Strong correlations between incoherent vortices

A. J. Jesus-Silva; J. M. Hickmann; E. J. S. Fonseca

doi:10.1364/OE.20.019708

1. Introduction

It’s well known that vortices in optical beams have an azimuthal phase dependence around a dark spot in form $e^{i m ϕ}$ , with $m$ an integer number also so-called topological charge (TC) [1], which determines the amount of the orbital angular momentum (OAM). Interesting enough, optical vortices with fractional topological charges have recently gained significant interest. The pioneer paper in this field was published by M. V. Berry [2]. Since then, several experimental works have been performed in order to explain the nature of this type of vortex [3–5]. One important characteristic is its instability in the propagation, which was explained by a propagation mechanism that involves an infinite chain of vortices annihilating in pairs during the propagation [2]. The vortex instability is characterized by the vortices breaking into fundamental unit, the vortices are spatially separated, in the Fourier plane.

Light possessing integer or fractional vortex has been widely studied in coherent systems where the phase is well-defined. However, partially coherent systems [6], where statistics are required to quantify the phase, can also present optical vortices in the correlation function, so-called coherence vortices. These are pairs of points where the spectral degree of coherence, a two-field correlation function, vanishes. In fact, coherence vortices involving correlation between beams with integer TC have unveiled a new research field [7–15]. But, for fractional TC no work has been reported, to the best of our knowledge.

In this paper, we explore various fundamental aspects of coherence vortices using the intensity correlation, also known as fourth-order field correlation: i) strong vortices correlation - the value of the equivalent TC obtained in the intensity correlation follows a correlation rule such that this value is bound to the TC associated to each incoherent beam; ii) stability of a coherence fractional vortices - a precise signature of an integer vortex was observed in the intensity correlation from two incoherent beams with each beam possessing fractional vortices; iii) non-localized azimuthal phase - a well-defined amount of OAM was observed in the intensity correlation when two incoherent beams possessing OAM were diffracted by different objects, the information of azimuthal phase was recovered of a distributed object. Our findings were supported by the theoretical analyses of the correlation function using the Gaussian-Schell correlator [12].

2. Theoretical results

For simplicity, we consider a vortex with the initial field amplitude of a Laguerre-Gauss (LG) beam [1]

E_{m} (r, ϕ) = E_{0} {| \frac{r}{w_{0}} |}^{| m |} \exp (- r^{2} / w {}_{0}^{2}) \exp (i m ϕ),

where

E_{0}

and

w_{0}

are the characteristic amplitude and beam size in the plane

z = 0

, respectively,

m

is the topological charge, and

ϕ

is the azimuthal phase.

The intensity correlation function is given by an ensemble average [6]

\begin{matrix} 〈 I_{1} ({\vec{r}}_{1}) I_{2} ({\vec{r}}_{2}) 〉 = 〈 E_{1}^{*} ({\vec{r}}_{1}) E_{2}^{*} ({\vec{r}}_{2}) E_{1} ({\vec{r}}_{1}) E_{2} ({\vec{r}}_{2}) 〉 \\ = 〈 I_{1} ({\vec{r}}_{1}) 〉 〈 I_{2} ({\vec{r}}_{2}) 〉 + {| Γ ({\vec{r}}_{1}, {\vec{r}}_{2}) |}^{2}, \end{matrix}

where

E_{1} ({\vec{r}}_{1})

and

E_{2} ({\vec{r}}_{2})

are the fields at the detection plane, and

〈 〉

denote the ensemble average.

〈 I_{1} ({\vec{r}}_{1}) 〉 〈 I_{2} ({\vec{r}}_{2}) 〉

corresponds to a background term, and

{| Γ ({\vec{r}}_{1}, {\vec{r}}_{2}) |}^{2}

is the squared modulus of the second-order field correlation [6],

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}) = 〈 E_{1} ({\vec{r}}_{1}) E_{2}^{*} ({\vec{r}}_{2}) 〉 .

We consider a random phase in the plane $z = 0$ whose statistical distribution produces a stationary light source which is well-described using a Gaussian-Schell correlator $C (| {\vec{r}}_{1} - {\vec{r}}_{2} |) = \exp (- {| {\vec{r}}_{1} - {\vec{r}}_{2} |}^{2} / l_{c}^{2})$ [6], where $l_{c}$ is the coherence length. By inserting Eq. (1) into Eq. (3) and using the Gaussian-Schell correlator, the second-order field correlation function of two OAM beams with topological charges of $m_{1}$ and $m_{2}$ may be written as:

\begin{matrix} Γ_{m_{1}, m_{2}} ({\vec{r}}_{1}, {\vec{r}}_{2}) = \frac{E_{0}^{2}}{w_{0}^{(| m_{1} | + | m_{2} |)}} r_{1}^{| m_{1} |} r_{2}^{| m_{2} |} \exp (i m_{Γ} ϕ_{Γ}) \\ \times \exp [- (r_{1}^{2} + r_{2}^{2}) / w_{0}^{2}] C (| {\vec{r}}_{1} - {\vec{r}}_{2} |), \end{matrix}

with

m_{Γ} ϕ_{Γ} = m_{1} ϕ_{1} - m_{2} ϕ_{2}

, where

m_{Γ}

refers to an effective topological charge (ETC) of the correlation function with effective azimuthal phase

ϕ_{Γ}

.

The angular spectrum of this correlation function is given by [6]:

{\tilde{Γ}}_{m_{1}, m_{2}} ({\vec{k}}_{1}, {\vec{k}}_{2}) = \iint A_{1} ({\vec{r}}_{1}) A_{2} ({\vec{r}}_{2}) Γ_{m_{1}, m_{2}} ({\vec{r}}_{1}, {\vec{r}}_{2}) \exp (i {\vec{k}}_{2} \cdot {\vec{r}}_{2} - i {\vec{k}}_{1} \cdot {\vec{r}}_{1}) d {\vec{r}}_{1} d {\vec{r}}_{2},

where

A_{i} ({\vec{r}}_{i})

(i = 1, 2)

is an aperture function.

The physical quantity ${\tilde{Γ}}_{m_{1}, m_{2}} (\vec{k}, - \vec{k})$ is particularly interesting for the study of the robustness of the optical vortex [11, 12]. When the coherence length $l_{c}$ is very small we can suppose that the Gaussian-Schell correlator reduce to $C (| {\vec{r}}_{1} - {\vec{r}}_{2} |) = I_{0} δ ({\vec{r}}_{1} - {\vec{r}}_{2}),$ where $I_{0}$ and $δ$ denote a positive constant and the two-dimensional Dirac delta function, respectively. In this case, Eq. (5) becomes

{\tilde{Γ}}_{m_{1}, m_{2}} = \frac{E_{0}^{2}}{w_{0}^{(| m_{1} | + | m_{2} |)}} \int A_{1} ({\vec{r}}_{1}) A_{2} ({\vec{r}}_{1}) r_{1}^{| m_{1} | + | m_{2} |} e^{[i (m_{1} - m_{2}) ϕ_{1}]} e^{[- 2 r_{1}^{2} / w_{0}^{2}]} e^{[- 2 i \vec{k} \cdot {\vec{r}}_{1}]} d {\vec{r}}_{1},

where we have used

{\vec{k}}_{2} = - {\vec{k}}_{1} = - \vec{k}

. And now the ETC is given by

m_{Γ} ϕ_{Γ} = (m_{1} - m_{2}) ϕ_{1}

.

3. Experimental setup

To experimentally verify the theoretical analysis, we used the intensity correlation [16–18] instead of the two-field correlation [12–14]. It is worth remembering that the two approaches are connected by Eq. (2). The experimental setup is illustrated in Fig. 1 . A CW Ar + laser operating at 514 nm illuminates through a 10X beam expander a phase computer generated hologram [19] that was displayed in a Hamamatsu model X10468-01 spatial light modulator (SLM) to prodce high-order LG modes. The hologram was imaged onto a rotating Ground Glass Disk (GGD) by a combination of lenses through a pinhole, working as a spatial filter. In same measurements, a pair of lenses, f₄ and f₅, was used to image the GGD surface onto the aperture, which can have any shape, here represented by $A_{1}$ and $A_{2}$ . In our case, we have used a distributed triangular aperture (see $A_{1}$ and $A_{2}$ in Fig. 1). The beam transmitted by the aperture was Fourier transformed by a f₆ lens and observed by a charge coupled device (CCD) camera.

Fig. 1 Experimental setup.

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It is important to notice that a coherent beam with a well-defined TC illuminates a GGD. However, it becomes spatially incoherent after being transmitted by the GGD, having a chaotic wave front, but still with a characteristic azimuthal phase $e^{i m ϕ}$ associated to it [12].

The rotating GGD and the CCD camera were synchronized to guarantee that two images can be acquired from the same point of the GGD, i.e., both LG beams with $m_{1}$ and $m_{2}$ are spatially incoherent, but correlated beams. We numerically performed one realization of the correlation between these two images and in the end we averaged over 100 realizations to obtain a resulting pattern in the intensity correlation. This procedure is similar to that of the compressive ghost imaging process [15].

4. Results and discussion

Figure 2 shows speckles patterns produced by illuminating the GGD with LG beams of different orders of m. Line (A) of Fig. 2 shows images of a CCD camera with $m_{1} = 4$ and $m_{2} = 0.5$ , with $A_{i}$ is equal to one, (i = 1,2). This case allows us to measure the spatial coherence length of $21.4 μ m$ . The same procedure was performed to obtain the results shown on the line (B) for m₁ = 2.5 and m₂ = - 0.5. However, these two images were recorded with $A_{1}$ and $A_{2}$ apertures (see Fig. 1) placed in the beam path, respectively. It is clear from Fig. 2 that no regular pattern can be observed. However, if we numerically correlate the corresponding images and average over 100 realizations, a well-defined pattern emerges, as shown in Figs. 3 and 4 .

Fig. 2 Speckles patterns recorded in single shot mode using a CCD camera. Line (A) shows images of LG beams with m₁ = 4 and m₂ = 0.5 after diffracting by GGD without A₁ and A₂ apertures in the beam path. Line (B) shows images of LG beams with m₁ = 2.5 and m₂ = - 0.5 after diffracting by GGD with A₁ and A₂ in the beams path, respectively.

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Fig. 3 Numerical intensity correlation averaged over 100 realization (First column), theoretical results of amplitude (second column) and phase (third column) of coherence function, Eq. (6). All results were obtained without considering A₁ and A₂ apertures.

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Fig. 4 Numerical intensity correlation averaged over 100 realization (First column), theoretical results of amplitude (second column) and phase (third column) of coherence function, Eq. (6). All results were obtained considering a distributed object (A₁ and A₂ apertures).

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Figure 3 illustrates experimental results (first line) and theoretical results of amplitude (second line) and phase (third line) of the function ${\tilde{Γ}}_{m_{1}, m_{2}}$ for the $A_{i} ({\vec{r}}_{i}) = 1$ , with i = 1, 2. In the first line, we have numerically performed intensity correlations between incoherent beams with OAM of different orders, $〈 I_{m_{1}} I_{m_{2}} 〉 .$ It is intriguing to notice that, from a spatially incoherent beam, similar to the images presented in Fig. 2, a well-formed pattern appears in the intensity correlation even for fractional values of m₁ and m₂. For the case $〈 I_{m_{1} = 4} I_{m_{2} = 0} 〉$ , we have a stable vortex with ETC equal to 4, in total agreement with the panel for $| {\tilde{Γ}}_{m_{1} = 4, m_{2} = 0} |$ . This result is confirmed by checking the phase diagram in the first column where a round trip phase (relative phase) of four times 2π is observed. On the other hand, unstable vortices are observed in the first line of columns two and three for fractional valuesof the ETC, i.e., for $〈 I_{m_{1} = 1.7} I_{m_{2} = 0} 〉$ , and $〈 I_{m_{1} = 4} I_{m_{2} = 0.5} 〉,$ respectively. However, the result shown for $〈 I_{m_{1} = 3.7} I_{m_{2} = 1.7} 〉$ is quite surprising. Even though each beam has non-integer values of m₁ and m₂, the vortex core in the center is stable (it doesn’t brake) and maintain its ETC, in this case, equal to two. This result is confirmed by observing the center of the phase diagram in the fourth column. However, a new ingredient appears in this panel. The phase diagram shows a ring dislocation. It emerges, by inspecting Eq. (6), due to the exponent in intensity distribution $(r_{1}^{| m_{1} | + | m_{2} |} = r_{1}^{5.7})$ be different from the ETC of the azimuthal phase $(\exp [i (m_{1} - m_{2}) ϕ_{1}] = \exp [i (2) ϕ_{1}])$ .

The physical interpretation behind Fig. 3 can be easily understood by Eq. (6), once that we can play with the TC associated to each incoherent beam, m₁ and m₂, according to $m_{Γ} = (m_{1} - m_{2})$ , to obtain an ETC value that can be different from $m_{1}$ and m₂ or even have fractional values. These results stress the fact that the OAM of incoherent fields obeys a well-defined vortices correlation rule.

A way to elucidate the strong correlation between coherence vortices is shown in Fig. 4. Now different apertures are aligned in the beam path. The apertures are illustrated in Fig. 1. In this case, two incoherent beams possessing topological charges $m_{1}$ and $m_{2}$ are diffracted by two apertures A₁ and A₂, respectively. It is interesting to point out that if we spatially superpose the aperture A₁ over A₂ we have an equilateral triangle. First line of Fig. 4 shows numerical intensity correlations between various combinations of m. Observe that following $m_{Γ} = m_{1} - m_{2}$ from Eq. (6), the ETC is equal to $m_{Γ} = 2$ for the first and third columns, $m_{Γ} = - 2$ for the second column, $m_{Γ} = 3$ for the panels in the fourth and fifth columns, and $m_{Γ} = 0$ for the sixth column . In fact, following Hickmann et all [20], the ETP can be obtained observing that $m_{Γ} = N - 1$ ( $N$ is a number of spots of any extern side of the triangle), in total agreement with the theory (second line of Fig. 4). A triangular aperture is a well know technique to determine the amount of OAM, unlike, for example, a square aperture [21]. It is worth point out that phase diagrams are the same to the panels of the third line of the fourth and fifth columns. It happens because the intensity distribution and azimuthal phase are the same (see Eq. (6)). In the last column observe the formation of a central spot as expected.

Finally, we would like to point out that the results obtained from arrangement of A₁ and A₂ apertures, forming a non-localized or distributed equilateral triangle, do not represent a nonlocal Bell measurement. However, the results presented in this paper may be of particular interest for quantum optics [22, 23].

5. Conclusions

In conclusion, we explored same fundamental aspects of coherence vortex using intensity correlation such as strong vortices correlation, stability of a coherence fractional vortex, and non-localized azimuthal phase. We have investigated the correlation between two spatially incoherent beams with different embedded phase singularities. A well defined topological charge was obtained in the intensity correlation following a correlation rule such that this value is bound to the topological charge associated to each incoherent beam. As a consequence of this rule, we observed a precise signature of an integer vortex in the intensity correlation from two incoherent beams with each beam possessing fractional vortices. And, we also showed that a well-defined amount of OAM can easily be identified in the intensity correlation when two incoherent beams possessing OAM were diffracted by a non-localized object. We have obtained excellent agreement between the theory using the Gaussian-Schell correlator and the experimental results. This paper may be useful in future works in singular statistical optics field.

Acknowledgments

The authors would like to thank the following organizations for financial support: CAPES Pró-equipamentos/PROCAD/PROCAD-NF, CNPq/MCT, FAPEAL, INCT- Fotônica para Telecomunicações, and INCT - Informação Quântica.

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Strong correlations between incoherent vortices

Abstract

1. Introduction

2. Theoretical results

3. Experimental setup

4. Results and discussion

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (6)

Optics Express