INTENSITY NOISE IN BALANCED DETECTION OF CORRELATED INCOHERENT SIGNALS

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1 INENSIY NOISE IN BALANCED DEECION OF CORRELAED INCOHEREN SIGNALS Mohammad Abtahi, Simon Ayotte, Julien Penon, Leslie A. Rusch Center for Optics, Photonics Laser, Dept. of Electrical Computer Eng., Laval University, Quebec (QC), GK 7P4, Canada {abtahi, sayotte, jpenon, gel.ulaval.ca ) ABSRAC We study the balanced detection of broadb incoherent optical signals - signals characterized by high intensity noise. We consider signals generated from a single incoherent source with overlapping, non-identical spectra but zero time delay. Our statistical analysis yields equations for the probability density function (PDF) of the balanced detector output for partially correlated input signals based on easily measured power spectral densities. We derive analytical expressions with extremely good prediction of measured values for correlation up to 95%. he analytic expressions can be used to characterize system performance, in particular, bit error rate for communications systems. KEY WORDS Balanced detection, probability density function, broadb incoherent light source, intensity noise.. Introduction Balanced detectors are widely used in many optical communication systems, in order to compensate the imperfections of fiber optics, to improve the system sensitivity or increase the signal to noise ratio (SNR) so on []-[8], in optical code division multiple access (OCDMA) they are used to cancel noise the multipleaccess interference (MAI). For example, balanced photodetector (BPD) is used in [5] for compensation of fiber dispersion in high speed long-haul SMF transmission. BPD suppresses the even order nonlinear distortions in radio-over-fiber (ROF) systems [6]. In [7], a balanced configuration is proposed for optical coherence tomography, where a higher SNR in comparison to the unbalanced configuration is obtained. A comparison of balanced photodiode to single-ended detection for the coherent optical receiver is provided in [8] to detect OOK signals at.5 5Gb/s. It is shown that the sensitivity is increased by at least 8 db. An optoelectronic phaselocked loop with balanced photodetection is proposed in [9] for clock recovery in high speed optical time-divisionmultiplexed systems. he use of BPD resulted in a recovered clock with better timing jitter performance. his article focuses particularly on the use of BPD for spectrally amplitude coding optical code division multiple access (SAC-OCDMA). he performance advantage of SAC-OCDMA over other version of OCDMA rests on the use of balanced detection to remove first order (in the mean) MAI. he special nature of OCDMA signals can lead to complex dependencies between signals in the BPD. Our analysis of the detection process in the presence of correlated signals can be used to predict the performance of aggressively spectrally efficient SAC- OCDMA systems [3]. In any system using incoherent broadb optical sources, the most important noise is signal power dependent intensity noise, where increasing the signal power does not improve the SNR. he intensity of the signal can be well modeled by a negative exponential distribution, the probability density function (PDF) of the integrated intensity of an unpolarized thermal source can be approximated by a gamma density function []. In this paper, we study the balanced detection of broadb incoherent optical signals with correlated intensity noise. All other noise sources such as thermal receiver noise dark current noise of the two photodetectors (PD) are independent from each other from the intensity noise. While intensity noise is the dominant noise source, we measure take into account all noise sources in our experiments, including RF amplifier noise the internal noise of measuring device. It is straight forward to theoretically manipulate the output of BPD when the associated noise of optical input signals are completely correlated or completely uncorrelated. In the case of complete correlation, the variation in the input signals are removed by BPD the output will be a dc-value representing the difference of the signals means. he PDF of the output can then be represented by a Dirac delta function which does not depend on the PDF of the input signals. On the other h, when the input signals are completely uncorrelated, the PDF of the output can be obtained by the convolution of the input PDFs. When the input signals are partially

2 correlated, the evaluation of the output PDF requires more information about the correlation of the input signals. Karhunen-Loève Series Expansion (KLSE) is a general method to analytically evaluate the BER in the optical systems. his method is used in [] [] to evaluate the BER in optically preamplified systems is generalized for DPSK systems in [3], taking account of optical amplifiers, interferometric demodulation direct detection. Accuracy of KLSE method is verified using Monte Carlo simulations [3]. However, in some cases the approximated methods are accurate enough to provide the PDF evaluating the BER. In this paper, we propose a simple approximated method to obtain the PDF of the BPD output. We examine in detail correlation in signals due to overlap in the PSDs of filtered signals we propose the algorithm to obtain the PDF of the BPD output. We validate our method by comparing experimental simulation results.. Optical Signals with Partially-Correlated Intensity Noises Consider an incoherent broadb optical source split filtered by two distinct filters. If the transfer functions of the filters are disjoint, the intensity noise generated by each optical signal is uncorrelated from the other. his is due to the phases of different frequency components being uncorrelated [4]. However, when the filters are not disjoint, the PSDs overlap partially, the associated intensity noises generated by the same optical source in the overlapping frequencies are correlated. he correlation between the two input signals can be measured by a correlation coefficient defined by: cov( I, I) ρ = () var( I) var( I) where I I are the intensities of the BPD input signals. he covariance of the intensities can be calculated from the well-known relation [5] var( I I) = var( I) + var( I) cov( I, I) where var( I ) var( I ) represent the variance of the intensity of the input signals. he PDF of the detected optical signals can be measured experimentally by a sampling oscilloscope. When only one input signal is connected to the BPD, the output PDF can be used to determine the variance of that signal, var( I i ). he variance of the intensity difference, var( I I), may be obtained from the PDF of the BPD output when both signals are connected to BPD inputs. hus, we can determine the correlation coefficient of the optical signals experimentally. When two signals have exactly the same PSD, the correlation coefficient is one, while two signals with disjoint PSD have zero correlation coefficient. We consider the partially correlated optical signals generated by passing a broadb light source through two optical filters having overlapping transfer functions. We use two tunable filters offset the center wavelengths to change the degree to which the spectra overlap, thus changing the degree to which the signals are correlated. Let e () t e () t represent the electric fields of optical signals with PSD of S ( ν ) S ( ν ), respectively, where ν is the optical frequency. he two optical signals are generated by filtering the same unpolarized thermal light source. As the PSD of a rom process is a positive real function, we can always decompose S ( ν ) S ( ν ) as: S( ν ) = SA( ν) + SC( ν) () S ( ν ) = SB( ν) + SC( ν) where by construction S A( ν ) SB ( ν ) are two disjoint PSDs. Consequently, the corresponding electric fields, i.e., ea() t eb () t, are independent. SC ( ν ) is the common part of the PSDs of S ( ν ) S ( ν ), with electric field of ec () t which depends statistically on e () A t eb () t. he detected signals at the BPD can be represented by: B = e () t dt (3) * = ea() t dt+ ec() t dt+ Re ea() t ec() t dt, B = e () t dt (4) * = eb() t dt + ec() t dt + Re eb() t ec() t dt where Re{.} is the real part function, represents complex conjugate operator. In (4) we have modeled the inherent low-pass nature of the PD by impulse response: t ht () = (5) elsewhere with an equivalent electrical bwidth of B /. Without lack of generality, we normalized the responsivity of the PDs to one in writing the previous relations. Let the instantaneous intensity be noted by I () () A t = ea t he electrical output of the balanced photodetector is the difference of (3) (4): Z = B B * (6) = I A() t dt IB() t dt+ Re [ ea() t eb()] t ec() t dt he common part, i.e., W = I () t dt, is eliminated by C C e 5

3 BBS Filter OA x4 Filter A Filter C Filter B x x DL x S BPD Balanced RF Detector Amp S Sampling Scope S( λ) S( λ) (a) Filter S DL BBS Filter OA x Filter S BPD Balanced Detector Sampling Scope SA( λ ) SC ( λ) SB( λ) A (b) Fig.. he schematic diagrams of experimental setups. balanced detection. In section V we examine experimentally the importance of the cross term (last term) in (6), representing the beating of common disjoint spectrums. We find this term can be neglected, reducing (6) to Z = I () t dt IB() t dt A WA he integrated intensities WB W A (7) W B in (7) are independent, as their optical power originate from distinct spectral regions. We see in (7) that the BPD output is essentially the detection of disjoint, uncorrelated components of the input signals. hus, the PDF of Z can be obtained by: fz( z) = fw ( f ) A WB (8) = fw ( x) f ( ) A W z+ x dx B where indicates convolution f W A (.) f W B (.) are the PDFs of W A W B, respectively. o take into account the effects of noises coming from the photodetectors, the RF amplifier measurement equipment, (8) should be convolved with the resultant PDF of all other noise sources f n (.), assumed to be independent of the input signals. 3. PDF of the Integrated Intensity We now address the specific nature of the PDF of the integrated intensities discussed in the previous section. he statistical properties of the integrated intensity of broadb thermal sources are well discussed in []. he PDF for the integrated intensity W, of an unpolarized thermal source is approximated by: M M W exp( MW / W) fw ( W) = ( M / W) (9) Γ( M ) where Γ (.) is the gamma function W is the average integrated optical power. M depends on the complex Fig.. he PSD of signals in the first setup: (a) PSD of signal S, (b) PSD of signal S (c) decomposition of them into disjoints common parts. degree of coherence of the light [] is frequently approximated by the signal to intensity noise ratio. When non-ideal photodetection electrical filtering are taken into account in a total effective electrical filter transfer function H ( f ), a more realistic M factor can be found by [, 4]: H() S( υ) dυ M = () S( υ) S( υ+ f) dυ H( f) df where f is the baseb frequency. Equation () allows the use of any arbitrary electrical profile of detector /or RF optical amplifier. In the case of ideal photodetection, H ( f ) in () is the Fourier transform of (5). In the case of optical electrical filters having rectangular shapes with optical electrical bwidths of B o B e, respectively, M is approximated by B / B. o e 4. Experimental Setup In order to validate the analysis of Section II, we developed the experimental setups shown in Fig.. he setup in Fig. (a) combines a common spectral b with each of two distinct bs to reproduce the system described by (3) (4). he setup in Fig. (b) also generates signals with overlapping spectra, but in this case tunable filters are used to allow us to experimentally vary the degree of correlation. In Fig. (a), a broadb source (BBS) is filtered by a wideb optical filter amplified by an erbium doped fiber amplifier. he amplified optical signal is then divided to three arms filtered by three different optical filters with central wavelengths of 539.8, 6

4 nm for filter A, B, C, respectively. he linewidth of filter A B is.5 nm, whereas that of filter C is nm. he optical couplers are used to combine these signals generate signals S S with PSDs given by (). In one branch, we used an adjustable delay line (DL) to balance the path length (i.e., zero relative time delay). As the power spectra of filters A B are disjoint, their signals are uncorrelated. herefore, the signals S S are partly correlated via the common spectrum from filter C. A New Focus 67- AC BPD with nominal 8 MHz bwidth a 5MHz RF amplifier are used to detect the signals. his setup permits us to validate the approximation in (7). Whereas the correlation coefficient of the signals is fixed in the first setup, we are able to change it in the second setup which is shown in Fig. (b). he same amplified broadb optical signal used in the previous setup is now filtered by two.5 nm optical filters. By varying the center wavelength of the second filter, the common spectrum in signals S S can be swept as a result, the correlation coefficient can be changed. he optical DL attenuator (A) are used to balance the amplitude the delay of the two branches. In this setup we used a New Focus 67-DC BPD without an RF amplifier. 5. Results Discussions We begin by using the setup of Fig. (a) to validate the approximation in (7) where we ignored the effects of the cross term representing the beating of common disjoint parts in the PSDs. he PSDs of S S are shown in Fig. (a) (b), respectively. he setup permits us to measure the PSD of the disjoint common parts as shown in Fig. (c). he PDF of the BPD output is measured by a sampling scope (Agilent 86A) in Eye Diagram mode. his test equipment can save intensity measurements in memory generate a histogram plot. When the number of samples is large (> 5 million in our tests) the histogram is an empirical estimate of the PDF of the input signal. Essentially the approximation in (7) neglects any contribution to the output PDF resulting from the presence of common spectral components, that is, there are no terms with index C in (7). We physically remove the middle filter in Fig. (a), the filter generating the common spectrum. We compare the PDF at the output with filter C present (6) with filter C removed (7). he measured PDFs are plotted in Fig. 3 in log scale. here is a very good match between the two PDFs. his means that the approximation in ignoring of the cross terms in (6) is accurate. We note that the common spectrum in S S as compared to the total power is high, so that the calculated correlation coefficient based Fig. 3. he PDF of the BPD output with without filter C. H(f) for BPD H(f) for BPD with RF Amp. Input Input Frequency (GHz) Fig. 4. he normalized frequency response of the BPDs used in setup (a) (b) in Fig.. on () is.69. Our goal in the second setup is to model the output of the PBD Z in closed form. We wish to exploit knowledge of the input spectra to parameterize our closed form PDF. We hypothesize that the output will be gamma distributed as Z is essentially the integrated intensity of a broadb incoherent source. However, the common section of the spectrum will be canceled during balanced detection, so the output can be parameterized with only information about the distinct, disjoint spectra of the input signals. We begin by comparing experimental predicted PDFs for the disjoint signals S A S B. o calculate the M factor, we first measured the frequency response of the BPD followed by an RF amplifier. he normalized frequency responses H ( f ) for setups of Fig. (a) (b) are shown in Fig. 4. Using the measured frequency response ( ) S ν H f the measured PSDs A ( ) SB ( ν ) of Fig., we calculated the M factor using (). We found the mean value of the detected output signal by W = P( RG) GAMP () where P = S( ν ) dν is the input optical power, R is the responsivity (in A/W), G is the internal transimpedance gain of balanced detector (in V/A), G is the gain of RF amplifier. Having the M factor AMP 7

5 SA SB able I. Comparison of the two definitions of the correlation coefficient. Δ λ (nm) ρ (measured) ρ PSD (>.5).5. Fig. 5. he measured simulated (gamma) distribution for PDF_A, PDF_B PDF of the BPD output. the signal s mean, the gamma density function can be obtained by (9). Next we characterized the other noises in the measurement process such as detector noise RF amplifier noise. We estimated the cumulative PDF of these noises by measuring the output histogram when disconnecting the input signals from the BPD. his noise PDF was convolved with the gamma PDF to predict the measured PDF. he measured PDF of disjoint signals S A S B as well as the estimated PDF are shown in Fig. 5. Note that when measuring the PDF of one signal, we simply disconnect the unused BPD input. hese results confirm that the gamma approximation is a good one, that we have good estimates of the measurement noise, M factor, H ( f ), signal means after photodetection. We will now describe how to predict the PDF of the BPD output from measurement of the input optical spectra. We approximate the PDF as gamma find the parameters of the gamma distribution by manipulating the measured PSD of the input signals using the following procedure: ) Decompose the PSD of input signals to disjoint common parts. ) Use the PSD of each disjoint part to calculate an M factor a mean value from () (), respectively. 3) Obtain the gamma PDF for each disjoint signal using (9) the appropriate M factor a mean value. 4) Convolve the PDFs per (8) to obtrain the PDF of the BPD output. 5) Convolve the resultant PDF with the noise PDF, f n. he calculated PDF for the BPD output is shown in Fig. 5 which provides a good match to the measured PDF. In the second experiment, we changed the degree of correlation of input signals by changing the common spectrum in the PSD of the signals. By decreasing the offset between center wavelengths of the filters, the common part of the PSD of the signals increases the correlation coefficient approaches one. Clearly the degree of correlation is determined by the common disjoints spectrum of the input signals. o see this relationship in more detail, we define a new parameter, ρ PSD which only depends on the PSD of the signals. We note that the photocurrent PSD is given by [4]: SI ( Ω ) = α S( ω) S( ω+ω) dω () where α is a constant. he total average output power is given by: I ( ) ( ) (3) P = S Ω H Ω dω where the integr represents the PSD of the filtered photocurrent. We define P ρ c PSD = (4) P+ P Here, P c, P P are the detected average power of the common spectrum, the total spectrum S the total spectrum S, respectively. able I, compares the ρ in (), based on the measured variances, to ρ PSD in (4), calculated for two signals with the same PSDs but with center wavelength offset of Δ λ. We increase overlapping spectra by decreasing Δ λ. In the experiment, we obtained the maximum correlation coefficient of.99. he small difference is due to slight mismatch in the spectral shape of filters. We conclude from able I that knowledge of the input PSD the transfer function H ( f ) are sufficient to predict the correlation coefficient. Fig. 6 (a) shows the PSD of input signals for four different overlapping spectra leading to a correlation coefficients of,.3, he corresponding measured simulated PDFs of the BPD output (in log scale) are shown in Fig. 6 (b). he difference between measurement simulation is negligible, the fitted PDF can be accurately used to estimate the BER of communication system. 6. Conclusion Having the PDF of the decision statistic is very important in the analysis of any communication systems, 8

6 ρ = ρ = Fig. 6. (a) he PSD of input signals (b) the measured simulated distribution of the BPD output for different correlation coefficients. as the bit error rate or other performance parameters can be estimated via the PDF. In this paper, we proposed experimentally validated a procedure that helps us to calculate the PDF of the balanced photodetector output when the input signals are partially correlated thermallike incoherent light. In general, the BPD output statistics can be easily found if the input signals are uncorrelated. We studied the degree of the correlation of input signals due to overlapping spectrum proposed the method in order to find the PDF of BPD output. Comparison of the measured simulated PDFs confirmed that the proposed method is efficient the approximation in our analysis is valid. hese results can be used to predict the performance of the aggressively spectrally efficient SAC-OCDMA systems, as well as other BPD applications where signals have significant correlation. References [] M. Kavehrad D. Zaccarin, Optical code-divisionmultiplexed systems based on spectral encoding of noncoherent sources, IEEE Journal of Lightwave echnology, 3(3), 995, [] S. Ayotte, M. Rochette, J. Magné, L. A. Rusch et S. LaRochelle, Experimental Verification Capacity Prediction of FE-OCDMA Using Superimposed FBG, IEEE Journal of Lightwave echnology, 3(), 5, [3] J. Penon, Z. A. El-Sahn, L. A. Rusch, S. LaRochelle, Spectral Amplitude Coded OCDMA Optimized for a Realistic FBG Frequency Response, submitted to IEEE Journal of Lightwave echnology [4] S. Kim, et al., -Gb/s temporally coded optical CDMA system using bipolar modulation/balanced detection, IEEE Photonics echnology Letters, 7(), 5. [5]. Kawanishi, et al., Gbit/s FSK transmission over 3 km SMF using group delay compensated balance detection, Optical Fiber Communication Conference OFC/NFOEC, Anaheim, CA, March 5. [6] B. Masella, X. Zhang, A Novel Single Wavelength Balanced System for Radio Over Fiber Links, IEEE Photonics ech. Letters, 8(), 6. [7] A. Gh. Podoleanu, Unbalanced versus balanced operation in an optical coherence tomography system, Applied Optics, 39(),. [8] C. Wree, et al., Optical Coherent Receivers for.5 5Gb/s, IEEE/LEOS Annual Meeting, Sydney, Australia, 5. [9] D.. K. ong, et al., Optoelectronic Phase-Locked Loop with Balanced Photodetection for Clock Recovery in High-Speed Optical ime-division-multiplexed Systems, IEEE Photonics ech. Letters, (8),. [] J. W. Goodman, Statistical Optics (New York: Wiley, ). [] E. Forestieri, Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape pre- postdetection filtering, IEEE Journal of Lightwave echnology, 8(),. [] C. Lawetz, J. Cartledge, Performance of optically preamplified receivers with Fabry-Perot optical filters, IEEE Journal of Lightwave echnology, 4 (), 996. [3] J. Wing, J. M. Kahn, Impact of chromatic polarization mode dispersion on DPSK systems using interferometric demodulation direct detection, IEEE Journal of Lightwave echnology, (), 4 [4] G.-H. Duan E. Georgiev, Non-white photodetection noise at the output of an optical amplifier: heory experiment, IEEE Journal of Quantum Electron., 37(8),, 8 4. [5] A. Papoulis, Probability, Rom Variables, Stochastic Processes (New York: McGraw-Hill, ). 9