2. Theory: introduction to G-LIA

In order to introduce the G-LIA technique, we need to provide an expression for the detected signal. We consider the simplest configuration of a 2-arm interferometer comprising a reference arm and a signal arm (cf. Figure 1). The system is illuminated by a monochromatic radiation. The detected signal intensity I(t) can be expressed as:

Figure 1. Pseudo-heterodyne approach. A quasi-linear phase modulation is achieved by a sawtooth modulation of the optical path using a piezo-actuated optical mirror in the reference arm. The detected intensity exhibits a sine modulation except during the flyback time of the mirror. Any phase change in the reference arm will produce a detectable phase shift of the observed quasi-sinusoidal pattern.

Interferometry Using Generalized Lock-in Amplifier (G-LIA): A Versatile Approach... http://dx.doi.org/10.5772/66657 213

$$I(t) \mathfrak{s} E\_r^2 + E\_s^2 + 2mE\_r E\_s \cos\left(\Delta\phi(t)\right) \tag{1}$$

where Es and Er are, respectively, the amplitude of the field of interest and the reference field impinging on the detector. The phase difference between the two fields is Δφ(t) while the factor m ≤ 1 in the interferometric term accounts for the interference contrast. Alternately, it is common to express the detected intensity as a function of the laser power P:

$$I(t) \propto P[1 + s \cos\left(\Delta\phi(t)\right)]\tag{2}$$

where s is proportional to the unknown signal amplitude Es, considered as constant during a measurement. The unknown spatial phase φ<sup>s</sup> of the signal field is also supposed to be constant during the measurement although a phase modulator can be included inside the signal arm. Any time dependence in Δφ(t) is therefore arbitrarily considered as coming from the reference field:

$$
\Delta\phi(t) = \phi\_{\mathcal{R}}(t) - \phi\_{\mathcal{S}} \tag{3}
$$

From the expression of I given by Eq. (2), different strategies can be proposed in order to recover amplitude and phase information. In order to extract the two unknowns (s, φS) from the signal I(t), the phase of one of the two beams can be modulated in time by a frequency shifter or another phase modulator. When the time dependence is linear φr(t) ∝ t as in the former case, (s, φS) are precisely determined by a Lock-in Amplifier (LIA) locked at the single frequency component present in I(t). For other functions of time such as a sine waveforme (i.e. φ<sup>R</sup> = α sin Ωt, where a is the modulation depth), the use of a conventional LIA is less trivial as the signal information is typically spread over a number of frequency components (nΩ/2π) having different weights. The G-LIA method was introduced to handle such cases, while keeping an extraction procedure very similar to a LIA. Hereafter, the two approaches are provided to highlight the similitudes and differences.

#### 2.1. Amplitude and phase determination using a standard LIA

#### 2.1.1. Case of linear phase modulation (LIA)

of cost, achromaticity or integration offers non-linear responses, that are even sometimes coupled with unwanted amplitude modulation. A critical question that arises is "How can we extract phase and amplitude information in an optimal way when non-linear phase

To solve this issue while keeping the benefits of high SNR, approaches have been proposed based on multiple lock-in detection at selected signal harmonics. These approaches were mainly employed in the case where the phase modulation is a sine function [1–6]. Such phase modulation is, for example, achieved using piezo-actuator, fiber stretchers, and other phase modulators where a sine excitation typically offers the best response. The multiple lock-in approach works fine but it is less direct and does not necessarily provide an optimal SNR or a straightforward implementation. Especially, if an amplitude modulation is present at the same frequency as that of the phase modulation. Alternately, the Generalized Lock-in Amplifier (G-LIA) technique was recently introduced [7] to solve this issue with a procedure similar to a single LIA operation. In this chapter, we first detail the principle of this method when operated in the simplest case where no amplitude modulation is present. Application is provided notably in the context of digital holography. Then we consider the case where there is an additional amplitude modulation in the signal field. The first case which is discussed is related to unbalanced interferometry where the phase modulation is achieved via a power modulation of the laser source. Finally, we also discuss the case of phase-

In order to introduce the G-LIA technique, we need to provide an expression for the detected signal. We consider the simplest configuration of a 2-arm interferometer comprising a reference arm and a signal arm (cf. Figure 1). The system is illuminated by a monochromatic

Figure 1. Pseudo-heterodyne approach. A quasi-linear phase modulation is achieved by a sawtooth modulation of the optical path using a piezo-actuated optical mirror in the reference arm. The detected intensity exhibits a sine modulation except during the flyback time of the mirror. Any phase change in the reference arm will produce a detectable phase shift

modulation is used?"

212 Optical Interferometry

sensitive near-field imaging.

of the observed quasi-sinusoidal pattern.

2. Theory: introduction to G-LIA

radiation. The detected signal intensity I(t) can be expressed as:

Considering the general Eq. (2), the use of LIA is direct when Δφ(t) is linearly modulated by φ<sup>R</sup> (t) = Ωt = 2πΔFt, where the phase modulation rate Ω can be induced in different ways. These ways include notably the use of a frequency shifter in one of the two arms (where ΔF is the frequency shift), a linear translation of one of the mirror (where ΔF is the associated Doppler shift), or a linear variation of the laser frequency<sup>1</sup> if the two arms are unbalanced. Amongst the cited methods, heterodyne measurement based on the use of frequency shifters is often considered as more favorable as a purely linear displacement of the mirror is hardly achievable in practise without alignment or coherence issues, while unbalanced

<sup>1</sup> In this case ΔF is function of the unbalance, that is, the optical path difference.

interferometry is subject to noise [8] induced by small wavelength fluctuation typically related to temperature drifts.

The two unknowns (s, φS) are simply the amplitude and phase of the sine waveform present in Eq. (1) which is modulated at the frequency shift ΔF. This determination is optimally carried out by a dual-output (X, Y) LIA locked at the frequency ΔF that is precisely provided by the modulator driver. Depending on the output, the detected signal I is multiplied by an in-phase or a quadrature sinusoids modulated at the same angular frequency ΔF and it is averaged over a time tint:

$$X(I) = \frac{1}{t\_{int}} \mathfrak{f}\_0^{t\_{int}} I \cos\left(\Omega t\right) d\text{tors } \cos\left(\phi\_S\right) \tag{4}$$

$$Y(I) = \frac{1}{t\_{int}} \mathfrak{f}\_0^{t\_{int}} I \sin\left(\Omega t\right) d\cos\sin\left(\phi\_S\right) \tag{5}$$

From these two outputs, the quantities s∝ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>X</sup><sup>2</sup> <sup>þ</sup> <sup>Y</sup><sup>2</sup> <sup>p</sup> and φ<sup>S</sup> ¼ atan 2ðX, YÞ are obtained with a signal to noise ratio that can be increased using longer integration time. We note that the two in-phase and quadrature sine waveforms are the LIA reference signals built from the frequency shift precisely provided to the LIA. If the LIA is not locked exactly at the angular frequency Ω, the measured values of φ<sup>s</sup> will drift in time, while the extracted signal amplitude will decay for long integration time.

### 2.1.2. Case of a non-linear phase modulation (LIA)

Achieving φ<sup>r</sup> (t) ∝ t is not possible for a number of phase modulators, notably because they have a finite range of phase modulation. Pseudo-heterodyne approaches were proposed long ago to circumvent this problem [9] by using a sawtooth modulation of the optical path, where the peak to peak amplitude of the sawtooth corresponds to an integer number of times 2π in term of phase. The approach is illustrated by Figure 1, in the case where the phase ramp is achieved by a piezo-actuated mirror in a balanced interferometer.

As can be seen, the detected intensity mimics the sinusoidal beating observed in heterodyne setups. Such approach is not widely used since errors are induced during the flyback time on the sawtooth edges, especially if the modulation is fast.

As mentioned, the use of sine modulation φ<sup>R</sup> = α sin Ωt is regarded as much more desirable as most of the modulator can operate better and faster when they are sinusoidal excited. It was early highlighted in this context [10] that for such modulation, the Fourier spectrum of the signal has harmonic sidebands coming from the interferometric term Imod ∝ P s cos (Δ φ (t)) in Eq. (2), as shown in Figure 2.

The amplitudes of these frequency components are obtained by developing the term in Imod ∝ P [cos (φs) cos (φR) + sin(φs) sin (φR)] and using the Jacobi-Anger expansion of cos (φR) (even harmonics) and sin (φR) (odd harmonics) [11]. From this expansion, we see that a LIA locked at an angular frequency harmonics mΩ, with m ≠ 0, gives:

$$X\_m(I) \propto s \cos \left(\phi\_S\right) \begin{vmatrix} J\_m(a) & \text{for } m \text{ even} \\ 0 & \text{for } m \text{ odd} \end{vmatrix} \tag{6}$$

$$Y\_m(I) \propto s \cdot \sin\left(\phi\_S\right) \begin{vmatrix} 0 & \text{for } m \text{ even} \\ J\_m(a) & \text{for } m \text{ odd} \end{vmatrix},\tag{7}$$

where Jm (a) is the m-th Bessel function. The amplitude and phase can then be extracted using both odd and even harmonics, for example, using s ∝ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 <sup>1</sup>=J 2 <sup>1</sup>ðaÞ þ <sup>Y</sup><sup>2</sup> <sup>2</sup>=J 2 <sup>2</sup>ðaÞ q and <sup>φ</sup><sup>S</sup> <sup>¼</sup> <sup>a</sup>tan 2� X1=J1ðaÞ, Y2=J2ðaÞ � :

Figure 2. Signals in the case of a sinusoidal phase modulation. (a) Top: example of detected intensity for different signal phases. Bottom: corresponding reference functions. (b) Schematic example of Fourier transform of and associated references and in the case of an arbitrary sine phase modulation and φ<sup>s</sup> ¼ π=4.

When only two harmonics m = (1, 2) are used, caution must be exercised in the choice of the modulation depth a in order to maximize the power density on the selected harmonics. The optimum value of a in this case is a = 2.19 rad (maximum of J<sup>1</sup> (a) <sup>2</sup> + J2(a) 2 ).

#### 2.2. G-LIA method

interferometry is subject to noise [8] induced by small wavelength fluctuation typically

The two unknowns (s, φS) are simply the amplitude and phase of the sine waveform present in Eq. (1) which is modulated at the frequency shift ΔF. This determination is optimally carried out by a dual-output (X, Y) LIA locked at the frequency ΔF that is precisely provided by the modulator driver. Depending on the output, the detected signal I is multiplied by an in-phase or a quadrature sinusoids modulated at the same angular frequency ΔF and it is averaged over

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>X</sup><sup>2</sup> <sup>þ</sup> <sup>Y</sup><sup>2</sup> <sup>p</sup>

signal to noise ratio that can be increased using longer integration time. We note that the two in-phase and quadrature sine waveforms are the LIA reference signals built from the frequency shift precisely provided to the LIA. If the LIA is not locked exactly at the angular frequency Ω, the measured values of φ<sup>s</sup> will drift in time, while the extracted signal amplitude will decay for

Achieving φ<sup>r</sup> (t) ∝ t is not possible for a number of phase modulators, notably because they have a finite range of phase modulation. Pseudo-heterodyne approaches were proposed long ago to circumvent this problem [9] by using a sawtooth modulation of the optical path, where the peak to peak amplitude of the sawtooth corresponds to an integer number of times 2π in term of phase. The approach is illustrated by Figure 1, in the case where the phase ramp is achieved by a piezo-actuated mirror in a balanced

As can be seen, the detected intensity mimics the sinusoidal beating observed in heterodyne setups. Such approach is not widely used since errors are induced during the flyback time on

As mentioned, the use of sine modulation φ<sup>R</sup> = α sin Ωt is regarded as much more desirable as most of the modulator can operate better and faster when they are sinusoidal excited. It was early highlighted in this context [10] that for such modulation, the Fourier spectrum of the signal has harmonic sidebands coming from the interferometric term Imod ∝ P s cos (Δ φ (t)) in

The amplitudes of these frequency components are obtained by developing the term in Imod ∝ P [cos (φs) cos (φR) + sin(φs) sin (φR)] and using the Jacobi-Anger expansion of cos (φR) (even

<sup>0</sup> I cos ðΩtÞdt∝s cos ðφSÞ (4)

<sup>0</sup> I sin ðΩtÞdt∝s sin ðφSÞ (5)

and φ<sup>S</sup> ¼ atan 2ðX, YÞ are obtained with a

<sup>X</sup>ðIÞ ¼ <sup>1</sup> tint ∫ tint

<sup>Y</sup>ðIÞ ¼ <sup>1</sup> tint ∫ tint

related to temperature drifts.

From these two outputs, the quantities s∝

2.1.2. Case of a non-linear phase modulation (LIA)

the sawtooth edges, especially if the modulation is fast.

long integration time.

interferometer.

Eq. (2), as shown in Figure 2.

a time tint:

214 Optical Interferometry

The main benefit of the G-LIA method is that all the weighted harmonics are used to retrieve phase and amplitude with an operation similar to that of a LIA. To introduce this method, we also remark that the interferometric term Imod ∝ P s cos (Δφ(t)) in Eq. (2) can be expressed as Imod ∝ P s [cos (φs) C(t) + sin (φs) S(t)], with C(t) = cos(φR) and S(t) = sin(φR). From this expression, it appears that C(t) and S(t) can be used as relevant reference signals within a modified LIA having the two following outputs:

$$X\_{\phi R}(I\_{mod}) = \frac{1}{t\_{int}} \mathfrak{f}\_0^{t\_{int}} I\_{mod} \mathbb{C}(t) \, dt = < I\_{mod} \mathbb{C}(t) > \,. \tag{8}$$

$$Y\_{\phi R}(I\_{mod}) = \frac{1}{t\_{int}} \mathbf{f}\_0^{t\_{int}} I\_{mod} S(t)dt = < I\_{mod} S(t)>\,,\tag{9}$$

These references contain the same frequency components than the interferometric term since Imod(t) is a function of C(t) and S(t). These frequency components are naturally weighted so that the contribution of stronger harmonics will be favored. Figure 2(b) exemplifies the case where the phase modulation function is a sine function.

We note that in the particular case where φr(t) = Ωt, C(t) = cos(Ωt) and S(t)=sin(Ωt) and the G-LIA operation degenerates to that of a standard LIA. More generally, by replacing the expression of Imod in Eqs. (8) and (9), we see that for any phase modulation we have:

$$X\_{\phi R}(I\_{\text{mod}}) \propto \mathcal{S}[k\_{\text{x}} \ast \cos \left(\phi\_{\text{s}}\right) + \dot{k\_{\text{x}}} \ast \sin \left(\phi\_{\text{s}}\right)],\tag{10}$$

$$Y\_{\phi R}(I\_{md})\infty[\dot{k\_y}^{'} \* \cos\left(\phi\_s\right) + k\_y \* \sin\left(\phi\_s\right)],\tag{11}$$

where kx = < C<sup>2</sup> (t) >, k′ <sup>x</sup> <sup>¼</sup> <sup>k</sup>′ <sup>y</sup> <sup>¼</sup> <sup>&</sup>lt; <sup>C</sup>ðtÞSðt<sup>Þ</sup> <sup>&</sup>gt;, and ky <sup>¼</sup> <sup>&</sup>lt; <sup>S</sup><sup>2</sup> ðtÞ > are constants that can be calculated numerically or analytically for the considered phase modulation. But for most of the phase modulation functions that can be used C(t) and S(t) are orthogonal, that is, k′ <sup>x</sup> ¼ k ′ <sup>y</sup> ¼ < SðtÞCðtÞ >¼ 0, so that the Xφ<sup>R</sup> and Yφ<sup>R</sup> outputs are:

$$X\_{\phi R}(I\_{mod})\infty k\_x \cos\left(\phi\_s\right),\tag{12}$$

$$Y\_{\phi R}(I\_{mod})\infty k\_{\mathcal{Y}}\sin\left(\phi\_s\right),\tag{13}$$

The G-LIA outputs are then similar to that of the LIA in the linear case (cf. Eqs. (4)–(5)). The difference is the presence of the additional proportionality constants kx and ky which need to be evaluated by calculating the average values <C<sup>2</sup> (t)> and <S<sup>2</sup> (t)>, respectively. In the case of sine phase modulation φ<sup>R</sup> = a sin Ωt, the constants have analytical expressions obtained from the integral representations of Bessel functions and simple trigonometric developments:

$$k\_x = 1 + l\_0(2a) \tag{14}$$

$$k\_y = 1 + l\_0(2a) \tag{15}$$

where J<sup>0</sup> is the Bessel function of first kind. However, it is difficult to extract Imod from I(t) to feed the G-LIA input as it requires to precisely remove the signal and reference field intensities from the detected intensity I(t). In general, it is not possible either to use directly I(t) to perform the G-LIA operation described above. The reason is that for phase modulation function such as sine or triangle, the useful term Imod also contains a DC component. In consequence the references C(t) and/or S(t) also contain a DC term so that the constant, non-interferometric term P in Eq. (2) is also detected by the G-LIA if I(t) is used directly. To avoid this problem, specific phase modulation depths for which both C(t) and S(t) do not have a DC component can be used.<sup>2</sup> For a sine phase modulation, this specific depth of modulation a corresponds to the zeros of the J0(a), for example, a = 2.405 rad. We note that there is no prejudice in term of signal to noise ratio using this a since all the harmonic contents is detected by the G-LIA operation.

<sup>X</sup>φ<sup>R</sup>ðImodÞ ¼ <sup>1</sup>

<sup>Y</sup>φ<sup>R</sup>ðImodÞ ¼ <sup>1</sup>

the phase modulation function is a sine function.

(t) >, k′

<sup>x</sup> <sup>¼</sup> <sup>k</sup>′

evaluated by calculating the average values <C<sup>2</sup>

where kx = < C<sup>2</sup>

216 Optical Interferometry

k′ <sup>x</sup> ¼ k ′ tint ∫ tint

tint ∫ tint

These references contain the same frequency components than the interferometric term since Imod(t) is a function of C(t) and S(t). These frequency components are naturally weighted so that the contribution of stronger harmonics will be favored. Figure 2(b) exemplifies the case where

We note that in the particular case where φr(t) = Ωt, C(t) = cos(Ωt) and S(t)=sin(Ωt) and the G-LIA operation degenerates to that of a standard LIA. More generally, by replacing the

<sup>y</sup> <sup>¼</sup> <sup>&</sup>lt; <sup>C</sup>ðtÞSðt<sup>Þ</sup> <sup>&</sup>gt;, and ky <sup>¼</sup> <sup>&</sup>lt; <sup>S</sup><sup>2</sup>

calculated numerically or analytically for the considered phase modulation. But for most of the phase modulation functions that can be used C(t) and S(t) are orthogonal, that is,

The G-LIA outputs are then similar to that of the LIA in the linear case (cf. Eqs. (4)–(5)). The difference is the presence of the additional proportionality constants kx and ky which need to be

phase modulation φ<sup>R</sup> = a sin Ωt, the constants have analytical expressions obtained from the

where J<sup>0</sup> is the Bessel function of first kind. However, it is difficult to extract Imod from I(t) to feed the G-LIA input as it requires to precisely remove the signal and reference field intensities from the detected intensity I(t). In general, it is not possible either to use directly I(t) to perform the G-LIA operation described above. The reason is that for phase modulation function such as sine or triangle, the useful term Imod also contains a DC component. In consequence the references C(t) and/or S(t) also contain a DC term so that the constant, non-interferometric term P in Eq. (2) is also detected by the G-LIA if I(t) is used directly. To avoid this problem, specific phase modulation depths for which both C(t) and S(t) do not have a DC component

integral representations of Bessel functions and simple trigonometric developments:

(t)> and <S<sup>2</sup>

expression of Imod in Eqs. (8) and (9), we see that for any phase modulation we have:

<sup>X</sup>φ<sup>R</sup>ðImodÞ∝S½kx � cos <sup>ð</sup>φsÞ þ <sup>k</sup>′

′

Yφ<sup>R</sup>ðImodÞ∝s½k

<sup>y</sup> ¼ < SðtÞCðtÞ >¼ 0, so that the Xφ<sup>R</sup> and Yφ<sup>R</sup> outputs are:

<sup>0</sup> ImodCðtÞ dt ¼< ImodCðtÞ > , (8)

<sup>0</sup> ImodSðtÞdt ¼< ImodSðtÞ > , (9)

<sup>y</sup> � cos ðφsÞ þ ky � sin ðφsÞ�, (11)

Xφ<sup>R</sup>ðImodÞ∝skx cos ðφsÞ, (12)

Yφ<sup>R</sup>ðImodÞ∝sky sin ðφsÞ, (13)

kx ¼ 1 þ J0ð2aÞ (14)

ky ¼ 1 þ J0ð2aÞ (15)

<sup>x</sup> � sin ðφsÞ�, (10)

ðtÞ > are constants that can be

(t)>, respectively. In the case of sine

Alternately, a satisfactory solution is to filter the detected intensity to remove all DC component from the signal. In fact, such operation is easy to do and is often highly desirable to directly remove the ambient light contribution in normal conditions [12]. In this case where the signal is filtered, the G-LIA operation is:

$$X\_{\phi R}(\check{I}) = <\check{I} \text{ C}(t) > \infty \,\mathrm{K}\_x \cos \left(\phi\_s\right), \tag{16}$$

$$Y\_{\phi \mathbb{R}}(\tilde{I}) = <\tilde{I}S(t)>\infty \, K\_y \sin \left(\phi\_s\right),\tag{17}$$

where the <sup>Ĩ</sup> denotes a DC-filtered quantity,3 that is, <sup>~</sup>IðtÞ ¼ <sup>I</sup>ðt<sup>Þ</sup> <sup>−</sup> <sup>&</sup>lt; <sup>I</sup>ðt<sup>Þ</sup> <sup>&</sup>gt;. The new proportionality constants are Kx <sup>¼</sup> <sup>&</sup>lt; <sup>C</sup><sup>~</sup> <sup>2</sup> <sup>ð</sup>t<sup>Þ</sup> <sup>&</sup>gt; and Ky <sup>¼</sup> <sup>&</sup>lt; <sup>S</sup>~<sup>2</sup> <sup>ð</sup>t<sup>Þ</sup> <sup>&</sup>gt;, where <sup>C</sup><sup>~</sup> <sup>ð</sup>tÞ ¼ <sup>C</sup>ðt<sup>Þ</sup> <sup>−</sup> <sup>&</sup>lt; <sup>C</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; and <sup>S</sup>~ðtÞ ¼ <sup>C</sup>ðt<sup>Þ</sup> <sup>−</sup> <sup>&</sup>lt; <sup>C</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; are used to evaluate these proportionality factors numerically or analytically.4 Amplitude and phase are then provided by:

$$\text{loss}\sqrt{\mathcal{X}\_{\phi R}^2(\tilde{I})/\mathcal{K}\_x^2} + y\_{\phi R}^2(\tilde{I})/\mathcal{K}\_y^2 \tag{18}$$

$$\phi\_s = a \tan 2 \left( \mathbf{X}\_{\phi \mathbb{R}}(\tilde{\mathbf{I}})/\mathcal{K}\_{\mathbf{x}}, \mathbf{Y}\_{\phi \mathbb{R}}(\tilde{\mathbf{I}})/\mathcal{K}\_{\mathbf{y}} \right) \tag{19}$$

In the useful case of a sine modulation of the form φ<sup>R</sup> = a sin Ωt, the constants Kx and Ky have the following analytical expressions:

$$K\_X = k\_x - f\_0^2(a) = 1 + f\_0(2a) - f\_0^2(a) \tag{20}$$

$$K\_Y = k\_y = 1 - f\_0(2a) \tag{21}$$

<sup>2</sup> Amplitude and phase are then determined using: s∝ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 <sup>φ</sup><sup>R</sup>ðIÞ=k 2 <sup>x</sup> <sup>þ</sup> <sup>Y</sup><sup>2</sup> φR ðIÞ=k 2 y q and <sup>φ</sup><sup>s</sup> <sup>¼</sup> a tan2ðX<sup>φ</sup><sup>R</sup> <sup>ð</sup>IÞ=k<sup>2</sup> <sup>x</sup>,Y<sup>φ</sup><sup>R</sup> <sup>ð</sup>IÞ=k<sup>2</sup> yÞ.

<sup>3</sup> An analog filter can be used. Alternately, it is possible to filter the DC component of the reference functions C(t) and S(t) only, or to filter both I and the references, with the same result. The operations <sup>&</sup>lt; <sup>~</sup>ICðt<sup>Þ</sup> <sup>&</sup>gt; , <sup>&</sup>lt; IC<sup>~</sup> <sup>ð</sup>t<sup>Þ</sup> <sup>&</sup>gt; and <sup>&</sup>lt; <sup>~</sup>IC<sup>~</sup> <sup>ð</sup>t<sup>Þ</sup> <sup>&</sup>gt; are theoretically equivalent. The interest of filtering both the signal and the references is that if the system operates at small modulation frequencies some filters may create a distortion of the modulated signal by changing the amplitudes of peaks and by creating phase shifts for the lowest frequency components. By filtering both the references and the signal, the distortion is similar for both the signal and references so that the distortion effect is cancelled out.

<sup>4</sup> A comment should also be made regarding the references C(t) and S(t). Building these references require the knowledge of φ ¼ a sin Ωt. In a number of setup φR can be monitored with sensors and it is then possible to take the sine and cosine of this quantity. The references can also be built numerically from the knowledge of the modulation depth a and frequency Ω, but C(t) and S(t) must be synchronized with φ<sup>R</sup> ¼ a sin Ωt. In other word, we should not use an ersatz φ′ <sup>R</sup> ¼ a sin ðΩt þ φoÞ as an argument for CðtÞ ¼ cosðφRÞ and SðtÞ ¼ sinðφRÞ. If a phase shit φ0 exists, a phase adjustment of the references or the modulation drive signal can be made. This phase shift can be measured by the phase output of a standard LIA locked at the frequency Ω.

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 numerically without difficulty for a variety of phase modulation functions.

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 calculated values.
