Photonic microwave oscillators E. Rubiola, K. Volyanskiy, H. Tavernier, Y. Kouomou Chembo, R. Bendoula, P. Salzenstein, J. Cussey, X. Jouvenceau, L. Larger FEMTO-ST Institute, Besançon, France CNRS and Université de Franche Comté Outline Phase noise and frequency stability Delay-line instrument Correlation instrument Delay line oscillator Nonlinear AM oscillations Optical resonators home page http:/ /rubiola.org
Phase and amplitude noise noise 2 Time Domain Phasor Representation amplitude fluctuation v ( t ) V 0 α ( t ) [volts] V 0 normalized ampl. fluct. α ( t ) [adimensional] √ V 0 / 2 t ampl. fluct. √ ( V 0 / 2) α ( t ) phase fluctuation v ( t ) ϕ ( t ) [rad] V 0 phase fluctuation phase time (fluct.) x ( t ) [seconds] ϕ ( t ) t √ V 0 / 2 v ( t ) = V 0 [1 + α ( t )] cos [ ω 0 t + ϕ ( t )] polar coordinates v ( t ) = V 0 cos ω 0 t + n c ( t ) cos ω 0 t − n s ( t ) sin ω 0 t Cartesian coordinates under low noise approximation It holds that α ( t ) = n c ( t ) ϕ ( t ) = n s ( t ) and | n c ( t ) | ≪ V 0 and | n s ( t ) | ≪ V 0 V 0 V 0
Phase noise & friends 3 processes not present random phase fluctuation random walk freq. in two-port devices S ϕ (f) f −4 b −4 S ϕ ( f ) = PSD of ϕ ( t ) power spectral density flicker freq. b −3 f −3 it is measured as S ϕ ( f ) = E { Φ ( f ) Φ ∗ ( f ) } white freq. b −2 f −2 S ϕ ( f ) ≈ � Φ ( f ) Φ ∗ ( f ) � m flicker phase. b −1 f −1 white phase b 0 L ( f ) = 1 2 S ϕ ( f ) dBc f f 2 / 2 ν 0 x S y (f) random fractional-frequency fluctuation h 2 f 2 h −2 f −2 S y = f 2 y ( t ) = ˙ ϕ ( t ) white phase random S ϕ ( f ) walk freq. ⇒ h −1 f −1 h 1 f ν 2 2 πν 0 h 0 0 flicker phase flicker freq. white freq. f Allan variance 2 σ ( τ ) y (two-sample wavelet-like variance) � 1 � 2 � freq. � σ 2 y ( τ ) = E flicker phase drift y k +1 − y k . 2 white phase flicker freq. random walk freq. white freq. approaches a half-octave bandpass filter (for white), ) 2 (2 π h 0 /2 τ 2ln(2)h −1 h −2 τ τ hence it converges for processes steeper than 1/f 6 E. Rubiola, Phase Noise and Frequency Stability in Oscillators , Cambridge 2008
Amplifier white noise 4 Noise figure F, Input power P 0 P=FkT 0 B RF spectrum S( ν ) P 0 B B V 0 cos ω 0 t ∑ g N e =FkT 0 ν 0 −f ν 0 ν 0 +f ν n rf ( t ) LSB USB S φ (f) low P 0 power law 0 b 0 = FkT 0 white � b i f i S ϕ = high P 0 phase noise P 0 P 0 i = − 4 f Cascaded amplifiers (Friis formula) The (phase) noise is chiefly that of the 1st stage F 1 F 2 F 3 g 1 g 2 g 3 The Friis formula applied to phase noise + ( F 2 − 1) kT 0 b 0 = F 1 kT 0 N = F 1 kT 0 + ( F 2 − 1) kT 0 + . . . + . . . P 0 g 2 P 0 g 2 1 1 H. T. Friis, Proc. IRE 32 p.419-422, jul 1944
Amplifier flicker noise 5 no carrier noise S(f) S(f) up-conversion near-dc flicker near-dc no flicker f f noise ω 0 = ? ω 0 stopband output bandwidth stopband output bandwidth t t a carrier near-dc noise v i ( t ) = V i e j ω 0 t + n ′ ( t ) + jn ′′ ( t ) the parametric nature of 1/f noise is hidden in n’ and n” substitute (careful, this hides the down-conversion) v o ( t ) = a 1 v i ( t ) + a 2 v 2 i ( t ) + . . . non-linear (parametric)amplifier expand and select the ω 0 terms � �� The noise sidebands are e j ω 0 t � n ′ ( t ) + jn ′′ ( t ) v o ( t ) = V i a 1 + 2 a 2 proportional to the input carrier get AM and PM noise α ( t ) = 2 a 2 ϕ ( t ) = 2 a 2 n ′ ( t ) n ′′ ( t ) The AM and the PM noise are a 1 a 1 independent of V i , thus of power
Delay line theory 6 Rubiola-Salik-Huang-Yu-Maleki, JOSA-B 22(5) p.987–997 (2005) τ d = 1.. 100 µ s Laplace transforms P λ phase (0.2−20 km) detector laser EOM Φ ( s ) = H ϕ ( s ) Φ i ( s ) v o (t) _ ∼ µ m τ d 0 1.55 R 0 20−40 out (calib.) dB analyz. 100 | H ϕ ( f ) | 2 = 4 sin 2 ( π f τ ) FFT 10 mW microwave mW input _0 τ∼ 52 dB 90° adjust power ampli S y ( f ) = | H y ( f ) | 2 S ϕ i ( s ) Note that here one arm is a microwave cable | H y ( f ) | 2 = 4 ν 2 f 2 sin 2 ( π f τ ) 0 Laplace transforms τ mixer −s e − Φ i (s) o (s) k ϕ Φ o (s) V = Φ o (s) k ϕ Σ + −s τ Φ o (s) ) Φ i (s) = (1−e • delay –> frequency-to-phase conversion • 10 GHz, 10 μ s 10 GHz, 10 μ s 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. 6.8 10 -6 /K cable: attenuation 0.8 dB/m, thermal coeff. ~ 10 -3 /K
White noise 7 P ( t ) = P (1 + m cos ω µ t ) intensity modulation i ( t ) = q η h ν P (1 + m cos ω µ t ) photocurrent � q η � 2 P µ = 1 2 m 2 R 0 P 2 microwave power h ν N s = 2 q 2 η shot noise h ν PR 0 thermal noise N t = FkT 0 shot thermal � 2 � 1 � � 2 � � h ν λ 2 h ν λ P + FkT 0 2 1 total white noise S ϕ 0 = m 2 (one detector) R 0 q η P η � 2 � 1 � � 2 � � h ν λ h ν λ P + FkT 0 S ϕ 0 = 16 1 total white noise (P/2 each detector) m 2 R 0 q η P η
Flicker (1/f) noise 8 experimentally determined (takes skill, time and patience) amplifier GaAs: b –1 ≈ –100 to –110 dBrad 2 /Hz, SiGe: b –1 ≈ –120 dBrad 2 /Hz photodetector b –1 ≈ –120 dBrad 2 /Hz Rubiola & al. IEEE Trans. MTT (& JLT) 54 (2) p.816–820 (2006) mixer b –1 ≈ –120 dBrad 2 /Hz contamination from AM noise (delay => de-correlation => no sweet point (Rubiola-Boudot, IEEE Transact UFFC 54(5) p.926–932 (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 + ( F 2 − 1) kT 0 b 0 = F 1 kT 0 b –1 � const. vs. P0 + . . . P 0 g 2 P 0 b –1 f –1 b 0 = FkT0 / P0 1 b 0 , higher P0 In a cascade, the 1/f noise just adds up b 0 , lower P0 m � ( b − 1 ) tot = ( b − 1 ) i f f' c f" c f c = ( b –1 / FkT0 ) P0 depends on P0 i =1
Single-channel instrument 9 Ampli JDS Uniphase Photodiode EOM JDS Uniphase SiGe ampli AML laser 1,5 µm DSC40S = 8-12GHz Analyseur FFT Contrôleur 2 km Fibre 2 Km 5 dBm (HP 3561A) de polarisation RF Att FFT 3dB DC sapphire oscillator LO Ampli DC 10 dBm phase Déphaseur Coupleur 10 dB 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)
Measurement of a sapphire oscillator 10 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 ! (" ! !"" ! !#" ! !%" ! !'" ! # $ % & !" !" !" !" !" )*+,-./0.12345 • 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
Basics of correlation spectrum measurements 11 a(t) phase noise measurements + DUT noise, a, b instrument noise x=c−a Δ Σ normal use c DUT noise analyzer c(t) − FFT background, a, b instrument noise − ideal case c = 0 no DUT y=c−b Δ Σ background, a, b instrument noise b(t) + with AM noise c ≠ 0 AM-to-DC noise S yx = E { Y X ∗ } W. K. theorem S yx = � Y X ∗ � m measured, m samples a , b and c are incorrelated expand X = C − A and Y = C − B Averaging on a sufficiently large S yx = S cc a , b , c independent number m of spectra is necessary � S yx = S cc + O ( 1 /m ) measured, m samples to reject the single-channel noise
Dual-channel (correlation) instrument 12 Salik, Yu, Maleki, Rubiola, Proc. Ultrasonics-FCS Joint Conf., Montreal, Aug 2004 p.303-306 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
Dual-channel (correlation) measurement 13 J.Cussey 20/02/07 Mesure200avg.txt –20 #" residual phase noise (cross-spectrum), ;<9/0=.*17.1>*-?@17.1A=9B.19C.011-/1*.@9*717.1!"DB12E?>*.1#FG5 short delay ( � � 0), m=200 averaged spectra, 6A.01H.*IE<.J1K!"F3419C.01-/1*.@9*717.1#"DB12E?>*.1%FG51 –40 %" unapplying the delay eq. with � =10 � s (2 km) –60 (" FFT –80 '" average effect FFT –100 !"" average effect S � (f), dBrad 2 /Hz –120 !#" FFT average –140 !%" effect –160 !(" –180 !'" Fourier frequency, Hz J.Cussey, feb 2007 10 1 10 2 10 3 10 4 10 5 the residual noise is clearly limited by the number of averaged spectra, m=200
Measurement of the optical-fiber noise 14 • 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
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