OAM-based optical wavelet using a single pixel detection system for probing dynamic environments with application to real-time measurements of strong atmospheric turbulence

Justin Free; Kunjian Dai; Liam Vanderschaaf; Michael Cox; J. Keith Miller; Richard J. Watkins; Eric G. Johnson

doi:10.1364/OE.474124

1. Introduction

Wavelets have been a tool in signal processing for several years. They were originally used for multi-resolution analysis of one-dimensional signals by localizing time and frequency information while retaining all the information of the signal [1,2]. Wavelets and multi-resolution analysis have been extended to two-dimensional space for applications in image processing and image compression [3,4]. Different types of two-dimensional wavelets have been introduced such as circular harmonic wavelets, steerable wavelets, and polar wavelets; these types of wavelets are preferred depending on the type of information that is of interest [5–7]. Wavelets have been used in the field of optics for signal processing, optical filters, analysis of interference patterns, and joint transform optical correlators [8–11]. However, to the authors’ knowledge, this paper is the first to describe an optical field as an OAM based wavelet for a real-time probing system capable of measuring a continuous wavelet transform of the environment. The basis of optical fields that are wavelets are asymmetric rings containing orbital angular momentum (OAM) [12]. Light possessing OAM has a spiral phase front and can be described by $\exp ({im\theta } )$, where m denotes the OAM charge and $\theta $ is the azimuthal coordinate. Light with OAM has been shown to be a preferred method for imaging or sensing objects with rotational symmetry or rotational velocity [13–16]. Beams with OAM can form a complete orthogonal set of modes. An example imaging system uses a 31 × 31 set of modes to measure the complex modal coefficients of various objects [16]. However, one problem associated with imaging and sensing with orthogonal modes of structured light is the mode generation rate; it takes 31 minutes to measure the complex coefficients of the 31 × 31 set of modes due to the limitations of the spatial light modulator in the experiment. For probing dynamic environments such as atmospheric turbulence, this measurement speed is unacceptable. This paper introduces an alternative probing method that does not rely on an orthogonal set of modes, but rather uses a continuous wavelet transform to measure the wavelet decomposition of a transmission function or the environment about a fixed radius in milliseconds or less. The wavelet beam generation system is based on the higher order Bessel beams integrated in time (HOBBIT) system, which consists of an acousto-optic deflector (AOD) and log-polar optics to generate spatially varying OAM beams with a mode switching rate of a few microseconds [17].

The probing system is applied to a three-dimensional turbulence volume to measure the wavelet decomposition of the turbulence propagation path about a fixed radius. OAM beams have applications in free-space optical communications; however, turbulence remains a problem for mode-division multiplexing due to the turbulence-induced modal coupling [18–20]. This probing system provides a tool to understand this modal coupling as it gives new insight into the localized OAM information in angular space. In addition, the probing method finds modes that have a high transmission through the turbulence to the target. The transmission of light through wave-guiding channels of turbulence has been studied and has applications in directed energy [21,22]. Previous work probes turbulence with HOBBIT beams and OAM to find the global OAM that propagates best to the target [23]. More insight is gained by probing with wavelets as it reveals the local OAM in space and the input modes that propagate to the target. The speed of the wavelet system also enables measurements of instantaneous realizations of turbulence, giving insight into the short time scales of turbulence [24].

This work introduces a two-dimensional wavelet with rotation and angular scaling operators that can be used to analyze the OAM information of a function at localized angular positions. The wavelet concept is applied experimentally to generate optical wavelet fields. Probing with optical wavelet fields is a new technique for remote optical probing. The optical wavelet generation system can scan the angular space about a fixed radius with a single wavelet function in less than 10µs. An entire wavelet transform of the environment about a fixed radius can be obtained in 3.8ms or less depending on the number of wavelet functions in the probing sequence. The phase of the wavelet transform can be recovered which enables reconstruction of the probed environment. The speed of the system can be applied to measure and characterize dynamic environments. We apply this system to atmospheric turbulence and obtain the wavelet transform of the turbulent propagation path. Localized OAM features of the turbulence are revealed in the wavelet transforms and many feature sizes are on the order of the Fried parameter. The time dynamics of the turbulence are analyzed by extracting relevant information from the wavelet transform, such as coherence time, wind velocity, and persistence of transmission channels. The data reveals new insight into the short time scale of turbulence.

2. Wavelet theory

Wavelets are a tool for multi-resolution analysis. To analyze local information of two-dimensional functions the wavelet must have finite support in spatial and frequency domains. This finite support allows the analysis and decomposition of the localized spatial and frequency information of the analyzed function [1,2]. Two-dimensional wavelets are square-integrable functions $\Psi \in {{\mathbf L}^{\mathbf 2}}({{{\mathbb R}^2}} )$ with finite energy and zero mean [7]. The ${{\mathbf L}^2}({{\mathbb R}^2})$ norm in polar coordinates $({r,\theta } )$ is defined as

(1)$$||\Psi ||_2^2 = \int\limits_0^\infty {\int\limits_0^{2\pi } {{{|{\Psi ({r,\theta } )} |}^2}rdrd\theta } } $$

with the requirement $||\Psi ||_2^2 < \infty $. A mother wavelet is a wavelet function from which we construct the set of analyzing wavelets. The choice of the mother wavelet function can be arbitrary as long as it is square integrable and has zero mean; however, the Morlet wavelet does not have zero mean. Although the Morlet wavelet does not have zero mean, it is still used because it has finite support and localized energy in the spatial and frequency domains. Similarly, the wavelet defined in this paper does not have zero mean; it has approximately zero mean and localized energy in angular space and the OAM spectrum. We demonstrate that despite not having zero mean, an alternative reconstruction formula can be derived. Other wavelet functions may be defined depending on the features that are desired to be analyzed; for the OAM-based wavelet we choose a simple azimuthal gaussian with azimuthal phase and a radial envelope. After scaling and rotating the mother wavelet we have a continuous set of basis functions for analyzing angular information about a fixed radius. Our mother wavelet is defined by

(2)$$\Psi ({r,\theta } )= \exp \left( { - \frac{{{{({r - {\rho_0}} )}^2}}}{{\rho_0^2\sigma_r^2}} - \frac{{{\theta^2}}}{{\sigma_\theta^2}} + j\theta } \right),$$

which describes an asymmetric ring in ${{\mathbb R}^2}$. In Eq. (2), ${\rho _0}$ defines the radius of the ring ${\rho _0}{\sigma _r}$, defines the radial Gaussian width, and ${\sigma _\theta }$ defines the angular Gaussian width. A visualization of this function is given in Fig. 1. We define scaling and rotation operators that act on the mother wavelet to create a family of wavelets in ${{\mathbf L}^2}({{{\mathbb R}^2}} )$. We are interested in localizing the spatial angular frequency (OAM information) of an arbitrary function in angular space. Therefore, from the mother wavelet we construct a set of wavelet functions

(3)$$\begin{aligned} {\Psi _{a,{\theta _0}}}(r,\theta ) &= \frac{1}{{\sqrt {|a |} }}\Psi \left( {r,\frac{{\theta - {\theta_0}}}{a}} \right)\\ \textrm{ } &= \frac{1}{{\sqrt {|a |} }}\exp \left( { - \frac{{{{({r - {\rho_0}} )}^2}}}{{\rho_0^2\sigma_r^2}} - \frac{{{{({\theta - {\theta_0}} )}^2}}}{{{a^2}\sigma_\theta^2}} + j\frac{{({\theta - {\theta_0}} )}}{a}} \right) \end{aligned}$$

with $\{{a \in {\mathbb R}|a \ne 0} \}$ and $\{{{\theta_0}|- \pi \le {\theta_0} \le \pi } \}$. The scaling parameter a scales the angular Gaussian width and azimuthal phase, and the parameter ${\theta _0}$ rotates the wavelets. The constant $1/\sqrt {|a |} $ preserves the ${{\mathbf L}^2}({{\mathbb R}^2})$ norm under the dilation and scaling operations. Note that Eq. (3) is separable in polar coordinates and the form of the angular part of Eq. (3) is similar to a Morlet wavelet [1,25]. Since we are defining optical fields as wavelets, if ${\Psi _{a,{\theta _0}}}$ describes the field of an optical beam in one plane, the OAM of that beam is related to the scaling parameter by $m = 1/a$, where m is the topological charge number. Therefore, the angular part of Eq. (3) is ideal for analyzing OAM information. Example functions in the continuous wavelet basis are given in Fig. 1, with ${\rho _0} = 17.5mm$, ${\sigma _r} = 0.42$, and ${\sigma _\theta } = 2$; these are the same parameters used in the experiment. In the experiment, ${\sigma _r}$ and ${\sigma _\theta }$ are determined by the dimensions of the input elliptical Gaussian beam and the log-polar mapping parameters. The probing radius of 17.5 mm was chosen to study the effects of strong turbulence. The convention to define strong turbulence is that the beam diameter divided by the Fried parameter is greater than one (D/r₀ > 1) [24]. Since the minimum Fried parameter in the experiment is r₀ = 3.8 mm, the maximum D/r₀ in the experiment is D/r₀ = 9.2.

Fig. 1. (a) Shows the intensity and phase of the mother wavelet in Eq. (2) with an illustration of the parameters, where in experiment ${\sigma _\theta } = 2$, ${\rho _0} = 17.5mm$, and ${\sigma _r} = 0.42$. (b) Shows selected functions in the continuous wavelet basis. Intensity and phase of Eq. (3) at different rotation angles and scales is shown. The topological charge number is also scaled and related to the scaling parameter by 1/a.

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The continuous wavelet transform decomposes a function $f(r,\theta ) \in {{\mathbf L}^2}({{\mathbb R}^2})$ onto the wavelet basis by measuring the inner product of every scaled and rotated wavelet with the function. In the experiment, the function f corresponds to the turbulence phase function, and the inner product is measured with a single lens and detector. The wavelet transform is defined as

(4)$$\begin{aligned} W(a,{\theta _0}) &= \left\langle {f,{\Psi _{a,{\theta_0}}}} \right\rangle \\ \textrm{ } &= \frac{1}{{\sqrt {|a |} }}\int_0^\infty {\int_0^{2\pi } {f(r,\theta ){\Psi ^\ast }\left( {r,\frac{{\theta - {\theta_0}}}{a}} \right)rdrd\theta } } , \end{aligned}$$

where $\left\langle { \cdot , \cdot } \right\rangle $ denotes the inner product in ${{\mathbf L}^2}({{\mathbb R}^2})$. Equation (4) can also be written as an angular convolution of f with each scaled wavelet. In the experiment we generate wavelet beams with a fixed radius ${\rho _0} = 17.5mm$. We do not probe multiple radial positions, but we have the angular decomposition of f about the radius ${\rho _0} = 17.5mm$ in terms of rotation/angular space (θ₀) and OAM (1/a).

Wavelet transforms are invertible, but the classical wavelet transform inversion formula requires that the wavelets satisfy an admissibility condition [1,7]. The admissibility condition states that the wavelet must have zero mean. The mean of this wavelet is solved as

(5)$$\frac{1}{{\sqrt {|a |} }}\int\limits_0^\infty {\int\limits_0^{2\pi } {\Psi \left( {r,\frac{{\theta - {\theta_0}}}{a}} \right)} } rdrd\theta = \rho _0^2{\sigma _r}{\sigma _\theta }\pi \frac{a}{{\sqrt {|a |} }}\exp \left( { - \frac{{\sigma_\theta^2}}{4}} \right).$$

For ${\sigma _\theta } > > 1$, the wavelet functions have approximately zero mean; however, they do not strictly satisfy the admissibility condition. Morlet wavelets also do not satisfy the admissibility condition but are commonly used regardless. An alternative reconstruction formula exists for continuous wavelet transforms without the use of the admissibility constant [26,27]. We derive the alternative reconstruction formula by first rewriting the wavelet transform in terms of the circular harmonic expansions of the wavelet and the turbulence phase function [28]. The circular harmonic expansion of a function f is given by

(6)$$f({r,\theta } )= \sum\limits_{n ={-} \infty }^\infty {{f_n}(r)\exp ({jn\theta } )} ,$$

where the coefficients of the expansion are given by

(7)$${f_n}(r) = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {f({r,\theta } )\exp ({ - jn\theta } )d\theta } .$$

Substituting the circular harmonic expansions of the wavelet and the turbulence phase into Eq. (4) we have

(8)$$\begin{aligned} W({a,{\theta_0}} )&= \int\limits_0^\infty {\sum\limits_{n ={-} \infty }^\infty {\sum\limits_{m ={-} \infty }^\infty {{f_n}(r)\Psi _m^\ast (r)rdr\int\limits_0^{2\pi } {\exp ({j\theta ({n - m} )} )d\theta } } } } \\ \textrm{ } &= 2\pi \sum\limits_{n ={-} \infty }^\infty {\int\limits_0^\infty {{f_n}(r)\Psi _n^\ast (r)rdr} } \\ \textrm{ } &= a{\sigma _\theta }\sqrt {\frac{\pi }{{|a |}}} \sum\limits_{n ={-} \infty }^\infty {\exp \left( { - \frac{{\sigma_\theta^2}}{4}{{({an - 1} )}^2} + jn{\theta_0}} \right)\int\limits_0^\infty {{f_n}(r)\exp \left( { - \frac{{{{({r - {\rho_0}} )}^2}}}{{\rho_0^2\sigma_r^2}}} \right)rdr} } . \end{aligned}$$

The derivation of the circular harmonic expansion of the wavelet can be seen in further detail in the supplementary material (See Supplement 1 ). If we approximate the radial Gaussian with a shifted delta function, we have

(9)$$\int\limits_0^\infty {{f_n}(r)\exp \left( { - \frac{{{{({r - {\rho_0}} )}^2}}}{{\rho_0^2\sigma_r^2}}} \right)rdr} \cong {f_n}({\rho _0}).$$

Therefore, the angular information of f about the radius ${\rho _0} = 17.5mm$ can be reconstructed from the complex wavelet coefficients $W(a,{\theta _0})$ using the reconstruction formula below,

(10)$$f({\rho _0},\theta ) = \frac{{ - j}}{{2\pi \textrm{erf}\left( {\frac{{{\sigma_\theta }}}{2}} \right)}}\int\limits_{\mathbb R} {\frac{\partial }{{\partial {\theta _0}}}\frac{1}{{\sqrt {|a |} }}} W(a,{\theta _0})da.$$

The full derivation can be found in the supplementary material (See Supplement 1 ). This reconstruction formula does not require that the wavelet satisfies the admissibility condition. Furthermore, it allows us to reconstruct localized features of the probing volume represented by f within a desired range of a or ${\theta _0}$. We can apply this reconstruction formula to the measurements since we are able to recover the phase of the wavelet transform.

The range of a in the experiment is ${a_{\min }} = 1/20$ and ${a_{\max }} = 1/2$. Since wavelets are used for multi-resolution analysis, there will be a theoretical resolution limit associated with each scaled wavelet function. We define the angular resolution ${\varepsilon _\theta }$ as the second moment of the wavelets in angular space given by the equation below,

(11)$$\varepsilon _\theta ^2 = \frac{1}{{||\Psi ||_2^2}}\int_0^\infty {\int_0^{2\pi } {{\theta ^2}{{|{{\Psi _a}(r,\theta )} |}^2}rdrd\theta } } ,$$

where $||\Psi ||_2^2 = {\sigma _r}{\sigma _\theta }\pi /2$ and ${\varepsilon _\theta } = a{\sigma _\theta }/2$ [1,7]. Similarly, we define the OAM resolution as the second moment of the OAM spectrum. The OAM spectrum of the wavelets is given by

(12)$$\begin{aligned} {G_n} &= 2\pi \int\limits_0^\infty {{{|{{\Psi _n}(r)} |}^2}rdr} \\ \textrm{ } &= \rho _0^2{\sigma _r}a\sigma _\theta ^2\sqrt {\frac{\pi }{8}} \exp \left( { - \frac{{\sigma_\theta^2}}{2}{{({an - 1} )}^2}} \right), \end{aligned}$$

where ${\Psi _n}(r)$ denotes the coefficients of the circular harmonic expansion of the scaled and rotated wavelets [29]. The center OAM can be calculated by

(13)$$m = \frac{1}{{||\Psi ||_2^2}}\int\limits_{ - \infty }^\infty {n{G_n}dn}, $$

where m is the first moment of the normalized OAM spectrum and is related to the scaling parameter by m = 1/a as expected. The second moment of the normalized OAM spectrum is given by

(14)$$\varepsilon _\phi ^2 = \frac{1}{{||\Psi ||_2^2}}\int_{ - \infty }^\infty {{{\left( {n - \frac{1}{a}} \right)}^2}{G_n}dn} ,$$

where ${\varepsilon _\phi }$ is solved as ${\varepsilon _\phi } = 1/(a{\sigma _\theta })$. The full derivation of Eqs. (11)–(14) can be found in the supplementary material (See Supplement 1). For wavelets with ${a_{\min }} = 1/20$ and ${\sigma _\theta } = 2$, the angular resolution is ${\varepsilon _\theta } = 0.05$ rad and the OAM resolution is ${\varepsilon _\phi } = 10$. For wavelets with ${a_{\max }} = 1/2$ and ${\sigma _\theta } = 2$, the angular and OAM resolution is ${\varepsilon _\theta } = 0.5$ rad and ${\varepsilon _\phi } = 1$, respectively. This resolution concept is known as the Heisenberg uncertainty of wavelets and will affect how the spatial versus the OAM information of the wavelet transform is localized.

3. Method

3.1 Generation method

Optical generation of the OAM based wavelets are implemented using the HOBBIT system [17]. The HOBBIT system is comprised of line generating optics, an AOD, and log-polar coordinate transform optics. The resulting beams are asymmetric rings with rapidly tunable OAM. With precise control of the RF signal applied to the AOD, we can scale and rotate the asymmetric beams while changing the OAM. The OAM is controlled by the frequency, the scale is controlled by the size of the Gaussian envelope on the frequency, and rotation is a natural result of the acoustic wave propagating across the AOD. The function of the AOD is critical to the generation system; the AOD provides high-speed amplitude and phase modulation. The AOD can also generate multiple coherent beams; this property is used to recover the phase of the wavelet transform. Figure 2 shows the experimental setup; a collimated 532nm laser is shaped into an elliptical Gaussian beam with three lenses. The resulting field entering the AOD can be expressed as

(15)$${E_i}(x,y) = \exp \left( { - \frac{{{x^2}}}{{\sigma_x^2}} - \frac{{{y^2}}}{{\sigma_y^2}} - j\frac{{2\pi }}{\lambda }({x\sin 2{\theta_B} + z\cos 2{\theta_B}} )} \right),$$

where λ is the wavelength, θ_B is the Bragg angle, and σ_x = 2.83mm and σ_y = 0.4mm are the 1/e² widths of the intensity in the x and y axis, respectively. The AOD is operated in the Bragg condition where the efficiency of light diffracted into the first order is maximized; the resulting beam’s amplitude and tilt angle can be controlled with the RF signal applied to the AOD. The signal applied to the AOD for wavelet generation is given by

(16)$$S(t )= \frac{1}{{\sqrt {|a |} }}\exp \left( { - \frac{{{t^2}}}{{{a^2}\sigma_t^2}} + j2\pi \left( {{f_A} + \frac{{\Delta f}}{a}} \right)t} \right),$$

where ${\sigma _t} = 2.94\mu s$ is the Gaussian width of the amplitude window, ${f_A} = 120MHz$ is the center frequency of the AOD, and $\Delta f = 108kHz$ is the frequency corresponding to a charge 1 OAM [17]. The field generated by this signal is given by

(17)$${E_d}(x,y) = \frac{1}{{\sqrt {|a |} }}\exp \left( { - \frac{{{x^2}}}{{\sigma_x^2}} - \frac{{{y^2}}}{{\sigma_y^2}} - \frac{{{{({x - tV} )}^2}}}{{{a^2}{V^2}\sigma_t^2}} - j\left( {\frac{{2\pi \Delta f}}{{aV}}({x - tV} )+ {k_z}z - 2\pi ({{f_c} + {f_A}} )t} \right)} \right)$$

Fig. 2. (a) Experimental setup showing the HOBBIT system for wavelet generation. A collimated 532 nm laser is shaped into an elliptical Gaussian beam with three lenses. The tilt angle and amplitude envelope of this elliptical Gaussian is controlled by the RF signals applied to the AOD. The resulting field is then mapped by the log-polar optics and propagated through the VTG. A simple detection system consisting of a single lens and detector is used. (b) Picture of the hardware showing the generation system and the VTG. (c) A turbulence perturbed beam for r₀ = 3.8 mm captured with a Phantom high-speed camera.

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In Eq. (17), $V = 650m/s$ is the acoustic velocity of the TeO₂ AOD, k_z is the z component of the k vector, and f_c is the center frequency of the laser. This field is then mapped by the log polar optics. The log polar transform is given by

(18)$$\begin{array}{l} u = A\theta ,\\ v ={-} A\ln \left( {\frac{r}{B}} \right). \end{array}$$

The log-polar coordinate transform maps coordinates (x = u, y = v) in the plane of the first log-polar optic to the polar coordinates (r, θ) in the plane of the second log-polar optic [30,31]. The constant parameters A and B are design parameters of the transform, where we have chosen $A = 0.955mm$ and $B = 1.5mm$. From this coordinate transform we have an expression for the field immediately exiting the system,

(19)$${E_\Psi } = \frac{1}{{\sqrt {|a |} }}\exp \left( { - \frac{{{{(r - {\rho_0})}^2}}}{{\rho_0^2\sigma_r^2}} - \frac{{{{(\theta - {\theta_0})}^2}}}{{{a^2}\sigma_\theta^2}} - \frac{{{\theta^2}}}{{{\sigma^2}}} - j\left( {\frac{{\theta - {\theta_0}}}{a} + {k_z}z - 2\pi ({f_c} + {f_A})t} \right)} \right),$$

where we have used the approximation $\ln (r/{\rho _0}) \cong r/{\rho _0} - 1$ for $0 \le r/{\rho _0} < 2$. In Eq. (19), ${\rho _0} = 17.5mm$ is the radius of the beam in the turbulence tunnel, ${\sigma _r} = {\sigma _y}/A = 0.42$, $\sigma = {\sigma _x}/A = 2.96$, ${\sigma _\theta } = V{\sigma _t}/A = 2$, and ${\theta _0}$ is the rotation parameter in the wavelet transform. The rotation of the wavelet is a function of time given by ${\theta _0} = tV/A$, which is related to the acoustic velocity of the AOD scaled by the log-polar A parameter. In our design the wavelet beam rotates π radians in 4.6µs. The scale of the wavelet is controlled by the width of the Gaussian envelope on the RF signal and the frequency of the RF signal. The field in Eq. (19) has the same form as Eq. (3) but with an additional angular Gaussian term $\exp ( - {\theta ^2}/{\sigma ^2})$ which is due to the Gaussian envelope of the beam incident on the log-polar optics. This term can be equalized out of the measurements to provide the same wavelet transform as in Eq. (4).

3.2 Detection method

We applied this system to measure the wavelet decomposition of a turbulence propagation path. Figure 2 shows the generation system, the variable turbulence generator (VTG), and the detection method. After generating the wavelet fields, we propagated the beams through a 60m turbulence path. Figure 2 displays the VTG from the transmitting side of the experimental setup. To achieve the desired 60m path length, the beam made three passes through the 20m VTG. By controlling the electrical current applied to a set of heating wires the VTG creates turbulence with a Fried Parameter varying from over 100cm to 0.38cm over a 60m path length. Experimental testing of the VTG showed that the observed turbulence closely follows theory [23,32]. The wavelet transform detection method consists of a single Fourier lens with focal length of 100mm and an avalanche photodetector APD430A2 with a 20µm pinhole placed one focal length away from the lens. The beam after propagating through the turbulence path can be described as

(20)$${\tilde{E}_\psi } = {E_\Psi }f(r,\theta )\ast{\ast} h(r,\theta ,z),$$

where $f(r,\theta )$ can be described as a turbulence phase screen and $h(r,\theta ,z)$ is the turbulence impulse response describing the propagation of the beam through the turbulence volume. Equation (20) describes the perturbed field as the result of the input field multiplied by a phase screen and then propagated through the volume, where ${\ast}{\ast} $ denotes a two-dimensional convolution. In the experiment we measured the on-axis component of the Fourier transform of this perturbed field. Let $\mathrm{\mathbb{F}}$ denote a Fourier transform and let $M(a,{\theta _0})$ denote the complex measurement. With a 20µm pinhole in the plane of our detector we measured the intensity of only the on-axis component of the Fourier transform of the perturbed field, described by

(21)$$\begin{aligned} {|{M(a,{\theta_0})} |^2} &= {|{\mathrm{\mathbb{F}}\{{{{\tilde{E}}_\psi }} \}(0,0)} |^2}\\ \textrm{ } &= {|{\mathrm{\mathbb{F}}\{{{E_\Psi }f(r,\theta )} \}({0,0} )\mathrm{\mathbb{F}}\{{h(r,\theta ,z)} \}({0,0} )} |^2}\\ \textrm{ } &= {\left|{\int\limits_0^\infty {\int\limits_0^{2\pi } {{E_\Psi }f(r,\theta )rdrd\theta } } \int\limits_0^\infty {\int\limits_0^{2\pi } {h({r,\theta ,z} )rdrd\theta } } } \right|^2}. \end{aligned}$$

where we have used Eq. (20) and the convolution property of Fourier transforms. The phase of $M(a,{\theta _0})$ was recovered using a direct measurement interferometry method. A sinusoidal frequency was added to the AOD signal in Eq. (16) to generate a reference beam with zero OAM. The phase recovery signal is given by the real part of the equation below:

(22)$$\begin{aligned} {S_\delta }(t )&= \frac{1}{{\sqrt {|a |} }}\exp \left( { - \frac{{{t^2}}}{{{a^2}\sigma_t^2}} + j2\pi \left( {{f_A} + \frac{{\Delta f}}{a}} \right)t} \right) + \exp ({j({2\pi {f_A}t + \delta } )} )\\ \textrm{ } &= S(t) + \exp ({j({2\pi {f_A}t + \delta } )} ). \end{aligned}$$

The phase of the wavelet transform was recovered with four power measurements with relative phase difference $\delta = \{{0,\pi /2,\pi ,3\pi /2} \}$ between the wavelet beam and reference beam. The field generated from the signal in Eq. (22) is given by

(23)$$\begin{aligned} {E_{ref}}(r,\theta ) &= {E_\Psi } + \exp (j\delta )\exp \left( { - \frac{{{{(r - {\rho_0})}^2}}}{{\rho_0^2\sigma_r^2}} - \frac{{{\theta^2}}}{{{\sigma^2}}} - j({{k_z}z - 2\pi ({{f_c} + {f_A}} )t} )} \right)\\ \textrm{ } &= {E_\Psi } + \exp (j\delta ){E_0}. \end{aligned}$$

Denote the mean of the turbulence impulse response by h₀. The four power measurements are given as follows:

(24)$$\begin{aligned} {P_1} &= {\left|{{h_0}\int\limits_0^\infty {\int\limits_0^{2\pi } {f(r,\theta )({{E_\psi } + {E_0}} )} } } \right|^2}\textrm{ }{P_2} = {\left|{{h_0}\int\limits_0^\infty {\int\limits_0^{2\pi } {f(r,\theta )({{E_\psi } + j{E_0}} )} } } \right|^2}\\ {P_3} &= {\left|{{h_0}\int\limits_0^\infty {\int\limits_0^{2\pi } {f(r,\theta )({{E_\psi } - {E_0}} )} } } \right|^2}\textrm{ }{P_4} = {\left|{{h_0}\int\limits_0^\infty {\int\limits_0^{2\pi } {f(r,\theta )({{E_\psi } - j{E_0}} )} } } \right|^2}. \end{aligned}$$

The phase was recovered by

(25)$${\tan ^{ - 1}}\left( {\frac{{{P_4} - {P_2}}}{{{P_1} - {P_3}}}} \right) = \angle \int\limits_0^\infty {\int\limits_0^{2\pi } {f(r,\theta ){E_\Psi }rdrd\theta } - \angle \int\limits_0^\infty {\int\limits_0^{2\pi } {f(r,\theta ){E_0}rdrd\theta } } } ,$$

where the angle symbol $\angle $ denotes the phase angle of a complex number; this resulted in the phase of $M(a,{\theta _0})$ referenced to the phase of the inner product of the reference beam and turbulence phase screen. Therefore, we recovered $M(a,{\theta _0})$ with five total measurements: one magnitude measurement and four relative phase measurements. After measuring $M(a,{\theta _0})$ the Gaussian term $\exp ( - {\theta ^2}/{\sigma ^2})$ in Eq. (19) was equalized to obtain the wavelet transform defined in Eq. (4). The wavelet transform is related to the measurement by

(26)$$\exp \left( {\frac{{{\theta_0}^2}}{{{\sigma^2}}}} \right)M(a,{\theta _0}) = W(a,{\theta _0})\int\limits_0^\infty {\int\limits_0^{2\pi } {h(r,\theta ,z)rdrd\theta } }$$

which resulted in the angular decomposition of the turbulence phase f(r,θ) about the probing radius of ${\rho _0} = 17.5mm$ multiplied by the mean of the turbulence propagation impulse response.

4. Experimental results

We measured the wavelet decomposition of the turbulent propagation path using wavelets with radius ${\rho _0} = 17.5mm$ and scale $a = ({ \pm 1/2, \pm 1/3,\ldots , \pm 1/20} )$ corresponding to OAM beams with $m = ({ \pm 2, \pm 3,\ldots , \pm 20} )$. The wavelet corresponding to an OAM of 1 was not included in the sequence since the entire angular Gaussian envelope could not be contained in the range of $[0,2\pi ]$. The turbulence strengths measured have a Fried parameter ranging from r₀ = 16.4mm to r₀ = 3.8mm. Each scaled wavelet was spaced 20µs apart. Five measurements are required to recover magnitude and phase of the wavelet transform. Rows of the wavelet transform correspond to the scale or OAM of the wavelet and columns of the wavelet transform correspond to the rotation angle of the wavelet. Measuring magnitude and phase of one row of the wavelet transform takes 100µs. With 38 total rows in the wavelet transform corresponding to 38 scaled wavelets it takes 3.8ms to measure magnitude and phase of the wavelet transform. During this measurement time the turbulence appeared frozen, and we obtained the wavelet transform for an instantaneous realization of turbulence. The detector was sampled at 125MSamples/sec and the time for one full rotation of a wavelet was 9.2µs, resulting in 1154 samples along the columns (θ₀) of the wavelet transform. An example continuous wavelet transform for an instantaneous realization of our strong turbulence case (r₀ = 3.8mm, D/r₀ = 9.2) is shown in Fig. 3. The magnitude-squared of the wavelet transform is called a scalogram. Scalograms and phase of the wavelet transform corresponding to positive and negative OAM are given. The area outside of the white-dashed line denotes the cone of influence, which we have defined as the rotation angle at which at least the 1/e value of the wavelet envelope is at the boundary [2]. Regions outside of the cone of influence will be influenced by boundary effects. The data was post-processed to account for the noise on the detector; a sliding mean smoothing filter of 10 points was applied to the data as well as a noise threshold of 19 mVolts. The noise threshold was defined as the mean of the noise multiplied by twice the standard deviation of the noise. From Fig. 3, one can see the localization of OAM information in angular space. The maximum points of the scalogram are called ridges. The ridges of the wavelet transform reveal most of the information contained within the scalograms [33,34]. Figure 3 also contains a reconstruction of the angular phase information of the turbulence phase screen about the radius $r = 17.5mm$ using Eq. (10). This represents the phase that one might use in an adaptive optics system to compensate for the zero-order turbulence distortion.

Fig. 3. (a) Wavelet transform of one realization of the strong turbulence case with r₀ = 3.8 mm. Magnitude squared (scalograms) and phase of the wavelet transform shown for both positive and negative values of OAM. (b) Reconstructed phase of the turbulence using Eq. (10). (c) Reconstructed phase wrapped around the probing radius for visualization.

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The wavelet transform reveals the angular information of the turbulence, and the input angular field that will propagate through the turbulence to the detector. We can demonstrate an example input field that will propagate through the turbulence to the detector by reconstructing a single ridge with Eq. (10). A single ridge was reconstructed to understand the spatial profile of the localized information that is common in most of the scalograms. The reconstruction of this ridge is shown in Fig. 4. The total angular information of the turbulence will be a linear combination of this reconstruction and the reconstruction of the rest of the wavelet transform. The reconstructed angular field was wrapped around the probing radius to demonstrate the reconstruction in two-dimensions. The angular width of the reconstructed field was approximately the size of r₀. Additionally, an OAM spectrum can be found by taking the Fourier transform of the reconstructed angular field. The OAM spectrum of the reconstructed ridge was similar to the OAM spectrum of a Gaussian beam the size of r₀ with OAM and shifted by the probing radius. The OAM spectrum of a Gaussian beam shifted by the probing radius is approximately given by $w/(2{\rho _0}\sqrt {2\pi } )\exp ( - {(nw/(8{\rho _0}))^2}),$ where w is the 1/e² width of the beam, ${\rho _0}$ is the amount of shift from the center of the axis, and n is the mode index [35]. For the given turbulence strength with ${r_0} = 3.8mm$, if the 1/e² width of the beam is $w = 2/\sqrt 2 {r_0}$ and ${\rho _0} = 17.5mm$, then the theoretical $1/{e^2}$ width of the spectrum is 26 and the width of the OAM spectrum of the reconstructed ridge is 23.2. If we consider that the reconstructed ridge is a Gaussian with OAM whose size is on the order of ${r_0}$, then the Gaussian distribution and the Gaussian width of the theoretical and reconstructed OAM spectrum closely match.

Fig. 4. (a) Reconstruction of one ridge of the wavelet transform from the strong turbulence case with r₀ = 3.8mm. Equation (10) is applied to the highlighted region of the wavelet transform of the positive OAM values. (b) Intensity and (c) phase of the reconstruction is given and is wrapped around the probing radius for illustrative purposes. (d) An OAM spectrum of the reconstruction can be obtained by taking the Fourier transform of the reconstruction. The width of the spectrum is similar to the width of the spectrum of a Gaussian the size of r₀ shifted by the probing radius.

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The probing method was used to measure the wavelet transform at different turbulence strengths. Figure 5 shows selected wavelet transforms for r₀ = 16.4mm, r₀ = 8.6mm, r₀ = 5.4mm, and r₀ = 3.8mm. It is clear from visually observing these wavelet transforms how the characteristics of the ridges change with the turbulence strength. As the turbulence strength increases, the angular width of the ridges decreases and the peak OAM value increases. The dynamics of the turbulence can also be observed from this probing method. For each turbulence strength, 400ms of continuous data was taken with 10ms of time between each wavelet transform measurement. Figure 6 shows an example of the time-evolution of the wavelet transform for the high turbulence strength case of r₀ = 3.8mm. Note that θ₀ = 0 corresponds to the bottom of the probing radius within the VTG and θ₀ = ±π/2 corresponds to the left and right of the probing radius within the VTG; the wavelet beams at these positions are shown in Fig. 1. The measured data shows that the ridges tend to move away from θ₀ = 0 towards θ₀ = ±π. The heat flow of the turbulence is in the vertical direction and thus we may infer that the motion of the ridges is related to the wind velocity.

Fig. 5. Selected scalograms of instantaneous turbulence realizations for different turbulence strengths: (left to right) r₀ = 16.4mm, r₀ = 8.6mm, r₀ = 5.4mm, and r₀ = 3.8mm.

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Fig. 6. Scalograms in a continuous set of data. Measurements of instantaneous turbulence realizations for r₀ = 3.8mm at relative time of 0ms, 10ms, 20ms, and 30ms are shown.

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A summary of the dynamic information obtained from the scalograms is given in Table 1. The first column of Table 1 lists the Fried parameter of the VTG and the associated C_n² values for each turbulence strength that was experimentally probed. These values were measured during the characterization of the VTG. The second column lists the measured ridge velocity for each turbulence strength. This information was obtained by integrating the OAM information of the scalograms, finding the peaks in the angular coordinate, and tracking the motion of these peaks over time. For the strong turbulence case the ridge velocity closely matches a wind speed simulation in COMSOL of 0.1m/s to 0.2m/s [23]. The same tracking method was used to measure the ridge persistence, which is defined as the amount of time for a ridge to be indistinguishable from the others or fade away. This measurement gave a broad range of persistence times; for the strong turbulence case, ridges persist from 10ms to over 100ms. The ridge persistence is information that is unique to the scalograms and indicates that eigenchannels can be tracked, can last for a long period of time, and may be predictable. The coherence time is obtained from the Fried parameter divided by the measured ridge velocity and is a measure of the amount of time for r₀ to move its own diameter. The theoretical coherence time for the strong turbulence case is 38ms. The ridge width was measured by integrating the OAM information of the scalograms, finding the peaks in the angular coordinates, and measuring the FWHM of these peaks. The ridge width is very close to the Fried parameter for each turbulence case. The ridge peak OAM was found by tracking every ridge in the angular coordinate, finding the peaks, and measuring the peak OAM value of the scalograms. This is another parameter that is unique to the scalograms and indicates the amount of OAM and spectral width required to send information through the turbulence. As turbulence strength increases, we expect the Fried parameter to get smaller. The experimental results indicate the smallest features in the wavelet transform are on the order of the Fried parameter. The smallest wavelet used in the experiment has a theoretical spatial resolution of 0.05 rad, or 0.875 mm in terms of arc length. Based on the theoretical spatial resolution and experimental results, we can still probe much stronger turbulence with the current system. If the Fried parameter of turbulence was less than 0.875 mm, we would need wavelets with smaller spatial resolution. We could achieve this either by including wavelets with smaller scale (higher OAM), or by defining a new set of wavelets with the same scale but smaller ${\sigma _\theta }$.

Table 1. Summary of mean values of dynamic information gathered from scalograms

View Table

5. Conclusions

This work has presented a two-dimensional wavelet capable of analyzing localized OAM information in angular space and time. The wavelet concept was applied experimentally to generate OAM based optical wavelet fields, and a new probing method was introduced enabling measurements of the wavelet transform of the environment. We applied the HOBBIT system to generate the wavelet basis with an AOD and log-polar optics. The use of an AOD results in a fast-probing method and can measure wavelet transforms in 3.8ms or less. The detection method is simple and requires a Fourier lens and single detector. The speed of the system enables measurements of dynamic environments such as turbulence. The OAM based wavelets were applied to turbulence for measurements of localized OAM information in angular space of the turbulence propagation path. This allows new insight into instantaneous turbulence characteristics. One insight gained from the wavelet transforms is the width of the ridges of the scalograms is approximately the size of the Fried parameter of the turbulence. The phase of an instantaneous turbulence realization was reconstructed and illustrates the phase that one might apply in an adaptive optics system to compensate for turbulence distortion. Some relevant features of the turbulence dynamics such as wind velocity and coherence time were extracted from the wavelet transforms. The wavelet transforms also reveal input beams that propagate to the target. Future work will involve probing the full turbulence volume by radially scaling the wavelets, orthogonalizing the wavelets for faster probing time, and exploring other types of wavelets such as spiral beams [36].

Funding

Office of Naval Research (N00014-20-1-2558).

Acknowledgments

We would like to thank Fletcher Blackmon, Mike Wardlaw, and Peter Morrison of ONR for their discussions and insights about the application and motivation of this work.

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.

Supplemental document

See Supplement 1 for supporting content.

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Fried Parameter (mm) and C_n² (m^-2/3)	Measured Ridge Velocity (m/s)	Measured Ridge Persistence (ms)	Measured coherence time (ms)	Ridge width (mm)	Ridge peak OAM
r₀= 16.4	0.06	102	329	14.8	2.96
C_n² = 2.7 × 10⁻¹³	0.06	102	329	14.8	2.96
r₀= 8.6	0.06	86	204	9.7	4.90
C_n² = 7.9 × 10⁻¹³	0.06	86	204	9.7	4.90
r₀= 5.4	0.09	63	73	7.1	6.75
C_n² = 1.7 × 10⁻¹²	0.09	63	73	7.1	6.75
r₀= 3.8	0.09	59	47	5.6	9.93
C_n² = 3.1 × 10⁻¹²	0.09	59	47	5.6	9.93

OAM-based optical wavelet using a single pixel detection system for probing dynamic environments with application to real-time measurements of strong atmospheric turbulence

Abstract

1. Introduction

2. Wavelet theory

3. Method

3.1 Generation method

3.2 Detection method

4. Experimental results

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (6)

Tables (1)

Equations (26)

Optics Express