1. Introduction

For many years, thermal detectors have been used

extensively to measure the output of high-power lasers

[1]. One of the most important properties of these

detectors is their optical response nonlinearity, i.e., the

deviation from an expected linear relationship between

the detector’s response and the incident optical power.

Although some thermal detectors have embedded

electrical heaters that can be used to characterize their

nonlinearity, for the majority of detectors the nonlinearity must be characterized with optical methods.

In the NIST high-power, continuous-wave (CW)

laser detector calibration laboratory, K-series calorimeters are used as primary standards for the absolute calibration of laser power and energy meters spanning a

laser power range of 5 W to 1000 W with a cumulative

energy input of 0.3 kJ to 3 kJ [2]. Since the calibration

of high-power laser detectors is costly and timeconsuming, calibrations at multiple power levels for a

single detector are not usually desirable. If a detector is

linear, i.e., its response is directly proportional to the

incident optical power, then a calibration needs to be

performed only at one power level within that specified

range. Unfortunately, most thermal detectors behave

linearly within a rather limited power range.

We have developed a system [3] for measuring the

nonlinearity of the detectors over the 1 W to multikilowatt power range. The system is based on an

attenuation method using a beamsplitter, a reflective

chopper wheel, and a translation stage to obtain a

quantitative measure of detector output as a function of

optical power input. Since a rotating chopper wheel

is involved, this system is only suitable for measuring

relatively slow detectors, such as thermal detectors,

and using CW lasers or pulsed lasers with high pulserepetition frequencies. By knowing the calibration

factor at a single power level as well as the nonVolume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology

429

[J. Res. Natl. Inst. Stand. Technol. 109, 429-434 (2004)]

Nonlinearity Measurements of High-Power

Laser Detectors at NIST

Volume 109 Number 4 July-August 2004

Xiaoyu Li, Thomas Scott,

Shao Yang, Chris Cromer, and

Marla Dowell

National Institute of Standards

and Technology,

Boulder, CO 80305

xiaoyu@boulder.nist.gov

tomrscott@wyoming.com

syang@boulder.nist.gov

cromer@boulder.nist.gov

mdowell@boulder.nist.gov

We briefly explain the fundamentals of

detector nonlinearity applicable to

both electrical and optical nonlinearity

measurements. We specifically discuss

the attenuation method for optical

nonlinearity measurement that the NIST

system is based upon, and we review the

possible sources of nonlinearity inherent to

thermal detectors used with high-power

lasers. We also describe, in detail, the

NIST nonlinearity measurement system,

in which detector responsivity can be

measured at wavelengths of 1.06 μm and

10.6 μm, over a power range from 1 W

to 1000 W. We present the data processing

method used and show measurement

results depicting both positive and

negative nonlinear behavior. The expanded

uncertainty of a typical NIST high-power

laser detector calibration including

nonlinearity characterization is about 1.3 %. 

Keywords: attenuation method; calibration; calorimeter; chopper wheel; highpower laser; nonlinearity measurement;

thermal detector. 

Accepted: September 5, 2004

Available online: http://www.nist.gov/jres
linearity behavior over a desired power range, the user

can then obtain a calibration factor for any desired

power level within this range. Consequently, this

combination of NIST services is the most cost-effective

calibration process for those customers who need

detector calibration at more than one power level.

2. Detector Nonlinearity

2.1 Fundamentals of Detector Nonlinearity

The definition of detector nonlinearity [4] can be

expressed in Eq. (1)

(1)

where R(V) = V/P = the responsivity of the detector

at an output of V, which is the voltage or reading of the power meter, at the input power or

energy of P,

R(Vr) = the responsivity at a reference point Vr. 

If a calibration is performed at Vc after the nonlinearity ∆NL(V;Vr

) is measured with reference point Vr different from Vc, the measured nonlinearity can be converted to ∆NL(V;Vc) by Eq. (2)

(2)

Equation (3) shows a calibration factor determined by

detector calibration procedure

(3)

where Pc = the power or energy of calibration,

Vc = the corresponding output of the detector.

If the detector is linear, F can be used for any output

V to determine the input power or energy P = V/F. For

a nonlinear detector, a correction factor CF(V) is

applied to the calibration factor F and we have a more

general expression in Eq. (4)

(4) 

When the nonlinearity is small, the most commonly

used function for the P-V relation is a polynomial of

the form in Eq. (5)

(5)

where bk = ak/a1.

a1 V is the linear term and the rest are nonlinear terms.

It is assumed here that the background reading is

adjusted to zero. We can further simplify the polynomial expression by dividing P(V) by a1 without

altering the nonlinearity in Eq. (6) 

(6) 

where p(V) = the normalized polynomial,

a1 = the calibration coefficient.

There are four commonly used measurement

methods of nonlinearity: (1) superposition, (2) attenuation, (3) correlation, and (4) AC-DC (or differential) [4].

Our system is based on the attenuation method and the

following discussion will focus on this method.

During the measurement process, the output voltages

of the detector V (without attenuation) and Vτ (with

attenuation) are recorded. Assuming the attenuation τ

of the attenuator is known and that we have a normalized polynomial relationship between p and V, then

Eq. (7) shows the ith measurement

(7)

Eliminating the unknown pi, we obtain Eq. (8)

(8)

By adjusting the source power, we can take sufficient

measurements, such that the coefficients of bk can be

obtained from the data by performing a linear least

square fitting. 

2.2 Possible Sources of the Thermal Detector

Nonlinearity

The majority of thermal detectors used for highpower CW laser measurements consist primarily of thin

metal discs with appropriate housings. The disk surface

(facing the incident laser beam) is typically coated

with highly absorbent material. A temperature sensor

(thermopile or other temperature sensing device) is

attached to the other side of the disc. The housing can

430

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Journal of Research of the National Institute of Standards and Technology

r

NL r

r

() ( ) (; ) , ( )RV RV V V

R V

− ∆ =

NL r NL c r

NL c

NL c r

(; ) ( ; ) (; ) , 1 (;)VV V V V V

V V

− = +∆ ∆ ∆

∆

c c FVP = / ,

, ( )

V P

F CF V = ×

1

1 2

( ) ( ),

n n k k

k k

k k

PV aV a V bV = == =+ ∑ ∑

1 2

( ) ( ) ( ),

n k

k

k

P V pV V bV

a =

= =+∑

2

2

,

n k

i i ii

k

n k

i i ki

k

p V bV

p V bV

=

=

 = + 





 = + 

∑

τ τ τ ∑

2

( ) ( ) 0,

n k k

i i ki i

k

V V bV V =

τ τ −+ − = τ τ ∑
431

be constructed in one of three different ways: (1) heat

sink with ambient-air convection cooling, (2) heat sink

with forced air (electrical fan) cooling, or (3) water

jacket cooling [5]. When the laser radiation strikes the

detector's surface, there is direct energy loss caused

by reflectance (specular and diffuse) at the detector’s

surface and this loss is usually in the range of 5 % to

25 % of the incident radiation. Although the amount of

optical reflectance of the detector surface varies with

incident laser wavelength, ideally it should be constant

over the entire power measuring range at a given laser

wavelength. The radiation absorbed by the detector’s

surface coating causes a temperature rise of the sensor

disc and then thermal sensors produce an electrical output that is proportional to this temperature change. The

resulting magnitude and distribution of laser-generated

thermal energy can directly affect the detector’s nonlinearity properties. 

The three major sources of detector nonlinearity are

(1) nonlinear behavior of the temperature sensor as a

function of temperature, (2) radiation loss from the

absorbing disc as the localized temperature varies

in response to different incident power levels, and

(3) temperature rise over time of the detector housing

due to insufficient cooling (this phenomenon is especially noticeable when the detector is exposed to laser

radiation over a long time period). Among the three

possible detector nonlinearity sources listed above,

nonlinearity sources (1) and (2) are the predominant

sources for most detectors. The nonlinearity source

(3) is a function of the exposure time, and for short

exposure times (customary operating condition for

most detectors) this nonlinearity contribution will be

quite small.

3. Nonlinearity Measurement System

Figure 1 is a schematic drawing of the NIST nonlinearity measurement system for characterizing highpower laser detectors. The laser beam passes through a 

beamsplitter oriented so that the first reflected beam

strikes a monitor detector and the primary beam is

transmitted to the test meter. A rotating optical chopper

wheel located on a translation stage between the beamsplitter and the test detector is moved in and out of the

laser beam. Because of its location, the monitor detector’s reading is only affected by the source power

levels, but not by the chopper wheel movement. The

following paragraphs give more details of the three key

optical elements used in our system.

3.1 Laser

A laser source (producing radiation at the desired

wavelength) with adjustable output power and shutter

is chosen to provide the required power range for the

nonlinearity measurement. The power is varied by

either adjusting the laser power itself (if the laser output power can be stabilized at all power levels) or by

using a variable high-power laser attenuator. Ideally the

output power stability should vary less than ± 0.5 % in

any 15 min time period as laser power fluctuations

directly affect the measurement uncertainty. An internal

or external shutter is used to turn the laser beam on and

off. Currently, our laboratory system employs a

1 kW CO2 cw laser and a 500 W Nd:YAG cw laser.

In addition, off-site laser sources can be used in conjunction with a portable beamsplitter/attenuator

system designed and built by NIST to perform these

measurements.

Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology

Fig. 1. Detector nonlinearity measurement system.
3.2 Beamsplitter

A beamsplitter and a monitor detector are used to

detect laser power fluctuations during the measurement. The beamsplitter ratio (i.e., the ratio of transmitted power to reflected power) does not have to be

known or even be constant over the entire power range

of the nonlinearity measurement. However, the ratio

does need to be thermally stable (with time) at each

fixed power level used. To minimize measurement

uncertainty, the detector used as a monitor should have

good spatial uniformity, proper optical aperture size,

and appropriate sensitivity. Because of the way it is

used, the monitor detector does not have to be linear

over the entire measurement power range, but it must

be stable over the exposure time at each fixed power

level. The time constant of the monitor detector should

be as close to the test detector's time constant as possible in order to accurately compensate for the fluctuation of the laser output.

3.3 Attenuator

A rotating optical chopper wheel mounted on a

moveable translation stage provides the consistent

attenuation required in this measurement technique.

The chopper wheel is essentially a thick aluminum disc

that has four open sectors precisely cut by EDM

(Electrical Discharge Machine). 45° beveled edges

were machined on the back surface along all the openings, but 0.5 mm straight edges from the front surface

were kept. The disc surface was first black anodized

and then the front surface was diamond-turned to produce a highly reflective surface to minimize the amount

of absorbed radiation. The transmitted beam, which

travels through the open sectors of the chopper wheel,

is incident on the test detector and the reflected beam is

incident onto a beam dump. The chopper wheel, thus,

acts as an attenuator for radiation going to the test

detector. The attenuation τ (Eq. (7), (8)) of the chopper

wheel is the ratio of the power in the attenuated beam

to the total laser power, or the duty cycle of the chopper. Since τ is a crucial parameter in the attenuation

method, we have used both mechanical and optical

methods to determine its value. The mechanical

method uses a precision optical comparator to determine the total angle of the radius opening, and the

optical method uses the NIST C-series calibration

system [6]. The results from the two methods agreed

within 0.1 %. The value of the in our system is about

70 %.

4. Measurement Procedures and Analysis

The nonlinearity system uses a computer controlled

data acquisition system to acquire and process the data.

Data acquisition instruments collect the output voltage of

the monitor detector and the test meter simultaneously.

The computer software written for this system controls

the translation stage and data acquisition system.

For each nonlinearity measurement, we typically

divide the desired measurement range into six equally

spaced power points. The laser source is adjusted to

produce each of the power points stepping from low to

high. At each power point we take five sets of data with

two measurements in each set of data. The first measurement is taken when the chopper wheel is out of the

laser beam, and the second measurement is taken when

the chopper wheel is inserted into the laser beam. The

background readings of the test meter and monitor

detector are always taken before the laser shutter is

opened. There are two different methods for taking the

detector’s background readings. For a water or fan

cooled test meter, in which the detector's housing

temperature is stable, we take the background reading

only once for each power level, so the laser shutter is

opened and closed only once during the five repeated

measurements. For a convection-cooled test meter, the

housing temperature will typically rise during a longterm exposure to a laser power, so we take background

readings before each individual measurement.

Consequently, the laser shutter is opened and closed

ten times for each power level.

As stated earlier, the thermal detector nonlinearity is

affected primarily by the magnitude and distribution of

laser generated thermal energy. In order to cover the

desired power range of 1 W to 1000 W, we perform the

nonlinearity measurements using a Nd:YAG laser at

1.06 μm and a CO2 laser at 10.6 μm. The Nd:YAG laser

produces a stable output from 1 W to 300 W whereas

the CO2 laser produces a stable output at powers of

300 W to 1000 W. Our measurement service for the

detector nonlinearity, thus, covers the range from 1 W

to 1000 W by using the two different lasers. Most high

power thermal detectors work at these two wavelengths.

The absolute calibration [7] of the test meter at the

desired laser wavelength and power level Pc [used in

Eq. (3)] can be done prior to or after the nonlinearity

measurement, since they are two independent measurements. The absolute calibration determines the detector

responsivity F defined in Eq. (3). If the test detector

output is in units of mV and the input power in units of

W, then the responsivity will have units of mV/W.

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433

The data from the detector nonlinearity measurements consist of pairs of measurements in which the

detector output is sampled with and without the

chopper wheel in the beam. After taking several sets of

these measurement pairs, the power level is changed

and the process is repeated. The first step of the data

processing is to correct the test meter’s readings by the

monitor detector’s readings to eliminate the effect of

laser instability during the measurement. Since the

attenuation τ of the chopper wheel is a known value,

the amount of experimental deviation from that value is

a measure of the test meter’s nonlinearity. In other

words, for a perfectly linear detector, measured attenuation of the chopper wheel using the test meter’s voltage outputs at different power levels should all equal τ.

In practice, very small discrepancies in measured

attenuations of the chopper wheel are expected due

to system noise, but these discrepancies should be

randomly distributed about τ when the detector response

is linear.

The detector responsivity as a function of laser

power is determined from a least-squares fit to the 

measurement data assuming a polynomial expression

for the relationship between detector output and

incident laser power. The absolute calibration result is

used as the reference point for the measurement data.

The curve fitting procedure requires that the difference

of the attenuation calculated from the fitted curve and

the actual τ be minimized. An optimum order of

the polynomial curve is selected considering the differences with respect to the noise.

Figures 2 and 3 plot the results from detector nonlinearity measurements performed on two detectors. The

horizontal axis shows incident optical power levels and

the vertical axis shows the detector’s responsivity in

mV/W. The detector in Fig. 2 shows negative nonlinear

behavior, or saturation in which the detector’s responsivity decreases as the incident optical power increases.

Figure 3 shows positive nonlinear behavior, or supralinearity, in which the responsivity increases as the

incident power increases. If each detector had been

perfectly linear, the responsivity at each power level

would be the same and, consequently, the data points

would all lie on a horizontal line in each graph.

Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology

Fig. 2. Negative nonlinear behavior of the detector.

Fig. 3. Positive nonlinear behavior of the detector.
5. Measurement Uncertainty

To establish uncertainty limits of detector nonlinearity measurements, uncertainty components are grouped

into two categories: Type A, whose magnitudes are

obtained statistically from a series of measurements,

and Type B, whose magnitudes are determined by

scientific judgment. [8]. There are two major components of Type B uncertainty in detector nonlinearity

measurements. The first is the laser/system instability

(about 0.5 %). Variation of the laser power, or of the

beam’s intensity profile can create several instability

effects in the detector nonlinearity measurement

system. For example, a rapid power shift may cause a

monitoring error due to unequal time constants of the

monitor detector and test meter, and changes in the high

order transverse mode content of the laser beam may

also generate monitoring error due to spatial nonuniformity of the monitor detector. The second Type B

component is the polynomial truncation (about 0.1 %)

performed in the polynomial curve fitting process.

There are two major components of uncertainty

arising from Type A effects in the detector nonlinearity

measurements. The first, stems from the chopper wheel

ratio measurements, which are normally performed at

three different radii along the open sector and, have a

typical standard deviation under 0.05 %. The second

Type A component is the test meter measurement

repeatability, which includes noise in the data from

both the test detector and the monitor detector. In addition the expanded uncertainty (k = 2) [9] of a typical

NIST high-power laser calibration is about 1 %, so that

the overall expanded uncertainty (k = 2) of a typical

NIST high-power laser calibration with the nonlinearity measurement is about 1.3 %. 

6. Conclusions

NIST has extended the power range capacity of its

high-power laser measurement program by developing

a conceptually simple, nonlinearity measurement

system for detector characterization. In conjunction

with the NIST absolute standard calorimeters, this new

system allows NIST to perform accurate power meter

calibration over a much larger power range than previously existed at our facility. The system has been

successfully demonstrated and used for power meter

calibrations for industrial customers. This technique

was recently used to perform an off-site detector

calibration for a U.S. industrial laser manufacturer

whose CO2 laser source was used with the NIST nonlinearity system described above to characterize the

nonlinearity of their detector over the power range of

400 W to 6000 W.

7. References

[1] H. Lara, Navigating the features and limitations of laser power

and energy meters, Laser Focus World 39 (1), 137-141 (2003).

[2] D. E. West and L. B. Schmidt, A System for Calibrating Laser

Power Meter for the Range 5-1000 W, NBS Technical Note 685

(1977).

[3] X. Li, T. R. Scott, C. L. Cromer, D. Keenan, F. Brandt, and

K. Möstl, Power measurement standards for high-power lasers:

comparison between the NIST and the PTB, Metrologia 37,

445-447 (2000).

[4] S. Yang, I. Vayshenker, X. Li, T. R. Scott, and M. Zander,

Optical Detector Nonlinearity: Simulation, NIST Technical

Note 1376 (1995).

[5] B. Moony and J. Buck, Thermopile Basics—A Technology

Update, Industrial Laser Review, 25-29, November 1996.

[6] D. E. West, W. E. Case, A. L. Rasmussen, and L. B. Schmidt, A

reference calorimeter for laser energy measurements, J. Res.

Natl. Bur. Stand. (U.S.) (Phys. And Chem.) 76A(1), 13 (1972).

[7] T. R. Scott, NBS Laser Power and Energy Measurements, Laser

Beam Radiometry, Proc. SPIE 888, 48-54 (1988).

[8] I. Vayshenker, X. Li, D. J. Livigni, T. R. Scott, and C. L.

Cromer, Optical Fiber Power Meter Calibrations at NIST, NIST

Special Publication, 250-54 (2000).

[9] Barry N. Taylor and Chris E. Kuyatt, Guidelines for Evaluating

and Expressing the Uncertainty of NIST Measurement Results,

NIST Tech. Note 1297 (1994).

About the authors: Xiaoyu Li is an electronics

engineer in the Optoelectronics Division of the NIST

Electronics and Electrical Engineering Laboratory.

Thomas Scott has since retired from NIST and was a

physicist in the Optoelectronics Division. Shao Yang is

an electronics engineer, Chris Cromer is a physicist,

and Marla Dowell is a physicist all in the Optoelectronics Division. The National Institute of Standards

and Technology is an agency of the Technology

Administration, U.S. Department of Commerce.

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