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A modified-weak-value-amplification(MWVA) technique of measuring the mirror'svelocity based on the Vernier-effect has been proposed. We have demonstratedwith sensitivity-enhanced and the higher signal-to-noise ratio(${\rm SNR}$) byusing two cascaded Michelson interferometers. These two interferometers arecomposed of similar optical structures. One interferometer with a fixed mirroracts as a fixed part of the Vernier-scale, while the other with a moving mirroracts as a sliding part of the Vernier-scale for velocity sensing. The envelopeof the cascaded interferometers shifts much more than a single one with acertain enhancement factor, which is related to the free space range differencebetween these two interferometers. In addition, we calculate the ${\rm SNR}$based on the Fisher information with both the MWVA technique and thetraditional-weak-value-amplification(TMVA) technique. The results show that the${\rm SNR}$ with our MWVA technique is larger than the the ${\rm SNR}$ with theTWVA technique within the range of our time measurement window. Our numericalanalysis proved that our MWVA technique is more efficient than the TWVAtechnique. And by using the principles of the Vernier-effect, it is applicativeand convenient to ulteriorly improving the sensitivity and ${\rm SNR}$ inmeasuring other quantities with the MWVA technique.

Improving the precision of weak-value-amplification with two cascaded Michelson interferometers based on Vernier-effect