Experimental methods

Transcription

1 Experimental methods for the measurement of phase noise and frequency stability May 10, 2007 Enrico Rubiola FEMTO-ST Institute, Besançon, France CNRS and Université de Franche Comté Phase noise & friends Saturated mixer Correlation (dual-channel) measurements Oscillator phase noise Calibration methods Bridge techniques AM noise Noise in systems Time-domain methods home page

2 1 introduction Phase noise & friends

3 1 introduction Clock signal affected by noise 3 Time Domain Phasor Representation V 0 v(t) amplitude fluctuation V 0 α(t) [volts] normalized ampl. fluct. α(t) [adimensional] t V 0 / 2 ampl. fluct. (V 0 / 2)α(t) V 0 v(t) phase fluctuation ϕ(t) [rad] phase time (fluct.) x(t) [seconds] phase fluctuation ϕ(t) t V 0 / 2 polar coordinates Cartesian coordinates v(t) = V 0 [1 + α(t)] cos [ω 0 t + ϕ(t)] v(t) = V 0 cos ω 0 t + n c (t) cos ω 0 t n s (t) sin ω 0 t under low noise approximation n c (t) V 0 and n s (t) V 0 It holds that α(t) = n c(t) and ϕ(t) = n s(t) V 0 V 0

4 1 introduction Phase noise & friends 4 random phase fluctuation S ϕ (f) = PSD of ϕ(t) power spectral density it is measured as S ϕ (f) = E {Φ(f)Φ (f)} S ϕ (f) Φ(f)Φ (f) m L(f) = 1 2 S ϕ(f) dbc (expectation) (average) S ϕ (f) random walk freq. b 4 f 4 flicker freq. b 3 f 3 white freq. b 2 f 2 b 1 f 1 signal sources only both signal sources and two-port devices flicker phase. white phase b 0 f x f 2 / ν2 0 random fractional-frequency fluctuation y(t) = ϕ(t) S y = f 2 2πν 0 ν0 2 S ϕ (f) S y (f) random walk freq. h 2 f 2 h 1 f 1 h 0 flicker freq. white freq. h 1 f h 2 f 2 flicker phase white phase f Allan variance (two-sample wavelet-like variance) { 1 ] } 2 σy(τ) 2 = E [y 2 k+1 y k approaches a half-octave bandpass filter (for white noise), hence it converges for processes steeper than 1/f. σ y 2 (τ) flicker phase white phase white freq. h 0 /2τ flicker freq. 2ln(2)h 1 E. Rubiola, The Leeson Effect Chap.1, arxiv:physics/ freq. drift random walk freq. (2π) 2 h 6 2 τ τ

5 1 introduction 5 Relationships between spectra and variances noise type S ϕ (f) S y (f) S ϕ S y σ 2 y(τ) mod σ 2 y(τ) white PM b 0 h 2 f 2 h 2 = b 0 ν 2 0 3f H h 2 (2π) 2 τ 2 2πτf H 1 3f H τ 0 h 2 (2π) 2 τ 3 flicker PM b 1 f 1 h 1 f h 1 = b 1 ν 2 0 [ ln(2πf H τ)] h 1 (2π) 2 τ h 1 τ 2 n 1 white FM flicker FM b 2 f 2 h 0 h 0 = b 2 ν 2 0 b 3 f 3 h 1 f 1 h 1 = b 3 ν h 0 τ h 0 τ 1 2 ln(2) h ln(2) h 1 random walk FM b 4f 4 h 2 f 2 h 2 = b 4 ν 2 0 (2π) 2 h 2 τ (2π)2 6 h 2 τ linear frequency drift ẏ 1 2 (ẏ)2 τ (ẏ)2 τ 2 f H is the high cutoff frequency, needed for the noise power to be finite.

6 1 introduction 6 Basic problem: how can we measure a low random signal (noise sidebands) close to a strong dazzling carrier? solution(s): suppress the carrier and measure the noise convolution (low-pass) s(t) h lp (t) distorsiometer, audio-frequency instruments time-domain product s(t) r(t T/4) traditional instruments for phase-noise measurement (saturated mixer) vector difference s(t) r(t) bridge (interferometric) instruments

7 2 saturated mixer Saturated mixer

8 E. Rubiola, Tutorial on the double-balanced mixer, arxiv/physics/ , 2 saturated mixer Double-balanced mixer 8 saturated multiplier => phase-to-voltage detector vo(t) = kφ φ(t) kill 2ν0 LO source LO input D2 mixer D1 RF input RF source R G v p (t) v i (t) R G D3 D4 IF out v o (t) IF load R L

9 2 saturated mixer Practical issues 9 needs a capacitive-input filter to recirculate the 2ω0 output signal ω i ω l RF LO IF dc and 2ω0 Z i filter load 50 Ω LO input v p (t) D2 D3 mixer D1 D4 RF input v i (t) IF out v o (t) actual phase-to-voltage conversion k! ! 50! 400 Output voltage, V Phase difference, degrees E. Rubiola, Tutorial on the double-balanced mixer, arxiv/physics/ , Phase-to-voltage gain, mv/rad DB0218LW2 term. to 10 kω M14A term. to 50 Ω Input power, dbm

10 2 saturated mixer Useful schemes 10 two-port device under test DUT a pair of two-port devices 3 db improved sensitivity DUT FFT FFT quadrature adjust DUT quadr. adj. the measurement of an amplifier needs an attenuator atten DUT FFT the measurement of a low-power DUT needs an amplifier, which flickers atten DUT FFT quadrature adjust quadrature adjust under test reference measure two oscillators best use a tight loop FFT measure an oscillator vs. a resonator under test reference resonator FFT phase lock quadr. adj.

11 2 saturated mixer Mixer limitations 11 kill 2ν0 1 Power narrow power range: ±5 db around Pnom = 7 13 dbm r(t) and s(t) should have ~ same P 2 Flicker noise due to the mixer internal diodes typical Sφ = 140 dbrad 2 /Hz at 1 Hz in average-good conditions 3 Low gain kφ ~ V/rad typ. 10 to 14 dbv/rad 4 White noise due to the operational amplifier S!(f), dbrad 2 /Hz 5 Takes in AM noise due to the residual power-to-offset conversion mixer background noise microwave HF-UHF mixer 1/f noise op-amp white noise frequency, Hz E. Rubiola, Tutorial on the double-balanced mixer, arxiv/physics/ ,

12 2 saturated mixer 12 The operational amplifier is often misused OP27: [3.2 nv/hz 1/2 ] / [0.2 V/rad] = 16 nrad/hz 1/2 ( 156 dbrad 2 /Hz) LT1028: [1.2 nv/hz 1/2 ] / [0.2 V/rad] = 2.4 nrad/hz 1/2 ( 164 dbrad 2 /Hz) Warning: if only one arm of the power supply is disconnected, the LT1028 may delivers a current from the input (I killed a $2k mixer in this way!) You may duplicate the low-noise amplifier designed at the FEMTO-ST Rubiola, Lardet-Vieudrin, Rev. Scientific Instruments 75(5) pp , May 2004

13 2 saturated mixer Mechanical stability 13 any flicker spectrum S(f) = h 1 /f can be transformed into the Allan variance σ 2 = 2 ln(2) h 1 (roughly speaking, the integral over one octave) a phase fluctuation is equivalent to a length fluctuation L = ϕ 2π λ = ϕ 2π c S L (f) = 1 c 2 ν 0 4π 2 ν0 2 S ϕ (f) 180 dbrad 2 /Hz at f = 1 Hz and ν 0 = 9.2 GHz (c = 0.8 c 0 ) is equivalent to S L = m 2 /Hz ( S L = m/ Hz) a residual flicker of 180 dbrad 2 /Hz at f = 1 Hz off the ν 0 = 9.2 GHz carrier (h 1 = ) is equivalent to a mechanical stability σ L = = m # don t think that s only engineering!!! # I learned a lot from non-optical microscopy # bulk solid matter is that stable

14 2 saturated mixer 14 Averaged spectra must be S! < f = 78'>7**:12 3 P " N 6**********789:12; '+ A*!"@B*789 >CD*35E*FG).H'> %-)*'+*.?>--@ >'9*3 >-DI)*.>I(J'>H)7 true PM noise plot 431!!L"@! 6!!EB@3; Rice representation!!b"@! 6!!KB@3;! v(t) = n=0!" $%&'()'*+'),&)-./0*12!" 3!" 4!" #!" 5 a n (t) cos(nω 0 t) b n (t) sin(nω 0 t) S v (nω 0 ) = [ a 2 n + b 2 n] /ω0 an(t) and bn(t) contain the noise in the ω0/2 band centered at ω0 stationary & ergodic process (means repeatable and reproducible): the statistics of all an(t) and bn(t) is the same average on m spectra: confidence of a point improves by 1/m 1/2 interchange ensemble with frequency: smoothness 1/m 1/2

15 3 correlation Correlation (dual-channel) measurements

16 3 correlation Correlation measurements 16 Two separate mixers measure the same DUT. Only the DUT noise is common a(t), b(t) > mixer noise c(t) > DUT noise DUT a(t) b(t) c(t) x = c a FFT y = c b basics of correlation S yx (f) = E {Y (f)x (f)} = E {(C A)(C B) } = E {CC AC CB + AB } = E {CC } S yx (f) = S cc (f) single-channel in practice, average on m realizations S yx (f) = Y (f)x (f) m S!(f) 1/"m = CC AC CB + AB m = CC m + O(1/m) 0 as 1/ m correlation frequency

17 3 correlation Example of correlation measurement MHz carrier!!k"c$ >!!MKCM?!!J"C$ >!!FKCM? S! 9 f : S " 9 f : ;<=78 ;<'/;..=78 $ N >...;<0=78? '3 *+,-)(./'0 plot 459 P " H.!#CI.;<0 /'0.!/6.!D /,-)(.2,%/)C!!I"C$ >!!KKCM?!!L"C$ >!!JKCM? %&''() k B T " =P " H.!!IJCI.;<>'/;..?=78 $!$""C$ >!!IKCM?!!" 1&2'+('.3'(42(,%56.78!" $!" # Noise of a by-step attenuator, measured at 100 MHz by correlation. The mixer is replaced with a bridge.

18 3 correlation Useful schemes 18 (ref) (ref) phase arm a DUT 2 port device arm b phase LO RF RF LO dc dc x y FFT analyzer DUT REF arm a arm b REF RF LO LO RF dc dc x y FFT analyzer REF phase lock phase and ampl. meter output (noise only) DUT REF arm a arm b LO RF RF LO dc dc phase lock x y FFT analyzer bridge a DUT 2 port device bridge b (ref) phase and ampl. Σ Σ Δ Δ µw µw RF RF LO LO dc dc x y FFT analyzer

19 E. Rubiola, R. Boudot, The effect of AM noise on correlation phase noise measurements, arxiv/physics/ correlation Pollution from AM noise 19 file am correl 2port (ref) (ref) phase arm a DUT 2 port device arm b phase AM LO RF RF LO AM AM dc dc x y FFT analyzer The mixer converts power into dc-offset, thus AM noise into dc-noise, which is mistaken for PM noise v(t) = kφ φ(t) + klo αlo + krf αrf rejected by correlation and avg not rejected by correlation and avg file am correl oscillator phase lock file am correl discrim DUT REF arm a arm b REF AM RF LO LO RF AM AM dc dc x y FFT analyzer REF DUT REF arm a arm b AM LO RF RF LO AM dc AM dc phase lock x y FFT analyzer

20 4 oscillators Oscillator phase noise

21 4 oscillators 21 A frequency discriminator can be used to measure the phase noise of an oscillator

22 4 oscillators Photonic delay line method 22 1 The delay line turns the oscillator frequency noise into phase noise, which is measured by the mixer. 2 The oscillator noise is calculated by unapplying the equation of the delay line 3 Photonic delay: the optical fiber exhibits low loss (0.2 db/km)

23 4 oscillators 23 Correlation dual-delay-line method semiconductor laser optical isolator electro-optic modulator photodetector microwave fiber delay amplifier DC amplifier microwave isolator splitter coupler! phase shifters! mixer F F T oscillator to be measured dual-channel FFT analyzer The only common part of the setup is the power splitter. Two completely separate systems measure the same oscillator under test

24 4 oscillators 24 Phase Locked Loop (PLL) Phase: the PLL is a low-pass filter S!2 " f # S!1 " f # $ %k o k! H c " f #% 2 4& 2 f 2 '%k o k! H c " f #% 2 Output voltage: the PLL is a high-pass filter S vo " f # S!1 " f # $ 4& f 2 k! 2 4& 2 f 2 '%k o k! H c " f #% 2 compare an oscillator under test to a reference low-noise oscillator or compare two equal oscillators and divide the spectrum by 2 (take away 3 db)

25 4 oscillators 25 Phase Locked Loop (PLL)

26 4 oscillators 26 A tight PLL shows many advantages but you have to correct the spectrum for the PLL transfer function

27 4 oscillators 27 Practical measurement of Sφ(f) with a PLL 1. Set the circuit for proper electrical operation a. power level b. lock condition (there is no beat note at the mixer out) c. zero dc error at the mixer output (a small V can be tolerated) 2. Choose the appropriate time constant 3. Measure the oscillator noise 4. At end, measure the background noise

28 4 oscillators 28 Warning: a PLL may not be what it seems

29 4 oscillators 29 PLL two frequencies The output frequency of the two oscillators is not the same. A synthesizer (or two synth.) is necessary to match the frequencies At low Fourier frequencies, the synthesizer noise is lower than the oscillator noise At higher Fourier frequencies, the white and flicker of phase of the synthesizer may dominate

30 4 oscillators 30 PLL low noise microwave oscillators With low-noise microwave oscillators (like whispering gallery) the noise of a microwave synthesizer at the oscillator output can not be tolerated. Due to the lower carrier frequency, the noise of a VHF synthesizer is lower than the noise of a microwave synthesizer. This scheme is useful with narrow tuning-range oscillator, which can not work at the same freq. to prevent injection locking due to microwave leakage

31 4 oscillators 31 Designing your own instrument is simple Standard commercial parts: double balanced mixer low-noise op-amp standard low-noise dc components in the feedback path commercial FFT analyzer Afterwards, you will appreciate more the commercial instruments: assembly instruction manual computer interface and software

32 5 calibration Calibration methods

33 5 calibration Calibration general procedure 33 1 adjust for proper operation: driving power and quadrature 2 measure the mixer gain kφ (volts/rad) > next 3 measure the residual noise of the instrument 4 measure the rejection of the oscillator noise Make sure that the power and the quadrature are the same during all the calibration process

34 5 calibration 34 Calibration measurement of kφ (phase mod.) Vm The reference signal can be: a) a tone: detect with the FFT, with a dual-channel FFT, or with a lock-in b) random or pseudo-random white noise tone: white noise Some FFTs have a white noise output Dual-channel FFTs calculate the transfer function H(f) 2 =SVm/SVd

35 5 calibration 35 Calibration measurement of kφ (rf signal) However often used, even in major laboratories this method is incorrect because: 1 the calibration signal yields AM and PM of equal depth, 2 the mixer shows a residual sensitivity to AM noise

36 5 calibration 36 Calibration measurement of kφ (rf signal) A reference rf noise is injected in the DUT path through a directional coupler However often used, this method is incorrect see previous slide

37 5 calibration 37 Primary calibration A modulated signal carrier V0 cos( ω 0 t) modulation xcos( ω 0 t) y sin(ω 0 t) RF input V0 cos( ω 0 t) carrier V cos( ω 0 t) 0 + RF output xcos( ω 0 t) y sin(ω 0 t) B C V 0 α= x amplitude modulated signal carrier V0 cos( ω 0 t) V 0 ϕ = y in phase modulation xcos( ω 0 t) I LO I Q modul x y RF Q modulation sidebands xcos( ω 0 t) y sin(ω 0 t) modul. input I Q modulator LO pump 90 RF output phase modulated signal carrier V0 cos( ω 0 t) ϕ y sin(ω 0 t) quadrature modulation I Q α rms = Px P 0 and ϕ rms = Py P 0 P0 = power of the carrier Px = power of the in-phase sidebands Py = power of the quadrature sidebands

38 6 bridge (interferometer) Bridge techniques

39 6 bridge (interferometer) Wheatstone bridge 39 adj. phase 0 LO RF IF equilibrium: Vd = 0 > carrier suppression synchronous detection: get vc(t) vs(t) (AM or PM noise) static error δz1 > some residual carrier real δz1 => in-phase residual carrier Vre cos(ω0t) imaginary δz1 => quadrature residual carrier Vim sin(ω0t) fluctuating error δz1 => noise sidebands real δz1 => AM noise vc(t) cos(ω0t) imaginary δz1 => PM noise vs(t) sin(ω0t)

40 6 bridge (interferometer) Synchronous detection 40

41 6 bridge (interferometer) 41 Bridge (interferometric) PM and AM noise measurement bridge detector and rejection of the master-oscillator noise yet, difficult for the measurement of oscillators

42 6 bridge (interferometer) 42 A bridge (interferometric) instrument can be built around a commercial instrument You will appreciate the computer interface and the software ready for use

43 6 bridge (interferometer) 43 Origin of flicker in the bridge In the early time of electronics, flicker was called contact noise

44 6 bridge (interferometer) 44 Coarse and fine adjustment of the bridge null are necessary

45 6 bridge (interferometer) 45 Flicker reduction, correlation, and closedloop carrier suppression can be combined R 0 =50 Ω power splitter CP1 atten atten atten inner interferometer DUT γ CP2 arbitrary phase γ Δ manual carr. suppr. var. att. & phase arbitrary phase x( t) 10 20dB coupl. pump rf virtual gnd null Re & Im CP3 LO R 0 R 0 atten RF I Q modul CP4 I Q RF I Q detect g ~ 40dB LO pump RF g ~ 40dB pump u 1 z 1 u 2 dual integr z 2 I Q detect LO channel b (optional) I v 1 readout Q v 2 G B matrix matrix I Q automatic carrier suppression control diagonaliz. D matrix channel a v 1 readout v 2 G B matrix matrix w 1 w 2 w 1 w 2 G: Gram Schmidt ortho normalization B: frame rotation FFT analyz. I Q detector/modulator I RF Q LO 90 0 E. Rubiola, V. Giordano, Rev. Scientific Instruments 73(6) pp , June 2002

46 6 bridge (interferometer) Example of results 46 CP2 isolation interferometer DUT resistive terminations isolation k B T 0 k B T 0 g g!!l"ak 5!!%LA%7!!M"AK 5!!KLA%7!!J"AK 5!!LLA%7!!="AK 5!!MLA%7 S " 1 + S! 1 f +,-./0 N f,-(2,33./0 # ,-6./07 (4 &'(()* 89:;*)32(6 k T B P 2<;3=#38>)&?(2?B'3(43&C2::A 2(63!2E3!F # " " Correlation-and-averaging rejects the thermal noise!#""ak 5!!JLA%7!!" G'D(9)(34()HD):&IE3/0!" #!" $!" %!" K Residual noise of the fixed-value bridge, in the absence of the DUT!!>DE$ 1!!$MEC3!!CDE$ 1!!%MEC3 S " S!,, f f & & '()*+ N '(-.'//)*+ # 1//////////'(2)*+3-0?7:@A8/.-2 P " L/!DED/'(2.F@/!G/?H8;B-. BI5/-0/;J.::E.-2/!.=/!K.:@A8/6:;.AE!!J"># :!!#I>%;!!H"># :!!II>%; S!, - S ", - f./) $ f./234 : /9234; N)< k T B 5678+*10)9 2P =1!!HH>!1./:)0.11;234 $ " " P " =1!%>!1./9 0?81&$@15A*'B)0 BC(1)<1'D077> 0)91!0F1!G 078+*1E7'0+>!!DDE$ 1!!>MEC3!!K"># :!!JI>%; '())*+!!GDE$ 1!!CMEC3!$""># :!!HI>%;!!MDE$ 1!!DMEC3 7:?B-628:B/:57?8!!" /0-8968:;<=/*+!" #!" $!" %!" >!$!"># L(E)6*)1<)*ME*7'NF134 :!!KI>%;!" $!" &!" %!" # Noise of a pair of HH-109 hybrid couplers measured at 100 MHz Residual noise of the fixed-value bridge. Same as above, but larger m

47 6 bridge (interferometer) ±45º detection 47 DUT noise without carrier UP reference DOWN reference cross spectral density S ud (f) = 1 2 n c (t) cos ω 0 t n s (t) sin ω 0 t u(t) = V P cos(ω 0 t π/4) d(t) = V P cos(ω 0 t + π/4) [ ] S α (f) S ϕ (f) -.!/01+"'.*-+" "'.*-+" 4"6")6-*. 4"6")6-*. #$ u #$!n s&'( t +-.&, "!% t( u!!k;f; A!!HKFKB!!;;F; A!!KKFKB S"@? f? <=>34 f <=)8<,,>34 # A,,,,,,,,,,,,,<=9>34B N )- 5*/67+,8)9 P " :,!%F!,<=9 8J6,K$H,5L+0C)8 CD',)-,0E8//F G%HI,<+C+0C*'/, 8/67+,(/087F n c &'()*+&, t t( "! %!"!"!!M;F; A!!;KFKB k B T " :,!!;%,<=9>34 d d!!n;f; A!!MKFKB!#";F; A!!NKFKB 0'))+7 &'()*+),-)+.(+/012,34!!"!" #!" $!" %!" H Residual noise, in the absence of the DUT Smart and nerdy, yet of scarce practical usefulness First used at 2 khz to measure electromigration on metals (H. Stoll, MPI)

48 6 bridge (interferometer) 48 The complete machine (100 MHz)

49 6 bridge (interferometer) 49 A 9 GHz experiment (dc circuits not shown)

50 6 bridge (interferometer) Advanced comparison 50 Comparison of the background noise S (f) dbrad 2 /Hz ϕ 140 real time correl. & avg saturated mixer interferometer double interferometer correl. sat. mix. mixer, interferometer nested interferometer saturated mixer residual flicker, by step interferometer residual flicker, fixed interferometer residual flicker, fixed interferometer residual flicker, fixed interferometer, ±45 detection Fourier frequency, Hz interferometer correl. saturated mixer double interf. measured floor, m=32k

51 7 AM noise AM noise

52 7 AM noise 52 Tunnel and Schottky power detectors rf in law: v = kd P ~60 Ω video out pf external 50 Ω to 100 kω The tunnel diode is actually a backward diode. The negative resistance region is absent. parameter Schottky tunnel input bandwidth up to 4 decades 1 3 octaves 10 MHz to 20 GHz up to 40 GHz vsvr max. 1.5:1 3.5:1 max. input power (spec.) 15 dbm 15 dbm absolute max. input power 20 dbm or more 20 dbm output resistance 1 10 kω Ω output capacitance pf pf gain 300 V/W 1000 V/W cryogenic temperature no yes electrically fragile no yes Measured detector gain, A 1 load resistance, Ω DZR124AA DT8012 (Schottky) (tunnel) output voltage, dbv Herotek DZR124AA s.no Schottky 3.2 kω 10 kω 100 Ω 320 Ω 1 kω input power, dbm output voltage, dbv Herotek DT8012 s.no Tunnel conditions: power 50 to 20 dbm ampli dc offset ampli dc offset kω 3.2 kω 1 kω Ω Ω -10 input power, dbm 0 10

53 7 AM noise Noise mechanisms 53 Shot noise SI (f ) = 2qI0 Rothe-Dahlke model of the amplifier detector rf in video out amplifier in v n noise free out ~60 Ω pf external 50 Ω to 100 kω i n Thermal noise SV (f ) = 4kBT0R Flicker (1/f ) noise is also present Never say that it s not fundamental, unless you know how to remove it In practice the amplifier white noise turns out to be higher than the detector noise and the amplifier flicker noise is even higher

54 7 AM noise Cross-spectrum method 54 v a (t) = 2k a P a α(t) + noise source under test power meter monitor P a P b v a v b dual channel FFT analyzer v b (t) = 2k a P b α(t) + noise The cross spectrum Sba(f ) rejects the single-channel noise because the two channels are independent. S ba (f) = 1 4k a k b P a P b S α (f) Sα (f) log/log scale 1 m single channel cross spectrum meas. limit Averaging on m spectra, the singlechannel noise is rejected by 1/2m A cross-spectrum higher than the averaging limit validates the measure The knowledge of the single-channel noise is not necessary f

55 7 AM noise Example of AM noise spectrum Wenzel E 100 MHz OCXO P0 = 10.2 dbm avg 2100 spectra Sα ( f ) db/hz Fourier frequency, Hz flicker: h 1 = Hz 1 ( db) σ α = Single-arm 1/f noise is that of the dc amplifier (the amplifier is still not optimized)

56 8 systems Noise in systems

57 8 systems 57 Additive (white) noise in amplifiers etc. Noise figure F Input power P0 power law S ϕ = 0 i= 4 b i f i V 0 cos ω 0 t n rf (t) g white phase noise b 0 = F kt 0 P 0 Sφ(f) low P0 high P0 P0 f Cascaded amplifiers (Friis formula) N = F 1 kt 0 + (F 2 1)kT 0 g As a consequence, (phase) noise is chiefly that of the 1st stage

58 8 systems 58 Parametric (flicker) noise in amplifiers etc. parametric up-conversion of the near-dc noise S(f) no carrier near-dc flicker S(f) noise up-conversion no flicker ω0 =? f ω0 f carrier + near-dc noise v i (t) = V i e jω 0t + n (t) + jn (t) v o (t) = a 1 v i (t) + a 2 v 2 i (t) +... non-linear amplifier substitute (careful, this hides the down-conversion) expand and select the ω0 terms v o (t) = V i { a 1 + 2a 2 [ n (t) + jn (t) ]} e jω 0t get AM and PM noise α(t) = 2 a 2 a 1 n (t) ϕ(t) = 2 a 2 a 1 n (t) independent of Vi (!) the parametric nature of 1/f noise is hidden in n and n Sφ(f) b 1 independent of P0 m cascaded amplifiers (b 1 ) cascade = m (b 1 ) i i=1 S ϕ = In practice, each stage contributes equally 0 i= 4 b i f i f

59 8 systems Frequency synthesis The ideal noise-free frequency synthesizer repeats the input time jitter 59 v i (t) ω i = ω 0 T = 2π/ω 0 t phase jitter ϕ i = 2π δt T ϕ i = ω 0 δt input jitter ϕi output jitter ϕ o = n d ϕ i v o (t) ω o = n d ω i time jitter x = δt phase jitter e jω 0t e j n d ω 0t ϕ o = n d ω 0δt t After division, the noise of the output buffer may be larger than the input-noise scaled down After multiplication, the scaled-up phase noise sinks energy from the carrier. At m 2.4, the carrier vanishes S ϕ (f) IN (1/d 2 ) OUT real buffer noiseless divider real buffer 1/d 2 input output signal output stage input/d 2 1/d 2 output signal output stage 1/d 2 f c input/d 2 f

60 8 systems 60 Saturation and sampling clipped waveform t gain 1 0 t Saturation is equivalent to reducing the gain Digital circuits, for example, amplify (linearly) only during the transitions

61 8 systems Photodiode white noise 61 intensity modulation photocurrent microwave power shot noise P (t) = P (1 + m cos ω µ t) i(t) = qη hν P (1 + m cos ω µt) P µ = 1 ( qη ) 2P 2 m2 2 R 0 hν N s = 2 q2 η hν P R 0 thermal noise N t = F kt 0 total white noise (one detector) S ϕ0 = 2 m 2 [ 2 hν λ η 1 P + F kt 0 R 0 ( hνλ qη ) 2 ( 1 P ) 2 ] Threshold power mw

62 8 systems Photodetector noise 62 infrared 1.32 µ m YAG laser monitor output (13dBm) 22dBm EOM 50% coupler power meter iso P! iso (!3dBm) photodiodes under test P µ (!26dBm) hybrid r(t)!90 s(t) phase & aten. (carrier suppression) 0 0!90 % & g=37db RF LO =6dB IF phase $ (detection of " or #) v(t) g =52dB FFT analyz GHz power MHz ampli PLL synth. microwave near!dc photodiode S α (1 Hz) S ϕ (1 Hz) estimate uncertainty estimate uncertainty HSD DSC30-1K QDMH unit db/hz db dbrad 2 /Hz db The noise of the amplifier is not detected Electron. Lett p (2003)

63 8 systems Photodetector noise 63 the photodetectors we measured are similar in AM and PM 1/f noise the 1/f noise is about -120 db[rad2]/hz other effects are easily mistaken for the photodetector 1/f noise environment and packaging deserve attention in order to take the full benefit from the low noise of the junction Figure 2: Example of measured spectra Sα (f ) and Sϕ (f ). W: waving a hand 0.2 m/s, 3 m far from the system B: background noise P: photodiode noise modulator (EOM) is rejected. The amplitude noise of the source is re to the same degree of the carrier attenuation in, as results from the g properties of the balanced bridge. This rejection applies to amplitude noi to the laser relative intensity noise (RIN). The power of the microwave source is set for the maximum modulation m, which is the Bessel function J1 ( ) that results from the sinusoidal respo the EOM. This choice also provides increased rejection of the amplitude n the microwave source. The sinusoidal response of the EOM results in har distortion, mainly of odd order; however, these harmonics are out of the s bandwidth. The photodetectors are operated with some 0.5 mw input which is low enough for the detectors to operate in a linear regime. This F: after bending a fiber, 1/f S: single spectrum, with optical A: averageaspectrum, withsuppression optical possible high carrier (50 60 db) in mistakes, mistakes which isaround stable f Figure 3: Examples of environment effects and experimental Figure 3: Examples ofisolators environment effects and experimental around noise can increase unpredictably connectors and no isolators connectors and no duration thethe measurement (half anb: Background hour), and noise also provides ab) high the corner. All plots instrument Background noise (spectrum B) re the Allofthe plots showshow the the instrument (spectrum background noise B: background noise B: corner. background noise RIN and of the thephotodiode noise of the (spectrum amplifier. The coherence and thelaser noise spectrum of pair P). Plot 1 spectrum len andof thethe noise spectrum of the Photodiode pair (spectrum P).noise Plot 1 spectrum P: photodiode P: photodiode noise P: photodiode noise W: experimentalist the experimentalist Waves aexperiment hand gently ( about 0.2 m/s), 3far m away far from the YAG laser used in our is andaway all optical W: the Waves a hand gently ( 0.2 m/s), 31mkm, from thethe sig

64 8 systems Physical phenomena in optical fibers 64 Birefringence. Common optical fibers are made of amorphous Ge-doped silica, for an ideal fiber is not expected to be birefringent. Nonetheless, actual fibers show birefringent behavior due to a variety of reasons, namely: core ellipticity, internal defects and forces, external forces (bending, twisting, tension, kinks), external electric and magnetic fields. The overall effect is that light propagates through the fiber core in a non-degenerate, orthogonal pair of axes at different speed. Polarization effects are strongly reduced in polarization maintaining (PM) fibers. In this case, the cladding structure stresses the core in order to increase the difference in refraction index between the two modes. Rayleigh scattering. This is random scattering due to molecules in a disordered medium, by which light looses direction and polarization. A small fraction of the light intensity is thereby back-scattered one or more times, for it reaches the fiber end after a stochastic to-and-fro path, which originates phase noise. In the early fibers it contributed 0.1 db/km to the optical loss. Bragg scattering. In the presence of monocromatic light (usually X-rays), the periodic structure of a crystal turns the randomness of scattering into an interference pattern. This is a weak phenomenon at micron wavelengths because the inter-atom distance is of the order of nm. Bragg scattering is not present in amorphous materials. Brillouin scattering. In solids, the photon-atom collision involves the emission or the absorption of an acoustic phonon, hence the scattered photons have a wavelength slightly different from incoming photons. An exotic form of Brillouin scattering has been reported in optical fibers, due to a transverse mechanical resonance in the cladding, which stresses the core and originates a noise bump on the region of MHz. Raman scattering. This phenomenon is somewhat similar to Rayleigh scattering, but the emission or the absorption of an optical phonon. Kerr effect. This effect states that an electric field changes the refraction index. So, the electric field of light modulate the refraction index, which originates the 2nd-order nonlinearity. Discontinuities. Discontinuities cause the wave to be reflected and/or to change polarization. As the pulse can be split into a pulse train depending on wavelength, this effect can turn into noise. Group delay dispersion (GVD). There exist dispersion-shifted fibers, that have a minimum GVD at 1550 nm. GVD compensators are also available. Polarization mode dispersion (PMD). This effect rises from the random birefringence of the optical fiber. The optical pulse can choose many different paths, for it broadens into a bell-like shape bounded by the propagation times determined by the highest and the lowest refraction index. Polarization vanishes exponentially along the light path. It is to be understood that PMD results from the vector sum over multiple forward paths, for it yields a well-shaped dispersion pattern. PMD-Kerr compensation. In principle, it is possible that PMD and Kerr effect null one another. This requires to launch the appropriate power into each polarization mode, for two power controllers are needed. Of course, this is incompatible with PM fibers. Which is the most important effect? In the community of optical communications, PMD is considered the most significant effect. Yet, this is related to the fact that excessive PMD increases the error rate and destroys the eye pattern of a channel. In the case of the photonic oscillator, the signal is a pure sinusoid, with no symbol randomness. My feeling is that Rayleigh scattering is the most relevant stochastic phenomenon.

65 8 systems 65 Rayleigh scattering Rayleigh scattering contributes some 0.1 db/km to the loss G. Agrawal, Fiber-optic communications systems, Wiley 1997 Stochastic scattering forward transmitted back scattered scattered twice

66 9 time-domain Time-domain methods

67 9 time-domain 67 Time-domain measurements direct measurement! 0 frequency counter Resolution limited by the counter quantization Example: resolution 100 ps => σy(1s) = Beat methods The beat mechanism keeps φ, for it amplifies the phase-time x by factor ν0/νb! 0! 0 +! b beat-note! b frequency counter Counter resolution improved by a factor ν0/νb Example: counter 1ns ν0=10mhz, νb=10hz σy(1s) = 10 9 /10 6 =10 15 [plus trigger noise] Need a frequency offset νb dual-mixer! 0! c! b stop counter Counter resolution improved by a factor ν0/νb Does not need the frequency offset νb The noise of the common oscillator cancels! b start! 0