## Abstract

A novel wavelength-insensitive differential laser Doppler velocimeter (LDV) without a grating has been proposed. The proposed LDV utilizes a position shift of the beam at the input plane according to wavelength change induced by Mach-Zehnder interferometers (MZIs). The gradual shift in the incident angle of the beam to the object is brought about with the combination of MZIs, lenses and apertures. The characteristics of the proposed structure are simulated using paraxial approximation of a lens system with an aperture. The simulation results indicate that almost wavelength-insensitive operation can be obtained by using the proposed structure without any grating element.

©2009 Optical Society of America

## 1. Introduction

Velocity measurement has been widely used in researches and industries to measure the velocity of a fluid flow or rigid object. The measurement by using differential laser Doppler velocimeter (LDV) has the advantage of contactless, small measuring volume giving excellent spatial resolution, and a linear response [1–4].

It is desirable to use a semiconductor laser as a lightsource of the LDV in order to reduce the cost of the system. However, typical inexpensive semiconductor lasers suffer from the problem of instability in the lasing wavelength due mainly to the dependence on temperature. In typical differential LDVs, the Doppler frequency shift at a monitoring point depends on the signal wavelength to be used. Hence, the wavelength instability in the semiconductor lasers causes measurement errors in the Doppler frequency shift.

To reduce the measurement error due to the wavelength change, some studies have been reported [5–8] utilizing the change in the incident angle according to wavelength shift. The differential LDV using a diffractive grating that utilizes the dependence of the diffraction angle on wavelength has been reported [5–7]. The authors have proposed the wavelength-insensitive LDV that uses arrayed waveguide gratings (AWGs) made by planar lightwave circuit (PLC) technology [8]. However, these types of LDVs need complicated grating elements such as a diffractive grating or a planar waveguide grating. A wavelength-insensitive LDV consisting of simpler optical elements is more desirable.

In this paper, a novel wavelength-insensitive differential LDV without any grating element has been proposed. The shift in the incident angle of the beam at the object is brought about with the combination of simple Mach-Zehnder interferometers (MZIs), lenses and apertures. The proposed LDV utilizes a position shift of the beam at the input plane according to the wavelength change induced by MZIs. The position shift of the beam induced by the MZI is converted into the shift in the incident angle to the object, and the aperture contributes to smoothing and continuously shifting the field distribution. In this paper, the principle of the proposed structure for wavelength-insensitive operation is described and its characteristics are simulated using paraxial approximation of a lens system with an aperture.

## 2. Principle

Figure 1 illustrates the optical system of the proposed wavelength-insensitive LDV. It consists of a laser, a power splitter, a photodetector (PD), and two sets of components, each of which consists of a mirror, an MZI, an aperture, and a lens. These two sets of the components having the same optical characteristics are symmetrically arranged. The MZIs can be fabricated using planar waveguides or fibers.

The input beam is split with a power splitter. Each beam is passing through the MZI, output from the output ports of the MZI according to the wavelength of the beam, passing through the aperture and the lens, and incident to the object. The beams are scattered on the object and detected by the PD as the beat of the light that depends on the Doppler shift due to the motion of the object.

In the differential LDV, the beat frequency detected by the PD, *F _{D}*, is expressed as [1,3,9]

*ψ*is the incident angle of the beam to the object, ${v}_{\perp}$ is the velocity of the object perpendicular to the bisector of the angles of the incident beams to the object, and

*λ*is the wavelength. From this equation, if

*ψ*appropriately changes depending on

*λ*, the wavelength-insensitive operation can be expected. When

*F*is to be insensitive to the wavelength around

_{D}*λ*=

*λ*

_{0}, the derivative of

*F*with respect to

_{D}*λ*should be zero at

*λ*=

*λ*

_{0}. From Eq. (1), this condition is expressed as

*ψ*

_{0}is the incident angle at

*λ*=

*λ*

_{0}.

In the proposed structure, *ψ* is changed with the system using an MZI, an aperture and a lens. The MZI is used to shift the position of the beam at the input plane of the each set of the components (i.e., at the outputs of the MZI) according to the wavelength. The beam is output as two beams from the output ports of the MZI. The ratio of the powers of these beams changes according to the wavelength.

Since the beams are output from the two discrete ports of the MZI, the field distribution at the input plane generally has two peaks. To obtain wavelength-insensitive operation, the field should be smoothed and continuously shifted at the output plane (i.e., around the measured point) as the change of the wavelength. The aperture contributes to smoothing and continuously shifting the field distribution at the output plane. The aperture acts as a spatial filter for truncating a part of the diffraction pattern of the beams from the MZI. The diffraction pattern of two beams with a displacement *d* becomes periodic as a consequence of interference during free propagation. Especially, when the spatial filter transmits just the pattern within one Brillouin zone determined by the arc subtended by the angle *λ/d* [10,11], i.e., the zone covering just one fringe of the periodic pattern, the field distribution at the output plane is bandlimited to a spatial frequency *a/*(*λz*) [12]. Here, 2*a* is the aperture width and *z* is the distance between the input plane and the aperture. This band limitation ensures smoothness in the field distribution.

The lens is used to convert position shift of the beam at the input plane into the shift in propagation angle at the output plane. Then, the incident angle *ψ* changes depending on the wavelength. The lens is also used to focus the beam.

## 3. Simulation Model

#### 3.1 Model using paraxial approximation of lens system

To simulate the characteristics of the proposed structure, a set of optical components shown in Fig. 1(b) is modeled using paraxial approximation of a lens system [13] with an aperture. For simplicity, we treat one-dimensional case. The model could be easily expanded to two-dimensional case.

Let the free spectral range of the MZI be Δ*λ _{FSR}* and the wavelength shift from the nominal wavelength

*λ*

_{0}be Δ

*λ*. The transfer function for the

*m*-th output port (

*m*= 1, 2) of the MZI,

*h*(

_{m}*λ*), is given by the product of transfer matrices of two couplers and two arms, expressed as [14]

*θ*is the amplitude coupling coefficient of the couplers and

*φ*is the constant phase term. Especially, when the couplers’ splitting ratio is 50:50, the transfer function is given by

Let the field distribution at the input plane be *u*
_{1}(*x*). The beam freely propagates with the distance *L*
_{1} and is incident to the aperture. The field distribution just before the aperture, *u*
_{2}(*x*), is given by the Fresnel diffraction of *u*
_{1}(*x*) under paraxial condition:

*k*= 2

*π/λ*is the wave number in the air. When the Fourier transform of

*f*(

*x*) is defined by

*u*

_{2}(

*x*) is given by

*U*

_{1}(

*x*) is the Fourier transforms of

*u*

_{1}(

*x*) and the function

*G*

_{1}(

*x*) is defined by

The beam passes through an aperture and a lens, and is focused at the output plane. The field distribution just after passing through the aperture, *u*
_{2}
*’*(*x*), is the product of *u*
_{2}(*x*) and an aperture function *p*(*x*) defined by

Here, the half width of the aperture just covering one Brillouin zone, *a*, is

The Fourier transform of *u*
_{2}
*’*(*x*) is expressed as the convolution of the Fourier transforms of *u*
_{2}(*x*) and *p*(*x*) as

*P*(

*x*) is the Fourier transform of

*p*(

*x*), expressed using a sinc function.

The wave front of the beam is converted by the lens. When the focal length of the lens is *f*, the field distribution just after the lens, *u*
_{3}(*x*), is given by

The field distribution of the beam at the output plane, *u*
_{4}(*x*), is given by the Fresnel diffraction of *u*
_{3}(*x*) with a propagation distance of *L*
_{2} as

Using Eqs. (7), (11) and (12), Eq. (13) is reduced to

*G*

_{2}(

*x*) is defined by

As a special case, when the output plane is set to the image plane defined by the thin-film equation, the following relation holds between *L*
_{1} and *L*
_{2} [15]:

For simplicity, we treat a one-dimensional Gaussian function as each input beam since a beam from a single-mode waveguide or fiber can be typically approximated well with a Gaussian function. Let the field distribution of a normalized one-dimensional off-axial Gaussian beam at the input plane be

*w*is the spot size. When the output ports of the MZI placed at the input plane are separated with

_{in}*d*, the field distribution at the input plane,

*u*

_{1}(

*x*), is given by the superposition of two off-axial Gaussian beams, i.e.,

*u*

_{1}(

*x*) =

*h*

_{1}(

*λ*)

*u*

_{0}(

*x*+

*d/*2) +

*h*

_{2}(

*λ*)

*u*

_{0}(

*x*-

*d/*2). The Fourier transform of

*u*

_{1}(

*x*) is then given by

By using Eqs. (14) and (18), the characteristics of the proposed differential LDV can be simulated.

#### 3.2 Design parameters

Table 1
shows the design parameters for the simulation. It is assumed to use MZIs made from silica waveguides with a relative index difference between a core and cladding, Δ, of 0.3% as input MZIs. The nominal wavelength *λ*
_{0} is set to 1.3 μm to be used as an example of simulation. It is also assumed that the core width, *W _{core}*, is 10 μm and the displacement of the cores,

*d*, is 20 μm at the output of the MZIs. The spot size of the fundamental mode,

*w*, is then assumed to be 4.74 μm under Gaussian approximation, derived from the approximated relation between the normalized core width

_{in}*D*(in this case,

*D*= 5.47) and the spot size

*w*expressed as [16]

_{in}The aperture width 2*a* is set to the width just covering one Brillouin zone determined by Eq. (10). The position of the output plane is set to around the image plane determined by Eq. (16).

## 4. Results and discussion

Figure 2
shows the contour plot of the calculated power distribution of the beam at the output plane for various relative wavelength deviation Δ*λ/*Δ*λ _{FSR}*. Here,

*f*= 2 mm and

*L*

_{1}=

*L*

_{2}= 4 mm. The constant phase term

*φ*was set so that the output powers from the two ports of the MZI are equally split when Δ

*λ/*Δ

*λ*= 0. The field distribution at the output plane has only one peak due to the effect of spatial band limitation by the aperture even if Δ

_{FSR}*λ/*Δ

*λ*is around zero where the field distribution has two peaks at the input plane. The position of the beam shifts continuously as the change of the wavelength.

_{FSR}To achieve wavelength-insensitive operation, the propagation angle of the beam should be changed depending on the wavelength change. Figure 3
shows the beam transition around the output plane of *L*
_{2} = 4 mm for *f* = 2 mm, *L*
_{1} = 4 mm, and Δ*λ/*Δ*λ _{FSR}* = –0.25 and –0.125. Figure 4
shows the propagation angle at the output plane as a function of the relative wavelength deviation Δ

*λ/*Δ

*λ*for

_{FSR}*f*= 2 mm and

*L*

_{1}=

*L*

_{2}= 4 mm. Here, the propagation angle is defined by the angle between the direction of propagation and the

*z*-axis. The propagation angle of the beam changes as the change of the wavelength

*λ*. It results from the conversion of the position shift of the beam at the input plane, caused by wavelength shift, into the shift in propagation angle by the lens. The change in the propagation angle is almost linear around Δ

*λ/*Δ

*λ*= 0 (i.e., around

_{FSR}*λ*=

*λ*

_{0}) where the output powers from the two ports of the MZI are almost equal.

From Fig. 4, the derivative of *ψ* with respect to *λ* can be characterized as Δ*λ _{FSR}*(

*dψ/dλ*) = 0.022 rad at

*λ*=

*λ*

_{0}for

*f*= 2 mm and

*L*

_{1}=

*L*

_{2}= 4 mm. Figure 5 plots the Δ

*λ*(

_{FSR}*dψ/dλ*) value at

*λ*=

*λ*

_{0}as a function of

*L*

_{2}for various

*L*

_{2}

*/f*under the condition Eq. (16). From Fig. 5, when the ratio

*L*

_{2}

*/f*is fixed, the Δ

*λ*(

_{FSR}*dψ/dλ*) value at

*λ*=

*λ*

_{0}is inversely proportional to the propagation length

*L*

_{2}because the propagation angle is also roughly inversely proportional to

*L*

_{2}. When

*L*

_{2}is fixed, the Δ

*λ*(

_{FSR}*dψ/dλ*) value at

*λ*=

*λ*

_{0}increases as the ratio

*L*

_{2}

*/f*increases because the position shift at the input plane is more effectively converted to the shift in the propagation angle as the decrease of the focal length

*f*.

Once the incident angle *ψ*
_{0} is fixed, Δ*λ _{FSR}* for wavelength-insensitive operation is derived from the condition Eq. (2) and the values from Fig. 5. When

*ψ*

_{0}= 30° for example, Δ

*λ*should be set to 48.6 nm for

_{FSR}*f*= 2 mm and

*L*

_{1}=

*L*

_{2}= 4 mm. Then, the dependence of

*F*

_{D}/v_{⊥}on wavelength can be calculated by using Eq. (1).

Figure 6
plots the deviation in *F _{D}/v*

_{⊥}due to the wavelength deviation Δ

*λ*=

*λ*–

*λ*

_{0}calculated by using Eq. (1) for Δ

*λ*= 48.6 nm,

_{FSR}*ψ*

_{0}= 30°,

*f*= 2 mm, and

*L*

_{1}=

*L*

_{2}= 4 mm. The deviation in

*F*

_{D}/v_{⊥}is determined as [

*F*

_{D}/v_{⊥}– (

*F*

_{D}/v_{⊥})|

_{λ}_{=}

_{λ}_{0}]

*/*(

*F*

_{D}/v_{⊥})|

_{λ}_{=}

_{λ}_{0}. For comparison, the data for a conventional LDV without MZIs are also shown in Fig. 6, calculated by using Eq. (1) for a constant incident angle

*ψ*=

*ψ*

_{0}. As shown in Fig. 6, the deviation in

*F*

_{D}/v_{⊥}for the proposed structure can be almost flat (within +/– 1.0 x 10

^{−4}), whereas the deviation in

*F*

_{D}/v_{⊥}for a conventional non-wavelength-insensitive LDV is within +/– 1.5 x 10

^{−3}when the wavelength deviation is within +/– 2 nm. If the wavelength deviation is within +/– 0.8 nm, the deviation in

*F*

_{D}/v_{⊥}can be reduced within +/– 1.0 x 10

^{−5}.

When slight increase of the deviation in *F _{D}/v*

_{⊥}is allowed near

*λ*=

*λ*

_{0}, the wavelength range in which the deviation in

*F*

_{D}/v_{⊥}is small can be enhanced by changing Δ

*λ*from the value determined from Fig. 5 and Eq. (2). It is understood that the increment of

_{FSR}*F*

_{D}/v_{⊥}with respect to

*λ*around

*λ*=

*λ*

_{0}is determined by Δ

*λ*for a constant Δ

_{FSR}*λ*(

_{FSR}*dψ/dλ*) value. Figure 7 plots the deviation in

*F*

_{D}/v_{⊥}for various Δ

*λ*. Here, all the other parameters are the same as those for Fig. 6. When Δ

_{FSR}*λ*= 43.5 nm, the wavelength range in which the deviation in

_{FSR}*F*

_{D}/v_{⊥}is below +/– 1.0 x 10

^{−4}can be enhanced to +/– 3 nm.

As described above, the deviation in *F _{D}/v*

_{⊥}under the existence of wavelength shift can be significantly reduced by using the proposed LDV with MZIs. It contributes to reducing measurement error due to wavelength change.

## 5. Conclusion

A novel wavelength-insensitive differential LDV without a grating has been proposed. The proposed LDV utilizes a position shift of the beam at the input plane according to wavelength change induced by MZIs. The gradual shift in the incident angle of the beam to the object is brought about with the combination of simple MZIs, lenses and apertures. As a result of the simulation using a model of a lens system with an aperture, it is found that the deviation in *F _{D}/v*

_{⊥}for the proposed structure can be significantly reduced to +/– 1.0 x 10

^{−4}, whereas the deviation in

*F*

_{D}/v_{⊥}for a conventional non-wavelength-insensitive LDV is +/– 1.5 x 10

^{−3}when the wavelength deviation is +/– 2 nm. It means that measurement errors due to wavelength change can be reduced without any grating element such as a diffractive grating or a planar waveguide grating.

## Acknowledgments

This work in part was supported by a research-aid fund of the Foundation for Technology Promotion of Electronic Circuit Board, a research-aid fund of the Suzuki Foundation, and Grant-in-Aid for Scientific Research (B) 21360196 (KAKENHI 21360196).

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