1. Introduction

Screw dislocations or vortices are classified as one of the categories for topological deformation of fields. They are associated with a topological charge, specified by an integer which determines the accumulated phase change in the closed loop around the dislocation center. In the case of optical beams, the Laguerre-Gaussian (LG) mode L${\mathrm{G}}_p^m$ is one of the modes of paraxial solutions to the wave equation and has screw phase dislocations. The main character of this mode is that its phase distribution of mφ (m; integer), which provides a helical shape for the wavefronts around the beam center with phase singularity. Here, φ is the azimuthal angle and m is the azimuthal index that represents topological charge, that is, the number of 2π cycles in phase about circumference. Another parameter p is the radial index that effectively represents the number of nodal rings about the beam axis. A peculiarity of beams with helical wavefronts lies in the presence of transverse energy circulation, which is termed an optical vortex. The optical vortex carries a well-defined orbital angular momentum (OAM) mħ per photon [1]. It is distinct from a spin angular momentum which is carried by a circularly-polarized light beam. Such a beam with the OAM recently attracted significant research interest because of its increasing applications, such as high-efficiency laser trapping [2, 3, 4], especially trapping for Bose-Einstein condensates (BECs) [5, 6, 7], microstructure rotation in laser tweezers and spanners [8, 9, 10], and quantum information using multidimensional entangled states [11, 12, 13]. Several nonlinear optical processes have been investigated using LG beams [14, 15], including ultrashort pulses [16]. Optical vortex solitons in self-defocusing media demonstrated many common properties with vortices in superfluids and in BECs [17].

The propagating complex electric field E_mp and its slowly-varying envelope u_mp of the L${\mathrm{G}}_p^m$ mode with frequency ω at time t are given by

respectively, where ρ and z denote cylindrical polar coordinates, k is wave number in vacuum, and $L_p^{\left|m\right|}$(x) is the generalized Laguerre polynomial defined by

Parameters R(z) and w(z) denote the radius of curvature of wavefronts and the beam size at a propagation distance z, as expressed by

with the Rayleigh range

The constant w ₀ is the beam waist. The parameter Φ _G denotes Gouy phase, which is known to be an additional phase shift for a focused and propagated beam, differing from that for a plane wave. It is given by

where Φ(z) is the fundamental Gouy phase for the Hermite-Gaussian HG₀₀ (TEM₀₀) beam. This Gouy phase is observed as an axial phase shift that a converging light beam experiences when it passes through a focal point in propagation. In recent papers, it has been shown that the Gouy phase shift is another manifestation of a general Berry’s phase [18, 19, 20, 21], which is an additional geometrical or topological phase acquired by a system after cyclic adiabatic evolution in parameter space. The Berry’s phase or geometric phase has been considered to be a promising tool for quantum computation [22, 23].

Recently, for single-cycle terahertz pulses [24, 25] or few-cycle femtosecond laser pulses [26], the Gouy phase shift has been directly observed both as a change of carrier-envelope phase. However, for continuous wave (cw) or many-cycle pulses, experimental observations of the Gouy phase shift have been based on interferometric measurements [27, 28]. A novel method demonstrated more recently was still relied on an interferometric measurement, where the Gouy phase shift was observed as a phase difference between TEM₀ and TEM₁ modes in one dimension, using a spatial mode interference-locking technique [29].

In the present paper, we present direct experimental observation of the Gouy phase shift for a propagated and focused LG beam by introducing an asymmetric defect. Having the ability to determine the value of the Gouy phase shift at a propagation distance, the technique that we used here is simple and useful without need of a conventional interference method. The present paper is organized as follows. First, the experimental observation of the Gouy phase shift for LG beams is described. Second, the experimental results are analyzed using slowly-varying envelope functions.

2. Experimental setup

An LG beam or an optical vortex was generated using two-dimensional-programmable spatial light modulator (SLM; HOLOEYE LC-R 2500, pixel number 1024×768, pixel size of 19 μm×19 μm) as shown in Fig. 1(a). The light source used was a linearly-polarized cw He-Ne laser (632.8 nm). The beam from the laser was reflectively diffracted by a hologram patterned on the SLM. A spiral pattern (typically as shown in Fig. 1(b)) on SLM enables us to eliminate easily the 0th-order diffraction beam using a configuration of optics and thus generate a well-defined LG beam with topological charge of m. For example, Fig. 1(c) shows the intensity profile of LG (m = 10 and p = 0) beam. Observation of only one ring guaranteed that the generated LG beam was in the well-defined p = 0 mode. In the present study, we introduced an asymmetric defect (defect angle π/3) of the hologram by SLM pattern like Fig. 1(d). This defect broke the circular symmetry. For the diffracted beam from the hologram, its beam profile becomes like a character of ‘C’ shown as Fig. 1(e). To clarify the beam profile evolution originating in the Gouy phase shift under propagation, the beam was focused by a convex lens with a focal length of f = 1000 mm after diffraction from the hologram and its transverse intensity profile was monitored by a charge-coupled device (CCD) camera at a propagation distances z as shown in Fig. 1(a). Here, z is the coordinate along the propagation direction and the focal point was put to be z = 0.

 

Fig. 1. (a) Experimental setup for spatial evolution measurement of a modified LG beam with an asymmetric defect, (b) a normal spiral hologram pattern, (c) a pure LG beam, (d) a spiral hologram pattern with an asymmetric defect (defect angle π/3), and (e) a modified LG beam with an asymmetric defect.

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3. Experimental results

We introduced an asymmetric defect to beams with (m, p) = (0,0) and (±10,0) by modifying hologram patterns. The spatial evolution of modified, that is, ‘C’-figured beams with m= 0 and ±10 was investigated. Figure 2 shows their intensity profiles at z =-55, -15, 0, +15, and 55 cm. Only at z=+55 cm, the 0th-order diffraction beams by SLM were not optically eliminated as shown at the center in Figs. 2(a) and 2(b).

For a modified-LG beam with (m, p) = (+10,0), the introduced defect was rotated along the ring from φ = 0 direction to φ ≃ +π direction with propagation distance z. The rotation was in the counterclockwise sense on CCD when observed from +z-direction. On the contrary, for a modified-LG beam with (m, p) = (-10,0), the defect was rotated in the clockwise sense, from φ = 0 direction to φ ≃-π direction, with increasing distance z.

In good contrast to these results, for a modified-LG beam with (m, p) = (0,0), which corresponds to the modified Hermite-Gaussian HG₀₀ mode, it is definitely found that the beam profile at z > 0 after passing the focal point was spatially inverted from that before the focal point at z<0. While the defect on the ring of the modified-LG beam with (m, p) = (0,0) was in the direction of φ = 0 in z < 0 region (before the focal point), it was in the direction of φ = ±π in z > 0 region (after the focal point). The detailed observation showed that the asymmetric defect was rotated in both clockwise and counterclockwise senses.

 

Fig. 2. Experimantally-observed spatial evolution of modified LG beams with an aymmetric defect for (a) (m, p) = (+10,0), (b) (m, p) = (-10,0) and (c) (m, p) = (0,0). The intensity profiles at z = -15, 0, and +15 cm are magnified 1.7, 2, and 1.7 times, respectively.

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We also investigated the spatial evolution of modified-LG beams with different (m,0) values. The rotation direction was found to be determined by the sign of the topological charge m. Namely, intensity profiles of modified-LG beams were rotated counterclockwise for m > 0 and clockwise for m < 0 when observed on CCD (from +z-direction), as indicated in Figs. 2(a) and 2(b), which was qualitatively in accordance with the result in Ref. [30]. Quantitatively the average azimuthal angles φ _D of the defect in the modified-LG beam ring for m = ±10 are plotted in Fig. 3. This behavior well agreed with the dependence of the fundamental Gouy phase sgn(m)Φ(z) = sgn(m)arctan(z/z _R) on propagation distance z, where

Hence, while the absolute value of the rotation angle was ~ π independent of the topological value m (in p = 0 cases), the rotation direction only depended on the sign of m. This rotational property of the defect is similar to that for dark solitons with weak nonlinearity [31]. In addition, we confirmed that the fundamental Gouy phase shift under propagation from the convex lens position to focal point was less than π/2, thanks to a finite propagation distance. The origin of this feature of modified-LG beams will be discussed in the next section.

In addition, we investigated the 3-dimensional (3D) trajectory [32, 33] of the asymmetric defect as well as rotation direction. Figure 4 shows the trajectories of the defect position projected on x-y plane as a function of propagation distance z, for modified LG beam with (m, p) = (+10,0) and (-10,0). The paths swept out by the defect are described as straight lines and the defect makes a uniform motion.

The advantages of our technique are not only direct Gouy-phase shift observation, but also the capability to evaluate the Rayleigh length z _R and beam waist w ₀. They are fundamental and significant parameters. However, it has been difficult to evaluate them directly from intensity profiles of LG beams, owing to their peculiar shape with central phase singularity. It has been a sharp contrast to the case for HG₀₀ beams. By fitting our experimental results to Eq. (6), the Rayleigh length z _R and beam waist w ₀ were evaluated to be 184 mm and 190 μm, respectively. Thus our technique is a powerful tool to obtain detailed information for LG beams.

 

Fig. 3. (a) Schematic drawing for an intensity profile of a modified LG beam with an aymmetric defect with the coordinates x, y, z, and average defect angle φ _D when observed from +z-direction on CCD. (b) Dependence of observed defect angle φ _D on the propagation distance z for modified LG beams with (m, p) = (+10,0) and (-10,0).

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Fig. 4. (a) 3D trajectories of the asymmetric defect for modified LG beams with (m, p) = (+10,0) and (-10,0) projected on x-y plane as a function of propagation distance z. (b) Relationship between coordinates x,y and x′, y′. (x′ ,y′, z) coordinates are obtained from (x,y, z) coordinates by a roation of Φ ₀ around the z axis. In our experimental configuration, we put φ _D = 0 at z = -65 cm with the initial phase Φ ₀.

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4. Discussions

4.1. Rotation of the defect

First, we discuss the rotation of the asymmetric defect in the x-y plane. The modified-LG beam with an asymmetric defect is not expressed by a pure LG mode u _mp (ρ, φ, z) but by a superposition of LG modes.

The slowly-varying envelope ${\mathrm{\Psi }}_{\mathit{\text{mp}}}^{\mathrm{C}}$(ρ, φ, z) of the modified (‘C’-figured) LG beam with an asymmetric defect can be approximately expressed by

where δφ _D is the effective angle of the defect area (1/e-intensity angle). The slowly-varying envelope ${\Psi }_{\mathit{\text{mp}}}^{\mathrm{C}}$(ρ, φ, z) is also described as a superposition of pure LG modes by

Here, C_{m′ p′} is an expansion coefficient which corresponds to the overlapping integral,

which can be shown to be constant independent of z.

The intensity profile of the modified LG beam at a propagation distance z is given by

where terms A,B,C,D, and E are defined by

for a modified L${\mathrm{G}}_p^m$ beam.

Here, we discuss the spatial evolution of propagating LG beams for three typical cases, (i) m ≠ 0 (|m| ≫ 1), p = 0, (ii) m = 0, p = 0, and (iii) m ≠ 0 (|m| ≫ 1), p ≠ 0 (p ≫ 1). Distributions of |C_mp |² as a function of indices m and p for modified LG beams, as expressed by Eq. (10), are depicted in Fig. 5. Figure 6 shows calculated spatial evolution of modified LG beams for three cases, using Eq. (11).

4.1.1. (i) m ≠ 0 (|m| ≫ 1) and p = 0 case

In case p = 0, that is, for ${\Psi }_{m0}^{\mathrm{C}}$, the pair of indices (m′, p′) of the expansion coefficients C_{m′ p′} (also (m″, p″) of the expansion coefficients C_{m″ p″} ) are distributed around (m,0). For example, the distribution of |C_{m′ p′} |² for m = 10 and p = 0 is shown in Fig. 5(a). Thus, we can see that the term including |C _m0|₂ in A and terms in B in Eq. (11) are the most dominant. The former terms are independent of φ. The latter terms have have a factor of cos Δ ≡ cos[(m′ - m″) φ -(|m′| - |m″|) Φ(z)]. When p = 0, terms of C and D are negligible. Terms of E are estimated to be small though they smear the intensity profile. When |m| ≫ 1, the index m′ and m″ are distributed mainly around m. Thus, we put

The factor cosΔ is therefore given by

Hence, the defect of the propagating beam along z-axis, which is expressed by φ - sgn(m)Φ(z) = const., is rotated counterclockwise or clockwise depending on the sign of m, namely, m > 0 or m < 0, respectively, as shown in Figs. 6(a) and 6(b). when observed on CCD.

4.1.2. (ii) m = 0 and p = 0 case

In this case, the pair of indices (m′, p′) of the expansion coefficients C_{m′ p′} are distributed around (0,0). For example, the distribution of |C_{m′ p′} |² for m = 0 and p = 0 is shown in Fig. 5(c). Thus the phase in Eq. (17) is given by

 

Fig. 5. Distributions of |C_mp |² as a function of indices m and p for modified LG beams with (a) (m, p) = (+10,0), (b) (m, p) = (-10,0) (c) (m, p) = (0,0) and (d) (m, p) = (+10,5).

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It is found that, in this case in good contrast to (i), the defect in the ring is rotated simultaneously in clockwise and counterclockwise senses. Both clockwise- and counterclockwise-rotated defects are broadened depending on the distribution of δm′ and δm″. It is clearly indicated in Figs. 2(c) and 6(c).

4.1.3. (iii) m ≠ 0 (|m| ≫ 1) and p ≠ 0 (p ≫ 1) case

In this case, the pair of indices (m′, p′) of the expansion coefficients C_{m′ p′} are distributed around (m, p). For example, the distribution of |C_{m′ p′} |² for m=+10 and p=+5 is shown in Fig. 5(d). It is noted that unlike p=0 case, C_{m′ p′} are distributed in the region of p′ < p. Thus, the situation is much complicated owing to contribution of D terms as well as A, B and E terms. The sense of rotation is still determined by the sign of m. Although D and E terms are small, they affect the intensity profile through Gouy phase shift including p′ and p″ values. The former makes the defect rotate more than π, while the latter less than π. Hence, the intensity profile has (p+1)- rings and rotational angles of the defects in the rings depend on the radial coordinate ρ. For example as shown in Fig. 6(d), when the rotated angle φ rot in the region of 0 < |φ _rot| < π/2, inner rings rotates faster than outer rings; in the region of π/2<|φ _rot|<π, vice versa. However, averages of total rotational angles are π for all rings, reflecting the Gouy phase shift.

It is noted that the present technique is more effective for larger |m|, because of the radial dependence for LG modes. The radius of maximum amplitude ρ _max is expressed by ${\rho }_{max}=\sqrt{\frac{m}{2}}w\left(z\right)$ for p = 0. Hence, for larger |m|, the displacement of ρ max between m+δm, m and/or m-δm is neglected, while the displacement of ρ max become larger for smaller |m| (≃ 1), resulting in less spatial overlapping of constituent LG modes.

 

Fig. 6. Calculated spatial evolution of modified LG beams with an aymmetric defect for (a) (m, p) = (+10,0), (b) (m, p) = (-10,0), (c) (m, p) = (0,0) and (d) (m, p) = (+10,+5) at propagation distances of z = -∞,-z _R,0,-z _R, and +∞. Dimensions of beams are scaled by w(z).

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4.2. 3D trajectory of the defect

Second we discuss the 3D trajectory of the asymmetric defect. Experimental results showed the asymmetric defect describes the straight and uniform path. This fact indicates that there is no additional force to the defect. Moreover, we can interpret that the fundamental Gouy phase shift corresponds to the defect rotation angle around the phase singular point of the vortex, that is, (x,y) = (0,0).

Now we analyze this 3D trajectory. The effective radius ρ _D of the asymmetric defect is expressed as $\sqrt{\frac{m}{2}}w\left(z\right)$ in the case where p = 0. As shown in Fig. 4(b), we put the azimuthal angle of the defect to be Φ _G + Φ ₀. Thereby the effective defect position in x-y plane is expressed by

With the help of the definition of w(z) in eq. (4), we obtained that

where x′,y′ are coordinates rotated by angle Φ ₀ around z-axis from (x,y) coordinates given by

Thus, it is derived that the trajectory of the defect describes the straight line and makes a uniform motion. Extending this context to other LG beams, it was found that trajectories of the defect (effective position) for any L${\mathrm{G}}_p^m$ modes have the same properties.

4.3. comparison with other analyses

Padgett et al. have calculated and investigated the Poynting vector of eigenmodes L${\mathrm{G}}_p^m$ [34]. They indicated that the rotation of the Poynting vector corresponds to the Gouy phase shift when p=0, and the Poynting vector of the inner ring is faster than that of outer ring(s) when p > 0. The intensity profile of knife-edged LG mode with small m was experimentally investigated and its behavior was investigated [30]. The knife-edged portion was azimuthally traversed owing to the Gouy phase shift, however it was difficult to estimate Gouy phase shift from their intensity profile. It was because of position ambiguity of the phase singular point of (x,y) =(0,0) resulting form their defect angle of π. Furthermore, the deformation of the profile became severe for smaller m.

It should be noted that the calculated Poynting vector in Ref. [34] is not appropriate to the LG beam with an asymmetric defect, or that it is applicable only in short range spatial evolution. It is attributed to the fact that the LG beam with an asymmetric defect is not a pure LG mode but is expressed by a superposition of LG modes. For example, there are six rings with a defect in Fig. 6(d) at z = -∞, Although the rings rotate according to the Poynting vector in z < 0 region, the defect blurs out at z = 0. This indicates that rings do not rotate independently with each other. Moreover, the defect disappears at z = 0 in the case of small m, which cannot be expected by only considering in the Poynting vector.

Here we discussed the case of p = 0 and large m introducing the appropriated defect angle of about π/3, only where it is optimal to evaluate Gouy phase shift. Thus our technique and analysis of the spatial evolution of optical vortices with an asymmetric defect is significant.

5. Conclusion

In conclusion, we directly observed the Gouy phase shift by measuring the intensity profiles of an L${\mathrm{G}}_p^m$ beam with an asymmetric defect when it was propagated around at the focal point. We quantitatively found that the rotation profile of a modified L${\mathrm{G}}_p^m$ beam manifests the Gouy phase effect where the rotation direction depends on only the sign of topological charge m (counterclockwise for m > 0; clockwise for m < 0 when observed from +z-direction). This profile measurement method of an LG beam with an asymmetric defect is a simple and useful technique for obtaining the information of fundamental Gouy phase shift, without need of a conventional interference method using two beams. Moreover, we analyzed the spatial evolution of the asymmetric defects in LG beams. A modified L${\mathrm{G}}_p^m$ beam with an asymmetric defect is expressed by the superposition of pure LG modes whose indices are distributed mainly around (m, p). This superposition well explains the rotational evolution of the defect in space. In addition, we showed experimentally and theoretically that the 3D trajectory of the defect describes a uniform straight line.