This work presents theoretical analysis on the cross correlation function (CCF) of partially coherent vortex beam (PCVB), where the relation of the number of the rings of CCF dislocations and orbital angular momentum (OAM) of PCVB is analyzed in detail. It is shown that rings of CCF dislocations do not always exist, and depend on the coherence length, the order of PCVB and location of observation plane, although the CCF indicates topological charge to some degree. Comprehensive analysis of the CCF of PCVB and numerical simulations all validate such phenomenon.
© 2014 Optical Society of America
The concept of vortex beam was first introduced by Nye and Berry in 1974 . In recent years, a lot of literature focuses the property and application of this special beam carrying orbital angular momentum [2–7]. Among the investigations, measurement of the topological charge (related with the OAM) of vortex beam becomes an important embranchment and most of them are focused on the coherent case [8–13]. An illuminative research on the CCF of PCVB was presented in . More recently, the relation between the topological charge of a partially coherent beam and its CCF has been studied theoretically and experimentally [15, 16]. Based on the analysis of CCF of PCVB, it is revealed by numerical simulation that the number of ring dislocations of the far-field CCF of PCVB is just equal to the value of its topological charge . Moreover, the detail of the spatial correlation singularity affected by both the radial and azimuthal mode indices is investigated in a partially coherent light field . However, it is noteworthy that the studies were focused on the case of far field. It is interesting to find that for very low coherence, the near-field CCF has clear ring dislocations, namely, the topological charge may be detected by the CCF in the near field rather than in the far field.
2. The CCF of PCVB during propagation
Generally, we assume a partially coherent beam carrying a Gaussian amplitude and mth-order vortex, of which the mutual correlation function (MCF) in the original plane can be described as18–20]Eq. (1) into Eq. (2) and by setting, the CCF of PCVB in the observation plane can be obtained after tedious integration
3 Correlation between CCF and topological charge of PCVB
In Eq. (3) there is an mth-order polynomial of , of which the number of zero points is related to the OAM of PCVB. In  the zero points are characterized by the rings of CCF dislocations. However, they are not always present. This work focuses on determining conditions that ensure existence of zero points in the CCF for different cases concerning the transverse coherence length and the observation distance.
3.1 The case while m = 1
For PCVB while m = 1, the CCF takes a simple form asEq. (5) that there exist two extrema with opposite sign in the distribution of CCF, which implies that a single zero point is located between the positions of the two extrema, regardless of the observation distance and coherence length, such that the OAM of the 1st order PCVB can be detected by CCF for all conditions. The radius of the ring dislocation is decided byFigure 1 shows the simulation result of normalized CCF of the 1st order PCVB and the parameters are chosen as: λ = 800nm, w = 2.5mm (ZR is the Rayleigh distance).
In the near field [Fig. 1(a)], the CCF with low coherence (Lc = 0.2w) shows high resolution for detection of the zero point (ring of dislocation) because the magnitude of two extrema are both considerable. While the coherence length increases beyond w, the zero point can only be revealed from the magnified drawing in the top right corner of Fig. 1(a). This phenomenon indicates that the normalized magnitude of the first extremum of the CCF becomes smaller with increasing of coherence length, resulting in obvious decreasing of resolution for detection of the zero point. In the field near Rayleigh distance [Figs. 1(b) and 1(c)], the CCF with low coherence (Lc = 0.2w) shows low resolution for detection of the zero point because the normalized magnitude of the second extremum approaches to zero. The CCF with moderate coherence length (Lc = w) possesses high resolution. While the coherence length increases to 3w, the first extremum drops and the resolution decreases, obviously. In the field far from Rayleigh distance [Fig. 1(d)], high coherence (Lc = 3w) dominates the resolution.
As a whole from Figs. 1(a)-1(d), it can be seen that the CCF of the 1st order PCVB shows high resolution to detect the zero point for the case of low coherence in near field, moderate coherence in the field near Rayleigh distance, and high coherence in the field far from Rayleigh distance, respectively.
3.2 The case of high order
For simplification, we first consider the case while m = 2. The CCF takes the form asFigure 2 shows the simulation result of normalized CCF of the 2nd order PCVB and the basic parameters are the same as in Fig. 1.
In the near field [Fig. 2(a)], the CCF with low coherence (Lc = 0.2w) still shows high resolution and it contains three extrema and two zero points. With increasing of coherence length, similar phenomenon arises for the case (Lc = w, 3w) as shown in Fig. 1(a). The magnitudes of the first two extrema of normalized CCF are very close to zero in the neighborhood of x-axis so that the zero points cannot be easily identified. In the field near and far from Rayleigh distance [Figs. 2(b)-2(d)], the CCF with moderate coherence length (Lc = w) possesses relatively high resolution. By comparing the case of Lc = 3w in Figs. 1(b)-1(d) and Figs. 2(b)-2(d), the similar characteristic can be found that, for larger coherence length, the CCFs of both the 1st and 2nd order PCVB need propagate longer distance to provide high resolution for the detection of zero points.
While considering higher order, the case becomes complex. For example m = 3, the CCF takes the form asEq. (9), which is the same as for the case while m>3. By numerous calculations we find that the characteristic of normalized CCF of higher order PCVB becomes more dependent on the coherence length. Simulation results are presented for the cases of low coherence, moderate coherence and high coherence in Figs. 3-5, respectively.
Figure 3 shows the case of low coherence. While the propagation length is 0.01ZR [Fig. 3(a)], the CCF of the 3rd order PCVB exhibits three zero points, however, the CCFs of the 4th and 5th order PCVB show no clear zero points. While the propagation length increases to 0.02ZR [Fig. 3(b)], it is interesting to find that the CCFs of the 3rd, 4th, 5th order PCVB show clearly 3,4,5 zero points, respectively. With the continuous increasing of propagation length [Figs. 3(c) and 3(d)], the CCFs (m = 3, 4, 5) only obviously display the anterior zero points and other zero points are blurry and almost missing. It can be observed that the CCF of the higher order PCVB with low coherence has high resolution of zero points in the near field and there exists an optimum observation plane [Fig. 3(b)].
For the case of moderate coherence (Lc = w), the CCFs of higher order PCVB show no clear zero points in the near field [Fig. 4(a)]. While the propagation length increases to 0.2ZR, the CCF of the3rd order PCVB takes the lead to show all zero points [Fig. 4(b)]. The CCFs of the 4th and 5th order PCVB follow to show the zero points with the increasing of propagation length, and while the propagation length increases to 0.5ZR, the CCFs show all the zero points obviously [Fig. 4(c)]). With the continuous increasing of propagation length, the CCFs can still show all the zero points but the resolution of the posterior zero points tends to decrease [Fig. 4(d)]. It can be concluded from Figs. 4(a)-4(d) that the best distance should be between the near field and Rayleigh distance to display all the zero points of CCF of higher order PCVB with moderate coherence.
The most troublesome is the case of higher order PCVB with high coherence (Lc = 3w), as described in . At the propagation length shorter than 0.5ZR, the CCFs show no clear zero points in Figs. 5(a) and 5(b). While the propagation length increases to 5ZR, the CCF of the 3rd PCVB takes the lead to show all zero points [Fig. 5(c)], which is a little similar to that in Fig. 4(b). However, with the continuous increasing of propagation length (z = 20ZR), the CCFs of the 4th and 5th order PCVB have no inclination to show the zero points at all. This characteristic does not change a little even at the propagation length beyond 100ZR (calculated, but not plotted in Fig. 5), which accords with that in .
To summarize, the following phenomena can be found for the 1st and 2nd order PCVB that: (I) for the low coherence case, all the zero point(s) of the CCF can be detected in the near field, the field near Rayleigh distance and the far field. The best choice is the near field; (II) for the moderate and high coherence cases, the best choice changes to the field near Rayleigh distance and the far field, respectively. However, for higher order PCVB with low coherence, all the zero points of CCF can only be detected in the near field; for higher order PCVB with moderate coherence, the best choice is the field below Rayleigh distance; with the further increasing of coherence, the method introduced for the case of coherent vortex beam can be used as a substitution to detect the topological charge.
This research was supported by the National Natural Science Foundations of China (Grant Nos. 61307001 and 61178015) and the National Natural Science Foundation of Fujian Province (Grant No.2013J05094).
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