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The generation and verification of Bessel-Gaussian beam based on coherent beam combining
Results in Physics 16 (2020) 102872
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The generation and verification of Bessel-Gaussian beam based on coherent beam combining
T
⁎
Tao Yua, Hui Xiaa, Wenke Xiea, , Guangzong Xiaob, Hongjian Lia a b
School of Physics and Electronics, Central South University, Changsha 410083, China College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Bessel Gaussian beam Coherent beam combing Power in bucket
We propose a method to generate Bessel-Gaussian (BG) beam by Gaussian beam array where beamlets are arranged along a circle. The phase difference between adjacent beamlets is a constant, and the total phase increment is equal to 2πl (l is integer). Based on coherent superposition of the Gaussian beam array, we proved that the vortex beam obtained by the coherent beam combining (CBC) scheme is Bessel Gaussian beam when the number of Gaussian beams is sufficiently large. The experimental results show that a combined BG (CBG) beam, despite existing side lobe, can still be generated by limited number of Gaussian beams. Therefore, the feasibility of generating vortex (CBG) beam by the CBC scheme has been proved theoretically and experimentally. Moreover, a CBG beam with expected beam radius is generated by controlling the ring radius of Gaussian beam array. Our theoretical predictions match with the experimental results. This work could provide a theoretical basis for practical implementation of generating vortex beams by CBC technology.
Introduction With spiral wavefront structure and annular intensity distribution, vortex beam carries Orbital Angular Momentum (OAM) [1], which has great application prospects in free space optical communication [2,3], optical imaging [4,5], optical manipulation [6] and laser material interaction [7,8]. In recent years, the generation and propagation of vortex beams have been investigated intensively [9–17]. In particular, high-power vortex beams are strongly required in many applications [18–20]. For example, free-space optical communication systems need enough signal power to compensate for the power loss caused by propagation [21]. However, the traditional generation method of vortex beam is extremely dependent on the spatial light modulator, spiral phase plate, diffraction phase holograms and other key devices, and the output power is limited. To overcome this challenge, many innovative methods for generating high power vortex beams have been presented. In 2014, Kim et al. used master-oscillator power-amplifier technology to generate Laguerre-Gaussian beam with high beam quality in the resonator cavity, and coupled the LG beam into a multimode ytterbium doped fiber for transmission and amplification, achieving 10.7w vortex beam output [22]. In 2015, Li demonstrated the amplification of a 1storder vortex beam through Ho:YAG rod amplifier. The simulation shows potentially high gain with more pump power due to the relatively longer length of active material [23,24]. Due to the annular ⁎
structure of vortex beam, the pump and seed beam overlap maybe different along the rod, which means the low amplification efficiency in both MOPA technology and pumping technology. Moreover, due to the thermal lens effect of the amplification system and the nonlinear effect at high power, the power of a single beam cannot be increased infinitely. Theoretically, the maximum output power of fiber laser pumped by semiconductor laser is 36KW without considering the mode instability effect [25,26]. Recently, in order to enhance output power, the CBC technology has gradually been applied to generating high power vortex beam [27–32]. Wang et al. proposed the generation of vortex beam by using the coherent-superposition of multi-beams in a radially symmetric configuration and the spatial evolution rules of the synthesized vortex beams was analyzed [27]. This work has inspired researchers to apply CBC technology to construct structured optical fields such as vortex beam. Chu et al. designed a phase control scheme for generating BesselGaussian beams based on the Stochastic Parallel Gradient Descent (SPGD) algorithm [31]. To eliminate phase noise and phase control errors, Hou et al. proposed a method to extract evaluation function from defocus plane, which optimized the SPGD phase control scheme [32]. Although many works have been carried out based on CBC technology, the theoretical analysis and verification of the vortex synthesis model have not been reported as far as we know. In this paper, based on
Corresponding author. E-mail address: [email protected] (W. Xie).
https://doi.org/10.1016/j.rinp.2019.102872 Received 18 November 2019; Received in revised form 6 December 2019; Accepted 6 December 2019 Available online 10 December 2019 2211-3797/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Results in Physics 16 (2020) 102872
T. Yu, et al.
the theory of diffraction optics, the approximate expression and conditions for generating BG beam of arbitrary order by the modulated Gaussian beam array coherent combing are deduced. Numerical results show that when the number of Gaussian beams is sufficient large, the synthesized CBG beam are consistent with the theoretical BG beams. The correctness of theoretical model is verified by the experimental results. The principle of CBG beam generation could provide theoretical foundation to generate other structured light fields by CBC technology.
Em (r , φ , z = 0; α ) = exp ⎛⎜− ⎝
(3) The coherent superposition of Gaussian beams can be expressed as M−1
E (r , φ , z = 0; α ) = ∑ Em (r , φ , z = 0; α ) m=0
= exp
Principle and method
(
r 2 + R2 − w2 0
M−1
) ∑ exp ⎡⎣ m=0
2Rr cos(φ − αm) ⎤ w02
⎦
exp(ilαm)
(4)
Considering the angle (Δα ) between adjacent beamlet
Bessel beam is the exact solution of the Helmholtz equation, which has infinite energy and cannot be generated in practice. By Gaussian amplitude restriction, the BG beam with finite energy is obtained, which has some properties as that of Bessel beam. In cylindrical coordinate system, the expression of the BG beam at the initial plane (z = 0) is given as [33]
2πm 2π ⎞= Δα = Δ ⎛ Δm M ⎝ M ⎠
(5)
As beam numbers M increases to sufficient large, the angelΔα approaches to infinitesimal dα,
2
r EBG (r , φ , z = 0) = exp ⎜⎛− 2 ⎞⎟ Jl (βr ) exp(ilφ) ⎝ w0 ⎠
r 2 + R2 ⎞ ⎡ 2Rr cos(φ − αm) ⎤ exp(ilα ) ⎟ exp m ⎢ ⎥ w02 ⎠ w02 ⎣ ⎦
dα = (1)
2π dm M
(6)
Therefore, the accumulation of Gaussian beams in Eq. (4) can be converted into the integration of angel α
where (r, φ) is a pair of polar coordinate, w0 is the waist width of Gaussian amplitude, Jl represents the lth-order Bessel function of the first kind, β is a scale factor. Taking BG beam as target beam, we consider coherent superposition of Gaussian beam array loaded with discrete vortex phase, whose centers are placed on a ring of radius R around the z-axis, and each subbeam has the same amplitude, waist width and linearly polarization state, as shown in Fig. 1. In rectangular Cartesian coordinate system, the complex amplitude of Gaussian beam at source plane is given by [27–29,34]
E (r , φ , z = 0; α ) ≈
M 2π
2π
∫ Em (r , φ, z = 0; α ) dα
(7)
0
Substitute Eq. (3) to Eq. (7), the optical field generated by the Gaussian beam array is reduced to
E (r , φ , z = 0; α ) =
(x − R cos αm)2 + (y − R sin αm)2 ⎤ Em (x , y, z = 0; α ) = exp ⎡− exp (ilαm) ⎢ ⎥ w02 ⎣ ⎦ (2)
M r 2 + R2 ⎞ exp ⎛⎜− ⎟ 2π w02 ⎠ ⎝
∫0
2π
exp ⎡ ⎢ ⎣
2Rr cos(φ − α ) ⎤ exp(ilα ) dα ⎥ w02 ⎦ (8)
Using standard integral and Bessel function identity [35],
here R, w0, lαm are the ring radius, waist width and initial phase, respectively. αm = 2πm/M is the angle between the center of the m-th Gaussian beam and the x-axis, the phase difference of adjacent subbeams is 2πl/M. Without loss of generality, an unimportant amplitude constant factor has been omitted. Using polar coordinate transformation, the Eq. (2) is turned to
Il (u) =
1 2π
∫0
2π
exp(u cos φ − ilφ) dφ
(9)
The optical field can be finally derived as
r 2 + R2 ⎞ ⎛ 2Rr ⎞ E (r , φ , z = 0) = M exp ⎜⎛− ⎟ Il ⎜ ⎟ exp(ilφ ) w02 ⎠ ⎝ w02 ⎠ ⎝
(10)
It can be noted from Eq. (10) that it is a combination of Modified Bessel function and Gaussian function, indicating that the optical field generated by the Gaussian beam array is a BG beam. Considering the complex amplitude of Gaussian beam at any plane,
(
Em (r , φ , z; α ) = exp − × exp ⎡ ⎣
r 2 + R2 w 2 (z )
ikRr cos(φ − αm) ⎤ R (z ) ⎦
) exp ⎡⎣
2Rr cos(φ − αm)
exp ⎡i arctan ⎣
w 2 (z )
⎤ exp ⎡− ⎦ ⎣
ik (r 2 + R2) ⎤ 2R (z ) ⎦
( ) − ikz⎤⎦ exp(ilα ) z f
m
(11) Similarly, by superimposing these Gaussian beams, the total complex amplitude can be written as M−1
E (r , φ , z; α ) =
∑
Em (r , φ , z; α )
(12)
m=0
Employing the same procedure as before, the optical field at any plane can be obtained by Eq. (9), (11), (12)
(
) I (βr ) exp (ilφ) ⎤ exp ⎡i arctan ( ) − ikz⎤ ⎦ ⎣ ⎦
E (r , φ , z; α ) = M exp − × exp ⎡− ⎣ Fig. 1. The geometric construction of Gaussian beam array.
where 2
ik (r 2 + R2) 2R (z )
r 2 + R2 w 2 (z )
l
z f
(13)
Results in Physics 16 (2020) 102872
T. Yu, et al.
Fig. 2. Simulation block diagram for generation the 2-nd CBG beam.
Fig. 3. Experimental setup: L, laser; BE, beam expander; BS1, BS2, beam splitters; A-SLM, P-SLM, amplitude spatial light modulator, phase spatial light modulator; M1, mirror; FL, lens; CCD, CCD camera.
To observe the far field of the CBG beam at relatively close distance during the experiment, a simple lens, as an optical Fourier transformer, is used to obtain the far field of target beam. This transformation for any arbitrary field E(r, φ) into E(ρ, θ) can be written in cylindrical coordinates as [36]
2
f2 2R ikR z , f = πw0 2/ λ, β = 2 w (z ) = w0 1 + ⎜⎛ ⎟⎞ , R (z ) = z + + z w ( z ) R (z ) f ⎝ ⎠ (14) Eq. (13) is still a combination of Gaussian and Bessel function, so the theoretical model of generating CBG beam at any plane is constructed. From Eq. (13), the CBG beams are determined by the beam parameters (waist width and wavelength), beam number and ring radius of Gaussian beam array. As shown in Fig. 1, given the parameters w0 and R at the initial plane, the number of Gaussian beams M ≤ πR/w0 if the space of adjacent sub-beams does not overlap. Therefore, the number of Gaussian beams can only be limited to a suitable number. The process of generating CBG beams is shown in Fig. 2, Gaussian beam array with appropriate number are arranged in a circle, and loading the discrete vortex phases. Then, the modulated Gaussian beams are superimposed coherently in free space. Finally, the CBG beam is obtained, which is similar to the BG beam structure.
E (ρ , θ) =
k i2πf
∞ 2π
∫ ∫ E (r , φ) exp ⎡⎢ ⎛− ikρf ⎞ rcos (θ − φ) ⎤⎥ rdrdφ 0
0
⎜
⎟
⎣⎝
⎠
⎦
(15)
Substitute (10) into (15), the expression of the CBG beam at focus plane can be written as
E (ρ , θ) =
kMil if
∞
( ) R2
( ) ( ) J (− ) rdr r2
exp − w 2 exp(ilθ) ∫ exp − w 2 Il =
0
0
kMw02
ρ exp ⎛− w2 ⎞ Jl ⎝ f⎠
2il + 1f
2
⎜
⎟
0
2Rr w02
l
kρr f
( ) exp(ilθ) Rkρ f
(16) The Gaussian beam waist at the focus plane is wf (=2f/kw0). Eq. (16) is also a combination of Gauss function and Bessel function, so the 3
Results in Physics 16 (2020) 102872
T. Yu, et al.
of the lens, the intensity distribution of near and far fields of 2nd-CBG beam composed of M = 16 are shown in Fig. 4(a) and (b). One can see that the near field of 2nd-order CBG beam exist obvious side lobes, but with increase of transmission distance, the stable annular optical field is formed by superposing these modulated sub-beams, and the side lobes decrease significantly. The near and far field intensity distribution of experimental 2nd-order CBG beam are obtained by using the experimental setup in Fig. 3, the optical field profile are shown in Fig. 4(d) and (e). The experimental and simulation results have the same optical field distribution. To illustrate the obtained CBG beams carry vortex phase, the interference experiment is carried out. It can be found from Fig. 4(f) that the interference fringes of CBG beam and plane wave are no longer vertical stripes of light and dark. The center stripe is divided into three, indicating that the detected beam have 4π phase variation. The detected beam is proved to carry 2nd order vortex phase, which is completely consistent with the topological charge of the target beam. Therefore, the generated CBG beam is a 2nd-order vortex beam. From the Eq. (16), the same parameters are set to obtain the theory BG beam, whose optical field profile at the focus plane is shown in Fig. 4(c). By comparing the CBG beam with the theory BG beam at focus plane, the optical field profile of CBG beam also has the same hollow ring structure as BG beam. Thus, the generated CBG beam has not only vortex phase but also intensity distribution modulated by Bessel function. So far, the theoretical model of CBG beam generation based on CBC scheme has been verified.
intensity profile of the CBG beam at focus plane still remain the BG beam structure. Experiment setup The used experimental setup to generate the CBG beam is shown in Fig. 3. The seed laser of 632.8 nm is expanded by a beam expander and then split into two channels by a 50:50 splitter. After the splitter, one of the beams is incident on an amplitude-type SLM (Holoeye LC2012) and the beam is divided into M Gaussian beams with the same parameters. Then the n-order discrete vortex phase is loaded into the Gaussian beam array by the phase-type SLM (Holoeye Pluto). To investigate the propagation of the CBG beam, a lens (f = 300 mm) is added to the experiment. By adjusting the distance between the lens and the CCD, the far field and near field profile of the CBG beam are obtained at the focus plane and off-focus plane, respectively. The other reference beam reflected by the mirror is used to interfere with the CBG beam. The topological charge of the CBG beam can be confirmed by observing the interference fringes in CCD. Power in the bucket (PIB) describes the energy encircled in an onaxis circular area at the receiver plane [37]. In CBC systems, the proportion of the power contained in the bucket and the total power can be used to describe the energy concentration at receiver plane, i.e. a 2π
PPIB
PIBa = = PIBb
∫ ∫ I (r , φ, z ) rdrdφ 0 0 b 2π
∫ ∫ I (r , φ, z ) rdrdφ 0 0
The sub beams of Gaussian beam array
(17)
where a is the half width of the bucket and b is the half width of the receiver plane. Due to the side lobe of CBG beam, the PPIB is taken as the performance index to evaluate the approximation degree of CBG beam and BG beam.
Although the theoretical model has been demonstrated that the generated CBG beam is BG beam, but the CBG beam exist side lobes, which is not completely matched with the theoretical BG beam. From the derivation of Eq. (11) and Eq. (13), the number of beamlets directly determine the approximation degree between CBG beam and theoretical BG beam. Therefore, the number of beams in the coherent combining system is studied in detail. In the case of given array ring radius R = 1.3 cm and w0 = 0.2 cm, the 2nd-order CBG beams composed of 8, 12, 16 and 20 sub-beams are studied. Simulation and experimental results are shown in Fig. 5(a)–(h) respectively. The experimental results are basically consistent with the simulation results. According to the variation of optical field in Fig. (5), the energy change of CBG beams
Results and discussion Verification of the CBG beam generation model To verify the correctness of the theoretical model mentioned above, the CBG beam and theoretical BG beam are investigated according to Eq. (11) and Eq. (13), respectively. According to the Fourier transform
Fig. 4. Generation of 2-nd CBG beam and theoretical BG beam. Simulation 2nd-CBG beam intensity profile: (a) near field (d = 270 mm) (b) far field (d = 300 mm) (c) Theoretical 2nd-BG beam. Experimental 2nd-CBG beam intensity profile: (d) near field (d = 270 mm) (e) far field (d = 300 mm) (f) Interference pattern of CBG beam. 4
Results in Physics 16 (2020) 102872
T. Yu, et al.
Fig. 5. 2nd-CBG beam with different composed sub-beams. Simulation 2nd-CBG beam: (a) M = 8 (b) M = 12 (c) M = 16 (d) M = 20; Experimental 2nd-CBG beam: (e) M = 8 (f) M = 12 (g) M = 16 (h) M = 20.
relatively large when composed beam numbers are small, the reason is that the CBG optical field has not yet formed a complete annular structure, and its energy is still mainly concentrated on the Gaussian array. By increasing the composed beam numbers, the energy of side lobe gradually transfer into the rings, so the PPIB increase simultaneously. When the number of beams increases sufficiently, the PPIB of the CBG beam is equal to the PPIB of the theoretical BG beam. This result further verify the correctness of the theoretical model proposed in this paper.
are mainly manifested as the change of side lobe and diffraction ring. Under the same ring radius, the 2nd-order CBG beam generated by the eight sub-beams does not form a ring structure, and the energy is still concentrated on the Gaussian beam array. With the increase of beam numbers, the energy of Gaussian beam array gradually transfers into the central main ring and diffraction ring of the CBG beam and the energy of the external side lobe decreases significantly. Moreover, during the process of energy transfer, the ring radius of the CBG beam can remain unchanged. Therefore, increasing the number of beams can significantly improve the similarity between CBG beam and standard BG beam. The PPIB of 2nd-order CBG beams composed of different beam numbers are calculated, which is used to quantitatively analyze the influence of number beams on the energy concentration of CBG beam, so the variation trend of intensity distribution of CBG beam can be revealed by the change of PPIB. As shown in Fig. 6(a), the barrel radius of PPIB is set as a (including the central main lobe and the first diffraction ring), the radius of whole optical field is b, where r is the radius of the central main lobe ring. Taking a as the barrel radius, the PPIB of 2nd-order CBG beams composed of 8 to 30 beams are calculated, and the results are shown in Fig. 6(b). We can found that the PPIB are
The ring radius of Gaussian beam array From Eq. (11) and Eq. (13), the complex amplitude of CBG beam is not only determined by the beam parameters and beam numbers, but also related to the ring radius R of Gaussian beam array. Since the beam numbers does not change the beam radius of CBG beam, we set the beam numbers close to πR/w0 in the following study. The 2nd-order CBG beam composed of Gaussian beam array with different ring radius is investigated, the simulation and experimental results are shown in Fig. 7(a)–(f) respectively. From Fig. 7, one can found that a CBG beams with a smaller ring radius r can be generated by increasing the radius R
Fig. 6. (a) Intensity profile of 2nd-BG beam (b) The PPIB of 2nd-order CBG beam. 5
Results in Physics 16 (2020) 102872
T. Yu, et al.
Fig. 7. 2nd-CBG beam generated by Gaussian beam array with different ring radius R. Simulation 2nd-CBG beam: (a) R = 1.1 cm (b) R = 1.3 cm (c) R = 1.55 cm; Experimental 2nd-CBG beam: (d) R = 1.1 cm (e) R = 1.3 cm (f) R = 1.55 cm.
feasibility of generating CBG beam by Gaussian beam array with finite number is demonstrated. Moreover, the CBG beam with suitable radius could be obtained by adjusting the ring radius of Gaussian beam array. This work could provide theoretical basis for CBC technology to generate high power vortex beams and expand the potential application of high power vortex beams. CRediT authorship contribution statement Tao Yu: Writing - original draft, Validation, Data curation. Hui Xia: Conceptualization, Writing - review & editing. Wenke Xie: Resources, Writing - review & editing. : . Guangzong Xiao: Funding acquisition, Software. Hongjian Li: Methodology, Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig. 8. The relationship between ring radius R of Gaussian beam array and the main lobe radius r.
Acknowledgements This work are supported by the Equipment Pre-research Foundation (No.61404150203), Hunan Provincial Key Laboratory of High Energy Laser Technology (No.GNJGJS04) and Hunan Engineering Research Center of Optoelectronic Inertial Technology (HN-NUDT1908).
of Gaussian beam array. In this paper, the relationship between the ring radius R and the main lobe radius r of CBG beam is studied in detail, and the results are shown in Fig. 8. The results show that the beam radius r of nth-order CBG beam is approximately inversely proportional to the ring radius R. Therefore, CBG beams with controllable radius can be obtained by adjusting the ring radius R, which may have great application potential in laser processing, optical tweezers and optical communication.
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Conclusion A theoretical model of CBG beam generated by the Gaussian beam array loaded with discrete vortex phase is constructed, and the correctness of the model is verified by experiments. By formula derivation, the conditions of generating theoretical BG beam based on CBC technology are clarified. Taking PPIB as the quality evaluation function, the 6
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The generation and verification of Bessel-Gaussian beam based on coherent beam combining
T
⁎
Tao Yua, Hui Xiaa, Wenke Xiea, , Guangzong Xiaob, Hongjian Lia a b
School of Physics and Electronics, Central South University, Changsha 410083, China College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Bessel Gaussian beam Coherent beam combing Power in bucket
We propose a method to generate Bessel-Gaussian (BG) beam by Gaussian beam array where beamlets are arranged along a circle. The phase difference between adjacent beamlets is a constant, and the total phase increment is equal to 2πl (l is integer). Based on coherent superposition of the Gaussian beam array, we proved that the vortex beam obtained by the coherent beam combining (CBC) scheme is Bessel Gaussian beam when the number of Gaussian beams is sufficiently large. The experimental results show that a combined BG (CBG) beam, despite existing side lobe, can still be generated by limited number of Gaussian beams. Therefore, the feasibility of generating vortex (CBG) beam by the CBC scheme has been proved theoretically and experimentally. Moreover, a CBG beam with expected beam radius is generated by controlling the ring radius of Gaussian beam array. Our theoretical predictions match with the experimental results. This work could provide a theoretical basis for practical implementation of generating vortex beams by CBC technology.
Introduction With spiral wavefront structure and annular intensity distribution, vortex beam carries Orbital Angular Momentum (OAM) [1], which has great application prospects in free space optical communication [2,3], optical imaging [4,5], optical manipulation [6] and laser material interaction [7,8]. In recent years, the generation and propagation of vortex beams have been investigated intensively [9–17]. In particular, high-power vortex beams are strongly required in many applications [18–20]. For example, free-space optical communication systems need enough signal power to compensate for the power loss caused by propagation [21]. However, the traditional generation method of vortex beam is extremely dependent on the spatial light modulator, spiral phase plate, diffraction phase holograms and other key devices, and the output power is limited. To overcome this challenge, many innovative methods for generating high power vortex beams have been presented. In 2014, Kim et al. used master-oscillator power-amplifier technology to generate Laguerre-Gaussian beam with high beam quality in the resonator cavity, and coupled the LG beam into a multimode ytterbium doped fiber for transmission and amplification, achieving 10.7w vortex beam output [22]. In 2015, Li demonstrated the amplification of a 1storder vortex beam through Ho:YAG rod amplifier. The simulation shows potentially high gain with more pump power due to the relatively longer length of active material [23,24]. Due to the annular ⁎
structure of vortex beam, the pump and seed beam overlap maybe different along the rod, which means the low amplification efficiency in both MOPA technology and pumping technology. Moreover, due to the thermal lens effect of the amplification system and the nonlinear effect at high power, the power of a single beam cannot be increased infinitely. Theoretically, the maximum output power of fiber laser pumped by semiconductor laser is 36KW without considering the mode instability effect [25,26]. Recently, in order to enhance output power, the CBC technology has gradually been applied to generating high power vortex beam [27–32]. Wang et al. proposed the generation of vortex beam by using the coherent-superposition of multi-beams in a radially symmetric configuration and the spatial evolution rules of the synthesized vortex beams was analyzed [27]. This work has inspired researchers to apply CBC technology to construct structured optical fields such as vortex beam. Chu et al. designed a phase control scheme for generating BesselGaussian beams based on the Stochastic Parallel Gradient Descent (SPGD) algorithm [31]. To eliminate phase noise and phase control errors, Hou et al. proposed a method to extract evaluation function from defocus plane, which optimized the SPGD phase control scheme [32]. Although many works have been carried out based on CBC technology, the theoretical analysis and verification of the vortex synthesis model have not been reported as far as we know. In this paper, based on
Corresponding author. E-mail address: [email protected] (W. Xie).
https://doi.org/10.1016/j.rinp.2019.102872 Received 18 November 2019; Received in revised form 6 December 2019; Accepted 6 December 2019 Available online 10 December 2019 2211-3797/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
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T. Yu, et al.
the theory of diffraction optics, the approximate expression and conditions for generating BG beam of arbitrary order by the modulated Gaussian beam array coherent combing are deduced. Numerical results show that when the number of Gaussian beams is sufficient large, the synthesized CBG beam are consistent with the theoretical BG beams. The correctness of theoretical model is verified by the experimental results. The principle of CBG beam generation could provide theoretical foundation to generate other structured light fields by CBC technology.
Em (r , φ , z = 0; α ) = exp ⎛⎜− ⎝
(3) The coherent superposition of Gaussian beams can be expressed as M−1
E (r , φ , z = 0; α ) = ∑ Em (r , φ , z = 0; α ) m=0
= exp
Principle and method
(
r 2 + R2 − w2 0
M−1
) ∑ exp ⎡⎣ m=0
2Rr cos(φ − αm) ⎤ w02
⎦
exp(ilαm)
(4)
Considering the angle (Δα ) between adjacent beamlet
Bessel beam is the exact solution of the Helmholtz equation, which has infinite energy and cannot be generated in practice. By Gaussian amplitude restriction, the BG beam with finite energy is obtained, which has some properties as that of Bessel beam. In cylindrical coordinate system, the expression of the BG beam at the initial plane (z = 0) is given as [33]
2πm 2π ⎞= Δα = Δ ⎛ Δm M ⎝ M ⎠
(5)
As beam numbers M increases to sufficient large, the angelΔα approaches to infinitesimal dα,
2
r EBG (r , φ , z = 0) = exp ⎜⎛− 2 ⎞⎟ Jl (βr ) exp(ilφ) ⎝ w0 ⎠
r 2 + R2 ⎞ ⎡ 2Rr cos(φ − αm) ⎤ exp(ilα ) ⎟ exp m ⎢ ⎥ w02 ⎠ w02 ⎣ ⎦
dα = (1)
2π dm M
(6)
Therefore, the accumulation of Gaussian beams in Eq. (4) can be converted into the integration of angel α
where (r, φ) is a pair of polar coordinate, w0 is the waist width of Gaussian amplitude, Jl represents the lth-order Bessel function of the first kind, β is a scale factor. Taking BG beam as target beam, we consider coherent superposition of Gaussian beam array loaded with discrete vortex phase, whose centers are placed on a ring of radius R around the z-axis, and each subbeam has the same amplitude, waist width and linearly polarization state, as shown in Fig. 1. In rectangular Cartesian coordinate system, the complex amplitude of Gaussian beam at source plane is given by [27–29,34]
E (r , φ , z = 0; α ) ≈
M 2π
2π
∫ Em (r , φ, z = 0; α ) dα
(7)
0
Substitute Eq. (3) to Eq. (7), the optical field generated by the Gaussian beam array is reduced to
E (r , φ , z = 0; α ) =
(x − R cos αm)2 + (y − R sin αm)2 ⎤ Em (x , y, z = 0; α ) = exp ⎡− exp (ilαm) ⎢ ⎥ w02 ⎣ ⎦ (2)
M r 2 + R2 ⎞ exp ⎛⎜− ⎟ 2π w02 ⎠ ⎝
∫0
2π
exp ⎡ ⎢ ⎣
2Rr cos(φ − α ) ⎤ exp(ilα ) dα ⎥ w02 ⎦ (8)
Using standard integral and Bessel function identity [35],
here R, w0, lαm are the ring radius, waist width and initial phase, respectively. αm = 2πm/M is the angle between the center of the m-th Gaussian beam and the x-axis, the phase difference of adjacent subbeams is 2πl/M. Without loss of generality, an unimportant amplitude constant factor has been omitted. Using polar coordinate transformation, the Eq. (2) is turned to
Il (u) =
1 2π
∫0
2π
exp(u cos φ − ilφ) dφ
(9)
The optical field can be finally derived as
r 2 + R2 ⎞ ⎛ 2Rr ⎞ E (r , φ , z = 0) = M exp ⎜⎛− ⎟ Il ⎜ ⎟ exp(ilφ ) w02 ⎠ ⎝ w02 ⎠ ⎝
(10)
It can be noted from Eq. (10) that it is a combination of Modified Bessel function and Gaussian function, indicating that the optical field generated by the Gaussian beam array is a BG beam. Considering the complex amplitude of Gaussian beam at any plane,
(
Em (r , φ , z; α ) = exp − × exp ⎡ ⎣
r 2 + R2 w 2 (z )
ikRr cos(φ − αm) ⎤ R (z ) ⎦
) exp ⎡⎣
2Rr cos(φ − αm)
exp ⎡i arctan ⎣
w 2 (z )
⎤ exp ⎡− ⎦ ⎣
ik (r 2 + R2) ⎤ 2R (z ) ⎦
( ) − ikz⎤⎦ exp(ilα ) z f
m
(11) Similarly, by superimposing these Gaussian beams, the total complex amplitude can be written as M−1
E (r , φ , z; α ) =
∑
Em (r , φ , z; α )
(12)
m=0
Employing the same procedure as before, the optical field at any plane can be obtained by Eq. (9), (11), (12)
(
) I (βr ) exp (ilφ) ⎤ exp ⎡i arctan ( ) − ikz⎤ ⎦ ⎣ ⎦
E (r , φ , z; α ) = M exp − × exp ⎡− ⎣ Fig. 1. The geometric construction of Gaussian beam array.
where 2
ik (r 2 + R2) 2R (z )
r 2 + R2 w 2 (z )
l
z f
(13)
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Fig. 2. Simulation block diagram for generation the 2-nd CBG beam.
Fig. 3. Experimental setup: L, laser; BE, beam expander; BS1, BS2, beam splitters; A-SLM, P-SLM, amplitude spatial light modulator, phase spatial light modulator; M1, mirror; FL, lens; CCD, CCD camera.
To observe the far field of the CBG beam at relatively close distance during the experiment, a simple lens, as an optical Fourier transformer, is used to obtain the far field of target beam. This transformation for any arbitrary field E(r, φ) into E(ρ, θ) can be written in cylindrical coordinates as [36]
2
f2 2R ikR z , f = πw0 2/ λ, β = 2 w (z ) = w0 1 + ⎜⎛ ⎟⎞ , R (z ) = z + + z w ( z ) R (z ) f ⎝ ⎠ (14) Eq. (13) is still a combination of Gaussian and Bessel function, so the theoretical model of generating CBG beam at any plane is constructed. From Eq. (13), the CBG beams are determined by the beam parameters (waist width and wavelength), beam number and ring radius of Gaussian beam array. As shown in Fig. 1, given the parameters w0 and R at the initial plane, the number of Gaussian beams M ≤ πR/w0 if the space of adjacent sub-beams does not overlap. Therefore, the number of Gaussian beams can only be limited to a suitable number. The process of generating CBG beams is shown in Fig. 2, Gaussian beam array with appropriate number are arranged in a circle, and loading the discrete vortex phases. Then, the modulated Gaussian beams are superimposed coherently in free space. Finally, the CBG beam is obtained, which is similar to the BG beam structure.
E (ρ , θ) =
k i2πf
∞ 2π
∫ ∫ E (r , φ) exp ⎡⎢ ⎛− ikρf ⎞ rcos (θ − φ) ⎤⎥ rdrdφ 0
0
⎜
⎟
⎣⎝
⎠
⎦
(15)
Substitute (10) into (15), the expression of the CBG beam at focus plane can be written as
E (ρ , θ) =
kMil if
∞
( ) R2
( ) ( ) J (− ) rdr r2
exp − w 2 exp(ilθ) ∫ exp − w 2 Il =
0
0
kMw02
ρ exp ⎛− w2 ⎞ Jl ⎝ f⎠
2il + 1f
2
⎜
⎟
0
2Rr w02
l
kρr f
( ) exp(ilθ) Rkρ f
(16) The Gaussian beam waist at the focus plane is wf (=2f/kw0). Eq. (16) is also a combination of Gauss function and Bessel function, so the 3
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of the lens, the intensity distribution of near and far fields of 2nd-CBG beam composed of M = 16 are shown in Fig. 4(a) and (b). One can see that the near field of 2nd-order CBG beam exist obvious side lobes, but with increase of transmission distance, the stable annular optical field is formed by superposing these modulated sub-beams, and the side lobes decrease significantly. The near and far field intensity distribution of experimental 2nd-order CBG beam are obtained by using the experimental setup in Fig. 3, the optical field profile are shown in Fig. 4(d) and (e). The experimental and simulation results have the same optical field distribution. To illustrate the obtained CBG beams carry vortex phase, the interference experiment is carried out. It can be found from Fig. 4(f) that the interference fringes of CBG beam and plane wave are no longer vertical stripes of light and dark. The center stripe is divided into three, indicating that the detected beam have 4π phase variation. The detected beam is proved to carry 2nd order vortex phase, which is completely consistent with the topological charge of the target beam. Therefore, the generated CBG beam is a 2nd-order vortex beam. From the Eq. (16), the same parameters are set to obtain the theory BG beam, whose optical field profile at the focus plane is shown in Fig. 4(c). By comparing the CBG beam with the theory BG beam at focus plane, the optical field profile of CBG beam also has the same hollow ring structure as BG beam. Thus, the generated CBG beam has not only vortex phase but also intensity distribution modulated by Bessel function. So far, the theoretical model of CBG beam generation based on CBC scheme has been verified.
intensity profile of the CBG beam at focus plane still remain the BG beam structure. Experiment setup The used experimental setup to generate the CBG beam is shown in Fig. 3. The seed laser of 632.8 nm is expanded by a beam expander and then split into two channels by a 50:50 splitter. After the splitter, one of the beams is incident on an amplitude-type SLM (Holoeye LC2012) and the beam is divided into M Gaussian beams with the same parameters. Then the n-order discrete vortex phase is loaded into the Gaussian beam array by the phase-type SLM (Holoeye Pluto). To investigate the propagation of the CBG beam, a lens (f = 300 mm) is added to the experiment. By adjusting the distance between the lens and the CCD, the far field and near field profile of the CBG beam are obtained at the focus plane and off-focus plane, respectively. The other reference beam reflected by the mirror is used to interfere with the CBG beam. The topological charge of the CBG beam can be confirmed by observing the interference fringes in CCD. Power in the bucket (PIB) describes the energy encircled in an onaxis circular area at the receiver plane [37]. In CBC systems, the proportion of the power contained in the bucket and the total power can be used to describe the energy concentration at receiver plane, i.e. a 2π
PPIB
PIBa = = PIBb
∫ ∫ I (r , φ, z ) rdrdφ 0 0 b 2π
∫ ∫ I (r , φ, z ) rdrdφ 0 0
The sub beams of Gaussian beam array
(17)
where a is the half width of the bucket and b is the half width of the receiver plane. Due to the side lobe of CBG beam, the PPIB is taken as the performance index to evaluate the approximation degree of CBG beam and BG beam.
Although the theoretical model has been demonstrated that the generated CBG beam is BG beam, but the CBG beam exist side lobes, which is not completely matched with the theoretical BG beam. From the derivation of Eq. (11) and Eq. (13), the number of beamlets directly determine the approximation degree between CBG beam and theoretical BG beam. Therefore, the number of beams in the coherent combining system is studied in detail. In the case of given array ring radius R = 1.3 cm and w0 = 0.2 cm, the 2nd-order CBG beams composed of 8, 12, 16 and 20 sub-beams are studied. Simulation and experimental results are shown in Fig. 5(a)–(h) respectively. The experimental results are basically consistent with the simulation results. According to the variation of optical field in Fig. (5), the energy change of CBG beams
Results and discussion Verification of the CBG beam generation model To verify the correctness of the theoretical model mentioned above, the CBG beam and theoretical BG beam are investigated according to Eq. (11) and Eq. (13), respectively. According to the Fourier transform
Fig. 4. Generation of 2-nd CBG beam and theoretical BG beam. Simulation 2nd-CBG beam intensity profile: (a) near field (d = 270 mm) (b) far field (d = 300 mm) (c) Theoretical 2nd-BG beam. Experimental 2nd-CBG beam intensity profile: (d) near field (d = 270 mm) (e) far field (d = 300 mm) (f) Interference pattern of CBG beam. 4
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T. Yu, et al.
Fig. 5. 2nd-CBG beam with different composed sub-beams. Simulation 2nd-CBG beam: (a) M = 8 (b) M = 12 (c) M = 16 (d) M = 20; Experimental 2nd-CBG beam: (e) M = 8 (f) M = 12 (g) M = 16 (h) M = 20.
relatively large when composed beam numbers are small, the reason is that the CBG optical field has not yet formed a complete annular structure, and its energy is still mainly concentrated on the Gaussian array. By increasing the composed beam numbers, the energy of side lobe gradually transfer into the rings, so the PPIB increase simultaneously. When the number of beams increases sufficiently, the PPIB of the CBG beam is equal to the PPIB of the theoretical BG beam. This result further verify the correctness of the theoretical model proposed in this paper.
are mainly manifested as the change of side lobe and diffraction ring. Under the same ring radius, the 2nd-order CBG beam generated by the eight sub-beams does not form a ring structure, and the energy is still concentrated on the Gaussian beam array. With the increase of beam numbers, the energy of Gaussian beam array gradually transfers into the central main ring and diffraction ring of the CBG beam and the energy of the external side lobe decreases significantly. Moreover, during the process of energy transfer, the ring radius of the CBG beam can remain unchanged. Therefore, increasing the number of beams can significantly improve the similarity between CBG beam and standard BG beam. The PPIB of 2nd-order CBG beams composed of different beam numbers are calculated, which is used to quantitatively analyze the influence of number beams on the energy concentration of CBG beam, so the variation trend of intensity distribution of CBG beam can be revealed by the change of PPIB. As shown in Fig. 6(a), the barrel radius of PPIB is set as a (including the central main lobe and the first diffraction ring), the radius of whole optical field is b, where r is the radius of the central main lobe ring. Taking a as the barrel radius, the PPIB of 2nd-order CBG beams composed of 8 to 30 beams are calculated, and the results are shown in Fig. 6(b). We can found that the PPIB are
The ring radius of Gaussian beam array From Eq. (11) and Eq. (13), the complex amplitude of CBG beam is not only determined by the beam parameters and beam numbers, but also related to the ring radius R of Gaussian beam array. Since the beam numbers does not change the beam radius of CBG beam, we set the beam numbers close to πR/w0 in the following study. The 2nd-order CBG beam composed of Gaussian beam array with different ring radius is investigated, the simulation and experimental results are shown in Fig. 7(a)–(f) respectively. From Fig. 7, one can found that a CBG beams with a smaller ring radius r can be generated by increasing the radius R
Fig. 6. (a) Intensity profile of 2nd-BG beam (b) The PPIB of 2nd-order CBG beam. 5
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T. Yu, et al.
Fig. 7. 2nd-CBG beam generated by Gaussian beam array with different ring radius R. Simulation 2nd-CBG beam: (a) R = 1.1 cm (b) R = 1.3 cm (c) R = 1.55 cm; Experimental 2nd-CBG beam: (d) R = 1.1 cm (e) R = 1.3 cm (f) R = 1.55 cm.
feasibility of generating CBG beam by Gaussian beam array with finite number is demonstrated. Moreover, the CBG beam with suitable radius could be obtained by adjusting the ring radius of Gaussian beam array. This work could provide theoretical basis for CBC technology to generate high power vortex beams and expand the potential application of high power vortex beams. CRediT authorship contribution statement Tao Yu: Writing - original draft, Validation, Data curation. Hui Xia: Conceptualization, Writing - review & editing. Wenke Xie: Resources, Writing - review & editing. : . Guangzong Xiao: Funding acquisition, Software. Hongjian Li: Methodology, Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig. 8. The relationship between ring radius R of Gaussian beam array and the main lobe radius r.
Acknowledgements This work are supported by the Equipment Pre-research Foundation (No.61404150203), Hunan Provincial Key Laboratory of High Energy Laser Technology (No.GNJGJS04) and Hunan Engineering Research Center of Optoelectronic Inertial Technology (HN-NUDT1908).
of Gaussian beam array. In this paper, the relationship between the ring radius R and the main lobe radius r of CBG beam is studied in detail, and the results are shown in Fig. 8. The results show that the beam radius r of nth-order CBG beam is approximately inversely proportional to the ring radius R. Therefore, CBG beams with controllable radius can be obtained by adjusting the ring radius R, which may have great application potential in laser processing, optical tweezers and optical communication.
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Conclusion A theoretical model of CBG beam generated by the Gaussian beam array loaded with discrete vortex phase is constructed, and the correctness of the model is verified by experiments. By formula derivation, the conditions of generating theoretical BG beam based on CBC technology are clarified. Taking PPIB as the quality evaluation function, the 6
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