Dear Paul, 

I believe the explanation for your reduced signal with double modulation is the following. The lockin measures the rms value of the first harmonic of your square wave modulation (which is ,let's say, from Vo to 0). The amplitude of your firrst harmonic is Vo/Pi, so that the rms value is Vo/(Pi*sqrt(2)) ~ Vo/4.4, which explains roughly a factor of 5 drop in signal when using the additional square wave probe modulation. 

I hope this is helpful. Regards,

Paulo Miranda

I'm not sure that addresses the point for two reasons. First, a LIA reports the rms of first harmonic of ANY input, so that even single modulation does not report the peak-to-peak value. Second, a 0-to-Vo square wave's first fourier component is (2*V0/pi)*sin(pi*duty-cycle). Assuming a 50% duty cycle, we arrive at 2*V0/pi. The rms output of the LIA would then be V0/2.2. 

I may not have been clear in my description. The reduction in signal I speak of is when comparing the double-modulation of both pump & probe scheme to a single-modulation of the pump beam only. So while the single modulation scheme yields V0/2.2, the double modulation scheme is yielding a 5x smaller signal. I do not understand why. 

A detail I neglected to mention is that the chopper I use cannot be fed an external reference and so is not synchronized to the laser when running at a sub-harmonic of the laser repetition rate.

Well I am struggling with similar question so I can propose the test. Since you know that when you chopping only pump you recording TA + pump scatter signal so the test is to set delay line to negative times (when probe precedes pump) in such setting TA signal is absent but all you have is pump scatter signal subtract one from another to know how much of TA signal from the system you really have. My meaning is that maybe this drop you see is merely pump scatter signal that is way stronger than TA. It happens in our research since our samples are solid and scattering a lot.

If you be so king tell me how much you increased time constant to get decent signal-to-noise in double chopping experiments?

Of course to see scatter signal you need to block probe beam. Scatter signal will have different phase factor (usually not 90deg so there is component of pump scatter in there) which allows you to estimate what fraction of total TA signal pump scatter really is. Then we can judge how much double chopping reduces signal intensity.

Rafal,

To answer your question: I make no change in the time constant. I simply average more scans (which is similar though not identical to a longer time constant) or pump harder, when possible.

To address your proposed experiment: The drop in signal is significantly greater than pump scatter. The proof is to compare measurements at wavelengths far from the pump wavelength. Here, due to the use of a monochromator, we see negligible pump scatter signal when chopping the pump-only. So the signal measured is purely TA. The decrease in signal I described above is seen here. So the decrease is not due to the elimination of pump scatter.

Another helpful bit of information. Japanese colleague working with TA signals has reported that use of non-phase locked choppers like SRS540 causes significant drop in TA signal amplitude and also lower phase stability. He did not provide any explanation to that phenomenon so I understand it was his observation.

Dear Paul and Rafal,

If you compare the first harmonic of the single modulation (w1) to the harmonic at w1+w2 (of a product of two square waves of different frequencies), there is a reduction by a factor of Pi. This does not explain your factor of 5.5, but then maybe this extra reduction from a factor of 3.1 to 5.5 may be due to the used of a nonsynchronized pulsed laser source, as the calculation was done for CW beams. With pulsed lasers, a few pulses will always get partially blocked by the chopper blades and not contribute to the TA signal. 

I hope this is useful... Regards, Paulo

Rafal, 

That does not surprise me. Asynchronous chopper means that some pulses are going to be partially blocked by the chopper blade. This likely adds noise. I've also noticed phase instability as well.

In my past experience, synchronous choppers seemed to me to amplify laser noise at sub-harmonics of the laser frequency. This was an observation years ago when I was in graduate school and may not have been a careful comparison. It may also depend on what are the limiting sources of noise in a particular system.

Another piece of the puzzle may also be related to the observed dependence of the LIA signal on chopper frequency. Even with single modulation I observe larger amplitude signals for lower frequency chopping. This may also have to do with using a chopper than is not synchronized to the laser rep rate.

However, I should point out that the different between single and double modulation that I observe is not a pure manifestation of this frequency-dependent phenomenon. The LIA  trigger received from the chopper is kept constant in the comparison I did.

Well, it's a very valuable discussion for me. I have a little confusion regarding double modulation. In pump-probe microscopy, double modulation technique result's excellent signal-to-noise ratio as compared to single modulation. 

In this case, pump and probe beam modulates at different frequencies and the demodulation signal detects in the lock-in amplifier with the sum  (f1+f2) or difference (f1-f2) frequency.

I am curious to know why do we recover the signal with the sum or difference frequency?

Please help to understand this concept.

Subir - In your description you chop the pump at F1 and the probe at F2. The reason for setting the LIA to the sum freq (F1+F2) is that this is the frequency coincident to both the pump and the probe. Therefore, you are suppressing noise present in both the pump and the probe and looking at the signal that arrises ONLY FROM BOTH PUMP and PROBE being present. As noise often goes at 1/f, ome benefit in SNR comes from using the sum freq as opposed to the diff freq. 

In my case, which is transient absorption, there is another benefit. The signal present from pump scattering at the pump wavelength would typically overwhelm any transient absorption signal in it's wavelength region. This limits experiments to regions far from the pump or could require cross polarization of pump and probe to reduce pump scatter, neither of which is convenient for certain experiments: e.g. anisotropy experiments or nearly degenerate pumping and probing. However, double modulation means we only see a signal due to BOTH pump and probe beams and NOT a signal arriving ONLY FROM THE PUMP.

However, this discussion topic / question is concerned with the observation that double modulation is actually decreasing the SNR in the example given in the original question. This goes counter to conventional wisdom and to the opening assumption of your question.