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Abstract
Orbital angular momentum (OAM) is a potential tool for remote sensing applications since amplitude/phase distributions can be decomposed into an OAM basis for analysis. We demonstrate the generation of a spatially asymmetric perfect vortex (APV) basis based on a pulsed 2D HOBBIT (Higher Order Bessel Beams Integrated in Time) system using two acousto-optic deflectors and optical coordinate transformation optics. Results are demonstrated for numerous radii and OAM charges as high as 20, with switching speeds greater than 400 kHz. The spatial APV basis is used to design different types of pulse trains for amplitude object pattern recognition and phase object wavefront sensing. Experimental results of sensing are provided for an amplitude object and a phase object to demonstrate the feasibility of the spatial APV on remote sensing tasks.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser beams carrying orbital angular momentum (OAM) offer a chance to manipulate light with a new degree of freedom, which shows great potential to encode and decode information. OAM-carrying beams are generally described by a phase term, $exp (i\ell \theta )$, where $\ell $ is the so-called OAM charge number and $\theta $ is the azimuthal coordinate [1]. The infinite integer OAM basis constructs a Hilbert space which has many advantages in fields like optical communications [2] and quantum key distribution [3]. In remote sensing involving Doppler frequency measurements [4] and optical imaging such as ghost imaging [5], the OAM beams also show great potential to enhance the resolution and detection speed. Recently, studies on fractional OAM provides insight into even higher degrees of freedom for laser beam manipulation and increasing the light spectrum efficiency [6,7]. Studies combining OAM with other dimensions such as using radial modes of Laguerre Gaussian beams for imaging applications have also stimulated interest [8].
New efforts of exploring the possibilities to discover new types of beams carrying specific intensity and phase patterns have been made to enrich the structured light family. Spatial-temporal manipulation is a promising way to manipulate light beams. In [9,10], a system exploiting Higher Order Bessel Beams integrated in Time (HOBBIT) was used to generate time-varying OAM beams of two types (perfect vortices and Bessel beams), with switching speeds in the MHz range. The OAM beams generated by this HOBBIT system show a one-to-one correspondence of fractional OAM charges which is different from traditional methods of generating fractional OAM such as a phase plate with fractional phase step or an integer phase plate shifting away from the axis. The related OAM nonlinear study also shows the global OAM conservation of fractional OAM [7]. The HOBBIT generated OAM beams were used for probing atmospheric turbulence to find eigenchannels in turbulent medium by using a continuous OAM spectrum [6]. In [11,12], two types of beams generated by the HOBBIT were used to measure a rotating phase plate and a rotating dense fog, which shows the potential for measuring rotating structures. A perfect vortex (PV) is a specific kind of structured light whose OAM information will not affect the intensity distribution and this characteristic offers the chance to observe light-matter interaction only considering the effects of OAM. The limitation of a PV may be the lack of radial information and the relatively slow OAM mode switching speed.
In this paper, the scheme of a spatially asymmetric perfect vortex (APV) basis generation based on a pulsed 2D HOBBIT system is reported along with remote sensing experimental results. An acousto-optic deflector (AOD) and log-polar optics [13] are two essential elements in the system. The log-polar optics work as a coordinate transformer which turns a horizontal linear phase into an azimuthal phase. The two AODs in the system are used to control the beam size and OAM, respectively. The examples of 11 radial modes and OAM charge as high as 20 are shown experimentally. By using an AOD, a mode switching speed of 406.25 kHz can be realized for this specific equipment, but can be extended to the MHz range. Different types of APV modes are used to create pulse trains for remote sensing of an amplitude object and phase object. Experimental results sensing an amplitude object containing several letters and a phase object consisting of concentric phase plates (CPP) which contains different OAM information with different radii are provided. Both the amplitude object and phase distributions can be decomposed into the spatial APV mode spectrum. The results fit well with the prediction and show the ability of the 2D HOBBIT system to be used for remote sensing applications.
2. Method
2.1 Spatial APV basis generation
An OAM basis can be used as information carriers and shows great potential in applications such as optical communications and remote sensing. The Laguerre Gaussian ($L{G_{p,\ell }}$, where p is the radial index and $\ell $ denotes OAM) modal basis is the most studied OAM-carrying beam because of its high dimensionality in both radial and azimuthal directions. One possible consideration about LG beams is the mode switching speed of current generation schemes based on spatial light modulators (SLM) or digital micromirror devices (DMD) is limited from tens of Hz to several kHz. This limitation may have an impact on applications requiring high speed mode switching. In order to generate an OAM basis with a high mode switching rate, we adopt the HOBBIT concept which is capable of OAM mode switching rates in the range of MHz for remote sensing. The generated OAM modes are described as an APV. A PV is one special kind of OAM-carrying beam whose beam sizes are OAM-independent. Ideal PV beams with different sizes are orthogonal to each other. Considering real PV beams have a ring width, we use a group of APVs having discrete ring radii to emulate the radial degree of freedom of LG basis. We introduce another AOD to the HOBBIT system to realize the manipulation in both radial and azimuthal dimensions, which provides for scaling in both the radial and azimuthal dimensions. The spatial APV basis can be described by
The HOBBIT system was described in our former work [6,7,9,10] which shows the ability to arbitrarily generate APV and Bessel-Gauss beams carrying time-varying OAM. The 2D HOBBIT system is shown in Fig. 1(a). The source used in the system is a 517 nm femtosecond (242 fs) pulsed laser (Coherent, Monaco), with a laser repetition rate of 1 kHz. The first AOD works as a vertical beam deflector adding a vertical linear phase to the incident beam. A cylindrical lens ${F_2}$ (${F_2} = 500$ mm) performs a vertical Fourier transform to the beam and in the focal plane, the vertical linear phase of the beam is transferred to a vertical shift. The second AOD adds a horizontal linear phase to the beam which is imaged to the same plane as the Fourier plane of ${F_2}$ by two cylindrical lenses ${F_1}$ and ${F_3}$ (${F_1} = 50$ mm and ${F_3} = 250$ mm). The two cylindrical lenses reshape the incident Gaussian beam and after the three cylindrical lenses system, the beam becomes an elliptical Gaussian beam carrying a horizontal linear phase. The log-polar optics contain two phase elements, one to perform an optical geometric transformation which is located in the back Fourier plane of ${F_2}$, and one which is in the Fourier plane of the first element to correct the phase distortion. According to the coordinate transformation, different vertical positions on the first log-polar optic correspond to different radii of the generated APVs. As shown in Fig. 1(b), two elliptical Gaussian beams illustrated with different colormaps are incident at different vertical positions on the first log-polar optic and along with propagation, the two beams are mapped to the second log-polar optic. For precise temporal manipulation, the laser pulse and the two AODs are synchronized by a trigger. Figures 1(c) and 1(d) show some examples of the generated APV with different beam sizes and multiple radial/OAM modes. Note that since we are using an ultrafast pulsed laser, the interference pattern can be observed in coherent OAM modes, compared to our former papers using continuous wave (CW) lasers [14] considering the Doppler effect. A complete theoretical model will be given in the following analysis.
Fig. 1. (a) 2D HOBBIT setup. (b) Log-polar coordinate mapping based on different incident beam positions. (c) The APV with different ${r_n}$. (d) Experimental examples for APV with different OAM and radial dimensions. The fidelity is calculated based on the intensity correlation between the experimental results and relative simulated modes using Eq. (7).
The RF signals applied to the two AODs can be described by
where ${c_n}$ and ${c_\ell }$ are coefficients of each frequency component in the RF signals, ${f_n}(t)$ and ${f_\ell }(t)$ are the frequency function in each sinusoidal wave, ${\phi _n}$ and ${\phi _\ell }$ are the initial phase for each wave. The frequency of the RF signal can be any combination of different frequency components which correspond to the generation of multiple modes. In the experiment, any arbitrary RF signal can be generated using an arbitrary waveform generator (AWG) based on the generic RF signal equations. In addition, one sinusoidal wave can be a function of time which corresponds to a chirp signal generating a phase acceleration/deceleration [6,10].The two RF signals allow for the spatiotemporal control of the generated beams. For the first AOD, it adds a vertical linear phase to the incident beam, and in the Fourier plane of ${F_2}$, the vertical shift ${v_n}$ represents the distance from the center of the optic to the beam and can be represented by
where ${\lambda _n}$ is the new wavelength after the first AOD, ${V_{at}} = 0.65\,{{mm} / {\mu s}}$ is the acoustic velocity, ${f_{AOD}} = 75\,MHz$ is the AOD central driving frequency.The relationship between the vertical shift ${v_n}$ and the ring radius ${r_n}$ of the APV is due to the log-polar coordinate transformation [9,13] and can be described by,
where $a = \frac{{6\,mm}}{{2\pi }}$ and $b = 1.5\,mm$ are the design parameters of the log-polar optics.For the second AOD, the horizontal linear phase is mapped to an azimuthal phase gradient which is demonstrated in detail in our former work [9]. The generated OAM can be described by
where $\eta = \frac{{{F_3}}}{{{F_1}}}$ is the magnification of the lens imaging system.After the log-polar optics, the generated near-field is an APV and can be described by
By manipulating ${v_n}$ in a 1 mm range from -500 µm to +500 µm with a step of 100 µm, the APV diameters directly after the log-polar optics change from 5.06 mm to 1.78 mm exponentially based on Eq. (5). The results in Fig. 1(c) are collected after a 20X reducing telescope which is not shown in the figure. we can generate multiple radial/OAM modes in a single pulse by summing sine waves with different frequencies to drive the two AODs. Relative examples are shown in Fig. 1(d). Specifically, the first 5 results show single radial modes carrying multiple OAM; the first mode contains ±5 OAM with a small beam diameter, the following 3 results contain ±3, ±15, and ±20 OAM with the same beam diameter, and the fifth result contains ±20 OAM with the largest beam diameter. The next 6 results show the generation of multiple radial/OAM modes; the first two results are OAM ±10 with 2 and 3 radial modes, and the last four results show OAM 0, ±3, ±5, and ±10 with the same 4 radial modes. The fidelity in Fig. 1(c) and Fig. 1(d) is calculated based on the intensity correlation between the experimental results and relative simulated modes using Eq. (7). Considering the laser power fluctuation and the RF power ratios for different frequencies, the calculated fidelity demonstrates strong modal quality generated by the 2D HOBBIT system. The examples show typical interference patterns of coherent OAM modes. The petal pattern also indicates an OAM generation of charge 20. Inherently, the optical transformation performed by the log-polar optics is designed for plane wave [13] which will introduce some errors to the incident beam carrying a phase gradient. This factor limits the maximum OAM that can be generated by the HOBBIT system. The APV beam size is also decided by the log-polar optics. When a long elliptical Gaussian line is wrapped to a small radius, more phase errors and aberrations are introduced due to the paraxial approximation of the log-polar design. Based on the fidelity calculation, we show that within the radius and OAM ranges mentioned above, the generated modes show very high beam quality.
The far-field, which forms an asymmetric Bessel-Gauss beam and is usually described as a summation of Bessel modes [10], is used to collect the correlation data since its beam size is OAM-dependent. For the APV with the same spatial distribution, due to the Fourier transform relation, the far-field beam size is dependent on the amount of OAM [6,9,14]. In Fig. 2, the intensity distributions of the APV and the relative asymmetric Bessel-Gauss beams carrying different OAM are shown. The asymmetry property of the APV is also shown in the far-field, which is one important feature of the asymmetric Bessel beams. The rotation of the asymmetric structure from near-field to far-field is due to the OAM spiral phase and the Gouy phase. The asymmetry of the generated APV with the current setup is 1.3 which is the ratio of the elliptical Gaussian beam size in the horizontal direction and the $2\pi a$ parameter of the log-polar optics which is 6 mm. With an asymmetry of 1.3, we decompose the APV modes into an integer OAM spectrum and get a mode purity over 97% which shows good mode orthogonality.
One advantage of the HOBBIT concept is rapid mode switching. Based on the RF signal loaded on the AOD, the HOBBIT system is capable of mode switching in the MHz range. The two parameters impacting the mode switching speed are the beam size incident at the AOD and the acoustic velocity. Specifically, for the 2D HOBBIT, the incident Gaussian beam diameter is 1.6 mm and the acoustic velocity of the AOD used in the setup is 0.65 mm/µs which allows mode switching up to 406.25 kHz. The mode switching speed can be further improved to MHz range by changing parameters of the setup. For example, by narrowing the beam size down by a factor of 2.5 and redesigning the lens system to reshape the beam to the same elliptical Gaussian beam, the mode switching speed could be increased to 1.01 MHz. Another option is to use an AOD having a much higher acoustic velocity, which will reduce the time for the RF signal traveling through the AOD and enable high mode switching rate. Compared to the dynamic properties of most media such as atmospheric turbulence or turbid underwater system, this mode switching is much faster which allows the real-time sensing of these media.
2.2 Binary amplitude pattern recognition using spatial APV pulse train
In this section, we use the spatial APV basis to create a pulse train containing radial and OAM information per pulse to sense the amplitude pattern. Different from the method using an interference pattern of the LG basis and a Gaussian beam [8], the benefit of using the APV basis based on HOBBIT is a self-reference by generating OAM 0 and OAM $\ell$ with the same radial index but an arbitrary phase difference. In order to measure the complex coefficient for one specific mode, four superpositions of the APV and its reference with different phase delays (0, 0.5$\pi $, $\pi $, 1.5$\pi $) are sent to the amplitude pattern. After the four beams interacting with the amplitude object, a power meter is used to collect four power values which determine the complex coefficient for one specific spatial mode. As shown in Fig. 3(a), a pulse train containing all the coherent modes is incident at an amplitude object, “OPTICA” whose letter area has a transmission rate of 1. The parameter $\varDelta t$ shown in the figure is the pulse repetition rate and is related to the mode switching speed. The amplitude pattern has a size of 80 mm by 30 mm, and the height of each letter is 10 mm. The object was moved horizontally across the light path. The intensity distributions of OAM 0 and +10 with their reference beam at a specific radial index are given as example. The transmitted power values, ${P_{n,\ell ,0}}$, ${P_{n,\ell ,0.5\pi }}$, ${P_{n,\ell ,\pi }}$, and ${P_{n,\ell ,1.5\pi }}$, are used to determine one complex coefficient for mode $AP{V_{n,\ell }}$. The complex coefficient ${C_{n,\ell }}$ can be described by
Fig. 3. (a) Spatial APV pulse train for amplitude pattern sensing. (b) Simulated and experimental reference without object. (c) Experimentally reconstructed pattern. Publisher Note: The Optica logo is a registered trademark of Optica. See https://optica.org/brand for appropriate logo use guidelines.
Then the amplitude object can be reconstructed by summing all the APV modes together,
A total number of 3280 pulses are included in the pulse train. The measurement used 20 radial modes (diameter from 3 mm to 12 mm changing linearly) and 41 integer OAM modes (-20 to +20). A 3X telescope is used to get the appropriate beam sizes which is not shown in the figure. Since the object is moving across the beam, a total number of 11 scans are taken to recover the whole pattern. A high-speed camera (Phantom Miro C110) is used to collect the beam and the gray-scale figure is used to calculate the power values. In the experiment, the laser pulse repetition rate of 1 kHz is used. At first the reference data is collected without any object in the light path. In Fig. 3(b), the comparison between simulation and experiment is given. The difference is due to the mode quality and power inequality in each pulse. In the experiment, the power can be controlled for each $AP{V_{n,\ell }}$ mode by changing the voltage for each RF signal. However, the response of the AOD to RF voltage is not linear and it is hard to balance the power for all the 3280 pulses. As a result, the mode quality and the power inequality will introduce some variance in the recovered pattern. The experimentally reconstructed pattern is shown in Fig. 3(c). We can see that the amplitude pattern is clearly recovered. In the experiment, the binary object is 3D printed and we use a vernier caliper to measure the spatial scale of the letters. Take the letter “O” as an example, the height of the letter and the gap on the left side of the letter “O” can be measured as the reference. The measured height and the gap size of the real object are 9.51 mm and 2.50 mm. The estimated height and gap size from Fig. 3(c) are 9.765 mm and 2.385 mm. The results show a relatively accurate measurement with this sensing procedure using spatial APV modes.
2.3 OAM phase plate measurement using spatial APV pulse train
The spatial APV basis can also be used to sense phase distributions. Since a wavefront can be decomposed into an OAM spectrum, PV basis can be used to analyze the phase pattern at a specific radial position, considering the beam size is OAM-independent. For a spatial APV basis, not only radially localized information can be decomposed, but also the asymmetric property allows the measurement of a continuous OAM spectrum by using fractional OAM [6,7]. Different from coherent modes in a single pulse for amplitude object sensing, pulses carrying a single OAM mode and radial index are used to build the pulse train for the phase measurement.
The experimental setup can be seen in Fig. 4(a). Each pulse in the train carries both radial and OAM information. Considering a static phase structure is measured, in this sensing scenario the same laser repetition rate of 1 kHz is used. One advantage of the APV is that it can carry OAM with a linear one-to-one correspondence [6,7], then both integer and fractional OAM can be used in the sensing procedure. In this paper, pulse trains containing two OAM ranges are designed. The first one contains integer OAM from -20 to +20 which can decompose a phase pattern into OAM eigen modes. The second one contains OAM from -5 to +5 with a step of 0.25 which can measure a continuous OAM spectrum. Both pulse trains include 10 radial modes and 41 OAM modes, as a result, a total number of 410 pulses are used in one scan. Considering the concentric phase plate has a relatively smaller inner section, we use a 2X reducing telescope to narrow the beam size down to an appropriate size. 10 radial modes whose radius linearly changes from 0.25 mm to 1 mm are chosen for sensing. Based on Eq. (4) and (5), the corresponding RF signals are designed. As shown in Fig. 4, the pulse train carrying fractional OAM is illustrated. The first group of 41 pulses have the same radial information and scan the OAM one by one. Then the second group of 41 pulses carry different radial information, but scan the same range of OAM. In Fig. 4(b), the phase structure of a CPP is given. The CPP is designed as radial combinations of spiral phase plates carrying different orders and it can be used for multiple OAM generation in applications like helical filamentation [15] and underwater communication [16]. Here we use “-1 + 2” to denote a CPP have an inter vortex order of -1 and an outer vortex order of +2. For the sensing experiment, we choose two separate concentric phase plates which have the same radial information but different vortex orders. The CPP has an inner plate radius of ${\rho _{in}} = $0.625 mm and an outside plate radius of ${\rho _{out}} = $2.5 mm. The vortex information for the two CPPs are -1 + 2 and -1 + 4. After the interaction between the pulse train and CPP, the light field is Fourier transformed by a lens and the far-field is captured by a high-speed camera (Phantom Miro C110). After imaging processing, the OAM sensing spectrum can be retrieved by measuring the correlation spot. The sensing speed is determined by the repetition rate of the laser source and the frame rate of the high-speed camera as well as the total number of pulses in the pulse train. The sensing speed can be improved by using a higher repetition rate and frame rate, in order to do real-time sensing on dynamic phase structure such as the turbulence. A digital aperture is used as a filter to calculate the power in the correlation when doing the imaging processing. The correlation measurement can be described by the following equation,
Fig. 4. (a) Experimental setup for sensing CPP using the spatial APV pulse train. (b) Transverse structure of a CPP.
The experimental results using the first APV pulse train are shown in Fig. 5. The reference data is taken without CPP and is shown in Fig. 5(a). As expected for the reference, an OAM spectrum centered at charge 0 can be observed. The measured OAM spectrum for the two CPPs are shown in Figs. 5(b) and (c). In Fig. 5(b), we can see the OAM spectrum includes two main regions: the first part centered at -1 and the second part centered at +2 which correspond well with the parameters of the CPP. Due to the radial edge diffraction, the OAM spectrums at the fifth and sixth radii show a combination of multiple peaks which indicates the radial information. We can conclude that the radial edge is between the fifth and sixth radii (0.58 mm and 0.67 mm), which corresponds to the designed parameter of the CPP, 0.625 mm. The measurement of CPP -1 + 4 shows similar OAM spectrum distribution which proves the ability of the spatial APV pulse train to measure radially localized phase distributions.
A continuous OAM spectrum will show more detailed phase information by using fractional OAM. From the measured results in Fig. 5 showing an OAM order smaller than 5, we designed a second pulse train that included fractional OAM from -5 to +5 with a step of 0.25. The same 10 radial modes are used and a total number of 410 pulses are in this pulse train, the same as the first one. The measurement results are shown in Fig. 6. Similar OAM spectrums are observed from the results which tell the OAM spectrum and radial information of the CPPs. The little shift on the spectrum for each radial mode is due to a slight misalignment of the system. The continuous OAM spectrum show and trace minor changes in phase distributions which has potential in sensing turbulence [6].
With the two designed pulse trains both including 410 pulses to sense the transverse phase structure, the measured phase can be decomposed into an OAM basis along the radial direction. The sensing speed is limited by the laser repetition rate, frame rate of the detector and the total number of pulses in the train. Instead of the fast camera used in our experiment, a single-pixel detector will further improve the sensing speed. Another limitation is due to the log-polar optics. When the incident beam is mapped to a smaller radius, for high OAM charge, the incident horizontal linear phase will shift the beam position on the second optics, causing more errors. A new log-polar design for high OAM generation and less shifting errors will be studied to improve the performance of the system.
3. Conclusions
In conclusion, this paper has introduced a spatial APV basis containing an APV with both radial and azimuthal dimensions. The key concept for creating the spatial APV basis is referred to as a 2D HOBBIT. This pulsed 2D HOBBIT system uses two AODs and a log-polar coordinate transformation to generate OAM states with controllable beam sizes. An example of 11 radial modes and OAM charge as high as 20 are demonstrated to show the mode generation ability of the 2D HOBBIT system. Since the RF signals are used to drive the AODs, the mode switching speed of the 2D HOBBIT is up to 406.25 kHz calculated with the acoustic velocity and the Gaussian beam size incident at the AOD. Using the spatial APV basis, two remote sensing schemes for measuring an amplitude object and phase distribution are experimentally demonstrated and the corresponding pulse trains for each method are designed. The relative pulse trains sensing amplitude and phase are designed. The sensing speed of the pulse train is related to the repetition rate of the laser source, the frame rate of the detector, and the total number of pulses in one scan. In this paper, it takes 3.28 ms and 0.41 ms to do one scan for measuring the binary amplitude object and the OAM phase plate using a laser repetition rate of 1 kHz, respectively. For all the pulse trains in this paper, with higher repetition rate and frame rate, more pulses carrying radial and fractional OAM information can be added for more accurate sensing. The spatial APV pulse train provides a tool to sense amplitude objects and transverse phase structures in real-time using a basis including radial and OAM information and shows potential applications in research such as atmospheric turbulence sensing, lidar, optical communication, and light-matter interaction.
Funding
Office of Naval Research (N00014-20-1-2558).
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.
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