Research on Bit Error Rate in the Presence of Local ... - IEEE Xplore

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 29, NO. 19, OCTOBER 1, 2011

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Research on Bit Error Rate in the Presence of Local Wavefront Aberration in Intersatellite Laser Communications Yuqiang Yang, Qiqi Han, Liying Tan, and Guangyu Zhang

Abstract—Local distortions in lenses of transmitter optical system could further deteriorate the performance of intersatellite laser communications (lasercom). Bit error rate (BER) for on–off keying system is investigated considering the combined influence of local distortion and pointing error, without considering the noise of the detector. It is shown that BER varies regularly with the characteristic parameters of local distortion. Peak-to-valley value that is usually used to evaluate lens machining precision is determined, and the key principles for selecting lenses are given. This paper can contribute to the lasercom system design. Index Terms—Bit error rate (BER), local distortion, optical communication.

I. INTRODUCTION

C

OMPARED to microwave communications, intersatellite laser communication (lasercom) has many advantages, such as smaller terminal size and weight, less power consumption, greater immunity to interference, larger data rate, and denser satellite orbit population; consequently, it provides an attractive alternate to microwave communication for both commercial and military applications. More and more researchers investigate it [1]–[14]. Lens surface errors are inevitable due to machining tolerance and nonuniform temperature. Although they tend to be much more global to the lens, local surface errors are often observed in optical tests. An example is shown in Fig. 1. When the beam transmits through the lens with local surface errors, its wavefront will have local distortions. In lasercom, local distortions will cause mutual alignment errors, degraded receiver power, and increased bit error rate (BER). In the earlier work [15]–[19], we proposed Gaussian model to represent local distortions and researched the effects on mutual alignment errors and receiver power. Manuscript received April 05, 2011; revised June 30, 2011; accepted July 26, 2011. Date of publication August 04, 2011; date of current version September 07, 2011. This work was supported in part by the National Natural Science Foundation of China under Grant 10904026, in part by the Heilongjiang Province Office of Education under Grant 12511116, and in part by the Natural Science Foundation of Heilongjiang Province of China under Grant F200831. Y. Yang and G. Zhang are with the Institute of Application Science, Harbin University of Science and Technology, Harbin 150080, China (e-mail: [email protected]; [email protected]). Q. Han and L. Tan are with the National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2011.2163813

Fig. 1. Local distortions in lenses observed in optical tests.

BER is an important parameter of lasercom. Without considering wavefront aberrations, Barry and Mecherle [8], Scott and Young [9], Koepf et al.[10], Chen and Gardner [11], and Arnon and Kopeika [12] studied the impact of mechanical vibration on the BER of lasercom. Considering the combined influence of mechanical vibration and wavefront aberrations, we researched the effect of primary aberrations on the BER [20]. To our best knowledge, there has been no investigation of the influence of local distortions on the BER yet. In this paper, the BER for on–off keying system is researched considering the combined effects of pointing vibration and local distortion, without considering the noise of the detector of APD. This paper has the following outline. Section 2 introduces the preliminaries. In Section 3, a BER as a function of local distortion parameters is given. Section 4 is devoted to numerical analysis. Section 5 summarizes our results. II. PRELIMINARIES A. Model of Local Distortions In our earlier work [15], local wavefront deformation, which is caused by local surface error in lenses, as shown in Fig. 2, is modeled by an ellipse Gaussian function as follows:

(1) where denotes the center amplitude of the ellipse Gaussian is the disfunction, and are the radii of the ellipse, + denotes the distortion center coordinate, and and (0,0). The formation tance between the coordinates

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Fig. 4. Relationship of the pointing error angle and receiver’s displacement. Fig. 2. Ellipse Gaussian model for the local distortion.

Fig. 5. Relationship of the optical intensity and receiver terminal’s displacement. Fig. 3. Formation of Gaussian local wavefront aberration.

of local phase aberration from the local surface error in lenses is shown in Fig. 3. Obviously, can be expressed as [15] (2) where is the refractive index of the lens, is the center depth of lens surface error, and is the wavelength.

immobile and the receiver terminal moves in the receiver plane. The relationship of pointing error and receiver terminal’s displacement is shown in Fig. 4. Receiver terminal’s displacements are distributed according to the probability density [20] (5) (6)

B. BER in the Presence of Wavefront Aberrations Satellite mechanical vibration causes beam pointing error. Generally, it is assumed that pointing errors in azimuth and elevation directions can be modeled as zero-mean Gaussian random variables with the following probability density functions [12]: (3) (4) where and are the pointing error and pointing standard deviation in the azimuth direction, respectively, and and are that in the elevation direction, respectively. Due to the relativity of movement, the transmitted beam’s angular vibration can be approached by supposing that the transmitted beam is

is the distance between transmitter terminal and rewhere ceiver terminal, and and are the pointing standard deviation in the azimuth direction and in the elevation direction, respectively. The relationship of the optical intensity at the receiver and the receiver terminal’s displacement is shown in Fig. 5. We can find that the optical intensity degrades with the receiver terminal’s displacement increasing. So the received power degrades too, which results in an increase of the BER. In laser communications, APD is generally used as the detector to detect the signal, and it has a threshold power . For on–off keying system, when the power received by APD is lower than the threshold power , the signal is judged as code “1,” otherwise it is judged as code “0.” If the code “1” is misjudged as code “0” or the code “0” is decoded as code “1,” a bit error appears. In general, the amount of codes “1” and “0” can

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be considered to be equal in a practical communication system, so the BER can be shown as

(7) where BER(0/1) denotes the rate of code “1” that is decoded as “0,” and BER(1/0) denotes the rate of code “0” that is misjudged as “1.” Due regardless of the noise of the detector, the code “0” will not be misjudged, i.e., (8) For on–off keying system with detector threshold power the error rate of code “1” is

,

(9) where denotes the optical power received by the detector, and is the probability of . Substituting (8) and (9) into (7), we can obtain

Fig. 6. Threshold area

truncated by a contour line of

.

(10) Due to the ultralong distance between the transmitter and the receiver terminals, the optical intensity at the entrance pupil of the receiver terminal is uniform. Consequently, the receiver power on the detector is proportional to the intensity at the entrance pupil, which can be shown as (11) where is the attenuation coefficient of the receiver optical system, and is the radius of the entrance pupil. Substituting (11) into (10), we can obtain

Equation (14) shows that when the parameters and are fixed, BER is determined by the integral area that is truncated by the intensity of and varies with the phase error of the transmitted beam. Here, the area is called as threshold area. When wavefront aberration appears, the threshold area will become smaller and maybe its center will move away from the coordinate center . Both changes will increase the BER and degrade the performance of the lasercom system. To simplify the problem, the threshold area is considered to be an ellipse and (14) can be rewritten as [20]

(12) where denotes the threshold intensity corresponding to the threshold intensity , and is the probability of . Fig. 6 shows the intensity distribution in the receiver plane with contour line of . According to (9), when the receiver terminal is not in the threshold area which is truncated out by the contour line of , the bit error appears. So (12) can be further written as [20] (13)

Substituting (5) and (6) into (13) and letting we can obtain [20]

,

(14)

(15) where and are the radii of the ellipse, is the distance between the ellipse center and the coordinate center, and erf is the error function. As proposed in [20], the ellipse radii of and can be determined by following processes that are shown in Fig. 7. 1) Choosing a symmetry axis or an approximate symmetry axis of the contour line of . 2) Plotting the tangent lines that are parallel to the approximate symmetry axis. The distance of the two tangent lines determines one radius of the ellipse with , where and are the coordinates of the cross point of the tangent lines and the approximate symmetry axis. 3) Plotting the tangent lines that are perpendicular to the approximate symmetry axis. Another radius of the ellipse is determined by , where and are the coordinates of the cross point of the tangent lines and their perpendicular line. When the ellipse is fixed, we can find the center of it. Then the

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aberrations, and the second part denotes the field caused by local wavefront distortion. Hence, comparing with the condition without wavefront aberrations, the variations of threshold area and BER are directly determined by the second part. Distortion area determines the integral range of the second part, so BER would increase as the local distortion becomes wider. Now we discuss BER varying with the distortion depth . Splitting the second part of (20) and substituting (1) into it, the following equation is obtained:

Fig. 7. Determination of the ellipse radii

and

.

distance between the ellipse center and the coordinate center is determined. (21) III. BER VERSUS DISTORTION PARAMETERS Because the distance between the transmitter and receiver terminals satisfies Fraunhofer diffraction, complex amplitude of the optical field at the receiver terminal with displacement of can be written as

As shown in (21), the first and second parts have no relation to the distortion depth , so BER variation with the depth is determined by the third part in which there is the aberration term . It is known that is a periodic function and the periodicity is . So it can be shown as

(16)

(22)

(17)

Because is a function of spatial coordinates of and , the BER changes periodically with RMS and the periodicity is . RMS of in the integral area can be expressed as

(18) Here, is the transmitted Gaussian beam with local phase error is the half-width of the transmitted beam, is the wave number , and is the radius of the transmitter antenna. Optical intensity at the receiver terminal can be shown as (19)

(23) Equation (23) shows that RMS depends proportionally on the distortion depth . Considering that the periodicity for RMS is , the periodicity for is (24)

The optical fields for (16) can be rewritten as

As shown in (24), the periodicity for BER varying with is determined by lens refractive index and the wavelength , and does not depend on other distortion parameters. For k9 china glass whose refractive index is , the periodicity is (25) (20) denotes the area of local distortion. Equation (20) where shows that the optical field consists of two parts. The first part denotes the field for the transmitted beam without wavefront

IV. NUMERICAL RESULTS AND ANALYSIS On the assumption that the entire distortion is on the lens, the affect of local deformation on the BER has been evaluated. In

YANG et al.: LOCAL WAVEFRONT ABERRATION IN INTERSATELLITE LASER COMMUNICATIONS

Fig. 8. BER as a function of distortion radius for different depth and center distance.

the numerical simulation process, the calculation uses the following values: the radii of the local distortion , the radius of the optical antenna mm, the truncation ratio ( is the waist of the transmitter Gaussian beam), the beam pointing standard deviation rad, the wavelength nm, the refractive index of lens , and the distance of the two satellites km. The maximum optical intensity at the receiver for the transmitted beam without wavefront aberrations is assumed to be , which can be assured by choosing suitable transmitted power and data rate of communication. Fig. 8 shows the BER as a function of the normalized radius for different center distance and center depth. denotes the BER for transmitted beam without wavefront aberrations; denotes the BER for transmitted beam with local wavefront aberrations. Comparing with the condition in which wavefront aberrations are not taken into account, the BER increases in the presence of local deformation, which means that the performance of lasercom is further deteriorated by local distortion. As expected, the BER monotonically increases with the increasing distortion radius, which means that the wider the local distortion is, the stronger the degradation of the system performance. BER as a function of the normalized center distance against radius and center depth is shown in Fig. 9. Obviously, the BER decreases monotonically as the distance increases. For the Gaussian transmitted beam, optical intensity on the local distortion decreases with center distance increasing; consequently, the effect of the local phase error on the BER lessens as the location of distortion deviates the center of the lens. Fig. 10 shows the BER versus the distortion center depth for different distortion radius and center distance. Obviously, the BER fluctuates like damped oscillation with increasing, and the fluctuation period is about . From (22), we find that the period depends only on the normalized center depth and the refractive index but has no relation to the distortion radii and the distortion center distance. Fig. 9 confirms the theoretic analysis. With the help of Fig. 9, we can see quite well that the peaks of the BER are independent on the distortion radii and the distortion center distance. However, the radii and the center distance determine the distortion area and position. Therefore, with the change of the radii or the center distance, the curves of the BER vary as a whole, leaving the shape of the curves invariable. In

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Fig. 9. BER versus distortion center distance for different radius and depth.

Fig. 10. BER versus distortion depth for different radius and center distance.

addition, the first peak of the curve is at the place of ; to reduce the influence of local distortions on the BER, distortion depth should be much less than . PV value is usually used to evaluate lens surface error, which is defined as the value of peak to valley. According to the definition of PV value, it can be assumed that PV value is two times larger than the distortion depth . Therefore, to reduce the influence of local distortion on the BER, the PV value of lenses should be much less than . V. CONCLUSION Influence of local lens surface errors on the BER is investigated, without considering the noise on the detector. It turns out that the performance of lasercom system will be further deteriorated by local distortions. BER depends on characteristic parameters of the local deformation. The key principles for selecting lens are determined as follows: 1) small radii and ; 2) long distance between the position of the distortion and the center of the lens; and 3) depth as small as possible. PV value of lens should be much less than . The results obtained here will be useful in the design of lasercom systems. REFERENCES [1] V. W. S. Chan, “Free-space optical communications,” J. Lightw. Technol., vol. 24, no. 12, pp. 4750–4762, Dec. 2006. [2] F. Cosson, P. Doubrere, and E. Perez, “Simulation model and on-ground performances validation of the PAT system for SILEX program,” in Proc. Int. Soc. Opt. Eng., 1991, vol. 1417, pp. 262–276. [3] B. Laurent and G. Planche, “SILEX overview after flight terminals campaign,” in Proc. Int. Soc. Opt. Eng., 1997, vol. 2990, pp. 10–22.

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[4] M. Renard, P. Dobie, J. Gollier, T. Heinrichs, P. Woszczyk, and A. Sobeczko, “Optical telecommunication performance of the qualification model SILEX beacon,” in Proc. Int. Soc. Opt. Eng., 1995, vol. 2381, pp. 289–300. [5] K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (laser utilizing communications equipment),” in Proc. Int. Soc. Opt. Eng., 1996, vol. 2699, pp. 114–120. [6] R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz–Gauss beams in turbulent atmosphere,” Opt. Exp., vol. 15, pp. 16328–16341, 2007. [7] W. Du, L. Tan, J. Ma, and Y. Jiang, “Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence,” Opt. Exp., vol. 18, pp. 5763–5775, 2010. [8] J. D. Barry and G. S. Mecherle, “Beam pointing error as a significant design parameter for satellite-borne free space optical communication system,” Opt. Eng., vol. 24, pp. 1049–1054, 1985. [9] P. W. Scott and P. W. Young, “Impact of temporal fluctuations of signal-to-noise ratio (burst error) on free-space laser communications system design,” in Proc. Int. Soc. Opt. Eng., 1986, vol. 616, pp. 174–182. [10] G. A. Koepf, R. Peters, and R. G. Marshalek, “Analysis of burst error occurrence on optical intersatellite link (ISL) design,” in Proc. Int. Soc. Opt. Eng., 1986, vol. 616, pp. 129–136. [11] C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun., vol. 37, no. 3, pp. 252–260, Mar. 1989. [12] S. Arnon and N. S. Kopeika, “Performance limitations of free-space optical communication satellite networks due to vibrations-analog case,” Opt. Eng., vol. 36, pp. 175–182, 1997. [13] M. Toyoshima, N. Takahashi, T. Jono, T. Yamawaki, K. Nakagawa, and A. Yamamoto, “Mutual alignment errors due to the variation of wave-front aberrations in a free-space laser communication link,” Opt. Exp., vol. 9, pp. 592–602, 2001. [14] J. F. Sun, L. R. Liu, M. J. Yun, and L. Y. Wan, “Mutual alignment errors due to wave-front aberrations in intersatellite laser communications,” Appl. Opt., vol. 44, pp. 4953–4958, 2005. [15] Y. Yang, L. Tan, and J. Ma, “Pointing and tracking errors due to localized distortion induced by a transmission-type antenna in intersatellite laser communications,” Appl. Opt., vol. 48, pp. 786–791, 2009. [16] Y. Yang, L. Tan, J. Ma, and J. Yu, “Effects of localized deformation induced by reflector antenna on received power,” Opt. Commun., vol. 282, pp. 396–400, 2009. [17] Y. Yang, L. Tan, and J. Ma, “Mutual alignment errors due to localized distortion in free-space laser communication links,” Opt. Commun., vol. 281, pp. 4180–4187, 2008.

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[18] L. Tan, Y. Yang, J. Ma, and J. Yu, “Pointing and tracking errors due to localized deformation in inter-satellite laser communication links,” Opt. Exp., vol. 16, pp. 13372–13380, 2008. [19] L. Tan, Y. Yang, J. Ma, and J. Yu, “Receiver power penalty due to localized distortion induced by transmission-type optical antenna in inter-satellite laser communication,” Opt. Eng., vol. 48, pp. 056001–056001-6, 2009. [20] Y. Yang, Q. Han, L. Tan, J. Ma, S. Yu, Z. Yan, J. Yu, and S. Zhao, “Influence of wave-front aberrations on bit error rate in inter-satellite laser communications,” Opt. Commun., vol. 284, pp. 3065–3069, 2011.

Yuqiang Yang received the B.S. degree in physics from Northeast Normal University, Changchun, China, in 2002, and the M.S. and Ph.D. degrees from the Harbin Institute of Technology, Harbin, China, in 2004 and 2010, respectively. He is currently at the Institute of Application Science, Harbin University of Science and Technology, Harbin. His research interests include free-space optical communications and laser beam propagation.

Qiqi Han received the M.S. and Ph.D. degrees from the Harbin Institute of Technology, Harbin, China, in 2000 and 2004, respectively. He is currently at the National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology. His current research interests include free-space optical communications, wavelet optics, and quantum communications.

Liying Tan received the M.S. and Ph.D. degrees from the Harbin Institute of Technology, Harbin, China, in 1995 and 2004, respectively. She is currently a Professor in the Department of Electronic Science and Technology, Harbin Institute of Technology. Her current research interests include free-space optical communications, wavelet optics, and quantum communications.

Guangyu Zhang received the M.S. degree from the Harbin University of Science and Technology, Harbin, China and the Ph.D. degree from the Harbin Institute of Technology, Harbin, in 2002 and 2007, respectively. He is currently a Professor at the Institute of Application Science, Harbin University of Science and Technology. His current research interests include quantum communication and nano-optics.