Orbital angular momentum mode detection of the combined vortex beam generated by coherent combining technology

Tao Yu; Hui Xia; Wenke Xie; Yiming Peng

doi:10.1364/OE.409122

1. Introduction

Owing to the helical phase structure, the vortex beams carrying OAM have performed impressively in many fields such as optical communication and optical manipulation since Allen firstly proposed in 1992 [1]. As the modes states with independent ‘twisting’ rate of optical vortices are mutually orthogonal, the vortex beams exhibit extraordinary advantages in improving the capacity of optical communication systems [2,3]. In addition, compared with the traditional gauss beam, the vortex beam exhibit excellent anti-disturbance ability when they propagate though the atmosphere disturbance [4,5]. In the field of optical manipulation, a large number of experiments have been proved that vortex beam can be used as optical tweezers to capture and manipulate micro-particles due to the unique light radiation pressure and trajectory [6,7]. These advantages mentioned above make vortex beam widely used in many fields. However, owing to the intrinsic beam expansion and power loss that cannot be ignored during propagation, the power of vortex beam generated experimentally cannot satisfy the requirements of practical applications, such as long-distance communication. Furthermore, high power vortex beams can be utilized in many applications, such as laser ablation [8], materials processing [9] and many other high-energy-physics applications, such as tubular plasma generation [10]. High power vortex beams remarkably expand the application range of traditional optical vortices. To date, many methods have been investigated to obtain high power vortex beams, including crystal laser [11], bar amplifier [12], fiber laser [13] and main oscillator power amplifier [14]. Unfortunately, owing to the limitation of nonlinear optical effects and thermal effects of material, the power of vortex beam generated by these methods still cannot realize kilowatt scale.

Coherent beam combination (CBC) has been proposed to break through the physic limit for single-channel laser beams. A few years ago, the tiled CBC system of fiber laser array designed by Ma et al. has increased the output power to kilowatt level with good performance and stable phase control ability [15]. Based on the traditional tiled CBC system, an additional discrete vortex phase array modulated on the fiber laser array, Zhi dong et al. obtained the CVB carrying OAM of ${\pm} $1 by converting the piston phase array into continuous spiral phase [16,17]. This scheme provides a new direction to break through the current low power bottleneck of vortex beam. However, in addition to the annular field distribution, the CVBs also have obvious side lobes. The research results of the Ref. [18] show that the side lobes of the CVBs can significantly reduce the power of the central vortex optical field, restricting the expansion of the CVBs to high power. Due to the limitation of the physical size of the device, the CVBs generated by the sub-beam array with low fill factor will have more obvious side lobes. Therefore, to obtain higher power CVB, it is necessary to investigate the reason for the side lobe of the CVB. Furthermore, an appropriate evaluation function is needed to evaluate the beam quality of the CVB.

In this paper, we indicated that the CVBs (including 0-order) generated by CBC technology are a multiplexing vortices optical field, whose phase have other helical phase components besides the target OAM mode. The side-lobe of the CVBs is the interference fringes of these BG mode. Obviously, the enhancement of the target topological charge purity can effectively improve the beam quality of the CVB, because the intensity of the sidelobe decreases or even disappears. Therefore, using the purity of the target OAM mode detected by DVG as an evaluation function, the beam quality of the CVB can be evaluated from the perspective of phase.

2. OAM mode analysis of the CVB

The concept of CVB generation scheme is to simulate the beam array as a single input Gaussian beam, and note that the beamlets should be coherent with each other and the array should have the same intensity distribution and phase. In addition, modulating the phase of sub-beam to form discrete vortex phase is the pivotal issue. With increasing propagation distance, the initially separated sub-beams completely overlap and the discrete vortex phase become a continuous vortex phase structure like a theory vortex beam.

The complex amplitude of M annularly arranged sub-beams loaded with discrete vortex phases can be simplified as [18–23]

(1)$$E(x,y,0) = \sum\limits_{m = 0}^{M - 1} {\exp [ - \frac{{{{(x - R\cos ({\alpha _m}))}^2} + {{(y - R\sin({\alpha _m}))}^2}}}{{w_0^2}}]} \exp (in{\alpha _m})$$

To calculate conveniently, the Eq (1) is rewritten in polar coordinate system as:

(2)$$E(x,y,0) = \exp ( - \frac{{{r^2} + {R^2}}}{{w_0^2}})\sum\limits_{m = 0}^{M - 1} {\exp [\frac{{2R\textrm{r}\cos (\varphi - {\alpha _m})}}{{w_0^2}}]} \exp (in{\alpha _m})$$

Where R is the ring radius of beam array, w₀ is the width of Gauss beam, (Rcos(α_m), Rsin(α_m)) is the coordinates of the m-th sub-beam, M is the number of sub-beams, α_m=2πm/M (m=1,2,3…M) is the angle between the center of the m-th Gaussian beam and the x-axis, as shown in Fig. 1(a). nα_m is the additional piston phase array and the phase difference of adjacent sub-beams is 2πn/M, as shown in Fig. 1(b). The modulated CBA will be combined in free space. Due to the diffraction effect, as the propagation distance increases, the sub-beams overlap with each other. At z=1 m plane, a hollow ring structure is formed in the center of the interference field, but the energy of the central light field accounts for a very low proportion, and the obvious discrete vortex phase distribution can still be found from Fig. 1(d). As the propagation distance increases to 10 m, the sub-beam arrays completely interfere with each other to form a stable vortex optical field, and the discrete vortex phase gradually forms a continuous vortex phase, as shown in Fig. 1(e) and (f).

Fig. 1. The parameters of Gauss beam array. (a) The geometric construction of Gaussian beam array; (b) the discrete vortex phase array. The propagation of 1-order CVB. (c) optical field of z=1 m;(d) phase distribution of z=1 m; (e) optical field of z=10 m;(f) phase distribution of z=10 m;

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According to the Bessel function expansion $\exp (x\cos\varphi ) = \sum\limits_{l ={-} \infty }^\infty {{I_l}} (x)\exp(il\varphi )$,Eq. (2) is reduced to

(3)$$E(r,\varphi ,0) = \exp ( - \frac{{{r^2} + {R^2}}}{{w_0^2}})\sum\limits_{m = 0}^{M - 1} {\sum\limits_{l ={-} \infty }^\infty {{I_l}} (\frac{{2Rr}}{{w_0^2}})\exp(il\varphi )} \exp [i(n - l){\alpha _m}]$$

When $\frac{{n - l}}{M} = p,(p = 0, \pm 1,\ldots )$, considering

(4)$$\sum\limits_{m\textrm{ = 0}}^{M - 1} {\exp (i\frac{{n - l}}{M}2m\pi )} = M$$

The amplitude of the modulated beam array rewrite as

(5)$$E(r,\varphi ,0) = M\exp ( - \frac{{{r^2} + {R^2}}}{{w_0^2}})\sum\limits_{p ={-} \infty }^\infty {{I_{n - pM}}(\frac{{2Rr}}{{w_0^2}})\exp[i(n - pM)\varphi ]}$$

To obtain the far field of CBG beam, lens as the typical Fourier transformer, this transformation for any arbitrary field E(r, φ) into E(ρ, θ) can be written in cylindrical coordinates as [24]

(6)$$E(\rho ,\theta ) = \frac{k}{{i2\pi f}}\int_0^\infty {\int_0^{2\pi } {E(r,\varphi )} } \exp [( - \frac{{ik\rho }}{f})rcos(\theta - \varphi )]rdrd\theta$$

Substituting the E (r, φ, 0) into Eq. (6), the optical field of the CVB at the focus plane is transformed to

(7)$$E(\rho ,\theta ) = \frac{{kMw_0^2}}{{2f}}\exp ( - \frac{{{\rho ^2}}}{{w_f^2}})\sum\limits_{p ={-} \infty }^\infty {{i^{n - pM - 1}}} {J_{n - pM}}(\frac{{Rk\rho }}{f})\exp[i(n - pM)\theta ]$$

Where f is the focus length of lens, w_f=2f/(kw₀). The relative weight P_l of the harmonic component of BG mode with charge n-pM in the Eq. (7) is defined by

(8)$${P_l} = \frac{{\int_0^\infty {{{|{E(\rho ,\theta )} |}^2}\rho d\rho } }}{{\sum\limits_{p ={-} \infty }^\infty {\int_0^\infty {{{|{E(\rho ,\theta )} |}^2}\rho d\rho } } }} = \frac{{{I_{n - pM}}(\frac{{{R^2}}}{{w_0^2}})}}{{\sum\limits_{p ={-} \infty }^\infty {{I_{n - pM}}(\frac{{{R^2}}}{{w_0^2}})} }}$$

Where I_l(x) is the modified Bessel function. We can summarize some valuable information from the theory analysis. The n-order CVB generated by M modulated beamlets is a complex vortex optical field, which is the result of coaxial interference of the BG beams carrying OAM of n ± Mp (p=0,±1,±2…). From the derivation of Eq. (2) and Eq. (3), it can be seen that the reason why the CVB can generated by the beam array is due to the circular symmetrical arrangement of the unit beams and the additional discrete vortex phase, regardless of the specific shape of the beam. Therefore, by changing the characteristics of unit beams, the coherent synthesis model of Gaussian laser arrays described in Eqs. (1)–(8) can be extended to other types of laser arrays, such as flat-top beams and truncated Gaussian beams. The amplitude of the beamlets only affect the relative weight of each spiral harmonic component of the CVB. Form the results of Eq. (8), the OAM mode components and its relative weights of CVBs are determined by the parameters of the modulated beam array, including the beam waist radius, the ring radius of the beam array, the number of beamlets and the target topology charge.

By giving the array parameters M=8, w₀=0.2 mm, R=1 mm, the CVBs with target charge of 0, 1 and 2 are generated, the corresponding relatively weight P_l are calculated by Eq. (8). The optical field and phase of CVBs and its relatively weight are shown on Fig. 2(a)–(i) respectively. From the optical field and phase map shown in Fig. 2(a)–(g), we can see that there are side lobes in the CVBs, which is different from the theoretical BG beam. As shown in Fig. 2(g)–(i), the CVB carries many OAM modes, and the corresponding relatively weight P_l are completely depended on the complex amplitude J_n_±pM (Rkρ/f) of these BG modes. According to properties of the modified Bessel function, I_pM(x)=I_-pM(x) and I_lower(x)>I_higher(x), the OAM spectral of 0-oreder CVB is symmetrically distributed on both sides of the coordinate axis, while the P_l of negative order OAM modes components in non-zero order CVB spectrum is higher than that positive OAM modes. This is also the reason why the P_l of n-order is getting lower and lower, as the topology charge n increase.

Fig. 2. The optical field and P_l of CVBs with different target charge. The CVB Optical field: (a) n=0 (b) n=1 (c) n=2; Relatively weight: (d) n=0 (e) n=1 (f) n=2.

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3. Detection of OAM spectrum

3.1 Dammann vortex grating spectrum analyzer

According to the above OAM spectrum analysis, the CVB can be understood as the multiplexing of multiple optical vortices. To verify the OAM spectrum theory, it is necessary to detect each OAM component and corresponding relative weight P_l. Dammam vortex grating (DVG) can generate equal power optical vortices at the desired diffraction orders, which can be widely used for multiplexing and demultiplexing of massive OAM channels. Consequently, the DVG is a feasible probe of the values and signs of the harmonic component of the CVB. Using the approach of Ref. [25,26], 5×5 DVG is designed to meet the requirement of the OAM detection, and the x and y axes of the DVG have intrinsic topological charges of l_x=1 and l_y=5, as shown in Fig. 3(a). The far field diffraction patterns of Gaussian beam propagating through the DVG is shown in Fig. 3(b), and Fig. 3(c) illustrates the location corresponding relationship between the diffraction order of DVG and the OAM states it can detect.

Fig. 3. (a) 2-D Dammam vortex grating (b) The diffraction field of the designed DVG (c) Schematic diagram of the generated vortex optical field array carrying OAM (d) The detection result of multiplexing optical vortices.

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Subsequently, a beam of multiplexing optical vortices, which consisting of −2 order and 2 order single OAM modes with intensity proportion of 1:4, is designed to verify the detection ability and precision of the designed DVG. When a multiplexing vortex beam incident on the Dammam vortex grating, it can be understood as the superposition of each vortex in the composite vortex field and the vortex beam array generated by the Dammam vortex grating. When there is an OAM mode in the multiplexing vortex beam that is opposite to the vortex beam array, the OAM mode at this diffraction order will degenerate into a Gauss-like spot, but at the same time, other OAM modes in the composite vortex beam will continue to superimpose this opposite OAM. Therefore, we will find that there are still other OAM modes in this diffraction order. According to the method using in Ref. [27] to calculate the relatively weight P_l of multiplexing optical vortices, we calculated the sum of the gray values of the bright spots of −2 and 2nd diffraction order in Fig. 3(d). The calculated result of intensity proportion is 0.9:4.1, which is basically consistent with the theory, indicating that the designed Dammam vortex grating can fully meet the detection demands.

3.2 Optical setup

Using spatial light modulator (SLM) as the core device, the experimental scheme for measuring the OAM spectrum of the CVB is shown in Fig. 4. The OAM detection scheme is mainly divided into two steps, including the generation of the CVB and the OAM mode detection. The collimated seed laser with a wavelength of 632.8 nm is expanded by the beam expander. Following that, the amplitude type SLM (A-SLM: Holoeye LC2002) is used as the beam splitter to split the expanded beam into 8 sub-beams arranged in a circle, waist radius of sub-beam w₀=0.2 mm, and the ring radius R=1 mm. Before sending them for a coherent combination, the generated sub beam array is modulated with the discrete vortex phase by the phase type SLM (P-SLM1: Holoeye Pluto), which can convert the array beam into a vortex beam. The final step of the experimental scheme is the OAM modes detection of the CVB. The generated CVB is passed through the P-SLM2 (Holoeye Pluto) loaded with the designed DVG. The transmitted beam is then focused by a 200 mm focal length converging lens to a CCD camera. Finally, the parallel detection of the OAM modes and relatively weight measurement of the CVB are realized by image processing of the diffraction field.

Fig. 4. Experimental setup: SL, seed laser; BE, beam expander; A-SLM, P-SLM, amplitude spatial light modulator, phase spatial light modulator; BS, beam splitters; FL, focus lens; CCD, CCD camera.

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4. Experiments results and discussion

For the convenience of experimental operation, the detection range of the designed DVG is set from −12 order to 12 order. Of course, adjusting the intrinsic charge l_x and l_y of the DVG can effectively increase the detection range. To prove the universality of the OAM mode analysis theory of the CVB, so two type of CVBs are generated by truncated Gaussian beam array and plane wave beam array in this experiment, and the OAM modes contained in these CVBs can be detected by DVG.

Generally, the Gaussian beam is partially truncated by annular aperture in CBC scheme [28,29]. So, we conducted the OAM mode detection of the CVB generated by the truncated Gaussian beam array. The OAM mode detection results of the truncated CVBs with 0 to 2 target topology charge are shown in Fig. 5. Due to the limited detection range of the DVG, the detection results show that the synthetic optic field carries at least three OAM modes. Moreover, the OAM spectrum of the truncated CVBs are obtained by calculating the sum of the gray value of these bright spots, the simulation and experimental results are shown in Fig. 5(g)–(i), and the optical field of 0–2 order CVBs are embedded in Fig. 5(g)–(i) respectively. Comparing the experiment detection OAM spectrum with simulation results, the error is below 6%, which concludes the validity of experimental results. From the detection results, we can find that the target OAM mode purity of n-order truncated CVBs is lower than the CVBs generated by theoretical Gaussian beam array. Taking the phase purity as the beam quality evaluation function, the beam quality of truncated CVBs is lower.

Fig. 5. (a) The DVG detection results of CVBs generated by truncated Gaussian beam array. Simulation diffraction field of the designed DVG (a) n=0 (b) n=1 (c) n=2; Experimental diffraction field of the designed DVG (d) n=0 (e) n=1 (f) n=2; The OAM spectrum of CVBs (g) n=0 (h) n=1 (i) n=2.

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Then, we also completed the OAM mode detection of the CVBs generated by the plane wave array, and the detection results are shown in Fig. 6. The experiment error of the relative weight P_l is less than 5%, indicating that the OAM experimental detection results are basically consistent with the simulation results. Comparing with the truncated CVBs, the synthesis optical field obtained by plane wave array have fewer sidelobe and higher target OAM mode purity. According to the OAM spectrum of these combined optical field generated by three types of beam arrays, we can see that the CVBs generated by the Gaussian beam array has the highest target OAM mode purity, the least side-lobes, and the best beam quality. So, the relatively weight P_l of target OAM mode can be used as a parameter to evaluate the beam quality of the combined field.

Fig. 6. (a) The DVG detection results of CVBs generated by plane wave array. Simulation diffraction field of the designed DVG (a) n=0 (b) n=1 (c) n=2; Experimental diffraction field of the designed DVG (d) n=0 (e) n=1 (f) n=2; The OAM spectrum of CVBs (g) n=0 (h) n=1 (i) n=2.

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So far, we have experimentally confirmed that the CVBs contain many helical harmonic components. What is curious is that there are variety of side lobes in CVBs. This phenomenon has not been adequately studied in depth. In order to enhance the understanding of the special structure of the CVB, the coaxial interference of the BG beams according to Eq. (7) is investigated, and the results are shown in Fig. 7. Referring to two vortex phase interferences with opposite topological charges, there will be two interference weakening and strengthening points within a 2π period, and the phase singularity will disappear, as shown in Fig. 7(c). From the phase map of BG beam, both the main lobe ring and the outer ring fringes of the BG beam possess vortex phase. When the −8th order BG beam J₋₈ (Rkρ/f) and the 8th order BG beam J₈ (Rkρ/f) interference with each other, both the central ring and outer fringes have 2M brightness and dark interference fringes, like petals, as shown in Fig. 7(a). Subsequently, the coaxial interference of ±8 (p=1) order BG beam and 0 order BG beam is conducted. The result shows that there are side-lobes caused by the interference of outer ring fringes, and there no phase singularity in the synthetic optical field. Analogously, the side-lobe of non–0th order CVB is also caused by the outer ring fringe interference of the BG beams. However, the helical harmonic components J_1±8 (Rkρ/f) do not carry the opposite OAM and the target helical harmonic component J₁ (Rkρ/f) exist phase singularities, so the 1-order CVB generated by the interference of these BG beams will carry OAM. Then, the coaxial interference of helical harmonic components J_n_±pM (Rkρ/f) (p=0,1, 2) is conducted. The correlation coefficient is used as the evaluation function to quantitatively analyze the degree of similarity between the coaxial interference optical fields of spiral harmonic component J_n_±pM (Rkρ/f) and the CVBs. As shown in Fig. 7(d), the results show that the interference pattern of helical harmonic components J_n_±pM (Rkρ/f) (p=0,1) is consisted with the theoretical CVBs show in Fig. 2(a) and (b), and the effect of helical harmonic J_n_±pM (Rkρ/f) (p=2) on the interference pattern can be ignored. The results of coaxial interference indicate that the relative weight of high-order spiral harmonic components in the CVB is very small, which is consistent with the spiral spectrum analysis results in Fig. 2. Obviously, the outer side lobes of the CVB is the interference fringes generated by the coaxial interference of low-order spiral harmonic components.

Fig. 7. The coaxial interference of BG beams with complex amplitude J_n_±_pM (Rkρ/f). (a) The coaxial interference of 0 order, ±1 order and ±2 order harmonic component of the CVB with target charge n=0; (b) The coaxial interference of 0 order, ±1 order and ±2 order harmonic component of the CVB with target charge n=1; (c) Coaxial interference of vortex beams with opposite topological charges.

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In summary, we systematically investigated the formation mechanism of the CVB generated by CBC technology. Based on the DVG detection results of these three CVBs, we can conclude that the OAM mode analysis theory of the CVB can be applied to a variety of coherent combination, and the purity of target OAM mode can be used as a parameter to evaluate the beam quality of the combined field. In addition, the beam radius of the vortex beam is proportional to the topological charge. The beam quality evaluation function that takes the optical field distribution as the main parameter, such as the power in the barrel, can no longer effectively evaluate the high-order vortex beam. In conventional tiled aperture CBC systems, stochastic parallel gradient descent (SPGD) algorithm is widely used in active phase locked control. Taking the target OAM mode purity as the beam quality evaluation parameter is an effective supplement to the current beam quality analysis methods of the combined optical field. We try to realize the closed-loop control of the CBA model by using the target topological OAM mode purity as the evaluation function of the SPGD algorithm, the results are shown in Fig. 8. As shown in Fig. 8(a) and (d), taking the degenerate Gaussian-like spot radius as the ring radius of the power in the barrel (PIB) and calculating the PIB, we can obtain the normalized OAM mode purity P_l of the CVB affected by phase noise as shown in Fig. 8(b) and (e). Then using the P_l as the evaluation function of the SPGD algorithm, we achieved the phase-locked output of the 1-st and 2-nd order CVB, the results shown in Fig. 8(c) and Fig. 8(f). Through parameter optimization and improved algorithm, the iteration rate of SPGD algorithm with P_l as the evaluation function will be remarkably improved. Moreover, the SPGD algorithm with P_l as the evaluation function can theoretically realize the closed-loop control of any-order CVB, which overcomes the embarrassment that PIB cannot effectively evaluate the quality of high-order vortex beams.

Fig. 8. The DVG detection results. (a) 1-nd order CVB with no phase noise; (b) 1-nd order CVB with phase noise; (d) 2-nd order CVB with no phase noise; (e) 2-nd order CVB with phase noise. Convergence curve of SPGD algorithm based on the purity of OAM mode detected by DVG. (c) n=1; (f) n=2.

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5. Conclusion

In this paper, the nth order (including zero order) CVBs generated by CBC are proved to be a multiplexing BG optical field. The OAM spectrum of the CVBs are experimentally detected by the designed DVG, the detection results demonstrated that the OAM mode analysis theory can be applied to the combined optical field generated by any type of array light source. Moreover, the OAM mode analysis theory reveals that the side-lobe of the CVB is the results of coaxial interference of multiple BG harmonic components. Therefore, to improve the beam quality of the synthesized optical field, it is necessary to improve the purity of the target BG mode and suppress other OAM mode components. We use the OAM mode purity of the CVB as the beam quality evaluation function, which provides inspiring guidance for improving the generation scheme of high-power light vortex beam based on the CBC technology.

Funding

Key Research and Development Program of Hunan Province (2019GK2181); Hunan Engineering Research Center of Optoelectronic Inertial Technology (HN-NUDT1908).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Orbital angular momentum mode detection of the combined vortex beam generated by coherent combining technology

Abstract

1. Introduction

2. OAM mode analysis of the CVB

3. Detection of OAM spectrum

3.1 Dammann vortex grating spectrum analyzer

3.2 Optical setup

4. Experiments results and discussion

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (8)

Equations (8)

Optics Express