# Smart Laser Interferometer with Electrically Tunable Lenses for Flow Velocity Measurements through Disturbing Interfaces

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## Abstract

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## 1. Motivation

## 2. Velocity Measurements through a Fluctuating Gas-Liquid Interface

#### 2.1. Preconsiderations

_{x}perpendicular to the interference fringe system orientation is resulting to

_{W}as the axial shift of the beam waist position relative to the center of the measurement volume and z

_{R}as the Rayleigh length of the Gaussian laser beams, which is defined by z

_{R}= πw

_{0}

^{2}/λ, where w

_{0}is the radius of laser beam waists [13].

- 0th order: Height or stroke of the air-water interface. For a lift of the interface a parallel shift of the beam occurs whereas the beam direction remains constant. The consequence is a shift of the position of the measurement volume, i.e., a dislocation of the measurement position. Furthermore a signal frequency jitter is introduced, if a stochastic fluctuation of the air-water interface occurs.
- 1st order: Tilt of the air-water interface. Due to refraction, a tilt of the interface will change the propagation direction of the laser beam. Going to the two-dimensional consideration, it has to be distinguished between a tilt in the plane spanned by the two partial beams δ
_{x}and a tilt in the direction normal to this plane δ_{y}.- (a)
- Tilt δ
_{x}in x-direction (tip): On one side a displacement of the measurement position results and on the other side there is a change in the intersection half angle θ of the interfering laser beams. Due to a change of the Rayleigh length z_{R}or the beam waist position z_{w}a variation of interference fringe spacing d results, see Equation (2). The standard deviation σ_{d}represents these variations. Using Equation (1) and the propagation law of statistical independent measurement, uncertainties of the fringe spacing σ_{d}and the signal frequency σ_{f}the relative velocity measurement uncertainty yields to [13]$$\frac{{\sigma}_{{v}_{x}}}{{v}_{x}}=\sqrt{{\left(\frac{{\sigma}_{f}}{f}\right)}^{2}+{\left(\frac{{\sigma}_{d}}{d}\right)}^{2}}$$ - (b)
- Tilt δ
_{y}in y-direction: A beam deflection in the y-direction normal to the plane spanned by the two Gaussian beams will result in skew rays. The reduced overlap of the partial laser beams results in lower interference visibility and signal-to-noise ratio (SNR). Only measurement signals with a sufficient SNR are considered for further evaluation. The corresponding validation rate is given by the ratio between the evaluable and all signals detected. It represents a crucial figure-of-merit of the LDV system, since on the one hand the maximum frequency bandwidth of the velocity fluctuations and on the other the effective measurement time is determined. In the worst case of negligible laser beam overlapping no measurements can be performed at all, represented by a validation rate of zero.

- 2nd order: Parabolic curvature of the interface. Due to refraction, a curvature of the interface induces a lens effect on the beam propagation. It changes the radius of the beam waist w
_{0}and the position z_{W}of the beam waist. As a consequence, the fringe spacing d is changed according to Equation (2), which in general enhances the velocity measurement uncertainty, see Equation (4). - 3nd order and higher orders: Distortions of the surface with high spatial frequency. The wavefront of the laser beams will be locally distorted, leading to inhomogeneities in the interference fringes. The fringe spacing can vary in all three directions. In consequence, the measurement uncertainty will increase, see Equation (4).

#### 2.2. Smart Laser Interferometer with Electrically Tunable Lenses

- A high defocus range of up to ±40 diopters can be covered.
- A 3 dB temporal bandwidth of over 60 Hz. It should be high enough to correct the distortions of the capillary waves at the current experiment.
- Due to its outer size of 20 mm diameter the tunable lens is ideal for being integrated into compact sensors.

- The focal lengths of the tunable lens for the first and second partial beams
- The deflections of the mirror in x-direction for the first and second partial beams
- The deflections of the mirror in y-direction for the first and second partial beams

**Figure 1.**Smart laser Doppler velocimeter (LDV) interferometer with two implemented electrically tunable lenses (TL) and two two-axis galvanometer mirrors (GM). The smart LDV compensates the wavefront distortions caused by the fluctuating air-water interface. A twin control loop was realized to control both partial laser beams simultaneously. In the overlapping area of the two laser beams a fringe system is generated. The scattered light from tracer particles moving through this fringe system is measured by the photodetector (PD). In result the flow velocity v is determined.

#### 2.3. Performance of the Distortion Correction

_{0}. According to Equation (1) the measured Doppler signal frequency is given by f = v

_{0}/d. In conclusion, variations of the signal frequency are caused mainly by changes of the fringe spacing d and therefore by the optical distortions from the fluctuating air-water interface. This procedure allows us to study the improvements by the used AO system, see Figure 2 and Figure 3.

**Figure 2.**Mean validation rate with and without distortion correction by the tunable lens (TL) in dependence of the mean amplitude of the distortion, i.e., height of the surface water wave.

**Figure 3.**Relative standard deviation of the velocity measurement with and without distortion correction by the tunable lens (TL) as a function of the mean amplitude of the distortion.

#### 2.4. Flow Velocity Measurements

**Figure 4.**Measurement of the flow profile of a submerged water nozzle by the smart LDV interferometer. The experiments have been conducted for a distortion amplitude, i.e., mean surface wave height of 50 µm. Both laser beams pass the free water surface, see Figure 1. They are controlled both by an adaptive optics (AO) system using electrically tunable lenses and galvanometer mirrors. (

**a**) Profile of the mean velocity with the velocity standard deviation as uncertainty bars. The velocity standard deviation is dominated mainly by the flow turbulence; (

**b**) Mean interference contrast; (

**c**) mean validation rate and (

**d**) confidence interval, all shown as a function of the position alongside the nozzle cross-section.

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Czarske, J.W.; Radner, H.; Leithold, C.; Büttner, L.
Smart Laser Interferometer with Electrically Tunable Lenses for Flow Velocity Measurements through Disturbing Interfaces. *Photonics* **2015**, *2*, 1-12.
https://doi.org/10.3390/photonics2010001

**AMA Style**

Czarske JW, Radner H, Leithold C, Büttner L.
Smart Laser Interferometer with Electrically Tunable Lenses for Flow Velocity Measurements through Disturbing Interfaces. *Photonics*. 2015; 2(1):1-12.
https://doi.org/10.3390/photonics2010001

**Chicago/Turabian Style**

Czarske, Jürgen W., Hannes Radner, Christoph Leithold, and Lars Büttner.
2015. "Smart Laser Interferometer with Electrically Tunable Lenses for Flow Velocity Measurements through Disturbing Interfaces" *Photonics* 2, no. 1: 1-12.
https://doi.org/10.3390/photonics2010001