Phase & Frequency Noise Metrology Enrico Rubiola Outline - PowerPoint PPT Presentation

Phase & Frequency Noise Metrology Enrico Rubiola Outline • Introduction • Measurement methods • Microwave photonics • Electronic and optical components • AM noise and RIN home page http://rubiola.org

Though frequency standards are 2 moving to optics (and beyond), RF and microwaves are inevitable S α | ϕ ( f ) − 164 = 19 dBm P 0 2 dB[rad ]/Hz avg 10 spectra single channel ν 0 = 9.2 GHz − 174 Microwave Narda CNA 8596 circulator s.no. 157 − 184 − 194 instrument noise Fourier frequency, Hz − 204 10 2 10 3 10 4 10 Lower phase noise is required

Phase noise & friends 3 v ( t ) = V p [1 + α ( t )] cos [1 + ϕ ( t )] random walk freq. signal sources only S ϕ (f) f −4 b −4 random phase fluctuation both signal sources flicker freq. S ϕ ( f ) = PSD of ϕ ( t ) b −3 f −3 and two-port devices power spectral density white freq. b −2 f −2 it is measured as flicker phase. b −1 f −1 white phase b 0 S ϕ ( f ) = E { Φ ( f ) Φ ∗ ( f ) } (expectation) f S ϕ ( f ) ≈ � Φ ( f ) Φ ∗ ( f ) � m (average) f 2 / 2 ν 0 L ( f ) = 1 x 2 S ϕ ( f ) dBc S y (f) h 2 f 2 h −2 f −2 white phase random walk freq. random fractional-frequency fluctuation h −1 f −1 h 1 f h 0 S y = f 2 flicker phase y ( t ) = ˙ ϕ ( t ) flicker freq. white freq. S ϕ ( f ) ⇒ f ν 2 2 πν 0 0 2 Allan variance σ ( τ ) y (two-sample wavelet-like variance) freq. � 1 � 2 � � flicker phase drift σ 2 y ( τ ) = E y k +1 − y k . white phase 2 flicker freq. random walk freq. white freq. ) 2 (2 π approaches a half-octave bandpass filter (for white noise), h 0 /2 τ 2ln(2)h −1 h −2 τ τ 6 hence it converges for processes steeper than 1/f

4 Mechanical stability S ϕ ( f ) 10 –18 rad 2 /Hz @ 1 Hz Any phase fluctuation can be h − 1 converted into length fluctuation / f L = 1 c f 2 π ν 0 b –1 = –180 dBrad 2 /Hz and ν 0 = 10 GHz is equivalent to L = 1 c S L = 1.46x10 –23 m 2 /Hz at f = 1 Hz 2 π ν 0 S L ( f ) 1.5x10 –23 m 2 /Hz @ 1 Hz Any flicker spectrum h –1 /f can be h − 1 / f converted into a flat Allan variance f σ 2 L = 2 ln(2) h − 1 σ 2 = 2 ln(2) h − 1 A residual flicker of –180 dBrad 2 /Hz at f = 1 Hz σ L ( τ ) off the 10 GHz carrier is equivalent to σ 2 = 2x10 –23 m 2 thus σ = 4.5x10 –12 m 4.5x10 –12 m for reference, the Bohr radius of the electron is R = 0.529 Å τ • Don’t think “this is just engineering” !!! • Learn from non-optical microscopy (bulk matter, 5x10 –14 m) • Careful DC section (capacitance and piezoelectricity) • The best advice is to be at least paranoiac

1 – Measurement methods

6 Correlation measurements x = c–a Two separate instruments measure the same DUT. Only the DUT noise is common c(t) DUT FFT a(t), b(t) –> instrument noise c(t) –> DUT noise y = c–b b(t) phase noise measurements basics of correlation DUT noise, a, b instrument noise normal use c DUT noise S yx ( f ) = E { Y ( f ) X ∗ ( f ) } background, a, b instrument noise ideal case c = 0 no DUT = E { ( C − A )( C − B ) ∗ } background, a, b instrument noise = E { CC ∗ − AC ∗ − CB ∗ + AB ∗ } with AM noise c ≠ 0 AM-to-DC noise = E { CC ∗ } 0 0 0 single-channel S yx ( f ) = S cc ( f ) S � (f) in practice, average on m realizations S yx ( f ) = � Y ( f ) X ∗ ( f ) � m 1/ � m = � CC ∗ − AC ∗ − CB ∗ + AB ∗ � m correlation 0 as 1/ √ m = � CC ∗ � m + O (1 /m ) frequency E. Rubiola, The magic of cross-spectrum measurements from DC to optics, http://rubiola.org

7 Cross-spectrum, increasing m |Re{Syx}| with C ≠ 0, #! #! #! #! m=1 g=0.32 m=2 g=0.32 m=4 g=0.32 m=8 g=0.32 |Sxx| |Sxx| |Sxx| |Sxx| # # # # |Re{Syx}| |Re{Syx}| |Re{Syx}| |Re{Syx}| !%# !%# !%# !%# |Scc| |Scc| |Scc| |Scc| !%!# !%!# !%!# !%!# frequency frequency frequency frequency !%!!# !%!!# !%!!# !%!!# ! "! #!! #"! $!! ! "! #!! #"! $!! ! "! #!! #"! $!! ! "! #!! #"! $!! #! #! #! #! m=16 g=0.32 m=32 g=0.32 m=64 g=0.32 m=128 g=0.32 |Sxx| |Sxx| |Sxx| |Sxx| # # # # |Re{Syx}| |Re{Syx}| |Re{Syx}| |Re{Syx}| !%# !%# !%# !%# |Scc| |Scc| |Scc| |Scc| !%!# !%!# !%!# !%!# frequency frequency frequency frequency !%!!# !%!!# !%!!# !%!!# ! "! #!! #"! $!! ! "! #!! #"! $!! ! "! #!! #"! $!! ! "! #!! #"! $!! #! #! #! # |Re{Syx}| m=256 g=0.32 m=512 g=0.32 m=1024 g=0.32 a v e r a g e |Sxx| |Sxx| |Sxx| # # # !%# |Re{Syx}| |Re{Syx}| |Re{Syx}| !%# !%# !%# d |Scc| |Scc| |Scc| e v i a t i o n !%!# !%!# !%!# &'()*+,)-./0 ! +)1 ! ## ! #!$2 ! !3#4 ! 05+6)789 m :%6;5'<(0=*0,/*$!!> !%!# # #! #!! #!!! frequency frequency frequency !%!!# !%!!# !%!!# ! "! #!! #"! $!! ! "! #!! #"! $!! ! "! #!! #"! $!! Increasing m: first, S yx decreases => single-channel noise rejection then, S xx shrinks => increased confidence level E. Rubiola, The magic of cross-spectrum measurements from DC to optics, http://rubiola.org

8 The thermal noise is rejected as any signal. The limit S φ = P 0 /kT does not apply A X = A + B 0º T 2 0º S yx = k (T 2 – T 1 ) / 2 180º B X = A – B T 1 0º X and Y are uncorrelated The cross spectrum is proportional to the temperature difference C. M. Allred, A precision noise spectral density comparator, J. Res. NBS 66C no.4 p.323-330, Oct-Dec 1962 Application to AM/PM noise: E. Rubiola, V. Giordano, Rev. Sci. Instrum. 71(8) p.3085-3091, Aug 2000

9 Carrier recirculation gas cell Invented by J. Hall for gas spectroscopy. The gain is increased by the number of times the light beam circulates in the cavity DUT Also works with RF/microwave carrier, provided the DUT be “transparent”. For small no. of roundtrips, gives the appearance of “real-time”

10 Bridge (interferometric) method fluctuating error δ Z => noise sidebands V 0 cos( � 0 t) ℜ { δ Z} => AM noise x(t) cos( ω 0 t) pump ℑ { δ Z} => PM noise –y(t) sin( ω 0 t) 0º –90º (microwave) x(t) bridge error amplifier – � (t) FFT � y(t) + hybrid junction Z Z DUT x(t) cos( � 0 t) – y(t) sin( � 0 t) Basic ideas • Carrier suppression => the error amplifier cannot flicker: it does know ω 0 • High gain, due to the (microwave) error amplifier • Low noise floor => the noise figure of the (microwave) error amplifier • High immunity to the low-frequency magnetic fields due to the microwave amplification before detecting • Rejection of the master oscillator’s noise • Detection is a scalar product => signal-processing techniques Derives from H. Sann, MTT 16(9) 1968, and F . Labaar, Microwaves 21(3) 1982 Later, E. Ivanov, MTT 46(10) oct 1998, and Rubiola, RSI 70(1) jan 1999

11 Actual block diagram E. Rubiola, V. Giordano, Rev. Sci. Instrum. 73(6) pp.2445-2457, June 2002 channel b (optional) v 1 w 1 readout I rf virtual gnd RF I − Q G B null Re & Im v 2 w 2 Q detect matrix matrix g ~ 40dB LO inner interferometer pump FFT x t ( ) CP1 CP2 R CP4 0 analyz. channel a DUT v 1 w 1 readout I CP3 Δ ’ RF R 0 =50 Ω I − Q G B γ v 2 w 2 Q detect matrix matrix g ~ 40dB LO 10 − 20dB R 0 coupl. G: Gram Schmidt ortho arbitrary phase pump atten normalization B: frame rotation power splitter manual carr. suppr. automatic carrier ’ I − Q detector/modulator atten γ ’ suppression control I diagonaliz. var. att. & phase atten RF RF u 1 z 1 Q I arbitrary phase pump LO I − Q dual D atten u 2 z 2 Q modul integr matrix LO − 90° 0° Concepts Im • Coarse and fine carrier suppression reduces the flicker noise Up • Scalar product gives v 1 (t) and v 2 (t) in Cartesian frame. Linear algebra v(t)/ � 2 v(t) fixes the arbitrary phase, gain asymmetry and quadrature defect Re null fluct • Closed-loop control of the carrier suppression works as a RF VGND v(t)/ � 2 Dn • Correlation is possible, using two amplifiers and two detectors • Correlating the signals detected on two orthogonal axes (±45º) S ud ( f ) = 1 � � S α ( f ) − S ϕ ( f ) eliminates the amplifier noise. Works with a single amplifier! 2