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°»EAV 0? NBS TECHNICAL NOTE 594-8

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress 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 consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent system of physical measurement; coordinates that system

with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Nuclear

Sciences - — Applied Radiation 2 — Quantum Electronics " — Electromagnetics 3 — Time

and Frequency '"' — Laboratory Astrophysics " — Cryogenics ". THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized

materials needed by industry, commerce, educational institutions, and Government; provides

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Environment * — Technical Evaluation and Application 4 — Fire Technology.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective utilization of automatic data processing equipment; and serves as the principal focus within

the executive branch for the development of Federal standards for automatic data processing

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divisions: Computer Services — Systems and Software — Computer Systems Engineering — Information Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and

a system of information analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services to ensure that the NBS staff has optimum

accessibility to the scientific information of the world. The Office consists of the following

organizational units: Office of Standard Reference Data — Office of Information Activities — Office of Technical

Publications — Library — Office of International Relations.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

3 Part of the Center for Radiation Research.

3 Located at Boulder, Colorado 80302.

* Part of the Center for Building Technology.
Tables of Diffraction Losses

W. B. Fussell

Heat Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

jftf**

°*£AU Of

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued June 1974
Library of Congress Catalog Number: 74-600088

National Bureau of Standards Technical Note 594-8

Nat. Bur. Stand. (U.S.), Tech. Note 594-8, 39 pages (June 1974) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1974

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. 03.46:594-8). Price 75 cents.
Preface

This is the eighth issue of a series of Technical Notes entitled

OPTICAL RADIATION MEASUREMENTS. The series will consist primarily of

reports of progress in, or details of, research conducted in radiometry

and photometry in the Optical Radiation Section of the Heat Division.

The level of presentation in OPTICAL RADIATION MEASUREMENTS will be

directed at a general technical audience. The equivalent of an undergraduate degree in engineering or physics, plus familiarity with the

basic concepts of radiometry and photometry [e.g. , G. Bauer, Measurement

of Optical Radiations (Focal Press, London, New York, 1965)], should be

sufficient for understanding the vast majority of material in this

series. Occasionally a more specialized background will be required.

Even in such instances,- however, a careful reading of the assumptions,

approximations, and final conclusions should permit the non-specialist

to understand the gist of the argument if not the details.

At times, certain commercial materials and equipment will be identified in this series in order to adequately specify the experimental

procedure. In no case does such identification imply recommendation or

endorsement by the National Bureau of Standards, nor does it imply that

the material or equipment identified is necessarily the best available

for the purpose.

Any suggestions readers may have to improve the utility of this

series are welcome.

Henry J. Kostkowski, Chief

Optical Radiation Section

National Bureau of Standards

iii
Contents

Page

1. Introduction 1

2. Tables of Diffraction Losses as Functions of

Wavelength and Geometry 3

3. Scaling the Diffraction Loss Tables. . 22

4. Effective Wavelengths to Use in the

Diffraction Loss Tables. 22

5. Sample Diffraction Loss Calculations . 23

5.1. Sample Diffraction Loss Calculations

for a Simple Case 23

5.2. Sample Diffraction Loss Calculations

for a Complex Case 23

6. Formulas Used for Computing the Diffraction Loss Tables. ... 26

7. Estimated Accuracy of the Diffraction

Loss Tables 29

8. References 30

Figures

Figure 1. Diffraction geometry used in the

diffraction loss tables 31

Figure 2. Diffraction geometry for the sample diffraction

loss calculations: complex case 32

Figure 3. Diffraction geometry used in the general

diffraction loss formulas 33

IV
Tables of Diffraction Losses*

W. B. Fussell

Tables of diffraction losses are given for

a range of typical experimental geometries for

wavelengths from 0.2 to 100 micrometers. The

scaling relationships for the diffraction losses

for varying wavelengths and geometries are also

given, and sample calculations are presented.

General formulas are given for the diffraction

losses; the formulas are derived from the

Kirchhoff scalar paraxial diffraction theory.

The accuracy of the tabulated values is estimated.

Key words: Diffraction; diffraction losses; Fresnel

diffraction; Kirchhoff diffraction theory; photometry; radiometry; scalar diffraction theory.

1. Introduction

With the improved precision and accuracy of radiometric measurements, diffraction losses have become significant. It is useful, therefore, to compute and tabulate diffraction losses for a range of typical

geometries and wavelengths. The Kirchhoff scalar paraxial diffraction

theory is used to calculate these losses. This is an approximate model

which evaluates the phase relationships over the diffracting aperture

(see fig. 1) for each elemental source area, for a given detection point

and wavelength; the resulting complex number is then integrated over the source area and the magnitude of the sum indicates the relative spectral

irradiance at the given detection point, compared with other detection

points on the detector area. The model assumes: a., all source points

radiate independently (that is, incoherent radiation); b., there are no polarization effects (that is, no vector effects); c. , off-axis angles

are small, and hence obliquity effects can be neglected. (Section 6 outlines the derivation of the equations used to compute the tables.)

The mathematical formulas used to compute the diffraction losses are refinements of the basic Fraunhofer on-axis diffraction formula (see

Blevin[l] l

) . The tabulated on-axis diffraction losses are estimated to

be accurate to within 10% mathematically; the off-axis diffraction losses

are estimated to be accurate to within 20%. (If the physical realities

of an experiment differ from the assumptions of the Kirchhoff model,

there will be additional errors besides those due tc the mathematical

approximations used to compute the tables; however, it is expected that

*Supported in part by the Calibration Coordination Group of the Department of Defense.

'•Figures in brackets indicate the literature references at the end of

this paper.
most situations in radiometry and photometry will be within the regime

of the Kirchhoff model. Blevin [1], for example, finds excellent

experimental agreement with the Kirchhoff model.) Thus, if it is

desired to calculate the off-axis diffraction loss for a given experiment

to within 0.1% of the spectral irradiance at the detector, then the

geometry of the experiment should be such that the tabulated diffraction

loss is less then 0.5% of the spectral irradiance at the detector, since

an error of 20% of 0.5% is equal to 0.1%.

The geometries and wavelengths selected for the diffraction loss

tables are:

a., wavelengths from 0.2 to 100 micrometers;

b.

,

source (or detector) diameters from 0.5 to 5 cm;

c.

,

source (or detector) -to- aperture distances from 5 to 20 cm;

d.

,

aperture diameters from 0.005 to 0.5 cm.

The geometry and terminology used in the diffraction loss tables is shown

in figure 1.

In general, if the circumference of the circle produced by projecting the aperture from every point on the detector (the geometry in this

report is assumed to be circularly symmetric in all cases) , onto the

plane of the source, lies within the source, then the radiation incident

on the detector is proportional to the source radiance (less diffraction

losses) . On the other hand, if the circumference of the circle produced

by projecting the aperture from every point of the source, onto the plane

of the detector, lies within the detector, then the radiation incident on

the detector is the total source radiation through the aperture (less

diffraction losses) . (The diffraction losses for a given configuration

are identical, whether the source is treated as a detector and the detector as a source, or vice versa; this is sometimes a conceptual advantage

in that it transforms a source radiance measurement into a total aperture

radiation measurement. ) In this report, the source radiance geometry

will always be meant unless it is explicitly stated that the total

aperture radiation geometry is under consideration.

For a given geometry, the diffraction loss in the plane of the detector is least on the axis; the diffraction loss increases steadily as the

distance from the axis increases (see sec. 6). Therefore, the diffraction loss realized with a circular detector increases steadily as the

detector radius increases. The diffraction losses listed in the following tables are for the on- axis case (the "point" detector) , and also for

the case of a detector that sees 90% of the diameter of the source (the

radius of such a detector is designated x ). These diffraction losses, — TTlcLX designated E". and E' r respectively, bracKet the loss for detector

radii between zero and x to within roughly ±20% for geometries where

the aperture diameter is much less than the source diameter, and more

accurately for ratios of the aperture diameter to the source diameter

2
larger than 0.1 (see the end of sec. 7).

2. Tables of Diffraction Losses as Functions

of Wavelength and Geometry

Terminology

X is the wavelength in micrometers.

d is the source diameter in cm (or the detector diameter, for

a total aperture radiation measurement; see sec. 1)

b is the source-aperture distance in cm.

D is the aperture diameter in cm.

v is the dimensionless quantity 7rDd(2bX)~.

E

1

. is the diffraction loss 1 for a point detector on- axis

whose distance from the aperture is at least 10 times

the source- aperture distance.

E

1 is the diffraction loss averaged over the area of a circular detector of radius x (see sec. 1 and fig. 1)

max

the radius x is defined to oe that radius for which

max

the field of view through the aperture covers the portion of the source disc whose ^diameter is 0.9 the source

diameter; the distance of the detector from the aperture

must be at least 10 times the source- aperture distance.

1 The diffraction losses E*. and E' are given as a percentage of the

irradiance that would be present at the detector in the absence of

diffraction. The mathematical formulas used to compute E

1

. and E'

, • ^ ^iJiin , -max

are given in section 6. Upper bounds for the errors in E . and E

T can be computed using the dimensionless quantity v, and formulas for

such computations are given in section 7.
X = 0^2 (urn)

d

=

0^5. (cm)

b D = .005 -01 .02 .05 .1 .2 .5

(cm) (cm)

v 39.3 78.5 157 393 785 1570 3930

($)E' . 1.62 0.8l 0.41 0.16 0.08 .05 _ min

($)E f 2.59 1.27 0.62 0.24 0.12 .06

max

v 19.6 39.3 78.5 196 393 785 I960

10 E' . 3. 24 1.62 0.81 0.33 0.17 0.10 _ man

E' 5.19 2.55 1.2U 0.47 0.23 0.13

max

v 9.82 19.6 39-3 98.2 196 393 982

20 E' . 6. 49 3.24 1.62 0.66 0.34 0.19 mm

E« ^8* 5.09 2.48 0.95 0.47 0.25

max

d = 1 (cm)

(cm;

v 78.5 157 314 785 1570 3140 7850

5 (#)E f

. 0.81 0.41 0.20 0.08 0.04 0.02 0.01 _ mm

($)E T 1.31 0.65 0.32 0.12 0.06 0.03 0.01

max

v 39.3 78.5 157 393 785 1570 3930

10 E' . 1.62 0.81 O.Ul 0.16 0.08 0.04 0.02

_ mm

E' 2.62 1.30 0.64 0.25 0.12 0.06 0.03

max

v 19.6 39.3 78.5 196 393 785 I960

20 E" . 3.24 1.62 0.81 0.33 0.16 0.06 0.04

_ mm

E

1 5-24 2.59 1.27 0.49 0.24 0.12 0.06

max

(Note: These values are upper bounds.)
X = 0^2_ (ym)

d = 2 (cm)

b D= .005 .01 .02 .05 .1 .2 .5

(cm) (cm)

v 157 3lM 628 1570 31^0 6280 15700

5 (%)E f

. O.il 0.20 0.10 O.OU 0.02 0.01 0.00 _ mm

{%)E 1 0.66 0.33 0.16 0.06 0.03 0.01 0.01

max

v 78.5 157 31 ^ 785 1570 31^0 7850

10 E' „ 0.8l O.il 0.20 0.08 0.0U 0.02 0.01

_ min

E

T 1.32 0.66 0.32 0.13 0.06 0.03 0.01

max

v 39.3 78.5 157 393 785 1570 3930

20 E' . 1.62 0.8l O.Ul 0.16 0.08 0.0^+ 0.02 mm

E' 2.64 1.31 O.65 0.25 0.12 0.06 0.02 max

d = 5 ( cm)

b

(cm)

V 393 785 1570 3930 7850 15700 39300

5 mm 0.16 0.08 0.0U 0.02 0.01 0.00 0.00

max

0.26 0.13 0.07 0.03 0.01 0.01 0.00

V 196 393 785 i960 3930 7850 19600

10 E' . mm 0.32 0.16 0.08 0.03 0.02 0.01 0.00

E' max

0.53 0.26 0.13 0.05 0.03 0.01 0.00

V 98.2 196 393 982 I960 3930 9820

20 E' . mm 0.65 0.32 0.16 0.06 0.03 0.02 0.01

E' 1.06 0.53 0.26 0.10 0.05 0.02 0.01

max
d = 0.5 (cm)

b

(cm)

D = (cm)

v U)E'

(%)E'

mm

max

005

15.7

U.05

6.48

01

31.4

2.03

3.18

.02

62.8

1.01

1.55

05

157

0.41

0.59

.2 .5

3l4 628 1570

0.21 0.12

0.29 0.16

10

20

V 7.85 15-7 31.4 78.5 157 314

E' . _ mm 8.11 4.05 2.03 0.82 0.42 0.24

E

1 max

^0* 6.37 3.10 1.19 0.59 0.32

V 3.93 7.85 15.7 39.3 78.5 157

E' . — 8.11 4.06 1.64 0.84 0.48 mm

i max

<20* 6.19 2.37 1.17 0.64

785

393

d = 1 (cm)

b

(cm)

V 31.4 62.8 126 314 628 1260 3140

5 U)E» . mm 2.03 1.01 0.51 0.20 0.10 0.05 0.03

(^)E'

max

3.28 1.62 0.80 0.31 0.15 0.07 0.03

V 15.7 31.4 62.8 157 314 628 1570

10 E' . _ mm 4.05 2.03 1.01 0.4l 0.20 0.11 0.05

E' max

6.55 3.24 1.59 0.6l 0.30 0.15 0.07

V 7.85 15-7 31.4 78.5 157 314 785

20 E' . _ mm 8.11 4.05 2.03 0.8l 0.4l 0.21 0.11

E

1 max

<20* 6.48 3.18 1.23 0.59 0.29 o.i4

'(Note: These values are upper bounds.)
\ = 0^5 (ym)

d = 2 (cm)

b

(cm)

10

20

D = (cm)

v

(^)E'

(%)E»

min

max

v

E'

E'

min

max

v

E'

E'

min

i max

,005

62.8

1.01

1.65

31. 4

2.03

3.30

15-7

4.05

6.59

.01

126

0.51

0.82

62.8

1.01

1.64

31. 4

2.03

3.28

,02

251

0.25

0.1+1

126

0.51

0.81

62.8

1.01

1.62

.05

628

0.10

0.16

3l4

0.20

0.32

157

o.ia

0.63

.1 .2

1260

0.05

0.08 o.o4

628

0.10

0.15

314

0.20

2510

0.03

1260

0.05

628

0.10

.5

6280

0.01

0.01

3l40

0.02

0.07 0.03

1570

0.04

0.31 0.15 0.06

b

(cm)

d = 5 ( cm)

(*)E«

(*)E

min

i max

157

0.1+1

0.66

311+ 0.20

0.33

628

0.10

0.16

1570

0.04

0.06

31 1+0 0.02

0.03

6280

0.01

15700

0.00

0.02 0.01

10

V 78.5 157 314 785 1570 3140 7850

E' . 0.81 0.1+1 0.20 0.08 o.o4 0.02 0.01 mm

E

1 max

1.32 0.66 0.33 0.13 0.06 0.03 0.01

20

V 39^3 78.5 157 393 785 1570 3930

E' . 1.62 0.81 0.4l 0.16 0.08 0.04 0.02 mm

max

2.65 1.32 0.66 0.26 0.13 0.06 0.02
X = 1 (ym)

d = 0^5 ( cm)

b D= .005 .01 .02 .05 .1 .2 .5

( cm) ( cm)

v 7.85 15-7 31.1* 78.5 157 314 785

(Jg)E' . 8.11 4.05 2.03 0.82 0.42 0.24 _ min

(%)E' ^0* 6.37 3.10 1.19 0.59 0.32

max

v 3.93 7.85 15.7 39.3 78,5 157 393

10 E' . - 8.11 4.06 1.64 0.84 0.48 _ mm

E' - <20* 6.19 2,37 1.17 0.64

max

v 1.96 3.93 7.85 19.6 39.3 78.5 196

20 E

1 . - 8.12 3.28 I.69 0.96 _ mm

E

1 - <20* 4.74 2.34 1.27

max

d = 1 (cm)

b

(cm)

v 15.7 31.4 62.8 157 314 628 1570

5 WE1 . 4.05 2.03 1.01 0.41 0.20 0.11 0.05 _ mm

(^)E' 6.55 3.24 1.59 0.6l 0.30 0.15 0.07

max

v 7.85 15.7 31.4 78.5 157 314 785

10 E' „ 8.11 4.05 2.03 0.81 0.41 0.21 0.11 _ mm.

E' <20* 6.48 3.18 1.23 0.59 0.29 0.14

max

v 3.93 7.85 15-7 39.3 78.5 157 393

20 E' . 8.11 4.05 1.63 0.82 0.42 0.22 _ mm

E» - <20* 6.37 2.45 1.19 0.59 0.28

max

'(Note: These values are upper bounds.)
X = 1 (ym)

d = 2 (cm)

b

(cm)

D =

(cm)

.005 01 .02 .05

10

20

v mm

max

v

E

1

I'

min

max

v

E' mm

max

31.4

2.03

3.30

15.7

4.05

6.59

7.85

8.11

<20*

62.8 126

1.01 0.51

1.64 o.8l

31.

52.8

2.03 1.01

3.28

15.7

4.05

6.55

1.62

31. 4

2.03

3.24

314

0.20

0.32

157

0.41

0.63

78.5

o.8i

1.26

628

0.10

0.15

314

0.20

0.31

157

0.41

o.6i

1260

0.05

0.07

628

0.10

0.15

314

0.20

0.30

3140

0.02

0.03

1570

0.04

0.06

785

0.09

0.12

b

(cm)

d = 5 (cm)

v mm

max

78.5

0.81

1.32

157

o.4i

0.66

314

0.20

0.33

785

0.08

0.13

1570

0.04

0.06

3140

0.02

0.03

7850

0.01

0.01

10

V 39.3 78.5 157 393 785 1570 3930

E» . 1.62 0.81 0.41 0.l6 0.08 0.04 0.02 mm

max

2.65 1.32 0.66 O.26 0.13 0.06 0.02

20

V 19.6 39-3 78.5 196 393 785 I960

E' . 3.24 1.62 0.81 0.32 0.16 0.08 0.03 mm

max

5.29 2.64 1.31 0.5; '.25 0.12 0.05

*(Note: These values are upper bounds.)
A_ = 2 (ym)

d = 0.5 (cm)

b D = .005 .01 .02 • 05 .1 .2

(cm) (cm)

V 3.93 7. 85 15.

39.3 78.5 157

5 (*)E« . mm

- 8.11 4.06 1.64 0.8U 0.48

U)E»

max

- <20* 6.19 2.37 1.17 0.64

V 1.96 3.93 T.85 19.6 39.3 78.5

10 E' . mm

- - a. 12 3.28 1.69 O.96

E' max

— — ^0* 4.74 2.34 1.27

V 0.982 1.96 3.93 9.82 19.6 39-3

20 E' . mm

- - - 6.55 3.38 1.93

E' max

d = 1 (cm)

<l8* 4.68 2.55

b

(cm)

'(Note: These values are upper bounds.)

393

196

98.2

v 7.85 15.7 31.4 78.5 157 314 785

(#)E f

• 8.11 4.05 2.03 0.81 0.41 0.21 0.11 _ mm

(%)E* <20* 6.48 3.18 1.23 0.59 0.29 0.14

max

v 3.93 7.85 15.7 39-3 78.5 157 393

10 E» . 8.11 4.05 1.63 0.82 0.42 0.22 _ mm

E

f - <20* 6.37 2.45 1.19 0.59 0.28

max

v 1.96 3.93 7.85 19.6 39-3 78.5 196

20 E' . - 8.11 3.25 1.64 0.84 0-.U3 _ mm

E* - <20* 4.90 2.37 1.17 0.56

max

10
A_ = 2 (urn)

d = 2 (cm)

b

(cm)

D =

(cm)

005 .01 02 05

(*)E« mm

max

15.7

4.05

6.59

31. 4

2.03

3.28

62.8

1.01

1.62

157

0.41

0.63

314

0.20

0.31

628

0.10

0.15

1570

o.o4

0.06

10

V 7.85 15-7 31.4 78-5 157 314 785

E' . 8.11 4.05 2.03 0.81 0.4l 0.20 0.09 mm

max

<20' 6.5* 3.24 1.26 0.61 0.30 0.12

20 v

E' mm

max

3.93 7.85

8.11

<20*

15-7

4.05

6.48

39-3

1.62

2.53

78.5

0.81

1.23

157

0.4l

0.59

393

0.17

0.24

b

(cm)

d = 5 ( cm)

(Jt)E' min

39.3

1.62

(^)E' 2.65

max

78.5

0.81

1.32

157

0.4l

0.66

393

0.16

0.26

785

0.08

0.13

1570

o.o4

0.06

3930

0.02

0.02

10 vE'

"tti mm

max

19.6

3.24

5.29

39.3

1.62

2.64

78.5

0.81

1.31

196

0.32

0.52

393

0.16

0.25

785

0.08

0.12

I960

0.03

0.05

20 v

E" mm

max

9.82

6.48

<18*

19.6

3.24

5.28

39.3

1.62

2.63

98.2

O.65

1.04

196

0.32

0.51

393 982

0.16 0.07

0.25 0.09

(NoU These values are upper bounds

11
A = 5 (ym)

d = 0.5 (cm)

b D= .005 -01 .02 .05 .1 .2 .5

(cm) (cm)

v 1.57 3.14 6.28 15.7 31.1+ 62.8 i.vi 5 (^)E' . _ mm

(^)E'

max

v 0.785 1-57 3.11+ 7.85 15.7 31.1+ 78.5

10 E

1

. min

E

1 max

v

20 E'

6.28

10.1

15.7

1+.09

31.1+

2.11

62.8

1.21

<23* 5.93 2.93 1.59

3.14 7.85

8.19

15.7

1+.22

31.1+

2.1+1 - <20* 5.85 3.19

1.57 3.93 7.85

8.1+1+ 15.71+.82

0.393 0.785 1-57 3.93 7.85 15.7 39-3

mm

E' - ^0* 6.37

max

d = 1 (cm)

b

(cm)

v 3.11+

5 (^)E' . _ mm

(#)E'

max

v 1.57

10 E' . mm

E»

max

v 0.785

20 E' . mm

6.28 12.6 31.1+ 62.8 126 314

10.1 5.07 2.03 1.02 0.53 0.27

<23* 7.96 3.06 1.1+8 0.73 0.35

3.14 6.28 15.7 31.1+ 62.8 157

- 10.1 4. Ob 2.0b 1.06 0.54

- <23* 6.13 2.96 1.46 0.70

1.57 3.14 7.85 15.7 31.1+ 78.5

- - 8.13 1+.09 2.11 1.08

E' - ^0* 5.93 2.93 1.39

max

*(Note: These values are upper bounds.)

12
1=5. (ym)

d = 2 (cm)

b

(cmj

10

20

D =

(cmj

mm

max

v

E' min

E' max

v

E' min

.005

6.28

10.1

<23*

3.14

1.57

.01

12.6

5.07

8.19

6.28

10.1

^3*

3.14

max

.02

25.1

2.53

4.05

12.6

5.07

8.11

6.28

10.1

^3*

• 05

62.8

1.01

1.58

31. 4

2.03

3.16

15.7

4.06

6.32

.1

126

0.51

0.77

b2.8

1.02

1.53

.2

251

0.26

0.37

126

0.51

0.74

.5

628

0.11

0.15

314

0.22

0.30

31.4 62.8 157

2.03 1.02 0.1+3

3.06 1.48 0.59

b

(cm)

d = 5 (cm)

10

20

v

(#)i

min

max

v

E'

E' mm

max

v

E

1

E' mm

15-7

4.05

6.61

7.85

8.11

<20*

3.93

max

31. 4

2.03

3.30

15-7

4.05

6.60

7.85

8.11

<20*

62.8

1.01

1.64

31. 4

2.03

3.26

15-7

4.05

6.57

157

0.41

0.65

78.5

0.8i

1.30

39.3

1.62

2.59

314

0.20

0.32

157

0.4l

0.64

78.5

0.8l

1.27

626

0.10

0.15

314

0.20

0.31

157

0.4l

0.62

1570

0.04

0.06

785

0.08

0.12

393

0.16

0.24

•(Note : These values are upper bounds.)

13
b D = .005

(cm) (cm)

V 0.785

5 (*)E» . mm

-

(^)E'

max

-

V 0.393

10 E' . mm

-

E

f max

—

V 0.196

20 E' . mm

-

E'

max

-

b

(cm)

_X = 10 (urn)

d = 0.5 ( cm)

.01 .02 .05 .1 .2 .5

1.57 3.14 7.85 15.7 31. 4 78.5

8.19 4.22 2.41

<20* 5.85 3.19

0.785 1.57 3.93 7.85 15.7 39.3

8.44 4.82

- - - <20* 6.37

0.393 0.785 1.96 3.93 7.85 19.6

9.65 -

_ <20*

d = 1 (cm)

v 1.57 3.14

5 (%)%' . _ mm

(^)E'

max

v 0.785 1-57

10 E* . _ mm

E' - max

v 0.393 0.785

20 E' . mm

E

! - - ^0* 5.85 2.78

max

6.28

10.1

15.7

4.06

31.4

2.05

62.8

1.06

157

0.54

<23* 6.13 2.96 1.46 0.70

3.14 7.85

8.13

15-7

4.09

31.4

2.11

78.5

1.08

- <20* 5.93 2.93 1.39

1.57 3.93 7.85

8.19

15.7

4.22

39-3

2.16

'(Note: These values are upper bounds.)

14
A

= 10 (ym)

d = 2_ (cm)

b _D = .005 .01 .02 .05 .1 .2 .5

(cm) (cm)

V 3.14 6.28 12.6 31. 4 62.8 126 311+

5 (^)E' . mm

- 10.1 5.07 2.03 1.02 0.51 0.22

max

— <23* 8.11 3.16 1.53 0.74 0.30

V 1.57 3.14 6.28 15.7 31. 4 62.8 157

10 E

1

. min

- - 10.1 4.06 2.03 1.02 0.1+3

E' max

— — ^3* 6.32 3.06 1.1+8 0.59

V 0.785 1.57 3. 11+ 7.85 15.7 31.1+ 78.5

20 E» . mm

- - - 8.11 4.06 2.05 0.86

E' - - - <20* 6.13 2.96 1.18

max

b

(cm)

d. = _5 ( cm)

v mm

max

7.85

8.11

<20*

15.7

1+.05

6.60

31.1+

2.03

3.28

76.5

0.01

1.30

157

0.41

0.64

314 785

0.21 0.08

0.31 0.12

10 v

E' min

max

3.93 7.85

8.11

<20*

15.7

1+.05

6.57

39-3

1.62

2.59

78.5

0.81

1.27

157

0.1+1

0.62

393

0.16

0.24

20 v

E'

I' mm

1.96 3.93

max

7.85

8.11

<20*

19.6

3.21+

5.19

39-3

1.62

2.55

76.5

0.81

1.24

196

0.33

0.1+7

*(Note : These values are upper bounds.)

15
A

= 20 (urn)

d = 0.5 (cm)

b _D = .005 .01 .02 .05 .1 .2 .5

(cm) (cm)

V 0.393 0.785 1.57 3.93 7.85 15.7 39.3

5 (#)E» . mm

- - - - 8.44 4.82 -

(*)E«

max

— — ~" — <20* 6.37 —

V 0.196 0.393 0.785 1.96 3.93 7.85 19.6

10 E» . mm

- - - - - 9.65 -

E

1 max

— — - - - <20* —

V o.09» 0.196 0.393 O.982 1.96 3.93 9.82

20 E

!

. _ min

— — — — — — —

E' max

- d = 1 (cm)

b

(cm)

V 0.785

5 (*)E» . mm

-

(*)E«

max

—

V 0.393

10 E' . mm

-

E*

max

—

V 0.196

20 E' . mm

-

E' max

-

1.57 3.14

0.785 1.57

0.393 O.785 1.96

7.85

8.13

15.7

4.09

31.4

2.11

78.5

1.08

<20* 5.93 2.93 1.39

3-93 7.85

8.19

15.7

4.22

39.3

2.16

- <20* 5.85 2.78

1.96 3.93 7.85

8.44

19.6

4.32

*(Note : These values are upper bounds.)

<20* 5.56

16
X = 20 (ym)

d = 2_ (cm)

D= .005 .01 .02 .05 .1 .2 .5

3-14

(cm) (cm)

V 1.57

5 WE' . _ min

-

(jOe 1 max

-

V 0.785

10 E' . min

-

E

1 max

—

V 0.393

20 E' . mm

-

E' max

-

1.57

0.785

6.28

10.1

15-7

1+.06

31. 4

2.03

62.8

1.02

157

0.43

<23* 6.32 3.06 1.1*8 0.59

3.14 7.85

8.11

15-7

4. 06

31 .4

2.05

78.5

0.86

- <20* 6.13 2.96 1.18

1.57 3.93 7.85

8.13

15-7

4.09

39-3

1.73

^0* 5.93 2.36

d = 5 (cm)

b

(cm)

V 3.93 7.85 15.7 39-3 78.5 157 393

5 WE' . mm

- 8.11 4.05 1.62 0.81 0.41 0.16

We' max

- ^0* 6.57 2.59 1.27 0.62 0.24

V 1.96 3.93 7.85 19.6 39.3 78.5 196

10 E* . mm

- - 8.11 3.24 1.62 0.81 0.33

E' max

— — <20* 5.19 2.55 1.24 0.47

V 0.982 1.96 3.93 9.82 19.6 39-3 98.2

20 E' . mm

- - - 6.49 3.24 1.62 0.66

E' — - - <18* 5.09 2.48 0.95

max

*(Kote : These values are upper bounds.)

17
X = 50 (ym)

d = 0__5 (cm)

b _D = .005 -01 .02 .05 .1 .2 .5

( cm) ( cm)

v 0.157 0.314 0.628 1.57 3.14 6.28 15.7

5 (XJE'^ ----- la.i - ^)E *max " - - - - <*• -

v 0.079 0.157 0.314 0.785 1.57 3. 14 7.85

10 E'

_ mm. - - - - ___

max

20 E

f

. mm

v 0.039 0.079 0.157 0.393 O.785 1.57 3.93

E

1 max

d = 1 (cm)

b

(cm)

V 0.314

5 (%)E' . mm

-

max

—

V 0.157

10 E» . mm

-

E*

max

—

V 0.079

20 E' . mm

-

E'

max

-

0.628 1.26 3.14

0.314 0.628 1.57

6.28

10.2

12.6

5.28

31.4

2.70

<2 3 * 7-32 3.48

3.14 6.28

10.6

15.7

5.40

<23* 6.95

0.157 0.314 0.785 1.57 3.14 7.85 _____ 10.8

<20*

*(Note : These values are upper bounds.)

18
b_ D = .005

(cm) (cm)

V 0.628

5 (*)E» . mm

-

t$)E" max

-

V 0.314

10 E' . mm

-

E' max

—

V 0.157

20 E' . mxn

-

E' max

-

b

(cm)

V 1.57

5 mm

max

-

V 0.785

10 E* . mm

- E

1 max

—

V 0.393

20 E' . mm

E' max

—

X = 50 (ym)

d = 2 (cm)

.01 .02 .05 .1 .2 .5

1.26 2.51

0.628 1.26

0.31 4 0.628

d = 5 (cm)

3.l4

1.57

0.785

*(Note : These values are upper "bounds.)

6.28 12.6 25.1 62.8

10.1 5.08 2.56 1.08

<£3* 7.66 3.70 1.1+8

3.14 6.28 12.6 31.4

- 10.2 5.12 2.16

- <23* 7.41 2.96

1.57 3.1^ 6.28 15.7

- - 10.2 4.32

^3* 5.91

6.28

10.1

15.7

4.05

31.4

2.03

62.8

1.01

157

0.41

<23* 6.48 3.18 1.55 0.59

3.14 7.85

8.11

15.7

4.05

31.4

2.03

78.5

0.82

- ^0* 6.37 3.10 1.19

1.57 3.93 7.85

8.11

15.7

4.06

39-3

1.64

^0* 6.19 2.37

19
X = 100_ (ym).

d = O^ (cm)

b D = .005 .01 .02 .05 .1 .2 .5

(cm) (cm)

v 0.079 0.157 0.31U O.785 1.57 3.1^ 7.85

5 (*)E* . mm

(*)E«

max

V

10 E' . mm

0.039 0.079 0.157 0.393 0.785 1.57 3.93

nin

max

v 0.020 0.039 0.079 0.196 0.393 O.785 1-96

20 E'_ mm. - - - ___

max

d = 1 (cm)

b

(cm)

v 0.157 0.31U 0.628 1.57 3.1U 6.28 15.7

5 (Jf)E' . - - 10.6 5.^0 _ mm

- <23* 6.95

0.079 0.157 ----- 0.314 0.785 1-57 3.1^ 7.85 _____ 10.8

<20*

0.039 0.079 0.157 0.393 0.785 1.57 3.93

nin

max

max

V

10 E

1

. min

E

1 max

V

20 E

1

. mm

*(Note : These values are upper bounds.)

20
X = 100 (ym)

d = 2_ (cm)

b _D = .005 .01 .02 .05 .1 .2 .5

(cm) (cm)

V 0.31U 0.628 1.26 3.14 6.28 12.6 31.4

5 min

- - - - 10.2 5.12 2.16

max

— — — — <23* <i6* 2.96

V 0.157 0.314 0.628 1.57 3.14 6.28 15.7

10 E' . mm

- - - - - 10.2 4.32

E» max

— — — — — <23* 5.91

V 0.079 0.157 0.314 0.785 1.57 3.14 7.85

20 E' . min

E'

— — - - - - 8.65

<?0*

max

a = 5 ( cm J

(cm)

V 0.785

5 (*)E« . mm

- U)E'

max

—

V 0.393

10 E' . mm

-

E' max

—

V O.196

20 E' . mm

-

E

1 max

-

1.57 3.14

0.785 1.57

0.393 O.785

7.85

8.11

15.7

4.05

31.4

2.03

78.5

0.82

<£0* 6.37 3.10 1.19

3.93 7.85

8.11

15.7

4.06

39-3

1.64

- <20* 6.19 2.37

1.96 3.93 7.85

8.12

19.6

3.28

'(Note: These values are upper bounds.)

<20* 4.74

21
E' . , E' A, b, D~ z (or d~ z

) ; min max —

3. Scaling the Diffraction Loss Tables

The range of the diffraction loss tables can be extended by using

scaling relationships (provided that the errors in the formulas used to

compute the diffraction losses do not become excessive in the extended

range) . It is clear from the formulas in section 6 for the quantities

E' . and E' r given in the tables, that the scaling relationships are simple only if the ratio of the aperture diameter to the source diameter,

Dd

-

*, is constant . Subject to this condition, the scaling relationships

are as follows

Quantity Scales as (constant Dd"" 1

) r2

)

Upper Bounds X

* 5

, lb

*5

, D

_1 (or d" 1 ).

(The upper bounds are the quantity denoted E(v,v,0) in sec. 5.)

4. Effective Wavelengths to Use in

the Diffraction Loss Tables

The effective wavelength, A -

, for computing the diffraction loss for

a given experimental geometry with a source of spectral radiance distribution S% (A) , is defined to be that wavelength which yields the average

spectral diffraction loss when substituted into the approximate Kirchhoff

scalar paraxial model (see sec. 1). Furthermore, if the approximate

formula for the effective diffraction loss for a circular detector, eq

(12) of section 6, is valid (see sec. 7 for a discussion of the mathematical errors in the formulas used in this report) , then the diffraction

loss scales proportionally to the wavelength, and an explicit equation

for A" can be derived in the form,

f^XdXS, (A)

, _ A 1 A fX

>l

dXs

x m '

where Al and A2 are the short- and longwave length limits to S^ (A)

A If S^ (A) is the Planck blackbody spectral radiance function [2] , denoted L (A,T) at temperature T, then A can be related to the temperature by approximate equation (derived by Blevin[i]),

A - = 5324/T (micrometers), (1)

if T is in degrees Kelvin. Thus A - is about 1.84A , the wavelength of

max maximum spectral radiance for a blackbody at temperature T.

loss for a source of spectral radiance distribution S (A) , is given by

A

The effective wavelength, A -

, for computing the luminous diffraction

for a soi

he equation,

22
/^XdXV(A)S (A)

/^dAV(A)S

x (A)'

where V(A) is the spectral luminous efficiency function for pho topic

vision and Al and A2 are the limits of the visible spectrum [3] . If

S

y (A) is the Planck blackbody spectral radiance function, then Blevin

[I] has shown that A - is 0.572 micrometers for a blackbody temperature

of 2856 K (CIE Illuminant A)

5. Sample Diffraction Loss Calculations

5.1. Sample Diffraction Loss Calculations for a Simple Case

A simple example is the following: Compute the average

diffraction loss over the face of a circular detector which views a 500 K

blackbody through a small aperture. The geometry is that of a source

radiance measurement (see fig. 1) . The diameter of the blackbody

aperture (d) is 1 cm; the distance from the blackbody aperture to the

diffracting aperture (b_) is 5 cm; the diameter of the diffracting

aperture (D) is 0.1 cm; the distance from the diffracting aperture to the

detector (a_) is 60 cm; the detector diameter (2x ) is 5 cm. Since the

blackbody temperature is 500 K, eq (1) shows that the effective diffraction wavelength \~ is 10.6 micrometers. Referring to the diffraction

loss tables (sec. 2) , it is seen that the tabulated wavelength closest to

10.6 micrometers is 10; at this wavelength, and at d = 1 cm, b_ = 5 cm,

D = 0.1 cm, the on-axis diffraction loss E". (for a_ at least 10b_, a con- dition which is met by this example) is found to be 2.05%; the corres- ponding area-average diffraction loss over the face of a detector of

radius x , E' , is found to be 2.96%. From the formula for X , max max max

x = 0.5[(0.9d-D)ab- 1 - D] , (2) max —

it is found that x is 4.8 cm; the detector radius is given above as 2.5 cm. Denoting tne desired average diffraction loss over the face of

the detector by the symbol E', it is reasonable to interpolate between

E

1

. and E

1 by the following area-weighting formula: mm max

E' = E' . +[E' - E' . ] (x /x ) 2

. (3) mxn max mm o max

Thus E' is found to be 2.30% at the wavelength of 10 micrometers; the

scaling table in section 3 shows that both E'. and E

1 scale proportionally to the wavelength, so the desired value of E' at a wavelength of

10.6 micrometers is therefore obtained by multiplying 2.30% by the ratio

10.6/10 = 1.06 to get 2.44 %.

5.2. Sample Diffraction Loss Calculations for a Complex Case

Figure 2 shows the essential geometry of a circularly symmetric

source-radiometer system currently in use at NBS. The source S is a

23
blackbody whose temperature is roughly 300 K; the source aperture SA

limits the radiating area; the radiometer aperture RA defines the solid

angle in which radiation is received from SA; the radiometer cavity RC

collects the radiation transmitted through RA. It is desired to calculate the diffraction loss for radiation from S transmitted through SA

and RA to RC.

This is really a 2-step diffraction problem; the total diffraction

loss DL is obtained from both:

a. , the diffraction loss for radiation from S transmitted through

SA to RA, denoted DL , and;

a

b. , the diffraction loss for radiation from SA transmitted through

RA to RC, denoted DL .

Thus the total diffraction loss, denoted DL, is

DL = DL + DL - DL DL , a. d a b

since the diffraction loss at the detector is given as a percentage of

the irradiance that would be present in the absence of diffraction.

The effective diffraction wavelength X~ for this problem is found

from the given temperature of 300 K and eq (1) of section 4 to be 17.7

micrometers. It is clear that the quantity of interest in computing DL

is the effective radiance of SA, compared with the radiance of S. On

the other hand, the quantity of interest in computing DL is the fraction

of the total radiation from SA, transmitted through RA, which is collected by RC. Therefore, in computing DL it is necessary to treat RC as the source and SA as the detector, since (as explained in sec. 1) the

diffraction loss formulas and the tables in this report all refer to the source radiance measurement geometry, and not to the total aperture

radiation measurement geometry.

Referring to figure 2, and using the terminology of the tables, it

is seen that the essential parameters for computing DL and DL are:

DL : X~ = 17.7 micrometers, d = 0.2 cm, b = 0.35 cm, D = 0.05 cm, a —

a_ =17.1 cm, x = 0.575 (source radiance measurement, see

fig. 3); °

DL : X" = 17.7 micrometers, d = 2.0 cm, b = 8.0 cm, D = 1.15 cm,

a_ =17.1 cm, x = 0.025 cm (total aperture radiation measurement, see fig. 3)

To compute DL , note that d = 0.2 cm is smaller than 0.5 cm, the

smallest tabulated value for d; therefore it is necessary to multiply d

and D (to keep the ratio Dd-1 constant) by a scaling factor 3 to use the

tables. Let d' = 3d be the scaled d and D' = 3D be the scaled D; if

24
3 = 10, then d' = 2.0 cm and D' = 0.5 cm, which are tabulated values.

Furthermore, b_ = 0.35 cm is much smaller than 5.0 cm, the smallest

tabulated value for b; therefore it is necessary to multiply b by a scaling factor a to use the tables. Let b' = ab be the scaled b_; if

a = 14.3, then b_' = 5.0 cm, a tabulated value.

In addition, the effective wavelength X~ = 17.7 micrometers is not

a tabulated wavelength. Therefore X~ is multiplied by a scaling factor

6 to use the tables. Let X~ ' = BX~ be the scaled A-

; if = 1.13, then

A - ' = 20 micrometers, a tabulated value.

Next compute x from eq (2) of section 5.1 to get x =3.15 cm,

4-v, 4- / rWS max

so that x /x = 0.IH.

o max

Referring to the diffraction loss tables for the values of E ' . and

E' for the scaled parameters, X~ ' = 20 micrometers, d' = 2.0 cm, ' D* = 0.5 cm, b_' = 5.0 cm (note that the condition that a. be at least 10b_

is met for the geometry of DL ) , it is found that E' . = 0.43% and

E

1 = 0.59% for the scaled parameters. To interpolate between E1

. and

E

4 to obtain the desired average diffraction loss E' over the radiometer aperture RA, for the scaled parameters, refer to eq (3) of section

5.1 and substitute the preceding values of E'. , E' , and x /x into

eq (3). The resulting value of E' = 0.435% for the scaled parameters is

essentially equal to E ' . . mm

The scaling process must now be reversed to obtain DL , the diffraction loss for the original unsealed parameters. Referring to the scaling

table in section 3, it is seen that

DL = E

1 (scaled)

2 a" 1 6- 1

,

or DL = 6. 19E' (scaled) = 2.69%. This value is very close to that calculated for DL from the more accurate formula, eq (12) of section 6,

2.70%.

a

Unfortunately DL cannot be obtained from the tables in their

present form. The values of E'. and E' in the tables are computed by min , . max . f , assuming that the detector- aperture distance a_ is much greater than the

source-aperture distance b. This clearly does not hold for the geometry

of DL (see fig. 2), since a_ = 17.1 cm (as explained above, since this

is a total aperture radiation measurement, the source is treated as the

detector and vice versa) and b_ = 8.0 cm, and therefore a_ = 2.14b and the

condition for the validity of the tables that a_ be at least 10b is not

met.

In addition, since Dd-1 = 0.575 for the geometry of DL , the values

of d and D cannot be scaled to fit the tables because Dd" 1 = 0.5 is the

largest value tabulated, and Dd-1 must be held constant in scaling.

In a situation of this kind, it is necessary to return to the

general formula given in section 6, eq (12) , for the average diffraction

25
loss over the detector disc. This formula is not subject to the restriction of the tables, that the aperture- detector distance a be at least 10

times the source-aperture distance b_. In eq_(12) the average diffraction loss over the detector disc is denoted E(u,v,w ), where u, v, and w are dimension less functions of the geometry and wavelength given by

eqs (5), (6), and (8), respectively. Substituting the values given above

for DL into eqs (5) , (6) , and (8) , u, v, and w are computed and then

substituted into eq (12) for E(u,v,w ) to get d£ = 0.87%.

Finally, therefore, the total diffraction loss, DL, from the source

S to the radiometer cavity RC, is found to be DL = 3.55% .

6. Formulas Used for Computing the Diffraction Loss Tables

The formulas used in computing the diffraction loss tables are derived from the basic Fresnel-Kirchhoff diffraction formula as given,

for example, in Born and Wolf [4].

For a source radiance measurement (as shown in fig. 3) , the diffraction losses increase with increasing detector radius x . It is felt

that a reasonable upper bound for the detector radius, for a source radiance measurement, is defined by the condition that the detector field of

view not extend beyond the inner portion of the source disc whose radius

is 0.9 of the source radius. If this upper bound is denoted x , it is

ITlcLX seen from figure 3 that x is given by eq (2) of section 5.1. The max minimum source diameter, for a source radiance measurement, is that which

makes x = 0; thus the minimum source diameter is [D(l + ba-1 )/0.9]. max —

For a total aperture radiation measurement, the diffraction losses

decrease with increasing detector radius. It is felt that a reasonable

lower bound for the detector radius, for a total aperture radiation

measurement, is defined by the condition that all the source radiation

which passes through the aperture - except for diffraction losses - be

incident upon that portion of the detector disc whose radius is 0.9 of

the detector radius. If this lower bound is denoted x . , it is seen from figure 3 that

x . = 0.5[(d+D) (0.9) _1 ab_1 + D]

nan —

Following the analysis of Blevin[l], it is found that the on-axis

diffraction loss at the center of the detector, denoted E(u,v,0), for

the source radiance geometry, is approximately

E(u,v,0) = ^[(v-u) -1 + (v+u) -1

], (4)

where u and v are the dimensionless quantities,

u = ^(a- 1 + b" 1

) (2X)" 1

, (5)

v = uDd(2bX) -1

. (6)

26
If the aperture-detector distance a_ is much greater than the sourceaperture distance b, then u is approximately

u . = TrD 2 (2bA) _1 = vDd" 1

; (7) mm —

note that u . is independent of a. min —

Now let E(u,v,w ) denote the diffraction loss at the off-axis point

in the detector plane whose radius is x (see fig. 3) , and define a third

dimensionless quantity,

w = ttDx (aA) -1

. (8) o o —

Blevin [1] has shown that E(u,v,w ) is approximately

E(u,v,w ) = 7T" 2 /^d6[(-u + w cosG + (v2 -w2 sin 2 e)°-5

) _1

+ (u + w cos0 +

o o o o

(v2 -w2 sin2

6) '5 )- 1

] , (9) o

and that E(u,v,w ) increases steadily from the minimum value E(u,v,0) as the radius of the off- axis detection point increases from x = (on axis)

out to the radius of the rim of the detector (which is also labeled x in

fig. 3)

°

Referring to eq (4) for E(u,v,0), it is seen that if the aperturedetector distance a_ is much greater than the source-aperture distance b_, then E(u,v,0) is approximately E (u . ,v,0) , denoted E'. , and min mm

E'. = 2(ttv)~ 1 (1 - D

2 d- 2

) -1

. (10) mm

Note that E ' . is independent of _a and that it is the minimum value of

E(u,v,0), considered as a function of a_, since E(u,v,0) decreases steadily to E'. as a increases (assuming the other parameters are held

, mm —

constant; .

Referring to eq (9) for E(u,v,w ), it is seen that the average

diffraction loss over the surface of a detector of radius x , denoted

E(u,v,w ), is given by the formula,

E(u,v,w ) = w

-2 ./" °2wdwE(u,v,w) . (11) o o Since the off-axis diffraction loss E(u,v,w ) increases steadily as the detection point moves away from the axis, it is clear that the

average diffraction loss over the disc of radius x also increases

steadily as x increases. Thus E(u,v,w ) attains its maximum value,

o o considered as a function of x , for a detector radius of x

o max

If the detector plane is now moved towards the aperture, holding the source diameter d, the source-aperture distance b_, and the aperture

27
diameter D constant, then a point will be reached for which x =0.

The value of —a at this point is denoted a . (see fig. 3) , and it is —min

seen that

a . = bD(0.9d - D)

-1

. —min —

It can be shown that the maximum value of E(u,v,w ) , the average

diffraction loss over the detector disc of radius x , considered as a function of the aperture-detector distance a, occurs at a . ; it can also —mm

w + u = 0.9v,

max

be shown that

where w is defined as max

w = ttDx (aX) _1

. max max —

Note that the preceding analysis assumes that the detector radius is x , and that the radius varies with x as the detector is moved

max , . . . max

towards the aperture.

Now consider the behavior of the average diffraction loss over the

disc of radius x as the detector plane is moved away from the

aperture. It is clear that u will approach its minimum value u . asymptotically in this case, and that consequently w will correspondingly approach its maximum value, considered as a function of a_, which

is v(0.9 - Dd-1

) . It can be shown that the average diffraction loss over

the disc of radius x , E(u,v,w ), attains its minimum value, con- sidered as a function of a, when a_ is much larger than b_ (and hence u is

approximately equal to u . and w is approximately equal to v(0.9 - Dd-1 )). This minimum value of E(u,v,w ) is denoted E

1 and is indemax max

pendent of a_.

Steel, De, and Bell [5] have derived a very useful and compact

approximate formula for E(u,v,w ), o [(v+w ) 2 - u

2

] E(u,v,w ) = (27TW ) -1 ln . (12)

[ (v-w ) z - u

z

] o

Thus E' can be expressed approximately by the formula,

(1.9-2Dd_1

) E' = [2Trv(0.9-Dd_1 )]- 1 ln [19 ]. (13)

max ,„ , „ __ i

(0.1+2Dd l

)

28
7. Estimated Accuracy of the

Diffraction Loss Tables

Steel, De, and Bell [5] show that an upper bound for the fractional

error in E ' . , as computed from eq (10) of section 6, is (2v)

-1

; thus

E' . is not given in the tables for values of v less than 5, in order to

limit the estimated error in the tabulated values to less than 10% (of

the value)

Similarly, Steel, De, and Bell [5] show that an upper bound for the

fractional error in E" , as computed from eq (13) of section 6, is

0.06 + 1.6v" ,• thus E is not given in the tables for values of v less

than 12, in order to limit the estimated error in the tabulated values to

less than 20% (of the value). However, an upper bound for E'_ is given

for values of v between 6 and 12; this upper bound is the diffraction

loss for the point on the axis at which the rim of the aperture appears

to coincide with the rim of the source; in other words, the field of view

through the aperture from "this point coincides with the source disc (in

fig. 3, this point is the distance a from the aperture) . Blevin [1]

shows that the diffraction loss for this point, denoted E(v,v,0), is given

by the approximate formula,

E(v,v,0) = (ttv) -0,5

.

Note that the ratio [E ' /E' . ] depends on Dd-1 only, and varies

. , - , , max mm approximately as follows

Dd~ l E ' /E '

— max min

0.0 1.64

0.1 1.45

0.2 1.39

0.3 1.35

0.4 1.32

0.5 1.29.

Thus E' ^ and E'. bracket the diffraction loss, for detector radii

between zero and x , to within ±20% roughly for the worst case,

Dd - 0, and more accurately for larger values of Dd-1

.

Furthermore, if the Kirchhoff scalar paraxial model does not

accurately represent the physical behavior of the experimental situation,

then the diffraction loss values in the tables will contain an additional

error besides those due to the mathematical approximations used to compute

the tables.

A good criterion for the validity of the Kirchhoff model, as Stratton

[6] points out, is that the diameter of the diffracting aperture must be

much larger than the wavelength. In the terminology of figure 3, if

29
D/X > 10,

then the Kirchhoff model is held to be valid; if this condition does not

hold, then the exact vector model may be required.

8. References

[1] Blevin, W. R. , Diffraction Losses in Radiometry and Photometry3 Metrologia 6_, 39 (April 1970) .

[2] See for example: Jamieson, J. A., et. al . , Infrared Physios and

Engineering^ p. 19 (McGraw-Hill Book Co., New York, 1963).

[3] The photopie spectral luminous efficiency function^ V(\) 3 is given

in: American Institute of Physics Handbook, 3rd ed. , chap, 6,

p. 184 (McGraw-Hill Book Co., New York, 1972).

[4] Born, M. , and Wolf, E. , Principles of 0ptics 3 2nd ed. , chap. 8,

p. 382 (The MacMillan Co., New York, 1964).

[5] Steel, w. H. , De, M. , and Bell, J. A., Diffraction Corrections in

Radiometry3 J. Opt. Soc, Am. 62_, 1099 (Sept. 1972) .

[6] Stratton, J. A., Electromagnetic Theory3 p. 463 (McGraw-Hill Book

Co. , New York, 1941)

30
Figure 1. Diffraction geometry used in the diffraction loss tables

31
Q

O

00

H

o

o

CM

o

~l

o

od

to

ro

Figure 2. Diffraction geometry for the sample diffraction loss calculations; complex case: A-A, optical axis; RA, radiometer aperture; RC, radiometer cavity; S, 300 K blackbody source; SA,

defining aperture for S (distances in cms; aperture diameters

magnified 10 times with respect to distances along the axis)

.

32
\

\

\

A

UJ

< 2 _l ^

QL

! o | i UJ

C/D X

bl h- c LU o

cr J— ZD < ^ o

o lL

r o

QQ LU LU

? *> 1— X

t < < f-

| • • h-

-01 LJ O fJL

h- (T o

1 O

t

*TMy

Figure 3. Diffraction geometry used in the general diffraction loss

formulas: A, aperture; B, detector; C, source; d, source

diameter; D, aperture diameter; x , detector radius; X-^ , maximum detector radius for a source radiance measurement;

xmin/ minimum detector radius for a total aperture radiation

measurement; a_, aperture-detector distance; ajai n > minimum

aperture-detector distance; a^, aperture-detector distance for

computing an upper bound for the diffraction loss; b, source- aperture distance.

33
{1BS-114A (rev. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN 594-8

2. Gov 't Accession

No.

4. TITLE AND SUBTITLE

OPTICAL RADIATION MEASUREMENTS:

Tables of Diffraction Losses

3. Recipient's Accession No.

5. Publ icat ion Date June 1974

6. Performing Organization ( ode

7. AUTHOR(S)

W. B. Fussell

Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project /Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

same as No. 9

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 74-60008!

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Tables of diffraction losses are given for a range of typical experimental

geometries for wavelengths from 0.2 to 100 micrometers. The scaling relationships

for the diffraction losses for varying wavelengths and geometries are also given,

and sample calculations are presented. General formulas are given for the

diffraction losses; the formulas are derived from the Kirchhoff scalar paraxial

diffraction theory. The accuracy of the tabulated values is estimated.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Diffraction; diffraction losses; Fresnel diffraction; Kirchhoff diffraction theory,

photometry; radiometry; scalar diffraction theory.

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