Transmitindo padrões de frequência atômicos por redes de fibras ópticas=Transmitting atomic frequency standards in optical fiber networks

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UNIVERSITY OF CAMPINAS

INSTITUTE OF PHYSICS “GLEB WATAGHIN”

Erick Abraham Lamilla Rubio

Transmitindo Padrões de Frequências Atômicos por

Redes de Fibras Ópticas

Transmitting Atomic Frequency Standards in

Optical Fiber Networks

CAMPINAS

2015

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ERICK ABRAHAM LAMILLA RUBIO

Transmitindo Padrões de Frequências Atômicos

por Redes de Fibras Ópticas

Transmitting Atomic Frequency Standards in

Optical Fiber Networks

Dissertação apresentada ao Instituto de Física “Gleb Wataghin” da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de mestre em física.

Dissertation presented to the Institute of Physics “Gleb Wataghin” of the University of Campinas in partial fulfillment of the requirements for the degree of Master of Science.

Supervisor/Orientador: Prof. Dr. Flávio Caldas Da Cruz

Co-supervisor/Coorientador: Prof. Dr. Luís Eduardo Evangelista de Araujo

ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA DISSERTAÇÃO DEFENDIDA PELO ALUNO ERICK ABRAHAM LAMILLA RUBIO, E ORIENTADA PELO PROF. DR. FLAVIO CALDAS DA CRUZ.

CAMPINAS

2015

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Apoio recebido pela:

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Processo No. 2013/15492-2

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Acknowledgement

First, immensely thanks God for giving me the lucidity and strength to finish this master work, my parents Melania Rubio Jimenez and Enrique Lamilla Navarrete and my fiancee Anais Burgos Rodriguez for trusting me.

I extend my sincere gratitude to Prof. Dr. Flavio Caldas Da Cruz for his guidance throughout the Master course and his contributions in the development of this work. It is my privilege to thank Prof. Dr. Daniel Varela Magalhaes and Prof. Dr. Stella Muller for their contributions, Engg. Ricardo Sis and the students Marcos Piau and Felipe Beretta by the contributions in the development of data acquisition systems, Engg. José Saquinaula Brito for his continued and untiring efforts and support in the experimental setup and I am extremely thankful to MsC. Shamaila Manzoor for her assistance in reviewing the semantic content of this work.

A special thanks to FAPESP, CAPES, CePOF, CNPq, Kyatera, FOTONICOM and Senescyt which provided financial support for this research project.

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Resumo

Neste trabalho foi feito um estudo experimental da transmissão de padrões de frequência atômicos através de uma rede de fibra óptica. Até onde sabemos este tipo de transmissão foi realizada pela primeira vez no Brasil. Utilizamos uma conexão de fibra óptica entre o Instituto de Física Gleb Wataghin (IFGW) e a Faculdade de Engenharia Elétrica e Computação (FEEC) da UNICAMP, correspondendo a uma distância de aproximadamente 2 km, e um comprimento total de fibra de 18 km. Frequências de RF derivadas de padrões de frequência de Rubídio e de um receptor GPS foram transmitidas e caracterizadas através de medidas de frequência, particularmente por gráficos de variância de Allan, e medidas da fase.

Palavras chaves: Padrões atómicos, link de fibra, relógios atómicos, Allan Deviation, instabilidade em frequência, desvio de fase, frequência de RF, frequência microondas, relógio de Rubídio, Rb AFS.

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Abstract

In this experimental work, transmission of an atomic frequency standard through an optical fiber network has been implemented for first time in Brazil, to the best of our knowledge. We have used a fiber link between the Institute of Physics (IFGW) and the Department of Electrical Engineering inside the campus of the University of Campinas (UNICAMP) corresponding to 18 km fiber link (2 km between buildings). Radio frequencies derived from a Rubidium standard and a GPS (Global Positioning System) receiver have been transmitted and characterized via phase and frequency measurements, particularly trough Allan deviation plots and phase measurements.

Keys words: Atomic Frequency standards, fiber link, atomic clocks, Allan Deviation, frequency instability, phase shift, RF frequency, microwaves frequency, Rubidium frequency standards.

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Lists of figures:

Chapter 2

Figure 2.1: Oscillating Signal parameters. Ref[1] ... 24 Figure 2.2: Top (1 in the figure): Instantaneous frequency is not stable from t2 to t3. Bottom (2 in the figure): Instantaneous frequency is stable throughout. Ref[1] ... 25 Figure 2.3: Comparison of frequency stability between an ideal oscillator and a real oscillator. ... 27 Figure 2.4: Slope characteristics of the five independent noise processes in the Allan deviation graph. Ref [6] ... 29 Figure 2.5: Power-law noise is indicated by a particular slope in the phase-noise measurements. Ref [6] ... 33 Figure 2.6: Measurement of the frequency difference (beat note) between oscillators can increase measurement precision. State-of-the-art oscillators can readily be measured by this method. Ref [1] ... 35 Figure 2.7: Measurement of the time difference between two oscillators. ... 36 Figure 2. 8: General scheme for the homodyne phase detection measurement. ... 37

Chapter 3

Figure 3. 1: Various experimental schemes for transfer of frequency references. ... 41 Figure 3. 2: Schematic of the RF transmission system from Physics Institute (IFGW) to Electrical Engineering department (FEEC) inside the campus of the University of Campinas (UNICAMP) ... 43

Chapter 4

Figure 4.1: Experimental setup assembled in our laboratory for characterization of frequency instability between oscillators and commercial AFS available in the lab according to the beat frequency method. ... 45 Figure 4.2: Experimental setup assembled in our laboratory for characterization of phase noise between oscillators and commercial AFS available in the lab according to the phase detection method. ... 46 Figure 4. 3: High stability RF/microwave oscillators from Physics laboratory (GLA). ... 46 Figure 4.4: Measured frequency instability between two Rb frequency standards (PRS10 vs. FS725). The graphic represents σ_y (τ)vs. τ (Allan deviation vs. integration time). ... 48

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Figure 4.5: Measured frequency instability between the 10 MHz local oscillator from GPS receiver and the Rb AFS (GPS Septentrio vs. FS725). ... 49 Figure 4.6: Measured frequency instability between the 10MHz voltage output from the synthesized signal generator and the Rb AFS (HP8648B signal generator vs. FS725). ... 50 Figure 4.7: Phase deviation between two Rb AFS (PRS10 vs. FS725) measured over a 12 h period... 51 Figure 4.8: Phase deviation between the 10 MHz output of GPS receiver without external reference and Rb AFS (GPS Septentrio vs. FS725) taken over a 12h period. ... 51 Figure 4.9: Phase deviation between the 10 MHz output signal of GPS referenced to an external AFS (FS725) and Rb AFS (GPS Septentrio vs. FS725) taken over a 12h period. ... 52 Figure 4.10: Phase deviation between the 10MHz output voltage of the synthesized signal generator (without external reference) and the Rb AFS (HP8648B signal generator vs. FS725) taken over a 12h period. ... 52 Figure 4.11: Phase deviation between the 10 MHz voltage signal from the synthesized signal generator referenced to an external AFS and Rb AFS (HP8648B signal generator vs. FS725) taken over a 12h period. ... 53 Figure 4.12: Experimental setup assembled in our laboratory for characterization of frequency instability and phase noise in light amplitude-modulated by Rb AFS. ... 54 Figure 4.13: Experimental setup assembled in our laboratory for characterization of frequency instability and phase noise: Amplitude-modulation stage and amplification stage... 55 Figure 4. 14: Measured frequency instability of the Rb AFS (FS725) after laser modulation, optical amplification and photo-detection. ... 56 Figure 4. 15: Phase deviation after laser modulation and photo-detection (AM+PD, Fig. 12) taken over a 12h period. ... 57 Figure 4.16: Phase deviation after optical amplification with the EDFA (AM+EDFA+PD) taken over a 12h period. ... 57 Figure 4.17: Experimental Setup for Rb AFS transmission through a commercial optical fiber roll. ... 58 Figure 4.18: Measured frequency instability of amplitude-modulated light transmitted through a commercial fiber roll according to the experimental setup shown in figure 4.17. ... 59 Figure 4.19: Phase deviations between Rb AFS after fiber roll taken over a 12h period according the experimental setup depicted in Fig. 4.17. ... 60 Figure 4.20: Map of the fiber link route with the geographic locations of the access points between IFGW laboratories and FEEC building. ... 61 Figure 4.21: Measured frequency instability of the Rb AFS (FS725) after optical fiber link (IFGW – FEEC round trip). ... 62

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Figure 4.22: Measurement of phase deviation after transmission of Rb AFS over optical fiber link (IFGW – FEEC round trip). ... 63 Figure 4.23: Measurement of phase deviation after transmission of Rb AFS over optical fiber link. - Phase deviation in the Rb AFS transmission taken over a 48-h period. ... 64

Chapter 5

Figure 5.1: Analysis of Figure 4.4. Measured frequency instability between AFS (PRS10 vs. FS725) with straight lines slope corresponding to White FM and Flicker FM noise. ... 66 Figure 5.2: Analysis of figure 4.5. Black: GPS receiver not referenced to an external AFS (FS725). Red: GPS receiver referenced to an external AFS (FS725). Straight lines slope corresponding to White FM and Flicker FM noise are shown. ... 67 Figure 5.3: Analysis of figure 4.6. Black: Signal generator not referenced to an external AFS (FS725). Red: Signal generator referenced to an external AFS (FS725). Straight lines slope corresponding to White FM and Flicker FM noise. ... 68 Figure 5. 4: Analysis of Figure 4.14. ... 70 Figure 5. 5: Figure 5.5: Analysis of Figure 4.15. ... 71 Figure 5.6: Phase deviation of Rb AFS transmission after fiber roll, showing long-term fractional frequency offset ~10-13 during AFS transmission. ... 72 Figure 5. 7: Phase noise between Rb AFS after fiber link (round trip), showing long-term fractional frequency offset ~10-11 during AFS transmission. ... 74

Appendix

Figure A.1: Phase detection scheme used for phase fluctuations measurements in this work ... 79

Figure C. 1: Experimental Setup for measuring link distance based on pulse train delay ... 84 Figure C. 2: IFGW-FEEC link distance measurement by pulse time delay method shown in figure C.1... 85

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Lists of tables

Table 2.1: Conversion table from time domain to frequency domain and from frequency domain to time domain for common kinds of interger power law spectral densities; fo= w h/2π is the

measurement system bandwidth. Measurement response should be within 3 dB from D.C. to fh

(3 dB down high-frequency cutoff is at fh). ... 34

Table 4. 1: Frequency instability of the oscillators and Rb AFS shown in Figure 4.3 according to the respective manual. ... 47

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Chapter 1: Introduction

In this chapter we will present the main reasons for carrying out this work and the importance of transmitting periodical signals with high frequency stability.

1.1. Motivation and context of this work

In this work we report the transmission, via an optical fiber link, of high precision and accuracy radio frequencies (RF) provided by an atomic frequency standard (AFS). The main objective is to transmit RF signals with the best frequency stability and low phase noise for use as reference in remote sites [1]. High quality oscillators and commercial atomic clocks, available in our laboratory, were chosen, providing frequency standards that were characterized according to methods which we will describe below [2].

Atomic clocks offer the potential to generate frequency references with lower instability and higher accuracy compared with any high performance oscillator. For accuracy, the best microwave frequency reference is provided by a cesium-fountain clock, which is able to measure the frequency of the hyperfine splitting of the cesium electronic ground state with an accuracy better than six parts in 1016 [3]. However, current researches have demonstrated that frequency references based on optical transitions in laser-cooled and trapped atoms and ions can provide uncertainties approaching one part in 1018, [4], [5], [6] [19], nearly three orders of magnitude better than the best microwave atomic clocks.

Frequency references with high stability and accuracy come from microwave or optical AFS or clocks. This work is focused on the transmission of AFS on optical fiber networks considering the advantages of this medium: low

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loss, low noise and scalability. Additionally, dissemination of frequency and time is explored in this work with a motivation for non-portable clocks, such as laboratory based optical clocks.

1.2. Atomic clocks and Atomic frequency Standard (AFS): Microwave

and Optical AFS

An oscillator is a device that produces a sinusoidal output, typically as a voltage or an electromagnetic field. We can use the frequency of the sinusoidal output of an oscillator as a parameter to measure other physical quantities, since frequency can be present in other physical quantities such as time, speed, distance, etc. An atomic clock consists of a high-quality oscillator, whose frequency is stabilized via a servo-mechanism to the frequency of an appropriate atomic transition, and a frequency counter that can measure such frequency. Time is measured by the counter through to the number of oscillation periods.

The re-definition of the SI second in 1967 as the amount of time in which 9,192,631,770 cycles of the frequency corresponding to the ground-state hyperﬁne splitting of an unperturbed Cs atom [7] changed the concept of time as measurement parameter in instrumentation. After the first successful demonstration of a Cs fountain clock, in the late 1980’s, it has been adopted as a primary frequency and time standard [3].

Other microwave frequency standards or atomic clocks with good frequency instabilities include the cryogenic sapphire oscillator with uncertainties approaching one part in 1015, the hydrogen maser with uncertainties approaching one part in 1013 [8], and the rubidium frequency standard with uncertainties

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In this experimental work we use a commercial Rubidium Frequency Standard (Rb AFS) that will be the reference for frequency instability measurements between oscillators. To have a theoretical review of this AFS, remember that rubidium is an alkali metal like cesium with similar electronic structure. There are two naturally occurring isotopes of rubidium 85Rb and 87Rb, with relative abundance of 72% and 28%. Similarly to Cesium, Rubidium atoms have one electron outside an inner core that has a ground-state hyperﬁne splitting of 6,834,682,612,8 Hertz. This very small splitting frequency arises from hyperfine interaction between the magnetic spins of the electron and the nucleus

[9].

A Rb AFS uses the resonance frequency of 87Rb to control the frequency of a quartz oscillator. Rb AFS works with a discharge lamp that emits red photons with energy corresponding to the resonance line transition of Rb. To simplify the discussion, we will assume that the light from the 87Rb discharge lamp consists of just two lines corresponding to transitions from a single excited state to the split ground state. A cell that contains 85Rb vapor acts as spectral filter which also has a split ground state and an isotopic shift relative to 87Rb. An important coincidence exists: one of the lines from the 87Rb discharge corresponds to one of the transitions in 85Rb. This will allow us to reduce the intensity of this line by passing the 87Rb discharge light through the 85Rb vapor.

Normally, atoms in the ground state will be equally distributed between the hyperfine states, as the splitting is much less than the thermal energy of the atoms in the vapor. This distribution is modified by the filtered light from the discharge, by a process called “optical pumping”. Suppose that the filter can

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completely remove one of the two discharge lines. The remaining light can be absorbed by 87Rb atoms in the resonance cell which are in the lower ground state, moving them to the upper state. When they decay from the upper state, they fall with equal probability into either ground state. As this continues, population will be moved from the lower ground state to the upper ground state. As the population in the lower ground state is decreased, the amount of light which reaches the photodetector will increase, as the number of atoms which can absorb the radiation is reduced. The photodetector measures how much of the beam is absorbed and its output is used to tune a quartz oscillator to a frequency that maximizes the amount of light absorption. The quartz oscillator is then locked to the resonance frequency of rubidium, and standard frequencies are derived from the quartz oscillator and provided as outputs [10].

Other kinds of atomic clocks, recently developed, are optical frequency standards based on optical transitions in ions or neutral atoms. These standards have much higher resonance frequency than atomic oscillators based on microwave transitions. The much higher Q leads to potentially much higher stability [10]. Today optical atomic clocks are being developed with atomic transitions in the optical spectrum, and examples include the wavelengths of 689 nm in Sr, 657nm in Ca, 578nm in Yb, for neutral atoms. For these clocks, an oscillator is a monochromatic laser with ultra-stable frequency locked to an atomic transition and a frequency counter with high resolution based on an “optical frequency comb” [11].

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1.3. Importance and use of atomic clocks

Atomic clocks maintain a continuous and a stable time scale. Their frequency stability allows us to conduct scientific studies in high precision applications in areas such as radio astronomy, geophysics, gravitational wave research, international centers for dissemination of weather, gravitational theories and atomic tests, test of fundamental physical principles, development of next-generation accelerator-based x-ray sources, long-baseline coherent radio telescope arrays, and the accurate mapping of the Earth’ geoid.

Optical atomic clocks have a complex system implementation and generally are not portable, so that transferring frequency reference remotely without introducing additional instability becomes important.

1.4. Motivation for ultrastable frequency reference transfer via fiber

networks

One motivation for ultrastable remote transfer of frequency references is the comparison of frequency references based on transitions in different atomic species. This would enable measurements of the time variation of fundamental constants, [12] such as the fine structure constant 𝛼 [13,14]. If α were changing over time, the frequencies of these transitions based on different atomic systems would change with respect to each other. Typically multiple clocks based on transitions in different atomic species are not built in the same laboratory; therefore, the ability to reliably transfer a frequency reference is crucial for these comparisons.

Another application is synchronization of equipment: transferring frequency reference is useful for applications that involve very tight timing

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synchronization among system components, where the distributed frequency reference to which all the components will be synchronized must have very high stability and low phase noise for short time scales. One such application is long-baseline interferometry for radio astronomy [15], where transfer of a frequency reference has been used to distribute a signal from a master oscillator to each telescope in an array of ≈ 60 radio telescopes over a distance of ≈ 20 km.

A traditional method for transferring frequency and time standards over long distances has been the global positioning system (GPS), which is the method used to compare the frequency and time standards of national laboratories around the world [16]. In this scheme, the transmitter and the receiver both compare their times simultaneously with that of a common GPS satellite that is in view of both. With knowledge about their relative distances to the satellite, their relative time difference can be determined, as can their relative frequency difference with subsequent measurements. Common-mode fluctuations in the path lengths to the satellite and the actual time of the satellite cancel out and do not play a role in the relative time or frequency measurement. By averaging for about a day it is possible to reach accuracies of one part in 1014[17]. GPS carrier phase and

two-way satellite time and frequency transfer (TWSTFT) techniques can push the frequency transfer instability to the low parts in 1015 in 1 day [18]. However, these techniques are limited by fluctuations in the paths that are not common mode. They do not provide the short-time stability necessary for synchronization applications, nor are they practical in situations such as the distribution of a frequency reference throughout a linear accelerator facility.

An interesting alternative for stable distribution of a frequency reference both in short and long-time is transmission over optical fibers. The frequency

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reference is encoded onto an optical carrier for the transmission over a fiber network. Remote users are then able to recover the frequency reference by decoding the received optical signal. One attractive feature of optical fibers is that an environmentally isolated fiber can be considerably more stable than free-space paths, as the traditional GPS method. In addition, the advantages that optical fibers offer like low loss and scalability are beneficial for a frequency distribution system. However, there are disadvantages in optical fiber links that must be taken into account such as the high sensitivity to thermal and mechanical variations, also transmission over optical fiber requires repeating at distance intervals, introducing certain amount noise in the signal to be transmitted. We also must consider the fact that fibers can be broken or have wrapped around curves of only a few centimeters radius.

AFS transmission over optical fiber has already been reported in experimental works which demonstrated transmission of microwave frequencies across hundreds of kilometres in fiber network, [1,19,20], with frequency instabilities below one part in 1014 in a second.

To the best of our knowledge, the present work of transmission of an atomic frequency standard through an optical fiber network has been accomplished by the first time in Brazil. Radio frequencies derived from a Rubidium standard and a GPS receiver have been transmitted through a 18 km fiber link (2 km between buildings) connecting the Institute of Physics and the Department of Electrical Engineering inside the campus of the University of Campinas (UNICAMP).

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Our initial goal was to use available scientific fiber test beds for transmission, such as the Kyatera network [21] (fibers not all underground) in local links and then extend the transmission between Campinas and Sao Carlos, in a 140 km distance. This would pave the way for longer, future transmissions, such as between Campinas/Sao Carlos and Inmetro at Xerem-Rio de Janeiro. However this objective was not possible due to unavailability of the links between Campinas and São Carlos.

1.5. Characterization methods used in transferring frequency

references

To characterize transmission of the AFS through the fiber link, two methods were used: Allan deviation method for characterizing frequency instabilities and phase-noise measurement for characterizing phase deviation.

The first method (Allan deviation) involves a measure of the fractional frequency deviations of an oscillator as a function of averaging time, while the second method (phase noise measurements), is especially useful for determining stability of a signal in short-time. Both methods will be explained in detail in the chapter 2.

References:

1 Jun Ye, Jin-Long Peng, “Delivery of high-stability optical and microwave frequency standards over an optical fiber network”, JILA, National Institute of Standards and Technology and University of Colorado, Boulder, J. Opt. Soc. Am. B Vol 20, No. 7, July (2003)

2 D.B Sullivan, D.W. Allan, D.A. Howe, F.L. Walls, “Characterization of Clocks and Oscillators”, NIST Technical Note 1337, (1990)

3 T.P. Heavner, S. R. Jefferts, E.A. Donley, J. H. Shirley, and T. E. Parker, “Optical-to-microwave frequency comparison with fractional uncertainty of 10-15” IEEE Trans, Instrum. Meas. 54, 842 (2005) 4 R.J. Rafac, B.C. Young, J.A. Beall, W.M. Itano, D.J. Wineland, and J. C. Bergquist, “Optical Frequency Standards and measurements”, Phys. Rev. Lett. 85, 2462, (2000)

5 G. Grosche, H. Schnatz, “Optical frequency transfer via 146 km fiber link with 10-19 relative accuracy”, Physikalisch-Technische Bundesanstalt and Max-Planck-Insitut für Quantenoptik, Optics Letters Vol. 34, August (2009).

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6 Stephan Falke, et all. “ A Strontium lattice clock with 3x10-17 inaccuracy and its frequency”, Physikalisch – Technische Bundesanstalt (PTB), New Journal of Physics 16 - 073023 (2014)

7 D.B. Sullivan “Time and frequency measurement at NIST: The first 100 years” IEEE International Frequency Control Symposium. NIST. pp.4- 17, (2001)

8 T. Parker, “Environmental Factors and Hydrogen maser Frequency Stability”Hydrogen maser. ST-5, Time and Frequency Division, IEEE Journal Vol. 46, No.3, May (1999)

9 Rubidium Frequency Standard SR FS725 Manual, Stanford Research Co. pp. 2-10 (2013) 10 http://tf.nist.gov/general/enc-c.htm

11 Jun Ye and Steven T. Cundiff, “Femtosecond Optical Frequency Comb: Principle, Operation, and Applications”, Eds, Springer, (2005).

12 S.G. Karshenboim, “Variation of fundamental constants”, Can. J. Phys. 78, 639 (2000)

13 J. D. Prestage, R. L. Tjoelker and L. Maleki, “Astrophysics, clocks and fundamental constants”, Phys. Rev. Letter. 74, 3511 (1995)

14 E. R. Hudson, H. J. Lewandowski, B. C. Sawyer, and J. Ye, “Cold molecule spectroscopy for constraining the evolution of the fine structure constant,” Phys. Rev. Lett. 96, 143004 (2006).

15 B. Shillue, S. AlBanna, and L. D’Addario, “IEEE International Topical Meeting on Microwave Photonics Tecnhical Digest”, volume 511 201-204. (2004)

16 J. Levine, “Introduction to time and Frequency metrology”, Rev. Sci. Instrum. 70, 2567 (1999) 17 J. L. Hall and T. W. Hansch, “ Femtosecond Optical Frequency Comb Technology: Principle, Operation, and Applications”, edited by J. Ye and S. T. Cundiff (Springer, New York, pp. 1 -11) (2005) 18 A. Bauch et al, “Comparison between frequency standards in Europe and the USA at the 10-15 uncertainly level”, Metrologia 43, 109 (2006)

19 O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86 km optical link with a resolution of 2x10-18 for RF frequency transfer” Eur. Phys. J. D 48, 35-41 (2008) 20 N.R. Newbury, P.A. Williams and W.C. Swann, “Coherent transfer of an optical carrier over 251 km”, Optics Letters, 32, No. 21, (2007)

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Chapter 2: Characterization of oscillators

and atomic clocks

In this chapter we will present the theoretical content for the characterization in frequency and phase of oscillators and atomic frequency standards. Parts of the text in this chapter have been taken from some of the references, as indicated below.

2.1. Properties of signal sources.

A sine wave signal generator produces an output voltage that changes periodically in time. Parameters that define an oscillating signal are: amplitude, wavelength,

period, frequency and phase deviation.

A sinusoidal wave voltage 𝑉(𝑡) without noise (ideal) is expressed as follows:

𝑉(𝑡) = 𝑉𝑜sin (2𝜋𝜈𝑜𝑡 + 𝜙𝑜)

Amplitude, 𝑉𝑜 is defined as the maximum voltage value produced by a

signal generator. Amplitude can be constant or can vary with time or position;

Wavelength, generally symbolized by λ is the distance between two consecutive

ridges or valleys, measured in units of length; period, T, is defined as the time required to complete one full cycle (2π radians of phase); frequency, 𝑓_𝑜 or 𝜈_𝑜, is defined as the number of cycles in one second (Hz), which is reciprocal of period and the phase deviation, 𝜙𝑜, that can be defined as the position of a point in time

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be constant or can vary with time or position. [1]. An ideal sine wave signal with all these parameters is shown in figure 2.1.

Figure 2.1: Oscillating Signal parameters. Ref[1]

In a measurement, sine wave voltage signals present fluctuations or noise in its amplitude and phase. The expression describing the output voltage 𝑉(𝑡) of a signal generator is given by:

𝑉(𝑡) = (𝑉_𝑜+ ε(t))sin (2𝜋𝜈_𝑜𝑡 + 𝜙(𝑡))

where, 𝑉_𝑜 is the peak voltage amplitude, (t)is the deviation amplitude that represents the amplitude noise,



_o is the frequency of the voltage signal and

) (t

 is the deviation of phase that represents phase noise. Both kinds of noise, amplitude and phase noise can be characterized in the time and frequency domain. This work will be centred in characterizing phase noise and its relationship with frequency stability.

For studying the performance of oscillators we will use the term “frequency stability” to relate the frequency differences of an oscillator compared with another one [1]. The general definition of “frequency stability” is associated with

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the degree to which an oscillating signal has the same frequency value for any interval Δt, over a specified period of time. Figure 2.2 shows a comparison between an unstable frequency and a stable frequency.

Figure 2.2: Top (1 in the figure): Frequency is not stable from t2 to t3. Bottom (2 in

the figure): frequency is stable throughout. Taken from Ref[1]

It is important to note that the measurement of “frequency stability” involves two oscillators, which must be compared. Frequency stability measured for only one oscillator simply is not possible. It is necessary to have another oscillator with higher performance that will be the reference for the oscillator under test. Those considerations help us to propose the parameters for frequency measurements and frequency instability measurements in a frequency oscillator:

Fractional frequency difference: 𝑦(𝑡) = 𝑣1(𝑡)−𝑣𝑜

𝑣𝑜 [1], defined as the

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Fractional frequency difference is related to frequency and phase fluctuations:

𝑦(𝑡) =

𝑣1(𝑡)−𝑣𝑜 𝑣𝑜

=

𝛿𝑣(𝑡) 𝑣𝑜

=

𝜑̇(𝑡) 2𝜋𝑣𝑜 [2]

,

where

𝜑

̇

_(𝑡)

_represents deviation of phase fluctuations between the oscillator under test and the reference oscillator.

Time deviation: 𝑥(𝑡) = ∫ 𝑦(𝑡𝑡 ′_)𝑑𝑡′

0 [1], defined as the relative temporal

deviation between oscillators, where 𝑥(𝑡) is calculated as the integral of 𝑦(𝑡) in a period of time, t. In the same manner as the fractional frequency difference, the time deviation between oscillators can be described as function of phase fluctuations 𝜑(𝑡) in this form: 𝑥(𝑡) = 𝜑(𝑡)

2𝜋𝑣𝑜 [2].

Average fractional frequency over a period τ: 𝑦̅(𝑡) =𝑥(𝑡+𝜏)−𝑥(𝑡) 𝜏 [1],

which is the fractional frequency difference between oscillators in a sample time, Δ𝑡 or “τ”. Since it is impossible to measure instantaneous frequency, average fractional frequency, 𝑦̅(𝑡), is necessary to establish gate times through which the oscillators are observed: picoseconds, seconds or days. Sample time or averaging time corresponds to the gate time of a frequency counter used to make the frequency measurements.

Figure 2.3 shows a comparison between the performance of an ideal oscillator and a real one as function of frequency stability parameters.

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Figure 2.3: Comparison of frequency stability between an ideal oscillator and a real oscillator.

2.2. Characterization methods

Here we will introduce two important characterization tools: Allan deviation and phase noise measurement. The reader will find a vast mathematical and conceptual discussion in the paper of Zhu and Hall [3] from which we have extracted much of the analysis presented below.

2.2.1. Allan deviation

Normally, the standard deviation 𝜎_𝑠𝑡 is the statistical quantity used to quantify the dispersion of a data group. The standard deviation is the numerical measure of the deviation of the data with respect to its mean value. That is, 𝜎_𝑠𝑡 indicates how each data series is away from the mean. However, classical standard deviation can be calculated only from stationary data, that is, for processes in which the mean and the standard deviation don´t change over time; and the results are independent of time. This leads to the idea of constant noise, i.e. its power spectrum density is

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constant (the power associated with each spectral noise component is uniformly distributed throughout the measurement bandwidth).

For stationary data, the traditional statistics tools as standard deviationconverge to specific values when the number of measurements increases considerably. With non-stationary data (such as those handled with the oscillators) we have a mean that changes each time that a new measure is added to the time series. Following the recommendation of the IEEE, Allan deviation is used to characterize the stability of frequency standards.

The Allan deviation, 𝜎𝑦(𝜏), is a standard, widely adopted, quantity for measuring

frequency fluctuations. The idea of using Allan deviation is to focus on consecutive frequency measurements, computing frequency difference in periods of time 𝜏. The fractional frequency differences between adjacent frequency measurements are then plotted as a function of τ, revealing how the oscillator frequency is fluctuating over various timescales. As a result, long-term noise processes at the different timescales can be studied and cleanly separated [4].

The Allan deviation for an averaging time τ is defined as: 𝜎_𝑦(𝜏) ≡ 〈1

2[𝑦̅(𝑡 + 𝜏) − 𝑦̅(𝑡)] 2_〉1/2

where 〈 〉 indicates a time average. 𝜎_𝑦(𝜏) can be estimated from a finite set of N consecutive average values of the centre frequency, 𝑣̅_𝑖, defined below, each averaged over a period τ [1]. An approximation of 𝜎𝑦(𝜏) can be expressed as

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𝜎𝑦(𝜏) ≈ [ 1 2(𝑁 − 1)𝑣𝑜2 ∑(𝑣̅𝑖 − 𝑣̅𝑖+1)2 𝑁−1 𝑖=0 ] 1/2

Allan deviation becomes a useful tool for characterizing an oscillator because the type of phase noise present during the measurement is revealed as a characteristic shape of 𝜎𝑦(𝜏) as a function of τ. Each type of phase noise in the

process is related to a particular slope on the graph 𝜎_𝑦(𝜏) vs. τ [6], as shown in figure 2.4.

Figure 2.4: Slope characteristics of the five independent noise processes in the Allan deviation graph. Taken from Ref [6]

We can make the following general remarks, taken from Ref [2], [10], about power-law noise processes:

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Random walk FM (𝝉𝟏/𝟐_{) noise is difficult to measure since it is usually very}

close to the carrier. Random walk FM usually relates to the oscillator’s physical environment. If random walk FM is a predominant feature of the spectral density plot then mechanical shock, vibration, temperature, or other environmental effects may be causing "random" shifts in the carrier frequency.

Flicker FM (𝝉𝟏/𝟐_{) noise is a noise whose physical cause is usually not fully}

understood but may typically be related to the physical resonance mechanism of an active oscillator or the design or choice of parts used for the electronics, or environmental properties. Flicker FM is common in high-quality oscillators, but may be masked by white FM (𝜏−1/2) or flicker PM (𝜏−1)) in lower-quality oscillators.

White FM (𝝉−𝟏/𝟐_{) noise is a common noise found in passive-resonator}

frequency standards. These contain a slave oscillator, often quartz, which is locked to a resonance feature of another device which behaves much like a high-Q filter. Cesium and rubidium AFS have white FM noise characteristics.

Flicker PM (𝝉−𝟏_{) noise may relate to a physical resonance mechanism in an}

oscillator, but it usually is added by noisy electronics. This type of noise is common, even in the highest quality oscillators, because in order to bring the signal amplitude up to a usable level, amplifiers are used after the signal source. Flicker PM noise may be introduced in these stages. It may also be introduced in a frequency multiplier. Flicker PM can be reduced with good low-noise amplifier design (e.g., using RF negative feedback) and hand-selecting transistors and other electronic components.

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White PM (𝝉−𝟏) noise is broadband phase noise and has little to do with the

resonance mechanism. It is probably produced by similar phenomena as flicker PM (𝜏−1) noise. Stages of amplification are usually responsible for white PM noise. This noise can be kept at a very low value with good amplifier design, hand-selected components, the addition of narrowband filtering at the output, or increasing, if feasible, the power of the primary frequency source.

Sinusoidal noise that is periodical noise from external oscillations which is

locked to the system and can modulate noises processes discussed above. In the Allan deviation plot, the amplitude of 𝜎_𝑦(𝜏) varies up and down depending on sampling time. This is because in the time domain the sensitivity to a periodic wave varies directly as the sampling interval. This effect (which is an alias effect) is a very powerful tool for filtering out a periodic wave imposed on a signal source. By sampling in the time domain at integer periods, one is virtually insensitive to the periodic (discrete line) term.

However, to reliably indicate the type of noise present via Allan deviation, it is crucial that there are no dead times between consecutive average frequency measurements. The presence of dead time, depending on the length, can change the slope during the computing of Allan deviation causing an error in the identification of noise, loss of coherence between data points, overlapping noises and in the worst case omitting existing noise. Finally, the Allan deviation is considered a main method for measuring frequency stability in an oscillator because it can be interpreted as the prediction of noise process in the frequency measurement as a function of averaging time.

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2.2.2. Phase noise

The second method of characterization for high performance oscillators used in this work is the phase noise measurement. The phase noise method consists in measuring directly phase fluctuations between oscillators. This measurement is useful for directly determining the coherence of a signal [7]. The phase noise measurement becomes significant when it shows abrupt and unwanted variations of the signal characteristic in the time domain commonly called “jitters”. To study phase noise, usually the time-domain picture is used instead of frequency-domain picture through a Fourier transform, giving a phase noise spectrum.

The phase noise spectrum is an excellent way to determine the coherence of a signal, because through the power spectral density (PSD) of phase fluctuations, 𝑆_𝜙(𝑓), it is possible to detect an existing power-law noise process over a range of relevant Fourier frequencies surrounding the signal carrier. Power spectral density of phase fluctuations, 𝑆_𝜙(𝑓), is represented as the mean square of phase fluctuations𝛿𝜙̃(𝑓) at Fourier frequency f from the carrier in a measurement bandwidth of 1 Hz [1]:

𝑆𝜙(𝑓) ≡ [𝛿𝜙̃(𝑓)] 2

[𝑟𝑎𝑑2

𝐻𝑧 ]

Similarly to the Allan deviation, the phase noise spectrum relates a particular noise process with the slope on the graph 𝑆𝜙(𝑓) vs. f for a particular range in

Fourier frequencies. Figure 2.5 shows the slope of the five independent noises processes linked to power spectral density of phase fluctuations.

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Figure 2.5: Power-law noise is indicated by a particular slope in the phase-noise measurements. Taken from Ref [6].

Knowing the spectral density of frequency fluctuations, 𝑆_𝑦(𝑓) = ∑+∞ℎ_𝛼𝑓𝛼

−∞ , where ℎ𝛼

is the amplitude of the noise process, 𝑓 is the value of the frequency over a period of time, and 𝛼 is the slope on a log-log plot for a given range of 𝑓, it is possible to connect mutually the Allan deviation 𝜎_𝑦(𝜏) with the spectral density of phase fluctuations 𝑆𝜙(𝑓). The table below lists the corresponding relations:

Noise Process White PM 2 Flicker PM 1 3𝑓ℎ (2𝜋)2_𝜏2_𝑓𝑜2 (2𝜋)2𝜏2𝑓2 1.038 + 3ln (𝑤ℎ𝜏) [1.038 + 3ln(𝑤_ℎ𝜏)]𝑓 (2𝜋)2_𝜏2_𝑓𝑜2 𝑓2 2𝜏𝑓𝑜2 𝑺𝒚(𝒇) = 𝒉𝜶𝒇𝜶 𝑺_𝒚(𝒇) = 𝑨𝝈_𝒚(𝝉)𝟐 𝝈_𝒚(𝝉)𝟐= 𝑩𝑺_𝝓(𝒇) 𝛼 = 𝐴 = 𝐵 = (2𝜋)2_𝜏2_𝑓2 3𝑓ℎ

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White FM 0

Flicker FM -1

Random Walk FM -2

Table 2.1: Conversion table from time domain to frequency domain and from frequency domain to time domain for common kinds of interger power law spectral densities; fo=wh/2π is the measurement system bandwidth.

Measurement response should be within 3 dB from D.C. to fh (3 dB down

high-frequency cutoff is at fh).

Now we can distinguish two types of characterization in terms of its application: Allan deviation for measuring fractional frequency fluctuations in different time scales and phase noise measurement for determining the coherence of frequency source [4]; the next point of analysis will focus on the experimental methods to quantify these characterization methods.

2.3. Common methods for frequency measurements.

There are different methods of measuring the frequency fluctuations in precision oscillators which do not include measuring the frequency directly in a frequency counter. The direct frequency counter technique is often very limiting because the number of resolvable digits on the counter are often inadequate for precision oscillators. Two common methods for frequency measurement used in this work will be shown: the beat frequency method and the time difference method [8].

2.3.1. Beat frequency method

This technique is also called heterodyne beat frequency measuring method. In this method (Figure 2.6), the voltage signal of the oscillator under test with frequency 𝑣1 is mixed in a product detector (mixer) with the voltage signal of a

reference oscillator with frequency 𝑣₀. The output of the product detector is

2𝜏 1 2𝑙𝑛2 ∗ 𝑓 2𝑙𝑛2 ∗ 𝑓3 𝑓𝑜2 6 (2𝜋)2_𝜏𝑓2 (2𝜋)2_𝜏𝑓4 6𝑓𝑜2

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proportional to the product of the signals of the oscillators. The output of the mixer in the frequency domain is a combination of the sum and difference of frequencies 𝑣₀ and 𝑣₁ [9]. We are interested in the difference of frequency, (𝛿𝑣(𝑡) = 𝑣_𝑏 in figure 2.6) and thus a low-pass filter is placed at the output of the mixer. The beat frequency is then amplified and sent to a frequency counter in order to compute the fractional frequency.

Figure 2.6: Measurement of the frequency difference (beat note) between oscillators can increase measurement precision. State-of-the-art oscillators can readily be measured by this method. Taken from Ref [1]

The disadvantages of using this frequency measuring method are: minimum τ determined by period of beat frequency is typically not adjustable; cannot compare oscillators near zero beat; cannot tell which oscillator has higher or lower frequency precision; dead time often is associated with these measurements.

2.3.2. Time difference method

As its name indicates, the time difference method is a direct measurement of time deviation between two oscillators 𝑥(𝑡). This method is very commonly used, but typically does not have the precision in the measurement more readily

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available as in the beat frequency method, but with appropriate counters and frequency dividers, it is possible to obtain a high precision measurement (precision of time difference measurements in the range of 10 ns to 10 ps).

The standard setup for this method is illustrated in Figure 2.7 obtained from

ref. [1]

Figure 2.7: Measurement of the time difference between two oscillators.

Usually after division by N to obtain 1 pulse-per-second, yields only moderate measurement performance compared to other methods. The technique is dependent on several properties of the counter and its trigger circuits. Taken from Ref [1].

The disadvantages of using this frequency measuring method are: limitation in signal-to-noise ratio; noise measurement is typically in excess of oscillator instabilities for τ up to and of the order of several thousand seconds; hence, typically it is not useful for short term stability analysis.

2.4. Common method for phase detection.

In this work we will only focus on one of the methods for measuring phase noise: the homodyne phase detection.

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2.4.1. Homodyne phase detection

The homodyne phase detection or commonly called the phase detector measurement technique converts the phase difference between two signals into a voltage.

The heart of this method is the phase detector. Two signals, at the same frequency are compared in order to measure their relative phase fluctuations, using a mixer for this purpose. The frequency sum (2f) term is eliminated using a low pass filter, staying only with the difference frequency at DC [10]. Voltage fluctuations appear superimposed on this DC signal that are proportional to the combined phase noise of the two oscillators. The average voltage output (DC voltage) is digitized and processed to obtain the phase noise information desired. Figure 2.8 illustrates the homodyne phase detection in general terms.

Figure 2. 8: General scheme for the homodyne phase detection measurement.

The phase detector consists of a mixer which multiplies the two signals. The frequency sum term in the mixer output is filtered off, staying only with a DC signal whose fluctuations are proportional to the combined phase noise between the two oscillators.

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The mathematical analysis, the electronic elements and the scheme for voltage-phase conversion used in the homodyne phase detection measurement are detailed in the Appendix A.

References:

1 D.W. Allan, D.A. Howe, F.L. Walls, “Characterization of Clocks and Oscillators”, NIST Technical Note 1337, (1990)

2 D.W. Allan, "Statistics of Atomic Frequency Standards," Proc. IEEE, 54, 221 (1966).

3 J. L. Hall and M. Zhu. In E. Arimondo, W. D. Phillips, and F. Strumia, “An introduction to phase-stable optical sources”, Laser Manipulation of Atoms and Ions, Proc. of the International School of Phys.Enrico Fermi: Course 118, pages 671–702. North Holland, Amsterdam, (1992).

4 Seth M. Foreman, “Femtosecond Frequency Combs for Optical Clocks and Timing Transfer”, PhD.Thesis, University of Colorado, (2007).

5 F. L. Walls. In J. L. Hall and J. Ye, “Phase noise issues in femtosecond lasers”, Laser Frequency Stabilization, Standards, Measurement, and Applications, volume 4269 of Proceedings of SPIE, pages 170–177. SPIE, Bellingham, Washington, (2001).

6 D.W. Allan, “Standard Terminology for fundamental frequency and time metrology”, National Bureau of Standards, Boulder, 42nd Annual Frequency Control symposium, (1988).

7 K. Blakkan, M. Soma, “A time domain method to measure oscillator phase noise”, University of Washington, 27 IEEE VLSI Test symposium, (2009).

8 NIST website: http://tf.nist.gov/phase/Properties/one.htm

9 James A. Barnes et all. “Characterization of frequency stability”, IEEE, Transactions on Instrumentation and measurement, IM-20. No. 2, (1971)

10 Esteban G. Najle, Robert M. Buckley, “Phase noise measurements utilizing a frequency down conversion/multiplier, direct spectrum measurement technique” , United States Patent, No. 5337014, (1994)

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Chapter 3: Frequency transfer in Optical

Fiber Networks: Experimental Setup

In this chapter we review the main experimental schemes for frequency transmission over optical fiber link.

3.1. Experimental schemes for frequency transfer

Depending on the of type frequency reference (microwave or optical) to be transmitted, it will appear noise process with more intensity during the transmission, hence the cancellation of noise is strictly linked to the scheme used for transmission and the type of frequency reference to transmit over the link.

For transmitting microwave references over fiber network in short distances, it is often used a passive transfer scheme, where the microwave reference is used to modulate the amplitude of a CW optical beam, which is then launched into the ﬁber for transfer. The remote user simply demodulates the sidebands on the optical carrier by photo-detection of the optical beam in order to recover the microwave information [1]. This scheme widely represented in the figure 3.1 (a) reaches performance to 10-15 between 100 and 1000 s.

For transmitting optical frequency standards, it is common to use a stable optical reference, where a portion of the optical beam is used as a reference arm, while the rest of the beam is first intensity-modulated by an acoustic-optical modulator AOM and then sent to the fiber network. A heterodyne beat signal between the round-trip optical beam and the local reference reveals the optical phase noise of the fiber link. The error signal is used to feed back to the AOM in order to compensate for the phase difference of the link [1]. This scheme widely

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represented in the figure 3.1 (b) reaches performance to 10 between 100 and 1000 s.

For transmitting optical frequency and microwave frequency standards simultaneously, frequency comb is used in the setup. In this case usually the group delay of the fiber is actively stabilized in order to stabilize the repetition rate of pulses exiting the fiber at the remote end, and can provide the practical issue of achieving an out-of-loop measurement of the excess instability induced by the fiber link on the frequency reference at the remote end of the system Ref [1]. This scheme widely presented in the figure 3.1 (c) reaches performance to 10-18 between 100 and 1000 s for both references, optical and microwave frequency references. Initial research experiments demonstrate the stability of the transfer over a long fiber link by simply collocating the “remote” end with the local reference in order to perform a direct comparison, before the full system is implemented at remote locations [2].

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Figure 3. 1: Various experimental schemes for transfer of frequency references. (a) Passive transfer of a microwave reference by modulating a diode laser. (b) Direct transfer of the optical carrier by stabilizing the optical phase of the ﬁber link. (c) Transfer of simultaneous microwave and optical frequency references by transferring an optical frequency comb. Either optical or microwave information from the round trip can be used to stabilize the group delay and/or optical phase delay of the ﬁber link. Taken from Ref[1]

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We will show in detail the technique of passive transfer in microwave frequency domain, which is the main objective of this work.

3.2.1. Passive transfer technique

Known as a simple method for transmitting frequency references, the passive transfer technique is represented by the description of the scheme (a) in the Figure 3.1. Ref [1]. In the passive transfer technique, the comparison between frequency references (original and transmitted signal) is accomplished without feedback for monitoring the fiber induced phase and frequency noise in the transmitted RF signal. The comparison is performed by use of two approaches discussed in detail in the Chapter 2: (1) by homodyne detection of the phase difference between the round-trip signal and the original source signal, and (2) by counting the frequency of a heterodyne beat between the round-trip signal and the original signal.

The passive transfer technique has been used to transmit microwave frequency standards in short distances of optical fiber [3, 4, 5] giving experimental results of accuracy in the order of 1015 parts in a second.

3.2.2. Passive microwave transfer for the Rubidium Standard FS725

We shall now apply this technique of passive transfer, the main objective of which is to transmit a high stability frequency signal from a commercial Rubidium Standard SRS FS 725[6] (details about accuracy and stability in Appendix 2) via optical fiber network. For the realization of our experimental setup we have relied upon the experimental setup of the rf transmission system from NIST to JILA [4].

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The transmission of a Rubidium Frequency Standard via fiber link from Institute of Physics (IFGW) to Department of Electrical Engineering (FEEC) inside the campus of the University of Campinas (UNICAMP) is accomplished by the setup shown in figure 3.2. A 40 mW, continuous-wave (cw), single-frequency laser at 1550 nm serves as the light source. The light is first coupled into an optical fiber, then into a high-speed intensity modulator (LiNbO3) which

can support modulation frequencies up to 10 GHz. The light that exits the modulator is amplified in an Erbium Doped Fiber Amplifier (EDFA) with adequate power to overcome transmission losses in the fiber link. Finally, amplified amplitude modulated light is launched into the fiber link.

EDFA

AM

Single-mode Laser Rubidium Frequency Standard RF signal To Fiber Link Intensity Modulator

Figure 3. 2: Schematic of the RF transmission system from Physics Institute (IFGW) to Electrical Engineering department (FEEC) inside the campus of the University of Campinas (UNICAMP)

References:

1 Seth M. Foreman, Kevin W. Holman, Darren D. Hudson, David J. Jones and Jun Ye, “Remote Transfer of ultrastable frequency references via fiber networks”, Rev. Sci. Instrum. 78, 021101 (2007)

2 F. Narbonneau, M. Lours, S. Bize, A. Clairon, G. Santarelli, O. Lopez, C. Daussy, A. Amy-Klein, and C. Chardonnet, “High-resolution microwave frequency dissemination on an 86- km urban optical link”, Rev. Sci. Instrum. 77, 064701 (2006).

3 K.Predehl, H. Schnatz, “A 920-Kilometer Optical Fiber Link for Frequency Metrology at the 19th Decimal Place”, Science 336, 441 (2012)

4 O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86 km optical link with a resolution of 2x10-18 for RF frequency transfer” Eur. Phys. J. D 48,35-41 (2008) 5 Jun Ye, Jin-Long Peng, Jason Jones, Kevin W.Holman, John L. Hall, “Delivery of high-stability optical and microwave frequency standards over an optical fiber network, J. Opt. Soc. Am. B Vol 20, 7, (2003) 6 Rubidium Frequency Standard SR FS725 Manual, Stanford Research Co. (2013)

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Chapter 4: Frequency transfer in fiber link:

Results

In this chapter we present the results obtained in the characterization of frequency and phase of different oscillators and AFS available in the laboratory, as well as the characterization of fiber link for transmission of Rb AFS.

4.1. Characterization of frequency and phase noise of different

oscillators and Rubidium frequency standards (Rb AFS) inside

the laboratory

The characterization of oscillators and AFS has been performed by the two methods presented in Chapter 2 in Figures 2.6 (heterodyne method) and 2.8 (homodyne method).

Measurement of frequency instability between oscillators and commercial AFS available in our laboratory using the experimental diagram of Figure 4.1 was conducted as follows: The reference oscillator in our laboratory is the Rb AFS model FS725 which delivers an output voltage at 5 MHz which is mixed with the voltage at 10 MHz from the oscillator under test. The mixer output is filtered by a Butterworth low-pass filter in order to obtain only the beat frequency at 5 MHz. The frequency counter which is also referenced to the Rb AFS FS725 is used to measure the beat frequency. Beat frequency is processed to show the frequency instability via Allan deviation graphs.

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Figure 4.1: Experimental setup assembled in our laboratory for characterization of frequency instability between oscillators and commercial AFS available in the lab according to the beat frequency method.

Measurement of phase noise between oscillators and commercial AFS available in our laboratory using the experimental diagram of Figure 4.2 was conducted as follows: The output voltage from the oscillator under test is multiplied (via mixer) with the output voltage generated by the Rb AFS model FS725 (reference). Both signals should have the same frequency (10 MHz). A low pass filter removes sum-frequency component to the mixer output, leaving only the difference-frequency component, which in this case would have zero frequency, that is, a continuous DC signal. If there is a phase difference between the signals after the filter, the data acquisition card will detect a certain voltage value that is proportional to the phase deviations between outputs of the oscillators.

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Figure 4.2: Experimental setup assembled in our laboratory for characterization of phase noise between oscillators and commercial AFS available in the lab according to the phase detection method.

For this work, the following oscillators were available in our lab (figure 4.3 and table 4.1):

Figure 4. 3: High stability RF/microwave oscillators from Physics laboratory (GLA).

Two Rb AFS (model FS725 and PR10 from Stanford Research Systems), a GPS receiver (model POLARx4GNSS from Septentrio Satellite Navigator) and a synthesized signal generator (model HP8648B from Agilent). Rb AFS PR10 is property of USP-Sao Carlos (Prof. Daniel Varela)

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FREQUENCY INSTABILITY SPECIFICATIONS BY

MANUFACTURER

Equipment

Frequency Instability

Aging (after 30 days)

Short term

Stability (Allan

Deviation)

Rb AFS

Model FS725

5x10

-11

Monthly

2x10

-11

1s

5x10

-10

yearly

1x10

-11

10s

5x10

-09

20 years (typical) 2x10

-12

100s

Rb AFS

Model PR10

5x10

-11

Monthly

2x10

-11

1s

5x10

-10

yearly

1x10

-11

10s

5x10

-09

20 years (typical) 2x10

-12

100s

GPS Receiver

PolaRx4 GNSS

2x10

-11

Monthly

2x10

-11

1s

2x10

-10

yearly

2x10

-11

10s

2x10

-09

20 years (typical) 8x10

-12

100s

Synthesized

Generator

Not defined

Table 4. 1: Frequency instability of the oscillators and Rb AFS shown in Figure 4.3 according to the respective manual.

Frequency instability between high-performance oscillators shown in the figure 4.3 was measured using the experimental setup of the beat frequency method described and shown in figure 4.1. For these measurements, the beat frequency was measured in the frequency counter in a gate time of 0,1s and then fractional frequency and Allan deviation measurements have been processed by the software Stable 32[1]. The synthesized signal generator and the GPS receiver can be referenced to an external output for improving their stability. Allan deviation plots for these oscillators show the two cases: referenced and not referenced to an external output. Measurements of frequency instability for these high-performance oscillators are displayed in figures 4.4, 4.5 and 4.6.

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Figure 4.4: Measured frequency instability between two Rb frequency standards (PRS10 vs FS725, 10 Mhz output). The plot represents σy (τ).

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Figure 4.5: Measured frequency instability between the 10 MHz local oscillator from GPS receiver and the Rb AFS (GPS Septentrio vs. FS725). Blue: Curve from Fig. 4.4 only for comparison. Black: GPS receiver not referenced to an external AFS (FS725). Red: GPS receiver referenced to an external AFS (FS725).

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Figure 4.6: Measured frequency instability between the 10MHz voltage output from the synthesized signal generator and the Rb AFS (HP8648B signal generator vs. FS725).

Blue: Curve from Fig. 4.4 only for comparison. Black: Synthesizer not referenced to an external AFS (FS725). Red: Synthesizer referenced to an external AFS (FS725).

Phase noise between high-performance oscillators shown in the figure 4.3 was measured using the experimental setup of the homodyne phase detection method (Fig. 4.2). In this case, the reference used is the output voltage at 10 MHz of the Rb AFS725 AFS. The oscillator under test will be the output voltage at 10MHz from the Rb AFS model PRS10, GPS receiver and synthesized signal generator respectively. Phase deviations among the respective oscillators are displayed in Figures 4.7, 4.8, 4.9, 4.10 and 4.11. It will be discussed in more detail each of these results in Chapter 5 of this work.

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Figure 4.7: Phase deviation between two Rb AFS (PRS10 vs. FS725) measured over a 12 h period.

Figure 4.8: Phase deviation between the 10 MHz output of GPS receiver without external reference and Rb AFS (GPS Septentrio vs. FS725) taken over a 12h period.

Transmitindo padrões de frequência atômicos por redes de fibras ópticas=Transmitting atomic frequency standards in optical fiber networks

UNIVERSITY OF CAMPINAS

INSTITUTE OF PHYSICS “GLEB WATAGHIN”

Erick Abraham Lamilla Rubio

Transmitindo Padrões de Frequências Atômicos por

Redes de Fibras Ópticas

Transmitting Atomic Frequency Standards in

Optical Fiber Networks

CAMPINAS

2015

ERICK ABRAHAM LAMILLA RUBIO

Transmitindo Padrões de Frequências Atômicos

por Redes de Fibras Ópticas

Transmitting Atomic Frequency Standards in

Optical Fiber Networks

CAMPINAS

2015

Acknowledgement

Resumo

Abstract

Lists of figures:

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Appendix

Lists of tables

Contents

Chapter 1: Introduction

1.1. Motivation and context of this work

1.2. Atomic clocks and Atomic frequency Standard (AFS): Microwave

and Optical AFS

1.3. Importance and use of atomic clocks

1.4. Motivation for ultrastable frequency reference transfer via fiber

networks

1.5. Characterization methods used in transferring frequency

references

References:

Chapter 2: Characterization of oscillators

and atomic clocks

2.1. Properties of signal sources.



𝑦(𝑡) =

=

=

,

𝜑

(𝑡)

2.2. Characterization methods

2.2.1. Allan deviation

2.2.2. Phase noise

2.3. Common methods for frequency measurements.

2.3.1. Beat frequency method

2.3.2. Time difference method

2.4. Common method for phase detection.

2.4.1. Homodyne phase detection

References:

Chapter 3: Frequency transfer in Optical

Fiber Networks: Experimental Setup

3.1. Experimental schemes for frequency transfer

3.2.1. Passive transfer technique

3.2.2. Passive microwave transfer for the Rubidium Standard FS725

AM

References:

Chapter 4: Frequency transfer in fiber link:

Results

4.1. Characterization of frequency and phase noise of different

oscillators and Rubidium frequency standards (Rb AFS) inside

the laboratory

FREQUENCY INSTABILITY SPECIFICATIONS BY

MANUFACTURER

Equipment

Frequency Instability

Aging (after 30 days)

Short term

Stability (Allan

Deviation)

Rb AFS

Model FS725

5x10

_(𝑡)