Phase noise, oscillators etc.

Transcription

1 Phase noise, oscillators etc. E. Rubiola, V. Giordano, K. Volyanskiy, H. Tavernier, Y. Kouomou Chembo, R. Bendoula, P. Salzenstein, J. Cussey, X. Jouvenceau, R. Boudot, L. Larger,... FEMTO-ST Institute, Besançon, France CNRS and Université de Franche Comté Outline Phase noise & friends Amplifier noise Correlation AM noise Bridge (interferometric) noise measurements Advanced methods Delay-line instrument Optical resonators Non-linear AM oscillations home page

2 1 Phase noise & friends

3 1 introduction Clock signal affected by noise X 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 X 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 freq. drift random walk freq. (2π) 2 h 6 2 τ τ

5 1 introduction X 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 X 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 Amplifier noise AM/PM noise additive parametric white local (flicker) environmental E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge 2008, ISBN

8 RF spectrum 2 amplifier noise Amplifier white noise 3 Noise figure F, Input power P0 S(ν ) P=FkT 0 B V 0 cos ω 0 t g B P 0 B N e =FkT 0 n rf (t) ν 0 f ν 0 ν 0 +f LSB USB ν power law S ϕ = 0 i= 4 b i f i white phase noise b 0 = F kt 0 P 0 Sφ(f) P0 low P0 high P0 f Cascaded amplifiers (Friis formula) The (phase) noise is chiefly that of the 1st stage F1 g1 F2 g2 F3 g3 The Friis formula applied to phase noise N = F 1 kt 0 + (F 2 1)kT 0 g b 0 = F 1kT 0 P 0 + (F 2 1)kT 0 P 0 g

9 2 amplifier noise Amplifier flicker noise 4 S(f) no carrier near-dc flicker S(f) noise up-conversion near-dc noise no flicker ω0 =? f ω0 f stopband output bandwidth stopband output bandwidth carrier 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 (parametric) amplifier near-dc noise substitute (careful, this hides the down-conversion) the parametric nature of 1/f noise is hidden in n and n expand and select the ω0 terms v o (t) = V i { a 1 + 2a 2 [ n (t) + jn (t) ]} e jω 0t The noise sidebands are proportional to the input carrier get AM and PM noise α(t) = 2 a 2 a 1 n (t) ϕ(t) = 2 a 2 a 1 n (t) The AM and the PM noise are independent of Vi, thus of power There is also a linear parametric model, which yields the same results

10 2 amplifier noise Amplifier flicker noise 5 S! (f), log-log scale b 1! const. vs. P 0 b 1 f 1 b 0 = FkT 0 / P 0 b 0, higher P 0 b 0, lower P 0 f' c f" c f f c = ( b 1 / FkT 0 ) P 0 depends on P 0 typical amplifier phase noise RATE GaAs HBT SiGe HBT Si bipolar microwave microwave HF/UHF fair good best unit dbrad 2 /Hz The phase flicker coefficient b 1 is about independent of power. Describing the 1/f noise in terms of fc is misleading because fc depends on the input power In a cascade, (b 1)tot does not depend of the amplifier order. Each stage contributes about equally b 1 is roughly proportional to the gain through the number of stages Paralleling m amplifier, (b 1)tot is divided by m

11 2 amplifier noise Amplifier flicker noise experiments 6 Phase noise, dbrad 2 /Hz Phase noise, dbrad 2 /Hz f, Hz P in =0dBm SiGe LPNT32 bias 2V, 20 ma R. Boudot 2006 = 15dBm P in = 10dBm P in = 5dBm P= 50dBm P= 60dBm P= 70dBm P= 80dBm P in Amplifier X H at 4.2 K Data from IEEE UFFC 47(6):1273 (2000) P= 80dBm P= 70dBm P= 60dBm P= 50dBm The 1/f phase noise b 1 is about independent of power The white noise b0 scales up/down as 1/P0, i.e., the inverse of the carrier power Fourier frequency, Hz

12 2 amplifier noise X Flicker noise in cascaded amplifiers S! (f), log-log scale b 1! const. vs. P 0 b 1 f 1 b 0 = FkT 0 / P 0 b 0, higher P 0 b 0, lower P 0 f' c f" c f f c = ( b 1 / FkT 0 ) P 0 depends on P 0 A B B A AB and BA have the same 1/f noise The phase flicker coefficient b 1 is about independent of power. Hence: in a cascade, (b 1)tot does not depend of the amplifier order in practice, in a cascade each stage contributes about equally m (b 1 ) tot = (b 1 ) i i=1 b 1 is roughly proportional to the gain through the number of stages

13 2 amplifier noise X Flicker in cascaded amplifiers experiments S (f), db.rad 2! /Hz!50!70!90!110!130 AML812PNB1901 (SiGe HBT) 15 Feb cascaded Amplifiers, Pin= 27.5dBm The expected flicker of a cascade increases by: 3 db, with 2 amplifiers 5 db, with 3 amplifiers!150 Single Amplifier, Pin= 4dBm! f, Hz S! (f), db.rad 2 /Hz!120!130!140!150!160 2Amps! Pin=!5dBm 1Amp! Pin=!5dBm Avantek UTC573, 10 MHz 23 Jan 07 3Amps! Pin=!24dBm!170 5!6 db NMS floor! f, Hz

14 2 amplifier noise Flicker noise in parallel amplifiers X u 1 (t) (t) A 1 v 1 ψ 1 input v i (t) m way power divider (t) u k A k v k (t) ψ k m way power combiner output v o (t) ϕ v m ψ m u m (t) (t) A m The phase flicker coefficient b 1 is about independent of power The flicker of a branch is not increased by splitting the input power At the output, the carrier adds up coherently b 1 = 1 [ ] b 1 the phase noise adds up statistically m Hence, the 1/f phase noise is reduced by a factor m Only the flicker noise can be reduced in this way Gedankenexperiment: join the m branches of a parallel amplifier forming a single large active device: the phase flickering is proportional to the inverse physical size of the amplifier active region branch

15 2 amplifier noise u 1 (t) (t) A 1 v 1 ψ 1 X Parallel amplifiers, mathematics input v i (t) m way power divider (t) u k A k v k (t) ψ k m way power combiner output v o (t) ϕ u k (t) = 1 m v i (t) v o (t) = 1 m m k=1 v k (t) v k (t) = 1 m V i {a 1 + 2a 2 [ n k (t) + jn k(t) ]} e j2πν 0t ψ k (t) = 2 a 2 a 1 n k(t) ϕ k (t) = 1 m V i 2a 2 n k (t) ej2πν 0t a 1 V i e j2πν 0t = 1 m 2a 2 n a k(t) 1 m 1 S ϕ (f) = m 2 4a2 2 a 2 S n (f) k 1 k=1 branch-amplifier input main output branch output branch branch output branches output v m ψ m u m (t) (t) A m S ϕ (f) = 1 m 4a2 2 a 2 1 S n (f) S ϕ (f) = 1 m S ψ(f) b 1 = 1 m [ b 1 ] branch m equal branches output

16 2 amplifier noise Flicker noise in parallel amplifiers X!50!70 JS2, 10 GHz 15 Feb S! (f), db.rad /Hz!90!110!130 Single amplifier Pin= 6 dbm Connecting two amplifiers in parallel, the expected flicker is reduced by 3 db!150 2 parallel amplifiers Pin= 5 dbm! f, Hz!50!70 AFS6, 10 GHz 27 Feb S! (f), db.rad /Hz!90!110!130 vibrations Single AFS6 Amplifier Pin= 45 dbm!150 2 Parallel AFS6 Amplifiers Pin= 33 dbm! f, Hz

17 2 amplifier noise X Flicker noise in parallel amplifiers 140 AML812PNA0901 (100mA) AML812PNB0801 (200mA) Phase noise, dbrad /Hz AML812PND0801 (800mA) AML812PNC0801 (400mA) Fourier frequency, Hz 10 Specification of low phase-noise amplifiers (AML web page) amplifier parameters phase noise vs. f, Hz gain F bias power AML812PNA AML812PNB AML812PNC AML812PND unit db db ma dbm dbrad 2 /Hz

18 2 amplifier noise X Environmental (parametric) noise in amplifiers temperature vibrations g phase A B etc. B A input carrier amplitude A time constant can be present φ = φa + φb and α = αa + αb regardless of the amplifier order Cascading m equal amplifiers, Sα(f) and Sφ(f) increase by a factor m 2. If the amplifier were independent, Sα (f) and Sφ(f) would increase only by a factor m. Cascaded amplifiers let z(t) = x(t) + y(t) Phase noise S z (f) = ZZ = (X + Y ) (X + Y ) = XX + Y Y + XY + Y X = S x + S y + S xy }{{} >0 + S yx }{{} >0

19 2 amplifier noise X Environmental effects in RF amplifiers f 5 f 5 Amplifier phase noise courtesy of J. Ackermann N8UR, comments on noise are of E. Rubiola f 5 TADD-1 Spectracom 8140T b 1 = db HP 5087A and TADD-1 10 MHz b 1 = 133 db b 1 is the 1/f noise coefficient in dbrad 2 /Hz (dbc/hz + 3 db) 8140T HP 5087A TADD-1 background b 1 = 142 db TADD-1 5 MHz b 1 = db It is experimentally observed that the temperature fluctuations cause a spectrum Sα(f) or Sφ(f) of the 1/f 5 type Yet, at lower frequencies the spectrum folds back to 1/f

20 3 Correlation

21 3 correlation Correlation measurements 8 a(t) x = c a DUT c(t) FFT b(t) 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) Two separate mixers measure the same DUT. Only the DUT noise is common a(t), b(t) > mixer noise c(t) > DUT noise phase noise measurements DUT noise, normal use background, ideal case a, b c a, b c = 0 background, with AM noise a, c b 0 single-channel instrument noise DUT noise instrument noise no DUT instrument noise AM-to-DC noise 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

22 3 correlation 9 Thermal noise compensation

23 3 correlation 10 Thermal noise compensation 100 MHz prototype, carrier power P o = 8 dbm mesured noise, dbrad 2 /Hz thermal floor injected noise, dbrad 2 /Hz

24 3 correlation 11 Example of correlation measurement 100 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.

25 3 correlation Useful schemes X (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

26 3 correlation Pollution from AM noise 12 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 E. Rubiola, R. Boudot, The effect of AM noise on correlation phase noise measurements, IEEE Tr.UFFC 54(5): May 2007, and arxiv/physics/

27 4 - AM noise E. Rubiola, arxiv/physics

28 4 AM noise 14 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

29 4 AM noise Noise mechanisms 15 Shot noise SI (f ) = 2qI0 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

30 4 AM noise Cross-spectrum method 16 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

31 4 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)

32 4 AM noise 18 AM noise of some sources source h 1 (flicker) (σ α ) floor Anritsu MG3690A synthesizer (10 GHz) db Marconi synthesizer (5 GHz) db Macom PLX GHz multipl db Omega DRV9R F 9.2 GHz DRO db Narda DBP-0812N733 amplifier (9.9 GHz) db HP 8662A no. 1 synthesizer (100 MHz) db HP 8662A no. 2 synthesizer (100 MHz) db Fluke 6160B synthesizer db Racal Dana 9087B synthesizer (100 MHz) db Wenzel D 100 MHz OCXO db Wenzel E no MHz OCXO db Wenzel E no MHz OCXO db worst best

33 5 Bridge method

34 5 bridge (interferometer) Wheatstone bridge 20 adj. phase RF LO IF 0 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)

35 5 bridge (interferometer) 21 Bridge (interferometric) PM and AM noise measurement bridge detector optional: I-Q detection High immunity to low-f magnetic fields and rejection of the master-oscillator noise yet, difficult for the measurement of oscillators

36 5 bridge (interferometer) X Synchronous detection

37 5 bridge (interferometer) 22 A bridge (interferometric) instrument can be built around a commercial instrument You will appreciate the computer interface and the software ready for use

38 6 Advanced methods

39 6 advanced methods Advanced flicker reduction 24 Origin of flicker in the bridge In the early time of electronics, flicker was called contact noise Coarse (by-step) and fine (continuous) adjustment of the bridge null are necessary

40 6 advanced methods X Coarse and fine adjustment of the bridge null are necessary

41 6 advanced methods 25 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. Sci. Instrum. 73(6) pp , June 2002

42 6 advanced methods Example of results 26 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

43 6 advanced methods 27 Microwave circulator in reverse mode (refers to the Pound scheme) 164 S α ϕ ( f ) 2 db[rad ]/Hz P 0 = 19 dbm avg 10 spectra single channel 174 Narda CNA 8596 s.no instrument noise Fourier frequency, Hz no post-processing is used to hide stray signals, like vibrations or the mains

44 6 advanced methods ±45º detection 28 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 nerdish, yet of scarce practical usefulness First used at 2 khz to measure electromigration on metals (H. Stoll, MPI)

45 6 advanced methods 29 The complete machine (100 MHz)

46 6 advanced methods 30 A 9 GHz experiment (dc circuits not shown)

47 6 advanced methods Advanced comparison 31 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

48 6 advanced methods Mechanical stability 32 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

49 7 Optical delay line

50 7 optical delay line Delay line theory Rubiola-Salik-Huang-Yu-Maleki, JOSA-B 22(5) p (2005) 34 laser 1.55 µm EOM P λ τ d = µ s ( km) _ mw 100 τ d (calib.) db mw 10 phase detector microwave input τ _ 0 90 adjust power ampli Note that here one arm is a microwave cable R 0 (t) v o out 52 db FFT analyz. Laplace transforms Φ(s) = H ϕ (s)φ i (s) H ϕ (f) 2 = 4 sin 2 (πfτ) S y (f) = H y (f) 2 S ϕ i (s) Φ i (s) e sτ Laplace transforms Σ + Φ o (s) mixer k ϕ V o (s) = k ϕ Φ o (s) H y (f) 2 = 4ν2 0 f 2 sin2 (πfτ) Φ o (s) = (1 e sτ) Φ i (s) delay > frequency-to-phase conversion works at any frequency long delay (microseconds) is necessary for high sensitivity the delay line must be an optical fiber fiber: attenuation 0.2 db/km, thermal coeff /K cable: attenuation 0.8 db/m, thermal coeff. ~ 10-3 /K 10 GHz, 10 μs

51 7 optical delay line White noise 35 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 [ shot 2 hν λ η 1 P + F kt 0 R 0 thermal ( ) 2 ( hνλ 1 qη P ) 2 ] total white noise (P/2 each detector) S ϕ0 = 16 m 2 [ hν λ η 1 P + F kt 0 R 0 ( hνλ qη ) 2 ( 1 P ) 2 ]

52 7 optical delay line Threshold power 36 S ϕ0 = 16 m 2 [ hν λ η 1 P + F kt 0 R 0 ( hνλ qη ) 2 ( 1 P ) 2 ] holds for two detectors thermal threshold power P λ,t = F kt 0 R 0 hν λ q 2 η shot new high-power photodetectors 5 10 mw

53 7 optical delay line Photodetector 1/f noise (1) X 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)

54 7 optical delay line Photodetector 1/f noise (2) X 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

55 7 optical delay line Flicker (1/f) noise 37 experimentally determined (takes skill, time and patience) amplifier GaAs: b to 110 dbrad 2 /Hz, SiGe: b dbrad 2 /Hz photodetector b dbrad 2 /Hz Rubiola & al. IEEE Trans. MTT (& JLT) 54(2) p (2006) mixer b dbrad 2 /Hz contamination from AM noise (delay => de-correlation => no sweet point (Rubiola-Boudot, IEEE Transact UFFC 54(5) p (2007) optical fiber The phase flicker coefficient b 1 is about independent of power in a cascade, (b 1)tot adds up, regardless of the device order The Friis formula applies to white phase noise S! (f), log-log scale b 1! const. vs. P 0 b 1 f 1 b 0 = FkT 0 / P 0 b 0, higher P 0 b 0, lower P 0 f' c f" c f f c = ( b 1 / FkT 0 ) P 0 depends on P 0 b 0 = F 1kT 0 P 0 + (F 2 1)kT 0 P 0 g 2 1 (b 1 ) tot = m (b 1 ) i i= In a cascade, the 1/f noise just adds up

56 7 optical delay line Single-channel instrument X JDS Uniphase =1,5 µm JDS Uniphase Photodiode DSC40S Ampli AML 8-12GHz laser EOM SiGe ampli Contrôleur de polarisation sapphire oscillator Att 3dB 2 km Fibre 2 Km RF LO 5 dbm DC Ampli DC 10 dbm Analyseur F (HP 3561A FFT Coupleur 10 db phase Déphaseur Ampli RF ISO ISO The laser RIN can limit the instrument sensitivity In some cases, the AM noise of the oscillator under test turns into a serious problem (got in trouble with an Anritsu synthesizer)

57 7 optical delay line Measurement of a sapphire oscillator 38 Mesure de bruit de phase oscillateur Saphir (Ampli Miteq) avec différents retards optiques "!#"!%" ;.<-*.19=.01!""> ;.<-*.19=.01&""> ;.<-*.19=.01!?> ;.<-*.19=.01#?> ;.<-*.19=.01%?>!'" 6!12781*97 # :345!("!!""!!#"!!%"!!'"!"!!" #!" $!" %!" & )*+,-./ The instrument noise scales as 1/τ, yet the blue and black plots overlap magenta, red, green => instrument noise blue, black => noise of the sapphire oscillator under test We can measure the 1/f 3 phase noise (frequency flicker) of a 10 GHz sapphire oscillator (the lowest-noise microwave oscillator) Low AM noise of the oscillator under test is necessary

58 7 optical delay line Dual-channel (correlation) instrument 39 Salik, Yu, Maleki, Rubiola, Proc. Ultrasonics-FCS Joint Conf., Montreal, Aug 2004 p uses cross spectrum to reduce the background noise requires two fully independent channels separate lasers for RIN rejection optical-input version is not useful because of the insufficient rejection of AM noise implemented at the FEMTO-ST Institute

59 7 optical delay line 40 Dual-channel (correlation) measurement J.Cussey 20/02/07 Mesure200avg.txt residual phase noise (cross-spectrum), short delay ("!0), m=200 averaged spectra, unapplying the delay eq. with "=10 "s (2 km) S!(f), dbrad 2 /Hz J.Cussey, feb 2007 FFT average effect FFT average effect FFT average effect Fourier frequency, Hz the residual noise is clearly limited by the number of averaged spectra, m=200

60 7 optical delay line 41 Measurement of the optical-fiber noise matching the delays, the oscillator phase noise cancels this scheme gives the total noise 2 (ampli + fiber + photodiode + ampli) + mixer thus it enables only to assess an upper bound of the fiber noise

61 7 optical delay line Phase noise of the optical fiber 42 The method enables only to assess an upper bound of the fiber noise b rad 2 /Hz for L = 2 km ( 113 dbrad 2 /Hz) We believe that this residual noise is the signature of the two GaAs power amplifier that drives the MZ modulator

62 7 optical delay line Delay-line oscillator 43 f L = ν 0 2Q Q eq = πν 0 τ f L = 1 4π 2 τ 2 Qeq= L=4km fl=8khz h 1 = b 3 /ν Leeson formula S ϕ (f) f 2 L f 2 S ψ(f) b 3 = ( 32 db) for f f L σ 2 y = 2 ln(2) h 1 σ y Allan deviation E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge 2008, ISBN

63 7 optical delay line Delay-line oscillator 44 0 Phase noise of the opto-electronic oscillator (4 km) S!(f), dbrad 2 /Hz our OEO b 3 =10 3 ( 30dB) Agilent E8257c, 10 GHz, low-noise opt. Wenzel OCXO 100MHz mult. to 10 GHz E. Rubiola, jun 2007 OEO: Kirill Volyanskiy, may frequency, Hz nm DFB CATV laser ( V/W Photodetector DSC 402 (R = 371 RF filter ν0 = 10 GHz, Q = 125 ( 22dB + RF amplifier AML812PNB1901 (gain expected phase noise b ( 32 db)

64 7 optical delay line In progress X input F shared fiber out F out B input B a single fiber is used for both directions environmental effects are the same some random effects are independent

65 8 Optical resonators

66 8 optical resonators Example of quartz small resonator 46

67 8 optical resonators Small resonators 47 Technology: dedicated leathe an air-spindle motor for lowest vibration (from a hard-disk test equipment) btw, can you figure out what a hard disk is? 3.5 & 7200 rpm => ~ 200 km/h 1 (μm) 2 bit area, 50 nm head disk distance air air air air air air Surface metrology: ready A few resonator already made quartz, 7 Mohs (technology training, not for serious oscillators) CaF2 4 Mohs, too soft for serious precision machining MgF2 (~6 Mohs) harder than CaF2, more suitable to machining Achieved Q=3x10 8 with MgF2 resonator (still low, but it goes with tapered-fiber coupling) Achieved stable coupling with tapered fiber

68 8 optical resonators Dedicated leathe 48

69 8 optical resonators Disk resonator surface characterization 49 disk surface surface rugosity (nm) Rt Ra Rq Rsk Rku Rp Rv Rz Rtm ( m)

70 8 optical resonators 50 coupling: prism-shaped optical fiber electron-beam microscope photo precision saw surface rugosity (nm) Rt Ra Rq Rsk Rku Rp Rv Rz Rtm ( m) Let technologists have fun with their weird equipment I don t think that the fiber machining is that critical (also experience)

71 8 optical resonators Raman oscillations 51 The Raman amplification is a quantum phenomenon of nonlinear origin that involves optical phonons. An amplifier inserted in a high-q cavity turns into an oscillator, like masers and lasers. Oscillation threshold ~ 1/Q 2 In CaF2 pumped at 1.56 μm, Raman oscillation occurs at 1.64 μm Due to the large linewidth, the Raman oscillation appears as a bunch of (noisy) spectral lines spaced by the FSR (12 GHz, or 100 pm in our case) Raman phonons modulate the optical properties of the crystal, which induces noise at the pump frequency (1.56 μm)

72 8 mm 8 optical resonators High temperature gradient mm CaF2 optical resonator cross section of the field region 1 μm 2 CaF2 thermal conductivity 9.5 W/mK dissipated power 300 μw wavelength 1.56 μm air temperature 300 K still air thermal conductivity 10 W/m 2 K simplification: the heat flow from the mode region is uniform bottom plane at a reference temperature inner bore at a reference temperature

73 8 mm 8 optical resonators Thermal effect on frequency mm CaF2 optical resonator wavelength 1.56 μm (ν0=192 THz) Q=5x10 9 > BW=40 khz a dissipated power of 300 μw shifts the resonant frequency by 1.2 MHz (6x10 9 ), i.e., 37.5 x BW time scale about 60 μs Q>10 11 is possible with CaF2 and other crystals!! laser scan calibration (2 MHz phase modulation)

74 8 optical resonators Low-power oscillator operation 54 Assume: λ = 1560 nm ρ = 0.8 A/W R = 50 Ohm (Pλ)peak = 2x10 5 W (20 μw) Shot noise (m=1) I RMS = 1 2 ρp λ S I = 2qI = 2q ρp λ SNR = 1 4 ρp λ q In practice, 131 dbrad 2 /Hz Thermal noise (m=1) I RMS = 1 2 ρp λ S I = 4kT R SNR = 1 8 4F kt or R ρ 2 P 2 λr kt In practice, 110 dbrad 2 /Hz with F=0 db (!!!) Thermal noise is dominant: below threshold, SNR ~ 1/Pλ 2 Thermal noise can be reduced (10 db or more?) using VGND amplifiers What about flicker of photodetectors with integrated VGND amplifier? Dramatic impact on the (phase) noise floor

75 8 optical resonators Small resonators X Let us dream diamond: probably chemical purity may be a problem (insufficient transparence) sapphire: think more about it (we can learn a lot from the microwave technology) Last-minute news MgF2 seems to have a turning point of the thermorefractive index 74 C, extraordinary wave 176 C, ordinary wave

76 9 Non-linear AM oscillations

77 9 non-linear AM Nonlinear model X

78 9 non-linear AM A complex envelope equation X

79 9 non-linear AM Stability of the oscillating solution X

80 9 non-linear AM A Hopf bifurcation X

81 9 non-linear AM Hopf bifurcation, observed X The Hopf bifurcation leads to the emergence of robust modulation side-peaks in the Fourier spectrum, which may drastically affect the phase noise performance of OEOs