5. Conclusion

We have detailed the principle of the G-LIA method, first in the case of pure phase modulations then in the case where the amplitude of the signal is also modulated. For pure phase modulations, the interest of the approach was illustrated in different contexts: position monitoring, sensing, and digital holography. In these experiments, the non-linear phase modulation was achieved by mirrors mounted on sinusoidally driven piezo-actuators. In this case, the main advantage of the G-LIA is to extract amplitude and phase information directly from all the harmonic contents created by the phase modulation function. While the examples only considered sine phase modulation functions which is often the most desirable one, the G-LIA also provides a unified treatment to handle arbitrary phase modulation function.

We have also detailed the case where an amplitude modulation can be present. This is notably the case in unbalanced interferometry where a non-negligible amplitude modulation can be perceived at the same frequency than the phase modulation. Experimentally, we considered the case of unbalanced interferometers where a fast sine phase modulation is provided by a current-driven single mode laser diode. A simple yet efficient setup was described to neutralize the impact of wavelength fluctuation on the system. Such approach offers the opportunity to develop simple and cost-efficient system without sacrificing precision. Finally, we discussed the case where the signal of interest is modulated in amplitude at a frequency different from that of the phase modulation. This case was detailed in the context of phase sensitive SNOM, where the low available signal requires to exploit all the available sidebands induced by the phase modulation. Notably, the condition to cancel the effect of the unmodulated background light was presented and attention was paid to the impact of the mechanical phase of the oscillating probe.

### Appendix A

The table (Figure 12) provides a summary of case handled by the G-LIA method.

Interferometry Using Generalized Lock-in Amplifier (G-LIA): A Versatile Approach... http://dx.doi.org/10.5772/66657 229


Figure 12. Summary table.

CðtÞ ¼ cos

SðtÞ ¼ cos

ðtÞ ¼ sin

ðtÞ ¼ sin

analytical expressions are given in Appendix A for the sine phase modulation φR(t).

also provides a unified treatment to handle arbitrary phase modulation function.

The table (Figure 12) provides a summary of case handled by the G-LIA method.

C0

S0

<sup>2</sup> <sup>&</sup>gt; cosðφsÞ, <sup>&</sup>lt; <sup>S</sup>ðt<sup>Þ</sup>

respectively, < CðtÞ

228 Optical Interferometry

5. Conclusion

oscillating probe.

Appendix A

 ΩAðtÞ 

 ΩAðtÞ 

 ΩAðtÞ 

 ΩAðtÞ 

The four outputs (X, Y, X′, Y′) can be evaluated numerically or analytically, they provide,

We have detailed the principle of the G-LIA method, first in the case of pure phase modulations then in the case where the amplitude of the signal is also modulated. For pure phase modulations, the interest of the approach was illustrated in different contexts: position monitoring, sensing, and digital holography. In these experiments, the non-linear phase modulation was achieved by mirrors mounted on sinusoidally driven piezo-actuators. In this case, the main advantage of the G-LIA is to extract amplitude and phase information directly from all the harmonic contents created by the phase modulation function. While the examples only considered sine phase modulation functions which is often the most desirable one, the G-LIA

We have also detailed the case where an amplitude modulation can be present. This is notably the case in unbalanced interferometry where a non-negligible amplitude modulation can be perceived at the same frequency than the phase modulation. Experimentally, we considered the case of unbalanced interferometers where a fast sine phase modulation is provided by a current-driven single mode laser diode. A simple yet efficient setup was described to neutralize the impact of wavelength fluctuation on the system. Such approach offers the opportunity to develop simple and cost-efficient system without sacrificing precision. Finally, we discussed the case where the signal of interest is modulated in amplitude at a frequency different from that of the phase modulation. This case was detailed in the context of phase sensitive SNOM, where the low available signal requires to exploit all the available sidebands induced by the phase modulation. Notably, the condition to cancel the effect of the unmodulated background light was presented and attention was paid to the impact of the mechanical phase of the

ðtÞ

<sup>2</sup> <sup>&</sup>gt; cosðφsÞ, <sup>&</sup>lt; <sup>S</sup><sup>0</sup>

<sup>2</sup> <sup>&</sup>gt; sinðφsÞ, <sup>&</sup>lt; <sup>C</sup><sup>0</sup>

cosðφRtÞ (29)

sinðφRtÞ (30)

cosðφRtÞ (31)

sinðφRtÞ (32)

ðtÞ

<sup>2</sup> <sup>&</sup>gt; sinðφs<sup>Þ</sup> whose
