Model of Four-Quadrant Photodetector
Xiaojun Tang, Junhua Liu, Liping Dang, Chen Jian
School of Electrical Engineering, Xi’an Jiaotong University, Xi’an ,710049, China
Abstract: Photodetector is a kind of novel sensor. In this paper, a complete mathematical model of the light path of four-quadrant photodetector has been modeled. In the model, the effects of the mounted position of four-quadrant photodetector on the conversion from light to electricity, and the structure parameters of photodetector are considered. The relation expressions between the light signals, including lighting intensity and the incidence angle, the mounted position and structure parameters of four-quadrant photodetector, and the output signals are given. By analyzing the mathematical model, four feature parameters are extracted. With the mathematical model, arbitrary samples can be made. And with the samples, the inverse model of detector is approached using Neural Network (NN). The measurement results of distance between photosurface and focus of photodetector indicate that the model is correct. And the base for the application of four-quadrant photodetector is founded.
Key-words: Four-quadrant photodetector; Mathematical model of light-path; Feature parameters; Inverse model
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1 Introduction
With the development of semiconductor technology, many photodetector emerge [1][2]. As one type of photodetector, four-quadrant photodetector has been widely applied in photoelectric tracking, photoelectric direction, photoelectric control and guide, and so on [3], and so on. For a sensor, the inverse model of it must be set before applied in practice. Generally, samples are made by calibration experimentation at first, and then, with the samples, the inverse model is modeled using NN or other approaches. It is well known that calibration experimentation is a very time-consuming process. Specially, more the sensed variables, more the necessary samples, and the necessary time for calibration experimentation rise exponentially. If the model of sensor can be constructed be analyzing the theory of sensor, it is a very convenient.
In this paper, the mathematical models of the light path of four-quadrant photodetector in special position are constructed at first, and then, on the basis of those models, the common model is given. This model of light path, from light to electricity, is called positive model of four-quadrant photodetector. In the positive model, the effects, on the conversion from light to electricity, of the position of four-quadrant photodetector and the structure parameters of four-quadrant photodetector are considered. The relation expressions about the light signal, including lighting intensity and the incidence angle, the position of four-quadrant photodetector, structure parameters of photodetector, and the electricity signals, the outputs, are given as:
ux(t)=f(φ,ψ,η,l,…) (1)
uy(t)=g(φ,ψ,η,l,…) (2)
where ux and uy symbolize the compositive outputs of four-quadrant photodetector. Sign φ, η and ψ symbolize the three dimensional position parameters, respectively. Sign l symbolizes the distance between the lens and the photosurface of photodetector
In practical measurement, what we observe are the outputs of detector and what need to be measured are some measured parameters. So the inverse function of detector, called as inverse model, must be constructed. In order to construct the inverse model, enough samples about mounted position parameters, structure parameters of detector and the outputs of four-quadrant photodetector are obtained with positive model at first. And then the inverse model is approached with these samples using neural network. In the end of this paper, the measurement results of distance between the focal plane and the photosurface of photodetector are given. In the measurement results, the maximal error is only 0.1dmm, which indicate that the mathematical model of the light path of four-quadrant photodetector is correct.
2 Analysis of light path
2.1 Light path of four-quadrant photodetector
The light path of four-quadrant photodetector is shown in Fig 1. Laser emitted by point laser source passes convex called as convex 1, and then becomes parallel light. The parallel light enters four-quadrant photodetector after having been reflected by reflector. The four-quadrant photodetector is made up of convex called as convex 2, barrel and photosurface. When working, the reflector wavers back and forth at even speed. The photosurface of detector is shown in Fig.2. In Fig.2, sign I, II, III and IV symbolize four quadrants of detector, respectively. Little circles denote beam spot on photosurface. And all the little circles symbolize the motion track of beam spot. Define the output voltage of four-quadrant of detector as uI, uII , uIII and uIV, and the output signals of detector as:
ux=(uI+ uIV)- (uII+ uIII) (3)
uy=(uI+ uII)- (uIII+ uIV) (4)
When the inclination between parallel light and reflector is 45˚, shown in Fig.1, the axis of four-quadrant photodetector is normal to the parallel light, and the beam spot is on the center of photosurface, we called this position of detector as fiducial position. It is obvious that uy will be 0 when detector is mounted in fiducial position because the beam spot will always share alike on both sides of axes x shown in Fig.2. And ux will change with the incidence angle of the light shining into the detector β, shown in Fig.1, because the area of beam spot lies in the left side of axis y doesn’t identically equal to that lies in the right side of axis y. Define R as the radius of convex 2, l as the distance between photosurface and convex 2, f as the focus of the convex 2. According to the light path, following equation holds
(5)
where r is the radius of the beam spot forming on the photosurface of the detector.
Define the distance between the center of beam spot and axis y as s. It is easy to obtain s=|ltanβ|. While sr, the area of beam spot at the right side of axis y, shown in Fig.3, is:
(6)
and the area of beam spot at the left side of axis y is:
Sl=πr2-Sr (7)
At this time, the output voltage signals of detector are:
(8)
uy=0 (9)
where ρ is the photometric brightness;
σ is the phototranslating coefficient;
cosβ is the efficient illumination, viz. projection of convex 2 in the cross section of the incident light;
τ is a sign: while the center of beam spot is at the right (up) side of axis y of photosurface, τ=1; while the center of beam spot is at the left (down) side of axis y of photosurface, τ=-1.
While s, in other words, whole beam spot is at one side of axis y absolutely. Then
ux=τρπr2σcosβ (10)
While detector is mounted in the fiducial position, the waveforms of outputs of detector are shown in Fig.4. Owing to β is little, β≤2˚, cosβ and two extremities of waveforms of ux are flat. Specially, substitute s=|ltanβ| and (5) into (10), following equation holds:
(11)
From equation (11), one finds that the slope of waveforms of ux at time β=0 is directly proportional to f-l, the distance between photosurface and focal plane of photodetector.
2.2 Special kinds of circumstance
In practical application, detector isn’t always in fiducial position. There may be some position errors on one or every direction. In this subsection, the input-output relation equations of photodetector under special position are discussed to indicate the relations between every kind of position error and the output signals of photodetector.
2.2.1 Pitching departure
When detector whirl around the cross section axis x, convex 2 moves on direction y (shown in Fig.1). This departure is called pitching departure. When pitching departure happens, the track of beam spot is that shown in Fig.5. In this case, owing to the center of beam spot isn’t symmetric about axis x, uy isn’t zero, but a constant value. The value lies on the magnitude of ψ, the angle of pitching alternation. The efficient illumination isn’t cosβ, but a function about β and ψ, and need to be considered renewedly. Commonly,ψ≤2˚, so the efficient illumination can be looked as 1.
The mounted length of detector (the length of the part of detector in the bracket of detector) is defined as 2m. From the point of view of geometry, the distance between center of beam spot and axis x in photosuarface can be written as:
t=(l-m)tanψ (12)
The geometry about incident light, axis x and axis y of convex 2 is shown in Fig.6. In Fig.6, AO denotes the incident light in case of β=0; A`O denotes the incident light in case of β0; AC is normal to axis y; A`D is normal to axis x; A`E is normal to plane CEDO; plane CEDO symbolizes convex 2; AO=A`D and AO//A`D; So ψ. From the point of view of geometry, following equations hold.
CO=AOtanψ (13)
DO=AOtanβ (14)
(15)
A`O=AOsecβ (16)
(17)
In Fig.6,is the adjacent angle of the included angle between convex 2 and the cross section of incident light of detector, and the efficient illumination is . In case of ψ=0, viz. pitching departure doesn’t happen, equation (17) becomes =π/2-β, hence fiducial position is a special case of pitching departure. Imitating equations (8) and (10), uy will be:
(18)
In this case, take the place of cosβ with, ux changes as:
(19)
Commonly, both β and ψ are less than 2˚, and the waveform of uy borders up on a line. From the waveforms of outputs of photodetector, shown in Fig.7, it is easy to make out that the changes of them in this case indicate the information of mounted position of detector is chiefly reflected on by.
2.2.2 Orientation departure
Corresponding to pitching departure, the departure is called orientation departure when detector whirls around the cross section axis y (shown in Fig.1). In this case, the track of beam spot is that shown in Fig.8. Because the departure angle ψ is on the same or negative direction of β, the track of beam spot doesn’t keep symmetry about axis y, but keep symmetry about axis x. So uy always be 0 and there is departure with the crossover point of ux. The efficient illumination is modified as cos(β+ψ), and ux is modified as:
(20)
The outputs of detector are shown in Fig.9. From Fig.9 and equation (20) and (21), we conclude that the information of orientation departure reflects on bx .
2.2.3 Rolling departure
When detector rolls round the axes of it (shown in Fig.1 and Fig.10), the departure is called rolling departure. In this case, the track of beam spot can be divided into two hefts, shown in Fig.11, one the direction x, the other on direction y. Symbolize the angle of rolling departure with η, corresponding to (8), (9) and (16), ux and uy are modified as:
(21)
(22)
In case of rolling departure, the output signals of detector are shown as Fig.12.Because there are hefts in direction x and y, the departure degree of rolling departure reflect primarily on the slopes on the zero point of ux and that of uy. This can be concluded according to equation (21), (22).
2.3 Generic departure
What discussed in subsection 2.2 are three special departures. In every case, only one free dimension is considered. In common conditions, two or three departures may happen at same time, we call the departure as generic departure. By analyzing the light path in generic departure using the same method used above, following equations are obtained:
(23)
where λ(t) symbolizes the efficient illumination, and
where φ denotes the position of departure. Specially, φ=0 or φ=π, orientation departure happens, φ=π/2 or φ=3π/2, pitching departure happens. ψ denotes the departure degree both of orientation departure and pitching departure. η denotes rolling departure degree.
3 Inverse model
3.1 Input of inverse model
For a four-quadrant photodetector, what one observes are ux and uy, but what need to be measured may be the structure parameters or the mounted position of four-quadrant photodetector. If measured parameters, such as the mounted position parameters φ, ψ, η, and the structure parameter l or f-l, and so on, need to be measured with ux and uy, following equations must be determined:
φ=h(ux , uy) ψ=i(ux , uy)
η=j(ux , uy) l=k (ux , uy) (24)
…………
In section 2, it has been discussed that ax, slope of waveforms of ux at time β=0, contain primarily the information of l, by and bx, shown in Fig.7 and Fig.9, respectively, contain primarily the information of φ and ψ, and the departure degree of rolling departure reflect primarily on slopes ax and ay, the slope on the zero point of uy. Hence, (24) can be modified as:
φ=h`(ax, ay, bx, by ) ψ=i`( ax, ay , bx, by)
η=j`(ax, ay , bx, by) l=k` (ax, ay, bx, by) (25)
…………
There is some noise in ux and uy. In order to reduce the effect of noise on calculating ax, ay, bx and by, here, we use following lines to approach the sections of ux and uy when -ΔttΔt, Δt is the set time length.
ux= axt+ bx
uy=ayt+ by (26)
The algorithm for approaching can be Least Square Method (LSM) [4]. Then, for the inverse model of four-quadrant photodetector, That use ax, ay , bx and by as the inputs of inverse may be more appropriate.
3.2 Approach inverse model using NN
In theory, artificial NN can approach arbitrary function with arbitrary precision. Backward Propagation Neural Network (BPNN) and Radial Basis Function Neural Network (RBFNN) are always used to approach the corresponding relation from the outputs to the inputs of measurement system. Genetic Neural Network (GNN) is the result of Genetic Arithmetic (GA) combining with Neural Network [5]. It overcomes the problem of local least. For the approaching of inverse model, here, a GNN is used. The architecture of NN is shown in Fig.13. In Fig.13, IW denotes the link weight matrix between input layer and hidden layer, b1 denotes the threshold vector of hidden layer, LW denotes the link weight matrix between hidden layer and output layer, and b2 denotes the threshold vector of output layer. Tansig function is selected as the transfer functions of hidden layer, and pureline function is selected as the activation functions of output layer. There are four input neurons and four output neurons.