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Increasing the resolution of a CCD-array radiation detector by positional modulation
INS11 kuw NTS & METHOOS "N PNYSMS RESEARCH
Nuclear Instruments and Methods in Physics Research A311 (1992) 293-295 North-Holland
Secti on A
Increasing the resolution of a CCD-array radiation detector by positional modulation Heikki Collan
Optoelectronics Laboratory, Helsinki University of Technology, SF-02150 Espoo, Finland Received 27 May 1991
Recently it was demonstrated [11 that, when taking topographic X-ray images of single crystal semiconductor samples with the help of synchrotron radiation, several distinct advantages are obtained if the traditional film detector is replaced by a direct-reading CCD-array photon detector . Apart from the state-of-the-art fine resolution CCD-array photon detectors, which are very expensive, typical commercially available CCD-array detectors have a pixel lattice modulus of several p,m. This translates directly into the positional resolution of the detector . By means of an X-ray film at least an order of magnitude better resolution can be obtained [21, albeit sometimes with a much longer exposure time . This fact would seem to restrict the use of moderately priced CCD-array detectors to cases where high positional resolution is of no interest . It is the purpose of this note to show that with a straight-forward experimental arrangement, i.e . by modulating the position of the detector, it is possible to considerably increase the positional resolution of a CCD-array detector.
1. Mathematical analysis
To put the discussion on a quantitative basis, we expand the function 1(x) as a power series in x,
As shown in fig. 1, we consider a linear CCD-array radiation detector, the pixel width of which is a and the linear lattice modulus L in the direction of the x-axis. This detector is used to sense a radiation intensity as a function of x, denoted by Ax). A pixel of the detector, positioned at x = xo, thus records a signal
I(x) = I(o) + IM(x -X O ,) + I(2)(x - xo)2/2! + I(3)( x - xo )3/3! + I( 4)( x -x(,)4/4! + . . .
S(xo) = J
x o~+a
x ~~
I(x) dx
which, to first approximation, is used to define I(x) at the point xO. Similarly, other pixels of the detector array define I(x) at the discrete points x =x o + L, xo + 2L, x o + 3L, etc. along the x-axis . In this manner the function JU), averagcU VV~,l Ll1\. citg, '- ..s, .~ defined only at these points with no further knowledge as to its variation elsewhere. If, say, L is typically in the range 10 to 50 Wtn, then this presents a severe limitation to the accuracy of recording Ax). Consider now what happens when the detector is shifted by a small distance S along the x-axis from the original position xO. Because of this shift the signal recorded, S(xo + S), will differ from S(xo) by an amount which is determined how I(x) changes with position .
with the derivatives I (` ~ evaluated at x = xo. We then put eq . (2) into eq . (1) and integrate over the pixel width a to obtain S(xo ) =1(o)a +I("(a2/2!) +1(2)(a3/3!) + J(3) ( a4/4!) + .. . , I(4)( as /5!) + Similarly, for the detector shifted by S, we obtain S(xo + S) =1so)a + I,,(> >(a2/2!) + Is2)( a3 /3!)
+1.(3)(a4/4!) + Is4 )(as/5!) + - . . , (4) where the index S refers to quantities evaluated at x =x o + S . By expanding the power series of eq . (2) now at x = x + S and by using this expansion in eq. (1) we find by subtraction that the difference AS(S) _ S(xo + S) - S(xo) equals AS(S) =S[1(ß)a +1(2) (a212!) +I(;'(a ;/3!) +1(4) a4/4!) + ( + ( 52/2!) [ I('-)a + I( 3 )( a2 /2!) +1(4)(a3/3!) + . . .
1
+(S3/3!)[I(3)a +I(4)(a2/2!) + . . .
0168-9002/92/$05 .00 IV) 1992 - Elsevier Science Publishers B.V . All rights reserved
I + ... .
H. Collan / Resolution of a CCD-array detector
294
By inserting the 10 ' obtained from eq. (8) into eq. (2) we have now obtained the representation for 1(x) through all x spanned by the detector array, and not only at the discrete points x + ?iL of the detector array lattice. 2. Conclusions
0
0
Fig. 1. Schematic drawing of a linear CCD-array detector. The linear array modulus in the x-direction of the detector is L and the pixel width of the detector is a. Also illustrated is the positional displacement modulation S, and the variation of the radiation intensity R x) with position .
Thus, not unexpectedly, OS(8) is given by a power series in S, AS(5) = SMS + S(2)132/2! +S(3)53 /3! +S(4)54/4! + . . . ,
(5') the prevailing coefficients of which S(') are determined experimentally by a fit to the intensity, measured for different 3, or by some other suitable algorithm. We thus have the matrix equation
s(t) =TI('),
(61 ., where, to order I(4) for example, T is the triangular matrix a a2/2! a3/3! a4/4! T= 0 a a2/2! a3/3! 0 0 a ' a 2 /2! 0 0 0 a From eq. (6) we thus have P') = T'S" .
It is clear that by the above procedure it is possible to increase the positional resolution of the CCD-detector roughly by an order of magnitude depending, of course, to which order S', [cf. eq . (5')] the difference AS(8) is recorded . In this manner the positional resolution of a CCD-array detector is no longer limited by the pixel size a or by the linear array modulus L of the detector but, rather, by how precisely the border-line of the sensitive area of each pixel is defined. Due to this the positional resolution obtainable with a CCDarray photon detector by far exceeds that of alternative electronic radiation detection systems [3-6]. A practical way of implementing the procedure of mechanical step displacements of the detector, described above, seems to be a piezo-electric transducer attached to the CCD-array detector, or some other well established micro-movement technology. The number of the positional steps 3, 23, 35, . . ., n3 is determined by the power in 5 to which the shifted signal S(x O~ + S) is desired [cf. eq. (4)] . The steps themselves are easily controlled by the same microprocessor which is used for data acquisition together with the CCD-array detector [1]. The procedure can be generalized also to 2-dimensional CCD-scanning [2]. Although implementation of a controlled positional shift of the CCD-detector adds to the complexity of a detector system, in many cases the possibility of using significantly cheaper low resolution CCD-array detectors, in stead of expensive high-resolution units, more than compensates for this. For instance, in situations of a very high radiation load, with the performance of a CCD-array detector expected to degrade in extended use and with a consequent limited detector lifetime, this approach may allow the use of disposable detectors. In addition to X-ray imaging, inceasing the resolution of a CCD-detector by nositional modulation would seem to suit also other fields of accelerator based experimental research where presently new tele-operational techniques are being developed .
(7)
Note added in proof
(8)
For the case of small CCD pixels and visual images, the generic sensor displacement technique, equivalent to part of the preient work, has previously been developed and shown to be fully implementable in practice
H. Collan / Resolution of a CCD-array detector
by Dr. Relmar Lenz, Institute for Telecommunication, Technical University of Munich, Arcisstr. 21 D-8000 Munich, Germany . See R. Lenz, Proc. 11th DAGM Symp., Informatik-Fachberichte 219 (Springer, Berlin, 1989) pp. 411-415. Last minute correspondence with Dr. Lenz is gratefully acknowledged. Acknowledgement Thanks are due to T. Tuomi, J. Partanen and K. Simomaa for supplying a preprint of ref. [1] prior to publication .
References
295
(1] T. Tuomi, J. Partanen and K. Simomaa. Nucl. Instr. and Meth. B61 (1991) 569. [2] M. Ichimura, K. Kirii and T. Shibata, Nucl. lnstr . and Meth. A300 (1991) 616. [3] W. Hartmann, G. Markewitz, U. Rettemaier and H.J. Queisser, Appl. Phys. Lett. 27 (1975) 308. [4] T. Tuomi, V. Kelhii and M. Blomberg, Nucl. Instr. and Meth. 208 (1983) 697. [5] J. Chikawa and I. Fujimoto, Appl. Phys. Lett. 13 (1968) 387. [6] S. Suzuki, M. Ando, K. Hayakawa, O. Nittono, H. Hashizurne, S. Kishino and K. Kohra, Nucl. Instr . and Meth. 227 (1984) 584.
Nuclear Instruments and Methods in Physics Research A311 (1992) 293-295 North-Holland
Secti on A
Increasing the resolution of a CCD-array radiation detector by positional modulation Heikki Collan
Optoelectronics Laboratory, Helsinki University of Technology, SF-02150 Espoo, Finland Received 27 May 1991
Recently it was demonstrated [11 that, when taking topographic X-ray images of single crystal semiconductor samples with the help of synchrotron radiation, several distinct advantages are obtained if the traditional film detector is replaced by a direct-reading CCD-array photon detector . Apart from the state-of-the-art fine resolution CCD-array photon detectors, which are very expensive, typical commercially available CCD-array detectors have a pixel lattice modulus of several p,m. This translates directly into the positional resolution of the detector . By means of an X-ray film at least an order of magnitude better resolution can be obtained [21, albeit sometimes with a much longer exposure time . This fact would seem to restrict the use of moderately priced CCD-array detectors to cases where high positional resolution is of no interest . It is the purpose of this note to show that with a straight-forward experimental arrangement, i.e . by modulating the position of the detector, it is possible to considerably increase the positional resolution of a CCD-array detector.
1. Mathematical analysis
To put the discussion on a quantitative basis, we expand the function 1(x) as a power series in x,
As shown in fig. 1, we consider a linear CCD-array radiation detector, the pixel width of which is a and the linear lattice modulus L in the direction of the x-axis. This detector is used to sense a radiation intensity as a function of x, denoted by Ax). A pixel of the detector, positioned at x = xo, thus records a signal
I(x) = I(o) + IM(x -X O ,) + I(2)(x - xo)2/2! + I(3)( x - xo )3/3! + I( 4)( x -x(,)4/4! + . . .
S(xo) = J
x o~+a
x ~~
I(x) dx
which, to first approximation, is used to define I(x) at the point xO. Similarly, other pixels of the detector array define I(x) at the discrete points x =x o + L, xo + 2L, x o + 3L, etc. along the x-axis . In this manner the function JU), averagcU VV~,l Ll1\. citg, '- ..s, .~ defined only at these points with no further knowledge as to its variation elsewhere. If, say, L is typically in the range 10 to 50 Wtn, then this presents a severe limitation to the accuracy of recording Ax). Consider now what happens when the detector is shifted by a small distance S along the x-axis from the original position xO. Because of this shift the signal recorded, S(xo + S), will differ from S(xo) by an amount which is determined how I(x) changes with position .
with the derivatives I (` ~ evaluated at x = xo. We then put eq . (2) into eq . (1) and integrate over the pixel width a to obtain S(xo ) =1(o)a +I("(a2/2!) +1(2)(a3/3!) + J(3) ( a4/4!) + .. . , I(4)( as /5!) + Similarly, for the detector shifted by S, we obtain S(xo + S) =1so)a + I,,(> >(a2/2!) + Is2)( a3 /3!)
+1.(3)(a4/4!) + Is4 )(as/5!) + - . . , (4) where the index S refers to quantities evaluated at x =x o + S . By expanding the power series of eq . (2) now at x = x + S and by using this expansion in eq. (1) we find by subtraction that the difference AS(S) _ S(xo + S) - S(xo) equals AS(S) =S[1(ß)a +1(2) (a212!) +I(;'(a ;/3!) +1(4) a4/4!) + ( + ( 52/2!) [ I('-)a + I( 3 )( a2 /2!) +1(4)(a3/3!) + . . .
1
+(S3/3!)[I(3)a +I(4)(a2/2!) + . . .
0168-9002/92/$05 .00 IV) 1992 - Elsevier Science Publishers B.V . All rights reserved
I + ... .
H. Collan / Resolution of a CCD-array detector
294
By inserting the 10 ' obtained from eq. (8) into eq. (2) we have now obtained the representation for 1(x) through all x spanned by the detector array, and not only at the discrete points x + ?iL of the detector array lattice. 2. Conclusions
0
0
Fig. 1. Schematic drawing of a linear CCD-array detector. The linear array modulus in the x-direction of the detector is L and the pixel width of the detector is a. Also illustrated is the positional displacement modulation S, and the variation of the radiation intensity R x) with position .
Thus, not unexpectedly, OS(8) is given by a power series in S, AS(5) = SMS + S(2)132/2! +S(3)53 /3! +S(4)54/4! + . . . ,
(5') the prevailing coefficients of which S(') are determined experimentally by a fit to the intensity, measured for different 3, or by some other suitable algorithm. We thus have the matrix equation
s(t) =TI('),
(61 ., where, to order I(4) for example, T is the triangular matrix a a2/2! a3/3! a4/4! T= 0 a a2/2! a3/3! 0 0 a ' a 2 /2! 0 0 0 a From eq. (6) we thus have P') = T'S" .
It is clear that by the above procedure it is possible to increase the positional resolution of the CCD-detector roughly by an order of magnitude depending, of course, to which order S', [cf. eq . (5')] the difference AS(8) is recorded . In this manner the positional resolution of a CCD-array detector is no longer limited by the pixel size a or by the linear array modulus L of the detector but, rather, by how precisely the border-line of the sensitive area of each pixel is defined. Due to this the positional resolution obtainable with a CCDarray photon detector by far exceeds that of alternative electronic radiation detection systems [3-6]. A practical way of implementing the procedure of mechanical step displacements of the detector, described above, seems to be a piezo-electric transducer attached to the CCD-array detector, or some other well established micro-movement technology. The number of the positional steps 3, 23, 35, . . ., n3 is determined by the power in 5 to which the shifted signal S(x O~ + S) is desired [cf. eq. (4)] . The steps themselves are easily controlled by the same microprocessor which is used for data acquisition together with the CCD-array detector [1]. The procedure can be generalized also to 2-dimensional CCD-scanning [2]. Although implementation of a controlled positional shift of the CCD-detector adds to the complexity of a detector system, in many cases the possibility of using significantly cheaper low resolution CCD-array detectors, in stead of expensive high-resolution units, more than compensates for this. For instance, in situations of a very high radiation load, with the performance of a CCD-array detector expected to degrade in extended use and with a consequent limited detector lifetime, this approach may allow the use of disposable detectors. In addition to X-ray imaging, inceasing the resolution of a CCD-detector by nositional modulation would seem to suit also other fields of accelerator based experimental research where presently new tele-operational techniques are being developed .
(7)
Note added in proof
(8)
For the case of small CCD pixels and visual images, the generic sensor displacement technique, equivalent to part of the preient work, has previously been developed and shown to be fully implementable in practice
H. Collan / Resolution of a CCD-array detector
by Dr. Relmar Lenz, Institute for Telecommunication, Technical University of Munich, Arcisstr. 21 D-8000 Munich, Germany . See R. Lenz, Proc. 11th DAGM Symp., Informatik-Fachberichte 219 (Springer, Berlin, 1989) pp. 411-415. Last minute correspondence with Dr. Lenz is gratefully acknowledged. Acknowledgement Thanks are due to T. Tuomi, J. Partanen and K. Simomaa for supplying a preprint of ref. [1] prior to publication .
References
295
(1] T. Tuomi, J. Partanen and K. Simomaa. Nucl. Instr. and Meth. B61 (1991) 569. [2] M. Ichimura, K. Kirii and T. Shibata, Nucl. lnstr . and Meth. A300 (1991) 616. [3] W. Hartmann, G. Markewitz, U. Rettemaier and H.J. Queisser, Appl. Phys. Lett. 27 (1975) 308. [4] T. Tuomi, V. Kelhii and M. Blomberg, Nucl. Instr. and Meth. 208 (1983) 697. [5] J. Chikawa and I. Fujimoto, Appl. Phys. Lett. 13 (1968) 387. [6] S. Suzuki, M. Ando, K. Hayakawa, O. Nittono, H. Hashizurne, S. Kishino and K. Kohra, Nucl. Instr . and Meth. 227 (1984) 584.