Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Cross-spectral densities with helical-Cartesian phases

Zhangrong Mei and Olga Korotkova

Open Access Open Access

Abstract

We introduce a class of planar, stationary sources whose cross-spectral densities carry a combination of helical and Cartesian phases. The helical phase is linear, separable in polar coordinates, resulting in a vortex-like average intensity with a dark area centered on the optical axis; the Cartesian counterpart is separable in the x- and y-coordinates and is responsible for asymmetric average intensity redistribution along the x- and y-axes. While endless possibilities exist for modeling of the Cartesian phase factor, in this paper we employ a superposition of linear phases with arbitrarily assigned weighing factors. Such construction is analytically and experimentally useful in general and, in particular, as we show, for generation of asymmetric vortex lattices.

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

1. Introduction

Two hot topics in modern physical optics are vortex beams and non-diffracting, deterministic beams both providing unprecedented opportunities in light evolution and light-matter interaction control. Vortex beams are radiated by sources with helical phase distribution, carry the optical angular momentum (OAM), and result in gradual appearance of the dark hollow center on the beam’s free-space evolution, as a consequence of destructive interference in all polar directions [1]. Vortex beams have already benefited optical micro-operation [2], optical communications [3], quantum information [4], etc. Apart from single optical vortices, optical vortex lattices are also of particular interest in capturing and manipulating a large number of particles [5], driving micro-optomechanical pumps [6], processing quantum information and micro-lithography [7].

On the other hand, non-diffracting beams first introduced in [8] and later developed to a very broad family (e. g. [9,10]) became of much use in applications where optical energy confinement and precise delivery is a necessity [11]. The analytical concept behind a non-diffracting beam is the finite support of its angular spectrum. A non-diffracting Airy beam introduced in [12] has presented even more exciting opportunity of lateral acceleration. The phase distribution of a source radiating to an Airy beam is cubic, in one or both Cartesian directions, resulting in complex self-interference effects and, hence, forming an axially asymmetric sequence of intensity maxima. There have been numerous studies regarding the combination of these two classes of beams (e.g. [13] and references wherein).

In the class of random, stationary beams the phase of the optical field is not observable while the phase of the second-order spatial correlation function, known as the Cross-Spectral Density (CSD), is an important quantity [14] that has been theoretically modeled and experimentally measured. Such optical beams are useful in certain optical systems because of their low speckle sensitively and little disturbances by turbulent media. The possibilities for modeling of a phase distribution of the CSD are much richer than those for that in deterministic beams. For instance, there exist at least three classes of structurally different random vortex beams [15]: with Rankine phase [16], with twisted phase [17] and with linear, separable phase [18]. In particular, numerous models of optical vortices formed in partially coherent light have also been extensively studied [19–22]. Compared with deterministic vortex beams, stationary vortex beams exhibit unique optical effects, such as the appearance of coherence singularities [23], beam reshaping, self-splitting [24], etc. Several recent studies showed that optical coherence vortex lattices could be generated by random sources with periodic degree of coherence [25,26].

Apart from the vortex-like phase, the source CSDs can possess an arbitrarily profiled, one-dimensional Cartesian phase, which was revealed only very recently [27,28] (with a single exception of a linear phase [29,30]). Such a phase can be modeled directly by auto-convolving any complex-valued, Hermitian function, with a magnitude having a finite L1 norm, which was termed a “sliding function” [27]. The presence of the Cartesian phase in the source CSD was shown to result in a variety of effects for the propagating beam, such as lateral shifting, asymmetric splitting, lateral acceleration, etc. Cartesian phase factor of the CSD was shown to affect the evolving beam much stronger than its magnitude [31]. The generalization of stationary sources with Cartesian phases from one to two dimensions appears to lead to two distinct types: with separable and non-separable phases [32,33].

The purpose of this paper is to introduce random sources with the CSD that is a combination of those with vortex and Cartesian phase factors. Since both classes are very rich we will restrict ourselves to a linear helical phase and a linear Cartesian phase. In order to illustrate the special power of this combination, we also choose to consider a linear superposition of linear Cartesian phases having different weighting functions. Such superposition has been previously shown to mimic non-linear Cartesian phases (aka cubic) but provides fine control of the radiated beam [34]. We must mention that the attempt of superposing the source CSDs with linear vortex phase and twisted phase (both carrying the OAM) has been made before [35] but it was not yet done with a vortex and a Cartesian factors, the former of which carries the OAM and the latter does not. After we establish such a combination for a single beam we explore its application to generating highly controllable two-dimensional optical arrays carrying vortex structures. While the linear vortex phase is capable of producing dark zones in the average intensity, the Cartesian phase is responsible splitting it and laterally shifting the obtained replicas.

2. Source-field models

We begin by recalling that the CSD across a typical planar source can be written as [36]

$${W^{(\textrm{0}) }}({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_1},{{\boldsymbol{\mathrm{\rho}} ^{\prime}}_2}) = \langle {E^ \ast }({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_1})E({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_2})\rangle ,$$
where ${{\boldsymbol{\mathrm{\rho}} ^{\prime}}_\textrm{1}}$ and ${{\boldsymbol{\mathrm{\rho}} ^{\prime}}_\textrm{2}}$ are the position vectors of two points in the source plane, E is the complex electric field, the angular brackets denote average over an ensemble of monochromatic realizations, and the asterisk stands for complex conjugate. Here, for brevity, we omit the explicit dependence of CSD on frequency.

Suppose that the complex electric field in the transverse plane of the source is a single optical vortex in a coherent scalar beam, which may be expressed as

$$E({\boldsymbol{\mathrm{\rho}} ^{\prime}}) = A(\rho ^{\prime})\exp (il\phi ^{\prime})\textrm{exp}(i\beta ),$$
where $\rho ^{\prime}$ and $\phi ^{\prime}$ are radial and azimuthal coordinates in the cylindrical coordinate system, $A(\rho ^{\prime})$ is a real function representing the profile of the beam envelope, l is the topological charge, and $\beta $ is an arbitrary phase constant. On substituting from Eq. (2) into Eq. (1), and assuming that the statistical distribution of random phase $\beta $ corresponds to a Schell-model correlator, we can express the CSD as
$${W^{(\textrm{0}) }}({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_1},{{\boldsymbol{\mathrm{\rho}} ^{\prime}}_2}) = A({\rho ^{\prime}_\textrm{1}})A({\rho ^{\prime}_\textrm{2}})\mu ({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_d})\exp [il({\phi ^{\prime}_2} - {\phi ^{\prime}_1})],$$
where $\mu ({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_d})$ is the Complex Degree of Coherence (CDC) defined as a function of separation between two source points ${{\boldsymbol{\mathrm{\rho}} ^{\prime}}_d} = {{\boldsymbol{\mathrm{\rho}} ^{\prime}}_1} - {{\boldsymbol{\mathrm{\rho}} ^{\prime}}_2}$.

We further assume that in the source CDC factorizes in Cartesian system as:

$$\mu ({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_d}) = \prod\limits_{\zeta = x^{\prime},y^{\prime}} {\nu ({\zeta _d})} ,$$
where ${\zeta ^{\prime}_d} = {\zeta ^{\prime}_1} - {\zeta ^{\prime}_2}$ and $(x^{\prime},y^{\prime})$ are the Cartesian coordinates of ${\boldsymbol{\mathrm{\rho}} ^{\prime}}$. Following [28] we can express the Schell-like source CDC $\nu ({\zeta _d})$ as a self-convolution of some complex-valued sliding function $g({\zeta ^{\prime}_d})$:
$$\nu ({\zeta _d}) = g({\zeta _d}) \otimes g({\zeta _d}),$$
where $\otimes$ stands for convolution. The sliding function $g({\zeta _d})$ belongs to the ${L^1}({\mathbb R})$ space, i.e. $\int_{\textrm{ - }\infty }^\infty {\textrm{|}g({{\zeta ^{\prime}}_d})\textrm{|d}{{\zeta ^{\prime}}_d} < \infty } $.

Any source’ CDC can be expressed as a linear combination of N CDCs belonging to the same class, i.e.,

$$\nu ({\zeta ^{\prime}_d}) = \frac{1}{{{C_{\zeta ^{\prime}}}}}\sum\limits_{n = 1}^{{N_{\zeta ^{\prime}}}} {{f_n}{\nu _n}({{\zeta ^{\prime}}_d})} = \frac{1}{{{C_{\zeta ^{\prime}}}}}\sum\limits_{n = 1}^{{N_{\zeta ^{\prime}}}} {{f_n}{g_n}({{\zeta ^{\prime}}_d}) \otimes {g_n}({{\zeta ^{\prime}}_d})} ,$$
where ${f_n}$ coefficients independent of the spatial position and ${C_\zeta } = {\sum\nolimits_{n = 1}^{{N_{\zeta ^{\prime}}}} f _n}$ is the normalization factor.

The complex-valued function ${g_n}({\zeta ^{\prime}_d})$ must be Hermitian [28]. Hence, if it is represented via its magnitude $g_n^M({\zeta ^{\prime}_d})$ and phase $g_n^P({\zeta ^{\prime}_d})$, i.e.,

$${g_n}({\zeta ^{\prime}_d}) = g_n^M({\zeta ^{\prime}_d})\exp [{ig_n^P({{\zeta^{\prime}}_d})} ],$$
then $g_n^M({\zeta ^{\prime}_d})$ and $g_n^P({\zeta ^{\prime}_d})$ must be an even function and an odd function, respectively.

Let us set $g_n^M({\zeta ^{\prime}_d})$ as a Gaussian function with the r.m.s. width ${\delta _{\zeta ^{\prime}}}$ and set $g_n^P({\zeta ^{\prime}_d})$ as a linear function, i.e.,

$${g_n}({\zeta ^{\prime}_d}) = \frac{1}{{\sqrt {{\delta _{\zeta ^{\prime}}}\sqrt \pi } }}\exp \left( { - \frac{{\zeta^{\prime 2}_d}}{{2\delta_{\zeta^{\prime 2}}}}} \right)\exp ({i{s_n}{{\zeta^{\prime}}_d}} ),$$
where ${s_n}$ is a sequence depending on index $n$. Then, the CDC along the $\zeta ^{\prime}$-direction obtained on substituting from Eq. (8) into Eq. (6), takes form
$$\nu ({\zeta ^{\prime}_d}) = \frac{1}{{{C_{\zeta ^{\prime}}}}}\sum\limits_{n = 1}^{{N_{\zeta ^{\prime}}}} {{f_n}\exp\left( { - \frac{{\zeta^{\prime 2}_d}}{{4\delta_{\zeta^{\prime 2}}}} + i{s_n}{{\zeta^{\prime}}_d}} \right)} .$$

The CSD may therefore be expressed by insertion of Eq. (9) into Eq. (4) and then into Eq. (3) to result in the formula:

$${W^{(\textrm{0}) }}({{\boldsymbol{\mathrm{\rho}} ^{\prime}}_1},{{\boldsymbol{\mathrm{\rho}} ^{\prime}}_2}) = A({\rho ^{\prime}_1})A({\rho ^{\prime}_2})\prod\limits_{\zeta ^{\prime} = x^{\prime},y^{\prime}} {\frac{1}{{{C_{\zeta ^{\prime}}}}}} \sum\limits_{n = 1}^{{N_{\zeta ^{\prime}}}} {{f_n}\exp\left( { - \frac{{\zeta^{\prime 2}_d}}{{4\delta_{\zeta^{\prime 2}}}} + i{s_n}{{\zeta^{\prime}}_d}} \right)} \exp [il({\phi ^{\prime}_2} - {\phi ^{\prime}_1})].$$

3. Propagation and far-field characteristics

The CSD of a partially coherent beam at a pair of points $({{{\boldsymbol{\mathrm{\rho}} }_{\bf 1}},z} )$ and $({{{\boldsymbol{\mathrm{\rho}} }_{\bf 2}},z} )$ in any transverse plane of the half-space $z > 0$ is related, under the paraxial approximation, to those in the source plane through the following propagation law [36]:

$$W({{{\boldsymbol{\mathrm{\rho}} }_{\bf 1}},{{\boldsymbol{\mathrm{\rho}} }_{\bf 2}},z} )= {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\!\!\!\int {{W^{(0 )}}} ({{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_{\bf 1}},{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_{\bf 2}}} )\exp \left\{ { - \frac{{\textrm{i}k}}{{2z}}[{{{({{{\boldsymbol{\mathrm{\rho}} }_{\bf 1}} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_{\bf 1}}} )}^2} - {{({{{\boldsymbol{\mathrm{\rho}} }_2} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_{\bf 2}}} )}^2}} ]} \right\}{\textrm{d}^2}{{\boldsymbol{\mathrm{\rho}} ^{\prime}}_1}{\textrm{d}^2}{{\boldsymbol{\mathrm{\rho}} ^{\prime}}_{\bf 2}},$$
where k is the wave number related to the wavelength $\lambda \textrm{ by }k\textrm{ = 2}\pi \textrm{/}\lambda $. Although the four-dimensional integral evaluation in Eq. (11) is generally computer memory intensive, it is possible to choose envelope functions and a smaller topological charge in Eq. (10) that allow Eq. (11) to be computed from two-dimensional integrals. For computational simplicity in the discussion below we consider an optical field in initial plane with an amplitude of a Laguerre-Gaussian mode:
$$A(\rho ^{\prime}) = {E_0}{(\rho ^{\prime}/\sigma )^l}\exp ( - {\rho ^{\prime 2}}/{\sigma ^2}),$$
where E0 and $\sigma $ are the characteristic amplitude and waist width, and with topological charge l=1. Substituting from Eq. (12) into Eq. (10) and then into Eq. (11), one obtains the formula
$$\begin{aligned} W({{\boldsymbol{\mathrm{\rho}} }_1},{{\boldsymbol{\mathrm{\rho}} }_2},z) &= {\left( {\frac{{{E_0}}}{{z\lambda \sigma }}} \right)^2}\exp \left[ { - \frac{{ik}}{{2z}}({\boldsymbol{\mathrm{\rho}} }_1^2 - {\boldsymbol{\mathrm{\rho}} }_2^2)} \right]\\ &\quad\times [{{F_a}(x^{\prime}){F_b}(y^{\prime}) + {F_b}(x^{\prime}){F_a}(y^{\prime}) - i{F_c}(x^{\prime}){F_d}(y^{\prime}) + i{F_d}(x^{\prime}){F_c}(y^{\prime})} ], \end{aligned}$$
where
$${F_a}(\zeta ^{\prime}) = \int\!\!\!\int {{{\zeta ^{\prime}}_1}{{\zeta ^{\prime}}_2}{T_{\zeta ^{\prime}}}\textrm{d}{{\zeta ^{\prime}}_1}\textrm{d}{{\zeta ^{\prime}}_2},} \quad\zeta ^{\prime} = x^{\prime},y^{\prime},$$
$${F_b}(\zeta ^{\prime}) = \int\!\!\!\int {{T_{\zeta ^{\prime}}}\textrm{d}{{\zeta ^{\prime}}_1}\textrm{d}{{\zeta ^{\prime}}_2},} \quad\zeta ^{\prime} = x^{\prime},y^{\prime},$$
$${F_c}(\zeta ^{\prime}) = \int\!\!\!\int {{{\zeta ^{\prime}}_1}{T_{\zeta ^{\prime}}}\textrm{d}{{\zeta ^{\prime}}_1}\textrm{d}{{\zeta ^{\prime}}_2},} \quad\zeta ^{\prime} = x^{\prime},y^{\prime},$$
$${F_d}(\zeta ^{\prime}) = \int\!\!\!\int {{{\zeta ^{\prime}}_2}{T_{\zeta ^{\prime}}}\textrm{d}{{\zeta ^{\prime}}_1}\textrm{d}{{\zeta ^{\prime}}_2},} \textrm{ }\zeta ^{\prime} = x^{\prime},y^{\prime},$$
and
$${T_{\zeta ^{\prime}}} = \exp [{ - (\zeta^{\prime 2}_{1} + \zeta^{\prime 2}_{2})/{\sigma^2} - ik(\zeta^{\prime 2}_{1} - \zeta^{\prime 2}_{2} - 2{\zeta_1}{{\zeta^{\prime}}_1} + 2{\zeta_2}{{\zeta^{\prime}}_2})/(2z)} ]\nu ({\zeta ^{\prime}_d}),\textrm{ }\zeta = x,y.$$

Then, the spectral density S and the CDC µ can be calculated by the expressions [36]

$$S({{\boldsymbol{\mathrm{\rho}} ,}z} )= W({{\boldsymbol{\mathrm{\rho}} },{\boldsymbol{\mathrm{\rho}} },z} ),$$
$$\mu ({{{\boldsymbol{\mathrm{\rho}} }_{\bf 1}},{{\boldsymbol{\mathrm{\rho}} }_{\bf 2}},z} )= W({{{\boldsymbol{\mathrm{\rho}} }_{\bf 1}},{{\boldsymbol{\mathrm{\rho}} }_{\bf 2}},z} )/\sqrt {S({{{\boldsymbol{\mathrm{\rho}} }_{\bf 1}},z} )S({{{\boldsymbol{\mathrm{\rho}} }_{\bf 2}},z} )} .$$
implying that the propagation characteristics of the vortex fields are closely related to their coherence properties in the source plane.

Let us set ${f_n} = 1$ and ${s_n} = (3n - 2){a_\zeta }$ as an arithmetic sequence with a common difference of $3{a_\zeta }$. Then, Eq. (9) is the sum of a geometric sequence with a common ratio $\exp ({i3{a_\zeta }{\zeta_d}} )$. In this case, by selecting appropriate values of source parameters, the far-field radiated by the source in Eq. (10) can have desirable features representing laterally self-shifted optical coherence vortex lattices with uniform spatial distribution. In all calculations of this paper, the source parameters are chosen as $\sigma = 1\textrm{mm}$, $\lambda = 632.8\textrm{nm}$ and while the rest of the values of parameters are indicated in the figure captions. Figures 1 and 2 show the evolution of spectral density S generated by source field in Eq. (10) with different values of parameters ${N_\zeta }$ in the x-z and y-z plane and the lateral distributions at several selected distances. As is seen from these figures, three interesting features are worth noting. First, the central axis of such beams is not along the z-axis during propagation, but is laterally shifted. That is, the light intensity distributions in any transverse plane shifts laterally and is not symmetrical about the x-axis and the y-axis. The amount and direction of the lateral shift are determined by parameters ${a_\zeta }$. Second, an intensity vortex in the source plane is gradually split into a vortex lattice distribution with the increase of the propagation distance. The dimensions of this lattice depend on the values of parameters ${N_\zeta }$. Third, those intensity vortices remain stable on propagation, except for those when ${N_\zeta } > 1$, the split vortices are destroyed in the near field due to the superposition of intensity of different modes.

 figure: Fig. 1.

Fig. 1. Evolution of the spectral density generated by source field in Eq. (10) with ${a_x} = 0$; ${a_y} = 2\pi /3$, ${\delta _x} = {\delta _y} = 5\textrm{mm}$ and ${N_x} = {N_y} = 1$.

Download Full Size | PDF

 figure: Fig. 2.

Fig. 2. Same as Fig. 1, but for ${a_x} = 2\pi /3$; ${a_y} ={-} 2\pi /3$ and ${N_x} = {N_y} = 2$.

Download Full Size | PDF

Next, we examine the change in the far-field characteristics of the fields generated by a source in Eq. (10) with different values of source parameters. The far-field plane is set at $z = 10{z_R}$, where ${z_R} = k{\sigma ^2}/2$ is the Rayleigh length. Figure 3 shows the far-field distributions of normalized spectral density as well as the magnitude and the phase of the corresponding CDC, evaluated numerically for different values of parameter ${N_\zeta }$. As we have shown, such sources produce far fields with vortex lattices and dimensions finely controlled by the source parameters ${N_\zeta }$. However, these optical coherent vortex lattices of different dimensionality possess almost the same coherent states. There are two coherent singularities on the line y = x that are symmetrical about the origin, and the corresponding phase is antisymmetric with respect to y = x having a jump from $- \pi $ to $\pi $ at the two singularities.

 figure: Fig. 3.

Fig. 3. Far-field distributions of the normalized spectral densities (left column), the magnitude (middle column) and phase (right column) of the corresponding CDC radiated by source field in Eq. (10) with ${a_x} = {a_y} = 2\pi /3$ and ${\delta _x} = {\delta _y} = 5\textrm{mm}$for different parameters ${N_x}$ and ${N_y}$. (a), (b), (c) ${N_x} = {N_y} = 1$; (d), (e), (f) ${N_x} = {N_y} = 2$; (g), (h), ${a_x} = {a_y} = 2\pi /3$(i) ${N_x} = 2,\textrm{ }{N_y} = 3$; (j), (k), (l) ${N_x} = {N_y} = 3$.

Download Full Size | PDF

To develop an understanding of the ways in which parameters ${a_\xi }$ affect the far-field characteristics we calculate the far-field spectral density, as well as the magnitude and the phase of the corresponding CDC for several values of ${a_\zeta }$ in Fig. 4. Without the phase factor in Eq. (9), i.e., when ${a_x} = {a_y} = 0$, this example corresponds to the top row of Fig. 4. We can see that in this case the light beam does not split, which corresponds to a general partially coherent vortex beam. When ${a_x} = 2\pi /3$ and ${a_y} = 0$, the light beam only splits along the x-axis direction, forming a 1D optical vortex lattice, and the coherent singularities are also located on the x-axis. As the value of ay gradually increases, the light beam gradually splits along the y-direction, and the position of the coherent singularities turns to the diagonal line y = x.

 figure: Fig. 4.

Fig. 4. Far-field distributions of the normalized spectral densities (left column), the magnitude (middle column) and phase (right column) of the corresponding CDC radiated by source field in Eq. (10) with ${N_x} = {N_y} = 2$, ${\delta _x} = {\delta _y} = 5\textrm{mm}$ for different parameters ${a_x}$ and ${a_y}$. (a), (b), (c) ${a_x} = {a_y} = 0$; (d), (e), (f) ${a_x} = 2\pi /3,\textrm{ }{a_y} = 0$; (g), (h), (i) ${a_x} = 2\pi /3,\textrm{ }{a_y} = 2\pi /5$; (j), (k), (l) ${a_x} = 2\pi /3,\textrm{ }{a_y} = \pi /2$.

Download Full Size | PDF

The dependence of the far-field spectral density, and of the magnitude and phase of the corresponding CDC on the initial coherence length ${\delta _\zeta }$ are shown in Fig. 5. The low-coherence intensity profile, Fig. 5(a), reveals no evidence of splitting beams and vortex cores. When the coherence length along a certain direction gradually increases, see Figs. 5(d) and 5(g), the beam splits and the central intensity of the beamlet decreases in this direction. Whereas the vortex lattice is easily distinguished in the high-coherence case, Fig. 5(j). Comparing the corresponding distributions of the degree of coherence, we notice that there are no singularities in the case of low coherence, as shown in Figs. 5(b) and 5(c). If the coherence length in y-direction increases, Figs. 5(e) and 5(f), two coherence singularities appear on the y-axis. When the coherence length in x-direction also gradually increases, the coherence singularities shift to the symmetry axis y = x.

 figure: Fig. 5.

Fig. 5. Far-field distributions of the normalized spectral densities (left column), the magnitude (middle column) and phase (right column) of the corresponding CDC radiated by source field in Eq. (10) with ${N_x} = {N_y} = 2$, ${a_x} = {a_y} = 2\pi/3$  for different parameters ${\delta _x}$ and ${\delta _y}$. (a), (b), (c) ${\delta _x}={\delta _y}=0.1{\rm{mm}}$; (d), (e), (f) ${\delta _x} = 0.1\textrm{mm},\textrm{ }{\delta _y} = 5\textrm{mm}$; (g), (h), (i) ${\delta _x} = 0.5\textrm{mm},$ ${\delta _y} = 5{\rm{mm}}$; (j), (k), (l)${\delta _x} = 2.5\textrm{mm},\textrm{ }{\delta _y} = 5\textrm{mm}$

Download Full Size | PDF

4. Conclusions

In summary, a new class of partially coherent sources of Schell type with both helical phase and Cartesian phase is introduced. While the helical phase governs the vortex properties of the light field the Cartesian phase factor, designed as the weighed sum of linear phases causes such sources to generate a vortex beam or its lattice-like distribution. The lattice’s dimension and the direction of its lateral shift can be flexibly controlled by weighting factors and signs. We have further illustrated how the initial coherence width and the Cartesian phase affect the far-field spectral density and the coherence state. The results suggest a new approach to generate controlled optical coherent vortex beams and their lattices that can meet the needs in multiple particle trapping, screening and manipulating.

Funding

National Natural Science Foundation of China (11974107, 11947410).

Disclosures

The authors declare no conflicts of interest.

References

1. L. Allen, M. W. Beijersbergen, R. J. C. 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]  

2. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010). [CrossRef]  

3. J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016). [CrossRef]  

4. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009). [CrossRef]  

5. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002). [CrossRef]  

6. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortices,” Opt. Express 12(6), 1144–1149 (2004). [CrossRef]  

7. B. C. Hiesmayr, M. J. de Dood, and W. Löffler, “Observation of four-photon orbital angular momentum entanglement,” Phys. Rev. Lett. 116(7), 073601 (2016). [CrossRef]  

8. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]  

9. J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Alternative formulation for invariant optical fields: Matthew beams,” Opt. Lett. 25(20), 1493–1495 (2000). [CrossRef]  

10. M. A. Bandres, J. C. Gutierrez-Vega, and S. Chavez-Cerda, “Parabolic non-diffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef]  

11. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments and applications,” Czech. J. Phys. 53(7), 537–578 (2003). [CrossRef]  

12. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]  

13. B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Measurement of acceleration and orbital angular momentum of Airy beam and Airy-vortex beam by astigmatic transformation,” Opt. Lett. 40(22), 5411–5414 (2015). [CrossRef]  

14. E. Wolf, “Significance and measurability of the phase of a spatially coherent optical field,” Opt. Lett. 28(1), 5–6 (2003). [CrossRef]  

15. G. Gbur, “Partially coherent vortex beams,” Proc. SPIE 10549, 1054903 (2018).

16. G. Gbur, T. Visser, and E. Wolf, “Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004). [CrossRef]  

17. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]  

18. G. Bogatyryova, C. Fel’de, P. Polyanskii, S. Ponomarenko, M. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003). [CrossRef]  

19. D. Palacios, I. Maleev, A. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004). [CrossRef]  

20. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef]  

21. Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014). [CrossRef]  

22. Z. Mei, “Modeling for partially spatially coherent vortex beams,” IEEE Photonics J. 9(6), 6102306 (2017). [CrossRef]  

23. T. V. Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79(5), 1–6 (2009).

24. Z. Mei, D. Zhou, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015). [CrossRef]  

25. X. Liu, L. Liu, X. Peng, L. Liu, F. Wang, and Y. Cai, “Partially coherent vortex beam with periodical coherence properties,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 138–144 (2019). [CrossRef]  

26. Z. Mei and O. Korotkova, “Sources for random arrays with structured complex degree of coherence,” Opt. Lett. 43(11), 2676–2679 (2018). [CrossRef]  

27. O. Korotkova and X. Chen, “Phase structuring of the complex degree of coherence,” Opt. Lett. 43(19), 4727–4730 (2018). [CrossRef]  

28. X. Chen and O. Korotkova, “Complex degree of coherence modeling with famous planar curves,” Opt. Lett. 43(24), 6049–6052 (2018). [CrossRef]  

29. Y. Chen, S. A. Ponomarenko, and Y. Cai, “Self-steering partially coherent beams,” Sci. Rep. 7(1), 39957 (2017). [CrossRef]  

30. H. Mao, Y. Chen, C. Liang, L. Chen, Y. Cai, and S. A. Ponomarenko, “Self-steering partially coherent vector beams,” Opt. Express 27(10), 14353–14368 (2019). [CrossRef]  

31. O. Korotkova, “Multi-Gaussian Schell-model source with a complex coherence state,” J. Opt. 21(4), 045607 (2019). [CrossRef]  

32. X. Chen and O. Korotkova, “Phase structuring of the 2D complex coherence states,” Opt. Lett. 44(10), 2470–2473 (2019). [CrossRef]  

33. L. Wan and D. Zhao, “Generalized partially coherent beams with nonseparable phase,” Opt. Lett. 44(19), 4714–4717 (2019). [CrossRef]  

34. Z. Mei and O. Korotkova, “Asymmetric coherence gratings,” Opt. Lett. 45(6), 1366–1369 (2020). [CrossRef]  

35. C. S. D. Stahl and G. Gbur, “Twisted vortex Gaussian Schell-model beams,” J. Opt. Soc. Am. A 35(11), 1899–1906 (2018). [CrossRef]  

36. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)
Fig. 1.
Fig. 1. Evolution of the spectral density generated by source field in Eq. (10) with ${a_x} = 0$; ${a_y} = 2\pi /3$, ${\delta _x} = {\delta _y} = 5\textrm{mm}$ and ${N_x} = {N_y} = 1$.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Same as Fig. 1, but for ${a_x} = 2\pi /3$; ${a_y} ={-} 2\pi /3$ and ${N_x} = {N_y} = 2$.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Far-field distributions of the normalized spectral densities (left column), the magnitude (middle column) and phase (right column) of the corresponding CDC radiated by source field in Eq. (10) with ${a_x} = {a_y} = 2\pi /3$ and ${\delta _x} = {\delta _y} = 5\textrm{mm}$for different parameters ${N_x}$ and ${N_y}$. (a), (b), (c) ${N_x} = {N_y} = 1$; (d), (e), (f) ${N_x} = {N_y} = 2$; (g), (h), ${a_x} = {a_y} = 2\pi /3$(i) ${N_x} = 2,\textrm{ }{N_y} = 3$; (j), (k), (l) ${N_x} = {N_y} = 3$.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Far-field distributions of the normalized spectral densities (left column), the magnitude (middle column) and phase (right column) of the corresponding CDC radiated by source field in Eq. (10) with ${N_x} = {N_y} = 2$, ${\delta _x} = {\delta _y} = 5\textrm{mm}$ for different parameters ${a_x}$ and ${a_y}$. (a), (b), (c) ${a_x} = {a_y} = 0$; (d), (e), (f) ${a_x} = 2\pi /3,\textrm{ }{a_y} = 0$; (g), (h), (i) ${a_x} = 2\pi /3,\textrm{ }{a_y} = 2\pi /5$; (j), (k), (l) ${a_x} = 2\pi /3,\textrm{ }{a_y} = \pi /2$.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Far-field distributions of the normalized spectral densities (left column), the magnitude (middle column) and phase (right column) of the corresponding CDC radiated by source field in Eq. (10) with ${N_x} = {N_y} = 2$, ${a_x} = {a_y} = 2\pi/3$  for different parameters ${\delta _x}$ and ${\delta _y}$. (a), (b), (c) ${\delta _x}={\delta _y}=0.1{\rm{mm}}$; (d), (e), (f) ${\delta _x} = 0.1\textrm{mm},\textrm{ }{\delta _y} = 5\textrm{mm}$; (g), (h), (i) ${\delta _x} = 0.5\textrm{mm},$ ${\delta _y} = 5{\rm{mm}}$; (j), (k), (l)${\delta _x} = 2.5\textrm{mm},\textrm{ }{\delta _y} = 5\textrm{mm}$

View in Article | Download Full Size | PDF

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

W ( 0 ) ( ρ ′ 1 , ρ ′ 2 ) = ⟨ E ∗ ( ρ ′ 1 ) E ( ρ ′ 2 ) ⟩ ,
E ( ρ ′ ) = A ( ρ ′ ) exp ⁡ ( i l ϕ ′ ) exp ( i β ) ,
W ( 0 ) ( ρ ′ 1 , ρ ′ 2 ) = A ( ρ 1 ′ ) A ( ρ 2 ′ ) μ ( ρ ′ d ) exp ⁡ [ i l ( ϕ 2 ′ − ϕ 1 ′ ) ] ,
μ ( ρ ′ d ) = ∏ ζ = x ′ , y ′ ν ( ζ d ) ,
ν ( ζ d ) = g ( ζ d ) ⊗ g ( ζ d ) ,
ν ( ζ d ′ ) = 1 C ζ ′ ∑ n = 1 N ζ ′ f n ν n ( ζ ′ d ) = 1 C ζ ′ ∑ n = 1 N ζ ′ f n g n ( ζ ′ d ) ⊗ g n ( ζ ′ d ) ,
g n ( ζ d ′ ) = g n M ( ζ d ′ ) exp ⁡ [ i g n P ( ζ ′ d ) ] ,
g n ( ζ d ′ ) = 1 δ ζ ′ π exp ⁡ ( − ζ d ′ 2 2 δ ζ ′ 2 ) exp ⁡ ( i s n ζ ′ d ) ,
ν ( ζ d ′ ) = 1 C ζ ′ ∑ n = 1 N ζ ′ f n exp ⁡ ( − ζ d ′ 2 4 δ ζ ′ 2 + i s n ζ ′ d ) .
W ( 0 ) ( ρ ′ 1 , ρ ′ 2 ) = A ( ρ 1 ′ ) A ( ρ 2 ′ ) ∏ ζ ′ = x ′ , y ′ 1 C ζ ′ ∑ n = 1 N ζ ′ f n exp ⁡ ( − ζ d ′ 2 4 δ ζ ′ 2 + i s n ζ ′ d ) exp ⁡ [ i l ( ϕ 2 ′ − ϕ 1 ′ ) ] .
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 ∫ ∫ W ( 0 ) ( ρ ′ 1 , ρ ′ 2 ) exp ⁡ { − i k 2 z [ ( ρ 1 − ρ ′ 1 ) 2 − ( ρ 2 − ρ ′ 2 ) 2 ] } d 2 ρ ′ 1 d 2 ρ ′ 2 ,
A ( ρ ′ ) = E 0 ( ρ ′ / σ ) l exp ⁡ ( − ρ ′ 2 / σ 2 ) ,
W ( ρ 1 , ρ 2 , z ) = ( E 0 z λ σ ) 2 exp ⁡ [ − i k 2 z ( ρ 1 2 − ρ 2 2 ) ] × [ F a ( x ′ ) F b ( y ′ ) + F b ( x ′ ) F a ( y ′ ) − i F c ( x ′ ) F d ( y ′ ) + i F d ( x ′ ) F c ( y ′ ) ] ,
F a ( ζ ′ ) = ∫ ∫ ζ ′ 1 ζ ′ 2 T ζ ′ d ζ ′ 1 d ζ ′ 2 , ζ ′ = x ′ , y ′ ,
F b ( ζ ′ ) = ∫ ∫ T ζ ′ d ζ ′ 1 d ζ ′ 2 , ζ ′ = x ′ , y ′ ,
F c ( ζ ′ ) = ∫ ∫ ζ ′ 1 T ζ ′ d ζ ′ 1 d ζ ′ 2 , ζ ′ = x ′ , y ′ ,
F d ( ζ ′ ) = ∫ ∫ ζ ′ 2 T ζ ′ d ζ ′ 1 d ζ ′ 2 ,   ζ ′ = x ′ , y ′ ,
T ζ ′ = exp ⁡ [ − ( ζ 1 ′ 2 + ζ 2 ′ 2 ) / σ 2 − i k ( ζ 1 ′ 2 − ζ 2 ′ 2 − 2 ζ 1 ζ ′ 1 + 2 ζ 2 ζ ′ 2 ) / ( 2 z ) ] ν ( ζ d ′ ) ,   ζ = x , y .
S ( ρ , z ) = W ( ρ , ρ , z ) ,
μ ( ρ 1 , ρ 2 , z ) = W ( ρ 1 , ρ 2 , z ) / S ( ρ 1 , z ) S ( ρ 2 , z ) .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use