The Sensitivity of Nonlinear Detectors

Georg Bauer

Starting from the general concept of sensitivity as the quotient of effect by cause, the differential spectraland relative spectral sensitivities of a radiation detector are derived. If the effect is not proportional tothe cause, it is shown that a relative spectral sensitivity without restrictions may be defined only by re-ferring to equal effect but not to equal cause. Some consequences of this condition are discussed.

1. Definition of SensitivityIt is usual to define as sensitivity s of a radiation

detector-radiation means here electromagnetic radia-tion in the optical wavelength region from 100 nm to1 mm-the quotient of the effect Y by the cause X. Inmost cases, the effect Y is an electrical quantity, forexample, a photocurrent, whereas cause X is representedby radiation power, irradiance or by the correspondingphotometric quantities luminous flux or illuminance'- 4:

s= Y/X. (1)

If Y is proportional to X, s has the same value for allvalues X, Y, and one may speak of a linear detector(Fig. 1). As soon as the relation between cause X andeffect Y ceases to be linear and the characteristic there-fore becomes a curve, there are several quantities thatmay be defined as sensitivities. First it is possible tokeep the definition according to Eq. (1); this quantityfurther on is called sensitivity. In addition, the slopeat a certain point of the characteristic may be intro-duced as differential sensitivity sA (Fig. 2):

s = dY/dX. (2)

Finally, certain practical applications demand thequotient of a greater change of effect by the correspond-ing change of cause, which may be called here intervalsensitivity (Fig. 3):

Si = (Y2 - Y 1)/(X 2 - XI). (3)

The interval X2-X1 may be a decade or more.As in plotting these relations one often uses the

logarithms of X and Y, the quantities introduced byEqs. (2) and (3) may also be referred to the logarithms ofX and Y. In order to avoid misunderstandings, theword logarithmic should be added.

The author is with the Physikalisch-Technische Bundesanstalt,33 Braunschweig, Germany.

Received 28 September 1967.

It should be said that effect means only the effectcaused by irradiation. Therefore, an effect existingwithout irradiation, for example, a dark current, must besubtracted from the whole effect. Thus for a nonlineardetector it is possible to define at least three differentsensitivities that in the linear case are identical with thedefinition of Eq. (1).

II. Spectral SensitivityIt is well known that the sensitivity of a radiation de-

tector generally depends on the wavelength of radiation.Therefore one introduces as sensitivity in the infinitesi-mal wavelength region dX belonging to the wavelengthX the spectral (monochromatic) sensitivity s(X) definedby the quotient

s(X) = [dY(X)/dX(X)] = [dY(X)/Xxdxl

= (Yxdx/Xxdx) = YN/Xx, (4)

where YX = dY(X)/dX; XX = dX(X)/dX. Here X(X)is the radiation in the spectral region 0 to X and Y(X) theeffect caused by this radiation. The derivative Xx =dX(X)/dX is, as usual, the spectral density of the quan-tity concerned, for example, the spectral (density of)radiation power, whereas the derivative Yx = dY(X)/d\means the spectral (density of the) effect belonging toXx. Now it is not usual to speak of a spectral (densityof) photocurrent I, = dI(X)/d\; but such a quantity isnot without sense, of course, only in connection with thespectral (density of) cause Xx.

This may be seen at once if, from the spectral causeXx, one derives the total cause X(X) in the spectral re-gion 0 to X by integration:

X(X) = fXxdx.Jo

By plotting X(X) and also the effect Y(X) created by itas functions of the upper integration limit X, one obtainstwo curves, the slopes of which at wavelength X areXx = dX(X)/dX and Y, = dY(X)ldX; thus, the quotients(X) = dY(X)/dX(X) equals Y/XN (Fig. 4).

June 1968 / Vol. 7, No. 6 / APPLIED OPTICS 1017

s= Y = tan a

X

cause X

relation between cause X and effect Y. Sen-sitivity s is constant.

XY tan et dy tan 3X dX

xcause X

Fig. 2. Nonlinear relation between cause X and effect Y.Sensitivity s and differential sensitivity s are different for each

point of the characteristic.

Written in this manner, s(X) is a differential sensitivityand in the case of a nonlinear detector depends on thevalue of X(X), which means the value of the radiationincident in the spectral region between 0 and X.

If one wishes to derive the sensitivity s for a wave-length region X to 2 from the spectral sensitivity s(X),this relation is obtained by integration:

s = S(Xl, X2) = Y(X1,X2)X(X=,X2)

rX2 I X2

(Xp YXd/f X dx

X2 /1lX2 XxdX). (5)

= Xxs(X)dX / lxi~A. 5

Equations (4) and (5) only hold for the case of linearity,as a nonlinear detector has a unique characteristicfor each wavelength, and up to now there exists no rela-tion between the different characteristics. If one wishesto apply these equations to a nonlinear detector, onehas to introduce the relative spectral sensitivity.

Ill. Relative Spectral SensitivityRelative spectral sensitivity s(X)roi is the ratio of the

value of s(X) at wavelength X and the value s(Xo) at afixed wavelength X0:

a, -

, X1 z

s(X)re = (X)/s(xo). (6)

For a nonlinear detector the spectral sensitivity atwavelength X and also at Xo depends on the magnitudeof the cause according to the characteristics. Thus,there is a variety of different values of spectral sen-sitivities belonging to each waveojngth: the questionis how to obtain a relative spectral sensitivity accordingto Eq. (6). This is possible if one introduces an addi-tional condition and refers either to equal cause or toequal effect.

Generally, the relative values s(X)re, obtained in thisway will still depend on the levels of cause or effect.If one desires to obtain a value of s(X)re,, which holdsfor all levels, there must exist a relation between thedifferent characteristics.

In order to deal with monochromatic characteristicsas it is necessary here, one must take a small but notinfinitesimal wavelength region AX. The value of thecause at any wavelength , therefore, is X(X) =XxAX and the effect belonging to this cause is AY(X).In practice, AX is given by the bandwidth of the mono-chromator or filter used, Xx by the intensity of the

Y2

= Y,

Y2-y1 =tanY2 -t n

X,

Y=f(XI

-A

cause X

Fig. 3. Nonlinear relation between cause X and effectInterval sensitivity s = (Y2 - Y)/(X2 - Xl).

Y.

dY(A) XIAI

dXU)

dA H-

Awavelength A

Fig. 4. Cause X(X) and effect Y(X) in relation with wavelengthX, The spectral sensitivity s(X) is the quotient of the spectraleffect Y by the spectral cause Xx. tan a = dX(X)/dX = XX;tan = dY(X)/dX = Yx; s(X) = dY(X)/dX(X) = Y/Xx.

1018 APPLIED OPTICS / Vol. 7, No. 6 / June 1968

Fig. 1. Linear

y

- i

N..

t

.!! Y

Z

t

s(A)",= A I = c IA)reL MI01

As for equal cause AX(X) = AX(Xo), there followsf[AX(X)] = fAX(Xo)]. According to Eq. (6), the re-sult is

AY(A)

A y 111

AXi2J= AX(l%)cause AX

Fig. 5. Relative spectral sensitivity s(X),ei for equal cause AX(X)= AX(Xo). Characteristic for Xo isf [X(XO)]; for X it is c(X)

f[AX(X) -

AYVA)

Fig. 6. RelativeAY(X) = Y(X,).

sl A), -- X-= -ArjX (A)

cause AX

spectral sensitivity s(X)rei for equal effectCharacteristic for Xo is f [AX(Xo)] ; for X it is

f [c(X)AX(?,)] .

(b)

S(X)rel = S(X)/S(X) = C,

s(X) = AY(X)/AX(X) = f[c(X)AX(X) /AX(X),

s(Xo) = AY(X0 )/AX(X 0) = fAX(X 0)l/AX(X0).

(11)

(12)

(13)

As for equal effect AY(X) = A Y(X0), there followsc(X)AX(X) = AX(X,). According to Eq. (6), the re-sult is

S(X)rel = s(X)/s(Xo) = c(X)- (14)

In both cases the result is the same, (X)rel = c(X), andthis value does not depend on the level of AX or AY,(Figs. 5 and 6)..

From Eqs. (7) and (8) it follows that by plotting on alogarithmic scale the characteristics become parallelcurves. If one refers to equal cause, they are displacedparallel to the AY axis, and if one refers to equal effect,they are displaced parallel to the AX axis (Fig. 7).

These relations here are derived only for the sensi-tivity defined according to Eq. (1). They hold also forthe differential sensitivity according to Eq. (2) and theinterval sensitivity according to Eq. (3).

IV. Physical Interpretation of Both Definitionsof Relative Spectral Sensitivity

According to the purely mathematical treatment,relative spectral sensitivity referring to equal cause (a)and to equal effect (b) are of equal rank, but they arenot compatible, although the result is the same. Nowthe question is what the physical meaning of each caseis.

One may start by postulating that the effect createdby radiation in a detector should not depend on thesequence of radiation with different wavelengths fallingon the detector. The effect must be the same, if thedetector is first irradiated with wavelength X0 and sec-

radiation source. If Xx is high enough, one may obtainany value AX(X) as high as desired, and the same holdsfor AY(X).

If f [AX(X0) ] is the characteristic for wavelength Xo,the function representing the characteristic for wave-length X is

(a) referring to equal cause: AY(X) = c(X)f[AX(X)l. (7)

(b) referring to equal effect: AY(X) = f[c(X)AX`(X)l. (8)

Here f may be any function, c(X) a parameter depend-ing only on wavelength X.

To prove this, we derive from Eqs. (7) and (8) thespectral sensitivities according to Eq. (4), taking intoaccount there is now the finite wavelength region AXand we have to write A instead of d. From therethe relative spectral sensitivity according to Eq. (6) isderived for both cases (a) and (b):

(a) s(X) AY(X)/AX(X) = c(X)f[AX(X)/AX(X), (9)

s(xo) AY(Xo)/AX(Xo) = f [AX(Xo)] /AX(Xo). (10)

a

Nf

_ 0,5t;

-

_;5

0,1

0,1 0,5

Ig(cause X/ X)

Fig. 7. Characteristics plotted logarithmically on each axis.Curve 0 for X0, I for X at equal cause, II for X at equal effect.The absolute value lgc(X) is equal in both cases; AXo, AYo

units of AX, AY.

June 1968 / Vol. 7, No. 6 / APPLIED OPTICS 1019

, o I

..' I -"r

z

I

a,

AY2

AY1

cause X

Fig. 8. Relative spectral sensitivity s(X)rc1 for equal effectAY(W = AY(Xo). First works AX,(Xo); equal effect AY, forAXI(X). If AX2(X) is added, the total effect is Y2 = A 4 ofFig. 9. AY 1 = f [AXX 0 )] = f[c(X)AXi(X)l; AX 1 (X) = AXIWo/c(A); A 2 = f[c(X)[AX 1(X) + AX2(X)l = f[AX1(XO) +

c(X)AX 2(X)l -

A X2 Ad AX2(A)

cause AX

Fig. 9. Relative spectral sensitivity 5(X)r 0 1 for equal effectAY(X) = AY(Xo). First works AX 2(X); equal effect AY 3 forAX2(XO). If AX,(Xo) is added, the total effect is AY 4 = A 2 ofFig. 8. AY3 = f[c(X)AX2(X)l = f[AX 2(Ao)l; AX2(XO) =c(N)AX 2(X) Al'4 = f[AX2(X0) + AX1(Xo)l = f[C(X)AX 2(X) +

AX (o) -

ondly with wavelength X or if one begins irradiatingwith wavelength X and then adds radiation of wave-length 0.

It is possible to show that this situation holds only ifthere is a relative spectral sensitivity at equal effect;if one refers to equal cause, the effects created in bothcases are different (Figs. and 9).

From a physical point of view this is understood ifone supposes a primary linear effect, depending onwavelength but not on the value of incident radiation,and a subsequent secondary nonlinear effect that isindependent of wavelength but depends on the value ofincident radiation. The primary effect for a thermaldetector is the absorptance, for a photoelectric detectorthe quantum yield. The secondary effect for a thermaldetector is caused by the nonlinear rise of heat losseswith increasing temperature for a photoelectric detectorby space charges. In this scheme it is essential thatas soon as we have an effect, it does not matter bywhich wavelength it has been caused. Only in this

manner is a relation between different characteristicspossible.

Now one could think that each monochromatic radi-ation XxdX works according to its own characteristic ifthe detector is irradiated with radiation consisting ofdifferent wavelengths. As independent of the finitevalue X the product Xxd becomes infinitely small bydiminishing d; this would mean that for each wave-length one would have to take the sensitivity belongingto the origin of the characteristic; in this case no devi-ation of linearity is permitted.

It can be seen therefrom that a relative spectralsensitivity in connection with nonlinear detectors isonly meaningful if one refers to equal effect. A relativespectral sensitivity based on equal cause may not beused in a general manner. This is in accordance withthe experimental results obtained, for example, with aphotovoltaic cell (Fig. 10).

One can see at once that the curves are displacedparallel to the AE axis and not parallel to the Al axis.According to Fig. 7, this means that there is a relativespectral sensitivity referring to equal photocurrentindependent of the value of the photocurrent, but notreferring to equal irradiance.

There are two more items to be pointed out. First,when measuring with monochromatic radiation, it isfor experimental reasons impossible to measure alwayswith equal effect. But one may convert results mea-sured at one level to another level if only a single char-acteristic is known. This characteristic may hold formonochromatic radiation or for radiation combined ofseveral wavelengths or even a continuum. Moreover,it does not matter whether radiant or photometricquantities are used, since the principal form of thecharacteristic does not depend on this choice (Fig. 11).If AX1 (N), AX,1 () cause the effect AY, and AY 2(X0),AX 2(X) the effect A Y2 , the following relation holds:

AXI(XO)/AX 1(X) = AX2 (XO)/AX 2(X) = c(X) = S(X)rel.

Therefore,

AX2(X) = AX 1(X)[AX 2(XO)/AX 1(Xo)].

10

1,0 -

0,1 -

0,01

(15)

(16)

576nm

I- II0,1 1,0 10

Lg( E )

100

Fig. 10. Characteristics of a photovoltaic cell plotted on log-arithmic scales. E irradiance in uW/cm2 , Al photocurrent in

PA.

1020 APPLIED OPTICS / Vol. 7, No. 6 / June 1968

rlllll F -u.ul ,nn"I

cause AX

spectral sensitivity s(X)rei foiConversion to equal effect by

characteristic.

Second, for other reasons it may also be impossibleto define a relative spectral sensitivity if one refers toequal effect. It may be that the primary effect is notthe same for all wavelengths, for example, if the absorp-tion depends on the depth of penetration for differentwavelengths and thus the starting conditions for thesecondary effect depend on wavelength, too.

It should be mentioned that with a linear detectorthe relative spectral sensitivity is the same whether onerefers to equal cause or to equal effect.

V. The Effect of Continuous Radiation on aNonlinear Detector

In calculating the sensitivity for a certain wavelengthregion according to Eq. (5), the following integral was inthe numerator:

>X2 fX2

Y = 1 X Yxd = f Xxs(X)dX. (17)

Up to here it has not been discussed whether thisrelation holds also for a nonlinear detector. If thereis a relative spectral sensitivity with respect to equaleffect, it may be written as

X2 X2

l = jf Xs(X)reI(Xo)dx = s(Xo) JX XXS(X)eldx. (18)

As s(Xo) does not depend on the wavelength, thisquantity may be put before the integral. But it is nec-essary to keep in mind that in the case of a nonlineardetector s(\o) is a function of the integral fXXS(N)reidX,which is equivalent to a monochromatic radiation ofwavelength No causing the same effect Y as the continuousradiation. The value of the integral therefore definesthe point on the characteristic curve for Xo, for which thevalue of s(No) must be taken.

VI. ActinityRadiations of different spectral distributions Z, N

generally cause different effects on a detector. Ifs(Z) and s(N) are the sensitivities according to Eq. (1)

or (5), belonging to the total radiations of Z and N, onemay define the actinity a (Z,N) as the ratio of these

f [c()AX(All sensitivities2' 3 :

a(Z,N) = s(Z)/s(N) (19)

One can see from Eq. (19) that the actinity is a kindof relative spectral sensitivity for radiation distributedover larger wavelength regions. If the detector is notlinear, one must also refer to equal effect, and the resultis

a(Z,N) = X(N)/X(Z), (20)

with Y(N) = Y(Z). Written this way, Eq. (20) holdsalso for nonlinear detectors.

requal effect VII. Radiation Evaluated According V(N) or Any_4 Other Function

As already mentioned, it is usual to take photo-metric quantities for the cause X also. A photometricquantity may be regarded as an effect on the photo-metric standard detector, the spectral sensitivity ofwhich is represented by the luminous efficiency Km V(N)of the radiation. This detector is linear by definition,as the nonlinear relation between light sensation andstimulating radiation is not taken into account. Strictlyspeaking, sensitivities using photometric quantitiesfor the cause are, therefore, already ratios of sensitivi-ties; for the real cause is always radiation energy.

If one deals with detectors, the spectral sensitivity ofwhich is matched to that of the human eye, precautionsneed to be taken. It goes without saying that theactinity of such a detector, referring to photometricquantities, does not depend on the spectral distributionof the radiation.

A further step consists in matching the radiationfalling on the detector according to any spectral sen-sitivity. This may be necessary if no unselective de-tector is at hand or if one wishes to take account ofsmall selectivities that cannot be avoided. This ispossible if this reference detector is linear.

If this condition is not fulfilled, difficulties arise,as one cannot measure for equal effect for both detectorsat the same time. In this case, an additional measure-ment, which is equivalent to measuring the two pointsof the characteristic of the reference detector, helps tosurmount this obstacle. The sensitivities and actinitiesmay be measured under these conditions, too.

References1. G. Bauer, Der Begriff der "Empfindlichkeit" bei Strahlungs-

empfangern fur den optischen Spektralbereich, PTB Mitteil-ungen (1967), S. 116

2. J. Eggert und A. Kiister, Empfindlichkeit photographischerSchichten bei Glihlampenbeleuchtung unter Beriicksichti-gung der Aktinitat der Sperrschicht-Photozellen. Die Photo-graphische Industrie, Heft 35 (1940), S. 516

3. H.-J. Helwig und J. Krochmann, tUber die Aktinitit einerStrahlung fur lichtempfindliche-Empffnger. Arch. Techn.Mess., V 434-10, Sept. 1966

4. J. Krochmann, Uber den Einfluss der Strahlungsfunktion aufdie Wirkung bei lichtempfindlichen Empfhngern. Lichttech-nik 17 79 A (1965), S.

June 1968. / Vol. 7, No. 6 / APPLIED OPTICS 1021

AY2

AY

Fig. 11. RelativeAY(X) = AY(Xo).

N.

a)

lll..l:. ;U- ".l.y

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