Time-of-flight anemometry using a displacement plate-beamsplitter

The first beam, which is reflected immediately, is denoted by the index i = 0. Each subsequent output beam, caused by partial tc $i\geqslant 1$ with the first transmitted beam being i = 1. The value of the separation distance between successive beams can be found through sequential applications of Snell's Law, and the result is well known. The plate thickness, ω, angle of incidence, β, and the ratio of material indices of refraction, ν, are related to the standoff distance, x, by, Wyant (1976)

$$\begin{align*} \nu\equiv\frac{n_2}{n_1}, \end{align*}$$

$$\begin{align} x=\frac{\omega \sin{2\beta}}{\nu\sqrt{1-\left(\frac{\sin{\beta}}{\nu}\right)^2}}. \end{align} \tag{ 1 }$$

By controlling the plate's thickness it is therefore possible to produce almost any beam separation at almost arbitrarily large or small distances from the beamsplitter because this equation holds true regardless of the diameter of the incoming beam, and it is relatively easy and inexpensive to produce large and flat glass plates. By making the incoming beam diameter large, possibly orders of magnitude greater than x, it is possible to keep the laser beam's numerical aperture large even for long range systems. As a result the beam spot size can be kept small relative to x, which produces higher signal to noise ratios and more precise measurements.

Since both optical surfaces are flat, if the beam is collimated each output beam's waveform is not distorted beyond distortions introduced by surface unevenness. The output beams are parallel to a very close approximation. This means the system's calibration is essentially the same regardless of depth. By utilizing the same lens for the laser and receiver (placed before the laser light reaches the beamsplitter), as is common for LDA and ToFA, the transmitter/receiver system becomes self-aligning.

However, one of the least attractive aspects of the displacement plate-beamsplitter in the context of ToFA is that if instead the beam is focused to a finite depth, this depth is altered for beams of order i > 0, and the quantity of this change depends on both the ray angle and the axis. This introduces defocusing, spherical aberration, astigmatism, and even coma. As a result, each beam will focus (in a slightly degraded manner) to its waist at a slightly different distance from the beamsplitter. Higher order beams will have a progressively degraded focus in the measurement volume as a result. Essentially, the system is self-aligning and focusing but the level of alignment and focus is not perfect and not independently tunable. Consider a ray of light that makes an angle of γ_s with the chief ray from the laser diode (the ray at the axis of symmetry of the incoming beam) in a plane normal to the plane depicted in figure 1. The subscript s here relates to the plane in which the ray deviates from the chief ray and does not imply it is only relevant to a certain polarity of light. Due to symmetry, no coma is possible in this axis. However, as was already known at the time but analyzed in mathematical and numerical detail by Fried and Turner (1970), passage of this angled ray through the flat glass will alter the eventual location where this ray crosses the chief ray. The difference in location between beams of different orders ($i = 0,\, i = 1$, etc) where this ray crosses the chief ray as measured in the direction of propagation (using the location where the chief ray leaves the first surface as the reference point) can be described by,

$$\begin{align} \Delta f_s=x \Delta i\left(\frac{2\cos{\gamma_s}}{\sin{2\beta}\sqrt{1-\left(\frac{\sin{\gamma_s}}{\nu}\right)^2}}-\tan{\beta}\right), \end{align} \tag{ 2 }$$

where $\Delta i$ is the difference in the index of the two beams being compared. The physical quantity this equation solves for is represented as $\Delta f$ in figure 1. Since this is a non-trivial but even function of the ray angle, spherical aberration is introduced. The reason this equation differs somewhat from Fried and Turner (1970) is because the beam is folded and placed on an angle.

Now consider a ray of light that makes an angle of γ_p , from above, with the chief ray (ie, it has a higher incidence angle) in the plane depicted by figure 1. For this ray, the difference in focal distance between it and the focal distance of the two beams (again in the direction of propagation) is given by,

$$\begin{align} {\scriptstyle \Delta f_p\,\,=\,\,x \Delta i\left[(\cot{\gamma_p}-\tan{\beta})\left(\left(\frac{\sin{(\beta-\gamma_p)}}{\sin{\beta}}\sqrt{\frac{1-\left(\genfrac{}{}{.3pt}{3}{\sin{\beta}}{\nu}\right)^2}{1-\left(\genfrac{}{}{.3pt}{3}{\sin{(\beta-\gamma_p)}}{\nu}\right)^2}}\right)-1\right)-\tan{\beta}\right]}. \end{align} \tag{ 3 }$$

This equation represents the same physical quantity as $\Delta f_s$, so it too is represented as $\Delta f$ in figure 1. However, because it comes from an angular deviation that is orthogonal to the one that causes $\Delta f_s$, its value is different. Since this equation is not even, coma is introduced. Since it is not identical to equation (2), astigmatism is introduced. As γ_s and γ_p tend to zero, the spherical aberration and coma disappear, but the defocus and astigmatism remain. Assuming $\Delta f_p$ is small for all rays in the beam relative to the depth of focus, the beams will focus to a diffraction limited sheet at the same depth. The depth of focus for a Gaussian beam can be written in terms of the laser's wavelength, λ, and the half apex angle, Θ

$$\begin{align} b=\frac{2\lambda}{\pi n_1 \Theta^2}. \end{align} \tag{ 4 }$$

Comparison of b and $\Delta f_p$ is critical for designing a system with sharp focus. The signal strength for laser anemometry depends directly on the beam's power density, and measurement uncertainty also decreases when the beams are thin compared to their separation. These equations imply the displacement plate-beamsplitter performs best when the numerical aperture is kept small enough to avoid aberrations but not so small that the beam waist is undesirably large.

The theory developed thus-far assumes a plate with perfect parallelism, perfect flatness, and immaculate surface quality is used. Real flat plates have none of these properties. The multiple reflections and two refractions experienced by higher order beams as well as the high angle of incidence present in this beamsplitter will amplify any imperfections in construction. This makes the allowable tolerance on the plate quite small. Luckily the methods available for producing flat plates, and especially flat surfaces, are among the most foundational and well refined in the fields of optics, metrology, and various industries. It is possible to obtain flat glass plates with extremely small tolerances that suffice to allow it to be used as a displacement plate-beamsplitter.

The required parallelism tolerance is the most difficult to achieve both because the effect of a small wedge angle is greatly amplified at large distances, and because it is more difficult to align two planes to each other than it is to produce a single pristine plane. Using the results of Beers (1974), we can derive a formula for the difference in angle between the 0th and ith beams, $\Delta \beta_i$. This quantity, represented as $\Delta \beta$, is shown in figure 1. Note that in this figure the angle shown is a negative value. For a beamsplitter with wedge angle A, one can re-derive a result first produced but not published by Beers (1974),

$$\begin{align} \Delta \beta_i = \beta-\arcsin{\left(\nu\sin{\left(\arcsin{\left(\frac{\sin{\beta}}{\nu}\right)}+2iA\right)}\right)}. \end{align} \tag{ 5 }$$

This formula shows high angles of incidence greatly increase the sensitivity to A. Consider the case of s-polar light, where β is tuned such that the power contained in the i = 0 and i = 1 beams are the same, ν is not approximately 1, and $iA\lt\lt1$. With these assumptions a convenient approximation for $\Delta \beta_i$ exists:

$$\begin{align} \Delta \beta_i \approx -2iA\xi. \end{align} \tag{ 6 }$$

More information on beam power matching and the definition of ξ are given in section 2.6. Intuitively this equation results in the conclusion that for practical designs $|\Delta \beta_i |\gtrsim 8iA$. If the beams converge or diverge too much their separation at the depth of the intended measurement volume will be too great. To achieve $x = 1\,\mathrm{mm}$ at 1 m measurement distance, $|A|$ should be 25 arcseconds at most to avoid the chance of the beams crossing in the measurement volume. At greater range, for smaller ω or higher β, if predictable calibration is desired (especially at multiple depths), or if more beams are kept, the requirement becomes even more strict. Optical flats and optical windows can be obtained with parallelism of <5 arcseconds without difficulty whereas lower parallelism tolerances come at an increasing premium. The parallelism of float glass varies significantly from batch to batch and even across a given pane, however we did not find it difficult to obtain float glass mirrors with parallelism in the vicinity of 1 arcsecond.

Surface flatness effects the beamsplitter's performance in three primary ways. Firstly each beam gets re-focused by the curvature of the beamsplitter's two faces. Secondly, beams of different orders interact in a slightly different way with the beamsplitter's slight curvature and thus they are not parallel to each other anymore. Finally, if the diameter of the beam at the displacement plate-beamsplitter is comparable to the distance over which the curvature changes, the complex surface unevenness profile will show up as further aberrations that will appear in the focused beams. To make analysis tractable it is convenient to assume the beamsplitter is locally spherical with radius $\mathcal{R}_1$ on the first surface and radius $\mathcal{R}_2$ on the second surface (concave down is positive). It is also convenient to use the best-fit parabola to mathematically approximate this spherical curvature. It is assumed that the axis of these surfaces are aligned, forming an optical axis, and that the chief ray strikes the first surface at the optical axis. Assuming the input beam is collimated, one can track the position and angle of a ray through the beamsplitter if the ray has a displacement of z from the chief ray (see figure 1). For the i = 0 beam, a reflection off of the first surface is experienced, so this is just a classic parabolic reflector (the math is fairly simple in this case, and both focus and aberration have been fully studied). For instance, for a parabolic reflector one can derive an equation for the intersection depth of a ray with the chief ray (assuming the reflector's height variation can be ignored and assuming surface slope is small),

$$\begin{align} f_0(z)\approx-\frac{\mathcal{R}_1\cos{\beta}}{2}+\frac{z}{2}\sin{(2\beta)}. \end{align} \tag{ 7 }$$

When β is small this reduces to the classic result for the focal length of a spherical mirror or best-fit parabolic mirror, $f_0\approx-\frac{\mathcal{R}_1}{2}$. Increasing β increases the focusing power of any slight curvature in the first surface as well as producing progressively greater coma.

The i = 1 beam is produced from refraction off of the first curved surface, reflection off of the second surface, and then refraction again off of the first surface. By tracing a ray through this path it is possible to obtain an expression for the angle between the reflected chief ray for the i = 0 beam (reflected at angle β) and the i = 1 transmitted ray (that started z above the chief ray). The angle between higher order beams can also be determined but the formulas become unwieldy. Using some trigonometry and using the same small slope and height assumptions as before, the angle difference $\Delta \beta_\mathrm {curv}$ is found to be,

$$\begin{align*} \kappa(z)\approx\arcsin{\left(\frac{\sin{\left(\beta+\frac{z\cos{\beta}}{\mathcal{R}_1}\right)}}{\nu}\right)}-\frac{z\cos{\beta}}{\mathcal{R}_1} \end{align*}$$

$$\begin{align*} d_1(z)\approx z\cos{\beta}+\omega\tan{\left(\kappa\right)} \end{align*}$$

$$\begin{align*} d_2(z)\approx\omega \tan{\left(\kappa+2\frac{d_1}{r_2}\right)} \end{align*}$$

$$\begin{align} & \Delta \beta_\mathrm {curv}(z)\approx\beta-\frac{d_1+d_2}{\mathcal{R}_1}\nonumber \\ & -\arcsin{\left(\nu \sin{\left(\kappa+d_1\left(\frac{1}{\mathcal{R}_2}-\frac{1}{\mathcal{R}_1}\right)+\left(\frac{d_1}{\mathcal{R}_2}-\frac{d_2}{\mathcal{R}_1}\right)\right)}\right)}. \end{align} \tag{ 8 }$$

The intermediate variable κ represents the angle the ray makes to the optical axis after entry into the glass. The variable d₁ represents the distance from the optical axis to where the ray hits the second surface. The variable d₂ is the difference in distance (from the optical axis) between where the ray of interest reached the first surface from within the glass and where it first reached the second surface. The quantity $\Delta \beta_\mathrm {curv}(0)$ represents the loss in parallelism due to imperfect flatness in the beamsplitter. Figure 1 shows $\Delta \beta_\mathrm {curv}(0)$ as $\Delta \beta$. It is important to note that much less parallelism error is introduced if $\mathcal{R}_1 = \mathcal{R}_2$ (about 5 arcseconds if $\mathcal{R}_1 = 50,000\omega$, regardless of ν, for β of roughly 75^∘) than if $\mathcal{R}_1 = -\mathcal{R}_2$ (about 100 arcseconds if $\mathcal{R}_1 = 50,000\omega$, regardless of ν, for β of roughly 75^∘). Due to the way it is manufactured, float glass tends to have curvature of the same sign on both surfaces whereas for optical flats this is not likely. Even though optical flats have vastly superior flatness, the types of manufacturing errors made by float glass are less costly so the performance gap is smaller than expected. The depth where the ray of interest crosses the chief ray in the i = 1 beam (measured along this chief ray from where it left the beamsplitter) can be found using,

$$\begin{align} & f_1(z)\approx \frac{d_1(z)+d_2(z)-d_1(0)-d_2(0)}{\Delta \beta_\mathrm {curv}(z)-\Delta \beta_\mathrm {curv}(0)}\cos{\left(\beta-\Delta \beta_\mathrm {curv}(0)\right)}\nonumber \\ &\qquad\quad +\sin{\beta}(d_1(z)+d_2(z)-d_1(0)-d_2(0)). \end{align} \tag{ 9 }$$

To a good approximation, the beam re-focusing effect described by f₁ is caused exclusively by reflections. This is true for beams of all orders, not just the first beam. The reason is because for beams i > 0 the beam encounters exactly two refractions that almost exactly cancel the focal effect of each other (one when entering the glass and one when leaving it). If $\mathcal{R}_1 = \mathcal{R}_2$ then behavior of a given ray is almost the same for the i = 0 and i = 1 beams ($\omega\gtrsim|f_0(z)-f_1(z)|$). As a result, it is a good approximation to assume $f_1(z)\approx f_0(z)$ and so any re-focusing effect caused by surface curvature will be eliminated almost entirely during initial alignment. This holds true even for fairly large z. Therefore, the only relevant and potentially significant error introduced by identical (first and second surface) local surface curvature in a displacement plate-beamsplitter is classical parabolic reflector coma. If the beam diameter at the displacement plate-beamsplitter is a significant fraction of the length scale over which the radius of curvature in the glass changes, complex surface-specific beam aberrations will occur. The magnitude of these more complex aberrations can be approximated using equations (7)–(9) as a starting point.

While surface quality can be an important optical consideration, we expect it to rarely be critical for this application despite the high angle of incidence and multiple reflections. The reason is both because the beam is not focused on the displacement plate-beamsplitter (so the power density is low, and imperfections are not focused into the measurement volume), and because it is more difficult to achieve acceptable levels of parallelism in the plates than acceptable levels of surface quality through polishing.

Considering $|A|$ and $\mathcal{R}_1\approx \mathcal{R}_2$ are generally more important than $|\mathcal{R}_1|$ in determining performance, properly sourced and inspected float glass should be competitive with precision ground optical flats in terms of performance despite their significantly lower cost. More generally this analysis shows that tolerancing issues should not be prohibitive even with designs that take advantage of the longer ranges this method enables.

The power in the $i\mathrm{th}$ beam, P_i , can be found by applying the Fresnel power equations to each reflection or transmission, accounting for the reflectivity of the second surface, R_m . When this is done, equation (10) results. Similar calculations have been performed by others for wedge beamsplitters, (Beers 1974)

$$\begin{align} P_i = \begin{cases} P_\mathrm TR & i=0\\ P_\mathrm TR_m^iR^{i-1}(1-R)^2 & i>0, \end{cases} \end{align} \tag{ 10 }$$

where $P_\mathrm T$ is the total power incident onto the displacement plate-beamsplitter. The analysis is greatly simplified by the fact that the fraction of light transmitted on each internal reflection at a glass-air interface, R, is equal to the fraction of light initially reflected off of the glass at the air-glass interface (Beers 1974). The Fresnel equations in the p and s-polar cases are used to find this fraction, R,

$$\begin{align} R = \begin{cases} \left[\frac{\frac{\cos{\beta}}{\nu}-\sqrt{1-\left(\frac{\sin{\beta}}{\nu}\right)^2}}{\frac{\cos{\beta}}{\nu}+\sqrt{1-\left(\frac{\sin{\beta}}{\nu}\right)^2}}\right]^2 & \mathrm s -\mathrm {polar}\\[8pt] \left[\frac{\nu\cos{\beta}-\sqrt{1-\left(\frac{\sin{\beta}}{\nu}\right)^2}}{\nu\cos{\beta}+\sqrt{1-\left(\frac{\sin{\beta}}{\nu}\right)^2}}\right]^2 & \mathrm p - \mathrm {polar}. \end{cases} \end{align} \tag{ 11 }$$

Note that the beamsplitter produces degenerate outputs for excessively large and excessively small angles of incidence since,

$$\begin{align*} \lim_{R\to 1} P_i=\left\{\begin{matrix} P_\mathrm T, \,\, i=0 \\ 0\,\, \forall i\neq 0 \end{matrix}\right. , \end{align*}$$

$$\begin{equation} \lim_{R\to 0} P_i=\left\{\begin{matrix} P_\mathrm TR_m, \,\, i=1 \\ 0\,\, \forall i\neq 1 \end{matrix}\right. . \end{equation} \tag{ 12 }$$

Under the assumption that the main source of noise is Gaussian and signal-independent (ie shot noise from unwanted background light entering the detector, or thermal noise), it is possible to use a CRB to find the minimum possible uncertainty in measured velocity resulting from a single particle passing through n Gaussian beams spaced x apart, each with $1/e^2$ beam radius r and power P_i . The relative 1 standard deviation precision in measurement velocity V is bounded by,

$$\begin{equation} \frac{\sqrt{\mathrm{Var}{[\hat{V}]}}}{V}\geqslant\frac{1}{\delta r_{S/N}\sqrt{\eta}}, \end{equation} \tag{ 13 }$$

where $\delta = x/r$ is the ratio between beam width and beam separation, and $r_{S/N} = \sum A_i/N_g$ is the signal to noise ratio. The value for $\sum A_i$ can be determined by summing the amplitudes of the Gaussian peaks observed in a return signal or it can be predicted from first principles using the laser beam's power output, the particle scattering parameters, and the definition given in the appendix. The noise N_g can be determined by calculating the standard deviation of the noise in a return signal. The variable η is the power utilization efficiency, which is a quantity that is specific to a given configuration that represents how well the beam power is utilized in producing Fischer Information for V. Our derivation is given in the appendix. It follows closely the derivation given by Fischer et al (not the original author for which Fischer Information is named), and matches their results for the special case of n = 2, $P_0 = P_1$, and when $P_\mathrm T = P_0+P_1$ (ie, equation (10) is not used) (Fischer et al 2010). The general case for η, which allows for any number of beams of any arbitrary power level so long as they have the same width and they are Gaussian is,

$$\begin{equation} \eta \equiv \frac{1}{2\sqrt{\pi}}\left[\sum_{i=0}^{n-1}\left(\frac{P_i}{P_T}\right)^2i^2-\frac{\left(\sum_{i=0}^{n-1}i\left(\frac{P_i}{P_\mathrm T}\right)^2\right)^2}{\sum_{i=0}^{n-1}\left(\frac{P_i}{P_\mathrm T}\right)^2}\right]. \end{equation} \tag{ 14 }$$

Important properties of η are that it is invariant to changes in laser power, invariant to spatial shifts, and invariant to spatial scale. This means η is an intrinsic quantity related to the geometry of the beam pattern itself. Under the assumptions leading to equation (10) (ie, our setup), and in the limit of infinite numbers of beams (which would occur if the measurement volume were infinitely wide), η can be expressed in terms of just R_m and the ratio of the power in the first transmitted beam to the reflected beam, $R_{1/0}$,

$$\begin{align} \eta=\frac{R_m^2(1-R)^4}{2\sqrt{\pi}(1-R_L^2)^3}\left(1 + R_L^2-\frac{R_m^2(1-R)^4}{R^2(1-R_L^2)+R_m^2(1-R)^4}\right). \end{align} \tag{ 15 }$$

The expression utilizes the definition $R_L\equiv R R_{1/0}$ where R is fully specified by R_m and $R_{1/0}$. In fact it can be shown that $R_L = R_m+R_{1/0}/2-\sqrt{R_{1/0}R_m+R_{1/0}^2/4}$. The expression for R_L follows from its definition and equation (10). A graph of $\sqrt{\eta}$ as a function of $R_{1/0}$ for $R_m = 1$ is given in figure 2.

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As illustrated in figure 2, putting more of the laser's energy in the first beam results in the greatest signal efficiency, however the cost of doing so is higher incidence angles are needed to achieve this state. Therefore the sensor will experience greater sensitivity to changes in angle (ie from vibration or setup error) as well as greater difficulty detecting the other peaks in the presence of other forms of noise. The performance degrades both when $R_{1/0}\gg 1$ and $R_{1/0}\ll 1$, as expected, since in either case almost all of the energy goes into a single output beam.

Compared to two-spot or two-sheet ToFA, our method has greater efficiency for all $R_{1/0}$, although this is because the higher order beams of our method increase the effective sampling volume: equation (15) assumes an unbounded measurement volume by considering an infinite number of beams. Often only the first few beams (at most three or four) are needed to achieve similar performance to that given in equation (15). This is shown by the $\sqrt{\eta}$ curves in figure 2 where only three of the generated beams are in the measurement volume: performance is similar to when all of the beams are in the measurement volume. Regardless of how many beams are kept in the measurement volume, when considering any number of beams truncated at n > 2, for a given x, the measurement volume of our method exceeds that of two-spot or two-sheet ToFA. Therefore, for a given volume two-spot ToFA can be more power efficient if the beam splitting method is highly efficient, but only by at most $3-\sqrt{5}$ if the second surface for our method is highly reflective (see equation (19) and subsequent discussion). Actually this means that our method is up to $76.4\%$ efficient at producing a two-spot beam by blocking off or otherwise ignoring the higher order beams, which is lower than most two-spot methods but not greatly so. Note that the $\sqrt{\eta}$ curve when two beams are in the measurement volume is not quite identical to the two-spot curve, even if re-scaled, because the two-spot curve does not include a power loss model that changes with $R_{1/0}$.

Matching the power in the first two beams often strikes a good balance between maximizing η and achieving reasonable signal profiles and incidence angles. It also allows detection of reverse flow if the third beam is kept. Indeed, figure 2 shows $R_{1/0} = 1$ achieves nearly optimal values of η regardless of how many beams are kept in the measurement volume. If just two beams are kept, the performance is very close to optimal. We now discuss various aspects of this special configuration both due to its theoretical advantages (especially if only the first two or even three beams are kept) and to illustrate various aspects of the beam profile in general.

It is possible to tune the angle of incidence, β, in order to match the power of the first reflection and the first transmission beams ($R_{1/0} = 1$). In fact, any power ratio between the i = 0 and i = 1 beams is possible in most cases. The tuning angle required to match power in the first two beams can be calculated algebraically. This is done by setting $P_0 = P_1$ using equation (10), substituting equation (11) for R, and solving for β. The result is,

$$\begin{equation*} \rho=\sqrt{\frac{1+4R_m}{R_m}}, \end{equation*}$$

$$\begin{align} \beta = \begin{cases} \arccos{\left[\frac{1}{2}\sqrt{\left(1-\nu^2\right)\left(2-\rho\right)}\right]} & \mathrm s - \mathrm {polar}\\[10pt] \arccos{\left[\sqrt{\frac{\left(1-\nu^2\right)\left(1-\nu^4\left(1+4R_m(2-\rho)\right)\right)}{1+\nu^8-2\nu^4(1+8R_m)}}\right]} & \mathrm p - \mathrm {polar} \end{cases}. \end{align} \tag{ 16 }$$

Notice that the beam power matching equations depend only on two variables: the index of refraction ratio at the beamsplitter's first surface, and the reflectivity of the second surface. These equations are plotted in figure 3. Notice that for the s-polar orientation there always exists a value of ν such that a desired matching angle β is achieved, while for p-polar light the matching angle achieves a well-defined extremum at just above ν = 1 and just below $\nu = 1/\xi$. If the beam spacing is strongly dependent on β, the system is more difficult to align and more susceptible to vibration. Also if $\Delta f_p$ is high relative to b, the level of defocusing between beams in the measurement volume will be significant. Since $\frac{\mathrm dx}{\mathrm d\beta}$ is smaller for β close to 45^∘ and $\Delta f_p$ is smaller near this angle too, the s-polar orientation has significant theoretical advantages, especially for high index materials (but not exceeding ν ≈ 3).

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It is not always possible to match the power of the two beams as indicated by the fact that these equations do not always return a real number. This is the case if the argument of the square root is negative or if the argument of the inverse cosine exceeds 1. For s-polar light, beam matching is only possible if $(1-\nu^2)(2-\rho)\geqslant 0$ and $(1-\nu^2)(2-\rho)\leqslant 4$, which implies $|\nu|\geqslant 1$ and $|\nu|\leqslant \sqrt{(\rho+2)/(\rho-2)}$. For positive indices of refraction, the combination of these two requirements can be expressed as,

$$\begin{align*} \xi=\sqrt{\frac{\rho+2}{\rho-2}}, \end{align*}$$

$$\begin{align} \nu \in [1,\xi]. \end{align} \tag{ 17 }$$

For p-polar light and positive indexes of refraction, for the same reason, it is only possible to match beam powers in the domain,

$$\begin{align} \nu \notin \left[\frac{1}{\xi},1\right]. \end{align} \tag{ 18 }$$

Therefore in the p-polar case there is a small region of index of refraction space where beam matching is possible and also $0\lt\nu \lt1$. This suggests a second potential configuration for beam splitting. The configuration is more compact than the conventional one since the angle of incidence is small. However it is only possible to achieve in special circumstances such as with infrared light and a Germanium-air interface since $\xi(R_m = 1)\approx 4.236$. See figure 4 for a depiction of both the conventional and second configuration.

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The amount of light contained in the first two beams when the beams are forced to have the same power is independent of polarization as well as ν. The rest of the light is absorbed, or goes into the periphery beams. When the beams are tuned to this angle, at each glass-air interface, the fraction of light reflected is given by the solution of $P_\mathrm TR = P_\mathrm TR_m(1-R))^2$ for R. This comes directly from setting $P_0 = P_1$ in equation (10). The solution for R under this constraint, denoted by $R_\mathrm {eq}$, is

$$\begin{align} R_\mathrm {eq}=1+\frac{1}{2R_m}-\frac{\rho}{2\sqrt{R_m}}. \end{align} \tag{ 19 }$$

The other root obtained is non-physical. Notice only one independent variable is present, R_m . The maximum amount of light contained in the first two beams, if they match powers, occurs when $R_m = 1$. According to equation (19), this means the maximum power in the first two beams when they are matched is $R_0+R_1 = 2R_\mathrm {eq} = 3-\sqrt{5}\approx 76.4\%$. Substituting $R_\mathrm {eq}$ for R into equation (10), it follows immediately that after the first two beams the light contained in each successive beam decreases geometrically as follows

$$\begin{align} P_{i+1}=R_\mathrm {eq}R_mP_i\quad \forall \quad i>0. \end{align} \tag{ 20 }$$

Again the only independent variable in equation (20) is R_m . Since this variable is generally close to 1, the power fraction in each beam is essentially universal under the condition that $R_{1/0} = 1$. The beam powers for $R_m = 1$ are shown in figure 5.

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A laser anemometer's temperature sensitivity comes from differences in the Doppler shift sensitivity in LDA (usually equivalent to a fringe spacing difference) or changes to the beam separation for standard ToFA methods (Kitchen et al 2003). We show theoretically that the temperature sensitivity is negligible in our setup using an unstabilized Sanyo 405 nm laser diode and soda-lime glass displacement plate-beamsplitter with 20^∘ C room temperature. All of the properties used for the calculations in this section are given in table 1.

Table 1. Laser and beamsplitter properties used to calculate temperature and coherence errors.

Parameter	Value
Rm	0.9
ν	1.54 Rubin (1985)
ω	2.8 mm
$\Delta \lambda$	1 nm
$\mathrm {FWHM}$	1 nm
$\alpha\equiv \frac{1}{\omega}\frac{\partial \omega}{\partial T}$	$9\times 10^{-6}\,^\circ \mathrm{C}^{-1}$ Ashby (2013)
λ	405 nm
$\frac{\partial\lambda}{\partial T}$	0.06 nm$\,^\circ \mathrm{C}^{-1}$
$\frac{\partial \nu}{\partial \lambda}$	$-10^{-4}\,$nm−1 Rubin (1985)
$\frac{\partial \nu}{\partial T}$	$2\times 10^{-6}\,^\circ \mathrm{C}^{-1}$ Jewell (1991)
β	82.9∘

The sensor's temperature dependence comes from a combination of changes to the index of refraction induced by lasing wavelength changes, glass expansion, and index of refraction changes with temperature. The values provided in table 1 are used to approximate the change in index of refraction with respect to a small change in temperature,

$$\begin{equation} \frac{\mathrm d\nu}{\mathrm dT}=\frac{\partial \nu}{\partial T} + \frac{\partial \nu}{\partial \lambda}\frac{\mathrm d\lambda}{\mathrm dT}. \end{equation} \tag{ 21 }$$

This means $\frac{\mathrm d\nu}{\mathrm dT} = -4\times 10^{-6}\,^\circ \mathrm{C}^{-1}$. This result can be used in conjunction with equation (1) to find the change in separation distance with respect to a small temperature change,

$$\begin{equation} \frac{\mathrm dx}{\mathrm dT}=\frac{\partial x}{\partial \omega}\frac{\partial\omega}{\partial T}+\frac{\partial x}{\partial \nu}\frac{\partial \nu}{\partial T}. \end{equation} \tag{ 22 }$$

For our setup this means $\frac{\mathrm dx}{\mathrm dT} = 8\,\textrm{nm}\,^\circ \mathrm{C}^{-1}$. The partial derivatives of x can be obtained by differentiating equation (1). Close to our linearization temperature, the temperature induced error rate in velocity readings is,

$$\begin{equation} \frac{\frac{\mathrm dV}{\mathrm dT}}{V}=\frac{\frac{\mathrm dx}{\mathrm dT}}{x}. \end{equation} \tag{ 23 }$$

For our setup a value of $\frac{\frac{\mathrm dV}{\mathrm dT}}{V} = 1\times 10^{-5}\,^\circ \mathrm{C}^{-1}$ is found. The proposed method is therefore essentially temperature insensitive because a 100 ^∘C temperature change will cause only a 0.1% measurement error.

The effect of the laser's incoherence on x is very important when using an unstabilized laser diode without temperature control because most diode lasers exhibit poor coherence properties on their own. Since our method uses geometric optics only, temporal coherence is not a factor. Non-ideal spectral coherence on the other hand can reduce both accuracy and precision. It could also potentially limit how small x can be made because dispersion broadening of the beam spacing could broaden the beams into each other for sufficiently small x (note however that dispersion does not change the degree to which the output beams are parallel). The change in beam separation for a given wavelength drift, which is also the peak width increase for a given spectral bandwidth (due to dispersion), is given as follows,

$$\begin{equation} \frac{\mathrm dx}{\mathrm d\lambda}=\frac{\partial x}{\partial \nu}\frac{\partial \nu}{\partial\lambda}. \end{equation} \tag{ 24 }$$

Since the change in beam spacing is directly proportional to $\frac{\partial \nu}{\partial\lambda}$, low dispersion is beneficial for this method. However, we have already stated the benefits of high ν when using an s-polar configuration, but high index of refraction materials generally have low Abbe numbers. The choice of plate material is a tradeoff. For our setup a value of $\frac{\mathrm dx}{\mathrm d\lambda} = 65$ is found. For a 1nm wavelength drift or a 1 nm FWHM spectral bandwidth (both are typical for a diode laser), the corresponding wavelength fluctuation or beam width increase is $\frac{\frac{\mathrm dx}{\mathrm d\lambda}}{x}\Delta\lambda = 0.01\%$ of x. As a comparison, two beam LDA has a fringe spacing relation of $x = \frac{\lambda}{2\sin{\theta/2}}$ where θ is the angle between the two beams, so $\frac{\frac{\mathrm dx}{\mathrm d\lambda}}{x}\Delta\lambda = \frac{\Delta \lambda}{\lambda} = 0.2\%$ and this is true regardless of the angle between the beams (in reality there are also other reasons to stabilize a diode laser for LDA) (Albrecht et al 2003). Beam width broadened and beam separation are therefore affected very little by dispersion resulting from the laser's lack of spectral coherence in our setup despite using a glass with moderate dispersion properties. Hence, even for unstabilized diode lasers we recommend optimizing ν rather than the Abbe number. Non-ideal spatial coherence limits how small the focal point can be and thus how small the measurement volume can be, but otherwise does not effect accuracy or precision directly. Since this method does not put strong restrictions on the light source's coherence it may even be possible to use non-lasing sources so long as their spectral radiance is sufficient to achieve good signal to noise ratios.