Abstract

We derive an exact expression for the continuous-wave signal-to-noise ratio of filtered incoherent light, which applies to arbitrary optical filtering and arbitrary receiver electrical response. We demonstrate a simple method for making accurate measurements of continuous-wave signal-to-noise ratio, and compare these measurements with values computed from independent measurements of optical-filter coherence time and receiver electrical response; our measurements and computed values agree to within ±0.2dB.

© 2017 Optical Society of America

1. INTRODUCTION

The noise in photoelectric mixing of incoherent light has been a subject of interest for a long time in spectroscopy [1,2], imaging optics [3], fiber-optic amplifiers [4], and communication systems based on spectrum slicing [57]. The signal-to-noise ratio (SNR) of filtered incoherent light measured using a photodetector with response time τD in comparison with the coherence time τC of the optical beam is given by [8]

SNR=i(t)2var(i(t))=1g(2)(0)D1=τDτC,
where i(t) is the time-dependent detector current, i(t) and var(i(t)) are the expected value and variance of that current, and g(2)(0)D is the second-order degree of coherence of the light beam averaged over the response time of the detector.

The analysis leading to Eq. (1) has been revisited in optical communications employing spectrum-sliced spontaneous emission [5,6,9], where the excess noise was estimated for incoherent light filtered with a variety of optical filter and receiver electrical bandwidth shapes, all under the same assumption: that the measurement time is much longer than the coherence time of the optical source. The resulting expression for SNR, consolidated from Refs. [6,7,10,11], and corrected for an arbitrary degree of coherence [3], can be written as

SNR(21+P2)·12BEτC,
where BE is the equivalent noise bandwidth (ENB) of the receiver (in Hertz), τC the optical coherence time, and P the degree of polarization. The ENB BE of the receiver is defined as
BE=0|H(f)|2df|H(0)|2,
where H(f) is the electrical transfer function of the receiver. The coherence time τC, commonly expressed with optical bandwidth BO=1/τC, can be computed from the optical-signal electric-field autocorrelation RO(τ) [3] or, equivalently, from the double-sided power-spectral density S(ω) of the optical field [7,10]:
τC=|RO(τ)RO(0)|2dτ=2π|S(ω)|2dω(S(ω)dω)2.

The degree of polarization P is defined as the ratio of optical powers of the polarized fraction to the total power in the optical beam [3]. Note that incoherent unpolarized light exhibits a larger SNR relative to polarized light for the same 2BEτC. The 2/(1+P2) term originates from combining two uncorrelated and incoherent light beams with varying degrees of polarization [3].

The essential condition for the validity of Eq. (2) is that the optical filter bandwidth BO is much greater than the electrical filter bandwidth BE, namely, Eq. (2) is valid for large SNR values (SNR1). We state this condition as

2BEτC1.

Equation (2) has been used for estimating spontaneous-spontaneous emission beat noise in spectrum-sliced wavelength division multiplexed (WDM) communication systems [5,12] and its impact on the bit-error-rate (BER) [13]. The key feature of Eq. (2) is that SNR is independent of light intensity and hence places an ultimate limit on link performance [5,6].

In most applications, the condition in Eq. (5) applies and the abovementioned Eq. (2) is sufficient for estimating the SNR of filtered incoherent light. As a consequence, the focus of previous reports has been on deriving expressions for coherence times [Eq. (4)] for specific optical channels, for example, Lorentz and Gaussian channels [10,11,14], combined with various electrical filters, such as integrate-and-dump receivers [3,10].

What has not been addressed is the range of validity of Eq. (2) and how to compute the SNR for arbitrary optical and electrical bandwidth characteristics when the condition in Eq. (5) has not been met. To answer this question, in Section 2, we provide a generalization of Eq. (2) that is exact for arbitrary optical and electrical characteristics. We then estimate the error in SNR resulting from using Eq. (2) rather than the generalized form for Gaussian and Lorentz line shapes. In Section 3, we derive this generalized expression, while in Section 4, we describe a simple measurement of continuous-wave noise and signal-to-noise ratio and show that the measured values of SNR match those computed from known optical and electrical characteristics to within ±0.2dB.

2. ARBITRARY OPTICAL AND ELECTRICAL FILTERS

For arbitrary optical power spectral density S(ω) and arbitrary electrical filter transfer function H(ω), the signal-to-noise ratio of incoherent light is given by

1SNR=(1+P22)·RE(τ)H2(0)|RO(τ)RO(0)|2dτ.

Here, RE(τ) is the autocorrelation of the electrical filter and RO(τ) the electric-field autocorrelation of the optical filter. The integral kernel is rather intuitive: it comprises a product of two functions, one of which contains the properties of the electrical filter, and the other the properties of the optical filter. When Eq. (5) is applied to Eq. (6), RO(τ) becomes much sharper than RE(τ) and effectively extracts RE(0) to the front of the integral and makes Eq. (6) converge to Eq. (2). In the other limit, when 2BEτC1, RE(τ) becomes sharp relative to RO(τ) and the integral converges to unity resulting in SNR=(1+P2)/2 as expected [3].

We use Eq. (6) to determine the range of validity of Eq. (2) for Gaussian and Lorentzian line shapes combined with a Gaussian electrical filter. We select a Gaussian electrical filter because of its similarity to Bessel filters, which are used in optical communications [15] due to their maximally flat group-delay characteristics.

The transfer function and autocorrelation of a Gaussian receiver with noise bandwidth BE is given by

|H(ω)|2=exp(πf24BE2),RE(τ)|H(0)|2=2BEexp(4πBE2τ2).

For a Gaussian line shape, we obtain optical power spectral density (PSD) and associated normalized autocorrelation function as

S(ω)=exp(ω2τC22π),|RO(τ)RO(0)|2=exp(πτ2τC2).

By inserting Eqs. (7) and (8) into Eq. (6) and integrating, we find the SNR to be

1SNR=(1+P22)2BEτC1+(2BEτC)2.

From here we see that Eq. (2) underestimates the SNR when 2BEτC becomes large. For example, when 2BEτC3/4, SNR will be underestimated by at least 20% (1 dB) when using Eq. (2) rather than Eq. (9).

For the Lorentzian line shape, we define optical PSD and associated square of the normalized autocorrelation function as

S(ω)=11+(ωτC)2,|RO(τ)RO(0)|2=exp(2|τ|τC).

Inserting Eqs. (7) and (10) into Eq. (6) and integrating, we obtain

1SNR=(1+P22)·exp(1π·(2BEτC)2)·erfc(1π·2BEτC).

From here we find that for 2BEτC1/2, the SNR is underestimated by 20% (1 dB) if one uses Eq. (2) rather than Eq. (11). The Lorentz line shape error is larger for a given 2BEτC value because the autocorrelation in Eq. (10) is sharper than in Eq. (8) and makes the integral in Eq. (6) more sensitive to the rate of electrical filter autocorrelation spreading. The above estimates of SNR illustrate that the accuracy of Eq. (2) depends on the actual shape of the optical and electrical transfer functions, namely, that knowing the product 2BEτC is not sufficient for accurate estimate of the SNR when the condition in Eq. (5) no longer holds.

In spectrum-sliced systems and other applications where incoherent light is filtered, the PSD is determined by the array-waveguide grating, which comes with a variety of transmission spectra (see examples in Fig. 4). In this case, rather than obtaining the SNR analytically (as shown above), one should measure the optical filter transmission and the receiver transfer function, and then numerically perform the integration in Eq. (6) to obtain the expected SNR.

In the next section, we derive Eq. (6), while in last section of this work, we demonstrate that measuring the optical filter and the electrical characteristics of the receiver can be used to accurately estimate the signal-to-noise ratio of filtered incoherent light using a simple method for continuous-wave SNR measurement.

3. DERIVATION OF EQUATION (6)

We study the optical signal-to-noise ratio (SNR) given by [3]:

SNR=(21+P2)·W2σW2,
where W is the filtered sample of expected light intensity I and σW is the associated noise power measured in the detector current (detector responsivity is calibrated out; explained in later text). Light intensity is treated as a wide-sense stationary stochastic process [16]. Relating SNR to bit-error rate (BER) requires knowing the intensity noise probability density function [17,18]. In Ref. [3], Goodman presented a filtered thermal-light treatment that was applicable to an arbitrary optical filter and an integrate-and-dump receiver (rectangular temporal measurement window). Following Goodman [3], we derive W and W2 as functions of the optical and electrical filter characteristics:
W=limA1AA/2A/2dth(ξt)I(ξ)dξ=h(x){limA1AA/2A/2I(x+t)dt}dx=I·h(x)dx=I·H(0).

Here, H(ω) is the Fourier transform of the electrical-filter impulse response h(t). To obtain the variance, we find the expected value of W2 by integrating the fluctuations over a long period of time:

W2=limA1AA/2A/2h(ηt)h(ξt)I(ξ)I(η)dηdξ.

Using the substitutions ξ=ξt and η=ηt, Eq. (14) is converted to

W2=h(η)h(ξ){limA1AA/2A/2I(t+ξ)I(t+η)dt}dηdξ,
where we recognize the inner integral as an autocorrelation of the light intensity,
RI(ξη)=I(ξ)I(η)=limA1AA/2A/2I(t+ξ)I(t+η)dt.

Note that the beam intensity is a stationary stochastic process. The expected value W2 is now

W2=h(η)h(ξ)RI(ξη)dηdξ.

Further simplification of the integral is obtained by recognizing that the h terms will yield another autocorrelation if we introduce τ=ξη. Setting

RE(τ)=h(η+τ)h(η)dη,
we finally obtain
W2=RE(τ)RI(τ)dτ.

The autocorrelation of the electrical filter RE(τ) is obtained in a straightforward fashion by measuring the frequency response H(ω) and applying Wiener–Khinchin theorem.

The autocorrelation of the intensity of the light beam is proportional to the degree of second-order coherence of light. For filtered incoherent light, the relationship between the first and second degree of coherence is given by [8]

g(2)(τ)=1+|g(1)(τ)|2,
where g(1)(τ) and g(2)(τ) are the first and second orders of coherence, respectively. The degrees of coherence can be related to the autocorrelations of electromagnetic wave intensity RI(τ) and electromagnetic field RO(τ), via
g(1)(τ)=E(t)E(t+τ)E2=RO(τ)RO(0),
g(2)(τ)=I(t)I(t+τ)I2=H2(0)W2RI(τ).

Combining Eqs. (20)–(22) we obtain the optical intensity autocorrelation,

RI(τ)=W2H2(0){1+|RO(τ)RO(0)|2}.

Inserting Eq. (23) into Eq. (19) and then combining Eqs. (13) and (12), we arrive at Eq. (6): a general relation for continuous-wave signal-to-noise ratio as a function of arbitrary optical and electrical filter transfer functions, and degree of polarization that does not require the assumption in Eq. (5). This is not the only way to derive Eq. (6). For example, one can arrive at Eq. (6) from the intermediate step (25) of reference [10] by performing two integrations and applying the Wiener–Khinchin theorem.

4. MEASUREMENT RESULTS

With the limits of Eq. (2) outlined, our goal is to show that SNR values computed from measured optical and electrical transfer functions can, with reasonable certainty, be used to estimate the actual SNR; our data experiments show to within ±0.2dB. To this end, we developed a simple measurement method for making accurate SNR measurements on continuous-wave optical signals.

Measurement of noise and signal-to-noise ratio of filtered incoherent light has been performed using an oscilloscope [7,14], RF spectrum analyzer [6,11,13,19], and evaluated from bit-error rate [17]. We base our continuous-wave noise measurement on the RINXOMA measurement approach from a 10-Gb Ethernet standard [15] which we adapt for continuous-wave measurement by using a separate optical power meter for the average optical power in addition to the RF power meter for the noise. The setup is illustrated in Fig. 1(a). As the source of incoherent light we use an unpolarized broadband light source (BLS) which emits amplified spontaneous emission from an erbium-doped fiber amplifier with an emission spectrum centered at 1550 nm and a 30-nm bandwidth.

 figure: Fig. 1.

Fig. 1. Characterization setups for (a) the SNR measurements and (b) evaluation of ENBs of electrical filters (VOA, variable optical attenuator; ISO, optical isolator; DET, optical detector; LP, electrical low-pass filter).

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The BLS output is first passed through a variable optical attenuator and then through an isolator before it is fed to one of the channels of an arrayed waveguide grating (AWG). The filtered signal is split on a coupler with nominal coupling ratio C12=C14=50%, with port 2 coupled to an optical detector with AC-coupled voltage output and port 4 to a calibrated optical power meter. The fluctuations in the optical signal, converted to electrical noise, are filtered using an electrical filter and fed to a high-bandwidth 26.5-GHz RF power meter. The power spectral density is normalized to its value at low frequency, as needed in Eq. (3). The SNR is then obtained through the ratio of input optical power and its noise fluctuations.

We relate the variance of the AC voltage to optical fluctuations through the optical-detector responsivity coefficient R, measured independently to be 1360±30 [V/W] with termination impedance Z0=50Ω. The detector is AC coupled, and to accurately determine R relative to the optical power meter, we applied a 2-MHz square wave signal on both the optical detector and the optical power meter and then inferred the relative calibration from the average to peak-to-peak values ratio. The dominant source of error in our SNR results is the uncertainty in the value of the optical-detector responsivity. The power PRF measured on the RF power meter is proportional to the sum of the optical signal variance σW2 and the other noise sources in the signal path referenced to the input of the optical head (equivalent input noise is denoted with σOE2): PRF σW2+σOE2. To evaluate W and σW2 simultaneously, we varied the incident optical power Popt using a variable optical attenuator through a wide range to eliminate the effect of the noise of the receiver (as is common in BER measurements). We extract the SNR value from the saturated part of the curve shown in Fig. 2, where the receiver noise is negligible. Furthermore, from the slope of the SNR versus the Popt curve shown in Fig. 2, we extracted the equivalent input noise to be σOE=27.6dBm. We then used this σOE2 value and the SNR value obtained from the saturated part of the curve shown in Fig. 2 to compute the expected shape of the entire relationship. This is shown along the measurement points noted as “computed” in Fig. 2. The excellent fit between shapes drawn by the measured and computed points implies that the optical noise of our receiver does not noticeably depend on intensity.

 figure: Fig. 2.

Fig. 2. Example of SNR measurement results for the unfiltered optical detector given for three different channel spacings.

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The accuracy of our noise measurement critically depends on the value of the responsivity which was accurately evaluated using the above-mentioned measurement. This difficulty could be alleviated if one were to use an optical head that is DC coupled. In this case, the responsivity would affect the noise and the average value of the light intensity equally.

We account for the difference in the coupling coefficients of the optical coupler splitting the average power and variance measurement. The accuracy of this coupling coefficient is important because any error ΔC/C alters the SNR by 2ΔC/C, as SNR is proportional to C/(1C). We calibrated the coupling coefficient C and also checked our results by making two measurements of SNR: one with the power meter connected to port 4 shown in Fig. 1(a), and another in which the coupler exit ports were exchanged. The geometric mean of the two obtained SNR values gives the correct SNR value that does not depend on the coupling coefficient or the loss in the optical coupler. For this method to work efficiently it is preferable that the coupling coefficient is around 50%, so that between the exchanges of the coupler ports, the SNR dependence on the equivalent input noise σOE2 and the power at which saturation (in Fig. 2) occurs remain similar. The SNR values determined from the saturated levels in Fig. 2 are listed in Table 1, noted with “M”.

Tables Icon

Table 1. Measurement Results for ENBs of Electrical Filters, Coherence Times of Optical Filters, and SNR Results Computed from Optical and Electrical Filter Characteristics (C) and from SNR Measurements (M); CS—Channel Separation

We characterized four different electrical filters: the unfiltered optical detector with nominal 3.5-GHz bandwidth, and then the same optical head followed by fourth-order Bessel–Thompson filters with nominal 3-dB bandwidths equal to 940 MHz, 120 MHz, and 94 MHz. Bessel–Thompson filters are used because they have maximally flat group-delay. The ENBs of the filters were measured using the unfiltered BLS as a noise source and measuring the transmitted noise spectra on a spectrum analyzer, as shown in Fig. 1(b). The ENBs computed from the data shown in Fig. 3 are listed in Table 1.

 figure: Fig. 3.

Fig. 3. Electrical filter transmission characteristics normalized to value at zero frequency: data (a)–(c) correspond to 94 MHz, 120 MHz, and 940 MHz low-pass filters inserted after the optical detector, respectively, while data (d) corresponds to the unfiltered optical detector.

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We used three different athermal AWGs as optical filters, with channel separation of 200 GHz, 100 GHz and 50 GHz. We measured the optical transmission S(ω) through a select channel by connecting the BLS to the AWG common port (Fig. 4). The coherence time was then computed by numerically integrating the spectra. The results are listed in Table 1.

 figure: Fig. 4.

Fig. 4. Transmission spectra through one of the channels of each AWG with noted channel separation in gigahertz.

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Finally, we compute the SNR for all combinations of filters (three optical filters and four electrical filters) from their measured τC and BE values. The degree of polarization P was kept at zero, inasmuch as the BLS was an unpolarized source. The computed SNR values are listed in Table 1, noted with “C”. The results in Table 1 show that our method of estimating the SNR of incoherent light from accurately measuring the optical and electrical bandwidth indeed works well for all the tested filters and AWGs. The difference between measured and estimated value is at most 0.2 dB.

For all filter combinations listed in Table 1, we find that 2BEτC<0.1 and hence the correction achieved with using Eq. (6) rather than Eq. (2) is less than the measurement uncertainty. However, a combination of, for example, an array-waveguide grating with a 25-GHz channel separation and a 10-Gbps line rate with a receiver filter with BE7.5GHz, would result in 2BEτC0.6 and a need to use Eq. (6) rather than Eq. (2).

As can be seen from the measurement results, the SNR improves with the increase of the optical filter bandwidth for incoherent light. This somewhat nonobvious phenomenon is expected and has been mathematically treated in multiple references [3,4,11,14]. All incoherent light components within the optical bandwidth BO mix with each other during photodetection and their down-converted intermodulation products add at baseband, but only those components whose frequency falls within the electrical bandwidth BE are detected. This detected baseband signal is hence a sum of approximately BO/BE signals with basically identical noise properties. Signal-to-noise ratio of a sum of noisy signals improves with the number of contributors because the variance of a sum of uncorrelated noisy signals increases with the number of those signals, while the square of the signal increases with the square of the number of contributors [16].

5. CONCLUSION

In conclusion, we derived an expression for the continuous-wave signal-to-noise ratio of filtered incoherent light which is exact for arbitrary optical and electrical filtering and degree of polarization. We used this expression to evaluate the range over which the conventional relationship Eq. (2) is valid. We furthermore confirmed experimentally that measuring the characteristics of the optical and electrical filters with sufficient accuracy can be used to accurately predict the SNR of incoherent light. To this end, we developed a simple method for measuring the SNR.

The importance of our general Eq. (6) lies with the fact that optical filters in WDM networks are becoming commensurate with the electrical filter bandwidths, and using the conventional 2BEτC1 approximation to evaluate spontaneous-spontaneous emission beating noise is reaching the limits of its applicability.

Funding

Hrvatska Zaklada za Znanost (HRZZ) (IP-11-2013-3425).

REFERENCES

1. E. Gerjouy, A. T. Forrester, and W. E. Parkins, “Signal-to-noise ratio in photoelectrically observed beats,” Phys. Rev. 73, 922–923 (1948). [CrossRef]  

2. A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955). [CrossRef]  

3. J. W. Goodman, Statistical Optics (Wiley, 1985).

4. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989). [CrossRef]  

5. M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988). [CrossRef]  

6. J.-S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photon. Technol. Lett. 5, 1458–1461 (1993). [CrossRef]  

7. G. J. Pendock and D. D. Sampson, “Transmission performance of high bit rate spectrum-sliced WDM systems,” J. Lightwave Technol. 14, 2141–2148 (1996). [CrossRef]  

8. R. Loudon, The Quantum Theory of Light (Oxford University, 1970).

9. E. Wong, K. L. Lee, and T. B. Anderson, “Directly modulated self-seeding reflective semiconductor optical amplifiers as colorless transmitters in wavelength division multiplexed passive optical networks,” J. Lightwave Technol. 25, 67–74 (2007). [CrossRef]  

10. J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. 14, 2197–2201 (1996). [CrossRef]  

11. G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37, 1008–1014 (2001). [CrossRef]  

12. C.-H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the home using a PON infrastructure,” J. Lightwave Technol. 24, 4568–4583 (2006). [CrossRef]  

13. W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005). [CrossRef]  

14. G. Kweon and J. Choi, “Dependence of the signal-to-noise ratio for a mixture of coherent and incoherent sources on the incoherent source spectral profile,” Opt. Rev. 10, 185–195 (2003). [CrossRef]  

15. “Relative intensity noise optical modulation (RINxOMA) measuring procedure,” IEEE Standard for Ethernet, IEEE Standard 802.3, Clause 52.9.6 (2015).

16. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).

17. W. Mathlouthi, M. Menif, and L. A. Rusch, “Beat noise effects on spectrum-sliced WDM,” Proc. SPIE 5260, 44–54 (2003). [CrossRef]  

18. D. Marcuse, “Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise,” J. Lightwave Technol. 9, 505–513 (1991). [CrossRef]  

19. A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004). [CrossRef]  

References

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  1. E. Gerjouy, A. T. Forrester, and W. E. Parkins, “Signal-to-noise ratio in photoelectrically observed beats,” Phys. Rev. 73, 922–923 (1948).
    [Crossref]
  2. A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
    [Crossref]
  3. J. W. Goodman, Statistical Optics (Wiley, 1985).
  4. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
    [Crossref]
  5. M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
    [Crossref]
  6. J.-S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photon. Technol. Lett. 5, 1458–1461 (1993).
    [Crossref]
  7. G. J. Pendock and D. D. Sampson, “Transmission performance of high bit rate spectrum-sliced WDM systems,” J. Lightwave Technol. 14, 2141–2148 (1996).
    [Crossref]
  8. R. Loudon, The Quantum Theory of Light (Oxford University, 1970).
  9. E. Wong, K. L. Lee, and T. B. Anderson, “Directly modulated self-seeding reflective semiconductor optical amplifiers as colorless transmitters in wavelength division multiplexed passive optical networks,” J. Lightwave Technol. 25, 67–74 (2007).
    [Crossref]
  10. J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. 14, 2197–2201 (1996).
    [Crossref]
  11. G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37, 1008–1014 (2001).
    [Crossref]
  12. C.-H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the home using a PON infrastructure,” J. Lightwave Technol. 24, 4568–4583 (2006).
    [Crossref]
  13. W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
    [Crossref]
  14. G. Kweon and J. Choi, “Dependence of the signal-to-noise ratio for a mixture of coherent and incoherent sources on the incoherent source spectral profile,” Opt. Rev. 10, 185–195 (2003).
    [Crossref]
  15. “Relative intensity noise optical modulation (RINxOMA) measuring procedure,” (2015).
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).
  17. W. Mathlouthi, M. Menif, and L. A. Rusch, “Beat noise effects on spectrum-sliced WDM,” Proc. SPIE 5260, 44–54 (2003).
    [Crossref]
  18. D. Marcuse, “Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise,” J. Lightwave Technol. 9, 505–513 (1991).
    [Crossref]
  19. A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004).
    [Crossref]

2007 (1)

2006 (1)

2005 (1)

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
[Crossref]

2004 (1)

A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004).
[Crossref]

2003 (2)

G. Kweon and J. Choi, “Dependence of the signal-to-noise ratio for a mixture of coherent and incoherent sources on the incoherent source spectral profile,” Opt. Rev. 10, 185–195 (2003).
[Crossref]

W. Mathlouthi, M. Menif, and L. A. Rusch, “Beat noise effects on spectrum-sliced WDM,” Proc. SPIE 5260, 44–54 (2003).
[Crossref]

2001 (1)

G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37, 1008–1014 (2001).
[Crossref]

1996 (2)

J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. 14, 2197–2201 (1996).
[Crossref]

G. J. Pendock and D. D. Sampson, “Transmission performance of high bit rate spectrum-sliced WDM systems,” J. Lightwave Technol. 14, 2141–2148 (1996).
[Crossref]

1993 (1)

J.-S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photon. Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

1991 (1)

D. Marcuse, “Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise,” J. Lightwave Technol. 9, 505–513 (1991).
[Crossref]

1989 (1)

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[Crossref]

1988 (1)

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

1955 (1)

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[Crossref]

1948 (1)

E. Gerjouy, A. T. Forrester, and W. E. Parkins, “Signal-to-noise ratio in photoelectrically observed beats,” Phys. Rev. 73, 922–923 (1948).
[Crossref]

Anderson, T. B.

Bickers, L.

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

Chen, W.

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
[Crossref]

Choi, J.

G. Kweon and J. Choi, “Dependence of the signal-to-noise ratio for a mixture of coherent and incoherent sources on the incoherent source spectral profile,” Opt. Rev. 10, 185–195 (2003).
[Crossref]

Chung, Y. C.

J.-S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photon. Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

DiGiovanni, D. J.

J.-S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photon. Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

Duan, G.-H.

G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37, 1008–1014 (2001).
[Crossref]

Evans, J. S.

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
[Crossref]

Forrester, A. T.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[Crossref]

E. Gerjouy, A. T. Forrester, and W. E. Parkins, “Signal-to-noise ratio in photoelectrically observed beats,” Phys. Rev. 73, 922–923 (1948).
[Crossref]

Georgiev, E.

G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37, 1008–1014 (2001).
[Crossref]

Gerjouy, E.

E. Gerjouy, A. T. Forrester, and W. E. Parkins, “Signal-to-noise ratio in photoelectrically observed beats,” Phys. Rev. 73, 922–923 (1948).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gudmundsen, R. A.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[Crossref]

Horning, S.

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

Hunwicks, A. R.

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

Ibsen, M.

A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004).
[Crossref]

Johnson, P. O.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[Crossref]

Kim, B. Y.

Kweon, G.

G. Kweon and J. Choi, “Dependence of the signal-to-noise ratio for a mixture of coherent and incoherent sources on the incoherent source spectral profile,” Opt. Rev. 10, 185–195 (2003).
[Crossref]

Lee, C.-H.

Lee, J.-S.

J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. 14, 2197–2201 (1996).
[Crossref]

J.-S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photon. Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

Lee, K. L.

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University, 1970).

Marcuse, D.

D. Marcuse, “Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise,” J. Lightwave Technol. 9, 505–513 (1991).
[Crossref]

Mathlouthi, W.

W. Mathlouthi, M. Menif, and L. A. Rusch, “Beat noise effects on spectrum-sliced WDM,” Proc. SPIE 5260, 44–54 (2003).
[Crossref]

McCoy, A. D.

A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004).
[Crossref]

Menif, M.

W. Mathlouthi, M. Menif, and L. A. Rusch, “Beat noise effects on spectrum-sliced WDM,” Proc. SPIE 5260, 44–54 (2003).
[Crossref]

Methley, S. G.

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

Olsson, N. A.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).

Parkins, W. E.

E. Gerjouy, A. T. Forrester, and W. E. Parkins, “Signal-to-noise ratio in photoelectrically observed beats,” Phys. Rev. 73, 922–923 (1948).
[Crossref]

Pendock, G. J.

G. J. Pendock and D. D. Sampson, “Transmission performance of high bit rate spectrum-sliced WDM systems,” J. Lightwave Technol. 14, 2141–2148 (1996).
[Crossref]

Reeve, M. H.

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

Richardson, D. J.

A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004).
[Crossref]

Rusch, L. A.

W. Mathlouthi, M. Menif, and L. A. Rusch, “Beat noise effects on spectrum-sliced WDM,” Proc. SPIE 5260, 44–54 (2003).
[Crossref]

Sampson, D. D.

G. J. Pendock and D. D. Sampson, “Transmission performance of high bit rate spectrum-sliced WDM systems,” J. Lightwave Technol. 14, 2141–2148 (1996).
[Crossref]

Shieh, W.

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
[Crossref]

Sorin, W. V.

Thomsen, B. C.

A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004).
[Crossref]

Tucker, R. S.

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
[Crossref]

Wong, E.

Yi, X.

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
[Crossref]

Zhao, W.

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

Electron. Lett. (1)

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Horning, “LED Spectral slicing for single-mode local loop applications,” Electron. Lett. 24, 389–390 (1988).
[Crossref]

IEEE J. Quantum Electron. (1)

G.-H. Duan and E. Georgiev, “Non-white photodetection noise at the output of an optical amplifier: theory and experiment,” IEEE J. Quantum Electron. 37, 1008–1014 (2001).
[Crossref]

IEEE Photon. Technol. Lett. (3)

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. S. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett. 17, 2484–2486 (2005).
[Crossref]

J.-S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for multichannel WDM applications,” IEEE Photon. Technol. Lett. 5, 1458–1461 (1993).
[Crossref]

A. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SO-based noise reduction,” IEEE Photon. Technol. Lett. 16, 680–682 (2004).
[Crossref]

J. Lightwave Technol. (6)

D. Marcuse, “Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise,” J. Lightwave Technol. 9, 505–513 (1991).
[Crossref]

G. J. Pendock and D. D. Sampson, “Transmission performance of high bit rate spectrum-sliced WDM systems,” J. Lightwave Technol. 14, 2141–2148 (1996).
[Crossref]

E. Wong, K. L. Lee, and T. B. Anderson, “Directly modulated self-seeding reflective semiconductor optical amplifiers as colorless transmitters in wavelength division multiplexed passive optical networks,” J. Lightwave Technol. 25, 67–74 (2007).
[Crossref]

J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. 14, 2197–2201 (1996).
[Crossref]

C.-H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the home using a PON infrastructure,” J. Lightwave Technol. 24, 4568–4583 (2006).
[Crossref]

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[Crossref]

Opt. Rev. (1)

G. Kweon and J. Choi, “Dependence of the signal-to-noise ratio for a mixture of coherent and incoherent sources on the incoherent source spectral profile,” Opt. Rev. 10, 185–195 (2003).
[Crossref]

Phys. Rev. (2)

E. Gerjouy, A. T. Forrester, and W. E. Parkins, “Signal-to-noise ratio in photoelectrically observed beats,” Phys. Rev. 73, 922–923 (1948).
[Crossref]

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691–1700 (1955).
[Crossref]

Proc. SPIE (1)

W. Mathlouthi, M. Menif, and L. A. Rusch, “Beat noise effects on spectrum-sliced WDM,” Proc. SPIE 5260, 44–54 (2003).
[Crossref]

Other (4)

“Relative intensity noise optical modulation (RINxOMA) measuring procedure,” (2015).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).

J. W. Goodman, Statistical Optics (Wiley, 1985).

R. Loudon, The Quantum Theory of Light (Oxford University, 1970).

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Figures (4)

Fig. 1.
Fig. 1. Characterization setups for (a) the SNR measurements and (b) evaluation of ENBs of electrical filters (VOA, variable optical attenuator; ISO, optical isolator; DET, optical detector; LP, electrical low-pass filter).
Fig. 2.
Fig. 2. Example of SNR measurement results for the unfiltered optical detector given for three different channel spacings.
Fig. 3.
Fig. 3. Electrical filter transmission characteristics normalized to value at zero frequency: data (a)–(c) correspond to 94 MHz, 120 MHz, and 940 MHz low-pass filters inserted after the optical detector, respectively, while data (d) corresponds to the unfiltered optical detector.
Fig. 4.
Fig. 4. Transmission spectra through one of the channels of each AWG with noted channel separation in gigahertz.

Tables (1)

Tables Icon

Table 1. Measurement Results for ENBs of Electrical Filters, Coherence Times of Optical Filters, and SNR Results Computed from Optical and Electrical Filter Characteristics (C) and from SNR Measurements (M); CS—Channel Separation

Equations (23)

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S N R = i ( t ) 2 var ( i ( t ) ) = 1 g ( 2 ) ( 0 ) D 1 = τ D τ C ,
S N R ( 2 1 + P 2 ) · 1 2 B E τ C ,
B E = 0 | H ( f ) | 2 d f | H ( 0 ) | 2 ,
τ C = | R O ( τ ) R O ( 0 ) | 2 d τ = 2 π | S ( ω ) | 2 d ω ( S ( ω ) d ω ) 2 .
2 B E τ C 1 .
1 S N R = ( 1 + P 2 2 ) · R E ( τ ) H 2 ( 0 ) | R O ( τ ) R O ( 0 ) | 2 d τ .
| H ( ω ) | 2 = exp ( π f 2 4 B E 2 ) , R E ( τ ) | H ( 0 ) | 2 = 2 B E exp ( 4 π B E 2 τ 2 ) .
S ( ω ) = exp ( ω 2 τ C 2 2 π ) , | R O ( τ ) R O ( 0 ) | 2 = exp ( π τ 2 τ C 2 ) .
1 S N R = ( 1 + P 2 2 ) 2 B E τ C 1 + ( 2 B E τ C ) 2 .
S ( ω ) = 1 1 + ( ω τ C ) 2 , | R O ( τ ) R O ( 0 ) | 2 = exp ( 2 | τ | τ C ) .
1 S N R = ( 1 + P 2 2 ) · exp ( 1 π · ( 2 B E τ C ) 2 ) · erfc ( 1 π · 2 B E τ C ) .
S N R = ( 2 1 + P 2 ) · W 2 σ W 2 ,
W = lim A 1 A A / 2 A / 2 d t h ( ξ t ) I ( ξ ) d ξ = h ( x ) { lim A 1 A A / 2 A / 2 I ( x + t ) d t } d x = I · h ( x ) d x = I · H ( 0 ) .
W 2 = lim A 1 A A / 2 A / 2 h ( η t ) h ( ξ t ) I ( ξ ) I ( η ) d η d ξ .
W 2 = h ( η ) h ( ξ ) { lim A 1 A A / 2 A / 2 I ( t + ξ ) I ( t + η ) d t } d η d ξ ,
R I ( ξ η ) = I ( ξ ) I ( η ) = lim A 1 A A / 2 A / 2 I ( t + ξ ) I ( t + η ) d t .
W 2 = h ( η ) h ( ξ ) R I ( ξ η ) d η d ξ .
R E ( τ ) = h ( η + τ ) h ( η ) d η ,
W 2 = R E ( τ ) R I ( τ ) d τ .
g ( 2 ) ( τ ) = 1 + | g ( 1 ) ( τ ) | 2 ,
g ( 1 ) ( τ ) = E ( t ) E ( t + τ ) E 2 = R O ( τ ) R O ( 0 ) ,
g ( 2 ) ( τ ) = I ( t ) I ( t + τ ) I 2 = H 2 ( 0 ) W 2 R I ( τ ) .
R I ( τ ) = W 2 H 2 ( 0 ) { 1 + | R O ( τ ) R O ( 0 ) | 2 } .

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