ii

Contents

1. INTRODUCTION ................. ................. ................. ................. . 1

2. HIGH-ACCURACY LASER POWER AND ENERGY METER CALIBRATION SYSTEM ........ 2

2.1 Calibration System Overview ................. ................. ................. . 2

2.2 Calibration Philosophy ................. ................. ................. ...... 4

2.3 Parameter Specification for End-User Uncertainty Assessment ................. ........ 5

3. SYSTEM DESIGN AND IMPLEMENTATION ................. ................. .......... 8

3.1 High-Accuracy Calibration System ................. ................. ............. 8

3.1.1 Data acquisition system ................. ................. .............. 8

3.1.2 Laboratory environment ................. ................. .............. 8

3.1.3 Mechanical systems ................. ................. ................ 10

3.1.4 Gas and cryogen handling systems ................. ................. .... 12

3.1.5 Optical system ................. ................. ................. ... 12

3.1.6 Laser power stabilization system ................. ................. ...... 16

3.1.7 The relative aperture transmittance correction ................. ............ 18

3.2 The LOCR Primary Standard ................. ................. ................. 19

3.2.1 Basic cryogenic radiometer operating principal ................. ........... 19

3.2.2 Description of the LOCR ................. ................. ............ 20

3.2.3 LOCR electrical calibration ................. ................. .......... 23

3.2.4 The NIST Brewster’s angle window ................. ................. ... 25

4. CALIBRATION PROCEDURE ................. ................. ................. ..... 28

4.1 Calibration System Operation ................. ................. ................ 28

4.1.1 Data acquisition software ................. ................. ............ 28

4.1.2 Optical component selection and placement ................. .............. 28

4.1.3 Laser source alignment ................. ................. .............. 29

4.1.4 Detector alignment ................. ................. ................. 31

4.1.5 Relative aperture transmittance measurement ................. ............. 33

4.1.6 Calibration system maintenance ................. ................. ...... 35

4.1.7 Quality assurance checks ................. ................. ............ 36

4.2 LOCR Operation ................. ................. ................. .......... 37

4.2.1 Vacuum and cryogenic systems ................. ................. ....... 37

4.2.2 Alignment and spatial uniformity measurement ................. ........... 42

4.2.3 Electrical calibration ................. ................. ............... 44

4.2.4 Window transmittance measurement ................. ................. ... 45

4.2.5 Data acquisition ................. ................. ................. . . 46

4.2.6 LOCR maintenance ................. ................. ................ 48

4.3 Calibration Factor Calculation ................. ................. ................ 49

4.3.1 Meter output analysis ................. ................. ............... 50

4.3.2 Applied power calculation ................. ................. ........... 51

4.3.3 Calibration factor calculation ................. ................. ......... 52

5. UNCERTAINTY ANALYSIS ................. ................. ................. ....... 54

5.1 Uncertainty of the LOCR Primary Standard ................. ................. ..... 55

5.1.1 Window transmittance ................. ................. .............. 56

5.1.2 Receiver absorptance ................. ................. ............... 56
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5.1.3 Receiver alignment ................. ................. ................. 57

5.1.4 Electrical calibration ................. ................. ............... 57

5.1.5 Electrical heating inequivalence ................. ................. ....... 57

5.1.6 Other LOCR uncertainties ................. ................. ........... 58

5.2 Uncertainty of the Applied Power ................. ................. ............. 59

5.2.1 Relative aperture transmittance ................. ................. ....... 59

5.2.2 Measurement repeatability of the applied power ................. ........... 59

5.2.3 Uncertainty of the applied energy ................. ................. ..... 60

5.3 Uncertainty of the Test Detector Calibration ................. ................. ..... 60

5.3.1 NIST amplifier gain ................. ................. ................ 60

5.3.2 NIST electrical measurement ................. ................. ......... 61

5.3.3 Quantization in digital test meters ................. ................. ..... 62

5.3.4 Measurement repeatability of the calibration factor ................. ........ 64

5.3.5 Uncertainty in the calibration factor for energy meters ................. ...... 64

5.4 Other Uncertainties ................. ................. ................. ........ 64

5.4.1 Test detector alignment ................. ................. ............. 64

5.4.2 Laser beam spatial characteristics ................. ................. ..... 65

5.4.3 Laser beam wavelength and spectrum ................. ................. . . 67

5.4.4 Environment specification ................. ................. ........... 68

REFERENCES ................. ................. ................. ................. .... 70

APPENDIX A. Glossary ................. ................. ................. ............. 72

APPENDIX B. Instructions for Submitting Meters for Calibration ................. .............. 76

APPENDIX C. Sample Calibration Report ................. ................. ................ 78

APPENDIX D. Gaussian Beam Model and Spatial Filter Pinhole Selection ................. ....... 86

D.1 Geometrical Optics Analysis ................. ................. ................. 86

D.2 Gaussian Beam Ray-Tracing Analysis ................. ................. ......... 87

D.3 Gaussian Beam Parameter Specification ................. ................. ........ 95

D.4 Spatial Filter Pinhole Selection ................. ................. ............... 96

APPENDIX E. Relative Aperture Transmittance: Estimation and Uncertainty ................. ..... 99

E.1 Measured and Theoretical Transmittance for Circular Apertures ................. ..... 100

E.2 Relative Transmittance Measurement for Circular Apertures ................. ........ 104

E.3 Estimation of the Relative Aperture Transmittance for Detectors ................. .... 107

APPENDIX F. System DMM and LOCR Electronics: Calibration and Correction ................. . 111

F.1 System DMM Calibration ................. ................. ................. . 111

F.2 Measurement of LOCR Electrical Power ................. ................. ....... 113

F.3 Calibration of LOCR Power ................. ................. ................. 115

F.4 Combined LOCR Power Measurement Correction ................. ................ 118

APPENDIX G. LOCR Optical Receiver Alignment: Uncertainty ................. ............... 119

G.1 Philosophy of Receiver Uniformity Measurement ................. ................ 119

G.2 Receiver Uniformity Measurement: Technique and Analysis ................. ........ 120

APPENDIX H. Window Transmittance: Estimation and Uncertainty ................. ........... 123
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H.1 Window Transmittance Measurement: Philosophy and Technique ................. ... 123

H.2 Transmittance Measurement: Uncertainty Analysis ................. ............... 125

H.3 Example Calculation of Transmittance ................. ................. ........ 128

APPENDIX I. Center-Weighted Spatial Uncertainty ................. ................. ....... 129

APPENDIX J. Estimate of LOCR Optical Receiver Absorptance ................. .............. 132

J.1 LOCR Receiver Absorptance at 633 nm ................. ................. ....... 132

J.2 LOCR Receiver Absorptance at Other Wavelengths ................. ............... 133

APPENDIX K. Weather Station Calibration and Uncertainty ................. ................. . 136

K.1 Internal and External Temperature Sensor Calibration ................. ............. 136

K.2 Barometric Pressure Sensor Calibration ................. ................. ....... 139

K.3 Relative Humidity Sensor Calibration ................. ................. ......... 142
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List of Illustrations

Figure 1. Calibration system overview. ................. ................. ................. ... 2

Figure 2. Current configuration of the high-accuracy calibration system. ................. .......... 7

Figure 3. High-accuracy calibration system laboratory table and enclosure. ................. ........ 9

Figure 4. High-accuracy calibration system’s laboratory table layout. ................. ............ 11

Figure 5. Detailed view of the optical source breadboard. ................. ................. .... 13

Figure 6. The apparent aperture of a detector that has no limiting aperture. ................. ....... 19

Figure 7. External view of the LOCR. ................. ................. ................. . . 20

Figure 8. Internal view of the LOCR. ................. ................. ................. ... 21

Figure 9. The LOCR’s optical receiver. ................. ................. ................. . 22

Figure 10. General functional schematic of the LOCR. ................. ................. ...... 22

Figure 11. Simplified LOCR electrical calibration circuit. ................. ................. .... 23

Figure 12. The NIST Brewster’s angle window mount. ................. ................. ...... 25

Figure 13. Main calibration process flow chart, for calibration at a single wavelength. ............... 27

Figure 14. Rotational and translational motions used during detector alignment. ................. ... 32

Figure 15. The NIST Brewster’s angle window mount, showing the vertical offset of the incident laser

beam. ................. ................. ................. ................. ..... 43

Figure 16. Typical LOCR illuminated output signal. The signal is dithered, so the quantization error is

randomized. ................. ................. ................. ................. 58

Figure 17. The output current from an illuminated photodiode, showing two discrete quantization levels.

................. ................. ................. ................. .......... 63

Figure D.1 Thin lens Gaussian optics model showing the 1550 nm optical source; mirrors, polarizers,

power modulator elements, and aperture stops are not shown. ................. ............ 86

Figure D.2 The size (radius) of the Gaussian beam in the detector plane versus the displacement of Lens

2 from the optimally collimated location. ................. ................. ........... 92

Figure D.3 The radius of curvature of the wave front in the detector plane versus the displacement of

Lens 2 from the optimally collimated location. ................. ................. ....... 93

Figure D.4 The distance from Lens 2 to the projected waist image (l

2

') versus the displacement of Lens 2

from the optimally collimated location. ................. ................. ............. 94

Figure D.5 The radius of curvature of the wave front in the detector plane versus the displacement in

Lens 2 from the optimally collimated location, when the 1550 nm laser is moved 1 m farther

away from Lens 1. ................. ................. ................. ............ 94

Figure E.1 Measured and theoretical relative aperture transmittance for small circular apertures, relative

to a circular reference aperture of 10 mm diameter, using a beam of 2 mm diameter. .......... 101

Figure E.2 Measured and theoretical relative aperture transmittance for large circular apertures, relative

to a circular reference aperture of 10 mm diameter, using a beam of 2 mm diameter. .......... 101

Figure K.1 Internal weather station temperature during a typical optical calibration. ................ 138

Figure K.2 External weather station temperature during a typical optical calibration. ............... 138

Figure K.3 The indicated barometric pressure acquired during a typical optical calibration. .......... 141

Figure K.4 The relative humidity measured during a typical optical calibration. ................. . . 143
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List of Acronyms

CO2................. ................. ................. ................. ... Carbon Dioxide

CW ................. ................. ................. ................. . . Continuous Wave

DMM ................. ................. ................. ................ Digital Multimeter

DUT ................. ................. ................. ................. Device Under Test

He ................. ................. ................. ................. .... Gaseous Helium

HEPA ................. ................. ................. ...... High Efficiency Particulate Air

LOCR ................. ................. ................ Laser Optimized Cryogenic Radiometer

LPC ................. ................. ................. ............. Laser Power Controller

N2................. ................. ................. ................. ... Gaseous Nitrogen

O2................. ................. ................. ................. ... Gaseous Oxygen

OSA ................. ................. ................. .......... Optical Spectrum Analyzer

ppm ................. ................. ................. ................. . . Parts per million

UPS ................. ................. ................. ........ Uninterruptible Power Supply
vii

High-Accuracy Laser Power and Energy Meter Calibration Service

David J. Livigni

National Institute of Standards and Technology

Boulder, CO 80303

This document describes the high-accuracy laser power and energy meter calibration

service provided by the National Institute of Standards and Technology (NIST). 

Calibrations are performed by direct substitution of a test detector with a cryogenic laser

radiometer, traceable to NIST electrical standards. The service currently supports

measurements with laser powers from 0.1 to 1.0 mW, at several vacuum wavelengths in

the range from 458 to 1550 nm. The expanded uncertainty (with a coverage factor of

k = 2) of calibrations based on the primary standard typically ranges from 0.02 to 0.05 %. 

A detailed description of the measurement system and uncertainty analysis is presented.

KEY WORDS: absolute power, calibration, cryogenic radiometer, laser power, optical

power.
1

1. INTRODUCTION

We have built a calibration system to meet the industry’s need for very high-accuracy measurements of

laser power and energy, at various (selected) wavelengths. The system employs a commercial cryogenic

radiometer as the primary standard. This commercial radiometer is referred to as the Laser Optimized

Cryogenic Radiometer (LOCR) and is one of several NIST cryogenic radiometers that are used as

primary standards for various types of high-accuracy optical measurements. The calibration system

provides measurements of detector responsivity as a function of laser power with an expanded uncertainty

of typically 0.03 % or less (with a coverage factor of k = 2) and provides traceability to SI units with

reduced uncertainty for customers requiring the highest level of accuracy. The new calibration system

and the procedures for detector calibration based on the LOCR are described within this document. Some

of the terminology used in this document is not commonly used, so the definitions are included in

Appendix A.

Optical power and energy measurements are traditionally tied to SI units through electrical standards. 

This is accomplished by means of optical radiometers and calorimeters, designed to allow accurate

comparison of absorbed optical power with dissipated electrical power. Electrical power is applied to an

optical receiver as a means to calibrate the induced temperature rise of the receiver as a function of

dissipated electrical power. Room-temperature devices of this type have a combined standard uncertainty

that is typically limited to a few tenths of a percent. The development of electrically calibrated cryogenic

radiometers operating near liquid-helium temperatures [1-3] has led to a more than tenfold increase in the

level of accuracy for optical power measurements. Indeed, most industrialized countries have at least one

cryogenic radiometer in their national standards laboratories.

Since 1967, NIST has built and maintained room-temperature, electrically calibrated laser calorimeters

for the calibration of laser power and energy meters for customers. A number of standard laser

calorimeters have been developed to provide measurements over a wide range of laser wavelengths

(ultraviolet to far-infrared), power levels (nanowatts to kilowatts), and energy levels (femtojoules to

megajoules). The combined standard uncertainty of measurements with these calorimeters is usually

limited to about 0.25 %, primarily due to their operation at room temperature. Their performance is

limited primarily by inequivalence between electrical and optical heating, due to factors such as: radiative

and convective cooling of the optical receiver, and its limited diffusivity, which result in the formation of

temperature gradients, and parasitic heating in the electrical heater leads. Commercial laser power and

energy meters have significantly improved over the last 10 years and customers now require lower

uncertainties. The LOCR calibration service was developed by NIST’s Optoelectronics Division to meet

select customers’ needs for higher accuracy.
2

Figure 1. Calibration system overview.

2. HIGH-ACCURACY LASER POWER AND ENERGY METER CALIBRATION SYSTEM

The goal of the high-accuracy calibration service is to provide laser power and energy meter calibrations

for the most demanding customers, with low uncertainty (< 0.1 %), performed in a stable and fully

documented environment, at an affordable cost and in a timely manner. The calibration system achieves

the goal by offering calibrations based on the highly accurate laser optimized cryogenic radiometer

(LOCR) primary standard, using a stabilized laser source, in an automated calibration system that allows

quick calibration of multiple detectors at the same time. The service will also be used internally by NIST

to improve the accuracy of routine calibrations.

2.1 Calibration System Overview

High-accuracy calibrations are possible because of the accuracy of the primary standard’s laser power

measurement and the power-stabilized laser source which allows for accurate transfer of the power

measurement to the customer’s detector. For brevity in the following discussion, the device being

calibrated, whether a detector head or complete power or energy meter, is called the device under test

(DUT). An overview of the calibration system is shown in Figure 1. The calibration system consists of a

power-stabilized laser source, two movable platforms, and a computer-controlled data-acquisition system. 

The laser source contains a polarized laser, interference filters to remove unwanted laser lines (when

necessary), a power stabilization system and an optical spatial filter. The DUT is calibrated using a direct

substitution method-the laboratory standard is physically substituted with the DUT. This method requires

a stable laser source power, so the power stabilization system is necessary.

The detectors are mounted on a motion-controlled platform that translates in a direction transverse to the

laser beam’s propagation axis, so that each detector can in turn be moved into the beam’s path, thereby
3

providing the direct substitution of the standard with the DUT. The LOCR primary standard and multiple

DUTs can be placed on the platform, allowing calibration of more than one DUT at a time. Each detector

is aligned so that when it is moved into the beam, its entrance or limiting aperture is at a specific location

along the beam’s longitudinal axis, so that each detector is exposed to the same beam profile. A trusted

check standard can be calibrated along with the DUT, as a check on quality control for the system.

The LOCR uses an optical receiver design that optimizes the capture of the collimated laser beam and

uses a Brewster’s angle window that reflects very little of the incident laser power. Polarized laser light

passes through the Brewster’s angle window with minimal attenuation and is absorbed in the optical

receiver, which converts the optical power to heat. The amount of heating power is then measured by

electrical substitution, so the device is a primary standard traceable to SI units through the electrical

substitution. The laser power that is applied to the DUT is interpolated in time from bracketing

measurements performed with the LOCR. The linear interpolation method is used to eliminate the effect

of linear drift in the applied optical power.

Energy calibrations are essentially the same as power calibrations, except that the period during which the

power is applied to the DUT is also measured. A digital counter records the period over which the optical

shutter is open. The applied energy is then calculated by multiplying the interpolated power by the period

during which the power is applied. The corrected rise calculation [4], used with many laser calorimeters

to measure energy, is also supported.

The calibration system’s laser source provides a polarized laser beam with stable continuous-wave (CW)

power at discrete laser wavelengths, but an optical chopper can be used with DUTs that require a chopped

signal, as shown in Figure 1. The beam’s absolute power is easily adjusted using the power stabilizer’s

electronics and external attenuators, and by adjusting the laser’s output power (when possible). When

operating at a power level of 1 mW, the stabilized source provides laser power with a standard deviation

of typically 0.001 %. Currently, 0.1 mW is the lowest power supported; but even lower powers can be

used, but with increasing uncertainty. The beam’s polarization axis is fixed in the vertical direction, but

the angle can be adjusted slightly by rotating the system’s polarizer.

The laser source produces a good quality Gaussian beam, the diameter of which is easily adjusted by

adjusting the optical system’s lenses. But, because of distortion, artifacts, and scattering produced by the

imperfect optical elements and finite apertures that are present in the system, the actual beams used are

only approximately Gaussian beams. The uncertainties introduced by the imperfect Gaussian beam are

evaluated and accounted for in the high-accuracy calibration system.

The range of laser wavelengths, approximately 425 to 1700 nm, the system can support is limited by the

power stabilizer and other optical elements. Currently we can provide lasers with vacuum wavelengths of

458.06, 476.62, 488.13, 496.65, 514.67, 632.99, 1064.42, 1343.09, and 1550.43 nm. New laser

wavelengths can be added, as required by customer demand. Support for ultraviolet and far-infrared

wavelengths is possible with additional optical equipment.

DUTs that output an analog voltage, have a GPIB interface, or have a direct-reading output are currently

supported. Support for devices with an RS-232 serial output will also be added in the next version of the

system. The calibration system can also measure sensor devices during the calibration. Examples of

sensor measurements include monitoring detector bias voltage and precision temperature measurements. 

Sensors and DUTs that produce an analog voltage are measured using the system’s digital multimeter

(DMM); GPIB devices are read directly by the data acquisition computer, but the operator must manually

enter data acquired from direct-reading devices. Also, an autonomous weather station is located in the
4

laboratory. The weather station logs air temperature, pressure, and humidity, so no additional sensors are

needed for these environmental parameters.

Calibrations at multiple power levels can be used to determine the DUT’s power linearity with a high

degree of accuracy, within the limited power range supported by the primary standard. Alternatively, the

customer can combine the absolute power with the DUT’s known power linearity, to adjust the

calibration factor for power levels other than the level used in the calibration. Similarly, calibrations at

multiple wavelengths can be used to determine the DUT’s spectral responsivity over a limited range.

2.2 Calibration Philosophy

Laser power and energy meters are calibrated without specific knowledge of many of the DUT’s

parameters. However, evaluation of many of the uncertainties associated with the DUT’s calibration does

require specific knowledge of some of the DUT’s parameters. Some examples are the DUT’s spectral

responsivity, power linearity, temperature coefficient, spatial uniformity, and sensitivity to beam

parameters such as polarization, incidence and divergence angles. Similarly, the uncertainty in the DUT’s

gain is unknown. To avoid having to gather the information from the customer or measure the DUT’s

properties ourselves, we specify in the calibration report the conditions present at the time of the

calibration.

Because of the DUT’s potential sensitivities to these parameters, the customer should provide more

detailed information than usual when submitting a detector for high-accuracy calibration. The customer

should specify what parameters they desire, such as beam parameters including diameter, divergence,

polarization angle, and power. Also important are parameters such as detector orientation, alignment

technique, cleanliness, electrical setup, and output signal processing method. The customer should also

specify what cleaning technique, if any, should be used with the detector; using an improper cleaning

technique can damage the detector or otherwise change its properties. Specific instructions for submitting

a detector for high-accuracy calibration are given in Appendix B.

There are two exceptions to this rule, in which the DUT’s uncertainty results from a property of the

calibration system that interacts with a known property of the DUT. The exceptions are for the correction

for the relative aperture transmittance correction and the DUT’s quantization uncertainty. The relative

aperture transmittance correction results from imperfections in the calibration system’s near-Gaussian

beam interacting with the fixed, finite entrance apertures of the standard and DUT. Quantization

uncertainty is caused by the calibration beam’s power varying less than the DUT’s quantization interval.

To calculate the correction for relative aperture transmittance, knowledge of the calibration system’s

actual beam profile and the standard and DUT’s apparent apertures is required. The correction is applied

by NIST because it is much easier to measure and correct the relative transmittance at the time of

calibration, than to provide sufficient information about the calibration beam’s profile and the standard’s

aperture to the customer.

Quantization is an issue with the digital electronics used in some DUTs. Some devices return a limited

number of digits in their power readings, which are insufficiently dithered. For such meters, the stability

of the calibration system’s beam power can result in multiple power measurements that have a standard

deviation of zero. In this case, the actual power can be anywhere within the DUT’s quantization interval,

so an uncertainty for its quantization error is assessed. For digital DUT’s in which the readings are

dithered, either internally or because of sufficient noise in the applied power, the quantization uncertainty

is reduced by averaging and becomes part of the measurement reproducibility. The necessity of assessing
5

a separate uncertainty for DUT quantization is determined by visual analysis of the readings acquired

during the DUT’s calibration.

2.3 Parameter Specification for End-User Uncertainty Assessment

Customers who desire high-accuracy calibrations typically use the DUT as a transfer standard in their

own laboratory to calibrate other detectors. The customer’s calibration system introduces uncertainty into

the calibrations they perform with the NIST-calibrated transfer standard. Given the low uncertainty in the

high-accuracy calibration provided by NIST, small sources of uncertainty that are usually negligible in

other calibrations can become significant and so must be accounted for. To aid the customer in assessing

these uncertainties, we specify parameters that are not usually provided with lower-accuracy calibrations.

An example high-accuracy calibration report is given in Appendix C. The parameters specified include a

description of the laser beam used in the calibration, details of the detector alignment, and the

environmental parameters measured by the autonomous weather station during the calibration. Some

quantities are derived, such as ideal Gaussian beam parameters for the applied laser beam and the local

air’s index of refraction.

The diameter of the laser beam used in the NIST calibration is important, because a change in the beam

size used with the detector may change its calibration factor. The beam diameter at the detector’s entrance

aperture, measured from its 1/e2 intensity points, is adjustable from about 0.5 to 4 mm. The diameter is

measured using a scanning-slit technique. The diameter of the beam on the detector’s actual

photosensitive element or elements can be different, depending on how far behind the entrance aperture

the elements are and how much the beam diverges. To provide the customer with enough information to

calculate how the beam changes with distance, we specify the derived full-width beam divergence angle

at the detector’s entrance aperture and the parameters necessary for modeling the approximately Gaussian

beam.

The uncertainties in the centering of the beam in the detector’s entrance aperture and in the angle of

incidence of the beam entering the detector are specified because they can impact where the beam falls on

the detector’s photosensitive element or elements. Uncertainty in the location of the beam in the

detector’s aperture can cause uncertainty in the detector’s responsivity, because of imperfections in the

detector’s spatial uniformity. The angle of incidence can cause increased uncertainty in the location of

the beam, depending on how far behind the entrance aperture the detector element or elements are, and

some detectors are sensitive to the angle of incidence of the light itself.

The orientation of the detector relative to the beam’s polarization angle is also specified, because some

detectors have a different responsivity for different polarizations. The polarization angle in the NIST

calibration system is approximately vertical relative to the optical bench. Statements such as “the detector

was mounted with its mounting hole down” and “the beam was vertically polarized” describe the

orientation. This information tells the customer the basic orientation of the DUT with regard to the beam

polarization, but if the DUT’s responsivity is sensitive to the beam’s polarization angle, additional

uncertainty analysis and a more precise measurement of the polarization angle relative to the detector may

be necessary. 

Laser wavelength and spectrum are specified, because some DUTs are spectrally sensitive. We use

single-longitudinal-mode lasers when possible, because they produce a very narrow spectral line, with a

center wavelength that can be measured very precisely. But multiple-longitudinal-mode lasers may also

be used because of their greater availability and lower cost. An upper bound for the half-power linewidth

of the main spectral line is stated, along with the center wavelength. The center wavelength of lasers with
6

a single dominant narrow spectral line (narrower than 15 THz, about 10 nm at 633 nm) is measured with

a precision wavelength meter. Some lasers produce multiple spectral lines; their spectra are usually

described in the text by stating the distance between the different lines and their relative intensities. 

When the main spectral line is too broad or other lines with sufficient intensity are present, the precision

wavelength meter cannot be used. The wavelength of such lasers is measured using an optical spectrum

analyzer, with a corresponding increase in uncertainty. If a laser’s spectrum is too complicated to be

described succinctly, a graph of the spectrum as measured with an optical spectrum analyzer can be

included in the report.

The laser’s wavelength in vacuum is specified, since it is invariant as opposed to the air wavelength,

which changes with the air’s index of refraction. Also, spectrally sensitive detectors that correct for

wavelength internally usually require the vacuum wavelength. The provided local air index can be used

to derive the air wavelength from the vacuum wavelength, if desired. Currently five lasers are supported,

a 633 nm He-Ne laser, 1064 nm, 1340 nm, and 1550 nm solid-state lasers, and a visible-wavelength

tunable argon-ion laser.

The temperature, barometric pressure, and relative humidity present during the NIST calibration are

measured using the system’s autonomous weather station, and are specified in the calibration report. 

Temperature usually has the greatest effect on the responsivity of most detectors. Therefore, if the

detector is used at a temperature different from that during the NIST calibration, an adjustment to its

calibration factor may be necessary, depending on the magnitude of the temperature difference, the

DUT’s temperature coefficient, and the customer’s uncertainty tolerance. Also some detectors, such as

the LOCR, are sensitive to temperature change, so the room temperature is held constant during the

calibration. The weather station measures the air temperature near the detectors, but a precision analog

temperature sensor can be placed in direct contact with the detector for a more precise measurement, if

desired. Similarly, a change in the relative humidity or pressure can also affect a detector’s responsivity. 

For example, high humidity can cause problems with detectors; such problems range from a slight change

in the detector’s absorptance, to condensation forming on cooled parts. The humidity at the NIST

laboratory in Boulder, Colorado is not controlled. The relative humidity in the laboratory is usually

between 20 and 50 %, but occasionally rises to higher levels.

The index of refraction of the air at the time of the calibration is calculated [5] and specified in the

calibration report. A change in the air’s refractive index can affect the performance of some DUTs,

usually by changing the amount of light that is reflected from the detector. While local changes in the

refractive air index are usually very small, the index can be significantly different at other locations. 

Since the Boulder branch of NIST is located at a relatively high altitude, the difference from that at sea

level is significant. The refractive index of air in Boulder is typically 1.00022. The index is calculated

using the measured environmental parameters, assuming typical values for other parameters, such as CO2

content. The uncertainty in the calculated index is dominated by the uncertainty in the measured

environmental parameters.

Additional parameters are specified for particular DUTs. For example, some detectors require application

of a bias voltage using an external power supply, and the magnitude of the applied voltage can affect the

detector’s responsivity. So if an external bias voltage is applied by NIST, the voltage is measured and

stated in the calibration report. Some detectors require a chopped signal. For these detectors, the chopper

frequency can be measured and specified. Similarly, when detectors require unusual tuning or alignment,

the procedures used to tune or align the device are also specified. If a NIST amplifier or measurement

device is used with the customer’s detector, the uncertainty of the NIST equipment is assessed and

specified. The uncertainty of customer-supplied equipment is not assessed.
7

Figure 2. Current configuration of the high-accuracy calibration system.
8

3. SYSTEM DESIGN AND IMPLEMENTATION

While conceptually simple, performing calibrations with the lowest possible uncertainty is difficult. 

Many sources of small uncertainties, typically considered negligible in less accurate calibration systems,

are not negligible in the high-accuracy calibration system. The high-accuracy system was designed to

minimize and account for the uncertainty caused by external influences.

3.1 High-Accuracy Calibration System

A block diagram of the high-accuracy calibration system is shown in Figure 2. Development of the

system was frozen in this configuration for introduction to the measurement community at the NewRad

97 conference [6] and so that special test calibrations could be performed for customers.

3.1.1 Data acquisition system: The data acquisition system is controlled by the process control computer,

which is programmed using Visual Basic. It controls all the motion control hardware, timing, acquisition,

and storage of the test meter power readings. Test meters with analog voltage, digital GPIB, and directreading outputs are supported. The system can also acquire data from sensor devices, such as temperature

probes or power supply voltages. The acquired data are stored in spreadsheet format and the calculations

are performed on the spreadsheet, so that the data processing is fully documented. The LOCR itself is

treated as a direct-reading meter that is manually operated using the separate LOCR computer.

3.1.2 Laboratory environment: The laboratory was designed to minimize environmental influences on the

calibration system. A precision room-temperature control system is used in the lab, which regulates the

room temperature to within a few tenths of a degree Celsius. An autonomous commercial weather station

is used to monitor and document the laboratory environment. The weather station measures and records

the air temperature, relative humidity, and barometric pressure using internal sensors. The weather

station also has an external air temperature sensor that is placed near the detectors to record the air

temperature near them. The weather station itself resides on the lab table, where it measures the

conditions near the laser source. A significant temperature gradient can exist between the internal and

external temperature sensors, depending in part on how much heat is produced by the laser on the lab

table. The weather station records the measured environmental parameters internally; the recorded data is

periodically transferred to the process control computer, at least once a day. The weather station’s

temperature sensors have a resolution of about 0.05° C. When necessary, greater temperature precision

can be obtained using stand-alone temperature sensors, monitored directly by the calibration system.

To keep dirt and dust off the lab table and optics, the table is enclosed and a High Efficiency Particulate

Air (HEPA) filter is used to force clean air into the enclosure. The lab table enclosure is shown in Figure

3; it provides a clean and dark optical environment for the detectors. One obvious sign of the enclosure’s

effectiveness is a lack of dust particles, which are commonly observed passing through visible laser

beams; such dust can add to the laser power instability as the particles drift through the beam. The air

flow through the enclosure is not laminar because eddies are formed inside the enclosure, but the

improvement in the cleanliness of the lab table is significant. The HEPA filter housing also contains

fluorescent lights for illumination of the lab table during setup and an air ionizer that prevents the buildup

of static charge.

A problem with static charge was noted when the few dust particles inside the enclosure were found to

stick to the optical elements. The particles can become charged and can cling to metal and glass. They

can be difficult to remove; rubbing with a brush treated with radioactive material is partially effective, but
9

Figure 3. High-accuracy calibration system laboratory table and enclosure.

washing the contaminated element was frequently required. The problem was greatly reduced by the

addition of the air ionizer, placed in the air flow below the HEPA filter.

To prevent light transmission, the enclosure’s side and top panels are made of black plastic. The inside of

the panels is painted with a flat-black paint and textured to reduce specular reflections. Between the

optical source and the detectors is a shield against scattered light. The shield is made of aluminum that

was sanded and painted with a flat-black paint. The shield prevents the light scattered by the elements of

the optical source from reaching the detectors. It also provides an approximately uniform dark field with

constant temperature around the beam. The top of the enclosure over the LOCR can be removed, so that

the liquid-helium transfer line can be inserted. When not in use, the liquid-nitrogen transfer line (not

shown) hangs from the top of the enclosure; nitrogen fills can be performed with the enclosure’s top in

place.

The enclosure panels behind and to one side of the LOCR are left open, to facilitate routing the numerous

cables onto the lab table, and to provide a passage for the clean air to escape without pressurizing the

enclosure. The airflow out of the enclosure caused by the HEPA filter is not noticeable, but is strong

enough to prevent dust from entering the opening. The HEPA filter runs continuously, except for when

the minor draft it creates causes excess instability in test detectors.

Not shown in Figure 3 is a small ventilation hood, which was recently added to the enclosure panel above

the LOCR. The hood draws the gasses released by the boiling cryogens out of the laboratory. Significant

oxygen depletion can occur when the large amounts of nitrogen and helium gas are released during

LOCR cryogen filling, which can be a serious safety hazard for the operator. Therefore the oxygen

content of the air is monitored during cryogen fill, to warn of any oxygen depletion that may occur. Air
10

flowing out of the ventilation hood is exchanged with fresh air from the building’s ventilation system, so

the gas concentrations return to normal atmospheric values after about an hour. A second independent

ventilation system with flexible ducts was also recently added. Located over the optical source plate, it

provides ventilation for air-cooled lasers.

Electrical power for the lab is isolated from the building by use of filtered surge suppressors, placed

before and after an uninterruptible power supply (UPS). A “true” UPS is used, so that equipment with

linear power supplies can be run from its inverter; the device continually runs its inverter, which produces

an approximately sinusoidal output. Devices called “back” UPSs should be avoided, because some

provide a simple square wave output when running from battery, which is sufficient for switching power

supplies like those in a computer, but can damage devices with a linear power supply. Also, all back

UPSs have a delay when switching from line power to battery power. The UPS used in the system also

employs an internal isolation transformer. The isolation transformer helps filter power line noise by

breaking the conductive path for low frequency noise and regenerates the line-neutral to earth-ground

connection.

Most of the laboratory’s electrical systems are powered by the UPS, except for devices that have a large

power consumption, such as the computer monitors, HEPA filter, turbomolecular pump, hot air gun, and

some of the lasers; these devices are protected only by surge suppressors. All of the electronics (except

the turbomolecular pump, hot air gun, and some lasers) are connected to a single power circuit and

ground to help reduce noise. It is not intended that the entire laboratory be run for long periods of time

off the UPS’s battery, but the UPS does keep the system from resetting during short outages. Also, the

UPS can provide continuous power for devices that need it, such as the ion pump, during longer outages. 

Filters in the surge suppressors and UPS help keep high-frequency conducted noise out of the laboratory

electronics.

Electrical shielding is provided to help reduce the amount of electrical noise in the system. The LOCR

bridge electronics are isolated from the common electrical ground and its cabling is well shielded. 

Laboratory electronics used for test detector output measurement are also well shielded; for example, the

system’s digital multimeter (DMM) utilizes a guard shield that is switched by the three-pole multiplexer

and propagated to analog voltage devices using shielded twisted-pair cabling. The multiplexor can also

apply earth ground, or ground straps located on the laboratory table can be used when necessary. The

effectiveness of the shielding is evident in the excellent measurement results produced by the system,

despite the electrical noise present in the lab.

3.1.3 Mechanical systems: The lab table layout is shown with greater detail in Figure 4. The optical

source resides on the elevated optical source breadboard, which is supported by five manually operated

leveling posts. The posts provide accurate height adjustment, with a resolution of 0.25 mm. The LOCR

is bolted to a small raised platform mounted on the detector breadboard. The raised platform reconciles

the height difference between the two breadboards. 

The detector breadboard is attached to a large, stepping-motor driven, ball-bearing linear translation

stage. The stage has a load carrying capability of 90 kg and the load (not including test detectors) is

approximately 75 kg. The LOCR itself weighs approximately 50 kg, when fully loaded with cryogens. 

Significant backlash in the detector stage’s motion can occur. To minimize the potential backlash,

motions are always concluded in a preferred direction. When motion in the opposite direction is required,

the target location is over-shot by a few millimeters and the stage moves back to the target in the

preferred direction. The preferred direction is also the direction in which the stage’s drive nut contacts its

fixed mount, so that there is no change in the absolute position even if the drive nut’s clamp loosens. The
11

Figure 4. High-accuracy calibration system’s laboratory table layout.

resulting unidirectional reproducibility is approximately ±0.04 mm, which is negligible compared to the

alignment uncertainty, which is usually a few tenths of a millimeters or more. 

A second motion-controlled stage, the chopper stage, is used to move an optical chopper into the beam’s

path, for detectors that need chopped signals. A third and forth stage, not shown in the figures, are

combined in an x-y configuration. The x-y stage will be used to perform experiments such as scanningslit beam diameter measurements, pinhole scans to measure the beam profile, precision detector centering,

and detector spatial uniformity scans. The process control software does not currently support the x-y

stage. 

To smooth their motion, all of the motion-controlled stages used in the system are driven with a ramping

acceleration. The maximum speed and acceleration time are set with user-accessible options in the

process control software. The detector stage’s ramp time of 4 s is long enough to prevent excessive

sloshing of the liquids in the cryogenic reservoirs, without resulting in motions so slow that the stage

takes excessive time to translate.

Laser power supplies are placed on the lab table around the optical source breadboard. During

calibrations, a movable table is placed behind the lab table, as shown in Figure 4. Detector electronics are

placed on the movable table. The movable table must be moved out of the way during liquid-helium fills

and turbomolecular pump operation.
12

3.1.4 Gas and cryogen handling systems: Both liquid and gaseous He and N2 are used in the HighAccuracy Calibration System. The cryogenic liquids are used to cool the LOCR. The N2

 gas is used

during LOCR pre-cooling and as a dry gas for venting and general cleaning purposes. The He gas is used

primarily for pressurizing the liquid-helium container when liquid is transferred.

Valves with quick disconnect fittings for the He and N2 gasses are conveniently located on the corner of

the lab table, as shown in Figure 4. A variety of extension lines with quick disconnect adapters can be

attached to the valves. Extensions include a high-pressure nozzle with trigger, a fitting designed to attach

to the liquid-helium container for pressurization, and a line with knurled adapter for attachment to the

rubber and plastic tubing used during LOCR pre-cooling and turbomolecular pump venting. Flexible

plastic tubing is used in the gas delivery system; the tubing is purged with clean gas before use. The

quick-disconnect fittings contain internal valves that close when the adapter are not connected, which

helps prevent contaminants from entering the gas system. The fittings and valves can be operated while

wearing insulated cryogenic gloves, which are worn whenever the cryogenic liquids are being handled.

The gas systems use high-pressure industrial-grade gas cylinders. Pressure regulators that are optimized

for the delivery of gas at relatively low pressure (200 kPa maximum) are used with the cylinders. The

gaseous He pressure regulator uses a metal diaphragm to prevent He leaks inside the regulator, and is

capable of regulating a small negative pressure, so that the liquid-helium temperature can be lowered by

applying a small vacuum to the LOCR’s liquid-helium reservoir. The He gas will slowly permeate

through the plastic tubing, so a vacuum is generated in the line when the valves at each end are closed. 

Valves that isolate the regulator from the plastic tubing are therefore always closed when the gas is not in

use, to prevent the regulator from being damage by the vacuum. When not in use, both gas cylinder’s

main valves are always closed and the pressure in the regulators released, for safety and to protect the

regulators.

The LOCR requires liquid helium and nitrogen, which are stored in commercial cryogenic liquid storage

containers. The liquid-nitrogen container holds 160 l and the liquid-helium container holds 100 l. When

performing calibrations, the liquid nitrogen is consumed in about two weeks and the liquid helium is

consumed in about one month. The lead time for obtaining full containers is about two days for liquid

nitrogen and two weeks for liquid helium. The liquid nitrogen is delivered through a flexible, insulated

transfer line about 6 m long, which has a device called a vapor-phase separator at the LOCR’s end. The

phase separator allows the N2 gas to escape while the liquid-nitrogen pours into the LOCR. The pressure

of the boiling liquid inside the storage container is sufficient to push the liquid-nitrogen through the

transfer line at an adequate rate. The liquid helium is delivered through a short, rigid, vacuum-insulated

transfer line. The liquid-helium storage container is pressurized using He gas, to move the liquid more

quickly through the transfer line. The liquid-helium transfer rate is closely monitored and controlled by

the application of He gas pressure, because the transfer becomes increasingly inefficient if the liquid is

transferred too slowly, or if the storage container is over-pressurized.

3.1.5 Optical system: The stabilized laser source is optimized to provide laser power that is very stable

with time. The laser source resides on the optical source breadboard, shown in detail in Figure 5. The

laser beam is vertically polarized and is delivered to the detector plane in an approximately Gaussian

beam. The detector plane, shown in Figure 4, is where the detector’s entrance or limiting aperture is

placed.

Some beam quality is sacrificed in favor of power stability. The laser beams have high coherence, so to

prevent interference from occurring, optical feedback must be avoided. The optical components are tilted

to prevent back-reflections from reaching the laser and to prevent the formation of interferometer cavities

within and between components. The tilt causes additional coma distortion in the lenses, but the
13

Figure 5. Detailed view of the optical source breadboard.

additional distortion is preferable to the power instability. Spherical doublet lenses are used to reduce the

distortion caused by the tilt, and are carefully adjusted so that the beam passes through their centers,

thereby minimizing the distortion. Other components, such as the polarizers and attenuators, are also

tilted to prevent back-reflections. The polarizer tilt is performed carefully to prevent exceeding the

polarizer’s acceptance angle. Parallel-plate beam splitters are used with the power monitor elements, but

the beam splitters are relatively thick, and are placed in the beam at an angle of about 45°, so that
14

interference does not occur in them. Thin optical elements are avoided, because even when tilted, a thin

element can still produce interference fringes.

The lasers used in the stabilized source must meet several criteria. Ideally, the lasers should produce a

linear polarization that is stable with time, a Gaussian laser beam, a single longitudinal mode or at least a

single spectral line, and a stable or slowly drifting power. A stable, linear polarization is desired because

polarization angle changes will become amplitude variations when the beam passes through Polarizer 1;

lasers with other polarizations would be acceptable so long as the component with a linear polarization

has a constant power. Lasers that produce a Gaussian beam are desired because Gaussian beams can be

accurately modeled and are fully described by specifying the beam’s waist location and diameter. Freespace lasers that produce only the TEM0,0 mode and laser beams delivered through a single-mode optical

fiber produce a Gaussian beam of adequate quality. But if an optical fiber is used, the fiber must preserve

the beam’s polarization and a collimating lens is needed. Lasers with a single longitudinal mode or at

least a single spectral line are desired, but filters can be used to clean up the spectrum of some sources

with multiple spectral lines. Lasers that produce a stable or slowly varying power are desired because the

laser power controller (LPC) cannot compensate for very large changes in beam power or for highfrequency fluctuations.

The lasers currently used produce approximately Gaussian beams with 1/e2 intensity diameter of about

0.25 to 1.0 mm at the laser’s output aperture and a divergence angle of a few milliradians. Larger or more

divergent beams can be significantly clipped by the circular entrance aperture, 4 mm in diameter, on the

LPC modulator and monitor, resulting in undesired diffraction artifacts. As shown in Figure 5, the

1550 nm laser uses an optical fiber with an attached collimating lens. The others lasers shown have a

free-space output. The layout shown in Figure 5 is typical, but other lasers are available, and when

desired, a longer beam path can be constructed using additional mirrors.

The optical system’s polarizers are calcite, so a very good extinction ratio of about 100,000 to 1 is

obtained. Polarizer 1 is a Glan-Laser calcite polarizer, which can withstand a relatively high peak power,

but has a narrow and asymmetric acceptance angle that is increasingly narrow at infrared wavelengths. 

To ensure that Polarizer 1 is placed so that the incident beam is close enough to normal incidence, its

back reflection is traced all the way back to the laser’s output aperture and the polarizer is angled just

enough to prevent its back reflection from entering the laser. Polarizer 2 is a Glan-Thompson calcite

polarizer, which has a larger acceptance angle, so it can be angled more than Polarizer 1 but it cannot

withstand as much power as the Glan-Laser polarizer.

Most of the components are mounted on carriers along rails 1 and 2. The carriers provide smooth

translation along the length of the rail, without rotation or translation in other directions. The carriers

used with the lenses and irises have a built-in translation stage that provides fine alignment control in the

horizontal plane, perpendicular to the length of the rail, without rotation or other translation. The optical

components are mounted to the rail carriers using post holders, which allow the components to be rotated

without translation. The two rails and the laser sources are at right angles to each other, and mirrors 1 and

2 are used with an incidence angle of 45° to help reduce mixing of the beam’s polarization. Protected

silver front-surface mirrors are usually used, but dielectric mirrors are also available when more efficient

transmission in the visible spectrum is needed. The beam is directed with the optics so that it travels over

the center of the rails at a constant height between components. Other components, such as the laser

mounts, attenuator 1, and any interference filters are mounted directly to the optical source breadboard.

Attenuator 1 and any interference filters are placed in front of the laser, mounted directly to the optical

source breadboard. To limit the amount of power at the filters, the majority of the attenuation is

performed with attenuator 1, before the interference filters. Attenuator 2 is used to further reduce the
15

power to about 2 mW at the input to the LPC modulator. Attenuator 1 uses neutral-density reflective

filters, and attenuator 2 uses absorbing glass filters; both are placed in the beam at a small angle to

prevent interference.

Lenses 1 and 2 and the mounted pinhole form a spatial filter. The lenses are tilted slightly to prevent back

reflections from entering the LPC modulator. The spherical, acromatic-doublet lenses are positioned so

that the beam passes through the approximate center of each lens, to reduce the coma distortion caused by

the tilt; the lenses can be used slightly off-center to steer the beam as necessary to compensate for

imperfections in Polarizer 1 and the LPC modulator. The two lenses are tilted in opposite directions to

prevent a horizontal translation of the transmitted beam. Acromatic-doublet lenses are used instead of

single-element lenses so that there are more refracting surfaces, to reduce spherical aberration over a

wider range of wavelengths. Lens 1 is placed at a distance from the LPC modulator to minimize the tilt

angle required to prevent the back-reflection from re-entering the modulator, but a tilt of a few degrees is

usually necessary because the spherical surface of the lens causes the back reflection to diverge. The

spherical lenses require less tilt and produce less aberration with the same tilt than comparable singleelement parabolic or “best form” lenses.

Several lens pairs, each with a diameter of 2.54 cm, are available with focal lengths of 50, 75, and

100 mm. The lenses are used throughout the visible to near infrared spectrum. The lenses have a single

layer of antireflection coating optimized for visible wavelengths, but they have sufficient transmission to

allow operation at infrared wavelengths. Irises 1 to 3 are used during beam alignment and are used to

block back reflections and scattered light after alignment, but are not used to restrict the extent of the

Gaussian beam (which would produce diffraction artifacts).

In the high-accuracy calibration system, the spatial filter [7], which consists of Lens 1, Lens 2, and the

pinhole, is used somewhat differently than usual. The beam illuminating Lens 1 is an approximately

Gaussian beam, produced by the laser source. Therefore, Lens 1 causes a Gaussian beam waist to form at

a location slightly behind the Fourier plane. A pinhole aperture is placed at the beam waist to effectively

remove any higher spatial frequencies present in the incident beam. Scattered optical power, which

surrounds the approximately Gaussian beam, is effectively removed by the spatial filter. The spatial filter

does not create a Gaussian beam from a non-Gaussian source, it helps remove only non-Gaussian

components.

The pinhole used in the spatial filter has a diameter at least twice the size of the 1/e2 intensity diameter of

the Gaussian beam waist, so that the intensity of the light at the edge of the pinhole is small and the

resulting diffraction produces minimal artifacts in the reconstructed beam. In practice, the result is that

the scattered power around the incident Gaussian beam is effectively removed, but the Gaussian beam

itself is essentially unchanged. This modified spatial filter can significantly improve the beam quality,

since there is usually significant scattered power produced by the mirrors, polarizer, attenuators, and the

laser source itself. But the quality of the Gaussian beam emanating from the modified spatial filter is

therefore only as good as the quality of the Gaussian beam produced by the laser source itself, so it is

important that the laser source itself is aligned well. The optical system can degrade the Gaussian beam’s

quality, but does not improve it, other than by removing the non-Gaussian components or scattered light.

To control the size and divergence of the beam in the detector plane, the location of Lens 2 is also varied

from that in the classic spatial filter. The lens focal lengths and the distance from the laser source to Lens

1 can also be varied to obtain other beam sizes and divergence angles, but varying the location of Lens 2

provides a fine control for the size and divergence of the resulting beam. Because the beam produced is a

good approximation to the ideal Gaussian beam, the behavior of the beam is accurately predicted using
16

the Gaussian beam model described in Appendix D. The size of the beam waist formed between the

lenses is also determined using the model.

The optical shutter uses a plastic-coated blade that opens and closes quickly, with rise times and delays

that are less than 10 ms. The amount of time the shutter is open is measured with the exposure counter,

which is a precision time-interval counter that is accurate to better then 0.001 % (for time intervals in

seconds). Asymmetry in the shutter switching time is not accounted for in the measurement, but is

usually negligible compared to the period during which the shutter is open. The period in which the

shutter is open is important for energy meters only, where the energy is the average power multiplied by

the period the shutter is open.

The final element in the optical source is a circular final aperture. The final aperture is mounted to the

shutter’s housing, so that its location relative to the laser beam is constant when the height of the optical

source is varied. The final aperture is a 4 mm diameter, painted ceramic aperture; it is used primarily to

block a stray beam that is produced by Polarizer 2. The final aperture must be accurately aligned, or it

can significantly clip the laser beam and cause diffraction artifacts in the detector plane. A larger final

aperture can be used to accommodate other beam diameters, but an aperture much larger than 4 mm may

not block the stray beam. The stray beam diverges from the main beam at a small angle, so the final

aperture must be small and placed sufficiently far away from Polarizer 2 so that the stray beam diverges

enough to be blocked by the aperture, hence its mount extends to the detector’s side of the scattered light

shield. The stray beam is visible when the 633 nm laser is used; it may not be detectable with IR cards at

infrared wavelengths, but probably is still present. If the final aperture were not present, the stray beam’s

divergence is typically such that it would be captured by a detector with a limiting aperture of 10 mm

diameter, but missed by a detector with a limiting aperture of 5 mm diameter. This effect would be

compensated for by the relative aperture transmittance correction, but the measurement uncertainty is

lower when the problem is eliminated by using the final aperture. Additionally, the stray beam may have

a different polarization from that of the main beam, which would complicate the LOCR Brewster’s angle

window transmittance and the measurement uncertainty analysis, if the beam were not blocked.

Because it is not much larger than the beam’s diameter, the final aperture does produce diffraction

artifacts in the detector plane, but the minor diffraction artifacts are preferable to the stray beam that

would otherwise be present. To minimize the diffraction artifacts, the final aperture’s diameter should be

at least twice as big as the 1/e2 intensity diameter of the beam at the aperture. Conversely, the small final

aperture can help block diffraction artifacts and scattering produced by the elements in the system that are

after the modified spatial filter. The precise effect of the final aperture on the beam profile in the detector

plane can be studied in detail using precision beam-profile scanning techniques, such as that recently

developed for measuring the mode field diameter of an optical fiber [8].

3.1.6 Laser power stabilization system: Laser power stabilization is provided by the laser power

controller (LPC). The LPC consists of three separate parts; the LPC modulator, the LPC monitor, and the

LPC electronics. The LPC modulator uses a liquid-crystal device to rotate the polarization of the

transmitted laser beam, by up to 90°. Polarizer 2 demodulates the beam, converting the polarization

modulation to an amplitude modulation. Polarizer 2 also removes any polarization mixing caused by the

intervening lenses of the spatial filter, ensuring that a strongly polarized beam is delivered to the LOCR

Brewster’s angle window. The LPC monitor is placed after Polarizer 2, so that it measures the power in

the demodulated beam. A parallel-plate beam splitter in the LPC monitor samples a small portion of the

beam and the sampled beam’s power is measured using an internal detector. The monitor’s power

measurement is fed to the LPC electronics, where a control system produces the modulator signal. The

LPC monitor’s beam splitter and detector combination are calibrated in the factory, so that the power

measured by the detector can be related to the absolute power in the transmitted beam.
17

The LPC modulator and LPC monitor are each housed in identical, precision-engineered enclosures. The

beam passes through the devices with no appreciable horizontal offset or angular deflection, but a small

vertical offset is produced. The enclosures have fixed circular apertures: a 4 mm diameter entrance and

5.5 mm diameter exit apertures. The beam must be small enough at these apertures to keep any resulting

diffraction artifacts to a minimum, so the beam should have a 1/e2 intensity diameter of one-half the

aperture diameter or less. This requirement is usually easily met at the LPC modulator, where the

Gaussian beam has a relatively small diameter; but the size of the apertures on the LPC monitor, which is

much closer to the detector plane, can limit the diameter of the largest usable beam. The LPC modulator

and monitor can however produce a slight wavelength-dependent angular deviation in the vertical plane. 

The offset produced by the LPC modulator is compensated for by slight steering with Lens 1. The

deviations are small, typically a few milliradians.

Two complete LPC systems are available in the high-accuracy calibration system. The visible

wavelength LPC uses Si detectors to measure the power and is designed to operate at wavelengths from

425 to 780 nm, with input laser power from 0.5 mW to 4 W. The infrared wavelength LPC uses InGaAs

detectors and is designed to operate at wavelengths from 950 to 1,700 nm, with input laser power from

0.5 mW to 2.0 W. Each modulator has a maximum optical transmission of 80 to 85 %, within the stated

wavelength range. The LPCs can be used outside their specified wavelength range, but the optical power

reported by the LPC electronics will be inaccurate and the stabilizer’s performance will be reduced. 

Therefore, operation over the wavelength range of 780 to 950 nm is possible with the available

equipment, albeit at reduced performance.

The LPC monitor uses a parallel-plate beam splitter, set at an angle of 45°. The beam splitter is oriented

so that it is near Brewster’s angle for the transmitted beam’s polarization, so very little of the power is

reflected to the monitor detector. The beam splitter is thick enough (a few millimeters) to prevent

interference from occurring. The small sample ratio limits the LPC to a usable minimum power of about

100 :W, lower powers become increasingly noisy because of noise in the monitor detector. Powers

lower than 100 :W can be achieved without adding excessive noise, by mounting the power monitor so

that it is rotated 90° around the beam’s propagation axis, then a greater percentage of the transmitted

beam is reflected to the monitor detector, improving its signal-to-noise ratio. The upper power range of

the LPC is ultimately limited by the damage level of the LPC modulator’s liquid crystal, but in practice

the lasers used limit the power to about 10 mW. Both the LPC modulator and monitor have internal

neutral density filters, which are used to attenuate the power reflected to the monitor detectors inside the

modules, when high powers are used. The LOCR cannot be used with powers much greater than 1 mW,

but transfer standards could be used with higher powers.

The spectral response of the monitor detectors used in the LPC is calibrated at the factory. The vacuum

wavelength of the beam being stabilized is entered into the LPC electronics, where the monitor’s spectral

response is corrected. The resulting power measurement, which is accurate to within a few percent, can

be displayed on the LPC electronics and read through its RS-232 port.

The LPC modulator can also function independently of the external LPC monitor. Each LPC modulator

module contains an internal polarizing beam splitter and internal monitor detector, so can be used without

the external polarizer and LPC monitor. The LPC modulator’s internal monitor is used during source

alignment, but to compensate for loss and polarization mixing in the spatial filter and loss in Polarizer 2,

the external LPC monitor is used during calibrations. Polarizer 2 has a greater extinction ratio than the

polarizer in the LPC modulator and is the last polarizer in the optical system, so it sets the polarization

angle of the beam at the detectors. Polarizer 2 can therefore be used to adjust the polarization angle to

match the orientation of the LOCR’s Brewster’s angle window. Using the current mount, rotating the

LOCR’s window mount is difficult and impossible to do with precision, while adjusting the beam
18

polarization with Polarizer 2 is simple. However, to minimize the power loss and maximize the dynamic

range of the system, the polarization angle of the laser source, Polarizers 1 and 2, and the LPC

modulator’s internal polarizer should all be set as close as possible to the same angle.

The LPC electronics is a control system that is effective at compensating for drift in the input laser power,

within its bandwidth of 5 kHz. Large noise spikes or drift can exceed the device’s modulation range, so

the LPC electronics package includes an error light that flashes when the modulator is driven out of

range. When operating properly, the power drift is typically less than 0.05 % peak-to-peak per hour, for

powers around 1 mW. The error light will illuminate continuously if the modulator is driven completely

out of range and no regulation is provided. A computer can be used to check the LPC error status and

transmission ratio, through the LPC electronic’s RS-232 port, to ensure it is operating within its range.

3.1.7 The relative aperture transmittance correction: The high-accuracy calibration system’s optical

source is designed to deliver to the detectors a good approximation of the theoretical Gaussian beam. 

Probably the most significant difference between the actual and theoretical Gaussian beams is that the

actual beam contains more power away from the beam’s center than the theoretical Gaussian beam does. 

The spatial filter helps reduce the scattered power from around the Gaussian beam, but some remains and

the subsequent optical elements can produce more. To compensate for this imperfection, the relative

aperture transmittance correction, called TRA, is applied.

A Gaussian beam has the majority of its power contained near the center of the beam and the beam’s

intensity drops rapidly at increasing distance from its center. Technically the beam has tails that extend to

infinity, but they become very small after a few beam radii. If a detector’s limiting aperture diameter is

several times the size of the Gaussian beam’s diameter, it is common to assume that the detector captures

the entire beam. However in the real optical system, there is significantly more power in the tails than in

the theoretical Gaussian beam, so the amount of power that is not captured by such detectors can be

significant in a high-accuracy calibration. Detectors with entrance apertures that are different sizes or

shapes can therefore be exposed to different absolute powers, even if their apertures are many times larger

than the beam.

The correction factor TRA compensates for the amount of power that is missed by the standard and test

detectors, so that the resulting detector responsivity calibration factor is based on the amount of power

that was actually delivered through each detector’s entrance aperture. Usually, the customer specifies a

beam diameter that is several times smaller than their detector’s aperture, with the assumption that

essentially all of a theoretical Gaussian beam will be captured. The correction TRA is consistent with this

assumption, and in this case, *TRA ! 1* is a measure of the imperfection in calibration system’s beam. But

more generally, the correction results in the same calibration factor being obtained for given test and

standard detector, independent of the beam’s actual intensity profile, so long as both the standard and test

detector’s responsivity is uniform and linear within their apertures. In practice, the TRA correction is

usually negligible compared to the overall uncertainty in the calibration, unless the test and standard

detectors have very different aperture sizes.

There is historical precedent for applying a correction for a detector’s aperture transmittance. The relative

aperture transmittance correction is similar in concept to the spillover monitor-based correction used with

an old device, the NIST BB calorimeter [9]. In the BB’s application, the total energy in the beam was

desired. It was evident that the BB’s optical receiver could miss a significant portion of the large multimode beam usually used with the calorimeter, so a spillover monitor (which has an entrance aperture

much larger than the calorimeter’s) was added to measure the amount of energy spilling over the

calorimeter’s aperture. Originally, the spillover monitor was used only as a quality control measure, to

determine wheather the calorimeter is capturing enough of the beam to meet its specified uncertainty. An
19

Figure 6. The apparent aperture of a detector that has no

limiting aperture.

uncertainty was assessed for the expected spillover and the monitor was used to determine whether the

actual amount of spillover was within the assessed uncertainty limit. The spillover was later corrected

for, to obtain lower calibration uncertainties.

The aperture referred to in this discussion is ideally the detector’s limiting aperture. But some detectors

have no physical limiting aperture and the size of their entrance aperture can be deceiving. With such

detectors, the optically responsive element inside can be smaller than the entrance aperture, so some light

entering their entrance aperture misses the detector, even if the light rays are perfectly collimated. For

these detectors, an equivalent limiting aperture called the apparent aperture is used. The apparent

aperture of such a detector is shown in Figure 6; the detector’s apparent aperture is clearly much smaller

than its approximately 12 mm diameter circular entrance aperture. The apparent aperture is simply the

50 % contour line in a spatial uniformity scan of the detector’s responsivity [10].

The 50 % responsivity contour is used because it inscribes the edge of the detector’s physical limiting

aperture, when one is present, so long as the beam used in the uniformity scan is small compared to the

aperture’s feature size. For example, when a scanning beam is centered over the edge of a relatively large

limiting aperture, half will fall on the detector and half will be blocked, so the responsivity drops 50 % 

from when the beam is centered in the aperture. Since the real scanning beam is not infinitely small

compared to the aperture, the measured apparent aperture will be somewhat smaller than the actual

limiting aperture. When the Gaussian scanning beam’s 1/e2 diameter is one-fifth the minimum aperture

diameter or less, the reduction in the size of the measured apparent aperture is considered sufficiently

small for the purpose of estimating a detector’s TRA, given that the uncertainty in the measurement of TRA

itself is relatively large. The process used to measure TRA is described in Section 4.1.5 and Appendix E.

3.2 The LOCR Primary Standard

The primary standard used in the high-accuracy calibration system is a laser-optimized cryogenic

radiometer, abbreviated as LOCR. Cryogenic radiometers provide the most accurate means of measuring

absolute optical power that is currently available. The uncertainty of the primary standard’s power

measurement directly limits the uncertainty obtainable with the calibration system, so a cryogenic

radiometer primary standard is necessary to

achieve the desired high-accuracy detector

responsivity calibration.

3.2.1 Basic cryogenic radiometer operating

principal: Cryogenic radiometers gain their

performance advantage over roomtemperature atmospheric-pressure devices

by improved diffusivity in the optical

receiver, reduced radiation from the optical

receiver, and reduced power dissipation in

the electrical heater leads. These factors

reduce the inequivalence between electrical

and optical heating, thereby improving the

accuracy of the electrical substitution. To

achieve these properties with the materials

commonly available today, the devices

operate usually near the liquid-helium

temperature of 4 K. Operation at cryogenic

temperature necessitates the use of ultra-
20

Figure 7. External view of the LOCR.

high vacuum inside the radiometer to insulate the cold interior from the external environment, which also

eliminates any convective heat flow inside the device, but necessitates the use of a window that allows the

light to enter the device.

Laser-optimized cryogenic radiometers such as the LOCR use an optical receiver design that optimizes

the capture of the collimated laser beam and use a Brewster’s angle window that reflects very little of the

incident laser power. Polarized laser light passes through the Brewster’s angle window with minimal

attenuation and is absorbed in the optical receiver, where the optical power is converted to heat. The

amount of heating power is then measured by electrical substitution, so the device is a primary standard,

traceable to SI units through the electrical substitution. 

3.2.2 Description of the LOCR: The LOCR has been previously described [6, 11, 12], so its internal

construction will be discussed only briefly here. The LOCR is a custom designed system that was built

by a commercial manufacturer [13]. An external view of the LOCR is shown in Figure 7, and Figure 8

shows an internal cross-section view. The LOCR uses both liquid-helium and liquid-nitrogen, held in

reservoirs that are exposed to atmospheric pressure. The hold time for liquid nitrogen is approximately

14 hours, so the nitrogen is refilled at least twice a day. The hold time for liquid helium is approximately

2.5 days, so the helium reservoir is filled twice a week (the cryogens expire over the weekend). The

cryogenic liquid hold times are affected by room temperature, and if the room is very warm, the hold

times are reduced. Vacuum inside the cryostat is provided by the cryo-pumping produced by the cold

temperatures, which is enhanced by the addition of the carbon getter attached to the liquid-helium

temperature heat shield, and by an ion pump that is located on top of the cryostat. A partial vacuum can

be applied to the helium reservoir to further reduce the temperature of the liquid, but the device has not

yet been operated in this mode, and the liquid-helium’s hold time would likely drop considerably.
21

Figure 8. Internal view of the LOCR.

As per the manufacturer’s recommendation, the LOCR is brought back to room temperature weekly. 

Leakages through the o-ring seals in the cryostat, valves, and window mount have the potential for

creating ice on the optical receiver, thus changing its optical properties, so the LOCR is warmed to room

temperature weekly to prevent any significant ice buildup. The warming also clears the carbon getter and

ion pump, allowing lower pressures to be maintained. Allowing the LOCR to warm up over the weekend

also eliminates the need for an automatic liquid-nitrogen filling system, or a technician coming in twice a

day to refill the nitrogen manually.

The dashed line in Figure 8 shows the location of the detector plane inside of the LOCR. The LOCR has

three circular apertures in front of its optical receiver. The detector plane is located at the final aperture,

which is the smallest (10 mm in diameter) and the closest to the optical receiver. The optical receiver

itself is slightly larger, so the final aperture is ideally the limiting aperture. By convention, the detectors

under test are placed on the detector breadboard so that their entrance or limiting aperture is in the

detector plane. 

Small imperfections in the LOCR’s construction and the optical system’s alignment necessitate tilting the

LOCR a few milliradians so that the beam passes through both the center of its window mount and the

center of its optical receiver. The tilt results in the LOCR’s apparent aperture being slightly smaller than

its limiting aperture and elliptical; in its current configuration, the LOCR’s apparent aperture measures

9.5 mm in diameter horizontally and 8.5 mm in diameter vertically.
22

Figure 9. The LOCR’s optical receiver.

Figure 10. General functional schematic of the LOCR.

Thermal modeling [11, 12] showed that

when operating at the nominal receiver

temperature of 5.4 K, the inequivalence

between electrical and optical heating is

negligible (0.0005 %), even when the

optical heating power is applied at an

extreme location on the receiver. 

Figure 9 shows the LOCR’s optical

receiver, with the normal and extreme

incident beam locations that were used in

the model. Since it is unlikely that the

optical power would be applied at such an extreme location, the actual inequivalence encountered is

likely to be much smaller than that determined by the modeling. Superconducting lead wires are used

with the LOCR’s resistive heaters to eliminate loss in the heater leads, thereby improving the accuracy of

the electrical heating power measurement.

To aid in describing the electrical operation of the LOCR, a simplified schematic, shown in Figure 10, is

used. As shown in the figure, the LOCR uses two precision digital temperature controllers, one to

regulate the heat sink near liquid-helium temperature, and one to regulate the optical receiver at slightly

higher temperature. The temperatures at the receiver and heat sink are sensed with germanium resistance

thermometers (GRTs). The GRT resistance is measured using precision alternating-current bridge

circuitry, located in the LOCR’s base. The temperature-control electronics apply power through resistive

heating in the thin-film resistors located on the heat sink and optical receiver, and use the GRTs for

feedback. The heat-sink temperature is regulated because the temperature of the liquid-helium well itself

can change with liquid level and atmospheric pressure.

Application of the optical power is controlled using an external shutter. Electrical substitution in the

LOCR is provided by measuring the drop in electrical heating power that is required to maintain the

optical receiver’s temperature, when the optical power is applied. This mode of operation is usually

called the active or dynamic mode. The optical power is therefore the difference between the two

absolute electrical powers; the electrical power when no optical power is applied (the background power)

and the electrical power when the optical power is applied (the illuminated power). The uncertainty in

the two electrical power measurements themselves can be larger than the uncertainty in the difference,

because some of the uncertainties in the power measurements are correlated, as described in Appendix F. 

In practice, the background

and illuminated electrical

powers are derived using the

dual-baseline measurement,

described in section 4.3.1, but

the underlying concept is the

same.

The active mode of operation

is advantageous because the

optical receiver is maintained

at the same temperature, so its

temperature and therefore its

thermal properties do not

change when the optical power

is applied. And because the
23

Figure 11. Simplified LOCR

electrical calibration circuit.

receiver’s temperature ideally never changes, the measurement of optical power can be performed

relatively quickly. The temperature controllers also have an “open” mode of operation, where they

simply apply a fixed amount of heating power. Open mode is used during LOCR electrical calibration,

but the LOCR is not intended to operate in open mode during optical power measurement.

One disadvantage of the active operating mode is that the device’s overall performance is directly tied to

the performance of its temperature-control systems, so temperature-control systems of very high quality

are required. The LOCR uses digital control systems that can track the step change that occurs when the

optical power is applied with no offset. At its typical operating temperature of 5.4° K, the thermal time

constant of the optical receiver itself is approximately 10 s.

Since the LOCR is a thermal device that measures essentially any radiation, changes in the background

temperature can affect the desired optical power measurement. Experiments have shown that the LOCR’s

temperature coefficient is approximately !950 nW/°C [11, 12], due to radiative heat transfer from the

window assembly to the optical receiver. Specifically, this is the change in receiver heater power that is

necessary to maintain the receiver at a constant temperature, for a given change in the window assembly

temperature. Since the window assembly is not temperature regulated, it follows room temperature; so

the above is essentially the room-temperature coefficient. Fortunately, a linear drift in the window

assembly temperature is compensated for by using the dual baseline and the power interpolation methods

used with the calibration system.

Other environmental influences also affect the LOCR, but are usually negligible compared to other

sources of uncertainty in the system. For example, since the LOCR’s cryogenic reservoirs are exposed to

the atmosphere, a change in atmospheric pressure will cause a change in the temperature of the cryogenic

liquid, which changes the temperature of the thermal shields connected to the reservoirs. The heat sink’s

temperature regulator isolates the optical receiver’s conductive heat flow path from the change in

reservoir temperature, but a thermal shield is connected directly to the liquid reservoir, so its temperature

can change, thereby changing the radiative heat transfer from the shield to the receiver. Similarly, the

temperature of the LOCR’s vapor-cooled liquid-nitrogen temperature shield changes with the level of the

liquid in the reservoir. And atmospheric humidity may change the window transmittance, as water vapor

is adsorbed on the surface of the window. Environmental parameters during the calibration are logged by

the weather station continuously, so that if other significant environmental effects are discovered later, a

post-processing correction can be applied.

3.2.3 LOCR electrical calibration: The LOCR’s temperature control electronics are calibrated in terms of

the amount of electrical power that they apply to the LOCR’s thin-film heaters. Imperfections in the

electronics cause an error in the electrical power that is reported by the temperature controller’s digital

electronics. To correct for the error in the reported power, the actual power delivered is measured

independently and a correction factor for the digital power reading is stored in the temperature control

electronics.

A simplified schematic of the LOCR’s electrical heating circuit is shown

in Figure 11. The resistor RS is the standard resistor, which has a known

resistance. The resistor RH is the thin-film resistor placed on the LOCR’s

optical receiver, through which the electrical heating power is delivered

to the receiver. The voltage V is applied by the temperature control

electronics and the current I flows through the resistors. It is unimportant

whether the temperature controller actually uses voltage or current

sources, since either will result in the voltage V across the resistors when

the current I flows through them. During electrical calibration, the
24

(1)

voltage drops across the resistors VS and VH are measured using the calibration system’s precision

multimeter. The power dissipated in the resistor RH, called PH, is given by:

The power PH determined using the measured voltages is then compared to the value of PH returned by

the temperature controller’s digital electronics; the difference is the correction factor for the digital

electronics. During calibration of the LOCR’s electronics, the correction factor for a large array of

applied powers is determined. The correction factors are analyzed and stored in the LOCR’s electronics

and subsequent measurement of electrical power are automatically corrected using the stored data. To

update the stored correction factors, the entire array of powers must be measured again, which is very

time consuming.

The stored calibration factors are accurate when the electronics are used in an environment similar to that

present during the calibration and for some period of time after the calibration. The non-zero temperature

coefficient of the electrical components cause the digital power reading to vary with temperature, and

humidity can affect some of the electrical components, causing a sensitivity to the environment. And

some of the electrical components, such as the precision resistors, can change with time and usage as they

age. Therefore, for the electrical calibration to be accurate, the calibration must be performed in an

environment similar to that which will be present during optical power measurements, and the

measurements must be performed a relatively short time after the electrical calibration (a 90-day limit on

the age of the electrical calibration is assumed in the uncertainty calculations).

Since the LOCR electrical calibration process is very time consuming, it is not efficient to perform a full

electrical calibration for each optical calibration that is performed. Instead, a relatively quick calibration

check is performed during each test detector calibration, at only the power levels of interest in the test

detector calibration.

In the calibration check, the power PH is measured as above, but the measured power is compared to the

corrected digital power, using the correction factor stored in the electronics from the previous full

electrical calibration. The difference between the measured and corrected digital powers is then the error

in the corrected digital power reading. Instead of using the resulting error to update the correction factor

stored inside the digital electronics, the error is simply recorded. Using this calibration check method, the

error in the digital power readings is determined at only the power levels that will be used in the

forthcoming high-accuracy calibration and the resulting error is corrected in post-processing. 

The calibration check must be performed at each nominal electrical power that is applied to the LOCR’s

receiver during the optical calibrations. For example, the nominal receiver power during background

illumination (no laser power) is 2.0 mW, so a calibration check at 2.0 mW is needed. A calibration check

is also needed for each laser power that will be used in the detector calibration. For example, if the test

detector is to be calibrated with laser powers of 1.0, 0.5, and 0.1 mW, the calibration check must also be

performed for receiver powers of 1.0, 1.5, and 1.9 mW, respectively.

Because of the LOCR’s room-temperature sensitivity, the receiver temperature that results from

the nominal 2.0 mW of electrical power can change somewhat with room temperature. Because the

electrical power is calibrated at only the nominal power levels, the nominal electrical power that is

delivered to the receiver is given priority over the specific receiver operating temperature. The receiver

regulation temperature used by the temperature control is therefore determined by applying the nominal
25

Figure 12. The NIST Brewster’s angle window

mount.

receiver electrical power using the temperature

controller’s open mode, allowing the receiver to

reach its steady-state temperature, and measuring

the resulting temperature. Then, the temperature

controller is set to maintain the measured

temperature. Fortunately, the room temperature

in the laboratory does not vary much from the

nominal room temperature of 23° C, so to date

the resulting receiver temperature has been in the

range of 5.405 ± 0.033 K, which is considered

acceptable since the thermal properties of the

receiver, modeled at 5.4 K, will not change

significantly over this range. The nominal

2.0 mW of electrical power that is applied to the

heat sink can be adjusted to optimize the receiver

temperature that results from application of the

nominal receiver heater power, but such an

adjustment has not been needed to date. If

unusual room temperatures are present, such as if

the room temperature is intentionally varied to

study the test detector’s temperature coefficient,

adjusting the power applied to the heat sink may

be necessary to keep the receiver temperature

within a reasonable range.

The heat sink and optical receiver temperature

controllers are essentially identical, so both are

calibrated the same way. However, only the optical receiver’s temperature control must be accurately

calibrated, because the absolute power applied to the heat sink is irrelevant to the measurement of optical

power. Therefore, it is not necessary to check the heat sink temperature controller’s calibration during

test detector calibration. The heat-sink temperature controller’s calibration is however checked

empirically when the LOCR is operated; a large change in the heat sink’s temperature for a given applied

power would indicate a problem.

The LOCR’s design was optimized for optical power measurements of about 1 mW, but the device can be

used over a range of powers, with a corresponding increase in its uncertainty. Optical receiver power

levels much above 4 mW cannot be used because the resulting receiver temperature will exceed the

superconducting wire’s critical temperature. The conductance of the heat link which connects the optical

receiver to the heat sink controls the magnitude of the receiver’s temperature rise over the heat sink

temperature, for a given receiver power. Unlike some other cryogenic radiometers, the LOCR’s heat

links are not intended to be changed, so cannot be easily tailored for operation at other power levels.

3.2.4 The NIST Brewster’s angle window: The LOCR Brewster’s angle window, shown in Figure 12,

was designed and built by NIST. The Brewster’s angle window is mounted independently from the

LOCR, on the window assembly, as shown in Figure 7. The window assembly rides on a transverse rail,

which is attached to the detector breadboard, in front of the LOCR. The rail allows the window assembly

to be precisely moved in front of the LOCR, with little tilt or other undesired translations. The rail is used

to move the window assembly away from in front of the LOCR, for window transmittance measurement.
26

The window mount is connected to a kinematic gimbal mount, that allows precise angular adjustment of

the window along vertical and horizontal axes. The window mount gimbal also holds the NW 40

clamping vacuum flange. The clamping flange allows easy removal of the window from the LOCR. 

Together, the window mount, window mount gimbal, rail carrier, and vacuum flange are called the

window assembly. The window assembly is rigid and heavily articulated, with adjustments for setting the

mount’s height as well as the angle of the Brewster’s angle window and vacuum flange location.

The window mount was designed to minimize the mechanical stress placed on the window, thereby

reducing any stress-induced birefringence that may occur inside the window. It accepts a common

51 mm diameter, 10 mm thick circular window, with loose tolerances. The window’s vacuum seal is

provided by a thin, soft (50 durometer) nitrile o-ring, which was chosen because it compresses

sufficiently under the approximately 18 kg of force exerted by atmospheric pressure when the back of the

window is exposed to vacuum. No mechanical clamp is used; atmospheric pressure alone compresses the

seal. Despite being adequately compressed, the o-ring has to be lightly lubricated with vacuum grease to

reduce the leakage to acceptable levels. A shoulder machined into the mount, underneath the window, is

intended to help limit any spread of the lubricant across the back side of the window by breaking the

capillary action. A butyl rubber o-ring is preferable to the nitrile o-ring because butyl would provide a

better seal with the relatively low compression, but the nitrile o-ring works adequately when lubricated. 

To prevent problems with multiple reflections, the inside of the mount is painted with a flat black,

vacuum-compatible paint. The mount has a circular clear aperture with a diameter of 19 mm.

The window currently used with the LOCR is well polished but uncoated and made of quartz glass. The

window is a circular plane-parallel window, 51 mm in diameter, 9.5 mm thick, with a wedge angle that is

less than or equal to 5 min of arc (1.45 mrad), a surface flatness of 8/10 at 633 nm, and a surface quality

of 10-5. The ability to rotate the window when it is on the mount is useful, because only a small section

of the window is actually used. So if there is an imperfection in the glass in the area that is being used,

the window can be rotated so that a better portion is used instead. However, the window should not be

rotated while the mount is evacuated, because a catastrophic rush-in of air will result if the vacuum seal is

broken. The mount will accommodate variations in the window’s diameter of about 1 mm and any

window thickness up to 10 mm.

Windows with smaller wedge angles are available at additional cost, but are unnecessary because the

additional attenuation caused by the small wedge is compensated for in the window transmittance

measurement. Surface quality and internal scatter centers typically have a much bigger impact on the

transmittance than the small wedge angle. The wedge reduces the window’s transmittance because the

laser’s incidence angle cannot be precisely Brewster’s angle on both sides of the window, but the

reduction in transmittance is accounted for in the window transmittance measurement. The change in the

window’s transmittance that occurs when vacuum is applied to the back of the window is accounted for in

the window transmittance uncertainty.

The window mount also has a tight-fitting dust cover, shown in Figure 12. The cover helps protects the

window from contamination when it is not in use and is used to aid in the window alignment, as described

in Section 4.2.4. The dust cover limits the maximum thickness of the window to about 1 cm, because a

thicker window will block the cover.
27

Figure 13. Main calibration process flow chart, for calibration at a single wavelength.
28

4. CALIBRATION PROCEDURE

Operation of the high-accuracy calibration system is too complex to explain here in detail, but complete

operating flowcharts for the high-accuracy calibration system are available in the form of an unpublished

document called the High-Accuracy Calibration System Operating Instructions [14]. In the operating

instructions, the calibration procedures are described as processes, using flow charts. The flow chart for

the main calibration process is shown in Figure 13. The flow chart shows the calibration process for

detector calibrations at a single wavelength; the entire process is repeated for additional wavelengths. 

Here, the operating procedures are broken-up into three fundamental components: operation of the

calibration system, operation of the LOCR primary standard, and calculation of the test detector

calibration factor.

4.1 Calibration System Operation

4.1.1 Data acquisition software: The calibration system is controlled with software that implements the

processes described in the operating instructions. Most of the system is controlled using software

implemented on the main process-control computer, which controls the system and acquires the data. 

Only preliminary data processing is performed during data acquisition, using nominal values for

parameters that are measured during the calibration. Final values are unavailable until after post

processing is complete.

Customizing the software for a particular test detector is accomplished using the software’s graphical

interface, which permits real-time editing of the commands sent to the detector, and control of how the

detector’s output is measured. The software monitors and displays the transactions to help the operator

debug the customized instructions. The program traps most of the errors that can occur and has many

other built-in checks to help ensure proper operation.

Currently the LOCR is controlled using software provided by its manufacturer, which is executed on a

separate computer. The LOCR is treated as a direct-reading detector in the main system software, with a

few special extensions that aid in coordinating the two computers and transferring the acquired data

between them. LOCR operation is described in section 4.2.

4.1.2 Optical component selection and placement: Component selection is based on the desired

wavelength of operation, the particular laser’s spectral characteristics, the desired absolute power, and the

desired detector-plane beam characteristics. The first requirements determine which laser source should

be used, and the choice of laser determines which interference filters (if any) and attenuators are needed. 

Determining which components are needed to produce the desired detector plane beam characteristics is

more difficult. The components of the optical source are shown in Figure 5.

A Gaussian-beam model is used to predict the detector-plane beam size and wavefront based on the

selection and placement of the optical components, before the components of the source are actually

assembled and aligned. Using the model, different configurations can be considered much more quickly

than would be possible if each configuration had to be tested experimentally. Therefore, whether or not

the calibration beam desired by the customer can be created using the existing optical components can be

quickly determined. Modeling of the Gaussian beam is performed using a ray-tracing analysis that

assumes the beam is a perfect Gaussian; a purely geometric analysis is insufficient and can be misleading. 

In Appendix D, the Gaussian-beam ray-tracing model is described and contrasted with a geometric

analysis.
29

When customers submit a detector for calibration, the minimum information they must provide is the

wavelength and power level of the desired calibration. However, they may also specify the desired beam

size, wavefront, or divergence. For example, they may request that a Gaussian beam waist with a specific

diameter occur at their detector’s input, or that the beam diverges or converges as it enters their detector. 

These parameters are controlled by the selection and location of the components used in the optical

system. While all combinations of detector-plane beam size and divergence are not possible, there is a

wide range of combinations that can be produced using the existing optical equipment.

The size of the beam in the detector plane is determined by a number of factors, including: the location of

the laser source relative to Lens 1, the size and location of the Gaussian beam waist produced by the

source; the focal lengths of Lens 1 and 2; the distance between Lens 1 and 2; and the distance from Lens

2 to the detector plane. Optimal collimation, which results in the smallest diameter beam in the far field, 

is achieved with a specific spacing between Lens 1 and 2. However, the beam can be varied significantly

from optimal collimation to obtain different detector-plane beam size, by adjusting the spacing between

the lenses. 

Initially, the lens spacing that results in optimal collimation is assumed. The lens focal lengths and

distance to the laser and detector plane are varied to obtain a detector plane beam that is close to what the

customer desires, using the limited set of lens focal lengths available, and laser and detector plane

distances that can be accommodated by the system. The detector-plane beam parameters are then finetuned by adjusting the spacing between the lenses. The fine-tuning allows fine adjustment of the

detector-plane beam size and divergence angle, but they are not independent variables, so one parameter

must be specified. For example, suppose that the customer wants the Gaussian beam waist to occur at

their detector’s entrance aperture. The lens spacing can be tuned to control the location of the beam

waist, but the beam’s size will also change with the lens spacing. If the resulting beam size is too far

from what the customer requires, lenses of different focal lengths must be used, or the laser or detectorplane distance adjusted. Usually the customer is more concerned about the beam’s size and the

divergence angle is unimportant so long as it is small.

However, the actual beam used in the system is an imperfect Gaussian beam and the optical elements can

cause distortion. The result of the imperfection is that the actual beam produced can be slightly different

from that predicted by the model. The fine-tuning provided by adjusting the lens spacing is usually

sufficient to accommodate any error resulting from the imperfections. In practice, the beams used in the

system are good approximations to true Gaussian beams and the lenses are nearly diffraction limited, so

the actual beam size and divergence is usually within a few percent of that predicted by theory. To help

ensure that the desired beam can be obtained, it is best to choose components and distances such that the

desired detector plane beam is well within the limits of the tuning provided by adjusting the lens spacing.

The predicted diameter of the beam waist that occurs between the lenses is used to select the size of the

pinhole that is used in the modified spatial filter. A circular pinhole aperture is used with a diameter that

is at least twice the theoretical Gaussian beam waist’s 1/e2 intensity diameter. The relatively large pinhole

minimizes the artifacts produced when the Gaussian beam waist is passed through the finite aperture and

effectively reduces the scattered power that surrounds the approximately Gaussian beam. It also reduces

the system’s sensitivity to the laser’s pointing stability.

4.1.3 Laser source alignment: Once the component values and distances are selected using the Gaussian

beam model, the components are assembled and aligned. Alignment of the laser source consists of

placing all the relevant components on the Optical Source Breadboard, shown in Figure 5, and aligning

them so that the Gaussian laser beam travels over the center of the rails, at a constant height between

components.
30

In the calibration report, the beam used in the calibration is described in several ways. The beam is

described by its 1/e2 intensity diameter and its full-width divergence angle, in the detector plane, and by

its Gaussian beam waist diameter and location relative to the detector plane. The parameters are

determined by direct measurement, or theoretical values are given. The detector-plane beam diameter is

always measured; the scanning-slit beam profiler is used to measure the diameter, and the average of the

vertical and horizontal width is stated. 

The Gaussian beam waist diameter and location are directly measured using the scanning-slit profiler, if a

real beam waist occurs at an accessible location. Sometimes the waist cannot be directly measured

because it occurs inside the optical source between Lens 2 and the Final Aperture, where it cannot be

accessed, or a virtual waist occurs. When the waist cannot be directly measured, theoretical values

obtained from the Gaussian beam model are used. Similarly, when the detector plane is in the far-field of

the Gaussian beam waist, its divergence is easily measured by performing two diameter measurements a

small distance apart. But if the detector plane is very near the beam waist, the divergence angle changes

with distance, so instead, the theoretical radius of curvature is used to calculate the divergence angle in

the detector plane. Any theoretical values given in the report are described as approximate, while

measured parameters are given with their expanded uncertainty.

After the optical components are placed on the rails and aligned as described, the location of Lens 2 is

adjusted while the detector plane beam is monitored, until the desired detector plane beam is produced. 

For example consider that a beam with a particular detector plane beam diameter is desired. In this case,

the scanning-slit beam profiler is placed in the detector plane to measure the beam’s 1/e2 intensity

diameter in real-time, as the lens spacing is adjusted as described below while the measured beam size is

monitored. However if a particular beam-waist location is desired, the waist location must be determined

between each adjustment of the lens spacing.

To adjust the detector plane beam size, the location of Lens 2 is first adjusted for optimal collimation,

which is when the far-field beam diameter is minimized. The adjustment is performed by observing the

size of the beam a long distance from Lens 2, while the lens is moved. A long distance is achieved by

bouncing the transmitted beam back to the operator with a mirror placed near the far wall. Then, the

location of Lens 2 is adjusted to achieve the desired detector-plane beam diameter.

The size of the beam in the detector plane is adjusted by varying the location of Lens 2 around the

location that results in optimal collimation. To increase the detector plane beam size, Lens 2 is moved

closer to Lens 1. To decrease the beam size, Lens 2 is moved farther from Lens 1. The beam diameter

varies nearly linearly with lens position over a small range, as described in detail in Appendix D. The

detector-plane beam diameter is measured using the scanning slit beam profiler while the lens location is

adjusted. If the beam is elliptical, the size is adjusted so that the average of the vertical and horizontal

diameters is the desired diameter. 

Imperfections in the Gaussian beam cause the beam’s actual behavior to differ slightly from the

theoretical results. The theoretical analysis usually predicts the beam’s minimum detector-plane beam

diameter, Gaussian beam-waist location and size, and the far-field divergence angle within about 10 % of

the measured results. The difference from the theoretical results is consistent with the beam’s quality; the

more Gaussian the beam is, the more accurate the theoretical results are. The beam quality is usually

limited by what is produced by the laser source itself. Therefore the quality of the laser mirror’s

alignment (for free-space laser beams), or the beam collimator (for single-mode optical fiber laser

sources), are very important and should be optimized to produce a Gaussian beam of the best quality

possible.
31

Currently, a commercial device is used to measure the calibration beam’s intensity profile. The

commercial device uses a scanning-slit beam diameter measurement technique [15]. The scanning-slit

technique accurately measures a Gaussian beam’s profile, so is appropriate for the approximately

Gaussian beams that are used in the high-accuracy calibration system. The device performs the

measurement very quickly, so is especially useful when adjusting the detector-plane beam diameter. The

1/e2 intensity diameter that is given in the calibration report is derived from measurements obtained with

the device.

A more careful scanning-slit beam profile measurement, performed with customized equipment, can have

greater accuracy than that obtained with the commercial scanning-slit beam profiler. More accurate

measurements of beam profile can be obtained by measuring the detector’s output more accurately and

using a precision motion-controlled stage to translate the slit. Such a precision scanning-slit system is

currently used in the spatial uniformity scanning system [10]; the precision system provides reproducible

beam diameter measurements with a standard deviation of typically 0.02 %. In contrast, the commercial

beam profiling system’s measurements based on a scanning slit are relatively noisy, with a typical

standard deviation of 0.4 %. The speed of the commercial device is helpful while the beam size is

initially adjusted, but a precision width measurement is more desirable for the calibration report. 

Precision scanning-slit or pinhole beam profiling measurements may be performed using the precision x-y

sub-stage, described in Section 3.3.

Once the laser source is aligned, the laser beam’s height is controlled by adjusting the height of the

Optical Source Breadboard; otherwise the beam’s direction is fixed and the detectors are aligned to the

beam’s position. The height adjustment of the source breadboard is necessary to accommodate the

LOCR, which has a fixed height. All other devices are aligned to the resulting beam height. Horizontal

positioning is aided by the motion-controlled detector stage.

After the beam is aligned and its height is adjusted, its location in the detector plane is fixed and

everything else must be aligned to that location. The beam power can subsequently be changed by

adjusting the LPC and attenuators, without changing the location of the beam in the detector plane. 

Therefore, the beam power can be adjusted without requiring detector realignment or beam parameter remeasurement. Changing the beam’s size necessitates remeasurement of the beam parameters that affect

measurement uncertainty, such as the relative aperture transmittance, LOCR window transmittance and

LOCR spatial uniformity. Similarly, changing the laser wavelength requires a complete realignment of

the laser source and detectors, and remeasurement of the beam parameters.

4.1.4 Detector alignment: After the optical source is aligned and the desired detector plane beam is

produced, the detectors are aligned. First the LOCR is aligned, but because the LOCR’s height is not

adjustable, the height of the optical source breadboard is adjusted instead. LOCR alignment is described

in detail in Section 4.2.2. After the LOCR is aligned, the test detectors are aligned.

To help describe the alignment procedure, the naming convention shown in Figure 14 is used. Usually

the detectors are mounted using a post and holder, as depicted in the figure. With such a mount, no pitch

or roll adjustments are available; the angles are set by the physical characteristics of the mount. However,

more complicated mounting platforms can have pitch and roll angle adjustments.

Unless the customer demands a specific alignment procedure, the test detectors are aligned so that the

beam’s incidence angle is as close to normal as possible relative to the plane of the detector’s entrance

aperture. The exception to this rule is when a strong back-reflection from the detector re-enters the

optical system, when the detector is aligned at normal incidence. In this case, the detector’s yaw angle is

changed enough to prevent the reflection from re-entering the optical system.
32

Figure 14. Rotational and translational motions used

during detector alignment.

The beam’s incidence angle is adjusted relative

to the detector’s physical entrance aperture. To

determine the beam’s incidence relative to the

aperture, the operator holds a glass flat up to

the aperture, so that a strong specular back

reflection is produced. The reflection from the

glass flat is compared to the incident beam to

adjust the incidence angle. The reflected beam

will fall on top of the incident beam when the

incident beam is at normal incidence relative to

the aperture.

Before the angular alignment begins, the

detector mount’s location is adjusted in the

longitudinal direction so that the detector’s

limiting aperture is in the detector plane. If the

location of the detector’s limiting aperture is

unclear or the detector has no limiting aperture, the detector’s entrance aperture is positioned in the

detector plane instead. Then the detector’s pitch angle is adjusted, if available. The pitch angle is

adjusted so that the beam reflected from the glass flat is in the same horizontal plane as the incident beam.

Next the detector’s height is adjusted, using a vertical translation. The vertical translation is usually

provided by sliding the post in its holder. Once the height is adjusted, a post collar is locked into position

so that the post can be rotated without changing its height. The height is adjusted so that the beam passes

through the vertical center of the detector’s entrance aperture. The aperture’s center can be located by eye

using a ruler, or using the alignment aid located on the detector’s aperture cover, if any. A more precise

alignment can be obtained by fitting the detector with an aperture cover that has a pinhole aperture in its

center. Then the vertical height is adjusted until the power passing through the pinhole aperture is

peaked. This technique is most commonly used with detectors intended for use with optical fibers, then

the optical fiber adapter itself is used as the pinhole aperture, to ensure that the collimated beam strikes

the detector in the same location that a fiber-coupled beam would.

After the detector’s height is adjusted, its yaw angle is adjusted. Usually, yaw is adjusted by rotating the

post in its holder; the post collar prevents any change in the detector’s height when the post is rotated. As

with the pitch angle, the yaw angle is adjusted by observing the reflection from the glass flat that is held

against the detector’s entrance aperture. The yaw angle is adjusted so that the back reflection is in the

same vertical plane as the incident beam.

Then the beam is centered in the detector’s aperture horizontally. The horizontal translation is performed

with the motion-controlled detector stage, using the y-axis of the joystick. As with the vertical centering,

the alignment is performed visually, using a pinhole aperture, or with whatever special technique the

customer requests.

Lastly, the detector’s roll angle is adjustment, if any, is performed. Usually, a roll adjustment is used only

with detectors that are sensitive to the polarization angle and have radial symmetry. With such detectors,

the roll angle can be adjusted without affecting the beam’s centering or incidence angle. If the roll

adjustment does affect the other parameters, the roll angle should be adjusted first instead of last. Usually

the detector’s roll angle is defined by the physical characteristics of its mount, so is not adjustable.
33

A conservative uncertainty estimate of the error in the detector’s alignment is assessed and stated in the

calibration report. Typical upper-bounds are 1 mm for visual centering using a ruler, 0.5 mm for visual

centering using an aperture cross-hair or other targeting device, and 0.1 mm for centering by peaking the

power transmitted through a pinhole aperture mounted on the detector. The alignment uncertainty is used

in conjunction with the detector’s spatial uniformity to determine the uncertainty due to an imperfect

alignment.

Similarly, a bound for the error in the laser beam’s incidence angle is also assessed and stated. The

difference between the laser beam’s actual incidence angle and normal incidence is determined by

holding the glass flat to the detector’s entrance aperture, locating the back reflection cast by the glass, and

determining the angle between the back reflection and the incident beam. The difference between the

actual incidence angle and normal incidence is half the angle between the incident and reflected beams. 

The measured angle is combined with its measurement uncertainty to give an upper bound for the

difference between the beam’s actual incidence angle and normal incidence, which is typically a few

milliradians. The direction of the angular alignment error is not specified.

4.1.5 Relative aperture transmittance measurement: After the optical source is aligned, and usually before

the test detectors are aligned, the relative aperture transmittance is measured. The measurements are

relevant only for a particular laser beam profile, so must be performed whenever the beam’s profile is

changed, such as when the optical system is adjusted for a different beam size. Once the optical source is

adjusted to deliver the desired beam profile, the relative transmittance is assumed to remain stable for the

duration of the calibration, so the measurement is performed only once during the calibration. Also, it is

assumed that the beam profile and hence the relative transmittance does not change when the beam power

is adjusted, so the power level can be adjusted without requiring another set of relative transmittance

measurements. A power level of 1 mW is usually used during transmittance measurement. The relative

aperture transmittance measurement process is discussed in detail in Appendix E, so is described only

briefly here.

To determine the relative aperture transmittance for a particular test detector, the shape and size of its

apparent aperture is needed. When the detector’s apparent aperture must be measured, because the

detector lacks a limiting aperture, a figure showing the measured apparent aperture is provided in the

calibration report. The detector’s apparent aperture is measured with a spatial uniformity scan. When a

physical limiting aperture is assumed, the assumed shape of the aperture is described in the report text;

such apertures are usually circular. A spatial uniformity scan of a detector with a physical limiting

aperture may or may not be performed, depending on the customer’s handling specifications. If the

customer allows the detector to be cleaned (by blowing with pressurized air, for example), uniformity

scans are performed to determine whether the detector is dirty, and if so, whether subsequent cleaning

was effective. In this case, a figure showing the remaining contamination, if any, may also be included in

the report. But a formal report of spatial uniformity is not provided as part of the high-accuracy

responsivity calibration.

The relative aperture transmittance is usually measured using a laboratory-standard detector and fixed

circular apertures. Whether or not the test detector itself is used to measure relative aperture

transmittance is dependent on its apparent aperture’s shape and size. If the test detector does not have a

physical limiting aperture, or if its aperture is larger than the largest fixed circular aperture, the test

detector itself is used to measure relative aperture transmittance.

The laboratory-standard detector that is used to measure relative aperture transmittance has an apparent

aperture that is at least as large as the largest fixed circular aperture, and has good spatial uniformity

within its aperture. A large single-element Si or Ge detector is used as the standard detector; the Si
34

detector is used with visible wavelengths, and the Ge detector is used with near-infrared wavelengths. 

The relative transmittance of the fixed circular apertures is measured using the standard detector. The

detector is angled slightly so that its back reflection does not re-enter the optical system. When placed in

front of the detector, the apertures are very close to the detector’s surface, so that no light that is

transmitted through the aperture is diffracted away from the detector. The apertures are painted with a

flat-black paint to help absorb any light that is reflected off the detector, so that it is not reflected back to

the detector.

First, the standard detector is aligned so that its entrance aperture is slightly behind the detector plane and

the beam is centered in its aperture. Then each fixed circular aperture is in turn placed in front of the

detector’s aperture, in the detector plane, so that the beam is centered in the aperture. One aperture,

usually the largest, a circular aperture of 1 cm diameter, is used as the reference; the transmittance of each

of the other apertures is measured relative to the transmittance of the reference.

Each fixed aperture’s transmittance relative to that of the reference aperture is measured. First the

reference aperture is placed in front of the standard detector and the power received by the detector is

recorded. Then the reference aperture is replaced by the aperture under test and the power received by the

detector is recorded. The transmittance of the test aperture relative to that of the reference aperture is then

the power recorded when the test aperture was used divided by the power recorded when the reference

aperture was used. The absolute beam power is assumed to be stable or vary linearly during the

measurement. To reduce the affect of any power drift, the power during the test aperture measurement is

interpolated in time using bracketing measurements made with the reference aperture, and multiple

measurements are averaged.

If all of the detectors involved in the calibration have true limiting apertures that are within the range of

the fixed circular apertures, no additional measurements are needed, even if a limiting aperture is noncircular. For example, the LOCR has a physical limiting aperture that is elliptical, with dimensions 8.5 by

9.5 mm. The transmittance of its aperture is therefore somewhere between the transmittances of the 8 mm

and 10 mm diameter circular apertures, so can be estimated using the transmittance data for the 8 and

10 mm diameter fixed apertures. Basically, the transmittance estimate for the elliptical aperture is the

average of the transmittance measurements for the 8 and 10 mm diameter circular apertures, with an

uncertainty that includes the change in transmittance between the 8 and 10 mm apertures, as described in

the appendix. In general, the circular apertures used in the estimate are (a) the largest centered circular

aperture that is completely enclosed within the detector’s limiting aperture, and (b) the smallest centered

circular aperture that fully encloses the detector’s limiting aperture.

Ultimately, the relative transmittance between the LOCR primary standard and each test detector being

calibrated is needed. Continuing the example, consider that a test detector with a circular entrance

aperture of 5 mm diameter is being calibrated. As described above, the transmittance of the LOCR’s

elliptical aperture, relative to that of the reference aperture, is the average of the 8 and 10 mm diameter

circular apertures, relative to the reference aperture. The transmittance of the test detector’s aperture

relative to that of the reference aperture is therefore the relative transmittance of the 5 mm diameter

circular aperture. Then the transmittance of the LOCR’s aperture relative to that of the test detector’s is

simply the transmittance of the LOCR’s aperture relative to the reference, divided by the transmittance of

the 5 mm diameter aperture relative to the reference.

The test detector is handled differently when it lacks a limiting aperture, or when the test detector’s

aperture is larger than the largest fixed circular aperture available. In these cases, the largest centered,

circular aperture that is fully enclosed by the test detector’s apparent aperture is used, along with the test

detector itself, to determine the detector’s relative transmittance. For example, the apparent aperture
35

shown in Figure 6 fully encloses a centered circular aperture with a diameter of 7.8 mm. Since circular

apertures of fractional sizes are not used, the circular aperture of 7 mm diameter is the largest that is fully

enclosed by the detector’s apparent aperture. Similarly, a test detector with a very large aperture fully

encloses the largest fixed circular aperture. The test detector itself is used to measure the transmittance of

its aperture relative to that of the centered circular aperture, by measuring the beam power when its

aperture is unobstructed, and dividing by the beam power measured when the centered circular aperture is

placed in front of the detector. This results in the transmittance of the detector’s aperture, relative to the

enclosed circular aperture. To determine the transmittance of the detector’s aperture relative to the

transmittance of the reference aperture, the transmittance relative to the enclosed aperture is multiplied by

the transmittance of the enclosed aperture relative to the reference aperture. Then the test detector’s

aperture transmittance relative to the standard’s is calculated as described above. Like the measurements

performed with the laboratory standard and fixed apertures, the measurement of the transmittance of the

test detector’s aperture relative to the enclosed circular aperture’s is performed using interpolation in

time, and multiple measurements are averaged.

In addition to the measurements described here, there are other measurements performed to determine the

uncertainty in the aperture transmittance measurement itself. And the measured uncertainties are

propagated as the different transmittance measurements are combined to obtain the final result. The

measurements and calculations are fully described in Appendix E.

4.1.6 Calibration system maintenance: Several of the components of the high-accuracy calibration

system require periodic maintenance. Maintenance of the system components that are independent of the

LOCR are discussed here. Maintenance of the LOCR primary standard and equipment used exclusively

with it is discussed in section 4.2.6.

The calibration system is powered through an uninterruptible power supply (UPS) that provides brownout protection as well as electrical isolation and noise filtering. The UPS has a lifetime that is limited

primarily by its internal battery, so the battery’s health must be monitored. The device performs a

monthly self-check and emits an audible and visible alarm if a problem is detected. The details of the

self-check and other status can be read from the UPS through its RS-232 serial interface. When problems

are detected, the UPS must be repaired or replaced.

The HEPA filter that filters the air inside the lab table enclosure also has a limited lifetime. The HEPA

filter element’s lifetime is determined in part by the amount of dirt present in the environment. Since the

filter cannot be inspected without disassembly of its enclosure, it is usually best to simply replace the

filter element periodically. The original filter was replaced after about five years of continuous operation;

at that time, the once-white filter element had a brown color. Since HEPA filter elements can

theoretically be operated until they are essentially black and impenetrable, the filter could probably have

been used longer. However, the rate of air flow through the filter decreases as the filter becomes clogged,

so a periodic replacement at the rate of once every five years is preferable.

The calibration system’s precision DMM requires periodic maintenance in the form of electrical

calibration. The DMM is the primary standard for all electrical measurements performed in the lab, so

the uncertainty of its measurements limits many of the other uncertainties in the calibration. The

accuracy of the DMM calibration is affected by temperature and time. Currently, the manufacture’s

uncertainty specification for a time period of 90 days and for a temperature range of the DMM’s

calibration temperature ±5° C is used in the uncertainty analysis. Therefore, the DMM must be

electrically calibrated within 90 days of when it is used to perform detector calibrations, and the

temperature during the calibration must be within ±5° C of the temperature present during its electrical

calibration. Currently, the DMM electrical calibration is performed in-house, and a matrix of measured
36

voltages and voltage error is provided. The measured voltage errors are corrected in post-processing, as

described in Appendix F.

The software used to control the calibration system will require maintenance in the form of adding new

features and debugging, in addition to migrating the system to newer hardware as computer technology

improves. To help in this maintenance, the computer programs are written using standard structured

techniques in a simple Basic language, so should be understandable by any computer programmer. The

user interface is graphical and intuitive, so an experienced laboratory technician should be able to operate

the system, after some training or experimentation. The software stores all the acquired data in standard

spreadsheet format, along with a description of the measurement performed. Most of the mathematical

calculations are performed on the spreadsheet, so that the analysis performed is documented.

4.1.7 Quality assurance checks: Throughout the calibration processes, several quality assurance checks

are performed. The checks are performed to identify problems before they become evident as bad

calibration results. For example, the optical source’s power stability is verified before any calibrations

are performed. The processes used in the calibration are documented in the High-Accuracy Calibration

System Operating Instructions [14]; strict adherence to the processes prevents missed steps that can add

unaccounted uncertainty to the calibration results.

In one of the more visible quality-assurance checks, a trusted transfer standard or “check” standard is

usually calibrated alongside every test detector calibration. If the check standard’s calibration changes, a

problem with the calibration system is indicated. Since the calibration system has the ability to calibrate

more than one detector at a time, calibrating the check standard alongside the customer’s test detector

consumes little additional time and resources.

In addition to the check standard, there are many other checks that occur during the calibration because

previously obtained results should be reproduced. If unusual results are obtained, a potential problem

with the calibration system is indicated. These system consistency checks are in some ways more useful

than the result of the check standard, because they can indicate a problem earlier in the calibration

process, while the entire calibration must be completed before the check standard’s result is known.

One example of a system consistency check is the electrical calibrations performed. In these checks, a

result consistent with previously obtained results is expected; if unusual results are obtained, a problem

with the system is indicated. For example, the absolute accuracy of the LOCR’s electronics is measured

during each calibration under the same environmental conditions that are present during the calibration. 

The calibration system’s DMM is the standard for all electrical measurements, it is electrically calibrated

within 90 days of the detector calibration. The electrical calibration factors for both devices should be

consistent, unless there is a reason for a change, such as a relatively large change in the environment or a

long time span between calibrations. Other similar consistency checks are performed during the

calibration, such as when the optical system’s stability is assessed, when the beam’s relative aperture

transmittance is measured, and when the LOCR’s uncertainties are assessed.

In perhaps the most visible and reassuring check, the system’s absolute performance is empirically tested

through comparison with other high-accuracy devices, usually other cryogenic radiometers. To date,

transfer standards have been used to compare the LOCR to the HACR cryogenic radiometer at 488, 515,

and 633 nm [16]; and the C-series calorimeter at 633 and 1550 nm [6]. Such comparisons will be

performed periodically.

When previously unaccounted uncertainties are identified, they are added to the error budget as soon as

possible. For example, a recent publication [17] showed that the spectral response of the optical receiver
37

in a device similar to the LOCR was not flat. Theoretical analysis based on the reflection characteristics

of the paint and the geometry of the receiver concluded that the spectral response should be essentially

flat from visible to near infrared wavelengths, so the optical receiver’s absorption was assumed to be

spectrally flat. But the recently measured results show that the spectral response is not as flat. Until

proven otherwise (the published variation may be the result of unaccounted measurement uncertainty), or

additional measurements are available, the published results will be used to reassess the uncertainty due to

spectral response in the LOCR. Similarly, after discovery that the calibration system’s beam contains

some non-Gaussian components, the relative aperture transmittance correction was added to better assess

the resulting uncertainty. And most recently, significant spatial non-uniformity in the LOCR’s optical

absorber was measured and accounted for. The uncertainty analysis is described in Section 5. While

including these uncertainties may make the system’s overall calibration uncertainty somewhat larger than

that advertised for other laboratories, it better estimates the true uncertainty in the calibration results.

In part because of the quality assurance checks, the final calibration factor and uncertainty cannot be

determined until after the calibrations are completed and all the measured corrections and uncertainties

are tallied. Fixed values for measured uncertainties that prove to be very stable with time may eventually

be used after a sufficiently large calibration history is obtained, but doing so would bypass the quality

assurance check performed when the parameter is remeasured for each calibration. Currently, using

typical values for the measured uncertainties gives a preliminary result with an uncertainty that is usually

less than 0.1 %, which is sufficient to help determine whether the system is functioning properly while the

calibrations are being performed.

To help ensure the accuracy of the calibration result, conservative uncertainty estimates are used when

there is doubt. For example, the LOCR window transmittance measurement is performed extremely

conservatively. Instead of measuring the transmittance at one location on the window one time, a matrix

of locations both before and after the calibrations is measured, to account for uncertainty in the window’s

alignment and changes in its transmittance that can occur during the calibration. If the window

transmittance changes significantly, the uncertainty of the calibration results will be increased

correspondingly, so the calibrations can be delivered with an uncertainty that is larger than expected, or

the calibrations can be repeated.

4.2 LOCR Operation

Operation of the LOCR primary standard is described in great detail in the manufacturer’s operating

instructions [13] and in the operating instructions for the high-accuracy calibration system [14], so is only

summarized here. Operation of the LOCR must be performed with great care, because mistakes or

missed steps can result in permanent damage to the device or injury to the operator.

4.2.1 Vacuum and cryogenic systems: The cryogenic liquids needed for LOCR operation present several

safety hazards. Oxygen (O2) depletion is probably the greatest hazard for the operator, so the O2 level

must be monitored. The O2 concentration of the lab air is monitored with a commercial O2 sensor that

displays the concentration and emits audible and visible alarms if the concentration drops to hazardously

low levels. The O2 depletion occurs because whenever cryogenic liquids are exposed to the relatively

warm atmosphere and equipment, the liquid boils and becomes a gas. The large amounts of N2 gas

released can easily displace enough of the O2 in the room to cause problems for the human operator

unless the air in the room is constantly replaced in an air exchange. In the high-accuracy calibration

system, the room air exchange is provided by a fan that blows the room air through a vent in the wall to

the outside of the building. To aid in the removal of the unwanted gasses from the laboratory, the air that

is exhausted is drawn from a vent located directly above the LOCR. Latex hoses are used to help direct
38

the gas to the vent, where appropriate. The ventilation fan’s speed is controlled from inside the laboratory

so that the air exchange rate can be increased when needed and reduced when not needed. 

The He gas causes less displacement of O2, because there is usually much less He gas released and the He

gas tends to escape from the room relatively quickly. However, even a small concentration of He in the

room’s air can change the air’s index of refraction. The air’s index is calculated from measurements of

the air pressure, temperature, and relative humidity; and the calculation assumes typical values for

concentrations of the gasses found in Earth’s atmosphere. The presence of excessive He or N2 gas in the

room air could therefore invalidate the calculation of the index of refraction. Fortunately, the location of

the vent directly above the LOCR helps eliminate most of the gasses released and the room air exchange

reduces the remaining excesses to negligible levels.

The cryogenic liquids present hazards for the operator in addition to the O2 depletion hazard. For

example, the operator can be burned or temporarily blinded if any cryogenic liquid contacts the skin or

eyes. To prevent these potential injuries, insulated cryogenic gloves and goggles are worn whenever the

cryogenic liquids are handled. Quality cryogenic gloves are used, which consist of a waterproof shell and

an insulated liner. The shell allows the gloves to withstand immersion in the cryogenic liquid for a short

time, while the insulated liner protects the skin from freezing. To protect the operator’s arms, a long

sleeve shirt or laboratory jacket is worn when handling the cryogenic liquids. Safety goggles that

completely cover the operator’s eyes are worn to prevent any splashed liquid from reaching the eyes.

The equipment itself can also be damaged by the cryogenic liquids. When the liquid nitrogen first

contacts the relatively warm metal, it boils furiously and small amounts of liquid can be sprayed around

the area. Also when the LOCR’s liquid-nitrogen reservoir is filled, there is usually some overfill that can

spill onto the detector breadboard. To protect the test detectors and other equipment located on detector

breadboard, a thin metal shield is used to deflect any sprayed liquid away from the equipment. The liquid

helium is a lesser problem because it boils so quickly that liquid helium is not observed outside of the

cryogenic containers.

The greatest danger to the equipment results from the tendency for ice to form on the cold surfaces. The

cryogenic liquids produce temperatures that are cold enough to condense atmospheric water vapor (and

other atmospheric constituents). The worst possible scenario is for an ice plug to form in the LOCR’s

cryogenic reservoir fill ports. If the ports are clogged, the pressure inside the reservoirs will buildup until

the ice plug is ejected, or the reservoir itself ruptures. There are no pressure relief valves on the LOCR’s

cryogen reservoirs. However, the LOCR’s cryostat was designed to minimize the potential for ice plugs

to form.

As shown in Figure 8, the LOCR’s liquid-nitrogen reservoir shields the liquid-helium reservoir from the

warm sides of the cryostat. Because of the reservoir’s exposure to the warm cryostat, the liquid-nitrogen

boils vigorously, even after the reservoir reaches its steady-state temperature. The rapid rate of boiling

produces a steady stream of N2 gas that helps to keep outside air from entering the reservoir and freezing. 

Additionally, the LOCR has two fill ports connected to its liquid-nitrogen reservoir, which provide

redundant paths for venting; if one port freezes, the rate of gas flow through the second port increases,

thereby lowering the risk of ice forming in the second port. In practice, latex tubing is placed in the

liquid-nitrogen reservoir’s fill ports to direct the escaping gas to the vent hood. To date, no icing inside

the latex tubes that vent the N2 gas has been observed, though ice does accumulate on the outside of the

tubing. However to ensure that a problem does not occur, the N2 gas vent hoses are inspected at least

twice a day, when the liquid-nitrogen reservoir is refilled. 
39

The liquid-helium reservoir is significantly different from the liquid-nitrogen reservoir and presents a

greater danger. The primary difference is that there is only one fill port on the liquid-helium reservoir and

its port is vacuum insulated so it is very cold. Also since the liquid-helium reservoir is well shielded by

the liquid-nitrogen reservoir and its heat shields, relatively little He gas is boiled off once the reservoir

reaches the temperature of liquid helium, so there is little flow of gas through the fill port. To prevent ice

from forming inside the liquid-helium fill port, a device called a vapor barrier is inserted into the port. 

The vapor barrier consists of three metal barriers that act as heat shields between the cold liquid helium

and the room-temperature atmosphere, as well as physical barriers to gas flow. The vapor barrier fits

underneath the liquid-helium reservoir’s cap, but the barrier must be removed during cryogenic liquid

fills. After filling the reservoir with liquid helium, the cap must be removed, the vapor barrier inserted,

and the cap replaced. Failure to insert the vapor barrier after a fill would very likely cause an ice plug to

form. 

The He gas produced by the boiling liquid vents through a horizontal fitting in the liquid-helium

reservoir’s cap, located above the vapor barrier. The fitting is designed so that a vacuum pump can be

attached and the pressure in the reservoir lowered, to lower the temperature of the liquid helium. In an

experiment, a length of latex hose was attached to the fitting to direct the escaping He gas to the vent

hood above, as is done with the N2 gas. While this technique is successful with the cold N2 gas which is

heavier than air, when used with the lighter-than-air He gas, the formation of an ice plug on the vapor

barrier is enhanced by the hose. Therefore a latex exhaust hose is not used on the He gas vent. However

even without a hose attached, the potential for formation of ice on the vapor barrier still exists, so the

liquid-helium fill port is inspected for the presence of ice at least twice a day. Also, ice formation in the

liquid-helium reservoir can be inferred by monitoring the rate of liquid-helium consumption; if ice is

being formed, the liquid helium will likely be consumed at a greatly accelerated rate and the liquidhelium temperature will increase if the pressure in the reservoir increases.

The storage container for liquid helium also requires special attention. Like the LOCR’s He reservoir, the

storage container also has only one fill port, which presents a similar icing hazard. The container is safe

when its fill port is properly closed and capped, but the port must be opened and uncapped to insert the

liquid-helium transfer line. While the port is open there is a danger of ice buildup and if an ice plug

forms, the pressure in the container can rise to dangerous levels. Fortunately the storage container has

several safety features which should prevent an explosion even if an ice plug does form, but the container

may still be damaged and the backup pressure reliefs should not be relied on. Instead any ice plug that

forms should be removed immediately. A long copper rod about 1 cm in diameter is used to remove the

ice plug; the rod is inserted into the fill port and used to push the ice plug into the container. The best

policy is to ensure that the container is properly capped when not in use and is uncapped for only a short

time, when necessary.

In addition to the icing that can occur inside the LOCR’s liquid cryogen fill ports, there is also an issue

with ice forming on the outside of the cryostat. The top of the cryostat tends to accumulate ice during

liquid-nitrogen fills. If the cryostat is allowed to freeze, its o-ring vacuum seal could also freeze and

would then likely rupture. The resulting inrush of air and subsequent ice formation would likely destroy

the LOCR, so care must be taken to ensure that o-ring seal does not freeze. To prevent such icing, a

heater designed to prevent water pipes from freezing is wrapped around the top of the cryostat. The

heater has an internal thermostat so should operate automatically, but it should be inspected to ensure that

it is operating correctly. As a backup, a commercial hot air gun is used to manually remove any ice

buildup. The hot air gun is also usually needed to warm the liquid-helium reservoir’s cap so that it can be

removed for fills, since the cap tends to become stuck after being exposed to the liquid-helium vapor for a

time.
40

Care must also be exercised when using the vacuum system, or damage can result. For example, if a

vacuum seal or a valve is opened at the wrong time, a inrush of air can occur, which would likely destroy

the LOCR’s fragile optical receiver. Also, the cleanliness of the turbo pump and its associated plumbing

must be maintained. To help prevent contamination, the vacuum system components are always handled

with gloved hands, and the pump is always vented with dry nitrogen gas after use. These and other

maintenance functions are discussed in greater detail in Section 4.2.6.

The first step in using the LOCR is to evacuate the cryostat and window assembly, using the

turbomolecular pump station. When the pressure in the LOCR’s cryostat drops below about 0.005 Pa, the

ion pump is started. The ion pump will be running hot at this pressure, with an unusually high operating

current. The heat will make the ion pump outgas, which will drive the pressure back up, which will make

the pump run even hotter, etc. in a positive feedback loop which will eventually result in destruction of

the ion pump or its power supply.

The ion pump power supply has a safety feature that shuts the pump off if it draws more current than a

preset limit. Unfortunately, the safety feature cannot be used while the pump is starting, because during

startup it draws much more current than the preset limit, so the safety feature must be disabled during

startup. After the pump is running with a stable pressure, the current will drop low enough to reenable the

safety feature. Then the safety feature is active for the duration of LOCR operation.

To prevent damaging the ion pump while the safety feature is disabled, the operator monitors the

pressure, and if the pressure rises to about 0.013 Pa, the ion pump is shut off and allowed to cool, while

the turbo pump removes the gasses that were released. When the pressure is reduced to about 0.005 Pa

again, the ion pump is restarted and the process is repeated. This cycle is typically repeated a few times,

until the ion pump can maintain the pressure without outgassing. Once the ion pump is running with a

stable pressure, the valve to the turbo pump is closed, the turbo pump is removed, and the LOCR’s liquidnitrogen and liquid-helium reservoirs are filled with liquid nitrogen. Introduction of the cryogenic liquids

will further reduce the pressure in the cryostat and take the load off the ion pump. 

The liquid nitrogen is transferred from the storage container to the LOCR using a flexible, insulated

transfer line, described in Section 3.1.4. The liquid-nitrogen reservoir is filled first. Before filling the

reservoir, the latex tubing is removed from the liquid-nitrogen fill ports. Then the liquid-nitrogen phase

separator at the end of the flexible liquid-nitrogen transfer line is inserted into one of the ports. A special

clear plastic pipe is placed over the unused fill port; the pipe helps direct the escaping gas to the vent

hood, while allowing visual inspection of the port. The flow of liquid-nitrogen is then started, the valve

on the liquid-nitrogen storage container is used to control the rate of liquid flow. When liquid is observed

overflowing from the exhaust port, the reservoir is full so the valve is closed to stop the flow. Then the

pipe and phase separator are removed and the latex tubing replaced.

The LOCR’s liquid-helium reservoir is pre-cooled using liquid nitrogen, so that less liquid helium is

consumed during the liquid-helium fill. First the reservoir cap and vapor barrier are removed. Then an

extension tube is inserted into the liquid-helium reservoir and the liquid-nitrogen phase separator is

inserted into the tube. The extension tube almost reaches the bottom of the reservoir, it is held off the

bottom by a metal clip which hooks onto the top of the fill port. Then the flow of liquid nitrogen is begun

and continues until excess liquid is observed flowing from the top of the fill port. When the fill is

completed, the vapor phase separator and extension tube are removed from the reservoir. At this time, the

reservoir cap is not reinstalled, so that the liquid nitrogen in the reservoir can be observed visually.

When the liquid nitrogen is first introduced to the LOCR’s warm cryogen reservoirs, the liquid will boil

furiously. After some period of time, about 30 minutes or so, the LOCR’s reservoirs are precooled to
41

liquid-nitrogen temperature and the liquid nitrogen’s rate of boiling will slow. When fully precooled, the

liquid nitrogen inside the well-insulated liquid-helium reservoir will practically stop boiling entirely. 

Then the liquid nitrogen can be blown out of the liquid-helium reservoir, so that the liquid helium can be

introduced.

To remove the liquid nitrogen from the liquid-helium reservoir, the liquid nitrogen is blown out of the

liquid-helium reservoir and into the liquid-nitrogen reservoir. A special liquid-nitrogen blow pipe is used

to remove the liquid. To use the blow tube, the reservoir cap is installed on the liquid-helium reservoir,

without the vapor barrier in place. Then the blow pipe is inserted into the cap, until it contacts the bottom

of the reservoir. The blow pipe’s end is cut at a slight angle and notched so that there is a opening for the

liquid to flow through. On the top of the blow pipe is a short length of latex tubing, which is inserted into

one of the liquid-nitrogen reservoir’s fill ports. Another length of latex tubing connects the vent on the

liquid-helium reservoir’s cap to the calibration system’s gas valves, shown in Figure 4, which are used to

control the flow of gas. Then gaseous N2 is blown into the reservoir through the tube, and the pressure

drives the liquid nitrogen up the blow pipe and into the liquid-nitrogen reservoir. Dry N2 gas is usually

used to blow out the liquid nitrogen, but He gas could also be used. It is very important that all the liquid

nitrogen be removed, or excessive amounts of liquid helium will be consumed as the remaining liquid

nitrogen is frozen solid. To ensure that all the liquid nitrogen is removed, a felt pad is inserted into the

liquid-helium reservoir to physically mopup any liquid remaining on the bottom of the reservoir. Then

the LOCR’s liquid-helium reservoir can be filled with liquid helium. It is important that these steps be

performed quickly, so that the liquid-helium reservoir does not warm up again after the liquid nitrogen is

removed. There should be enough room in the liquid-nitrogen reservoir to hold all of the liquid that is

blown out of the liquid-helium reservoir; if not, the excess is discarded.

The liquid helium is transferred to the LOCR using a rigid, vacuum-insulated transfer line. To control the

rate of flow of the liquid-helium through the transfer line, the liquid-helium storage container is

pressurized with He gas. A special adapter is used to connect the calibration system’s gas valves to the

liquid-helium container and the system’s gas valves are used to control the flow of gas. At first, the

warmth of the liquid-helium transfer line will vaporize the liquid-helium inside the line, so only He gas

will flow. The transfer line is not inserted into the LOCR until liquid begins to flow through the transfer

line, which may take a few minutes. Then the transfer line is inserted into the LOCR’s liquid-helium

reservoir and the reservoir is filled. The amount of liquid helium consumed in the filling process is

directly dependent on the operator’s skill, a sloppy fill will consume much more liquid helium than a

properly performed fill. The most efficient transfer occurs with a certain rate of flow of liquid helium

through the transfer line; either too fast or too slow a flow can result in excessive consumption of liquid

helium. So the operator must carefully monitor the rate of liquid-helium flow, by observing the plume

that occurs at the liquid-helium reservoir’s vent, and control the flow rate by the application of gas

pressure. The fullness of the liquid-helium reservoir can be determined by observing the exhaust plume,

or with the liquid-helium level sensor that is built into the reservoir’s cap. The level sensor provides a

direct reading of the amount of liquid-helium in the reservoir, but an experienced operator can identify

when liquid is being ejected from the vent port by the look of the exhaust plume, which becomes denser

and more feathery when containing liquid helium. After the reservoir is filled, the liquid-helium transfer

line and reservoir cap are removed. Then the vapor barrier is inserted into the cap and the cap is slowly

lowered onto the reservoir’s fill port.

Once the liquid-helium reservoir is filled with liquid helium, the LOCR’s internal temperature will

quickly drop to near liquid-helium temperature and the LOCR’s temperature-control electronics will

begin to function. The electronics will not function at higher temperatures, primarily because the wire

leads do not become superconducting until their temperature is well below liquid-nitrogen temperature.
42

4.2.2 Alignment and spatial uniformity measurement: The first step in aligning the LOCR is to center the

laser beam in its optical receiver. The LOCR is first coarsely aligned so that at least part of the laser

beam enters the optical receiver. The LOCR alignment target can be used for the coarse alignment, if the

target was previously aligned as described below, otherwise the LOCR is coarsely aligned. After at least

part of the beam enters the LOCR’s receiver, the LOCR is translated horizontally, using the motioncontrolled platform, and the optical power is measured and recorded at discrete locations along a

horizontal line that passes through the receiver. The acquired powers are then normalized to the

maximum and the 50 % points are located. The center of the receiver is then interpolated from the 50 %

points, and the LOCR is translated so that the beam is centered horizontally. This process is repeated to

center the beam vertically in the receiver, but instead of translating the LOCR vertically, the height of the

optical source breadboard is adjusted. Then the horizontal center is located again, to determine whether

the source breadboard shifted during vertical adjustment, and to measure the apparent receiver diameter. 

The distance between the 50 % points defines LOCR’s apparent aperture, which is approximately an

ellipse with a diameter of 9.5 mm and a height of 8.5 mm. When the alignment is completed, the window

is adjusted to Brewster’s angle and the laser’s incidence angle relative to the body of the LOCR is

adjusted.

To adjust the window to Brewster’s angle for a particular wavelength of light, the window mount’s

gimbals are adjusted so that the intensity of the reflection from the window’s first surface is nulled. The

null is determined visually, for wavelengths that are visible, using a white target. The intensity of the

beam on the target is minimized; if the beam is properly polarized and the window is good, the reflection

will be completely nulled. With infrared wavelengths, a single-element germanium detector is used to

measure the reflected power. The reflection off the back side of the window is not used, because its

intensity can change when vacuum is applied; and when in air, its minimum can occur at an angle 

different from that of the first surface’s, because of the window’s nonzero wedge angle. The change in

transmittance that occurs when vacuum is applied to the back of the window is accounted for in the

uncertainty analysis of window transmittance. The relatively thick window used with the LOCR ensures

that the reflections from the front and back of the window are well separated.

The incidence angle of the laser light entering the LOCR is important, because if this angle is not correct,

the beam’s extent can be limited by apertures other than the LOCR’s final aperture. To adjust the

incidence angle, first the optical receiver is aligned and the window is adjusted to Brewster’s angle, then

the location of the beam on the window mount’s dust cover is observed. The beam should strike the dust

cover on its vertical centerline and approximately 4 mm below its horizontal centerline. This vertical

offset is caused by displacement of the beam by the window, as shown in Figure 15. A graphical analysis

was performed for the figure using Snell’s law [18], using the index of refraction for fused silica at a

wavelength of 633 nm [19], and the window at Brewster’s angle in air. The index for fused silica is

adequate because quartz is crystalline silica. Fused silica’s index varies with wavelength, from 1.47 at

400 nm to 1.44 at 2,000 nm, and the change in Brewster’s angle for the different wavelengths does not

significantly change the offset by more than about 1 mm. An earlier prototype of the NIST window

mount is shown in the figure, but the differences from the final version do not affect the offset

calculation. However, the offset can change if a window with a different thickness is used.

To force the incident beam to strike the desired vertical location on the window’s dust cover, the pitch of

the LOCR is adjusted by placing metal shims under the LOCR’s bridge electronics enclosure. A pitch

angle of approximately 7 mrad is required in the current setup. The pitch is necessary because of

imperfections in the welds and vacuum seals used in the long string of vacuum components that connect

the window assembly to the cryostat (shown in Figure 7), and imperfections in the polarization angle and

window mount. Similarly, the horizontal location of the beam on the dust cover is set by adjusting the

yaw of the LOCR. Since the adjustments of pitch and yaw angle will likely change where the beam
43

Figure 15. The NIST Brewster’s angle window mount, showing the vertical offset of the

incident laser beam.

strikes the optical receiver, the LOCR’s receiver must be realigned after the angles are adjusted, and then

the window Brewster’s angle and the LOCR’s incidence angle readjusted as necessary, in an iterative

process. Fortunately, the angle at which the beam is launched from the optical source is very

reproducible, so the adjustment of incidence angle has to be performed only once, whenever the LOCR’s

location on the detector platform is changed.

The resulting incidence angle is double-checked during the window transmittance measurement by

observing where the transmitted beam is inside the window mount’s vacuum flange; the transmitted beam

should be at the center of the flange. Also, if the incidence angle is wrong, the window transmittance at

some of the points in the nine-point transmittance matrix may differ significantly from the others, because

of partial clipping of the beam by the apertures in the window mount.

One unfortunate result of the necessary adjustments of pitch and yaw is that the center of the cryogenic

apertures inside the LOCR, shown in Figure 8, are driven out-of-line with the beam’s propagation axis. 

Apertures other than the final aperture therefore partially limit the extent of the beam, so the final aperture

is not necessarily the limiting aperture. One result is that the LOCR’s apparent aperture is slightly smaller

than its final aperture. Since the pitch adjustment is more severe than the yaw adjustment, the vertical

width of the apparent aperture is reduced more than the horizontal width, and the LOCR’s apparent

aperture is therefore approximately elliptical. The error caused by this misalignment is accounted for in

the uncertainty of the relative aperture transmittance correction, since conservative bounding circular

apertures are used in the LOCR’s relative aperture transmittance calculation; bounding apertures of 8 and

10 mm are used when the LOCR’s apparent aperture measures 8.5 x 9.5 mm, as described in Section

4.1.5.

Error in matching the beam’s polarization angle to the orientation of the slope of the Brewster’s angle

window can result in the beam being centered on the window mount’s dust cover, but not in the mount’s

vacuum flange. This difficulty arises because the window mount must be driven significantly off the
44

LOCR’s optical axis when its gimbal mount is used to effect a roll. The window mount does not have a

roll-angle adjustment; its roll angle is set when the mount is bolted together. To reduce this problem, the

slope of the window, when its gimbal mount’s adjustments are in their neutral position, must match the

beam’s polarization angle (i.e., a line that is normal to the window must lie in the plane of the beam’s

polarization). Since the rotational angle of the window mount is fixed, it is easiest to adjust the beam’s

polarization axis by adjusting the roll angle of Polarizer 2. However, the polarizer’s roll can be adjusted

by only a small amount or the dynamic range of the power stabilizer will be significantly reduced, so the

window mount must be bolted together accurately. During assembly of the window mount, a precision

level is used to accurately set the window mount’s orientation, using the flat areas machined into the top

and sides of the mount. The flat on the side of the mount is visible in Figure 12.

Once the LOCR is properly aligned, the device’s orientation is locked down with four bolts that connect

the LOCR to its platform. Then the LOCR’s alignment target is aligned to the beam, after translating the

LOCR horizontally a known amount (typically 10 cm), using the motion-controlled detector breadboard. 

Then if the radiometer’s alignment is subsequently lost, the alignment target can be used to rapidly realign the device.

The alignment of the LOCR is performed carefully, but there is always some uncertainty in the resulting

location of the laser beam in the optical receiver. If the receiver’s absorptance has some spatial

nonuniformity, the uncertainty in location results in an uncertainty in the absorbed power. To assess this

uncertainty, measurements of the spatial uniformity of the LOCR’s optical absorber are performed after

the LOCR is fully aligned. The spatial uniformity is estimated by measuring the power in a stable beam,

with the beam located at slightly different locations around the receiver’s center. After the influence of

random power variation, window transmittance and aperture transmittance are removed, the remaining

variation in the measured power is due to nonuniformity in the LOCR’s optical receiver’s absorptance.

Currently, the LOCR receiver’s spatial uniformity is estimated by using a small array of samples taken in

a horizontal line through the receiver’s nominal center. The array is assumed to be a random sampling of

the uniformity around the receiver’s center, so is analyzed using standard statistical techniques. The

current measurement technique and uncertainty analysis is discussed in detail in Appendix G. The

measurement technique will likely be improved in the future, primarily by measuring a matrix of points

instead of a linear array. Results obtained to date show that there is significant nonuniformity in the

receiver’s absorption, which appears to be worse when light with shorter wavelengths is used. The

nonuniformity is also likely a function of beam diameter, with larger beams sensing less nonuniformity

than smaller beams, because the larger beam integrates the responsivity over a larger area than the smaller

beam. Because of the dependance on beam profile and wavelength, the LOCR’s nonuniformity is

measured each time the beam size or wavelength is changed.

4.2.3 Electrical calibration: The LOCR is electrically calibrated once during each optical calibration, in

an environment similar to that encountered during the optical calibration. What is calibrated is the

amount of heating power that is applied to the LOCR receiver’s electrical heater. In the electrical

calibration, the error in the LOCR’s electronics is measured at the power levels that are used in the optical

calibration. The measured error in the amount of power delivered is corrected for in post-processing.

Currently, software provided by the LOCR’s manufacturer is used to perform the electrical calibration. 

The manufacturer provided two programs, “ACRCAL.EXE” and “TCALVER.EXE”. The software’s

operating instructions are well documented in the manufacturer’s Operating Instructions [20], so will not

be repeated in detail here. Both programs measure the error in the LOCR’s electronics; the main

difference is that “ACRCAL.EXE” updates the electronics to remove the error, while “TCALVER.EXE”

only measures the error. However, the program “ACRCAL.EXE” is not usually used, because it is very
45

time-consuming; instead “TCALVER.EXE” is used to measure the error at only the power levels that are

actually used in the optical calibration. While using “TCALVER.EXE” adds complexity to the postprocessing in the form of an electrical calibration correction factor, the correction factor would be

required in any case because the system DMM itself also has measurement error that must be corrected.

Both electrical calibration programs use the system DMM to measure the power applied by the LOCR’s

electronics, as described in Section 3.2.3. In addition to the error in the applied power that is measured

during the LOCR electrical calibration, there is also error in the DMM’s voltage readings themselves, and

there is error in the DMM’s measurement of the LOCR’s standard resistor. These errors are all corrected

at the same time, in post-processing when the LOCR electrical calibration correction factor is generated. 

How the LOCR’s electrical calibration correction factor is used in the optical calibration is discussed in

Section 4.3.3, and a detailed description of how the correction is calculated is given in Appendix F. Also

described in the appendix is a similar correction that is performed for test detectors that utilize the

system’s DMM to measure their outputs.

The LOCR electrical calibration must be performed while the LOCR is fully operational. Therefore the

LOCR must be at operating temperature and the window assembly must be in place with the gate valve

open. The LOCR and its electronics must also be allowed to reach their steady-state temperature for the

given environmental conditions and the environmental conditions should be the same as those present

during the optical calibrations. Usually the electrical calibration is performed after the optical

calibrations, but the electrical calibration can be performed at any time during the calibration process

when the above conditions are present and stable.

Performing the electrical calibration requires connecting the system DMM to the LOCR electronics and

connecting the LOCR Computer to the DMM’s GPIB interface. Then the calibration program is executed

on the LOCR Computer and the operator follows the on-screen prompts. The operator is required to

operate manual switches on the LOCR electronics during the calibration, so the process cannot be

automated. These programs will have to be implemented on the main process-control computer before

the LOCR computer can be eliminated from the system.

4.2.4 Window transmittance measurement: LOCR window transmittance is measured extremely

conservatively. Instead of measuring the transmittance at one location on the window one time, we

measure a matrix of locations both before and after the calibrations, to account for uncertainty in the

window’s alignment and changes in its transmittance that can occur during the calibration. If the window

transmittance changes significantly during the calibration, the uncertainty of the calibration results will be

increased correspondingly, so the calibration can either be delivered with an accurate but larger

uncertainty, or the calibration can be repeated.

The window assembly must be removed from the LOCR for transmittance measurement. A large gate

valve isolates the window assembly from the body of the LOCR; the valve allows the assembly to be

removed while there is a vacuum in the LOCR. A small pumping port and valve are located between the

gate valve and the window assembly, so that the assembly can be evacuated when reattached to the

LOCR, before the gate valve is opened. When the window assembly is placed back on the LOCR after

transmittance measurement, it is first attached and then evacuated with the turbomolecular pump, before

the gate valve is opened again. The pressure inside the window assembly must be reduced using the turbo

pump, so that a inrush of air does not occur when the gate valve is opened. A pressure of 0.005 Pa or less

should be attained before the gate valve is opened.

The LOCR must first be aligned to the laser beam and the window Brewster’s angle adjusted, before the

window assembly is removed from the LOCR for transmittance measurement. Then the dust cover is
46

placed over the window mount and rotated to its keyed position. A strip of adhesive paper is attached to

the front of the dust cover, the precise location where the laser beam strikes the cover is then marked on

the paper. When the assembly is removed from the LOCR for transmittance measurement, the laser is

aligned to the location marked on the paper. The transmittance is measured in a nine-point rectangular

array around the nominal beam location.

When the window is reattached to the LOCR, the laser beam should strike the same location, but

imperfections in the window assembly and rotation of the vacuum flange when it is tightened can change

the location by a small amount. The change in location change is minimized by using the same o-ring

and clamp, in the same orientation that they were in before the window assembly was removed. To

compensate for any remaining change in location, the transmittance is measured both before and after the

LOCR is used for calibrations, in an initial and final measurement. For the final measurement, the

location of the beam on the dust cover is again marked, so that if the location of the beam moved due to

the imperfections in the mount, the nominal center location is more accurate in the final measurement

than in the initial measurement. The measurements are combined to obtain the transmittance value used in

the calibration, and a conservative uncertainty is derived. How the measurements are combined is

described in Appendix H. The paper target documents any shift in location of the beam on the window.

4.2.5 Data acquisition: In the main calibration program that runs on the Process Control Computer, the

LOCR is treated as a special direct-reading detector, that is manually operated using the LOCR

Computer. Data are transferred from the LOCR Computer to the Process Control Computer using a

floppy disk. The LOCR is operated using software called “ACR.EXE”, provided by its manufacturer. 

The manufacturer’s software will eventually be replaced by software in the main calibration program,

allowing fully automated operation of the LOCR, as described in Section 3.3. Operation of the

manufacturer’s software is described in the User’s Manual provided by the manufacturer [20], so will not

be described in detail here.

The operator must enter several parameters into the “ACR.EXE” program, the most significant of which

are the operating point and the control loop parameters. The operating point is defined by the desired

temperatures of the heat sink and optical receiver. As previously described, the LOCR is normally

operated with heat sink and receiver powers of 2 mW each. The regulation temperatures are chosen so

that the desired power is delivered. The absolute heat sink power may be varied slightly to adjust the

resulting receiver temperature, but a nominal power of 2 mW is always delivered to the receiver for the

purpose of electrical calibration, as described in Sections 3.2.3 and 4.2.3.

There are three preset sets of control-loop parameters that were determined by the manufacturer and are

called “fast,” “medium,” and “slow.” The “slow” controller parameters require the longest amount of

time for the applied power to reach steady-state, but then the power readings have the lowest noise. With

the “fast” parameters, the steady-state power is reached more quickly, but the subsequent power readings

are more noisy. The “medium” parameters are a compromise between the “fast” and “slow” parameters. 

These parameters can be modified by the operator, but to date only the preset values only have been used.

Which control loop parameters are used with the optical receiver’s temperature controller depends

primarily on the applied optical power. A large optical power requires a large change in the electrical

power applied to the optical receiver, while a relatively small optical power requires little change in the

receiver’s electrical power. The period from when a change in the optical power occurs to when the

applied electrical power stabilizes is called the settling period. In general, a settling period of 40 s is

desired. This settling period provides an adequate balance between how much time the measurement

takes and how much the absolute optical power drifts between measurements. When a relatively large

optical power is applied, the electrical power that is applied to the optical receiver must change
47

significantly. For example, to measure 1 mW of optical power, the electrical receiver power must drop

from 2.0 to 1.0 mW; a change of 50 %. Similarly, when a relatively small optical power is applied, the

electrical power has to change relatively little. For example, to measure 0.1 mW of optical power, the

electrical power changes from 2.0 to 1.9 mW; a change of only 5 %.

When the change in electrical power is small, the applied electrical power stabilizes within the desired

settling period even when the “slow” controller parameters are used, and the subsequent power readings

have the lowest possible noise. However when the change in electrical power is large, the electrical

power will take more than the desired settling period to stabilize when the “slow” parameters are used, so

the “fast” parameters are used instead, at the expense of increased noise in the measurement of electrical

power. The increased noise due to using the “fast” settings with large optical powers is not a problem

because while the noise is increased, the signal is also increased, so roughly the same signal-to-noise ratio

is obtained with both power levels. The “medium” controller parameters are used with intermediate

optical powers as necessary, the parameters used are determined experimentally at the time of the

calibration by applying the optical power in a test run and observing the resulting settling period.

The temperature-controller parameters used with the heat sink’s temperature controller must also be

selected. Limited experimentation with different heat-sink parameters has been performed with the

general conclusion that the system works adequately when the same parameters are used with both

controllers. So when the receiver is used with its “fast” parameters, the heat sink is also used with its

“fast” parameters. Note that the actual values of the controller parameters, determined by the

manufacturer, are different for the two temperature controllers because the heat sink’s thermal response is

different from the receiver’s.

One additional parameter must be adjusted by the operator at the time of the optical calibration. This

parameter, the phase delay, is adjusted in real time while observing the AC bridge signals with an

oscilloscope. The bridge phase is adjusted so that the AC component in the oscilloscope trace is

minimized. The phase delay of both the receiver and heat-sink temperature controllers must be adjusted;

the adjustment is stable so long as the environmental conditions are stable. Several other parameters are

available, such as the AC bridge frequency, but the default values for these parameters work adequately.

When optical power is measured, the ACR.EXE program requires the operator to initiate the

measurement, and to open and close the shutter at the appropriate times. The main Process Control 

Computer also requires operator input at these times; it takes the file names and keeps track of the timing

to coordinate its operation with that of the LOCR Computer. After a set of optical calibrations are

performed, the operator transfers the acquired data from the LOCR Computer to the Process Control

Computer using a floppy disk, and the Process Control Computer converts the data to its internal format. 

The acquired data are stored in spreadsheet format along with the data for the other detectors and sensors,

and preliminary calibration results are calculated.

The LOCR data are analyzed with the dual-baseline technique which is described later, in Section 4.3.1. 

Basically the technique requires acquisition of the detector’s baseline before the detector is exposed to

optical power, acquisition of data while the optical power is applied, and acquisition of a second baseline

taken after the optical power is removed. The ACR.EXE program averages the baselines internally and

returns only the average value and standard deviation for the baselines, so the actual data acquired during

the baselines are unavailable. Data acquired when the LOCR is exposed to the optical power are however

available. But since the baseline points are unavailable, the LOCR data are stored in the spreadsheet with

a format slightly different from that used for other detectors. The primary difference is that in the

spreadsheet the ACR.EXE program’s average for the baselines is used instead of calculating the baselines

directly from the raw data.
48

The period over which data are sampled after stabilizing is called the rating period. With the LOCR, a

rating period long enough for the acquisition of 30 samples is usually used. Since the ACR.EXE program

samples the LOCR at a rate of approximately once every 1.23 s, the rating period lasts for about 37 s. 

Therefore, 30 samples are used in the calculation of the average baselines and illuminated power level.

4.2.6 LOCR maintenance: Several of the systems needed to support the LOCR require periodic

maintenance. Many of the maintenance steps are necessary to prevent potential damage to the equipment

or operator injury. Some maintenance steps are very important, such as inspection of the liquid-helium

reservoir vent, which is needed to prevent a potentially catastrophic failure. The required maintenance

steps and safety procedures are documented in the High Accuracy Calibration System Operating

Instructions [14], so only the most significant events are covered here. However when the LOCR is not

in operation, it requires practically no maintenance. Shutting down the LOCR on the weekends therefore

eliminates any requirements for weekend maintenance, in addition to providing the other benefits

previously mentioned.

As described in Section 4.2.1, periodic monitoring of the cryogenic and related systems is necessary to

protect the equipment from potential damage from ice buildup. The liquid-helium and liquid-nitrogen

reservoirs’ fill ports must be checked for ice formation at least twice a day. If an ice plug forms in the

liquid-helium reservoir’s fill port and the plug is not removed when the vapor barrier is removed, it is

necessary to push the plug into the reservoir using the copper rod previously described. Ice that forms on

the outside of the ports is removed by hand, but the heat gun can be used if necessary.

In addition to checking the liquid cryogen fill ports for ice, the reservoirs must also be periodically

refilled. The level of liquid nitrogen in the reservoir is checked visually, by looking into one of the

liquid-nitrogen fill ports with a flashlight; the liquid nitrogen looks like boiling water. A liquid-nitrogen

level sensor is available, but it is less convenient than directly inspecting the level by eye. The liquid

helium cannot be seen since it is obscured by a cloud of cold gas, so its level is monitored using the

superconducting level sensor that is integrated into the liquid-helium reservoir’s cap. The gauge is

activated for only short periods because it produces a small amount of heat when active, which consumes

liquid helium. The rate of cryogen consumption is very predictable, so the level checks are performed

primarily to check that the consumption is as expected. Checking the levels at least twice a day also

provides an opportunity to inspect the fill ports for ice.

Refilling the liquid-nitrogen and liquid-helium reservoirs is essentially the same as the initial fill,

described in Section 4.2.1. The main difference is that the liquid-helium reservoir pre-cooling step is not

needed, since the reservoir is already at liquid-helium temperature. As in the initial fill, the top of the

cryostat tends to accumulate ice during the cryogen refill. Therefore the top of the cryostat must be

monitored during the refill and any ice buildup removed. The pipe heater attached to the cryostat should

remove the ice, but additional heat can be applied by using the heat gun as necessary to keep the cryostat

ice-free.

In general, the liquid-nitrogen will be consumed in about 17 hours, so the reservoir is refilled at least

twice a day. The liquid helium lasts two to three days; since the LOCR is shut down on the weekend, this

rate of consumption usually requires only one liquid-helium refill per week. However high room

temperatures, excessive motion, and low levels of liquid nitrogen can cause the liquid helium to be

consumed faster; in which case two refills may be needed during the week to continue operation on

Friday. 

To determine when cryogen storage containers need to be replaced, the amount of liquid remaining in the

containers must be monitored. The liquid-nitrogen containers usually have an integrated level gauge, and
49

the amount of liquid remaining can also be inferred by the container’s weight. However, the liquidhelium container does not have a level sensor, so a device called a thistle tube is usually used to measure

the liquid level. Like the transfer line, the thistle tube is inserted into the container’s fill port. To prevent

a buildup of ice inside the port, the level should be measured as quickly as possible. There is little change

in the liquid-helium container’s weight form full to empty, so its weight cannot be used to judge the

liquid level. But an empty container can be quickly identified by checking its primary pressure relief

valve; if no gas escapes when the valve is held open, the container is empty.

The vacuum systems used with the LOCR also require periodic maintenance. The turbomolecular pump

station is the single most expensive and delicate vacuum component. The turbomolecular pump contains

a turbine that turns at a very high-speed on a bearing that is intended to be periodically replaced. The

bearing, located at the bottom of the turbomolecular pump’s body, contains a pad that is impregnated with

a high-quality lubricant. The manufacturer recommends that pad be replaced periodically because the

lubricant breaks down with use and time and looses its protective properties, damaging the bearing itself. 

The bearing in the calibration system’s station has not yet been changed, but the pump is used only

occasionally and currently has been operated for much less than 2,000 hours. The lubricating oil in the

pump station’s rotary-vane backing pump must also be replaced periodically. The oil in the pump station

has been changed once and the remaining supply of new oil is stored with the pump station. The waste

oil is toxic, so must be disposed of properly. Last, there is an oil-trap filter located on the rotary vane

pump’s exhaust that should also be changed periodically. A spare filter is kept with the station, but the

original has not yet been worn enough to warrant replacement. Located on top of the pumping station is a

log in which the pump station’s usage and maintenance is recorded. Consult the pump station’s

documentation for details about the replacement intervals.

In addition to the manufacturer’s maintenance requirements stated above, care is also taken to maintain

the cleanliness of the vacuum components. To prevent the accumulation of oils on the vacuum lines and

seals, the vacuum components are always handled with gloves and never with bare hands. Care is also

taken to avoid scratching the sealing surfaces. To prevent the buildup of water vapor on the components,

after the pump is used, it is vented with dry nitrogen gas, and the vacuum parts are sealed. Keeping the

pump clean results in shorter pump-down time and a lower initial pressure. Similarly, whenever the

vacuum inside the cryostat is released, the cryostat itself is vented using dry N2 gas. Care must be taken

while venting to prevent pressurizing the cryostat, which could activate its over-pressure burst valve.

To prevent damaging the LOCR’s ion pump, it is important that the pump is turned off before the

cryogenic liquids in reservoirs expire. When the liquids expire, the reservoir temperature will rise very

quickly and the gasses that were cryopumped by the cold reservoirs is released. The resulting sudden

increase in pressure can damage the ion pump. The pump’s power supply has safety features that shut the

pump down before a catastrophic failure, but the electrodes inside the pump will be prematurely worn by

the relatively high pressure, so it is important that the pump be manually shut off before the cryogens

expire.

4.3 Calibration Factor Calculation

Two types of optical detector are supported in the calibration system: power and energy meters. Pulsepower and pulse-energy meters are not supported, so all the meters mentioned here are assumed to be

continuous-wave (CW) meters, or chopped CW meters. Power meters measure the time-averaged optical

power to which the detector is exposed; energy meters measure the total energy injected into the detector. 

Since the calibration system’s absolute power is very stable, the energy can be accurately determined

from the average power, by multiplying the average power by the period over which the power was

applied. Conversely, the average power can be determined from the total energy by dividing the total
50

(2)

(3)

energy by the period over which the power was applied. In the high-accuracy calibration system, the

average power is measured. The period over which the power was applied is also accurately measured,

for conversion from average power to total energy, and vise versa, as necessary.

4.3.1 Meter output analysis: Three methods of determining the average power from a power meter’s

output are supported; the single-baseline, dual-baseline, and no-baseline methods. The two methods that

use baselines assume that at the start of data acquisition the meter is in steady-state with only background

illumination. All three methods use three timing parameters called the sampling period, the settling

period, and the rating period. The sampling period is the period between samples of the meter’s output. 

The settling period is the minimum period allowed for the meter to reach steady-state after a change in the

applied optical power occurs. The settling period is usually several meter time constants long, so that the

meter rises sufficiently close to its ultimate steady-state output. The rating period is the minimum period

during which the detector’s output is measured after reaching steady state. With some extremely slow

detectors, waiting several time constants is not practical, so the customer specifies a shorter settling

period. A short settling period is reasonable so long as the meter is consistently used with the shorter

interval. The requested settling and rating periods are minima, because the actual number of samples

acquired is the requested settling period divided by the sampling period, rounded up to the next whole

number. So if a settling period of 1 s is requested and the sampling period is 4 s, a settling period of 4 s is

used. 

The meter output readings that are acquired during the rating period are averaged and the average is used

in the responsivity calculation. The average meter output (O) is given by

where there are N samples of the meter’s output o, within the rating period. Depending on the meter, the

average output can have units such as volts, amps, watts, or simply counts. Output readings are recorded

during the settling periods, but the values are not used in the power calculations.

The dual-baseline method is used most often, because it compensates for linear drift in the meter’s output. 

With this method, one rating period of samples is acquired before the shutter is opened; the average

output during this period is called the initial baseline, or OB1. After the optical power has been applied for

a settling period, another rating period of samples is acquired; the average output during this time period

is called the signal, OS. Then the shutter is closed, another settling period passes, then the final rating

period of samples is acquired. The average output during this time period is called the final baseline or

OB2.

In the dual-baseline method, the baseline during the illuminated period is interpolated from the bracketing

baseline measurements; then the change in the meter’s output due to the illumination, called OM, is given

by

The dual-baseline method is the method usually used with the LOCR, since the two baselines help

compensate for its drift with room temperature.
51

(5)

(4)

The single baseline method is used with some meters, typically because they have negligible drift,

respond slowly and so have a long settling period, have asymmetrical rise and fall times, or do not return

to the same baseline after the illumination. The single-baseline method is essentially the same as the

dual- baseline method, except that data acquisition is terminated when the shutter is closed, and 

With the no-baseline method, the shutter is opened at the start of data acquisition. Then one settling

period is waited and one rating period of samples is acquired for measurement of the signal; then the

shutter is closed and the acquisition is terminated. Therefore no baselines are acquired and the change in

the meter’s output is simply the signal, so OM = OS.

Energy-meter output is processed differently from power-meter output. Their outputs are typically a

single number that represents the amount of energy measured. Laser calorimeters use a string of equally

spaced samples that are processed using the corrected rise algorithm [4]. Performing the corrected rise

calculation is complicated, so will not be repeated here; but the data for the corrected rise are acquired in

a manner that is essentially identical to the dual-baseline method that is used with power meters. In either

case, the resulting energy meter output is a number, called OM, that is a measure of the applied energy.

The test meter’s output OM is proportional to the average applied power (or energy), but does not

necessarily have units of watts (or joules). A scale factor called G may be used to convert the meter

output to the desired units. For example, a photodiode-based detector produces an output current that is

proportional to the applied optical power. If its output current is amplified using a current-to-voltage

converter, the meter’s measured OM will have units of volts. But the customer may want the detector

calibrated in terms of amps per watt, so that the calibration factor is independent of the amplifier used. 

To convert the voltage output back to a current, the measured output voltage is divided by the converter’s

gain G (in volts per amp, in this case), to give the meter output in terms of amps. Whether or not a scale

factor G is necessary is determined by the units of the calibration factor that the customer desires. If units

other than the raw units of OM are desired, an appropriate gain term G must be used. Similarly, if NIST

electronics such as a voltmeter are used to measure the meter’s output, a correction for error in the NIST

electronics, called kT, is assessed. Since G is usually an amplifier gain, the scaled and corrected detector

output, called OD, is usually given by 

If a scale factor is not needed, G is assigned a value of 1, or left out entirely, so that OD = kT OM.

Similarly, if NIST electrical measurements are not used with the test meter, the term kT is omitted and

OD = OM.

The absolute power that is measured by the detector is usually desired, so G is usually chosen to give OD

in terms of watts. In the case of the LOCR primary standard, OM already has units of watts, so G = 1 and

the measured power, called P, is given by: P = kTOD = kTOM. Special correction factors for the LOCR,

described below, are functionally equivalent to the correction factor kT.

4.3.2 Applied power calculation: The absolute average power that is applied to the test meter is

interpolated from bracketing power measurements that are performed with the standard. The

interpolation is done in time, using the midpoints of the power injection intervals as the time references. 

The bracketing primary standard power measurements, called P1 and P2, are acquired at times t

1

 and t

2

,
52

(6)

(7)

(8)

(9)

respectively. The test meter was illuminated at time tM, where t

1

 < tM < t

2

. Using linear interpolation, the

average power at the time of the test meter illumination, called PS, is given by

The powers P1 and P2 are measured at the LOCR’s optical absorber. The derived power PS is therefore

the power that would be measured by the LOCR at the time of the test meter calibration. To determine

the absolute power that the test meter is exposed to, several correction factors are needed. The LOCR

window transmittance (TW) is the ratio of the optical power transmitted by the LOCR’s window, to the

amount of power incident on the window. The LOCR receiver absorptance (AR) is the ratio of the amount

of optical power absorbed by the LOCR’s optical receiver, to the incident power. The relative aperture

transmittance (TA) is the ratio of the amount of power that is transmitted by the test meter’s entrance

aperture, to the power transmitted by the LOCR’s entrance aperture. The LOCR electrical calibration

correction factor (kL) corrects for the offset in the LOCR’s electrical power measurement. Therefore, the

power that is applied to the test meter (PA) is given by

The LOCR correction factors TW, TA, and kL are derived, respectively, from measurements of the LOCR’s

window transmittance, the LOCR’s relative aperture transmittance, and DMM and LOCR electronics

electrical calibrations. The correction factor AR is derived from previous measurements and other

research, not from measurements performed at the time of the calibration.

The period during which the optical power was injected into the test meter is called the injection period

(t

I

). The injection period is measured using a precision time-interval counter that is connected to the

optical shutter’s controller. When an energy meter is calibrated, the applied energy, given by EA = PAC t

I

,

is used in place of the average power.

4.3.3 Calibration factor calculation: To determine the calibration factor (CF) for a power meter, usually

the detector’s output (OD) is divided by the applied power (PA), so

The calibration factor for energy meters is similar, where the applied power (PA) is replaced by the

applied energy (EA), so

Equations (8) and (9) give the calibration factor in terms of detector output per unit power (or energy). 

Therefore, if the meter output is divided by CF, the resulting values will agree with those of the NIST

measurement system. However, some customers desire the calibration factor in terms of the applied

power (or energy) per unit of the detector’s output. Then, the detector output is multiplied by the

calibration factor. In this case, the reciprocal of eq (8) or (9) is used to determine the calibration factor. 
53

The specific equation used is modified as necessary to give the calibration factor in the units that the

customer desires. In any case, the equation used is defined explicitly in the calibration report.
54

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5. UNCERTAINTY ANALYSIS

To achieve high accuracy, it is desirable to minimize all the sources of uncertainty as much as possible. 

But to keep the service affordable to the customer, uncertainty must also be measured in a reasonable

amount of time. So in some cases a trade-off is made between the lowest possible uncertainty and an

uncertainty that can be estimated with a reasonable amount of time and effort. Towards this end, we may

accept a more conservative but simpler uncertainty analysis, even though a smaller uncertainty may be

obtainable with more effort. There may be insufficient information to perform a more accurate analysis,

or a complex data analysis involved, so a conservative but simpler estimate of uncertainty may be used

instead. In some cases, even conservative estimates are so small that they are negligible compared to

other terms, so putting more effort into a better analysis would not be advantageous. Additional

measurements and analysis may be performed in the future to reduce the dominant uncertainties.

The analysis for many of the uncertainty terms is provided in appendixes. The analysis is located in

appendixes to (1) keep the body of this document a reasonable size, and (2) because if the details of the

measurements and analysis change in the future, only the appendixes will need to be updated. Only the

higher-level concepts are discussed here, with details left to the appendixes.

Uncertainty estimates for the values given in the NIST calibration report are assessed using the following

technique, which is derived from the official NIST guidelines [21]. Each component of uncertainty that

contributes to the uncertainty of the measurement result is represented by an estimated standard deviation,

termed the standard uncertainty u. The combined standard uncertainty uc, which represents the estimated

standard deviation of the result, is obtained by combining the individual standard uncertainties. The

standard uncertainties are estimated from Type A evaluations, where magnitudes are obtained statistically

from a series of measurements; and from Type B evaluations, whose magnitudes are based on scientific

judgement using all the relevant information available. 

The Type A uncertainty components are assumed to be independent and, consequently, the standard

deviation, Sr, for each component is

where the xi values represent the individual measurements and N is the number of xi values used for a

particular Type A component. The standard uncertainty of each Type A component is the standard

deviation of the mean, given by Sr/N

1⁄2

. The total standard uncertainty of the Type A components is

[E(Sr

2

/N)]1⁄2, where the summation is carried out over all Type A components.

Most of the Type B components are assumed to be independent and have rectangular or uniform

distributions (that is, each has an equal probability of being within the region, ±*s, and zero probability of

being outside that region). If the distribution is rectangular, the standard uncertainty for each Type B

component is equal to *s/31⁄2, and the total standard uncertainty for all Type B components is (EFs

2

)

1⁄2

,

where the summation is carried out for all Type B components.

The combined standard uncertainty is determined by combining the total Type A and Type B standard

uncertainties in quadrature; the expanded uncertainty is obtained by multiplying this result by a coverage

factor of k = 2. The expanded uncertainty U is then
55

(11)

Table 1. Example values for the LOCR primary standard’s uncertainty components.

Component of Uncertainty Correction Factor Standard Uncertainty (%)

Window Transmittance (TW, uTW) 0.99962 0.0138

Receiver Absorptance (AR, uAR) 0.99987 0.0070

Receiver Alignment (uLA) 1.00000 0.0075

Electrical Calibration (kL, ukL) 0.99998 0.0034

Heating Inequivalence (uHI) 1.00000 0.0002

Combined Standard Uncertainty (uPS) 0.0175

The values used to calculate the NIST expanded uncertainty (shown in Table 1) are listed in Table 2. The

number of decimal places used in reporting the mean values of the calibration factor listed in Table 1 was

determined by expressing the expanded NIST uncertainty to two significant digits.

5.1 Uncertainty of the LOCR Primary Standard

Only calibrations based on the LOCR primary standard are considered here; the uncertainty that results

from the use of a transfer or other primary standard is not discussed. The uncertainty terms for other

standards can be very different from the LOCR’s. So if a standard other than the LOCR is used with the

system, additional uncertainty analysis and documentation are necessary.

The uncertainties assessed for the LOCR are the result of years of experimentation and analysis and may

differ from the uncertainty terms used in other laboratories for cryogenic radiometer-based calibrations. 

But the LOCR’s combined standard uncertainty is consistent with published results from other quality

laboratories [22]. When operated at an optical power of 1 mW, The LOCR usually meets its design goal

of absolute power measurement with a standard uncertainty of 0.01 %.

The combined standard uncertainty of LOCR power measurements varies from calibration to calibration,

because several of the uncertainty components are measured when the calibration is performed. After a

large number of calibrations have been performed, a statistical analysis of the measured uncertainties can

be performed to determine probable values. Measurement of the uncertainties each time a calibration is

performed provides an additional quality assurance check. The combined standard uncertainty of LOCR

power measurement has typically varied from 0.01 % to 0.04 %. The combined standard uncertainty

cannot be calculated until the calibration runs are complete and after the final window transmittance is

measured. 

In each LOCR power measurement, the absolute optical power PL applied to the LOCR is given by the

LOCR’s raw power measurement multiplied by the LOCR’s electrical calibration factor kL and divided by

the product of the LOCR’s window transmittance TW and the LOCR’s receiver absorptance AR. 

Additional uncorrected uncertainty components that contribute to the uncertainty in the LOCR’s optical

power measurement are the uncertainty in the LOCR’s receiver alignment uLA, and that of the LOCR’s
56

(12)

heating inequivalence uHI. Example LOCR-based power measurement uncertainties are given in Table 1

and the uncertainty components are described below. The combined standard uncertainty uPL of the

LOCR primary standard’s power measurement is given by

5.1.1 Window transmittance: The LOCR’s window transmittance correction factor is measured. The

window transmittance is always less than one because some of the optical power is lost in the LOCR’s

window assembly, through absorption, scattering, or reflection. The window transmittance is the ratio of

the power transmitted to the optical receiver when the window assembly is in the beam, to the power

transmitted to the receiver when the assembly is removed.

To measure its transmittance, the window assembly is removed from the LOCR. To measure the power

with and without the window in the beam, a test detector is used in place of the LOCR’s optical receiver. 

The measurement is complicated by the fact that whenever the window assembly is reattached to the

LOCR after a measurement of transmittance, the vacuum flanges may compress differently, causing the

location where the light strikes the window to change slightly. To account for this effect, and for the

difficulty of aligning the window during transmittance measurement exactly like it is aligned on the

LOCR, a matrix of transmittance measurements is performed. And because the window’s transmittance

can change during the calibration, its transmittance is measured both before and after the calibrations are

performed. 

Currently, a nine-point square matrix, centered about the nominal beam location, is acquired in each

measurement and a simple statistical analysis is performed. The measurements are combined by use of

the technique described in Appendix H. In the future, uncertainty in the window transmittance may be

reduced by using a center-weighted technique, described in Appendix I. Also, to reduce the uncertainty

in the window’s alignment, the visual alignment technique currently used may be replaced with an

alignment technique that uses a translatable pinhole aperture detector.

The window transmittance at each location is measured with a technique that combines the window

absorptance, scattering loss, and reflectance into a single measurement of window transmittance [3]. The

technique relies on duplicating the field of view and reflection characteristics of the LOCR's optical

receiver with the test detector, duplicating the window mount alignment, and assuming that the window's

properties do not change when it is removed from the cryogenic radiometer's vacuum. The result of the

analysis of transmittance is the absolute window transmittance TW and its standard uncertainty uTW,

derived in Appendix H.

5.1.2 Receiver absorptance: Receiver absorptance is a correctable imperfection that occurs because some

of the power that enters the LOCR’s optical receiver is not absorbed and converted to heat. A small

amount of the incident optical power is reflected from the receiver. The LOCR’s receiver is designed to

absorb the vast majority of the collimated laser power, but a small amount is lost. The amount of power

that is lost depends primarily on the wavelength of the light; the effect of beam profile and receiver

alignment are covered in the relative aperture transmittance correction and receiver alignment

uncertainties, respectively.

The LOCR’s manufacturer provided an accurate measurement of the receiver’s absorptance at a

wavelength of 633 nm. They also provided measurements of the reflectance of the black paint used on

the receiver, over a wide range of wavelengths. Since the paint’s reflectance is essentially constant over

the range of wavelengths considered, it is reasonable to assume that the receiver’s absorptance is also
57

constant over the given wavelength range, and many laboratories do consider the absorptance to be

constant. But actual measurement of receiver absorptance, performed on a receiver similar to the

LOCR’s, show that the absorptance is indeed wavelength-dependent [9].

In keeping with our desire to provide conservative uncertainty estimates, the results of the actual

absorptance measurements are used to assess the absorptance of the LOCR’s optical receiver at

wavelengths other than 633 nm. The receiver absorptance correction factor AR and its associated standard

uncertainty uAR are derived in detail in Appendix J. To reduce the uncertainty in the future, the LOCR

may be disassembled and its optical receiver’s absorptance measured directly. The direct measurement

should be performed periodically anyway, because the receiver absorptance may change with time as the

paint ages.

Ice can build up on the optical receiver during use because of atmospheric gasses leaking through the

LOCR’s imperfect vacuum seals and freezing on the cold surface. Ice deposited on the receiver would

seriously affect its absorptance. To avoid this problem, the LOCR is warmed to room temperature each

week. The LOCR’s carbon getter is also purged by the warming.

5.1.3 Receiver alignment: The term receiver (or LOCR) alignment describes the uncertainty that results

from the inability to perfectly align the LOCR’s optical receiver during use and during measurement of

the receiver absorptance. There is insufficient information about the receiver’s spatial uniformity and the

receiver alignments to make a correction. The uncertainty can also vary with beam diameter and

wavelength.

Measuring the LOCR receiver’s spatial uniformity is extremely difficult, because of the device’s slow

response time, the long time it takes to translate the beam vertically, and the system’s imperfect stability

of beam power and environment. The result is drift in the measurement of relative responsivity that

usually exceeds the underlying uniformity being measured. Therefore, rather than attempting to map the

receiver’s uniformity, its uniformity is estimated by using a small number of samples taken at discrete

locations spaced horizontally on the receiver. The technique currently used is described in Appendix G; a

value for the uncertainty uLA due to the LOCR’s alignment is returned. The measurement is performed

each time the LOCR is aligned, so it also serves as a check on quality assurance.

5.1.4 Electrical calibration: The LOCR measures optical power by electrical substitution, so the

uncertainty in its electrical measurements directly impact the uncertainty in the measured optical power. 

The error in the electrical measurements is determined by calibrating the electronics, so that the electrical

measurements can be corrected. A single multiplicative correction factor kL with standard uncertainty ukL

is used to correct the optical power reading. Calculation of the correction factor and its uncertainty is

described in Appendix F.

The electrical calibration correction factor and its uncertainty are usually calculated in post-processing,

after the calibrations are complete. The magnitude of the correction and its uncertainty are usually

negligible compared to other components, except for when the LOCR is used with low optical powers. 

Since the error in the LOCR’s electronics is measured for each calibration, it also provides a check on

quality assurance for the LOCR because any large change in the error would signify a potential problem

with the radiometer itself.

5.1.5 Electrical heating inequivalence: The manufacturer measured the inequivalence between electrical

and optical heating by comparing the response when electrical power is delivered to the main heater,

which is close to where the optical power is absorbed, to the response when electrical power is delivered

to the spare heater at the front of the optical receiver. The electrical power required by each heater to
58

Figure 16. Typical LOCR illuminated output signal. The signal is dithered, so the quantization

error is randomized.

achieve the same receiver temperature was compared. The manufacturer performed five measurements;

the mean disagreement between the two heaters was 0.0017 %, with a standard deviation of 0.002 %.

This manufacturer’s measurement is consistent with values used by most other calibration laboratories,

but some leave it out entirely [22]. Thermal modeling of the LOCR [11, 12] showed that the actual

worst-case electrical heating inequivalence is less than 0.0002 %, which is much smaller than the

uncertainty in the inequivalence measurement itself. Therefore the result from the thermal modeling is

used as the standard uncertainty of the heating inequivalence, uHI.

5.1.6 Other LOCR uncertainties: Other sources of uncertainty in the LOCR are negligible when

compared to the dominant uncertainties, or are random and are therefore included in the measurement

repeatability. Two such sources of uncertainty that can be significant, digital quantization and nonlinear

drift, are described below.

Because they use a finite number of bits, all digital systems have a finite quantization error, and the

LOCR is no exception. If the readings are properly rounded, the quantization error is bounded by plus or

minus half of the least significant bit. However, in systems designed like the LOCR, the quantization

interval is smaller than the noise in the system, so that the noise randomizes the quantization. A simple

but conservative approach to estimating the quantization error is used. The device’s illuminated output

signal is analyzed by eye and if significant quantization is observed, then a worst-case uncertainty for the

quantization error is assessed. But if the variation in the signal is significantly larger than the apparent

quantization interval, the quantization is assumed to be randomized and is therefore included in the

measurement repeatability, so no additional uncertainty for the quantization is assessed.

A typical illuminated output signal from the LOCR is shown in Figure 16; the variation in the signal is

clearly much larger than the apparent quantization interval of 10 nW, so the quantization is sufficiently

randomized and no quantization uncertainty is assessed. Quantization uncertainty is also assessed for the

test detector, as described in section 5.3.3.
59

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The LOCR’s background power readings drift with room temperature and other environmental

influences, as discussed in section 3.2. Therefore the dual-baseline measurement technique, discussed in

section 4.3.1, is used with the LOCR; the technique corrects for linear drift. However if there are

components of higher order in the drift, the interpolated baseline can be in error, causing error in the

calculated power. If the resulting error in the calculated power is not randomized, it will not be reduced

by averaging multiple measurements. For example, if a higher-order drift is always in the same direction,

a systematic error can result even with the dual-baseline method. However, if for example the room

temperature is drifting randomly, the error in the interpolated baseline will also be randomized, and is

therefore included in the measurement repeatability.

To help ensure that the drift is random or at least minimized, the laboratory environment is carefully

controlled (when possible) and monitored. The laboratory air temperature, a significant contributor to

LOCR drift, is controlled. Uncontrolled parameters, such as the humidity and pressure, are monitored;

and if a significant non-random drift is encountered during a calibration, additional calibration runs

should be performed. Another source of drift in the LOCR is drift in the temperature of its liquidnitrogen-cooled heat shield. The liquid nitrogen temperature shield is vapor-cooled, so its temperature is

sensitive to the level of the liquid. This drift accelerates rapidly as the liquid-nitrogen expires and in the

past has interfered with measurements. So care must be taken to not operate the LOCR when its liquidnitrogen level is becoming low. 

5.2 Uncertainty of the Applied Power

Equation (12) gives the combined standard uncertainty uPS of each individual LOCR power measurement. 

As described in eq (6), two power measurements are interpolated linearly in time to determine the power

that the LOCR would have measured at the time the test detector was illuminated. Ideally, the applied

optical power is stable, so PA = P1 = P2 = PL, and the times are irrelevant. To simplify the uncertainty

analysis, the power is assumed to be stable and any remaining uncertainty random, so that the uncertainty

in the interpolated power is the same as uPL, described in section 5.1. Equation (7) describes how the

interpolated LOCR power PA is used to calculate the power that is applied to the test detector. The

uncertainty in the applied power is the uncertainty in the LOCR power measurements combined with the

uncertainty of the new terms uTA and uRP, which are described below. The resulting standard uncertainty

of the power applied to the test detector uAP is given by

5.2.1 Relative aperture transmittance: The relative aperture transmittance is a correctable error caused by

(a) imperfections in the approximately Gaussian beam that is used in the system and (b) the finite limiting

apertures of the detectors. The correction compensates for any difference in the amount of optical power

that is applied to the standard and test detectors. Estimation of the correction factor TA and its standard

uncertainty uTA are described in Appendix E.

5.2.2 Measurement repeatability of the applied power: The measurement repeatability is an

uncorrectable error, with an uncertainty that is calculated using a Type A evaluation of the results of

multiple applied power measurements, which are assumed to vary randomly. Therefore, the mean of the

measurements of PA is the best estimate of the absolute power that was applied to the test detector, and

the standard deviation of the mean of the measurements of PA is the standard uncertainty. The standard

uncertainty uRP is the standard deviation of the mean.
60

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5.2.3 Uncertainty of the applied energy: When an energy meter is calibrated, the uncertainty in the

applied energy is used. The uncertainty in the applied energy is the same as that for the applied power

except that the uncertainty uIT of the measured injection period is also included in the quadrature sum

given in eq (13), and the measurement repeatability of the applied energy is used in place of the

measurement repeatability of the applied power.

5.3 Uncertainty of the Test Detector Calibration

The uncertainty in the test detector’s calibration factor essentially the uncertainty in the applied power,

combined with additional uncertainties that are associated with the test detector. However, because the

random component of uncertainty in the calibration factor can be smaller than the random component of

uncertainty in the applied power, the uncertainty in the calibration factor can be larger or smaller than the

uncertainty in the applied power. Equation (8) gives the calculated calibration factor (CF) for an example

power meter.

In accordance with our calibration philosophy, uncertainties inherent to the test meter are not evaluated. 

But there are three additional cases in which an uncertainty will be assessed, because the uncertainty is

due to NIST equipment or imperfections in the calibration system. The three cases are for: amplifier gain

uncertainty uG; uncertainty uTE in the test detector electrical measurement; and uncertainty uDQ due to

quantization in the test meter’s output. These uncertainties are not appropriate for all test meters; gain

uncertainty is an issue only when a NIST amplifier is used, electrical measurement uncertainty is

appropriate only if NIST equipment is used to measure the test detector’s output, and quantization is an

issue only with digital test meters.

When the component of uncertainty is inappropriate for the particular test meter, its uncertainty is

assigned a value of zero, or the component is left out completely. The combined standard uncertainty uCF

of the calibration factor is

The new components of uncertainty introduced in eq (14) are described below. The combined standard

uncertainty in the applied power and calibration factor for an example test meter calibration are shown in

Table 2. The example is from an actual 100 :W calibration of a test detector with a digital output, so

gain and electrical measurement uncertainty are inappropriate, but a small uncertainty due to quantization

is assessed.

5.3.1 NIST amplifier gain: When a NIST amplifier is used to amplify or convert a test detector’s output,

the uncertainty uG of the amplifier’s gain is assessed. For example, a NIST current-to-voltage converter is

frequently used to amplify and convert the output current from a photodiode-based detector to a voltage. 

Since the converter is not an actual part of the customer’s meter and is therefore not always used with the

detector, its uncertainty must be assessed.

Parameters describing the NIST amplifier are also included in the calibration report. For example, the

current-to-voltage converter can load the detector so its input characteristics, such as input impedance and

voltage burden, are quoted. These parameters are measured by NIST and an upper bound is stated. For 
61

Table 2. Example uncertainties for the applied power and calibration factor calculation. 

Component of Uncertainty Correction Factor Standard Uncertainty (%)

Window Transmittance (TW, uTW) 0.99962 0.0138

Receiver Absorptance (AR, uAR) 0.99987 0.0070

LOCR Receiver Alignment (uLA) 1.00000 0.0075

Electrical Calibration (kL, ukL) 0.99998 0.0034

Heating Inequivalence (uHI) 1.00000 0.0002

Relative Aperture Transmittance (TA, uTA) 1.00000 0.0011

Measurement Repeatability of the Applied Power (uRP)* 0.0072 

Combined Standard Uncertainty of the Applied Power (uAP) 0.0190

Ammeter Quantization (uDQ) 1.00000 0.0007

Measurement Repeatability of the Calibration Factor (uRC)* 0.0075 

Combined Standard Uncertainty of the Calibration Factor (uCF) 0.0191

* - Type A evaluation with N = 6.

(15)

the example NIST current-to-voltage converter, when the converter is used with gains of up to

100,000 V/A, the input impedance is less than 0.1 S, and the voltage burden is less than 1 mV.

5.3.2 NIST electrical measurement: If the test detector produces an analog electrical signal such as a

voltage that is measured by NIST electrical equipment, the uncertainty in the NIST equipment must be

assessed. For example, consider a test detector that produces an analog voltage that is measured by the

calibration system’s DMM. Evaluation of the uncertainty in the system’s DMM is described in relation to

the LOCR electrical calibration in Appendix F; assessment of test detector electrical measurement

uncertainty is similar. The error and the uncertainty in the voltage measurements are used to obtain the

correction factor kT and uncertainty ukT for the measured detector output.

For example, consider a test detector that produces an analog voltage that is processed with the dualbaseline method. To simplify the analysis, the two baselines are assumed to be equal; in practice the two

baselines are usually very close together. The baselines have an average voltage vB and the illuminated

signal has an average voltage vS. The voltmeter has an absolute error eB with uncertainty uB when

measuring voltage vB, and an absolute error eS with uncertainty of uS when measuring voltage vS. These

parameters are determined from the voltmeter’s calibration, as described in Appendix F. The voltmeter

error is such that the actual voltage is the measured voltage plus the voltage error.

When the dual-baseline method is used, the meter output is calculated using eq (3). Given the voltmeter

errors, a multiplicative correction factor can be calculated:

The uncertainties uS and uB are assumed to be uncorrelated, which is reasonable if |vS| >> |vB|. Even

though two baselines are used, the baselines are ideally identical, so the averaging of the two baselines in

eq (3) is ignored and the uncertainty in the baseline is simply uB. Therefore, the combined standard
62

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(17)

(18)

uncertainty uKT in the correction factor kT is given by

When using the single-baseline method from eq (4), the correction factor and uncertainty are the same as

with the dual-baseline method, because the baseline averaging in the dual-baseline method is ignored. 

When using the no-baseline method, only the voltage measurement error at vS is relevant, so the

correction factor becomes

When estimating the uncertainty for the no-baseline method, a term for the voltage offset in the

measurement system’s cabling and connectors must be assessed. The baseline methods are differential

measurements, so the offset is canceled; but with the no-baseline method (and any other nondifferential

voltage measurements), the offset must be assessed. To ensure the quality of the system’s cabling and

connectors, the offset is measured periodically, as a quality-control measure. In the measurements, when

a high-quality electrical short terminates the cables, the absolute value of the offset voltage is consistently

below 1 :V. Therefore, using a worst-case bound of ±10 :V for the offset voltage is probably very

conservative, yet does not significantly overestimate the uncertainty because the absolute value of vS is

usually greater than 1 V. The voltage offset is not corrected, because it is not measured before each

calibration and can change with time and temperature. The resulting standard uncertainty in kT is

5.3.3 Quantization in digital test meters: Because of the calibration system’s power stability, quantization

in digital test meters can lead to uncertainty in its power measurement if there is insufficient dithering in

the meter itself. As described in section 5.1.6, the LOCR’s output is sufficiently dithered, so that any

error due to quantization is randomized. The test meter’s output must also be inspected, to determine

whether an uncertainty for its quantization must be assessed.

To determine whether quantization is an issue, the meter’s sampled output is observed. In general, if a

meter outputs a digital power reading with a fixed number of digits, it may have quantization error. But

meters with an analog output can also be quantized, so a more general rule is used: if the meter’s output is

quantized, the sampled readings will occur at discrete levels, with no readings occurring between the

levels. Determining whether or not the output is sufficiently dithered is more speculative, but in

accordance with the desire to return a conservative estimate of uncertainty, it is better to include the

uncertainty when unnecessary than to omit it when necessary. So a simple but conservative rule is used

to determine whether quantization error should be assessed: if the samples in the rating period are all at

one or two levels, an uncertainty for the potential quantization error should be assessed. The worst-case

scenario is when all the sampled readings are at a single level, with a standard deviation of zero.

Figure 17 shows the sampled readings from a test meter with potential quantization error. Since all the

readings occur at only two levels, the output is considered insufficiently dithered to randomize the

quantization error, so an uncertainty for the quantization is assessed. In the example, the power level was

intentionally adjusted so that two levels are visible. Because about half of the samples occur at each level

in the example, the uncertainty in the average is actually less than the assessed quantization uncertainty. 

However, had the applied optical power been slightly different, all of the readings could have occurred at
63

Figure 17. The output current from an illuminated photodiode, showing two discrete quantization

levels.

(19)

a single level, then the assessed uncertainty due to quantization is accurate. To be conservative, an

uncertainty for the quantization is assessed, even though in cases like that shown in Figure 17, the actual

uncertainty due to quantization is reduced by averaging.

A bounded uncertainty estimate is used to quantify the quantization error. The meter’s output is assumed

to be properly rounded to the nearest quantization interval. Therefore, the largest possible error in the

quantized output reading is one-half of the quantization interval Q. The quantization error has a

rectangular distribution, so the standard uncertainty due to quantization, uQ, is

Depending on the output processing method used, the samples from multiple rating periods may have to

be observed. In all methods, the rating period samples acquired when the detector is illuminated must be

considered. When a processing method with baselines is used, quantization in the baseline rating periods

must also be considered.

When the no-baseline processing method is used, the standard uncertainty due to quantization is equal to

that for the illuminated rating period only, which is given by eq (19). When the single-baseline method is

used, the uncertainty for both the illuminated and background rating periods must be considered. Using

eq (19), define uQB as the standard uncertainty due to quantization in the baseline rating periods and uQS as

the uncertainty in the illuminated rating period. Both uncertainties are uncorrelated and since the baseline

is subtracted from the signal, the combined standard uncertainty, uDQ, is the quadrature sum of the

standard uncertainties uQB and uQS. When the dual-baseline method is used, the two baselines are

essentially the same, so the standard uncertainty due to quantization is the same as that for the singlebaseline method.

The effect of quantization error in the baselines can be reduced by allowing the meter to auto-range; then

the size of the quantization interval in the baseline will be much smaller than the quantization interval

while illuminated, if the meter has higher ranges of gain available. Then the contribution of the
64

quantization in the baseline will be insignificant compared to the quantization when the detector is

illuminated, so the combined uncertainty is essentially the same as that for the no-baseline method.

5.3.4 Measurement repeatability of the calibration factor: The measurement repeatability of the

calibration factor is essentially the same as the measurement repeatability of the applied power, described

in section 5.2.2, except that the calculated calibration factors are used in the analysis, instead of the

calculated applied powers. The measurement repeatability uncertainty is the result of a Type A

evaluation of the measured calibration factors, so the average is the best estimate, and the standard

uncertainty is the standard deviation of the mean.

5.3.5 Uncertainty in the calibration factor for energy meters: When an energy meter is calibrated, the

uncertainty in the injection period must be included in the uncertainty analysis. The uncertainty in the

calibration factor for energy meters is the same as that for power meters, given in eq (14), except that the

standard uncertainty in the injection period, called uIT, must also be included in the quadrature sum.

5.4 Other Uncertainties

Many additional parameters are specified in the calibration report to fully characterize the conditions

present during the calibration. Each parameter is specified by an estimation with expanded uncertainty, a

conservative upper bound, or an approximation based on a model. Parameters that vary during the

calibration, such as room temperature, are specified with an uncertainty that includes the variation. The

intent of these uncertainty statements is to document the conditions present at the time of the calibration;

they only incidently affect the calibrated quantities, so do not warrant a complex measurement procedure

or detailed uncertainty analysis. However, this information must be documented in order for the highaccuracy calibration to be reproduced.

5.4.1 Test detector alignment: The uncertainty in the centering of the laser beam in the test detector’s

entrance aperture is specified, along with a measure of the beam’s incidence angle in relation to the test

detector’s entrance aperture. The orientation of the detector relative to the beam’s polarization angle is

also specified. A specific alignment procedure, described in section 4.1.4, is used so that the alignment

can be performed reproducibly.

The uncertainty in the centering of the laser beam in the test detector’s entrance aperture is based on

experiments performed by the laboratory operator and can differ for other operators. In the experiment,

the operator aligns a detector a few times and the variation in the beam’s final location on the detector is

determined. A conservative upper bound for the error in the alignment is then derived, by selecting a

value significantly larger than any variation that occurred in the experiment. The resulting upper bound is

given in the calibration report. A more formal experiment would have to be performed and documented

to obtain accurate standard uncertainties for the alignment.

Three different centering techniques are used, each with its own uncertainty bound. The coarsest

technique involves visually interpolating the center of the entrance aperture using an external scale such

as a ruler. Experiments have shown this type of alignment is performed with an uncertainty of less than

1 mm, for a beam of 1 to 2 mm diameter (in general, smaller beams can be centered more accurately than

larger beams). If the detector has a visual alignment aid such as a crosshair on its aperture cover, the

beam can be centered more accurately. Experiments have shown a centering accuracy of 0.5 mm is

conservative for beams of 1 to 2 mm diameter when such alignment aids are used. The uncertainty of the

cross hair’s centering on the aperture cover, and slop in the aperture cover itself, must also be considered,

but are usually less than the operator’s visual limitations. 
65

A better alignment technique uses a small aperture as an alignment aid, such as a small hole in the center

of the aperture cover, or the hole through the center of an attached optical fiber adapter. The beam is

centered over the hole by using the detector itself to peak the power transmitted through the hole. Again,

the accuracy of the hole’s centering in the aperture cover must also be included in the uncertainty

assessment. Experiments have shown when using the common screw-on FC optical fiber adapter as the

alignment aid, the detector is centered to better than 0.2 mm when a beam of 1 to 2 mm diameter is used. 

Most of this uncertainty with this technique is due to uncertainty in the centering of the FC connector in

the screw-on adapter and slop in its screw threads. Therefore it is best to use the same FC adapter in the

calibration that is used with the detector in practice.

The result of the centering uncertainty analysis is given as a bound in a statement such as: “With power

meter x, the beam was aligned visually to the center of the aperture cover’s cross hairs, with an

uncertainty of less than 0.5 mm.” The description is changed to match the actual alignment technique

used, as appropriate. If the customer specifies a particular alignment procedure, their procedure is used

and no uncertainties are assessed.

The incidence angle of the laser beam as it enters the test detector is also reported. Usually, the incidence

angle is set as close to normal incidence as possible. But to sufficiently document the conditions during

the calibration, the difference in the beam’s incidence angle from the normal relative to a surface on the

detector is measured and stated in the report. The surface on the detector that is used to judge what is

normal, such as the detector’s face plate or entrance aperture, is also specified.

When a detector that is intended for use in optical fiber systems is aligned, the beam is centered using the

hole in its optical fiber adapter, and the incidence angle is adjusted using the normal for the fiber adapter

itself. Then the calibration system’s collimated beam travels straight down the center of the fiber adapter

and strikes the detector’s active surface at the same location that a fiber-coupled beam would.

The error in the incidence angle is determined by holding a glass flat against the specified detector’s

surface and locating the back reflection from the glass flat in relation to the incident beam. Then a simple

geometric calculation is performed to determine the difference in the incidence angle from the detector

surface’s normal. Conservative uncertainty bounds for the geometric distance measurements are used to

determine bounds for the error in the incidence angle calculation, from which the standard uncertainty is

derived using a Type B evaluation, and the expanded uncertainty with a coverage factor of 2 is specified. 

The result is given in a statement such as: “the beam’s incidence angle was approximately 0.7 ± 0.2 mrad

from normal relative to the detector’s entrance aperture.” Information about the direction of the error in

the incidence angle is not currently provided. But in general, the error in the incidence angle is usually in

the vertical direction, because no pitch adjustment is available with most detector mounts. If an

exceptionally accurate incidence angle is required, such as when an optical fiber adapter is used as

previously described, a special mount with pitch adjustment can be used with the detector.

The laser beam polarization angle’s relation to the test detector is described by specifying the detector’s

orientation relative to horizontal. The polarization angle is described as vertical. Some characteristic of

the detector is used to describe its orientation on the table, such as: “the detector was mounted so that the

label on its faceplate was up” or “the detector was mounted on a post holder, with its threaded mounting

screw down.” The relation between the polarization angle and the test detector’s orientation is thus

defined, but no numerical measure is given. Currently the actual polarization angle is not known

accurately, so the precise polarization angle relative to the detector orientation cannot be specified.

5.4.2 Laser beam spatial characteristics: To fully characterize the laser beam used in the calibration,

several spatial beam parameters are specified in the calibration report. A theoretical Gaussian beam
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(21)

(22)

model is combined with measured parameters to provide values for beam’s diameter and divergence at

the detector’s entrance aperture, and theoretical values for the Gaussian beam waist’s diameter and

location.

Since some of these parameters are theoretical values that are not directly measured, they are described as

“approximate” in the calibration report, and no uncertainty is specified. The uncertainty of the theoretical

values is dependent on how well the actual laser beam matches a true Gaussian beam, which is quantified

by the beam’s M

2

 parameter [23]. Currently, the beam’s M

2

 parameter is not routinely measured, but

occasional measurement of the theoretical waist diameter and far-field divergence have been consistent

with the theoretical values to better than 10 %. So using a standard uncertainty of 10 % with the

parameters should be conservative, but the figure cannot be guaranteed without measurement, so

approximate values are reported with no estimate of uncertainty.

Using a few measured parameters, the Gaussian beam model for the system’s laser beam is derived. The

Gaussian beam model is described in Appendix D. The model returns the final beam waist diameter and

location relative to the detector plane, and the beam’s characteristics in the detector plane.

The beam’s diameter is measured using an autonomous scanning-slit beam profiler, which measures the

beam’s 1/e2 intensity diameter. The device is relatively noisy, so the uncertainty in measurements of its

beam diameter is dominated by random error, which is evaluated using a Type A analysis. Therefore the

mean of the diameter measurements is used as the measured beam diameter, called D, and the standard

uncertainty that results from the random variation in the measurement, called uA, is the standard deviation

of the mean.

Additional error in the scanning-slit measurement technique is caused by the ratio of the slit’s width to the

beam diameter [10]. The error in the scanning-slit diameter measurement is less than 0.17 % when:

Where w is the slit width and D is the 1/e2 intensity beam diameter. The error of 0.17 % is one-sided, the

width measured with the scanning slit is always larger than the actual width. And the error is smaller

when the ratio of the slit width to beam diameter is smaller, so the error of 0.17 % is the upper bound on

the error, so long as eq (20) is true.

Highly accurate measurements of beam diameter are not necessary, so the error in the diameter

measurement that results from the finite slit width is not corrected; instead the maximum error is used as

an uncertainty bound. Therefore, when eq (20) holds true, the standard uncertainty uS due to the finite slit

diameter is

Additionally, the manufacturer provides the uncertainty uM of the scanning-slit device’s calibration. For

example, since the beam profiler’s slit width is 5 :m, beams with a diameter of only 0.1 mm can be

measured without increasing uS. The uncertainties uA, uS, and uM are uncorrelated, so the combined

standard uncertainty in D, called uD, is given by
67

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Displayed Resolution

(nm)

Estimated Standard Uncertainty uI

(pm)

Implied Optical Source Bandwidth

(THz)

0.0001 0.06 #0.0015

0.001 0.58 #0.015 

0.01 5.8 #0.15 

0.1 58.0 #1.5 

1.0 580.0 #15.0 

Table 3. Precision wavelength meter’s estimated uncertainty and source bandwidth estimate. 

Because of imperfections in the polarized laser beam and coma distortion caused by the tilt in the spatial

filter lenses, the beam produced by the optical source is usually elliptical, with its long axis in the

horizontal direction. Both vertical and horizontal diameters are measured, and the average of the vertical

and horizontal diameters is the nominal beam diameter.

Call the two diameter measurements DV and DH, and their associated standard uncertainties uDV and uDH,

each given by eq (22). Then the average diameter is DA = (DV + DH) / 2. We assume that the

uncertainties uDV and uDH are uncorrelated, and that DV and DH are the bounds of a rectangular

distribution. The standard uncertainty uDA in DA is given by

5.4.3 Laser beam wavelength and spectrum: Uncertainty in the wavelength measurement is dependent

on the device that is used to measure the wavelength. The precision wavelength meter has the lowest

uncertainty, but can be used only with sources that produce a relatively narrow spectral line. The

wavelength of broader sources and sources that produce multiple spectral lines is measured using an

optical spectral analyzer.

The precision wavelength meter automatically determines its inherent accuracy, and infers the optical

source’s linewidth and mode structure. The device limits the resolution of its wavelength display in

accordance with its inherent accuracy. Using the displayed resolution as an uncertainty bound, the

equivalent standard uncertainty uI inherent in the devices wavelength measurement is the displayed

resolution divided by .

The wavelength measurement repeatability uncertainty uR is also assessed using a Type A evaluation of

multiple wavelength measurements. If the source’s center wavelength is unstable, or the source has

multiple spectral lines, uR can be prohibitively large, so the optical spectrum analyzer is used to measure

the wavelength instead. Since the uncertainty components are uncorrelated, the combined standard

uncertainty u8 for the wavelength measurement is the quadrature sum of uI and uR.

The wavelength meter’s display resolution, standard uncertainty, and implied optical source bandwidth

are listed in Table 3. The wavelength meter measures and corrects for the air’s index and can return the

wavelength in air or vacuum; the wavelength in vacuum is reported. Where )< is the implied optical
68

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source bandwidth, 8 is the vacuum wavelength, and C is the speed of light in vacuum, an upper-bound

for the one-half power spectral linewidth in vacuum ()8) is given by

By observing the wavelength meter’s “trigger” and “window” output, whether the implied bandwidth is

for a single spectral line or multiple spectral lines can be inferred. If there are multiple spectral lines, the

implied bandwidth can be for either the dominant spectral line, or for the half-power bandwidth of all the

lines combined, depending on their relative intensities. In the former case, a stable wavelength will be

returned by the device, but in the latter case, the measured wavelength will probably be erratic, as the

mode structure of the multi-spectral-line source changes with time. For such multiline sources, the

measurement repeatability of the wavelength meter’s measurements will be excessive, so the optical

spectrum analyzer is used instead of the wavelength meter. Therefore, the wavelength meter is used for

wavelength and bandwidth measurement only when the source has a single spectral line, or multiple lines

with one dominant line. The source spectrum is then specified in the report with a statement such as: “The

laser radiation was contained in a single spectral line having a half-power linewidth of ()8 C 109) nm or

less.” Note that a “single spectral line” can actually contain multiple longitudinal modes that are too close

together for the wavelength meter to resolve. So the source cannot be stated as having a single

longitudinal mode unless its is known to have only a single mode for other reasons, such as the laser’s

geometry or its feedback structure.

When an optical spectrum analyzer (OSA) is used instead of the wavelength meter, because the source

has multiple spectral lines or an unstable center wavelength, the wavelength and linewidth are much more

uncertain. The near-infrared wavelength OSA currently used has a standard uncertainty in its wavelength

measurements of 0.13 nm. The visible wavelength OSA has a standard uncertainty in its wavelength

measurements of 0.29 nm. To determine the combined standard uncertainty for the OSA wavelength

measurement, this uncertainty is added in quadrature with the measurement repeatability uncertainty,

from a Type A evaluation of multiple measurements. Measurement of spectral linewidth using the OSA

are limited by the devices resolution of about 0.1 nm. When used with sources that contain multiple

spectral lines, the OSA allows documentation of the structure of the spectral lines, including their relative

intensity. The spectral content of such sources is described using the OSA measurements. For example,

although a particular 1319 nm laser produces three spectral lines, one is dominant; so an accurate

wavelength meter measurement is possible, but the spectral line structure is defined using the OSA

analysis. The laser is then described with a statement such as: “The wavelength of the laser is

1319.52 ± 0.13 nm. Within this wavelength region the laser has three spectral lines, each with half-power

linewidth of less than 0.1 nm.” For lasers with more complex spectra, the verbal description can be

supplemented with a graph of the spectrum as measured by the OSA.

5.4.4 Environment specification: The air temperature, pressure, and humidity during the test detector

calibration are monitored using the autonomous weather station and specified in the calibration report

along with their expanded uncertainties. Also, the index of refraction of the air surrounding the test

detector is also calculated from these parameters and specified in the report.

The autonomous weather station records readings once every minute. The weather station measures two

air temperatures, the temperature at the weather station’s location, and the temperature in the detector

plane where its external temperature sensor is placed. Using its internal sensors, temperature, humidity

and pressure are measured near the optical source, on the opposite side of the scattered light shield from

the detectors. Pressure and humidity do not vary significantly between the two locations, but the
69

Uncertainty

Component

Nominal Value Partial Derivative Standard Uncertainty

Temperature (K) 298.11 !7.5 C 10-7 0.34

Pressure (Pa) 83879.0 2.7 C 10-9 406.0 

Relative Humidity (%) 30.5 !1.1 C 10-8 5.0 

Table 4. Components of uncertainty used in the calculation of the index of refraction of the air.

temperature can. The temperature difference between the detector plane and the optical source is used to

estimate the size of the temperature gradient uG in the vicinity of the detectors. The uncertainty in the

weather station’s measurements is determined by calibrating the instrument, as described in Appendix K. 

The variation in the parameter measured during the calibration is included in the uncertainty, as described

in the appendix.

The numerical refractive index of the air is necessary for calculating the air wavelength of the laser light,

and can affect the test detector’s absorptance. Because of the altitude of the laboratory, the air’s index of

refraction is typically 1.00022, which is significantly lower than usually assumed. The index of the air

around the detectors is calculated from the detector plane air temperature, pressure, relative humidity, and

the laser’s wavelength [5]. Other parameters used in the calculation, enhancement factor (1.00364),

saturation water vapor pressure (2693 Pa), and compressibility (0.999686), are based on typical local

values for pressure (85,000 Pa), temperature (22° C), relative humidity (30 %), and CO2 abundance

(0.0003). The resulting index is relatively insensitive to these parameters, so using the typical values is

sufficient.

The uncertainty in the index calculation is determined using the law of propagation of uncertainties [21],

assuming that the components of uncertainty are uncorrelated. The equations used are complex, so 

numerical partial derivatives are used instead of an analytic solution. Example values and the result of the

numerical partial derivative calculation are given in Table 4. The nominal index and its expanded

uncertainty are given in the calibration report. Using a coverage factor of k = 2, the resulting index of

refraction for the example parameters is 1.0002239 ± 0.0000023. Uncertainty in the model used in the

index calculation is much smaller, typically around 2 C 10-8 [5], so can be ignored; the uncertainty in the

measured temperature, pressure, and humidity dominate. 
70

REFERENCES

Note. Brand and company names are used here only so that the system can be duplicated, and imply

neither endorsement by the National Institute of Standards and Technology (NIST), nor that the materials

or products identified are necessarily the best available for the purpose.

[1] J. E. Martin, N. P. Fox, and P. J. Key, “A cryogenic radiometer for absolute radiometric

measurements,” Metrologia 21, 147-155 (1985).

[2] T. R. Gentile, J. M. Houston, J. E. Hardis, C. L. Cromer, and A. C. Parr, “The NIST highaccuracy cryogenic radiometer,” Appl. Opt. 35, 1056-1068 (1996).

[3] P. V. Foukal, C. Hoyt, H. Kochling, and P. Miller, “Cryogenic absolute radiometers as laboratory

irradiance standards, remote sensing detectors, and pyroheliometers,” Appl. Opt. 29, 988-993

(1990).

[4] E. D. West, “Data analysis for isoperibol laser calorimetry,” Nat. Bur. Stand. (U.S.) Technical

Note 396 (1971).

[5] F. E. Jones, “The refractivity of air,” Nat. Bur. Stand. (U.S.) J. Res. 86, 27-32 (1981).

[6] D. J. Livigni, C. L. Cromer, T. R. Scott, B. C. Johnson, and Z. M. Zhang, “Thermal

characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia

35, 819-827 (1998).

[7] E. Hecht, Optics, 3rd ed., pp. 604-610 (Addison Wesley Longman, New York, NY, 1998).

[8] M. Young, “Mode-field diameter of single-mode fiber by far-field scanning,” Appl. Opt. 37,

5605-5619 (1998).

[9] G. E. Chamberlain, P. A. Simpson, and R. L. Smith, “Improvements in a calorimeter for highpower CW lasers,” IEEE Trans. instrum. meas. IM-27, 81-86 (1978).

[10] D. J. Livigni and X. Li, “Spatial uniformity of optical detector responsivity,” Proc. NCSL

Workshop and Symposium, Chicago, IL., 337-352 (1994).

[11] B. C. Johnson, Heat transfer analysis and modeling of a cryogenic radiometer, M.S. Thes.

University of Florida (1997).

[12] B. C. Johnson, A. R. Kumar, Z. M. Zhang, D. J. Livigni, C. L. Cromer, and T. R. Scott, “Heat

transfer analysis and modeling of a cryogenic laser radiometer,” J. Thermophys. Heat Trans. 12,

575-581 (1998).

[13] Cambridge Research and Instrumentation, Inc., 21 Erie Street, Cambridge, MA 02139, USA.

[14] D. J. Livigni, “High-Accuracy Calibration System Operating Instructions,” unpublished, Natl.

Inst. Stand. Technol. (1998).

[15] L. D. Dickson, “Ronchi ruling method for measuring Gaussian beam diameter,” Opt. Eng. 18, 70-

75 (1979).
71

[16] J. M. Houston, and D. J. Livigni, “Comparison of two cryogenic radiometers at NIST,” Natl. Inst.

Stand. Technol. J. Res. 106(4): 641 (2001).

[17] O. R. Touayar, J. M. Coutin, and J. Bastie, “Measurement of the reflectance of the INM

cryogenic radiometer cavity at several wavelengths,” Metrologia 35, 387-391 (1998).

[18] M. Young, Optics and Lasers including Fibers and Optical Waveguides, 4th ed., Chapter 2, Eq.

(2.1) (Springer, New York, 1994).

[19] Malitson, I.H. “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc.

Am. 55(10), 1205–1209 (October 1965).

[20] “Cryogenic Radiometer Operating Instructions,” Cambridge Research and Instrumentation, Inc.,

21 Erie Street, Cambridge, MA 02139, USA.

[21] B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty in NIST

measurement results,” Natl. Inst. Stand. Technol. Technical Note 1297 (1994).

[22] R. Goebel, M. Stock, and R. Köhler, “Report on the international comparison of cryogenic

radiometers, based on transfer detectors,” BIPM Report BIPM-2000/9 (2000).

[23] "Lasers and laser-related equipment -- Test methods for laser beam parameters -- Beam widths,

divergence angle and beam propagation factor," ISO 11146 (1999).

[24] Ref. 7, pp. 249-250.

[25] Ref. 18, Section 2.2.4, pp. 14-15.

[26] J. T. Verdeyen, Laser Electronics, 3rd ed., Section 3.6. Verydin, ABCD matrix, pp. 76-79, 

(Prentice Hall, NJ, 1995).

[27] F. Barnes, “ECEN 4045 Lecture Notes,” University of Colorado, 1991.

[28] A. E. Siegman, Lasers, pp. 665-670 (University Science Books, CA, 1986).

[29] Ref. 18, Section 10.2.6, pp. 260-261

[30] Ref. 7, pp. 587.

[31] Ref. 18, Section 7.2.4, pp. 158-159.

[32] Ref. 18, Section 8.3.3, pp. 205-208.

[33] Ref. 7, Section 4.6, pp. 109-120.

[34] HP 3458A multimeter operating, programming, and configuration manual, Appendix A, HewlettPackard Company (1994).

[35] Davis Instruments Precision Weather Instruments Catalog, Davis Instruments, Weather Monitor

II specifications (1996).
72

APPENDIX A. Glossary

Some of the more unusual or potentially misunderstood terms and abbreviations used throughout this

document are defined. In this section, terms that are defined here are shown in italics.

Apparent Aperture: The apparent aperture is an estimate of the size and shape of a detector’s equivalent

limiting aperture. The apparent aperture is determined using a spatial uniformity scan of the detector. 

When the spatial uniformity scan data is scaled to 100 % in the detector’s center, the 50 % contour line

describes the detector’s apparent aperture. With a detector that has a true limiting aperture, the apparent

aperture is consistent with the limiting aperture; at the edge of the aperture, half the beam is blocked and

the other half enters the detector, so the response is reduced to 50 % of the response in the detector’s

center. The apparent aperture accurately describes the limiting aperture when the diameter of the

scanning beam is much smaller than the diameter of the limiting aperture; if the beam is too large, the

apparent aperture will be somewhat smaller than the true limiting aperture. 

However, some detectors do not have a true limiting aperture, their entrance aperture is larger than the

detector’s photosensitive element. For example, the apparent aperture of a detector with no limiting

aperture is shown in Figure 6. The 50 % contour line shown in the figure describes the detector’s

apparent aperture. The detector has a deceptively large physical entrance aperture, which is circular, and

has a diameter of slightly more than 12 mm. The apparent aperture is much smaller and is not circular. 

Knowledge of the detector’s apparent aperture is necessary for calculation of the relative aperture

transmittance.

Detector and Meter: The terms detector and meter are sometimes used interchangeably to describe a

device that measures optical power or energy. Specifically, detector refers to the part of the device into

which the optical power enters and is sometimes called the “head.” Meter specifically refers to the

electronic package into which the detector plugs, and which provides the power or energy readout. Some

devices are a single package so there is no distinction. We make this distinction because we will

sometimes calibrate a customer’s detector using our own meter electronics; in this case we calibrate only

the detector part of the device.

Detector Alignment: When a detector is aligned to the laser beam, the terms pitch, yaw, roll, and

translation are used. The terms are depicted in Figure 14. The motions are made relative to the beam’s

propagation axis, and the direction of down as defined by the direction of gravity, where the beam’s

propagation axis is perpendicular to the direction of down. A vertical translation is a movement of the

detector, without any angular change, along an axis that is parallel to the direction of down. A horizontal

translation is a movement, without any angular change, in a direction that is perpendicular to the direction

of down. During detector alignment, translations are usually performed within the detector plane, which

is a plane to which the beam propagation axis is normal, at a prescribed location along the propagation

axis. A horizontal translation in the detector plane is defined as along the x-axis, with x increasing to the

right, when facing the direction of the beam’s propagation. A vertical translation in the detector plane is

defined as along the y-axis, with y increasing in the direction opposite of down. A translation without

rotation that is along the beam’s propagation axis is a longitudinal translation, and is considered a

translation along the z-axis, with z increasing with the distance from the optical source.

Pitch, yaw, and roll are angular adjustments, in which the detector is rotated about an axis. Pitch is a

rotation around an axis that is perpendicular to the direction of the beam’s propagation, in the horizontal

direction, or perpendicular to the direction of down. Yaw is a rotation around an axis that is parallel to

the direction of down. Roll is a rotation around an axis that is parallel to the beam’s propagation axis. 

The location of the rotational axes with respect to the detector is unimportant, as is any translation which
73

results from the rotations, because when a detector is aligned, the rotations are adjusted before the

translations. However to aid in the visualization of the rotations, the rotational axes can be considered to

pass through the center of the detector’s mass, as shown in Figure 14.

Detector Breadboard: The detector breadboard is the large platform to which the LOCR and test

detectors are mounted. The detector breadboard is mounted to the motion-controlled detector stage,

which provides translation along the horizontal axis, perpendicular to the propagation axis of the beam. 

Detector Plane: The detector plane is a plane perpendicular to the propagation axis of the beam, where

the limiting or input aperture of the detectors are placed. The diameter, profile, and ideally the

divergence angle of the laser beam in the detector plane are specified. When the divergence angle is not

measured, it is estimated using Gaussian beam analysis. The location of the detector plane along the laser

beam’s propagation axis is defined by the location of the entrance or limiting aperture of the standard. A

detector’s entrance aperture is by convention the hole in the detector’s face-plate where the light enters. 

With the LOCR standard, the best location for the detector plane is not obvious. Currently the detector

plane is placed where the LOCR’s final internal aperture is, because it is the primary limiting aperture.

Device Under Test (DUT): The device under test is a generic term that refers to the device being

calibrated, whether it is a simple detector or complete meter of power or energy.

Gaussian Beam: The high-accuracy calibration system uses Gaussian beams, produced by single

transverse mode (TEM00) lasers or a single-mode optical fiber cores. Ideal Gaussian beams have a

Gaussian intensity profile, and their propagation through space can be described by specifying the beam’s

wavelength, direction, waist location, waist diameter, and the refractive index of the surrounding media. 

The actual beams used in the system are only approximately Gaussian, but Gaussian Beam theory

effectively predicts the beam’s behavior. Measurements performed to confirm the theoretical results

show that in the high-accuracy calibration system, the theory predicts the beam diameter to within a few

percent. The diameter of a Gaussian beam is usually specified by the 1/e2 intensity diameter. The quality

of a Gaussian beam is usually specified with the m

2

 parameter, which is equal to unity for a perfect

Gaussian beam and is greater than unity for imperfect beams.

Laser Power Controller (LPC): The Laser Power Controller is a commercial device used to stabilize the

laser beam’s absolute power. The LPC consists of three main parts and their connecting cables; the LPC

electronics, the LPC modulator and the LPC monitor. The modulator uses a liquid-crystal-based device

to rotate the polarization axis of the transmitted laser beam. The beam is then demodulated using a

polarizer and the demodulated power is measured with the monitor. The electronics package controls the

modulator and monitor, and returns the power measured by the monitor and the modulator’s transmission

ratio. The LPC modulator itself contains an internal polarizer and monitor detector and can function alone

without the LPC monitor. The external LPC monitor is used to compensate for loss and polarization

mixing in the spatial filter and other intervening optical elements. The monitor that is used by the LPC

electronics is determined by the device into which the monitor cable is plugged. Two LPC’s are used in

the high accuracy calibration system, one is designed for use with visible wavelengths, the other is

optimized for infrared wavelengths.

Optical Source Breadboard: The large breadboard that supports the optical source optics is called the

optical source breadboard. The breadboard is supported on five manually operated leveling posts; the

leveling posts allow adjustment of the laser beam’s height above the laboratory table.

Primary Standard: A standard that is directly traceable to physical standards, usually electrical standards. 

The primary standard should be well documented and understood from a physics standpoint and ideally
74

uses simple processes that can be accurately modeled. Cryogenic radiometers like the LOCR are

currently the best primary standard available for optical power measurement, in terms of absolute

accuracy.

Relative Aperture Transmittance: The relative aperture transmittance for a test detector is the ratio of

the amount of power transmitted by the test detector’s aperture to the amount of power transmitted by the

standard’s aperture, when the detectors are illuminated by the beam used in the calibration system. The

ratio is used to correct the amount of power delivered to the test detector, thereby accounting for any

differences in the aperture shapes or sizes. The relative aperture transmittance is either measured directly

for each test detector, or is estimated using measurements performed with circular apertures. The

detector’s apparent aperture is used to estimate its relative aperture transmittance.

Sensors: Sensors are devices that are sampled between sets of optical detector measurements. They are

used for special purposes, such as monitoring a detector’s temperature, or measuring a detector’s bias

voltage. Sensors are not required for calibrations.

Spatial Filter: The spatial filter is an optical device that consists of two lenses and a pinhole and is used

to limit the spatial frequencies present in a beam. The device is commonly used to convert a uniform

illumination into an approximately Gaussian beam. In the high-accuracy calibration system, the optical

source itself produces an approximately Gaussian beam, and a spatial filter is used to remove scattered

light and other artifacts from around the Gaussian beam. To produce the desired beam size in the

detector plane, the location of the second lens in the spatial filter is adjusted slightly around the optimal

collimation distance. The result of the lens adjustment is accurately predicted by Gaussian beam theory. 

To further control the beam size and parameters, lenses of different focal lengths can be used.

Spatial Uniformity Scan: The spatial uniformity scan of a detector is a map of the detector’s responsivity

to a given beam, versus the location of the beam on the detector. The beam location has two dimensions

and is in the plane of the detector’s entrance aperture. The responsivity is sampled at discrete locations,

with a sample spacing fine enough to prevent aliasing of the underlying features. The resulting spatial

uniformity scan data are usually presented with a wire-frame or contour map; a contour map showing the

apparent aperture of a detector is presented in Figure 6. The data can also be presented as a ray-traced

surface or matrix of numbers, and statistical analysis can be performed on the data to quantify the

detector’s uniformity. The spatial uniformity scan data are accurate only when the detector is used with a

similar beam; the detector’s uniformity can be different for other beams.

The spatial uniformity of the standard is important in the high-accuracy calibration system, because the

detector’s alignment is imperfect. Uncertainty in the detector’s alignment causes uncertainty in the

detector’s responsivity; the uncertainty in responsivity that results from the spatial uncertainty is

dependent on the detector’s spatial uniformity. Since the detectors are aligned only once during a

calibration, this uncertainty does not contribute randomly, so must be accounted for separately. The

uncertainty that results from imperfect alignment of the test detectors is not accounted for in the

calibration, but should be added by the end-user when the calibrated test detector is used as a standard in

their laboratory.

Standard: The standard is the meter which is used as the reference in a calibration. The standard can be

a primary standard or a transfer standard.

Test Detector: The device that is being calibrated is called the test detector, the device used as the

reference in the calibrations is the standard. Each test detector consists ideally of a meter and a detector,
75

but the detector alone is sometimes calibrated. The test detector can be an optical meter of either power

or energy.

Transfer Standard: A laboratory standard that may not be suitable as a primary standard, but is traceable

to a primary standard through calibration. Transfer standards are sometimes used in place of the primary

standard in calibrations, because they are usually quicker and easier to use, but they also add additional

uncertainty to the calibration. Transfer standards are sometimes called secondary standards.

Turbomolecular pump: The turbomolecular pump, sometimes abbreviated as turbo pump, is a selfcontained high-vacuum pumping station that is used as a roughing pump for the LOCR. The station

includes (a) a turbomolecular pump, which is a high-speed turbine and is the main pumping mechanism,

(b) the control electronics, and (c) a rotary-vane backing pump. Pressure gauges are attached to the

vacuum system plumbing above the turbomolecular pump; the plumbing serves as a foreline when the

LOCR is evacuated, so the gauges indirectly measure the pressure in the LOCR’s cryostat during

evacuation. Protection circuitry and electrically controlled valves make the pump safe to use, but care

must be taken to prevent damage to the pump and to keep the inside of the vacuum system components

clean and dry. Some oil vapor is released by the rotary-vane backing pump, so the pump exhaust is

vented to the outside of the building.
76

APPENDIX B. Instructions for Submitting Meters for Calibration

1. Select the calibration service from the NIST Calibration Group Calibration Services Users Guide SP

250 Appendix Fee Schedule. The guide is available in the following ways:

A. From calibration services office:

Calibration Services

National Institute of Standards and Technology

100 Bureau Drive, Stop 2330

Gaithersburg, MD 20899-2330

301-975-2002 phone

301-869-3548 fax

calibrations@nist.gov

B. From the NIST calibration service web site:

http://ts.nist.gov/ts/htdocs/230/233/calibrations/.

The high-accuracy laser power and energy calibration service is Service ID Number 42162S, “Special

Test for High Accuracy Laser and Optical Fiber Power Measurements,” which is listed in the “Lasers and

Optoelectronic Components Used with Lasers” portion of the “Optical Radiation Measurements” section. 

The service is listed in chapter 7 section E of the guide, and at the web site:

http://ts.nist.gov/ts/htdocs/230/233/calibrations/optical-rad/laseroptoelectronic.htm.

2. Set a calibration date by contacting the personnel responsible for the desired calibration service, given

with the description of the calibration services.

3. Specific technical information is required from the customer. If some of the information is omitted, a

best-guess assumption will be used and documented in the calibration report. When the high-accuracy

calibration service is desired, the following information is requested:

A. The desired laser wavelength. We currently support vacuum wavelengths of 458.06,

476.62, 488.13, 496.65, 514.67, 632.99, 1064.42, 1343.09 and 1550.43 nm.

B. The desired power level in the range from 0.1 to 1.0 mW. Other power levels are

negotiable.

C. The desired beam size to be used with the detector. We use approximately circular

Gaussian-profile beams. Please specify the 1/e2 intensity width desired at the detector’s

input aperture. The divergence angle or radius of curvature at the detector’s entrance

aperture can also be specified, but the beam size may have to be changed to support the

desired divergence.

D. Detector cleanliness is important for high-accuracy calibrations, since a speck of dust

on the detector can cause a calibration error larger than the stated uncertainty. So NIST

will normally blow out the detector with filtered compressed air. The customer should
77

inform NIST if their detector can not be cleaned in this way, then the cleaning step will

be skipped for their detector. 

E. The desired detector orientation. The system uses a polarized beam, so detector

orientation can be significant.

F. The desired detector alignment technique. How the detector should be aligned

relative to the laser beam should be specified.

G. A complete description of how the meter should be setup before the calibration, such

as the desired initialization routine for any coupled electronics.

H. A complete description of how the meter’s output data should be read and processed,

including desired time intervals. We can use single-, dual-, or no-baseline analysis

techniques or the corrected-rise algorithm. The customer should specify which technique

to use, and the desired settling, rating, sampling, and injection intervals.

I. Instructions on the use of any ancillary components needed for the meter, such as

external temperature controllers or optical choppers.

J. Exceptions to the above, including using other power levels and wavelengths, and

using NIST meters and equipment with customer’s detector, will be considered on a percustomer basis.

4. Shipping instructions. Meters sent to NIST for calibration should be shipped in well-padded foam or

otherwise mechanical-shock-insulated case, appropriate for return shipment back to the customer. 

Operation instructions or instruction manuals should be included, as well as the customer-chosen set-up

parameters for instrument functions. The customer should include all cables and connectors that are

necessary to calibrate the equipment as specified. Exceptions and special cases will be considered on a

per-customer basis.
78

APPENDIX C. Sample Calibration Report

A sample calibration report is shown on the following pages. The report details a calibration where the

customer provided three photodiode-based detectors and one ammeter, which were calibrated as a unit

(each detector was in turn calibrated using the single ammeter). The report for other types of meters will

differ slightly. For confidentiality, places where customer-specific information would be given are shown

as a description within square brackets; the symbol # is used to signify where an alpha-numeric entry

would occur. The bottom-centered page number is an artifact of this document and would not appear in

an actual report.

The calibration equation and uncertainty analysis terms given in the report can change slightly depending

on the specifics of the calibration. However, the calibration equation is always consistent with eq (8) or

eq (9), depending on whether the meter measures power or energy. In the example, each detector is

combined with the ammeter to form a single unit that measures optical power. The ammeter is read

through its GPIB output, so no analog voltage is measured. Therefore the calibration equation given in

the report is the same as eq (8), with G = 1, OM = IM, and kT = 1. Had a NIST device, such as a current-tovoltage converter with analog voltage output, been used with the detectors, G and kT

 would be included in

the calibration equation and in the uncertainty analysis.

Other information that changes in each report includes the measured optical power and detector

calibration factors, the environmental conditions, the descriptions of the optical source, and the specifics

of the detector alignment. Depending on the specifics of the calibration, additional measurements may be

performed, such as a bias voltage measurement.

Each report is for calibrations performed at a single wavelength. Calibrations performed at additional

wavelengths are reported separately, since each wavelength is a separate calibration, performed at a

different time, possibly with a different laser. The environment parameters, beam parameters, detector

parameters, and voltmeter uncertainty can therefore differ for each wavelength. To avoid the complexity

of detailing all this different information in a single report, calibrations at additional wavelengths are

reported separately.
Folder No. & NIST ID: [######-##] & [######]

Date of Report: [Report completion date]

Test Number: [#####]

Reference: [Purchase order number, submission date] Page # of #

79

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

ELECTRONICS & ELECTRICAL ENGINEERING LABORATORY

Boulder, Colorado 80305

Report of Calibration

for

LASER POWER METER

[Manufacturer name] Model: [model #], Serial No: 1, NIST ID: [#####]

[Manufacturer name] Model: [model #], Serial No: 2, NIST ID: [#####]

[Manufacturer name] Model: [model #], Serial No: 3, NIST ID: [#####]

with [Manufacturer name] PICOAMMETER

Model [####], Serial No: [#####]

Submitted By:

[Customer Name]

[Customer Address line 1]

[Customer Address line 2]

Measurement Summary

The laser power measurement system supplied by the customer consisted of a picoammeter, a

connecting cable with switch box, and three detectors called [model #]. Each detector was calibrated

by comparison to the NIST Laser Optimized Cryogenic Radiometer (LOCR). The measurement results

are provided in Table 1. The comparison was performed by direct substitution of the standard with the

test detector, using the calibration system shown in Figure 1. Each detector was calibrated, in turn,

Detector Serial

Number N

Applied Power Result

Average

(mW)

Expanded

Uncertainty

(%), (k = 2)

Average

(A/W)

Standard

Deviation

(%)

Expanded

Uncertainty

(%), (k = 2)

1 12 1.00063 0.023 -0.50937 0.0087 0.023

1 6 0.099671 0.025 -0.50945 0.0127 0.025

2 6 1.00077 0.022 -0.50924 0.0042 0.023

2 6 0.099692 0.023 -0.50932 0.0070 0.024

3 6 1.00085 0.022 -0.50831 0.0071 0.023

3 6 0.099689 0.024 -0.50832 0.0090 0.024

Table 1: Calibration Results (Wavelength = 633 nm).
LASER POWER METER

Model: [Manufacturer, Model #], Serial No: 1, NIST ID: [#####]

Model: [Manufacturer, Model #], Serial No: 1, NIST ID: [#####]

Model: [Manufacturer, Model #], Serial No: 1, NIST ID: [#####]

[Manufacturer name] Picoammeter, Model [###], Serial No: [#####]

Folder No. & NIST ID: [######-##] & [######]

Date of Report: [Report completion date]

Test Number: [#####]

Reference: [Purchase order number, submission date] Page # of #

80

Figure 1. Measurement system diagram.

using the supplied picoammeter and connecting cable with switch box. The calibration system used a

power stabilizer to stabilize the absolute power and a spatial filter to remove scattered light. The

optical power applied to the test detector was calculated by interpolating between power measurements

performed with the standard before and after the test detector measurement, and then applying the

appropriate correction factors. 

The calibration was performed using a helium-neon gas laser, with a nominal center vacuum

wavelength of 632.99217 ± 0.00017 nm. The laser radiation was contained in a single spectral line

having a half-power linewidth of less than 0.004 nm. The laser beam was vertically polarized and had

an approximately Gaussian intensity profile with 1/e2 diameter of 1.97 ± 0.10 mm at the detector’s

entrance aperture. The full-width beam divergence was approximately 0.7 mrad at the entrance. The

beam was approximately equivalent to an ideal Gaussian beam, having a waist of diameter 1.0 mm,

located 2.1 m in front of the detector’s entrance aperture.

The detectors were oriented so that the NIST ID tag was at the top, and the electrical output connector

was on the right side, when facing the detector’s entrance aperture. The end of the detector displaying

the manufacturer’s label was used as the entrance. The top of the detector was leveled to within ± 0.2°. 

The detectors were aligned to the beam visually by adjusting the detector so that the beam passed

through both the front and back irises, when these openings were minimized. The detectors were then

finely aligned so that with the irises open, the transmitted beam had a diamond shape with an

approximately uniform illumination, and so that the irises closed around the center of the transmitted
LASER POWER METER

Model: [Manufacturer, Model #], Serial No: 1, NIST ID: [#####]

Model: [Manufacturer, Model #], Serial No: 1, NIST ID: [#####]

Model: [Manufacturer, Model #], Serial No: 1, NIST ID: [#####]

[Manufacturer name] Picoammeter, Model [###], Serial No: [#####]

Folder No. & NIST ID: [######-##] & [######]

Date of Report: [Report completion date]

Test Number: [#####]

Reference: [Purchase order number, submission date] Page # of #

81

beam. All of the switches on the connecting cable’s switch box were turned on, except for the bias

switch, which was turned off. Both the front and back irises were fully open during the calibrations.

Each detector’s output was measured using the [Manufacturer, model #] picoammeter’s GPIB port. 

Three strings were used to initialize the instrument: “L0X”, “C0G1T5W0Y3X” and “R0X”. These

commands activated auto-ranging, among other things. To initiate a reading, the command “X” was

sent. Each detector’s output was processed with a dual-baseline method using 20 s minimum rating

periods and 1 s minimum settling periods. A sampling rate of 1 sample per second was used. The

environment near the detectors during the calibration was: temperature = 21.63 ± 0.72° C, pressure =

836 ± 11 hPa, relative humidity = 35.0 ± 6.2 %, and air index of refraction = 1.0002200 ± 0.0000036. 

The tolerances given in the text are expanded uncertainties with a coverage factor of k = 2.

The calibration factor for the detector was calculated by dividing the detector’s output current by the

optical power delivered to the detector. Four correction factors were used: the LOCR window

transmittance (TW), the LOCR receiver absorptance (AR), the relative aperture transmittance (TA), and

the LOCR electrical calibration (kL). The detector’s calibration factor (CF) in amps/watt is given by

the equation:

where PS is the applied power in watts, interpolated from bracketing primary standard measurements,

and IM is the detector output in amps. If the output current of the detector is divided by the calibration

factor listed in Table 1, the resulting values will agree with those of the NIST measurement system.

Uncertainty Assessment

Uncertainty estimates for the values given in the NIST calibration report are assessed using the NIST

Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results (Technical

Note 1297). Uncertainty estimates for the values given in the NIST calibration report are assessed

using the following technique, which is derived from the official NIST guidelines. Each component of

uncertainty that contributes to the uncertainty of the measurement result is represented by an estimated

standard deviation, termed the standard uncertainty u. The combined standard uncertainty uc, which

represents the estimated standard deviation of the result, is obtained by combining the individual

standard uncertainties. The standard uncertainties are estimated from Type A evaluations, where

magnitudes are obtained statistically from a series of measurements; and from Type B evaluations,

whose magnitudes are based on scientific judgement using all the relevant information available. 

The Type A uncertainty components are assumed to be independent; and consequently, the standard

deviation, Sr, for each component is
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82

(10)

(11)

where the xi values represent the individual measurements and N is the number of xi values used for a

particular Type A component. The standard uncertainty of each Type A component is the standard

deviation of the mean, given by Sr/N

1⁄2

. The total standard uncertainty of the Type A components is

[E(Sr

2

/N)]1⁄2, where the summation is carried out for all Type A components.

Most of the Type B components are assumed to be independent and have rectangular or uniform

distributions (that is, each has an equal probability of being within the region, ±*s, and zero probability

of being outside that region). If the distribution is rectangular, the standard uncertainty for each

Type B component is equal to *s/31⁄2, and the total standard uncertainty for all Type B components is

(EFs

2

)

1⁄2

, where the summation is carried out for all Type B components.

The combined standard uncertainty is determined by combining the Type A and Type B standard

deviations in quadrature; the expanded uncertainty is obtained by multiplying this result by a coverage

factor of k = 2. The expanded uncertainty U is then 

The values used to calculate the NIST expanded uncertainty (shown in Table 1) are listed in Table 2. 

The number of decimal places used in reporting the mean values of the calibration factor listed in

Table 1 were determined by expressing the expanded NIST uncertainty to two significant digits.

The correction factors, their normalized standard uncertainties, and the combined standard uncertainty

for the applied power and the calibration factor calculation are given in Tables 2 and 3, for nominal

power levels of 1.0 and 0.1 mW, respectively. The combined standard uncertainty of the applied

power is the quadrature sum of the common uncertainties and the applied power measurement’s

repeatability; the combined standard uncertainty of the calibration factor is the quadrature sum of the

common uncertainties, the calibration factor measurement’s repeatability, and the test detector

ammeter quantization uncertainty. The two measurement repeatabilities differ slightly, depending on

the amount of test detector noise and the effectiveness of the power interpolation. An uncertainty for

quantization in the ammeter was assessed, because the ammeter’s five-digit readings varied

insufficiently to randomize the uncertainty.
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83

Component of

Uncertainty

Detector Serial Number

1 2 3

Correction

Factor

Std. Unc.

(%)

Correction

Factor

Std. Unc.

(%)

Correction

Factor

Std. Unc.

(%)

LOCR Receiver

Absorptance (AR)

0.99992 0.0002 0.99992 0.0002 0.99992 0.0002

LOCR Window

Transmittance (TW )

0.99970 0.0072 0.99970 0.0072 0.99970 0.0072

LOCR Electrical

Calibration (kL)

1.00000 0.0030 1.00000 0.0030 1.00000 0.0030

LOCR Alignment 1.00000 0.0075 1.00000 0.0075 1.00000 0.0075

LOCR Electrical

Heating Inequivalence

1.00000 0.0002 1.00000 0.0002 1.00000 0.0002

Relative Aperture

Transmittance (TRA)

1.00003 0.0033 1.00003 0.0033 1.00003 0.0033

Measurement Repeatability of the

Applied Power

 0.0016* 0.0013** 0.0016**

Combined Standard Uncertainty of the

Applied Power

0.0115 0.0114 0.0115

Ammeter Quantization 1.00000 0.0006 1.00000 0.0006 1.00000 0.0006

Measurement Repeatability of the

Calibration Factor

 0.0025* 0.0017** 0.0029**

Combined Standard Uncertainty of the

Calibration Factor

0.0116 0.0115 0.0117

* - Type A evaluation with N = 12. ** - Type A evaluation with N = 6.

Table 2. Uncertainty components for 1 mW calibrations.
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84

Component of

Uncertainty

Detector Serial Number

1 2 3

Correction

Factor

Std. Unc.

(%)

Correction

Factor

Std. Unc.

(%)

Correction

Factor

Std. Unc.

(%)

LOCR Receiver

Absorptance (AR)

0.99992 0.0002 0.99992 0.0002 0.99992 0.0002

LOCR Window

Transmittance (TW )

0.99970 0.0072 0.99970 0.0072 0.99970 0.0072

LOCR Electrical

Calibration (kL)

1.00000 0.0032 1.00000 0.0032 1.00000 0.0032

LOCR Alignment 1.00000 0.0075 1.00000 0.0075 1.00000 0.0075

LOCR Electrical

Heating Inequivalence

1.00000 0.0002 1.00000 0.0002 1.00000 0.0002

Relative Aperture

Transmittance (TRA)

1.00003 0.0033 1.00003 0.0033 1.00003 0.0033

Measurement Repeatability of the

Applied Power

 0.0051* 0.0019* 0.0040*

Combined Standard Uncertainty of the

Applied Power

0.0125 0.0116 0.0121

Ammeter Quantization 0.0006 0.0006 0.0006

Measurement Repeatability of the

Calibration Factor

 0.0052* 0.0029* 0.0037*

Combined Standard Uncertainty of the

Calibration Factor

0.0125 0.0118 0.0120

* - Type A evaluation with N = 6.

Table 3. Uncertainty components for 0.1 mW calibrations.
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85

For the Director,

National Institute of Standards and Technology Report Reviewed By:

Marla L. Dowell, Group Leader John H. Lehman, Project Leader 

Sources and Detectors Group Sources and Detectors Group

Optoelectronics Division Optoelectronics Division

Report Prepared / Calibrated By:

David J. Livigni, Electronics Engineer

Sources and Detectors Group

Optoelectronics Division
86

Figure D.1 Thin lens Gaussian optics model showing the 1550 nm optical source; mirrors,

polarizers, power modulator elements, and aperture stops are not shown.

(D!1)

APPENDIX D. Gaussian Beam Model and Spatial Filter Pinhole Selection

The optical elements considered in the Gaussian beam model are shown in Figure D.1. Elements that do

not affect the shape of the beam delivered to the detector plane, such as the beam steering mirrors,

polarizers, and power stabilizer components, are not modeled. Aperture stops, except for the spatial filter

pinhole, are also not shown; diffraction affects caused by the finite apertures are not considered in the

model because the apertures are assumed to be sufficiently larger than the beam so that any resulting

artifacts are small. Any such artifacts that do occur are compensated by the relative aperture

transmittance correction.

The laser beams used in the high-accuracy calibration system are approximately Gaussian beams. 

Gaussian beams have a Gaussian intensity profile, and propagate in a predictable manner from their waist,

where the wavefront is planar. Diffraction causes the beam to spread and diverge as it propagates from

the waist, with smaller beams diverging more than larger beams. In classical geometrical optics analysis,

diffraction is ignored, which leads to a simple but potentially inaccurate model. A Gaussian beam raytracing analysis leads to a much more complex but also more accurate solution. In the following sections,

both geometrical and Gaussian beam models are derived and contrasted, to show why the Gaussian model

is necessary. The spatial filter pinhole selection criterion is also described.

D.1 Geometrical Optics Analysis

Using a geometrical optics analysis, the lens system shown in Figure D.1 is very simple. The distance l

0

is chosen so that a geometrically collimated beam propagates from Lens 0. Because diffraction is ignored

in geometrical optics analysis, the geometrically collimated beam has a constant size and a planar

wavefront at all distances beyond the lens. When the spacing between Lens 1 and Lens 2 is such that

l

1

' = f

1

, and l

2

 = f

2

, the geometrically collimated beam passes through the lenses with no change, except for

its size. The geometrically collimated beam then propagates from Lens 2 to the detector plane without

change.

More specifically, using geometrical optics, the diameter DO of the collimated beam propagating from

Lens 2 is related to the diameter of the collimated beam D input to Lens 1 by the following equation,

where f

1

 is the focal length of Lens 1, and f

2

 is the focal length of Lens 2:
87

(D!2)

The resulting magnification of the collimated beam is given by M = DO/D = f

2

/f

1

. The diameter of the

beam output by the two lenses of the spatial filter can therefore be adjusted by selecting the ratio of the

focal lengths of the lenses. This equation can be used to select Lens 1 and Lens 2 from the limited

selection of lenses, so that D0 is close to the desired beam diameter at the detector plane. The size of the

beam in the detector plane can be further adjusted by deviating from geometrical collimation, by

adjusting the distance between Lens 1 and Lens 2.

Lens 1 and 2 form a compound thick-lens system. When the thin-lens approximation is used for the

elements of the compound lens, the focal length of the compound lens (fC) is given by the following

equation [24, 25], where d is the distance between the lenses:

When the beam input to the compound lens is geometrically collimated, the beam output by the

compound lens is also geometrically collimated when d = f

1 + f2

, and fC = 4. The size of the geometrically

collimated beam does not change as the beam propagates away from Lens 2, so the beam diameter in the

detector plane is DO. Increasing the lens spacing d brings the focal length of the compound lens in from

minus infinity, so the beam converges towards the detector plane and has a smaller diameter there. As d

is increased farther, the minimum beam diameter occurs when the beam is focused on the detector plane. 

The beam diameter in the detector plane can therefore be controlled by adjusting the lens spacing d, at the

expense of the beam’s collimation. In practice, d is not increased beyond the point where the beam

focuses on the detector plane. To obtain a larger beam, d is decreased from the collimated length, the

focal point comes in from plus infinity, so the beam diverges towards the detector plane, and the resulting

beam diameter in the detector plane is larger.

While this geometrical analysis accurately describes the mechanisms used in the optical system,

diffraction causes the actual detector plane beam diameter and radius of curvature to be different from

those predicted by the geometrical. Therefore a more complex model which accounts for diffraction is

needed.

D.2 Gaussian Beam Ray-Tracing Analysis

Since the actual optical system uses approximately Gaussian beams, an analysis using Gaussian beams

that includes diffraction affects is more accurate than the geometrical analysis. A ray-tracing analysis

using the ABCD law for Gaussian beams [26] is used to model the system. The model predicts the

behavior of the real system with good accuracy, because the laser beams used in the system are a good

approximation of a true Gaussian beam. Measurements performed with the actual system confirm that the

Gaussian beam analysis predicts the system’s true performance to within a few percent.

A Gaussian beam is defined by its waist, where the beam diameter is a local minimum, and the radius of

curvature of the wave front is infinite (the wave front is planar). As the beam propagates away from the

waist, diffraction causes the beam to spread and diverge. The way the beam propagates from the waist is

known, so only the waist location, size, and direction of propagation are needed to completely model the

beam. A true Gaussian beam has a Gaussian intensity profile everywhere along its length, with radial

symmetry around its propagation axis. The ray-tracing analysis follows the 1/e2 intensity point of the

beam profile; the 1/e2 intensity radius (w0) is used to define the beam’s width at the waist. 

Each lens in the system is analyzed separately as a region or cell, and each region is represented by its

own transform matrix. Transform matrixes for the separate regions are combined by multiplying the
88

(D!3)

matrixes together. The beam travels down the z-axis, which is zero at the input to the lens in the last

region. The transform matrix is accurate for z $ 0 (in the region at and past the lens); the matrix for the

previous region is used to predict the behavior before the lens. 

To simplify the analysis, the thin-lens approximation is used. The thin-lens approximation is adequate

because the distances between the principal planes of the thick lenses is small compared to the lens

spacing, and the beam spreading caused by diffraction within the lenses is small compared to the beam

diameter. An analysis that includes the thick lenses can be performed using dedicated optical analysis

software, but details about the doublet lens’s type of glasses and radiuses of curvature would be required.

Each lens in the optical system and the space around it is considered separately, as a cell. Each lens is

presented with an object, which is the beam waist produced by the optical source, or the image waist

produced by the previous cell. Each lens projects an image of its object waist. Positive lenses are used,

so the image waist is usually a real image, projected on the side of the lens opposite the object. The raytracing analysis starts at the optical source’s beam waist, where the beam parameter q is given by

1/q = !j(80/Bnw0

2

), where 80 is the vacuum wavelength of the light, and n is the index of refraction of the

surrounding media. The lenses are in air, so n = 1.0003 is sufficient for these calculations, though a value

of 1.0002 is more accurate for the actual laboratory at high altitude.

The lenses and aperture stops must have a clear aperture large enough to pass the Gaussian beam with

negligible clipping, or else the Gaussian beam will be distorted. A rule of thumb is to use components

with a clear aperture diameter at least 1.5 times the 1/e2 beam diameter at the component, so as to

intercept at least 99 % of the incoming intensity [26]. In the real system, a more conservative factor is

used to reduce any degradation of the approximately Gaussian beam, so aperture diameters at least three

times the beam diameter are preferred. One exception is the final aperture, which is small enough to

block the stray beam produced by the system’s polarizer. Other exceptions are the fixed apertures on the

power stabilizer components.

The lenses used in the calibration system have a clear aperture of over 20 mm, and so their diameters are

10 times that of the 2 mm diameter beam typically encountered at the lenses. However, the fixed

apertures on the LPC modulator and monitor may cause problems with larger beams, since their input

aperture diameter is 4 mm, and their output aperture diameter is 6 mm. The largest beam that can be

passed through the optical system is limited by these apertures.

For each cell m, the beam propagates through air a distance of lm from the object beam waist to the thin

lens, which has focal length fm. The object beam waist has a 1/e2 beam radius of w0m. After the lens, the

beam propagates through air an arbitrary distance z. Transmission matrices for the three regions in the

cell, the object space, the thin lens, and the image space, are combined to form a transmission matrix for

the cell. For the cell transmission matrix, z = 0 occurs at the input to the last region, which is the output

of the thin lens in this case. The beam size and radius of curvature at an arbitrary distance past the thin

lens are derived from the cell transmission matrix, using the ABCD law for Gaussian beams. It is

convenient to define a characteristic length z0m for each cell m, where

The analysis gives the following general solutions for the size (wm(z)) of the transmitted beam at a

distance of z from lens m:
89

(D!4)

(D!5)

(D!7)

(D!6)

The radius of curvature of the wave front (Rm(z)) at a distance of z from lens m is given by

The image waist occurrs a distance of l'm from the lens, on the opposite side of the lens from the object. 

A negative l'm means that a virtual image waist occurs on the same side of the lens as the object waist. 

The location of the image waist is determined by solving for where the radius of curvature is infinite

(1/Rm(z) = 0). The resulting image distance is given by

Equation (D!6) can be expressed in the following form, which is similar to the classic lens equation [25],

derived using geometrical optics analysis. This equation may therefore be considered a modified lens

equation for Gaussian beams [27]:

The sign of the second term differs from that in the referenced lens equation because of differences in the

sign conventions used. Here, the distance from the thin lens to the object (lm) is considered positive,

while in the reference the distance is considered negative; the difference results in the sign change in the

referenced equation. Also, since the thin lens is in air, fm' = fm.

Equation (D!7) shows that when |lm - fm| >> z0m

2

, the system conforms to the geometrical optics analysis,

and the Gaussian beam waist is imaged like any other object. But in the region around lm = fm, the

geometrical optics solution has appreciable error. All the lenses in the calibration system are used with

lm . fm, so the Gaussian beam optics solution is needed to obtain accurate results. It should be noted that

when the above equations are numerically evaluated, the semi quadratic form shown in eq (D!6) is

preferable to the form shown in eq (D!7) because the latter has a division by zero when lm = fm, which can

cause problems in the numerical computations and can cause excessive roundoff error when lm is close to

fm, and 1/(lm - fm) is very large.
90

(D!8)

Several interesting results are obtained from the Gaussian beam analysis, perhaps the most significant of

which is how a Gaussian beam is collimated. A geometrically collimated beam is usually thought of as

having an infinite radius of curvature, or planar wave front and finite extent at all distances. The

Gaussian beam waist is the only place in a Gaussian beam which meets this criterion if the requirement of

finite extent is neglected. So in a sense, all Gaussian beams are collimated, at least at their waist. Around

the waist, the beam is considered collimated within the Rayleigh range of the waist [28], which is given

by zR = z0m. Since all Gaussian beams are collimated near their waist, of what distance should the lenses

of the spatial filter be spaced to collimate the transmitted Gaussian beam? 

The transmitted Gaussian beam is considered optimally collimated when it has the lowest divergence in

the far field, which occurs when lm = fm . When lm = fm, the object beam waist is placed at the primary

focal point of lens m, where the point source is placed in geometrical collimation. With such an optimally

collimated Gaussian beam, the image waist is projected to the lens’s secondary focal point, and not to

infinity as predicted by the geometrical analysis. This condition results in the largest image beam waist,

which diverges the least since large Gaussian beam waists diverge more slowly than smaller waists, so in

the far field the smallest diameter beam and smallest divergence angle is produced when the waist is

largest. The largest beam waist also produces the longest Rayleigh range. However, in the real system,

since the detector plane is not very far from the lenses, a smaller diameter detector plane beam can usually

be produced when Lens 2 is in a position slightly different from the optimal collimation location, similar

to the way the focused beam is smaller than the collimated beam in the geometrical optics analysis. The

size of the optimally collimated beam waist image is given by wCm = wm(fm), where lm = fm, or

Another interesting result of the Gaussian beam analysis is how under certain circumstances (when 

|lm ! fm| >> z0m

2

), the Gaussian beam waist is imaged in accordance with geometric optics. In this case,

the lenses in the spatial filter can be considered as a compound lens with a variable focal length, as

previously described, that images the laser source’s waist. Then the image and object distances are

related by the focal length of the compound lens, in accordance with the classic lens equation, and the

image distance can be increased by increasing the object distance. This result is significant in that it

shows how in the calibration system, the final image beam waist can be projected farther from Lens 2 by

moving the laser source farther from Lens 1. Therefore by moving the laser source farther away, it is

possible to project the image waist all the way to the detector plane and beyond. If the image waist is

projected to the detector plane, the detector will be illuminated with a true planar wave front. This effect

is demonstrated later in this section; the analysis uses the Gaussian beam result, because using the

geometrical optics analysis on the compound lens is greatly complicated by the shift in the location of the

principle planes of the compound lens when the focal length is adjusted (the compound lens cannot be

considered “thin” when the laser and detector plane are at reasonable distances). However, understanding

the relationship to geometric optics does provide some insight into the seemingly bizarre behavior of the

Gaussian beam. 

Another interesting result of this analysis is that a Gaussian beam waist image can never be projected to

infinity. Doing so would require an infinite object waist distance, in which case diffraction would also

result in an infinitely large beam. Since an infinite object distance cannot be achieved, this condition

cannot be produced in the laboratory.

The high-accuracy calibration system’s optical system is shown with Gaussian beams in Figure D.1. In

the figure, the rays shown follow the 1/e2 intensity points of the Gaussian beam. The figure shows all the
91

lenses in the system, the image and object distances (lm and l'm) for each cell m, and the distances d1 (the

distance between the laser’s output and Lens 1) and d2 (the distance from Lens 2 to the detector plane). 

The distance d1 is different for each laser, but d2 is considered to be constant even though it changes

slightly when Lens 2's position is adjusted, because d2 is much greater than the amount of adjustment. 

The mirrors, polarizers, and other elements of the optical system are not shown in the figure, because they

do not significantly affect the Gaussian beam.

The 1550 nm optical fiber coupled laser is depicted in Figure D.1; this laser has a single-mode,

polarization-maintaining optical fiber pigtail that is permanently attached to the laser’s gain medium. The

single-mode optical fiber core produces a very nearly Gaussian profile at the end of the optical fiber

where the core is cleaved [29]. The mode field diameter of the single-mode optical fiber core is used as

the Gaussian beam waist diameter [8]. A collimating lens (Lens 0 in Figure D.1) is used to generate an

optimally collimated, free-space beam from the optical fiber. Optimal collimation is achieved by

positioning Lens 0 so that the end of the optical fiber’s core is at the lens’s primary focal point; then the

image waist is formed at the secondary focal point.

The other lasers used in the system have a free-space output with a single transverse mode, TEM00, which

has a flux density that is ideally Gaussian over the beam’s cross section [30]. With these lasers, the beam

waist generated inside the laser’s resonant cavity is analogous to the waist in the optical fiber core, and

the laser’s collimating optic is analogous to Lens 0. The fiber-coupled laser is used in this analysis

because its beam waist diameter and collimating lens focal length are known, where they are unknown for

the free-space lasers. The free-space lasers are analyzed with the same technique, by estimating the

values for cell 0. For these lasers, the output beam diameter specified by the manufacturer is assumed to

be the 1/e2 intensity diameter of the waist image produced by the laser’s collimating optic. And the

distance l'

0

 is considered negligible compared to d1 so l

1 . d1, and the waist is assumed to occur at the

laser’s output. The analysis begins in cell 1, using one-half of the manufacturer’s specified beam

diameter as w01. The beam diameter at Lens 1 (D in Figure D.1) is measured, and the values of l

1

and/or

w01 are adjusted so that the theoretical value of D is consistent with the measured value of D. The

discrepancy between the theoretical and measured values of D is due in part to imperfections in the

approximately Gaussian beam, as well as to the assumptions made about w00 and l'

0

, so the adjustment is

usually necessary even with optical fiber-coupled sources.

The parameters of interest for each cell in Figure D.1 are determined as follows. In cell 0, the initial

Gaussian beam waist (w00) is produced at the end of the optical fiber, and has width determined from the

optical fiber’s mode-field diameter, which for the 1550 nm laser is 9.5 :m. The equivalent Gaussian

beam waist radius is thus 4.75 :m, (one-half of the 1/e2 diameter of the mode field). Lens 0 is an antireflection coated glass aspherical lens with focal length of 6.2 mm, so f

0

 = 6.2 mm. The collimator is

assumed to optimally collimate the beam, so l

0

 = l'

0

 = f

0

. The image beam waist’s size is w01 = wC0, from

eq (D!8) with m = 0, and w01 is the input to cell 1. The diameter at the input of Lens 1 (D) can be

calculated using the matrix for cell 0 with a distance of z = d1, so D = 2w0(d1), which is equivalent to the

matrix for cell 1 with z = 0 or D = 2w1(0). In cell 1, the object waist distance is given by: l

1

 = d1 - l'

0

, and

the image distance (l'

1

 ) is given by eq (D!6) with m = 1. The size of the image waist projected by Lens 1

is given by w02 = w1( l'

1

 ) = wC1. The spatial filter pinhole is placed around w02, where it primarily limits

the field of view of the system; ideally, the Gaussian beam is not affected by the pinhole, as previously

described. The size of the spatial filter pinhole does not affect the spatial filter’s numerical aperture or fnumber, which is a function of the lens diameter and focal length.

Lens 2 is initially placed so that the transmitted beam is optimally collimated, by placing the lens one

focal length away from the spatial filter pinhole, or minimizing the spot size in the far field. Precise

positioning of Lens 2 is not necessary at this point in the alignment, so approximate optimal collimation is
92

Figure D.2 The size (radius) of the Gaussian beam in the detector plane versus the

displacement of Lens 2 from the optimally collimated location.

sufficient. When optimally collimated, the object and image waist distances are equal to the focal length

of the lens, so l

2

 = l'

2

 = f

2

. In the detector plane, the beam size (wdp) and radius of curvature (Rdp) are given

by eq (D!10) and eq (D!11) with m = 2 and z = d2. However, the system is seldom used with the output

beam optimally collimated; in practice the detector plane beam size is adjusted by changing the distance

between Lens 2 and w02, which is l

2

.

The result of adjusting l

2

 on the beam size and radius of curvature in the detector plane (wdp and Rdp), can

be determined by substituting, in eq (D!4) and eq (D!5), l

2

 with f

2

 + *, and z with d2. Then, * is the

amount Lens 2 is moved from the optimally collimated position. The result of the substitution on wdp and

Rdp is shown in Figures D.2 and D.3, using real values for the parameters: 80 = 1550 nm, n = 1.00027,

w00 = 4.75 :m, f

0

 = l

0

 = 6.2 mm, d1 = 1.04 m, f

1

 = f

2

 = 0.1 m, l

2

 = f

2

 + *, and d2 = 1.28 m. While these

curves can change significantly with different sources or lenses, all of the lasers currently used in the

system have a similar relationship between * and wdp, which is convenient for producing the 2 mm

diameter beam in the detector plane (wdp = 1 mm). To date, a detector plane beam size of only a 2 mm

has been used with the system.

Figure D.2 shows that a nearly linear relationship between * and wdp results. However, Figure D.3 shows

that if the detector plane beam size is reduced to its minimum, a shorter radius of curvature results. How

the projected waist image distance (l'

2

) varies with * is shown in Figure D.4, using eq (D!6) with m = 2

and l

2

 = f

2

 + *. The largest waist is projected to l'

2

 = f

2

 when * = 0 (the beam is optimally collimated). 

The smallest wdp is obtained with a positive *, which projects a smaller beam waist closer to the detector

plane. Increasing * farther makes the image waist even smaller, but moves it back towards the lens,

resulting in an increased detector-plane beam diameter, but with a shorter radius of curvature. So to

increase the detector-plane beam size, a negative * is used, which causes a smaller waist to be projected

farther from the detector plane, and the smaller waist diverges more, resulting in a larger beam in the

detector plane. A sufficiently large negative * causes the waist image to be projected to the left of Lens 2,

resulting in a virtual waist image. However with a negative *, the resulting detector plane beam has a
93

Figure D.3 The radius of curvature of the wave front in the detector plane versus the

displacement of Lens 2 from the optimally collimated location.

longer radius of curvature than it would with a large positive *, so using a negative * is the preferred

method for expanding the beam.

Using an increasingly negative * will further expand the size of the detector plane beam, but at the

expense of a shorter radius of curvature. Additionally, the physical layout of the lens and pinhole limit

how negative * can be. So if a larger beam is required, it is better to adjust other parameters in the system

(such as the lens focal lengths or d1), and keep the system near the point of optimal collimation by

limiting the range of * that is used. The fixed apertures on the LPC monitor element will ultimately limit

the maximum beam size that can be supported without excessive diffraction artifacts. Similarly, if a

detector plane beam size that is smaller than that achieved with a positive * is required, other parameters

must be adjusted to reduce the minimum beam size.

The longest radius of curvature shown in Figure D.3 occurs when * is adjusted so that the image waist is

projected the longest distance, as shown in Figure D.4 for the same *. The radius of curvature in this

configuration is long, because the beam waist is projected close to the detector plane. If the waist could

be projected a little farther so that it occurs at the detector plane, the radius of curvature would approach

infinity. Recall the result which showed that in certain circumstances, the spatial filter lens system can be

considered a compound lens that produces a geometric image of the laser’s waist, so by increasing the

object waist distance, the image waist distance should also increase. This result is confirmed by the

Gaussian beam analysis; increasing the distance between the laser’s output and Lens 1 (d1 in Figure D.1),

does indeed increase the distance to the final image waist (l

2

), for certain lens displacements *. For the

1550 nm laser source, increasing d1 by 1 m results in the detector plane radius of curvature shown in

Figure D.5. In this configuration, the waist can actually be projected behind the detector plane, so it

crosses the plane twice, with * .0.5 mm, and * .8 mm.
94

Figure D.4 The distance from Lens 2 to the projected waist image (l

2

') versus the displacement

of Lens 2 from the optimally collimated location.

Figure D.5 The radius of curvature of the wave front in the detector plane versus the

displacement in Lens 2 from the optimally collimated location, when the 1550 nm laser is moved

1 m farther away from Lens 1.

The absolute value of the radius of curvature is shown in Figure D.5, so that a logarithmic axis could be

used; the radius of curvature is actually negative when * is between 0.5 and 8 mm, because the waist

occurs behind the detector plane, and the rays are therefore converging as they approach the detector

plane, so the radius of curvature is negative. The detector-plane variation in beam size has a form very
95

(D!9)

similar to that shown in Figure D.2, except that the beam radius varies from 0.35 to 3.5 mm. As * is

increased from zero, the first time the waist crosses the detector plane it has a diameter of approximately 3

mm. As * is increased farther, the second time the waist crosses the detector plane its diameter is

approximately 0.8 mm. A similar situation occurs when the laser is placed in its original location and a

lens of 50 mm focal length is used as Lens 1, in place of the lens with 100 mm focal length.

This analysis shows that a wide range of detector plane beam diameters and radii of curvatures can be

obtained by adjusting the location of Lens 2. However, if the range of adjustment is not sufficient, other

parameters such as the lenses’ focal lengths and distance to the laser can be adjusted. However, the result

of changing the lens focal length and laser distance is complex, so an ad hoc, trial-and-error approach is

used to determine which configuration adequately provides the desired beam characteristics. Dedicated

mathematics software is used to evaluate the Gaussian beam equations that predict the behavior of the

system, so that several different configurations can be tested virtually before the actual system is aligned. 

After the desired configuration is chosen, the system is aligned and * adjusted to achieve the desired

beam diameter in the detector plane. If the beam waist is being projected to the detector plane, * is

adjusted to position the location of the beam waist, instead of the beam size.

The actual lens displacement * that is used in the system is determined experimentally by measuring the

detector plane beam diameter while * is adjusted. The theoretical value for * that is used in the final

Gaussian beam analysis is determined using the measured detector-plane beam diameter, so the

theoretical * is consistent with the average measured detector plane beam diameter. Imperfections in the

approximately Gaussian beam and lenses cause the actual system to behave slightly differently from the

theoretical system, but measurements performed to date show that the actual system usually performs to

within a few percent of the theoretical system.

D.3 Gaussian Beam Parameter Specification

Several parameters describing the Gaussian beam are provided in the calibration report. The incidence

angle of the beam, relative to the detector’s faceplate or entrance aperture is measured and reported. The

laser beam’s incidence angle is described as the difference from normal incidence. The incidence angle is

measured by holding a glass flat to the detector’s entrance aperture or faceplate, and locating the beam

reflected from the glass flat. The difference between the incidence angle and normal incidence (2I) is

calculated by measuring the distance between the incident and reflected beam (dI), at a distance (DI) in

front of the detector, using the equation:

The approximation is accurate when 2I is small. The incidence angle is minimized during detector

alignment, so is usually very small (the incidence angle is usually a fraction of a milliradian).

The full-width beam divergence angle at the detector’s entrance aperture is specified in the calibration

report to allow for simple geometrical modeling. To support a better Gaussian optics model, the distance

from the detector plane to the final beam waist (ddp in Figure D.1), and the final beam waist radius (w03 in

Figure D.1) are also specified in the report. These parameters are determined by the Gaussian beam

model using the theoretical lens displacement *, which is consistent with the average measured detector

plane beam diameter. Again, measurements have shown that the actual parameters usually agree within a

few percent with the theoretical parameters. 
96

(D!11)

(D!10)

(D!12)

The full-width beam divergence angle in the detector plane (2dp) can be derived from the radius of

curvature in the detector plane (Rdp), given by eq (D!5) with m = 2, l

2

 = f

2

 + *, and z = d2. The equation

relating 2dp to Rdp is given by the following; the approximation is accurate when 2dp is small:

When the beam waist occurs sufficiently far from the detector plane (ddp >> z03), the detector plane beam

divergence can be measured directly, by measuring the beam size in the detector plane (wdp), and at a

location a short distance d beyond the detector plane (wdp+d). The detector plane beam divergence is then

given by

The approximation is accurate for small 2dp. Similarly, the radius of curvature can be determined directly

from the measured diameters:

However, the analysis used to derive eq (D!11) and eq (D!12) is based on the assumption that the

divergence angle is constant for d2 < z < d2 + d. The equations are therefore accurate in the far-field of the

beam waist, where the divergence angle is approximately constant. Nearer to the beam waist the

divergence is changing rapidly. So unless d is small compared to the Rayleigh range of the waist (z03), the

result can be erroneous. Therefore when the waist occurs near the detector plane, the beam divergence

from the Gaussian analysis is usually more accurate. 

When the detector plane is sufficiently into the far-field of the beam waist, the result for divergence angle

from either eq (D!10) or eq (D!11) is reported; they typically differ by at most only a few percent. But

the waist size (w03) and distance (ddp) that are reported are the result of using the theoretical * in eq (D!4)

and eq (D!5). This is the mode in which the system has been used to date. However in the future the

beam waist may be projected closer to or behind the detector plane, as previously described. In these

cases, instead of adjusting the position of Lens 2 to obtain the desired detector plane beam size, the lens

position is adjusted to obtain the desired waist location. The waist can be located by moving the beam

size measurement device along the z-axis until the local minimum is located. When the waist location

coincides with the location of the detector plane, the measured detector plane beam size is also the waist

size (w03 = wdp), the divergence angle is zero (2dp = 0), and the distance from the beam waist to the

detector plane is zero (ddp = 0).

D.4 Spatial Filter Pinhole Selection

Lens 1 and 2 form a classic spatial filter [7], where the spatial characteristics of the incident beam are

broken up into their Fourier components in the transform plane. A pinhole aperture is placed in the center

of the transform plane, where it ideally passes only the spatial mode of lowest order. However, when

used with coherent light sources, diffraction around the edge of the pinhole causes undesired artifacts,
97

(D!13)

(D!14)

which complicates the selection of the pinhole size. In the high-accuracy calibration system, a relatively

large pinhole is used to minimize the degradation of the laser beam, already only approximately Gaussian.

The calibration system’s spatial filter consists of Lens 1, Lens 2, and the pinhole mount with precision

pinhole installed. Both lenses are positive, so using geometrical optics analysis, the pinhole is placed at

the secondary focal point of Lens 1, where a plane wave incident on the lens is focused. The pinhole

itself is a circular aperture. Lens 2 is placed so that the focused spot and pinhole are at its primary focal

point, so that the image of the focused spot is projected to infinity. This arrangement can be used with

lenses of dissimilar focal lengths to expand or contract the beam; in common beam-expander devices for

example, a microscope objective is used as Lens 1, with a much longer focal length Lens 2. But in the

high-accuracy calibration system, the focal lengths of both lenses is the same or similar, because no

magnification is desired. Lenses of relatively long focal length are used to provide sufficient working

distances.

The image formed in the focal plane of Lens 1 is the two-dimensional Fourier transform of the spatial

frequencies present in the beam that is incident on the lens. The secondary focal plane of Lens 1 is

therefore called the transform plane. The pinhole is placed in the transform plane, centered around the

secondary focal point of the lens. Only the lowest spatial frequencies pass through the pinhole; higher

frequencies are blocked. Lens 2 provides the inverse transform, which reconstructs only the lowest

frequencies of the incident beam. The arrangement therefore forms a spatial low-pass filter. While the

concept is simple, selecting the best diameter of pinhole to use is more complex.

Diffraction of the coherent light at the edge of the pinhole distorts the reconstructed beam. Truncation of

the Gaussian beam by finite apertures, such as the spatial filter pinhole, results in diffraction artifacts that

resemble Fresnel zone plates, being superimposed on top of the ideal Gaussian profile [28, 31]. 

Therefore, selection of the spatial filter pinhole size is a trade-off between effective spatial filtering, and

minimizing the diffraction artifacts produced by the pinhole. As a compromise, a pinhole is used with

diameter that is at least 2.3 times the 1/e2 intensity diameter of the Gaussian beam waist that occurs in the

spatial filter. Then, approximately 100 % of the incident power is transmitted, and the ripples caused by

diffraction have a maximum amplitude of approximately 1 % [28]. Therefore, the minimum pinhole

diameter DP is given by the following, where w02 is the radius of the Gaussian beam waist that is formed

in the spatial filter:

The Gaussian beam waist radius w02 can be determined using several methods. The least uncertain

method is direct measurement of the beam waist diameter using the scanning-slit beam profiler; then

imperfections in the Gaussian beam are included in the width measurement. But if the beam waist is too

small or the power density too high for the beam profiler, the theoretical waist size from the Gaussian

beam analysis is used. The following equation is commonly used to estimate the focused Gaussian beam

waist radius [26, 32, 33], where D is the measured 1/e2 intensity diameter at Lens 1. In this equation, the

incident beam waist is assumed to occur at Lens 1, where D is measured; since in the real system the

waist does not occur at the lens, the relationship is approximate:
98

In practice, the spatial filter pinhole is selected from a small number of available pinhole sizes, with

diameters typically spaced 10 to 100 :m apart. Selecting too large a pinhole is better than selecting too

small a pinhole, since we are more concerned with minimizing diffraction artifacts then with filtering the

already approximately Gaussian beam. The calculated diameter DP is used as a minimum, so the smallest

pinhole that is at least as large as DP is used, unless DP is only slightly larger than a smaller pinhole, in

which case the smaller pinhole is used. Pinholes from various manufacturers are used, but all have a

precision pinhole cut in a thin metal substrate. The pinholes are visually inspected for damage before use,

but the power levels usually used in the calibration system will not damage the pinholes, even when the

beam is focused.

The pinhole diameter DP from eq (D!13) is somewhat larger than that recommended by most spatial filter

manufacturers, but selection of a pinhole potentially larger than necessary has other benefits. Selecting a

larger pinhole provides more room for error in the estimate of w02. Also, pointing instability in the laser

can make the location of the focused spot change slightly with time; so using a larger pinhole helps

prevent the drift from causing a severe error in pinhole alignment. The larger pinhole is also easier to

align, with more tolerance for alignment error. To minimize the alignment error, a precision pinhole

mount with differential micrometers is used.

When precision beam profile measurements with sufficient dynamic range are available, the actual beam

profile in the detector plane can be measured, and the effect of different spatial filter pinhole diameters

better assessed. Other filtering techniques may also be explored. A scanning pinhole will likely be used

for the precision measurements of profile, but a good camera system may also be usable over a limited

range of wavelengths.
99

APPENDIX E. Relative Aperture Transmittance: Estimation and Uncertainty

A Gaussian laser beam can result in power meters with different sized or shaped limiting apertures to

absorb different amounts of power, because of the difference in their aperture’s transmittance. To

minimize the discrepancy, detectors with limiting apertures that are large compared to the beam size are

usually used. For example, a circular limiting aperture with a diameter that is twice the beam’s 1/e2

intensity diameter will transmit 99.97 % of the total power in a true Gaussian beam. Since a loss of

0.03 % of the incident power is significant in high-accuracy calibrations, a larger relative aperture size is

usually used; a circular aperture three times the size of the beam diameter transmits all of the incident

power in an ideal Gaussian beam, to within a fraction of a part per million. Unfortunately, the real beam is

only approximately Gaussian, and much more power is blocked by the aperture than predicted by the

ideal Gaussian model. With the actual laser beams used in the high-accuracy calibration system, a

measurable power loss typically occurs even with circular apertures that have a diameter that is four times

the 1/e2 beam diameter. To compensate for any such discrepancy in the power absorbed by the standard

and test detectors, the relative aperture transmittance correction is applied to the test detector calibration

result.

Determining the relative aperture transmittance for a test detector requires knowledge of the test

detector’s apparent aperture. When the shape of the detector’s limiting aperture is in question, a spatial

uniformity scan of the test detector is necessary to determine the detector’s apparent aperture. The

aperture is in question when the detector does not have an obvious physical limiting aperture. If a spatial

uniformity scan is required but the customer has not ordered a formal scan (which is a separate special

test), a quick scan with sufficient resolution for determination of the detector’s apparent aperture is

performed as part of the high-accuracy calibration. When such a scan is needed, a figure showing the

detector’s apparent aperture is included in the report, but the actual uniformity scan data is not provided. 

The apparent aperture of a detector with no limiting aperture is shown in Figure 6 (of the main

document).

When NIST is able to perform sufficiently precise measurements of beam profile, the beam profile itself

can be supplied to the end user. However, the beam profile cannot currently be measured with sufficient

precision to accurately estimate the small correction for relative aperture transmittance. Currently a

measurement technique that uses fixed, circular apertures is used to estimate the relative transmittance. In

the future, a measurement technique using a scanning detector with a pinhole aperture could be used to

measure the beam profile with sufficient precision for the measured profile and the detector’s spatial

uniformity to be combined to calculate the relative transmittance without requiring the measurements of

circular aperture transmittance.

The additional power present in the real beam results from imperfections in the optical system that

include diffraction artifacts produced by the finite apertures in the optical system and laser source, light

scattered by the air and the optical elements, aberration caused by imperfect lenses, and multiple beams

caused by multiple reflections in optical elements such as the polarizers. And in addition to the

diffraction caused by its finite apertures, imperfections in the laser’s internal alignment can distort the

beam at the source. Even with optical fiber-coupled lasers, the beam produced at the end of the fiber core

is not purely Gaussian, but can contain cladding modes and scattered power [8]. Because of these

shortcomings, there is enough power away from the center of the beam to cause problems at the

uncertainty levels of interest in the high-accuracy calibration system.

The aperture referred to in this discussion is the detector’s apparent aperture. For detectors that have true

limiting apertures, the apparent aperture is the same as the limiting aperture. However, not all detectors

have physical limiting apertures. For detectors that have no physical limiting aperture, the apparent
100

(E!5)

(E!1)

(E!2)

(E!3)

(E!4)

aperture estimates an equivalent limiting aperture. With these detectors, the apparent aperture may not be

exactly equivalent to a limiting aperture, because with a true limiting aperture, the detector has no

response to light outside the boundary defined by the aperture. But when there is no true limiting

aperture the detector may have some response to light outside its apparent aperture. Also, even if a

detector does have a true limiting aperture, its responsivity may not be uniform within the boundaries of

its apparent aperture. The relative aperture transmittance for these detectors is measured in such a way

that helps accounts for these effects, as described later.

E.1 Measured and Theoretical Transmittance for Circular Apertures

The calibration system’s beam has significantly more power away from its center than a true Gaussian

beam would, so a correction based on calculations using the theoretical Gaussian profile is not accurate;

therefore measured results are needed. When the calculated relative aperture transmittance for the

theoretical Gaussian beam is compared to the measured results for the real beam, the difference is clearly

evident. The relative aperture transmittance for a true Gaussian is derived first, for comparison with the

measured results.

A Gaussian profile beam has an intensity profile IG(r) given by the following equation [32], where w is

the 1/e2 beam radius, r is the distance from the center of the beam, and A is an arbitrary scale factor:

The amount of power P that falls within a circular aperture with radius R, centered around the beam, is

given by

When eq (E!2) is solved analytically, the following is obtained [28]:

The relative aperture transmittance is expressed as TRA(R,C), which means the transmittance of aperture R

relative to the transmittance of aperture C. For centered, circular apertures, TRA(R,C) is the ratio of the

power transmitted by a circular aperture with radius R to the amount of power transmitted by the circular

reference aperture with radius C:

When eq (E!3) is combined with eq (E!4) and simplified, the following relationship for the relative

aperture transmittance of the theoretical Gaussian beam, in centered circular apertures, is obtained:
101

Figure E.1 Measured and theoretical relative aperture transmittance for small circular

apertures, relative to a circular reference aperture of 10 mm diameter, using a beam of 2 mm

diameter.

Figure E.2 Measured and theoretical relative aperture transmittance for large circular

apertures, relative to a circular reference aperture of 10 mm diameter, using a beam of 2 mm

diameter.

In the high-accuracy calibration system, the relative aperture transmittance is measured. Figures E.1 and

E.2 show measured results along with the calculated result from eq (E!5) for the true Gaussian beam. 

Figure E.1 shows the results for apertures 1 to 5 mm in diameter, Figure E.2 shows the results for

apertures 5 to 10 mm in diameter. The results are also listed in Table E.1, where the standard uncertainty

of the measurements is also given. The transmittance shown is for a beam of nominally 2 mm diameter in

a centered, circular aperture, and is relative to a circular reference aperture of 10 mm diameter.

Figure E.1 shows the relative aperture transmittance for apertures with diameters from 1 to 5 mm. In
102

practice a smaller beam would probably be used with such small detectors, but the figure clearly shows

the significant difference between the measured and true Gaussian beam transmittance. Figure E.2 shows

the relative aperture transmittance for more practical aperture sizes, where essentially all of a true

Gaussian beam would be transmitted; again the difference between the measured and Gaussian beam

results is clearly evident. The difference between the measured transmittance for the different

wavelengths is usually small, but the transmittance of the 1319 nm wavelength is exceptionally low,

possibly due to increased scattering and distortion caused by the two narrow-band interference filters that

are used with the laser. The two 633 nm measurements show that the relative aperture transmittance can

change slightly when the system is realigned. When a sufficient history of measurements is available, it

may be possible to use the average of previous transmittance measurements instead of measuring the

transmittance each time the system is used. However for quality control purposes, even when a

transmittance history is available, it is likely that some measurements will be performed each time the

optics are re-aligned, to ensure that the current alignment is comparable to the historic alignments.

Not shown in the measured data is the relative aperture transmittance for apertures that are larger than the

reference aperture. In this case, the relative aperture transmittance can be >1, because the larger aperture

can transmit more power than can the reference.

Aperture

Diameter

(mm)

Laser Beam (2 mm 1/e2 intensity diameter)

Gaussian 1550 nm* 1319 nm* 633 nm* 633 nm #2 515 nm 488 nm

1 0.393469 0.3519

±0.0013

0.3247

±0.0011

0.31272

±0.00021

N/A N/A N/A

2 0.864665 0.7693

±0.0014

0.7678

±0.0014

0.77806

±0.00044

N/A N/A N/A

3 0.988891 0.95012

±0.00089

0.95670

±0.00087

0.96020

±0.00015

N/A N/A N/A

4 0.999665 0.99424

±0.00046

0.99206

±0.00046

0.992424

±0.000095

N/A 0.9984

±0.0020

N/A

5 0.999996 0.99880

±0.00025

0.99755

±0.00024

0.998884

±0.000053

0.99941

± 0.00033

0.99950

±0.00063

0.99968

±0.00030

6 1.000000 0.99952

±0.00016

0.99921

±0.00016

0.999809

±0.000035

0.999852

±0.000049

0.999892

±0.000084

0.999897

±0.000040

7 1.000000 0.99972

±0.00013

0.99966

±0.00013

0.99977

±0.00011

0.999922

±0.000046

0.999932

±0.000043

0.999925

±0.000022

8 1.000000 0.99988

±0.00011

0.999785

±0.000083

0.999951

±0.000026

0.999963

±0.000036

0.999965

±0.000039

0.999955

±0.000012

9 1.000000 1.000000

±0.000062

0.999840

±0.000070

0.999974

±0.000023

0.999982

±0.000050

0.999974

±0.000032

0.999982

±0.000011

10 1.000000 1.000000

±0.000062

1.000000

±0.000070

1.000000

±0.000025

1.000000

±0.000035

1.000000

±0.000030

1.000000

±0.000009

N/A - Not Available.

 * - Measured using a slightly different process and uncertainty analysis.

Table E.1 Theoretical and measured relative aperture transmittance for circular apertures and a

2 mm diameter beam. The standard uncertainty of the measured results is shown.
103

The measured relative aperture transmittance for three additional wavelengths (488 nm, 515 nm, and

633 nm #2) are shown in Figure E.2; they were acquired more recently than the others, and better reflect

the current uncertainty in the measurement. The other measurements were acquired with an older

measurement process, and have a less conservative uncertainty analysis. Also, the lower uncertainty in

the older measurements is also due in part to the more stable laboratory environment that was present

during the earlier measurements. The transmittance of the smaller apertures was not measured in the

more recent acquisitions, because of environmental instability, which necessitated additional

measurements of the larger aperture’s transmittance to obtain a sufficiently small uncertainty in the

average. Also, the transmittance of the small apertures was not needed for the calibrations performed.

The measured data show that the magnitude of the relative aperture transmittance correction can be very

significant at the uncertainty levels of interest in the high-accuracy calibration system, where the goal is a

standard uncertainty of 0.01 %. For example the magnitude of the correction, 100 % C |1 ! TRA|, can be

calculated for the transmittance results. The average magnitude of the correction for an 8 mm test

detector aperture relative to a standard with an aperture of 10 mm diameter, when using a beam of 2 mm

diameter, is 0.008 %, which is almost as large as the entire desired error budget. And the average

measured correction for a test detector aperture of 5 mm diameter increases to 0.1 %, about 10 times the

desired error budget! Therefore, ignoring the relative transmittance can cause a significant and

unaccounted error in high-accuracy calibrations, whenever the standard and test detector have dissimilar

apertures.

Even when the theoretical Gaussian beam is considered, the error that results from ignoring the relative

aperture transmittance can be prohibitive. As shown by the data for the theoretical Gaussian in Table 1,

using a circular aperture that is 1.5 times the Gaussian beam diameter (a 3 mm diameter aperture with a

2 mm diameter beam) would result in a loss of 1.1 % of the power that is delivered through a 10 mm

diameter aperture. An aperture with a diameter that is two times the beam diameter will miss 0.035 % of

the power, which is again greater than the desired standard uncertainty. And the situation with the real

beam is worse; using the measured results, an aperture three times the beam diameter will typically miss

0.03 % of the power that is transmitted through a 10 mm diameter aperture. 

The relative aperture transmittance for the theoretical Gaussian beam becomes insignificant (at the

uncertainty levels of interest) for apertures with a diameter two and a half times the beam diameter or

more. But the measured transmittance for the actual laboratory beam can still be significant even when

the aperture is more than four times the beam size. The measured data in Table 1 show that the average

magnitude of the correction even for a relatively large aperture of 9 mm diameter (which is four and a

half times the 2 mm beam size) is 0.004 %, approximately half of the desired error budget. Therefore, the

correction for relative aperture transmittance is applied to all detectors, even those with relatively large

apertures. 

Since the relative aperture transmittance is always applied, the magnitude of the uncertainty in the

correction can limit the accuracy of the calibration system. For the more recent measurements shown in

Table 1, the standard uncertainty of the correction typically exceeds 0.01 % for apertures smaller than

three times the beam diameter. Therefore, test detectors with an apparent aperture that has a diameter

greater than three times the beam’s diameter are recommended. Even if both the standard and test

detector have identical but small apertures, the uncertainty may still be prohibitive, primarily because of

the uncertainty that results from imperfections in the centering of the beam in the aperture.

Uncertainty in the measurement of relative aperture transmittance results from two primary sources:

uncertainty in the centering of the beam in the apertures, and instability in the beam power and detector

responsivity. The uncertainty in the aperture centering is determined experimentally, by physically
104

translating the external circular aperture in front of the transmittance measurement detector, and

measuring the change in the response of the stationary detector. An upper bound is placed on the aperture

misalignment; a 0.5 mm bound is currently used. The detector response when the aperture is well

centered is compared with the response when the aperture is offset by the misalignment bound, the

change in the response is then used as a bound for the centering uncertainty. Uncertainty that results from

instability in the source power and detector is random, so is reduced by averaging multiple measurements. 

The main source of instability in the system is temperature fluctuations, which affect all of the elements

of the calibration system to various degrees. Therefore, measurements with a lower uncertainty are

obtained when the environment is more stable.

The detector is assumed to be perfectly uniform within its apparent aperture; variations in its spatial

uniformity can affect the measured result, but the uncertainty caused by the detector’s uniformity is

assumed to be small compared to the uncertainty in the measurement. To minimize the uncertainty

resulting from detector nonuniformity, the detector used for the circular aperture transmittance

measurements have a spatial uniformity variation less than 1 % within their apparent aperture. When the

test detector itself is used to measure its own aperture’s transmittance, its uniformity can cause additional

uncertainty in the measurement. In this case, the uncertainty will be at least partially accounted for when

the end user adds the uncertainty due to the test detector’s alignment, which is based in part on the spatial

uniformity of the detector. 

The result of imperfect detector uniformity can be better quantified once the actual beam profile is

obtained with sufficient resolution; currently, an actual beam profile with sufficient dynamic range to

resolve the details of the power distribution in the beam tails is not available. However, a theoretical

example can help. Assume that 99.9 % of the beam’s power is concentrated in the center of the detector,

and 0.1 % is in a second beam that is produced by a multiple reflection. When a large aperture is placed

in front of the detector, the second beam enters the detector. But when a small aperture is used, the

second beam is blocked. The transmittance of the smaller aperture relative to the larger aperture is then

99.9 / 100 = 0.999. Consider that because of nonuniformity in the detector, the second beam falls on an

area where the responsivity is 1 % lower than that in the detector’s center. The ratio would then be

99.9 / (99.9 + 0.1 C 0.99) = 0.99901, for a change of 0.001 %, which is negligible.

E.2 Relative Transmittance Measurement for Circular Apertures

The relative aperture transmittance for circular apertures is currently measured using a detector with an

apparent aperture that is at least as large as the largest circular reference aperture (which has a 10 mm

diameter), and has good spatial uniformity within its apparent aperture. The detector must also be

sensitive to the beam’s wavelength, and have low noise and good stability over a period of a few minutes. 

The circular apertures are placed in front of the large detector for the measurement; the detector itself is

held stationary. If the detector has a built-in, circular, limiting aperture, then the built-in aperture can be

used as the reference aperture, for the purpose of the transmittance measurements.

Using a detector with a single, flat, optically sensitive element that can be placed very close to (or in

contact with) the circular apertures ensures that the detector is illuminated by all of the transmitted power,

at the expense of its field of view. Placing the detector a distance behind the aperture can result in some

of the transmitted power being diffracted away from the detector by the aperture’s edge, so is less

desirable. However, sometimes trap detectors are used even though their detecting element(s) must be

some distance behind the aperture, because their superior spatial uniformity and reduced or eliminated

back-reflection is more important than the small loss of diffracted power that may result; using trap

detectors with apparent apertures that are larger than the reference aperture helps reduce any potential

power loss. Since the resulting configuration can have a large field of view, a dark and unchanging
105

(E!6)

background around the beam is necessary. When a single-element detector or trap with a back reflection

is used, it is placed in the beam at a small angle to prevent the back-reflection from interfering with the

incident beam. Currently, a Si trap detector with a large apparent aperture is used with visible

wavelengths, and a large single-element Ge detector is used at near-infrared wavelengths.

The relative transmittance of the aperture that is being measured is determined by dividing the amount of

power transmitted with the measured aperture by the amount of power transmitted with the reference

aperture, when both apertures are in turn illuminated by the same stable beam. The calculated

transmittance is therefore relative to the transmittance of the reference aperture, and is appropriate only

for the particular beam used in the measurement. The circular apertures are placed in the detector plane,

where the detector’s entrance apertures are also placed, so that the circular apertures are illuminated by

the same profile beam that illuminates the entrance apertures of the detectors being calibrated. Since the

measured circular apertures are all smaller than the reference aperture, their relative transmittance is

always less than one. Therefore, a measurement result that returns a transmittance that is greater than one

can be corrected to a value of 1. Apertures that are larger than the reference aperture can have a relative

transmittance that is greater than 1.

The relative transmittance of several fixed, precision, circular apertures is measured with the system. The

apertures are made of a thin ceramic material with precision circular holes. To prevent any optical power

from penetrating the diffusely translucent ceramic material, the ceramic is painted with a black epoxy

spray paint for opaqueness, and then further coated with a quality flat-black paint to reduce reflections. 

The precision apertures have diameters from 1 to 10 mm, in 1 mm steps. The 10 mm diameter aperture is

usually used as the reference aperture.

The differences in the transmittance of the apertures is usually very small, especially for the larger

apertures; therefore the measurement of relative transmittance must be performed very carefully. For

example, the magnitude of the optical power drift is typically larger than the change in transmittance from

one large aperture to another. However, the relative transmittance measurement is performed relatively

quickly; a single measurement can be performed in less than a minute. And the drift in the optical power

over such a time period is essentially linear, so linearly interpolating the power between bracketing

reference aperture power measurements effectively reduces the effect of the power drift. To perform the

interpolation, the period during which the power measurements occur is needed.

The transmittance of the smaller precision apertures relative to the reference aperture is determined with a

series of measurements that are performed in such a way that the effects of drift in detector responsivity

and beam power are minimized. The detector is illuminated throughout the measurement. First, the

detector output P1 is measured at time t

1

, using a no-baseline technique, with the reference aperture

having diameter C in place (or with no additional aperture, if the detector’s own circular limiting aperture

with diameter C is used as the reference). Then the aperture under test with diameter D is placed on the

detector, and the detector’s response P2 is measured at time t

2

. Then the test aperture is removed and the

reference replaced, and the response P3 is measured at time t

3

. Interpolating linearly in time, the relative

transmittance is:

The measurement of TRA(D,C) is repeated several times, and the mean and standard deviation are

calculated. The mean estimates the relative transmittance, and the standard deviation of the mean,

uM(D,C), is used in the uncertainty estimate. The measurements are performed rapidly to minimize the

amount of drift encountered, so P1 . P3.
106

(E!7)

(E!8)

(E!9)

The measurements of the powers P1, P2, and P3 contain uncertainties due to the imperfect centering of the

apertures around the laser beam. The centering uncertainty is estimated in a separate experiment,

described below, which provides an estimate of the standard uncertainty uP(X) due to imperfect centering

in aperture X. In eq (E!6), the centering uncertainty in the measurement of P2 is uP(D), and the

uncertainty in P1 and P3 is uP(C). The uncertainties are uncorrelated, and approximating eq (E-6) with

TRA(D,C) = P(D) / P(C), the standard uncertainty uRA(D,C) in TRA(D,C) is:

The uncertainty due to the imperfect centering of the circular apertures around the beam is also measured. 

It is assumed that the apertures are centered around the beam with an error of less than 0.5 mm;

experimentation has shown that the error in the aperture centering is less than this amount, since an error

of 0.5 mm is clearly discernable by eye, even at infrared wavelengths (using an IR card). The

measurement is performed to determine how much the aperture transmittance changes when the aperture

is misaligned by this amount.

To measure the transmittance change due to the aperture centering error, the circular aperture with

diameter d is placed in the detector plane, and centered around the beam as best as possible. The aperture

is placed on a micrometer-driven mount so that it can be translated by a precise amount in the horizontal

direction, and a detector is placed behind the aperture as in the primary transmittance measurement

previously described. As before, the aperture is constantly illuminated with the stabilized beam. When

the aperture is centered the detector’s response is measured using a no-baseline technique and recorded as

P1 at time t

1

. Then the aperture is translated 0.5 mm in the positive horizontal direction, and the detector’s

response P2 is measured at time t

2

. The aperture is then returned to center and the response P3 is

measured at time t

3

. The aperture is translated in the negative horizontal direction, and the detector’s

response P4 is measured at time t

4

. Finally the aperture is again returned to the centered position, and the

detector’s response P5 is recorded at time t

5

. Using linear interpolation, the relative transmittance change

due to the positive translation, called R+(d), is given by

The relative transmittance change for a negative translation, called R!(D), is calculated similarly:

The measurements of R+(d) and R!(d) are repeated several times so that the mean and standard deviation

can be calculated; the resulting means estimate the relative transmittance change, and the standard

deviation of the means, called uM+(d ) and uM!(d ), are used to estimate the uncertainty.

In equation (E!7), the uncertainty uP has units of power, so the relative measurements must be converted. 

To simplify the conversion, the small drift in the power with time is ignored, and the nominal power

when the aperture is centered is called P0 = P1 = P3 = P5. Then the absolute change in the transmitted

power due to the offset in the aperture’s centering is estimated by P0(R±(d) ! 1), and the absolute standard

deviation of the mean becomes P0uM±(d ).
107

(E!10)

Because the actual centering uncertainty is assumed to be less than the ±0.5 mm translation used in the

measurement, the quantities P0(R+(d) ! 1) and P0(R!(d) ! 1) estimate the bounds of the change in

transmitted power due to centering uncertainty. A rectangular probability distribution is assumed. The

quantities can be positive or negative, because of the asymmetrical distribution of non-Gaussian power

around the main Gaussian beam, such as that produced by scattering and multiple reflections, and error in

the initial aperture centering. The resulting combined standard uncertainties due to the horizontal

translations, called uC+(d) and uC!(d), are given by

The greater of uC+(d) and uC!(d) is used as the standard uncertainty, uP(d), due to the imperfect centering

of the circular aperture with diameter d around the beam. Using the greater uncertainty estimate helps

compensate for any error in the initial centering of the aperture during the measurement of centering

uncertainty.

Even when the transmittance measurement detector’s built-in aperture is used to measure relative aperture

transmittance, its aperture centering uncertainty must be calculated using an external aperture and a

detector large enough to capture the entire transmitted beam, even when the aperture is displaced by

0.5 mm. If the detector with built-in aperture were used for the centering uncertainty measurement itself

by translating the entire detector, the detector’s spatial nonuniformity would affect the result when the

detector is translated along with its aperture. Since the detector’s spatial nonuniformity is likely to be

much larger than the effect of the misaligned aperture, an external aperture that can be translated

independently of the detector must be used.

E.3 Estimation of the Relative Aperture Transmittance for Detectors

When a detector has a true limiting aperture that is circular and is the same size as one of the measured

circular apertures, the results of the measurements of circular aperture transmittance are used to determine

the detector’s transmittance relative to the standard. When a detector has a limiting aperture that is larger

than the reference aperture, or when the detector has no true limiting aperture (such as that shown in

Figure 6), the test detector’s aperture transmittance is usually measured directly. Alternately, the test

detector’s transmittance can be estimated using bounding circular apertures. Bounding circular apertures

are also used to estimate the transmittance of noncircular limiting apertures. Also, for the purpose of the

calibration, an external circular aperture that fits within the detector’s apparent aperture can be placed on

the detector to form a true circular limiting aperture. However, the technique of adding an aperture to the

detector is not usually used because it entails a modification of the test detector.

The transmittance of the test detector’s aperture relative to the standard detector’s aperture is used to

correct the test detector calibration result. Three techniques, described below, are used to estimate test

detector’s relative transmittance. Each technique results in the estimated transmittance of the test

detector’s aperture, relative to the transmittance TRA(D,S) of the standard detector’s aperture, with

combined standard uncertainty uRA(D,S). In this context, D is the test detector’s aperture, and S is the

standard detector’s aperture, but D and S do not necessarily represent the diameter of a circular aperture. 

The standard detector’s transmittance relative to the reference aperture’s transmittance TRA(S,C), and its

associated standard uncertainty uRA(S,C), are needed. These parameters are determined using a technique

similar to one of the three described below, except instead of calculating the transmittance of the test
108

(E!11)

(E!12)

(E!13)

detector’s aperture relative to the standard’s aperture, the transmittance of the standard’s aperture relative

to the reference aperture is determined. For the LOCR’s elliptical aperture, method 2 is used.

1. Estimation of the relative transmittance for detectors with a circular limiting aperture that is the same

size as a measured circular aperture. 

For these detectors, the measurement of the precision circular aperture’s transmittance is used to estimate

the transmittance of the detector’s aperture, relative to the standard’s aperture. Given that the detector has

a circular limiting aperture with diameter D, and the reference is a circular aperture with diameter C, the

relative transmittance of the detector’s aperture TRA(D,C) is determined by performing the measurement

described by eq (E!6). The corresponding combined standard uncertainty, uRA(D,C), is given by

eq (E!7).

The standard detector’s transmittance TRA(S,C) relative to the reference aperture’s transmittance, and its

associated standard uncertainty uRA(S,C), are previously determined. Then the transmittance TRA(D,S) of

the test detector’s aperture relative to the standard’s aperture is given by

The standard uncertainty in TRA(D,S) is given by

2. Estimation of the relative transmittance for detectors with a noncircular or odd-sized circular limiting

aperture that is bounded by measured circular apertures.

The noncircular or odd-sized circular aperture’s relative transmittance is estimated using the measured

transmittance of bounding, centered, circular apertures. Technique 3 is preferred for detectors that have

an irregular apparent aperture but no limiting aperture, but this technique can be used with such detectors

if necessary. The bounding apertures are the largest centered, circular aperture that is enclosed by the

detector’s apparent aperture, and the smallest centered, circular aperture that encloses the detector’s

apparent aperture. For example, the LOCR has a centered, approximately elliptical, limiting aperture with

dimensions 8.5 by 9.5 mm; therefore the bounding circular apertures have diameters of 8 and 10 mm. 

The measured transmittance of the bounding circular apertures are combined to give an estimate for the

transmittance of the noncircular aperture. Given that the bounding circular apertures have diameter D1

and D2 with mean relative transmittance TRA(D1,C) and TRA(D2,C) from eq (E!6); the estimated

transmittance TRA(D,C) for the detector is given by the average (midpoint) of TRA(D1,C) and TRA(D2,C), or

The combined standard uncertainty uRA(D,C) of the transmittance estimate is determined by applying the

law of propagation of uncertainty, and summing in quadrature with a term for the transmittance bounds. 

The resulting combined standard uncertainty is given by
109

(E!14)

(E!15)

The transmittance of the detector’s aperture relative to the standard’s aperture can then be calculated

using eq (E!11), with uncertainty calculated using eq (E!12).

3. Estimation of the relative transmittance for detectors with large apertures, and detectors that have no

true limiting aperture.

The relative transmittance of detectors with large or unlimited apertures is measured directly, using a 

centered, circular aperture of an intermediate size. This technique is best for detectors that have no

limiting aperture, because it includes the effects of the detector’s partial response outside of its apparent

aperture. However if the detector has a very slow or noisy response, method 2 can be used instead to

obtain a more timely or less uncertain measurement. There is a small difference between the calculations

for detectors with no limiting aperture, and those for detectors with a large aperture; the case of the

detector with no limiting aperture, like that shown in Figure 6, is described first.

To measure the transmittance directly, the detector’s response with no external aperture is compared to its

response when the intermediate-sized external circular aperture is placed in front of the detector. The

intermediate external circular aperture, which can be smaller than or equal to the reference aperture, must

have a diameter equal to or smaller than the largest centered circular aperture that is completely enclosed

in the detector’s apparent aperture. The intermediate aperture is called I. To measure the detector

aperture’s transmittance relative to the intermediate aperture’s transmittance, the detector is first placed

with its entrance aperture in the detector plane, and is aligned to the beam in the same way that it will be

aligned during absolute responsivity calibration. The mount with the intermediate aperture is then placed

immediately in front of the detector, and is centered around the beam. The detector is then illuminated,

and its transmittance TRA(D,I) relative to the intermediate aperture is measured as previously described for

eq (E!6), except that aperture I replaces aperture C, and aperture D is the detector’s unmodified aperture.

The combined standard uncertainty uRA(D,I) in the mean TRA(D,I) is the same as that given in eq (E!7),

with a few changes to the uncertainty components. For uRA(D,I), the uncertainty uM(D,C) is replaced by

uM(D,I), and the aperture centering term uC(C) is replaced by uC(I). To estimate the centering uncertainty

term for the unmodified detector’s aperture, uC(D), the smallest centered circular aperture that fully

encloses the detector’s apparent aperture, called D2, is used. However the centering uncertainty for the

detector’s aperture can be larger than uC(D2), because of (a) the detector’s partial response outside its

apparent aperture, and (b) the asymmetrical power distribution outside the central Gaussian beam. To

help compensate for the effect, the centering uncertainty for D2 and all larger apertures are considered,

and the largest centering uncertainty is used as uC(D) in eq (E!7). The centering uncertainty is generally

smaller for larger apertures, but because of the asymmetrical power distribution is not necessarily so. 

While it is possible for the actual uC(D) to be larger, this estimation is considered adequate because the

centering uncertainty for large apertures is usually insignificant compared to the uncertainty in the

centering of the smaller intermediate aperture.

Estimation of the detector’s unlimited aperture transmittance relative to the standard’s aperture

transmittance, called TRA(D,S), is complicated by the addition of the intermediate aperture. The relative

transmittance is given by
110

(E!16)

(E!17)

(E!18)

The combined standard uncertainty uRA(D,S) is given by

When the detector’s apparent aperture is larger than the reference aperture, the reference aperture itself

can be used as the intermediate aperture, reducing the uncertainty in the measurement. Then, the relative

transmittance is

The combined standard uncertainty uRA(D,S) is given by:
111

(F!1)

(F!2)

(F!3)

APPENDIX F. System DMM and LOCR Electronics: Calibration and Correction

All of the electrical systems that are directly involved in the detector calibration are traceable to the highaccuracy calibration system’s digital multimeter (DMM), which is traceable to NIST standards. The

system DMM itself is calibrated using an internal NIST calibration system, which returns the offset in the

DMM’s indicated readings. Since the offset in the DMM is reported but not nulled, a correction must be

applied for every critical measurement that is performed with the DMM. This method of calibration adds

significant complexity to electrical measurements performed in the laboratory. The correction is not

applied at the time of the measurement, instead the indicating readings are recorded, and the correction is

applied in post-processing where the calculations are fully documented. This technique also has the

advantage that the acquisition system records readings that are consistent with those displayed on the

device.

F.1 System DMM Calibration

The system DMM is calibrated using an internal NIST calibration system. The system measures the

offset between the DMM’s indicated voltage or resistance, and the actual voltage or resistance. The offset

measurements are performed for each of the DMM’s inputs and ranges, as requested. Only voltage and

resistance measurements are needed so the DMM’s current ranges are not calibrated. Generally, voltage

measurements from 0 to ±10 V using the 10 0.V scale are calibrated, and the offset for a resistance of

1,000 S is determined. The actual voltage V is the sum of the indicated voltage reading VI and voltage

offset VC(VI):

The offset is measured for discrete voltages; offset values for voltages that fall between the calibrated

values are linearly interpolated from the bracketing measured values. The bracketing calibrated voltages

are V1 and V2, where V1 < VI < V2. The interpolation is performed as follows:

where Vc(V) is the calibrated offset voltage for indicated voltage V.

Similarly, the standard uncertainty uVC(VI) of VC is also linearly interpolated from the standard

uncertainties uC(V) of the bracketing values:

In addition to the uncertainty uC(V) of the calibration, the DMM experiences additional uncertainty with

temperature and time. The difference between the operating temperature and the calibration temperature,

and the interval since the calibration, add uncertainty to the DMM’s measurements. The DMM’s

manufacturer has assessed and detailed these uncertainties in the DMM’s operating manual [34]. The

uncertainty is described in three parts: one relative to the DMM reading, one relative to the DMM range,

and one relative to the reading that describes the traceability to NIST standards. The uncertainties are

given as relative uncertainties in parts-per-million (ppm). The traceability to NIST electrical standards is

included in uC(V), so only the first two uncertainties (ppm reading and ppm range) are relevant. The

stated uncertainties are valid only for the operating mode and conditions specified in the document; the

modes used in the high-accuracy calibration system conform to the specified requirements. The
112

Indicated 

Voltage 

VI (V)

Voltage

Correction

VC (:V)

Standard Uncertainties

uVC (:V) uVS (:V) uV (:V)

 1.00 !2.70 3.0 2.7 4.0

 2.00 !6.28 5.9 5.0 7.7

 3.00 !9.30 8.9 7.4 11.5

 4.00 !12.72 11.8 9.8 15.3

 5.00 !16.55 15.0 12.1 19.3

 6.00 !19.08 17.7 14.5 22.9

 7.00 !22.40 20.7 16.9 26.7

 8.00 !26.40 24.0 19.2 30.8

 9.00 !29.25 27.0 21.6 34.6

10.00 !34.00 30.0 24.0 38.4

11.00 !35.86 33.0 26.3 42.2

11.76 !38.45 35.3 28.1 45.1

Table F.1 DMM voltage calibration values, for the front input terminals, and the 10 V range.

(F!4)

uncertainty specifications include the device’s temperature coefficient for a range of ±5° C about the

calibration temperature. The laboratory temperature is normally within this range, so adding additional

uncertainty is not usually needed. If the temperature is out of range, additional uncertainty is added to

both the ppm range and ppm reading used in the example below.

Because of aging and drift in the DMM’s internal components, the amount of uncertainty in the

instrument’s calibration increases with time, so the device’s performance is specified for time intervals of

24 hours, 90 days, 1 year, and 2 years. The DMM is usually calibrated within 90 days of its use in the

calibration laboratory, so the 90-day specification is usually used.

The uncertainties given by the manufacturer are an expanded uncertainty, but the coverage factor used is

unclear. Therefore to be conservative, the specifications are assumed to be the limits of a rectangular

interval, so are divided by the square-root of 3 to obtain the standard uncertainty. This is much more

conservative than assuming a coverage factor of 2 or 3, and therefore dividing by 2 or 3 to get the

standard uncertainty.

For example, when DC volts are measured using the DMM’s 10 V range within 90 days of the previous

calibration, the specified accuracy (uncertainty) is 4.1 ppm of reading plus 0.05 ppm of range, so the

standard uncertainty uVS is

The combined standard uncertainty uV of the DC voltage measurement is then the quadrature sum of the

uncorrelated standard uncertainties uVS and uVC.
113

Indicated

Resistance 

RR (S)

Resistance

Offset

RC (mS)

Standard Uncertainty

uRC (mS) uRS (mS) uR (mS) uR (%)

 1000 !174.8 4.3 2.7 5.0 0.00050

Table F.2 DMM resistance calibration values for the front input terminals and for the 1000 S

range.

(F!5)

Example calibration results are shown for the DMM’s front input terminals in Table F.1. Only positive

voltages were calibrated in the example, but negative voltages are usually also calibrated. The

uncertainty components uVC and uVS and combined standard uncertainty uV are also given in the table. 

The 90-day DMM accuracy specifications are used to calculate uVS.

In addition to the voltage calibrations shown on the table, the DMM’s front terminal, four-wire resistance

measurement was also calibrated at 1,000 S, as shown in Table F.2. The uncertainty in the resistance is

assessed in the same way as the uncertainty in the voltage. Given in the table are the indicated resistance

RR, the resistance offset RC, the standard uncertainty uRS due to the NIST calibration, the DMM’s specified

uncertainty uRS, and the combined standard uncertainty uR of the resistance measurement. The combined

standard uncertainty is given both in milliohms, and as a percentage of the indicated resistance. The

specified uncertainty is the 90-day specification for the DMM’s 1,000 S resistance range: 8 ppm of the

reading, plus 0.5 ppm of the range. The DMM calibrations were performed at TC = 23 ± 1°C.

F.2 Measurement of LOCR Electrical Power

The LOCR’s internal electronics are calibrated by comparing its internally measured power to the power

measured by the system DMM. Correction factors for the internal electronics are then stored in the

LOCR. But this type of calibration is very time consuming, so is performed infrequently. However, the

LOCR’s calibration cannot be assumed constant with time and changes in the environment, so its

calibration is checked for quality assurance whenever optical calibrations are performed, under the same

conditions that exist during the optical calibration. The LOCR’s calibration is only checked, it is not

internally recalibrated to null the offset. Instead, the offset in the LOCR’s internal calibration is

measured. The correction for the LOCR’s electrical calibration is combined with the correction for the

DMM’s measured voltage and resistance, to determine a single correction factor for the measured optical

power.

During LOCR calibration, the system DMM is used to measure the power delivered to the LOCR

receiver’s heater. The electrical circuit consists of two resistors in series: the LOCR’s heater resistor RH,

and the LOCR’s standard resistor, RS. Both resistors have a resistance of approximately 1,000 S. The

precise resistance of the standard resistor, and the voltage drop around the heater and standard resistors,

called VH and VS, respectively, are measured with the DMM. First, RS is measured with no power applied

to the circuit. Then the LOCR’s electronics apply a current, and VH and VS are measured to determine the

power PH delivered to the heater. The heater power is given by
114

(F!6)

(F!7)

(F!8)

(F!9)

(F!10)

The LOCR heater’s resistance is within 1 % of the standard resistor’s resistance; so to simplify the

following uncertainty analysis, they are considered equal. Thus RH = RS = R, VH = VS = V, and the applied

electrical power can be simplified to

The amount of absorbed optical power is determined by the change in electrical power that occurs when

the optical power is applied, therefore the uncertainty in the difference between two measured electrical

powers is needed. The LOCR’s electronics maintain the receiver at a constant temperature by applying

electrical power to the heater, so the change in applied electrical power necessary to maintain the constant

temperature when optical power is applied, is equal to the amount of absorbed optical power. When

operating with no absorbed optical power, the electrical system applies power P1 to the receiver to

maintain its temperature at the desired level. A nominal unilluminated power of 2 mW is used, so

P1 = 2 mW. When optical power is absorbed, the electrical power must drop to P2 to maintain the same

temperature. The amount of optical power absorbed, called PO, is therefore given by the following, where

V1 is the voltage applied when no optical power is present, and V2 is the voltage applied when the optical

power is present:

The offset in the indicated voltages and resistance cause an offset in the power calculated from eq (F!4). 

The offset in the power can be determined by adding the absolute offsets dV1, dV2, and dR to V1, V2, and

R, respectively. The optical power can then be expressed by

Since the DMM’s readings have an offset, we desire the more accurate PO(VC1,VC2,RC), where VC1, VC2,

and RC are interpolated from the calibration summarized in Tables F.1 and F.2, but PO(0,0,0) is what is

actually measured. We desire a correction factor k1 such that PO(VC1,VC2,RC) = k1C PO(0,0,0). The

correction factor is given by

The uncertainty in k1 is determined using the law of propagation of uncertainty. The uncertainty uk1 in k1, 

is given by

The term C represents the components that are correlated. When V1  V2, the uncertainties are assumed to

be uncorrelated, so C = 0. However, when V1 . V2, the uncertainties in the voltages are correlated, and

omitting C will result in an inflated uncertainty estimate.
115

(F!11)

(F!12)

Optical

Power 

PO

 (mW)

Electrical

Power 

PM

(mW)

Voltages CorrelationResult Standard Uncertainty uk1 x 106

VM

(V)

VCM

(:V)

uVM

(:V)

kVC k1 Uncorrelated100 %

Correlated

Estimated

Correlation

All 2.00 1.414 !4.183 5.533 N/A N/A N/A N/A N/A

1.00 1.00 1.000 !2.700 4.000 0.657001 1.000168 18 9 13

0.25 1.75 1.323 !3.856 5.195 0.933503 1.000168 84 9 23

0.10 1.90 1.378 !4.055 5.400 0.974212 1.000168 215 9 36

0.01 1.99 1.411 !4.170 5.519 0.997876 1.000168 2,605 10 120 

N/A - Not applicable

Table F.3. Examples of the correction factor k1 and its uncertainty, for several optical powers.

The covariance term in the law of propagation of uncertainty is estimated by u(V1, V2) = kVCuV1uV2, where

the term kVC is the estimated voltage correlation. The other covariance terms are zero since the

uncertainty in R is uncorrelated. In the limiting case where V1 = V2, the uncertainties are perfectly

correlated so VC1 = VC2, and kVC = 1; when there is no correlation, kVC = 0. Noting that PO(0,0,0)2 has been

factored out of the denominator, C is then given by

The correlation in the voltages is estimated from

When kVC < 0, the voltages are considered uncorrelated, and kVC = 0 is used instead.

The correction factor and uncertainty, calculated using the example data given in Tables F.1 and F.2, are

shown in Table F.3. Uncertainty estimates are given for uncorrelated voltages, 100 % correlated voltages,

and using the estimated voltage correlation. In practice, the result for the estimated voltage correlation is

used, but the other results are given to show by how much the uncertainty changes for the different

degrees of correlation.

F.3 Calibration of LOCR Power

The LOCR’s internal electrical heater power measurements are calibrated using the “ACRCAL.EXE”

program. This program calibrates the LOCR’s internal electronics over their entire operating range, so is

very time consuming, and environmental drift may occur during the lengthy calibration. To better assess

the LOCR’s uncertainty at the time of an actual optical calibration, the uncertainty of its internal

electronics is assessed using the calibration verification program “TCALVER.EXE”. The calibration is

verified at only the power levels of interest in the detector calibration, and in an environment consistent

with that present during the detector calibrations. The verification program returns the offset in the

internal electronic’s power measurement, based on measurements performed with the system DMM, as

described in the previous section. The offsets are used to generate a correction factor k2 for the LOCR’s

internal electronics and its combined standard uncertainty uk2. The measurement is performed several
116

Nominal Power 

P

(mW)

Indicated Power

(mW)

Measured Offset

(nW)

Average Power

Offset PC(P)

(nW)

Standard

Uncertainty uPC(P)

(nW)

2.00

2.02 !14.1

!14.7 1.0

2.00 !14.0

1.98 !15.8

1.90

1.92 !13.4

!14.6 2.1

1.90 !13.3

1.88 !16.9

1.75

1.77 !17.7

!14.0 3.6

1.75 !10.5

1.73 !13.9

1.00

1.01 !10.1

!11.3 1.1

1.00 !12.0

0.99 !11.9

Table F.4 Typical LOCR power calibration check results.

times to assess the random component of the uncertainty, but the uncertainty in the electrical

measurements themselves is accounted for elsewhere, in k1, so the indicated electrical powers are assumed

to be accurate here.

When the LOCR is used to measure a single optical power, the offset of its its internal electronics at two

power levels, P1 and P2, are needed. Since P1 is always the same (2 mW), only one additional offset

needs to be measured for each additional power level. However, the actual absolute power levels can

fluctuate for several reasons, including drift in the environment and the absolute optical power, so the

LOCR’s calibration is assessed for a region of power levels that includes the potential drift. The drift in

the nominal absolute powers is limited to an amount dP; if either power level P1 or P2 change from their

nominal values by more than dP, the LOCR and/or optical source is readjusted. The power drift dP can

therefore be considered as the bound of a rectangular distribution around the nominal power P. A dP of

1 % is easily achieved; at higher powers, a dP of 0.1 % can be achieved.

For each nominal power P, the LOCR calibration offset is measured at several points within the interval

P ± dP, and a statistical analysis is performed. To complete the calibration in a reasonable amount of

time, the power offset is usually measured at only three levels, P, P + dP, and P ! dP. The offset at each

level is measured at least once, and the mean offset and standard deviation of the mean are calculated for

all the measurements within the interval. The mean offset PC(P) for indicated power P is such that the

corrected power is P + PC(P). The standard deviation of the offsets is the standard uncertainty uPC(P) of

the offset; the standard deviation of mean is not used because we desire the variation that occurs with the

variation in the nominal power, not the uncertainty of the nominal power measurement. The results for a

typical LOCR calibration at several power levels are shown in Table F.4; a dP of 1 % was used. In the
117

(F!13)

(F!14)

(F!15)

(F!16)

(F!17)

Optical Power

(mW)

kPC k2 Standard Uncertainty uk2 C 106

Uncorrelated Correlated Estimated

1.00 0.738426 0.999997 1 0 1

0.25 0.951220 0.999997 15 10 11

0.10 0.993174 0.999999 23 11 11

0.01 0.999319 0.999999 149 11 12

Table F.5 Example LOCR power calibration correction factor and uncertainty.

table, a single measurement at each indicated power is shown, but in practice the measurements should be

performed multiple times. For simplicity, such multiple measurements are not pre-averaged; but if it

becomes desirable to reduce this uncertainty, pre-averaging could be employed.

In optical power calibrations, the change in indicated power is calculated. In this case, the optical power,

PO, is a expressed as the following function:

To correct for the offset in the indicated optical power, the indicated electrical powers in eq (F!13) are

replaced by the indicated powers P plus the power corrections PC(P), from Table F.4. We desire the

multiplicative correction factor k2 that converts the indicated optical power to the corrected optical power,

so

Using the law of propagation of uncertainty, the combined standard uncertainty uk2 of k2 is given by

As before, C represents the components that are correlated. The covariance term is estimated by 

uPC(P1, P2) = kPCuPC(P1)uPC(P2), where kPC is the estimated correlation in the powers. In the limiting case

where P1 = P2 the uncertainty is perfectly correlated, so kPC = 1; and when the uncertainty is uncorrelated,

kPC = 0. The correlation term C is then given by

The correlation in the powers is estimated by

When kPC < 0, the powers are considered uncorrelated and kPC = 0 is used instead.
118

(F!18)

(F!19)

Optical Power

(mW)

kL

Standard Uncertainty ukL (%)

Uncorrelated Correlated Estimated

1.00 1.000165 0.0018 0.0009 0.0013

0.25 1.000165 0.0085 0.0013 0.0025

0.10 1.000167 0.0216 0.0014 0.0038

0.01 1.000167 0.2609 0.0015 0.0121

Table F.6 Combined LOCR electrical calibration correction factor and uncertainty.

The correction factor k2 and its uncertainty are given in Table F.5 using the data from Table F.4. The

values for a nominal power of 1.99 mW, used in the 0.01 mW calculations, are interpolated from the

results for powers of 1.9 and 2.0 mW. Three uncertainty estimates are given; the uncorrelated result

assumes no correlation in the powers, the correlated result assumes 100 % correlation, and the estimated

correlation uses the estimated value of kPC. In practice the estimated correlation result is used, but the

other results are given to show by how much the uncertainty changes for the different degrees of

correlation.

F.4 Combined LOCR Power Measurement Correction

The combined correction factor for the LOCR’s power measurements is the superposition of the DVM

voltage and resistance measurement corrections, and the LOCR calibration correction. Therefore the 

combined correction factor kL is given by

The combined standard uncertainty ukL of kL is estimated using the law of propagation of uncertainty,

assuming that the uncertainties are uncorrelated:

The correction factor and its standard uncertainty are given in Table F.6. Two uncertainty estimates are

given; the uncorrelated result assumes that the uncertainties are uncorrelated, the correlated result

assumes that the uncertainty in the voltages and powers are perfectly correlated, and the estimated result

uses the estimated correlation. The estimated result is used in practice, and the other results are given to

show by how the uncertainty changes for different degrees of correlation. The 0.01 mW example shows

that the LOCR electronics can accurately measure such low powers, but random uncertainty in the optical

power measurement itself would likely be prohibitively large.
119

APPENDIX G. LOCR Optical Receiver Alignment: Uncertainty

It is usually assumed that with cryogenic radiometers, the optical receiver’s absorptance is uniform within

its apparent aperture. However, a recent publication [8] suggests that difficulties encountered in

measurement of optical receiver absorptance may be due, at least in part, to spatial nonuniformity in the

receiver’s absorptance. Subsequent experiments performed on the LOCR confirmed that there is

significant non-uniformity in its optical receiver’s absorptance. This nonuniformity, combined with the

imperfect centering of the laser beam within the receiver, adds additional uncertainty to power

measurements made with the device. In the high-accuracy calibration system, this uncertainty is

accounted for in the LOCR alignment uncertainty.

G.1 Philosophy of Receiver Uniformity Measurement

Unlike the measurements performed for the external publication, the LOCR measurements were

performed with the LOCR in normal operating mode, with its optical receiver at normal operating

temperature and pressure. Since a greater nonuniformity was encountered in the LOCR measurements

than was inferred in the publication, the nonuniformity may increase when the optical receiver is at

cryogenic temperature. The source of the nonuniformity in absorptance may be due to spatial variations

in the paint thickness, adhesion, consistency, or contamination. It may also be due to cracks in the paint

that open wider when the receiver is at cryogenic temperature. These explanations are speculation, but no

matter what the source of the nonuniformity, the measurements show that a significant nonuniformity is

present. Since the magnitude of the nonuniformity can be affected by the size and wavelength of the laser

beam used, the LOCR’s uniformity is measured after absolute power calibration are performed, using the

same laser beam that was used in the calibrations. When a sufficient history of uniformity measurements

is acquired, it may be possible to eliminate the measurement and use historical values. However,

performing the measurement provides another level of quality assurance, since the receiver uniformity

may be degraded at any time.

The LOCR optical receiver’s spatial uniformity is estimated by measuring relative power at several

different locations around the center of its optical receiver, while the LOCR is in normal operating mode. 

Since the optical power and LOCR’s thermal sensitivity are constant, any variation in the measured

responsivity must be due to variation in the amount of power absorbed by the receiver. The amount of

power absorbed can vary for several reasons: nonuniformity in the LOCR’s window transmittance, a

change in the aperture transmittance as the beam location within the aperture is varied, or nonuniformity

in the receiver’s absorptance. When the first two sources of variation are removed from the measured

variation, any remaining variation must therefore be due to nonuniformity in the receiver’s absorptance. 

The nonuniformity in receiver absorptance results in uncertainty in absolute power measurements because

the location of the beam in the LOCR’s receiver (the LOCR’s alignment) is itself uncertain. The location

of the beam during the absorptance measurement is unknown, but the location was most likely not exactly

the same location used in the absolute power measurements. A conservative alignment uncertainty of

1 mm is used, because such a large error in the centering of the beam in the receiver would be obvious,

and therefore should not occur in practice. We are therefore concerned with the variation in the optical

receiver’s absorptance within a centered, circular area with a radius of 1 mm. This radius could be

reduced to perhaps 0.5 mm in the future, if new receiver absorptance measurements are performed with a

know uncertainty in centering.

Because of limitations in the current calibration system, and the necessity of calibrating the detector in a

timely manner, the variation in responsivity within the entire circular area is not completely measured. 

Instead, a statistical sampling of the relative absorptance within the area is used to estimate the variation
120

(G!1)

within the entire circular region. A vertical translation of the beam’s location in the receiver cannot be

performed. To translate the beam vertically, the entire optical source breadboard must be raised or

lowered. Currently, this involves manually adjusting the five leveling posts that support the optical

source breadboard. Performing the manual adjustment requires a significant amount of time, and the

beam’s location in the detector plane swings wildly as the posts are adjusted one at a time. The time

required to adjust the posts, in addition to the time required for the LOCR to stabilize after the wild swing

in beam location, is prohibitive because the drift in the absolute power during this time is too large to be

effectively compensated for by linear interpolation. Therefore, vertical translations cannot currently be

used in the measurement of receiver uniformity. In the future, if the manual leveling posts are replaced

with motion-controlled jack stands, the vertical translation could be performed quickly, and the receiver’s

uniformity more completely measured.

G.2 Receiver Uniformity Measurement: Technique and Analysis

The amount of nonuniformity is small compared to the amount of drift in the optical power, so is difficult

to measure. To minimize the affect of the drift in the absolute power, a relative measurement is

performed quickly, typically in less than one minute. As described in Appendix E, the drift in the optical

power over such a time period is essentially linear; linearly interpolating the power between bracketing

measurements effectively reduces the effect of the power drift. To perform the interpolation, the time that

the power measurements occur is also needed.

The LOCR’s responsivity relative to its responsivity when the beam is located at the receiver’s nominal

center is measured. The beam is first located at the nominal center, and the LOCR’s response P1 is

measured at time t

1

. Then, the beam is translated to location (x,y) on the receiver, and the response P2 is

measured at time t

2

. Finally, the beam is translated back to the nominal center, and the response P3 is

measured at time t

3

. By interpolating the nominal center response in time, the responsivity at location

(x,y), relative to the response at the nominal center, is given by r(x,y), where

The responsivity r(x,y) is measured several times, and the average of r(x,y), R(x,y), is used as the final

responsivity at location (x,y). The uncertainty uR(x,y) in the relative responsivity is dominated by the

measurement reproducibility, so is given by the standard deviation of the mean of R(x,y). If the

measurement reproducibility is reduced in the future, assessment of the uncertainty in the power

measurements themselves may be needed; but the uncertainty in the powers are strongly correlated, so the

resulting uncertainty contribution would be small, as described in Appendix F.

Since a vertical translation in the beam’s location cannot be performed at this time, samples taken at

horizontal translations only are used. Currently measurements acquired at five locations, spaced 0.5 mm

apart on a horizontal line through the receiver’s nominal center are used. The locations are x = !1, !0.5,

0, +0.5, and +1 mm; the nominal center is located at x = 0 mm. The y-locations are always zero, so in the

following description, the second subscript is dropped and R(x) = R(x,y) = R(x,0). 

To obtain the receiver absorptance, spatial variation in the aperture and window transmittance must be

removed from R(x). Variations caused by the aperture translation are measured to assess the uncertainty

in the aperture transmittance measurement, described in Appendix E; spatial variation in the window

transmittance is measured during window transmittance measurement, described in Appendix H. The

corresponding window transmittance values are called TW(x), with standard uncertainty uTW(x), and the
121

(G!2)

(G-3)

aperture transmittance values are called TA(x), with standard uncertainty uTA(x). Like the responsivity

measurements, the transmittances are also measured relative to the transmittance at the nominal center, so

each measurement has a value of 1 at x = 0. The receiver absorptance AR(x) is then given by

The combined standard uncertainty uAR(x) of the relative receiver absorptance measurement at location x

is given by

The terms TW(x) and TA(x) are measured during the final measurement of window transmittance, and the

measurements of relative aperture transmittance, respectively. The relative aperture transmittance TA(x) is

measured in the same way as the relative aperture transmittance’s aperture centering uncertainty,

described in Appendix E, but two additional measurements are required. For the relative aperture

transmittance measurement, the relative transmittance for only an aperture displacement of ±0.5 mm is

measured; for the receiver uniformity measurement, additional measurements at a displacement of

±1.0 mm are also needed. Since the LOCR’s receiver has an elliptical apparent aperture with major and

minor axes of approximately 8.5 by 9.5 mm, the results for a 9 mm diameter circular aperture are used.

Similarly, additional window transmittance measurements similar to those described in Appendix G are

needed. For this purpose, the change in window transmittance relative to the transmittance at the

window’s nominal center is needed, not the absolute transmittance measured for the window

transmittance calculation. Instead of measuring the absolute transmittance, by moving the window in and

out of the beam, the transmittance at the displaced location is measured relative to the transmittance when

the window is nominally centered, by displacing the window and returning it to the nominal center

location. An analysis analogous to that described in Appendix E for the aperture centering uncertainty is

used. This measurement is performed only during the final window transmittance measurement, using the

same nominal window center location that was used in the calibrations. Both the aperture displacement

and the window displacement are measured several times, and the means of the measurements are used as

TA(x) and TW(x). The standard deviation of the means are the standard uncertainties uTA(x) and uTW(x).

The uncertainty in LOCR alignment was discovered recently, so the additional measurements described

above were not performed during earlier calibrations. Therefore only the receiver uniformity results from

later calibrations, performed at visible wavelengths, are available. In all future test detector calibrations,

the aperture offset and relative window transmittance measurements described above will be performed.

Since the locations of the nominal center used in the calibration runs and receiver absorptance

measurement are unknown, a correction for the LOCR’s alignment cannot be applied. Only the combined

standard uncertainty due to the LOCR’s alignment, called uLA, is needed. Using a “between and within”

analysis, two components of uncertainty, called uLA1 and uLA2, contribute to uLA. The standard uncertainty

uLA1 is the standard deviation of the 5 values of R(x). The standard uncertainty uLA2 is the average of the

uncertainties uR(x), for x = !1.0, !0.5, 0.5, and 1.0 (the uncertainty uR(0) is omitted because the
122

x = !1.0 mm x = !0.5 mm x = 0* x = 0.5 mm x = 1.0 mm

Value Std. Unc. Value Std. Unc. Value Value Std. Unc. Value Std. Unc.

8 = 633 nm

R(x) 0.999900 0.000020 0.999902 0.000025 1.000000 1.000181 0.000023 1.000226 0.000023

TW(x) 0.999974 0.000013 0.999987 0.000012 1.000000 1.000021 0.000012 1.000043 0.000011

TA(x) 0.999980 0.000009 0.999986 0.000006 1.000000 1.000002 0.000006 1.000008 0.000009

AR(x) 0.999946 0.000025 0.999929 0.000028 1.000000 1.000157 0.000027 1.000176 0.000027

8 = 515 nm

R(x) 0.999780 0.000001 0.999924 0.000015 1.000000 1.000033 0.000005 0.999975 0.000008

TW(x) 0.999924 0.000012 0.999962 0.000011 1.000000 0.999982 0.000011 0.999964 0.000010

TA(x) 0.999984 0.000001 0.999996 0.000001 1.000000 1.000001 0.000002 0.999995 0.000001

AR(x) 0.999872 0.000012 0.999965 0.000019 1.000000 1.000050 0.000012 1.000016 0.000013

8 = 458 nm

R(x) 1.000302 0.000033 1.000365 0.000034 1.000000 0.999874 0.000047 0.999850 0.000020

TW(x) 1.000038 0.000018 1.000019 0.000011 1.000000 1.000048 0.000011 1.000095 0.000009

TA(x) 1.000003 0.000001 1.000003 0.000001 1.000000 1.000000 0.000001 1.000000 0.000001

AR(x) 1.000261 0.000038 1.000343 0.000036 1.000000 0.999826 0.000048 0.999754 0.000022

* - Values are measured relative to the value at x = 0, so there is no u ncertainty at x = 0.

Table G.1 Example LOCR alignment uncertainty data.

8 (nm) uLA1 uLA2 uLA

633 0.000117 0.000027 0.000120

515 0.000068 0.000014 0.000069

458 0.000260 0.000036 0.000262

Table G.2 Example LOCR alignment uncertainty calculation.

uncertainty at location x = 0 is unknown). The combined standard uncertainty uLA is then given by the

quadrature sum of uLA2 and uLA2.

Example calculations, using the data from Table G.1, are shown in Table G.2. Measured values of uLA

have varied from approximately 0.007 to 0.03 % at different wavelengths; the uncertainty is therefore

very significant, and can even dominate the calibration uncertainty at some wavelengths. These values

are for a beam diameter of 2 mm beam. The combined standard uncertainty uLA is likely to increase for

smaller diameter beams, and decrease for larger beams. Environmental instability contributed to the size

of the uncertainty, especially at 458 nm.
123

APPENDIX H. Window Transmittance: Estimation and Uncertainty

The transmittance of the Brewster’s angle window is the ratio of the amount of optical power that is

incident on the cryogenic radiometer’s optical receiver when the radiometer’s window is in place, relative

to the amount of power incident on the receiver when the window is removed. A theoretical

transmittance for the LOCR Brewster’s angle window can be calculated from the wavelength of the laser

light and the properties of the window’s glass, but as with the relative aperture transmittance,

imperfections in the real system cause the theoretical calculation to be in error by an amount that is

significant in the high-accuracy calibration system. Therefore, like the relative aperture transmittance, the

window transmittance must be measured. The window transmittance is measured each time the LOCR is

used for a calibration because the transmittance depends on many factors, which include the window’s

type, surface quality, alignment, orientation, contamination, scatter centers, and other imperfections;

along with the specifics of the laser beam, such as its wavelength, profile, and radius of curvature.

H.1 Window Transmittance Measurement: Philosophy and Technique

The transmittance of the Brewster’s angle window is measured using a test detector with little or no backreflection and which has an apparent aperture that is at least as large as the LOCR’s. The window

assembly is mounted at the same location relative to the optical source as when it is on the LOCR, and the

test detector is mounted at the same relative location as the LOCR’s optical receiver. The test detector is

fitted with two external aperture stops that simulate the apertures present in the LOCR; one aperture stop

simulates the LOCR’s limiting aperture, and the second simulates the first aperture in the LOCR’s

cryogenic shield. The test detector therefore has the same field of view as the LOCR’s receiver. The test

detector and aperture stops are mounted on a jig that contains a jack stand that allows adjustment of the

assembly’s height, without other translations. The window assembly is mounted on the transverse rail,

which allows the window to be moved in front of the test detector’s jig without other translations. The

window is adjusted to Brewster’s angle by minimizing the intensity of the reflection from the front-side

of the window; the reflection from the back of the window is not used. 

To determine the transmittance of the window, the power incident on the test detector when the window

is in place is measured and divided by the power incident on the test detector when the window is

removed. Since there is a vertical (and slight angular) offset of the transmitted beam when the window is

in place, the jig’s jack stand is adjusted to keep the beam in the center of the apertures. The jig also

contains a translation stage for lateral adjustment, if necessary. To reduce the effect of drift in the applied

power and detector responsivity, the measurement is performed quickly, and the power present at the time

of the measurement with the window in place is interpolated from bracketing measurements made without

the window, as previously described in relation to the power calibration and relative aperture

transmittance measurements. The transmittance measurement is repeated several times, and a mean and

standard deviation of mean are calculated.

Since there is always some loss in the window, the transmittance should be less than one. Therefore, a

measured transmittance that is greater than one indicates a problem with the measurement; using a

detector with a strong back reflection can cause such a problem. When using a three-element reflection

trap as the test detector, the NIST window mount’s transmittance was frequently greater than one until the

inside of the mount was painted black. Using a transmission trap in the measurement helps ensure that

any remaining problem with back reflections is eliminated.

The uncertainty in the window transmittance measurement is dominated by three factors: drift in the

transmittance during the calibration, uncertainty in where the beam is located on the window, and noise in

the transmittance measurement itself. These uncertainties are measured as described in the next section. 
124

Other potential sources of uncertainty are insignificant compared to these factors, so are not included in

the uncertainty analysis. Two such insignificant uncertainties are: imperfection in the scattered optical

power measurement, and the change in transmittance due to the vacuum in the cryogenic radiometer;

these factors are discussed first.

Calibration laboratories usually use one of two philosophies to account for the optical power that is

scattered by the window. The technique used in the high-accuracy calibration system accounts for the

scattered power with the transmittance measurement itself. Other laboratories directly measure the

amount of scattered power, and generate a correction factor and uncertainty for the scattered power that is

applied to the transmittance measurement. Measuring the scattered power requires additional sensors

located inside the cryogenic radiometer, so devices without the additional sensors, such as the LOCR,

cannot use the method.

With the LOCR, optical power that is scattered by the window is accounted for in the transmission

measurement itself, by duplicating the field of view of the LOCR’s optical receiver with the test detector

that is used in the window transmission measurement [3]. Therefore, any scattered power that misses the

LOCR’s optical receiver also misses the detector used in the transmittance measurement. The loss due to

scattering is therefore measured along with the window transmittance itself. Any difference between the

two scattering losses, such as the additional scattering caused by the air, which is present in the

transmittance measurement but not when the window is on the LOCR, is negligible when compared to the

primary uncertainties in the measurement. One remaining source of error is scattered power that is

reflected to the optical receiver by the walls of the connecting steel vacuum lines, which are present when

the window mount is on the LOCR, but are absent during the transmittance measurement. Painting the

inside of these tubes with a vacuum-compatible flat-black paint, or duplicating the tubing in the

transmittance measurement would eliminate this concern; currently any error resulting from the potential

reflection is assumed to be negligible.

Another potential source of uncertainty in the window transmittance measurement arises from the fact

that the transmittance is measured with the window in air, but when on the LOCR, the back-side of the

window is exposed to vacuum. The reflection from the second surface of the window is therefore slightly

different when the window is on and off the LOCR. The theoretical change in the transmittance can be

determined using the Fresnel equations to estimate the window’s transmittance to light with polarization

angle that is parallel to the plane of incidence [33]. The following results were produced using a quartzglass index of 1.457, and a wavelength of 633 nm. To obtain a worst-case result, an air index of 1.0003,

which is appropriate at sea-level but is slightly higher than the actual index present in the NIST-Boulder

laboratory, is used in the calculations. Assuming that the window’s front and back surfaces are perfectly

parallel, when the window is in air and the light’s incidence angle is at Brewster’s angle, there is no

reflection from either the front or the back of the window (for the parallel polarization). When the back

of the window is exposed to vacuum, the transmittance changes by approximately !3 C 10-6 %. The

situation is somewhat worse for the actual window, which has a small wedge angle. The windows

currently used with the LOCR have a wedge angle of no greater than 5 minutes of arc (about 1.5 mrad). 

Assuming the worst-case wedge angle orientation, when the front-side of the window is at Brewster’s

angle, applying vacuum to the back changes the transmittance by about ±6 C 10-5 %, which is negligible

when compared to the other uncertainty terms, which have a typical magnitude of 10-3 %, so the effect is

ignored. The presence of the wedge also reduces the overall transmittance slightly, by approximately

3 C 10-4 % for the worst-case scenario; depending on the orientation of the wedge angle relative to the

polarization angle, the loss can be smaller. The window’s nonzero wedge angle can therefore reduce the

maximum theoretical window transmittance, but any reduction is accounted for by the transmittance

measurement. The presence of a surface contaminant on the glass can have a much greater effect, as

described later.
125

The presence of the wedge and the vacuum on the back of the window also cause the direction of the

transmitted beam to deviate slightly from the direction of the incident beam. For the perfectly parallel

window, the vacuum on the back of the window causes the transmitted beam to be deviated by about

0.4 mrad relative to the incident beam; the worst-case deviation for the wedged window is about 4 mrad. 

But any resulting deviation is compensated for during the LOCR and test detector alignment, so does not

significantly affect the accuracy of the measurement. The thickness of the window also causes a vertical

translation of the transmitted beam, as described in the LOCR alignment section. The potential for the

intensity of the reflection from the back of the window to change when the window is removed from the

LOCR’s vacuum is the reason why only the reflection from the window’s first surface is used to adjust

Brewster’s angle; the intensity of the reflection from the front-side of the window is constant when the

window is both on and off the LOCR. Stress-induced birefringence in the window glass would

complicate this analysis, but the thick quartz window and low-stress window mount used are assumed to

minimize any such birefringence to negligible proportions.

H.2 Transmittance Measurement: Uncertainty Analysis

The window transmittance and uncertainty used in the detector calibration are derived from multiple

measurements of window transmittance performed both before and after the calibration runs, using the

analysis described below. This analysis is the result of a trade-off between performing timely

calibrations, and performing extremely accurate measurements of window transmittance. Given that

some other laboratories measure the window transmittance at only a single location on the window, either

before or after the calibrations are performed, this analysis is conservative in comparison. The

uncertainty in the window transmittance measurement is dominated by three factors: drift in the

transmittance during the calibration, uncertainty in where the beam is located on the window, and noise in

the transmittance measurement itself. In the high-accuracy calibration system, all of these factors are

included in the uncertainty analysis of window transmittance.

The transmittance drift during the calibration is frequently the largest uncertainty component. The

transmittance measured after the calibration is almost always higher than that measured before the

calibrations. The drift is likely due to an absorbate, such as water vapor, accumulating on the back-side of

the window, when the window is not in use. When the window is then used on the LOCR, the absorbate

is slowly released into the LOCR’s internal vacuum, and the window’s transmittance increases. To

account for the affect, the window’s transmittance is measured both before and after the calibrations are

performed. To help reduce the magnitude of the drift, the window is placed on the LOCR and exposed to

vacuum for some time before the initial transmittance measurement is performed. If the LOCR is used

more often in the future, the back of the window will be exposed to the atmosphere for less time, and the

drift should be reduced.

The window transmittance measurement is complicated by the fact that the transmittance is not uniform

over the surface of the window. Also, there is uncertainty in the exact location where the laser beam

strikes the window, as described in section 3.2.4. To account for the uncertainty in the beam’s location, a

matrix of transmittance measurements around the nominal beam location is acquired. A nine-point square

matrix with a 1 mm spacing is used. The 1 mm spacing is used because the shift in the beam’s location

on the window, due to imperfections in the mounting hardware, is typically 1 mm or less. However, the

beam diameter at the window can be smaller than 1 mm, so the matrix can be considered a statistical

sampling of the transmittance, rather than a complete mapping of the transmittance in the region.

The result of the two transmittance measurements is two arrays of nine transmittance means, called

TW(x,y), with their associated standard deviation of mean, called uW(x,y). As described in section 3.2.4,

the nominal beam location in the first set of transmittance measurements can be different from the
126

(H!1)

(H!2)

(H!3)

(H!4)

(H!5)

nominal beam location in the second set of measurements, because of imperfections in the mounting

hardware. Therefore, the transmittance value at a location (x,y) in the two matrices may not represent the

same physical location on the window. The location shift would have to be reconciled if a more complex

analysis is used, such as the center-weighted technique described in Appendix F. However, in the simple

analysis described below, the location shift is ignored. A center-weighted analysis, such as that described

in Appendix F, would likely result in a smaller estimate of combined standard uncertainty.

Currently, a simple “between and within” approach is used to analyze the transmittance data. The actual

window transmittance present during the calibrations is assumed to be between the two mean

transmittances and within their uncertainties. The transmittance estimate from the first set of

measurements is called TW1, with standard uncertainty uW1. The results from the second set of

measurement are called TW2 and uW2. The transmittance estimates and uncertainties are derived below. 

The transmittance TW at the time of the test detector calibration is assumed to lie somewhere between TW1

and TW2, with a rectangular probability distribution, so

Two sources contribute uncertainty to TW. The drift in transmittance from TW1 to TW2 contributes the

standard uncertainty uD. Assuming a rectangular probability distribution,

The uncertainty in TW1 and TW2 also contributes uncertainty TW. Assuming the uncertainties are

uncorrelated, the contribution’s standard uncertainty uM is given by

The combined standard uncertainty uTW is

The transmittance estimates TW1 and TW2 and their associated uncertainties are determined similarly. Call

T(1) to T(9) the average transmittance measured at each of the nine locations, and u(1) to u(9) the

standard uncertainties associated with each measurement. Several transmittance measurements are

performed at each location N; and the average measurement is used as TWN, and the standard deviation of

the mean of the measurements is the standard uncertainty uSN.

For N = 1 to 2, the estimated transmittance TWN for the each sample set is given by the mean of the

samples:
127

(H!6)

(H!7)

(H!8)

Location 

(x,y)

mm

Initial Measurement Final Measurement

Transmittance Standard

Uncertainty

Transmittance Standard

Uncertainty

(!1,1) 0.999746 0.000020 0.999703 0.000009

(0,1) 0.999657 0.000015 0.999808 0.000004

(1,1) 0.999582 0.000013 0.999805 0.000008

(!1,0) 0.999742 0.000017 0.999754 0.000013

(0,0) 0.999686 0.000014 0.999780 0.000013

(1,0) 0.999703 0.000020 0.999823 0.000011

(!1,!1) 0.999725 0.000011 0.999768 0.000014

(0,!1) 0.999691 0.000014 0.999779 0.000006

(1,!1) 0.999736 0.000015 0.999798 0.000004

Table H.1 Example measured window transmittance TW(x,y), and standard uncertainty uW(x,y).

The uncertainty in TWN contains contributions from both the measured variation and the uncertainty in the

measurement. The standard deviation of the nine samples is used as the standard uncertainty uSN in TWN

that is due to variation in the transmittance,

The uncertainty contribution uMN due to the uncertainty in the measurements themselves is determined by

using the law of propagation of uncertainty, assuming the uncertainties are uncorrelated:

The combined standard uncertainty uWN in TWN is the quadrature sum of the two uncorrelated uncertainty

components:

The transmittance estimate and standard uncertainty for each of the two measurements sets is determined

in this way, and designated, respectively, TW1 and uW1, and TW2 and uW2.
128

Initial Transmittance

Measurement

Final Transmittance

Measurement

Combined Transmittance

Measurement

TW1 uW1 TW2 uW2 TW uW

0.999696 0.000052 0.999780 0.000036 0.999738 0.000040

Table H.2 Analysis of the example window transmittance data.

H.3 Example Calculation of Transmittance

The following example uses actual window transmittance measurement data acquired with a 633 nm

wavelength laser. Table H.1 shows window transmittance data acquired before the calibrations were

performed (initial measurement), and the data acquired after the calibrations were complete (final

measurement). There was a shift in the nominal beam location of 1 mm in the negative-x direction, but

the shift does not affect the calculations, and the location given in the table is the unshifted location. 

Three measurements were performed at each location; the transmittance is the mean of the measurements,

and the standard deviation of the mean is the standard uncertainty. Table H.2 shows the results of the

final estimation of window transmittance, its uncertainty components, and combined standard uncertainty. 

When expressed relative to the transmittance, the combined standard uncertainty is 0.0040 %.
129

(I!1)

(I!2)

(I!3)

(I!4)

APPENDIX I. Center-Weighted Spatial Uncertainty

Dr. Chih-Ming Wang, NIST, Statistical Engineering Division

Assume that the transmittance at (x, y), t(x, y), can be modeled approximately by a smooth curve, say a

quadratic form, within a given window; that is,

where $0, ..., $5 are to be estimated. Define region R = {(x,y); (x ! x0)

2

 + (y ! y0)

2 # r02

}. Then the

observed transmission, centered at (x0,y0) with beam diameter 2 r0, is given by

where w(x,y) is the weight function.

We consider only the Gaussian weight. The truncated bivariate Gaussian density, centered at (x0,y0)

having standard deviation F and circular equal-probability contours, is given by

Since the sum of the weights must be 1, i.e.,

c is found to be (1 ! e

-2)2BF2

. Also, an e

2

 beam implies w(0,0)/w(0,r) = e

2

 or F = r0/2.

With this Gaussian weight function, we have 

where k = (1 ! 3e-2)/(4 ! 4e!2) = 0.1717412. The result was obtained by transforming the integral using

polar coordinates. The result allows us to estimate $i by linear regression with dependant variable

(x0,y0,r0) and independent variables x0, y0, x0

2

 + kr0

2

, x0y0, and y0

2

 + kr0

2

.

The data in the following example used a Gaussian beam of 2 mm in diameter. The wavelength was

1550.4 nm. The centers for the nine-measurement spatial experiment were located at (!1, 1), (!1, 0),

(!1, !1), (0, !1), (1, 1), (1, 0), and (1, !1).

The beam size used was 2 mm, i.e., r0 = 1 mm. The vector of dependent variables and the matrix of

independent variables for this example are
130

(I!5)

(I!6)

Beam Diameter (mm)

1 2 3 4 5

bias !0.000004 0.0 0.000007 0.000016 0.000028

ub 0.1288 0.0 0.2147 0.5152 0.9016

Table I.1 Bias and ub.

The parameter estimates are obtained by

The variance-covariance matrix of the parameters can also be calculated.

Inferences can be made based on the model just derived. For example, if we are interested in estimating

the bias and the uncertainty of the transmission at (0,0) with various beam sizes, the difference in

transmission between beam sizes r0 and 1 is

The standard deviation of this difference is given by

The following table displays the bias and ub for beam diameter 2r0 = 1 to 5 mm, where is the

uncertainty of the transmission measurements.
131

(I!7)

We can also evaluate the uncertainty in estimating spatial transmission due to errors in locating (x0,y0). 

The variance, assuming independence of x0 and y0, is given by

A small experiment can be conducted to estimate var(x0), @@@, var(y0

2

).
132

N Reflectance R(N) Absorptance A(N)

1 0.000069 0.999931

2 0.000087 0.999913

3 0.000084 0.999916

4 0.000082 0.999918

5 0.000081 0.999919

6 0.000083 0.999917

7 0.000069 0.999931

8 0.000084 0.999916

9 0.000083 0.999917

10 0.000083 0.999917

Mean (RAVE and AAVE) 0.000081 0.999919

Standard Deviation of the Mean

(uTA)

0.000002 0.000002

Table J.1 LOCR receiver absorptance at 633 nm.

APPENDIX J. Estimate of LOCR Optical Receiver Absorptance

The absorptance of the LOCR’s optical receiver was measured at a wavelength of 633 nm by the device’s

manufacturer [13]. The receiver’s absorptance at other wavelengths is estimated using measurements

performed on a similar optical receiver by an outside laboratory [17]. Theoretical analysis using

reflection data for the paints used on the receiver show that the device should be essentially spectrally flat

over the supported wavelength range (visible to infrared), so the single absorptance measurement is

commonly used. Unfortunately, the results of direct measurement of the absorptance at other

wavelengths, performed and published by the outside laboratory, show that the absorptance is not

spectrally flat over the wavelength region of interest to within the uncertainty of the 633 nm absorptance

measurement. Imperfections in the paint, such as thick spots which could become more glossy, pooling

in the corners, contamination, or voids in the paint could cause such a behavior.

J.1 LOCR Receiver Absorptance at 633 nm

The manufacturer measured the absorptance measurements of the LOCR’s optical receiver at a

wavelength of 633 nm. Specifically, the reflection from the receiver was measured using an integrating

sphere and the absorptance was calculated from the reflectance. The receiver was illuminated with an

approximately Gaussian beam, produced by a spatially filtered, single-transverse-mode helium-neon

laser. Multiple reflectance measurements, called R(N), were acquired by comparing the receiver’s

absorptance with that of a reflectance standard. The measurement has a systematic error with a standard

uncertainty of approximately 2 % of the reflectance.

In each measurement, the equivalent absorptance A(N) is given by A(N) = (1 ! R(N)). The absorptance

data is analyzed with a Type A evaluation, which returns the mean absorptance AAVE and the standard
133

(J!1)

Receiver Absorptance

at 633 nm 

(AR633)

Type A Uncertainty

Estimate 

(uTA)

Type B Uncertainty

Estimate 

(uTB)

Combined Standard

Uncertainty

(uR633)

0.999919 0.000002 0.000002 0.000003

Table J.2 Receiver absorptance uncertainty calculation.

deviation of the mean uA. The reflectance measurements and absorptance calculation are shown in Table

J.1, where the Type A evaluation is summarized.

The standard uncertainty contributed by the systematic error, called uB, is 2 % of the average reflection

RAVE, so uB = 0.02 C RAVE. The receiver absorptance at 633 nm is called AR633, and has combined standard

uncertainty uR633, where

The combined standard uncertainty calculation is summarized in Table J.2.

J.2 LOCR Receiver Absorptance at Other Wavelengths

The absorptance of the LOCR’s optical receiver at wavelengths other than 633 nm is estimated using a

between-and-within analysis of absorptance measurement results published by an outside laboratory [17]. 

The LOCR’s optical receiver is identical to the measured receiver except for its scale; the LOCR’s

receiver is twice as large as the measured receiver; the LOCR’s receiver has a diameter of approximately

10 mm, while the measured receiver has a diameter of approximately 5 mm. The receivers have

otherwise the same geometry and were constructed in the same way, using the same techniques and

materials, by the same manufacturer. The spectral performance of the two receivers is therefore assumed

to be comparable, so the measurement results for the smaller receiver are also appropriate for the LOCR’s

larger receiver. The laboratories’ measured absorptance of the smaller receiver at 633 nm is consistent

with the manufacturer’s measurement at 633 nm, so the manufacturer’s measurement of the absorptance

of the LOCR’s receiver at 633 nm is also assumed to be accurate. The manufacturer’s absorptance

measurements show that the LOCR’s larger receiver has a slightly higher absorptance than the smaller

receiver, which is likely due to the larger receiver capturing more of the laser beam that was used in the

absorptance measurement.

The outside laboratory performed their absorptance measurements in a manner similar to that used by the

manufacturer to measure the absorptance at 633 nm; a spatially filtered, power stabilized, approximately

Gaussian laser beam was injected into the center of the receiver, and the power reflected from the receiver

was measured using an integrating sphere. The laboratory stated that their laser beams had a 1/e2

intensity diameter of less than 1 mm; the size of the beam used in the manufacturer’s measurement is not

specified in the literature, but was described in conversations as having a diameter of approximately

1 mm. The measurements are therefore assumed to be equivalent.

The outside laboratory measured their receiver’s absorptance at several wavelengths from 325 to

1550 nm, in addition to the 633 nm wavelength. We assume that these measurements are typical for the

receiver within the wavelength region, and therefore represent a random statistical sampling of the

receiver’s absorptance within this wavelength region. The measurements can therefore be used to

estimate the absorptance of the LOCR’s receiver within this wavelength region. These assumptions

cannot be proven with the information currently available, but analysis of the paint and receiver geometry
134

show that the absorptance should vary smoothly throughout the wavelength region, with no resonance

peaks or dips; therefore the assumption that the measurements are a valid statistical sampling of the

absorptance should be valid. This analysis is considered adequate given the limited amount of

absorptance information available, and is much better than assuming that the absorptance measured by the

manufacturer at 633 nm is appropriate for the entire range of wavelengths. In the future, absorptance

measurements may be performed on the actual LOCR receiver to verify the assumptions and reduce the

uncertainty in the measurement.

In reference [17] it was noted that there was a spatial variation in the optical receiver’s absorptance in that

a small change in the location of the beam in the receiver caused the measured absorptance to change. 

They stated that this variation was a major component of the uncertainty in their absorptance

measurements, and speculated that much of the measured variation in the absorption was due to spatial

nonuniformity in the receiver’s absorptance, or the presence of optical power in addition to that in the

approximately Gaussian beam. Both of these factors probably influenced their measurements, as both

have been observed in the high-accuracy calibration system. The relative aperture transmittance

measurements described in Appendix F show that there is significant optical power in addition to that in

the Gaussian beam, and spatial nonuniformity is evident in the evaluation of the receiver alignment

uncertainty, described in Appendix I. The uncertainty that results from these imperfections likely

contributed to the apparent variation in the receiver absorptance as described in the publication. The

uncertainty due to these imperfections may also be double-counted in this analysis, but until the receiver

absorptance measurements are reproduced using the high-accuracy calibration system where these

uncertainties are assessed independently; this analysis results in an uncertainty that is at least

conservative. 

The outside laboratories’ measurement results consist of a nominal reflectance measured at several

wavelengths, called R(8), with associated standard uncertainty uR(8). The reflection data are converted to

absorptance data as described for the manufacturer’s measurement, resulting in the nominal absorptance

A(8), and its standard uncertainty uA(8). The nominal absorptance over the entire wavelength range,

called AOL, is the mean of A(8). The measured variation in the absorptance is accounted for in the

standard uncertainty called uA3, which is the standard deviation of A(8); the standard deviation of mean is

not used here because we are interested in the standard deviation of the absorptance over the range of

wavelengths, not the uncertainty in the average AOL. The uncertainty in the outside laboratories’

measurements is accounted for in the uncertainty component called uA4, which is the average of the

uncertainties uA(8).

Since the LOCR’s receiver has a higher absorptance at 633 nm than the outside laboratories, the results of

the outside laboratories’ measurements must be scaled for use with the LOCR. The scale factor kA is the

LOCR’s absorptance at 633 nm divided by the absorptance of the outside laboratories’ receiver at

633 nm, so kA = AR633 / A(633 nm). Then the nominal absorptance of the LOCR’s receiver over the entire

wavelength range is AR = kAC AOL. Since the conversion from reflection to absorptance involves a

subtraction, the uncertainty is unchanged, so the uncertainties uR(8) and uA(8) are equal. The combined

standard uncertainty in AOL, uAOL, is then given by the quadrature sum of the three uncorrelated

uncertainties, uA3, uA4, and uR633. The uncertainty in AR is uAR = kAC uAOL. The results of these

calculations are given in Table J.3. When the LOCR is used at a wavelength of 633 nm, the

manufacturer’s absorptance measurement is used, so AR633 and uAR633 are used as AR and uAR. When the

LOCR is used at any other wavelength within the range from 325 to 1550 nm, AR and uAR from Table J.3

are used instead.
135

Wavelength

(nm)

Reflectance Absorptance

R(8) uR(8) A(8) uA(8)

325 0.000123 0.000016 0.999877 0.000016

488 0.000231 0.000030 0.999769 0.000030

515 0.000177 0.000019 0.999823 0.000019

633 0.000123 0.000013 0.999877 0.000013

1550 0.000244 0.000009 0.999756 0.000009

Standard Deviation of Absorptance, and Mean Uncertainty 

(uA3 and uA4)

0.000057 0.000017

Mean Receiver Absorptance and Standard Uncertainty

(AOL and uAOL)

0.999820 0.000060

LOCR Receiver Absorptance and Standard Uncertainty

(AR and uAR)

0.999862 0.000060

Table J.3 LOCR receiver absorptance at wavelengths other than 633 nm.
136

APPENDIX K. Weather Station Calibration and Uncertainty

Environmental parameters are measured using a commercial weather station [35]. To obtain accurate

measurements with the weather station, its temperature and barometric pressure functions are calibrated

using the following techniques. The station’s relative humidity sensor is not calibrated, but its uncertainty

is estimated, as described below. The standard uncertainty of the weather station measurements is

combined with other uncertainties to determine the combined standard uncertainty of the measured

parameter, as described in section 5.4.4. For convenience, quantization in the weather station’s readings

is accounted for in the calibration uncertainty. Examples of the measurements acquired during a typical

calibration are shown, along with the uncertainty analysis for the example data.

The weather station is fully autonomous and records the environmental readings internally. The stored

readings are downloaded to the main computer for processing and archiving as desired. Additional

precision analog temperature sensors are available, which are monitored by the main computer directly,

through the system’s DMM. These sensors are calibrated in the same manner as the weather station’s

temperature sensors, except that they have no quantization error.

Currently, the weather station is calibrated as well as possible using available standards. If better

standards become available, especially for pressure and humidity, the calibration uncertainties will likely

be reduced. However, the uncertainty of the temperature standard (a precision mercury thermometer) is

probably adequate given the size of the thermal gradients and temperature fluctuations encountered in

typical optical calibrations.

K.1 Internal and External Temperature Sensor Calibration

The weather station has two temperature sensors, an internal sensor which measures the temperature on

the optical source breadboard where the weather station is located, and an external sensor that measures

the temperature in the detector plane. When the calibration system is not in use, the sensors are placed

together, in close proximity to a precision mercury thermometer, which serves as the primary standard for

the calibration of the temperature sensor. Several thermometer and weather station readings are recorded

over time, and used to derive a correction factor for the weather station’s indicated readings. Since the

time constants of the three sensors is different, the sensors are ideally allowed to reach steady-state in a

stable environment. Any error due to drift in the environment during the measurement period is

randomized by acquiring the sensor data at various times throughout the day.

The correction factors for the temperature sensors are occasionally entered into the weather station’s

internal memory, so that the station’s indicated readings are more accurate. Detector calibrations are

performed continually to derive the external correction factor, which accounts for ageing and drift in the

temperature sensors since the previous update. To obtain the most accurate correction possible, the

external correction factors are derived from measurements acquired within a few weeks before optical

calibrations are performed,. Whenever the station’s internal correction factors are changed, the external

calibration factor must, of course, be calculated anew.

The standard uncertainty of each temperature sensor’s calibration factor has two parts: the standard

uncertainty of the primary standard, called uT1; and the measurement repeatability of the sensor

calibration, called uT2. The combined standard uncertainty of the temperature sensor calibration, called

ukT, is therefore the quadrature sum of the two independent standard uncertainties, uT1 and uT2.

An additive correction factor called kT is derived for each temperature sensor. The correction factor is

such that the actual temperature is the indicated temperature plus kT. The correction factor is derived
137

Internal Temperature

Sensor

External Temperature

Sensor

Measured Offsets, dT (°C) !0.20 0.00

!0.25 !0.05

!0.10 !0.10

!0.20 !0.10

!0.15 !0.05

!0.10 !0.10

!0.10 !0.10

!0.05 N/A

Number of Measurements, N 8 7 

Mean, kT (°C) !0.144 !0.071

Standard Deviation of the Mean, uT2 (°C) 0.024 0.015

Mercury Thermometer Standard Uncertainty,

uT1 (°C)

 0.058 0.058

Combined Standard Uncertainty, ukT (°C) 0.063 0.060

N/A - Not available.

Table K.1. Example temperature sensor calibration.

using a Type A analysis of the sensor calibration data. For each sensor measurement, the difference

between the standard’s temperature and the station’s indicated temperature is given by the amount dT. 

The correction factor for the sensor, kT, is then the mean of the offsets dT, and the standard deviation of

the mean is the measurement repeatability, uT2.

The uncertainty of the temperature standard, a precision mercury thermometer, is assumed to be bounded

by ±0.1° C, the finest division on the thermometer’s scale. Therefore the standard uncertainty of the

primary temperature standard is uT1 = 0.1° C / . Example calibration data and the calculated

calibration factor and uncertainties are given in Table K.1 for both the internal and external temperature

sensors. 

The average of the temperature during the optical calibration at the weather station and in the vicinity of

the external sensor is desired. Call TI the mean internal temperature, TE the mean external temperature,

and kI and kE the internal and external temperature correction factors with standard uncertainties ukI and

ukE. The correction factors and their uncertainty contribution kI, ukI, kE, and ukE are equivalent to kT and

ukT from Table K.1, for the internal and external sensors. The mean temperature in the detector plane

(where the external sensor is placed), called TDP, is given by TDP = TE + kE. Also, the mean temperature at

the weather station, called TWS, is given by TWS = TI + kI. An example of the weather station’s internal and

external temperature readings, acquired during an actual calibration of optical power, are shown in

Figures K.1 and K.2. 
138

Figure K.1 Internal weather station temperature during a typical optical calibration.

Figure K.2 External weather station temperature during a typical optical calibration.

The standard uncertainty of the temperature at the weather station TWS is the quadrature sum of three

standard uncertainties: the standard uncertainties of the internal temperature sensor calibration ukI; the

measured temperature variation, called uVI; and the standard uncertainty due to quantization in the

weather station’s digital output, called uTQ. The measured temperature variation uVI is given by the

standard deviation of the measured internal temperatures; standard deviation of the mean is not used,

since we are assessing the amount of variation. The digitized output, shown in Figures K.1 and K.2, is

clearly quantized, with an interval of about 0.1° C. We assume that the indicated readings are rounded off

so that the actual reading is within ±0.05° C of the indicated reading; therefore the standard uncertainty

due to the station’s quantization is uTQ = 0.05 / ° C. 

The detector plane temperature TDP has three analogous uncertainties and one additional term. As

described in section 5.4.4, the uncertainty in the detector plane temperature uDP includes a term for the

longitudinal temperature gradient, called uG. This temperature gradient is much less significant since the

ventilation system for the argon-ion laser was improved, but is retained for completeness. The transverse

temperature gradient, evident as the periodic fluctuation in Figure K.2, is accounted for in the
139

Internal Temperature

(°C)

External Temperature

(°C)

Calibration factors kI, kE !0.144 !0.071 

Calibration factor standard uncertainty ukI, ukE 0.063 0.060

Mean indicated temperature TI, TE 25.216 25.930 

Corrected temperature TWS, TDP 25.072 24.859 

Standard uncertainty due to temperature variation 

uVI, uVE

 0.037 0.079

Output quantization standard uncertainty, uTQ 0.029 0.029

Standard uncertainty due to temperature gradient 

uG

0.014 0.014

Combined standard uncertainty of the corrected

temperature uWS, uDP

 0.080 0.104

Table K.2 Calculated temperatures and uncertainties for the example data.

(K!1)

(K!2)

measurement variation uVE. The longitudinal temperature gradient is estimated as 1/10 the temperature

difference between the internal and external temperature sensors. The two sensors are separated by about

1 m, and a typical test detector spans about 0.1 m, so assuming that the gradient varies linearly, the

temperature gradient across the detector is bounded by: ±(TDP ! TWS) / 10. Assuming a rectangular

probability distribution, the estimated standard uncertainty is

This estimation is conservative because in the real system, most of the temperature gradient occurs across

the scattered light shield located between the two sensors, which acts as a heat shield in this case. Since

the uncertainties are uncorrelated, the combined standard uncertainty of the detector plane temperature

measurement is the quadrature sum of the uncertainty components, so

The calculated temperatures and uncertainties for the example data are shown in Table K.2.

K.2 Barometric Pressure Sensor Calibration

The weather station’s barometric pressure sensor is calibrated by comparison to nearby laboratories

operated by the National Center for Atmospheric Research (NCAR). The two laboratories, the NCAR

Mesa Laboratory and the NCAR Foothills Laboratory, post real-time local weather information to the

internet, which include barometric pressure readings. The sites are accessible at the internet addresses

www.atd.ucar.edu/cgi-bin/weather.cgi?site=ml and www.atd.ucar.edu/cgi-bin/weather.cgi?site=fl. 
140

(K!3)

Barometric Pressure

Measured Offset, PWS ! PWI (hPa) 0.36

0.96

0.06

0.56

0.56

1.25

0.75

Number of Measurements, N 7 

Mean Offset, kP (hPa) 0.64

Standard Deviation of Mean, ukP (hPa) 0.15

Table K.3 Weather station barometer calibration.

The two NCAR laboratories are located within a few miles of the high-accuracy calibration system, at

altitudes above and below the system. 

The pressure at the calibration system is interpolated from the pressure at the two nearby NCAR

laboratories. The interpolation is performed assuming that the pressure changes linearly with altitude. 

According to the web sites, the NCAR Mesa laboratories’ weather station is located at an altitude of

AM = 1,885 m, and the Foothills laboratories’ weather station is located at an altitude of AF = 1,625 m. 

The calibration system’s weather station is located at an altitude of AWS = 1,652 m above sea level, using

the National Vertical Datum of 1929; the NCAR laboratory altitudes are assumed to be relative to the

same datum, since the stated altitudes are consistent with the altitude given on topographical maps of the

area, which are based on this datum. The pressure at the calibration system’s weather station PWS, is

derived from the Foothills laboratories’ pressure PF, and the Mesa laboratories’ pressure, called PM, by the

following equation:

Deviations from eq (K!3) caused by local weather systems, sensor time constants, diurnal fluctuations, or

other influences are randomized by sampling the pressures at various times throughout the day, over

many days, and so are accounted for in the measurement repeatability. 

The pressure readings provided by the weather station are called the indicated pressure, PWI. We desire

an additive calibration factor called kP, such that the actual laboratory pressure PWS = PWI + kP. Using a

Type A analysis, the best value of kP is the mean of PWS ! PWI, for the recorded values of PWS and PWI, and

the standard deviation of the mean is the uncertainty ukP Example pressure calibration data are given in

Table K.3, along with the calculated calibration factor and uncertainty.

The uncertainty in PWS that is contributed by the uncertainty in the parameters used in eq (K!3), called uB,

is determined using the law of propagation of uncertainty. Since the uncertainties are uncorrelated, the

combined standard uncertainty is given by
141

(K!4)

AWS (m) AF (m) AM (m) PM (hPa) PF (hPa) uB (hPa)

0.269 !0.241 ! 0.028 0.896 0.896

uX 5 2 2 3 3 4.06

Table K.4 Uncertainty in the pressure calibration that is due to uncertainty in the parameters.

Figure K.3 The indicated barometric pressure acquired during a typical optical calibration.

To evaluate uB, the uncertainty of the stated altitudes and NCAR pressure readings are needed. We

assume a standard uncertainty for the NCAR altitudes of 2 m, so uAM = uAF = 2 m. We use a standard

uncertainty of 3 m for the calibration laboratories’ altitude, so uAWS = 5 m. The NCAR web pages state

that the accuracy of their barometers is 3 hPa, we assume that this is their standard uncertainty, so

uPM = uPF = 3 hPa. The NCAR barometers are compensated for temperatures from 0 to 50° C, so only

when the ambient temperature at both NCAR sites is within this range are the performed calibrations used

in the calculation of kP and ukP; therefore the temperature at the NCAR sites is also recorded.

In Table K.4, numerically derived partial derivatives are used to evaluate eq (K!4). The derivatives are

numerically derived using nominal values for the parameters: AWS = 1652 m, AF =1625 m, AM = 1885 m,

PF = 700 hPa, and PM = 770 hPa.

The measured variation in the pressure during the calibration also contributes uncertainty to the

laboratory pressure PWS. The pressures recorded by the weather station during the calibration are

averaged, then the mean pressure is PWI, and the standard deviation is the uncertainty due to the measured

variation, called uPV. Example pressure measurements acquired during an actual calibration are shown in

Figure K.3. While the data shown in Figure K.3 are adequately dithered, quantization in the output is

assessed for completeness. From the figure, the quantization interval is 0.1 hPa, so assuming that the

readings are properly rounded off, the standard uncertainty due to quantization is uPQ = 0.05 / hPa.
142

(K!5)

Barometric Pressure (hPa)

Calibration factor, kP 0.64

Calibration factor standard uncertainty, ukP 0.15

Indicated pressure, PI 838.67 

Corrected pressure, P 839.31 

Combined standard uncertainty due to parameters, uB 4.06

Measured pressure variation, uPV 0.16

Output quantization standard uncertainty, uPQ 0.03

Combined standard uncertainty of the corrected pressure, uP 4.07

Table K.5 Calculated barometric pressure and uncertainty for the example data.

The nominal pressure during the calibration, called P, is the sum of the mean indicated pressure, PWI, and

the pressure calibration factor, kP. The combined standard uncertainty in P, called uP, is the quadrature

sum of the independent uncertainties ukP, uB, uPV, and uPQ:

The resulting pressure and uncertainty for the example data are given in Table K.5. The data clearly show

that the uncertainty in the corrected pressure is dominated by the uncertainty in the in the parameters used

in the pressure calibration.

As with the system weather station’s temperature sensors, the calibration factor for barometric pressure

can be entered into the station so that its indicated readings are accurate. If the station’s internal

calibration factor is changed, the external calibration factor must be calculated anew. Calculating the

correction factor continually compensates for aging and drift in the system weather station’s barometer.

K.3 Relative Humidity Sensor Calibration

Currently, the weather station’s relative humidity sensor is not calibrated, primarily because of the lack of

an available standard for relative humidity. Typically, relative humidity sensors are calibrated by

enclosing the sensor in an airtight box with known relative humidity. A known relative humidity can be

generated using solutions of certain salts. The weather station’s sensor could be placed in such an

enclosure for calibration, or a transfer standard that was calibrated using such a method could be used. 

Currently, the relative humidity measurements are not considered important enough to go through such a

calibration. Of importance to most detectors is whether the relative humidity is high or low, the precise

relative humidity is not generally needed. The calculation of the air’s index of refraction is also relatively

insensitive to the humidity. Therefore, the weather station manufacturer’s published specification for the

accuracy of the relative humidity sensor is considered adequate.

The manufacturer states that the accuracy of the weather station’s internal relative humidity sensor is

±5 %, from 10 to 90 %. We conservatively assume that this is a standard uncertainty, therefore the

standard uncertainty of the humidity sensor, called uHS = 5 %. There is no reason to believe that this
143

Figure K.4 The relative humidity measured during a typical optical calibration.

specification is invalid, because the published specifications for the temperature and pressure sensors

seem conservative when compared to the actual performance of the sensors. The accuracy specification

for both the internal and external temperature sensors is stated as ±0.5° C; the calibrated sensor

uncertainty is typically less than 0.1° C, so the published specification seems conservative. The accuracy

of the barometric pressure sensor is stated as 1.7 hPa. Due to the inaccuracy of the pressure standard used

to calibrate the sensor, it cannot be determined definitively that the sensor is actually performing better

than this amount, but when compared to calibrations performed with a constant external temperature at

the NCAR sites, the pressures are in significantly better agreement than implied by the published

accuracy specification. This is likely because the main source of uncertainty in the NCAR pressure

sensors is the external temperature. Therefore we assume that the published specification for the pressure

sensor’s accuracy is conservative. 

Further, while the weather station is designed to internally store a calibration factor for the temperature

and pressure sensors, no support for an internal humidity sensor calibration factor is provided. So it

appears that the manufacturer does not believe that the sensor needs calibration to meet its published

specification. Therefore we assume that the published humidity sensor specification is accurate, when the

relative humidity is between 10 and 90 %. The relative humidity in the calibration laboratory is never

likely to be outside this range. Therefore, we use the published accuracy specification as the standard

uncertainty, and uHS = 5 %.

Example relative humidity sensor data are shown in Figure K.4. As with the other environmental sensors,

the nominal relative humidity during an optical calibration, RH, is given by the mean of the relative

humidity measurements. Since the sensor is not calibrated, no correction is applied to the indicated

readings. The standard uncertainty in RH, called uRH, is given by the quadrature sum of three independent

standard uncertainties: the sensor calibration uncertainty uHS; the measured variation, called uHM; and the

standard uncertainty due to quantization in the digital output, called uHQ. 

The measured variation uHM is given by the standard deviation of the humidity measurements. From

Figure K.4, the digitized output is clearly quantized by about 1 %, therefore assuming that the values are

properly rounded off, the standard uncertainty due to output quantization uHQ = 0.5 / %. The

calculated relative humidity and uncertainty for the example data are given in Table K.6.
144

Relative Humidity (%)

Mean relative humidity, RH 31.08

Measured relative humidity variation, uHM 0.27

Standard uncertainty of the sensor calibration, uHS 5.00

Output quantization standard uncertainty, uHQ 0.58

Combined standard uncertainty of the relative humidity, uRH 5.04

Table K.6 Calculated relative humidity and uncertainty for the example data.