December 15, 2013 / Vol. 38, No. 24 / OPTICS LETTERS
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Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam Xianlong Liu,1 Yan Shen,1 Lin Liu,1 Fei Wang,1,2 and Yangjian Cai1,* 1
School of Physical Science and Technology, Soochow University, Suzhou 215006, China 2
e-mail: [email protected] *Corresponding author: [email protected] Received October 22, 2013; accepted November 6, 2013; posted November 8, 2013 (Doc. ID 199924); published December 6, 2013 We carry out experimental measurement of the scintillation index of a partially coherent beam-carrying vortex phase (i.e., Gaussian–Schell model vortex beam) propagating through thermally induced turbulence. It is demonstrated that a Gaussian–Schell model vortex beam has appreciably smaller scintillation than a Gaussian–Schell model beam, which will be useful in free-space optical communication. © 2013 Optical Society of America OCIS codes: (030.1640) Coherence; (030.1670) Coherent optical effects; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/OL.38.005323
It is well known that one of the fundamental limitations for free-space optical communication is the scintillation in a laser beam induced by atmospheric turbulence [1,2]. Scintillation is caused from distortion of the phase structure of the field by the turbulence. Thus, it is extremely important to develop a method to reduce turbulenceinduced scintillation. It has been revealed that one can reduce turbulence-induced scintillation by use of a partially coherent beam [3,4] or a coherent beam with special beam properties (i.e., special beam profile, phase, and polarization) [5–10]. By modulating the beam profile, polarization, correlation function, and phase of a partially coherent beam, one can expect to further reduce the turbulence-induced scintillation [10–16]. In [16], it is predicted in theory that a twisted Gaussian–Schell model (GSM) beam (i.e., partially coherent beam carrying twist phase) has smaller scintillation than a GSM beam in a turbulent atmosphere. One may expect that a Gaussian– Schell model vortex (GSMV) beam (i.e., partially coherent beam carrying vortex phase) also has an advantage over a GSM beam for reducing turbulence-induced scintillation. A GSMV beam was generated in experiments just recently, and it was found that a GSMV beam is useful for trapping particles [17]. In this Letter, we carry out experimental study of the scintillation of a GSMV beam propagating through thermally induced turbulence. Some useful results are found. Figure 1 shows our experimental setup for generating a GSMV beam, measuring its spatial coherence width, and measuring the scintillation index of the generated beam propagating through thermally induced turbulence. Part I of Fig. 1 shows the experimental setup for generating a GSMV beam with controllable spatial coherence. A laser beam generated by a He–Ne laser (λ ≈ 632.8 nm) is reflected by a reflecting mirror, then it passes through the thin lens L1 and illuminates the rotating ground-glass disk (RGGD), producing a partially coherent beam with Gaussian statistics. After passing through the thin lens L2 and the Gaussian amplitude filter (GAF), the generated partially coherent beam becomes a GSM beam. L2 is used to collimate the generated partially coherent beam, and the GAF is used to transform its intensity into a Gaussian 0146-9592/13/245323-04$15.00/0
profile. After passing through a spiral phase plate (SPP) with a transmission function Tφ expilφ, which is located just behind the GAF, the generated GSM beam becomes a GSMV beam. The cross-spectral density (CSD) of the generated GSM beam is expressed as [18] 2 r r2 r − r 2 ΓGSM r1 ; r2 exp − 1 2 2 − 1 2 2 ; 4σ 0 2σ g
(1)
where r1 ≡ x1 ; y1 and r2 ≡ x2 ; y2 are two arbitrary transverse position vectors, σ 0 and σ g denote the transverse beam width and spatial coherence width, respectively. In our experiment, σ 0 is determined by the transmission function of the GAF and equals to 1 mm. σ g is determined by the beam spot size on the RGGD and the roughness of the RGGD together. In our experiment, the roughness of the RGGD is fixed, and we mainly modulate σ g through varying the beam spot size on the RGGD, which is controlled by varying the distance
Fig. 1. Experimental setup for generating a GSMV beam, measuring its spatial coherence width, and measuring the scintillation index of the generated beam propagating through thermally induced turbulence. RM, reflecting mirror; L1 , L2 , L3 , L4 , thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SPP, spiral phase plate; BS1 , BS2 , beam splitters; D1 , D2 , single photon detectors; ECC, electronic coincidence circuit; CCD, charge-coupled device; PC, personal computer. © 2013 Optical Society of America
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between L1 and the RGGD. The SPP only adds a vortex phase to the GSM beam, and it doesn’t alter the beam width and the spatial coherence width; thus we can express the CSD of the generated GSMV beam as follows [17]: 2 r r2 r − r 2 ΓGSMV r1 ; r2 exp − 1 2 2 − 1 2 2 4σ 0 2σ g × exp ilφ1 − φ2 ;
(2)
where φi arctanyi ∕xi , l is the topological charge. The experimental setup for measuring the spatial coherence width σ g of the generated GSM or GSMV beam is shown in part II of Fig. 1. The generated GSM beam is split into two beams by the beam splitter (BS1 ). The reflected beam is further split into two distinct imaging optical paths by the beam splitter (BS2 ), and the two beams from BS2 arrive at the single photon detectors D1 and D2 , which scan the transverse planes of u1 and u2 , respectively. The thin lens L3 with focal length f 3 is located between BS1 and BS2 , and the distances from the GAF to L3 and from L3 to D1 and D2 are 2f 3 (i.e., 2f imaging system). Thus the spatial coherence width of the beam in the plane of u1 or u2 is the same with that of the beam in the source plane (just behind the GAF). The output signals from D1 and D2 are sent to an electronic coincidence circuit (ECC) to measure the normalized intensity fluctuation of the beam defined as g2 u1 ; u2 ; τ
hΔIu1 ; tΔIu2 ; t τi ; hIu1 ; tihIu2 ; t τi
(3)
where ΔIu1 ; t Iu1 ; t − hIu1 ; ti;
(4)
ΔIu2 ; t τ Iu2 ; t τ − hIu2 ; t τi:
(5)
Here the angular brackets represent the time average, Iu1 ; t and Iu2 ; t τ denote the instantaneous intensities, and τ denotes the delay time of the photon flux of two optical paths, which is controlled by the ECC. Applying the Gaussian moment theorem [19], g2 u1 ; u2 ; τ with τ 0 of the generated GSM beam can be simplified as g2 u1 ; u2 ; 0 exp−u1 − u2 2 ∕σ 2g :
(6)
If we fix D2 at u2 0, and D1 scans along u1 , we can obtain the distribution of the normalized intensity fluctuation g2 u1 ; 0; τ 0 with τ 0 from the ECC. From the curve of the theoretical fit for the experimental data, we can obtain the value of σ g [20]. Furthermore, if we fix D1 and D2 at u1 u2 0 and vary the delay time τ, we can obtain the distribution of g2 0; 0; τ versus τ. As shown in [21], the distribution of g2 0; 0; τ also has a Gaussian profile, i.e., g2 0; 0; τ exp−τ2 ∕τ2s ;
where τs represents the characteristic time of the intensity fluctuation of the beam induced by the RGGD. From the curve of the theoretical fit for the experimental data, we can obtain the value of τs . The experimental setup for measuring the scintillation index of the generated GSM or GSMV beam propagating through the thermal-induced turbulence is shown in part III of Fig. 1. The transmitted beam from the BS1 first passes through a 35 cm × 50 cm electric hot plate, which is used to produce thermal turbulence through convection, then passes through a collection lens L4 with focal length f 4 15 cm, and finally arrives at the chargecoupled device (CCD), which is located in the geometrical focal plane (receiver plane) and is used to record the signal by a data-acquisition system. The beam center of the transmitted beam is about 2 cm above the electric hot-plate surface. The distance from the left side of the hot plate to L4 is about 1.5 m. In our experiment, the strength of the turbulence is modulated by the temperature T of the hot plate. Figure 2 shows our experimental results of the intensity distributions of the generated GSM beam and GSMV beam (l 1; 2) with σ g 1.2 mm at z 10 cm and in the receiver plane. One finds that the intensity distribution of a GSM beam is always of Gaussian profile on propagation. The intensity distribution of a GSMV beam (l 1; 2) has a Gaussian beam profile in the source plane, while it has a dark hollow beam profile at a certain propagation distance due to its intrinsic vortex phase, which induces a phase singularity in the beam center. In the receiver plane, the intensity distribution of the GSMV beam has a Gaussian profile again. Note the intensity distribution of the GSMV beam in the receiver plane is closely determined by its initial coherence width σ g , and it can exhibit Gaussian or flat-topped or a dark hollow beam profile with the increase of σ g (σ g > 1.2 mm) [16]. In our experiment, the intensity distribution of the GSMV beam in the receiver plane is always of a Gaussian beam profile through choosing a suitable value of σ g (σ g < 1.2 mm). In order to measure the scintillation index of the generated GSM or GSMV beam, the characteristic time τs of the intensity fluctuation of the beam induced by the RGGD, the characteristic time τa of the intensity
(7)
Fig. 2. Experimental results of the intensity distributions of the generated GSM beam and GSMV beam (l 1; 2) with σ g 1.2 mm at z 10 cm (a)–(c) and in the receiver plane (d)–(f).
December 15, 2013 / Vol. 38, No. 24 / OPTICS LETTERS
fluctuation of the beam induced by the thermal turbulence, and the integration time τd of the detector should satisfy the relation τs ≪ τd ≪ τa (i.e., slow detection). Under the condition of slow detection, the detector is insensitive to the intensity fluctuation of the source beam induced by the RGGD. The intensity fluctuation g2 0; 0; τ of a Gaussian beam after passing through thermally induced turbulence can be approximately modeled as [22] g2 0; 0; τ A exp−τ∕τa :
(8)
Here A is a constant. τa is mainly controlled by the temperature of the hot plate. Figure 3 shows our experimental results of the characteristic time τs and τa versus the initial coherence length σ g and the temperature of the hot plate, respectively. From Fig. 3(a), one finds that the value of τs is nearly independent of σ g within the range 0.4 mm < σ g < 1.2 mm, while the value of τs increases rapidly with the decrease of σ g when σ g < 0.4 mm. The value of τs varies from 6.7 to 14 μs. From Fig. 3(b), one sees that the value of τa exhibits monotonic growth with the decrease of the temperature of the hot plate, and its value varies from 65 to 50 ms when the temperature varies from 80°C to 160°C. In our experiment, the integration time of the detector is set as τd 20 ms, the condition of “slow detection” is satisfied. The scintillation index of a beam is defined as [2] m2c hI 2 x; yi∕hIx; yi2 − 1;
(9)
where Ix; y, hIx; yi, and hI 2 x; yi represent the instantaneous intensity, the average intensity, and the intensity correlation function of the beam in the receiver plane. In our experiment, the CCD in the receiver plane captures 4000 pictures totally, and each frame represents one realization of the beam cross section, and each realization is represented as a matrix I n x; y, where x and y are pixel spatial coordinates, and n denotes each realization and ranges from 1 to 4000. The coordinates (x¯ n ; y¯ n ) of the centroid of each realization are given as x¯ n
XX i
y¯ n
j
XX i
xi I n xi ; yj ∕
j
XX i
yi I n xi ; yj ∕
(10)
I n xi ; yj :
(11)
The intensity at the centroid of each realization is given as I n x¯ n ; y¯ n . The scintillation index of a beam at the centroid in the receiver plane can be obtained as X I 2n x¯ n ; y¯ n ∕N I¯ 2 − 1; (12) m2c N
where N denotes the number of total frames, and I¯ represents the average intensity at the centroid over all frames, i.e., X I n x¯ n ; y¯ n ∕N: (13) I¯ N
All experimental data are processed by MATLAB. Figure 4 shows our experimental results of the scintillation index of a GSMV beam (l 1; 2) at the centroid versus the initial spatial coherence width. For the convenience of comparison, the corresponding results of a GSM beam are also shown in Fig. 4. One finds from Fig. 4 that the scintillation index of a GSM or GSMV beam decreases with the decrease of σ g , which means that a partially coherent beam with a low value of σ g is less affected by the turbulence as expected [2–4]. When σ g > 0.35 mm, the scintillation index of a GSMV beam is smaller than that of a GSM beam, and the scintillation index decreases with the increase of the topological charge, which implies that a GSMV beam has an advantage over a GSM beam for reducing turbulence-induced scintillation. When σ g < 0.35 mm, the scintillation index of a GSMV beam is almost the same as that of a GSM beam and almost doesn’t vary as σ g varies. We may explain this phenomenon by the fact that the scintillation index of the GSMV beam is influenced by the coherence and the vortex phase together. When σ g is small, the GSMV beam has already evolved into a GSM beam in the entrance side of turbulence, and the influence of the coherence on the scintillation index plays a dominant role, while the influence of the vortex phase is negligible. Thus the scintillation index of the GSMV beam with extremely low coherence is similar to that of a GSM beam. Figure 5 shows our experimental results of the scintillation index of a GSM or GSMV beam at the centroid
j
XX i
I n xi ; yj ;
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j
Fig. 3. Experimental results of (a) the characteristic time τs versus the initial coherence width σ g and (b) the characteristic time τa versus the temperature of the hot plate.
Fig. 4. Experimental results of the scintillation index of a GSM or GSMV beam (l 1; 2) at the centroid versus the initial coherence width.
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Jiangsu Province under Grant No. 11KJB140007, the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
Fig. 5. Experimental results of the scintillation index of a GSM or GSMV beam (l 1; 2) at the centroid versus the temperature of the hot plate for two different values of σ g .
versus the temperature of the hot plate for two different values of σ g . One finds that the advantage of a GSMV beam over a GSM beam for reducing the turbulenceinduced scintillation also is affected by the strength of the turbulence. The scintillation index exhibits monotonic growth with the increase of the temperature of the hot plate. The advantage of a GSMV beam over a GSM beam is enhanced with the increase of the temperature of the hot plate (i.e., the strength of turbulence increases). When the temperature is small, the scintillation index of a GSMV beam with l 1 is almost the same with that of a GSMV beam with l 2. Therefore, it is necessary for us to take the coherence, topological charge, and the strength of turbulence into consideration in practical applications. In conclusion, we have carried out an experimental study of the scintillation index of a GSMV beam propagating through thermally induced turbulence and compared with that of a GSM beam. Our experimental results have clearly shown that a GSMV beam has an advantage over a GSM beam for reducing turbulence-induced scintillation under suitable conditions. Our results will be useful in free-space optical communication. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11274005 and 11104195, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Universities Natural Science Research Project of
References 1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2. 2. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001). 3. S. J. Wang, Y. Baykal, and M. A. Plonus, J. Opt. Soc. Am. 73, 831 (1983). 4. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, Opt. Spektrosk. 54, 1054 (1983). 5. Y. Baykal and H. T. Eyyuboğlu, Appl. Opt. 45, 3793 (2006). 6. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, Opt. Lett. 32, 2405 (2007). 7. Y. Gu, O. Korotkova, and G. Gbur, Opt. Lett. 34, 2261 (2009). 8. Y. Gu and G. Gbur, Opt. Lett. 35, 3456 (2010). 9. Y. Gu and G. Gbur, Opt. Lett. 37, 1553 (2012). 10. W. Chen, J. W. Haus, and Q. Zhan, Opt. Express 17, 17829 (2009). 11. Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, Appl. Opt. 48, 1943 (2009). 12. O. Korotkova, Opt. Commun. 281, 2342 (2008). 13. Y. Gu and G. Gbur, Opt. Lett. 38, 1395 (2013). 14. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, Opt. Commun. 305, 57 (2013). 15. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, Appl. Phys. Lett. 103, 091102 (2013). 16. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, Opt. Lett. 37, 184 (2012). 17. F. Wang, S. Zhu, and Y. Cai, Opt. Lett. 36, 3281 (2011). 18. P. De Santis, F. Gori, G. Guattari, and C. Palma, Opt. Commun. 29, 256 (1979). 19. L. Mandel and E. Wolf, eds., Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 36–38, 428. 20. F. Wang and Y. Cai, J. Opt. Soc. Am. A 24, 1937 (2007). 21. A. Kumar, J. Banerji, and R. P. Singh, Opt. Lett. 35, 3841 (2010). 22. C. Aime, J. Borgnino, F. Martin, R. Petrov, and G. Ricort, J. Opt. Soc. Am. A 3, 1001 (1986).
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Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam Xianlong Liu,1 Yan Shen,1 Lin Liu,1 Fei Wang,1,2 and Yangjian Cai1,* 1
School of Physical Science and Technology, Soochow University, Suzhou 215006, China 2
e-mail: [email protected] *Corresponding author: [email protected] Received October 22, 2013; accepted November 6, 2013; posted November 8, 2013 (Doc. ID 199924); published December 6, 2013 We carry out experimental measurement of the scintillation index of a partially coherent beam-carrying vortex phase (i.e., Gaussian–Schell model vortex beam) propagating through thermally induced turbulence. It is demonstrated that a Gaussian–Schell model vortex beam has appreciably smaller scintillation than a Gaussian–Schell model beam, which will be useful in free-space optical communication. © 2013 Optical Society of America OCIS codes: (030.1640) Coherence; (030.1670) Coherent optical effects; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/OL.38.005323
It is well known that one of the fundamental limitations for free-space optical communication is the scintillation in a laser beam induced by atmospheric turbulence [1,2]. Scintillation is caused from distortion of the phase structure of the field by the turbulence. Thus, it is extremely important to develop a method to reduce turbulenceinduced scintillation. It has been revealed that one can reduce turbulence-induced scintillation by use of a partially coherent beam [3,4] or a coherent beam with special beam properties (i.e., special beam profile, phase, and polarization) [5–10]. By modulating the beam profile, polarization, correlation function, and phase of a partially coherent beam, one can expect to further reduce the turbulence-induced scintillation [10–16]. In [16], it is predicted in theory that a twisted Gaussian–Schell model (GSM) beam (i.e., partially coherent beam carrying twist phase) has smaller scintillation than a GSM beam in a turbulent atmosphere. One may expect that a Gaussian– Schell model vortex (GSMV) beam (i.e., partially coherent beam carrying vortex phase) also has an advantage over a GSM beam for reducing turbulence-induced scintillation. A GSMV beam was generated in experiments just recently, and it was found that a GSMV beam is useful for trapping particles [17]. In this Letter, we carry out experimental study of the scintillation of a GSMV beam propagating through thermally induced turbulence. Some useful results are found. Figure 1 shows our experimental setup for generating a GSMV beam, measuring its spatial coherence width, and measuring the scintillation index of the generated beam propagating through thermally induced turbulence. Part I of Fig. 1 shows the experimental setup for generating a GSMV beam with controllable spatial coherence. A laser beam generated by a He–Ne laser (λ ≈ 632.8 nm) is reflected by a reflecting mirror, then it passes through the thin lens L1 and illuminates the rotating ground-glass disk (RGGD), producing a partially coherent beam with Gaussian statistics. After passing through the thin lens L2 and the Gaussian amplitude filter (GAF), the generated partially coherent beam becomes a GSM beam. L2 is used to collimate the generated partially coherent beam, and the GAF is used to transform its intensity into a Gaussian 0146-9592/13/245323-04$15.00/0
profile. After passing through a spiral phase plate (SPP) with a transmission function Tφ expilφ, which is located just behind the GAF, the generated GSM beam becomes a GSMV beam. The cross-spectral density (CSD) of the generated GSM beam is expressed as [18] 2 r r2 r − r 2 ΓGSM r1 ; r2 exp − 1 2 2 − 1 2 2 ; 4σ 0 2σ g
(1)
where r1 ≡ x1 ; y1 and r2 ≡ x2 ; y2 are two arbitrary transverse position vectors, σ 0 and σ g denote the transverse beam width and spatial coherence width, respectively. In our experiment, σ 0 is determined by the transmission function of the GAF and equals to 1 mm. σ g is determined by the beam spot size on the RGGD and the roughness of the RGGD together. In our experiment, the roughness of the RGGD is fixed, and we mainly modulate σ g through varying the beam spot size on the RGGD, which is controlled by varying the distance
Fig. 1. Experimental setup for generating a GSMV beam, measuring its spatial coherence width, and measuring the scintillation index of the generated beam propagating through thermally induced turbulence. RM, reflecting mirror; L1 , L2 , L3 , L4 , thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SPP, spiral phase plate; BS1 , BS2 , beam splitters; D1 , D2 , single photon detectors; ECC, electronic coincidence circuit; CCD, charge-coupled device; PC, personal computer. © 2013 Optical Society of America
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between L1 and the RGGD. The SPP only adds a vortex phase to the GSM beam, and it doesn’t alter the beam width and the spatial coherence width; thus we can express the CSD of the generated GSMV beam as follows [17]: 2 r r2 r − r 2 ΓGSMV r1 ; r2 exp − 1 2 2 − 1 2 2 4σ 0 2σ g × exp ilφ1 − φ2 ;
(2)
where φi arctanyi ∕xi , l is the topological charge. The experimental setup for measuring the spatial coherence width σ g of the generated GSM or GSMV beam is shown in part II of Fig. 1. The generated GSM beam is split into two beams by the beam splitter (BS1 ). The reflected beam is further split into two distinct imaging optical paths by the beam splitter (BS2 ), and the two beams from BS2 arrive at the single photon detectors D1 and D2 , which scan the transverse planes of u1 and u2 , respectively. The thin lens L3 with focal length f 3 is located between BS1 and BS2 , and the distances from the GAF to L3 and from L3 to D1 and D2 are 2f 3 (i.e., 2f imaging system). Thus the spatial coherence width of the beam in the plane of u1 or u2 is the same with that of the beam in the source plane (just behind the GAF). The output signals from D1 and D2 are sent to an electronic coincidence circuit (ECC) to measure the normalized intensity fluctuation of the beam defined as g2 u1 ; u2 ; τ
hΔIu1 ; tΔIu2 ; t τi ; hIu1 ; tihIu2 ; t τi
(3)
where ΔIu1 ; t Iu1 ; t − hIu1 ; ti;
(4)
ΔIu2 ; t τ Iu2 ; t τ − hIu2 ; t τi:
(5)
Here the angular brackets represent the time average, Iu1 ; t and Iu2 ; t τ denote the instantaneous intensities, and τ denotes the delay time of the photon flux of two optical paths, which is controlled by the ECC. Applying the Gaussian moment theorem [19], g2 u1 ; u2 ; τ with τ 0 of the generated GSM beam can be simplified as g2 u1 ; u2 ; 0 exp−u1 − u2 2 ∕σ 2g :
(6)
If we fix D2 at u2 0, and D1 scans along u1 , we can obtain the distribution of the normalized intensity fluctuation g2 u1 ; 0; τ 0 with τ 0 from the ECC. From the curve of the theoretical fit for the experimental data, we can obtain the value of σ g [20]. Furthermore, if we fix D1 and D2 at u1 u2 0 and vary the delay time τ, we can obtain the distribution of g2 0; 0; τ versus τ. As shown in [21], the distribution of g2 0; 0; τ also has a Gaussian profile, i.e., g2 0; 0; τ exp−τ2 ∕τ2s ;
where τs represents the characteristic time of the intensity fluctuation of the beam induced by the RGGD. From the curve of the theoretical fit for the experimental data, we can obtain the value of τs . The experimental setup for measuring the scintillation index of the generated GSM or GSMV beam propagating through the thermal-induced turbulence is shown in part III of Fig. 1. The transmitted beam from the BS1 first passes through a 35 cm × 50 cm electric hot plate, which is used to produce thermal turbulence through convection, then passes through a collection lens L4 with focal length f 4 15 cm, and finally arrives at the chargecoupled device (CCD), which is located in the geometrical focal plane (receiver plane) and is used to record the signal by a data-acquisition system. The beam center of the transmitted beam is about 2 cm above the electric hot-plate surface. The distance from the left side of the hot plate to L4 is about 1.5 m. In our experiment, the strength of the turbulence is modulated by the temperature T of the hot plate. Figure 2 shows our experimental results of the intensity distributions of the generated GSM beam and GSMV beam (l 1; 2) with σ g 1.2 mm at z 10 cm and in the receiver plane. One finds that the intensity distribution of a GSM beam is always of Gaussian profile on propagation. The intensity distribution of a GSMV beam (l 1; 2) has a Gaussian beam profile in the source plane, while it has a dark hollow beam profile at a certain propagation distance due to its intrinsic vortex phase, which induces a phase singularity in the beam center. In the receiver plane, the intensity distribution of the GSMV beam has a Gaussian profile again. Note the intensity distribution of the GSMV beam in the receiver plane is closely determined by its initial coherence width σ g , and it can exhibit Gaussian or flat-topped or a dark hollow beam profile with the increase of σ g (σ g > 1.2 mm) [16]. In our experiment, the intensity distribution of the GSMV beam in the receiver plane is always of a Gaussian beam profile through choosing a suitable value of σ g (σ g < 1.2 mm). In order to measure the scintillation index of the generated GSM or GSMV beam, the characteristic time τs of the intensity fluctuation of the beam induced by the RGGD, the characteristic time τa of the intensity
(7)
Fig. 2. Experimental results of the intensity distributions of the generated GSM beam and GSMV beam (l 1; 2) with σ g 1.2 mm at z 10 cm (a)–(c) and in the receiver plane (d)–(f).
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fluctuation of the beam induced by the thermal turbulence, and the integration time τd of the detector should satisfy the relation τs ≪ τd ≪ τa (i.e., slow detection). Under the condition of slow detection, the detector is insensitive to the intensity fluctuation of the source beam induced by the RGGD. The intensity fluctuation g2 0; 0; τ of a Gaussian beam after passing through thermally induced turbulence can be approximately modeled as [22] g2 0; 0; τ A exp−τ∕τa :
(8)
Here A is a constant. τa is mainly controlled by the temperature of the hot plate. Figure 3 shows our experimental results of the characteristic time τs and τa versus the initial coherence length σ g and the temperature of the hot plate, respectively. From Fig. 3(a), one finds that the value of τs is nearly independent of σ g within the range 0.4 mm < σ g < 1.2 mm, while the value of τs increases rapidly with the decrease of σ g when σ g < 0.4 mm. The value of τs varies from 6.7 to 14 μs. From Fig. 3(b), one sees that the value of τa exhibits monotonic growth with the decrease of the temperature of the hot plate, and its value varies from 65 to 50 ms when the temperature varies from 80°C to 160°C. In our experiment, the integration time of the detector is set as τd 20 ms, the condition of “slow detection” is satisfied. The scintillation index of a beam is defined as [2] m2c hI 2 x; yi∕hIx; yi2 − 1;
(9)
where Ix; y, hIx; yi, and hI 2 x; yi represent the instantaneous intensity, the average intensity, and the intensity correlation function of the beam in the receiver plane. In our experiment, the CCD in the receiver plane captures 4000 pictures totally, and each frame represents one realization of the beam cross section, and each realization is represented as a matrix I n x; y, where x and y are pixel spatial coordinates, and n denotes each realization and ranges from 1 to 4000. The coordinates (x¯ n ; y¯ n ) of the centroid of each realization are given as x¯ n
XX i
y¯ n
j
XX i
xi I n xi ; yj ∕
j
XX i
yi I n xi ; yj ∕
(10)
I n xi ; yj :
(11)
The intensity at the centroid of each realization is given as I n x¯ n ; y¯ n . The scintillation index of a beam at the centroid in the receiver plane can be obtained as X I 2n x¯ n ; y¯ n ∕N I¯ 2 − 1; (12) m2c N
where N denotes the number of total frames, and I¯ represents the average intensity at the centroid over all frames, i.e., X I n x¯ n ; y¯ n ∕N: (13) I¯ N
All experimental data are processed by MATLAB. Figure 4 shows our experimental results of the scintillation index of a GSMV beam (l 1; 2) at the centroid versus the initial spatial coherence width. For the convenience of comparison, the corresponding results of a GSM beam are also shown in Fig. 4. One finds from Fig. 4 that the scintillation index of a GSM or GSMV beam decreases with the decrease of σ g , which means that a partially coherent beam with a low value of σ g is less affected by the turbulence as expected [2–4]. When σ g > 0.35 mm, the scintillation index of a GSMV beam is smaller than that of a GSM beam, and the scintillation index decreases with the increase of the topological charge, which implies that a GSMV beam has an advantage over a GSM beam for reducing turbulence-induced scintillation. When σ g < 0.35 mm, the scintillation index of a GSMV beam is almost the same as that of a GSM beam and almost doesn’t vary as σ g varies. We may explain this phenomenon by the fact that the scintillation index of the GSMV beam is influenced by the coherence and the vortex phase together. When σ g is small, the GSMV beam has already evolved into a GSM beam in the entrance side of turbulence, and the influence of the coherence on the scintillation index plays a dominant role, while the influence of the vortex phase is negligible. Thus the scintillation index of the GSMV beam with extremely low coherence is similar to that of a GSM beam. Figure 5 shows our experimental results of the scintillation index of a GSM or GSMV beam at the centroid
j
XX i
I n xi ; yj ;
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j
Fig. 3. Experimental results of (a) the characteristic time τs versus the initial coherence width σ g and (b) the characteristic time τa versus the temperature of the hot plate.
Fig. 4. Experimental results of the scintillation index of a GSM or GSMV beam (l 1; 2) at the centroid versus the initial coherence width.
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OPTICS LETTERS / Vol. 38, No. 24 / December 15, 2013
Jiangsu Province under Grant No. 11KJB140007, the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
Fig. 5. Experimental results of the scintillation index of a GSM or GSMV beam (l 1; 2) at the centroid versus the temperature of the hot plate for two different values of σ g .
versus the temperature of the hot plate for two different values of σ g . One finds that the advantage of a GSMV beam over a GSM beam for reducing the turbulenceinduced scintillation also is affected by the strength of the turbulence. The scintillation index exhibits monotonic growth with the increase of the temperature of the hot plate. The advantage of a GSMV beam over a GSM beam is enhanced with the increase of the temperature of the hot plate (i.e., the strength of turbulence increases). When the temperature is small, the scintillation index of a GSMV beam with l 1 is almost the same with that of a GSMV beam with l 2. Therefore, it is necessary for us to take the coherence, topological charge, and the strength of turbulence into consideration in practical applications. In conclusion, we have carried out an experimental study of the scintillation index of a GSMV beam propagating through thermally induced turbulence and compared with that of a GSM beam. Our experimental results have clearly shown that a GSMV beam has an advantage over a GSM beam for reducing turbulence-induced scintillation under suitable conditions. Our results will be useful in free-space optical communication. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11274005 and 11104195, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Universities Natural Science Research Project of
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