Technical Note on the MTF of CCD Sensors

Transcription

1 Techncal Note on the MTF of CCD Sensors A unque MTF cannot be defned for a CCD usng the mathematcal structure of the classcal theory, whch s only applcable to lnearly and unformly respondng devces wth spatally contnuous mage surfaces. The perodc, pxellated, structure of the CCD sensor surface samples any nput mage at a fxed spatal frequency. Ths creates alasng and destroys the spatal contnuty of test mages employed to defne the MTF. All MTF data related to a CCD, or any perodcally structured devce, should only be used for system desgn or devce comparsons wth extreme cauton. However by specfyng and declarng a detaled measurement procedure an operatonal defnton of a (pseudo) MTF can be devsed. Ths allows repeatable measurements to be made. Such measurements can be compared wth confdence and f used wth care and a declared qualfcaton, they can provde useful gudance for both CCD qualty control and system desgn. INTRODUCTION General The MTF performance acheved at the output of a CCD s determned by: (a) CCD devce type (b) Qualty of manufacture (c) Test condtons (d) Actual MTF measurement procedure. Clearly the MTF specfcaton for a gven CCD devce type must defne all the sgnfcant test condtons and the exact procedure to be employed. The MTF data provded by dlgently executng the specfed procedures may then be used for devce qualty (relatve) assessment. However extendng the use of such data to system desgn, e.g. by convolvng t wth the MTFs of other system components, should only be done wth extreme cauton. Comparson of MTFs measured usng dfferent procedures wll probably be futle. Ths s because there s no unque MTF for an magng sensor comprsng an array of dscrete lght sensng elements (pxels) such as a CCD. CCD Devce Types The achevable MTF performance s controlled by a large number of desgn features ncludng: (a) Front or back surface llumnaton (b) Ant-reflecton coatngs on the slcon [a] (c) Back surface recombnaton treatment [a] (d) Charge transfer electrode structure (e) Pxel ptches n row and column drectons (f) Interlacng technque, f any (g) Ant-bloomng structures, f any (h) Actve slcon thckness and resstvty () Package wndow and ant-reflecton coatngs. The desgn features wll dctate the choce of operatng condtons, e.g. appled potentals, clock rates etc. In practce these may be modfed by the effects of manufacturng tolerances. Qualty of Manufacture Va the effects of varatons n all the as manufactured desgn features, (see CCD devce types), both from one CCD to another and across the magng surface of any one CCD sample, there wll be correspondng changes n the MTF. Thus t s mportant to specfy the postons on the CCD at whch MTF s to be measured. Test Condtons For a gven type of CCD, a large number of test condtons must be specfed and precsely controlled durng any test n order to obtan repeatable MTF measurements. These condtons nclude: (a) Wavelength of the test mage rradance (b) Numercal aperture of the lens projectng the test mage on to the CCD (c) Maxmum and mnmum sgnal levels n the test mage and the form of the mage pattern (d) CCD substrate potental, Vss (e) Hgh and low clock potentals appled to all charge transfer electrodes (f) Clockng scheme, waveforms etc and clock rates whch nfluence CTI (Charge Transfer Ineffcency) (g) Interlace, f any, especally when MTF n column drecton s to be measured (h) Temperature of the CCD () Sgnal processng scheme, ncludng reference level clampng technque (j) Locaton(s) of the test mage on the surface. It s advantageous to specfy test condtons whch represent those mposed by the end user. [a] On back surface llumnated devces e2v technologes lmted, Waterhouse Lane, Chelmsford, Essex CM1 2QU England Telephone: +44 (0) Facsmle: +44 (0) e-mal: enqures@e2vtechnologes.com Internet: Holdng Company: e2v holdngs lmted e2v technologes nc. 4 Westchester Plaza, PO Box 1482, Elmsford, NY USA Telephone: (914) Facsmle: (914) e-mal: enqures@e2vtechnologes.us # e2v technologes lmted 2003 A1A-CCDTN105 Issue 5, June /5977

2 MTF Measurement Procedure A unque MTF can be defned for an magng devce whch responds lnearly, unformly and contnuously across ts mage plane to the ncdent mage rradance. The MTF s defned theoretcally n terms of the devce s response to a large area snusodally modulated parallel bar pattern, used as an nput test mage. The value of the MTF at the spatal frequency of the test pattern, s the rato of the modulaton at the correspondng frequency n the devce output to the modulaton n the nput pattern. The modulaton tself s also a rato, namely the ampltude of the snusodal rradance or response varaton dvded by the mean rradance or mean response level respectvely, see Appendx. The perodc, pxellated, structure of the CCD sensor surface samples any nput mage at a fxed spatal frequency. Ths creates alasng (beat frequency modulaton patterns across the sensor output) and destroys the spatal unformty and contnuty of the test mages, n partcular bar patterns. The apparent MTF derved from such a dstorted output vares rapdly wth poston and a unque MTF cannot be defned. The local MTF s determned by the phase of the test pattern wth respect to the pxel array and s also a functon of the angle between the rows or columns of pxels and the bars of the pattern. However t s possble to specfy and declare a detaled measurement procedure that forms an operatonal defnton of a (pseudo) MTF, whch can be measured wth excellent repeatablty. Frst t should be ponted out that the theoretcal measurement of MTF would ental use of a separate test mage (chart) for each spatal frequency requred. Constructon of snusodally modulated charts s not easy but use of the readly constructed square wave modulated charts s not worth dscussng because they do not overcome the alasng problem ether. Lne Response Functon and Computed Statc MTFs The MTF can be derved (Fourer Transform) from the Lne Spread Functon, LSF, of an optcal system. The LSF of an mage sensor s meanngless but a Lne Response Functon, L (x), can be defned and measured, (see LINE RESPONSE FUNCTION). Then from one measurement of L (x) any desred form of (pseudo) MTF at any spatal frequency can be calculated. Outlne procedures for measurement and calculaton of MTF are gven n Lne Response Functon and Lne Response Functon to MTF (n) Transformaton. Usng the Lne Response Functon, L (x), t s possble to calculate the deal MTF that mght be observed as a functon of the phase (poston) of the deal parallel bar test pattern, wth respect to the pxel array of a CCD. The maxmum statc MTF [b] s easly calculated for those spatal frequences where (n 0 /n) s an nteger and n 0 = 1/(2p), the Nyqust frequency. Here p s the pxel ptch n a drecton perpendcular to the bars of the test pattern. No useful and unambguous mage structure s avalable n the CCD output for n4n 0. In fact no useful MTF data s really avalable n the CCD output for n 5 n 0 /4. The mnmum statc MTF [b] may also be calculated and s a fracton of the maxmum statc MTF whch falls rapdly to zero at n 0. A mean statc MTF can be calculated assumng that all phases of pattern are equally lkely. [b] Termnology follows T L Wllams, Electro Optcal Systems Desgn Conference Brghton, March 1971 These statc MTFs have been referred to as pseudo MTFs because any measurement attempted usng the theoretcally deal method would not necessarly yeld any of these values. In partcular measurement of the mnmum value s extremely unrelable. Snce the pseudo MTFs are derved by precsely defned mathematcal transformatons of the Lne Response Functon L (x), any one of them would suffce for specfcaton purposes. The mean and mnmum statc MTFs are fractons of the maxmum value, whch are functons of the parameter (n/n 0 ) alone and ndependent of the CCD or maxmum MTF tself (see Fg. 1). Thus the maxmum statc MTF forms a useful pseudo MTF whch s a functon of L (x), n and n 0 alone. 1.0 MTF M n 0.5 Maxmum [Note 1] Mean [Note 2] Mnmum n 0 = Nyqust (spatal) frequency M n = Maxmum statc MTF at each spatal frequency n, as computed from lne response functon n/n Notes [1] The results presented were calculated gven that the bars of the pattern were parallel to the CCD pxel columns. The MTF calculated from the measured lne response functon at e2v technologes s the maxmum value M obtaned for the optmum phase of the pattern wth respect to the pxel columns. [2] The mean MTF was calculated wth all phases assumed equally probable. Fg. 1 Spatal varatons of statc MTF obtaned wth a snusodally modulated bar pattern nput test mage Specfed MTFs Alasng s ncreasngly mportant at spatal frequences exceedng (n 0 /2). At the Nyqust lmt, n 0, the perceved modulaton can vary from zero to a maxmum value as the test pattern s shfted by half a pxel ptch across the CCD. In spte of the fact that real mage detal components at the Nyqust spatal frequency lmt are subject to extreme alasng errors, MTF at the Nyqust lmt s almost nvarably ncluded n CCD performance specfcatons. It s advsable n these cases also to specfy the MTF at other spatal frequences, e.g. (n 0 /4). Fortunately the verner technque for measurng lne response functons, (see Verner Technque), permts repeatable and accurate estmates to be made of the MAXIMUM statc pseudo MTF at n 0 and at other lower spatal frequences. It s mportant to note that the Lne Response Functon measurng technque elmnates the effects of velng glare almost completely. Velng glare can degrade the measurements of MTF attempted usng parallel bar test patterns. An objectve assessment of velng glare as t affects a CCD must be the subject of a separate measurement. It wll be powerfully nfluenced by the CCD desgn and n partcular by the wndow and ts ant-reflecton coatngs or by the use of a fbre-optc couplng plate and by the numercal aperture of the optcal system projectng the nput mage. CCDTN105, page 2 # e2v technologes

3 LINE RESPONSE FUNCTION General The CCD s an almost deal sensor for capturng the lne mage profle produced by another optcal system, ready for analyss to yeld MTF (n). It provdes extremely precse spatal samplng of the mage, provded the CCD pxel ptch s suffcently fne wth respect to the wdth of the lne mage produced. When the CCD tself must be charactersed t makes no sense to thnk n terms of lne mage profles and the pont of vew must be nverted so as to consder ts lne response functon. Lne Response Functon The CCD lne response functon, L (x) for a sngle pxel, s defned as the relatve response to an nfntesmally narrow rradated lne mage stuated at a dstance x from the centre of the pxel. The lne mage s algned parallel to the pxel column f x s measured along the pxel rows or vce versa. The unts chosen for x are mean pxel ptches, p. Thus, actual dstances are X = px (mm). The functon L (x) s normalsed for convenence to have unt area under the curve, thus: b $ L (x).dx = (1) a Note that (a, b), the lower and upper lmts of ntegraton, are chosen to nclude all non-zero values of the functon L (x) whch are dstngushable from the output nose. Condtons The CCD sensor and the camera system n whch t operates wll determne L (x). Thus, when L (x) s measured, all the test condtons, ncludng a precse specfcaton of the camera system, must be defned. All condtons deally should be representatve of those to whch the system wll be subjected n ts ntended applcaton. The response of the CCD used to generate L (x) s the pxel sgnal defned as total pxel output less the dark current, both ntegrated over one frame perod. In practce, a number of pxel sgnals wll be averaged over many frames n order to reduce the effects of nose. The averaged pxel sgnals used to derve L (x) must be lnearly related to the pxel nput rradance throughout the range of sgnals of nterest. All sgnals must be substantally less than the saturaton or ant-bloomng thresholds and sgnfcantly greater than the total nose level. Samplng the Lne Response Functon of the CCD The deal lne response s rectangular wth a wdth equal to the pxel ptch n the drecton of nterest. A well desgned and manufactured CCD wll have a lne response functon wth a wdth at 50% of peak response (FWHM) almost equal to ts pxel ptch. However, n practce, the response wngs can extend to ether sde of the orgn by one or two pxel ptches. Ths lne broadenng wll ncreasngly degrade the MTF (n) at the hgher spatal frequences. The precse shape of the lne response functon and hence the MTF (n) acheved, s also nfluenced by the wavelength of the ncdent lght and the mode of operaton of the CCD (e.g. depleton depths, etc.) as well as by ts desgn and constructon. If the nput test mage (llumnated narrow lne) s located at a dstance x from the centre of the pxel of nterest, then the adjacent pxels are located at dstances of (x + ). Here = + 1, +2, etc. (NB x s measured n unts of pxel ptches). If all pxels n the measurement area of the CCD are assumed to be dentcal [c], then a row of pxels wll provde samples of lne response functon at ntervals of one pxel ptch. Such ntervals are too coarse and addtonal ntermedate samples must be obtaned to construct a useful response functon. Clearly, mechancally dsplacng the lne mage by a small fracton of a pxel ptch (mcron movements) between successve frames (exposures) would enable the acquston of enough response data at ntermedate postons to construct a suffcently accurate response functon. Verner Technque An alternatve statc technque has been devsed and s now used for routne MTF (n) measurements at e2v technologes. Ths s the VERNIER technque n whch the mechancal dsplacement of the lne mage has been acheved by tltng the lne slghtly. Successve pxel rows sample the response functon at the pxel ptch but each (row) set of samples s dsplaced by a fracton of a pxel [d] per row. By nterleavng these samples, a suffcently closely spaced set of samples of the lne spread functon can be obtaned. The well focused lne mage s rotated onto approxmately the desred angle when t s dsplaced by one pxel ptch over an nterval of say 8 rows, see Fg PIXEL PITCHES Test stmulus: dealsed lne mage of neglgble wdth Notes [1] In ths example a 16 x 16 pxel regon s desgnated on the sensor, e.g. enclosed by a one pxel wdth border sutably hghlghted on the mage dsplay. The test mage s focused as shown wth a slope of 1 n 8 wth respect to the pxel columns or rows. [2] An almost deal sensor wll yeld a response as llustrated above. The shaded pxels ndcate partal responses because the lne energy s shared between adjacent pxels. Fg. 2 Verner samplng technque to measure the Lne Response Functon of a CCD [c] [d] In practce, pxel to pxel response non-unformtes wll ntroduce a fxed pattern nose on the samples. The currently used fracton s 1 / 8. # e2v technologes CCDTN105, page 3

4 The pxel sgnals obtaned from rows contanng the upper and lower column crossng ponts are shown n Fg. 3. Pxel sgnals from the central row, on whch the lne mage crosses the centre of a pxel, e.g. at x = 0, are shown n the mddle of Fg. 3. In practce a sgnal weghted best ft algorthm s employed to compute the slope of the lne and ts wdth, whch are dsplayed for the test operator, to focus the lne and set ts angle. Hgher row, number j = +4 Central row, number j = 0 LINE RESPONSE FUNCTION TO MTF (n) TRANSFORMATION Foundatons It can be shown readly that the modulaton transfer functon s gven by: MTF(n) =H 7 2 I 1 +I (2) I 1 and I 2 are functons of the spatal frequency, n, defned by the followng ntegrals of the lne response functon L (x): x 2 I 1 = $ L (x)cos x 1 [ pnx n 0 ] dx (3) x 2 I 2 = $ L (x)sn x 1 [ pnx n 0 ] dx (4) The lmts x 1 and x 2 are set such that all non-zero values of L (x) are ncluded n the range: Lower row, number j = 74 x 1 5 x 5 x (5) The frequency n 0 s the Nyqust lmt for the CCD, whch s defned as: n 0 = 1/(2p) (mm 71 ) (6) where p s the pxel ptch n mm. Symmetrcal Lne Responses Note that f L (x) s an even functon symmetrcal about x = 0 such that: Fg. 3 Pxel sgnals L (x) = L (7x) (7) then the ntegral I 2 = 0 and MTF(n) =I (8) Generally equaton (7) s satsfed and hence equaton (8) s applcable to many CCDs. If operatng condtons sgnfcantly degrade the Charge Transfer Effcency then the symmetrcal form of the CCD lne response wll be dstorted and equaton (8) would not apply. It s standard practce at e2v technologes always to calculate I 1 and I 2 and to use equaton (2). CCDTN105, page 4 # e2v technologes

5 APPENDIX Modulaton The contrast between two specfed areas of an mage wth rradances E 1 and E 2, s defned here [e] by the rato (E 1 7 E 2 ) (E 1 +E 2 ) The Modulaton Transfer Functon, MTF, descrbes the response of an optcal devce or system to an deal nput test mage comprsng a pattern of alternate parallel lght and dark bars wth a snusodal varaton of rradance scanned n a drecton perpendcular to the pattern bars. The rradance n ths deal test mage s represented by a contnuous functon of the form E(n, X)=e 0 +e(n)cos(2pnx) () Where: E(n, X) = Image rradance n = Spatal frequency of the pattern (cycles/mm) X = Dstance across the bars (mm) e 0 = Ampltude of the zero frequency component,.e. mean rradance e(n) = Ampltude of the rradance varaton at frequency n. The modulaton n such a pattern s the contrast (defnton above) between the areas of maxmum and mnmum rradance n adjacent lght and dark bars of the pattern. Clearly these rradances are E 1 =e 0 +e(n) and E 2 =e 0 7 e(n), hence the modulaton M (n) s gven by the rato M (n) = [ e(n) e 0 ] () If the devce or system responds lnearly and unformly [f] to the nput mage rradance wthn a specfed area of the mage, then here the output response may also be descrbed by a contnuous functon R(n 1,X 1 )=r 0 +r(n 1 )cos(2pn 1 X 1 + b).... () Where n 1 = Spatal frequency at output correspondng to n at the nput (as modfed by the mage magnfcaton) The modulaton n ths output response s M o (n 1 )= r(n1 ) r (v) 0 The effcency wth whch nput modulaton of the deal test mage s transferred to the output s descrbed by the rato MTF(n) = M o(n 1 ) M (n) Clearly at n = 0 the nput and output modulatons are unty by defnton and hence the MTF at n = 0 s also unty. In real devces or systems wth fnte mage felds, t s mpossble to examne the response at zero spatal frequency. Arbtrary choces must be made and the MTF s normalsed to unty at the lowest convenent spatal frequency. Ths can ntroduce sgnfcant dfferences between varous measurement technques, caused by the effects of velng glare. In a televson system the couplng of vdeo sgnals representng varous spatal frequences n the mage s AC. The zero or low frequency mean level s not DC coupled and s determned by clampng the dark sgnal to some arbtrary level, whch drectly affects the modulaton measured. The MTF of an electro-optc mage sensng devce, e.g. a CCD, can only be determned f the electronc offsets are removed and a clearly defned dark level s establshed wthn the area of the mage tested. X 1 = Dstance across the bars of the output mage (mm) b = Phase of the output pattern at frequency n 1 and at the orgn chosen for X 1. [e] [f] In other contexts dfferent contrast defntons may be employed. That s wth spatally unform transmsson or gan. Whlst e2v technologes has taken care to ensure the accuracy of the nformaton contaned heren t accepts no responsblty for the consequences of any use thereof and also reserves the rght to change the specfcaton of goods wthout notce. e2v technologes accepts no lablty beyond that set out n ts standard condtons of sale n respect of nfrngement of thrd party patents arsng from the use of tubes or other devces n accordance wth nformaton contaned heren. # e2v technologes Prnted n England CCDTN105, page 5