Autocorrelation properties of fully coherent beam with and without orbital angular momentum

Yuanjie Yang; Yuan Dong; Chengliang Zhao; Yi-dong Liu; Yangjian Cai

doi:10.1364/OE.22.002925

1. Introduction

Vortices are inherent to any wave phenomena, and orbital angular momentum (OAM) is a natural property of various types of vortices with helical phase [1]. An optical vortex is a beam of light whose phase twists like a corkscrew around its axis of propagation. Optical vortex with an azimuthal phase structure exp(iλφ) may carry an OAM of λħ per photon, where λ is an integer number and denotes the topological charge (azimuthal index) of the field, and φ is azimuthal coordinate [2]. The vortex beams have been used in many fields, such as free-space information transfer and communications [3–5], quantum information processing and quantum cryptography [6–8], and optical manipulation [9, 10]. Therefore, the study of optical vortices is a fundamental theme of modern optics.

Recently, the spatial correlation singularity of partially coherent vortex field has been studied extensively both in theory [11–14] and in experiment [13, 15]. The correlation singularity is a “virtual” feature of the wavefield, since it cannot be associated with any nulls of intensities but only with nulls in the two-point coherence function [12]. It was shown that when the spatial coherence is low, the cross-correlation function maintains a ring dislocation [13]. Furthermore, in [13] Palacios et al. showed that the ring dislocation of cross-correlation function will disappear as the spatial coherence increases. To our knowledge, the investigation on the autocorrelation properties of the fully coherent beam has not been reported yet.

On the other hand, determining the radial and azimuthal mode indices (p, λ) of a vortex remains an intriguing problem in both the quantum and classical domain [16]. A wide array of diffractive structures have been used to measure the azimuthal index λ of individual Laguerre-Gaussian (LG) beams from the diffraction pattern [17–21] with assumption that the radial mode index p of the incident light field is equal to zero. More recently, Prabhakar et al. [22] proposed a method for determining the topological charge of a vortex beam through measuring the Fourier transform (FT) of its intensity, and it is shown that the number of the dark rings in the FT of the intensity (DRITFOI) is equal to the topological charge of the vortex. However, it is noted that the radial index p is also equal to zero as well in [22]. In fact, the radial index itself adds a new degree of freedom that may be exploited for quantum communication [16]. Therefore, it is necessary to determine both the azimuthal mode and the radial mode in optical vortices.

In this paper, we show that the existence of ring dislocations in the spatial autocorrelation function of a full coherent light is analogous to those in a partially coherent vortex field. We present analytical expressions, numerical simulations, and a heuristic argument that explains how both the radial and the azimuthal indices (p, λ) affect the autocorrelation properties. Our experimental results show a good agreement with the theoretical results. We also show a way to measure the topological charge of light beams when the radial mode p is taken into account.

2. Fourier transform of the intensity in the source plane and the far-field spatial autocorrelation function

Let’s assume that the field of a fully coherent beam in the source plane is written as u(x, y). Then we can get its field U(X, Y) in the far zone through Fourier transform, i.e.,

U (X, Y) = FT {u (x, y)} .

It is known that autocorrelation is the cross-correlation of a variable with itself and the spatial field autocorrelation means that a dependency exists between values of a field in the neighboring or proximal locations in values of a field across the observation plane. In optics, the autocorrelation function is given by [23]

χ (X, Y) = U^{*} (- X, - Y) \otimes U (X, Y),

where * denotes the complex conjugate and ⊗ represents the convolution notation.

According to the Wiener-Khinchin theorem (autocorrelation theorem) [23], Eq. (2) can be written as

χ (X, Y) = FT {u^{*} (x, y) u (x, y)} = FT {I (x, y)} .

Equation (3) shows that we can get the autocorrelation function in the far field through the Fourier transform of the intensity in the source plane.

In the cylindrical coordinate system, the field of a fully coherent Laguerre-Gauss beam with topological charge λ in the source plane z = 0 is given as

u {(ρ, ϕ, 0)}_{p, ℓ} \propto ρ^{| ℓ |} L_{p}^{| ℓ |} (\frac{2 ρ^{2}}{w_{0}^{2}}) \exp (- \frac{ρ^{2}}{w_{0}^{2}}) \exp (- i ℓ ϕ) \exp (i θ),

where w is the waist width, L_p^l(•) is the associated Laguerre polynomial, p is the radial mode index, λ is the azimuthal mode index, ρ and φ are radial and azimuthal coordinates, respectively, and θ is an arbitrary phase.

To exploit the circular symmetry of ${| u (ρ, φ, 0) |}^{2}$ , we can use the Fourier–Bessel transform [23] in a system of polar coordinates,

χ (ξ) = 2 π \int_{0}^{\infty} {| u (ρ, φ, 0) |}^{2} J_{0} (2 π ξ ρ) ρ d ρ .

where ξ is spatial frequency. Substituting Eq. (4) into Eq. (5), one can obtain the following expression for the far-field autocorrelation function (or the FT of the intensity in the source plane),

\begin{array}{l} χ (ξ) = 2 π \int_{0}^{\infty} ρ^{2 | ℓ | + 1} \exp (- \frac{2 ρ^{2}}{w_{0}^{2}}) {(L_{p}^{| ℓ |} (\frac{2 ρ^{2}}{w_{0}^{2}}))}^{2} J_{0} (2 π ξ ρ) d ρ \\ = \frac{π w_{0}^{2 | ℓ | + 2}}{2^{| ℓ | + 1}} \exp (- \frac{π^{2} w_{0}^{2} ξ^{2}}{2}) \sum_{s = 0}^{p} \sum_{t = 0}^{p} \frac{{(- 1)}^{s + t} (s + t + ℓ)!}{s! t!} \\ \times (\begin{matrix} p + | ℓ | \\ p - s \end{matrix}) (\begin{matrix} p + | ℓ | \\ p - t \end{matrix}) L_{s + t + | ℓ |} (\frac{π^{2} w_{0}^{2} ξ^{2}}{2}) \\ = \frac{π w_{0}^{2 | ℓ | + 2}}{2^{| ℓ | + 1}} \exp (- \frac{π^{2} w_{0}^{2} ξ^{2}}{2}) \sum_{s = 0}^{p} \sum_{t = 0}^{p} \frac{{(- 1)}^{s + t} (s + t + | ℓ |)!}{s! t!} \\ \times \frac{(p + | ℓ |)!}{(s + | ℓ |)! (p - s)!} \frac{(p + | ℓ |)!}{(t + | ℓ |)! (p - t)!} L_{s + t + | ℓ |} (\frac{π^{2} w_{0}^{2} ξ^{2}}{2}) \\ = \frac{π w_{0}^{2 | ℓ | + 2}}{2^{| ℓ | + 1}} \exp (- \frac{π^{2} w_{0}^{2} ξ^{2}}{2}) \sum_{s = 0}^{p} \sum_{t = 0}^{p + | ℓ |} {(- 1)}^{s + t} \frac{(s + t + | ℓ |)!}{s! t!} \\ \times \frac{(p + | ℓ |)!}{(s + t + | ℓ |)! s! (p - s)!} \frac{(p + | ℓ |)!}{t! (p + | ℓ | - t)!} {(\frac{π^{2} w_{0}^{2} ξ^{2}}{2})}^{s + t} \\ = \frac{π w_{0}^{2 | ℓ | + 2}}{2^{| ℓ | + 1}} \frac{(p + | ℓ |)!}{p!} \exp (- \frac{π^{2} w_{0}^{2} ξ^{2}}{2}) L_{p} (\frac{π^{2} w_{0}^{2} ξ^{2}}{2}) L_{p + | ℓ |} (\frac{π^{2} w_{0}^{2} ξ^{2}}{2}) . \end{array}

where

(\begin{matrix} n \\ m \end{matrix}) = \frac{n!}{(n - m)! m!}

denotes the binomial coefficient. In above derivations, the following expansion and integral formulae [24] have been used

L_{n}^{α} (x) = \sum_{m = 0}^{n} {(- 1)}^{m} (\begin{matrix} n + α \\ n - m \end{matrix}) \frac{x^{m}}{m!},

\begin{array}{l} \int_{0}^{\infty} x^{μ} e^{- α x^{2}} J_{v} (x y) d x = \frac{y^{v} Γ (\frac{1}{2} (μ - v + 1))}{2^{v + 1} α^{\frac{1}{2} (μ + v + 1)}} \exp (- \frac{y^{2}}{4 α}) L_{\frac{1}{2} (μ - v - 1)}^{v} (\frac{y^{2}}{4 α}) . \\ [Re α > 0, Re (μ + v) > - 1] \end{array}

From Eq. (6) one can study the far-field autocorrelation properties of a LG beam. It is well known that the FT is used to decompose an image into its sine and cosine components. Accordingly, the input intensity profile ${| u (ρ, φ, 0) |}^{2}$ represents the image in the spatial domain, while the output of the transformation in Eq. (5) is the image in the Fourier or frequency domain. In the Fourier domain image, each point represents a particular frequency contained in the spatial domain image. According to the properties of Laguerre polynomials, one sees from Eq. (6) that the number of the zeros in the FT of the intensity, or the far-field autocorrelation function, is dependent on both the azimuthal index λ and the radial index p.

3. Numerical results and analyses

Numerical calculations can be carried out by using the analytical expressions Eq. (6), and the far-field autocorrelation functions of a LG beam are shown in Fig. 1. From Fig. 1 we can see that dark ring dislocations appear in the far-field autocorrelation function, and the number of dark rings is dependent on both the radial and azimuthal indices. Moreover, one can find that a relationship between the number (N) of dark rings of autocorrelation function and the radial and azimuthal mode indices (p, λ), namely, $N = 2 p + | ℓ |$ , is available except Figs. 1(d) and 1(f). This phenomenon can be understood from Eq. (6) clearly. According to the properties of Laguerre polynomial, we know that the function $L_{p} (π^{2} w_{0}^{2} ξ^{2} / 2)$ and $L_{p + ℓ} (π^{2} w_{0}^{2} ξ^{2} / 2)$ have p and $N = p + | ℓ |$ zeros, respectively. Therefore, the number of zeros of $L_{p} (π^{2} w_{0}^{2} ξ^{2} / 2) L_{p + ℓ} (π^{2} w_{0}^{2} ξ^{2} / 2)$ is equal to $N = 2 p + | ℓ |$ . Therefore, the number of dark rings in the far-field autocorrelation function is $N = 2 p + | ℓ |$ consequently. In some special cases, the position of zero of $L_{p} (π^{2} w_{0}^{2} ξ^{2} / 2)$ appears close to that of $L_{p + | ℓ |} (π^{2} w_{0}^{2} ξ^{2} / 2)$ , which will cause the adjacent two dark rings too close to be recognized. As can be seen in Fig. 1(d), the first order dark ring and the second order dark ring are so close that it is not easy to distinguish them. In Fig. 1(f) the fourth dark ring looks wider than others in elsewhere, which in fact shows there are two dark rings together here.

Fig. 1 Contour graph of the far-field autocorrelation function with different azimuthal index λ and radial index p. (a) p = 1, λ = 1; (b) p = 1, λ = 2; (c) p = 1, λ = 3; (d) p = 2, λ = 1; (e) p = 2, λ = 2; (f) p = 2, λ = 3;

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It is noted that from Eq. (6) that one can study both the vortex beam with a Gaussian host (p = 0) [22] and the non-vortex beam (λ = 0), as two special cases of the general results. When p = 0, Eq. (6) can be rewritten as

χ_{l} (ξ) = \frac{π w_{0}^{2 | ℓ | + 2}}{2^{| ℓ | + 1}} | ℓ |! \exp (- \frac{π^{2} w_{0}^{2} ξ^{2}}{2}) L_{| ℓ |} (\frac{π^{2} w_{0}^{2} ξ^{2}}{2}) .

From Eq. (9) we can see that when the radial mode index is equal to zero, the number of the dark rings of the far-field autocorrelation function, or the number of dark rings of the DRITFOI, is equal to the topological charge λ definitely. It is noted that Eq. (9) is consistent with the results of Eq. (10) in [22].

For the case of non-vortex beam, the FT of its intensity can be obtained from Eq. (6) by letting λ = 0 and is expressed as

χ_{p} (ξ) = \frac{π w_{0}^{2}}{2} \exp (- \frac{π^{2} w_{0}^{2} ξ^{2}}{2}) {(L_{p} (\frac{π^{2} w_{0}^{2} ξ^{2}}{2}))}^{2} .

Equation (10) shows that for the beam without OAM, namely, λ = 0, the number of the dark rings in the far-field autocorrelation function is identical to the value of the radial index p definitely. The distribution of the far-field autocorrelation function for non-vortex beams with p = 2 and 5 are shown in Fig. 2, from which we see that there are 2 and 5 dark rings, respectively. Therefore, the dark rings of the far-field autocorrelation function are not the sole property of optical vortex, and for a priori unknown beam we cannot determine the topological charge by observing the pattern of FT of its intensity only.

Fig. 2 Contour graph of the far-field autocorrelation function, or FT, for non-vortex beams with (a) p = 2, and (b) p = 5.

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From Eqs. (6), (9) and (10), one can obtain the following the relationship among the number of dark rings, the azimuthal index λ and the radial index p

N = {\begin{matrix} 2 p + | ℓ | \\ p \end{matrix}, \begin{matrix} ℓ \neq 0 \\ ℓ = 0 \end{matrix} .

From a physical point of view, the dark rings in field autocorrelation function mean the disappearance of the correlation. It is also shown that the influence of radial index p of an optical vortex on the number of dark rings of the far-field autocorrelation function is more significant than that of the non-vortex beam.

We now discuss whether the FT method in [22] is still available to determine the topological charge λ of a vortex beam when the radial mode p is not equal to zero. It is well known that the radial mode index p of LG beams denotes the number of dark rings in the intensity profile. Therefore, one can directly determine the radial index p of a LG beam from its intensity profile. Moreover, if there is a dark center in the intensity profile, which shows that the azimuthal index (topological charge) is not equal to zero. Then we can carry out the FT of the intensity by using MATLAB and measure the number of the dark rings, namely, N, in the FT pattern. Therefore, the topological charge can be determined by recalling the relationship $| ℓ | = N - 2 p$ as shown in Eq. (11).

4. Experiments

Now we carry out experimental demonstration to verify the aforementioned theoretical analysis. The experimental setup to generate a vortex beam and measure its topological charge (the azimuthal index) is shown in Fig. 3. A beam emitted from the diode-pumped solid-state laser ( $λ = 532 nm$ ) first passes through a neutral density filter (NDF) which is used to reduce its power, then it goes toward a reflective mirror (RM) and the reflected beam passes through a beam expander (BE). The beam from the BE is sent toward a spatial light modulator (SLM, BQ-SLM1024), which acts as a grating designed by a computer-generated hologram. The first-order diffraction pattern of the beam from the SLM, selected out by a circular aperture (CA), is exactly a vortex beam with mode indices p and λ. After passing through the thin lens (L), the generated coherent LG beam arrives at a beam profile analyzer (BPA), which is used to measure the focused intensity profile. The final intensity image of the LG beam recorded by the BPA can be stored in a computer (PC₂) and its FT can be processed in MATLAB.

Fig. 3 Schematic of the experimental setup to generate a LG beam and measure its topological charge (the azimuthal index). DPSSL, diode-pumped solid-state laser ( $λ = 532 nm$ ); NDF, neutral density filter; RM, reflective mirror; BE, beam expander; SLM, spatial light modulator; CGH, computer-generated hologram; CA, circular aperture; L, thin lens; BPA, beam profile analyzer; PC₁ and PC₂, personal computer.

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Figure 4(a) shows our experimental results of the focused intensity profile of a LG beam with p = 1 and λ = 2, and its FT pattern that is processed in MATLAB is shown in Fig. 4(b). The corresponding theoretical results of the intensity and its FT patterns are shown in Fig. 4(c) and 4(d), respectively. From Fig. 4, one sees that our experimental results agree well with the theoretical results. Therefore, for a given LG beam, we can determine its radial index from intensity profile, and determine its azimuthal mode index (topological charge) from the Fourier transform of the intensity profile, step by step. In other words, the FT method proposed in [22] is still available for determining the topological charge of vortex beam with high order radial mode index. However, we should measure the radial mode index from the intensity profile before determining the topological charge. Moreover, we have to recognize the number of dark rings in the FT of intensity carefully, because the adjacent two dark rings are very close and it is difficult to distinguish them in some special cases.

Fig. 4 Experimental results of the focused intensity profile (a) and its FT pattern (b), and the corresponding theoretical results (c) and (d) for a LG beam with p = 1 and λ = 2.

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5. Conclusions

In conclusion, the autocorrelation properties of the fully coherent have been studied, and we have demonstrated the existence of ring dislocations in the spatial autocorrelation function of full coherent light beams theoretically and experimentally. The relationships among the radial and azimuthal mode indices and the number of the ring dislocations of far-field autocorrelation function for beams with and without OAM are given. According to the relationship, we have proposed a two-step method to determine the topological charge of a vortex beam with nonzero radial mode index, which was verified in experiment. It is noted that our study is carried out under the assumption that no misalignment exists in our optical system. In the practical applications, misalignment in optical system exists more or less, thus the effect of a misalignment and other practically problems should be considered, and this is a topic for further study. Compared with the simultaneous determination method, the strong advantage of our two-step method is its simplicity.

Acknowledgments

We acknowledge the support by the National Natural Science Foundation of China under Grant nos. 61008009, 11274005, 61205122 and 11374222, the Fundamental Research Funds for the Central Universities under grant no. ZYGX2010J112, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Huo Ying Dong Education Foundation of China under Grant No. 121009, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Autocorrelation properties of fully coherent beam with and without orbital angular momentum

Abstract

1. Introduction

2. Fourier transform of the intensity in the source plane and the far-field spatial autocorrelation function

3. Numerical results and analyses

4. Experiments

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (11)

Optics Express