3.1.1. Displacement measurement

where the negative extra term in Kx comes from the filtering of the DC component which is not zero for C(t). As shown in Figure 3, for certain phase modulation amplitude the two constants are identical and approximately equal to unity. This is obtained for a ≈ 2.814 rad, but any phase modulation can be used. A better signal to noise ratio (SNR) is naturally achieved when there is no DC component in C(t) and S(t), since this part is filtered. These cases correspond to the zeros of J0(a) as previously mentioned (e.g. a = 2.405 rad), however the SNR is nearly optimum for a continuous range of values above a = 2 rad. Other analytical expressions can be given, for example, for a triangular modulation [7], however the constants estimation can be made

In this section we review and present several results of interferometric measurements performed with the G-LIA approach described in the previous section. Results include measurement with a point detector reported elsewhere and interferometric measurement with 2D

Figure 3. Proportionality factors and used in a G-LIA working with a sine phase modulation as a function of the phase modulation depth. The analytical evaluations are plotted in solid lines; the markers correspond to the numerically

Figure 4 shows measurement results adapted from the Ref. [7], where the G-LIA can be used with or without filtering to monitor an arbitrary displacement (here a triangle-shaped dis-

numerically without difficulty for a variety of phase modulation functions.

3. Application examples

placement).

calculated values.

218 Optical Interferometry

detector in the framework of holographic measurement.

3.1. Measurement with a point detector

The setup is shown on Figure 4(a). In this example, the phase modulation is a sine function φ<sup>R</sup> = a sin Ωt, with a modulation depth approximately equals to a = 2.405 rad. As explained before, with this value it is not necessary to filter the detected intensity to extract amplitude and phase from the G-LIA operations X = <I C(t)> and Y = <I S(t)>. However, it is a common practise to filter the DC signal directly after the detector to get rid of the environmental light condition before acquisition in order to optimize the analog to digital conversion. In the Michelson configuration shown here, the value of a corresponds to a peak to valley oscillation of the reference mirror of about 38.3% of the wavelength. If no position sensor is present on the reference mirrors, several methods can be used to achieve the desired phase modulation depth a used in the references C(t) and S(t), notably by inspecting the signal I(t) whose shape changes continuously with increasing value of a. Alternately the modulation depth can be precisely adjusted to recover precisely a controlled displacement of the signal mirror without affecting the amplitude output.

### 3.1.2. Sensing

Determining the phase rather than the amplitude is known to offer potential advantage in term of sensitivity in optical sensing systems [13]. More precisely, the phase detection coupled with surface plasmon resonance (SPR) is known to improve the measurement sensitive by one to several order of magnitude depending on the exact system geometry. Many different designs on combining interferometry or heterodyne detection on Kretschmann configuration-based SPR sensor have been done [14, 15].

Figure 5 shows the demonstration setup used in [7] to demonstrate the applicability of the G-LIA for phase sensitive sensing application. The setup is similar to that of Figure 4(a), except that an I<sup>2</sup> gas cell was added in the reference arm. The phase modulation is still achieved with a piezo-actuated mirror and the wavelength of a laser diode emitting near 660 nm is ramped over 4pm across an absorption line of the gas. To avoid phase drift induced by the wavelength ramp, the length of the arms must be precisely balanced, since even minute wavelength fluctuation can create phase fluctuation in unbalanced interferometer. This balanced setting can be achieved by ramping the laser diode wavelength outside an absorption band, and adjusting the position the reference mirror until the phase output of the G-LIA remains constant.

Figure 5. (a) Interferometric measurement with a single detector, applied to gas sensing. (b) Phase-sensitive detection of an absorption line. The obtained spectrum is adapted from Ref [7].

As can be seen, the phase varies more abruptly at the absorption peak center. However, the benefit of measuring the phase for monitoring a gas concentration is not clear since the amplitude has similar variation on the two sides of the absorption peak which indicates a similar sensitivity than the phase if the detection is made where the slope is maximum on the amplitude.

The interest of phase sensitive detection in SPR-based measurement is more obvious. In fact, strong plasmonic resonances can be reached by carefully adjusting the opto-geometrical parameters of the plasmonic layer in order to obtain very sharp phase variation across a resonance. One possible combination of phase sensitivity SPR bio-sensor using G-LIA for phase extraction is proposed in Figure 6, where a cuvette is put on a plasmonic chip to convey a fluid on the surface of a plasmonic chip. A coupling prism makes it possible to satisfy the Kretschmann condition for which the reflectivity of a p-polarized incident beam reaches a minimum corresponding the excitation of the plasmon-polariton surface mode. In order to have a stable phase, immune to wavelength fluctuations, the length of the two arms are made equal. Figure 6(b) presents the numerically calculated complex reflectivity as a function of the incident beam angle in the case of a glass coupling prism coated by a gold layer of thickness h covered by water and excited by a red laser. The best coupling angle is close to α = 70° in the provided examples.

As can be seen, the phase variation across the resonance can be made very sharp by adjusting the metal thickness h. It should be noted however that this fast variation is associated with a strong attenuation of the reflected beam and a compromise between signal level and sensitivity can be made depending on the available laser power. For example, on Figure 6(b) the reflection is about 1.6% for h = 48 nm at Kretschmann angle, but it drops to 2.5‰ for h = 50 nm. If we consider a reasonable phase resolution of 10−<sup>3</sup> rad, simple calculation show that the case h = 50 nm shown on the figure leads to sensitivity slightly better than 8×10−<sup>7</sup> RIU (refractive index unit). On the other hand, if the thickness is slightly inferior, the RIU sensitivity drops rapidly (about 2×10−<sup>6</sup> for h = 48 nm). To obtain a similar sensitivity with the only amplitude signal, the noise level on the amplitude measured at the maximum slope should be smaller than 10−<sup>3</sup> %, which is hardly achievable.

Figure 6. Example of possible experimental setup for phase sensing based on an SPR chip. (b) Simulation of the complex reflectivity (magnitude and phase) for a light beam impinging a gold layer with a thickness h as a function of the incident angle. The wavelength is 670 nm.

### 3.2. Digital holography

can be achieved by ramping the laser diode wavelength outside an absorption band, and adjusting the position the reference mirror until the phase output of the G-LIA remains

As can be seen, the phase varies more abruptly at the absorption peak center. However, the benefit of measuring the phase for monitoring a gas concentration is not clear since the amplitude has similar variation on the two sides of the absorption peak which indicates a similar sensitivity than the phase if the detection is made where the slope is maximum on the

Figure 5. (a) Interferometric measurement with a single detector, applied to gas sensing. (b) Phase-sensitive detection of

an absorption line. The obtained spectrum is adapted from Ref [7].

The interest of phase sensitive detection in SPR-based measurement is more obvious. In fact, strong plasmonic resonances can be reached by carefully adjusting the opto-geometrical parameters of the plasmonic layer in order to obtain very sharp phase variation across a resonance. One possible combination of phase sensitivity SPR bio-sensor using G-LIA for phase extraction is proposed in Figure 6, where a cuvette is put on a plasmonic chip to convey a fluid on the surface of a plasmonic chip. A coupling prism makes it possible to satisfy the Kretschmann condition for which the reflectivity of a p-polarized incident beam reaches a minimum corresponding the excitation of the plasmon-polariton surface mode. In order to have a stable phase, immune to wavelength fluctuations, the length of the two arms are made equal. Figure 6(b) presents the numerically calculated complex reflectivity as a function of the incident beam angle in the case of a glass coupling prism coated by a gold layer of thickness h covered by water and excited by a red laser. The best coupling angle is close to α = 70° in the

As can be seen, the phase variation across the resonance can be made very sharp by adjusting the metal thickness h. It should be noted however that this fast variation is associated with a strong attenuation of the reflected beam and a compromise between signal level and sensitivity can be made depending on the available laser power. For example, on Figure 6(b) the reflection is about 1.6% for h = 48 nm at Kretschmann angle, but it drops to 2.5‰ for h = 50 nm.

constant.

220 Optical Interferometry

amplitude.

provided examples.

In digital holography, the holograms of a sample object are recorded on a 2D detector such as a Charge-coupled Device (CCD) or a Complementary Metal-Oxide-Semiconductor (CMOS) camera. Such system can notably be used as an optical profilometer, or for sensing applications [1, 6, 16]. Figure 7(a) presents the experimental setup of a lensless, compact, digital microscope working with the G-LIA extraction method.

Figure 7. (a) Lensless digital holography setup. A diaphgram may be to select the central zone of interest in the sample. (b) Amplitude and phase of a grating of straight and tilted slits made in a steel surface.

In the provided example, a metallic grid of slit is imaged in amplitude and phase. The Lead Zirconate Titanate (PZT) oscillates in the reference arm at 10 Hz to generate the phase modulation function φ<sup>R</sup> = a sin Ωt at a wavelength λ = 640 nm in the reference arm. During the piezo oscillation, a video is recorded at a frame rate of 120 Hz. Then, amplitude and phase of the detected field is obtained on the camera by performing a G-LIA operation on each pixel using a program code. In this 2D case, the operation was not made in real-time because of the nonnegligible processing time (In the order of 1 min or less depending on the computing resource). Once the detected complex field is retrieved, the associated plane wave spectrum can be obtained by Fourier transform. Then each plane wave can be back-propagated numerically to any position before the CCD [17], typically up to the sample plane or surface.

In this example, the raw signal I(X,Y) is processed by the numeric, software-based, G-LIA, without filtering. For a correct operation it is then mandatory to use an amplitude modulation a of about 2.405 rad. Figure 7(b) shows the amplitude and phase of the complex field which is back-propagated up to the sample plane. As we are dealing with a rough surface, the recorded phase has a speckle-like distribution. Because the illumination direction is normal to the sample surface, the system is strongly sensitive to any out-of-plane displacement of the structure. To give an idea of the system sensitivity, the sample is slightly rotated. By subtracting the phase image before rotation to the phase image after rotation, we obtain the phase-shift associated with the out-of-plane displacements, as shown in Figure 8.

Figure 8. Effect of slight rotation on the holographic images. By subtracting the complex field after rotation s (2)(X,Y) from the complex field before rotation s (1)(X,Y), the out of plane displacement is revealed.
