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Optimization of sensor arrays for beam position estimation
Sensors and Actuators 87 Ž2000. 11–18 www.elsevier.nlrlocatersna
Optimization of sensor arrays for beam position estimation P.A. Sorichetti a,),1, C.L. Matteo b,2 , A.J. Marzocca b a b
Departamento de Fısica, Facultad de Ingenierıa, ´ ´ UniÕersidad de Buenos Aires, AÕ. Paseo Colon ´ 850, 1063 Buenos Aires, Argentina Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, UniÕersidad de Buenos Aires, Pabellon ´ ´ I, Cdad. UniÕersitaria, 1428 Buenos Aires, Argentina Received 4 December 1999; received in revised form 3 May 2000; accepted 5 May 2000
Abstract A design procedure is presented to optimize linear arrays of sensors for the estimation of the position of an incident beam. The procedure is based on the adjustment of the position and gain of the elements of the array in order to approximate the desired response Žin the least squares sense.. The optimization takes into account the shape of the beam profile, and provides adequate results for a wide range of beam diameters. An application is made to a linear array of circular sensors illuminated by a beam with a rectangular profile. Optimized designs are given for arrays of 2, 4 and 6 sensors. The experimental results were in good agreement with the calculations. The relative error was found to be less than "2% and the performance of the system was insensitive to background light. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Sensor; Array; Optimization; Beam position estimation
1. Introduction The estimation of the position of an incident beam of light has several applications w1,2x, for example, in the measurement of linear or angular displacements ŽFig. 1.. Simple photoelectric sensors, both in single and differential configurations, have often been used to detect the displacement of a beam with respect to a reference position, for instance, as part of a feedback loop andror to provide a rough estimation of the position of the beam w3x. Position sensitive detectors ŽPSD. w4x and charge-coupled-devices ŽCCD. w5–7x are at present employed to give much higher accuracy and resolution in a number of applications, including the precision alignment of heavy machinery, topographic leveling, and the measurement of small displacements in large buildings. The PSD is a continuous structure, which can locate the position of a
) Corresponding author. Tel.: q54-11-4342-2950; fax: q54-11-43425286. E-mail address: [email protected] ŽP.A. Sorichetti.. 1 P.A. Sorichetti at present is also affiliated with Professional Communications S.A., Rivadavia 755, 1002 Buenos Aires, Argentina. 2 C.L. Matteo belongs to the Argentine National Council of Scientific and Technological Research ŽCONICET., Argentina.
beam spot with very high spatial resolution and at high speed. On the other hand, the CCD works as a discrete array with a very large number of sensing elements, whose output can be post-processed in order to give a highly accurate estimation of the position of the illuminating beam. However, the added cost and complexity of CCD devices and the related post-processing hardware and software are a disadvantage, particularly for less demanding applications where only a moderate precision is required. Moreover, the CCD is much slower than the PSD due to the time needed to scan the array, which may be of the order of 1 ms. On the contrary, the CCD may detect and locate several beams simultaneously, while the PSD requires some sort of control at the sources to discriminate between beams. Commercially available photodetector arrays are also an alternative. Linear arrays with a fairly large number of sensors can be found w8x, with typical geometries usually limited to arrays of equidistant sensors. Therefore, integrated arrays with a large number of sensors make possible, in principle, to provide a good estimation of the position of the illuminating beam, even in the case of multiple beams w9x. However, processing the output of a large number of sensors results again in the requirement of complex and costly circuits.
0924-4247r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 0 0 . 0 0 4 6 8 - 4
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P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
Žb. All the sensors operate within their linear range. Žc. The displacement of the illuminating beam is aligned with the axis of the array. Žd. The irradiance profile of the illuminating beam is assumed to be uniform in the direction perpendicular to the axis of the array.
Fig. 1. Measurement of linear and angular displacements with sensor arrays.
In this work, a different approach to the problem is sought. An optimization procedure is presented in order to get the optimum response from a linear array with a limited number of discrete sensors. To minimize the complexity of the processing circuit, the output of the array results from a weighted sum of the output of the individual sensors. However, instead of considering an array of equidistant sensors, the procedure includes as optimization variables the positions of the sensors within the array, in addition to the relative weights of the sensor outputs. Calculations show that the additional degrees of freedom provided by removing the requirement of equidistance between sensors gives the optimized array a much better performance than conventional arrays of the same number of elements Ži.e., arrays with equidistant sensors, in which only the output weights may be optimised., while keeping the cost and complexity well below of systems based on CCDs or large integrated arrays. This improvement is confirmed by the comparison of the experimental results of optimized arrays and conventional Žequidistant. arrays. It is important to remark that the optimization procedure is not limited to the positioning of light beams. In addition, it may also be applied to other types of sensors, such as acoustic Žultrasonic. transducers, particularly when the integration of a large number of sensors would be impractical.
2. Selection criteria for the sensors of the array and the illuminating beam The optimization procedure makes a limited number of assumptions on the characteristics of the sensors and the illuminating beam. Ža. The responsivity of each sensor is uniform over all its area.
It must be remarked that procedure does not make any specific assumptions on the type of photodetector used. As long as the stated assumptions are met, any kind of sensor might in principle be used. Therefore, the selection of a particular type of sensor may be made considering other practical considerations Žcost, size, spectral response, temporal response, etc... In particular, semiconductor photodiodes Že.g., silicon and germanium devices. are widely available and quite adequate for the implementation of the optimized sensor arrays discussed in this work. The optimization procedure has no specific requirements for the selection of the source of the illuminating beam, as long as the beam spot is linear in shape, with constant intensity in the direction perpendicular to axis of the array. The main constraints are set by the geometry of the beam and the irradiance profile Žas indicated in Section 7 below., but they may be easily met by the use of incandescent, LED or laser sources, suitably diaphragmed, together with a cilindrical lens. Therefore, the selection of the source may be made taking into account the spectral range of the detectors used and other practical considerations such as size, cost, etc.
3. Modeling of the response of an element of the array Even though the general expression of the output of a sensor involves the use of a two-dimensional convolution of the beam profile and the responsivity function of the sensor, with the stated hypothesis the modeling of the response of an array element is reduced to a one-dimensional integral. Therefore, the output of the jth sensor of the array is given by Vj Ž a . s
X max j
HX min
f j Ž x y sj . g Ž x y a. d x ,
Ž 1.
j
where a is the displacement of the beam along the axis of the array with respect to its reference position Žassumed to be at x s 0., Vj Ž a. is the output of the sensor, f j Ž x . is the response of the sensor when illuminated by an infinitesimally thin beam located at x, and g Ž x . is the irradiance profile of the illuminating beam, assumed constant in the direction perpendicular to the x-axis. The sensor is located at x s s j and its lateral extension is given by X max j and X min j Žsee Fig. 2..
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
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Fig. 2. Geometrical disposition of the sensors and the illuminating beam.
It is worth noting that the elements of the array need not be identical, each element being described by its response function f j Ž x ..
4. Optimization procedure If the array consists of N elements, the output of the array can be obtained as a linear combination of the outputs of the sensors, that is N
Vo Ž a . s
Ý Ck Vk Ž a. .
Ž 2.
ks1
The output function Vo Ž a. of a given array will therefore depend on the beam irradiance profile g Ž x . and also on a set of 2 N parameters: the positions of the sensors, sk , and the weights Ck of the sensors outputs. The proposed optimization procedure for an array consists in choosing the optimum values of the 2 N parameters sk and Ck for Vo Ž a. to give the best approximation Žin the least squares sense. to the desired output function F Ž a., within the limits a max and a min . The optimized array will give a minimum value of the error residue E Es
amax
Ha
Ž Vo Ž x . y F Ž x . .
2
d x.
illuminating beam Žgiven, for instance, by a change of the distance between the array and the source of the beam.. It should be noted that although in many cases F Ž a. is chosen as a linear function, this needs not to be the case. Any reasonably well-behaved function, continuous and monotonous, might, in principle, be selected. An important property of optimized arrays with an even number of sensors may be obtained from symmetry considerations. If the output function of the array, F Ž a., is chosen as an odd function, i.e., F Ž a. s yF Žya., for example, as a linear function where F Ž a. s ka, it is easy to show that the positions of the sensors must be made symmetrical with respect to the center of the array. Moreover, the weights of symmetrically placed pairs of sensors should be equal and of opposite signs. In consequence, the effects of a uniform background illumination on the optimized are cancelled out, as long as all the sensors of the array Žand the associated processing circuit. remain within their linear range. This is an important characteristic from the point of view of applications, since in many cases a linear output function is desired and also the arrays must operate in the presence of background illumination.
Ž 3.
min
The minimization of the residue E as a function of the parameters may be carried out by any standard minimization technique, such as Newton or Conjugate-Gradient methods w10x. In general, several local minima of E may exist for a given array. This gives the possibility of selecting values of the parameters that give a good adjustment and also satisfy additional requirements, such as minimizing the influence of the variations of the spatial scale of the
5. Response function of a circular sensor To perform the optimization procedure outlined above, it is convenient to normalize all linear dimensions to a characteristic length of the sensors to be used, for instance, the radius of the active area in the case of circular sensors. In the remaining sections of this work, all linear dimensions will be normalized in this way, unless explicitly indicated.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
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Therefore, for a circular sensor of unit radius Ž r s 1., centered at the origin, the response function f j Ž x . as defined in Section 3 above, is f j Ž x . s 2'1 y x 2 .
Ž 4.
Therefore, the output of the jth element of the array is given by S jq1
HS y1 (1 y Ž x y s .
Vj Ž a . s 2
2
j
where Pn and Q n are polynomials, given by: for n s 0 x P0 Ž x ,bj . s 2
Ž 9.
Q 0 Ž bj . s 1 for n s 1
g Ž x y a. d x ,
Ž 5. 1
j
where X min j s S j y1 and X max j s S j q 1. The response function of the sensor, f j Ž x ., may be thus considered as the kernel of the integral Ž5..
P1 Ž x ,bj . s y
y
bj x
3
x2
Ž 10 .
q
2
3
Q1 Ž bj . s bj for n s 2
6. Power series expansion for the profile of the illuminating beam In order to carry out the optimization procedure, it is useful to determine an analytic approach to evaluate the output function V Ž a. for a given function g Ž x .. It is important to note that integral Ž5. may be evaluated analytically for the circular sensor kernel, if the function g Ž x y a. can be expressed as a power series in Ž x y a. inside the interval where it is non-zero:
°Ý B Ž x y a. g Ž x y a . s~ ¢ 0
n
n
where Bn are the expansion coefficients of the function g and d1 and d 2 are the lateral limits of the illuminating beam. Instead of evaluating a full series expansion, in many cases an adequate approximation to the function g Ž x y a. may be obtained by fitting a low order Ž n - 3 . polynomial in Ž x y a., for instance by the least squares technique. The integral of Eq. Ž5. may be calculated analytically by integrating each term of Eq. Ž6. with the kernel of Eq. Ž4., using the following indefinite integrals n
q
Ž y1 q 4 bj2 . x y 2 bj x 2 q x 3
3
8
3
4
y1 q 4 bj2 4
The expressions for higher values of n may be easily carried out with a symbolic integration program, if a better polynomial approximation to g Ž x y a. is desired.
2
j
d x,
In this work, only the first term of the power series expansion Ž n s 0. of the irradiance profile g Ž x y a. will be retained. That is, the irradiance profile of the illuminating beam is considered to be constant along the x-axis. Therefore, the function g Ž x . may be written as gŽ x. s
½
1 0
yd - x - d otherwise
Ž 7.
In Ž x ,bj .
p y 2
~
Q n Ž bj .
s 2 Pn Ž x ,bj . '1 y x 2 q Q n Ž bj . arcsin Ž x . p Q Žb . 2 n j
¢
x F y1 < x
Ž 8.
Ž 12 .
where d is the half-width of the beam. It must be noted, as indicated in Section 4, that the sensor radius is taken as unity. Taking into account Eqs. Ž5., Ž8., Ž9. and Ž12., the
where bj s a y s j . These functions, which are defined for x within the wy1,1x interval, may be extended analytically for other values as
°
Ž 11 .
7. Application to an array of circular sensors
otherwise
Ž 6.
H Ž x y b . '1 y x
Q 2 Ž bj . s
2a
yd1 - Ž x y a . - d 2
n
In Ž x ,bj . s 2
P2 Ž x ,bj . s
Fig. 3. Response function for a circular sensor.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
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Table 1 Gain and position of the array elements 2 Element array 4 Element array 6 Element array
Gain Position Gain Position Gain Position
y4.189 y4.751
y1.000 y1.877 y1.000 y1.842 y1.000 y1.267
y1.894 y3.649 y2.662 y2.974
output function Vj Ž a. with the beam half-width d as a parameter, can be written as Vj Ž a . s I0 Ž a q d,bj . y I0 Ž a y d,bj . .
Ž 13 .
Fig. 3 shows Vj as a function of bj for different values of d. The optimization procedure described in Section 2 above, has been applied to arrays of 2, 4 and 6 sensors. For the optimization procedure the target function was taken as linear: F Ž x . s kx with unity slope Ž k s 1., for a beam half-width d s 1, Ži.e., equal to the radius of the active area of the sensors.. The limit values for the position of the illuminating beam center, w a min ,a max x, were chosen as wy1.3,1.3x, wy2,2x and wy3,3x for 2, 4 and 6 sensors, respectively. These limiting values are found numerically, in each case, to give solutions with both good linearity and linear range for the given number of sensors. In particular, the limiting values wy1.3,1.3x for the array with two sensors are interesting, since they clearly reflect the improvement over an equidistant ŽAside to sideB . design: satisfactory results are obtained for w a min ,amax x greater than wy1,1x. It is easily understood that, given the symmetry of the problem, it is necessary to calculate the weights and positions of only one half of the sensors of the array, for instance, for those corresponding to x ) 0, as indicated in Section 4. Furthermore, the weights of the central pair of sensors are chosen as y1 and q1. The position and gain of the elements, together with the residue of the fitting E are given in Table 1. To illustrate the dependence of the results with the width of the illuminating beam, the output of the array
Fig. 4. Output for a six-element array.
1.000 1.877 1.000 1.842 1.000 1.267
1.894 3.649 2.662 2.974
4.189 4.751
Residue 0.007 Residue 0.004 Residue 0.064
with 6 elements is plotted in Fig. 4, for different values of the illuminating beam half-width, d. In addition, Fig. 5 shows the slope of the response of the array and the residue E as a function of d, for the same array. It may be seen that increasing d with respect to the initial value adopted for the optimization also results in a linear response, where the slope k is directly proportional to d. In addition, the value of the residue of the fitting process remains fairly constant over the whole range of beam diameters. However, it must be taken into account that there is a slight reduction of the linear response zone is observed at higher values of d. Similar results for arrays with 2, 4 and 6 sensors, for values of d from 1 to 3, are given in Table 2. These results show the robustness of the design procedure, which gives the arrays added flexibility for operation. For instance, the arrays may be used at different distances of the illuminating source. 8. Effects of non-ideal characteristics of sensors and processing circuits Real sensors have several characteristics Že.g., darkness current, imperfect matching, etc.. that must be taken into account in the implementation of an optimized array. Moreover, electronic processing circuits also have limitations Ži.e., offset voltages, finite bandwidth, thermal and shot noise, 1rf noise, etc... However, the impact of most of these non-ideal characteristics and limitations on the optimization procedure may be circumvented by adequate design techniques, as briefly discussed in this section.
Fig. 5. Parameters of the linear fit of the six-element array response.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
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Table 2 Slope of the array output and fitting residue vs. half-width of the illuminating beam d
k
E
2 4 6 2 4 6 Elements Elements Elements Elements Elements Elements 1 1 1.5 1.566 2 1.718 2.5 3
1 1.449 1.997 2.531 2.818
1 1.489 2.004 2.519 2.952
0.007 0.241 0.421
0.004 0.324 0.003 0.322 0.506
0.064 0.334 0.113 0.327 0.664
The effect of darkness current of the sensors is to superimpose a constant term into the output signal Vo Ž a., as it may be easily seen from Eq. Ž2.. This may be compensated, together with any eventual offset voltages of the processing circuit, by shifting the level of the output signal. Furthermore, the need for this adjustment may be eliminated by periodically modulating the amplitude of the illuminating beam, for instance, by means of a chopper wheel or, in practical applications, by direct Aon –offB modulation Žas in the case of LED or laser sources.. In this way, the effects of darkness currents and offset voltages may be easily eliminated by high-pass filtering in the processing circuit, thus blocking DC and low frequency components. The output signal is taken as the envelope of the rectified signal, after processing. This has the additional advantage of eliminating the excess low frequency noise Ž1rf noise. of the electronic circuit if the modulation frequency is chosen above the A1rf cornerB frequency of the electronic devices Žtypically, a few tens of hertz.. It should be remarked that the optimization procedure may be applied without changes to an amplitude modulated beam, as long as the bandwidth of the system is large enough to process the modulation frequency. Fluctuations in darkness currents, thermal and shot noise of electronic circuits fall within the cathegory of Awhite noiseB, with zero average value and constant noise power per unit bandwidth. The amplitude of these fluctuations may be characterised by their root mean square ŽRMS. value. Since the output signal is obtained as a linear combination of the outputs of the sensors, the RMS value of the fluctuations in the output signal SŽ Vo . may be easily calculated from the RMS value of the fluctuations of the output of the individual sensors SŽ Vj ., including the appropiate weights Ck S Ž Vo . s
)
N
2 Ý Ž Ck S Ž Vk . .
.
Ž 14 .
ks1
Eq. Ž14., referred to the input of the sensor system, makes possible to estimate the minimum intensity level of the illuminating beam for a given signal-to-noise ratio. This in turn conditions the achievable spatial resolution of the system. If necessary, the signal-to-noise ratio may be
improved by low-pass filtering of the output signal, up to the limit imposed by the required response time to changes in the position of the illuminating beam. With regard to the imperfect matching of the sensors in the array, the effects of any differences in responsivities may be easily corrected when calibrating the array, by adjusting the relative gain of each sensor in the processing circuit. This may be easily done, given the linearity of Eq. Ž2., by adjusting the gain of each sensor separately. Every sensor in the array is successively illuminated by a beam of constant intensity while keeping the other sensors in darkness, the gain of the illuminated sensor being adjusted so that the output of the circuit is proportional to the required value of the weight coefficient of the sensor under adjustment Žthis proportionality constant is arbitrary, but must be the same for the adjustment of all the sensors in the array..
9. Experimental results In order to verify the results of the optimization procedure, arrays with an even number of sensors were built using Siemens TP65 photodiodes. The characteristics of the sensors are given in Table 3. The radius of the active area of the sensors, i.e., the characteristic length to which all linear dimensions are normalized, is 7.15 mm. The photodiodes were used in the zero reverse bias configuration, followed by transimpedance amplifiers built with National Semiconductor LF353 dual operational amplifiers. The output of the preamplifiers were summed by a National Semiconductor LF351 operational amplifier, with weights set by variable resistors at the input of the summing node. The output of the circuit may be shifted by adding a constant voltage adjusted by a potentiometer. The electronic circuit is given in Fig. 6 for an array of two sensors Žthe extension for more sensors is straightforward., and was designed taking into account the remarks in Section 8. All the fixed resistors used are of the metal film type and the variable resistors and potentiometers Žused to set the coefficients Ck and to shift the output of the circuit. are linear 10-turn potentiometers. Responsivity mismatches between detectors were easily compensated using the procedure outlined in Section 8.
Table 3 Characteristics of the sensors Manufacturer Model External diameter Diameter of the active area Forward voltage @ 1 mA Dark current at y5 V reverse bias Uniformity of the response
Siemens TP65 15.5 mm 14.3 mm 0.41 V 1.5 mA "3% Žmeasured with a spot of 0.2 mm in diameter.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
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Fig. 6. Electronic circuit for an array of two sensors.
Both incandescent lamps and LEDs were used as light sources with good results. Background light Žnormal indoors lighting. was not a problem, as long as the system was kept within linear range, as could be expected from the symmetry properties indicated in Section 4. The position of the sensors within the array was measured with an error of 0.1 mm. The relative positioning of the light beam and the sensor array was measured with an error of 0.1 mm. The output signal was measured with an error of 0.2%, using a time constant of about 20 ms. The output voltage for the system in darkness was adjusted to less than 1 mV. Under these conditions, the observed peak noise level at the output was also under 0.1 mV. The results obtained were in excellent agreement with the calculations of Section 7. In Fig. 7, the output of an optimized array with two sensors is shown. For an illuminating beam width of 15 mm the linear response zone extended from y15 to q15 mm, while for a beam width of 20 mm the limits of the linear range were y10 and
q10 mm. Within the linear response range, the measured output was within "2% with respect to the linear function fit. The results were not appreciably affected by the presence of background lighting. As estimated from the observed linear output range and noise level of the circuit, the spatial resolution is better than 10 mm. In order to compare the performance of the optimized arrays with the conventional geometry, equidistant arrays with the same sensors were built, in which the photodiodes were placed Aside by sideB and only the weight coefficients were optimized, using the same electronic processing circuit. The results, shown in Fig. 8 for an array of two sensors, agree with the model predictions and show, as expected much poorer linearity than the arrays optimized both in weights and positions of the sensors. The observed values depart noticeably from the linear fit around the origin Ž x s 0.. Furthermore, the limits of the monotonous part of the response are wy9,q 9 mmx and wy7,q 7 mmx for beam widths of 15 and 20 mm, respectively.
Fig 7. Output of an optimized array with two sensors for two different illuminating beam widths.
Fig 8. Output of a conventional Aside by sideB array with two sensors for two different illuminating beam widths.
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P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
10. Summary and discussion An optimization procedure for sensor arrays is presented. The desired output function is approximated by a linear combination of the outputs of the array elements. The position and gain of the elements are the parameters for the optimization, given the irradiance profile of the illuminating beam. The shape of the sensor determines the kernel of the integral which gives the output of each element for a given irradiance profile of the illuminating beam. The integral may be carried out analytically, for the circular sensor kernel, if the profile of the illuminating beam is expanded as a power series. Expressions are given for the terms of order 0, 1 and 2. In many cases, a reasonable approximation to the beam profile may be achieved by fitting a low order polynomial instead of using a full series expansion. It is important to remark that the same procedure is applied irrespective of the number of sensors in the array. Arrays with an even number of sensors are especially important, since their symmetry properties make them insensitive to background lighting as long as the system is kept within its linear range. The optimization procedure is applied to arrays of 2, 4 and 6 circular sensors. Only the zero-order term of the expansion of the irradiance profile is retained, that is, the array is considered as if illuminated by a beam with rectangular profile. Satisfactory results are obtained for a wide range of beam widths. In the example developed in this work, the slope of the linear response of the array is found to be directly proportional to the width of the illuminating beam, for a range of 3:1 in excess of the initial beam width adopted for the optimization of the array. Moreover, the value of the residue of the fitting does not increase appreciably over the whole range of beam widths. These results show the flexibility and robustness in the operation of the arrays designed with this procedure, making it possible, for instance, to operate the array at different distances from
the illuminating source. The experimental results for optimized arrays were in excellent agreement with the calculations. The observed linearity was well within "2% with respect to the linear function fit and the results were not appreciably affected by the presence of background lighting. To make a comparison, equidistant arrays with the same sensors were built, in which the photodiodes were placed Aside by sideB and only the weight coefficients were optimized, using the same electronic processing circuit. The results agree well with the model predictions and show, as expected much poorer linearity than the arrays optimized both in weights and positions of the sensors.
Acknowledgements This work was supported in part by a grant from Fundacion ´ Antorchas and by the Argentine National Council of Scientific and Technological Research ŽCONICET..
References w1x J.A. Jamieson, R.H. McFee, G.N. Plass, R.H. Grube, R.G. Richards, Infrared Physics and Engineering, Mc Graw-Hill, New York, 1963, Chap. 12. w2x C. Torras, Computer Vision and Industrial Applications, SpringerVerlag, New York, 1992, p. 104. w3x R.H. Kingston, Detection of Optical and Infrared Radiation, Springer-Verlag, Berlin, 1978, Chap. 11. w4x H.J. Woltring, IEEE Trans. Electron. Dev. 22 Ž1975. 581–586. w5x R.J. Keyes, Optical and Infrared Detectors, Springer-Verlag, Berlin, 1980, Chap. 6. w6x W. Gopel, J. Hesse, J.N. Zemel, Sensors, Weinheim 6 Ž1992. 233. w7x G. Williams, J. Janesick, Cameras with CCDs capture new markets, Laser Focus World, Detector Handbook, vol. 33, 1996, pp. S5–S9. w8x Hamamatsu Photonics, Product Catalogue, 1992, pp. 1–6. w9x M. Baba, T. Konishi, Y. Furui, Meas. Sci. Technol. 10 Ž1999. 531–537. w10x M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming, Wiley, New York, 1993, Chap. 8.
Optimization of sensor arrays for beam position estimation P.A. Sorichetti a,),1, C.L. Matteo b,2 , A.J. Marzocca b a b
Departamento de Fısica, Facultad de Ingenierıa, ´ ´ UniÕersidad de Buenos Aires, AÕ. Paseo Colon ´ 850, 1063 Buenos Aires, Argentina Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, UniÕersidad de Buenos Aires, Pabellon ´ ´ I, Cdad. UniÕersitaria, 1428 Buenos Aires, Argentina Received 4 December 1999; received in revised form 3 May 2000; accepted 5 May 2000
Abstract A design procedure is presented to optimize linear arrays of sensors for the estimation of the position of an incident beam. The procedure is based on the adjustment of the position and gain of the elements of the array in order to approximate the desired response Žin the least squares sense.. The optimization takes into account the shape of the beam profile, and provides adequate results for a wide range of beam diameters. An application is made to a linear array of circular sensors illuminated by a beam with a rectangular profile. Optimized designs are given for arrays of 2, 4 and 6 sensors. The experimental results were in good agreement with the calculations. The relative error was found to be less than "2% and the performance of the system was insensitive to background light. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Sensor; Array; Optimization; Beam position estimation
1. Introduction The estimation of the position of an incident beam of light has several applications w1,2x, for example, in the measurement of linear or angular displacements ŽFig. 1.. Simple photoelectric sensors, both in single and differential configurations, have often been used to detect the displacement of a beam with respect to a reference position, for instance, as part of a feedback loop andror to provide a rough estimation of the position of the beam w3x. Position sensitive detectors ŽPSD. w4x and charge-coupled-devices ŽCCD. w5–7x are at present employed to give much higher accuracy and resolution in a number of applications, including the precision alignment of heavy machinery, topographic leveling, and the measurement of small displacements in large buildings. The PSD is a continuous structure, which can locate the position of a
) Corresponding author. Tel.: q54-11-4342-2950; fax: q54-11-43425286. E-mail address: [email protected] ŽP.A. Sorichetti.. 1 P.A. Sorichetti at present is also affiliated with Professional Communications S.A., Rivadavia 755, 1002 Buenos Aires, Argentina. 2 C.L. Matteo belongs to the Argentine National Council of Scientific and Technological Research ŽCONICET., Argentina.
beam spot with very high spatial resolution and at high speed. On the other hand, the CCD works as a discrete array with a very large number of sensing elements, whose output can be post-processed in order to give a highly accurate estimation of the position of the illuminating beam. However, the added cost and complexity of CCD devices and the related post-processing hardware and software are a disadvantage, particularly for less demanding applications where only a moderate precision is required. Moreover, the CCD is much slower than the PSD due to the time needed to scan the array, which may be of the order of 1 ms. On the contrary, the CCD may detect and locate several beams simultaneously, while the PSD requires some sort of control at the sources to discriminate between beams. Commercially available photodetector arrays are also an alternative. Linear arrays with a fairly large number of sensors can be found w8x, with typical geometries usually limited to arrays of equidistant sensors. Therefore, integrated arrays with a large number of sensors make possible, in principle, to provide a good estimation of the position of the illuminating beam, even in the case of multiple beams w9x. However, processing the output of a large number of sensors results again in the requirement of complex and costly circuits.
0924-4247r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 0 0 . 0 0 4 6 8 - 4
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P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
Žb. All the sensors operate within their linear range. Žc. The displacement of the illuminating beam is aligned with the axis of the array. Žd. The irradiance profile of the illuminating beam is assumed to be uniform in the direction perpendicular to the axis of the array.
Fig. 1. Measurement of linear and angular displacements with sensor arrays.
In this work, a different approach to the problem is sought. An optimization procedure is presented in order to get the optimum response from a linear array with a limited number of discrete sensors. To minimize the complexity of the processing circuit, the output of the array results from a weighted sum of the output of the individual sensors. However, instead of considering an array of equidistant sensors, the procedure includes as optimization variables the positions of the sensors within the array, in addition to the relative weights of the sensor outputs. Calculations show that the additional degrees of freedom provided by removing the requirement of equidistance between sensors gives the optimized array a much better performance than conventional arrays of the same number of elements Ži.e., arrays with equidistant sensors, in which only the output weights may be optimised., while keeping the cost and complexity well below of systems based on CCDs or large integrated arrays. This improvement is confirmed by the comparison of the experimental results of optimized arrays and conventional Žequidistant. arrays. It is important to remark that the optimization procedure is not limited to the positioning of light beams. In addition, it may also be applied to other types of sensors, such as acoustic Žultrasonic. transducers, particularly when the integration of a large number of sensors would be impractical.
2. Selection criteria for the sensors of the array and the illuminating beam The optimization procedure makes a limited number of assumptions on the characteristics of the sensors and the illuminating beam. Ža. The responsivity of each sensor is uniform over all its area.
It must be remarked that procedure does not make any specific assumptions on the type of photodetector used. As long as the stated assumptions are met, any kind of sensor might in principle be used. Therefore, the selection of a particular type of sensor may be made considering other practical considerations Žcost, size, spectral response, temporal response, etc... In particular, semiconductor photodiodes Že.g., silicon and germanium devices. are widely available and quite adequate for the implementation of the optimized sensor arrays discussed in this work. The optimization procedure has no specific requirements for the selection of the source of the illuminating beam, as long as the beam spot is linear in shape, with constant intensity in the direction perpendicular to axis of the array. The main constraints are set by the geometry of the beam and the irradiance profile Žas indicated in Section 7 below., but they may be easily met by the use of incandescent, LED or laser sources, suitably diaphragmed, together with a cilindrical lens. Therefore, the selection of the source may be made taking into account the spectral range of the detectors used and other practical considerations such as size, cost, etc.
3. Modeling of the response of an element of the array Even though the general expression of the output of a sensor involves the use of a two-dimensional convolution of the beam profile and the responsivity function of the sensor, with the stated hypothesis the modeling of the response of an array element is reduced to a one-dimensional integral. Therefore, the output of the jth sensor of the array is given by Vj Ž a . s
X max j
HX min
f j Ž x y sj . g Ž x y a. d x ,
Ž 1.
j
where a is the displacement of the beam along the axis of the array with respect to its reference position Žassumed to be at x s 0., Vj Ž a. is the output of the sensor, f j Ž x . is the response of the sensor when illuminated by an infinitesimally thin beam located at x, and g Ž x . is the irradiance profile of the illuminating beam, assumed constant in the direction perpendicular to the x-axis. The sensor is located at x s s j and its lateral extension is given by X max j and X min j Žsee Fig. 2..
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
13
Fig. 2. Geometrical disposition of the sensors and the illuminating beam.
It is worth noting that the elements of the array need not be identical, each element being described by its response function f j Ž x ..
4. Optimization procedure If the array consists of N elements, the output of the array can be obtained as a linear combination of the outputs of the sensors, that is N
Vo Ž a . s
Ý Ck Vk Ž a. .
Ž 2.
ks1
The output function Vo Ž a. of a given array will therefore depend on the beam irradiance profile g Ž x . and also on a set of 2 N parameters: the positions of the sensors, sk , and the weights Ck of the sensors outputs. The proposed optimization procedure for an array consists in choosing the optimum values of the 2 N parameters sk and Ck for Vo Ž a. to give the best approximation Žin the least squares sense. to the desired output function F Ž a., within the limits a max and a min . The optimized array will give a minimum value of the error residue E Es
amax
Ha
Ž Vo Ž x . y F Ž x . .
2
d x.
illuminating beam Žgiven, for instance, by a change of the distance between the array and the source of the beam.. It should be noted that although in many cases F Ž a. is chosen as a linear function, this needs not to be the case. Any reasonably well-behaved function, continuous and monotonous, might, in principle, be selected. An important property of optimized arrays with an even number of sensors may be obtained from symmetry considerations. If the output function of the array, F Ž a., is chosen as an odd function, i.e., F Ž a. s yF Žya., for example, as a linear function where F Ž a. s ka, it is easy to show that the positions of the sensors must be made symmetrical with respect to the center of the array. Moreover, the weights of symmetrically placed pairs of sensors should be equal and of opposite signs. In consequence, the effects of a uniform background illumination on the optimized are cancelled out, as long as all the sensors of the array Žand the associated processing circuit. remain within their linear range. This is an important characteristic from the point of view of applications, since in many cases a linear output function is desired and also the arrays must operate in the presence of background illumination.
Ž 3.
min
The minimization of the residue E as a function of the parameters may be carried out by any standard minimization technique, such as Newton or Conjugate-Gradient methods w10x. In general, several local minima of E may exist for a given array. This gives the possibility of selecting values of the parameters that give a good adjustment and also satisfy additional requirements, such as minimizing the influence of the variations of the spatial scale of the
5. Response function of a circular sensor To perform the optimization procedure outlined above, it is convenient to normalize all linear dimensions to a characteristic length of the sensors to be used, for instance, the radius of the active area in the case of circular sensors. In the remaining sections of this work, all linear dimensions will be normalized in this way, unless explicitly indicated.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
14
Therefore, for a circular sensor of unit radius Ž r s 1., centered at the origin, the response function f j Ž x . as defined in Section 3 above, is f j Ž x . s 2'1 y x 2 .
Ž 4.
Therefore, the output of the jth element of the array is given by S jq1
HS y1 (1 y Ž x y s .
Vj Ž a . s 2
2
j
where Pn and Q n are polynomials, given by: for n s 0 x P0 Ž x ,bj . s 2
Ž 9.
Q 0 Ž bj . s 1 for n s 1
g Ž x y a. d x ,
Ž 5. 1
j
where X min j s S j y1 and X max j s S j q 1. The response function of the sensor, f j Ž x ., may be thus considered as the kernel of the integral Ž5..
P1 Ž x ,bj . s y
y
bj x
3
x2
Ž 10 .
q
2
3
Q1 Ž bj . s bj for n s 2
6. Power series expansion for the profile of the illuminating beam In order to carry out the optimization procedure, it is useful to determine an analytic approach to evaluate the output function V Ž a. for a given function g Ž x .. It is important to note that integral Ž5. may be evaluated analytically for the circular sensor kernel, if the function g Ž x y a. can be expressed as a power series in Ž x y a. inside the interval where it is non-zero:
°Ý B Ž x y a. g Ž x y a . s~ ¢ 0
n
n
where Bn are the expansion coefficients of the function g and d1 and d 2 are the lateral limits of the illuminating beam. Instead of evaluating a full series expansion, in many cases an adequate approximation to the function g Ž x y a. may be obtained by fitting a low order Ž n - 3 . polynomial in Ž x y a., for instance by the least squares technique. The integral of Eq. Ž5. may be calculated analytically by integrating each term of Eq. Ž6. with the kernel of Eq. Ž4., using the following indefinite integrals n
q
Ž y1 q 4 bj2 . x y 2 bj x 2 q x 3
3
8
3
4
y1 q 4 bj2 4
The expressions for higher values of n may be easily carried out with a symbolic integration program, if a better polynomial approximation to g Ž x y a. is desired.
2
j
d x,
In this work, only the first term of the power series expansion Ž n s 0. of the irradiance profile g Ž x y a. will be retained. That is, the irradiance profile of the illuminating beam is considered to be constant along the x-axis. Therefore, the function g Ž x . may be written as gŽ x. s
½
1 0
yd - x - d otherwise
Ž 7.
In Ž x ,bj .
p y 2
~
Q n Ž bj .
s 2 Pn Ž x ,bj . '1 y x 2 q Q n Ž bj . arcsin Ž x . p Q Žb . 2 n j
¢
x F y1 < x
Ž 8.
Ž 12 .
where d is the half-width of the beam. It must be noted, as indicated in Section 4, that the sensor radius is taken as unity. Taking into account Eqs. Ž5., Ž8., Ž9. and Ž12., the
where bj s a y s j . These functions, which are defined for x within the wy1,1x interval, may be extended analytically for other values as
°
Ž 11 .
7. Application to an array of circular sensors
otherwise
Ž 6.
H Ž x y b . '1 y x
Q 2 Ž bj . s
2a
yd1 - Ž x y a . - d 2
n
In Ž x ,bj . s 2
P2 Ž x ,bj . s
Fig. 3. Response function for a circular sensor.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
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Table 1 Gain and position of the array elements 2 Element array 4 Element array 6 Element array
Gain Position Gain Position Gain Position
y4.189 y4.751
y1.000 y1.877 y1.000 y1.842 y1.000 y1.267
y1.894 y3.649 y2.662 y2.974
output function Vj Ž a. with the beam half-width d as a parameter, can be written as Vj Ž a . s I0 Ž a q d,bj . y I0 Ž a y d,bj . .
Ž 13 .
Fig. 3 shows Vj as a function of bj for different values of d. The optimization procedure described in Section 2 above, has been applied to arrays of 2, 4 and 6 sensors. For the optimization procedure the target function was taken as linear: F Ž x . s kx with unity slope Ž k s 1., for a beam half-width d s 1, Ži.e., equal to the radius of the active area of the sensors.. The limit values for the position of the illuminating beam center, w a min ,a max x, were chosen as wy1.3,1.3x, wy2,2x and wy3,3x for 2, 4 and 6 sensors, respectively. These limiting values are found numerically, in each case, to give solutions with both good linearity and linear range for the given number of sensors. In particular, the limiting values wy1.3,1.3x for the array with two sensors are interesting, since they clearly reflect the improvement over an equidistant ŽAside to sideB . design: satisfactory results are obtained for w a min ,amax x greater than wy1,1x. It is easily understood that, given the symmetry of the problem, it is necessary to calculate the weights and positions of only one half of the sensors of the array, for instance, for those corresponding to x ) 0, as indicated in Section 4. Furthermore, the weights of the central pair of sensors are chosen as y1 and q1. The position and gain of the elements, together with the residue of the fitting E are given in Table 1. To illustrate the dependence of the results with the width of the illuminating beam, the output of the array
Fig. 4. Output for a six-element array.
1.000 1.877 1.000 1.842 1.000 1.267
1.894 3.649 2.662 2.974
4.189 4.751
Residue 0.007 Residue 0.004 Residue 0.064
with 6 elements is plotted in Fig. 4, for different values of the illuminating beam half-width, d. In addition, Fig. 5 shows the slope of the response of the array and the residue E as a function of d, for the same array. It may be seen that increasing d with respect to the initial value adopted for the optimization also results in a linear response, where the slope k is directly proportional to d. In addition, the value of the residue of the fitting process remains fairly constant over the whole range of beam diameters. However, it must be taken into account that there is a slight reduction of the linear response zone is observed at higher values of d. Similar results for arrays with 2, 4 and 6 sensors, for values of d from 1 to 3, are given in Table 2. These results show the robustness of the design procedure, which gives the arrays added flexibility for operation. For instance, the arrays may be used at different distances of the illuminating source. 8. Effects of non-ideal characteristics of sensors and processing circuits Real sensors have several characteristics Že.g., darkness current, imperfect matching, etc.. that must be taken into account in the implementation of an optimized array. Moreover, electronic processing circuits also have limitations Ži.e., offset voltages, finite bandwidth, thermal and shot noise, 1rf noise, etc... However, the impact of most of these non-ideal characteristics and limitations on the optimization procedure may be circumvented by adequate design techniques, as briefly discussed in this section.
Fig. 5. Parameters of the linear fit of the six-element array response.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
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Table 2 Slope of the array output and fitting residue vs. half-width of the illuminating beam d
k
E
2 4 6 2 4 6 Elements Elements Elements Elements Elements Elements 1 1 1.5 1.566 2 1.718 2.5 3
1 1.449 1.997 2.531 2.818
1 1.489 2.004 2.519 2.952
0.007 0.241 0.421
0.004 0.324 0.003 0.322 0.506
0.064 0.334 0.113 0.327 0.664
The effect of darkness current of the sensors is to superimpose a constant term into the output signal Vo Ž a., as it may be easily seen from Eq. Ž2.. This may be compensated, together with any eventual offset voltages of the processing circuit, by shifting the level of the output signal. Furthermore, the need for this adjustment may be eliminated by periodically modulating the amplitude of the illuminating beam, for instance, by means of a chopper wheel or, in practical applications, by direct Aon –offB modulation Žas in the case of LED or laser sources.. In this way, the effects of darkness currents and offset voltages may be easily eliminated by high-pass filtering in the processing circuit, thus blocking DC and low frequency components. The output signal is taken as the envelope of the rectified signal, after processing. This has the additional advantage of eliminating the excess low frequency noise Ž1rf noise. of the electronic circuit if the modulation frequency is chosen above the A1rf cornerB frequency of the electronic devices Žtypically, a few tens of hertz.. It should be remarked that the optimization procedure may be applied without changes to an amplitude modulated beam, as long as the bandwidth of the system is large enough to process the modulation frequency. Fluctuations in darkness currents, thermal and shot noise of electronic circuits fall within the cathegory of Awhite noiseB, with zero average value and constant noise power per unit bandwidth. The amplitude of these fluctuations may be characterised by their root mean square ŽRMS. value. Since the output signal is obtained as a linear combination of the outputs of the sensors, the RMS value of the fluctuations in the output signal SŽ Vo . may be easily calculated from the RMS value of the fluctuations of the output of the individual sensors SŽ Vj ., including the appropiate weights Ck S Ž Vo . s
)
N
2 Ý Ž Ck S Ž Vk . .
.
Ž 14 .
ks1
Eq. Ž14., referred to the input of the sensor system, makes possible to estimate the minimum intensity level of the illuminating beam for a given signal-to-noise ratio. This in turn conditions the achievable spatial resolution of the system. If necessary, the signal-to-noise ratio may be
improved by low-pass filtering of the output signal, up to the limit imposed by the required response time to changes in the position of the illuminating beam. With regard to the imperfect matching of the sensors in the array, the effects of any differences in responsivities may be easily corrected when calibrating the array, by adjusting the relative gain of each sensor in the processing circuit. This may be easily done, given the linearity of Eq. Ž2., by adjusting the gain of each sensor separately. Every sensor in the array is successively illuminated by a beam of constant intensity while keeping the other sensors in darkness, the gain of the illuminated sensor being adjusted so that the output of the circuit is proportional to the required value of the weight coefficient of the sensor under adjustment Žthis proportionality constant is arbitrary, but must be the same for the adjustment of all the sensors in the array..
9. Experimental results In order to verify the results of the optimization procedure, arrays with an even number of sensors were built using Siemens TP65 photodiodes. The characteristics of the sensors are given in Table 3. The radius of the active area of the sensors, i.e., the characteristic length to which all linear dimensions are normalized, is 7.15 mm. The photodiodes were used in the zero reverse bias configuration, followed by transimpedance amplifiers built with National Semiconductor LF353 dual operational amplifiers. The output of the preamplifiers were summed by a National Semiconductor LF351 operational amplifier, with weights set by variable resistors at the input of the summing node. The output of the circuit may be shifted by adding a constant voltage adjusted by a potentiometer. The electronic circuit is given in Fig. 6 for an array of two sensors Žthe extension for more sensors is straightforward., and was designed taking into account the remarks in Section 8. All the fixed resistors used are of the metal film type and the variable resistors and potentiometers Žused to set the coefficients Ck and to shift the output of the circuit. are linear 10-turn potentiometers. Responsivity mismatches between detectors were easily compensated using the procedure outlined in Section 8.
Table 3 Characteristics of the sensors Manufacturer Model External diameter Diameter of the active area Forward voltage @ 1 mA Dark current at y5 V reverse bias Uniformity of the response
Siemens TP65 15.5 mm 14.3 mm 0.41 V 1.5 mA "3% Žmeasured with a spot of 0.2 mm in diameter.
P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
17
Fig. 6. Electronic circuit for an array of two sensors.
Both incandescent lamps and LEDs were used as light sources with good results. Background light Žnormal indoors lighting. was not a problem, as long as the system was kept within linear range, as could be expected from the symmetry properties indicated in Section 4. The position of the sensors within the array was measured with an error of 0.1 mm. The relative positioning of the light beam and the sensor array was measured with an error of 0.1 mm. The output signal was measured with an error of 0.2%, using a time constant of about 20 ms. The output voltage for the system in darkness was adjusted to less than 1 mV. Under these conditions, the observed peak noise level at the output was also under 0.1 mV. The results obtained were in excellent agreement with the calculations of Section 7. In Fig. 7, the output of an optimized array with two sensors is shown. For an illuminating beam width of 15 mm the linear response zone extended from y15 to q15 mm, while for a beam width of 20 mm the limits of the linear range were y10 and
q10 mm. Within the linear response range, the measured output was within "2% with respect to the linear function fit. The results were not appreciably affected by the presence of background lighting. As estimated from the observed linear output range and noise level of the circuit, the spatial resolution is better than 10 mm. In order to compare the performance of the optimized arrays with the conventional geometry, equidistant arrays with the same sensors were built, in which the photodiodes were placed Aside by sideB and only the weight coefficients were optimized, using the same electronic processing circuit. The results, shown in Fig. 8 for an array of two sensors, agree with the model predictions and show, as expected much poorer linearity than the arrays optimized both in weights and positions of the sensors. The observed values depart noticeably from the linear fit around the origin Ž x s 0.. Furthermore, the limits of the monotonous part of the response are wy9,q 9 mmx and wy7,q 7 mmx for beam widths of 15 and 20 mm, respectively.
Fig 7. Output of an optimized array with two sensors for two different illuminating beam widths.
Fig 8. Output of a conventional Aside by sideB array with two sensors for two different illuminating beam widths.
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P.A. Sorichetti et al.r Sensors and Actuators 87 (2000) 11–18
10. Summary and discussion An optimization procedure for sensor arrays is presented. The desired output function is approximated by a linear combination of the outputs of the array elements. The position and gain of the elements are the parameters for the optimization, given the irradiance profile of the illuminating beam. The shape of the sensor determines the kernel of the integral which gives the output of each element for a given irradiance profile of the illuminating beam. The integral may be carried out analytically, for the circular sensor kernel, if the profile of the illuminating beam is expanded as a power series. Expressions are given for the terms of order 0, 1 and 2. In many cases, a reasonable approximation to the beam profile may be achieved by fitting a low order polynomial instead of using a full series expansion. It is important to remark that the same procedure is applied irrespective of the number of sensors in the array. Arrays with an even number of sensors are especially important, since their symmetry properties make them insensitive to background lighting as long as the system is kept within its linear range. The optimization procedure is applied to arrays of 2, 4 and 6 circular sensors. Only the zero-order term of the expansion of the irradiance profile is retained, that is, the array is considered as if illuminated by a beam with rectangular profile. Satisfactory results are obtained for a wide range of beam widths. In the example developed in this work, the slope of the linear response of the array is found to be directly proportional to the width of the illuminating beam, for a range of 3:1 in excess of the initial beam width adopted for the optimization of the array. Moreover, the value of the residue of the fitting does not increase appreciably over the whole range of beam widths. These results show the flexibility and robustness in the operation of the arrays designed with this procedure, making it possible, for instance, to operate the array at different distances from
the illuminating source. The experimental results for optimized arrays were in excellent agreement with the calculations. The observed linearity was well within "2% with respect to the linear function fit and the results were not appreciably affected by the presence of background lighting. To make a comparison, equidistant arrays with the same sensors were built, in which the photodiodes were placed Aside by sideB and only the weight coefficients were optimized, using the same electronic processing circuit. The results agree well with the model predictions and show, as expected much poorer linearity than the arrays optimized both in weights and positions of the sensors.
Acknowledgements This work was supported in part by a grant from Fundacion ´ Antorchas and by the Argentine National Council of Scientific and Technological Research ŽCONICET..
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