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We have proposed an approach to the interference phase extraction in the homodyne laser interferometry. The method employs a series of computational steps to reconstruct the signals for quadrature detection from an interference signal from a non-polarising interferometer sampled by a simple photodetector. The complexity trade-off is the use of laser beam with frequency modulation capability. It is analytically derived and its validity and performance is experimentally verified. The method has proven to be a feasible alternative for the traditional homodyne detection since it performs with comparable accuracy, especially where the optical setup complexity is principal issue and the modulation of laser beam is not a heavy burden (e.g., in multi-axis sensor or laser diode based systems).

The laser interferometry represents the most precise technique for measurement of geometrical quantities today. Among currently known methods, it provides the best resolution and accuracy and ultimate (theoretically an infinite) dynamic range. The basic resolution of the laser interferometer corresponds to the wavelength of the laser source, but for many applications such an accuracy is insufficient. The resolution is further improved by various methods, commonly referred to as fringe subdivision or detection techniques [

Generally speaking, the detection methods usually expand the basic concept of interferometric measurement by adding extra information to the interference signal that helps to extract the phase information with an increased precision [

In this paper, we present a novel detection technique, related to a traditional homodyne detection, that relies on a digital computation instead of rather complex optical processing. It employs a frequency modulated laser source on the input of a non-polarising laser interferometer and a simple photodetector that observes the output. Then a combination of synchronous demodulation, linear transforms, mixing, phase shifting and trigonometric transforms is used to reconstruct the same pair of quadrature signals. The detection is further improved by a scale linearization techniques that mitigate the measurement error introduced when the method is applied in a real measurement system.

The presented approach brings several advantages, especially reduced demands on the precise construction, reliability and cost of laser interferometry based measurement systems. Our aim was not to present a novel technique that would allow for better precision, accuracy or bandwidth—especially the bandwidth is limited by the frequency modulation bandwidth of the laser source (we deal with DC—100 Hz in our experimentation). Instead, we proposed a method that achieves a performance comparable to that of the current methods,

The rest of this paper is organised as follows. Section 2 summarises the principles of the homodyne detection and presents an analytic description of the proposed method. Section 3 describes the experiments we have carried out to validate the performance of our method in comparison with a homodyne detection, while Section 4 presents the results and holds the discussion. Finally, Section 5 summarises and concludes the paper.

In a typical Michelson setup with homodyne detection, the interferometer setup contains a polarising beam splitter to produce the measurement and reference beams with a different polarisation plane. As a result, there is a complex light beam compound from the waves with mutually perpendicular polarisation on the output of the interferometer rather than a simple interference. That compound beam is further processed by an optoelectronic detection system, usually referred to as a quadrature detection unit (schematically shown in _{x}_{y}

The immediate value of the interference phase _{x}_{y}

One of our objectives was to propose a method that would be simpler in terms of optical hardware complexity. This complexity is represented by, e.g., the polarising beam splitters, the corner reflectors, semi-transparent mirrors and the demands for optical system adjustment. This is why using exclusively non-polarising interferometer with planar reflectors and a simple photodetector can be considered interesting direction towards cost-effective measurement systems, despite the necessity of laser beam modulation. Several extensions of the basic scheme of the homodyne detection are currently used in modern measurement systems, e.g., the use of planar mirrors [

The next section presents the analytical description of the interference phenomenon and the signals produced by the quadrature detection unit. Note that the laser beam is considered as a set of monochromatic plane waves in vacuum since an ideal source of coherent light is assumed.

The electromagnetic wave generated by a laser source can be described at a given point by a vector of electric field

The electric field at the source is given by the real part of the complex function
_{0} is amplitude and

Hence the phase can be calculated from the known angular frequency as

The propagation of the wave from the source to the point

The intensity observed by a detector placed at the point

Now assume there are two waves propagating along a single axis (practically representing two waves that travel along two arms of the interferometer). At the output of the interferometer the two waves are recombined, the interference occurs and the compound wave, usually denoted as an interference signal, is observed. Let _{1} be the distance travelled from the source to the detector by the wave that propagates through the reference arm and _{2} the distance through the measurement arm. Then the electric field of the compound wave at the detector is
_{1} = _{1}/_{2} = _{2}/

The intensity at the point of observation can be calculated using

Changing the variables _{1} → _{2} − _{1} =

We see that the observed intensity depends only on amplitudes of two interfering waves and their phase difference. The latter is caused by the path difference _{2} − _{1} between the paths travelled by two interfering waves.

When the laser source works on single frequency _{0} is the initial phase. Similarly, the shifted phase of the second wave is

The intensity _{x}_{2} − _{1})/_{2} of the measurement arm.

The intensity _{y}_{x}

On the basis of analytic model of the quadrature signals, the next subsection presents the approach to computational reconstruction of quadrature signals from a signal observed by simple detector.

As mentioned above, we employ a frequency modulated laser beam instead of a monochromatic one. Similar approach has been already employed, e.g., in phase-shifting and phase-modulation based interferometry [

Particularly, we apply the sinusoidal frequency modulation. First, we denote the mean value of angular frequency of the wave _{s}_{m}

Similarly, the phase of the second wave shifted by the time delay

The intensity _{x}

This expression can be further simplified on the basis of the following considerations. Due to the typical dimensions of commonly used interferometers and a large value of the speed of light, the time delay

The phase difference is then

Inserting this expression into _{x}

The intensity _{y}_{x}

We see that the observed intensity is variable even if the time delay

The intensities _{x}_{y}_{m}

Since the absolute value _{1} + _{2} and the factor
_{x}_{y}_{x}_{y}

Then we start with normalised intensities

Using standard formulae for trigonometric functions we can rewrite them to

Since the maximum absolute value of the term sin(Ω_{m}τ

Both observed intensities _{x}_{y}_{x}_{y}

The first term of _{x}_{s}τ_{s}_{x}_{m}τ_{s}τ_{m}τ_{x}

For further considerations we make the following assumption. When the mirror in measurement arm is moving we demand that the function sin(Ω_{s}τ

For illustration, considering the motion of the mirror in measurement arm with a constant speed _{s}v

The mean values 〈_{x}_{y}_{s}τ_{s}τ

We see that under the stated assumptions, the mean values of observed intensities when the frequency is modulated are identical to the intensities observed when the frequency is constant, see

Now we would like to show the procedure that allows us to calculate the mean value of intensity 〈_{y}_{x}_{x}

Calculation of their difference Δ_{x}

Next, multiplying the difference Δ_{x}

Finally, from comparison with _{d}

We see that the derived signal _{d}_{y}_{y}_{x}

The last part of the novel detection method, presented in the final part of this section, addresses the issues with the practical implementation.

We have identified several sources of inaccuracies related to a practical implementation of the proposed method:

the measured intensities _{x}_{y}

the currently available laser sources are liable to a residual amplitude modulation (RAM) [

the computation over discrete representation of the analog signals causes additional inaccuracy, e.g., numeric calculation of derivation, round-off error.

Note that the list of influences is far from being complete, nonetheless these were the most significant ones that we met and found important to be addressed.

Currently, the problems with scale linearity (i) are well explored [

The RAM effect (ii) is superimposed onto the effect of the laser frequency modulation. As a consequence, there occurs an additional phase shift between _{x}_{y}

The phase shift is typically corrected by the scale linearization techniques, nonetheless our preliminary experimentation indicated that a compensation of the phase shift between the quadrature signals before the elliptic-fitting based linearization technique leads to better overall performance of our method. On the basis of this experience, we have employed both techniques: the phase shift removal is described in the following section and for the scale linearization technique we used an existing method [

The principle of our phase shift removal method is the following: after calculation of _{y}_{x}_{a}_{y}

The following description presents the method for a general case, _{x}_{y}

We start again with intensities _{x}_{y}_{x}, K_{y}_{x}, I_{y}_{x}_{y}_{xy}_{x}_{y}

The value _{xy}

We illustrate the functions _{x}_{y}_{x}_{y}_{x}_{y}_{a}_{x}_{y}_{x}_{y}

The geometrical meaning of quantity _{xy}_{x}_{y}_{xy}_{x}_{y}_{x}, K_{y}, K_{xy}_{a}_{a}

It should be emphasised that the function _{xy}_{a}_{a}_{x}_{y}_{xy}_{a}_{a}_{xy}_{x}, K_{y}_{a}

For this purpose we now need to transform intensities _{x}, I_{y}

Here
_{x}_{x}, I_{y}_{a}

Solving the second equation with respect to

The transformed intensities
_{a}

Note that the algorithm is principally based on a single data-point determined from the measured data (maximum value of _{x}, I_{y}_{x}_{y}_{a}

We have carried out several experiments to verify the performance of the proposed technique. We have assembled an optical setup that allowed us to make the displacement measurements at the nanometre scale referenced to an interferometer with the homodyne detection technique,

The reference was physically provided by a calibrated quadrature detection unit in combination with the support logic. The reference delivers the reference phase information with measurement uncertainty of

The experimental setup, schematically shown in

A solid-state laser (frequency doubled Nd:YAG at 532 nm with linearly polarised beam; PROMETHEUS by Innolight, Germany) was used. The output beam was collimated to particular diameter (3 mm) that optimally fits the detection unit and the photodetector. A half-wave plate (

A non-polarising splitter (NP) is used to form two detection branches: the testing branch that involves our novel detection method with an ordinary photodetector (PD) and the reference one that uses the traditional detection unit (denoted QDU).

The bandwidth of both detection methods is limited by the combination of several factors: response of the photodetector, signal amplifiers, analog-to-digital conversion sample rate and processing capability. In our particular case, the available bandwidth was 50 kHz for the QDU and 100 Hz for the novel method (

The signal generation (for laser source modulation) and acquisition was done by a dedicated DAQ device (U2531 by Agilent). The generation part produced synchronised sine and cosine wave—the sine controlled the modulation, the cosine was synchronously sampled (acquired) together with the output of the QDU and the PD output and used for the quadrature signal reconstruction. The reason to generate the modulation signal and its phase shifted copy and then sample the copy again was to achieve precise synchronisation between the generated outputs and signal inputs.

The measurement arm hardware was successively placed into different positions (measurement points)

The outputs from the quadrature detection unit _{x}, I_{y}_{p}_{in}

We have inspected the difference between the interference phase information extracted by the reference and by the testing method, referred to as the phase determination error (PDE). In the reference branch, the X- and Y-axis signals from the quadrature detection unit were recorded and the scale linearization was applied. The photodetector output _{p}_{x}

A total of 60 measurement cycles in 20 positions was completed, where each cycle contains 890 phase readings. A sample evaluation of a single cycle is shown in

There is an apparently good coincidence between the quadrature signals from reference and testing detection branch (compare

The linearity of our method is summarised in

Several related issues were omitted within the experimentation, e.g., the refraction index of air influence [

The results indicate that the proposed method is a suitable alternative to the traditional homodyne detection, especially in situation when the accuracy is not the most critical aim but the construction simplicity and scalability is. There are several issues that need to be considered to judge the novel approach: its dynamic range, the trade-off between optical complexity and the requirement for laser modulation, the response bandwidth limitations and the uncertainty introduced by the computations.

As the homodyne detection has theoretically infinite dynamic range, it allows for the measurement with the same precision even on long distances (several kilometres). The practical limitation is given only by the construction issues, laser stability and influences of the environment. The novel method is principally limited by the available depth of laser frequency modulation [

There are also higher demands posed on the laser source—there is the need for the laser beam frequency modulation. The modulation bandwidth limits the response of the measurement systems since the duration of at least one complete modulation period (five in case of our experiments) is processed to produce a single phase reading. Consequently, velocity of measurement is significantly slower than the optical processing. The demand for laser with a fast modulation capability can be fulfilled, to a certain extent, by cost-effective semiconductor lasers [

On the other hand, the method requires less complex optical setup to achieve comparable results—there is no need for polarising beam splitters and semi-transparent mirrors within the optical lineup. As a consequence, the method also scales well, e.g., when several measurement axes would be required within a measurement system. In such a case, the added complexity per each added axis is minimal in comparison to the traditional means—it has been already demonstrated in a practical application for surface diagnostics [

In our experiments, the modulation bandwidth was limited by available high-voltage amplifiers to 200 phase readings per seconds (corresponds to 1 kHz modulation). With the particular laser source, a 100 × higher response would be achievable; further enhancement would be possible using and acousto-optic modulator.

In the basic principle, similar computational methods are also sensitive to the noise issues. In case of our novel method, the issue is well mitigated by a massive averaging and suitable signal filtering without increased demands on system performance as the operations are essential parts of the phase extraction method.

There were also several assumptions we stated (in sake of simplicity) in the physical description of the novel approach. We assume the time delay ^{−9} and the term fir Ω^{−6}. We also assume that the product of _{m}τ_{m}_{m}τ_{m}τ_{s}v^{−7}m/s, the Ω = 1kHz, thus for the mean frequency _{s}^{3} ≫ 7.5 × 10^{−6}.

We have proposed and verified a novel detection technique that represents a novel approach to the homodyne detection in laser interferometry.

A detailed analytic description of the principles has been presented and the validity and performance were evaluated by experimental means.

The experimental results indicate that the novel approach performs comparably to the traditional detection technique. Since it requires simpler optical hardware setup—e.g., exclusively a non-polarising interferometer, single photodetector, but frequency modulated laser beam and computational power—it can be considered a suitable alternative to the techniques that rely on additional optical processing.

The authors wish to express thanks for the support of the GACR, project GAP102/10/1813, the research intent RVO: 68081731 and EU supported projects No. CZ. 1.05/2.1.00/01.0017 and CZ. 1.07/2.3.00/30.0054. Experimental tasks were supported by the Ministry of Industry and Commerce, projects FR-TI2/705 and FR-TI1/241.

Polarising laser interferometer with quadrature detection unit for homodyne detection: the interferometer employs the polarising divider, the reference corner cube reflector CC_{ref}_{meas}_{x}_{y}

Detection of the unwanted phase shift _{a}_{x}, I_{y}_{x}, K⃗_{y}_{a}_{x}_{y}_{xy}_{xy}_{a}

Experimental setup scheme: the interferometer employs the polarising divider, the reference corner cube CC_{ref}_{meas}

Evaluation of single experimental cycle: the X- and Y-axis signals from quadrature detection unit, _{x}_{y}_{x}_{d}

Experimental results: the red points show the periodic error, _{i}

Experimental results: the red points show the dependence of the mean phase determination error on the path length difference; the blue error bars indicate the interval of radius of standard deviation around the mean PDE at each corresponding point _{i}