- Email: [email protected]
Hypergeometric Gaussian beam and its propagation in turbulence
Optics Communications 285 (2012) 4194–4199
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Hypergeometric Gaussian beam and its propagation in turbulence Halil Tanyer Eyyubo˘glu a,n, Yangjian Cai b,1 a b
C - ankaya University, Electronic and Communication Engineering Department, Eskis- ehir yolu 29. km, Yenimahalle, 06810 Ankara, Turkey School of Physical Science and Technology, Soochow University, Suzhou 215006, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 January 2012 Received in revised form 23 June 2012 Accepted 3 July 2012 Available online 24 July 2012
We study propagation characteristics of hypergeometric Gaussian beam in turbulence. In this context, we formulate the receiver plane intensity using extended Huygens–Fresnel integral. From the graphical results, it is seen that, after propagation, hypergeometric Gaussian will in general assume the shape of a dark hollow beam at topological charges other than zero. Increasing values of topological charge will make the beam broader with steeper walls. On the other hand, higher values of hollowness parameter will contract into a narrower shape. Raising the topological charge or the hollowness parameter individually will cause outer rings to appear. Both increased levels of turbulence and longer propagation distances will accelerate the beam evolution and help reach the final Gaussian shape sooner. At lower wavelengths, there will be less beam spreading. & 2012 Elsevier B.V. All rights reserved.
Keywords: Hypergeometric Gaussian beam Beam propagation Turbulent atmosphere
1. Introduction As a solution of paraxial wave equation, hypergeometric beam was first presented in [1], later to be introduced within the context of circular beams [2]. Similar to beams like Bessel and Laguerre beams, hypergeometric beam carries a topological charge, thus being a member of singular optical beam family. A fully coherent hypergeometric beam would exhibit non-diffractive behavior upon propagation in free space. But this would call for a source beam of infinite power. One way of achieving a practically realizable beam is to modulate the hypergeometric beam with a Gaussian exponential. This way, we obtain the hypergeometric Gaussian beam. The relevant studies in this area are cited below. Generation and particular cases of hypergeometric Gaussian beams were discussed and analyzed in terms of intensity and diffraction plots in [3]. Near field and far field profiles of hypergeometric Gaussian beams and its dependence on source parameters were examined in [4]. In [5] for a different type of beam, called hypergeometric Gaussian-II beam, tighter focusing was found to be possible. For communication purposes, it is foreseen in several works that, data can be carried on the topological charge [6–8]. In this respect hypergeometric beams would seem to be an alternative candidate. For instance in [6,8], for vortex beam type,
n
Corresponding author. Tel.: þ90 312 2331322; fax: þ90 312 2848043. E-mail addresses: [email protected] (H.T. Eyyubo˘glu), [email protected] (Y. Cai). 1 Tel.: þ86 512 69153532; fax: þ 86 512 69153532. 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.07.020
Laguarre Gaussian beam was chosen and the information carrying capability of the topological chage was analyzed theoretically and experimentally. Meanwhile, the operational principles of an optical vortex detector were put foward in [7]. Furthermore, another application was envisaged for hypergeometric beam propagating through a fiber of hyperbolic index [9]. In this article, we analyze the propagation properties of hypergeometric Gaussian beams travelling in turbulent atmosphere. In particular, we attempt to find the dependence of the receiver intensity profile on source and propagation conditions. Noting that such a study has not been undertaken up to now, it is contemplated that our findings will be useful for short haul horizontal optical links operating in atmospheric turbulence.
2. Formulation of hypergeometric Gaussian beam on source and receiver planes Exempting terms that have no influence in shaping the profile, the source field of a hypergeometric Gaussian beam can be expressed as [1–3] 2 s us ðs, yÞ ¼ sp þ ja exp 2 jny ð1Þ
as
In Eq. (1), s and y are the radial coordinates, p and a are the parameters that determine the hollowness, as the Gaussian source size, n is the topological charge. It is clear from Eq. (1) that, for a being real, the parameters a and n will have no effect on the intensity profile when on the source plane. This way we can say that the source intensity is essentially governed by p and as,
˘ H.T. Eyyuboglu, Y. Cai / Optics Communications 285 (2012) 4194–4199
looking like a Gaussian beam when p ¼0 and it resembles a dark hollow beam for p40. For the hypergeometric Gaussian beam given in Eq. (1), the average intensity on the receiver plane can be found using the extended Huygens–Fresnel integral in the following manner [10]: Z
1
Rr ðr, fÞ ¼
2
l D
2
1
Z 2p Z 0
0
1
0
Z 2p 0
us ðs1 , y1 Þuns ðs1 , y1 Þs1 s2
2 exp r2 s1 þ s22 2s1 s2 cosðy1 y2 Þ t jp 2rs1 cosðfy1 Þ þs21 þ2rs2 cosðfy2 Þs22 ds1 dy1 ds2 dy2 exp lD
ð2Þ Here l stands for the wavelength, D is the axial distance between 2 the source and the receiver. rt ¼ 0:1586ðC 2n l DÞ3=5 represents the coherence length of the spherical wave and C 2n is structure constant. The first exponential in Eq. (2) takes into account the effects of the turbulent medium, while the second exponential describes the diffraction effects.Upon substituting for us(s, y) in Eq. (2) from Eq. (1) and solving the double integration over the first group of variables indexed with 1, Eq. (2) will transform into
pGð0:5n þ 0:5p þ 0:5ja þ 1Þ
Rr ðr, fÞ ¼
l0:5n0:5p0:5ja þ 1 n!
n þ p þ ja þ 2 s ðl 2t Dj
a
r
Z
Z 2p
1
þ p þ ja þ 2 0:5n þ 0:5p þ 0:5ja1 rn D t 2 prt a2s þ la2s DÞ0:5n þ 0:5p þ 0:5ja þ 1
s2 lDs2 exp ðjy2 Þ
0
0
n jpr2t r exp ðjfÞ 1 F 1 ½½0:5n þ 0:5p þ 0:5ja þ 1,n þ1,z s2 s2 jp 2rs2 cosðfy2 Þs22 ds2 dy2 exp 22 22 þ jny2 þ lD rt as ð3Þ where 1F1() is the confluent hypergeometric function and 2 2 2 2 4 2 2 t s r l s D s2 þ2j l 2t Dðl 2t Dj
2
p r a
z¼
a
r
l 2t 2s Drs2 cosðfy2 Þ 2 2 þ l 2 DÞ t s s
p r a pr a a
r
ð4Þ
From this point onwards, it is impossible to proceed analytically. But from the computation point of view, it is more convenient to modify Eq. (3) so that the confluent hypergeometric function is replaced by modified Bessel function. With this conversion, Eq. (3) now reads
pGð0:5n þ 0:5p þ0:5jaþ 1Þ
Rr ðr, fÞ ¼
l
n þ p þ ja þ 2 s ðl 2t Dj
a
i¼0 1
s2 lDs2 exp ðjy2 Þ
0
jpr2t r exp ðjfÞ exp
a
ðnpja1Þi ðpja1Þi ð0:5n0:5p0:5ja0:5þ iÞ i!ðn þ 1Þi
Z 2p
0
n!G1 ð0:5n0:5p0:5ja0:5Þ
n þ p þ ja þ 2 0:5n þ 0:5p þ 0:5ja1 D t 2 2 þ l 2 DÞ0:5n þ 0:5p þ 0:5ja þ 1 t s s
r pr a
r
p þX ja þ 1
Z
0:5n0:5p0:5ja þ 1
n
ð0:25zÞ0:5n þ 0:5p þ 0:5ja þ 0:5 I0:5n0:5p0:5ja0:5 þ i ð0:5zÞ
s2 s2 jp 22 22 þ jny2 þ 2rs2 cosðfy2 Þs22 0:5z ds2 dy2 lD rt as
ð5Þ
Eqs. (4) or (5) can be used to study the propagation of hypergeometric beam in turbulence, but they are cumbersome in the sense that they do not reveal the dependence of receiver intensity on source and propagation parameters in a comprehensible manner. For this reason, it is imperative that the free space receiver field expression is supplied as well, which can be obtained from Eq. (2) by letting rt go to infinity and reducing Eq. (2) to a double integral. This way, the free space field of the hypergeometric
4195
Gaussian beam will become n þ 1 jp rn ur ðr, fÞ ¼ 0:5n0:5p0:5ja l n!D 0:5n þ 0:5p þ 0:5ja þ 1 la2s Gð0:5n þ 0:5pþ 0:5ja þ 1Þ lDjpa2s 2pjD jpr 2 exp þ jnf þ F ½0:5n þ 0:5p þ 0:5ja þ 1,n þ 1, l D 1 1
2 2 2 p as r ð6Þ lDðlDjpa2s Þ It is worth mentioning that Eq. (6) agrees well with Eq. (5) of [4].
3. Results and discussions As commented above n, the topological charge cannot have an effect on source intensity profile. On the other hand, the parameter a has to be pure imaginary, in order produce mathematically computable results as seen from Eqs. (3) and (5). It is also clear that a and p always appear together in these equations. For convenience therefore, we drop a assuming that its presence is embedded into p and source beam hollowness is represented entirely by p. Furthermore, to avoid the creation of singularities by the Gamma function that is in the numerator of the first term in Eq. (5), for receiver intensities, we opt for fractional values of p. In the graphs to follow, we have used as ¼1 cm and the Cartesian counterparts of s, y and r,f respectively denoted by the symbols sx, sy and rx, ry. The subscript N that is placed in the label of the vertical axis of the graphs indicates that the intensity profiles are normalized as us ðs, yÞ 2 Rr ðr, fÞ h h SsN ðs, yÞ ¼ ð7Þ 2 i , RrN ðr, fÞ ¼ 2 i max us ðs, yÞ max us ðs, yÞ Fig. 1 displays the variation of source intensity with p. As seen from this figure, pa0 furnishes the hypergeometric beam with an appearance similar to dark hollow beam [10,11], where increasing p values will lead to more hollowness. In particular, these larger values of p also cause the surrounding wall of the dark hollow beam to be steeper. Needless to say p ¼0 gives a pure Gaussian beam as the subplot a of Fig. 1 shows. Now we turn to receiver intensity illustrations. By fixing the parameter p and the propagation distance to p ¼3.6, and D ¼1 km, Fig. 2 exhibits the intensity variation against changes in n. It is important to realize that here the dark hollow nature of the beam is controlled by n. To understand the formation of profiles in Fig. 2, we refer to Eq. (6). Here we see that at n ¼0, there will not be a zero crossing at on-axis point, due to the presence of the term rn. Additionally due to the selection of propagation distance, the corresponding Bessel function expansion of the hypergeometric function will be more in terms of Bessel function of first kind rather than the modified kind. Thus ring formations will be observed towards the outer edges. As Fig. 2 illustrates, n 40 will give rise to dark hollow beam profiles where, the surrounding walls will become steeper with increasing n values. Our tests showed that at n 40, keeping n fixed and raising p caused the formation of more outer rings and the same would happen if the roles of n and p were interchanged. But simultaneous rises in both n and p would submerge the outer rings. The detailed plots of these variations are not included here to save space. In Fig. 3, we examine the variation of the receiver intensity profile with respect to p, while n is kept at n¼ 1. As seen from this figure, as p increases, the receiver beam contracts more and the peak intensity levels are raised as well. Such an occurrence seems to be in line with the appearance in Fig. 1, where we have seen that higher values of p make the source beam broader.
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˘ H.T. Eyyuboglu, Y. Cai / Optics Communications 285 (2012) 4194–4199
Fig. 1. Source plane intensity variation of hypergeometric Gaussian beam at p ¼0, 1, 2, and 10.
Fig. 2. Receiver plane intensity variation of hypergeometric Gaussian beam at n¼ 0, 1, 3, and 4.
And broader beams are known to suffer less diffraction, during propagation [12]. It is possible to detect from the subplot d of Fig. 3, that keeping n constant and allowing p to increase, will make the outer rings more visible outer rings, as explained above. Fig. 4 explores the impact of turbulence on intensity profile by changing the structure constant, C 2n . To this end, we fix source parameters as n ¼1, p¼3.8, and l ¼1.55 mm and use the same propagation distance as those of Figs. (2) and (3). Fig. 4 shows that, increased levels of turbulence expressed by higher values of C 2n , accelerates the beam evolution process. This way dark hollow beam structure approaches the eventual Gaussian beam profile sooner. Such an observation is in line with those noted in [13,14]. Fig. 5 is constructed to assess how the vortexes with various topological charges disappear against the different levels of turbulence by cutting the three dimensional intensity profile along the slanted axis. Here we see that central dark region is widened as the toplogical charge value increases. On the other hand, going to higher levels of turbulence gradually eliminates
this central dark region all together.In Fig. 6, the variation of intensity profile against propagation distance is considered. Here we take n and p to be n ¼2, and p ¼2.8. Further to better reveal the beam evolution, we select propagation ranges of 0.5, 1, 3, and 6 km and set the structure constant to C 2n ¼ 5 1014 m2=3 , Increased propagation distances will create more turbulence, thus in Fig. 6, a process similar to Fig. 4 is observed, which means that with increasing propagation distance, beam evolution is speeded up such that we approach more towards the eventual Gaussian shape. Although not displayed here explicitly, it was found that higher values of p would accelerate this process, while the higher values of n would do the opposite and retard the process. Finally in Fig. 7, we investigate the impact of source wavelength. To do that, we freeze all other parameters and vary the wavelength as l ¼0.5, 1, 1.5, and 2 mm. Comparing the subplots from a to d in Fig. 7, it can be said that if the other parameters are held constant, lower wavelengths will give rise to less beam spreading. This is reasonable, since at the settings of
˘ H.T. Eyyuboglu, Y. Cai / Optics Communications 285 (2012) 4194–4199
Fig. 3. Receiver plane intensity variation of hypergeometric Gaussian beam at p ¼ 1.2, 2.2, 3.2, and 4.2.
Fig. 4. Receiver plane intensity variation of hypergeometric Gaussian beam at C 2n ¼ 1016 , 1014 , 5 1014 , and 1012 m2=3 .
Fig. 5. Receiver plane intensity variation of hypergeometric Gaussian beam at n¼ 1, 2, 3, and 4 and C 2n ¼ 1015 , and 1013 m2=3 .
4197
4198
˘ H.T. Eyyuboglu, Y. Cai / Optics Communications 285 (2012) 4194–4199
Fig. 6. Receiver plane intensity variation of hypergeometric Gaussian beam at D ¼ 0.5, 1, 3, and 6 km.
Fig. 7. Receiver plane intensity variation of hypergeometric Gaussian beam at l ¼ 0.5, 1, 1.5, and 2 mm.
Fig. 7, diffractive rather than turbulence effects will still be dominating. Again this observation is in conformity with previously analyzed cases [13].
4. Conclusion For a turbulent medium, propagation aspects of hypergeometric Gaussian beams are investigated. It is found that the source intensity profile is governed only by the hollowness parameter, while both the topological charge and the hollowness parameter are instrumental in shaping the receiver intensity profile. It is further observed that the higher values of these parameters give rise to the formation of outer rings. Longer propagation distance and larger structure constant values contribute positively to the beam evolution process so that the eventual Gaussian shape is achieved faster. Finally it is seen that the use of higher wavelengths will result in more beam spreading. As mentioned in Section 1, it is possible to propose the use of the vortex property of hypergeometric Gaussian beams in
communications and make a comparison of performance between all available vortex beams. However this requires a more detailed study. For instance it is reported in [15] that topological charge of a vortex beam (in the form of Laguerre Gaussian beam) will fluctuate as the beam propagates in turbulence. Therefore we plan to include such a topic in our future studies.
Acknowledgment Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science Foundation of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081 and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
˘ H.T. Eyyuboglu, Y. Cai / Optics Communications 285 (2012) 4194–4199
References [1] V.V. Kotlyar, R.V. Skidanov, S.N. Khonina, V.A. Sofier, Optics Letters 32 (2007) 742. [2] M.A. Bandres, J.C. Guttierrez-Vega, Optics Letters 33 (2008) 177. [3] V.V. Kotlyar, A.A. Kovalev, R.V. Skidanov, S.N. Khonina, J. Turunen, Applied Optics 47 (2008) 6124. [4] V.V. Kotlyar, A.A. Kovalev, Journal of the Optical Society of America A 25 (2008) 262. [5] E. Karimi, B. Piccirillo, R. Marrucci, E. Santamato, Optics Express 16 (2008) 21069. [6] G. Gbur, R.K. Tyson, Journal of the Optical Society of America A 25 (2008) 225.
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[7] V.P. Aksenov, I.V. Izmailov, F.Y. Kanev, B.N. Poizner, Optics Communications 285 (2012) 905. [8] G. Gibson, J. Courtial, M.J. Padgett, M. Vasnetsov, V. Pas’ko, S.M. Barnett, S. Franke-Arnold, Optics Express 12 (2004) 5448. [9] B. de, L. Bernardo, F. Moraes, Optics Express 19 (2011) 11264. [10] Y. Cai, H.T. Eyyubo˘glu, Y. Baykal, Optics and Lasers in Engineering 48 (2010) 1019. [11] H.T. Eyyubo˘glu, Optics and Laser Technology 40 (2008) 156. [12] H.T. Eyyubo˘glu, Applied Physics B 88 (2007) 259. [13] H.T. Eyyubo˘glu, Optics Communications 245 (2005) 37. [14] H.T. Eyyubo˘glu, Y. Baykal, E. Sermutlu, Optics Communications 265 (2006) 399. [15] T. Wang, J. Pu, Z. Chen, Optics Communications 282 (2009) 1255.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Hypergeometric Gaussian beam and its propagation in turbulence Halil Tanyer Eyyubo˘glu a,n, Yangjian Cai b,1 a b
C - ankaya University, Electronic and Communication Engineering Department, Eskis- ehir yolu 29. km, Yenimahalle, 06810 Ankara, Turkey School of Physical Science and Technology, Soochow University, Suzhou 215006, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 January 2012 Received in revised form 23 June 2012 Accepted 3 July 2012 Available online 24 July 2012
We study propagation characteristics of hypergeometric Gaussian beam in turbulence. In this context, we formulate the receiver plane intensity using extended Huygens–Fresnel integral. From the graphical results, it is seen that, after propagation, hypergeometric Gaussian will in general assume the shape of a dark hollow beam at topological charges other than zero. Increasing values of topological charge will make the beam broader with steeper walls. On the other hand, higher values of hollowness parameter will contract into a narrower shape. Raising the topological charge or the hollowness parameter individually will cause outer rings to appear. Both increased levels of turbulence and longer propagation distances will accelerate the beam evolution and help reach the final Gaussian shape sooner. At lower wavelengths, there will be less beam spreading. & 2012 Elsevier B.V. All rights reserved.
Keywords: Hypergeometric Gaussian beam Beam propagation Turbulent atmosphere
1. Introduction As a solution of paraxial wave equation, hypergeometric beam was first presented in [1], later to be introduced within the context of circular beams [2]. Similar to beams like Bessel and Laguerre beams, hypergeometric beam carries a topological charge, thus being a member of singular optical beam family. A fully coherent hypergeometric beam would exhibit non-diffractive behavior upon propagation in free space. But this would call for a source beam of infinite power. One way of achieving a practically realizable beam is to modulate the hypergeometric beam with a Gaussian exponential. This way, we obtain the hypergeometric Gaussian beam. The relevant studies in this area are cited below. Generation and particular cases of hypergeometric Gaussian beams were discussed and analyzed in terms of intensity and diffraction plots in [3]. Near field and far field profiles of hypergeometric Gaussian beams and its dependence on source parameters were examined in [4]. In [5] for a different type of beam, called hypergeometric Gaussian-II beam, tighter focusing was found to be possible. For communication purposes, it is foreseen in several works that, data can be carried on the topological charge [6–8]. In this respect hypergeometric beams would seem to be an alternative candidate. For instance in [6,8], for vortex beam type,
n
Corresponding author. Tel.: þ90 312 2331322; fax: þ90 312 2848043. E-mail addresses: [email protected] (H.T. Eyyubo˘glu), [email protected] (Y. Cai). 1 Tel.: þ86 512 69153532; fax: þ 86 512 69153532. 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.07.020
Laguarre Gaussian beam was chosen and the information carrying capability of the topological chage was analyzed theoretically and experimentally. Meanwhile, the operational principles of an optical vortex detector were put foward in [7]. Furthermore, another application was envisaged for hypergeometric beam propagating through a fiber of hyperbolic index [9]. In this article, we analyze the propagation properties of hypergeometric Gaussian beams travelling in turbulent atmosphere. In particular, we attempt to find the dependence of the receiver intensity profile on source and propagation conditions. Noting that such a study has not been undertaken up to now, it is contemplated that our findings will be useful for short haul horizontal optical links operating in atmospheric turbulence.
2. Formulation of hypergeometric Gaussian beam on source and receiver planes Exempting terms that have no influence in shaping the profile, the source field of a hypergeometric Gaussian beam can be expressed as [1–3] 2 s us ðs, yÞ ¼ sp þ ja exp 2 jny ð1Þ
as
In Eq. (1), s and y are the radial coordinates, p and a are the parameters that determine the hollowness, as the Gaussian source size, n is the topological charge. It is clear from Eq. (1) that, for a being real, the parameters a and n will have no effect on the intensity profile when on the source plane. This way we can say that the source intensity is essentially governed by p and as,
˘ H.T. Eyyuboglu, Y. Cai / Optics Communications 285 (2012) 4194–4199
looking like a Gaussian beam when p ¼0 and it resembles a dark hollow beam for p40. For the hypergeometric Gaussian beam given in Eq. (1), the average intensity on the receiver plane can be found using the extended Huygens–Fresnel integral in the following manner [10]: Z
1
Rr ðr, fÞ ¼
2
l D
2
1
Z 2p Z 0
0
1
0
Z 2p 0
us ðs1 , y1 Þuns ðs1 , y1 Þs1 s2
2 exp r2 s1 þ s22 2s1 s2 cosðy1 y2 Þ t jp 2rs1 cosðfy1 Þ þs21 þ2rs2 cosðfy2 Þs22 ds1 dy1 ds2 dy2 exp lD
ð2Þ Here l stands for the wavelength, D is the axial distance between 2 the source and the receiver. rt ¼ 0:1586ðC 2n l DÞ3=5 represents the coherence length of the spherical wave and C 2n is structure constant. The first exponential in Eq. (2) takes into account the effects of the turbulent medium, while the second exponential describes the diffraction effects.Upon substituting for us(s, y) in Eq. (2) from Eq. (1) and solving the double integration over the first group of variables indexed with 1, Eq. (2) will transform into
pGð0:5n þ 0:5p þ 0:5ja þ 1Þ
Rr ðr, fÞ ¼
l0:5n0:5p0:5ja þ 1 n!
n þ p þ ja þ 2 s ðl 2t Dj
a
r
Z
Z 2p
1
þ p þ ja þ 2 0:5n þ 0:5p þ 0:5ja1 rn D t 2 prt a2s þ la2s DÞ0:5n þ 0:5p þ 0:5ja þ 1
s2 lDs2 exp ðjy2 Þ
0
0
n jpr2t r exp ðjfÞ 1 F 1 ½½0:5n þ 0:5p þ 0:5ja þ 1,n þ1,z s2 s2 jp 2rs2 cosðfy2 Þs22 ds2 dy2 exp 22 22 þ jny2 þ lD rt as ð3Þ where 1F1() is the confluent hypergeometric function and 2 2 2 2 4 2 2 t s r l s D s2 þ2j l 2t Dðl 2t Dj
2
p r a
z¼
a
r
l 2t 2s Drs2 cosðfy2 Þ 2 2 þ l 2 DÞ t s s
p r a pr a a
r
ð4Þ
From this point onwards, it is impossible to proceed analytically. But from the computation point of view, it is more convenient to modify Eq. (3) so that the confluent hypergeometric function is replaced by modified Bessel function. With this conversion, Eq. (3) now reads
pGð0:5n þ 0:5p þ0:5jaþ 1Þ
Rr ðr, fÞ ¼
l
n þ p þ ja þ 2 s ðl 2t Dj
a
i¼0 1
s2 lDs2 exp ðjy2 Þ
0
jpr2t r exp ðjfÞ exp
a
ðnpja1Þi ðpja1Þi ð0:5n0:5p0:5ja0:5þ iÞ i!ðn þ 1Þi
Z 2p
0
n!G1 ð0:5n0:5p0:5ja0:5Þ
n þ p þ ja þ 2 0:5n þ 0:5p þ 0:5ja1 D t 2 2 þ l 2 DÞ0:5n þ 0:5p þ 0:5ja þ 1 t s s
r pr a
r
p þX ja þ 1
Z
0:5n0:5p0:5ja þ 1
n
ð0:25zÞ0:5n þ 0:5p þ 0:5ja þ 0:5 I0:5n0:5p0:5ja0:5 þ i ð0:5zÞ
s2 s2 jp 22 22 þ jny2 þ 2rs2 cosðfy2 Þs22 0:5z ds2 dy2 lD rt as
ð5Þ
Eqs. (4) or (5) can be used to study the propagation of hypergeometric beam in turbulence, but they are cumbersome in the sense that they do not reveal the dependence of receiver intensity on source and propagation parameters in a comprehensible manner. For this reason, it is imperative that the free space receiver field expression is supplied as well, which can be obtained from Eq. (2) by letting rt go to infinity and reducing Eq. (2) to a double integral. This way, the free space field of the hypergeometric
4195
Gaussian beam will become n þ 1 jp rn ur ðr, fÞ ¼ 0:5n0:5p0:5ja l n!D 0:5n þ 0:5p þ 0:5ja þ 1 la2s Gð0:5n þ 0:5pþ 0:5ja þ 1Þ lDjpa2s 2pjD jpr 2 exp þ jnf þ F ½0:5n þ 0:5p þ 0:5ja þ 1,n þ 1, l D 1 1
2 2 2 p as r ð6Þ lDðlDjpa2s Þ It is worth mentioning that Eq. (6) agrees well with Eq. (5) of [4].
3. Results and discussions As commented above n, the topological charge cannot have an effect on source intensity profile. On the other hand, the parameter a has to be pure imaginary, in order produce mathematically computable results as seen from Eqs. (3) and (5). It is also clear that a and p always appear together in these equations. For convenience therefore, we drop a assuming that its presence is embedded into p and source beam hollowness is represented entirely by p. Furthermore, to avoid the creation of singularities by the Gamma function that is in the numerator of the first term in Eq. (5), for receiver intensities, we opt for fractional values of p. In the graphs to follow, we have used as ¼1 cm and the Cartesian counterparts of s, y and r,f respectively denoted by the symbols sx, sy and rx, ry. The subscript N that is placed in the label of the vertical axis of the graphs indicates that the intensity profiles are normalized as us ðs, yÞ 2 Rr ðr, fÞ h h SsN ðs, yÞ ¼ ð7Þ 2 i , RrN ðr, fÞ ¼ 2 i max us ðs, yÞ max us ðs, yÞ Fig. 1 displays the variation of source intensity with p. As seen from this figure, pa0 furnishes the hypergeometric beam with an appearance similar to dark hollow beam [10,11], where increasing p values will lead to more hollowness. In particular, these larger values of p also cause the surrounding wall of the dark hollow beam to be steeper. Needless to say p ¼0 gives a pure Gaussian beam as the subplot a of Fig. 1 shows. Now we turn to receiver intensity illustrations. By fixing the parameter p and the propagation distance to p ¼3.6, and D ¼1 km, Fig. 2 exhibits the intensity variation against changes in n. It is important to realize that here the dark hollow nature of the beam is controlled by n. To understand the formation of profiles in Fig. 2, we refer to Eq. (6). Here we see that at n ¼0, there will not be a zero crossing at on-axis point, due to the presence of the term rn. Additionally due to the selection of propagation distance, the corresponding Bessel function expansion of the hypergeometric function will be more in terms of Bessel function of first kind rather than the modified kind. Thus ring formations will be observed towards the outer edges. As Fig. 2 illustrates, n 40 will give rise to dark hollow beam profiles where, the surrounding walls will become steeper with increasing n values. Our tests showed that at n 40, keeping n fixed and raising p caused the formation of more outer rings and the same would happen if the roles of n and p were interchanged. But simultaneous rises in both n and p would submerge the outer rings. The detailed plots of these variations are not included here to save space. In Fig. 3, we examine the variation of the receiver intensity profile with respect to p, while n is kept at n¼ 1. As seen from this figure, as p increases, the receiver beam contracts more and the peak intensity levels are raised as well. Such an occurrence seems to be in line with the appearance in Fig. 1, where we have seen that higher values of p make the source beam broader.
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Fig. 1. Source plane intensity variation of hypergeometric Gaussian beam at p ¼0, 1, 2, and 10.
Fig. 2. Receiver plane intensity variation of hypergeometric Gaussian beam at n¼ 0, 1, 3, and 4.
And broader beams are known to suffer less diffraction, during propagation [12]. It is possible to detect from the subplot d of Fig. 3, that keeping n constant and allowing p to increase, will make the outer rings more visible outer rings, as explained above. Fig. 4 explores the impact of turbulence on intensity profile by changing the structure constant, C 2n . To this end, we fix source parameters as n ¼1, p¼3.8, and l ¼1.55 mm and use the same propagation distance as those of Figs. (2) and (3). Fig. 4 shows that, increased levels of turbulence expressed by higher values of C 2n , accelerates the beam evolution process. This way dark hollow beam structure approaches the eventual Gaussian beam profile sooner. Such an observation is in line with those noted in [13,14]. Fig. 5 is constructed to assess how the vortexes with various topological charges disappear against the different levels of turbulence by cutting the three dimensional intensity profile along the slanted axis. Here we see that central dark region is widened as the toplogical charge value increases. On the other hand, going to higher levels of turbulence gradually eliminates
this central dark region all together.In Fig. 6, the variation of intensity profile against propagation distance is considered. Here we take n and p to be n ¼2, and p ¼2.8. Further to better reveal the beam evolution, we select propagation ranges of 0.5, 1, 3, and 6 km and set the structure constant to C 2n ¼ 5 1014 m2=3 , Increased propagation distances will create more turbulence, thus in Fig. 6, a process similar to Fig. 4 is observed, which means that with increasing propagation distance, beam evolution is speeded up such that we approach more towards the eventual Gaussian shape. Although not displayed here explicitly, it was found that higher values of p would accelerate this process, while the higher values of n would do the opposite and retard the process. Finally in Fig. 7, we investigate the impact of source wavelength. To do that, we freeze all other parameters and vary the wavelength as l ¼0.5, 1, 1.5, and 2 mm. Comparing the subplots from a to d in Fig. 7, it can be said that if the other parameters are held constant, lower wavelengths will give rise to less beam spreading. This is reasonable, since at the settings of
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Fig. 3. Receiver plane intensity variation of hypergeometric Gaussian beam at p ¼ 1.2, 2.2, 3.2, and 4.2.
Fig. 4. Receiver plane intensity variation of hypergeometric Gaussian beam at C 2n ¼ 1016 , 1014 , 5 1014 , and 1012 m2=3 .
Fig. 5. Receiver plane intensity variation of hypergeometric Gaussian beam at n¼ 1, 2, 3, and 4 and C 2n ¼ 1015 , and 1013 m2=3 .
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Fig. 6. Receiver plane intensity variation of hypergeometric Gaussian beam at D ¼ 0.5, 1, 3, and 6 km.
Fig. 7. Receiver plane intensity variation of hypergeometric Gaussian beam at l ¼ 0.5, 1, 1.5, and 2 mm.
Fig. 7, diffractive rather than turbulence effects will still be dominating. Again this observation is in conformity with previously analyzed cases [13].
4. Conclusion For a turbulent medium, propagation aspects of hypergeometric Gaussian beams are investigated. It is found that the source intensity profile is governed only by the hollowness parameter, while both the topological charge and the hollowness parameter are instrumental in shaping the receiver intensity profile. It is further observed that the higher values of these parameters give rise to the formation of outer rings. Longer propagation distance and larger structure constant values contribute positively to the beam evolution process so that the eventual Gaussian shape is achieved faster. Finally it is seen that the use of higher wavelengths will result in more beam spreading. As mentioned in Section 1, it is possible to propose the use of the vortex property of hypergeometric Gaussian beams in
communications and make a comparison of performance between all available vortex beams. However this requires a more detailed study. For instance it is reported in [15] that topological charge of a vortex beam (in the form of Laguerre Gaussian beam) will fluctuate as the beam propagates in turbulence. Therefore we plan to include such a topic in our future studies.
Acknowledgment Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science Foundation of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081 and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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