Lock-in amplifiers

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

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

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. Product Two sine waves Sum 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

5. Signal freq approaches ref freq • Beat frequency approaches DC as signal freq approaches ref freq Reference Signal freq vs ref freq 1 1.05 1.1 1.15 1.2 1.25 Mixer outputs

6. Phase sensitive detection • Signal freq matches reference freq • Reference = sin(2pft) • Signal = sin(2pft+ f) • f is signal phase shift • Product = cos(f) - cos(2pft) DC part Signal phase shift f 0 0.2 p 0.4 p 0.6 p 0.8 p p Reference wave Product waveforms -- signal times reference

7. Demodulated signal Lock-in amplifier After mixer Mixer Low pass filter Input Output Buffer After mixer & low pass Voltage time Reference 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

8. Ideal 6 db/oct 12 db/oct log gain 18 db/oct frequency 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

9. Mixer Input Output Reference Phase shift f Phase control • Reference has phase control • Can vary from 0 to 360° • Arbitrary input signal phase • Tune reference phase to give maximum DC output

10. Reference options System Lock-in amplifier Mixer Signal • 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 System Lock-in amplifier Mixer Signal Reference VCO PLL Integrate

11. Analog mixer Multiplying mixer • Direct multiplication • accurate • not enough dynamic range • weak signal buried in noise • Switching mixer • big dynamic range • but also demodulates harmonics Switching mixer Harmonic content of square wave 1 1/3 1/5 1/7 1/9

12. bias source drain gate n p Signal voltage 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 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. F(x) x Lock-in amps in servos Take derivative with lock-in • 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 • 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 Input signal F(x) x Lock-in output (derivative) Zero crossing at resonance

17. Fabry-Perot Laser PD LIA Acoustic noise reference Sum & HV 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

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