## Abstract

The angular acceleration of a spinning object can be estimated by probing the object with Laguerre-Gauss (LG) beams and analyzing the rotational Doppler frequency shift of returned signals. The frequency shift is time dependent because of the change of the rotational angular velocity over time. The detection system is built to collect the beating signals of LG beams back-scattered from a non-uniform spinning body. Then a time-frequency analysis method is proposed to study the evolution of the angular velocity in time. The experimental results of different angular accelerations of the rotator are consistent with expectations. The measurement errors of different probe beams with various topological charges from *l* = ± 10 to *l* = ± 100 are also investigated.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The rotational Doppler effect [1] which arises from the rotational motion and orbital angular momentum (OAM), is attracting more and more attention in recent years. This kind of Doppler effect was predicted by G. Nienhuis [2] and then verified by Allen and Padgett with a rotating Dove prism [3,4]. The complex amplitude of an OAM beam comprises a transverse angular phase profile exp(i*lφ*), where *l* is the topological charge and *φ* is the azimuthal angle [5]. A spinning object with a rough surface will induce a Doppler frequency shift in OAM beams reflected parallelly to the rotation axis [6]. And the physical mechanism of such frequency shift is theoretically studied [7–12]. The relationship between rotational and linear Doppler shifts in terms of time-evolving phase or the momentum and energy conservation is also studied [13]. Furthermore, the rotational frequency shift associated with OAM is different from that with intrinsic spin angular momentum (SAM) [14–18]. The SAM arises from the circular polarization of light and results in the frequency shift equivalent to 2Ω, where Ω is the rotational angular velocity. However, OAM arises from helical phase fronts described by the helical term exp(i*lφ*) and results in the frequency shift equivalent to *l*Ω, which can improve measurement accuracy. In addition, the rotational Doppler effect can also be resulted from the dynamic evolution of geometric phase, associated with the variation of total angular momentum [19,20]. Recently the rotational Doppler effect associated with OAM gets much attention in angular velocity measurement [21–29], fluid flow vorticity measurement [30,31], optical sensor [32,33] and so on [34–37].

Most of the previous papers investigate the detection of the rotational angular velocity of an object with a constant speed. Angular acceleration is also an important parameter which describes the change in the rotational angular velocity. The measurement of the angular acceleration plays an important role in many domains such as inertial navigation, rotator detection, etc. There are many kinds of angular acceleration sensors based on various physical mechanisms. Such sensors can be broadly classified as direct and indirect measurement. The fiber optic gyroscope (FOG) [38] exploiting the Sagnac effect, is a well-established class of sensor. However, not much attention has been paid to detection of the angular acceleration based on the optical rotational Doppler effect.

In this paper, a remote sensing scheme of angular acceleration is reported. We extend the approach of the angular acceleration detection by analyzing the beat frequency signals of OAM beams back-scattered from a non-uniform spinning body. Although the rotational angular velocity Ω(*t*) is time-dependent, the OAM components in scattered light won’t change over time and the changing rate of the frequency shift with time is proportional to angular acceleration. And then the time-frequency analysis method is proposed for analyzing intensity modulation signals. Different angular accelerations of the rotator and different OAM modes of the probe beams are employed.

## 2. Angular Doppler frequency shift due to non-uniform rotator

Strictly the frequency shift characteristics of an optical vortex due to non-uniform rotators are fundamental problems associated with Maxwell’s equations and special relativity. However, the rotational speed is far lower than the velocity of light in practice. Ignoring the relativistic effects, we can describe this phenomenon with the classical diffraction theory. The phase distribution of an object can be written as Ф(*r*, *θ*) *=* 4π*h*(*r,θ*)/*λ*, where *h*(*r,θ*) describes the roughness of the surface and *λ* is the wavelength. Then, the modulation function in Fourier expansion form can be expressed as $\text{exp}\left(i\text{\u0424}\left(r,\theta \right)\right)={{\displaystyle \sum}}^{\text{}}{A}_{n}\left(r\right)\mathrm{exp}\left(in\theta \right),$ where *n* is an integer. *A _{n}*(

*r*) is the complex amplitude of the

*n*-th harmonic and ${{\displaystyle \sum}}^{\text{}}{\left|{A}_{n}\left(r\right)\right|}^{2}$ = 1. Then, the modulated function of the non-uniform spinning object should be written as:

*a*(

*t*) is the angular acceleration and Ω

_{0}is the initial angular velocity.

When an OAM beam expressed by *B*(*r*)exp(-*i*2π*ft*)exp(*ilθ*) is incident on the non-uniform spinning object, the scattered light can be written as:

*l*+

*n*. The frequency shift of every OAM mode can be written as:

*φ*(

*t*) is the time-evolving phase.

From Eq. (4), we know the OAM components in the scattered light don’t change over time although the angular velocity Ω(*t*) is time-dependent, which is consistent with [39]. And the frequency shift is proportional to *n* and Ω(*t*). As time goes on, the frequency shift value is changed because of the time-dependent parameter Ω(*t*). From Eqs. (3) and (4), the angular acceleration *a*(*t*) satisfies:

*a*(

*t*) can be deduced from the changing rate of the frequency shift.

In order to understand the evolution of the frequency shift in time due to the non-uniform rotation, Fig. 1 describes the physical process of an OAM beam passing through a special rotating surface with a spiral phase distribution for simplicity. When an OAM beam with the topological charge *l* = + 2 passes through a spiral phase plate (1st SPP) with a non-uniform rotational angular velocity Ω(*t*), the topological charge of the transmitted beam turns into + 3 resulting from the spiral phase structure of SPP, meanwhile the frequency shift changes over time. Different colors of output OAM beams represent different frequencies of output beams at different times. The frequency shift satisfies Eqs. (4) and (5). In fact, the effect of the spinning object on the incident light is similar to the acousto-optical effect. The spinning object completes the time modulation [40,41] of the OAM beam in term of the frequency domain. When *a*(*t*) = 0, the frequency shift in Eq. (4) reduces to a constant. Besides, the SPP in Fig. 1 is assumed to be perfect, ignoring the internal reflections in the device which have been studied in [42–44].

## 3. Experimental setup

We assume that the detection system is based on the beating effect. 2-fold multiplexed OAM beams with fields *E _{0}* = ${\sum}_{s=1,2}{B}_{s}\left(r\right)$exp(i

*l*)exp(-i2π

_{s}θ*ft*) are incident on the non-uniform spinning object. Modulated by the rough surface, the light scattered from the spinning object contains many OAM modes. Behind the mode filter, the fundamental mode (

*l*= 0) is selected and the interference pattern is collected by a photodetector. The frequency of the beating signals can be written as:

*f*(

*t*) = (

*l*

_{1}–

*l*

_{2})Ω(

*t*)/(2π). Taking the derivative of both sides with respect to time, the relationship between the angular acceleration

*a*(

*t*) and the frequency of the beating signals

*f*(

*t*) reads:

The experimental setup is shown in Fig. 2. Gaussian beams with the wavelength of 1.6μm are generated by a laser diode and then collimated. The half wave plate (HWP) and the polarized beam splitter (PBS1) behind the collimator are used to change the power and polarization of the beam, to match the requirement of the pure-phase modulation of SLM (Holoeye, PLUTO-TELCO-013-C). The SLM is encoded by the computer generate hologram (CGH) to transform Gaussian beams into 2-fold multiplexed LG beams with the topological charges of *l*_{1} and *l*_{2}. Then the generated LG beams are selected as the first diffraction order by an iris diaphragm together with a 4-*f* system (L1 and L2, *f* = 100mm). A 45° quarter wave plate in front of the surface is to ensure the scattered light could be reflected by PBS2. Then the scattered light is collected using a lens L4 (*f* = 50mm) and recorded by a photodiode (THORLABS, PDA20CS). The output signals of the photodiode are analyzed by an oscilloscope and a computer. The angular velocity of the rotator is controlled by a motion controller, which offers a variable voltage for the stepper motor. For the sake of observing the intensity profile of probe beams, an infrared CCD camera (Xenics, Bobcat-320-staris) in conjunction with a lens L3 (*f* = 300mm) is located behind the beam splitter.

## 4. Detecting angular velocity with the time-frequency analysis

We firstly produced the 2-fold multiplexed LG beams with the topological charges of + *l* and -*l*. The intensity distribution is recorded by a CCD in Fig. 3. Then we set the rotator at a constant rotational angular velocity Ω(*t*) = 94.25 rad/s, and input 2-fold multiplexed LG modes + 50 and −50 to verify the beating effect. The returned signal is presented in Fig. 4(a) and the frequency spectrum is shown as Fig. 4(c). The location of the peak in the frequency domain is 1500 Hz which is proportional to the rotational angular velocity. Substituting *f* = 1500 Hz into Eq. (4), the rotational angular velocity is obtained as Ω(*t*) = 94.25 rad/s. Obviously the experimental results meet well with expectations.

Then, we designed the rotator rotating at a non-uniform angular velocity which can be described as Eq. (2). We set Ω_{0} = 6.28 rad/s and *a*(*t*) = 12.95 rad/s^{2}. The returned signal in 0-0.4 s corresponding to a non-uniform angular velocity is shown in Fig. 4(b) and the frequency spectrum in 0-10 s is shown in Fig. 4(d). Due to the non-uniform angular velocity, the returned signals in Fig. 4(b) are non-stationary signals. As a result, the frequency spectrum contains lots of different frequency components. The frequency ranges from 0 to 2.5 kHz in Fig. 4(d), which means that the rotational angular velocity varies from 0 to 94.25 rad/s. However, we can’t obtain the evolution of the angular velocity in time from this frequency spectrum.

To get the angular acceleration, the time-frequency analysis method is necessary when the returned signals are non-stationary signals. The common ways are Short-time Fourier transform (STFT), Gabor-Wigner distribution function and so on [45]. In this paper, we use the STFT [46] to analyze the non-stationary signals. The STFT is a Fourier-related transformation used to determine the sinusoidal frequency of a signal as it changes over time. For a signal, *g*(*t*), the STFT is given by

*s*(

*u*) is the Gaussian window function. In practice, the procedure for computing STFT is to divide a longer time signal

*g*(

*t*) into shorter segments of equal length by use of window function

*s*(

*u*) and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment and then one can plot the changingspectrums as a function of time.

Figure 5 shows the analysis results, where the left and right are the simulated and experimental results separately. The yellow line describes the frequency changes over time and the changing rate of the frequency with respect to time can be described by the slope Δ*f*/Δ*t* ≈206.16 Hz/s. And then the angular acceleration *a*(*t*) can be obtained according to Eq. (6). After calculation, *a*(*t*) is equal to 12.95 rad/s^{2}, which is consistent with the theoretical value.

The time-frequency analysis results of different kinds of rotational motion are displayed in Figs. 6(a)–6(c), corresponding to the uniform rotation, uniformly accelerated rotation, uniformly decelerated rotation, respectively. Figure 6(d) describes the object spins faster and faster at first, and then keeps a constant angular velocity. Immediately the angular velocity decreases uniformly, and then reaches a lower constant angular velocity. After that it starts a new cycle.

In order to verify the proposed scheme sufficiently, we measured various angular accelerations of the rotator. The measured data as shown in Fig. 7 are in good agreement with theory. The maximum relative error of the measurement is less than 2%. This indicates that the detection is effective for different angular accelerations.

To verify the relationship among the rate of change of the frequency with respect to time and topological charge *l* given in Eq. (6), various OAM modes are employed to detect the non-uniform rotator. We set Ω_{0} = 11.39 rad/s and *a*(*t*) = 12.95 rad/s^{2}. Figures 8(a) and 8(b) show thetime-frequency spectrums of the OAM beams with *l* = ± 60 and *l* = ± 70 respectively. The slope can be described by Δ*f*/Δt ≈240.11 Hz/s for *l* = ± 60 and 271.80 Hz/s for *l* = ± 70. Then the angular acceleration *a*(*t*) ≈12.57 rad/s^{2} for *l* = ± 60 and 12.20 rad/s^{2} for *l* = ± 70 can be acquired. Such phenomenon indicates that the changing rate of frequency with respect to time is proportional to the topological charge of the probe beam. Hence, the higher changing rate can be achieved by employing the LG beams with larger topological charges.

The measurement errors of different probe beams with various topological charges from *l* = ± 10 to *l* = ± 100 are shown in Fig. (9). One can see the experimental results are consistent with the theoretical values. The biggest measurement error is approximately equal to 2 Hz/s, when the topological charges equal to ± 10. The rest of the measurement errors are less than 2 Hz/s. Thus, the maximum relative error of the measurement is less than 10%, when the topological charges from *l* = ± 10 to *l* = ± 100.

## 5. Discussions and conclusions

Although OAM beams have good performance in detecting the angular acceleration, misalignments between the light axis and rotation axis still exist, caused by the unstable velocity of the rotator. The effects of possible slight misalignments will broaden the OAM spectrum [47], which result in the crosstalk between different modes, as well as the increase of measurement error. To mitigate this effect, a feasible solution may be to ensure that the light axis and rotation axis are coaxial by use of anti-shake device.

Another point should be noted that in the process of short-time Fourier transform, the length of Gaussian window function *s*(*u*) determines the frequency resolution and time resolution in time-frequency spectrums. The longer the window, the higher the frequency resolution. In contrast, the shorter the window, the better the time resolution. Hence, according to requirements, we must have a balance between frequency and time resolutions in practice. The length of Gaussian window function *s*(*u*) is set to 0.008 s in our experiment. Then the frequency resolution 12 Hz and time resolution 0.008 s can be obtained.

In conclusion, a detection scheme of the angular acceleration based on the rotational Doppler effect is presented. The angular acceleration is proportional to the changing rate of the frequency shift. To measure the angular acceleration, a time-frequency method is proposed for analyzing beat frequency signals. Different kinds of beat frequency signals are analyzed in time and frequency domains. The angular acceleration can be obtained from the change of the beating signal’s frequency. Different angular accelerations of the rotator and different OAM modes of the probe beams are employed. The experimental results agree with the theoretical estimation.

## Funding

National Natural Science Foundation of China (NSFC) (11834001); National Postdoctoral Program for Innovative Talents of China (BX20190036); China Postdoctoral Science Foundation (2019M650015); Graduate Technological Innovation Project of Beijing Institute of Technology (2018CX10020).

## References

**1. **M. Padgett, “Electromagnetism: like a speeding watch,” Nature **443**(7114), 924–925 (2006). [CrossRef] [PubMed]

**2. **G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. **132**, 8–14 (1996). [CrossRef]

**3. **J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the Rotational Frequency Shift Imparted to a Rotating Light Beam Possessing Orbital Angular Momentum,” Phys. Rev. Lett. **80**(15), 3217–3219 (1998). [CrossRef]

**4. **J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light Beam,” Phys. Rev. Lett. **81**(22), 4828–4830 (1998). [CrossRef]

**5. **L. Allen, M. W. Beijersbergen, R. J. 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] [PubMed]

**6. **M. P. 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] [PubMed]

**7. **H. Zhou, D. Fu, J. Dong, P. Zhang, and X. Zhang, “Theoretical analysis and experimental verification on optical rotational Doppler effect,” Opt. Express **24**(9), 10050–10056 (2016). [CrossRef] [PubMed]

**8. **I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Y. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. **28**(14), 1185–1187 (2003). [CrossRef] [PubMed]

**9. **A. Belmonte and J. P. Torres, “Optical Doppler shift with structured light,” Opt. Lett. **36**(22), 4437–4439 (2011). [CrossRef] [PubMed]

**10. **Z. Liu, Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, and S. Wen, “Geometric phase Doppler effect: when structured light meets rotating structured materials,” Opt. Express **25**(10), 11564–11573 (2017). [CrossRef] [PubMed]

**11. **I. V. Basistiy, A. Y. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational Doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. **76**(8), 486–489 (2002). [CrossRef]

**12. **F. C. Speirits, M. P. Lavery, M. J. Padgett, and S. M. Barnett, “Optical angular momentum in a rotating frame,” Opt. Lett. **39**(10), 2944–2946 (2014). [CrossRef] [PubMed]

**13. **L. Fang, M. J. Padgett, and J. Wang, “Sharing a Common Origin Between the Rotational and Linear Doppler Effects,” Laser Photonics Rev. **11**(6), 1700183 (2017). [CrossRef]

**14. **B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. **31**(1), 1–3 (1979). [CrossRef]

**15. **R. Simon, H. J. Kimble, and E. C. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: An optical experiment,” Phys. Rev. Lett. **61**(1), 19–22 (1988). [CrossRef] [PubMed]

**16. **F. Bretenaker and A. L. Floch, “Energy exchanges between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. **65**(18), 2316–2319 (1990). [CrossRef] [PubMed]

**17. **O. Faucher, E. Prost, E. Hertz, F. Billard, B. Lavorel, A. A. Milner, V. A. Milner, J. Zyss, and I. S. Averbukh, “Rotational Doppler effect in harmonic generation from spinning molecules,” Phys. Rev. A (Coll. Park) **94**(5), 051402 (2016). [CrossRef]

**18. **O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing molecular spinning via the rotational Doppler effect,” Nat. Photonics **7**(9), 711–714 (2013). [CrossRef]

**19. **Y. Liu, Y. Ke, J. Zhou, H. Luo, and S. Wen, “Manipulating the spin-dependent splitting by geometric Doppler effect,” Opt. Express **23**(13), 16682–16692 (2015). [CrossRef] [PubMed]

**20. **Y. Liu, Z. Liu, J. Zhou, X. Ling, W. Shu, H. Luo, and S. Wen, “Measurements of Pancharatnam-Berry phase in mode transformations on hybrid-order Poincaré sphere,” Opt. Lett. **42**(17), 3447–3450 (2017). [CrossRef] [PubMed]

**21. **S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Non-diffractive Bessel-Gauss beams for the detection of rotating object free of obstructions,” Opt. Express **25**(17), 20098–20108 (2017). [CrossRef] [PubMed]

**22. **C. Rosales-Guzmán, N. Hermosa, A. Belmonte, and J. P. Torres, “Direction-sensitive transverse velocity measurement by phase-modulated structured light beams,” Opt. Lett. **39**(18), 5415–5418 (2014). [CrossRef] [PubMed]

**23. **M. Zhao, X. Gao, M. Xie, W. Zhai, W. Xu, S. Huang, and W. Gu, “Measurement of the rotational Doppler frequency shift of a spinning object using a radio frequency orbital angular momentum beam,” Opt. Lett. **41**(11), 2549–2552 (2016). [CrossRef] [PubMed]

**24. **S. Xiao, L. Zhang, D. Wei, F. Liu, Y. Zhang, and M. Xiao, “Orbital angular momentum-enhanced measurement of rotation vibration using a Sagnac interferometer,” Opt. Express **26**(2), 1997–2005 (2018). [CrossRef] [PubMed]

**25. **D. B. Phillips, M. P. Lee, F. C. Speirits, S. M. Simpson, M. P. J. Lavery, M. J. Padgett, and G. M. Gibsion, “Rotational Doppler velocity to probe the angular velocity of spinning microparticles,” Phys. Rev. A **90**, 011801 (2014). [CrossRef]

**26. **R. Neo, S. Leon-Saval, J. Bland-Hawthorn, and G. Molina-Terriza, “OAM interferometry: the detection of the rotational Doppler shift,” Opt. Express **25**(18), 21159–21170 (2017). [CrossRef] [PubMed]

**27. **M. P. J. Lavery, S. M. Barnett, F. C. Speirits, and M. J. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica **1**(1), 1–4 (2014). [CrossRef]

**28. **M. Seghilani, M. Myara, I. Sagnes, B. Chomet, R. Bendoula, and A. Garnache, “Self-mixing in low-noise semiconductor vortex laser: detection of a rotational Doppler shift in backscattered light,” Opt. Lett. **40**(24), 5778–5781 (2015). [CrossRef] [PubMed]

**29. **W. H. Zhang, J. S. Gao, D. Zhang, Y. He, T. Xu, R. Fickler, and L. Chen, “Free-Space Remote Sensing of Rotation at the Photon-Counting Level,” Phys. Rev. Appl. **10**(4), 044014 (2018). [CrossRef]

**30. **A. Ryabtsev, S. Pouya, A. Safaripour, M. Koochesfahani, and M. Dantus, “Fluid flow vorticity measurement using laser beams with orbital angular momentum,” Opt. Express **24**(11), 11762–11767 (2016). [CrossRef] [PubMed]

**31. **A. Belmonte, C. Rosalesguzmán, and J. P. Torres, “Measurement of flow vorticity with helical beams of light,” Optica **2**(11), 1002–1005 (2015). [CrossRef]

**32. **F. Xia, Y. Zhao, H. Hu, and Y. Zhang, “Optical fiber sensing technology based on Mach-Zehnder interferometer and orbital angular momentum beam,” Appl. Phys. Lett. **112**(22), 221105 (2018). [CrossRef]

**33. **S. Shi, D. Ding, Z. Zhou, Y. Li, W. Zhang, and B. Shi, “Magnetic-field-induced rotation of light with orbital angular momentum,” Appl. Phys. Lett. **106**(26), 261110 (2015). [CrossRef]

**34. **T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. Tedford, and S. Weinfurtner, “Rotational superradiant scattering in a vortex flow,” Nat. Phys. **13**(9), 833–836 (2017). [CrossRef]

**35. **G. Li, T. Zentgraf, and S. Zhang, “Rotational Doppler effect in nonlinear optics,” Nat. Phys. **12**(8), 736–740 (2016). [CrossRef]

**36. **H. L. Zhou, D. Z. Fu, J. J. Dong, P. Zhang, D. X. Chen, X. L. Cai, F. L. Li, and X. L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. **6**(4), e16251 (2017). [CrossRef] [PubMed]

**37. **Y. Zhai, S. Fu, R. Zhang, C. Yin, H. Zhou, J. Zhang, and C. Gao, “The radial Doppler effect of optical vortex beams induced by a surface with radially moving periodic structure,” J. Opt. **21**(5), 054002 (2019). [CrossRef]

**38. **B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. **9**(2), 57–79 (2003). [CrossRef]

**39. **V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic laser beams with a large orbital angular momentum,” Opt. Express **26**(1), 141–156 (2018). [CrossRef] [PubMed]

**40. **A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfaces and Lorentz non-reciprocity,” Opt. Mater. Express **5**(11), 2459–2467 (2015). [CrossRef]

**41. **D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics **11**(12), 774–783 (2017). [CrossRef]

**42. **Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. **54**(11), 111306 (2015). [CrossRef]

**43. **Y. S. Rumala and A. E. Leanhardt, “Multiple-beam interference in a spiral phase plate,” J. Opt. Soc. Am. B **30**(3), 615–621 (2013). [CrossRef]

**44. **Y. S. Rumala, “Wave transfer matrix for a spiral phase plate,” Appl. Opt. **54**(14), 4395–4402 (2015). [CrossRef] [PubMed]

**45. **E. Sejdić, I. Djurović, and J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digit. Signal Process. **19**(1), 153–183 (2009). [CrossRef]

**46. **P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. **15**(2), 70–73 (1967). [CrossRef]

**47. **M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, “Analysis of orbital angular momentum of a misaligned optical beam,” New J. Phys. **7**(1), 46 (2005). [CrossRef]