锁相放大器
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8 6 Rb spectrum
Rb cell Laser
Photodiode
4
2
0 Frequency
1/f, ―pink‖, ―flicker‖ noise
Ubiquitous Mysterious: There is no generally accepted explanation for 1/f noise More noise at low frequency
Signal from noise
A lock-in amplifier is used to extract signal from noise It detects signal based on modulation at some known frequency Premise:
Vref cost
phase is variable oscillator frequency same as modulation frequency
Multiply modulated signal by REF :
VsigVref VS t cost cost
Measuring something
Need to measure at high frequency, where noise is low Modulate signal, look for component oscillating at modulation frequency
Rb spectrum 8
Noise
Typical photodetector noise spectrum
Photodector intensity -90 On resonance
Power spectral density (dB)
Bad noise -100 here…
-110 Off resonance
-120
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
less noise at reduce bandwidth
Narrow Band Detection
Let’s return to our generic lock-in example. Suppose that instead of being a pure sine wave, the input is made up of signal plus noise. The PSD and low pass filter only detect signals whose frequencies are very close to the lock-in reference frequency. Noise signals, at frequencies far from the reference, are attenuated at the PSD output by the low pass filter (neither ωnoise − ωref nor ωnoise + ωref are close to DC). Noise at frequencies very close to the reference frequency will result in very low frequency AC outputs from the PSD (|ωnoise − ωref| is small). Their attenuation depends upon the low pass filter bandwidth and rolloff. A narrower bandwidth will remove noise sources very close to the reference frequency; a wider bandwidth allows these signals to pass. The low pass filter bandwidth determines the bandwidth of detection. Only the signal at the reference frequency will result in a true DC output and be unaffected by the low pass filter. This is the signal we want to measure.
Mathematical description
Signal VS(t) varies relatively slowly
e.g. absorption spectrum scan over 10 seconds
Modulate at relatively high frequency (e.g. chopper): Vsig VS t cost Reference (local oscillator) of fixed amplitude:
Measuring something
Common task: measure light intensity, e.g. absorption spectrum Need very low intensity to reduce broadening Noise becomes a problem
Signal Experiment Low-pass filter Buffer Output
Mod
Reference
External modulator: true ―lock-in‖
Lock-in amplifier
Vref Vsig VS t cost cost n(t ) cost 1 VS t cos 1 VS t cos2t n(t ) cost 2 2
Third term – noise – at frequency Low-pass filter, frequency less than /2, leaves signal components We win twice:
Biblioteka Baidu
1 VS t cos 1 VS t cos2t 2 2
Second term at high frequency (2)
Low-pass filter (cutoff ~ /2 or lower)
Vsig Vref filter 1 VS t cos 2
Note phase-sensitive detection!
Noise
Laser frequency noise
Noise reduces with frequency (1/f noise is major problem) Shift signal to higher frequency Noise within given bandwidth reduces as we measure at higher frequency
-130 Dark noise -140
90Hz f here...
Better here!
-150 1.0 10 100 1kHz Frequency (Hz)
…and here!
10kHz 100kHz
dc measurements: • broad-spectrum (bad) • at low frequency (bad)
dbV/sqrt(Hz)
-40
-50
-60
-70
-80
-90
-100 1.0 10.0 100.0 1000.0 Frequency (Hz) 10000.0 100000.0
With noise
Signal has noise:
Vsig VS t cost n(t )
Multiply reference by modulated signal:
滤波带宽—噪声等效带宽
采用的滤波器斜率是18dB，积分时间设置为T=100 ms。可以计算出锁相放大器的滤波带宽即相应的噪声 等效带宽为 ENBW=3/32T=0.94 Hz。
Using PSD oscillator to modulate
Phase-sensitive detector
Mixer
DON’T EVER MEASURE DC!
Note log scales!
The principle
Fundamental law of communication theory:Wiener-Khinchin theorem Reduction of noise imposed upon a useful signal with frequency f0, is proportional to the square root of the bandwidth of a bandpass filter, centre frequency f0
6
Rb cell Laser
Photodiode
4
2
Chopper Lock-in amplifier
0
8
6
4
2
0
Frequency
Why Use a Lock-In?
Suppose the signal is a 10 nV sine wave at 10 kHz. Clearly amplification is required to bring the signal above the noise. A good low-noise amplifier will have about 5 nV/√Hz of input noise. If the amplifier bandwidth is 100 kHz and the gain is 1000, we can expect our output to be 10 μV of signal (10 nV × 1000) and 1.6 mV of broadband noise (5 nV/√Hz × √100 kHz × 1000). If we follow the amplifier with a band pass filter with a Q=100 centered at 10 kHz, any signal in a 100 Hz bandwidth will be detected (10 kHz/Q). The noise in the filter pass band will be 50 μV (5 nV/√Hz ×√100 Hz × 1000), and the signal will still be 10 μV. Now try following the amplifier with a phase-sensitive detector (PSD). The PSD can detect the signal at 10 kHz with a bandwidth as narrow as 0.01 Hz! In this case, the noise in the detection bandwidth will be 0.5 μV (5 nV/√Hz × √.01 Hz × 1000), while the signal is still 10 μV. The signal-to-noise ratio is now 20.
Rb cell Laser
Photodiode
4
2
0 Frequency
1/f, ―pink‖, ―flicker‖ noise
Ubiquitous Mysterious: There is no generally accepted explanation for 1/f noise More noise at low frequency
Signal from noise
A lock-in amplifier is used to extract signal from noise It detects signal based on modulation at some known frequency Premise:
Vref cost
phase is variable oscillator frequency same as modulation frequency
Multiply modulated signal by REF :
VsigVref VS t cost cost
Measuring something
Need to measure at high frequency, where noise is low Modulate signal, look for component oscillating at modulation frequency
Rb spectrum 8
Noise
Typical photodetector noise spectrum
Photodector intensity -90 On resonance
Power spectral density (dB)
Bad noise -100 here…
-110 Off resonance
-120
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
less noise at reduce bandwidth
Narrow Band Detection
Let’s return to our generic lock-in example. Suppose that instead of being a pure sine wave, the input is made up of signal plus noise. The PSD and low pass filter only detect signals whose frequencies are very close to the lock-in reference frequency. Noise signals, at frequencies far from the reference, are attenuated at the PSD output by the low pass filter (neither ωnoise − ωref nor ωnoise + ωref are close to DC). Noise at frequencies very close to the reference frequency will result in very low frequency AC outputs from the PSD (|ωnoise − ωref| is small). Their attenuation depends upon the low pass filter bandwidth and rolloff. A narrower bandwidth will remove noise sources very close to the reference frequency; a wider bandwidth allows these signals to pass. The low pass filter bandwidth determines the bandwidth of detection. Only the signal at the reference frequency will result in a true DC output and be unaffected by the low pass filter. This is the signal we want to measure.
Mathematical description
Signal VS(t) varies relatively slowly
e.g. absorption spectrum scan over 10 seconds
Modulate at relatively high frequency (e.g. chopper): Vsig VS t cost Reference (local oscillator) of fixed amplitude:
Measuring something
Common task: measure light intensity, e.g. absorption spectrum Need very low intensity to reduce broadening Noise becomes a problem
Signal Experiment Low-pass filter Buffer Output
Mod
Reference
External modulator: true ―lock-in‖
Lock-in amplifier
Vref Vsig VS t cost cost n(t ) cost 1 VS t cos 1 VS t cos2t n(t ) cost 2 2
Third term – noise – at frequency Low-pass filter, frequency less than /2, leaves signal components We win twice:
Biblioteka Baidu
1 VS t cos 1 VS t cos2t 2 2
Second term at high frequency (2)
Low-pass filter (cutoff ~ /2 or lower)
Vsig Vref filter 1 VS t cos 2
Note phase-sensitive detection!
Noise
Laser frequency noise
Noise reduces with frequency (1/f noise is major problem) Shift signal to higher frequency Noise within given bandwidth reduces as we measure at higher frequency
-130 Dark noise -140
90Hz f here...
Better here!
-150 1.0 10 100 1kHz Frequency (Hz)
…and here!
10kHz 100kHz
dc measurements: • broad-spectrum (bad) • at low frequency (bad)
dbV/sqrt(Hz)
-40
-50
-60
-70
-80
-90
-100 1.0 10.0 100.0 1000.0 Frequency (Hz) 10000.0 100000.0
With noise
Signal has noise:
Vsig VS t cost n(t )
Multiply reference by modulated signal:
滤波带宽—噪声等效带宽
采用的滤波器斜率是18dB，积分时间设置为T=100 ms。可以计算出锁相放大器的滤波带宽即相应的噪声 等效带宽为 ENBW=3/32T=0.94 Hz。
Using PSD oscillator to modulate
Phase-sensitive detector
Mixer
DON’T EVER MEASURE DC!
Note log scales!
The principle
Fundamental law of communication theory:Wiener-Khinchin theorem Reduction of noise imposed upon a useful signal with frequency f0, is proportional to the square root of the bandwidth of a bandpass filter, centre frequency f0
6
Rb cell Laser
Photodiode
4
2
Chopper Lock-in amplifier
0
8
6
4
2
0
Frequency
Why Use a Lock-In?
Suppose the signal is a 10 nV sine wave at 10 kHz. Clearly amplification is required to bring the signal above the noise. A good low-noise amplifier will have about 5 nV/√Hz of input noise. If the amplifier bandwidth is 100 kHz and the gain is 1000, we can expect our output to be 10 μV of signal (10 nV × 1000) and 1.6 mV of broadband noise (5 nV/√Hz × √100 kHz × 1000). If we follow the amplifier with a band pass filter with a Q=100 centered at 10 kHz, any signal in a 100 Hz bandwidth will be detected (10 kHz/Q). The noise in the filter pass band will be 50 μV (5 nV/√Hz ×√100 Hz × 1000), and the signal will still be 10 μV. Now try following the amplifier with a phase-sensitive detector (PSD). The PSD can detect the signal at 10 kHz with a bandwidth as narrow as 0.01 Hz! In this case, the noise in the detection bandwidth will be 0.5 μV (5 nV/√Hz × √.01 Hz × 1000), while the signal is still 10 μV. The signal-to-noise ratio is now 20.