## Abstract

A stationary beam forming an Airy-like spectral density in the far field is analyzed theoretically and experimentally. The Schell-model source that radiates such a beam is an extended version of a recently introduced source [O. Korotkova, *et al*., Opt. Lett. **43**, 4727 (2018) [CrossRef] ; X. Chen, *et al*., Opt. Lett. **44**, 2470 (2019) [CrossRef] , in 1D and 2D, respectively]. We show, in particular, that the source degree of coherence, being the fourth-order root of a Lorentz-Gaussian function and having linear and cubic phase terms, may be either obtained from the Fourier transform of the far-field Airy-like pattern or at the source using the sliding function method. The spectral density of the beam is analyzed on propagation through paraxial ABCD optical systems, on the basis of the generalized Collins integral, by means of the derived closed-form expression. We show that the distribution of the side lobes in the Airy beam spectral density can be controlled by the parameters of the source degree of coherence. Further, an experiment involving a spatial light modulator (SLM) is carried out for generation of such a beam. We experimentally measure the complex degree of coherence of the source and observe the gradual formation of a high-quality Airy-like spectral density towards the far field. In addition, the trajectory of the intensity maxima of the beam after a thin lens is studied both theoretically and experimentally. The random counterpart of the classic, deterministic Airy beam may find applications in directed energy, imaging, beam shaping, and optical trapping.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, there has been a growing interest in spatial structuring of the (complex-valued) degrees of coherence (DOCs) of partially coherent beams, due to their importance in optical science and potential applications in directed energy, beam shaping, free-space optical communications and optical trapping (c.f. [1–16] for papers published only during last two years). The more established and notable modal partially coherent beams with controllable DOCs and experimental procedures for their generation are overviewed in [17,18]. In general, partially coherent beams can be divided into three categories: those with uniform, non-uniform and twisted correlations. In uniformly-correlated sources (also known as Schell-model sources), the two-point DOC functions solely depend on the separation distance between two points. The most notable members of this class are multi-Gaussian correlated Schell-model beams [19,20], Bessel- and Laguerre-Gaussian correlated Schell-model beams [21,22], and cosine-Gaussian correlated beams [23]. Through controlling the DOCs, the radiated Schell-model beams are capable of producing flat-topped, dark-hollow and four beamlets profiles in the far field, which is useful for beam shaping and other optical manipulations. The second category of partially coherent sources contains those with spatially varying coherence: the values of the two-point DOC functions may also depend, in addition to the separation distance, on the specific points’ choice. The best examples of such sources are the nonuniformly correlated beams [24], partially coherent beams with circular/radial coherence [11,16], as well as the pseudo Schell-model beams [8]. Owing to point-dependent source coherence properties, these beams may self-focus both on and off axis, on free-space propagation [24]. The third category is the variety of beams with the twist phase [25–27], capable of rotation of the formed spectral density, as the beam propagates. This category can be regarded as a special case of that with spatially-varying correlations but it does stand alone, due to the presence in it of the orbital angular momentum (OAM).

However, most of the aforementioned studies were concerned with a particular case of the real-valued DOC distributions (apart from beams with twist phase). When the DOC is complex-valued, its non-trivial phase can be shown to break the Cartesian symmetry in the propagating spectral density (e.g., [28,29]). Special attention has been recently paid to structuring of the phase of the DOC from the source plane, on the basis of the newly introduced “sliding function” technique [30–32]. The essence of the method is in self-convolution of any Hermitian function making it possible to readily obtain unlimited classes of the Schell-model sources, which, otherwise, is somewhat hard to directly model due to the non-negative definiteness restriction. One particular outcome of the sliding function method was the model for the DOC involving a cubic phase leading to the Airy-like spectral density profile in the far field [30,31]. It was also illustrated in [31] that other powers of the phase function may result in similar multi-lobe distributions. In addition, it was shown in [33] that the magnitude and the phase of the sliding function do not need to be chosen independently and can be intrinsically related by a planar curve. Also, in [34] it was demonstrated that a non-linear phase of the DOC substantially dominates its magnitude distribution on forming the far-field spectral densities.

In 1979, Berry and Balazs theoretically demonstrated that the Airy wave packet is a solution of the Schrödinger equation for a free particle [35]. Later, Gori and co-workers investigated the time-dependent Schrödinger equation for a particle moving under the action of uniform force, and demonstrated that the Airy wave function is also a solution in this case [36]. Due to the analogy between the Schrödinger equation in quantum mechanics and the paraxial wave equation in optics, the possible extension of the Airy wave function into optical regime was also discussed in [36]. In 2007, Siviloglou et. al first theoretically predicted and experimentally observed the Airy waves/beams in optical regime [37,38]. Since then, considerable attention has been paid to such deterministic Airy optical waves.

One of the intriguing propagation characteristics of Airy beams is that its field structure takes the curved parabolic trajectory (self-acceleration) along the propagation axis even in the absence of any refractive index potential [38–40]. Owing to their unique features, the Airy beams have found important applications in optical manipulation [41–43], plasma guidance [44], and optical imaging [45,46]. In addition, Airy beams are the diffraction-free solutions of the paraxial wave equation in one or two transverse dimensions. Thus the ideal Airy beam is propagation-invariant and possesses self-healing properties [47], which is very important for propagation in turbulence or in other random environments. The detailed review for the Airy waves and their applications can be found in [48] and references therein. The ideal Airy beams carry infinite amount of energy and therefore cannot be realized experimentally. The practically accessible, experimentally synthesized Airy beams are truncated by suitable, typically exponentially decaying functions. Although such Airy beams can preserve non-diffracting and self-accelerating characteristics over a certain intermediate distance, their far-field beam profiles will degrade to Gaussian-like shape due to the diffraction effects imposed by the truncating screen.

The aim of this paper is to introduce a more general analytical model (as compared with that in [30] and [31]) for a DOC producing the Airy spectral density profile in the far field (but having Gaussian shape in the source plane), that exactly matches its deterministic counterpart [38]. The derived DOC is a complex-valued function whose amplitude is the fourth-order root of a Lorentz-Gaussian function and its phase contains a cubic, a Gouy, and a linear phase terms. For this, two methods, the inverse Fourier transform and the sliding function method are applied and shown to produce equivalent results. The propagation of such a partially coherent beam through an ABCD optical system is investigated and shows that the Airy-like spectral density pattern is formed in the far field or in the focal plane. Furthermore, a random source radiating to the Airy-like far-field spectral density is experimentally generated via an optical arrangement involving a single spatial light modulator (SLM). The complex DOC is then measured using the generalized Hanbury Brown and Twiss (HBT) type experiment. The focusing properties of the source and the trajectory of the main lobe passing through a single lens depending on the parameters of the source DOC are also examined experimentally.

## 2. Theoretical model of a random source generating Airy-like spectral density in the far field

Let us consider a scalar, stochastic beam-like field, propagating from source plane ($z = 0$) to the half space ($z \;>\; 0$). According to the optical coherence theory in the space-frequency domain [49], the second-order statistics of the field in the source plane are characterized by the cross-spectral density (CSD) function specified at two points $\mathbf {r}_1$ and $\mathbf {r}_2$

*Modified Complex Lorentz-Gaussian Correlated*(MCLGC) source. A straightforward comparison shows that, apart from the two last phase factors, and some re-definition of parameters, the DOC in Eq. (8) has a similar form to that of [31]. It follows from Eq. (8) that the CSD function is separable with respect to $(x_1, x_2)$ and $(y_1, y_2)$, and the spectral density $S_0(\mathbf {r}) = W_0(\mathbf {r},\mathbf {r})$ in the source plane is of a Gaussian profile.

Let us examine in greater detail the structure of the DOC in Eq. (8). Its magnitude is a fourth-order square root of a Lorentz-Gauss function being even. The phase contains three terms. The first one is a cubic phase which plays an important role for generating an Airy beam pattern in the far field. The second term represents the phase delay associated with the difference of $x_1$ and $x_2$, similar to the Gouy phase but depending on the difference of transverse coordinates (we may term it as the DOC Gouy phase). The last one is the tilt phase determined the mean propagation axis of the beam in the defined coordinate system. Further, if we expand the second phase term $\varphi (x)$ in the Taylor series: $\varphi (x) = x - x^3 /3 + x^5/5 -\cdots$, it also contains the tilt phase term: $x_d/4a_0^{3/2} \delta _0$. By combining the tilt phases in the second and the third phase terms, we obtain the mean propagation direction of the MCLGC beam with respect to the $z$-axis to be $\theta _{x(y)} = (1/4a_0^{3/2} - a_0^{3/2}) /k\delta _0$, where $k = 2\pi /\lambda$ is the wavenumber of a light beam with $\lambda$ being the wavelength.

We now illustrate that the same model can be obtained by the sliding function $g(\mathbf {r}_d)$ method of [31] if we set

withFigure 1 illustrates the density plots of the square of the modulus, the real part, and the imaginary part, of the DOC in Eq. (7) or Eq. (11), in the source plane. The parameters used in the calculation are chosen to be $a_0=0.1$, $b_0=0.12$ mm, and $\delta _0=0.4$ mm. One finds that the square of the modulus of the DOC is similar to a diamond shape but with long tails along the $x$- and $y$- directions. Due to the cubic phase and the lateral Gouy phase modulation, both real and imaginary parts of the DOC exhibit complex profiles. It follows from Eq. (8) that the real part is an even function which is symmetric with respect to the origin, while the imaginary part is an odd function which guarantees the non-negativity of the beam’s spectral density during propagation.

## 3. Second-order statistics of the MCLGC beam passing through a paraxial ABCD optical system

In this section, we first derive the CSD function of the MCLGC beam propagating through a paraxial ABCD optical system. The evolution of the spectral density passing through a thin lens focusing system is then studied in detail through numerical examples.

Within the accuracy of paraxial approximation, the propagation of the spectral density (equal-point CSD function) of a partially coherent beam from the source plane ($z = 0$) to the output plane through a paraxial ABCD optical system can be treated by the extended Collins integral [51,52]:

As a numerical example, we consider a simple optical system in which a beam passes through a thin lens located at the source plane. Under this circumstance, the elements of the transfer matrix are $A = 1-z/f$, $B = z$, $C = -1/f$, and $D = 0$. Figure 2 illustrates the normalized spectral density $S(\boldsymbol {\rho }, z)/[S(\boldsymbol {\rho }, z)]_{\textrm {max}}$ of the MCLGC beam at several propagation distances after the lens with focal length $f = 250$ mm. The term $[S(\boldsymbol {\rho }, z)]_{\textrm {max}}$ denotes the maximum spectral density at the propagation distance $z$. In the calculation, other parameters are chosen to be $a_0 = 0.1$, $b_0 = 0.12$ mm, $\delta _0 = 0.40$ mm, $\sigma _0 = 1.0$ mm, and $\lambda _0 = 632.8$ nm. It can be seen from Fig. 2 that the beam profile gradually transforms from Gaussian to Airy-like, as it propagates from the source plane to distance $z = 250$ mm (in the far field). However, before ($z = 225$ mm) or after ($z = 270$ mm) the focal plane, the Airy beam profile is more or less blurred. From the propagation formula in Eq. (15), one can see that the spectral density is the convolution of the $p(\mathbf {v})$ function with the Gaussian filter $f_{x(y)}$, and therefore, function $f_{x(y)}$ acts as a point spread function that modified the “resolution” of the output Airy beam pattern [$p(\mathbf {v})$ function]. The width of the Gaussian function $f_{x(y)}$ plays a critical role in determining the spectral density of the MCLGC beam during propagation. The smaller the width of function $f_{x(y)}$ is, the higher resolution of the Airy beam profile in the output plane is. In Fig. 3, we plot the variation of the width of function $f_{x(y)}$, i.e., $w_d = \sqrt {2} k \sigma _0 \Lambda /B c_0$, defined as the $1/\mathrm {e}$ point of $f_{x(y)}$, as it varies with propagation distance $z$. For comparison, the width of $p(\mathbf {v})$ is also plotted in Fig. 3, which is defined as the width from the on-axis point to $1/\mathrm {e}$ point for the truncation function $\exp (2a_0x/b_0)$, for $x\;<\;0$. It can be clearly seen that the width of function $f_{x(y)}$ is much narrower near the focal region, the minimum width occurs at the propagation $z=f$ that is located in the focal plane, implying that the Airy beam pattern reaches the maximum resolution.

Figure 4 shows the normalized spectral density of the MCLGC beams in the focal plane for different values of parameters $a_0$ and $\delta _0$. The far-field spectral densities are closely dependent on these parameters. As parameter $a_0$ increases, the side lobes are gradually suppressed. This is because the exponential function shown in Eq. (4) decays much faster with the increase of $a_0$. As a result, only the main lobe remains the same for large values of $a_0$. When the coherence width $\delta _0$ increases, the Airy profile is gradually blurred. The reason is that when other parameters are fixed, the value of parameter $w_d$ is proportional to the coherence width, i.e., $w_d \propto \delta _0$ . Therefore, the width $w_d$ increases with the increase of $\delta _0$, resulting in the low quality of the Airy beam pattern.

## 4. Experimental setup for generating the MCLGC beam, measuring the DOC, and its propagation characteristics

In this section, we carry out the experiment for generation of the MCLGC beam and measuring the real and imaginary parts of its DOC utilizing the generalized Hanbury Brown Twiss (HBT) experiment. After that, we examine the measured propagation characteristics of the beam after its passing through a thin lens.

Up to now, several methods have been proposed for experimental realization of partially coherent beams with prescribed DOCs. One of the methods relies on modulation of the intensity profile of light beams in the Fourier plane where the rotating ground glass is located [22]. On the basis of the generalized van Cittert-Zernike theorem, the DOC in the source plane is the Fourier transform of the light intensity pattern in the Fourier plane. Another widely used approach is to apply an ensemble of random phase screens realizations with a prescribed correlation function, which is used to turn a coherent beam to a random beam [53–58]. The random phase screens can be synthesized using a computer program in advance and loaded dynamically on spatial light modulators (SLM). The second method is much more flexible than the first one and we adopt it to generate the MCLGC beam.

Figure 5 shows the experimental setup for the generation of the MCLGC beam and the measurement of the real and imaginary parts of the DOC. A $x$-polarized Gaussian beam generated from a He-Ne laser ($\lambda =632.8$ nm) is first divided into two parts by a beam splitter ($\textrm {BS}_1$). The reflected part passes through a neural density filter (NDF) and a quarter wave plate (QWP) whose fast axis forms an angle of $\pi /4$ with the $x$-axis. Thus, the output beam after the QWP becomes the circularly polarized beam and the phase difference between the $x$- and $y$- polarized component is $\pi /2$. Such the beam is further expanded by a beam expander ($\textrm {BE}_1$) and reflected by a mirror ($\textrm {M}_2$), acting as a reference which is used to measure the real and the imaginary parts of the DOC of the generated random beam. The transmitted part is first expanded by a beam expander ($\textrm {BE}_2$), then goes towards a SLM (LC2012, Holoeye). In this application, the SLM acts as a thin screen on which a series of random phase screens is loaded dynamically. To generate the high-quality MCLGC beam, we apply the transverse shift version of the power spectrum [$p(\mathbf {v})$ function] to synthesize the random phase screens. The synthesizing process has been reported in [58]. After the SLM, the partially coherent light passes through a linear polarizer (LP) whose transmission axis is along $x$-axis and a half wave plate (HWP) whose fast axis forms an angle of $\pi /8$ with the $x$-axis. After the HWP the beam becomes the $\pi /4$-linear polarized beam, and then enters a two-lens system consisting of lenses $\textrm {L}_1$ and $\textrm {L}_2$. Both focal lengths of $\textrm {L}_1$ and $\textrm {L}_2$ are 250 mm. A circular aperture located in the focal plane of $\textrm {L}_1$ is used to block unwanted diffraction orders, only allowing the first diffraction order to pass. In the plane after $\textrm {L}_2$, the generated beam can be regarded as the MCLGC source. Once the beam is generated, it is split into two parts by a beam splitter (BS$_2$), and the transmitted part is focused by a thin lens $\textrm {L}_3$ with focal length $f_3 = 250$ mm. A CCD detector (D$_1$) is mounted on the transition stage along the $z$-direction, being used to examine the focusing properties of the MCLGC beam. The reflected part of the beam is mixed with the reference beam from the reference path, and then is divided into reflected part and transmitted part by a polarization beam splitter (PBS), finally arrives at a CCD$_2$ (D$_2$) and a CCD$_3$ (D$_3$), respectively. A lens L$_4$ is located between BS$_2$ and PBS for imaging the MCLGC source in the detector plane with unit magnification, i.e., both the distance from the source plane (dashed line) to the L$_4$ and from the L$_4$ to D$_2$/D$_3$ is $2f_4$. Under this circumstance, the electric field of one realization in the D$_2$ and D$_3$ plane can be expressed as $E_2(\mathbf {r}_1) = E_x^s (\mathbf {r}_1) + E_x^r (\mathbf {r}_1)$ and $E_3(\mathbf {r}_2) = E_y^s (\mathbf {r}_2) + E_y^r (\mathbf {r}_2)$, respectively. $E_j^s$ and $E_j^r$, ($j = (x, y)$) denote the $x(y)$ components of the electric filed of the random beam and reference beam, respectively. Through performing the auto intensity-intensity correlation and cross intensity-intensity correlation between two detectors, the real and imaginary part of DOC of the random field can be extracted. The detailed information for measuring the DOC can be found in [59].

Figure 6 presents the experimental results of the square of the amplitude as well as of the real and imaginary parts of the DOC of the generated source. It is shown that the modulus of square of DOC indeed exhibits the diamond-like shape with rectangular symmetry. The real and imaginary parts display the even and odd function characteristics. Through theoretical fitting, the parameters measured from the experiment are about $a_0 = 0.1$, $b_0 = 0.12$ mm, and $\delta _0 = 0.4$ mm. The experimental results agree well with the theoretical predictions shown in Fig. 1.

Figure 7 illustrates the measured average spectral density of the generated beam at different propagation distances after passing through the thin lens L$_3$. Note that in this measurement, the reference beam is blocked by a hard aperture in the reference path. The beam profile is captured by the CCD with averaging over $2000$ frames. As expected, the spectral density gradually evolves from the Gaussian shape to the Airy pattern in the focal plane ($z = 250$ mm) due to the effect of the initial coherence structure. After the focal plane, the Airy beam profile is gradually blurred [see in Fig. 7(f)].

As discussed in [37,38], the peculiar feature of the Airy beams is that the trajectory of the main lobe (intensity maxima) follows a parabola of a free falling body during free-space propagation. In the experiment, we also examine the lateral shift of the intensity maxima of the MCLGC beam with the propagation distance after passing through a lens for three different initial parameters $a_0$ and $\delta _0$. The experimental results (circular dots) are plotted in Fig. 8. Such shift is defined as the distance from the intensity maxima to the centroid of the beam at that plane. For comparison, the theoretical results (solid lines) are also plotted in Fig. 8. It is shown that the lateral shifts somewhat deviate from the parabola of a free-falling body, and are closely related to the initial beam parameters such as $a_0$ and the coherence width $\delta _0$. Especially for the case of $a_0 = 0.2$, $\delta _0 = 0.19$ mm [see in Fig. 8(c)], there is a bump near the distance in the range from 20 cm to 25 cm. This feature provides one an efficient way to control the trace of intensity maxima through modulating the initial beam parameters.

## 5. Summary

In conclusion, we have analyzed theoretically and experimentally a partially coherent beam whose DOC in the source plane is the fourth-order root of the Lorentz-Gaussian function associated with the cubic phase, named a MCLGC beam, and derived the propagation formula for its passage through an ABCD optical system. In particular, we have established a close relation between the MCLGC and its recently introduced version [31]. We have used two methods for finding the DOC of the MCLGC, the inverse Fourier transform method and the sliding function method, to confirm obtaining identical results. It was shown that the beam gradually acquires the Airy beam pattern towards the far field or in the focal plane. The number of side lobes of the far-field Airy beam can be controlled by the initial parameters of the DOC in the source plane. Furthermore, we experimentally generated the MCLGC beams through loading specially designed random phase screens on an SLM. We have also investigated the focusing properties of the MCLGC beams passing through a single lens, and observed a high-quality Airy beam in the focal plane. The trajectories of the intensity maxima (main lobe of Airy beam) in the focusing process with different beam parameters were also studied experimentally. The introduced MCLGC beams have potential applications to directed energy, optical trapping and particle guiding.

## Appendix A

To derive the analytical expression for the CSD function shown in Eqs. (7) and (8), we first insert Eqs. (4) and (6) into Eq. (3), it becomes

## Appendix B: Derivation of the DOC from the convolution of sliding functions

The sliding function in Eq. (9) is a separable function with respect to $x_d$ and $y_d$, and therefore, the convolution of the $g_x$ and $g_y$ can be treated separately. In this appendix, we only consider the convolution of $g_x$, the result for the $g_y$ is the same.

The convolution of the sliding function $g_x$ takes the form

## Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11525418, 11774251, 11874046, 11904247, 11974218, 91750201); Innovation Group of Jinan (2018GXRC010); China Postdoctoral Science Foundation (2019M661915); Natural Science Foundation of Shandong Province (ZR2019QA004); Priority Academic Program Development of Jiangsu Higher Education Institutions; Natural Science Research of Jiangsu Higher Education Institutions of China (19KJB140017); Qinglan Project of Jiangsu Province of China.

## Disclosures

The authors declare no conflicts of interest.

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