OPTICAL DETECTION SYSTEM CALIBRATION

Abstract

According to a first aspect of the present invention there is provided a method of measuring the optical reflectance R of a target using a detection system comprising a light emitter and a light detector spaced apart from one another. The method comprises illuminating the target with the light emitter, detecting light reflected from the target using the light detector, wherein the light detector provides an electrical output signal S.sub.S indicative of the intensity of the detected light, and determining the optical reflectance R of the target according to (Formula 1), where R.sub.R is the spectral reflectance of a reference standard, S.sub.R is the detector electrical output signal with the reference standard in place, S.sub.H is the detector electrical output signal with no target in front of the light emitter and light detector, and M is a calibration factor.

[00001]$\begin{array}{cc}R=M.Math.R_R\frac{s_s-s_H}{M.Math..Math.s_R-s_H.Math.-R_R.Math.s_R-s_s.Math.},& \left(1\right)\end{array}$

Claims

1. A method of measuring the optical reflectance R of a target using a detection system comprising a light emitter and a light detector spaced apart from one another, the method comprising: illuminating the target with the light emitter; detecting light reflected from the target using the light detector, wherein the light detector provides an electrical output signal S.sub.S indicative of the intensity of the detected light; and determining the optical reflectance R of the target according to $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]},$ where R.sub.R is the spectral reflectance of a reference standard, S.sub.R is the detector electrical output signal with the reference standard in place, S.sub.H is the detector electrical output signal with no target in front of the light emitter and light detector, and M is a calibration factor.

2. A method according to claim 1 and comprising: determining a height z of the target above the detection system; using said height z to determine a height scaling factor η(z); scaling the determined reflectance using the scaling factor according to $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]}.Math.\eta \left(z\right).$

3. A method according to claim 1, wherein said reflectance R is measured for one or more wavelengths λ, according to: $R\left(\lambda \right)=M\left(\lambda \right).Math.R_R\left(\lambda \right)\frac{s_S\left(\lambda \right)-s_H\left(\lambda \right)}{M\left(\lambda \right).Math.\left[s_R\left(\lambda \right)-s_H\left(\lambda \right)\right]-R_R\left(\lambda \right)\left[s_R\left(\lambda \right)-s_S\left(\lambda \right)\right]}$

4. A method according to claim 1, wherein said electrical output signals are integrated over a wavelength range, and said reflectances represent reflectances averaged over the wavelength range.

5. A method according to claim 1, said step of determining the optical reflectance R of the target comprising obtaining from a memory values for each of R.sub.R, S.sub.R, S.sub.H and M, and evaluating the equation $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]}.$

6. A method according to claim 1, wherein said step of determining the optical reflectance R of the target comprises using said output signal S.sub.S as a look-up to a look-up table, the look-up table being populated with reflectance values evaluated using the equation $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]}$ and values for each of R.sub.R, S.sub.R, S.sub.H and M.

7. A method according to claim 1, wherein the calibration factor M is dependent upon a target material type.

8. A method of obtaining calibration data for a detection system configured to use the method of claim 1 to measure optical reflectance, the method comprising: operating a calibration detection system comprising a light emitter and a light detector spaced apart from one another, for a plurality of targets of known optical reflectances R and obtaining the respective electrical output signals of the detector S.sub.R; and solving the following equation to determine a calibration factor M; $S_R=k.Math.I_0.Math.\frac{R}{M}\frac{1}{1-\frac{R}{M}}.$

9. A method according to claim 8, wherein: said calibration detection system is the detection system for which calibration data is being obtained, or said calibration detection system is a different system from the detection system for which calibration data is being obtained and the method comprises providing the calibration factor M to the detection system for which calibration data is being obtained.

10. A method according to claim 8, wherein the detection system for which calibration data is being obtained is a detection system configured to use the method of claim 3, the method comprising performing the steps of operating and solving for one or more wavelengths, and determining a value of M(λ) for each wavelength.

11. A method according to claim 8, wherein the detection system is configured to use the method of claim 2, and comprising repeating the steps of operating and solving, for a plurality of different heights z above the calibration detection system, to determine values of M and η(z) for respective heights.

12. An optical reflectance measurement system comprising a chassis and, mounted to the chassis, a light emitter and a light detector, the system further comprising a processor or processors configured to: cause the light emitter to illuminate the target; cause the light detector to detect light reflected from the target, wherein the light detector provides an electrical output signal S.sub.S indicative of the intensity of the detected light; and determine the optical reflectance R of the target according to $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]},$ where R.sub.R is the spectral reflectance of a reference standard, S.sub.R is the detector electrical output signal with the reference standard in place, S.sub.H is the detector electrical output signal with no target in front of the light emitter and light detector, and M is a calibration factor.

13. An optical reflectance measurement system according to claim 12 and comprising a proximity sensor for determining a height z of the target above the detection system, said processor or processors configured to: use said height z to determine a height scaling factor η(z); scale the determined reflectance using the scaling factor according to $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]}.Math.\eta \left(z\right).$

14. An optical reflectance measurement system according to claim 12, said processor or processors configured to determine said reflectance R for one or more wavelengths λ, according to: $R\left(\lambda \right)=M\left(\lambda \right).Math.R_R\left(\lambda \right)\frac{s_S\left(\lambda \right)-s_H\left(\lambda \right)}{M\left(\lambda \right).Math.\left[s_R\left(\lambda \right)-s_H\left(\lambda \right)\right]-R_R\left(\lambda \right)\left[s_R\left(\lambda \right)-s_S\left(\lambda \right)\right]}$

15. An optical reflectance measurement system according to claim 12 and comprising a memory, wherein, either, said processor or processors is configured to obtain from said memory, values for each of R.sub.R, S.sub.R, S.sub.H and M, and to evaluate the equation $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]},$ or said processor or processors is configured to use said output signal S.sub.S as a look-up to a look-up table stored in said memory, the look-up table being populated with reflectance values evaluated using the equation $R=M.Math.R_R\frac{s_S-s_H}{M.Math.\left[s_R-s_H\right]-R_R\left[s_R-s_S\right]}$ and values for each of R.sub.R, S.sub.R, S.sub.H and M.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0027] FIG. 1 illustrates schematically an optical reflectance detection system used to measure the reflectance of a target;

[0028] FIG. 2a shows simulated results for an optical reflectance detection system, plotting detected signal S intensity versus a number of rays N having the right angles to hit the detector;

[0029] FIG. 2b shows a portion of the results of FIG. 2a with the signal S scale magnified for lower values of S;

[0030] FIG. 3 shows signal intensity versus reflectance characteristics for compensated and an uncompensated detection systems;

[0031] FIG. 4 illustrates schematically a reflectance detection system;

[0032] FIG. 5 is a flow diagram illustrating a reflectance detection method; and

[0033] FIG. 6 is a flow diagram illustrating a method of obtaining compensation data for a reflectance detection system;

[0034] FIG. 7 illustrates schematically a number of accepted modes;

[0035] FIG. 8 illustrates a simplified maximum light path scheme; and

[0036] FIG. 9 illustrates the FoI and FoV for a simplified reflectance detection system.

DETAILED DESCRIPTION

[0037] An approach to measuring the optical, spectral reflectance of a target will now be described and which takes into account the loss of light intensity arising from multiple reflection paths between the emitter and the detector of the system. This novel approach makes use of the assumption that each ray of light, originating from the emitter, is scattered at each reflection into a number M of rays of equal intensity, whilst the reflection reduces the total intensity by a factor equal to the reflectance of the reflecting surface R(λ).

[0038] For the purpose of illustration, it is assumed that the chassis is metallic and has a reflectance R.sub.2(λ)=1, whilst the target has a reflectance R.sub.1(λ)=R(λ). Moreover, upon each reflection from the target and the chassis, the light intensity I.sub.D arriving at the detector will be reduced by the factor M. So, if the total number of reflections between emitter and detector is j, the total effect of the scattering will be to reduce the intensity at the detector by M to the power j, i.e. M.sup.j. If it is assumed that N is the number of rays that have the right angle θ.sub.n to hit the detector and I.sub.0 is the light intensity of the emitter, then:

[00014]$\begin{array}{cc}{I}_D=I_0.Math.{.Math.}_{j=1}^N\frac{{R\left(\lambda \right)}^j}{M^j}\mathrm{where}:& \left(2\right)\end{array}$$\begin{array}{cc}{\theta }_n={\tan }^{-1}\left(\frac{2.Math.n.Math.h}{d}\right)& \left(3\right)\end{array}$

[0039] As a result, the analogue (electrical) signal produced by the detector will be:

[00015]$\begin{array}{cc}{S}_{\mathrm{NM}}=k.Math.I_0.Math.{.Math.}_{j=1}^N\frac{R^j}{M^j}+\gamma & \left(4\right)\end{array}$

where k is a multiplicative factor (taking into account the responsivity of the photodetector, potential filter responses and/or interposed media) and γ takes into account the amount of diffused light that reaches the detector, the dark counts and the cross-talk.

[0040] FIG. 2a shows simulated data for an optical detection system, illustrating the signal intensity (x-axis) versus the number of rays N having the right angles to hit the detector, for a range of assumed values of M, 1 to 10. FIG. 2b shows the same data but with the x-axis expanded and showing only smaller values of signal intensity (up to 1.6 on the scale). It will be apparent from these figures that the signal intensity tends to saturate for an increasing N. As equation (4) is a geometric series and is convergent (since R/M<1), one may substitute the series with its sum; the sum of the series is already reached for N=6 and M=2. In reality, the amount of allowed rays must be very high (theoretically infinite) and certainly greater than six, meaning:

[00016]$\begin{array}{cc}{S}_{\mathrm{NM}}\sim S_{\infty M}=k.Math.I_0.Math.\frac{R}{M}\frac{1}{1-\frac{R}{M}}+\gamma & \left(5\right)\end{array}$

[0041] The advantage of equation (5) is that is straightforward to compensate for the non-linear effect, by removing the offset γ and multiplying the remainder by the factor (1−R/M). This is illustrated by FIG. 3 which shows the normalized integral signal intensity versus (actual) target reflectance where M is assumed to have a value of 1.686, with the upper trace (broken line) showing the uncompensated data and the lower trace (solid line) showing the compensated data.

[0042] Considering now the signal in terms of its wavelength response (as is necessary in the case of spectrometers) and relating it to the target reflectance, the commonly used linear formulation is:

s(λ)=k.Math.I.sub.0(λ).Math.R(λ)   (6)

[0043] This assumption of linearity with reflectance of material under test (equation (1)) in such a configuration creates a potential flaw in the calibration algorithm. Using instead the formulation of equation (4), equation (6) becomes:

[00017]$\begin{array}{cc}s\left(\lambda \right)=k.Math.I_0\left(\lambda \right).Math.{.Math.}_{j=1}^N\frac{R^j\left(\lambda \right)}{M^j\left(\lambda \right)}& \left(7\right)\end{array}$

[0044] Replacing the series with the sum in line with equation (5) we get:

[00018]$\begin{array}{cc}s\left(\lambda \right)=k.Math.I_0\left(\lambda \right).Math.\frac{R\left(\lambda \right)}{M\left(\lambda \right)}\frac{1}{1-\frac{R\left(\lambda \right)}{M\left(\lambda \right)}}& \left(8\right)\end{array}$

[0045] This can be reformulated to provide an expression for determining the target reflectance:

[00019]$\begin{array}{cc}R\left(\lambda \right)=M\left(\lambda \right).Math.R_R\left(\lambda \right)\frac{s_S\left(\lambda \right)-s_H\left(\lambda \right)}{M\left(\lambda \right).Math.\left[s_R\left(\lambda \right)-s_H\left(\lambda \right)\right]-R_R\left(\lambda \right)\left[s_R\left(\lambda \right)-s_S\left(\lambda \right)\right]}& \left(9\right)\end{array}$

NB. In equations (6) to (8), s(λ) is considered to be only the signal arising from reflectance at the target and includes dark count and cross-talk. By providing calibration, the dark count and corss-talk are removed. In equation (9), S.sub.H(λ). corresponds to the offset factor γ of equation (5).

[0046] The parameter M provides a means to quantify the non-linearity of an optical system such as that illustrated in FIG. 1. The smaller the value of M, the more non-linear the optical system is. If M tends to infinity, the system is linear with reflectance and equation (9) is equal to equation (1). The use of equation (9) linearizes every optical system response. Therefore, by prior calibration in terms of M (and γ), every optical system can be used no matter how “non-linear” a case it represents.

[0047] The procedure to compensate the integral signal (integral over the wavelength if coming from a spectrometer) coming from the detector may be described as follows: [0048] 1. Acquisition of signals for respective targets with different reflectances. [0049] a. If the output detector signal is a spectrum, the integral over the wavelengths might be used (to provide an average value of M over the wavelength range). [0050] b. Alternatively, deduce M for each wavelength. [0051] 2. Fit the obtained data pairs [reflectance, detector signal] with equation (5). [0052] 3. Extract the parameters γ and M from the fitting (step 2)

[0053] It will be appreciated that changing the target height with respect to the detections system (i.e. h in FIG. 1), the optical field changes as the light intensity arriving to the target, as well as the amount of light coming from the multi-paths interaction, will be different. Consequently, a system calibrated with equation (1) will only provide a satisfactory result (subject to the non-linearities produced by multi-path reflections) for a given target height if calibration is performed for that height. If however we perform calibration using equation (5) however, at a target height that is different for the calibration target height, the system will still show a linear behaviour, only on a different M.

[0054] This is provided for by equation (9) being dependent upon the target height (variable z);

[00020]$\begin{array}{cc}R\left(\lambda \right)=M\left(\lambda ,z\right).Math.R_R\left(\lambda \right)\frac{s\left(\lambda \right)-s_H\left(\lambda \right)}{M\left(\lambda ,z\right).Math.\left[s_R\left(\lambda \right)-s_H\left(\lambda \right)\right]-R_R\left(\lambda \right)\left[s_R\left(\lambda \right)-s\left(\lambda \right)\right]}.Math.\eta \left(z\right)& \left(10\right)\end{array}$

Where η(z) is the ratio of S(z.sub.REF)/S(z) (S integral over the wavelength of s(λ)), and z.sub.REF is a height of reference.

[0055] The procedure for additionally calibrating a system for height is as follows: [0056] 4. Repeat steps 1, 2, 3, for different heights in order to obtain M(z) for each height. [0057] 5. Obtain the ratio η(z) for each height by measuring S(z)

[0058] It will be appreciated that M(z) increases linearly with height until reaching a saturating value. The greater the height of the target above the system, the lower the contribution of multi-paths reflections will be to the detected signal and therefore the higher M will be. On the other hand, η(z) has a non-linear behaviour and it increases with increasing z (approximately according to a second degree polynomial of second grade).

[0059] In normal use, the height of the target above the system can be determined using, for example an on-board distance sensor and equation (10) can be used to determine the reflectance of the target, using the values M and η for that height as well as γ which is independent of height.

[0060] The approach described above to calibrate a detection system and to calculate reflectance essentially enable every system response to be considered linear with the reflectance. Furthermore, height compensation is easier to apply. The approaches allow for the equalization of output signals provided by multiple sensors located at different positions but still illuminated by the same source. This is an interesting case that is particularly useful when a system includes multiple detectors acting as miniaturized spectrometers, the detectors having overlapping wavelength ranges. As the approaches effectively eliminate the effect of the chassis holding the detector and emitter, optical detection modules can be sold without any restriction on the end use case.

[0061] Further advantages achieved by the described approaches may include: [0062] In the case of miniaturized spectrometers, the measured and processed spectra may have much less dispersion from device to device, either sharing the same chassis or using different chassis. [0063] They may improve the measurement precision. [0064] Height compensation may be applied. [0065] Spectra/signals at different heights can be compensated in order to provide consistent results.

[0066] The described approaches may be implemented into a system by embedding appropriate algorithms in the software/firmware of those systems, which measure and consequently provide the calculated reflectance of a target. After determining the parameters M(λ) by a priori calibration with reference reflectance targets, they may be stored in a register (memory location), along with the arrays R.sub.R(λ), S.sub.R(λ), and S.sub.H(λ). When a measurement is performed on a target, the raw signals generated by the photodetector, S.sub.S(λ), are processed according to equation (9) to provide the target reflectance R(λ). Where height compensation is provided for, and the detection system is capable of measuring the target height (e.g. using a proximity sensor), the parameter η(z) is determined during calibration and stored into the system. The raw signal is then processed according to equation (10). Alternatively the parameter η(z) is determined during calibration and equation (11) is applied to the raw signal. It will nonetheless be appreciated that the processing operations may be distributed, with certain tasks being performed outside of the device within which the detection module is provided.

[0067] FIG. 4 illustrate an optical reflectance detection system comprising a light emitter 3, light detector 4 and chassis 5. In addition, the system comprises a processor 8 and memory 9, where the processor 8 is configured to operate as discussed above. The memory 9 may store the various parameters required to evaluate equation (9), or equation (10) in which case the system also comprises a proximity sensor 10. Alternatively, the memory 9 may store a pre-calculated look up table mapping sensor output values to reflectances.

[0068] FIGS. 5 describes various steps (S1 to S3) associated with determining a reflectance value for a target, whilst FIG. 6 describes steps (S100 to S200) associated with calibrating the system, either using the system itself to perform the calibration or using a calibration system having the same characteristics.

[0069] It will be appreciated that the M(λ) and η(z) parameter calculation may be performed using an on-board computer of the device comprising the reflectance detector, e.g. on a smartphone. Alternatively, raw or pre-processed data may be sent via a network connection to a server or the like at which the parameter calculation is performed, with the calculated parameters being returned to the device where they are stored for later use.

[0070] Systems into which the reflectance detector can be integrated include, for example, MEMS-based miniaturized spectrometers, tuneable light sources (UV, visible or infrared).

[0071] It will be appreciated by the person of skill in the art that various modifications may be made to the above described embodiments without departing from the scope of the present invention.

Additional Technical Information

[0072] To better understand the proposals presented above, reference may be made to the following detailed technical discussion.

[0073] FIG. 7 illustrates certain accepted modes. [0074] I vs ζ

[0075] Three main mechanisms are taken into account for studying the behavior of intensity versus the height (or better ζ=z/d). These mechanisms limit the amount of light that reaches the detector (PD) and are: [0076] Maximum number of reflections (N.sub.M) [0077] Maximum light path [0078] Minimum acceptance angle [0079] Maximum number of reflections

[0080] Allowed modes are not infinite. In reality, modes which have a high value of N will undergo a high number of reflections and if their exiting intensity is low, they will not contribute to the signal read by the PD. In other words, for a certain target reflectance R, the maximum number of accepted modes N has a superior limit N.sub.M, which is imposed by the detector sensitivity.

[0081] Assuming that the PD is only able to read intensities ‘I’ higher than a certain threshold ‘I.sub.0’, a generic mode, in the case of no attenuation through the medium the following inequality must hold:

[00021]$\begin{array}{cc}I=I_S.Math.{\left(\frac{R}{M}\right)}^N\ge I_0.fwdarw.N\le .Math.\frac{\ln \left(\frac{I_0}{I_S}\right)}{\ln \left(\frac{R}{M}\right)}.Math.=N_M& \left(11\right)\end{array}$

[0082] This relationship is important, because it relates the PD sensitivity I.sub.0, the emitted intensity from the illuminating source I.sub.S, the target reflectance R, and the parameter M. The function └ ┘ represents the floor function. [0083] Maximum Light path (θ.sub.max)

[0084] We assume that an optical ray cannot travel longer than a certain distance, because of attenuation through the medium. Therefore, although the number of reflections that it must undergo is equal or lower than N.sub.M, that mode does not reach the PD if the optical path is longer than a certain limit.

[0085] Let us suppose that FIG. 8a depicts the situation in which a ray has reached the maximum light path (at z.sub.1, θ.sub.max). If the height is increased to z.sub.2 (FIG. 8b), and we keep the same angle θ.sub.max (N=3), light has to travel a longer path, therefore the same mode or higher is not allowed. Only inferior modes (N1, N2) may reach up to the PD, if the light path is shorter than the limit.

[0086] The longest path sets a limiting angle θ.sub.max at a certain elevation, beyond which light will not reach the PD.

[00022]$\begin{array}{cc}{I}_0=I_S.Math.\left(\frac{R}{M}\right).Math.e^{-\alpha .Math.r_{\max }}& \left(12\right)\end{array}$$\begin{array}{cc}{r}_{\max }=\frac{z}{\mathrm{sen}\left({\theta }_{\max }\right)}2N& \left(13\right)\end{array}$

[0087] Substituting (13) in (12):

[00023]$\begin{array}{cc}\ln \left(\frac{I_0}{Is}\right)=N.Math.\ln \left(\frac{R}{M}\right)-\alpha .Math.r_{\max }=N.Math.\ln \left(\frac{R}{M}\right)-\frac{\alpha .Math.z}{\mathrm{sen}\left({\theta }_{\max }\right)}2N& \left(14\right)\end{array}$$\begin{array}{cc}\mathrm{sen}\left({\theta }_{\max }\right)=\frac{2\alpha .Math.z.Math.N}{N.Math.\ln \left(\frac{R}{M}\right)-\ln \left(\frac{I_0}{I_S}\right)}& \left(15\right)\end{array}$

In equation (15), the medium is also taken into account by the attenuation coefficient α, N≤N.sub.M. [0088] θ.sub.min

[0089] FIG. 9 illustrates FoI and FoV simplified scheme. Because of the characteristic field of view (FoV) of the PD and the field of illumination (FoI) of the emitter, there is a minimum angle θ.sub.min, below which light cannot exit from the illuminator or enter to the PD. Although modes below such a limit would be allowed, the PD will read no power.

[0090] Referring to FIG. 8,

[00024]$\begin{array}{cc}{\theta }_{\min }=\max \left\{{\theta }_1,{\theta }_2\right\}=\left\{\frac{\pi }{2}-{\theta }_V,\frac{\pi }{2}-{\theta }_I\right\}& \left(16\right)\end{array}$ [0091] Effective Modes (N.sub.max−N.sub.min)

[0092] The three limiting mechanisms provide the number of possible modes for specific system settings (I.sub.0, I.sub.S, R, M, α, FoI, FoV, ζ), N.sub.max−N.sub.min. The amount of rays will clearly determine the amount of light sensed by the PD. By plotting the expression:

θ.sub.n=tan.sup.−1(2ζn)   (17)

it is possible to easily interpret how the limiting factors determine N.sub.max−N.sub.min, which are given by the interceptions with the lines θ=θ.sub.max and θ=θ.sub.min and the curve θ.sub.n(ζ). By increasing ζ the curves move towards the y-axis becoming steeper, the modes below θ.sub.max are more concentrated toward 90°, so N.sub.max is smaller.

[0093] By decreasing ζ, the difference N.sub.max−N.sub.min increases, as does the intensity. The maximum intensity is reached when ζ is equal to ζ.sub.T. Below this value, the intensity decreases again because N.sub.M is reached by angles that are smaller than θ.sub.max. At the same time, N.sub.min (minimum angle, from which the optical ray reaches the PD) increases, the difference N.sub.max−N.sub.min decreases. The intensity will reach its minimum value when ζ=ζ.sub.0.

[0094] From eq. (16) we can deduce the values of tan(θ.sub.max), tan(θ.sub.min) by substituting N.sub.M to n, and ζ.sub.T, ζ.sub.0 to ζ, respectively.

tan(θ.sub.max)=2ζ.sub.TN.sub.M   (18)

tan(θ.sub.min)=2ζ.sub.0N.sub.M   (19)

[0095] It is clear that the number of rays is limited and it is no longer legitimate to use the sum from 1 to ∞, but rather from N.sub.min to N.sub.max. For simplicity, the attenuation of each ray is excluded. It is only considered that modes beyond θ.sub.max are excluded.

[00025]$\begin{array}{cc}{.Math.}_{j=N_{\min }}^{N_{\max }}{\left(\frac{R}{M}\right)}^j={.Math.}_{j=0}^{N_{\max }}{\left(\frac{R}{M}\right)}^j-{.Math.}_{j=0}^{N_{\min -1}}{\left(\frac{R}{M}\right)}^j& \left(20\right)\end{array}$

sums of the first N.sub.max and N.sub.min−1 terms of a geometric series are:

[00026]$\begin{array}{cc}{.Math.}_{j=0}^{N_{\max }}{\left(\frac{R}{M}\right)}^j=\frac{1-{\left(\frac{R}{M}\right)}^{N_{\max }+1}}{1-\left(\frac{R}{M}\right)}& \left(21\right)\end{array}$$\begin{array}{cc}{.Math.}_{j=0}^{N_{\min }-1}{\left(\frac{R}{M}\right)}^j=\frac{1-{\left(\frac{R}{M}\right)}^{N_{\min }}}{1-\left(\frac{R}{M}\right)}& \left(22\right)\end{array}$$\begin{array}{cc}{.Math.}_{j=N_{\min }}^{N_{\max }}{\left(\frac{R}{M}\right)}^j=\frac{\frac{R}{M}}{1-\left(\frac{R}{M}\right)}\left[{\left(\frac{R}{M}\right)}^{N_{\min }-1}-{\left(\frac{R}{M}\right)}^{N_{\max }}\right]& \left(23\right)\end{array}$$\left(\mathrm{Please}\mathrm{note}M\mathrm{does}\mathrm{not}\mathrm{change}\mathrm{with}\zeta \right).$$\begin{array}{cc}{.Math.}_{j=N_{\min }}^{N_{\max }}{\left(\frac{R}{M}\right)}^j=\frac{\frac{R}{M}}{1-\left(\frac{R}{M}\right)}\left[{\left(\frac{R}{M}\right)}^{.Math.\frac{\tan \left({\theta }_{\min }\right)}{2\zeta }.Math.-1}-{\left(\frac{R}{M}\right)}^{.Math.\frac{\tan \left({\theta }_{\max }\right)}{2\zeta }.Math.}\right]& \left(24\right)\end{array}$

[0096] Therefore, what it is really changing with ζ is the weight of the following factor on the rest of the function.

[00027]$\begin{array}{cc}{\left(\frac{R}{M}\right)}^{N_{\min }-1}-{\left(\frac{R}{M}\right)}^{N_{\max }}={\left(\frac{R}{M}\right)}^{.Math.\frac{\tan \left({\theta }_{\min }\right)}{2\zeta }.Math.-1}-{\left(\frac{R}{M}\right)}^{.Math.\frac{\tan \left({\theta }_{\max }\right)}{2\zeta }.Math.}={\left(\frac{R}{M}\right)}^{.Math.\frac{{\zeta }_0}{\zeta }.Math.N_M-1}-{\left(\frac{R}{M}\right)}^{.Math.\frac{{\zeta }_T}{\zeta }.Math.N_M}& \left(25\right)\end{array}$

[0097] The higher the ζ the lower the modes. At higher elevations, the curve intensity versus reflectance becomes linear for a certain range, when theoretically only one ray is allowed to hit the PD. On the contrary, when ζ is low the number of modes increases and the terms

[00028]${\left(\frac{R}{M}\right)}^n$

become very small.

[0098] As already explained, below ζ.sub.T, the intensity decreases (verified experimentally) because no higher modes than N.sub.M are allowed and either the emitter or the PD have a field of illumination (FoI) and a field of view (FoV) respectively. When ζ=ζ.sub.0 the intensity approaches zero. Both ζ.sub.T and ζ.sub.0 depends on R/M as expected.

[0099] Exerimental results demonstrate that the model under the conditions described holds well.