intensity noise in balanced detection of correlated incoherent signals

INTENSITY NOISE IN BALANCED DETECTION OF CORRELATED INCOHERENT SIGNALS Mohammad Abtahi, Simon Ayotte, Julien Penon, and Leslie A. Rusch Center for Optics, Photonics and Laser, Dept. of Electrical and Computer Eng., Laval University, Quebec (QC), G1K 7P4, Canada {abtahi, sayotte, jpenon, rusch}@ gel.ulaval.ca ) [9] for clock recovery in high speed optical time-divisionmultiplexed systems. The use of BPD resulted in a recovered clock with better timing jitter performance. This article focuses particularly on the use of BPD for spectrally amplitude coding optical code division multiple access (SAC-OCDMA). The 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. The 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 SACOCDMA systems [3]. In any system using incoherent broadband optical sources, the most important noise is signal power dependent intensity noise, where increasing the signal power does not improve the SNR. The intensity of the signal can be well modeled by a negative exponential distribution, and the probability density function (PDF) of the integrated intensity of an unpolarized thermal source can be approximated by a gamma density function [10]. In this paper, we study the balanced detection of broadband incoherent optical signals with correlated intensity noise. All other noise sources such as thermal receiver noise and dark current noise of the two photodetectors (PD) are independent from each other and from the intensity noise. While intensity noise is the dominant noise source, we measure and take into account all noise sources in our experiments, including RF amplifier noise and 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 and the output will be a dc-value representing the difference of the signals’ means. The 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 hand, 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

ABSTRACT We study the balanced detection of broadband 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%. The analytic expressions can be used to characterize system performance, in particular, bit error rate for communications systems. KEY WORDS Balanced detection, probability density function, broadband incoherent light source, intensity noise.

1. 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) and so on [1]-[8], and in optical code division multiple access (OCDMA) they are used to cancel noise and 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 2.5 and 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

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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. This method is used in [11] and [12] to evaluate the BER in optically preamplified systems and is generalized for DPSK systems in [13], taking account of optical amplifiers, interferometric demodulation and direct detection. Accuracy of KLSE method is verified using Monte Carlo simulations [13]. However, in some cases the approximated methods are accurate enough to provide the PDF and 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 and we propose the algorithm to obtain the PDF of the BPD output. We validate our method by comparing experimental and simulation results.

generated by passing a broadband light source through two optical filters having overlapping transfer functions. We use two tunable filters and offset the center wavelengths to change the degree to which the spectra overlap, thus changing the degree to which the signals are correlated. Let e1 (t ) and e2 (t ) represent the electric fields of optical signals with PSD of S1 (ν ) and S 2 (ν ) , respectively, where ν is the optical frequency. The two optical signals are generated by filtering the same unpolarized thermal light source. As the PSD of a random process is a positive real function, we can always decompose S1 (ν ) and S 2 (ν ) as: S1 (ν ) = S A (ν ) + SC (ν ) S 2 (ν ) = S B (ν ) + SC (ν )

(2)

where by construction S A (ν ) and S B (ν ) are two disjoint PSDs. Consequently, the corresponding electric fields, i.e., eA (t ) and eB (t ) , are independent. SC (ν ) is the common part of the PSDs of S1 (ν ) and S 2 (ν ) , with

2. Optical Signals with Partially-Correlated Intensity Noises

electric field of eC (t ) which depends statistically on eA (t ) and eB (t ) . The detected signals at the BPD can be represented by:

Consider an incoherent broadband optical source split and 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. This is due to the phases of different frequency components being uncorrelated [14]. However, when the filters are not disjoint, the PSDs overlap partially, and the associated intensity noises generated by the same optical source in the overlapping frequencies are correlated. The correlation between the two input signals can be measured by a correlation coefficient defined by: cov( I1 , I 2 ) ρ= (1) var( I1 ) var( I 2 )

T

B1 = ∫ e1 (t ) dt 2

0

⎧⎪ T ⎫⎪ = ∫ eA (t ) dt + ∫ eC (t ) dt + 2 Re ⎨ ∫ eA (t )eC* (t ) dt ⎬ , 0 0 ⎩⎪ 0 ⎭⎪ T

T

2

(3)

2

and T

B2 = ∫ e2 (t ) dt 2

0

(4) T ⎪⎧ ⎪⎫ * = ∫ eB (t ) dt + ∫ eC (t ) dt + 2 Re ⎨ ∫ eB (t )eC (t )dt ⎬ 0 0 ⎩⎪ 0 ⎭⎪ 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: ⎧1 0 ≤ t ≤ T (5) h(t ) = ⎨ ⎩0 elsewhere T

T

2

where I1 and I 2 are the intensities of the BPD input signals. The covariance of the intensities can be calculated from the well-known relation [15] var( I1 − I 2 ) = var( I1 ) + var( I 2 ) − 2 cov( I1 , I 2 ) where var( I1 ) and var( I 2 ) represent the variance of the intensity of the input signals. The 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 ) . The

2

with an equivalent electrical bandwidth of Be ≈ 1/ T . 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 )

variance of the intensity difference, var( I1 − I 2 ) , may be obtained from the PDF of the BPD output when both signals are connected to BPD inputs. Thus, 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

2

The electrical output of the balanced photodetector is the difference of (3) and (4): Z = B1 − B2 T T T ⎪⎧ ⎪⎫ (6) = ∫ I A (t ) dt − ∫ I B (t ) dt + 2 Re ⎨ ∫ [eA (t ) − eB (t )] eC* (t )dt ⎬ 0 0 ⎩⎪ 0 ⎭⎪

The common part, i.e., WC = ∫ I C (t ) dt , is eliminated by T

0

205

Filter A

S1 (λ ) 2x1

Filter

DL

Filter C OA

BBS

1x4

S1 BPD1 Balanced RF Detector Amp

1x2

Filter B 2x1

Sampling Scope

S2

S 2 (λ )

(a) Filter S1

DL

Filter BBS

OA

1x2

BPD2 Balanced Detector

Filter S2

Sampling Scope

S A (λ )

S B (λ )

SC (λ )

ATT (b)

Fig. 1. The schematic diagrams of experimental setups. Fig. 2. The PSD of signals in the first setup: (a) PSD of signal

balanced detection. In section V we examine experimentally the importance of the cross term (last term) in (6), representing the beating of common and disjoint spectrums. We find this term can be neglected, reducing (6) to T

PSD of signal

WA

and (c) decomposition of them into disjoints and common parts.

degree of coherence of the light [10] and is frequently approximated by the signal to intensity noise ratio. When non-ideal photodetection and 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 [2, 14]:

T

Z = ∫ I A (t ) dt − ∫ I B (t ) dt 0 0   

  

S2

S1 , (b)

(7)

WB

The integrated intensities WA and WB 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. Thus, the PDF of Z can be obtained by: f Z ( z ) = fWA ⊗ (− fWB ) (8) = ∫ fWA ( x) fWB ( z + x) dx

2

+∞ ⎞ 2⎛ 2 H (0) ⎜ ∫ S (υ ) dυ ⎟ ⎝ −∞ ⎠ M = +∞ +∞ (10) ⎛ ⎞ 2 ∫ ⎜ ∫ S (υ )S (υ + f ) dυ ⎟⎠ H ( f ) df −∞ ⎝ −∞ where f is the baseband frequency. Equation (10) allows the use of any arbitrary electrical profile of detector and/or RF optical amplifier. In the case of ideal photodetection, H ( f ) in (10) is the Fourier transform of (5). In the case of optical and electrical filters having rectangular shapes with optical and electrical bandwidths of Bo and Be , respectively, M is approximated by

where ⊗ indicates convolution and fWA (.) and fWB (.) are the PDFs of WA and WB , respectively. To take into account the effects of noises coming from the photodetectors, the RF amplifier and 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.

Bo / Be .

4. Experimental Setup 3. PDF of the Integrated Intensity

In order to validate the analysis of Section II, we developed the experimental setups shown in Fig. 1. The setup in Fig. 1(a) combines a common spectral band with each of two distinct bands to reproduce the system described by (3) and (4). The setup in Fig. 1(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. 1(a), a broadband source (BBS) is filtered by a wideband optical filter and amplified by an erbium doped fiber amplifier. The amplified optical signal is then divided to three arms and filtered by three different optical filters with central wavelengths of 1539.80,

We now address the specific nature of the PDF of the integrated intensities discussed in the previous section. The statistical properties of the integrated intensity of broadband thermal sources are well discussed in [10]. The PDF for the integrated intensity W ≥ 0 , of an unpolarized thermal source is approximated by: W M −1 exp(− MW / W ) (9) fW (W ) = ( M / W ) M Γ( M ) where Γ(.) is the gamma function and W is the average integrated optical power. M depends on the complex

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1543.07 and 1541.32 nm for filter A, B, and C, respectively. The linewidth of filter A and B is 0.25 nm, whereas that of filter C is 1 nm. The optical couplers are used to combine these signals and generate signals S1 and S 2 with PSDs given by (2). 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 and B are disjoint, their signals are uncorrelated. Therefore, the signals S1 and S 2 are partly correlated via the common spectrum from filter C. A New Focus 1617AC BPD with nominal 800 MHz bandwidth and a 500MHz RF amplifier are used to detect the signals. This 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. 1(b). The same amplified broadband optical signal used in the previous setup is now filtered by two 0.25 nm optical filters. By varying the center wavelength of the second filter, the common spectrum in signals S1 and S 2 can be swept and as a result, the correlation coefficient can be changed. The optical DL and attenuator (ATT) are used to balance the amplitude and the delay of the two branches. In this setup we used a New Focus 1617-DC BPD without an RF amplifier.

Fig. 3. The PDF of the BPD output with and without filter C. 1.2 Input 1 1.0

Input 2 H(f) for BPD1 with RF Amp.

0.8

0.6 H(f) for BPD2 0.4

0.2 0 0

0.25

0.5

0.75

1.0

1.25

1.5

Frequency (GHz)

Fig. 4. The normalized frequency response of the BPDs used in setup (a) and (b) in Fig. 1.

5. Results and Discussions

on (1) is 0.69. 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 broadband 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 and predicted PDFs for the disjoint signals S A and S B . To calculate the M factor, we first measured the frequency response of the BPD followed by an RF amplifier. The normalized frequency responses H ( f ) for setups of Fig. 1(a) and 1(b) are shown in Fig. 4. Using the measured frequency response H ( f ) and the measured PSDs S A (ν ) and

We begin by using the setup of Fig. 1(a) to validate the approximation in (7) where we ignored the effects of the cross term representing the beating of common and disjoint parts in the PSDs. The PSDs of S1 and S 2 are shown in Fig. 2(a) and 2(b), respectively. The setup permits us to measure the PSD of the disjoint and common parts as shown in Fig. 2(c). The PDF of the BPD output is measured by a sampling scope (Agilent 86100A) in Eye Diagram mode. This test equipment can save intensity measurements in memory and 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. 1(a), the filter generating the common spectrum. We compare the PDF at the output with filter C present (6) and with filter C removed (7). The measured PDFs are plotted in Fig. 3 in log scale. There is a very good match between the two PDFs. This means that the approximation in ignoring of the cross terms in (6) is accurate. We note that the common spectrum in S1 and S 2 as compared to the total power is high, so that the calculated correlation coefficient based

S B (ν ) of Fig. 2, we calculated the M factor using (10).

We found the mean value of the detected output signal by W = P0 ( RG0 )GAMP (11) where P0 = ∫

+∞

−∞

S (ν ) dν is the input optical power, R is

the responsivity (in A/W), G0 is the internal transimpedance gain of balanced detector (in V/A), and GAMP is the gain of RF amplifier. Having the M factor

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Table I. Comparison of the two definitions of the correlation coefficient.

ρ (measured)

Δλ (nm)

SA

0 0.06 0.1 0.14 0.18 0.24 0.3 0.4 1 (> 0.5)

SB

ρ PSD

0.994 0.938 0.812 0.677 0.498 0.302 0.170 0.052 0.015

0.999 0.944 0.813 0.681 0.496 0.302 0.165 0.038 0.0001

spectrum of the input signals. To 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 [14]:

Fig. 5. The measured and simulated (gamma) distribution for PDF_A, PDF_B and PDF of the BPD output.

and 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 and RF amplifier noise. We estimated the cumulative PDF of these noises by measuring the output histogram when disconnecting the input signals from the BPD. This noise PDF was convolved with the gamma PDF to predict the measured PDF. The measured PDF of disjoint signals S A and 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. These results confirm that the gamma approximation is a good one, and that we have good estimates of the measurement noise, M factor, H ( f ) , and 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 and find the parameters of the gamma distribution by manipulating the measured PSD of the input signals using the following procedure: 1) Decompose the PSD of input signals to disjoint and common parts. 2) Use the PSD of each disjoint part to calculate an M factor and a mean value from (10) and (11), respectively. 3) Obtain the gamma PDF for each disjoint signal using (9) and the appropriate M factor and 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 . The 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 and the correlation coefficient approaches one. Clearly the degree of correlation is determined by the common and disjoints

S I ( Ω) = α

+∞

∫ S (ω )S (ω + Ω) dω

(12)

−∞

where α is a constant. The total average output power is given by: +∞

P=

∫ S ( Ω) H (Ω) I

2

dΩ

(13)

−∞

where the integrand represents the PSD of the filtered photocurrent. We define 2 Pc ρ PSD = (14) P1 + P2 Here, Pc , P1 and P2 are the detected average power of the common spectrum, the total spectrum S1 and the total spectrum S 2 , respectively. Table I, compares the ρ in (1), based on the measured variances, to ρ PSD in (14), calculated for two signals with the same PSDs but with center wavelength offset of Δλ . We increase overlapping spectra by decreasing Δλ . In the experiment, we obtained the maximum correlation coefficient of 0.99. The small difference is due to slight mismatch in the spectral shape of filters. We conclude from Table I that knowledge of the input PSD and 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 0, 0.3, 0.8 and 0.93. The corresponding measured and simulated PDFs of the BPD output (in log scale) are shown in Fig. 6 (b). The difference between measurement and simulation is negligible, and 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,

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ρ =

ρ =

Fig. 6. (a) The PSD of input signals and (b) the measured and simulated distribution of the BPD output for different correlation coefficients.

detection, IEEE Photonics Technology Letters, 17(2), 2005. [5] T. Kawanishi, et al., 10 Gbit/s FSK transmission over 130 km SMF using group delay compensated balance detection, Optical Fiber Communication Conference OFC/NFOEC, Anaheim, CA, March 2005. [6] B. Masella, and X. Zhang, A Novel Single Wavelength Balanced System for Radio Over Fiber Links, IEEE Photonics Tech. Letters, 18(1), 2006. [7] A. Gh. Podoleanu, Unbalanced versus balanced operation in an optical coherence tomography system, Applied Optics, 39(1), 2000. [8] C. Wree, et al., Optical Coherent Receivers for 2.5 and 5Gb/s, IEEE/LEOS Annual Meeting, Sydney, Australia, 2005. [9] D. T. K. Tong, et al., Optoelectronic Phase-Locked Loop with Balanced Photodetection for Clock Recovery in High-Speed Optical Time-Division-Multiplexed Systems, IEEE Photonics Tech. Letters, 12(8), 2000. [10] J. W. Goodman, Statistical Optics (New York: Wiley, 2000). [11] E. Forestieri, Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering, IEEE Journal of Lightwave Technology, 18(11), 2000. [12] C. Lawetz, J. Cartledge, Performance of optically preamplified receivers with Fabry-Perot optical filters, IEEE Journal of Lightwave Technology, 14 (11), 1996. [13] J. Wing, J. M. Kahn, Impact of chromatic and polarization mode dispersion on DPSK systems using interferometric demodulation and direct detection, IEEE Journal of Lightwave Technology, 22(2), 2004 [14] G.-H. Duan and E. Georgiev, Non-white photodetection noise at the output of an optical amplifier: Theory and experiment, IEEE Journal of Quantum Electron., 37(8), 2001, 1008–1014. [15] A. Papoulis, Probability, Random Variables, and Stochastic Processes (New York: McGraw-Hill, 2002).

as the bit error rate or other performance parameters can be estimated via the PDF. In this paper, we proposed and 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 and proposed the method in order to find the PDF of BPD output. Comparison of the measured and simulated PDFs confirmed that the proposed method is efficient and the approximation in our analysis is valid. These 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 [1] M. Kavehrad and D. Zaccarin, Optical code-divisionmultiplexed systems based on spectral encoding of noncoherent sources, IEEE Journal of Lightwave Technology, 13(3), 1995, 534–545. [2] S. Ayotte, M. Rochette, J. Magné, L. A. Rusch et S. LaRochelle, Experimental Verification and Capacity Prediction of FE-OCDMA Using Superimposed FBG, IEEE Journal of Lightwave Technology, 23(2), 2005, 724731. [3] J. Penon, Z. A. El-Sahn, L. A. Rusch, and S. LaRochelle, Spectral Amplitude Coded OCDMA Optimized for a Realistic FBG Frequency Response, submitted to IEEE Journal of Lightwave Technology [4] S. Kim, et al., 10-Gb/s temporally coded optical CDMA system using bipolar modulation/balanced

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