transfer standard for the spectral density of relative intensity noise of optical fiber sources near...

750 J. Opt. Soc. Am. B/Vol. 18, No. 6 /June 2001 G. E. Obarski and J. D. Splett

Transfer standard for the spectral densityof relative intensity noise

of optical fiber sources near 1550 nm

Gregory E. Obarski and Jolene D. Splett

National Institute of Standards and Technology, Boulder, Colorado 80303

Received May 29, 2000; revised manuscript received December 11, 2000

We have developed a transfer standard for the spectral density of relative intensity noise (RIN) of optical fibersources near 1550 nm. Amplified spontaneous emission (ASE) from an erbium-doped fiber amplifier (EDFA),when it is optically filtered over a narrow band (,5 nm), yields a stable RIN spectrum that is practically con-stant to several tens of gigahertz. The RIN is calculated from the power spectral density as measured with acalibrated optical spectrum analyzer. For a typical device it is 2110 dB/Hz, with uncertainty <0.12 dB/Hz.The invariance of the RIN under attenuation yields a considerable dynamic range with respect to rf noise lev-els. Results are compared with those from a second method that uses a distributed-feedback laser (DFB) thathas a Poisson-limited RIN. Application of each method to the same RIN measurement system yieldsfrequency-dependent calibration functions that, when they are averaged, differ by <0.2 dB. © 2001 OpticalSociety of America

OCIS codes: 120.4800, 270.2500, 060.2330, 060.2320.

1. INTRODUCTIONIncreasing demand for greater bandwidth in optical fibercommunications has brought to fruition laser transmit-ters and optical fiber amplifiers with low-magnitude rela-tive intensity noise (RIN) and noise figures, respectively.The noise figure of an optical amplifier can depend on thesignal input RIN.1 One can use a RIN measurement sys-tem that employs a rf spectrum analyzer in combinationwith a low-RIN laser (RIN , 2160 dB/Hz) to determinethe spectral density of the noise figure of a fiber amplifier.Such electrical methods are important because contribu-tions of multiple path interference, which are neglectedwhen optical techniques are used, are included in thenoise figure measurement.1–5 RIN can limit system per-formance in optical communications systems.6–9 Onecan also use it to determine the resonance frequency, theintrinsic bandwidth, and the damping factor of lasers.10

The appearance of commercial distributed-feedback(DFB) lasers and diode-pumped Nd:YAG lasers that haveRINs with values of ,2170 dB/Hz requires sensitive, ac-curate measurement systems. Thus we developed atransfer standard for precise calibration of RIN measure-ment systems that employ rf spectrum analyzers. Thestandard is an erbium-doped fiber amplifier (EDFA) towhose output are coupled a linear polarizer and a narrow-band filter. We characterize a typical device for wave-lengths near 1550 nm and rf frequencies in the 0.1–1.1-GHz range. The spectral density of RIN, however, ispractically constant from zero to many tens of gigahertz,rendering it suitable for calibrations at even greaterbandwidths.

The amplitude noise of the standard is amplified spon-taneous emission (ASE) in the EDFA. Such fluctuationsobey Bose–Einstein statistics, which can be approximatedby thermal or Gaussian light.11 The invariance of the

0740-3224/2001/060750-12$15.00 ©

spectral density of the RIN of thermal light under attenu-ation yields high accuracy and a considerable dynamicrange with respect to rf noise levels; it also allows for theuse of optical fiber connectors. The RIN of a typical de-vice is 2110 dB/Hz, with an uncertainty of ,0.15 dB/Hz.To confirm the properties of the standard, we developed asecond calibration method based on a DFB laser that hasPoisson-limited RIN (see Section 8 below). Application ofeach method to a RIN measurement system yieldsfrequency-dependent calibration functions (denoted Kap-pas herein) that are nearly identical in shape and whoseequivalence can be derived by application of the RINequations for each method to calibration of the RIN sys-tem. The difference between the Kappas of the twomethods, averaged over a 1-GHz bandwidth, falls into the0.04–0.2-dB range. Statistical tests demonstrate thatthe variability in Kappa of the two methods is similar.Thus the RIN standard compares favorably with that of asecond method, i.e., the use of a Poisson-limited laser.

We denote the two EDFAs and the three filters used inthis study EDFA1 and EDFA2, and F1, F2, and F3, re-spectively. A standard formed by EDFA2, F3, and linearpolarizer P is written as EDFA2 1 F3, with the P omit-ted; the polarizer is assumed present unless otherwisespecified.

2. DEFINITION AND MEASUREMENT OFRELATIVE INTENSITY NOISERIN9,12–14 can be precisely calculated from the autocorre-lation integral of optical power fluctuations divided by to-tal power squared. These temporal fluctuations can alsobe expressed in terms of their frequency spectrum bymeans of the Fourier transform of the autocorrelation in-tegral. An optical source of output power P(t) and fluc-

2001 Optical Society of America

G. E. Obarski and J. D. Splett Vol. 18, No. 6 /June 2001/J. Opt. Soc. Am. B 751

tuation dP(t) has a total RIN, RINT , given by the ratio ofthe mean square of the fluctuation to the square of the av-erage power:

RINT 5^dP~t !2&

^P~t !&2 , (1)

where the time average ^dP(t)2& arises from the autocor-relation function ^dP(t)dP(t 1 t)& evaluated at timet 5 0. The total RIN can be represented in the fre-quency domain by definition of a RIN spectral density, towhich we refer simply as the RIN. Then RINT is also theintegral of the RIN, R(n), over all frequencies:

RINT 5 E0

`

R~n!dn, (2)

where n is the optical frequency (in hertz). The spectraldensity of the RIN, R(n), is derived by application of theWiener–Khintchine theorem to the autocorrelation func-tion l(t). By equating the right-hand sides of Eqs. (1)and (2), one can show that R(n) is

R~n! 5 2 E2`

`

l~t!exp~i2pnt!dt. (3)

Thus we define the RIN of an optical source to be the RINspectral density R(n) and the integral of the RIN to be thetotal RIN.

For classical states of light, the spectral density of am-plitude fluctuations has a minimum of the standard Pois-son limit, to which may be added some excess noise.Thus the RIN spectral density also consists of PoissonRIN and excess RIN. Whereas the excess RIN propa-gates unchanged through the system, the Poisson RIN de-pends on system losses. Expressed in terms of RIN, thestandard quantum limit is the Poisson RIN. The totalRIN is the sum of the spectral integrals of excess RIN andPoisson RIN.15 Thus it is unitless. To develop a RINstandard we characterize two different RIN sources. Thefirst source (RIN standard) is ASE from an EDFA, whichcan be approximated by thermal light. The secondsource is a laser with Poisson-limited RIN.

For light that obeys a Poisson distribution, the vari-ance in the photon number is proportional to the photonnumber. Thus the square of the optical power fluctua-tions is proportional to the optical power. When it is rep-resented by a single-sided noise spectrum, the PoissonRIN at a single optical frequency n8 is 2hn8/P0 , where his Planck’s constant, n is photon frequency, and P0 is op-tical power. In electrical units it is 2q/i, where q is the

Fig. 1. Basic RIN measurement system.

electron charge, i 5 hqP0 /hn is the photocurrent, and his the photodetector quantum efficiency. Thus the Pois-son RIN in a real detection circuit increases as 1/h.

A RIN measurement system is shown in Fig. 1. Thebeam traverses a lossy medium, such as an attenuator,and is collected by the photodetector. R(n) is to be deter-mined at reference plane A before any losses. The Pois-son RIN increases at plane B as a result of losses andagain at plane C because of inefficiency in the photodetec-tion process. System efficiency can be combined into onefactor, hT 5 h(1 2 L), where L is the fractional loss be-fore the detector. The excess RIN, however, propagatesunchanged through the system. If RC(n) is the mea-sured RIN at plane C, then the RIN is

RA~n! 5 RC~n! 2 ~2q/i !~1 2 hT!. (4)

We determine the excess RIN, Rex(n), by subtracting themeasured Poisson or shot-noise RIN, 2q/i, from Rc(n),giving

Rex~n! 5 RC~n! 2 2q/i. (5)

Equation (5) is equivalent to subtracting the Poisson RIN,represented here as 2hn8/P0 , from the RIN at plane A:

Rex~n! 5 RA~n! 2 2hn8/P0 , (6)

where P0 is the laser power at plane A and n8 is the laserfrequency.

Measuring RIN in the electrical domain entails using abias tee to send the dc photocurrent to an ammeter whilethe ac noise is amplified and then displayed on a rf elec-trical spectrum analyzer (ESA). We denote the electricalfrequency (referenced to the baseband) by f. Then theRIN is the noise power per unit bandwidth, dPe( f ),weighted with the electrical frequency-dependent calibra-tion function k( f ) of the detection system and divided bythe electrical dc power Pe . Thus

R~ f ! 5k~ f !dPe~ f !

Pe, (7)

where dPe( f ) is the noise after subtraction of the thermalnoise floor. Note that k( f ) is proportional to the fre-quency response of the system and can be obtained by useof a broadband, flat source of known RIN.

3. FORM OF THE RELATIVE INTENSITYNOISE STANDARD AND A SECONDARYMETHODThe RIN transfer standard is shown in Fig. 2. The ASEfrom an EDFA is fed through a linear polarizer (P) fol-lowed by an optical filter (F) of 1–3-nm bandwidth, cen-tered near 1550 nm.16,17 The EDFA has a built-in opticalisolator and was chosen for its semirugged qualities. Tosimplify operation for the customer, we arranged for asimplified front panel display with only an on–off powerswitch. The linear polarizer eliminates the potential forpolarization imbalance between orthogonally polarizedmodes that might occur from fiber bending. The excessRIN of this device is much greater than the Poisson RIN,so the total RIN can be considered excess RIN. Theshape and the bandwidth of the filter determine the RIN.


The optical power spectral density (OPSD) from the stan-dard is measured by a diffraction-grating type of opticalspectrum analyzer (OSA). For a single measurement,the OSA is set to average four sweeps on high sensitivitywith the resolution bandwidth set at the minimum of 0.05nm. High-quality threaded, physical-contact optical-fiber connectors and single-mode fiber are used. Points1–4 represent locations where connectors were routinelydisconnected to change components. Connector matingcan change the power transmitted through the fiber linkbut not the RIN, which is invariant under attenuation.

4. PRINCIPLESA. Spectral Density of the Relative Intensity NoiseStandardThe spectral density of the RIN standard, as approxi-mated by thermal or Gaussian light, must be obtainedfrom a combination of measurement and theory-basedcalculation, as it cannot be measured directly. Althoughthe spectral density theory has been developed,11,18 weneed an expression for the spectral density of RIN explic-itly that correctly accounts for all scale factors.19 Weshow that the RIN can be calculated from the OPSD asmeasured on an OSA. Contributions from the Poisson orstandard quantum limit are considered negligible com-pared with those of the ASE.

It is well known that from the autocorrelation of thephotocurrent fluctuations arises the frequency spectrumof the current fluctuations in terms of measurablequantities.11,18 By application of the properties of the de-gree of coherence of the light field, the spectral density ofthe RIN can be expressed as an autocorrelation of a powerspectral density function. For completeness we give theresults for both polarized and unpolarized light, althoughwe use the latter for the standard. For light that obeysGaussian statistics, all space–time correlation functionsof the field can be represented by ones of lower order.11

An important result is that the second-order complex de-gree of coherence g (t), from which l(t) is derived, can befound from the first-order space–time correlation func-tion. Thus the normalized intensity correlations for un-polarized thermal light at a single point in space can bederived from the degree of coherence:

^DP~t !DP~t 1 t!& 5 ~^P&2/2!ug ~t!u2, (8)

whereas for polarized thermal light

^DP~t !D~t 1 t!& 5 ^P&2ug ~t!u2. (9)

Fig. 2. Form of the RIN transfer standard; measurement of itsOPSD with a calibrated OSA.

For unpolarized thermal light the autocorrelation of theintensity fluctuations in the time domain is

l~t! [^DP~t !DP~t 1 t!&

^P&2 5ug ~t!u2

2. (10)

Now the autocorrelation function of a stationary randomprocess and the spectral density of the process form aFourier-transform pair (Wiener–Khintchine theorem).Thus there exists a Fourier transform of g (t) in the fre-quency domain that is the normalized spectral density ofthe optical field18:

f~n! 5 E2`

`

g ~t!exp~22pint!dt. (11)

Inasmuch as the Fourier transform in the frequency do-main of the optical fields is zero for negative frequencies,one can show that f(n) is single sided.18 This is a naturalresult because f(n) is the power spectrum or average en-ergy density at a single point in space. Thus the inverseof f(n) is

g ~t! 5 E0

`

f~n!exp~2pint!dn. (12)

From Eqs. (3) and (10), the RIN is

R~n! 5 E2`

`

ug ~t!u2exp~2pint!dt

5 E0

`

f~m!f~m 1 n!dm. (13)

If S(n) is the power spectral density as measured by theOSA, and P is the total power in the spectrum, thenf(n) 5 S(n)/P, and the RIN for unpolarized thermallight is

R~ f ! 5 E0

` S~n!S~n 1 f !dn

P2 . (14)

For polarized thermal light we need only introduce thesame factor of 2 that appears in Eq. (9) for the autocorre-lation of the intensity fluctuations:

R~ f ! 5 2 E0

` S~n!S~n 1 f !dn

P2 . (15)

This result establishes the RIN standard of physicaltheory.

B. Numerical ExpressionsWe measured the power spectral density with a grating-type OSA that has a wavelength scale. However, we cal-culated the RIN in frequency units. Data points on thewavelength scale, l, were converted to frequencies withln 5 c. Then each wavelength interval was converted toa frequency interval over the whole wavelength span.The RIN was calculated from Eq. (15).

We now develop a numerical expression for the RIN inthe wavelength representation to illustrate the measure-ment process and because the uncertainty in the RIN wasevaluated directly in wavelength units. If dpj(l j) is the


optical power measured at wavelength l j , where theresolution bandwidth is Bj(l j), then the power spectraldensity Sj(l j) at l j is

Sj~l j! 5dPj~l j!

Bj~l j!. (16)

In the wavelength representation,

RIN~L! > ~2/P2!E0

Z

S~l!S~l 1 L!dl, (17)

where P is the total optical power, l is the wavelength, Zis the measurement span, and L is the shifting parameterin units of wavelength. The integral is not exact becausel and n are inversely proportional. The resolution band-width, B(l), is converted to frequency units from theequation Dn/n 5 2Dl/l. Thus B(n) 5 1 Hz corre-sponds to B(l) 5 l2/c in wavelength units, which can re-sult in a considerable error in the RIN when the opticalpower is finite over .5 nm or more of the wavelengthspan.

Rewriting both the autocorrelation of the power spec-tral density and the optical power as sums, one can showthat the RIN evaluated at some arbitrary wavelengthshift L i is

RIN~L i! 5

2 (j

dPj~l j!

Bj~l j!

dPi1j~l i1j!

Bi1j~l i1j!Dl j

F(j

dPj~l j!

Bj~l j!Dl jG 2 , (18)

where Dl is the wavelength interval defined by division ofthe span by the number of data points minus 1. Theshifting parameter does not appear explicitly but is de-fined to exist at the arbitrary shifting value of L i 5 l i .Because the RIN of the standard is practically constantfrom zero shift to many tens of gigahertz, we can set L i5 0, which implies that i 5 0.

The total uncertainty in the RIN can be calculated fromthe appropriate versions of Eq. (18) as determined by thesignificant uncertainties in the parameters measured bythe OSA. For example, the resolution bandwidth B(l) ofour OSA turns out to be practically constant over thewavelength range of interest; thus it cancels in the RINequation. If the wavelength distortion is also negligible,then Dl j 5 Dl and Bj(l j) 5 Bi1j(l i1j) 5 B for all i andj. One can then show that at zero shifting the RIN re-duces to

RIN 5

2 (j

dPj~l j!2

DlF(j

dPj~l j!G2 . (19)

5. PROPERTIES OF THE RELATIVEINTENSITY NOISE STANDARDA. PolarizationWhen unpolarized thermal light is used, fiber bendingcan unbalance the power transmitted for orthogonally po-larized modes and change the RIN by a finite amount.We attempted to measure this effect by changing the cur-

vature of the fiber that links the various components butfound the effect to be nearly negligible. Nonetheless, weadd a polarizer to eliminate potential effects that mightarise under various conditions. Furthermore, we haveconfirmed that the RIN for polarized thermal light in-creases by a factor of 2 compaired with that of unpolar-ized light. Measurement of the RIN with and withoutthe linearly polarizing isolator gave a difference of 3.003dB in noise power measured with an ESA. This resultcompares well with the 3.0103-dB difference predicted bytheory. One need not calibrate the RIN system to per-form this measurement because the calibration functioncancels in the equations that define the difference in theRIN of the two states; it is necessary only that the mea-surement be performed at the same ESA settings.

B. Absence of RippleA low-amplitude ripple of tens of megahertz was previ-ously observed in EDFA’s and attributed to multiple pathinterference.20 To test for ripple we visually inspectedthe RIN spectrum with the ESA at several frequencyspans. Figures 3(a) and 3(b) show the amplitude noiseover the ranges of 0.5–0.6 GHz (data at intervals of.0.167 MHz) and 700–850 MHz (data at intervals of.0.25 MHz), respectively. Both figures are devoid ofripple. Sampling other frequency intervals, such as 500–600, 100–250, and 400–550 MHz (not shown), also

Fig. 3. Absence of ripple on the rf noise from the RIN standardover frequency bands (a) 0.5–0.6 GHz and (b) 0.7–0.85 GHz.


yielded no ripple. To further verify the absence of ripple,we note that a comparison of the calibration functions ob-tained from the standard and from the Poisson laser, anindependent source without ripple (see Section 10 below),shows that the functions are nearly identical. Thus thebuilt-in optical isolator effectively shields the EDFA fromfeedback.

C. Relative Intensity Noise of Various Filters ofDifferent Transmittance ShapesWe solved the exact RIN spectral density equation for fil-ters that had Gaussian, Lorentzian, and rectanglarshapes. For all three cases, RIN( f ) 5 RIN(0) up to ahigh frequency, and RIN(0) is inversely proportional tothe bandwidth.17 RIN calculations give similar resultsfor filters of intermediate shape. However, the RIN of afilter with an asymmetric shape may behave differently.

Figure 4 shows the OPSD from EDFA2 with three dif-ferent filters, F1–F3. Filter F1 is rectangular in shapeand has a 3-dB bandwidth of 1.37 nm. Filter F2 is some-what rectangular but spreads more quickly toward thewings; it has a bandwidth of 3.42 nm. Although filter F3spreads quickly toward the wings, it has a 3-dB band-width of 1.32 nm. Figure 5 shows that the RIN obtainedwith all three filters is constant from zero to tens of giga-hertz. The RIN from F1 falls off the fastest. Increasingthe bandwidth (B) from 1.37 to 3.42 nm decreases theRIN. For F3, however, much of the power density existsfar beyond the 3-dB points, so the contribution of the filtershape outweighs the 1/B dependence. The RIN of twoadditional filters (not shown) that have symmetricalshapes also followed a 1/B dependence. Figure 6 com-

Fig. 4. OPSD of the three filters, F1 (1.37 nm), F2 (3.42 nm),and F3 (1.32 nm) used in this study.

Fig. 5. RIN of F1–F3 from near zero to 1000 GHz with EDFA2.

pares the RIN obtained from filter F1 connected toEDFA1 and to EDFA2. For both combinations the RINis practically constant to several tens of gigahertz. Thedifference in RIN of ;0.012 dB is very small, which maydemonstrate that filter shape alone does not determinethe RIN. The similar shape of the curves occurs becauseeach curve arises from the same filter. The filter band-width is probably narrow enough to compensate for a po-tentially nonzero slope of the spectral density of a typicalEDFA, which might occur over a wavelength interval ofseveral nanometers about the center wavelength of thefilter. If such an effect were significant, the RIN wouldnot be constant.

D. Use of Fiber Connectors and LossesThe components that constitute a RIN standard can bejoined by either fusion splicing or the more convenient butlossier fiber connectors. The effect of disconnecting andreconnecting components (on the RIN standard in aggre-gate) is a small change in the transmitted power spectraldensity. As the RIN is practically invariant under at-tenuation, we have chosen threaded, physical-contact op-tical fiber connectors.

6. CALIBRATION OF THE OPTICALSPECTRUM ANALYZEREquation (15) determines the RIN of the standard fromthe OPSD measured by the OSA (after the wavelength isconverted to frequency units). The numerical equationsfor the RIN indicate that the OSA must be calibrated forresolution bandwidth, absolute wavelength and distor-tion, and spectral responsivity.

A. Resolution BandwidthWe determined resolution bandwidth over a wide wave-length range by two distinct methods, using severalnarrow-linewidth lasers (linewidth, !instrument resolu-tion bandwidth). Among these were a tunable laser(linewidth, ,1 MHz) in the 1540–1580 nm range, with awavelength measured by a high-accuracy wavemeter (1part in 106), and a 1523-nm He–Ne laser (linewidth, ,50MHz; previously developed as a secondary wavelengthstandard). In one method, the response of the OSA to anarrow-linewidth laser (or to delta-function input) is readdirectly from the OSA. The single-wavelength input is

Fig. 6. Comparison of the RIN from F1 with EDFA1 andEFDA2.


broadened by a monochrometer, and the resolution band-width is equated with the 3-dB bandwidth as read di-rectly from the OSA. We refer to the broadened curve asthe slit function. Figure 7 shows that the slit function forour OSA is fairly smooth from the peak value to ;25 dBbelow peak value, below which it becomes quite irregular.The 1-nm span is centered at 1551 nm, the resolutionbandwidth is set at 0.05 nm, and the tunable laser is setat 3-mW optical power. In the second method, the noiseequivalent bandwidth (denoted herein b) is calculatedand used as the definition of resolution bandwidth. If l0is the center wavelength of an arbitrarily shaped slitfunction T(l) of peak value T(l0), then

b 51

T~l0!E T~l!dl, (20)

where integration is performed over the entire slit func-tion with the background noise subtracted out. For aperfectly rectangular slit, either method gives the exactanswer, whereas, for a distorted slit, b is more meaning-ful because it accounts for the same slit geometry.

For each method the tunable laser was set at a prede-termined wavelength and the slit function displayed onthe OSA. The two methods produce results that agree,as shown in Fig. 8. For either method, the resolutionbandwidth is practically constant over the broad wave-length range. For the 3-dB method, the average resolu-tion bandwidth (bw) is 0.041 nm, with a standard devia-tion (std dev) of 0.001 nm. For the noise equivalentbandwidth, the average is 0.045 nm, with the same stan-dard deviation. We use the latter result because our slitfunction is irregular.

B. Wavelength Error and DistortionThe wavelength scale suffers distortion that manifests it-self as a nonuniform distribution of wavelength points.We determined the absolute error and the distortion inthe wavelength scale by comparing the wavelength from anarrow-linewidth tunable laser as read by the OSA withthe known wavelength as read by a precision wavemeter.OSA data were averaged over four sweeps. For example,a wavelength setting of the tunable laser at 1560 nmyielded a value of 1559.975 nm when the wavelength wasread by the wavemeter. The corresponding OSA wave-length reading was 1559.94 nm. The difference betweenthe two results gives an uncertainty of 0.035 nm, whichfalls well within the manufacturer’s specifications forwavelength, 60.5 nm over the 350–1750-nm range.Wavelength error was measured for many points aboutboth 1548 and 1560 nm. For the 21 points shown in Fig.9, the standard deviation is 0.042 nm (in agreement withcalibration results obtained by the manufacturer at spe-cific wavelengths). For example, application of a He–Nelaser of 1523.1-nm wavelength (in air) yielded a wave-length reading of 1523.13 nm on the OSA. This is an er-ror of 0.03 nm, in agreement with our average error of0.042 nm.

C. Combined Effects of Attenuation, Pump Current,and Power Spectral Density LevelsBecause the thermal RIN is a ratio of powers squared,and as such is invariant under attenuation, absolute

power measurements are not required and thus calibra-tion of the OSA is simplified, specifically for thermal RINmeasurements. However, the potential effect on the RINof a nonuniform power spectral density across the wave-length span of the OSA must be considered. In simpli-fied numerical equation (19) for the RIN, a constant lin-ear scaling error, say, c, would cancel if it were uniformlyapplied at every noise power, dPj(l j), over the entirewavelength span. In general, we expect c to arise mainlyfrom the spectral responsivity of the OSA’s photodetectorcombined with potential attenuation effects of wave-length dispersion from components or connectors in the fi-ber transmission line of the standard. As such, c is ameasure of the total wavelength or frequency responseof the OSA. On some level of fineness, we expect thatc j 5 c j(l j), so the uncertainty in noise power will notcancel. Thus we determined the uncertainty of spectralresponsivity and attenuation in the aggregate and as-signed the result as an upper bound for the spectral re-sponsivity alone. In so doing, we determined the effect ofc j without measuring it explicitly.

The uncertainty that is due to noise power measure-ments was determined first by attenuation of the inputRIN signal from the standard to fall at predetermined lev-els on the OSA screen. Attenuation of the RIN standardfrom 0 to 10 dB is shown in Fig. 10(a). The average RIN

Fig. 7. Shape of the slit function of the OSA for a narrow-linewidth, tunable laser.

Fig. 8. Measurement of OSA resolution bandwidth versuswavelength as determined from the 3-dB bandwidth and thenoise equivalent bandwidth for a narrow-linewidth, tunable la-ser.


Fig. 9. Comparison of wavelengths measured by the OSA withthe true wavelength in (a) the 1548-nm region and (b) the1560-nm region.

Fig. 10. RIN of EDFA1 1 F1 versus (a) attenuation and (b)pump current.

is 1.0296 3 10211 Hz21 (2109.87 dB/Hz), with a stan-dard deviation (Std Dev) of 0.0008 3 10 211 Hz21 (0.003dB/Hz). To first order this result determines that theRIN is nearly invariant under practical attenuationstrengths that may arise from component and connectorlosses. Figure 10(b) shows the variation in RIN withpump current at zero attenuation. The average RIN is1.0291 3 10211 Hz21 (109.87 dB/Hz), and the standarddeviation is 0.0005 3 10211 Hz21 (0.002 dB/Hz). Next wedetermined the RIN for a variety of pump currents, at-tenuation levels, and intensity levels, as listed in Table 1.For example, the effect of a variable attenuation could bestudied with the spectral density held constant on theOSA screen by compensation of the pump current. Forother measurements the intensity level was allowed tovary. Column 3 of Table 1 shows the intensity relative tothe maximum, which occurs at zero attenuation andmaximum pump current (214 mA). Thus the maximumintensity is represented by 0 dB, whereas 4 dB below themaximum appears as 24 dB. Such varied conditionsagain yielded nearly the same average value for the RIN,1.0297 3 10211 Hz21 (2109.87 dB/Hz), with a standarddeviation of 0.002 3 10211 Hz21 (0.008 dB/Hz), as whenthe intensity levels were allowed to vary on the screen.Thus the uncertainty in the RIN that is due to the uncer-tainties in spectral responsivity and attenuation in theaggregate is indeed very small.

7. UNCERTAINTY IN RIN OF THETRANSFER STANDARDThe total uncertainty in the RIN standard is determinedfrom the random error or repeatability of measurements21

Table 1. RIN for Various Values of Pump Current,Attenuation, and Intensitya

PumpCurrent (mA)

Attenuation(dB)

IntensityRelative ToMaximum,Imax (dB)

RIN(310212 Hz21)

214 0 0 10.296214 0 0 10.286187 0 21 10.294148 0 23 10.288137 0 24 10.298

100.5 0 210 10.286214 1 21 10.286214 3 23 10.304214 4 24 10.294214 10 210 10.308

187 3 24 10.308149 1 24 10.288113 3 210 10.298185 10 211 10.306186 10 211 10.310

a To determine the effect of a nonlinear spectral responsivity, we ad-justed some combinations of pump current and attenuation to give equalvalues of intensity on the OSA screen.


and by the contributions of the OSA parameters to theOPSD measurement. The OSA’s contribution is deter-mined by application of the propagation of errors to thenumerical RIN formulas of Section 6. The parametersthat contribute significant errors are resolution band-width, wavelength distortion (absolute wavelength andwavelength interval), and spectral responsivity. Table 2lists the various contributions to the total uncertainty.The quantities UREP , ULIN , Ul , and URBW are the uncer-tainties that arise from the repeatability, linearity, wave-length distortion, and resolution bandwidth, respectively.The specific and combined standard uncertainties areshown in Table 2 for each filter. The RIN of the standardis the average of all RIN measurements used in comput-ing UREP . For filter F1 with 40 observations, ^RIN&5 1.0325 3 10211 Hz21. To compute the combined stan-dard uncertainty of the average RIN, we add UREP , ULIN ,Ul , and URBW in quadrature. Because all four sources ofuncertainty use measured data (as opposed to manufac-turer’s specifications, for example) to compute their val-ues, they are all considered Type A uncertainties. Al-though different amplifiers were used to quantify UREPand ULIN , this should not influence the uncertainty esti-mates. The expanded uncertainty, U, is twice the com-bined standard uncertainty (U 5 2uc), which representsan approximate 95% confidence interval for the averageRIN.22 The combined standard uncertainty and ex-panded uncertainty for filters F1 and F2 are listed inTable 2. For F1 the combined standard uncertainty(0.014475 3 10211 Hz21) is 1.4% of the average RIN(1.0325 3 10211 Hz21). The average RIN for filter F2,^RIN& 5 6.3712 3 10212 Hz21, was based on 20 observa-tions used to compute UREP . The combined standard un-certainty (0.053564 3 10212 Hz21) is 0.84% of the averageRIN.

8. POISSON-LIMITED LASERRecall that any optical source can be rendered Poissonlimited by sufficient attenuation23 because the Poisson

Table 2. Combined Standard Uncertaintyin the RIN of the Standard (EDFA2

with Filters F1 and F2)

Standard

UncertaintyType

(A or B)

StandardUncertainty

(31/Hz)

StandardUncertainty

(dB)

EDFA2 1 F1UREP A 0.000561 3 10211 0.002ULIN A 0.001180 3 10211 0.005Ul A 0.012315 3 10211 0.051URBW A 0.000419 3 10211 0.002

Combined, uc – 0.014475 3 10211 0.060Expanded, U – 0.029 3 10211 0.12EDFA2 1 F2

UREP A 0.0068325 3 10212 0.005ULIN A 0.011804 3 10212 0.008Ul A 0.025354 3 10212 0.017URBW A 0.009581 3 10212 0.006

Combined, uc – 0.053564 3 10212 0.036Expanded, U – 0.10713 3 10212 0.072

RIN increases with attenuation whereas the excess RINremains invariant. Thus the Poisson limit is reachedwhen the ratio of excess RIN to Poisson RIN vanishes.For an optical field at wavelength l and power P0 thePoisson RIN is

RP~v! 5 Rp 52hc

lP0' const., (21)

where c is the speed of light and h is Planck’s constant.A Poisson-limited source offers two distinct advantagesover a low-excess-RIN source. First, the Poisson RINmanifests itself in the electrical detection circuit as thesimple quantity 2q/i. Second, a RIN system can be cali-brated with a Poisson laser, without explicit knowledge ofthe RIN. Calibration of the RIN system yields afrequency-dependent calibration function that we havedenoted Kappa (it is a dimensionless quantity, preciselydefined in Section 9).

Application of relation (21) to a series of RIN measure-ments under attenuation demonstrates that the Poissonlimit has been practically reached. Figure 11 shows therf-noise power (in arbitrary units) at 400 MHz plottedversus dc voltage. Because Poisson-noise power is mani-fest in the electrical detection circuit as 2qvDV 5 2qV ina 1-Hz bandwidth (where V is voltage), the data shouldyield a straight line.24 The voltage values range from9.39 to 23.66 mV, with corresponding attenuations from18 to 14 dB. The first curve is a second-order polynomialfitted to all data points. The second curve is a linear fit ofall points except V 5 23.66 mV (which corresponds toonly 12-dB attenuation). The two curves rapidly con-verge to a straight line with increasing attenuation (de-creasing voltage). At a voltage of 11.82 mV, where theattenuation is 17 dB, the curves are straight lines. Thisis the normal operating regime for this source. A second-order polynomial fit of all points except V 5 23.66 mVyields a coefficient of nearly zero (slightly negative) forthe second-order term. Thus the excess RIN componentis too small to be resolved. (We note that the manufac-turer specifies RIN < 2172 dB/Hz at the operating powerof 27.82 mW. At 17-dB attenuation, accounting for all

Fig. 11. Electrical noise power of a laser at 400 MHz versus dcvoltage.


losses, the Poisson RIN would be 2149.2 dB/Hz and theexcess RIN would remain <2172 dB/Hz. Thus the ratioof excess RIN to Poisson RIN would be <0.02 dB, wellwithin the noise.)

9. CALIBRATION OF THE RELATIVEINTENSITY NOISE SYSTEM BYTWO DISTINCT METHODSWe developed two independent methods with which tocalibrate a RIN measurement system, each using a preci-sion RIN source. We can use a comparison of the resultsto evaluate the RIN standard. Because the distributionsof photon number differ for the two methods, so too willthe calibration functions. By formulating the RIN ofeach source both outside and inside the RIN system, wederive an exact equation that relates the system’s re-sponse to the two independent sources. First we note afew simplifications. The amplitude fluctuations for thestandard can be considered second-order only, that is, de-void of Poisson light. The laser RIN is practically Pois-son limited, as was shown above.

A. Calibration Equations for the Standard and thePoisson LaserWe now derive the frequency-dependent calibration func-tions that govern the response of a RIN system to the in-put RIN of the standard and the Poisson laser, denotingthem ka(v) and kp(v), respectively. Then we derive anequivalent Kappa that is common to both methods. LetdPa and dPp be the rf-noise powers measured by the ESAfrom the standard and the laser and Pa and Pp be the cor-responding electrical dc powers. Let Ra be the RIN ofthe standard as calculated from the measured OPSD bythe OSA and Rp be the RIN of the laser. Then, for thestandard,

Ra~v! 5 Ra 5 ka~v!dPa~v!

Pa' const. (22)

Solving for ka gives

ka~v! 5RaPa

dPa~v!. (23)

From expression (21), the Poisson RIN of the laser is2hc/lP0 . Thus

Rp 5 kp~v!dPp~v!

Pp5

2hc

lP0' const., (24)

kp~v! 52hc

lP0

Pp

dPp~v!. (25)

B. Equivalent Calibration FunctionsTo determine an equivalent Kappa that is common toboth methods, recall that the Poisson RIN increases inthe electrical detection circuit by an amount 1/h. There-fore an equivalent optical RIN for the laser that wouldyield a Kappa that is equivalent to that for the standard,with all other conditions kept constant, is

Rp

h5 kp~v!

dPp~v!

Pp, (26)

where now ka 5 kp . Solving for kp gives

kp~v! 5RpPp

hdPp~v!. (27)

Now the photocurrent in the electrical circuit is i 5 rP05 hqP/hn, where q is the electron charge and r and hare the responsivity and the quantum efficiency, respec-tively, of the photodetector. The dc voltage is V 5 ir,where r is resistance. When these expressions are sub-stituted for current, and expression (24) for Poisson RIN,the quantum efficiency cancels, and we obtain

kp~v! 52qV

dPp~v!. (28)

Therefore

k~v! 5RaPa

dPa~v!5

2qV

dPp~v!, (29)

where we have now dropped the subscripts on Kappa.This analysis shows that kp can be determined withoutknowledge of the laser RIN or the quantum efficiency ofthe photodetector (measured at 0.77 at the operatingwavelength of 1555.9 nm). For an actual calibration, thenoise floor of the ESA is subtracted out.

10. COMPARISON OF THE STANDARDAND THE POISSON LASERA. Agreement of Calibration ResultsTo evaluate the accuracy of the RIN standard we used itto calibrate our RIN measurement system and comparedthe results with those obtained with the laser. On agiven day we determined Kappa for the standard and thelaser as a pair of sequential measurement sets taken inthe least time possible (which is ;2 min); this is the timeneeded to store the data for the standard, disconnect thestandard, connect the laser, and collect the new data.This procedure ensures nearly identical environmentalconditions (such as temperature and background rf fields)for the two methods. Stray rf signals can appear unpre-dictably in a data spectrum and arise from mobile tele-

Fig. 12. Comparison of Kappas with EDFA2 and filters F1–F3.


phone sources, and the magnitude of the signals may varyduring the time of a measurement. They must be aver-aged by manual inspection of the data. (The ESA dataare averaged from nine sweeps, which takes ;20 s.) Weused a variety of components to constitute a workingstandard. Among these were the six combinations thatarise from the two EDFAs and the three filters discussedpreviously (Subsection 5.D). Figure 12 shows the resultsfor Kappa obtained with EDFA2 with three filters of dif-ferent shapes and bandwidths (1.32, 1.37, and 3.42 nm).The shape and overlap of the three sets of data confirmsthat our calculation of the RIN scales with filter band-width (BW) and shape, as expected from theory. Thesimilarity of the three Kappas indicates that calibrationof the RIN system is independent of the magnitude of theRIN and the shape of the filter. Next we compared thelaser with F2 connected to EDFA1 or EDFA2. The re-sults, shown in Fig. 13, are again similar.

To compare calibration methods quantitatively, we de-termined the simple average of the difference among Kap-pas recorded at all frequencies over the spectrum. Letka,i and kp,i be the Kappas of the standard and of the la-ser at the ith frequency, fi , respectively, and Di the dif-ference, Di 5 ka,i 2 kp,i . If Z is the average differenceobtained by summing over the 601 points in a spectrum,then

Z 5( Di

6015

(~ka,i 2 kp,i!

601. (30)

Figure 14 shows a typical distribution of values of D ob-tained by comparison of EDFA1 1 F2 with the laser.The differences range from 20.7 to 1 dB, with an averageof 0.058 dB and a standard deviation of 0.3 dB. Table 3compares calibration results obtained from various com-binations of EDFAs and filters with one another and with

Fig. 13. Comparison of Kappas from the laser, EDFA1 1 F2,and EDFA2 1 F2.

Fig. 14. Difference between Kappas from the laser and fromEDFA1 1 F2. The data range from 21 to 11 dB. The averagedifference is 0.058 dB.

Table 3. Comparison of Calibration Results Obtained with Various Combinations of EDFAs and Filterswith One Another and with the Laser

Sources ComparedZ 5

S Di

6015

S~ka,i 2 kp,i!

601Standard Deviationsof the Differences, D

Filters $F1 and F3%

EDFA1 1 F3 with EDFA2 1 F3 20.036 0.30EDFA1 1 F3 with DFB(3) 0.094 0.299EDFA2 1 F3 with DFB(3) 0.13 0.311EDFA1 1 F1 with EDFA2 1 F1 0.041 0.198EDFA1 1 F1 with DFB(1) 0.113 0.248EDFA2 1 F1 with DFB(1) 0.071 0.249EDFA1 1 F1 with EDFA1 1 F3 0.074 0.232EDFA2 1 F1 with EDFA2 1 F3 0.025 0.252DFB(1) and DFB(3) 0.0061 0.285

Filters $F2 and F3%

EDFA1 1 F2 with EDFA2 1 F2 20.074 0.349EDFA1 1 F2 with DFB(2) 0.058 0.302EDFA2 1 F2 with DFB(2) 0.143 0.35EDFA1 1 F3 with EDFA2 1 F3 20.056 0.295EDFA1 1 F3 with DFB(3) 0.076 0.271EDFA2 1 F3 with DFB(3) 0.131 0.301EDFA1 1 F2 with EDFA11F3 0.105 0.268EDFA2 1 F2 with EDFA2 1 F3 0.073 0.333DFB(2) and DFB(3) 0.051 0.28


Table 4. Minimum and Maximum Values of Z Obtained by Comparison of Different Sources

Combinations of Sources and Filters

Z 5S Di

6015

S~ka,i 2 kp,i!

601Standard Deviations of the

Differences, D

Z Minimum Z Maximum Z Minimum Z Maximum

EDFA1 1 F1 with DFB 0.001 0.113 0.248 0.188EDFA1 1 F2 with DFB 0.058 0.161 0.302 0.316EDFA1 1 F3 with DFB 0.076 0.103 0.271 0.269EDFA2 1 F1 with DFB 0.036 0.118 0.25 0.254EDFA2 1 F2 with DFB 0.001 0.268 0.333 0.358EDFA2 1 F3 with DFB 0.13 0.152 0.311 0.279

EDFA2 1 F1 with EDFA2 1 F3 0.025 – 0.252 –EDFA1 1 F2 with EDFA2 1 F2 0.074 0.202 0.349 0.327DFB with DFB 0.006 0.112 0.285 0.295

the laser. The sources appear in column 1; Z and thestandard deviations of the values of D appear in columns3 and 4 and 5 and 6, respectively. Groups $F1 and F3%and $F2 and F3% were taken on different days. The Kap-pas obtained from all sources are similar. The averagedifferences between single data sets are quite small. Thesmallest variation occurs for two EDFAs that use thesame filter or for the laser with itself (DFB and DFB).Table 4 lists the minimum and maximum values of Z fortypical comparisons, with measurements taken on differ-ent days. For example, a comparison of EDFA1 1 F1with the laser yields 0.001 dB for Z minimum and 0.113dB for Z maximum. Nearly all comparisons give similarresults. We conclude that the two independent methodsagree, in that the two methods applied to the same RINmeasurement system will yield calibration functions thatdiffer by a small amount.

B. Relative Stability of Two EDFAs Compared with thatof the Poisson LaserOne can use the data in Tables 3 and 4 to compare theaccuracy and the relative stability of EDFA1 and EDFA2with respect to the laser. For a specific EDFA with eachof the three filters we determine the average value of Z.For example, values of Z obtained by comparison ofEDFA1 1 F1, EDFA1 1 F2, and EDFA 1 F3 with thelaser are averaged in the group EDFA1 1 Fx . Similarresults occur for EDFA2 1 Fx . EDFA1 with average Z5 0.088 dB is slightly more stable than EDFA2 with av-erage Z 5 0.103 dB. EDFA2 is chosen for the standardbecause it is more rugged. The average of standard de-viations, in decibels, is 0.046 for EDFA1 1 Fx and 0.077for EDFA2 1 Fx .

C. Statistical TestsThe independence of the standard and the laser as cali-bration sources suggests application of statistical tests tocompare their Kappas. Figure 15 compares the averageKappa for EDFA2 1 F1 with that of the laser, with eachpoint representing the average obtained from data takenon four different days. The Kappa of the laser is slightlylower. One can use the normalized standard deviation ofthe four values of Kappa at each frequency to compare thevariation in Kappa for the two methods. A scatter plot

for the laser compared with EDFA2 1 F1 (Fig. 16) indi-cates that the standard deviations are similar over theentire frequency range. The reader is referred to a re-cent National Institute of Standards and Technology spe-cial publication for a thorough treatment of the experi-mental data.25

Fig. 15. Comparison of Kappa from EDFA2 1 F1 and the la-ser. Data were averaged over four data sets obtained on differ-ent days.

Fig. 16. Comparison of the frequency dependence of the normal-ized standard deviation of the four data sets for Kappa fromEDFA2 1 F1 and the corresponding four data sets from the la-ser.


11. CONCLUSIONA transfer standard has been developed for the spectraldensity of the RIN of optical fiber sources in a 1550-nmrange. The noise source is ASE from an EDFA that iscoupled to a narrow-band optical filter and a linear polar-izer. The RIN is obtained from a combination of mea-surement and theory, but it is also traceable to a secondlaboratory standard based on a laser’s having Poisson-limited RIN. The standard is useful for accurate calibra-tion of broadband RIN measurement systems.

APPENDIX A: CALIBRATION OF ANELECTRICAL SPECTRUM ANALYZERA RIN measurement system is calibrated with the trans-fer standard only, for a constant setting of the rf spectrumanalyzer functions, such as sweep time, resolution andvideo bandwidth, and internal attenuation. Changing afunction setting to accomodate a signal requires a newcalibration because the Kappa obtained may differslightly from the original. In addition, each ESA has aunique scale fidelity factor that accounts for the verticalposition of the noise signal on the screen. RIN measure-ments for which the rf-amplitude noise differs must beweighted with this factor, which, like Kappa, is frequencydependent. The scale fidelity factor can be obtained byuse of a rf signal generator, an attenuator, the ESA, and arf power meter. Connector losses should be included andcontrol settings kept the same as for a RIN measurement.The rf power is recorded at two positions on the ESA, withand without the attenuator. The scale fidelity correctionfor signals of various magnitudes can be expressed as acorrection to Kappa at each frequency separately.25

ACKNOWLEDGMENTSWe are grateful to the following National Institute ofStandards and Technology scientists: Wayne Itano forcarefully verifying the mathematical expressions for theRIN of thermal light, Paul Hale for many stimulating dis-cussions of various aspects of this research, and JohnKitching, Richard Jones, and Jack Wang for valuable dis-cussions. Jack Dupre of Agilent Technologies providedinsight into calibration methods for optical spectrum ana-lyzers.

G. E. Obarski’s e-mail address [email protected].

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8. I. Joindot, C. Boisrobert, and G. Kuhn, ‘‘Laser RIN calibra-tion by extra noise injection,’’ Electron. Lett. 25, 1052–1053(1989).

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19. L. Mandel, Dept. of Physics and Astronomy, University ofRochester, Rochester, N.Y. 14627 (personal communica-tions, June, 1999). Professor Mandel acknowledges an er-ror of a factor of 2 in Eq. (9.8.26) of their text; see Ref. 11.The factor of 1/2 in front of the middle integral should beremoved.

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transfer standard for the spectral density of relative intensity noise of optical fiber sources near...

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