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
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
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
1
of the light path of four-quadrant photodetector is
correct.
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
2 Analysis of light path
2.1 Light path of four-quadrant photodetector
rR
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
f l
f
(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 s<r,
the area of beam spot at the right side of axis y, shown
in Fig.3, is:
S r  r 2 arccos( s )  s r 2  s 2
r
(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:
u x   (S r  S l ) cos 
  [2r 2 arccos( s )  2s r 2  s 2  r 2 ] cos 
r
(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  r , in other words, whole beam spot is at one
β/2
Parallel light
Reflector
Dot laser
source
z
x
Condensing lens
y
Convex lens
l
f
Middle
section
2m
Photosurface
Wall of Facula
detector
x
y
Detector
bracket
(b)
Axis
(a)
Fig.1 Sketch map of device for adjusting photovoltaic conversion coefficient of photodetector (a) Light path (b) Coordinate system
x shown in Fig.2. And ux will change with the
incidence angle of the light shining into the detector β,
side of axis y absolutely. Then
ux=τρπr2σcosβ
2
(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  1 and two extremities
of waveforms of ux are flat. Specially, substitute
s=|ltanβ| and (5) into (10), following equation holds:
du x
d
 0
  R ( f  l ) / f
2
and need to be considered renewedly. Commonly,ψ≤2˚,
so the efficient illumination can be looked as 1.
y
(11)
Ⅱ
Ⅰ
Ⅲ
Ⅳ
s
r
x
Fig.2 The photoconductive
surface of detector
Ⅰ
Ⅲ
Ⅳ
x
t
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.
y
Ⅱ
Fig.5 The track of focal spot when pitching departure happen
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 β>0; 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 CAO  ψ. From the point of view of geometry,
following equations hold.
CO=AOtanψ
(13)
DO=AOtanβ
(14)
y
x
Fig.3 Position of focal spot
while s<r
Fig.4 Waveforms of outputs of detector mounted in fiducial position
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,
EO  CO 2  DO 2
(15)
A`O=AOsecβ
(16)
A`OE  arccos
tan 2   tan 2 
EO
 arccos
A`O
sec 
(17)
In Fig.6, A`OE 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 sin A`OE . In case of ψ=0, viz. pitching departure
doesn’t happen, equation (17) becomes A`OE =π/2-β,
hence fiducial position is a special case of pitching
departure. Imitating equations (8) and (10), uy will be:
 
u   [r 2  2r 2 arccos t  2t r 2  t 2 ]
r
 y

(18)
tr
  sin A`OE

2
u y   r sin A`OE
tr


In this case, take the place of cosβ with sin A`OE , ux
changes as:
u x   [2r 2 arccos( s )  2s r 2  s 2  r 2 ]  sin A`OE (19)
r
3
information of orientation departure reflects on bx .
A`
A
β
ψ
D
o
C
x
E
y
Fig.6 The geometry about incident light, axis x and axis y of convex
2
Fig.9 Waveforms of outputs in case of orientation departure
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:
Fig.7 Waveforms of outputs of detector in case of pitching
departure
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:
u   [2r 2 arccos(s cos )  r 2
r
 x

2
2
s cos  r
  2s r  ( s cos ) ] cos 

2
s cos  r
u x  r cos 

u   [2r 2 arccos(s sin  )  r 2
r
 y

2
2
s sin   r (22)
  2s r  ( s cos ) ] cos 

2
s sin   r
u y   r cos 

y
z
 
u   [2r 2 arccos s  2 s r 2  s 2  r 2 ]
r
 x

sr
  cos(   )

2
u x   r cos(   )
sr


(21)
y`
η
x` x
y
(20)
Fig.10 Rolling departure
Ⅱ
Ⅰ
Ⅲ
Ⅳ
x
Fig.11 Track of focal spot in
case of rolling departure
y
Ⅱ
Ⅰ
Ⅲ
Ⅳ
x
Fig.8 the track of focal spot in case of orientation departure
Fig.12 Waveforms of outputs of detector in case of rolling departure
The outputs of detector are shown in Fig.9. From Fig.9
and equation (20) and (21), we conclude that the
In case of rolling departure, the output signals of detector
are shown as Fig.12.Because there are hefts in direction
4
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 -Δt<t<Δt, 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.
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:
MO2 sin KMO2 

2
/  1)
u x   (t ) R (arccos
R
2
(23)

O
N
sin

KNO

2
u   (t ) R 2 (arccos 2
/  1)
 y
R
2
where λ(t) symbolizes the efficient illumination, and
cos2   {sin   cos[  arcsin(sin  sin  )
 arccos(sin  sin  )]}2  cos2[arccos(sin  sin  )   ]
MO2  l  tan sec2  sec2 (   )  tan 2 


O2 N 

sec sin 

sin(  arccos(sin cos(   ))) 
f O3 R sin(RO3 S       )
f l
sin(   )
2
RO3 S  arccos
O3 R 2  O3 S 2  RS 2
2O3 R  O3 S
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
O3 R  O3 S 2  RS 2  2O3 S  RS cos(  )
RS  l sec
sin 
sin(    arccos(sin cos(   )))
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
5
of output layer. There are four input neurons and four
output neurons.
only 0.1dmm. The measurement results show that
the measurement precision is high.
(2) The model of four-quadrant photodetector is about
the relation between the structure parameters,
position and the feature parameters of the outputs
of four-quadrant photodetector. So the all the
structure parameters and the position of
four-quadrant photodetector can be measured using
this model.
(3) There are only four feature parameters, and only
four measured parameters can be measured using
this model.
(4) According to the idea of modeling four-quadrant
photodetector, models of other photodetectors, such
as three-quadrant photodetector, six-quadrant
photodetector, can be constructed in the same method.
Fig.13 The architecture of the selected model of NN
After built the neural network, set the training
performance as 110 8 , and train it with 1000 group
of samples, obtained from positive model of
four-quadrant by altering ψ, φ, η and l, for 400 epochs.
A set of samples is made up with a set of input vectors
ax, bx, cx, and dx, and a set of output vectors ψ, φ, η, l, in
other words, the errors between the real inputs and the
outputs of NN are very little. So the NN can be treated
as the inverse model of four-quadrant photodetector.
References:
[1]E.Huseynov,
N.Ismailov,
S.R.Samedov,
etc,
IR-Detectors
Based
on
IN2O3-Anode
Oxide-CDxHg1-xTe, International Journal of Infrared
and Millimeter Waves, 23(9), (2002):1337-1345;
[2]K. Hirakawa, S.-W. Lee, Ph. Lelong, etc,
High-sensitivity modulation-doped quantum dot
infrared photodetectors, Microelectronic Engineering,
63 (2002) :185–192;
[3]Feng Longling, Simple Analysis of Signal-processing
Skill for Four–quadrant Opto-electronic Detective
System, OPTICAL TECHNOLOGY, (3), 1995:12-17;
[4] Jin Yun, Yuan, Numerical methods for generalized
least squares problems, Journal of Computational
and Applied Mathematics, 66(1-2), (1996):571-584
[5] Zhang Y, Liu JH, Zhang YH, Tang XJ, Cross
Sensitivity Reduction of Gas Sensor Using Genetic
Neural Network, Optical Engineering, 41(3)
(2002):615-625.
4 Measurement results
In order to validate the correctness of the model of
four-quadrant photodetector, here, f-l, one of
architecture parameters of four-quadrant photodetector,
is measured. The corresponding expected values and
the measured results are shown in table 1. From table 1,
one finds that the maximal measurement error is only
0.1dmm, denoted with underline. In other words, both
the positive model and the inverse model of
four-quadrant are correct and can be used for
measurement of structure parameters and the position
of four-quadrant photodetector.
Table 1 Measurement result and the corresponding expected value
Expected value (dmm)
26.00 26.00 27.00 27.00
Measurement value (dmm)
26.06 25.95 26.93 27.01
Expected value (dmm)
28.00 28.00 29.00 29.00
Measurement value (dmm)
28.10 27.99 29.01 29.08
5. Conclusion
From above discussion and measurement results,
following conclusion can be drawn:
(1) For the measurement of f-l, the maximal error is
6