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PUBLICATIONS

AlllOM iOMait

NisrUnited States Department of Commerce

Technology Administration

National Institute of Standards and Technology

IM/ST Technical Note 1376

Optical Detector Nonlinearity

Simulation

Shao Yang

Igor Vayshenker

Xiaoyu Li

Thonnas R. Scott

Mike Zander

Optical Detector Nonlinearity:

Simulation

Shao Yang

Igor Vayshenker

Xiaoyu Li

Thomas R. Scott

Mike Zander

Optoelectronics Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

May 1995

^ \ Me.

*

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar. Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1376, 40 pages (May 1995)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1995

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
CONTENTS

1. Introduction 2

2. Definition and Basic Expressions 3

2.1 Definition 3

2.2 Polynomial Expression 6

3. Measurement Methods 8

3.1 Superposition Methods 9

3.1.1 Integral-Step Superposition 10

3.1.2 Modified Superposition 12

3.1.3 Triplet Superposition 12

3.2 Attenuation Method 14

3.3 Differential Method 16

4. Computer Simulation 19

4.1 Results 20

4.1.1 Systematic Error 21

4.1.2 Random Uncertainty 23

4.1.3 Combined Uncertainty 23

4.1.4 Effect of Reference Point 28

4.1.5 Other Results 28

5. Conclusions 32

6. References 34

Appendix 35

iii

Optical Detector Nonlinearity: Simulation

Shao Yang', Igor Vayshenker, Xiaoyu Li, and Thomas R. Scott

National Institute of Standards and Technology

Optoelectronic Division

Boulder, Colorado 80303

W. Michael Zander

Army Primary Standards Laboratory

USATA

Redstone Arsenal, Alabama

Abstract

We have developed a unified mathematical treatment for five commonly used

methods of measuring optical detector nonlinearity. We performed computer simulations

to compare these methods for different measurement conditions and data processing

options. The triplet and differential methods gave the best overall results, and third- and

fourth-order polynomial representations of the measurement results will yield the least total error for a common practical measurement system.

Keywords: nonlinearity, optical power calibration, optical detector, simulation

'Shao Yang is with Ohmeda Medical Systems, Inc., Louisville, Colorado 80027
1. Introduction

Optical detector linearity has always been an important issue for people who are concerned about the accuracy of an optical radiation measurement. The calibration of an

optical power meter from a characterized optical power transfer standard is often

performed at one power. To extend the calibration to the full dynamic range of the power

meter, it is essential to measure the meter linearity. Even when a power meter is used for

relative measurements such as attenuation measurements of a large magnitude, the

knowledge of the meter nonlinearity will improve the measurement accuracy. Many

laboratories have studied the problem of detector linearity and have developed various

methods to measure this property. Each method has advantages and disadvantages relative

to their resulting accuracy, precision, and effectiveness. However, we found it difficult to

choose a best method suitable for optical power meters by simply comparing the published

results. This was due to the fact that each of the previous individual efforts used their own

method of analysis, and results were reported with expressions using different definitions

of detector nonlinearity; it was, therefore, impossible to make a direct comparison of these

published results. The purpose of this work is to investigate theoretically some of the

major issues that have an impact on the accurate measurement of detector nonlinearity.

These issues are related to definition, measurement methods, data uncertainties, and data

processing.

It is important to adopt a single definition and use a unified mathematical treatment

to describe the nonlinearity of a detector measured by different methods. Only then can we compare these methods on a common basis. For this purpose, we employed a specific

definition of detector nonlinearity and derived useful expressions based on this definition

for data processing and comparison of different methods. These mathematical expressions

and derivations are presented in section 2 of this paper. Section 3 describes, in terms of

the basic expressions laid out in section 2, the five methods we considered. These are the

methods now most commonly used by various laboratories. We also derived relationships

between the definitions used by the different laboratories.

The comparison study was done using computer simulations. Section 4 describes

how the simulations were performed and gives results of the simulations. The computer

simulations studied issues common to the five methods, such as effects due to type and

magnitude of data error, number of data points, and the order of the curve-fitting

polynomial. For this analysis, all five methods were compared under the same conditions.

Where appropriate, issues specific to individual methods were also studied in the

simulations. We found that the quality of results produced by these methods is significantly

affected by the characteristics of the detector being studied. Also, the choice of polynomial

order used for data processing has a substantial effect on the systematic error and random

uncertainty of the results, and, thus, there is an optimum polynomial order for each specific

condition. The results of the simulations helped us to choose the method, the design of

measurement system, and the optimum data processing to achieve a desired accuracy.
To measure optical radiation, a detector always needs accompanying equipment such

as electronic circuits and display instruments. It is not always possible to separate the

detector from these elements, so we characterized not just the detector, but the detection

system. Because the underlying principle is the same whether we measure the nonlinearity

of a single detector or a detection system, we will simply use the word 'detector' in the

remainder of this paper.

2. Defimtion and Basic E}q)ressioiis

2.1. Definition

In the Draft International Standard of Calibration of Optical Power Meters [1], a document of the International Electrotechnical Commission (lEC), nonlinearity is defined

as "Departure from linear response, i.e., the difference between the response at an arbitrary

power level and the response at the reference power level, divided by the response at the

reference power level." The reference power is usually the calibration power P^. This

definition can be expressed as

where R(P) = V/P is the responsivity of the detector at optical power P incident on the

detector, the subscript c indicates calibration, and V is the detector output, which can be

electric current or voltage, or the display reading from a power meter. The relationship

between the detector output and the input power is described by the response function

V = V(P). (2)

The responsivity R(P) can thus be written as

R(P) = ^. (3)

Using eq (3), we express the nonlinearity of eq (1) in terms of the response function:

V(P)P

A^(P;PJ = -2J-1 - 1. (4)

V(P )P

Because the purpose of measuring the detector nonlinearity is to estimate the

accuracy of or to make corrections to optical radiation measurements, nonlinearity of
V(P,)

V(PJ

p Pm p.

INPUT POWER

Figure 1. Response function.

eq (1) can equivalently be expressed as the error it causes in the measurement of optical

power:

P - P

A = _^ (5)

where P^ = V/R(Pg) = V(P)PyV(Pc) is the measured optical power, which is dependent

on calibration as is illustrated in the response curve of a detector in figure 1. Once the

detector nonlinearity is measured, a correction can be made to obtain the actual power P

from the measured power P^:

P =

1 + A

P.(l - M- (6)

NL

The approximate equality in eq (6) holds because A^l is usually very small (no more than

a few percent) and the higher order terms can be ignored in the Taylor series expansion

of (1 + A^)-\
The inverse of the response function is called the conversion function, illustrated in

figure 2, which converts the output V to the input power P and is written as P = P(V). (7)

In the real measurement, it is more convenient to use the conversion function than the

response function. Equation (7) may be used in eq (1) to express nonlinearity A,^ in terms

of V and the conversion function:

Anl(V;V) = P(Ve)v

P(V)V

- 1. (8)

There are two types of nonlinearity. The first is the saturation type, where the

responsivity R(V) decreases with increasing input power P, or output V. The second type

is called supralinearity [2], where the responsivity increases with the input power, or the

output. The two types of nonlinearity can therefore be distinguished by the sign of the first derivative of the responsivity dR(P)/dP or, from eq (1), by the sign of the first derivative

P(V)/ P(VJ

V V,

OUTPUT

Figure 2. Conversion function.

5
of nonlinearity Anl(P;PJ or Anl(V;V^)

dA^(V;V)

dV

dA^(V;V)

dV

< saturation type;

(9)

> supralinear type.

It is often necessary to characterize the nonlinearity of a detector before calibrating

it. However, the definition of nonlinearity A^l is referred to the calibration power P^ (or

calibration output VJ. Since the output V is directly measurable while the power P is derived from V through calibration, using nonlinearity expressions in terms of V makes it possible to characterize the nonlinearity of a detector without calibration. We can use a dummy calibration output, or reference output, V^ for the expression of nonlinearity

Anl(V;VJ. If later the detector is calibrated at power P^ where the output is V^, the

detector nonlinearity referred to V^ can simply be calculated from the nonlinearity referred

to V^ through the relation

Anl(V;V ) = A^(V;V ) - A^(V ;V ). (10)

Again this relation holds with the assumption that nonlinearity is very small. Since V^ is not a real calibration point, it can be 0, representing a limiting case: A^(V;0) = lim[A^(V;V)]

v—o

(11)

_ F(0)V _ J P(V)

where P'(0) is the derivative of the conversion function P(V) at V=0.

2.2. Polynomial Expression

We now have all the basic expressions for characterizing the nonlinearity of a

detector. But we still need a specific form of the conversion function (or response

function) to really do so. Because the conversion function cannot generally be expressed

in a closed form, a polynomial is frequently used to represent it [3,4]:

P(V) = Y^ a^VK (12)

When the nonlinearity is small, a polynomial can represent the conversion function well.
The responses of the commonly used photodiodes, Si, Ge, or InGaAs, are exponential with

respect to input power, but they are often used in nearly short-circuited or reverse-biased

configurations to achieve a more linear response. As a result, their response can be

approximated as a polynomial produced by Taylor series expansion shown in eq (12), where

the zero-order term is not included because we assume that the dark output of the detector

is always adjusted to 0. If the dark output is not 0, we can add a zero-order term in the

polynomial.

The nonlinearity A^l of eq (8) now becomes

Anl(V;VJ = -t l'(V'- - Vj-') (13)

k-2 ^1

when the nonlinearity is very small.

The factor aja^ in eq (13) suggests that we can further simplify the polynomial

expression by dividing all the coefficients a^. by the first coefficient a^ without altering the

nonlinearity Aj^. The polynomial thus obtained is called normalized polynomial and is denoted by p(V):

p(V)=V^J2\V\ (14)

k:c2

where b,^=at/ai. The normalized conversion function p(V) can be turned to the general

conversion function P(V) through the determination of the coefficient a^ by means of

calibration. If the calibration power is P^ and its corresponding output is V^, we have

a, = -^. (15)

' P(V)

Equation (15) holds for conversion functions P(V) and p(V) in general. P(V) is the

calibrated conversion function, p(V) the uncalibrated conversion function, and a^ the

calibration factor or correction factor (fig. 3). In terms of the normalized polynomial, eq

(13) becomes

^nl(V;v ) = -J: \(y'-' - vJ-Y (16)

k=2
(^

o

PL.

P(Vc) P(V)

OUTPUT

Figure 3. Calibrated and uncalibrated conversion functions.

3. Measurement Methods

Methods for the measurement of detector nonlinearity can be divided into two

groups: dependent and independent measurements. Dependent methods always involve the

known values or the measurement of some physical quantities other than the output of the

detector under test. Typical methods in this category are the inverse square-law method

[5], Bouguer's law method [5], Beer's law method [5], and the attenuation method [3,6].

Independent methods rely solely on the measurement of the output of the device under

test. The superposition (addition) method and the differential method belong to this

group. We favor independent methods over dependent methods since they are more direct.

Three variations of the superposition method, the differential method, and the attenuation

method from the dependent group are considered in this study. The attenuation method

is probably the only dependent method still being used. This is because of the simplicity

of the setup and, perhaps, because the other quantity involved in this method is an optical

parameter, the transmittance of an optical filter. The attenuation method can also be

handled as an independent method if the transmittance of the filter is treated as an unknown parameter in the data processing. Both the dependent and independent

8
treatments of the attenuation method are studied in this work.

3.1. Superpositioii Method

The superposition method uses two types of systems: beam superposition [3,4,7-10]

and aperture superposition [11-14]. Figures 4 and 5 depict these two systems. The beam

superposition method, shown in figure 4 uses a single light source with a beam splitter to

get two beams from the source and then the two beams are recombined on device under

test (DUT). Precautions must be taken to avoid interference between the two combined

beams in this design. A more recent design is an optical fiber system which includes optical

fiber attenuators, couplers, and splitters. There are many designs of apertures used in the

aperture superposition method, which is mainly used for the integral-step superposition

method.

source 1 shutter ^S

variable

fUter

DVM

DUT

shutter

variable filter

source 2

Figure 4. Beam superposition method.

multiaperture

shutter

source

lens

DVM

lens

shutter

Figure 5. Aperture superposition method.
3.1.1. Integral-Step Superposition Method

The power from one of the two beams, Beam 1, is adjusted to an arbitrary power

Pj, which is usually at the lower end of the measurement range, when the other beam.

Beam 2, is blocked [7,8]. The corresponding output from the detector is Vj. Then Beam

1 is blocked, and Beam 2 is opened and adjusted so that the detector output is exactly V^,

which indicates that the incident power from Beam 2 is also P^. Then both beams are opened and combined. The power incident onto the detector is 2Pi. The detector output

is recorded as Vj. V2 may not equal IV-^ because of detector nonlinearity. Repeating the

superposition steps, we can get any integral multiples of the power unit P^ and their

corresponding outputs of the detector. This is illustrated in Table I.

Table 1. Illustration of integral-step superposition.

Beam 1 Beam 2 Beam 1 + Beam 2

Pi ^\^ Vi =^ Pi Pi -I- Pi = 2Pi =» V2

Vi =» Pi V2 =j. 2Pi Pi + 2Pi = 3Pi ^ V3

With n measurements made, we obtain n sets of data (j, Vj), where j is power in

integers of an arbitrary unit. When n is sufficiently large, an uncalibrated conversion curve

in the form of normalized polynomial can be obtained by linear least-squares curve fitting,

A second definition of nonlinearity can also be used with integral-step superposition

measurement [9-14]. When this definition is used, the powers from the two individual

beams in every step are always adjusted the same and equal to the combined power of the

previous step. Thus the powers, individual or combined, are always in powers of 2 of the

initial arbitrary unit power, as illustrated in table 2.

If we define the nonlinearity as NL, then for measurements as shown on the first

line of table 2:

10
Table 2. Illustration of integral-step superposition.

Beam 1 Beam 2 Beam 1 + Beam 2

Pi=^ Vi Vi=^ Pi P, + P, = 2P, =. V2

V2 =^ 2P, V, =^ 2P, 2Pi + 2Pi = 4Pi =$ V4

V, =^ 4P, V4 =^ 4P, 4Pi + 4Pi = 8P1 =* Vg

: ;

(V, - V,) - v.

NL =

(17)

2V(P:)

J V(2PJ

Similar expressions hold for measurements at other powers. Using eq (5) and switching

from expressions in terms of P to expressions in terms of V, we obtain the following

relation between NL and A^lI

j^ V(Vn;V ) - A,,(Vi;VJ

1 - A^(V.;VJ

(18) = A^(V„;V ) - A^(V,;VJ

= V(V„;V,) = A^(2P,;P,).

This result indicates that the nonlinearity NL is, indeed, the nonlinearity A^l at the

combined beam power with reference to the individual beam power.

Integral-step superposition is straightforward. The data sets are pairs of power and

output, and the powers are integral multiples of an arbitrary unit. However, the integral

multiples of the power are judged by reading the detector output. Therefore the accuracy

of those multiples cannot be higher than that of the output V. In addition, uncertainies

in the powers of the earlier steps will be carried to later steps and hence there is an

accumulation of uncertainty.

11
3.1.2. Modified Superposition Method

In the modified superposition method [4] the input powers from the two beams vary

arbitrarily and the outputs are recorded. Using the normalized conversion polynomial, we

have, for the ith measurement of a single beam:

Pi = v, .I:b,v^ (19)

k=2

where the P; are arbitrary unknowns and the V; are their corresponding measured outputs.

Because for each measurement we have one unknown power, there are always more

unknowns than equations, and the coefficients of the polynomial can never be determined

from the measurement data. However, when we use superposition of all combinations of

the individual measurements from the two beams, we will get new equations without

increasing the number of unknowns:

Pi^Pi'V, .J^b.vJ, (20)

k=2

where subscripts i denotes the ith power from beam 1, j the jth power from beam 2, and

ij their combination. If we have m individual measurements from beam 1 and n measurements from beam 2, we will have m x n combinations of new measurements and

m X n new equations with no new unknowns. When enough measurements are made,

there will be more equations than unknowns, and the coefficients b^ can be obtained by

linear least-squares fitting.

Although data points in approximately equal increments of power are preferred, no

exact increments are required. This is a big improvement over the integral-step method

because it will greatly ease the measurement process and eliminate some sources of

uncertainty.

3.1.3. Triplet Superposition Method

In the triplet superposition method [3], a group of three measurements is made, two

for individual powers from each beam and one for the combination of the two. Then the

powers of the two individual beams are varied arbitrarily and combined again. For the ith

group of measurements, we have a set of three equations:

12
(21)

k>2

k>2

where the Pj are arbitrary unknowns and the V^ are measured output values. A new

equation

(Vsi - V,, - vj . Y.KC^'i - < - <) = (22)

k=2

is formed by eliminating the unknown P; in the group of the three equations eq (21).

Again the coefficients b^ can be obtained by linear least-squares fitting when enough data

sets have been measured.

There are many similarities between the modified superposition and the triplet

superposition methods. Data taken for the modified superposition method can also be

arranged in groups of triple measurements if data for individual beams are used several

times in different groups. Then they can be processed the same way as the triplet method.

Likewise, data taken in the triplet method can be processed as data taken in the modified

superposition method by keeping the powers as unknowns. However, the two measurement

procedures are different in one respect. In the triplet measurement, the three

measurements in one group are usually made in immediate succession. In the modified

superposition method, because all combinations of individual power from the two beams

are needed, it is not always possible to make all the individual and combined measurements

in immediate succession. The triplet procedure reduces the uncertainty that might occur due to the slow drift of the output of the light source. If a fixed pattern of measurements

is used, this source drift will cause some systematic error in the measurement for the

modified superposition method. One means to reduce this error is to reverse the

measurement procedure and use the average value of the forward and backward

measurements for each datum. Another way is to randomize the steps of the measurement

of individual and combined powers and thus convert this drift error to a random

uncertainty and further reduce it by increasing the number of data points taken. The two

procedures of taking data and of data processing can therefore be combined in four ways,

and we may choose any one of them. We considered two different methods for the triplet

method in our study.

13
source shutter BS filter

variable

filter

DVM

DUT

Figure 6. Attenuation method.

3.2. Attenuatioii Method

Figure 6 shows a schematic of the attenuation method [3]. Ideally, the filter has a

fixed density that is constant over the entire measurement power range. It is inserted in

the optical path at different powers, and the outputs of the attenuated and unattenuated

beams are measured. Expressing the ith measurement in the normalized conversion

polynomial, we have

k=2 (23)

k=2

where r is the transmittance of the neutral density filter, and V^ and V; are measured

outputs with and without the filter. The unknown P; is eliminated by forming the equation

(V^ -r^d -E^(Vh -rV\) =0, (24)

k=2

The coefficients b^^ can be obtained by linear least-squares fitting.

The fitting results rely on the value of transmittance r being used. It is in general

a measured quantity and therefore has uncertainty. This is an extra source of uncertainty

specific to this method. The common way of measuring the transmittance is similar to this

attenuation method described above. The magnitude of r is obtained by

14
V

r = V

(25)

The transmittance r in eq (25) is indeed an approximate value of r measured at power P

and biased by detector nonlinearity. We can further express eq (25) as

rrP) = Yl = Xfc^ = ^y(rP)P

(26)

^ ^ V V(P) V(P)rP*

Then, using eq (4), we have

^(V) = ^ = T[l . A^(tP;P)]

= ^[1 - A^(V;V)]

(27)

Because 1TMW = ^ and ^""[AnlC^'^)] = 0' the true value of r is the value when the

V-O V-.0 output V is 0, or the input power P is 0: V V

lim(_:) = lim(_:) = r. (28)

P_»o V v— V

Using eq (16) in eq (27), we obtain for polynomial expression

V

V k-2 (29)

n-1 = r ^Jl^^iv' - v!;),

k=l

where r and c^ are unknowns to be determined by curve fitting, r is the zero-order term

of this polynomial. The coefficients b,^ of the normalized conversion polynomial are obtained by \ = c^Jt. Thus r and the conversion polynomial are obtained at the same

time from the measurement. This approach of the attenuation method falls in the

independent category, where the prior knowledge of the transmittance r is not necessary.

Nonlinearity measured by the attenuation method is sometimes expressed as the

percent change of the transmittance measured at output V or input power P with respect

to the transmittance measured at a reference output V^ or input power P,:

15
NL =

Using eq (27), we can relate this nonlinearity NL to the nonlinearity Aj^ as NL » -[A^(V;V) - A^(V,;V„)],

(30)

(31)

where again it is assumed that the nonlinearity is very small so its higher-order terms can be ignored. If r is unknown and if V=P", where a is a constant, then the attenuation

method should not be used, because V/V=r".

The most attractive feature of the attenuation method is that it needs only one light source and thus simplifies the measurement setup.

3.3. Differential Method

A schematic diagram of the measurement [15] is shown in figure 7. Figure 8 is an

illustration of the method by means of the conversion curve. A small constant ac power

AP is superimposed onto a varying dc power P. A dc meter and an ac meter are used to

measure the ac output h(V) = kAV(V) at different dc output V, where k is used because

different meters are used for dc and ac readings. Because AP is constant,

h(V)«AV(V)/AP. When AP is sufficiently small, we have

dc n

shutter ^^

riable

liter shutter

- chopper

source 2

ac I

Figure 7. Differential method.

16
,. AV 1 lim — = AP^oAP dP(V)/dV

(32)

h(V) is then approximately proportional to the reciprocal of the derivative of the

conversion curve at V:

h(V) = kAV «

kAP a

dP(V)/dV dP(V)/dV

(33)

where a = kAP. The conversion curve is obtained by integrating the reciprocal of h(V):

P(V) = } ^^(^%V' = f-^dV,

ii dVV ihrvM

(34)

I. h(VO

where V is a dummy variable.

When polynomial expressions are used, we first obtain a polynomial for the

reciprocal of h(V) by curve fitting:

n-l

h(V) h

' (35)

P-P(V)

VV AV(V)

OUTPUT

Figure 8. Differential method.

17
Then we obtain the conversion polynomial by integration:

n-l P(V) = r aJ^c^V^MV

(36)

= i:a,v',

k-1

where a^=a(\,i/k. Dividing every term by a^, we obtain the normalized conversioqj

polynomial with coefficients b,,=at/ai=c,,.i/kCo:

n p(V) = V + J^b.V^ (37)

k-2

This calculation is based on the assumption that the intensity of the input ac signal

is constant during the measurement, which ensures that a = KAP is constant. In a real

measurement, however, this assumption does not always hold, and a is not constant. Error

will be introduced if we still treat a as a constant. In order to reduce the effect of source

intensity drift, a modified procedure of taking and processing the data is used. Immediately

after each h(V) is measured, the dc signal is blocked and the ac output at dc input is measured. Because any ac light signal has a dc component, the dc output is not when

the dc signal is blocked. It is the dc component Vq of the ac output. Thus the ac output

measured is h(Vo) when dc is blocked. The ratio of h(Vo)/h(V) is then taken; that is.

h(Vo) _ dP(V)/dV .3g.

h(V) dP(V)/dV|^.^*

The conversion function is then

P(v) = r^£(X) KVo),,„

i dV Lv h(VO

dV'

.

(39)

dP(V)

Comparing this equation with eq (34), we see that the factor dV has replaced a.

iv.v. Because this factor is a function of the dc component V^ of the ac output, it still varies with

the changing intensity of the input ac. However, we will show in Appendix that the effect

of ac drift on this factor is much less than that on a.

18
The ac input power cannot be made very small due to considerations of signal-tonoise ratio and the lowest range of the ac meter. Thus, the measured h(V) does not

represent the true derivative of the conversion curve; thus an error will result. This

problem is also studied by computer simulation and presented in the next chapter.

4. Computer Simulation

The purpose of this study is to consider what variations the experiment and data

processing will affect the final result. We will consider five different methods, two kinds

of data uncertainties at three different levels, and four different polynomial orders. It is not always possible to vary some of the conditions deliberately on a particular measurement

system, whereas any situation can be simulated on the computer. Nevertheless, not

everything can be studied by the simulation, especially systematic errors due to a specific

measurement setup. Our study is limited to the effects of random data uncertainty,

systematic error due to the truncating of polynomials, their combined effect, and their

effect on the different methods. We also studied issues specific to some individual

methods, such as the accuracy of the transmittance t obtained from the attenuation

measurement and the effect of finite ac power input for the differential method.

The simulations were performed in the following way. We assumed a polynomial

of finite order as an original response curve. A real conversion curve would need a polynomial of infinite order to represent it. The inverse function of an arbitrary polynomial

of finite order is generally a polynomial of infinite order. In this sense, the conversion

function, which is the inverse of our assumed response function, resembles the conversion

function of a real detector and we can study the error due to the omitting of higher order

terms. Because we knew the exact response function, the exact nonlinearity A^l was known

and used to calculate the error in the result.

From the assumed response polynomial, we created data of incident powers and

their corresponding outputs for different methods. The power covers a range of one order

of magnitude. The data were then processed in the way described in the previous chapter

for each method, and conversion polynomials of orders from 2 to 5 were obtained. The

resultant curve does not coincide with the original curve. The difference is the systematic

error (we know it exactly, so it is not an uncertainty) due to the truncation of the

polynomial and may also be due to some specific technical problems related with individual

method, for example, the finite input power in the differential method.

We then introduced random uncertainties with chosen standard deviation into the

output data. The random uncertainties have a Gaussian distribution. For the integral-step

superposition method, accumulated uncertainties were also added to the integers

representing powers. We then processed the data thus created to obtain the conversion

curves. Next we calculated the nonlinearities from the fitted curve and from the original

response curve. The difference between the two calculated values is the error caused by

both the truncation of the polynomial and the random uncertainty of the data.

19
This prcx^ess was repeated 50 times for each case and standard deviation a of the

nonlinearity difference was calculated:

a =

^ 50 (40) 3Q^v NLj

where A^l is the original nonlinearity and Aj^ is the resultant nonlinearity of the jth

measurement. If we denote the average nonlinearity of the 50 measurements as A ,we

can further express the standard deviation a as

a =

1 50 _

(41)

This shows that a is actually a combination of the systematic error and the random

uncertainty. The first term represents the systematic error due to truncation of the

polynomial, and the second term is the standard deviation caused by data random

uncertainty. We call o total uncertainty or combined uncertainty.

Three types of data random uncertainties represent different measurement

conditions. The first is when all the data in the same output range of the measurement

have the same absolute standard deviation. This is the case when the uncertainty is due

to the least count of the digital meter or detector dark current noise. Over one decade of

the input power, the relative uncertainty of the data varies by one order of magnitude. The

second type is where all the data have equal relative standard deviation. A typical example

is noise due to source intensity fluctuation. The third type is uncertainty due to shot noise.

In this case, the uncertainty is proportional to the square root of the power. Our

simulation considers the first two cases only because the magnitude of the uncertainty in

the third case lies somewhere between the first two cases.

4.1. Results

Results are presented in figures 9 through 21 that follow. When different methods

are compared on the same plot, SI stands for integral-step superposition method, SM for

modified superposition method, ST for triplet superposition method, AT-1 for attenuation

method with known transmittance, AT-2 for attenuation method with unknown

transmittance, and DF for differential method. The response functions we assumed in the

simulations are listed in table 3. The percentages in table 3 are nonlinearities calculated

at P = 1 with a dummy calibration power at 0. These percentages are used as indicators

of these functions when they appear in the same plot. When only one response function

is used, it is always the first function listed above. Data are created between P = 0.1 and

20
Table 3. Assumed response function used in the simulation.

3%: V = 2P + 0.05P^ + O.OOSP^ + 0.002?^

1%: V = P - 0.03P^ + 0.05P' - O.OIP*

0.5%: V = P + 0.013P2 - 0.002P^ - 0.006P^

0.15%: V = 2P + 0.005P2 - 0.003P^ + O.OOIP*

P = 1, The dummy calibration power is always unless otherwise stated. The

transmittance is 0.5 for the attenuation method.

4.1.1. Systematic Error

Figure 9 shows the systematic error due to truncation of the polynomial for the

methods considered. All the methods show the same trend, that the systematic error is less

when higher-order polynomial is used. The three superposition methods and the

attenuation method with known transmittance r have almost the same systematic error, while the differential method and the attenuation method without known t have larger

errors. That the differential method has a larger error is due to the fact that for the same

polynomial order of the final conversion function to be compared with other methods, we

are actually fitting the derivative of the conversion polynomial which has one less order

than the conversion polynomial, as is indicated by eq (35). As for the attenuation method

with unknown r, eq (29) shows that all the coefficients b,, are affected by the fitted value

of T, the zero-order term in the fitting polynomial, from which the final conversion

polynomial is obtained by dividing each coefficient by r. We noticed that each fitting

parameter usually has a higher error than that of the complete curve. The larger error in

T caused the larger error in the conversion curve of this method than those in the other

methods.

Because Figure 9 was obtained for one particular response curve that we have

assumed, one may ask whether it is representative for any response curve. To answer this

question, we conducted the same simulation with four different response curves (see table

3), each having a different nonlinearity. The result for the integral-step method is shown

in figure 10, from which we can draw the following conclusions. The magnitude of

systematic error due to truncation of the fitting polynomial is different for different

response functions. Although the main trend remains the same, the slope from one order

of polynomial to the next can be quite different, which depends on the relative significance

of each term in the conversion polynomial.

21
1E-02

1E-08

2 3 4

POLYNOMIAL ORDER

-BSI -^SM -e-ST A-AT-l-»AT-2«-DF

Figure 9. Systematic error due to polynomial truncation.

lE-01

g lE-02 - Pi

g lE-03 I ^ lE-04 4

Pi

< lE-05 4

5 lE-06 4

g lE-07 -

lE-08

3 4

POLYNOMIAL ORDER

e3% -^-1% -e-0.5% 7^0.15%

Figure 10. Systematic error of integral-step superposition method

for the four different response polynomials listed in table 3.

22
4.1.2. Random Uncertainty

The effect of random data uncertainty is demonstrated in figure 11. First we notice

that all the curves are the same, that is, uncertainty of the result increases with increasing

number of order of fitting polynomial. The sign of the slope is opposite to that induced

by systematic error. The reason is that the lower-order polynomial has less freedom in

fitting. It represents the general shape of the conversion curve more than the details of the

curve. It is therefore less influenced by any single data point. As a result, lower-order

polynomials are less affected by the random uncertainties of the data and thus yield results

with lower deviations.

The attenuation method with unknown transmittance t and the integral-step

superposition method yield significantly larger uncertainties than all the other methods, and

the differential method gives the least uncertainty. In the attenuation method, r is one of

the fitting parameters, and it is more sensitive to the data random uncertainty than the

entire fitting curve. The larger random uncertainty of r is then carried into all the b^

through eq (29). The final conversion curve, which is constructed from all the b^, bears a higher uncertainty than those directly obtained from the fitting for other methods. In the case of the integral-step superposition method, the high standard deviation is due to the

fact that there are random uncertainties in both measured variables, the output V and the

input power, and the random uncertainty accumulates as the power increases.

The differential method yields the least random uncertainty of all because the fitting

curve, which is the derivative of the conversion curve, has one less polynomial order than

the fitting curves of the other methods, which are the conversion curves. For this reason,

the differential method has a higher standard deviation. If, in figure 11, we move the curve

for the differential method horizontally to the left a distance of one polynomial order, it will almost overlap with the curves representing the superposition methods and the

attenuation method with known r. This suggests that the cause of the lower uncertainty

of this method is that the fitting polynomial in this method has actually one less order than

the final conversion polynomial, which is obtained directly fi-om fitting in other methods.

The differential method also has two independent variables, DC output V and AC

output h(V). However, the uncertainty due to V is much less than that due to h(V) if the

two variables have the same relative uncertainties. Simulations of this method are presented in figure 12.

4.1.3. Combined Uncertainty

When we consider the systematic error and random uncertainty at the same time,

we have the combined uncertainty o of eq (41), demonstrated in figure 13. Because the

systematic error and the random uncertainty behave differently with respect to polynomial

order, there is an optimum polynomial order where the combined uncertainty is minimum.

In the case shown in figure 13, the optimum is the third order. For polynomial orders

higher than this optimum one, the random uncertainty is higher than the systematic error,

23
10

0.1

0.01

0.001

3 4 5

POLYNOMIAL ORDER

SI -^SM ST AT-l-ffl-AT-2-&DF

Figure 11. Nonlinearity uncertainty due to data random

uncertainty. The data have an equal absolute random uncertainty

which is 1 percent of the value at the middle of the range.

O

lE-07

2 3 4 5

POLYNOMIAL ORDER

Figure 12. Differential method nonlinearity uncertainty due to

data random uncertainty in ac or dc or both, compared to

systematic error.

24
and the combined uncertainty behaves the same as random uncertainty, while for the lower

orders, the systematic error dominates and the combined uncertainty behaves the same as systematic error.

The systematic error, the random uncertainty, and the combined uncertainty are demonstrated in a different way in figure 14, where curves for different data uncertainties are presented for the attenuation method. When the data uncertainty is highest, systematic

error is always lower than random uncertainty for all the polynomial orders shown. When

data uncertainty decreases, the random uncertainty falls until at the second order it is almost the same as systematic error. Further reducing the random uncertainty of the data

makes the combined uncertainty determined by systematic error at second order and by

random uncertainty at higher orders; therefore, the total uncertainty is minimized at the

third-order polynomial.

In figure 15, four curves representing combined uncertainties for four different

response functions with different nonlinearity uncertainty are depicted. These are the same

four situations represented in figure 10. Figure 15 shows that the behavior of the random

uncertainty does not rely on the form of the response function. Therefore, the conclusions

about the behavior of the random uncertainty can be applied to any real detectors, even

if they do not have similar response functions. But when systematic error dominates, as

in the second order and third order, the results differ significantly for the four cases. Since

combined uncertainty can never be made lower than the systematic error, we suggest using

polynomials whose systematic error is well below the desired uncertainty and using other

techniques, such as data averaging, to further reduce the random uncertainty.

Figures 11 through 15 are for data uncertainty of equal absolute uncertainty. Figure

16 is the combined uncertainty when the data have an equal relative uncertainty of 0.01

percent. It shows the same behavior as the case of the equal absolute uncertainty shown

in figure 13. However, if we look carefully at the scales, we see that equal relative

uncertainty yields a lower combined uncertainty than the equal absolute uncertainty when

the latter has the same relative uncertainty at the mid-range. At the lower end of the

range, data of equal relative uncertainty have lower relative uncertainty than data of equal

absolute uncertainty. We are considering nonlinearity A^(V;0) whose reference linear curve is the tangent of the conversion curve at 0, which is primarily determined by the

behavior of the lower end of the conversion curve. Higher relative uncertainty at the lower

end thus will result in higher deviation of the reference linear curve, which in turn worsens

the overall uncertainty of the nonlinearity.

25
0.1 -

0.01

0.001

0.0001

3 4 5

POLYNOMIAL ORDER

SI SM ST AT-1 -ffl- AT-2 DF

Figure 13. Combined nonlinearity uncertainty. The data have an

equal absolute uncertainty which is 0.1 percent of the data value

in the middle of the range. The third-order polynomial gives the

least combined uncertainty.

g lE-08

3 4 5

POLYNOMIAL ORDER

0.01% -©-0.1% 1%

Figure 14. Nonlinearity uncertainty for attenuation method with

known transmittance when data have random uncertainty and is dominated by random error for high polynomial order. It is limited by systematic error for the low polynomial orders.

26
0.0001

3 4 5

POLYNOMIAL ORDER

e3% V-1% 00.5% -z^O.15%

Figure 15. Combined error for four different response

polynomials listed in table 3. This is the result for triplet method

and the data random uncertainty is 0.01 percent.

1E-02 -

Q lE-05

7^ 3 4 5

POLYNOMIAL ORDER

SI -^SM ST AT-l-»-AT-2-e-DF

Figure 16. Combined nonlinearity uncertainty when data have

the same 0.01 percent relative random uncertainty.

27
4.1.4. Effect of Reference Point

From section 2, we know that we can express the nonlinearity uncertainty with

reference to any assumed calibration point without doing the calibration and later get the

nonlinearity with reference to the real calibration point by eq (15). However, the standard

deviation of the result using different dummy calibration points may not be the same.

Figure 17 illustrates this effect of reference point for the triplet method. Since the data are taken between 0.1 and 1 times power interval, results using a reference point within that

range have standard deviation significantly less than those with reference outside the data

range. The best reference point is at the middle of the power range.

4.1.5. Other Results

Figure 18 demonstrates that, when the data have random uncertainty, increasing the

number of data points will reduce the resultant uncertainty linearly by VN, where N is the

number of data points. When the data are without uncertainty, increasing the number of

data points will have small effect.

Figure 19 shows the uncertainty of transmittance (t) obtained from the attenuation

method by eq (34). One alternative way of using the attenuation method is to first measure

the transmittance r of the filter in the system, ff we are able to make the positioning of

the filter very repeatable, we can measure it many times, say 100 times, and obtain the

magnitude of r whose uncertainty is reduced to a tolerable level. Then we use this value

of T in the attenuation method. Figure 20 gives the effect of the deviation of the value of

T from the true value. We can determine from this figure what uncertainty is tolerable in

the measurement of r in order to achieve the target accuracy of the nonlinearity

measurement.

We mentioned in the description of the differential method that the finite input of

the ac signal will cause uncertainty in the result. Results of simulations of this effect are depicted in figure 21. When the ac input is one-hundredth of the maximum dc of the

measurement range, the resulting systematic error and random uncertainty will almost be

the same as those when we are measuring the true derivative of the conversion function, or when the ac input is 0. When the ac input is one-tenth of the maximum dc input, the

uncertainties are noticeably different from the true derivative case. Because larger ac input

means better signal-to-noise ratio in the ac measurement, a compromise must be made in

the measurement between the extra uncertainty due to the finite ac input and the signal-tonoise ratio.

28
0.01

2 3 4 5

POLYNOMIAL ORDER

B-Pc=0 •e-Pc=0.2-^Pc=0.5-z^Pc=0.9-ffl-Pc=L2

Figure 17. Nonlinearity error for different reference points for

triplet superposition method.

Pi

O

lE-01

lE-02 -

lE-03 -

S lE-04 -

^ lE-05 - O

lE-06

Q- Q-

*V

-Q

4 6

SQUARE ROOT OF N

8

OO -^O.OP/oO-O.P/o 7^1%

Figure 18. Nonlinearity error versus the square root of the

number of data points.

29
lE+00

3 4 5

POLYNOMIAL ORDER

eSTD.=0 -^STD=0.01%^STD=0.1% T^STD=1%

Figure 19. Error in the transmittance obtained from the

attenuation method due to polynomial truncation and data

random uncertainty.

lE-01

S lE-02 4 ^ lE-03 4

2 lE-04 4

<

Z lE-05 4

J

g lE-06 -

;z; lE-07

2E-07 2E-06 2E-05 0.0002 0.002

RELATIVE ERROR IN TAU

e2ND ORDER -^ 3RD ORDER

-& 4TH ORDER -z^ 5TH ORDER

Figure 20. Nonlinearity error due to inaccurate transmittance

used in the attenuation method.

30
lE-02

O

W

t-H <

lE-03 -

g lE-04 - o

lE-05

2 3 4 5

POLYNOMIAL ORDER

edP=0

^dP=0.01 OdP=0.

1

Figure 21. Nonlinearity error due to finite ac input in the differential method.

31
5, Conclusions

(1) A unified mathematical treatment has been established for all the measurement

methods. Although detector nonlinearity is expressed with reference to the calibration

point, it can be measured without first calibrating the detector. The nonlinearity is expressed with reference to any dummy calibration point which can later be converted to

nonlinearity with reference to the real calibration point. A reference point at the middle

of the measurement range is recommended, and reference points outside the data range

should be avoided.

(2) Under the same conditions assumed in the simulations, all the superposition

methods and the attenuation method, with the transmittance known accurately, yield the

same accuracies, while the differential method shows a slightly higher systematic error and

less random uncertainty than those methods. The attenuation method with unknown

transmittance gives the largest systematic error and random uncertainty; the attenuation

method with a known transmittance (with uncertainty) is not able to yield results as accurate as the superposition methods and the differential method. The integral-step

superposition method has large random uncertainties in its results due to the accumulation

of uncertainty in the data and consequently, requires a careful adjustment of the input

power to avoid even higher uncertainties. Therefore, it is not a desirable choice. Because

source drift may have a larger impact on the modified superposition method, we are left with the choice between the triplet superposition method and the differential method.

However, the assumption of identical measurement conditions often does not hold for all these methods in a real measurement and it may be necessary to compare the methods

under different conditions. Furthermore, effects other than those considered in our

simulation may play an important role in determining the measurement accuracy. For

example, attenuation method uses the simplest measurement setup, which will reduce the

number of sources of potential uncertainty.

(3) The various methods have systematic errors and random uncertainties which

behave differently depending on the fitting polynomial order. A compromise usually must

be made. Because we can reduce the random uncertainty by increasing the number of data

points and the number of measurements, keeping systematic errors well below the target

accuracy usually is a priority. Systematic error depends very much on the real response

function of the detector, which is not known. A fourth-order polynomial is recommended

if the nonlinearity is very small (at or less than 0.1 percent) and the data standard deviation

is very small (at or less than 0.05 percent). If the data standard deviation is not very small

and the detector nonlinearity is at or above 1 percent, a third-order polynomial may be

used. We suggest the following criterion when it is not obvious which polynomial order

should be used. The same measurement data may be treated by both the third- and fourthorder polynomials. If the difference between the two results is less than the sum of the

standard deviations of each of them, the two results are considered in agreement with each

other statistically, and the third-order polynomial should be used. When the two results

do not agree in this sense, truncation error may have caused the difference and therefore

32
the fourth-order polynomial should be used. If the system has large noise and nonlinearity, a fifth- or higher-order polynomial will yield a higher random uncertainty. Using too many

data points or repetitions of measurements to reduce the random uncertainty is not always

practical and may cause other technical problems; thus, fifth- or higher-order polynomials

are not recommended unless the measurement system can produce data with extremely low

random uncertainties and very small detector nonlinearities are to be measured.

This work is supported by the Calibration Coordination Group (CCG) of the Department

of Defense with the lead agency for this project being the U.S. Naval Warfare Assessment

Division, Corona, California. Matt Young, Robert Gallawa, and Steven Mechels of NIST

reviewed the manuscript; the authors thank them for their comments.

33
6. References

I] "Calibration of fiber-optic power meters," Draft International Standard, lEC TC 86,

1992.

2] A.R. Schaefer, E.F. Zalewski, and Jon Geist, "Silicon detector nonlinearity and

related effects, Appl. Opt. 22, 1232-1236 (1983).

3] L. Coslovi and F. Righini, "Fast determination of the nonlinearity of photodetectors,"

Appl. Opt. 19, 3200-3203 (1980).

4] R.D. Sanders and J.B. Schumaker, "Automated radiometric linearity tester," Appl.

Opt. 23, 3504-3506 (1984).

5] C.L. Sanders, "Accurate measurements of and corrections for nonlinearities in

radiometers," J. Res. Nat. Bur. Stand. (U.S.) 76A, 437-453 (1972).

6] H.J. Keegan, J.C. Schleter, and D.B. Judd, "Glass filters for checking performance

of spectrophotometer-integrator systems of color measurements," J. Res. Nat. Bur.

Stand. (U.S.) 66A, 203-221 (1962).

7] D.E. Erminy, "Scheme for obtaining integral and fractional multiples of a given

radiance," J. Opt. Soc. Amer. 53, 1448-1449 (1963).

8] A. Reule, "Testing spectrophotometer linearity," Appl. Opt. 7, 1023-1028 (1968).

9] H.J. Jung, "Spectral nonlinearity characteristics of low noise silicon detectors and their

applications to accurate measurements of radiant flux ratios," Metrologia 15, 173-181

(1979).

10] J. Fischer and L. Fu, "Photodiode nonlinearity measurement with an intensity stabilized

laser as a radiation source," Appl. Opt. 32, 4187-4190 (1993).

II] K.D. Mielenz and K.L. Ekerle, "Spectrophotometer linearity testing using double

aperture method," Appl. Opt. 11, 2295-2303 (1972).

12] W. Budde, "Multidecade linearity measurements on Si photodiodes," Appl. Opt. 18,

1555-1558 (1979).

13] C.L. Sanders, "A photocell linearity tester," Appl. Opt. 1, 207-211 (1962).

14] K.D. Stock, "Calibration of fiber optical power meters at PTB," Inst. Phys. Conf. Ser.

No. 92, Int. Conf. Optical Radiometry, NPL, London, 12-13 April, 1988.

34
[15] R.G. Frehlich, "Estimation of the nonlinearity of a photodetector," Appl. Opt. 31,

5926-5929 (1992).

Appendix

We compare the variation of a;=kAP and

dP(V)

dV

with the change of AP, where

iv-v„

Vo is the dc component of the ac output AV when the dc input P = 0. We can show

immediately that

Sa ^ 5AP

~a "aF

(Al)

Denoting

dP(V)

dV iv.v„ as P'(Vo)» w6 h^ve

dF(V„) =P'^(V„)dV„,

where P"(Vo) is the second derivative of P(V) at Vp. Then

dP'(V„) P- (V„)

P'(Vo) P'(Vo)

-dV

P'(Vo)

^dV ^ _dP

dP tv=v

(A2)

(A3)

Using P = P(V), we can further write

dP'(V„) P'^(V„)P(V„)/dP^

P'(Vo) P'(Vo) F(V„) vP. v=v.

(A4)

We denote P = Pq when V = Vq. Because Pq is the dc component of the ac input AP, it

is proportional to the amplitude of the ac input AP and d(AP)/AP = dP/P. Thus we have

35
dP^(V„) P^^(V„)P(V„)

P^(Vo) P^(Vo) P^(V„)

dAP^

AP

(A5)

p-p.

This result shows that the effect of the variation of input ac power is reduced (or

magnified) by a factor of ^" (Vo)P(Vo)

Using the normalized polynomial of eq (19) and

[P^(Vo)]^

the fact that nonlinearity is very small, we approximate this factor as

5:b,k(k-i)v;k-l

k>2

Comparing this with eq (20), we can roughly estimate that

dP„ dP^(V„) dP„

2A^(V„;0) ° < -^ < 10A^(V„;0) r

^0 ^ Wo)

(A6)

For small nonlinearity that is no more than a few percent, the effect of ac power drift is reduced at least by an order of magnitude.

36

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