1. Introduction

Laser Doppler Velocimetry (LDV) is a well established, minimally invasive flow measurement technique, capable of providing high velocity resolution data over a large range of flow conditions. LDV technique offers significant advantages over conventional electromechanical velocimeters because it can operate in a non-contact manner. Conceived in mid 1960’s, by Yeh and Cumming [1], the LDV technique has found widespread applications in various scientific fields, especially in the field of medicine, where it can be used for blood flow measurements [2–4]. The field of LDV witnessed a rapid development in the 1980’s, aided by the advances made in optical communication industry. A number of LDV configurations have been reported and LDV since then has developed into a major industry [4–9].

A typical choice of laser sources for LDV is either a He-Ne laser or a semiconductor diode laser. He-Ne lasers can provide a narrow spectral width, but have rather limited penetration depth (due to high absorption and scattering losses), and are not entirely safe for imaging flow of retinal blood vessels. Laser diodes on the other hand are compact, and efficient, but have a rather broad spectral width (typically in MHz), resulting in limited velocity resolution. This is a critical problem, since the spectral width is directly related to the resolution of velocity measurement. An all-fiber laser source can offer the combined advantages of both He-Ne and semiconductor diode lasers, as it can be compact, efficient, ultra-narrow spectral width, and can provide higher levels of safety as well as greater penetration depth by operating in the eye-safe regime (~1550 nm). Recent works in all-fiber lasers have demonstrated linewidths as narrow as 1 kHz [10], which theoretically could provide a velocity resolution of ~0.0008 m/sec. To the best of our knowledge, no one has ever used an all fiber laser source for LDV. In our previous work, we demonstrated a tunable, stable, ultra-narrow linewidth, single longitudinal mode operations in an all-fiber laser system [11]. In this letter, we propose and experimentally demonstrate a novel fiber-optic confocal laser Doppler velocimeter design using the previously described ultra-narrow linewidth, all-fiber laser source. The confocal design can provide an extremely high longitudinal resolution (<5μm), thereby allowing the possibility of exceptionally precise measurements of spatial blood flow velocity distribution.

2. Experimental setup and theory

In a conventional LDV sensor, the moving sample is illuminated by a monochromatic light and the backscattered Doppler shifted light signal is mixed with a reference beam on a photodetector. The measured frequency shift, Δν(x, y, z) , is related to the velocity distribution function of the sample, V(x, y,z) , by the following dependence

where θ is the angle between the incident and viewing direction, λ is the wavelength of the monochromatic laser output, and ϕ is the angle between the sample velocity vector and the bisector of the viewing and incident direction. However, in our common path design where the incident and viewing direction is same and the laser beam direction vector is same as the velocity vector, the above expression simply reduces to

In our LDV scheme, we have used a common path approach and hence the reference beam is derived from the 4% Fresnel reflection from the cleaved fiber end, similar to the approach used by Nishihara et al. [5]. While this scheme considerably eases the alignment issues, and provides common mode rejection, it faces the shortcoming of the working distance being limited to half the coherence length of the source [12]. This might be a crucial issue if working with semiconductor laser diodes, however, the working distance can be greatly enhanced by at least two orders of magnitude if we use a narrow linewidth fiber laser (and hence longer coherence length).

 

Fig. 1. Schematic configuration of the fiber optic confocal Doppler flowmeter using a narrow linewidth all-fiber laser source.

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The schematic layout of our proposed confocal laser Doppler velocimeter using a fiber laser source is shown in Fig. 1. The all-fiber laser is based on a standard Erbium-doped fiber (EDF) ring laser setup, with two Sagnac loop filters providing a polarization independent, tunable, and stable filter function to obtain a narrow linewidth lasing source [11]. He-Ne laser is used to align the confocal optics. An isolator is provided at the output of the fiber laser to prevent any feedback. The output of the fiber laser is then modulated by an electro-optic modulator (at around 5 MHz), which acts as a frequency shifter and allows the system to differentiate between the forward and reverse motion. The modulated output is then fed into the port 1 of the optical circulator and exits at port 2 to the probe arm. The reference is provided by the partial back reflection (Fresnel reflection) at the fiber-tip/air interface. The diverging light exiting from the fiber is collimated by a microscopic objective lens and then focused on to the sample by another microscopic objective lens. We used a small electro-acoustic transducer (sample with a partially reflective membrane) driven by a sinusoidal RF signal to Doppler shift the incident light. Light scattered from the vibrating membrane of the transducer is collected by the microscopic objective and then coupled back into the fiber probe arm. The combined Doppler shifted signal as well as the reference light is fed back into the circulator at the port 2, and then routed to a photodetector via port 3. The beat frequencies caused by the Doppler shift are detected by an RF spectrum analyzer with a full scan range of 3 GHz.

The confocal design has the capability of measuring the velocity of a very small sample volume. The relative amplitude distribution function of the Doppler shifted signal generated by a moving sample is sampled by a tightly focused Gaussian beam. Besides offering extremely high spatial resolution in lateral as well as in longitudinal direction, the confocal design is much more effective in collecting the backscattered signal as compared to the designs where a single high numerical aperture lens is used. The confocal design is capable of optimizing the signal collection by matching the numerical apertures of the single mode fiber (SMF) and the microscopic objective at the non focusing end. The circulator based design is easy to implement and its directional property can also circumvent any feedback to the laser source. It is power conserving compared to the conventional 50/50 coupler based designs, where 75% of the laser output power is left unutilized. Although circulator based design offers a limited option to tune the power levels of the reference signal, it can be optimized by using fiber tips coated with low reflectivity coatings.

3. Performance analysis and measurement results

The confocal LDV performance is studied for various combinations of microscopic objective lenses. We used a 10X microscopic objective lens to collimate the beam, and a series of 5X, 10X, 20X, and 40X microscopic objective lens, respectively, on the focusing end. In order to find out the longitudinal resolution, we did a simple experiment by introducing a mechanical chopper between the two microscopic objectives and using a mirror on a translational stage as a sample. The confocal design would allow a good collection of signal only within the Rayleigh range region at the beam waist. After aligning the mirror normal to the beam direction, we translated the mirror in longitudinal direction and measured the signal level as a function of distance from the beam waist (f ₂-L ₂) for various combinations of microscopic objectives on the other end. The solid curves in the Fig. 2 show the experimentally measured normalized signal power as a function of longitudinal displacement from the beam waist. Due to the symmetry, only the positive half is shown. As expected, we observed a large Rayleigh range (>100μm) for a 5X objective lens (Fig. 2(a)) and a small Rayleigh range (<10μm) for a tightly focusing 40X objective lens (Fig. 2(d)). Hence, we can just change the microscopic objective lens at the focusing end to obtain the desired accuracy for precise localized measurements.

It is well known that the light-field distribution of the fundamental mode in a single mode fiber (SMF) can be well approximated by a Gaussian profile [13]. Hence, in order to study the system performance and verify our experimental measurements, we perform Gaussian beam propagation analysis by using the ABCD law. Assuming a Gaussian profile of the input beam exiting from the fiber tip, we can calculate, by using the ABCD law, the shape of the return beam at the same fiber tip after it is reflected by the sample membrane. The q-parameters describing the input and output Gaussian beams of the optical system outlined by the dashed rectangle in Fig. 1 are related to each other by

where A, B, C and D are the elements of the matrix describing the confocal optical system, and are found to be

where f ₁ and f ₂ are the focal lengths of the two microscopic objective lenses, L ₁ is the distance between the two lenses, and L ₂ is the distance between the sample membrane and the f ₂ lens (Fig. 1). Simplifications to the above expressions can be achieved if we assume L ₁= f ₁+ f ₂:

Because the input beam exiting from the fiber tip has a flat wavefront, q₁ can be written as:

where w ₀ and R ₁ are the beam waist and the radius of curvature of the input Gaussian beam, respectively. By substituting Eqs. (5–6) into (3), the q-parameter of the return beam can be easily written as

Since we also have

by comparing the real and imaginary parts of Eqs. (7) and (8), we can obtain the radius of curvature R ₂ and the waist w ₂ of the output beam, respectively:

For a well aligned confocal optical system, the optical power collected by the fiber tip can be approximated as the integral of the beam intensity over the fiber core area at the fiber tip plane.

where P ₀ is the power carried by the beam, R_s is the sample reflectivity, and s is the circular fiber core area with radius r ₀ . The radius of input beam exiting from the fiber tip can be approximated to be equal to the fiber core radius (r ₀=w ₀). The power collection will be maximum when the sample lies in the focal plane (f ₂ = L ₂ and w ₂ = w ₀). The normalized power collection ratio (PCR) can be calculated as follows:

We used this equation to fit the experimental data in Fig. 2. The simulation results (dashed curves) shown in Fig. 2 agree very well with the experimental results (solid curves).

 

Fig. 2. Experimental (solid) and simulation (dashed) results of longitudinal resolution (FWHM) of the fiber optic confocal setup for various combinations of microscopic objective lenses. The measured FWHM was found to be (a) 5X: 102 μm, (b) 10X: 28.4 μm , (c) 20X: 11 μm, and (d) 40X: 4.4 μm, respectively

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The spectral bandwidth of our fiber laser is measured to be less than 10 kHz, which corresponds to a velocity measurement resolution of 0.0075m/s. To obtain Doppler shifted measurements, we first used a 5X objective lens to focus the beam and replaced the mirror with a tiny electro-acoustic transducer. The transducer is placed in such a manner that the position of the beam waist coincides with the center of the transducer membrane. The velocity of the transducer membrane follows a periodic profile (starting from zero velocity at the maximum displacement on one side, passing the equilibrium point with maximum velocity, and then again attaining the zero velocity at the maximum displacement on the other side) as it is driven by a sinusoidal RF signal. Since the size of the beam is much smaller than the size of the acoustic transducer, we can ignore the x and y velocity distribution when the beam is focused on the center of the transducer membrane and the ideal velocity distribution function can be written as

where Δd is the maximum displacement of the transducer membrane. However, we observed during the experiments, the movement of the transducer is distorted and asymmetric. Thus, Eq. (13) is used as a guide for our observation and does not represent the true velocity distribution of the transducer. The light backscattered by the vibrating transducer membrane produced almost uniform amplitude Doppler shifted frequencies over the whole range of transducer membrane motion (Fig. 3). The sharp edge of the Doppler shifted frequency band corresponds to the maximum velocity (V_Max) of the transducer membrane. The span of the Doppler shifted frequencies increased as we increased the voltage of the RF signal driving the transducer while keeping the driving frequency constant at 1 kHz, which shows a direct correlation between the maximum velocity of the membrane and the range of Doppler shifted frequencies (Fig. 3). Figure 3(a) shows the photodetector spectrum in the absence of transducer membrane motion. The maximum velocity can be calculated by using Eq. (2) and the measured maximum Doppler shift. For example, the maximum velocity corresponding to a 470 kHz Doppler shift (Fig. 3(d)) is 0.35m/sec. The corresponding maximum displacement of the transducer membrane was calculated to be around 85 μm.

 

Fig. 3. Doppler shifted frequency measurements of the transducer membrane vibrations at 1 kHz for different levels of applied RF voltage. a) photodetector spectrum in absence of transducer membrane motion. The maximum velocity and the maximum displacement were calculated to be, b) 0.126 m/s, 31 μm, c) 0.215 m/s, 53 μm, and d) 0.35 m/s, 87 μm, respectively.

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When the beam is focused on to the center of the vibrating membrane where the velocity is the maximum, although you obtain the maximum amount of the signal for the Δν_max , other Doppler frequencies get coupled back to the fiber as well because of the varying velocity distribution profile within the finite Rayleigh range. Since the return beam waist w ₂ depends on the transducer membrane position, L ₂, by combining Eq. (10) and (13), one can obtain the Doppler frequency dependent beam waist as shown here.

As one can see from the above Equation, if f ₂ is small compared to Δd, once can obtain a high longitudinal velocity resolution. Based on Eqn. (14), we can obtain the dynamic Doppler frequency dependent PCR where

 

Fig. 4. Doppler frequency spectra of the measured and simulation (thick lines) results. a). 5X objective lens: Doppler shifted frequencies corresponding to the whole range of the transducer membrane motion with almost uniform amplitude are observed. b). 40X objective lens: Doppler shifted frequencies from a highly localized region (<5μm) can be extracted, thereby providing a high longitudinal resolution.

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This Equation is used to fit the Doppler shifted frequency spectrum in Fig. 4, which shows the measured and simulation results (thick lines) of the received signal levels as functions of the Doppler shifted frequencies. They agree with each other amazingly well considering all the simplification assumptions we have made. We can observe that the Rayleigh range (or longitudinal resolution) of the beam focused by a 5X objective lens (Fig. 2(a)) and the maximum displacement of the transducer membrane are of the same order and hence the corresponding Doppler shifted frequency spectrum is fairly uniform over the whole range of membrane motion (Fig. 4(a)). However, if we use a 40X objective, it should be able to provide us with much precise information about the velocity of the sample as a function of displacement. Our assumption is verified as we are able to observe the clearly differentiable peaks in the Doppler shifted frequencies, depending upon the relative position of the focal plane and the transducer membrane (Fig. 4(b)). The peak moved clearly as we varied the distance between the transducer membrane and the focus point of the beam. The measured values of Doppler shifted frequency peaks were used to find out the velocity profile of the transducer membrane as a function of displacement (Fig. 5). The theoretical curve in figure 5 is based on Eq. (13) and it is meant to serve as a guide to eye. As mentioned earlier, we can observe that the velocity distribution of the transducer membrane is slightly distorted and asymmetric.

 

Fig. 5. Theoretical (solid line) and experimental (points with error bars) velocity distribution of the elctroacoustic transducer membrane as a function of displacement from its mean position. The error bars corresponds to the longitudinal resolution of the confocal Doppler velocimeter.

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4. Conclusions and future scope

In summary, we have proposed and demonstrated a novel high resolution fiber optic confocal LDV design based on an ultra-narrow linewidth all-fiber laser source. Our circulator based LDV design is novel, easier to implement, and power conserving. The confocal design can provide high spatial as well as longitudinal resolution. The ABCD matrix analysis of the confocal design has been in very good agreement with all the experimental results. The sensitivity of this LDV system can be further improved by using low reflectivity coated fiber tips to provide lower levels of reference signal.