Analysis of orbital angular momentum of a misaligned optical beam

We report an analysis of the orbital angular momentum of an optical beam misaligned with respect to a reference axis. Both laterally displaced and angularly deflected Laguerre–Gaussian beams are represented in terms of the superposition of azimuthal harmonics with well-defined orbital angular momentum. Simultaneous parallel displacement and angular tilt cause the coupling between azimuthal harmonics and therefore change the projection of the orbital angular momentum on the reference axis. Rotation of beams around the reference axis was simulated by attributing corresponding rotational frequency shifts to the components.


Introduction
As is well known in optics, the photon, being a quantum of electromagnetic radiation with frequency ω, possesses energyhω, momentumhω/c and spin, which attains the value ±h and is directed along the axis of propagation. The concept of photon orbital angular momentum (OAM) [1] has attracted much attention since Allen et al showed, in 1992, the possibility of separating spin and OAM contributions to the total angular momentum for paraxial light beams, namely those with helical phase structure of a wavefront [2,3]. In contrast to the spin angular momentum, the OAM of a single photon can take any integer value inh units, including zero. The origin of the OAM can be explained by the orientation of a Poynting vector, possessing a nonzero azimuthal component. For instance, Laguerre-Gaussian (LG) modes LG l p with the azimuth index l and phase dependence in the form exp(ilϕ + ikz) (where ϕ is the azimuthal angle and z the distance along the beam propagation) are examples of beams with helicoidal wavefronts. Geometrically, the integer l is the winding number of the wavefront helix. The light energy circulation around the beam axis produces the so-called optical vortex (OV) [4,5].Any OV beam with circular symmetry of intensity distribution and azimuth phase dependence exp(imϕ) can be represented as an azimuthal harmonic with a quantized amount of the OAM per photon mh [2]. Hereafter, we shall refer to the winding number of a helicoid m as an orbital number.
The past decade of research efforts demonstrated the effect of OAM transfer from a light beam to small particles captured within a beam, causing their rotation [6], and the rotational Doppler effect (RDE) associated with helicoidal wavefront beams [7,8]. The possibility of obtaining photons possessing OAM from the quantum noise [9] was shown in frequency downconversion experiments. Expressive results of a quantum-mechanical experiment with entangled OAM states were reported recently [10]. The separation of different OAM states was shown to be possible in space, as in the frequency domain [11,12]. The direct experimental scheme for indicating the OAM of a single photon was reported [13]. Moreover, management with photon OAM becomes possible [14].
It is natural to define OAM of a light beam with respect to its optical axis. With this choice of the reference axis, any paraxial beam can be represented as a superposition of LG modes, which are eigenmodes of the paraxial wave equation and, moreover, eigenmodes of the quantum mechanical angular momentum operator [15]. Therefore, they produce a discrete OAM spectrum, along the azimuth index of the mode (orbital number). As was shown in [16], a lateral displacement of a symmetric beam does not change its OAM. However, for an individual photon, this operation can change the photon OAM state related to the reference axis. In this sense, the pure OAM state of an individual photon in a LG mode transforms to the superposition of OAM states in a displaced coordinate frame [14]. Generally, the OAM of a light beam depends on the choice of the reference axis [14,17] and an unambiguous determination of the reference axis for a nonsymmetric beam, e.g. a spatially truncated beam, is not possible.
The goal of the present communication is to show on an example of LG beams how a simple beam transformation generates a discrete OAM spectrum. For a lateral displacement and angular deflection, we represent the transformed beam in the form of an infinite set of azimuthal harmonics in the form of BG beams (or a more complicated structure), with a well-defined OAM. In both cases, the projection of OAM on the axis gives the intrinsic OAM of the mode, irrespective of the angle of deflection or the displacement distance. The transformation of a simultaneous shift and the tilt of a beam give quite different results. We demonstrate the coupling of OAM components, which varies the OAM projection from the intrinsic OAM of the beam.
We discuss the rotational frequency shift appearing in the azimuthal components in the case of the misaligned beam rotation around the reference axis. In order to illustrate the discrete OAM spectrum, we performed a numerical simulation of the off-axis beam rotation by attributing a corresponding frequency shift to the OAM components.

Lateral displacement of the LG beam
A general formula relating the decomposition of an optical beam into a set of azimuthal harmonics with respect to the arbitrary reference axis is given in [14]. Below we will show practical results for displaced Gaussian and LG beams.
Let us consider a Gaussian beam with amplitude E 0 and waist transversal dimension w 0 , so the amplitude distribution in the waist attains the form ∼E 0 exp [−(x 2 + y 2 )/w 2 0 ]. In the coordinate frame (figure 1), which is laterally displaced with respect to the optical axis of the beam, the parallel displacement is determined by the values x 0 and y 0 . Therefore, at z = 0, the displaced Gaussian beam can be written as In polar coordinates, by letting x 0 = δ cos θ and y 0 = δ sinθ (as shown in figure 1), we obtain This form can be represented through the decomposition of the second exponent as where I m is the modified Bessel function of integer order m [18]. In this way, we come to an infinite set of BG beams [19] with the azimuth phase dependence exp(imϕ). As a result of these transformations, the laterally displaced Gaussian beam appears as a superposition of axial BG beams. Each BG component possesses the corresponding orbital number m. The use of the BG basis is an alternative to the LG-mode basis, and they can be easily transformed into each other, if necessary. The result of a superposition of the BG beams is shown in figure 2. A single zero-order BG component looks like an axial beam (figure 2(a)), while the superposition of the zero-order BG beam and plus/minus first-order beams definitely produces an off-axis spot (figure 2(b)). The result of a superposition of 11 components, from m = −5 to 5, is shown in figure 2(c), with high accuracy restores the displaced Gaussian beam.
Each BG component represents an azimuthal harmonic with the orbital number m and therefore mh OAM per photon [2]. The circular symmetry of the intensity distribution ensures the pure OAM state of a photon belonging to each component. Figure 3 shows the calculated power spectrum of the OAM components (part of the total beam power carrying with the component) for the case of the displacement δ equal to the waist parameter w 0 of the Gaussian envelope. As is seen, the power of the components with the numbers |m| > 3 is negligible (figure 3(a)). Larger displacement broadens the spectrum, as shown in figure 3(b). The calculated dispersion of the spectrum was found to be proportional to the displacement, in the form where P m are normalized power weights of the m components. The same operation can be applied for the high-order LG mode with amplitude parameter E LG . Let us consider a mode LG l 0 at the waist z = 0, which is shifted parallel to the Z-axis: In cylindrical coordinates, we get, by decomposition of the Gaussian term into a set of BG components, Using the binomial decomposition, we come to the result for the displaced beam where are amplitudes of the OAM components, and C n l are binomial coefficients. The obtained OAM spectrum is symmetric with respect to m = l. This ensures that the total OAM, as an arithmetic  sum of the contribution of all components, is the same as that for the initial (nondisplaced) beam. The same is valid, of course, for a Gaussian beam (l = 0). As an example, the OAM power spectrum is shown in figure 4 for a displaced LG 1 0 mode, for two displacement values. In contrast to the Gaussian beam, the broadening of the spectrum is accompanied by a depletion of the central component ( figure 4(b)).

Angular deflection of a beam
In a similar way, we can consider a Gaussian beam propagating at some angle γ to the reference axis, as shown in figure 5. The angular deviation of a beam from its initial direction of propagation also results in the appearance of discrete OAM components. Below, we shall restrict ourselves to a range of moderate angles of deflection. The restriction on the deflection angles follows from the structure of a paraxial BG beam description, which assumes that the transversal component of a wavevector k ρ is significantly smaller than the longitudinal one, i.e. k 2 ρ k 2 z , and therefore Within this limitation, we can represent a deflected Gaussian beam as the beam passing through a phase wedge with the transmission function exp[iαρ cos(ϕ − η)], where α is related to the deflection angle γ, α = k ρ = k sinγ and η is the azimuth angle of the deflection. Then, we can rewrite the transmission function by a decomposition in terms of the Bessel functions and obtain a description of the deflected beam: The obtained decomposition of the deflected Gaussian beam to the set of BG beams possesses the symmetric OAM spectrum, shown in figure 6 for the deflection angle γ = 2.5 × 10 −4 (the parameter α = 2.5 mm −1 ). Free space propagation of the beam will be described as the interference of these components taking into account their Gouy phase shifts [20]. The equation for free space propagation of a BG beam of the order m has the form where z R is the Rayleigh range, (z) is the Gouy phase shift and R(z) is the wavefront spherical curvature attributed to the Gaussian envelope of the beam: (z) = arctan(z/z R ), R(z) = z + z 2 R /z.  With these axial BG beams, we can construct the deflected beam via superposition (9) at any distance from the deflection element.
The analysis of the OAM spectrum of a deflected beam, carrying initially non-zero OAM in a pure state lh per photon (e.g. LG LG l 0 mode), is also straightforward. With the use of the phase wedge function decomposition (9), we get or The complex amplitude of an OAM spectral component B ml is These amplitude distributions do not correspond to a solution of the wave equation, and therefore it will be necessary to represent them through a set of LG modes for a description of their free space propagation. However, the phase dependence in equation (12) ensures a pure OAM state in each component. Figure 7 shows the corresponding spectrum for a deflected LG 3 0 beam with the deflection angle γ = 2.5 × 10 −4 .
Some important conclusions can be obtained immediately: the OAM spectrum is symmetric around m = l. Another consequence is that the total angular momentum carried by the OAM components is exactly the same as that for the initial beam. In this sense, the projection of the OAM of the deflected beam does not coincide with a geometrical projection, which depends on the angle γ.

Combination of the tilt and lateral displacement
In any practical situation, a light beam can experience both tilt and displacement (figure 8) with respect to the reference axis. The transformation of an LG mode, which experiences a (here δ is the displacement, α the tilt parameter and θ and η are azimuth angles of shift and tilt, respectively). After mathematical operations, similar to that used above, we get The complex amplitude of an OAM spectral component C ml (ρ) is where The resulting OAM spectrum will no longer be symmetric, except for special combinations of angles θ and η. It is evident that even a Gaussian beam may achieve nonzero OAM with respect to the reference axis. The skew beam attains an extrinsic OAM, due to . OAM power spectrum of LG 1 0 mode, which is displaced by a distance δ = 0.5w 0 to the origin and tilted by an angle γ = 10 −4 . (a) Azimuth parameters are θ = 0, η = π/2; (b) θ = π/2, η = 0.
the appearance of a transversal linear momentum. The total OAM projection will be a superposition of the intrinsic and extrinsic ones. Moreover, the contribution of the extrinsic OAM depends on both the angles θ and η, and therefore beam rotation around the reference axis will vary the OAM projection to the axis. Figure 9 shows OAM spectra for two cases, opposite in the sense of the contributions of an extrinsic OAM and the intrinsic one to the total OAM.

Off-axis beam rotation and RDE
To resolve the OAM spectrum experimentally, one can use the so-called rotational frequency shift or RDE, which appears for rotating beams, bearing nonzero OAM [6,7,11]. High-frequency rotation of a displaced or tilted LG beam around the reference axis will generate sidebands around its optical frequency with the frequency separation equal to the frequency of the beam rotation. However, to detect the sidebands, for instance with a Fabry-Perot interferometer, extremely high angular velocities are required [11]. At the moment, the direct experiment looks rather complicated, but we can perform a reciprocal operation by computer simulation. Each OAM component in equation (3) is supposed to attain a corresponding frequency shift m , along the transformation θ = t, which results in the rotation of the light spot as a whole with the angular velocity . Moreover, it is not necessary to use the whole infinite spectrum to get the proper rotation, as higher BG components do not influence the angular velocity of the spot rotation. The animation shows the off-axis beam rotation.
Continuous rotation of the displaced LG mode around the Z-axis, expressed mathematically as θ = t, generates a frequency shift equal to (m − l) for the m component. The conclusion is that the only difference between input and output photon OAM state can generate a frequency shift in the OAM component by a rotating element. Another important conclusion is that the RDE spectrum is symmetric around the zero-frequency shift in this case, independent of l. The reason is that the central component in the OAM spectrum with orbital number m = l attains no frequency shift, while the component with m = l + n attains n shift for any integer n. Figure 10 shows the RDE spectra, which correspond to the OAM spectra of displaced Gaussian and LG beams, shown in figures 3 and 4. Initially, the monochromatic beam becomes polychromatic, with sidebands around the optical frequency of the beam. The amplitude of the sidebands depends on the beam displacement, while the frequency separation equals the frequency of the beam rotation. The beam rotation will not change the total beam energy, since the influences of positive and negative RDE frequency shifts will balance each other. Therefore, we can expect zero phase variation in rotating the displaced LG beam. (We note, however, that this result is valid only for the initially pure OAM state of the beam.) In the case of a tilted Gaussian or LG beam, the corresponding RDE spectrum is also symmetric around zero-frequency shift, as figure 11 shows for the LG 3 0 mode. The origin of the frequency shift is easily understandable from equation (12), since the rotation of the deflection element produces azimuthal angle variation η = t, thus generating the frequency shift (m − l) for the m component. In a computer simulation, this frequency shift can be attributed to the OAM components to simulate the rotation of the beam.
As a consequence, the rotation of the deflected beam around the Z-axis will not change the beam energy and will not influence the phase of the beam. The same is valid, of course, for a Gaussian beam (l = 0).
A combination of the shift and tilt of the beam radically changes the situation. The OAM projection to the reference axis is no longer equal to the intrinsic OAM of the beam and depends on the beam orientation. Rotation of the beam causes variation of the OAM projection on the axis, and the RDE spectrum becomes rather complicated.  Figure 11. RDE spectrum of a rotating deflected LG 3 0 mode, corresponding to its OAM spectrum (figure 7), in units, on the horizontal axis.

Conclusions
In two simple examples with an easily understandable geometry, we have shown how a spectrum of OAM components originates in operations of a beam's lateral displacement or tilt even for a Gaussian beam that possesses zero OAM. An off-axis beam in this case can be represented as a superposition of axial BG beams, bearing nonzero OAM values. The relative weights of these components reflect the probabilities for an individual photon to get the corresponding OAM value quantized inh units. The presence of the OAM components can be detected through the beam rotation, when they will attain rotational Doppler frequency shifts. Reciprocally, attributing the corresponding frequency shifts to the OAM components in the shape of BG beams, we obtain the composed beam rotation in a computer simulation. In this way, we get a confirmation of the representation of the OAM of a displaced or deflected beam through a discrete spectrum of OAM components.
For a deflected beam possessing nonzero OAM (LG mode), we found that the total OAM carried by azimuthal axial components (OAM projection) has the same absolute value as the input beam, in contrast to geometrical projection of the OAM vector. This result ensures that zero torque is acquired by the deflection element (phase wedge) and ensures zero energy variation of the beam set into rotation.
The situation may be more complicated with a simultaneous shift and tilt of a beam. The reason is that the deflected beam attains a transversal linear momentum, whose cross product with the radius vector generates an extrinsic OAM, the projection of which to the reference axis is also quantized. This additional contribution to the OAM depends on the beam orientation and therefore varies during the beam rotation around the reference axis.
We also mention the general rule for the OAM projection to an arbitrary axis. For any paraxial optical beam, created even with specific properties of noninteger topological charge of an embedded OV [21], the OAM spectrum will reveal a discrete (generally infinite) set of components with integer per photon OAM inh units. The same is valid for an anisotropic OV, e.g. formed by transmission of an OV beam through a cylindrical lens. This case was analysed in [22], where the RDE effect was predicted for OV beams.
We note also the importance of the obtained results for practical application in optical communications, based on the OAM principle [23].