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Signals and noiseFrequency dependence of noise• Low frequency ~ 1 / f
– example: temperature (0.1 Hz) , pressure (1 Hz), acoustics (10 -- 100 Hz)
• High frequency ~ constant = white noise– example: shot noise, Johnson noise, spontaneous emission
noise• Total noise depends strongly on signal freq
– worst at DC, best in white noise region• Problem -- most signals at DC
log(Vnoise)
log(f )
Noise amplitude
1/f noise
0
White noise
0.1 1 10 100 1kHz
log(
Vno
ise)
log(f )
Total noise in 10 Hz bandwidth
1/f noise
0
White noise
0.1 1 10 100 1kHz
Signal at DC
log(
Vno
ise)
log(f )
1/f noise
0
White noise
0.1 1 10 100 1kHz
Signal at 1 kHz
10 Hz
10 Hz
Lock-in amplifiers• Shift signal out to higher frequencies• Approach:• Modulate signal, but not noise, at high freq
– no universal technique -- art– example: optical chopper wheel, freq modulation
• Detect only at modulation frequency– Noise at all other frequencies averages to zero– Use demodulator and low-pass filter
Demodulation / Mixing• Multiply input signal by sine wave• Sum and difference freq generated• Compare to signal addition -- interference• Signal frequency close to reference freq
– low freq beat– DC for equal freq sine waves– DC output level depends on relative phase
Two sine waves
Product
Sum
Signal freq approaches ref freq• Beat frequency approaches DC as signal freq approaches ref freq
1
1.05
1.1
1.15
1.2
1.25
Signal freqvs ref freq
Reference
Mix
er o
utpu
ts
Phase sensitive detection• Signal freq matches reference freq• Reference = sin(2ft) • Signal = sin(2ft + )
– is signal phase shift• Product = cos() - cos(2ft)
Signalphaseshift
0
0.2
0.4
0.6
0.8
Reference wave
Prod
uct w
avef
orm
s--
sig
nal t
imes
ref
eren
ceDC part
Low pass filterRemoves noise• Example -- modulate above 1/f noise
– noise slow compared to reference freq– noise converted to slowly modulated sine wave– averages out to zero over 1 cycle
• Low pass filter integrates out modulated noise – leaves signal alone
Reference
Input Output Mixer Low pass
filter Buffer
Lock-in amplifier Demodulated signal
After mixer
Vol
tage
time
After mixer & low pass
Typical LIA low pass filters• For weak signal buried in noise
• Ideal low pass filter blocks all except signal
• Approximate ideal filter with cascaded low pass filters
18 db/oct
12 db/oct
6 db/oct
Ideal
loggain
frequency
Phase control• Reference has phase control
• Can vary from 0 to 360°
• Arbitrary input signal phase
• Tune reference phase to give maximum DC output
Reference
Phaseshift
Input Output Mixer
Reference options• Option 1 -- Internal reference
– best performance
– stable reference freq
• Option 2 -- External reference
• System generates reference– ex: chopper wheel
• Lock internal ref to system ref– use phase locked loop (PLL)
– source of name “lock-in amplifier”
Reference
Signal Mixer
Lock-in amplifier System
Reference
Signal Mixer
Lock-in amplifier System
VCO
Integrate
PLL
Analog mixer• Direct multiplication
– accurate
– not enough dynamic range
– weak signal buried in noise
• Switching mixer– big dynamic range
– but also demodulates harmonics
Multiplying mixer
Switching mixer
Harmonic content of square wave
1
1/31/5 1/7 1/9
Switching mixer design• Sample switching mixer• Back-to-back FETs
– example: 1 n-channel & 1 p-channel– feed signal to one FET, inverted signal to second FET
• Apply square wave to gates– upper FET conducts on positive part of square wave– lower FET conducts on negative part
Switching mixer circuit
p
n
Signal voltage
source draingate
bias
n-channel FET
Signals with harmonic content• Option 1: Use multi-switch mixer
– approximate sine wave
– cancel out first few harmonic signals
• Option 2: Filter harmonic content from signal– bandpass filter at input
– Q > 100
Lock-in amp with input filter
Digital mixers
• Digitize input with DAC• Multiply in processor• Advantages:
– Accurate sine wave multiplication– No DC drift in low pass filters– Digital signal enhancement
• Problems:– Need 32 bit DAC for signals buried in noise– Cannot digitize 32 bits at 100 kHz rates
• Should be excellent for slow servos– Ex: tele-medicine, temperature controllers– Digital processing can compensate for certain system time delays ?
Lock-in amps in servos• Lock to resonance peak
– Servos only lock to zero– Need to turn peak into zero
• Take derivative of lineshape– modulate x-voltage– F(x)-voltage amplitude like derivative
• Use lock-in amp to extract amplitude of F(x)– “DC” part of mixer output– filter with integrator, not low-pass
x
F(x)
Take derivative with lock-in
No fundamental• only 2 f signal
Lock-in amps for derivative• Lock-in turns sine wave signal into DC voltage
• At peak of resonance– no signal at modulation freq
– lock-in output crosses zero
• Discriminant– use to lock
x
F(x)
Input signal
Lock-inoutput
(derivative)
Zero crossingat resonance
Effect of modulation on lineshape• Start with resonance lineshape• Intensity vs PZT voltage: I = I0 exp( -V2)
• Modulate voltage: V= V0 sin (2 f t)
• Modified lineshape
• Analog to numerical derivatives• Derivative is: I’ = I(V+ V) - I(V) / V
– Set V = 1
• Modulation replaces V= V0 sin (2 f t)• Derivative is sine wave part
– Assumes is V0 small
V
I
t
t
V
I
Modulation amplitude
0.05 linewidth
0.1
0.2
0.5 linewidth
1
2
Effect of modulation amplitude• For large modulation amps
– Distortion and broadening• Modulation like a noise source
– Always use minimum necessary
Expanded scan
Modulation amplitude
0.1 linewidth
0.2
0.5 linewidth
1
2
Mixer outputs• Maximum mixer output
– modulation ~ 1 linewidth– saturates and broadens
Mixer out0.1 linewidth
0.2
0.5
1
2
Fabry-Perot servo• Lock to peak transmission of high Q Fabry-Perot etalon• Use lock-in amp to give discriminant
– No input bandpass -- or low Q < 2• Bandpass rolloff usually 2-pole or greater
– No low pass filter -- replace with integrator• Low pass filter removes noise• Need noise to produce correction
• Design tips– reference freq must exceed servo bandwidth by factor of ~ 10– but PZT bandwidth is servo limiter– use PZT resonance for modulation
Acoustic noise
Laser
Fabry-Perot
PD LIA
Sum& HV
reference
