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Abstract
The creation and detection of spatial modes of light with transient orbital angular momentum (OAM) properties is of critical importance in a number of applications in sensing and light matter interactions. Most methods are limited in their frequency response as a result of their modulation techniques. In this paper, a new method is introduced for the coherent detection of transient properties of OAM using a single pixel detector system for the creation of an OAM spectrogram. This technique is based on the ideas utilized in acousto-optic based optical correlators with log-polar optical elements for the creation and detection of higher order bessel beams integrated in time (HOBBIT) at MHz data rates. Results are provided for beams with time varying OAM, coherent combinations, and transient scattering by phase objects.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Orbital angular momentum (OAM) has been utilized in a number of applications based largely on its ability to exploit properties of a large dimensional Hilbert Space [1], i.e. optical communications [2], quantum cryptography and entanglement [3,4], particle manipulation [5] and remote sensing [6] etc. Optical beams carrying OAM have a characteristic helical wave-front, $\textrm{exp} (i\ell \theta )$, where $\ell $ is the so-called OAM charge number and $\theta $ is the azimuthal coordinate [7]. In general, $\ell $ is restricted to be of integer value and in the case where $\ell $ assumes a fractional value, it can be considered as a superposition of beams carrying integer OAM components. The beams carrying OAM are different from plane waves, since the Poynting vector of OAM beams are known to spiral along the propagation direction, which is a property that can be exploited for the applications which can encode these features associated with OAM. However, the challenge still persists as to how one can efficiently encode and decode the OAM onto beams in discrete and/or continuous time varying functions of OAM at speeds competitive with the desired applications.
Optical phase elements are usually used to generate and detect OAM. Examples include: Spiral phase plates (SPP) [8], diffractive gratings [9,10], and cylindrical lenses [11–13]. All are static methods for the generation and detection of OAM, which are quite simple to implement. Dynamic methods of generation and detection rely on pixelated Spatial light modulators (SLMs) and deformable mirrors (DMs); however, refresh rates of SLMs tend to be limited 10’s of Hz and pixelated DMs can reach speeds in the kHz range, but both are limited in resolution based on their space bandwidth product. Recently, alternative approaches have been introduced for sorting OAM states. These methods include: cascaded Mach-Zehnder interferometers [14], a multi-plane Laguerre–Gaussian (LG) mode sorter [15], a mode sorter exploiting rotational Doppler effects [16], a cascaded tunable resonator [17] and a Gouy phase radial mode sorter [18]. In many cases, mode sorters are utilized based on an inverse log-polar optical system with a detector array [19–21]. In a recent paper, a set of static phase masks and an acousto-optic modulator are used to generate a discrete number of OAM states for correlation, with a switching rate of up to 500 kHz [22]. Many of these methods have proven to be effective for a subset of OAM detection applications; however, there is still a desire for continuous control for the detection of time varying OAM consisting of integer and fractional states at speeds in excess of 100’s kHz.
In order to address this desired detection of fractional and integer time varying OAM states, this paper introduces a real-time OAM generation and detection method using log-polar optics and an acousto-optic deflector (AOD). The log-polar optics are used for the transformation between from a rectangular input to a ring shaped output that maps a linear phase to OAM spiral phase. Two phase elements are included in this system, one to do an optical geometric transformation, and the second element to correct the phase distortion from the mapping [19–21]. The coordinate transformation has been used in image processing for rotation invariant pattern recognition [23]. Acousto-optic devices are well known for laser beam modulation and beam shaping [24–26]. Combining these two technologies and exploiting the different driving radio frequency (RF) signals, the AOD works as a time varying linear phase generator on the transmitter side and phase correlator on the receiver (mode sorter) side. Due to the inherent properties of the RF control on the creation and detection ends of the system, the resulting architecture yields a real time system for both the creation and detection of time varying OAM. Additionally, the encoded OAM has specific Doppler frequencies as a result of the momentum conservation of the AOD, which can be used to extract relative phase information from coherent combinations of OAM. This paper provides a summary of this technique for real-time OAM generation and detection of single and multiple coherent states of OAM. The OAM transmitter can dynamically generate time varying beams with integer and fractional OAM based on our previous applications [27,28]. The detection system is basically a reverse of this system in order to provide coherent detection to verify the generation and interaction of single and time varying beams with fractional and integer OAM values at MHz rates. Next, coherent combinations are encoded and decoded to verify that coherent combinations of beams carrying OAM can be generated and detected. This is useful for applications in sensing and communications with coherent combinations of beams carrying OAM [29]. Lastly, an example is provided for the detection of time varying OAM as a result of the scattering with a rotating phase plate.
2. Method
The time varying OAM generation system is illustrated in Fig. 1(a) and is based on our previous work for the creation of higher order bessel beams integrated in time (HOBBIT) [30]. In this system, a linear polarized Gaussian beam travels through an AOD, which is driven by an RF signal $S(t)$. Changes in $S(t)$ will result in a deflection of the laser beam to different directions, which correspond to linear phases based on a change in the Bragg angle of the AOD. This linear phase is then mapped to a spiral phase as known as OAM by the log-polar optics. One key feature is based on the incident beam being converted to an elliptical Gaussian beam with a lens system (${F_1} = 50\,mm,\,{F_2} = 100\,mm,\,{F_3} ={-} 50\,mm$ as shown in the figure) to convert the elliptical Gaussian distribution to an appropriate width, which is suitable for the log-polar optics requirement and eliminate the interference effect from the phase discontinuity so one can realize both integer and fractional values of OAM. After the log-polar optics, the complex amplitude distribution is called the near-field distribution, which can be converted to the far-field using a lens or free space propagation. This far-field distribution are the HOBBIT beams as shown in Fig. 1(c). This time signal $S(t)$ can be represented in its general form as follows:
where, ${c_n}$ is the weighting factor used to control the amplitude of each sinusoidal component, ${f_n}(t)$ is the time varying frequency function for each sine wave, and ${\phi _n}$ is the initial phase of each sine wave. When a laser beam passes through the transmitter AOD, the general Bragg angle is expressed as follows: where, ${\lambda _0} = \frac{c}{{{f_c} + {f_{AOD}}}}$ is the Doppler shifted wavelength due to the transmitter AOD central driving frequency ${f_{AOD}} = 125\,MHz$, c is speed of light, ${f_c}$ is the incident light frequency, and ${V_{at}}$ is the acoustic velocity of transmitter AOD. The light deflected at the Bragg condition corresponds to an OAM charge of $\ell $=0. Changes from this Bragg frequency correspond to the +/- OAM values. In this setup, the frequency step for an integer change in OAM by 1 unit, the corresponding frequency change from the Bragg frequency is 0.3611 $MHz$. Considering the lens system, $\Delta {\theta _{\ell n}}$ can be described as
Fig. 1. (a) The HOBBIT OAM transmitter. (b) The reverse-HOBBIT OAM receiver. (c) Intensity distribution of HOBBIT beams (far-field).
Therefore, each sine wave with a function of ${f_n}(t)$ will be used to encode a time-varying OAM${\ell _n}(t)$. This can then be used to generate a complex field carrying coherent combinations of arbitrarily time-varying OAM with a summation of sine waves at the desired frequencies. The complex amplitude of the near field can be expressed as follows:
The reverse-HOBBIT OAM receiver setup is similar to the HOBBIT OAM transmitter but with a reverse scheme and is shown in Fig. 1(b). In this case, the log-polar optics are used in reverse order as compared to the transmitter side. In many cases, this configuration of OAM sorting has been used for detection as in [19]. In this case, the ring-shape OAM beams are transformed back to a line-shape with a horizontal linear phase distribution. This linear phase is then used with a lens function to measure the shifts corresponding to the OAM mode number on a detector or CCD array. In order to do the mode sorting in real-time, a second AOD (receiver AOD) is driven by a time signal to deflect light. Because of the limited aperture size of receiver AOD and the line-shape width after the log-polar optics, a reducing telescope (${F_1}^{\prime} = 500\,mm$ and ${F_2}^{\prime} = 50\,mm$) is adopted to narrow the optical beam. By using a Fourier lens to couple the light into a multi-mode fiber which acts as an aperture, a time signal of this cross correlation can be mapped with a single optical detector to create an OAM spectrogram with time.
This optical receiver sees an optical beam with a time varying OAM phase, which can be represented by a complex phase distribution $\sum\limits_n {\textrm{exp} ({ - i{\ell_n}(t)\theta } )\textrm{exp} [{i2\pi ({{f_c} + {f_n}(t)} )t + i{\phi_n}} ]} $. After the inverse log-polar optics, the transformed linear phase can be described as follows:
Then new linear phase function ${t_2}$ is sent through the receiver AOD and which cross-correlates with the time signal loaded on the receiver AOD. This time signal has a similar form as $S(t)$ and is described in Eq. (10). This signal is different from $S(t)$, since $R(t)$ only includes one sinusoidal wave which is a linear chirp time signal
where, ${d_n}$ is the weighting factor used to equalize the deflected power of each frequency component in the sinusoidal chirp, $g(t)$, which is time varying frequency function. This time varying signal added to the receiver AOD is considered a phase grating and has a corresponding the transmittance function described as follows:Obviously, Eq. (13) is very similar to Eq. (5) in the transmitter part of the system. The frequency increment corresponding to an OAM step of 1 is 11.6667 $MHz$ on the receiver AOD. When the correlation happens, the deflected beam is coupled into the fiber. The real-time correlation is illustrated in Fig. 2, where the blue curve is the incident OAM to the receiver, and the red lines correspond to the RF chirp signal on the receiver AOD. Because of the relationship between OAM and frequency, the correlation happens when the blue and red curves intersect. In reality, the chirp signal is much shorter than the illustration, and the correlation points comprise the real-time OAM measurement. In the case of multiple OAM incident beams, the fields carrying different OAM will have different light frequencies caused by the Doppler frequency shift of the transmitter AOD. In this case, the interference of the deflected beams will show oscillations on the waveform. The Doppler frequency shift caused by the receiver AOD will not affect the measurement because $R(t)$ only include one sinusoidal waveform which will add an identical frequency shift to any OAM signal. This receiver also works for regular LG modes but without any oscillation in the results, if they are not generated using an AOD with Doppler frequency shifts. Because the sweeping of signal $R(t)$ is a much faster than the OAM varying in the incident light, we can do the cross-correlation and the OAM measurement in real-time.
Fig. 2. Illustration of the correlation of the incident OAM and the RF chirp signal on the receiver side. The actual receiver chirp signal changes faster than the time varying OAM and the correlation points (black points) reflect the OAM change in real-time.
3. Experimental results
In this section, experimental results are provided for the creation and detection of optical beams with time varying OAM, detections of coherent combinations of beams with OAM and the interaction of HOBBIT beams with a rotating phase vortex plate to visualize the transient effects of OAM interactions.
Time varying OAM: In order to verify the detection of time varying OAM, a series of time varying OAM beams were created using the HOBBIT system as described. Figure 3(a) shows the detector results measuring different beams with integer OAM values of −2, −1, 0, +1, and +2. The difference between the amplitude is due to the diffraction efficiency for each OAM charge number is slightly different. Figure 3(b) gives an OAM spectrogram, which is a mapping of the OAM to time for 100 $\mu s$ with an incident OAM charge 0 when $S(t)$ is a single frequency sine wave. In this case, $R(t)$ is a linear 1 $\mu s$ chirp signal which can measure, or correlate, OAM from −4 to +4. This scanning rate on the receiver corresponds to an OAM measurement at a MHz rate. When considering a transient OAM signal generated with $S(t)$ using only one signal with a changing $f(t)$, one can generate an arbitrary time-varying OAM beam. Figure 4 shows the dynamic OAM measurement results for single beams with transient OAM signals with different functional forms of $f(t)$. In this case, $R(t)$ was a chirped signal at a 1 $\mu s$ scan rate. In this case, by mapping $\ell (t)$ to $g(t)$, the OAM was measured every 1 $\mu s$. These results verify that time varying OAM can be encoded into the HOBBIT beams and their corresponding detection is ideally matched to this architecture. Further optimization in $R(t)$ can also lead to further enhancements in the detection process.
Fig. 3. Single OAM charge number measurement results. (a) Integer OAM results from −2 to +2. (b) The mapping of OAM and time with a range of 100 $\mu s$. Each column in (b) is a Gaussian-like shape as shown in (a) collected by an oscilloscope with a 1 $\mu s$ scan rate.
Fig. 4. Temporal generation and verification of OAM with an OAM transmitter and receiver. Four kinds of time varying OAM functions are designed and verified. The relative functions are given in each figure above.
Coherent combinations of OAM: If one considers more than one OAM beam, $n = 2$ with time varying properties, then coherent combinations of OAM beams are feasible. Detection of these beams will require the extraction of both phase and amplitude in the spectrogram by itself or with subsequent processing. In this case, the receiver will be used to sort the OAM amplitude and phase as a function of time in the OAM spectrogram. Coherent combinations have been utilized for a number of applications and are frequently mapped to a Poincaré sphere equivalent representing different types of relative amplitudes and phases as provide in [29,32]. In this case, a point ${U_a}$ on the Poincaré sphere can be expressed by Eq. (14)
In order to determine where a point resides on the Poincaré sphere equivalent, a series of measurements are performed similar to the Stokes measurements for polarization states and a coordinate value is determined. If one considers a coherent combination of OAM charge numbers $\ell$=${\pm} 1$, then the coherent combination of the generated signal for the transmitter time signal $S(t)$ using its sinusoidal form can be expressed as follows:
In this case, the transmitter AOD driving frequencies for the OAM values of $\ell$=1 and −1 are ${f_1} = \textrm{124}\textrm{.6389}\,MHz$ and ${f_{ - 1}} = 125.3611\,MHz$. The relative phase differences are provided by $\phi $ = 0, ${\pi / 4}$ and ${\pi / 2}$. The amplitudes take on values of ${c_1},{c_{ - 1}} \in [{0,1} ]$, where the poles are pure states of OAM beams for 1 or −1. Given these values, 6 points on the sphere were selected and encoded onto the transmitter with 6 time periods sequentially performed from the transmitter. These time intervals were not restricted in anyway, but were selected to demonstrate the fidelity of the OAM spectrogram for illustrative purposes. In this experiment, a designed $R(t)$ which is a linear 100 $\mu s$ chirp is used so that enough resolution in the beat frequency is included in the data for subsequent processing. The OAM spectrogram measurement results are shown in Fig. 5(b). In this case, the changes in the position from a −1 to +1 are clearly evident in time windows 1 and 2. Time windows 3–6 represent different combinations of the relative phase between OAM beams with the same power. The close detail on the OAM spectrogram exhibits the Doppler beat frequency of 722 kHz between the two +/- 1 OAM beams as predicted. Further examination of this is performed by take a 1D FFT on the columns of the OAM spectrogram and studying the amplitude and corresponding phase as shown in Figs. 5(c) and 5(d). In fact, the Doppler beat frequency is clearly defined in Fig. 5(c) and the relative phases in Fig. 5(d) are to within 0.01 $\pi $ of their actual values. Although only a few points were selected as an example of the Poincaré sphere equivalent, the approach can easily be extended to include all points in the 3D space. Moreover, the method can be further extended to uncovering the amplitude and phase of more complex combinations of two or more HOBBIT beams using a single detector.
Fig. 5. Poincaré sphere creation and detection. (a) Poincaré sphere representing phase and amplitude. Six points are selected and labeled on the sphere in each msec time interval. (b)–(d)Experimental OAM, frequency and phase spectrogram. In (b)–(d), the 6 time intervals are examples representing the 6 points on the Poincaré sphere. One column in time window 6 is plotted in (b) to show the waveform oscillation because of the Doppler frequency shift.
Transient OAM scattering: The HOBBIT OAM transmitter and reverse-HOBBIT OAM receiver can be used to sense perturbation in the propagation path or medium. For example, if one considers a rotating vortex phase plate (charge −2) placed in the light path, then the OAM will experience a transient impact on the measured OAM as a result of the decentered optical vortex phase plate. In this example, a vortex phase plate is placed at the far-field plane where the HOBBIT beam is formed and fixed on a mechanical chopper as in Fig. 6(a). When the chopper rotates, the beam scans the circumference and the result creates a phase perturbation on the transmitted beam. The chopper was rotated at 20 Hz and the laser beam scanned through the center of the phase plate in less than 1 $ms$. Figure 6(b) shows the result by using an OAM charge number of $\ell$=−1 in the 500 $\mu m$HOBBIT beam. The position of the singularity from the phase plate will determine the mapping results, since there will be an oscillation in the measured OAM as a result of the beam moving through the center of the phase plate. As a result, an interesting artifact appears in the 0.4 to 0.6 $ms$ time window due to the symmetry of the log-polar optics. The results are actually showing the OAM spectrum as one would expect [33].
Fig. 6. Sensing the phase change from a rotating vortex phase plate of order −2. (a) The experimental scheme. (b) OAM mapping with time by using OAM −1 as input. The OAM spectrogram only gives a short time period when the incident beam scans across the center of the phase plate.
In the present setup, the reverse-HOBBIT OAM receiver is limited by the aperture size of the receiver AOD. As a result, the interference will be unobservable when using large OAM because the fiber may not be able to collect enough the power corresponding to the charge number which will limit the measurement of the phase information. An AOD with a larger aperture size will solve this problem. An AOD with a larger aperture size will also reduce the space occupied by the reducing telescope and make the setup more compact. When detecting conventional OAM modes like LG modes, the mode sorter still realizes a real-time measurement. The limitation may exist when dealing with multiple OAM modes. The resolution is the same as the regular way when log-polar elements are used.
4. Conclusion
In conclusion, this paper has introduced a new technique to measure OAM spectrograms for a space time mapping of OAM based on a single pixel cross-correlator system. The system utilizes well known log-polar optics for geometrical optical transformations and acousto-optic deflectors with RF signal processing. Furthermore, the log-polar optics realize the transformation between linear phase and OAM spiral phase in both transmit and receive configurations. The AODs are ideally matched for encoding information on linear phase and or combinations of linear phase deflections, which are possible at extremely high data rates as compared to SLMs and DMs. These fundamental building blocks are ideally suited for optical correlator based systems [34]. However, in this work, the AOD works as a linear phase generator on the HOBBIT OAM transmitter side and a phase correlator on the reverse-HOBBIT receiver side. The HOBBIT OAM transmitter generates OAM at rates in excess of $MHz$ with arbitrarily time-varying single OAM or multiple coherent OAM beams. This property is verified using the reverse-HOBBIT OAM receiver. This arbitrary control of the OAM can also be used to exploit torque or even jerk in the angular momentum for particle manipulation, which could provide new functionalities to the field of particle trapping and manipulation.
Another salient feature associated with this system, is the ability to encode OAM with an induced Doppler frequency shift that can be used to “tag” the OAM beams for more complex measurements for investigating transient effects in the medium. This Doppler frequency shift is easily detected by our OAM receiver and extracted using digital signal processing techniques on the resulting OAM spectrogram. An example based on a coherent combination using a Poincaré sphere equivalent mapping was also measured using our transmitter and receiver to demonstrate the amplitude and phase measurements using a single detector. This system also provides a way to study the impact of OAM beams travel through a turbulent atmosphere [28]. As an example, a phase perturbation caused by a rotating vortex phase plate was detected by using the OAM transmitter and receiver. Although simple in its form, coherent combinations can also be explored utilizing the OAM spectrogram. Future research in this subject will be dedicated to the understanding of interactions between OAM beams and corresponding linear and nonlinear interactions. Although detection was primarily focused on the angular decomposition of the structured light beam, it would be interesting to explore measurements on other structured light beams with non-cylindrical symmetry such as Ince-Gauss or Mathieu-Gauss beams. Additionally, this technique could be further expanded to consider changes in the radial direction. However, this system has now made it feasible to create and detect combinations of integer and/or fractional OAM for a variety of applications that can exploit multiplexing and de-multiplexing of higher order bessel beams and their subsequent interactions. The real time measurement and interpretation of the OAM spectrogram will prove to be a valuable method in understanding transient effects on OAM and their interactions with media.
Funding
Office of Naval Research (N00014-16-1-3090, N00014-17-1-2779, N00014-20-1-2037, N00014-20-1-2558).
Disclosures
The authors declare no conflicts of interest.
References
1. V. Potoček, F. M. Miatto, M. Mirhosseini, O. S. Magaña-Loaiza, A. C. Liapis, D. K. L. Oi, R. W. Boyd, and J. Jeffers, “Quantum Hilbert Hotel,” Phys. Rev. Lett. 115(16), 160505 (2015). [CrossRef]
2. 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). [CrossRef]
3. M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015). [CrossRef]
4. R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012). [CrossRef]
5. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef]
6. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013). [CrossRef]
7. L. Allen, M. Beijersbergen, R. 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). [CrossRef]
8. M. W. Beijerbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]
9. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef]
10. K. Dai, C. Gao, L. Zhong, Q. Na, and Q. Wang, “Measuring OAM states of light beams with gradually-changing-period gratings,” Opt. Lett. 40(4), 562–565 (2015). [CrossRef]
11. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009). [CrossRef]
12. S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016). [CrossRef]
13. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27(8), 11236–11251 (2019). [CrossRef]
14. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef]
15. N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019). [CrossRef]
16. H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl.6(4), e16251 (2017). [CrossRef]
17. S. Wei, S. K. Earl, J. Lin, S. S. Kou, and X. Yuan, “Active sorting of orbital angular momentum states of light with a cascaded tunable resonator,” Light: Sci. Appl. 9(1), 10 (2020). [CrossRef]
18. X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy Phase Radial Mode Sorter for Light: Concepts and Experiments,” Phys. Rev. Lett. 120(10), 103601 (2018). [CrossRef]
19. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient Sorting of Orbital Angular Momentum States of Light,” Phys. Rev. Lett. 105(15), 153601 (2010). [CrossRef]
20. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4(1), 2781 (2013). [CrossRef]
21. C. Wan, J. Chen, and Q. Zhan, “Tailoring optical orbital angular momentum spectrum with spiral complex field modulation,” Opt. Express 25(13), 15108–15117 (2017). [CrossRef]
22. B. Braverman, A. Skerjanc, N. Sullivan, and R. W. Boyd, “Fast generation and detection of spatial modes of light using an acousto-optic modulator,” Opt. Express 28(20), 29112–29121 (2020). [CrossRef]
23. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64(8), 1092–1099 (1974). [CrossRef]
24. M. A. Dugan, J. X. Tull, and W. S. Warren, “High-resolution acousto-optic shaping of unamplified and amplified femtosecond laser pulses,” J. Opt. Soc. Am. B 14(9), 2348–2358 (1997). [CrossRef]
25. T. Häfner, J. Strauß, C. Roider, J. Heberle, and M. Schmidt, “Tailored laser beam shaping for efficient and accurate microstructuring,” Appl. Phys. A 124(2), 111 (2018). [CrossRef]
26. A. Zunino, S. Surdo, and M. Duocastella, “Dynamic Multifocus Laser Writing with Acousto-Optofluidics,” Adv. Mater. Technol. 4, 1900623 (2019). [CrossRef]
27. K. Dai, W. Li, K. S. Morgan, Y. Li, J. K. Miller, R. J. Watkins, and E. G. Johnson, “Second-harmonic generation of asymmetric Bessel-Gaussian beams carrying orbital angular momentum,” Opt. Express 28(2), 2536–2546 (2020). [CrossRef]
28. R. J. Watkins, K. Dai, G. White, W. Li, J. K. Miller, K. S. Morgan, and E. G. Johnson, “Experimental probing of turbulence using a continuous spectrum of asymmetric OAM beams,” Opt. Express 28(2), 924–935 (2020). [CrossRef]
29. K. Morgan, Y. Li, W. Li, J. K. Miller, R. J. Watkins, and E. G. Johnson, “Multilevel quadrature amplitude multiplexing using coherently coupled orbital angular momentum modes,” Opt. Express 26(9), 12180–12190 (2018). [CrossRef]
30. W. Li, K. S. Morgan, Y. Li, J. K. Miller, G. White, R. J. Watkins, and E. G. Johnson, “Rapidly tunable orbital angular momentum (OAM) system for higher order Bessel beams integrated in time (HOBBIT),” Opt. Express 27(4), 3920–3934 (2019). [CrossRef]
31. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22(2), 289–298 (2005). [CrossRef]
32. B. Jack, A. M. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. G. Ireland, S. M. Barnett, and M. J. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A 81(4), 043844 (2010). [CrossRef]
33. B. Berger, M. Kahlert, D. Schmidt, and M. Assmann, “Spectroscopy of fractional orbital angular momentum states,” Opt. Express 26(24), 32248–32258 (2018). [CrossRef]
34. T. Turpin, “Spectrum analysis using optical processing,” Proc. IEEE 69(1), 79–92 (1981). [CrossRef]
