**5. Evaluation of the uncertainty of such a measurement system**

This evaluation is based on a previous work [13]. For the evaluation of the uncertainty, we use the main guideline delivered by the institution in charge of international metrology rules, *Bureau International des Poids et Mesures* (BIPM) in the guide "Evaluation of measure‐ ment data — Guide to the expression of uncertainty in measurement" [14]. According to this guideline, the uncertainty in the result of a measurement generally consists of several com‐ ponents which may be grouped into two categories according to the way in which their nu‐ merical value is estimated. The first category is called "type A". It is evaluated by statistical methods such as reproducibility, repeatability, special consideration about Fast Fourier Transform analysis, and the experimental standard deviation. The components in category A are characterized by the estimated variances. Second family of uncertainties contributions is for those which are evaluated by other mean. They are called "type B" and due to various components and temperature control. Experience with or general knowledge of the behav‐ ior and properties of relevant materials and instruments, manufacturer's specifications, data provided in calibration and other certificates (noted BR), uncertainties assigned to reference data taken from handbooks. The components in category B should be characterized by quantities which may be considered as approximations to the corresponding variances, the existence of which is assumed.

ments. Main aspect is that operators are the same. In our case, operator don't change. It

Uncertainty term due to the number of samples is noted A3. It depend on how many points are chosen for each decades. For this term, we check how works the Fast Fourier Transform

According to equation (5), it can then be considered that the whole statistical contribution is

Our system is not yet formally traceable to any standard, according to how the phase noise is determined. Phase noise measurement generally don't need to be referenced to a standard as the method is intrinsic. So the data provided in calibration and other certificates, noted BR, are not applicable. It results that we can take 0 dB as a good approximation of BR.

Influence of the gain of the DC amplifier, noted BL1, is determined to be less than 0.04 dB.

Temperature effects, noted BL2, are less than 0.1 dB as optical fiber regulation system of

Resolution of instruments, noted BL3, is determined by the value read on each voltmeter when we need to search minimum and maximum for the modulator but also for wattmeter.

During the measurement, what influence could bring the DUT, generally an oscillator to be characterized? We call BL4 the influence of the variation of the DUT. It stays negligible while the variation stay limited. It results that we can keep 0 dB as a reasonable value for

Uncertainty on the determination of the coefficient Kφ (noted BL5) dependent for the slope

For the contribution of the use of automatic/manual range, noted BL6, we can deduce from experimental curves that this influence is no more than 0.02 dB. In our case, all these terms

The influence of the chosen input power value of the DUT, noted BL7, is negligible as the input power in normal range i. e. between -4 dBm and +6 dBm, has a very limited influence.

Other elementary terms still have lower contributions. BL is the arithmetic sum of each ele‐

*A* = √(∑ *Ai* ²) (5)

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means a first approximation we can take 0 dB for this uncertainty.

Finally, statistical contribution can be considered as:

better than 0.69 dB.

**5.2. "Type B". Other means**

temperature in turned on.

Resolution is then no worse than 0.1 dB.

BL4 if the DUT is not too unstable.

were found lower than repeatability.

expressed in Volt/rad is found to be lower than 0.08dB.

mentary contribution. It is determine to be 0.38 dB.

We experimentally observe an influence. It stay better than 0.02 dB.

(FFT) analyzer, it leads to an elementary term of uncertainty less than 0.1 dB.

#### **5.1. "Type A". Statistical contributions**

Uncertainty on £(f) strongly depends on propagation of uncertainties through the transfer function as deduced from equation (3). We see here that "type A" is the main contribution. For statistical contributions aspects, the global uncertainty is strongly related to repeatability of the measurements. Repeatability (noted A1) is the variation in measurements obtained by one person on the same item and under the same conditions. Repeatability conditions in‐ clude: the same measurement procedure, the same observer, the same measuring instru‐ ment, used under the same conditions, repetition over a short period of time, the same location. We switch on all the components of the instrument and perform a measurement keeping the data of the curve. Then we need to switch them off and switch them on again to obtain another curve. We must repeat this action several times until we have ten curves. Ele‐ mentary term of uncertainty for repeatability is experimentally found to be equal to 0.68 dB.

Other elementary terms of statistical contributions still have lower contributions.

Reproducibility, noted A2, is the variability of the measurement system caused by differen‐ ces in operator behavior. Mathematically, it is the variability of the average values obtained by several operators while measuring the same item. The variability of the individual opera‐ tors are the same, but because each operator has a different bias, the total variability of the measurement system is higher when three operators are used than when one operator is used. When the instrument is connected, there is no change of each component or device in‐ side. That makes this term negligible because, for example, the amplifier is never replaced by another one. We remark that, if one or more of the components would be replaced, it will be necessary to evaluate the influence of the new component on the results of measure‐ ments. Main aspect is that operators are the same. In our case, operator don't change. It means a first approximation we can take 0 dB for this uncertainty.

Uncertainty term due to the number of samples is noted A3. It depend on how many points are chosen for each decades. For this term, we check how works the Fast Fourier Transform (FFT) analyzer, it leads to an elementary term of uncertainty less than 0.1 dB.

Finally, statistical contribution can be considered as:

$$A = \sqrt[4]{\sum A i^2} \tag{5}$$

According to equation (5), it can then be considered that the whole statistical contribution is better than 0.69 dB.

### **5.2. "Type B". Other means**

**5. Evaluation of the uncertainty of such a measurement system**

existence of which is assumed.

344 Optoelectronics - Advanced Materials and Devices

**5.1. "Type A". Statistical contributions**

This evaluation is based on a previous work [13]. For the evaluation of the uncertainty, we use the main guideline delivered by the institution in charge of international metrology rules, *Bureau International des Poids et Mesures* (BIPM) in the guide "Evaluation of measure‐ ment data — Guide to the expression of uncertainty in measurement" [14]. According to this guideline, the uncertainty in the result of a measurement generally consists of several com‐ ponents which may be grouped into two categories according to the way in which their nu‐ merical value is estimated. The first category is called "type A". It is evaluated by statistical methods such as reproducibility, repeatability, special consideration about Fast Fourier Transform analysis, and the experimental standard deviation. The components in category A are characterized by the estimated variances. Second family of uncertainties contributions is for those which are evaluated by other mean. They are called "type B" and due to various components and temperature control. Experience with or general knowledge of the behav‐ ior and properties of relevant materials and instruments, manufacturer's specifications, data provided in calibration and other certificates (noted BR), uncertainties assigned to reference data taken from handbooks. The components in category B should be characterized by quantities which may be considered as approximations to the corresponding variances, the

Uncertainty on £(f) strongly depends on propagation of uncertainties through the transfer function as deduced from equation (3). We see here that "type A" is the main contribution. For statistical contributions aspects, the global uncertainty is strongly related to repeatability of the measurements. Repeatability (noted A1) is the variation in measurements obtained by one person on the same item and under the same conditions. Repeatability conditions in‐ clude: the same measurement procedure, the same observer, the same measuring instru‐ ment, used under the same conditions, repetition over a short period of time, the same location. We switch on all the components of the instrument and perform a measurement keeping the data of the curve. Then we need to switch them off and switch them on again to obtain another curve. We must repeat this action several times until we have ten curves. Ele‐ mentary term of uncertainty for repeatability is experimentally found to be equal to 0.68 dB.

Other elementary terms of statistical contributions still have lower contributions.

Reproducibility, noted A2, is the variability of the measurement system caused by differen‐ ces in operator behavior. Mathematically, it is the variability of the average values obtained by several operators while measuring the same item. The variability of the individual opera‐ tors are the same, but because each operator has a different bias, the total variability of the measurement system is higher when three operators are used than when one operator is used. When the instrument is connected, there is no change of each component or device in‐ side. That makes this term negligible because, for example, the amplifier is never replaced by another one. We remark that, if one or more of the components would be replaced, it will be necessary to evaluate the influence of the new component on the results of measure‐

Our system is not yet formally traceable to any standard, according to how the phase noise is determined. Phase noise measurement generally don't need to be referenced to a standard as the method is intrinsic. So the data provided in calibration and other certificates, noted BR, are not applicable. It results that we can take 0 dB as a good approximation of BR.

Influence of the gain of the DC amplifier, noted BL1, is determined to be less than 0.04 dB.

Temperature effects, noted BL2, are less than 0.1 dB as optical fiber regulation system of temperature in turned on.

Resolution of instruments, noted BL3, is determined by the value read on each voltmeter when we need to search minimum and maximum for the modulator but also for wattmeter. Resolution is then no worse than 0.1 dB.

During the measurement, what influence could bring the DUT, generally an oscillator to be characterized? We call BL4 the influence of the variation of the DUT. It stays negligible while the variation stay limited. It results that we can keep 0 dB as a reasonable value for BL4 if the DUT is not too unstable.

Uncertainty on the determination of the coefficient Kφ (noted BL5) dependent for the slope expressed in Volt/rad is found to be lower than 0.08dB.

For the contribution of the use of automatic/manual range, noted BL6, we can deduce from experimental curves that this influence is no more than 0.02 dB. In our case, all these terms were found lower than repeatability.

The influence of the chosen input power value of the DUT, noted BL7, is negligible as the input power in normal range i. e. between -4 dBm and +6 dBm, has a very limited influence. We experimentally observe an influence. It stay better than 0.02 dB.

Other elementary terms still have lower contributions. BL is the arithmetic sum of each ele‐ mentary contribution. It is determine to be 0.38 dB.

#### **5.3. Estimation of the global uncertainty of this system**

According to the Guide to the expression of uncertainty in measurement, uncertainty at 1 sigma interval of confidence is calculated as follows:

$$
\mu\_c = \sqrt{\left(A^2 + BR^2 + BL^2\right)} \tag{6}
$$

**6. Conclusion**

higher microwave operating frequencies.

(ANR) grant "ANR 2010 BLAN 0312 02".

(accessed 5 June 2012)., 401-410.

*ics Letters*, 30(18), 1525-1526.

Address all correspondence to: patrice.salzenstein@femto-st.fr

**Acknowledgements**

**Author details**

Patrice Salzenstein\*

TO-ST),, France

**References**

The author wish that the phase noise optoelectronic system presented in this chapter is use‐ ful for those who want to understand how the phase noise can be experimentally deter‐ mined. We detailed performances and consideration about estimation of the uncertainty to show the main advantage of such developed instrument for metrology or telecommunica‐ tion applications and characterizations of compact OEO's operating in X-band. With high performance better than -170 dBc/Hz at 10 kHz from the 10 GHz carrier, it is interesting to underline that it is possible to determine the phase noise of a single oscillator in X-band with a global uncertainty set to be better than ±2 dB. This system is to be extended at lower and

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Author would like to thank his colleagues from FEMTO-ST, Besançon, Ms. Nathalie Chol‐ ley, Engineer, Dr. Abdelhamid Hmima and Dr. Yanne K. Chembo, CNRS senior researcher, for fruitful discussions. Author also acknowledges the French National Research Agency

Centre National de la Recherche Scientifique (CNRS), Unité Mixte de Recherche (UMR), Franche Comté Electronique Mécanique Thermique Optique Sciences et Technologies (FEM‐

[1] Salzenstein, P. (2011). Optoelectronic Oscillators. In: Sergiyenko O. (ed.) Optoelec‐ tronic Devices and Properties. Rijeka: InTech;. . Available from http://www.intechop‐ en.com/books/optoelectronic-devices-and-properties/optoelectronic-oscillators

[2] Yao, X. S., & Maleki, L. (1994). High frequency optical subcarrier generator. *Electron‐*

[3] Volyanskiy, K., Cussey, J., Tavernier, H., Salzenstein, P., Sauvage, G., Larger, L., & Rubiola, E. (2008). Applications of the optical fiber to the generation and to the meas‐

We deduce from equation (6) that uncertainty at 1 sigma, noted uc, is better than 0.79 dB.

Table 1 summarizes how is deduced the global uncertainty for the spectral density of phase noise at 1 sigma.


**Table 1.** Budget of uncertainties.

Its leads to a global uncertainty of U = 2xuc = 1.58 dB at 2 sigma.

For convenience and to have an operational uncertainty in case of degradation or drift of any elementary term of uncertainty, it is wise to degrade the global uncertainty. That's why we choose to keep U = 2 dB at 2 sigma for a common use of the optoelectronic system for phase noise determination. According to this evaluation of the uncertainty at 1 sigma, it leads to 1.58 at two sigma. It concretely means that it is possible to determine the phase noise of a single oscillator in X-band with a global uncertainty set to be better than ±2 dB.
