1. Introduction

In 1998, C. M. Bender and S. Boettcher demonstrated that the eigenvalues of a non-Hermitian Hamiltonian can all be real-valued if the Hamiltonian has Parity-Time ($\mathcal {PT}$) symmetry [1]. This discovery was the beginning of a new formulation of non-relativistic quantum mechanics by relaxing the Hermitian restriction for the observable quantities. Such formulation gained strong support after the discovery of a new inner product for the Hilbert space [2]. Thus, by relaxing the Hermitian constraint, and using this new definition for inner product, all remaining postulates of ordinary quantum mechanics, such as unitary evolution, are automatically satisfied [3–7]. However, verifying that a given Hamiltonian is $\mathcal {PT}$-symmetric is not sufficient to guarantee that its eigenvalues are all real-valued. To give a simple example, the class of $\mathcal {PT}$-symmetric Hamiltonians $H(\alpha ) = p^2 - (ix)^{\alpha }$, with $p$ and $x$ being momentum and position operators, has a real-valued spectrum only if $\alpha \geq 2$ [1] (the case $\alpha = 2$, representing the usual harmonic oscillator, is known as the exceptional point). Thus, $\mathcal {PT}$ symmetry is not sufficient to guarantee the reality of eigenvalues. In the parametric region $\alpha < 2$, we say that the system is in a broken phase, while for $\alpha \geq 2$ the system is in unbroken. The $\mathcal {PT}$-symmetric formulation of non-relativistic quantum mechanics is thus only valid for the unbroken phases of the Hamiltonian. Sufficient and necessary conditions for the eigenvalues to be real were obtained by Mostafazadeh, shortly after the publication of the seminal work by Bender [8–10].

Due to the analogy existing between the time-dependent Schrödinger equation and the (generalized) paraxial wave equation in optics, it has become possible to model, in some respects, the quantum behavior of a particle under the action of a non-Hermitian Hamiltonian, by using classical optical fields [11]. In fact, the first experimental observation of $\mathcal {PT}$-symmetry enabled phenomena came from the field of optics [12]. The complex potential $V(\boldsymbol {r})$ in quantum mechanics becomes equivalent to a complex refractive index $n(\boldsymbol {r})$ in optics. The imaginary part of the refractive index is related to the gain or loss present in the material and the $\mathcal {PT}$ symmetry condition requires that the imaginary part $n_i(\boldsymbol {r})$ of $n(\boldsymbol {r})$ be odd under inversion, $n_i(-\boldsymbol {r}) = -n_i(\boldsymbol {r})$, implying that gain and loss must be equal balanced [11]. A number of new optical effects have been discovered in the following years such as unidirectional invisibility [13], a $\mathcal {PT}$-symmetric coherent perfect absorber laser [14], birefringence and power oscillations in periodic lattices [15], to cite a few.

Despite the large number of new studies related to non-Hermitian photonics, the majority of results are based on the assumption that the optical field is highly coherent. It is well known that the state of optical (spatial and temporal) coherence affects wave propagation and its interaction with matter [16–18]. The effect of classical coherence on non-Hermitian optical systems has just begun to be explored in scattering systems with deterministic [19–23] and random [24] materials. It was shown that gain and loss strongly influences the coherence properties of the scattered field, changing its directional [20,24] and spectral [21,22] properties. In fact, it was demonstrated in [23] that Gaussian-Schell beams with a low spatial coherence can induce large resonant peaks in the transmitted and reflected amplitudes of a double-layer structure having $\mathcal {PT}$ symmetry. These resonances disappear in the case of states with high coherence, thus making this effect unique for beams with a low degree of spatial coherence.

Here, we investigate the problem of scattering of partially coherent vortex beams by a $\mathcal {PT}$ dipole. After its discovery by Allen et. al. in 1992 [25], coherent beams carrying well-defined orbital angular momentum (OAM) have been considered in both classical and quantum protocols [26,27] due to the new physical properties that these beams possess, including topological charge [28], polarization [29] and a variety of spatial field distribution [30]. In contrast to fully coherent beams, partially coherent light is known to possess properties which may be used to improve applications such as free-space communication [31], atom cooling [32], imaging [33] and optical trapping [34]. Vortex beams with partial coherence can have both, conventional and unconventional correlation functions and therefore, can be described by Gaussian-correlated and non-Gaussian-correlated Schell models and even non-Schell models [35,36]. Concerning the latter, models describing partially coherent vortex beams carrying arbitrary topological charge [37] and arbitrary radial orders [38] have been derived and used for proper characterization of their phase singularities and OAM behaviour. A $\mathcal {PT}$ dipole is simply a pair of point particles, one having gain and the other having loss, that interacts with the incident field. Scattering of light from the $\mathcal {PT}$ dipole has been considered recently under deterministic [39] and stochastic [20] light illumination. Our objective is to understand how the scattered radiation behaves when the incident wave has a nontrivial phase and amplitude distributions. This is, in general, a very difficult problem and we must rely on some approximations. Therefore, the first Born approximation is assumed throughout the paper along with the scalar approximation for the fields. Due to the singular nature of the potential [Eq. (2) below], we are able to obtain analytical solutions to the problem, which we will now explore.

2. Scattering theory for partially coherent fields

Consider a non-Hermitian point dipole, having $\mathcal {PT}$ symmetry, described by the refractive index $n(\boldsymbol {r},\omega )$. In what follows, we omit the frequency dependence of all quantities, since we are only considering monochromatic incident field. In the context of scattering, it is usually more convenient to deal with the scattering potential function $F(\boldsymbol {r})$, defined by

Let us consider a stochastic optical beam $u_i(\boldsymbol {r})$, described by the cross-spectral density $W_i(\boldsymbol {r}_1,\boldsymbol {r}_2) = \langle u_i^*(\boldsymbol {r}_1)u_i(\boldsymbol {r}_2)\rangle _{\omega }$, propagating in the $z$ direction, gets scattered by the dipole. The average $\langle \cdot \rangle _{\omega }$ being taken over an ensemble of monochromatic realizations. This light-dipole interaction generates an outgoing, scattered, radiation field $u_s(\boldsymbol {r})$ described by the cross-spectral density $W_s(\boldsymbol {r}_1,\boldsymbol {r}_2) = \langle u_s^*(\boldsymbol {r}_1)u_s(\boldsymbol {r}_2)\rangle _{\omega }$. Once the cross-spectral density of the scattered field is known, the spectral density $S_s(\boldsymbol {r}) = W_s(\boldsymbol {r},\boldsymbol {r})$ and the complex degree of coherence $\mu _s(\boldsymbol {r}_1,\boldsymbol {r}_2)$, defined by

3. Partially coherent vortex beams

To describe the incident beam, we consider a monochromatic, spatially coherent, Laguerre-Gaussian beam with azimuthal order $\pm m$ ($m>0$) and null radial order at $z = 0$,

In the above expressions, $\delta$ is the coherence parameter arising from the beam wander model by assuming a Gaussian probability distribution for the beam’s center position. A large value of $\delta$ corresponds to less (transverse) spatial coherence. The spatial correlation length $\Delta$ can be seen to be determined by

In this model, the correlation length $\Delta$ has a minimum limiting value of $\sqrt {2}w_0$ as $\delta \rightarrow \infty$. From (7), we obtain the spectral density $S_i(\boldsymbol {\rho })$ at $z = 0$,

4. Results and discussion

We are now in a position to obtain the cross-spectral density $W_s(\hat {s}_1,\hat {s}_2)$ for the scattered radiation. Direct substitution of (7) into (4) gives

We will restrict ourselves to the particular case where the incident vortex beam has topological charge $m = 1$. In this case, the spectral density $S_i(\boldsymbol {\rho })$ is given by

Figure 1 illustrates the incident vortex beam along with the $\mathcal {PT}$ dipole with separation distance $ka = 15$. It should be pointed out that $S_i(\boldsymbol {\rho })$ given by (16) is never zero in the finite plane, so the dipole is always interacting with the incident beam. The spectral density $S_s(\hat {s}) = W_s(\hat {s},\hat {s})$ of the scattered field is obtained from (13):

It can be seen from the structure of (17) that the spectral density for the scattered field depends mainly on the sum of two terms, $S_G(\hat {s})$ and $S_1(\hat {s})$, the former being independent of the topological charge $m$ and therefore carrying no phase information for the scattering process. The second term is responsible for the effects relating to the vortex structure of the incident beam. However, both terms depend on the coherence properties of the incident field. Therefore, we expect a competition between $S_G(\hat {s})$ and $S_1(\hat {s})$ to dictate the directional properties of the scattered radiation. In order to evaluate the relative contributions of these two terms in (17), in Fig. 2 we plot the spectral density $S_s(\hat {s})$, for a fixed direction $\hat {s}$, as a function of the scaled coherence parameter $k\delta$. In the high coherence regime (small $k\delta$), the factor $S_1(\hat {s})$ is dominant over $S_G(\hat {s})$ and the directional properties of the scattered radiation are mainly dictated by the $S_1(\hat {s})$ term. On the other hand, for a low spatial coherence (large $k\delta$), the Gaussian distribution term $S_G(\hat {s})$ dominates although not as significantly as in the first case. Nevertheless, Fig. 2 demonstrates that both, the coherence and non-Hermitian parameters, strongly influence the properties of the scattered field.

 

Fig. 1. Spectral density of the incident vortex beam at $z = 0$ with $m=1$ [16]. The $\mathcal {PT}$ dipole is also shown (white dots) with distance between scatterers given by $2ka=30$. Parameters used: $k\delta =1$ and $kw_0=100$.

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Fig. 2. Plots of $S_G(\hat {s})$ (dashed lines) and $S_1(\hat {s})$ (solid lines) as a function of the normalized coherence parameter $k\delta$ for three values of the non-Hermitian parameter $\gamma$ and fixed $s_x = \sin \theta _0\cos \phi _0$. The case $\gamma = 0$ being the Hermitian configuration (no loss or gain present). Parameters used: $\sigma =1$, $kw_0=100$, $\theta _0=\pi /2$, $\phi _0=0$.

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Since we are mainly interested in the directional properties of the scattered radiation for a fixed frequency $\omega$, and how the coherence and non-Hermitian parameters influence its behavior, parts (a), (b) and (c) of Fig. 3 show a polar diagram of the spectral density $S_s(\hat {s})$ in the $(x,z)$ plane for increasing values of the scaled coherence parameter $k\delta$ in the Hermitian scenario with $\gamma = 0$. The first geometrical aspect of these distributions is that they are all symmetric with respect to the inversion $\boldsymbol {r}_p \rightarrow -\boldsymbol {r}_p$, where $\boldsymbol {r}_p$ is a vector in the $(x,z)$ plane. An interesting phenomenon takes place when $k\delta \sim 15$. It is seen from part (b) of Fig. 3 that radiation is scattered in all directions. This behavior is typical for scattering systems where the incident field is highly incoherent [20]. However, this is not the case here since by increasing (or decreasing) $k\delta$ a little further, we recover the interference fringes in the scattered light [part (c) of Fig. 3]. For this effect to happen, there must be an almost exact cancellation between the terms $S_G$ and $S_1$ that renders $S_s(\hat {s})$ constant. This could only happen if both terms are of the same order. Whether this is the case, can be verified by inspecting Fig. 2 and realizing that the point $k\delta = 15$ lies in the region where the solid and dashed black lines cross each other, as expected. Part (d) of Fig. 3 shows the degree of spectral coherence [3] for the symmetric scattering directions $s_{x1} = \sin \theta \cos \phi$ and $s_{x2} = -\sin \theta \cos \phi$. It is evident from this plot that the spatial coherence of the scattered radiation strongly depends upon the coherence properties of the incident light. There are a few directions shown in the plot where the scattered radiation is completely spatially coherent ($|\mu _s| = 1$).

 

Fig. 3. Hermitian spectral density, described by (17), with $m=1$ for (a) $k\delta =1$, (b) $k\delta =15$ and (c) $k\delta =100$. (d) Far-zone degree of coherence $|\mu ^s(\theta ,\phi =0)|$ for $k\delta =1$ (black), $k\delta =15$ (blue) and $k\delta =100$ (orange). Parameters used: $\sigma =1$, $\gamma =0$ and $kw_0=100$.

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By allowing the two point scatterers to absorb and emit radiation in a balanced way, thus forming a $\mathcal {PT}$ dipole, we have found that the symmetry property $\boldsymbol {r}_p \rightarrow -\boldsymbol {r}_p$ for the scattered light is no longer valid. This asymmetry is a common signature of non-Hermitian optical effects, highlighting the presence of some preferred directions. Indeed, Fig. 4 shows the same plots as in Fig. 3 but this time with $\gamma = 1$. First, the all-directional-emission distribution of radiation is also present when gain and loss are active in the system, as can be seen in part (b) of Fig. 4. This suggests that the almost exact cancellation between the terms $S_G$ and $S_1$ is a robust effect. Although the plots of Figs. 3 and 4 were produced in the $(x,z)$ plane, we verified that this effect occurs for all scattered directions. In other words, radiation is being emitted uniformly in three dimensions with $S_s(\hat {s})$ being almost constant for all directions. Second, the changes in the scattered properties are most pronounced in the $\theta = \pi /2$, $3\pi /2$ directions. This is expected since positions of the point particles are located in the $x$-axis [2]. Therefore, one can control the shape of the scattered radiation by tuning the gain and loss properties of the dipole. Part (d) of Fig. 4 shows the modulus of the spectral degree of coherence $\mu _s$ for the same values of $k\delta$. Again, the spatial correlation properties of the scattered light are strongly dependent upon the gain/loss and the coherence parameter of the incident field. The exact relation between all these quantities is directly obtained from the analytic solutions but they are not particularly illuminating.

 

Fig. 4. Non-Hermitian spectral density with $m=1$ and [17] for (a) $k\delta =1$, (b) $k\delta =15$ and (c) $k\delta =100$. (d) Far-zone degree of coherence $|\mu ^s(\theta ,\phi =0)|$ for $k\delta =1$ (black), $k\delta =15$ (blue) and $k\delta =100$ (orange). Parameters used: $\sigma =1$, $\gamma =1$ and $kw_0=100$.

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To verify that the ability of the point particles to absorb and give energy to the field can influence the statistical properties of radiated light, in Fig. 5 we show the spectral degree of coherence $\mu _s(\theta )$ for three values of $\gamma$ and $k\delta$. We chose the symmetric directions $s_{x1} = \sin \theta \cos \phi$ and $s_{x2} = -\sin \theta \phi$ for the $x$ component of the pair of positions $\boldsymbol {r}_1$ and $\boldsymbol {r}_2$. Clearly, the gain and loss present in the system drastically alter the spatial coherence properties of scattered light. For some point pairs, $\mu (\hat {s}_1,\hat {s}_2)$ is found to have complete spatial coherence ($|\mu _s|=1$) while, in general, the radiated field is partially coherent.

 

Fig. 5. Far-zone degree of coherence $|\mu ^s(\theta ,\phi =0)|$ with (a) $k\delta =1$, (b) $k\delta =15$ and (c) $k\delta =100$ for $\gamma =0$ (solid black), $\gamma =0.5$ (dashed blue) and $\gamma =1$ (dash-dot orange). Parameters used: $\sigma =1$ and $kw_0=100$.

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The physical reason behind the strong relationship between the spectral degree of coherence of the scattered light and the gain/loss properties of the material still remains an open problem. From a more fundamental point of view, a deeper understanding of the interaction between partially coherent radiation and “non-Hermitian matter” can be obtained (at least in principle) by appealing to a microscopic theory. Such an approach has been considered in the past to study the microscopic origin of the Wolf effect, where the correlations between the electric dipoles present in the system are responsible for the shift in the center frequency of the spectrum of the emitted radiation [40] (see also [41]). Progress in this direction is currently under investigation and will be reported elsewhere.

5. Conclusion

We studied the scattering of partially coherent vortex beams by localized point scatterers possessing PT-symmetry. Under the first Born approximation, together with the asymptotic approximation for the Green’s function of the Helmholtz equation, a general expression was obtained for the far-zone cross-spectral density. It was found that the spectral density is mainly given by the sum of two terms, $S_G(\hat {s})$ and $S_m(\hat {s})$, associated with the Gaussian and OAM contributions, respectively. The balance between these terms depends strongly on the coherence parameter $\delta$, where one can observe three main regimes. When the incident beam is highly coherent, $S_m(\hat {s})$ dominates the overall spectral density, while $S_G(\hat {s})$ dominates if the incident beam is poorly coherent. Between these regimes, when both $S_G(\hat {s})$ and $S_m(\hat {s})$ assume comparable values, the scattered profile resemble that of an incoherent system, where light gets scattered in all directions in the three-dimensional space. The same behaviour occurs when turn on the gain/loss property of the point scatterers $(\gamma \neq 0)$, but light gets scattered in some preferred directions. The far-zone spectral degree of coherence shows us that both, the coherence parameter of the incident beam and the gain/loss parameter of the scatterers, drastically change the statistical properties of the scattered radiation, where at some points it possesses complete spatial coherence.

In summary, the results presented here show that one can have a precise control of directionality for the scattered light by changing the spatial coherence properties of the incident beam. Such control can be enhanced if the scatterers have gain and loss, enabling on demand engineering of the scattered radiation pattern. These results may find applications for optical sensing and probing in turbid media.