Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments

Victor V. Kotlyar; Alexey A. Kovalev; Alexey P. Porfirev

doi:10.1364/OE.27.011236

1. Introduction

In recent years, optical vortices and laser beams with orbital angular momentum (OAM) have been actively studied, having found many practical uses in wireless optical communications, quantum information, probing atmosphere, and micromanipulation [1,2]. Modern reviews [3,4] and monographs [5] dealing with these topics have recently been published. Thus, developing simple and high-efficiency techniques for measuring the OAM of such laser beams has been a topical issue. For paraxial laser beams, OAM can be measured using multi-order diffractive optical elements [6,7], diffractive transformation optics [8], metasurfaces [9], cylindrical or astigmatic lenses [10–15], interferograms [16–18], triangular apertures [19] and annular diffraction gratings [20]. While not all of the above-listed techniques are suited for measuring both integer and fractional OAM, an arbitrary OAM can be measured via measuring the intensity of light with a cylindrical lens and computing intensity moments [13,14,21].

It has been proposed [21,22] that the OAM of a paraxial laser beam should be measured using intensity moments. Having been inspired by the idea, here we propose a modernized and simplified version of the technique proposed in Ref [21], which enables the OAM to be identified from a single measurement of the intensity distribution in an arbitrary transverse plane of the beam of interest. The technique allows both integer and fractional OAM to be measured. The only limitation of the techniques is that it is suited only for laser beams that can be fairly accurately approximated by a finite superposition of Laguerre-Gauss (LG) modes (n, 0). Another technique is based on the approach described in Refs [13,14], which describe the use of a single cylindrical lens for measuring the OAM of radially symmetric beams. Meanwhile, for arbitrary laser beams two cylindrical lenses need to be used. It is worth noting that the technique described in Refs [13,14]. is also an approximation (see Appendix). We found the conditions when this technique is almost accurate (error is less than 1%).

In this work, we suggest for the first time measuring the fractional OAM by registering only one intensity distribution and without using any optical elements. An additional advantage of the proposed method is that besides the full OAM it determines the OAM-spectrum of a beam. In addition, we generalize a proposed in [13,14] method of fractional OAM measurement for asymmetric paraxial beams.

2. Determining OAM from single measurement of intensity distribution

Similar to Ref [21], we assume the complex amplitude of a laser beam in a plane perpendicular to the optical axis z to be given by

E (r, φ, z) = \sum_{n = - N}^{N} C_{n} \exp (i n φ) Ψ_{n} (r, z),

where (r, φ) are polar coordinates, C_n are sought-for constants, Ψ_n(r, z) is the radial amplitude of a LG mode (n, 0) [23]:

Ψ_{n} (r, z) = \frac{- i w_{0}}{w (z)} {(\frac{r \sqrt{2}}{w (z)})}^{n} \exp [\frac{- r^{2}}{w^{2} (z)} + \frac{i k r^{2}}{2 R (z)} - i (2 n + 1) arc \tan (\frac{z}{z_{0}})],

_{$z_{0} = k w_{0}^{2} / 2$} is the Rayleigh range,

w (z) = w_{0} {(1 + z^{2} / z_{0}^{2})}^{1 / 2}

is the Gaussian beam radius and

R (z) = z (1 + z_{0}^{2} / z^{2})

is the curvature radius of the Gaussian beam wavefront. Although the finite sum in Eq. (1) is not suitable for approximating an arbitrary function, many vortex laser beams found in practice, e.g. in optical communication lines [24], are described by Eq. (1).

The basis functions of Eq. (2) can be replaced by the radial functions of quasimodal Bessel-Gauss (BG) beams [25]:

Ψ_{1 n} (r, z) = q^{- 1} (z) J_{n} (\frac{α r}{q (z)}) \exp [\frac{- r^{2}}{w_{0}^{2} q (z)} + i k z - \frac{i α^{2} z}{2 k q (z)}],

where q(z) = 1 + iz/z₀, J_n(x) is the n-th order Bessel function and α is Bessel function's scale parameter. Unlike Eq. (1), with the functions in Eq. (3) having two scale parameters, (α, w₀), their use for approximating the light field is preferable.

The intensity of the field in Eq. (1) is given by

I (r, φ, z) = {| E (r, φ, z) |}^{2} = \sum_{n = - N}^{N} \sum_{m = - N}^{N} C_{n} {\bar{C}}_{m} \exp [i (n - m) φ] Ψ_{n} (r, z) {\bar{Ψ}}_{m} (r, z),

where

\bar{C}, \bar{Ψ}

are complex conjugate values. Let us integrate both parts of Eq. (4) with respect to the angular variable:

\tilde{I} (r, z) = \int_{0}^{2 π} I (r, φ, z) d φ = 2 π \sum_{n = - N}^{N} {| C_{n} |}^{2} {| Ψ_{n} (r, z) |}^{2} .

Using CCD-array-aided measurements of the light field intensity I(r, φ, z) and summing it up on 2N radii r_m, Eq. (5) is reduced to a set of 2N × 2N linear algebraic equations:

{\tilde{I}}_{m} = \sum_{n = - N}^{N} M_{n m} x_{n}, m = - N, - (N - 1), ..., (N - 1), N,

where

{\tilde{I}}_{m} = \tilde{I} (r_{m}, z)

is the angle-averaged intensity on the radii r_m,

M_{n m} = {| Ψ_{n} (r_{m}, z) |}^{2}

are theoretically known values of squared modules of the basis functions, e. g. see Eq. (2) or (3), and x_n = |C_n|² are the sought-for squared modules of the expansion coefficients in Eq. (1). Solving the system (6) by, for example, inverse matrix method, we can derive the square modules of all coefficients in Eq. (1). Note that for the system (6) to have an unambiguous solution, the radial basis functions in Eq. (1) need to meet the following condition:

| Ψ_{n} (r, z) | \neq | Ψ_{- n} (r, z) |, n \in [- N, N]

. This condition can be satisfied by choosing the basis functions in Eq. (1), for instance, in the form:

Ψ_{n} (r, z = 0) = A n \exp [- {(r - r_{n})}^{2} / w_{0}^{2}],

where r_n = r₀ + nΔ, n ∈ [–N, N], Δ = r₀/N, $A n$ is a normalizing factor.

For the basis functions given by Eq. (2) and (3), the system (6) can be unambiguously solved at n ∈ [–N, 0] or n ∈ [0, N].

Let us now derive a relationship for determining the OAM of the paraxial light field (1). May the on-axis projection J_z of the total OAM vector and the total light field energy W are given by the well-known relations [7]:

J_{z} = Im \int_{0}^{\infty} \int_{0}^{2 π} \bar{E} (r, φ, z) (\frac{\partial E (r, φ, z)}{\partial φ}) r d r d φ,

W = \int_{0}^{\infty} \int_{0}^{2 π} E (r, φ, z) \bar{E} (r, φ, z) r d r d φ .

Substituting (1) into (7) and (8) yields the normalized total OAM of the beam (1):

\frac{J_{z}}{W} = \frac{\sum_{n = - N}^{N} n {| C_{n} |}^{2} I_{n}}{\sum_{n = - N}^{N} {| C_{n} |}^{2} I_{n}},

where

I_{n} = \int_{0}^{\infty} {| Ψ_{n} (r, z) |}^{2} r d r

is the power of each constituent beam in the linear combination (1). Equation (9) shows that for the OAM to be derived it will suffice to measure just squared modules of the coefficients in Eq. (1), without knowing their phases. The squared modules of the coefficients of Eq. (1) can be derived by solving the set of Eqs. (6): x_n = |C_n|², with the powers I_n of all constituent beams calculated using the known expansion basis: Eq. (2) or Eq. (3), with regard to the above considerations.

2.1 Calculating fractional OAM of superposition of LG modes

Next, using Eq. (9) we can engineer a light field with the desired OAM value. For instance, a superposition of Laguerre-Gauss modes of Eq. (2) with orders (0,2) and (0, 4) can be selected, producing the normalized OAM equal to 3 and 3.5, thus generating the field in the form:

E (r, φ, z = 0) = [\frac{C_{2}}{\sqrt{π w_{0}^{2}}} \exp (2 i φ) {(\frac{r \sqrt{2}}{w_{0}})}^{2} + \frac{C_{4}}{\sqrt{12 π w_{0}^{2}}} \exp (4 i φ) {(\frac{r \sqrt{2}}{w_{0}})}^{4}] \exp (- \frac{r^{2}}{w_{0}^{2}}) .

By introducing the factors 1/(πw₀²)^1/2 and 1/(12πw₀²)^1/2 the energy of both modes is equated. Equation (9) suggests that at C₂ = 1, C₄ = 1, we have J_z/W = 3, and at C₂ = 1, C₄ = 3^1/2 we have J_z/W = 3.5.

The numerical simulation also relies on the Fresnel transform. The simulation parameters are: the wavelength, λ = 532 nm, the waist radius, w₀ = 1 mm, computation domain: –R ≤ x, y ≤ R, (R = 5 mm), and the increment of the coordinates, Δx = Δy = 20 µm.

Figure 1 depicts distributions of intensity [Fig. 1(a,c)] and phase [Fig. 1(b,d)] in the initial plane z = 0, for superposition of the LG modes with the normalized OAM equal to 3 [Fig. 1(a,b)] and 3.5 [Fig. 1(c,d)].

Fig. 1 Two distributions of (a,c) intensity and (b,d) phase in the initial plane z = 0 for superposition of LG modes whose normalized OAM equals (a,b) 3 and (c,d) 3.5.

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For the beams shown in Fig. 1, OAM was also calculated using Eqs. (6) and (9), assuming that the beam contained five modes with the topological charges ranging from m = 1 to m = 5. The radii r_m were chosen in the form mΔr (m = 1, ..., 5), thus providing that 80% of the incident beam energy were found within a radius of 5Δr [Fig. 2(a)], i.e. $\int_{0}^{5 Δ r} {| E |}^{2} r d r d φ = 0.8 \int_{0}^{\infty} {| E |}^{2} r d r d φ$ . In our case 5Δr ≈1.7w₀ in Fig. 1(a,b) and 5Δr ≈1.8w₀ in Fig. 1(c,d).

Fig. 2 Five radii of the rings that were utilized to calculate the intensity for solving the set of Eqs. (6) for the beam in (a) Figs. 1(a, b) (a) and (b) Figs. 1(c, d).

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For the beam in Fig. 1(a, b) the superposition (1) was calculated for the coefficients C₁ = 0; C₂ = 1; C₃ = 0; C₄ = 1; C₅ = 0. By solving the set of Eqs. (6), the squared modules of the coefficients were found to be |C₁|² = 0.001; |C₂|² = 0.98; |C₃|² = 0.05; |C₄|² = 0.94; C₅ = 0.03. With the theoretical value of the normalized OAM being J_z/W = 3.00, the value calculated from Eq. (9) was 3.01.

For the beam in Fig. 1(c, d) the superposition (1) was calculated for the coefficients C₁ = 0; C₂ = 1; C₃ = 0; C₄ = 3^1/2; C₅ = 0. With the radii r_m in Fig. 2(b) calculated in a similar way, the solution of Eqs. (6) the squared modules of the coefficients were found to be |C₁|² = −0.002; |C₂|² = 1.03; |C₃|² = −0.09; |C₄|² = 3.12; and |C₅|² = −0.06. With the theoretical value of the beam normalized OAM being J_z/W = 3.50, Eq. (9) gives a value of 3.51. The error in determining the OAM using the numerical solution of the system in Eq. (6) is less than 1%.

2.2 Experimental determining OAM of superposition of LG modes

Figure 3 shows an experimental setup for determining the OAM of various superpositions of the LG modes. Collimated and expanded laser beam from a solid state laser illuminated the display of a spatial light modulator SLM HOLOEYE PLUTO VIS (1920 × 1080 pixels, each pixel of 8 μm). The phase mask on the modulator was obtained by encoding the complex amplitudes distribution of a given superposition of the LG modes. The reflected and modulated laser beam was spatially filtered by lenses L₂ (f₂ = 500 mm) and L₃ (f₃ = 150 mm) and a diaphragm D, which blocked the light unmodulated by the modulator. Intensity distributions of the generated superpositions of the LG beams were recorded by the camera placed in the focal plane of the lens L₂ (the plane conjugate to the plane of the modulator display z = 0). Figure 4 shows examples of the generated intensity distributions for different superpositions of the LG modes. Using Eqs. (6) and (9). OAM of such beams were calculated. It was supposed that the beam consists of 5 modes with their topological charges from m = 1 to m = 5. The Gaussian beam had the waist radius of ω₀ = 190 μm.

Fig. 3 Optical setup for the experimental determination of OAM of superpositions of the LG modes: Laser is a solid state laser (λ = 532 nm), MO is a microobjective (8 × , NA = 0.3), PH is a pinhole (the hole diameter is 40 μm), L₁, L₂ and L₃ are spherical lenses (f₁ = 350 mm, f₂ = 500 mm, f₃ = 150 mm), SLM is a spatial light modulator HOLOEYE PLUTO-VIS, M is a mirror, D is a diaphragm, Cam is a camera ToupCam U3CMOS08500KPA.

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Fig. 4 Experimentally generated intensity distributions of various superpositions of LG modes registered by the camera (Fig. 3): LG_0,2 + LG_0,4 (a), LG_0,2 + 1/3^1/2LG_0,4 (b), LG_0,2 + 1/2^1/2LG_0,5 (c), LG_0,1 + LG_0,5 (d). Image size is 1500 × 1500 μm.

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For the beam in Fig. 4(a), the coefficients in the superposition in Eq. (1) are equal to C₁ = 0; C₂ = 1; C₃ = 0; C₄ = 1; C₅ = 0. Solution of the system in Eq. (6) gives the following values: C₁ = −0.31i; C₂ = 0.95; C₃ = −0.71i; C₄ = 1.00; C₅ = −0.21i. Theoretical value of the normalized OAM of such a superposition of LG modes is J_z/W = 3.00. Calculation by Eq. (9) using the experimentally obtained intensity distribution gives the value 3.17.

For the beam in Fig. 4(b), the coefficients in the superposition in Eq. (1) are equal to C₁ = 0; C₂ = 1; C₃ = 0; C₄ = 1/3^1/2; C₅ = 0. Solution of the system in Eq. (6) gives the following values: C₁ = −0.31i; C₂ = 1.00; C₃ = −0.77i; C₄ = 0.75; C₅ = 0.19i. Theoretical value of the normalized OAM of such a superposition of LG modes is J_z/W = 2.50. Calculation by Eq. (9) using the experimentally obtained intensity distribution gives the value 2.81.

For the beam in Fig. 4(c), the coefficients in the superposition in Eq. (1) are equal to C₁ = 0; C₂ = 1; C₃ = 0; C₄ = 0; C₅ = 1/2^1/2. Solution of the system in Eq. (6) gives the following values: C₁ = −0.24i; C₂ = 1.00; C₃ = −0.53i; C₄ = −0.23i; C₅ = 0.78. Theoretical value of the normalized OAM of such a superposition of LG modes is J_z/W = 3.00. Calculation by Eq. (9) using the experimentally obtained intensity distribution gives the value 3.23.

For the beam in Fig. 4(d), the coefficients in the superposition in Eq. (1) are equal to C₁ = 1; C₂ = 0; C₃ = 0; C₄ = 0; C₅ = 1. Solution of the system in Eq. (6) gives the following values: C₁ = 1.00; C₂ = 0.40; C₃ = −0.48i; C₄ = 0.32; C₅ = 0.97. Theoretical value of the normalized OAM of such a superposition of LG modes is J_z/W = 3.00. Calculation by Eq. (9) using the experimentally obtained intensity distribution gives the value 2.91.

Table 1 shows that for a superposition of small number of LG modes the OAM measurement error is about 10%. Here we investigated a simple method for determining the OAM by using a single intensity distribution. In principle, no any optical elements are required for this method. Below we study more accurate and more universal method for determining the OAM, though using this method requires measuring two intensity distributions.

Table 1. Comparison of OAM values obtained theoretically by Eq. (9), by solving the system in Eq. (6) and experimentally by using the intensity distributions in Fig. 4

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3. Determining OAM from intensity measurements in focus of two cylindrical lenses

In Refs [13,14]. it was proposed that OAM should be determined from a single measurement of the intensity in the focus of a cylindrical lens. But this method is only suited for optical vortices with a circularly symmetric radial component of the beam, whose amplitude is given by

E_{n} (r, φ, z) = \exp (- i n φ) Ψ (r, z),

where (r, φ, z) are the cylindrical coordinates, and for optical vortices with fractional topological charge, with the integer n in Eq. (10) replaced by a real-valued μ:

E_{μ} (r, φ, z) = \exp (- i μ φ) Ψ (r, z) .

Below, we describe a simple generalization of the above technique allowing OAM of an arbitrary paraxial beam to be determined from two intensity distribution measurements in the Fourier planes of two cylindrical lenses positioned perpendicularly to each other and placed in identical beam branches outgoing from a splitting cube (Fig. 5).

Fig. 5 Optical setup for measuring the OAM with a couple of cylindrical lenses: a Laser, PH - an opaque screen with circular aperture, L1, L2, L3 - spherical lenses, SLM - a spatial light modulator, F - a spatial filter to cut off the non-modulated zero diffraction order, BS - a beam-splitter to divide the beam in two identical beams, CL1, CL2 - perpendicularly positioned cylindrical lenses, CCD1, CCD2 - CCD-arrays, and PC - a computer.

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In the Cartesian coordinates, Eq. (7) can be rewritten as

J_{z} = Im \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \bar{E} (x, y, z) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E (x, y, z) d x d y .

Substituting in (13) the complex amplitude E(x, y, z) expressed as a 1D Fourier transform separately for each coordinate:

E (x, y) = \sqrt{\frac{i k}{2 π f}} \int_{- \infty}^{\infty} {\hat{E}}_{x} (η, y) \exp (\frac{i k η x}{f}) d η,

E (x, y) = \sqrt{\frac{i k}{2 π f}} \int_{- \infty}^{\infty} {\hat{E}}_{y} (x, ξ) \exp (\frac{i k ξ y}{f}) d ξ,

where k is the wavenumber of light and f is the focal length of the cylindrical lenses, yields:

J_{z} = \frac{k}{f} (\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x ξ {| {\hat{E}}_{y} (x, ξ, z) |}^{2} d x d ξ - \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} η y {| {\hat{E}}_{x} (η, y, z) |}^{2} d η d y),

where (in Eqs. (14)-(16))

{\hat{E}}_{y} (x, ξ, z), {\hat{E}}_{x} (η, y, z)

are 1D Fourier transforms in the Cartesian coordinates y and x. In Eq. (16), the coordinates x, ξ, y, η are taken along the Cartesian axes in the focal plane of the cylindrical lenses. Note that with the Fourier transform of Eqs. (14) and (15) not taken along the x- and y-coordinates, these remain the same as in the cylindrical lenses' plane. Thus, OAM can be determined using either Eq. (16) via measuring two intensity distributions in the focal planes of two cylindrical lenses, or Eqs. (6) and (9) via measuring a single intensity distribution. The advantage of the approach based on Eq. (16) over that based on Eqs. (6) and (9) is that the former is suited for arbitrary paraxial light fields and not only for light fields described by superposition of a finite number of LG modes (n, 0) of Eq. (2) or BG modes of Eq. (3). With the relation (16) being approximate, for the derivation of the exact relation see the Appendix section. The exact formula (28) does not allow the OAM to be calculated from the experimental data. Meanwhile, the approximate relation (16) offers a near accurate result (with a 1 - 4% error) for the paraxial beams.

3.1 Paraxial laser beams with fractional OAM

While optical vortices can carry both integer and fractional OAM, the fractional OAM vortices can be of different nature. For example, a detailed analysis of the structure of an optical vortex with fractional topological charge was conducted in [26], Eq. (12). An exact formula to describe the fractional OAM of such vortices was derived in Ref [27]. For non-integer µ, such an optical vortex was shown to carry fractional OAM and be devoid of an isolated on-axis intensity null (singularity), having instead a chain of isolated intensity nulls with topological charges + 1 and −1 near a transverse coordinate axis. Upon free-space propagation, these isolated intensity nulls can be mutually destroyed [26]. Similar behavior is characteristic of asymmetric Bessel beams [28], Bessel-Gauss beams, and Lommel modes [5]. All these beams carry fractional OAM and have a chain of isolated intensity nulls with topological charges + 1 and −1 in the transverse plane.

On the other hand, the fractional-OAM beam can be generated as a linear combination of a finite number of conventional Laguerre-Gauss modes (Eqs. (1), (2)) or Bessel-Gauss modes (Eq. (3)). In this case, as it propagates, the beam can retain its structure (with due account of scale and rotation). Such beams have a finite number of isolated intensity nulls equal to the number of terms N in Eq. (1).

Using quantum formalism, a relationship to describe the fractional OAM of the beam (12) has been derived [27]. Below, we derive a similar relationship for conventional, rather than quantum, vortices. A fractional-OAM optical vortex can be expanded in terms of integer optical vortices:

E_{μ} (r, φ, z) = \exp (- i μ φ) Ψ (r, z) = \frac{e^{i π μ} \sin π μ}{π} Ψ (r, z) \sum_{n = - \infty}^{\infty} \frac{e^{i n φ}}{μ - n} .

Substituting the right-hand side of Eq. (17) into (7) yields:

J_{z} = W \frac{\sin^{2} (π μ)}{π^{2}} \sum_{n = - \infty}^{\infty} \frac{n}{{(μ - n)}^{2}},

where W is the beam energy (power) derived from Eq. (8). The series in the right-hand side of Eq. (18) can be reduced to a reference series [29]

\sum_{n = 1}^{\infty} \frac{n^{2}}{{(n^{2} \pm a^{2})}^{2}} = \frac{π}{4 a} [\pm {\begin{cases} \coth π a \\ \cot π a \end{cases}} \mp a {\begin{cases} {csch}^{2} π a \\ \csc^{2} π a \end{cases}}],

using which the final relationship for the normalized OAM of the field (12) can be written as [27]:

\frac{J_{z}}{W} = μ - \frac{\sin 2 π μ}{2 π} .

From Eq. (20), OAM is seen to be equal to the topological charge μ only for integer and half-integer values of μ. In Ref [13], the relationship (20) was verified experimentally for n < 3.

3.2 Gaussian optical vortex with fractional topological charge

Let us now derive a relationship to describe the complex amplitude in the Fresnel zone for a Gaussian beam having passed through a spiral phase plate (SPP) with fractional topological charge. In the initial plane, the beam amplitude is given by

E_{μ g} (r, ϕ, z = 0) = \exp (i μ ϕ - \frac{r^{2}}{w_{0}^{2}}) .

Then, using a Fresnel transform, we find that at a distance z from the initial plane the complex amplitude of the field (21) takes the form:

E_{μ g} (ρ, θ, z) = (\frac{- i z_{0}}{z q_{1} (z)}) (\frac{e^{i μ π} \sin μ π}{\sqrt{2 π}}) e^{\frac{i k ρ^{2}}{2 z}} x e^{- x} \sum_{n = - \infty}^{\infty} \frac{{(- i)}^{n} e^{i n θ}}{μ - n} [I_{(n - 1) / 2} (x) - I_{(n + 1) / 2} (x)],

where

q_{1} (z) = 1 - \frac{i z_{0}}{z}, x = {(\frac{z_{0}}{z})}^{2} {(\frac{ρ}{w_{0}})}^{2} (\frac{1}{8 q_{1} (z)}),

I_ν(x) is a modified Bessel function of integer and half-integer order. Note that Eq. (22) has been derived using a reference integral [29]:

\int_{0}^{\infty} e^{- p r^{2}} r J_{n} (c r) d r = \frac{c \sqrt{π}}{8 p^{3 / 2}} e^{- \frac{c^{2}}{8 p}} [I_{(n - 1) / 2} (\frac{c^{2}}{8 p}) - I_{(n + 1) / 2} (\frac{c^{2}}{8 p})] .

From Eq. (22), an intensity null is seen to occur on the optical axis (ρ = 0), while the entire intensity pattern is not circularly symmetric. With the modified Bessel functions in Eq. (22) described by a complex argument, the argument takes real values only in the far-field diffraction zone (z >> z₀). The OAM of the field of Eq. (22) is the same as that of the initial field of Eq. (21) and is equal to Eq. (20).

3.3 Bessel optical vortex carrying fractional topological charge

Below, we discuss another example of an optical vortex carrying a fractional topological charge. May a SPP be illuminated by a narrow annular beam whose complex amplitude in the initial plane is

E_{μ b} (r, ϕ, z = 0) = r_{0} \exp (i μ ϕ) δ (r - r_{0}),

where r₀ is the bright ring radius and δ(x) is the Dirac delta function. Then, at a distance z from the initial plane, the complex amplitude of the Bessel beam with fractional topological charge will be given by

E_{μ b} (ρ, θ, z) = (\frac{- i k r_{0}^{2}}{z}) (\frac{e^{i μ π} \sin μ π}{π}) \exp [\frac{i k}{2 z} (ρ^{2} + r_{0}^{2})] \sum_{n = - \infty}^{\infty} \frac{{(- i)}^{n} e^{i n θ}}{μ - n} J_{n} (\frac{k r_{0} ρ}{2 z}),

where J_n(x) is the n-th order Bessel function. The field (26) and the field (25) have the same OAM, which is described by Eq. (20). It is worth noting that the field (26) in general has no on-axis (ρ = 0) intensity null. The intensity null in Eq. (26) occurs only when µ is integer. Actually, for non-integer µ, in Eq. (26) the only non-zero on-axis term (ρ = 0) is the term J₀(0) = 1. If, however, µ is integer this term also equals zero, because the sine function in Eq. (26) takes a zero value.

An optical vortex with fractional topological charge in Eq. (12) is described by superposition of a countable number of optical vortices with integer topological charge. And vice versa, the light field (1) is composed of a finite number of optical vortices with integer topological charge. For the field (1) to carry fractional OAM it would suffice to keep just two terms. Actually, if in Eq. (1) the power of the constituent beams in Eq. (1) is normalized to unit magnitude, $I_{n} = \int_{0}^{\infty} {| Ψ_{n} (r, z) |}^{2} r d r = 1$ , and the sum of squared modules of the coefficients is set equal to unity: $\sum_{n = - N}^{N} {| C_{n} |}^{2} = 1$ , Eq. (9) for two terms with topological charges n and m takes the form (W = 1):

J_{z} = n a + m (1 - a), a = {| C |}^{2} \leq 1.

From Eq. (27), it is seen that if a is an irrational number smaller than unity it would suffice to superimpose two optical vortices with integer topological charge to generate a fractional-OAM light beam.

3.4 Calculating OAM by using intensity distributions in foci of two cylindrical lenses

In this section, we numerically verify proposed technique for determining OAM based on Eq. (20), using as illustration the field (22), which is generated with the aid of the phase μφ formed on the liquid-crystal display of a SLM.

The modeling is based on the numerical calculation of a Fresnel transform. The simulation parameters are: wavelength of light, λ = 532 nm, Gaussian beam waist, w₀ = 1 mm, focal length of the cylindrical lens, f = 1 m, calculation domain: –R ≤ x, y ≤ R, (R = 5 mm), z = f, and discretization step on both coordinates, Δx = Δy = 20 µm.

Shown in Fig. 6 are phase patterns in the initial plane z = 0 [Fig. 6(a,d,g,k)] and intensity patterns in the focus of the lens whose axis is parallel to the x-axis [Fig. 6(b,e,h,l)] and y-axis [Fig. 6(c,f,i,m)] for a Gaussian beam having passed through a SPP with the topological charge 3.00 [Fig. 6(a-c)], 3.25 [Fig. 6(d-f)], 3.50 [Fig. 6(g-i)], and 3.75 [Fig. 6(j-l)]. The intensity patterns in the focal plane of the cylindrical lens [Fig. 6(b,c,e,f,h,i,k,l)] are shown in the domain –R/2 ≤ x, y ≤ R/2.

Fig. 6 Phase patterns (a,d,g,j) in the initial plane z = 0 and intensity patterns in the focal plane of the cylindrical lens whose axis is parallel to the x-axis (b,e, h,k) and y-axis (c,f,i,l) for a Gaussian beam having passed through a SPP with the topological charge 3.00 (a-c), 3.25 (d-f), 3.50 (g-i), and 3.75 (j-l).

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According to Eq. (20), for the above-indicated values of the topological charge, the normalized OAM should be equal to 3.00; 3.09, 3.50, and 3.91, respectively. Actually, by substituting in Eq. (16) intensity values corresponding to the intensity distribution in Fig. 6, we obtain the following OAM values: 2.98 (1.50 + 1.48), 3.06 (1.46 + 1.60), 3.45 (1.70 + 1.75), and 3.87 (2.00 + 1.87). The respective error is 0.7%, 1%, 1.4%, and 1%. Hereafter, OAM is presented as a two-term sum in compliance with Eq. (16). This serves to demonstrate a different, albeit similar, contribution of these terms into OAM. In Refs [13,14]. the contribution of the two terms in Eq. (16) into OAM was assumed to be the same in magnitude and opposite in sign.

In a similar way, measurement of the large-valued fractional OAM can be numerically simulated using a Fresnel integral. Considering that in this case the optical vortex has a stronger divergence the computation domain R needs to be increased, while the increments Δx and Δy need to be made smaller. The simulation parameters are: wavelength, λ = 532 nm, waist radius, w₀ = 1 mm, focal length of the cylindrical lens, f = 1 m, computation domain, –R ≤ x, y ≤ R, (R = 10 mm), z = f, the increment on both coordinates, Δx = Δy = 10 µm.

Figure 7 shows phase patterns [Fig. 7(a)] in the initial plane z = 0 and intensity patterns in the focus of the cylindrical lens whose axis is parallel to the x-axis [Fig. 7(b)] and y-axis [Fig. 7(c)] for a Gaussian beam having passed through a SPP with the topological charge 30.3.

Fig. 7 (a) Phase pattern in the initial plane z = 0 and intensity pattern in the focus of the cylindrical lens whose axis is parallel to the (b) x-axis and (c) y-axis for a Gaussian beam having passed through a SPP with topological charge 30.3.

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According to Eq. (20), for the topological charge 30.3, the normalized OAM should be equal to 30.15. Substituting the intensity values from Fig. 7 into Eq. (16), we find that OAM is 28.87 (14.31 + 14.56), with the error amounting to 4%.

When determining the OAM of superposition of two LG modes, the error is essentially lower because such beams contain just two angular harmonics, unlike beams generated by a SPP with fractional topological charge, containing an infinite number of angular harmonics. Plots in Fig. 8 show in which way the normalized power of individual harmonics (OAM-spectrum) depends on the topological charge of (a) a Gaussian beam having passed through the SPP (12) with fractional topological charge μ = 3.5 and (b) superposition of the LG modes (10) with indices (0,2) and (0,4), whose normalized OAM also equals 3.5.

Fig. 8 Normalized power of the angular harmonics against the topological charge of (a) a Gaussian beam having passed through a SPP with topological charge μ = 3.5 and (b) superposition of the LG modes (0, 2) and (0, 4), with its normalized OAM equal to 3.5.

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From Fig. 8 it is seen that while having different angular spectrum, beams with integer topological charge can carry the same OAM. The fewer angular harmonics in the beam, the more accurately is the OAM determined by measuring two intensities in the foci of the cylindrical lenses. The proposed technique offers a 4% accuracy when measuring a fractional OAM under 30, with the accuracy depending on the size of the computational domain. For instance, the intensity in Fig. 7 was calculated in a 10-mm × 10-mm domain. By properly increasing the computational domain, the error of OAM determination can be further decreased. However, in practice the accuracy cannot be improved in this way as when measuring low intensities under 0.01 of the maximum, the accuracy is not improved due to decreasing signal-to-noise ratio.

Now we confirm our method for asymmetric crescent-shaped Laguerre-Gaussian beams [30]. Complex amplitude of these beams in the initial plane reads as E(x, y, z = 0) = [(x – aw₀) + (y – iaw₀)]ⁿexp(–s/w₀²)L_mⁿ(2s/w₀²), where (x, y, z) are the Cartesian coordinates, L_mⁿ(x) is the associated Laguerre polynomial, w₀ is the Gaussian beam waist radius, a is the asymmetry parameter, n is the vortex topological charge, m defines the number of light rings (or arcs), s = (x – aw₀)² + (y – iaw₀)². Figures 9(a,b) show intensity distributions of the symmetric (a = 0) and asymmetric (a = 0.25) Laguerre-Gaussian beams. We used the following parameters for calculations: wavelength λ = 532 nm, Gaussian beam waist radius w₀ = 1mm, Laguerre polynomial indices (m, n) = (3, 5), calculation area -R ≤ x, y ≤ R (R = 10 mm) (4096 × 4096 points). Figures 9(c, d) show intensity patterns in the focus of the cylindrical lens whose axis is parallel to the x-axis [Fig. 9(c)] and y-axis [Fig. 9(d)]. Using these intensity patterns, the OAM was calculated. Similarly, the OAM was computed for several other values of the asymmetry parameter. Figure 9(e) shows the theoretical dependence of the OAM on the asymmetry parameter (black curve) (Eq. (22) in [30]) and the values computed via Eqs. (14)-(16) by using the intensity distributions in the foci of the cylindrical lenses (red circles).

Fig. 9 Intensity distribution of the symmetric (a = 0) (a) and asymmetric (a = 0.25) (b) Laguerre-Gaussian beams, intensity distributions of the asymmetric Laguerre-Gaussian beam in the foci of the cylindrical lens (c, d), and the dependence of the OAM vs the asymmetry parameter (e). Red circles are the values obtained by using Eqs. (14)-(16).

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3.5 Experimental determination of OAM by measuring two intensity distributions

The experimental study was conducted using an optical setup in Fig. 5. An initial Gaussian laser beam of wavelength 532 nm was expanded and collimated with a 40-µm circular pinhole PH. The collimated light beam was normally incident on a transmission SLM (a 1024 × 768-pixel HOLOEYE LC 2012, with 36-µm pixels and a 36.9 × 27.6-mm operating region). Spherical lenses of focal lengths 350, 250, and 150 mm were utilized as lenses L1, L2, and L3. Using an optical configuration composed of lenses L2, L3 and a spatial filter F, the light field from the SLM was filtered, thus cutting off an inoperative zero diffraction order. The mutually perpendicular cylindrical lenses CL1 and CL2 had a 500-mm focal length. Note that the use of long-focus cylindrical lenses in the numerical simulation and the experiment stems from the requirement of the paraxial focusing. The said lenses were imaged onto the SLM-display, with the Gaussian beam waist being about 6.5 mm. The intensity pattern in the focus of the cylindrical lenses CL1 and CL2 was recorded with 3264 × 2448-pixel CCD-cameras CCD1 and CCD2 (with a 1.67 µm × 1.67 µm pixel).

Figure 10 depicts intensity distributions obtained in the foci of the lenses CL1 and CL2 by the SLM-aided optical vortices at different values of the topological charge (3; 3.25; 3.5; 3.75, and 30.3). In this case, respective values of the normalized OAM derived from Eq. (16) based on the intensity measurements (Fig. 10) were found to be 3.02; 3.10; 3.47; 3.96, and 27.78. These results are in good agreement with the numerical simulation results.

Fig. 10 Intensity distribution in the foci of the cylindrical lenses CL1 and CL2, given that their axes are parallel to the (a-e) x-axis and (f-j) y-axis for a Gaussian beam having passed through a SPP whose topological charge equals (a,f) 3.00, (b,g) 3.25, (c,h) 3.50, (d,i) 3.75, and (e,j) 30.3.

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For comparison, Table 2 gives OAM values numerically and experimentally derived using two cylindrical lenses. For the beams outgoing from a SPP with a fractional topological charge of 3.00; 3.25; 3.50; 3.75; 30.3, Table 2 gives theoretical OAM values derived from Eq. (20), alongside values obtained from Eq. (16) using numerically and experimentally derived intensity distributions.

Table 2. OAM values calculated theoretically using Eq. (20) and numerically using Eq. (16) based on numerically and experimentally derived intensity distributions

View Table | View all tables in this article

According to Table 2, for the topological charges less than 30, the experimental error in determining the OAM of a fractional-topological-charge laser beam by measuring two intensity distributions in the foci of two cylindrical lenses rotated 90 degrees relative to each other does not exceed 8%.

Figure 11 shows theoretical dependence of the OAM on the topological charge m as well as several experimental OAM values. According to the experiments, the measurement error increases with the topological charge. Maximal error was nearly 10% for m = 34.1, while for m < 20 the error was several percents.

Fig. 11 Experimentally measured OAM (red rods) for several fractional topological charges and the theoretical dependence of the OAM on the topological charge (green curve). Length of the rods corresponds to the tolerance of the experiments. The numbers near the plot show the topological charges for which the OAM was measured.

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4. Conclusion

In this work, we have proposed and studied numerically and experimentally two simple and efficient approaches to determining OAM of paraxial laser beams. With the first one, the intensity measured in the Fresnel zone is then numerically averaged over angle at discrete radii before deriving squared modules of the light field expansion coefficients via solving a linear set of equations. With the other technique, two intensity distributions are measured in the Fourier plane of a pair of cylindrical lenses with their axes positioned perpendicularly, before calculating the first-order moments of the measured intensities. The experimental error grows almost linearly from about 1% when determining small fractional OAM (up to 4), and to about 10% for large fractional OAM (up to 34).

5 Appendix 1

Considering that cylindrical lenses perform a Fresnel transform along one Cartesian coordinate, with free space performing a Fresnel transform along the second Cartesian coordinate, Eqs. (14) and (15) are rearranged to

E (x, y) = \frac{i k}{2 π f} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\hat{E}}_{x} (ξ, η) \exp {- \frac{i k}{2 f} [ξ^{2} + {(η - y)}^{2}] + i \frac{k}{f} x ξ} d ξ d η,

E (x, y) = \frac{i k}{2 π f} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\hat{E}}_{y} (ξ, η) \exp {- \frac{i k}{2 f} [{(ξ - x)}^{2} + η^{2}] + i \frac{k}{f} y η} d ξ d η .

Then, OAM takes the form:

\begin{array}{l} J_{z} = - i \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E^{*} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E (x, y) d x d y = \\ = \frac{k^{2}}{2 π f^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} η {\hat{E}}_{y}^{*} (ξ^{'}, η) {\hat{E}}_{y} ({ξ^{'}}^{'}, η) d ξ^{'} d {ξ^{'}}^{'} d η \times \\ \times \exp {\frac{i k}{2 f} ({ξ^{'}}^{2} - {ξ^{'}}^{'}^{2}) - i \frac{k}{f} x (ξ^{'} - {ξ^{'}}^{'})} x d x - \\ - \frac{k^{2}}{2 π f^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ {\hat{E}}_{x}^{*} (ξ, η^{'}) {\hat{E}}_{x} (ξ, {η^{'}}^{'}) d ξ d η^{'} d {η^{'}}^{'} \times \\ \times \exp {\frac{i k}{2 f} ({η^{'}}^{2} - {η^{'}}^{'}^{2}) - \frac{i k}{f} y (η^{'} - {η^{'}}^{'})} y d y . \end{array}

Taking the integrals in (30), we obtain:

\begin{array}{l} J_{z} = - i \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} η {\hat{E}}_{y} (ξ, η) [\frac{\partial}{\partial ξ} {\hat{E}}_{y}^{*} (ξ, η) + \frac{i k ξ}{f} {\hat{E}}_{y}^{*} (ξ, η)] d ξ d η + \\ + i \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ {\hat{E}}_{x} (ξ, η) [\frac{\partial}{\partial η} {\hat{E}}_{x}^{*} (ξ, η) + \frac{i k η}{f} {\hat{E}}_{x}^{*} (ξ, η)] d ξ d η . \end{array}

Or, in a different form:

\begin{array}{l} J_{z} = \frac{k}{f} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ η {| {\hat{E}}_{y} (ξ, η) |}^{2} d ξ d η - \frac{k}{f} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ η {| {\hat{E}}_{x} (ξ, η) |}^{2} d ξ d η - \\ - i \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} η {\hat{E}}_{y} (ξ, η) \frac{\partial}{\partial ξ} {\hat{E}}_{y}^{*} (ξ, η) d ξ d η + i \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ξ {\hat{E}}_{x} (ξ, η) \frac{\partial}{\partial η} {\hat{E}}_{x}^{*} (ξ, η) d ξ d η . \end{array}

The difference between Eqs. (32) and (16), which describe the OAM, is in the third and fourth terms that cannot be derived experimentally. However, from the numerical simulation and experiment, for paraxially propagating beams, the contribution of these two terms to the OAM is seen to be insignificant (several percent).

Funding

RF Ministry of Science and Higher Education (State assignment); Russian Science Foundation (17-19-01186); Russian Foundation for Basic Research (18-29-20003).

Acknowledgements

Section 2 “Determining the OAM from a single measurement of the intensity distribution” was supported by the Russian Science Foundation (17-19-01186), Section 3 “Determining OAM from intensity measurements in the focus of two cylindrical lenses” was supported by the Russian Foundation for Basic Research (18-29-20003), Appendix 1 was supported by RF Ministry of Science and Higher Education (State assignment).

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Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments

Abstract

1. Introduction

2. Determining OAM from single measurement of intensity distribution

2.1 Calculating fractional OAM of superposition of LG modes

2.2 Experimental determining OAM of superposition of LG modes

3. Determining OAM from intensity measurements in focus of two cylindrical lenses

3.1 Paraxial laser beams with fractional OAM

3.2 Gaussian optical vortex with fractional topological charge

3.3 Bessel optical vortex carrying fractional topological charge

3.4 Calculating OAM by using intensity distributions in foci of two cylindrical lenses

3.5 Experimental determination of OAM by measuring two intensity distributions

4. Conclusion

5 Appendix 1

Funding

Acknowledgements

References

Cited By

Figures (11)

Tables (2)

Equations (33)

Optics Express

Superpositions of LG modes of Eq. (2)	Calculation by Eq. (9)	Solution of the system in Eq. (6)	Experiment (Fig. 4)	Standard deviation, %
LG_0,2 + LG_0,4	3.0	3.01	3.17	5.6
LG_0,2 + 1/3^1/2LG_0,4	2.5	2.51	2.81	12.4
LG_0,2 + 1/2^1/2LG_0,5	3.0	-	3.23	7.6
LG_0,1 + LG_0,5	3.0	-	2.91	3.0

μ	3.00	3.25	3.50	3.75	30.3
OAM, Eq. (20)	3.00	3.09	3.50	3.91	30.15
OAM, numerically, Eq. (16)	2.98	3.06	3.45	3.87	28.87
Error, numerically, %	0.7	1.0	1.4	1.0	4.25
OAM, experiment, Eq. (16)	3.02	3.10	3.47	3.96	27.78
Error, experimental, %	0.7	0.3	0.9	1.3	7.9