4. G-LIA in the case of amplitude and phase-modulated signal

In some important cases, the signal field is modulated both in phase and amplitude. The modulated term Imod ∝ P s cos (Δφ(t)) in the detected intensity can then be expressed as:

$$I\_{\text{mod}} \mathfrak{a} f(t) \cos \left( \Delta \phi(t) \right) \tag{22}$$

where f(t) accounts for the amplitude modulation function. We can mention two relevant examples in the context of phase-sensitive nanoscopy and unbalanced interferometry.

### 4.1. Unbalanced interferometry

0

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

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

any position before the CCD [17], typically up to the sample plane or surface.

Figure 8. Effect of slight rotation on the holographic images. By subtracting the complex field after rotation s

(1)(X,Y), the out of plane displacement is revealed.

(2)(X,Y) from

Figure 8.

222 Optical Interferometry

the complex field before rotation s

In interferometers having a path unbalance, a phase modulation can be efficiently induced by a wavelength modulation of the emission wavelength. For this purpose a spectrally single-mode laser diode working at a central wavelength λ<sup>0</sup> can be used. The wavelength modulation is typically obtained by a current modulation which is associated with a power modulation. In air, the phase modulation is related to the small wavelength modulation δλ(t) by: <sup>φ</sup>RðtÞ ¼ <sup>4</sup>πΔ<sup>l</sup> δλðt<sup>Þ</sup> λ2 , where Δl is the length difference between the two arms.

For modulation frequency below the MHz range, the change in wavelength is considered to be primarily due to a change of temperature that increases with the current. Therefore, using a sawtooth function to create a quasi-linear phase change is usually not an excellent choice, as the thermal inertia of the system prevents the wavelength to precisely follow the driving excitation. On the other hand, a sine power modulation will typically induce the desired sine wavelength and phase modulation. In this case, the detected intensity within an unbalanced interferometer is:

$$I\infty P\_0 \left(1 + \mu \sin\left(\Omega t\right)\right) \left(1 + s \cos\left(\phi\_r - \phi\_s\right)\right) \tag{23}$$

with <sup>φ</sup>R(t) = <sup>a</sup> sin (φR), where <sup>a</sup> <sup>¼</sup> <sup>4</sup>πΔla<sup>λ</sup> λ2 0 with a<sup>λ</sup> corresponding to the depth of wavelength modulation.

It is clear that in the case where the amplitude modulation is small (µ≪1 and µ≪s), the amplitude modulation disappears and the G-LIA method can be applied directly using the signal <sup>~</sup>Iðt<sup>Þ</sup> where the DC component is filtered, with the references <sup>C</sup>(t) = cos (φR) and <sup>S</sup>(t) = sin (φR). The operations are identical to those given by Eqs. (16)–(21). However, while µ can be small compared to 1, the signal s of interest can also be very small in some experiments and setups so that µ≪s cannot be satisfied in general. To see how to handle this problem, we can express the filtered intensity <sup>~</sup>IðtÞ:

$$\ddot{I}\cos[\cos\left(\varphi\_r - \varphi\_s\right)\left(\mu\sin\left(\Omega t\right) + 1\right)]\_{\text{filtered}} + \mu\sin\left(\Omega t\right) \tag{24}$$

where we have normalized the detected intensity by the constant laser power factor. The brackets indicate the quantity is filtered from its DC component. We see that the main issue comes from the modulated term outside the bracket which is independent from the signal s. A direct solution to get rid of this unwanted term is to use references without harmonic component at Ω. This is achieved by choosing J1(a) = 0 (e.g. a = 3.832 rad). With this choice of a, the G-LIA operation given by Eqs. (16)–(24) gives excellent results provided that µ is kept reasonably small in comparison with 1 (e.g. for µ = 0.1 the maximum error on the G-LIA X and Youtput is about 2%).5 This condition is reasonably achieved in many cases. For example, if we consider a standard single mode laser diode such as a Vertical Cavity Surface Emitting Laser (VCSEL) with a tunability of 0.5 nm/mA and a bias current of 1 mA. We see that a current modulation of 4.4% is sufficient to achieve a = 3.8 rad for an unbalance of 10 mm at a wavelength of 850 nm. The corresponding power modulation µ depends on its P(I) but will be typically smaller than 0.1.

### 4.1.1. Improved unbalanced configuration

Despite its advantages in term of cost, unbalanced interferometry is not currently widely used. The main reason is also related to the extreme sensitivity of the system to minute wavelengths changes. Figure 9(a) represents a compensation scheme to solve this issue. The idea is to illuminate the interferometer with a linear polarization at 45° with respect to the horizontal and vertical axis and to discriminate the two s and p polarization using polarization beam splitters. An additional signal arm is equipped with a fixed mirror in order to measure the phase fluctuation induced by any wavelength drifts in time. The light impinging on this mirror is s-polarized and is selectively detected by the photodiode PD1, using a polarization beam splitter in reflection. On the other hand, the p-polarized light impinging on the piezo-actuated mirror is reflected back onto the second photodiode (PD2). Both amplitude and phases are recorded with the above described G-LIA operation.

Figure 9. (a) Unbalanced interferometer with an extra arm for wavelength drifts compensation. The sine phase modulation is induced by a power modulation of the VCSEL laser source. BS: Beam splitter. (b) Actual displacement of the piezo actuated mirror (red); measured displacement without drift compensation (dotted blue); phase fluctuation induced by the intentional wavelength fluctuation (black line), and final measurement (dashed black line) obtained by subtracting the black line to the blue dotted line.

<sup>5</sup> We note that the cases where µ is too large to be neglected can be handled exactly without approximation but it requires to know µ in order to determine analytically or numerically all the coefficients of the G-LIA outputs (4 in this case). In general, the percentage of power modulation µ can be measured without difficulty. The condition J1(a) = 0 is still required.

Figure 9(b) shows a controlled triangular displacement which is correctly determined despite the presence of intentional wavelength drifts. In this experiment, the wavelength of the VCSEL is driven sinusoidally at about 10 kHz to create the phase modulation. The important wavelength drifts are artificially created by adding a low frequency sine to this excitation signal. The compensation is obtained by plotting the phase of the p-polarized light minus the phase of the s-polarized light which is coming from the fixed mirror. Both signal phases are obtained by the G-LIA method with a approximately equal to 3.83 rad.

Such system is really interesting in term of performance since VCSELs are very affordable laser sources that can be driven at very sinusoidally at very high speed. In the described experiment the phase modulation frequency was only limited by the acquisition card used to perform the G-LIA measurement.

#### 4.2. Phase-sensitive nanoscopy

comes from the modulated term outside the bracket which is independent from the signal s. A direct solution to get rid of this unwanted term is to use references without harmonic component at Ω. This is achieved by choosing J1(a) = 0 (e.g. a = 3.832 rad). With this choice of a, the G-LIA operation given by Eqs. (16)–(24) gives excellent results provided that µ is kept reasonably small in comparison with 1 (e.g. for µ = 0.1 the maximum error on the G-LIA X and Youtput is about 2%).5 This condition is reasonably achieved in many cases. For example, if we consider a standard single mode laser diode such as a Vertical Cavity Surface Emitting Laser (VCSEL) with a tunability of 0.5 nm/mA and a bias current of 1 mA. We see that a current modulation of 4.4% is sufficient to achieve a = 3.8 rad for an unbalance of 10 mm at a wavelength of 850 nm. The corresponding power modulation µ depends on its P(I) but will be typically smaller than 0.1.

Despite its advantages in term of cost, unbalanced interferometry is not currently widely used. The main reason is also related to the extreme sensitivity of the system to minute wavelengths changes. Figure 9(a) represents a compensation scheme to solve this issue. The idea is to illuminate the interferometer with a linear polarization at 45° with respect to the horizontal and vertical axis and to discriminate the two s and p polarization using polarization beam splitters. An additional signal arm is equipped with a fixed mirror in order to measure the phase fluctuation induced by any wavelength drifts in time. The light impinging on this mirror is s-polarized and is selectively detected by the photodiode PD1, using a polarization beam splitter in reflection. On the other hand, the p-polarized light impinging on the piezo-actuated mirror is reflected back onto the second photodiode (PD2). Both amplitude and phases are

Figure 9. (a) Unbalanced interferometer with an extra arm for wavelength drifts compensation. The sine phase modulation is induced by a power modulation of the VCSEL laser source. BS: Beam splitter. (b) Actual displacement of the piezo actuated mirror (red); measured displacement without drift compensation (dotted blue); phase fluctuation induced by the intentional wavelength fluctuation (black line), and final measurement (dashed black line) obtained by subtracting the

We note that the cases where µ is too large to be neglected can be handled exactly without approximation but it requires to know µ in order to determine analytically or numerically all the coefficients of the G-LIA outputs (4 in this case). In general, the percentage of power modulation µ can be measured without difficulty. The condition J1(a) = 0 is still required.

4.1.1. Improved unbalanced configuration

224 Optical Interferometry

recorded with the above described G-LIA operation.

black line to the blue dotted line.

5

A modulation of the amplitude at an angular frequency Ω<sup>A</sup> can be introduced in the signal arm to discriminate the amplitude-modulated signal from other unwanted contributions able to interfere with the reference field. This is the case in near-field nanoscopy where the signal light is coming from a near-field probe in interaction with a surface, oscillating at an angular frequency Ωprobe. The situation is depicted in Figure 10.

Figure 10. Phase-sensitive nanoscopy experiment based on G-LIA, where the phase modulation is ϕ<sup>R</sup> ¼ a sin Ωt. (a) Basic setup. (b) Illustration of the frequency spectra of (top) the detected intensity and (bottom) the references signal used in the extraction process. In this example Ω<sup>A</sup> = 2Ωprobe. The condition (e.g. a = 2.405 rad) removes the unwanted peaks from the references.

In Figure 10, the near-field head is included in the signal arm of a Michelson interferometer. Alternately the near-field microscope can be used in the signal arm of a Mach-Zhender which is well adapted to the characterization of waveguiding photonic devices as in [18–20]. Here, the sample is scanned under a nano-tip which is precisely positioned in the focus spot of an objective lens. The light backscattered by the oscillating probe operating in tapping mode contains information on the local optical properties of the sample. This backscattered light can have a rich harmonic content due to its near-field interaction with a sample. The amplitude modulation function appearing in Imod is therefore fðtÞ ¼ cte þ s<sup>1</sup> cosðΩprobet þ Φ1Þ þ s<sup>2</sup> cosð2Ωprobet þ Φ2Þ þ :::

We note that Imod in this case refers to the interference term between the near-field signal from the probe and the phase-modulated reference signal. However other parasitic fields can be backscattered by the probe-sample system. When using a standard, cantilevered, AFM probe, the oscillation amplitude is typically larger than the probe radius. In this case it is often interesting to detect the contribution from a higher harmonics Ω<sup>A</sup> = kΩprobe as the unwanted part of the light which is modulated by the shaft of the probe (rather than the apex) only contributes to the first harmonic(s). This unwanted contribution modulated by the probe shaft can be referred as modulated background contribution (MBC) in contrast with the unmodulated background contribution (UBC) coming, for example, from the sample backscattering (nanodusts, roughness, …) which is also unwanted. In less stringent configurations, especially when the oscillation is small compared to the tip radius (the near field varies almost linearly on the excursion of the probe), the MBC is not perceived6 while keeping k=1.

As shown in Figure 10(b), because of the interference between the probe signal and the reference field, the signal is split into sidebands at kΩprobe ± pΩ where Ω characterize the phase modulation frequency. In order to collect the information spread throughout all these sidebands, the G-LIA can use the following references:

$$\mathbf{C}(t) = f\_{\Omega A}(t) \cos(\phi\_R t) \tag{25}$$

$$S(t) = f\_{\Omega\_{\Lambda}}(t) \sin(\phi\_{\mathcal{R}} t) \tag{26}$$

where f ΩAðtÞ ¼ cosðΩAt þ ψÞ is the amplitude carrier at the frequency of interest ΩA. But even in this favorable case where the MBC is easily excluded, the UBC can be especially detrimental and has been described by several authors. UBC however can be efficiently removed with interferometric detection. As noted elsewhere, the UBC is perceived because the unmodulated background interferes with the modulated signal from the tip on the detector creating signal peaks at the harmonics frequencies kΩprobe. Therefore a direct solution consists in excluding the frequencies Ω<sup>A</sup> = kΩprobe from the references C(t) and S(t). In the case of a sine modulation, the condition a = 2.405 rad (J0(a) = 0) is sufficient to exclude the unwanted frequencies component by removing the possible DC contribution in cos(φRt).

First examples of phase-sensitive near-field imaging based on G-LIA can be found in Ref. [7]. In Figure 11, a simple demonstration experiment is made by using a bare tuning oscillating fork to modulate part of the signal at an angular frequency ΩA. The trace of the experimentally detected signal intensity exhibits clearly a slow modulation due the phase modulation (φ<sup>R</sup> = a sin Ωt with a frequency of 1 kHz) as well as a sine amplitude oscillation at the oscillation frequency of the tuning fork Ω<sup>A</sup> (at about 32 kHz). An additional coverglass can be added to

<sup>6</sup> Such case occurs when using elongated probes like tungsten probes, mounted on tuning fork working in tapping mode. The elongated shape minimizes the possible modulation of the background light, while an oscillation amplitude of few nanometers can also prevent a detectable modulation of the background light. In some other case where a Mach-Zehnder interferometer is used, only the apex of the probe can be illuminated (e.g.when imaging waveguiding structures). In general, reducing the amplitude of modulation of the probe reduces the background contributions more efficiently than the near-field contribution.

check the system immunity to UBC. In this example, the references are those given by Eqs. (25)–(26) with f ΩAðtÞ ¼ cosðΩAt þ ψÞ where ψ is adjusted to be in phase with the fork oscillation, resulting in a maximized amplitude signal (not shown here). With a = 2.405 rad to exclude the UBC, the two G-LIA outputs provide:

We note that Imod in this case refers to the interference term between the near-field signal from the probe and the phase-modulated reference signal. However other parasitic fields can be backscattered by the probe-sample system. When using a standard, cantilevered, AFM probe, the oscillation amplitude is typically larger than the probe radius. In this case it is often interesting to detect the contribution from a higher harmonics Ω<sup>A</sup> = kΩprobe as the unwanted part of the light which is modulated by the shaft of the probe (rather than the apex) only contributes to the first harmonic(s). This unwanted contribution modulated by the probe shaft can be referred as modulated background contribution (MBC) in contrast with the unmodulated background contribution (UBC) coming, for example, from the sample backscattering (nanodusts, roughness, …) which is also unwanted. In less stringent configurations, especially when the oscillation is small compared to the tip radius (the near field varies almost linearly on the

As shown in Figure 10(b), because of the interference between the probe signal and the reference field, the signal is split into sidebands at kΩprobe ± pΩ where Ω characterize the phase modulation frequency. In order to collect the information spread throughout all these side-

where f ΩAðtÞ ¼ cosðΩAt þ ψÞ is the amplitude carrier at the frequency of interest ΩA. But even in this favorable case where the MBC is easily excluded, the UBC can be especially detrimental and has been described by several authors. UBC however can be efficiently removed with interferometric detection. As noted elsewhere, the UBC is perceived because the unmodulated background interferes with the modulated signal from the tip on the detector creating signal peaks at the harmonics frequencies kΩprobe. Therefore a direct solution consists in excluding the frequencies Ω<sup>A</sup> = kΩprobe from the references C(t) and S(t). In the case of a sine modulation, the condition a = 2.405 rad (J0(a) = 0) is sufficient to exclude the unwanted frequencies component

First examples of phase-sensitive near-field imaging based on G-LIA can be found in Ref. [7]. In Figure 11, a simple demonstration experiment is made by using a bare tuning oscillating fork to modulate part of the signal at an angular frequency ΩA. The trace of the experimentally detected signal intensity exhibits clearly a slow modulation due the phase modulation (φ<sup>R</sup> = a sin Ωt with a frequency of 1 kHz) as well as a sine amplitude oscillation at the oscillation frequency of the tuning fork Ω<sup>A</sup> (at about 32 kHz). An additional coverglass can be added to

Such case occurs when using elongated probes like tungsten probes, mounted on tuning fork working in tapping mode. The elongated shape minimizes the possible modulation of the background light, while an oscillation amplitude of few nanometers can also prevent a detectable modulation of the background light. In some other case where a Mach-Zehnder interferometer is used, only the apex of the probe can be illuminated (e.g.when imaging waveguiding structures). In general, reducing the amplitude of modulation of the probe reduces the background contributions more efficiently than

CðtÞ ¼ f <sup>Ω</sup><sup>A</sup>ðtÞcosðφRtÞ (25)

SðtÞ ¼ f <sup>Ω</sup><sup>A</sup> ðtÞsinðφRtÞ (26)

excursion of the probe), the MBC is not perceived6 while keeping k=1.

bands, the G-LIA can use the following references:

226 Optical Interferometry

by removing the possible DC contribution in cos(φRt).

6

the near-field contribution.

$$X =  \infty \mathbf{k}\_{\mathbf{x}} \mathbf{s}\_1 \cos \left(\phi\_s\right), \text{with } \mathbf{k}\_{\mathbf{x}} = \langle \mathbb{C}^2(t) \rangle \tag{27}$$

$$\mathcal{Y} =  \mathfrak{a}k\_y s\_1 \cos \left(\phi\_s\right) \text{ with } k\_y = \langle \mathcal{S}^2(t) \rangle \tag{28}$$

where s<sup>1</sup> corresponds to the amplitude of signal field modulated at ΩA. For a sine phase modulation, an analytical expression can be derived for the proportionality constants kx and ky.These expressions can be found in the summary table given in Appendix A.

Figure 11. (a) Demonstration setup with φ<sup>R</sup> = a sin Ωt. An UBC can be added via a glass coverslip in the signal arm. (b) Recovered signal phase for a triangular displacement of the signal mirror.

Figure 11(b) shows the phase determined with this method when a triangular phase modulation having a peak to peak phase modulation depth of about 2.0 rad is induced by the signal mirror. The signal phase is precisely retrieved.

The value of ψ includes the mechanical phase shift existing between the driving signal and the actual motion of the fork. In fact, in a near-field experiment, this shift can vary from one position to another on the sample depending on the material in interaction with the probe. Depending on the system, the value of ψ in f ΩAðtÞ ¼ cosðΩAt þ ψÞ is not necessarily known if only the driving signal is accessible. We note that the retrieved optical phase value is not affected by the value of ψ, which only affects the amplitude, but the SNR can be strongly decreased if we omit ψ.

If ψ is unknown, the G-LIA can be applied twice to solve this issue with two quadrature amplitude modulation functions. In other word, we can calculate for outputs signals <sup>ð</sup>X, <sup>Y</sup>, <sup>X</sup>′ , Y′ Þ ¼< I∗ref erenceðtÞ >, using, respectively, the following references:

$$\mathbf{C}(t) = \cos\left(\Omega\_A(t)\right)\cos(\phi\_R t) \tag{29}$$

$$S(t) = \cos\left(\Omega\_A(t)\right)\sin(\phi\_R t)\tag{30}$$

$$\mathbf{C}'(t) = \sin\left(\Omega\_A(t)\right)\cos(\phi\_R t) \tag{31}$$

$$S'(t) = \sin\left(\Omega\_A(t)\right)\sin(\phi\_R t)\tag{32}$$

The four outputs (X, Y, X′, Y′) can be evaluated numerically or analytically, they provide, respectively, < CðtÞ <sup>2</sup> <sup>&</sup>gt; cosðφsÞ, <sup>&</sup>lt; <sup>S</sup>ðt<sup>Þ</sup> <sup>2</sup> <sup>&</sup>gt; sinðφsÞ, <sup>&</sup>lt; <sup>C</sup><sup>0</sup> ðtÞ <sup>2</sup> <sup>&</sup>gt; cosðφsÞ, <sup>&</sup>lt; <sup>S</sup><sup>0</sup> ðtÞ <sup>2</sup> <sup>&</sup>gt; sinðφs<sup>Þ</sup> whose analytical expressions are given in Appendix A for the sine phase modulation φR(t).
