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A multi-point laser Doppler velocimeter (LDV) using the arrayed waveguide gratings (AWGs) with small wavelength sensitivity (less than 1/10 of that for a conventional LDV without the AWGs) is proposed, in which velocities at different points in the depth direction can be simultaneously measured with compact optical systems. The design and characteristics of the proposed LDV are investigated with the model using the grating equation of the AWGs. From our simulation results, the wavelength sensitivity for multiple measured points can be reduced to less than 1/10 of that for a conventional LDV without an AWG.
©2010 Optical Society of America
1. Introduction
In many researches and industries, velocity measurement has been an essential technology. The velocity measurement using differential laser Doppler velocimeters (LDVs) has the advantages of contactless measurement, a linear response, and a small measuring volume giving an excellent spatial resolution.
In some applications such as fluid flow in narrow pipes, the velocity distribution in depth direction should be measured. In this case, the velocities of different points in the depth direction should be simultaneously measured. For the purpose, several types of LDVs for multi-point measurement have been proposed [1–3]. However, these multi-point LDVs consist of bulk optical systems which have large sizes and complexity of assembly, and are often affected by environmental disturbances due to a large optical path length. Hence, integrated LDVs have been highly demanded. Several integrated differential LDVs have been proposed [4–6] as applications of integrated optical sensors although these LDVs are used for single-point measurement.
It is desirable to use a semiconductor laser in an integrated LDV so that several optical elements can be integrated within a small area at low cost. However, typical inexpensive semiconductor lasers suffer from the problem of instability in lasing wavelength due mainly to the temperature dependence. The instability in the lasing wavelength causes measurement errors in the Doppler frequency shift measured by the LDVs because the Doppler frequency shift generally depends on the signal wavelength of the input laser beam. To reduce the measurement errors due to the wavelength change, some studies have reported the differential LDVs that use a diffractive grating and utilize the dependence of the diffraction angle on wavelength [3,7,8]. The authors have proposed the wavelength-insensitive LDV that uses the arrayed waveguide gratings (AWGs) [6] or the Mach-Zehnder interferometers [9] based on planar lightwave circuit (PLC) technology [10–17]. Since these LDVs are used for single-point measurement, multi-point velocity measurement with small wavelength sensitivity is desirable for practical applications.
In this paper, we propose a multi-point differential LDV with small wavelength sensitivity using the AWGs. With the proposed LDV, velocities at multiple points in depth direction can be simultaneously measured. The design and characteristics of the proposed LDV are discussed.
2. Principle
2.1 Structure
Figure 1 illustrates the optical circuit of the proposed multi-point differential LDV. The LDV consists of laser diodes (LDs) with different lasing wavelengths as light sources, power splitters, two laser-side AWGs with the same design, cylindrical lenses, a detector-side AWG, and photodetectors (PDs). The beam from each laser is split with a the power splitter. Each spilt beam is incident on the input slab of each laser-side AWG, phase-shifted through its waveguide array, output with diffraction, and incident on the measured point. The beam is scattered on the object at the measured point, inputted to the detector-side AWG, diffracted according to its wavelength, and detected by the PD as the beat of the beam depending on the Doppler shift due to the motion of the object. The cylindrical lenses are used to collimate the beams in the vertical direction. The beams with different wavelengths are focused on different focusing points by the laser-side AWGs, and detected by different PDs by the detector-side AWG. Hence, the velocities at multiple measured points can be simultaneously measured. The laser-side AWGs are also used to diffract the beam whose diffraction angle is changed depending on the wavelength in order to reduce the dependence of the measured Doppler shift on the wavelength. This change in the diffraction angle contributes to reducing wavelength sensitivity [6].
To accurately detect the Doppler shift according to velocity, diffracted beams from two laser-side AWGs should be focused around the cross point of these beams (i.e., the measured point). In other words, the focal points should be aligned in the depth direction. To align the focal points of the diffracted beams with different wavelengths, the distances between positions of each input waveguides connected to the input slab and the input of the waveguide array are optimized for each laser-side AWG. The design will be described in Section 2.2.
2.2 Modeling
In order to investigate the design and characteristics of the circuit, a model for the optical circuit of the proposed LDV is derived. Figure 2 illustrates the schematic diagram of one of the laser-side AWGs. In Fig. 2, the z axis is set to along the depth direction and the x axis is set to perpendicular to the z axis.
2.2.1 Grating equation of AWGs
Let the grating order of laser-side AWGs be m, the effective refractive indices for the waveguides in the array and the slabs be na and ns respectively, the difference in waveguide length between the adjacent waveguides in the array be ΔL, and the interval of the waveguides in the array at slab-to-array interface be d. The diffraction angle of the beam from the each AWG, θ, is expressed based on the grating equation [11,13] as
where ϕ is the input angle from the input waveguide to the waveguide array and λ is the wavelength of the input beam. When the diffraction angle of the beam with the wavelength λ 0 is normally incident on the input of the array and normally diffracted from the output of the array, the relation λ 0 = naΔL/m is satisfied. Substituting this relation into Eq. (1) yieldsThe output aperture of the array is directed at the angle of ψout. The incident angle of the beam to the measured point is then given by ψout + θ. When the centers of the output sides of the two waveguide arrays are separated with the distance of ΔxAWG, the position of the measured point in the depth direction, zmeas, is given by
2.2.2 Wavelength-insensitive condition
Here, we consider the condition for wavelength-insensitive operation. The Doppler shift frequency detected at the PD, FD, is expressed as [5]
where v ⊥ is the x-component of velocity. When wavelength-insensitive characteristics are to be obtained around the wavelength of λ 1, the derivative of FD with respect to λ should be zero at λ = λ 1. From Eq. (4), this condition is expressed aswhere θ 1 is the diffraction angle for the wavelength λ 1. From the derivative of Eq. (2) with respect to λ and Eq. (5), the condition for wavelength-insensitive operation around the wavelength of λ 1 is given byThen, the relation among the design parameters m, d and ψout are derived from Eq. (6) as the condition for small wavelength sensitivity.
2.2.3 Design for focal point alignment
Here, in order to align the focal points of the diffracted beams from different input waveguides, the distance between the position of each input waveguide connected to the input slab and the input of the waveguide array in the laser-side AWG, Lin, is derived.
Suppose that a Gaussian beam with a spot size win is emitted into the input slab. The waveguides in the array are connected to the input slab along the arc with a radius Roin. Under the paraxial approximation, the distortion of the phase at the input of the waveguide array, excluding the linear phase change, is expressed based on the paraxial approximation [18] as
where k is the wave number defined as k = 2π/λ, ξ is the position along the slab-array interface, and Rin is defined asAfter the beam propagates the waveguide array, the distortion is reproduced at the output of the waveguide array. The distortion causes the change of the radius of the wave front of the laser. When the waveguides in the array are connected to the output slab along the arc with a radius Roout, the wave front lies on the arc with the radius Rout given by the following relation:
When the spot size of the focusing point is wout, the diffracted beam is focused on the plane with the distance of Lout from the waveguide array, which satisfies the following relation:
In a typical AWG design, the spot sizes win and wout are much smaller than Lin and Lout. Then, Lout is given by the following relation:
2.2.4 Limitation of input and diffracted angles
If the beam is diffracted to the direction out of the central Brillouin zone [19,20], because its power is significantly decreased and the beam becomes no longer the main beam. The input beam to the waveguide array also suffers considerable loss when the beam direction is out of the central Brillouin zone. Hence, the input angle ϕ and diffraction angle θ for each input wavelength must be within the central Brillouin zone. The angle of the edge of the central Brillouin zone is determined by the following equation [19,20]:
To obtain the wavelength-insensitive operation at λ = λ 1, the condition Eq. (6) should be satisfied. As a special case for the wavelength insensitive operation that λ 1 = λ 0 and θ 1 = 0, ΔθB is expressed as
3. Simulation results and discussion
3.1 Relation among design parameters
In this section, the relation among the design parameters is determined. Here, we assume ns = 1.45 to use silica-based materials based on PLC technology as the optical circuit. The wavelength λ 0 is set to 1.3 μm to be used as an example in our simulation.
The range of the positions of the measured points in the depth direction is one of the important characteristics for the proposed LDV. Figure 3 shows the relation between the array aperture angle ψout and the extreme range of positions of the measured points, ΔzB, defined by the range of positions of the measured points projected with the central Brillouin zone, ΔθB, for various grating orders m. The values ΔzB relative to ΔxAWG are plotted in this figure. The range ΔzB is calculated from Eq. (3) by substituting –ΔθB and + ΔθB into θ. Here, ΔθB should satisfy Eq. (13) for the wavelength-insensitive condition at the wavelength of λ 0. The relation between ΔzB and ψout is not so simple because ΔθB depends on ψout as the condition Eq. (13). ΔzB decreases as ψout increases and is minimized at around ψout = 45°. ΔzB also decreases for large m when ψout is fixed. The waveguide interval d also depends on ψout for each value of m under the condition Eq. (6). In a typical design for reducing coupling loss between the waveguide array and the slab, the gap width of the waveguides in the array at the array-to-slab interface should be narrowed. Hence, too large d results in large core widths in the waveguides in the array at the array-to-slab interface, which easily leads to an undesirable higher-order mode excitation. On the other hand, too small d may result in large mutual coupling between the waveguides, which easily leads to crosstalk [21]. These facts should be taken into account for determining the design parameters. As an example for the design parameters for the silica-based waveguides with Δ of 1.5%, the relation between ψout and ΔzB/ΔxAWG under the condition of d = 10 μm is also plotted in the dotted line in Fig. 3. In this case, when m = 2 for an example, ψout becomes 10.17° under the condition Eq. (6).
Fig. 3 Relation between array aperture angle ψout and relative extreme range ΔzB/ΔxAWG for various m. The relation between ψout and ΔzB/ΔxAWG under the condition of d = 10 μm is also plotted in the dotted line.
Figure 4 shows the relation between the relative position of the measured point in the depth direction, zmeas/ΔxAWG, and the input wavelength λ for various input angles ϕ. Here, we assume m = 2, d = 10 μm and ψout = 10.17° as a set of the design parameters which satisfy the condition Eq. (6). When the input angle θ is not changed, the measured position zmeas/ΔxAWG is also not changed as Eq. (3). The lines for the constant θ are also plotted in Fig. 4. The condition of angles ϕ and θ within ± ΔθB is indicated as the area within the dotted line. The positions of measured points can be derived from this figure once a set of input angles ϕ and the input wavelengths λ is determined. Table 1 shows an example of the parameters ϕ, λ, Lin and zmeas for 5-point velocity measurement. Lin is calculated from Eq. (11). Here, we assume ΔxAWG = 30 mm and Roin = Roout = 30 mm. In this case, the measured points are arranged over the range of 25.77 mm.
Fig. 4 Relation between relative position of measured point zmeas/ΔxAWG and input wavelength λ for various ϕ. m = 2, d = 10 μm and ψout = 10.17°. The condition of angles ϕ and θ within ± ΔθB is indicated as the area within the dotted line.
Table 1. Example of Parameters ϕ, λ, Lin and zmeas
3.2 Wavelength sensitivity
Figure 5 shows the absolute value of deviation in FD/v ⊥ for the proposed multi-point LDV due to the wavelength deviation for m = 2, d = 10 μm and ψout = 10.17°. The parameters shown in Table 1 are used as the nominal input wavelengths and the input angles. The deviation in FD/v ⊥ is defined as [FD/v ⊥ – (FD/v ⊥)|λ = λn]/(FD/v ⊥)|λ = λn, where λn is the nominal input wavelength for each measured point. For comparison, the deviation for a conventional differential LDV without the AWG is also shown in Fig. 5. The absolute value of the deviation in FD/v ⊥ for λn = 1.3 μm (i.e., the wavelength of the beam from the central input port) for the proposed structure can be significantly reduced to less than 10−4 of that for a conventional LDV, as well as the single-point LDV using the AWGs we previously proposed [6]. It should be remarked that the deviation for λn = 1.2 μm and 1.4 μm (i.e., beams from marginal input ports) can be also reduced to less than 1/10 of that for a conventional LDV. It indicates that the wavelength sensitivity for all of the measured points can be reduced by using the proposed structure.
Fig. 5 Absolute value of deviation in FD/v ⊥ due to wavelength deviation for m = 2, d = 10 μm and ψout = 10.17°. λn = 1.2, 1.3 and 1.4 μm. The deviation for a conventional differential LDV without an AWG is also plotted.
With typical differential LDVs as the proposed one, the velocities at the measurement volumes (i.e., at the measured points) are measured even in the case of turbulent flow. By using the proposed multi-point LDV, the distribution of multi-point one-dimensional velocities in turbulent flow can be measured.
4. Conclusion
We have proposed a multi-point LDV using the AWGs with small wavelength sensitivity. With the proposed the LDV, velocities at different points in the depth direction can be simultaneously measured in compact optical systems. As an example of the design for the silica-based waveguides for 5-point velocity measurement, the measured points can be arranged over the range of 25.77 mm, which would be enough to measure the velocity distribution in some applications such as fluid flow in narrow pipes. From our simulation results using the model based on the grating equation of the AWGs, we found that the wavelength sensitivity for all of the measured points can be reduced by using the proposed structure. The proposed LDV would be applicable to many researches and industries.
Acknowledgments
This work in part was supported by a research-aid fund of the Nakatani Foundation of Electronic Measuring Technology Advancement and a research-aid fund of the Suzuki Foundation.
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