Bit error rates for general beams

Serap Altay Arpali,* Halil T. Eyyuboğlu, and Yahya BaykalÇankaya University, Electronic and Communication Engineering Department, Öğretmenler Cad.

No: 14 Yüzüncüyıl 06530 Balgat Ankara, Turkey

*Corresponding author: saltay@cankaya.edu.tr

Received 2 June 2008; revised 7 October 2008; accepted 11 October 2008;posted 14 October 2008 (Doc. ID 96942); published 3 November 2008

In order to analyze the effect of beam type on free space optical communication systems, bit error rate(BER) values versus signal-to-noise ratio (SNR) are calculated for zero order and higher order generalbeam types, namely for Gaussian, cos-Gaussian, cosh-Gaussian, and annular beams. BER analysis isbased on optical scintillation using log-normal distribution for the intensity, which is valid in weak atmo-spheric turbulence. BERs for these beams are plotted under variations of propagation length, source size,wavelength of operation, and order of the beam. According to our graphical outputs, at small source sizesand long propagation distances, the smallest BER value is obtained for the annular beam. On the otherhand, at large source size and small propagation distance, the smallest BER value is obtained for the cos-Gaussian beam. Moreover, our study of the order of the beam shows that higher order beams have lowerBER values than the zero order beams at longer propagation distances. But this drop compared with theorder seems to be incremental. © 2008 Optical Society of America

OCIS codes: 200.2605, 010.1300, 010.1330, 290.5930.

1. Introduction

While an optical wave propagates through the atmo-sphere, turbulence causes intensity fluctuations inthe optical wave. The performance of laser communi-cation systems is degraded by these fluctuations. Fora laser communication system, after choosing a cer-tain beam and a communication range, link perfor-mance is characterized by its bit error rate (BER),which is related to SNR [1]. Research on the link per-formance is focused on the probability of error, whichin the case of bit modulated transmission is abbre-viated as BER [2–7]. Tyson and Canning measuredthe BER in a laser communication system andshowed that improved performance could be ob-tained with the implementation of low-order adap-tive optics in a free space link [8]. Ricklin andDavidson investigated average BER for a GaussianSchell beam in weak turbulence [9], demonstratingthat average BER in free space optic system couldbe reduced by the use of a partially coherent trans-mitted signal beam [10]. Korotkova et al. developed a

model for the calculation of the SNR and BER of thecommunication link partially coherent Gaussianbeamwith diffuser at the transmitter and slow detec-tion system in weak and strong atmospheric turbu-lence [11]. Vetelino et al. worked on fade statisticsand aperture averaging for Gaussian beam obtainedfrom experimental data and compared to the resultstheoretical predictions based on the log-normal andgamma-gamma distributions of intensity [12]. To ourknowledge the BER of higher order beam types hasnot been so far investigated in the literature. In thisstudy we investigate the BER for zero order andhigher order beam types in weak atmospheric turbu-lence, utilizing the scintillation index of the higherorder general beam formula developed earlier [13].We analyze the BER trends for zero order and higherorder Gaussian, cos-Gaussian, cosh-Gaussian, andannular beams against SNR variations using thebeam source and propagation parameters and the or-der of beams. The log-normal distribution for the in-tensity and on–off keying modulation in a directdetection optical receiver are employed in the calcu-lation of BER. In our previous study [14], the inves-tigated source beam types of this study are clearlydefined and illustrated. In Ref. [14], a table is

0003-6935/08/325971-05$15.00/0© 2008 Optical Society of America

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provided that shows how to set the source para-meters to generate the intensity profiles of allthe beam types such as Gaussian, cos-Gaussian,cosh-Gaussian, and annular and their higher ordercounterparts.

2. Signal-to-Noise Ratio

In the absence of atmospheric turbulence, signal-to-noise ratio (SNR0) is defined by the ratio of the de-tector signal current is to the standard deviation ofthe detector noise σN [1],

SNR0 ¼ isσN

: ð1Þ

In the presence of atmospheric turbulence, for pointdetectors, the mean signal-to-noise ratio hSNRi in-corporates SNR0 as well as intensity fluctuationsdue to atmospheric turbulence and will be givenby [1]

hSNRi ¼ SNR0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPSOhPSi þ σ2ISNR2

0

q ; ð2Þ

where PSO is the on-axis signal power in the absenceof atmospheric turbulence, hPSi is the on-axis aver-aged power of the signal propagating in the atmo-sphere, and σ2I is the on-axis scintillation index.For a point detector, the powers PSO and hPSi referto the on-axis intensity values of general beam de-rived earlier [14].

3. Bit Error Rate

In the presence of atmospheric turbulence, averageBER is [15],

hBERi ¼ 12

Z∞

0pIðuÞerfc

�hSNRiu2

ffiffiffi2

p�du; ð3Þ

where pIðuÞ is the probability density function of theintensity fluctuations, erfcðxÞ is the complementaryerror function, and u ¼ s=hisi is the normalized sig-nal with unity mean. In this study hBERi is investi-gated in weak atmospheric turbulence, thereforepIðuÞ becomes a log-normal distribution with unitymean defined as follows [1,16]:

pIðuÞ ¼1

uσIffiffiffiffiffiffi2π

p exp�−

�lnðuÞ þ 1

2 σ2I�2

2σ2I

�; u > 0;

ð4Þwhere σ2I indicates the on-axis scintillation index. Toproduce numerical results in Section 4, Eq. (4) isevaluated using the formulation of the scintillationindex reported in our earlier work [13], and the re-sults are then inserted into Eq. (3) to obtain hBERivalues.

4. Results and Discussion

In this section, we analyze hBERi against hSNRiusing Eq. (3) for higher order Gaussian, cos-Gaussian, cosh-Gaussian, and annular beams. Forthis purpose, our plots incorporate the hBERi varia-tion of these beams against the variations in propa-gation length, source size, wavelength, and order ofbeams. In the numerical evaluations of Eq. (3), thescintillation index formula in Ref. [13] is utilized.For further details and in particular the formulationand the notation concerning the source beams usedin this study, the reader is requested to consultRef. [13]. In Ref. [13], scintillation indices of all beamtypes were evaluated at on-axis positions of thetransverse receiver plane. Correspondingly, hBERicalculations of this study will also be applicable foron-axis points. For all on-axis scintillation indices,structure constant C2

n is taken as 10−14 m−2=3, the dis-placement parameters are taken for cos-Gaussianbeam as Vx ¼ 50m−1, and for cosh-Gaussian beamthey are taken as Vx ¼ 50im−1. For all cases exceptthe annular beams, source sizes are equal, i.e.,αs ¼ αs1 ¼ αs2; however for the annular case, the firstbeam source size is greater than the second beamsource size, i.e., αs1 > αs2. From Eq. (2), hSNRi valuesare related to SNR0 and scintillation index.

In Fig. 1, hBERi variation is displayed againsthSNRi for the zero order case (n ¼ 0, m ¼ 0) ofGaussian, cos-Gaussian, cosh-Gaussian, and annu-lar beams. In this figure, the propagation length isL ¼ 3km, the source size is αs ¼ 2 cm, and the wave-length is λ ¼ 1:55 μm. From Fig. 1 it is seen that theannular beam has the lowest hBERi. Similarly inFig. 2 hBERi is analyzed by just reducing the propa-gation length to L ¼ 2:5km and keeping the otherparameters the same as Fig. 1. In this figure, thecos-Gaussian beam has the lowest hBERi value.Comparing Figs. 1 and 2, it is observed that whenthe propagation length is reduced, hBERi decreases

Fig. 1. hBERi of zero order (n ¼ 0, m ¼ 0) Gaussian, cos-Gaus-sian, cosh-Gaussian, and annular beams versus hSNRi at selectedvalues of propagation parameters for L ¼ 3km, αs ¼ 2 cm.

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as expected, but simultaneously the scintillationrankings of the beams are changed. In Fig. 3, onlythe source size is raised to αs ¼ 3 cm. Here, the smal-lest hBERi is obtained for the cos-Gaussian beam.Examining Figs. 2 and 3 together, we see that in thisrange of source sizes, hBERi increases for all thebeams with growing source sizes.In Fig. 4, we take the n ¼ 0, m ¼ 0, L ¼ 3km,

αs ¼ 1 cm, and λ ¼ 1:55 μm. For the given parametersof Fig. 4, the differences between hBERi values ofbeams gradually diminish, with the best system per-formance being obtained for the annular beam.Figures 5 and 6 show hBERi variation of hSNRi

for higher order Gaussian, cosh-Gaussian, cos-Gaussian, and annular beams at n ¼ 2, m ¼ 0,n ¼ 4, andm ¼ 0. In both of the figures, the propaga-tion parameters are the same as in Fig. 4. Accordingto Figs. 5 and 6, for this range of parameters, the

annular beam attains the lowest hBERi values.When Figs. 5 and 6 are compared among themselves,for even orders of higher order beams, despite therise in the order, we observe that only a slight de-crease in hBERi occurs. This observation is in linewith our earlier findings of the slow variation ofthe scintillation index against rising orders [13]. Thisslight decrease also continues to exist beyond therange even of orders considered here. Higher orderbeams start on the source plane with a multitudeof sidelobes arranged around on-axis point. As thehigher order beams propagate in turbulence, thesesidelobes tend to merge towards the center, thusmaking the on-axis intensity greater than that ofthe fundamental mode. This way at the chosensource parameters and propagation lengths asillustrated by Fig. 2 of Ref. [13], the scintillations ofhigher order beams tend to be lower than the

Fig. 2. hBERi of zero order (n ¼ 0, m ¼ 0) Gaussian, cos-Gaus-sian, cosh-Gaussian, and annular beams versus hSNRi at selectedvalues of propagation parameters for L ¼ 2:5km, αs ¼ 2 cm.

Fig. 3. hBERi of zero order (n ¼ 0, m ¼ 0) Gaussian, cos-Gaus-sian, cosh-Gaussian, and annular beams versus hSNRi at selectedvalues of propagation parameters for L ¼ 2:5km, αs ¼ 3 cm.

Fig. 4. hBERi of zero order (n ¼ 0, m ¼ 0) Gaussian, cos-Gaus-sian, cosh-Gaussian, and annular beams versus hSNRi at selectedvalues of propagation parameters for L ¼ 3km, αs ¼ 1 cm.

Fig. 5. hBERi of higher order (n ¼ 2, m ¼ 0) Gaussian, cos-Gaus-sian, cosh-Gaussian, and annular beams versus hSNRi at selectedvalues of propagation parameters for L ¼ 3km, αs ¼ 1 cm.

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scintillations of the fundamental mode. Since hBERiis directly related to the amount of scintillations, asseen in Figs. 4–6, we observe that the higher orderbeams possess less hBERi.Finally, Fig. 7 shows how the hBERi variation is

affected by the wavelength variation, for the higherorder case of n ¼ 4, m ¼ 0 for the Gaussian, cos-Gaussian, cosh-Gaussian, and annular beams. WhenFigs. 6 and 7 are analyzed regarding wavelengths, itis seen that hBERi values increase for smaller wave-lengths. In this figure, the hBERi curve of the cos-Gaussian beam remains below the hBERi curves ofthe other beams.

5. Conclusion

In this study, we calculated and analyzed BER in di-rect detection optical receivers for general beamsusing on–off keying modulation. Log-normal distrib-uted intensity is used for BER calculations since the

system performances are investigated in weakatmospheric turbulence. The acceptable BER valuesmatching those encountered in atmospheric opticalcommunication links were determined by utilizingthe appropriate source beam and propagation para-meters. The investigated general beam types arezero order and higher order cos-Gaussian, cosh-Gaussian, and annular beams, where the Gaussianbeam case is the limiting case. We observe thatthe annular beam has the lowest BER value at smallsource size and long propagation distance. On theother hand, at large source size and short propaga-tion distance, the cos-Gaussian beam has the lowestBER value. Similar to the scintillation index varia-tions, a slight change is observed in BER values,while increasing the beam orders. It is found thatwithin our range of examinations an increase inthe source size basically gives rise to higher BERvalues for all the investigated beams. Additionally,as expected, BER values are found to increase withextended propagation distance. Again, for all thebeams, lower wavelengths yield higher BER values,due to the well established theory of turbulence.

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Fig. 6. hBERi of higher order (n ¼ 4, m ¼ 0) Gaussian, cos-Gaus-sian, cosh-Gaussian, and annular beams versus hSNRi at selectedvalues of propagation parameters for L ¼ 3km, αs ¼ 1 cm.

Fig. 7. hBERi of higher order (n ¼ 4, m ¼ 0) Gaussian, cos-Gaus-sian, cosh-Gaussian, and annular beams versus hSNRi at selectedvalues of propagation parameters for L ¼ 2km, λ ¼ 0:85 μm.

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