- Email: [email protected]
Scintillation analysis of hypergeometric Gaussian beam via phase screen method
Optics Communications 309 (2013) 103–107
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Scintillation analysis of hypergeometric Gaussian beam via phase screen method Halil T. Eyyuboğlu n Çankaya University, Department of Electronic and Communication Engineering, Eskişehir Yolu, 29 km, Yenimahalle, 06810 Ankara, Turkey
art ic l e i nf o
a b s t r a c t
Article history: Received 26 April 2013 Received in revised form 3 July 2013 Accepted 4 July 2013 Available online 18 July 2013
We give a scintillation treatment of hypergeometric Gaussian beams via the use of random phase screens. In particular, we analyse the on-axis, point-like and aperture averaged power scintillation characteristics of this beam that cannot be undertaken easily by analytic means. Within the range of examined source and propagation parameters, our evaluations show that there will be less scintillation, with increasing hollowness at small source sizes and zero topological charge. At larger source sizes or topological charges, this is reversed and decreasing hollowness will reduce scintillation. More or less the same trend is observed for aperture averaging such that at small source sizes and zero topological charge, increased hollowness will result in lower scintillation. At larger source size and topological charges, there will be a transition from the case of smaller values of hollowness giving rise to less scintillation at smaller aperture openings to the case of larger values of hollowness giving rise to less scintillation at larger aperture openings. In general nonzero topological charges will produces more scintillations, both in onaxis and aperture averaged cases. & 2013 Elsevier B.V. All rights reserved.
Keywords: Random phase screen Scintillation Hypergeometric Gaussian beam
1. Introduction Analytic formulation of scintillation induced by turbulent atmosphere requires lengthy derivations, whether Rytov approximation or extended Huygens–Fresnel approach is adopted. This difficulty is alleviated by switching to the random phase screen method, since there finding scintillation is reduced to the simple act of summations and divisions. In the use of random phase screen, there is the added advantage that we can deal with almost any beam type. Finally it is quite easy with random phase screen to extend the single receiver point scintillation calculations to aperture averaged scintillation results, which would again pose enormous difficulties, if it were to be tackled analytically. In this study we have opted for a beam whose scintillation characteristics are difficult to analyse by analytic means and besides have not investigated up to now. The use of random phase screen to model propagation in turbulence is now well established through quite a number of publications. In two of the earlier works, by adopting a varying power law spectra, the effect of inner and outer scales on intensity variance (scintillation), intensity spectra were analysed, covering strong turbulence regimes for plane and spherical waves [1,2]. For different types of inner scales, the intensity variance results of
n
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0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.024
random phase screen computations and analytic calculations were reported to agree within 2% in [3]. Design considerations of a Kolmogorov phase screen based on the accurate approximation of the phase were discussed in [4]. Errors originating from the use of finite grid size, finite transverse plane dimensions and the separation of phase screens along the propagation path were investigated quantitatively in [5] and expressed for plane and spherical waves. In [6], it was found that simulation of scintillation, beam spreading and reflective speckle patterns via phase screen approach produced results in good agreement with analytic predictions and experimental data. Using a Gaussian beam source, comparisons of random phase screen results with the analytic ones were made in [7] for scintillations, beam spreading, beam wander, coherence diameters, variance and autocorrelation of the beam intensity. Due to the great deal of attention devoted to the subject, two books were published recently covering theoretical backgrounds as well as implementation details of numerical simulation of the optical beam propagation [8,9]. The random phase screen method is also used for scintillation performances of specialized beams. In this context, pseudo-Bessel correlated beams were studied in [10], where it was demonstrated that with the choice of appropriate coherence parameter, these beams could offer lower scintillation both in weak and strong turbulence. In another study [11], the irradiance pattern, degree of polarisation and scintillation index of radially polarised vortex beam represented by the lowest order Laguerre Gaussian beam were investigated and the vector vortex beam was found to posses
104
H.T. Eyyuboğlu / Optics Communications 309 (2013) 103–107
lower scintillation than the scalar counterpart. In [12], again the use of vortex beam of elliptical profile was reported to result in lower scintillation by adjusting the ratio of minor axis to major axis to smaller values. Based on the use of the Gaussian Schell model beam in wave optics simulations, the suitability of gamma and lognormal probability density functions were examined against analytic predictions and it was seen that in weak fluctuation regime, the results of wave optics simulations followed the gamma distribution model, but in strong fluctuations case, the results approximated better to the lognormal probability density function [13]. Scintillation properties of the Gaussian Schell model beam was also investigated in [14] along with the beam spreading and beam wander effects, where it was observed that lower beam coherence would lead to reductions in scintillations and relative beam spreading, but would have less impact on beam wander. As an alternative to the existing random phase screen techniques, a new model named “sparse spectrum” has been proposed and claimed to offer substantial reduction in computation times [15]. It has long been known that as the receiver aperture size is enlarged, aperture averaging will occur and then we will obtain reductions in scintillations [16–20]. But the formulation of aperture averaged scintillation poses extreme difficulties, since it requires the evaluation of covariance function. Up to now, it has been formulated only for plane, spherical and Gaussian beams. In [21], the aperture averaged scintillation for partially coherent Gaussian beam was investigated in relation to correlation time of the source phase variations and the integration time of the detector and it was found that scintillation advantages of partially coherent Gaussian beam could be realized only when the ratio of these two parameters was less than a certain threshold. The generation, the free space and turbulence propagation characteristics of hypergeometric Gaussian beams were analysed in [22–24]. Hypergeometric Gaussian beam is considered to be a member of the vortex beam family. In [25–27], it was proposed that the vortex feature of such beams could be used as a type of modulation. In this study, we perform scintillation analysis on hypergeometric Gaussian beams. Here our motivation is to understand the advantages that may be offered by the use of such a beam in optical links. It is well known that scintillation causes a degradation in the probability of error performance of laser communication systems [20]. Therefore any improvement in that direction would be well accepted. The scintillation characteristics of hypergeometric Gaussian beams are explored with this particular aim in mind. The random phase screen method is used, thus allowing both the point-like (on-axis) and aperture averaged scintillations to be evaluated with ease.
For a source field us ðsx ; sy Þ, the Hygens–Fresnel integral delivers the receiver field ur ðr x ; r y ; LÞ as follows: jkexpðjkLÞ 2πL
Z
1 1
2 i þ r y sy
Z
1 1
dsx dsy us ðsx ; sy Þexp
Z ur ðrx ; r y ; LÞ ¼ us ðsx ; sy Þ⊗hðr x ; r y Þ ¼
1 1
Z
1 1
dsx dsy us ðsx ; sy Þhðr x sx ; r y sy Þ
ð2Þ where hðr x sx ; r y sy Þ denotes the collection of terms from Eq. (1) such that h 2 i jkexpðjkLÞ jk ð3Þ exp ðr x sx Þ2 þ r y sy hðr x sx ; r y sy Þ ¼ 2πL 2L The relationship established on the left hand side of Eq. (2) can also be converted into ur ðr x ; r y ; LÞ ¼ F1 fF½us ðsx ; sy ÞF½hðr x ; r y Þg ¼ F1 fF½us ðsx ; sy ÞHðf x ; f y Þg
ð4Þ
where F is the Fourier transform operator and
2jπ 2 L 2 2 fx þ fy H f x ; f y ¼ F h r x ; r y ¼ expðjkLÞexp k h i 2 2 ¼ expðjkLÞexp jπλL f x þ f y
ð5Þ
From Eq. (4), we conclude that the receiver field can be found by taking the Fourier transform of source field us sx ; sy , then multiplying this by the Fourier transform of the transfer function h r x ; r y , eventually inverse transforming this product. So all together, this is a three step operation. Although Eq. (2) demonstrates that the same could be achieved in one step, i.e., with a single convolution operation, in practical computation, it is seen that for the level of resolution required, the three step Fourier transform method, based on FFT functions available in most software packages, is much faster than the one step convolution implementation. It is known that a propagating beam will expand due to diffractive effects. To compensate for this, a scaling factor expressing the ratio between receiver and source transverse plane coordinates must be embedded into Eq. (3) [8]. The above development covers propagation in free space, i.e., a medium free of turbulence. To include the effects of turbulence, the entire propagation length of L is split up into N shorter intervals of ΔL ¼ L=N and a thin phase screen plane is placed at each interval. These phase screen planes are then used to create random phase distribution over the transverse plane. In this manner, the field at the nth plane is linked to the one at the ðn1Þth plane as ur r x ; r y ; nΔL ¼ F1 F us r x ; r y ; ðn1ÞΔL exp jϕ r x ; r y H f x; f y ð6Þ
2. Description of the random phase screen method
ur ðr x ; r y ; LÞ ¼
operator ⊗, as shown below:
jk h ðr x sx Þ2 2L ð1Þ
where k is the wave number and related to the source wavelength λ as k ¼ 2π=λ, Lis the on- axis receiver plane location, r x ; r y and sx ; sy are the transverse coordinates of the source and receiver sides respectively. Eq. (1) can be interpreted as a two dimensional convolution integral and can be written using the convolution
The random phase distribution ϕðr x ; r y Þ is retrieved from one of the phase power spectral density functions adopted. In this study, we have selected von-Karman spectrum, thus the phase power spectral density function will be h i 2 2 2 exp 1:1265l0 f x þ f y 2 2 11=3 Φ f x ; f y ¼ 0:0036k LC n L0 ð7Þ h i11=6 2 2 L20 f x þ f y þ 1 where L0 ; l0 denote the outer and inner scales of turbulence, C 2n is the refractive index structure constant. With the utilisation of Eq. (7), the field given in Eq. (6) will incorporate the turbulence induced effects of this particular spectrum. Here the choice of vonKarman spectrum is justifiable, since we cannot go to the limit of zero inner scale due to finite grid spacing used [7].
H.T. Eyyuboğlu / Optics Communications 309 (2013) 103–107
Scintillation index for a particular receiver plane location of r x ; r y is defined as D E I 2r r x ; r y ; L 2 b rx ; ry ; L ¼ ð8Þ 2 1 Ir rx ; ry ; L The receiver plane intensity, I r r x ; r y ; L ¼ ur r x ; r y ; L unr r x ; r y ; L is obtained from Eq. (6), by going through all free space propagation intervals and the phase screens which model the turbulence via the random phase ϕ r x ; r y and finally reaching the setting of n ¼ N. The average receiver intensity, I r r x ; r y ; L is generated by summing the individually realized intensity terms, I r r x ; r y ; L and then D dividing E the result by the number of realisations. For the on the other hand, the square of the individually I 2r r x ; r y ; L realized intensities are summed with again the averaging operation being applied at the end. If an analytic derivation was to be undertaken to arrive at the scintillation index, lengthy derivations would emerge regardless of Rytov or extended Huygens–Fresnel approach being chosen. With the use of random phase screens, the procedure is reduced to above mentioned summations and divisions. Another important parameter in the measure of scintillation is the aperture averaged scintillation, which can be defined over a circular receiver area of radius Ra as follows: D E Z 0:5pffiffiπ Ra Z 0:5pffiffiπ Ra P r 2 ðLÞ 2 b ðLÞ ¼ 1 ; P ð L Þ ¼ I r r x ; r y ; L dr x dr y r 2 pffiffi pffiffi 0:5 π Ra 0:5 π Ra P r ðL Þ ð9Þ It should be stressed that the random phase screen method is particularly instrumental in the calculation of aperture averaged scintillation as well, since such scintillations are hardly available analytically apart from simple cases of plane, spherical waves and Gaussian beams [16–21].
105
For the graphical displays, we fix the operating wavelength and the refractive index structure constant as λ ¼ 1:55 μm; C 2n ¼ 1015 m2=3 . Fig. 1 illustrates the variation of on-axis scintillation of hypergeometric Gaussian beam against propagation lengths up to L ¼ 5:5 kmfor three values of p, namely 0,2,5 at αs ¼ 1 cm; n ¼ 0. Here, because of the fluctuating nature of the random phase screen results, polynomial fitted curves of fourth degree are also plotted to better reveal the trend against the changing values of p. This way, it is seen from Fig. 1 that scintillation reductions are obtained by lowering the value of p, when n ¼ 0. This was also confirmed by observing the scintillation curves at values of p other than those plotted in Fig. 1. It is worth noting that in Eq. (10), the settings of n ¼ 0; p ¼ 0 yield the fundamental Gaussian beam. In this respect, there is close agreement between the curve of p ¼ 0 in Fig. 1 and the analytic results of [29]. Next we go to a higher value of n. In this manner, Fig. 2 exhibits the on-axis scintillation variations at αs ¼ 1 cm; n ¼ 2, against the same propagation range of Fig. 1. Here rather high scintillations are encountered. Besides at this value of n, scintillations seem to rise with larger p values, i.e., the opposite of that observed in Fig. 1. This was seen to the case for almost all n 4 0. In Fig. 3, we examine the variation of on-axis scintillation index against Gaussian source size at n ¼ 0; L ¼ 3 km for the selected values of p ¼ 0; 2; 5. The curves of Fig. 3 reflect the typical behaviour of scintillation index variation against source size reported in the literature [29]. At source sizes up to roughly αs ¼ 2 cm, the curves of p ¼ 0; 2; 5 appear to follow the trend in Fig. 1. But this trend changes from there onwards and at larger source sizes, scintillation seems to be rising more rapidly particularly with increasing p. The topological charge is changed to n ¼ 2 in Fig. 4, where similar to Fig. 2, high levels of scintillations are present.
3. Numeric results and discussions In this section, we present numeric results obtained for scintillations of a hypergeometric Gaussian beam whose source plane field is given by 2 s us ðs; θÞ ¼ sp exp 2 jnθ ð10Þ αs where s2 ¼ s2x þ s2y , thus s; θ constitute the radial coordinates of the source plane. p is the hollowness parameter, αs is the Gaussian source size, n is the topological charge. It is clear from Eq. (10) that on the source plane, hypergeometric beam will resemble a dark hollow beam for p 40, while this hollowness will also be controlled by n after propagation [24]. In our scintillation runs, square source and receiver planes with grid size of 512 512 are used. The propagation length L is divided into 20 intervals. This means that we have employed multiple random phase screens rather than a single one. This way, each propagation interval has a much reduced length, so that the grid (sample) size on the plane of the screens could be kept at reasonable numbers, preventing excessive rises in computation time [8]. Secondly the choice of multiple screens is appropriate in the sense that such a choice acknowledges the fact that random phase screens are the physical counterpart of the concept of multiple scatters, utilised in the solution of parabolic fourth moment equation [28]. Averaging is taken over 500 realisations. The on-axis scintillation values based on the evaluation of Eq. (8), are obtained over a 6 6 square grid area. In all relevant cases, this corresponded to a point-like receiver aperture area, by ensuring that the radius of the equivalent circular pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi area of the on-axis 6 6 square grid remained below 0:5λL=π [20].
Fig. 1. Variations of on-axis scintillation against propagation length at αs ¼ 1 cm; n ¼ 0 for different values of p. Fitted curves are in colour and marked with circles.
Fig. 2. Variations of on-axis scintillation against propagation length at αs ¼ 1 cm; n ¼ 2 for different values of p. Fitted curves are in colour and marked with circles.
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H.T. Eyyuboğlu / Optics Communications 309 (2013) 103–107
Fig. 3. Variations of on-axis scintillation against source size at n ¼ 0; L ¼ 3 km for different values of p. Fitted curves are in colour and marked with dots.
Fig. 4. Variations of on-axis scintillation against source size at n ¼ 2; L ¼ 3 km for different values of p. Fitted curves are in colour and marked with circles.
Additionally, Fig. 4 reveals that at the value of n ¼ 2, the rise of scintillation with rising values of p is confined to small source sizes. As seen from Fig. 4, this trend is reversed at large source sizes and higher values of p start to yield less scintillation. The next four graphs concern aperture averaged scintillations. We start with the settings of αs ¼ 1 cm; n ¼ 0; L ¼ 3 km in Fig. 5 and explore the variation of aperture averaged scintillation against the aperture radius Ra as given in Eq. (9) at different p. As expected, aperture averaged scintillation falls rapidly with increases in aperture radius. The ordering of curves in Fig. 5 in terms of p appears to be the same as Fig. 1, that is, the larger p values result in lower aperture averaged scintillations. Fig. 6 shows the situation of aperture averaged scintillation after switching to n ¼ 3. Here it is seen that at small aperture radii, aperture averaged scintillation is higher for larger values of p, however, as the aperture radius is increased, this appearance turns the other way around and smaller values of p begin to offer higher scintillation. Particularly the curves of p ¼ 0; 1; 2 manage this turning by slightly bending upwards around the middle aperture radius sizes. Finally two more graphs are exhibited for aperture averaged scintillations. In Fig. 7, we have chosen the fixed settings as αs ¼ 5 cm; n ¼ 0; L ¼ 5 km. Here contrary to the case of Fig. 5 and in line with Fig. 3, the rule of smaller p values possessing lower scintillation applies, with the differentiation between the beams being somewhat less. Fig. 8 illustrates the situation after setting n ¼ 5, where it is seen that in comparison to the turnings of Fig. 6, the transition to the case of larger p values offering less scintillation, takes place at smaller aperture radii. When compared to Fig. 7, the merging of all curves towards the same low aperture averaged scintillation levels lags behind to some extent in Fig. 8. In general however, for all beam and propagation parameters examined, we witness, in Figs. 5–8, the clear advantage of enlarging the aperture radius in order to acquire low scintillations.
Fig. 5. Variations of aperture averaged scintillation against aperture radius at αs ¼ 1 cm; n ¼ 0; L ¼ 3 km for different values of p.
Fig. 6. Variations of aperture averaged scintillation against aperture radius at αs ¼ 1 cm; n ¼ 3; L ¼ 3 km for different values of p.
Fig. 7. Variations of aperture averaged scintillation against aperture radius at αs ¼ 5 cm; n ¼ 0; L ¼ 5 km for different values of p.
Fig. 8. Variations of aperture averaged scintillation against aperture radius at αs ¼ 5 cm; n ¼ 5; L ¼ 5 km for different values of p.
H.T. Eyyuboğlu / Optics Communications 309 (2013) 103–107
4. Conclusion We have evaluated the scintillation characteristics of hypergeometric Gaussian beams using the random phase screen method. Our findings indicate that, for on-axis scintillation, larger values of p, the hollowness parameter, result in lower scintillation at smaller Gaussian source sizes and zero topological charge, i.e., n ¼ 0. This situation is reversed however for n 4 0 or at larger Gaussian sources sizes. There, lower on-axis scintillation is obtained at smaller values of p. When aperture averaged scintillation is considered, it is found that at small source sizes and zero topological charge, larger values of p will lead to lower scintillations. For n 4 0 however, at small aperture sizes, i.e., nearer to point-like receivers, scintillation is lower at smaller values of p. A crossover occurs towards the larger aperture sizes, where scintillation becomes lower at larger p values. At larger source sizes, longer propagation distances and at higher topological charges, this crossover is brought to smaller aperture sizes. In general, both on-axis and aperture averaged scintillations are seen to act in a similar manner with changes in beam and propagation parameters. Another generalisation is that topological charges larger than zero will tend to result in higher scintillations. References [1] J.M. Martin, S.M. Flatte, Applied Optics 27 (1988) 2111. [2] J.M. Martin, S.M. Flatte, Journal of the Optical Society of America A: Optics, Image Science, and Vision 7 (1990) 838. [3] C.A. Davis, D.L. Walters, Applied Optics 33 (1994) 8406.
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[4] R.G. Lane, A. Glindemann, J.C. Dainty, Waves in Random Media 2 (1992) 209. [5] W.A. Coles, J.P. Filice, R.G. Frehlich, M. Yadlowsky, Applied Optics 34 (1995) 2089. [6] D.H. Nelson, D.L. Walters, E.P. MacKerrow, M.J. Schmitt, C.R. Quick, W.M. Porch, R.R. Petrin, Applied Optics 39 (2000) 1857. [7] A. Belmonte, Applied Optics 39 (2000) 5426. [8] J.D. Schmidt, Numerical Simulation of Optical Wave Propagation with examples in MATLAB, SPIE Press, Bellingham, Washington, USA, 2010. [9] D. Voelz, Computational Fourier Optics A MATLAB Tutorial, Washington, SPIE Press, Bellingham, Washington, USA, 2011. [10] Y. Gu, G. Gbur, Journal of the Optical Society of America A 27 (2010) 2621. [11] W. Cheng, J.H. Haus, Q. Zhan, Optics Express 17 (2009) 17829. [12] X. Liu, J. Pu, Optics Express 18 (2011) 26444. [13] X. Xiao, D. Voelz, Applied Optics 48 (2009) 167. [14] X. Qian, W. Zhu, R. Rao, Optics Express 17 (2009) 3782. [15] M. Charnotskii, Journal of the Optical Society of America A 30 (2013) 479. [16] D.L. Fried, Journal of the Optical Society of America 55 (1965) 1427. [17] D.L. Fried, Journal of the Optical Society of America 57 (1967) 169. [18] R.F. Lutomirski, H.T. Yura, Journal of the Optical Society of America 59 (1969) 1247. [19] J.H. Churside, Applied Optics 30 (1991) 1982. [20] L.C. Andrews, R.L. Phillips, Laser Beam Propagation through Random Media, SPIE Press, Bellingham, Washington, USA, 2005. [21] Z. Chen, S. Yu, L. Jian, C. Zheng, G. Wu, W. Gu, Optics Communications 298–299 (2013) 8. [22] V.V. Kotlyar, S.N. Khonina, R.V. Skidanov, V.A. Soifer, Optics Communications 274 (2007) 8. [23] M.A. Bandres, D. Lopez-Mago, J.C. Gutierrez-Vega, Journal of Optics 12 (2010) 1. [24] H.T. Eyyuboğlu, Y. Cai, Optics Communications 285 (2012) 4194. [25] G. Gibson, J. Courtial, M.J. Padgett, M. Vasnetsov, V. Pas'ko, S.M. Barnett, S. Franke-Arnold, Optics Express 12 (2004) 5448. [26] G. Gbur, R.K. Tyson, Journal of Optical Society of America A 25 (2008) 225. [27] V.P. Aksenov, I.V. Izmailov, F.Y. Kanev, B.N. Poizner, Optics Communications 285 (2012) 905. [28] B.J. Uscinski, Journal of Optical Society of America A 2 (1985) 2077. [29] H.T. Eyyuboğlu, Y. Baykal, X. Ji, Applied Physics B-Laser and Optics 98 (2010) 857.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Scintillation analysis of hypergeometric Gaussian beam via phase screen method Halil T. Eyyuboğlu n Çankaya University, Department of Electronic and Communication Engineering, Eskişehir Yolu, 29 km, Yenimahalle, 06810 Ankara, Turkey
art ic l e i nf o
a b s t r a c t
Article history: Received 26 April 2013 Received in revised form 3 July 2013 Accepted 4 July 2013 Available online 18 July 2013
We give a scintillation treatment of hypergeometric Gaussian beams via the use of random phase screens. In particular, we analyse the on-axis, point-like and aperture averaged power scintillation characteristics of this beam that cannot be undertaken easily by analytic means. Within the range of examined source and propagation parameters, our evaluations show that there will be less scintillation, with increasing hollowness at small source sizes and zero topological charge. At larger source sizes or topological charges, this is reversed and decreasing hollowness will reduce scintillation. More or less the same trend is observed for aperture averaging such that at small source sizes and zero topological charge, increased hollowness will result in lower scintillation. At larger source size and topological charges, there will be a transition from the case of smaller values of hollowness giving rise to less scintillation at smaller aperture openings to the case of larger values of hollowness giving rise to less scintillation at larger aperture openings. In general nonzero topological charges will produces more scintillations, both in onaxis and aperture averaged cases. & 2013 Elsevier B.V. All rights reserved.
Keywords: Random phase screen Scintillation Hypergeometric Gaussian beam
1. Introduction Analytic formulation of scintillation induced by turbulent atmosphere requires lengthy derivations, whether Rytov approximation or extended Huygens–Fresnel approach is adopted. This difficulty is alleviated by switching to the random phase screen method, since there finding scintillation is reduced to the simple act of summations and divisions. In the use of random phase screen, there is the added advantage that we can deal with almost any beam type. Finally it is quite easy with random phase screen to extend the single receiver point scintillation calculations to aperture averaged scintillation results, which would again pose enormous difficulties, if it were to be tackled analytically. In this study we have opted for a beam whose scintillation characteristics are difficult to analyse by analytic means and besides have not investigated up to now. The use of random phase screen to model propagation in turbulence is now well established through quite a number of publications. In two of the earlier works, by adopting a varying power law spectra, the effect of inner and outer scales on intensity variance (scintillation), intensity spectra were analysed, covering strong turbulence regimes for plane and spherical waves [1,2]. For different types of inner scales, the intensity variance results of
n
Tel.: +90 312-2331322; fax: +90 312-2331026. E-mail address: [email protected]
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.024
random phase screen computations and analytic calculations were reported to agree within 2% in [3]. Design considerations of a Kolmogorov phase screen based on the accurate approximation of the phase were discussed in [4]. Errors originating from the use of finite grid size, finite transverse plane dimensions and the separation of phase screens along the propagation path were investigated quantitatively in [5] and expressed for plane and spherical waves. In [6], it was found that simulation of scintillation, beam spreading and reflective speckle patterns via phase screen approach produced results in good agreement with analytic predictions and experimental data. Using a Gaussian beam source, comparisons of random phase screen results with the analytic ones were made in [7] for scintillations, beam spreading, beam wander, coherence diameters, variance and autocorrelation of the beam intensity. Due to the great deal of attention devoted to the subject, two books were published recently covering theoretical backgrounds as well as implementation details of numerical simulation of the optical beam propagation [8,9]. The random phase screen method is also used for scintillation performances of specialized beams. In this context, pseudo-Bessel correlated beams were studied in [10], where it was demonstrated that with the choice of appropriate coherence parameter, these beams could offer lower scintillation both in weak and strong turbulence. In another study [11], the irradiance pattern, degree of polarisation and scintillation index of radially polarised vortex beam represented by the lowest order Laguerre Gaussian beam were investigated and the vector vortex beam was found to posses
104
H.T. Eyyuboğlu / Optics Communications 309 (2013) 103–107
lower scintillation than the scalar counterpart. In [12], again the use of vortex beam of elliptical profile was reported to result in lower scintillation by adjusting the ratio of minor axis to major axis to smaller values. Based on the use of the Gaussian Schell model beam in wave optics simulations, the suitability of gamma and lognormal probability density functions were examined against analytic predictions and it was seen that in weak fluctuation regime, the results of wave optics simulations followed the gamma distribution model, but in strong fluctuations case, the results approximated better to the lognormal probability density function [13]. Scintillation properties of the Gaussian Schell model beam was also investigated in [14] along with the beam spreading and beam wander effects, where it was observed that lower beam coherence would lead to reductions in scintillations and relative beam spreading, but would have less impact on beam wander. As an alternative to the existing random phase screen techniques, a new model named “sparse spectrum” has been proposed and claimed to offer substantial reduction in computation times [15]. It has long been known that as the receiver aperture size is enlarged, aperture averaging will occur and then we will obtain reductions in scintillations [16–20]. But the formulation of aperture averaged scintillation poses extreme difficulties, since it requires the evaluation of covariance function. Up to now, it has been formulated only for plane, spherical and Gaussian beams. In [21], the aperture averaged scintillation for partially coherent Gaussian beam was investigated in relation to correlation time of the source phase variations and the integration time of the detector and it was found that scintillation advantages of partially coherent Gaussian beam could be realized only when the ratio of these two parameters was less than a certain threshold. The generation, the free space and turbulence propagation characteristics of hypergeometric Gaussian beams were analysed in [22–24]. Hypergeometric Gaussian beam is considered to be a member of the vortex beam family. In [25–27], it was proposed that the vortex feature of such beams could be used as a type of modulation. In this study, we perform scintillation analysis on hypergeometric Gaussian beams. Here our motivation is to understand the advantages that may be offered by the use of such a beam in optical links. It is well known that scintillation causes a degradation in the probability of error performance of laser communication systems [20]. Therefore any improvement in that direction would be well accepted. The scintillation characteristics of hypergeometric Gaussian beams are explored with this particular aim in mind. The random phase screen method is used, thus allowing both the point-like (on-axis) and aperture averaged scintillations to be evaluated with ease.
For a source field us ðsx ; sy Þ, the Hygens–Fresnel integral delivers the receiver field ur ðr x ; r y ; LÞ as follows: jkexpðjkLÞ 2πL
Z
1 1
2 i þ r y sy
Z
1 1
dsx dsy us ðsx ; sy Þexp
Z ur ðrx ; r y ; LÞ ¼ us ðsx ; sy Þ⊗hðr x ; r y Þ ¼
1 1
Z
1 1
dsx dsy us ðsx ; sy Þhðr x sx ; r y sy Þ
ð2Þ where hðr x sx ; r y sy Þ denotes the collection of terms from Eq. (1) such that h 2 i jkexpðjkLÞ jk ð3Þ exp ðr x sx Þ2 þ r y sy hðr x sx ; r y sy Þ ¼ 2πL 2L The relationship established on the left hand side of Eq. (2) can also be converted into ur ðr x ; r y ; LÞ ¼ F1 fF½us ðsx ; sy ÞF½hðr x ; r y Þg ¼ F1 fF½us ðsx ; sy ÞHðf x ; f y Þg
ð4Þ
where F is the Fourier transform operator and
2jπ 2 L 2 2 fx þ fy H f x ; f y ¼ F h r x ; r y ¼ expðjkLÞexp k h i 2 2 ¼ expðjkLÞexp jπλL f x þ f y
ð5Þ
From Eq. (4), we conclude that the receiver field can be found by taking the Fourier transform of source field us sx ; sy , then multiplying this by the Fourier transform of the transfer function h r x ; r y , eventually inverse transforming this product. So all together, this is a three step operation. Although Eq. (2) demonstrates that the same could be achieved in one step, i.e., with a single convolution operation, in practical computation, it is seen that for the level of resolution required, the three step Fourier transform method, based on FFT functions available in most software packages, is much faster than the one step convolution implementation. It is known that a propagating beam will expand due to diffractive effects. To compensate for this, a scaling factor expressing the ratio between receiver and source transverse plane coordinates must be embedded into Eq. (3) [8]. The above development covers propagation in free space, i.e., a medium free of turbulence. To include the effects of turbulence, the entire propagation length of L is split up into N shorter intervals of ΔL ¼ L=N and a thin phase screen plane is placed at each interval. These phase screen planes are then used to create random phase distribution over the transverse plane. In this manner, the field at the nth plane is linked to the one at the ðn1Þth plane as ur r x ; r y ; nΔL ¼ F1 F us r x ; r y ; ðn1ÞΔL exp jϕ r x ; r y H f x; f y ð6Þ
2. Description of the random phase screen method
ur ðr x ; r y ; LÞ ¼
operator ⊗, as shown below:
jk h ðr x sx Þ2 2L ð1Þ
where k is the wave number and related to the source wavelength λ as k ¼ 2π=λ, Lis the on- axis receiver plane location, r x ; r y and sx ; sy are the transverse coordinates of the source and receiver sides respectively. Eq. (1) can be interpreted as a two dimensional convolution integral and can be written using the convolution
The random phase distribution ϕðr x ; r y Þ is retrieved from one of the phase power spectral density functions adopted. In this study, we have selected von-Karman spectrum, thus the phase power spectral density function will be h i 2 2 2 exp 1:1265l0 f x þ f y 2 2 11=3 Φ f x ; f y ¼ 0:0036k LC n L0 ð7Þ h i11=6 2 2 L20 f x þ f y þ 1 where L0 ; l0 denote the outer and inner scales of turbulence, C 2n is the refractive index structure constant. With the utilisation of Eq. (7), the field given in Eq. (6) will incorporate the turbulence induced effects of this particular spectrum. Here the choice of vonKarman spectrum is justifiable, since we cannot go to the limit of zero inner scale due to finite grid spacing used [7].
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Scintillation index for a particular receiver plane location of r x ; r y is defined as D E I 2r r x ; r y ; L 2 b rx ; ry ; L ¼ ð8Þ 2 1 Ir rx ; ry ; L The receiver plane intensity, I r r x ; r y ; L ¼ ur r x ; r y ; L unr r x ; r y ; L is obtained from Eq. (6), by going through all free space propagation intervals and the phase screens which model the turbulence via the random phase ϕ r x ; r y and finally reaching the setting of n ¼ N. The average receiver intensity, I r r x ; r y ; L is generated by summing the individually realized intensity terms, I r r x ; r y ; L and then D dividing E the result by the number of realisations. For the on the other hand, the square of the individually I 2r r x ; r y ; L realized intensities are summed with again the averaging operation being applied at the end. If an analytic derivation was to be undertaken to arrive at the scintillation index, lengthy derivations would emerge regardless of Rytov or extended Huygens–Fresnel approach being chosen. With the use of random phase screens, the procedure is reduced to above mentioned summations and divisions. Another important parameter in the measure of scintillation is the aperture averaged scintillation, which can be defined over a circular receiver area of radius Ra as follows: D E Z 0:5pffiffiπ Ra Z 0:5pffiffiπ Ra P r 2 ðLÞ 2 b ðLÞ ¼ 1 ; P ð L Þ ¼ I r r x ; r y ; L dr x dr y r 2 pffiffi pffiffi 0:5 π Ra 0:5 π Ra P r ðL Þ ð9Þ It should be stressed that the random phase screen method is particularly instrumental in the calculation of aperture averaged scintillation as well, since such scintillations are hardly available analytically apart from simple cases of plane, spherical waves and Gaussian beams [16–21].
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For the graphical displays, we fix the operating wavelength and the refractive index structure constant as λ ¼ 1:55 μm; C 2n ¼ 1015 m2=3 . Fig. 1 illustrates the variation of on-axis scintillation of hypergeometric Gaussian beam against propagation lengths up to L ¼ 5:5 kmfor three values of p, namely 0,2,5 at αs ¼ 1 cm; n ¼ 0. Here, because of the fluctuating nature of the random phase screen results, polynomial fitted curves of fourth degree are also plotted to better reveal the trend against the changing values of p. This way, it is seen from Fig. 1 that scintillation reductions are obtained by lowering the value of p, when n ¼ 0. This was also confirmed by observing the scintillation curves at values of p other than those plotted in Fig. 1. It is worth noting that in Eq. (10), the settings of n ¼ 0; p ¼ 0 yield the fundamental Gaussian beam. In this respect, there is close agreement between the curve of p ¼ 0 in Fig. 1 and the analytic results of [29]. Next we go to a higher value of n. In this manner, Fig. 2 exhibits the on-axis scintillation variations at αs ¼ 1 cm; n ¼ 2, against the same propagation range of Fig. 1. Here rather high scintillations are encountered. Besides at this value of n, scintillations seem to rise with larger p values, i.e., the opposite of that observed in Fig. 1. This was seen to the case for almost all n 4 0. In Fig. 3, we examine the variation of on-axis scintillation index against Gaussian source size at n ¼ 0; L ¼ 3 km for the selected values of p ¼ 0; 2; 5. The curves of Fig. 3 reflect the typical behaviour of scintillation index variation against source size reported in the literature [29]. At source sizes up to roughly αs ¼ 2 cm, the curves of p ¼ 0; 2; 5 appear to follow the trend in Fig. 1. But this trend changes from there onwards and at larger source sizes, scintillation seems to be rising more rapidly particularly with increasing p. The topological charge is changed to n ¼ 2 in Fig. 4, where similar to Fig. 2, high levels of scintillations are present.
3. Numeric results and discussions In this section, we present numeric results obtained for scintillations of a hypergeometric Gaussian beam whose source plane field is given by 2 s us ðs; θÞ ¼ sp exp 2 jnθ ð10Þ αs where s2 ¼ s2x þ s2y , thus s; θ constitute the radial coordinates of the source plane. p is the hollowness parameter, αs is the Gaussian source size, n is the topological charge. It is clear from Eq. (10) that on the source plane, hypergeometric beam will resemble a dark hollow beam for p 40, while this hollowness will also be controlled by n after propagation [24]. In our scintillation runs, square source and receiver planes with grid size of 512 512 are used. The propagation length L is divided into 20 intervals. This means that we have employed multiple random phase screens rather than a single one. This way, each propagation interval has a much reduced length, so that the grid (sample) size on the plane of the screens could be kept at reasonable numbers, preventing excessive rises in computation time [8]. Secondly the choice of multiple screens is appropriate in the sense that such a choice acknowledges the fact that random phase screens are the physical counterpart of the concept of multiple scatters, utilised in the solution of parabolic fourth moment equation [28]. Averaging is taken over 500 realisations. The on-axis scintillation values based on the evaluation of Eq. (8), are obtained over a 6 6 square grid area. In all relevant cases, this corresponded to a point-like receiver aperture area, by ensuring that the radius of the equivalent circular pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi area of the on-axis 6 6 square grid remained below 0:5λL=π [20].
Fig. 1. Variations of on-axis scintillation against propagation length at αs ¼ 1 cm; n ¼ 0 for different values of p. Fitted curves are in colour and marked with circles.
Fig. 2. Variations of on-axis scintillation against propagation length at αs ¼ 1 cm; n ¼ 2 for different values of p. Fitted curves are in colour and marked with circles.
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Fig. 3. Variations of on-axis scintillation against source size at n ¼ 0; L ¼ 3 km for different values of p. Fitted curves are in colour and marked with dots.
Fig. 4. Variations of on-axis scintillation against source size at n ¼ 2; L ¼ 3 km for different values of p. Fitted curves are in colour and marked with circles.
Additionally, Fig. 4 reveals that at the value of n ¼ 2, the rise of scintillation with rising values of p is confined to small source sizes. As seen from Fig. 4, this trend is reversed at large source sizes and higher values of p start to yield less scintillation. The next four graphs concern aperture averaged scintillations. We start with the settings of αs ¼ 1 cm; n ¼ 0; L ¼ 3 km in Fig. 5 and explore the variation of aperture averaged scintillation against the aperture radius Ra as given in Eq. (9) at different p. As expected, aperture averaged scintillation falls rapidly with increases in aperture radius. The ordering of curves in Fig. 5 in terms of p appears to be the same as Fig. 1, that is, the larger p values result in lower aperture averaged scintillations. Fig. 6 shows the situation of aperture averaged scintillation after switching to n ¼ 3. Here it is seen that at small aperture radii, aperture averaged scintillation is higher for larger values of p, however, as the aperture radius is increased, this appearance turns the other way around and smaller values of p begin to offer higher scintillation. Particularly the curves of p ¼ 0; 1; 2 manage this turning by slightly bending upwards around the middle aperture radius sizes. Finally two more graphs are exhibited for aperture averaged scintillations. In Fig. 7, we have chosen the fixed settings as αs ¼ 5 cm; n ¼ 0; L ¼ 5 km. Here contrary to the case of Fig. 5 and in line with Fig. 3, the rule of smaller p values possessing lower scintillation applies, with the differentiation between the beams being somewhat less. Fig. 8 illustrates the situation after setting n ¼ 5, where it is seen that in comparison to the turnings of Fig. 6, the transition to the case of larger p values offering less scintillation, takes place at smaller aperture radii. When compared to Fig. 7, the merging of all curves towards the same low aperture averaged scintillation levels lags behind to some extent in Fig. 8. In general however, for all beam and propagation parameters examined, we witness, in Figs. 5–8, the clear advantage of enlarging the aperture radius in order to acquire low scintillations.
Fig. 5. Variations of aperture averaged scintillation against aperture radius at αs ¼ 1 cm; n ¼ 0; L ¼ 3 km for different values of p.
Fig. 6. Variations of aperture averaged scintillation against aperture radius at αs ¼ 1 cm; n ¼ 3; L ¼ 3 km for different values of p.
Fig. 7. Variations of aperture averaged scintillation against aperture radius at αs ¼ 5 cm; n ¼ 0; L ¼ 5 km for different values of p.
Fig. 8. Variations of aperture averaged scintillation against aperture radius at αs ¼ 5 cm; n ¼ 5; L ¼ 5 km for different values of p.
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4. Conclusion We have evaluated the scintillation characteristics of hypergeometric Gaussian beams using the random phase screen method. Our findings indicate that, for on-axis scintillation, larger values of p, the hollowness parameter, result in lower scintillation at smaller Gaussian source sizes and zero topological charge, i.e., n ¼ 0. This situation is reversed however for n 4 0 or at larger Gaussian sources sizes. There, lower on-axis scintillation is obtained at smaller values of p. When aperture averaged scintillation is considered, it is found that at small source sizes and zero topological charge, larger values of p will lead to lower scintillations. For n 4 0 however, at small aperture sizes, i.e., nearer to point-like receivers, scintillation is lower at smaller values of p. A crossover occurs towards the larger aperture sizes, where scintillation becomes lower at larger p values. At larger source sizes, longer propagation distances and at higher topological charges, this crossover is brought to smaller aperture sizes. In general, both on-axis and aperture averaged scintillations are seen to act in a similar manner with changes in beam and propagation parameters. Another generalisation is that topological charges larger than zero will tend to result in higher scintillations. References [1] J.M. Martin, S.M. Flatte, Applied Optics 27 (1988) 2111. [2] J.M. Martin, S.M. Flatte, Journal of the Optical Society of America A: Optics, Image Science, and Vision 7 (1990) 838. [3] C.A. Davis, D.L. Walters, Applied Optics 33 (1994) 8406.
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