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<=> NBS SPECIAL PUBLICATION 260-66

U.S. DEPARTMENT OF COMMERCE /National Bureau of Standards

1

9

Standard Reference Materials:

Didymium Glass Filters for

Calibrating the Wavelength

Scale of Spectrophotometers - SRM 2009, 2010, 2013, and 2014
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in" trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system ot physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods ot measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

Absolute Physical Quantities2 — Radiation Research — Thermodynamics and

Molecular Science — Analytical Chemistry — Materials Science. THE NATIONAL ENGINEERING LABORATORY provides technology and technical ser- vices to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement capabilities;

provides engineering measurement traceability services; develops test methods and proposes

engineering standards and code changes; develops and proposes new engineering practices;

and develops and improves mechanisms to transfer results of its research to the ultimate user. The Laboratory consists of the following centers:

Applied Mathematics — Electronics and Electrical Engineering 2 — Mechanical

Engineering and Process Technology 2 — Building Technology — Fire Research —

Consumer Product Technology — Field Methods.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

guidelines, and managing Federal participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

provides the technical foundation for computer-related policies of the Federal Government. The Institute consists of the following centers:

Programming Science and Technology — Computer Systems Engineering.

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
William H. Venable, Jr. and Kenneth L. Eckerle

mom 13 1979

Standard Reference Materials:

Didymium Glass Filters for Calibrating the

Wavelength Scale of Spectrophotometers - SRM 2009, 2010, 2013, and 2014

\ ^1 H

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Luther H. Hodges, Jr., Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued October 1979
Library of Congress Catalog Card Number: 79 600122

National Bureau of Standards Special Publication 260-66

Nat. Bur. Stand. (U.S.), Spec. Publ. 260-66. 85 pages (Oct. 1979) CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1979

le by the Superintendent of Documents, U.S. Government Printing Office. Washington, D C 20402

Stock No. 003-003-02127-0 Price $3.50 (Add 25 percent additional for other than U.S. mailing).
PREFACE

Standard Reference Materials (SRM's) as defined by the

National Bureau of Standards are wel 1 -character!* zed materials,

produced in quantity and certified for one or more physical

or chemical properties. They are used to assure the accuracy

and compatibility of measurements throughout the nation.

SRM's are widely used as primary standards in many diverse

fields in science, industry, and technology, both within the

United States and throughout the world. They are also used

extensively in the fields of environmental and clinical analysis. In many applications, traceability of quality control

and measurement processes to the national measurement system

are carried out through the mechanism and use of SRM's. For

many of the nation's scientists and technologists it is

therefore of more than passing interest to know the details

of the measurements made at NBS in arriving at the certified

values of the SRM's produced. An NBS series of papers, of

which this publication is a member, called the NBS Special

Publication - 260 Series is reserved for this purpose.

This 260 Series is dedicated to the dissemination of

information on different phases of the preparation, measurement, certification and use of NBS-SRM's. In general, much

rrore detail will be found in these papers than is generally

allowed, or desirable, in scientific journal articles. This

enables the user to assess the validity and accuracy of the

measurement processes employed, to judge the statistical

analysis, and to learn details of techniques and methods

utilized for work entailing the greatest care and accuracy.

These papers also should provide sufficient additional information not found on the certificate so that new applications

in diverse fields not foreseen at the time the SRM was originally issued will be sought and found.

Inquiries concerning the technical content of this paper

should be directed to the author(s). Other questions concerned with the availability, delivery, price, and so forth

will receive prompt attention from:

Office of Standard Reference Materials

National Bureau of Standards

Washington, D.C. 20234

George A. Uriano, Chief

Office of Standard Reference Materials

i i i
OTHER NBS PUBLICATIONS IN THIS SERIES

Catalog of NBS Standard Reference Materials

(1979-80 edition), R. W. Seward, ed., NBS

Spec. Publ. 260 (April 1979) $3.00* SNO03-O03-

02048-6

Michaelis, R. E., and Wyman, L. L., Standard

Reference Materials: Preparation of White Cast

Iron Spectrochemical Standards, NBS Misc.

Publ. 260-1 (June 1964). COM74-1 1061**

Michaelis, R. E., Wyman, L. L., and Flitsch, R., Standard Reference Materials: Preparation of

NBS Copper-Base Spectrochemical Standards,

NBS Misc. Publ. 260-2 (October 1964). COM74-

11063**

Michaelis, R. E., Yakowitz, H., and Moore, G. A.,

Standard Reference Materials: Metallographic

Characterization of an NBS Spectrometric LowAlloy Steel Standard, NBS Misc. Publ. 260-3

(October 1964). COM74-1 1060**

Hague, J. L., Mears, T. W., and Michaelis, R. E., Standard Reference Materials: Sources of

Information, NBS Misc. Publ. 260-4 (February

1965). COM74-11059

Alvarez, R., and Flitsch, R., Standard Reference

Materials: Accuracy of Solution X-Ray Spectrometric Analysis of Copper-Base Alloys, NBS

Misc. Publ. 260-5 (March 1965). PB168068**

Shultz, J. I., Standard Reference Materials:

Methods for the Chemical Analysis of White

Cast Iron Standards, NBS Misc. Publ. 260-6

(July 1975). COM74-11068**

Bell, R. K., Standard Reference Materials:

Methods for the Chemical Analysis of NBS

Copper-Base Spectrochemical Standards, NBS

Misc. Publ. 260-7 (October 1965). COM74-

11067**

Richmond, M. S., Standard Reference Materials:

Analysis of Uranium Concentrates at the

National Bureau of Standards, NBS Misc. Publ.

260-8 (December 1965). COM74-1 1066**

Anspach, S. C, Cavallo, L. M., Garfinkel, S. B., Hutchinson, J. M. R., and Smith, C. N., Standard Reference Materials: Half Lives of Materials

Used in the Preparation of Standard Reference

Materials of Nineteen Radioactive Nuclides

Issued by the National Bureau of Standards,

NBS Misc. Publ. 260-9 (November 1965).

COM74-11065**

Yakowitz, H., Vieth, D. L., Heinrich, K. F. J., and

Michaelis, R. E., Standard Reference Materials:

Homogeneity Characterization on NBS Spectrometric Standards II: Cartridge Brass and LowAlloy Steel, NBS Misc. Publ. 260-10 (December

1965). COM74-11064**

Napolitano, A., and Hawkins, E. G., Standard

Reference Materials: Viscosity of Standard

Lead-Silica Glass, NBS Misc. Publ. 260-11

(November 1966). NBS Misc. Publ. 260-11**

Yakowitz, H., Vieth, D. L., and Michaelis, R. E., Standard Reference Materials: Homogeneity

Characterization of NBS Spectrometric Standards III: White Cast Iron and Stainless Steel Powder Compact, NBS Misc. Publ. 260-12

(September 1966). NBS Misc. Publ. 260-12**

Spijkerman, J. L., Snediker, D. K., Ruegg, F. C,

and DeVoe, J. R., Standard Reference Materials: Mossbauer Spectroscopy Standard for the Chemical Shift of Iron Compounds, NBS Misc.

Publ. 260-13 (July 1967). NBS Misc. Publ.

260-13**

Menis, O., and Sterling, J. T., Standard Reference

Materials: Determination of Oxygen in Ferrous

Materials - SRM 1090, 1091, and 1092, NBS

Misc. Publ. 260-14 (September 1966). NBS Misc.

Publ. 260-14**

Passaglia, E., and Shouse, P. J. Standard Refer- ence Materials: Recommended Method of Use

of Standard Light-Sensitive Paper for Calibrating Carbon Arcs Used in Testing Textiles for Colorfastness to Light, NBS Misc. Publ. 260-15

(June 1967). (Replaced by NBS Spec. Publ.

260-41.)

Yakowitz, H., Michaelis, R. E., and Vieth, D. L., Standard Reference Materials: Homogeneity

Characterization of NBS Spectrometric Standards IV: Preparation and Microprobe Characterization of W-20% MO Alloy Fabricated by

Powder Metallurgical Methods, NBS Spec.

Publ. 260-16 (January 1969). COM74-1 1062**

Catanzaro, E. J., Champion, C. E., Garner, E. L., Marinenko, G., Sappenfield, K. M., and Shields,

W. R. Standard Reference Materials: Boric

Acid; Isotopic and Assay Standard Reference

Materials, NBS Spec. Publ. 260-17 (February

1970). Out of Print

iv
Geller, S. B., Mantek, P.A., and Cleveland, N.

G., Standard Reference Materials: Calibration of NBS Secondary Standard Magnetic Tape

(Computer Amplitude Reference) Using the

Reference Tape Amplitude Measurement

"Process A, "NBS Spec. Publ. 260-18 (November

1969). (See NBS Spec. Publ. 260-29.)

Paule, R. C, and Mandel, J., Standard Reference

Materials: Analysis of Interlaboratory Measurements on the Vapor Pressure of Gold (Certification of Standard Reference Material 745). NBS

Spec. Publ. 260-19 (January 1970). PB190071**

Paule, R. C, and Mandel, J., Standard Reference

Materials: Analysis of Interlaboratory Measurements on the Vapor Pressures of Cadmium and

Silver, NBS Spec. Publ. 260-21 (January 1971).

COM74-11359**

Yakowitz, H., Fiori, C. E., and Michaelis, R. E., Standard Reference Materials: Homogeneity

Characterization of Fe-3 Si Alloy, NBS Spec.

Publ. 260-22 (February 1971). COM74-1 1357**

Napolitano, A., and Hawkins, E. G., Standard

Reference Materials: Viscosity of a Standard

Borosilicate Glass, NBS Spec. Publ. 260-23

(December 1970). COM71-00157**

Sappenfield, K. M., Marineko, G., and Hague, J. L., Standard Reference Materials: Comparison

of Redox Standards, NBS Spec. Publ. 260-24

(January 1972). COM72-50058**

Hicho, G. E., Yakowitz, H., Rasberry, S. D., and

Michaelis, R. E., Standard Reference Materials:

A Standard Reference Material Containing

Nominally Four Percent Austenite, NBS Spec.

Publ. 260-25 (February 1971). COM74-1 1356**

Martin, J. F., Standard Reference Materials:

National Bureau of Standards-US Steel Corportion Joint Program for Determining Oxygen and

Nitrogen in Steel, NBS Spec. Publ. 260-26

(February 1971). 85 cents* SN003-003-00786-2

Garner, E. L., Machlan, L. A., and Shields, W. R.,

Standard Reference Materials: Uranium

Isotopic Standard Reference materials, NBS

Spec. Publ. 260-27 (April 1971). COM74-

11358**

Heinrich, K. F. J., Myklebust, R. L., Rasberry, S. D., and Michaelis, R. E., Standard Reference

Materials: Preparation and Evaluation of

SRM's 481 and 482 Gold-Silver and GoldCopper Alloys for Microanalysis, NBS Spec.

Publ. 260-28 (August 1971). COM71-50365**

Geller, S. B., Standard Reference Materials: Calibration of NBS Secondary Standard Magnetic

Tape (Computer Amplitude Reference) Using

the Reference Tape Amplitude Measurement

"Process A-Model 2," NBS Spec. Publ. 260-29

(June 1971). COM71-50282

Gorozhanina, R. S., Freedman, A. Y., and

Shaievitch, A. B. (translated by M. C. Selby),

Standard Reference Materials: Standard

Samples Issued in the USSR (A Translation

from the Russian). NBS Spec. Publ. 260-30 (June

1971). COM71-50283**

Hust, J. G., and Sparks, L. L., Standard Reference

Materials: Thermal Conductivity of Electrolytic Iron SRM 734 from 4 to 300 K, NBS Spec. Publ.

260-31 (November 1971). COM71-50563**

Mavrodineanu, R., and Lazar, J. W., Standard

Reference Materials: Standard Quartz Cuvettes,

for High Accuracy Spectrophotometry, NBS

Spec. Publ. 260-32 (December 1973). 55 cents* SN003-003-01213-1

Wagner, H. L., Standard Reference Materials:

Comparison of Original and Supplemental

SRM 705, Narrow Molecular Weight Distri- bution Polystyrene, NBS Spec. Publ. 260-33

(May 1972). COM72-50526**

Sparks, L. L., and Hust, J. G., Standard Reference

Materials: Thermoelectric Voltage, NBS Spec.

Publ. 260-34, (April 1972). COM72-50371**

Sparks, L. L., and Hust, J. G., Standard Refer- ence Materials: Thermal Conductivity of

Austenitic Stainless Steel, SRM 735 from 5

to 280 K, NBS Spec. Publ. 260-35 (April 1972.)

35 cents* COM72-50368**

Cali, J. P., Mandel, J., Moore, L. J., and Young, D.

S., Standard Reference Materials: A Referee

Method for the Determination of Calcium in Serum, NBS SRM 915, NBS Spec. Publ. 260-36

(May 1972). COM72-50527**

Shultz, J. I. Bell., R. K. Rains, T. C, and Menis,

O., Standard Reference Materials: Methods of

Analysis of NBS Clay Standards, NBS Spec.

Publ. 260-37 (June 1972). COM72-50692**

Richmond, J. C, and Hsia, J. J., Standard Refer- ence Materials: Preparation and Calibration of

Standards of Spectral Specular Reflectance,

NBS Spec. Publ. 260-38 (May 1972). COM72-

50528**

Clark, A. F., Denson, V.A., Hust, J. G., and

Powell, R. L., Standard Reference Materials

The Eddy Current Decay Method for Resistivity

Characterization of High-Purity Metals, NBS

Spec. Publ. 260-39 (May 1972). COM72-50529**
McAdie, H. G., Garn, P.D., and Menis, O., Standard Reference Materials: Selection of Thermal

Analysis Temperature Standards Through a Cooperative Study (SRM 758, 759, 760), NBS

Spec. Publ. 260-40 (August 1972.) COM72-

50776**

Wood, L. A., and Shouse, P. J., Standard

Reference Materials: Use of Standard LightSensitive Paper for Calibrating Carbon Arcs

Used in Testing Textiles for Colorfastness to Light, NBS Spec. Publ. 260-41 (August 1972)

COM72-50775**

Wagner, H. L. and Verdier, P. H., eds., Standard

Reference Materials: The Characterization of

Linear Polyethylene, SRM 1475, NBS Spec.

Publ. 260-42 (September 1972). COM72-

50944**

Yakowitz, H., Ruff, A. W., and Michaelis, R. E., Standard Reference Materials: Preparation and

Homogeneity Characterization of an Austenitic

Iron-Chromium-Nickel Alloy, NBS Spec. Publ.

260^3 (November 1972). COM73-50760**

Schooley, J. F., Soulen, R. J., Jr., and Evans, G. A.,

Jr., Standard Reference Materials: Preparation

and Use of Superconductive Fixed Point

Devices, SRM 767, NBS Spec. Publ. 260-44

(December 1972). COM73-50037**

Greifer, B., Maienthal, E. J. Rains, T. C, and

Rasberry, S. D., Standard Reference Materials:

Powdered Lead-Based Paint, SRM 1579, NBS

Spec. Publ. 260-45 (March 1973). COM73-

50226**

Hust, J. G., and Giarratano, P. J., Standard Refer- ence Materials: Thermal Conductivity and

Electrical Resistivity Standard Reference Materials: Austenitic Stainless Steel, SRM's 735 and

798, from 4 to 1200 k, NBS Spec. Publ. 260-46

(March 1975). SN003-003-0 1278-5

Hust, J. G., Standard Reference Materials: Electrical Resistivity of Electrolytic Iron, SRM 797,

and Austenitic Stainless Steel, SRM 798, from 5 to 280 K, NBS Spec. Publ. 260^17 (February

1974). COM74-50176**

Mangum, B. W., and Wise, J. A., Standard Refer- ence Materials: Description and Use of Precision

Thermometers for the Clinical Laboratory,

SRM 933 and SRM 934, NBS Spec. Publ. 260-48

(May 1974). 60 cents* SN003-003-0 1278-5

Carpenter, B. S., and Reimer, G. M., Standard

Reference Materials Calibrated Glass Standards for Fission Track Use, NBS Spec. Publ.

260-49 (November 1974). SN003-003-0 1344-7

Hust, J. G., and Giarratano, P. J., Standard Refer- ence Materials: Thermal Conductivity and Electrical Resistivity Standard Reference Materials: 11 Electrolytic Iron, SRM's 734 and 797 from 4 to 1 1000 K, NBS Spec. Publ. 260-50 (June 1975).

$1.00* SN003-003-0 1425-7

Mavrodineanu, R., and Baldwin, J. R., Standard

| Reference Materials: Glass Filters As a Standard

Reference Material for Spectrophotometry;

Selection; Preparation; Certification; Use-SRM

930, NBS Spec. Publ. 260-51 (November 1975).

$1.90* SN003-003-01481-8

Hust, J. G., and Giarratano, P. J., Standard Refer- ence Materials: Thermal Conductivity and Electrical Resistivity Standard Reference Materials

730 and 799, from 4 to 3000 K, NBS Spec. Publ.

260-52 (September 1975). $1.05* SN003-003-

01464-8

Durst, R. A., Standard Reference Materials: j Standardization of pH Measurements, NBS | Spec. Publ. 260-53 (Under Revision).

Burke, R. W., and Mavrondineanu, R. Standard

Reference Materials: Certification and Use of

Acidic Potassium Dichromate Solutions as an

Ultraviolet Absorbance Standard, NBS Spec.

Publ. 260-54 (August 1977). $3.00* SN003-003-

01828-7

Ditmars, D. A., Cezairliyan, A., Ishihara, S., and

Douglas, T. B., Standard Reference Materials:

Enthalpy and Heat Capacity; Molybdenum

SRM 781, from 273 to 2800 K, NBS Spec. Publ.

260-55 (September 1977). $2.20* SN003-003-

01836-8

Powell, R. L., Sparks, L. L., and Hust, J. G.,

Standard Reference Materials: Standard

Thermocouple Materials, Pt.67: SRM 1967, NBS

Spec. Publ. 260-56 (February 1978). $2.20*

SN003-003-0 18864

Cali, J. P. and Plebanski, T., Guide to United

States Reference Materials, NBS Spec. Publ.

260-57 (February 1978). $2.20* SN003-003-

01883-0

Barnes, J. D., and Martin, G. M., Standard Refer- ence Materials: Polyester Film for Oxygen Gas

Transmission Measurements SRM 1470, NBS

Spec. Publ. 260-58 (June 1979) $2.00* SN003-

003-02077

Chang, T., and Kahn, A. H., Standard Reference

Materials: Electron Paramagentic Resonance

Intensity Standard: SRM 2601, NBS Spec.

Publ. 260-59 (August 1978) $2.30* SN003-

003-01975-5

vi
Velapoldi, R. A., Paule, R. C, Schaffer, R.,

Mandel, J., and Moody, J. R., Standard Refer- ence Materials: A Reference Method for the Determination of Sodium in Serum, NBS Spec.

Publ. 260-60 (August 1978). $3.00* SN003-003-

01978-0

Verdier, P. H., and Wagner, H. L., Standard Refer- ence Materials: The Characterization of Linear

Polyethylene (SRM 1482, 1483, 1484), NBS

Spec. Publ. 260-61 (December 1978). $1.70*

SN003-O03-O2006-1

Soulen, R. J., and Dove, R. B., Standard Reference

Materials: Temperature Reference Standard

for Use Below 0.5 K (SRM 768), NBS Spec.

Publ. 260-62 (April 1979). $2.30* SN003-003-

02047-8

Velapoldi, R. A., Paule, R. C, Schaffer, R.,

Mandel, J., Machlan, J. L., and Gramlich, J. W.,

Standard Reference Materials: A Reference

Method for the Determination of Potassium in Serum, NBS Spec. Publ. 260-63 (May 1979).

$3.75* SN003-003-02068

Velapoldi, R. A., and Mielenz, K. D., Standard

Reference Materials: A Fluorescence Standard

Reference Material Quinine Sulfate Dihydrate

(SRM 936), NBS Spec. Publ. 260-64 (in press).

Marinenko, R. B., Heinrich, K. F. J., and Ruegg,

F. C, Standard Reference Materials: MicroHomogeneity Studies of NBS Standard Reference

Materials, NBS Research Materials, and Other

Related Samples, NBS Spec. Publ. 260-65

(September 1979). $3.50* SN-003-003-021 14-1 Venable, W. H., Jr. and Eckerle, K. L., Standard

Reference Materials: Didymium Glass Filters for Calibrating the Wavelength Scale of Spectrophotometers (SRM 2009, 2010, 2013), NBS Spec.

Publ. 260-66 (in press).

Velapoldi, R. A., Paule, R. C, Schaffer, R., Mandel,

J., Murphy, T. J., and Gramlich, J. W., Standard

Reference Materials: A Reference Method for the Determination of Chloride in Serum, NBS Spec.

Publ. 260-67 (in press).

* Send order with remittance to: Superintendent

of Documents, US Government Printing

Office, Washington, DC 20402. Remittance

from foreign countries should include an

additional one-fourth of the purchase price for postage. ** May be ordered from: National Technical

Information Services (NTIS), Springfield,

Virginia 22151.

vi i
CONTENTS

Page

1. INTRODUCTION 2

1.1 Description of Materials 2 1.2 The General Approach to Calibrating Wavelength 2 1.3 Extending Use to Reflectance Spectrophotometers

and Densitometers 2 1.4 Illustrative Examples Using the General Electric

Recording Spectrophotometer 3

2. USING SRM DIDYMIUM GLASS FILTERS 3

2.1 Discrete Wavelength Corrections from Transmi ttance Minima . . 3 2.1.1 Locating the Wavelengths of Minimum Transmi ttance

on the Instrument Scale 3 2.1.1.1 Direct Search . ,5

2.1.1.2 Graphical Extrapolation 5 2.1.1.3 Curve Fitting . . . 5 2.1.2 Obtaining Correct Wavelengths for the

Transmi ttance Minima 10

2.1.3 Determining the Additive Correction 23

2.2 Discrete Wavelength Corrections from Transmi ttance

Inflect ion Points 23

2.2.1 Locating Wavelengths of the Points of Inflection

on the Instrument Scale 23

2.2.2 Obtaining the Correct Wavelengths for

Inflect ion Points '. . . 23

2.2.3 Determining the Additive Correction 27

2.3 Producing a Wavelength Calibration Curve 27

2.3.1 Choosing a Fitting Function 27

2.3.2 Fitting with no Correction for Systematic Errors ... 28

2.3.3 Fitting with Correction for Systematic Errors .... 28

2.3.3.1 Correcting for Systematic Errors by

Graphical Fitting 28

2.3.3.2 Correcting for Systematic Errors by

Calculation 30

3. CERTIFICATION AND MEASUREMENT UNCERTAINTIES 33

3.1 Introduction 33

3.2 Uncertainties in the NBS Measurements 33

3.2.1 Wavelength Calibration of the Reference

Spectrophotometer 33

3.2.2 The Effects of the Sensitivity Function of

the Reference Spectrophotometer 34

3.2.3 Transmi ttance Measurements 34

3.2.3.1 High Precision Transmi ttance

Measurements 34

3.2.3.2 Lower Precision Transmi ttance

Measurements 39

3.2.4 Synthesizing Data for Other Passbands 39

vi i i
Page

3.2.4.1 Symmetrical Triangular Passbands 43

3.2.4.2 A Family of Trapezoidal Passbands 43

3.2.4.3 A Family of Asymmetrical Triangular Passbands. 43

3.2.4.4 Data Constructed from Real Passbands 43

3.2.5 Locating Minima by Polynomial Fit 47

3.2.6 Random Noise in Wavelength-at-Minimum Data 47

3.2.7 Obtaining the Wavelengths of the Inflection

Points 47

3.2.8 Sensitivity of Inflection Determination to

Passband 47

3.2.9 Temperature Effects 51 3.2.10 Effects due to Filter Thickness 51 3.2.11 Effects due to Polarization of Incident Radiation . . . 51 3.2.12 Effects due to Changes in the Condition of the

Filter Surface 51

3.3 Uncertainties Introduced by the User 51

3.3.1 Effects of Instrument Sensitivity Function 54

3.3.2 Effects of Passband Shape and Width 54

3.3.3 Uncertainties Connected with Transmittance 55

3.3.4 Other Causes of Systematic Uncertainty 55

3.3.5 The Effect of Random Measurement Errors 55

3.4 Illustrated Summary of Uncertainty Estimations 55

4. MAKING USE OF THE CALIBRATION CURVE 56

APPENDIX A BACKGROUND AND TERMINOLOGY . 59

A.l Introduction 59

A. 2 Spectral Transmittance 59

A. 3 The Spectral Passband (Function) of Wavelength 59

A. 3.1 Determining the Geometrical Passband Experimentally . . 60

A. 3. 2 Determining the Instrument Sensitivity

Experimentally 60

A. 3. 3 Calculation of the Spectral Passband (Function)

from Experimental Data 63

A. 4 Wavelength Calibration 63

A. 4.1 Measuring Spectral Transmittance 63

A. 4. 2 The Passband Centroid Wavelength 64

A. 4. 2.1 Centroid Shift due to Instrument

Sensitivity 6 5

A. 4. 3 The Significance of the Passband Centroid

Wa vel ength 65

A. 5 Characterizing the Passband 70

A. 5.1 Bandwidth 70

A. 5. 2 Effective Bandwidth 70

REFERENCES 72

ix
LIST OF TABLES

Table Page

I Systematic Direct Search for Wavelength of Minimum

Transmi ttance 6

II Wavelength of Minimum Transmittance by Graphical

Extrapolation . ..... ..... 8

III Illustration of Fitting Data with a Parabola by the

Method of Least. Squares 11

IV Determination of Wavelength Corrections for General

Electric Spectrophotometer from Wavelengths at

Transmittance Minima ..... . 12

V Wavelengths at Minimum Transmittance, ^MIN' ^ or Master D-l

at the indicated ban dwidths 13

VI Wavelengths and Transmi ttances at Nine Selected Points

of Inflection for Master Filter D-l 24

VII Wavelength Corrections for the General Electric

Spectrophotometer from Inflection Points on the

Spectral Transmittance Curve . 26

VIII Elements and Wavelengths of the Atomic Lines Used for

the Wavelength Calibration: a. Lines used with the .25 mm aperture; and b. Lines used with the 1 mm apertures . . . . 35

IX Transmittance of Master Filter D-l Measured with the

Reference Spectrophotometer ... 40

X Weighting Factors Used to Construct Asymmetrical Triangular

Passband Functions from a Sum of 1.5 nm Symmetrical

Passbands 45

XI Determination of Wavelength Corrections for General Electric

Spectrophotometer from Wavelength at Transmittance Minima ... 48

XII Comparison of Wavelengths at Minimum Transmittance

Obtained by 1 . 5 and 0.15 nm Scan Intervals , . . . . 49

XIII Random Uncertainty for Certified Values as Obtained from Four

Sets of Measurements on Master D-l Using the Lower-Precision

Measurement Sequence 50

XIV Maximum Departure of Fitted Wavelength Correction Curve

from Correct Wavelength Correction Curve for Various

Correction Curve Fitting Functions and Passbands 52

XV Measured Values for Wavelength at Minimum Transmittance

for Filter D-l for Three Different Temperatures 53

AI Calculation of Centroid Wavelength 57

All Additive Wavelength Corrections and Effective Bandwidths

Determined for the General Electric Recording Spectrophotometer from Line Source Data 68

x
LIST OF FIGURES

Figure Page

1 Spectral transmi ttance of a typical didymium glass

filter 4

2 Systematic direct search for wavelength of minimum

transmi ttance 7

3 Graphical extrapolation to wavelength of minimum

transmi ttance 9

4a Transmi ttance minima, 1^., as a function of bandwidth 15

4b Transmi ttance minima, ^IN' as a function of bandwidth 16

4c Transmi ttance minima, Ljj,, as a function of bandwidth 17

4d Transmi ttance minima, ^jmi as a function of bandwidth 18

4e Transmi ttance minima, X^^, a s a function of bandwidth 19

4f Transmi ttance minima, X^j^, a s a function of bandwidth 20

4.g Transmi ttance minima,

j ^

, as a function of bandwidth 21

4h Transmi ttance minima, ^j|\j> as a function of bandwidth 22

5 Interpolating to find the wavelength A

s for point of

inflect ionA 25

6 Fitting additive corrections with no corrections for

systematic errors in the data 29

7 Graphical fitting of additive correction data with

corrections for systematic errors in bandwidth and

tran smi ttance 31

8 Least squares fitting results 32

9 Wavelength calibration curves for reference spectrophotometer 36

10 Spectral sensitivity function of the NBS reference

spectrophotometer 37

11 Shift of passband centroid, A c

- A g

, due to the slope of the

sensitivity function for the NBS reference spectrophotometer 38

12 Typical normalized geometrical passband G(A

$ ,A) for the

NBS reference spectrophotometer used to evaluate these

SRM 42

13 Illustration of the approximate construction of a symmetric triangular passband between 9 and 10.6

nm in bandwidth 44

14 Construction of approximation of General Electric

Recording Spectrophotometer passband from sum of 1.5 nm triangular passbands 46

xi
LIST OF FIGURES (Continued)

Fi gure Page

15 Wavelength calibration of NBS reference spectrophotometer

for diffuse transmi ttance and reflectance 57

Al Experimental evaluation of G(.X

S ,X) near X

g = 435 nm 61

A2 Experimental evaluation of S ( X ) 62

A3 Passband as determined experimentally for X

s = 435.8 nm . . . . 66

xii
DIDYMIUM GLASS FILTERS FOR CALIBRATING

THE WAVELENGTH SCALE OF SPECTROPHOTOMETERS - SRM 2009, 2010, 2013, and 2014

William H. Venable, Jr. and Kenneth L. Eckerle

Radiometric Physics Division

Center for Radiation Research

National Bureau of Standards

Washington, DC 20234

This publication describes the use of didymium glass filters as wavelength standards.

These filters are supplied as Standard Reference Materials (SRM). The Standard Reference

Materials are labeled 2009, 2010, 2013 and 2014 depending on size and method of calibration.

The certification and uncertainties are also discussed. An appendix with background material

and terminology is included. Wavelengths of minimum transmittance and inflection points in

the transmittance curve are given. They provide a convenient basis for calibrating the

passband centroid of spectrophotometers with bandwidths between 1.5 and 10.5 nm.

Key Words: Bandwidth; didymium glass filter; passband centroid; spectrophotometer; transmittance; wavelength standard.

1
1. INTRODUCTION

1 .1 Description of Materials

Standard Reference Materials SRM 2009, 2010, 2013 and 2014 are optical glass filters

containing rare earth oxides which have distinct absorption bands. These filters are intended to be used in setting the wavelength scale of a spectrophotometer which is capable

of continuous scan in wavelength and which has a bandwidth* from 1.5 to 10.5 nanometers.

Wavelength data for this purpose are supplied for points of minimum transmittance and for

points of inflection in the transmittance curve in the wavelength range 400 nm to 750 nm. As an aid to interpreting the wavelength data, transmittance data are also supplied in the

wavelength range 380 to 780.5 nm. However, since the transmittance of the filters depends

to some extent upon the condition of the filter surfaces and upon the thickness of the

filter, these filters should not be used as standards of transmittance.

The principal differences between these Standard Reference Materials are the size and

the measurements made on the specific filter. All of the filters are approximately 3 mm

thick. SRM 2009 and 2013 are approximately 1 cm wide by 3 cm high and are supplied in a holder which fits in the place of a standard analytical cuvette. SRM 2010 and 2014 are in the form of squares approximately 5.1 cm by 5.1 cm. The SRM 2009 and 2010 come from a set

of filters all made to common specifications from the same melt of glass. These filters

are not measured individually, but data in tables supplied separately with these filters

give the average of measurements made on two representative samples from the set. Each of

the SRM 2013 and 2014 filters is supplied with data which were obtained from measurements of

that specific filter. Thus the data provide a more accurate representation of the optical

properties of the individual filter, but the main advantage which is gained is in terms of more concrete assurance that the filter's performance does not depart significantly from

that indicated in the data. It is recommended that SRM 2013 and 2014 be purchased only in cases in which assurance in the evaluation of an instrument's wavelength scale is highly

critical. SRM 2009 and 2010 are recommended for most applications.

1 . 2 The General Approach to Calibrating Wavelength

The wavelength scale of an instrument is calibrated by using one or both of two sets of

characteristic points on the spectral transmittance curve of the filter -- points of minimum

transmittance and points of inflection. At each point, a correction factor can be established consisting of the difference between what the instrument wavelength scale should read

and what it actually reads. A curve can be fitted to these discrete correction factors

which will provide wavelength scale correction data at any wavelength. In choosing the type

of curve which is fitted to the data, one should take into account the way the wavelength

drive mechanism works in the instrument being calibrated.

1 . 3 Extending Use to Reflectance Spectrophotometers and Densitometers

In addition to being used to calibrate the wavelength scales of transmittance spectrophotometers, these SRM filters can be used to calibrate the wavelength scale of spectrophotometers designed to measure reflectance and of spectrophotometers with output in terms

of optical density (negative logarithm of transmittance). To calibrate a reflectance

spectrophotometer, a spectrally neutral reflecting sample is placed in the sample position

and the SRM filter is placed at some point in the path of the beam of radiation incident on

*See Appendix, Sec. A.4.1.

2
the sample from the monochromator or coming from the sample through the monochromator depending respectively upon whether or not the sample illumination is monochromatic. In the case of double beam instruments, one must always be sure that the filter is placed in the

sample beam of radiation at a point where the sample and reference beams are separate.

From this point on, the wavelength calibration procedures for the reflectance and transmittance instruments are identical. In the case of instruments with output in terms of

optical density, it is recommended that the optical density values be changed to transmittance values by an appropriate mathematical transformation and then that the same procedure

be used as for an instrument which reads transmittance directly.

1 . 4 Illustrative Examples Using the General Electric Recording Spectrophotometer

Throughout this Special Publication, the Procedures being discussed will be applied by

way of illustration to calibrating the wavelength scale of a 1934 General Electric Recording

Spectrophotometer [1 ] . That instrument was chosen because the technology of its day produced more noticeable departures from idealized performance than will be found in a very high

quality modern instrument. As a result, many of the points being illustrated are more

obvious. The caption "ILLUSTRATION" will appear at the top of all figures and tables which

apply to this illustration in order to distinguish them from figures and tables which apply

to the filters in general.

2. USING SRM DIDYMIUM GLASS FILTERS

This section provides an illustrated set of instructions for using the SRM didymium

glass filters to calibrate the wavelength scale of a spectrophotometer. Subsection 2.1 describes the process for determining discrete corrections to the wavelength scale using

points of minimum transmittance and Subsection 2.2 contains a similar description for the

points of inflection. Procedures for fitting these correction data to obtain a continuous

wavelength scale correction curve are discussed in Subsection 2.3.

2.1 Discrete Wavelength Corrections from Transmittance Minima

Using transmittance minima for wavelength calibration is very straightforward in

principle. The spectral transmitance of the filter is measured and the wavelengths on the

instrument scale are noted. The difference obtained by subtracting the wavelength at minimum transmittance as observed on the instrument scale from the "correct" value for the

corresponding wavelength provides an additive wavelength correction. The difficulty in this

approach, as can be seen in Section 2.1.2, is that the "correct" value for the wavelength

of a transmittance measurement depends upon the nature of the instrument passband function*,

a quantity whose values are usually not well known. However, the procedure is simple to apply and provides adequate wavelength calibration data for many applications. A method is

described in Section 2.3 by means of which some adjustment for lack of knowledge of the

instrument passband can be made during the fitting of the final correction curve.

2.1.1 Locating the Wavelengths of Minimum Transmittance on the Instrument Scale

The transmittance of the filters as a function of wavelength is shown in Figure 1. for

two different instrument bandwidths representing the extremes for which data are supplied.

For the 1 . 5 nm bandpass there are fifteen distinct points at which the transmittance is

locally at a minimum. These minima, indicated by arrows in the figure, have been numbered

consecutively from shorter to longer wavelengths for convenience. For an instrument which

has a larger bandwidth than 1.5 nm, the lack of resolution may obscure some of these minima.

At 10.5 nm bandpass, the number of distinct minima is reduced to six.

Transmittance data near the points of minimum transmittance should be obtained by

using the procedure for measuring transmittance which is normally prescribed for the particular instrument. In any procedure for gathering data to locate the wavelength scale

reading at minimum transmittance in which a continuous scan of the wavelength is used, the rate of scan must be slow enough so that the data do not depend upon the rate of scan. A

*See Appendix. Sec. A.

3
400 500 600 700 Wavelength (nm)

Figure 1: Spectral transmi ttance of a typical didymium glass filter. Numbers

indicate the principal points of minimum transmi ttance and letters

indicate the principal points of inflection.

4
mechanical or electronic lag in the instrument response can cause a shift of the measured

wavelength at minimum transmittance in the direction of scan. For example if the scan is from shorter to longer wavelength, an overly high scan rate will cause a shift in the

measured wavelength at minimum transmittance toward longer wavelengths. The extent of this

shift will depend upon the shape of the transmittance minimum being examined, the speed of

scan and the temporal response of the instrument. It is not the purpose of these SRM

filters either to provide a way to evaluate these factors or to provide a means to correct

them.

In the following three sections are given three commonly used methods for locating the

point on the wavelength scale at which the minimum transmittance occurs. Before one performs

a complete calibration, it is advisable to make several repeated independent experimental

determinations of the wavelength of minimum transmittance at one of the minimum points. This

will provide an estimate of the uncertainty in locating these wavelengths of minimum transmittance for the instrument under consideration and may prevent the forming of unwarranted

conclusions about the accuracy of calibration.

2.1.1.1 Direct Search

If the instrument readout is sufficiently steady, one can simply set and reset the

wavelength scale until the minimum transmittance is located. During such a search, the

wavelength setting should always be approached from the same direction on the scale as when

scanning in normal use in order to avoid errors due to backlash in the wavelength-setting

mechanism. A systematic means of locating the minimum-transmittance point by direct search

is illustrated in Table I and Figure 2 using data from the General Electric spectrophotometer. In this procedure, the transmittance is measured at four wavelengths spaced at equal

intervals over a range which is chosen just large enough to certainly include the wavelength

of minimum transmittance. If the wavelength interval chosen contains a single minimum, the

lowest measured transmittance of the four will occur at either the second or third wavelength

in the sequence. The two wavelengths flanking the wavelength at which the transmittance was

the lowest are then taken as the end points of a second interval and the transmittance at

two additional wavelengths equally spaced in that interval can be measured to provide a new sequence of four wavelengths which are equally spaced over an interval which includes the

wavelength of minimum transmittance and which is only 2/3 the extent of the original interval. Each repetition of this process allows the interval containing the wavelength of

minimum transmittance to be reduced by a factor of 2/3 or less with at most only two additional measurements of transmittance, and the process can be repeated until noise or lack of

resolution in the data makes further reduction of the interval impossible.

2.1.1.2 Graphical Extrapolation

For instruments in which the output is in the form of a chart recording, interpolation

to the wavelength of minimum transmittance can be made from a graph which is made with high

resolution both with respect to wavelength and transmittance. This method can also be used

with digital data which has been graphed and through which a representative curve has been

drawn as will be done in the illustration, but the results depend to some extent upon the

choice of the representative curve. A series of lines of constant transmittance are drawn

through the curve and the segment of each of these lines which is bounded by the two intersections with the curve is bisected. A smooth curve is then drawn through the center points

of the line segments and extrapolated downward to pass through the transmittance data curve.

The intersection of these two curves is assumed to occur at the wavelength of minimum transmittance. This procedure is illustrated in Figure 3 and Table II using the same data as in

Figure 2. It is based on the hypothesis that a small enough section of curve containing the

minimum point of that curve can be approximated by a parabola, which is symmetric about the

minimum point. A disadvantage of this procedure as compared to the direct method is that it

does not provide a good estimate of the uncertainty of the wavel ength-at-minimum value which

is obtained.

2.1.1.3 Curve Fitting

When data of measured transmittance of the filter as a function of wavelength scale

5
ILLUSTRATION

Table I. Systematic Direct Search for Wavelength of Minimum Transmittance in the following

table and Figure 2, the first five steps of a systematic search for the wavelength

of minimum transmittance are illustrated. Notice how the convergence rate has

been made less than 2/3 per step by careful attention to the data.

Comments Data Wavelength Transmittance

First step - Interval of wave- lengths from 513 to 540 nm chosen arbitrarily

513

522

531 540

.569

.496

.419

.733

Second step - Interval of wavelengths from 522 to 540 nm includes lowest transmittance

measured so far at 531

Third step - Interval of wavelengths from 522 to 531 nm includes lowest transmittance

measured so far at 528. Only

one measurement needed

522

528

534

540

522

525

528

531

.496 (measured in first step)

.397

.499

.733 (measured in first step)

.496 (measured in first step)

.434

.397 (measured in second step)

.419 (measured in first step)

Fourth step - Interval of wavelengths from 525 to 531 includes

lowest transmittance measured so far at 528

Fifth step - Interval of wavelengths from 528 to 529 is bounded by the two lowest trans- mittances measured so far.

525

527

529

531

528

528.33

528.67

529

,434 (measured in third step)

.404

,397

.419 (measured in first step)

.397 (measured in second step)

,396

.396

,397 (measured in fourth step)

The next interval would logically be chosed to extend from 528.33 to 528.67 nm. However,

in this region, there is no change in the measured transmittance. Therefore one would

assign the value 528.5 +0.2 nm to the wavelength at minimum transmittance obtained in this manner.

6
7
ILLUSTRATION

Table II. An illustration of finding the wavelength of minimum transmittance by graphical

extrapolation (See Figure 3). In this case, the instrument is being calibrated with respect

to its digital readout, and therefore, the data are in digital form. The following points

from the data list are plotted in the figure as solid circles

These data points were connected with a smooth curve by means of a French curve. Points of

equal transmittance were connected with horizontal line segments (at .005 transmittance intervals in this case) and the center of each line segment was found (indicated by x in Figure 3). A draftsman's curve was again used to connect these center points, and the intersection of this curve with the original data curve (indicated by an open circle in Figure 3) was taken to fall at the point of minimum transmittance. From this figure, the wavelength

of minimum transmittance is found to be 528.5 nm.

Wavelength Transmittance

525

526

527

528

529

530

531 532

0.434

.416

.404

.397

.397

.405

.420

.442

8
525 526 527 528 529 530 531 532

Wavelength nm

Figure 3: Graphical extrapolation to wavelength of minimum transmi ttance.

9
readings can be taken in digital form, the wavelength at minimum can be found by fitting a curve to the data and determining the minimum point of the curve. If a number of data are taken over a small wavelength range including the minimum transmittance point, a reasonable

fitting function for the measured transmittance Tm Us

) is a parabola of the form:

2

t (xj = a + b\ c + c\~. m s s s (1)

where A g is the wavelength scale reading

Once the fitting coefficients a, b, and c have been obtained, the wavelength at minimun

A can be determined as sm

If the data must be taken over a range so great that it cannot be reasonably fitted by

a quadratic of the form given in (1) above, then functions in higher powers of A must be

used to fit the data. The choice of the degree of the fitting function must be made sen- sibly in terms of the number of data points, the noise in the data, and the resolution of

the wavelength and transmittance readouts. The higher order fitting procedure used to determine the values of the data supplied with the filters is discussed in Section 3.2. The

wavelength at minimum transmittance as determined by curve fitting for each of the six

applicable minima are listed for the General Electric Recording Spectrophotometer in the

second column of Table IV.

2.1.2 Obtaining Correct Wavelengths for the Transmittance Minima

The wavelength at which the centroid of the instrument passband is located for each

minimum transmittance depends upon the shape and width of the passband. Table V gives the

wavelength at which the minimum transmittance of Master Filter D-l will occur for triangular

passbands with width at half-height which are even multiples of 0.3 nm. (Master Filter D-l is a selected filter which has optical properties falling near the center of the range for

the measured set of Standard Reference Material didymium filters.) Interpolation to obtain

data for a triangular passband with a half-height bandwidth falling anywhere in the range

1.5 to 10.5 nm can be obtained by interpolating in Table V, using the curves in Figure 4a-h

as a guide for interpolation. The triangular passband occurs in the case of a well-focussed

prism or grating monochromator in which the exit slit and the image of the entrance slit at the exit slit are both the same width, a common arrangement for the monochromators used in spectrophotometers. These data can also be applied to other passband shapes through the use of the equivalent triangular bandwidth concept defined in Section A. 4. 2, or the wavelength

at minimum transmittance can be calculated by means of the procedure given in Section 3.2.4

specifically for the particular passband being used. The General Electric Recording Spectrophotometer is commonly assumed to have a triangular passband width of 10 nm[2]. By

interpolating in Table V, the wavelength at minimum transmittance for a 10-nm triangular

passband was obtained for each of the six applicable minima as given in the third column of

Table IV.

10
ILLUSTRATION

Table III

The following data were fitted with a parabola by using a least squares fitting routine on

a digital computer.

Wavelength Transmittance

527 0.404

528 .397

529 .397

530 .405

From this fitting, the coefficients in equation (1) were determined to be:

Coefficient Value

a 1047.26

b -3.96194

c 3.74857 x 10" 3

The wavelength of minimum transmittance given by equation (2) in this case is:

\ = 528.46

sm

11
ILLUSTRATION

Table IV

Determination of wavelength corrections for General Electric Spectrophotometer from

wavelengths at transmittance minima.

Minimum

Number

Instrument

Wavel ength

at Minimum*

(nm)

Correct

Wavelength

at Minimum # (nm)

Additive

Correction

(\ nm) ***** j

3 440.66 441 .43 0.77

6 475.07 475.94 0.87

8 528.42 529.02 0.60

10 594.85 584.78 -0.07

13 683.78 684.66 0.88

14 740.67 743.43 2.76

*0btained from a 4th degree polynomial fitting of seven data points at 1-nm intervals.

(Sec. 2.1.1.3)

# Based on the assumption of a triangular passband of 10-nm bandwidth. (Sec. 2.1.2)

t See Section 2.1.3

12
TABLE V Wavelengths at minimum transmi ttance , a Bandwidth

MIN

for Master D-l at the indicated bandwidths

1 .5 nm l1 . Ro nmlull 2.1 nm 2.4 nm 2.7 nm 3.0 nm 3.3 nm 3.6 nm 0 .K nmMill

ad? . JU 4n? ?Rc o ad? HU£ i 7 40? HUl. mUj AmHU 1 Rfi AmHU 1 fi7 4D1 HU 1 m 4mHU 1 Rfi . OD AD1 HU 1 J L- 431 HO I . 4QHJ 431 H O 1 • OO 431 HO 1 . 0 3 431 H J 1 . RROO A3? Hoc. . 1 0 A3? HOC . ARHO 4AD 1 7 44D HHU . c \ 44n HHU . C?RO 44fl HHU ?Q AAfl HHU . OO33 AAD HHU 3Q 44n HHU . ARHO 440 HHU fil 44D HHU . ou 44R RR 445 • U I 44R . 44HH 44HH JR . o37/ AAR HHJ ?7 HHO 44R . 1

1 R0 47? 71 i / 1 47? * U _/ 47? . u / 47? H / C fiR 47? H / C . U 1 H47?/ C . R7o / 47? H/ C . RRoo 47? H/ C fil A7? • DO 478 Tf O . 77 47R RR AIR QQ 47Q H / V . ORUO A7Q H/ 3 . 1

1 7/ 47Q H / J . ?RCO H47Q/ 3 ?fi 47Q H / 3 479 > c o

ZLA . lt

lt RlO \ ^O 47 ri ^ ro . OORR Rl 3 . RRJO RlO 1 O3 . uu ORl1 03 . fi4 DH RlO 1 30 f>Q J 1 O • 7R/ o

J L. J 67 . / u . rrO J R?Q • QR_/ U R3n J ou . OAUH R3n o ou • 11 ?C R3n 0 ou 1 ? R3D o o u 1 1 o ou . ORuo

572.,67 R7? O 1 c ou 572 .93 573 .04 573 .15 573 .25 573 .46 573 .67 R73 o / o * o z> 585,,33 585 .39 585 .44 585 .48 585 .52 585 .55 585 .59 585 .63 585 .58

623,.62 623 .68 . 74/ H R? 623 .90 624 .00 629.,62 629 .58 629 .56 629 .52 629 .49 629 .44 629 .35 629 .24 629 .09 684..70 684 .71 fiRA OOH 7

fiRA OOH 7? 684 .72 684 .73 684 .74 684 .73 684 .74

739,,91 739 .94 739 .96 739 .97 739 .98 739 .98 740 .04 740 .10 740 .16 748,,34 748 .31 7/1 Q / HO 97 7/1 P / Ho 9/1

. £H 748 .20 748 .15

4.2 nm 4.5 nm 4.8 nm 5.1 nm 5 . 4 nm 5.7 nm 6.0 nm 6.3 nm 6.6 nm

401,50 401 .48 HU 1 . HO ini HU 1 /l 9 . HL 401 .40 401 .38 401 .36 401 ,31 401,.25

441,10 441 .58 /I /1

44 1 . yo /I /14H£ 1 Q . i y AA? HH£ . . O 1 442,,36 44? HHC "3R . OO 442,,38 442,,36

472,,74 472 .94 -• -

479,,23 479 .20 /I7Q 4/ y . 1

/I 7 Q H/O . yo 478,,76 478, 52 478,,26 478. 05 477..86 513.,81 513,.88 0 1 O .. yo 0 1 H . Uo 514,,11 514. 20 514,,31 514. 42 514. 56

530,,04 530,.00 529,.92 529 .83 529, 73 529. 63 529,.54 529. 48 529. 43

574.,05 574,.18 R74 O / H , 3R 0R7A/ H

r ^ 574.,71 574. 89 575. 07 575. 28 575. 52

585..73 585..78 585,.82 585 .86 585. 91 585. 97 586. 03 586. 06 586. 08

628..88 628,.55 628..15 627,,61 627. 09 627. 01 626. 99 626. 97 626. 97

684.,75 684,,76 684,,76 fiRA DOH , , 77/ / 684. 77 684. 77 684. 77 684. 77 684. 77 740. 21 740,,27 740,,36 740,,47 740. 60 740. 75 740. 92 741. 11 741

31

6.9 nm 7.2 nm 7 .5 nm 7.8 nm 8.1 nm 8.4 nm 8.7 nm 9. 0 nm 9.3 nm

401. 18 401.,09 400,,98 ADO HUU , , R7o / 400. 75 400. 59 400. 43 400. 24

442. 32 442,,28 442,.23 44? HHl. , 1

. 1 O 442. 12 442. 06 441,98 44190 441.79

477. 68 477,,51 477,,35 All H / / , 1 Q 477.,04 476, 90 476,.71 476,.52 476..34 514. 73 514,,97 515.,39

529. 38 529,,35 529.,32 529 ,29 529,,26 529,,23 529 .20 529 .17 529 .13 575. 80 576.,14 576,,52

586. 07 586,,05 586.,01 585 ,93 585,.83 585,,69 585 .54 585,.37 585 .20

626. 97 626,,98 626,,99 626 .97 626,,94

684. 77 684..76 684,,75 684..75 684,,74 684,,73 684 .72 684..71 684 .69 741. 54 741,,77 742,,01 742 ,23 742,,43 742,,62 742 .80 742 .96 743 .11

Minimum No.

1234

5

6

7

10

111213

1415

12

34

56

78

910

1112

1314

15

12

34

567

8

9 10

111213

14

15

13
TABLE V (Continued)

Bandwidth 9.6nm 9.9nm 10.2nm 10.5nm

Minimum No.

2

34 441

.

,66 441

,

49 441

.

,28 441

,

,04

56

7 476.,17 476,,00 475,.83 475,.66

8

9 529.,08 529,.04 528,.98 528,,92

10 585,,03 584,,84 584,.65 584,.45

111213 684,,68 684 ,66 684,.65 684,.63

14 743,,25 743 .39 743,,52 743 .64

15

14
Figure 4a: Transmi ttance Minima, x MTN , as a function of bandwidth

15
4460

l I I i i i i i r

I 234 56 789 10 II BANDPASS (nm)

Figure 4b: Transmittance Minima,

in* as a functl0n of bandwidth

16
E 476.5 —

Figure 4c: Transmi ttance Minima, as a function of bandwidth

17
Figure 4d: Transmittance Minima, XMTN , as a function of bandwidth

18
Figure 4e: Transmittance Minima, XMTN , as a function of bandwidth

19
Figure 4f: Transmi ttance Minima, x MTN , as a function of bandwidth

20
Figure 4g: Transmittance Minima, AMTN , as a function of bandwidth

21
22
2.1.3 Determining the Additive Correction

Once the point on the instrument wavelength scale at which a minimum occurs and the

wavelength at which that same minimum should occur are both known, an additive correction

6A can be calculated straightforwardly as:

SA = X c

- A

s (3)

where

A c is the wavelength at which the minimum transmittance should occur and

A is the wavelength reading on the instrument scale at which the minimum

transmittance is found.

These corrections are used as the basis for determining an overall wavelength correction as described in Section 2.3. The additive correction values for our General Electric Recording

Spectrophotometer as determined from the wavelengths at minimum transmittance are listed in

the fourth column of Table IV.

2 . 2 Discrete Wavelength Corrections from Transmittance Inflection Points

A second set of characteristic points on the spectral transmittance curve which can be

used to provide wavelength calibration data are the inflection points. Not all inflection

points are satisfactory for this purpose. What is desired is that the transmittance curve

should be nearly linear over a range comparable to the width of the largest passband for

which the wavelength calibration is to be used. If the transmittance curve is completely

asymmetric about the inflection point, it follows that all instruments with symmetric passbands will measure the same transmittance value at the point of inflection regardless of

bandwidth or passband shape. Nine points on the didymium filter spectral transmittance

curve which very nearly have this property have been selected, and these are denoted by

1 etters in Figure 1

2.2.1 Locating Wavelengths of the Points of Inflection on the Instrument Scale

Before the points of inflection can be located, one must have the instrument approximately within calibration in order to be sure that the correct part of the transmittance

curve is being measured. This can be done by producing a spectral transmittance curve with

the instrument and comparing it with Figure 1 or by locating the wavelengths of minimum

transmittance as described in Section 2.1. Once a wavelength can be approximately selected,

one refers to Table VI for the wavel ength-transmittance pairs corresponding to each of the

selected points of inflection. The setting on the instrument wavelength scale at which the

given transmittance occurs is then determined. Since the transmittance curve is very nearly

a straight line at these points, linear interpolation is an adequate way to locate the

wavelength. This is illustrated with data from the General Electric Recording Spectrophotometer in Figure 5. In this figure, a graph was made of the measured spectral transmittance near point of inflection A. The transmittance at point of inflection A was found

from Table VI to be .7562, and the wavelength at which this transmittance occurs on the

instrument wavelength scale was found from the curve in Figure 5 to occur at 404.25 nm. The second column of Table VII lists the wavelengths so determined for each of the nine

selected inflection points.

2.2.2 Obtaining the Correct Wavelengths for Inflection Points

The wavelengths listed in Table VI are independent of the instrument or filter. However,

this does not mean that the wavelength calibration will be free of error. The spectral transmittance curve has a large but finite slope at each of the points of inflection. Therefore,

since there may be an error in the transmittance measurement due to instrument non-linearity

or to dirt or some other surface film on the filter, and since the transmittance of a particular filter probably differs from Master Filter D-l because of a difference in thickness or

in concentration of didymium, the transmittance value given in Table IV is approximate for a specific filter. Consequently, the wavelength of the point of inflection on the instrument

23
Table VI

Wavelengths and Transmittances at Nine Selected

Points of Inflection for Master Filter D-l

Point Wavelength

Identification Transmittance (nm)

A .7562 406.38

B .7293 429.42

C .6514 449.45

D .6743 484.77

E .5801 536.52

F .4008 568.16

G .3346 598.99

H .4730 733.45

I .4215 756.48

24
— ILLUSTRATION —

404.25

403 404 405

Wavelength nm

406

Figure 5: Interpolating to find the wavelength A g for point of inflection A

25
ILLUSTRATION

Table VII

Wavelength corrections for the General Electric Spectrophotometer

from inflection points on the spectral transmittance curve

int

ID InstrumentWavelength

Correct

Wavel ength

Additive

Correction

Slope of

Curve

A 404.25 OO 2.13 .012

B 427.,62 A 90 A 9 1.80 -.019

C 447.,42 449. 45 2.03 .009

nU 483,,16 484. 77 1 . 0 1 099

E 535,.52 536..52 1 .00 .042

F 567 .23 568..16 0.93 -.078

G 597 .59 598.,99 1 .40 .044

H 731,.46 733,.45 1.99 -.065

I Off Scale

26
scale as determined in Section 2.2.1 will approximate. Fortunately, the wavelength error introduced in this way is small, and also a very effective correction can be made for them

if they are introduced by errors in transmittance measurements if one carries out the

fitting procedure for the calibration curve. This procedure is described in Section 2.3.

2.2.3 Determining the Additive Correction

The additive correction 6A is determined from the inflection point data in exactly the same way as it is obtained from the minimum point data as described in Section 2.1.2. These

corrections for the General Electric Recording Spectrophotometer appear in the fourth

column of Table VII. Comparing these additive corrections with those obtained from transmittance minima (See Table IV) shows differences which can be reconciled by the curve fitting procedures described in Section 2.3.

2.3 Producing a Wavelength Calipration Curve

The additive wavelength corrections obtained by the procedures in Sections 2.1 and 2.2

contain information on which a spectral wavelength correction can be based. However, these

data also contain both systematic and random errors. The systematic errors in the corrections derived from the minimum transmittance data arise mainly from the lack of knowledge

about the instrument passband. From Figure 4, it can be seen that an error in determining

the bandwidth will produce different errors in the wavelength assigned to each of the

transmittance minima, but the relative magnitude of these errors is quite predictable.

Thus, these errors should be considered as systematic. Similarly, an error in measured

transmittance will produce a set of errors of various sizes in the correction factors

determined from the points of inflection, but the relative magnitudes of these errors are predictable from the transmittance curves. In addition, there are random errors in the

additive correction data just as in any other experimental data. These errors arise both

from random errors in the transmittance measurements, and possibly from some randomness in

the performance of the wavelength scanning mechanism itself. The goal of the wavelength

calibration is usually to obtain a correction curve which is a relatively simple function

of wavelength based on reasonable assumptions concerning possible causes of systematic

error in the wavelength-scale setting of the instrument. For this reason, it is desirable

to make a rational approach to fitting the correction data rather than making wavelength

corrections by simple linear interpolation. In the following parts of this section,

methods for fitting curves will be given by which one can take advantage of what is known

about the systematic errors in order to obtain a reliable correction function.

2.3.1 Choosing a Fitting Function

As in any curve-fitting effort, it is best to choose the simplest fitting function

which is consistent with the physical system being described. If the mathematical equations

governing the motion of the wavelength drive can be expressed in terms of parameters representing settings of the various adjustments in the instrument wavelength drive mechanism,

there can be a distinct advantage in using such an equation. Not only will the fit obtained

be more representative of the instrument's actual performance, but also clues may be provided as to how to adjust the wavelength mechanism to correct for the wavelength error. In lieu of such equations, a linear or quadratic power series expansion in wavelength is usually sufficient to provide an empirical fit to the data which will be as accurate as the

random errors in the data will permit. A fitting function involving more than three parameters is probably too complicated and will tend to follow excursions in the data which do not represent real systematic wavelength setting errors. For the purposes of the general

discussion which follows, the function used to represent the wavelength correction will be

denoted by C(a,b,c;A) where a, b, and c represent three fitting parameters and A represents

the wavelength. When the data from the General Electric Recording Spectrophotometer are treated in illustration, a simple quadratic function of wavelength will be used since it is readily followed in calculations. Thus, for these illustrations we will use as a correction

function

C(a,b,c;A) = a + bA + cA

2 (4)

27
2.3.2 Fitting with no Correction for Systematic Errors

One can make the assumption that the bandwidth has been properly estimated and that

there is no transmi ttance error introduced either because of instrument nonlinearity or

by differences between the filters. If this assumption is made, the correction function

C(a,b,c;x) can simply be fitted to the additive correction data. This has been done for

the data obtained from the General Electric Recording Spectrophotometer and the results of

least squares fitting are shown in Figure 6. The fit of the curve representing all of the

data is not very satisfactory, and the separate fits of the data from the transmittance

minima and the data from the points of inflection appear to indicate two separate but

similar families of data. Therefore, it is reasonable to assume that there may be a systematic error in the transmittance, in the bandwidth estimate, or in both and that perhaps

a better fit can be obtained if the possibility of such systematic errors is taken into account.

2.3.3 Fitting with Correction for Systematic Errors

The fitting process used in Section 2.3.2 above can be represented symbolically in terms of a set of simultaneous equations having the general form

5X

i = C(a,b,c;x.) (5)

or, in the case of the illustration, the specific quadratic form

6X. = a + bx

i + ex.

2

(6)

The index i is used to denote a particular point (either minimum or inflection point),

6X. denotes the additive correction corresponding to that point, and x. the wavelength of

that point as it appears on the instrument wavelength scale. (Note that in this type of

expression in which a small error in a variable is expressed as a function of the variable

itself, either the corrected or uncorrected value of the variable can be used equally well.

In this case, x. might just as well be the "correct" wavelength.) The object of the

fitting procedure, whether it is done graphically or by calculations is to select values

for the parameters a, b, and c such that the fit to the data is most satisfactory. When

the fitting is done by calculation, the most satisfactory fitting is usually taken to be

the one for which the sum of the squares of the deviations from the correction curve

C(a,b,c;x) of the additive correction data is minimum, i.e. a "least squares fit" is obtained. In order to take possible systematic errors into account, one needs simply to add two more terms to each of the simultaneous equations represented in (4) to obtain the

expressions

:

6X. = C(a,b,c;x) + d-AX./AW + e-AX./AT (7)

In this expression (ax. /aW) represents rate at which the "correct" wavelength changes

with respect to triangular bandwidth at point i. This can be obtained from Figure 4 for

the points of minimum transmittance and is assumed to be zero for the points of inflection,

since these points are supposed to occur at the same wavelength regardless of bandwidth.

The value of the new fitting parameter d which is obtained through the fitting process will

be the systematic error in the estimation of the equivalent triangular bandwidth. The

quantity ax, /At is the rate at which the wavelength measured for the characteristic point

changes with measured transmittance. This quantity can be evaluated from the experimental

transmittance data at the point of inflection and is zero at the points of minimum trans- mittance. The value of the parameter e which is obtained through the fitting process will

be the systematic error in the transmittance measurement. Notice that since the trans- mittances at the inflection points are all of roughly the same magnitude, the cause of the

transmittance error is not important. To a first order of correction, representing the error in transmittance by a constant is satisfactory.

2.3.3.1 Correcting for Systematic Errors by Graphical Fitting

In order to illustrate the fitting procedure, the data from our General Electric

Recording Spectrophotometer will be used again. First, the fitting will be done graphically

in order to help promote a feeling for what is being done in the fitting procedure and then

28
o

I I L

00 500 600 700

Wavelength nm

Figure 6: Fitting additive corrections with no corrections

for systematic errors in the data.

29
the fitting will be done analytically using a least squares fitting done with the aid of a computer. In this illustration, we are not seeking to fit any particular form of C(a,b,c;x),

but will simply try to find a "satisfactory" curve which passes through two sets of data, one representing wavelength at minimum data taken with a particular effective triangular

passband and the other representing data taken with a constant correction of the measured

transmittance. The different sets of additive corrections for the wavelengths of minimum

transmittance are obtained in exactly the manner described in Section 2.1.3 with various

assumptions being made about the bandwidth. The different sets of additive corrections

for the wavelengths at the points of inflection are obtained by assuming a certain error

in the measured transmittance and adjusting the instrument wavelength values to "correct"

for that error in measured transmittance. The amount of correction required at each inflection point is calculated from the slope of the spectral transmittance curve at that

inflection point. The goal of the fitting is to find a pair of sets of points, one set for transmittance minima and one set for inflection points through which a curve can be

passed in a satisfying manner. This type of fitting is illustrated for the General

Electric Recording Spectrophotometer in Figure 7. In the figure, the additive correction

data for the wavelengths at minimum transmittance are shown as squares, with the triangular

bandwidth shown as a number inside the square. The additive correction data for the inflection points are shown as circles. T'ie circles with a negative sign indicate the

additive corrections which would be needed if the transmittance scale reads too low

and the circles with the positive sign indicate the additive corrections which would be

needed if the transmittance scale reads 0.005 too high. By visual estimation, it was determined that a good fit could be obtained by assuming that the effective triangular bandwidth was 8.5 nm and that the transmittance scale was reading approximately 0.001 too high. These conclusions about the bandwidth and transmittance are confirmed by independent

evidence, as will be shown in the next section. The corresponding additive corrections

are shown on Figure 7 as solid dots, and the curve was drawn by visual estimation through

these points. It should be pointed out that in this visual fitting, more weight was given

to the additive corrections obtained from the points of inflection than to those obtained

from the minima since the uncertainties in the wavelengths of minimum transmittance due to uncertainties in the shape of the passband are considerably greater than the corresponding

uncertainties in the wavelengths of the points of inflection. Similar account should be

taken in fitting the data by calculation.

2.3.3.2 Correcting for Systematic Errors by Calculation

If, in the example of the General Electric Recording Spectrophotometer, we use the

quadratic form for the fitting function, we can write equation (7) as

6A. = a + bA

i + CA.J + d(AA.j/AW) + e(AX./Ax) (8)

The value of X. can be taken either as the nominal wavelength of the characteristic point

or as the wavelength of the characteristic point as read on the instrument scale. Values

for the average slope over the range of bandwidths from 8 to 10 nm was obtained from the curves in Figure 4 and used as AX. /AW. The value of ax, /At at each wavelength was taken

from the transmittance data and is the inverse of the slope of the transmittance curve as given in the fifth column of Table VII. A computer least-squares-fitting program was used

to produce a least-squares best-fit solution to the set of simultaneous equations represented by expression (8 ). The results, appearing in Figure 8, were very similar to those obtained graphically in the preceding section, and the scatter of the data is con- sistent with what would be expected from the three-slit cam-operated wavelength mechanism

used in the monochromator of the General Electric Spectrophotometer. In this illustration,

fitting by calculation was very straightforward, but this may not always be the case. It

is always a good idea to roughly fit the data graphically in order to see what range of

correction for bandwidth and transmittance seems reasonable. The bandwidth and transmittance adjustments should never be allowed to be larger than is consistent with what is known about the instrument. In the case of the General Electric Spectrophotometer, it was assumed that the transmittance error had to be small since the instrument transmittance

scale had recently been very carefully calibrated and the filter which was measured was

the same one from which the transmittance data in Table VI were taken. It was also known

from the line source experiments described in Appendix A that the passband was trapezoidal

30
t r

I I I L

400 500 600 700

Wavelength nm

Figure 7: Graphical fitting of additive correction data with corrections

for systematic errors in bandwidth and transmittance.

31
i I I — ILLUSTRATION —

O From fitting

From line source data (See Table AH)

X From calculations based on passband X

shape ( See 3.2.4.4 )

E

c

o

a>

k_

h_o

o

Q)>

a

<

400 500 600

Wavelength nm

700

Figure 8: Least squares fitting results using equation 10. (The line-source

and passband calculated data were not included in the fitting.)

32
rather than triangular, and the effective triangular bandwidth for that trapezoidal -shaped

passband is, as described in Appendix A, expected to be approximately 8.5 run, somewhat

narrower than the actual half-height bandwidth of the trapezoid. The results shown in

Figure 8 are consistent with this knowledge, since the fitting indicated an effective

triangular bandwidth of 8.4 nm and a transmittance error of only 0.0025. Notice the good

agreement of the data resulting from this fitting with the data which were obtained independently from line sources (square points in Figure 8) and with the data which were

obtained from wavelengths of minimum transmittance curves generated from the reference

instrument data using the measured passband shapes for the General Electric Recording

Spectrophotometer (cross points in Figure 8).

3. Certification and Measurement Uncertainties

3.1 Introduction

As early as 1938, Jena BG-11 didymium filters were used at the National Bureau of

Standards for in-house calibrations of the General Electric Recording Spectrophotometer [3]

which was the predecessor of the one used as an illustration in this special publication.

At that time, an uncertainty of 0.2 to 0.3 nm was placed on the wavelength calibration

being obtained for that instrument. In the February 1941 meeting of the optical Society

of America, it was announced that Corning 5120 didymium glass filters were available from

NBS for calibrating the wavelength scale of spectrophotometers . [4] In a later presentation

to the Optical Society of America[5] a single wavelength at each transmittance minimum was given which was to be used with 4, 8, and 10 nm bandwidths. From Figure 4, it is apparent

that there is a systematic shift of any given wavelength of transmittance with bandwidth,

so that the rather large error limits of +1 nm may have been assigned to cover that shift.

However, no written evidence has survived"the intervening 34 years to confirm that viewpoint.

The filters were issued during those 34 years "for use with recording spectrophotometers

with 10 nm bandwidth" and the wavelength uncertainty was given as +1 nm. They have provided a stable, reliable, and moderately accurate means of wavelength calibration over

that period of time.

In 1971 a reference spectrophotometer was constructed at NBS which made available a highly accurate transmittance measurement capability. [6] After the initial demand for

measurements on the new instrument was satisfied, it was decided to upgrade the measurements on the didymium filters as much as could reasonably be used and to offer them as Standard Reference Materials. Through the use of the new instrument and the application

of high-speed digital computation to the analysis of the data, it has been possible to provide the data and techniques given in this Special Publication. Using these data and

techniques, wavelength scale of a high quality spectrophotometer with bandwidth in the

range 1.5 nm to 10.5 nm can be calibrated.

3.2. Uncertainties in the NBS Measurements

There are uncertainties associated with calibrating and operating the NBS instruments

as well as uncertainties associated with the data analysis used to provide the numerical

tables which appear in this Special Publication and in the data sheets which go with each

filter. The user is in no position to reduce these uncertainties, and therefore they will

determine a lower limit «n the uncertainty which he can achieve in his calibration. To provide an understanding of these uncertainties, the calibrations, measurement procedures,

and computations used to provide the NBS data for these filters are described in the re- maining part of Section 3.2.

3.2.1. Wavelength Calibration of the Reference Spectrophotometer.

The wavelength scale of the reference spectrophotometer on which the filters were

measured was calibrated using line sources as described in Section A. 3. One of the difficulties encountered in this type of calibration is that the presence within the passband

of weaker neighboring lines tends to make determining the centroid wavelength for a particular line very difficult. Therefore, the wavelength calibration of the reference spectrophotometer was carried out in two steps. First, calibration was made at high resolution,

33
for which a large number of lines could be used, in order to determine the general shape of

the wavelength calibration curve for fitting purposes. Then, with the spectrophotometer

set for a larger bandwidth, a calibration was made using a smaller number of lines and

fitting with the same type of curve. For the high resolution determination, .25 mm diameter entrance and exit ports were used in the monochromator for a half-height bandwidth of

approximately 0.38 nm. The centroid values were obtained from measurements made at intervals of 0.2 nm. The sources and line wavelengths used for this determination are listed in Part a. of Table VIII and the corresponding additive corrections are shown in Figure 9. From these data it was concluded that a straight line would provide an adequate fit within

the scatter of the points. Next, the entrance and exit ports which were to be used for

measuring the didymium filters were installed. These ports were 1 mm in diameter and

yielded a half-height bandwidth of approximately 1.5 nm. A wavelength calibration was again carried out using the spectral lines listed in Part b. of Table VIII. In these

determinations, the location of the centroid was calculated from measurements made at 0.04

nm intervals. The additive corrections for this calibration and the final wavelength

calibration curve are also shown in Figure 9. The uncertainty in wavelength setting

associated with this calibration curve is estimated to be a random 0.02 nm uncertainty

associated with the wavelength drive mechanism superimposed upon a 0.02 nm systematic

uncertainty associated with obtaining the calibration curve. 3.2.2 The Effects of the Sensitivity Function of the Reference Spectrophotometer

The sensitivity function of the reference spectrophotometer is not a constant, but has

the form illustrated in Figure 10. The general trend of the curve as determined by the

detector sensitivity and the source spectral power distribution is approximately that shown

for the 90° polarization (electric vector parallel to grating grooves). In the 0° polarization configuration, Wood's anomalies introduced by the monochromator grating produce

additional complexities in the curve. The effect of the sensitivity function on the cen- troid of the bandpass of the reference instrument, as determined by the method described in Section A. 4. 2.1, is shown in Figure 11. Since the bandwidth is relatively narrow, the

centroid shift due to the slope of the sensitivity function is quite small and was not included as a correction to the wavelength data in this Special Publication or in the wave- length data supplied with the filters. For applications in which a wavelength error of

this magnitude is important, a direct calibration using line sources as described in

Section A. 3 should be carried out rather than using the SRM filters.

3.2.3 Transmi ttance Measurements

Transmittance measurements were made with the reference spectrophotometer using pro- cedures similar to those described in reference 6. Measurements were made at two levels

(high and low) of precision. High precision measurements were made on three selected

filters which will be called the "master filters". The data from these filters, which are

from the same melt as the 2013 and 2014 filters, were used in the analysis from which the

procedures for using the filters were derived. Two master filters from the melt used for

2009 and 2010 filters were also measured with high precision, and the average of the data

from these two filters are supplied as representative data for the melt. In addition,

special sets of high precision measurements were used to explore for relatively small

effects on the filter caused by outside influences such as temperature changes and cleaning

of the filter surfaces. Measurements of a slightly lower precision which could be made in

a significantly shorter time were used to supply the individual sets of data which are issued with each SRM 2013 and 2014.

3.2.3.1 High Precision Transmittance Measurements

The high precision measurements were made on sets of three filters as follows: The

filters were mounted in a filter wheel which positioned them in the instrument beam auto- matically and the following time-symmetric measurement sequence was used at each wavelength:

0-1-2-3-0-3-2-1-0-1-2-3-0-3-2-1-0

where 0 represents measurement of a signal for the beam with no filter in place and 1, 2,

and 3 represent the signals obtained with each of the three filters in place. Between each

measurement, a reading was made of the background signal with a light trap inserted in the

34
Table VIII

The elements and wavelengths of the atomic lines used for the wavelength calibration

a. Lines used with the .25 mm aperture.

Element Wavelength Element Wavelength

(nm) (nm)

Cs 455.536 He 388.865

459.318 501.568

Rb 420.185

421 .556

Hg 546.075

Rl 780.023

587.562

667.815

780.023 Ne 640.225

794.760 692.947

724.517

Hg 275.278

296.728 Cd 467.816

334.148 479.992

404.657 508.582

407.784 643.847

435.834

491 .604

546.075 Zn 213.856

576.960

579.066

b. Lines used with the 1 mm apertures

Element Wavelength

Zn 213.856

He 388.865

35
36
37
uju 441.11s pioJ^uaQ

38
beam just ahead of the detector unit. Each measurement was made by integration of the output current over a ten-second period, and a ten-second settling time was allowed before the

measurement of each signal except for the background where only a brief wait was found to be

needed. The background signal from the photomultipl ier was subtracted from each of the

other signal values and the resulting values were then corrected for a very slight instrument non-linearity. The resulting data were fitted to straight lines with respect to time

by means of a least-squares fitting routine in which it was assumed that the time drift in a signal was due to a drift in gain and, therefore, that the slope of each of the fitted lines

was proportional to the average signal. The ratio of the appropriate signals represented by

the lines at any one time are taken to be the transmittance values. A typical value of the

random uncertainty associated with a 95% level of confidence for these transmittance measurements was 0.0002 for x-.9. Transmittance data taken at 1.5 nm intervals on one of the

master filters are given in Table IX. The high precision measurements described above

required approximately 20 minutes measuring time per wavelength for a set of three filters.

3.2.3.2 Lower Precision Transmittance Measurements

Since measurements are made at each of 268 different wavelengths, it was not economically feasible to use that procedure on all of the replica filters. Therefore a shortened

measurement sequence of the form

0-0-1-1-2-2-3-3-0-0

was used. This required much less motion of the filter wheel as well as having fewer total

readings. A background reading was taken only at the beginning and the end of the sequence,

the "settling" time was reduced from 10 seconds to 2.5 seconds, and the current integration

time was reduced from 10 seconds to 1 second. Using this sequence, the measurement time was reduced to 90 seconds per wavelength for a set of three filters. The uncertainty in transmittance associated with a 95% level of confidence for these measurements was found to be

0.0009 for x-.9.

3.2.4 Synthesizing Data for Other Passbands

In the mode used for measuring these filters, the reference spectrophotometer was equipped with circular apertures rather than the usual slits. Consequently, the geometrical

passband (see Section A. 3) is not quite triangular although, as can be seen from Figure 12,

it very closely resembles a triangular passband with a bandwidth of 1.5 nm. Therefore, the

transmittance data in Table IX and the corresponding transmittance data supplied for each

particular SRM can, for most practical purposes, be regarded as being from a triangular

passband of 1.5 nm bandwidth and will be so treated in the remainder of this section. The

transmittance which would be measured for other passbands that can be constructed as the

weighted sum of 1.5 nm triangular passbands can be calculated from a table of transmittance

data such as Table IX by using a series of weighting factors' B. which are related to the

shape of a passband of bandwidth greater than 1.5 nm that is sampled at 1.5 nm intervals

(see Sections 3.2.4.1, f f ) . The measured transmittance t can be calculated by taking the

weighted average of a series of transmittance values from Table IX using the B. as weighting

factors

2(N-1)

E B.-AU+l)

x

- 1=0

, (9)

T

j " 2T¥T5

E B. i=0 1

The centroid wavelength for the transmittance measurement so represented is

X = X . + AX (10)

where X. is the wavelength at which x. was measured and AX is the distance from X- of the

passband centroid which can be calculated as

39
Table IX

Transmittance of Master Filter D-l measured with the reference spectrophotometer. The

bandwidth was approximately 1.5 nm. A typical passband is shown in Figure 12.

CAUTION : THESE TRANSMITTANCE VALUES ARE GIVEN ONLY TO ILLUSTRATE A TYPICAL

SET OF TRANSMITTANCE DATA. THESE SRM FILTERS SHOULD NOT BE USED

AS TRANSMITTANCE STANDARDS.

Wavelength Transmittance Wavelength Transmittance Wavelength Transmittance

380. 00 0. 5660 458. 00 0. 6973 536. 00 0. 5565

381

.

50 0. 5898 459. 50 0. 6759 537. 50 0. 6419

383. 00 0. 6161 461 . 00 0. 6575 539. 00 0. 7046

384. 50 0. 6396 462. 50 0. 6490 540. 50 0. 7544

386. 00 0. 6606 464. 00 0. 6480 542. 00 0. 7935

387. 50 0. 6806 465. 50 0. 6433 543. 50 0. 8206

389. 00 0. 6996 467. 00 0. 6314 545. 00 0. 8386

390. 50 0. 7153 468. 50 0. 6170 546. 50 0. 8508

392. 00 0. 7300 470. 00 0. 6026 548. 00 0. 8586

393. 50 0. 7409 471

.

50 0. 5849 549 50 0 8623

395. 00 0. 7482 473. 00 0. 5752 551 00 0 8628

396. 50 0. 7458 474. 50 0. 5965 552. 50 0 8623

398. 00 0. 7280 476. 00 0. 6061 554 00 0 8619

399. 50 0 7059 477 50 0 5710 555 50 0 8609

401

.

00 0 6944 479 00 0 5618 557 00 0 8575

402 50 0 6826 480 50 0 5702 558 50 0 8511

404 00 0 7233 482 00 0 5870 560 00 0 8409

405 50 0 7453 483 50 0 6317 561 50 0 8246

407 00 0 7638 485 00 0 6864 563 00 0 7983

408 50 0 7921 486 50 0 7270 564 50 0 7550

410 00 0 8079 488 00 0 7579 566 00 0 6744

411 50 0 8142 489 50 0 7849 567 50 0 5176

413 00 0 8178 491 00 0 8059 569 00 0 2723

414 50 0 8205 492 50 0 8171 570 50 0 0868

416 00 0 8208 494 00 0 8204 572 00 0 0308

417 50 0 8214 495 50 0 8186 573 50 0 0305

419 00 0 8228 497 00 0 8123 575 00 0 0411

420 50 0 8290 498 50 0 8019 576 50 0 0459

422 00 0 8283 500 00 0 7886 578 00 0 0603

423 50 0 8254 501 50 0 7733 579 50 0 0808

425 00 0 8274 503 00 0 7582 581 00 0 0803

426 50 0 8269 504 50 0 7438 582 50 0 0470

428 00 0 8105 506 00 0 7279 584 00 0 0197

429 50 0 7279 507 50 0 7037 585 50 0 0133

431 00 0 6281 509 00 0 6607 587 00 0 0200

432 50 0 6404 510 50 0 6014 588 50 0 0364

434 00 0 6474 512 00 0 5529 590 00 0 0559

435 50 0 6408 513 50 0 5332 591 50 0.0672

437 .00 0 6270 515 00 0 5503 593 00 0 0870

438 .50 0 .6112 516 50 0 5793 594 50 0 1331 440 .00 0 .6035 518 00 0 5978 596 00 0 2078

441 .50 0 .6072 519 50 0 6082 597 50 0 2860

443 .00 0 .6128 521 .00 0 .5920 599 00 0 3429

444 .50 0 .6077 522 .50 0 .5297 600 50 0 3997

446 .00 0 .6051 524 .00 0 .4546 602 .00 0 4635

447 .50 0 .6194 525 .50 0 .4261 603 50 0 .5256

449 .00 0 .6461 527 .00 0 .4165 605 .00 0 5781

450 .50 0 .6723 528 .50 0 .3872 606 .50 0 .6133

452 .00 0 .6877 530 .00 0 .3805 608 .00 0 .6331 453 .50 0 .6982 531 .50 0 .3894 609 .50 0 .6519

455 .00 0 .7084 533 .00 0 .4054 611 .00 0 .6860

456 .50 0 .7097 534 .50 0 .4648 612 .50 0 .7390

40
Table IX (Continued)

Wavelength Transmittance Wave! ength Transmittance

614.00 0.7986 702.50 0.8939

615.50 0.8445 704.00 0.8955

617.00 0.8717 705.50 0.8967

618.50 0.8814 707.00 0.8975

620.00 0.8811 708.50 0.8982

621 .50 0.8744 710.00 0.8981

623.00 0.8685 711 .50 0.8974

624.50 0.8688 713.00 0.8966

626.00 0.8713 714.50 0.8945

627.50 0.8710 716.00 0.8922

629.00 0.8683 717.50 0.8889

630.50 0.8687 719.00 0.8846

632.00 0.8733 720.50 0.8784

633.50 0.8795 722.00 0.8712

635.00 0.8851 723.50 0.8614

636.50 0.8879 725.00 0.8473

638.00 0.8896 726.50 0.8274

639.50 0.8918 728.00 0.7978

641 .00 0.8936 729.50 0.7508

642.50 0.8951 731 .00 0.6784

644.00 0.8961 732.50 0.5701

645.50 0.8968 734.00 0.4264

647.00 0.8972 735.50 0.2891

648.50 0.8971 737.00 0.1983

650.00 0.8968 738.50 0.1661

651 .50 0.8970 740.00 0.1617

653.00 0.8980 741 .50 0.1663

654.50 0.8988 743.00 0.1881

656.00 0.8987 744.50 0.2135

657.50 0.8985 746.00 0.2144

659.00 0.8979 747.50 0.1977

660.50 0.8970 749.00 0.1965

662.00 0.8956 750.50 0.2225

663.50 0.8927 752.00 0.2640

665.00 0.8882 753.50 0.3124

666.50 0.8823 755.00 0.3673

668.00 0.8757 756.50 0.4274

669.50 0.8696 758.00 0.4871

671.00 0.8652 759.50 0.5410

672.50 0.8613 761 .00 0.5863

674.00 0.8579 762.50 0.6242

675.50 0.8530 764.00 0.6592

677.00 0.8444 765.50 0.6924

678.50 0.8323 767.00 0.7225

680.00 0.8180 768.50 0.7479

681 .50 0.8011 770.00 0.7674

683.00 0.7821 771 .50 0.7807

684.50 0.7680 773.00 0.7885

686.00 0.7764 774.50 0.7919

687.50 0.7949 776.00 0.7897

689.00 0.8108 777.50 0.7801

690.50 0.8274 779.00 0.7623

692.00 0.8434 780.50 0.7370

693.50 0.8579

695.00 0.8692

696.50 0.8779

698.00 0.8845

699.50 0.8892

701 .00 0.8920

41
-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

X CN - X CENTROID -*

Figure 12: Typical Normalized geometrical passband G(x ,x) for the

NBS reference spectrophotometer used to evaluate these SRM

42
e i-B, AX = 1.5

E B.

[nra] (IT)

The synthesizing of such data for particular passband shapes which were used in the development of the procedures given in this Special Publication will be discussed briefly in the

following subsections.

3.2.4.1 Symmetrical Triangular Passbands

Triangular passband functions having a bandwidth which is an even multiple of 1.5 nm were constructed exactly from the weighted sum of 1.5 nm bandwidth triangular functions

spaced at 1.5 nm intervals. To construct such a triangular passband of width 1.5N, the

summation index i must range from 0 to 2(N-1). The B. are evaluated as

For example, for a 4.5 nm bandwidth, N = 3 and we have B~ = 1 , B-, = 2, B~ = 3, B., = 2, and

B, = 1 . From inspection or from the use of (11) it can He seen that the centroid wavelength is 3 nm higher than A., i.e. at the wavelength for which B

2 is the multiplier. The

data for the bandwidths, 1.5, 3, 4.5, 6, 7.5, 9 and 10.5 nm given in Table V were obtained

from this type of exact triangular construction. The interpolation used to obtain the curves in Figure 4 and the remaining data in Table V was done by using approximations of

triangular passbands which were not exact multiples of 1.5 nm. The approximation which was

used is illustrated in Figure 13 for bandwidth 1.5Y where 6<Y<7. In this case, the largest

i is 2 ( N- 1

) where N is the next integer larger than Y. The value of each B. is proportional

to the corresponding height on the triangle and multiplies the corresponding transmi ttance

value as before except for the end points corresponding to Bq and B^ in this case. At these points, the height of the triangle B' or B-j 2

) is reduced by the fraction of the triangle which falls outside the passband being constructed (shaded areas) and is multiplied

by the transmittance value linearly interpolated to the wavelength corresponding to the

centroid (point a or point b) of the unshaded portion of the triangle contained in the passband being approximated.

3.2.4.2 A Family of Trapezoidal Passbands

A set of transmittance data were obtained for a family of trapezoidal passbands with

bandwidths an odd multiple of 1.5 nm by making B Q through B

f ._, equal for odd N. The characteristic shape of these trapezoids is that the upper base is"3 nm shorter than the lower

base. Data were calculated for such trapezoids having bandwidths from 1.5 to 19.5 nm. These

data were used in evaluating the effect of passband shape on the wavelength calibrations

obtained from points of inflection.

3.2.4.3 A Family of Asymmetrical Triangular Passbands

A set of asymmetrical passbands were constructed for further extreme-case evaluation

of the use of the inflection points for wavelength calibration. These were designed so that

they could be exactly constructed from 1.5 nm triangles at 1.5 nm spacing and so that their

centroids fell at one of the measured wavelengths. The B. values and the i corresponding to the centroid are listed in Table X.

3.2.4.4 Data Constructed from Real Passbands

A set of B. values can be constructed to represent an approximation to a real passband

which would be obtained by adding 1.5 nm triangles. This approximation corresponds to choosing points on the passband curve at 1.5 nm intervals and connecting them with straight

lines as illustrated for the passband of the General Electric Recording Spectrophotometer

in Figure 14. With the B. so obtained, equation (11) can be used to calculate the transmittance which would be measured with such an instrument and therefore to obtain the correct

B, = i + 1 for i < N (12a)

and

B. - (2N-l)-i for N<i<2(N-l) 02b)

43
<0, CD • EE +-> >> "O

to "1

3

rc M— to Oco

•I-

+J E o3

S4->

l/lO

Coo T3

OJ ro

+-> <o cn E

•r— c X CD o CD s- s

CL +-> Cl

«3 J3

GJ T3

.c £

+-> -O

M— CO O 00

fC

£1 Q- O

S-

-t-> ra

to

S-

+-> cnE

(0

s-

+->

44
Table X

Weighting factors used to construct asymmetrical triangular passband functions from a sum of 1.5-nm symmetrical passbands (See Section 3.2.4.3)

Weighting

Factor 1 2 Passband

3 Number4 5 6

B„

0

1 4 1 7 1 10

Bl 2 3 2 6 2 9

Bo 3 2 3 5 3 8

Bo

3 4 1 4 4 4 7

B/i4

5 3 5 6

B

5 6 2 6 5

B

6

7 1 7 4

B

7 8 3

B8

9 2

B

9 10 1

Centroid i 2 1 4 2 6 3

45
46
wavelength at minimum for that instrument. The correct wavelengths at the minima and the

corresponding additive corrections which were obtained by this method are listed in Table XI and the additive corrections are plotted in Figure 8.

3.2.5 Locating Minima by Polynomial Fit

Since the transmittance data were obtained at discrete wavelengths taken at intervals

of 1.5 nm, polynomial fitting was used to estimate the location of the point of minimum

transmittance to a higher degree of precision than 1.5 nm. Polynomials of various degrees

from 2 to 6 in wavelength were fitted to various numbers of data points surrounding each of

the transmittance minima for each of the three master filters in order to test for consistency between the various fitting approximations. The transmittance near the wavelength

minima of each master filter was also measured at intervals of 0.15 nm and the minima were

found by fitting these data with a fourth degree polynomial in wavelength. From comparing

these results, it appeared that a sixth degree polynomial fitted to seven points at 1.5 nm intervals gave the most consistent data. The comparison between the wavelengths at minimum

which were found from sixth degree polynomial fitting to those which were obtained from the

data taken at 0.15 nm intervals is made in Table XII. From this table the uncertainty in

locating the wavelength of minimum transmittance due to the curve-fitting procedure can be

estimated. This uncertainty is the largest uncertainty of all of the uncertainties which

are attributable completely to the NBS measurements.

3.2.6 Random Noise in Wavelength-at-Minimum Data

In order to test the effects of random noise in the transmittance measurements on the

determination of the wavelengths of minimum transmittance, four independent sets of measurements were made on master filter D-l using the lower precision sequence of measurements described in Section 3.2.3.3. For each set of transmittance data, the wavelength of minimum

transmittance was calculated for each of seven different triangular passbands at each recognizable point of minimum transmittance. From the four independent wavel ength-at-mi nimum

values, a standard deviation in the wavelength of minimum transmittance was calculated for

each minimum at each bandwidth. These values are shown in Table XIII and indicate that in

general the contribution to the wavelength uncertainty from this type of noise is less than

the uncertainty introduced by the polynomial fitting process. (It should be noted when com- paring Tables XII and XIII that the values in Table XII were calculated from data obtained

with the high precision measurement sequence.)

3.2.7 Obtaining the Wavelengths of the Inflection Points

Transmittance data for each of seven different symmetric triangular passbands with

bandwidths ranging from 1.5 nm to 10.5 nm were constructed as described in Section 3.2.4.1.

In principle, the transmittance measured at an inflection point should be independent of the

bandwidth, and linear interpolation should be very accurate at an inflection point. Therefore, linear interpolation was used to determine the transmittance for each of the seven passbands at any wavelength in the vicinity of a point of inflection. The point of inflection was found by a computer search routine which sought out the wavelength at which the sum of the squares of the deviations of the seven transmittances from their mean value was

a minimum. The wavelength and transmittance values at the inflection points given in

Table VI and in the corresponding data table accompanying each filter were found in this manner.

3.2.8 Sensitivity of Inflection Determination to Passband

The wavelengths of minimum transmittance depend fairly strongly on the second moment

of the passband function and to some extent on other characteristics of the passband shape.

By their very nature, the wavelength determinations at the inflection points are not supposed to depend strongly upon passband shape or width. This was tested by developing transmittance data for each of 19 different passbands (7 symmetric triangles, 6 trapezoids, and

6 skew triangles) and determining for each inflection point the wavelength corresponding to the appropriate inflection point transmittance from Table VI. The fitting procedure from

section 2.3.3.2 was then applied to the data to produce a calibration curve. The fitting

was done for a constant, for a line, and for a quadratic function. Since the data for the

various passbands was generated from the same data used to obtain Table VI, it follows that

47
Table XI

Determination of wavelength corrections for General Electric Spectrophotometer from

wavelength at transmi ttance minima

Minimum

Number

Instrument

Wavel ength

at Minimum*

(nm)

Correct

Wavel ength

at Minimum*

(nm)

Additive

Correction

(nm)

3 440.66 442.30 1 .64

6 475.07 476.54 1.47

8 528.42 529.19 .77

10 584.85 585.71 .86

13 683.78 684.63 .85 :

14 740.67 743.08 2.41

* Obtained from a 4th degree polynomial fitting of seven data points at

1 nm intervals.

# Based on transmittance values calculated from measured bandpass function

approximation as described in Section 3.2.4.4.

40
Table XII

Comparison of wavelength at minimum transmittance obtained by fitting polynomial of

degree 6 to 7 transmittance values measured at 1.5 nm intervals to the wavelength at minimum transmittance obtained by fitting a polynomial of degree 4 to the indicated

number of transmittance values measured at 0.15 nm intervals. All measurements were

made of master filter D-l with 1.5 nm bandwidth.

Mi nimum

Number

Number

of Data Points

(0.15 nm)

Minimum

Wavelength of Transmittance

x . ( .15 nm) x . (1 .5 nm) min v

' min v

'

Di fference

x . ( 1 . 5 nm

x . ( . 1 5 nm

1 5 402 .43 402 .36 -0.07

2 5 431 .43 431 .49 0.06

3 10 440 .22 440 .17 -0.05

4 9 445 .52 445 .58 0.06

5 8 472 .79 472 .71 -0.08

6 8 478 .62 478 .77 0.15

7 7 513 .52 513 ,44 -0.08

8 7 529 .46 529,.67 0.21

9 5 572 .72 572,,67 -0.05

1 0 5 585 .39 585.,33 -0.06

11 8 623 .67 623. 61 -0.06

12 7 629,.64 629. 62 -0.02

T3- 9 684.,75 684. 70 -0.05

14 9 740..16 739. 91 -0.25

15 8 748,,30 748. 34 0.04

49
en

LOo oo oo ooo o ! o o o o

01 CU 3 .c >

T3o

-oc

rej o

enCMo

LOoo oo oo o oo

4-> CI CD E U E 113 E T3

s= +-> 3

r- -i- LO ~o E • c

l/l i— 03 4- SZ CO O n3

sC -Q

re) E

CQ 3 .— CM

O

QJ res LO CTi o o •^1- oo en E CQ LO CM 1 o 1 O o 1 CTi o CD O 1 o 1 1 o o o o o 1 o o o 1 S- ~o 3 cu CD o o o o o o o o o o i/i <_> 4-*

res c re) CD CD CJ E nb T3

4— cu O l/l co cn ro CM oo ro o 00 +-> S- o CM 1 C\J 1 o o ro 1 4-> c o O 1 o 1 1 o o o o o o o 1 CU cu 4- LO

00 em

O o o o o o o o o o

S- s- o 3 3 o (/> 4-J 4- 03 (0 1—1

0) 1—1 E E > O CD

O o CM Cn o o r-^

X sQ LO CM LO 1 ro o o 1 1^ CM 1 4- o o 1 o 1 o o o o o o 1 o o 1 CD "O

-o oo s_ o o o o o o o o o o o -Q cu re! ro U "O

1—

obtai

r-pre c

ro

-t-> (.0

CD OO o ro cn CM ro LO o o o ro LO

on 3 o CM o LO CM CM

re) o

CO o o o o o o o o o o o o o o

(/) o o o o o o o o o o o o o o o

cu LO ro ro LO CM CM ^1- oo o cn cn o LO CM CM CM o o LO o CM

l£ o o o o o o o o o o o CM o o o

4-J o o o o o o o o o o o o o o o

co o co ro LO LO CO LO CM LO

ro LO LO 1

<* LO LO ro LO LO CO Cn CM CM o LO CM 00 ro CTl CM LO CO en CTl co o CO r~« CM LO CM CM co CO

"3" "3- LO LO LO LO LO LO LO

50
if there were no passband effects the fitting should result in a constant value of zero for

the correction curve. The maximum departure of the fitted wavelength correction curve from

zero is given in Table XIV for each class of passband shape and for each degree of polynomial. It can be seen that the errors become larger as the passbands depart from the symmetric triangles from which Table VI was derived and when more freedom is allowed in the

fitting function.

3.2.9 Temperature Effects

Most of the measurements described in this Special Publication were made at an ambient

temperature of 25 + 0.5 C. In order to determine the effect of changes in temperature of

the filter upon the wavelength at minimum transmittance, we mounted one of the master filters (Master #3) in a special filter holder in which the temperature of the filter and the

surrounding air could be set with an accuracy of +0.2 C and held constant to within 0.1 C for the duration of an entire set of measurements? Measurements were made at 18.3 C,

25.0 C, and 32.0 C with data taken at 1.5 nm intervals using the high precision measurement

sequence described in Section 3.2.3.1. Transmittance values were calculated for seven different symmetrical triangular passbands, and the wavelength at minimum transmittance was determined by fitting a polynomial of degree 6 to 7 transmittance values closest to the

wavelength at minimum. The wavelengths at minimum transmittance obtained in each case are reported in Table XV. A repeat of the measurement at 32.0 C showed these data to be re- peatable to the point that the last figures appear significant. However, none of the

differences due to changes in temperature can be assumed to be significant in view of the

uncertainty in locating the wavelength at minimum transmittance as described in Section

3.2.5.

3.2.10 Effects due to Filter Thickness

The three master filters, D-l, D-2, and Master #3, covered the full range of filter

thicknesses from 2.84 mm to 3.00 mm. In no case did the wavelength of minimum transmittance differ from filter to filter by more than 0.02 nm and in most cases the difference

was less than 0.01 nm. It is probable that the observed differences were due to the un- certainty in locating the minimum by polynomial fitting. Differences in filter thickness

produced no observable effect on the wavelength of the inflection points.

3.2.11 Effects due to Polarization of Incident Radiation

No significant differences in the transmittance data from high precision measurements

were observed under a 90° rotation of the plane of polarization of the incident light.

3.2.12 Effects due to Changes in the Condition of the Filter Surface.

The transmittance of the didymium filters change slightly in time even in a reasonably

controlled environment such as an air conditioned laboratory. This is due to a change in

the surface of the filter. For example, over a period of approximately six months after

its surfaces were cleaned, one of the master filters was observed to decrease in transmittance by Mo at 400 nm and by 0.2% at 775 nm. After the filter was cleaned, transmittance

returned to its original value. This change in transmittance does not affect the use of

the filter for wavelength calibrations, since the change in transmittance causes no significant change in the wavelengths of minimum transmittance and is taken into account in

the fitting procedure used in connection with the points of inflection (see Section 2.3.3).

If the filters are kept free from obvious contamination, they probably will not need

cleaning. However, if cleaning becomes necessary, it is recommended that they be wetted

with water, cleaned with lens tissue wetted with a solution of mild soap, rinsed with distilled water, rinsed with isopropyl alcohol, and rinsed again with distilled water.

3.3 Uncertainties Introduced by the User

The uncertainties discussed in the preceding section are those introduced in the calibration of the NBS instruments, in the measurements made with them, and in the subsequent

data analysis. By identifying these sources of uncertainty and by placing an upper limit

on them, it is possible to provide a certification of the values assigned to the filters

and to the data supplied in the tables in this special publication. However, if the user

is to calibrate the wavelength scale of his instrument, he must also make a number of

51
Table XIV

Maximum departure of fitted wavelength correction curve from correct (constant at

zero) wavelength correction curve for various correction curve fitting functions

and passbands.

Magnitude of Maximum Wavelength Error (nm)

Fitting Function

Passband

symmetric triangle

bandwidth (nm)

(see 3.2.4.1)

1.5

3 4.5

67.5

9 10.5

Constant

.0.02

.01 .0.02

.02

.01

Linear

.01 .03

.01 .02

.04 .03

.04

Quadratic

.08

.04

.02

.05

.06

.04

.07

symmetric trapezoid

bandwidth (nm)

(see 3.2.4.2)

4.

7. 10.

13.

16, 19.

.03

.01 .03

.05

.03

.07

.05

.02

.10

.09

.03

.32

.11 .06

.18

.18

.14

.77

asymmetric triangles

passband number

from Table BIV

(see 3.2.4.3)

12

34

5

6

.09

.06

.20

.20

.34 .37

.16

.10

.30

.34 .49

.67

.27 .14

.46

.40

.72

.84

5?
Table XV Measured values for wavelength at minimum transmittance for filter D-l for three

different temperatures.

Band Temperature

BANDPASS

Number (Cel sius) 1 .5 nm 3.0 nm 4.5 nm D . U nml Mil 7.5 nm 9.0 nm 10.5 nm

1 18.3 402. 41 401

71 401

51 401 40 401. 04 400. 34 25 402. 37 401

67 401 . 48 zim H-U 1 oO 400 98 400. 23

32 402 39 401

68 401

48 zim jj 400 95 400. 17

2 18.3 431. 51 432 44

25 431

50 432 45

32 431

53 432 49

3 18.3 440. 15 440 45 441

50 442 34 442 20 441 87 441 .00

25 440. 15 440 38 441

56 AA? 0 / 442 22 441 89 441 .02

32 440 20 440. 42 441

62 AA? ACt 442 23 441 90 441 .04

4 18.3 445. 58 445 15

25 445. 58 445 14

32 445 58 445 14

5 18.3 472. 73 472 58 472. 91 25 472. 72 472 58 472. 95

32 472. 74 472 61 472. 98

6 18.3 478 75 479 25 479 21 478 27 477 37 476 52 475.65

25 478 78 479 25 479 20 A7P. C\J 477 35 476 51 475.66

32 478 80 479 27 479 21 A7P. H- / O c u 477 36 476 53 475.68

7 18.3 513 43 513 58 513 85 514 26 515 24

25 513 44 513 60 513 88 ORl1 AH j i 515 40

32 51 3 48 51 3 64 51 3 93 Rid o37/ 51 5 58

8 18.3 529 56 530 06 529 95 529 50 529 29 529 14 528.89

25 529 67 530 12 530 00 DC 3 J*T 529 32 529 16 528.92

32 529 64 530 1 5 530 02 R?Q OO 529 34 529 18 528.94

9 18.3 572 66 573 22 574 16 575 04 576 49

25 572 67 573 25 574 19 0 / 0 Uo 576 .53 32 572 68 573 25 574 20 D/O I I 576 58

10 18.3 585 31 585 51 585 74 586 00 585 .97 585 33 584.38

25 585 33 585 55 585 78 OoD 586 .01 585 37 584.44

32 585 35 585 55 585 78 586 03 585 40 584.48

11 18.3 623 63 624 03

25 623 60 624 01 32 623 67 624 02

12 18.3 629 64 629 44 628 51 626 98 627 00

25 629 64 629 45 628 58 OC 1 mUj 627 01 32 629 65 629 46 628 57 0(1/ DA 627 00

13 18.3 684 68 684 70 684 72 684 73 684 71 684 66 684.57

25 684 71 684 73 684 76 Do4 77 684 75 684 71 684.63

32 684 71 684 74 684 78 Dot fin ou 684 78 684 74 684.66

14 18.3 739 81 739 93 740 21 740 85 741 92 742 88 743.57

25 739 .91 739 .98 740 27 740 .92 742 .01 742 97 743.65

32 739 .99 740 .01 740 29 740 95 742 05 743 00 743.68

15 18.3 748 28 748 10

25 748 34 748 .15 32 748 .30 748 .12

53
measurements as described in Section 2 of this special publication, and in so doing he introduces additional uncertainties into the calibration process. Many of these sources of

uncertainty have already been discussed in connection with the various measurements or are similar to those for the NBS measurements as described in Section 3.2. In the remainder of

this section, the uncertainties introduced in the use of these standard reference filters

will be listed and ways to estimate them will be discussed.

3.3.1 Effects of Instrument Sensitivity Function*

If there is a substantial rate of change of the instrument sensitivity function with

wavelength, the effect would be to cause the actual passband centroid to be shifted away

from the centroid of the geometrical passband. In most instruments, the wavelength setting

mechanism is not designed to take this factor into account, and therefore a wavelength calibration error is introduced. However, a calibration made with an SRM filter calibrates

the instrument completely with respect to the centroid of passband, including the effects

of the instrument sensitivity function. For this reason, one need know nothing about the

instrument sensitivity function to obtain a correct calibration at each of the calibration

points. If the instrument sensitivity is a smooth function of wavelength as, for example,

in the 90° polarization curve in Figure 10 or the sensitivity curve in Figure A2 , a smooth curve fitted to these exact calibration points would be a correct calibration curve for the

whole instrument. If, on the other hand, the sensitivity curve is not smooth, as, for

example, in the 0° polarization curve in Figure 10, a smooth curve through correct calibration points might not be a good representation of the general calibration curve if the

calibration points fell on parts of the sensitivity curve which did not represent its

general trend. This effect is illustrated in Figure 11 and in the accompanying discussion.

If the sensitivity function is unknown, one must assume that the calibration curve derived

from the use of the filters is representative. If something is known about the sensitivity

curve, an estimate can be made of the amount of shift of the centroid due to the slope of

the sensitivity curve. In particular, for a triangular geometrical passband the displacement

dX of the passband centroid from the center of the geometrical passband is given by

dX=-*f- (13)

where a is the slope of the sensitivity curve normalized at the center of the triangle and

w is the width at half-height of the geometrical passband. If the slope and the effective

triangular bandwidth are both known, equation (13) can be used to provide a very good estimate of the magnitude of the centroid shift due to slope in the sensitivity curve. 3.3.2 Effects of Passband Shape and Width

In the case of the calibrations by points of inflection, the passband shape and width

have little effect upon the wavelength calibration which is obtained. The values given in the tables of transmittances and wavelengths for the inflection points were obtained from

the average of seven triangular passbands ranging from 1.5 to 10.5 nanometers in width as described in Section 3.2.7. The magnitude of the wavelength error which results from assuming

the average value to be correct for a given bandwidth is given in Table XIV for several

shapes and widths of passbands. It can be seen that except for the very wide and very asym- :- metric passbands, the maximum error in the resulting calibration curves is rather small. The

effects of passband width and shape on the wavelength of minimum transmittance, on the

other hand, is quite large. Errors due to this can be minimized in a case in which the bandpass shape is known by generating transmittance data from the known passband shape and then

finding the wavelengths at minimum transmittance corresponding to that transmittance data.

This procedure is described in Section 3.2.4.4. Advantage can be taken of the stability of

the inflection point data to make use of the minimum transmittance points even when the

bandwidth is only approximately known. This is done through the curve fitting procedure

symbolized by equation (7) and discussed in Section 2.3.3. In this equation, the only

assumption made about the bandwidth is that it is constant. Determining the effective

bandwidth (d in equation 7) consumes a portion of the information contained in the measured

*The instrument sensitivity function is sometimes called the responsivity factor.

I 54
wavelength of minimum transmittance data, but the net result of fitting in this manner is to improve the calibration curve. This improvement depends, of course, on the extent to which the bandwidth is actually constant.

3.3.3 Uncertainties Connected with Transmittance

An error in measured transmittance can occur for any number of reasons. The instrument

which is to have its wavelength scale calibrated may be off calibration on its transmittance

scale because of an improper gain setting, an improper zero setting, or a non-linearity in

the detection system. Such an error is usually constant with respect to wavelength and

will make no difference at all in the wavelength of minimum transmittance. If the constant

error is small, the wavelengths determined from the inflection points will only be slightly

in error, and this error will be reduced further by the fitting procedure described in

Section 2.3.3. This also holds true for wavelength errors which are introduced because the

actual transmittance of the filters has changed by a constant amount at all wavelengths. If

the transmittance errors are not constant with respect to wavelength, the fitting procedure

in Section 2.3.3 will not reduce the error much. Such non-constant wavelength errors can

be introduced through stray radiation of another wavelength in the user's instrument or through contamination of the filter, which is usually more effective at the short wavelength

end of the spectrum than at the long wavelength end. Additional terms could be put into

the fitting expression, equation (7), but this is not recommended. If a non-constant error

in transmittance measurements is suspected because of difficulties in obtaining a good fit

with expression (7), the filters should be cleaned and the instrument should be checked for

stray light.

3.3.4 Other Causes of Systematic Uncertainty

Two other parameters are potentially causes of systematic uncertainties in the measurement. One of them, polarization, either of the incident radiation or in the instrument's

detection system, has a negligible effect. The other, temperature, produces no noticeable

effect on the wavelength at minimum transmittance. The wavelength and transmittance at the

inflection point is measurably affected by changes in temperature, but the effect does not follow a regular pattern. In most cases, the effect of temperature is small, however, with

a change in inflection point wavelength of less than 0.1 nm over the temperature range

18 °C to 32 °C.

3.3.5 The Effect of Random Measurement Errors

The most reliable way to determine the effect upon the wavelength calibration of

random errors introduced in using the filters is to make several repeated calibrations and

to look for differences in the results. Evaluating the random error in each of the measurement parameters and calculating the net effect upon the wavelength calibration would be

time consuming and subject to error. However, if the random error in the calibration is found to be too large, the latter procedure is useful for locating the principle causes of

random uncertainty.

3.4 Illustrated Summary of Uncertainty Estimations

The systematic uncertainty was experimentally evaluated for the 1.5 nm bandpass data

as discussed in Section 3.2. It was also possible to experimentally evaluate the uncertainty

for 10 nm triangular bandpass data using the NBS Reference Spectrophotometer for Diffuse

Reflectance and Transmittance. [7] This instrument has a 10 nm triangular bandpass defined

by the dispersion of the grating and the entrance and exit apertures in the monochromator

The instrument was calibrated by the method discussed below in Section A. 3.1 and using 5 atomic lines and also taking account of the instrument sensitivity functions as described

in Section A. 3. 2. The atomic lines used were Hg 435.834 nm, Hg 546.075 nm, He 587.562 nm, He 667.815 nm, Cd 508.582 nm, and Cd 643.847 nm. A linear least squares fit through the

data for the 6 atomic lines was corrected by the instrument sensitivity function to obtain

a calibration curve.

Then, the instrument was calibrated using the didymium glass filter Master 3. It was found that the corrections obtained from the 6 didymium minima with 10 nm bandpass never differed from the calibration curve by more than 0.25 nm, and the corrections obtained from

55
the points of inflection never differ by more than 0.2 nm from the calibration curve. We list these as indications for the uncertainty at 10 nm bandpass. However, it is possible

to use techniques to improve the results obtained from the didymium glass filter as described

in the following paragraph.

The results of fitting the additive correction data under three assumptions are shown

in Figure 15. In the first case, equation (8) was used with a least squares fitting to obtain the curve shown as a dashed line. A second curve, shown as a dotted line, was obtained by adding an extra term in X

3

. to equation (8). In both of these cases, it was assumed that nothing was known about possible causes of instrument wavelength error. The

difference between the two curves was simply in terms of the complexity of the fitting

function. Actually, information was available which could be used to produce a better

calibration curve. The instrument sensitivity function had been measured, and so it was possible through the use of equation (13) to determine the contribution of this function to the additive correction. This additive correction, shown as the lower alternate dot-dash

curve in Figure 15, was subtracted from each piece of experimental additive correction data

and the resulting points were fitted with equation (8) with the constraint that c = 0, i.e.

a corrected linear fit since gratings have a linear dispersion. The sensitivity function

correction was then added back to this line to give the solid curve in Figure 15. This curve is essentially the sensitivity correction curve shifted by 0.45 nm in the positive

direction. Although any one of the three calibration curves is reasonable in light of the

random noise in the data, the solid curve is to be preferred since it was based on a great

deal of additional information. The agreement with the calibration curve obtained from the

atomic lines was always within the 0.1 nm.

The uncertainties for bandpasses between 1.5 and 10.5 nm have not been explored by

comparisons with atomic lines. Although the authors believe that they are probably not much different from the other uncertainties since the fitted polynomials did not oscillate

near the minima, we have not mathematically proven it and to measure each minimum at each

bandpass is not economically feasible. It was pointed out to us by an NBS mathematician

that spline fitting yields itself to numerical error analysis*, and if further work is done, this avenue will be explored.

4. Making Use of the Calibration Curve

If the additive corrections which are determined in the calibration process turn out

to be approximately the same size as or smaller than the uncertainties in the calibration,

then it is usually advisable to simply assume the instrument wavelength scale to be accurate enough. If the additive corrections are significant, the proper course of action depends strongly upon the particular instrument being calibrated. One of two approaches is often taken in making wavelength corrections. The first approach is to make physical

adjustments on the instrument to correct its wavelength scale reading. After each such

adjustment, the filter should be remeasured to determine that the wavelength shift is in the right direction and by the right amount. The second approach is to leave the instrument alone and to simply calculate the correct instrument wavelength scale setting from the

correction curve. As can be seen from equation (3) the correct wavelength scale setting X

s to obtain a desired centroid wavelength A is given by subtracting the additive correction

SA from X . This procedure is convenientc when the wavelength setting is done by hand or under the direct control of a computer. If transmi ttance measurements are made at regular

wavelength intervals and the transmi ttance is a slowly varying function of wavelength, the

additive correction can be used to calculate the transmittance value t at the nth correct

wavelength value from the measured transmittance values by means of linear interpolation:

*Dr. Walter Liggett, Private Communication

56
9

Additive correction due to

sensitivity function

. Quadratic fit

Cubic fit

• INFLECTION POINTS

% \

A TRANSMITTANCE MINIMA •

«

1 s 500 600

Wavelength nm

700

Figure 15: Wavelength Calibration of NBS Reference Spectrophotometer

for Diffuse Transmittance and Reflectance.

57
where A . is the wavelength scale value at which measurement number i in the series was made

and x'^is the corresponding measured value of the transmi ttance.

58
APPENDIX A

Background and Terminology

A. 1 Introduction

In the main body of this Special Publication, the way in which the didymium glass SRM

filters are to be used to calibrate the wavelength scale of a spectrophotometer is described

in detail. Although a satisfactory calibration can usually be obtained by simply following

the procedure as described, a more complete understanding of the problem of wavelength

calibrations will enable one to make better use of the information which can be obtained

from the filters. For this reason, a more detailed discussion of principles underlying

wavelength calibration is presented in this appendix and, where possible, these principles

are also illustrated with data from the General Electric Recording Spectrophotometer which

was used to illustrate the main text.

A. 2 Spectral Transmi ttance

Since the filters are used by making measurements of their spectral transmi ttance

this quantity should first be carefully discussed. In principle, the spectral transmittance t(x) at a given wavelength x incident on the filter which is transmitted by it.

If one were to experimentally measure this quantity using very nearly monochromatic radiation such as that from a laser, one would find that the measured spectral transmittance

curve is shaped by interference effects arising from reflections from the filter's surfaces

as well as from absorption of radiation by the rare earths within the glass. Since the

interference effects can depend strongly upon the way the instrument projects the radiation upon the filter and can also depend strongly upon the minute details of the thickness

and surface shape of the filter, spectral transmittance so defined would not be a satisfactory quantity upon which to base wavelength calibrations. Instead, we will first define

measured spectral transmittance t(x) as the fraction of radiation from a band of radiation

centered at X which is transmitted by the filter, where the band of radiation is sufficiently broad that interference effects are averaged out. The spectral transmittance

t(x) can then be defined as the extrapolation to zero bandwidth of measured transmittances

made with narrower and narrower bandwidths. Under this definition, the dominating property shaping the spectral transmittance curve would be the absorption of the radiation in

the rare earth oxide in the glass, and it could be correctly considered to be a property

of the glass. In calibrating an instrument, one would like to adjust the instrument so that over a large variety of randomly selected measurements the rms deviation between

measured spectral transmittance and spectral transmittance would be minimized. Once this

is done, one would also like to know to what other instruments a particular instrument is equivalent, that is to say, what other instruments would consistently give the same differences between the measured spectral transmittance and the spectral transmittance. Both

of these aspects of calibration will be considered here.

A. 3 The Spectral Passband* (Function) of Wavelength

The instruments with which these SRM filters are intended to be used made measurements

using a mixture radiation from a finite range of wavelengths. We define a function

B(x ,x), which we will call the spectral passband (function), to describe, as a function of

wavilength x, the way the mixture of radiation is effectively used when the instrument

measures transmittance with its wavelength scale set at x . It is convenient to think of

the passband as being made up of the product of two functions S(x) and G(x ,x) which represent two measurable aspects of the passband.

B(X

S ,X) = G(x

s ,x) S(X) (Al)

The function S(x), which we will call the instrument sensitivity, indicates the relative

efficiency of the measurement with respect to wavelength and results from the combined

*Not to be confused with the bandwidth AX = A 2

- X-j (of the passband).

59
relative wavelength dependency of the light emitted from the source, of the amount of light

absorbed or scattered as it passes through the monochromator along a path for which it is expected to be transmitted, and of the sensitivity of the detector. The function

G(x ,x), which we will call the geometrical passband, describes the way in which, by specific dispersion or absorption, the monochromator limits the radiation it passes to a small

range around A . It is interesting to note that B(x ,x) can be very precisely defined in terms of its role in relating measured spectral tranimi ttance to actual spectral transmittance, but it is nearly impossible to measure directly. However, S(x) and G(x ,x), which

are not very precisely defined, can be measured rather directly in a way in which their

product will provide a representation of B(x ,x) which is sufficiently good for the purposes

of wavelength calibration. Therefore, before we show how B(x ,x) is involved in describing

the measured transmittance, we will show how S(x) and G(x ,x) can be experimentally determi ned.

A. 3.1 Determining the Geometrical Passband Experimentally

This experimental method of determining the geometrical passband is based on the

assumption that the geometrical passband does not change its shape very rapidly with

changing x , so that, over a range of x equal in magnitude to the range over which

B(x ,x) is non-zero for a fixed x , the curve G(x ,x) can be regarded as shifting its

position along the wavelength axis without changing shape. In mathematical terms this can

be represented as

G(X-j ,X.j+AX) = G(A-j-AA,X-| ) (A2)

where X-, is a specific wavelength and ax represents a specific displacement in wavelength.

To make use of this result to determine G(x ,x) experimentally, the usual light source is removed from the monochromator and replaced by a stable source of monochromatic radiation

such as low pressure gaseous discharge lamp. Under these conditions, the instrument sensitivity is held constant and when the wavelength scale is scanned with no sample in the instrument, the instrument produces a signal which is proportional to G(x ,x) for fixed x. As an illustration, the results are shown in Figure Ala for a measurement made with the General

Electric Spectrophotometer using the mercury line at 435.8 nm. Equation (A2) above can be

used to produce the curve of G(x ,x) for fixed x shown in Figure Alb. Note that the correct wavelength scale on the instrument and true

s physical wavelength can be tied together

at the wavelength of the line source, since at this point both should have the same numerical value. It might be pointed out that for this determination, a different detector than

the one in the instrument can be used if it is more convenient, since all the measurements

are made with radiation at one wavelength. However, care should be taken that the line source and the substitute detector, if one is used, bear the same geometrical relationships

to the monochromator as the operating source and detector. For example, the radiation from

the line source should fill the monochromator entrance optics in the same way as the operating source. Otherwise a change in G(x ,x) from that for ordinary use may be caused by

the change in geometry.

s

A. 3. 2 Determining the Instrument Sensitivity Experimentally

To determine S(x) experimentally, one can observe the detector output as the instrument

is scanned over its full wavelength range using its normal source and detector and with no sample in the measuring position. The detector output must be measured in such a way that

a linear relationship between the detector output and the amount of radiation striking it at a given wavelength is obtained. In a dual beam spectrophotometer such as the General

Electric instrument it is generally not possible to use the instrument output scale for

obtaining the detector output, since what is usually shown on the output scale of such an instrument is a number proportional to the ratio of the signals from the two beams rather

than the signals themselves. For this determination, we stopped the chopping mechanism at the point where a maximum amount of radiation was in the sample beam, disconnected the detector from the instrument electronics, and read the detector current directly with a picoameter. The resulting plot of detector current as a function of wavelength scale

reading X s is shown in Figure A2a.

To establish the relationship between this function and S(x), one must take into

account the particular monochromator which is being used. In the case of the General

60
X nanometers

Figure Al : Experimental evaluation of G(X ,A) near A

g = 435 nm

61
400 500 600 700

Wavelength nm

Figure A2: Experimental Evaluation of S(x)

62
Electric instrument, the mo no chroma tor uses prisms for the dispersive elements and has the

widths of the entrance and exit slits automatically adjusted in constant proportion to each

other by means of cams to maintain a G(x ,x) nearly constant both in width and shape.

Under these circumstances, the detector Signal at x is proportional to the product S(x)

times the slit width at that wavelength scale setting. That is to say, if S(x) were constant, one would expect the function shown in Figure A2a to be proportional to the slit

width. Therefore, if the function in Figure A2a is divided by the instrument slit width

at each wavelength, shown in Figure A2b, the resulting function will be the approximation

to S(x) shown in Figure A2c. This approximation of S(x) is crude both because of the noise

and lack of resolution of the data in Figure A2a and because of the approximations used in

deriving the relationship between the data and S(x). However, the influence of the S(x) curve on the wavelength calibration is small and thus the uncertainties in wavelength

calibration induced by the uncertainties in S(x) will be very small.

A. 3. 3 Calculation of the Spectral Passband (Function) from Experimental Data

With experimental functions proportional to G(x ,x) and S(x) available, a function

proportional to B(x ,x) can be constructed by multiplying G(x ,x) by S(x) at each wavelength x. In Figure A3, a section of the S(x) curve in the region of 435.8 nm is shown

normalized to 1 at 435.8 nm, the passband B(x ,x) obtained by multiplying this portion of

the S(x) curve by the G(x ,x) curve shown in

s Figure Alb. The G(x ,x) curve is shown in

the same figure to show how the shape of the geometrical passband Ts modified by the sen- sitivity function to produce the passband. In the course of this determination of the

passband, the results of all measurements have been placed on an arbitrary scale and no effort has been made to define or determine an associated constant of proportionality.

The reason for this is that the constant of proportionality cancels in the expression

for measured transmittance. The aspect of the passband in which we are most interested

from the wavelength calibration standpoint is the shape of the passband as a function of

wavelength at each setting of the instrument wavelength x s<

A. 4 Wavelength Calibration

In general terms a wavelength calibration is an effort to relate the wavelength scale

on an instrument to a wavelength representative of the passband. There are any number of

possible choices for the wavelength to represent the passband. One common choice of a representative wavelength, for example, is the wavelength at which the geometric passband

reaches its peak value. This choice is not unreasonable, and this wavelength can be

readily determined for a number of points x on the wavelength scale with the aid of line sources by the method described in Section A. 3.1. However, the analysis of the way the

instrument is used to measure transmittance which is to be presented in this section re- veals that the wavelength of the centroid of the passband is a better representative point.

In this Special Publication, therefore, a wavelength calibration is by definition the re- lationship between an instrument wavelength scale reading and the wavelength of the cor- responding passband centroid. Since an understanding of the reasoning behind this definition is fundamental both to the understanding of the meaning of a transmittance measurement as well as to the use of these SRM filters, we will give a brief discussion of the

nature of transmittance measurements and the reason for choosing the centroid wavelength as the wavelength to represent the passband.

A. 4.1 Measuring Spectral Transmittance

The goal of a transmittance measurement is to determine the spectral transmittance

t(x) of an object as described in Section A.l. For the preponderance of practical applications, however, it is not advisable to measure spectral transmittance using highly mono- chromatic radiation. The interference effects which pose a problem in interpreting the

measurements have already been discussed in A.l. In a wide variety of applications such as measuring color, luminous transmittance, or effective radiant energy transfer for solar

energy applications, for example, one is interested in integrated effects over a large

range of wavelengths. In order to be assured that the entire wavelength range is covered

with no gaps, the wavelengths at which measurements are made should be spaced closely

enough so that the passbands overlap at the half-height points. In this way, the entire

63
spectral transmittance curve receives nearly equal weighting when the integrals are taken.

In order to produce this kind of uniform coverage with nearly monochromatic radiation, much more data would have to be taken than can usually be justified from the detailed shape of

the spectral transmittance curve, and, because of the lower total amount of radiation

involved in each measurement at narrow bands, the corresponding noise level of each measured

transmittance would be higher.

Even though the spectral transmittance measurements may not be done with monochromatic

radiation, it is convenient to represent the limiting ratio in terms of an imaginary interference-free measurement done with monochromatic radiation in which the radiation incident

on the detector without a filter in place produces a signal I_.('X) and the same radiation

passing through the filter produces a signal I. (a) . For such an idealized experiment, the

spectral transmittance x(x) would be the ratio of the two signals

t(x) = I t (x)/I

Q

(X) (A3)

(It should be mentioned that here as in the other portions of this special publication, we have deliberately avoided discussing the geometrical parameters of the measurement. These

are, of course, important to an accurate measurement, but it will be assumed that they are properly handled and we will concentrate on the wavelength-related aspects of the measurement. )

We can now relate the measured spectral transmittance x (x) for a given instrument to the actual spectral transmittance t(x) through the use of anmequation similar to (A3)

above. In this case, however, the signals are proportional to integrals related to the

passband.

(A

s

) = ky*B(x

s ,x)dX (A4)

and

I t

(x

s

) = ky*B(x

s ,x) T (x)dx (A5)

where the constant of proportionality k is the same in both expressions and the integrals

are taken over the entire range for which B(x ,x) is non-zero. The measured transmittance

is, therefore

y*B(x

s ,x)x(x)dx

><*s> = f-^xW (A6)

Notice that the constant of proportionality cancels in this definition so that it is not important to determine it or to normalize the expression for B(x ,x) for the purposes of

this discussion. The measured transmittance is seen to be a weighted average of the

idealized spectral transmittance in which the instrument's passband is the weighting

function. In addition to being a definition of measured transmittance, equation (A6) forms

the basis for a very exact definition of the passband of a given instrument. By definition

the passband B(x ,x) is the weighting function which correctly related the measured spectral transmittance to the actual spectral transmittance for all arbitrarily chosen x(x).

A. 4. 2 The Passband Centroid Wavelength

We have chosen to represent the wavelength of the passband by its centroid wavelength

or first moment wavelength, as it is sometimes called. The centroid wavelength X c (x ) of

the passband, when the wavelength scale is set at x , can be defined in terms of the

passband function as:

h{\ c ,x)xdx UU = VV ~ /*B(x ,x)dx js ,a,aua

(A7)

/' 7',

, Since the passband represents the effective contribution of radiation at wavelength x to the transmittance measurement, the centroid wavelength can in a very real sense be thought

64
of as the wavelength of the effective center of the passband. As an example of how the

centroid can be located from data obtained from a real instrument, consider the passband

illustrated in Figure A3. The values of B(435.8,x) can be read from the curve at uniform

intervals and the integrals in equation (A7) approximated by the appropriate sums. The

results of such a calculation made with intervals of 1 nm are summarized in Table AI. In that calculation, the centroid wavelength was found to be at approximately 437.5 nm, which

is 1.7 nm greater than the 435.8 nm indicated by the wavelength scale. Therefore, in this

region of the spectrum, we must add 1.7 nm to the wavelength indicated by the instrument

scale in reporting the wavelength of spectral transmi ttance data measured with the General

Electric Spectrophotometer. A similar calibration was made for each of three other line sources, with the results shown in Table All. Notice that we have calibrated the instrument directly from the measurements reported in Section A. 2 and did not need to use the SRM

filters at all. However, a similar calibration can be obtained from transmittance measurements on the SRM filters without the inconvenience of having to switch sources in the

instrument and make special measurements of the detector output. It is their convenience

which makes the SRM wavelength filters a valuable tool for calibrating the wavelength

seal e.

A. 4. 2.1 Centroid Shift due to Instrument Sensitivity

Quite often it is the case that the sensitivity function does not change greatly over

the region for which the geometrical passband is not zero, and therefore, the shape of the

passband resembles the shape of the geometrical passband. This is true, for example in the

illustration in Figure A3. In this case, a computational tool can be developed which is useful in locating the centroid wavelength when the centroid of the geometrical passband is known, as it would be, for example in the case of a symmetric triangular passband. If the

instrument function is sufficiently smooth to be well represented by a straight line in the

region of the passband, then the sensitivity factor, normalized at the centroid of the

geometrical passband can be represented as:

S = 1 + a(A-A ) (A7a)

where a is the slope of the curve so normalized and X is the centroid wavelength of the

geometrical passband. The centroid wavelength corresponding to the passband is obtained by

using equations (Al) and (A7a) in (A7) to obtain an expression for the centroid wavelength

y*G(x

s ,A)[1+a(x-x

q )]AdA

(A7b)

T

xc^x s^ "

: ,(x

s ,x)[l+a(x-x

g )]xdx

By making use of the fact that Tg(a ,x)-[x-Xg]dx is zero by definition of the centroid

wavelength A of the geometrical pasiband, one can reduce expression (A7b) to the

following form:

x c (A

s

) = A g (x

s

) + ayG(x

s ,x)(x-x

g

) 2 dx/y*G(x

s ,x)dx (A7c!

In words, this expression says that the centroid wavelength of the passband (function) is the centroid wavelength of the geometrical passband plus the product of the slope of the

normalized instrument sensitivity with the second moment of the geometrical passband taken

about its own centroid. For a symmetric triangular geometric passband, the second moment

about its centroid, which is the center in this case, is W

2 /6 as is indicated in equation

(15) in the main text. Therefore for example, if one knows that he has a triangular geometrical passband with a bandwidth of W nm and knows from experimental measurement of the

instrument sensitivity function that the slope of the normalized instrument sensitivity

function is a per nm, he knows that the centroid of the passband lies a W

2 /6 from the peak

of the geometrical passband as might be determined from scanning line sources. This can sometimes be a useful calculation shortcut.

A. 4. 3 The Significance of the Passband Centroid Wavelength

The main objective in calibrating the instrument wavelength scale would be to minimize

the difference between t (a ) and t(a) to the greatest possible extent commensurate with a given passband. A secona and related objective would be to minimize the difference between

65
425 435 445

Wavelength nm

Figure A3: Passband as determined experimentally for X = 435.8 nm

66
Table AI

Calculation of Centroid Wavelength * C US

)

A_ = 435.8

Wave! ength Passband

X B(435.8,\)

446.6 1.7

445.6 3.5

444.6 34.1

443.6 77

442.6 123.7

441 .6 163

440.6 191 .1

439.6 215.5

438.6 224.2

437.6 222.5

436.6 219

435.6 214.7

434.6 208.4

433.6 181.5

432.6 137.1

431 .6 71.7

430.6 34

429.6 7.5

Sums 2330.2

A c (435.8)

f

B(435.8,A)-A-dA

B(435.8,A)dA

1019525

2330.2

Product

B(435.8,A)-A

737

1570

15145

34173

54759

71971

84202

94729

98314

97378

95631

93513

90570

78688

59312

30964

14639

3237

1019525

A (435.8) = 437.5

67
Table All

Additive Wavelength Corrections and Effective Bandwidths Determined for

the General Electric Recording Spectrophotometer from Line Source Data

Lamp Li ne Addi ti ve Ef f ecti ve Equi val ent

Gas Wavelength Correction Bandwidth Triangular

(nm) (nm) (nm) Bandwidth*(nm)

Hg 435.8 1.7 3.4 8.4

Hg 546.1 .7 3.4 8.4

He 587.6 .9 3.2 7.8

Cd 643.8 .7 3.5 8.6

*Half-height bandwidth of triangular bandpass which has the same effective bandwidth as is given in the preceding column.

68
the x (a ) measured by two instruments having approximately the same passband. A wavelength

calibration in which the reported value of the instrument wavelength scale is corrected so that it is given in terms of the passband centroid wavelength is correct to the greatest

extent possible for measurements made on arbitrarily selected samples, i.e. for arbitrary

t(a). This can be demonstrated by considering the instrument to be set at wavelength A

g and expressing t(a) in terms of the centroid wavelength a C US

). This can be shown by

expressing the spectral transmittance on a new scale as x\l) where

L = A-A

C (A

S

) (A8)

The transmittance can be expressed as a power series in the new wavelength, generated by

Taylor's series expansion or by some other means, as

t(L) = T Q + t-j L + x

2 L

2 + x 3

L

3 + ...(9)

which can be expressed in summation notation as

x(L) = E t.-L

1 (A9)

i=o For the well-behaved spectral transmittance functions usually encountered with real

materials, the magnitude of the coefficients x. will decrease rather rapidly for increasing i. If equation (A6) is rewritten using the new wavelength scale and the ex- pansion for the spectral transmittance given in equation (A9), the measured transmittance

can be shown to be

- Tb^.qlV ( fll0 ) VV " To ,!2

T i>B(i

s ,L)<iL

The term for i = 0 is just x , which from equation (A9) can be seen to be the spectral

transmittance for L = 0, tha? is, the spectral transmittance at the passband centroid

wavelength. The term for i = 1 is identically zero because the multiplier of x, defines

the centroid wavelength, which on the L scale is zero by definition, i.e. L = 0 when

A = A (a ) according to equation (A8). The magnitude and sign of the sum of the remaining

terms

c in equation (A10) depends upon the functional form of the particular spectral transmittance function x(a) which is being considered. The important things which can be seen

from this expansion at this point in the discussion are that if the wavelength scale is

calibrated with respect to the passband centroid wavelength, the dominant term in the measured transmittance is the spectral transmittance at the scale wavelength, and there will

be no error introduced by the slope of the spectral transmittance curve. Put in other

words, to the extent that the portion of the spectral transmittance function falling within

The passband is a straight line, any instrument with its wavelength scale calibrated to

its passband centroid wavelength will measure the correct spectral transmittance. By

placing other restrictions on the instrument with regard to its bandwidth and passband

shape, it is possible to further control the magnitude of the summation of terms at the

right side of equation ( Al 0 ) and obtain better instrument-to-instrument agreement.

It should be pointed out that if the passband is symmetric around its centroid its

central peak value (if it has one) will fall at the centroid wavelength. In this case, calibration to the peak wavelength and calibration to the centroid wavelength would be

identical and equation (A10) would apply in both cases. However, if the peak wavelength

is not at the centroid, calibration to the peak wavelength will result in contribution to the measured transmittance at each point at which the slope of the spectral transmittance

curve is not zero.

The results of this section could be summarized from the point of view of the series

expansion of the transmittance in terms of increasingly complex spectral transmittance

functions as follows:

1) if the spectral transmittance is constant in a given region, any wavelength calibration

69
can be used,

2) if the spectral transmittance is linear in a given region, calibration with respect to the centroid wavelength will produce an exact measurement of the spectral transmittance and

therefore will produce inter-instrument agreement, and

3) if the transmittance function is more complex than linear, there will be certain passband-related differences between the spectral transmittance and the measured spectral

transmittance which cannot be resolved by wavelength calibration.

For further improvement in inter-instrument agreement beyond that which can be achieved

by wavelength calibration, the shape and width of the passband must be taken into account.

This aspect of the measurement will be discussed briefly in the remainder of this appendix.

A. 5 Characterizing the Passband

For any instrument, the passband is completely characterized by the function B(A ,X)

introduced in Section A. 3. However, it would be more convenient if there were one numBer

which could be used to characterize the passband which would relate significantly to the

measurement of spectral transmittance in such a way as to promote inter-instrument agreement. The concept of bandwidth, and especially of effective bandwidth, provides such a characterizing parameter.

A. 5.1 Bandwidth

In an instrument for spectral measurements, the passband will usually be non-zero over

only a limited range of wavelengths. It is convenient to have a concise way to describe

just how large this non-zero range is, and therefore the concept of "half-height bandwidth"

or, for simplicity, "bandwidth" is customarily introduced. This is defined as the difference between the shortest wavelength at which B(x ,x) is at half its maximum value and

the longest wavelength at which B(x ,x) is at half ils maximum value. In the case of the

passband illustrated in Figure A3, the passband is half its maximum value at 432 nm and at 443 nm. Therefore, the bandwidth in this case is defined to be 11 nm. It is this bandwidth which should govern the spacing of data points in a set of measurements in which one

is planning to perform an integration and therefore wants a nearly uniform weighting of the

spectral transmittance over the entire spectrum.

A. 5. 2 Effective Bandwidth

Another concept of bandwidth can be introduced which is more significant than halfheight bandwidth in terms of inter-instrument agreement for spectral transmittance measurements. This bandwidth, which will be called "effective bandwidth", is defined as the

square root of the second moment of the passband with respect to wavelength taken about the

centroid wavelength, i.e.

where

W

eq

- M

2 (All)

B(X.L)L 2 dL "o =

d / 1 i \m (A12) '2 B(x

s ,L)dL

The symbols in equation (A12) are defined in the discussion connected with equation (A10),

and it is from equation (A10) that the significance of the effective bandwidth can be seen. This is the instrument-dependent portion of the first, and usually the largest, of

the terms which are summed to obtain the departure of the measured transmittance from

the idealized spectral transmittance. As a matter of fact, in a very large number of

practical cases in which the transmittance is not an extremely rapidly varying function of

wavelength, this term completely dominates the summation. Therefore, two instruments

which have the same effective bandwidth and which are properly calibrated for wavelength

with respect to the centroid wavelength will produce measured spectral transmittance values

which are in agreement. Thus, the first, and generally the only, additional step which

70
needs to be taken in principle to assure inter-instrument agreement is to require that all instruments in a given group have the same effective bandwidth. In Table All, the effective bandwidth of the General Electric Spectrophotometer was calculated for each of the

four spectral lines for which passband data were obtained as described in Section A. 3. In the final column of the table is given the half-height bandwidth of a triangular passband

which would have the same effective bandwidth. It can be seen that these data confirm the

results of Section 2.3.3 in which it was found that the best wavelength calibration curve

fit for the General Electric Spectrophotometer was obtained from the triangular bandpass

data by using the data for 8.4 nm half-height bandwidth.

71
REFERENCES

General Electric Hardy-type spectrophotometer Number 732986 originally installed at

NBS in 1943 (NBS Serial Number 7735). Certain commercial instruments and materials

are identified in this paper in order to specify adequately the procedure. In no case does such identification imply endorsement or evaluation by the National Bureau

of Standards.

B.S. Pritchard and W.A. Holmwood, "New Recording Spectrophotometer", J. Opt. Soc.

Am. 45, 690 (1955).

Kasson S. Gibson and Harry J. Keegan, "Calibration and Operation of the General

Electric Recording Spectrophotometer of the National Bureau of Standards", J. Opt.

Soc. Am. 28, 372 (1938).

Kasson S. Gibson and Harry J. Keegan, "Use of Didymium Glasses for Wavelength Calibration of Recording Spectrophotometers" (abstract), J. Opt. Soc. Am. 3j_, 462 (1941).

Harry J. Keegan and Kasson S. Gibson, "On the Use of Working Standards of Didymium

and Vitrolite Glasses for Spectrophotometric Measurements." (abstract), J. Opt. Soc.

Am. 34, 770 (1944).

K. D. Mielenz and K. L. Eckerle, "Design, Construction, and Testing of a New High

Accuracy Spectrophotometer", NBS Technical Note 729, (1972).

William H. Venable, Jr., Jack J. Hsia, and Victor R. Weidner, "Development of an NBS Reference Spectrophotometer for Diffuse Transmittance and Reflectance", NBS

Technical Note 594-11 , (1976).

72
NBS-114A (rev. 9-78)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. SP 260-66

2. Gov't Accession No, 5. fteeipiertt's Accession Wo,

4. TITLE AND SUBTITLE Didymium Glass Filters for Calibrating the Wavelength

Scale of Spectrophotometers - SRM 2009, 2010, 2013,

and 2014

5. Publication Date October 1979

6. Performing Organization Code

7. AUTHOR(S)

William H. Venable, Jr.

8. Performing Organ. Report No. and Kenneth L. Eckerle

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, state, ZIP)

Same as Item 9

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number

Document describes a computer program; SF-185, FIPS Software Summary, is attached. 79-600122

16. ABSTRACT (A 200-word or less factual sl literature survey, mention it here.) nary ot most s i gni ficant information, ff document includes a si gni ficant bibliography or

This publication presents the use of didymium glass filters as Standard

Reference Materials (SRM). The standard reference materials are labeled

2009, 2010, 2013 and 2014 depending on size and method of calibration. The

certification and uncertainties are also discussed. An appendix with background material and terminology is included. Wavelengths of minimum transmi ttance and inflection points in the transmi ttance curve are certified

to be at specified wavelength and provide a convenient method of calibrating

the passband centroid of spectrophotometers with bandwidths between 1.5

and 10.5 nm.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper namt

sePB rs t e d by „.mi co/o„,j Bandwidth; didymium glass filter; passband centroid;

spectrophotometer; transmi ttance; wavelength standard

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