reflected resulting beams are directed to the same position on SLM 2. Subsequently, the six resulting beams are coherently combined using SLM 2. Here, for each frequency, a designed grating pattern on SLM 2 could help to control the output direction of the light beam after SLM 2. Six different grating patterns can be designed to ensure the same output direction for all the six frequencies. Therefore, the final phase pattern of SLM 2 is designed based on a combination of these six grating patterns. Since all the six frequencies shine on the same position on SLM 2 and they have the same output direction, these six beams would become colinear after SLM 2 and the 4f imaging system. Figure 2(c) and (d) show an example of phase patterns on SLM 1 and 2 when θ = 45.2°. The six components of the ST OAM wave packet are imaged from the plane of the SLM 1 to an output plane by using a 4f system comprising of two lenses with the focal length of 1000 mm and 300 mm, respectively. Due to demagnification induced by the 4f system, the ST spectrum of the OAM wave packet is modified. Finally, an infrared camera is put at the output plane of the 4f system to detect the interferogram between the generated OAM wave packet and the reference pulse for off-axis digital holography.
Fig. 2. (a) Experimental setup for the generation and detection of near-diffraction-free OAM wave packets with a controllable group velocity. EDFA: erbium-doped fiber amplifier; PC: polarization controller; Col.: collimator; BS: beam splitter; G.: grating; SLM: spatial light modulator. (b) Optical spectrum of the selected six frequency lines after the waveshaper and the EDFA. (c) Phase pattern on SLM 1 for spatial modulation. (d) Phase pattern on SLM 2 for frequency combining. Here θ is selected to be 45.2° as an example.
3.2 Off-axis digital holography for 2D transverse complex profile measurements at different times
We utilize off-axis holography to measure the 2D transverse complex profile (i.e., amplitude and phase) of the generated ST OAM wave packets [36]. In our experimental setup, after a time delay stage, an off-axis reference Gaussian pulsed beam (beam waist ∼ 3.5 mm) with the same six frequencies is incident on the infrared camera with a tilted angle. Firstly, we record the off-axis interferogram between the large reference pulse and the ST OAM wave packet. Secondly, a 2D Fourier transform of the interferogram is performed to obtain the spatial frequency spectrum. Subsequently, the 1st-order diffraction is filtered out, and then shifted to the center of the spatial frequency spectrum. Finally, a 2D inverse Fourier transform of the shifted spatial frequency spectrum is performed, and the spatial amplitude and phase profiles of the ST OAM wave packet at a given $\Delta t$ is retrieved. By scanning the time delay stage, 2D transverse complex profiles (i.e., $E({x,y} )$) at different $\Delta t$ can be further retrieved.
3.3 Mode purity calculation
Since LG modes are a complete 2D orthogonal modal basis set, a structured beam can generally be decomposed into LG modes with different ℓ and p values. It should be noted that if the mode purity is analyzed by only considering the fundamental helical modes (i.e., $\textrm{exp}({i\ell\varphi} )$), the calculated coefficients would be dependent on radial coordinate. Therefore, a complete analysis of a structured beam can be obtained by decomposing it into a complete set of modes having a well-defined radial dependence [37]. Here, we can analyze the modal power coupling to other ℓ modes by decomposing the 2D transverse complex profiles of the ST OAM wave packet in the 2D LG modal basis using ${C_{\ell ,p}} = \mathrm{\int\!\!\!\int }E({x,y} )\cdot LG_{\ell ,p}^\ast ({x,y} )dxdy$., where $E({x,y} )$ and $L{G_{\ell ,p}}({x,y} )$ are the measured 2D complex fields of ST OAM wave packet and the theoretical complex field of an $L{G_{\ell ,p}}$ mode, respectively. In addition, both $E({x,y} )$ and $L{G_{\ell ,p}}({x,y} )$ are normalized electrical fields. The ratio of optical power coupling to the $L{G_{l,p}}$ mode is given by ${|{{C_{\ell ,p}}} |^2}$. Subsequently, we calculate the mode purity of the generated ST OAM wave packet (ℓ = ℓ0) at Δt = 0 and z = 0 in three steps: (i) we first calculate the ${|{{C_{\ell ,p}}} |^2}$ using the integral at ℓ = ℓ0 when p value is varied from 0 to 50 and the beam waist of all the LG components is selected as a certain value ${\omega _{ST}}$; (ii) we then calculate the mode purity (M) of the ST OAM wave packet (ℓ = ℓ0) for this ${\omega _{ST}}$ using; (iii) we vary the ${\omega _{ST}}$ within a given range and then calculate M value based on the previous two steps. Finally, we can find the maximum M value, which is considered as the mode purity for the ST OAM wave packet. Here, Bessel modes are not selected to analyze the mode purity. This is because Bessel modes are characterized by (ℓ, kr) pair, in which kr is a continuous parameter. Therefore, Bessel modes with the same ℓ but different kr values might not be orthogonal with each other. As a result, if the generated OAM wave packet is decomposed into different Bessel modes, the calculated mode purity might be larger than 100% after summing all the Bessel components.
3.4 Group velocity measurements
First, by tuning the time delay stage within a certain range, the intensity profiles ${|{E({x = 0,y,\; t} )} |^2}$ with $\Delta t$ at $z\; = \; 0$ for ST OAM wave packets can be measured based on the off-axis holography. When the camera is moved in the z direction and a distance Δz is introduced to the generated ST OAM wave packet, the intensity profiles can be retrieved again using the same tuning range on the time delay stage. Here, the reference pulse is assumed to be luminal [21]. If the group velocity of ST OAM wave packet is vg that is different from c, there would be time delay shift of the intensity peak (${t_s}$) compared with the reference point at $z\; = \; 0$. In the superluminal case (${v_g} > c$), the intensity peak of the ST OAM wave packet would be shifted to the left of the reference point, which corresponds to the case of Fig. 1(b). In the subluminal case (${v_g} < c$), the intensity peak of the ST OAM wave packet would be shifted to the right of the reference point, which corresponds to the case of Fig. 1(c). Based on measured Δz and ${t_s}$, when considering the tilted angle of the reference pulse ($\alpha $) in our scheme, the group velocity of the ST OAM wave packet can be calculated by ${v_g} = \; \; ({\Delta z \cdot \textrm{cos}\alpha } )/({\Delta z/c\; - \Delta {t_s}\; } )$.
4. Results and discussion
4.1 Measured profiles of 2D ST OAM wave packets with different OAM orders
First, we experimentally generate three different ST OAM wave packets with different OAM orders (ℓ = +1, +2, or +3). As an example, the value of θ is selected as 45.2°, and all the Bessel components have the same amplitude and phase for the pulse generation. The designed ST spectrum of such OAM wave packets is shown in Fig. 3(a). It should be noted that the ST spectra (i.e., the correlation between f and ${k_r}$) of the OAM wave packets are the same for different OAM orders. Subsequently, we measure the ST profiles of the generated OAM wave packets based on the interferometric setup using a reference pulse with a large beam waist. By tuning the time delay between the resulting OAM wave packets and reference pulse, the profiles at different times can be detected. Figure 3(b1-d1) presents measured intensity profiles ${|{E({x,y = 0,\; z = 0,\; t} )} |^2}$ with the reference time delay ($\Delta t$) for ST OAM wave packets with OAM order of +1, +2, and +3, respectively. Since the frequency spacing between the neighboring frequency lines is ∼192 GHz, the time period of the OAM wave packets is ∼5.2 ps. It should be noted that since the reference pulse is luminal, the reference time delay ($\Delta t$) in Fig. 3 should be the same as the moving time frame (τ) in Fig. 1. By tuning the time delay stage, a central pulse peak can be found, and it is then considered as the point of $\Delta t = 0$ here. The low intensity level at transverse position at $x = y = 0$ is due to the characteristic ring shape of OAM beams. In addition, the transverse profiles at $\Delta t = 0$ (i.e., the position of the pulse intensity peak) are also presented in Fig. 3(b2-d2) for OAM +1, OAM +2, and OAM +3 wave packets, respectively. The number of 2π phase shifts in the azimuthal direction of their phase profiles verifies their corresponding OAM orders. Moreover, the modal purities of these three OAM wave packets at $\Delta t = 0$ are 91.31%, 89.59%, and 86.58%, respectively,
Fig. 3. (a) The designed spatio-temporal spectrum of the OAM wave packets at θ = 45.2°. The measured intensity profiles ${|{E({x,y = 0,z = 0,t} )} |^2}$ with reference time delay (Δt) and the 2D transverse profiles at Δt = 0 for (b) OAM +1, (c) OAM +2, and (d) OAM +3 wave packets.
4.2 Controlling the group velocity of an ST OAM wave packet
We utilized the interferometric setup to measure the group velocity (vg) of the generated OAM wave packet. First, by tuning the time delay stage within a given range in the reference branch, we measured the intensity profiles of the ST OAM wave packet with $\Delta t$ at $z = 0$. Here, we take the central intensity peak (i.e., $\Delta t = 0$) of the ST OAM wave packet as the reference point for the measurement of the group velocity. Subsequently, the camera is moved to a different z position ($z > 0$) and the intensity profiles of the ST OAM wave packet with Δt is measured again by tuning the time delay stage within the same range. Since the reference pulse and the OAM wave packet have different group velocities, there would be an intensity peak shift between the two pulse envelopes. As mentioned before, the intensity peak of the ST OAM wave packet would be shifted to the left and right of the reference point for superluminal and subluminal cases, respectively. Consequently, the group velocity of the OAM wave packet can be calculated based on Δz and time delay shift ($\Delta {t_s}$). As an example, Fig. 4(a-b) present the measured intensity profiles ${|{E({x,y = 0,\; t} )} |^2}$ of the ST OAM wave packet (ℓ = +1) at $z = 0$ and 20 mm when θ = 45.2°. The value of the $\Delta {t_s}$ is measured to be ∼ 0.616 ps, corresponding to vg = 1.0069c, which is in good agreement with the expected value of vg = 1.007c. We also experimentally generated and detected such ST OAM wave packets with other θ values to control the group velocities. Table 1 shows theoretical and measured group velocities of OAM wave packets for different θ values. We see that experimental values are close to theoretical ones (vg = c·tanθ). In addition, the disagreement might be due to the imperfect generation of Bessel beams. Therefore, based on our scheme, the group velocity can be tuned from 0.9933c to 1.0069c by varying θ and the ST spectrum.
Fig. 4. (a) The measured intensity profiles ${|{E({x,y = 0,t} )} |^2}$ with Δt at (a1) z = 0 and (b) z = 20 mm when θ = 45.2° for the OAM +1 wave packet.
Table 1. The theoretical and measured group velocities of OAM wave packets for different θ values.
4.3 Near-diffraction-free properties of the ST OAM wave packet
We also investigate the near-diffraction-free properties of the ST OAM wave packet. With the reference pulse turned-off, the camera could capture the time-averaged intensity profile ($I({x,z} )= 1/T\int\nolimits_0^T {{{|{E({x,z,t} )} |}^2}dt}$) for ST OAM wave packets, and such profiles at different z positions can be captured by scanning the camera in the z direction. As an example, experimental time-averaged intensity profiles for ST OAM wave packets (ℓ = +1, +2, or +3) at different z positions when θ = 45.2° are presented in Fig. 5(a). Here, zR, defined as the distance over which the radius of maximum intensity r (Imax) of the ring is increased by a factor of $\sqrt 2 $, is used to characterize the diffraction properties. We note that the zR values for ST OAM +1, OAM +2, and OAM +3 wave packets are 56, 56, and 44 mm, respectively. It is observed that the zR value for OAM +3 wave packet is smaller. This might be due to that for the same kr value, the Bessel modes with a higher OAM order tends to have larger inner rings. In our experiment, specific phase patterns are used to generate the desired Bessel modes. Therefore, with the same limited-size aperture, the higher the OAM order is, the less rings the phase pattern tends to have. As a result, the effects of limited-size aperture induced by SLM 1 to its diffraction-free performance of the generated Bessel modes with higher OAM orders would be more significant. Consequently, it is expected that as the OAM order of the ST wave packet increases, the zR value might be decreased. Moreover, we validate the measured results by conducting numerical simulation. In the simulation model, all the six frequencies and their corresponding ${k_r}$ values that are used for wave packet generation are the same as in the experimental case. In addition, replicating the experimental demonstration, a limited-size aperture (the aperture radius is 1.38 mm) is added to all Bessel beams. By coherently combining these six Bessel beams in the model, the ST OAM wave packet is generated, whose propagation is simulated based on Kirchhoff–Fresnel diffraction integral [38,39]. It is found that the simulated intensity profiles and the zR values are close to the experimental ones. For comparison, we also generate the OAM wave packets by coherently combining these six frequencies, each carrying a Laguerre Gaussian (LG) mode with the same ℓ and p (p = 0) values. Here, the r (Imax) values of such LG-based OAM wave packets are the same as those of the ST OAM wave packets. We observe that the ST OAM wave packets based on Bessel modes have larger zR values, compared with LG-based OAM wave packets, which might be due to the near-diffraction-free properties of the Bessel modes that are used for generating ST OAM wave packets. Furthermore, the effect of the transmitter aperture size on the diffraction is shown in Fig. 6. Such a limited-size aperture is performed by adding a circular shape to each of the six phase patterns on SLM 1. The radii of the circular apertures on SLM 1 are defined as the aperture size here. We see that as the aperture size decreases, zR also becomes smaller. This is due to that an ideal Bessel beam needs to have infinite extent in the transverse direction and requires an infinite amount of energy. Here, each Bessel beam is generated by transmitting a Gaussian beam through a SLM, which can be considered as an approximation to the ideal Bessel beams (i.e., Bessel-Gaussian beam). As a result, the propagation-invariant distance of such a Bessel-Gaussian beam is related to the transmitter aperture size [40,41].
Fig. 5. The experimental and simulated time averaged intensity profile $\left( {I({x,y} )= 1/T\mathop \int \nolimits_0^T {{|{E({x,z,t} )} |}^2}dt} \right)$ at different z positions for OAM wave packets. (a) Experimental time averaged intensity profiles for ST OAM wave packets (ℓ = +1, +2, or +3) at different z positions when θ = 45.2°. (b) Corresponding simulated time averaged intensity profiles. (c) Experimental time averaged intensity profiles for OAM wave packets (ℓ = +1, +2, or +3) generated by LG modes at different z positions. zR is defined as the distance over which the radius of maximum intensity r (Imax) of the ring is increased by a factor of $\sqrt 2 $. Here, the aperture radius is 1.38 mm.
Fig. 6. The effect of the transmitter aperture size on the diffraction of the ST OAM wave packet when θ = 45.2°. zR values with various aperture sizes for different ST OAM wave packets (ℓ = +1, +2, or +3).
5. Conclusion
We have experimentally generated and detected near-diffraction-free 2D ST OAM wave packets (ℓ = 1, 2, or 3) having a controllable group velocity from 0.9933c to 1.0069c. By using a diffraction grating and SLMs to build up specific space-time correlations, 2D OAM wave packets with different ℓ values and group velocities can be generated. Moreover, the diffraction performance of such ST OAM wave packets is also investigated, which verifies its near-diffraction-free properties. Such interesting ST OAM wave packets may find applications in imaging [42], nonlinear optics [43,44], optical communications [45]. In addition, our approach and scheme might also provide some insights for generating other various interesting ST beams.
It is worth mentioning the following points: (i) limitations on controlling group velocities. The maximum superluminal and minimum subluminal group velocities that can be achieved are limited by the pixel size of the SLM. This is because when the difference between θ and 45° increases, the theoretical ${k_r}$ values for the six frequencies become larger based on our designed space-time correlation and the formula. Moreover, for a given Bessel beam, its central core spot size (r0) is inversely proportional to the ${k_r}$ value. Consequently, if ${k_r}$ value is large, r0 will become very small and there will be less pixels that could be used to control the beam profile. In this case, the mode purity of each Bessel beam might be limited; (ii) different frequency spacing. It should be noted that a smaller frequency spacing of the comb lines might induce a larger controlling range of the group velocity. This is due to that the limitation in (i) is mainly determined by the largest ${k_r}$ value among the six frequencies. For a given ${k_{z,0}}$, when the frequency range is smaller, the largest ${k_r}$ value would also become smaller. However, with a smaller frequency spacing, it might be harder to spatially separate these frequencies by a diffraction grating; (iii) luminal group velocity. In our approach, when θ = 45°, the r0 value for each frequency would become infinite. This means we may need an infinitely large SLM to generate the desired Bessel beam. Therefore, a perfect luminal case might not be achieved based on our scheme; and (iv) comb linewidth. The linewidth of our optical Kerr frequency comb is ∼100 kHz, which might induce little spectral uncertainty in the space-time correlation.
Funding
Defense University Research Instrumentation Program (DURIP) (FA9550-20-1-0152); Qualcomm Innovation Fellowship (QIF); Vannevar Bush Faculty Fellowship sponsored by the Basic Research Office of the Assistant Secretary of Defense (ASD) for Research and Engineering (R&E) and funded by the Office of Naval Research (ONR) (N00014-16-1-2813); Office of Naval Research through a MURI award N00014-20-1-2789.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. The MATLAB codes used for the simulations in this study are not publicly available at this time but may be obtained from the authors upon reasonable request.
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