# US 9,518,869 B2 · Electronic method for extracting the amplitude and phase of a signal in a synchronous detector and application thereof in interferometric circuits

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- Number: US 9,518,869 B2
- Document ID: US9518869B2
- Status: Granted
- Kind: B2
- Application number: 14/313,990
- Filed: 2014-06-24
- Published: 2015-03-12
- Granted: 2016-12-13
- Earliest claimed: 2013-09-11
- Assignees: Universite de Technologies de Troyes
- Applicants: Universite De Technologie De Troyes
- Inventors: Aurelien Pierre Bruyant; Julien Vaillant; Abeer Al Mohtar
- Web page: https://patentgrep.com/patent/US9518869B2

## CPC

- G01J 9/02: by interferometric methods (using interferometers for measuring optically the linear dimensions of objects G01B9/02)
- G01B 9/02002: using two or more frequencies
- G01B 9/02041: {characterised by particular imaging or detection techniques}
- G06F 17/10: Complex mathematical operations {(function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544)}
- G01J: MEASUREMENT OF INTENSITY, VELOCITY, SPECTRAL CONTENT, POLARISATION, PHASE OR PULSE CHARACTERISTICS OF INFRARED, VISIBLE OR ULTRAVIOLET LIGHT; COLORIMETRY; RADIATION PYROMETRY (light sources F21, H01J, H01K, H05B; investigating properties of materials by optical means G01N)
- G01B: MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

The invention concerns an electronic method for extracting the amplitude ES and the phase φS of an electrical signal in a synchronous detector, the signal containing a modulated part Imod of the form Imod ∝ ES f(t)*cos(φS−φR(t)), where φR(t) and f(t) are two known temporal modulation functions. The method comprises the following steps: multiplying the signal by two reference signals C(t) and S(t) constructed from φR(t) and f(t); integrating the resulting signals over a time tint; determining the amplitude and phase of said signal from the quantities X and Y resulting from the previous integrations. The method is characterised in that: said electrical signal is multiplied by C(t) and S(t) that can be decomposed on the same set of frequencies as those present in Imod. The invention also concerns the application of the above method in interferometric circuits and the use and execution of the above method.

## Claims

### Claim 1

Electronic method for extracting the amplitude ES and phase φS of an electrical signal Idet in a lock-in amplifier, said electrical signal Idet containing a modulated part Imod of the form Imod∝ES f(t)*cos(φS−φR(t)), where φR(t) and f(t) are two known temporal modulation functions, comprising the following steps:
multiplying said electrical signal by two reference signals C(t) and S(t) constructed from known temporal modulation functions φR(t) and f(t);
integrating the two signals resulting from the two multiplications defined at the previous step during a time tint;
determining the amplitude and phase of said signal from the quantities X and Y resulting from the integrations defined at the previous step;
the method is characterised in that:
said electrical signal is multiplied by two references signals C(t) and S(t) that can be decomposed on the same set of frequencies as those present in Imod,
wherein the electrical signal Idet is detected using a detector coupled to a signal arm and a reference arm that are optically coupled to a light source in an interferometric arrangement.

### Claim 2 (depends on 1)

Electronic extraction method according to claim 1, wherein the two references signals C(t) and S(t) are defined by C∝cos(φR)*f(t) and S∝sin(φR)*f(t).

### Claim 3 (depends on 2)

Electronic extraction method according to claim 2, wherein the temporal amplitude modulation function f(t) in question is periodic of period 2π/ΩA during the integration time tint, or constant during the integration time tint, and wherein the temporal phase modulation function φR(t) in question is periodic during the integration time tint of period 2π/Ω.

### Claim 4 (depends on 3)

Electronic extraction method according to claim 3, wherein the quantities X and Y are linked to ES*cos(φS) and ES*sin(φS) by two proportionality coefficients determined analytically or numerically.

### Claim 5 (depends on 2)

Electronic extraction method according to claim 2, in which the known temporal modulation function φR(t) is sinusoidal.

### Claim 6 (depends on 2)

Electronic extraction method according to claim 2, in which the temporal modulation function is triangular.

### Claim 7 (depends on 1)

Electronic extraction method according to claim 1, in which the known temporal modulation functions φR and f(t) are used for producing the two reference signals making it possible to extract the amplitude and phase of the signal of interest with an optimum signal to noise ratio.

### Claim 8 (depends on 7)

Electronic extraction method according to claim 7, in which the known temporal modulation function φR(t) is exact, imposed or measured.

### Claim 9 (depends on 1)

Electronic extraction method according to claim 1, where the periodic modulation function f(t) in question is reduced to a single sinusoidal component of period 2π/ΩA , without any harmonic.

### Claim 10 (depends on 1)

Method according to claim 1, where the signal Idet and/or the reference signals C(t) and S(t) are filtered of their continuous component or certain frequency components just before the multiplication step.

### Claim 11 (depends on 1)

Execution of the method according to claim 1 in interferometric setups using analog components or digital cards.

### Claim 12 (depends on 1)

Execution of the method according to claim 1 in interferometric setups using software solutions.

### Claim 13

An interferometer comprising:
a light source;
a signal arm optically coupled to the light source to receive light from the light source;
a reference arm optically coupled to the light source to also receive the light from the light source; and
a lock-in amplifier optically coupled via a detector to the signal arm and the reference arm to detect an electrical signal Idet, the electrical signal Idet containing a modulated part Imod of the form Imod∝ES*f(t)*cos(φS−φR(t)), where φR(t) and f(t) are two known temporal modulation functions and where Es and φS correspond to amplitude and phase signals, the lock-in amplifier being programmed to:
multiply said electrical signal by two references signals C(t) and S(t) constructed from known temporal modulation functions φR(t) and f(t);
integrate the two signals resulting from the two multiplications defined at the previous step during a time tint; and
determine the amplitude and phase of said signal from the quantities X and Y resulting from the integrations defined at the previous step,
wherein said electrical signal is multiplied by two references signals C(t) and S(t) that can be decomposed on the same set of frequencies as those present in Imod.

### Claim 14 (depends on 13)

The interferometer according to claim 13, wherein the two reference signals C(t) and S(t) are defined by C∝cos(φR)*f(t) and S∝sin(φR)*f(t).

### Claim 15 (depends on 14)

The interferometer according to claim 14, wherein the temporal amplitude modulation function f(t) in question is periodic of period 2π/ΩA during the integration time tint, or constant during the integration time tint, and wherein the temporal phase modulation function φR(t) in question is periodic during the integration time tint of period 2πΩ.

### Claim 16 (depends on 15)

The interferometer according to claim 15, wherein the quantities X and Y are linked to ES*cos(φS) and ES*sin(φS) by two proportionality coefficients determined analytically or numerically.

### Claim 17 (depends on 13)

The interferometer according to claim 13, comprising furthermore a local probe microscope where the amplitude modulation angular frequency ΩA is a harmonic of the mechanical oscillation angular frequency of the probe.

### Claim 18 (depends on 17)

The interferometer according to claim 17, where the modulation depth of φR is chosen for filtering the contribution of certain non-desired modulated components possibly present in said signal, typically at the angular frequency ΩA and the harmonics thereof.

### Claim 19 (depends on 17)

The interferometer according to claim 17, for collecting information on the local interaction existing between the probe and a sample.

### Claim 20 (depends on 13)

The interferometer according to claim 13, for performing a phase resolved Fourier analysis.