Bit error rate analysis of Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams with the help of random phase screens Halil T. Eyyuboğlu Electronics and Communications Department, Çankaya University, Eskişehir Yolu 29. km, Yenimahalle, 06810 Ankara, Turkey ([email protected]) Received 27 February 2014; revised 2 May 2014; accepted 7 May 2014; posted 9 May 2014 (Doc. ID 207260); published 10 June 2014

Using the random phase screen approach, we carry out a simulation analysis of the probability of error performance of Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams. In our scenario, these beams are intensity-modulated by the randomly generated binary symbols of an electrical message signal and then launched from the transmitter plane in equal powers. They propagate through a turbulent atmosphere modeled by a series of random phase screens. Upon arriving at the receiver plane, detection is performed in a circuitry consisting of a pin photodiode and a matched filter. The symbols detected are compared with the transmitted ones, errors are counted, and from there the probability of error is evaluated numerically. Within the range of source and propagation parameters tested, the lowest probability of error is obtained for the annular Gaussian beam. Our investigation reveals that there is hardly any difference between the aperture-averaged scintillations of the beams used, and the distinctive advantage of the annular Gaussian beam lies in the fact that the receiver aperture captures the maximum amount of power when this particular beam is launched from the transmitter plane. © 2014 Optical Society of America OCIS codes: (010.0010) Atmospheric and oceanic optics; (140.0140) Lasers and laser optics; (010.1330) Atmospheric turbulence; (010.3310) Laser beam transmission; (140.3295) Laser beam characterization. http://dx.doi.org/10.1364/AO.53.003758

1. Introduction

An important performance criterion of an atmospheric optical link is the probability of error performance, which is also named as bit error rate (BER) in the case of binary symbol transmission. In addition to thermal noise and shot noise, this performance parameter is also affected by scintillations. An array detector configuration was proposed in [1] to reduce such scintillation effects. A BER performance analysis for short-range optical links was carried out in [2]. In another study [3], for all turbulence regimes, on and off keying and amplitude-shift keying (ASK) modulation types, horizontal, and vertical links, it was demonstrated that removal of Zernike modes by the use of adaptive optics would help reduce 1559-128X/14/173758-06$15.00/0 © 2014 Optical Society of America 3758

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scintillation, thus bringing performance improvement. Some of these theoretical predictions were later confirmed with a laboratory experimental setup [4]. The positive influence of reduced source beam spatial coherence on the BER performance of an optical link was investigated in [5]. Performance evaluations of spatial diversity receivers were made in [6] and it was shown that BER improvement was possible by employing an increased number of collecting lenses. An extensive BER formulation covering the different probability density function (pdf) models of irradiance fluctuations, fade analysis, and the related graphical displays are offered in [7]. Using a lognormal Rician pdf of irradiance fluctuations, a BER analysis for a binary phase shift keying modulation scheme was conducted in [8]. For an optical link that incorporates the transmit diversity, variations of BER were explored in [9] against selected turbulence levels, normalized jitters, and normalized

beam widths. The impact of strong turbulence on BER was presented in [10] in terms of system state parameters. In another work [11], the performance of a subcarrier intensity-modulated free-space optical link operating over a K-distributed turbulence channel was considered; BER, outage probability, and channel capacity expressions were derived. For non-Kolmogorov turbulence, the performance of a phase compensated coherent free-space optical link was evaluated against variations in compensation modes, normalized receiver diameter, exponent parameter of the spectrum, and outer scale of turbulence [12]. The random phase screen concept is quite a handy model to the study of propagation of beams in turbulence. There are two main benefits of this approach. First, we can treat almost any source beam and easily create various propagation conditions with this model. Second, it is one step closer to physical reality. Being almost source beam and propagation environment independent, the random phase screen setup will directly alleviate the difficulties faced in the cumbersome analytic derivations. The precautions that have to be taken in its implementation, application examples, evaluation of several beam statistics, and comparison of outcomes with analytic results are detailed in quite a number of literature sources [13–16]. Up to now, the use of random phase screens for beams other than fundamental Gaussian beams has been quite limited. Recently, a random phase screen setup was used to evaluate the scintillations of various beams [17–20], where in [20] close agreement between analytic and random phase screen results was also demonstrated. Based on the confidence of these previous results, here in this paper, we attempt to investigate the BER behavior of the same beams by converting the random phase screen setup into a fully equipped free-space optical link. This is done by incorporating an electrical message signal that modulates the light source in a binary ASK mode. On the receiver side, a pin photodiode is employed for demodulation. Then matched filtering is applied and threshold detection is made; finally errors are counted to arrive at a probability of error or BER. Here, our motivation is to explore and reveal if there are any advantages to be gained from the use of beams other than a fundamental Gaussian beam. Our present findings indicate that there is such a possibility and this is achieved by using an annular Gaussian beam. In communications, a lot of effort is devoted to acquire BER improvements by adjusting the source parameters suitably [21,22]. Particularly, BER improvements that are obtained by keeping the source power constant are of extreme importance. This is exactly the goal achieved in the present paper via the use of an annular Gaussian beam. It should be emphasized additionally that in BER analysis, the existing literature has not gone much beyond fundamental Gaussian beams and the related theoretical formulation. Hence, comparison between the BER performances of different beam types has not been an

issue. Furthermore, the use of random phase screens to attain BER results is, to the best of our knowledge, new and original. The present work is, therefore, expected to simulate further research in similar areas. 2. Formulation for BER Analysis

We assume a receiver employing a pin photodiode. The electrical current output from such a diode for an optical power incidence of Pr is I r  RPr ;

(1)

where R is the known as the responsivity of the diode, being measured in amperes/watt. The current delivered flows through a load resistor, RL , and constitutes the message signal received. In this process, including the circuitry that is to come after the load resistor, two noise currents are created whose variances can be expressed as [23] σ 2SN  2qI r  I d Bw ;

σ 2TN  4kB T K Bw F N ∕RL ; (2)

where the first variance, i.e., σ 2SN , is due to shot noise, whereas the second variance, i.e., σ 2TN , is that of thermal noise. It is known that thermal noise has a Gaussian pdf, whereas the pdf of shot noise can be approximated to Gaussian [23]. In Eq. (2), q is electron charge, I d is the dark current of the pin diode (negligible compared with I r ), Bw denotes the (electrical) bandwidth of the receiver, kB is Boltzmann constant, T K is the working temperature of the load resistor in Kelvin degrees, and F N is the overall noise figure of the receiver unit. From Eqs. (1) and (2), it is possible to define the signal-to-noise ratio (SNR) in terms of normalized current quantities as Ir  SNR p 2qI r  I d Bw  4kB T K Bw F N ∕RL  Ir :  q 2 σ SN  σ 2TN

(3)

Equation (3) refers to an instantaneous SNR, but I r fluctuates due to the randomness caused by the atmospheric turbulence present in the propagation medium between the transmitter and the receiver. This way, we can speak about an average SNR such that hI r i hSNRi  p 2qhI r i  I d Bw  4kB T K Bw F N ∕RL hI r i  q : σ 2SN  σ 2TN

(4)

From Eq. (1), it is clear that the fluctuations in I r will stem from the incident power fluctuations. Such variations are described by gamma–gamma 10 June 2014 / Vol. 53, No. 17 / APPLIED OPTICS

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us;θ

2 X

Ai exp−s2 ∕α2si expcos θsin θDi s; (5)

i1

where s and θ are the source plane radial coordinates, αsi is the source size, Ai is the amplitude factor, and Di is the displacement parameter. By suitable adjustment of αsi , Ai , and Di , Eq. (5) will yield fundamental Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams as detailed elsewhere [24]. In this study, our aim is to assess the probability of error performance of these beams under equal p source power condition. To this end, we set αs  2 cm; this gives a source power of Ps  0.1π  0.314 mW. The beams will deliver this same source power with the settings listed in Table 1. 3. Simulation Arrangements, Results, and Discussion

The simulation environment is depicted in Fig. 1. As shown in the figure, the transmitter consists of a laser-light source intensity-modulated by an electrical message signal. The beam emitted from the light source is then launched into a turbulent atmosphere where it experiences phase fluctuations, which in turn give rise to intensity fluctuations. The propagation path of the atmospheric turbulence is modeled by a series of random phase screens, whose properties are derived from the phase power spectral density. On the receiver plane, with the help of a pin photodiode, these fluctuations as well as the modulation impressed upon the laser source at the transmitter are translated into electrical current variations [see Eq. (1)] and the resulting current flows through a load resistor, RL . As stated above, in this process, shot and thermal noise are added to this current. Eventually, from Table 1. Parameter Settings of Gaussian, Annular Gaussian, Cos Gaussian, and Cosh Gaussian Beams for Equal Source Power Condition

Beam Name Parameter Gaussian A1 A2 αs1 αs2 D1 D2

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1 p0 2 cm 0 0

Annular Gaussian

Cos Gaussian

Cosh Gaussian

1 0.6 0.4 p−0.4486  p0.6  p0.4  2∕0.6 p cm p2  cm p2  cm 1.5 2 cm 2 cm 2 cm 0 0.687 j cm−1 0.614 cm−1 0 −0.687 j cm−1 −0.614 cm−1

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Receiver aperture

Ls = 10 cm

L r=

pin photodiode

m 58 c

Detector Matched filter Decision device

m 10 c

Transmitter plane

RL

Ls=

Random phase screens

Load resistor

Ra = 2.56 cm

Detected symbols

4 km

Turbulent atmosphere

Lr = 58 cm

Laser light source

Electrical message signal

or lognormal pdfs. Thus, from Eq. (3), we arrive at a probability error formulation by integrating over the entire range of selected pdfs [7]. Here, the effects of atmospheric fluctuations are incorporated into our model by the random phase screen setup; consequently, in the hSNRi given by Eq. (4). In this manner, the probability of error, Pe (BER), is simply computed by counting the number of errors and dividing it by the total number of symbols transmitted, thus skipping the lengthy integration stages. The field of the source beams used in the simulations is formulated as

Receiver plane

Comparison Error counting

Pe (BER)

Fig. 1. Pictorial representation of the simulation environment.

the voltage induced across the load resistor, a detector circuitry composed of a matched filter and a decision (threshold) device makes a decision as to which symbol was transmitted. This threshold is set to the mean of symbols received. The probability of error, Pe BER, is computed by comparing the symbols detected with the ones transmitted. For simulations, the following arrangements were made: (a) Two-level electrical message signaling was used, where symbol duration was set to 1 ns. This way, the bandwidth requirement on the receiver side was approximated to Bw  1 GHz. The electrical message signal intensity-modulated the laser-light source in the ASK mode. The low level of this modulation was set to 10% of its high level with the aim of avoiding the nonlinear region of the laser diode [23]. (b) The side lengths of the transverse transmitter and receiver planes, i.e., Ls  10 cm, Lr  58 cm, were chosen such that on each plane, the footprints of the beams were well covered. Both planes and the intermediate random phase screens (20 in total) had fixed grid sizes of 512 × 512, with the growing grid spacing in proportion to increasing distance. The propagation distance between the transmitter and the receiver was held constant at 4 km; the wavelength of operation was fixed to 1.55 μm. Responsivity was taken as R  0.5, while the load resistance, noise figure, and working temperature were, respectively, set to RL  50 Ω, F N  2, T K  293 K. (c) For each count of BER, a total of 105 randomly generated (electrical) symbols were transmitted. From the literature [25], it is known that with such a number of symbols, we can estimate the probability of error values down to Pe  10−4 within a confidence interval of 90%. To extend the probability of error count down to Pe  10−6 , the symbols received were repeated 100 times. Such an act is justifiable, since the coherence length of the atmosphere is much longer than the selected symbol rate [9]. Thus, on the receiver side, following the pin photodiode, 107 Gaussian distributed noise samples, whose variances were as defined in Eq. (2), were added to the received symbols. In the simulation environment in Fig. 1, the level of atmospheric turbulence is varied by changing the structure constant in the range C2n :0 → 10−13 m−2∕3 . The corresponding Pe (BER) results are displayed

-1

10

0

Aperture averaged scintillation

10

-2

Pe (BER)

10

cos Gaussian beam

-3

10

annular Gaussian beam -4

10

-5

10

Gaussian beam

cosh Gaussian beam cos Gaussian beam -1

10

annular Gaussian beam

-2

Gaussian beam

10

cosh Gaussian beam -6

10

-3

-16

-15

10

-14

10

10

10

-13

10

-16

-15

10

C 2 in m -2/3 (structure constant)

-14

10

10

-13

10

2 -2/3 (structure constant) Cn in m

n

Fig. 2. BER plots of equal-source-power Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams against variations in structure constant.

Fig. 4. Aperture-averaged scintillation plots of equal-sourcepower Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams against variations in structure constant.

in Fig. 2. For these results, a fixed receiver aperture opening with a radius of Ra  2.56 cm was used. As shown in Fig. 2, under the circumstances arranged, the best (lowest) probability of error performance is offered by the annular Gaussian beam, which is closely followed by the cosh Gaussian beam; then comes the Gaussian beam and finally, the worst performance is demonstrated by the cos Gaussian beam. To understand the underlying mechanism leading to these results, we further examine the variations in the other parameters. Figure 3 shows the hSNRi variations against the same range of structure constant values for the four beams in question. Here, it can be observed that the order of hSNRi curves for all the beams is perfectly consistent with the ones encountered in Fig. 2. That is, the beams exhibiting high hSNRi values in Fig. 3 will yield low Pe values in Fig. 2, whereas the opposite will occur for beams with a low hSNRi. In this context, the annular Gaussian beam has the highest hSNRi curve in Fig. 3, and thus, the lowest Pe curve in Fig. 2. It is clear from Eqs. (1), (2), and (4) that hSNRi is governed by the amount of power received and the associated fluctuations, i.e., scintillations. To assess which one has a distinctive role, variations of aperture-averaged scintillation and variations of received power against changes in structure constant values are plotted in Figs. 4 and 5, respectively. It can be seen in Fig. 4 that the annular Gaussian beam seems to have the lowest apertureaveraged scintillation, particularly at small structure

constant values. But when considered as a whole, the aperture-averaged scintillation differences between the beams are found to be minimal. On the other hand, Fig. 5 reveals that the amounts of power captured for the different beams vary substantially. There it can be seen that the greatest amount of power is captured if the annular Gaussian beam is launched from the transmitter plane. In Fig. 5, the ordering of the beams in terms of received power is in perfect agreement with the trends observed in Figs. 2 and 3. Finally, in Fig. 6, we explore the variation of BER against changes in the receiver aperture radius when the structure constant is kept fixed at C2n  10−15 m−2∕3 . Here, on the horizontal axis, we have marked the location of the aperture radius used in the simulation runs in Figs. 2–5, i.e., Ra  2.56 cm. Figure 6 demonstrates that the BER advantage of the annular Gaussian beam over the other beams will continue to exist at even larger aperture radii. In short, we can say that the BER performance superiority of the annular Gaussian beam arises not because of the different scintillation characteristics of the beams, but is rather related to the better power concentration capability of this beam on-axis during propagation. To verify the above findings to some extent, we have made comparisons of the results obtained here with those available in the literature. In [26] and [27], the BER performance of similar beams was investigated representing the atmospheric turbulence

annular Gaussian beam

0.055

< P > (received power in mW)

12

cosh Gaussian beam < SNR > in dB

10

8

Gaussian beam 6

cos Gaussian beam

r

4

2

annular Gaussian beam

0.05 0.045 0.04 0.035

cosh Gaussian beam

0.03 0.025 0.02 0.015

Gaussian beam

0.01

cos Gaussian beam 0.005 -16

10

-15

10

-14

10

-13

10

Cn2 in m -2/3 (structure constant)

Fig. 3. hSNRi plots of equal-source-power Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams against variations in structure constant.

-16

-15

10

10

-14

10

-13

10

C 2 in m -2/3 (structure constant) n

Fig. 5. Received power plots of equal-source-power Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams against variations in structure constant. 10 June 2014 / Vol. 53, No. 17 / APPLIED OPTICS

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2

R = 2.56 cm

-1

Cn = 10

a

10

-15

m-2/3

Gaussian beam -2

annular Gaussian beam

Pe (BER)

10

-3

10

cosh Gaussian beam

-4

10

cos Gaussian beam

-5

10

1

2

3

4

5

6

7

8

9

10

Aperture radius (R in cm) a

Fig. 6. BER plots of equal-source-power Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams against variations in receiver aperture radius.

by a lognormal pdf and using on the receiver side an on-axis scintillation index and a point detector. In the cited works, the BER performance curves were plotted for Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams of the same Gaussian source size, thus, for unequal source beam powers. As described above, in the present study, equal source beam powers and finite receiver apertures are considered. Despite these differences and the differences in propagation distance, source sizes, and displacement parameters, it is worth noting that our results displayed in Figs. 2 and 3 agree quite well with their the closest counterparts in [26] and [27], namely Fig. 4 in [26] and Fig. 1(b) in [27]. This agreement covers the ordering of BER performance curves as well as the corresponding BER values for the common range of SNRs. This way, both from our results and from the figures mentioned in the cited works, we see that an annular Gaussian has the lowest BER performance; then come cosh Gaussian and Gaussian beams, and finally, a cos Gaussian has the worst BER behavior. On the other hand, at C2n  10−14 m−2∕3 , we read from Fig. 2 that the probability of error for the annular Gaussian beam is Pe  0.133. According to Fig. 3, C2n  10−14 m−2∕3 corresponds to hSNRi ≈ 10 dB. In Fig. 4 of [26] and Fig. 1(b) of [27], for the SNR setting of 10 dB, we read the probability of error as Pe ≈ 0.08. Here, the discrepancy is to be attributed to the use of a shorter propagation distance in [26] and [27], i.e., L  3 km, and a longer propagation distance in the present study, i.e., L  4 km. 4. Conclusion

Through a random phase screen setup equipped with the appropriate input and output sections, we have evaluated the probability of error performance of an optical communication link by employing different equal-power source beams such as Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams. For the range of selected source and propagation parameters, our results clearly indicate that the annular Gaussian beam has the lowest probability of error performance; this is then followed by the cosh Gaussian beam and the Gaussian beam. The worst performance is exhibited by the cos Gaussian beam. 3762

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Our analysis has shown that there are minute differences between the aperture-averaged scintillations values of these beams. In this sense, the performance advantage and the distinction of the annular Gaussian are based heavily on the amount of power captured by the receiver aperture. This means, given the equal-power source beams of Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams listed in Table 1, the power captured by the receiver aperture is maximized when an annular Gaussian beam is launched from the transmitter plane and this maximization leads to the best probability of error performance among the beams examined. At the moment, it is difficult to comment whether the present findings will apply to source and propagation settings other than the ones utilized in this study, simply because such results are derived from excessively time-consuming simulation runs. We stipulate to undertake such additional studies in future. It is well in order to point out that use of different beam types and probability of error performance assessment taking into account the aspects of scintillation as well as captured power is a subject not well covered in the existing literature. Particularly bearing in mind the amount of BER improvement brought about by the use of a beam other than a fundamental Gaussian beam, we will endeavor to check the practicality of this result in our currently running free-space optical link project at the nearest opportunity. References 1. A. Chaman-Motlagh, V. Ahmadi, and Z. Ghassemlooy, “A modified model of the atmospheric effects on the performance of FSO links employing single and multiple receivers,” J. Mod. Opt. 57, 37–42 (2010). 2. B. I. Erkmen and J. H. Shapiro, “Performance analysis for near-field atmospheric optical communications,” in Proceedings of Global Telecommunications Conference (GLOBECOM) (IEEE, 2004), pp. 318–324. 3. R. K. Tyson, “Bit-error rate for free-space adaptive optics laser communications,” J. Opt. Soc. Am. A 19, 753–758 (2002). 4. R. K. Tyson, D. E. Canning, and J. S. Tharp, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005). 5. J. C. Ricklin and F. M. Davidson, “Atmospheric optical communication with a Gaussian Schell beam,” J. Opt. Soc. Am. A 20, 856–866 (2003). 6. L. C. Andrews and R. L. Phillips, “Free space optical communication link and atmospheric effects: single aperture and arrays,” Proc. SPIE 5338, 265–275 (2005). 7. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Chap. 5. 8. F. Yang and J. Cheng, “Coherent free-space optical communications in lognormal-Rician turbulence,” IEEE Commun. Lett. 16, 1872–1875 (2012). 9. A. Garcia-Zambrana, B. Castillo-Vazquez, and C. CastilloVazquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma–gamma atmospheric turbulence channels with pointing errors,” Opt. Express 20, 2096–2109 (2012). 10. G. K. Rodrigues, V. G. A. Carneiro, A. R. da Cruz, and M. T. M. R. Giraldi, “Evaluation of the strong turbulence impact over free-space optical links,” Opt. Commun. 305, 42–47 (2013). 11. K. Prabu, S. Bose, and D. S. Kumar, “BPSK based subcarrier intensity modulated free space optical system in combined

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18. W. Cheng, J. H. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17, 17829–17836 (2009). 19. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19, 26444–26450 (2011). 20. H. T. Eyyuboğlu, “Estimation of aperture averaged scintillations in weak turbulence regime for annular, sinusoidal and hyperbolic Gaussian beams using random phase screen,” Opt. Laser Technol. 52, 96–102 (2013). 21. B. Sklar, Digital Communications Fundamentals and Applications (Prentice-Hall, 2002), Chap. 5. 22. J. G. Proakis and M. Salehi, Fundamentals of Communication Systems (Pearson, 2005), Chaps. 8, 9. 23. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002), Chap. 4. 24. H. T. Eyyuboğlu, “Annular cosh and cos Gaussian beams in strong turbulence,” Appl. Phys. B 103, 763–769 (2011). 25. M. C. Jeruchim, “Techniques for estimating the bit error rate in the simulation of digital communication systems,” IEEE J. Sel. Areas Commun. 2, 153–170 (1984). 26. S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Bit error rates for general beams,” Appl. Opt. 47, 5971–5975 (2008). 27. S. A. Arpali and Y. Baykal, “Bit error rates for focused general-type beams,” PIERS Online 5, 633–636 (2009).

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Bit error rate analysis of Gaussian, annular Gaussian, cos Gaussian, and cosh Gaussian beams with the help of random phase screens.

Using the random phase screen approach, we carry out a simulation analysis of the probability of error performance of Gaussian, annular Gaussian, cos ...
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