1. Lock-in amplifiers http://www.lockin.de/

2. Signals and noise Frequency 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(V noise ) log(  f ) Noise amplitude 1/f noise 0 White noise 0.1 1 10 100 1kHz log(V noise ) log(  f ) Total noise in 10 Hz bandwidth 1/f noise 0 White noise 0.1 1 10 100 1kHz Signal at DC log(V noise ) log(  f ) 1/f noise 0 White noise 0.1 1 10 100 1kHz Signal at 1 kHz 10 Hz

3. 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

4. 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

5. 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 freq vs ref freq Reference Mixer outputs

6. Phase sensitive detection Signal freq matches reference freq Reference = sin(2  ft) Signal = sin(2  ft +  ) –  is signal phase shift Product = cos(  ) - cos(2  ft) Signal phase shift  0 0.2  0.4  0.6  0.8    Reference wave Product waveforms -- signal times reference DC part

7. Low pass filter Removes 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 InputOutput MixerLow pass filter Buffer Lock-in amplifier Demodulated signal After mixer Voltage time After mixer & low pass

8. 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 log gain frequency

9. 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 Phase shift  InputOutput Mixer

10. 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 amplifierSystem Reference Signal Mixer Lock-in amplifierSystem VCO Integrate PLL

11. 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 1/3 1/5 1/71/9

12. 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 sourcedrain gate bias n-channel FET

13. 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

14. 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 Digital mixers –Good for slow signals –High signal to noise or low accuracy

15. 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

16. 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-in output (derivative) Zero crossing at resonance

17. 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

18. Digital mixers in servos May be okay for low precision, medium speed servo –Not for fast servos -- ex: laser frequency stabilization –Not for high accuracy -- ex: laser gyro Should be excellent for slow servos –Ex: tele-medicine, temperature controllers –Digital processing can compensate for system time delay