Lockin amplifiers. A short tutorial by R. Scholten


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1 Lockin amplifiers A short tutorial by R. cholten
2 Measuring something Common task: measure light intensity, e.g. absorption spectrum Need very low intensity to reduce broadening Noise becomes a problem Rb spectrum 8 Rb cell Photodiode 6 Laser 4 0 Frequency
3 The principle Fundamental law of communication theory:wienerkhinchin theorem Reduction of noise imposed upon a useful signal with frequency f 0, is proportional to the square root of the bandwidth of a bandpass filter, centre frequency f 0
4 Noise Typical photodetector noise spectrum 90 Photodector intensity Power spectral density (db) Bad noise 00 here Off resonance Dark noise 90Hz Δf here... On resonance Better here! 50 and here! khz 0kHz 00kHz Frequency (Hz) dc measurements: broadspectrum (bad) at low frequency (bad)
5 Measuring something Need to measure at high frequency, where noise is low Modulate signal, look for component oscillating at modulation frequency Laser Rb cell Photodiode 8 Rb spectrum 6 Chopper Lockin amplifier 4 0 Frequency
6 ignal from noise A lockin amplifier is used to extract signal from noise It detects signal based on modulation at some known frequency Premise: much noise at low frequency (e.g. dc), less noise at high frequency measure within narrow spectral range, reduce noise bandwidth Hence shift measurement to high frequency Figures from Bentham Instruments document 5.0 Lockin amplifiers
7 Demodulator or PD (phasesensitive detector) Demodulator Input Mixer Lowpass filter Output Buffer Reference
8 Mathematical description ignal (t) varies relatively slowly e.g. absorption spectrum scan over 0 seconds Modulate at relatively high frequency ω (e.g. chopper): Reference (local oscillator) of fixed amplitude: phase φ is variable oscillator frequency ω same as modulation frequency Multiply modulated signal by REF : sig ref = = = sig ( t) cos t = cos( ω t +φ) ω ref ( t) cosωt cos( ωt + φ) () t cosφ + () t cos( ωt + φ) econd term at high frequency (ω) Lowpass filter (cutoff ~ ω/ or lower) sig ref filter = ( t) cosφ Note phasesensitive detection!
9 ome details imple trig ref sig = = = = = = = () t cosωt cos( ωt + φ) + n( t)cos( ωt + φ) () t cosωt[ cosωt cosφ sinωt sinφ] +... ()[ t cos ωt cosφ cosωt sinωt sinφ] +... ()( t [ + cos ωt ) cosφ sin ωt sinφ] +... ()[ t cosφ + cos ωt cosφ sin ωt sinφ] +... () t cosφ + ()[ t cos ωt cosφ sin ωt sinφ] () t cosφ + () t cos( ωt + φ) + n( t)cos( ωt + φ) +...
10 Noise Noise reduces with frequency (/f noise is major problem) Laser frequency noise hift signal to higher frequency Noise within given bandwidth reduces as we measure at higher frequency db/sqrt(hz) Frequency (Hz)
11 With noise ignal has noise: sig ( t) cos t n( t) = ω + Multiply reference by modulated signal: ref sig = = ( t) cosωt cos( ωt + φ) + n( t)cos( ωt + φ) () t cosφ + () t cos( ωt + φ) + n( t)cos( ωt + φ) Third term noise at frequency ω Lowpass filter, frequency less than ω/, leaves signal components We win twice: less noise at ω reduce bandwidth
12 Using PD oscillator to modulate Phasesensitive detector Experiment ignal Mixer Buffer Lowpass filter Output Mod Reference
13 External modulator: true lockin Lockin amplifier Experiment ignal Mixer vco Buffer Lowpass filter Output Ref PLL = phaselocked loop Integrator
14 Further details True lockin amp can work with external oscillator for Reference: Input reference from external experiment Use phaselockedloop to generate stable local oscillator Lockin amp has variable postmultiplier (lowpass) filter Time constants: what time constant is appropriate? hapes (6 th, th, order): which is best? If input signal has harmonics (e.g. due to imperfect modulation) then will detect spurious signal Use input filter to minimise Dynamic reserve?
15 Other applications Often use lockin to measure response function of actuator (or similar) Twochannel lockin measure signal and phase Phase resonances Phasesensitive detector Inphase and quadrature mixers Actuator (e.g. piezo) ignal Mod Ref
16 Experiments Photodiode + LED R FFT spectrum analyser Oscilloscope witch LED on/off, e.g. with hand to block HP function generator to modulate LED And/or chopper R lockin amp
17 The other half of the story Frequency modulation
18 Derivatives: Lockin amps & feedback servos o far, we have modulated amplitude, and used LIA to demodulate PD (lockin) = fancy bandpass filter?! Can also use frequency modulation (like FM radio) Let s measure (ω) i.e. a spectrum, where we slowly vary ω(t) ( ) t Frequencymodulate: ω t = ω 0 + Ω0 cosω where Ω is the modulation (Fourier) frequency Using Taylorseries expansion: ( ω) = ( ω ) + ( Ω cosωt) d dω ω= ω 0 0 Note two things immediately: dc component is same as unmodulated spectrum ac component is proportional to derivative of spectrum
19 Extract derivative with PD/lockin amp We now multiply our signal by our reference, as before: sig = ( ω ) + ( Ω cosωt) +... = cos( Ωt +φ) 0 d dω ω= ω 0 0 Note modulation at Ω, and fixed ω 0 i.e. slowly varying laser frequency ref sig ref = =... = cos cos d dω ( Ωt + φ) + Ω cosωt cos( Ωt + φ) d dω d dω ( Ωt + φ) + Ω cos( Ωt + φ) + Ω cosφ Again: lowpass filter (cutoff ~ ω/ or lower) sig ref Ω 0 d dω cosφ We have a measurement proportional to the derivative Measurement changes sign if slope changes sign: dispersion Note: modulation depth Ω 0 must not be larger than peak in spectrum! Higherorder terms in Taylor expansion: can measure nd deriv, 3 rd deriv, etc.
20 Lockin amplifiers and feedback servos Example: Lorentzian peak in atomic absorption spectrum On centre: small double frequency maller slope = smaller signal ω PD out Note opposite phase: cosφ <0 ω ω
21 Lockin amps in servos Modulated output from detector t t t t Demodulated output from lockin t ω PD out sig ref Ω 0 d φ dω cos ω
22 Yet other half of the story preadspectrum
23 PRB: Lockin on steroids! Lockin uses small part of spectrum Can use broad spectrum and still separate signal from noise Pseudorandom bit sequence preadspectrum communications computer 80. wireless, etc. CDMA telephones Modems ecurity/encryption Acoustics
24 preadspectrum modulation PREADPECTRUM MODULATION all frequencies present simultaneously in modulation function phases adjusted so that components add in quadrature truly random phases cause excursions out of range use pseudorandom functions 0 frequencies random phases
25 preadspectrum history Also famous as first nude in cinemarelease movie! Hedy Lamarr (93000), composer George Antheil ( ) patented submarine communication device ynchronized frequency hopping to evade jamming Original mechanical action based upon pianolas Used today in GP, cellphones, digital radio
26 Binary pseudorandom sequences clock input Clock D HIFT REGITER time output n sequence length bits with n bit shift register AUTOCORRELATION = n ( ) ±
27 PRB: Lockin on steroids! Generate signal in pseudorandom bit sequence, for example: 6bit (64bits long) bit (56 bits long): Record signal Multiply by PRB (autocorrelate) ery much like a lockin! But uses broad spectrum Computer Experiment ignal Mod Record/average Reference Multiplier Output
28 3bit (89 bits long) ML single scan
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